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Fitted Value Iteration Methods for Bicausal Optimal Transport

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  • Erhan Bayraktar
  • Bingyan Han

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.

Suggested Citation

  • Erhan Bayraktar & Bingyan Han, 2023. "Fitted Value Iteration Methods for Bicausal Optimal Transport," Papers 2306.12658, arXiv.org, revised Nov 2023.
    • Handle: RePEc:arx:papers:2306.12658
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    References listed on IDEAS

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    1. Julio Backhoff-Veraguas & Daniel Bartl & Mathias Beiglböck & Manu Eder, 2020. "Adapted Wasserstein distances and stability in mathematical finance," Finance and Stochastics, Springer, vol. 24(3), pages 601-632, July.
    2. Alois Pichler & Michael Weinhardt, 2022. "The nested Sinkhorn divergence to learn the nested distance," Computational Management Science, Springer, vol. 19(2), pages 269-293, June.
    3. Tobias Achterberg & Robert E. Bixby & Zonghao Gu & Edward Rothberg & Dieter Weninger, 2020. "Presolve Reductions in Mixed Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 473-506, April.
    4. Langer, Sophie, 2021. "Approximating smooth functions by deep neural networks with sigmoid activation function," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    5. Julio Backhoff-Veraguas & Daniel Bartl & Mathias Beiglbock & Manu Eder, 2019. "Adapted Wasserstein Distances and Stability in Mathematical Finance," Papers 1901.07450, arXiv.org, revised May 2020.
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