  In many online learning or multi-armed bandit problems, the taken actions or
pulled arms are ordinal and required to be monotone over time. Examples include
dynamic pricing, in which the firms use markup pricing policies to please early
adopters and deter strategic waiting, and clinical trials, in which the dose
allocation usually follows the dose escalation principle to prevent dose
limiting toxicities. We consider the continuum-armed bandit problem when the
arm sequence is required to be monotone. We show that when the unknown
objective function is Lipschitz continuous, the regret is $O(T)$. When in
addition the objective function is unimodal or quasiconcave, the regret is
$\tilde O(T^{3/4})$ under the proposed algorithm, which is also shown to be the
optimal rate. This deviates from the optimal rate $\tilde O(T^{2/3})$ in the
continuous-armed bandit literature and demonstrates the cost to the learning
efficiency brought by the monotonicity requirement.
