Fast rates for online learning in Linearly Solvable Markov Decision Processes

We study the problem of online learning in a class of Markov decision processes known as \emphlinearly solvable MDPs. In the stationary version of this problem, a learner interacts with its environment by directly controlling the state transitions, attempting to balance a fixed state-dependent cost and a certain smooth cost penalizing extreme control inputs. In the current paper, we consider an online setting where the state costs may change arbitrarily between consecutive rounds, and the learner only observes the costs at the end of each respective round. We are interested in constructing algorithms for the learner that guarantee small regret against the best stationary control policy chosen in full knowledge of the cost sequence. Our main result is showing that the smoothness of the control cost enables the simple algorithm of \emphfollowing the leader to achieve a regret of order $\log^2 T$ after $T$ rounds, vastly improving on the best known regret bound of order $T^3/4$ for this setting.