Gradient Descent can Learn Less Over-parameterized Two-layer Neural Networks on Classification Problems

Atsushi Nitanda, Geoffrey Chinot, Taiji Suzuki

Abstract: Recently, several studies have proven the global convergence and generalization abilities of the gradient descent method for two-layer ReLU networks. Most studies especially focused on the regression problems with the squared loss function, except for a few, and the importance of the positivity of the neural tangent kernel has been pointed out. However, the performance of gradient descent on classification problems using the logistic loss function has not been well studied, and further investigation of this problem structure is possible. In this work, we demonstrate that the separability assumption using a neural tangent model is more reasonable than the positivity condition of the neural tangent kernel and provide a refined convergence analysis of the gradient descent for two-layer networks with smooth activations. A remarkable point of our result is that our convergence and generalization bounds have much better dependence on the network width in comparison to related studies. Consequently, our theory significantly enlarges a class of over-parameterized networks with provable generalization ability, with respect to the network width, while most studies require much higher over-parameterization.

Keywords: gradient descent, neural network, over-parameterization