Problem
Research questions and friction points this paper is trying to address.
distributed optimization
signum-based functions
convergence rate
optimality gap
non-Lipschitz
Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning: Trade-off Between Con-vergence Rate and Optimality Gap
🤖 AI Summary
This work investigates the optimality gap arising from the use of non-Lipschitz sign functions to accelerate convergence in distributed machine learning. Focusing on discrete-time distributed regression tasks, the study proposes an optimization algorithm that integrates consensus control with the sign function. It systematically demonstrates, for the first time, that while this mechanism substantially enhances convergence speed, it inevitably introduces a steady-state residual and a gap between the attained solution and the true optimum of the objective function. Through rigorous theoretical analysis and extensive simulations, the paper establishes a precise trade-off between convergence rate and optimization accuracy, thereby providing crucial theoretical foundations for the design of future distributed algorithms for constrained optimization and 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
non-Lipschitz optimization
signum-based functions
convergence rate
optimality gap
distributed machine learning
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