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REVIEW 5 major objections 5 minor 7 references

On characterizing optimal learning trajectories in a class of learning problems

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.16521 v2 pith:IHVOKBB6 submitted 2025-01-27 math.OC stat.ML

Getachew K Befekadu This is my paper · ORCID
classification math.OCstat.ML MSC 49K1549L2065M60
keywords dynamicprogrammingGalerkinmethodgeneralizationHamiltonianfunctionoptimalcontrolproblemlearningtrajectoriesPontryagin'smaximumprinciplesuccessiveapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

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.

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 (5)
  1. [Section 2, Proposition 2.1 and Eq. (2.2)] 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.
  2. [Section 1, Eq. (1.4)] 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.
  3. [Section 2, Proposition 2.1] 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.
  4. [Section 2, 'A successive Galerkin approximation method'] 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.
  5. [Section 1(d) and Remark 1] 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.
minor comments (5)
  1. [Abstract] The abstract contains several broken words, such as 'characteriz ing', 'tha t', and 'paramete rs', and should be carefully proofread.
  2. [Footnote 5] 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.
  3. [Eq. (2.5)] 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).
  4. [Algorithm, Step 1] 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.
  5. [Section 1, Eq. (1.4)] 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.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: central claim rests on standard optimal control; one definitional restatement of θ* and non-load-bearing self-citations.

  1. self definitional [Section 1(c), footnote 5; Proposition 2.1, Eq. (2.3)]
    "serves as a measure for evaluating the quality of the estimated optimal parameter θ∗ = θǫ(T) ... Moreover , the optimal estimated parameters θ∗ can be recovered from θ∗ = ¯θǫ(T)."

    In Section 1(c), the paper already defines the optimal estimated parameter as θ∗ = θ^ε(T). Proposition 2.1 then presents the identity θ∗ = ¯θ^ε(T) as a recoverable consequence of the maximum principle. This is the same equality by construction, so Eq. (2.3) restates the definition rather than deriving a new result. The step is tautological but not load-bearing: the main characterization in Proposition 2.1 does not depend on this identity.

full rationale

The derivation chain is not circular in any load-bearing way. The necessary conditions (1.3)-(1.5) are imported from Pontryagin's maximum principle [3] and the maximum-principle/dynamic-programming relationship is imported from Clarke and Vinter [6], both external to the authors; the self-citations [1] and [4] appear only in explanatory footnotes and do not supply the premises of Proposition 2.1. Proposition 2.1, if it were correct, would be an application of standard optimal-control theory rather than an assumption of its own conclusion. The only definitional step is Eq. (2.3), which repeats the earlier definition θ* = θ^ε(T); this is tautological but harmless to the central characterization. The paper's serious concerns are mathematical correctness, such as Eq. (2.2) not being the Hamiltonian maximization condition for the stated dynamics, which is a correctness issue rather than a circularity issue.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The framework rests on standard optimal control theory and on the author's prior formulation [1],[4]; the main gap is the unproved iff proposition and the unstated smoothness requirements.

free parameters (4)
  • ε = small positive parameter
    Controls the strength of the control term in the dynamics; assumed small but value unspecified.
  • σ = σ = c‖y_i‖∞
    Noise level in the perturbed dataset; c is a weighted noise level such as 1%, 5%, or 10%, chosen by hand to provide dithering.
  • γ_ij = γ_ij ∈ [0, 1]
    Step sizes in the successive approximation update; chosen by hand, no guidance given.
  • c_ij = unknown coefficients
    Galerkin coefficients for the control expansion, to be determined by the algorithm.
assumptions (5)
  • standard math Pontryagin's maximum principle provides necessary optimality conditions for the optimal control problem.
    Invoked in Section 1 to derive the Euler-Lagrange equations (1.3)-(1.5), citing Pontryagin et al. [3].
  • standard math Dynamic programming principle and the Hamilton-Jacobi-Bellman equation characterize the value function.
    Used in Section 2 to relate the adjoint variable to the gradient of the value function, citing Fleming-Rishel [5] and Clarke-Vinter [6].
  • domain assumption Existence of an optimal control and a compact reachable set for the weakly-controlled system.
    Assumed in Section 1(d)(iii) and (d)(iv), with a citation to Liberzon [2] for Filippov's theorem.
  • ad hoc to paper The value function V(t,θ) is smooth and continuously differentiable so that −p(t) = ∇_θ V(t,θ) holds.
    Eq (2.4) in Section 2; Remark 1 acknowledges this may fail, and proposes a differential inclusion substitute that is not developed.
  • ad hoc to paper The loss function ℓ and the training objective J0 are twice continuously differentiable in θ.
    Required for the Hessian terms ∇²J0 in the adjoint equation (1.4) and the maximum principle (2.2), but not stated among the general assumptions in Section 1(d).

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Cite this review

Pith. "Pith review of On characterizing optimal learning trajectories in a class of learning problems." pith.science (2026). https://pith.science/paper/IHVOKBB6

@misc{pith2026250116521,
  author       = {Pith},
  title        = {Pith review of: On characterizing optimal learning trajectories in a class of learning problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IHVOKBB6}},
  note         = {Machine review of arXiv:2501.16521}
}
read the original abstract

In this brief paper, we provide a mathematical framework that exploits the relationship between the maximum principle and dynamic programming for characterizing optimal learning trajectories in a class of learning problem, which is related to point estimations for modeling of high-dimensional nonlinear functions. Here, such characterization for the optimal learning trajectories is associated with the solution of an optimal control problem for a weakly-controlled gradient system with small parameters, whose time-evolution is guided by a model training dataset and its perturbed version, while the optimization problem consists of a cost functional that summarizes how to gauge the quality/performance of the estimated model parameters at a certain fixed final time w.r.t. a model validating dataset. Moreover, using a successive Galerkin approximation method, we provide an algorithmic recipe how to construct the corresponding optimal learning trajectories leading to the optimal estimated model parameters for such a class of learning problem.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages

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    work page Pith review arXiv doi:10.48550/arxiv.2412.08772 2024

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    A successive approximation method in functional spaces for hierarchical optimal control problems and its application to learning

    G.K. Befekadu . A successive approximation method in functional spaces for hierarchical optimal control problems and its application to learning. arXiv:2410.20617 [math.OC] , 2024. Available at https://doi.org/10.48550/arXiv.2410.20617

    work page Pith review arXiv doi:10.48550/arxiv.2410.20617 2024

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    Liberzon

    D. Liberzon . Calculus of variations and optimal control theory. Princeton University Press, Princeton, NJ, 2012

    work page 2012

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    Pontryagin, V

    L.S. Pontryagin, V. Boltianski, R. Gamkrelidze, & E. Mitchtchenko . The mathematical theory of optimal processes . John Wiley & Sons, New York, 1962

    work page 1962

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    H. Fleming & R.W. Rishel . Deterministic and stochastic optimal control . Springer, New York, 1975

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    F.H. Clarke & R.B. Vinter . The relationship between the maximum principle and dynamic programming. SIAM J. Control Optim. , vol. 25(5), 1291--1311, 1987

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  7. [7]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.checka empty 'skip 'new.block if FUNCTION field.or.null duplicate empty pop "" 'skip ...

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