arXiv subjects

Guillaume Lauga

Publications and source records attributed to Guillaume Lauga.

10 recordsLinked to original sources

arXiv2026

Proximal basin hopping: global optimization with guarantees

Global optimization is a challenging problem, with plenty of algorithms displaying empirical success, but scarce theoretical backing. In this work, we propose a new theoretical framework called Proximal Basin Hopping (PBH), carefully tailored to combine proximal optimization and local minimization. We use it to construct a practical algorithm that converges to the global minimizer with high probability, when using a finite amount of samples. Proximal Basin Hopping outperforms well known algorithms with theoretical backing on standard synthetic hard functions, and real problems such as fitting scaling laws for deep learning. Furthermore, the higher the dimension, the better the performance gap.

cs.LG↗

arXiv2026

Characterizations of inexact proximal operators

Proximal operators are now ubiquitous in non-smooth optimization. Since their introduction in the seminal work of Moreau, many papers have shown their effectiveness on a wide variety of problems, culminating in their use to construct convergent deep learning methods. The characterization of these operators for non-convex penalties was completed recently in [Gribonval et al, A characterization of proximity operators, 2020]. In this paper, we propose to follow this line of work by characterizing inexact proximal operators, thus providing an answer to what constitutes a good approximation of these operators. We propose several definitions of approximations and discuss their regularity, approximation power, and their fixed points. Equipped with these characterizations, we investigate the convergence of proximal algorithms in the presence of errors that may be non-summable and/or non-vanishing. In particular, we look at the proximal point algorithm, and at the forward-backward, Peaceman-Rachford and Douglas-Rachford algorithms when we minimize the sum of a weakly convex function (whose proximal operator is approximated) and a strongly convex function.

math.OC↗

arXiv2026

A block-coordinate descent framework for non-convex composite optimization. Application to sparse precision matrix estimation

Block-coordinate descent (BCD) is the method of choice to solve numerous large scale optimization problems, however their theoretical study for non-convex optimization, has received less attention. In this paper, we present a new block-coordinate descent (BCD) framework to tackle non-convex composite optimization problems, ensuring decrease of the objective function and convergence to a solution. This framework is general enough to include variable metric proximal gradient updates, proximal Newton updates, and alternated minimization updates. This generality allows to encompass three versions of the most used solvers in the sparse precision matrix estimation problem, deemed Graphical Lasso: graphical ISTA, Primal GLasso, and QUIC. We demonstrate the value of this new framework on non-convex sparse precision matrix estimation problems, providing convergence guarantees and up to a $100$-fold reduction in the number of iterations required to reach state-of-the-art estimation quality.

cs.LG↗

arXiv2025

A flexible block-coordinate forward-backward algorithm for non-smooth and non-convex optimization

Block coordinate descent (BCD) methods are prevalent in large scale optimization problems due to the low memory and computational costs per iteration, the predisposition to parallelization, and the ability to exploit the structure of the problem. The theoretical and practical performance of BCD relies heavily on the rules defining the choice of the blocks to be updated at each iteration. We propose a new deterministic BCD framework that allows for very flexible updates, while guaranteeing state-of-the-art convergence guarantees on non-smooth nonconvex optimization problems. While encompassing several update rules from the literature, this framework allows for priority on updates of particular blocks and correlations in the block selection between iterations, which is not permitted under the classical convergent stochastic framework. This flexibility is leveraged in the context of multilevel optimization algorithms and, in particular, in multilevel image restoration problems, where the efficiency of the approach is illustrated.

math.OC↗

arXiv2025

A multilevel approach to accelerate the training of Transformers

In this article, we investigate the potential of multilevel approaches to accelerate the training of transformer architectures. Using an ordinary differential equation (ODE) interpretation of these architectures, we propose an appropriate way of varying the discretization of these ODE Transformers in order to accelerate the training. We validate our approach experimentally by a comparison with the standard training procedure.

cs.LG↗

arXiv2024

Sparsity in neural networks can improve their privacy

This article measures how sparsity can make neural networks more robust to membership inference attacks. The obtained empirical results show that sparsity improves the privacy of the network, while preserving comparable performances on the task at hand. This empirical study completes and extends existing literature.

cs.LG↗

arXiv2024

Can sparsity improve the privacy of neural networks?

Sparse neural networks are mainly motivated by ressource efficiency since they use fewer parameters than their dense counterparts but still reach comparable accuracies. This article empirically investigates whether sparsity could also improve the privacy of the data used to train the networks. The experiments show positive correlations between the sparsity of the model, its privacy, and its classification error. Simply comparing the privacy of two models with different sparsity levels can yield misleading conclusions on the role of sparsity, because of the additional correlation with the classification error. From this perspective, some caveats are raised about previous works that investigate sparsity and privacy.

cs.CR↗

arXiv2024

IML FISTA: A Multilevel Framework for Inexact and Inertial Forward-Backward. Application to Image Restoration

This paper presents a multilevel framework for inertial and inexact proximal algorithms, that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems and we show that it considerably accelerates the convergence of the corresponding one-level (i.e. standard) version of the methods, for large-scale images.

math.OC↗

arXiv2024

A multilevel framework for accelerating uSARA in radio-interferometric imaging

This paper presents a multilevel algorithm specifically designed for radio-interferometric imaging in astronomy. The proposed algorithm is used to solve the uSARA (unconstrained Sparsity Averaging Reweighting Analysis) formulation of this image restoration problem. Multilevel algorithms rely on a hierarchy of approximations of the objective function to accelerate its optimization. In contrast to the usual multilevel approaches where this hierarchy is derived in the parameter space, here we construct the hierarchy of approximations in the observation space. The proposed approach is compared to a reweighted forward-backward procedure, which is the backbone iteration scheme for solving the uSARA problem.

math.OC↗

arXiv2022

Multilevel fista for image restoration

This paper presents a multilevel FISTA algorithm, based on the use of the Moreau envelope to build the correction brought by the coarse models, which is easy to compute when the explicit form of the proximal operator of the considered functions is known. This approach is supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We evaluate our approach on image restoration problems and we show that it outperforms classical FISTA for large-scale images.

math.OC↗