Vovk, Vladimir (2005) Competitive on-line learning with a convex loss function.
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We consider the problem of sequential decision making under uncertainty in which the loss caused by a decision depends on the following binary observation. In competitive on-line learning, the goal is to design decision algorithms that are almost as good as the best decision rules in a wide benchmark class, without making any assumptions about the way the observations are generated. However, standard algorithms in this area can only deal with finite-dimensional (often countable) benchmark classes. In this paper we give similar results for decision rules ranging over an arbitrary reproducing kernel Hilbert space. For example, it is shown that for a wide class of loss functions (including the standard square, absolute, and log loss functions) the averageloss of the master algorithm, over the first $N$ observations, does not exceed the average loss of the best decision rule with a bounded norm plus $O(N^{-1/2})$. Our proof technique is very different from the standard ones and is based on recent results about defensive forecasting. Given the probabilities produced by a defensive forecasting algorithm, which are known to be well calibrated and to have good resolution in the long run, we use the expected loss minimization principle to find a suitable decision.
This is a Submitted version This version's date is: 11/6/2005 This item is not peer reviewed
https://repository.royalholloway.ac.uk/items/94379268-b041-6d33-2625-0415144e8ef0/6/
Deposited by Research Information System (atira) on 22-Jul-2014 in Royal Holloway Research Online.Last modified on 22-Jul-2014
26 pages