  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.
