Event cameras offer high temporal resolution and low latency, making them ideal sensors for high-speed robotic applications where conventional cameras suffer from motion blur. However, their widespread adoption in robot learning is severely bottlenecked by the computational cost of simulating high-frequency event data during online training. In this work, we present Approximate Imitation Learning, a novel framework that fundamentally resolves this bottleneck, reducing policy training time for complex, agile drone flight from 52.44 hours to just 1.86 hours - a 28x computational speedup. Our key insight is to separate representation learning from policy search. We first leverage a large-scale offline dataset to learn a task-specific representation space. Subsequently, the policy is fine-tuned through online interactions that rely solely on lightweight state information, completely eliminating the need to render events during the active policy search phase. This training paradigm drastically reduces development overhead and enables event-based control policies to scale to complex environments. Furthermore, our approach eliminates the reliance on standard cameras or intermediate representations during deployment, mapping events directly to control commands. In simulation, our method matches or exceeds the performance of standard imitation learning baselines that require full online event rendering. Finally, we successfully validate the framework in the real world, demonstrating that a policy trained via this ultra-efficient paradigm enables a quadrotor to fly through highly cluttered environments at remarkable speeds of up to 9.8 m/s.