Finite Sample Properties of Adaptive Markov Chains via Curvature

TL;DR

This paper develops concentration inequalities and finite sample bounds for adaptive Markov chains using Ricci curvature, demonstrating improved performance of equi-energy samplers over other methods after a learning period.

Contribution

It introduces a novel theoretical framework for finite sample analysis of adaptive Markov chains using Ricci curvature, with applications to equi-energy algorithms.

Findings

Finite sample bounds established for adaptive Markov chains.

Quantitative comparison showing equi-energy sampler's superiority.

Rigorous proofs of improved finite sample properties after learning period.

Abstract

Adaptive Markov chains are an important class of Monte Carlo methods for sampling from probability distributions. The time evolution of adaptive algorithms depends on past samples, and thus these algorithms are non-Markovian. Although there has been previous work establishing conditions for their ergodicity, not much is known theoretically about their finite sample properties. In this paper, using a notion of discrete Ricci curvature for Markov kernels introduced by Ollivier, we establish concentration inequalities and finite sample bounds for a class of adaptive Markov chains. After establishing some general results, we give quantitative bounds for `multi-level' adaptive algorithms such as the equi-energy sampler. We also provide the first rigorous proofs that the finite sample properties of an equi-energy sampler are superior to those of related parallel tempering and Metropolis-Hastings samplers after a learning period comparable to their mixing times.