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

Nonconvex Stochastic Optimization Nonconvex stochastic optimization problems arise in many machine learning problems, including deep learning. The stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradients with a momentum method in which a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates toward a global minimum. SGHMC has shown empirical success in practice for solving nonconvex stochastic optimization problems. In "Global convergence of stochastic gradient Hamiltonian Monte Carlo for nonconvex stochastic optimization: Nonasymptotic performance bounds and momentum-based acceleration," Gao, Gürbüzbalaban, and Zhu provide, for the first time, the finite-time performance bounds for the global convergence of SGHMC in the context of both population and empirical risk minimization problems and show that acceleration with momentum is possible in the context of global nonconvex stochastic optimization. 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. [ABSTRACT FROM AUTHOR]