Fitted value iteration methods for bicausal optimal transport
This work addresses a domain-specific problem in optimal transport for researchers and practitioners dealing with time-series or adapted data structures, representing an incremental improvement in computational methods.
The paper tackles the problem of computing bicausal optimal transport with adapted couplings by developing a fitted value iteration method, which demonstrates superior scalability over linear programming and adapted Sinkhorn methods as the time horizon increases while maintaining acceptable accuracy.
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.