Sequential randomized prediction of an arbitrary binary sequence is
investigated. No assumption is made on the mechanism of generating the bit
sequence. The goal of the predictor is to minimize its relative loss (or
regret), that is, to make almost as few mistakes as the best
“expert” in a fixed, possibly infinite, set of experts. We point
out a surprising connection between this prediction problem and empirical
process theory. First, in the special case of static (memoryless) expert, we
completely characterize the minimax regret in terms of the maximum of an
associated Rademacher process. Then we show general upper and lower bounds on
the minimax regret in terms of the geometry of the class of experts. As main
examples, we determine the exact order of magnitude of the minimax regret for
the class of autoregressive linear predictors and for the class of Markov
experts.
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