Tabular and Deep Learning for the Whittle Index
This work addresses the challenge of efficiently solving RMABPs, which are important for resource allocation in areas like healthcare and communications, though it is incremental as it builds on prior Whittle index heuristics with new algorithmic implementations.
The authors tackled the problem of learning Whittle indices for Restless Multi-Armed Bandit Problems by developing QWI and QWINN, two reinforcement learning algorithms that converge faster than existing methods, with QWI proving convergence to real indices and QWINN showing stability in deep Q-network schemes.
The Whittle index policy is a heuristic that has shown remarkably good performance (with guaranteed asymptotic optimality) when applied to the class of problems known as Restless Multi-Armed Bandit Problems (RMABPs). In this paper we present QWI and QWINN, two reinforcement learning algorithms, respectively tabular and deep, to learn the Whittle index for the total discounted criterion. The key feature is the use of two time-scales, a faster one to update the state-action Q -values, and a relatively slower one to update the Whittle indices. In our main theoretical result we show that QWI, which is a tabular implementation, converges to the real Whittle indices. We then present QWINN, an adaptation of QWI algorithm using neural networks to compute the Q -values on the faster time-scale, which is able to extrapolate information from one state to another and scales naturally to large state-space environments. For QWINN, we show that all local minima of the Bellman error are locally stable equilibria, which is the first result of its kind for DQN-based schemes. Numerical computations show that QWI and QWINN converge faster than the standard Q -learning algorithm, neural-network based approximate Q-learning and other state of the art algorithms.