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This paper unifies various statistical inference methods, including conformal prediction and the classical bootstrap, under a single framework based on approximate data invariance. By establishing theoretical coverage guarantees that interpolate between finite-sample and asymptotic regimes, it provides a rigorous bridge between heuristic data augmentation practices and formal statistical theory. The approach offers a versatile toolkit for constructing confidence intervals across diverse data types without requiring strict group symmetry assumptions.
Data augmented bootstrap: Unifying confidence interval construction by approximate invariance
arXiv:2606.09049v1 Announce Type: cross Abstract: 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.
Abstract: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.
Methodology (stat.ME); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
arXiv:2606.09049 [stat.ME]
(or arXiv:2606.09049v1 [stat.ME] for this version)
https://doi.org/10.48550/arXiv.2606.09049 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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