Andreas Maurer, Massimiliano Pontil
We prove concentration inequalities for functions of independent random variables {under} sub-gaussian and sub-exponential conditions. The utility of the inequalities is demonstrated by an extension of the now classical method of Rademacher complexities to Lipschitz function classes and unbounded sub-exponential distribution.
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On the Bernstein-Hoeffding method
我们证明了Bernstein-Hoeffding方法可以用于更大的一类广义矩。这一类包括指数的时刻，其属性发挥了关键作用，在证明一个著名的不等式的瓦西里Hoeffding，为独立和有界的随机变量，其平均值被假定为已知的总和。因此，我们可以推广和改进这个不等式。我们表明，Hoeffding的界限是最优的，在更广泛的意义。我们的方法允许获得“失踪”的因素Hoeffding的不平等，其存在的动机是由中心极限定理。后来的结果是一个相当弱的版本的定理，是由于米歇尔Talagrand。利用伯恩斯坦多项式理论的思想，证明了Bernstein-Hoeffding方法适用于具有随机变量高阶矩信息的情形.此外，我们考虑的条件分布的随机变量的额外信息下的方法的性能，最后，我们表明，该方法减少到马尔可夫不等式时，采用非负和无界的随机变量。
Hoeffding and Bernstein inequalities for weighted sums of exchangeable random variables
本文为可交换随机变量的加权和建立了Hoeffding和Bernstein型集中不等式，统一了独立同分布与可交换两种情形。
We consider Markov chain with spectral gap in $L^2$ space. Assume that $f$ is a bounded function. Then the probabilities of large deviations of average along trajectory satisfy Hoeffding's-type inequalities. These bounds depend only on the stationary mean, spectral gap and the end-points of support of $f$.
It is well known that Hoeffding's inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding's inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz \cite{r1}. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding's inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding's inequality and its refinement one by Hertz…
Concentration inequalities using approximate zero bias couplings with applications to Hoeffding's statistic under the Ewens distribution
本文利用近似零偏耦合技术，推导了Hoeffding统计量在Ewens分布下的集中不等式，并通过模拟验证了其在θ≤1时的有效性。
Concentration inequalities using higher moments information
针对已知随机变量前 p 阶矩的情形，本文提出更紧且具闭式形式的推广 Hoeffding 与 Bennett 不等式，并通过矩母函数新界及凸性分析加以证明。
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.
A short operator proof of Hoeffding inequalities for Markov chains
本文用两个关键观察（算子分解与秩一扰动）给出马尔可夫链加性泛函及随机矩阵Hoeffding不等式的简短算子证明，恢复并改进现有结果。
In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M_n=X_1+... +X_n of bounded independent random variables X_j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) …
In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).
Bernstein type's concentration inequalities for symmetric Markov processes
利用\cite{GLWY}中引入的迁移信息不等式方法，建立了经验平均$\frac 1t \int_0^t g(X_s)ds$的Bernstein型浓度不等式，其中$g$是对称马尔可夫过程$(X_t)$的无界可观测量.提出了三种方法：函数不等式方法;李雅普诺夫函数方法;以及通过泊松方程解的Lipschitz范数的方法。几个应用程序和例子进行了研究。
For self-normalized martingales with conditionally symmetric differences, de la Peña [A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27, No.1, 537-564] established the Gaussian type exponential inequalities. Bercu and Touati [Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18: 1848-1869] extended de la Peña's inequalities to martingales with differences heavy on left. In this paper, we establish Bernstein type exponential inequalities for self-normalized martingales with differences bounded from below. Moreove…
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。
Concentration Inequalities for Bounded Random Vectors
