Boundary Geometry Determines Optimal Shallow ReLU Approximation under Weighted Coefficient Constraints

Abstract

We study how boundary geometry affects $L^q$ approximation by shallow ReLU neural networks on convex bodies $K\subset\mathbb R^d$. The target classes are unit balls of weighted variation spaces that can enlarge classical spaces by incorporating domain geometry into constraints on output coefficients. We establish upper and lower bounds for arbitrary convex bodies and determine the optimal polynomial approximation exponents for all planar convex bodies and their products with polytopes, including cubes and circular cylinders. For $C^2$ convex bodies, we determine the optimal exponents in a dimension-dependent range of $q$ extending beyond $2$. When all principal curvatures are positive, the result extends to every finite $q\ge2$. For these classes, the dependence of the optimal exponent on the domain is captured by a single geometric growth exponent associated with small boundary caps. We also identify sharp weight thresholds for retaining the classical approximation exponent and improve previous weights on Euclidean balls while preserving the classical $L^2$ approximation rate. When the target is given as a finite ReLU network, the upper bounds can be realized by a compressed network using only the original neurons, without increasing the weighted coefficient sum. These results show that boundary geometry can change the optimal exponent even when dimension, error norm, and weighting rule are fixed.