In this work, we develop and analyze a Gradient Descent (GD) based solution, called Alternating GD and Minimization (AltGDmin), for efficiently solving the low rank matrix completion (LRMC) in a federated setting. LRMC involves recovering an $n \times q$ rank-$r$ matrix $\Xstar$ from a subset of its entries when $r \ll \min(n,q)$. Our theoretical guarantees (iteration and sample complexity bounds) imply that AltGDmin is the most communication-efficient solution in a federated setting, is one of the fastest, and has the second best sample complexity among all iterative solutions to LRMC. In addition, we also prove two important corollaries. (a) We provide a guarantee for AltGDmin for solving the noisy LRMC problem. (b) We show how our lemmas can be used to provide an improved sample complexity guarantee for AltMin, which is the fastest centralized solution.

翻译：在本工作中，我们针对联邦场景下的低秩矩阵补全问题，提出并分析了一种基于梯度下降的解决方案——交替梯度下降最小化算法。低秩矩阵补全旨在从部分观测条目中恢复一个$n \times q$的秩-$r$矩阵$\Xstar$，其中$r \ll \min(n,q)$。我们的理论保证（迭代复杂度与样本复杂度界）表明，AltGDmin是联邦场景中通信效率最高的解决方案，同时也是求解LRMC问题最快的迭代算法之一，其样本复杂度在所有迭代解法中位居第二。此外，我们还证明了两个重要推论：（a）为AltGDmin求解含噪LRMC问题提供了理论保证；（b）展示了如何利用我们的引理为AltMin——这一最快的集中式解决方案——提供改进的样本复杂度保证。