Note: fractal dimension and logarithmic loss unpredictability

Authors:
John M. Hitchcock
Affiliations:
Department of Computer Science, Iowa State University, 226 Atanasoff Hall, Ames IA
Venue:
Theoretical Computer Science
Year:
2003

Citing 5
Cited 18

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Abstract

We show that the Hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet. Using computable, feasible, and finite-state predictors, this equivalence also holds for the computable, feasible, and finite-state dimensions. Combining this with recent results of Fortnow and Lutz (Proc. 15th Ann. Conf. on Comput. Learning Theory (2002) 380), we have a tight relationship between prediction with respect to logarithmic loss and prediction with respect to absolute loss.