Fast rates for online learning in Linearly Solvable Markov Decision Processes
This provides a significant improvement in regret bounds for online control in a specific class of MDPs, though it is incremental as it builds on existing methods for known bottlenecks.
The paper tackles the problem of online learning in linearly solvable Markov decision processes with arbitrarily changing state costs, showing that following the leader achieves a regret of order log^2 T, improving on the previous best bound of T^{3/4}.
We study the problem of online learning in a class of Markov decision processes known as linearly 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 following 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.