Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games

Significance Mean-field games (MFGs) is an emerging field that models large populations of agents. They play a central role in many disciplines, such as economics, data science, and engineering. Since many applications come in the form of high-dimensional stochastic MFGs, numerical methods that use spatial grids are prone to the curse of dimensionality. To this end, we exploit the variational structure of potential MFGs and reformulate it as a generative adversarial network (GAN) training problem. This reformulation allays a bit the curse of dimensionality when solving high-dimensional MFGs in the stochastic setting, by avoiding spatial grids or uniform sampling in high dimensions, and instead utilizes the structure of the MFG and its connection with GANs. We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean-field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are not approachable with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex–concave saddle-point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems.