Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling

Abstract

We provide a new convergence analysis of stochastic gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave. At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain. Under certain conditions on the target distribution, we prove that Oe(d4є−2) stochastic gradient evaluations suffice to guarantee є-sampling error in terms of the total variation distance, where d is the problem dimension. This improves existing results on the convergence rate of SGLD [Raginsky et al., 2017, Xu et al., 2018]. We further show that provided an additional Hessian Lipschitz condition on the log-density function, SGLD is guaranteed to achieve є-sampling error within Oe(d15/4є−3/2) stochastic gradient evaluations. Our proof technique provides a new way to study the convergence of Langevin based algorithms, and sheds some light on the design of fast stochastic gradient based sampling algorithms.