Reinforcement Learning for Adaptive MCMC

A vast literature on algorithms, tips, and tricks is testament to the success of Markov chain Monte Carlo (MCMC), which remains the most popular approach to numerical approximation of probability distributions characterised up to an intractable normalisation constant. Yet the breadth of methodology also presents a difficulty in selecting an appropriate algorithm for a specific task. The goal of adaptive MCMC is to automate, as much as possible, the design of a fast-mixing Markov transition kernel. To achieve this, one alternates between observing the performance of the current transition kernel, and updating the transition kernel in a manner that is expected to improve its future performance (Andrieu and Thoms, 2008). Though the online adaptation of a Markov transition kernel in principle sacrifices the ergodicy of MCMC, there are several ways to prove that ergodicity is in fact retained if the transition kernel converges fast enough (in an appropriate sense) to a sensible limit.