Detecting an Odd Restless Markov Arm with a Trembling Hand

TL;DR

This paper develops a theoretical framework for quickly identifying an odd Markov arm among multiple restless arms with a trembling hand, deriving asymptotic bounds and near-optimal strategies for the problem.

Contribution

It introduces the first asymptotic lower bound and near-optimal strategies for identifying an odd restless Markov arm with trembling hand errors.

Findings

Derived the first asymptotic lower bound on detection time.

Constructed strategies approaching the lower bound in vanishing error regime.

Extended analysis to restless Markov arms, unlike prior i.i.d. or rested models.

Abstract

In this paper, we consider a multi-armed bandit in which each arm is a Markov process evolving on a finite state space. The state space is common across the arms, and the arms are independent of each other. The transition probability matrix of one of the arms (the odd arm) is different from the common transition probability matrix of all the other arms. A decision maker, who knows these transition probability matrices, wishes to identify the odd arm as quickly as possible, while keeping the probability of decision error small. To do so, the decision maker collects observations from the arms by pulling the arms in a sequential manner, one at each discrete time instant. However, the decision maker has a trembling hand, and the arm that is actually pulled at any given time differs, with a small probability, from the one he intended to pull. The observation at any given time is the arm that is actually pulled and its current state. The Markov processes of the unobserved arms continue to evolve. This makes the arms restless. For the above setting, we derive the first known asymptotic lower bound on the expected time required to identify the odd arm, where the asymptotics is of vanishing error probability. The continued evolution of each arm adds a new dimension to the problem, leading to a family of Markov decision problems (MDPs) on a countable state space. We then stitch together certain parameterised solutions to these MDPs and obtain a sequence of strategies whose expected times to identify the odd arm come arbitrarily close to the lower bound in the regime of vanishing error probability. Prior works dealt with independent and identically distributed (across time) arms and rested Markov arms, whereas our work deals with restless Markov arms.