Enhancing Convergence of Decentralized Gradient Tracking under the KL Property

TL;DR

This paper proves convergence rates for a decentralized optimization algorithm under the Kurdyka-Łojasiewicz property, applicable to nonconvex functions with practical relevance, extending centralized convergence guarantees to multi-agent networks.

Contribution

It establishes the first convergence guarantees for decentralized gradient tracking algorithms under the KL property, covering various exponents and matching centralized algorithm performance.

Findings

Convergence to stationary solutions with R-linear rate for 0,1/2]

Sublinear convergence rate for 0,1]

Finite or R-linear convergence when =0

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.