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A leading hypothesis for the surprising generalization of neural networks isthat 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 theconditions on depth and initialization for which this phenomenon arises in deeplinear models. Our main theoretical contribution is a dynamical depthseparation result, proving that while shallow models can exhibit incrementallearning dynamics, they require the initialization to be exponentially smallfor these dynamics to present themselves. However, once the model becomesdeeper, the dependence becomes polynomial and incremental learning can arise inmore natural settings. We complement our theoretical findings by experimentingwith deep matrix sensing, quadratic neural networks and with binaryclassification using diagonal and convolutional linear networks, showing all ofthese models exhibit incremental learning.

The Implicit Bias of Depth: How Incremental Learning Drives Generalization