Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss

TL;DR

This paper analyzes how gradient descent on wide two-layer neural networks with logistic loss implicitly finds max-margin classifiers, explaining their strong generalization and statistical benefits in classification tasks.

Contribution

It characterizes the implicit bias of gradient flow as a max-margin classifier in a non-Hilbertian space, revealing benefits of overparameterization and low-dimensional structures.

Findings

Max-margin classifiers emerge as the limit of gradient flow.

Implicit bias leads to strong generalization bounds.

Numerical experiments confirm theoretical predictions.

Abstract

Neural networks trained to minimize the logistic (a.k.a. cross-entropy) loss with gradient-based methods are observed to perform well in many supervised classification tasks. Towards understanding this phenomenon, we analyze the training and generalization behavior of infinitely wide two-layer neural networks with homogeneous activations. We show that the limits of the gradient flow on exponentially tailed losses can be fully characterized as a max-margin classifier in a certain non-Hilbertian space of functions. In presence of hidden low-dimensional structures, the resulting margin is independent of the ambiant dimension, which leads to strong generalization bounds. In contrast, training only the output layer implicitly solves a kernel support vector machine, which a priori does not enjoy such an adaptivity. Our analysis of training is non-quantitative in terms of running time but we prove computational guarantees in simplified settings by showing equivalences with online mirror descent. Finally, numerical experiments suggest that our analysis describes well the practical behavior of two-layer neural networks with ReLU activation and confirm the statistical benefits of this implicit bias.