M. Grendar, Jr. and M. Grendar, ``Chernoff's bound forms,'' in Bayesian inference and Maximum Entropy methods in Science and Engineering, edited by Ch. Williams, AIP Conference Proceedings 659, Melville, New York, 2003, pp. 67-72
发表信息：
In: Bayesian inference and Maximum Entropy methods in Science and Engineering, Ch. Williams (ed.), AIP (Melville), 67-72, 2003
Chernoff's bound binds a tail probability (ie. $Pr(X \ge a)$, where $a \ge EX$). Assuming that the distribution of $X$ is $Q$, the logarithm of the bound is known to be equal to the value of relative entropy (or minus Kullback-Leibler distance) for $I$-projection $\hat P$ of $Q$ on a set $\mathcal{H} \triangleq \{P: E_PX = a\}$. Here, Chernoff's bound is related to Maximum Likelihood on exponential form and consequently implications for the notion of complementarity are discussed. Moreover, a novel form of the bound is proposed, which expresses the value of the Chernoff's bound directly in terms of the $I$-projection (or generalized $I$-projection).
A Chernoff-type distribution is a nonnormal distribution defined by the slope at zero of the greatest convex minorant of a two-sided Brownian motion with a polynomial drift. While a Chernoff-type distribution is known to appear as the distributional limit in many non-regular statistical estimation problems, the accuracy of Chernoff-type approximations has remained largely unknown. In the present paper, we tackle this problem and derive Berry-Esseen bounds for Chernoff-type limit distributions in the canonical non-regular statistical estimation problem of isotonic (or monotone) regression. The …
This paper describes the construction of a lower bound for the tails of general random variables, using solely knowledge of their moment generating function. The tilting procedure used allows for the construction of lower bounds that are tighter and more broadly applicable than existing tail approximations.
This paper develops an optimal Chernoff type bound for the probabilities of large deviations of sums $\sum_{k=1}^n f (X_k)$ where $f$ is a real-valued function and $(X_k)_{k \in \mathbb{Z}_{\ge 0}}$ is a finite state Markov chain with an arbitrary initial distribution and an irreducible transition probability matrix satisfying a mild assumption on its positivity pattern, related to the function $f$ being considered. The novelty lies in this being a non-asymptotic finite sample bound. Further, our bound is optimal in the large deviations sense, attaining a constant prefactor and an exponential …
This paper develops sharp bounds on moments of sums of k-wise independent bounded random variables, under constrained average variance. The result closes the problem addressed in part in the previous works of Schmidt et al. and Bellare, Rompel. We The work also discuss discusses other applications of independent interests, such as asymptotically sharp bounds on binomial moments.
We utilize operational methods to generalize the Chernoff inequality and prove a new result that relates the moment bound to strictly absolute monotonic functions. We show that the Chernoff bound is part of a continuum of probability bounds.
The Chernoff bound is an important inequality relation in probability theory. The original version of the Chernoff bound is to give an exponential decreasing bound on the tail distribution of sums of independent random variables. Recent years, several works have been done by extending the original version of the Chernoff bound to high-dimensional random objects, e.g., random matrices, or/and to consider the relaxation that there is no requirement of independent assumptions among random objects. In this work, we generalize the matrix expander Chernoff bound studied by Garg et al. at work: A Mat…
In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions for sums of independent random variables and such bound is applied at different fields in science and engineering. In this work, we generalize the conventional Chernoff bound from the summation of independent random variables to the summation of dependent random T-product tensors. Our main tool used at this work is majorization technique. We first apply majorizaton method to establish norm inequalitites for T-product tensors and these norm inequalities are used to derive T-product tensor expan…
The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equality. Here we will show that through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.
A lower bound for the Gaussian Q-function is presented in the form of a single exponential function with parametric order and weight. We prove the lower bound by introducing two functions, one related to the Q-function and the other similarly related to the exponential function, and by obtaining inequalities that indicate the sign of the difference of the two functions.
We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a regular (aperiodic and irreducible) finite Markov chain. Specially, consider a random walk on a regular Markov chain and a Hermitian matrix-valued function on its state space. Our result gives exponentially decreasing bounds on the tail distributions of the extreme eigenvalues of the sample mean matrix. Our proof is based on the matrix expander (regular undirected graph) Chernoff bound [Garg et al. STOC '18] and scalar Chernoff-Hoeffding bounds for Markov chains [Chung et al. STACS '12]. Our matrix Chernoff…
We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of non-contracting chains. In particular, geometric ergodicity provides a simple and versatile framework. This leads to a short, elementary proof of a general concentration inequality for Markov and hidden Markov chains (HMM), which supercedes some of the known results and easily extends to other processes such as Markov trees. As applications, we give a Dvoretzky-Kiefer-Wolfowitz-type inequality and a uniform Chernoff bound. All of our bounds are dimension-free and hold for cou…
We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an expander, confirming a conjecture due to Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the Golden-Thompson inequality which improves in some ways the inequality of Sutter, Berta, and Tomamichel, and may be of independent interest, as well as an adaptation of an argument for the scalar case due to Healy. Secondarily, we also provide a generic reduction showing that any concentration inequality for vector-valued martingales implies a concentration inequality for t…
In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n \times p} be two matrices and \eps>0. We approximate the product A^\top B using two down-sampled sketches, \tilde{A}\in\RR^{t\times m} and \tilde{B}\in\RR^{t\times p}, where t\ll n such that \norm{\tilde{A}^\top \tilde{B} - A^\top B} \leq \eps \norm{A}\norm{B} with high probability. We use two different sampling procedures for constructing \tilde{A} and \tilde{B}; one of them is done by i.i.d. non-uniform sampling rows from A and B and the othe…
We prove the first Chernoff-Hoeffding bounds for general nonreversible finite-state Markov chains based on the standard L_1 (variation distance) mixing-time of the chain. Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step random walk (v_1,...,v_t) on M starting from the stationary distribution pi has expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the L_1 mixing-time of M. We show that the probability of X deviating f…
Strongly Rayleigh distributions are a class of negatively dependent distributions of binary-valued random variables [Borcea, Branden, Liggett JAMS 09]. Recently, these distributions have played a crucial role in the analysis of algorithms for fundamental graph problems, e.g. Traveling Salesman Problem [Gharan, Saberi, Singh FOCS 11]. We prove a new matrix Chernoff bound for Strongly Rayleigh distributions. As an immediate application, we show that adding together the Laplacians of $ε^{-2} \log^2 n$ random spanning trees gives an $(1\pm ε)$ spectral sparsifiers of graph Laplacians with high pro…
For a wide class of monotonic functions $f$, we develop a Chernoff-style concentration inequality for quadratic forms $Q_f \sim \sum\limits_{i=1}^n f(η_i) (Z_i + δ_i)^2$, where $Z_i \sim N(0,1)$. The inequality is expressed in terms of traces that are rapid to compute, making it useful for bounding p-values in high-dimensional screening applications. The bounds we obtain are significantly tighter than those that have been previously developed, which we illustrate with numerical examples.
