We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.
We address the problem of non-parametric density estimation under the additional constraint that only privatised data are allowed to be published and available for inference. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local α-differential privacy and provide a lower bound on the rate of convergence over Besov spaces B s pq under mean integrated L r -risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. In order to fulfil the privacy requirement, we suggest adding suitably scaled Laplace noise to empirical wavelet coefficients. Upper bounds within (at most) a logarithmic factor are derived under the assumption that α stays bounded as n increases: A linear but non-adaptive wavelet estimator is shown to attain the lower bound whenever p ≥ r but provides a slower rate of convergence otherwise. An adaptive non-linear wavelet estimator with appropriately chosen smoothing parameters and thresholding is shown to attain the lower bound within a logarithmic factor for all cases.Date: March 6, 2019. 2010 Mathematics Subject Classification. 62G07 (primary), and 62G20 (secondary).
58p.We consider the estimation of a bounded regression function with nonparametric heteroscedastic noise. We are interested by the true and empirical excess risks of the least-squares estimator on a nite-dimensional vector space. For these quantities, we give upper and lower bounds in probability that are optimal at the rst order. Moreover, these bounds show the equivalence between the true and empirical excess risks when, among other things, the least-squares estimator is consistent in sup-norm towards the projection of the regression function onto the considered model. Consistency in sup-norm is then proved for suitable histogram models and more general models of piecewise polynomials that are endowed with a localized basis structure
We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a weighted Poincaré inequality with the weight being the Stein kernel, that indeed exists and is unique in this case. Furthermore, we prove weighted log-Sobolev and asymmetric Brascamp-Lieb type inequalities related to Stein kernels. We also show that existence of a uniformly bounded Stein kernel is sufficient to ensure a positive Cheeger isoperimetric constant. Then we derive new concentration inequalities. In particular, we prove generalized Mills' type inequalities when a Stein kernel is uniformly bounded and sub-gamma concentration for Lipschitz functions of a variable with a sub-linear Stein kernel. When some exponential moments are finite, a general concentration inequality is then expressed in terms of Legendre-Fenchel transform of the Laplace transform of the Stein kernel. Along the way, we prove a general lemma for bounding the Laplace transform of a random variable, that should be useful in many other contexts when deriving concentration inequalities. Finally, we provide density and tail formulas as well as tail bounds, generalizing previous results that where obtained in the context of Malliavin calculus.
In the first part of this paper, we show that the small-ball condition, recently introduced by [Men15], may behave poorly for important classes of localized functions such as wavelets, piecewise polynomials or trigonometric polynomials, in particular leading to suboptimal estimates of the rate of convergence of ERM for the linear aggregation problem. In a second part, we recover optimal rates of covergence for the excess risk of ERM when the dictionary is made of trigonometric functions. Considering the bounded case, we derive the concentration of the excess risk around a single point, which is an information far more precise than the rate of convergence. In the general setting of a L 2 noise, we finally refine the small ball argument by rightly selecting the directions we are looking at, in such a way that we obtain optimal rates of aggregation for the Fourier dictionary.Keywords: empirical risk minimization, linear aggregation, small-ball property, concentration inequality, empirical process theory. * Research partly supported by the french Agence Nationale de la Recherche (ANR 2011 BS01 010 01 projet Calibration). Recently, [LM16b] have shown that ERM is suboptimal for the linear aggregation problem in general, in the sense that there exist a dictionary S and a pair (X, Y ) of random variables for which the rate of ERM (drastically) deteriorates, even in the case where the response variable Y and the dictionary are uniformly bounded.On the positive side, [LM16b] also made a breakthrough by showing that if a so-called small-ball condition is achieved with absolute constants, uniformly over
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