Decentralized Riemannian Gradient Descent on the Stiefel Manifold

arXiv2102.07091MaRDI QIDQ6360549

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Publication date: 14 February 2021

Abstract: We consider a distributed non-convex optimization where a network of agents aims at minimizing a global function over the Stiefel manifold. The global function is represented as a finite sum of smooth local functions, where each local function is associated with one agent and agents communicate with each other over an undirected connected graph. The problem is non-convex as local functions are possibly non-convex (but smooth) and the Steifel manifold is a non-convex set. We present a decentralized Riemannian stochastic gradient method (DRSGD) with the convergence rate of $mathcalO(1/sqrtK)$ to a stationary point. To have exact convergence with constant stepsize, we also propose a decentralized Riemannian gradient tracking algorithm (DRGTA) with the convergence rate of $mathcalO(1/K)$ to a stationary point. We use multi-step consensus to preserve the iteration in the local (consensus) region. DRGTA is the first decentralized algorithm with exact convergence for distributed optimization on Stiefel manifold.

Has companion code repository: https://github.com/chenshixiang/Decentralized_Riemannian_gradient_descent_on_Stiefel_manifold

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