发表机构
University of North Carolina at Chapel Hill; California Institute of Technology（北卡罗来纳大学教堂山分校; 加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对朗之万正则化SVGD，建立了其定量目标收敛性与时不变混沌传播的理论结果，推导了相关界并对比了两种有限时间机制的性能。
AI 中文摘要
我们建立了朗之万正则化斯坦变分梯度下降（Langevin-regularized Stein variational gradient descent）的定量目标收敛性与时不变混沌传播。斯坦因相互作用相对于约束朗之万漂移不一定很小，且通常不会产生收缩的粒子耦合。在平均场层面，斯坦因项和朗之万项在核诱导斯坦因几何与2-瓦塞斯坦几何中耗散相同的相对熵，产生平方核斯坦因差异（squared kernel Stein discrepancy）和相对费舍尔信息。在目标满足对数-索伯列夫不等式的条件下，这给出了指数型最终迭代收敛。我们还推导了相对于乘积目标的有限粒子熵恒等式，给出了经验测度在多项式采样误差范围内的时间指数型收敛。对于混沌传播，我们开发了两种互补的有限时间方法：同步耦合结合非线性平均场扩散的指数矩估计，得到瓦塞斯坦距离和核斯坦因差异（KSD）的显式单指数界；移动乘积熵给出相对于演化平均场乘积律的联合律相对熵控制，并通过熵超加性和集中性，得到固定边际相对熵、总变差界及经验KSD估计，在初始律满足额外T₂不等式时还给出瓦塞斯坦界。将这些有限时间估计与对数截止时间处的目标收敛性结合，得到经验KSD和W₂²的期望多项式时不变混沌传播率，以及固定边际总变差和W₂²的对应率，所有界均在物理时间内控制最终迭代。我们还比较了两种有限时间机制，确定了各自给出更优多项式指数的区域。
英文摘要
We establish quantitative convergence to the target and uniform-in-time propagation of chaos for Langevin-regularized Stein variational gradient descent. The Stein interaction need not be small relative to the confining Langevin drift and does not generally yield a contractive particle coupling. At the mean-field level, the Stein and Langevin components dissipate the same relative entropy in the kernel-induced Stein and $2$-Wasserstein geometries, producing the squared kernel Stein discrepancy and relative Fisher information. Under a log-Sobolev inequality for the target, this yields exponential last-iterate convergence. We also derive a finite-particle entropy identity relative to the product target, giving exponential-in-time convergence of the empirical measure up to polynomial sampling errors. For propagation of chaos, we develop two complementary finite-time approaches. A synchronous coupling, combined with exponential moment estimates for the nonlinear mean-field diffusion, yields explicit single-exponential bounds in Wasserstein distance and kernel Stein discrepancy (KSD). Moving-product entropy gives joint-law relative entropy control relative to the evolving mean-field product law and, through entropy superadditivity and concentration, fixed-marginal relative entropy and total variation bounds and empirical KSD estimates. Under an additional $T_2$ inequality for the initial law, it also yields Wasserstein bounds. Combining these finite-time estimates with target convergence at a logarithmic cutoff time gives polynomial uniform-in-time propagation of chaos rates in expectation for empirical KSD and $W_2^2$, and for fixed-marginal total variation and $W_2^2$. All bounds control the last iterate in physical time. We also compare the two finite-time mechanisms and identify regimes in which each gives the sharper polynomial exponent.
Comments54 pages
