Bias-Variance Tradeoff in a Sliding Window Implementation of the Stochastic Gradient Algorithm

TL;DR

This paper introduces a framework for analyzing stochastic gradient algorithms using asymptotic normality, and proposes the SW-SGD method which reduces mean squared error compared to standard SGD on convex problems.

Contribution

The paper develops a new analysis framework for biased gradient estimators and introduces the SW-SGD algorithm with proven convergence and improved MSE performance.

Findings

SW-SGD achieves lower MSE than SGD on quadratic and convex problems.

The framework effectively characterizes the distribution of iterates considering bias and covariance.

Numerical results confirm the superiority of SW-SGD over traditional SGD.

Abstract

This paper provides a framework to analyze stochastic gradient algorithms in a mean squared error (MSE) sense using the asymptotic normality result of the stochastic gradient descent (SGD) iterates. We perform this analysis by taking the asymptotic normality result and applying it to the finite iteration case. Specifically, we look at problems where the gradient estimators are biased and have reduced variance and compare the iterates generated by these gradient estimators to the iterates generated by the SGD algorithm. We use the work of Fabian to characterize the mean and the variance of the distribution of the iterates in terms of the bias and the covariance matrix of the gradient estimators. We introduce the sliding window SGD (SW-SGD) algorithm, with its proof of convergence, which incurs a lower MSE than the SGD algorithm on quadratic and convex problems. Lastly, we present some numerical results to show the effectiveness of this framework and the superiority of SW-SGD algorithm over the SGD algorithm.