Ikhlef Bechar
We introduce a Bernstein-type inequality which serves to uniformly control quadratic forms of gaussian variables. The latter can for example be used to derive sharp model selection criteria for linear estimation in linear regression and linear inverse problems via penalization, and we do not exclude that its scope of application can be made even broader.
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In this article we present a Bernstein inequality for sums of random variables which are defined on a spatial lattice structure. The inequality can be used to derive concentration inequalities. It can be useful to obtain consistency properties for nonparametric estimators of conditional expectation functions.
We establish a Bernstein-type inequality for a class of stochastic processes that include the classical geometrically $ϕ$-mixing processes, Rio's generalization of these processes, as well as many time-discrete dynamical systems. Modulo a logarithmic factor and some constants, our Bernstein-type inequality coincides with the classical Bernstein inequality for i.i.d.~data. We further use this new Bernstein-type inequality to derive an oracle inequality for generic regularized empirical risk minimization algorithms and data generated by such processes. Applying this oracle inequality to support …
In this article we present a Bernstein inequality for sums of random variables which are defined on a graphical network whose nodes grow at an exponential rate. The inequality can be used to derive concentration inequalities in highly-connected networks. It can be useful to obtain consistency properties for nonparametric estimators of conditional expectation functions which are derived from such networks.
We present some extensions of Bernstein's concentration inequality for random matrices. This inequality has become a useful and powerful tool for many problems in statistics, signal processing and theoretical computer science. The main feature of our bounds is that, unlike the majority of previous related results, they do not depend on the dimension $d$ of the ambient space. Instead, the dimension factor is replaced by the "effective rank" associated with the underlying distribution that is bounded from above by $d$. In particular, this makes an extension to the infinite-dimensional setting po…
We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.
Sharp Matrix Empirical Bernstein Inequalities
提出了两个针对具有有界特征值的对称随机矩阵的锐利闭式经验伯恩斯坦不等式，其偏差能紧凑适应未知方差，分别适用于独立矩阵样本均值和停时下鞅依赖的均值估计量。
We prove a new Bernstein-type inequality for the log-likelihood function of Bernoulli variables. In contrast to classical Bernstein's inequality and Hoeffding's inequality when applied to the log-likelihood, the new bound is independent of the parameters of the Bernoulli variables and therefore does not blow up as the parameters approach 0 or 1. The new inequality strengthens certain theoretical results on likelihood-based methods for community detection in networks and can be applied to other likelihood-based methods for binary data.
We use the generic chaining device proposed by Talagrand to establish exponential bounds on the deviation probability of some suprema of random processes. Then, given a random vector $ξ$ in $\R^{n}$ the components of which are independent and admit a suitable exponential moment, we deduce a deviation inequality for the squared Euclidean norm of the projection of $ξ$ onto a linear subspace of $\R^{n}$. Finally, we provide an application of such an inequality to statistics, performing model selection in the regression setting when the errors are possibly non-Gaussian and the collection of models…
A concentration result for quadratic form of independent subgaussian random variables is derived. If the moments of the random variables satisfy a "Bernstein condition", then the variance term of the Hanson-Wright inequality can be improved. The Bernstein condition is satisfied, for instance, by all log-concave subgaussian distributions.
Bernstein-type and Bennett-type inequalities for unbounded matrix martingales
本文研究无界矩阵值鞅的集中性，在 $ψ_α$-Orlicz 范数有界条件下建立 Bernstein 型与 Bennett 型不等式，并以有效秩改进维度依赖。
A Bernstein-type Inequality for High Dimensional Linear Processes with Applications to Robust Estimation of Time Series Regressions
针对超高维、肥尾且序列相关的时间序列数据，本文提出新的 Bernstein 型不等式，并用于稳健时间序列回归和肥尾 VAR 估计，在温和条件下允许维数随样本量指数增长并保持一致性。
We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $π$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{é} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernst…
Bernstein's inequalities for general Markov chains
本文为一般马尔可夫链的函数建立伯恩斯坦不等式，通过分析扰动转移核算子范数并结合凸分析推导，将经典独立情形纳入特例，应用于MCMC积分估计等问题获更紧偏差界。
We provide a general theoretical framework to derive Bernstein-von Mises theorems for matrix functionals. The conditions on functionals and priors are explicit and easy to check. Results are obtained for various functionals including entries of covariance matrix, entries of precision matrix, quadratic forms, log-determinant, eigenvalues in the Bayesian Gaussian covariance/precision matrix estimation setting, as well as for Bayesian linear and quadratic discriminant analysis.
