Stochastic Bias-Reduced Gradient Methods

TL;DR

This paper introduces a low-bias, low-cost stochastic gradient estimator using multilevel Monte Carlo, enabling more efficient optimization and smoothing techniques with broad applications.

Contribution

It develops a novel estimator for the minimizer of Lipschitz strongly-convex functions, improving stochastic optimization efficiency and enabling dimension-free smoothing.

Findings

Achieves bias $oldsymbol{ ext{δ}}$, variance $O( ext{log}(1/δ))$, and cost $O( ext{log}(1/δ))$ in estimation.

Improves optimization of the maximum of $N$ functions, matching lower bounds up to logarithmic factors.

Enables nearly linear-time, differentially-private non-smooth stochastic optimization.

Abstract

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an estimator of $x_\star$ with bias $\delta$, variance $O(\log(1/\delta))$, and an expected sampling cost of $O(\log(1/\delta))$ stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau-Yoshida envelope of any Lipschitz convex function, allowing us to perform dimension-free randomized smoothing. We demonstrate the potential of our estimator through four applications. First, we develop a method for minimizing the maximum of $N$ functions, improving on recent results and matching a lower bound up to logarithmic factors. Second and third, we recover state-of-the-art rates for projection-efficient and gradient-efficient optimization using simple algorithms with a transparent analysis. Finally, we show that an improved version of our estimator would yield a nearly linear-time, optimal-utility, differentially-private non-smooth stochastic optimization method.