Retraction-Free Decentralized Non-convex Optimization with Orthogonal Constraints

In this paper, we investigate decentralized non-convex optimization with orthogonal constraints. Conventional algorithms for this setting require either manifold retractions or other types of projection to ensure feasibility, both of which involve costly linear algebra operations (e.g., SVD or matrix inversion). On the other hand, infeasible methods are able to provide similar performance with higher computational efficiency. Inspired by this, we propose the first decentralized version of the retraction-free landing algorithm, called $\mathbf{\text{D}}$ecentralized $\mathbf{\text{R}}$etraction-$\mathbf{\text{F}}$ree $\mathbf{\text{G}}$radient $\mathbf{\text{T}}$racking (DRFGT). We theoretically prove that DRFGT enjoys the ergodic convergence rate of $\mathcal{O}(1\mathrm{/}K)$, matching the convergence rate of centralized, retraction-based methods. We further establish that under a local Riemannian PŁ condition, DRFGT achieves a much faster linear convergence rate. Numerical experiments demonstrate that DRFGT performs on par with the state-of-the-art retraction-based methods with substantially reduced computational overhead.