{"id":"82e24f3b-163c-4c0b-adc8-fadd4016ed7e","arxiv_id":"2604.18453","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A parametric analysis method translates quadratic regularization effects in data-driven LQR from auxiliary variables to system quantities for better interpretability and reduced computation.","lead":"The paper introduces an explainability framework for quadratic regularization in direct data-driven LQR by translating regularization costs from auxiliary variables into interpretable system quantities. This approach also allows removing auxiliary variables to lower computational cost, shown through simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Parametric translation of quadratic regularization costs from auxiliaries to system quantities may not preserve exact optimality in the data-driven LQR problem without additional data conditions.","rationale":"The reader's weakest assumption directly identifies the equivalence-preserving property of the mapping as the critical unverified step. Because the abstract alone supplies no derivation or counter-example check, the concrete numerical equivalence test on a low-dimensional instance is the minimal experiment that would confirm or refute whether the translation is exact inside the data-driven constraint set. If the test passes, the claim strengthens; if it fails, the elimination step introduces an unstated approximation.","tokens_in":1563,"tokens_out":378,"duration_ms":19879,"concrete_test":"Generate a minimal controllable scalar LQR instance, collect a single persistently exciting data trajectory of length T=20, formulate the regularized direct data-driven problem with auxiliary variables, solve for the optimal cost and input sequence; then apply the paper's parametric mapping to obtain the equivalent system-cost problem, eliminate auxiliaries, and re-solve. Check whether the two optimal costs and input sequences agree to machine precision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim rests on the parametric effect allowing exact translation of regularization penalties from auxiliary variables (e.g., trajectory or slack variables in the data-driven formulation) onto effective system matrices or costs, which then permits elimination of those auxiliaries while keeping the same optimizer. For this to be load-bearing, the reparameterized problem must be mathematically equivalent to the original regularized one for all feasible data sets. In direct data-driven LQR the feasible set is defined implicitly by the data Hankel matrices and any persistency-of-excitation or rank conditions; a cost translation that ignores how regularization interacts with these linear constraints can shift the optimum even if the algebraic mapping looks clean on paper.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1704,"tokens_out":438,"duration_ms":19182,"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":[{"comment":"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.","section":"Derivation of the parametric effect (likely §3–4)"}],"minor_comments":[{"comment":"The abstract summarizes the contribution but contains no equations, explicit mapping, or quantitative results, which hinders immediate technical assessment.","section":"Abstract"},{"comment":"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.","section":"Numerical examples"}],"recommendation":"major_revision","confidential_remarks":"The provided abstract and reader's notes indicate that the manuscript may lack the detailed derivation and data-condition analysis needed to substantiate the equivalence claim; the full text should be examined for any hidden assumptions on the data matrices."},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"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."}],"tokens_in":1133,"tokens_out":355,"duration_ms":21310,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a parametric translation that converts regularization penalties on the auxiliary trajectory or slack variables into equivalent adjustments on the effective system costs or matrices. This keeps the optimizer unchanged while removing those variables from the problem, which cuts computational load and gives a clearer picture of what the regularizer actually does to the closed-loop behavior in the data-driven setting.","headline":"The paper reinterprets quadratic regularization in direct data-driven LQR by mapping its effect from auxiliary variables onto system quantities, which aids intuition and lets you drop auxiliaries, but the exact equivalence under Hankel constraints is not yet airtight.","tokens_in":2201,"tokens_out":162,"would_cite":false,"duration_ms":12854,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Quadratic regularization in direct data-driven LQR translates costs from auxiliary variables to system quantities.","keywords":["data-driven control","LQR","quadratic regularization","parametric effect","explainability","auxiliary elimination","computational complexity"],"falsifier":"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.","tokens_in":2457,"feed_emoji":"","tokens_out":413,"duration_ms":34948,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quadratic regularization in data-driven LQR mapped to system costs","feed_subtitle":"This translation provides intuitive interpretations and lets auxiliary variables be dropped to simplify the optimization.","key_machinery":"the parametric effect of regularization, which re-expresses regularization penalties defined on auxiliary data-driven variables as penalties on the estimated system parameters","core_discovery":"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.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Quadratic reg mapped to system costs in data-driven LQR","Data-driven LQR regularization translated into system terms","Regularization effects on system matrices in direct LQR","Eliminating aux vars by mapping reg to system quantities in LQR"],"cache_read_input_tokens":64,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic reg mapped to system costs in data-driven LQR","Data-driven LQR regularization translated into system terms","Regularization effects on system matrices in direct LQR","Eliminating aux vars by mapping reg to system quantities in LQR"]},"model":"grok-4.3","cost_usd":0.007335,"raw_usage":{"total_tokens":3288,"prompt_tokens":493,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":73349500,"prompt_tokens_details":{"text_tokens":493,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2730,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":493,"tokens_out":65,"duration_ms":21479,"temperature":1.0,"reasoning_tokens":2730,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T03:42:24.571486+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}