Optimal Tradeoffs Between Network Size and Parameter Magnitude in Neural Approximation and Minimax Regression

arXiv:2609.25710v1 Announce Type: cross Abstract: The statistical accuracy of neural networks depends on both their approximation power and the complexity of the class fitted from data. While increasing network size is a natural way to improve approximation, parameter magnitude provides another resource whose role must be quantified in both respects. We establish a sharp width--magnitude tradeoff at fixed depth using one elementary bounded $1$-Lipschitz Dyadic--Triangular Activation. For the unit $\beta$-H\"older ball on $[0,1]^d$ with $0<\beta\leq1$, the optimal $L^p$ approximation error for $0