Wasserstein gradient flows describe the evolution of probability distributions driven by energy minimization in the space of measures. The particles (or agents, or samples) move to reduce a collective energy functional, and the flow is the gradient descent in the Wasserstein metric — the metric that measures the cost of transporting one distribution to another.
The problem: many energy functionals of interest are non-convex in Wasserstein space. Gradient descent on non-convex landscapes gets stuck in local minima. The standard remedy in Euclidean optimization — adding noise (stochastic gradient descent) — has a Wasserstein analogue (adding diffusion), but diffusion spreads the distribution rather than escaping local minima in a controlled way.
The paper introduces feedback control for local convexification. Instead of adding noise globally, the method applies a control field that locally convexifies the energy landscape along the flow's trajectory. The control field is computed from the Hessian of the energy at the current distribution and applies only enough force to make the local landscape convex — not globally, not everywhere, just along the direction the flow is heading.
The result: convergence guarantees for non-convex Wasserstein gradient flows, with the control cost bounded by the degree of non-convexity encountered along the trajectory. If the flow happens to pass through convex regions, no control is needed. If it encounters non-convex regions, the control kicks in precisely enough to maintain descent.
The method turns a hard global problem (making a non-convex functional convex) into an easy local problem (making the functional convex along the current direction). The price is online computation of the Hessian; the benefit is guaranteed convergence without globally modifying the problem.