Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs

Learning population dynamics from unpaired temporal marginals is an ill-posed inverse problem that requires structural assumptions on the underlying dynamics. We introduce _Discrete Action Matching_ (DAM), a finite-state analogue of Action Matching based on discrete Wasserstein geometry on graphs. At the population level, DAM selects the canonical minimum-kinetic-energy graph velocity associated with a prescribed marginal evolution and a chosen transport geometry. For general admissible positively $1$-homogeneous mobilities, we show that the unknown density dependence of the discrete kinetic energy reduces to neighboring density ratios, which can be estimated directly from unpaired marginal samples. This yields a sample-based objective requiring neither explicit probability vectors nor marginal time derivatives, together with a graph-supported continuous-time Markov realization of the learned probability current. Experiments on controlled synthetic dynamics and real mouse gastrulation data demonstrate accurate held-out marginal prediction across diverse graph structures and finite-data regimes.