Local Rademacher complexity bounds based on covering numbers

This paper provides a general result on controlling local Rademacher complexities, which captures in an elegant form to relate complexities with constraints on expected norms to the corresponding ones with constraints on empirical norms. This result is convenient to apply and could yield refined local Rademacher complexity bounds for function classes satisfying general entropy conditions. We demonstrate the power of our complexity bounds by applying them to simplify the derivation of effective generalization error bounds.