# Distributionally Robust Control with End-to-End Statistically Guaranteed Metric Learning

arXiv: https://arxiv.org/abs/2510.10214 · HTML: https://mlsanity.com/paper/2510.10214/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 42 references that are in the map.
- First posted on arXiv 11 Oct 2025.

## Builds on (most cited first, 29 of 29)

- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework](https://mlsanity.com/paper/1912.12970/) (2019)
- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [Wasserstein Distributionally Robust Stochastic Control: A Data-Driven Approach](https://mlsanity.com/paper/1812.09808/) (2018)
- [Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach](https://mlsanity.com/paper/2006.04361/) (2020)
- [Solution Refinement at Regular Points of Conic Problems](https://mlsanity.com/paper/1811.02157/) (2018)
- [Distributionally Robust Chance Constrained Data-enabled Predictive Control](https://mlsanity.com/paper/2006.01702/) (2020)
- [Learning Decision-Focused Uncertainty Sets in Robust Optimization](https://mlsanity.com/paper/2305.19225/) (2023)
- [Data-driven ambiguity sets with probabilistic guarantees for dynamic processes](https://mlsanity.com/paper/1909.11194/) (2019)
- [Wasserstein Distributionally Robust Motion Control for Collision Avoidance Using Conditional Value-at-Risk](https://mlsanity.com/paper/2001.04727/) (2020)
- [Distributional Reinforcement Learning with Regularized Wasserstein Loss](https://mlsanity.com/paper/2202.00769/) (2022)
- [Distributional robustness in minimax linear quadratic control with Wasserstein distance](https://mlsanity.com/paper/2102.12715/) (2021)
- [Distributionally Robust Optimal and Safe Control of Stochastic Systems via Kernel Conditional Mean Embedding](https://mlsanity.com/paper/2304.00644/) (2023)
- [A dynamic game approach to distributionally robust safety specifications for stochastic systems](https://mlsanity.com/paper/1701.06260/) (2017)
- [Differentiable Robust Model Predictive Control](https://mlsanity.com/paper/2308.08426/) (2023)
- [Constraint-Tightening and Stability in Stochastic Model Predictive Control](https://mlsanity.com/paper/1511.03488/) (2015)
- [Stochastic MPC with Offline Uncertainty Sampling](https://mlsanity.com/paper/1606.06056/) (2016)
- [A General Framework for Learning-Based Distributionally Robust MPC of Markov Jump Systems](https://mlsanity.com/paper/2106.00561/) (2021)
- [Wasserstein Distributionally Robust Control of Partially Observable Linear Stochastic Systems](https://mlsanity.com/paper/2212.04644/) (2022)
- [Minimax Q-learning Control for Linear Systems Using the Wasserstein Metric](https://mlsanity.com/paper/2010.06794/) (2020)
- [Online Learning Based Risk-Averse Stochastic MPC of Constrained Linear Uncertain Systems](https://mlsanity.com/paper/2011.11441/) (2020)
- [Ambiguity Tube MPC](https://mlsanity.com/paper/2206.09085/) (2022)
- [Data-driven distributionally robust MPC for systems with uncertain dynamics](https://mlsanity.com/paper/2209.08869/) (2022)
- [Distributionally Robust LQG control under Distributed Uncertainty](https://mlsanity.com/paper/2306.05227/) (2023)
- [Distributionally Robust Infinite-horizon Control: from a pool of samples to the design of dependable controllers](https://mlsanity.com/paper/2312.07324/) (2023)
- [Distributionally Robust Control for Chance-Constrained Signal Temporal Logic Specifications](https://mlsanity.com/paper/2409.03855/) (2024)
- [Data-Driven Distributionally Robust Mixed-Integer Control through Lifted Control Policy](https://mlsanity.com/paper/2502.05469/) (2025)

## Abstract (arXiv)

Wasserstein distributionally robust control (DRC) recently emerges as a principled paradigm for handling uncertainty in stochastic dynamical systems. However, it constructs data-driven ambiguity sets via uniform distribution shifts before sequentially incorporating them into downstream control synthesis. This segregation between ambiguity set construction and control objectives inherently introduces a structural misalignment, which undesirably leads to conservative control policies with sub-optimal performance. To address this limitation, we propose a novel end-to-end finite-horizon Wasserstein DRC framework that integrates the learning of anisotropic Wasserstein metrics with downstream control tasks in a closed-loop manner, thus enabling ambiguity sets to be systematically adjusted along performance-critical directions and yielding more effective control policies. This framework is formulated as a bilevel program: the inner level characterizes dynamical system evolution under DRC, while the outer level refines the anisotropic metric leveraging control-performance feedback across a range of initial conditions. To solve this program efficiently, we develop a stochastic augmented Lagrangian algorithm tailored to the bilevel structure. Theoretically, we prove that the learned ambiguity sets preserve statistical finite-sample guarantees under a novel radius adjustment mechanism, and we establish the well-posedness of the bilevel formulation by demonstrating its continuity with respect to the learnable metric. Furthermore, we show that the algorithm converges to stationary points of the outer level problem, which are statistically consistent with the optimal metric at a non-asymptotic convergence rate. Experiments on both numerical and inventory control tasks verify that the proposed framework achieves superior closed-loop performance and robustness compared against state-of-the-art methods.