Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains

Aymeric Dieuleveut 1, 2 Alain Durmus 3 Francis Bach 1
1 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : We consider the minimization of an objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step-size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged SGD iterates that outlines the dependence on initial conditions, the effect of noise and the step-size, as well as the lack of convergence in the general (non-quadratic) case. For this analysis, we bring tools from Markov chain theory into the analysis of stochastic gradient and create new ones (similar but different from stochastic MCMC methods). We then show that Richardson-Romberg extrapolation may be used to get closer to the global optimum and we show empirical improvements of the new extrapolation scheme.
Type de document :
Pré-publication, Document de travail
2017
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Contributeur : Aymeric Dieuleveut <>
Soumis le : mercredi 19 juillet 2017 - 19:02:31
Dernière modification le : jeudi 11 janvier 2018 - 06:28:04

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  • HAL Id : hal-01565514, version 1
  • ARXIV : 1707.06386

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Aymeric Dieuleveut, Alain Durmus, Francis Bach. Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains. 2017. 〈hal-01565514〉

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