Barrier Smoothing for Nonsmooth Convex Minimization

Volkan Cevher, Yen-Huan Li, Quoc Tran Dinh
Ieee, 2014
Article de conférence

This paper proposes a smoothing technique for nonsmooth convex minimization using self-concordant barriers. To illustrate the main ideas, we compare our technique and the proximity smoothing approach (Nesterov2005) via the classical gradient method on both the theoretical and numerical aspects. While the barrier smoothing approach maintains the sublinear-convergence rate, it affords a new analytic step size, which significantly enhances the practical convergence of the gradient method as compared to proximity smoothing.

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés

Chargement

Publications associées

Chargement

Volkan Cevher, Yen-Huan Li, Quoc Tran Dinh
Ieee, 2014
Article de conférence

Résumé

Official source

https://infoscience.epfl.ch/record/197508?ln=fr

À propos de ce résultat

Concepts associés (8)

Concepts associés

Chargement

Publications associées (2)

Publications associées

Chargement

Path-following gradient-based decomposition algorithms for separable convex optimization

Quoc Tran Dinh

A new decomposition optimization algorithm, called path-following gradient-based decomposition, is proposed to solve separable convex optimization problems. Unlike path-following Newton methods considered in the literature, this algorithm does not require any smoothness assumption on the objective function. This allows us to handle more general classes of problems arising in many real applications than in the path-following Newton methods. The new algorithm is a combination of three techniques, namely smoothing, Lagrangian decomposition and path-following gradient framework. The algorithm decomposes the original problem into smaller subproblems by using dual decomposition and smoothing via self-concordant barriers, updates the dual variables using a path-following gradient method and allows one to solve the subproblems in parallel. Moreover, compared to augmented Lagrangian approaches, our algorithmic parameters are updated automatically without any tuning strategy. We prove the global convergence of the new algorithm and analyze its convergence rate. Then, we modify the proposed algorithm by applying Nesterov's accelerating scheme to get a new variant which has a better convergence rate than the first algorithm. Finally, we present preliminary numerical tests that confirm the theoretical development.

Springer Verlag2014

Chargement

Stochastic Zeroth-Order Optimisation Algorithms with Variance Reduction

Ahmad Ajalloeian

Introduction of optimisation problems in which the objective function is black box or obtaining the gradient is infeasible, has recently raised interest in zeroth-order optimisation methods. As an example finding adversarial examples for Deep Learning models (Chen et al. (2017); Moosavi-Dezfooli et al. (2016)) is one of the most common applications in which zeroth-order methods could be used. These optimisation methods use only function values at certain points to estimate the gradient. Most current approaches iteratively sample a random search direction along which they compute an estimation of the gradient (Nesterov and Spokoiny (2017); Conn et al. (2009); Wibisono et al. (2012)). However, due to the high variance in the search direction, these methods usually need d times more iterations than the standard gradient methods, where d is the dimensionality of the problem. So it seems that the main effort for improving the zeroth-order methods should be in reducing the variance of the gradient estimate. In this work we will analyse the gradient-free oracle which uses random directions sampled form a Gaussian distribution. Our analysis shows that in smooth and strongly convex setting, we have a convergence rate of O( d/T) which clearly shows the dependency to the dimension of the problem. Furthermore we propose some variance reduction methods to make the zeroth-order optimisation faster. We experiment our proposed methods in Python to compare their convergence in stochastic and non-stochastic setting. Our empirical results show that in a setting that number of allowed function evaluation is fixed, using a variance reduction method (e.g. momentum) can make the convergence of zeroth-order methods happen faster.

2019

Chargement

Algorithme du gradient stochastique

L'algorithme du gradient stochastique est une méthode de descente de gradient (itérative) utilisée pour la minimisation d'une fonction objectif qui est écrite comme une somme de fonctions différentiab

Vitesse de convergence des suites

En analyse numérique — une branche des mathématiques — on peut classer les suites convergentes en fonction de leur vitesse de convergence vers leur point limite. C'est une manière d'apprécier l'effica

Algorithme du gradient

Lalgorithme du gradient, aussi appelé algorithme de descente de gradient, désigne un algorithme d'optimisation différentiable. Il est par conséquent destiné à minimiser une fonction réelle différentia