Fixed-architecture networks of width O(D) and depth O(r) approximate Hölder functions with parameter magnitude log P = O(ε^{-2D/(r+γ)} log(1/ε)) via CRT encoding.
arXiv preprint arXiv:2305.00608 , year=
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abstract
We study the properties of differentiable neural networks activated by rectified power unit (RePU) functions. We show that the partial derivatives of RePU neural networks can be represented by RePUs mixed-activated networks and derive upper bounds for the complexity of the function class of derivatives of RePUs networks. We establish error bounds for simultaneously approximating $C^s$ smooth functions and their derivatives using RePU-activated deep neural networks. Furthermore, we derive improved approximation error bounds when data has an approximate low-dimensional support, demonstrating the ability of RePU networks to mitigate the curse of dimensionality. To illustrate the usefulness of our results, we consider a deep score matching estimator (DSME) and propose a penalized deep isotonic regression (PDIR) using RePU networks. We establish non-asymptotic excess risk bounds for DSME and PDIR under the assumption that the target functions belong to a class of $C^s$ smooth functions. We also show that PDIR achieves the minimax optimal convergence rate and has a robustness property in the sense it is consistent with vanishing penalty parameters even when the monotonicity assumption is not satisfied. Furthermore, if the data distribution is supported on an approximate low-dimensional manifold, we show that DSME and PDIR can mitigate the curse of dimensionality.
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Smoothly activated DNNs (feedforward and residual) achieve non-asymptotic uniform convergence rates that mitigate the curse of dimensionality by adaptively using hierarchical composition structure of the target function.
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On Explicit Super-Expressive Approximation for Neural Networks
Fixed-architecture networks of width O(D) and depth O(r) approximate Hölder functions with parameter magnitude log P = O(ε^{-2D/(r+γ)} log(1/ε)) via CRT encoding.
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Mitigating the Curse of Dimensionality in Uniform Convergence of Deep Neural Networks via Smooth Activations
Smoothly activated DNNs (feedforward and residual) achieve non-asymptotic uniform convergence rates that mitigate the curse of dimensionality by adaptively using hierarchical composition structure of the target function.
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