On Sparsity in Overparametrised Shallow ReLU Networks
This addresses a key gap in neural network theory for practitioners by providing guarantees for finite-width training in shallow networks, though it is incremental as it builds on existing regularization methods.
The paper tackles the challenge of ensuring finite-neuron solutions in overparametrized shallow ReLU networks by analyzing implicit regularization via noise injection and variation-norm regularization, proving both minimize to functions with a finite number of neurons regardless of overparametrization.
The analysis of neural network training beyond their linearization regime remains an outstanding open question, even in the simplest setup of a single hidden-layer. The limit of infinitely wide networks provides an appealing route forward through the mean-field perspective, but a key challenge is to bring learning guarantees back to the finite-neuron setting, where practical algorithms operate. Towards closing this gap, and focusing on shallow neural networks, in this work we study the ability of different regularisation strategies to capture solutions requiring only a finite amount of neurons, even on the infinitely wide regime. Specifically, we consider (i) a form of implicit regularisation obtained by injecting noise into training targets [Blanc et al.~19], and (ii) the variation-norm regularisation [Bach~17], compatible with the mean-field scaling. Under mild assumptions on the activation function (satisfied for instance with ReLUs), we establish that both schemes are minimised by functions having only a finite number of neurons, irrespective of the amount of overparametrisation. We study the consequences of such property and describe the settings where one form of regularisation is favorable over the other.