# Local Risk Bounds for Statistical Aggregation

arXiv: https://arxiv.org/abs/2306.17151 · HTML: https://mlsanity.com/paper/2306.17151/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 77 references that are in the map.
- First posted on arXiv 29 Jun 2023.

## Builds on (most cited first, 23 of 23)

- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Reasoning About Generalization via Conditional Mutual Information](https://mlsanity.com/paper/2001.09122/) (2020)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Minimax Analysis of Active Learning](https://mlsanity.com/paper/1410.0996/) (2014)
- [Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions](https://mlsanity.com/paper/1301.6677/) (2013)
- [Second-order Quantile Methods for Experts and Combinatorial Games](https://mlsanity.com/paper/1502.08009/) (2015)
- [Learning with Square Loss: Localization through Offset Rademacher Complexity](https://mlsanity.com/paper/1502.06134/) (2015)
- [Optimal learning with Bernstein Online Aggregation](https://mlsanity.com/paper/1404.1356/) (2014)
- [PAC-Bayes, MAC-Bayes and Conditional Mutual Information: Fast rate bounds that handle general VC classes](https://mlsanity.com/paper/2106.09683/) (2021)
- [Towards a Unified Information-Theoretic Framework for Generalization](https://mlsanity.com/paper/2111.05275/) (2021)
- [MetaGrad: Adaptation using Multiple Learning Rates in Online Learning](https://mlsanity.com/paper/2102.06622/) (2021)
- [Refined Error Bounds for Several Learning Algorithms](https://mlsanity.com/paper/1512.07146/) (2015)
- [An improper estimator with optimal excess risk in misspecified density estimation and logistic regression](https://mlsanity.com/paper/1912.10784/) (2019)
- [On the Generalization Ability of Online Learning Algorithms for Pairwise Loss Functions](https://mlsanity.com/paper/1305.2505/) (2013)
- [Empirical entropy, minimax regret and minimax risk](https://mlsanity.com/paper/1308.1147/) (2013)
- [Distribution-Free Robust Linear Regression](https://mlsanity.com/paper/2102.12919/) (2021)
- [The Sample Complexity of Learning Linear Predictors with the Squared Loss](https://mlsanity.com/paper/1406.5143/) (2014)
- [Online-to-PAC Conversions: Generalization Bounds via Regret Analysis](https://mlsanity.com/paper/2305.19674/) (2023)
- [Exponential Tail Local Rademacher Complexity Risk Bounds Without the Bernstein Condition](https://mlsanity.com/paper/2202.11461/) (2022)
- [On aggregation for heavy-tailed classes](https://mlsanity.com/paper/1502.07097/) (2015)
- [Localization, Convexity, and Star Aggregation](https://mlsanity.com/paper/2105.08866/) (2021)
- [A Regret-Variance Trade-Off in Online Learning](https://mlsanity.com/paper/2206.02656/) (2022)

## Built on by (most cited first, 3 of 3)

- [Ratio Covers of Convex Sets and Optimal Mixture Density Estimation](https://mlsanity.com/paper/2602.16142/) (2026)
- [Aggregation with Exponential Weights is Optimal in Expectation](https://mlsanity.com/paper/2607.02247/) (2026)
- [Handling Covariate Mismatch in Collaborative Linear Prediction](https://mlsanity.com/paper/2602.02083/) (2026)

## Abstract (arXiv)

In the problem of aggregation, the aim is to combine a given class of base predictors to achieve predictions nearly as accurate as the best one. In this flexible framework, no assumption is made on the structure of the class or the nature of the target. Aggregation has been studied in both sequential and statistical contexts. Despite some important differences between the two problems, the classical results in both cases feature the same global complexity measure. In this paper, we revisit and tighten classical results in the theory of aggregation in the statistical setting by replacing the global complexity with a smaller, local one. Some of our proofs build on the PAC-Bayes localization technique introduced by Catoni. Among other results, we prove localized versions of the classical bound for the exponential weights estimator due to Leung and Barron and deviation-optimal bounds for the Q-aggregation estimator. These bounds improve over the results of Dai, Rigollet and Zhang for fixed design regression and the results of Lecu'e and Rigollet for random design regression.