Improved Approximation of Linear Threshold Functions

We prove two main results on how arbitrary linear threshold functions f(x)=sign(w⋅x−θ) 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 ϵ -close to a threshold function depending only on Inf(f)² ⋅ poly(1/ϵ) 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 (Friedgut in Combinatorica 18(1):474–483, 1998), which states that every Boolean function f is ϵ -close to a function depending only on 2^O(Inf(f)/ϵ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f)² +1/ϵ²) many variables are required for ϵ -approximating threshold functions. Our second result is a proof that every n-variable threshold function is ϵ -close to a threshold function with integer weights at most poly(n)⋅2^{O ~ (1/ϵ 2/3)} . This is an improvement, in the dependence on the error parameter ϵ , on an earlier result of Servedio (Comput Complex 16(2):180–209, 2007) which gave a poly(n)⋅2 ^{O ~ (1/ϵ 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.