Universal Prediction of Individual Sequences

Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. In this report, the problem of predicting the next outcome of an individual binary sequence using finite memory [1], is considered. The problem of sequential prediction, deprived of any probabilistic assumption, is deeply connected with the information-theoretic problem of compressing an individual data sequence. A pioneering research in this field was carried out by Ziv [2] and Lempel and Ziv [3], who solved the problem of compressing an individual data sequence almost as well as the best finite-state automaton. As shown by Feder, Merhav, and Gutman [1], the LempelZiv compressor can be used as a randomized forecaster [4, Chap 4] (for the absolute loss [4, Chap 8]) with a vanishing per-round regret against the class of all finite-state experts, a surprising result considering the rich structure of this class. In addition, Feder, Merhav, and Gutman devise, for the same expert class, a forecaster with a convergence rate better than the rate provable for the LempelZiv forecaster (see also [5] for further results along these lines). Another important contribution of [1] is introducing the Markov experts as a class of predictors. We now make comparisons of the problem with other prediction problems in fields of gambling, compression and complexity.