A connection between Tempering and Entropic Mirror Descent

TL;DR

This paper establishes a theoretical link between tempering in Sequential Monte Carlo methods and entropic mirror descent, providing convergence analysis and new adaptive tempering strategies based on optimization principles.

Contribution

It reveals that tempering SMC corresponds to entropic mirror descent on the reverse KL divergence, offering a new optimization perspective and practical adaptive algorithms.

Findings

Temperings correspond to entropic mirror descent on the reverse KL divergence.

Convergence rates for tempering iterates are derived.

Adaptive tempering rules outperform existing benchmarks.

Abstract

This paper explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.