Comparing Decentralized Gradient Descent Approaches and Guarantees

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.