How the Experts Algorithm Can Help Solve LPs Online

TL;DR

This paper presents a primal-dual online algorithm for solving packing/covering linear programs with random column order, achieving near-optimal bounds using online learning and martingale techniques.

Contribution

It introduces a modular primal-dual algorithm leveraging regret minimization for dual solutions, matching known bounds for online LP approximation.

Findings

Achieves $(1+\epsilon)$-approximation with $\Omega(\epsilon^{-2} \log m)$ right-hand sides.

Uses online learning to construct dual solutions in a black-box manner.

Employs martingale concentration to handle correlations in the algorithm.

Abstract

We consider the problem of solving packing/covering LPs online, when the columns of the constraint matrix are presented in random order. This problem has received much attention and the main focus is to figure out how large the right-hand sides of the LPs have to be (compared to the entries on the left-hand side of the constraints) to allow $(1+\epsilon)$-approximations online. It is known that the right-hand sides have to be $\Omega(\epsilon^{-2} \log m)$ times the left-hand sides, where $m$ is the number of constraints. In this paper we give a primal-dual algorithm that achieve this bound for mixed packing/covering LPs. Our algorithms construct dual solutions using a regret-minimizing online learning algorithm in a black-box fashion, and use them to construct primal solutions. The adversarial guarantee that holds for the constructed duals helps us to take care of most of the correlations that arise in the algorithm; the remaining correlations are handled via martingale concentration and maximal inequalities. These ideas lead to conceptually simple and modular algorithms, which we hope will be useful in other contexts.