Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization

This problem arises in many cases in machine learning, most notably in large-scale (mini-batch) Bayesian inference (Welling and Teh, 2011, Ahn et al., 2012) and nonconvex stochastic optimization (Raginsky et al., 2017). For the setting of Bayesian inference, one is interested in sampling from a posterior probability measure where U corresponds to the sum of the log-likelihood and the log-prior. For the nonconvex optimization, U(·) is the nonconvex cost function to be minimized. For large values ofβ, a sample from the target measure (1) is an approximate minimizer of the potential U (Raginsky et al., 2017). Consequently, nonasymptotic error bounds for the schemes, which are designed to sample from (1), can be used to obtain guarantees for Bayesian inference or nonconvex optimization. Sampling from a measure of the form (1) is also central in statistical physics (Binder et al., 1993), most notably in molecular dynamics Haile (1992).