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

The paper studies the finite‑sample approximation of a Hessian‑guided perturbed Wasserstein gradient flow (PWGF), which adds Gaussian perturbations to Wasserstein gradient descent to escape saddle points in nonconvex problems. It analyzes when an interacting‑particle approximation remains accurate over growing time horizons, showing that accumulated negative curvature can amplify errors while subsequent positive curvature can damp them. Under regularity assumptions and a fixed perturbation schedule, the authors prove high‑probability tracking bounds for both particles and objective values, construct a population‑first coupling to handle state‑dependent jumps, and verify the theory in a variance‑plus‑cosine model and a regularized matrix‑factorization setting.

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