Non-geodesically-convex optimization in the Wasserstein space
This addresses optimization challenges in probability measure spaces for researchers in computational mathematics and machine learning, but appears incremental as it modifies an existing method for a more general setting.
The paper tackles optimization problems in the Wasserstein space with nonconvex objective functions, deriving convergence insights for a novel semi Forward-Backward Euler scheme under nonconvex and nonsmooth regimes.
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.