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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.49

L1 Regression with Lewis Weights Subsampling

Authors: Aditya Parulekar, Advait Parulekar, and Eric Price

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract

We consider the problem of finding an approximate solution to 𝓁₁ regression while only observing a small number of labels. Given an n × d unlabeled data matrix X, we must choose a small set of m ≪ n rows to observe the labels of, then output an estimate β̂ whose error on the original problem is within a 1 + ε factor of optimal. We show that sampling from X according to its Lewis weights and outputting the empirical minimizer succeeds with probability 1-δ for m > O(1/(ε²) d log d/(ε δ)). This is analogous to the performance of sampling according to leverage scores for 𝓁₂ regression, but with exponentially better dependence on δ. We also give a corresponding lower bound of Ω(d/(ε²) + (d + 1/(ε²)) log 1/(δ)).

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Aditya Parulekar, Advait Parulekar, and Eric Price. L1 Regression with Lewis Weights Subsampling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 49:1-49:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{parulekar_et_al:LIPIcs.APPROX/RANDOM.2021.49,
  author =	{Parulekar, Aditya and Parulekar, Advait and Price, Eric},
  title =	{{L1 Regression with Lewis Weights Subsampling}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{49:1--49:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.49},
  URN =		{urn:nbn:de:0030-drops-147422},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.49},
  annote =	{Keywords: Active regression, Lewis weights}
}

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L1 Regression with Lewis Weights Subsampling

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