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Convergence and Error Bounds for Universal Prediction of Nonbinary Sequences.pdf

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Convergence and Error Bounds for Universal Prediction of Nonbinary Sequences

a r X i v : c s / 0 1 0 6 0 3 6 v 1 [ c s .L G ] 1 5 J u n 2 0 0 1 Technical Report IDSIA-07-01, 26. February 2001 Convergence and Error Bounds for Universal Prediction of Nonbinary Sequences Marcus Hutter IDSIA, Galleria 2, CH-6928 Manno-Lugano, Switzerland marcus@idsia.ch 1 http://www.idsia.ch/～marcus Keywords Bayesian sequence prediction; Solomonoff induction; Kolmogorov complexity; learn- ing; universal probability; finite non-binary alphabet; convergence; error bounds; games of chance; partial and delayed prediction; classification. Abstract Solomonoff’s uncomputable universal prediction scheme ξ allows to predict the next symbol xk of a sequence x1...xk?1 for any Turing computable, but otherwise un- known, probabilistic environment μ. This scheme will be generalized to arbitrary environmental classes, which, among others, allows the construction of computable universal prediction schemes ξ. Convergence of ξ to μ in a conditional mean squared sense and with μ probability 1 is proven. It is shown that the average number of prediction errors made by the universal ξ scheme rapidly converges to those made by the best possible informed μ scheme. The schemes, theorems and proofs are given for general finite alphabet, which results in additional complications as compared to the binary case. Several extensions of the presented theory and results are outlined. They include general loss functions and bounds, games of chance, infinite alphabet, partial and delayed prediction, classification, and more active systems. 1This work was supported by SNF grant 2000-61847.00 to Ju?rgen Schmidhuber. 1 Introduction The Bayesian framework is ideally suited for studying induction problems. The probability of observing xk at time k, given past observations x1...xk?1, can be computed with Bayes’ rule if the generating probability distribution μ, from which sequences x1x2x3... are drawn, is known. The problem, however, is that in many cases one does not even have a reasonable es