# Non-vacuous Generalization Bounds for Deep Neural Networks without any modification to the trained models

arXiv: https://arxiv.org/abs/2503.07325 · HTML: https://mlsanity.com/paper/2503.07325/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 61 references that are in the map.
- First posted on arXiv 10 Mar 2025.

## Builds on (most cited first, 30 of 37)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Information-theoretic analysis of generalization capability of learning algorithms](https://mlsanity.com/paper/1705.07809/) (2017)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Non-Vacuous Generalization Bounds at the ImageNet Scale: A PAC-Bayesian Compression Approach](https://mlsanity.com/paper/1804.05862/) (2018)
- [Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent](https://mlsanity.com/paper/2006.08157/) (2020)
- [Robust Large Margin Deep Neural Networks](https://mlsanity.com/paper/1605.08254/) (2016)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [Stability and Generalization of Learning Algorithms that Converge to Global Optima](https://mlsanity.com/paper/1710.08402/) (2017)
- [High probability generalization bounds for uniformly stable algorithms with nearly optimal rate](https://mlsanity.com/paper/1902.10710/) (2019)
- [Data-Dependent Stability of Stochastic Gradient Descent](https://mlsanity.com/paper/1703.01678/) (2017)
- [PAC-Bayes Compression Bounds So Tight That They Can Explain Generalization](https://mlsanity.com/paper/2211.13609/) (2022)
- [Non-Vacuous Generalization Bounds for Large Language Models](https://mlsanity.com/paper/2312.17173/) (2023)
- [Unlocking Tokens as Data Points for Generalization Bounds on Larger Language Models](https://mlsanity.com/paper/2407.18158/) (2024)
- [PAC-Bayes Generalisation Bounds for Heavy-Tailed Losses through Supermartingales](https://mlsanity.com/paper/2210.00928/) (2022)
- [Robustness Implies Generalization via Data-Dependent Generalization Bounds](https://mlsanity.com/paper/2206.13497/) (2022)
- [Non-Vacuous Generalisation Bounds for Shallow Neural Networks](https://mlsanity.com/paper/2202.01627/) (2022)
- [Rate-Distortion Theoretic Generalization Bounds for Stochastic Learning Algorithms](https://mlsanity.com/paper/2203.02474/) (2022)
- [A General Framework for the Practical Disintegration of PAC-Bayesian Bounds](https://mlsanity.com/paper/2102.08649/) (2021)
- [Instance-Dependent Generalization Bounds via Optimal Transport](https://mlsanity.com/paper/2211.01258/) (2022)

## Built on by (most cited first, 2 of 2)

- [Bound to Disagree: Generalization Bounds via Certifiable Surrogates](https://mlsanity.com/paper/2602.23128/) (2026)
- [Certification Frontiers for Gaussian LoRA: Independent Priors, Posterior Risk, and Prediction-Preserving Balancing](https://mlsanity.com/paper/2609.32271/) (2026)

## Abstract (arXiv)

Understanding and certifying the behavior of modern deep neural networks remains a fundamental challenge in reliable machine learning. We introduce a new class of data-dependent generalization bounds that apply directly to trained models, without any modification. In particular, we present an exactly computable bound that is non-vacuous across all evaluated networks, including ImageNet-scale models with 600M parameters. This this is the first work showing that meaningful generalization guarantees are achievable even for large, unaltered deep networks. Our approach reveals that generalization is governed by the interaction between the trained model and the geometry of the data distribution. We decompose the generalization error into two interpretable components: a distributional complexity term, capturing how the data mass is distributed across the input space, and local model-behavior terms, capturing the network's behavior within individual regions. This joint dependence identifies where and why generalization gaps arise. Empirically, some components of our bound are highly predictive of the true test error, and the bound tightens when the partition aligns with the intrinsic data geometry, highlighting data-dependent local regularity as a key driver of generalization.