arXiv subjects

Nicolas Verzelen

Publications and source records attributed to Nicolas Verzelen.

At least 19 recordsLinked to original sources

arXiv2026

The Sampling Complexity of Condorcet Winner Identification in Dueling Bandits

We study best-arm identification in stochastic dueling bandits under the sole assumption that a Condorcet winner exists, i.e., an arm that wins each noisy pairwise comparison with probability at least $1/2$. We introduce a new identification procedure that exploits the full gap matrix $\Delta_{i,j}=q_{i,j}-\tfrac12$ (where $q_{i,j}$ is the probability that arm $i$ beats arm $j$), rather than only the gaps between the Condorcet winner and the other arms. We derive high-probability, instance-dependent sample-complexity guarantees that (up to logarithmic factors) improve the best known ones by leveraging informative comparisons beyond those involving the winner. We complement these results with new lower bounds which, to our knowledge, are the first for Condorcet-winner identification in stochastic dueling bandits. Our lower-bound analysis isolates the intrinsic cost of locating informative entries in the gap matrix and estimating them to the required confidence, establishing the optimality of our non-asymptotic bounds. Overall, our results reveal new regimes and trade-offs in the sample complexity that are not captured by asymptotic analyses based only on the expected budget.

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arXiv2026

Low-degree Lower bounds for clustering in moderate dimension

We study the fundamental problem of clustering $n$ points into $K$ groups drawn from a mixture of isotropic Gaussians in $\mathbb{R}^d$. Specifically, we investigate the requisite minimal distance $\Delta$ between mean vectors to partially recover the underlying partition. While the minimax-optimal threshold for $\Delta$ is well-established, a significant gap exists between this information-theoretic limit and the performance of known polynomial-time procedures. Although this gap was recently characterized in the high-dimensional regime ($n \leq dK$), it remains largely unexplored in the moderate-dimensional regime ($n \geq dK$). In this manuscript, we address this regime by establishing a new low-degree polynomial lower bound for the moderate-dimensional case when $d \geq K$. We show that while the difficulty of clustering for $n \leq dK$ is primarily driven by dimension reduction and spectral methods, the moderate-dimensional regime involves more delicate phenomena leading to a "non-parametric rate". We provide a novel non-spectral algorithm matching this rate, shedding new light on the computational limits of the clustering problem in moderate dimension.

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arXiv2026

Nonparametric Kernel Clustering with Bandit Feedback

Clustering with bandit feedback refers to the problem of partitioning a set of items, where the clustering algorithm can sequentially query the items to receive noisy observations. The problem is formally posed as the task of partitioning the arms of an N-armed stochastic bandit according to their underlying distributions, grouping two arms together if and only if they share the same distribution, using samples collected sequentially and adaptively. This setting has gained attention in recent years due to its applicability in recommendation systems and crowdsourcing. Existing works on clustering with bandit feedback rely on a strong assumption that the underlying distributions are sub-Gaussian. As a consequence, the existing methods mainly cover settings with linearly-separable clusters, which has little practical relevance. We introduce a framework of ``nonparametric clustering with bandit feedback'', where the underlying arm distributions are not constrained to any parametric, and hence, it is applicable for active clustering of real-world datasets. We adopt a kernel-based approach, which allows us to reformulate the nonparametric problem as the task of clustering the arms according to their kernel mean embeddings in a reproducing kernel Hilbert space (RKHS). Building on this formulation, we introduce the KABC algorithm with theoretical correctness guarantees and analyze its sampling budget. We introduce a notion of signal-to-noise ratio for this problem that depends on the maximum mean discrepancy (MMD) between the arm distributions and on their variance in the RKHS. Our algorithm is adaptive to this unknown quantity: it does not require it as an input yet achieves instance-dependent guarantees.

