{"id":"7179175d-8734-4777-aad9-3b469a27acbe","arxiv_id":"2606.23827","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hessian-augmented polynomial regression using PMP-derived data reduces sample complexity for approximating value functions and recovering feedback laws in deterministic optimal control problems.","lead":"The paper develops a supervised learning method to approximate value functions for nonlinear optimal control by generating training data that includes values, gradients, and Hessians from the Pontryagin Maximum Principle and associated Riccati equations. This augmentation allows polynomial regression to achieve similar accuracy with substantially fewer samples than value-only approaches.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the Riccati-to-Hessian step, but that step is theoretically standard for smooth deterministic problems and the paper's numerical results appear to confirm its practical utility. Because the full text supplies the missing implementation and comparison details that prompted the original UNVERDICTED verdict, and no load-bearing flaw surfaces, the assessment does not shift.","tokens_in":1657,"tokens_out":334,"duration_ms":24730,"concrete_test":"Recompute the approximation error and closed-loop cost for the highest-dimensional example using exactly the same hyperbolic-cross basis and sample counts reported in the paper; if the second-order augmentation still yields at least a 5-fold reduction in required trajectories relative to value-only regression, the headline empirical claim holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an empirical one: that augmenting polynomial regression with value/gradient/Hessian data generated via PMP + Riccati reduces the number of trajectories needed for a given accuracy. The construction relies on the standard fact that the second variation along an optimal trajectory yields the Hessian of the value function (via the Riccati equation), which is then used to supply additional linear equations in the weighted least-squares fit. The manuscript validates this on a sequence of control-affine problems with increasing state dimension and reports the claimed sample-complexity improvement together with closed-loop performance gains. No internal inconsistency, unstated regularity assumption that fails in the tested regimes, or mismatch between the Riccati data and the true value-function derivatives is apparent.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a data-driven method for approximating value functions of deterministic optimal control problems with nonlinear control-affine dynamics. It solves the Pontryagin Maximum Principle optimality system from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function (Hessians obtained via the matrix Riccati equation along optimal trajectories). These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians supplying additional linear equations per sample. Feedback laws are recovered analytically from the learned value function. In high dimensions a partial Hessian strategy is used to control data-generation cost. The approach is validated on control-affine problems of increasing state dimension, where second-order augmentation is reported to improve accuracy and closed-loop performance with up to an order-of-magnitude reduction in required training samples relative to value-only regression.","tokens_in":1801,"tokens_out":645,"duration_ms":19789,"significance":"If the reported empirical gains hold under broader testing, the method provides a concrete way to reduce sample complexity when learning HJB value functions by systematically incorporating second-order information generated from standard PMP+Riccati theory. This is potentially useful for moderate-to-high-dimensional deterministic control problems where trajectory data is expensive to obtain. The construction is parameter-free in the sense that no new constants are fitted to define the result, and the use of hyperbolic-cross polynomial bases together with the analytic feedback recovery are clear strengths.","major_comments":[{"comment":"§5 (numerical experiments): the reported order-of-magnitude sample-complexity reduction is shown only relative to value-only regression on the same polynomial basis; no comparison is supplied against other established HJB approximation techniques (e.g., deep neural network solvers or other sparse-grid methods), which limits assessment of whether the second-order augmentation yields a genuine advance beyond the immediate baseline.","section":"§5"},{"comment":"§4.2 (regression formulation): the weighting factors that balance the value, gradient, and Hessian equations in the least-squares objective are introduced without an explicit rule or sensitivity study; because the central claim rests on the benefit of the additional linear equations, the lack of justification for these weights is load-bearing for reproducibility of the reported gains.","section":"§4.2"}],"minor_comments":[{"comment":"The abstract states quantitative gains but the main text should cross-reference the specific tables or figures that contain the error metrics and sample counts supporting those claims.","section":"Abstract"},{"comment":"Notation for the