Title: Query-Efficient Correlation Clustering with Noisy Oracle
Venue: NeurIPS 2024
Accepted: yes
Keywords: Correlation Clustering; Online Learning; Pure Exploration of Multi-armed Bandits
Abstract: We study a general clustering setting in which we have $n$ elements to be clustered, and we aim to perform as few queries as possible to an oracle that returns a noisy sample of the weighted similarity between two elements. Our setting encompasses many application domains in which the similarity function is costly to compute and inherently noisy. We introduce two novel formulations of online learning problems rooted in the paradigm of Pure Exploration in Combinatorial Multi-Armed Bandits (PE-CMAB): fixed confidence and fixed budget settings. For both settings, we design algorithms that combine a sampling strategy with a classic approximation algorithm for correlation clustering and study their theoretical guarantees. Our results are the first examples of polynomial-time algorithms that work for the case of PE-CMAB in which the underlying offline optimization problem is NP-hard.

Query-Efficient Correlation Clustering with Noisy Oracle

Abstract

We study a general clustering setting in which we have n elements to be clustered, and we aim to perform as few queries as possible to an oracle that returns a noisy sample of the weighted similarity between two elements. Our setting encompasses many application domains in which the similarity function is costly to compute and inherently noisy. We introduce two novel formulations of online learning problems rooted in the paradigm of Pure Exploration in Combinatorial Multi-Armed Bandits (PE-CMAB): fixed confidence and fixed budget settings. For both settings, we design algorithms that combine a sampling strategy with a classic approximation algorithm for correlation clustering and study their theoretical guarantees. Our results are the first examples of polynomial-time algorithms that work for the case of PE-CMAB in which the underlying offline optimization problem is NP-hard.

1 Introduction

Given a set V = [ n ] of n objects and a pairwise similarity measure s : ( V 2 ) → [0 , 1] (where ( V 2 ) is the set of unordered pairs of elements of V , and the value closer to 1 means higher similarity), the goal of Correlation Clustering [7] is to cluster the objects so that, to the best possible extent, similar objects are put in the same cluster and dissimilar objects are put in different clusters. Assuming that cluster identifiers are represented by natural numbers, a clustering C can be represented as a function glyph[lscript] : V → N , where each cluster is a maximal set of objects sharing the same label. The objective is to minimize the following cost:

glyph[negationslash]

The intuition underlying the above problem definition is that if two objects x and y are dissimilar, expressed by a small value of s( x, y ) , yet they are assigned to the same cluster, we should incur a high cost. Conversely, if s( x, y ) is high, indicating that x and y are very similar, but they are assigned to different clusters, we should also incur a high cost.

Two key features make correlation clustering quite suitable in real-world applications. Firstly, it does not require the number of clusters as part of the input; instead, it automatically finds the optimal number, performing model selection. Secondly, it only requires the pairwise information without

assuming any specific structure of the data. This reasonably eliminates the need for domain knowledge about complex data. Correlation clustering has been applied to a wide range of problems across various domains, including duplicate detection and similarity joins [34, 46], spam detection [12, 72], co-reference resolution [66], biology [9, 14], image segmentation [54], social network analysis [13], and clustering aggregation [42].

Correlation clustering is NP-hard even in the simplest formulations [7, 75], and minimizing the cost function in (1) is APX-hard [20]; thus, we cannot expect a polynomial-time approximation scheme. Nevertheless, there are a number of constant-factor approximation algorithms for various settings [1, 3, 7, 20, 22, 30-32]. For the formulation of (1), Ailon et al. [3] presented KwikCluster , a simple 5-approximation algorithm. The algorithm randomly picks a pivot v ∈ V and constructs a cluster by taking all the vertices similar to v ; then, the algorithm removes the cluster and repeats the process until V is fully clustered. The simplicity and theoretical guarantees of KwikCluster have produced a lot of variations in different scenarios [13, 29, 61, 69, 76, 79].

In practice, preparing the similarity function involves costly measurements . Given n items to be clustered, Θ( n 2 ) similarity computations are needed to prepare the input to correlation clustering algorithms. Moreover, computing the similarity s( x, y ) might have additional expenses (e.g., human effort or financial resources) besides the mere computational cost. To mitigate these issues, some query-efficient methods have been proposed based on the active learning framework [11, 15, 40]. In this framework, the similarity function is initially unknown but an oracle that returns the true similarity in { 0 , 1 } for a pair of objects is sequentially queried. In particular, these studies provided a randomized algorithm that, given a budget T of queries, attains a solution whose expected cost is at most 3 · OPT+ O ( n 3 T ) , where OPT is the optimal value of the problem.

However, the above methods for query-efficient correlation clustering have significant limitations. Indeed, all the aforementioned works [11, 15, 40] only consider the binary similarity of { 0 , 1 } , while the similarity between two objects are often non-binary in real-world scenarios. For example, in biological sciences, protein-protein interaction networks are commonly analyzed, where the strength of the interactions among proteins is represented as a real-valued similarity [68]. As another example, in entity resolution, i.e., a task central to data integration [80], real-valued similarity is used to indicate the likelihood of matches of objects instead of binary decisions. Therefore, allowing the similarity to be real-valued in the interval [0 , 1] would be more practical and flexible. Furthermore, the above works assume the access to the strong oracle that returns the true value of s( x, y ) ( = 0 or 1 ), while evaluating s( x, y ) might be inherently noisy , due to error-prone experiments, noisy measurements, or biased judgments. In the above first example the strength of the interactions among proteins is often measured based on biological experiments involving unavoidable noise, while in the second example the likelihood of matches of objects is usually obtained based on biased human judgements.

In this paper, we focus on the challenging scenario where (i) the underlying similarity measure can take any real value in [0 , 1] rather than being binary, and (ii) we can only query a noisy oracle that provides inaccurate evaluations of the weighted similarity s( x, y ) . The goal of this paper is to devise clustering algorithms that perform as few queries on s( x, y ) as possible to an oracle that returns noisy answers to s( x, y ) . In pursuit of this goal, we introduce two novel formulations based on multi-armed bandits problems, both of which achieve a reasonable trade-off between the number of queries to the oracle and the quality of solutions.

While our problem formulations are novel, recent prior work has explored related issues. Silwal et al. [76] proposed a practical model using the strong oracle along with a cheaper but inaccurate oracle. Their algorithm achieves a cost of 3 · OPT+ glyph[epsilon1]n 2 using n + O ( γ glyph[epsilon1] ) queries to the strong oracle, where γ > 0 is the error level of noisy oracle and glyph[epsilon1] > 0 is the additive error. However, they still focus on the binary similarity and there is no guarantee on the query upper bound for the noisy oracle. Unlike theirs, our models are designed to handle the weighted similarity and do not rely on any strong oracle. Aronsson and Chehreghani [4, 5] studied a non-persistent noise model where the oracle returns the true value of s( x, y ) with probability 1 -γ and a noisy value otherwise. Their algorithm handles a general weighted similarity but provides neither query complexity nor approximation guarantee.

1.1 Our contributions

In this paper, we study the problem of query-efficient correlation clustering with noisy oracles , where the similarity function s : ( V 2 ) → [0 , 1] is initially unknown , and only noisy feedback instead of

the true similarity s( x, y ) is observed when querying a pair of objects ( x, y ) . In this scenario, it is desired to achieve a reasonable trade-off between the number of queries to the oracle and the cost of clustering. To this end, we introduce two formulations of online learning problems rooted in the paradigm of Pure Exploration of Combinatorial Multi-Armed Bandits (PE-CMAB). In the fixed confidence setting (Problem 1), given a confidence level δ ∈ (0 , 1) , the learner aims to find a well-approximate solution with probability at least 1 -δ while minimizing the number of queries required to determine the output. Conversely, in the fixed budget setting (Problem 2), given a querying budget T , the learner aims to maximize the probability that the output is a well-approximate solution. Our contributions can be summarized as follows:

For Problem 1, we design KC-FC (Algorithm 1), which effectively combines threshold bandits with KwikCluster . We prove that given confidence level δ ∈ (0 , 1) , KC-FC finds a solution whose expected cost is at most 5 · OPT+ glyph[epsilon1] with probability at least 1 -δ , where OPT is the optimal value of the problem, and provide the upper bound of the number of queries (Theorem 1).

We design KC-FB (Algorithm 3) for Problem 2, which adaptively determines the number of queries for each pair of objects based on KwikCluster . We prove that the error probability of the expected cost being worse than 5 · OPT+ glyph[epsilon1] decreases exponentially with budget T (Theorem 2).

We empirically validate our theoretical findings by demonstrating that KC-FC and KC-FB outperform baseline methods in terms of the sample complexity and cost of clustering, respectively.

