Improved Approximation of Linear Threshold Functions

Ilias Diakonikolas, Rocco A. Servedio

Abstract: We prove two main results on how arbitrary linear threshold functions f(x) = sign(w ldr x - thetas) over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is isin-close to a threshold function depending only on Inf(f) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ldr poly (1/isin) many variables, where Inf(f) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut's well-known theorem [Fri98], which states that every Boolean function f is isin-close to a function depending only on 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(Inf(f)/isin)</sup> many variables, for the case of threshold functions. We complement this upper bound by showing that OmegaInf(f) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + 1/isin <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) many variables are required for isin-approximating threshold functions. Our second result is a proof that every n-variable threshold function is isin-close to a threshold function with integer weights at most poly(n) ldr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2Omacr(1/isin</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/3</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">)</sup> This is a significant improvement, in the dependence on the error parameter isin, on an earlier result of [Ser07] which gave a poly(n) ldr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2Omacr(1/isin</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">)</sup> bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original [Ser07] result, and extends to give low-weight approximators for threshold functions under a range of probability distributions beyond just the uniform distribution.