On the Proof of Global Convergence of Gradient Descent for Deep ReLU Networks with Linear Widths

This paper studies the global convergence of gradient descent for deep ReLU networks under the square loss. For this setting, the current state-of-the-art results show that gradient descent converges to a global optimum if the widths of all the hidden layers scale at least as $\Omega(N^8)$ ($N$ being the number of training samples). In this paper, we discuss a simple proof framework which allows us to improve the existing over-parameterization condition to linear, quadratic and cubic widths (depending on the type of initialization scheme and/or the depth of the network).