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arXiv:2505.20979v1 [cs.SD] 27 May 2025MelodySim: Measuring Melody-aware Music Similarity for Plagiarism Detection Tongyu Lu∗1, Charlotta-Marlena Geist∗2, Jan Melechovsky1, Abhinaba Roy1, Dorien Herremans1 1Singapore University of Technology and Design 2Otto von Guericke University Magdeburg [email protected], [email protected], [email protected], [email protected], [email protected] ABSTRACT We propose MelodySim , a melody-aware music simi- larity model and dataset for plagiarism detection. First, we introduce a novel method to construct a dataset with focus on melodic similarity. By augmenting Slakh2100; an ex- isting MIDI dataset, we generate variations of each piece while preserving the melody through modifications such as note splitting, arpeggiation, minor track dropout (ex- cluding bass), and re-instrumentation. A user study con- firms that positive pairs indeed contain similar melodies, with other musical tracks significantly changed. Second, we develop a segment-wise melodic-similarity detection model that uses a MERT encoder and applies a triplet neu- ral network to capture melodic similarity. The resultant decision matrix highlights where plagiarism might occur. Our model achieves high accuracy on the MelodySim test set. 1. INTRODUCTION In recent years, the popularity of generative music models has rapidly increased. With the rise of commercial models such as Suno1and Udio2, as well as open source models like Mustango [1] and MusicGen [2], the question of artist- protection question arises. There currently is an ongoing discussion as well as legal battles on how artists should be compensated for the use of their music as training data [3], e.g. Recording Industry Association of America (RIAA) vs. Udio and Suno (June 2024)3. In addition, the mu- sic generated by these models might plagiarize the original training data. In this work, we develop tools that may help with melody-related plagiarism detection. When generative models are trained on (often improp- erly licensed) copyrighted data, it becomes a strong pos- sibility that the generated music plagiarizes the original training data. In particular, diffusion models have shown to be prone to replicate their training data, as shown by [4, 5] on the image generation task. Artists have made public outcries showcasing examples of their work or style repli- cated by generative models4. In literature, we noticed that *These authors contributed equally to this work. 1https://suno.com 2https://udio.com 3https://shorturl.at/YlQ0P 4https://shorturl.at/j5tJbgenerative AI models are typically evaluated in terms of their ability to predict similarly to the input data (accu- racy) rather in terms of the originality of the generated out- put [6]. At the moment there is no clear legal precedent or ruling to tackle the copyright issues on the input data, how- ever, we can examine resulting plagiarism by the output of the generative models. Finding and confirming music plagiarism in general is a complex task. When deciding on plagiarism cases, [7] highlight the necessity of individually considering each case. An automatic plagiarism detection tool could help speed up the process of both flagging new plagiarism cases, as well as confirming expert opinions in existing lawsuits. Such tools might even be integrated into the music gener- ation models themselves to avoid plagiarized output in the first case. This task, however, is not trivial, as there is no generally accepted, objective definition of what plagiarized	https://arxiv.org/abs/2505.20979v1
music is. In an analysis of 17 lawsuits, [7] observed that the melody was prioritized when deciding on plagiarism, followed by the ‘overall impression’ of the music. This leads us to believe that there is a need for a melody-aware music similarity tool. The existing work on melody simi- larity metrics, however, is limited to the field of symbolic music (MIDI) [8–10]. To be able to deal with real-life court case data and generated music, we develop an audio based melody-aware similarity model in this work. This task is arguably more challenging than using symbolic music, due to the overlay of multiple audio signals, as well as the lack of data to train the model. The contributions of this work include the creation of a novel dataset, MelodySim, which contains 1568 full length instrumental songs, with three additional variations per song, resulting in a total of 6,272 files. These variations contain slight melodic changes in terms of altered pitches, note durations, speed and instrumentation. These changes are subtle such that the resulting matching tracks can still be construed as ‘plagiarized’ in terms of melody. This data is then used to train a Triplet Neural Network with a MERT encoder [11], that minimizes the distance in representation of matching segments and maximizes the distance between different segments. This results in a melody-aware similar- ity embedding that is then used in a classification model to directly predict matching segments. In the following sections, we first provide an overview of related work. Section 3 then describes how we have created the MelodySim dataset. This is followed by a de- scription of the melody-aware triplet neural network that we developed to predict similar music fragments as well as the full-song plagiarism detection method. Finally, the results of our model on the training set as well as an in- the-wild plagiarism dataset are presented, followed by a conclusion. 2. RELATED WORK In this section, we provide a brief overview of how music plagiarism has been defined in existing literature. We then discuss related work on music similarity detection models. 2.1 What is plagiarism? When developing a model for plagiarism, we have to ask ourselves: which elements of music count towards plagia- rism? A lot of popular rock, pop, and folk music shares the same 3-chord progression: I-IV-V , and has a similar drum track, making these elements non-eligible copyright infringement. This leaves other musical features such as melody and timbre as potential sources for plagiarism. Currently, there does not exist a fixed rule set that de- fines plagiarism in music. In a study by [7], 17 music pla- giarism lawsuits were analyzed. The authors observed that the melody was clearly prioritized when deciding on pla- giarism, but always paired with another parameter which in most cases appeared to be the rhythm. Huber also stated that melody is the most discussed aspect in legal disputes, second to ‘overall impression’, which can be considered as the composition of various musical characteristics. Based on this, we decided to build a melody-aware sim- ilarity metric, that not yet looks at the	https://arxiv.org/abs/2505.20979v1
melody, but also encodes the music in general through MERT-features [11]. To achieve this, we carefully constructed a new dataset by thoroughly altering musical features in different levels of detail while maintaining the main melody, as explained in Section 3. 2.2 Automatic Music Similarity Detection Most existing work on melody similarity detection is in the symbolic domain. Much of this work is not necessar- ily developed towards plagiarism detection, but could have other goals such as melody retrieval [8], repeated (exact) sample detection [12]. For a more comprehensive historic overview of music similarity models, the reader is referred to [13]. For instance, [8] developed a music similarity model for that was trained on the Meertens Tune Collections dataset [14]. Their recurrent neural network models al- lowed them to consider melody recommendation as a rank- ing problem of similarity. More recently, [15] present a way to generate an originality report, which includes an originality score (based on cardinality) to evaluate how much a generative symbolic music model copies from its training set. These metrics are then used to inform an early stopping mechanism that cuts of training when the optimal level or ‘originality’ is reached on the validation set, thuspreventing the transformer from generating music that is too similar to its training data. In [9], an image-based approach for solving the task of plagiarism detection based on musical features such as rhythm and melody similarity. The authors used the Lakh MIDI dataset [16] and represented the MIDI into 8-bar units and grayscale images. Generated simulated plagia- rism cases were then generated by reversing and removing operations on note and rhythm vectors as well as note se- quences. This work only considered monophonic instru- mental songs. An interesting in-the-wild dataset for plagiarism in the Music Copyright Infringement Cases (MCIC) dataset [10]. The dataset contains music pairs from 116 court cases ( denied: 66, infringed: 32, settled: 18) in both MIDI as well as score form. In the domain of audio similarity research, [17] and [18] developed similarity techniques based on spectrograms and fingerprinting to tackle plagiarism detection. These methods require high computational power with a large fin- gerprint database and tend to result in low accuracy with decreasing audio quality and higher noise level [9, 12]. The resulting similarity relies on general acoustic features extracted from spectrograms and does not directly distin- guish between specific musical characteristics like melody, rhythm or timbre. Another audio-based similarity approach is the Music Replication Assessment (MiRA) tool [19], which includes several similarity metrics for raw audio. In their experi- ments, embedding-based metrics showed the most promis- ing results in terms of robustness and sensitivity. The scope of their work, however, is limited to exact replications in music audio. Their dataset was generated by putting a frac- tion of a reference track into a random point of a target track. The problem of finding reused samples in other songs was tackled in [12]. Their deep learning approach uses a siamese-based convolutional neural network (CNN) with mel-spectrograms and a triplet loss. Their similarity score based on the resulting embeddings consisted	https://arxiv.org/abs/2505.20979v1
of a combina- tion of Euclidean distance, cosine similarity and the Pear- son correlation. The model was trained on the WhoSam- pled5dataset. The task of finding replicated samples is also limited to finding exact repetitions. In this work, we aim to improve upon such an approach by including note-level variations to make the algorithm more robust. Our work build upon the gaps in literature by providing the first open, large-scale synthetic audio dataset for au- dio plagiarism. Each song contains three variations with slight music theory-informed melody changes, that con- tains many small melody variations (altered pitches, note durations, speed and instrumentation) while significantly altering the other tracks and timbre. The subtleness of the melody changes ensures that the paired tracks may be con- strued as plagiarized. This new dataset then allows us to train a triplet neural network-based melody-aware similar- ity model for plagiarism detection directly on audio. 5www.whosampled.com 3. MELODYSIM DATASET In order to be able to create a strong melody-aware em- bedding for audio music, we need a suitable dataset to train on. We used various MIDI and audio augmentations to cre- ateMelodySim , a new audio dataset which contains three variations for each song. These variations aim to keep the melody constant (except for tiny changes for robustness), and change other aspects such as removing tracks, chang- ing instruments, inverting chords, changing the tempo and transposing the composition as shown in Figure 1. We thus aim to capture melodic similarity between otherwise dif- ferent songs, as melody is one of the main plagiarism cri- teria [7]. We used 1568 MIDI files from the Slakh2100 dataset [20] as a base dataset to start the augmentations. In the following subsections the augmentation procedure is described in detail. The final dataset consists of 6,272 full-length audio music files, consisting of original pieces with three additional versions for each piece. The dataset and augmentation code are available online6. 3.1 Step 1 - Melody track identification For each of the multi-track MIDI files, we first identify the melody track by training a machine learning model. Our best performing model is a gradient boosting classi- fier model following the approach presented in [21]. A re- fined CMU Computer Music Analysis Dataset7was used for training the model, where we manually relabeled a por- tion of this dataset after noticing a number of incorrect la- bels. The refined dataset is available online8. Taking [21] as a reference, a number of adjustments were made to the input features, that lead to improved re- sults. First, additional track features including polyphony rate and note activation density were added. Secondly, apart from the features from the current track under in- spection, average features of other tracks in the same MIDI file were also computed and added to the classification inputs as reference-features.Through cross-validation, fi- nally a histogram-based gradient boosting model was se- lected as our model, which reached an accuracy of 97% on the validation split of CMU. Through manual inspections, we found that the model generalized well on Slakh2100. Our melody track identification model is available as open	https://arxiv.org/abs/2505.20979v1
source9. 3.2 Step 2 - MIDI-level augmentations Now that we have identified the melody track in Step 1, we are able to perform a number of MIDI augmentations on both the instrument- and note-level. Instrument replacement: For each of the MIDI tracks a new instrument are considered. We first group the MIDI instrument indices (from 1 to 128) into ensembles (pianos, guitars, high-register strings, low-register strings, etc.), 6https://huggingface.co/datasets/amaai-lab/melodySim 7https://www.cs.cmu.edu/music/data/melody-identification/ 8https://huggingface.co/datasets/amaai-lab/melodySim 9https://huggingface.co/amaai-lab/MelodySimand then reassign the track instruments with the following rules: 1. with probability 0.2, retain the instrument as it is; 2. if not, then with probability 0.7, change the instrument to another one in its ensemble (e.g., replacing piano with e-piano); 3. otherwise replace the instrument with another one in a different ensemble with similar pitch register; 4. ensure coupled tracks (e.g., piano tracks) to be applied with the same replacement policy; 5. avoid different instrument tracks being replaced into the same instrument. Track removal: 1. with a probability drawn from a uniform distribution of [0.1, 0.5], for each track, mute the track; 2. with a probability of 0.5, mute the percussion track; 3. never mute the melody tracks (identified), bass tracks and other important tracks (vocals, piano or guitar companies). Note splitting: With a probability Pn, split the current note of typical duration (whole notes, half notes, quarter notes) into two of half the original duration. Pnis drawn from a uniform distribution of [0.3, 0.85] for each track n. Chord inversion: For each track, detect block chords (concurrently played notes) consisting of 3 or 4 notes. For each such chord, with a probability Pn, shift the top notes an octave down or the bottom notes an octave up. Pnis drawn from a uniform distribution of [0.3, 0.85] for each track n. Chord argpeggiation: For each track, detect block chords that are in regular durations (1x/2x/3x/4x of quar- ter note). With a probability Pn, split the chord into an arpeggio (consisting of equally-placed chord notes) with the same total duration as the original chord. Pnis drawn from a uniform distribution of [0.3, 0.85] for each track n. 3.3 Step 3 - Audio-level augmentations After augmenting the MIDI files, the resulting audio files are obtained by synthesizing with the Musyng soundfont. Then, a set of audio augmentations (as depicted in Fig- ure 1) is applied to further diversify the different versions, in particular: •Pitch shift: The audio track is pitch-shifted by a random integer of semitones in the range of [-4, 4]. •Time shift: The whole track is shifted by a random time from a range of [-3, 3] seconds. This time shift is used when matching the positive pairs later on. •Tempo change: The audio track’s tempo is altered by a random factor in the range of [0.9, 1.1]. The resulting audio files are then cut into 10 sec long segments each being saved with representative track name, version index, and segment index. 4. MUSIC SIMILARITY MODEL Using the newly created MelodySim, we train a triplet neural network model [22] that that enables the creation of melody-sensitive embeddings of music audio, and	https://arxiv.org/abs/2505.20979v1
the computation of the distance or similarity between these embeddings. MIDIMelody track identification1. Non -melody stem removal 2. Instrument replacement1. Note splitting 2. Chord inversion 3. Arpeggiation1. Pitch shift 2. Time shift 3. Segmenting 4. Tempo changeAudioNote -level Instrument -level SynthesizeAudio MIDI Augmentations Audio Augmentations(augmented )Figure 1 . The proposed melody-aware augmentation pipeline used for constructing MelodySim dataset by augmenting Slakh MIDI. anchor positive negativeMERT Encoder MERT Encoder MERT Encoder1D Resnet 1D Resnet 1D ResnetClassifier𝐱anc 𝐱pos 𝐱negsgabs 𝐱anc−𝐱pos sgabs 𝐱anc−𝐱neg Classifier𝐲same 𝐲diff Training shared parameters shared parametersshared parameters waveform inputs MERT embedding sequences melody -aware embeddings sigmoid distance / similarity segment 1 segment 2MERT Encoder MERT Encoder1D Resnet 1D Resnet𝐱1 𝐱2Classifier abs 𝐱1−𝐱2𝐲 similarity =1−𝐲 Inferenceℒtriplet 𝐱anc,𝐱pos,𝐱neg ℒBCE 𝐲same ,0+ℒBCE 𝐲diff,1 shared parameters shared parameters Figure 2 . The proposed architecture for training and inference. sg[·]means "stop gradient" and abs(·)notates element-wise absolute function. 4.1 Triplet dataset To train a triplet neural network model, we reformulate MelodySim into triplets, consisting of an anchor sample, a positive sample similar to the anchor, and a negative sam- ple dissimilar to the anchor. We construct the positive pairs by combining time-aligned segments from the original and augmented tracks. The negative pairs are formed using inter-song segments. The example below illustrates a triplet (anchor ,positive ,negative )structure: anchor =Track 00125/version 0/segment02, positive =Track 00125/original /segment02, negative =Track 00007/version 2/segment12. Each triplet consists of an anchor data sample, a pos- itive data sample that shares the same melody but varies in other characteristics (such as texture, tempo, or instru- mentation), and a negative data sample that differs in both melody and other features. This triplet construction en- sures the model can learn to differentiate between similar and dissimilar musical excerpts based on melody.4.2 Triplet Neural Network As shown in Figure 2, the music similarity model is a triplet neural network (TNN) consisting of a MERT en- coder, a ResNet backbone and a classifier head. The similarity model starts with a MERT encoder, a pretrained state of the art model open source on hugging- face by [11]. For capacity limitation reasons the audio files were fed into the more compact MERT-v1-95M version of the feature extractor and stored as encodings, before using them as input for the adaption network in the training pro- cess. In order to reduce memory load the output features of MERT were postprocessed with a moving average with size=10, stride=10 over the time token axis and the selected hidden states were limited to h3, h6, h9, h12. After MERT encoding, a sequence of trainable 1D con- volutional residual blocks is applied as an adaption net- work. An average pooling layer is applied at the end of 1D Resnet to aggregate the information over time dimension, getting a fixed dimension embedding for its correspond- ing MERT embedding sequence. In a training step, all three components in a triplet (anchor ,positive ,negative ) run through the MERT encoder (parameters frozen) and the 1D Resnet (parameters trainable), getting the corre- sponding embeddings, i.e., xanc,xpos,xneg. To integrate melody-aware information, we update the 1D Resnet pa- rameters through backward propagation	https://arxiv.org/abs/2505.20979v1
with triplet loss, which is defined as follows: Ltriplet (xanc,xpos,xneg) = max d(xanc,xpos)−d(xanc,xneg) +α,0 where α= 1.0is the margin, d(xi, yi) =∥xi−yi∥2is the Euclidean distance. Finally, a fully-connected classifier is maintained at the end in measuring the sigmoid distance between embed- dings, with output scaling in range [0,1]. In each triplet, we inspect a "same case", namely (xanc,xpos)and a "different case" (xanc,xneg). The classifier takes abs(xanc−xpos) andabs(xanc−xneg)as inputs, giving ysame,ydiffas outputs respectively. To train the classifier, we backward propagate the Binary Cross Entropy (BCE) loss LBCE(ysame,0) +LBCE(ydiff,1) = mean(log(1 −ysame) + log ydiff) In this way, we train the classifier with balanced labels. In addition, we remove the gradient of classifier inputs (i.e., abs(xanc−xpos)andabs(xanc−xneg)) during training to avoid BCE loss confusing the 1D Resnet. In Figure 2 we usesg[·]to show the "stop gradient" operation. During inference, we utilize the similarity model as a Siamese Neural Network: forwarding both input audio segments with MERT encoder and 1D Resnet respectively in the same manner, getting the absolute difference and fi- nally get the classification result. Note that the final output of the inference pipeline is similarity (also falls in range [0,1]) instead of sigmoid distance. 4.3 Plagiarism identification Note that the TNN mentioned in the previous section com- putes similarity between two music segments. However, it remains a problem to decide whether or not two entire pieces are plagiarized. In view of this, we compute the similarity matrix and design a rule-based decision strategy. Given two pieces, we segment them into 10-sec win- dows [w(i) 1, i= 1, ..., N 1],[w(j) 2, j= 1, ..., N 2]in the same way as when constructing the MelodySim dataset. If we notate the similarity model (inference mode) with sij=f(w(i) 1,w(j) 2) we can finally get a similarity matrix S= [sij]∈[0,1]N1×N2 Next, we threshold (default γ= 0.5) the similarity ma- trix, getting a decision matrix D=u(S−γ) where u(·)is the unit step function. Summing up rows or columns of D, we have plagia- rized counts in both directions, namely d(i) 1→2=X jDij,d(j) 2→1=X iDijIf we further define a sensitivity (how many similar seg- ments in piece 2 are enough to determine that the segment in piece 1 is plagiarized and vice versa), we can finally ob- tain the proportion of plagiarized segments in d1→2,d2→1. In our testing cases, we set the maximum proportion to be 0.2, which means that "if both pieces have number of pla- giarized segments larger than 20% of the total segments, then the two pieces have plagiarism relationship". 5. EXPERIMENTS 5.1 Experimental setup The similarity model was trained on 95% of the MelodySim dataset, reserving the remaining 5% for eval- uation purposes. The training process was executed on a single Nvidia V100 GPU for a duration of 7.3 hours, with a batch size of 512. During the training phase, all MelodySim training tracks were traversed, and anchors were randomly selected along with their corresponding positive and negative samples. To enhance diversity, each track was loaded 4 times per epoch. The training regimen encompassed a total of 797 epochs. To thoroughly test the model, we utilize	https://arxiv.org/abs/2505.20979v1
the 78 pieces from the test split. Specifically, we construct 546 = 7 ×78 positive pairs , where the factor 7comes from all combi- nations among versions along with self-comparison, i.e., {(orig,orig),(orig,ver1) ,(orig,ver2) , ...,(ver2 ,ver3)}. Correspondingly, we select an equal number of negative pairs to maintain a balanced test set. These negative pairs are formed by combining excerpts from different tracks and randomly sampling from all possible combinations. 5.2 Objective evaluation We performed objective evaluation of the melody- similarity classifier that detects positive pairs in the dataset. We present the similarity matrices between selected exam- ples and report the classification metrics on the test set. A selection of similarity matrices is depicted in Figure 3. Table 1 shows the classification results on the test set. Table 1 . Classification metrics on test split. Precision Recall F1 Different 1.00 0.94 0.97 Similar 0.94 1.00 0.97 Average 0.97 0.97 0.97 Accuracy 0.97 The similarity matrix reveals that the model effectively captures melodic similarity, accurately reflecting the prox- imity between music audio segments. However, we no- tice that the positive pairs tend to have large-scale acti- vations like Figure 3 shows. This shows that the model may not only be sensitive in melody, but also the music texture (if the model is only sensitive in melody, then the positive pair similarity matrix should present the repeating pattern). In addition, some of the negative pair similarity Track01880 -version 3Track01976 -version 1Track01889 -originalTrack01889 -version2 differentsimilar Figure 3 . Examples of similarity matrices, a positive pair (top) and a negative pair (bottom) from the test split are demonstrated. matrices shows horizontal or vertical activations, meaning that "one or several adjacent segments in piece 1 may be similar to all windows in piece 2", which is not likely in real case. This reflects to some inherent problem on the similarity model as a black box. Observed from the classi- fication metrics, we would say that our model fits well on MelodySim, reaching 97% accuracy as well as F1 score, which indicates that the detection on positive pairs and negative pairs is balanced. 5.3 Subjective evaluation of dataset To assess the efficiency of our MelodySim dataset, we con- ducted a listening study. A total of 12 participants listened to 12 audio pairs and rated the overall similarity, melodic similarity, and similarity of non-melodic aspects on a 7- point Likert scale [23]. The results, depicted in Table 2, confirm that the proposed augmentations mainly alter non- melodic aspects of the music.Table 2 . MelodySim dataset listening study results; as- pects are rated on a 7-point Likert Scale; reported Mean Opinion Score with 95% Confidence interval. Aspect Positive pairs Negative pairs Overall similarity 4.23±0.80 2 .00±0.68 Melodic similarity 4.53±0.84 1 .90±0.90 Non-melodic similarity 3.94±0.53 2 .27±0.22 6. DISCUSSION AND LIMITATIONS The task of targeted augmentation to preserve melody but alter other attributes is not simple due to a few factors. First, identifying melody is not always straightforward, as some files include multiple melody tracks, or have melody being played in some parts of the song by an otherwise non-melodic track. This	https://arxiv.org/abs/2505.20979v1
makes it difficult to craft a sim- ple rule for melody identification, which could sometimes result, for instance, in a part of the melody missing, or a non-melody track being treated as a melody track, thus being always present after passing through the augmenta- tion pipeline. Furthermore, melody identification rules can be genre-dependent. In this paper, we offer a good base- line melody identification model, which can be further im- proved in future work. When constructing positive and negative pairs, we did not consider the possibility of pairing two segments from the same song at different time marks. The probability of a repeating motive in the same song is too high and would require a similarity metric to automatically identify such similar segments. However, using segments of the same song as either positive pairs (with matching melody, but slightly varied background, for instance, when the song culminates vs when it starts), or as negative pairs (when the melody played is different, e.g., verse vs chorus), would benefit the training of the similarity model further. Future work will focus on further augmentation im- provement, and include more analyses, potentially with real-life plagiarism cases. 7. CONCLUSION We present the MelodySim dataset, an open source audio dataset and model for melody-aware music similarity and plagiarism detection. MelodySim was constructed through a set of targeted midi and audio augmentations such that it contains original tracks as well as three variations that have a comparable melody, but vary in terms of other musical aspects (arpeggiated chords, changed instruments, missing tracks, etc.). The similarity in terms of melody and other musical aspects was verified through a listening study. We also propose a melody-aware similarity model. This model consists of a MERT-encoder, combined with a ResNet backbone and classification head. We employ a triplet neural network architecture for training the model on the MelodySim dataset. In an objective evaluation, we show that the model performs admirably in detecting vari- ations of songs in the test set. 8. ACKNOWLEDGMENT This work has received support from SUTD’s Kickstart Initiative under grant number SKI 2021 04 06 and MOE under grant number MOE-T2EP20124-0014. 9. REFERENCES [1] J. Melechovsky, Z. Guo, D. Ghosal, N. Majumder, D. Herremans, and S. Poria, “Mustango: Toward con- trollable text-to-music generation,” in Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Hu- man Language Technologies (Volume 1: Long Papers) , 2024, pp. 8286–8309. [2] J. Copet, F. Kreuk, I. Gat, T. Remez, D. Kant, G. Syn- naeve, Y . Adi, and A. Défossez, “Simple and control- lable music generation,” Advances in Neural Informa- tion Processing Systems , vol. 36, pp. 47 704–47 720, 2023. [3] M. Wei, M. Modrzejewski, A. Sivaraman, and D. Her- remans, “Prevailing research areas for music ai in the era of foundation models,” arXiv preprint arXiv:2409.09378 , 2024. [4] G. Somepalli, V . Singla, M. Goldblum, J. Geiping, and T. Goldstein, “Diffusion art or digital forgery? inves- tigating data replication in diffusion models,” in Proc. of the IEEE/CVF Conf. on computer vision and pattern recognition , 2023,	https://arxiv.org/abs/2505.20979v1
pp. 6048–6058. [5] N. Carlini, J. Hayes, M. Nasr, M. Jagielski, V . Sehwag, F. Tramer, B. Balle, D. Ippolito, and E. Wallace, “Ex- tracting training data from diffusion models,” in 32nd USENIX Security Symposium , 2023, pp. 5253–5270. [6] B. L. Sturm, M. Iglesias, O. Ben-Tal, M. Miron, and E. Gómez, “Artificial intelligence and music: open questions of copyright law and engineering praxis,” in Arts, vol. 8, no. 3. MDPI, 2019, p. 115. [7] J. Huber, D. Müllensiefen, and R. Kopiez, “V on der" armseligen allerweltsfloskel" zur" schöpferischen eige- nart": Eine analyse deutscher gerichtsentscheidungen zu plagiaten in der musik von 1966 bis 2020,” Ph.D. dissertation, Hochschule für Musik, Theater und Me- dien Hannover, 2020. [8] F. Karsdorp, P. van Kranenburg, and E. Manjavacas, “Learning similarity metrics for melody retrieval,” in Proc. of the 20th Int. Society for Music Information Re- trieval Conf. , 2019, pp. 478–485. [9] K. Park, S. Baek, J. Jeon, and Y .-S. Jeong, “Music pla- giarism detection based on siamese cnn,” Hum.-Cent. Comput. Inf. Sci , vol. 12, pp. 12–38, 2022. [10] S. Park, H. Kim, J. Pak, and J. Kim, “Quantitative anal- ysis of melodic similarity in music copyright infringe- ment cases,” in International Society for Music Infor- mation Retrieval Conference . International Society for Music Information Retrieval, 2024.[11] Y . Li, R. Yuan, G. Zhang, Y . Ma, X. Chen, H. Yin, C. Xiao, C. Lin, A. Ragni, E. Benetos et al. , “Mert: Acoustic music understanding model with large-scale self-supervised training,” arXiv:2306.00107 , 2023. [12] G. Kasif and G. Thondilege, “Exploring music simi- larity through siamese cnns using triplet loss on music samples,” in 2023 Int. Research Conf. on Smart Com- puting and Systems Engineering (SCSE) , vol. 6. IEEE, 2023, pp. 1–8. [13] P. Knees and M. Schedl, “A survey of music similarity and recommendation from music context data,” ACM Transactions on Multimedia Computing, Communica- tions, and Applications (TOMM) , vol. 10, no. 1, pp. 1–21, 2013. [14] P. Van Kranenburg, M. de Bruin, L. P. Grijp, and F. Wiering, “The meertens tune collections,” Meertens Online Reports , vol. 2014, no. 1, 2014. [15] Z. Yin, F. Reuben, S. Stepney, and T. Collins, ““a good algorithm does not steal–it imitates”: The originality report as a means of measuring when a music gener- ation algorithm copies too much,” in Artificial Intel- ligence in Music, Sound, Art and Design: 10th Int. Conf., EvoMUSART 2021, Part of EvoStar . Springer, 2021, pp. 360–375. [16] T. Bertin-Mahieux, D. P. Ellis, B. Whitman, and P. Lamere, “The million song dataset.” in Ismir , vol. 2, no. 9, 2011, p. 10. [17] N. Borkar, S. Patre, R. S. Khalsa, R. Kawale, and P. Chakurkar, “Music plagiarism detection using audio fingerprinting and segment matching,” in 2021 Smart Technologies, Communication and Robotics (STCR) . IEEE, 2021, pp. 1–4. [18] A. López-García, B. Martínez-Rodríguez, and V . Liern, “A proposal to compare the similarity be- tween musical products. one more step for automated plagiarism detection?” in Int. Conf. on Mathematics and Computation in Music . Springer, 2022, pp. 192–204. [19] R. Batlle-Roca,	https://arxiv.org/abs/2505.20979v1
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arXiv:2505.20993v1 [cs.CL] 27 May 2025Who Reasons in the Large Language Models? Jie Shao Jianxin Wu∗ National Key Laboratory for Novel Software Technology, Nanjing University, China School of Artificial Intelligence, Nanjing University, China [email protected], [email protected] Abstract Despite the impressive performance of large language models (LLMs), the pro- cess of endowing them with new capabilities—such as mathematical reasoning— remains largely empirical and opaque. A critical open question is whether reasoning abilities stem from the entire model, specific modules, or are merely artifacts of overfitting. In this work, we hypothesize that the reasoning capabilities in well- trained LLMs are primarily attributed to the output projection module ( o_proj ) in the Transformer’s multi-head self-attention (MHSA) module. To support this hypothesis, we introduce Stethoscope for Networks (SfN), a suite of diagnostic tools designed to probe and analyze the internal behaviors of LLMs. Using SfN, we provide both circumstantial and empirical evidence suggesting that o_proj plays a central role in enabling reasoning, whereas other modules contribute more to fluent dialogue. These findings offer a new perspective on LLM interpretability and open avenues for more targeted training strategies, potentially enabling more efficient and specialized LLMs. 1 Introduction Although large language models (LLMs) [ 29,6,41,5] have exhibited great success and potential in various aspects, developing new capabilities for LLMs [ 53,17,37,14] is still a trial and error experimentation process in most cases. For example, one of the most exciting milestones is LLMs that can reason [ 18,13,39], e.g., solving complicated mathematical problems using a reasoning sequence that is agreeable by human experts. This success, however, is still in the black-box style. Currently, there are two primary approaches to inspiring reasoning capabilities in LLMs. For the most advanced models [ 13,51], reinforcement learning method (for example, PPO [ 36], DPO [ 30], or GRPO [ 37]) is commonly adopted to enhance the model’s ability to solve complex mathematical or programming problems in a step-by-step manner [ 48]. A more efficient alternative involves supervised fine-tuning (SFT): by providing the backbone LLM with well-prepared, diverse, and step-by-step reasoning traces—often generated through handcrafted examples or existing reasoning models [ 54,25,13,51]—the model surprisingly acquires reasoning abilities after training. However, despite the practical success of this method, the underlying mechanism remains largely unexplained. It is still unclear why or how this ability emerges. Several potential explanations may account for this phenomenon: Case 1 Is it the LLM in its entirety (i.e., the union of all its weights) that leads to this capability, such that this miracle is not explainable? Case 2 Or, is there certain module(s) in it that should be praised for this success, such that we can advance our understanding of LLMs? ∗Corresponding author. Preprint. Under review. Chat: {q, k, v}_proj & MLP Reason: o_projStethoscope for Networks 🩺 1. The Delta Stethoscope2. The Merge Stethoscope3. The Freeze Stethoscope Base Model Reasoning Model 💯minus ❓ ❓ Base Model 🧊 🔥SFT Reasoning Model Reasoning Model 💯 Base Model Merged Model merge4. The Destruction Stethoscope chat chat LLMChatReasonConjecture: Figure 1: Stethoscope for Networks. SfN is a framework designed to identify which components of an LLM give	https://arxiv.org/abs/2505.20993v1
