# Efficient Distribution Learning with Error Bounds in Wasserstein Distance

arXiv: https://arxiv.org/abs/2602.08063 · HTML: https://mlsanity.com/paper/2602.08063/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 8 Feb 2026.

## Builds on (most cited first, 14 of 14)

- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Sampling-Based Robust Control of Autonomous Systems with Non-Gaussian Noise](https://mlsanity.com/paper/2110.12662/) (2021)
- [The cost of using exact confidence intervals for a binomial proportion](https://mlsanity.com/paper/1303.1288/) (2013)
- [Data-driven ambiguity sets with probabilistic guarantees for dynamic processes](https://mlsanity.com/paper/1909.11194/) (2019)
- [High-Confidence Data-Driven Ambiguity Sets for Time-Varying Linear Systems](https://mlsanity.com/paper/2102.01142/) (2021)
- [Efficient Uncertainty Propagation with Guarantees in Wasserstein Distance](https://mlsanity.com/paper/2506.08689/) (2025)
- [Efficient Strategy Synthesis for Switched Stochastic Systems with Distributional Uncertainty](https://mlsanity.com/paper/2212.14260/) (2022)
- [Data-Driven Strategy Synthesis for Stochastic Systems with Unknown Nonlinear Disturbances](https://mlsanity.com/paper/2406.09704/) (2024)

## Abstract (arXiv)

The Wasserstein distance has emerged as a key metric to quantify distances between probability distributions, with applications in various fields, including machine learning, control theory, decision theory, and biological systems. Consequently, learning an unknown distribution with non-asymptotic and easy-to-compute error bounds in Wasserstein distance has become a fundamental problem in many fields. In this paper, we devise a novel algorithmic and theoretical framework to approximate an unknown probability distribution mathbbP from a finite set of samples by an approximate discrete distribution widehatmathbbP while bounding the Wasserstein distance between mathbbP and widehatmathbbP. Our framework leverages optimal transport, nonlinear optimization, and concentration inequalities. In particular, we show that, even if mathbbP is unknown, the Wasserstein distance between mathbbP and widehatmathbbP can be efficiently bounded with high confidence by solving a tractable optimization problem (a mixed integer linear program) of a size that only depends on the size of the support of widehatmathbbP. This enables us to develop intelligent clustering algorithms to optimally find the support of widehatmathbbP while minimizing the Wasserstein distance error. On a set of benchmarks, we demonstrate that our approach outperforms state-of-the-art comparable methods by generally returning approximating distributions with substantially smaller support and tighter error bounds.