A global convergence theory for deep ReLU implicit networks via over-parameterization

Tianxiang Gao, Hailiang Liu, Jia Liu, Hridesh Rajan, Hongyang Gao

Keywords: Deep learning, Deep implicit learning, deep equilibrium model, gradient descent, stochastic gradient descent, over-parameterization

Abstract: Implicit deep learning has received increasing attention recently due to the fact that it generalizes the recursive prediction rule of many commonly used neural network architectures. Its prediction rule is provided implicitly based on the solution of an equilibrium equation. Although a line of recent empirical studies has demonstrated its superior performances, the theoretical understanding of implicit neural networks is limited. In general, the equilibrium equation may not be well-posed during the training. As a result, there is no guarantee that a vanilla (stochastic) gradient descent (SGD) training nonlinear implicit neural networks can converge. This paper fills the gap by analyzing the gradient flow of Rectified Linear Unit (ReLU) activated implicit neural networks. For an $m$ width implicit neural network with ReLU activation and $n$ training samples, we show that a randomly initialized gradient descent converges to a global minimum at a linear rate for the square loss function if the implicit neural network is over-parameterized. It is worth noting that, unlike existing works on the convergence of (S)GD on finite-layer over-parameterized neural networks, our convergence results hold for implicit neural networks, where the number of layers is infinite.

One-sentence Summary: For a ReLU activated implicit neural network with infinitely many layers, we prove that randomly initialized gradient descent with a fixed step-size converges to a global minimum at a linear rate if the width m is the squared sample size n.