rise to specific abilities. By comparing weight changes and observing behaviors under controlled module merging, tuning, or destruction, SfN provides interpretable insights into the origin of capabilities like reasoning. Case 3 Or in the worst scenario, is reasoning an illusion (e.g., by overfitting to certain types of data), such that we have overestimated the potentials of LLMs? A definitive answer to any of the above questions will be extremely valuable to guiding the future direction of LLM research. Even a hypothesis or conjecture supported by circumstantial evidences will be highly enlightening, too, let alone when convincing empirical evidences are available. To this end, our hypothesis is that Case 2 holds in LLMs that reason well. To be more precise, we hypothesize that it is the output projection’s parameters ( o_proj ) in the Transformer [ 43]’s multi-head self-attention (MHSA) module that is in charge of reasoning in an LLM. To support our hypothesis, we propose a few techniques for diagnosing LLM’s behaviors, in particular, the potential functionalities and impacts of various modules in it. We call these techniques Stethoscope for Networks, or SfN (summarized and illustrated in Figure 1). Starting from reasoning-enhanced models, we argue that the weight differences between a base LLM and its fine-tuned counterpart (e.g., for reasoning tasks) provide firsthand and crucial evidence for understanding internal changes. We refer to this approach as the Delta Stethoscope. In addition, we introduce two novel and previously unexplored methods within the SfN framework: the Merge Stethoscope and the Destruction Stethoscope. The Merge Stethoscope replaces specific modules in a base model with those from a reasoning model. Surprisingly, the resulting variant can maintain fluent dialogue and demonstrate improved reasoning ability in some cases. This phenomenon offers strong clues about the origin and localization of reasoning capability in LLMs. The Destruction Stethoscope, in contrast, systematically disables individual modules and observes the resulting behavior to infer the functional roles of each component. We also propose the Freeze Stethoscope, which selectively freezes parts of the model during fine-tuning. By controlling which modules are updated, we provide convincing empirical support for earlier insights and clues into the localization of reasoning within LLMs. With different gadgets we propose in SfN, we provide not only sanity check level tests for our hypothesis, but also more convincing circumstantial supports and even direct empirical evidences. In short, the contributions in this paper are two-fold: •With various diagnosis evidence (SfN), we are confident in hypothesizing that the output projection o_proj is mainly responsible for the reasoning in LLMs. The impact of this finding include not only potential ways to improve LLM that reasons (e.g., training much faster), but may generalize to produce better LLMs for other tasks (e.g., for a vertical LLM designed specifically for a domain). Our further conjecture is that other modules combined together lead to lucid conversations, but o_proj is less important in conversational ability. •The proposed Stethoscope for Networks (SfN) gadgets are a set of tools that are useful in understanding modern LLMs and even other networks, which have the potential to enhance our understanding of LLM or	https://arxiv.org/abs/2505.20993v1
deep neural network and may lead to alternative routes for further deep learning research. 2 2 Key Hypothesis: Output Projection is the Key for Reasoning To present our findings, we start by introducing necessary background information and notations, while discussions on related work are deferred to Section 5. Modern LLMs [ 41,5,29] mostly consist of many Transformer blocks. A Transformer [ 43] block is composed of a multi-head self-attention (MHSA) module and a multi-layer perceptron (MLP) module. Components in MHSA include various projections, such as those for computing Q, K and V , denoted as q_proj ,k_proj , and v_proj , respectively. The output projection ( o_proj ) produces MHSA’s output. Components in the MLP are mainly linear projections: up, down, and gate [ 16,41,5] projections, denoted as up_proj ,down_proj , and gate_proj , respectively. The computation process is defined as: xattn=wo Softmax(wqx)(wkx)⊤ √ d (wvx) xmlp=wdown σ(wgatex)⊙(wupx)(1) For simplicity, we omit residual connections and present the computation at the token level, without using matrix or vectorized notation. Other essential components not explicitly included in equa- tion 1 include rotary positional embeddings (RoPE)[ 38], input embeddings ( embed_tokens ), layer normalization[4] ( layernorm ), and the language modeling head ( lm_head ). LetAbe an LLM with weak or no reasoning ability. By carefully procuring a dataset of reasoning examples [ 13,25,51], one can cleanse and improve the quality of the dataset into the training data D, and then finetune the existing model Aby using techniques such as SFT. The resulting LLM, model B, exhibits strong reasoning capabilities. For example, in commonly adopted practices, the base LLM Ais typically a widely used open-source model such as Qwen2.5-Math-1.5B, 7B or Qwen2.5-14B, 32B [ 52]. The reasoning model Bdenotes a publicly available reasoning-enhanced variant, such as DeepSeek-R1-Distill-Qwen-1.5B, 7B, 14B, 32B [ 13], which comes with a clearly specified base model and well-documented training procedure. Models that are either not open-sourced [ 13,39], or open-sourced without sufficient training details [ 40] or access to the base model [ 51], are not discussed in this paper. 2.1 The Delta Stethoscope In the above scenario, it is obvious that AandBshare exactly the same network architecture and structure, with their sole difference being the weights (parameters) inside various components. Suppose w(A)(w(B)) denotes the set of weights for all modules in A(B). Then, it is natural to conclude that to understand the difference between AandB(i.e., reasoning or not), we should focus on the difference between w(A)andw(B). Hence, we propose our first Stethoscope for Network. Assumption 1 (The Delta Stethoscope) Suppose AandBare two LLMs with weak and strong reasoning ability, respectively, and Bis obtained by finetuning from A. Then w(B)−w(A)contains essential information if we want to pinpoint the source of the reasoning ability in B. For each component X(e.g. X=q_proj ), we compute the ℓ2norm of the weight difference, ∥wX(B)−wX(A)∥ℓ2, and visualize the results across all the blocks in Figure 2. For simplicity and due to space constraints, we present three representative comparisons: Ais Qwen2.5-Math-1.5B [ 53] or Qwen2.5-14B, 32B [ 52] and Bis DeepSeek-R1-Distill-Qwen-1.5B, 14B, 32B [ 13]. Additional results	https://arxiv.org/abs/2505.20993v1
for other model sizes (7B and 8B) are provided in the appendix and exhibit similar patterns. For the 1.5B models, the signal is less clear, but o_proj still exhibits a distinct pattern compared to q,k,v_proj —showing the largest change within the attention module and the second-largest across the entire model. As model size increases to 14B and 32B, this trend becomes more pronounced. In both cases, the most notable observation is that when X=o_proj , theℓ2norm is at least two times larger than any other component, indicating the substantial changes in this module during reasoning enhancement. In Figure 3, we further analyze the distribution of relative weight changeswX(B)−wX(A) wX(A)for each linear module. To improve clarity and visual appeal, we plot the distribution every 5 layers and clip values in the range [−1.0,1.0]to mitigate the influence of outliers. The vertical axis represents the 3 Qwen2.5-14B vs. DeepSeek-R1-Distill-Qwen-14BQwen2.5-32B vs. DeepSeek-R1-Distill-Qwen-32BQwen2.5-Math-1.5B vs. DeepSeek-R1-Distill-Qwen-1.5B q_proj k_proj v_proj o_proj up_proj gate_proj down_proj q_proj k_proj v_proj o_proj up_proj gate_proj down_proj q_proj k_proj v_proj o_proj up_proj gate_proj down_proj(Layer)Figure 2: Per-module L2 distance of linear weights between models AandB.Notably, the o_proj module shows the second-largest change in 1.5B models, and the largest in 14B and 32B models, highlighting its potential importance for reasoning. Similar trends are observed in 7B and 8B models (see appendix). Qwen2.5-32B vs. DeepSeek-R1-Distill-Qwen-32BQwen2.5-14B vs. DeepSeek-R1-Distill-Qwen-14B MLP.down_proj MLP.gate_proj MLP.up_proj self_attn.k_proj self_attn.o_proj self_attn.q_proj self_attn.v_projQwen-2.5-Math-1.5B vs. DeepSeek-R1-Distill-Qwen-1.5B Figure 3: Layer-wise distribution of relative weight changes between models AandB.While most modules display a unimodal distribution, the o_proj module uniquely exhibits a bimodal distribution, highlighting its distinctive behavior. Consistent patterns are observed across models of other sizes, with detailed results provided in the appendix. frequency. A striking and consistent finding is that all linear modules—except o_proj —exhibit a unimodal distribution centered around zero, whereas o_proj uniquely displays a clear bimodal pattern , highlighting its distinct role. Both observations hold consistently across model sizes and base models: o_proj exhibits the largest or second-largest weight shift, and the overall weight difference patterns remain strikingly similar. Therefore, it is reasonable to guess that the output projection o_proj plays a pivotal role in curating B’s reasoning ability. We are, however, not aware of o_proj ’s specific role: is it solely responsible for reasoning? Or, is it collaborating with another module(s)? Or, in the worst scenario, is this difference in ∥wX(B)−wX(A)∥ℓ2andwX(B)−wX(A) wX(A)coincidental? 4 Level I: NonsenseOutput lacks coherence or meaning.Q: Can you write a short paragraph about the importance of reading books? A1: [[[[[[ … / A2: the importance of course, the importance of course, of course, the importance of course, of course …Level II: IrrelevantGrammatically correct, but contextually oﬀ-topic.Q: Can you write a short paragraph about the importance of reading books? A: Can you provide me with some data?Level III: Context-AwareUnderstands the question, but may apply ﬂawed reasoning.Q: Every morning, Aya … This morning, if she walks at s+1/2 kilometers per hour, how many minutes will the walk take? A: First, let's set up the equations … Simplify and solve: ; … Final answer: 12144s−st−2t=2524s−st60=9Level IV: Correct and Logical ReasoningFully understands the context and provides	https://arxiv.org/abs/2505.20993v1
a valid solution.Q: Every morning, Aya … This morning, if she walks at s+1/2 kilometers per hour, how many minutes will the walk take? A: First, the problem says that … Subtract: Then: Today: speed is 3 km/h, walk = 180 min, total = 180 + 24 = 204 minutes. 9(1s−1s+2)=1.6→s(s+2)=11.25→s=2.53.6+t60=4→t=24 ❌ ⚠ 🤔 ✅Q: Can you write a short paragraph about the importance of reading books? A: Absolutely! Reading is a crucial skill that can help students improve … and understand the implications of these skills. Figure 4: Four levels of responses generated by the LLM . From level I to level IV , the model exhibits stronger language organization and logical reasoning skills. Each example includes a question (e.g., a math problem from AIME or a typical user-issued request) and the corresponding response generated by the LLM. 2.2 The Merge Stethoscope We design another gadget, the Merge Stethoscope, to answer this question. Suppose an LLM Mis formed by merging models AandB, that is, Mhas the same structure as AandB, while a subset of its modules’ parameters come from Aand the rest from B. In a conversational or reasoning task, what will the output of Mlook like? We can imagine 4 levels of different output, as Level I A sequence of random or nonsense tokens. Level II A sequence that looks like normal sentences, but does not fit into the context of the task. Level III A sequence that is meaningful sentences that match the task’s context well but will fail to reason in difficult problems. Level IV A sequence that reasons—and reasons correctly in most cases. Figure 4 shows examples of level I to IV outputs. It is worth highlighting that Misrudely merged fromAandBwithout any further tuning . Hence, the intuitive conjecture will be that Mwill produce level I output (i.e., ushering meaningless tokens). However, if model M, when merged in a specific configuration, is capable of producing level IV outputs for questions that model Afails to solve, then the specially merged components are likely critical for reasoning. Assumption 2 (The Merge Stethoscope) Suppose Mis created by merging the output projection (o_proj ) weights of B(which has strong reasoning ability) and all other components of A(which is weak in reasoning), and further suppose that Mhas stronger reasoning ability compared to A. Then, we assume o_proj is crucial in achieving reasoning in LLMs. We attempt a minimal or atomic merge by replacing only the o_proj modules in model A= Qwen2.5-Math-1.5B [53] with that of model B=DeepSeek-R1-Distill-Qwen-1.5B [13], keeping all other components unchanged. Although we initially expected the resulting model to produce level I or level II outputs, the results turn out to be surprising. On the AIME 2024 benchmark [ 19], the merged model M1achieves level IV performance on several questions that model Acannot solve. As shown in Table 1, the merged model not only yields correct reasoning and answers, but also tends to generate longer and more detailed responses compared to A. In contrast, replacing other modules such as {q,k,v}_proj andmlpleads to performance degradation. For example, model M2, which replaces	https://arxiv.org/abs/2505.20993v1
{q,k,v}_proj , produces level III outputs, while M3, which replaces mlp, deteriorates to level I. Only replacing o_proj results in a correct reasoning process and a correct answer, as illustrated in Figure 5. This striking difference motivates our further investigation in Section 3. 5 ModelReplaced ModuleAIME 2024Average Tokens A(Q-1.5B) - 0.067 2421 M1 o_proj 0.200 5418 M2 {q,k,v}_proj 0.000 2058 M3 mlp 0.000 15532 B(D-1.5B) - 0.233 11892 Table 1: AIME 2024 accuracy of the base model, the reasoning model, and their merged variants. Each merged model is constructed by replacing specific modules in model Awith the corresponding module from model B. Q: Every morning, Aya does a 9 kilometer walk … if she walks at s+1/2 kilometers per hour, how many minutes will the walk take?: To solve this problem, we need to determine … So, the walk will take 204 minutes, including the 24 minutes at the coﬀee shop. The ﬁnal answer is 204.M1: To solve this problem … output 12.0000000000000. The output indicates that the time taken for the walk is 12 minutes. So, the ﬁnal answer is 12.M2: … walking speeds increase speeds faster walking speeds increase walking speeds faster walking speeds faster walking …M3Figure 5: Examples of outputs generated by merged models. OnlyM1produces both a valid reasoning process and the correct answer. These results clearly show that the merged model Mhas a stronger reasoning capacity than A, despite thatMis sutured from two completely different models and has never being finetuned. Now we feel confident in our assumption that o_proj is the key component responsible for reasoning in LLMs. 2.3 The Freeze Stethoscope As models AandBscale up (e.g., to 7B parameters), merging components such as q,k,v_proj or mlpstill results in significant performance degradation. However, unfortunately, merging o_proj no longer brings notable improvements in solving complex mathematical problems—although it does not harm accuracy, and still increases the generated output length. Our analysis of \|\|wX(B)−wX(A)\|\|ℓ2suggests that this is due to a substantial mismatch in normal- ization parameters (that is, layernorm modules) between AandBat larger scales, compared to smaller models (e.g. 1.5B). Even when we merge both o_proj andlayernorm parameters from B, the resulting model Mstill fails to reason effectively, probably because the remaining parameters of Aare incompatible with the normalization parameters of B. To investigate this hypothesis in larger LLMs, we introduce the Freeze Stethoscope. Assumption 3 (The Freeze Stethoscope) Suppose that an LLM Fis obtained by supervised fine- tuning using the dataset D.Fis initialized from A, and both o_proj and normalization components are tuned while other components are frozen. If Fexhibits strong reasoning ability, then we assume thato_proj is crucial in achieving reasoning in LLMs even in large-scale models. It is worth noting that embed_tokens andlm_head are also tuned.2Normalization module pa- rameters are unfrozen by default. We adopt the pipeline of s1 [ 25] as our baseline, which uses the base model A=Qwen2.5-32B-Instruct and the dataset D=s1K containing 1,000 high-quality reasoning traces. The results are shown in Table 2, where our model F4corresponds to model Bin Assumption 3. We do notstrictly follow the training or testing setup of s1,	https://arxiv.org/abs/2505.20993v1
primarily due to limited computational resources and the lack of an exact testing recipe to reproduce the reported results. However, our objective is not to optimize accuracy via testing tricks or prompt tuning, but to highlight the effectiveness of o_proj tuning compared to full-parameter tuning. For fair comparison, we adopt the “Budget Forcing Wait 2x” setting from s1 and retain all configurations without hyperparameter tuning. Using this simplest possible experimental setup, Table 2 clearly shows that simply tuning o_proj andlayernorm (model F2)) leads to strong reasoning ability, while at the same time only tuning layernorm (model F1) harms the reasoning of the LLM. Further unfreezing the parameters of {q,k,v}_proj (model F3) yields little additional gain or even negative impact. The training loss curves are shown in Figure 6. When all parameters including MLP are unfrozen, the model exhibits clear signs of overfitting, likely using the large MLP capacity to memorize the training set. In contrast, tuning only o_proj yields a smoother and more stable curve. Combined 2Without tuning these components, finetuning failed to converge. 6 0.10.20.30.4 050100150200250300Loss Steps0.10.20.30.4 050100150200250300Loss Steps32B14BF1F2F3F4Figure 6: Training loss curves for fine-tuning Qwen2.5-14B,32B-Instruct on reasoning tasks. Different models unfreeze different sets of parameters, as detailed in Table 2. Model Fintuned Modules #Param (B) Steps/s AIME 2024 Math 500 GPQA Diamond A(Q-32B) - - - 0.167 0.836 0.485 F1 Emb + Head 1.5 0.055 0.200 0.756 0.444 F2 Emb + Head + o_proj 3.2 0.052 0.367 0.890 0.520 F3 Emb + Head + {q,k,v,o}_proj 5.6 0.044 0.300 0.886 0.525 F4(B) All 32.8 0.015 0.367 0.906 0.591 A(Q-14B) - - - 0.133 0.810 0.449 F1 Emb + Head 1.5 0.106 0.133 0.722 0.414 F2 Emb + Head + o_proj 2.8 0.099 0.266 0.848 0.485 F3 Emb + Head + {q,k,v,o}_proj 3.7 0.081 0.233 0.854 0.490 F4(B) All 14.7 0.053 0.266 0.872 0.530 Table 2: Reasoning performance of different fine-tuning strategies on Qwen2.5-{14B, 32B}- Instruct. Emb denotes embed_tokens , Head denotes lm_head , and Attn denotes the entire MHSA. #Param refers to the number of trainable parameters, Steps/s indicates training speed, and the last three columns report commonly used metrics for evaluating reasoning models. with its competitive performance, this suggests that the model learns to reason rather than simply memorize. Hence, we are now prepared and feel supported to propose our key hypothesis: Hypothesis 1 (Outstanding Output Projection) In an LLM that reasons well, we hypothesize that the output projection ( o_proj ) component is the single or at least the most important module that dominates its reasoning ability. With carefully chosen tuning strategy and hyperparameters, there is reason to believe that tuning only o_proj (+LN) can reach the level of model Bin terms of reasoning performance. And, beyond exhibiting reasoning abilities, Table 2 also shows that tuning only o_proj (+LN) has other significant advantages: e.g., significantly faster finetuning (3 times faster) and smaller GPU memory consumption. These advantages will become more established when larger LLMs are tuned. 3 Conjecture: Conversation Hinges on Other Modules but Not Output We are mainly concerned with two abilities of LLMs: conversation and reasoning, which	https://arxiv.org/abs/2505.20993v1
map to level III and IV in our categorization of LLM’s outputs, respectively. Our Hypothesis 1 is on reasoning, but are there one module or several modules accounting for lucid conversations? In this section, we further propose a new stethoscope to diagnose this question and raise our conjectures accordingly. 3.1 The Destruction Stethoscope Our previous stethoscopes follow a “constructive proof” style, while now we resort to the “proof by contradiction” style. If one module in an LLM is “destructed”, and the LLM can still produce level III conversation outputs, then we have good reasons to guess that this module is not important in conversational ability; while it is important if the LLM ceases to dialogue regularly. Assumption 4 (The Destruction Stethoscope) Suppose a module Xis destructed (i.e., its normal functionality is disabled by some destruction method) in an LLM A. We denote the resulting LLM as 7 Destruction MethodModuleOutput LevelDestruction MethodModuleOutput Level Zeroq_proj I ReInitq_proj I k_proj I k_proj I v_proj III v_proj II o_proj III o_proj III up_proj I up_proj I gate_proj I gate_proj I down_proj I down_proj I Remove - I Table 3: Output levels of different modules under the three destruction methods: Zero ,ReInit , andRemove .All experiments are based on Qwen2.5-32B with destruction applied to specific layers. D. Then, the fact that Dcontinues (or ceases to) produce level III output (meaningful sentences in the conversation’s context) indicates whether Xis important for conversational abilities or not. We propose 3 destructors to destroy a module: Zero Set all parameters within Xto 0. ReInit Re-initialize all parameters inside Xusing Gaussian random numbers (mean=0, std=0.02). Remove Remove the entire layer. TheZero destructor is often equivalent to setting the output activation of Xto zeros (e.g., in a linear module like o_proj ). We want to emphasize that ReInit incurs more serious damages to an LLM thanZero does. Zero may change activations to zero, but ReInit exerts random effects (i.e., noise) to LLM activations. What is more important, these random effects will act as input to the next Transformer block and the noise is quickly amplified. Hence, level I or II output is expected when X is destroyed (especially when reinitialized) in a large number of Transformer blocks. 3.2 Conjectures Concerning the Conversation Capability For model Qwen2.5-32B with 64 layers, we observe that destroying modules in early or late layers— where input and output representations are more sensitive—consistently yields level I outputs. To avoid this, we restrict destruction to blocks 5–30. This range is empirically chosen, as affecting more layers often causes all outputs to degrade to level I, making distinctions between modules impossible. The experimental results are presented in Table 3. Specifically, we destroy selected modules and analyze the corresponding output. The Remove destructor removes the transformer layers as a whole. Note that the results are not statistics computed in many different experiments—it only reflects the conversation illustrated in Figure 4, but we observed similar patterns for other conversations. Table 3 reveals distinct roles of modules in conversation. Notably, o_proj —crucial for reasoning— appears unimportant for conversation. In contrast, all MLP components (	https://arxiv.org/abs/2505.20993v1
up_proj ,down_proj , gate_proj ) are essential. Within MHSA, q_proj andk_proj are important, while v_proj plays a minor role. Based on these (admittedly weaker) observations, we propose the following conjecture. Conjecture 1 (Division of Labor) Based on current observations, an LLM can be roughly divided as two sets of modules: output projection ( o_proj ) and all others, where o_proj is mainly responsi- ble for reasoning and other modules for conversation. Then, output projection plays a unique role if this conjecture holds. Hence, we further propose another conjecture for it. Conjecture 2 (Output Projection Plugin) With conversational capabilities provided by other (frozen) modules, output projections may act as a plugin. For example, one set of o_proj for reasoning, and another set of o_proj for migrating an LLM to a vertical domain. 8 4 Potential Implications and Applications This paper mainly diagnoses LLMs from a theoretical, highly abstract perspective. However, our hypothesis and conjectures can also have highly practical implications and applications as long as they are correct or at least partially hold. •Fast and better reasoning LLMs . By finetuning only o_proj , we can potentially find a better reasoning LLM with much faster training and much smaller GPU memory footprint. •Integrating non-reasoning and reasoning LLMs. There is a recent trend to integrate chatting and reasoning LLMs into one model [ 51]. When we finetune a base LLM into a reasoning one using the previous procedure, they only differ in o_proj ,layernorm , embed_tokens andlmhead , which occupy only 10% of model size. Hence, the two LLMs are easily loaded as one LLM with two sets of these module for different purposes. •Vertical LLMs . Similarly, when equipped with different output projection plugins, one may adeptly obtain vertical LLMs for different domains. •Understanding deep neural networks. The proposed Stethoscopes for Networks might be useful gadgets to understand other deep models, and new stethoscopes can be further developed. They will be potentially useful in diagnosing existing networks and even in providing alternative directions to future deep learning research. 5 Related Work Large Language Models. Modern LLMs such as GPT [ 29,6], LLaMA [ 41,42], Qwen [ 5,52], and other representative models [ 7,20] adopt an auto-regressive architecture and have demonstrated impressive capabilities across a wide range of natural language processing tasks, including question answering [ 32,22], summarization [ 26,27], and translation [ 50]. These models are typically trained on large-scale corpora using next-token prediction objectives, and their performance has been shown to scale with model size [ 21]. Further improvements in alignment and usability have been achieved through instruction tuning [ 28,9,46] and reinforcement learning from human feedback (RLHF) [8, 30], enabling more controllable and helpful dialogue generation. Reasoning Models. While LLMs exhibit emergent reasoning abilities [ 47], recent efforts have further enhanced these capabilities through fine-tuning and architectural modifications [ 35,55]. Chain-of-thought prompting [ 48] encourages intermediate reasoning steps, improving performance in arithmetic tasks, while self-consistency decoding [ 45] improves robustness by sampling multiple reasoning paths. Inspired by OpenAI’s o1[18], most advanced models now employ reinforcement learning [ 36,30] to generate long	https://arxiv.org/abs/2505.20993v1
reasoning traces with sparse rewards. This leads to significant improvements, particularly in complex math, code, and other professional domains [ 13,51]. Despite these advances, the origin and location of reasoning ability in LLMs remain underexplored. Interpretability of LLMs. Understanding the inner workings of LLMs has attracted growing interest. Prior efforts include attention visualization [ 44], probing [ 15], and model editing [ 24,34], with the aim of interpreting internal representations. Other studies decompose the behavior of the model into attribute functions to specific modules [ 11]. The "Physics of Language Models" series [ 1,2,3] investigates LLMs through controlled setups to reveal empirical and universal laws that dictate LLM behavior. However, these studies often exclude the most advanced models or focus on narrow, synthetic settings, offering limited insight into real-world models. Their findings provide little practical guidance for understanding reasoning in state-of-the-art models. 6 Conclusions This work investigates a fundamental question in understanding large language models (LLMs): Is there a component or several components that are responsible for achieving the reasoning ability in LLMs? If the answer is affirmative, which components are responsible for the improvement? We hypothesize that the output projection ( o_proj ) module plays a central role in enabling reasoning capabilities. To support this, we propose Stethoscope for Networks (SfN) , a diagnostic framework 9 that encompasses several probing techniques. Through the proposed Delta ,Merge ,Freeze , and Destruction stethoscopes, we observe consistent patterns indicating that o_proj is critical for reasoning, while other modules primarily support conversational fluency. These findings open new directions for efficient and modular LLM training. Our findings are primarily based on a limited set of model families and reasoning benchmarks, and may not generalize to all architectures or tasks. Some diagnostic results rely on qualitative assessments rather than statistical validation. Furthermore, while the role of o_proj is empirically highlighted, a theoretical understanding of its function in reasoning remains to be established. Acknowledgments and Disclosure of Funding This work was partly supported by the National Natural Science Foundation of China under Grant 62276123 JW proposed the assumptions (Stethoscopes for Networks), hypothesis and conjectures. JS started this line of research in our group, proposed the Zero destructor, and effectively supported our main findings with experimental results. JW and JS wrote the paper. We thank Ke Zhu for discussions. A Experimental Details We primarily utilize open-sourced models to conduct experiments in this work. Given that DeepSeek- R1 is one of the most widely adopted reasoning models, and its authors have released a series of distilled models based on R1 [ 13], including both the specified base and finetuned reasoning models, we adopt their configurations in our study. Specifically, we use the DeepSeek-R1-Distill-Qwen [ 13] models with sizes of 1.5B, 7B, 14B, 32B and 70B as our reasoning models, and select Qwen2.5- Math-1.5B, 7B [ 53], LLaMA3.1-8B [ 12], Qwen2.5-14B, 32B [ 52] or Llama-3.3-70B-Instruct [ 12] as base models. All models are loaded and run using the Transformers library [49]. Our evaluation framework is based on the lm-evaluation-harness package [ 10]. To accelerate inference, we use vLLM [ 23]	https://arxiv.org/abs/2505.20993v1
as the backend, which may slightly affect performance due to backend-specific optimizations. In the Merge Stethoscope experiments, we observe that the “chat” interface often generates irrelevant or nonsensical responses, while the “generate” interface produces coherent and contextually appropriate outputs. We suspect this discrepancy arises from misinterpreted system prompts. Therefore, we rely on the “generate” interface and implement a custom evaluation toolkit. For the Freeze Stethoscope experiments, we build on the codebase of s1[ 25]. We use a learning rate of 1e-5, weight decay of 1e-4, a batch size of 16, and train for 5 epochs. Due to hardware limitations (i.e., lack of access to 16 H100 GPUs), we leverage DeepSpeed[ 33] with ZeRO Stage 3[31] to enable efficient training. The base model used here is Qwen2.5-32B-Instruct[ 52]. Evaluation is again conducted with lm-evaluation-harness, following the modified pipeline by the authors of s1, which disables generation of the end-of-thinking token and optionally appends the string “Wait” to the reasoning trace to encourage model reflection. We adopt the Budget Forcing “Wait” ×2 as our default testing configuration. All visualization and inference experiments on 1.5B–14B models are conducted on a single NVIDIA A100 GPU. For training and evaluating 32B-70B models, we use a cluster of 8 NVIDIA A100 GPUs. Training typically takes around 6 hours, while testing on a single dataset usually requires about 2 hours. B More Experimental Results In the main paper, we present visualization results for the 1.5B, 14B, and 32B models. Here, we supplement those results by providing additional visualizations for the 7B, 8B, and 70B models. Following the Delta Stethoscope pipeline, we visualize both the absolute weight shift \|wX(B)− wX(A)\|ℓ2and the relative weight shiftwX(B)−wX(A) wX(A). The absolute weight shifts are shown in Figure 7, and the relative weight shifts are presented in Figure 8. The trends observed in the main paper remain consistent across these additional models. Notably, o_proj consistently exhibits the 10 Qwen2.5Math–7B vs. DeepSeek-R1-Distill-Qwen-7BLlama-3.1-8B vs. DeepSeek-R1-Distill-Llama-8BLlama3.3-70B-Instruct vs. DeepSeek-R1-Distill-Llama-70BFigure 7: Per-module L2 distance of linear weights between models AandB.Notably, the o_proj module shows the largest in 7B, 8B and 70B models, highlighting its potential importance for reasoning. Llama3.3-70B-Instruct vs. DeepSeek-R1-Distill-Llama-70BLlama-3.1-8B vs. DeepSeek-R1-Distill-Llama-8B MLP.down_projMLP.gate_projMLP.up_projself_attn.k_projself_attn.o_projself_attn.q_projself_attn.v_projQwen2.5Math–7B vs. DeepSeek-R1-Distill-Qwen-7B Figure 8: Layer-wise distribution of relative weight changes between models AandB.While most modules display a unimodal distribution, the o_proj module uniquely exhibits a bimodal distribution, highlighting its distinctive behavior. largest weight shift, with the effect being especially pronounced in the 70B model. Moreover, o_proj is the only module that displays a bimodal distribution in the relative weight shift. C Statistical Significance and Broader Impacts We report appropriate information regarding the statistical significance of our experiments. While we do not primarily focus on classical significance tests such as p-values, we provide multiple forms of empirical evidence—such as consistent module-specific weight shifts, response-level comparisons under controlled manipulations, and loss curves under different tuning strategies—that collectively establish the robustness of our findings. These analyses serve as a practical alternative to traditional error bars or confidence intervals and help substantiate our key claims. This research has both promising benefits and important risks to consider. On the positive side,	https://arxiv.org/abs/2505.20993v1
the proposed Stethoscope for Networks (SfN) framework provides a novel set of tools for interpreting 11 LLMs, especially by localizing specific capabilities—such as reasoning—to individual components like the output projection (o_proj). These tools may significantly improve our understanding of LLMs, enabling more transparent, modular, and efficient model development. For instance, if reasoning abilities can be enhanced by tuning a small subset of parameters, it could greatly reduce computational costs and increase accessibility for developing domain-specific or lightweight models. However, this line of work also carries potential risks. Precisely identifying and isolating reasoning- related components might lower the barrier for targeted manipulation, such as unauthorized transfer or removal of reasoning abilities across models. This could facilitate misuse scenarios, including capability extraction, tampering, or model theft. Furthermore, while the diagnostic methods proposed aim to support interpretability, there is a risk that they may be overinterpreted, leading to an inflated sense of model transparency that does not generalize across architectures or tasks. References [1]Zeyuan Allen-Zhu and Yuanzhi Li. Physics of language models: Part 3.1, knowledge storage and extraction. arXiv preprint arXiv:2309.14316 , 2023. [2]Zeyuan Allen-Zhu and Yuanzhi Li. Physics of language models: Part 3.2, knowledge manipula- tion. arXiv preprint arXiv:2309.14402 , 2023. [3] Zeyuan Allen-Zhu and Yuanzhi Li. Physics of language models: Part 3.3, knowledge capacity scaling laws. arXiv preprint arXiv:2404.05405 , 2024. [4]Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450 , 2016. [5]Jinze Bai, Shuai Bai, Yunfei Chu, Zeyu Cui, Kai Dang, Xiaodong Deng, Yang Fan, Wenbin Ge, Yu Han, Fei Huang, et al. Qwen technical report. arXiv preprint arXiv:2309.16609 , 2023. [6]Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neural information processing systems , 33:1877–1901, 2020. [7]Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Nina Mielke, Alec Radford, et al. Palm: Scaling language modeling with pathways. arXiv preprint arXiv:2204.02311 , 2022. [8]Paul F Christiano, Jan Leike, Tom B Brown, Miljan Martic, Shane Legg, and Dario Amodei. Deep reinforcement learning from human preferences. In Advances in neural information processing systems , volume 30, 2017. [9]Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Xin Wang, Xingyu Yuan, Adams Yu, Sharan Narang, et al. Scaling instruction-finetuned language models. arXiv preprint arXiv:2210.11416 , 2022. [10] Leo Gao, Jonathan Tow, Baber Abbasi, Stella Biderman, Sid Black, Anthony DiPofi, Charles Foster, Laurence Golding, Jeffrey Hsu, Alain Le Noac’h, Haonan Li, Kyle McDonell, Niklas Muennighoff, Chris Ociepa, Jason Phang, Laria Reynolds, Hailey Schoelkopf, Aviya Skowron, Lintang Sutawika, Eric Tang, Anish Thite, Ben Wang, Kevin Wang, and Andy Zou. The language model evaluation harness, 07 2024. [11] Mor Geva, Tal Schuster, and Jonathan Berant. Transformer feed-forward layers are key-value memories. arXiv preprint arXiv:2012.14913 , 2021. [12] Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, et al. The llama 3 herd of models. arXiv preprint arXiv:2407.21783 , 2024. [13] Daya Guo, Dejian Yang, Haowei	https://arxiv.org/abs/2505.20993v1
