Proximal Average Approximated Incremental Gradient Method for Composite Penalty Regularized Empirical Risk Minimization *

Abstract Proximal average (PA) is an approximation technique proposed recently to handle nonsmooth composite regularizer in empirical risk minimization problem. For nonsmooth composite regularizer, it is often difficult to directly derive the corresponding proximal update when solving with popular proximal update. While traditional approaches resort to complex splitting methods like ADMM, proximal average provides an alternative, featuring the tractability of implementation and theoretical analysis. Nevertheless, compared to SDCA-ADMM and SAG-ADMM which are examples of ADMM-based methods achieving faster convergence rate and low per-iteration complexity, existing PA-based approaches either converge slowly (e.g. PA-ASGD) or suffer from high per-iteration cost (e.g. PA-APG). In this paper, we therefore propose a new PA-based algorithm called PA-SAGA, which is optimal in both convergence rate and per-iteration cost, by incorporating into incremental gradient-based framework

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Efficient Methods For Large-Scale Empirical Risk Minimization

Empirical risk minimization (ERM) problems express optimal classifiers as solutions of optimization problems in which the objective is the sum of a very large number of sample costs. An evident obstacle in using traditional descent algorithms for solving this class of problems is their prohibitive computational complexity when the number of component functions in the ERM problem is large. The main goal of this thesis is to study different approaches to solve these large-scale ERM problems. We begin by focusing on incremental and stochastic methods which split the training samples into smaller sets across time to lower the computation burden of traditional descent algorithms. We develop and analyze convergent stochastic variants of quasi-Newton methods which do not require computation of the objective Hessian and approximate the curvature using only gradient information. We show that the curvature approximation in stochastic quasi-Newton methods leads to faster convergence relative to first-order stochastic methods when the problem is ill-conditioned. We culminate with the introduction of an incremental method that exploits memory to achieve a superlinear convergence rate. This is the best known convergence rate for an incremental method. An alternative strategy for lowering the prohibitive cost of solving large-scale ERM problems is decentralized optimization whereby samples are separated not across time but across multiple nodes of a network. In this regime, the main contribution of this thesis is in incorporating second-order information of the aggregate risk corresponding to samples of all nodes in the network in a way that can be implemented in a distributed fashion. We also explore the separation of samples across both, time and space, to reduce the computational and communication cost for solving large-scale ERM problems. We study this path by introducing a decentralized stochastic method which incorporates the idea of stochastic averaging gradient leading to a low computational complexity method with a fast linear convergence rate. We then introduce a rethinking of ERM in which we consider not a partition of the training set as in the case of stochastic and distributed optimization, but a nested collection of subsets that we grow geometrically. The key insight is that the optimal argument associated with a training subset of a certain size is not that far from the optimal argument associated with a larger training subset. Based on this insight, we present adaptive sample size schemes which start with a small number of samples and solve the corresponding ERM problem to its statistical accuracy. The sample size is then grown geometrically and use the solution of the previous ERM as a warm start for the new ERM. Theoretical analyses show that the use of adaptive sample size methods reduces the overall computational cost of achieving the statistical accuracy of the whole dataset for a broad range of deterministic and stochastic first-order methods. We further show that if we couple the adaptive sample size scheme with Newton\u27s method, it is possible to consider subsequent doubling of the training set and perform a single Newton iteration in between. This is possible because of the interplay between the statistical accuracy and the quadratic convergence region of these problems and yields a method that is guaranteed to solve an ERM problem by performing just two passes over the dataset

