Global Minimizers of $ell^p$-Regularized Objectives Yield the Sparsest ReLU Neural Networks

arXiv:2505.21791v1 Announce Type: new
Abstract: Overparameterized neural networks can interpolate a given dataset in many different ways, prompting the fundamental question: which among these solutions should we prefer, and what explicit regularization strategies will provably yield these solutions? This paper addresses the challenge of finding the sparsest interpolating ReLU network — i.e., the network with the fewest nonzero parameters or neurons — a goal with wide-ranging implications for efficiency, generalization, interpretability, theory, and model compression. Unlike post hoc pruning approaches, we propose a continuous, almost-everywhere differentiable training objective whose global minima are guaranteed to correspond to the sparsest single-hidden-layer ReLU networks that fit the data. This result marks a conceptual advance: it recasts the combinatorial problem of sparse interpolation as a smooth optimization task, potentially enabling the use of gradient-based training methods. Our objective is based on minimizing $ell^p$ quasinorms of the weights for $0