发表机构
Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America; FinTech Thrust, Hong Kong University of Science and Technology (Guangzhou), Guangzhou, Guangdong, People’s Republic of China（美利坚合众国佛罗里达州塔拉哈西市佛罗里达州立大学数学系; 中华人民共和国广东省广州市香港科技大学（广州）金融科技方向）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出惩罚非可逆Langevin算法，结合平方距离惩罚与斜对称扰动，在紧凸集上高效采样，并证明其收敛性与非可逆加速，实验验证于多种约束采样任务。
AI 中文摘要
我们提出了惩罚非可逆Langevin算法，用于从$\pi(x)\propto e^{-f(x)}\mathbf 1_{\mathcal C}(x)$中采样，其中$\mathcal C\subset\mathbb R^d$是一个紧凸集。这些算法将平方距离惩罚与常数或兼容的状态依赖斜对称扰动相结合，这些扰动保持惩罚吉布斯分布不变。对于光滑的、可能非凸的$f$，我们在对数Sobolev不等式下推导了全梯度算法的非渐近全变差界。当无偏随机梯度可用时，我们在全局收缩和全漂移在自适应二次度量下的Lipschitz条件下建立了$2$-Wasserstein界。对于固定的惩罚参数，相对于惩罚吉布斯分布的误差指数衰减到$\mathcal{O}(\sqrt{\eta})$邻域，其中$\eta$是步长。我们还限制了惩罚吉布斯分布与约束目标之间的差异。在二维二次模型中，我们通过将斜对称扰动调整到由惩罚引起的曲率不平衡来建立非可逆加速。在目标精度和较小曲率固定且初始Wasserstein距离一致有界的条件下，调整斜对称扰动将充分的欧拉迭代界从曲率比值的线性改进为对数。数值实验在约束贝叶斯回归、分类、神经网络和截断采样上评估了算法，并在随机二次模型中检验了加速机制。
英文摘要
We propose penalized nonreversible Langevin algorithms for sampling from $π(x)\propto e^{-f(x)}\mathbf 1_{\mathcal C}(x)$, where $\mathcal C\subset\mathbb R^d$ is a compact convex set. The algorithms combine a squared distance penalty with constant or compatible state dependent skew symmetric perturbations that preserve the penalized Gibbs distribution. For smooth, possibly nonconvex $f$, we derive nonasymptotic total variation bounds for the full gradient algorithm under a log Sobolev inequality. When unbiased stochastic gradients are available, we establish $2$-Wasserstein bounds under global contraction and Lipschitz conditions on the full drift in an adapted quadratic metric. For a fixed penalty parameter, the error relative to the penalized Gibbs distribution decays exponentially to an $\mathcal{O}(\sqrtη)$ neighborhood, where $η$ is the stepsize. We also bound the discrepancy between the penalized Gibbs distribution and the constrained target. In a two dimensional quadratic model, we establish nonreversible acceleration by tuning the skew perturbation to the curvature imbalance induced by penalization. With the target accuracy and smaller curvature fixed and initial Wasserstein distances uniformly bounded, tuning the skew perturbation improves the sufficient Euler iteration bound from linear to logarithmic in the curvature ratio. Numerical experiments evaluate the algorithms on constrained Bayesian regression, classification, neural networks, and truncated sampling, and examine the acceleration mechanism in a stochastic quadratic model.
Comments68 pages, 8 figures
