On Transportation of Mini-batches: A Hierarchical Approach

Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a novel mini-batching scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds the optimal coupling between mini-batches and it can be seen as an approximation to a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the BoMb-OT when the regularized parameter goes to infinity. Finally, we present the new algorithms of the BoMb-OT in various applications, such as deep generative models and deep domain adaptation. From extensive experiments, we observe that the BoMb-OT achieves a favorable performance in deep learning models such as deep generative models and deep domain adaptation. In other applications such as approximate Bayesian computation, color transfer, and gradient flow, the BoMb-OT also yields either a lower quantitative result or a better qualitative result than the m-OT.