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We derive sharper probabilistic concentration bounds for the Monte CarloEmpirical Rademacher Averages (MCERA), which are proved through recent resultson the concentration of self-bounding functions. Our novel bounds arecharacterized by convergence rates that depend on data-dependent characteristicquantities of the set of functions under consideration, such as the empiricalwimpy variance, an essential improvement w.r.t. standard bounds based on themethods of bounded differences. For this reason, our new results are applicableto yield sharper bounds to (Local) Rademacher Averages. We also derive improvednovel variance-dependent bounds for the special case where only one vector ofRademacher random variables is used to compute the MCERA, through theapplication of Bousquet's inequality and novel data-dependent bounds to thewimpy variance. Then, we leverage the framework of self-bounding functions toderive novel probabilistic bounds to the supremum deviations, that may be ofindependent interest.

Sharper convergence bounds of Monte Carlo Rademacher Averages through Self-Bounding functions