Today's robots can learn the human's reward function online, during the current interaction. This real-time learning requires fast but approximate learning rules; when the human's behavior is noisy or suboptimal, today's approximations can result in unstable robot learning. Accordingly, in this paper we seek to enhance the robustness and convergence properties of gradient descent learning rules when inferring the human's reward parameters. We model the robot's learning algorithm as a dynamical system over the human preference parameters, where the human's true (but unknown) preferences are the equilibrium point. This enables us to perform Lyapunov stability analysis to derive the conditions under which the robot's learning dynamics converge. Our proposed algorithm (StROL) takes advantage of these stability conditions offline to modify the original learning dynamics: we introduce a corrective term that expands the basins of attraction around likely human rewards. In practice, our modified learning rule can correctly infer what the human is trying to convey, even when the human is noisy, biased, and suboptimal. Across simulations and a user study we find that StROL results in a more accurate estimate and less regret than state-of-the-art approaches for online reward learning. See videos here: https://youtu.be/uDGpkvJnY8g

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