Abstract
Problem-solving and reasoning involve mental exploration and navigation in sparse relational spaces. A physical analogue is spatial navigation in structured environments such as a network of burrows. Recent experiments with mice navigating a labyrinth show a sharp discontinuity during learning, corresponding to a distinct moment of "sudden insight" when mice figure out long, direct paths to the goal. This discontinuity is seemingly at odds with reinforcement learning (RL), which involves a gradual build-up of a value signal during learning. Here, we show that biologically plausible RL rules combined with persistent exploration generically exhibit discontinuous learning. In tree-like structured environments, positive feedback from learning on behavior generates a "reinforcement wave" with a steep profile. The discontinuity occurs when the wave reaches the starting point. By examining the nonlinear dynamics of reinforcement propagation, we establish a quantitative relationship between the learning rule, the agent's exploration biases, and learning speed. Predictions explain existing data and motivate specific experiments to isolate the phenomenon. Additionally, we characterize the exact learning dynamics of various RL rules for a complex sequential task.