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.
See @cite_26 for a proof of this theorem and for a precise definition of the function @math . In other words, Talagrand's result improves upon Hoeffding's by inserting a "missing" factor of order @math in the Hoeffding bound. Notice that Talagrand's result holds true for @math , for some absolute constant @math whose value does seem to be known. Talagrand (see @cite_26 , page 692) mentions that one can obtain a rather small numerical value for @math , but numerical computations are left to others with the talent for it. One of the purposes of this paper is to improve upon Hoeffding's inequality by obtaining "missing" factors with exact numerical values for the constants. Part of Bentkus' paper performs comparisons between @math and tails of binomial and Poisson random variables. A crucial idea in the results of @cite_25 is to compare @math with means of particular functions of certain random variables. In particular, in the proof of Theorem @math in @cite_25 one can find the following result.
@cite_26 @cite_25
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Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the @math workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60-62]. In this model there are @math processors @math , each of which is initially given a private input bit @math . The goal is for @math to learn the value of @math , for some specified function @math , using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176-180] gave a noise-resistant protocol that allows @math to learn the entire input with constant probability in @math broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of @math on the depth of a tree that learns the entire input. While the above lower bound is for an @math -bit function, we also show an @math lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least @math for a constant parameter @math (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of @math on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) @math lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts. The method of stable random projections is popular in data stream computations, data mining, information retrieval, and machine learning, for efficiently computing the lα (0 We propose algorithms based on (1) the geometric mean estimator, for all 0 • The general sample complexity bound for α ≠ 1,2. For α = 1, [27] provided a nice argument based on the inverse of Cauchy density about the median, leading to a sample complexity bound, although they did not provide the constants and their proof restricted e to be "small enough." For general α ≠ 1, 2, however, the task becomes much more difficult. [27] provided the "conceptual promise" that the sample complexity bound similar to that for α = 1 should exist for general α, if a "non-uniform algorithm based on t-quantile" could be implemented. Such a conceptual algorithm was only for supporting the arguments in [27], not a real implementation. We consider this is one of the main problems left open in [27]. In this study, we propose a practical algorithm based on the geometric mean estimator and derive the sample complexity bound for all 0 • The practical and optimal algorithm for α = 0+ The l0 norm is an important case. Stable random projections can provide an approximation to the l0 norm using α → 0+. We provide an algorithm based on the harmonic mean estimator, which is simple and statistically optimal. Its tail bounds are sharper than the bounds derived based on the geometric mean. We also discover a (possibly surprising) fact: in boolean data, stable random projections using α = 0+ with the harmonic mean estimator will be about twice as accurate as (l2) normal random projections. Because high-dimensional boolean data are common, we expect this fact will be practically quite useful. • The precise theoretical analysis and practical implications We provide the precise constants in the tail bounds for both the geometric mean and harmonic mean estimators. We also provide the variances (either exact or asymptotic) for the proposed estimators. These results can assist practitioners to choose sample sizes accurately. We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm that makes @math nonadaptive queries to a function @math , always accepts a monotone function, and rejects a function that is @math -far from being monotone with constant probability.