Tracking the gradients using the Hessian: A new look at variance reducing stochastic methods

TL;DR

This paper introduces a Hessian-based approach to enhance variance reduction in stochastic optimization, improving convergence speed through better control variates and efficient Hessian approximations.

Contribution

It proposes a novel Hessian-tracking modification of SVRG, with efficient Hessian approximations, leading to faster convergence in stochastic methods.

Findings

Faster theoretical convergence close to the optimum.

Effective Hessian approximations using diagonal and low-rank matrices.

Demonstrated improvements across diverse problems.

Abstract

Our goal is to improve variance reducing stochastic methods through better control variates. We first propose a modification of SVRG which uses the Hessian to track gradients over time, rather than to recondition, increasing the correlation of the control variates and leading to faster theoretical convergence close to the optimum. We then propose accurate and computationally efficient approximations to the Hessian, both using a diagonal and a low-rank matrix. Finally, we demonstrate the effectiveness of our method on a wide range of problems.