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Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points
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Proposes perturbed Wasserstein gradient flow (PWGF), which injects Hessian-directed Gaussian-process vector-field perturbations at approximate first-order stationary points to escape saddles and reach (ε,δ)-second-order stationary points in polynomial time.
Problem
Wasserstein gradient flow over probability measures converges only to first-order stationary points for nonconvex functionals, which may be saddle points; there is no second-order guarantee for measure optimization.
Approach
Perturb the measure by pushing it forward along a Gaussian-process random vector field whose kernel is the square of the Wasserstein Hessian (K_mu = H_mu^2), so the noise is directed toward the negative-curvature direction; a function-decrease check after WGF decides whether a stationary point is a saddle.
Key results
- PWGF reaches an (ε,δ)-second-order stationary point in t = Õ(ΔF(1/ε² + 1/δ⁴)) time steps with probability 1−ζ (continuous-time theorem).
- Discrete-time PWGF attains the same Õ(ΔF(1/ε² + 1/δ⁴)) step complexity.
- For strictly benign objectives, discrete-time PWGF converges to an α-neighborhood of a global optimum in the same polynomial number of steps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Hessian-guided Gaussian-process perturbations near saddle points to WGF | Wasserstein gradient flow · General nonconvex probability measure optimization | objective decrease after perturbation and WGF | Helped |
| Use a Gaussian-process kernel constructed from the Wasserstein Hessian instead of a fixed kernel | PWGF · Approximate saddle points in probability measure optimization | objective decrease after perturbation and WGF | Helped |
Limitations
- Computing the Hessian-based kernel and Gaussian process samples is computationally costly; the paper suggests stochastic Hessian approximations as future work.
- Benignity of general distributional objectives is not analyzed; guarantees rely on strict benignity assumptions.
- Assumes smoothness and Lipschitz conditions on the Wasserstein gradient and Hessian (L1, L2, L3), and the saddle-detection step relies on a time-threshold heuristic.
Experimental setup
Baselines: Wasserstein gradient flow isotropic perturbation in measure space
Builds on: perturbed gradient descent Wasserstein gradient flow mean-field Langevin dynamics
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