Enhancing Convergence of Decentralized Gradient Tracking under the KL Property
This work provides convergence guarantees for decentralized optimization in nonconvex settings, which is incremental but important for applications like machine learning with constraints.
The paper tackles the problem of decentralized multiagent optimization over networks for nonconvex functions with the KL property, establishing that the SONATA algorithm achieves R-linear convergence for certain KL exponents and sublinear rates for others, matching centralized methods in most cases.
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-Łojasiewicz (KL) property, with given exponent $θ\in [0,1)$. 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 $θ\in (0,1/2]$, the sequence generated by SONATA converges to a stationary solution of the problem at R-linear rate;$ \textbf{(ii)} $when $θ\in (1/2,1)$, sublinear rate is certified; and finally $\textbf{(iii)}$ when $θ=0$, 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 $θ=0$. Numerical results validate our theoretical findings.