We focus on a class of non-smooth optimization problems over the Stiefel manifold in the decentralized setting, where a connected network of $n$ agents cooperatively minimize a finite-sum objective function with each component being weakly convex in the ambient Euclidean space. Such optimization problems, albeit frequently encountered in applications, are quite challenging due to their non-smoothness and non-convexity. To tackle them, we propose an iterative method called the decentralized Riemannian subgradient method (DRSM). The global convergence and an iteration complexity of $\mathcal{O}(\varepsilon^{-2} \log^2(\varepsilon^{-1}))$ for forcing a natural stationarity measure below $\varepsilon$ are established via the powerful tool of proximal smoothness from variational analysis, which could be of independent interest. Besides, we show the local linear convergence of the DRSM using geometrically diminishing stepsizes when the problem at hand further possesses a sharpness property. Numerical experiments are conducted to corroborate our theoretical findings.

翻译：本文关注去中心化场景下一类定义在Stiefel流形上的非光滑优化问题：由$n$个智能体组成的连通网络协同最小化一个有限和形式的目标函数，其中每个分量在欧氏空间中均为弱凸函数。此类优化问题尽管在应用中频繁出现，但由于其非光滑性与非凸性而极具挑战性。为解决该问题，我们提出一种名为去中心化黎曼次梯度方法（DRSM）的迭代算法。借助变分分析中具有独立研究价值的近端光滑性这一有力工具，我们建立了该算法的全局收敛性，并得到了驱使某自然平稳性测度降至$\varepsilon$以下所需的迭代复杂度$\mathcal{O}(\varepsilon^{-2} \log^2(\varepsilon^{-1}))$。此外，当所处理的问题进一步具备尖锐性性质时，我们证明了采用几何衰减步长的DRSM可实现局部线性收敛。通过数值实验验证了本文的理论发现。