无海森矩阵的高分辨率蒙特卡洛采样的改进分析
Improved Analysis for Hessian-free High-resolution Monte Carlo Sampling
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中文总结 AI 辅助
本文针对无海森矩阵的高分辨率（HFHR）蒙特卡洛采样，建立了其动力学的显式收缩率，推导了HFHRMC算法的非渐近收敛界与迭代复杂度，相关结果优于现有工作，并通过数值实验验证了正α的益处。
中文摘要 AI 辅助
无海森矩阵的高分辨率（HFHR）动力学为机器学习中的采样问题，在欠阻尼朗之万动力学（ULD）的基础上添加了可逆位置扩散项。我们在位置庞加莱不等式、加权海森与拉普拉斯界以及紧索伯列夫嵌入的条件下，针对势函数不一定为凸的情况，建立了HFHR动力学的显式定量收缩率。适配的时间增广庞加莱不等式给出的显式率优于欠阻尼朗之万动力学的收缩率。我们还给出了弱解构造以及支撑该论证的发散引理的自包含谱证明。对于基于HFHR动力学离散化方案的HFHR蒙特卡洛（HFHRMC）算法，我们利用路径空间吉萨诺夫论证，得到了总变差距离下的非渐近收敛界和显式迭代复杂度。该界对所有α≥0和γ>0均成立，且在ULD端点处保持正则性。优化迭代复杂度界会在有限精度下得到一个正的、依赖于精度的位置扩散参数，其主导高阶精度与优化后的ULD端点的主导高阶精度一致。我们的迭代复杂度界优于现有的HFHR算法相关工作，还提供了包括真实数据贝叶斯学习问题在内的数值实验，以说明正α的作用及其益处。
英文摘要
Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every $α\geq0$ and $γ>0$ and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive $α$ and its benefit.
发表机构
- School of Mathematics and Statistics, Donghua University（东华大学数学与统计学院）
- Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
- School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
- Department of Mathematics, Florida State University（佛罗里达州立大学数学系）
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