T. Klein, E. Rio
In this paper we give optimal constants in Talagrand's concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables. Our approach is based on the entropy method introduced by Ledoux.
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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.
In this paper we revisit Talagrand's proof of concentration inequality for empirical processes. We give a different shorter proof of the main technical lemma that guarantees the existence of a certain kernel. Our proof provides the almost optimal value of the constant involved in the statement of this lemma.
We establish some quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moments estimates for solutions of certain diffusive partial differential equations.
Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension
本文研究了经验测度在Wasserstein距离中的集中性，通过覆盖维度扩展了Fournier和Guillin的结果，适用于R^d空间。
Recently in Gao and Stoev (2018) it was established that the concentration of maxima phenomenon is the key to solving the exact sparse support recovery problem in high dimensions. This phenomenon, known also as relative stability, has been little studied in the context of dependence. Here, we obtain bounds on the rate of concentration of maxima in Gaussian triangular arrays. These results are used to establish sufficient conditions for the uniform relative stability of functions of Gaussian arrays, leading to new models that exhibit phase transitions in the exact support recovery problem. Fina…
We introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional. As an application we give a new easy proof of Talagrand's concentration inequality for empirical processes, where besides symmetrization we use only Talagrand's concentration inequality on the discrete cube {-1,+1}^n. As another application of this technique we prove new Vapnik-Chervonenkis type inequalities. For example, for VC-classes of fu…
Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lév…
Concentration of empirical entropy and transfer entropy for non-regular chains with unbounded memory
针对无界记忆平稳链，在合并耦合假设下证明熵率与转移熵率插件估计量的非渐近集中不等式，并在均匀非零性与指数混合条件下控制偏差，获得一致性与集中性，适用于非全局连续的g函数。
This paper studies concentration inequalities for functions of locally dependent random variables. We show that the usual definition of local dependence does not imply concentration for general Hamming Lipschitz functions. We define hypergraph dependence, which is a special case of local dependence, and show that it implies concentration if the maximal neighborhood size is small. We prove concentration in Hamming distance, Talagrand distance, and for self-bounding functions of a particular type under this dependence structure.
Concentration inequalities for maximal displacement of random walks on groups of polynomial growth
本文研究了具有多项式增长的群上随机游走最大位移的集中不等式，证明了在特定群类上随机游走的扩散性质。
Anti-Concentration Inequalities for the Difference of Maxima of Gaussian Random Vectors
本文推导了两个高斯随机向量最大值之差的无维度反集中界，在退化协方差下成立且条件尖锐，并应用于推导离散经验过程最大化器的中心极限定理。
We consider a system of particles experiencing diffusion and mean field interaction, and study its behaviour when the number of particles goes to infinity. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around its limit. The method is based on a coupling argument, strong integrability estimates on the paths in Holder norm, and some general concentration result for the empirical measure of identically distributed independent paths.
We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_n$ be a Hermitian random matrix of size $n\times n$ that can be split into independent blocks of the size at most $d_n=o(n^2)$. We prove that under some mild conditions on the distribution of the entries of $X_n$, the empirical spectral measure of $X_n$ concentrates around its mean. The main theorem is a strengthening of a recent result by Kemp and Zimmerman, where the size of blocks grows as $o(\log n)$. As an application, we are able to upgrade th…
Let $\mathcal{F}$ be a class of measurable functions on a measurable space $(S,\mathcal{S})$ with values in $[0,1]$ and let \[P_n=n^{-1}\sum_{i=1}^nδ_{X_i}\] be the empirical measure based on an i.i.d. sample $(X_1,...,X_n)$ from a probability distribution $P$ on $(S,\mathcal{S})$. We study the behavior of suprema of the following type: \[\sup_{r_n<σ_Pf\leq δ_n}\frac{|P_nf-Pf|}{ϕ(σ_Pf)},\] where $σ_Pf\ge\operatorname {Var}^{1/2}_Pf$ and $ϕ$ is a continuous, strictly increasing function with $ϕ(0)=0$. Using Talagrand's concentration inequality for empirical processes, we establish concentration…
We prove a Bennett-type concentration bound for suprema of empirical processes based on sampling without replacement and a corresponding bound in the case of an arbitrary Hoeffding statistics. We improve on the previous results of such type, providing a sharper concentration profile.
While effective concentration inequalities for suprema of empirical processes exist under boundedness or strict tail assumptions, no comparable results have been available under considerably weaker assumptions. In this paper, we derive concentration inequalities assuming only low moments for an envelope of the empirical process. These concentration inequalities are beneficial even when the envelope is much larger than the single functions under consideration.
The goal of this paper is to establish relative perturbation bounds, tailored for empirical covariance operators. Our main results are expansions for empirical eigenvalues and spectral projectors, leading to concentration inequalities and limit theorems. One of the key ingredients is a specific separation measure for population eigenvalues, which we call the relative rank, giving rise to a sharp invariance principle in terms of limit theorems, concentration inequalities and inconsistency results. Our framework is very general, requiring only $p > 4$ moments and allows for a huge variety of dep…
Concentration of the bootstrap empirical process, with applications to statistical inference
本文研究了基于可交换权重的自助经验过程的集中性，推导了其上界不等式，并应用于均值估计的置信区域和两样本置换检验的非渐近界。
We provide explicit nonasymptotic estimates for the rate of convergence of empirical means of Markov chains, together with a Gaussian or exponential control on the deviations of empirical means. These estimates hold under a "positive curvature" assumption expressing a kind of metric ergodicity, which generalizes the Ricci curvature from differential geometry and, on finite graphs, amounts to contraction under path coupling.
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.
