arXiv subjects

Guillaume Braun

Publications and source records attributed to Guillaume Braun.

11 recordsLinked to original sources

arXiv2026

Overfitting of Spectral Gradient Descent: How Matrix Geometry shapes Generalization and Implicit Bias

We study the generalization of spectral gradient descent (SpecGD) in overparameterized matrix classification with corrupted labels. Each input combines a shared low-rank signal with a rank-one sample-specific perturbation, referred to as a shortcut, that enables memorization but does not generalize. We contrast collapsed shortcuts, which share a singular direction, with dispersed shortcuts, which occupy distinct singular directions. Changing only this geometry can reverse the relative generalization of GD and SpecGD: collapsed shortcuts can favor SpecGD, while dispersed shortcuts can favor GD. In the dispersed regime, exact shortcut orthogonality eliminates the signal from the late-stage SpecGD direction, while vanishing random correlations collectively generate a small but generalization-relevant signal through a second-order effect. To identify the direction selected by SpecGD, which the spectral max-margin problem alone does not determine, we combine a refined analysis of its dual with the exponentiated-gradient dynamics of normalized loss weights. Finally, we show that a single SpecGD step can already interpolate and generalize well, while continued training converges to a direction with substantially worse generalization.

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arXiv2026

Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval

Spectral gradient methods, such as the Muon optimizer, modify gradient updates by preserving directional information while discarding scale, and have shown strong empirical performance in deep learning. We investigate the mechanisms underlying these gains through a dynamical analysis of a nonlinear phase retrieval model with anisotropic Gaussian inputs, equivalent to training a two-layer neural network with the quadratic activation and fixed second-layer weights. Focusing on a spiked covariance setting where the dominant variance direction is orthogonal to the signal, we show that gradient descent (GD) suffers from a variance-induced misalignment: during the early escaping stage, the high-variance but uninformative spike direction is multiplicatively amplified, degrading alignment with the true signal under strong anisotropy. In contrast, spectral gradient descent (SpecGD) removes this spike amplification effect, leading to stable alignment and accelerated noise contraction. Numerical experiments confirm the theory and show that these phenomena persist under broader anisotropic covariances.

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arXiv2026

Neuron Block Dynamics for XOR Classification with Zero-Margin

The ability of neural networks to learn useful features through stochastic gradient descent (SGD) is a cornerstone of their success. Most theoretical analyses focus on regression or on classification tasks with a positive margin, where worst-case gradient bounds suffice. In contrast, we study zero-margin nonlinear classification by analyzing the Gaussian XOR problem, where inputs are Gaussian and the XOR decision boundary determines labels. In this setting, a non-negligible fraction of data lies arbitrarily close to the boundary, breaking standard margin-based arguments. Building on Glasgow's (2024) analysis, we extend the study of training dynamics from discrete to Gaussian inputs and develop a framework for the dynamics of neuron blocks. We show that neurons cluster into four directions and that block-level signals evolve coherently, a phenomenon essential in the Gaussian setting where individual neuron signals vary significantly. Leveraging this block perspective, we analyze generalization without relying on margin assumptions, adopting an average-case view that distinguishes regions of reliable prediction from regions of persistent error. Numerical experiments confirm the predicted two-phase block dynamics and demonstrate their robustness beyond the Gaussian setting.

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arXiv2025

Fast Escape, Slow Convergence: Learning Dynamics of Phase Retrieval under Power-Law Data

Scaling laws describe how learning performance improves with data, compute, or training time, and have become a central theme in modern deep learning. We study this phenomenon in a canonical nonlinear model: phase retrieval with anisotropic Gaussian inputs whose covariance spectrum follows a power law. Unlike the isotropic case, where dynamics collapse to a two-dimensional system, anisotropy yields a qualitatively new regime in which an infinite hierarchy of coupled equations governs the evolution of the summary statistics. We develop a tractable reduction that reveals a three-phase trajectory: (i) fast escape from low alignment, (ii) slow convergence of the summary statistics, and (iii) spectral-tail learning in low-variance directions. From this decomposition, we derive explicit scaling laws for the mean-squared error, showing how spectral decay dictates convergence times and error curves. Experiments confirm the predicted phases and exponents. These results provide the first rigorous characterization of scaling laws in nonlinear regression with anisotropic data, highlighting how anisotropy reshapes learning dynamics.

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arXiv2025

Learning a Single Index Model from Anisotropic Data with vanilla Stochastic Gradient Descent

We investigate the problem of learning a Single Index Model (SIM)- a popular model for studying the ability of neural networks to learn features - from anisotropic Gaussian inputs by training a neuron using vanilla Stochastic Gradient Descent (SGD). While the isotropic case has been extensively studied, the anisotropic case has received less attention and the impact of the covariance matrix on the learning dynamics remains unclear. For instance, Mousavi-Hosseini et al. (2023b) proposed a spherical SGD that requires a separate estimation of the data covariance matrix, thereby oversimplifying the influence of covariance. In this study, we analyze the learning dynamics of vanilla SGD under the SIM with anisotropic input data, demonstrating that vanilla SGD automatically adapts to the data's covariance structure. Leveraging these results, we derive upper and lower bounds on the sample complexity using a notion of effective dimension that is determined by the structure of the covariance matrix instead of the input data dimension.

