Improved Approximation of Linear Threshold Functions

Abstract

We prove two main results on how arbitrary linear threshold functions $${f(x) = {\rm sign}(w \cdot x - \theta)}$$ 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 $${\epsilon}$$ -close to a threshold function depending only on $${{\rm Inf}(f)^2 \cdot {\rm poly}(1/\epsilon)}$$ many variables, where $${{\rm Inf}(f)}$$ denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut's well-known theorem (Friedgut in Combinatorica 18(1):474---483, 1998), which states that every Boolean function f is $${\epsilon}$$ -close to a function depending only on $${2^{O({\rm Inf}(f)/\epsilon)}}$$ many variables, for the case of threshold functions. We complement this upper bound by showing that $${\Omega({\rm Inf}(f)^2 + 1/\epsilon^2)}$$ many variables are required for $${\epsilon}$$ -approximating threshold functions. Our second result is a proof that every n-variable threshold function is $${\epsilon}$$ -close to a threshold function with integer weights at most $${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.}$$ This is an improvement, in the dependence on the error parameter $${\epsilon}$$ , on an earlier result of Servedio (Comput Complex 16(2):180---209, 2007) which gave a $${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}}$$ 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 result of Servedio (Comput Complex 16(2):180---209, 2007) and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.