Convergence and Alignment of Gradient Descent with Random Backpropagation Weights

TL;DR

This paper analyzes the mathematical properties of feedback alignment, a biologically plausible alternative to backpropagation, showing convergence and the conditions for alignment in neural networks, and providing insights into biologically inspired learning algorithms.

Contribution

It provides a rigorous analysis of feedback alignment's convergence and alignment properties, highlighting the role of regularization in neural network training with random weights.

Findings

Error converges to zero exponentially fast in overparameterized networks.

Regularization is necessary for parameter alignment with random weights.

Results suggest feedback alignment can perform comparably to backpropagation.

Abstract

Stochastic gradient descent with backpropagation is the workhorse of artificial neural networks. It has long been recognized that backpropagation fails to be a biologically plausible algorithm. Fundamentally, it is a non-local procedure -- updating one neuron's synaptic weights requires knowledge of synaptic weights or receptive fields of downstream neurons. This limits the use of artificial neural networks as a tool for understanding the biological principles of information processing in the brain. Lillicrap et al. (2016) propose a more biologically plausible "feedback alignment" algorithm that uses random and fixed backpropagation weights, and show promising simulations. In this paper we study the mathematical properties of the feedback alignment procedure by analyzing convergence and alignment for two-layer networks under squared error loss. In the overparameterized setting, we prove that the error converges to zero exponentially fast, and also that regularization is necessary in order for the parameters to become aligned with the random backpropagation weights. Simulations are given that are consistent with this analysis and suggest further generalizations. These results contribute to our understanding of how biologically plausible algorithms might carry out weight learning in a manner different from Hebbian learning, with performance that is comparable with the full non-local backpropagation algorithm.