Probably Approximately Correct Learning in Adversarial Environments With Temporal Logic Specifications

Reinforcement learning (RL) algorithms have been used to learn how to implement tasks in uncertain and partially unknown environments. In practice, environments are usually uncontrolled and may affect task performance in an adversarial way. In this paper, we model the interaction between an RL agent and its potentially adversarial environment as a turn-based zero-sum stochastic game. The task requirements are represented both qualitatively as a linear temporal logic specification, and quantitatively as a reward function. For each case in which the specification is realizable and can be equivalently transformed into a deterministic Buchi automaton, we show that there always exists a memoryless almost-sure winning strategy that is epsilon-optimal for the discountedd sum objective for any arbitrary positive epsilon. We propose a probably approximately correct (PAC) learning algorithm that learns such a strategy efficiently in an online manner with a priori unknown reward functions and unknown transition distributions. To the best of our knowledge, this is the first result on PAC learning in stochastic games with independent quantitative and qualitative objectives.