The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.

翻译：随机梯度梯度 Langevin 动力学是解决抽样问题和若干机器学习应用中出现的非碳化优化的最基本算法之一。 特别是, 其差异减少的版本如今引起了特别关注。 在本文中, 我们研究了两种此类变种, 即: 蒸发差异减少的渐变Langevin 动力学和 蒸发递递增变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变。 我们变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变,变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变变