Benign Overfitting without Linearity: Neural Network Classifiers Trained by Gradient Descent for Noisy Linear Data
This provides theoretical justification for benign overfitting in nonlinear neural networks, addressing a key empirical observation in machine learning.
The paper tackles the problem of understanding benign overfitting in neural networks by analyzing two-layer networks trained with gradient descent on noisy linear data, showing they achieve zero training error and minimax optimal test error even with adversarial label corruption.
Benign overfitting, the phenomenon where interpolating models generalize well in the presence of noisy data, was first observed in neural network models trained with gradient descent. To better understand this empirical observation, we consider the generalization error of two-layer neural networks trained to interpolation by gradient descent on the logistic loss following random initialization. We assume the data comes from well-separated class-conditional log-concave distributions and allow for a constant fraction of the training labels to be corrupted by an adversary. We show that in this setting, neural networks exhibit benign overfitting: they can be driven to zero training error, perfectly fitting any noisy training labels, and simultaneously achieve minimax optimal test error. In contrast to previous work on benign overfitting that require linear or kernel-based predictors, our analysis holds in a setting where both the model and learning dynamics are fundamentally nonlinear.