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 $\tilde O(d^4\epsilon^{-2})$ stochastic gradient evaluations suffice to guarantee $\epsilon$-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 $\epsilon$-sampling error within $\tilde O(d^{15/4}\epsilon^{-3/2})$ 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.

本文提供一种新的收敛分析方法，使用辅助的时间可逆Markov链来分析随机梯度Langevin动力学（SGLD）对一类难以log-concave优化分布进行采样的效果。在目标分布的一定条件下，我们证明了SGLD的收敛速度优于过去的研究成果，并提供了一些关于Langevin算法收敛和快速随机梯度采样算法设计方面的新思路。

非对数凹采样的随机梯度 Langevin 动力学的更快收敛