ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport

We present a new approach for Neural Optimal Transport (NOT) training procedure, capable of accurately and efficiently estimating optimal transportation plan via specific regularization on dual Kantorovich potentials. The main bottleneck of existing NOT solvers is associated with the procedure of finding a near-exact approximation of the conjugate operator (i.e., the c-transform), which is done either by optimizing over non-convex max-min objectives or by the computationally intensive fine-tuning of the initial approximated prediction. We resolve both issues by proposing a new, theoretically justified loss in the form of expectile regularisation which enforces binding conditions on the learning process of dual potentials. Such a regularization provides the upper bound estimation over the distribution of possible conjugate potentials and makes the learning stable, completely eliminating the need for additional extensive fine-tuning. Proposed method, called Expectile-Regularised Neural Optimal Transport (ENOT), outperforms previous state-of-the-art approaches on the established Wasserstein-2 benchmark tasks by a large margin (up to a 3-fold improvement in quality and up to a 10-fold improvement in runtime). Moreover, we showcase performance of ENOT for varying cost functions on different tasks such as image generation, showing robustness of proposed algorithm. OTT-JAX library includes our implementation of ENOT algorithm https://ott-jax.readthedocs.io/en/latest/tutorials/ENOT.html