Variance Reduced Distributed Non-Convex Optimization Using Matrix Stepsizes

TL;DR

This paper introduces variance-reduced matrix-stepsized algorithms for distributed non-convex optimization, demonstrating improved convergence and communication efficiency over existing methods through theoretical analysis and empirical validation.

Contribution

It proposes two variance-reduced algorithms, det-MARINA and det-DASHA, that enhance the performance of matrix-stepsized gradient descent in distributed non-convex settings.

Findings

det-MARINA and det-DASHA outperform existing methods in iteration complexity

The new algorithms achieve better communication efficiency

Theoretical and empirical results confirm improved convergence

Abstract

Matrix-stepsized gradient descent algorithms have been shown to have superior performance in non-convex optimization problems compared to their scalar counterparts. The det-CGD algorithm, as introduced by Li et al. (2023), leverages matrix stepsizes to perform compressed gradient descent for non-convex objectives and matrix-smooth problems in a federated manner. The authors establish the algorithm's convergence to a neighborhood of a weighted stationarity point under a convex condition for the symmetric and positive-definite matrix stepsize. In this paper, we propose two variance-reduced versions of the det-CGD algorithm, incorporating MARINA and DASHA methods. Notably, we establish theoretically and empirically, that det-MARINA and det-DASHA outperform MARINA, DASHA and the distributed det-CGD algorithms in terms of iteration and communication complexities.