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arXiv2025

Statistical and computational challenges in ranking

We consider the problem of ranking $n$ experts according to their abilities, based on the correctness of their answers to $d$ questions. This is modeled by the so-called crowd-sourcing model, where the answer of expert $i$ on question $k$ is modeled by a random entry, parametrized by $M_{i,k}$ which is increasing linearly with the expected quality of the answer. To enable the unambiguous ranking of the experts by ability, several assumptions on $M$ are available in the literature. We consider here the general isotonic crowd-sourcing model, where $M$ is assumed to be isotonic up to an unknown permutation $\pi^*$ of the experts - namely, $M_{\pi^{*-1}(i),k} \geq M_{\pi^{*-1}(i+1),k}$ for any $i\in [n-1], k \in [d]$. Then, ranking experts amounts to constructing an estimator of $\pi^*$. In particular, we investigate here the existence of statistically optimal and computationally efficient procedures and we describe recent results that disprove the existence of computational-statistical gaps for this problem. To provide insights on the key ideas, we start by discussing simpler and yet related sub-problems, namely sub-matrix detection and estimation. This corresponds to specific instances of the ranking problem where the matrix $M$ is constrained to be of the form $\lambda \mathbf 1\{S\times T\}$ where $S\subset [n], T\subset [d]$. This model has been extensively studied. We provide an overview of the results and proof techniques for this problem with a particular emphasis on the computational lower bounds based on low-degree polynomial methods. Then, we build upon this instrumental sub-problem to discuss existing results and algorithmic ideas for the general ranking problem.

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arXiv2025

Phase Transition for Stochastic Block Model with more than $\sqrt{n}$ Communities (II)

A fundamental theoretical question in network analysis is to determine under which conditions community recovery is possible in polynomial time in the Stochastic Block Model (SBM). When the number $K$ of communities remains smaller than $\sqrt{n}$ --where $n$ denotes the number of nodes--, non-trivial community recovery is possible in polynomial time above, and only above, the Kesten--Stigum (KS) threshold, originally postulated using arguments from statistical physics. When $K \geq \sqrt{n}$, Chin, Mossel, Sohn, and Wein recently proved that, in the \emph{sparse regime}, community recovery in polynomial time is achievable below the KS threshold by counting non-backtracking paths. This finding led them to postulate a new threshold for the many-communities regime $K \geq \sqrt{n}$. Subsequently, Carpentier, Giraud, and Verzelen established the failure of low-degree polynomials below this new threshold across all density regimes, and demonstrated successful recovery above the threshold in certain moderately sparse settings. While these results provide strong evidence that, in the many community setting, the computational barrier lies at the threshold proposed in~Chin et al., the question of achieving recovery above this threshold still remains open in most density regimes. The present work is a follow-up to~Carpentier et al., in which we prove Conjecture~1.4 stated therein by: \\ 1- Constructing a family of motifs satisfying specific structural properties; and\\ 2- Proving that community recovery is possible above the proposed threshold by counting such motifs.\\ Our results complete the picture of the computational barrier for community recovery in the SBM with $K \geq \sqrt{n}$ communities. They also indicate that, in moderately sparse regimes, the optimal algorithms appear to be fundamentally different from spectral methods.

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arXiv2025

Phase Transition for Stochastic Block Model with more than $\sqrt{n}$ Communities

