Improved Analysis for Hessian-free High-resolution Monte Carlo Sampling

2026-08-25

AI Summary

该研究通过引入可逆位置扩散改进了欠阻尼朗之万动力学，用于解决机器学习中的采样问题，并提供了更优的收敛率和迭代复杂度界限。

Abstract

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.

Problem

• Hessian-free
• High-resolution
• Monte Carlo
• Contraction rate
• Underdamped Langevin dynamics

Innovation

• Hessian-free High-resolution Dynamics
• Quantitative Contraction Rate
• Path-space Girsanov Argument
• Non-asymptotic Convergence Bound
• Iteration Complexity