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111An Analysis of Fusion Functions for Hybrid Retrieval SEBASTIAN BRUCH, Pinecone, USA SIYU GAI∗,University of California, Berkeley, USA AMIR INGBER, Pinecone, Israel We study hybrid search in text retrieval where lexical and semantic search are fused together with the intuition that the two are complementary in how they model relevance. In particular, we examine fusion by a convex combination (CC) of lexical and semantic scores, as well as the Reciprocal Rank Fusion (RRF) method, and identify their advantages and potential pitfalls. Contrary to existing studies, we find RRF to be sensitive to its parameters; that the learning of a CC fusion is generally agnostic to the choice of score normalization; that CC outperforms RRF in in-domain and out-of-domain settings; and finally, that CC is sample efficient, requiring only a small set of training examples to tune its only parameter to a target domain. CCS Concepts: •Information systems →Retrieval models and ranking ;Combination, fusion and federated search .

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federated search . Additional Key Words and Phrases: Hybrid Retrieval, Lexical and Semantic Search, Fusion Functions 1 INTRODUCTION Retrieval is the first stage in a multi-stage ranking system [ 1,2,43], where the objective is to find the top-𝑘set of documents, that are the most relevant to a given query 𝑞, from a large collection of documentsD. Implicit in this task are two major research questions: a) How do we measure the relevance between a query 𝑞and a document 𝑑∈D?; and, b) How do we find the top- 𝑘documents according to a given similarity metric efficiently ? In this work, we are primarily concerned with the former question in the context of text retrieval. As a fundamental problem in Information Retrieval ( IR), the question of the similarity between queries and documents has been explored extensively. Early methods model text as a Bag of Words ( BoW ) and compute the similarity of two pieces of text using a statistical measure such as

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Words ( BoW ) and compute the similarity of two pieces of text using a statistical measure such as the term frequency-inverse document frequency (TF-IDF) family, with BM25 [ 32,33] being its most prominent member. We refer to retrieval with a BoW model as lexical search and the similarity scores computed by such a system as lexical scores . Lexical search is simple, efficient, (naturally) “zero-shot, ” and generally effective, but has important limitations: It is susceptible to the vocabulary mismatch problem and, moreover, does not take into account the semantic similarity of queries and documents [ 5]. That, it turns out, is what deep learning models are excellent at. With the rise of pre-trained language models such as BERT [ 8], it is now standard practice to learn a vector representation of queries and documents that does capture their semantics, and thereby, reduce top- 𝑘retrieval to the problem of finding 𝑘nearest neighbors in the resulting vector space [ 9,12,15,31,39,42]—where closeness is measured using

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vector similarity or distance. We refer to this method as semantic search and the similarity scores computed by such a system as semantic scores . Hypothesizing that lexical and semantic search are complementary in how they model relevance, recent works [ 5,12,13,18,19,41] began exploring methods to fusetogether lexical and semantic retrieval: For a query 𝑞and ranked lists of documents 𝑅Lexand𝑅Semretrieved separately by lexical and semantic search systems respectively, the task is to construct a final ranked list 𝑅Fusion so as to improve retrieval quality. This is often referred to as hybrid search . ∗Contributed to this work during a research internship at Pinecone. Authors’ addresses: Sebastian Bruch, [email protected], Pinecone, New York, NY, USA; Siyu Gai, [email protected],

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University of California, Berkeley, Berkeley, CA, USA; Amir Ingber, Pinecone, Tel Aviv, Israel, [email protected]:2210.11934v1 [cs.IR] 21 Oct 2022

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111:2 Sebastian Bruch, Siyu Gai, and Amir Ingber It is becoming increasingly clear that hybrid search does indeed lead to meaningful gains in retrieval quality, especially when applied to out-of-domain datasets [ 5,39]—settings in which the semantic retrieval component uses a model that was not trained or fine-tuned on the target dataset. What is less clear and is worthy of further investigation, however, is how this fusion is done. One intuitive and common approach is to linearly combine lexical and semantic scores [ 12,19,39]. If𝑓Lex(𝑞,𝑑)and𝑓Sem(𝑞,𝑑)represent the lexical and semantic scores of document 𝑑with respect to query𝑞, then a linear (or more accurately, convex) combination is expressed as 𝑓Convex = 𝛼𝑓Sem+(1−𝛼)𝑓Lexwhere 0≤𝛼≤1. Because lexical scores (such as BM25) and semantic scores

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(such as dot product) may be unbounded, often they are normalized with min-max scaling [ 15,39] prior to fusion. A recent study [ 5] argues that convex combination is sensitive to its parameter 𝛼and the choice of score normalization.1They claim and show empirically, instead, that Reciprocal Rank Fusion ( RRF) [6] may be a more suitable fusion as it is non-parametric and may be utilized in a zero-shot manner. They demonstrate its impressive performance even in zero-shot settings on a number of benchmark datasets. This work was inspired by the claims made in [ 5]; whereas [ 5] addresses how various hybrid methods perform relative to one another in an empirical study, we re-examine their findings and analyze why these methods work and what contributes to their relative performance. Our contributions thus can best be summarized as an in-depth examination of fusion functions and their behavior. As our first research question (RQ1), we investigate whether the convex combination fusion is a reasonable choice and study its sensitivity to the normalization protocol. We show that, while

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a reasonable choice and study its sensitivity to the normalization protocol. We show that, while normalization is essential to create a bounded function and thereby bestow consistency to the fusion across domains, the specific choice of normalization is a rather small detail: There always exist convex combinations of scores normalized by min-max, standard score, or any other linear transformation that are rank-equivalent. In fact, when formulated as a per-query learning problem, the solution found for a dataset that is normalized with one scheme can be transformed to a solution for a different choice. We next investigate the properties of RRF. We first unpack RRF and examine its sensitivity to its parameters as our second research question (RQ2)—contrary to [ 5], we adopt a parametric view ofRRF where we have as many parameters as there are retrieval functions to fuse, a quantity that is always one more than that in a convex combination. We find that, in contrast to a convex combination, a tuned RRFgeneralizes poorly to out-of-domain datasets. We then intuit that, because RRF is a function of ranks , it disregards the distribution of scores and, as such, discards useful

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information. Observe that the distance between raw scores plays no role in determining their hybrid score—a behavior we find counter-intuitive in a metric space where distance does matter. Examining this property constitutes our third and final research question (RQ3). Finally, we empirically demonstrate an unsurprising yet important fact: Tuning 𝛼in a convex combination fusion function is extremely sample-efficient, requiring just a handful of labeled queries to arrive at a value suitable for a target domain, regardless of the magnitude of shift in the data distribution. RRF, on the other hand, is relatively less sample-efficient and converges to a relatively less effective retrieval system. We believe our findings, both theoretical and empirical, are important and pertinent to the research in this field. Our analysis leads us to believe that the convex combination formulation is theoretically sound, empirically effective, sample-efficient, and robust to domain shift. Moreover,

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research in this field. Our analysis leads us to believe that the convex combination formulation is theoretically sound, empirically effective, sample-efficient, and robust to domain shift. Moreover, 1c.f. Section 3.1 in [ 5]: “This fusion method is sensitive to the score scales ...which needs careful score normalization” (emphasis ours).

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An Analysis of Fusion Functions for Hybrid Retrieval 111:3 unlike the parameters in RRF, the parameter(s) of a convex function are highly interpretable and, if no training samples are available, can be adjusted to incorporate domain knowledge. We organized the remainder of this article as follows. In Section 2, we review the relevant literature on hybrid search. Section 3 then introduces our adopted notation and provides details of our empirical setup, thereby providing context for the theoretical and empirical analysis of fusion functions. In Section 4, we begin our analysis by a detailed look at the convex combination of retrieval scores. We then examine RRF in Section 5. In Section 6, we summarize our observations and identify the properties a fusion function should have to behave well in hybrid retrieval. We then conclude this work and state future research directions in Section 7. 2 RELATED WORK A multi-stage ranking system is typically comprised of a retrieval stage and several subsequent re- ranking stages, where the retrieved candidates are ordered using more complex ranking functions [ 2, 38]. Conventional wisdom has that retrieval must be recall-oriented while improving ranking quality

