Abstract

This thesis is devoted to on-line learning. An on-line learning algorithm receives elements of a sequence one by one and tries to predict every element before it arrives. The performance of such an algorithm is measured by the discrepancies between its predictions and the outcomes. Discrepancies over several trials sum up to total cumulative loss. The starting point is the Aggregating Algorithm (AA). This algorithm deals with the problem of prediction with expert advice. In this thesis the existing theory of the AA is surveyed and some further properties are established. The concept of predictive complexity introduced by V. Vovk is a natural development of the theory of prediction with expert advice. Predictive complexity bounds the loss of every algorithm from below. Generally this bound does not correspond to the loss of an algorithm but it is optimal ‘in the limit’. Thus it is an intrinsic measure of ‘learnability ’ of a finite sequence. It is similar to Kolmogorov complexity, which is a measure of the descriptive complexity of a string independent of a particular description method. Different approaches to optimality give

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