Active Information Acquisition for Linear Optimization

TL;DR

This paper introduces algorithms for efficiently acquiring missing information in linear programs, enabling approximate solutions with minimal sampling by focusing on unknown parameters in objectives or constraints.

Contribution

It provides an information-theoretic bound for unknown objective parameters and an efficient adaptive algorithm for unknown constraints, improving sample efficiency in linear optimization.

Findings

Matching upper and lower bounds for sample complexity in objective parameter case.

An efficient adaptive algorithm for constraint parameters with proven sample bounds.

Simulation results showing improved performance over naive methods.

Abstract

We consider partially-specified optimization problems where the goal is to actively, but efficiently, acquire missing information about the problem in order to solve it. An algorithm designer wishes to solve a linear program (LP), $\max \mathbf{c}^T \mathbf{x}$ s.t. $\mathbf{A}\mathbf{x} \leq \mathbf{b}, \mathbf{x} \ge \mathbf{0}$, but does not initially know some of the parameters. The algorithm can iteratively choose an unknown parameter and gather information in the form of a noisy sample centered at the parameter's (unknown) value. The goal is to find an approximately feasible and optimal solution to the underlying LP with high probability while drawing a small number of samples. We focus on two cases. (1) When the parameters $\mathbf{c}$ of the objective are initially unknown, we take an information-theoretic approach and give roughly matching upper and lower sample complexity bounds, with an (inefficient) successive-elimination algorithm. (2) When the parameters $\mathbf{b}$ of the constraints are initially unknown, we propose an efficient algorithm combining techniques from the ellipsoid method for LP and confidence-bound approaches from bandit algorithms. The algorithm adaptively gathers information about constraints only as needed in order to make progress. We give sample complexity bounds for the algorithm and demonstrate its improvement over a naive approach via simulation.