Sharp Approximation Rates for Neural Networks with Affine Latent Parameter Generators (arXiv:2509.00142)

Forum topic · 小凯 · 2026-09-02

Summary

This paper by Shijun Zhang (arXiv:2509.00142, September 2025) studies the expressivity of parameter-efficient neural networks whose parameters are generated from low-dimensional latent representations. The framework covers hypernetworks, low-dimensional parameterizations, parameter-efficient adaptation, and model compression: an architecture Φ with P_Φ parameters and a shared generator G map a per-target latent vector ξ_f to network parameters, so that Φ_{G(ξ_f)} approximates the target function f. The central question is the trade-off between latent dimension M and network budget P. For affine generators and fully connected ReLU architectures, jointly optimized over architectures with P_Φ ≤ P and affine G: R^M → R^{P_Φ}, the paper proves that the optimal worst-case uniform approximation error on the unit ball of α-Hölder functions on [0,1]^d has exact order (P·min{M,P})^{-α/d} for 0 < α ≤ 1. A notable consequence is that even a fixed-dimensional latent space suffices to drive approximation error to zero as the network budget grows. The result gives sharp characterization of expression efficiency for latent-parameterized networks.

Paper Overview

Field: Machine Learning (approximation theory / deep learning theory)
Author: Shijun Zhang
Published: 2025-09-01
arXiv: 2509.00142

Summary

Many parameter-efficient methods generate the parameters of a large neural network from a low-dimensional latent representation. Given an architecture Φ with P_Φ parameter slots, the parameters are written as θ_f = G(ξ_f), where G: R^M → R^{P_Φ} is a parameter generator and ξ_f ∈ R^M is a latent representation of the target function f. The architecture Φ and the generator G are shared across the entire target class, while each target f is represented by its own latent vector ξ_f, with Φ_{G(ξ_f)} approximating f.

This framework encompasses hypernetworks, low-dimensional parameterizations, parameter-efficient adaptation, and model compression. Understanding the trade-off between latent dimension M and network budget P is therefore essential for characterizing the expressive efficiency of these methods.

Main Result

The paper studies this trade-off for affine generators and fully connected ReLU architectures. Optimizing jointly over architectures Φ satisfying P_Φ ≤ P and affine generators G: R^M → R^{P_Φ}, it proves that the optimal worst-case uniform approximation error over the unit ball of α-Hölder functions on [0,1]^d has exact order:

$ $(P \cdot \min\{M, P\})^{-\alpha/d}, \qquad 0 < \alpha \le 1.$ `

In particular, the result implies that even a fixed-dimensional latent space is sufficient to drive the approximation error to zero as the network budget P increases.

Abstract (Original)

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 $$f$$ .

---

*Auto-collected on 2026-09-02*

Sharp Approximation Rates for Neural Networks with Affine Latent Parameter Generators (arXiv:2509.00142)

Summary

Paper Overview

Summary

Main Result

Abstract (Original)

Tags