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Samplers and Extractors for Unbounded Functions

Author Rohit Agrawal



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Rohit Agrawal
  • John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Acknowledgements

The author would like to thank Jarosław Błasiok for suggesting the problem of constructing subgaussian samplers and for helpful discussions and feedback, Salil Vadhan for many helpful discussions and his detailed feedback on this writeup, and the anonymous reviewers for their helpful comments and feedback.

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Rohit Agrawal. Samplers and Extractors for Unbounded Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 59:1-59:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.59

Abstract

Błasiok (SODA'18) recently introduced the notion of a subgaussian sampler, defined as an averaging sampler for approximating the mean of functions f from {0,1}^m to the real numbers such that f(U_m) has subgaussian tails, and asked for explicit constructions. In this work, we give the first explicit constructions of subgaussian samplers (and in fact averaging samplers for the broader class of subexponential functions) that match the best known constructions of averaging samplers for [0,1]-bounded functions in the regime of parameters where the approximation error epsilon and failure probability delta are subconstant. Our constructions are established via an extension of the standard notion of randomness extractor (Nisan and Zuckerman, JCSS'96) where the error is measured by an arbitrary divergence rather than total variation distance, and a generalization of Zuckerman’s equivalence (Random Struct. Alg.'97) between extractors and samplers. We believe that the framework we develop, and specifically the notion of an extractor for the Kullback-Leibler (KL) divergence, are of independent interest. In particular, KL-extractors are stronger than both standard extractors and subgaussian samplers, but we show that they exist with essentially the same parameters (constructively and non-constructively) as standard extractors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Pseudorandomness and derandomization
  • Mathematics of computing → Information theory
Keywords
  • averaging samplers
  • subgaussian samplers
  • randomness extractors
  • Kullback-Leibler divergence

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