Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?

Felix Schur, Niklas Pfister, Peng Ding, Sach Mukherjee, Jonas Peters
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:108536-108572, 2026.

Abstract

In many applications, practical constraints prevent measuring covariates and outcomes on the same units, resulting in unpaired data. We study the problem of estimating causal effects under hidden confounding in the following unpaired data setting: we observe some covariates $X$ and an outcome $Y$ under different experimental conditions (environments) but do not observe them jointly – we either observe $X$ or $Y$. Under appropriate regularity conditions, the problem can be cast as an instrumental variable (IV) regression with the environment acting as a (possibly high-dimensional) instrument. When there are many environments but only a few observations per environment, standard two-sample IV estimators fail to be consistent. We propose a GMM-type estimator based on cross-fold sample splitting of the instrument–covariate sample that also applies in standard IV settings. We prove that it is consistent as the number of environments grows but the sample size per environment remains constant. We further extend the method to sparse causal effects via $\ell_1$-regularized estimation and post-selection refitting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-schur26a, title = {Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?}, author = {Schur, Felix and Pfister, Niklas and Ding, Peng and Mukherjee, Sach and Peters, Jonas}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {108536--108572}, 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/schur26a/schur26a.pdf}, url = {https://proceedings.mlr.press/v306/schur26a.html}, abstract = {In many applications, practical constraints prevent measuring covariates and outcomes on the same units, resulting in unpaired data. We study the problem of estimating causal effects under hidden confounding in the following unpaired data setting: we observe some covariates $X$ and an outcome $Y$ under different experimental conditions (environments) but do not observe them jointly – we either observe $X$ or $Y$. Under appropriate regularity conditions, the problem can be cast as an instrumental variable (IV) regression with the environment acting as a (possibly high-dimensional) instrument. When there are many environments but only a few observations per environment, standard two-sample IV estimators fail to be consistent. We propose a GMM-type estimator based on cross-fold sample splitting of the instrument–covariate sample that also applies in standard IV settings. We prove that it is consistent as the number of environments grows but the sample size per environment remains constant. We further extend the method to sparse causal effects via $\ell_1$-regularized estimation and post-selection refitting.} }
Endnote
%0 Conference Paper %T Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible? %A Felix Schur %A Niklas Pfister %A Peng Ding %A Sach Mukherjee %A Jonas Peters %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-schur26a %I PMLR %P 108536--108572 %U https://proceedings.mlr.press/v306/schur26a.html %V 306 %X In many applications, practical constraints prevent measuring covariates and outcomes on the same units, resulting in unpaired data. We study the problem of estimating causal effects under hidden confounding in the following unpaired data setting: we observe some covariates $X$ and an outcome $Y$ under different experimental conditions (environments) but do not observe them jointly – we either observe $X$ or $Y$. Under appropriate regularity conditions, the problem can be cast as an instrumental variable (IV) regression with the environment acting as a (possibly high-dimensional) instrument. When there are many environments but only a few observations per environment, standard two-sample IV estimators fail to be consistent. We propose a GMM-type estimator based on cross-fold sample splitting of the instrument–covariate sample that also applies in standard IV settings. We prove that it is consistent as the number of environments grows but the sample size per environment remains constant. We further extend the method to sparse causal effects via $\ell_1$-regularized estimation and post-selection refitting.
APA
Schur, F., Pfister, N., Ding, P., Mukherjee, S. & Peters, J.. (2026). Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:108536-108572 Available from https://proceedings.mlr.press/v306/schur26a.html.

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