What is the relationship between Q-Learning and fixed-point convergence?

Q-Learning converges to a fixed point of the Bellman optimality operator under certain conditions.

The Bellman optimality operator is a contraction mapping, meaning repeated application converges to a unique fixed point representing optimal Q-values. Q-learning approximates this process through stochastic updates using sampled transitions. Convergence is guaranteed in tabular settings under conditions like sufficient exploration and decaying learning rates, but not necessarily in deep Q-learning due to function approximation instability.