On the Low-Complexity of Fair Learning for Combinatorial Multi-Armed Bandit

Combinatorial Multi-Armed Bandit with fairness constraints is a framework where multiple arms form a super arm and can be pulled in each round under uncertainty to maximize cumulative rewards while ensuring the minimum average reward required by each arm. The existing pessimistic-optimistic algorithm linearly combines virtual queue-lengths (tracking the fairness violations) and Upper Confidence Bound estimates as a weight for each arm and selects a super arm with the maximum total weight. The number of super arms could be exponential in the number of arms in many scenarios. In wireless networks, due to interference constraints the number of super arms can grow exponentially with the number of arms. Evaluating all the feasible super arms to find the one with the maximum total weight can incur extremely high computational complexity in the pessimistic-optimistic algorithm. To tackle this issue, we develop a low-complexity fair learning algorithm based on the so-called pick-and-compare approach that involves randomly picking$M$feasible super arms to evaluate. By setting$M$to a constant, the number of comparison steps in the pessimistic-optimistic algorithm can be reduced to a constant, thereby significantly reducing the computational complexity. The theoretical analysis shows that our low-complexity design sacrifices fairness and regret performance only marginally. Finally, we validate our theoretical results through extensive simulations.