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Competitive on-line learning with a convex loss function (2005)
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BibTeX
@TECHREPORT{Vovk05competitiveon-line,
author = {Vladimir Vovk},
title = {Competitive on-line learning with a convex loss function},
institution = {},
year = {2005}
}
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Abstract
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 average loss 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. 1
Keyphrases
competitive on-line learning decision rule convex loss function average loss proof technique loss function wide benchmark class suitable decision defensive forecasting algorithm similar result standard one first observation sequential decision good resolution benchmark class standard algorithm standard square defensive forecasting bounded norm following binary observation long run wide class expected loss minimization principle master algorithm recent result log loss function arbitrary reproducing kernel hilbert space