It is worth noting that our approximation guarantees in Theorems 1 and 2 match the approximation ratio 5 of KwikCluster [3], where s : ( V 2 ) → [0 , 1] is known in advance, up to the additive error glyph[epsilon1] > 0 . These results are not achievable using existing PE-CMAB algorithms due to the NP-hardness of correlation clustering. In the standard PE-CMAB, a learner aims to identify the best action that maximizes the linear reward from the combinatorial decision set D ⊆ 2 [ m ] with m -base arms. Existing algorithms for PE-CMAB (e.g., [23, 25, 35, 52, 82]) rely on the assumption that the offline problem is polynomial-time solvable. Redesigning them to obtain a well-approximate solution while running efficiently is quite challenging, as the exact optimization of the offline problem is crucial to achieving statistical validity and a correctness guarantee for the output. Ours are the first polynomialtime algorithms that work for the case of PE-CMAB where the underlying offline optimization is NP-hard, filling a critical gap in existing PE-CMAB algorithms, which is of independent interest.

1.2 Related work

Correlation clustering with noisy input. The bulk of the literature on noisy correlation clustering (see Section 4.6 of Bonchi et al. [10]) considers the binary similarity and assumes that there is the ground-truth clustering but some of the s( x, y ) are wrong: they are 0 instead of 1, or vice versa. The seminal work by Bansal et al. [7] and Joachims and Hopcroft [49] provided the bounds on the error with which correlation clustering recovers the ground truth under a simple probabilistic model over graphs. Mathieu and Schudy [64] studied the model starting from an arbitrary partition of the n elements into clusters, where s( x, y ) is perturbed independently with probability p , and a more general model with the adversary. They proposed an algorithm that achieves some approximation ratio and manages to approximately recover the ground truth. Chen et al. [28] extended the framework to sparse Erd˝ os-Rényi random graphs and obtained an algorithm that conditionally recovers the ground truth. Finally, Makarychev et al. [62] overcame some limitations of Mathieu and Schudy [64] and Chen et al. [28]; they assumed very little about the observations and gave two approximation algorithms. Unlike the above models, ours is based on online learning with an unknown distribution with mean of s( x, y ) , which is in general not binary, and does not assume any ground-truth clustering.

Combinatorial multi-armed bandits. Multi-Armed Bandit (MAB) is a classical decision-making model [57, 59, 73]: There are m possible actions (called arms ), whose expected reward µ i for each i ∈ [ m ] is unknown. At each round, a learner chooses an arm to pull and observes a stochastic reward sampled from an unknown probability distribution. The most popular objective is to minimize the cumulative regret [16, 18]. Another popular objective is to identify the arm with the maximum expected reward. This problem, called the Best Arm Identification (BAI) or Pure Exploration (PE) in MAB, has also received much attention [6, 8, 17, 21, 24, 37, 38, 41, 48, 53]. The model of Combinatorial Multi-Armed Bandits (CMAB) is a generalization of MAB [19, 26], where an interested subset of arms forms a certain combinatorial structure such as a spanning tree, matching, or path. Since its introduction by Chen et al. [25], the study of PE-CMAB has been actively pursued

in various settings [23, 35, 36, 47, 50, 52, 55, 56, 67, 78, 82]. Notably, Gullo et al. [44] addressed regret minimization for correlation clustering by adapting UCB-type algorithms. However, regret minimization in CMAB is quite different from pure exploration framework when working with approximation oracles (i.e., offline approximation algorithms) for solving NP-hard problems. For regret minimization, we can incorporate approximation oracles with the UCB framework, consistent with the optimization under uncertainty principle (e.g., [26, 27, 81]). However, in pure exploration, the lack of uniqueness of α -approximate solutions makes it difficult to determine the stopping condition in the FC setting. In the FB setting, the Combinatorial Successive Accept Reject algorithm proposed by Chen et al. [25] iteratively solves the so-called Constrained Oracle problem, which is often NP-hard, as later addressed in Du et al. [35]. We anticipate a similar NP-hard problem in correlation clustering, requiring a different approach.

Other clustering settings. Ailon et al. [2] and Saha and Subramanian [74] studied correlation clustering with same-cluster queries , where all similarities of ( V 2 ) are known in advance and their query is further allowed to access the optimal clustering. Our setting differs significantly as we are interested in the case where similarities are unknown and only noisy similarity values are received rather than same-cluster queries. Finally, it is worth mentioning that Xia and Huang [83] and Gupta et al. [45] proposed a MAB approach for clustering reconstruction with noisy same-cluster queries [58, 65, 70, 71, 77]. However, this clustering reconstruction problem does not directly offer any algorithmic result for correlation clustering. The detailed comparison is deferred to Appendix A.

2 Problem statements

Here we formally define our formulations of PE-CMAB for correlation clustering. Our problem instances are characterized by ( V, s) , where V = [ n ] is the set of elements to be clustered and s : ( V 2 ) → [0 , 1] is the pairwise similarity function, which is unknown to the learner. Define the set of unordered pairs as E = ( V 2 ) with m := | E | .

At each round t = 1 , 2 , . . . , a learner will pull (i.e., query) one arm (i.e., pair of elements in V ) from action space E = ( V 2 ) based on past observations. After pulling e ∈ E , the learner can observe the random feedback X t ( e ) , which is independently sampled from an unknown distribution such as Bernoulli or R -sub-Gaussian with unknown mean s( e ) ∈ [0 , 1] . 1 After some exploration rounds, the learner must identify a well-approximate solution. Let OPT(s) be the optimal value of the offline problem minimizing the cost function (1) and let C out be the output by an algorithm. For α ≥ 1 and glyph[epsilon1] > 0 , we say C out to be an ( α, glyph[epsilon1] ) -approximate solution if cost s ( C out ) ≤ α · OPT(s) + glyph[epsilon1] . We study the following two formulations: Fixed Confidence (FC) and Fixed Budget (FB) settings.

Problem 1 (Fixed confidence setting) . Let α ≥ 1 . Given a confidence level δ ∈ (0 , 1) and additive error glyph[epsilon1] > 0 , the learner aims to guarantee that the output C out is an ( α, glyph[epsilon1] ) -approximate solution with probability at least 1 -δ . The evaluation metric of an algorithm is the sample complexity, i.e., the number of queries to the oracle the learner uses.

Problem 2 (Fixed budget setting) . Let α ≥ 1 . Given a querying budget T and additive error glyph[epsilon1] > 0 , the learner aims to maximize the probability that the output C out is an ( α, glyph[epsilon1] ) -approximate solution.

Note that the case of α = 1 corresponds to the standard PE-CMAB formulations. However, as the offline problem minimizing the cost function (1) is APX-hard [20], we cannot expect any polynomialtime algorithm that can handle α = 1 in the above formulations.

3 Fixed confidence setting

In this section, we design KC-FC (Algorithm 1) for Problem 1, built on a novel combination of KwikCluster (detailed in Algorithm 4 in Appendix B) and techniques of threshold bandits. The key idea of the proposed method is to first identify pairs with seemingly high similarity, which are then passed to KwikCluster to produce a high-quality clustering.

For the first phase, we leverage one of the variants of MAB, called the threshold bandits [51, 60, 63], which is defined as follows: Given a confidence level δ and m -arms, the learner must return the set of good arms, i.e., arms whose expected rewards are greater than a given threshold θ > 0 , as soon as possible, and stops when the learner believes that there is no remaining good arm, w.p. at least 1 -δ . TB-HS (detailed in Algorithm 2) is our key procedure, which is designed for identifying seemingly high similarity pairs. Note that, if we naively use the existing algorithm by Kano et al. [51] for threshold bandits where the set of arms is E = ( V 2 ) and the threshold is θ = 0 . 5 , the algorithm is not even guaranteed to terminate; the resulting sample complexity becomes infinitely large if s( e ) = s( e ′ ) for different e, e ′ ∈ E or if there exists e ∈ E with s( e ) = 0 . 5 , which may frequently happen in practice. Our strategy to avoid such an unbounded sample complexity is to allow TB-HS to misidentify pairs of elements with similarity close to 0.5, taking advantage of the fact that the output accuracy can be guaranteed despite such misidentification.

