离散空间上采样的哈密顿动力学
Hamiltonian dynamics for sampling on discrete spaces
浏览论文内容
中文总结 AI 辅助
该研究提出适用于离散空间的非可逆哈密顿蒙特卡洛动力学，推导其标度极限，设计模拟方案并通过数值示例验证，相比可逆采样器有加速效果。
中文摘要 AI 辅助
我们开发了一类适用于离散状态空间的非可逆哈密顿蒙特卡洛（Hamiltonian Monte Carlo）动力学通用方法。该方法为离散状态补充了连续动量变量，且无需对离散状态空间进行连续嵌入。我们建立了目标分布的不变性条件，并从遍历性、指数收缩性、渐近方差和弛豫时间等方面研究了所得过程。随后，我们推导了在日益精细的格点和高维超立方体上的标度极限，在两种场景下，合适的重标度均收敛到连续空间中的哈密顿动力学，这表明相较于可逆采样器，该方法存在从扩散到弹道的加速效应，即便对于标准非可逆方法会变为扩散的异质目标分布亦是如此。在可获取当前状态所有邻居处的目标分布的前提下，所提动力学可在连续时间中精确模拟。为降低计算成本，我们提供了基于分裂、τ-跳跃（τ-leaping）和梯度近似的近似方案。我们在涉及高维超立方体、混合模型和置换的示例上对所提框架进行了数值验证。
英文摘要
We develop a general class of non-reversible Hamiltonian Monte Carlo dynamics on discrete state spaces. The method augments the discrete state with a continuous momentum variable and does not require a continuous embedding of the discrete state space. We establish conditions for invariance of the target distribution and study the resulting processes in terms of ergodicity, exponential contractivity, asymptotic variance and relaxation time. We then derive scaling limits on increasingly fine lattices and high-dimensional hypercubes. In both settings, suitable rescalings converge to Hamiltonian dynamics in continuous space, revealing a diffusive-to-ballistic speed-up over reversible samplers, even for heterogeneous target distributions where standard non-reversible methods become diffusive. The proposed dynamics can be simulated exactly in continuous time, given access to the target distribution at all neighbours of the current state. To reduce computational cost, we provide approximation schemes based on splitting, $τ$-leaping and gradient approximations. We illustrate the proposed framework numerically on examples involving high-dimensional hypercubes, mixture models and permutations.
