scholarly journals APPROXIMATE RESULTS FOR A GENERALIZED SECRETARY PROBLEM

2011 ◽  
Vol 25 (2) ◽  
pp. 157-169 ◽  
Author(s):  
Chris Dietz ◽  
Dinard van der Laan ◽  
Ad Ridder
Keyword(s):  
Optimal Policy ◽  
Random Order ◽  
Secretary Problem ◽  
Backward Induction ◽  
Accurate Approximation ◽  
Long Time

A version of the classical secretary problem is studied, in which one is interested in selecting one of the b best out of a group of n differently ranked persons who are presented one by one in a random order. It is assumed that b ≥ 1 is a preassigned number. It is known, already for a long time, that for the optimal policy, one needs to compute b position thresholds (for instance, via backward induction). In this article we study approximate policies that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as n → ∞) results, which show that the double-level policy is an extremely accurate approximation.

Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

2021 ◽  
Author(s):  
José Correa ◽  
Paul Dütting ◽  
Felix Fischer ◽  
Kevin Schewior
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Optimal Solution ◽  
Random Order ◽  
Secretary Problem ◽  
Maximum Value ◽  
Optimal Stopping Theory ◽  
Long Time ◽  
Object Of Study ◽  
Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

scholarly journals Labelling Training Samples Using Crowdsourcing Annotation for Recommendation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Qingren Wang ◽  
Min Zhang ◽  
Tao Tao ◽  
Victor S. Sheng
Keyword(s):  
Supervised Learning ◽  
Random Order ◽  
Ground Truth ◽  
Training Sample ◽  
Machine Intelligence ◽  
High Quality ◽  
Inference Algorithms ◽  
Training Samples ◽  
Expert Annotation ◽  
Long Time

The supervised learning-based recommendation models, whose infrastructures are sufficient training samples with high quality, have been widely applied in many domains. In the era of big data with the explosive growth of data volume, training samples should be labelled timely and accurately to guarantee the excellent recommendation performance of supervised learning-based models. Machine annotation cannot complete the tasks of labelling training samples with high quality because of limited machine intelligence. Although expert annotation can achieve a high accuracy, it requires a long time as well as more resources. As a new way of human intelligence to participate in machine computing, crowdsourcing annotation makes up for shortages of machine annotation and expert annotation. Therefore, in this paper, we utilize crowdsourcing annotation to label training samples. First, a suitable crowdsourcing mechanism is designed to create crowdsourcing annotation-based tasks for training sample labelling, and then two entropy-based ground truth inference algorithms (i.e., HILED and HILI) are proposed to achieve quality improvement of noise labels provided by the crowd. In addition, the descending and random order manners in crowdsourcing annotation-based tasks are also explored. The experimental results demonstrate that crowdsourcing annotation significantly improves the performance of machine annotation. Among the ground truth inference algorithms, both HILED and HILI improve the performance of baselines; meanwhile, HILED performs better than HILI.

Optimal selection based on relative ranks with a random number of individuals

1979 ◽  
Vol 11 (4) ◽  
pp. 720-736 ◽  
Author(s):  
Jacqueline Gianini-Pettitt
Keyword(s):  
Selection Procedure ◽  
Random Order ◽  
Random Variable ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Asymptotic Results ◽  
Fixed Integer ◽  
Bounded Random Variable ◽  
The Individual ◽  
Number Of Individuals

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.

scholarly journals Optimal Stopping Rule for the No-Information Duration Problem with Random Horizon

2013 ◽  
Vol 45 (04) ◽  
pp. 1028-1048 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Classical Problem ◽  
Random Order ◽  
Random Variable ◽  
Planning Horizon ◽  
Secretary Problem ◽  
Main Concern ◽  
Asymptotic Results ◽  
Optimal Rule ◽  
Random Horizon

As a version of the secretary problem, Ferguson, Hardwick and Tamaki (1992) considered an optimal stopping problem called the duration problem. The basic duration problem is the classical duration problem, in which the objective is to maximize the time of possession of a relatively best object when a known number of rankable objects appear in random order. In this paper we generalize this classical problem in two directions by allowing the number N (of available objects) to be a random variable with a known upper bound n and also allowing the objects to appear in accordance with Bernoulli trials. Two models can be considered for our random horizon duration problem according to whether the planning horizon is N or n. Since the form of the optimal rule is in general complicated, our main concern is to give to each model a sufficient condition for the optimal rule to be simple. For N having a uniform, generalized uniform, or curtailed geometric distribution, the optimal rule is shown to be simple in the so-called secretary case. The asymptotic results, as n → ∞, will also be given for these priors.