我们得到了有界随机向量的简单的集中不等式，推广了Hoeffding关于有界纯量随机变量的不等式.作为应用，我们将一般结果应用于多项式和Dirichlet分布，得到了多元浓度不等式。
Bernstein-type Inequalities for Markov Chains and Markov Processes: A Simple and Robust Proof
该研究用初等方法提出了适用于一般非可逆离散时间马尔可夫链的新Bernstein型偏差不等式，仅需满足迭代Poincaré不等式，鲁棒性更强且方法可推广至连续时间马尔可夫过程。
We consider the three dimensional array $\mathcal{A} = \{a_{i,j,k}\}_{1\le i,j,k \le n}$, with $a_{i,j,k} \in [0,1]$, and the two random statistics $T_{1}:= \sum_{i=1}^n \sum_{j=1}^n a_{i,j,σ(i)}$ and $T_{2}:= \sum_{i=1}^{n} a_{i,σ(i),π(i)}$, where $σ$ and $π$ are chosen independently from the set of permutations of $\{1,2,\ldots,n \}.$ These can be viewed as natural three dimensional generalizations of the statistic $T_{3}=\sum_{i=1}^{n} a_{i,σ(i)}$, considered by Hoeffding \cite{Hoe51}. Here we give Bernstein type concentration inequalities for $T_{1}$ and $T_{2}$ by extending the argument f…
Efficient Concentration with Gaussian Approximation
针对有界随机变量，提出具有渐近最优宽度、有限样本有效性和次高斯衰减的可计算集中不等式，通过零偏耦合和Stein交换对方法紧界非均匀Kolmogorov和Wasserstein距离，构建高效置信区间和经验Berry-Esseen界。
We consider the stochastic integrals of multivariate point processes and study their concentration phenomena. In particular, we obtain a Bernstein type of concentration inequality through Doléans-Dade exponential formula and a uniform exponential inequality using a generic chaining argument. As applications, we obtain a upper bound for a sequence of discrete time martingales indexed by a class of functionals, and so derive the rate of convergence for nonparametric maximum likelihood estimators, which is an improvement of earlier work of van de Geer.
Strong Converse Inequalities for Bernstein Polynomials with Explicit Asymptotic Constants
本文通过克拉夫丘克多项式表示、逆矩估计、中心矩与全变差距离的精确估计以及迭代概率表示，建立了伯恩斯坦多项式具有显式渐近常数的强逆不等式。
Hoeffding-Type Concentration Bounds for Exchangeable Random Variables
针对可交换随机变量序列，提出基于de Finetti混合测度支撑中分布的最大和最小均值的Hoeffding型浓度不等式，填补了有限样本均值与总体均值之间的空白。
The Entropy method provides a powerful framework for proving scalar concentration inequalities by establishing functional inequalities like Poincare and log-Sobolev inequalities. These inequalities are especially useful for deriving concentration for dependent distributions coming from stationary distributions of Markov chains. In contrast to the scalar case, the study of matrix concentration inequalities is fairly recent and there does not seem to be a general framework for proving matrix concentration inequalities using the Entropy method. In this paper, we initiate the study of Matrix value…
The Bernstein-Orlicz norm and deviation inequalities
我们引入了两个新的概念，旨在研究的经验过程。首先，我们引入一个新的Orlicz范数，我们称之为Bernstein-Orlicz范数。这种新的范数插值亚高斯和亚指数尾部行为。特别是，我们展示了如何使用这个规范来简化推导的偏差不等式的上确界的随机变量的集合。其次，我们介绍了沿着树的链接和泛型链接。这些简化了众所周知的链接和通用链接的概念。然后作为特例研究了经验过程的上确界。我们表明，沿着一棵树的链接可以使用熵与括号。最后，我们建立了无界情形下经验过程的偏差不等式。
Exponential inequalities for the distribution tails of canonical U- and V-statistics of $ρ$-mixing observations
本文给出了基于满足$ρ$ -混合的平稳观测序列样本的任意阶正则（退化）U-和V-统计量分布尾的Hoeffding型不等式.
Concentration Inequalities for Sums of Markov Dependent Random Matrices
本文针对一般状态空间上的时间依赖矩阵值函数，利用非交换矩母函数和谱方法，推导出 Markov 链随机矩阵和最大特征值的 Hoeffding 与 Bernstein 型集中不等式，推广了已有结果并建立了与动力系统及 Banach 空间理论的联系。
We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.
