On agnostic learning with {0,*,1}-valued and real-valued hypotheses

We consider the problem of classification using a variant of the agnostic learning model in which the algorithm's hypothesis is evaluated by comparison with hypotheses that do not classify all possible instances. Such hypotheses axe formalized as functions from the instance space X to (0, *, 1), where * is interpreted as "don't know". We provide a characterization of the sets of (0, *, 1)-valued functions that are learnable in this setting. Using a similar analysis, we improve on sufficient conditions for a class of real-valued functions to be agnostically learnable with a particular relative accuracy; in particular, we improve by a factor of two the scale at which scale-sensitive dimensions must be finite in order to imply learnability.