Decentralized Douglas-Rachford Splitting Methods for Smooth Optimization over Compact Submanifolds

We study decentralized smooth optimization problems over compact submanifolds. Recasting it as a composite optimization problem, we propose a decentralized Douglas-Rachford splitting algorithm (DDRS). When the proximal operator of the local loss function does not have a closed-form solution, an inexact version of DDRS (iDDRS), is also presented. Both algorithms rely on careful integration of the nonconvex Douglas-Rachford splitting algorithm with gradient tracking and manifold optimization. We show that our DDRS and iDDRS achieve the convergence rate of $\mathcal{O}(1/k).$ The main challenge in the proof is how to handle the nonconvexity of the manifold constraint. To address this issue, we utilize the concept of proximal smoothness for compact submanifolds. This ensures that the projection onto the submanifold exhibits convexity-like properties, which allows us to control the consensus error across agents. Numerical experiments on the principal component analysis are conducted to demonstrate the effectiveness of our decentralized DRS compared with the state-of-the-art ones.