arXiv subjects

Guillaume Garrigos

Publications and source records attributed to Guillaume Garrigos.

At least 19 recordsLinked to original sources

arXiv2026

Muon Does Not Converge on Convex Lipschitz Functions

Muon and its variants have shown strong empirical performance in a variety of deep learning tasks. Existing convergence analyses of Muon rely on smoothness assumptions, though arguably the most successful function class for developing deep learning methods (such as AdaGrad, Shampoo, Schedule-Free and more) has been the class of convex and Lipschitz functions. In this paper we question whether the classical convex Lipschitz model is a useful one for understanding Muon. Our answer is no. We show that Muon does not converge on the class of convex and Lipschitz functions, regardless of the choice of learning rate schedule. We also show that error feedback restores convergence of Muon and all the non-Euclidean subgradient methods with momentum. However, this theoretical fix using error feedback degrades the performance of Muon in two representative settings for image classification (CIFAR-10) and language modeling (nanoGPT on FineWeb-Edu 10B). Our conclusion is that convex Lipschitz theory, despite having a prominent role in the design of practical methods for deep learning, is not the most suited one for Muon. This suggests that Muon's success must come from structure absent from this model, most plausibly related to smoothness.

cs.LG↗

arXiv2026

Bias-Optimal Bounds for SGD: A Computer-Aided Lyapunov Analysis

The non-asymptotic analysis of Stochastic Gradient Descent (SGD) typically yields bounds that decompose into a bias term and a variance term. In this work, we focus on the bias component and study the extent to which SGD can match the optimal convergence behavior of deterministic gradient descent. Assuming only (strong) convexity and smoothness of the objective, we derive new bounds that are bias-optimal, in the sense that the bias term coincides with the worst-case rate of gradient descent. Our results hold for the full range of constant step-sizes $γL \in (0,2)$, including critical and large step-size regimes that were previously unexplored without additional variance assumptions. The bounds are obtained through the construction of a simple Lyapunov energy whose monotonicity yields sharp convergence guarantees. To design the parameters of this energy, we employ the Performance Estimation Problem framework, which we also use to provide numerical evidence for the optimality of the associated variance terms.

math.OC↗

arXiv2025

Tracking the Median of Gradients with a Stochastic Proximal Point Method

There are several applications of stochastic optimization where one can benefit from a robust estimate of the gradient. For example, domains such as distributed learning with corrupted nodes, the presence of large outliers in the training data, learning under privacy constraints, or even heavy-tailed noise due to the dynamics of the algorithm itself. Here we study SGD with robust gradient estimators based on estimating the median. We first derive iterative methods based on the stochastic proximal point method for computing the median gradient and generalizations thereof. Then we propose an algorithm estimating the median gradient across iterations, and find that several well known methods are particular cases of this framework. For instance, we observe that different forms of clipping allow to compute online estimators of the median of gradients, in contrast to (heavy-ball) momentum, which corresponds to an online estimator of the mean. Finally, we provide a theoretical framework for an algorithm computing the median gradient across samples, and show that the resulting method can converge even under heavy-tailed, state-dependent noise.

stat.ML↗

arXiv2025

Last-Iterate Complexity of SGD for Convex and Smooth Stochastic Problems

Most results on Stochastic Gradient Descent (SGD) in the convex and smooth setting are presented under the form of bounds on the ergodic function value gap. It is an open question whether bounds can be derived directly on the last iterate of SGD in this context. Recent advances suggest that it should be possible. For instance, it can be achieved by making the additional, yet unverifiable, assumption that the variance of the stochastic gradients is uniformly bounded. In this paper, we show that there is no need of such an assumption, and that SGD enjoys a $\tilde O \left( T^{-1/2} \right)$ last-iterate complexity rate for convex smooth stochastic problems.

math.OC↗

arXiv2025

Analysis of an Idealized Stochastic Polyak Method and its Application to Black-Box Model Distillation

We provide a general convergence theorem of an idealized stochastic Polyak step size called SPS$^*$. Besides convexity, we only assume a local expected gradient bound, that includes locally smooth and locally Lipschitz losses as special cases. We refer to SPS$^*$ as idealized because it requires access to the loss for every training batch evaluated at a solution. It is also ideal, in that it achieves the optimal lower bound for globally Lipschitz function, and is the first Polyak step size to have an $O(1/\sqrt{t})$ anytime convergence in the smooth setting. We show how to combine SPS$^*$ with momentum to achieve the same favorable rates for the last iterate. We conclude with several experiments to validate our theory, and a more practical setting showing how we can distill a teacher GPT-2 model into a smaller student model without any hyperparameter tuning.

