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Preserving Randomness for Adaptive Algorithms

Authors William M. Hoza , Adam R. Klivans

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Author Details

William M. Hoza
  • Department of Computer Science, University of Texas at Austin, Austin, TX, USA
Adam R. Klivans
  • Department of Computer Science, University of Texas at Austin, Austin, TX, USA

Funding

  • Hoza, William M.: Supported by the NSF GRFP under Grant DGE-1610403 and by a Harrington Fellowship from the University of Texas at Austin.

Cite AsGet BibTex

William M. Hoza and Adam R. Klivans. Preserving Randomness for Adaptive Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.43

Abstract

Suppose Est is a randomized estimation algorithm that uses n random bits and outputs values in R^d. We show how to execute Est on k adaptively chosen inputs using only n + O(k log(d + 1)) random bits instead of the trivial nk (at the cost of mild increases in the error and failure probability). Our algorithm combines a variant of the INW pseudorandom generator [Impagliazzo et al., 1994] with a new scheme for shifting and rounding the outputs of Est. We prove that modifying the outputs of Est is necessary in this setting, and furthermore, our algorithm's randomness complexity is near-optimal in the case d <= O(1). As an application, we give a randomness-efficient version of the Goldreich-Levin algorithm; our algorithm finds all Fourier coefficients with absolute value at least theta of a function F: {0, 1}^n -> {-1, 1} using O(n log n) * poly(1/theta) queries to F and O(n) random bits (independent of theta), improving previous work by Bshouty et al. [Bshouty et al., 2004].

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • pseudorandomness
  • adaptivity
  • estimation

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