Fractal dimension and logarithmic loss unpredictability

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.

[1]  Neri Merhav,et al.  Universal Prediction , 1998, IEEE Trans. Inf. Theory.

[2]  Meir Feder,et al.  Gambling using a finite state machine , 1991, IEEE Trans. Inf. Theory.

[3]  Ludwig Staiger,et al.  A Tight Upper Bound on Kolmogorov Complexity and Uniformly Optimal Prediction , 1998, Theory of Computing Systems.

[4]  Boris Ryabko,et al.  The Complexity and Effectiveness of Prediction Algorithms , 1994, J. Complex..

[5]  Jack H. Lutz,et al.  Dimension in complexity classes , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[6]  Neri Merhav,et al.  Universal prediction of individual sequences , 1992, IEEE Trans. Inf. Theory.

[7]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[8]  Lance Fortnow,et al.  Prediction and dimension , 2002, J. Comput. Syst. Sci..

[9]  Boris Ryabko.  The nonprobabilistic approach to learning the best prediction , 2001, Electron. Trans. Artif. Intell..

[10]  Claus-Peter Schnorr,et al.  Endliche Automaten und Zufallsfolgen , 1972, Acta Informatica.

[11]  Jack H. Lutz,et al.  Finite-state dimension , 2001, Theor. Comput. Sci..

[12]  F. Hausdorff.  Dimension und äußeres Maß , 1918 .