Learning in the Presence of Corruption

Abstract: In supervised learning one wishes to identify a pattern present in a joint distribution $P$, of instances, label pairs, by providing a function $f$ from instances to labels that has low risk $\mathbb{E}_{P}\ell(y,f(x))$. To do so, the learner is given access to $n$ iid samples drawn from $P$. In many real world problems clean samples are not available. Rather, the learner is given access to samples from a corrupted distribution $\tilde{P}$ from which to learn, while the goal of predicting the clean pattern remains. There are many different types of corruption one can consider, and as of yet there is no general means to compare the relative ease of learning under these different corruption processes. In this paper we develop a general framework for tackling such problems as well as introducing upper and lower bounds on the risk for learning in the presence of corruption. Our ultimate goal is to be able to make informed economic decisions in regards to the acquisition of data sets. For a certain subclass of corruption processes (those that are \emph{reconstructible}) we achieve this goal in a particular sense. Our lower bounds are in terms of the coefficient of ergodicity, a simple to calculate property of stochastic matrices. Our upper bounds proceed via a generalization of the method of unbiased estimators appearing in recent work of Natarajan et al and implicit in the earlier work of Kearns.