{"id":"d10d029c-4b65-48ab-9742-7b67843b10f5","arxiv_id":"2501.16521","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper frames learning as an optimal control problem and claims an iff characterization of optimal learning trajectories via the maximum principle, plus a Galerkin successive approximation algorithm.","lead":"This paper suggests viewing model training as an optimal control problem, where a small control term steers the gradient descent trajectory to minimize the validation error. The authors propose a theoretical characterization of the optimal training path and a numerical recipe for finding it.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.1 is unsupported: Eq (2.2) is not the maximum-principle condition for the stated control problem, and Eq (1.4) does not match the derivative of D, so the central iff claim fails as stated.","rationale":"The paper's stated goal is to characterize optimal learning trajectories by relating the maximum principle to dynamic programming. The only substantive result is Proposition 2.1, and it is asserted without proof. My check focuses on the internal correctness of that proposition rather than on external novelty or on the C^2 regularity assumption flagged by the reader. The reader's weakest_assumption (C^2 smoothness) is legitimate but not the decisive failure: the equations are algebraically inconsistent with the stated dynamics even for C^infinity losses. The D matrix's derivative is incorrectly transcribed, and Eq (2.2) does not encode the Hamiltonian maximization required by Pontryagin's principle. Therefore the central claim is not established, and the reader's REJECT verdict is appropriate. I give the paper credit for clearly specifying the control system, cost functional, and algorithmic skeleton, but the missing proof of Proposition 2.1 and the misstated PMP condition leave the central theorem unsupported. No further adjustment to the verdict is needed.","tokens_in":5393,"tokens_out":8618,"duration_ms":89242,"concrete_test":"Scalar analytic check: take J0(theta) = theta^4/4, so grad J0 = theta^3, D = theta^6, grad^2 J0 = 3 theta^2. With U = [-1,1] and eps = 1, the true Hamiltonian is H = p(-theta^3 + theta^6 u), so the PMP maximum condition gives u = sign(p), and the correct adjoint ODE is pdot = (3 theta^2 - 6 theta^5 u) p. Equations (1.4)/(2.1) instead give pdot = (3 theta^2 - 3 theta^8 u) p. Substitute both into Eq (2.2): the left side is p(-theta^3 + theta^6 u), while the right side becomes sup_v p(-3 theta^2 + 3 theta^8 v). For p = 1, u = 1, theta = 2 these evaluate to 56 and 756 respectively. Re-deriving these two lines independently will show that Eq (2.2) is not the Pontryagin condition for the stated problem, falsifying the claimed equivalence even in a smooth one-dimensional case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Proposition 2.1, which asserts an if-and-only-if characterization of optimal trajectories via the adjoint equation (2.1) and the maximum-principle condition (2.2). This step is not valid. For the stated dynamics (1.2), the Hamiltonian is H^eps(theta,p,u) = <p, -grad J0(theta,Z^(1)) + eps D(theta,Ztilde^(1)) u>. Pontryagin's principle requires u(t) to maximize H^eps at almost every t, i.e. ubar(t) in argmax_{v in U} <p(t), eps D(theta(t),Ztilde^(1)) v>, with pdot = -partial H^eps/partial theta and p(T) = -grad Phi(theta(T),Z^(2)). Equation (2.2) instead imposes a different maximization involving -grad^2 J0(theta,Z^(1)) + eps D(theta,Ztilde^(1)) grad^2 J0(theta,Ztilde^(1)) u; this is neither H^eps nor partial H^eps/partial theta, so it is not the first-order necessary condition for (1.1)-(1.2). The internal derivation is also inconsistent: D(theta,Ztilde^(1)) is the diagonal matrix of squared first partials of J0, so partial D/partial theta_k has entries 2 (partial J0/partial theta_i)(partial^2 J0/partial theta_i partial theta_k). Eq (1.4) writes the control term as eps u^T grad^2 J0(theta,Ztilde) D(theta,Ztilde) p, which is not this derivative and mixes a vector equation with a scalar term. Proposition 2.1 is asserted without proof and rests entirely on these equations, so the proposed characterization is not established. The C^2 regularity issue is secondary: even with all functions C^infinity, the maximum-principle condition is misstated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates a learning problem as an optimal control problem for the weakly-controlled gradient system (1.2), with a terminal cost Φ(θ(T), Z^(2)) that evaluates the model on a validation dataset. In Section 2, Proposition 2.1 asserts an if-and-only-if characterization of optimal trajectories via the adjoint equation (2.1) and the maximum-principle condition (2.2), and claims that the optimal parameter is recovered by θ* = θ̄^ε(T). The paper