46,547 research outputs found

    Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions

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    The Average Sensitivity of an Intersection of Half Spaces

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    We prove new bounds on the average sensitivity of the indicator function of an intersection of kk halfspaces. In particular, we prove the optimal bound of O(nlog(k))O(\sqrt{n\log(k)}). This generalizes a result of Nazarov, who proved the analogous result in the Gaussian case, and improves upon a result of Harsha, Klivans and Meka. Furthermore, our result has implications for the runtime required to learn intersections of halfspaces

    The Correct Exponent for the Gotsman-Linial Conjecture

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    We prove a new bound on the average sensitivity of polynomial threshold functions. In particular we show that a polynomial threshold function of degree dd in at most nn variables has average sensitivity at most n(log(n))O(dlog(d))2O(d2log(d)\sqrt{n}(\log(n))^{O(d\log(d))}2^{O(d^2\log(d)}. For fixed dd the exponent in terms of nn in this bound is known to be optimal. This bound makes significant progress towards the Gotsman-Linial Conjecture which would put the correct bound at Θ(dn)\Theta(d\sqrt{n})

    Three Puzzles on Mathematics, Computation, and Games

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    In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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