Every Linear Threshold Function has a Low-Weight Approximator

Given any linear threshold function f on n Boolean variables, we construct a linear threshold function g which disagrees with f on at most an epsiv fraction of inputs and has integer weights each of magnitude at most radicn middot 2^{Omacr(1/epsiv2)}. We show that the construction is optimal in terms of its dependence on n by proving a lower bound of Omega(radicn) on the weights required to approximate a particular linear threshold function. We give two applications. The first is a deterministic algorithm for approximately counting the fraction of satisfying assignments to an instance of the zero-one knapsack problem to within an additive plusmnepsiv. The algorithm runs in time polynomial in n (but exponential in 1/epsiv²). In our second application, we show that any linear threshold function f is specified to within error epsiv by estimates of its Chow parameters (degree 0 and 1 Fourier coefficients) which are accurate to within an additive error of plusmn1/(nmiddot2 ^{Omacr(1/epsiv2)}. This is the first such accuracy bound which is inverse polynomial in n (previous work of Goldberg gave a 1/quasipoly(n) bound), and gives the first polynomial bound (in terms of n) on the number of examples required for learning linear threshold functions in the "restricted focus of attention" framework