  In Apprenticeship Learning (AL), we are given a Markov Decision Process (MDP)
without access to the cost function. Instead, we observe trajectories sampled
by an expert that acts according to some policy. The goal is to find a policy
that matches the expert's performance on some predefined set of cost functions.
We introduce an online variant of AL (Online Apprenticeship Learning; OAL),
where the agent is expected to perform comparably to the expert while
interacting with the environment. We show that the OAL problem can be
effectively solved by combining two mirror descent based no-regret algorithms:
one for policy optimization and another for learning the worst case cost. To
this end, we derive a convergent algorithm with $O(\sqrt{K})$ regret, where $K$
is the number of interactions with the MDP, and an additional linear error term
that depends on the amount of expert trajectories available. Importantly, our
algorithm avoids the need to solve an MDP at each iteration, making it more
practical compared to prior AL methods. Finally, we implement a deep variant of
our algorithm which shares some similarities to GAIL \cite{ho2016generative},
but where the discriminator is replaced with the costs learned by the OAL
problem. Our simulations demonstrate our theoretically grounded approach
outperforms the baselines.
