Daily Papers

Large language models (LLMs) show strong performance across natural language processing (NLP), mathematical reasoning, and programming, and recent large reasoning models (LRMs) further emphasize explicit reasoning. Yet their computational limits, particularly spatial complexity constrained by finite context windows, remain poorly understood. While recent works often focus on problems within the NP complexity class, we push the boundary by introducing a novel benchmark grounded in two PSPACE-complete regular expression (regex) problems: equivalence decision (RegexEQ) and minimization (RegexMin). PSPACE-complete problems serve as a more rigorous standard for assessing computational capacity, as their solutions require massive search space exploration. We perform a double-exponential space exploration to construct a labeled dataset of over a million regex instances with a sound filtering process to build the benchmark. We conduct extensive evaluations on 6 LLMs and 5 LRMs of varying scales, revealing common failure patterns such as verbosity and repetition. With its well-defined structure and quantitative evaluation metrics, this work presents the first empirical investigation into the spatial computational limitations of LLMs and LRMs, offering a new framework for evaluating their advanced reasoning capabilities. Our code is available at https://github.com/hyundong98/RegexPSPACE .

The process of revising (or constructing) a policy at execution time -- known as decision-time planning -- has been key to achieving superhuman performance in perfect-information games like chess and Go. A recent line of work has extended decision-time planning to imperfect-information games, leading to superhuman performance in poker. However, these methods involve solving subgames whose sizes grow quickly in the amount of non-public information, making them unhelpful when the amount of non-public information is large. Motivated by this issue, we introduce an alternative framework for decision-time planning that is not based on solving subgames, but rather on update equivalence. In this update-equivalence framework, decision-time planning algorithms replicate the updates of last-iterate algorithms, which need not rely on public information. This facilitates scalability to games with large amounts of non-public information. Using this framework, we derive a provably sound search algorithm for fully cooperative games based on mirror descent and a search algorithm for adversarial games based on magnetic mirror descent. We validate the performance of these algorithms in cooperative and adversarial domains, notably in Hanabi, the standard benchmark for search in fully cooperative imperfect-information games. Here, our mirror descent approach exceeds or matches the performance of public information-based search while using two orders of magnitude less search time. This is the first instance of a non-public-information-based algorithm outperforming public-information-based approaches in a domain they have historically dominated.

In this manuscript, we show that any neural network with any activation function can be represented as a decision tree. The representation is equivalence and not an approximation, thus keeping the accuracy of the neural network exactly as is. We believe that this work provides better understanding of neural networks and paves the way to tackle their black-box nature. We share equivalent trees of some neural networks and show that besides providing interpretability, tree representation can also achieve some computational advantages for small networks. The analysis holds both for fully connected and convolutional networks, which may or may not also include skip connections and/or normalizations.

Implementing Boolean functions with circuits consisting of logic gates is fundamental in digital computer design. However, the implemented circuit must be exactly equivalent, which hinders generative neural approaches on this task due to their occasionally wrong predictions. In this study, we introduce a generative neural model, the "Circuit Transformer", which eliminates such wrong predictions and produces logic circuits strictly equivalent to given Boolean functions. The main idea is a carefully designed decoding mechanism that builds a circuit step-by-step by generating tokens, which has beneficial "cutoff properties" that block a candidate token once it invalidate equivalence. In such a way, the proposed model works similar to typical LLMs while logical equivalence is strictly preserved. A Markov decision process formulation is also proposed for optimizing certain objectives of circuits. Experimentally, we trained an 88-million-parameter Circuit Transformer to generate equivalent yet more compact forms of input circuits, outperforming existing neural approaches on both synthetic and real world benchmarks, without any violation of equivalence constraints.

Robust Markov Decision Processes (MDPs) and risk-sensitive MDPs are both powerful tools for making decisions in the presence of uncertainties. Previous efforts have aimed to establish their connections, revealing equivalences in specific formulations. This paper introduces a new formulation for risk-sensitive MDPs, which assesses risk in a slightly different manner compared to the classical Markov risk measure (Ruszczy\'nski 2010), and establishes its equivalence with a class of regularized robust MDP (RMDP) problems, including the standard RMDP as a special case. Leveraging this equivalence, we further derive the policy gradient theorem for both problems, proving gradient domination and global convergence of the exact policy gradient method under the tabular setting with direct parameterization. This forms a sharp contrast to the Markov risk measure, known to be potentially non-gradient-dominant (Huang et al. 2021). We also propose a sample-based offline learning algorithm, namely the robust fitted-Z iteration (RFZI), for a specific regularized RMDP problem with a KL-divergence regularization term (or equivalently the risk-sensitive MDP with an entropy risk measure). We showcase its streamlined design and less stringent assumptions due to the equivalence and analyze its sample complexity.

A fundamental problem in combinatorial optimization is identifying equivalent formulations, which can lead to more efficient solution strategies and deeper insights into a problem's computational complexity. The need to automatically identify equivalence between problem formulations has grown as optimization copilots--systems that generate problem formulations from natural language descriptions--have proliferated. However, existing approaches to checking formulation equivalence lack grounding, relying on simple heuristics which are insufficient for rigorous validation. Inspired by Karp reductions, in this work we introduce quasi-Karp equivalence, a formal criterion for determining when two optimization formulations are equivalent based on the existence of a mapping between their decision variables. We propose EquivaMap, a framework that leverages large language models to automatically discover such mappings, enabling scalable and reliable equivalence verification. To evaluate our approach, we construct the first open-source dataset of equivalent optimization formulations, generated by applying transformations such as adding slack variables or valid inequalities to existing formulations. Empirically, EquivaMap significantly outperforms existing methods, achieving substantial improvements in correctly identifying formulation equivalence.

Ensuring alignment of language models' outputs with human preferences is critical to guarantee a useful, safe, and pleasant user experience. Thus, human alignment has been extensively studied recently and several methods such as Reinforcement Learning from Human Feedback (RLHF), Direct Policy Optimisation (DPO) and Sequence Likelihood Calibration (SLiC) have emerged. In this paper, our contribution is two-fold. First, we show the equivalence between two recent alignment methods, namely Identity Policy Optimisation (IPO) and Nash Mirror Descent (Nash-MD). Second, we introduce a generalisation of IPO, named IPO-MD, that leverages the regularised sampling approach proposed by Nash-MD. This equivalence may seem surprising at first sight, since IPO is an offline method whereas Nash-MD is an online method using a preference model. However, this equivalence can be proven when we consider the online version of IPO, that is when both generations are sampled by the online policy and annotated by a trained preference model. Optimising the IPO loss with such a stream of data becomes then equivalent to finding the Nash equilibrium of the preference model through self-play. Building on this equivalence, we introduce the IPO-MD algorithm that generates data with a mixture policy (between the online and reference policy) similarly as the general Nash-MD algorithm. We compare online-IPO and IPO-MD to different online versions of existing losses on preference data such as DPO and SLiC on a summarisation task.

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust learning, it's surprising that we still lack a unified theory; traditional proofs of the equivalence tend to be disparate, and rely on strong model-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining the equivalence of realizable and agnostic learnability: a three-line blackbox reduction that simplifies, unifies, and extends our understanding across a wide variety of settings. This includes models with no known characterization of learnability such as learning with arbitrary distributional assumptions and more general loss functions, as well as a host of other popular settings such as robust learning, partial learning, fair learning, and the statistical query model. More generally, we argue that the equivalence of realizable and agnostic learning is actually a special case of a broader phenomenon we call property generalization: any desirable property of a learning algorithm (e.g. noise tolerance, privacy, stability) that can be satisfied over finite hypothesis classes extends (possibly in some variation) to any learnable hypothesis class.

We present a complete formalization in Isabelle/HOL of the object part of an equivalence between L-mosaics and bounded join-semilattices, employing an AI-assisted methodology that integrates large language models as reasoning assistants throughout the proof development process. The equivalence was originally established by Cangiotti, Linzi, and Talotti in their study of hypercompositional structures related to orthomodular lattices and quantum logic. Our formalization rigorously verifies the main theoretical result and demonstrates the mutual inverse property of the transformations establishing this equivalence. The development showcases both the mathematical depth of multivalued algebraic operations and the potential for AI-enhanced interactive theorem proving in tackling complex formalization projects.

We consider strategy proof mechanisms for facility location which maximize equitability between agents. As is common in the literature, we measure equitability with the Gini index. We first prove a simple but fundamental impossibility result that no strategy proof mechanism can bound the approximation ratio of the optimal Gini index of utilities for one or more facilities. We propose instead computing approximation ratios of the complemented Gini index of utilities, and consider how well both deterministic and randomized mechanisms approximate this. In addition, as Nash welfare is often put forwards as an equitable compromise between egalitarian and utilitarian outcomes, we consider how well mechanisms approximate the Nash welfare.

We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.

