We give improved constants for data dependent and variance sensitive confidence bounds, called empirical Bernstein bounds, and extend these inequalities to hold uniformly over classes of functions whose growth function is polynomial in the sample size n. The bounds lead us to consider sample variance penalization, a novel learning method which takes into account the empirical variance of the loss function. We give conditions under which sample variance penalization is effective. In particular, we present a bound on the excess risk incurred by the method. Using this, we argue that there are situations in which the excess risk of our method is of order 1/n, while the excess risk of empirical risk minimization is of order 1/ √ n. We show some experimental results, which confirm the theory. Finally, we discuss the potential application of our results to sample compression schemes.
Maurer and Pontil REF substitute population variance by its empirical counterpart in the excess-risk bound and introduce sample variance penalization as a regularizer.
Empirical Bernstein Bounds and Sample Variance Penalization
COLT
Mathematics Computer Science