Unbiased least squares regression via averaged stochastic gradient descent

TL;DR

This paper introduces an unbiased stochastic gradient descent estimator for online least squares regression that achieves low excess risk with efficient computation, based on a novel randomized multilevel Monte Carlo approach.

Contribution

It proposes a new unbiased estimator for the regression parameters that improves upon traditional methods by reducing bias and computational complexity.

Findings

Unbiased estimator achieves O(1/k) excess risk.

Estimator requires no prior knowledge of Hessian or true parameters.

Numerical experiments validate theoretical results.

Abstract

We consider an on-line least squares regression problem with optimal solution $\theta^*$ and Hessian matrix H, and study a time-average stochastic gradient descent estimator of $\theta^*$. For $k\ge2$, we provide an unbiased estimator of $\theta^*$ that is a modification of the time-average estimator, runs with an expected number of time-steps of order k, with O(1/k) expected excess risk. The constant behind the O notation depends on parameters of the regression and is a poly-logarithmic function of the smallest eigenvalue of H. We provide both a biased and unbiased estimator of the expected excess risk of the time-average estimator and of its unbiased counterpart, without requiring knowledge of either H or $\theta^*$. We describe an "average-start" version of our estimators with similar properties. Our approach is based on randomized multilevel Monte Carlo. Our numerical experiments confirm our theoretical findings.