Accelerated Diffusion-Based Sampling by the Non-Reversible Dynamics with Skew-Symmetric Matrices

Futoshi Futami,Issei Sato,Naonori Ueda,Tomoharu Iwata

doi:10.3390/e23080993

Futoshi Futami, Issei Sato + Show 2 more

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https://doi.org/10.3390/e23080993

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Journal: Entropy	Publication Date: Jul 30, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Kyoto Seika University, The University of Tokyo

Abstract

Langevin dynamics (LD) has been extensively studied theoretically and practically as a basic sampling technique. Recently, the incorporation of non-reversible dynamics into LD is attracting attention because it accelerates the mixing speed of LD. Popular choices for non-reversible dynamics include underdamped Langevin dynamics (ULD), which uses second-order dynamics and perturbations with skew-symmetric matrices. Although ULD has been widely used in practice, the application of skew acceleration is limited although it is expected to show superior performance theoretically. Current work lacks a theoretical understanding of issues that are important to practitioners, including the selection criteria for skew-symmetric matrices, quantitative evaluations of acceleration, and the large memory cost of storing skew matrices. In this study, we theoretically and numerically clarify these problems by analyzing acceleration focusing on how the skew-symmetric matrix perturbs the Hessian matrix of potential functions. We also present a practical algorithm that accelerates the standard LD and ULD, which uses novel memory-efficient skew-symmetric matrices under parallel-chain Monte Carlo settings.

Highlights

We considered the ensemble version of underdamped Langevin dynamics (ULD) (parallel ULD (PULD))
We studied skew acceleration for Langevin dynamics (LD) and ULD from practical viewpoints and concluded that the improved eigenvalues of the perturbed Hessian matrix caused acceleration and derived the explicit condition for acceleration
We described a novel ensemble sampling method, which couples multiple SGLD or stochastic gradient Hamilton Monte Carlo (SGHMC) with memory-efficient skew matrices

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Sampling is one of the most widely used techniques for the approximation of posterior distribution in Bayesian inference [1]. Markov Chain Monte Carlo (MCMC) is widely used to obtain samples. In MCMC, Langevin dynamics (LD) is a popular choice for sampling from high-dimensional distributions. Each sample in LD moves toward a gradient direction with added Gaussian noise. LD efficiently explore around a mode of a target distribution using the gradient information without being trapped by local minima thanks to added Gaussian noise. Many previous studies theoretically and numerically proved

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