Kimang Khun | Searcharxiv

arXiv2023

Testing Indexability and Computing Whittle and Gittins Index in Subcubic Time

Whittle index is a generalization of Gittins index that provides very efficient allocation rules for restless multi-armed bandits. In this work, we develop an algorithm to test the indexability and compute the Whittle indices of any finite-state restless bandit arm. This algorithm works in the discounted and non-discounted cases, and can compute Gittins index. Our algorithm builds on three tools: (1) a careful characterization of Whittle index that allows one to compute recursively the kth smallest index from the $(k - 1)$th smallest, and to test indexability, (2) the use of the Sherman-Morrison formula to make this recursive computation efficient, and (3) a sporadic use of the fastest matrix inversion and multiplication methods to obtain a subcubic complexity. We show that an efficient use of the Sherman-Morrison formula leads to an algorithm that computes Whittle index in $(2/3)n^3 + o(n^3)$ arithmetic operations, where $n$ is the number of states of the arm. The careful use of fast matrix multiplication leads to the first subcubic algorithm to compute Whittle or Gittins index: By using the current fastest matrix multiplication, the theoretical complexity of our algorithm is O(n^2.5286 ). We also develop an efficient implementation of our algorithm that can compute indices of Markov chains with several thousands of states in less than a few seconds.

cs.CC↗

arXiv2022

Reinforcement Learning for Markovian Bandits: Is Posterior Sampling more Scalable than Optimism?

We study learning algorithms for the classical Markovian bandit problem with discount. We explain how to adapt PSRL [24] and UCRL2 [2] to exploit the problem structure. These variants are called MB-PSRL and MB-UCRL2. While the regret bound and runtime of vanilla implementations of PSRL and UCRL2 are exponential in the number of bandits, we show that the episodic regret of MB-PSRL and MB-UCRL2 is $\tilde{O}(S\sqrt{nK})$ where $K$ is the number of episodes, $n$ is the number of bandits and $S$ is the number of states of each bandit (the exact bound in S, n and K is given in the paper). Up to a factor $\sqrt S$, this matches the lower bound of $Ω(\sqrt{SnK})$ that we also derive in the paper. MB-PSRL is also computationally efficient: its runtime is linear in the number of bandits. We further show that this linear runtime cannot be achieved by adapting classical non-Bayesian algorithms such as UCRL2 or UCBVI to Markovian bandit problems. Finally, we perform numerical experiments that confirm that MB-PSRL outperforms other existing algorithms in practice, both in terms of regret and of computation time.

cs.LG↗