## A Global Convergence Theory for Deep ReLU Implicit Networks via Over-Parameterization

#### By: Tianxiang Gao, Hailiang Liu, Jia Liu, Hridesh Rajan, and Hongyang Gao

Download Paper#### Abstract

Implicit deep learning has received increasing attention recently, since 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 many recent studies have experimentally demonstrates 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.

#### ACM Reference

Gao, T. et al. 2022. A Global Convergence Theory for Deep ReLU Implicit Networks via Over-Parameterization. *ICLR’22: The 10th International Conference on Learning Representations* (Apr. 2022).

#### BibTeX Reference

```
@inproceedings{gao22global,
author = {Tianxiang Gao and Hailiang Liu and Jia Liu and Hridesh Rajan and Hongyang Gao},
title = {A Global Convergence Theory for Deep ReLU Implicit Networks via Over-Parameterization},
booktitle = {ICLR'22: The 10th International Conference on Learning Representations},
location = {Virtual},
month = {April 25-April 29},
year = {2022},
entrysubtype = {conference},
abstract = {
Implicit deep learning has received increasing attention recently, since
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 many recent
studies have experimentally demonstrates 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.
}
}
```