Non-Vacuous Generalisation Bounds for Shallow Neural Networks
This work addresses the challenge of providing rigorous generalization guarantees for neural networks, which is a foundational problem in machine learning theory, though it is incremental as it focuses on specific network types and activations.
The paper tackles the problem of deriving non-vacuous generalization bounds for shallow neural networks with specific activations, achieving bounds that are empirically non-vacuous on MNIST and Fashion-MNIST datasets.
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.