Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond
This work addresses sampling efficiency for machine learning and statistics, offering incremental improvements in convergence rates for strongly log-concave distributions.
The paper tackles the problem of sampling from probability distributions using Markov chain Monte Carlo by discretizing continuous-time dynamics, specifically analyzing stochastic Runge-Kutta methods for overdamped Langevin diffusion. It achieves convergence in $ ilde{\mathcal{O}}(dε^{-2/3})$ iterations for strongly convex potentials, improving upon prior gradient-based methods.
Sampling with Markov chain Monte Carlo methods often amounts to discretizing some continuous-time dynamics with numerical integration. In this paper, we establish the convergence rate of sampling algorithms obtained by discretizing smooth Itô diffusions exhibiting fast Wasserstein-$2$ contraction, based on local deviation properties of the integration scheme. In particular, we study a sampling algorithm constructed by discretizing the overdamped Langevin diffusion with the method of stochastic Runge-Kutta. For strongly convex potentials that are smooth up to a certain order, its iterates converge to the target distribution in $2$-Wasserstein distance in $\tilde{\mathcal{O}}(dε^{-2/3})$ iterations. This improves upon the best-known rate for strongly log-concave sampling based on the overdamped Langevin equation using only the gradient oracle without adjustment. In addition, we extend our analysis of stochastic Runge-Kutta methods to uniformly dissipative diffusions with possibly non-convex potentials and show they achieve better rates compared to the Euler-Maruyama scheme in terms of the dependence on tolerance $ε$. Numerical studies show that these algorithms lead to better stability and lower asymptotic errors.