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.

本研究提出了改进的PCS算法ILESS，它可以在不需要任何假设的情况下实现快速拒绝率，并证明了Hanneke不一致系数的多对数界与任何PCS学习算法（如ILESS）存在快速拒绝率以及新的基于不一致性的主动学习算法ActiveiLESS的指数加速之间具有等价性。

不可知选择性分类主动学习与不一致系数的关系