Poisson Midpoint Method for Log Concave Sampling: Beyond the Strong Error Lower Bounds
Rishikesh Srinivasan, Dheeraj Nagaraj
TL;DR
This paper introduces the Poisson Midpoint Method (PLMC) for discretizing Langevin dynamics to sample from strongly log-concave targets $\pi(x) \propto e^{-f(x)}$ in $\mathbb{R}^d$ and demonstrates a cubic improvement in the $\varepsilon$-dependence of convergence in the $W_2$ metric compared to Euler–Maruyama. By performing batch-based, randomized midpoint updates and leveraging a sharp coupling bound in $W_2$, the authors establish explicit oracle complexities: overdamped PLMC achieves $\tilde{O}\left(\frac{\kappa^{4/3}+\kappa d^{1/3}}{\varepsilon^{2/3}}\right)$ and underdamped PLMC achieves $\tilde{O}\left(\frac{\kappa^{7/6} d^{1/6}}{\varepsilon^{1/3}} + \frac{\kappa^{(11p+6)/(8p+6)} d^{p/(4p+3)}}{\varepsilon^{(p+2)/(4p+3)}}\right)$ for arbitrary integer $p$, with a purely $\varepsilon$-dependent rate of $\tilde{O}(\varepsilon^{-1/3})$ when $p\ge 3$. The results show a separation between weak error convergence and strong $L^2$ lower bounds, particularly for the underdamped case, and contribute a framework for achieving faster sampling in the strongly log-concave setting. The work positions PLMC as a leading discretization for Langevin-based sampling with favorable scalability in accuracy and opens avenues for tightening moment bounds to further improve dependence on problem parameters.
Abstract
We study the problem of sampling from strongly log-concave distributions over $\mathbb{R}^d$ using the Poisson midpoint discretization (a variant of the randomized midpoint method) for overdamped/underdamped Langevin dynamics. We prove its convergence in the 2-Wasserstein distance ($W_2$), achieving a cubic speedup in dependence on the target accuracy ($ε$) over the Euler-Maruyama discretization, surpassing existing bounds for randomized midpoint methods. Notably, in the case of underdamped Langevin dynamics, we demonstrate the complexity of $W_2$ convergence is much smaller than the complexity lower bounds for convergence in $L^2$ strong error established in the literature.