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arXiv:2505.20997v1 [cs.LG] 27 May 2025BIPNN: L EARNING TO SOLVE BINARY INTEGER PROGRAMMING VIAHYPERGRAPH NEURAL NETWORKS Sen Bai Changchun University of Science and Technology, China [email protected] Yang Changchun University of Science and Technology, China [email protected] Xin Bai Huawei Technologies Co. Ltd China [email protected] Zhang Changchun University of Science and Technology, China [email protected] Jiang Changchun University of Science and Technology, China [email protected] May 28, 2025 ABSTRACT Binary (0-1) integer programming (BIP) is pivotal in scientific domains requiring discrete decision- making. As the advance of AI computing, recent works explore neural network-based solvers for integer linear programming (ILP) problems. Yet, they lack scalability for tackling nonlinear chal- lenges. To handle nonlinearities, state-of-the-art Branch-and-Cut solvers employ linear relaxations, leading to exponential growth in auxiliary variables and severe computation limitations. To over- come these limitations, we propose BIPNN (Binary Integer Programming Neural Network), an unsupervised learning framework to solve nonlinear BIP problems via hypergraph neural networks (HyperGNN). Specifically, (I)BIPNN reformulates BIPs-constrained, discrete, and nonlinear ( sin, log,exp) optimization problems-into unconstrained, differentiable, and polynomial loss functions. The reformulation stems from the observation of a precise one-to-one mapping between polynomial BIP objectives and hypergraph structures, enabling the unsupervised training of HyperGNN to op- timize BIP problems in an end-to-end manner. On this basis, (II)we propose a GPU-accelerated and continuous-annealing-enhanced training pipeline for BIPNN. The pipeline enables BIPNN to optimize large-scale nonlinear terms in BIPs fully in parallel via straightforward gradient descent, thus significantly reducing the training cost while ensuring the generation of discrete, high-quality solutions. Extensive experiments on synthetic and real-world datasets highlight the superiority of our approach. 1 Introduction For decades, binary integer programming (BIP)—a powerful mathematical tool characterized by discrete binary decision variables (0 or 1)—is of critical importance in numerous domains, such as operational optimization [ 1,2,3], quantum computing [ 4,5,6], computational biology [ 7,8], materials science and computational chemistry [ 9,10]. However, BIP is known to be NP-complete [11], making large-scale BIP instances computationally intractable. Along with AI computing shines in scientific discovery, the potential of neural network-based IP solvers has emerged in recent years. To address integer linear programming (ILP) problems, MIP-GNN [ 12] leverages graph neural networks (GNN) to improve the performance. Another GNN&GBDT-guided framework [ 13] for large-scale ILP problems can save up 99% of running time in achieving the same solution quality as SCIP [ 14], a leading IP solver. However, these neural network-based ILP solvers lack scalability for nonlinear BIPs. To handle nonlinearities, state-of-the-art Branch-and-Cut solvers (e.g., SCIP [ 15]) rely on linear relaxation, which introduces a number of auxiliary variables. Once linearized, these problems are solved using linear programming APREPRINT - M AY28, 2025 ：H 1e 2e 3e 4e 5e 6.0 2.0 3.0 6.0 2.0 3.0 6.0 1.0 1e 2e 3e 4e 5e 6.0 2.0 3.0 1.0 6.0 2.0 3.0 6.0 1.0 6.0 1.0 ：x Matrix Incidence 1e 2e 3e 4e 5e 1v 2v 3v 4v 1x 2x 3x 4x 6.0 2.0 3.0 1.0 ... ... ... ... Tensors Random Sigmoid 2x 1x 3x 4x 5 3 231 12 xx exx 4 43 21 1 2 8451.0 x xx	https://arxiv.org/abs/2505.20997v1
xx x   4 3 7183.1 231 2 1   xx x x 321 4 43 21 1 2 8451.0 xxx x xx xx x   Loss PUBO min 1x 2x 4x 3x ion Reformulat Polynomial Problem BIP ion Reformulat ned Unconstrai 1 2 11x 12x 03x 04x HyperGNN optimize to Training Converge 4 43 21 1 2 ) (sin x xx xx x   s.t. s.t. }1,0{ix }1,0{ix 4,3,2,1i 4,3,2,1i Hypergraph 1v 2v 2e 3e 5e 1e 3v 4v 4e ... ,, ,PUBO     kjikjiijk jijiij iii xxxQ xxQ xQ O   x 1 x ColMB PUBO   TQH- H O Solutions ： Framework BIPNN The Optimizer based- Network Neural ： Workflow Training 4e HyperGNN ]1,0[ix Relaxation Output optimize Train to Loss PUBO loss PUBO d accelerate- GPU loss PUBO d accelerate- GPU : Example Output ：H-1 1e 2e 3e 4e 5e 0 1 ：TQ ColM 1eQ 2eQ 3eQ 4eQ 5eQ 0 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 solutions to Converge Optimizer based- Network Neural min min min min Annealing⨀ 1 1.0 ：H- H 1 x B ⨀ ：HBx ⨀ 12.0 03.0 04.0 a b c d Figure 1: The BIPNN framework. (LP) solvers (e.g., the Simplex method1). Consequently, large-scale nonlinear BIPs often suffer from prohibitive computational costs. As BIP solvers continue to evolve, linearization remains indispensable for making nonlinearities more tractable for BIP solvers. These limitations motivate us to develop a streamlined and general-purpose BIP solver to advance the state of the art. To profoundly adapt to real-world applications, our work grapples with challenges arising from neural networks’ unique characteristics beyond linearization-based methods, as summarized below: Challenge 1 . Meticulously modeling nonlinear terms in BIP objectives and constraints; Challenge 2 . Utilizing GPU’s parallel computing capability. To this end, in this work we propose BIPNN (Binary Integer Programming Neural Network), an unsupervised BIP solver that bridges the gap between nonlinear BIP and deep neural networks. Our overarching idea stems from the observation of one-to-one mapping correspondence between polynomial BIP objectives and hypergraph structures (upper right of Fig. 1). As depicted in Fig. 1, our framework consists of three phases: 1) In the first phase, we employ broadly applicable penalty term method to convert constrained BIP problems into polynomial unconstrained binary optimization (PUBO2) formalism. To handle exponential and trigonometric terms, we propose a novel transformation to represent them in the form of polynomials. These refined polynomial objectives are adaptable to neural network-based solvers when applied as loss functions. 1To be precise, the Simplex method is designed to solve linear programming (LP) problems in polynomial time, meaning they belong to the class P [16]. 2The mathematical formulation PUBO is well-known in quantum computing, for modeling complex optimization problems in a way	https://arxiv.org/abs/2505.20997v1
quantum computers may solve efficiently. 2 APREPRINT - M AY28, 2025 2) In the second phase, we leverage hypergraph neural networks (HyperGNN) to address Challenge 1 , capturing high-order correlations between binary decision variables, or in other words the polynomial terms in the refined PUBO objective. By applying a relaxation strategy to the PUBO objective to generate a differentiable loss function with which we train the HyperGNN in an unsupervised manner. 3) Nevertheless, when we train these HyperGNNs to minimize the PUBO objectives, we encounter severe obstacles of low computational efficiency in these polynomial losses with numerous variables. In the third phase, leveraging GPUs, we further propose an algorithm to address Challenge 2 via matrix operations on the incidence matrices of hypergraphs. In summary, we contribute: 1) BIPNN, an unsupervised HyperGNN-based solver that allows learning approximate BIP solutions in an end-to-end differentiable way with strong empirical performance. 2) An empirical study of the performance of BIPNN on synthetic and real-world data, demonstrating that unsupervised neural network solvers outperform classic BIP solvers such as SCIP and Tabu in tackling large-scale nonlinear BIP problems. 3) Large-scale nonlinear optimization has long been challenging due to its inherent complexity and scalability issues. We advance this field by employing several nonlinearity modeling methods for BIP, including the polynomial reformulation and unconstrained reformulation. These methods provide instructive guidance for unsupervised neural network-based solvers. 2 Notations and Definitions In the following, we will formulate the BIP problem and articulate the definition of hypergraphs. Definition 1 (Formulation of BIP). Non-linear BIP is an optimization problem where the decision variables x= (x1, x2, ..., x m)are restricted to binary values ( 0or1), and the objective function OBIPor constraints (or both) are nonlinear. Below is the general formulation. min OBIP=f(x) s.t. gk(x)≤0 for all k= 1,2, . . . , K ql(x) = 0 for all l= 1,2, . . . , L xi∈ {0,1}for all i= 1,2, . . . , n(1) where f(x),gk(x)andql(x)are nonlinear functions of the decision variables x. □ Definition 2 (Hypergraph). A hypergraph is defined by G= (V, E), where V={v1, v2, ..., v \|V\|}stands for a set of vertices and E={e1, e2, ..., e \|E\|}denotes a set of hyperedges. Each hyperedge ej∈Eis a subset of V. A hypergraph Gcan be represented by the incidence matrix (Fig. 1 at the bottom) H∈ {0,1}\|V\|×\|E\|, where Hij= 1ifvi∈ej, or otherwise Hij= 0. □ 3 BIPNN: HyperGNN-based Optimizer for PUBO-formulated BIP For easier comprehension of our approach, in this section we first elaborate how to solve an unconstrained, PUBO- formulated BIP problem as depicted in Eq. 2. Then, in Sec. 4, we will show how to transform a general BIP problem with constraints and nonlinear terms into PUBO formalism. 3.1 Modeling PUBO-formulated BIPs via Hypergraphs BIPNN employs a HyperGNN-based optimizer (upper right of Fig. 1) to solve PUBO-formulated BIP problems. Inspired by the binary characteristic of variables, we can reformulate general BIPs as PUBO problems through the polynomial reformulation in Sec.4.1 and unconstrained reformulation in Sec.4.2. A PUBO problem is to optimize the cost function: OPUBO =X iQixi+X i,jQijxixj+X i,j,kQijkxixjxk+···	https://arxiv.org/abs/2505.20997v1
(2) where xi∈ {0,1}are binary descision variables and the set of all decision variables is denoted by x= (x1, x2,···, xm). As shown in Fig. 2, for ease of representation, a PUBO objective OPUBO withnterms can be decomposed into two components: the PUBO matrix Q= [Q1, Q2, ..., Q n], and nlinear or polynomial terms such as xi,xixj, orxixjxk. 3 APREPRINT - M AY28, 2025 3215 44 433 212 11 xxxQxQxxQxxQxQ O PUBO 4 4~xe 321 5~ xxxe ],,,,[5 4 3 2 1 QQQQQ matrix Incidence 1e 2e 3e 4e 5e 1x 2x 3x 4x ： matrix PUBO ： terms Polynomial 1 1~xe 43 3~xxe 21 2~xxe 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 1x 2x 2e 3e 5e 1e 3x 4x 4e ：H Figure 2: Modeling PUBO-formulated BIPs via hypergraphs. In this way, we discover multi-variable interactions in OPUBO can be modeled as a hypergraph G= (V, E), where \|E\|=n, and each hyperedge e∈Eencodes a single descision variable xior a polynomial term such as xixjor xixjxk. 3.2 Neural Network-based Optimizer The training workflow of the neural network-based optimizer is illustrated at the bottom of Fig. 1. HyperGNN Architecture. Initially, for a PUBO-transformed hypergraph G= (V, E), HyperGNNs take the incidence matrix HofGand a randomly initialized X(0)∈Rm×das inputs. Subsequently, BIPNN applies the sigmoid function to produce the output vector x= (x1, x2,···, xm), where xi∈[0,1]are the relaxation of decision variables xi∈ {0,1}. The HyperGNN model operates as follows: x= sigmoid(HyperGNN( H, X(0))) (3) where HyperGNN is a multi-layer hypergraph convolutional network, such as HGNN+ [ 17], HyperGCN [ 18], or UniGCN [19]. Training to Optimize. As an unsupervised learning model, BIPNN relaxes the PUBO objective OPUBO into a differentiable loss function and trains to optimize it. Specifically, OPUBO can be expressed by the output xand the incidence matrix Has depicted in Fig. 1. We aim to find the optimal solution xs= argmin OPUBO (x, H). As training progresses, xi∈xwill gradually converge to binary solutions. GPU-accelerated Training. For a large-scale BIP problem, numerous polynomial terms in OPUBO lead to a high computational cost. To address this, an intuitive idea is to leverage GPU-supported matrix operations to accelerate training. However, PUBO problems lack a straightforward matrix formulation. To this end, we propose GPU-accelerated PUBO objective as follows. OPUBO = ColM( x⊙(B)H+ (1−H))QT(4) where xis the output of HyperGNN, His the incidence matrix, and Q= [Q1, Q2, ..., Q n]is the PUBO matrix. More concretely, x⊙(B)Hdenotes the element-wise Hadamard product with broadcasting between m-dimensional vector x and matrix H∈Rm×n. We add 1−Honx⊙(B)Hto fill zero-valued elements with 1. Based on this operation, we use the column-wise multiplication denoted by ColM on the first dimension of the matrix obtained by x⊙(B)H+ (1−H). Through the ColM operation we obtain an n-dimensional vector, of which each element represents a polynomial term inOPUBO . The final loss function is computed by scaling each polynomial term with its respective coefficient Qi. The detailed explanation is illustrated in Fig. 1. Time Complexity Analysis. Forx∈Rm,Q∈R1×n, andH∈Rm×n, the time complexity of	https://arxiv.org/abs/2505.20997v1
Eq. 4 is O(m×n). For GPU-accelerated training, element-wise operations such as Hadamard product are fully parallelizable. Column-wise product over mleads to time complexity O(logm). Thus, the theoretical best GPU time complexity is O(logm). Utilizing Tcores, the realistic GPU time complexity is O(m×n T). Annealing Strategy. To achieve unsupervised learning, BIPNN relaxes PUBO problems into continuous space. The differentiable relaxation of discrete decision variables sometimes leads to continuous solutions xi∈[0,1]. To address this, we employ the continuous relaxation annealing (CRA) [ 20] method. Specifically, BIPNN uses the following loss function: OPUBO = ColM( x⊙(B)H+ (1−H))QT+ϕ(x), where ϕ(x) =γPn i=1(1−(2xi−1)α)is the penalty term, γcontrols the penalty strength and αis an even integer. We initialize γ <0and gradually increase it to a positive value as training progresses. The annealing strategy enhances the performance of BIPNN in three aspects, (i)In the high-temperature phase ( γ <0), it smooths the HyperGNN, preventing it from getting trapped in local optima; (ii)In the low-temperature phase ( γ >0), it enforces the discreteness of solutions; (iii)It effectively accelerates the training process. 4 APREPRINT - M AY28, 2025 4 BIPNN: Polynomial & Unconstrained Reformulation of BIP In this section, we explain how to reformulate nonlinear BIPs as unconstrained and polynomial optimization problems, which are compatible with our neural network-based optimizer. 4.1 Polynomial Reformulation of BIP Our approach is inspired by the observation that for any binary variable, a nonlinear term such as excan be exactly fitted by a polynomial equivalent h(x) =ax+b, such that h(x) =exforx∈ {0,1}. That is, h(x) = (e−1)x+ 1, where h(0) = 1 andh(1) = e. To handle univariate nonlinearities, including trigonometric, logarithmic, and exponential terms (e.g., sinx,logx, andex), we have the following transformation: h(x) = (h(1)−h(0))x+h(0). For multivariate terms such as exixjandsin(xixj), where xixj∈ {0,1}, we can perform the transformation as follows: h(Q i∈Sxi) = (h(1)−h(0))Q i∈Sxi+h(0). BIPNN employs a more general method to handle more intricate multivariate nonlinear terms (such as sin(xi+xj)). For a set of binary decision variables x1, x2, ..., x n, a non-linear function h(x1, x2, ..., x n)can be transformed into the polynomial forms as follows. h(x1, x2, ..., x m) =X S⊆{1,2,...,m}cSY i∈Sxi (5) By setting up a system of equations based on all possible combinations of x1, x2, ..., x m, we can determine the coefficients cSto precisely fit h(x1, x2, ..., x m)by leveraging simple inclusion-exclusion principle (refer to Appendix A) as below. cS=X T⊆S(−1)\|S\|−\|T\|f(T) (6) where f(T)represents the function value when the variables in the subset Tare1and the others are 0. For each subset S, it needs to calculate 2\|S\|values of f(T). □ As an example, we have sin(x1+x2) = 0 .8415x1+ 0.8415x2−0.7737x1x2. A toy example of sin(x1+x2+x3) is illustrated in Appendix A. To be noticed, polynomial reformulation of all nonlinear terms in a BIP objective is not necessary. If the transformation becomes overly complex, we may opt to retain the original nonlinear term and directly incorporate it as part of the loss function of HyperGNN. 4.2 Unconstrained Reformulation of BIP We propose a novel penalty method to transform the constrained BIP problem into an unconstrained form.	https://arxiv.org/abs/2505.20997v1
In penalty methods [ 21,22], unconstrained reformulation is achieved by adding "penalty terms" to the objective function that penalize violations of constraints. A well-constructed penalty term must be designed such that it equals 0if and only if the constraint is satisfied, and takes a positive value otherwise. Specifically, given a BIP problem in Eq. 1, for inequality constraints gk(x)≤0, we have penalty terms Pk(x) =λk·(max (0 , gk(x)))2, for equality constraints ql(x) = 0 , we have penalty terms Ql(x) =µl·(ql(x))2, where λk, µlare sufficiently large penalty coefficients. By combining all terms into a single objective function, we have an unconstrained BIP objective: min OBIP=f(x) +KX k=1λk·(max (0 , gk(x)))2+LX l=1µl·(ql(x))2(7) As part of the loss function of BIPNN, OBIPmust be differentiable to enable gradient-based optimization. However, max (0 , gk(x))is not a continuously differentiable function, thus finding an appropriate penalty term is crucial. We propose two methods to address this issue: 1)ReLU -based Penalty . We can use ReLU( gk(x))2= (max(0 , gk(x)))2to handle constraints. This is a general method for a large number of variables xiin a constraint gk(x). 2)Polynomial Penalty . In the following, we present an algorithm to construct polynomial penalty terms with 2∆time complexity for gk(x), where ∆is the number of variables in constraint gk(x). For binary variables, do there exist polynomial penalty terms that correspond to BIP constraints? To answer this question, we have the following discussion. For x1+ 2x2−2≤0, we observe that the violating subset {x1= 1, x2= 1} corresponds to polynomial penalty term λ(x1x2). For another constraint x1+ 3x2−2≤0, the violating subsets {x1= 0, x2= 1}and{x1= 1, x2= 1}correspond to the polynomial penalty term λ(x2+x1x2)orλx2. Through an in-depth analysis, we propose a novel method to transform nonlinear BIP constraints into polynomial penalty terms. To 5 APREPRINT - M AY28, 2025 43 32 31 21 4 3 2 1 2 3 2 2 2 xx xx xx xx x x x x −−−−+++ 1x 2x 4x 3x 1e 2e 3e 1x 2x 4x 3x 1e 2e 3e min BIPNN problemcut max− 1x 2x 4x 3x 1e 2e 3e 1x 2x 4x 3x 1e 2e 3e 4e 1x 2x 4x 3x 1e 2e 3e 4e Loss PUBO modeling hypergraph optimize to training？？ Figure 3: To solve the hypergraph max-cut problem, BIPNN generates a new hypergraph structure. However, both of these hypergraphs can be utilized for training the HyperGNN model. handle an inequality constraint g(x)≤0for the BIP problem in Eq. 1, our method consists of three steps (to see a toy example, refer to Appendix B): (i)Initially, we express the constraint g(x)≤0as a boolean indicator function: ψ(x) =1ifg(x)>0 (violation ) 0otherwise (feasible ), then define minimal violation subsets Vas the smallest variable combinations causing constraint violations: V= S⊆ {1, ..., n} ψ(x) = 1 when xi= 1∀i∈Sandxj= 0∀j /∈S (8) eachS∈ V cannot be reduced further without eliminating the violation. (ii)Generate a penalty term for each minimal violation subset S∈ V: P(x) =λX S∈VY i∈Sxi (9) where λis the penalty coefficient. (iii)Combine each term into the BIP objective function: min OBIP=f(x) +P(x) (10) In the worst	https://arxiv.org/abs/2505.20997v1
case, when an enumeration method is used in step (i), it requires calculating 2∆subsets, where ∆is the number of variables in constraint g(x). Nevertheless, in most real-world problems (e.g. max-cut, and maximal independent set or MIS) involving graphs, the variables associated with each constraint often exhibit locality. □ The polynomial penalty method facilitates to incorporate penalty terms to PUBO objectives and use GPU-accelerated training pipeline to solve BIPs. As far as we know, only a few number of constraint/penalty pairs [ 22] associated have been identified in existing literature. Our work significantly expands the potential application domains of the penalty method. 5 Discussion Feasible Solutions. Firstly, a PUBO problem always has feasible solutions. The feasible set is the entire space of binary variable combinations, since there are no constraints to exclude any combination. Every possible binary assignment xi∈ {0,1}is inherently feasible. Secondly, the feasibility of a nonlinear BIP problem depends on the constraint compatibility—whether there exists at least one binary variable assignment x∈ {0,1}mthat satisfies all nonlinear constraints simultaneously. In BIPNN, we determine the existence of feasible solutions through (i)Training-phase feasibility check: if all penalty terms (e.g., constraint violations) converge to zero during training, feasible solutions exist; otherwise, the problem is infeasible. (ii)Post-training verification: we sample candidate solutions from the trained model and explicitly verify whether they satisfy all constraints. The Effectiveness of BIPNN’s Hypergraph Generation Mechanism. As depicted in Fig. 3, when BIPNN is applied to solve combinatorial optimization (CO) problems on hypergraphs, it generates an alternative hypergraph structure. However, both of the hypergraphs can be used as the input of BIPNN. A critical question arises: which type of hypergraph structure achieves better performance when applied to HyperGNN? The main difference between these two hypergraphs is that the hypergraph generated by BIPNN breaks down the original hypergraph’s high-order hyperedges into numerous low-order ones. We argue that BIPNN training with the original hypergraph structure is more computationally efficiency, while BIPNN-generated hypergraph structure leads to more optimal solutions. In Sec. 6.3, we will empirically compare the solution quality of both methods. 6 APREPRINT - M AY28, 2025 (a) SCIP, d= 4. (b) SCIP, d= 6. (c) SCIP, d= 4. (d) SCIP, d= 6. (e) Tabu, d= 4. (f) Tabu, d= 6. Figure 4: Comparison of BIPNN and existing BIP solvers. dis the degree of polynomial terms in BIP objective functions. (a)(b) show the solving time required for BIPNN and SCIP to obtain the same solution. (c)(d) show the ratio of the solutions of BIPNN to SCIP; (e)(f) illustrate the ratio of the solutions of BIPNN to Tabu; Runtime is restricted to 1 hour. 6 Experimental Results In this section, we describe our empirical experiments on BIPNN and baseline optimization tools. Benchmarks . To evaluate BIPNN on BIP problems with diverse scales, the datasets are generated using DHG library3. To evaluate the quality of solutions and computational efficiency of BIPNN, datasets of varying scales are generated in three steps: Initially, DHG library is applied to generate hypergraph structures (where \|E\|= 2\|V\|). Subsequently, a random coefficient is assigned to each hyperedge (representing a polynomial term)	https://arxiv.org/abs/2505.20997v1
to generate PUBO objective functions. Thereafter, several constraints (penalty terms) were randomly incorporated into the PUBO objectives. To demonstrate the effectiveness of BIPNN on real-world settings, we also conduct experiments on the hypergraph max-cut problem (refer to Appendix C), a well-known BIP problem benchmark. Moreover, we conduct experiments on publicly-available hypergraph datasets (refer to Appendix D). Baseline Methods. In our experiments, the baseline methods include optimization techniques and tools such as SCIP [14], Tabu search [23]. Implementation Details . Experiments are conducted on an Intel Core i9-12900K CPU with 24 cores, and an NVIDIA GeForce RTX 3090 GPU with 24 G of memory. We adopt two-layer HGNN+ [ 17] as the HyperGNN model for the experiments. 6.1 Comparison with Linearization-based BIP Solvers SCIP. SCIP is an exact solver based on the branch-and-cut algorithm. Theoretically, given sufficient time and computational resources, SCIP guarantees an exact solution. However, for large-scale problems, due to time constraints, SCIP may terminate prematurely and return the approximate solution. To conduct the experiment, we generate a specific BIP instance for each size of variables. Specifically, for a BIPNN-generated hypergraph, the number of vertices (variables) \|V\|ranges from 200to3000 . The degrees of vertices are set to 4(Fig. 4a) and 6(Fig. 4b) respectively. Fig. 4a and Fig. 4b show the comparison of the solving time for BIPNN and SCIP. We evaluate the solving time taken by BIPNN to obtain the best approximate solution and the time required by SCIP to find the same solution. Experimental 3https://deephypergraph.readthedocs.io/en/latest/index.html 7 APREPRINT - M AY28, 2025 Table 1: The solutions of graph/hypergraph max-cut problems ( 1-hour time limit). Method BAT EAT UAT DBLP CiteSeer Primary High Cora SCIP 655 3,849 7,899 2,869 3,960 7,603 4,599 1,215 Tabu 652 3,972 8,402 2,710 3,717 8,500 5,160 1,360 BIPNN 651 3,978 8,407 2,801 3,852 8,509 5,216 1,384 (a)d= 4. (b)d= 6. (c)d= 4. (d)d= 6. Figure 5: Comparison of the quality of solutions and time efficiency of BIPNN when it applys its generated hypergraph structure or the original hypergraph structure to solve hypergraph max-cut problems. dis the degree of polynomial terms in BIP objective functions. (a)(b) show the numbers of cuts; (c)(d) show the solving time. results demonstrate that the solving time of BIPNN grows linearly and slowly with increasing problem size, while SCIP’s solving time exhibits exponential growth. This trend becomes more pronounced when the degree of polynomial terms is 6. Moreover, we impose a 1-hour time limit and evaluate the solution quality of BIPNN and SCIP across varying scales of BIP instances. Fig. 4c and Fig. 4d show the comparative ratio of solutions obtained by BIPNN and SCIP. Specifically, the comparative ratio is defined asOs BIPNN Os SCIP, where Os BIPNN andOs SCIP are the solutions obtained by BIPNN and SCIP. Experimental results demonstrate that BIPNN starts outperforming SCIP when the number of variables exceeds 2,500 when d= 4. As the problem size increases, BIPNN’s solutions increasingly outperform SCIP’s solutions. For d= 6, BIPNN outperforms SCIP when the number of vertices exceeds 1,000. Tabu Search. Tabu search is a heuristic method that typically provides approximate solutions.	https://arxiv.org/abs/2505.20997v1
We also impose a 1-hour time limit and evaluate the difference in solution quality for Tabu when the degrees of polynomial terms are set to 4and6. The number of vertices (variables) \|V\|in the hypergraph generated by BIPNN ranges from 200to5,000. Experimental results are depicted in Fig. 4e ( d= 4) and Fig. 4f ( d= 6). As shown in the figures, BIPNN achieves the performance comparable to Tabu when the number of variables exceeds 1,000. When the number of variables exceeds 2,500, BIPNN significantly outperforms Tabu as the variable count increases further. 6.2 Comparison on Real-world Datasets We compare our method against baseline methods on real-world graph and hypergraph datasets, including BAT, EAT, UAT, DBLP, CiteSeer, Primary, High, and Cora (refer to Appendix D). Graph datasets include BAT, EAT, UAT, DBLP, and CiteSeer. Hypergraph datasets include Primary, High, and Cora. Graph and hypergraph max-cut problems are selected as the BIP problem benchmarks. We impose 1hour time limit and evaluate the number of cuts obtained by BIPNN, SCIP, and Tabu. As depicted in Tab. 1, SCIP achieved the best performance on three graph datasets, while BIPNN achieved the best performance on two graph datasets and all three hypergraph datasets. In summary, compared to the graph max-cut problem, due to higher degree of polynomial terms in the objective function of the hypergraph max-cut problem, BIPNN tends to achieve better performance on hypergraph datasets. 6.3 Comparative Analysis on Hypergraph Generation Mechanism In Sec. 5 and Fig. 3, we propose to evaluate the effectiveness of BIPNN’s hypergraph generation mechanism by comparing the effects of its generated hypergraph structures against the original hypergraph structures in a hy- pergraph CO problem. In this section, we select hypergraph max-cut as benchmark and conduct experiments to evaluate the performance of BIPNN under both of the hypergraph structures. Experimental results are depicted in 8 APREPRINT - M AY28, 2025 Fig. 5. The number of variables ranges from 100to2000 . The degrees of polynomial terms dare set to d= 4 andd= 6 respectively. We perform 10 tests each time and record the average value of the cut numbers. As illustrated in Fig. 5a and Fig. 5b, the hypergraph structure generated by BIPNN can identify more cuts in com- parison. However, as depicted in Fig. 5c and Fig. 5d, when the parameter dis larger, the number of hyperedges (polynomial terms in PUBO objectives) in the hypergraph structure generated by BIPNN increases sharply, leading to significantly higher computational costs. The results align with the theoretical analysis we presented in Sec. 5. Figure 6: Comparison of the training time for BIPNN with or without GPU accelerated algo- rithm for PUBO losses.6.4 Ablation Study GPU Acceleration. The superior time efficiency of BIPNN is primarily attributed to the GPU-accelerated algorithm employed in computing large-scale PUBO loss functions. Fig. 6 shows a comparison of the training times for BIPNN with or without the GPU-accelerated algorithm. We evaluate the training time of BIPNN on the hypergraph max-cut problem. The number of variables ranges from 200to1000 . The degree of polynomial terms is set to 4. We train BIPNN for	https://arxiv.org/abs/2505.20997v1
a fixed number of 1000 epochs. As Fig. 6 illustrates, when GPU acceleration is applied to compute the PUBO loss function, the training time does not exhibit significant growth with an increasing number of variables. In contrast, without GPU acceleration, the training time increases rapidly as the number of variables rises. Annealing Strategy. We validate the effectiveness of the annealing strategy of BIPNN on the hypergraph max-cut problem. The experiments are conducted on Cora with 1,330vertices. The metrics include the number of cuts and discreteness of variables. The penalty strength γ is set to −2.5initially and its value is gradually increased during training. The value of γreaches 0after500epochs and continued to increase thereafter. As illustrated in Fig. 7, the annealing strategy ensures BIPNN to get better solutions while guaranteeing all variables to converge to discrete values. It demonstrates that negative γvalues enable BIPNN to escape local optima, thereby discovering better solutions. Moreover, when γis set to positive values, it facilitates the convergence of variables toward discrete values. 7 Conclusion Figure 7: Quality and discrete- ness of solutions with or without the annealing strategy.This work proposes BIPNN, a novel neural network solver for nonlinear BIP prob- lems. It reformulates nonlinear BIPs into PUBO cost functions, which correspond to hypergraph structures. On this basis, these PUBO cost functions are used as loss functions for HyperGNNs, enabling the model to solve BIPs in an unsupervised training manner. Compared with existing BIP solvers (e.g., SCIP) that rely on linearization, BIPNN reduces the training cost by optimizing nonlinear BIPs via straightforward gradient descent. Empirical results demonstrate that BIPNN achieves state-of-the-art performance in learning approximate solutions for large-scale BIP problems. 9 APREPRINT - M AY28, 2025 References [1]Yan Qiao, Yanjun Lu, Jie Li, Siwei Zhang, Naiqi Wu, and Bin Liu. An efficient binary integer programming model for residency time-constrained cluster tools with chamber cleaning requirements. IEEE Transactions on Automation Science and Engineering , 19(3):1757–1771, 2021. [2]Theodore P Papalexopoulos, Christian Tjandraatmadja, Ross Anderson, Juan Pablo Vielma, and David Belanger. Constrained discrete black-box optimization using mixed-integer programming. In International Conference on Machine Learning , pages 17295–17322. PMLR, 2022. [3]Libin Wang, Han Hu, Qisen Shang, Haowei Zeng, and Qing Zhu. Struc- turedmesh: 3-d structured optimization of façade components on photogram- metric mesh models using binary integer programming. IEEE Transactions on Geoscience and Remote Sensing , 62:1–12, 2024. [4]Giacomo Nannicini, Lev S Bishop, Oktay Günlük, and Petar Jurcevic. Optimal qubit assignment and routing via integer programming. ACM Transactions on Quantum Computing , 4(1):1–31, 2022. [5]Akshay Ajagekar, Kumail Al Hamoud, and Fengqi You. Hybrid classical- quantum optimization techniques for solving mixed-integer programming prob- lems in production scheduling. IEEE Transactions on Quantum Engineering , 3:1–16, 2022. [6]Lei Fan and Zhu Han. Hybrid quantum-classical computing for future network optimization. IEEE Network , 36(5):72–76, 2022. [7]Mercè Llabrés, Gabriel Riera, Francesc Rosselló, and Gabriel Valiente. Align- ment of biological networks by integer linear programming: virus-host protein- protein interaction networks. BMC bioinformatics , 21(Suppl 6):434, 2020. [8]Jianshen Zhu, Naveed Ahmed Azam, Fan Zhang, Aleksandar Shurbevski, Kazuya Haraguchi, Liang Zhao, Hiroshi Nagamochi, and Tatsuya Akutsu. A novel method for	https://arxiv.org/abs/2505.20997v1
inferring chemical compounds with prescribed topological substructures based on integer programming. IEEE/ACM Transactions on Computational Biology and Bioinformatics , 19(6):3233–3245, 2021. [9]Vladimir V Gusev, Duncan Adamson, Argyrios Deligkas, Dmytro Anty- pov, Christopher M Collins, Piotr Krysta, Igor Potapov, George R Darling, Matthew S Dyer, Paul Spirakis, et al. Optimality guarantees for crystal structure prediction. Nature , 619(7968):68–72, 2023. [10] Georgia Stinchfield, Joshua C Morgan, Sakshi Naik, Lorenz T Biegler, John C Eslick, Clas Jacobson, David C Miller, John D Siirola, Miguel Zamarripa, Chen Zhang, et al. A mixed integer linear programming approach for the design of chemical process families. Computers & Chemical Engineering , 183:108620, 2024. [11] Richard M Karp. Reducibility among combinatorial problems . Springer, 2010. [12] Elias B Khalil, Christopher Morris, and Andrea Lodi. Mip-gnn: A data-driven framework for guiding combinatorial solvers. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 36, pages 10219–10227, 2022. [13] Huigen Ye, Hua Xu, Hongyan Wang, Chengming Wang, and Yu Jiang. Gnn&gbdt-guided fast optimizing framework for large-scale integer program- ming. In International conference on machine learning , pages 39864–39878. PMLR, 2023. [14] Stephen Maher, Matthias Miltenberger, João Pedro Pedroso, Daniel Rehfeldt, Robert Schwarz, and Felipe Serrano. PySCIPOpt: Mathematical programming in python with the SCIP optimization suite. In Mathematical Software – ICMS 2016 , pages 301–307. Springer International Publishing, 2016. [15] Tobias Achterberg. Scip: solving constraint integer programs. Mathematical Programming Computation , 1:1–41, 2009. 10 APREPRINT - M AY28, 2025 [16] Narendra Karmarkar. A new polynomial-time algorithm for linear program- ming. In Proceedings of the sixteenth annual ACM symposium on Theory of computing , pages 302–311, 1984. [17] Yue Gao, Yifan Feng, Shuyi Ji, and Rongrong Ji. Hgnn+: General hyper- graph neural networks. IEEE Transactions on Pattern Analysis and Machine Intelligence , 45(3):3181–3199, 2022. [18] Naganand Yadati, Madhav Nimishakavi, Prateek Yadav, Vikram Nitin, Anand Louis, and Partha Talukdar. Hypergcn: A new method for training graph convo- lutional networks on hypergraphs. Advances in neural information processing systems , 32, 2019. [19] Jing Huang and Jie Yang. Unignn: a unified framework for graph and hy- pergraph neural networks. In the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI) , 2021. [20] Yuma Ichikawa. Controlling continuous relaxation for combinatorial opti- mization. Advances in Neural Information Processing Systems (NeurIPS) , 37:47189–47216, 2024. [21] Jorge Nocedal and Stephen J Wright. Numerical optimization . Springer, 1999. [22] Fred Glover, Gary Kochenberger, Rick Hennig, and Yu Du. Quantum bridge analytics i: a tutorial on formulating and using qubo models. Annals of Operations Research , 314(1):141–183, 2022. [23] Fred Glover and Manuel Laguna. Tabu search . Springer, 1998. 11 APREPRINT - M AY28, 2025 A A toy example of the polynomial reformulation of BIP (Sec. 4). Forsin(x1+x2+x3), where x1, x2, x3∈ {0,1}, we can construct a polynomial to precisely fit the function, such that it matches sin(x1+x2+x3)for all combinations of x1, x2, x3∈ {0,1}. For multiple binary variables, the polynomial can be generalized as: P(x1, x2, x3) =a1x1+a2x2+a3x3+b12x1x2+b13x1x3+b23x2x3+cx1x2x3+d (11) Based on all possible combinations of x1, x2, x3, we can set up the following equations: 1) When x1= 0, x2= 0,	https://arxiv.org/abs/2505.20997v1
