Variational Representations of Annealing Paths: Bregman Information under Monotonic Embedding

TL;DR

This paper explores variational representations of annealing paths in MCMC, extending known divergence minimization results to quasi-arithmetic means under monotonic transformations, linking divergence functionals with intermediate densities.

Contribution

It generalizes the minimization of expected Bregman divergence to quasi-arithmetic means with monotonic embeddings, enriching the theoretical understanding of annealing paths.

Findings

Extended Bregman divergence minimization to quasi-arithmetic means.

Linked divergence functionals with intermediate densities in annealing.

Provided a unified framework for analyzing annealing paths using monotonic embeddings.

Abstract

Markov Chain Monte Carlo methods for sampling from complex distributions and estimating normalization constants often simulate samples from a sequence of intermediate distributions along an annealing path, which bridges between a tractable initial distribution and a target density of interest. Prior works have constructed annealing paths using quasi-arithmetic means, and interpreted the resulting intermediate densities as minimizing an expected divergence to the endpoints. To analyze these variational representations of annealing paths, we extend known results showing that the arithmetic mean over arguments minimizes the expected Bregman divergence to a single representative point. In particular, we obtain an analogous result for quasi-arithmetic means, when the inputs to the Bregman divergence are transformed under a monotonic embedding function. Our analysis highlights the interplay between quasi-arithmetic means, parametric families, and divergence functionals using the rho-tau representational Bregman divergence framework, and associates common divergence functionals with intermediate densities along an annealing path.