Restricted Spectral Gap Decomposition for Simulated Tempering Targeting Mixture Distributions

TL;DR

This work develops a discrete-time, $\,\Omega^0$-restricted spectral-gap decomposition for simulated tempering when paired with a generic local MCMC kernel, enabling lower bounds on the mixing rate for mixture targets. By introducing a latent component structure $p_i(x) = \sum_j w_{(i,j)} p_{(i,j)}(x)$ and a projected chain $\overline{M}$, the authors relate the global convergence to fast local mixing within components and to inter-component transitions. Applied to simulated tempering with Metropolis--Hastings for Gaussian mixtures, they prove that sampling from $P^*$ can be achieved in time polynomial in the mode separation distance $D$, logarithmic in the accuracy $\varepsilon$, and exponential in dimension $d$ (with a more favorable, poly-time bound in fixed $d$). The paper also provides a simulation study showing STMH outperforms standard MH in multimodal settings, corroborating the theoretical scaling. Overall, this framework yields tractable spectral-gap-based guarantees for tempered sampling of mixtures and informs potential improvements via alternative local samplers like MALA or proximal methods.

Abstract

Simulated tempering is a widely used strategy for sampling from multimodal distributions. In this paper, we consider simulated tempering combined with an arbitrary local Markov chain Monte Carlo sampler and present a new decomposition theorem that provides a lower bound on the restricted spectral gap of the algorithm for sampling from mixture distributions. By working with the restricted spectral gap, the applicability of our results is extended to broader settings such as when the usual spectral gap is difficult to bound or becomes degenerate. We demonstrate the application of our theoretical results by analyzing simulated tempering combined with random walk Metropolis--Hastings for sampling from mixtures of Gaussian distributions. Our complexity bound scales polynomially with the separation between modes, logarithmically with $1/\varepsilon$, where $\varepsilon$ denotes the target accuracy in total variation distance, and exponentially with the dimension $d$.

Figures (1)

• Figure 1: Left: number of steps until $\|\widehat{\mu}\| < 0.1$ versus $D^2$. Right: $\log^2\left(1 / \|\widehat{\mu}\|\right)$ versus the number of steps $N$ for $D = 30$.

Theorems & Definitions (68)

• Definition 1: Restricted Spectral Gap
• Definition 2: Simulated Tempering Markov Chain
• Remark 1
• Remark 2
• Definition 3
• Lemma 1
• proof
• Remark 3
• Theorem 2
• proof