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              <p>We study  a generalization of  the multi-armed bandit problem  with multiple plays where there is  a cost associated with pulling each  arm and the agent has a budget at each time that dictates how much she can expect  to spend.  We  derive an asymptotic  regret lower bound  for any uniformly efficient  algorithm in our  setting. We  then study a  variant of Thompson sampling  for Bernoulli rewards  and a  variant of KL-UCB  for both single-parameter  exponential  families   and  bounded,  finitely  supported rewards. We show  these algorithms are asymptotically optimal,  both in rateand  leading  problem-dependent constants,  including  in  the thick  margin setting where multiple arms fall on the decision boundary.</p>
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