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## Sequence Prediction based on Monotone Complexity

Author:Marcus Hutter (2003) Comments:15 pages Subj-class:Probability Theory; Compexity; Learning Reference:Proceedings of the 16th Annual Conference on Learning Theory (COLT 2003) pages 298-312 Report-no:IDSIA-09-03 and cs.AI/0306036 Paper:LaTeX - PostScript - PDF - Html/Gif Slides:PostScript - PDF

Keywords:Sequence prediction; Algorithmic Information Theory; Solomonoff's prior; Monotone Kolmogorov Complexity; Minimal Description Length; Convergence; Self-Optimizingness.

Abstract:This paper studies sequence prediction based on the monotone Kolmogorov complexityKm=-logm, i.e. based on universal MDL.mis extremely close to Solomonoff's priorM, the latter being an excellent predictor in deterministic as well as probabilistic environments, where performance is measured in terms of convergence of posteriors or losses. Despite this closeness toM, it is difficult to assess the prediction quality ofm, since little is known about the closeness of their posteriors, which are the important quantities for prediction. We show that for deterministic computable environments, the "posterior" and losses ofmconverge, but rapid convergence could only be shown on-sequence; the off-sequence behavior is unclear. In probabilistic environments, neither the posterior nor the losses converge, in general.

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@InProceedings{Hutter:03unimdl, author = "Marcus Hutter", title = "Sequence Prediction based on Monotone Complexity", booktitle = "Proceedings of the 16th Annual Conference on Learning Theory ({COLT-2003})", series = "Lecture Notes in Artificial Intelligence", editor = "B. Sch{\"o}lkopf and M. K. Warmuth", publisher = "Springer", address = "Berlin", pages = "506--521", year = "2003", http = "http://www.hutter1.net/ai/unimdl.htm", url = "http://arxiv.org/abs/cs.AI/0306036", ftp = "ftp://ftp.idsia.ch/pub/techrep/IDSIA-09-03.ps.gz", keywords = "Sequence prediction; Algorithmic Information Theory; Solomonoff's prior; Monotone Kolmogorov Complexity; Minimal Description Length; Convergence; Self-Optimizingness", abstract = "This paper studies sequence prediction based on the monotone Kolmogorov complexity $\Km=-\lb m$, i.e.\ based on universal MDL. $m$ is extremely close to Solomonoff's prior $M$, the latter being an excellent predictor in deterministic as well as probabilistic environments, where performance is measured in terms of convergence of posteriors or losses. Despite this closeness to $M$, it is difficult to assess the prediction quality of $m$, since little is known about the closeness of their posteriors, which are the important quantities for prediction. We show that for deterministic computable environments, the ``posterior'' and losses of $m$ converge, but rapid convergence could only be shown on-sequence; the off-sequence behavior is unclear. In probabilistic environments, neither the posterior nor the losses converge, in general.", }

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