Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond

Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)

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Xuechen Li, Yi Wu, Lester Mackey, Murat A. Erdogdu


Sampling with Markov chain Monte Carlo methods typically 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 It\^o diffusions exhibiting fast $2$-Wasserstein 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 $\tilde{\mathcal{O}}(d\epsilon^{-2/3})$ 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. Additionally, 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 on the dependence on tolerance $\epsilon$. Numerical studies show that these algorithms lead to better stability and lower asymptotic errors.