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.

翻译：我们研究了基于无向图建模的网络上的去中心化多智能体优化问题。该优化问题包含最小化一个非凸光滑函数与一个凸扩展值函数之和，后者用于对解施加约束或额外结构（例如稀疏性、低秩性）。我们进一步假设目标函数满足Kurdyka-{\L}ojasiewicz (KL) 性质，其指数为给定值 $\theta\in [0,1)$。该KL性质被多个实际应用中的（非凸）函数所满足，例如源于机器学习应用；在集中式设置中，该性质使得算法能够获得强收敛性保证。本文针对著名的基于去中心化梯度跟踪的算法SONATA，建立了同类型的收敛性结果。具体而言：$\textbf{(i)}$ 当 $\theta\in (0,1/2]$ 时，SONATA生成的序列以R线性速率收敛至问题的一个平稳解；$\textbf{(ii)}$ 当 $\theta\in (1/2,1)$ 时，可证明其具有次线性收敛速率；最后 $\textbf{(iii)}$ 当 $\theta=0$ 时，迭代序列要么在有限步内收敛，要么以R线性速率收敛。除了 $\theta=0$ 的情况外，这与集中式近端梯度算法的收敛行为一致。数值实验结果验证了我们的理论发现。