Bounding Expected Sup-Norms of Empirical and Rademacher Processes

Abstract

In what follows, we will use a number of bounds on expectation of suprema of empirical and Rademacher processes. Because of symmetrization inequalities, the problems of bounding expected suprema for these two stochastic processes are equivalent. The bounds are usually based on various complexity measures of function classes (such as linear dimension, VC-dimension, shattering numbers, uniform covering numbers, random covering numbers, bracketing numbers, generic chaining complexities, etc). It would be of interest to develop the bounds with precise dependence on such geometric parameters as the L2.P /-diameter of the class. Combining the bounds on expected suprema with Talagrand’s concentration inequalities yields exponential inequalities for the tail probabilities of sup-norms.