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.
"“Local” vs. “global” parameters—breaking the Gaussian complexity barrier." Ann. Statist. 45 (5) 1835 - 1862, October 2017. https://doi.org/10.1214/16-AOS1510