The Intersection of Two Halfspaces Has High Threshold Degree

The threshold degree of a Boolean function f: {0, 1}ⁿ ¿ {-1, +1} is the least degree of a real polynomial p such f(x) ¿ sgn p(x). We construct two halfspaces on {0,1}ⁿ whose intersection has threshold degree ¿(¿(n)), an exponential improvement on previous lower bounds. This solves an open problem due to Klivans (2002) and rules out the use of perceptronbased techniques for PAC learning the intersection of two halfspaces, a central unresolved challenge in computational learning. We also prove that the intersection of two majority functions has threshold degree ¿(log n), which is tight and settles a conjecture of O´Donnell and Servedio (2003). Our proof consists of two parts. First, we show that for any Boolean functions f and g, the intersection f(x) ¿ g(y) has threshold degree O(d) if and only if ¿f - F||_¿ + ||g - G||_¿ < 1 for some rational functions F, G of degree O(d). Second, we settle the least degree required for approximating a halfspace and a majority function to any given accuracy by rational functions. Our technique further allows us to make progress on Aaronson´s challenge (2008) and contribute strong direct product theorems for the threshold degree of composed Boolean functions of the form F(f₁, ..., f_n). Essentially the only previous technique for analyzing the threshold degree was symmetrization (1969).