Algorithm details. Let ̂ s t ( e ) be the empirical mean of the similarity for each pair e ∈ E kept at round t . Let N t ( e ) be the number of queries of e ∈ E that has been pulled by the end of round t . TB-HS maintains the confidence bound defined as rad t ( e ) := √ log(4 mN t ( e ) 2 /δ ) 2 N t ( e ) for each e ∈ E . The arm selection at round t is based on the Lower-Confidence-Bound (LCB) score, i.e., s t ( e ) := ̂ s t ( e ) -rad t ( e ) and the Upper-Confidence-Bound (UCB) score, i.e., s t ( e ) := ̂ s t ( e )+rad t ( e ) . We pull the arm ˆ e g t with the highest LCB (line 5) and the arm ˆ e b t with the lowest UCB (line 6). Then ˆ e g t will be added to ̂ G glyph[epsilon1] if its LCB is no less than 0 . 5 -glyph[epsilon1] , and ˆ e b t will be added to ̂ B glyph[epsilon1] if its UCB is no greater than 0 . 5 + glyph[epsilon1] . TB-HS continues this procedure until every e ∈ E is added to either ̂ G glyph[epsilon1] or ̂ B glyph[epsilon1] . Our main algorithm KC-FC invokes TB-HS to compute ̂ G glyph[epsilon1] ′ with parameter glyph[epsilon1] ′ = glyph[epsilon1] 12 m . Then it carries out KwikCluster using the predicted similarity by ̂ G glyph[epsilon1] ′ as follows. Until an unclustered element exists, it picks one pivot element p r uniformly at random, builds a cluster C r around it by adding those among the unclustered elements that seemingly have high similarity with a pivot p r (based on ̂ G glyph[epsilon1] ′ ), and removes all the elements in C r from the list of unclustered elements.

Analysis. For a given glyph[epsilon1] ∈ (0 , 0 . 5) , we define the following sets, which appear only in the theoretical analysis and are unknown to the learner: E [0 . 5 ± glyph[epsilon1] ] := { e ∈ E : | 0 . 5 -s( e ) | ≤ glyph[epsilon1] } , E (0 . 5+ glyph[epsilon1], 1] := { e ∈ E : s( e ) > 0 . 5 + glyph[epsilon1] } , and E [0 , 0 . 5 -glyph[epsilon1] ) := { e ∈ E : s( e ) < 0 . 5 -glyph[epsilon1] } . For glyph[epsilon1] ∈ (0 , 0 . 5) , we introduce the definition of the gaps that characterize our sample complexity:

where ∆ e := | s( e ) -0 . 5 | for e ∈ [ m ] and ∆ min := min e ∈ [ m ] ∆ e .

Now we present our theorem, guaranteeing that KC-FC finds a (5 , glyph[epsilon1] ) -approximate solution with high probability and provides an upper bound of the number of queries, i.e., the sample complexity.

Theorem 1. Given a confidence level δ ∈ (0 , 1) and additive error glyph[epsilon1] > 0 , KC-FC (Algorithm 1) guarantees that

and letting glyph[epsilon1] ′ = glyph[epsilon1] 12 m , the sample complexity T is

Furthermore, KC-FC runs in time polynomial in n .

Proof Sketch. For the outputs ̂ G glyph[epsilon1] and ̂ B glyph[epsilon1] of TB-HS (Algorithm 2) with parameters glyph[epsilon1] ∈ (0 , 0 . 5) and δ ∈ (0 , 1) , by using the Hoeffding inequality and the procedure of TB-HS (lines 8 and 10), it is easy to see that E (0 . 5+ glyph[epsilon1], 1] ⊆ ̂ G glyph[epsilon1] and E [0 , 0 . 5 -glyph[epsilon1] ) ⊆ ̂ B glyph[epsilon1] w.p. at least 1 -δ . Consider the similarity function ˜ s : E → [0 , 1] such that for each e ∈ E , ˜ s( e ) = s( e ) if e ∈ E [0 , 0 . 5 -glyph[epsilon1] ) ∪ E (0 . 5+ glyph[epsilon1], 1] , and ˜ s e otherwise, where ˜ s e is an arbitrary value that satisfies | s( e ) -˜ s e | < 2 glyph[epsilon1] . Noticing that KCFC corresponds to KwikCluster associated with a certain choice of ˜ s (i.e., ˜ s e for e ∈ E [0 . 5 ± glyph[epsilon1] ] ), we can show that E [cost s ( C out )] ≤ 5 · OPT (s) + 12 glyph[epsilon1] | E [0 . 5 ± glyph[epsilon1] ] | for the output C out , providing the approximation guarantee. The rest of the proof requires the analysis of the upper bound of the number of queries that TB-HS used to stop. This can be done based on a prior analysis of threshold bandits [51], while carefully handling glyph[epsilon1] > 0 . The complete proof for analysis is given in Appendix C.

For the time complexity, each iteration of sub-routine TB-HS takes O ( m ) steps in a naive implementation or amortized O (log T ) steps if we manage arms using two heaps corresponding to LCB/UCB values, and the other procedure in KC-FC runs in time polynomial in n .

Comparison with existing PE-CMAB methods in the FC setting. Existing methods for PECMAB (e.g., [23, 25, 35, 82]) are limited by their reliance on the polynomial-time solvability of the underlying offline problem. If we use an efficient approximation algorithm in those existing methods, their stopping conditions no longer have a guarantee of the quality of the output. Specifically, such existing methods use the LUCB-type strategy, and its stopping condition requires the exact computation of the empirical best solution and the second empirical best solution to check if the current estimation is enough or not. When we only have an approximate oracle (i.e., approximation algorithm), such existing stopping conditions are no longer valid, and the algorithm is not guaranteed to stop. In contrast, KC-FC runs in time polynomial in n while ensuring sample complexity and approximation guarantee. We also note that ∆ e , the distance between s( e ) and 0 . 5 , interestingly characterizes our sample complexity, as we show that the learning task boils down to identifying E (0 . 5+ glyph[epsilon1] ′ , 1] and E [0 , 0 . 5 -glyph[epsilon1] ′ ) thanks to the behavior of KC-FC - they leverage the property that by accurately estimating the mean of the base arms (i.e., pairs of elements), we can maintain the approximation guarantee of KwikCluster in the offline setting with small additive error.

Statistical efficiency. In the noise-free setting, ( n 2 ) queries are sufficient, while in the noisy setting, there is even no trivial upper bound on the sample complexity to achieve some desired approximation guarantee (e.g., our (5 , glyph[epsilon1] ) -approximation). Note that the value of ˜ ∆ e,glyph[epsilon1] defined in (2) always has the following lower bound: ˜ ∆ e,glyph[epsilon1] = ∆ e + glyph[epsilon1]/ 2 ( > 0) if glyph[epsilon1]/ 2 ≥ ∆ min holds and ˜ ∆ e,glyph[epsilon1] = ∆ e + glyph[epsilon1] -∆ min ≥ glyph[epsilon1] ( > 0) otherwise. Therefore, our sample complexity T given in Theorem 1 is always bounded, contrasting existing results for threshold bandits [51]. The naive sampling algorithm ( Uniform-FC in Appendix E) requires O ( n 6 glyph[epsilon1] 2 log n δ ) samples to achieve the (5 , glyph[epsilon1] ) -approximation w.p. at least 1 -δ . KC-FC achieves a much better sample complexity than Uniform-FC , as ∑ e ∈ E ˜ ∆ -2 e,glyph[epsilon1] ′ = ∑ e ∈ E (∆ e + glyph[epsilon1] ′ 2 ) -2 glyph[lessmuch] n 6 glyph[epsilon1] 2 when ∆ min ≤ glyph[epsilon1] ′ 2 glyph[lessmuch] ∆ e for most e ∈ E , which is often the case in practice. To the best of our knowledge, lower bounds on the sample complexity related to PE-CMAB are known only for the following settings: (i) the time complexity of algorithms can be exponential, or (ii) the underlying offline problem is assumed to be polynomial-time solvable and to have the unique correct (namely optimal) solution [25, 35, 39]. Deriving an effective lower bound on the number of samples required to guarantee an approximate solution is particularly challenging because it necessitates dealing with multiple correct solutions [33], while most existing approaches rely on the uniqueness of the correct solution. Evaluating the necessity of the second term n 2 max { ∆ min , glyph[epsilon1] ′ 2 } 2 and investigating a lower bound for our case are crucial and remain important future work. However, it is worth noting that the additional term is independent of a dominating term involving log 1 δ .

Remark. If we utilize TB-HS within the loop (Algorithm 5 in Appendix B), the algorithm achieves (5 , glyph[epsilon1] ) -approximation guarantee with probability at least 1 -δ , and the sample complexity T is:

where k is the total number of loops in Algorithm 5, glyph[epsilon1] ′ r := glyph[epsilon1]/ (12 | I V r ( p r ) | ) , I V r ( p r ) ⊆ E represents the set of pairs between the pivot p r selected in phase r and its neighbors in V r , and ∆ min ,r := min e ∈ I Vr ( p r ) ∆ e . When k glyph[lessmuch] n , the above sample complexity can be better than that of Theorem 1. However, it should be noted that the symbols related to r and the total number of loops k , especially instance-dependent gaps ˜ ∆ e,glyph[epsilon1] ′ r , are all random variables. In contrast, the current Theorem 1 does not contain any random variables. Specifically, the significant term related to log δ -1 is characterized by the gap ˜ ∆ e,glyph[epsilon1] or ∆ e , which represents the distance from 0.5 and not a random variable.

