Higher Order Langevin Monte Carlo Algorithm

Abstract

A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density $\hat{\pi }$ on ${\mathbb{R}}^d$ known up to a normalizing constant. Moreover, $-\log \hat{\pi }$ is assumed to have a locally Lipschitz gradient and its third derivative is locally Hölder continuous with exponent $\beta \in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+\beta /2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-\log \hat{\pi }$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.

arXiv comment47 pages

References in corpus (9)

• Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
• Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients
• A note on tamed Euler approximations
• User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient
• Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients
• Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes
• Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes
• On explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients
• High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm

Cited by in corpus (8)

• Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond
• Optimized Population Monte Carlo
• Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization
• Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization
• On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method
• Polygonal Unadjusted Langevin Algorithms: Creating stable and efficient adaptive algorithms for neural networks
• Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
• Non-asymptotic estimates for TUSLA algorithm for non-convex learning with applications to neural networks with ReLU activation function