aiqbench analysis
This research establishes rigorous error bounds for ReLU neural networks approximating Sobolev functions under path norm constraints, extending previous uniform approximation results to derivative-based metrics. The findings clarify how network width and depth influence convergence rates in $W^{1,p}$-norms, offering theoretical guarantees for deep learning models handling smooth function classes without relying on restrictive smoothness assumptions.
Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks
arXiv:2609.19937v1 Announce Type: cross Abstract: 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.
Abstract: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.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2609.19937 [stat.ML]
(or arXiv:2609.19937v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2609.19937 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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