  We consider the classical problem of learning rates for classes with finite
VC dimension. It is well known that fast learning rates up to
$O\left(\frac{d}{n}\right)$ are achievable by the empirical risk minimization
algorithm (ERM) if low noise or margin assumptions are satisfied. These usually
require the optimal Bayes classifier to be in the class, and it has been shown
that when this is not the case, the fast rates cannot be achieved even in the
noise free case. In this paper, we further investigate the question of the fast
rates under the misspecification, when the Bayes classifier is not in the class
(also called the agnostic setting).
  First, we consider classification with a reject option, namely Chow's reject
option model, and show that by slightly lowering the impact of hard instances,
a learning rate of order $O\left(\frac{d}{n}\log \frac{n}{d}\right)$ is always
achievable in the agnostic setting by a specific learning algorithm. Similar
results were only known under special versions of margin assumptions. We also
show that the performance of the proposed algorithm is never worse than the
performance of ERM.
  Based on those results, we derive the necessary and sufficient conditions for
classification (without a reject option) with fast rates in the agnostic
setting achievable by improper learners. This simultaneously extends the work
of Massart and N\'{e}d\'{e}lec (Ann. of Statistics, 2006), which studied this
question in the case where the Bayesian optimal rule belongs to the class, and
the work of Ben-David and Urner (COLT, 2014), which allows the misspecification
but is limited to the no noise setting. Our result also provides the first
general setup in statistical learning theory in which an improper learning
algorithm may significantly improve the learning rate for non-convex losses.
