A Homogenization Approach for Gradient-Dominated Stochastic Optimization

TL;DR

This paper introduces SHSODM, a second-order stochastic optimization method leveraging gradient dominance and homogenization, offering improved convergence, reduced computational cost, and robustness in non-convex machine learning and reinforcement learning tasks.

Contribution

The paper proposes SHSODM, a novel second-order method for gradient-dominated stochastic functions, with theoretical analysis and empirical validation showing advantages over existing methods.

Findings

SHSODM matches best-known sample complexity for gradient-dominated problems.

It avoids cubic regularization, reducing computational costs.

Numerical experiments show superior performance in RL tasks.

Abstract

Gradient dominance property is a condition weaker than strong convexity, yet sufficiently ensures global convergence even in non-convex optimization. This property finds wide applications in machine learning, reinforcement learning (RL), and operations management. In this paper, we propose the stochastic homogeneous second-order descent method (SHSODM) for stochastic functions enjoying gradient dominance property based on a recently proposed homogenization approach. Theoretically, we provide its sample complexity analysis, and further present an enhanced result by incorporating variance reduction techniques. Our findings show that SHSODM matches the best-known sample complexity achieved by other second-order methods for gradient-dominated stochastic optimization but without cubic regularization. Empirically, since the homogenization approach only relies on solving extremal eigenvector problem at each iteration instead of Newton-type system, our methods gain the advantage of cheaper computational cost and robustness in ill-conditioned problems. Numerical experiments on several RL tasks demonstrate the better performance of SHSODM compared to other off-the-shelf methods.