  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}.