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38]. Conventional wisdom has that retrieval must be recall-oriented while improving ranking quality may be left to the re-ranking stages, which are typically Learning to Rank ( LtR) models [ 16,23, 28,38,40]. There is indeed much research on the trade-offs between recall and precision in such multi-stage cascades [ 7,20], but a recent study [ 44] challenges that established convention and presents theoretical analysis that suggests retrieval must instead optimize precision . We therefore report both recall andNDCG [ 10], but focus on NDCG where space constraints prevent us from presenting both or when similar conclusions can be reached regardless of the metric used. One choice for retrieval that remains popular to date is BM25 [ 32,33]. This additive statistic computes a weighted lexical match between query and document terms: It computes, for each query term, the product of its “importance” (i.e., frequency of a term in a document, normalized by document and global statistics such as average length) and its propensity—a quantity that is inversely proportionate to the fraction of documents that contain the term—and adds the scores of

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inversely proportionate to the fraction of documents that contain the term—and adds the scores of query terms to arrive at the final similarity or relevance score. Because BM25, like other lexical scoring functions, insists on an exact match of terms, even a slight typo can throw the function off. This vocabulary mismatch problem has been subject to much research in the past, with remedies ranging from pseudo-relevance feedback to document and query expansion techniques [ 14,29,35]. Trying to address the limitations of lexical search can only go so far, however. After all, they additionally do not capture the semantic similarity between queries and documents, which may be an important signal indicative of relevance. It has been shown that both of these issues can be remedied by Transformer-based [ 37] pre-trained language models such as BERT [ 8]. Applied to the ranking task, such models [ 24,26–28] have advanced the state-of-the-art dramatically on benchmark datasets [25]. The computationally intensive inference of these deep models often renders them too ineffi- cient for first-stage retrieval, however, making them more suitable for re-ranking stages. But by

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cient for first-stage retrieval, however, making them more suitable for re-ranking stages. But by cleverly disentangling the query and document transformations into the so-called dual-encoder architecture, where, in the resulting design, the “embedding” of a document can be computed independently of queries, we can pre-compute document vectors and store them offline. In this way, we substantially reduce the computational cost during inference as it is only necessary to obtain the vector representation of the query during inference. At a high level, these models project queries and documents onto a low-dimensional vector space where semantically-similar points

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obtain the vector representation of the query during inference. At a high level, these models project queries and documents onto a low-dimensional vector space where semantically-similar points stay closer to each other. By doing so we transform the retrieval problem to one of similarity search or Approximate Nearest Neighbor (ANN) search—the 𝑘nearest neighbors to a query vector are the desired top-𝑘documents. This ANN problem can be solved efficiently using a number of algorithms such as FAISS [ 11] or Hierarchical Navigable Small World Graphs [ 21] available as open source

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111:4 Sebastian Bruch, Siyu Gai, and Amir Ingber packages or through a managed service such as Pinecone2, creating an opportunity to use deep models and vector representations for first-stage retrieval [ 12,42]—a setup that we refer to as semantic search. Semantic search, however, has its own limitations. Previous studies [ 5,36] have shown, for example, that when applied to out-of-domain datasets, their performance is often worse than BM25. Observing that lexical and semantic retrievers can be complementary in the way they model relevance [ 5], it is only natural to consider a hybrid approach where lexical and semantic similarities both contribute to the makeup of final retrieved list. To date there have been many studies [ 12,13,17–19,39,41,45] that do just that, where most focus on in-domain tasks with one exception [ 5] that considers a zero-shot application too. Most of these works only use one of the many existing fusion functions in experiments, but none compares the main ideas comprehensively.

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many existing fusion functions in experiments, but none compares the main ideas comprehensively. We review the popular fusion functions from these works in the subsequent sections and, through a comparative study, elaborate what about their behavior may or may not be problematic. 3 SETUP In the sections that follow, we study fusion functions with a mix of theoretical and empirical analysis. For that reason, we present our notation as well as empirical setup and evaluation measures here to provide sufficient context for our arguments. 3.1 Notation We adopt the following notation in this work. We use 𝑓o(𝑞,𝑑):Q×D→ Rto denote the score of document𝑑∈D to query𝑞∈Qaccording to the retrieval system o∈O. Ifois a semantic retriever, Sem, thenQandDare the space of (dense) vectors in R𝑑and𝑓Semis typically cosine similarity or inner product. Similarly, when ois a lexical retriever, Lex,QandDare high-dimensional sparse

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vectors in R|𝑉|, with|𝑉|denoting the size of the vocabulary, and 𝑓Lexis typically BM25. A retrieval system ois the spaceQ×D equipped with a metric 𝑓o(·,·)—which need not be a proper metric. We denote the set of top- 𝑘documents retrieved for query 𝑞by retrieval system oby𝑅𝑘 o(𝑞). We write𝜋o(𝑞,𝑑)to denote the rank of document 𝑑with respect to query 𝑞according to retrieval system o. Note that,𝜋o(𝑞,𝑑𝑖)can be expressed as the sum of indicator functions: 𝜋o(𝑞,𝑑𝑖)=1+∑︁ 𝑑𝑗∈𝑅𝑘o(𝑞)1𝑓o(𝑞,𝑑𝑗)>𝑓o(𝑞,𝑑𝑖), (1)

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where 1𝑐is1when the predicate 𝑐holds and 0otherwise. In words, and ignoring the subtleties introduced by the presence of score ties, the rank of document 𝑑is the count of documents whose score is larger than the score of 𝑑. Hybrid retrieval operates on the product space ofÎo𝑖with metric 𝑓Fusion :Î𝑓o𝑖→R. Without loss of generality, in this work, we restrictÎo𝑖to be Lex×Sem. That is, we only consider the problem of fusing two retrieval scores, but note that much of the anlysis can be trivially extended to the fusion of multiple retrieval systems. We refer to this hybrid metric as a fusion function. A fusion function 𝑓Fusion is typically applied to documents in the union of retrieved sets U𝑘(𝑞)=Ð o𝑅𝑘 o(𝑞), which we simply call the union set . When a document 𝑑in the union set is not present in

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one of the top- 𝑘sets (i.e.,𝑑∈U𝑘(𝑞)but𝑑∉𝑅𝑘 o𝑖(𝑞)for some o𝑖), we compute its missing score (i.e.,𝑓o𝑖(𝑞,𝑑)) prior to fusion. 2http://pinecone.io

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An Analysis of Fusion Functions for Hybrid Retrieval 111:5 3.2 Empirical Setup Datasets : We evaluate our methods on a variety of publicly available benchmark datasets, both in in-domain and out-of-domain, zero-shot settings. One of the datasets is the MS MARCO Passage Retrieval v1 dataset [ 25], a publicly available retrieval and ranking collection from Microsoft. It consists of roughly 8.8million short passages which, along with queries in natural language, originate from Bing. The queries are split into train, dev, and eval non-overlapping subsets. We use the train queries for any learning or tuning and evaluate exclusively on the small dev query set (consisting of 6,980labeled queries) in our analysis. Included in the dataset also are relevance labels. We additionally experiment with 8datasets from the BeIR collection [ 36]3: Natural Questions (NQ, question answering), Quora (duplicate detection), NFCorpus (medical), HotpotQA (question answering), Fever (fact extraction), SciFact (scientific claim verification), DBPedia (entity search),

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andFiQA (financial). For details of and statistics for each dataset, we refer the reader to [36]. Lexical search : We use PISA [ 22] for keyword-based lexical retrieval. We tokenize queries and documents by space and apply stemming available in PISA—we do not employ any other preprocessing steps such as stopword removal, lemmatization, or expansion. We use BM25 with the same hyperparameters as [5] (k1= 0.9and b= 0.4) to retrieve the top 1000 candidates. Semantic search : We use the all-MiniLM-L6-v2 model checkpoint available on HuggingFace4 to project queries and documents into 384-dimensional vectors, which can subsequently be used for indexing and top- 𝑘retrieval using cosine similarity. This model has been shown to achieve competitive quality on an array of benchmark datasets while remaining compact in size and efficient to infer5, thereby allowing us to conduct extensive experiments with results that are competitive with existing state-of-the-art models. This model has been fine-tuned on a large number of datasets,

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exceeding a total of 1 billion pairs of text, including NQ, MS MARCO Passage, and Quora . As such, we consider all experiments on these three datasets as in-domain, and the rest as out-of-domain. We use the exact search for inner product algorithm ( IndexFlatIP ) from FAISS [ 11] to retrieve top 1000 approximate nearest neighbors. Evaluation : Unless noted otherwise, we form the union set for every query from the candidates retrieved by the lexical and semantic search systems. We then compute missing scores where required, compute 𝑓Fusion on the union set, and re-order according to the hybrid scores. We then measure Recall@ 1000 and NDCG@ 1000 to quantify ranking quality, as recommended by Zamani et al. [ 44]. On SciFact andNFCorpus , we evaluate Recall and NDCG at rank cutoff 100due to the small size of these two collections. Note that, we choose to evaluate deep (i.e., with a larger rank cut-off) rather than shallow metrics per the discussion in [ 39] to understand the performance of

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cut-off) rather than shallow metrics per the discussion in [ 39] to understand the performance of each system more completely. 4 ANALYSIS OF CONVEX COMBINATION OF RETRIEVAL SCORES We are interested in understanding the behavior and properties of fusion functions. In the remainder of this work, we study through that lens two popular methods that are representative of existing ideas in the literature, beginning with a convex combination of scores. As noted earlier, most existing works use a convex combination of lexical and semantic scores as follows:𝑓Convex(𝑞,𝑑)=𝛼𝑓Sem(𝑞,𝑑)+(1−𝛼)𝑓Lex(𝑞,𝑑)for some 0≤𝛼≤1. When𝛼=1the above collapses to semantic scores and when it is 0, to lexical scores. 3Available at https://github.com/beir-cellar/beir 4https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2 5c.f. https://sbert.net for details.

