Abstract
We consider applying stochastic approximation (SA) methods to solve nonsmooth variational inclusion problems. Existing studies have shown that the averaged iterates of SA methods exhibit asymptotic normality, with an optimal limiting covariance matrix in the local minimax sense of Hájek and Le Cam. However, no methods have been proposed to estimate this covariance matrix in a nonsmooth and potentially non-monotone (nonconvex) setting. In this paper, we study an online batch-means covariance matrix estimator introduced in Zhu et al.(2023). The estimator groups the SA iterates appropriately and computes the sample covariance among batches as an estimate of the limiting covariance. Its construction does not require prior knowledge of the total sample size, and updates can be performed recursively as new data arrives. We establish that, as long as the batch size sequence is properly specified (depending on the stepsize sequence), the estimator achieves a convergence rate of order $O(\sqrt{d}n^{−1/8+\epsilon})$ for any $\epsilon>0$, where $d$ and $n$ denote the problem dimensionality and the number of iterations (or samples) used. Although the problem is nonsmooth and potentially non-monotone (nonconvex), our convergence rate matches the best-known rate for covariance estimation methods using only first-order information in smooth and strongly-convex settings. The consistency of this covariance estimator enables asymptotically valid statistical inference, including constructing confidence intervals and performing hypothesis testing.
Sen Na
Assistant Professor in ISyE
Sen Na is an Assistant Professor in the School of Industrial and Systems Engineering at Georgia Tech. Prior to joining ISyE, he was a postdoctoral researcher in the statistics department and ICSI at UC Berkeley. His research interests broadly lie in the mathematical foundations of data science, with topics including high-dimensional statistics, graphical models, semiparametric models, optimal control, and large-scale and stochastic nonlinear optimization. He is also interested in applying machine learning methods to problems in biology, neuroscience, and engineering.