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 Fq is the set of all absolutely continuous functions f from [0,1] to R such that ∥f′∥q≤1, and opt(Fq,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(Fq,m) = Θ(√logm), and that opt(F2,m)≤(√log2 m)/2 + O(1), matching a known lower bound of (√log2 m)/2 - O(1) to within an additive constant. © 2000 Elsevier Science B.V. All rights reserved.