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111:6 Sebastian Bruch, Siyu Gai, and Amir Ingber An interesting property of this fusion is that it takes into account the distribution of scores. In other words, the distance between lexical (or semantic) scores of two documents plays a significant role in determining their final hybrid score. One disadvantage, however, is that the range of 𝑓Sem can be very different from 𝑓Lex. Moreover, as with TF-IDF in lexical search or with inner product in semantic search, the range of individual functions 𝑓omay depend on the norm of the query and document vectors (e.g., BM25 is a function of the number of query terms). As such any constant 𝛼 is likely to yield inconsistently-scaled hybrid scores. The problem above is trivially addressed by applying score normalization prior to fusion [ 15,39]. Suppose we have collected a union set U𝑘(𝑞)for𝑞, and that for every candidate we have computed both lexical and semantic scores. Now, consider the min-max scaling of scores 𝜙mm:R→[0,1] below:

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below: 𝜙mm(𝑓o(𝑞,𝑑))=𝑓o(𝑞,𝑑)−𝑚𝑞 𝑀𝑞−𝑚𝑞, (2) where𝑚𝑞=min𝑑∈U𝑘(𝑞)𝑓o(𝑞,𝑑)and𝑀𝑞=max𝑑∈U𝑘(𝑞)𝑓o(𝑞,𝑑). We note that, min-max scaling is thede facto method in the literature, but other choices of 𝜙o(·)in the more general expression below: 𝑓Convex(𝑞,𝑑)=𝛼𝜙Sem(𝑓Sem(𝑞,𝑑))+( 1−𝛼)𝜙Lex(𝑓Lex(𝑞,𝑑)), (3)

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are valid as well so long as 𝜙Sem,𝜙Lex:R→Rare monotone in their argument. For example, for reasons that will become clearer later, we can redefine the normalization by replacing the minimum of the set with the theoretical minimum of the function (i.e., the maximum value that is always less than or equal to all values attainable by the scoring function, or its infimum) to arrive at: 𝜙tmm(𝑓o(𝑞,𝑑))=𝑓o(𝑞,𝑑)−inf𝑓o(𝑞,·) 𝑀𝑞−inf𝑓o(𝑞,·). (4) As an example, when 𝑓Lexis BM25, then its infimum is 0. When𝑓Semis cosine similarity, then that quantity is−1. Another popular choice is the standard score (z-score) normalization which is defined as follows:

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Another popular choice is the standard score (z-score) normalization which is defined as follows: 𝜙z(𝑓o(𝑞,𝑑))=𝑓o(𝑞,𝑑)−𝜇 𝜎, (5) where𝜇and𝜎denote the mean and standard deviation of the set of scores 𝑓o(𝑞,·)for query𝑞. We will return to normalization shortly, but we make note of one small but important fact: In cases where the variance of lexical (semantic) scores in the union set is 0, we may skip the fusion step altogether because retrieval quality will be unaffected by lexical (semantic) scores. The case where the variance is arbitrarily close to 0, however, creates challenges for certain normalization functions. While this would make for an interesting theoretical analysis, we do not study this particular setting in this work as, empirically, we do observe a reasonably large variance among scores in the union set on all datasets using state-of-the-art lexical and semantic retrieval functions. 4.1 Suitability of Convex Combination

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4.1 Suitability of Convex Combination A convex combination of scores is a natural choice for creating a mixture of two retrieval systems, but is it a reasonable choice? It has been established in many past empirical studies that 𝑓Convex with min-max normalization often serves as a strong baseline. So the answer to our question appears to be positive. Nonetheless, we believe it is important to understand precisely why this fusion works. We investigate this question empirically, by visualizing lexical and semantic scores of query- document pairs from an array of datasets. Because we operate in a two-dimensional space, observing the pattern of positive (where document is relevant to query) and negative samples in a plot can reveal a lot about whether and how they are separable and how the fusion function behaves. To

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An Analysis of Fusion Functions for Hybrid Retrieval 111:7 (a) MS MARCO (b)Quora (c)NQ (d)FiQA (e)HotpotQA (f)Fever Fig. 1. Visualization of the normalized lexical ( 𝜙tmm(𝑓Lex)) and semantic ( 𝜙tmm(𝑓Sem)) scores of query- document pairs sampled from the validation split of each dataset. Shown in red are up to 20,000positive samples where document is relevant to query, and in black up to the same number of negative samples. Adding a lexical (semantic) dimension to query-document pairs helps tease out the relevant documents that would be statistically indistinguishable in a one-dimensional semantic (lexical) view of the data—when samples are projected onto the 𝑥(𝑦) axis. that end, we sample up to 20,000positive and up to the same number of negative query-document pairs from the validation split of each dataset, and illustrate the collected points in a scatter plot in Figure 1.

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111:8 Sebastian Bruch, Siyu Gai, and Amir Ingber (a)𝛼=0.6 (b)𝛼=0.8 Fig. 2. Effect of 𝑓Convex on pairs of lexical and semantic scores. From these figures, it is clear that positive and negative samples form clusters that are, with some error, separable by a linear function. What is different between datasets is the slope of this separating line. For example, in MS MARCO, Quora , and NQ, which are in-domain datasets, the separating line is almost vertical, suggesting that the semantic scores serve as a sufficiently strong signal for relevance. This is somewhat true of FiQA . In other out-of-domain datasets, however, the line is rotated counter-clockwise, indicating a more balanced weighting of lexical and semantic scores. Said differently, adding a lexical (semantic) dimension to query-document pairs helps tease out the relevant documents that would be statistically indistinguishable in a one-dimensional semantic (lexical) view of the data. Interestingly, across all datasets, there is a higher concentration of negative samples where lexical scores vanish.

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of negative samples where lexical scores vanish. This empirical evidence suggests that lexical and semantic scores may indeed be complementary— an observation that is in agreement with prior work [ 5]—and a line may be a reasonable choice for distinguishing between positive and negative samples. But while these figures shed light on the shape of positive and negative clusters and their separability, our problem is not classification but ranking . We seek to order query-document pairs and, as such, separability is less critical and, in fact, not required. It is therefore instructive to understand the effect of a particular convex combination on pairs of lexical and semantic scores. This is visualized in Figure 2 for two values of 𝛼in𝑓Convex . The plots in Figure 2 illustrate how the parameter 𝛼determines how different regions of the plane are ranked relative to each other. This is a trivial fact, but it is nonetheless interesting to map these patterns to the distributions in Figure 1. In-domain datasets, for example, form a pattern of

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positives and negatives that is unsurprisingly more in tune with the 𝛼=0.8setting of𝑓Convex than 𝛼=0.6. 4.2 Role of Normalization We have thus far used min-max normalization to be consistent with the literature. In this section, we ask the question first raised by Chen et al. [ 5] on whether and to what extent the choice of nor- malization matters and how carefully one must choose the normalization protocol. In other words, we wish to examine the effect of 𝜙Sem(·)and𝜙Lex(·)on the convex combination in Equation (3). Before we begin, let us consider the following suite of functions:

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An Analysis of Fusion Functions for Hybrid Retrieval 111:9 •𝜙mm: Min-max scaling of Equation (2); •𝜙tmm: Theoretical min-max scaling of Equation (4); •𝜙z: z-score normalization of Equation (5); •𝜙mm−Lex: Min-max scaling of lexical scores, unnormalized semantic scores; •𝜙tmm−Lex: Theoretical min-max normalized lexical scores, unnormalized semantic scores; •𝜙z−Lex: z-score normalized lexical scores, unnormalized semantic scores; and, •𝐼: The identity transformation, leaving both semantic and lexical scores unnormalized. We believe these transformations together test the various conditions in our upcoming arguments. Let us first state the notion of rank-equivalence more formally: Definition 4.1. We say two functions 𝑓and𝑔arerank-equivalent on the setUand write𝑓𝜋=𝑔, if the order among documents in a set Uinduced by 𝑓is the same as that induced by 𝑔.

