A Bootstrap Perspective on Stochastic Gradient Descent

TL;DR

This paper explores how stochastic gradient descent (SGD) improves generalization in machine learning by implicitly regularizing solution variability through bootstrap-like sampling, supported by empirical and theoretical analysis.

Contribution

It introduces a bootstrap perspective on SGD, showing that it implicitly regularizes the trace of the gradient covariance matrix to enhance generalization.

Findings

SGD favors solutions robust under resampling.

Explicit regularization of gradient covariance improves test performance.

SGD controls solution sensitivity to sampling noise.

Abstract

Machine learning models trained with \emph{stochastic} gradient descent (SGD) can generalize better than those trained with deterministic gradient descent (GD). In this work, we study SGD's impact on generalization through the lens of the statistical bootstrap: SGD uses gradient variability under batch sampling as a proxy for solution variability under the randomness of the data collection process. We use empirical results and theoretical analysis to substantiate this claim. In idealized experiments on empirical risk minimization, we show that SGD is drawn to parameter choices that are robust under resampling and thus avoids spurious solutions even if they lie in wider and deeper minima of the training loss. We prove rigorously that by implicitly regularizing the trace of the gradient covariance matrix, SGD controls the algorithmic variability. This regularization leads to solutions that are less sensitive to sampling noise, thereby improving generalization. Numerical experiments on neural network training show that explicitly incorporating the estimate of the algorithmic variability as a regularizer improves test performance. This fact supports our claim that bootstrap estimation underpins SGD's generalization advantages.