Ai-Sampler: Adversarial Learning of Markov kernels with involutive maps

TL;DR

This paper introduces Ai-Sampler, a novel method for training Markov chain transition kernels using involutive neural networks to improve sampling efficiency and mixing, with a focus on reversible kernels and their properties.

Contribution

It proposes a new parameterization and training approach for Markov kernels leveraging involutive neural networks to ensure detailed balance and enhance sampling performance.

Findings

Reversible neural networks can construct involutive Metropolis-Hastings kernels.

Training minimizes total variation distance between stationary and empirical distributions.

Reversibility implies $C_2$-equivariance of the discriminator, restricting its function space.

Abstract

Markov chain Monte Carlo methods have become popular in statistics as versatile techniques to sample from complicated probability distributions. In this work, we propose a method to parameterize and train transition kernels of Markov chains to achieve efficient sampling and good mixing. This training procedure minimizes the total variation distance between the stationary distribution of the chain and the empirical distribution of the data. Our approach leverages involutive Metropolis-Hastings kernels constructed from reversible neural networks that ensure detailed balance by construction. We find that reversibility also implies $C_2$-equivariance of the discriminator function which can be used to restrict its function space.