# Statistical Learning Guarantees for Group-Invariant Barron Functions

arXiv: https://arxiv.org/abs/2509.23474 · HTML: https://mlsanity.com/paper/2509.23474/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 49 references that are in the map.
- First posted on arXiv 27 Sep 2025.

## Builds on (most cited first, 30 of 31)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Generalization Error of Invariant Classifiers](https://mlsanity.com/paper/1610.04574/) (2016)
- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [Provably Strict Generalisation Benefit for Invariance in Kernel Methods](https://mlsanity.com/paper/2106.02346/) (2021)
- [Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives](https://mlsanity.com/paper/2305.08466/) (2023)
- [Nonparametric regression using over-parameterized shallow ReLU neural networks](https://mlsanity.com/paper/2306.08321/) (2023)
- [Complexity Measures for Neural Networks with General Activation Functions Using Path-based Norms](https://mlsanity.com/paper/2009.06132/) (2020)
- [Phase Diagram of Initial Condensation for Two-layer Neural Networks](https://mlsanity.com/paper/2303.06561/) (2023)
- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks](https://mlsanity.com/paper/2404.18769/) (2024)
- [On the Implicit Bias of Linear Equivariant Steerable Networks](https://mlsanity.com/paper/2303.04198/) (2023)
- [Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms](https://mlsanity.com/paper/2508.05141/) (2025)
- [Statistical Guarantees of Group-Invariant GANs](https://mlsanity.com/paper/2305.13517/) (2023)
- [Homotopy Relaxation Training Algorithms for Infinite-Width Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/2309.15244/) (2023)
- [Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences](https://mlsanity.com/paper/2410.01244/) (2024)

## Built on by (most cited first, 1 of 1)

- [On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions](https://mlsanity.com/paper/2511.12398/) (2025)

## Abstract (arXiv)

We investigate the generalization error of group-invariant neural networks within the Barron framework. Our analysis shows that incorporating group-invariant structures introduces a group-dependent factor δG,Gamma,σ ≤ 1 into the approximation rate. When this factor is small, group invariance yields substantial improvements in approximation accuracy. On the estimation side, we establish that the Rademacher complexity of the group-invariant class is no larger than that of the non-invariant counterpart, implying that the estimation error remains unaffected by the incorporation of symmetry. Consequently, the generalization error can improve significantly when learning functions with inherent group symmetries. We further provide illustrative examples demonstrating both favorable cases, where δG,Gamma,σ≈ |G|-1, and unfavorable ones, where δG,Gamma,σ≈ 1. Overall, our results offer a rigorous theoretical foundation showing that encoding group-invariant structures in neural networks leads to clear statistical advantages for symmetric target functions.