In this paper, we study the learning of safe policies in the setting of reinforcement learning problems. This is, we aim to control a Markov Decision Process (MDP) of which we do not know the transition probabilities, but we have access to sample trajectories through experience. We define safety as the agent remaining in a desired safe set with high probability during the operation time. We therefore consider a constrained MDP where the constraints are probabilistic. Since there is no straightforward way to optimize the policy with respect to the probabilistic constraint in a reinforcement learning framework, we propose an ergodic relaxation of the problem. The advantages of the proposed relaxation are threefold. (i) The safety guarantees are maintained in the case of episodic tasks and they are kept up to a given time horizon for continuing tasks. (ii) The constrained optimization problem despite its non-convexity has arbitrarily small duality gap if the parametrization of the policy is rich enough. (iii) The gradients of the Lagrangian associated with the safe-learning problem can be easily computed using standard policy gradient results and stochastic approximation tools. Leveraging these advantages, we establish that primal-dual algorithms are able to find policies that are safe and optimal. We test the proposed approach in a navigation task in a continuous domain. The numerical results show that our algorithm is capable of dynamically adapting the policy to the environment and the required safety levels.
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