The Implicit Bias of Depth: How Incremental Learning Drives Generalization
This work provides theoretical insights into the generalization behavior of deep learning models, which is a foundational problem in machine learning, though it is incremental in nature as it builds on existing hypotheses about implicit bias.
The paper tackles the problem of understanding why neural networks generalize well by investigating the hypothesis that gradient descent biases models toward simple solutions through incremental learning dynamics. It proves that deeper models require less restrictive initialization for incremental learning to occur, showing a polynomial dependence on depth compared to an exponential one for shallow models.
A leading hypothesis for the surprising generalization of neural networks is that the dynamics of gradient descent bias the model towards simple solutions, by searching through the solution space in an incremental order of complexity. We formally define the notion of incremental learning dynamics and derive the conditions on depth and initialization for which this phenomenon arises in deep linear models. Our main theoretical contribution is a dynamical depth separation result, proving that while shallow models can exhibit incremental learning dynamics, they require the initialization to be exponentially small for these dynamics to present themselves. However, once the model becomes deeper, the dependence becomes polynomial and incremental learning can arise in more natural settings. We complement our theoretical findings by experimenting with deep matrix sensing, quadratic neural networks and with binary classification using diagonal and convolutional linear networks, showing all of these models exhibit incremental learning.