then proposes a successive Galerkin approximation algorithm to construct these trajectories. The central result is stated without proof, and the equations on which it rests contain internal inconsistencies, so the paper does not establish its main claims.","tokens_in":5867,"tokens_out":7913,"duration_ms":72258,"significance":"The ambition of connecting Pontryagin's maximum principle with dynamic programming for learned trajectories is reasonable and the idea of using a small random perturbation of the training set to create a controlled term is interesting. However, the correctness of the characterization is the entire contribution, and the manuscript's central equation (2.2) is not the maximum-principle condition for the stated control problem, while the adjoint equation (1.4) is dimensionally inconsistent. Because Proposition 2.1 is stated without proof and the numerical component is only described programmatically, the paper in its current form provides neither a valid theorem nor a validated algorithm. The manuscript is clearly organized and lists its assumptions, but the technical content does not support the conclusions.","major_comments":[{"comment":"Equation (2.2) is not the maximum-principle condition for the optimal control problem (1.1)-(1.2). With the Hamiltonian defined in Eq. (1.6), Pontryagin's principle requires u(t) to maximize ⟨p(t), εD(θ(t),Z̃^(1)) v⟩ over v ∈ U for almost every t, together with the adjoint equation ṗ(t) = -∂H^ε/∂θ. Equation (2.2) instead maximizes ⟨-∇²J0(θ̄^ε(t),Z^(1)) + εD(θ̄^ε(t),Z̃^(1))∇²J0(θ̄^ε(t),Z̃^(1)) u(t), p^ε(t)⟩, which is neither the Hamiltonian nor a derivative of it. No derivation connects this expression to the first-order necessary conditions of (1.1)-(1.2), so the 'if and only if' claim in Proposition 2.1 is unsupported. Moreover, even a correct PMP condition is only necessary in general; the converse direction would require additional sufficiency assumptions such as convexity, which are not stated.","section":"Section 2, Proposition 2.1 and Eq. (2.2)"},{"comment":"The adjoint equation (1.4) is dimensionally inconsistent and does not match the derivative of the Hamiltonian. With D(θ,Z̃^(1)) = diag((∂J0/∂θ_i)^2), the k-th component of ∂(D(θ)u)/∂θ_k is 2ε∑_{i=1}^p u_i (∂J0/∂θ_i)(∂²J0/(∂θ_i ∂θ_k)). Equation (1.4) writes the control term as ε u^T ∇²J0(θ^ε(t),Z̃^(1)) D(θ^ε(t),Z̃^(1)) p^ε(t), which is a scalar, not a vector, and is not the correct derivative. This issue also propagates to Eq. (2.1), where the bracketed expression mixes a p×p matrix with a row vector. Since the adjoint system is a core ingredient of Proposition 2.1 and of the algorithm, the formulation is not well-defined.","section":"Section 1, Eq. (1.4)"},{"comment":"The central result is stated without proof. The paragraph following Eqs. (1.2)-(1.5) says the optimality conditions follow from Pontryagin's principle, but Proposition 2.1's converse direction and the recovery formula θ* = θ̄^ε(T) are asserted rather than derived. In a paper whose main contribution is this characterization, an unproved 'if and only if' statement is a load-bearing gap. The footnote mentioning 'numerical works' does not provide evidence for the theorem.","section":"Section 2, Proposition 2.1"},{"comment":"The algorithmic update rule c^{k+1}_{ij} = c^k_{ij} + γ_{ij} δH^ε/δc_{ij} uses a functional derivative δH^ε/δc_{ij} that is never defined. Since u(t) = CΨ(t), one could interpret this as a partial derivative, but no expression or derivation is given. Furthermore, the stopping criterion ‖∑_{i,j} δH^ε/δc_{ij}‖ ≤ ε_tol is not shown to imply convergence to any point satisfying the maximum-principle conditions, and no convergence analysis or numerical experiment is included. Thus the proposed recipe is not justified as a method for computing the optimal learning trajectory.","section":"Section 2, 'A successive Galerkin approximation method'"},{"comment":"The regularity assumptions are insufficient for the quantities used in the paper. Section 1(d)(ii) only assumes Φ is locally Lipschitz, but the adjoint terminal condition p^ε(T) = -∇Φ and the maximum-principle condition (2.2) require differentiability of Φ and second differentiability of J0. These hypotheses are not stated. Additionally, Eq. (2.4) postulates smoothness of the value function V, which the paper itself acknowledges in Remark 1 is generally false; the proposed replacement by a differential inclusion is mentioned but not used in the subsequent derivation. These regularity gaps further undermine the linking of the maximum principle to dynamic programming.","section":"Section 1(d) and Remark 1"}],"minor_comments":[{"comment":"The abstract contains several broken words, such as 'characteriz ing', 'tha t', and 'paramete rs', and should be carefully