We derive simple but nearly tight upper and lower bounds for the binomial lower tail probability (with straightforward generalization to the upper tail probability) that apply to the whole parameter regime. These bounds are easy to compute and are tight within a constant factor of $89/44$. Moreover, they are asymptotically tight in the regimes of large deviation and moderate deviation. By virtue of a surprising connection with Ramanujan's equation, we also provide strong evidences suggesting that the lower bound is tight within a factor of $1.26434$. It may even be regarded as the natural lowe…
Chernoff bound is a fundamental tool in theoretical computer science. It has been extensively used in randomized algorithm design and stochastic type analysis. Discrepancy theory, which deals with finding a bi-coloring of a set system such that the coloring of each set is balanced, has a huge number of applications in approximation algorithms design. Chernoff bound [Che52] implies that a random bi-coloring of any set system with $n$ sets and $n$ elements will have discrepancy $O(\sqrt{n \log n})$ with high probability, while the famous result by Spencer [Spe85] shows that there exists an $O(\s…
An alternate form for the binomial tail is presented, which leads to a variety of bounds for the central tail. A few can be weakened into the corresponding Chernoff and Slud bounds, which not only demonstrates the quality of the presented bounds, but also provides alternate proofs for the classical bounds.
A halfspace is a function $f\colon\{-1,1\}^n \rightarrow \{0,1\}$ of the form $f(x)=\mathbb{1}(a\cdot x>t)$, where $\sum_i a_i^2=1$. We show that if $f$ is a halfspace with $\mathbb{E}[f]=ε$ and $a'=\max_i |a_i|$, then the degree-1 Fourier weight of $f$ is $W^1(f)=Θ(ε^2 \log(1/ε))$, and the maximal influence of $f$ is $I_{\max}(f)=Θ(ε\min(1,a' \sqrt{\log(1/ε)}))$. These results, which determine the exact asymptotic order of $W^1(f)$ and $I_{\max}(f)$, provide sharp generalizations of theorems proved by Matulef, O'Donnell, Rubinfeld, and Servedio, and settle a conjecture posed by Kalai, Keller …
We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,...,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,...,Y_r$.
Using coupling techniques based on Stein's method for probability approximation, we revisit classical variance bounding inequalities of Chernoff, Cacoullos, Chen and Klaassen. Taking advantage of modern coupling techniques allows us to establish novel variance bounds in settings where the underlying density function is unknown or intractable. Applications include bounds for asymptotically Gaussian random variables using zero-biased couplings, bounds for random variables which are New Better (Worse) than Used in Expectation, and analysis of the posterior in Bayesian statistics.
We consider a multivariate distributional recursion of sum-type as arising in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.
This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Lapalace transform method and Lieb's concavity theorem from matrices to tensors, and apply these tools to generalize the classical bounds associated with the names Chernoff, Bennett, and Bernstein from the scalar to the tensor setting. Tail bounds for the norm of a sum of random rectangular tensors are also derived from corollaries of random Hermitian tensors cases. The pro…
Rutgers University（罗格斯大学）；University of Pennsylvania（宾夕法尼亚大学）
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimension is large or infinite. Some applications to principal component analysis and approximate matrix multiplication are given to illustrate the utility of the new bounds.
This paper is the Part II of a serious work about T product tensors focusing at establishing new probability bounds for sums of random, independent, T product tensors. These probability bounds characterize large deviation behavior of the extreme eigenvalue of the sums of random T product tensors. We apply Lapalace transform method and Lieb concavity theorem for T product tensors obtained from our Part I paper, and apply these tools to generalize the classical bounds associated with the names Chernoff, and Bernstein from the scalar to the T product tensor setting. Tail bounds for the norm of a …
In recent years, tensors have been applied to different applications in science and engineering fields. In order to establish theory about tail bounds of the tensors summation behavior, this work extends previous work by considering the tensors summation tail behavior of the top $k$-largest singular values of a function of the tensors summation, instead of the largest/smallest singular value of the tensors summation directly (identity function) explored in Shih Yu's work: Convenient tail bounds for sums of random tensors. Majorization and antisymmetric tensor product tools are main techniques …
This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Benn…