4 Fixed budget setting

In this section, we investigate Problem 2 and design KC-FB (Algorithm 3). KC-FB is inspired by the successive reject algorithm [6] and exploits KwikCluster to determine the number of queries for each pair adaptively.

Algorithm. KC-FB proceeds in at most n phases and maintains the subset of elements V r ⊆ V in each phase r ∈ [ n ] starting with V 1 = V . We denote the set of pairs that can be formed with v in V r by I V r ( v ) := {{ v, u } ∈ ( V r 2 ) : u ∈ V r } . In each phase r , the algorithm chooses the pivot p r uniformly at random from V r , and pulls each e ∈ I V r ( p r ) for appropriately determined τ r times. Based on the empirical mean ̂ s r ( e ) := ∑ τ r k =1 X k ( e ) /τ r for each e ∈ I V r ( p r ) , it finds one cluster C r = { p r } ∪ Γ V r ( p r , ̂ s r ) , where Γ V r ( p r , ̂ s r ) := { u ∈ V r : ̂ s r ( p r , u ) > 0 . 5 } , and updates V r +1 ← V r \ C r . This procedure will be continued until | V r | = 0 and finally the algorithm outputs C out consisting of all clusters computed. Updating the number of pulls τ r (line 8) is a key to prove the statistical property. Intuitively, τ r represents a pre-fixed budget of queries when e ∈ ( V r 2 ) would be pulled: In the initial phase, we allocate τ 1 := glyph[floorleft] T/m glyph[floorright] to each e ∈ ( V 1 2 ) . Notice that the surplus, the sum of the pre-fixed budgets of pairs that have been removed without being queried, is τ 1 · ( | ( V 1 2 ) | - | ( V 2 2 ) | -( | V 1 | -1) ) , because the number of pairs that have been removed in this phase is | ( V 1 2 ) | - | ( V 2 2 ) | , and among those pairs, the number of pairs that have been actually pulled by the algorithm is ( | V 1 | -1) . This surplus is additionally redistributed equally to each e ∈ ( V 2 2 ) . This will be also done for the remaining phases r = 2 , . . . , n .

Algorithm 3 KwikCluster with Fixed Budget ( KC-FB )

Input : Budget T > 0 , set V of n objects, additive error

Analysis. The following theorem states that KC-FB outputs a well-approximate solution with high probability. The proof of Theorem 2 is deferred to Appendix D.

Theorem 2. For glyph[epsilon1] > 0 , define the minimal gap ∆ min ,glyph[epsilon1] as

Then, KC-FB (Algorithm 3) uses at most T queries to output C out that satisfies

Assuming that each query takes O (1) time, the time complexity of KC-FB is O ( T + n 2 ) .

Proof Sketch. We can show the random event Pr[ ⋂ n r =1 E r ] occurs with high probability, where E r := {∀ e ∈ I V r ( p r ) , | s( e ) -̂ s r ( e ) | < max { glyph[epsilon1], ∆ e }} for each phase r ∈ [ n ] (See Lemma 8 in Appendix D.1). Under the assumption of such estimation success event ⋂ n r =1 E r , by utilizing the unique feature of KwikCluster , we can maintain the approximation guarantee of KwikCluster in the noise-free setting up to additive error (See Lemma 9 in Appendix D.2). Simply combining these lemmas with adjusted parameter glyph[epsilon1] ′ ∈ (0 , 0 . 5) , defined as glyph[epsilon1] 6 max { 1 , | E [0 . 5 ± glyph[epsilon1] ] |} if glyph[epsilon1] < 0 . 5 and glyph[epsilon1] 6 m otherwise, will conclude the proof (See Appendix D.3 for details).

The parameter δ ∈ (0 , 1) in (3) represents the error probability of C out being worse than any (5 , glyph[epsilon1] ) -approximate solution, and it decays exponentially to the querying budget T . A larger parameter ∆ min ,glyph[epsilon1] provides the better guarantee; KC-FB performs better when the similarity function clearly expresses similarity ( +1 ) or dissimilarity ( -1 ), as min e ∈ E ∆ e tends to be large.

To evaluate the significance of our results, we analyze the uniform sampling algorithm ( Uniform-FB in Appendix E); Uniform-FB queries each e ∈ E uniformly glyph[floorleft] T/m glyph[floorright] times to obtain ̂ s( e ) , and then applies any α -approximation algorithm to instance ( V, ̂ s) of the offline problem minimizing (1). We see that the error probability that the output is not an ( α, glyph[epsilon1] ) -approximate solution is bounded by O ( n 2 exp ( -Tglyph[epsilon1] 2 α 2 n 6 )) . In contrast, KC-FB adaptively allocates the budget to the remaining pairs, which enables us to query essential pairs of elements, i.e., pairs whose estimated similarity values affect the behavior of cluster construction, more times than glyph[floorleft] T/m glyph[floorright] . This leads to a better performance in the cost of clustering in practice (see Section 5).

Comparison with existing PE-CMAB methods in the FB setting. In the literature of PE-CMAB, the FB setting presents even more computational challenges and a scarcity of theoretical results. The current state-of-the-art algorithms [6, 25, 36] suffer from one or more of the following issues: (i) inability to handle a partition structure in correlation clustering, (ii) requiring exponential running time, and (iii) lacking any approximation guarantees when the underlying problem is NP-hard. By leveraging the properties of KwikCluster , our approach ensures the polynomial-time complexity of O ( T + n 2 ) while guaranteeing that the probability of obtaining a well-approximate solution exponentially increases with the budget T , along with instance-dependent analysis.

5 Experimental evaluation

In this section, we evaluate the performance of our proposed algorithms, KC-FC and KC-FB , using various datasets, providing empirical evidence to support our theoretical findings.

Datasets. We use publicly-available realworld graphs presented in Table 1. In the FC setting, to observe the behavior of the sample complexity with respect to the hidden minimum gap ∆ min in (2), we generate our instances as follows. For each graph, we vary the lower bound on ∆ min , which we denote by LB ∆ min , in { 0 . 10 , 0 . 15 , 0 . 20 , . . . , 0 . 50 } . For each pair

of vertices u, v , we set s( u, v ) = uniform[0 . 5 + LB ∆ min , 1] if u, v have an edge in the graph, and s( u, v ) = uniform[0 , 0 . 5 -LB ∆ min ] otherwise, where uniform[ a, b ] is the value drawn from the interval [ a, b ] uniformly at random. On the other hand, in the FB setting, we employ a more realistic setting: For each graph, our problem instance is generated by embedding the vertices into a d -dimensional Euclidean space using node2vec [43], obtaining a vector vec( v ) ∈ R d for each vertex v . Specifically, we used the publicly-available Python module of node2vec 2 with default parameter settings (particularly d = 64 ). Then, define the unknown similarity of each pair of vertices u, v as s( u, v ) = sim cos (vec( u ) , vec( v )) -min _ cos max _ cos -min _ cos ∈ [0 , 1] , where min _ cos and max _ cos are the minimal and maximal cosine similarities, respectively, among all pairs of vertices. We note that max _ cos > min _ cos holds for all instances. In all experiments, noisy feedback when querying a pair e ∈ E is generated by a Bernoulli distribution with mean s( e ) .

Baselines. We compare our methods with Uniform-FC in the FC setting and Uniform-FB in the FB setting, whose pseudocode and full analysis are given in Appendix E. Uniform-FC pulls each e ∈ E for glyph[ceilingleft] 18 m 2 glyph[epsilon1] 2 log 2 m δ glyph[ceilingright] times and employs KwikCluster with respect to the empirical similarity, while Uniform-FB is its adaption to the FB setting. Moreover, we compare the cost of clustering of our algorithms with that of KwikCluster having access to the unknown (true) similarity, which is regarded as the stronger baseline than other KwikCluster -based methods for the binary case [11, 15, 40, 76].

Machine and code. The experiments were performed on a machine with Apple M1 Chip and 16 GB RAM. The code was written in Python 3, which is available online. 3

Performance of KC-FC . We evaluate the performance of algorithms in terms of not only the cost of clustering but also the sample complexity. In both KC-FC and Uniform-FC , we set glyph[epsilon1] = √ n allowing each element to make only 1 / √ n mistakes, and δ = 0 . 01 following a standard choice in PE-MAB. Taking into account the limited scalability of the algorithms, we only use the instances with n < 1,000. In particular, as will be shown later, Uniform-FC requires a large number of samples, which makes the algorithm prohibitive even for quite small instances. Therefore, we do not run the algorithm and just report the sample complexity, which can be calculated without running it. For each LB ∆ min , we run both KC-FC and KwikCluster having access to the unknown similarity 100 times and report the average value and the standard deviation.

