arXiv：cs.LG· Ryotaro Kawata, Atsushi Nitanda, Taiji Suzuki·· 4 小时前AI 评分29
Hessian 引导扰动 Wasserstein 梯度流的有限样本近似分析
Finite-Sample Approximation of Hessian-Guided Perturbed Wasserstein Gradient Flows
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研究 Hessian 引导扰动 Wasserstein 梯度流（PWGF）在有限粒子近似下、随时间跨度增长时的跟踪精度。分析表明，沿参考路径累积的负曲率会放大近似误差，随后的正曲率可抑制其影响，从而刻画了短暂不稳定仍可保证长期精确跟踪的情形。作者在方差加余弦模型和正则化矩阵分解模型中验证了曲率条件，并给出了高概率下的粒子与目标值跟踪界。
正文
Abstract:Wasserstein gradient flow extends gradient descent to probability measures. Its Hessian-guided perturbed variant (PWGF) adds Gaussian perturbations to escape saddle points in nonconvex problems. We investigate when its approximation by finitely many interacting particles remains accurate over growing time horizons. Our analysis retains the curvature accumulated along the population-driven reference path: negative curvature can amplify approximation errors, while subsequent positive curvature can damp their influence. This captures favorable scenarios in which temporary instability is compatible with accurate tracking over growing horizons. Under regularity assumptions and a prescribed common perturbation schedule, we prove particle and objective-value tracking bounds on a high-probability event for reference paths satisfying explicit conditions on accumulated curvature. To handle state-dependent Gaussian jumps, we construct a population-first coupling that preserves the reference particles' conditional independence and reduces jump errors to covariance comparison. We verify the conditions in a variance-plus-cosine model, where curvature recovery yields a growing-horizon tracking guarantee. We also establish local attraction, transverse descent, and positive second variation in two regions of a regularized matrix-factorization model, motivating a positive-negative-positive curvature pattern.
| Subjects: | Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.10218 [cs.LG] |
| (or arXiv:2610.10218v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.10218 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Ryotaro Kawata [view email]
[v1]
Wed, 7 Oct 2026 15:13:56 UTC (509 KB)
来源：arXiv：cs.LG · arxiv.org