Global Convergence of Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Stochastic Optimization: Non-Asymptotic Performance Bounds and Momentum-Based Acceleration
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Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradient with momentum where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. Many works reported its empirical success in practice for solving stochastic non-convex 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 asymptotic global convergence properties of SGHMC are well known, its finite-time performance is not well-understood. In this work, we provide finite-time performance bounds for the global convergence of SGHMC for solving stochastic non-convex optimization problems with explicit constants. Our results lead to non-asymptotic guarantees for both population and empirical risk minimization problems. For a fixed target accuracy level ε, on a class of non-convex problems, we obtain iteration complexity bounds for SGHMC that can be tighter than those for SGLD up to a square root factor. These results show that acceleration with momentum is possible in the context of non-convex optimization algorithms.
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