STIMULUS: Achieving Fast Convergence and Low Sample Complexity in Stochastic Multi-Objective Learning
This addresses a bottleneck in multi-objective learning for ML and engineering applications, offering improved efficiency with theoretical guarantees.
The paper tackles the problem of slow convergence and high sample complexity in stochastic multi-objective optimization by proposing the STIMULUS algorithm, which achieves O(1/T) convergence for non-convex settings and state-of-the-art sample complexities like O(n + sqrt(n)ε^{-1}).
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 $O (\exp{-μT})$ for strongly convex settings, where $T$ is the total number of iteration rounds. Additionally, we achieve the state-of-the-art $O \left(n+\sqrt{n}ε^{-1}\right)$ sample complexities for non-convex settings and $O\left(n+ \sqrt{n} \ln ({μ/ε})\right)$ for strongly convex settings, where $ε>0$ 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.