We propose a variant of Hamiltonian Monte Carlo (HMC), called the Repelling-Attracting Hamiltonian Monte Carlo (RAHMC), for sampling from multimodal distributions. The key idea that underpins RAHMC is a departure from the conservative dynamics of Hamiltonian systems, which form the basis of traditional HMC, and turning instead to the dissipative dynamics of conformal Hamiltonian systems. In particular, RAHMC involves two stages: a mode-repelling stage to encourage the sampler to move away from regions of high probability density; and, a mode-attracting stage, which facilitates the sampler to find and settle near alternative modes. We achieve this by introducing just one additional tuning parameter -- the coefficient of friction. The proposed method adapts to the geometry of the target distribution, e.g., modes and density ridges, and can generate proposals that cross low-probability barriers with little to no computational overhead in comparison to traditional HMC. Notably, RAHMC requires no additional information about the target distribution or memory of previously visited modes. We establish the theoretical basis for RAHMC, and we discuss repelling-attracting extensions to several variants of HMC in literature. Finally, we provide a tuning-free implementation via dual-averaging, and we demonstrate its effectiveness in sampling from, both, multimodal and unimodal distributions in high dimensions.

翻译：我们提出一种哈密顿蒙特卡洛（HMC）的变体，称为排斥-吸引哈密顿蒙特卡洛（RAHMC），用于从多峰分布中采样。支撑RAHMC的核心思想在于偏离传统HMC所依赖的哈密顿系统的保守动力学，转而采用共形哈密顿系统的耗散动力学。具体而言，RAHMC包含两个阶段：模式排斥阶段，鼓励采样器远离高概率密度区域；以及模式吸引阶段，促进采样器发现并稳定在替代模式附近。我们通过引入仅一个额外调节参数——摩擦系数——来实现这一目标。所提方法能够自适应目标分布的几何结构（如模式和密度脊），并可在与传统HMC相比几乎无计算开销的情况下生成跨越低概率障碍的提议。值得注意的是，RAHMC无需目标分布的任何额外信息或对先前访问模式的记忆。我们建立了RAHMC的理论基础，并讨论了其在文献中多种HMC变体上的排斥-吸引扩展。最后，我们通过双平均法提供了一种免调参的实现，并展示了该方法在高维多峰与单峰分布采样中的有效性。