x3= 0:P(0,0,0) = d= sin(0) = 0 . Thus, d= 0. 2) When x1= 0, x2= 0, x3= 1:P(0,0,1) = a3= sin(1) ≈0.8415 . Thus, a3= 0.8415 . 3) When x1= 0, x2= 1, x3= 0:P(0,1,0) = a2= sin(1) ≈0.8415 . Thus, a2= 0.8415 . 4) When x1= 1, x2= 0, x3= 0:P(1,0,0) = a1= sin(1) ≈0.8415 . Thus, a1= 0.8415 . 5) When x1= 0, x2= 1, x3= 1:P(0,1,1) = a2+a3+b23= sin(2) ≈0.9093 . Substituting a2= 0.8415 anda3= 0.8415 :b23=−0.7737 . 6) When x1= 1, x2= 0, x3= 1:P(1,0,1) = a1+a3+b13= sin(2) ≈0.9093 Substituting a1= 0.8415 anda3= 0.8415 :b13=−0.7737 . 7) When x1= 1, x2= 1, x3= 0:P(1,1,0) = a1+a2+b12= sin(2) ≈0.9093 Substituting a1= 0.8415 anda2= 0.8415 :b12=−0.7737 8) When x1= 1, x2= 1, x3= 1:P(1,1,1) = a1+a2+a3+b12+b13+b23+c= sin(3) ≈0.1411 . Substituting known values: c=−0.0623 . Based on the above calculations, the polynomial is: P(x1, x2, x3) = 0 .8415( x1+x2+x3)−0.7737( x1x2+x1x3+x2x3)−0.0623x1x2x3 (12) B A toy example of the unconstrained reformulation of BIP (Sec. 4). For a nonlinear constraint with exponential term g(x):2x1+ex2+ 3x1x3≤5, where x1, x2, x3∈ {0,1}, we can find the minimal violation subsets Vbased on all possible combinations of x1, x2, x3. 1) When x1= 0, x2= 0, x3= 0:g(x) = 1≤5, feasible. 2) When x1= 0, x2= 0, x3= 1:g(x) = 1≤5, feasible. 3) When x1= 0, x2= 1, x3= 0:g(x) =e≤5, feasible. 4) When x1= 1, x2= 0, x3= 0:g(x) = 3≤5, feasible. 5) When x1= 0, x2= 1, x3= 1:g(x) =e≤5, feasible. 6) When x1= 1, x2= 0, x3= 1:g(x) = 6≥5, violation. 7) When x1= 1, x2= 1, x3= 0:g(x) =e+ 2≤5, feasible. 8) When x1= 1, x2= 1, x3= 1:g(x) = 5 + e≥5, violation (not minimal). Identified minimal violation subsets: {x1, x3}. Thus, P(x) =λ(x1x3) (13) Final BIP objective: OBIP=f(x) +λ(x1x3) (14) C The hypergraph max-cut problem. The max-cut problem of a hypergraph G= (V, E)involves partitioning the vertex set into two disjoint subsets such that the number of hyperedges crossing the partitioned blocks is maximized. 12 APREPRINT - M AY28, 2025 PUBO Form. The hypergraph max-cut problem on Gcan be formulated by optimizing a PUBO objective as follows: min Omax−cut=X e∈E(1−Y i∈exi−Y i∈e(1−xi)) (15) where xi∈ {0,1}are binary decision variables. For a simple example illustrated in Fig. 3, the original hypergraph consists of three hyperedges: {x1, x2},{x3, x4}, and {x1, x2, x3}. Thus, the max-cut objective of Gis to minimize 2x1+ 2x2+ 2x3+x4−3x1x2−x1x3−x2x3−2x3x4. BIPNN typically generates a new hypergraph structure with five hyperedges, {x1, x2},{x3, x4},{x1, x3}, and{x2, x3}, to solve this PUBO objective. we found that both hypergraphs can be utilized for HyperGNN training in BIPNN framework. D Datasets. Table 2: Summary statistics of five real-world graphs: the number of vertices \|V\|, the number of edges \|E\|. Three hypergraphs: the number of vertices \|V\|, the number of hyperedges \|E\|, the size of the hypergraphP e∈E\|e\|. Graphs \|V\| \| E\| Hypergraphs \|V\| \| E\|P e∈E\|e\| BAT 131 1,003 Primary 242 12,704 30,729 EAT 399 5,993 High 327 7,818 18,192 UAT 1,190 13,599 Cora 1,330 1,503 4,599 DBLP 2,591 3,528 CiteSeer 3,279	https://arxiv.org/abs/2505.20997v1
arXiv:2505.21012v1 [cs.LG] 27 May 2025FEDERATED INSTRUMENTAL VARIABLE ANALYSIS VIA FEDERATED GENERALIZED METHOD OF MOMENTS Geetika, Somya Tyagi, Bapi Chatterjee∗ Department of Computer Science and Engineering, IIIT Delhi New Delhi, India {geetikai, somya23005, bapi}@iiitd.ac.in ABSTRACT Instrumental variables (IV) analysis is an important applied tool for areas such as healthcare and consumer economics. For IV analysis in high-dimensional settings, the Generalized Method of Moments (GMM) using deep neural networks offers an efficient approach. With non-i.i.d. data sourced from scattered decentralized clients, federated learning is a popular paradigm for training the models while promising data privacy. However, to our knowledge, no federated algorithm for either GMM or IV analysis exists to date. In this work, we introduce federated instrumental variables analysis ( FEDIV) via federated generalized method of moments ( FEDGMM ). We formulate FEDGMM as a federated zero-sum game defined by a federated non-convex non-concave minimax optimization problem, which is solved using federated gradient descent ascent ( FEDGDA ) algorithm. One key challenge arises in theoretically characterizing the federated local optimality. To address this, we present properties and existence results of clients’ local equilibria via FEDGDA limit points. Thereby, we show that the federated solution consistently estimates the local moment conditions of every participating client. The proposed algorithm is backed by extensive experiments to demonstrate the efficacy of our approach. Keywords Federated Learning ·Generalized Method of Moments ·Instrumental Variables Analysis ·Causal Inference 1 Introduction Federated Learning (FL) (McMahan et al., 2017) over scattered clients without data sharing is now an established paradigm for training Machine Learning (ML) models. The data privacy makes it attractive for applications to healthcare (Nguyen et al., 2022; Antunes et al., 2022; Oh and Nadkarni, 2023), finance and banking (Byrd and Polychroniadou, 2020; Long et al., 2020), smart cities and mobility (Zheng et al., 2022; Gecer and Garbinato, 2024), drug discovery (Oldenhof et al., 2023) and many others (Ye et al., 2023). However, the existing research in FL primarily focuses on supervised learning (Kairouz et al., 2021), which struggles to predict the outcomes due to confounding variables not observed in training data. For example, consider the Nature Medicine report by Dayan et al. (2021) on a global-scale FL to predict the effectiveness of oxygen administration (a treatment variable) to COVID-19 patients in the emergency rooms while maintaining their privacy. It is known that COVID-19 revival rates are highly influenced by lifestyle-related factors such as obesity and diabetes (Wang, Sato, and Sakuraba, 2021), other co-morbidities (Russell, Lone, and Baillie, 2023), and the patients’ conditions at the emergency care admission time (Izcovich et al., 2020). Arguably, the Dayan et al. (2021)’s approach may over- or under-estimate the effects of oxygen treatment. ∗This work is supported in part by the Indo-French Centre for the Promotion of Advanced Research (IFCPAR/CEFIPRA) through the FedAutoMoDL project, the Infosys Center for Artificial Intelligence (CAI) at IIIT-Delhi through the Scalable Federated Learning project. Geetika is partially supported by the INSPIRE fellowship No: DST/INSPIRE Fellowship/[IF220579] offered by the Department of Science & Technology (DST), Government of India. Bapi Chatterjee also acknowledges support by Anusandhan National Research Foundation under project	https://arxiv.org/abs/2505.21012v1
SRG/2022/002269. Federated IV Analysis via Federated GMM, Geetika et al. One can address the above issue by observing and accommodating every confounding latent factor that may influence the outcome. Thus, it may require that obesity, diabetes, overall health at the time of admission, and even genetic factors are accommodated; for example, using a technique such as matching (Kallus, 2020b; Kallus, 2020a). It may potentially render the treatment variable undergo a randomized controlled trial such as A/B testing (Kohavi et al., 2013) on decentralized, scattered, and possibly private data. However, to our knowledge, these techniques are yet unexplored in the realms of FL. Alternatively, one could assume conditional independence between unobserved confounders and the treatment variable, for example, the works by Shalit, Johansson, and Sontag (2017) and Imai and Li (2023), etc. However, this may not be a fair approach for an application such as the federated estimation of effectiveness of oxygen therapy (Dayan et al., 2021). To elaborate, Liang et al. (2023) suggests the hypoxia-inducible factors (HIF) – a protein that controls the rate of transcription of genetic information from DNA to messenger RNA by binding to a specific DNA sequence (Latchman, 1993) – plays a vital role in oxygen consumption at the cellular level. The machine learning model developed by FL implementation of Dayan et al. (2021) would miss the crucial counterfactual scenarios, such as HIF levels among patients undergoing oxygen therapy impacting morbidity outcomes, should it assume conditional independence between effects of oxygen treatment and every confounder. Such variables can be often traced in applications such as industry-scale federated drug discovery by AstraZeneca (Oldenhof et al., 2023). Instrumental variables (IV) provide a workaround to both the above issues under the assumption that the latent confounding factor influences only the treatment variable but does not directly affect the outcome. In the above example, the measure of HIF works as an instrumental variable that affects oxygen treatment as in its effective organ-level consumption but does not directly affect the mortality of the COVID-19 patient (Dayan et al., 2021). IV can play an important role in a federated setting as the influence assumption between the confounders and the treatment variables will remain local to the clients. IV analysis has been comprehensively explored in econometrics (Angrist and Krueger, 2001; Angrist and Pischke, 2009) with several decades of history such as works of Wright (1928) and Reiersøl (1945). Its efficiency is now accepted for learning even high-dimensional complex causal relationships such as one in image datasets (Hartford et al., 2017; Bennett, Kallus, and Schnabel, 2019). Naturally, the growing demand of FL entails designing methods for federated IV analysis, which, to our knowledge, is yet unexplored. In the centralized deep learning setting, Hartford et al. (2017) introduced an IV analysis framework, namely D EEPIV, which uses two stages of neural networks training – first for the treatment prediction and the second with a loss function involving integration over the conditional treatment distribution. The two-stage process has precursors in applying least square regressions in the two phases (Angrist and Pischke, 2009)[4.1.1]. In the same setting, another approach	https://arxiv.org/abs/2505.21012v1
for IV analysis applies the generalized method of moments (GMM) (Wooldridge, 2001). GMM is a celebrated estimation approach in social sciences and economics. It was introduced by Hansen (1982), for which he won a Nobel Prize in Economics (Steif et al., 2014). Building on (Wooldridge, 2001), Bennett, Kallus, and Schnabel (2019) introduced deep learning models to GMM estimation; they named their method DEEPGMM . Empirically, DEEPGMM outperformed DEEPIV.DEEPGMM is solved as a smooth zero-sum game formulated as a minimax optimization problem. Prior to DEEPGMM , Lewis and Syrgkanis (2018) also employed neural networks for GMM estimation. Their method, called the adversarial generalized method of moments ( AGMM ), also formulated the problem as a minimax optimization to fit a GMM criterion function over a finite set of unconditional moments. DEEPGMM differs from AGMM in using a weighted norm to define the objective function. The experiments in (Bennett, Kallus, and Schnabel, 2019) showed that DEEPGMM outperformed AGMM for IV analysis, and both won against DEEPIV. Nonetheless, to our knowledge, none of these methods have a federated counterpart. Minimax optimization has been studied in federated settings (Sharma et al., 2022; Wu et al., 2024), which potentially provides an underpinning for federated GMM. However, beyond the algorithm and its convergence results, there are a few key challenges: (A)For non-i.i.d. client-local data, describing common federated GMM estimators is not immediate. It requires characterizing a synchronized model state that fit moment conditions of every client. (B)To show that the dynamics of federated minimax optimization retrieves an equilibrium solution of the federated zero-sum game as a limit point. And, (C)Under heterogeneity, to establish that the federated game equilibria also satisfies the equilibrium requirements of every client thereby consistently estimating the clients’ local moments. In this work, we address the above challenges. Our contributions are summarized as the following: 2 Federated IV Analysis via Federated GMM, Geetika et al. 1.We introduce FEDIV: federated IV analysis. To our knowledge, FEDIVis the first work on IV analysis in a federated setting. 2. We present FEDDEEPGMM2– a federated adaptation of D EEPGMM of Bennett, Kallus, and Schnabel (2019) to solve F EDIV. F EDDEEPGMM is implemented as a federated smooth zero-sum game. 3.We show that the limit points of a federated gradient descent ascent ( FEDGDA ) algorithm include the equilibria of the zero-sum game. 4.We show that an equilibrium solution of the federated game obtained at the server consistently estimates the moment conditions of every client. 5.We experimentally validate our algorithm. The experiments show that even for heterogenous data, FEDDEEPGMM has convergent dynamics analogous to the centralized D EEPGMM algorithm. 1.1 Related work The federated supervised learning has received algorithmic advancements guided by factors such as tackling the system and statistical heterogeneities, better sample and communication complexities, model personalization, differential privacy, etc. An inexhaustible list includes FEDPROX (Li et al., 2020), SCAFFOLD (Karimireddy et al., 2020), FEDOPT (Reddi et al., 2020), LPP-SGD (Chatterjee, Kungurtsev, and Alistarh, 2024), PFEDME(T Dinh, Tran, and Nguyen, 2020), DP-SCAFFOLD (Noble, Bellet, and Dieuleveut, 2022), and others. By contrast, federated learning with confounders, which typically forms a	https://arxiv.org/abs/2505.21012v1
causal learning setting, is a relatively under-explored research area. V o et al. (2022a) presented a method to learn the similarities among the data sources translating a structural causal model (Pearl, 2009) to federated setting. They transform the loss function by utilizing Random Fourier Features into components associated with the clients. Thereby they compute individual treatment effects (ITE) and average treatment effects (ATE) by a federated maximization of evidence lower bound (ELBO). V o et al. (2022b) presented another federated Bayesian method to estimate the posterior distributions of the ITE and ATE using a non-parametric approach. Xiong et al. (2023) presented maximum likelihood estimator (MLE) computation in a federated setting for ATE estimation. They showed that the federated MLE consistently estimates the ATE parameters considering the combined data across clients. However, it is not clear if this approach is applicable to consistent local moment conditions estimation for the participating clients. Almodóvar, Parras, and Zazo (2024) applied FedAvg to variational autoencoder (Kingma, Welling, et al., 2019) based treatment effect estimation TEDV AE (Zhang, Liu, and Li, 2021). However, their work mainly focused on comparing the performance of vanilla FedAvg with a propensity score-weighted FedAvg in the context of federated implementation of TEDV AE. Our work differs from the above related works in the following: (a)we introduce IV analysis in federated setting, and, we introduce federated GMM estimators, which has applications for various empirical research (Wooldridge, 2001), (b)specifically, we adopt a non-Bayesian approach based on a federated zero-sum game, wherein we focus on analysing the dynamics of the federated minimax optimization and characterize the global equilibria as a consistent estimator of the clients’ moment conditions. Our work also differs from federated minimax optimization algorithms: Sharma et al. (2022), Shen et al. (2024), Wu et al. (2024), and Zhu et al. (2024), where the motivation is to analyse and improve the non-asymptotic convergence under various analytical assumptions on the objective functions. We primarily focus on deriving the equilibrium via the limit points of the federated GDA algorithm. 2 Preliminaries We model our basic terminologies after (Bennett, Kallus, and Schnabel, 2019) for a client-local setting. Consider a distributed system as a set of Nclients [N]with datasets Si={(xi j, yi j)}ni j=1,∀i∈[N]. We assume that for a client i∈ [N], the treatment and outcome variables xi jandyi j, respectively, are related by the process Yi=gi 0(Xi)+ϵi, i∈[N]. We assume that each client-local residual ϵihas zero mean and finite variance, i.e. E[ϵi] = 0,E[(ϵi)2]<∞.Furthermore, we assume that the treatment variables Xiare endogenous on the clients, i.e. E[ϵi\|Xi]̸= 0,and therefore, gi 0(Xi)̸= E[Yi\|Xi]. We assume that the treatment variables are influenced by instrumental variables Zi,∀i∈[N]so that P(Xi\|Zi)̸=P(Xi). (1) 2Wu et al. (2023) used F EDGMM as an acronym for federated Gaussian mixture models. 3 Federated IV Analysis via Federated GMM, Geetika et al. Furthermore, the instrumental variables do not directly influence the outcome variables Yi,∀i∈[N]: E[ϵi\|Zi] = 0. (2) Note that, assumptions 1, 2 are local to the clients, thus, honour the data-privacy requirements of a federated learning task. In this setting, we aim to discover a common or	https://arxiv.org/abs/2505.21012v1
global causal response function that would fit the data generation processes of each client without centralizing the data. More specifically, we learn a parametric function g0(.)∈G:= {g(., θ)\|θ∈Θ}expressed as g0:=g(., θ0)forθ0∈Θ, defined by g(., θ0) =1 NNX i=1gi(., θ0). (3) The learning process essentially involves estimating the true parameter θ0byˆθ. To measure the performance of the learning procedure, we use the MSE of the estimate ˆg:=g(.,ˆθ)against the true g0averaged over the clients. 3 Federated Deep Generalized Method of Moments We adapt DEEPGMM (Bennett, Kallus, and Schnabel, 2019) in the local setting of a client i∈[N]. For a self-contained reading, we include the description here. 3.1 Client-local Deep Generalized Method of Moments (D EEPGMM) GMM estimates the parameters of the causal response function using a certain number of moment conditions . Define the moment function on a client i∈[N]as a vector-valued function fi:R\|Z\|→Rmwith components fi 1, fi 2, . . . , fi m. We consider the moment conditions as parametrized functions {fi j}m j=1∀i∈[N]with the assumption that their expectation is zero at the true parameter values. More specifically, using equation (2), we have E[fi j(Zi)ϵi] = 0,∀j∈[m],∀i∈[N], (4) We assume that mmoment conditions {fi j}m j=1at each client i∈[N]are sufficient to identify a unique federated estimate ˆθtoθ0. With (4), we define the moment conditions on a client i∈[N]as ψ(fi j;θ) = 0 ,∀j∈[m],where (5) ψ(fi;θ) =E[fi(Zi)ϵi] =E[fi(Zi)(Yi−gi(Xi;θ)). In empirical terms, the sample moments for the i-th client with nisamples are given by ψni(fi;θ) =Eni[fi(Z)ϵi] =1 niniX k=1fi(Zi k)(Yi k−gi(Xi k;θ)), (6) where ψni(fi;θ) = ψni(fi 1;θ), ψni(fi 2;θ), . . . , ψ ni(fi m;θ) is the moment condition vector, and ψni(fi j;θ) =1 niniX k=1fi j(Zi k)(Yi k−gi(Xi k;θ)). (7) Thus, for empirical estimation of the causal response function gi 0at client i∈[N], it needs to satisfy ψni(fi j;θ0) = 0 ,∀i∈[N]andj∈[m] (8) atθ=θ0. Equation (8) is reformulated as an optimization problem given by min θ∈Θ∥ψni(fi 1;θ), ψni(fi 2;θ), . . . , ψ ni(fi m;θ)∥2, (9) where we use the Euclidean norm ∥w∥2=wTw. Drawing inspiration from Hansen (1982), DEEPGMM used a weighted norm, which yields minimal asymptotic variance for a consistent estimator ˜θ, to cater to the cases of (finitely) large number of moment conditions. We adapt their weighted norm ∥w∥2 ˜θ=wTC−1 ˜θw, to a client-local setting via the covariance matrix C˜θdefined by C˜θ jl=1 niniX k=1fi j(Zi k)fi l(Zi k)(Yi k−gi(Xi k;˜θ))2. (10) 4 Federated IV Analysis via Federated GMM, Geetika et al. Now considering the vector space Vof real-valued functions, ψni(fi;θ) = ψni(fi 1;θ), ψni(fi 2;θ), . . . , ψ ni(fi m;θ) is a linear operator on Vand C˜θ(fi, hi) =1 niniX k=1fi(Zi k)hi(Zi k)(Yi k−gi(Xi k;˜θ))2(11) is a bilinear form. With that, for any subset Fi⊂ V, we define a function Ψni(θ,Fi,˜θ) = sup fi∈Fiψni(fi;θ)−1 4C˜θ(fi, fi), which leads to the following optimization problem. Lemma 1 (Lemma 1 of (Bennett, Kallus, and Schnabel, 2019)) .With the weighted norm defined by equation (10), and forFi=span({fi j}m j=1) ∥ψni(fi 1;θ), ψni(fi 2;θ), . . . , ψ ni(fi m;θ)∥2 ˜θ= Ψ ni(θ,Fi,˜θ). (12) Thus, a weighted reformulation	https://arxiv.org/abs/2505.21012v1
of (9) is given by θGMM∈arg min θ∈ΘΨni(θ,Fi,˜θ). (13) As the data-dimension grows, the function class Fiis replaced with a class of neural networks of a certain architecture, i.e.Fi={fi(z, τ) :τ∈ T } . Similarly, let Gi={gi(x, θ) :θ∈Θ}be another class of neural networks with varying weights. With that, define Ui ˜θ(θ, τ) :=1 niniX k=1fi(Zi k, τ) Yi k−gi(Xi k;θ) −1 4niniX k=1 fi(Zi k, τ)2 Yi k−gi(Xi k;θ)2(14) Then (13) is reformulated as the following θDGMM∈arg min θ∈Θsup τ∈TUi ˜θ(θ, τ).(15) Equation (15) forms a zero-sum game, whose equilibrium solution is shown to be a true estimator to θ0under a set of standard assumptions; see Theorem 2 in (Bennett, Kallus, and Schnabel, 2019). 3.2 Federated Deep GMM (F EDDEEPGMM) The federated generalized method moment ( FEDDEEPGMM ) needs to find the global moment estimators for the causal response function to fit data on each client. Thus, the federated counterpart of equation (5) is given by ψ(f;θ) =Ei[E[fi(Zi)(Yi k−gi(Xi;θ)]] = 0 , (16) where the expectation Eiis over the clients. In this work, we consider full client participation . Thus, for the empirical federated moment estimation, we formulate: ψn(f;θ) =1 NNX i=1ψni(fi;θ) =1 NNX i=11 niniX k=1fi(Zi k)(Yi k−gi(Xi k;θ)) (17) With that, the federated moment estimation problem following (13) is formulated as: θFedDeepGMM∈arg min θ∈Θ∥ψn(f;θ)∥2 ˜θ, (18) where ∥w∥˜θ=w⊤C−1 ˜θxis the previously defined weighted-norm with inverse covariance as weights. In general cases, we do not have explicit knowledge of the moment conditions of various clients. We propose FEDDEEPGMM , a “deep" reformulation of the federated optimization problem based on the neural networks of a given architecture shared among clients and is shown to have the same solution as the federated GMM problem formulated earlier. Lemma 2. LetF=span{fi j\|i∈[N], j∈[m]}. An equivalent objective function for the federated moment estimation optimization problem (18) is given by: ∥ψN(f;θ)∥2 ˜θ= sup fi∈F ∀i∈[N]1 NNX i=1 ψni(fi;θ)−1 4C˜θ(fi;fi) ,where (19) ψni(fi;θ) :=1 niniX k=1fi(Zi k)(Yi k−gi(Xi k;θ)),andC˜θ(fi, fi) :=1 niniX k=1(fi(Zi k))2(Yi k−gi(Xi k;˜θ))2. 5 Federated IV Analysis via Federated GMM, Geetika et al. The detailed proof is similar to Lemma 1 and is given in Appendix C.1. The federated zero-sum game is then defined by: ˆθFedDeepGMM∈arg min θ∈Θsup τ∈TU˜θ(θ, τ) :=1 NNX i=1Ui ˜θ(θ, τ), (20) where Ui ˜θ(θ, τ)is defined in equation (14). The federated GMM formulation by a zero-sum game defined by a federated minimax optimization problem (20) provides the global estimator as its equilibrium solution. We solve (20) using the federated gradient descent ascent (F EDGDA) algorithm described next. 3.3 Federated Gradient Descent Ascent (F EDGDA) Algorithm An adaptation of the standard gradient descent ascent algorithm to federated setting is well-explored: (Deng and Mahdavi, 2021; Sharma et al., 2022; Shen et al., 2024; Wu et al., 2024). The clients run the gradient descent ascent algorithm for several local updates and then the orchestrating server synchronizes them by collecting the model states, averaging them, and broadcasting it to the clients. A detailed description is included as a pseudocode in Appendix B. Similar to (Bennett, Kallus, and Schnabel, 2019), we note that the	https://arxiv.org/abs/2505.21012v1
federated minimax optimization problem (20) is not convex-concave on (θ, τ). The convergence results of variants of FEDGDA (Sharma et al., 2022; Shen et al., 2024; Wu et al., 2024) assume that U˜θ(θ, τ)is non-convex on θand satisfies a µ−Polyak Łojasiewicz (PL) inequality on τ, see assumption 4 in (Sharma et al., 2022). PL condition is known to be satisfied by over-parametrized neural networks (Charles and Papailiopoulos, 2018; Liu, Zhu, and Belkin, 2022). The convergence results of our method will follow (Sharma et al., 2022). We include a formal statement in Appendix B. However, beyond convergence, we primarily aim to show that an optimal solution will consistently estimate the moment conditions of the clients, which we do next. 4 Federated Equilibrium Solutions In this section, we present our main results, which establish the existence and characterize the federated equilibrium solution. 4.1 Federated Sequential Game As minimax is not equal to maximin in general for a non-convex-non-concave problem, it is important to model the federated game as a sequential game (Jin, Netrapalli, and Jordan, 2020) whose outcome would depend on what move – maximization or minimization – is taken first. We use some results from Jin, Netrapalli, and Jordan (2020), which we include here for a self-contained reading. We start with the following assumptions: Assumption 1. Client-local objective Ui ˜θ(θ, τ)∀i∈[N]is twice continuously differentiable for both θandτ. Thus, the global objective U˜θ(θ, τ)is also a twice continuously differentiable function. Assumption 2 (Smoothness) .The gradient of each client’s local objective, ∇Ui ˜θ(θ, τ), is Lipschitz continuous with respect to both θandτ. For all i∈[N], there exist constants L >0such that: ∥∇θUi ˜θ(θ1, τ1)− ∇ θUi ˜θ(θ2, τ2)∥ ≤L∥(θ1, τ1)−(θ2, τ2)∥,and ∥∇τUi ˜θ(θ1, τ1)− ∇ τUi ˜θ(θ2, τ2)∥ ≤L∥(θ1, τ1)−(θ2, τ2)∥, ∀(θ1, τ1),(θ2, τ2). Thus, U˜θ(θ, τ)isL-Lipschitz smooth. Assumption 3 (Gradient Dissimilarity) .The heterogeneity of the local gradients with respect to (w.r.t.) θandτis bounded as follows: ∥∇θUi ˜θ(θ, τ)− ∇ θU˜θ(θ, τ)∥ ≤ζi θ ∥∇τUi ˜θ(θ, τ)− ∇ τU˜θ(θ, τ)∥ ≤ζi τ, where ζi θ, ζi τ≥0are the bounds that quantify the degree of gradient dissimilarity at client i∈[N]. Assumption 4 (Hessian Dissimilarity) .The heterogeneity in terms of hessian w.r.t. θandτis bounded as follows: ∥∇2 θθUi ˜θ(θ, τ)− ∇2 θθU˜θ(θ, τ)∥σ≤ρi θ, ∥∇2 ττUi ˜θ(θ, τ)− ∇2 ττU˜θ(θ, τ)∥σ≤ρi τ, ∥∇2 θτUi ˜θ(θ, τ)− ∇2 θτU˜θ(θ, τ)∥σ≤ρi θτ, ∥∇2 τθUi ˜θ(θ, τ)− ∇2 τθU˜θ(θ, τ)∥σ≤ρi τθ, where ρi θ, ρi τ, ρi θτ,andρi τθ≥0quantify the degree of hessian dissimilarity at client i∈[N]by spectral norm ∥.∥σ. 6 Federated IV Analysis via Federated GMM, Geetika et al. Assumptions 3 and 4 provide a measure of data heterogeneity across clients in a federated setting. We assume that ζ′s andρ′sare bounded. In the special case, when ζandρ’s are all 0, then the data is homogeneous across clients. We adopt the notion of Stackelberg equilibrium for pure strategies, as discussed in (Jin, Netrapalli, and Jordan, 2020), to characterize the solution of the minimax federated optimization problem for a non-convex non-concave function U˜θ(θ, τ)for the sequential game where min-player goes first and the max-player goes second. To avoid ambiguity between the	https://arxiv.org/abs/2505.21012v1
adjectives of the terms global/local objective functions in federated learning and the global/local nature of minimax points in optimization, we refer to a global objective as the federated objective and a local objective as the client’s objective. Definition 1 (Local minimax point) .[Definition 14 of (Jin, Netrapalli, and Jordan, 2020)] Let U(θ, τ)be a function defined over Θ× T and let hbe a function satisfying h(δ)→0asδ→0. There exists a δ0, such that for any δ∈(0, δ0],and any (θ, τ)such that ∥θ−ˆθ∥ ≤δand∥τ−ˆτ∥ ≤δ, then a point (ˆθ,ˆτ)is a local minimax point of U, if ∀(θ, τ)∈Θ× T, it satisfies: U˜θ(ˆθ, τ)≤U˜θ(ˆθ,ˆτ)≤ max τ′:∥τ′−ˆτ∥≤h(δ)U˜θ(θ, τ′), (21) With that, the first-order & second-order necessary conditions for local minimax points are as below. Lemma 3 (Propositions 18, 19, 20 of (Jin, Netrapalli, and Jordan, 2020)) .Under assumption 1, any local minimax point satisfies the following conditions: •First-order Necessary Condition: A local minimax point (θ, τ)satisfies: ∇θU˜θ(θ, τ) = 0 and∇τU˜θ(θ, τ) = 0 . •Second-order Necessary Condition: A local minimax point (θ, τ)satisfies: ∇2 ττU˜θ(θ, τ)⪯0.Moreover, if ∇2 ττU˜θ(θ, τ)≺0, thenh ∇2 θθU˜θ− ∇2 θτU˜θ ∇2 ττU˜θ−1∇2 τθU˜θi (θ, τ)⪰0. •Second-order Sufficient Condition: A stationary point (θ, τ)that satisfies ∇2 ττU˜θ(θ, τ)≺0, and h ∇2 θθU˜θ− ∇2 θτU˜θ ∇2 ττU˜θ−1∇2 τθU˜θi (θ, τ)≻0 guarantees that (θ, τ)is a strict local minimax. Now, in order to define the federated approximate equilibrium solutions, we first define an approximate local minimax point. Definition 2 (Approximate Local minimax point) .[An adaptation of definition 34 of (Jin, Netrapalli, and Jordan, 2020)] Let U(θ, τ)be a function defined over Θ×T and let hbe a function satisfying h(δ)→0asδ→0. There exists aδ0, such that for any δ∈(0, δ0],and any (θ, τ)such that ∥θ−ˆθ∥ ≤δand∥τ−ˆτ∥ ≤δ, then a point (ˆθ,ˆτ)is an ε-approximate local minimax point of U, if it satisfies: U˜θ(ˆθ, τ)−ε≤U˜θ(ˆθ,ˆτ)≤ max τ′:∥τ′−ˆτ∥≤h(δ)U˜θ(θ, τ′) +ε, (22) We aim to achieve approximate local minimax points for every client as a solution of the federated minimax optimization. With that, we characterize the federated solution as the following. Definition 3 (E-Approximate Federated Equilibrium Solutions) .LetE={εi}N i=1be the approximation error vector for clients [N]. LetUi ˜θ(θ, τ)be a function defined over Θ×T for a client i∈[N]. AnE-approximate federated equilibrium point (ˆθ,ˆτ)that is an εi-approximate local minimax point for every clients’ objective Ui ˜θ, where the federated objective isU˜θ(θ, τ) :=1 NPN i=1Ui ˜θ(θ, τ), must follow the conditions below: 1.εi- First-order Necessary Condition: The point (ˆθ,ˆτ)must be an εistationary point for every client i∈[N], i.e., ∥∇θUi ˜θ(ˆθ,ˆτ)∥ ≤εi,and ∥∇τUi ˜θ(ˆθ,ˆτ)∥ ≤εi. 2.Second-Order εiNecessary Condition: The point (ˆθ,ˆτ)must satisfy the second-order conditions: ∇2 ττUi ˜θ(ˆθ,ˆτ)⪯ −εiI, andh ∇2 θθUi ˜θ− ∇2 θτUi ˜θ ∇2 ττU˜θ−1∇2 τθUi ˜θi (ˆθ,ˆτ)⪰εiI. 7 Federated IV Analysis via Federated GMM, Geetika et al. 3.Second-Order εiSufficient Condition: Anεistationary point (θ, τ)that satisfies ∇2 ττUi ˜θ(ˆθ,ˆτ)≺ −εiI, and h ∇2 θθU˜θ− ∇2 θτU˜θ ∇2 ττU˜θ−1∇2 τθU˜θi (ˆθ,ˆτ)≻εiI guarantees that (ˆθ,ˆτ)is a strict local minimax point ∀i∈[N]that satisfies εiapproximate equilibrium as in definition 2. We now state the main theoretical result of our work in the following theorem. Theorem 1. Under assumptions 1, 2, 3 and	https://arxiv.org/abs/2505.21012v1
4, a minimax solution (ˆθ,ˆτ)of federated optimization problem (20) that satisfies the equilibrium condition as in definition 1: U˜θ(ˆθ, τ)≤U˜θ(ˆθ,ˆτ)≤ max τ′:∥τ′−ˆτ∥≤h(δ)U˜θ(θ, τ′), is anE-approximate federated equilibrium solution as defined in 3, where the approximation error εifor each client i∈[N]lies in: max{ζi θ, ζi τ} ≤εi≤min{α−ρi τ, β−Bi} for ρi τ < α and Bi> β , such that α := λmax ∇2 ττU˜θ(ˆθ,ˆτ) ,β := λminh ∇2 θθU˜θ− ∇2 θτU˜θ ∇2 ττU˜θ−1∇2 τθU˜θi (ˆθ,ˆτ) andBi:=ρi θ+Lρi θτ1 \|λmax(∇2ττUi ˜θ)\|+Lρi τθ1 \|λmax(∇2ττUi ˜θ)\|+ L2ρi τ1 \|λmax(∇2ττUi ˜θ)·λmax(∇2ττU˜θ)\|. The proof of theorem 1 is given in Appendix C.2. Note that when data is homogeneous (i.e., for each client i,ζi θ,ζi τ,ρi τ andBiare all zeroes), each client satisfies an exact local minimax equilibrium. Remark 1. In Theorem 1, note that if the interval [max{ζi θ, ζi τ},min{α−ρi τ, β−Bi}]is empty, i.e. max{ζi θ, ζi τ}> min{α−ρi τ, β−Bi}, then no such εiexists and (ˆθ,ˆτ)fails to be a local εiapproximate equilibrium point for that clients. It may happen in two cases: 1.The gradient dissimilarity ζi θ, ζi τis too large indicating high heterogeneity, then (ˆθ,ˆτ)- the solution to the federated objective would fail to become an approximate equilibrium point for the clients. It is a practical consideration for a federated convergence facing difficulty against high heterogeneity. 2.Ifα≈ρi τorβ≈Bi, indicating that the client’s local curvature structure significantly differs from the global curvature. In this case, the clients’ objectives may be flatter or even oppositely curved compared to the global model, that is, the objectives are highly heterogeneous. Now we state the result on the consistency of the estimator of the clients’ moment conditions. Theorem 2 (Consistency) .[Adaptation of Theorem 2 of (Bennett, Kallus, and Schnabel, 2019)] Let ˜θnbe a data- dependent choice for the federated objective that has a limit in probability. For each client i∈[N], define mi(θ, τ,˜θ) := fi(Zi;τ)(Yi−g(Xi;θ))−1 4fi(Zi;τ)2(Yi−g(Xi;˜θ))2,Mi(θ) = supτ∈TE[mi(θ, τ,˜θ)]and ηi(ϵ) := infd(θ,θ0)≥ϵMi(θ)−Mi(θ0)for every ϵ >0. Let (ˆθn,ˆτn)be a solution that satisfies the approximate equilibrium for each of the client i∈[N]as sup τ∈TUi ˜θ(ˆθn, τ)−εi−op(1)≤Ui ˜θ(ˆθn,ˆτn)≤inf θ∈Θmax τ′:∥τ′−ˆτn∥≤h(δ)Ui ˜θ(θ, τ′) +εi+op(1), for some δ0, such that for any δ∈(0, δ0],and any θ, τsuch that ∥θ−ˆθ∥ ≤δand∥τ−ˆτ∥ ≤δand a function h(δ)→0asδ→0. Then, under similar assumptions as in Assumptions 1 to 5 of (Bennett, Kallus, and Schnabel, 2019), the global solution ˆθnis a consistent estimator to the true parameter θ0, i.e. ˆθnp− →θ0when the approximate error εi<ηi(ϵ) 2for every ϵ >0for each client i∈[N]. The assumptions and the proof of Theorem 2 are included in Appendix C.3. Remark 2. Theorem 2 formalizes a tradeoff between data heterogeneity and the consistency of the global estimator in federated learning. If the approximation error εiis large for a client i∈[N], then the solution ˆθnmay fail to consistently estimate the true parameter of client i. In contrast, when data across clients have similar distribution (i.e., case for low heterogeneity), the federated optimal model ˆθnis consistent across clients. Now, we discuss that the limit points of FEDGDA will retrieve the local minimax points of the federated optimization problem. 8 Federated IV Analysis via Federated GMM, Geetika et	https://arxiv.org/abs/2505.21012v1
al. 4.2 Limit Points of F EDGDA Letα1=η γ, α2=ηbe the learning rates for gradient updates to θandτ, respectively. For details, refer to Algorithm 1 in Appendix B. Without loss of generality the F EDGDA updates are: θt+1=θt−η1 γ1 NX i∈[N]RX r=1∇θUi ˜θ(θi t,r, τi t,r)andτt+1=τt+η1 NX i∈[N]RX r=1∇τUi ˜θ(θi t,r, τi t,r) (23) We call it γ-FEDGDA , where γis the ratio of α1toα2. Asη→0corresponds to FEDGDA -flow, under the smoothness ofUi ˜θ, Assumption 3 and for some fixed R, FEDGDA-flow becomes: dθ dt=−1 γR∇θU˜θ(θ, τ) +OR γζθ ,anddτ dt=R∇τU˜θ(θ, τ) +O(Rζτ). (24) We further elaborate on F EDGDA-flow in Appendix D.1. Proposition 1. Given the Jacobian matrix for γ−FEDGDA flow as J=−1 γR∇2 θθU˜θ(θ, τ)−1 γR∇2 θτU˜θ(θ, τ) R∇2 τθU˜θ(θ, τ) R∇2 ττU˜θ(θ, τ) , a point (θ, τ)is a strictly linearly stable equilibrium of the γ−FEDGDA flow if and only if the real parts of all eigenvalues of Jare negative, i.e., Re(Λ j)<0for all j. Proposition 1 essentially defines a strictly linearly stable equilibrium of the γ−FEDGDA flow. The proof follows a strategy similar to (Jin, Netrapalli, and Jordan, 2020). With that, let γ-FGDA be the set of strictly linearly stable points of the γ-FEDGDA flow,LocMinimax be the set of local minimax points of the federated zero-sum game. Define ∞ − FGDA := lim sup γ→∞γ− FGDA :=∩γ0>0∪γ>γ 0γ− FGDA ,and ∞ − FGDA := lim inf γ→∞γ− FGDA :=∪γ0>0∩γ>γ 0γ− FGDA . We now state the theorem that establishes the stable limit points of ∞-FGDA as local minimax points, up to some degenerate cases. This theorem ensures that solutions to a minimax problem obtained using FEDGDA in the limit γ→ ∞ correspond to equilibrium points. Theorem 3. Under Assumption 1, LocMinimax ⊂ ∞ − FGDA ⊂∞ − FGDA ⊂ L ocMinimax ∪ A, where A:={(θ, τ)\|(θ, τ)is stationary and ∇2 ττU˜θ(θ, τ)is degenerate }. Moreover, if the hessian ∇2 ττU˜θ(θ, τ)is smooth, thenAhas measure zero in Θ× T ⊂ Rd×Rk. Essentially, Theorem 3 states that the limit points of FEDGDA are the local minimax solutions, and thereby the equilibrium solution of the federated zero-sum game at the server, up to some degenerate cases with measure 0. The proof of Theorem 3 is included in Appendix D.2. Theorems 1, 2, and 3 together complete the theoretical foundation of the pipeline in our work. Obtaining the equilibrium solution of the federated zero-sum game at the server via the FEDGDA limit points, using Theorem 1 we get E- approximate federated equilibrium solutions, wherefrom we obtain clients’ approximate local minimax. Finally, applying Theorem 2 we retrieve the consistent estimators for GMM at the clients. 5 Experiments In the experiments, we extend the experimental evaluations of (Bennett, Kallus, and Schnabel, 2019) to a federated setting. We discuss this benchmark choice further in Appendix A. More specifically, we evaluate the ability of FEDGMM to fit low and high dimensional data to demonstrate that it converges analogous to the centralized algorithm DEEPGMM. Similar to (Bennett, Kallus, and Schnabel, 2019), we assess two scenarios in regards to ((X, Y), Z): (a)The instrumental and treatment variables ZandXare	https://arxiv.org/abs/2505.21012v1
