Phase Transitions and Cyclic Phenomena in Bandits with Switching Constraints

We consider the classical stochastic multi-armed bandit problem with a constraint on the total cost incurred by switching between actions. We prove matching upper and lower bounds on regret and provide near-optimal algorithms for this problem. Surprisingly, we discover phase transitions and cyclic phenomena of the optimal regret. That is, we show that associated with the multi-armed bandit problem, there are phases defined by the number of arms and switching costs, where the regret upper and lower bounds in each phase remain the same and drop significantly between phases. The results enable us to fully characterize the trade-off between regret and incurred switching cost in the stochastic multi-armed bandit problem, contributing new insights to this fundamental problem. Under the general switching cost structure, the results reveal a deep connection between bandit problems and graph traversal problems, such as the shortest Hamiltonian path problem.