  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.
