Smoothed Picard Hamiltonian Monte Carlo

2026-09-07

AI Summary

本文提出了一种新的低精度采样器——平滑Picard哈密顿蒙特卡洛，通过结合高斯平滑、Picard迭代和高阶离散化方法来解决对数凹目标的高效采样问题。

Abstract

We develop a new low-accuracy sampler, called \emph{smoothed Picard Hamiltonian Monte Carlo}, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target $\pi \propto \exp(-V)$ in dimension $d$ satisfying $0 \prec \alpha I \preceq \nabla^2 V \preceq \beta I$, with condition number $\kappa := \beta/\alpha$, smoothed Picard HMC returns a sample with $\sqrt \alpha\,W_2(\cdot,\pi) \le \varepsilon$ using $\widetilde O(\kappa^2 + \kappa^{7/6} d^{1/6}/\varepsilon^{1/3})$ gradient queries. We also prove stronger $W_q$ bounds, and then develop an algorithmic framework, the recursive warm start generator, to upgrade these $W_q$ bounds to stronger divergence guarantees. This produces a warm start for the proximal bouncy particle sampler, introduced in a companion work, leading to a high-accuracy log-concave sampler with complexity $\widetilde O((\kappa^{7/6} d^{1/6} + \kappa^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))$.

Problem

• log-concave
• sampler
• Wasserstein distance
• gradient queries

Innovation

• smoothed Picard Hamiltonian Monte Carlo
• Gaussian smoothing
• Picard iteration
• higher-order discretization
• log-concave sampling