English

Logarithmic Regret in Multisecretary and Online Linear Programs with Continuous Valuations

Machine Learning 2023-08-30 v6 Computer Science and Game Theory

Abstract

I study how the shadow prices of a linear program that allocates an endowment of nβRmn\beta \in \mathbb{R}^{m} resources to nn customers behave as nn \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 Θ(1/n)\Theta(1/n). I use these results to prove that the expected regret in \cites{Li2019b} online linear program is Θ(logn)\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(lognloglogn)O(\log n \log \log n) to O(logn)O(\log n), and extend \cites{Lueker1995} Ω(logn)\Omega(\log n) lower bound to the multi-dimensional setting. I illustrate my new techniques with a simple analysis of \cites{Arlotto2019} multisecretary problem.

Keywords

Cite

@article{arxiv.1912.08917,
  title  = {Logarithmic Regret in Multisecretary and Online Linear Programs with Continuous Valuations},
  author = {Robert L. Bray},
  journal= {arXiv preprint arXiv:1912.08917},
  year   = {2023}
}
R2 v1 2026-06-23T12:50:23.986Z