Concept drift means that the concept about which data is obtained may shift from time to time, each time after some minimum permanence. Except for this minimum permanence, the concept shifts may not have to satisfy any further requirements and may occur infinitely often. Within this work is studied to what extent it is still possible to predict or learn values for a data sequence produced by drifting concepts. Various ways to measure the quality of such predictions, including martingale betting strategies and density and frequency of correctness, are introduced and compared with one another. For each of these measures of prediction quality, for some interesting concrete classes, (nearly) optimal bounds on permanence for attaining learnability are established. The concrete classes, from which the drifting concepts are selected, include regular languages accepted by finite automata of bounded size, polynomials of bounded degree, and sequences defined by recurrence relations of bounded size. Some important, restricted cases of drifts are also studied, for example, the case where the intervals of permanence are computable. In the case where the concepts shift only

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This article investigates algorithmic learning, in the limit, of correct programs for recursive functions f from both inputÂoutput examples of f and several interesting varieties of approximate additional (algorithmic) information about f. Specifically considered, as such approximate additional information about f, are Rose's frequency computations for f and several natural generalizations from the literature, each generalization involving programs for restricted trees of recursive functions which have f as a branch. Considered as the types of trees are those with bounded variation, bounded width, and bounded rank. For the case of learning final correct programs for recursive functions, EX-learning, where the additional information involves frequency computations, an insightful and interestingly complex combinatorial characterization of learning power is presented as a function of the frequency parameters. For EX-learning (as well as for BC-learning, where a final sequence of correct programs is learned), for the cases of providing the types of additional information considered in this paper, the maximal probability is determined such that the entire class of recursive functions is learnable with that probability. ] 1997 Academic Press

The present w ork investigates Gold style algorithmic learning from inputoutput examples whereby the learner has access to oracles as additional information. Furthermore this access has to be robust, that means that a single learning algorithm has to succeed with every oracle which meets a given speci cation. The rst main result considers oracles of the same Turing degree: Robust learning with any oracle from a given degree does not achieve more than learning without any additional information.