Accelerated Multiple Wasserstein Gradient Flows for Multi-objective Distributional Optimization

TL;DR

This paper introduces an accelerated algorithm for multi-objective distributional optimization in Wasserstein space, achieving faster convergence rates and better practical performance than previous methods.

Contribution

It proposes A-MWGraD, an accelerated variant of MWGraD, with theoretical convergence guarantees and a practical kernel-based discretization for improved efficiency.

Findings

A-MWGraD achieves O(1/t^2) convergence for convex objectives.

A-MWGraD outperforms MWGraD in numerical experiments.

The method demonstrates improved sampling efficiency and convergence speed.

Abstract

We study multi-objective optimization over probability distributions in Wasserstein space. Recently, Nguyen et al. (2025) introduced Multiple Wasserstein Gradient Descent (MWGraD) algorithm, which exploits the geometric structure of Wasserstein space to jointly optimize multiple objectives. Building on this approach, we propose an accelerated variant, A-MWGraD, inspired by Nesterov's acceleration. We analyze the continuous-time dynamics and establish convergence to weakly Pareto optimal points in probability space. Our theoretical results show that A-MWGraD achieves a convergence rate of O(1/t^2) for geodesically convex objectives and O(e^{-\sqrt{\beta}t}) for $\beta$-strongly geodesically convex objectives, improving upon the O(1/t) rate of MWGraD in the geodesically convex setting. We further introduce a practical kernel-based discretization for A-MWGraD and demonstrate through numerical experiments that it consistently outperforms MWGraD in convergence speed and sampling efficiency on multi-target sampling tasks.