  We consider a continuous-time multi-arm bandit problem (CTMAB), where the
learner can sample arms any number of times in a given interval and obtain a
random reward from each sample, however, increasing the frequency of sampling
incurs an additive penalty/cost. Thus, there is a tradeoff between obtaining
large reward and incurring sampling cost as a function of the sampling
frequency. The goal is to design a learning algorithm that minimizes regret,
that is defined as the difference of the payoff of the oracle policy and that
of the learning algorithm. CTMAB is fundamentally different than the usual
multi-arm bandit problem (MAB), e.g., even the single-arm case is non-trivial
in CTMAB, since the optimal sampling frequency depends on the mean of the arm,
which needs to be estimated. We first establish lower bounds on the regret
achievable with any algorithm and then propose algorithms that achieve the
lower bound up to logarithmic factors. For the single-arm case, we show that
the lower bound on the regret is $\Omega((\log T)^2/\mu)$, where $\mu$ is the
mean of the arm, and $T$ is the time horizon. For the multiple arms case, we
show that the lower bound on the regret is $\Omega((\log T)^2 \mu/\Delta^2)$,
where $\mu$ now represents the mean of the best arm, and $\Delta$ is the
difference of the mean of the best and the second-best arm. We then propose an
algorithm that achieves the bound up to constant terms.
