Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning: Trade-off Between Con-vergence Rate and Optimality Gap

• URL: http://arxiv.org/abs/2608.01220v1
• Date: Sun, 02 Aug 2026 13:06:12 GMT
• Title: Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning: Trade-off Between Con-vergence Rate and Optimality Gap
• Abstract summary: Non-Lipschitz continuous optimization algorithms have been proposed to improve the slow convergence rate of the existing linear solutions.<n>We investigate the distributed regression problem and check its convergence rate by applying both linear and non-Lipschitz signum-based functions.<n>Our results show that although adopting signum-based func-tions may give faster convergence, it results in large optimality gaps.
• Score: 9.045698110081686
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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