Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT
Abstract
We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein's method and Lindeberg's argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense.
Keywords
Cite
@article{arxiv.1904.02130,
title = {Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT},
author = {Andreas Anastasiou and Krishnakumar Balasubramanian and Murat A. Erdogdu},
journal= {arXiv preprint arXiv:1904.02130},
year = {2019}
}