New degree bounds for polynomial threshold functions

A real multivariate polynomial p(x1, . . . , xn) is said to sign-represent a Boolean function f : {0, 1}n {-1, 1} if the sign of p(x) equals f(x) for all inputs x {0, 1}n. We give new upper and lower bounds on the degree of 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. Running Head: New Degree Bounds for PTFs Mathematics Subject Classification Codes: 68Q17, 68Q32 A preliminary version of these results appeared as [23]. This work was done while at the Department of Mathematics, MIT, Cambridge, MA, and while supported by NSF grant 99-12342. Corresponding author. Support...

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: Ryan O'Donnell, Rocco A. Servedio
• Keywords: Algorithms, Degree Lower Bounds, NSF Mathematical Sciences, STOC 2003, Upper Bounds