The performance of Metropolis-Hastings algorithms is highly sensitive to the choice of step size, and miss-specification can lead to severe loss of efficiency. We study algorithms with randomized step sizes, considering both auxiliary-variable and marginalized constructions. We show that algorithms with a randomized step size inherit weak Poincaré inequalities/spectral gaps from their fixed-step-size counterparts under minimal conditions, and that the marginalized kernel should always be preferred in terms of asymptotic variance to the auxiliary-variable choice if it is implementable. In addition we show that both types of randomization make an algorithm robust to tuning, meaning that spectral gaps decay polynomially as the step size is increasingly poorly chosen. We further show that step-size randomization often preserves high-dimensional scaling limits and algorithmic complexity, while increasing the optimal acceptance rate for Langevin and Hamiltonian samplers when an Exponential or Uniform distribution is chosen to randomize the step size. Theoretical results are complemented with a numerical study on challenging benchmarks such as Poisson regression, Neal's funnel and the Rosenbrock (banana) distribution.

翻译：Metropolis-Hastings算法的性能对步长选择高度敏感，参数设定不当会导致效率严重下降。本文研究具有随机步长的算法，同时考虑辅助变量构造与边缘化构造。我们证明，在最小条件下，具有随机步长的算法能够从其固定步长对应版本继承弱庞加莱不等式/谱隙特性，并且若可实现，边缘化核在渐近方差方面始终优于辅助变量选择。此外，我们证明两种随机化方式均能增强算法对参数调节的鲁棒性，即当步长选择严重失当时，谱隙仅以多项式速率衰减。我们进一步证明，步长随机化通常能保持高维尺度极限与算法复杂度，同时当选择指数分布或均匀分布对步长进行随机化时，能够提高Langevin采样器与Hamiltonian采样器的最优接受率。理论结果通过数值实验在具有挑战性的基准测试中得到验证，包括泊松回归、Neal漏斗分布和Rosenbrock（香蕉）分布。