"Local" vs. "Global" parameters-breaking the Gaussian complexity barrier

We show that if F is a convex class of functions that is L-sub-Gaussian, the error rate of learning problems generated by independent noise is equivalent to a fixed point determined by "local" covering estimates of the class (i.e., the covering number at a specific level), rather than by the Gaussian average, which takes into account the structure of F at an arbitrarily small scale. To that end, we establish new sharp upper and lower estimates on the error rate in such learning problems.