Decentralized Optimization Over the Stiefel Manifold by an Approximate Augmented Lagrangian Function
Lei Wang, Xin Liu
Abstract
In this paper, we focus on the decentralized optimization problem over the Stiefel manifold, which is defined on a connected network of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> agents. The objective is an average of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> local functions, and each function is privately held by an agent and encodes its data. The agents can only communicate with their neighbors in a collaborative effort to solve this problem. In existing methods, multiple rounds of communications are required to guarantee the convergence, giving rise to high communication costs. In contrast, this paper proposes a decentralized algorithm, called DESTINY, which only invokes a single round of communications per iteration. DESTINY combines gradient tracking techniques with a novel approximate augmented Lagrangian function. The global convergence to stationary points is rigorously established. Comprehensive numerical experiments demonstrate that DESTINY has a strong potential to deliver a cutting-edge performance in solving a variety of testing problems.