  Approximate learning machines have become popular in the era of small
devices, including quantised, factorised, hashed, or otherwise compressed
predictors, and the quest to explain and guarantee good generalisation
abilities for such methods has just begun. In this paper we study the role of
approximability in learning, both in the full precision and the approximated
settings of the predictor that is learned from the data, through a notion of
sensitivity of predictors to the action of the approximation operator at hand.
We prove upper bounds on the generalisation of such predictors, yielding the
following main findings, for any PAC-learnable class and any given
approximation operator. 1) We show that under mild conditions, approximable
target concepts are learnable from a smaller labelled sample, provided
sufficient unlabelled data. 2) We give algorithms that guarantee a good
predictor whose approximation also enjoys the same generalisation guarantees.
3) We highlight natural examples of structure in the class of sensitivities,
which reduce, and possibly even eliminate the otherwise abundant requirement of
additional unlabelled data, and henceforth shed new light onto what makes one
problem instance easier to learn than another. These results embed the scope of
modern model compression approaches into the general goal of statistical
learning theory, which in return suggests appropriate algorithms through
minimising uniform bounds.
