  Implicit deep learning has recently become popular in the machine learning
community since these implicit models can achieve competitive performance with
state-of-the-art deep networks while using significantly less memory and
computational resources. However, our theoretical understanding of when and how
first-order methods such as gradient descent (GD) converge on
\textit{nonlinear} implicit networks is limited. Although this type of problem
has been studied in standard feed-forward networks, the case of implicit models
is still intriguing because implicit networks have \textit{infinitely} many
layers. The corresponding equilibrium equation probably admits no or multiple
solutions during training. This paper studies the convergence of both gradient
flow (GF) and gradient descent for nonlinear ReLU activated implicit networks.
To deal with the well-posedness problem, we introduce a fixed scalar to scale
the weight matrix of the implicit layer and show that there exists a small
enough scaling constant, keeping the equilibrium equation well-posed throughout
training. As a result, we prove that both GF and GD converge to a global
minimum at a linear rate if the width $m$ of the implicit network is
\textit{linear} in the sample size $N$, i.e., $m=\Omega(N)$.

Machine Learning;Optimization and Control;Machine Learning