In this paper, departing from the existing literature on IL for MFGs, we introduce a new solution concept called the Nash imitation gap.

In this paper, departing from the existing literature on IL for MFGs, we introduce a new solution concept called the Nash imitation gap.

We explore the problem of imitation learning (IL) in the context of mean-field games (MFGs), where the goal is to imitate the behavior of a population of agents following a Nash equilibrium policy according to some unknown payoff function. IL in MFGs presents new challenges compared to single-agent IL, particularly when both the reward function and the transition kernel depend on the population distribution. In this paper, departing from the existing literature on IL for MFGs, we introduce a new solution concept called the Nash imitation gap. Then we show that when only the reward depends on the population distribution, IL in MFGs can be reduced to single-agent IL with similar guarantees. However, when the dynamics is population-dependent, we provide a novel upper-bound that suggests IL is harder in this setting. To address this issue, we propose a new adversarial formulation where the reinforcement learning problem is replaced by a mean-field control (MFC) problem, suggesting progress in IL within MFGs may have to build upon MFC.

The vastness of the string landscape presents a serious computational challenge. This immensity stems from the multitude of choices for the internal manifolds on which string theory is compactified (or for non-geometric constructions, choices of conformal field theory). Even with a fixed compactification manifold, additional discrete choices--such as bundle or brane configurations, and the quantized fluxes threaded through internal cycles--further enlarge the space of solutions. Despite its vastness, the string landscape is conjectured to be finite, in the sense that there are only finitely many low energy effective field theories with a fixed, finite energy cutoff that are consistent with quantum gravity [1-3]. The finiteness of the landscape is both an important premise in the program of landscape statistics [1] and argued to be a universal property of quantum gravity [2]. It is however only when we restrict to very small regions of the landscape, e.g., intersecting D-brane models in a specific Calabi-Yau orientifold, that an exact number of solutions is known [4] (though it was shown earlier that the number is finite [5]). Compactifications of string theory on Calabi-Yau manifolds stand out as an especially well-motivated class of solutions for data mining the landscape. In particular, Calabi-Yau threefolds yield four-dimensional vacuum configurations of superstring theory that can potentially accommodate realistic particle physics coupled to gravity [6].

We establish the convergence of the deep actor-critic reinforcement learning algorithm presented in [Angiuli et al., 2023a] in the setting of continuous state and action spaces with an infinite discrete-time horizon. This algorithm provides solutions to Mean Field Game (MFG) or Mean Field Control (MFC) problems depending on the ratio between two learning rates: one for the value function and the other for the mean field term. In the MFC case, to rigorously identify the limit, we introduce a discretization of the state and action spaces, following the approach used in the finite-space case in [Angiuli et al., 2023b]. The convergence proofs rely on a generalization of the two-timescale framework introduced in [Borkar, 1997]. We further extend our convergence results to Mean Field Control Games, which involve locally cooperative and globally competitive populations. Finally, we present numerical experiments for linear-quadratic problems in one and two dimensions, for which explicit solutions are available.

Human trajectory data is crucial in urban planning, traffic engineering, and public health. However, directly using real-world trajectory data often faces challenges such as privacy concerns, data acquisition costs, and data quality. A practical solution to these challenges is trajectory generation, a method developed to simulate human mobility behaviors. Existing trajectory generation methods mainly focus on capturing individual movement patterns but often overlook the influence of population distribution on trajectory generation. In reality, dynamic population distribution reflects changes in population density across different regions, significantly impacting individual mobility behavior. Thus, we propose a novel trajectory generation framework based on a diffusion model, which integrates the dynamic population distribution constraints to guide high-fidelity generation outcomes. Specifically, we construct a spatial graph to enhance the spatial correlation of trajectories. Then, we design a dynamic population distribution aware denoising network to capture the spatiotemporal dependencies of human mobility behavior as well as the impact of population distribution in the denoising process. Extensive experiments show that the trajectories generated by our model can resemble real-world trajectories in terms of some critical statistical metrics, outperforming state-of-the-art algorithms by over 54%.

Mean field games (MFGs) have emerged as a powerful framework for modeling interactions in large-scale multi-agent systems. Despite recent advancements in reinforcement learning (RL) for MFGs, existing methods are typically limited to finite spaces or stationary models, hindering their applicability to real-world problems. This paper introduces a novel deep reinforcement learning (DRL) algorithm specifically designed for non-stationary continuous MFGs. The proposed approach builds upon a Fictitious Play (FP) methodology, leveraging DRL for best-response computation and supervised learning for average policy representation. Furthermore, it learns a representation of the time-dependent population distribution using a Conditional Normalizing Flow. To validate the effectiveness of our method, we evaluate it on three different examples of increasing complexity. By addressing critical limitations in scalability and density approximation, this work represents a significant advancement in applying DRL techniques to complex MFG problems, bringing the field closer to real-world multi-agent systems.