Decentralized Sum-of-Nonconvex Optimization

• URL: http://arxiv.org/abs/2402.02356v1
• Date: Sun, 4 Feb 2024 05:48:45 GMT
• Title: Decentralized Sum-of-Nonconvex Optimization
• Authors: Zhuanghua Liu and Bryan Kian Hsiang Low
• Abstract summary: We consider the optimization problem of the sum-of-non function, i.e., a guarantee function that is the average non-consensus number. We propose an accelerated decentralized first-order algorithm by techniques of gradient, tracking into the rate, and multi-consensus.
• Score: 42.04181488477227
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the optimization problem of minimizing the sum-of-nonconvex function, i.e., a convex function that is the average of nonconvex components. The existing stochastic algorithms for such a problem only focus on a single machine and the centralized scenario. In this paper, we study the sum-of-nonconvex optimization in the decentralized setting. We present a new theoretical analysis of the PMGT-SVRG algorithm for this problem and prove the linear convergence of their approach. However, the convergence rate of the PMGT-SVRG algorithm has a linear dependency on the condition number, which is undesirable for the ill-conditioned problem. To remedy this issue, we propose an accelerated stochastic decentralized first-order algorithm by incorporating the techniques of acceleration, gradient tracking, and multi-consensus mixing into the SVRG algorithm. The convergence rate of the proposed method has a square-root dependency on the condition number. The numerical experiments validate the theoretical guarantee of our proposed algorithms on both synthetic and real-world datasets.

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