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

Author: Philip M. Long.

Source: Theoretical Computer Science Vol. 241, No. 1-2, 2000, 25-35.

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) = (), and that opt(F2,m) ()/2 + O(1), matching a known lower bound of ()/2 - O(1) to within an additive constant.

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