A Chain Rule for the Expected Suprema of Bernoulli Processes

Sharp bounds on the suprema of Bernoulli processes are ubiquitous in applications of probability theory. For example, in statistical learning theory, they arise as so-called Rademacher complexities that quantify the effective "size" of a function class in the context of a given learning task [1]. To evaluate the generalization error bound of a learning algorithm, one usually needs to obtain a good estimate on the (empirical) Rademacher complexity of a suitable function class. In many scenarios, the function class at hand is composite, and separate assumptions are imposed on the constituent classes forming the composition. Thus, it is desirable to obtain a bound that takes into account the properties of the function classes in the composition.