Principal Components Bias in Over-parameterized Linear Models, and its Manifestation in Deep Neural Networks
This work addresses the problem of understanding learning dynamics in deep neural networks for researchers, providing incremental insights into biases affecting convergence order.
The paper tackles the phenomenon of neural networks learning image classifications in the same order by analyzing over-parameterized deep linear models, revealing that convergence is exponentially faster along larger principal components of the data, termed Principal Components bias (PC-bias). It shows empirically that PC-bias streamlines learning order in linear and non-linear networks, especially early on, and explains benefits of early stopping and slower convergence with random labels.
Recent work suggests that convolutional neural networks of different architectures learn to classify images in the same order. To understand this phenomenon, we revisit the over-parametrized deep linear network model. Our analysis reveals that, when the hidden layers are wide enough, the convergence rate of this model's parameters is exponentially faster along the directions of the larger principal components of the data, at a rate governed by the corresponding singular values. We term this convergence pattern the Principal Components bias (PC-bias). Empirically, we show how the PC-bias streamlines the order of learning of both linear and non-linear networks, more prominently at earlier stages of learning. We then compare our results to the simplicity bias, showing that both biases can be seen independently, and affect the order of learning in different ways. Finally, we discuss how the PC-bias may explain some benefits of early stopping and its connection to PCA, and why deep networks converge more slowly with random labels.