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The results are depicted in Figures 1 and 2. As can be seen, the sample complexity of KC-FC is much smaller than that of Uniform-FC . In fact, the sample complexity of Uniform-FC makes the algorithm

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prohibitive even for very small instances. Moreover, consistent with the theoretical analysis, as (the lower bound LB ∆ min on) ∆ min increases, the sample complexity of KC-FC becomes smaller. This desirable property is not possessed by Uniform-FC . Remarkably, looking at Figure 2, we see that KC-FC outputs a clustering whose quality is comparable with that of KwikCluster having access to the unknown similarity.

Performance of KC-FB . Here we evaluate the performance of KC-FB . For small instances with n < 1,000, we vary T in { n 2 . 1 , n 2 . 2 , . . . , n 3 . 0 } and observe the cost of clustering with respect to the budget T . For large instances with n ≥ 1,000, we fix T = n 2 . 2 for scalability. For each instance and T , we run both KC-FB and Uniform-FB 100 times and report the average value and the standard deviation. As KwikCluster having access to the unknown similarity is independent of T , we just run it 100 times for each instance.

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The results are shown in Figure 3 and Table 2. As can be seen, KC-FB outperforms the baseline method Uniform-FB . In fact, for all instances and almost all values of T , KC-FB outputs a better clustering than that of Uniform-FB . We can see that this superiority comes from the fact that KC-FB estimates the unknown similarity better than Uniform-FB thanks to its sophisticated sampling strategy. Indeed, KwikCluster having access to the unknown similarity showcases the best performance, verifying the importance of the precise estimation of the unknown similarity.

6 Conclusions

We studied the online learning problems of correlation clustering, where the similarity function is initially unknown and only noisy feedback is observed. For the FC setting, we devised KC-FC and proved the upper bound of the number of queries required to find a clustering whose cost is at most 5 · OPT+ glyph[epsilon1] with high probability. For the FB setting, we devised KC-FB and showed that the error probability of the expected cost being worse than 5 · OPT+ glyph[epsilon1] decays exponentially with budget T . Importantly, our algorithms are the first examples of PE-CMAB with NP-hard offline problems. One future work, yet a significant challenge, is to derive information-theoretic lower bounds of PE-CMAB in the case where the offline problem is NP-hard. Investigating other variants of correlation clustering or exploring the case where the variance of random feedback differs across pairs, namely heteroscedastic noise, would also be worthwhile directions.

Acknowledgment

The work of Yuko Kuroki is supported by Japan Science and Technology Agency (JST) Strategic Basic Research Programs PRESTO 'R&D Process Innovation by AI and Robotics: Technical Foundations and Practical Applications' grant number JPMJPR24T2, and was partially supported by Microsoft Research Asia and JST Strategic Basic Research Programs ACT-X grant number JPMJAX200E while she was at The University of Tokyo. The authors would like to thank the anonymous reviewers for their insightful comments and useful feedback.

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Appendix

A Additional comparison with other clustering models

Cluster recovering with noisy same-cluster queries. Another line of studies has focused on clustering reconstruction with noisy same-cluster queries, which was first proposed by Mazumdar and Saha [65] and further investigated by Larsen et al. [58], Peng and Zhang [70], Pia et al. [71], Tsourakakis et al. [77] and Xia and Huang [83]. In this model, given a set of n elements, the goal is to recover the underlying ground-truth k -clustering by asking pairwise queries to an oracle, which tells us if the two elements belong to the same cluster, but whose answer is correct only with probability 1 2 + δ 2 . Recently, Gupta et al. [45] first considered a noisy and inconsistent oracle, in contrast to a consistent oracle that returns the same answer when queried. Although this clustering reconstruction problem shares the intuition with correlation clustering, there are also important differences which do not allow to transfer algorithmic results from one to the other. Firstly, in correlation clustering the input information might be inconsistent (e.g., a is very similar to b , which is very similar to c , but a is not similar to c ), instead in clustering reconstruction this is not possible: if a is in the same cluster of b and b is in the same cluster of c , then a is in the same cluster of c . Secondly, the aim is to reconstruct the exact underlying clustering, while we aim at minimizing the cost function in (1). Lastly and more importantly, the number of clusters k is part of the input to the problem, while in correlation clustering it is unknown. Therefore, the theoretical results and techniques for solving the clustering reconstruction problem cannot be directly applied to correlation clustering.

Query-based correlation clustering. As fully discussed in the main text, our work lies in queryefficient correlation clustering, for which we utilize the methodology of PE-CMAB. Table 3 summarizes how existing query-based settings differ from our attempt. Note that the oracle in Aronsson and Chehreghani [4, 5] is assumed to return the true value of s( x, y ) with probability 1 -γ and a noisy value with probability γ , which is different from our models; we only observe a random variable independently sampled from an unknown distribution with mean s( x, y ) ∈ [0 , 1] . While one might consider using majority voting on repeated queries to handle noise when the underlying distribution is Bernoulli, this approach lacks any approximation guarantees and query complexity bounds. Moreover, for R -sub-Gaussian noise, majority voting is not well-defined. Instead, using sample mean estimates, as done in PE-MAB methods, is standard. Our approach leverages these principles, ensuring a (5 , glyph[epsilon1] ) -approximation guarantee with fewer queries and statistical guarantees.

B Pseudocode of KwikCluster and Algorithm 5

We detail the pseudocode of KwikCluster [3] in Algorithm 4. The approximation guarantee of KwikCluster is 5 for the weighted similarity case (and 3 for the restricted binary similarity case). We also detail the pseudocode of Algorithm 5, which sequentially uses TB-HS at each phase in the framework of KC-FC .

Algorithm 5 KC-FC variant with sequential use of TB-HS

C Analysis of KC-FC

In this section, we provide a complete proof of Theorem 1 in Section 3. In particular, Lemma 4 guarantees the accuracy of the subroutine TB-HS , and based on this, Lemma 5 assures the (5 , glyph[epsilon1] ) -approximation using the properties of KwikCluster . Moreover, Lemma 6 establishes an upper bound crucial for the sample complexity via novel analysis dependent on glyph[epsilon1] and ∆ min (Lemma 7). Finally, by combining the sample complexity required by subroutine TB-HS (Lemma 6) and the output guarantee of KC-FC (Lemma 5), Theorem 1 is demonstrated.

C.1 Basic lemmas

We first introduce the Hoeffding inequality, which will be frequently used in our proof. Note that we consider Bernoulli distribution for the sake of simplicity, but our results would carry on for R -sub-Gaussian distribution by simply adjusting the Hoeffding inequality for the case accordingly.

Lemma 1 (Hoeffding inequality for bounded random variables) . Let X 1 , . . . , X k be k independent random variables such that, E [ X i ] = µ and a ≤ X i ≤ b for each i ∈ [ k ] . Let ¯ X = 1 k ∑ k i =1 X i denote the average of these random variables. Then, for any λ > 0 , we have

The next lemma presents the probability that some random event happens, which will be used later.

Lemma 2. Let ̂ s e,k be the empirical mean of the rewards when e has been pulled k times. For each e ∈ [ m ] and k , define the random event E e,k as follows:

Let E k be the random event that for all e ∈ [ m ] , the random event E e,k happens. Then we have

Proof of Lemma 2. We have

where the first inequality follows from the Hoeffding inequality (Lemma 1). Therefore, by taking the union bound, we have

C.2 Approximation guarantee

We provide the following lemmas for guaranteeing the quality of the output.

Lemma 3. Let glyph[epsilon1] ∈ (0 , 0 . 5) and δ ∈ (0 , 1) . TB-HS (Algorithm 2) with parameter glyph[epsilon1], δ outputs, with probability at least 1 -δ , ̂ G glyph[epsilon1] and ̂ B glyph[epsilon1] such that

Proof of Lemma 3. The proof is almost straightforward by the procedure of the algorithm, line 8 and 10, as follows. By Lemma 2, we have Pr[ ⋂ ∞ k =1 E k ] ≥ 1 -δ . Now we assume that ⋂ ∞ k =1 E k happens. Let t > 0 be the stopping time, where every e ∈ E has been added to either ̂ G glyph[epsilon1] or ̂ B glyph[epsilon1] . For e ∈ ̂ G glyph[epsilon1] , from the stopping condition and the random event ⋂ ∞ k =1 E k , it is easy to see that s( e ) ≥ ̂ s t ′ ( e ) -rad t ′ ( e ) ≥ 0 . 5 -glyph[epsilon1] , where t ′ denotes the round that arm e was added to ̂ G glyph[epsilon1] . For e ∈ ̂ B glyph[epsilon1] , it is also easy to see that s( e ) ≤ ̂ s t ′ ( e ) + rad t ′ ( e ) ≤ 0 . 5 + glyph[epsilon1] , where t ′ denotes the round that arm e was added to ̂ B glyph[epsilon1] . The third condition is obvious from the stopping condition of the algorithm.

