Multi-armed Bandits with Constrained Arms and Hidden States

The problem of rested and restless multi-armed bandits with constrained availability of arms is considered. The states of arms evolve in Markovian manner and the exact states are hidden from the decision maker. First, some structural results on value functions are claimed. Following these results, the optimal policy turns out to be a \textit{threshold policy}. Further, \textit{indexability} of rested bandits is established and index formula is derived. The performance of index policy is illustrated and compared with myopic policy using numerical examples.Comment: 17 pages, 2 figure

Constrained Restless Bandits for Dynamic Scheduling in Cyber-Physical Systems

This paper studies a class of constrained restless multi-armed bandits (CRMAB). The constraints are in the form of time varying set of actions (set of available arms). This variation can be either stochastic or semi-deterministic. Given a set of arms, a fixed number of them can be chosen to be played in each decision interval. The play of each arm yields a state dependent reward. The current states of arms are partially observable through binary feedback signals from arms that are played. The current availability of arms is fully observable. The objective is to maximize long term cumulative reward. The uncertainty about future availability of arms along with partial state information makes this objective challenging. Applications for CRMAB abound in the domain of cyber-physical systems. First, this optimization problem is analyzed using Whittle's index policy. To this end, a constrained restless single-armed bandit is studied. It is shown to admit a threshold-type optimal policy and is also indexable. An algorithm to compute Whittle's index is presented. An alternate solution method with lower complexity is also presented in the form of an online rollout policy. Further, upper bounds on the value function are derived in order to estimate the degree of sub-optimality of various solutions. The simulation study compares the performance of Whittle's index, online rollout, myopic and modified Whittle's index policies.Comment: 14 pages, 2 figure

Sequential Decision Making with Limited Observation Capability: Application to Wireless Networks

This work studies a generalized class of restless multi-armed bandits with hidden states and allow cumulative feedback, as opposed to the conventional instantaneous feedback. We call them lazy restless bandits (LRB) as the events of decision-making are sparser than events of state transition. Hence, feedback after each decision event is the cumulative effect of the following state transition events. The states of arms are hidden from the decision-maker and rewards for actions are state dependent. The decision-maker needs to choose one arm in each decision interval, such that long term cumulative reward is maximized. As the states are hidden, the decision-maker maintains and updates its belief about them. It is shown that LRBs admit an optimal policy which has threshold structure in belief space. The Whittle-index policy for solving LRB problem is analyzed; indexability of LRBs is shown. Further, closed-form index expressions are provided for two sets of special cases; for more general cases, an algorithm for index computation is provided. An extensive simulation study is presented; Whittle-index, modified Whittle-index and myopic policies are compared. Lagrangian relaxation of the problem provides an upper bound on the optimal value function; it is used to assess the degree of sub-optimality various policies