How many strings are easy to predict?

Authors:
Yuri Kalnishkan;Vladimir Vovk;Michael V. Vyugin
Affiliations:
Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK
Venue:
Information and Computation
Year:
2005

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Abstract

It is well known in the theory of Kolmogorov complexity that most strings cannot be compressed; more precisely, only exponentially few (Θ(2n-m)) binary strings of length n can be compressed by m bits. This paper extends the 'incompressibility' property of Kolmogorov complexity to the 'unpredictability' property of predictive complexity. The 'unpredictability' property states that predictive complexity (defined as the loss suffered by a universal prediction algorithm working infinitely long) of most strings is close to a trivial upper bound (the loss suffered by a trivial minimax constant prediction strategy). We show that only exponentially few strings can be successfully predicted and find the base of the exponent.