Inexact JKO and proximal-gradient algorithms in the Wasserstein space

TL;DR

This paper analyzes the convergence of inexact JKO and proximal-gradient algorithms in Wasserstein spaces, providing theoretical guarantees for their use with approximate solutions in practical scenarios.

Contribution

It offers the first rigorous convergence analysis of inexact JKO and proximal-gradient methods in Wasserstein spaces, accounting for computational errors.

Findings

Weak convergence is maintained under controlled inexactness.

Convergence guarantees extend to both Wasserstein distance errors and energy evaluation errors.

The results support practical implementation of these algorithms with approximate solutions.

Abstract

This paper studies the convergence properties of the inexact Jordan-Kinderlehrer-Otto (JKO) scheme and proximal-gradient algorithm in the context of Wasserstein spaces. The JKO scheme, a widely-used method for approximating solutions to gradient flows in Wasserstein spaces, typically assumes exact solutions to iterative minimization problems. However, practical applications often require approximate solutions due to computational limitations. This work focuses on the convergence of the scheme to minimizers for the underlying functional and addresses these challenges by analyzing two types of inexactness: errors in Wasserstein distance and errors in energy functional evaluations. The paper provides rigorous convergence guarantees under controlled error conditions, demonstrating that weak convergence can still be achieved with inexact steps. The analysis is further extended to proximal-gradient algorithms, showing that convergence is preserved under inexact evaluations.