Large language models (LLMs) achieve remarkable performance in code generation tasks. However, a significant performance disparity persists between popular programming languages (e.g., Python, C++) and others. To address this capability gap, we leverage the code translation task to train LLMs, thereby facilitating the transfer of coding proficiency across diverse programming languages. Moreover, we introduce OORL for training, a novel reinforcement learning (RL) framework that integrates on-policy and off-policy strategies. Within OORL, on-policy RL is applied during code translation, guided by a rule-based reward signal derived from unit tests. Complementing this coarse-grained rule-based reward, we propose Group Equivalent Preference Optimization (GEPO), a novel preference optimization method. Specifically, GEPO trains the LLM using intermediate representations (IRs) groups. LLMs can be guided to discern IRs equivalent to the source code from inequivalent ones, while also utilizing signals about the mutual equivalence between IRs within the group. This process allows LLMs to capture nuanced aspects of code functionality. By employing OORL for training with code translation tasks, LLMs improve their recognition of code functionality and their understanding of the relationships between code implemented in different languages. Extensive experiments demonstrate that our OORL for LLMs training with code translation tasks achieves significant performance improvements on code benchmarks across multiple programming languages.

Equipping agents with the capacity to justify made decisions using supporting evidence represents a cornerstone of accountable decision-making. Furthermore, ensuring that justifications are in line with human expectations and societal norms is vital, especially in high-stakes situations such as healthcare. In this work, we propose the use of a debate-based reward model for reinforcement learning agents, where the outcome of a zero-sum debate game quantifies the justifiability of a decision in a particular state. This reward model is then used to train a justifiable policy, whose decisions can be more easily corroborated with supporting evidence. In the debate game, two argumentative agents take turns providing supporting evidence for two competing decisions. Given the proposed evidence, a proxy of a human judge evaluates which decision is better justified. We demonstrate the potential of our approach in learning policies for prescribing and justifying treatment decisions of septic patients. We show that augmenting the reward with the feedback signal generated by the debate-based reward model yields policies highly favored by the judge when compared to the policy obtained solely from the environment rewards, while hardly sacrificing any performance. Moreover, in terms of the overall performance and justifiability of trained policies, the debate-based feedback is comparable to the feedback obtained from an ideal judge proxy that evaluates decisions using the full information encoded in the state. This suggests that the debate game outputs key information contained in states that is most relevant for evaluating decisions, which in turn substantiates the practicality of combining our approach with human-in-the-loop evaluations. Lastly, we showcase that agents trained via multi-agent debate learn to propose evidence that is resilient to refutations and closely aligns with human preferences.

As a seemingly self-explanatory task, problem-solving has been a significant component of science and engineering. However, a general yet concrete formulation of problem-solving itself is missing. With the recent development of AI-based problem-solving agents, the demand for process-level verifiability is rapidly increasing yet underexplored. To fill these gaps, we present a principled formulation of problem-solving as a deterministic Markov decision process; a novel framework, FPS (Formal Problem-Solving), which utilizes existing FTP (formal theorem proving) environments to perform process-verified problem-solving; and D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment. The expressiveness, soundness and completeness of the frameworks are proven. We construct three benchmarks on problem-solving: FormalMath500, a formalization of a subset of the MATH500 benchmark; MiniF2F-Solving and PutnamBench-Solving, adaptations of FTP benchmarks MiniF2F and PutnamBench. For faithful, interpretable, and human-aligned evaluation, we propose RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification. We evaluate four prevalent FTP models and two prompting methods as baselines, solving at most 23.77% of FormalMath500, 27.47% of MiniF2F-Solving, and 0.31% of PutnamBench-Solving.

The current literature on AI-advised decision making -- involving explainable AI systems advising human decision makers -- presents a series of inconclusive and confounding results. To synthesize these findings, we propose a simple theory that elucidates the frequent failure of AI explanations to engender appropriate reliance and complementary decision making performance. We argue explanations are only useful to the extent that they allow a human decision maker to verify the correctness of an AI's prediction, in contrast to other desiderata, e.g., interpretability or spelling out the AI's reasoning process. Prior studies find in many decision making contexts AI explanations do not facilitate such verification. Moreover, most tasks fundamentally do not allow easy verification, regardless of explanation method, limiting the potential benefit of any type of explanation. We also compare the objective of complementary performance with that of appropriate reliance, decomposing the latter into the notions of outcome-graded and strategy-graded reliance.

When robots learn reward functions using high capacity models that take raw state directly as input, they need to both learn a representation for what matters in the task -- the task ``features" -- as well as how to combine these features into a single objective. If they try to do both at once from input designed to teach the full reward function, it is easy to end up with a representation that contains spurious correlations in the data, which fails to generalize to new settings. Instead, our ultimate goal is to enable robots to identify and isolate the causal features that people actually care about and use when they represent states and behavior. Our idea is that we can tune into this representation by asking users what behaviors they consider similar: behaviors will be similar if the features that matter are similar, even if low-level behavior is different; conversely, behaviors will be different if even one of the features that matter differs. This, in turn, is what enables the robot to disambiguate between what needs to go into the representation versus what is spurious, as well as what aspects of behavior can be compressed together versus not. The notion of learning representations based on similarity has a nice parallel in contrastive learning, a self-supervised representation learning technique that maps visually similar data points to similar embeddings, where similarity is defined by a designer through data augmentation heuristics. By contrast, in order to learn the representations that people use, so we can learn their preferences and objectives, we use their definition of similarity. In simulation as well as in a user study, we show that learning through such similarity queries leads to representations that, while far from perfect, are indeed more generalizable than self-supervised and task-input alternatives.

As AI systems progress, we rely more on them to make decisions with us and for us. To ensure that such decisions are aligned with human values, it is imperative for us to understand not only what decisions they make but also how they come to those decisions. Reasoning language models, which provide both final responses and (partially transparent) intermediate thinking traces, present a timely opportunity to study AI procedural reasoning. Unlike math and code problems which often have objectively correct answers, moral dilemmas are an excellent testbed for process-focused evaluation because they allow for multiple defensible conclusions. To do so, we present MoReBench: 1,000 moral scenarios, each paired with a set of rubric criteria that experts consider essential to include (or avoid) when reasoning about the scenarios. MoReBench contains over 23 thousand criteria including identifying moral considerations, weighing trade-offs, and giving actionable recommendations to cover cases on AI advising humans moral decisions as well as making moral decisions autonomously. Separately, we curate MoReBench-Theory: 150 examples to test whether AI can reason under five major frameworks in normative ethics. Our results show that scaling laws and existing benchmarks on math, code, and scientific reasoning tasks fail to predict models' abilities to perform moral reasoning. Models also show partiality towards specific moral frameworks (e.g., Benthamite Act Utilitarianism and Kantian Deontology), which might be side effects of popular training paradigms. Together, these benchmarks advance process-focused reasoning evaluation towards safer and more transparent AI.

We propose a novel approach for learning interchangeable tokens in language models to obtain an extendable vocabulary that can generalize to new tokens. Our method is designed to address alpha-equivalence, the principle that renaming bound variables in a syntactic expression preserves semantics. This property arises in many formal languages such as temporal logics, in which all proposition symbols represent the same concept but are distinguishable from each other. To handle such tokens, we develop a dual-part embedding approach. The first part is shared across all interchangeable tokens, thereby enforcing that they represent the same core concept. The second part is randomly generated for each token, which enables distinguishability. We evaluate our method in a Transformer encoder-decoder model on two tasks: solving linear temporal logic formulae and copying with extendable vocabulary. Our method demonstrates promising generalization capabilities in addition to introducing a favorable inductive bias for alpha-equivalence.

With the release of the o1 model by OpenAI, reasoning models adopting slow thinking strategies have gradually emerged. As the responses generated by such models often include complex reasoning, intermediate steps, and self-reflection, existing evaluation methods are often inadequate. They struggle to determine whether the LLM output is truly equivalent to the reference answer, and also have difficulty identifying and extracting the final answer from long, complex responses. To address this issue, we propose xVerify, an efficient answer verifier for reasoning model evaluations. xVerify demonstrates strong capability in equivalence judgment, enabling it to effectively determine whether the answers produced by reasoning models are equivalent to reference answers across various types of objective questions. To train and evaluate xVerify, we construct the VAR dataset by collecting question-answer pairs generated by multiple LLMs across various datasets, leveraging multiple reasoning models and challenging evaluation sets designed specifically for reasoning model assessment. A multi-round annotation process is employed to ensure label accuracy. Based on the VAR dataset, we train multiple xVerify models of different scales. In evaluation experiments conducted on both the test set and generalization set, all xVerify models achieve overall F1 scores and accuracy exceeding 95\%. Notably, the smallest variant, xVerify-0.5B-I, outperforms all evaluation methods except GPT-4o, while xVerify-3B-Ib surpasses GPT-4o in overall performance. These results validate the effectiveness and generalizability of xVerify.

We consider a fair division model in which agents have positive, zero and negative utilities for items. For this model, we analyse one existing fairness property - EFX - and three new and related properties - EFX_0, EFX^3 and EF1^3 - in combination with Pareto-optimality. With general utilities, we give a modified version of an existing algorithm for computing an EF1^3 allocation. With -alpha/0/alpha utilities, this algorithm returns an EFX^3 and PO allocation. With absolute identical utilities, we give a new algorithm for an EFX and PO allocation. With -alpha/0/beta utilities, this algorithm also returns such an allocation. We report some new impossibility results as well.