hyperbolic-cross index sets and the partial-Hessian selection mask could be introduced earlier (e.g., in §3) rather than only in the experimental section to improve readability.","section":"§3"},{"comment":"Figure captions should explicitly state the number of Monte-Carlo repetitions or random seeds used for each regression run so that variability of the reported accuracy improvements can be assessed.","section":"Figures in §5"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading, positive assessment of the contribution, and constructive comments. We address the two major comments point by point below.","responses":[{"response":"We agree that the experiments isolate the benefit of Hessian augmentation relative to value-only regression on the identical polynomial basis and hyperbolic-cross index set. This design choice was deliberate to highlight the effect of the additional linear equations supplied by gradients and Hessians. Broader benchmarking against deep neural network HJB solvers or other sparse-grid techniques would indeed provide a more complete picture of the method's standing, but would have required substantial extra implementation and compute that lay outside the scope of the present work. In the revision we will add an explicit limitations paragraph in §5 acknowledging this and identifying such comparisons as a natural direction for follow-up research.","revision_made":"partial","referee_comment":"[§5] §5 (numerical experiments): the reported order-of-magnitude sample-complexity reduction is shown only relative to value-only regression on the same polynomial basis; no comparison is supplied against other established HJB approximation techniques (e.g., deep neural network solvers or other sparse-grid methods), which limits assessment of whether the second-order augmentation yields a genuine advance beyond the immediate baseline."},{"response":"The referee correctly notes that an explicit, reproducible rule for the weighting coefficients is needed. In the current manuscript the weights are chosen to normalize each residual type (value, gradient, Hessian) by its typical magnitude across the training set so that the three blocks contribute on comparable scales; however, this procedure is only described informally. We will revise §4.2 to state the normalization rule in full detail and will add a short sensitivity study (varying the relative weights by factors of 0.1–10) demonstrating that the reported accuracy gains remain stable within a reasonable range around the chosen values.","revision_made":"yes","referee_comment":"[§4.2] §4.2 (regression formulation): the weighting factors that balance the value, gradient, and Hessian equations in the least-squares objective are introduced without an explicit rule or sensitivity study; because the central claim rests on the benefit of the additional linear equations, the lack of justification for these weights is load-bearing for reproducibility of the reported gains."}],"tokens_in":1424,"tokens_out":491,"duration_ms":15845,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's main result is that feeding value, gradient, and Hessian data generated from the Pontryagin system plus the matrix Riccati equation into a weighted least-squares fit over sparse polynomial bases on hyperbolic crosses reduces the number of trajectories required while preserving or improving closed-loop performance.\n\nWhat is new is the specific use of the second-order Riccati information to add linear constraints per sample. Earlier supervised approaches stopped at value or first-derivative data; the extra equations from the Hessian improve the fit outside the training set. The construction stays within standard PMP and Riccati theory, with no fitted constants that would make the result circular.\n\nThe experiments are the strongest part. They run the method on control-affine problems with increasing state dimension, report the sample-complexity reduction, and check the resulting feedback laws in closed loop. The partial-Hessian strategy for high dimensions is a sensible engineering choice that keeps data generation tractable.\n\nThe soft spots are modest and mostly about scope. All tests stay inside control-affine dynamics, so it is not yet clear how the gains translate when the control enters nonlinearly. The regression weights and the precise hyperbolic-cross truncation are chosen to work on these examples; a reader would want to see how sensitive the reported order-of-magnitude improvement is to those choices. No load-bearing inconsistencies show up in the setup or the reported numbers.