Abstract

The threshold degree of a function f: X → {−1, +1}, X ⊂ R n, is the least degree of a polynomial � with f (x) ≡ sgn �(x). This notion has numerous applications in complexity theory and learning theory. Analyzing the threshold degree is a challenge, with few techniques currently available. We develop a novel technique for estimating the threshold degree for a broad and natural class of problems. Specifically, fix nonconstant functions f: X → {−1, +1} and g: Y → {−1, +1} on any finite or compact infinite sets X, Y ⊂ R n. We prove that the conjunction f (x) ∧ g(y) has threshold degree Θ(d) if and only if there exist degree-Θ(d) rational functions F (x) and G (y) with sup X | f − F | + sup Y |g − G | < 1. The “if ” part is simple and well-known, and our contribution is to prove its converse. Our results extend to conjunctions f1 ∧ f2 ∧ · · · ∧ fk of any Boolean functions f1, f2,..., fk and further to compositions h ( f1, f2,..., fk) for various h such as halfspaces and read-once formulas. As an application, we prove the conjecture of O’Donnell and Servedio (2003) that the intersection of two majorities has threshold degree Θ(log n). We discuss several other applications in communication complexity and learning. At the heart of our proof is a novel method for analyzing polynomials �(x, y) that sign-represent a given function f (x) ∧g(y), whereby we recover rational approximants of f and g from any such �. This recovery crucially uses LP duality and succeeds regardless of how tightly x and y are coupled inside �. Also employed here are classical results on compact sets of inequalities and a new, optimal lower bound for the rational approximation of MAJORITY of any given order.

Citations