Competing with Wild Prediction Rules

We consider the problem of on-line prediction competitive with a benchmark class of continuous but highly irregular prediction rules. It is known that if the benchmark class is a reproducing kernel Hilbert space, there exists a prediction algorithm whose average loss over the first N examples does not exceed the average loss of any prediction rule in the class plus a “regret term” of O(N−1/2). The elements of some natural benchmark classes, however, are so irregular that these classes are not Hilbert spaces. In this paper we develop Banach-space methods to construct a prediction algorithm with a regret term of O(N$^{\rm -1/{\it p}}$), where p∈(2,∞) and p–2 reflects the degree to which the benchmark class fails to be a Hilbert space.