A Fast-Pivoting Algorithm for Whittle’s Restless Bandit Index

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)n3+O(n2) 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.

Testing Indexability and Computing Whittle and Gittins Index in Subcubic Time

“…This iterative update is very close to the one used in [5,38] for discounted restless bandit. This results in an algorithm that has the same total complexity as Subroutine 3 (of 𝑛 3 +𝑂(𝑛 2 ) or (2⇑3)𝑛 3 +𝑂(𝑛 2 ) with or without the indexability test) because both algorithms will have computed the same values of 𝑋 𝑘 𝑖 𝑗 .…”

“…Hence, the total complexity of this algorithm is 𝑛 3 +𝑜(𝑛 3 ) if we test indexability and (2⇑3)𝑛 3 +𝑜(𝑛 3 ) if we do not test indexability. Without testing the indexability, our algorithm has the same main complexity as [38]. However, the algorithm of [38] computes Whittle index only for an arm that is PCL-indexable.…”

“…Without testing the indexability, our algorithm has the same main complexity as [38]. However, the algorithm of [38] computes Whittle index only for an arm that is PCL-indexable. Hence we can claim our algorithm is the first algorithm that computes Whittle index with cubic complexity for all indexable restless bandits.…”

“…In the section below, we show how to reduce the cost by avoiding the computation of 𝑋 ℓ 𝑖𝜎 𝑘 when ℓ is much smaller than 𝑘. We comment more on the differences with [5,38] in Appendix D and in particular we explain why our approach can be tuned into a subcubic algorithm while (15) cannot. The main computational burden of Algorithm 2 is concentrated on two lines: on Line 3 where we compute 𝑋 1 by solving a linear system, and on Line 17 where we compute 𝑋 𝑘+1 ∶𝜎 𝑘 from 𝑋 1 ∶𝜎 𝑘 .…”

“…Finally, we present a detailed numerical study on the performance of our algorithm and compare it with some raw data taken from [38]. The reported results show that our algorithm is very efficient in computing Whittle index and testing indexability.…”