New degree bounds for polynomial threshold functions
journal contributionposted on 01.01.1980 by Ryan O'Donnell, Rocco Servedio
Any type of content formally published in an academic journal, usually following a peer-review process.
We give new upper and lower bounds on the degree of real multivariate polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of superconstant depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved constructively; we give explicit dual solutions to the necessary linear programs