Logarithmic Regret in Multisecretary and Online Linear Programs with Continuous Valuations

• URL: http://arxiv.org/abs/1912.08917v6
• Date: Mon, 28 Aug 2023 16:46:50 GMT
• Title: Logarithmic Regret in Multisecretary and Online Linear Programs with Continuous Valuations
• Authors: Robert L. Bray
• Abstract summary: I study how the shadow prices of a linear program that allocates an endowment of $nbeta in mathbbRm$ resources to $n$ customers behave as $n rightarrow infty$. I use these results to prove that the expected regret in citesLi 2019b online linear program is $Theta(log n)$, both when the customer variable distribution is known upfront and must be learned on the fly.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: I study how the shadow prices of a linear program that allocates an endowment of $n\beta \in \mathbb{R}^{m}$ resources to $n$ customers behave as $n \rightarrow \infty$. I show the shadow prices (i) adhere to a concentration of measure, (ii) converge to a multivariate normal under central-limit-theorem scaling, and (iii) have a variance that decreases like $\Theta(1/n)$. I use these results to prove that the expected regret in \cites{Li2019b} online linear program is $\Theta(\log n)$, both when the customer variable distribution is known upfront and must be learned on the fly. I thus tighten \citeauthors{Li2019b} upper bound from $O(\log n \log \log n)$ to $O(\log n)$, and extend \cites{Lueker1995} $\Omega(\log n)$ lower bound to the multi-dimensional setting. I illustrate my new techniques with a simple analysis of \cites{Arlotto2019} multisecretary problem.

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