In order to better understand feature learning in neural networks, we propose a framework for understanding linear models in tangent feature space where the features are allowed to be transformed during training. We consider linear transformations of features, resulting in a joint optimization over parameters and transformations with a bilinear interpolation constraint. We show that this optimization problem has an equivalent linearly constrained optimization with structured regularization that encourages approximately low rank solutions. Specializing to neural network structure, we gain insights into how the features and thus the kernel function change, providing additional nuance to the phenomenon of kernel alignment when the target function is poorly represented using tangent features. In addition to verifying our theoretical observations in real neural networks on a simple regression problem, we empirically show that an adaptive feature implementation of tangent feature classification has an order of magnitude lower sample complexity than the fixed tangent feature model on MNIST and CIFAR-10.

Modern Deep Neural Networks (DNNs) exhibit impressive generalization properties on a variety of tasks without explicit regularization, suggesting the existence of hidden regularization effects. Recent work by Baratin et al. (2021) sheds light on an intriguing implicit regularization effect, showing that some layers are much more aligned with data labels than other layers. This suggests that as the network grows in depth and width, an implicit layer selection phenomenon occurs during training. In this work, we provide the first explanation for this alignment hierarchy. We introduce and empirically validate the Equilibrium Hypothesis which states that the layers that achieve some balance between forward and backward information loss are the ones with the highest alignment to data labels. Our experiments demonstrate an excellent match with the theoretical predictions.

We approach the problem of implicit regularization in deep learning from a geometrical viewpoint. We highlight a regularization effect induced by a dynamical alignment of the neural tangent features introduced by Jacot et al, along a small number of task-relevant directions. This can be interpreted as a combined mechanism of feature selection and model compression. By extrapolating a new analysis of Rademacher complexity bounds for linear models, we motivate and study a heuristic complexity measure that captures this phenomenon, in terms of sequences of tangent kernel classes along the optimization paths.