Recent development in high-dimensional statistical inference has necessitated concentration inequalities for a broader range of random variables. We focus on sub-Weibull random variables, which extend sub-Gaussian or sub-exponential random variables to allow heavy-tailed distributions. This paper presents concentration inequalities for independent sub-Weibull random variables with finite Generalized Bernstein-Orlicz norms, providing generalized Bernstein's inequalities and Rosenthal-type moment bounds. The tightness of the proposed bounds is shown through lower bounds of the concentration ineq…
Bernstein-type dimension-free concentration for self-normalised martingales
本文提出了一种无维的伯恩斯坦型尾界不等式，用于自归一化鞅，其中归一化使用可预测的二次变分，半径取决于观测协方差的信息增益。应用包括为具有自适应选择的希尔伯特值协变量的逻辑回归提供椭球置信序列，以及为希尔伯特臂逻辑带宽提供实例自适应的后悔界。
Let $\pa{X_{t}}_{t\in T}$ be a family of real-valued centered random variables indexed by a countable set $T$. In the first part of this paper, we establish exponential bounds for the deviation probabilities of the supremum $Z=\sup_{t\in T}X_{t}$ by using the generic chaining device introduced in Talagrand (2005). Compared to concentration-type inequalities, these bounds offer the advantage to hold under weaker conditions on the family $\pa{X_{t}}_{t\in T}$. The second part of the paper is oriented towards statistics. We consider the regression setting $Y=f+\eps$ where $f$ is an unknown vector…
We present Hoeffding-type and Bernstein-type inequalities for right-censored data. The inequalities bound the difference between an inverse of the probability of censoring weighting (IPCW) estimator and its expectation. We first discuss the asymptotic properties of the estimator and provide conditions for its efficiency. We present standard, data dependent, and uniform Hoeffding-type inequalities. We then present a Bernstein-type inequality. Finally, we show how to apply these inequalities in an empirical risk minimization setting.
Sharp Empirical Bernstein Bounds for the Variance of Bounded Random Variables
针对有界随机变量方差，提出在常数条件方差和均值下无需独立同分布假设的尖锐经验伯恩斯坦不等式，适用于批量与序贯设定，渐近最优且可推广至希尔伯特空间。
Policy learning "without" overlap: Pessimism and generalized empirical Bernstein's inequality
本文提出悲观策略学习（PPL）算法，通过优化置信下界而非点估计，在不依赖一致重叠假设的条件下实现离线策略学习，并利用广义经验伯恩斯坦不等式为自适应收集数据建立次优性上界，理论和实验均验证了其有效性。
Empirical Bernstein in smooth Banach spaces
针对有界向量值随机变量，提出一种利用方差经验估计量的新经验Bernstein不等式，适用于2-光滑可分Banach空间，渐近达到与Bernstein不等式相同的置信集宽度。
Bernstein Concentration Inequalities for Tensors via Einstein Products
本文提出了一种将伯恩斯坦矩阵集中不等式推广到一般阶随机张量的方法，通过爱因斯坦乘积建立矩阵与张量之间的强联系，从而利用已有的矩阵结果。
Semiparametric Bernstein-von Mises theorems from Stein's method
本文提出基于 Stein 方法与信息几何框架的半参数 Bernstein-von Mises 定理证明新策略，给出非渐近上界，并应用于高斯白噪声与直方图模型中的各类泛函，通过泛函修正消除后验偏差。
Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement beyond the unstructured Rosenthal inequality of quadratic forms when dimension of the vectors increases without bound. Applications to test the block diagonal structures and detect the sparsity in the high-dimensional covariance matrix are presented.
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