Fitted value iteration methods for bicausal optimal transport

TL;DR

This paper introduces a fitted value iteration (FVI) approach for solving bicausal optimal transport by casting it as a dynamic program and approximating value functions with neural networks. It provides finite-sample guarantees through a concentrability coefficient and (local) Rademacher complexity, and shows that common neural-network architectures satisfy the required approximation properties, enabling rigorous analysis. Empirically, FVI scales favorably to long horizons and higher dimensions, outperforming linear programming and adapted Sinkhorn methods in challenging settings while maintaining acceptable accuracy. The work advances scalable, learning-based methods for temporally structured OT and suggests avenues for variance reduction and more general cost structures.

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.

Figures (1)

• Figure 1: A non-recombining binomial tree.

Theorems & Definitions (41)

• Definition 3.2
• Lemma 3.3
• Definition 3.4
• Lemma 3.8
• Theorem 3.10
• Corollary 3.11
• Definition 3.12
• Definition 3.13
• Lemma 3.15
• Theorem 3.16