A fast-pivoting algorithm for Whittle's restless bandit index

TL;DR

This paper introduces a novel fast-pivoting algorithm that efficiently computes Whittle's index for large-state restless bandits, significantly reducing computational complexity and runtime compared to existing methods.

Contribution

The paper presents a new algorithm leveraging parametric simplex methods and pattern analysis to compute Whittle's index with improved speed for large state spaces.

Findings

Substantial runtime speed-ups demonstrated in numerical experiments

Algorithm reduces complexity to approximately (2/3) n^3 operations after initialization

Effective for large-state restless bandit problems

Abstract

The Whittle index for restless bandits (two-action semi-Markov decision processes) provides an intuitively appealing optimal policy for controlling a single generic project that can be active (engaged) or passive (rested) at each decision epoch, and which can change state while passive. It further provides a practical heuristic priority-index policy for the computationally intractable multi-armed restless bandit problem, which has been widely applied over the last three decades in multifarious settings, yet mostly restricted to project models with a one-dimensional state. This is due in part to the difficulty of establishing indexability (existence of the index) and of computing the index for projects with large state spaces. This paper draws on the author's prior results on sufficient indexability conditions and an adaptive-greedy algorithmic scheme for restless bandits to obtain a new fast-pivoting algorithm that computes the $n$ Whittle index values of an $n$-state restless bandit by performing, after an initialization stage, $n$ steps that entail $(2/3) n^3 + O(n^2)$ arithmetic operations. This algorithm also draws on the parametric simplex method, and is based on elucidating the pattern of parametric simplex tableaux, which allows to exploit special structure to substantially simplify and reduce the complexity of simplex pivoting steps. A numerical study demonstrates substantial runtime speed-ups versus alternative algorithms.