Related papers: Uniform concentration bounds for frequencies of ra…
We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$,…
This note is concerned with weakly interacting stochastic particle systems with possibly singular pairwise interactions. In this setting, we observe a connection between entropic propagation of chaos and exponential concentration bounds for…
Concentration inequalities, a major tool in probability theory, quantify how much a random variable deviates from a certain quantity. This paper proposes a systematic convex optimization approach to studying and generating concentration…
Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the…
The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the…
We give a new proof of VC bounds where we avoid the use of symmetrization and use a shadow sample of arbitrary size. We also improve on the variance term. This results in better constants, as shown on numerical examples. Moreover our bounds…
Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.
The method to derive uniform bounds with Gaussian and Rademacher complexities is extended to the case where the sample average is replaced by a nonlinear statistic. Tight bounds are obtained for U-statistics, smoothened L-statistics and…
Vapnik--Chervonenkis' theorem is a seminal result in machine learning. It establishes sufficient conditions for empirical probabilities to converge to theoretical probabilities, uniformly over families of events. It also provides an…
We establish concentration inequalities for random dynamical systems (RDSs), assuming that the observables of interest are separately Lipschitz. Under a weak average contraction condition, we obtain deviation bounds for several random…
We derive simple concentration inequalities for bounded random vectors, which generalize Hoeffding's inequalities for bounded scalar random variables. As applications, we apply the general results to multinomial and Dirichlet distributions…
We give a concentration inequality based on the premise that random variables take values within a particular region. The concentration inequality guarantees that, for any sequence of correlated random variables, the difference between the…
This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-free to distribution-dependent, from sub-Gaussian to…
We present novel bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These are nearly optimal in various precise senses, including a kind of instance-optimality. Our data-dependent convergence guarantees…
We develop novel empirical Bernstein inequalities for the variance of bounded random variables. Our inequalities hold under constant conditional variance and mean, without further assumptions like independence or identical distribution of…
The inequality of Vapnik and Chervonenkis controls the expectation of the function by its sample average uniformly over a VC-major class of functions taking into account the size of the expectation. Using Talagrand's kernel method we prove…
Concentration inequalities quantify the deviation of a random variable from a fixed value. In spite of numerous applications, such as opinion surveys or ecological counting procedures, few concentration results are known for the setting of…
We study the Vapnik-Chervonenkis (VC) density of definable families in certain stable first-order theories. In particular we obtain uniform bounds on VC density of definable families in finite U-rank theories without the finite cover…
We provide finite-sample distribution approximations, that are uniform in the parameter, for inference in linear mixed models. Focus is on variances and covariances of random effects in cases where existing theory fails because their…
In many contemporary statistical and machine learning methods, one needs to optimize an objective function that depends on the discrepancy between two probability distributions. The discrepancy can be referred to as a metric for…