arXiv subjects

Laurent Massoulié

Publications and source records attributed to Laurent Massoulié.

At least 19 recordsLinked to original sources

arXiv2026

Polylogarithmic Collective Tree Exploration

We study asynchronous collective tree exploration, where $k$ agents with unrestricted communication start at the root of an unknown tree and discover edges online. At each step, an adversary chooses which agent moves. We give a deterministic algorithm that explores any tree with $n$ nodes and depth $D$ in at most \[ 2n+O\left(k\log^2(k)D\right) \] moves, matching known lower bounds up to a constant factor. As a direct consequence, we obtain a near-optimal competitive ratio of $O(\log^2 k)$ for synchronous collective tree exploration, where all agents move at each round. The proof relies on a multiscale power regularizer that may be of independent interest.

cs.DS↗

arXiv2026

Phase Transition in Convex Relaxations for Graph Alignment

We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+σ^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $σ= o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(Π^\star)$, i.e., $\|X^\star-Π^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-Π^\star\|_F^2$ transitions from $o(n)$ for $σ= \tilde{o}(n^{-1/2})$ to $Ω(n)$ for $σ= \tildeΩ(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

stat.ML↗

arXiv2026

Improved Analysis of the Accelerated Noisy Power Method with Applications to Decentralized PCA

We analyze the Accelerated Noisy Power Method, an algorithm for Principal Component Analysis in the setting where only inexact matrix-vector products are available, which can arise for instance in decentralized PCA. While previous works have established that acceleration can improve convergence rates compared to the standard Noisy Power Method, these guarantees require overly restrictive upper bounds on the magnitude of the perturbations, limiting their practical applicability. We provide an improved analysis of this algorithm, which preserves the accelerated convergence rate under much milder conditions on the perturbations. We show that our new analysis is worst-case optimal, in the sense that the convergence rate cannot be improved, and that the noise conditions we derive cannot be relaxed without sacrificing convergence guarantees. We demonstrate the practical relevance of our results by deriving an accelerated algorithm for decentralized PCA, which has similar communication costs to non-accelerated methods. To our knowledge, this is the first decentralized algorithm for PCA with provably accelerated convergence.

stat.ML↗

arXiv2026

The feasibility of multi-graph alignment: a Bayesian approach

We establish thresholds for the feasibility of random multi-graph alignment in two models. In the Gaussian model, we demonstrate an "all-or-nothing" phenomenon: above a critical threshold, exact alignment is achievable with high probability, while below it, even partial alignment is statistically impossible. In the sparse Erdős-Rényi model, we rigorously identify a threshold below which no meaningful partial alignment is possible and conjecture that above this threshold, partial alignment can be achieved. To prove these results, we develop a general Bayesian estimation framework over metric spaces, which provides insight into a broader class of high-dimensional statistical problems.

math.ST↗

arXiv2026

Learning with Shallow Neural Networks on Cluster-Structured Features

The success of deep learning in high-dimensional settings is often attributed to the presence of low-dimensional structure in real-world data. While standard theoretical models typically assume that this structure lies in the target function, projecting unstructured inputs onto a low-dimensional subspace, data such as images, text or genomic sequences exhibit strong spatial correlations within the input space itself. In this paper, we propose a tractable model to study how these correlations affect the sample complexity of learning with gradient descent on shallow neural networks. Specifically, we consider targets that depend on a small number of latent Boolean variables, and input features grouped into clusters and correlated with the latent variables. Under an identifiability assumption, we show that for a layerwise gradient-descent variant, the sample complexity scales with the number of hidden variables and, when the signal-to-noise ratio is sufficiently high, is independent of the input dimension, up to logarithmic terms. We empirically test our theoretical findings on both synthetic and real data.

cs.LG↗

arXiv2026

Asymmetric graph alignment and the phase transition for asymmetric tree correlation testing

Graph alignment - identifying node correspondences between two graphs - is a fundamental problem with applications in network analysis, biology, and privacy research. While substantial progress has been made in aligning correlated Erdős-Rényi graphs under symmetric settings, real-world networks often exhibit asymmetry in both node numbers and edge densities. In this work, we introduce a novel framework for asymmetric correlated Erdős-Rényi graphs, generalizing existing models to account for these asymmetries. We conduct a rigorous theoretical analysis of graph alignment in the sparse regime, where local neighborhoods exhibit tree-like structures. Our approach leverages tree correlation testing as the central tool in our polynomial-time algorithm, MPAlign, which achieves one-sided partial alignment under certain conditions. A key contribution of our work is characterizing these conditions under which asymmetric tree correlation testing is feasible: If two correlated graphs $G$ and $G'$ have average degrees $λs$ and $λs'$ respectively, where $λ$ is their common density and $s,s'$ are marginal correlation parameters, their tree neighborhoods can be aligned if $ss' > α$, where $α$ denotes Otter's constant and $λ$ is supposed large enough. The feasibility of this tree comparison problem undergoes a sharp phase transition since $ss' \leq α$ implies its impossibility. These new results on tree correlation testing allow us to solve a class of random subgraph isomorphism problems, resolving an open problem in the field.

