In the field of sampling algorithms, MCMC (Markov Chain Monte Carlo) methods are widely used when direct sampling is not possible. However, multimodality of target distributions often leads to slow convergence and mixing. One common solution is parallel tempering. Though highly effective in practice, theoretical guarantees on its performance are limited. In this paper, we present a new lower bound for parallel tempering on the spectral gap that has a polynomial dependence on all parameters except $\log L$, where $(L + 1)$ is the number of levels. This improves the best existing bound which depends exponentially on the number of modes. Moreover, we complement our result with a hypothetical upper bound on spectral gap that has an exponential dependence on $\log L$, which shows that, in some sense, our bound is tight.

翻译：在采样算法领域，MCMC（马尔可夫链蒙特卡洛）方法在无法直接采样时被广泛使用。然而，目标分布的多模态性常常导致收敛和混合速度缓慢。一种常见解决方案是并行采样。尽管在实践中非常有效，但其性能的理论保证仍然有限。本文针对并行采样提出了谱间隙的新下界，该下界除 $\log L$ 外对所有参数呈多项式依赖（其中 $(L + 1)$ 为层数）。这改进了现有最佳下界（该下界对模态数量呈指数依赖）。此外，我们通过一个假设性的谱间隙上界补充了结果，该上界对 $\log L$ 呈指数依赖，表明在某种意义上我们的下界是紧的。