Optimising two-block averaging kernels to speed up Markov chains

TL;DR

This paper develops methods to select optimal two-block partitions in Markov chains to accelerate mixing, linking objectives like KL divergence and Frobenius distance to spectral properties, and proposes algorithms for practical approximation.

Contribution

It introduces a structured combinatorial optimization framework for two-block partition selection, connecting objectives to spectral decay rates and proposing efficient approximation algorithms.

Findings

Optimal cuts significantly reduce total variation distance to stationarity.

Proposed algorithms effectively approximate optimal partitions.

Numerical experiments demonstrate practical efficiency of methods.

Abstract

We study the problem of selecting optimal two-block partitions to accelerate the mixing of finite Markov chains under group-averaging transformations. The main objectives considered are the Kullback-Leibler (KL) divergence and the Frobenius distance to stationarity. We establish explicit connections between these objectives and the induced projection chain. In the case of the KL divergence, this reduction yields explicit decay rates in terms of the log-Sobolev constant. For the Frobenius distance, we identify a Cheeger-type functional that characterises optimal cuts. This formulation recasts two-block selection as a structured combinatorial optimisation problem admitting difference-of-submodular decompositions. We further propose several algorithmic approximations, including majorisation-minimisation and coordinate descent schemes, as computationally feasible alternatives to exhaustive combinatorial search. Our numerical experiments reveal that optimal cuts under the two objectives can substantially reduce total variation distance to stationarity and demonstrate the practical effectiveness of the proposed approximation algorithms.