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L1 Regression with Lewis Weights Subsampling

Authors Aditya Parulekar, Advait Parulekar, Eric Price

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Aditya Parulekar
  • Department of Computer Science, University of Texas at Austin, TX, USA
Advait Parulekar
  • Department of Electrical and Computer Engineering, University of Texas at Austin, TX, USA
Eric Price
  • Department of Computer Science, University of Texas at Austin, TX, USA


The authors wish to thank the anonymous reviewers for several comments that improved the paper.

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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)


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)).

Subject Classification

ACM Subject Classification
  • Computing methodologies → Active learning settings
  • Active regression
  • Lewis weights


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