both low-dimensional. In this case, we use 1- dimensional synthetic datasets corresponding to the following functions: (a) Absolute :g0(x) =\|x\|, (b) Step : g0(x) = 1 {x≥0}, (c)Linear :g0(x) =x. 9 Federated IV Analysis via Federated GMM, Geetika et al. To generate the synthetic data, similar to (Bennett, Kallus, and Schnabel, 2019; Lewis and Syrgkanis, 2018) we apply the following generation process: Y=g0(X) +e+δ andX=Z(1)+Z(2)+e+γ (25) (Z(1), Z(2))∼Uniform ([−3,3]2) ande∼ N(0,1), γ, δ ∼ N(0,0.1) (26) (b)ZandXare low-dimensional or high-dimensional or both. First, ZandXare generated as in (25,26). Then for high-dimensional data, we map ZandXto an image using the mapping: Image (x) =Dataset (round (min (max(1 .5x+ 5,0),9))), where (round (min(max(1 .5x+ 5,0),9))) returns an integer between 0 and 9. Essentially, the function Dataset (.)randomly selects an image following its index. We use datasets FEMNIST (Federated Extended MNIST) and CIFAR10 (Caldas et al., 2018) for images of size 28×28and3×32×32, respectively. Thus, we have the following cases: (a) Dataset z:X=Xlow, Z=Image (Zlow), (b)Dataset x:Z=Zlow, X=Image (Xlow), and (c) Dataset x,z:Z=Image (Zlow),X=Image (Xlow), where Dataset takes values FEMNIST and CIFAR10 and the superscript lowindicates the values generated using the process in low-dimensional case. (Bennett, Kallus, and Schnabel, 2019) used Optimistic Adam ( OA DAM ), a variant of Adam (Kingma, 2015) based stochastic gradient descent ascent algorithm (Daskalakis et al., 2018), which applies mirror descent based gradient updates. It guarantees the last iteration convergence of a GAN (Goodfellow et al., 2014) training problem. It is known that a well-tuned SGDoutperforms Adam in over-parametrized settings (Wilson et al., 2017), closely resembling ourFEDGMM implementation, where the size of neural networks often exceeds the data available on the clients. Considering that, we explored the comparative performance of GDA andSGDA against OA DAM for a centralized DEEPGMM implementation. Note that GDA also aligns with the analytical discussion presented in Section (4). We then implemented the federated versions of each of these methods and benchmarked them for solving the federated minimax optimization problem for the FEDDEEPGMM algorithm. For high-dimensional scenarios, we implement a Actual Causal Effect DeepGMM-OAdam DeepGMM-SGDA FedDeepGMM-SGDA DeepGMM-GDA FedDeepGMM-GDA (a)Absolute (b)Step (c)Linear Figure 1: Estimated ˆgcompared to true gin low-dimensional scenarios convolutional neural network (CNN) architecture to process images, while for low-dimensional scenarios, we use a multilayer perceptron (MLP). Code is available at https://github.com/dcll-iiitd/FederatedDeepGMM . Estimations DEEPGMM - OAdamDEEPGMM - GDAFDEEPGMM - GDADEEPGMM - SGDAFDEEPGMM - SGDA Absolute 0.03±0.01 0.013±.01 0.4±0.01 0.009±0.01 0.2±0.00 Step 0.3±0.00 0.03±0.00 0.04±0.01 0.112±0.00 0.23±0.01 Linear 0.01±0.00 0.02±0.00 0.01±0.00 0.03±0.00 0.04±0.00 FEMNIST x 0.50±0.00 1.11±0.01 0.21±0.02 0.40±0.01 0.19±0.01 FEMNIST x,z0.24±0.00 0.46±0.09 0.19±0.03 0.14±0.02 0.20±0.00 FEMNIST z 0.10±0.00 0.42±0.01 0.24±0.01 0.11±0.02 0.23±0.01 CIFAR10 x 0.55±0.30 0.19±0.01 0.25±0.03 0.20±0.08 0.22±0.08 CIFAR10 x,z 0.40±0.11 0.24±0.00 0.24±0.03 0.19±0.03 0.22±0.02 CIFAR10 z 0.13±0.03 0.13±0.01 1.70±2.60 0.24±0.01 0.52±0.60 Table 1: The averaged Test MSE with standard deviation on the low- and high-dimensional scenarios. Non-i.i.d. data. We sample the train, test and validation sets similar to (Bennett, Kallus, and Schnabel, 2019). For the low-dimensional scenario, we sample n= 20000 points for each train, validation, and test set, while, for the 10 Federated IV Analysis via Federated GMM, Geetika et al.	https://arxiv.org/abs/2505.21012v1
high-dimensional scenario, we have n= 20000 for the train set and n= 10000 for the validation and test set. To set up a non-i.i.d. distribution of data between clients, samples were divided amongst the clients using a Dirichlet distribution DirS(α)(Wang et al., 2019), where αdetermines the degree of heterogeneity across Sclients. We used DirS(α) = 0 .3for each train, test, and validation samples. Hyperparameters. We perform extensive grid-search to tune the learning rate. For FEDSGDA , we use a minibatch-size of 256. To avoid numerical instability, we standardize the observed Yvalues by removing the mean and scaling to unit variance. We perform five runs of each experiment and present the mean and standard deviation of the results. Observations and Discussion. In figure (1), we first observe that SGDA andGDA algorithms perform at par with OA DAM to fit the DEEPGMM estimator. It establishes that hyperparameter tuning is effective. With that, we further observe that the federated algorithms efficiently fit the estimated function to the true data-generating process competitive to the centralized algorithms even though the data is decentralized and non-i.i.d.. Thus, it shows that the federated algorithm converges effectively. In Table 1 we present the test mean squared error (MSE) values. The MSE values indicate that the federated implementation achieves competitive convergence to their centralized counterpart. These experiments establish the efficacy of our method. An Open Problem In this work, we characterized the equilibrium solutions of federated zero-sum games in consideration of local minimax solutions for non-convex non-concave minimax optimization problems. Regardless of the analytical assumptions over the objective, the mixed strategy solutions for zero-sum games exist. However, unlike the pure strategy solutions, where the standard heterogeneity considerations over gradients and Hessians across clients, translates a local minimax solution for the federated objective to approximate local solutions for the clients, it is not immediate how a mixed strategy solution as a probability measure can be translated to that for clients. It leaves an interesting open problem to characterize the mixed startegy solutions for federated zero-sum games. References Almodóvar, Alejandro, Juan Parras, and Santiago Zazo (2024). “Propensity Weighted federated learning for treatment effect estimation in distributed imbalanced environments”. In: Computers in Biology and Medicine 178, p. 108779 (cit. on p. 3). Angrist, Joshua D and Alan B Krueger (2001). “Instrumental variables and the search for identification: From supply and demand to natural experiments”. In: Journal of Economic perspectives 15.4, pp. 69–85 (cit. on p. 2). Angrist, Joshua D and Jörn-Steffen Pischke (2009). Mostly harmless econometrics: An empiricist’s companion . Princeton university press (cit. on p. 2). Antunes, Rodolfo Stoffel et al. (2022). “Federated learning for healthcare: Systematic review and architecture proposal”. In:ACM Transactions on Intelligent Systems and Technology (TIST) 13.4, pp. 1–23 (cit. on p. 1). Bennett, Andrew, Nathan Kallus, and Tobias Schnabel (2019). “Deep generalized method of moments for instrumental variable analysis”. In: Advances in neural information processing systems 32 (cit. on pp. 2–6, 8–10, 16, 22). Byrd, David and Antigoni Polychroniadou (2020). “Differentially private secure multi-party computation for federated learning in financial applications”. In: Proceedings of the First ACM International Conference	https://arxiv.org/abs/2505.21012v1
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the following: (i)CAUSAL RFF (Vo et al., 2022a) and FEDCI(Vo et al., 2022b). The aim of CAUSAL RFF (V o et al., 2022a) is to estimate the conditional average treatment effect (CATE) and average treatment effect (ATE), whereas FEDCI(V o et al., 2022b) aims to estimate individual treatment effect (ITE) and ATE. For this, (V o et al., 2022a) consider a setting of Y,W, andXto be random variables denoting the outcome, treatment, and proxy variable, respectively. Along with that, they also consider a confounding variable Z. However, their causal dependency builds on the dependence of each of Y,W, and XonZbesides dependency of YonW. Consequently, to compute CATE and ATE, they need to estimate the conditional probabilities p(wi\|xi),p(yi\|xi, wi),p(zi\|xi, yi, wi),p(yi\|wi, zi), where the superscript irepresents a client. Their experiments compare the estimates of CATE and ATE with the Bayesian baselines (Hill, 2011), (Shalit, Johansson, and Sontag, 2017), (Louizos et al., 2017), etc. in a centralized setting without any consideration of data decentralization or heterogeneity native to federated learning. Further, they compare against the same baselines in a one-shot federated setting, where at the end of training on separate data sources independently, the predicted treatment effects are averaged. Similar is the experimental evaluation of (V o et al., 2022b). By contrast, the setting of IV analysis as in our work does not consider dependency of the outcome variable Yon the confounder Z, though the treatment variable Xcould be endogenous and depend on Z. For us, computing the treatment effects and thereby comparing it against these works is not direct. Furthermore, it is unclear, if the approach of (V o et al., 2022a) and (V o et al., 2022b), where the predicted inference over a number of datasets is averaged as the final result, would be comparable to our approach where the problem is solved using a federated maximin optimization with multiple synchronization rounds among the clients. For us, the federated optimization subsumes the experimental of comparing the average predicted values after independent training with the predicted value over the entire data. This is the reason that our centralized counterpart i.e. DEEPGMM (Bennett, Kallus, and Schnabel, 2019), do not experimentally compare against the baselines of (V o et al., 2022a) and (V o et al., 2022b). In summary, for us the experimental benchmarks were guided by showing the efficient fit of the GMM estimator in a federated setting. (ii)TEDVAE (Zhang, Liu, and Li, 2021). As mentioned above, their aim was to showcase the advantage of a weighted averaging over the vanilla averaging of FedAvg. By contrast, our experiments tried to showcase that even in a federated setting, the maximin optimization converges analogous to the centralized counterpart. B Federated Gradient Descent Ascent Algorithm Description Algorithm 1 FEDGDA running on a federated learning server to solve the minimax problem (20) Server Input : initial global estimate θ1, τ1; constant local learning rate α1, α2; total Nclients Output : global model states θT+1, τT+1 1:forsynchronization round t= 1, . . . , T do 2: server sends θt, τtto all clients 3: foreachi∈[N]in parallel do 4: θi	https://arxiv.org/abs/2505.21012v1
t,1←θt,τi t,1←τt 5: forr= 1,2, . . . , R do 6: θi t,r+1=θi t,r−α1∇θfi(θi t,r, τi t,r) 7: τi t,r+1=τi t,r+α2∇τfi(θi t,r, τi t,r) 8: end for 9: (∆θi t,∆τt)←(θi t,R+1−θt, τi t,R+1−τt) 10: end for 11: (∆θt,∆τt)←1 NP i∈[N](∆θi t,∆τi t) 12: θt+1←(θt+ ∆θt),τt+1←(τt+ ∆τt) 13:end for 14:return θT+1;τT+1 16 Federated IV Analysis via Federated GMM, Geetika et al. We adapt the proof of Theorem 1 in (Sharma et al., 2022) for the SGDA algorithm proposed in (Deng and Mahdavi, 2021) for the F EDGDA algorithm 1 for smooth non-convex- PL problems. Assumption 5 (Polyak Łojaisiewicz (PL) condition in τ).The function U˜θsatisfyies µ−PLcondition in τ,µ >0, if for any fixed θ,arg maxτ′U˜θ(θ, τ′)̸=ϕand∥∇τU˜θ(θ, τ)∥2≥2µ max τ′U˜θ(θ, τ′)−U˜θ(θ, τ) . Theorem 4. Let the local loss functions Ui ˜θfor all i∈ {1,2, . . . , N }satisfy assumption 2 and 3. The federated objective function satisfies assumption 5. Suppose α2≤1 8LR,α1 α2≤1 8κ2,where κ=L µis the condition number. Let ¯θT+1is drawn uniformly at random from {θt}T+1 t=1, then the following holds: ∥∇˜Φ(¯θT+1)∥2≤ O κ2∆˜Φ α2R(T+ 1) +O κ2(R−1)2[α2 2ζ2 τ+α2 1ζ2 θ] , where ∇˜Φ(.) := max τU˜θ(., τ)is the envelope function, ∆˜Φ:=˜Φ(θ0)−minθ˜Φ(θ),andζθ:=1 NPN i=1ζi θ, ζτ:= 1 NPN i=1ζi τ. Using α1=O 1 κ2q N R(T+1) ,α2=Oq N R(T+1) ,∥∇˜Φ(¯θT+1)∥2can be bounded as O κ2∆˜Φp NR(T+ 1)+κ2(R−1)2NR(ζ2 θ+ζ2 τ) R(T+ 1)! . Although the original assumption uses the supremum of average squared deviations, say ζ′ θandζ′ τ, we use per-client dissimilarity bounds ζi θ, ζi τand upper bound their quantity as ζ′ θ2≤1 NPN i=1(ζi θ)2:=ζθ2andζ′ τ2≤1 NPN i=1(ζi τ)2:= ζτ2. Since there is no stochasticity, we used the bounded variance σ= 0. For details, refer to proof of Theorem 1 in (Sharma et al., 2022). C Proofs C.1 Proof of Lemma 2 Lemma 4 (Restatement of Lemma 2) .LetF=span{fi j\|i∈[N], j∈[m]}. An equivalent objective function for the federated moment estimation optimization problem (18) is given by: ∥ψN(f;θ)∥2 ˜θ= sup fi∈F ∀i∈[N]1 NNX i=1 ψni(fi;θ)−1 4C˜θ(fi;fi) ,where (27) ψni(fi;θ) :=1 niniX k=1fi(Zi k)(Yi k−gi(Xi k;θ)),andC˜θ(fi, fi) :=1 niniX k=1(fi(Zi k))2(Yi k−gi(Xi k;˜θ))2. Proof. Letψ= (1 NPN i=1ψni(fi 1;θ),1 NPN i=1ψni(fi 2;θ), . . . ,1 NPN i=1ψni(fi m;θ)). We know that ∥v∥2=v⊤C−1 ˜θvand the associated dual norm is obtained as ∥v∥∗= sup∥v∥≤1v⊤v=v⊤C˜θv. Using the definition of the dual norm, ∥ψ∥= sup ∥v∥∗≤1v⊤ψ ∥ψ∥2= sup ∥v∥∗≤∥ψ∥v⊤ψ ∥ψ∥2= sup v⊤C˜θv≤∥ψ∥2v⊤ψ. (28) We now find the equivalent dual optimization problem for (28). The Lagrangian of the constrained maximization problem (28) is given as L(v, λ) =v⊤ψ+λ(v⊤C˜θv− ∥ψ∥2),where λ≤0. To maximize L(v, λ)w.r.t. v, put∂L ∂v=ψ+ 2λC˜θv= 0to obtain v=−1 2λC−1 ˜θψ. 17 Federated IV Analysis via Federated GMM, Geetika et al. When ∥ψ∥>0,v= 0satisfies the Slater’s condition as a strictly feasible interior point of the constraint v⊤C˜θv− ∥ψ∥2≤0. Thus, strong duality holds. Substituting v=−1 2λC−1 ˜θψin the Lagrangian gives L∗(λ) =−1 2λψ⊤C−1 ˜θψ+1 4λψ⊤C−1 ˜θψ−λ∥ψ∥2 =−∥ψ∥2 4λ−λ∥ψ∥2. Hence, the dual becomes ∥ψ∥2=infλ<0{L∗(λ)}. Thus, the equivalent dual optimization problem for (28) is given as ∥ψ∥2= inf λ<0 −∥ψ∥2 4λ−λ∥ψ∥2 . (29) Putting∂L ∂λ=∥ψ∥2 4λ2− ∥ψ∥2= 0gives λ=−1 2.Thus, due to strong duality ∥ψ∥2=	https://arxiv.org/abs/2505.21012v1
supvL(v,−1 2) = supvv⊤ψ− 1 2(v⊤C˜θv− ∥ψ∥2). Rewriting it1 2∥ψ∥2= supvv⊤ψ−1 2v⊤C˜θvand substituting u= 2v ∥ψ∥2= sup uu⊤ψ−1 4u⊤C˜θu. Using change of variables u→v ∥ψ∥2= sup vv⊤ψ−1 4v⊤C˜θv. Now, we want to find a function form for the optimization problem mentioned above. Consider a finite-dimensional functional spaces Fi=span{fi 1, fi 2, . . . , fi m}for each client i. Hence, for fi∈ Fi fi=mX j=1vjfi j. Since all the clients share the same neural network architecture, we define a global functional space Fas F=span{fi j\|i∈[N], j∈[m]}. Therefore, vcorresponds to fisuch that fi=NX c=1mX j=1vi jfc j,where vi j=vjifc=i 0 ifc̸=i Hence, v⊤ψ=1 NNX i=1mX j=1vjψni(fi j;θ) =1 NNX i=11 niniX k=1fi(Zi k)(Yi k−gi(Xi k;θ)). Similarly, v⊤C˜θv=mX p=1mX q=1vpvq[C˜θ]pq =mX p=1mX q=1vpvq1 NNX i=11 niniX k=1fi p(Zi k)fi q(Zi k)(Yi k−gi(Xi k;˜θ)) =1 NNX i=11 niniX k=1mX p=1vpfi p(Zi k)mX q=1vqfi q(Zi k)(Yi k−gi(Xi k;˜θ))2 =1 NNX i=11 niniX k=1(fi(Zi k))2(Yi k−gi(Xi k;˜θ))2 =1 NNX i=1C˜θ(fi, fi). 18 Federated IV Analysis via Federated GMM, Geetika et al. Thus, applying the Riesz Representation theorem using the representations v⊤ψ=1 NPN i=1ψni(fi;θ)andv⊤C˜θv= 1 NPN i=1C˜θ(fi, fi), we can write the objective in functional form as ∥ψ∥2= sup fi∈F ∀i∈[N]1 NNX i=1 ψni(fi;θ)−1 4C˜θ(fi, fi) . This gives us the desired result. C.2 Proof of Theorem 1 Theorem 5 (Restatement of Theorem 1) .Under assumptions 1, 2, 3 and 4, a minimax solution (ˆθ,ˆτ)of feder- ated optimization problem (20) that satisfies the equilibrium condition as in definition 1: U˜θ(ˆθ, τ)≤U˜θ(ˆθ,ˆτ)≤ max τ′:∥τ′−ˆτ∥≤h(δ)U˜θ(θ, τ′),is anE-approximate federated equilibrium solution as defined in 3, where the approx- imation error εifor each client i∈[N]lies in: max{ζi θ, ζi τ} ≤ εi≤min{α−ρi τ, β−Bi}forρi τ< α and Bi> β, such that α:= λmax ∇2 ττU˜θ(ˆθ,ˆτ) ,β:=λminh ∇2 θθU˜θ− ∇2 θτU˜θ ∇2 ττU˜θ−1∇2 τθU˜θi (ˆθ,ˆτ) and Bi:=ρi θ+Lρi θτ1 \|λmax(∇2ττUi ˜θ)\|+Lρi τθ1 \|λmax(∇2ττUi ˜θ)\|+L2ρi τ1 \|λmax(∇2ττUi ˜θ)·λmax(∇2ττU˜θ)\|. Proof. The pure-strategy Stackelberg equilibrium for the federated objective is: U˜θ(ˆθ, τ)≤U˜θ(ˆθ,ˆτ)≤ max τ′:∥τ′−τ∗∥≤h(δ)U˜θ(θ, τ′), (30) We want to show that the ϵi- approximate equilibrium for each client’s objective Ui ˜θalso hold individually. The first-order necessary condition for (30) to hold is ∇θU˜θ(ˆθ,ˆτ) = 0 and∇τU˜θ(ˆθ,ˆτ) = 0 . Thus, ∇θU˜θ(ˆθ,ˆτ) 2 = 0. Consider ∇θU˜θ(ˆθ,ˆτ) 2 = ∇θU˜θ(ˆθ,ˆτ)− ∇ θUi ˜θ(ˆθ,ˆτ) +∇θUi ˜θ(ˆθ,ˆτ) 2 = ∇θU˜θ(ˆθ,ˆτ)− ∇ θUi ˜θ(ˆθ,ˆτ) 2 + ∇θUi ˜θ(ˆθ,ˆτ) 2 + 2 ∇θU˜θ(ˆθ,ˆτ)− ∇ θUi ˜θ(ˆθ,ˆτ)⊤ ∇θUi ˜θ(ˆθ,ˆτ) Rearranging 2 ∇θUi ˜θ(ˆθ,ˆτ)− ∇ θU˜θ(ˆθ,ˆτ)⊤ ∇θUi ˜θ(ˆθ,ˆτ) − ∇θUi ˜θ(ˆθ,ˆτ) 2 = ∇θU˜θ(ˆθ,ˆτ)− ∇ θUi ˜θ(ˆθ,ˆτ) 2 ∇θUi ˜θ(ˆθ,ˆτ) 2 −2 ∇θU˜θ(ˆθ,ˆτ)⊤ ∇θUi ˜θ(ˆθ,ˆτ) = ∇θU˜θ(ˆθ,ˆτ)− ∇ θUi ˜θ(ˆθ,ˆτ) 2 Using gradient heterogeneity assumption (3) on R.H.S. ∇θU˜θ(ˆθ,ˆτ)− ∇ θUi ˜θ(ˆθ,ˆτ) 2 ≤(ζi θ)2 Thus, we obtain ∇θUi ˜θ(ˆθ,ˆτ) ≤ζi θ.Similarly, ∇τUi ˜θ(ˆθ,ˆτ) ≤ζi τ. In the special case, when ζi θ= 0andζi τ= 0, thus we will have ∇θUi ˜θ(ˆθ,ˆτ) 2 = ∇τUi ˜θ(ˆθ,ˆτ) 2 = 0for all i∈[N], which gives ∇θUi ˜θ(ˆθ,ˆτ) =∇τUi ˜θ(ˆθ,ˆτ) = 0 for all clients i. Next, we prove that each client satisfies the second-order necessary condition approximately. Since (ˆθ,ˆτ)satisfy the equilibrium condition (30), the second-order necessary condition holds for the global function U˜θ, i.e.∇2	https://arxiv.org/abs/2505.21012v1
ττU˜θ(ˆθ,ˆτ)⪯0. We now prove that ∇2 ττUi ˜θ(ˆθ,ˆτ)⪯0. Using assumption 1, the hessian is symmetric. Thus, ∇2 ττU˜θ(ˆθ,ˆτ)⪯0implies λmax(∇2 ττU˜θ(ˆθ,ˆτ))≤0, where λmax is the largest eigenvalue of the hessian. Suppose, λmax(∇2 ττU˜θ(ˆθ,ˆτ)) =−α, for some α≥0. 19 Federated IV Analysis via Federated GMM, Geetika et al. We can write ∇2 ττUi ˜θ(ˆθ,ˆτ) =∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ) +∇2 ττU˜θ(ˆθ,ˆτ). Using a corollary of Weyl’s theorem (Horn and Johnson, 2012) for real symmetric matrices AandB,λmax(A+B)≤ λmax(A) +λmax(B). Hence, λmax(∇2 ττUi ˜θ(ˆθ,ˆτ))≤λmax(∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ)) +λmax(∇2 ττU˜θ(ˆθ,ˆτ)). Thus, λmax(∇2 ττUi ˜θ(ˆθ,ˆτ))≤λmax(∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ))−α. Since the spectral norm of a real symmetric matrix A is given as ∥A∥σ= max {\|λmax(A)\|,\|λmin(A)\|}. Under hessian heterogeneity assumption 4 ∥∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ)∥σ= max λmax(∇2 ττUi ˜θ(θ, τ)− ∇2 ττU˜θ(θ, τ)) , λmin(∇2 ττUi ˜θ(θ, τ)− ∇2 ττU˜θ(θ, τ)) ≤ρi τ. By definition of the spectral norm ∥∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ)∥σ=λmax(∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ)), λmax(∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ))≤maxn λmax(∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ)) , λmin(∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ)) o ≤ρi τ. Thus, λmax(∇2 ττUi ˜θ(ˆθ,ˆτ))≤λmax(∇2 ττUi ˜θ(ˆθ,ˆτ)− ∇2 ττU˜θ(ˆθ,ˆτ))−α≤ρi τ−α, where ρi τ≥0. Hence, ∇2 ττUi ˜θ(ˆθ,ˆτ)⪯(ρi τ−α)I. When ρi τ≤α, then∇2 ττUi ˜θ(ˆθ,ˆτ)⪯0. Now, since (ˆθ,ˆτ)satisfy the equilibrium condition (30), thus ∇2 ττU˜θ(ˆθ,ˆτ)≺0and the Schur complement of ∇2 ττU˜θ(ˆθ,ˆτ)is positive semi-definite. Now when ρi τ< α , it follows from above that ∇2 ττUi ˜θ(ˆθ,ˆτ)≺0, hence ∇2 ττUi ˜θ(ˆθ,ˆτ)−1 exists. Now, we need to show that Schur complement of ∇2 ττUi ˜θ(ˆθ,ˆτ)is positive semi-definite. Since, S(ˆθ,ˆτ) :=h ∇2 θθU˜θ− ∇2 θτU˜θ ∇2 ττU˜θ−1∇2 τθU˜θi (ˆθ,ˆτ)≻0. Define Si:= ∇2 θθUi ˜θ− ∇2 θτUi ˜θ ∇2 ττUi ˜θ−1 ∇2 τθUi ˜θ . We aim to prove λmin(Si)≥0to show Siis positive semidefinite (PSD). Analogous to the above part, using corollary to Weyl’s theorem, we have λmin(Si−S) +λmin(S)≤λmin(Si). Letλmin(S) =β, where β≥0. Moreover, ∥Si−S∥σ= max λmax(Si−S) , λmin(Si−S) , thus λmin(Si− S)≥ −∥ Si−S∥σ. Thus, we have −∥(Si−S)∥σ+β≤λmin(Si). We can write Si−Sas Si−S= (∇2 θθUi ˜θ− ∇2 θθU˜θ)−h (∇2 θτUi ˜θ− ∇2 θτU˜θ)(∇2 ττUi ˜θ)−1∇2 τθUi ˜θ +∇2 θτU˜θ(∇2 ττUi ˜θ)−1(∇2 τθUi ˜θ− ∇2 τθU˜θ) +∇2 θτU˜θ (∇2 ττUi ˜θ)−1−(∇2 ττU˜θ)−1 ∇2 τθU˜θi . 20 Federated IV Analysis via Federated GMM, Geetika et al. Hence, ∥Si−S∥σ≤ ∥∇2 θθUi ˜θ− ∇2 θθU˜θ∥σ+∥(∇2 θτUi ˜θ− ∇2 θτU˜θ)(∇2 ττUi ˜θ)−1∇2 τθUi ˜θ∥σ\| {z } T1 +∥∇2 θτU˜θ(∇2 ττUi ˜θ)−1(∇2 τθUi ˜θ− ∇2 τθU˜θ)∥σ\| {z } T2 +∥∇2 θτU˜θ (∇2 ττUi ˜θ)−1−(∇2 ττU˜θ)−1 ∇2 τθU˜θ∥σ \| {z } T3. Note that the eigenvalue of (∇2 ττUi ˜θ)−1isλ (∇2 ττUi ˜θ)−1 =1 λ(∇2ττUi ˜θ), hence ∥(∇2 ττUi ˜θ)−1∥σ=1 \|λmax(∇2ττUi ˜θ)\|as ∇2 ττUi ˜θis negative definite. By Assumption 2, each client’s function UiisL-Lipschitz thus ∥∇2Ui ˜θ∥σ≤L. Since the Hessian ∇2Ui ˜θis a block matrix of the form: ∇2Ui ˜θ=∇2 θθUi ˜θ∇2 θτUi ˜θ ∇2 τθUi ˜θ∇2 ττUi ˜θ , The norm of Hessian is at least the norm of one of its components ∥∇2 θθUi ˜θ∥σ≤L,∥∇2 θτUi ˜θ∥σ≤L,∥∇2 τθUi ˜θ∥σ≤L,∥∇2 ττUi ˜θ∥σ≤L. Thus, each Hessian block is individually bounded by L. Additionally, UisL-Lipschitz too. Using Assumption 4, bounding T1 T1=∥(∇2 θτUi ˜θ− ∇2 θτU˜θ)(∇2 ττUi ˜θ)−1∇2 τθUi ˜θ∥σ ≤ ∥(∇2	https://arxiv.org/abs/2505.21012v1
θτUi ˜θ− ∇2 θτU˜θ)∥σ· ∥(∇2 ττUi ˜θ)−1∥σ· ∥∇2 τθUi ˜θ∥σ ≤Lρi θτ1 \|λmax(∇2ττUi ˜θ)\| Similarly, bounding T2 T2=∥∇2 θτU˜θ(∇2 ττUi ˜θ)−1(∇2 τθUi ˜θ− ∇2 τθU˜θ)∥σ ≤ ∥∇2 θτU˜θ∥σ· ∥(∇2 ττUi ˜θ)−1∥σ· ∥(∇2 τθUi ˜θ− ∇2 τθU˜θ)∥σ ≤Lρi τθ1 \|λmax(∇2ττUi ˜θ)\| Lastly we bound T3, it is easy to verify that A−1−B−1=A−1(B−A)B−1 T3=∥∇2 θτU˜θ (∇2 ττUi ˜θ)−1−(∇2 ττU˜θ)−1 ∇2 τθU˜θ∥σ ≤ ∥∇2 θτU˜θ∥σ· ∥(∇2 ττUi ˜θ)−1−(∇2 ττU˜θ)−1∥σ· ∥∇2 τθU˜θ∥σ =∥∇2 θτU˜θ∥σ· ∥(∇2 ττUi ˜θ)−1(∇2 ττU˜θ− ∇2 ττUi ˜θ)(∇2 ττU˜θ)−1∥σ· ∥∇2 τθU˜θ∥σ ≤ ∥∇2 θτU˜θ∥σ· ∥(∇2 ττUi ˜θ)−1∥σ· ∥∇2 ττU˜θ− ∇2 ττUi ˜θ∥σ· ∥(∇2 ττU˜θ)−1∥σ· ∥∇2 τθU˜θ∥σ ≤L2ρi τ1 \|λmax(∇2ττUi ˜θ)·λmax(∇2ττU˜θ)\| Using bounds for T1, T2andT3, we can obtain a bound on ∥Si−S∥σ≤Bi, where Bi=ρi θ+Lρi θτ1 \|λmax(∇2ττUi ˜θ)\|+ Lρi τθ1 \|λmax(∇2ττUi ˜θ)\|+L2ρi τ1 \|λmax(∇2ττUi ˜θ)·λmax(∇2ττU˜θ)\|. Consider ρi= max {ρi θ, ρi τθ, ρi θτ, ρi τ}. Hence, Bi≤ ρi 1 +L λmax(∇2ττUi ˜θ) 2 +1 λmax(∇2ττU˜θ . Hence, we obtain λmin(Si)≥ −Bi+β, where λmax(S) = βsuch that β≥0. Hence, we obtain ∇2 θθUi ˜θ− ∇2 θτUi ˜θ ∇2 ττUi ˜θ−1 ∇2 τθUi ˜θ (ˆθ,ˆτ)⪰ (β−Bi)I. When β≥Bi, then Siis positive semi-definite. When Bi= 0 , hence 21 Federated IV Analysis via Federated GMM, Geetika et al. ∇2 θθUi ˜θ− ∇2 θτUi ˜θ ∇2 ττUi ˜θ−1 ∇2 τθUi ˜θ (ˆθ,ˆτ)⪰βI, thus it will be positive semidefinite. When ρi τ< α andβ > Bi, then the suuficient condition for εi-approximate equilibrium is satisfied. And we obtain the result. Thus, for each client i, any approximation error εithat satisfies: max{ζi θ, ζi τ} ≤εi≤min{α−ρi τ, β−Bi}. forρi τ< α andBi> β, then (ˆθ,ˆτ)is anεi-approximate local equilibrium point for client i. C.3 Consistency C.3.1 Assumptions We first state the assumptions that are necessary to establish the consistency of the estimated parameter. Assumption 6 (Identification) .θ0is the unique θ∈Θsuch that ψ(fi;θ) = 0 for all fi∈ F, where i∈[n]. Assumption 7 (Absolutely Star Shaped) .For every fi∈ Fiand\|c\| ≤1, we have cfi∈ Fi. Assumption 8 (Continuity) .For any x,gi(x;θ), fi(x;τ)are continuous in θandτ, respectively for all i∈[N]. Assumption 9 (Boundedness) .Yi,supθ∈Θ\|gi(X;θ)\|,supτ∈T\|fi(Z;τ)\|are bounded random variables for all i∈[N]. Assumption 10 (Bounded Complexity) .FiandGihave bounded Rademacher complexities: 1 2niX ξi∈{−1,+1}niEsup τ∈T1 niniX k=1ξifi(Zk;τ)→0,1 2niX ξi∈{−1,+1}niEsup θ∈Θ1 niniX k=1ξigi(Xk;θ)→0. C.3.2 Proof of Theorem 2 Theorem 6 (Restatement of of Theorem 2) .Let˜θnbe a data-dependent choice for the federated objective that has a limit in probability. For each client i∈[N], define mi(θ, τ,˜θ) :=fi(Zi;τ)(Yi−g(Xi;θ))−1 4fi(Zi;τ)2(Yi−g(Xi;˜θ))2, Mi(θ) = supτ∈TE[mi(θ, τ,˜θ)]andηi(ϵ) :=infd(θ,θ0)≥ϵMi(θ)−Mi(θ0)for every ϵ >0. Let(ˆθn,ˆτn)be a solution that satisfies the approximate equilibrium for each of the client i∈[N]as sup τ∈TUi ˜θ(ˆθn, τ)−εi−op(1)≤Ui ˜θ(ˆθn,ˆτn)≤inf θ∈Θmax τ′:∥τ′−ˆτn∥≤h(δ)Ui ˜θ(θ, τ′) +εi+op(1), for some δ0, such that for any δ∈(0, δ0],and any θ, τsuch that ∥θ−ˆθ∥ ≤δand∥τ−ˆτ∥ ≤δand a function h(δ)→0asδ→0. Then, under similar assumptions as in Assumptions 1 to 5 of (Bennett, Kallus, and Schnabel, 2019), the global solution ˆθnis a consistent estimator to the true parameter θ0, i.e. ˆθnp− →θ0when the approximate error εi<ηi(ϵ) 2for every ϵ >0for each client i∈[N]. Proof. The proof follows from the result of Bennett, Kallus, and Schnabel (2019) that established the consistency of the D EEPGMM estimator. First,	https://arxiv.org/abs/2505.21012v1
we define the following terms for the ease of analysis: mi(θ, τ,˜θ) =fi(Zi;τ)(Yi−g(Xi;θ))−1 4fi(Zi;τ)2(Yi−g(Xi;˜θ))2 Mi(θ) = sup τ∈TE[mi(θ, τ,˜θ)] Mni(θ) = sup τ∈TEni[mi(θ, τ,˜θn)] Note that ˜θnis a data-dependent sequence for the global model. Practically, the previous global iterate is used as ˜θ. Thus, we can define for the federated setting ˜θn=1 NPN i=1˜θni. Let’s assume ˜θnp− →˜θ. 22 Federated IV Analysis via Federated GMM, Geetika et al. Claim 1: supθ\|Mni(θ)−Mi(θ)\|p− →0. sup θ\|Mni(θ)−Mi(θ)\|= sup θ sup τ∈TEni[mi(θ, τ,˜θn)]−sup τ∈TE[mi(θ, τ,˜θ)] ≤sup θ,τ Eni[mi(θ, τ,˜θn)]−E[mi(θ, τ,˜θ)] ≤sup θ,τ Eni[mi(θ, τ,˜θn)]−E[mi(θ, τ,˜θn)] + sup θ,τ E[mi(θ, τ,˜θn)]−E[mi(θ, τ,˜θ)] ≤sup θ1,θ2,τ Eni[mi(θ1, τ, θ 2)]−E[mi(θ1, τ, θ 2)] + sup θ,τ E[mi(θ, τ,˜θn)]−E[mi(θ, τ,˜θ)] We will now handle the two terms in the above equation separately. We will take the first term and call it B1. For mi(θ, τ,˜θn), we constitute its empirical counterpart mi k(θ, τ,˜θn) = fi(Zi k;τ)(Yi k−gi(Xi k;θ))−1 4fi(Zi k;τ)2(Yi k−gi(Xi k;˜θ))2and using mi k′(θ, τ,˜θ′ n)with ghost variables ˜θ′ nfor symmetrization and ϵkaski.i.d. Rademacher random variables , we obtain E[B1] =E" sup θ1,θ2,τ 1 niniX k=1mi k(θ1, τ, θ 2)−Eh mi k′(θ1, τ, θ′ 2)i # ≤E" sup θ1,θ2,τ 1 niniX k=1 mi k(θ1, τ, θ 2)−mi k′(θ1, τ, θ′ 2) # ≤E" sup θ1,θ2,τ 1 niniX k=1ϵk mi k(θ1, τ, θ 2)−mi k′(θ1, τ, θ′ 2) # ≤2E" sup θ1,θ2,τ 1 niniX k=1ϵkmi k(θ1, τ, θ 2) # ≤2E" sup θ,τ 1 niniX k=1ϵkfi(Zi k;τ)(Yi k−gi(Xi k;θ)) # +1 2E" sup θ,τ 1 niniX k=1ϵkfi(Zi k;τ)2(Yi k−gi(Xi k;˜θ))2 # ≤2E" sup θ,τ 1 niniX k=1ϵk1 2fi(Zi k;τ)2+1 2(Yi k−gi(Xi k;θ))2 # +1 2E" sup θ,τ 1 niniX k=1ϵk1 2fi(Zi k;τ)4+1 2(Yi k−gi(Xi k;˜θ))4 # ≤E" sup θ,τ 1 niniX k=1ϵkfi(Zi k;τ)2 # +E" sup θ,τ 1 niniX k=1ϵk(Yi k−gi(Xi k;θ))2 # +1 4E" sup θ,τ 1 niniX k=1ϵkfi(Zi k;τ)4 # +1 4E" sup θ,τ 1 niniX k=1ϵk(Yi k−gi(Xi k;˜θ))4 # Using boundedness assumption 9, we consider the mapping from fi(Zi k;τ)andgi(Xi k;˜θ)to the summation terms in the last inequality as Lipschitz functions, hence for any functional class FiandL- Lipschitz function ϕ,Rni(ϕ◦fi)≤ LRni(Fi),where Rni(Fi)is the Rademacher complexity of class Fi. Hence, E[B1]≤L(Rni(Gi) +Rni(Fi)). Using assumption 10, E[B1]→0.LetB′ 1be a modified value of B, after changing the j-th value of Xi, ZiandYi values, using assumption 9 on boundedness, we obtain the bounded difference inequality: sup X1:ni,Z1:ni,Y1:ni,X′ j,Z′ j,Y′ j\|B1−B′ 1\| ≤ sup θ1,θ2,τ,X 1:ni,Z1:ni,Y1:ni,X′ j,Z′ j,Y′ j\|1 ni mi j(θ1, τ, θ 2)−mi′ j(θ1, τ, θ 2) \| ≤b ni, 23 Federated IV Analysis via Federated GMM, Geetika et al. where bis some constant. Using McDiarmid’s Inequality, we have P(\|B1−E[B1]\| ≥ϵ0)≤2 exp −2niϵ2 0 c2 . And E[B1]→0, we have B1p− →0. Now, we will handle B2. For that B2= sup θ,τ Eh mi(θ, τ,˜θn)i −Eh mi(θ, τ,˜θ)i = sup θ,τ E fi(Zi;τ)(Yi−g(Xi;θ))−1 4fi(Zi;τ)2(Yi−g(Xi;˜θn))2 −E fi(Zi;τ)(Yi−g(Xi;θ))−1 4fi(Zi;τ)2(Yi−g(Xi;˜θ))2 = sup θ,τ1 4 Eh fi(Zi;τ)2(Yi−g(Xi;˜θn))2i −Eh fi(Zi;τ)2(Yi−g(Xi;˜θ))2i = sup θ,τ1 4 Eh fi(Zi;τ)2(Yi−g(Xi;˜θn))2i +Eh fi(Zi;τ)2(Yi−g(Xi;˜θ))2i −Eh fi(Zi;τ)2(Yi−g(Xi;˜θ))2i −Eh fi(Zi;τ)2(Yi−g(Xi;˜θ))2i ≤1 4sup τ E fi(Zi;τ)2ωn Here, ωn= (Yi−g(Xi;˜θn))2−(Yi−g(Xi;˜θ))2 . Due to our assumption, ˜θnp− →˜θ, thus ωnp− →0due to Slutsky’s and continuous mapping	https://arxiv.org/abs/2505.21012v1
theorem. Since, fi(Z;τ)is uniformly bounded, thus for some constant b′>0, we have B2≤b′ 4sup τ1 NNX i=1\|E[ωn]\| ≤b′ 4sup τ1 NNX i=1E[\|ωn\|] Based on the boundedness assumption, we can verify that ωnis bounded, hence using Lebesgue Dominated Convergence Theorem, we can conclude that E[\|ωn\|]→0. Thus, using the convergence of B1andB2, we have supθ\|Mni(θ)−Mi(θ)\|p− →0for each i∈[N]. Claim 2: for every ϵ >0, we have infd(θ,θ0)≥ϵMi(θ)> Mi(θ0). Mi(θ0)is the unique minimizer of Mi(θ). By assumption (6) and (7), θ0is the unique minimizer of supτE[fi(Zi;τ)(Yi−gi(X;θ))]such that supτE[fi(Zi;τ)(Yi−gi(X;θ))] = 0 . Thus, any other value of θwill have at least one τsuch that this expectation is strictly positive. M(θ0) = 0 andM(θ0) = supτ−1 4fi(Zi;τ)2(Yi− gi(X;θ))2, the function whose supremum is being evaluated is non-positive but can be set to zero by assump- tion (7) by taking the zero function of fi. Let for any other θ′̸=θ0, letfi′be a function in Fisuch that E[fi(Z)(Yi−gi(X;θ′))]>0. If we have E[fi′(Z)2(Yi−gi(X;˜θ))2] = 0 , then Mi(θ′)>0. Else, consider cfi′for any c∈(0,1). Using assumption (7), cfi′∈ Fi, thus Mi(θ′) = supfi∈FiEh fi(Zi)(Yi−g(Xi;θ′))−1 4fi(Zi)2(Yi−g(Xi;˜θ))2i ≤cEh fi′(Zi)(Yi−g(Xi;θ′))i −c2 4Eh fi′(Zi)2(Yi−g(Xi;˜θ))2i This is quadratic in cand is positive when cis sufficiently small, thus Mi(θ′)>0. We now prove claim 2 using contradiction. Let us assume claim 2 is false, i.e. for some ϵ > 0, we have infθ∈B(θ0,ϵ)Mi(θ) =Mi(θ0),where B(θ0, ϵ)c={θ\|d(θ, θ0)≥ϵ}., since θ0is the unique minimizer of Mi(θ) by assumption (6). Thus, there must exist some sequence (θ1, θ2, . . .)inB(θ0, ϵ)csuch that Mi(θn)→Mi(θ0). By construction, B(θ0, ϵ)cis closed and the corresponding limit parameters θ∗= lim n→∞θn∈B(θ0, ϵ)cmust satisfy 24 Federated IV Analysis via Federated GMM, Geetika et al. Mi(θ∗) =Mi(θ0)using assumption (8). But d(θ∗, θ0)≥ϵ >0,thusθ∗̸=θ0. This contradicts that θ0is the unique minimizer of Mi(θ); hence, claim 2 is true. Claim 3: For the third part, we know that ˆθnsatisfies the εi- approximate equilibrium condition, given as: Eni[mi(ˆθn, τ,˜θn)]−εi≤Eni[mi(ˆθn,ˆτn,˜θn)]≤ max τ′:∥τ′−ˆτn∥≤h(δ)Eni[mi(θ, τ′,˜θn)] +εi, for a function h(δ)→0asδ→0and some δ0, such that for any δ∈(0, δ0],and any θ, τsuch that ∥θ−ˆθ∥ ≤δand ∥τ−ˆτ∥ ≤δ. Assume that this is true with op(1), hence sup τEni[mi(ˆθn, τ,˜θn)]−εi−op(1)≤Eni[mi(ˆθn,ˆτn,˜θn)]≤inf θmax τ′:∥τ′−ˆτn∥≤h(δ)Eni[mi(θ, τ′,˜θn)] +εi+op(1), . Now, since Mni(ˆθn) =supτEni[mi(ˆθn, τ,˜θn)]. Hence, infθ max τ′:∥τ′−ˆτn∥≤h(δ)Eni[mi(θ, τ′,˜θn)≤infθsupτEni[mi(θ, τ′,˜θn)] =infθMni(θ)≤Mni(θ0) Thus, we have Mni(ˆθn)−εi−op(1)≤Eni[mi(ˆθn,ˆτn,˜θn)]≤Mni(θ0) +εi+op(1). We have proven all three conditions until now. From the first and second condition, since \|Mni(θ0)−Mi(θ0)\|p− →0, hence Mni(ˆθn)≤Mi(θ0) + 2εi+op(1). Hence, we obtain Mi(ˆθn)−Mi(θ0)≤Mi(ˆθn)−Mni(ˆθn) + 2εi+op(1) ≤sup θ\|Mi(ˆθ)−Mni(ˆθ)\|+ 2εi+op(1) ≤2εi+op(1) Hence, we obtain Mi(ˆθn)−Mi(θ0)−2εi≤Mi(ˆθn)−Mni(ˆθn) +op(1) ≤sup θ\|Mi(ˆθ)−Mni(ˆθ)\|+op(1) ≤op(1) Since, let ηi(ϵ) :=infd(θ,θ0)≥ϵMi(θ)−Mi(θ0). Hence, whenever d(ˆθn, θ0)≥ϵ, we have Mi(ˆθn)−Mi(θ0)≥ηi(ϵ). Thus, P[d(ˆθn, θ0)≥ϵ]≤P[Mi(ˆθn)−Mi(θ0)≥ηi(ϵ)] =P[Mi(ˆθn)−Mi(θ0)−2εi≥ηi(ϵ)−2εi]. For every ϵ >0, we have ηi(ϵ)>0from claim 2, and Mi(ˆθn)−Mi(θ0)−2εi=op(1). Thus, ηi(ϵ)−2εi>0when εi<ηi(ϵ) 2. We have that for every ϵ >0andεi<ηi(ϵ) 2, the RHS probability converges to 0, thus d(ˆθn, θ0) =op(1), hence ˆθn converges in probability to θ0for each client i∈[N]. D Limit Points of F EDGDA We first discuss the γ- FEDGDA flow. D.1 F EDGDA Flow The F EDGDA updates can be written as θt+1=θt−η1 γ1 NX i∈[N]RX r=1 ∇θU˜θ(θt, τt) + (∇θUi ˜θ(θi t,r, τi t,r)− ∇ θUi ˜θ(θt, τt)) +(∇θUi ˜θ(θt, τt)− ∇ θU˜θ(θt,	https://arxiv.org/abs/2505.21012v1
τt)) τt+1=τt+η1 NX i∈[N]RX r=1 ∇τU˜θ(θt, τt) + (∇τUi ˜θ(θi t,r, τi t,r)− ∇ τUi ˜θ(θt, τt)) +(∇τUi ˜θ(θt, τt)− ∇ τU˜θ(θt, τt)) 25 Federated IV Analysis via Federated GMM, Geetika et al. Rearranging the terms and taking the continuous-time limit as η→0 lim η→0θt+1−θt η= lim η→0−1 γ1 NX i∈[N]RX r=1 ∇θU˜θ(θt, τt) + (∇θUi ˜θ(θi t,r, τi t,r)− ∇ θUi ˜θ(θt, τt)) +(∇θUi ˜θ(θt, τt)− ∇ θU˜θ(θt, τt)) lim η→0τt+1−τt η= lim η→01 NX i∈[N]RX r=1 ∇τU˜θ(θt, τt) + (∇τUi ˜θ(θi t,r, τi t,r)− ∇ τUi ˜θ(θt, τt)) +(∇τUi ˜θ(θt, τt)− ∇ τU˜θ(θt, τt)) We obtain the gradient flow equations as dθ dt=−R γ1 NX i∈[N] ∇θU˜θ(θ(t), τ(t)) −R γ1 NX i∈[N] ∇θUi ˜θ(θi(t), τi(t))− ∇ θUi ˜θ(θ(t), τ(t)) −R γ1 NX i∈[N] ∇θUi ˜θ(θ(t), τ(t))− ∇ θU˜θ(θ(t), τ(t))) , (31) dτ dt=R1 NX i∈[N] ∇τU˜θ(θ(t), τ(t)) +R1 NX i∈[N] ∇τUi ˜θ(θi(t), τi(t))− ∇ τUi ˜θ(θ(t), τ(t)) +R1 NX i∈[N] ∇τUi ˜θ(θ(t), τ(t))− ∇ τU˜θ(θ(t), τ(t)) . (32) Using Assumption 3 R γ1 NX i∈[N](∇θUi ˜θ(θ(t), τ(t))− ∇ θU˜θ(θ(t), τ(t))) ≤R γζθ R1 NX i∈[N](∇τUi ˜θ(θ(t), τ(t))− ∇ τU˜θ(θ(t), τ(t))) ≤Rζτ Thus, R γ1 NX i∈[N](∇θUi ˜θ(θ(t), τ(t))− ∇ θU˜θ(θ(t), τ(t))) =OR γζθ R1 NX i∈[N](∇τUi ˜θ(θ(t), τ(t))− ∇ τU˜θ(θ(t), τ(t))) =O(Rζτ) Since Ui ˜θis Lipschitz smooth by assumption 2, we have R γ1 NX i∈[N](∇θUi ˜θ(θi(t), τi(t))− ∇ θUi ˜θ(θ(t), τ(t))) ≤LR γ1 NX i∈[N]∥(θi(t), τi(t))−(θ(t), τ(t)∥, R1 NX i∈[N](∇τUi ˜θ(θi(t), τi(t))− ∇ τUi ˜θ(θ(t), τ(t))) ≤LR1 NX i∈[N]∥(θi(t), τi(t))−(θ(t), τ(t))∥. Substituting these bounds into Equations (31) and (32), we obtain R γ1 NX i∈[N](∇θUi ˜θ(θi(t), τi(t))− ∇ θUi ˜θ(θ, τ)) =O LR γ1 NX i∈[N]∥(θi(t), τi(t))−(θ(t), τ(t)∥ , R1 NX i∈[N](∇τUi ˜θ(θi(t), τi(t))− ∇ τUi ˜θ(θ, τ)) =O LR1 NX i∈[N]∥(θi(t), τi(t))−(θ(t), τ(t))∥ . 26 Federated IV Analysis via Federated GMM, Geetika et al. Since the local update follows θi(t) =θ(t)−η γRX j=1∇θUi ˜θ(θi j(t), τi j(t)), τi(t) =τ(t) +ηRX j=1∇τUi ˜θ(θi j(t), τi j(t)), Using bounded gradient assumption, i.e. ∥∇θUi ˜θ(θ, τ))∥2≤Gθand∥∇τUi ˜θ(θ, τ))∥2≤Gτfor all i, asη→0andR is fixed and finite, the deviation ∥(θi(t), τi(t))−(θ(t), τ(t))∥vanish, leading to dθ dt=−1 γR∇θU˜θ(θ(t), τ(t)) +OR γζθ , dτ dt=R∇τU˜θ(θ(t), τ(t)) +O(Rζτ). D.2 Proof of Theorem 3 Proof. LetA=∇2 θθU˜θ(θ, τ),B=∇2 ττU˜θ(θ, τ)andC=∇2 θτU˜θ(θ, τ).Consider ϵ=1 γ, thus for sufficiently small ϵ (hence a large γ), the Jacobian Jof F EDGDA for a point (θ, τ)is given as: Jϵ=R−ϵA−ϵC C⊤B . Using Lemma 6, Jϵhasd1+d2complex eigenvalues {Λj}d1+d2 j=1 such that \|Λj+ϵµj\|=o(ϵ) 1 ≤j≤d1 \|Λj+d1−νj\|=o(1), 1≤j≤d2,(33) where {µj}d1 j=1and{νj}d2 j=1are the eigenvalues of matrices R(A−CB−1C⊤)andRBrespectively. We now prove the theorem statement: LocMinimax ⊂ ∞ − FGDA ⊂ ∞ − FGDA ⊂ L ocMinimax ∪ {(θ, τ)\|(θ, τ)is stationary and ∇2 ττU˜θ(θ, τ)is degenerate }. By definition of lim sup andlim inf , we know that ∞ − FGDA ⊂∞ − FGDA . Now we show LocMinimax ⊂ ∞ − FGDA . Consider a strict local minimax point (θ, τ), then by sufficient condition it follows that: B≺0,and A−CB−1C⊤≻0. Thus, RB≺0,andR(A−CB−1C⊤)≻0, where Ris always positive. Hence, {νj}d1 j=1<0and{µj}d2 j=1<0. Using equations 33, for some small ϵ0< ϵ,Re(Λ j)<0for all j.Thus, (θ, τ)is	https://arxiv.org/abs/2505.21012v1
a strict linearly stable point of 1 ϵ-FEDGDA. Now, we show ∞ − FGDA ⊂ L ocMinimax ∪ {(θ, τ)\|(θ, τ)is stationary and ∇2 ττU˜θ(θ, τ)is degenerate }.Consider (θ, τ)a strict linearly stable point of1 ϵ-FEDGDA , such that for some small ϵ,Re(Λ j)<0for all j.By equation 33, assuming B−1exists RB≺0,and R(A−CB−1C⊤)⪰0. Since, Ris positive, thus B≺0,andA−CB−1C⊤⪰0.Let’s assume A−CB−1C⊤has0as an eigenvalue. Thus, there exists a unit eigenvector wsuch that A−CB−1C⊤w= 0. Then, Jϵ·(w,−B−1C⊤w)⊤=R−ϵA−ϵC C⊤B ·w −B−1C⊤w =0. Thus,Jϵhas0as its eigenvalue, which is a contradiction because for strict linearly stable point Re(Λ j)<0for all j. Thus,A−CB−1C⊤≻0. Hence, (θ, τ)is a strict local minimax point. 27 Federated IV Analysis via Federated GMM, Geetika et al. LetG:Rd×Rk→Rbe the function defined as: G(θ, τ) = det( ∇2 ττU˜θ(θ, τ)).Let’s assume that ∇2 ττU˜θ(θ, τ)is smooth, thus the determinant function is a polynomial in the entries of the Hessian, which implies that Gis a smooth function. Since ∇2 ττU˜θ(θ, τ) = 0 implies at least one eigenvalue of ∇2 ττU˜θ(θ, τ)is zero, thus det(∇2 ττU˜θ(θ, τ)) = 0 . We aim to show that the set A={(θ, τ)\|(θ, τ)is stationary and det(∇2 ττU˜θ(θ, τ)) = 0} has measure zero in Rd×Rk. A point q∈Rd×Rkis aregular value ofGif for every (θ, τ)∈G−1(q), the differential dG(θ, τ)is surjective. Otherwise, qis acritical value . The differential of Gis given by: ∇G(θ, τ) = Tr Adj(∇2 ττU˜θ)· ∇(∇2 ττU˜θ) .Ifdet(∇2 ττU˜θ(θ, τ)) = 0 , then the Hessian ∇2 ττU˜θis singular. This causes its adjugate matrix to lose rank, leading to a degeneracy in ∇G(θ, τ), making dG(θ, τ)not surjective . Thus, every (θ, τ)satisfying G(θ, τ) = 0 is a critical point of G, meaning that 0is acritical value ofG. By Sard’s theorem, the set of critical values of a smooth function has measure zero in the codomain. Since Gis smooth, the set of critical values of GinRhas measure zero. In particular, since 0is a critical value of G, the set: G−1(0) ={(θ, τ)\|det(∇2 ττU˜θ(θ, τ)) = 0}has measure zero in Rd+k. Since the set of degenerate ∇2 ττU˜θ(θ, τ)is precisely G−1(0), we conclude that Lebesgue measure (A) = 0 .Thus, the set of stationary points where the Hessian ∇2 ττU˜θ(θ, τ)is singular has measure zero in Rd×Rk. Lemma 5. (Zedek, 1965) Given a polynomial pn(z) :=Pn k=0akzk,where an̸= 0, an integer m≥nand a number ϵ >0, there exists a number δ >0such that whenever the m+ 1complex numbers bk,0≤k≤m, satisfy the inequalities \|bk−ak\|< δ for0≤k≤n, and\|bk\|< δ forn+ 1≤k≤m, then the roots βk,1≤k≤m, of the polynomial qm(z) :=Pm k=0bkzkcan be labeled in such a way as to satisfy, with respect to the zeros αk,1≤k≤n, ofpn(z), the inequalities \|βk−αk\|< ϵ for1≤k≤n, and\|βk\|>1/ϵ forn+ 1≤k≤m. Lemma 6. For any symmetric matrix A∈Rd1×d1,B∈Rd2×d2, any rectangular matrix C∈Rd1×d2and a scalar R, assume that Bis non-degenerate. Then, matrix R−ϵA−ϵC C⊤B hasd1+d2complex eigenvalues {Λj}d1+d2 j=1 with following form for sufficiently small ϵ: \|Λj+ϵµj\|=o(ϵ) 1 ≤j≤d1 \|Λj+d1−νj\|=o(1), 1≤j≤d2, where {1 Rµj}d1 j=1and{1 Rνj}d2 j=1are the eigenvalues of matrices A−CB−1C⊤andBrespectively. The proof follows from Lemma 5 by a similar argument as in (Jin, Netrapalli, and Jordan, 2020)	https://arxiv.org/abs/2505.21012v1
