Finite-Sample Approximation of Hessian-Guided Perturbed Wasserstein Gradient Flows

2026-10-07

AI Summary

This study addresses the accuracy degradation of finite-particle approximations to Hessian-guided perturbed Wasserstein gradient flows over long time horizons. To this end, it proposes an analytical framework that preserves the curvature of reference trajectories, revealing the intrinsic mechanisms by which negative curvature amplifies approximation errors while positive curvature suppresses them. Furthermore, a population-first coupling technique is introduced to handle state-dependent Gaussian jumps, and a variance-plus-cosine model is established to verify the underlying conditions. The primary contribution lies in rigorously proving high-probability error bounds for particle tracking of target distributions. These theoretical guarantees are validated within matrix factorization models, demonstrating local attractivity and alternating positive-negative curvature patterns, thereby ensuring high-precision tracking over extended time horizons.

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.

Problem

• Wasserstein gradient flow
• finite-sample approximation
• nonconvex optimization
• interacting particles
• saddle point escape

Innovation

• Wasserstein gradient flow
• finite-sample approximation
• accumulated curvature
• population-first coupling
• Hessian-guided perturbation