Polynomial Mixing Times of Simulated Tempering for Mixture Targets by Conductance Decomposition

TL;DR

This paper provides the first polynomial-time complexity guarantees for simulated tempering algorithms applied to mixture models of log-concave components, improving understanding of their efficiency in high-dimensional sampling tasks.

Contribution

It introduces a novel conductance decomposition approach to analyze the mixing times of simulated tempering with MALA and random-walk Metropolis, establishing polynomial bounds.

Findings

Polynomial-time guarantee for simulated tempering with MALA

Improved complexity bounds for simulated tempering with random-walk Metropolis

Optimal inverse-temperature ladder parameters up to logarithmic factors

Abstract

We study the theoretical complexity of simulated tempering for sampling from mixtures of log-concave components differing only by location shifts. The main result establishes the first polynomial-time guarantee for simulated tempering combined with the Metropolis-adjusted Langevin algorithm (MALA) with respect to the problem dimension $d$, maximum mode displacement $D$, and logarithmic accuracy $\log \epsilon^{-1}$. The proof builds on a general state decomposition theorem for $s$-conductance, applied to an auxiliary Markov chain constructed on an augmented space. We also obtain an improved complexity estimate for simulated tempering combined with random-walk Metropolis. Our bounds assume an inverse-temperature ladder with smallest value $\beta_1 = O(D^{-2})$ and spacing $\beta_{i+1}/\beta_i = 1 + O( d^{-1/2} )$, both of which are shown to be asymptotically optimal up to logarithmic factors.