  While a large body of empirical results show that temporally-extended actions
and options may significantly affect the learning performance of an agent, the
theoretical understanding of how and when options can be beneficial in online
reinforcement learning is relatively limited. In this paper, we derive an upper
and lower bound on the regret of a variant of UCRL using options. While we
first analyze the algorithm in the general case of semi-Markov decision
processes (SMDPs), we show how these results can be translated to the specific
case of MDPs with options and we illustrate simple scenarios in which the
regret of learning with options can be \textit{provably} much smaller than the
regret suffered when learning with primitive actions.
