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arxiv: 2604.18453 · v1 · submitted 2026-04-20 · 📡 eess.SY · cs.SY· math.OC

On the Effect of Quadratic Regularization in Direct Data-Driven LQR

Manuel Kl\"adtke , Feiran Zhao , Florian D\"orfler , Moritz Schulze Darup This is my paper

Pith reviewed 2026-05-10 03:42 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords data-driven controlLQRquadratic regularizationparametric effectexplainabilityauxiliary eliminationcomputational complexity
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The pith

Quadratic regularization in direct data-driven LQR translates costs from auxiliary variables to system quantities.

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

This paper develops an explainability concept for direct data-driven linear quadratic regulation with quadratic regularization. It focuses on the parametric effect of regularization to shift the penalties from auxiliary variables onto the system quantities themselves. This shift gives clear interpretations of what the regularization achieves in terms of the actual dynamics and inputs. It also permits removing the auxiliary variables from the problem, which lowers the computational burden of solving for the controller. The method is validated through simulation examples that match the performance of the standard formulation.

Core claim

The paper establishes that the parametric effect of regularization in direct data-driven LQR maps the quadratic regularization terms applied to auxiliary variables into equivalent regularization terms on the system matrices and vectors. This mapping keeps the optimization outcome unchanged, offers intuitive explanations in system terms, and allows the auxiliary variables to be eliminated, reducing the dimensionality and complexity of the resulting optimization problem.

What carries the argument

the parametric effect of regularization, which re-expresses regularization penalties defined on auxiliary data-driven variables as penalties on the estimated system parameters

Load-bearing premise

That the costs of quadratic regularization can be mapped parametrically from auxiliary variables to system quantities while exactly preserving the original optimization result in the data-driven LQR setting.

What would settle it

A concrete data set and LQR problem where the controller obtained after mapping and eliminating auxiliaries differs from the one solved with the original auxiliary-variable formulation.

Figures

Figures reproduced from arXiv: 2604.18453 by Feiran Zhao, Florian D\"orfler, Manuel Kl\"adtke, Moritz Schulze Darup.

Figure 1
Figure 1. Figure 1: Effect of H∗ 1 for different regularizations shown via the deviation between Acl and ALS + BLSK for λ ∈ [10−6 , 106 ]. -3 -2 -1 0 1 0 2 4 6 8 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Effect of H∗ 3 for the case {3} shown via phase por￾traits and (disturbance-free) example trajectories of x(k + 1) = Aclx(k) . Table I: Mean computation times over 10 runs on an Intel Core i7. ℓ = 30 ℓ = 60 ℓ = 90 ℓ = 120 tr(GP G) 1.31 s 2.60 s 11.97 s 44.26 s {1, 2, 3} 0.51 s 0.51 s 0.47 s 0.49 s tr(Π⊥GP G⊤) 1.66 s 4.57 s 22.59 s 62.28 s {1} 0.52 s 0.50 s 0.46 s 0.48 s trajectories of the synthesized clos… view at source ↗
read the original abstract

This paper proposes an explainability concept for direct data-driven linear quadratic regulation (LQR) with quadratic regularization. Our perspective follows the parametric effect of regularization, an analysis approach that translates regularization costs from auxiliary variables to system quantities, enabling intuitive interpretations. The framework further enables the elimination of auxiliary variables, thereby reducing computational complexity. We demonstrate the effectiveness of our approach and the identified effect of regularization via simulations.

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

1 major / 2 minor

Summary. The paper proposes an explainability framework for direct data-driven LQR under quadratic regularization. It analyzes the parametric effect of regularization to translate penalties from auxiliary variables (e.g., trajectory or slack variables) onto effective system quantities, yielding intuitive interpretations and permitting elimination of auxiliaries to reduce computational complexity. The approach is validated through simulations.

Significance. If the proposed translation is shown to be exactly equivalent to the original regularized problem (i.e., the reparameterized optimizer coincides for all data sets satisfying the implicit Hankel-matrix constraints), the work would offer both interpretability gains and practical complexity reduction in data-driven control. The simulation results supply initial empirical support, but the overall significance hinges on whether the mapping holds without additional unstated restrictions on data rank or persistency of excitation.

major comments (1)
  1. [Derivation of the parametric effect (likely §3–4)] The central claim that regularization costs can be parametrically mapped from auxiliary variables to system quantities while preserving the exact optimizer requires an explicit proof of equivalence. The reparameterized problem must remain mathematically identical to the original for every feasible data set; any interaction between the translated cost and the linear constraints imposed by the data Hankel matrices could shift the solution even if the algebraic mapping appears clean.
minor comments (2)
  1. [Abstract] The abstract summarizes the contribution but contains no equations, explicit mapping, or quantitative results, which hinders immediate technical assessment.
  2. [Numerical examples] Simulation section should report concrete details: system order, data length, persistency-of-excitation verification, and quantitative metrics (e.g., closed-loop cost or regret) comparing the original and reduced formulations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and for highlighting the need to strengthen the equivalence claim in our parametric analysis of quadratic regularization for data-driven LQR. We address the major comment below and will revise the manuscript accordingly to improve rigor and clarity.

read point-by-point responses

  1. Referee: The central claim that regularization costs can be parametrically mapped from auxiliary variables to system quantities while preserving the exact optimizer requires an explicit proof of equivalence. The reparameterized problem must remain mathematically identical to the original for every feasible data set; any interaction between the translated cost and the linear constraints imposed by the data Hankel matrices could shift the solution even if the algebraic mapping appears clean.

    Authors: We agree that an explicit proof of equivalence is required to fully substantiate the central claim. The manuscript derives the parametric mapping via algebraic completion of squares on the quadratic regularization terms, translating penalties on auxiliary trajectory and slack variables into effective costs on the system matrices and initial state. However, we acknowledge that the current presentation does not include a standalone theorem verifying that the reparameterized optimizer coincides exactly with the original under the Hankel-matrix constraints for arbitrary feasible data sets. In the revised version we will add a formal proof (new Theorem in Section 3) showing that the first-order optimality conditions remain identical: the gradient of the translated cost, when augmented by the same Lagrange multipliers associated with the linear data constraints, recovers the original KKT system. This accounts for potential interactions with the Hankel constraints by construction, without introducing additional restrictions beyond the standard persistency-of-excitation and rank conditions already stated in the paper. The simulation results are consistent with this equivalence, but the proof will be provided explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the parametric translation framework for quadratic regularization.

full rationale

The paper derives an explainability concept by following the parametric effect of regularization to translate costs from auxiliary variables to system quantities in the direct data-driven LQR setting. This enables interpretations and elimination of auxiliaries while preserving the optimization outcome. No load-bearing step reduces by construction to a self-definition, a fitted input renamed as prediction, or a self-citation chain whose validity depends on the present result. The translation is presented as an algebraic reparameterization of the regularized problem that remains equivalent under the data Hankel constraints, constituting an independent derivation rather than a tautology. The framework is self-contained against the standard data-driven LQR formulation without requiring external unverified premises for its core claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated validity of the parametric translation in the data-driven LQR setting.

pith-pipeline@v0.9.0 · 5369 in / 1105 out tokens · 23463 ms · 2026-05-10T03:42:24.571486+00:00 · methodology

discussion (0)

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Reference graph

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