STIMULUS: Achieving Fast Convergence and Low Sample Complexity in Stochastic Multi-Objective Learning

TL;DR

This paper introduces STIMULUS, a new stochastic multi-objective optimization algorithm that achieves fast convergence and low sample complexity, with enhanced versions using adaptive batching for practical efficiency.

Contribution

The paper proposes STIMULUS and STIMULUS-M algorithms with recursive gradient estimation, achieving state-of-the-art convergence rates and sample complexities in multi-objective optimization.

Findings

Achieves $O(1/T)$ convergence in non-convex settings.

Attains $O(n+rac{ ext{sqrt}(n)}{ ext{epsilon}})$ sample complexity.

Enhanced versions with adaptive batching reduce full gradient evaluations.

Abstract

Recently, multi-objective optimization (MOO) has gained attention for its broad applications in ML, operations research, and engineering. However, MOO algorithm design remains in its infancy and many existing MOO methods suffer from unsatisfactory convergence rate and sample complexity performance. To address this challenge, in this paper, we propose an algorithm called STIMULUS( stochastic path-integrated multi-gradient recursive e\ulstimator), a new and robust approach for solving MOO problems. Different from the traditional methods, STIMULUS introduces a simple yet powerful recursive framework for updating stochastic gradient estimates to improve convergence performance with low sample complexity. In addition, we introduce an enhanced version of STIMULUS, termed STIMULUS-M, which incorporates a momentum term to further expedite convergence. We establish $O(1/T)$ convergence rates of the proposed methods for non-convex settings and $O (\exp{-\mu T})$ for strongly convex settings, where $T$ is the total number of iteration rounds. Additionally, we achieve the state-of-the-art $O \left(n+\sqrt{n}\epsilon^{-1}\right)$ sample complexities for non-convex settings and $O\left(n+ \sqrt{n} \ln ({\mu/\epsilon})\right)$ for strongly convex settings, where $\epsilon>0$ is a desired stationarity error. Moreover, to alleviate the periodic full gradient evaluation requirement in STIMULUS and STIMULUS-M, we further propose enhanced versions with adaptive batching called STIMULUS+/ STIMULUS-M+ and provide their theoretical analysis.