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.

翻译：在截断线性回归中，样本$(x,y)$仅在结果$y$落入某个生存集$S^\star$内时被观测到，目标是估计未知的$d$维回归系数$w^\star$。该问题在统计学和机器学习领域具有悠久的研究历史，可追溯至(Galton, 1897; Tobin, 1958)的工作，近期研究见(Daskalakis et al., 2019; 2021; Lee et al., 2023; 2024)等文献。然而尽管历史久远，现有研究大多局限于$S^\star$精确已知的特殊情形。更具实际意义的场景——即$S^\star$未知且需从数据中学习——仍然悬而未决：事实上，目前仅有的算法需要对特征向量分布（如高斯性）施加强假设，且即使满足假设，达到$\varepsilon$精度仍需$d^{\mathrm{poly} (1/\varepsilon)}$的运行时间。本研究首次提出针对未知生存集的截断线性回归算法，该算法仅要求特征向量满足次高斯条件，即可在$\mathrm{poly} (d/\varepsilon)$时间内运行。我们的算法依赖于一种新颖的子程序，该程序能在特定平滑性条件下，通过仅访问正例样本（无需任何反例）高效学习有界数量区间的并集。这一学习保证为仅正例PAC学习的研究体系增添了新成果，可能具有独立的学术价值。