Unconditional lower bounds for learning intersections of halfspaces

摘要

We prove new lower bounds for learning intersections of halfspaces, one of the most important concept classes in computational learning theory. Our main result is that any statistical-query algorithm for learning the intersection of \(\sqrt{n}\) halfspaces in n dimensions must make \(2^{\varOmega (\sqrt{n})}\) queries. This is the first non-trivial lower bound on the statistical query dimension for this concept class (the previous best lower bound was n Ω(log n)). Our lower bound holds even for intersections of low-weight halfspaces. In the latter case, it is nearly tight.