Bernstein-like Concentration and Moment Inequalities for Polynomials of Independent Random Variables: Multilinear Case
我们证明了独立随机变量的多线性多项式$f$超过其平均值的概率是最多$e^{-λ^2 / (R^q Var(f))}$足够小的$λ$，其中$R$是一个绝对常数。这与人们从中心极限定理中所期望的相匹配（直到指数中的常数）。我们的方法处理各种类型的随机变量，包括高斯，布尔，指数和泊松。以前的工作由金武和Schudy-Sviridenko给出的界限相同的形式，涉及较少的自然参数的地方的方差。
We establish concentration inequalities for Lipschitz functions of dependent random variables, whose dependencies are specified by forests. We also give concentration results for decomposable functions, improving Janson's Hoeffding-type inequality for the summation of graph-dependent bounded variables. These results extend McDiarmid's bounded difference inequality to the dependent cases.
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。
Concentration inequalities for disordered models
我们使用的Hoeffding不等式的推广显示浓度的无序钉扎模型的自由能的结果，假设只有一个有限的指数矩的障碍。我们还证明了随机环境中有向聚合物的一些浓度不等式，并利用这些不等式在聚合物测度下建立了聚合物末端位置的一个大偏差结果.
Various New Inequalities for Beta Distributions
该研究针对 Beta 分布给出尾不等式、Hoeffding 与 Bernstein 型指数不等式、高斯不等式及近似等新结果。
This note contains some estimates for tail spaces on Hamming cube. We use analytic paraproduct operator for that purpose. We also show several types of Bernstein--Markov inequalities for Banach space valued functions on Hamming cube. Here the novelty is in getting rid of some irritating logarithms and in proving Bernstein--Markov inequalities for $|\nabla f|_X$ rather than for $Δ^{1/2} f$ for $X$ valued polynomials on Hamming cube.
Dimension-free tail inequalities for sums of random matrices
本文导出了与显式矩阵维数无关的随机矩阵和的指数尾不等式。这些都类似于矩阵版本的Bauffff界和伯恩斯坦不等式，除了显式矩阵维数被一个迹量所取代，即使当维数很大或无限大时，这个迹量也可以很小。给出了主成分分析和近似矩阵乘法的一些应用以说明新界的实用性。
The Bernstein inequality is a tight upper bound on tail probabilities for independent random variables. Freedman extended the Bernstein inequality to martingales with differences bounded from above, and then Dzhaparidze and van Zanten generalized Freedman's result to non-bounded locally square integrable martingales. In this paper, we derive some Dzhaparidze-van Zanten type inequalities for self-normalized martingales with square integrable and non-square integrable differences.
We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi and Massart (2000,2003), we show that our result can be applied to the concentration of random functions, Erdös-Rényi random graph, and maxima of empirical processes. Several interesting Gaussian concentration inequalities have been obtained.
Concentration inequalities are widely used for analyzing machine learning algorithms. However, current concentration inequalities cannot be applied to some of the most popular deep neural networks, notably in natural language processing. This is mostly due to the non-causal nature of such involved data, in the sense that each data point depends on other neighbor data points. In this paper, a framework for modeling non-causal random fields is provided and a Hoeffding-type concentration inequality is obtained for this framework. The proof of this result relies on a local approximation of the non…
A note on exponential inequalities for the distribution tails of canonical von Mises' statistics of dependent observations
对于满足φ-混合条件的平稳随机变量序列的样本，得到了任意阶标准von Mises统计量分布尾的Hoeffding型指数不等式.本文削弱了Borisov和Volodko（2009）的论文“Exponential inequalities for the distributions of canonical U- and V-statistics of dependent observations”中对系数φ（）的限制。同时，对文中的相应证明进行了修正.
