Gradient-trained Weights in Wide Neural Networks Align Layerwise to Error-scaled Input Correlations

Recent works have examined how deep neural networks, which can solve a variety of difficult problems, incorporate the statistics of training data to achieve their success. However, existing results have been established only in limited settings. In this work, we derive the layerwise weight dynamics of infinite-width neural networks with nonlinear activations trained by gradient descent. We show theoretically that weight updates are aligned with input correlations from intermediate layers weighted by error, and demonstrate empirically that the result also holds in finite-width wide networks. The alignment result allows us to formulate backpropagation-free learning rules, named Align-zero and Align-ada, that theoretically achieve the same alignment as backpropagation. Finally, we test these learning rules on benchmark problems in feedforward and recurrent neural networks and demonstrate, in wide networks, comparable performance to backpropagation.