Lemma 4. Let glyph[epsilon1] ∈ (0 , 0 . 5) and δ ∈ (0 , 1) . Let ̂ G glyph[epsilon1] and ̂ B glyph[epsilon1] be the output of TB-HS (Algorithm 2) with parameters glyph[epsilon1], δ . Then, with probability at least 1 -δ , we have that (i) every e ∈ E (0 . 5+ glyph[epsilon1], 1] is included in ̂ G glyph[epsilon1] , and (ii) every e ∈ E [0 , 0 . 5 -glyph[epsilon1] ) is included in ̂ B glyph[epsilon1] .

Proof of Lemma 4. We have Pr[ ⋂ ∞ k =1 E k ] ≥ 1 -δ by Lemma 2 again, and we assume that ⋂ ∞ k =1 E k happens. Consider any e ∈ E (0 . 5+ glyph[epsilon1], 1] . Suppose that e is not included in ̂ G glyph[epsilon1] . Then, from Lemma 3, we see that e ∈ ̂ B glyph[epsilon1] , and thus s( e ) ≤ 0 . 5+ glyph[epsilon1] , which contradicts the fact that e ∈ E (0 . 5+ glyph[epsilon1], 1] . Therefore, e is included in ̂ G glyph[epsilon1] . Similarly, consider any e ∈ E [0 , 0 . 5 -glyph[epsilon1] ) . Suppose that e is not included in ̂ B glyph[epsilon1] . Then, from Lemma 3, we see that e ∈ ̂ G glyph[epsilon1] , and thus s( e ) ≥ 0 . 5 -glyph[epsilon1] , which contradicts the fact that e ∈ E [0 , 0 . 5 -glyph[epsilon1] ) . Therefore, e is included in ̂ B glyph[epsilon1] .

Based on Lemma 4, we prove the following key lemma.

Lemma 5 (Approximation guarantee) . Let glyph[epsilon1] ∈ (0 , 0 . 5) and δ ∈ (0 , 1) . With probability at least 1 -δ , the output C out of KC-FC (Algorithm 1), where subroutine TB-HS (Algorithm 2) is invoked with parameters glyph[epsilon1], δ , is a (5 , 12 glyph[epsilon1] | E [0 . 5 ± glyph[epsilon1] ] | ) -approximate solution for instance ( V, s) of the offline problem minimizing (1) .

Proof of Lemma 5. By Lemma 4, we have E (0 . 5+ glyph[epsilon1], 1] ⊆ ̂ G glyph[epsilon1] and E [0 , 0 . 5 -glyph[epsilon1] ) ⊆ ̂ B glyph[epsilon1] w.p. at least 1 -δ . Construct the similarity function ˜ s : E → [0 , 1] such that for each e ∈ E ,

where ˜ s e is an arbitrary value that satisfies | s( e ) -˜ s e | < 2 glyph[epsilon1] . Consider running KwikCluster with the similarity ˜ s . Let C ′ out be the output of this algorithm. Then we have

Noticing that KC-FC corresponds to the above algorithm associated with a certain choice of ˜ s (i.e., ˜ s e for e ∈ E [0 . 5 ± glyph[epsilon1] ] ), we have the lemma.

C.3 Sample complexity analysis and proof of Theorem 1

We prove the following main lemma to evaluate the sample complexity of TB-HS . Let m g be the number of pairs (i.e., arms) whose similarity is no less than the threshold 0 . 5 . Without loss of generality, we assume that E = [ m ] indexed as s(1) ≥ · · · ≥ s( m g ) ≥ 0 . 5 > s( m g +1) ≥ · · · ≥ s( m ) in whole analysis.

Lemma 6 (Sample complexity) . The upper bound of the sample complexity of TB-HS (Algorithm 2) with parameters glyph[epsilon1] ∈ (0 , 0 . 5) and δ ∈ (0 , 1) is

To prove Lemma 6, we begin with the following lemma. Recall that ˜ ∆ e,glyph[epsilon1] and ∆ min are defined by (2).

Lemma 7. Let glyph[epsilon1] ∈ (0 , 0 . 5) and δ ∈ (0 , 1) . Define

Let s e,k := ̂ s e,k -√ log(4 mk 2 /δ ) 2 k , and s e,k := ̂ s e,k + √ log(4 mk 2 /δ ) 2 k , where ̂ s e,k is the empirical mean of the rewards when e has been pulled k times. If k ≥ k e holds, then

It also holds that

Proof of Lemma 7. Suppose that

Then, for each e ∈ [ m g ] , we have

where the last inequality follows from the Hoeffding equality (Lemma 1). Now we show, via a similar analysis of Lemma 2 in Kano et al. [51], that for k ≥ k e , it indeed holds that

Let c e := ( ∆ e +min { glyph[epsilon1] -∆ min , glyph[epsilon1] 2 }) 2 for simplicity, Then we can rewrite k ≥ k e as

for some t ≥ log 5 m/δ c e > 1 . Then we have

where e is the base of natural logarithms and log t ≤ t -1 is used. Therefore, t ≥ log 5 √ m/δ c e is sufficient to fulfill (5).

The second statement of Lemma 7 can easily be shown by adapting the proof of Lemma 3 in Kano et al. [51]. For each e ∈ [ m g ] , we have

For e ∈ [ m ] \ [ m g ] , we omit the proof, as the analysis is essentially the same as the case for e ∈ [ m g ] .

Proof of Lemma 6. Let a ( t ) ∈ ( V 2 ) denote the selected pair (i.e., arm) by the algorithm in round t . Then we have

where the third inequality follows from the fact that event { a ( t ) = e, N t ( e ) = k } occurs for at most one t ∈ N . For e ∈ [ m g ] , we have

Similarly, for each e ∈ [ m ] \ [ m g ] , we have

which concludes the proof.

Proof of Theorem 1. Finally, we are ready to complete the proof of Theorem 1. In KC-FC , TB-HS is run with parameter glyph[epsilon1] ′ = glyph[epsilon1] 12 m and confidence δ . Therefore, by Lemma 5 for glyph[epsilon1] ′ , δ , we have the approximation guarantee:

The sample complexity of KC-FC is equal to that of TB-HS with parameters glyph[epsilon1] ′ , δ , which is given by Lemma 6 for glyph[epsilon1] ′ , δ .

D Analysis of KC-FB

In this section, we prove Theorem 2 in Section 4.

D.1 Basic analysis of some random event and its occurrence probability

The following lemma states that ̂ s r for phase r ∈ [ n ] is well-estimated with high probability. The proof is almost straightforward from the Hoeffding inequality (Lemma 1) and union bounds.

Lemma 8. Let glyph[epsilon1] ∈ (0 , 0 . 5) . Given a phase r ∈ [ n ] , we define the random event

Then, we have

Proof. Wefirst evaluate Pr[ | ̂ s r ( e ) -s( e ) | ≥ max { glyph[epsilon1], ∆ e } ] for a fixed phase r ∈ [ n ] and e ∈ I V r ( p r ) . Each e ∈ I V r ( p r ) has been pulled at least glyph[floorleft] T/m glyph[floorright] times because the initial budget for the pair was set to τ 1 = glyph[floorleft] T/m glyph[floorright] and the budget has not decreased in the later iterations. Then we have

where the second inequality follows from Lemma 1. Taking a union bound for r ∈ [ n ] and all e ∈ I V r ( p r ) , we further have

where the third inequality follows from (7).

D.2 Theoretical guarantee of the output

Next we prove a key lemma that provides the theoretical guarantee of the output C out of KC-FB . Lemma 9. Let glyph[epsilon1] ∈ (0 , 0 . 5) . Under the assumption that ⋂ n r =1 E r happens, the output C out of KC-FB is a (5 , 6 glyph[epsilon1] | E [0 . 5 ± glyph[epsilon1] ] | ) -approximate solution for instance ( V, s) of the offline problem minimizing (1) .

