# Data-driven ambiguity sets with probabilistic guarantees for dynamic processes

arXiv: https://arxiv.org/abs/1909.11194 · HTML: https://mlsanity.com/paper/1909.11194/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2019.

## Built on by (most cited first, 9 of 9)

- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Distributionally Robust Policy and Lyapunov-Certificate Learning](https://mlsanity.com/paper/2404.03017/) (2024)
- [Online Optimization and Ambiguity-based Learning of Distributionally Uncertain Dynamic Systems](https://mlsanity.com/paper/2102.09111/) (2021)
- [Designing Ambiguity Sets for Distributionally Robust Optimization Using Structural Causal Optimal Transport](https://mlsanity.com/paper/2510.00599/) (2025)
- [Distributionally Robust Control with End-to-End Statistically Guaranteed Metric Learning](https://mlsanity.com/paper/2510.10214/) (2025)
- [Efficient Distribution Learning with Error Bounds in Wasserstein Distance](https://mlsanity.com/paper/2602.08063/) (2026)
- [Data-driven robust Markov decision processes on Borel spaces: performance guarantees via an axiomatic approach](https://mlsanity.com/paper/2603.08979/) (2026)
- [A Unified Perspective on Conformal Prediction and Wasserstein Distributionally Robust Optimization for Uncertainty Quantification](https://mlsanity.com/paper/2608.29789/) (2026)

## Abstract (arXiv)

Distributional ambiguity sets provide quantifiable ways to characterize the uncertainty about the true probability distribution of random variables of interest. This makes them a key element in data-driven robust optimization by exploiting high-confidence guarantees to hedge against uncertainty. This paper explores the construction of Wasserstein ambiguity sets in dynamic scenarios where data is collected progressively and may only reveal partial information about the unknown random variable. For random variables evolving according to known dynamics, we leverage assimilated samples to make inferences about their unknown distribution at the end of the sampling horizon. Under exact knowledge of the flow map, we provide sufficient conditions that relate the growth of the trajectories with the sampling rate to establish a reduction of the ambiguity set size as the horizon increases. Further, we characterize the exploitable sample history that results in a guaranteed reduction of ambiguity sets under errors in the computation of the flow and when the dynamics is subject to bounded unknown disturbances. Our treatment deals with both full- and partial-state measurements and, in the latter case, exploits the sampled-data observability properties of linear time-varying systems under irregular sampling. Simulations on a UAV detection application show the superior performance resulting from the proposed dynamic ambiguity sets.