arXiv:2505.21025v1 [cs.SD] 27 May 2025SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 1 Text-Queried Audio Source Separation via Hierarchical Modeling Xinlei Yin, Xiulian Peng, Xue Jiang, Zhiwei Xiong, Yan Lu Abstract —Target audio source separation with natural lan- guage queries presents a promising paradigm for extracting arbitrary audio events through arbitrary text descriptions. Ex- isting methods mainly face two challenges, the difficulty in jointly modeling acoustic-textual alignment and semantic-aware separation within a blindly-learned single-stage architecture, and the reliance on large-scale accurately-labeled training data to compensate for inefficient cross-modal learning and separation. To address these challenges, we propose a hierarchical decom- position framework, HSM-TSS , that decouples the task into global-local semantic-guided feature separation and structure- preserving acoustic reconstruction. Our approach introduces a dual-stage mechanism for semantic separation, operating on distinct global and local semantic feature spaces. We first perform global-semantic separation through a global semantic feature space aligned with text queries. A Q-Audio architecture is employed to align audio and text modalities, serving as pre- trained global-semantic encoders. Conditioned on the predicted global feature, we then perform the second-stage local-semantic separation on AudioMAE features that preserve time-frequency structures, followed by acoustic reconstruction. We also propose an instruction processing pipeline to parse arbitrary text queries into structured operations, extraction orremoval , coupled with audio descriptions, enabling flexible sound manipulation. Our method achieves state-of-the-art separation performance with data-efficient training while maintaining superior semantic con- sistency with queries in complex auditory scenes. Index Terms —text-queried audio source separation, hierarchi- cal modeling, audio representation learning. I. I NTRODUCTION REAL-world environmental sounds typically comprise di- verse audio events from multiple sources. Target sound separation, which isolates specific sound components from mixtures across domains like speech [1], [2], [3], general audio [4], and music [5], conventionally relies on single-source training samples and focuses on separating predefined source types [6]. Recent advances in universal sound separation (USS) [7] have expanded this capability to arbitrary sound sources in real-world recordings. However, the inherent complexity of separating overlapping audio events with varying char- acteristics persists as a fundamental challenge. Text-queried Xinlei Yin and Zhiwei Xiong are with the University of Science and Technology of China (email: xyxl [email protected]; zwx- [email protected]). Xue Jiang is with the School of Information and Communication Engi- neering, Communication University of China, Beijing 100024, China (e-mail: [email protected]). Xiulian Peng and Yan Lu are with the Microsoft Research Asia, Beijing 100080, China (e-mail: [email protected]; [email protected]). This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. This paper is the result of an open-source project starting from Jan. 2024. Separator Encoder DecoderText Separator (global -semantic) Separator (local -semantic) DecoderEncoder (global -semantic)Text Encoder (local -semantic)(a) Previous single -stage separation structure (b) Proposed hierarchical modeling structureacoustic token global -semantic token local -semantic toke nFig. 1. Comparison between previous blind single-stage and our hierarchical modeling frameworks. target sound extraction (TSE), which uses natural language descriptions to selectively separate sounds, has emerged as a promising solution. Unlike audio-visual [8], [9] and audio- queried [7], [10],	https://arxiv.org/abs/2505.21025v1
[11] methods, it provides greater flexibility in query formulation. It also outperforms label-queried [12], [13], [14] methods by eliminating the need for fixed label categories, thereby supporting queries of any type and facilitating open- domain generalization. The principal challenge in text-queried target sound ex- traction lies in establishing robust cross-modal associations between linguistically diverse queries and intricate acoustic patterns. Natural language instructions may contain temporally ordered event sequences (e.g. “A man talks to himself happily before playing the violin.”) or detailed sound characteristics (e.g. heavy rain, high/low frequency engine), requiring precise alignment with corresponding audio segments in potentially overlapping mixtures. Recently, an increasing number of studies have explored this task, which can mainly be divided into two categories: mask-based separators [15], [16], [17], [4] and conditional generative models [18], [19], [20]. The dominant mask-based approaches typically employ separation networks to estimate a multiplicative time-frequency mask through conditioned U- Net architectures. These methods incorporate a conditioning signal into the audio U-Net with a query network to predict the target mask. The other stream, the emerging generative paradigm by transformer [21] or diffusion [22] models, for- mulates separation as a specialized audio editing task [18]. Despite their varied frameworks, these methods all em- SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 2 ploy a joint optimization of cross-modal understanding and acoustic separation within a single-stage end-to-end frame- work (shown in Figure 1 (a)). This predominantly data-driven approach could potentially lead to training instability and an increased risk of overfitting, necessitating the use of large- scale audio-text datasets. The fundamental limitations persist when operating on the spectrograms [4] or V AEs’ latent [18] representations, as they lack explicit semantics. This forces models to resolve ambiguities through acoustic details alone, often resulting in under-separation or inconsistent audio events. Furthermore, commonly used weakly-labeled datasets like AudioSet [23] and VGGSound [24] contain significant label noise and irrelevant audio content, which can confuse the trained models and further complicate text instruction understanding. In this work, we decompose the text-queried audio separa- tion into semantic-feature-level manipulation at different levels followed by acoustic reconstruction. Different from acoustic representations (e.g., spectrograms or V AE latents), we attempt to operate separation in a more compact global-local semantic feature space. The global semantic space aligns audio and text, inside which we can extract target semantic hints. In the local semantic space that preserves not only event semantics but also local time-frequency details for reconstruction, we can extract more fine-grained semantic features conditioned on previous-stage predicted global hints. Such a fine-grained feature is then utilized to reconstruct the waveform through a generative model. This hierarchical modeling with dual-stage separation (shown in Figure 1 (b)) effectively separates target audio events that align well with the text query without using large weakly-labeled datasets. Our contributions can be summarized as follows: 1) We propose a hierarchical modeling paradigm, HSM- TSS, to separate audios on different feature levels queried by text. The decomposition of global and fine- grained local semantic representations relieves the bur- den of learning audio-text alignment and extracting semantically-correct audio events simultaneously, effec- tively	https://arxiv.org/abs/2505.21025v1
increasing the data efficiency and our model generalizability. 2) We pretrain a text-audio aligned audio representation, Q- Audio , through contrastive learning, which outperforms the commonly used CLAP [25], [26] in several bench- marks. 3) We design an instruction parsing pipeline with large language models (LLMs) to turn arbitrary textual queries into structured separation types, extraction orremoval , and targeted audio descriptions, better facilitating query understanding with bidirectional sound manipulation. 4) We conduct experiments on several source separation benchmarks and demonstrate that the proposed hierar- chical modeling achieves state-of-the-art performance by both signal-level and semantic-level metrics. II. R ELATED WORK A. Unconditional sound separation Unconditional audio separation aims to decompose mixed audio signals into individual sources without relying on exter-nal queries or prior knowledge. Early research primarily fo- cuses on domain-specific tasks, such as speech separation [6], [3] and music source separation [27]. These methods address label ambiguity problem by permutation invariant training (PIT) [28] that permutes predictions to match target signals during training, and often require post-processing to select target sources from separated outputs. However, they typically rely on single-source training data, limiting the scalability to real-world mixtures. To mitigate this, mixture invariant training (MixIT) [29] introduces unsupervised learning with multi-source mixtures. However, it tends to over-separate and requires post-selection. Subsequent work combines MixIT with pre-trained sound classifiers [30], yet these classifiers still require single-source annotations. Other approaches, like MixPIT [31], attempted direct prediction from mixtures but encountered under-separation problems. Weakly supervised methods [32] utilized sound classifiers but were constrained by fixed label categories. These limitations underscore the challenge of achieving open-domain separation without single- source supervision or post-processing. B. Text-queried sound separation Text-queried audio separation leverages natural language descriptions or labels to guide the extraction of target sounds from mixtures. Early methods explored label-queried separa- tion [12], [13], which relies on predefined class labels and struggles with generalization to unseen categories. A paradigm shift emerged with language-queried audio source separation (LASS), which allowed for flexible separation using free-form text queries. Pioneering efforts like LASS-Net [15] introduced end-to-end training with audio-text pairs, while CLIPSep [16] utilized contrastive language-image pretraining (CLIP) [33] to align text and audio embeddings through a visual modality, achieving zero-shot separation. Recent advancements have incorporated multi-modal supervision [34] and hybrid training [35] to tackle data scarcity. For instance, AudioSep [4] scaled training with large audio-language datasets and showcased ro- bust open-domain generalization. Despite these advancements, challenges remain in handling linguistic diversity (e.g., para- phrased descriptions) and noisy real-world data. Alternative approaches, like audio-visual separation [36], [9], used visual cues but were sensitive to occlusions or off-screen sounds. Conversely, text-based methods possess broader applicability, as natural language offers a scalable and intuitive interface for specifying target sources. C. Audio editing with generative models The most common approach for audio editing is to train specialized models for particular tasks, like style transfer [37], [38] or inpainting [39], [40]. Recently, some works have studied general audio editing following human instructions. AUDIT [18] designs several editing tasks involving adding, removal, inpainting, replacement, and super-resolution based on latent diffusion models. UniAudio [19] is	https://arxiv.org/abs/2505.21025v1
an audio founda- tion model which takes adding, dropping and super-resolution as general audio editing tasks. In terms of this, target sound separation can be seen as a sub-task of instruction-based SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 3 audio editing. However, they still leveraged common acoustic representations, like V AEs’ latent or neural codec tokens, and typically suffered from under-separation or over-separation problems. Instead, our approach investigates the hierarchical semantic modeling paradigm with pretrained representations at different feature levels, with which it achieves superior performance with carefully synthesized training data with accurate labels. D. Audio representation learning Recent advancements in self-supervised learning (SSL) have significantly shaped audio representation learning, empower- ing models to extract meaningful features from raw audio data without labeled supervision. These approaches can be broadly categorized into predictive, contrastive, and masked predictive modeling techniques, each addressing distinct challenges in audio processing. Predictive modeling approaches, such as autoregressive predictive coding (APC) [41], [42], have been pivotal in learning sequential audio representations by predict- ing future elements of a sequential audio input based on past data. In masked predictive approach, MAE-AST [43] adapted the masked autoencoders (MAEs) [44] for audios, combining discriminative and generative objectives for training. It splits audio spectrogram into patches, mask a portion of them, and use transformers to reconstruct the masked segments. The work AudioMAE [45], further investigated shifting windows and local self-attention mechanisms to enhance modeling. Unlike predictive and masked approaches, contrastive learning has largely been utilized to learn cross-modal representations such as CLIP [33] in computer vision. It has also been adapted for audio , as seen in CLAP [26], [25], which unifies general audio and text into a joint embedding space. These multi- modal approaches provide weak supervision for tasks like multi-source sound separation, further enriching audio repre- sentation learning. Motivated by BLIP-2 [46] that leverages a Q-Former for vision-language representation learning, we introduce a lightweight module Q-Audio as a bridge between text and audio with frozen audio and text encoders in our hierarchical modeling. III. T HE PROPOSED FRAMEWORK A. Overview Letxmix∈RLdenote the audio mixture of the target audio xtgt∈RLand an interfering audio xother∈RL, given by xmix=xtgt+xother. (1) Lis the audio length. The goal is to separate the target audio source xtgtbased on a given text instruction T. This separation process can be formalized as H: (xmix,T)7→xtgt, where H denotes the separation model. Conventional TSE methods [15], [16], [17], [4] typically rely on a purely data-driven approach through a single-stage end-to-end blind learning architecture, where semantic and acoustic information are entangled in the mixture and it struggles to distinguish target audio events from the interference. Inspired by audio and speech generative mod- els [22], [47], we reformulate the TSE task as a regeneration process by H: (xmix,T)7→S7→xtgt. Here, an intermediatefeature Sis introduced to decouple semantic separation with acoustic reconstruction. Such an intermediate representation Sis both semantic and information-rich in containing local semantics to reconstruct the target audio. We take it as local semantic feature in our approach. To provide more robust semantic guidance, we further decompose the semantic feature	https://arxiv.org/abs/2505.21025v1
Sby extracting a semantic- only global feature Gon top of it, which captures high-level audio event descriptions (e.g., dog barking ) without spatial or acoustic details, and aligns well with the text feature space. Comparatively, Sis more fine-grained, which integrates both semantic representation (e.g. what happened in the audio ) and acoustic properties (e.g., pitch or spatial positioning ) neces- sary for waveform reconstruction. This motivates a hierarchi- cal framework that divides the feature-level separation into global-semantic andlocal-semantic stages, with the predicted global feature ˆGacting as a conditioning input to refine the extraction of the local feature ˆS. The decomposition process can be denoted as H: (xmix,T)7→G7→S7→xtgt. This hierarchical design generates audio representations across three distinct levels: the global-semantic level (via ˆG), the local-semantic level (via ˆS), and the acoustic-only level (viaxtgt). Consequently, our proposed framework is structured into three sequential components: H1: (xmix,T)7→ˆG, (2) H2: (xmix,T,ˆG)7→ˆS, (3) H3:ˆS7→xtgt. (4) H1,H2,H3denote global-semantic separation, local-semantic separation and acoustic decoder, respectively. As depicted in Figure 2 (b), each stage yields a specialized representation tailored to its role in the separation pipeline. The acoustic decoder maps the local-semantic feature ˆSto the acoustic domain and then reconstructs the target waveform using a neural codec decoder. In the following sections III-C, III-D and III-E, we will describe these three modules in detail. To enhance the model’s ability to interpret diverse instruc- tions, we leverage a pre-trained LLM to decompose the text input into two components: the task type Ttaskand the caption Tcap, as shown in the input of Figure 2. By isolating the captions from task types, we can exploit the strengths of audio- language contrastive learning that aligns text and audio in a shared representation space. We define two separation tasks: removal and extraction , allowing the captions to describe either the audio events to be removed or those to be retained, thereby increasing the flexibility of user instructions. This process is elaborated in Section III-F. Compared to single-stage methods, this hierarchical frame- work offers several advantages. It disentangles semantic and acoustic processing to mitigate the impact of interference, enables modular optimization of each stage, and improves ro- bustness in scenarios where target and interfering audio events overlap significantly. By integrating instruction decomposition with staged feature extraction, our approach achieves precise and flexible text-queried sound separation. SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 4 Fig. 2. Overview of our proposed hierarchical modeling and separation frameworks. (a) Two-level audio feature representation. (b) Text-queried two-level separation. B. Hierarchical audio representations 1) Local semantic representation: We leverage the Au- dioMAE [45], a self-supervised learned audio representation, for the local semantic feature S. Unlike discrete neural codec codes that prioritizes acoustic fidelity but lacks semantic rep- resentativeness, or contrastively learned cross-modal features CLAP that lose details, AudioMAE balances acoustic details and semantic information by predicting masked acoustic fea- tures. This makes it a good local-semantic representation, clustering semantically similar audio events while retaining spatial-temporal structure. AudioMAE takes a Vision Transformer (ViT)-based encoder-decoder structure. It takes the log mel spectrogram X∈RT×Fof an audio signal xas input, splits it	https://arxiv.org/abs/2505.21025v1
into P×Ppatches, and embeds them into a Cs-dimensional latent space, yielding features of a shapeT P×F P×Cs.TandFare the number of frames and mel-frequency bins, respectively. Its masked autoencoder design uses an asymmetric structure, pairing large encoders with small decoders and scales well for linear probing, thanks to a high masking ratio that reduces encoding complexity [44]. Following MW-MAE [48], we adopt a small 4-layer transformer-based decoder with a width of 384 and 8 attention heads and pretrain it on AudioSet [23]. 2) Global semantic representation: While AudioMAE cap- tures both semantic and acoustic details, it lacks alignment with text feature space, limiting cross-modal understanding. To address this, we introduce Q-Audio as the cross-modal global semantic feature extractor, inspired by BLIP-2 [46]. As depicted in Figure 2 (a), Q-Audio bridges features from a frozen local semantic audio encoder (our pretrained Au- dioMAE) and a frozen FLAN-T5 text encoder [49] into a shared space. It comprises two feature extractors that utilize a transformer structure and share self-attention layers. The audio part extracts semantic components from AudioMAE via a self-attention followed by a cross-attention, and the text feature extractor utilizes only self-attention layers. We adopt one learnable query embedding to interact with the frozen audio features through cross-attention layers. The query alsointeracts with the text branch in self-attention layers through masking. This structure distills global semantic information from the local-semantic representation while alleviating text comprehension demands through FLAN-T5, which is different from the original BLIP-2 design [46]. This design makes the extractor an audio-language aligner with a much lower com- plexity than the original one. What’s more, extracting global feature G∈R1×Cgfrom the local-semantic representation S∈RT P×F P×Csfurther ensures the correlation and consis- tency between two-level representations, better facilitating our hierarchical modeling and separation design. During the learning phase, the main objective is to en- able the learnable query to extract audio representations that are most informative for the corresponding text input. We adopt the original optimization design in [46] with three training objectives: audio-language contrastive learning, audio- language matching, and audio-grounded text generation, each leveraging a different attention masking mechanism to regulate the interaction between the learnable query and the text branch in self-attention layers. As shown in Figure 2, we use the audio branch of Q-Audio as the global-semantic audio encoder, and the text branch as the text encoder for the two-level separation. C. Global-semantic separation The first stage H1: (xmix,T)7→ˆGof our separation framework operates in the Q-Audio feature space to separate semantic representations of the target audio. We use Q-Audio as the mixture audio and text encoders and employ a 6-layer non-autoregressive (NAR) transformer to predict the target semantic feature Gtgt∈R1×Cg. This prediction is conditioned on the concatenation of the mixture audio feature Gmix∈ R1×Cg, the text feature Tcap∈RN×Ct, and the task token Ttask.Nis the sequence length of text tokens and Ctis the token dimension. The semantic alignment between audio and text feature inputs enables the transformer to accurately model audio-text relationships and guide the separation process. The optimization involves two objectives: a similarity loss and an L1 loss. The similarity loss is	https://arxiv.org/abs/2505.21025v1
defined as the cosine SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 5 similarity between target Gtgtand predicted features ˆG, which is given by Lsim= 1−cos(ˆG, G tgt). (5) The L1 loss is the L1-norm distance between two features given by LL1=\|\|ˆG−Gtgt\|\|1. (6) The total loss is a weighted combination of these two terms with weights λ1andλ2, ensuring both vector-wise and element-wise accuracy. It is defined as follows Lglobal =λ1Lsim+λ2LL1. (7) The predicted ˆGserves as a semantic guidance for the subse- quent local-semantic separation stage. D. Local-semantic separation The second stage H2: (xmix,T,ˆG)7→ˆSoperates in the more fine-grained AudioMAE feature space. This stage takes the mixture audio feature Smix∈RT P×F P×Csby AudioMAE encoder as input to predict the target feature Stgt∈RT P×F P×Cs. The previous-stage output ˆG, text feature Tcap∈RN×Ct, and the task token Ttask are concatenated as the conditioning input, as shown in Figure 2. During training, ground-truth global audio feature Gtgtis used instead of the prediction ˆG. We leverage the L1-based regression loss for this stage optimization, which is Llocal=\|\|ˆS−Stgt\|\|1, (8) Where ˆSis the predicted output. To mitigate possible error propagation between two stages, we further conduct joint fine- tuning by Equations 7 and 8 after optimizing each stage inde- pendently. We introduce a switcher to control the conditioning input of the second stage, selecting ground-truth Gtgtwith a probability of Pgt, and prediction ˆGwith a probability of Ppred = 1−Pgt. This joint fine-tuning optimizes the two stages end-to-end, minimizing inconsistencies and improving separation accuracy. E. Acoustic decoder After the two-stage separation, the last part H3:ˆS7→xtgt, the acoustic decoder, aims to reconstruct the waveform from the predicted local semantic feature. In light of the power of generative models, we leverage auto-regressive transformer for this stage. Specifically, we encode xtgtinto discrete acoustic tokens Ausing a neural audio codec, and leverage an au- toregressive transformer to generate audio tokens conditioned onˆS, followed by a neural codec decoder to reconstruct the waveform, similar to that in UniCodec [50]. 1) Autoregressive audio token generation: As illustrated in Figure 3, an autoregressive transformer decoder is employed to convert local semantic tokens into acoustic tokens, given by p(A\|Stgt;θ) =TaY t=0p(at\|a<t, Stgt;θ), (9) Fig. 3. Overview of the acoustic decoder. The autoregressive transformer generates acoustic tokens by our neural codec TF-Codec, conditioned on local semantic features. where Tadenotes the acoustic token sequence length, and at is the t-th frame token of A.a−1andaTaare start and end tokens, respectively. θdenotes the network parameters. The decoder is trained using a teacher-forcing approach with a cross-entropy loss by Lce=−logTaY t=0p(at\|a<t, Stgt;θ). (10) The generated tokens are then used by the neural codec de- coder to reconstruct waveforms. During inference, predictions are made token by token, using each predicted token as input for the next. 2) Reconstruction via TF-Codec: TF-Codec [51] is a low- latency neural speech codec designed for high-quality speech at low bitrates. We retrain a non-predictive version of TF- Codec to adapt to general audio for acoustic tokens. It employs a VQ-V AE framework, including an encoder, a group vector quantizer, and a decoder optimized end to end. To	https://arxiv.org/abs/2505.21025v1
ensure high perceptual quality for diverse general audio, we apply adversarial training with a multi-scale mel discriminator [19] to replace the original single-scale frequency-domain discrimi- nator in [51]. Other training losses are the same as that in [51]. Instead of training from scratch, we finetune from the pre- trained TF-Codec speech codec model for better performance. Similar to UniCodec [50], in autoregressive token generation of Equation 9, all groups of t-th step are simultaneously generated in a single stage, leading to a short token length. F . Instruction semantic parsing In text-guided audio separation, users typically provide de- scriptive prompts to retain some audio components or remove some undesired audio events. This instruction is arbitrary and may target at single or multiple sound sources with specified characteristics. To facilitate this open-vocabulary text prompts with bidirectional operations, our framework introduces dual separation operations: removal and extraction , allowing the SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 6 Fig. 4. The pipeline of processing arbitrary text instructions. model to isolate target audio through complementary ap- proaches. We also utilize pre-trained LLMs to parse text prompts into this task type and a target audio description. For example, an input “ Could you help me separate the sound of a dog barking and the background music? ” is decomposed into a task type “ extraction ” and an audio description “ a dog barking and the background music ”, enabling distinct processing of operation intent and acoustic context. This process offers two critical advantages. It enhances the prompt understanding capability of the separation model. What’s more, the decomposition of event description from operation types makes it feasible to leverage audio-language contrastive learning to extract aligned audio-text features, im- proving the separation model’s ability to capture cross-modal correlations. We introduce a dual-channel text conditioning mechanism for this decomposed processing. The task type is encoded using pretrained token representations from T5’s vocabulary, producing continuous task tokens Ttask∈RL×Ct, where Lis the token length and Ctis the embedding dimen- sion. Simultaneously, the event description is processed by the semantic text encoder, denoted as Tcap∈RN×Ct, where N is the length of the caption sequence. These two streams of tokens are concatenated along the sequence dimension to form the final text conditioning input [Ttask, Tcap]∈R(L+N)×Ct, as shown in Figure 2 (b). IV. E XPERIMENTAL SETTING A. Dataset and preprocessing 1) Semantic separation: We utilize audio data from Au- dioCaps [52], ESC-50 [53], Clotho v2 [54], FSD50K [55] and WavCaps [56] due to their relatively accurate and diverse labels or captions. Each audio contains single or multiple audio events with or without overlaps. AudioCaps is a sub- set of AudioSet [23] with handcrafted captions and we use 45K audio clips. Clotho v2 provides human-annotated cap- tions, and we use 3,839 training clips, each accompanied by five captions. ESC-50 comprises 2,000 environmental audio recordings, evenly distributed across 50 semantic classes, including natural sounds, non-speech human sounds, domestic sounds, and urban noises. FSD50K contains approximately 40K audio clips with 200 classes, varying in duration from 0.3 to 30 seconds. WavCaps includes four	https://arxiv.org/abs/2505.21025v1
subsets: AudioSet SL,Freesound, SoundBible, and BBC Sound Effect. We use the entire AudioSet SL subset, exclude Freesound with inaccurate captions, and filter the SoundBible and BBC Sound Effect sub- sets by removing audios longer than 60 seconds. Ultimately, we select 121K audio clips from WavCaps. All audios are resampled to 16 kHz and we normalize all of them to 10 seconds by cropping or padding. In total, we compile 230K ten-second audio clips with class labels or captions as the source data for generating mixtures. For semantic separation, we employ triplet data (mixed audio, target audio, instruction) for supervised training. In- structions are generated using two task types and audio descriptions. We generate bidirectional instructions for each triplet and randomly select one during training, balancing the model’s ability to interpret both target and undesired audio events. Given that most source audios contain multiple audio events, when randomly selecting two samples to create each mixture, we follow the rule that the separated two audios do not have any overlap in audio classes. As text-audio pairs from various datasets feature diverse audio descriptions (e.g. FSD50K and ESC-50 provide different fixed sets of class labels, while AudioCaps, WavCaps and Clotho v2 offer detailed captions), it is necessary to unify the class labels to distinguish audio event types during audio mixing. We employ GPT-41[57] to semantically categorize different audio types of different datasets and summarize audio event tags from captions. For example, “ rain” and “ raindrop ” are the same type and GPT-4 extracts tags “ man speech, thump, frog croak ” from the original caption “ An adult male speaks while thumps occur in the background, then frogs croak .” It is noted that these unified tags are only used in audio mixing to avoid mixing audios including the same categories. The instruction part will keep the original caption and labels to preserve the diversity. In total, we create 1.7M mixture-target audio pairs for training, containing approximately 4500 hours. 2) Q-Audio and AudioMAE: To pretrain the Q-Audio mod- ule, we use the same datasets as that in Section IV-A1, i.e. AudioCaps, ESC-50, Clotho v2, FSD50K and WavCaps, which provide detailed captions or class labels paired with each audio. Following the approaches of AudioMAE and MW- MAE [48], we train the MAE model using AudioSet-2M [23], encompassing both balanced and unbalanced subsets. 3) Acoustic decoder: The acoustic decoder is trained using audios only without text labels. We gather data from AudioSet for general sounds, the Jamendo dataset for music [58], and a multilingual speech dataset with 6k hours for clean speech [50] that combines the Libri-Light medium subset [59], the multi- lingual speech subsets of the ICASSP 2021 DNS Challenge corpus [60], and A VSpeech [61]. All audios are resampled to 16 kHz. We randomly sample 2,000 hours from each category. During TF-Codec training, audios are cropped to 3-second segments. For autoregressive audio token generation module, we set the maximum duration of each training audio to 10 seconds and perform padding masks on audio shorter than 10 seconds. 1https://chatgpt.com/ SUBMITTED TO IEEE TRANSACTIONS ON AUDIO,	https://arxiv.org/abs/2505.21025v1
SPEECH, AND LANGUAGE PROCESSING 7 B. Evaluation benchmark and metrics 1) Evaluation benchmark: We compile evaluation data from the test sets of AudioCaps, FSD50K, and Clotho v2, for general audio assessment. The mixing strategy is the same as that used for training. As shown in Table I, we also create a “3 Sets” that combine three datasets with an equal number of randomly sampled audio clips from each source dataset. Mixing sources from different datasets can account for distribution biases, leading to better approximations of real- world recordings. The average SNR between two components of the mixture is controlled within [-15dB, 15dB]. TABLE I DETAILS OF TEST SETS Dataset Num.sources Caption Duration(sec) Num.pairs AudioCaps 952 ✓ 10 500 Clotho 1045 ✓ 15-30 500 FSD50K 10231 ✗ 5-30 500 3 Sets 4000 ✓ 10 1000 Additionally, we assess zero-shot audio separation perfor- mance on MUSIC dataset [8], which comprises 536 video recordings of individuals playing musical instruments across 11 classes (e.g., accordion, acoustic guitar, cello). Following [16], we downloaded 46 video recordings from the test split, randomly selected two clips from different instrument classes and mixed them into 10-second segments, producing 500 audio pairs. The average SNR in mixing is within [-5dB, 5dB]. 2) Objective metrics: We evaluate our separation perfor- mance using log spectral distance (LSD), peak signal-to- noise ratio (PSNR), and Kullback-Leibler divergence (KL), following the AudioLDM eval audio generation evaluation pipeline2. LSD quantifies the difference between spectrograms of the predicted and target samples. PSNR measures the logarithmic ratio of the maximum possible signal power to the mean squared error between the predicted and target signals. KL divergence measures the similarity between the predicted and target audio with the label calculated by an audio tagging model. To further evaluate the semantic correction of predicted audios, we leverage the CLAP score [62], which calculates audio-text similarity based on CLAP models [26], [25]. The MSCLAP [26] is chosen for this measurement in our exper- iment. As CLAP tends to poorly capture temporal semantics and complex multi-source audio semantics [63], we further introduce another semantic score, AFSim , by calculating a cosine similarity between predicted and target signals on their semantic embeddings based on large audio language models (AudioLLM). Specifically, we leverage Audio Flamingo [64], an audio LLM with advanced audio understanding capabilities, extract the feature from the penultimate (last second) layer by using the captioning prompt, and perform a mean-pooling to get the final embedding AF∈R1×2048, which is taken as the semantic embedding for our AFSim score measurement. The superior semantic representation capability of the AF over CLAP is demonstrated in Table VI in audio captioning evaluation. 2https://github.com/haoheliu/audioldm evalC. Implementation details For the Q-Audio module, we use a two-layer transformer with two self-attention layers and two cross-attention layers. The loss weights of the three losses are all set to 1.0. We choose FLAN-T5-base as the text encoder. The global- semantic separation module takes a 6-layer NAR transformer, with the Q-Audio audio and text encoders kept frozen during training. λ1andλ2are all set to 1.0. We train the local-semantic separation module with a 12- layer	https://arxiv.org/abs/2505.21025v1