Proof. Let ̂ E ⊆ E be the set of pairs that have been pulled in the algorithm. For e = { u, v } ∈ ̂ E , let r e be the phase, in which either u or v is selected as a pivot. Construct the weight ˜ s : E → [0 , 1] such

that for each e ∈ E ,

Consider running KwikCluster (Algorithm 4) with the similarity ˜ s while respecting the selection of pivots p r of KC-FB , that is, in the r -th iteration, the algorithm selects the pivot p r if it exists. In the first iteration, the algorithm can select the pivot p 1 and construct the cluster { p 1 }∪ Γ V 1 ( p 1 , ˜ s) . By the definition of ˜ s and the assumption of the lemma, we have Γ V 1 ( p 1 , ˜ s) = Γ V 1 ( p 1 , ̂ s 1 ) . In fact, for any element u in V 1 (except for p 1 ), we see that ˜ s( p 1 , u ) > 0 . 5 if and only if ̂ s( p 1 , u ) > 0 . 5 . Therefore, the cluster produced is exactly the same as C 1 in KC-FB . In the second iteration, the algorithm can select p 2 because p 2 was not contained in the cluster of the first iteration, and by applying the same argument as above, we see that the cluster of this iteration is exactly the same as C 2 in KC-FB . The later iterations can be handled in the same way. Therefore, we see that the output of the above algorithm coincides with that of KC-FB .

Then it suffices to show that the output of the above algorithm has the desired approximation guarantee. Let C ′ out be the output of the above algorithm. Recalling that KC-FB picks pivot p t uniformly at random, we have

where the first and third inequalities follow from the fact that s( e ) and ˜ s( e ) may be different only for e ∈ E [0 . 5 ± glyph[epsilon1] ] ( ∩ ̂ E ) and the difference there is at most glyph[epsilon1] from the assumption of the lemma.

D.3 Proof of Theorem 2

Proof of Theorem 2. For glyph[epsilon1] > 0 , define glyph[epsilon1] ′ ∈ (0 , 0 . 5) as glyph[epsilon1] 6 max { 1 , | E [0 . 5 ± glyph[epsilon1] ] |} if glyph[epsilon1] < 0 . 5 and glyph[epsilon1] 6 m otherwise. By Lemma 8 for glyph[epsilon1] ′ , we have that

where E ′ r is the random event for phase r ∈ { 1 , . . . , n } that is defined by (6) with glyph[epsilon1] ′ . Therefore, using Lemma 9 for glyph[epsilon1] ′ , we can see that the output C out of KC-FB is a (5 , glyph[epsilon1] ) -approximate solution for instance ( V, s) of the offline problem minimizing (1) w.p. at least 1 -2 n 3 exp ( -2 T min e ∈ E max { glyph[epsilon1] ′ , ∆ e } 2 n 2 )

Finally, we can easily confirm that KC-FB does not exceed the given budget T due to the algorithm procedure of line 8, which concludes the proof.

E Uniform sampling algorithms

Here we provide the complete description and analysis of the naive uniform-sampling algorithms for both the FC setting (Algorithm 6) and the FB setting (Algorithm 7). Note that in the FC setting, no feasible stopping conditions are known from previous studies to guarantee that the output is an approximate solution, even with uniform or arbitrary sampling strategies. Therefore existing analysis of uniform sampling given in Chen et al. [25] is not applicable to our case with offline optimization being NP-hard.

First, we show a basic analysis of the cost of clustering when the estimate ̂ s is close to the unknown similarity s .

Lemma 10. Let glyph[epsilon1] ∈ (0 , 0 . 5) . Assume that | s( e ) -̂ s( e ) | ≤ glyph[epsilon1] for every e ∈ E . Let C out be the output of any α -approximation algorithm for instance ( V, ̂ s) of the offline problem minimizing (1) . Then C out is an ( α, ( α +1) glyph[epsilon1]m ) -approximate solution for instance ( V, s) of the offline problem.

Algorithm 6 Uniform sampling in the FC setting ( Uniform-FC )

Input : Set V of n objects, confidence level δ , additive error glyph[epsilon1] > 0

1 T ( e ) ←glyph[ceilingleft] ( α +1) 2 m 2 2 glyph[epsilon1] 2 log 2 m δ glyph[ceilingright] for each e ∈ E ;

2 Sample each e ∈ E for T ( e ) times and compute empirical mean ̂ s( e ) ;

3 ˆ C ← solution of an approximation algorithm for instance ( V, ̂ s) of the offline problem minimizing (1);

4 return C out := ˆ C

Algorithm 7 Uniform sampling in the FB setting ( Uniform-FB )

Input : Set V of n objects, budget T

1 Sample each e ∈ E for glyph[floorleft] T/m glyph[floorright] times and compute the empirical mean ̂ s( e ) ;

2 ˆ C ← solution of an approximation algorithm for instance ( V, ̂ s) of the offline problem minimizing (1);

3 return C out := ˆ C

Proof. We have

Next we evaluate the performance of Algorithm 6.

Proposition 1. Given a confidence level δ ∈ (0 , 1) and an additive error glyph[epsilon1] ∈ (0 , 0 . 5) , the uniform sampling algorithm with an α -approximation oracle for the FC setting (Algorithm 6) outputs C out that satisfies

and the upper bound of the number of samples is

Proof. As the algorithm samples each e ∈ E for T ( e ) times, by the Hoeffding inequality (Lemma 1), it holds that

Note that T ( e ) ≥ ( α +1) 2 m 2 2 glyph[epsilon1] 2 log 2 m δ gives

Therefore, by taking a union bound, we have

By Lemma 10 for glyph[epsilon1] := glyph[epsilon1] ( α +1) m , we see that C out is an ( α, glyph[epsilon1] ) -approximate solution for instance ( V, s) of the offline problem minimizing (1) w.p. at least 1 -δ , as desired.

The next proposition evaluates the performance of Algorithm 7.

Proposition 2. Given a sampling budget T and additive error glyph[epsilon1] ∈ (0 , 0 . 5) , the uniform sampling algorithm with an α -approximation oracle for the FB setting (Algorithm 7) outputs C out that satisfies

Proof. As e ∈ E has been pulled at least glyph[floorleft] T m glyph[floorright] times, by Lemma 1, we have

