# A PAC-Bayesian Tutorial with A Dropout Bound

arXiv: https://arxiv.org/abs/1307.2118 · HTML: https://mlsanity.com/paper/1307.2118/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2013.

## Built on by (most cited first, 30 of 43)

- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Information-Theoretic Generalization Bounds for Stochastic Gradient Descent](https://mlsanity.com/paper/2102.00931/) (2021)
- [Sparse MoE as the New Dropout: Scaling Dense and Self-Slimmable Transformers](https://mlsanity.com/paper/2303.01610/) (2023)
- [Generalization in Graph Neural Networks: Improved PAC-Bayesian Bounds on Graph Diffusion](https://mlsanity.com/paper/2302.04451/) (2023)
- [Robust Fine-Tuning of Deep Neural Networks with Hessian-based Generalization Guarantees](https://mlsanity.com/paper/2206.02659/) (2022)
- [Generalization Bounds for Meta-Learning via PAC-Bayes and Uniform Stability](https://mlsanity.com/paper/2102.06589/) (2021)
- [A New Family of Generalization Bounds Using Samplewise Evaluated CMI](https://mlsanity.com/paper/2210.06422/) (2022)
- [Implicit regularization of dropout](https://mlsanity.com/paper/2207.05952/) (2022)
- [How Tight Can PAC-Bayes be in the Small Data Regime?](https://mlsanity.com/paper/2106.03542/) (2021)
- [Information Complexity and Generalization Bounds](https://mlsanity.com/paper/2105.01747/) (2021)
- [Online-to-PAC Conversions: Generalization Bounds via Regret Analysis](https://mlsanity.com/paper/2305.19674/) (2023)
- [Noise Stability Optimization for Finding Flat Minima: A Hessian-based Regularization Approach](https://mlsanity.com/paper/2306.08553/) (2023)
- [Learning Augmentation Distributions using Transformed Risk Minimization](https://mlsanity.com/paper/2111.08190/) (2021)
- [PAC-Bayesian Domain Adaptation Bounds for Multiclass Learners](https://mlsanity.com/paper/2207.05685/) (2022)
- [Evaluated CMI Bounds for Meta Learning: Tightness and Expressiveness](https://mlsanity.com/paper/2210.06511/) (2022)
- [Stochastic Modified Equations and Dynamics of Dropout Algorithm](https://mlsanity.com/paper/2305.15850/) (2023)
- [Robustness to Pruning Predicts Generalization in Deep Neural Networks](https://mlsanity.com/paper/2103.06002/) (2021)
- [Is K-fold cross validation the best model selection method for Machine Learning?](https://mlsanity.com/paper/2401.16407/) (2024)
- [Scalable Multi-Objective and Meta Reinforcement Learning via Gradient Estimation](https://mlsanity.com/paper/2511.12779/) (2025)
- [Structured Dropout Variational Inference for Bayesian Neural Networks](https://mlsanity.com/paper/2102.07927/) (2021)
- [PAC-Bayes Bounds for Bandit Problems: A Survey and Experimental Comparison](https://mlsanity.com/paper/2211.16110/) (2022)
- [Weight Expansion: A New Perspective on Dropout and Generalization](https://mlsanity.com/paper/2201.09209/) (2022)
- [On the Importance of Gradient Norm in PAC-Bayesian Bounds](https://mlsanity.com/paper/2210.06143/) (2022)
- [What training reveals about neural network complexity](https://mlsanity.com/paper/2106.04186/) (2021)
- [Task-Driven Detection of Distribution Shifts with Statistical Guarantees for Robot Learning](https://mlsanity.com/paper/2106.13703/) (2021)
- [PAC Bayesian Performance Guarantees for Deep (Stochastic) Networks in Medical Imaging](https://mlsanity.com/paper/2104.05600/) (2021)
- [Towards Data-Algorithm Dependent Generalization: a Case Study on Overparameterized Linear Regression](https://mlsanity.com/paper/2202.06054/) (2022)
- [A General framework for PAC-Bayes Bounds for Meta-Learning](https://mlsanity.com/paper/2206.05454/) (2022)
- [Tighter Risk Bounds for Mixtures of Experts](https://mlsanity.com/paper/2410.10397/) (2024)
- [Dropout in Training Neural Networks: Flatness of Solution and Noise Structure](https://mlsanity.com/paper/2111.01022/) (2021)

## Abstract (arXiv)

This tutorial gives a concise overview of existing PAC-Bayesian theory focusing on three generalization bounds. The first is an Occam bound which handles rules with finite precision parameters and which states that generalization loss is near training loss when the number of bits needed to write the rule is small compared to the sample size. The second is a PAC-Bayesian bound providing a generalization guarantee for posterior distributions rather than for individual rules. The PAC-Bayesian bound naturally handles infinite precision rule parameters, L2 regularization, em provides a bound for dropout training, and defines a natural notion of a single distinguished PAC-Bayesian posterior distribution. The third bound is a training-variance bound --- a kind of bias-variance analysis but with bias replaced by expected training loss. The training-variance bound dominates the other bounds but is more difficult to interpret. It seems to suggest variance reduction methods such as bagging and may ultimately provide a more meaningful analysis of dropouts.