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For example, when 𝜙Sem(𝑥)=𝑎𝑥+𝑏and𝜙Lex(𝑥)=𝑐𝑥+𝑑are linear transformations of scores for some positive coefficients 𝑎,𝑏and real intercepts 𝑏,𝑐, then they can be reduced to the following rank-equivalent form: 𝑓Convex(𝑞,𝑑)𝜋=(𝑎𝛼)𝑓Sem(𝑞,𝑑)+𝑐(1−𝛼)𝑓Lex(𝑞,𝑑). In fact, letting 𝛼′=𝑐𝛼/[𝑐𝛼+𝑐(1−𝛼)]transforms the problem to one of learning a convex combination of the original scores with a modified weight. This family of functions includes 𝜙mm, 𝜙z, and𝜙tmm, and as such solutions for one family can be transformed to solutions for another normalization protocol. More formally:

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normalization protocol. More formally: Lemma 4.2. For every query, given an arbitrary 𝛼, there exists a 𝛼′such that the convex combination of min-max normalized scores with parameter 𝛼is rank-equivalent to a convex combination of z-score normalized scores with 𝛼′, and vice versa. Proof. Write𝑚oand𝑀ofor the minimum and maximum scores retrieved by system o, and𝜇o and𝜎ofor their mean and standard deviation. We also write 𝑅o=𝑀o−𝑚ofor brevity. For every document𝑑, we have the following: 𝛼𝑓Sem(𝑞,𝑑)−𝑚Sem 𝑅Sem+(1−𝛼)𝑓Lex(𝑞,𝑑)−𝑚Lex 𝑅Lex𝜋=𝛼 𝑅sem𝑓Sem(𝑞,𝑑)+1−𝛼

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𝑅Lex𝑓Lex(𝑞,𝑑) 𝜋=1 𝜎Sem𝜎Lex𝛼 𝑅Sem𝑓Sem(𝑞,𝑑)+1−𝛼 𝑅Lex𝑓Lex(𝑞,𝑑)−𝛼 𝑅Sem𝜇Sem−1−𝛼 𝑅Lex𝜇Lex 𝜋=𝛼 𝑅Sem𝜎Lex𝑓Sem(𝑞,𝑑)−𝜇Sem 𝜎Sem
+1−𝛼 𝑅Lex𝜎Sem𝑓Lex(𝑞,𝑑)−𝜇Lex 𝜎Lex
, where in every step we either added a constant or multiplied the expression by a positive constant, both rank-preserving operations. Finally, setting 𝛼′=𝛼 𝑅Sem𝜎Lex/(𝛼

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𝑅Sem𝜎Lex/(𝛼 𝑅Sem𝜎Lex+1−𝛼 𝑅Lex𝜎Sem) completes the proof. The other direction is similar. □ The fact above implies that the problem of tuning 𝛼for a query in a min-max normalization regime is equivalent to learning 𝛼′in a z-score normalized setting. In other words, there is a one-to-one relationship between these parameters, and as a result solutions can be mapped from one problem space to the other. However, this statement is only true for individual queries and does not have any implications for the learning of the weight in the convex combination over an entire collection of queries. Let us now consider this more complex setup.

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111:10 Sebastian Bruch, Siyu Gai, and Amir Ingber The question we wish to answer is as follows: Under what conditions is 𝑓Convex with parameter 𝛼and a pair of normalization functions (𝜙Sem,𝜙Sem)rank-equivalent to an 𝑓′ Convexof a new pair of normalization functions (𝜙′ Sem,𝜙′ Lex)with weight 𝛼′? That is, for a constant 𝛼with one normalization protocol, when is there a constant 𝛼′that produces the same ranked lists for every query but with a different normalization protocol? The answer to this question helps us understand whether and when changing normalization schemes from min-max to z-score, for example, matters. We state the following definitions followed by a theorem that answers this question. Definition 4.3. We say𝑓:R→Ris a𝛿-expansion with respect to 𝑔:R→Rif for any𝑥and𝑦

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in the domains of 𝑓and𝑔we have that|𝑓(𝑦)−𝑓(𝑥)|≥𝛿|𝑔(𝑦)−𝑔(𝑥)|for some𝛿≥1. For example, 𝜙mm(·)is an expansion with respect to 𝜙tmm(·)with a factor 𝛿that depends on the range of the scores. As another example, 𝜙z(·)is an expansion with respect to 𝜙mm(·). Definition 4.4. For two pairs of functions 𝑓,𝑔:R→Rand𝑓′,𝑔′:R→R, and two points 𝑥and 𝑦in their domains, we say that 𝑓′expands with respect to 𝑓more rapidly than𝑔′expands with respect to𝑔, with a relative expansion rate of𝜆≥1, if the following condition holds:

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|𝑓′(𝑦)−𝑓′(𝑥)| |𝑓(𝑦)−𝑓(𝑥)|=𝜆|𝑔′(𝑦)−𝑔′(𝑥)| |𝑔(𝑦)−𝑔(𝑥)|. When𝜆is independent of the points 𝑥and𝑦, we call this relative expansion uniform : |Δ𝑓′|/|Δ𝑓| |Δ𝑔′|/|Δ𝑔|=𝜆,∀𝑥,𝑦. As an example, if 𝑓and𝑔are min-max scaling and 𝑓′and𝑔′are z-score normalization, then their respective rate of expansion is roughly similar. We will later show that this property often holds empirically across different transformations. Theorem 4.5. For every choice of 𝛼, there exists a constant 𝛼′such that the following functions are

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rank-equivalent on a collection of queries 𝑄: 𝑓Convex =𝛼𝜙(𝑓Sem(𝑞,𝑑))+( 1−𝛼)𝜔(𝑓Lex(𝑞,𝑑)), and 𝑓′ Convex =𝛼′𝜙′(𝑓Sem(𝑞,𝑑))+( 1−𝛼′)𝜔′(𝑓Lex(𝑞,𝑑)), if for the monotone functions 𝜙,𝜔,𝜙′,𝜔′:R→R,𝜙′expands with respect to 𝜙more rapidly than 𝜔′ expands with respect to 𝜔with a uniform rate 𝜆. Proof. Consider a pair of documents 𝑑𝑖and𝑑𝑗in the ranked list of a query 𝑞such that𝑑𝑖is

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ranked above 𝑑𝑗according to 𝑓Convex . Shortening 𝑓o(𝑞,𝑑𝑘)to𝑓(𝑘) ofor brevity, we have that: 𝑓(𝑖) Convex>𝑓(𝑗) Convex=⇒𝛼 (𝜙(𝑓(𝑖) Sem)−𝜙(𝑓(𝑗) Sem)) | {z } Δ𝜙𝑖 𝑗+(𝜔(𝑓(𝑗) Lex)−𝜔(𝑓(𝑖) Lex)) | {z } Δ𝜔𝑗𝑖 >𝜔(𝑓(𝑗) Lex)−𝜔(𝑓(𝑖) Lex) This holds if and only if we have the following: (

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Lex) This holds if and only if we have the following: ( 𝛼>1/(1+Δ𝜙𝑖 𝑗 Δ𝜔𝑗𝑖),ifΔ𝜙𝑖 𝑗+Δ𝜔𝑗𝑖>0, 𝛼<1/(1+Δ𝜙𝑖 𝑗 Δ𝜔𝑗𝑖),otherwise.(6) Observe that, because of the monotonicity of a convex combination and the monotonicity of the normalization functions, the case Δ𝜙𝑖 𝑗<0andΔ𝜔𝑗𝑖>0(which implies that the semantic and lexical scores of 𝑑𝑗are both larger than 𝑑𝑖) is not valid as it leads to a reversal of ranks. Similarly,

