# Recursive PAC-Bayes: A Frequentist Approach to Sequential Prior Updates with No Information Loss

arXiv: https://arxiv.org/abs/2405.14681 · HTML: https://mlsanity.com/paper/2405.14681/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 28 references that are in the map.
- First posted on arXiv 23 May 2024.

## Builds on (most cited first, 15 of 15)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [A Strongly Quasiconvex PAC-Bayesian Bound](https://mlsanity.com/paper/1608.05610/) (2016)
- [PAC-Bayes Un-Expected Bernstein Inequality](https://mlsanity.com/paper/1905.13367/) (2019)
- [PAC-Bayes with Backprop](https://mlsanity.com/paper/1908.07380/) (2019)
- [A unified recipe for deriving (time-uniform) PAC-Bayes bounds](https://mlsanity.com/paper/2302.03421/) (2023)
- [How Tight Can PAC-Bayes be in the Small Data Regime?](https://mlsanity.com/paper/2106.03542/) (2021)
- [PAC-Bayes Generalisation Bounds for Heavy-Tailed Losses through Supermartingales](https://mlsanity.com/paper/2210.00928/) (2022)
- [More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity](https://mlsanity.com/paper/2306.12214/) (2023)
- [Chebyshev-Cantelli PAC-Bayes-Bennett Inequality for the Weighted Majority Vote](https://mlsanity.com/paper/2106.13624/) (2021)
- [Tighter PAC-Bayes Bounds Through Coin-Betting](https://mlsanity.com/paper/2302.05829/) (2023)
- [Split-kl and PAC-Bayes-split-kl Inequalities for Ternary Random Variables](https://mlsanity.com/paper/2206.00706/) (2022)
- [Tighter PAC-Bayes Generalisation Bounds by Leveraging Example Difficulty](https://mlsanity.com/paper/2210.11289/) (2022)
- [A Note on the Chernoff Bound for Random Variables in the Unit Interval](https://mlsanity.com/paper/2205.07880/) (2022)

## Built on by (most cited first, 4 of 4)

- [The Alignment Bottleneck](https://mlsanity.com/paper/2509.15932/) (2025)
- [Deep Actor-Critics with Tight Risk Certificates](https://mlsanity.com/paper/2505.19682/) (2025)
- [Block-Sample MAC-Bayes Generalization Bounds](https://mlsanity.com/paper/2602.12605/) (2026)
- [Certification Frontiers for Gaussian LoRA: Independent Priors, Posterior Risk, and Prediction-Preserving Balancing](https://mlsanity.com/paper/2609.32271/) (2026)

## Abstract (arXiv)

PAC-Bayesian analysis is a frequentist framework for incorporating prior knowledge into learning. It was inspired by Bayesian learning, which allows sequential data processing and naturally turns posteriors from one processing step into priors for the next. However, despite two and a half decades of research, the ability to update priors sequentially without losing confidence information along the way remained elusive for PAC-Bayes. While PAC-Bayes allows construction of data-informed priors, the final confidence intervals depend only on the number of points that were not used for the construction of the prior, whereas confidence information in the prior, which is related to the number of points used to construct the prior, is lost. This limits the possibility and benefit of sequential prior updates, because the final bounds depend only on the size of the final batch. We present a novel and, in retrospect, surprisingly simple and powerful PAC-Bayesian procedure that allows sequential prior updates with no information loss. The procedure is based on a novel decomposition of the expected loss of randomized classifiers. The decomposition rewrites the loss of the posterior as an excess loss relative to a downscaled loss of the prior plus the downscaled loss of the prior, which is bounded recursively. As a side result, we also present a generalization of the split-kl and PAC-Bayes-split-kl inequalities to discrete random variables, which we use for bounding the excess losses, and which can be of independent interest. In empirical evaluation the new procedure significantly outperforms state-of-the-art.