Chernoff–Hoeffding Bounds for Applications with Limited Independence

Schmidt, Jeanette P.; Siegel, Andrew F.; Srinivasan, Aravind

doi:10.1137/s089548019223872x

Cited by 288 publications

(217 citation statements)

References 25 publications

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“…In general, these bounds are applicable to sums of negatively associated, identically distributed random variables. Their precise derivation can be found in various papers and standard textbooks (e.g., see [20,21,22,23]). …”

Section: Phase-error Ratementioning

confidence: 99%

Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication

2006

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We investigate the error tolerance of quantum cryptographic protocols using d-level systems. In particular, we focus on prepare-and-measure schemes that use two mutually unbiased bases and a key-distillation procedure with two-way classical communication. For arbitrary quantum channels, we obtain a sufficient condition for secret-key distillation which, in the case of isotropic quantum channels, yields an analytic expression for the maximally tolerable error rate of the cryptographic protocols under consideration. The difference between the tolerable error rate and its theoretical upper bound tends slowly to zero for sufficiently large dimensions of the information carriers.

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Section: Phase-error Ratementioning

confidence: 99%

Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication

2006

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“…k-wise independence is a limited notion of independence where any set of k (or fewer) random variables are assumed to be independent from a total of n random variables (k ≤ n). Bounds assuming k-wise independence are given in [37] which extend the ideas given by Chernoff and Hoeffding. The use of these bounds is in applications where pseudo-randomness is assumed.…”

Section: Related Workmentioning

confidence: 80%

Distribution-free bounds for relational classification

Dhurandhar

Dobra

2011

Knowl Inf Syst

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Statistical Relational Learning (SRL) is a sub-area in Machine Learning which addresses the problem of performing statistical inference on data that is correlated and not independently and identically distributed (i.i.d.) -as is generally assumed. For the traditional i.i.d. setting, distribution free bounds exist, such as the Hoeffding bound, which are used to provide confidence bounds on the generalization error of a classification algorithm given its hold-out error on a sample size of N . Bounds of this form are currently not present for the type of interactions that are considered in the data by relational classification algorithms. In this paper we extend the Hoeffding bounds to the relational setting. In particular, we derive distribution free bounds for certain classes of data generation models that do not produce i.i.d. data and are based on the type of interactions that are considered by relational classification algorithms that have been developed in SRL. We conduct empirical studies on synthetic and real data which show that these data generation models are indeed realistic and the derived bounds are tight enough for practical use.

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“…We use the following extension of a standard result [40,Theorem 4]. 15 Technically, we must pad U with zeros in the locations specified by T (i. e., U i = 0 for i ∈ T ) to obtain the right length.…”

Section: Compute Appropriate Values Of Parameters Satisfying 13mentioning

confidence: 99%