  A selective classifier (f,g) comprises a classification function f and a
binary selection function g, which determines if the classifier abstains from
prediction, or uses f to predict. The classifier is called
pointwise-competitive if it classifies each point identically to the best
classifier in hindsight (from the same class), whenever it does not abstain.
The quality of such a classifier is quantified by its rejection mass, defined
to be the probability mass of the points it rejects. A "fast" rejection rate is
achieved if the rejection mass is bounded from above by O(1/m) where m is the
number of labeled examples used to train the classifier (and O hides
logarithmic factors). Pointwise-competitive selective (PCS) classifiers are
intimately related to disagreement-based active learning and it is known that
in the realizable case, a fast rejection rate of a known PCS algorithm (called
Consistent Selective Strategy) is equivalent to an exponential speedup of the
well-known CAL active algorithm.
  We focus on the agnostic setting, for which there is a known algorithm called
LESS that learns a PCS classifier and achieves a fast rejection rate (depending
on Hanneke's disagreement coefficient) under strong assumptions. We present an
improved PCS learning algorithm called ILESS for which we show a fast rate
(depending on Hanneke's disagreement coefficient) without any assumptions. Our
rejection bound smoothly interpolates the realizable and agnostic settings. The
main result of this paper is an equivalence between the following three
entities: (i) the existence of a fast rejection rate for any PCS learning
algorithm (such as ILESS); (ii) a poly-logarithmic bound for Hanneke's
disagreement coefficient; and (iii) an exponential speedup for a new
disagreement-based active learner called ActiveiLESS.
