Title: Why does the two-timescale Q-learning converge to different mean field solutions? A unified convergence analysis

Abstract: We revisit the unified two-timescale Q-learning algorithm as initially introduced by Angiuli et al. \cite{angiuli2022unified}. This algorithm demonstrates efficacy in solving mean field game (MFG) and mean field control (MFC) problems, simply by tuning the ratio of two learning rates for mean field distribution and the Q-functions respectively. In this paper, we provide a comprehensive theoretical explanation of the algorithm's bifurcated numerical outcomes under fixed learning rates. We achieve this by establishing a diagram that correlates continuous-time mean field problems to their discrete-time Q-function counterparts, forming the basis of the algorithm. Our key contribution lies in the construction of a Lyapunov function integrating both mean field distribution and Q-function iterates. This Lyapunov function facilitates a unified convergence of the algorithm across the entire spectrum of learning rates, thus providing a cohesive framework for analysis.