A Linearized Alternating Direction Multiplier Method for Federated Matrix Completion Problems
Patrick Hytla, Tran T. A. Nghia, Duy Nhat Phan, Andrew Rice
TL;DR
This work tackles federated matrix completion where private user data are distributed across $p$ clients. It introduces FedMC-ADMM, which integrates the Alternating Direction Method of Multipliers with randomized block-coordinate updates and alternating proximal gradient steps to address multi-block, nonconvex, and nonsmooth objectives while preserving privacy. The authors prove almost sure subsequential convergence to stationary points with a rate of $O(K^{-1/2})$ and a communication complexity of $O(ε^{-2})$ rounds to reach an $ε$-stationary point. Empirical results on MovieLens 1M/10M and Netflix show that FedMC-ADMM achieves faster convergence and lower test RMSE than FedMAvg, with better scalability as data size grows. Overall, the paper advances privacy-preserving, communication-efficient federated optimization for matrix completion and sets the stage for extensions to other FL tasks with nonconvex, nonsmooth, multi-block structures.
Abstract
Matrix completion is fundamental for predicting missing data with a wide range of applications in personalized healthcare, e-commerce, recommendation systems, and social network analysis. Traditional matrix completion approaches typically assume centralized data storage, which raises challenges in terms of computational efficiency, scalability, and user privacy. In this paper, we address the problem of federated matrix completion, focusing on scenarios where user-specific data is distributed across multiple clients, and privacy constraints are uncompromising. Federated learning provides a promising framework to address these challenges by enabling collaborative learning across distributed datasets without sharing raw data. We propose \texttt{FedMC-ADMM} for solving federated matrix completion problems, a novel algorithmic framework that combines the Alternating Direction Method of Multipliers with a randomized block-coordinate strategy and alternating proximal gradient steps. Unlike existing federated approaches, \texttt{FedMC-ADMM} effectively handles multi-block nonconvex and nonsmooth optimization problems, allowing efficient computation while preserving user privacy. We analyze the theoretical properties of our algorithm, demonstrating subsequential convergence and establishing a convergence rate of $\mathcal{O}(K^{-1/2})$, leading to a communication complexity of $\mathcal{O}(ε^{-2})$ for reaching an $ε$-stationary point. This work is the first to establish these theoretical guarantees for federated matrix completion in the presence of multi-block variables. To validate our approach, we conduct extensive experiments on real-world datasets, including MovieLens 1M, 10M, and Netflix. The results demonstrate that \texttt{FedMC-ADMM} outperforms existing methods in terms of convergence speed and testing accuracy.