Generalization Bounds

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Sep 17, 2026 · stat.MLJ/K move · Enter open · S save

Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks

Xianjun Li, Yunfei Yang

School of Statistics and Data Science, Capital University of Economics and Business, Beijing, China. · School of Mathematics (Zhuhai) and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Zhuhai, China.

Recent studies have shown that smooth functions can be well approximated by ReLU neural networks with path norm constraint on the weights. We extend these results from uniform approximation to approximation in Sobolev norm. Specifically, we analyze how well Sobolev functions in

W^{n,p}

can be approximated by neural networks with width

W

, depth

L

and path norm bounded by

K

, when the approximation error is measured in the

W^{1,p}

-norm. For shallow networks with depth

L=1

, we derive the approximation error bound

\mathcal{O}(\max\{W^{-(n-1)/d}, K^{-(n-1)/(s-n)}\})

, when the smoothness index satisfies

n<s=(d+3)/2

and the input is

d

-dimensional. For deep networks, we remove the restriction on the smoothness by showing that the approximation bound

\mathcal{O}(K^{-(n-1)/(d+d/p+1)})

holds if the width

W

and depth

L

are sufficiently large.

Logarithmic Sobolev Inequalities Rectified Linear Unit Networks Neural-Network Approximation Approximation Generalization Bounds Neural Network

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