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An Analysis of Fusion Functions for Hybrid Retrieval 111:11 the opposite case Δ𝜙𝑖 𝑗>0andΔ𝜔𝑗𝑖<0always leads to the correct order regardless of the weight in the convex combination. We consider the other two cases separately below. Case 1: Δ𝜙𝑖 𝑗>0andΔ𝜔𝑗𝑖>0. Because of the monotonicity property, we can deduce that Δ𝜙′ 𝑖 𝑗>0andΔ𝜔′ 𝑗𝑖>0. From Equation (6), for the order between 𝑑𝑖and𝑑𝑗to be preserved under the image of 𝑓′ Convex, we must therefore have the following: 𝛼′>1/(1+Δ𝜙′ 𝑖 𝑗 Δ𝜔′ 𝑗𝑖). By assumption, using Definition 4.4, we observe that:

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By assumption, using Definition 4.4, we observe that: Δ𝜙′ 𝑖 𝑗 Δ𝜙𝑖 𝑗≥Δ𝜔′ 𝑗𝑖 Δ𝜔𝑗𝑖=⇒Δ𝜙′ 𝑖 𝑗 Δ𝜔′ 𝑗𝑖≥Δ𝜙𝑖 𝑗 Δ𝜔𝑗𝑖. As such, the lower-bound on 𝛼′imposed by documents 𝑑𝑖and𝑑𝑗of query𝑞,𝐿′ 𝑖 𝑗(𝑞), is smaller than the lower-bound on 𝛼,𝐿𝑖 𝑗(𝑞). Like𝛼, this case does not additionally constrain 𝛼′from above (i.e., the upper-bound does not change: 𝑈′

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the upper-bound does not change: 𝑈′ 𝑖 𝑗(𝑞)=𝑈𝑖 𝑗(𝑞)=1). Case 2: Δ𝜙𝑖 𝑗<0,Δ𝜔𝑗𝑖<0. Once again, due to monotonicity, it is easy to see that Δ𝜙′ 𝑖 𝑗<0and Δ𝜔′ 𝑗𝑖<0. Equation (6) tells us that, for the order to be preserved under 𝑓′ Convex, we must similarly have that: 𝛼′<1/(1+Δ𝜙′ 𝑖 𝑗 Δ𝜔′ 𝑗𝑖). Once again, by assumption we have that the upper-bound on 𝛼′is a translation of the upper-bound on𝛼to the left. The lower-bound is unaffected and remains 0. For𝑓′

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For𝑓′ Convexto induce the same order as 𝑓Convex among all pairs of documents for all queries in 𝑄, the intersection of the intervals produced by the constraints on 𝛼′has to be non-empty: 𝐼′≜Ù 𝑞[max 𝑖 𝑗𝐿′ 𝑖 𝑗(𝑞),min 𝑖 𝑗𝑈′ 𝑖 𝑗(𝑞)]=[max 𝑞,𝑖 𝑗𝐿′ 𝑖 𝑗(𝑞),min 𝑞,𝑖 𝑗𝑈′ 𝑖 𝑗(𝑞)]≠∅. We next prove that 𝐼′is always non-empty to conclude the proof of the theorem.

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We next prove that 𝐼′is always non-empty to conclude the proof of the theorem. By Equation (6) and the existence of 𝛼, we know that max 𝑞,𝑖 𝑗𝐿𝑖 𝑗(𝑞)≤min 𝑞,𝑖 𝑗𝑈𝑖 𝑗(𝑞). Suppose that documents 𝑑𝑖and𝑑𝑗of query𝑞1maximize the lower-bound, and that documents 𝑑𝑚and𝑑𝑛of query𝑞2minimize the upper-bound. We therefore have that: 1/(1+Δ𝜙𝑖 𝑗 Δ𝜔𝑗𝑖)≤1/(1+Δ𝜙𝑚𝑛 Δ𝜔𝑛𝑚)=⇒Δ𝜙𝑖 𝑗 Δ𝜔𝑗𝑖≥Δ𝜙𝑚𝑛

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Δ𝜔𝑛𝑚 Because of the uniformity of the relative expansion rate, we can deduce that: Δ𝜙′ 𝑖 𝑗 Δ𝜔′ 𝑗𝑖≥Δ𝜙′ 𝑚𝑛 Δ𝜔′𝑛𝑚=⇒max 𝑞,𝑖 𝑗𝐿′ 𝑖 𝑗(𝑞)≤min 𝑞,𝑖 𝑗𝑈′ 𝑖 𝑗(𝑞). □ It is easy to show that the theorem above also holds when the condition is updated to reflect a shift of lower- and upper-bounds to the right, which happens when 𝜙′contracts with respect to 𝜙 more rapidly than 𝜔′does with respect to 𝜔. The picture painted by Theorem 4.5 is that switching from min-max scaling to z-score nor-

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malization or any other linear transformation that is bounded and does not severely distort the distribution of scores, especially among the top-ranking documents, results in a rank-equivalent function. At most, for any given value of the ranking metric of interest such as NDCG, we should observe a shift of the weight in the convex combination to the right or left. Figure 3 illustrates this

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111:12 Sebastian Bruch, Siyu Gai, and Amir Ingber (a) MS MARCO (b)Quora (c)HotpotQA (d)FiQA Fig. 3. Effect of normalization on the performance of 𝑓Convex as a function of 𝛼on the validation set. effect empirically on select datasets. As anticipated, the peak performance in terms of NDCG shifts to the left or right depending on the type of normalization. The uniformity requirement on the relative expansion rate, 𝜆, in Theorem 4.5 is not as strict and restrictive as it may appear. First, it is only necessary for 𝜆to be stable on the set of ordered pairs of documents as ranked by 𝑓Convex : |Δ𝜙′ 𝑖 𝑗|/|Δ𝜙𝑖 𝑗| |Δ𝜔′

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|Δ𝜔′ 𝑗𝑖|/|Δ𝜔𝑗𝑖|=𝜆,∀(𝑑𝑖,𝑑𝑗)st𝑓Convex(𝑑𝑖)>𝑓Convex(𝑑𝑗). Second, it turns out, close to uniformity (i.e., when 𝜆isconcentrated around one value) is often sufficient for the effect to materialize in practice. We observe this phenomenon empirically by fixing the parameter 𝛼in𝑓Convex with one transformation and forming ranked lists, then choosing another transformation and computing its relative expansion rate 𝜆on all ordered pairs of documents. We show the measured relative expansion rate in Figure 4 for various transformations. Figure 4 shows that most pairs of transformations yield a stable relative expansion rate. For example, if𝑓Convex uses𝜙tmmand𝑓′

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example, if𝑓Convex uses𝜙tmmand𝑓′ Convexuses𝜙mm—denoted by 𝜙tmm→𝜙mm—for every choice of 𝛼,

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An Analysis of Fusion Functions for Hybrid Retrieval 111:13 (a) MS MARCO (b)Quora (c)HotpotQA (d)FiQA Fig. 4. Relative expansion rate of semantic scores with respect to lexical scores, 𝜆, when changing from one transformation to another, with 95% confidence intervals. Prior to visualization, we normalize values of 𝜆 to bring them into a similar scale—this only affects aesthetics and readability, but is the reason why the vertical axis is not scaled. For most transformations and every value of 𝛼, we observe a stable relative rate of expansion where 𝜆concentrates around one value for the vast majority of queries. the relative expansion rate 𝜆is concentrated around a constant value. This implies that any ranked list obtained from 𝑓Convex can be reconstructed by 𝑓′ Convex. Interestingly, 𝜙z−Lex→𝜙mm−Lexhas a comparatively less stable 𝜆, but removing normalization altogether (i.e., 𝜙mm−Lex→𝐼) dramatically

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distorts the expansion rates. This goes some way to explain why normalization and boundedness are important properties. In the last two sections, we have answered RQ1: Convex combination is an appropriate fusion function and its performance is not sensitive to the choice of normalization so long as the transfor- mation has reasonable properties. Interestingly, the behavior of 𝜙tmmappears to be more robust to the data distribution—its peak remains within a small neighborhood as we move from one dataset to another. We believe the reason 𝜙tmm-normalized scores are more stable is because it has one fewer data-dependent statistic in the transformation (i.e., minimum score in the retrieved set is

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111:14 Sebastian Bruch, Siyu Gai, and Amir Ingber Table 1. Recall@1000 and NDCG@1000 (except SciFact andNFCorpus where cutoff is 100) on the test split of various datasets for lexical and semantic search as well as hybrid retrieval using RRF [5] (𝜂=60) and TM2C2 ( 𝛼=0.8). The symbols‡and∗indicate statistical significance ( 𝑝-value <0.01) with respect to TM2C2 and RRF respectively, according to a paired two-tailed 𝑡-test. Recall NDCG Dataset Lex. Sem. TM2C2 RRF Lex. Sem. TM2C2 RRF Oraclein-domainMS MARCO 0.836‡∗0.964‡∗0.974 0.969‡0.309‡∗0.441‡∗0.454 0.425‡0.547

