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.

中文翻译：

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通过近似增广拉格朗日函数对 Stiefel 流形进行分散优化

在本文中，我们关注 Stiefel 流形上的分散优化问题，该问题定义在连接网络上$d$代理。目标是平均$d$本地函数，每个函数都由代理私有持有并对其数据进行编码。代理只能与他们的邻居进行协作以解决这个问题。在现有的方法中，需要多轮通信来保证收敛，导致通信成本高。相比之下，本文提出了一种去中心化算法，称为 DESTINY，每次迭代仅调用单轮通信。DESTINY 将梯度跟踪技术与一种新颖的近似增强拉格朗日函数相结合。全局收敛到静止点是严格建立的。综合数值实验表明，DESTINY 具有强大的潜力，可以在解决各种测试问题方面提供前沿性能。