# Dynamic Programming in Probability Spaces via Optimal Transport

arXiv: https://arxiv.org/abs/2302.13550 · HTML: https://mlsanity.com/paper/2302.13550/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2023.

## Built on by (most cited first, 6 of 6)

- [Learning diffusion at lightspeed](https://mlsanity.com/paper/2406.12616/) (2024)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [Decision-Dependent Stochastic Optimization: The Role of Distribution Dynamics](https://mlsanity.com/paper/2503.07324/) (2025)
- [Go With the Flow: Fast Diffusion for Gaussian Mixture Models](https://mlsanity.com/paper/2412.09059/) (2024)
- [Markov Decision Processes of the Third Kind: Learning Distributions by Policy Gradient Descent](https://mlsanity.com/paper/2602.06567/) (2026)
- [Causal Optimal Coupling for Gaussian Input-Output Distributional Data](https://mlsanity.com/paper/2604.01406/) (2026)

## Abstract (arXiv)

We study discrete-time finite-horizon optimal control problems in probability spaces, whereby the state of the system is a probability measure. We show that, in many instances, the solution of dynamic programming in probability spaces results from two ingredients: (i) the solution of dynamic programming in the "ground space" (i.e., the space on which the probability measures live) and (ii) the solution of an optimal transport problem. From a multi-agent control perspective, a separation principle holds: The "low-level control of the agents of the fleet" (how does one reach the destination?) and "fleet-level control" (who goes where?) are decoupled.