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

TL;DR

This paper establishes a global convergence theory for over-parameterized deep ReLU implicit neural networks, demonstrating that gradient descent converges linearly to a global minimum even with infinitely many layers.

Contribution

It provides the first theoretical guarantee of convergence for implicit neural networks with infinite layers under over-parameterization.

Findings

Gradient descent converges linearly to a global minimum.

Convergence holds for networks with infinitely many layers.

Results apply to over-parameterized ReLU implicit neural networks.

Abstract

Implicit deep learning has received increasing attention recently due to the fact that it generalizes the recursive prediction rules 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 \textit{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 \textit{infinite}.