Every Linear Threshold Function has a Low-Weight Approximator

Authors:
Rocco A. Servedio
Affiliations:
Columbia University, USA
Venue:
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Year:
2006

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Abstract

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 \in fraction of inputs and has integer weights each of magnitude at most \sqrt n· 2 x O(1/ \in^2). We show that the construction is optimal in terms of its dependence on n by proving a lower bound of O(\sqrt n) on the weights required to approximate a particular linear threshold function.