# A Distributionally Robust Approach to Regret Optimal Control using the Wasserstein Distance

arXiv: https://arxiv.org/abs/2304.06783 · HTML: https://mlsanity.com/paper/2304.06783/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 30 references that are in the map.
- First posted on arXiv 13 Apr 2023.

## Builds on (most cited first, 13 of 13)

- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization](https://mlsanity.com/paper/1706.02586/) (2017)
- [Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator](https://mlsanity.com/paper/1805.07194/) (2018)
- [Regret-optimal Estimation and Control](https://mlsanity.com/paper/2106.12097/) (2021)
- [Data-driven distributionally robust LQR with multiplicative noise](https://mlsanity.com/paper/1912.09990/) (2019)
- [Distributional robustness in minimax linear quadratic control with Wasserstein distance](https://mlsanity.com/paper/2102.12715/) (2021)
- [Safe Control with Minimal Regret](https://mlsanity.com/paper/2203.00358/) (2022)
- [Chordal decomposition in operator-splitting methods for sparse semidefinite programs](https://mlsanity.com/paper/1707.05058/) (2017)
- [A System Level Approach to Regret Optimal Control](https://mlsanity.com/paper/2202.13763/) (2022)
- [System Level Synthesis](https://mlsanity.com/paper/1904.01634/) (2019)

## Built on by (most cited first, 3 of 3)

- [Wasserstein Distributionally Robust Regret Optimization](https://mlsanity.com/paper/2504.10796/) (2025)
- [Distributionally Robust Regret Optimal Control Under Moment-Based Ambiguity Sets](https://mlsanity.com/paper/2512.10906/) (2025)
- [Wasserstein Distributionally Robust Risk-Sensitive Estimation via Conditional Value-at-Risk](https://mlsanity.com/paper/2604.18546/) (2026)

## Abstract (arXiv)

This paper proposes a distributionally robust approach to regret optimal control of discrete-time linear dynamical systems with quadratic costs subject to a stochastic additive disturbance on the state process. The underlying probability distribution of the disturbance process is unknown, but assumed to lie in a given ball of distributions defined in terms of the type-2 Wasserstein distance. In this framework, strictly causal linear disturbance feedback controllers are designed to minimize the worst-case expected regret. The regret incurred by a controller is defined as the difference between the cost it incurs in response to a realization of the disturbance process and the cost incurred by the optimal noncausal controller which has perfect knowledge of the disturbance process realization at the outset. Building on a well-established duality theory for optimal transport problems, we derive a reformulation of the minimax regret optimal control problem as a tractable semidefinite program. Using the equivalent dual reformulation, we characterize a worst-case distribution achieving the worst-case expected regret in relation to the distribution at the center of the Wasserstein ball. We compare the minimax regret optimal control design method with the distributionally robust optimal control approach using an illustrative example and numerical experiments.