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Implicit deep learning has received increasing attention recently due to thefact that it generalizes the recursive prediction rules of many commonly usedneural network architectures. Its prediction rule is provided implicitly basedon the solution of an equilibrium equation. Although a line of recent empiricalstudies has demonstrated its superior performances, the theoreticalunderstanding of implicit neural networks is limited. In general, theequilibrium 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 thegap by analyzing the gradient flow of Rectified Linear Unit (ReLU) activatedimplicit neural networks. For an $m$-width implicit neural network with ReLUactivation and $n$ training samples, we show that a randomly initializedgradient descent converges to a global minimum at a linear rate for the squareloss function if the implicit neural network is \textit{over-parameterized}. Itis worth noting that, unlike existing works on the convergence of (S)GD onfinite-layer over-parameterized neural networks, our convergence results holdfor implicit neural networks, where the number of layers is \textit{infinite}.

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