The classical binary classification problem is investigated when it is known in advance that the posterior probability function (or regression function) belongs to some class of functions. We introduce and analyze a method which effectively exploits this knowledge. The method is based on...

Given $n$ independent replicates of a jointly distributed pair $(X,Y) \in {\cal R}^d \times {\cal R}$, we wish to select from a fixed sequence of model classes ${\cal F}_1, {\cal F}_2, \ldots$ a deterministic prediction rule $f: {\cal R}^d \to {\cal R}$ whose risk is small. We investigate the...

We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a stationary and ergodic random process then the average...

We show that every finite N-player normal form game possesses a correlated equilibrium with a precise lower bound on the number of outcomes to which it assigns zero probability. In particular, the largest games with a unique fully supported correlated equilibrium are two-player games; moreover,...

We obtain minimax lower bounds on the regret for the classical two--armed bandit problem. We provide a finite--sample minimax version of the well--known log $n$ asymptotic lower bound of Lai and Robbins. Also, in contrast to the log $n$ asymptotic results on the regret, we show that the minimax...

We consider adaptive sequential lossy coding of bounded individual sequences when the performance is measured by the sequentially accumulated mean squared distortion. The encoder and the decoder are connected via a noiseless channel of capacity $R$ and both are assumed to have zero delay. No...

We obtain minimax lower and upper bounds for the expected distortion redundancy of empirically designed vector quantizers. We show that the mean squared distortion of a vector quantizer designed from $n$ i.i.d. data points using any design algorithm is at least $\Omega (n^{-1/2})$ away from the...

We present simple procedures for the prediction of a real valued sequence. The algorithms are based on a combination of several simple predictors. We show that if the sequence is a realization of a bounded stationary and ergodic random process then the average of squared errors converges, almost...

Minimax lower bounds for concept learning state, for example, that for each sample size $n$ and learning rule $g_n$, there exists a distribution of the observation $X$ and a concept $C$ to be learnt such that the expected error of $g_n$ is at least a constant times $V/n$, where $V$ is the VC...

We continue the development of a method for the selection of a bandwidth or a number of design parameters in density estimation. We provide explicit non-asymptotic density-free inequalities that relate the $L_1$ error of the selected estimate with that of the best possible estimate, and study in...