Where do learning signals come from when there is no ground truth in post-training? We propose turning exploration into supervision through Compute as Teacher (CaT), which converts the model's own exploration at inference-time into reference-free supervision by synthesizing a single reference from a group of parallel rollouts and then optimizing toward it. Concretely, the current policy produces a group of rollouts; a frozen anchor (the initial policy) reconciles omissions and contradictions to estimate a reference, turning extra inference-time compute into a teacher signal. We turn this into rewards in two regimes: (i) verifiable tasks use programmatic equivalence on final answers; (ii) non-verifiable tasks use self-proposed rubrics-binary, auditable criteria scored by an independent LLM judge, with reward given by the fraction satisfied. Unlike selection methods (best-of-N, majority, perplexity, or judge scores), synthesis may disagree with the majority and be correct even when all rollouts are wrong; performance scales with the number of rollouts. As a test-time procedure, CaT improves Gemma 3 4B, Qwen 3 4B, and Llama 3.1 8B (up to +27% on MATH-500; +12% on HealthBench). With reinforcement learning (CaT-RL), we obtain further gains (up to +33% and +30%), with the trained policy surpassing the initial teacher signal.

The optimized certainty equivalent (OCE) is a family of risk measures that cover important examples such as entropic risk, conditional value-at-risk and mean-variance models. In this paper, we propose a new episodic risk-sensitive reinforcement learning formulation based on tabular Markov decision processes with recursive OCEs. We design an efficient learning algorithm for this problem based on value iteration and upper confidence bound. We derive an upper bound on the regret of the proposed algorithm, and also establish a minimax lower bound. Our bounds show that the regret rate achieved by our proposed algorithm has optimal dependence on the number of episodes and the number of actions.

Aligning generative models with human preference via RLHF typically suffers from overoptimization, where an imperfectly learned reward model can misguide the generative model to output undesired responses. We investigate this problem in a principled manner by identifying the source of the misalignment as a form of distributional shift and uncertainty in learning human preferences. To mitigate overoptimization, we first propose a theoretical algorithm that chooses the best policy for an adversarially chosen reward model; one that simultaneously minimizes the maximum likelihood estimation of the loss and a reward penalty term. Here, the reward penalty term is introduced to prevent the policy from choosing actions with spurious high proxy rewards, resulting in provable sample efficiency of the algorithm under a partial coverage style condition. Moving from theory to practice, the proposed algorithm further enjoys an equivalent but surprisingly easy-to-implement reformulation. Using the equivalence between reward models and the corresponding optimal policy, the algorithm features a simple objective that combines: (i) a preference optimization loss that directly aligns the policy with human preference, and (ii) a supervised learning loss that explicitly imitates the policy with a (suitable) baseline distribution. In the context of aligning large language models (LLM), this objective fuses the direct preference optimization (DPO) loss with the supervised fune-tuning (SFT) loss to help mitigate the overoptimization towards undesired responses, for which we name the algorithm Regularized Preference Optimization (RPO). Experiments of aligning LLMs demonstrate the improved performance of RPO compared with DPO baselines. Our work sheds light on the interplay between preference optimization and SFT in tuning LLMs with both theoretical guarantees and empirical evidence.

As large language models (LLMs) are increasingly applied to scientific reasoning, the complexity of answer formats and the diversity of equivalent expressions make answer verification a critical yet challenging task. Existing verification studies in scientific domains suffer from two major limitations: (a) the absence of systematic evaluation standards and insufficient disciplinary coverage, which hinders their comprehensive assessment; and (b) heavy reliance on cumbersome rule design or prompt engineering, which reduces their effectiveness in complex reasoning scenarios or limits their cross-disciplinary generalization. To address these challenges, we propose solutions at both the data and model levels. On the data side, we construct SCI-VerifyBench, a cross-disciplinary benchmark covering mathematics, physics, biology, chemistry, and general scientific QA. The benchmark is built from real LLM responses and enhanced with domain-specific equivalence transformations that generate challenging and realistic data. Model-based and expert annotations ensure both quality and diversity, enabling rigorous evaluation of verification ability. On the model side, we emphasize the importance of reasoning for verification and introduce SCI-Verifier, a unified reasoning-augmented verifier for scientific domains. Through post-training, SCI-Verifier demonstrates strong logical reasoning and equivalence judgment capabilities while maintaining concise and stable outputs. Together, SCI-VerifyBench and SCI-Verifier provide a principled framework for scientific verification, offering both systematic evaluation and practical pathways to enhance the reliability and applicability of LLMs in scientific domains.

People's decision-making abilities often fail to improve or may even erode when they rely on AI for decision-support, even when the AI provides informative explanations. We argue this is partly because people intuitively seek contrastive explanations, which clarify the difference between the AI's decision and their own reasoning, while most AI systems offer "unilateral" explanations that justify the AI's decision but do not account for users' thinking. To align human-AI knowledge on decision tasks, we introduce a framework for generating human-centered contrastive explanations that explain the difference between AI's choice and a predicted, likely human choice about the same task. Results from a large-scale experiment (N = 628) demonstrate that contrastive explanations significantly enhance users' independent decision-making skills compared to unilateral explanations, without sacrificing decision accuracy. Amid rising deskilling concerns, our research demonstrates that incorporating human reasoning into AI design can foster human skill development.

Large Language Models (LLMs) were shown to struggle with long-term planning, which may be caused by the limited way in which they explore the space of possible solutions. We propose an architecture where a Reinforcement Learning (RL) Agent guides an LLM's space exploration: (1) the Agent has access to domain-specific information, and can therefore make decisions about the quality of candidate solutions based on specific and relevant metrics, which were not explicitly considered by the LLM's training objective; (2) the LLM can focus on generating immediate next steps, without the need for long-term planning. We allow non-linear reasoning by exploring alternative paths and backtracking. We evaluate this architecture on the program equivalence task, and compare it against Chain of Thought (CoT) and Tree of Thoughts (ToT). We assess both the downstream task, denoting the binary classification, and the intermediate reasoning steps. Our approach compares positively against CoT and ToT.

A common approach for aligning language models to human preferences is to first learn a reward model from preference data, and then use this reward model to update the language model. We study two closely related problems that arise in this approach. First, any monotone transformation of the reward model preserves preference ranking; is there a choice that is ``better'' than others? Second, we often wish to align language models to multiple properties: how should we combine multiple reward models? Using a probabilistic interpretation of the alignment procedure, we identify a natural choice for transformation for (the common case of) rewards learned from Bradley-Terry preference models. This derived transformation has two important properties. First, it emphasizes improving poorly-performing outputs, rather than outputs that already score well. This mitigates both underfitting (where some prompts are not improved) and reward hacking (where the model learns to exploit misspecification of the reward model). Second, it enables principled aggregation of rewards by linking summation to logical conjunction: the sum of transformed rewards corresponds to the probability that the output is ``good'' in all measured properties, in a sense we make precise. Experiments aligning language models to be both helpful and harmless using RLHF show substantial improvements over the baseline (non-transformed) approach.

We present a novel unified bilevel optimization-based framework, PARL, formulated to address the recently highlighted critical issue of policy alignment in reinforcement learning using utility or preference-based feedback. We identify a major gap within current algorithmic designs for solving policy alignment due to a lack of precise characterization of the dependence of the alignment objective on the data generated by policy trajectories. This shortfall contributes to the sub-optimal performance observed in contemporary algorithms. Our framework addressed these concerns by explicitly parameterizing the distribution of the upper alignment objective (reward design) by the lower optimal variable (optimal policy for the designed reward). Interestingly, from an optimization perspective, our formulation leads to a new class of stochastic bilevel problems where the stochasticity at the upper objective depends upon the lower-level variable. To demonstrate the efficacy of our formulation in resolving alignment issues in RL, we devised an algorithm named A-PARL to solve PARL problem, establishing sample complexity bounds of order O(1/T). Our empirical results substantiate that the proposed PARL can address the alignment concerns in RL by showing significant improvements (up to 63\% in terms of required samples) for policy alignment in large-scale environments of the Deepmind control suite and Meta world tasks.

The Machine Learning for Combinatorial Optimization (ML4CO) NeurIPS 2021 competition aims to improve state-of-the-art combinatorial optimization solvers by replacing key heuristic components with machine learning models. On the dual task, we design models to make branching decisions to promote the dual bound increase faster. We propose a knowledge inheritance method to generalize knowledge of different models from the dataset aggregation process, named KIDA. Our improvement overcomes some defects of the baseline graph-neural-networks-based methods. Further, we won the 1st Place on the dual task. We hope this report can provide useful experience for developers and researchers. The code is available at https://github.com/megvii-research/NeurIPS2021-ML4CO-KIDA.

We present a fine-grained theoretical analysis of the performance gap between reinforcement learning from human feedback (RLHF) and direct preference optimization (DPO) under a representation gap. Our study decomposes this gap into two sources: an explicit representation gap under exact optimization and an implicit representation gap under finite samples. In the exact optimization setting, we characterize how the relative capacities of the reward and policy model classes influence the final policy qualities. We show that RLHF, DPO, or online DPO can outperform one another depending on the type of model mis-specifications. Notably, online DPO can outperform both RLHF and standard DPO when the reward and policy model classes are isomorphic and both mis-specified. In the approximate optimization setting, we provide a concrete construction where the ground-truth reward is implicitly sparse and show that RLHF requires significantly fewer samples than DPO to recover an effective reward model -- highlighting a statistical advantage of two-stage learning. Together, these results provide a comprehensive understanding of the performance gap between RLHF and DPO under various settings, and offer practical insights into when each method is preferred.

