Learning Lipschitz Functions by GD-trained Shallow Overparameterized ReLU Neural Networks
This addresses the inconsistency issue in neural network training for noisy, non-smooth functions, providing a theoretical foundation for early stopping rules in practical applications.
The paper tackles the problem of learning Lipschitz, nondifferentiable, bounded functions with additive noise using overparameterized shallow ReLU neural networks trained by Gradient Descent, showing that early-stopped GD achieves consistency and minimax optimal rates for excess risk.
We explore the ability of overparameterized shallow ReLU neural networks to learn Lipschitz, nondifferentiable, bounded functions with additive noise when trained by Gradient Descent (GD). To avoid the problem that in the presence of noise, neural networks trained to nearly zero training error are inconsistent in this class, we focus on the early-stopped GD which allows us to show consistency and optimal rates. In particular, we explore this problem from the viewpoint of the Neural Tangent Kernel (NTK) approximation of a GD-trained finite-width neural network. We show that whenever some early stopping rule is guaranteed to give an optimal rate (of excess risk) on the Hilbert space of the kernel induced by the ReLU activation function, the same rule can be used to achieve minimax optimal rate for learning on the class of considered Lipschitz functions by neural networks. We discuss several data-free and data-dependent practically appealing stopping rules that yield optimal rates.