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L1 Regression with Lewis Weights Subsampling

Authors Aditya Parulekar, Advait Parulekar, Eric Price



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Author Details

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

Acknowledgements

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

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.49

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)).

Subject Classification

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

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