On additive averaging kernels for finite Markov chains

TL;DR

This paper investigates additive mixtures of Markov kernels, deriving explicit formulas and optimization methods to improve convergence rates for Markov chain sampling, with applications demonstrated on the Curie-Weiss model.

Contribution

It introduces a structured approach to optimize additive Markov kernels using spectral and combinatorial techniques, enhancing convergence speed.

Findings

Explicit trace formulas for Frobenius norm minimization.

Cheeger-type functional characterizes optimal partitions.

Numerical experiments show improved convergence with tuned parameters.

Abstract

We study additive mixtures of Markov kernels of the form $A_\alpha = \alpha P + (1-\alpha)G$, where $\alpha \in [0,1]$, $P$ is a baseline sampler and $G$ is a Gibbs kernel induced by a partition of the state space. We first motivate the study of $A_\alpha$, which can be interpreted as the projection of a lifted Markov chain. We then consider the minimisation of distance to stationarity under two objectives: the squared Frobenius norm and the Kullback-Leibler (KL) divergence. For the Frobenius objective, we derive explicit trace formulas and identify a Cheeger-type functional that characterises optimal two-block partitions. This yields a structured combinatorial optimisation problem admitting a difference-of-submodular decomposition, enabling efficient approximation via majorisation-minimisation. We also obtain geometric decay rates governed by the absolute spectral gap of $P$. For the KL divergence, we establish convexity-based bounds showing that the divergence of $A_\alpha$ is controlled by those of both $P$ and $G$, thereby reducing partition selection to the Gibbs component. Numerical experiments on the Curie-Weiss model demonstrate that suitable choice of both the partition and the parameter $\alpha$ can significantly accelerate convergence in total variation distance. We observe a consistent trade-off between local exploration and global averaging, with intermediate values of $\alpha$ achieving the best performance across regimes.