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 $π^*$ of the experts - namely, $M_{π^{*-1}(i),k} \geq M_{π^{*-1}(i+1),k}$ for any $i\in [n-1], k \in [d]$. Then, ranking experts amounts to constructing an estimator of $π^*$. 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 $λ\mathbf 1\{S\times T\}$ 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.

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

Predictions from statistical physics postulate that recovery of the communities in Stochastic Block Model (SBM) 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, as long as the number $K$ of communities remains smaller than $\sqrt{n}$, where $n$ is the number of nodes in the observed graph. Failure of low-degree polynomials below the KS threshold was also proven when $K=o(\sqrt{n})$. When $K\geq \sqrt{n}$, Chin et al.(2025) recently prove that, in a sparse regime, community recovery in polynomial time is possible below the KS threshold by counting non-backtracking paths. This breakthrough result lead them to postulate a new threshold for the many communities regime $K\geq \sqrt{n}$. In this work, we provide evidences that confirm their conjecture for $K\geq \sqrt{n}$: 1- We prove that, for any density of the graph, low-degree polynomials 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 sparse regime of~Chin et al., but also in some (but not all) moderately sparse regimes by essentially counting clique occurence in the observed graph.

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.

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.

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.

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.

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.

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.

Covariance Adaptive Best Arm Identification

We consider the problem of best arm identification in the multi-armed bandit model, under fixed confidence. Given a confidence input $\delta$, the goal is to identify the arm with the highest mean reward with a probability of at least 1 -- $\delta$, while minimizing the number of arm pulls. While the literature provides solutions to this problem under the assumption of independent arms distributions, we propose a more flexible scenario where arms can be dependent and rewards can be sampled simultaneously. This framework allows the learner to estimate the covariance among the arms distributions, enabling a more efficient identification of the best arm. The relaxed setting we propose is relevant in various applications, such as clinical trials, where similarities between patients or drugs suggest underlying correlations in the outcomes. We introduce new algorithms that adapt to the unknown covariance of the arms and demonstrate through theoretical guarantees that substantial improvement can be achieved over the standard setting. Additionally, we provide new lower bounds for the relaxed setting and present numerical simulations that support their theoretical findings.

Active Ranking of Experts Based on their Performances in Many Tasks

We consider the problem of ranking n experts based on their performances on d tasks. We make a monotonicity assumption stating that for each pair of experts, one outperforms the other on all tasks. We consider the sequential setting where in each round, the learner has access to noisy evaluations of actively chosen pair of expert-task, given the information available up to the actual round. Given a confidence parameter $\delta$ $\in$ (0, 1), we provide strategies allowing to recover the correct ranking of experts and develop a bound on the total number of queries made by our algorithm that hold with probability at least 1 -- $\delta$. We show that our strategy is adaptive to the complexity of the problem (our bounds are instance dependent), and develop matching lower bounds up to a poly-logarithmic factor. Finally, we adapt our strategy to the relaxed problem of best expert identification and provide numerical simulation consistent with our theoretical results.