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Sinkhorn Distributionally Robust State Estimation via System Level Synthesis

Yulin Feng, Xianyu Li, Steven X. Ding, Hao Ye, Chao Shang

TL;DR

This work develops Sinkhorn distributionally robust state estimation (DRSE) within the System Level Synthesis framework by replacing the Wasserstein ambiguity set with an entropy-regularized Sinkhorn set. It provides the first finite-sample probabilistic guarantee for the Sinkhorn ambiguity set, and shows that Sinkhorn DRSE interpolates between $H_2$ design and Wasserstein DRSE as the entropic parameter $\epsilon$ varies. The min-max problem is reformulated as a finite-dimensional convex program via duality, and a tailored Frank-Wolfe algorithm with convergence guarantees solves it efficiently by exploiting a compact feasible set. Numerical case studies demonstrate reduced conservatism and improved out-of-sample performance, especially in small-sample regimes, while maintaining computational tractability through a first-order method. Overall, the approach offers a flexible, data-driven, and robust alternative for state estimation under distributional uncertainty in dynamical systems.

Abstract

In state estimation tasks, the usual assumption of exactly known disturbance distribution is often unrealistic and renders the estimator fragile in practice. The recently emerging Wasserstein distributionally robust state estimation (DRSE) design can partially mitigate this fragility; however, its worst-case distribution is provably discrete, which deviates from the inherent continuity of real-world distributions and results in over-pessimism. In this work, we develop a new Sinkhorn DRSE design within system level synthesis scheme with the aim of shaping the closed-loop errors under the unknown continuous disturbance distribution. For uncertainty description, we adopt the Sinkhorn ambiguity set that includes an entropic regularizer to penalize non-smooth and discrete distributions within a Wasserstein ball. We present the first result of finite-sample probabilistic guarantee of the Sinkhorn ambiguity set. Then we analyze the limiting properties of our Sinkhorn DRSE design, thereby highlighting its close connection with the generic $\mathcal{H}_2$ design and Wasserstein DRSE. To tackle the min-max optimization problem, we reformulate it as a finite-dimensional convex program through duality theory. By identifying a compact subset of the feasible set guaranteed to enclose the global optimum, we develop a tailored Frank-Wolfe solution algorithm and formally establish its convergence rate. The advantage of Sinkhorn DRSE over existing design schemes is verified through numerical case studies.

Sinkhorn Distributionally Robust State Estimation via System Level Synthesis

TL;DR

This work develops Sinkhorn distributionally robust state estimation (DRSE) within the System Level Synthesis framework by replacing the Wasserstein ambiguity set with an entropy-regularized Sinkhorn set. It provides the first finite-sample probabilistic guarantee for the Sinkhorn ambiguity set, and shows that Sinkhorn DRSE interpolates between design and Wasserstein DRSE as the entropic parameter varies. The min-max problem is reformulated as a finite-dimensional convex program via duality, and a tailored Frank-Wolfe algorithm with convergence guarantees solves it efficiently by exploiting a compact feasible set. Numerical case studies demonstrate reduced conservatism and improved out-of-sample performance, especially in small-sample regimes, while maintaining computational tractability through a first-order method. Overall, the approach offers a flexible, data-driven, and robust alternative for state estimation under distributional uncertainty in dynamical systems.

Abstract

In state estimation tasks, the usual assumption of exactly known disturbance distribution is often unrealistic and renders the estimator fragile in practice. The recently emerging Wasserstein distributionally robust state estimation (DRSE) design can partially mitigate this fragility; however, its worst-case distribution is provably discrete, which deviates from the inherent continuity of real-world distributions and results in over-pessimism. In this work, we develop a new Sinkhorn DRSE design within system level synthesis scheme with the aim of shaping the closed-loop errors under the unknown continuous disturbance distribution. For uncertainty description, we adopt the Sinkhorn ambiguity set that includes an entropic regularizer to penalize non-smooth and discrete distributions within a Wasserstein ball. We present the first result of finite-sample probabilistic guarantee of the Sinkhorn ambiguity set. Then we analyze the limiting properties of our Sinkhorn DRSE design, thereby highlighting its close connection with the generic design and Wasserstein DRSE. To tackle the min-max optimization problem, we reformulate it as a finite-dimensional convex program through duality theory. By identifying a compact subset of the feasible set guaranteed to enclose the global optimum, we develop a tailored Frank-Wolfe solution algorithm and formally establish its convergence rate. The advantage of Sinkhorn DRSE over existing design schemes is verified through numerical case studies.
Paper Structure (10 sections, 8 theorems, 83 equations, 5 figures, 1 algorithm)

This paper contains 10 sections, 8 theorems, 83 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Sinkhorn DRSE Design Problem
  5. Finite-Sample Probabilistic Guarantee
  6. Tractable Reformulation of Sinkhorn DRSE Problem
  7. Limiting Properties
  8. First-Order Solution Algorithm
  9. Simulation Case Study
  10. Conclusion

Key Result

Theorem 1

Under Assumptions assu_light and assu_nodirac, for a prescribed confidence level $\eta \in (0,1)$ and sample size $N$, one can construct an ambiguity set $\mathcal{D}_{\epsilon}(\hat{\mathbb{P}}_N,\theta_N(\eta))$ by setting the ball radius to $\theta_N(\eta)$, such that with the radius where $c_1>0$ and $c_2>0$ are constants depending only on $(\psi(a),a,n_\xi)$, and

Figures (5)

  • Figure 1: The optimal objective values of the $\mathcal{H}_2$ estimation the Wasserstein DRSE, the Sinkhorn DRSE \ref{['eq_reform']} using a grid of hyperparameters $(\theta,\epsilon)$.
  • Figure 2: Out-of-sample performance of the $\mathcal{H}_2$ estimation, the Wasserstein DRSE, the Sinkhorn DRSE \ref{['eq_reform']} using a grid of hyperparameters $(\theta,\epsilon)$ across $20,000$ Monte Carlo simulations.
  • Figure 3: The first dimension $\hat{\xi}_{i,1}$ of the samples and the worst-case univariate marginal distribution $\tilde{\mathbb{P}}_1$ of \ref{['eq_reform']} with $\theta=1$ within $4$ standard deviations of $\hat{\xi}_{i,1}$.
  • Figure 4: The upper and lower bounds of Algorithm \ref{['alg1']} in solving \ref{['eq_reform']} with $\theta=1$
  • Figure 5: The average MSE of the Sinkhorn DRSE with $(\theta=0.035,~\epsilon=10^{-3.8})$ and the Wasserstein with $\theta=0.035$ as well as solution time of Algorithm \ref{['alg1']} using different sample sizes $N$.

Theorems & Definitions (21)

  • Remark 1
  • Definition 1: Wasserstein distance, kantorovich1958space
  • Definition 2: Wasserstein ambiguity set, mohajerin2018data
  • Definition 3: Sinkhorn distance, wang2025sinkhorn
  • Definition 4: Sinkhorn ambiguity set, mohajerin2018data
  • Theorem 1
  • proof
  • Remark 2
  • Corollary 1
  • Theorem 2
  • ...and 11 more