Multiple Wasserstein Gradient Descent Algorithm for Multi-Objective Distributional Optimization

TL;DR

This paper introduces MWGraD, a particle-based algorithm for multi-objective distributional optimization that iteratively updates distributions using Wasserstein gradients, with theoretical analysis and experimental validation.

Contribution

The paper proposes MWGraD, a novel iterative particle-based algorithm for multi-objective distributional optimization using Wasserstein gradients, with theoretical guarantees and empirical results.

Findings

MWGraD effectively minimizes multiple objectives simultaneously.

Theoretical analysis confirms convergence properties.

Experimental results demonstrate superior performance on synthetic and real datasets.

Abstract

We address the optimization problem of simultaneously minimizing multiple objective functionals over a family of probability distributions. This type of Multi-Objective Distributional Optimization commonly arises in machine learning and statistics, with applications in areas such as multiple target sampling, multi-task learning, and multi-objective generative modeling. To solve this problem, we propose an iterative particle-based algorithm, which we call Muliple Wasserstein Gradient Descent (MWGraD), which constructs a flow of intermediate empirical distributions, each being represented by a set of particles, which gradually minimize the multiple objective functionals simultaneously. Specifically, MWGraD consists of two key steps at each iteration. First, it estimates the Wasserstein gradient for each objective functional based on the current particles. Then, it aggregates these gradients into a single Wasserstein gradient using dynamically adjusted weights and updates the particles accordingly. In addition, we provide theoretical analysis and present experimental results on both synthetic and real-world datasets, demonstrating the effectiveness of MWGraD.