# Data augmented bootstrap: Unifying confidence interval construction by approximate invariance

arXiv: https://arxiv.org/abs/2606.09049 · HTML: https://mlsanity.com/paper/2606.09049/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 94 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 4 of 4)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [A Survey on Mixup Augmentations and Beyond](https://mlsanity.com/paper/2409.05202/) (2024)
- [Large sample analysis of the median heuristic](https://mlsanity.com/paper/1707.07269/) (2017)
- [On a Probability Inequality for Order Statistics with Applications to Bootstrap, Conformal Prediction, and more](https://mlsanity.com/paper/2604.15229/) (2026)

## Built on by (most cited first, 1 of 1)

- [Conformal Prediction for Dyadic Regression Under Complex Missingness](https://mlsanity.com/paper/2606.11136/) (2026)

## Abstract (arXiv)

We propose the data augmented bootstrap (DAB), a framework for constructing confidence intervals from approximately invariant transformations of the data. As special cases, DAB recovers popular methods that rely on exact group symmetries, such as conformal prediction, wild bootstrap for Maximum Mean Discrepancy U-statistics and the recently proposed SymmPI. Meanwhile, DAB also recovers the classical bootstrap method, which exploits the dataset's approximate invariance under uniform sampling of data indices as the dataset size grows. For all DAB methods, we establish theoretical coverage results that interpolate between finite-sample and asymptotic guarantees according to the strength of the invariance, and without assuming a group structure. The approximate invariance is measured in the Kolmogorov distance and, for statistics that satisfy Gaussian universality, reduces to conditional mean and variance matching. This allows us to incorporate data augmentation (DA), a widely used machine learning heuristic based on approximate invariances, into known statistical methods. We empirically test the performance of incorporating DA into bootstrap, wild bootstrap and conformal prediction for simulated settings as well as for image, language and scientific data.