Mirror descent method for stochastic multi-objective optimization

TL;DR

This paper introduces the Multi-gradient Stochastic Mirror Descent (MSMD) method for stochastic multi-objective optimization, providing convergence guarantees and outperforming existing methods in generating Pareto fronts and neural network training.

Contribution

The paper proposes a novel MSMD method that uses stochastic mirror descent for SMOO, with analytical solutions and convergence guarantees, advancing optimization techniques in machine learning.

Findings

MSMD achieves sublinear convergence rates under various step setups.

MSMD outperforms weighted sum and state-of-the-art SMOO methods in Pareto front quality.

The method is effective in neural network training with competitive results.

Abstract

Stochastic multi-objective optimization (SMOO) has recently emerged as a powerful framework for addressing machine learning problems with multiple objectives. The bias introduced by the nonlinearity of the subproblem solution mapping complicates the convergence analysis of multi-gradient methods. In this paper, we propose a novel SMOO method called the Multi-gradient Stochastic Mirror Descent (MSMD) method, which incorporates stochastic mirror descent method to solve the SMOO subproblem, providing convergence guarantees. By selecting an appropriate Bregman function, our method enables analytical solutions of the weighting vector and requires only a single gradient sample at each iteration. We demonstrate the sublinear convergence rate of our MSMD method under four different inner and outer step setups. For SMOO with preferences, we propose a variant of MSMD method and demonstrate its convergence rate. Through extensive numerical experiments, we compare our method with both stochastic descent methods based on weighted sum and state-of-the-art SMOO methods. Our method consistently outperforms these methods in terms of generating superior Pareto fronts on benchmark test functions while also achieving competitive results in neural network training.