  We consider the classical stochastic multi-armed bandit problem with a
constraint that limits the total cost incurred by switching between actions to
be no larger than a given switching budget. For this problem, we prove matching
upper and lower bounds on the optimal (i.e., minimax) regret, and provide
efficient rate-optimal algorithms. Surprisingly, the optimal regret of this
problem exhibits a non-conventional growth rate in terms of the time horizon
and the number of arms. Consequently, we discover surprising "phase
transitions" regarding how the optimal regret rate changes with respect to the
switching budget: when the number of arms is fixed, there are equal-length
phases, where the optimal regret rate remains (almost) the same within each
phase and exhibits abrupt changes between phases; when the number of arms grows
with the time horizon, such abrupt changes become subtler and may disappear,
but a generalized notion of phase transitions involving certain new
measurements still exists. The results enable us to fully characterize the
trade-off between the regret rate and the 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 interesting connections between bandit problems and graph traversal
problems, such as the shortest Hamiltonian path problem.
