Abstract: We focus on a specific class of shallow neural networks with a single hidden layer, namely those with $L_2$-normalised data and either a sigmoid-shaped Gaussian error function (``erf'') activation or a Gaussian Error Linear Unit (GELU) activation. For these networks, we derive new generalisation bounds through the PAC-Bayesian theory; unlike most existing such bounds they apply to neural networks with deterministic rather than randomised parameters. Our bounds are empirically non-vacuous when the network is trained with vanilla stochastic gradient descent on MNIST and Fashion-MNIST.