Differentiating Bisimulation Metrics: A Framework for Parametric Markov Chain Fitting via Bicausal Optimal Transport

Sergio Calo, Amy Zhang, Javier Segovia-Aguas, Anders Jonsson

Many problems in sequential decision-making, such as imitation learning from observations, state-space compression, world-model learning, and sim-to-real transfer, can be reduced to learning a model such that a notion of distance with respect to the target process is minimized. We consider this general framework and consider the bisimulation metric, equivalently Bicausal Optimal Transport (BOT), as the notion of distance to minimize. We show that BOT, since it can be formulated as a linear program (LP), is differentiable with respect to the model dynamics. We then derive an exact closed-form gradient via the envelope theorem applied to the LP saddle point. The result is a general algorithm, Differentiable Bicausal Optimal Transport (D-BOT), that can be applied to each of the problems above. The proposed algorithm learns the best model by alternating between distance computation and gradient steps. We apply D-BOT for three different settings: state-space compression, parametric model learning, and imitation learning from observations (ILfO). We show empirical results that confirm the viability of all three instantiations.