Predictions from statistical physics postulate that recovery of the communities in the Stochastic Block Model (SBM) with a fixed number $K$ of communities is possible in polynomial time above, and only above, the Kesten-Stigum (KS) threshold. This conjecture has given rise to a rich literature, proving that non-trivial community recovery is indeed possible in SBM above the KS threshold. Failure of low-degree polynomials (LDP) below the KS threshold was also proven, as long as $K\ll \sqrt{n}$, where $n$ is the number of nodes in the observed graph. When $K\geq \sqrt{n}$, Chin et al.(2025) recently proved that, in a \emph{sparse regime}, community recovery in polynomial time is possible below the KS threshold by counting non-backtracking paths. This breakthrough led them to postulate a new threshold for the many-communities regime $K\geq \sqrt{n}$. In this work, we provide evidence supporting their conjecture:\\ 1- We prove that, for \emph{any graph density}, LDP fail to recover communities below the threshold postulated by Chin et al.(2025) ;\\ 2- We prove that community recovery is possible in polynomial time above the postulated threshold, not only in the \emph{sparse regime} considered in Chin et al.~(2025), but also in \emph{moderately sparse regimes}, by counting occurrences of some specific motifs inspired by the LDP analysis.\\ In particular, counting self-avoiding paths of length $\log(n)$, which is closely related to spectral algorithms based on the Non-Backtracking operator, is optimal only in the sparse regime. More complex motifs based on the blow-up of a cycle must be considered in denser regimes.

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arXiv2025

Model-free algorithms for fast node clustering in SBM type graphs and application to social role inference in animals

We propose a novel family of model-free algorithms for node clustering and parameter inference in graphs generated from the Stochastic Block Model (SBM), a fundamental framework in community detection. Drawing inspiration from the Lloyd algorithm for the $k$-means problem, our approach extends to SBMs with general edge weight distributions. We establish the consistency of our estimator under a natural identifiability condition. Through extensive numerical experiments, we benchmark our methods against state-of-the-art techniques, demonstrating significantly faster computation times with the lower order of estimation error. Finally, we validate the practical relevance of our algorithms by applying them to empirical network data from behavioral ecology.

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arXiv2025

Low-degree lower bounds via almost orthonormal bases

Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a planted distribution $\mathbb{P}'$ against a null distribution $\mathbb{P}$ with independent components -- the standard approach is to bound the advantage using an $\mathbb{L}^2(\mathbb{P})$-orthonormal family of polynomials. However, this method breaks down for estimation tasks or more complex testing problems where $\mathbb{P}$ has some planted structures, so that no simple $\mathbb{L}^2(\mathbb{P})$-orthogonal polynomial family is available. To address this challenge, several technical workarounds have been proposed [SW22,SW25], though their implementation can be delicate. In this work, we propose a more direct proof strategy. Focusing on random graph models, we construct a basis of polynomials that is almost orthonormal under $\mathbb{P}$, in precisely those regimes where statistical-computational gaps arise. This almost orthonormal basis not only yields a direct route to establishing low-degree lower bounds, but also allows us to explicitly identify the polynomials that optimize the low-degree criterion. This, in turn, provides insights into the design of optimal polynomial-time algorithms. We illustrate the effectiveness of our approach by recovering known low-degree lower bounds, and establishing new ones for problems such as hidden subcliques, stochastic block models, and seriation models.

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arXiv2025

Computational barriers for permutation-based problems, and cumulants of weakly dependent random variables

In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the performance of low-degree polynomials. The seminal work of arXiv:2008.02269 connects low-degree lower bounds to multivariate cumulants. Prior works arXiv:2308.15728, arXiv:2506.13647 leverage independence among latent variables to bound cumulants. However, such approaches break down for problems with latent structure lacking independence, such as those involving random permutations. To address this important restriction, we develop a technique to upper-bound cumulants under weak dependencies, such as those arising from sampling without replacement or random permutations. To show-case the effectiveness of our approach, we uncover evidence of statistical-computational gaps in multiple feature matching and in seriation problems.

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arXiv2025

Computational lower bounds in latent models: clustering, sparse-clustering, biclustering

In many high-dimensional problems, like sparse-PCA, planted clique, or clustering, the best known algorithms with polynomial time complexity fail to reach the statistical performance provably achievable by algorithms free of computational constraints. This observation has given rise to the conjecture of the existence, for some problems, of gaps -- so called statistical-computational gaps -- between the best possible statistical performance achievable without computational constraints, and the best performance achievable with poly-time algorithms. A powerful approach to assess the best performance achievable in poly-time is to investigate the best performance achievable by polynomials with low-degree. We build on the seminal paper of Schramm and Wein (2022) and propose a new scheme to derive lower bounds on the performance of low-degree polynomials in some latent space models. By better leveraging the latent structures, we obtain new and sharper results, with simplified proofs. We then instantiate our scheme to provide computational lower bounds for the problems of clustering, sparse clustering, and biclustering. We also prove matching upper-bounds and some additional statistical results, in order to provide a comprehensive description of the statistical-computational gaps occurring in these three problems.

