⚡ electrical engineering

Sinkhorn Ambiguity Sets for Distributionally Robust Control: Convexity, Weak Compactness, and Tractability

This paper establishes the convexity and weak compactness of Sinkhorn ambiguity sets to formulate a tractable convex optimization framework for distributionally robust linear quadratic control, offering a data-efficient alternative to Wasserstein-based methods that effectively combines observed data with prior knowledge.

Original authors: Riccardo Cescon, Andrea Martin, Giancarlo Ferrari-Trecate

Published 2026-05-06

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Riccardo Cescon, Andrea Martin, Giancarlo Ferrari-Trecate

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a large airplane, the Boeing B-747, trying to fly from point A to point B. You want to get there efficiently, but there's a problem: the wind is unpredictable.

In the old days of flying (classical control), engineers assumed they knew the wind perfectly. They would say, "The wind will be exactly 10 mph from the north," and plan the flight accordingly. But in reality, the wind is messy. Sometimes it's 5 mph, sometimes 20, and sometimes it gusts in weird directions. If you plan for the "average" wind, a sudden strong gust could push your plane off course, violating safety rules.

On the other hand, if you try to be super safe (robust control), you assume the wind will be the absolute worst possible scenario imaginable—like a hurricane hitting the plane from every angle at once. You would fly very slowly and take a huge detour to be safe. While you'd never crash, you'd waste a lot of fuel and time. This is too conservative.

The Middle Ground: Distributionally Robust Control (DRC)
This paper proposes a smarter way to fly. Instead of guessing the exact wind or fearing the absolute worst, the engineers say: "We have some data from past flights (samples), but we know it's not perfect. Let's assume the real wind is somewhere in a 'cloud' of possibilities around our data."

This "cloud" is called an Ambiguity Set. The goal is to design a flight path that works well even if the wind turns out to be the worst version inside that specific cloud.

The Problem with the Old "Cloud" (Wasserstein Distance)
For a while, the best way to define this cloud was using a mathematical tool called the Wasserstein distance. However, this tool has a weird quirk. If your data comes from a few specific wind samples (like 5 recorded gusts), the "worst-case" wind the computer calculates is also just a few specific points. It's like saying the worst possible wind is only a gust from the north or only a gust from the south, ignoring the fact that real wind is a smooth, continuous flow that can blow from anywhere in between. This forces the plane to be overly cautious because the math thinks the wind is "spiky" and unpredictable in a way nature isn't.

The New Solution: The "Sinkhorn" Cloud
The authors of this paper introduce a new tool called the Sinkhorn discrepancy. Think of this as a new way to draw the "cloud" of uncertainty.

The Metaphor: Imagine your data is a few drops of ink on a piece of paper.
- The Wasserstein method says the worst-case scenario is just a few other drops of ink. It doesn't fill in the space between them.
- The Sinkhorn method says, "Let's assume the ink spreads out smoothly like water." It allows the worst-case wind to be a smooth, continuous flow that covers the whole sky, not just a few dots.

This is a huge advantage because real-world noise (like wind turbulence) is usually smooth and continuous, not made of isolated dots. By using Sinkhorn, the computer can say, "Okay, the wind might be a smooth breeze from the northeast," rather than just "North" or "East." This leads to a flight plan that is safer than the "average" plan but less wasteful than the "worst-case" plan.

What Did They Prove?
The paper does three main things:

Mathematical Safety: They proved that these new "Sinkhorn clouds" are mathematically well-behaved. They are "convex" (shaped like a smooth bowl, not a jagged rock) and "compact" (they don't stretch out to infinity). This is important because it means the math problems used to design the flight path can actually be solved by computers without getting stuck.
Solving the Puzzle: They showed how to turn the complex problem of "finding the worst wind in the cloud" into a standard, solvable math problem (convex programming). This means engineers can actually use this method on real computers.
The Test Drive: They tested this on a simulation of a Boeing B-747 flying through realistic wind turbulence.
- The "average" flight plan crashed (violated safety constraints).
- The "super safe" plan was too conservative.
- The new Sinkhorn plan kept the plane safe and used less fuel than the super safe plan. It found the sweet spot.

