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

Zhuqing Liu, Chaosheng Dong, Michinari Momma, Simone Shao, Shaoyuan Xu, Haibo Yang, Jia Liu

Primary Area: optimization

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Keywords: multi-objective optimization, sample complexity, variance reduction, momentum

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Abstract: Recently, multi-objective optimization (MOO) problems have received increasing attention due to their wide range of applications in various fields, such as machine learning (ML), operations research, and many engineering applications. However, MOO algorithm design remains in its infancy and many existing MOO methods suffer from unsatisfactory convergence performance. To address this challenge, in this paper, we propose an algorithm called STIMULUS (**ST**ochastic path-**I**ntegrated **MUL**ti-graident rec**U**rsive e**S**timator), 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. This methodology improves convergence performance by reducing the variance in multi-gradient estimation, leading to more stable convergence paths. In addition, we introduce an enhanced version of STIMULUS, termed STIMULUS-M, which incorporates the momentum term to further expedite convergence. One of the key contributions of this paper is the theoretical analysis for both STIMULUS and STIMULUS-M, where we establish an $\mathcal{O}(\frac{1}{T})$ convergence rate for both methods, which implies a state-of-the-art sample complexity of $O\left(n+\sqrt{n}\epsilon^{-1}\right)$ under non-convexity settings. In the case where the objectives are strongly convex, we further establish a linear convergence rate of $\mathcal{O}(e^{-\mu T})$ of the proposed methods, which suggests an even stronger $\mathcal{O}\left(n+ \sqrt{n} \ln ({\mu/\epsilon})\right)$ sample complexity. Moreover, to further 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. Our extensive experimental results verify the efficacy of our proposed algorithms and their superiority over existing methods.

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Submission Number: 8491