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NQ 0.886‡∗0.978‡∗0.985 0.984 0.382‡∗0.505‡0.542 0.514‡0.637 Quora 0.992‡∗0.999 0.999 0.999 0.800‡∗0.889‡∗0.901 0.877‡0.936zero-shotNFCorpus 0.283‡∗0.314‡∗0.348 0.344 0.298‡∗0.309‡∗0.343 0.326‡0.371 HotpotQA 0.878‡∗0.756‡∗0.884 0.888 0.682‡∗0.520‡∗0.699 0.675‡0.767

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FEVER 0.969‡∗0.931‡∗0.972 0.972 0.689‡∗0.558‡∗0.744 0.721‡0.814 SciFact 0.900‡∗0.932‡∗0.958 0.955 0.698‡∗0.681‡∗0.753 0.730‡0.796 DBPedia 0.540‡∗0.408‡∗0.564 0.567 0.415‡∗0.425‡∗0.512 0.489‡0.553 FiQA 0.720‡∗0.908 0.907 0.904 0.315‡∗0.467‡0.496 0.464‡0.561 replaced with minimum feasible value regardless of the candidate set). In the remainder of this work, we use 𝜙tmmand denote a convex combination of scores normalized by it by TM2C2 for

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brevity. 5 ANALYSIS OF RECIPROCAL RANK FUSION Chen et al. [ 5] show that RRF performs better and more reliably than a convex combination of normalized scores. RRF is computed as follows: 𝑓RRF(𝑞,𝑑)=1 𝜂+𝜋Lex(𝑞,𝑑)+1 𝜂+𝜋Sem(𝑞,𝑑), (7) where𝜂is a free parameter. The authors of [ 5] take a non-parametric view of RRF, where the parameter𝜂is set to its default value 60, in order to apply the fusion to out-of-domain datasets in a zero-shot manner. In this work, we additionally take a parametric view of RRF, where as we elaborate later, the number of free parameters is the same as the number of functions being fused together, a quantity that is always larger than the number of parameters in a convex combination.

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together, a quantity that is always larger than the number of parameters in a convex combination. Let us begin by comparing the performance of RRF and TM2C2 empirically to get a sense of their relative efficacy. We first verify whether hybrid retrieval leads to significant gains in in-domain and out-of-domain experiments. In a way, we seek to confirm the findings reported in [ 5] and compare the two fusion functions in the process. Table 1 summarizes our results. We note that, we set RRF’s𝜂to60per [ 5] but tuned TM2C2’s 𝛼 on the validation set of the in-domain datasets and found that 𝛼=0.8works well for the three datasets. In the experiments leading to Table 1, we fix 𝛼=0.8and evaluate methods on the test split of the datasets. Per [ 5,39], we have also included the performance of an oracle system that uses a per-query 𝛼, to establish an upper-bound—the oracle knows which value of 𝛼works best for any given query.

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any given query. Our results show that hybrid retrieval using RRF outperforms pure-lexical and pure-semantic retrieval on most datasets. This fusion method is particularly effective on out-of-domain datasets,

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An Analysis of Fusion Functions for Hybrid Retrieval 111:15 (a) in-domain (b) out-of-domain Fig. 5. Difference in NDCG@ 1000 of TM2C2 and RRF (positive indicates better ranking quality by TM2C2) as a function of 𝛼. When𝛼=0the model is rank-equivalent to lexical search while 𝛼=1is rank-equivalent to semantic search. rendering the observation of [ 5] a robust finding and asserting once more the remarkable perfor- mance of RRF in zeros-shot settings. Contrary to [ 5], however, we find that TM2C2 significantly outperforms RRF on all datasets in terms of NDCG, and does generally better in terms of Recall. Our observation is consistent with [39] that TM2C2 substantially boosts NDCG even on in-domain datasets. To contextualize the effect of 𝛼on ranking quality, we visualize a parameter sweep on the validation split of in-domain datasets in Figure 5(a), and for completeness, on the test split of

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out-of-domain datasets in Figure 5(b). These figures also compare the performance of TM2C2 with RRF by reporting the difference between NDCG of the two methods. These plots show that there always exists an interval of 𝛼for which𝑓TM2C2≻𝑓RRFwith≻indicating better rank quality. 5.1 Effect of Parameters Chen et al. rightly argue that because RRF is merely a function of ranks, rather than scores, it naturally addresses the scale and range problem without requiring normalization—which, as we showed, is not a consequential choice anyway. While that statement is accurate, we believe it introduces new problems that must be recognized too. The first, more minor issue is that ranks cannot be computed exactly unless the entire collection Dis ranked by retrieval system ofor every query. That is because, there may be documents that appear in the union set, but not in one of the individual top- 𝑘sets. Their true rank is therefore unknown, though is often approximated by ranking documents within the union set. We take this approach when computing ranks.

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approach when computing ranks. The second issue is that, unlike TM2C2, RRF ignores the raw scores and discards information about their distribution. In this regime, whether or not a document has a low or high semantic score does not matter so long as its rank in 𝑅𝑘 Semstays the same. It is arguable in this case whether rank is a stronger signal of relevance than score, a measurement in a metric space where distance matters greatly. We intuit that, such distortion of distances may result in a loss of valuable information that would lead to better final ranked lists.

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111:16 Sebastian Bruch, Siyu Gai, and Amir Ingber (a) MS MARCO (b)Quora (c)NQ (d)FiQA (e)HotpotQA (f)Fever Fig. 6. Visualization of the reciprocal rank determined by lexical ( 𝑟𝑟(𝜋Lex)=1/(60+𝜋Lex)) and semantic (𝑟𝑟(𝜋Sem)=1/(60+𝜋Sem)) retrieval for query-document pairs sampled from the validation split of each dataset. Shown in red are up to 20,000positive samples where document is relevant to query, and in black up to the same number of negative samples. To understand these issues better, let us first repeat the exercise in Section 4.1 for RRF. In Figure 6, we have plotted the reciprocal rank (i.e., 𝑟𝑟(𝜋o)=1/(𝜂+𝜋o)with𝜂=60) for sampled query-

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document pairs as before. From the figure, we can see that samples are pulled towards one of the poles at(0,0)and(1/61,1/61). The former attracts a higher concentration of negative samples while the latter positive samples. While this separation is somewhat consistent across datasets,

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An Analysis of Fusion Functions for Hybrid Retrieval 111:17 (a) MS MARCO (b)HotpotQA Fig. 7. Difference in NDCG@1000 of 𝑓RRFwith distinct values 𝜂Lexand𝜂Sem, and𝑓RRFwith𝜂Lex=𝜂Sem=60 (positive indicates better ranking quality by the former). On MS MARCO, an in-domain dataset, NDCG improves when 𝜂Lex>𝜂Sem, while the opposite effect can be seen for HotpotQA , an out-of-domain dataset. the concentration around poles and axes changes. Indeed, on HotpotQA andFever there is a higher concentration of positive documents near the top, whereas on FiQA and the in-domain datasets more positive samples end up along the vertical line at 𝑟𝑟(𝜋Sem)=1/61, indicating that lexical ranks matter less. This suggests that a simple addition of reciprocal ranks does not behave consistently across domains.

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consistently across domains. We argued earlier that RRF is parametric and that it, in fact, has as many parameters as there are retrieval functions to fuse. To see this more clearly, let us rewrite Equation (7) as follows: 𝑓RRF(𝑞,𝑑)=1 𝜂Lex+𝜋Lex(𝑞,𝑑)+1 𝜂Sem+𝜋Sem(𝑞,𝑑). (8) We study the effect of parameters on 𝑓RRFby comparing the NDCG obtained from RRF with a particular choice of 𝜂Lexand𝜂Semagainst a realization of RRF with𝜂Lex=𝜂Sem=60. In this way, we are able to visualize the impact on performance relative to the baseline configuration that is typically used in the literature. This difference in NDCG is rendered as a heatmap in Figure 7 for select datasets—figures for all other datasets show a similar pattern. As a general observation, we note that NDCG swings wildly as a function of RRF parameters.

