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APPROX
2015
Springer

Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness

5 years 7 months ago
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness
Low-degree polynomial approximations to the sign function underlie pseudorandom generators for halfspaces, as well as algorithms for agnostically learning halfspaces. We study the limits of these constructions by proving inapproximability results for the sign function. First, we investigate the derandomization of Chernoff-type concentration inequalities. Schmidt et al. (SIAM J. Discrete Math. 1995) showed that a tail bound of δ can be established for sums of Bernoulli random variables with only O(log(1/δ))-wise independence. We show that their results are tight up to constant factors. Secondly, the “polynomial regression” algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be efficiently learned with respect to log-concave distributions on Rn in the challenging agnostic learning model. The power of this algorithm relies on the fact that under log-concave distributions, halfspaces can be approximated arbitrarily well by low-degree polynomials. In contrast,...
Mark Bun, Thomas Steinke
Added 16 Apr 2016
Updated 16 Apr 2016
Type Journal
Year 2015
Where APPROX
Authors Mark Bun, Thomas Steinke
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