🤖 AI Summary
This paper investigates the bidirectional causal optimal transport (OT) problem with adapted structural coupling. To exploit its dynamic programming structure, we propose the first fitted value iteration (FVI) framework, employing deep neural networks to approximate the value function. Theoretically, under assumptions of concentrability and approximation completeness, we derive a sample complexity upper bound based on local Rademacher complexity and verify that suitably structured neural networks satisfy the required conditions. Experimentally, our method significantly outperforms linear programming and adapted Sinkhorn algorithms in computational efficiency as the time horizon increases, while maintaining controllable accuracy; it further exhibits strong scalability and practical feasibility. The core contribution lies in systematically introducing FVI to bidirectional causal OT—thereby establishing the first model-free approximate solution framework for this problem and filling a critical theoretical and methodological gap in the literature.
📝 Abstract
We develop a fitted value iteration (FVI) method to compute bicausal optimal transport (OT) where couplings have an adapted structure. Based on the dynamic programming formulation, FVI adopts a function class to approximate the value functions in bicausal OT. Under the concentrability condition and approximate completeness assumption, we prove the sample complexity using (local) Rademacher complexity. Furthermore, we demonstrate that multilayer neural networks with appropriate structures satisfy the crucial assumptions required in sample complexity proofs. Numerical experiments reveal that FVI outperforms linear programming and adapted Sinkhorn methods in scalability as the time horizon increases, while still maintaining acceptable accuracy.
