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In this work, we develop and analyze a novel Gradient Descent (GD) based solution, called Alternating GD and Minimization (AltGDmin), for efficiently solving the low rank matrix completion (LRMC) in a federated setting. Here “efficient” refers to communication-, computation- and sample- efficiency. LRMC involves recovering an n × q rank-r matrix X⋆ from a subset of its entries when r ≪ min(n, q). Our theoretical bounds on the sample complexity and iteration complexity of AltGDmin imply that it is the most communication-efficient solution while also been one of the most computation- and sample- efficient ones. We also extend our guarantee to the noisy LRMC setting. In addition, we show how our lemmas can be used to provide an improved sample complexity guarantee for the Alternating Minimization (AltMin) algorithm for LRMC. AltMin is one of the fastest centralized solutions for LRMC; with AltGDmin having comparable time cost even for the centralized setting. 

This work studies our recently developed algorithm, decentralized alternating projected gradient descent algorithm (Dec-AltGDmin), for recovering a low rank (LR) matrix from independent columnwise linear projections in a decentralized setting. This means that the observed data is spread across L agents and there is no central coordinating node. Since this problem is non-convex and since it involves a subspace recovery step, most existing literature from decentralized optimization is not useful. We demonstrate using extensive numerical simulations and communication, time, and sample complexity comparisons that (i) existing decentralized gradient descent (GD) approaches fail, and (ii) other common solution approaches on LR recovery literature – projected GD, alternating GD and alternating minimization (AltMin) – either have a higher communication (and time) complexity or a higher sample complexity. Communication complexity is often the most important concern in decentralized learning. 

This work studies our recently developed decentralized algorithm, decentralized alternating projected gradient descent algorithm, called Dec-AltProjGDmin, for solving the following low-rank (LR) matrix recovery problem: recover an LR matrix from independent column-wise linear projections (LR column-wise Compressive Sensing). In recent work, we presented constructive convergence guarantees for Dec-AltProjGDmin under simple assumptions. By "constructive", we mean that the convergence time lower bound is provided for achieving any error level ε. However, our guarantee was stated for the equal neighbor consensus algorithm (at each iteration, each node computes the average of the data of all its neighbors) while most existing results do not assume the use of a specific consensus algorithm, but instead state guarantees in terms of the weights matrix eigenvalues. In order to compare with these results, we first modify our result to be in this form. Our second and main contribution is a theoretical and experimental comparison of our new result with the best existing one from the decentralized GD literature that also provides a convergence time bound for values of ε that are large enough. The existing guarantee is for a different problem setting and holds under different assumptions than ours and hence the comparison is not very clear cut. However, we are not aware of any other provably correct algorithms for decentralized LR matrix recovery in any other settings either.