A Homogenization Approach for Gradient-Dominated Stochastic Optimization

• URL: http://arxiv.org/abs/2308.10630v3
• Date: Wed, 29 May 2024 07:22:58 GMT
• Title: A Homogenization Approach for Gradient-Dominated Stochastic Optimization
• Authors: Jiyuan Tan, Chenyu Xue, Chuwen Zhang, Qi Deng, Dongdong Ge, Yinyu Ye,
• Abstract summary: We propose a homogeneous second-order descent method (SHSOD) for functions enjoying gradient dominance. Our findings show that SHSODM matches the best-known sample complexity achieved by other second-order methods for gradient-dominated optimization.
• Score: 6.1144486886258065
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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