If machine learning models were to achieve superhuman abilities at various reasoning or decision-making tasks, how would we go about evaluating such models, given that humans would necessarily be poor proxies for ground truth? In this paper, we propose a framework for evaluating superhuman models via consistency checks. Our premise is that while the correctness of superhuman decisions may be impossible to evaluate, we can still surface mistakes if the model's decisions fail to satisfy certain logical, human-interpretable rules. We instantiate our framework on three tasks where correctness of decisions is hard to evaluate due to either superhuman model abilities, or to otherwise missing ground truth: evaluating chess positions, forecasting future events, and making legal judgments. We show that regardless of a model's (possibly superhuman) performance on these tasks, we can discover logical inconsistencies in decision making. For example: a chess engine assigning opposing valuations to semantically identical boards; GPT-4 forecasting that sports records will evolve non-monotonically over time; or an AI judge assigning bail to a defendant only after we add a felony to their criminal record.

Inverse Reinforcement Learning (IRL) is a powerful set of techniques for imitation learning that aims to learn a reward function that rationalizes expert demonstrations. Unfortunately, traditional IRL methods suffer from a computational weakness: they require repeatedly solving a hard reinforcement learning (RL) problem as a subroutine. This is counter-intuitive from the viewpoint of reductions: we have reduced the easier problem of imitation learning to repeatedly solving the harder problem of RL. Another thread of work has proved that access to the side-information of the distribution of states where a strong policy spends time can dramatically reduce the sample and computational complexities of solving an RL problem. In this work, we demonstrate for the first time a more informed imitation learning reduction where we utilize the state distribution of the expert to alleviate the global exploration component of the RL subroutine, providing an exponential speedup in theory. In practice, we find that we are able to significantly speed up the prior art on continuous control tasks.

We study the problem of (learning) algorithm comparison, where the goal is to find differences between models trained with two different learning algorithms. We begin by formalizing this goal as one of finding distinguishing feature transformations, i.e., input transformations that change the predictions of models trained with one learning algorithm but not the other. We then present ModelDiff, a method that leverages the datamodels framework (Ilyas et al., 2022) to compare learning algorithms based on how they use their training data. We demonstrate ModelDiff through three case studies, comparing models trained with/without data augmentation, with/without pre-training, and with different SGD hyperparameters. Our code is available at https://github.com/MadryLab/modeldiff .

The prevalence and importance of algorithmic two-sided marketplaces has drawn attention to the issue of fairness in such settings. Algorithmic decisions are used in assigning students to schools, users to advertisers, and applicants to job interviews. These decisions should heed the preferences of individuals, and simultaneously be fair with respect to their merits (synonymous with fit, future performance, or need). Merits conditioned on observable features are always uncertain, a fact that is exacerbated by the widespread use of machine learning algorithms to infer merit from the observables. As our key contribution, we carefully axiomatize a notion of individual fairness in the two-sided marketplace setting which respects the uncertainty in the merits; indeed, it simultaneously recognizes uncertainty as the primary potential cause of unfairness and an approach to address it. We design a linear programming framework to find fair utility-maximizing distributions over allocations, and we show that the linear program is robust to perturbations in the estimated parameters of the uncertain merit distributions, a key property in combining the approach with machine learning techniques.

The observed similarities in the behavior of humans and Large Language Models (LLMs) have prompted researchers to consider the potential of using LLMs as models of human cognition. However, several significant challenges must be addressed before LLMs can be legitimately regarded as cognitive models. For instance, LLMs are trained on far more data than humans typically encounter, and may have been directly trained on human data in specific cognitive tasks or aligned with human preferences. Consequently, the origins of these behavioral similarities are not well understood. In this paper, we propose a novel way to enhance the utility of LLMs as cognitive models. This approach involves (i) leveraging computationally equivalent tasks that both an LLM and a rational agent need to master for solving a cognitive problem and (ii) examining the specific task distributions required for an LLM to exhibit human-like behaviors. We apply this approach to decision-making -- specifically risky and intertemporal choice -- where the key computationally equivalent task is the arithmetic of expected value calculations. We show that an LLM pretrained on an ecologically valid arithmetic dataset, which we call Arithmetic-GPT, predicts human behavior better than many traditional cognitive models. Pretraining LLMs on ecologically valid arithmetic datasets is sufficient to produce a strong correspondence between these models and human decision-making. Our results also suggest that LLMs used as cognitive models should be carefully investigated via ablation studies of the pretraining data.

As AIGC has impacted our society profoundly in the past years, ethical issues have received tremendous attention. The most urgent one is the AIGC copyright dilemma, which can immensely stifle the development of AIGC and greatly cost the entire society. Given the complexity of AIGC copyright governance and the fact that no perfect solution currently exists, previous work advocated copyleft on AI governance but without substantive analysis. In this paper, we take a step further to explore the feasibility of copyleft to alleviate the AIGC copyright dilemma. We conduct a mixed-methods study from two aspects: qualitatively, we use a formal what-if analysis to clarify the dilemma and provide case studies to show the feasibility of copyleft; quantitatively, we perform a carefully designed survey to find out how the public feels about copylefting AIGC. The key findings include: a) people generally perceive the dilemma, b) they prefer to use authorized AIGC under loose restriction, and c) they are positive to copyleft in AIGC and willing to use it in the future.

Reinforcement learning from human feedback (RLHF) has become essential for improving language model capabilities, but traditional approaches rely on the assumption that human preferences follow a transitive Bradley-Terry model. This assumption fails to capture the non-transitive nature of populational human preferences. Nash learning from human feedback (NLHF), targeting non-transitive preferences, is a problem of computing the Nash equilibrium (NE) of the two-player constant-sum game defined by the human preference. We introduce Extragradient preference optimization (EGPO), a novel algorithm for NLHF achieving last-iterate linear convergence to the NE of KL-regularized games and polynomial convergence to the NE of original games, while being robust to noise. Unlike previous approaches that rely on nested optimization, we derive an equivalent implementation using gradients of an online variant of the identity preference optimization (IPO) loss, enabling more faithful implementation for neural networks. Our empirical evaluations demonstrate EGPO's superior performance over baseline methods when training for the same number of epochs, as measured by pairwise win-rates using the ground truth preference. These results validate both the theoretical strengths and practical advantages of EGPO for language model alignment with non-transitive human preferences.

Reinforcement learning has been widely applied to enhance the reasoning capabilities of large language models. Extending the inference limits of smaller models has become a prominent research focus. However, algorithms such as Group Relative Policy Optimization (GRPO) suffer from a clear drawback: the upper bound of a model's rollout responses is entirely determined by the model itself, preventing the acquisition of knowledge from samples that are either all incorrect or all correct. In this paper, we introduce Group Contrastive Policy Optimization (GCPO), a method that incorporates external standard reference answers. When the model cannot solve a problem, the reference answer supplies the correct response, steering the model toward an unequivocally accurate update direction. This approach offers two main advantages: (1) it improves training efficiency by fully utilizing every sample; (2) it enables the model to emulate the problem solving strategy of the reference answer during training, thereby enhancing generalization in reasoning. GCPO achieves outstanding results across multiple benchmark datasets, yielding substantial improvements over the baseline model. Our code is available at: https://github.com/AchoWu/GCPO.

Seven years ago, researchers proposed a postprocessing method to equalize the error rates of a model across different demographic groups. The work launched hundreds of papers purporting to improve over the postprocessing baseline. We empirically evaluate these claims through thousands of model evaluations on several tabular datasets. We find that the fairness-accuracy Pareto frontier achieved by postprocessing contains all other methods we were feasibly able to evaluate. In doing so, we address two common methodological errors that have confounded previous observations. One relates to the comparison of methods with different unconstrained base models. The other concerns methods achieving different levels of constraint relaxation. At the heart of our study is a simple idea we call unprocessing that roughly corresponds to the inverse of postprocessing. Unprocessing allows for a direct comparison of methods using different underlying models and levels of relaxation.

We reveal and address the frequently overlooked yet important issue of disguised procedural unfairness, namely, the potentially inadvertent alterations on the behavior of neutral (i.e., not problematic) aspects of data generating process, and/or the lack of procedural assurance of the greatest benefit of the least advantaged individuals. Inspired by John Rawls's advocacy for pure procedural justice, we view automated decision-making as a microcosm of social institutions, and consider how the data generating process itself can satisfy the requirements of procedural fairness. We propose a framework that decouples the objectionable data generating components from the neutral ones by utilizing reference points and the associated value instantiation rule. Our findings highlight the necessity of preventing disguised procedural unfairness, drawing attention not only to the objectionable data generating components that we aim to mitigate, but also more importantly, to the neutral components that we intend to keep unaffected.

In this paper, we conduct a study to utilize LLMs as a solution for decision making that requires complex data analysis. We define Decision QA as the task of answering the best decision, d_{best}, for a decision-making question Q, business rules R and a database D. Since there is no benchmark that can examine Decision QA, we propose Decision QA benchmark, DQA. It has two scenarios, Locating and Building, constructed from two video games (Europa Universalis IV and Victoria 3) that have almost the same goal as Decision QA. To address Decision QA effectively, we also propose a new RAG technique called the iterative plan-then-retrieval augmented generation (PlanRAG). Our PlanRAG-based LM generates the plan for decision making as the first step, and the retriever generates the queries for data analysis as the second step. The proposed method outperforms the state-of-the-art iterative RAG method by 15.8% in the Locating scenario and by 7.4% in the Building scenario, respectively. We release our code and benchmark at https://github.com/myeon9h/PlanRAG.