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Crucially, performance improves off-diagonal, where the parameter takes on different values for the semantic and lexical components. On MS MARCO, shown in Figure 7(a), NDCG improves when 𝜂Lex>𝜂Sem, while the opposite effect can be seen for HotpotQA , an out-of-domain dataset. This can be easily explained by the fact that increasing 𝜂ofor retrieval system oeffectively discounts the contribution of ranks from oto the final hybrid score. On in-domain datasets where the semantic model already performs strongly, for example, discounting the lexical system by increasing 𝜂Lex leads to better performance. Having observed that tuning RRF potentially leads to gains in NDCG, we ask if tuned parameters generalize on out-of-domain datasets. To investigate that question, we tune RRF on in-domain datasets and pick the value of parameters that maximize NDCG on the validation split of in-domain datasets, and measure the performance of the resulting function on the test split of all (in-domain and out-of-domain) datasets. We present the results in Table 2. While tuning a parametric RRF does

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111:18 Sebastian Bruch, Siyu Gai, and Amir Ingber (a)𝜂Lex=60,𝜂Sem=60 (b)𝜂Lex=10,𝜂Sem=4 (c)𝜂Lex=3,𝜂Sem=5 Fig. 8. Effect of 𝑓RRFwith select configurations of 𝜂Lexand𝜂Semon pairs of ranks from lexical and semantic systems. When 𝜂Lex>𝜂Sem, the fusion function discounts the lexical system’s contribution. Table 2. Mean NDCG@1000 (NDCG@100 for SciFact andNFCorpus ) on the test split of various datasets for hybrid retrieval using TM2C2 ( 𝛼=0.8) and RRF (𝜂Lex,𝜂Sem). The symbols‡and∗indicate statistical significance ( 𝑝-value <0.01) with respect to TM2C2 and baseline RRF ( 60,60) respectively, according to a paired two-tailed 𝑡-test. NDCG

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paired two-tailed 𝑡-test. NDCG Dataset TM2C2 RRF(60,60)RRF(5,5)RRF(10,4)in-domainMS MARCO 0.454 0.425‡0.435‡∗0.451∗ NQ 0.542 0.514‡0.521‡∗0.528‡∗ Quora 0.901 0.877‡0.885‡∗0.896∗zero-shotNFCorpus 0.343 0.326‡0.335‡∗0.327‡ HotpotQA 0.699 0.675‡0.693∗0.621‡∗ FEVER 0.744 0.721‡0.727‡∗0.649‡∗ SciFact 0.753 0.730‡0.738‡0.715‡∗

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DBPedia 0.512 0.489‡0.489‡0.480‡∗ FiQA 0.496 0.464‡0.470‡∗0.482‡∗ indeed lead to gains in NDCG on in-domain datasets, the tuned function does not generalize well to out-of-domain datasets. The poor generalization can be explained by the reversal of patterns observed in Figure 7 where 𝜂Lex>𝜂Semsuits in-domain datasets better but the opposite is true for out-of-domain datasets. By modifying𝜂Lexand𝜂Semwe modify the fusion of ranks and boost certain regions and discount others in an imbalanced manner. Figure 8 visualizes this effect on 𝑓RRFfor particular values of its parameters. This addresses RQ2. 5.2 Effect of Lipschitz Continuity In the previous section, we stated an intuition that because RRF does not preserve the distribution of raw scores, it loses valuable information in the process of fusing retrieval systems. In our final

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of raw scores, it loses valuable information in the process of fusing retrieval systems. In our final research question, RQ3, we investigate if this indeed matters in practice.

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An Analysis of Fusion Functions for Hybrid Retrieval 111:19 (a) in-domain (b) out-of-domain Fig. 9. The difference in NDCG@1000 of 𝑓SRRF and𝑓RRFwith𝜂=60(positive indicates better ranking quality bySRRF ) as a function of 𝛽. The notion of “preserving” information is well captured by the concept of Lipschitz continuity.6 When a function is Lipschitz continuous with a small Lipschitz constant, it does not oscillate wildly with a small change to its input. RRF does not have this property because the moment one lexical (or semantic) score becomes larger than another the function makes a hard transition to a new value. We can therefore cast RQ3 as a question of whether Lipschitz continuity is an important property in practice. To put that hypothesis to the test, we design a smooth approximation of RRF using known techniques [4, 30]. As expressed in Equation (1), the rank of a document is simply the sum of indicators. It is

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thus trivial to approximate this quantity using a generalized sigmoid with parameter 𝛽:𝜎𝛽(𝑥)= 1/(1+exp(−𝛽𝑥)). As𝛽approaches 1, the sigmoid takes its usual Sshape, while 𝛽→∞ produces a very close approximation of the indicator. Interestingly, the Lipschitz constant of 𝜎𝛽(·)is, in fact, 𝛽. As𝛽increases, the approximation of ranks becomes more accurate, but the Lipschitz constant becomes larger. When 𝛽is too small, however, the approximation breaks down but the function transitions more slowly, thereby preserving much of the characteristics of the underlying data distribution. RRF being a function of ranks can now be approximated by plugging in approximate ranks in Equation (7), resulting in SRRF: 𝑓SRRF(𝑞,𝑑)=1 𝜂+˜𝜋Lex(𝑞,𝑑)+1

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𝜂+˜𝜋Lex(𝑞,𝑑)+1 𝜂+˜𝜋Sem(𝑞,𝑑), (9) where ˜𝜋o(𝑞,𝑑𝑖)=0.5+Í 𝑑𝑗∈𝑅𝑘o(𝑞)𝜎𝛽(𝑓o(𝑞,𝑑𝑗)−𝑓o(𝑞,𝑑𝑖)). By increasing 𝛽we increase the Lipschitz constant of 𝑓SRRF. This is the lever we need to test the idea that Lipschitz continuity matters and that functions that do not distort the distributional properties of raw scores lead to better ranking quality.

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that functions that do not distort the distributional properties of raw scores lead to better ranking quality. 6A function 𝑓is Lipschitz continous with constant 𝐿if||𝑓(𝑦)−𝑓(𝑥)||𝑜≤𝐿||𝑦−𝑥||𝑖for some norm||·|| 𝑜and||·|| 𝑖 on the output and input space of 𝑓.

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111:20 Sebastian Bruch, Siyu Gai, and Amir Ingber (a) in-domain (b) out-of-domain Fig. 10. The difference in NDCG@1000 of 𝑓SRRF and𝑓RRFwith𝜂=5(positive indicates better ranking quality bySRRF ) as a function of 𝛽. Figures 9 and 10 visualize the difference between SRRF and RRF for two settings of 𝜂selected based on the results in Table 2. As anticipated, when 𝛽is too small, the approximation error is large and ranking quality degrades. As 𝛽becomes larger, ranking quality trends in the direction ofRRF. Interestingly, as 𝛽becomes gradually smaller, the performance of SRRF improves over theRRF baseline. This effect is more pronounced for the 𝜂=60setting of RRF, as well as on the out-of-domain datasets. While we acknowledge the possibility that the approximation in Equation (9) may cause a change

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While we acknowledge the possibility that the approximation in Equation (9) may cause a change in ranking quality, we expected that change to be a degradation, not an improvement. However, given we do observe gains by smoothing the function, and that the only other difference between SRRF and RRF is their Lipschitz constant, we believe these results highlight the role of Lipschitz continuity in ranking quality. For completeness, we have also included a comparison of SRRF, RRF, and TM2C2 in Table 3. 6 DISCUSSION The analysis in this work motivates us to identify and document the properties of a well-behaved fusion function, and present the principles that, we hope, will guide future research in this space. These desiderata are stated below. Monotonicity : When𝑓ois positively correlated with a target ranking metric (i.e., ordering documents in decreasing order of 𝑓omust lead to higher quality), then it is natural to require that𝑓Hybrid be monotone increasing in its arguments. We have already seen and indeed used this

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property in our analysis of the convex combination fusion function. It is trivial to show why this property is crucial. Homogeneity : The order induced by a fusion function must be unaffected by a positive re- scaling of query and document vectors. That is: 𝑓Hybrid(𝑞,𝑑)𝜋=𝑓Hybrid(𝑞,𝛾𝑑)𝜋=𝑓Hybrid(𝛾𝑞,𝑑)where 𝜋=denotes rank-equivalence and 𝛾>0. This property prevents any retrieval system from inflating its contribution to the final hybrid score by simply boosting its document or query vectors.

