Raoul-Martin Memmesheimer

Carlo Wenig, Raoul-Martin Memmesheimer, Christian Klos

The ability to train spiking neural networks is essential for modeling biological neural networks as well as for neuromorphic computing. However, for the extensively used leaky integrate-and-fire (LIF) neurons, arbitrarily small parameter changes can induce spike (dis)appearances that disrupt subsequent activity, leading to unstable neural representations and permanently silent neurons during exact spike-based gradient descent. Recent work shows that a class of neuron models, which includes the quadratic integrate-and-fire (QIF) neuron, avoids these discontinuities and enables continuous and even smooth spike-based gradient descent. However, it remains unclear whether these advantages translate into practice. Here, we demonstrate that they do so via a controlled comparison between networks of LIF and QIF neurons on the popular Spiking Heidelberg Digits dataset. Specifically, in a first step, we perform a thorough hyperparameter search to optimize both models, revealing a clear performance advantage of QIF neurons. In a second step, we visualize the loss and gradient landscapes. Consistent with their inferior performance, we find that the loss landscapes of LIF neurons, which are discontinuous, appear more fragmented and the related gradients more erratic. An analysis of the landscapes of single samples indicates that these features arise from changes in the temporal order of spikes, which often cause disruptive spike (dis)appearances. Overall, our results advocate replacing LIF neurons with neuron models exhibiting continuous spiking dynamics, such as QIF neurons, for gradient descent training.

Luke Eilers, Raoul-Martin Memmesheimer, Sven Goedeke

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.