# Towards Initialization-dependent and Non-vacuous Generalization Bounds for Overparameterized Shallow Neural Networks

arXiv: https://arxiv.org/abs/2604.00505 · HTML: https://mlsanity.com/paper/2604.00505/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 43 of its 57 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Builds on (most cited first, 30 of 43)

- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Towards moderate overparameterization: global convergence guarantees for training shallow neural networks](https://mlsanity.com/paper/1902.04674/) (2019)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks](https://mlsanity.com/paper/1805.12076/) (2018)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [Theoretical insights into the optimization landscape of over-parameterized shallow neural networks](https://mlsanity.com/paper/1707.04926/) (2017)
- [Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks](https://mlsanity.com/paper/1909.12292/) (2019)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)
- [Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent](https://mlsanity.com/paper/2006.08157/) (2020)
- [Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?](https://mlsanity.com/paper/1812.10004/) (2018)
- [How Much Over-parameterization Is Sufficient to Learn Deep ReLU Networks?](https://mlsanity.com/paper/1911.12360/) (2019)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Generalization in Deep Networks: The Role of Distance from Initialization](https://mlsanity.com/paper/1901.01672/) (2019)
- [Compression based bound for non-compressed network: unified generalization error analysis of large compressible deep neural network](https://mlsanity.com/paper/1909.11274/) (2019)
- [On the Optimization and Generalization of Multi-head Attention](https://mlsanity.com/paper/2310.12680/) (2023)
- [Stability & Generalisation of Gradient Descent for Shallow Neural Networks without the Neural Tangent Kernel](https://mlsanity.com/paper/2107.12723/) (2021)
- [Measuring Generalization with Optimal Transport](https://mlsanity.com/paper/2106.03314/) (2021)

## Abstract (arXiv)

Overparameterized neural networks often show a benign overfitting property in the sense of achieving excellent generalization behavior despite the number of parameters exceeding the number of training examples. A promising direction to explain benign overfitting is to relate generalization to the norm of distance from initialization, motivated by the empirical observations that this distance is often significantly smaller than the norm itself. However, the existing initialization-dependent complexity analyses measure the distance from initialization by the Frobenius norm, and often imply vacuous bounds in practice for overparamterized models. In this paper, we develop initialization-dependent complexity bounds for shallow neural networks with general Lipschitz activation functions. Our bounds depend on the path-norm of the distance from initialization, which are derived by introducing a new peeling technique to handle the challenge along with the initialization-dependent constraint. We also develop a lower bound tight up to a constant factor. Finally, we conduct empirical comparisons and show that our generalization analysis implies non-vacuous bounds for overparameterized networks.