Generalization Bounds for Neural Networks with Sparse Connectivity

Abstract

Empirical evidence suggests that large neural networks rarely make effective use of all available parameters, with learned solutions exhibiting pronounced sparsity. Despite this ubiquity, existing generalization theory only partially explains how such sparsity influences statistical performance. In this paper, we derive non-asymptotic excess risk bounds for deep neural networks whose layer-wise weight matrices are restricted by $\ell_p$ quasi-norm ($0<p\leq 1$) constraints. Our analysis gives rise to a novel result that interpolates between classical norm-based generalization guarantees and parameter-counting bounds. Remarkably, in the extreme sparsity limit as the quasi-norm order $p$ approach zero, our analysis reveals a transition to a square-root dependence on the number of \emph{active} (non-zero) parameters, resulting in bounds that are strictly tighter than existing parameter-counting approaches. Experiments demonstrate that our bounds exhibit a milder dependence on initial architectural choice at the over-parametrization regime, effectively capturing the generalization benefit of the sparsity arising from training on natural data.