Det-CGD: Compressed Gradient Descent with Matrix Stepsizes for Non-Convex Optimization

Hanmin Li, Avetik Karagulyan, Peter Richtárik

Keywords: Optimization, First-order optimization, Non-convex optimization, Distributed optimization

TL;DR: We propose two matrix stepsized sketch gradient descent algorithms for minimizing matrix-smooth non-convex objectives.

Abstract:

This paper introduces a new method for minimizing matrix-smooth non-convex objectives through the use of novel Compressed Gradient Descent (CGD) algorithms enhanced with a matrix-valued stepsize.
The proposed algorithms are theoretically analyzed first in the single-node and subsequently in the distributed settings.
Our theoretical results reveal that the matrix stepsize in CGD can capture the objective's structure and lead to faster convergence compared to a scalar stepsize.
As a byproduct of our general results, we emphasize the importance of selecting the compression mechanism and the matrix stepsize in a layer-wise manner, taking advantage of model structure.
Moreover, we provide theoretical guarantees for free compression, by designing specific layer-wise compressors for the non-convex matrix smooth objectives. Our findings are supported with empirical evidence.

Submission Number: 3488