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This research investigates the trade-off between convergence speed and optimality gaps in distributed machine learning by comparing linear methods with non-Lipschitz signum-based functions. The study reveals that while discontinuous algorithms can accelerate training, they often introduce significant steady-state errors in discrete-time regression tasks. These findings offer critical insights for engineers designing scalable ML systems where balancing rapid adaptation against final solution accuracy is essential.
Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning: Trade-off Between Con-vergence Rate and Optimality Gap
arXiv:2608.01220v1 Announce Type: cross Abstract: In recent years, the prevalence of large-scale data-sets and the demand for sophisti-cated learning models have necessitated the development of efficient distributed ma-chine learning (ML) solutions. Convergence speed is a critical factor influencing the practicality and effectiveness of these distributed frameworks. Recently, non-Lipschitz continuous optimization algorithms have been proposed to improve the slow conver-gence rate of the existing linear solutions. The use of signum-based functions is previ-ously considered in consensus and control literature to reach fast convergence in the prescribed time and also to provide robust algorithms to noisy/outlier data. However, as shown in this work, these algorithms lead to an optimality gap and steady-state re-sidual of the objective function in discrete-time setup. This motivates us to investigate the distributed optimization and ML algorithms in terms of trade-off between conver-gence rate and optimality gap. In this direction, we specifically consider the distributed regression problem and check its convergence rate by applying both linear and non-Lipschitz signum-based functions. We check our distributed regression approach by extensive simulations. Our results show that although adopting signum-based func-tions may give faster convergence, it results in large optimality gaps. The findings pre-sented in this paper may contribute to and advance the ongoing discourse of similar distributed algorithms, e.g., for distributed constrained optimization and distributed estimation.
Abstract:In recent years, the prevalence of large-scale data-sets and the demand for sophisti-cated learning models have necessitated the development of efficient distributed ma-chine learning (ML) solutions. Convergence speed is a critical factor influencing the practicality and effectiveness of these distributed frameworks. Recently, non-Lipschitz continuous optimization algorithms have been proposed to improve the slow conver-gence rate of the existing linear solutions. The use of signum-based functions is previ-ously considered in consensus and control literature to reach fast convergence in the prescribed time and also to provide robust algorithms to noisy/outlier data. However, as shown in this work, these algorithms lead to an optimality gap and steady-state re-sidual of the objective function in discrete-time setup. This motivates us to investigate the distributed optimization and ML algorithms in terms of trade-off between conver-gence rate and optimality gap. In this direction, we specifically consider the distributed regression problem and check its convergence rate by applying both linear and non-Lipschitz signum-based functions. We check our distributed regression approach by extensive simulations. Our results show that although adopting signum-based func-tions may give faster convergence, it results in large optimality gaps. The findings pre-sented in this paper may contribute to and advance the ongoing discourse of similar distributed algorithms, e.g., for distributed constrained optimization and distributed estimation.
Systems and Control (eess.SY); Machine Learning (cs.LG); Signal Processing (eess.SP); Optimization and Control (math.OC)
arXiv:2608.01220 [eess.SY]
(or arXiv:2608.01220v1 [eess.SY] for this version)
https://doi.org/10.48550/arXiv.2608.01220 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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