Subsampled Stochastic Variance-Reduced Gradient Langevin Dynamics

Stochastic variance-reduced gradient Langevin dynamics (SVRG-LD) was recently proposed to improve the performance of stochastic gradient Langevin dynamics (SGLD) by reducing the variance of the stochastic gradient. In this paper, we propose a variant of SVRG-LD, namely SVRG-LD+, which replaces the full gradient in each epoch with a subsampled one. We provide a nonasymptotic analysis of the convergence of SVRG-LD+ in 2-Wasserstein distance, and show that SVRG-LD+ enjoys a lower gradient complexity(1) than SVRG-LD, when the sample size is large or the target accuracy requirement is moderate. Our analysis directly implies a sharper convergence rate for SVRG-LD, which improves the existing convergence rate by a factor of kappa(1/6)n(1/6), where kappa is the condition number of the log-density function and n is the sample size. Experiments on both synthetic and real-world datasets validate our theoretical results.