Learning theory for conditional risk minimization

In this work we study the learnability of stochastic processes with respect to the conditional risk, i.e. the existence of a learning algorithm that improves its next-step performance with the amount of observed data. We introduce a notion of pairwise discrepancy between conditional distributions at different times steps and show how certain properties of these discrepancies can be used to construct a successful learning algorithm. Our main results are two theorems that establish criteria for learnability for many classes of stochastic processes, including all special cases studied previously in the literature.