Linear Regression with Unknown Truncation Beyond Gaussian Features

• URL: http://arxiv.org/abs/2602.12534v1
• Date: Fri, 13 Feb 2026 02:29:54 GMT
• Title: Linear Regression with Unknown Truncation Beyond Gaussian Features
• Authors: Alexandros Kouridakis, Anay Mehrotra, Alkis Kalavasis, Constantine Caramanis,
• Abstract summary: In truncated linear regression, samples $(x,y)$ are shown only when the outcome $y$ falls inside a certain survival set $Sstar$.<n>We give the first algorithm for truncated linear regression with unknown survival set that runs in $mathrmpoly (d/varepsilon)$ time.
• Score: 66.1580269813084
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In truncated linear regression, samples $(x,y)$ are shown only when the outcome $y$ falls inside a certain survival set $S^\star$ and the goal is to estimate the unknown $d$-dimensional regressor $w^\star$. This problem has a long history of study in Statistics and Machine Learning going back to the works of (Galton, 1897; Tobin, 1958) and more recently in, e.g., (Daskalakis et al., 2019; 2021; Lee et al., 2023; 2024). Despite this long history, however, most prior works are limited to the special case where $S^\star$ is precisely known. The more practically relevant case, where $S^\star$ is unknown and must be learned from data, remains open: indeed, here the only available algorithms require strong assumptions on the distribution of the feature vectors (e.g., Gaussianity) and, even then, have a $d^{\mathrm{poly} (1/\varepsilon)}$ run time for achieving $\varepsilon$ accuracy. In this work, we give the first algorithm for truncated linear regression with unknown survival set that runs in $\mathrm{poly} (d/\varepsilon)$ time, by only requiring that the feature vectors are sub-Gaussian. Our algorithm relies on a novel subroutine for efficiently learning unions of a bounded number of intervals using access to positive examples (without any negative examples) under a certain smoothness condition. This learning guarantee adds to the line of works on positive-only PAC learning and may be of independent interest.

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