In Summary
This paper gives engineers a new, better tool to handle uncertainty. By using the Sinkhorn discrepancy, they can create a "cloud" of possible future events that looks more like the real world (smooth and continuous) rather than a jagged collection of data points. This allows them to design control systems (like autopilots) that are safe enough to prevent accidents but efficient enough to save energy, all while being mathematically guaranteed to work.

Technical Summary: Sinkhorn Ambiguity Sets for Distributionally Robust Control

Problem Statement

Classical stochastic control relies on the assumption that the true probability distribution of uncertainty is known, an assumption often violated in practice where only finite samples are available. While robust control addresses uncertainty via worst-case analysis, it often yields overly conservative solutions. Distributionally Robust Optimization (DRO) offers a middle ground by optimizing against the worst-case distribution within an "ambiguity set" centered on a nominal distribution.

However, existing DRO approaches based on the Wasserstein distance (a standard Optimal Transport metric) face significant limitations. When the nominal distribution is discrete (e.g., an empirical distribution from data), the resulting worst-case distribution is also discrete. This fails to capture the continuous nature of many physical noise processes. Furthermore, achieving high-probability guarantees with Wasserstein sets often requires large radii when data is scarce, leading to excessive conservatism.

This paper addresses these limitations by proposing a Distributionally Robust Control (DRC) framework utilizing Sinkhorn ambiguity sets. These sets are defined via the Sinkhorn discrepancy, an entropy-regularized variant of Optimal Transport. The core challenge is to establish the theoretical properties of these sets and derive a tractable reformulation for linear quadratic control problems with probabilistic safety constraints.

Methodology

1. Theoretical Foundations of Sinkhorn Ambiguity Sets

The authors first characterize the geometric and topological properties of the Sinkhorn ambiguity set, denoted as $\mathcal{B}_{\rho, \epsilon}(P)$ . This set contains all probability distributions $Q$ such that the Sinkhorn discrepancy $W_c^\epsilon(Q, P)$ is less than or equal to a radius $\rho$ .

Convexity: The paper proves that the Sinkhorn ambiguity set is convex. This is a critical property for ensuring the tractability of optimization problems defined over these sets.
Weak Compactness: Under standard assumptions on the transportation cost (specifically, that it grows sufficiently fast), the authors establish that the Sinkhorn ambiguity set is weakly compact. This property is essential for guaranteeing the existence of worst-case distributions and for applying minimax theorems.
Comparison with Wasserstein: Unlike Wasserstein sets, where the worst-case distribution is discrete if the center is discrete, the Sinkhorn discrepancy allows the worst-case distribution to share the support of a reference measure $\nu$ . By choosing a continuous reference measure (e.g., a Gaussian), the worst-case distribution remains continuous, offering a more realistic modeling of uncertainty.

2. Reformulation of Distributionally Robust CVaR Constraints

The control problem involves Conditional Value-at-Risk (CVaR) constraints to ensure safety with a specified confidence level. The authors leverage the convexity and weak compactness of the Sinkhorn set to interchange the supremum (over distributions) and infimum (over risk thresholds) in the definition of worst-case CVaR.

They derive a strong dual formulation for the worst-case expected cost and CVaR constraints.
When the loss function is piecewise affine (common in polytopic constraints) and the reference measure is Gaussian, the infinite-dimensional optimization over distributions is reduced to a finite-dimensional convex optimization problem.
This reformulation recovers known Wasserstein results in the limit as the regularization parameter $\epsilon \to 0$ .

3. Tractable Control Synthesis via System Level Synthesis (SLS)

The paper applies these results to a finite-horizon linear quadratic control problem with probabilistic safety constraints.