proofread.","section":"Abstract"},{"comment":"The footnote says Φ(θ, Z^(2)) is defined 'w.r.t. the model training dataset Z^(2)', but Z^(2) is the validation dataset; this terminology should be corrected to avoid confusion.","section":"Footnote 5"},{"comment":"The infimum in Eq. (2.5) is written over Φ(θ̄^ε(T)) without explicit dependence on the control; it should be made clear that the infimum is taken over the set of admissible controls (or over the corresponding terminal states).","section":"Eq. (2.5)"},{"comment":"In the formula for ū^k_i(t), the basis function is written as ψ_i(t), but the basis was introduced as ψ_j(t); this index inconsistency should be fixed.","section":"Algorithm, Step 1"},{"comment":"The notation in Eq. (1.4) is ambiguous: the term 'u^T ∇²J0 D p' should be written with explicit indices so that the vector nature of each term is clear, if it is ever intended to be a vector.","section":"Section 1, Eq. (1.4)"}],"recommendation":"reject","confidential_remarks":"The paper's central result is not established. The errors in Eqs. (1.4) and (2.2) are not typographical: they affect the main theorem and the algorithm. Correcting them would require a substantial rewrite of the statement and proof of Proposition 2.1, together with explicit regularity and sufficiency conditions and a convergence analysis. As it stands, the manuscript does not meet the bar for publication in a mathematical journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Befekadu paper. The reader's take is right: reject. The central Proposition 2.1 is asserted without proof, and the equations around it don't survive contact.\n\nWhat is genuinely there: the paper is clearly organized, and the idea of adding a control term shaped by squared partial derivatives to a gradient system, with a validation loss as terminal cost, is a reasonable thing to want to study. The author also points to the right classical literature (Pontryagin, Clarke-Vinter). That part is fine.\n\nBut the core math is broken. For the dynamics in (1.2), the Hamiltonian is <p, -grad J0 + eps D u>, and Pontryagin's condition is that u maximizes that expression pointwise. Equation (2.2) instead maximizes a different object involving grad^2 J0 and D grad^2 J0 u, which is neither the Hamiltonian nor any standard first-order condition for this problem. On top of that, the adjoint equation (1.4) has a dimension mismatch: the control term eps u^T grad^2 J0 D p is scalar, not a vector, so the equation as written is not a valid vector ODE. These aren't typos that don't matter; they are the load-bearing parts of the claimed characterization. Proposition 2.1 is stated without proof, so there is no way to check whether some corrected version might hold. The Galerkin update rule uses an undefined functional derivative delta H / delta c and has no convergence analysis, and there are no numerical results. Footnote 1 says numerics exist but are reported elsewhere.\n\nThe self-citations to [1] and [4] are relevant, but that's not the issue. The issue is that the central result is not established. The C^2 regularity gap the reader flagged is secondary; even with smooth functions the maximum-principle condition in (2.2) is misstated.\n\nI don't see a path to salvage this without rewriting the main results and adding proofs. It is not ready for peer review in its current form. If the author fixes the adjoint equation, states the correct PMP condition, and proves the iff claim, there might be a usable paper about a specialized optimal-control view of learning. Right now, I wouldn't cite it or bring it to reading group.\n\nRecommendation: desk reject, with an explanation that the central equations need correction and proof before resubmission.","headline":"The paper frames learning as a weakly-controlled gradient system, but the central iff characterization is unproved and the maximum-principle condition is misstated, so the main result fails as stated.","tokens_in":6297,"tokens_out":3314,"would_cite":false,"duration_ms":32958,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49K15","49L20","65M60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a trajectory-control pair in a weakly controlled gradient learning system is optimal if and only if the adjoint state meets a maximum-principle condition, with the optimal parameters given by the trajectory's…","keywords":["dynamic programming","Galerkin method","generalization","Hamiltonian function","optimal control problem","optimal learning trajectories","Pontryagin's maximum principle","successive approximation method"],"falsifier":"Solve the forward-backward system (1.3)-(1.4) together with the maximum-principle condition (2.2) for a one-dimensional quadratic example, say $J_0(\\theta)=(\\theta-a)^2/2$ and $\\Phi(\\theta)=(\\theta-b)^2/2$, and compare two