Approximation Rates of Shallow Neural Networks: Barron Spaces, Activation Functions and Optimality Analysis

• URL: http://arxiv.org/abs/2510.18388v1
• Date: Tue, 21 Oct 2025 08:08:35 GMT
• Title: Approximation Rates of Shallow Neural Networks: Barron Spaces, Activation Functions and Optimality Analysis
• Authors: Jian Lu, Xiaohuang Huang,
• Abstract summary: It focuses on the dependence of the approximation rate on the dimension and the smoothness of the function being approximated within the Barron function space.<n>We establish optimal approximation rates in various norms for functions in Barron spaces and Sobolev spaces, confirming the curse of dimensionality.
• Score: 7.106210679849991
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness of the function being approximated within the Barron function space. We examine the approximation rates of ReLU$^{k}$ activation functions, proving that the optimal rate cannot be achieved under $\ell^{1}$-bounded coefficients or insufficient smoothness conditions. We also establish optimal approximation rates in various norms for functions in Barron spaces and Sobolev spaces, confirming the curse of dimensionality. Our results clarify the limits of shallow neural networks' approximation capabilities and offer insights into the selection of activation functions and network structures.

Related papers

• Provable wavelet-based neural approximation [0.0]
  We develop a wavelet-based theoretical framework for analyzing the universal approximation capabilities of neural networks.<n>We derive sufficient conditions on activation functions to ensure that the associated neural network approximates any functions in the given space, along with an error estimate.
  arXiv  Detail & Related papers  (2025-04-23T13:02:37Z)
• Approximation properties of neural ODEs [5.828989070109041]
  We prove the universal approximation property (UAP) of shallow neural networks in the space of continuous functions.<n>In particular, we constrain the Lipschitz constant of the neural ODE's flow map and the norms of the weights to increase the network's stability.
  arXiv  Detail & Related papers  (2025-03-19T21:11:28Z)
• Dimension-independent learning rates for high-dimensional classification problems [53.622581586464634]
  We show that every $RBV2$ function can be approximated by a neural network with bounded weights. We then prove the existence of a neural network with bounded weights approximating a classification function.
  arXiv  Detail & Related papers  (2024-09-26T16:02:13Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• Approximation and interpolation of deep neural networks [0.0]
  In the overparametrized regime, deep neural network provide universal approximations and can interpolate any data set. In the last section, we provide a practical probabilistic method of finding such a point under general conditions on the activation function.
  arXiv  Detail & Related papers  (2023-04-20T08:45:16Z)
• Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization [73.80101701431103]
  The linearized-Laplace approximation (LLA) has been shown to be effective and efficient in constructing Bayesian neural networks. We study the usefulness of the LLA in Bayesian optimization and highlight its strong performance and flexibility.
  arXiv  Detail & Related papers  (2023-04-17T14:23:43Z)
• Optimal Approximation Complexity of High-Dimensional Functions with Neural Networks [3.222802562733787]
  We investigate properties of neural networks that use both ReLU and $x2$ as activation functions. We show how to leverage low local dimensionality in some contexts to overcome the curse of dimensionality, obtaining approximation rates that are optimal for unknown lower-dimensional subspaces.
  arXiv  Detail & Related papers  (2023-01-30T17:29:19Z)
• Going Beyond Linear RL: Sample Efficient Neural Function Approximation [76.57464214864756]
  We study function approximation with two-layer neural networks. Our results significantly improve upon what can be attained with linear (or eluder dimension) methods.
  arXiv  Detail & Related papers  (2021-07-14T03:03:56Z)
• Deep neural network approximation of analytic functions [91.3755431537592]
  entropy bound for the spaces of neural networks with piecewise linear activation functions. We derive an oracle inequality for the expected error of the considered penalized deep neural network estimators.
  arXiv  Detail & Related papers  (2021-04-05T18:02:04Z)
• Sequential Subspace Search for Functional Bayesian Optimization Incorporating Experimenter Intuition [63.011641517977644]
  Our algorithm generates a sequence of finite-dimensional random subspaces of functional space spanned by a set of draws from the experimenter's Gaussian Process. Standard Bayesian optimisation is applied on each subspace, and the best solution found used as a starting point (origin) for the next subspace. We test our algorithm in simulated and real-world experiments, namely blind function matching, finding the optimal precipitation-strengthening function for an aluminium alloy, and learning rate schedule optimisation for deep networks.
  arXiv  Detail & Related papers  (2020-09-08T06:54:11Z)
• Approximation with Neural Networks in Variable Lebesgue Spaces [1.0152838128195465]
  This paper concerns the universal approximation property with neural networks in variable Lebesgue spaces. We show that, whenever the exponent function of the space is bounded, every function can be approximated with shallow neural networks with any desired accuracy.
  arXiv  Detail & Related papers  (2020-07-08T14:52:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.