Logarithmic Regret in Multisecretary and Online Linear Programs with Continuous Valuations

TL;DR

This paper analyzes the behavior of shadow prices in large-scale linear programs with resource allocation, showing they concentrate, converge to normal distributions, and lead to a logarithmic regret bound in online decision problems.

Contribution

It establishes new concentration and distributional results for shadow prices, tightening regret bounds in online linear programs and extending classical lower bounds to multi-dimensional cases.

Findings

Shadow prices concentrate and are normally distributed as resources grow large.

Expected regret in online linear programs is proven to be Θ(log n).

Extends lower bounds to multi-dimensional resource allocation settings.

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.