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While robust optimal control theory provides a rigorous framework to compute robot control policies that are provably safe, it struggles to scale to high- dimensional problems, leading to increased use of deep learning for tractable synthesis of robot safety. Unfortunately, existing neural safety synthesis methods often lack convergence guarantees and solution interpretability. In this paper, we present Minimax Actors Guided by Implicit Critic Stackelberg (MAGICS), a novel adversarial reinforcement learning (RL) algorithm that guarantees local convergence to a minimax equilibrium solution. We then build on this approach to provide local convergence guarantees for a general deep RL-based robot safety synthesis algorithm. Through both simulation studies on OpenAI Gym environ- ments and hardware experiments with a 36-dimensional quadruped robot, we show that MAGICS can yield robust control policies outperforming the state- of-the-art neural safety synthesis methods.