Complexity of randomized algorithms for underdamped Langevin dynamics

Abstract

We establish an information complexity lower bound of randomized algorithms for simulating underdamped Langevin dynamics. More specifically, we prove that the worst $L^2$ strong error is of order $\Omega (\sqrt{d}{N}^{-3/2})$, for solving a family of $d$-dimensional underdamped Langevin dynamics, by any randomized algorithm with only $N$ queries to $\nabla U$, the driving Brownian motion and its weighted integration, respectively. The lower bound we establish matches the upper bound for the randomized midpoint method recently proposed by Shen and Lee [NIPS 2019], in terms of both parameters $N$ and $d$.

arXiv comment27 pages; some revision (e.g., Sec 2.1), and new supplementary materials in Appendices

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• High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm
• On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness
• On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method
• The query complexity of sampling from strongly log-concave distributions in one dimension
• Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration
• Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo
• Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations
• Higher Order Generalization Error for First Order Discretization of Langevin Diffusion