Optimal Stopping Rule for the No-Information Duration Problem with Random Horizon

2013 ◽  
Vol 45 (4) ◽  
pp. 1028-1048
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Classical Problem ◽  
Random Order ◽  
Random Variable ◽  
Planning Horizon ◽  
Secretary Problem ◽  
Main Concern ◽  
Asymptotic Results ◽  
Optimal Rule ◽  
Random Horizon

As a version of the secretary problem, Ferguson, Hardwick and Tamaki (1992) considered an optimal stopping problem called the duration problem. The basic duration problem is the classical duration problem, in which the objective is to maximize the time of possession of a relatively best object when a known number of rankable objects appear in random order. In this paper we generalize this classical problem in two directions by allowing the number N (of available objects) to be a random variable with a known upper bound n and also allowing the objects to appear in accordance with Bernoulli trials. Two models can be considered for our random horizon duration problem according to whether the planning horizon is N or n. Since the form of the optimal rule is in general complicated, our main concern is to give to each model a sufficient condition for the optimal rule to be simple. For N having a uniform, generalized uniform, or curtailed geometric distribution, the optimal rule is shown to be simple in the so-called secretary case. The asymptotic results, as n → ∞, will also be given for these priors.

Olfactory Evoked Potentials and Contingent Negative Variation Simultaneously Recorded for Diagnosis of Smell Disorders

1993 ◽  
Vol 102 (1) ◽  
pp. 6-10 ◽  
Author(s):  
Heike Auffermann ◽  
Frank Mathe ◽  
Günther Gerull ◽  
Dieter Mrowinski
Keyword(s):  
Evoked Potentials ◽  
Contingent Negative Variation ◽  
Time Window ◽  
Random Order ◽  
Discrimination Threshold ◽  
The Other ◽  
Single Session ◽  
Negative Voltage ◽  
Long Time ◽  
Smell Disorders

Objective diagnosis of olfaction can be performed by registration of cortical olfactory evoked potentials (OEP) and of contingent negative variation (CNV). The CNV is a negative voltage developing at the vertex after discrimination of one of two smells while the patient is expecting a second stimulus. By an adequate procedure, including a long time window for averaging (2.56 seconds) with appropriate filters, the two tests can be performed simultaneously in a single session of less than 10 minutes. Anosmia is determinable by both OEP and CNV, although CNV shows less variability. On the other hand, CNV requires attention and some cooperation of the patient. Parosmia is accessible by CNV only; two odor qualities presented in random order have to be distinguished. Hyposmia can also be detected; just above the discrimination threshold, CNV amplitudes tend to be large — even enhanced — whereas OEP amplitudes may still be undetectable.

scholarly journals Improved Online Algorithms for Knapsack and GAP in the Random Order Model

Algorithmica ◽  
2021 ◽  
Author(s):  
Susanne Albers ◽  
Arindam Khan ◽  
Leon Ladewig
Keyword(s):  
Knapsack Problem ◽  
Competitive Ratio ◽  
Randomized Algorithm ◽  
Input Sequence ◽  
Random Order ◽  
Random Permutation ◽  
Secretary Problem ◽  
Order Model ◽  
Sequential Approach ◽  
Online Setting

AbstractThe knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set. We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 (Kesselheim et al. in SIAM J Comput 47(5):1939–1964, 2018). Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem. The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 (Kesselheim et al. in SIAM J Comput 47(5):1939–1964, 2018).

Optimal selection based on relative ranks with a random number of individuals

1979 ◽  
Vol 11 (04) ◽  
pp. 720-736 ◽  
Author(s):  
Jacqueline Gianini-Pettitt
Keyword(s):  
Selection Procedure ◽  
Random Order ◽  
Random Variable ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Asymptotic Results ◽  
Fixed Integer ◽  
Bounded Random Variable ◽  
The Individual ◽  
Number Of Individuals

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.

scholarly journals On a Competitive Secretary Problem with Deferred Selections

2021 ◽  
Author(s):  
Tomer Ezra ◽  
Michal Feldman ◽  
Ron Kupfer
Keyword(s):  
Decision Maker ◽  
Random Order ◽  
Secretary Problem ◽  
Implicit Assumption ◽  
Multiple Agents ◽  
Winning Probability ◽  
Full Characterization ◽  
Multi Agent ◽  
And Performance

We study the secretary problem in multi-agent environments. In the standard secretary problem, a sequence of arbitrary awards arrive online, in a random order, and a single decision maker makes an immediate and irrevocable decision whether to accept each award upon its arrival. The requirement to make immediate decisions arises in many cases due to an implicit assumption regarding competition. Namely, if the decision maker does not take the offered award immediately, it will be taken by someone else. We introduce a novel multi-agent secretary model, in which the competition is explicit. In our model, multiple agents compete over the arriving awards, but the decisions need not be immediate; instead, agents may select previous awards as long as they are available (i.e., not taken by another agent). If an award is selected by multiple agents, ties are broken either randomly or according to a global ranking. This induces a multi-agent game in which the time of selection is not enforced by the rules of the games, rather it is an important component of the agent's strategy. We study the structure and performance of equilibria in this game. For random tie breaking, we characterize the equilibria of the game, and show that the expected social welfare in equilibrium is nearly optimal, despite competition among the agents. For ranked tie breaking, we give a full characterization of equilibria in the 3-agent game, and show that as the number of agents grows, the winning probability of every agent under non-immediate selections approaches her winning probability under immediate selections.

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