controls that both satisfy the conditions; if they lead to different terminal costs, the if-and-only-if claim in Proposition 2.1 is false.","tokens_in":5217,"feed_emoji":"🎯","tokens_out":14291,"duration_ms":117767,"temperature":0.7,"pith_summary":"This paper is a mathematical study of a class of learning problems in which training is modeled as a weakly controlled gradient system with a small control parameter, and the quality of the estimated parameters is measured by a validation loss at a fixed final time. The paper's aim is to characterize which learning trajectories are optimal for this terminal-cost problem: a trajectory-control pair is optimal if and only if the adjoint state associated with it satisfies a maximum-principle condition, and in that case the optimal model parameters are the endpoint of the trajectory. The paper also connects this characterization to the dynamic-programming value function and provides a successive Galerkin approximation algorithm for constructing such trajectories numerically. If the characterization is correct, it gives a principled criterion for optimality of a training run and a path from validation-aware optimal control back to estimated parameters.","feed_headline":"Optimal learning trajectories pass a maximum-principle test","feed_subtitle":"If the adjoint state meets the condition, the trajectory's endpoint is the optimal model parameter.","key_machinery":"The load-bearing object is the weakly-controlled gradient system $\\dot{\\theta}^\\varepsilon=-\\nabla J_0(\\theta^\\varepsilon,Z^{(1)})+\\varepsilon D(\\theta^\\varepsilon,\\tilde Z^{(1)})u(t)$, where $D$ is the diagonal matrix of squared partial derivatives of the loss with respect to each parameter, evaluated on a noise-perturbed version of the training set. Optimality is expressed through the Hamiltonian $H^\\varepsilon(\\theta,p,u)=\\langle p,-\\nabla J_0(\\theta,Z^{(1)})+\\varepsilon D(\\theta,\\tilde Z^{(1)})u\\rangle$, from which the forward state equation and backward adjoint equation (1.3)-(1.4) are derived. The identity that carries the argument is $-p^\\varepsilon(t)=\\nabla_\\theta V(t,\\bar\\theta^\\varepsilon(t))$, linking the adjoint state to the gradient of the value function from dynamic programming; this bridge is what allows the maximum-principle condition (2.2) to characterize optimal trajectories. The constructive part expands the control in a finite Galerkin basis, $u(t)=C\\Psi(t)$, and updates the coefficient matrix by successive approximation until the Hamiltonian variation vanishes.","core_discovery":"The central claim is Proposition 2.1: under the general assumptions of Section 1, the trajectory-control pair $(\\bar\\theta^\\varepsilon,\\bar u)$ of the weakly-controlled gradient system is optimal for the problem $J^\\varepsilon[u]=\\Phi(\\theta^\\varepsilon(T),Z^{(2)})\\to\\min$ if and only if the solution $p^\\varepsilon$ of the adjoint system with terminal condition $p^\\varepsilon(T)=-\\nabla\\Phi(\\bar\\theta^\\varepsilon(T),Z^{(2)})$ satisfies the maximum-principle condition (2.2) almost everywhere on $[0,T]$. In that case the optimal estimated parameter is recovered as $\\theta^*=\\bar\\theta^\\varepsilon(T)$. The paper further claims that along the optimal trajectory the adjoint state coincides with the negative gradient of the dynamic-programming value function, $-p^\\varepsilon(t)=\\nabla_\\theta V(t,\\bar\\theta^\\varepsilon(t))$, so the two classical optimality principles agree in this setting. The result turns the search for good model parameters into a forward-backward boundary value problem whose solution is the optimal trajectory.","pith_inferences":["One extension the paper leaves implicit is to use the maximum-principle condition as a training diagnostic: along any trajectory, the size of the violation of (2.2) would measure how far the trajectory is from the optimal one.","Because the perturbed dataset enters only through the control-coupling matrix $D$, the noise level in $\\tilde Z^{(1)}$ controls how much influence the control can exert; tuning that noise level may offer a practical trade-off between training fidelity and validation performance.","The success of the Galerkin scheme depends on the choice of basis $\\{\\psi_j\\}$ and the step sizes $\\gamma_{ij}$, both left unspecified; error bounds in the number of basis functions or a principled basis choice would be a natural next step."],"forward_implications":["Optimal parameters in this class of problems can be computed by solving a forward-backward system: the state equation forward, the adjoint equation backward, with the control updated by the maximum condition, and the optimal estimate is the final state $\\bar\\theta^\\varepsilon(T)$.","A candidate training trajectory can be checked for optimality by computing its adjoint