Uniform ergodicity and spectral gap of importance-tempered MCMC with heavy-tailed targets (arxiv.org)
arXiv:2506.19186v2 Announce Type: replace-cross
Abstract: We study the importance-tempered random-walk Metropolis (RWM) algorithm on $\mathbb{R}^d$, which targets the tempered density $\pi_\beta(x) \propto \pi(x)^\beta$ with $\beta \in (0,1)$ and corrects for the resulting bias by importance weighting. The dynamics of this sampler can be described by a continuous-time Markov chain whose embedded chain is the tempered RWM. We prove that when the tails of $\pi$ decay at rate $\|x\|^{-\gamma}$, this continuous-time chain is uniformly ergodic if and only if $d/\gamma < \beta < (\gamma - 2)/\gamma$. Further, the global spectral gap of this chain can be controlled via the spectral gap of the chain restricted to a ball with radius growing polynomially in $d$. Our theory and simulation studies show that, with a suitable choice of $\beta$, importance tempering can substantially reduce sensitivity to initialization and significantly improve efficiency over the standard RWM.
Abstract: We study the importance-tempered random-walk Metropolis (RWM) algorithm on $\mathbb{R}^d$, which targets the tempered density $\pi_\beta(x) \propto \pi(x)^\beta$ with $\beta \in (0,1)$ and corrects for the resulting bias by importance weighting. The dynamics of this sampler can be described by a continuous-time Markov chain whose embedded chain is the tempered RWM. We prove that when the tails of $\pi$ decay at rate $\|x\|^{-\gamma}$, this continuous-time chain is uniformly ergodic if and only if $d/\gamma < \beta < (\gamma - 2)/\gamma$. Further, the global spectral gap of this chain can be controlled via the spectral gap of the chain restricted to a ball with radius growing polynomially in $d$. Our theory and simulation studies show that, with a suitable choice of $\beta$, importance tempering can substantially reduce sensitivity to initialization and significantly improve efficiency over the standard RWM.
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