A Chain Rule for the Expected Suprema of Bernoulli Processes
This work addresses theoretical challenges in probability and machine learning for researchers, offering an incremental extension from Gaussian to Bernoulli processes.
The paper tackles the problem of bounding the expected supremum of Bernoulli processes indexed by transformed sets, extending prior Gaussian process results to Bernoulli processes. It provides an upper bound based on properties of the index set and function class, leveraging recent boundedness results.
We obtain an upper bound on the expected supremum of a Bernoulli process indexed by the image of an index set under a uniformly Lipschitz function class in terms of properties of the index set and the function class, extending an earlier result of Maurer for Gaussian processes. The proof makes essential use of recent results of Bednorz and Latala on the boundedness of Bernoulli processes.