In Machine Learning, a benchmark refers to an ensemble of datasets associated with one or multiple metrics together with a way to aggregate different systems performances. They are instrumental in (i) assessing the progress of new methods along different axes and (ii) selecting the best systems for practical use. This is particularly the case for NLP with the development of large pre-trained models (e.g. GPT, BERT) that are expected to generalize well on a variety of tasks. While the community mainly focused on developing new datasets and metrics, there has been little interest in the aggregation procedure, which is often reduced to a simple average over various performance measures. However, this procedure can be problematic when the metrics are on a different scale, which may lead to spurious conclusions. This paper proposes a new procedure to rank systems based on their performance across different tasks. Motivated by the social choice theory, the final system ordering is obtained through aggregating the rankings induced by each task and is theoretically grounded. We conduct extensive numerical experiments (on over 270k scores) to assess the soundness of our approach both on synthetic and real scores (e.g. GLUE, EXTREM, SEVAL, TAC, FLICKR). In particular, we show that our method yields different conclusions on state-of-the-art systems than the mean-aggregation procedure while being both more reliable and robust.

Proximal Policy Optimization (PPO)-based Reinforcement Learning from Human Feedback (RLHF) is essential for aligning large language models (LLMs) with human preferences. It requires joint training of an actor and critic with a pretrained, fixed reward model for guidance. This approach increases computational complexity and instability due to actor-critic interdependence. Additionally, PPO lacks access to true environment rewards in LLM tasks, limiting its adaptability. Under such conditions, pretraining a value model or a reward model becomes equivalent, as both provide fixed supervisory signals without new ground-truth feedback. To address these issues, we propose Decoupled Value Policy Optimization (DVPO), a lean framework that replaces traditional reward modeling with a pretrained global value model (GVM). The GVM is conditioned on policy trajectories and predicts token-level return-to-go estimates. By decoupling value model from policy training (via frozen GVM-driven RL objectives), DVPO eliminates actor-critic interdependence, reducing GPU memory usage by 40\% and training time by 35\% compared to conventional RLHF. Experiments across benchmarks show DVPO outperforms efficient RLHF methods (e.g., DPO) while matching state-of-the-art PPO in performance.

We consider the regret minimization task in a dueling bandits problem with context information. In every round of the sequential decision problem, the learner makes a context-dependent selection of two choice alternatives (arms) to be compared with each other and receives feedback in the form of noisy preference information. We assume that the feedback process is determined by a linear stochastic transitivity model with contextualized utilities (CoLST), and the learner's task is to include the best arm (with highest latent context-dependent utility) in the duel. We propose a computationally efficient algorithm, CoLSTIM, which makes its choice based on imitating the feedback process using perturbed context-dependent utility estimates of the underlying CoLST model. If each arm is associated with a d-dimensional feature vector, we show that CoLSTIM achieves a regret of order tilde O( dT) after T learning rounds. Additionally, we also establish the optimality of CoLSTIM by showing a lower bound for the weak regret that refines the existing average regret analysis. Our experiments demonstrate its superiority over state-of-art algorithms for special cases of CoLST models.

Modern reinforcement learning (RL) systems capture deep truths about general, human problem-solving. In domains where new data can be simulated cheaply, these systems uncover sequential decision-making policies that far exceed the ability of any human. Society faces many problems whose solutions require this skill, but they are often in domains where new data cannot be cheaply simulated. In such scenarios, we can learn simulators from existing data, but these will only ever be approximately correct, and can be pathologically incorrect when queried outside of their training distribution. As a result, a misalignment between the environments in which we train our agents and the real-world in which we wish to deploy our agents is inevitable. Dealing with this misalignment is the primary concern of zero-shot reinforcement learning, a problem setting where the agent must generalise to a new task or domain with zero practice shots. Whilst impressive progress has been made on methods that perform zero-shot RL in idealised settings, new work is needed if these results are to be replicated in real-world settings. In this thesis, we argue that doing so requires us to navigate (at least) three constraints. First, the data quality constraint: real-world datasets are small and homogeneous. Second, the observability constraint: states, dynamics and rewards in the real-world are often only partially observed. And third, the data availability constraint: a priori access to data cannot always be assumed. This work proposes a suite of methods that perform zero-shot RL subject to these constraints. In a series of empirical studies we expose the failings of existing methods, and justify our techniques for remedying them. We believe these designs take us a step closer to RL methods that can be deployed to solve real-world problems.

In this work, we study the simple yet universally applicable case of reward shaping in value-based Deep Reinforcement Learning (DRL). We show that reward shifting in the form of the linear transformation is equivalent to changing the initialization of the Q-function in function approximation. Based on such an equivalence, we bring the key insight that a positive reward shifting leads to conservative exploitation, while a negative reward shifting leads to curiosity-driven exploration. Accordingly, conservative exploitation improves offline RL value estimation, and optimistic value estimation improves exploration for online RL. We validate our insight on a range of RL tasks and show its improvement over baselines: (1) In offline RL, the conservative exploitation leads to improved performance based on off-the-shelf algorithms; (2) In online continuous control, multiple value functions with different shifting constants can be used to tackle the exploration-exploitation dilemma for better sample efficiency; (3) In discrete control tasks, a negative reward shifting yields an improvement over the curiosity-based exploration method.

Reward models are key in reinforcement learning from human feedback (RLHF) systems, aligning the model behavior with human preferences. Particularly in the math domain, there have been plenty of studies using reward models to align policies for improving reasoning capabilities. Recently, as the importance of reward models has been emphasized, RewardBench is proposed to understand their behavior. However, we figure out that the math subset of RewardBench has different representations between chosen and rejected completions, and relies on a single comparison, which may lead to unreliable results as it only see an isolated case. Therefore, it fails to accurately present the robustness of reward models, leading to a misunderstanding of its performance and potentially resulting in reward hacking. In this work, we introduce a new design for reliable evaluation of reward models, and to validate this, we construct RewardMATH, a benchmark that effectively represents the robustness of reward models in mathematical reasoning tasks. We demonstrate that the scores on RewardMATH strongly correlate with the results of optimized policy and effectively estimate reward overoptimization, whereas the existing benchmark shows almost no correlation. The results underscore the potential of our design to enhance the reliability of evaluation, and represent the robustness of reward model. We make our code and data publicly available.

Is it possible to understand or imitate a policy maker's rationale by looking at past decisions they made? We formalize this question as the problem of learning social welfare functions belonging to the well-studied family of power mean functions. We focus on two learning tasks; in the first, the input is vectors of utilities of an action (decision or policy) for individuals in a group and their associated social welfare as judged by a policy maker, whereas in the second, the input is pairwise comparisons between the welfares associated with a given pair of utility vectors. We show that power mean functions are learnable with polynomial sample complexity in both cases, even if the comparisons are social welfare information is noisy. Finally, we design practical algorithms for these tasks and evaluate their performance.

As artificial intelligence (AI) systems play an increasingly prominent role in human decision-making, challenges surface in the realm of human-AI interactions. One challenge arises from the suboptimal AI policies due to the inadequate consideration of humans disregarding AI recommendations, as well as the need for AI to provide advice selectively when it is most pertinent. This paper presents a sequential decision-making model that (i) takes into account the human's adherence level (the probability that the human follows/rejects machine advice) and (ii) incorporates a defer option so that the machine can temporarily refrain from making advice. We provide learning algorithms that learn the optimal advice policy and make advice only at critical time stamps. Compared to problem-agnostic reinforcement learning algorithms, our specialized learning algorithms not only enjoy better theoretical convergence properties but also show strong empirical performance.

In reinforcement learning from human feedback, it is common to optimize against a reward model trained to predict human preferences. Because the reward model is an imperfect proxy, optimizing its value too much can hinder ground truth performance, in accordance with Goodhart's law. This effect has been frequently observed, but not carefully measured due to the expense of collecting human preference data. In this work, we use a synthetic setup in which a fixed "gold-standard" reward model plays the role of humans, providing labels used to train a proxy reward model. We study how the gold reward model score changes as we optimize against the proxy reward model using either reinforcement learning or best-of-n sampling. We find that this relationship follows a different functional form depending on the method of optimization, and that in both cases its coefficients scale smoothly with the number of reward model parameters. We also study the effect on this relationship of the size of the reward model dataset, the number of reward model and policy parameters, and the coefficient of the KL penalty added to the reward in the reinforcement learning setup. We explore the implications of these empirical results for theoretical considerations in AI alignment.

The fairness of machine learning-based decisions has become an increasingly important focus in the design of supervised machine learning methods. Most fairness approaches optimize a specified trade-off between performance measure(s) (e.g., accuracy, log loss, or AUC) and fairness metric(s) (e.g., demographic parity, equalized odds). This begs the question: are the right performance-fairness trade-offs being specified? We instead re-cast fair machine learning as an imitation learning task by introducing superhuman fairness, which seeks to simultaneously outperform human decisions on multiple predictive performance and fairness measures. We demonstrate the benefits of this approach given suboptimal decisions.

