We study the problem of online learning Markov Decision Processes (MDPs) when both the transition distributions and loss functions are chosen by an adversary. We present an algorithm that, under a mixing assumption, achieves O(\sqrt{T\log \Pi } \log \Pi) regret with respect to a comparison set of policies \Pi . The regret is independent of the size of the state and action spaces. When expectations over sample paths can be computed efficiently and the comparison set \Pi has polynomial size, this algorithm is efficient. We also consider the episodic adversarial online shortest path problem.

We study the problem of online learning Markov Decision Processes (MDPs) when both the transition distributions and loss functions are chosen by an adversary. We present an algorithm that, under a mixing assumption, achieves O(\sqrt{T\log \Pi } \log \Pi) regret with respect to a comparison set of policies \Pi . The regret is independent of the size of the state and action spaces. When expectations over sample paths can be computed efficiently and the comparison set \Pi has polynomial size, this algorithm is efficient. We also consider the episodic adversarial online shortest path problem.

We study the problem of online learning Markov Decision Processes (MDPs) when both the transition distributions and loss functions are chosen by an adversary. We present an algorithm that, under a mixing assumption, achieves $O(\sqrt{T\log \Pi } \log \Pi)$ regret with respect to a comparison set of policies $\Pi$. The regret is independent of the size of the state and action spaces. When expectations over sample paths can be computed efficiently and the comparison set $\Pi$ has polynomial size, this algorithm is efficient. We also consider the episodic adversarial online shortest path problem.

We study the problem of learning Markov decision processes with finite state and action spaces when the transition probability distributions and loss functions are chosen adversarially and are allowed to change with time. We introduce an algorithm whose regret with respect to any policy in a comparison class grows as the square root of the number of rounds of the game, provided the transition probabilities satisfy a uniform mixing condition. Our approach is efficient as long as the comparison class is polynomial and we can compute expectations over sample paths for each policy. Designing an efficient algorithm with small regret for the general case remains an open problem.