Core Concepts
Prior diffusion in Langevin algorithms can achieve dimension-independent convergence for a broader class of target distributions beyond log-concavity.
Abstract
The paper discusses the dimension dependency of computational complexity in high-dimensional sampling problems. It introduces the modified Langevin algorithm with prior diffusion to achieve dimension-independent convergence for a broader class of target distributions beyond log-concavity. The analysis focuses on the convergence of KL divergence with different step size schedules, providing insights into faster sampling algorithms. The content is structured as follows:
Abstract
Introduction
Sampling from unnormalized distribution
Langevin algorithms and their popularity
Sampling algorithms categorized by dimension dependency
Freund et al.'s suggestion on dimension-independent convergence
Prior diffusion for Gaussian mixtures
Theoretical results on KL convergence with fixed and varying step sizes
Proof sketch and discussion on specific examples
Conclusions and future work
Stats
Freund et al. (2022) suggest that the convergence rate of the modified Langevin dynamics only depends on the trace of log-likelihood Hessian.
The convergence rate of LAPD only depends on the number of mixture components K and the radius of means Rµ.
Quotes
"Understanding the dimension dependency of computational complexity in high-dimensional sampling problem is a fundamental problem."
"LAPD can be considered as a more general version of ULA and is able to achieve a faster convergence by properly tuning m."