No Free Lunch: Non-Asymptotic Analysis of Prediction-Powered Inference

Pranav Mani, Peng Xu, Zachary Chase Lipton, Michael Oberst
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:86296-86335, 2026.

Abstract

Prediction-Powered Inference (PPI) is a popular strategy for combining gold-standard and possibly noisy pseudo-labels to perform statistical estimation. Prior work has shown an asymptotic free lunch for PPI++, an adaptive form of PPI, showing that the asymptotic variance of PPI++ is always less than or equal to the variance obtained from using gold-standard labels alone. Notably, this result holds regardless of the quality of the pseudo-labels. In this work, we demystify this result by conducting an exact finite-sample analysis of the estimation error of PPI++ on the mean estimation problem. We give a no free lunch result, characterizing the settings (and sample sizes) where PPI++ has provably worse estimation error than using gold-standard labels alone. Specifically, PPI++ will outperform if and only if the correlation between pseudo- and gold-standard is above a certain level that depends on the number of labeled samples ($n$). In some cases our results simplify considerably: For Gaussian data, for instance, the correlation must be at least $1/\sqrt{n - 2}$ in order to see improvement. More broadly, by providing exact non-asymptotic expressions for the variance of PPI++ under sample splitting, we aim to empower practitioners to transparently reason about the benefits of PPI++ in specific applications. In experiments, we illustrate that our theoretical findings hold on real-world datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-mani26b, title = {No Free Lunch: Non-Asymptotic Analysis of Prediction-Powered Inference}, author = {Mani, Pranav and Xu, Peng and Lipton, Zachary Chase and Oberst, Michael}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {86296--86335}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/mani26b/mani26b.pdf}, url = {https://proceedings.mlr.press/v306/mani26b.html}, abstract = {Prediction-Powered Inference (PPI) is a popular strategy for combining gold-standard and possibly noisy pseudo-labels to perform statistical estimation. Prior work has shown an asymptotic free lunch for PPI++, an adaptive form of PPI, showing that the asymptotic variance of PPI++ is always less than or equal to the variance obtained from using gold-standard labels alone. Notably, this result holds regardless of the quality of the pseudo-labels. In this work, we demystify this result by conducting an exact finite-sample analysis of the estimation error of PPI++ on the mean estimation problem. We give a no free lunch result, characterizing the settings (and sample sizes) where PPI++ has provably worse estimation error than using gold-standard labels alone. Specifically, PPI++ will outperform if and only if the correlation between pseudo- and gold-standard is above a certain level that depends on the number of labeled samples ($n$). In some cases our results simplify considerably: For Gaussian data, for instance, the correlation must be at least $1/\sqrt{n - 2}$ in order to see improvement. More broadly, by providing exact non-asymptotic expressions for the variance of PPI++ under sample splitting, we aim to empower practitioners to transparently reason about the benefits of PPI++ in specific applications. In experiments, we illustrate that our theoretical findings hold on real-world datasets.} }
Endnote
%0 Conference Paper %T No Free Lunch: Non-Asymptotic Analysis of Prediction-Powered Inference %A Pranav Mani %A Peng Xu %A Zachary Chase Lipton %A Michael Oberst %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-mani26b %I PMLR %P 86296--86335 %U https://proceedings.mlr.press/v306/mani26b.html %V 306 %X Prediction-Powered Inference (PPI) is a popular strategy for combining gold-standard and possibly noisy pseudo-labels to perform statistical estimation. Prior work has shown an asymptotic free lunch for PPI++, an adaptive form of PPI, showing that the asymptotic variance of PPI++ is always less than or equal to the variance obtained from using gold-standard labels alone. Notably, this result holds regardless of the quality of the pseudo-labels. In this work, we demystify this result by conducting an exact finite-sample analysis of the estimation error of PPI++ on the mean estimation problem. We give a no free lunch result, characterizing the settings (and sample sizes) where PPI++ has provably worse estimation error than using gold-standard labels alone. Specifically, PPI++ will outperform if and only if the correlation between pseudo- and gold-standard is above a certain level that depends on the number of labeled samples ($n$). In some cases our results simplify considerably: For Gaussian data, for instance, the correlation must be at least $1/\sqrt{n - 2}$ in order to see improvement. More broadly, by providing exact non-asymptotic expressions for the variance of PPI++ under sample splitting, we aim to empower practitioners to transparently reason about the benefits of PPI++ in specific applications. In experiments, we illustrate that our theoretical findings hold on real-world datasets.
APA
Mani, P., Xu, P., Lipton, Z.C. & Oberst, M.. (2026). No Free Lunch: Non-Asymptotic Analysis of Prediction-Powered Inference. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:86296-86335 Available from https://proceedings.mlr.press/v306/mani26b.html.

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