Agnostic Boosting
We extend the boosting paradigm to the realistic setting of agnostic learning, that is, to a setting where the training sample is generated by an arbitrary (unknown) probability distribution over examples and labels. We define a β-weak agnostic learner with respect to a hypothesis class F as follows: given a distribution P it outputs some hypothesis h ∈ F whose error is at most erP (F) + β, where erP (F) is the minimal error of an hypothesis from F under the distribution P (note that for some distributions the bound may exceed a half). 
 
We show a boosting algorithm that using the weak agnostic learner computes a hypothesis whose error is at most max{c1(β)er(F)c2(β), Ɛ}, in time polynomial in 1/Ɛ. While this generalization guarantee is significantly weaker than the one resulting from the known PAC boosting algorithms, one should note that the assumption required for β-weak agnostic learner is much weaker. In fact, an important virtue of the notion of weak agnostic learning is that in many cases such learning is achieved by efficient algorithms.