Learning Geometric Concepts via Gaussian Surface Area
Adam R. Klivans, Ryan W. O’Donnell, Rocco A. Servedio · 2008
We study the learnability of sets in Ropfnunder the Gaussian distribution, taking Gaussian surface area as the "complexity measure" of the sets being learned. Let CSdenote the class of all (measurable) sets with surface area at most S. We first show that the class CSis learnable to any constant accuracy in time nO(S2), even in the arbitrary noise ("agnostic'') model. Complementing this, we also show that any learning algorithm for CSinformation-theoretically requires 2Omega(S2)examples for learning to constant accuracy. These results together show that Gaussian surface area essentially characterizes the computational complexity of learning under the Gaussian distribution. Our approach yields several new learning results, including the following (all bounds are for learning to any constant accuracy): The class of all convex sets can be agnostically learned in time 2O~(radicn)(and we prove a 2Omega(radicn)lower bound for noise-free learning). This is the first subexponential time algorithm for learning general convex sets even in the noise-free (PAC) model. Intersections of k halfspaces can be agnostically learned in time nO(logk)(cf. Vempala's nO(k)time algorithm for learning in the noise-free model).Cones (with apex centered at the origin), and spheres witharbitrary radius and center, can be agnostically learned in time poly(n).