state and verifying that the maximum-principle condition (2.2) holds almost everywhere.","Generalization performance, as measured by the validation cost at the final time, is the terminal cost of the control problem, so steering the weakly controlled gradient flow is the same as improving generalization.","The identity $-p^\\varepsilon(t)=\\nabla_\\theta V(t,\\bar\\theta^\\varepsilon(t))$ makes the adjoint state interpretable as the sensitivity of the optimal value with respect to the current parameter, connecting the two classical optimality conditions."],"supporting_citations":[{"why":"Supplies Pontryagin's maximum principle, from which the necessary optimality conditions (1.3)-(1.5) and the maximum-principle condition (2.2) are derived.","marker":"[3]"},{"why":"Establishes the relationship between the maximum principle and dynamic programming, the basis for the adjoint/value-function identity (2.4) and its nonsmooth variant in Remark 1.","marker":"[6]"},{"why":"Provides the optimal-control background and the Filippov-type reachable-set argument used in the general assumptions guaranteeing bounded solutions.","marker":"[2]"},{"why":"Provides the successive approximation method in functional spaces that the Galerkin algorithm adapts for constructing optimal trajectories.","marker":"[4]"},{"why":"Introduces the weakly-controlled gradient system with small parameters that defines the learning dynamics studied here.","marker":"[1]"},{"why":"Supplies the dynamic-programming regularity discussion referenced when the value function is not smooth, supporting Remark 1.","marker":"[5]"}],"fun_headline_variants":["Maximum principle maps optimal learning path","Adjoint state fixes the optimal learning route","Optimal learning: max principle meets dynamic programming","A test for optimal trajectories in learning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The training loss must have continuous second derivatives with respect to the model parameters, because the adjoint equation and the maximum-principle condition both contain second-derivative terms; this regularity is used throughout the derivation but never stated as an explicit assumption.","fun_headline_variants_meta":{"raw":{"variants":["Maximum principle maps optimal learning path","Adjoint state fixes the optimal learning route","Optimal learning: max principle meets dynamic programming","A test for optimal trajectories in learning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000134,"raw_usage":{"total_tokens":1121,"prompt_tokens":905,"completion_tokens":216,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":163}},"tokens_in":521,"tokens_out":216,"duration_ms":3514,"temperature":1.0,"reasoning_tokens":163,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T12:37:20.159496+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the forward-backward system (1.3)-(1.4) together with the maximum-principle condition (2.2) for a one-dimensional quadratic example, say $J_0(\\theta)=(\\theta-a)^2/2$ and $\\Phi(\\theta)=(\\theta-b)^2/2$, and compare two controls that both satisfy the conditions; if they lead to different terminal costs, the if-and-only-if claim in Proposition 2.1 is false.","supporting_citations":[{"cited_title":"Pontryagin, V","cited_arxiv_id":null,"evidence_quote":"Supplies Pontryagin's maximum principle, from which the necessary optimality conditions (1.3)-(1.5) and the maximum-principle condition (2.2) are derived."},{"cited_title":"Clarke & R.B","cited_arxiv_id":null,"evidence_quote":"Establishes the relationship between the maximum principle and dynamic programming, the basis for the adjoint/value-function identity (2.4) and its nonsmooth variant in Remark 1."},{"cited_title":"Liberzon","cited_arxiv_id":null,"evidence_quote":"Provides the optimal-control background and the Filippov-type reachable-set argument used in the general assumptions guaranteeing bounded solutions."},{"cited_title":"A successive approximation method in functional spaces for hierarchical optimal control problems and its application to learning","cited_arxiv_id":"2410.20617","evidence_quote":"Provides the successive approximation method in functional spaces that the Galerkin algorithm adapts for constructing optimal trajectories."},{"cited_title":"On improving generalization in a class of learning problems with the method of small parameters for weakly-controlled optimal gradient systems","cited_arxiv_id":"2412.08772","evidence_quote":"Introduces the weakly-controlled gradient system with small parameters that defines the learning dynamics studied here."},{"cited_title":"Fleming & R.W","cited_arxiv_id":null,"evidence_quote":"Supplies the dynamic-programming regularity discussion referenced when the value function is not smooth, supporting Remark 1."}],"review_version":1}