We consider a binary classification problem under group fairness constraints, which can be one of Demographic Parity (DP), Equalized Opportunity (EOp), or Equalized Odds (EO). We propose an explicit characterization of Bayes optimal classifier under the fairness constraints, which turns out to be a simple modification rule of the unconstrained classifier. Namely, we introduce a novel instance-level measure of bias, which we call bias score, and the modification rule is a simple linear rule on top of the finite amount of bias scores.Based on this characterization, we develop a post-hoc approach that allows us to adapt to fairness constraints while maintaining high accuracy. In the case of DP and EOp constraints, the modification rule is thresholding a single bias score, while in the case of EO constraints we are required to fit a linear modification rule with 2 parameters. The method can also be applied for composite group-fairness criteria, such as ones involving several sensitive attributes.

Machine learning models are increasingly used to automate decisions that affect humans - deciding who should receive a loan, a job interview, or a social service. In such applications, a person should have the ability to change the decision of a model. When a person is denied a loan by a credit score, for example, they should be able to alter its input variables in a way that guarantees approval. Otherwise, they will be denied the loan as long as the model is deployed. More importantly, they will lack the ability to influence a decision that affects their livelihood. In this paper, we frame these issues in terms of recourse, which we define as the ability of a person to change the decision of a model by altering actionable input variables (e.g., income vs. age or marital status). We present integer programming tools to ensure recourse in linear classification problems without interfering in model development. We demonstrate how our tools can inform stakeholders through experiments on credit scoring problems. Our results show that recourse can be significantly affected by standard practices in model development, and motivate the need to evaluate recourse in practice.

Large language models~(LLMs) are expected to be helpful, harmless, and honest. In alignment scenarios such as safety, confidence, and general preference alignment, binary preference data collection and reward modeling are resource-intensive but essential for transferring human preference. In this work, we explore using the similarity between sampled generations and high-quality reference answers as an alternative reward function choice for LLM alignment. Similarity reward circumvents binary preference data collection and reward modeling when unary high-quality reference answers are available. We introduce RefAlign, a versatile REINFORCE-style alignment algorithm that does not rely on reference or reward models. RefAlign utilizes similarity metrics, such as BERTScore between sampled generations and reference answers as surrogate rewards. Beyond general human preference optimization, RefAlign can be readily extended to diverse scenarios, such as safety and confidence alignment, by incorporating the similarity reward with task-related objectives. In various scenarios, RefAlign demonstrates comparable performance to previous alignment methods without binary preference data and reward models.

State-of-the-art AI systems can be significantly improved without expensive retraining via "post-training enhancements"-techniques applied after initial training like fine-tuning the system to use a web browser. We review recent post-training enhancements, categorizing them into five types: tool-use, prompting methods, scaffolding, solution selection, and data generation. Different enhancements improve performance on different tasks, making it hard to compare their significance. So we translate improvements from different enhancements into a common currency, the compute-equivalent gain: how much additional training compute would be needed to improve performance by the same amount as the enhancement. Our non-experimental work shows that post-training enhancements have significant benefits: most surveyed enhancements improve benchmark performance by more than a 5x increase in training compute, some by more than 20x. Post-training enhancements are relatively cheap to develop: fine-tuning costs are typically <1% of the original training cost. Governing the development of capable post-training enhancements may be challenging because frontier models could be enhanced by a wide range of actors.

Large Language Models (LLMs) can acquire extensive world knowledge through pre-training on large corpora. However, due to exposure to low-quality data, LLMs may exhibit harmful behavior without aligning with human values. The dominant approach for steering LLMs towards beneficial behavior involves Reinforcement Learning with Human Feedback (RLHF), with Proximal Policy Optimization (PPO) serving as the default RL optimizer. Despite its effectiveness, PPO has limitations when optimizing rewards trained from comparison-based loss. Primarily, PPO is not invariant to equivalent reward functions containing identical preference information due to the need to calibrate the reward scale. Additionally, PPO's necessity for token-wise updates introduces complexity in both function approximation and algorithm design compared to trajectory-wise optimization. This paper proposes a new framework, reinforcement learning with relative feedback, and a novel trajectory-wise policy gradient algorithm, Pairwise Proximal Policy Optimization (P3O) that operates directly on comparative rewards. We show theoretically that P3O is invariant to equivalent rewards and avoids the complexity of PPO. Empirical evaluations demonstrate that P3O outperforms PPO in the KL-Reward trade-off and can align with human preferences as well as or better than prior methods. In summary, this work introduces a simpler yet effective approach for aligning LLMs to human preferences through relative feedback.

The concept of causal abstraction got recently popularised to demystify the opaque decision-making processes of machine learning models; in short, a neural network can be abstracted as a higher-level algorithm if there exists a function which allows us to map between them. Notably, most interpretability papers implement these maps as linear functions, motivated by the linear representation hypothesis: the idea that features are encoded linearly in a model's representations. However, this linearity constraint is not required by the definition of causal abstraction. In this work, we critically examine the concept of causal abstraction by considering arbitrarily powerful alignment maps. In particular, we prove that under reasonable assumptions, any neural network can be mapped to any algorithm, rendering this unrestricted notion of causal abstraction trivial and uninformative. We complement these theoretical findings with empirical evidence, demonstrating that it is possible to perfectly map models to algorithms even when these models are incapable of solving the actual task; e.g., on an experiment using randomly initialised language models, our alignment maps reach 100% interchange-intervention accuracy on the indirect object identification task. This raises the non-linear representation dilemma: if we lift the linearity constraint imposed to alignment maps in causal abstraction analyses, we are left with no principled way to balance the inherent trade-off between these maps' complexity and accuracy. Together, these results suggest an answer to our title's question: causal abstraction is not enough for mechanistic interpretability, as it becomes vacuous without assumptions about how models encode information. Studying the connection between this information-encoding assumption and causal abstraction should lead to exciting future work.

Recent work on neural algorithmic reasoning has investigated the reasoning capabilities of neural networks, effectively demonstrating they can learn to execute classical algorithms on unseen data coming from the train distribution. However, the performance of existing neural reasoners significantly degrades on out-of-distribution (OOD) test data, where inputs have larger sizes. In this work, we make an important observation: there are many different inputs for which an algorithm will perform certain intermediate computations identically. This insight allows us to develop data augmentation procedures that, given an algorithm's intermediate trajectory, produce inputs for which the target algorithm would have exactly the same next trajectory step. Then, we employ a causal framework to design a corresponding self-supervised objective, and we prove that it improves the OOD generalisation capabilities of the reasoner. We evaluate our method on the CLRS algorithmic reasoning benchmark, where we show up to 3times improvements on the OOD test data.

Extensive studies have been devoted to privatizing general-domain Large Language Models (LLMs) as Domain-Specific LLMs via feeding specific-domain data. However, these privatization efforts often ignored a critical aspect: Dual Logic Ability, which is a core reasoning ability for LLMs. The dual logic ability of LLMs ensures that they can maintain a consistent stance when confronted with both positive and negative statements about the same fact. Our study focuses on how the dual logic ability of LLMs is affected during the privatization process in the medical domain. We conduct several experiments to analyze the dual logic ability of LLMs by examining the consistency of the stance in responses to paired questions about the same fact. In our experiments, interestingly, we observed a significant decrease in the dual logic ability of existing LLMs after privatization. Besides, our results indicate that incorporating general domain dual logic data into LLMs not only enhances LLMs' dual logic ability but also further improves their accuracy. These findings underscore the importance of prioritizing LLMs' dual logic ability during the privatization process. Our study establishes a benchmark for future research aimed at exploring LLMs' dual logic ability during the privatization process and offers valuable guidance for privatization efforts in real-world applications.

In arena-style evaluation of large language models (LLMs), two LLMs respond to a user query, and the user chooses the winning response or deems the "battle" a draw, resulting in an adjustment to the ratings of both models. The prevailing approach for modeling these rating dynamics is to view battles as two-player game matches, as in chess, and apply the Elo rating system and its derivatives. In this paper, we critically examine this paradigm. Specifically, we question whether a draw genuinely means that the two models are equal and hence whether their ratings should be equalized. Instead, we conjecture that draws are more indicative of query difficulty: if the query is too easy, then both models are more likely to succeed equally. On three real-world arena datasets, we show that ignoring rating updates for draws yields a 1-3% relative increase in battle outcome prediction accuracy (which includes draws) for all four rating systems studied. Further analyses suggest that draws occur more for queries rated as very easy and those as highly objective, with risk ratios of 1.37 and 1.35, respectively. We recommend future rating systems to reconsider existing draw semantics and to account for query properties in rating updates.

Quantizing the weights of large language models (LLMs) from 16-bit to lower bitwidth is the de facto approach to deploy massive transformers onto more affordable accelerators. GPTQ emerged as one of the standard methods for one-shot post-training quantization at LLM scale. Yet, its inner workings are described as a sequence of ad-hoc algebraic updates that obscure any geometric meaning or worst-case guarantees. In this work, we show that, when executed back-to-front (from the last to first dimension) for a linear layer, GPTQ is mathematically identical to Babai's nearest plane algorithm for the classical closest vector problem (CVP) on a lattice defined by the Hessian matrix of the layer's inputs. This equivalence is based on a sophisticated mathematical argument, and has two analytical consequences: (i) the GPTQ error propagation step gains an intuitive geometric interpretation; (ii) GPTQ inherits the error upper bound of Babai's algorithm under the no-clipping condition. Taken together, these results place GPTQ on firm theoretical footing and open the door to importing decades of progress in lattice algorithms towards the design of future quantization algorithms for billion-parameter models.

