# Overfitting and Generalizing with (PAC) Bayesian Prediction in Noisy Binary Classification

arXiv: https://arxiv.org/abs/2603.22644 · HTML: https://mlsanity.com/paper/2603.22644/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 23 Mar 2026.

## Builds on (most cited first, 12 of 12)

- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)
- [PAC-Bayesian Theory Meets Bayesian Inference](https://mlsanity.com/paper/1605.08636/) (2016)
- [Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors](https://mlsanity.com/paper/1712.09376/) (2017)
- [PAC-Bayes Analysis Beyond the Usual Bounds](https://mlsanity.com/paper/2006.13057/) (2020)
- [Deterministic PAC-Bayesian generalization bounds for deep networks via generalizing noise-resilience](https://mlsanity.com/paper/1905.13344/) (2019)
- [Risk Bounds for the Majority Vote: From a PAC-Bayesian Analysis to a Learning Algorithm](https://mlsanity.com/paper/1503.08329/) (2015)
- [A PAC-Bayesian Analysis of Randomized Learning with Application to Stochastic Gradient Descent](https://mlsanity.com/paper/1709.06617/) (2017)
- [Tight Bounds on the Binomial CDF, and the Minimum of i.i.d Binomials, in terms of KL-Divergence](https://mlsanity.com/paper/2502.18611/) (2025)
- [Quantifying Overfitting along the Regularization Path for Two-Part-Code MDL in Supervised Classification](https://mlsanity.com/paper/2503.02110/) (2025)

## Abstract (arXiv)

We consider a PAC-Bayes type learning rule for binary classification, balancing the training error of a randomized ''posterior'' predictor with its KL divergence to a pre-specified ''prior''. This can be seen as an extension of a modified two-part-code Minimum Description Length (MDL) learning rule, to continuous priors and randomized predictions. With a balancing parameter of λ=1 this learning rule recovers an (empirical) Bayes posterior and a modified variant recovers the profile posterior, linking with standard Bayesian prediction (up to the treatment of the single-parameter noise level). However, from a risk-minimization prediction perspective, this Bayesian predictor overfits and can lead to non-vanishing excess loss in the agnostic case. Instead a choice of λ gg 1, which can be seen as using a sample-size-dependent-prior, ensures uniformly vanishing excess loss even in the agnostic case. We precisely characterize the effect of under-regularizing (and over-regularizing) as a function of the balance parameter λ, understanding the regimes in which this under-regularization is tempered or catastrophic. This work extends previous work by Zhu and Srebro [2025] that considered only discrete priors to PAC Bayes type learning rules and, through their rigorous Bayesian interpretation, to Bayesian prediction more generally.