Hoeffding's inequality for sums of weakly dependent random variables
我们提供了一个系统的方法来处理以下问题。设$X_1,\ldots,X_n$是$[0,1]$值随机变量，可能是相关的.它们的总和显著大于平均值的概率的上限是多少？在独立随机变量的情况下，一个基本的工具，界定这种概率是由瓦西里Hoeffding。在本文中，我们考虑类似的Hoeffding的结果的总和相依随机变量，我们有一定的信息，其依赖结构。我们证明了一个结果，产生浓度不等式的几个概念之间的弱相依随机变量。此外，我们还得到了一个新的浓度不等式的总和，可能依赖，$[0,1]$值的随机变量，$X_1,\ldots,X_n$，满足以下条件：存在常数$γ\in (0,1)$和$δ\in (0,1]$，使得对于每个子集$A\subseteq \{1,\ldots,n\}$，我们有$\mathbb{E}\left[\prod_{i\in A} X_i \prod_{i\notin A}(1-X_i) \right]\leq γ^{|A|} δ^{n-|A|}$，其中$|A|$表示的基数$A$ .我们的方法适用于几个弱相依随机变量的和，如鞅差序列的和，$k$ -独立随机变量的和以及$U$ -统计量。最后，我们讨论了随机图理论的一些应用。
Matrix Concentration: Order versus Anti-order
本文针对现有矩阵集中不等式多基于Loewner反序、仅能给出最大特征值上尾界的问题，通过推导新的矩阵马尔可夫不等式，建立了Loewner序下的各类对应不等式，得到紧密度比现有最大特征值界高d倍的最小特征值上尾界。
For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $ϕ$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle ϕ(λ_{1}) μ(\{x \in Ω: f(x) \geq λ_{1} \}) + \sum_{k=2}^{n}\left(ϕ(λ_{k})- ϕ(λ_{k-1})\right) μ(\{x \in Ω: f(x) \geq λ_{k}\}) ,$ where $0 < λ_1 < λ_2 \cdots λ_n < \infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé-Chebyshev, Cantelli and Hoeffding inequalities are obtained.
Using the renewal approach we prove Bernstein-like inequalities for additive functionals of geometrically ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables. The coefficient in the sub-Gaussian part of our estimate is the asymptotic variance of the additive functional, i.e. the variance of the limiting Gaussian variable in the Central Limit Theorem for Markov chains. This refines earlier results by R. Adamczak and W. Bednorz, which were obtained under the additional assumption of strong aperiodicity of the chain.
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 obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\leq p<\infty$ there exists a constant $C_p>0$ such that for any $k\geq 1$ and all polynomials $P$ on $\mathbb{R}^{k}$ we have $$ \| \nabla P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)} \leq C_p (\mathrm{deg}\, P)^{\frac{1}{2}+\frac{1}π\arctan\left(\frac{|p-2|}{2\sqrt{p-1}}\right)}\|P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)}, $$ where $\mathrm{d}γ_k$ is the standard Gaussian measure on $\mathbb{R}^{k}$. We also show that under some mild growth assumptions on any function $B \in C^{2}((0,\infty))\c…
Matrix concentration inequalities via the method of exchangeable pairs
本文给出了随机矩阵谱范数的指数集中不等式和多项式矩不等式。分析需要一个矩阵扩展的标量浓度理论开发的Sourav Chatterjee使用斯坦的方法交换对。当应用到一个独立的随机矩阵的总和，这种方法产生矩阵推广的经典不等式由于Hoeffding，伯恩斯坦，Khintchine和罗森塔尔。相同的技术提供了相关随机矩阵和相关随机变量的更一般的矩阵值函数的和的界。
Dimension-free Markov--Bernstein inequalities for product measures
针对乘积概率测度下的多项式，研究无维数Markov-Bernstein不等式，在Gauss情形下对p≥4得到与次数相关的界，并推广到均匀分布、单峰密度乘积及Freud测度乘积。
Finite sample bounds for barycenter estimation in geodesic spaces
研究测地线空间中给定独立同分布数据时估计分布重心的问题，在特定条件下建立期望和高概率的有限样本误差界，推广相关不等式并为重心计算算法提供统计保证。
Since Kilmer et al. introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors, T-product tensors have been applied to many fields in science and engineering, such as low-rank tensor approximation, signal processing, image feature extraction, machine learning, computer vision, and the multi-view clustering problem, etc. However, there are very few works dedicated to exploring the behavior of random T-product tensors. This work considers the problem about the tail behavior o…
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…
Exponential and moment inequalities for U-statistics
得到了2阶（广义）标准U-统计量的Bernstein型指数不等式，并将独立随机变量和的Rosenthal和Hoffmann-Jørgensen不等式推广到核为非负或标准的任意阶（广义）U-统计量
Constructive approaches to concentration inequalities with independent random variables
本文提出系统性凸优化方法研究独立随机变量浓度不等式，通过变分方法扩展经典矩母函数至一阶矩条件，并开发基于平方和近似的多项式方法处理高阶矩条件，进而改进Hoeffding、Bennett和Bernstein不等式，提供更优的最坏情况保证。
We derive concentration inequalities for empirical means $\frac{1}{t} \int_0^t f(X_s) ds$ where $X_s$ is an irreducible Markov jump process on a finite state space and $f$ some observable. Using a Feynman-Kac semigroup we first derive a general concentration inequality. Then, based on this inequality we derive further concentration inequalities. Hereby we use three different methods; perturbation theory, Poincaré inequalities and information inequalities. We also obtain a Bernstein type concentration inequality.
