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
2 LIENS - Laboratoire d'informatique de l'école normale supérieure
3 LTCI - Laboratoire Traitement et Communication de l'Information

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.

Mots clés: Stocahstic gradient descent; Markov chains

Domaines: Machine Learning [stat.ML]; Optimisation et contrôle [math.OC]

Aymeric Dieuleveut, Alain Durmus, Francis Bach. Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains. 2017. ⟨hal-01565514v1⟩