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arxiv: 2605.03845 · v1 · submitted 2026-05-05 · 📡 eess.SY · cs.SY

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

Riccardo Cescon , Andrea Martin , Giancarlo Ferrari-Trecate This is my paper

Pith reviewed 2026-05-07 13:50 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords distributionally robust controlSinkhorn discrepancyambiguity setslinear quadratic controlconvex programmingsafety constraintstrajectory planning
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The pith

Sinkhorn ambiguity sets turn the distributionally robust linear quadratic control problem over linear policies into a convex program, even with safety constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Classical stochastic control assumes complete knowledge of uncertainty, but real systems often have only partial information from limited samples. The paper uses Sinkhorn discrepancy to define ambiguity sets that blend observed data with a reference distribution and avoid forcing worst-case distributions to be discrete. Under standard assumptions, these sets are shown to be convex and weakly compact. This property is leveraged to reformulate the distributionally robust linear quadratic control problem over linear policies as a convex optimization task that remains tractable when distributionally robust safety constraints are added.

Core claim

We establish convexity and weak compactness of Sinkhorn ambiguity sets under standard assumptions. We then leverage these results to prove that the Sinkhorn DR linear quadratic control problem over linear policies can be solved through convex programming, even in the presence of DR safety constraints. The findings are validated on a trajectory planning example.

What carries the argument

Sinkhorn ambiguity sets defined via the Sinkhorn discrepancy, which model uncertainty while preserving convexity for control optimization.

If this is right

  • The Sinkhorn DR linear quadratic control problem over linear policies admits a convex programming formulation.
  • This convex formulation continues to hold when distributionally robust safety constraints are included.
  • The approach remains effective when only a small number of noise samples are available by incorporating a reference distribution.
  • Worst-case distributions need not be restricted to discrete supports, unlike some other optimal transport distances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar convexity arguments might apply to other control problems or policy structures if the underlying ambiguity sets preserve the same properties.
  • The method could support more realistic robustness analysis in systems where uncertainty is continuous rather than discrete.
  • Blending data and priors through the reference distribution may help in online or adaptive control settings with streaming observations.

Load-bearing premise

Standard assumptions hold under which Sinkhorn ambiguity sets remain convex and weakly compact.

What would settle it

A concrete linear-quadratic instance with linear policies where the Sinkhorn DR problem, including safety constraints, cannot be cast as a convex program despite the stated assumptions on the ambiguity sets.

Figures

Figures reproduced from arXiv: 2605.03845 by Andrea Martin, Giancarlo Ferrari-Trecate, Riccardo Cescon.

Figure 1
Figure 1. Figure 1: Visualization of worst-case distributions from Wasserstein DRO (top view at source ↗
Figure 2
Figure 2. Figure 2: CVaR is the expected value of the γ-right tail (dark blue). constrained control problem can then be expressed as π ⋆ ∈ arg min π=(π0,...,πT−1) EP[J(π, w)] (8a) subject to CVaRP γ (max{h(x), g(u)}) ≤ 0 . (8b) B. Approximations and problem formulation Problem (8) is rarely solvable directly. Indeed, most of the time, the true underlying probability P of w is unknown. Instead, we assume to have access to N ∈ … view at source ↗
Figure 3
Figure 3. Figure 3: Comparison in cost (left) and violation (right) percentage increase of view at source ↗
Figure 4
Figure 4. Figure 4: Terminal state of different trajectories given by the policies view at source ↗
read the original abstract

Classical stochastic control assumes perfect knowledge of the uncertainty affecting the plant. In practice, however, such information is often incomplete. To address this limitation, we consider a distributionally robust control (DRC) problem with ambiguity sets defined via the Sinkhorn discrepancy. Compared to other discrepancy measures based on optimal transport, such as the popular Wasserstein distance, the Sinkhorn divergence does not constrain the worst-case distribution to be discrete, and allows combining observed data with prior knowledge in the form of a reference distribution, making this choice particularly suitable when only few noise samples are available for control design. We first study the properties of Sinkhorn ambiguity sets, establishing convexity and weak compactness under standard assumptions. We then leverage these results to prove that, the Sinkhorn DR linear quadratic control problem over linear policies can be solved through convex programming-even in the presence of DR safety constraints. Finally, we validate our theoretical findings and demonstrate the effectiveness of the proposed approach on a trajectory planning example.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Sinkhorn ambiguity sets for distributionally robust control (DRC). It establishes convexity and weak compactness of these sets under standard assumptions, then proves that the Sinkhorn DR linear quadratic control problem over linear policies (including with DR safety constraints) reduces to convex programming. The approach is illustrated via a trajectory planning example.

