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On the Global Optimality of Linear Policies for Sinkhorn Distributionally Robust Linear Quadratic Control

Riccardo Cescon, Andrea Martin, Giancarlo Ferrari-Trecate

Abstract

The Linear Quadratic Gaussian (LQG) regulator is a cornerstone of optimal control theory, yet its performance can degrade significantly when the noise distributions deviate from the assumed Gaussian model. To address this limitation, this work proposes a distributionally robust generalization of the finite-horizon LQG control problem. Specifically, we assume that the noise distributions are unknown and belong to ambiguity sets defined in terms of an entropy-regularized Wasserstein distance centered at a nominal Gaussian distribution. By deriving novel bounds on this Sinkhorn discrepancy and proving structural and topological properties of the resulting ambiguity sets, we establish global optimality of linear policies. Numerical experiments showcase improved distributional robustness of our control policy.

On the Global Optimality of Linear Policies for Sinkhorn Distributionally Robust Linear Quadratic Control

Abstract

The Linear Quadratic Gaussian (LQG) regulator is a cornerstone of optimal control theory, yet its performance can degrade significantly when the noise distributions deviate from the assumed Gaussian model. To address this limitation, this work proposes a distributionally robust generalization of the finite-horizon LQG control problem. Specifically, we assume that the noise distributions are unknown and belong to ambiguity sets defined in terms of an entropy-regularized Wasserstein distance centered at a nominal Gaussian distribution. By deriving novel bounds on this Sinkhorn discrepancy and proving structural and topological properties of the resulting ambiguity sets, we establish global optimality of linear policies. Numerical experiments showcase improved distributional robustness of our control policy.

Paper Structure

This paper contains 11 sections, 8 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Analysis of the Sinkhorn DR LQG problem
  5. Purified output re-parametrization
  6. Construction of a lower bound for d*
  7. Construction of an upper bound for p*
  8. Existence of a Nash equilibrium
  9. Numerical Experiments
  10. Conclusion
  11. Appendix

Key Result

Proposition 1

(Tightness for normal distributions). For any $\mathbb{P}_1 \sim \mathcal{N}(0, \Sigma_1)$ and $\mathbb{P}_2 \sim \mathcal{N}(0, \Sigma_2)$ with $\Sigma_1, \Sigma_2 \in \mathbb{S}_{++}^d$, the optimal coupling for the entropy-regularized problem eq:sinkhorn is Gaussian and is given by $\gamma_0 \sim Moreover, it holds that $W_{\epsilon}(\mathbb{P}_1, \mathbb{P}_2) = G_\epsilon(\Sigma_1, \Sigma_2)$

Figures (1)

  • Figure 1: Comparison between the control cost incurred by the nominal LQG controller (red histograms, in the background) and the proposed Sinkhorn DR LQG policy (green histograms, in the foreground) over $5000$ disturbance realizations drawn from the nominal distribution (on the left) and the respective worst-case distribution in the Sinkhorn ambiguity set (on the right). Dotted vertical lines represent theoretical mean values.

Theorems & Definitions (12)

  • Definition 1: KL divergence, kuhn2025distributionally
  • Definition 2: Sinkhorn discrepancy, sinkhorn
  • Remark 1
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Proposition 4
  • Lemma 1
  • ...and 2 more