On-line regression competitive with reproducing kernel Hilbert spaces

We consider the problem of on-line prediction of real-valued labels, assumed bounded in absolute value by a known constant, of new objects from known labeled objects. The prediction algorithm's performance is measured by the squared deviation of the predictions from the actual labels. No stochastic assumptions are made about the way the labels and objects are generated. Instead, we are given a benchmark class of prediction rules some of which are hoped to produce good predictions. We show that for a wide range of infinite-dimensional benchmark classes one can construct a prediction algorithm whose cumulative loss over the first N examples does not exceed the cumulative loss of any prediction rule in the class plus O(sqrt(N)); the main differences from the known results are that we do not impose any upper bound on the norm of the considered prediction rules and that we achieve an optimal leading term in the excess loss of our algorithm. If the benchmark class is "universal" (dense in the class of continuous functions on each compact set), this provides an on-line non-stochastic analogue of universally consistent prediction in non-parametric statistics. We use two proof techniques: one is based on the Aggregating Algorithm and the other on the recently developed method of defensive forecasting.Comment: 37 pages, 1 figur

Competitive on-line learning with a convex loss function

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.Comment: 26 page