In this paper, we discuss that an observable-based single-system Copenhagen and entanglement-based two-system von Neumann measurement protocols in quantum theory can be made equivalent by considering the second part of the two-system scheme to be a Dirac-type negative sea filling up the first system. Based on this equivalence, and by considering the universe as a computational process, the choice of the apparatus state in the two-system protocol can be identified with the choice of the observable in the single-system scheme as negative sea filling up the observable universe. In particular, the measuring party's state is considered to be evolving backwards in time to the big bang as a nondeterministic computational process, which chooses the acceptable path as a time-reversal process of irreversible computation. The suggested model proposes that the prepared microstate of the universe, or reality, corresponds to the observer's choice, therefore, subjective reality. Thus, this effectively provides a specific description of the subjective universe model previously proposed, which is based on the symmetry breakdown between the Schrodinger and the Heisenberg pictures of quantum theory.

AI explanations are often mentioned as a way to improve human-AI decision-making, but empirical studies have not found consistent evidence of explanations' effectiveness and, on the contrary, suggest that they can increase overreliance when the AI system is wrong. While many factors may affect reliance on AI support, one important factor is how decision-makers reconcile their own intuition -- beliefs or heuristics, based on prior knowledge, experience, or pattern recognition, used to make judgments -- with the information provided by the AI system to determine when to override AI predictions. We conduct a think-aloud, mixed-methods study with two explanation types (feature- and example-based) for two prediction tasks to explore how decision-makers' intuition affects their use of AI predictions and explanations, and ultimately their choice of when to rely on AI. Our results identify three types of intuition involved in reasoning about AI predictions and explanations: intuition about the task outcome, features, and AI limitations. Building on these, we summarize three observed pathways for decision-makers to apply their own intuition and override AI predictions. We use these pathways to explain why (1) the feature-based explanations we used did not improve participants' decision outcomes and increased their overreliance on AI, and (2) the example-based explanations we used improved decision-makers' performance over feature-based explanations and helped achieve complementary human-AI performance. Overall, our work identifies directions for further development of AI decision-support systems and explanation methods that help decision-makers effectively apply their intuition to achieve appropriate reliance on AI.

A major barrier to deploying current machine learning models lies in their non-reliability to dataset shifts. To resolve this problem, most existing studies attempted to transfer stable information to unseen environments. Particularly, independent causal mechanisms-based methods proposed to remove mutable causal mechanisms via the do-operator. Compared to previous methods, the obtained stable predictors are more effective in identifying stable information. However, a key question remains: which subset of this whole stable information should the model transfer, in order to achieve optimal generalization ability? To answer this question, we present a comprehensive minimax analysis from a causal perspective. Specifically, we first provide a graphical condition for the whole stable set to be optimal. When this condition fails, we surprisingly find with an example that this whole stable set, although can fully exploit stable information, is not the optimal one to transfer. To identify the optimal subset under this case, we propose to estimate the worst-case risk with a novel optimization scheme over the intervention functions on mutable causal mechanisms. We then propose an efficient algorithm to search for the subset with minimal worst-case risk, based on a newly defined equivalence relation between stable subsets. Compared to the exponential cost of exhaustively searching over all subsets, our searching strategy enjoys a polynomial complexity. The effectiveness and efficiency of our methods are demonstrated on synthetic data and the diagnosis of Alzheimer's disease.

We study the problem of designing mechanisms for information acquisition scenarios. This setting models strategic interactions between an uniformed receiver and a set of informed senders. In our model the senders receive information about the underlying state of nature and communicate their observation (either truthfully or not) to the receiver, which, based on this information, selects an action. Our goal is to design mechanisms maximizing the receiver's utility while incentivizing the senders to report truthfully their information. First, we provide an algorithm that efficiently computes an optimal incentive compatible (IC) mechanism. Then, we focus on the online problem in which the receiver sequentially interacts in an unknown game, with the objective of minimizing the cumulative regret w.r.t. the optimal IC mechanism, and the cumulative violation of the incentive compatibility constraints. We investigate two different online scenarios, i.e., the full and bandit feedback settings. For the full feedback problem, we propose an algorithm that guarantees mathcal O(sqrt T) regret and violation, while for the bandit feedback setting we present an algorithm that attains mathcal O(T^{alpha}) regret and mathcal O(T^{1-alpha/2}) violation for any alphain[1/2, 1]. Finally, we complement our results providing a tight lower bound.

We consider multi-agent reinforcement learning (MARL) for cooperative communication and coordination tasks. MARL agents can be brittle because they can overfit their training partners' policies. This overfitting can produce agents that adopt policies that act under the expectation that other agents will act in a certain way rather than react to their actions. Our objective is to bias the learning process towards finding reactive strategies towards other agents' behaviors. Our method, transfer empowerment, measures the potential influence between agents' actions. Results from three simulated cooperation scenarios support our hypothesis that transfer empowerment improves MARL performance. We discuss how transfer empowerment could be a useful principle to guide multi-agent coordination by ensuring reactiveness to one's partner.

Predictable behavior from scaling advanced AI systems is an extremely desirable property. Although a well-established literature exists on how pretraining performance scales, the literature on how particular downstream capabilities scale is significantly muddier. In this work, we take a step back and ask: why has predicting specific downstream capabilities with scale remained elusive? While many factors are certainly responsible, we identify a new factor that makes modeling scaling behavior on widely used multiple-choice question-answering benchmarks challenging. Using five model families and twelve well-established multiple-choice benchmarks, we show that downstream performance is computed from negative log likelihoods via a sequence of transformations that progressively degrade the statistical relationship between performance and scale. We then reveal the mechanism causing this degradation: downstream metrics require comparing the correct choice against a small number of specific incorrect choices, meaning accurately predicting downstream capabilities requires predicting not just how probability mass concentrates on the correct choice with scale, but also how probability mass fluctuates on specific incorrect choices with scale. We empirically study how probability mass on the correct choice co-varies with probability mass on incorrect choices with increasing compute, suggesting that scaling laws for incorrect choices might be achievable. Our work also explains why pretraining scaling laws are commonly regarded as more predictable than downstream capabilities and contributes towards establishing scaling-predictable evaluations of frontier AI models.

In this position paper, we argue that instead of morally aligning LLMs to specific set of ethical principles, we should infuse generic ethical reasoning capabilities into them so that they can handle value pluralism at a global scale. When provided with an ethical policy, an LLM should be capable of making decisions that are ethically consistent to the policy. We develop a framework that integrates moral dilemmas with moral principles pertaining to different foramlisms of normative ethics, and at different levels of abstractions. Initial experiments with GPT-x models shows that while GPT-4 is a nearly perfect ethical reasoner, the models still have bias towards the moral values of Western and English speaking societies.

Logical reading comprehension is a challenging task that entails grasping the underlying semantics of text and applying reasoning to deduce the correct answer. Prior researches have primarily focused on enhancing logical reasoning capabilities through Chain-of-Thought (CoT) or data augmentation. However, previous work constructing chain-of-thought rationales concentrates solely on analyzing correct options, neglecting the incorrect alternatives. Addtionally, earlier efforts on data augmentation by altering contexts rely on rule-based methods, which result in generated contexts that lack diversity and coherence. To address these issues, we propose a Premise-Oriented Data Augmentation (PODA) framework. This framework can generate CoT rationales including analyses for both correct and incorrect options, while constructing diverse and high-quality counterfactual contexts from incorrect candidate options. We integrate summarizing premises and identifying premises for each option into rationales. Subsequently, we employ multi-step prompts with identified premises to construct counterfactual context. To facilitate the model's capabilities to better differentiate the reasoning process associated with each option, we introduce a novel thought-path contrastive learning method that compares reasoning paths between the original and counterfactual samples. Experimental results on three representative LLMs demonstrate that our method can improve the baselines substantially across two challenging logical reasoning benchmarks (ReClor and LogiQA 2.0). The data and code are released at https://github.com/lalalamdbf/TPReasoner.

Recent benchmarks have probed factual consistency and rhetorical robustness in Large Language Models (LLMs). However, a knowledge gap exists regarding how directional framing of factually true statements influences model agreement, a common scenario for LLM users. AssertBench addresses this by sampling evidence-supported facts from FEVEROUS, a fact verification dataset. For each (evidence-backed) fact, we construct two framing prompts: one where the user claims the statement is factually correct, and another where the user claims it is incorrect. We then record the model's agreement and reasoning. The desired outcome is that the model asserts itself, maintaining consistent truth evaluation across both framings, rather than switching its evaluation to agree with the user. AssertBench isolates framing-induced variability from the model's underlying factual knowledge by stratifying results based on the model's accuracy on the same claims when presented neutrally. In doing so, this benchmark aims to measure an LLM's ability to "stick to its guns" when presented with contradictory user assertions about the same fact. The complete source code is available at https://github.com/achowd32/assert-bench.

Decision-making is a dynamic process requiring perception, memory, and reasoning to make choices and find optimal policies. Traditional approaches to decision-making suffer from sample efficiency and generalization, while large-scale self-supervised pretraining has enabled fast adaptation with fine-tuning or few-shot learning in language and vision. We thus argue to integrate knowledge acquired from generic large-scale self-supervised pretraining into downstream decision-making problems. We propose Pretrain-Then-Adapt pipeline and survey recent work on data collection, pretraining objectives and adaptation strategies for decision-making pretraining and downstream inference. Finally, we identify critical challenges and future directions for developing decision foundation model with the help of generic and flexible self-supervised pretraining.

