Kernel Stein Discrepancy thinning: a theoretical perspective of pathologies and a practical fix with regularization

TL;DR

This paper analyzes the limitations of Stein thinning in MCMC post-processing, provides a theoretical understanding of its issues, and introduces a regularized version with strong guarantees and practical efficiency.

Contribution

It offers a theoretical analysis of Stein thinning pathologies and proposes a regularized algorithm with proven guarantees and improved performance.

Findings

Regularized Stein thinning alleviates empirical pathologies.

Theoretical guarantees support the effectiveness of the proposed method.

Extensive experiments demonstrate high efficiency of the regularized approach.

Abstract

Stein thinning is a promising algorithm proposed by (Riabiz et al., 2022) for post-processing outputs of Markov chain Monte Carlo (MCMC). The main principle is to greedily minimize the kernelized Stein discrepancy (KSD), which only requires the gradient of the log-target distribution, and is thus well-suited for Bayesian inference. The main advantages of Stein thinning are the automatic remove of the burn-in period, the correction of the bias introduced by recent MCMC algorithms, and the asymptotic properties of convergence towards the target distribution. Nevertheless, Stein thinning suffers from several empirical pathologies, which may result in poor approximations, as observed in the literature. In this article, we conduct a theoretical analysis of these pathologies, to clearly identify the mechanisms at stake, and suggest improved strategies. Then, we introduce the regularized Stein thinning algorithm to alleviate the identified pathologies. Finally, theoretical guarantees and extensive experiments show the high efficiency of the proposed algorithm. An implementation of regularized Stein thinning as the kernax library in python and JAX is available at https://gitlab.com/drti/kernax.