Optimal Underdamped Langevin MCMC Method

Part of Advances in Neural Information Processing Systems 34 (NeurIPS 2021)

Authors

Zhengmian Hu, Feihu Huang, Heng Huang

Abstract

In the paper, we study the underdamped Langevin diffusion (ULD) with strongly-convex potential consisting of finite summation of $N$ smooth components, and propose an efficient discretization method, which requires $O(N+d^\frac{1}{3}N^\frac{2}{3}/\varepsilon^\frac{2}{3})$ gradient evaluations to achieve $\varepsilon$-error (in $\sqrt{\mathbb{E}{\lVert{\cdot}\rVert_2^2}}$ distance) for approximating $d$-dimensional ULD. Moreover, we prove a lower bound of gradient complexity as $\Omega(N+d^\frac{1}{3}N^\frac{2}{3}/\varepsilon^\frac{2}{3})$, which indicates that our method is optimal in dependence of $N$, $\varepsilon$, and $d$. In particular, we apply our method to sample the strongly-log-concave distribution and obtain gradient complexity better than all existing gradient based sampling algorithms. Experimental results on both synthetic and real-world data show that our new method consistently outperforms the existing ULD approaches.