• DOI:10.48550/arXiv.2205.03246
• Corpus ID: 248562732

What Makes A Good Fisherman? Linear Regression under Self-Selection Bias

@article{Cherapanamjeri2022WhatMA,
  title={What Makes A Good Fisherman? Linear Regression under Self-Selection Bias},
  author={Yeshwanth Cherapanamjeri and Constantinos Daskalakis and Andrew Ilyas and Manolis Zampetakis},
  journal={ArXiv},
  year={2022},
  volume={abs/2205.03246}
}

• Yeshwanth Cherapanamjeri, C. Daskalakis, +1 author M. Zampetakis
• Published 6 May 2022
• Mathematics
• ArXiv

In the classical setting of self-selection, the goal is to learn k models, simultaneously from observations ( x ( i ) , y ( i ) ) where y ( i ) is the output of one of k underlying models on input x ( i ) . In contrast to mixture models, where we observe the output of a randomly selected model, here the observed model depends on the outputs themselves, and is determined by some known selection criterion. For example, we might observe the highest output, the smallest output, or the median output of the k models. In known-index self-selection, the identity of the observed model output is observable; in unknown-index self-selection, it is not. Self-selection has a long history in Econometrics (see, e.g., the works of Roy [Roy51], Gronau [Gro74], Lewis [Lew74], or Heckman [Hec74]) and applications in various theoretical and applied fields, including treatment effect estimation, imitation learning, learning from strategically reported data, and learning from markets at disequilibrium. In this work, we present the first computationally and statistically efficient estimation algorithms for the most standard setting of this problem where the models are linear. In the known-index case, we require poly ( 1/ ε , k , d ) sample and time complexity to estimate all model parameters to accuracy ε in d dimensions, and can accommodate quite general selection crite-ria. In the more challenging unknown-index case, even the identifiability of the linear models (from infinitely many samples) was not known. We show three results in this case for the commonly studied max self-selection criterion: (1) we show that the linear models are indeed identifiable,