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arXiv2025

Clustering Items through Bandit Feedback: Finding the Right Feature out of Many

We study the problem of clustering a set of items based on bandit feedback. Each of the $n$ items is characterized by a feature vector, with a possibly large dimension $d$. The items are partitioned into two unknown groups such that items within the same group share the same feature vector. We consider a sequential and adaptive setting in which, at each round, the learner selects one item and one feature, then observes a noisy evaluation of the item's feature. The learner's objective is to recover the correct partition of the items, while keeping the number of observations as small as possible. We provide an algorithm which relies on finding a relevant feature for the clustering task, leveraging the Sequential Halving algorithm. With probability at least $1-\delta$, we obtain an accurate recovery of the partition and derive an upper bound on the budget required. Furthermore, we derive an instance-dependent lower bound, which is tight in some relevant cases.

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arXiv2024

Optimal level set estimation for non-parametric tournament and crowdsourcing problems

Motivated by crowdsourcing, we consider a problem where we partially observe the correctness of the answers of $n$ experts on $d$ questions. In this paper, we assume that both the experts and the questions can be ordered, namely that the matrix $M$ containing the probability that expert $i$ answers correctly to question $j$ is bi-isotonic up to a permutation of it rows and columns. When $n=d$, this also encompasses the strongly stochastic transitive (SST) model from the tournament literature. Here, we focus on the relevant problem of deciphering small entries of $M$ from large entries of $M$, which is key in crowdsourcing for efficient allocation of workers to questions. More precisely, we aim at recovering a (or several) level set $p$ of the matrix up to a precision $h$, namely recovering resp. the sets of positions $(i,j)$ in $M$ such that $M_{ij}>p+h$ and $M_{i,j}<p-h$. We consider, as a loss measure, the number of misclassified entries. As our main result, we construct an efficient polynomial-time algorithm that turns out to be minimax optimal for this classification problem. This heavily contrasts with existing literature in the SST model where, for the stronger reconstruction loss, statistical-computational gaps have been conjectured. More generally, this shades light on the nature of statistical-computational gaps for permutations models.

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arXiv2024

Seriation of Toeplitz and latent position matrices: optimal rates and computational trade-offs

In this paper, we consider the problem of seriation of a permuted structured matrix based on noisy observations. The entries of the matrix relate to an expected quantification of interaction between two objects: the higher the value, the closer the objects. A popular structured class for modelling such matrices is the permuted Robinson class, namely the set of matrices whose coefficients are decreasing away from its diagonal, up to a permutation of its lines and columns. We consider in this paper two submodels of Robinson matrices: the Toeplitz model, and the latent position model. We provide a computational lower bound based on the low-degree paradigm, which hints that there is a statistical-computational gap for seriation when measuring the error based on the Frobenius norm. We also provide a simple and polynomial-time algorithm that achives this lower bound. Along the way, we also characterize the information-theory optimal risk thereby giving evidence for the extent of the computation/information gap for this problem.

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arXiv2024

Active clustering with bandit feedback

We investigate the Active Clustering Problem (ACP). A learner interacts with an $N$-armed stochastic bandit with $d$-dimensional subGaussian feedback. There exists a hidden partition of the arms into $K$ groups, such that arms within the same group, share the same mean vector. The learner's task is to uncover this hidden partition with the smallest budget - i.e., the least number of observation - and with a probability of error smaller than a prescribed constant $\delta$. In this paper, (i) we derive a non-asymptotic lower bound for the budget, and (ii) we introduce the computationally efficient ACB algorithm, whose budget matches the lower bound in most regimes. We improve on the performance of a uniform sampling strategy. Importantly, contrary to the batch setting, we establish that there is no computation-information gap in the active setting.

