We consider extensions of Hoeffding's " exponential method " approach for obtaining upper estimates on the probability that a sum of independent and bounded random variables is significantly larger than its mean. We show that the exponential function in Hoeffding's approach can be replaced with any function which is non-negative, increasing and convex. As a result we generalize and improve upon Hoeffding's inequality. Our approach allows to obtain " missing factors " in Hoeffding's inequality. The later result is a rather weaker version of a theorem that is due to Michel Talagrand. Moreover, we characterize the class of functions with respect to which our method yields optimal concentration bounds. Finally, using ideas from the theory of Bernstein polynomials, we show that similar ideas apply under information on higher moments of the random variables.
There exists quite some work dedicated to improving Hoeffding's bound. See for example the work of Bentkus @cite_25 , Pinelis @cite_10 , Siegel @cite_23 and Talagrand @cite_26 , just to name a few references. Let us bring the reader's to attention the following two results that are extracted from the papers of Talagrand ( @cite_26 , Theorem @math ) and Bentkus ( @cite_25 , Theorem @math ). Talagrand's paper focuses on obtaining some "missing" factors in Hoeffding's inequality whose existence is motivated by the Central Limit Theorem (see @cite_26 , Section @math ). These factors are obtained by combining the Bernstein-Hoeffding method together with a technique (i.e. suitable change of measure) that is used in the proof of Cram 'er's theorem on large deviations, yielding the following.
@cite_26 @cite_10 @cite_25 @cite_23
2037982184 2124357902 2963702702 2609098951
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) by me. In this article, I introduce a new type of inequality. Namely, I show that P M n ≥ x ≤ cP S n ≥ x , where c is an absolute constant and S n = e 1 + ... + e n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x ∈ Ρ where the survival function x → P S n ≥ x has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate P S n ≥ x special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere. In mining data streams the most popular tool is the Hoeffding tree algorithm. It uses the Hoeffding's bound to determine the smallest number of examples needed at a node to select a splitting attribute. In the literature the same Hoeffding's bound was used for any evaluation function (heuristic measure), e.g., information gain or Gini index. In this paper, it is shown that the Hoeffding's inequality is not appropriate to solve the underlying problem. We prove two theorems presenting the McDiarmid's bound for both the information gain, used in ID3 algorithm, and for Gini index, used in Classification and Regression Trees (CART) algorithm. The results of the paper guarantee that a decision tree learning system, applied to data streams and based on the McDiarmid's bound, has the property that its output is nearly identical to that of a conventional learner. The results of the paper have a great impact on the state of the art of mining data streams and various developed so far methods and algorithms should be reconsidered. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem.