Efficient Federated Low Rank Matrix Completion

TL;DR

This paper introduces AltGDmin, a gradient descent-based algorithm for federated low rank matrix completion, offering improved communication efficiency, speed, and sample complexity guarantees, including for noisy data.

Contribution

The paper presents AltGDmin, a novel federated algorithm for LRMC with theoretical guarantees on efficiency, speed, and sample complexity, including for noisy scenarios.

Findings

AltGDmin is the most communication-efficient federated LRMC solution.

AltGDmin achieves one of the fastest convergence rates among iterative LRMC methods.

The method provides improved sample complexity guarantees for both federated and centralized LRMC.

Abstract

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.