Advantage Actor-critic (A2C) and Proximal Policy Optimization (PPO) are popular deep reinforcement learning algorithms used for game AI in recent years. A common understanding is that A2C and PPO are separate algorithms because PPO's clipped objective appears significantly different than A2C's objective. In this paper, however, we show A2C is a special case of PPO. We present theoretical justifications and pseudocode analysis to demonstrate why. To validate our claim, we conduct an empirical experiment using Stable-baselines3, showing A2C and PPO produce the exact same models when other settings are controlled.

In many real-world sequential decision-making problems, an action does not immediately reflect on the feedback and spreads its effects over a long time frame. For instance, in online advertising, investing in a platform produces an instantaneous increase of awareness, but the actual reward, i.e., a conversion, might occur far in the future. Furthermore, whether a conversion takes place depends on: how fast the awareness grows, its vanishing effects, and the synergy or interference with other advertising platforms. Previous work has investigated the Multi-Armed Bandit framework with the possibility of delayed and aggregated feedback, without a particular structure on how an action propagates in the future, disregarding possible dynamical effects. In this paper, we introduce a novel setting, the Dynamical Linear Bandits (DLB), an extension of the linear bandits characterized by a hidden state. When an action is performed, the learner observes a noisy reward whose mean is a linear function of the hidden state and of the action. Then, the hidden state evolves according to linear dynamics, affected by the performed action too. We start by introducing the setting, discussing the notion of optimal policy, and deriving an expected regret lower bound. Then, we provide an optimistic regret minimization algorithm, Dynamical Linear Upper Confidence Bound (DynLin-UCB), that suffers an expected regret of order mathcal{O} Big( d sqrt{T}{(1-rho)^{3/2}} Big), where rho is a measure of the stability of the system, and d is the dimension of the action vector. Finally, we conduct a numerical validation on a synthetic environment and on real-world data to show the effectiveness of DynLin-UCB in comparison with several baselines.

Unfair treatment and discrimination are critical ethical concerns in AI systems, particularly as their adoption expands across diverse domains. Addressing these challenges, the recent introduction of the EU AI Act establishes a unified legal framework to ensure legal certainty for AI innovation and investment while safeguarding public interests, such as health, safety, fundamental rights, democracy, and the rule of law (Recital 8). The Act encourages stakeholders to initiate dialogue on existing AI fairness notions to address discriminatory outcomes of AI systems. However, these notions often overlook the critical role of Socio-Economic Status (SES), inadvertently perpetuating biases that favour the economically advantaged. This is concerning, given that principles of equalization advocate for equalizing resources or opportunities to mitigate disadvantages beyond an individual's control. While provisions for discrimination are laid down in the AI Act, specialized directions should be broadened, particularly in addressing economic disparities perpetuated by AI systems. In this work, we explore the limitations of popular AI fairness notions using a real-world dataset (Adult), highlighting their inability to address SES-driven disparities. To fill this gap, we propose a novel fairness notion, Socio-Economic Parity (SEP), which incorporates SES and promotes positive actions for underprivileged groups while accounting for factors within an individual's control, such as working hours, which can serve as a proxy for effort. We define a corresponding fairness measure and optimize a model constrained by SEP to demonstrate practical utility. Our results show the effectiveness of SEP in mitigating SES-driven biases. By analyzing the AI Act alongside our method, we lay a foundation for aligning AI fairness with SES factors while ensuring legal compliance.

We investigate the mechanism design problem faced by a principal who hires multiple agents to gather and report costly information. Then, the principal exploits the information to make an informed decision. We model this problem as a game, where the principal announces a mechanism consisting in action recommendations and a payment function, a.k.a. scoring rule. Then, each agent chooses an effort level and receives partial information about an underlying state of nature based on the effort. Finally, the agents report the information (possibly non-truthfully), the principal takes a decision based on this information, and the agents are paid according to the scoring rule. While previous work focuses on single-agent problems, we consider multi-agents settings. This poses the challenge of coordinating the agents' efforts and aggregating correlated information. Indeed, we show that optimal mechanisms must correlate agents' efforts, which introduces externalities among the agents, and hence complex incentive compatibility constraints and equilibrium selection problems. First, we design a polynomial-time algorithm to find an optimal incentive compatible mechanism. Then, we study an online problem, where the principal repeatedly interacts with a group of unknown agents. We design a no-regret algorithm that provides mathcal{O}(T^{2/3}) regret with respect to an optimal mechanism, matching the state-of-the-art bound for single-agent settings.

In their recent Nature Human Behaviour paper, "Emergent analogical reasoning in large language models," (Webb, Holyoak, and Lu, 2023) the authors argue that "large language models such as GPT-3 have acquired an emergent ability to find zero-shot solutions to a broad range of analogy problems." In this response, we provide counterexamples of the letter string analogies. In our tests, GPT-3 fails to solve even the easiest variants of the problems presented in the original paper. Zero-shot reasoning is an extraordinary claim that requires extraordinary evidence. We do not see that evidence in our experiments. To strengthen claims of humanlike reasoning such as zero-shot reasoning, it is important that the field develop approaches that rule out data memorization.

Standard practice within Reinforcement Learning from Human Feedback (RLHF) involves optimizing against a Reward Model (RM), which itself is trained to reflect human preferences for desirable generations. A notable subject that is understudied is the (in-)consistency of RMs -- whether they can recognize the semantic changes to different prompts and appropriately adapt their reward assignments -- and their impact on the downstream RLHF model. In this paper, we visit a series of research questions relevant to RM inconsistency: (1) How can we measure the consistency of reward models? (2) How consistent are the existing RMs and how can we improve them? (3) In what ways does reward inconsistency influence the chatbots resulting from the RLHF model training? We propose Contrast Instructions -- a benchmarking strategy for the consistency of RM. Each example in Contrast Instructions features a pair of lexically similar instructions with different ground truth responses. A consistent RM is expected to rank the corresponding instruction and response higher than other combinations. We observe that current RMs trained with the standard ranking objective fail miserably on Contrast Instructions compared to average humans. To show that RM consistency can be improved efficiently without using extra training budget, we propose two techniques ConvexDA and RewardFusion, which enhance reward consistency through extrapolation during the RM training and inference stage, respectively. We show that RLHF models trained with a more consistent RM yield more useful responses, suggesting that reward inconsistency exhibits a trickle-down effect on the downstream RLHF process.

Despite the recent success of representation learning in sequential decision making, the study of the pure exploration scenario (i.e., identify the best option and minimize the sample complexity) is still limited. In this paper, we study multi-task representation learning for best arm identification in linear bandits (RepBAI-LB) and best policy identification in contextual linear bandits (RepBPI-CLB), two popular pure exploration settings with wide applications, e.g., clinical trials and web content optimization. In these two problems, all tasks share a common low-dimensional linear representation, and our goal is to leverage this feature to accelerate the best arm (policy) identification process for all tasks. For these problems, we design computationally and sample efficient algorithms DouExpDes and C-DouExpDes, which perform double experimental designs to plan optimal sample allocations for learning the global representation. We show that by learning the common representation among tasks, our sample complexity is significantly better than that of the native approach which solves tasks independently. To the best of our knowledge, this is the first work to demonstrate the benefits of representation learning for multi-task pure exploration.

The prevalent deployment of learning from human preferences through reinforcement learning (RLHF) relies on two important approximations: the first assumes that pairwise preferences can be substituted with pointwise rewards. The second assumes that a reward model trained on these pointwise rewards can generalize from collected data to out-of-distribution data sampled by the policy. Recently, Direct Preference Optimisation (DPO) has been proposed as an approach that bypasses the second approximation and learn directly a policy from collected data without the reward modelling stage. However, this method still heavily relies on the first approximation. In this paper we try to gain a deeper theoretical understanding of these practical algorithms. In particular we derive a new general objective called PsiPO for learning from human preferences that is expressed in terms of pairwise preferences and therefore bypasses both approximations. This new general objective allows us to perform an in-depth analysis of the behavior of RLHF and DPO (as special cases of PsiPO) and to identify their potential pitfalls. We then consider another special case for PsiPO by setting Psi simply to Identity, for which we can derive an efficient optimisation procedure, prove performance guarantees and demonstrate its empirical superiority to DPO on some illustrative examples.

In machine learning, the notion of multi-armed bandits refers to a class of online learning problems, in which an agent is supposed to simultaneously explore and exploit a given set of choice alternatives in the course of a sequential decision process. In the standard setting, the agent learns from stochastic feedback in the form of real-valued rewards. In many applications, however, numerical reward signals are not readily available -- instead, only weaker information is provided, in particular relative preferences in the form of qualitative comparisons between pairs of alternatives. This observation has motivated the study of variants of the multi-armed bandit problem, in which more general representations are used both for the type of feedback to learn from and the target of prediction. The aim of this paper is to provide a survey of the state of the art in this field, referred to as preference-based multi-armed bandits or dueling bandits. To this end, we provide an overview of problems that have been considered in the literature as well as methods for tackling them. Our taxonomy is mainly based on the assumptions made by these methods about the data-generating process and, related to this, the properties of the preference-based feedback.