Decentralized Optimization Over the Stiefel Manifold by an Approximate Augmented Lagrangian Function

TL;DR

This paper introduces DESTINY, a decentralized optimization algorithm over the Stiefel manifold that reduces communication rounds to one per iteration, combining gradient tracking with an approximate augmented Lagrangian for efficient convergence.

Contribution

The paper presents DESTINY, a novel decentralized algorithm that minimizes communication rounds by integrating gradient tracking with an approximate augmented Lagrangian on the Stiefel manifold.

Findings

DESTINY converges to stationary points under certain conditions.

Numerical experiments show DESTINY outperforms existing methods in efficiency.

The method significantly reduces communication costs in decentralized optimization.

Abstract

In this paper, we focus on the decentralized optimization problem over the Stiefel manifold, which is defined on a connected network of $d$ agents. The objective is an average of $d$ 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.