Bounds for Rademacher Processes via Chaining

Abstract

We study Rademacher processes where the coefficients are functions evaluated at fixed, but arbitrary covariables. Specifically, we assume the function class under consideration to be parametrized by the standard cocube in l dimensions and we are mainly interested in the high-dimensional, asymptotic situation, that is, l as well the number of Rademacher variables n go to infinity with l much larger than n. We refine and apply classical entropy bounds and Majorizing Measures, both going back to the well known idea of chaining. That way, we derive general upper bounds for Rademacher processes. In the linear case and under high correlations, we further improve on these bounds. In particular, we give bounds independent of l for highly correlated covariables.