An adaptive <i>O</i>(log <i>n</i>)‐optimal policy for the online selection of a monotone subsequence from a random sample

Arlotto, Alessandro; Wei, Yehua Dennis; Xie, Xinchang

doi:10.1002/rsa.20728

Cited by 8 publications

(15 citation statements)

References 18 publications

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“…, X n } but not the succession in which the items are revealed in the course of observation. Furthermore, we settle the conjecture from [4] by showing that the performance of the policy used there is indeed within O(1) from the maximum v n .…”

Section: Introductionmentioning

confidence: 57%

“…A more sophisticated policy was introduced by Arlotto et al [4]. In contrast to the stationary policy of Samuels and Steele, the acceptance window here is variable.…”

Section: Asymptotically Optimal Policiesmentioning

confidence: 99%

“…The lower bound was shown recently in [4]. By assessing a suboptimal selection policy with an acceptance window which depends on both the size of the last selection and the number of items yet to be observed, Arlotto et al suggested that the optimality gap in (2) can be further tightened.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

Seksenbayev

2021

Acta Appl Math

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We study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size $n$ n . In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length $k$ k from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with $O(1)$ O ( 1 ) remainder in the first problem and $O(k)$ O ( k ) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible.

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Section: Introductionmentioning

confidence: 57%

“…A more sophisticated policy was introduced by Arlotto et al [4]. In contrast to the stationary policy of Samuels and Steele, the acceptance window here is variable.…”

Section: Asymptotically Optimal Policiesmentioning

confidence: 99%

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Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

Seksenbayev

2021

Acta Appl Math

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“…Because the weights as well as the subsequence elements are both uniformly distributed on the unit interval, these two actions happen with the same probability. For this subsequence-selection problem, Arlotto et al (2015Arlotto et al ( , 2018 prove that the expected performance ν * n of the best online policy satisfies the estimate ν…”

mentioning

confidence: 99%

Logarithmic Regret in the Dynamic and Stochastic Knapsack Problem with Equal Rewards

Arlotto

Xie

2020

Stochastic Systems

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We study a dynamic and stochastic knapsack problem in which a decision maker is sequentially presented with items arriving according to a Bernoulli process over n discrete time periods. Items have equal rewards and independent weights that are drawn from a known nonnegative continuous distribution F. The decision maker seeks to maximize the expected total reward of the items that the decision maker includes in the knapsack while satisfying a capacity constraint and while making terminal decisions as soon as each item weight is revealed. Under mild regularity conditions on the weight distribution F, we prove that the regret—the expected difference between the performance of the best sequential algorithm and that of a prophet who sees all of the weights before making any decision—is, at most, logarithmic in n. Our proof is constructive. We devise a reoptimized heuristic that achieves this regret bound.

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“…The upper bound appeared in [5] in the context of a sequential knapsack problem and was generalised in [9] for the problem with random sample size. The lower bound appeared recently in Arlotto et al [3]. To derive (1.1) Samuels and Steele [13] employed a stationary policy which accepts the ith item each time X i exceeds the previous selection by no more than 2/n; this policy, however, falls by O(n 1/4 ) below the upper bound (1.2).…”

Section: Introductionmentioning

confidence: 99%

Refined Asymptotics in the Online Selection of an Increasing Subsequence

Seksenbayev¹

2018

Preprint

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Let vn be the maximum expected length of an increasing subsequence, which can be selected by an online nonanticipating policy from a random sample of size n. Refining known estimates, we obtain an asymptotic expansion of vn up to a O(1) term. The method we use is based on detailed analysis of the dynamic programming equation, and is also applicable to the online selection problem with observations occurring at times of a Poisson process.

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An adaptive O(log n)‐optimal policy for the online selection of a monotone subsequence from a random sample

Cited by 8 publications

References 18 publications

Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

Logarithmic Regret in the Dynamic and Stochastic Knapsack Problem with Equal Rewards

Refined Asymptotics in the Online Selection of an Increasing Subsequence

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