Enhancing Convergence of Decentralized Gradient Tracking under the KL Property

• URL: http://arxiv.org/abs/2412.09556v1
• Date: Thu, 12 Dec 2024 18:44:36 GMT
• Title: Enhancing Convergence of Decentralized Gradient Tracking under the KL Property
• Authors: Xiaokai Chen, Tianyu Cao, Gesualdo Scutari,
• Abstract summary: We establish convergence of the same type for the notorious decentralized gradient-tracking-based algorithm SONATA. This matches the convergence behavior of centralized-gradient algorithms except when. thetain (1/2,1)$, sub rate is certified; and. textbf(iii)$ when $thetain (1/2,1)$, sub rate is certified; and. textbf(iii)$ when $thetain (1/2,1)$, sub rate is certified.
• Score: 10.925931212031692
• License:
• Abstract: We study decentralized multiagent optimization over networks, modeled as undirected graphs. The optimization problem consists of minimizing a nonconvex smooth function plus a convex extended-value function, which enforces constraints or extra structure on the solution (e.g., sparsity, low-rank). We further assume that the objective function satisfies the Kurdyka-{\L}ojasiewicz (KL) property, with given exponent $\theta\in [0,1)$. The KL property is satisfied by several (nonconvex) functions of practical interest, e.g., arising from machine learning applications; in the centralized setting, it permits to achieve strong convergence guarantees. Here we establish convergence of the same type for the notorious decentralized gradient-tracking-based algorithm SONATA. Specifically, $\textbf{(i)}$ when $\theta\in (0,1/2]$, the sequence generated by SONATA converges to a stationary solution of the problem at R-linear rate;$ \textbf{(ii)} $when $\theta\in (1/2,1)$, sublinear rate is certified; and finally $\textbf{(iii)}$ when $\theta=0$, the iterates will either converge in a finite number of steps or converges at R-linear rate. This matches the convergence behavior of centralized proximal-gradient algorithms except when $\theta=0$. Numerical results validate our theoretical findings.

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