Learning Policies for Markov Decision Processes from Data

We consider the problem of learning a policy for a Markov decision process consistent with data captured on the state-actions pairs followed by the policy. We assume that the policy belongs to a class of parameterized policies which are defined using features associated with the state-action pairs. The features are known a priori, however, only an unknown subset of them could be relevant. The policy parameters that correspond to an observed target policy are recovered using $\ell_1$-regularized logistic regression that best fits the observed state-action samples. We establish bounds on the difference between the average reward of the estimated and the original policy (regret) in terms of the generalization error and the ergodic coefficient of the underlying Markov chain. To that end, we combine sample complexity theory and sensitivity analysis of the stationary distribution of Markov chains. Our analysis suggests that to achieve regret within order $O(\sqrt{\epsilon})$, it suffices to use training sample size on the order of $\Omega(\log n \cdot poly(1/\epsilon))$, where $n$ is the number of the features. We demonstrate the effectiveness of our method on a synthetic robot navigation example.