# Uniform Generalization Bounds for Overparameterized Neural Networks

arXiv: https://arxiv.org/abs/2109.06099 · HTML: https://mlsanity.com/paper/2109.06099/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 49 references that are in the map.
- First posted on arXiv 13 Sep 2021.

## Builds on (most cited first, 26 of 26)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Finite-Time Analysis of Kernelised Contextual Bandits](https://mlsanity.com/paper/1309.6869/) (2013)
- [On Information Gain and Regret Bounds in Gaussian Process Bandits](https://mlsanity.com/paper/2009.06966/) (2020)
- [Neural Tangents: Fast and Easy Infinite Neural Networks in Python](https://mlsanity.com/paper/1912.02803/) (2019)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (2019)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [Neural Thompson Sampling](https://mlsanity.com/paper/2010.00827/) (2020)
- [Deep Equals Shallow for ReLU Networks in Kernel Regimes](https://mlsanity.com/paper/2009.14397/) (2020)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [On the Similarity between the Laplace and Neural Tangent Kernels](https://mlsanity.com/paper/2007.01580/) (2020)
- [Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS](https://mlsanity.com/paper/2009.10683/) (2020)
- [Optimal Order Simple Regret for Gaussian Process Bandits](https://mlsanity.com/paper/2108.09262/) (2021)
- [Sparse Gaussian Processes with Spherical Harmonic Features](https://mlsanity.com/paper/2006.16649/) (2020)
- [Regularization Matters: A Nonparametric Perspective on Overparametrized Neural Network](https://mlsanity.com/paper/2007.02486/) (2020)
- [Batched Neural Bandits](https://mlsanity.com/paper/2102.13028/) (2021)
- [A Short Note on the Relationship of Information Gain and Eluder Dimension](https://mlsanity.com/paper/2107.02377/) (2021)

## Built on by (most cited first, 21 of 21)

- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Mind the spikes: Benign overfitting of kernels and neural networks in fixed dimension](https://mlsanity.com/paper/2305.14077/) (2023)
- [Kernelized Reinforcement Learning with Order Optimal Regret Bounds](https://mlsanity.com/paper/2306.07745/) (2023)
- [Improved Convergence Rates for Sparse Approximation Methods in Kernel-Based Learning](https://mlsanity.com/paper/2202.04005/) (2022)
- [Federated Neural Bandits](https://mlsanity.com/paper/2205.14309/) (2022)
- [An Optimization-based Algorithm for Non-stationary Kernel Bandits without Prior Knowledge](https://mlsanity.com/paper/2205.14775/) (2022)
- [Improved Regret Bounds for Gaussian Process Upper Confidence Bound in Bayesian Optimization](https://mlsanity.com/paper/2506.01393/) (2025)
- [Delayed Feedback in Kernel Bandits](https://mlsanity.com/paper/2302.00392/) (2023)
- [Wide Bayesian neural networks have a simple weight posterior: theory and accelerated sampling](https://mlsanity.com/paper/2206.07673/) (2022)
- [Gaussian Process Upper Confidence Bound Achieves Nearly-Optimal Regret in Noise-Free Gaussian Process Bandits](https://mlsanity.com/paper/2502.19006/) (2025)
- [Sample Complexity of Kernel-Based Q-Learning](https://mlsanity.com/paper/2302.00727/) (2023)
- [Provably and Practically Efficient Neural Contextual Bandits](https://mlsanity.com/paper/2206.00099/) (2022)
- [Adaptation to Misspecified Kernel Regularity in Kernelised Bandits](https://mlsanity.com/paper/2304.13830/) (2023)
- [Training-Free Neural Active Learning with Initialization-Robustness Guarantees](https://mlsanity.com/paper/2306.04454/) (2023)
- [Determinant Estimation under Memory Constraints and Neural Scaling Laws](https://mlsanity.com/paper/2503.04424/) (2025)
- [Tighter Regret Lower Bound for Gaussian Process Bandits with Squared Exponential Kernel in Hypersphere](https://mlsanity.com/paper/2602.17940/) (2026)
- [Beyond ReLU: How Activations Affect Neural Kernels and Random Wide Networks](https://mlsanity.com/paper/2506.22429/) (2025)
- [Spectral Estimation with Free Decompression](https://mlsanity.com/paper/2506.11994/) (2025)
- [Sparsity-depth Tradeoff in Infinitely Wide Deep Neural Networks](https://mlsanity.com/paper/2305.10550/) (2023)
- [Optimization dependent generalization bound for ReLU networks based on sensitivity in the tangent bundle](https://mlsanity.com/paper/2310.17378/) (2023)

## Abstract (arXiv)

An interesting observation in artificial neural networks is their favorable generalization error despite typically being extremely overparameterized. It is well known that the classical statistical learning methods often result in vacuous generalization errors in the case of overparameterized neural networks. Adopting the recently developed Neural Tangent (NT) kernel theory, we prove uniform generalization bounds for overparameterized neural networks in kernel regimes, when the true data generating model belongs to the reproducing kernel Hilbert space (RKHS) corresponding to the NT kernel. Importantly, our bounds capture the exact error rates depending on the differentiability of the activation functions. In order to establish these bounds, we propose the information gain of the NT kernel as a measure of complexity of the learning problem. Our analysis uses a Mercer decomposition of the NT kernel in the basis of spherical harmonics and the decay rate of the corresponding eigenvalues. As a byproduct of our results, we show the equivalence between the RKHS corresponding to the NT kernel and its counterpart corresponding to the Mat'ern family of kernels, showing the NT kernels induce a very general class of models. We further discuss the implications of our analysis for some recent results on the regret bounds for reinforcement learning and bandit algorithms, which use overparameterized neural networks.