Smoothing Gradient Tracking for Decentralized Optimization over the Stiefel Manifold with Non-smooth Regularizers

Abstract: Recently, decentralized optimization over the Stiefel manifold has attacked tremendous attentions due to its wide range of applications in various fields. Existing methods rely on the gradients to update variables, which are not applicable to the objective functions with non-smooth regularizers, such as sparse PCA. In this paper, to the best of our knowledge, we propose the first decentralized algorithm for non-smooth optimization over Stiefel manifolds. Our algorithm approximates the non-smooth part of objective function by its Moreau envelope, and then existing algorithms for smooth optimization can be deployed. We establish the convergence guarantee with the iteration complexity of $\mathcal{O} (\epsilon^{-4})$. Numerical experiments conducted under the decentralized setting demonstrate the effectiveness and efficiency of our algorithm.