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 $d$.

翻译：模拟回火是采样多峰分布的一种广泛使用策略。本文考虑将模拟回火与任意局部马尔可夫链蒙特卡洛采样器相结合，提出一个新的分解定理，该定理为从混合分布采样的算法提供了受限谱隙的下界。通过研究受限谱隙，我们结果的适用性可扩展到更广泛的场景，例如当常规谱隙难以界定或出现退化时。我们通过分析模拟回火结合随机游走Metropolis-Hastings算法对高斯混合分布的采样，展示了理论结果的应用。我们的复杂度界随模态间距离呈多项式缩放，随$1/\varepsilon$呈对数缩放（其中$\varepsilon$表示总变差距离的目标精度），并随维度$d$呈指数缩放。