In this paper we study the regularity of paths in terms of properties of admissible nets. We show the right concentration inequality above the modulus of continuity. Using the approach we prove the Bernstein type inequality for the empirical processes. Therefore we obtain the best form of concentration for processes studied recently by Mendelson and Paouris. Results of this type are of importance in the compressed sensing theory.
We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with non-asymptotic concentration bounds for the Wasserstein distance between the $n$-player Na…
Maximal Inequalities for Empirical Processes under General Mixing Conditions
本文在一般混合过程中研究样本均值的上界，提出基于浓度速率和新复杂度度量的界，揭示混合速度影响浓度速率并导致相变，应用于获得新的Glivenko-Cantelli型结果。
In this paper, we consider graphon particle systems with heterogeneous mean-field type interactions and the associated finite particle approximations. Under suitable growth (resp. convexity) assumptions, we obtain uniform-in-time concentration estimates, over finite (resp. infinite) time horizon, for the Wasserstein distance between the empirical measure and its limit, extending the work of Bolley--Guillin--Villani.
We derive concentration inequalities for maxima of empirical processes associated with Poisson point processes. The proofs are based on a careful application of Ledoux's entropy method. We demonstrate the utility of the obtained concentration inequalities by application to adaptive intensity estimation.
Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.
New Vapnik and Chervonenkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on the maximal deviation over classes of Borel sets within a low probability region. The constants are explicit, enabling numerical comparisons.
Concentration of measure for Graphon particle system
本文研究了图论粒子系统中的测度集中性，通过均场型相互作用和图论结构，推导了有限粒子近似在有限时间内的指数集中估计。
We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening. The system consists of $n$ particles in $(0,\infty)$ that move at unit speed to the left. Each time a particle hits the boundary point $0$, it is removed from the system along with a second particle chosen uniformly from the particles in $(0,\infty)$. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density $f_0(x) \in L^1_+(0,\infty…
A simple lemma bounds $\mathrm{s.d.}(T)/\mathbb{E} T$ for hitting times $T$ in Markov chains with a certain strong monotonicity property. We show how this lemma may be applied to several increasing set-valued processes. Our main result concerns a model of first passage percolation on a finite graph, where the traversal times of edges are independent Exponentials with arbitrary rates. Consider the percolation time $X$ between two arbitrary vertices. We prove that $\mathrm{s.d.}(X)/\mathbb{E} X$ is small if and only if $Ξ/\mathbb{E} X$ is small, where $Ξ$ is the maximal edge-traversal time in th…
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。
On concentration of the empirical measure for radial transport costs
本文研究径向传输代价下经验测度与真实测度间最优传输代价的集中性，通过环带划分结合经验过程理论与紧支撑结果，推广并改进了 Fournier–Guillin 估计。
In this note, we show that the relative entropy of an empirical distribution of $n$ samples drawn from a set of size $k$ with respect to the true underlying distribution is exponentially concentrated around its expectation, with central moment generating function bounded by that of a gamma distribution with shape $2k$ and rate $n/2$. This improves on recent work of Bhatt and Pensia (arXiv 2021) on the same problem, who showed such a similar bound with an additional polylogarithmic factor of $k$ in the shape, and also confirms a recent conjecture of Mardia et al. (Information and Inference 2020…
Concentration Inequalities for Sub-Weibull Random Tensors
本文提出了一种针对子韦布分布随机张量的集中不等式理论，通过广义最大不等式和鞅分析，建立了欧几里得函数的集中界限，并揭示了亚高斯与重尾区域之间的相变现象。
Sharp Concentration of Simple Random Tensors
本文研究简单随机张量之和的集中现象，利用 generic chaining 技术证明与维度无关的锐偏差界，并将二次及乘积经验过程的经典结果推广到高阶。
Barycenters (aka Fréchet means) were introduced in statistics in the 1940's and popularized in the fields of shape statistics and, later, in optimal transport and matrix analysis. They provide the most natural extension of linear averaging to non-Euclidean geometries, which is perhaps the most basic and widely used tool in data science. In various setups, their asymptotic properties, such as laws of large numbers and central limit theorems, have been established, but their non-asymptotic behaviour is still not well understood. In this work, we prove finite sample concentration inequalities (na…
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
Concentration of Non-Isotropic Random Tensors with Applications to Learning and Empirical Risk Minimization
针对现代学习任务中维度瓶颈问题，本文研究非各向同性数据分布，基于椭球度量熵非渐近估计和链式论证提出含有效维度的一致集中界，改进分布式优化统计预处理结果并引入非各向同性随机平滑，覆盖线性模型ERM等函数。
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。
Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes
针对图依赖观测数据，提出一种结合图适应依赖系数与块分区着色的耦合构造，导出经验过程的极大不等式，并应用于网络自回归、非线性局部传播等模型。
Sharp Concentration of Simple Random Tensors II: Asymmetry
本文研究了简单随机张量的尖锐集中不等式，发现非对称张量在有效秩跨越临界阈值时会出现额外对数因子。通过扩展通用链技术到高阶设置，建立了非对称张量的集中界。
Concentration inequalities for random dynamical systems
本文在弱平均收缩条件下，建立随机动力系统中观测量的集中不等式，研究时间平均同步、经验测度等随机量的偏差界，并给出有限时间Lyapunov指数的集中不等式。
The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spect…
Concentration of Empirical First-Passage Times
针对欠采样数据中首次通过统计量难以可靠推断的问题，本章研究可逆 Markov 过程经验首次通过时间的有限样本集中性，给出样本均值偏差界、非渐近置信区间及极值波动双侧界。
Concentration Inequalities for Output Statistics of Quantum Markov Processes
本研究运用谱方法、微扰技术等结合非交换$L_2$理论，推导了量子马尔可夫过程测量统计量的多类集中界，拓展了经典马尔可夫链相关结果的适用范围，还探讨了其在参数估计等方向的潜在应用与关联。
We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is re…
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