On Bayes Methods for On-Line Boolean Prediction

Abstract

We examine a general Bayesian framework for constructing on-line prediction algorithms in the experts setting. These algorithms predict the bits of an unknown Boolean sequence using the advice of a finite set of experts. In this framework we use probabilistic assumptions on the unknown sequence to motivate prediction strategies. However, the relative bounds that we prove on the number of prediction mistakes made by these strategies hold for any sequence. The Bayesian framework provides a unified derivation and analysis of previously known prediction strategies, such as the Weighted Majority and Binomial Weighting algorithms. Furthermore, it provides a principled way of automatically adapting the parameters of Weighted Majority to the sequence, in contrast to previous ad hoc doubling techniques. Finally, we discuss the generalization of our methods to algorithms making randomized predictions. Introduction.A fundamental problem in learning theory is to predict the bits of an unknown Boolean sequence. The problem is uninteresting when the algorithm is required to minimize its worst-case number of mistakes over all sequences, as no algorithm can do better than random guessing. A richer problem results if the algorithm is given a (finite) set of models and the sequence is reasonably close to that generated by one of the models. Now interesting "relative" mistake bounds that depend on the distance between the unknown Boolean sequence and the closest model can be proven. This is sometimes referred to as the "experts" setting, since the models can be viewed as "experts" providing "advice" to the algorithm. Variants and extensions of this experts setting have been extensively studied by Littlestone and Warmuth [10], Vovk [12], CesaBianchi et al.[2], [3], Haussler et al. [6], and others in the area of computational learning theory. Here we use a Bayesian approach to derive prediction algorithms with good performance in the experts settings. A crucial aspect of this work is that although the algorithms are derived by making probabilistic assumptions about the generation of the sequence to be predicted, they are analyzed in the adversarial experts setting.In this experts setting, a "master algorithm" attempts to predict, one by one, the bits of an unknown sequence. Before predicting each bit, the master is allowed to listen to the "advice" provided by a pool of N experts. After each bit is revealed, the master

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“…This bound improves on a result obtained by Cesa-Bianchi, Helmbold, and Panizza (1998), which was essentially the best robust bound to date. In the case of randomized binary classification we adapt the p-norm algorithm to learn with respect to an infinite pool of experts.…”

“…Bounds for algorithms that do not need K are provided by Cesa-Bianchi et al (1997) and Cesa-Bianchi, Helmbold, and Panizza (1998). In the first paper the authors exploit a sophisticated doubling trick to repeatedly re-estimate the best η for Weighted Majority.…”

“…A competing algorithm for binary classification tasks that also does not need parameters to be set is a so-called apobayesian algorithm described by Littlestone and Mesterharm (1997)-it is closely related to the algorithm studied by Cesa-Bianchi, Helmbold, and Panizza (1998). That algorithm functions roughly as a variant of Winnow that learns its parameters, though the parameters do not appear as such in the algorithm.…”

General convergence results for linear discriminant updates

“…Aside from the regression case, another different but seemingly related learning model is the so-called expert case (e.g., Littlestone & Warmuth, 1989;Vovk, 1990;Cesa-Bianchi et al, 1997;Cesa-Bianchi, Helmbold & Panizza, 1996), where there is only a single relevant attribute (that is, perfect classification can be accomplished by a discriminant with a single non-zero weight). Algorithms for learning general linear discriminants can be of course still be applied in this case, so it would be interesting to see if our analysis adds any additional insight.…”

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“…For further information on the theory of prediction with expert advice, the reader can also consult, e.g., (Auer & Long, 1994;Cesa-Bianchi, Helmbold, & Panizza, 1996;Feder, Merhav, & Gutman, 1992;Yamanishi, 1995).…”