  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.
