Global Convergence of Stochastic Gradient Hamiltonian Monte Carlo for Nonconvex Stochastic Optimization: Nonasymptotic Performance Bounds and Momentum-Based Acceleration

• 成果类型: Article; Early Access
• 署名作者: Gao, Xuefeng; Gurbuzbalaban, Mert; Zhu, Lingjiong
• 署名单位: Chinese University of Hong Kong; Rutgers University System; Rutgers University New Brunswick; State University System of Florida; Florida State University
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2021.2162
• 发表日期: 2021
• 关键词: Langevin dynamics; stochastic gradient methods; momentum-based acceleration; Nonconvex Optimization; Empirical Risk Minimization; Gibbs sampling

摘要

Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradients with momentum where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates toward a global minimum. Many works report its empirical success in practice for solving stochastic nonconvex optimization problems; in particular, it has been observed to outperform overdamped Langevin Monte Carlo-based methods, such as stochastic gradient Langevin dynamics (SGLD), in many applications. Although the asymptotic global convergence properties of SGHMC are well known, its finite-time performance is not well understood. In this work, we study two variants of SGHMC based on two alternative discretizations of the underdamped Langevin diffusion. We provide finite-time performance bounds for the global convergence of both SGHMC variants for solving stochastic nonconvex optimization problems with explicit constants. Our results lead to nonasymptotic guarantees for both population and empirical risk minimization problems. For a fixed target accuracy level on a class of nonconvex problems, we obtain complexity bounds for SGHMC that can be tighter than those available for SGLD.