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

TL;DR

This work analyzes simulated tempering for sampling from mixtures of log-concave components with location shifts, establishing polynomial-time mixing guarantees when coupled with either RWM or MALA. The core approach is a state-decomposition argument for $s$-conductance, implemented via auxiliary chains on augmented spaces to relate the full tempered chain to tractable projected and restricted chains. A carefully chosen inverse-temperature ladder with $\beta_1 \le \frac{1}{4LD^2}$ and $\frac{\beta_{i+1}}{\beta_i} = 1 + O(d^{-1/2})$ yields asymptotically optimal dependence on the dimension $d$ and mode displacement $D$, up to logarithmic factors. The results significantly advance theoretical understanding of simulated tempering with mixture targets and point to practical ladder designs for high-dimensional non-log-concave targets.

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 $\log ε^{-1}$. 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 $β_1 = O(D^{-2})$ and spacing $β_{i+1}/β_i = 1 + O( d^{-1/2} )$, both of which are shown to be asymptotically optimal up to logarithmic factors.

Theorems & Definitions (58)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Theorem 2.4
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof