Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling
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We establish 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 Õ(d^4ϵ^-2) stochastic gradient evaluations suffice to guarantee ϵ-sampling error in terms of the total variation distance, where d is the problem dimension, which 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 ϵ-sampling error within Õ(d^15/4ϵ^-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.
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