Title: Imitation Learning for Mean Field Games with Correlated Equilibria. Abstract: Imitation learning (IL) aims at achieving optimal actions by learning from demonstrated behaviors without knowing the reward function and transition kernels. Conducting IL with a large population of agents is challenging as agents' interactions grow exponentially with respect to the population size. Mean field theory provides an efficient tool to study multi-agent problems by aggregating information on the population level. While the approximation is tractable, it is non-trivial to restore mean field Nash equilibria (MFNE) from demonstrations. Importantly, there are many real-world problems that cannot be explained by the classic MFNE concept; this includes the traffic network equilibrium induced from the public routing recommendations and the pricing equilibrium of goods generated on the E-commerce platform.   In both examples, correlated devices are introduced to the equilibrium due to the intervention from the platform. To accommodate this, we propose a novel solution concept named adaptive mean field correlated equilibrium (AMFCE) that generalizes MFNE. On the theory side, we first prove the existence of AMFCE, and establish a novel framework based on IL and AMFCE with entropy regularization (MaxEnt-AMFCE) to recover the AMFCE policy from real-world demonstrations. Signatures from the rough path theory are then applied to characterize the mean-field evolution. A significant benefit of MaxEnt-AMFCE is that it can recover both the equilibrium policy and the correlation device from data. We test our MaxEnt-AMFCE against the state-of-the-art IL algorithms for MFGs on several tasks (including a real-world traffic flow prediction problem), results justify the effectiveness of our proposed method and show its potential to predicting and explaining large population behavior under correlated signals. 