Scale-Insensitive Neural Network Significance Tests

Abstract

This paper develops a scale-insensitive framework for neural network significance testing, substantially generalizing existing approaches through three key innovations. First, we replace metric entropy calculations with Rademacher complexity bounds, enabling the analysis of neural networks without requiring bounded weights or specific architectural constraints. Second, we weaken the regularity conditions on the target function to require only Sobolev space membership $H^s([-1,1]^d)$ with $s > d/2$, significantly relaxing previous smoothness assumptions while maintaining optimal approximation rates. Third, we introduce a modified sieve space construction based on moment bounds rather than weight constraints, providing a more natural theoretical framework for modern deep learning practices. Our approach achieves these generalizations while preserving optimal convergence rates and establishing valid asymptotic distributions for test statistics. The technical foundation combines localization theory, sharp concentration inequalities, and scale-insensitive complexity measures to handle unbounded weights and general Lipschitz activation functions. This framework better aligns theoretical guarantees with contemporary deep learning practice while maintaining mathematical rigor.