We present a new general concentration-of-measure inequality and illustrate its power by applications in random combinatorics. The results find direct applications in some problems of learning theory.

We introduce a simple new hypothesis testing procedure, which, based on an independent sample drawn from a certain density, detects which of $k$ nominal densities is the true density is closest to, under the total variation (L_{1}) distance. We obtain a density-free uniform exponential bound for...

Let a class $\F$ of densities be given. We draw an i.i.d.\ sample from a density $f$ which may or may not be in $\F$. After every $n$, one must make a guess whether $f \in \F$ or not. A class is almost surely testable if there exists such a testing sequence such that for any $f$, we make...

We investigate on-line prediction of individual sequences. Given a class of predictors, the goal is to predict as well as the best predictor in the class, where the loss is measured by the self information (logarithmic) loss function. The excess loss (regret) is closely related to the redundancy...

We derive a new inequality for uniform deviations of averages from their means. The inequality is a common generalization of previous results of Vapnik and Chervonenkis (1974) and Pollard (1986). Using the new inequality we obtain tight bounds for empirical loss minimization learning.

For the standard kernel density estimate, it is known that one can tune the bandwidth such that the expected L1 error is within a constant factor of the optimal L1 error (obtained when one is allowed to choose the bandwidth with knowledge of the density). In this paper, we pose the same problem...

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...