Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descent

TL;DR

This paper proves the non-asymptotic validity of the multiplier bootstrap for confidence sets in SGD, achieving fast approximation rates without covariance estimation, and provides the first such bounds.

Contribution

It introduces a non-asymptotic bootstrap validity framework for SGD, surpassing previous asymptotic results and avoiding covariance matrix approximation.

Findings

Bootstrap approximation rate up to 1/√n

First non-asymptotic bound for bootstrap in SGD

Faster rates than Polyak-Juditsky CLT

Abstract

In this paper, we establish the non-asymptotic validity of the multiplier bootstrap procedure for constructing the confidence sets using the Stochastic Gradient Descent (SGD) algorithm. Under appropriate regularity conditions, our approach avoids the need to approximate the limiting covariance of Polyak-Ruppert SGD iterates, which allows us to derive approximation rates in convex distance of order up to $1/\sqrt{n}$. Notably, this rate can be faster than the one that can be proven in the Polyak-Juditsky central limit theorem. To our knowledge, this provides the first fully non-asymptotic bound on the accuracy of bootstrap approximations in SGD algorithms. Our analysis builds on the Gaussian approximation results for nonlinear statistics of independent random variables.