[Paper Review] Causal transport in discrete time and applications

Julio Backhoff‐Veraguas, Mathias Beiglböck+2 more|arXiv (Cornell University)|Jun 13, 2016

Point processes and geometric inequalities28 references3 citations

TL;DR

This paper establishes a dynamic programming principle for causal optimal transport in discrete time, linking it to general optimal transport problems. It identifies conditions under which the Knothe-Rosenblatt rearrangement acts as a causal analogue to the Brenier map, and derives transport-information inequalities for the nested distance, enabling discrepancy analysis in stochastic programs.

ABSTRACT

Loosely speaking, causal transport plans are a relaxation of adapted processes in the same sense as Kantorovich transport plans extend Monge-type transport maps. The corresponding causal version of the transport problem has recently been introduced by Lassalle. Working in a discrete time setup, we establish a dynamic programming principle that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. Based on this recursive principle, we give conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal analogue to the Brenier's map. Moreover, these considerations provide transport-information inequalities for the nested distance between stochastic processes pioneered by Pflug and Pichler, and so serve to gauge the discrepancy between stochastic programs driven by different noise distributions.

Motivation & Objective

• To develop a dynamic programming framework for causal optimal transport in discrete time, extending classical optimal transport to causally constrained plans.
• To characterize conditions under which the Knothe-Rosenblatt rearrangement serves as a causal counterpart to the Brenier map in the Monge problem.
• To derive transport-information inequalities for the nested distance between stochastic processes, enabling error quantification in multistage stochastic programming.
• To clarify the relationship between causal and bicausal optimal transport problems, particularly when they coincide or differ.
• To investigate the existence and structure of causal Monge maps, showing that such maps may fail to exist even when classical Monge maps do.

Proposed method

• Formalizes causal transport plans as joint distributions satisfying a time-adapted conditional independence constraint, ensuring that future decisions depend only on past information.
• Derives a dynamic programming principle by recursively minimizing cost over conditional distributions, reducing the global problem to sequential optimization.
• Establishes a dual formulation for the causal optimal transport problem and proves zero duality gap under suitable conditions.
• Applies the recursive structure to show that under separable costs and independent marginals, the Knothe-Rosenblatt map solves the causal Monge problem.
• Uses counterexamples to demonstrate that causality and bicausality may yield different optimal values when either the independence or separability assumptions fail.
• Employs the nested distance as a metric to quantify discrepancies between stochastic programs driven by different noise distributions, linking it to transport inequalities.

Experimental results

Research questions

• RQ1 Under what conditions does the Knothe-Rosenblatt rearrangement solve the causal Monge problem in discrete time?
• RQ2 When do the causal and bicausal optimal transport problems yield identical optimal values, and what structural assumptions are required?
• RQ3 How can the dynamic programming principle be used to recursively solve causal optimal transport problems with general cost functions?
• RQ4 What transport-information inequalities can be derived for the nested distance between stochastic processes using causal transport?
• RQ5 Can causal Monge maps exist when classical Monge maps do, and what conditions prevent their existence?

Key findings

• The dynamic programming principle reduces the causal optimal transport problem to a sequence of conditional minimization problems, enabling recursive solution construction.
• When the source measure is a product of its marginals and the cost is separable, the Knothe-Rosenblatt rearrangement provides an optimal causal transport map.
• In Example 7.1, the optimal causal value is 0.15, while the optimal bicausal value is 0.19, demonstrating a strict gap when the cost is non-separable.
• In Example 7.2, the optimal causal value is 2.528 and the bicausal value is 2.72, showing a gap even under separable quadratic costs when the source measure is not a product of marginals.
• Example 7.3 shows that no causal Monge map exists for certain measures even when a non-causal Monge map does, due to temporal information constraints.
• The paper establishes that the monotone regression condition from Rüschendorf (1985) is insufficient to guarantee equality between causal and bicausal problems, even for separable costs.