Fitted value iteration methods for bicausal optimal transport

TL;DR

This paper introduces a fitted value iteration approach for bicausal optimal transport, leveraging neural networks and theoretical analysis to improve scalability and efficiency over existing methods.

Contribution

The paper develops a novel FVI method for bicausal OT, providing theoretical guarantees and demonstrating superior scalability with neural network approximations.

Findings

FVI outperforms linear programming in scalability

Neural networks satisfy key assumptions for sample complexity

Method maintains accuracy with increasing time horizon

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.