cs.LG↗

arXiv2024

Handbook of Convergence Theorems for (Stochastic) Gradient Methods

This is a handbook of simple proofs of the convergence of gradient and stochastic gradient descent type methods. We consider functions that are Lipschitz, smooth, convex, strongly convex, and/or Polyak-Łojasiewicz functions. Our focus is on ``good proofs'' that are also simple. Each section can be consulted separately. We start with proofs of gradient descent, then on stochastic variants, including minibatching and momentum. Then move on to nonsmooth problems with the subgradient method, the proximal gradient descent and their stochastic variants. Our focus is on global convergence rates and complexity rates. Some slightly less common proofs found here include that of SGD (Stochastic gradient descent) with a proximal step, with momentum, and with mini-batching without replacement.

math.OC↗

arXiv2024

Multivariate Online Linear Regression for Hierarchical Forecasting

In this paper, we consider a deterministic online linear regression model where we allow the responses to be multivariate. To address this problem, we introduce MultiVAW, a method that extends the well-known Vovk-Azoury-Warmuth algorithm to the multivariate setting, and show that it also enjoys logarithmic regret in time. We apply our results to the online hierarchical forecasting problem and recover an algorithm from this literature as a special case, allowing us to relax the hypotheses usually made for its analysis.

stat.ML↗

arXiv2023

Provable convergence guarantees for black-box variational inference

Black-box variational inference is widely used in situations where there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs: namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides rigorous guarantees that methods similar to those used in practice converge on realistic inference problems.

cs.LG↗

arXiv2023

Function Value Learning: Adaptive Learning Rates Based on the Polyak Stepsize and Function Splitting in ERM

Here we develop variants of SGD (stochastic gradient descent) with an adaptive step size that make use of the sampled loss values. In particular, we focus on solving a finite sum-of-terms problem, also known as empirical risk minimization. We first detail an idealized adaptive method called $\texttt{SPS}_+$ that makes use of the sampled loss values and assumes knowledge of the sampled loss at optimality. This $\texttt{SPS}_+$ is a minor modification of the SPS (Stochastic Polyak Stepsize) method, where the step size is enforced to be positive. We then show that $\texttt{SPS}_+$ achieves the best known rates of convergence for SGD in the Lipschitz non-smooth. We then move onto to develop $\texttt{FUVAL}$, a variant of $\texttt{SPS}_+$ where the loss values at optimality are gradually learned, as opposed to being given. We give three viewpoints of $\texttt{FUVAL}$, as a projection based method, as a variant of the prox-linear method, and then as a particular online SGD method. We then present a convergence analysis of $\texttt{FUVAL}$ and experimental results. The shortcomings of our work is that the convergence analysis of $\texttt{FUVAL}$ shows no advantage over SGD. Another shortcomming is that currently only the full batch version of $\texttt{FUVAL}$ shows a minor advantages of GD (Gradient Descent) in terms of sensitivity to the step size. The stochastic version shows no clear advantage over SGD. We conjecture that large mini-batches are required to make $\texttt{FUVAL}$ competitive. Currently the new $\texttt{FUVAL}$ method studied in this paper does not offer any clear theoretical or practical advantage. We have chosen to make this draft available online nonetheless because of some of the analysis techniques we use, such as the non-smooth analysis of $\texttt{SPS}_+$, and also to show an apparently interesting approach that currently does not work.

cs.LG↗

arXiv2023

Online Inventory Problems: Beyond the i.i.d. Setting with Online Convex Optimization

We study multi-product inventory control problems where a manager makes sequential replenishment decisions based on partial historical information in order to minimize its cumulative losses. Our motivation is to consider general demands, losses and dynamics to go beyond standard models which usually rely on newsvendor-type losses, fixed dynamics, and unrealistic i.i.d. demand assumptions. We propose MaxCOSD, an online algorithm that has provable guarantees even for problems with non-i.i.d. demands and stateful dynamics, including for instance perishability. We consider what we call non-degeneracy assumptions on the demand process, and argue that they are necessary to allow learning.

math.OC↗

arXiv2023

Square distance functions are Polyak-Łojasiewicz and vice-versa

This short note gathers known results to state that the squared distance function to a (nonconvex) closed set of an Euclidean space is Polyak-Łojasiewicz. As a fuzzy reciprocate, we also recall that every Polyak-Łojasiewicz function can be bounded from below by the squared distance function to its set of minimizers.