cs.IT↗

arXiv2026

Unbiased Approximate Vector-Jacobian Products for Efficient Backpropagation

In this work we introduce methods to reduce the computational and memory costs of training deep neural networks. Our approach consists in replacing exact vector-jacobian products by randomized, unbiased approximations thereof during backpropagation. We provide a theoretical analysis of the trade-off between the number of epochs needed to achieve a target precision and the cost reduction for each epoch. We then identify specific unbiased estimates of vector-jacobian products for which we establish desirable optimality properties of minimal variance under sparsity constraints. Finally we provide in-depth experiments on multi-layer perceptrons, BagNets and Visual Transfomers architectures. These validate our theoretical results, and confirm the potential of our proposed unbiased randomized backpropagation approach for reducing the cost of deep learning.

cs.LG↗

arXiv2026

The value of random zero-sum games

We study the value of a two-player zero-sum game on a random matrix $M\in \mathbb{R}^{n\times m}$, defined by $v(M) = \min_{x\inΔ_n}\max_{y\in Δ_m}x^T M y$. In the setting where $n=m$ and $M$ has i.i.d. standard Gaussian entries, we prove that the standard deviation of $v(M)$ is of order $\frac{1}{n}$. This confirms an experimental conjecture dating back to the 1980s. We also investigate the case where $M$ is a rectangular Gaussian matrix with $m = n+λ\sqrt{n}$, showing that the expected value of the game is of order $\fracλ{n}$, as well as the case where $M$ is a random orthogonal matrix. Our techniques are based on probabilistic arguments and convex geometry. We argue that the study of random games could shed new light on various problems in theoretical computer science.

math.PR↗

arXiv2025

Performance of Gaussian Boson Sampling on Planted Bipartite Clique Detection

We investigate whether Gaussian Boson Sampling (GBS) can provide a computational advantage for solving the planted biclique problem, which is a graph problem widely believed to be classically hard when the planted structure is small. Although GBS has been heuristically and experimentally observed to favor sampling dense subgraphs, its theoretical performance on this classically hard problem remains largely unexplored. We focus on a natural statistic derived from GBS output: the frequency with which a node appears in GBS samples, referred to as the node weight. We rigorously analyze whether this signal is strong enough to distinguish planted biclique nodes from background nodes. Our analysis characterizes the distribution of node weights under GBS and quantifies the bias introduced by the planted structure. The results reveal a sharp limitation: when the planted biclique size falls within the conjectured hard regime, the natural fluctuations in node weights dominate the bias signal, making detection unreliable using simple ranking strategies. These findings provide the first rigorous evidence that planted biclique detection may remain computationally hard even under GBS-based quantum computing, and they motivate further investigation into more advanced GBS-based algorithms or other quantum approaches for this problem.

quant-ph↗

arXiv2025

Computational thresholds in high-dimensional statistics: the case of graph alignment

In this article we consider the graph alignment problem from the perspective of high-dimensional statistics: we aim to estimate an unknown permutation $π^*$ from the observation of two correlated random adjacency matrices $A_1$, $A_2$. We establish the following computational thresholds. For $A_1$, $A_2$ the adjacency matrices of two correlated Erdős-Rényi random graphs ${\mathcal {G}}(n,p)$ in the sparse regime with average degree $λ:=np= O(1)$ and edge correlation parameter $s\in(0,1)$, we identify a critical threshold $s^*(λ)$ for $s$ above which a message-passing, local algorithm succeeds at alignment, and below which no local algorithm succeeds. This result crucially depends on an associated model of correlated random trees. We then consider the case where $A_1$, $A_2$ are two correlated Gaussian Wigner matrices with correlation parameter $s=1/\sqrt{1+σ^2}$ for some noise parameter $σ$. For a fast spectral algorithm, we identify the critical scaling for noise parameter $σ$ at which the fraction of entries of $π^*$ correctly recovered goes from $1-o(1)$ to $o(1)$. We next consider the convex relaxation approach which obtains the doubly stochastic matrix $X$ that minimizes $\|X A_1 -A_2 X\|_F$. We obtain upper and lower bounds on the critical noise parameter $σ$ at which a simple post-processing of $X$ correctly recovers a fraction $1-o(1)$ of entries of $π^*$. We finally identify promising future directions on i) computational thresholds for spectral methods and convex relaxation methods of practical interest, and ii) impossibility results for broad classes of algorithms, notably low degree polynomial algorithms and local search algorithms.

