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 -close to a threshold function depending only on Inf(f)2 · poly(1/ ) 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 -close to a function depending only on 2O(Inf(f)/ ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f)2 + 1/ 2) many variables are required for -approximating threshold functions. Our second result is a proof that every n-variable threshold function is -close to a threshold function with integer weights at most poly(n)·2 ˜O(1/ 2/3). This is a significant improvement, in the dependence on the e...

• Type: Conference
• Year: 2009
• Where: COCO
• Authors: Ilias Diakonikolas, Rocco A. Servedio
• Keywords: COCO 2009, Linear Threshold Functions, N-variable Threshold Function, Theoretical Computer Science, Threshold Functions