Simplicity bias and optimization threshold in two-layer ReLU networks

TL;DR

This paper theoretically investigates how two-layer ReLU networks tend to favor simpler solutions over interpolating data, especially beyond an optimization threshold, which improves generalization in complex tasks.

Contribution

It introduces a theoretical framework explaining the simplicity bias and the optimization threshold in overparametrized two-layer ReLU networks, highlighting their impact on generalization.

Findings

Networks often converge to simpler solutions rather than interpolating data.

Early alignment phase causes neurons to align towards specific directions.

Simplicity bias enhances generalization beyond the interpolation regime.

Abstract

Understanding generalization of overparametrized neural networks remains a fundamental challenge in machine learning. Most of the literature mostly studies generalization from an interpolation point of view, taking convergence of parameters towards a global minimum of the training loss for granted. While overparametrized architectures indeed interpolated the data for typical classification tasks, this interpolation paradigm does not seem valid anymore for more complex tasks such as in-context learning or diffusion. Instead for such tasks, it has been empirically observed that the trained models goes from global minima to spurious local minima of the training loss as the number of training samples becomes larger than some level we call optimization threshold. While the former yields a poor generalization to the true population loss, the latter was observed to actually correspond to the minimiser of this true loss. This paper explores theoretically this phenomenon in the context of two-layer ReLU networks. We demonstrate that, despite overparametrization, networks often converge toward simpler solutions rather than interpolating the training data, which can lead to a drastic improvement on the test loss with respect to interpolating solutions. Our analysis relies on the so called early alignment phase, during which neurons align towards specific directions. This directional alignment, which occurs in the early stage of training, leads to a simplicity bias, wherein the network approximates the ground truth model without converging to the global minimum of the training loss. Our results suggest that this bias, resulting in an optimization threshold from which interpolation is not reached anymore, is beneficial and enhances the generalization of trained models.