math.PR↗

arXiv2025

Graph Alignment via Birkhoff Relaxation

We consider the graph alignment problem, wherein the objective is to find a vertex correspondence between two graphs that maximizes the edge overlap. The graph alignment problem is an instance of the quadratic assignment problem (QAP), known to be NP-hard in the worst case even to approximately solve. In this paper, we analyze Birkhoff relaxation, a tight convex relaxation of QAP, and present theoretical guarantees on its performance when the inputs follow the Gaussian Wigner Model. More specifically, the weighted adjacency matrices are correlated Gaussian Orthogonal Ensemble with correlation $1/\sqrt{1+σ^2}$. Denote the optimal solutions of the QAP and Birkhoff relaxation by $Π^\star$ and $X^\star$ respectively. We show that $\|X^\star-Π^\star\|_F^2 = o(n)$ when $σ= o(n^{-1.25})$ and $\|X^\star-Π^\star\|_F^2 = Ω(n)$ when $σ= Ω(n^{-0.5})$. Thus, the optimal solution $X^\star$ transitions from a small perturbation of $Π^\star$ for small $σ$ to being well separated from $Π^\star$ as $σ$ becomes larger than $n^{-0.5}$. This result allows us to guarantee that simple rounding procedures on $X^\star$ align $1-o(1)$ fraction of vertices correctly whenever $σ= o(n^{-1.25})$. This condition on $σ$ to ensure the success of the Birkhoff relaxation is state-of-the-art.

stat.ML↗

arXiv2025

In-depth Analysis of Low-rank Matrix Factorisation in a Federated Setting

We analyze a distributed algorithm to compute a low-rank matrix factorization on $N$ clients, each holding a local dataset $\mathbf{S}^i \in \mathbb{R}^{n_i \times d}$, mathematically, we seek to solve $min_{\mathbf{U}^i \in \mathbb{R}^{n_i\times r}, \mathbf{V}\in \mathbb{R}^{d \times r} } \frac{1}{2} \sum_{i=1}^N \|\mathbf{S}^i - \mathbf{U}^i \mathbf{V}^\top\|^2_{\text{F}}$. Considering a power initialization of $\mathbf{V}$, we rewrite the previous smooth non-convex problem into a smooth strongly-convex problem that we solve using a parallel Nesterov gradient descent potentially requiring a single step of communication at the initialization step. For any client $i$ in $\{1, \dots, N\}$, we obtain a global $\mathbf{V}$ in $\mathbb{R}^{d \times r}$ common to all clients and a local variable $\mathbf{U}^i$ in $\mathbb{R}^{n_i \times r}$. We provide a linear rate of convergence of the excess loss which depends on $σ_{\max} / σ_{r}$, where $σ_{r}$ is the $r^{\mathrm{th}}$ singular value of the concatenation $\mathbf{S}$ of the matrices $(\mathbf{S}^i)_{i=1}^N$. This result improves the rates of convergence given in the literature, which depend on $σ_{\max}^2 / σ_{\min}^2$. We provide an upper bound on the Frobenius-norm error of reconstruction under the power initialization strategy. We complete our analysis with experiments on both synthetic and real data.

cs.LG↗

arXiv2025

Asynchronous Collective Tree Exploration: a Distributed Algorithm, and a new Lower Bound

We study the problem of collective tree exploration in which a team of $k$ mobile agents must collectively visit all nodes of an unknown tree in as few moves as possible. The agents all start from the root and discover adjacent edges as they progress in the tree. Communication is distributed in the sense that agents share information by reading and writing on whiteboards located at all nodes. Movements are asynchronous, in the sense that the speeds of all agents are controlled by an adversary at all times. All previous competitive guarantees for collective tree exploration are either distributed but synchronous, or asynchronous but centralized. In contrast, we present a distributed asynchronous algorithm that explores any tree of $n$ nodes and depth $D$ in at most $2n+O(k^2 2^kD)$ moves, i.e., with a regret that is linear in $D$, and a variant algorithm with a guarantee in $O(k/\log k)(n+kD)$, i.e., with a competitive ratio in $O(k/\log k)$. We note that our regret guarantee is asymptotically optimal (i.e., $1$-competitive) from the perspective of average-case complexity. We then present a new general lower bound on the competitive ratio of asynchronous collective tree exploration, in $Ω(\log^2 k)$. This lower bound applies to both the distributed and centralized settings, and improves upon the previous lower bound in $Ω(\log k)$.

