Decentralized Weakly Convex Optimization Over the Stiefel Manifold

We focus on a class of non-smooth non-convex optimization problems over the Stiefel manifold in the decentralized setting, where a connected network of $n$ agents cooperatively minimizes 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 to a stationary point and an iteration complexity of $\mathcal{O}(\varepsilon^{-2}\log^{2}(\varepsilon^{-1}))$ for finding an $\varepsilon$-stationary point are established using a new result we developed concerning the Lipschitz continuity of a certain manifold proximal mapping, which could be of independent interest. Besides, when the problem at hand further possesses a sharpness property, we show that DRSM with geometrically diminishing stepsizes enjoys a local linear rate of convergence. Lastly, we conduct numerical experiments to verify our theoretical findings.