In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.
This paper is a continuation work of Ren et al. (2026) aiming to further devise q-learning algorithms for mean-field control (MFC) with controlled common noise. Based on the relaxed control formulation, we first establish the martingale condition of the value function and the Iq-function by evaluating along the conditional state distributions generated by all test policies. As the data in the relaxed control formulation are not observable in practice, we quantify the error incurred when they are replaced by the observable ones in the exploratory formulation under discretely sampled actions. This, together with a two-layer fixed point characterization of an optimal policy in Ren et al. (2026), allows us to propose several algorithms including the Actor-Critic q-learning algorithm, in which the policy is updated in the Actor-step based on the iteration rule induced by the improved Iq-function, and the value function and Iq-function are updated in the Critic-step based on the martingale orthogonality condition using the data from the exploratory formulation. We also establish the convergence of the inner iterations in the Actor-step in an infinite-horizon linear quadratic (LQ) framework. In two examples, within and beyond LQ framework, our q-learning algorithms are implemented with satisfactory performance.