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Title:
A Combinatorial Proof of the Chernoff-Hoeffding Bound, With Applications to Direct-Product Theorems
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Abstract:
We give a simple combinatorial proof of the Chernoff-Hoeffding concentrationbound for sums of independent Boolean random variables. Unlike the standardproofs, our proof does not rely on the method of higher moments, but rather usesan intuitive counting argument. In addition, this new proof is constructive in thefollowing sense: if the given random variables fail the concentration bound, thenwe can efficiently find a subset of the variables that are statistically dependent.As easy corollaries, we also get concentration bounds for [0, 1]-valued randomvariables, martingales (Azuma’s inequality), and expander walks (from the hittingproperty of expander walks).We also give applications of these concentration results to direct product theoremsin complexity. In many areas of complexity theory, we want to make a somewhathard problem into a reliably hard problem. An immediate motivation for this typeof hardness amplification comes from cryptography and security. For example,CAPTCHAs are now widely used to distinguish human users from artificiallyintelligent “bots”. However, current CAPTCHAs are neither impossible for abot to solve some significant fraction of the time, nor reliably solved by humanusers. Can we make a CAPTCHA both reliably hard for bots and reliably easyfor humans?The main tools for hardness amplification are direct product theorems, where weshow that a feasible solver’s probability of solving many instances of a problem issignificantly smaller than their probability of solving a single instance. In otherwords, if f(x) is hard for a constant fraction of x’s, fk(x1, ..xk) = f(x1) â—¦ f(x2).. â—¦f(xk) is hard on almost every tuple x1, ...xk. While intuitive, direct producttheorems are not always true and do not not always have the intuitive parameterseven if true.We give simple reductions between the different varieties of direct product the-orems: the xor lemma, standard direct product and thresholded direct product.(This builds on recent work of Unger).Joint work with Russell Impagliazzo.
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