Apprentissage à grande échelle et applications

This thesis presents my main research activities in statistical machine learning aftermy PhD, starting from my post-doc at UC Berkeley to my present research position atInria Grenoble. The first chapter introduces the context and a summary of my scientificcontributions and emphasizes the importance of pluri-disciplinary research. For instance,mathematical optimization has become central in machine learning and the interplay betweensignal processing, statistics, bioinformatics, and computer vision is stronger thanever. With many scientific and industrial fields producing massive amounts of data, theimpact of machine learning is potentially huge and diverse. However, dealing with massivedata raises also many challenges. In this context, the manuscript presents differentcontributions, which are organized in three main topics.Chapter 2 is devoted to large-scale optimization in machine learning with a focus onalgorithmic methods. We start with majorization-minimization algorithms for structuredproblems, including block-coordinate, incremental, and stochastic variants. These algorithmsare analyzed in terms of convergence rates for convex problems and in terms ofconvergence to stationary points for non-convex ones. We also introduce fast schemesfor minimizing large sums of convex functions and principles to accelerate gradient-basedapproaches, based on Nesterov’s acceleration and on Quasi-Newton approaches.Chapter 3 presents the paradigm of deep kernel machine, which is an alliance betweenkernel methods and multilayer neural networks. In the context of visual recognition, weintroduce a new invariant image model called convolutional kernel networks, which is anew type of convolutional neural network with a reproducing kernel interpretation. Thenetwork comes with simple and effective principles to do unsupervised learning, and iscompatible with supervised learning via backpropagation rules.Chapter 4 is devoted to sparse estimation—that is, the automatic selection of modelvariables for explaining observed data; in particular, this chapter presents the result ofpluri-disciplinary collaborations in bioinformatics and neuroscience where the sparsityprinciple is a key to build intepretable predictive models.Finally, the last chapter concludes the manuscript and suggests future perspectives.Ce mémoire présente mes activités de recherche en apprentissage statistique après mathèse de doctorat, dans une période allant de mon post-doctorat à UC Berkeley jusqu’àmon activité actuelle de chercheur chez Inria. Le premier chapitre fournit un contextescientifique dans lequel s’inscrivent mes travaux et un résumé de mes contributions, enmettant l’accent sur l’importance de la recherche pluri-disciplinaire. L’optimisation mathématiqueest ainsi devenue un outil central en apprentissage statistique et les interactionsavec les communautés de vision artificielle, traitement du signal et bio-informatiquen’ont jamais été aussi fortes. De nombreux domaines scientifiques et industriels produisentdes données massives, mais les traiter efficacement nécessite de lever de nombreux verrousscientifiques. Dans ce contexte, ce mémoire présente différentes contributions, qui sontorganisées en trois thématiques.Le chapitre 2 est dédié à l’optimisation à large échelle en apprentissage statistique.Dans un premier lieu, nous étudions plusieurs variantes d’algorithmes de majoration/minimisationpour des problèmes structurés, telles que des variantes par bloc de variables,incrémentales, et stochastiques. Chaque algorithme est analysé en terme de taux deconvergence lorsque le problème est convexe, et nous montrons la convergence de ceux-civers des points stationnaires dans le cas contraire. Des méthodes de minimisation rapidespour traiter le cas de sommes finies de fonctions sont aussi introduites, ainsi que desalgorithmes d’accélération pour les techniques d’optimisation de premier ordre.Le chapitre 3 présente le paradigme des méthodes à noyaux profonds, que l’on peutinterpréter comme un mariage entre les méthodes à noyaux classiques et les techniquesd’apprentissage profond. Dans le contexte de la reconnaissance visuelle, ce chapitre introduitun nouveau modèle d’image invariant appelé réseau convolutionnel à noyaux, qui estun nouveau type de réseau de neurones convolutionnel avec une interprétation en termesde noyaux reproduisants. Le réseau peut être appris simplement sans supervision grâceà des techniques classiques d’approximation de noyaux, mais est aussi compatible avecl’apprentissage supervisé grâce à des règles de backpropagation.Le chapitre 4 est dédié à l’estimation parcimonieuse, c’est à dire, à la séléction automatiquede variables permettant d’expliquer des données observées. En particulier, cechapitre décrit des collaborations pluri-disciplinaires en bioinformatique et neuroscience,où le principe de parcimonie est crucial pour obtenir des modèles prédictifs interprétables.Enfin, le dernier chapitre conclut ce mémoire et présente des perspectives futures