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We consider the problem of transmitting at the optimal rate over a rapidly-varying wireless channel with unknown statistics when the feedback about channel quality is very limited. One motivation for this problem is that, in emerging wireless networks, the use of mmWave bands means that the channel quality can fluctuate rapidly and thus, one cannot rely on full channel-state feedback to make transmission rate decisions. Inspired by related problems in the context of multi-armed bandits, we consider a well-known algorithm called Thompson sampling to address this problem. However, unlike the traditional multi-armed bandit problem, a direct application of Thompson sampling results in a computational and storage complexity that grows exponentially with time. Therefore, we propose an algorithm called Modified Thompson sampling (MTS), whose computational and storage complexity is simply linear in the number of channel states and which achieves at most logarithmic regret as a function of time when compared to an optimal algorithm which knows the probability distribution of the channel states.