  State-of-the-art neural network training methods depend on the gradient of
the network function. Therefore, they cannot be applied to networks whose
activation functions do not have useful derivatives, such as binary and
discrete-time spiking neural networks. To overcome this problem, the activation
function's derivative is commonly substituted with a surrogate derivative,
giving rise to surrogate gradient learning (SGL). This method works well in
practice but lacks theoretical foundation. The neural tangent kernel (NTK) has
proven successful in the analysis of gradient descent. Here, we provide a
generalization of the NTK, which we call the surrogate gradient NTK, that
enables the analysis of SGL. First, we study a naive extension of the NTK to
activation functions with jumps, demonstrating that gradient descent for such
activation functions is also ill-posed in the infinite-width limit. To address
this problem, we generalize the NTK to gradient descent with surrogate
derivatives, i.e., SGL. We carefully define this generalization and expand the
existing key theorems on the NTK with mathematical rigor. Further, we
illustrate our findings with numerical experiments. Finally, we numerically
compare SGL in networks with sign activation function and finite width to
kernel regression with the surrogate gradient NTK; the results confirm that the
surrogate gradient NTK provides a good characterization of SGL.
