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

arXiv · AI, language, vision and robotics · article · Sep 22, 2026 · UTC

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 $β$-Hölder ball on $[0,1]^d$ with $0<β\leq1$, the optimal $L^p$ approximation error for $0<p<\infty$ is of order $[N^2\log(eNT)]^{-β/d}$ when the ne

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