# Online Learning to Transport via the Minimal Selection Principle

arXiv: https://arxiv.org/abs/2202.04732 · HTML: https://mlsanity.com/paper/2202.04732/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 9 Feb 2022.

## Builds on (most cited first, 17 of 17)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Scale-Free Online Learning](https://mlsanity.com/paper/1601.01974/) (2016)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Follow the Leader If You Can, Hedge If You Must](https://mlsanity.com/paper/1301.0534/) (2013)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [Scale-Free Algorithms for Online Linear Optimization](https://mlsanity.com/paper/1502.05744/) (2015)
- [Estimating Certain Integral Probability Metric (IPM) is as Hard as Estimating under the IPM](https://mlsanity.com/paper/1911.00730/) (2019)
- [Reversible Gromov-Monge Sampler for Simulation-Based Inference](https://mlsanity.com/paper/2109.14090/) (2021)

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

- [Coin Sampling: Gradient-Based Bayesian Inference without Learning Rates](https://mlsanity.com/paper/2301.11294/) (2023)
- [Denoising Diffusions with Optimal Transport: Localization, Curvature, and Multi-Scale Complexity](https://mlsanity.com/paper/2411.01629/) (2024)
- [Efficient Transferable Optimal Transport via Min-Sliced Transport Plans](https://mlsanity.com/paper/2511.19741/) (2025)
- [Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points](https://mlsanity.com/paper/2509.16974/) (2025)
- [Blessings and Curses of Covariate Shifts: Adversarial Learning Dynamics, Directional Convergence, and Equilibria](https://mlsanity.com/paper/2212.02457/) (2022)
- [Linear Bandits beyond Inner Product Spaces, the case of Bandit Optimal Transport](https://mlsanity.com/paper/2502.07397/) (2025)

## Abstract (arXiv)

Motivated by robust dynamic resource allocation in operations research, we study the textitOnline Learning to Transport (OLT) problem where the decision variable is a probability measure, an infinite-dimensional object. We draw connections between online learning, optimal transport, and partial differential equations through an insight called the minimal selection principle, originally studied in the Wasserstein gradient flow setting by citetAmbrosio2005. This allows us to extend the standard online learning framework to the infinite-dimensional setting seamlessly. Based on our framework, we derive a novel method called the textitminimal selection or exploration (MSoE) algorithm to solve OLT problems using mean-field approximation and discretization techniques. In the displacement convex setting, the main theoretical message underpinning our approach is that minimizing transport cost over time (via the minimal selection principle) ensures optimal cumulative regret upper bounds. On the algorithmic side, our MSoE algorithm applies beyond the displacement convex setting, making the mathematical theory of optimal transport practically relevant to non-convex settings common in dynamic resource allocation.