math.OC↗

arXiv2022

SAN: Stochastic Average Newton Algorithm for Minimizing Finite Sums

We present a principled approach for designing stochastic Newton methods for solving finite sum optimization problems. Our approach has two steps. First, we re-write the stationarity conditions as a system of nonlinear equations that associates each data point to a new row. Second, we apply a Subsampled Newton Raphson method to solve this system of nonlinear equations. Using our approach, we develop a new Stochastic Average Newton (SAN) method, which is incremental by design, in that it requires only a single data point per iteration. It is also cheap to implement when solving regularized generalized linear models, with a cost per iteration of the order of the number of the parameters. We show through extensive numerical experiments that SAN requires no knowledge about the problem, neither parameter tuning, while remaining competitive as compared to classical variance reduced gradient methods (e.g. SAG and SVRG), incremental Newton and quasi-Newton methods (e.g. SNM, IQN).

math.OC↗

arXiv2020

Convergence of the Forward-Backward Algorithm: Beyond the Worst Case with the Help of Geometry

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.

math.OC↗

arXiv2019

Accelerated iterative regularization via dual diagonal descent

We propose and analyze an accelerated iterative dual diagonal descent algorithm for the solution of linear inverse problems with general regularization and data-fit functions. In particular, we develop an inertial approach of which we analyze both convergence and stability. Using tools from inexact proximal calculus, we prove early stopping results with optimal convergence rates for additive data-fit terms as well as more general cases, such as the Kullback-Leibler divergence, for which different type of proximal point approximations hold.

math.OC↗

arXiv2019

Model Consistency for Learning with Mirror-Stratifiable Regularizers

Low-complexity non-smooth convex regularizers are routinely used to impose some structure (such as sparsity or low-rank) on the coefficients for linear predictors in supervised learning. Model consistency consists then in selecting the correct structure (for instance support or rank) by regularized empirical risk minimization. It is known that model consistency holds under appropriate non-degeneracy conditions. However such conditions typically fail for highly correlated designs and it is observed that regularization methods tend to select larger models. In this work, we provide the theoretical underpinning of this behavior using the notion of mirror-stratifiable regularizers. This class of regularizers encompasses the most well-known in the literature, including the $\ell_1$ or trace norms. It brings into play a pair of primal-dual models, which in turn allows one to locate the structure of the solution using a specific dual certificate. We also show how this analysis is applicable to optimal solutions of the learning problem, and also to the iterates computed by a certain class of stochastic proximal-gradient algorithms.

math.OC↗

arXiv2018

Sparse Multiple Kernel Learning: Support Identification via Mirror Stratifiability

In statistical machine learning, kernel methods allow to consider infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done by solving an optimization problem depending on a data fit term and a suitable regularizer. In this paper we consider feature maps which are the concatenation of a fixed, possibly large, set of simpler feature maps. The penalty is a sparsity inducing one, promoting solutions depending only on a small subset of the features. The group lasso problem is a special case of this more general setting. We show that one of the most popular optimization algorithms to solve the regularized objective function, the forward-backward splitting method, allows to perform feature selection in a stable manner. In particular, we prove that the set of relevant features is identified by the algorithm after a finite number of iterations if a suitable qualification condition holds. The main tools used in the proofs are the notions of stratification and mirror stratifiability.

math.OC↗

arXiv2017

Thresholding gradient methods in Hilbert spaces: support identification and linear convergence

We study $\ell^1$ regularized least squares optimization problem in a separable Hilbert space. We show that the iterative soft-thresholding algorithm (ISTA) converges linearly, without making any assumption on the linear operator into play or on the problem. The result is obtained combining two key concepts: the notion of extended support, a finite set containing the support, and the notion of conditioning over finite dimensional sets. We prove that ISTA identifies the solution extended support after a finite number of iterations, and we derive linear convergence from the conditioning property, which is always satisfied for $\ell^1$ regularized least squares problems. Our analysis extends to the the entire class of thresholding gradient algorithms, for which we provide a conceptually new proof of strong convergence, as well as convergence rates.

math.OC↗

arXiv2017

Iterative regularization via dual diagonal descent

In the context of linear inverse problems, we propose and study a general iterative regularization method allowing to consider large classes of regularizers and data-fit terms. The algorithm we propose is based on a primal-dual diagonal {descent} method. Our analysis establishes convergence as well as stability results. Theoretical findings are complemented with numerical experiments showing state of the art performances.

math.OC↗