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arXiv2024

Minimax optimal seriation in polynomial time

We consider the seriation problem, whose goal is to recover a hidden ordering from a noisy observation of a permuted Robinson matrix. We establish sharp minimax rates under average-Lipschitz conditions that strictly extend the bi-Lipschitz framework of [Giraud et al., 2023]. We further design a polynomial-time algorithm that attains these optimal rates, thereby resolving two open questions raised in [Giraud et al., 2023]. Finally, our analysis extends to a broader class of matrices beyond those generated by exact permutations.

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arXiv2024

One-Bit Total Variation Denoising over Networks with Applications to Partially Observed Epidemics

This paper introduces a novel approach for epidemic nowcasting and forecasting over networks using total variation (TV) denoising, a method inspired by classical signal processing techniques. Considering a network that models a population as a set of $n$ nodes characterized by their infection statuses $Y_i$ and that represents contacts as edges, we prove the consistency of graph-TV denoising for estimating the underlying infection probabilities $\{p_i\}_{ i \in \{1,\cdots, n\}}$ in the presence of Bernoulli noise. Our results provide an important extension of existing bounds derived in the Gaussian case to the study of binary variables -- an approach hereafter referred to as one-bit total variation denoising. The methodology is further extended to handle incomplete observations, thereby expanding its relevance to various real-world situations where observations over the full graph may not be accessible. Focusing on the context of epidemics, we establish that one-bit total variation denoising enhances both nowcasting and forecasting accuracy in networks, as further evidenced by comprehensive numerical experiments and two real-world examples. The contributions of this paper lie in its theoretical developments, particularly in addressing the incomplete data case, thereby paving the way for more precise epidemic modelling and enhanced surveillance strategies in practical settings.

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arXiv2024

Computation-information gap in high-dimensional clustering

We investigate the existence of a fundamental computation-information gap for the problem of clustering a mixture of isotropic Gaussian in the high-dimensional regime, where the ambient dimension $p$ is larger than the number $n$ of points. The existence of a computation-information gap in a specific Bayesian high-dimensional asymptotic regime has been conjectured by arXiv:1610.02918 based on the replica heuristic from statistical physics. We provide evidence of the existence of such a gap generically in the high-dimensional regime $p \geq n$, by (i) proving a non-asymptotic low-degree polynomials computational barrier for clustering in high-dimension, matching the performance of the best known polynomial time algorithms, and by (ii) establishing that the information barrier for clustering is smaller than the computational barrier, when the number $K$ of clusters is large enough. These results are in contrast with the (moderately) low-dimensional regime $n \geq poly(p, K)$, where there is no computation-information gap for clustering a mixture of isotropic Gaussian. In order to prove our low-degree computational barrier, we develop sophisticated combinatorial arguments to upper-bound the mixed moments of the signal under a Bernoulli Bayesian model.

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arXiv2023

Optimal rates for ranking a permuted isotonic matrix in polynomial time

We consider a ranking problem where we have noisy observations from a matrix with isotonic columns whose rows have been permuted by some permutation $\pi$ *. This encompasses many models, including crowd-labeling and ranking in tournaments by pair-wise comparisons. In this work, we provide an optimal and polynomial-time procedure for recovering $\pi$ * , settling an open problem in [7]. As a byproduct, our procedure is used to improve the state-of-the art for ranking problems in the stochastically transitive model (SST). Our approach is based on iterative pairwise comparisons by suitable data-driven weighted means of the columns. These weights are built using a combination of spectral methods with new dimension-reduction techniques. In order to deal with the important case of missing data, we establish a new concentration inequality for sparse and centered rectangular Wishart-type matrices.

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