NAR transformer which has 8 attention heads, and a hidden dimenion of 768. During two-stage joint fine-tuning, we utilize the same training data and fintune the parameters of two NAR transformers, with other modules kept frozen. Pgt is set to 0.1. The loss weights of Lglobal andLlocal are set to 0.1 and 1.0, respectively. The acoustic decoder includes two parts. The autoregressive transformer has 12 layers with 8 attention heads and a hidden dimension of 1024. We take TF-Codec with a bitrate of 6 kbps with a causal setting, similar to that in [50]. V. R ESULTS In this section, we evaluate the performance of our HSM- TSS approach and its modules including global-local repre- sentations and neural audio codec. A. Evaluation results on general audio We compare the separation performance of our HSM-TSS with several text-queried audio separation methods, LASS- Net [15], AudioSep [4], CLIPSep [16] and BiModalSep [17], which are all frequency-domain mask-based approaches. LASS-Net employs a pre-trained BERT as the text query en- coder and ResUNet as the separation model, while AudioSep further integrates a CLAP text encoder as the query network and trains on a much larger dataset, yielding substantial performance gains. CLIPSep adopts CLIP as the query encoder and a Sound-of-Pixels (SOP)-based separation model, trained on approximately 500 hours of noisy audio-visual data from the VGGSound [24] dataset using hybrid vision-text supervi- sion. BiModalSep introduces a weakly-supervised approach and leverages bi-modal semantic similarity via CLAP to align single-source language prompts with audio predictions, achieving robust separation without curated single-source data. We evaluate their performance using the official open-sourced models on our benchmarks. To show the effectiveness of our hierarchical modeling, we also compare with a single-separation-stage variant of our HSM-TSS, termed “Ours single” in Table II. It removes the first global-semantic separation stage and only leverages the local-semantic separation stage. The text encoder is FLAN-T5 without Q-Audio in this variant. We also provide results for not only “extraction” as used in four compared methods but also “removal” task types that produce the same target audios. As shown in Table II, the proposed HSM-TSS, noted as “Ours”, outperforms both baseline methods and its single- separation-stage variant across various metrics. When com- pared to four methods, LASSNet, CLIPSep, AudioSep, and BiModalSep, our approach consistently achieves higher scores SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 8 TABLE II OBJECTIVE EVALUATION OF SEPARATION FOR GENERAL AUDIO Model Train.Data 3 Sets Clotho AudioCaps FSD50K (hrs) KL (↓) LSD (↓) PSNR (↑) KL(↓) LSD (↓) PSNR (↑) KL(↓) LSD (↓) PSNR (↑) KL(↓) LSD (↓) PSNR (↑) LASS-Net 17 2.577 3.170 16.42 2.713 1.696 18.21 2.446 2.276 17.64 2.474 4.528 15.17 CLIPSep 550 2.320 3.197 14.38 2.616 1.634 18.43 2.738 2.464 17.07 2.967 4.475 17.15 BiModalSep 670 1.634 3.118 19.49 1.819 1.613 19.71 1.441 2.256 19.44 1.789 4.428 18.89 AudioSep 14100 1.027 3.037 22.50 1.191 1.616 21.57 1.002 2.142 21.17 1.172 4.313 23.66 Extraction Ours single 600 0.999 2.878 25.50 1.144 1.395 23.80 0.960 1.916 23.49 0.948 4.197 27.11 Ours 600 0.924 2.848 25.77 1.089 1.378 24.05 0.910 1.884 23.93	https://arxiv.org/abs/2505.21025v1
0.889 4.156 27.34 Removal Ours single 600 1.079 2.869 25.50 1.184 1.383 23.98 1.148 1.919 23.66 1.069 4.209 26.65 Ours 600 1.007 2.852 25.66 1.112 1.373 23.94 1.105 1.908 23.88 1.041 4.179 27.00 TABLE III SEMANTIC EVALUATION FOR GENERAL AUDIO Model 3 Sets Clotho AudioCaps FSD50K AFSim (↑) CLAP (↑) AFSim (↑) CLAP (↑) AFSim (↑) CLAP (↑) AFSim (↑) CLAP (↑) GT - 0.481 - 0.437 - 0.575 - 0.466 LASS-Net 0.615 0.268 0.590 0.314 0.641 0.264 0.592 0.223 CLIPSep 0.514 0.229 0.487 0.257 0.519 0.214 0.525 0.203 BiModalSep 0.678 0.420 0.674 0.385 0.713 0.501 0.631 0.387 AudioSep 0.730 0.357 0.720 0.401 0.752 0.376 0.699 0.305 Extraction Ours single 0.744 0.420 0.734 0.379 0.763 0.500 0.721 0.428 Ours 0.752 0.436 0.737 0.393 0.767 0.511 0.729 0.437 Removal Ours single 0.738 0.419 0.730 0.377 0.755 0.486 0.718 0.420 Ours 0.741 0.428 0.733 0.380 0.763 0.492 0.720 0.425 in the three metrics, especially for the 3 Sets benchmark, highlighting its effectiveness in separating target audio events from complex mixtures. In contrast to the single-separation- stage setting, our dual-stage separation allows for progressive refinement of audio features from coarse to fine, leading to more precise separation of target sounds, as demonstrated in Table II. We can also see that our model with “removal” in- struction achieves comparable scores with “extraction”, show- ing the bidirectional operation capability of our method. Table III shows the evaluation results on semantic consis- tency of the output audios with instructions. For “removal” setting, we use the same audio description as “extraction” to calculate the CLAP score. We can see that our method con- sistently outperforms other methods and the single-separation- stage variant in both AFSim and CLAP scores, showing its superior capability to follow instructions with diverse audio descriptions. For all these comparisons, we only use the simple “extract” and “removal” instruction templates as the compared methods are not designed for open vocabulary separation instruction inputs but only target audio captions. Our method has good arbitrary instruction following capability, as shown in our demo page3. B. Zero-shot performance on unseen datasets We perform zero-shot evaluation on mixtures of music instruments from the MUSIC dataset, as we do not use any specialized music data during training. We can see from Table IV that our method outperforms all other methods in all met- rics. It’s worth noting that the audio clips with clear musical instrument labels in our training data primarily come from 3https://hsm-tss.github.ioTABLE IV EVALUATION ON ZERO -SHOT MUSIC DATASET Model KL (↓) LSD (↓) PSNR (↑) CLAP (↑) AFSim (↑) GT - - - 0.490 - LASS-Net 4.180 2.066 13.23 0.126 0.503 CLIPSep 2.179 2.554 14.07 0.372 0.657 BiModalSep 2.607 2.250 14.35 0.374 0.671 AudioSep 0.535 1.624 22.01 0.342 0.804 Extraction Ours single 0.585 1.400 23.22 0.463 0.799 Ours 0.501 1.375 23.85 0.470 0.811 FSD50K, accounting for less than 5% of the entire dataset, significantly lower than the proportion in AudioSep’s training dataset, AudioSet and VGGSound, that contain rich YouTube- sourced music instrument data. The superior performance of our HSM-TSS method demonstrates its strong generalization capability across diverse music	https://arxiv.org/abs/2505.21025v1
content. C. Visualization 1) t-SNE visualization: To show how the global-semantic separation performs, we visualize the extracted features from this stage with t-SNE [65]. In Figure 5, each color shows a sound event class and we present the ground-truth global audio semantic feature and the separated output with different markers. It can be observed that our global-semantic separation model effectively extracts target audio features with good clustering. 2) Visualization of attention maps: To show how the pre- dicted global semantic feature helps the local-semantic separa- tion stage, we visualize the attention map from the last layer of the NAR transformer. In Figure 6, the bottom two subfigures SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 9 Fig. 5. t-SNE visualization of global-semantic features: ground-truth vs. predicted. Fig. 6. Visualization of attention weights. The upper subfigures are the spectrograms of the mixed and target audios, respectively. The bottom two subfigures show the attention weights of the global audio feature attending to local AudioMAE patches for ground-truth and predicted global feature inputs, respectively. show the attention weights of the global feature attending to all 512 local semantic patches in temporal order, with the upper one takes ground-truth global feature as input and the bottom one uses previous global-semantic separation stage output feature. We can observe that the high peaks happen where target audio event occurs, which well demonstrates the semantic alignment of predicted global feature with target audio. 3) Visualization of separation outputs: We show the spec- trograms for mixture input, target audio and the separated audio with text queries for both “extraction” and “removal” commands in Figure 7. We can see that the spectrograms from separated output audios closely resemble the target one, and the two commands achieve similar results, consistent with our objective evaluation results.D. Hierarchical representations 1) Global semantic feature evaluation: To evaluate the cross-modal alignment and semantic representation capbility of our Q-Audio, we follow the settings of LAION-CLAP [25] on AudioCaps and Clotho datasets. Table V shows the text-to- audio retrieval results. Compared to the widely used LAION- CLAP [25] and MSCLAP [26], Q-Audio achieves better performance, underscoring its capability in audio-language modeling. Table VI shows the audio captioning evaluation, where a linear mapping layer and a pretrained GPT2 [66] are used as the downstream model to generate audio captions. Here we only train the downstream model and keep audio fea- ture extraction frozen. Although our Q-Audio is not pretrained on the captioning task as CLAP models, it still achieves higher scores on the three metrics, showing its excellent semantic representation capability. TABLE V TEXT-TO-AUDIO RETRIEVAL PERFORMANCE Model Clotho AudioCaps R@1 R@5 R@10 R@1 R@5 R@10 LAION-CLAP 14.6 37.3 49.9 20.4 49.7 64.3 MSCLAP 15.6 38.6 51.3 25.4 55.6 69.7 Q-Audio 17.1 41.4 54.1 26.6 56.8 70.4 TABLE VI AUDIO CAPTIONING ON AUDIO CAPS AND CLOTHO Model METEOR (↑) SPICE (↑) SPIDEr (↑) LAION-CLAP 0.115 0.152 0.311 MSCLAP 0.116 0.160 0.326 Q-Audio 0.118 0.164 0.329 AF 0.121 0.170 0.336 2) Local semantic feature evaluation: We evaluate our pretrained AudioMAE model on the HEAR benchmark [67] and three tasks,	https://arxiv.org/abs/2505.21025v1
ESC-50, SC-5h and NS-5h, where only the downstream model is trained. As shown in Table VII, our pretrained model achieves good performance, outperforming the official AudioMAE-B [45]. TABLE VII EVALUATION OF AUDIO MAE ONHEAR BENCHMARK Model ESC-50 (↑) SC-5h (↑) NS-5h (↑) AudioMAE-B [45] 57.6 33.9 61.4 Our pretrained AudioMAE 83.7 77.1 66.0 E. Neural codec performance In Table VIII, we assess the reconstruction quality of general audio by our neural codec TF-Codec. We randomly sample 1000 audios from AudioSet validation set for evaluation. All models have a bitrate of 6 kbps. We can see that although our general audio TF-codec has a low latency by using a causal structure and much less parameters, it achieves good performance, superior than EnCodec [68]. DAC [69] achieves the best performance but it is much heavier with a non-causal structure. It should be noted that we utilize TF-Codec just as a proof of concept, and future works may utilize any non-causal codec with superior performances. SUBMITTED TO IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 10 Fig. 7. Visualization of mixture, target and the separated audio by our method. TABLE VIII NEURAL CODEC PERFORMANCE ON GENERAL AUDIO AT 6KBPS Model #param Causal MEL-D (↓) LSD(↓) VISQOL (↑) EnCodec 14.85M ✓ 1.047 4.058 4.209 DAC 74.17M ✗ 0.630 3.556 4.521 Our TF-Codec 7.63M ✓ 0.797 3.588 4.375 VI. C ONCLUSION AND FUTURE WORKS In this study, we propose a hierarchical modeling and sepa- ration framework for text-queried audio source separation, de- coupling multi-level semantic feature separation and acoustic reconstruction. Leveraging Q-Audio for global-semantic mod- eling on top of AudioMAE for structure-preserving representa- tions, our model achieves superior separation performance and semantic correctness, outperforming existing methods. The instruction parser enhances flexibility in handling diverse text queries with frozen LLMs. In future work, we will scale up the model in training data coverage which incorporates speech as well, and explore more fantastic editing options besides separation. REFERENCES [1] Y . Liu and D. Wang, “Divide and conquer: A deep casa approach to talker-independent monaural speaker separation,” IEEE/ACM Transac- tions on Audio, Speech, and Language Processing , vol. 27, no. 12, pp. 2092–2102, 2019. [2] ——, “A casa approach to deep learning based speaker-independent co- channel speech separation,” in 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , 2018, pp. 5399– 5403. [3] Y . Luo and N. Mesgarani, “Conv-tasnet: Surpassing ideal time– frequency magnitude masking for speech separation,” IEEE/ACM trans- actions on audio, speech, and language processing , vol. 27, no. 8, pp. 1256–1266, 2019. [4] X. Liu, Q. Kong, Y . Zhao, H. Liu, Y . Yuan, Y . Liu, R. Xia, Y . Wang, M. D. Plumbley, and W. Wang, “Separate anything you describe,” IEEE/ACM Transactions on Audio, Speech, and Language Processing , 2024.[5] Q. Kong, Y . Cao, H. Liu, K. Choi, and Y . Wang, “Decoupling magnitude and phase estimation with deep resunet for music source separation,” inInternational Society for Music Information Retrieval Conference , 2021. [Online]. Available: https://api.semanticscholar.org/CorpusID: 237491546 [6] D. Wang and J. Chen, “Supervised speech separation based on	https://arxiv.org/abs/2505.21025v1
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arXiv:2505.21026v1 [eess.SY] 27 May 2025IEEE TRANSACTIONS ON CYBERNETICS, APRIL 2025 1 Multi-Mode Process Control Using Multi-Task Inverse Reinforcement Learning Runze Lin , Junghui Chen , Biao Huang ,Fellow, IEEE , Lei Xie , Hongye Su ,Senior Member, IEEE Abstract —In the era of Industry 4.0 and smart manufacturing, process systems engineering must adapt to digital transformation. While reinforcement learning offers a model-free approach to process control, its applications are limited by the dependence on accurate digital twins and well-designed reward functions. To address these limitations, this paper introduces a novel framework that integrates inverse reinforcement learning (IRL) with multi-task learning for data-driven, multi-mode control design. Using historical closed-loop data as expert demonstra- tions, IRL extracts optimal reward functions and control policies. A latent-context variable is incorporated to distinguish modes, enabling the training of mode-specific controllers. Case studies on a continuous stirred tank reactor and a fed-batch bioreactor validate the effectiveness of this framework in handling multi- mode data and training adaptable controllers. Index Terms —multi-mode process, data-driven controller de- sign, deep reinforcement learning, inverse reinforcement learning, multi-task learning. I. I NTRODUCTION IN the era of Industry 4.0 and smart manufacturing, intel- ligent process control systems have become increasingly pivotal. The integration of AI into science and engineering has ushered in a new paradigm, with Deep Reinforcement Learning (DRL) offering transformative potential for modern process industries. In recent years, there has been growing interest in applying DRL to process control, leading to sig- nificant advancements in both continuous and batch process applications [1], [2]. A key development in this area is the incorporation of transfer learning into DRL frameworks, enhancing the safety, robustness, and practical deployment of process control systems [3]–[8]. Current research has yet to fully address key challenges in applying DRL to process control, such as the high costs of trial-and-error learning, low sample efficiency, and exploration instability. Industrial settings have amassed vast amounts of historical closed-loop data from PLC or DCS-control opera- tions. However, traditional control strategies like MPC, as well as advanced algorithms such as DRL, have not fully harnessed This work was supported in part by the National Key R&D Program of China under Grant 2022YFB3305903, and the National Science and Technology Council, Taiwan under Grant NSTC 113-2221-E-033-003. R. Lin, L. Xie, and H. Su are with the State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]; [email protected]; [email protected]). J. Chen is with the Department of Chemical Engineering, Chung- Yuan Christian University, Taoyuan 32023, Taiwan, R.O.C. (e-mail: ja- [email protected]). B. Huang is with the Department of Chemical and Materials Engi- neering, University of Alberta, Edmonton, AB T6G 2G6, Canada (e-mail: [email protected]).the valuable insights embedded in this industrial big data. Extracting control patterns from real-world closed-loop data could serve as a robust foundation for DRL transfer learning [9], [10]. Learning from demonstrations involves various techniques for training Reinforcement Learning (RL) controllers using expert demonstrations [11], with imitation learning being a widely used approach. In industrial production, abundant historical closed-loop data can be utilized to derive controller priors through imitation	https://arxiv.org/abs/2505.21026v1
learning [12]. These RL controllers can then be fine-tuned via transfer learning in real-world processes, improving the safety of DRL training. A basic method, behavior cloning, fits “state xt-action ut” pairs from expert trajectories. In contrast, Inverse RL (IRL), particularly Adversarial IRL (AIRL) [13], offers a more sophisticated approach by framing learning from demonstrations as a proba- bilistic inference problem [14]. Nevertheless, conventional IRL methods struggle to address the multi-mode nature of process control, where varying operating modes lead to distinct data distributions, complicating their application in multi-mode controller design. Operating processes across different modes is far from ideal, yet it remains a common and realistic challenge that poses significant difficulties for DRL. PSE researchers are well- acquainted with multi-mode processes, particularly in fields such as data-driven process monitoring, soft sensing, and fault diagnosis, where multi-mode modeling is frequently required [15]. This necessitates the development of models tailored to distinct data distributions. Common approaches include Kernel Principal Component Analysis, Kernel Partial Least Squares, Gaussian Mixture Models, Variational Autoencoders, and InfoGAN [16]. These methods typically utilize latent variables to capture the variability across operating modes and data distributions in the modeling process. However, while these methods have advanced modeling capabilities, they fall short in addressing the complexities of optimizing and controlling multi-mode systems. Robust control, though capable of managing variability, is often overly conservative. Approaches like controller fusion, weighting or gain scheduling show promise but require manual design for each mode and tailored integration strategies. These methods also struggle with generalization, particularly when confronted with extrapolated conditions or unforeseen modes. Addition- ally, MPC relies on predefined models to capture process dy- namics, but in multi-mode scenarios, model-plant mismatches frequently arise, leading to suboptimal control performance. Although different modes involve distinct characteristics, many share common features since only certain operatingThis work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. 2 IEEE TRANSACTIONS ON CYBERNETICS, APRIL 2025 conditions or materials vary. This observation leads to an important question: Can concepts from multi-mode modeling be leveraged to design a multi-mode controller using latent variables? Given that IRL offers a robust probabilistic infer- ence framework for offline, data-driven control design [9], this paper proposes augmenting IRL with latent variables and multi-task learning. The approach enables the learning of multi-mode controller priors from historical data across differ- ent modes, creating a universal controller architecture within the IRL framework. This facilitates multi-mode adaptability, as multi-task learning integrates prior knowledge from various modes [17]. Therefore, the multi-mode controller can quickly adapt to new, unseen scenarios. This paper introduces a novel framework for developing a multi-mode process controller using multi-task IRL. The proposed method incorporates adversarial reward learning, variational inference, and MaxEnt IRL to train a purely data- driven controller capable that can adapt to various operating modes. To the best of our knowledge, this is the first study to apply multi-task IRL for designing multi-mode process controllers in a data-driven context. This approach offers several advantages: 1) reducing the safety risks associated with DRL’s	https://arxiv.org/abs/2505.21026v1
direct interaction with the environment, 2) fully utilizing the latent controller features embedded in historical industrial data, and 3) addressing the complexities of multi- mode control design through multi-task learning. The remainder of this paper is structured as follows: Section II reviews the preliminaries, including RL, IRL, and AIRL. Section III outlines the motivation and problem statement. In Section IV, the proposed methodology is presented. It details the multi-task IRL approach for addressing the multi-mode process control problem. Section V covers the experimental setup and result analysis. Finally, Section VI concludes by summarizing the key contributions of this work. II. P RELIMINARIES A. Markov decision process (MDP) and RL The MDP in RL is defined by a tuple (X,U, pX, η, r, γ ), where XandUrepresent the state and action spaces respectively, pX:X×U×X → [0,1]denotes the state transition probability of the next state xt+1∈ X given the current state xt∈ X and action ut∈ U,η:X → P (X)is the initial state distribution, r:X × U → Rspecifies the reward function rt∆=r(xt, ut), andγ∈(0,1]is the discount factor. Let πdenote the control policy that selects the optimal action ut∼π(ut\|xt)based on the current state xt, and the state-action marginal distribution induced by the policy is represented as ρπ(xt, ut). The objective of RL is to learn a control policy that max- imizes long-term (discounted) rewards through interactions with the environment Ein a trial-and-error manner. Formally, the standard RL problem is defined as follows: π∗= arg max πTX t=1E(xt,ut)∼ρπ[r(xt, ut)] (1)B. Maximum entropy RL The maximum entropy (MaxEnt) RL objective is defined as: π∗= arg max πTX t=1E(xt,ut)∼ρπ[r(xt, ut) +H(π(ut\|xt))](2) where H(π) =Eπ[−logπ(u\|x)]is the entropy-regularization term for the control policy. Unlike traditional RL, which focuses solely on maximizing the expected sum of rewards, MaxEnt RL aims to develop a policy that maximizes the likelihood of exploring favorable conditions to achieve the primary objective outlined in Eq. (1). This approach allows the learned policy to account for sub-optimal or stochastic behaviors that conventional RL may overlook, thereby pro- viding valuable insights for inverse RL and enhancing transfer learning through exploration. In addition to Eq. (2), there is another explanation and derivation of MaxEnt RL. Define the trajectory τ∆= {x1:T, u1:T}as a sequence of state-action pairs generated by a specific control policy π(ut\|xt). The trajectory distribution under this policy πcan be expressed as follows: pπ(τ) =η(x1)TY t=1p(xt+1\|xt, ut)π(ut\|xt) (3) In the context of probabilistic graphical model, MaxEnt RL reframes the RL control problem as an inference problem that can be addressed using the probability theory. Assuming the existence of a ground-truth reward function r(xt, ut), the objective of MaxEnt RL is to learn a policy from the following optimal trajectory distribution: p(τ)∆=1 Zη(x1)TQ t=1p(xt+1\|xt, ut) exp( r(xt, ut)) ∝ η(x1)TQ t=1p(xt+1\|xt, ut) expTP t=1r(xt, ut) (4) where the partition function Z =R η(x1)Q tp(xt+1\|xt, ut) exp( r(xt, ut))dτ acts as a normalizer enforcing p(τ)∈[0,1]andR p(τ)dτ= 1 . Now let us consider the KL divergence between pπ(τ)and p(τ)as follows: −DKL(pπ(τ)\|\|p(τ)) =Eτ∼pπ(τ)[logp(τ)−logpπ(τ)] =E(xt,ut)∼pπ(xt,ut)[logη(x1) +TX t=1(logp(xt+1\|xt, ut) +r(xt, ut))−logZ −logη(x1)−TX t=1(logp(xt+1\|xt, ut)−logπ(ut\|xt))# =Eπ" TX t=1r(xt, ut)−logπ(ut\|xt)! −logZ# ∝TX t=1E(xt,ut)∼pπ(xt,ut)[r(xt, ut)−logπ(ut\|xt)] =TX	https://arxiv.org/abs/2505.21026v1
t=1E(xt,ut)∼ρπ[r(xt, ut) +H(π(ut\|xt))].(5) LIN et al. : MULTI-MODE PROCESS CONTROL USING MULTI-TASK INVERSE REINFORCEMENT LEARNING 3 Therefore, the objective of MaxEnt RL is to maximize the entropy-regularized long-term (discounted) rewards as shown in Eq. (2). From another perspective, this is equivalent to minimizing the KL divergence between 1) the target (optimal) trajectory distribution p(τ)induced by r(xt, ut), and 2) the trajectory distribution pπ(τ)generated by the MaxEnt RL policy πthat is to be learned. The relationship presented in Eq. (5) can be formally expressed as follows: arg min πDKL(pπ(τ)\|\|p(τ)) = arg max πTX t=1E(xt,ut)∼ρπ[r(xt, ut) +H(π(ut\|xt))].(6) C. Maximum entropy inverse RL Inverse RL aims to infer the intent (reward function) of an expert by observing its behaviors, specifically through optimal expert demonstrations (trajectories). MaxEnt IRL is a classical IRL method within the above MaxEnt RL framework [18], which simultaneously learns the reward function and the policy based on the trajectories pπE(τ)generated by a particular expert policy πE. Formally, the objective is to infer a reward function rθ(xt, ut)parametrized by θ, using the optimal trajectory distribution pθ(τ), similar to Eq. (4): pθ(τ)∆=p(τ\|θ) =1 Zθη(x1)TQ t=1p(xt+1\|xt, ut) exp( rθ(xt, ut)) ∝ η(x1)TQ t=1p(xt+1\|xt, ut) expTP t=1rθ(xt, ut) (7) where the partition function is defined as: Zθ∆=Z η(x1)Y tp(xt+1\|xt, ut) exp( rθ(xt, ut))dτ (8) Then the objective of MaxEnt IRL is to address the maximum likelihood estimation (MLE) problem as follows: arg min θDKL(pπE(τ)\|\|pθ(τ)) = arg max θEpπE(τ)[logpθ(τ)] =Eτ∼πE"TX t=1rθ(xt, ut)# −logZθ (9) D. Adversarial inverse RL (AIRL) However, the term Zθin the MLE problem in Eq. (9) can become computationally intractable when the state-action spaces are large or even continuous. Additionally, MaxEnt IRL requires explicit knowledge of the environment’s dynamics, which is often infeasible. To address these issues, adversarial IRL (AIRL) [13] is proposed to cast optimization of Eq. (9) as a GAN problem. This approach also extends conventional methods that learn from entire trajectories into just learning over single state-action pairs. The discriminator Dθin the AIRL algorithm is chosen as a particular form: Dθ(x, u) =exp{rθ(x, u)} exp{rθ(x, u)}+πω(u\|x)(10) where rθ(x, u)is the learned reward function and πω(u\|x) is the corresponding policy induced by the reward rθ. In Eq.(10), πω(u\|x)is precomputed as a filled-in value for Dθ. The discriminator aims to distinguish between the samples from the expert demonstrations and those generated by the current policy πω. In the AIRL algorithm, the policy πωis trained to maximize Eρπω[logDθ(x, u)−log(1−Dθ(x, u))], which is equivalent to maximizing the objective of an MaxEnt RL policy as follows: Eπω"TX t=1log(Dθ(xt, ut))−log(1−Dθ(xt, ut))# =Eπω"TX t=1rθ(xt, ut)−logπω(ut\|xt)# (11) Therefore, the objective of the discriminator in the GAN- inspired IRL framework aligns precisely with learning the reward function. Simultaneously, the policy (generator) is adjusted to make it increasingly difficult for the discriminator to distinguish between expert demonstrations and samples generated by the policy. It can be demonstrated that, when trained to optimality, the learned reward function rθ(x, u)can recover the ground-truth reward up to a constant, provided that the true reward is solely a function of state [13]. III. P ROBLEM STATEMENT A. Multi-mode process control problem The general state-space representation	https://arxiv.org/abs/2505.21026v1
of a control system can be described as follows: xt+1∼p(xt+1\|xt, ut) (12) ut∼p(ut\|xt, ω)∆=πω(ut\|xt) (13) where p(ut\|xt, ω)is the conditional distribution of actions explicitly denoted as a ω-parameterized policy πω(ut\|xt)to emphasize the role of the control policy. Utilizing the system dynamics model and controller de- scribed above, the evolution trajectory of the MDP unfolds from the initial state as follows: p(τ) =p(x1, ut, . . . , x T, uT\|ω) =η(x1)TY t=1p(xt+1\|xt, ut)πω(ut\|xt).(14) However, in process control scenarios, many controlled processes naturally exhibit multi-mode behaviors. These pro- cesses often operate under a variety of modes or working conditions, which may include different optimization and control setpoints, varying feed compositions (recipes), distinct system parameters, and even different equipment scales. Such variability leads to different dynamic models for each operat- ing mode. In this paper, the term “multi-mode processes” is used broadly to describe processes characterized by multiple distinct models from their corresponding operating scenarios. Formally, each mode within a multi-mode control system possesses unique characteristics, reflecting inherent differences in their dynamics p(xt+1\|xt, ut). Nevertheless, these operating modes also share certain common features, which adhere to an underlying probability distribution p(·\|·). Unlike the gen- eral control systems described in Eqs. (12)-(13), the optimal 4 IEEE TRANSACTIONS ON CYBERNETICS, APRIL 2025 Fig. 1. Multi-task inverse reinforcement learning framework for designing multi-mode process control systems. controllers for each mode in multi-mode processes are not necessarily consistent. This inconsistency necessitates the de- velopment of rational and suitable mathematical formulations to accurately describe the control problem for multi-mode processes. Given a process system with M∈N+operating modes, each associated with a different optimal or near-optimal con- troller πE∆={π1 E, π2 E,···, πM E}, where the optimal controller for mode mis denoted as πm E, the multi-mode controller πE will generate Mdifferent but structurally similar trajectory distributions under the dynamics specific to each mode. Fig. 1 presents the overall concept of the proposed framework. In multi-mode control systems, multiple expert controllers correspond to different operating modes, leading to distinct distributions of expert trajectories. While the trajectory of each mode captures its unique dynamic behavior, the overall closed-loop process control system exhibits inherent simi- larities across modes due to shared system architecture and underlying trajectory distributions. Therefore, the multi-mode process control problem is to learn a universal controller πω(ut\|xt)that can adapt to various control objectives across different operating modes. This paper aims to leverage historical industrial closed-loop big data and multi-task IRL to learn controller priors from mode- specific trajectory distributions, enabling efficient few-shot adaptation during implementation. In other words, πω∆= {π1 ω, π2 ω,···, πM ω}should be capable of providing optimal or near-optimal control policies for all pm(xt+1\|xt, ut), i.e., πω.=πEshould be approximately satisfied for each mode. B. Context-conditional multi-task controller learning problem With the vast amount of historical closed-loop operational data available in industrial production, leveraging this multi-mode data for offline controller design presents both significant challenges and exciting opportunities. Developing an initial controller from such data could greatly enhance the practi- cality and safety of DRL applications. This paper proposes a framework for designing a multi-mode process controller	https://arxiv.org/abs/2505.21026v1
based on IRL. The objective is to develop a fully closed-loop- data-driven controller that can effectively adapt to different operating modes. As analyzed in Section III-A, assuming the controlled system operates under Mdistinct modes, corresponding to M distinct optimal or near-optimal controllers, the operation of this multi-mode process control system will generate a largevolume of historical closed-loop data. Over time, this data will encapsulate Mdifferent trajectory distributions, which are implicitly and structurally embedded within the database. The next step is to formally define the problem: how can a multi-task IRL approach be employed to effectively solve the multi-mode process control challenge? This work introduces the concept of contextual policy to re- frame the multi-task RL (IRL) for multi-mode process control as the problem of solving a context-conditional MDP using a latent context model. In a multi-mode process control system with structurally similar dynamic models and controllers, the MDP is augmented by a latent context variable, capturing dependencies across operating modes. This approach enables the development of a universally structured controller that can adapt to all modes while accounting for the unique characteristics of each individual mode. Letπ(ut\|xt, z)represent the controller for each operating mode. Unlike conventional RL and IRL, where the optimal control policy is defined solely based on the current state xt, this framework incorporates a mode-specific representation. The latent context variable zis introduced as an additional explicit dependency in the conditional policy π(·\|·, z)for each mode. By incorporating probabilistic latent variable model- ing, the generative model for the expert trajectory τz E∆= {x1:T, u1:T}zunder the z-dependent operating mode can be expressed as: x1∼η(x1), z∼p(z), ut∼π(ut\|xt, z), xt+1∼p(xt+1\|xt, ut) (15) The joint distribution of the latent variable zand the trajectory τz Ecan be expressed as: pπE(z, τ) =p(z)pπE(τ\|z) (16) The marginal distribution of the overall historical dataset, which consists of data from different modes, can be repre- sented as: pπE(τ) =Z ZpπE(z, τ)dz=Z Zp(z)pπE(τ\|z)dz (17) where p(z)represents the prior distribution of the latent context variable (i.e., the mode-indicating variable), η(x1)and p(xt+1\|xt, ut)respectively denote the probability distributions for the initial state and state transition, and pπE(τ\|z)represents the context-conditional trajectory distribution. The objective is to learn the control policies πω∆= {π1 ω, π2 ω,···, πM ω}for each mode, along with the correspond- ing optimality or rationality descriptions (which, in the context of DRL, represent the reward functions) based on the overall marginal distribution pπE(τ). However, the probabilistic latent variable formulation above abstracts a complex multi-mode process control problem. In real-world industrial processes, the necessary assumptions may not hold. Specifically, there is often limited or no knowledge of the prior distribution p(z) representing the mode-indicating variable zin Eq. (15) and Eq. (17), and the conditional trajectory distribution pπE(τ\|z)for different modes may be unknown. In other words, in practice, only the marginal distribution pπE(τ)of the entire set of expert trajectories is observable, which makes learning from multi- mode demonstrations particularly challenging. LIN et al. : MULTI-MODE PROCESS CONTROL USING MULTI-TASK INVERSE REINFORCEMENT LEARNING 5 IV. M ETHODOLOGY A. From single-task to multi-task MDP To apply a multi-task IRL approach for learning controllers from historical closed-loop	https://arxiv.org/abs/2505.21026v1
operating data of multi-mode pro- cesses, the conventional single-mode MDP definition must be extended. Specifically, the original MDP is modified and augmented by introducing a conditional term based on z∈ Z, where Zrepresents the value space of the latent context variable z. Consequently, each MDP component—except for the state transition probability determined by system dynam- ics—will now include an additional dependency on z. This enables the MDP to effectively capture the varying character- istics of different operating modes within the process. Based on the latent context model, the context-conditional policy is now defined as π:X × Z → P (U), and the corresponding reward function is modified to r:X ×U×Z → R. This generalized MDP formulation enables the controller design for each mode to be conditioned on the latent context z. In other words, with different predetermined values of z, the multi-task IRL agent can be trained in a mode-specific manner for various control tasks that share common structures or feature spaces, allowing for efficient adaptation across multiple operating modes. Building upon the MaxEnt RL framework in Eq. (2), in multi-mode scenarios, the optimal context-conditional policy can be calculated as: πz E←π∗= arg max πEz∼p(z),(x1:T,u1:T)∼pπ(·\|z) "TX t=1r(xt, ut, z)−logπ(ut\|xt, z)#(18) where r(xt, ut, z)is the mode-specific reward function, and−logπ(ut\|xt, z)is the entropy-regularization term for the contextual policies. Analogous to Eq. (3), the context- conditional distribution for the z-th expert trajectory pπE(τ\|z) in Eq. (17) can be formulated as follows: τz E∼pπE(τ\|z) =pπE(x1:T, u1:T\|z) =η(x1)TY t=1p(xt+1\|xt, ut)πE(ut\|xt, z)(19) In summary, the multi-mode process control problem in this paper is reframed as a context-conditioned multi-task IRL training approach that effectively learns from historical multi- mode closed-loop big data. This approach utilizes a set of multi-task demonstrations τEi.i.d. sampled from the marginal distribution pπE(τ)defined by Eq. (17) and Eq. (19), i.e., τE∼pπE(τ) =R Zp(z)pπE(τ\|z)dz =R Zp(z)η(x1)TQ t=1p(xt+1\|xt, ut)πE(ut\|xt, z)dz(20) The context-conditioned multi-task IRL approach aims to uncover historical multi-mode patterns and subsequently learn a multi-mode controller prior in a purely data-driven man- ner. This methodology enables the identification of distinct operating modes and their corresponding control strategies, supporting the development of a robust controller capable of adapting to varying conditions using only historical data.B. Latent context inference model for multi-task learning Since both the reward function and the control policy are conditioned on zin the generalized MDP, estimating this latent variable is essential for multi-task IRL-based controller learning. To achieve this, a probabilistic inference model q(z\|τ)should be introduced to approximate the true posterior distribution p(z\|τ=τz E), which is generally inaccessible. By denoting the inference model as a variational approximation for calculating z, the context-conditional reward function rIRL(x, u, z )(which needs to be learned) can be determined using the inferred z. Specifically, when given a set of demonstrations sampled from the mode prior and the conditional distribution z∼ p(z), τz E∼pπE(τ\|z), the inference model can be employed to estimate the latent context variable ˆz∼q(z\|τz E). Once the mode-indicating variable ˆzis inferred, it can be substituted into the learned reward rIRL(x, u, ˆz). The DRL agent, guided by this context-conditional reward, should then generate policies that	https://arxiv.org/abs/2505.21026v1
