# On the Importance of Gradient Norm in PAC-Bayesian Bounds

arXiv: https://arxiv.org/abs/2210.06143 · HTML: https://mlsanity.com/paper/2210.06143/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 76 references that are in the map.
- First posted on arXiv 12 Oct 2022.

## Builds on (most cited first, 30 of 47)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (2017)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (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)
- [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)
- [Fixup Initialization: Residual Learning Without Normalization](https://mlsanity.com/paper/1901.09321/) (2019)
- [Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks](https://mlsanity.com/paper/1805.12076/) (2018)
- [On the properties of variational approximations of Gibbs posteriors](https://mlsanity.com/paper/1506.04091/) (2015)
- [Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates](https://mlsanity.com/paper/1911.02151/) (2019)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [Data-Dependent Stability of Stochastic Gradient Descent](https://mlsanity.com/paper/1703.01678/) (2017)
- [Sharpened Generalization Bounds based on Conditional Mutual Information and an Application to Noisy, Iterative Algorithms](https://mlsanity.com/paper/2004.12983/) (2020)
- [PAC-Bayesian Theory Meets Bayesian Inference](https://mlsanity.com/paper/1605.08636/) (2016)
- [PACOH: Bayes-Optimal Meta-Learning with PAC-Guarantees](https://mlsanity.com/paper/2002.05551/) (2020)
- [Structured and Efficient Variational Deep Learning with Matrix Gaussian Posteriors](https://mlsanity.com/paper/1603.04733/) (2016)
- [Generalization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints](https://mlsanity.com/paper/1707.05947/) (2017)

## Built on by (most cited first, 6 of 6)

- [Layer Collaboration in the Forward-Forward Algorithm](https://mlsanity.com/paper/2305.12393/) (2023)
- [A PAC-Bayesian Link Between Generalisation and Flat Minima](https://mlsanity.com/paper/2402.08508/) (2024)
- [CR-SAM: Curvature Regularized Sharpness-Aware Minimization](https://mlsanity.com/paper/2312.13555/) (2023)
- [PAC-Bayes-Chernoff bounds for unbounded losses](https://mlsanity.com/paper/2401.01148/) (2024)
- [Optimizing Learning Rate Schedules for Iterative Pruning of Deep Neural Networks](https://mlsanity.com/paper/2212.06144/) (2022)
- [A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning](https://mlsanity.com/paper/2605.26341/) (2026)

## Abstract (arXiv)

Generalization bounds which assess the difference between the true risk and the empirical risk, have been studied extensively. However, to obtain bounds, current techniques use strict assumptions such as a uniformly bounded or a Lipschitz loss function. To avoid these assumptions, in this paper, we follow an alternative approach: we relax uniform bounds assumptions by using on-average bounded loss and on-average bounded gradient norm assumptions. Following this relaxation, we propose a new generalization bound that exploits the contractivity of the log-Sobolev inequalities. These inequalities add an additional loss-gradient norm term to the generalization bound, which is intuitively a surrogate of the model complexity. We apply the proposed bound on Bayesian deep nets and empirically analyze the effect of this new loss-gradient norm term on different neural architectures.