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arXiv2024

Seeded graph matching for the correlated Gaussian Wigner model via the projected power method

In the \emph{graph matching} problem we observe two graphs $G,H$ and the goal is to find an assignment (or matching) between their vertices such that some measure of edge agreement is maximized. We assume in this work that the observed pair $G,H$ has been drawn from the Correlated Gaussian Wigner (CGW) model -- a popular model for correlated weighted graphs -- where the entries of the adjacency matrices of $G$ and $H$ are independent Gaussians and each edge of $G$ is correlated with one edge of $H$ (determined by the unknown matching) with the edge correlation described by a parameter $σ\in [0,1)$. In this paper, we analyse the performance of the \emph{projected power method} (PPM) as a \emph{seeded} graph matching algorithm where we are given an initial partially correct matching (called the seed) as side information. We prove that if the seed is close enough to the ground-truth matching, then with high probability, PPM iteratively improves the seed and recovers the ground-truth matching (either partially or exactly) in $\mathcal{O}(\log n)$ iterations. Our results prove that PPM works even in regimes of constant $σ$, thus extending the analysis in (Mao et al. 2023) for the sparse Correlated Erdos-Renyi(CER) model to the (dense) CGW model. As a byproduct of our analysis, we see that the PPM framework generalizes some of the state-of-art algorithms for seeded graph matching. We support and complement our theoretical findings with numerical experiments on synthetic data.

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arXiv2024

VEC-SBM: Optimal Community Detection with Vectorial Edges Covariates

Social networks are often associated with rich side information, such as texts and images. While numerous methods have been developed to identify communities from pairwise interactions, they usually ignore such side information. In this work, we study an extension of the Stochastic Block Model (SBM), a widely used statistical framework for community detection, that integrates vectorial edges covariates: the Vectorial Edges Covariates Stochastic Block Model (VEC-SBM). We propose a novel algorithm based on iterative refinement techniques and show that it optimally recovers the latent communities under the VEC-SBM. Furthermore, we rigorously assess the added value of leveraging edge's side information in the community detection process. We complement our theoretical results with numerical experiments on synthetic and semi-synthetic data.

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arXiv2023

Strong Consistency Guarantees for Clustering High-Dimensional Bipartite Graphs with the Spectral Method

In this work, we focus on the Bipartite Stochastic Block Model (BiSBM), a popular model for bipartite graphs with a community structure. We consider the high dimensional setting where the number $n_1$ of type I nodes is far smaller than the number $n_2$ of type II nodes. The recent work of Braun and Tyagi (2022) established a sufficient and necessary condition on the sparsity level $p_{max}$ of the bipartite graph to be able to recover the latent partition of type I nodes. They proposed an iterative method that extends the one proposed by Ndaoud et al. (2022) to achieve this goal. Their method requires a good enough initialization, usually obtained by a spectral method, but empirical results showed that the refinement algorithm doesn't improve much the performance of the spectral method. This suggests that the spectral achieves exact recovery in the same regime as the refinement method. We show that it is indeed the case by providing new entrywise bounds on the eigenvectors of the similarity matrix used by the spectral method. Our analysis extend the framework of Lei (2019) that only applies to symmetric matrices with limited dependencies. As an important technical step, we also derive an improved concentration inequality for similarity matrices.

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arXiv2023

Minimax Optimal Clustering of Bipartite Graphs with a Generalized Power Method

Clustering bipartite graphs is a fundamental task in network analysis. In the high-dimensional regime where the number of rows $n_1$ and the number of columns $n_2$ of the associated adjacency matrix are of different order, existing methods derived from the ones used for symmetric graphs can come with sub-optimal guarantees. Due to increasing number of applications for bipartite graphs in the high dimensional regime, it is of fundamental importance to design optimal algorithms for this setting. The recent work of Ndaoud et al. (2022) improves the existing upper-bound for the misclustering rate in the special case where the columns (resp. rows) can be partitioned into $L = 2$ (resp. $K = 2$) communities. Unfortunately, their algorithm cannot be extended to the more general setting where $K \neq L \geq 2$. We overcome this limitation by introducing a new algorithm based on the power method. We derive conditions for exact recovery in the general setting where $K \neq L \geq 2$, and show that it recovers the result in Ndaoud et al. (2022). We also derive a minimax lower bound on the misclustering error when $K = L$ under a symmetric version of our model, which matches the corresponding upper bound up to a factor depending on $K$.

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arXiv2022

An iterative clustering algorithm for the Contextual Stochastic Block Model with optimality guarantees

Real-world networks often come with side information that can help to improve the performance of network analysis tasks such as clustering. Despite a large number of empirical and theoretical studies conducted on network clustering methods during the past decade, the added value of side information and the methods used to incorporate it optimally in clustering algorithms are relatively less understood. We propose a new iterative algorithm to cluster networks with side information for nodes (in the form of covariates) and show that our algorithm is optimal under the Contextual Symmetric Stochastic Block Model. Our algorithm can be applied to general Contextual Stochastic Block Models and avoids hyperparameter tuning in contrast to previously proposed methods. We confirm our theoretical results on synthetic data experiments where our algorithm significantly outperforms other methods, and show that it can also be applied to signed graphs. Finally we demonstrate the practical interest of our method on real data.

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arXiv2021

Clustering multilayer graphs with missing nodes

Relationship between agents can be conveniently represented by graphs. When these relationships have different modalities, they are better modelled by multilayer graphs where each layer is associated with one modality. Such graphs arise naturally in many contexts including biological and social networks. Clustering is a fundamental problem in network analysis where the goal is to regroup nodes with similar connectivity profiles. In the past decade, various clustering methods have been extended from the unilayer setting to multilayer graphs in order to incorporate the information provided by each layer. While most existing works assume - rather restrictively - that all layers share the same set of nodes, we propose a new framework that allows for layers to be defined on different sets of nodes. In particular, the nodes not recorded in a layer are treated as missing. Within this paradigm, we investigate several generalizations of well-known clustering methods in the complete setting to the incomplete one and prove some consistency results under the Multi-Layer Stochastic Block Model assumption. Our theoretical results are complemented by thorough numerical comparisons between our proposed algorithms on synthetic data, and also on real datasets, thus highlighting the promising behaviour of our methods in various settings.

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