On Asymptotically Tight Tail Bounds for Sums of Geometric and Exponential Random Variables

Jin, Yaonan; Li, Yingkai; Wang, Yining; Zhou, Yuan

doi:10.48550/arxiv.1902.02852

Cited by 2 publications

(2 citation statements)

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“…Lemma A.3 can be proved by following Appendix D in [Agrawal et al, 2019]. Similar inequalities with constants smaller than 48 were shown in [Jin et al, 2019;Janson, 2018].…”

Section: Appendices a Concentration Inequalitiesmentioning

confidence: 52%

Fully Gap-Dependent Bounds for Multinomial Logit Bandit

Yang

2020

Preprint

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We study the multinomial logit (MNL) bandit problem, where at each time step, the seller offers an assortment of size at most K from a pool of N items, and the buyer purchases an item from the assortment according to a MNL choice model. The objective is to learn the model parameters and maximize the expected revenue. We present (i) an algorithm that identifies the optimal assortment S * within O( N i=1 ∆ −2 i ) time steps with high probability, and (ii) an algorithm that incurs O( i / ∈S * K∆ −1 i log T ) regret in T time steps. To our knowledge, our algorithms are the first to achieve gap-dependent bounds that fully depends on the suboptimality gaps of all items. Our technical contributions include an algorithmic framework that relates the MNL-bandit problem to a variant of the top-K arm identification problem in multi-armed bandits, a generalized epoch-based offering procedure, and a layer-based adaptive estimation procedure.

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“…Lemma A.3 can be proved by following Appendix D in [Agrawal et al, 2019]. Similar inequalities with constants smaller than 48 were shown in [Jin et al, 2019;Janson, 2018].…”

Section: Appendices a Concentration Inequalitiesmentioning

confidence: 52%

Fully Gap-Dependent Bounds for Multinomial Logit Bandit

Yang

2020

Preprint

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“…The Chernoff-Hoeffding inequality only applies to bounded random variables. Corollary 5.5 inLattimore and Szepesvári (2020) andJin et al (2019) establish concentration inequalities for random variables with sub-Gaussian distributions. By contrast, our result is more general since it only requires the random variable to have bounded first and second moments.4 APPLICATION: DYNAMIC POLICY CHOICE GAME…”

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confidence: 97%

Misspecified Beliefs about Time Lags

Li,

Pei

2020

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We examine the long-term behavior of a Bayesian agent who has a misspecified belief about the time lag between actions and feedback, and learns about the payoff consequences of his actions over time. Misspecified beliefs about time lags result in attribution errors, which have no long-term effect when the agent's action converges, but can lead to arbitrarily large long-term inefficiencies when his action cycles. Our proof uses concentration inequalities to bound the frequency of action switches, which are useful to study learning problems with history dependence. We apply our methods to study a policy choice game between a policy-maker who has a correctly specified belief about the time lag and the public who has a misspecified belief.

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On Asymptotically Tight Tail Bounds for Sums of Geometric and Exponential Random Variables

Abstract: In this note we prove bounds on the upper and lower probability tails of sums of independent geometric or exponentially distributed random variables. We also prove negative results showing that our established tail bounds are asymptotically tight.

Cited by 2 publications

References 4 publications

Fully Gap-Dependent Bounds for Multinomial Logit Bandit

Fully Gap-Dependent Bounds for Multinomial Logit Bandit

Misspecified Beliefs about Time Lags

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