In this paper, we present a framework for enabling autonomous vehicles to interact with cyclists in a manner that balances safety and optimality. The approach integrates Hamilton-Jacobi reachability analysis with deep Q-learning to jointly address safety guarantees and time-efficient navigation. A value function is computed as the solution to a time-dependent Hamilton-Jacobi-Bellman inequality, providing a quantitative measure of safety for each system state. This safety metric is incorporated as a structured reward signal within a reinforcement learning framework. The method further models the cyclist's latent response to the vehicle, allowing disturbance inputs to reflect human comfort and behavioral adaptation. The proposed framework is evaluated through simulation and comparison with human driving behavior and an existing state-of-the-art method.