Taking a union bound for all e ∈ E , we have

By Lemma 10, when for all e ∈ E , | ̂ s( e ) -s( e ) | < glyph[epsilon1] ( α +1) m , we have cost s ( C out ) ≤ α · OPT(s) + glyph[epsilon1] , which concludes the proof.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers a correlation clustering setting with an unknown similarity function but with the ability to query a noisy oracle, i.e., the oracle may yield noisy feedback instead of the true similarity of the queried pair of objects. In this scenario, the goal is to achieve a reasonable trade-off between the number of queries to the oracle and the cost of the clustering. In the considered setting, the authors introduce two novel problem formulations within the paradigm of pure exploration of combinatorial multi-armed bandits (PE-CMAB): fixed confidence and fixed budget settings. For both problems, the paper introduces two novel algorithms (KC-FC and KC-FB) and analyzes their theoretical guarantees. Some experiments on real-world graph datasets with some artificially generated features validate the theoretical findings of the work by demonstrating that in both settings the proposed methods outperform a baseline method in terms of sample complexity and cost of the clustering.\n\nSTRENGTHS:\nS1) Good originality: the paper studies two novel problem formulations for correlation clustering with unknown similarity values and an oracle with noisy feedback.\n\nS2) The settings considered in this work are well-motivated in the Introduction.\n\nS3) The paper is overall well-written and very well-contextualized relative to prior work (>80 references) on similar topics.\n\nS4) The proposed algorithms are theoretically analyzed.\n\nWEAKNESSES:\nW1) The authors considered the most basic formulation for correlation clustering, where the costs of placing two objects in the same or different clusters sum to 1, and there is a cost for every pair of objects (Equation 1). The authors neither motivate the choice of this specific setting nor discuss the more general case where such costs are not related and some pairs are missing from the objective function. A discussion on how/if the results presented in this paper can be extended to the more general correlation clustering problem would have been appreciated.\n\nW2) Limited number of baselines used in the experiments. Besides their proposed method, the authors only tested against the baseline KwikCluster, which knows the true similarities, and Uniform-FB (Uniform-FC), which is defined by the authors themselves. The lack of comparison with existing methods for general-purpose PE-CMAB algorithms for the FB and FC settings is notable. The authors justify this choice by stating that the theoretical analysis of such algorithms does not hold in the context of correlation clustering considered in this paper (lines 220-227). However, comparing the proposed approach with existing general PE-CMAB approaches would have provided a more convincing argument about the ineffectiveness of existing approaches in practice for the considered problem. This represents a missed opportunity to demonstrate the necessity of their proposed algorithms with practical evidence.\n\nW3) Limited reproducibility: the source code is not provided with the submission.\n\nW4) Missing proof sketch for Theorem 2 in the main paper. As done for Theorem 1, the authors should include a proof sketch for Theorem 2 as well.\n\nW5) The paper is not clear in the following aspects:\n\n- It is not clear why the authors decided to use different methods to generate the similarity values for the FC and FB settings. In particular, why not use the same strategy adopted for the FC setting for the FB setting as well, to observe, by varying Delta_{min}, how often the algorithm yields the desired approximate solution, i.e., estimate the probability in Equation 3?\n\n- The authors specify how the similarity scores, hidden from the learning algorithm, are generated. However, the main paper does not explain how the noisy feedback is provided to the learner when querying a particular pair of objects. Are the defined similarities the mean of a Bernoulli or R-sub-Gaussian distribution? Are the noisy feedbacks generated by sampling from these distributions? Please specify this in the experimental evaluation section.\n\n- Some parameter values are not justified in the paper: why is epsilon set to sqrt{n}, delta = 0.001, and T = n^{2.2}? The latter is said to be for scalability, but why n^{2.2} and not n^{2.1}?\n\n- The motivation for using only instances with n < 1000 nodes for the experiments in Figures 1 and 2 is unconvincing. The authors could have used better hardware for the experiments, not just a notebook.\n\n- The authors do not run Uniform-FC algorithms in the experiments but report the sample complexity, which can be calculated without running it. How is this done? This is not clear and should be explicitly stated on line 325.\n\nMinors:\n- The font size in Figures 1, 2, and 3 is too small and difficult to read.\n- Algorithm 2, line 7: please specify that the updates are done for edges \\bar{e}_t^g and \\bar{e}_t^b to improve clarity.\n- State in the paragraph \"Algorithm details\" that the sets G and B correspond intuitively to the sets of \"good\" and \"bad\" pairs, respectively. This is only stated in the algorithm.\n- Line 323, “as it will be shown later” is too general. Please be more specific (using pointers) if it refers to something shown in the Figures or the Supplemental Material.\n\nQUESTIONS:\nQ1) In the more general case of correlation clustering (see W1), the KwikCluster algorithm does not provide any theoretical guarantees (even in the offline setting), thus the theoretical guarantees of KC-FC and KC-FB algorithms should not hold either, is it correct? If yes, how can your proposed algorithms be modified to account for such a general setting? \n\nQ2) Besides the missing theoretical guarantees of existing “general-purpose” PE-CMAB on the specific PE-CMAB instance considered in this paper, are there any other reasons that make the applicability of such algorithms unfeasible in practice in the context of the problems considered in this paper? (See also W2)\n\nQ3) How is the noisy feedback provided to the learner in the experiments when querying a particular pair of objects?\n\nQ4) How is the sample complexity of the Uniform-FC algorithm computed without running it?\n\nQ5) In the experiments, why is epsilon set to sqrt(n), delta to 0.001, and T to n^2.2?\n\nQ6) Why did you decide to use different methods to generate the similarity values for the FC and FB settings. As an example, why not use the same strategy adopted for the FC setting for the FB setting as well, to observe, by varying Delta_{min}, how often the algorithm yields the desired approximate solution, i.e., estimate the probability in Equation 3?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n**[Setting]**\nThis paper studies correlation clustering using a noisy oracle. The learner queries a pair of items and receives a noisy estimate of their similarity in the range [0, 1] from an oracle. The goal is to either:\n1. [Fixed confidence setting] Minimize the number of queries and return a clustering with cost at most $\\alpha OPT + \\epsilon$ with high probability\n2. [Fixed budget setting] Minimize the probability of returning a clustering with cost more than $\\alpha OPT + \\epsilon$ for a given query budget $T$\n\n**[Contributions]**\n1. An algorithm (KC-FC) for the fixed confidence setting that makes roughly $\\tilde{O}(n^2/\\Delta_1^2)$ queries ($\\Delta_1$ being a custom gap-based metric) and returns a clustering with cost at most $5 OPT + \\epsilon$ with high probability.\n2. An algorithm (KC-FB) for the fixed budget setting that makes $T$ queries and returns clustering with cost at most $5 OPT + \\epsilon$ with probability at least $1 - O(n^3 \\exp(-2T\\Delta_2^2/n^2))$, where $\\Delta_2$ is another custom gap-based metric.\n3. Empirical validation that the algorithms work as expected\n\nSTRENGTHS:\n1. This paper considers an oracle that returns values in range [0, 1] instead of a binary response, as in existing works. \n2. Results for both fixed confidence and fixed budget settings are included.\n3. The paper is well-written and easy to follow. \n4. Experiments show that the cost of recovered clustering in the fixed-confidence setting is roughly the same as the cost of clustering obtained from KwikCluster if it is given the \"ground-truth\" similarity scores.\n\nWEAKNESSES:\n1. I am not entirely convinced by the strength of the theoretical results in the fixed confidence setting. Why not move sampling inside the clustering loop in Algorithm 1? Keep sampling $\\lbrace u, v_r \\rbrace$ for all unclustered nodes $u$ and the chosen pivot $v_r$ until $s(u, v_r) > 0.5 - \\epsilon$ or $s(u, v_r) < 0.5 + \\epsilon$ can be determined with high confidence. Then move nodes into the same cluster as $v_r$ if $s(u, v_r) > 0.5 - \\epsilon$. This would require $\\tilde{O}(n / \\Delta^2)$ queries per round $r$, leading to overall complexity of $\\tilde{O}(nk / \\Delta^2)$ instead of $\\tilde{O}(n^2 / \\Delta^2)$. Algorithm 3 already does something similar.\n2. The results will benefit from more context. For example, when $T = \\theta(n^2)$, the fixed budget bound becomes meaningless. How does this result compare to the case of noisy correlation clustering with only one sample per edge (e.g., Mathieu and Schudy, 2010 and similar results)? In the absence of a lower bound, such context would be very valuable.\n3. More context can also be added to experiments. For example, it would be useful to threshold the oracle's response and compare the query complexity with an approach for binary responses? The naïve baselines seem too simplistic.\n\nQUESTIONS:\nPlease respond to the points under weaknesses\n\nLIMITATIONS:\nYes, the authors addressed the lack of a lower bound.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-06"}

comments: SUMMARY:
This paper introduces algorithms for correlation clustering with noisy, expensive similarity functions. The authors present two formulations in the PE-CMAB framework: fixed confidence and fixed budget. Their proposed algorithms, KC-FC and KC-FB, combine sampling with KwikCluster approximation.

STRENGTHS:
The paper addresses the realistic and challenging scenario of noisy and costly similarity functions, which differs from the traditional assumption that provides the exact similarities value.

The theoretical analysis looks solid. The problem setting is motivated by practical applications where computing similarities are expensive and noisy. The paper provides a strong practical evaluation. 

The paper is clearly written and easy to read.

WEAKNESSES:
The paper could benefit from a more detailed discussion of the limitations of the current to achieve 3-approximation.

QUESTIONS:
What is the main challenge to extend this approach for 3-approximation algorithms?

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-07

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper copes with the problem of active (weighted) correlation clustering when oracle queries are corrupted by random noise. \n\nAuthors frame two variants of the problem where high probability guarantees are required: the fixed-confidence and the fixed budget settings; algorithms for the these settings are provided in order. \n\nTheoretical results are supported by numerical experiments.\n\nSTRENGTHS:\nThe subject considered in this paper is of interest, as also confirmed by the vast literature on this problem: indeed, correlation clustering (CC) is one of the most popular clustering framework.\n\nThe proposed algorithms are simple to understand and easy to implement. One aspect that should be clarified is the precise runtime of KC-FC which I recommend to include in the statement of Theorem 1. \n\nThe performance guarantees in Th. 1 is nice in being instance dependent.\n\nWEAKNESSES:\nRelated works: it is not clear to me why authors only refers to [40] when mentioning the SOTA bounds for active CC in the noiseless setting, when also [15] nature essentially the same performance guarantee.\n\nAs for Theorem 1, the guarantees does not recover the 3 apx. factor when the true similarities are binary, instead settles for the much worse 5 apx. factor. \n\nFor binary queries, it seems to me that a simple strategy that ask roughly log(n) times queries for each edge queried (e.g.. the choice of kwick cluster) and takes the majority vote, would be equivalent to an algorithm running on the true similarity and should feature a 3 apx. factor. Could authors comment on that? A similar question holds for similarities in [0,1], how about replacing the majority voting above with a simple average, and then running a standard algorithm on the top?\n\nIn the numerical evaluation it may be good to consider adding the above baselines in problems where the similarities are binary.\n\nMy main concerns revolves around the relevance of the contribution provided by this paper, if you can provide convincing arguments, I'm willing to raise my score.\n\nI'm (partially) satisfied with the authors answer, and increase my score accordingly.\n\nQUESTIONS:\nN/A\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-19"}