Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations

Abstract

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them is significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. Moreover, our method automatically and dynamically adapts the ranks during training to achieve a desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.

Disentangling Mixed Classes of Covariability in Large-Scale Neural Data

“…Our analytical results provide intuition for our observation that the rank of learning dynamics is limited by task complexity. This supports previous findings of low (matrix) rank weight changes in RNNs [13] and in deep networks [45,46,47]. Our results on the numerical matrix rank also has interesting ties to work that uses rank compression for more efficient training in deep networks [45], and for numerical solutions to systems with time-varying dynamics [48].…”