\n\nThe work is aimed at people who need practical, sample-efficient approximations for high-dimensional deterministic optimal control. Anyone already using polynomial regression or PMP-based data generation for HJB problems will see a clear incremental advance. It is worth sending to peer review because the central empirical claim is supported by concrete experiments on a sequence of problems and the underlying theory is standard and correctly applied.","headline":"Hessian data from the Riccati equation along PMP trajectories cuts the samples needed for accurate hyperbolic-cross polynomial approximation of HJB value functions by up to 10x on the control-affine examples.","tokens_in":2278,"tokens_out":441,"would_cite":true,"duration_ms":23160,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Hessian data from Riccati equations along PMP trajectories cuts the samples needed for accurate value-function approximation by up to an order of magnitude.","keywords":["Hamilton-Jacobi-Bellman equations","optimal control","Pontryagin Maximum Principle","Riccati equation","supervised learning","polynomial regression","value function approximation","feedback synthesis"],"falsifier":"On a fixed high-dimensional test problem, run the regression with and without the Hessian equations using identical numbers of trajectories and measure whether the closed-loop cost and L2 error of the value function differ by more than the observed variance across random initial-condition sets.","tokens_in":2578,"feed_emoji":"📉","tokens_out":672,"duration_ms":15427,"temperature":0.7,"pith_summary":"The paper presents a supervised learning method that generates training data for Hamilton-Jacobi-Bellman value functions by solving the Pontryagin Maximum Principle system from many initial states. Each trajectory supplies not only the value and gradient but also the Hessian, computed via an auxiliary matrix Riccati equation. These second-order quantities are inserted as additional linear equations into a weighted least-squares fit over sparse polynomial bases on hyperbolic cross index sets. The resulting polynomial yields an analytic feedback law. Numerical tests on control-affine systems show that the extra derivative information produces lower approximation error and better closed-loop performance than value-only or gradient-only regression at the same sample count.","feed_headline":"Hessian augmentation cuts samples 10x for HJB value functions","feed_subtitle":"Riccati-derived second derivatives supply extra equations that let sparse polynomial regression generalize from far fewer PMP trajectories.","key_machinery":"Augmentation of the weighted least-squares system with gradient and Hessian rows derived from PMP trajectories and the associated Riccati equation, performed over sparse polynomial bases on hyperbolic cross index sets.","core_discovery":"Incorporating Hessian information obtained from the matrix Riccati equation along optimal trajectories into a polynomial regression reduces the number of required training samples by up to an order of magnitude while improving both pointwise accuracy of the learned value function and the performance of the recovered feedback law.","pith_inferences":["The same derivative-augmentation idea could be tested on other basis families that admit cheap evaluation of gradients and Hessians.","If the Riccati-derived Hessians remain reliable under moderate noise, the approach might extend to nearby stochastic problems where a similar linearization exists.","The reduction in sample count suggests that second-order information could be leveraged in other regression-based HJB solvers that currently rely only on value or gradient data."],"forward_implications":["Feedback laws are recovered by direct differentiation of the learned polynomial without additional optimization.","A partial Hessian strategy keeps data-generation cost manageable when the state dimension is large.","Second-order augmentation yields measurable gains in both approximation accuracy and closed-loop performance on problems of increasing dimension.","The method applies to deterministic control-affine dynamics and produces globally defined polynomial approximations."],"fun_headline_variants":["Hessian from Riccati reduces HJB sample requirements","Riccati Hessians lower sample needs for HJB approximation","Hessian data from Riccati enables fewer HJB samples","Matrix Riccati Hessians augment HJB polynomial regression","Second-order PMP data reduces HJB sample complexity"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The matrix Riccati equation solved along PMP trajectories supplies accurate second derivatives of the true value function.","fun_headline_variants_meta":{"raw":{"variants":["Hessian from Riccati reduces HJB sample requirements","Riccati Hessians lower sample needs for HJB approximation","Hessian data from Riccati enables fewer HJB samples","Matrix Riccati Hessians augment HJB polynomial regression","Second-order PMP data reduces HJB sample complexity"]},"model":"grok-4.3","cost_usd":0.007825,"raw_usage":{"total_tokens":3534,"prompt_tokens":593,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":78249500,"prompt_tokens_details":{"text_tokens":593,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2860,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":593,"tokens_out":81,"duration_ms":26290,"temperature":1.0,"reasoning_tokens":2860,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T07:05:02.851817+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"On a fixed high-dimensional test problem, run the regression with and without the Hessian equations using identical numbers of trajectories and measure whether the closed-loop cost and L2 error of the value function differ by more than the observed variance across random initial-condition sets.","supporting_citations":[],"review_version":1}