cs.DS↗

arXiv2025

Adaptive collaboration for online personalized distributed learning with heterogeneous clients

We study the problem of online personalized decentralized learning with $N$ statistically heterogeneous clients collaborating to accelerate local training. An important challenge in this setting is to select relevant collaborators to reduce gradient variance while mitigating the introduced bias. To tackle this, we introduce a gradient-based collaboration criterion, allowing each client to dynamically select peers with similar gradients during the optimization process. Our criterion is motivated by a refined and more general theoretical analysis of the All-for-one algorithm, proved to be optimal in Even et al. (2022) for an oracle collaboration scheme. We derive excess loss upper-bounds for smooth objective functions, being either strongly convex, non-convex, or satisfying the Polyak-Lojasiewicz condition; our analysis reveals that the algorithm acts as a variance reduction method where the speed-up depends on a sufficient variance. We put forward two collaboration methods instantiating the proposed general schema; and we show that one variant preserves the optimality of All-for-one. We validate our results with experiments on synthetic and real datasets.

stat.ML↗

arXiv2025

Learnable Adaptive Time-Frequency Representation via Differentiable Short-Time Fourier Transform

The short-time Fourier transform (STFT) is widely used for analyzing non-stationary signals. However, its performance is highly sensitive to its parameters, and manual or heuristic tuning often yields suboptimal results. To overcome this limitation, we propose a unified differentiable formulation of the STFT that enables gradient-based optimization of its parameters. This approach addresses the limitations of traditional STFT parameter tuning methods, which often rely on computationally intensive discrete searches. It enables fine-tuning of the time-frequency representation (TFR) based on any desired criterion. Moreover, our approach integrates seamlessly with neural networks, allowing joint optimization of the STFT parameters and network weights. The efficacy of the proposed differentiable STFT in enhancing TFRs and improving performance in downstream tasks is demonstrated through experiments on both simulated and real-world data.

cs.SD↗

arXiv2025

Meta-learning of shared linear representations beyond well-specified linear regression

Motivated by multi-task and meta-learning approaches, we consider the problem of learning structure shared by tasks or users, such as shared low-rank representations or clustered structures. While all previous works focus on well-specified linear regression, we consider more general convex objectives, where the structural low-rank and cluster assumptions are expressed on the optima of each function. We show that under mild assumptions such as \textit{Hessian concentration} and \textit{noise concentration at the optimum}, rank and clustered regularized estimators recover such structure, provided the number of samples per task and the number of tasks are large enough. We then study the problem of recovering the subspace in which all the solutions lie, in the setting where there is only a single sample per task: we show that in that case, the rank-constrained estimator can recover the subspace, but that the number of tasks needs to scale exponentially large with the dimension of the subspace. Finally, we provide a polynomial-time algorithm via nuclear norm constraints for learning a shared linear representation in the context of convex learning objectives.

cs.LG↗

arXiv2025

Concentration of Non-Isotropic Random Tensors with Applications to Learning and Empirical Risk Minimization

Dimension is an inherent bottleneck to some modern learning tasks, where optimization methods suffer from the size of the data. In this paper, we study non-isotropic distributions of data and develop tools that aim at reducing these dimensional costs by a dependency on an effective dimension rather than the ambient one. Based on non-asymptotic estimates of the metric entropy of ellipsoids -- that prove to generalize to infinite dimensions -- and on a chaining argument, our uniform concentration bounds involve an effective dimension instead of the global dimension, improving over existing results. We show the importance of taking advantage of non-isotropic properties in learning problems with the following applications: i) we improve state-of-the-art results in statistical preconditioning for communication-efficient distributed optimization, ii) we introduce a non-isotropic randomized smoothing for non-smooth optimization. Both applications cover a class of functions that encompasses empirical risk minization (ERM) for linear models.

stat.ML↗

arXiv2024

Barely Random Algorithms and Collective Metrical Task Systems

We consider metrical task systems on general metric spaces with $n$ points, and show that any fully randomized algorithm can be turned into a randomized algorithm that uses only $2\log n$ random bits, and achieves the same competitive ratio up to a factor $2$. This provides the first order-optimal barely random algorithms for metrical task systems, i.e., which use a number of random bits that does not depend on the number of requests addressed to the system. We discuss implications on various aspects of online decision-making such as: distributed systems, advice complexity, and transaction costs, suggesting broad applicability. We put forward an equivalent view that we call collective metrical task systems where $k$ agents in a metrical task system team up, and suffer the average cost paid by each agent. Our results imply that such a team can be $O(\log^2 n)$-competitive as soon as $k\geq n^2$. In comparison, a single agent is always $Ω(n)$-competitive.

cs.DS↗