Policy Restriction: The authors restrict the search space to affine feedback policies.
SLS Parametrization: Instead of optimizing controller gains directly, they employ the System Level Synthesis (SLS) framework. This shifts the design variables to the closed-loop system response maps ( $\Phi, \phi$ ), which linearly map disturbances to states and inputs.
Convex Program: By combining the SLS parametrization with the dual reformulation of the Sinkhorn DR constraints, the authors demonstrate that the entire control synthesis problem can be cast as a convex optimization program (specifically, a conic program when $\epsilon > 0$ ).
Feasibility Conditions: The paper provides explicit conditions on the ambiguity set radius $\rho$ required for the problem to be feasible.

Key Contributions

Topological Properties: The paper establishes the convexity and weak compactness of Sinkhorn ambiguity sets. These are presented as results of independent interest to the optimal transport literature, serving as a foundation for solving distributionally robust constrained control problems.
Tractable Reformulation: The authors derive a finite-dimensional, convex reformulation for the Sinkhorn Distributionally Robust Linear Quadratic (DR-LQ) control problem. This formulation handles both the objective function and probabilistic safety constraints (CVaR) simultaneously.
Continuous Support Characterization: The work highlights the ability of Sinkhorn sets to maintain continuous support for worst-case distributions when a continuous reference measure is used, contrasting this with the discrete support limitation of Wasserstein-based DRO.
Numerical Validation: The proposed method is validated on a trajectory planning problem for a Boeing B-747 aircraft subject to wind turbulence modeled by a Dryden filter.

Results and Experimental Findings

The numerical experiments compare three controllers:

$\pi_{emp}$ : A certainty-equivalence controller using the empirical distribution (ignoring uncertainty).
$\pi_{W}$ : A Wasserstein-based DRO controller.
$\pi_{S}$ : The proposed Sinkhorn-based DRO controller.

Key Observations:

Performance vs. Conservatism: The empirical controller ( $\pi_{emp}$ ) exhibited high control costs and significant constraint violations on unseen test data. Both robust controllers ( $\pi_{W}$ and $\pi_{S}$ ) significantly outperformed the empirical baseline.
Reduced Conservatism: The Sinkhorn controller ( $\pi_{S}$ ) demonstrated reduced conservatism compared to the Wasserstein controller ( $\pi_{W}$ ). While $\pi_{S}$ incurred a slightly higher constraint violation rate than $\pi_{W}$ for very small radii, it consistently met the required safety thresholds while achieving substantially lower control costs.
Trajectory Behavior: Visualizations of terminal states showed that $\pi_{S}$ trajectories remained closer to the boundary of the feasible region compared to $\pi_{W}$ , indicating that the Sinkhorn ambiguity set allows for a less conservative utilization of the safety margin without compromising safety guarantees.
Computational Complexity: The problem is solved using a golden section search over the dual variable $\lambda$ , with each iteration solving a conic optimization problem. The approach is computationally feasible for the tested system dimensions.

Significance and Claims

The paper claims that the proposed approach offers a principled way to balance robustness and performance in control design when data is scarce. By utilizing the Sinkhorn discrepancy, the method:

Avoids Discrete Artifacts: It prevents the worst-case distribution from collapsing into a discrete set of points, thereby better capturing the continuous nature of physical noise.
Integrates Prior Knowledge: It allows for the smooth integration of prior knowledge (via the reference measure $\nu$ ) with empirical data.
Enables Convex Optimization: It transforms a potentially intractable infinite-dimensional distributionally robust problem into a tractable convex program, even in the presence of safety constraints.

The authors position this work as a step toward more effective distributionally robust control, particularly in safety-critical applications where the trade-off between safety guarantees and performance is critical. They note that while the current work focuses on linear systems and affine policies, the theoretical properties of Sinkhorn sets established here are foundational for future extensions to nonlinear systems and gradient-based solution methods.