Empirical Bernstein Bounds and Sample Variance Penalization

TL;DR

This paper introduces improved empirical Bernstein bounds and a novel sample variance penalization method that adaptively accounts for variance, leading to faster convergence rates in certain learning scenarios.

Contribution

It provides tighter variance-sensitive confidence bounds and proposes a new learning method with theoretical guarantees on excess risk reduction.

Findings

Sample variance penalization can achieve excess risk of order 1/n.

Empirical Bernstein bounds are extended to function classes with polynomial growth.

Experimental results support the theoretical advantages of the proposed method.

Abstract

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 functionswhose 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/sqrt/{n}. We show some experimental results, which confirm the theory. Finally, we discuss the potential application of our results to sample compression schemes.