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This paper establishes sharp theoretical bounds for the approximation error of neural networks using affine latent parameterizations, offering rigorous insights into the tradeoff between network budget and latent dimensionality. The results demonstrate that even fixed-dimensional latent spaces can achieve vanishing error as computational resources scale, providing a mathematical foundation for efficient model compression and adaptation techniques.
Sharp Approximation Rates for Neural Networks with Affine Latent Parameterizations
arXiv:2608.31157v1 Announce Type: new Abstract: Many parameter-efficient methods generate the parameters of a large neural network from a low-dimensional latent representation. Given an architecture $\Phi$ with $P_\Phi$ parameter slots, we write $\boldsymbol{\theta}_f=\mathcal{G}(\boldsymbol{\xi}_f)$, where $\mathcal{G}\colon\mathbb{R}^M\to\mathbb{R}^{P_\Phi}$ is a parameter generator and $\boldsymbol{\xi}_f\in\mathbb{R}^M$ is a latent representation of the target function $f$. The architecture $\Phi$ and the generator $\mathcal{G}$ are shared across the entire target class, while each target $f$ is represented by its own latent vector $\boldsymbol{\xi}_f$, with $\Phi_{\mathcal{G}(\boldsymbol{\xi}_f)}$ approximating $f$. This framework encompasses hypernetworks, low-dimensional parameterizations, parameter-efficient adaptation, and model compression. Understanding the tradeoff between the latent dimension $M$ and the network budget $P$ is therefore fundamental to characterizing the expressive efficiency of these methods. We study this tradeoff for affine generators and fully connected ReLU architectures. More precisely, optimizing jointly over architectures $\Phi$ satisfying $P_\Phi\leq P$ and affine generators $\mathcal{G}:\mathbb{R}^M\to \mathbb{R}^{P_\Phi}$, we prove that the optimal worst-case uniform approximation error over the unit ball of $\alpha$-H\"older functions on $[0,1]^d$, where $0<\alpha\leq1$, has the sharp order $ \bigl(P\min\{M,P\}\bigr)^{-\alpha/d}. $ In particular, our result shows that even a fixed-dimensional latent space suffices to achieve vanishing approximation error as the network budget increases.
Abstract:Many parameter-efficient methods generate the parameters of a large neural network from a low-dimensional latent representation. Given an architecture $\Phi$ with $P_\Phi$ parameter slots, we write $\boldsymbol{\theta}_f=\mathcal{G}(\boldsymbol{\xi}_f)$, where $\mathcal{G}\colon\mathbb{R}^M\to\mathbb{R}^{P_\Phi}$ is a parameter generator and $\boldsymbol{\xi}_f\in\mathbb{R}^M$ is a latent representation of the target function $f$. The architecture $\Phi$ and the generator $\mathcal{G}$ are shared across the entire target class, while each target $f$ is represented by its own latent vector $\boldsymbol{\xi}_f$, with $\Phi_{\mathcal{G}(\boldsymbol{\xi}_f)}$ approximating $f$. This framework encompasses hypernetworks, low-dimensional parameterizations, parameter-efficient adaptation, and model compression. Understanding the tradeoff between the latent dimension $M$ and the network budget $P$ is therefore fundamental to characterizing the expressive efficiency of these methods. We study this tradeoff for affine generators and fully connected ReLU architectures. More precisely, optimizing jointly over architectures $\Phi$ satisfying $P_\Phi\leq P$ and affine generators $\mathcal{G}:\mathbb{R}^M\to \mathbb{R}^{P_\Phi}$, we prove that the optimal worst-case uniform approximation error over the unit ball of $\alpha$-Hölder functions on $[0,1]^d$, where $0<\alpha\leq1$, has the sharp order $ \bigl(P\min\{M,P\}\bigr)^{-\alpha/d}. $ In particular, our result shows that even a fixed-dimensional latent space suffices to achieve vanishing approximation error as the network budget increases.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2608.31157 [cs.LG]
(or arXiv:2608.31157v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.31157 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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