A Hoeffding Inequality for Finite State Markov Chains and its Applications to Markovian Bandits

Moulos, Vrettos

doi:10.1109/isit44484.2020.9173931

Cited by 7 publications

(5 citation statements)

References 34 publications

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“…We then apply Hoeffding's inequality for Markov chains to prove the optimality of R for a batch S k of M sequences. The following result is needed (see [28,Corollary 1] or [29]).…”

Section: B the Structure Of µ Implies Theoremmentioning

confidence: 99%

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Optimal Reference for DNA Synthesis

Elishco¹,

Huleihel²

2022

Preprint

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In the recent years, DNA has emerged as a potentially viable storage technology. DNA synthesis, which refers to the task of writing the data into DNA, is perhaps the most costly part of existing storage systems. Accordingly, this high cost and low throughput limits the practical use in available DNA synthesis technologies. It has been found that the homopolymer run (i.e., the repetition of the same nucleotide) is a major factor affecting the synthesis and sequencing errors. Quite recently, [26] studied the role of batch optimization in reducing the cost of large scale DNA synthesis, for a given pool S of random quaternary strings of fixed length. Among other things, it was shown that the asymptotic cost savings of batch optimization are significantly greater when the strings in S contain repeats of the same character (homopolymer run of length one), as compared to the case where strings are unconstrained.Following the lead of [26], in this paper, we take a step forward towards the theoretical understanding of DNA synthesis, and study the homopolymer run of length k 1. Specifically, we are given a set of DNA strands S, randomly drawn from a natural Markovian distribution modeling a general homopolymer run length constraint, that we wish to synthesize. For this problem, we prove that for any k 1, the optimal reference strand, minimizing the cost of DNA synthesis is, perhaps surprisingly, the periodic sequence ACGT. It turns out that tackling the homopolymer constraint of length k 2 is a challenging problem; our main technical contribution is the representation of the DNA synthesis process as a certain constrained system, for which string techniques can be applied.

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“…We then apply Hoeffding's inequality for Markov chains to prove the optimality of R for a batch S k of M sequences. The following result is needed (see [28,Corollary 1] or [29]).…”

Section: B the Structure Of µ Implies Theoremmentioning

confidence: 99%

“…Lemma 4 (Hoeffding's inequality for Markov chains [28]) Let {X i } i be an irreducible Markov chain on a finite state space V , with initial distribution π (the stationary distribution).…”

Section: B the Structure Of µ Implies Theoremmentioning

confidence: 99%

Optimal Reference for DNA Synthesis

Elishco¹,

Huleihel²

2022

Preprint

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“…In a uniformly ergodic Markov model, for example, regeneration can happen in geometric time. Related works include Moulos (2020) that establishes a Heoffding inequality without assuming bounded regenerative time but on a finite state space Markov chain; and Bertail and Cio lek (2017) that also establishes a Hoeffding inequality of a different form.…”

Section: Regenerative Processes and Convergence Ratementioning

confidence: 99%

Online Regenerative Learning

Shen¹

2022

Preprint

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We study a type of Online Linear Programming (OLP) problem that maximizes the objective function with stochastic inputs. The performance of various algorithms that analyze this type of OLP is well studied when the stochastic inputs follow some i.i.d distribution. The two central questions to ask are: (i) can the algorithms achieve the same efficiency if the stochastic inputs are not i.i.d but still stationary, and (ii) how can we modify our algorithms if we know the stochastic inputs are trendy, hence not stationary. We answer the first question by analyzing a regenerative type of input and show the regrets of two popular algorithms are bounded by the same orders as their i.i.d counterparts. We discuss the second question in the context of linearly growing inputs and propose a trend-adaptive algorithm. We provide numerical simulations to illustrate the performance of our algorithms under both regenerative and trendy inputs.

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“…Let X be a random variable and a ∈ R. According to the Markov inequality [34], we have the following:…”

Section: Analysis Of the Constructionmentioning

confidence: 99%

Fair and Efficient Robust Secret Sharing Scheme against Rushing Adversaries

Zhang

Yang

Fan

et al. 2022

Security and Communication Networks

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Robust secret sharing (RSS) is an extension of secret sharing, which can reconstruct a secret correctly even if any t shares are incorrect. The existing scheme would not effectively achieve fairness. Moreover, even for an optimal scheme, RSS still has the problem that one party is verified by multiple parties, leading to expensive communication costs. In this work, we construct a blockchain-aided RSS scheme that can ensure decentralization and fairness. The central building block for our scheme to handle a rushing adversary is a bulletin board we implement on the InterPlanetary File System. Furthermore, we design a monetary penalty mechanism to impose real penalties on corrupt parties. Each participant either uploads his share correctly or loses his deposit. In addition, our scheme eliminates considerable communication between participants. Compared with the previous schemes, our scheme has a lower communication complexity, which is close to O ˜ n . We conduct experiments to show the performance of our scheme. To our knowledge, this is the first implementation of a fair RSS scheme. For shares with a length of 128 bits, the time for each participant to execute the verification phase is 470 ms.

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A Hoeffding Inequality for Finite State Markov Chains and its Applications to Markovian Bandits

Cited by 7 publications

References 34 publications

Optimal Reference for DNA Synthesis

Optimal Reference for DNA Synthesis

Online Regenerative Learning

Fair and Efficient Robust Secret Sharing Scheme against Rushing Adversaries

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