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Logarithmic Regret in Multisecretary and Online Linear Programs with Continuous Valuations

16 December 2019
R. Bray
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Abstract

I study how the shadow prices of a linear program that allocates an endowment of nβ∈Rmn\beta \in \mathbb{R}^{m}nβ∈Rm resources to nnn customers behave as n→∞n \rightarrow \inftyn→∞. 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 Θ(1/n)\Theta(1/n)Θ(1/n). I use these results to prove that the expected regret in \cites{Li2019b} online linear program is Θ(log⁡n)\Theta(\log n)Θ(logn), 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⁡nlog⁡log⁡n)O(\log n \log \log n)O(lognloglogn) to O(log⁡n)O(\log n)O(logn), and extend \cites{Lueker1995} Ω(log⁡n)\Omega(\log n)Ω(logn) 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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