Minimax Optimization without Spurious Solutions in Optimal Transport Learning

Using neural networks to learn optimal transport (OT) maps via minimax optimization has gained increasing interest in generative modeling. However, the widely used semi-dual OT maximin formulation is known to suffer from spurious global solutions that do not correspond to valid transport maps, often addressed by regularizing the objective function. We provide an alternative explanation by showing that these spurious solutions are not stationary points of the associated loss, implying that their appearance in practice stems from the optimizer's failure to converge to stationary points in nonconvex-nonconcave settings. However, when the $c$-concavity constraints are imposed on the potential, commonly to stabilize training, we show that these spurious solutions become stationary. Based on these observations, we propose solving the semi-dual OT problem without $c$-concavity constraints, using optimizers that can reliably converge to stationary points corresponding to OT maps in nonconvex-nonconcave settings, such as a two-timescale extragradient (TTS-EG) method. At the formulation level, we further show that a Lagrangian-based OT minimax formulation provably eliminates spurious global solutions. Our empirical results demonstrate that TTS-EG reliably recovers OT maps under both semi-dual and Lagrangian-based OT formulations without any regularization.