Abstract:Theoretical studies show that for any differentiable function on a compact domain, there exists a neural network that approximates both the function values and gradients. However, such a result cannot be used in practice since it assumes real parameters and exact internal operations. In contrast, real implementations only use a finite subset of reals and machine operations with round-off errors. In this work, we investigate whether a similar result holds for neural networks under floating-point arithmetic, when the gradient with respect to the input is computed by the automatic differentiation algorithm $D^\mathtt{AD}$. We first show that given a floating-point function $\phi$ (e.g., a loss function), arbitrary function values and gradients can be represented by a floating-point network $f$ and $D^\mathtt{AD}(\phi\circ f)$, respectively. We further extend this result: given $\phi_1,\dots,\phi_n$, $D^\mathtt{AD}(\phi_i\circ f)$ can simultaneously represent arbitrary gradients while $f$ represents the target values, under mild conditions. Our results hold for practical activation functions, e.g., $\mathrm{ReLU}$, $\mathrm{ELU}$, $\mathrm{GeLU}$, $\mathrm{Swish}$, $\mathrm{Sigmoid}$, and $\mathrm{tanh}$.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.01702 [cs.LG] |
| (or arXiv:2605.01702v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.01702 arXiv-issued DOI via DataCite (pending registration) |
From: Yeachan Park [view email]
[v1]
Sun, 3 May 2026 04:06:41 UTC (89 KB)
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