Langevin Monte Carlo with SPSA-approximated Gradients

Abstract

In sampling problems, gradient-based sampling schemes, like Langevin Monte Carlo (LMC), are widely used due to the short burn-in process compared with non-gradient-based sampling methods like the Metropolis-Hastings method. On the other hand, the application of LMC is limited to whether the gradients are accessible. To extend the application scenario of LMC, we propose a sampling algorithm LMC-SPSA. The method approximates the gradients of the target log density and applies the approximated gradients in Langevin Monte Carlo. We prove the convergence in distribution of LMC-SPSA by proving the Wasserstein distance converging to zero. Numerical experiments are conducted to verify the performance of LMC-SPSA.