发表机构
Massachusetts Institute of Technology; Yale University; Duke University（麻省理工学院; 耶鲁大学; 杜克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出平滑Picard哈密顿蒙特卡洛采样器，结合高斯平滑、Picard迭代和高阶离散化，实现低精度对数凹采样，并通过递归热启动生成器升级为高精度采样，显著降低复杂度。
AI 中文摘要
我们开发了一种新的低精度采样器，称为平滑Picard哈密顿蒙特卡洛，它结合了高斯平滑、Picard迭代和高阶离散化。对于维度d中满足0≺αI⪯∇²V⪯βI的对数凹目标π∝exp(-V)，条件数κ:=β/α，平滑Picard HMC使用O~(κ²+κ^{7/6}d^{1/6}/ε^{1/3})次梯度查询返回满足√α W₂(·,π)≤ε的样本。我们还证明了更强的W_q界，然后开发了一个算法框架，即递归热启动生成器，将这些W_q界升级为更强的散度保证。这为伴随工作中引入的近端弹跳粒子采样器提供了热启动，从而产生了一个复杂度为O~((κ^{7/6}d^{1/6}+κ^{1/2}d^{1/4})polylog(1/ε))的高精度对数凹采样器。
英文摘要
We develop a new low-accuracy sampler, called smoothed Picard Hamiltonian Monte Carlo, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target $π\propto \exp(-V)$ in dimension $d$ satisfying $0 \prec αI \preceq \nabla^2 V \preceq βI$, with condition number $κ:= β/α$, smoothed Picard HMC returns a sample with $\sqrt α\,W_2(\cdot,π) \le \varepsilon$ using $\widetilde O(κ^2 + κ^{7/6} d^{1/6}/\varepsilon^{1/3})$ 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 $\widetilde O((κ^{7/6} d^{1/6} + κ^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))$.
Commentsv2: Sharpened some bounds
