Improved bounds about on-line learning of smooth functions of a single variable

Abstract

We consider the complexity of learning classes of smooth functions formed by bounding different norms of a function's derivative. The learning model is the generalization of the mistake-bound model to continuous-valued functions. Suppose F_q is the set of all absolutely continuous functions f from [0, 1] to R such that ∥f′∥_q ≤1, and opt(F_q, m) is the best possible bound on the worst-case sum of absolute prediction errors over sequences of m trials. We show that for all q ≥ 2, opt(F_q, m)=Θ(√log m), and that opt(F₂, m)=√log m/2 ±O(1).