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

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}.

Accepted by ICLR 2022

References in corpus (5)

• Very Deep Convolutional Networks for Large-Scale Image Recognition
• Deep Equilibrium Models
• Scalable Differentiable Physics for Learning and Control
• On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers
• On the Proof of Global Convergence of Gradient Descent for Deep ReLU Networks with Linear Widths