A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds

This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches at present for tackling such composite optimization problems. The first, subgradient-based approaches, rely on subgradient information of the objective function to update variables, achieving an iteration complexity of $O(\epsilon^{-4}\log^2(\epsilon^{-2}))$. The second, smoothing approaches, involve constructing a smooth approximation of the nonsmooth regularization term, resulting in an iteration complexity of $O(\epsilon^{-4})$. This paper proposes a proximal gradient type algorithm that fully exploits the composite structure. The global convergence to a stationary point is established with a significantly improved iteration complexity of $O(\epsilon^{-2})$. To validate the effectiveness and efficiency of our proposed method, we present numerical results from real-world applications, showcasing its superior performance compared to existing approaches.