We study the problem of sampling from a target distribution $π(q)\propto e^{-U(q)}$ on $\mathbb{R}^d$, where $U$ can be non-convex, via the Hessian-free high-resolution (HFHR) dynamics, which is a second-order Langevin-type process that has $e^{-U(q)-\frac12|p|^2}$ as its unique invariant distribution, and it reduces to kinetic Langevin dynamics (KLD) as the resolution parameter $α\to0$. The existing theory for HFHR dynamics in the literature is restricted to strongly-convex $U$, although numerical experiments are promising for non-convex settings as well. We focus on studying the convergence of HFHR dynamics when $U$ can be non-convex, which bridges a gap between theory and practice. Under a standard assumption of dissipativity and smoothness on $U$, we adopt the reflection/synchronous coupling method. This yields a Lyapunov-weighted Wasserstein distance in which the HFHR semigroup is exponentially contractive for all sufficiently small $α>0$ whenever KLD is. We further show that, under an additional assumption that asymptotically $\nabla U$ has linear growth at infinity, the contraction rate for HFHR dynamics is strictly better than that of KLD, with an explicit gain. As a case study, we verify the assumptions and the resulting acceleration for three examples: a multi-well potential, Bayesian linear regression with $L^p$ regularizer and Bayesian binary classification. We conduct numerical experiments based on these examples, as well as an additional example of Bayesian logistic regression with real data processed by the neural networks, which illustrates the efficiency of the algorithms based on HFHR dynamics and verifies the acceleration and superior performance compared to KLD.

翻译：本文研究从目标分布$π(q)\propto e^{-U(q)}$（定义于$\mathbb{R}^d$上）的采样问题，其中$U$可能为非凸函数。我们采用无Hessian高分辨率动力学——一种二阶朗之万型随机过程，其唯一不变分布为$e^{-U(q)-\frac12|p|^2}$，且当分辨率参数$α\to0$时退化为经典动能朗之万动力学。现有文献中关于HFHR动力学的理论分析仅限于$U$为强凸函数的情形，尽管数值实验在非凸场景中亦展现出良好前景。本文重点研究$U$为非凸函数时HFHR动力学的收敛性，以弥合理论与应用之间的鸿沟。在$U$满足标准耗散性与光滑性假设的前提下，我们采用反射/同步耦合方法，构建了Lyapunov加权Wasserstein距离度量。在此度量下，只要KLD具有指数收缩性，则对所有充分小的$α>0$，HFHR半群同样呈现指数收缩特性。进一步地，在附加假设“$\nabla U$在无穷远处具有渐近线性增长”条件下，我们证明HFHR动力学的收缩率严格优于KLD，并给出显式的加速增益。作为案例研究，我们在三类模型中验证了理论假设与加速效果：多势阱系统、$L^p$正则化贝叶斯线性回归以及贝叶斯二分类问题。基于这些案例及额外进行的真实数据神经网络贝叶斯逻辑回归实验，我们开展了系统的数值研究。实验结果不仅验证了基于HFHR动力学算法的采样效率，更通过对比KLD凸显了其加速优势与卓越性能。