Deviation inequalities for sums of weakly dependent time series
本文给出了弱相依时间序列部分和的Bernstein型偏差不等式。独立案例的损失进行了仔细研究。我们给出非混合的例子，这样的动力系统和伯努利位移，我们的偏差不等式。证明是基于块技术和不同的耦合参数。
Bernstein-Type Bounds for Beta Distribution
本文为Beta分布建立了尖锐的Bernstein型指数集中不等式，通过由超几何表示导出的二阶中心矩递推改进已有次高斯和次伽马界。
Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et al. proved a new Bernstein-type concentration inequality based on martingale theory. This work presents a new proof of this inequality based on the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first i…
We obtain moderate deviations theorems and exponential (Bernstein type) concentration inequalities for "nonconventional" sums of the form $S_N=\sum_{n=1}^N (F(ξ_{q_1(n)},ξ_{q_2(n)},...,ξ_{q_\ell(n)})-\bar F)$.
Probability and moment inequalities for additive functionals of geometrically ergodic Markov chains
该研究针对几何遍历马尔可夫链的加性泛函，建立了适用范围更广的矩与Bernstein型不等式，推广了独立随机变量的相关结果，且不等式显式依赖条件中的常数。
In the present paper we prove that the probabilities of the Pólya urn distribution (with negative replacement) satisfy a monotonicity property similar to that of the binomial distribution (Pólya urn distribution with no replacement). As a consequence, we show that the random variables with Pólya urn distribution (with negative replacement) are stochastically ordered with respect to the parameter giving the initial distribution of the urn. An equivalent formulation of this result shows that the new Bernstein operator recently introduced in [3] is a monotone operator. The proofs are probabilisti…
We derive tight and computable bounds on the bias of statistical estimators, or more generally of quantities of interest, when evaluated on a baseline model P rather than on the typically unknown true model Q. Our proposed method combines the scalable information inequality derived by P. Dupuis, K.Chowdhary, the authors and their collaborators together with classical concentration inequalities (such as Bennett's and Hoeffding-Azuma inequalities). Our bounds are expressed in terms of the Kullback-Leibler divergence R(Q||P) of model Q with respect to P and the moment generating function for the …
Sharp large deviation results for sums of independent random variables
我们证明了满足Bernstein条件的独立随机变量和的大偏差概率的精确界。在某些情况下，一个这样的界限非常接近标准高斯定律的尾部;其他界限通过在Talagrand（1995）的精神中添加缺失因子来改进Bennett和Hoeffding的不等式。我们还完成了Talagrand的不等式，给出了一个下界相同的形式，导致一个平等。结果，我们得到了类似于Cramér（1938），Bahadur-Rao（1960）和Sakhanenko（1991）的大偏差展开式。我们还证明了我们的界可以用来改进Pinelis（2014）最近的一个不等式。
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…
Concentration Inequalities for Exchangeable Tensors and Matrix-valued Data
本文提出了一种针对可交换张量和矩阵值数据的集中不等式方法，改进了现有结果并应用于多因素模型和联邦平均
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