Covering numbers for deep ReLU networks with applications to function approximation and nonparametric regression

Authors

Weigutian Ou and Helmut Bölcskei

Reference

Foundations of Computational Mathematics, 2026, to appear.

Abstract

Covering numbers of (deep) ReLU networks have been used to characterize approximation-theoretic performance, to upper-bound prediction error in non-parametric regression, and to quantify classification capacity. These results rely on covering number upper bounds obtained via explicit constructions of coverings. Lower bounds on covering numbers do not appear to be available in the literature. The present paper fills this gap by deriving tight (up to multiplicative constants) lower and upper bounds on the metric entropy (i.e., the logarithm of the covering numbers) of fully connected networks with bounded weights, sparse networks with bounded weights, and fully connected networks with quantized weights. The tightness of these bounds yields a fundamental understanding of the impact of sparsity, quantization, bounded versus unbounded weights, and network output truncation. Moreover, the bounds allow one to characterize fundamental limits of neural network transformation, including network compression, and lead to sharp upper bounds on the prediction error in nonparametric regression through deep networks. In particular, we remove a log^6(n)-factor from the best known sample complexity rate for estimating Lipschitz functions via deep networks, thereby establishing optimality. Finally, we identify a systematic relation between optimal nonparametric regression and optimal approximation through deep networks, unifying numerous results in the literature and revealing underlying general principles.

Keywords

Deep neural networks, quantized networks, metric entropy, approximation theory, non-parametric regression

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