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Rapid convergence of tempering chains to multimodal Gibbs measures

Seungjae Son

Abstract

We study the spectral gaps of parallel and simulated tempering chains targeting multimodal Gibbs measures. In particular, we consider chains constructed from Metropolis random walks that preserve the Gibbs distributions at a sequence of harmonically spaced temperatures. We prove that their spectral gaps admit polynomial lower bounds of order $11$ and $12$ in terms of the low target temperature. The analysis applies to a broad class of potentials, beyond mixture models, without requiring explicit structural information on the energy landscape. The main idea is to decompose the state space and construct a Lyapunov function based on a suitably perturbed potential, which allows us to establish lower bounds on the local spectral gaps.

Rapid convergence of tempering chains to multimodal Gibbs measures

Abstract

We study the spectral gaps of parallel and simulated tempering chains targeting multimodal Gibbs measures. In particular, we consider chains constructed from Metropolis random walks that preserve the Gibbs distributions at a sequence of harmonically spaced temperatures. We prove that their spectral gaps admit polynomial lower bounds of order and in terms of the low target temperature. The analysis applies to a broad class of potentials, beyond mixture models, without requiring explicit structural information on the energy landscape. The main idea is to decompose the state space and construct a Lyapunov function based on a suitably perturbed potential, which allows us to establish lower bounds on the local spectral gaps.

Paper Structure

This paper contains 20 sections, 34 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Informal statement of main result
  3. Motivation and Background
  4. Plan of the paper
  5. Main result and the key lemmas
  6. Assumptions
  7. Main result
  8. Key lemmas
  9. Construction of Lyapunov function (Proof of Lemma \ref{['l:lyapunov']})
  10. Construction of the perturbed potential $\hat{U}$
  11. Estimates required for the Lyapunov condition
  12. Proofs for the Lyapunov estimates
  13. Proofs for the properties of $\hat{P}$
  14. Spectral gap of local chain (Proof of Lemmas \ref{['l:sg-restricted']} and \ref{['l:large-temperature-gap']})
  15. Proof of Lemma \ref{['l:sg-restricted']}
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $U\colon \mathbb{T}^d \to \mathbb{R}$ be a regular double-well potential with wells of equal depth (but not necessarily the same shape). Then there exist constants $\eta, c_1, c_2, \bar{C}_\mathrm{BV}, c_d > 0$ such that the following holds. For any $0 < \underline \varepsilon < \bar{\varepsilon and choose $\paren*{1/\varepsilon_k}_{k=0}^N$ to be linearly spaced. For each $k$, define Then the

Theorems & Definitions (70)

  • Definition 1.1: Spectral gap
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.4: Parallel tempering chain
  • Definition 2.5: (Lazy) Metropolis random walk
  • Theorem 2.6
  • Definition 2.7: Restriction of a Markov chain
  • Lemma 2.8: Lyapunov drift for a perturbed potential
  • Lemma 2.9: Spectral gap of restricted chain for small $\varepsilon$
  • ...and 60 more