closely resemble those driven by the true underlying reward r(x, u, z ). Successfully training the multi-task IRL agent with latent dependencies equips the IRL-based process controller to effectively manage scenarios characterized by multi-mode behaviors. C. Multi-task IRL using context-conditional probabilistic in- ference The next step involves analyzing how to address the multi-task IRL problem to facilitate the learning of a multi-mode process controller. Drawing from the MLE approach for MaxEnt IRL outlined in Section II (Eq. (9)), and considering the previously defined multi-task IRL problem based on latent context model, the context-conditional trajectory distribution, parameterized by the reward parameter θ, can be derived as follows: τz θ∼pθ(τ\|z) =pθ(x1:T, u1:T\|z) =1 Zθ[η(x1)p(xt+1\|xt, ut)] expTP t=1rθ(xt, ut, z)(21) where the conditional input zis inferred using an inference model qψ(z\|τ)parametrized by ψ,rθis the reward function in the multi-task IRL, while Zθis the partition function. To de- velop a multi-mode process controller, this framework ensures that the context-conditional trajectory distribution accurately captures the dependencies imposed by the latent context. Therefore, the primary goal of multi-task IRL is to solve an MLE problem: arg min θEp(z)[DKL(pπE(τ\|z)\|\|pθ(τ\|z))] = arg max θEp(z),pπE(τ\|z)[logpθ(τ\|z)] = arg max θEz∼p(z),τ∼πz E"TX t=1rθ(xt, ut, z)# −logZθ(22) To achieve this, various IRL algorithms, such as AIRL, can be employed to minimize the KL divergence between the optimal or near-optimal expert trajectory distribution and the θ-induced trajectory distribution. The underlying objective of IRL agent is to match the trajectory distribution of τ={x1:T, u1:T}. 6 IEEE TRANSACTIONS ON CYBERNETICS, APRIL 2025 However, the challenge arises because every trainable term in Eq. (22) is conditioned on the latent context z, which is not directly related to the IRL optimization process. This limitation complicates the learning process, as the IRL agent struggles to effectively optimize distinct modes without a well-defined association between trajectory distributions and latent contexts. In other words, an explicit correlation between theθ-induced trajectory τand the latent variable zmust be enforced to distinguish the conditional reward function and the corresponding policy within each mode. Building on the principles of InfoGAN [16], mutual infor- mation (MI) between zandτcan be applied as a constraint or correlation measure to enhance the dependency between the latent context and the resulting trajectory. Within the framework of multi-task IRL, a higher MI value indicates a stronger correlation, thereby improving the interpretability of z in relation to the trajectory τz. The MI under joint distribution pθ(z, τ)is calculated as follows: Ipθ(z;τ) =H(z)− H(z\|τ) =H(z) +Z zZ τpθ(z, τ) logpθ(z, τ)dzdτ =H(z) +Z zZ τp(z)pθ(τ\|z) logpθ(z\|τ)dzdτ =H(z) +Ez∼p(z),τ∼pθ(τ\|z)[logpθ(z\|τ)] =Ez∼p(z),τ∼pθ(τ\|z)[logpθ(z\|τ)−logp(z)](23) Since pθ(z\|τ)is the posterior distribution that is unknown, the probabilistic inference model qψ(z\|τ)can serve as a variational approximation of this posterior, and then Eq. (23) can be reformulated as: Ipθ(z;τ) =H(z) +Ez∼p(z),τ∼pθ(τ\|z)[logpθ(z\|τ)] =H(z) +Ez∼p(z),τ∼pθ(τ\|z)[DKL(pθ\|\|qψ)\|{z} ≥0+ log qψ(z\|τ)] ≥ H(z) +Ez∼p(z),τ∼pθ(τ\|z)[logqψ(z\|τ)] =Ez∼p(z),τ∼pθ(τ\|z)[logqψ(z\|τ)−logp(z)] =LI(pθ, qψ) (24) where LI(pθ, qψ)is the variational lower bound of the MI. The following analysis will outline the solution of the multi-task IRL problem. In this context, Eq. (22) represents the primary objective based on the MaxEnt principle, while Eq. (24) introduces an additional regularization term for the latent context variable. Accordingly, the	https://arxiv.org/abs/2505.21026v1
overall optimization objective can be formulated as: min θ,ψEp(z)[DKL(pπE(τ\|z)\|\|pθ(τ\|z))] −α·Ipθ(z;τ) +β·Epθ(τ)[DKL(pθ(z\|τ)\|\|qψ(z\|τ))](25) The first term aims to align the conditional distributions between the closed-loop expert trajectories and the IRL trajec- tories generated by the θ-parameterized reward function and the corresponding RL policy. This alignment represents the primary objective of the context-conditional multi-task IRL problem within the MaxEnt reward learning framework. The second term seeks to maximize the MI between the context and the corresponding trajectory, ensuring that the informationembedded in the latent context zis retained throughout the training process [16]. Finally, the third term addresses the alignment of the variational inference approximation qψ(z\|τ) with the true posterior pθ(z\|τ)of the latent context, which is required to train the inference model qψ. For simplicity and without loss of generality, the tunable hyperparameters can be treated as constants α=β= 1. Consequently, Eq. (25) can be expressed as: min θ,ψEp(z)[DKL(pπE(τ\|z)\|\|pθ(τ\|z))] +Ez∼p(z),τ∼pθ(τ\|z) logp(z) pθ(z\|τ)+ logpθ(z\|τ) qψ(z\|τ) ≡max θ,ψ−Ep(z)[DKL(pπE(τ\|z)\|\|pθ(τ\|z))] +Ez∼p(z),τ∼pθ(τ\|z)[logqψ(z\|τ)−logp(z)]\| {z } LI(pθ,qψ) = max θ,ψ−Ep(z) DKL(pπE(τ\|z)\|\|pθ(τ\|z))−logp(z)\|{z} regardless of θ,ψ  +Ez∼p(z),τ∼pθ(τ\|z)logqψ(z\|τ) = max θ,ψ−Ep(z)[DKL(pπE(τ\|z)\|\|pθ(τ\|z))] +Ez∼p(z),τ∼pθ(τ\|z)logqψ(z\|τ) = max θ,ψ−Ep(z)[DKL(pπE(τ\|z)\|\|pθ(τ\|z))] +Linfo(θ, ψ). (26) D. Practical implementation for solving multi-task IRL At this stage, the optimization objective in Eq. (26) remains intractable, as it is not feasible to approximate the prior distri- bution p(z)and the conditional trajectory distribution pθ(τ\|z) by directly sampling from the marginal distribution pπE(τ) (i.e., expert trajectory distribution), which encompasses multi- mode process characteristics. Fortunately, with the latent context inference model qψ(z\|τ), an estimate over the sampled expert trajectory can be used to approximate the prior distribution p(z)as follows: τE∼pπE(τ), z∼p(z).=qψ(z\|τE) (27) And the conditional trajectory distribution pθ(τ\|z)can be sampled from trajectories generated during the training process of the forward DRL agent within the inner loop of the multi- task IRL algorithm. This is feasible because, if the forward DRL policy πωis trained to optimality, the resulting trajectory distribution pπ∗ω(τ\|z)induced by the optimal policy π∗ ωwill match the conditional trajectory distribution pθ(τ\|z)[19]. With the approximately sampled p(z)andpθ(τ\|z), the second term in Eq. (26) can be optimized with respect to θ andψ. For the first KL divergence minimization term, any adversarial IRL algorithms such as AIRL can be applied. The only adjustment needed is to augment the RL state with an additional input dependency—the latent context z. This modification allows the IRL policy to be conditioned to πω(ut\|xt, z), where ⟨x, z⟩serves as the augmented MDP state in the practical implementation of the algorithm. LIN et al. : MULTI-MODE PROCESS CONTROL USING MULTI-TASK INVERSE REINFORCEMENT LEARNING 7 Fig. 2. Flowchart of the proposed multi-task inverse reinforcement learning scheme. Based upon the above analysis, the overall objective of the context-conditional multi-task IRL algorithm can be expressed as follows: min ωmax θ,ψEpπE(τ),qψ(z\|τE),ρπω(x,u\|z)log(1−Dθ(x, u, z )) +EτE∼pπE(τ),z∼qψ(z\|τE)log(Dθ(x, u, m )) +Linfo(θ, ψ) (28) where Dθ(x, u, z ) =exp{rθ(x, u, z )} exp{rθ(x, u, z )}+πω(u\|x, z)(29) Therefore, the multi-task learning procedure for addressing the multi-mode process control problem can be summarized in Fig. 2 and Algorithm 1. This approach involves three neural networks: the Generator (policy), the Discriminator (reward) and the Inference Network (mode indicator). Unlike conven- tional IRL, the proposed	https://arxiv.org/abs/2505.21026v1
approach incorporates an additional input dependency, i.e., the latent context z, resulting in an augmented MDP state ⟨x, z⟩that is used to train both the multi-mode policy and the reward. The process begins with the Inference Network estimating the mode-specific latent context of sampled trajectories. This inferred context is then fed into both the policy and the reward function. The GAN-inspired multi-task IRL framework utilizes a Generator to create virtual data corresponding to the current multi-mode policy, while the Discriminator defines the reward function by distinguishing expert data from the generated virtual data. During training, the outermost loop follows a GAN structure, where the Gener- ator, acting as a DRL agent, updates the weights of the context- conditional policy based on the latest reward. The internal loop consists of any standard DRL algorithm, coupled with a parallel mode inference module, ensuring efficient learning across multiple modes. From the process control perspective, once the training on multi-mode historical closed-loop data is completed, the multi-task IRL-based controller can serve as an initialized multi-mode controller, enabling adaptability across those unseen operating modes in transfer learning settings. V. R ESULTS &DISCUSSION In this section, the proposed multi-mode controller learning method is applied to two distinct cases. The first case involves a fed-batch bioreactor, where the modes are characterized by unique system dynamics. The second case examines a continuous reactor, with modes defined by varying temperature setpoints.Algorithm 1 Multi-task IRL training procedure Input: Expert trajectories DE={τj E}; Initial parameters of fθ, πω, qψ. repeat Sample two batches of unlabeled demonstrations: τE, τ′ E∼ DE Infer a batch of latent context variables from the sampled demonstra- tions: z∼qψ(z\|τE) Sample trajectories Dfrom πω(τ\|z), with the latent context variable fixed during each rollout and included in D. Update ψto increase Linfo(θ, ψ)with gradients in Eq. (26), with samples from D. Update θto increase Linfo(θ, ψ)with gradients in Eq. (26), with samples fromD. Update θto decrease the binary classification loss: E(x,u,z )∼D[∇θlogDθ(x, u, z )] + Eτ′ E∼DE,z∼qψ(z\|τ′ E)[∇θlog(1−Dθ(x, u, z ))] Update ωwith “forward” RL to increase the following objective: E(x,u,z )∼D[logDθ(x, u, z )] until Convergence Output: Learned inference model qψ(z\|τ), reward function fθ(x, u, z ) and policy πω(u\|x, z). A. Case 1: A fed-batch bioreactor (batch profile optimization) To validate the feasibility of the multi-task IRL solution in recovering the reward function and addressing imitation learning control policies, this case study uses the same photo- production system as the numerical example presented in [20]. The process involves a fed-batch bioreactor that necessitates solving a batch-to-batch optimization problem. Two distinct operating modes are defined, each corresponding to a unique set of internal system parameters. To implement this, the dynamic model is modified to incorporate a mode variable: dy1 dt=− u1+ 0.5u2 1 y1+u2 dy2 dt=u1y1−k·u2y1(30) where u1, u2are the manipulated variables (i.e., light and an inflow rate) and y1, y2are the outlet concentrations of the reactant and product, respectively. The mode variable kis: k=0.5,Mode 1 0.7,Mode 2(31) The batch operation time course is normalized to 1, with control actions constrained within the interval [0, 5]. The objective is	https://arxiv.org/abs/2505.21026v1
to design a control policy that adjusts the system inputs u1, u2to maximize the product concentration y2 at the end of the batch operation. Positive reward feedback is only provided at the end of the batch operation, with rewards set to penalize excessive action changes at all other intervals. The reward function is defined as follows: rt=−0.01× ∥u(t+ 1)−u(t)∥1, t = 0,1,···, T−1 rT=y2(T). (32) The aim is to train an IRL agent to autonomously discover the optimal control policy from expert trajectories, ensuring that the recovered reward function aligns with the true reward function as described. It is important to note that the experimental setup is highly challenging for both RL and IRL due to the sparsity of rewards, since positive feedback is granted only at the end of the episode. In this context, if the multi-task IRL agent fails to accurately infer the reward structures across different modes, it risks generating ineffective control policies. This may, in turn, 8 IEEE TRANSACTIONS ON CYBERNETICS, APRIL 2025 Fig. 3. Typical batch optimization profiles of the TRPO expert demonstrations (left: Mode 1 k= 0.5; right: Mode 2 k= 0.7). Fig. 4. Batch optimization profiles of the successfully trained multi-task IRL agent based on the TRPO expert demonstrations (left: Mode 1 k= 0.5; right: Mode 2 k= 0.7). lead to inaccurate reward estimates, perpetuating a feedback loop that hinders learning. Therefore, this setup serves as an ideal testbed for rigorously assessing the effectiveness of the multi-task IRL approach. First, a DRL-based expert policy is developed using the Trust Region Policy Optimization (TRPO) algorithm to maxi- mize the reward function outlined in Eq. (32). In the RL setup, the state is represented as X∆= [y1, y2]T, and the action as U∆= [u1, u2]T. During training, each episode randomly selects an operating mode as the environment. The trained TRPO agent then generates expert demonstrations across both modes. Specifically, 2,112 expert trajectories are collected, with each comprising 20 samples. These trajectories are shuffled to sim- ulate the mixed data sources typical in industrial production involving multiple devices. The typical expert trajectories for the two modes, k= 0.5andk= 0.7, are illustrated in Fig. 3, while the trajectories generated by the successfully trained multi-task IRL agent are displayed in Fig. 4. The results indicate that the multi-task IRL agent effectively recovers both the control policy and reward function from historical multi-mode data. At the endpoint of the batch trajectories, the product concentrations for each mode dif- fer, reflecting the agent’s ability to learn the batch-to-batch optimization patterns unique to each operating mode. It is worth noting that, because the multi-task IRL method learns the reward function from historical data, it assumes expert demonstrations as optimal. Consequently, the recovered reward values are slightly lower than those of the expert, which is expected. This approach, centered on purely offline training of the multi-mode controller, is designed to fully leverage the prior knowledge embedded in expert behaviors. Such learned controller prior(s) can provide a strong foundation for subsequent transfer learning applications. FC TCVC ρC TCV CA T AO FC TCiCoolantFeedF CAi Ti	https://arxiv.org/abs/2505.21026v1
F CA TProductTT 21b TC 21TC 21 TY 21 I PmTsetFig. 5. Sketch of the CSTR control system. B. Case 2: A benchmark CSTR process (continuous control) 1) System description and problem formulation: To demon- strate the effectiveness of the proposed method in continuous control scenarios, a continuous stirred tank reactor (CSTR) process is selected as the test case, depicted in Fig. 5. This reactor operates as a jacketed, non-adiabatic tank that facili- tates a single irreversible and exothermic first-order reaction. The primary control objective is to maintain the reaction temperature Tnear the target setpoint Tsetby adjusting the valve opening m, which modulates the coolant feed flow rate. Further details on the first-principles model and system parameters are provided in [3]. Unlike the batch-to-batch optimization case, continuous processes require dynamic control solutions. Designing control strategies for continuous systems is considerably more com- plex than for batch profile optimization. In this scenario, the IRL agent faces increased difficulty due to the slow dynamic characteristics inherent to continuous process control. This necessitates a larger number of samples to effectively learn the reward function and establish a robust control policy, while also increasing instability in the training process. In the experiment, two distinct modes are introduced to represent the multi-mode control scenario: Mode 1: Setpoint 88→90°C, and Mode 2: Setpoint 88→86°C. Expert trajec- tories from these modes are combined and randomly shuffled to mimic the multi-mode nature of industrial big data. The IRL agent is tasked with learning the characteristics of the expert controller from a multi-mode dataset with varied distributions, aiming to recover a reward function that can explain expert behavior(s). Two types of expert sources are used to validate the effectiveness of the multi-task IRL approach. The first type consists of expert demonstrations directly generated by a DRL agent. The second type uses multi-mode industrial closed-loop data as the basis for expert trajectories. In this case, the RL state is represented as X∆= [CA, T, T C, b, Tset−T]T, and the action as U∆= [m]. For the multi-mode CSTR control problem, the temperature setpoint serves as the mode-indicating variable, which is unknown to the agent. Consequently, the inference model is critical for distinguishing between different operating modes within the encapsulated mode-specific information. LIN et al. : MULTI-MODE PROCESS CONTROL USING MULTI-TASK INVERSE REINFORCEMENT LEARNING 9 Fig. 6. Typical control performances of the TRPO expert demonstrations (left: Mode 1 Tset= 90 ; right: Mode 2 Tset= 86 ). Fig. 7. Control performances of the successfully trained multi-task IRL agent based on the TRPO expert demonstrations (left: Mode 1 Tset= 90 ; right: Mode 2 Tset= 86 ). 2) DRL agent as optimal policy for generating expert demonstrations: As in Case 1, the TRPO algorithm and the reward function outlined in [3], [7] are employed to train the DRL-based expert. A total of 2,112 expert trajectories are collected across both modes. The typical TRPO expert trajectories for each mode are displayed in Fig. 6, while the trajectories generated by the trained multi-task IRL agent are shown in Fig. 7. These results are obtained by deploying	https://arxiv.org/abs/2505.21026v1
the trained IRL controller directly into the environment for validation. While a small residual error remains after the controller stabilizes, the overall performance meets acceptable standards. Furthermore, in industrial applications, the pre- trained controller can be fine-tuned in real-world settings to facilitate Sim2Real transfer learning [4], [21], potentially enhancing control performance further. 3) Historical closed-loop operation data as expert trajec- tories: To demonstrate the practical engineering potential and feasibility of using multi-task IRL for multi-mode process control, unknown expert demonstrations derived from histori- cal closed-loop operating data will now serve as the training source. Compared to using RL agents as expert(s), adopting a classical control scheme (such as PID) as the expert controller offers a more compelling test. This is because the behavior patterns of RL agents and IRL agents are relatively similar, making the trajectory distribution generated by RL well-suited for IRL learning. In contrast, traditional PI control operates as a feedback control law based on error and cumulative error, presenting a notable challenge for IRL in accurately capturing its control characteristics. In this experiment, a well-tuned PI controller for the CSTR system, is employed to generate closed-loop operating data for training the IRL agent. To introduce variability and stochas- ticity in the expert trajectories, white noise is applied to the Fig. 8. Typical control performances of the PI expert demonstrations (left: Mode 1 Tset= 90 ; right: Mode 2 Tset= 86 ). Fig. 9. Control performances of the successfully trained multi-task IRL agent based on the PI expert demonstrations (left: Mode 1 Tset= 90 ; right: Mode 2Tset= 86 ). inlet concentration. A total of 2,112 expert trajectories are recorded, each containing 300 samples (sampling interval Ts = 10 seconds, corresponding to a total duration of 3,000 seconds for the system). The typical PI controller trajectories for the two modes are depicted in Fig. 8, indicating that the PI controller maintains satisfactory control performance. The closed-loop expert demonstrations described above are used for multi-task IRL training. The control performance of the trained multi-task IRL-based controller across the two modes is presented in Fig. 9. The results indicate that the multi-task IRL agent effectively imitates expert behaviors and naturally adapts to varying modes. As in the previous case, if additional refinement is necessary, Sim2Real transfer learning can be employed to minimize model-plant mismatches. VI. C ONCLUSION This paper presents a novel multi-task IRL approach aimed at addressing the multi-mode process control problem, with the primary objective of extracting controller patterns and RL value information from closed-loop big data encompassing multiple operating modes. To accomplish this, latent variables, commonly used in multi-mode modeling, are integrated to identify different mode indicators. Specifically, by introducing a latent context variable, the proposed method first establishes a mathematical framework to represent the conditional policy and trajectory distribution. Subsequently, techniques such as MaxEnt IRL, mutual information regularization, and varia- tional inference are employed to optimize context-conditional rewards and policies. Experimental results demonstrate that the proposed method effectively learns a universal controller that can adapt to various scenarios based on multi-mode historical closed-loop data. This promising approach offers	https://arxiv.org/abs/2505.21026v1
a probabilis- tic inference-based solution for data-driven controller design 10 IEEE TRANSACTIONS ON CYBERNETICS, APRIL 2025 and underscores the potential of context-conditional latent variable modeling techniques in the development of multi- mode process controllers. REFERENCES [1] R. Nian, J. Liu, and B. Huang, “A review on reinforcement learning: Introduction and applications in industrial process control,” Computers & Chemical Engineering , vol. 139, p. 106886, 2020. [2] J. Shin, T. A. Badgwell, K.-H. Liu, and J. H. Lee, “Reinforcement learning – overview of recent progress and implications for process control,” Computers & Chemical Engineering , vol. 127, pp. 282–294, 2019. [3] R. Lin, J. Chen, L. Xie, and H. Su, “Accelerating reinforcement learning with case-based model-assisted experience augmentation for process control,” Neural Networks , vol. 158, pp. 197–215, 2023. [4] R. Lin, Y . Luo, X. Wu, J. Chen, B. Huang, H. Su, and L. Xie, “Surrogate empowered Sim2Real transfer of deep reinforcement learning for ORC superheat control,” Applied Energy , vol. 356, p. 122310, 2024. [5] L. Zhang, R. Lin, L. Xie, W. Dai, and H. Su, “Event-triggered con- strained optimal control for organic rankine cycle systems via safe reinforcement learning,” IEEE Transactions on Neural Networks and Learning Systems , vol. 35, no. 5, pp. 7126–7137, 2024. [6] Y . Shi, R. Lin, X. Wu, Z. Zhang, P. Sun, L. Xie, and H. Su, “Dual-mode fast DMC algorithm for the control of ORC based waste heat recovery system,” Energy , vol. 244, p. 122664, 2022. [7] R. Lin, J. Chen, L. Xie, and H. Su, “Accelerating reinforcement learning with local data enhancement for process control,” in 2021 China Automation Congress (CAC) , Conference Proceedings, pp. 5690–5695. [8] H. Chang, Q. Chen, R. Lin, Y . Shi, L. Xie, and H. Su, “Controlling pressure of gas pipeline network based on mixed proximal policy optimization,” in 2022 China Automation Congress (CAC) , Conference Proceedings, pp. 4642–4647. [9] R. Lin, J. Chen, B. Huang, L. Xie, and H. Su, Developing Purely Data- Driven Multi-Mode Process Controllers Using Inverse Reinforcement Learning . Elsevier, 2024, vol. 53, pp. 2731–2736. [10] M. Mowbray, R. Smith, E. A. Del Rio-Chanona, and D. Zhang, “Using process data to generate an optimal control policy via apprenticeship and reinforcement learning,” AIChE Journal , vol. 67, no. 9, p. e17306, 2021. [11] S. Adams, T. Cody, and P. A. Beling, “A survey of inverse reinforcement learning,” Artificial Intelligence Review , 2022. [12] J. Ho and S. Ermon, “Generative adversarial imitation learning,” in Proceedings of the 30th International Conference on Neural Information Processing Systems , ser. NIPS’16. Red Hook, NY , USA: Curran Associates Inc., 2016, p. 4572–4580. [13] J. Fu, K. Luo, and S. Levine, “Learning Robust Rewards with Adversarial Inverse Reinforcement Learning,” arXiv e-prints , p. arXiv:1710.11248, Oct. 2017. [14] S. Levine, “Reinforcement Learning and Control as Probabilistic Infer- ence: Tutorial and Review,” arXiv e-prints , p. arXiv:1805.00909, May 2018. [15] L. Yao, B. Shen, L. Cui, J. Zheng, and Z. Ge, “Semi-supervised deep dynamic probabilistic latent variable model for multimode process soft sensor application,” IEEE Transactions on Industrial Informatics , vol.	https://arxiv.org/abs/2505.21026v1
19, no. 4, pp. 6056–6068, 2023. [16] X. Chen, Y . Duan, R. Houthooft, J. Schulman, I. Sutskever, and P. Abbeel, “InfoGAN: Interpretable representation learning by infor- mation maximizing generative adversarial nets,” in Advances in Neural Information Processing Systems , vol. 29, 2016, Conference Proceedings. [17] Y . Zhang and Q. Yang, “A survey on multi-task learning,” IEEE Transactions on Knowledge and Data Engineering , vol. 34, no. 12, pp. 5586–5609, 2022. [18] B. D. Ziebart, A. Maas, J. A. Bagnell, and A. K. Dey, “Maximum entropy inverse reinforcement learning,” in Proceedings of the 23rd national conference on Artificial intelligence (AAAI) , vol. 3. AAAI Press, Conference Proceedings, p. 1433–1438. [19] L. Yu, T. Yu, C. Finn, and S. Ermon, “Meta-inverse reinforcement learning with probabilistic context variables,” in Advances in Neural Information Processing Systems , vol. 32. Curran Associates, Inc., 2019. [20] P. Petsagkourakis, I. O. Sandoval, E. Bradford, D. Zhang, and E. A. del Rio-Chanona, “Reinforcement learning for batch bioprocess optimiza- tion,” Computers & Chemical Engineering , vol. 133, p. 106649, 2020. [21] R. Lin, J. Chen, L. Xie, H. Su, and B. Huang, “Facilitating Re- inforcement Learning for Process Control Using Transfer Learning: Perspectives,” arXiv e-prints , p. arXiv:2404.00247, Mar. 2024. Runze Lin received the B.S. degree in automation from the College of Control Science and Engineer- ing, Zhejiang University, Hangzhou, China, in 2020, where he is currently pursuing the Ph.D. degree in control science and engineering with the State Key Laboratory of Industrial Control Technology, China. From 2022 to 2023, he was a Visiting Scholar with the University of Alberta, Edmonton, AB, Canada. His research interests include reinforcement learn- ing, transfer learning, process control, data analytics, industrial big data and its applications. Junghui Chen received the B.S. degree from the Department of Chemical Engineering, Chung Yuan Christian University, Taoyuan, Taiwan, in 1982, the M.S. degree from the Department of Chemical Engi- neering, National Taiwan University, Taipei, Taiwan, in 1984, and the Ph.D. degree from the Department of Chemical Engineering, The University of Ten- nessee at Knoxville, Knoxville, TN, USA, in 1995. He is currently a Full Professor with Chung Yuan Christian University. His research interests are pro- cess system engineering, including process design for operability, nonlinear control, process monitoring and diagnosis, control loop performance assessment, batch control, model predictive control, data mining and analytics, and iterative learning design. Biao Huang (Fellow, IEEE) received the B.S. and M.S. degrees in automatic control from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 1983 and 1986, respectively, and the Ph.D. degree in process control from the University of Alberta, Edmonton, AB, Canada, in 1997. He joined the University of Alberta in 1997 as an Assistant Professor with the Department of Chemi- cal and Materials Engineering, where he is currently a Full Professor. He was an NSERC Industrial Research Chair in control of oil sands processes and the AITF Industry Chair in process control from 2013 to 2018. His research interests include data analytics, process control, system identification, control performance assessment, Bayesian methods, and state estimation. He has applied his expertise	https://arxiv.org/abs/2505.21026v1
extensively in industrial practice. Lei Xie received the B.S. and Ph.D. degrees from Zhejiang University, China, in 2000 and 2005, re- spectively. From 2005 to 2006, he was a Postdoctoral Re- searcher with the Berlin University of Technology and an Assistant Professor from 2005 to 2008. He is currently a Professor with the Department of Control Science and Engineering, Zhejiang University. His research interests focus on the interdisciplinary area of statistics and system control theory. Hongye Su (Senior Member, IEEE) received the B.S. degree in industrial automation from the Nan- jing University of Chemical Technology, Jiangsu, China, in 1990, and the M.S. and Ph.D. degrees in industrial automation from Zhejiang University, Hangzhou, China, in 1993 and 1995, respectively. From 1995 to 1997, he was a Lecturer with the Department of Chemical Engineering, Zhejiang University. From 1998 to 2000, he was an Associate Professor with the Institute of Advanced Process Control, Zhejiang University, where he is currently a Professor with the Institute of Cyber-Systems and Control. His current research interests include robust control, time-delay systems, and advanced process control theory and applications.	https://arxiv.org/abs/2505.21026v1
arXiv:2505.21027v1 [cs.LG] 27 May 2025TabAttackBench : A Benchmark for Adversarial Attacks on Tabular Data Zhipeng Hea,b,∗, Chun Ouyanga,b, Lijie Wenc, Cong Liud, Catarina Moreirae,b,f aSchool of Information Systems, Queensland University of Technology, Brisbane, Australia bCenter for Data Science, Queensland University of Technology, Brisbane, Australia cSchool of Software, Tsinghua University, Beijing, China dNOVA Information Management School, NOVA University of Lisbon, Lisboa, Portugal eData Science Institute, University of Technology, Sydney, Australia fINESC-ID/Instituto Superior T´ ecnico, University of Lisboa, Lisboa, Portugal Abstract Adversarial attacks pose a significant threat to machine learning models by inducing in- correct predictions through imperceptible perturbations to input data. While these attacks have been extensively studied in unstructured data like images, their application to tab- ular data presents new challenges. These challenges arise from the inherent heterogeneity and complex feature interdependencies in tabular data, which differ significantly from those in image data. To address these differences, it is crucial to consider imperceptibility as a key criterion specific to tabular data. Most current research focuses primarily on achieving effective adversarial attacks, often overlooking the importance of maintaining imperceptibil- ity. To address this gap, we propose a new benchmark for adversarial attacks on tabular data that evaluates both effectiveness and imperceptibility. In this study, we assess the effectiveness and imperceptibility of five adversarial attacks across four models using eleven tabular datasets, including both mixed and numerical-only datasets. Our analysis explores how these factors interact and influence the overall performance of the attacks. We also compare the results across different dataset types to understand the broader implications of these findings. The findings from this benchmark provide valuable insights for improving the design of adversarial attack algorithms, thereby advancing the field of adversarial machine learning on tabular data. Keywords: Adversarial attack, Tabular data, Benchmark, Machine learning, Robustness Preprint submitted to arXiv May 28, 2025 1. Introduction In recent years, the field of machine learning has seen substantial advancements, lead- ing to the deployment of models across a wide range of applications. However, with these advancements comes increasing concern about the robustness and security of models, par- ticularly in the context of adversarial attacks. Adversarial attacks involve the intentional manipulation of input data to deceive machine learning models, causing incorrect or mis- leading outputs [1]. This area of research has drawn significant attention as researchers strive to understand and mitigate the vulnerabilities in various types of data and mod- els. For instance, adversarial attacks on image data can cause misclassification of objects, which is concerning for applications like autonomous driving, surveillance, and facial recog- nition systems [2]. Similarly, Natural Language Processing (NLP) models are susceptible to attacks that can alter the meaning of sentences or generate misleading summaries, impact- ing applications in sentiment analysis, machine translation, and chatbots [3]. Additionally, speech recognition systems can be tricked by adversarial audio inputs, leading to incorrect transcriptions or commands, which has serious implications for virtual assistants and voice- controlled devices [4]. By addressing the vulnerabilities in these types of data, researchers aim to develop more robust and secure machine learning systems across various domains. 1.1. Challenges in Adversarial Attacks	https://arxiv.org/abs/2505.21027v1
on Tabular Data Tabular data, structured yet rich in semantics, heterogeneity, and interdependencies, is prevalent in domains such as finance, healthcare, and e-commerce. These datasets of- ten contain vital information used for decision-making processes, predictive modelling, and anomaly detection. Despite their significance, machine learning models trained on tabular data (which can be referred to as tabular data models) remain underexplored regarding the vulnerabilities to adversarial attacks. ∗Corresponding author Email addresses: [email protected] (Zhipeng He), [email protected] (Chun Ouyang), [email protected] (Lijie Wen), [email protected] (Cong Liu), [email protected] (Catarina Moreira) 2 The potential impact of adversarial attacks on tabular data models is profound. Such attacks can compromise the integrity and reliability of machine learning models, resulting in misclassification and potentially severe consequences for applications relying on precise pre- diction and data-driven decisions. The vulnerabilities of tabular data models to adversarial attacks are particularly notable due to the unique characteristics of tabular data. Unlike image or text data, where each data point is typically represented as pixels or words, tab- ular data presents a different challenge due to its varied nature. For example, consider a dataset containing customer information for a bank loan application. It includes categori- cal variables like marital status andemployment type , numerical variables such as income , and possibly missing values in fields like previous loans . Additionally, these features often exhibit diverse distributions; for instance, income might follow a skewed distribution, while employment type is categorical. These complexities make applying adversarial attacks to tabular data more intricate compared to image or text data. 1.2. Benchmarking Adversarial Attacks on Tabular Data An important aspect of advancing adversarial attack research is the establishment of benchmarks. These benchmarks function as standardised tests that evaluate the robustness of machine learning models against adversarial attacks. They provide a common ground for comparing different approaches and methodologies, thereby facilitating the development of more robust models. While considerable progress has been made in understanding ad- versarial attacks on image [5] and text [6], there remains a relatively underexplored area: adversarial attacks on tabular data. Our work addresses this gap by introducing a new benchmark specifically designed for attacks on tabular data. Existing benchmarks, as summarised in Table 1, primarily focus on evaluating attacks on image, graph, and time-series data, covering a range of adversarial techniques such as black-box attacks [9], patch-based attacks [10], and transferability-based attacks [7]. These benchmarks typically evaluate adversarial robustness using metrics like attack success rate, adversarial accuracy, and norm-based metrics (e.g., ℓ∞,ℓ2) to quantify the strength of the adversarial perturbations. While these metrics are well-suited for image data, where 3 Table 1: Overview of existing benchmarks on adversarial attacks across different data types, attack types, and evaluation metrics; our work introduces a new benchmark for attacks on tabular data. Benchmark Data Type Attack Type Evaluation Metric Jin et al. [7] Image Transferable Attacks Attack Transferability Score Dong et al. [8] Image White-box, Black-box Attacks Adversarial Accuracy, ℓ∞Norm Zheng et al. [9] Image Black-box Attacks Attack Success Rate, Query Count Croce et al. [5] Image ℓ∞,ℓ2Norm-based Attacks ℓ∞,ℓ2Norm, Corruption Robustness Hingun et al. [10] Image Patch-based Attacks	https://arxiv.org/abs/2505.21027v1

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