Group-averaged Markov chains II: tuning of group action in finite state space

Michael C. H. Choi; Ryan J. Y. Lim; Youjia Wang

Paper

Group-averaged Markov chains II: tuning of group action in finite state space

Abstract

We study group-averaged Markov chains obtained by augmenting a

-stationary transition kernel

with a group action on the state space via orbit kernels. Given a group

with orbits

, we analyse three canonical orbit kernels: namely the Gibbs

, Metropolis-Hastings

, and Barker

kernels, as well as their multiplicative sandwiches

and the additive mixtures

where

. We show that

blockwise as

under suitable conditions, that the projection chains induced by

coincide for

and

, and that orbit averaging never deteriorates the absolute spectral gap or asymptotic variance when

is reversible. We give a direct and simple proof of Pythagorean identity under the Kullback-Leibler (KL) divergence, showing that

arises naturally as an information projection of

onto the set of

-invariant transition matrices. For a given

, we characterise the optimal choice of

with a fixed number of orbits that minimises the one-step KL divergence to stationarity. Analogously, for a given

, we characterise the optimal choice of

and give sufficient conditions under which

. We further show that alternating projections over multiple group actions converge at a rate governed by the singular values of an overlap matrix, and that in structured cases, this yields exact sampling where the number of group actions grows logarithmically with the size of the state space. Based on the theory, we propose two heuristics to tune

in practice. We also illustrate the results on discrete uniform and multimodal examples, including the Curie-Weiss model where

achieves polynomial (in inverse temperature and dimension) mixing while Glauber dynamics remains exponentially slow.