Non-geodesically-convex optimization in the Wasserstein space

2024-06-01

arXiv.org

AI Summary

This work addresses optimization and sampling in the Wasserstein space for non-geodesically convex—and even non-convex—objectives, particularly where standard convexity along generalized geodesics fails (e.g., when the log-density is difference-of-convex). To this end, we propose the semi-implicit forward-backward Euler scheme, the first method to establish rigorous convergence guarantees under non-geodesic convexity and potential nonsmoothness of the objective. Our approach integrates Wasserstein gradient flow modeling, difference-of-convex (DC) analysis, and the variational principle of generalized geodesics, systematically refining the classical forward-backward Euler discretization. Theoretical results provide multiple convergence assurances—including energy dissipation, distance contraction in Wasserstein metric, and weak convergence of probability measures—substantially broadening the scope of non-convex distributional sampling and learning. This framework delivers a novel theoretical foundation for high-dimensional non-convex Bayesian inference and generative modeling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Wasserstein Space
• Differential Convexity
• Non-Geodesic Convex Optimization

Innovation

• Wasserstein space
• semi-forward-backward Euler scheme
• non-geodesic convex optimization