  Mean-Field Game (MFG) serves as a crucial mathematical framework in modeling
the collective behavior of individual agents interacting stochastically with a
large population. In this work, we aim at solving a challenging class of MFGs
in which the differentiability of these interacting preferences may not be
available to the solver, and the population is urged to converge exactly to
some desired distribution. These setups are, despite being well-motivated for
practical purposes, complicated enough to paralyze most (deep) numerical
solvers. Nevertheless, we show that Schr\"odinger Bridge - as an
entropy-regularized optimal transport model - can be generalized to accepting
mean-field structures, hence solving these MFGs. This is achieved via the
application of Forward-Backward Stochastic Differential Equations theory,
which, intriguingly, leads to a computational framework with a similar
structure to Temporal Difference learning. As such, it opens up novel
algorithmic connections to Deep Reinforcement Learning that we leverage to
facilitate practical training. We show that our proposed objective function
provides necessary and sufficient conditions to the mean-field problem. Our
method, named Deep Generalized Schr\"odinger Bridge (DeepGSB), not only
outperforms prior methods in solving classical population navigation MFGs, but
is also capable of solving 1000-dimensional opinion depolarization, setting a
new state-of-the-art numerical solver for high-dimensional MFGs. Our code will
be made available at https://github.com/ghliu/DeepGSB.

Machine Learning;Computer Science and Game Theory;Machine Learning;Optimization and Control