Non-geodesically-convex optimization in the Wasserstein space

TL;DR

This paper investigates optimization in the Wasserstein space with nonconvex objectives, introducing a semi Forward-Backward Euler scheme that converges under broad nonconvex conditions, expanding understanding of such algorithms.

Contribution

It presents a novel semi Forward-Backward Euler method for non-geodesically-convex optimization in Wasserstein space, with convergence analysis in general nonconvex regimes.

Findings

Convergence of the semi Forward-Backward Euler scheme in nonconvex settings.

Extension of analysis to nonsmooth and difference-of-convex objectives.

Applicable to sampling problems with difference-of-convex log-target distributions.

Abstract

We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is nonconvex along generalized geodesics. Specifically, the objective exhibits some difference-of-convex structure along these geodesics. The setting also encompasses sampling problems where the logarithm of the target distribution is difference-of-convex. We derive multiple convergence insights for a novel semi Forward-Backward Euler scheme under several nonconvex (and possibly nonsmooth) regimes. Notably, the semi Forward-Backward Euler is just a slight modification of the Forward-Backward Euler whose convergence is -- to our knowledge -- still unknown in our very general non-geodesically-convex setting.