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In edge computing, users' service profiles are migrated due to user mobility. Reinforcement learning (RL) frameworks have been proposed to do so, often trained on simulated data. However, existing RL frameworks overlook occasional server failures, which although rare, impact latency-sensitive applications like autonomous driving and real-time obstacle detection. Nevertheless, these failures (rare events), being not adequately represented in historical training data, pose a challenge for data-driven RL algorithms. As it is impractical to adjust failure frequency in real-world applications for training, we introduce FIRE, a framework that adapts to rare events by training a RL policy in an edge computing digital twin environment. We propose ImRE, an importance sampling-based Q-learning algorithm, which samples rare events proportionally to their impact on the value function. FIRE considers delay, migration, failure, and backup placement costs across individual and shared service profiles. We prove ImRE's boundedness and convergence to optimality. Next, we introduce novel deep Q-learning (ImDQL) and actor critic (ImACRE) versions of our algorithm to enhance scalability. We extend our framework to accommodate users with varying risk tolerances. Through trace driven experiments, we show that FIRE reduces costs compared to vanilla RL and the greedy baseline in the event of failures.