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An Analysis of Fusion Functions for Hybrid Retrieval 111:21 Table 3. Mean NDCG@1000 (NDCG@100 for SciFact andNFCorpus ) on the test split of various datasets for hybrid retrieval using TM2C2 ( 𝛼=0.8),RRF (𝜂), and SRRF( 𝜂,𝛽). The parameters 𝛽are fixed to values that maximize NDCG on the validation split of in-domain datasets. The symbols ‡and∗indicate statistical significance ( 𝑝-value <0.01) with respect to TM2C2 and RRF respectively, according to a paired two-tailed 𝑡-test. NDCG Dataset TM2C2 RRF(60) SRRF ( 60,40)RRF(5) SRRF ( 5,100)in-domainMS MARCO 0.454 0.425‡0.431‡∗0.435‡0.431‡∗

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NQ 0.542 0.514‡0.516‡0.521‡0.517‡ Quora 0.901 0.877‡0.889‡∗0.885‡0.889‡∗zero-shotNFCorpus 0.343 0.326‡0.338‡∗0.335‡0.339‡ HotpotQA 0.699 0.675‡0.695∗0.693‡0.705‡∗ FEVER 0.744 0.721‡0.725‡0.727‡0.735‡∗ SciFact 0.753 0.730‡0.740‡0.738‡0.740‡ DBPedia 0.512 0.489‡0.501‡∗0.489‡0.492‡ FiQA 0.496 0.464‡0.468‡0.470‡0.469‡

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Boundedness : Recall that, a convex combination without score normalization is often ineffective and inconsistent because BM25 is unbounded and that lexical and semantic scores are on different scales. To see this effect we turn to Figure 11. We observe in Figure 11(a) that, for in-domain datasets, adding the unnormalized lexical scores using a convex combination leads to a severe degradation of ranking quality. We believe this is because of the fact that the semantic retrieval model, which is fine-tuned on these datasets, already produces ranked lists of high quality, and that adding the lexical scores which are on a very different scale distorts the rankings and leads to poor performance. In out-of-domain experiments as shown in Figure 11(b), however, the addition of lexical scores leads to often significant gains in quality. We believe this can be explained exactly as the in-domain observations: The semantic model generally does poorly on out-of-domain datasets while the lexical retriever does well. But because the semantic scores are bounded and relatively small, they do not significantly distort the rankings produced by the lexical retriever.

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produced by the lexical retriever. To avoid that pitfall, we require that 𝑓Hybrid be bounded:|𝑓Hybrid|≤𝑀for some𝑀>0. As we have seen before, normalizing the raw scores addresses this issue. Lipschitz Continuity : We argued that because RRF does not take into consideration the raw scores, it distorts their distribution and thereby loses valuable information. On the other hand, TM2C2 (or any convex combination of scores) is a smooth function of scores and preserves much of the characteristics of its underlying distribution. We formalized this idea using the notion of Lipschitz continuity: A larger Lipschitz constant leads to a larger distortion of retrieval score distribution. Interpretability and Sample Efficiency : The question of hybrid retrieval is an important topic inIR. What makes it particularly pertinent is its zero-shot applicability, a property that makes deep models reusable , reducing computational costs and emissions as a result [ 3,34], and enabling resource-constrained research labs to innovate. Given the strong evidence supporting the idea that

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resource-constrained research labs to innovate. Given the strong evidence supporting the idea that hybrid retrieval is most valuable when applied to out-of-domain datasets [ 5], we believe that 𝑓Hybrid should be robust to distributional shifts and should not need training or fine-tuning on target

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111:22 Sebastian Bruch, Siyu Gai, and Amir Ingber (a) in-domain (b) out-of-domain Fig. 11. The difference in NDCG of convex combination of unnormalized scores and a pure semantic search (positive indicates better ranking quality by a convex combination) as a function of 𝛼. datasets. This implies that either the function must be non-parametric, that its parameters can be tuned efficiently with respect to the training samples required, or that they are highly interpretable such that their value can be guided by expert knowledge. In the absence of a truly non-parametric approach, however, we believe a fusion that is more sample-efficient to tune is preferred. Because convex combination has fewer parameters than the fully parameterized RRF, we believe it should have this property. To confirm, we ask how many training queries it takes to converge to the correct 𝛼on a target dataset. Figure 12 visualizes our experiments, where we plot NDCG of RRF (𝜂=60) and TM2C2 with

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𝛼=0.8from Table 1. Additionally, we take the train split of each dataset and sample from it progressively larger subsets (with a step size of 5%), and use it to tune the parameters of each function. We then measure NDCG of the tuned functions on the test split. For the depicted datasets as well as all other datasets in this work, we observe a similar trend: With less than 5%of the training data, which is often a small set of queries, TM2C2’s 𝛼converges, regardless of the magnitude of domain shift. This sample efficiency is remarkable because it enables significant gains with little labeling effort. Finally, while RRF does not settle on a value and its parameters are sensitive to the training sample, its performance does more or less converge. However, the performance of the fully parameterized RRF is still sub-optimal compared with TM2C2. In Figure 12, we also include a convex combination of fully parameterized RRF terms, denoted by RRF-CC and defined as:

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by RRF-CC and defined as: 𝑓RRF(𝑞,𝑑)=(1−𝛼)1 𝜂Lex+𝜋Lex(𝑞,𝑑)+𝛼1 𝜂Sem+𝜋Sem(𝑞,𝑑), (10) where𝛼,𝜂Lex, and𝜂Semare tunable parameters. The question this particular formulation tries to answer is whether adding an additional weight to the combination of the RRF terms affects retrieval quality. From the figure, it is clear that the addition of this parameter does not have a significant impact on the overall performance. This also serves as additional evidence supporting the claim that Lipschitz continuity is an important property.

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An Analysis of Fusion Functions for Hybrid Retrieval 111:23 (a) MS MARCO (b)Quora (c)HotpotQA (d)Fever Fig. 12. Sample efficiency of TM2C2 and the parameterized variants of RRF (single parameter where 𝜂Sem= 𝜂Lex, and two parameters where we allow different values of 𝜂Semand𝜂Lex, and a third variation that is a convex combination of RRF terms defined in Equation 10). We sample progressively larger subsets of the validation set (with a step size of 5%), tune the parameters of each function on the resulting set, and evaluate the resulting function on the test split. These figures depict NDCG@1000 as a function of the size of the tuning set, averaged over 5trials with the shaded regions illustrating the 95% confidence intervals. For reference, we have also plotted NDCG on the test split for RRF (𝜂=60) and TM2C2 with 𝛼=0.8from Table 1. 7 CONCLUSION

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7 CONCLUSION We studied the behavior of two popular functions that fuse together lexical and semantic retrieval to produce hybrid retrieval, and identified their advantages and pitfalls. Importantly, we inves- tigated several questions and claims in prior work. We established theoretically that the choice of normalization is not as consequential as once thought for a convex combination-based fusion function. We found that RRF is sensitive to its parameters. We also observed empirically that convex combination of normalized scores outperforms RRF on in-domain and out-of-domain datasets—a finding that is in disagreement with [5].

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111:24 Sebastian Bruch, Siyu Gai, and Amir Ingber We believe that a convex combination with theoretical minimum-maximum normalization (TM2C2) indeed enjoys properties that are important in a fusion function. Its parameter, too, can be tuned sample-efficiently or set to a reasonable value based on domain knowledge. In our experiments, for example, we found the range 𝛼∈[0.6,0.8]to consistently lead to improvements. While we observed that a line appears to be appropriate for a collection of query-document pairs, we acknowledge that that may change if our analysis was conducted on a per-query basis—itself a rather non-trivial effort. For example, it is unclear if bringing non-linearity to the design of the fusion function or the normalization itself leads to a more accurate prediction of 𝛼on a per-query basis. We leave an exploration of this question to future work. We also note that, while our analysis does not exclude the use of multiple retrieval engines as input, and indeed can be extended, both theoretically and empirically, to a setting where we have

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input, and indeed can be extended, both theoretically and empirically, to a setting where we have more than just lexical and semantic scores, it is nonetheless important to conduct experiments and validate that our findings generalize. We believe, however, that our current assumptions are practical and are reflective of the current state of hybrid search where we typically fuse only lexical and semantic retrieval systems. As such, we leave an extended analysis of fusion on multiple retrieval systems to future work. ACKNOWLEDGMENTS We benefited greatly from conversations with Brian Hentschel, Edo Liberty, and Michael Bendersky. We are grateful to them for their insight and time. REFERENCES [1] Nima Asadi. 2013. Multi-Stage Search Architectures for Streaming Documents . University of Maryland. [2]Nima Asadi and Jimmy Lin. 2013. Effectiveness/Efficiency Tradeoffs for Candidate Generation in Multi-Stage Retrieval Architectures. In Proceedings of the 36th International ACM SIGIR Conference on Research and Development in Information Retrieval (Dublin, Ireland). 997–1000.

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