AI 中文总结
该研究针对非强凸场景，分析有限批次粒子平均场变分推断算法的Wasserstein稳定性，证明其迭代收敛性并给出相关缺陷估计与基准构造。
AI 中文摘要
我们研究用于平均场变分推断（MFVI）的可实现有限批次粒子算法，将其作为投影Wasserstein动力学的完全离散随机近似。目标势是全局光滑的，但不必是强凸的。我们通过曲率缺陷\\( \mathfrak d_\alpha(x,y) = \bigl[\alpha\\|x-y\\|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+ \\)量化收缩性的偏离，该缺陷是一步欧拉收缩估计中的加性损失。我们证明了一个非渐近Wasserstein稳定性界，该界分离了初始化、乘积经验近似、有限批次漂移误差、时间离散化以及耦合轨迹上累积的缺陷。在\\( \mathfrak d_\alpha\leq\beta \\)的一致界下，粒子迭代保持在任意MFVI极小值点的\\( O(\sqrt{\beta/\alpha}) \\)范围内，误差源于粒子数、批次大小和步长的显式取值。证明使用了一个平稳比较阵列，其总体律为MFVI极小值点，而粒子级律为随机乘积经验测度，并显式控制了所得的投影漂移偏差。我们还给出了按坐标的缺陷估计、与维度无关的投影漂移敏感性的结构条件，构造了一个具有闭式MFVI极小值的任意维光滑非凸基准，并解释了多项式增长漂移为何需要对未驯服显式格式进行修改。
英文摘要
We study the implementable finite-batch particle algorithm for mean-field variational inference as a fully discrete stochastic approximation of the projected Wasserstein dynamics. The target potential is globally smooth but need not be strongly convex. The departure from contractivity is quantified by the curvature defect \[ \mathfrak d_α(x,y) = \bigl[α\|x-y\|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+, \] which is the additive loss in the one-step Euler contraction estimate. We prove a non-asymptotic Wasserstein stability bound that separates initialization, product-empirical approximation, finite-batch drift error, time discretization, and the defects accumulated along the coupled trajectories. Under the uniform bound $\mathfrak d_α\leqβ$, the particle iterates remain within $O(\sqrt{β/α})$ of any MFVI minimizer, up to explicit errors in the particle number, batch size, and step size. The proof uses a stationary comparison array whose population law is an MFVI minimizer but whose particle-level law is a random product empirical measure, and it controls the resulting projected-drift discrepancy explicitly. We also give coordinatewise defect estimates and structural conditions for dimension-independent projected-drift sensitivity, construct an arbitrary-dimensional smooth nonconvex benchmark with a closed-form MFVI minimizer, and explain why polynomially growing drifts require a modification of the untamed explicit scheme.
Comments39 pages, 8 figures