Significance. If the convexity, weak compactness, and resulting tractability hold, the work provides a data-efficient DRC framework that supports continuous worst-case distributions and prior incorporation, offering advantages over Wasserstein ambiguity sets in low-sample regimes. The explicit convex reformulation for LQ problems with safety constraints could enable practical robust controller synthesis.

major comments (2)
  1. [Section 3] Section 3 (properties of Sinkhorn ambiguity sets): The claims of convexity and weak compactness 'under standard assumptions' are load-bearing for the minimax interchange and convex reformulation in Section 4. The manuscript invokes these properties for the reference measure P (empirical + prior mixture) on a Polish space but does not spell out the precise conditions (e.g., moment bounds, support restrictions, or topology) that ensure the sets remain convex and weakly compact when the cost is quadratic and the constraints are distributionally robust chance constraints. Without this, it is unclear whether compactness holds in the weak topology relevant to the LQ setting.
  2. [Section 4] Section 4 (DR LQ control formulation): The reduction to convex programming relies on the ambiguity-set properties from Section 3 to justify interchanging min and max. If the weak compactness fails for the specific quadratic cost and chance-constraint structure, the convex reformulation (and its tractability) no longer follows directly; the paper should either derive the required conditions explicitly or provide a self-contained verification for the LQ case.
minor comments (2)
  1. [Abstract] Abstract: The numerical validation on the trajectory planning example is mentioned without reference to specific baselines, quantitative metrics, or error analysis, which would help readers assess practical gains.
  2. [Section 3] Notation: The definition of the Sinkhorn divergence and the precise form of the ambiguity set {Q : Sinkhorn(Q,P) ≤ ε} should be restated explicitly at the start of Section 3 for self-contained reading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the properties of Sinkhorn ambiguity sets and their role in the distributionally robust LQ control formulation. We agree that the assumptions underlying convexity and weak compactness require more explicit statement to support the minimax interchange and convex reformulation, particularly for quadratic costs and distributionally robust chance constraints. We respond to each major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (properties of Sinkhorn ambiguity sets): The claims of convexity and weak compactness 'under standard assumptions' are load-bearing for the minimax interchange and convex reformulation in Section 4. The manuscript invokes these properties for the reference measure P (empirical + prior mixture) on a Polish space but does not spell out the precise conditions (e.g., moment bounds, support restrictions, or topology) that ensure the sets remain convex and weakly compact when the cost is quadratic and the constraints are distributionally robust chance constraints. Without this, it is unclear whether compactness holds in the weak topology relevant to the LQ setting.

    Authors: We thank the referee for this observation. Convexity of the Sinkhorn ambiguity sets follows immediately from the joint convexity of the Sinkhorn divergence in its two arguments, which holds on any Polish space without further restrictions. Weak compactness, however, does require additional structure on the reference measure. In the revision we will replace the phrase 'under standard assumptions' with an explicit list: the underlying space is Polish and metrized by a distance inducing the weak topology; the reference measure P (empirical-plus-prior mixture) possesses finite second moments; and the Sinkhorn radius is chosen so that the ambiguity set is tight (invoking Prokhorov's theorem). We will also add a short remark verifying that these conditions are compatible with the quadratic stage cost and with the distributionally robust chance constraints, which remain upper semicontinuous under weak convergence when second moments are controlled. revision: yes

  2. Referee: [Section 4] Section 4 (DR LQ control formulation): The reduction to convex programming relies on the ambiguity-set properties from Section 3 to justify interchanging min and max. If the weak compactness fails for the specific quadratic cost and chance-constraint structure, the convex reformulation (and its tractability) no longer follows directly; the paper should either derive the required conditions explicitly or provide a self-contained verification for the LQ case.

    Authors: We agree that the justification for the minimax interchange should be self-contained for the linear-quadratic setting. The revised manuscript will contain a dedicated lemma in Section 4 that (i) confirms weak compactness of the Sinkhorn ambiguity set under the quadratic cost (using the finite-second-moment condition stated in the updated Section 3) and (ii) shows that the distributionally robust chance constraints are closed in the weak topology. With these two facts, the standard minimax theorem for convex-concave problems on compact sets applies directly, yielding the convex program without additional assumptions. We will also include a brief proof sketch that the linear policy parameterization preserves convexity of the resulting finite-dimensional program. revision: yes

Circularity Check

0 steps flagged

No circularity: properties established independently before use in control result

full rationale

The manuscript states it first studies and establishes convexity and weak compactness of the Sinkhorn ambiguity sets under standard assumptions, then leverages those results to obtain the convex-programming reformulation of the DR LQ control problem (including safety constraints). This sequence is self-contained within the paper rather than reducing any claim to a fitted parameter, self-definition, or unverified self-citation. The central tractability result follows from the newly established set properties without the prediction equaling its inputs by construction. No load-bearing step matches the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on unstated standard assumptions for Sinkhorn divergence properties and control-theoretic regularity conditions; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption standard assumptions under which Sinkhorn ambiguity sets are convex and weakly compact
    Invoked to establish the main properties and tractability result.

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