Bridging the gap between constant step size stochastic gradient descent and Markov chains

Abstract

We consider the minimization of a strongly convex 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 (nonquadratic) case. For this analysis we bring tools from Markov chain theory into the analysis of stochastic gradient. 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.