Hoeffding’s inequality for supermartingales

Fan, Xiequan; Grama, Ion; Liu, Quansheng

doi:10.1016/j.spa.2012.06.009

Cited by 64 publications

(58 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Over BR graphs, this property is lost, and in particular the node degrees are dependent and non-identically distributed. In order to overcome this difficulty, we appeal to the martingale structure of the degrees and to a maximal martingale inequality [34], which allow us to prove (15), and then (14). Using now ( 13), ( 14) and the triangle inequality we get:…”

Section: Resultsmentioning

confidence: 99%

Learning Bollobás-Riordan Graphs Under Partial Observability

Cirillo

Matta

Sayed

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

View full text Add to dashboard Cite

This work examines the problem of learning the topology of a network (graph learning) from the signals produced at a subset of the network nodes (partial observability). This challenging problem was recently tackled assuming that the topology is drawn according to an Erdős-Rényi model, for which it was shown that graph learning under partial observability is achievable, exploiting in particular homogeneity across nodes and independence across edges. However, several real-world networks do not match the optimistic assumptions of homogeneity/independence, for example, high heterogeneity is often observed between very connected nodes (hubs) and scarcely connected peripheral nodes. Random graphs with preferential attachment were conceived to overcome these issues. In this work, we discover that, over first-order vector autoregressive systems with a stable Laplacian combination matrix, graph learning is achievable under partial observability, when the network topology is drawn according to a popular preferential attachment model known as the Bollobás-Riordan model.

show abstract

Section: Resultsmentioning

confidence: 99%

Learning Bollobás-Riordan Graphs Under Partial Observability

Cirillo

Matta

Sayed

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

View full text Add to dashboard Cite

show abstract

“…Theorem 8.3 (Corollary 2.2 in [FGL12]). Let (ξ i , F i ) 0≤i≤n be a supermartingale difference sequence such that ξ 0 = 0 and ξ i ≤ 1 for each i ∈ {1, .…”

Section: Reduction To the Simplified Protocolmentioning

confidence: 99%

A Cryptographic Test of Quantumness and Certifiable Randomness from a Single Quantum Device

Brakerski

Christiano

Mahadev

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

We give a protocol for producing certifiable randomness from a single untrusted quantum device that is polynomial-time bounded. The randomness is certified to be statistically close to uniform from the point of view of any computationally unbounded quantum adversary, that may share entanglement with the quantum device. The protocol relies on the existence of post-quantum secure trapdoor claw-free functions, and introduces a new primitive for constraining the power of an untrusted quantum device. We show how to construct this primitive based on the hardness of the learning with errors (LWE) problem, and prove that it has a crucial adaptive hardcore bit property. The randomness protocol can be used as the basis for an efficiently verifiable "test of quantumness", thus answering an outstanding challenge in the field.

show abstract

“…Moreover, in Section 7, we will apply a version of the Inequality for submartingales, where the bound only holds for the lower tail of Y i . This formulation (with slightly stronger constants) can be found in [14].…”

Section: Martingale Concentration Inequalitiesmentioning

confidence: 99%

The Kőnig graph process

Kamčev

Krivelevich

Morrison

et al. 2020

Random Struct Algorithms

View full text Add to dashboard Cite

Say that a graph G has property  if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set N ∶= (n 2) and let e 1 , e 2 , … e N be a uniformly random ordering of the edges of K n , with n an even integer. Let G 0 be the empty graph on n vertices. For m ≥ 0, G m+1 is obtained from G m by adding the edge e m+1 exactly if G m ∪{e m+1 } has property . We analyze the behavior of this process, focusing mainly on two questions: What can be said about the structure of G N and for which m will G m contain a perfect matching? KEYWORDS matching, perfect matching, random graph, random process, vertex cover 1 INTRODUCTION The modern study of random graph processes began in 1959 with the inaugural papers of Erdős and Rényi [10, 11]. Given a uniformly random permutation e 1 , … , e N of E(K n), they studied the evolution and properties of the graph G n,m with edge set {e 1 , … , e m }, which is now known as the Erdős-Rényi random graph. This work has since grown into a well-established research area with many important applications in theoretical computer science, statistical physics, and other branches of mathematics [5, 17, 20]. An important variant of the standard Erdős-Rényi process, often referred to as the random greedy process, is the following. Given a graph property , preserved by the removal of edges, begin with an empty n-vertex graph and at each step add an edge chosen uniformly at random from those that do not violate property . The random greedy process was first considered by Ruciński and Wormald [27] (in the case of bounded degree) and, following discussions of Bollobás and Erdős, by Erdős, Suen and Winkler in 1995 [13] (in the case of triangle-freeness). Their motivation was defining and analyzing a natural probability measure on the set of -maximal graphs.

show abstract

Hoeffding’s inequality for supermartingales

Abstract: We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.

Cited by 64 publications

References 32 publications

Learning Bollobás-Riordan Graphs Under Partial Observability

Learning Bollobás-Riordan Graphs Under Partial Observability

A Cryptographic Test of Quantumness and Certifiable Randomness from a Single Quantum Device

The Kőnig graph process

Contact Info

Product

Resources

About