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REVIEW 3 major objections 5 minor 56 references

Kernel-Based Optimal Control: An Infinitesimal Generator Approach

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that learning the diffusion generator from drift snapshots turns stochastic optimal control into a finite ODE.

desk verdict Novel generator-regression kernel HJB with a real infinite-horizon gap; deserves referee time but needs major revision. read the letter →

arxiv 2412.01591 v3 pith:NUQCJSJV submitted 2024-12-02 math.OC cs.LGcs.ROcs.SYeess.SYstat.ML

Petar Bevanda , Nicolas Hoischen , Tobias Wittmann , Jan Brüdigam , Sandra Hirche , Boris Houska This is my paper · ORCID
classification math.OCcs.LGcs.ROcs.SYeess.SYstat.ML MSC 49L2093E2046E2247B32
keywords stochasticoptimalcontrolinfinitesimalgeneratorreproducingkernelHilbertspaceHamilton-Jacobi-BellmanequationFokker-Planck-Kolmogorovoperatorridgeregressionlearningdata-driven
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

The paper claims that the stochastic optimal control problem for a control-affine nonlinear system with small noise can be solved directly from data, without fitting a model in a physical state space or gridding the state space. The load-bearing object is the adjoint infinitesimal generator (the Fokker-Planck-Kolmogorov operator) of the controlled diffusion, estimated in a reproducing kernel Hilbert space from pointwise drift measurements under zero and one-hot inputs. The authors prove that the action of this generator on kernel sections is a kernel matrix whose entries require only drift values, kernel gradients, and kernel Laplacians, and they substitute that matrix into a convex Hamilton-Jacobi-Bellman recursion. This substitution reduces the infinite-dimensional HJB final-value problem to an $N$-dimensional ordinary differential equation whose terminal value produces an explicit feedback policy. A sympathetic reader would care because, if the claim is right, approximately globally optimal feedback policies become computable from modest state-derivative data rather than from a high-dimensional PDE discretization.

What carries the argument

The load-bearing identity is the kernel target matrix $(K_\pi)_{ij}$ of Lemma 1, which encodes the restriction of the adjoint infinitesimal generator $L_\pi^*$ to the canonical feature maps $k(\cdot,x)$. The first term is the drift-gradient (transport) contribution and the second term, $\epsilon$ times the trace of the kernel Hessian, is the diffusion contribution. The derivative reproducing property of the kernel converts what would be an infinite-dimensional Hilbert-Schmidt regression into finite matrix constructions using only the Gram matrix and derivative evaluations of the kernel. This is what allows the Hamilton-Jacobi-Bellman recursion to collapse to an $N$-dimensional ordinary differential equation.

What would settle it

Run the method on a one-dimensional Ornstein-Uhlenbeck process whose optimal policy is known analytically, and compare the learned policy against the analytic one while varying how the drift derivatives in the dataset are obtained (exact, finite-difference, or noisy). If the finite-difference version does not converge to the analytic solution as $N$ grows and $\epsilon$ shrinks, then the derivative-snapshot requirement of Assumption 2, not the kernel regularity condition, is the decisive limitation.

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Extended reading notes

Core claim

The paper's central claim is that the adjoint infinitesimal generator $L_\pi^*$ of the controlled diffusion $dX_t=(f(X_t)+G(X_t)\pi(X_t))dt+\sqrt{2\epsilon}\,dW_t$ can be learned nonparametrically in a reproducing kernel Hilbert space, and that the learned generator suffices to solve the stochastic optimal control problem. The key identity (Lemma 1) states that for a smooth symmetric positive-definite kernel $k$, the action of $L_\pi^*$ on the kernel section $k(\cdot,x^{(j)})$, evaluated at $x^{(i)}$, is $(K_\pi)_{ij}=\langle f(x^{(i)})+G(x^{(i)})\pi(x^{(i)}),\,\nabla_{x^{(i)}}k(x^{(i)},x^{(j)})\rangle+\epsilon\,\mathrm{Tr}(\nabla^2_{x^{(i)}}k(x^{(i)},x^{(j)}))$. With $A=K_\gamma^{-1}K_0$ and $B_m=K_\gamma^{-1}(K_{e_m}-K_0)$, and with $D_r$ the dual control penalty $D_r(\lambda)=\min_{u\in U}\{r(u)+\langle\lambda,u\rangle\}$, the infinite-dimensional HJB recursion becomes the finite-dimensional final-value problem $-\dot v=Av+q+D_r(Bv)$, $v(T)=0$, on $(0,T)$, and Algorithm 1 sets $\hat\pi^\star(x)=u^\star(\langle Bv_0,k(x)\rangle)$. The paper asserts that this $N$-dimensional ODE, built entirely from Gram matrices and derivative evaluations, approximates the infinite-dimensional solution and yields approximately globally optimal feedback policies.

Load-bearing premise

The method only works as stated if the dataset contains exact instantaneous derivative measurements of the drift vector field at the sampled states under zero control and under each one-hot input; when those derivatives are approximated by finite differences, as in the authors' simulated-robot experiments, the resulting bias in the learned generator is left unquantified.

Editorial extensions

If this is right

  • The stochastic optimal control problem reduces to integrating an $N$-dimensional ordinary differential equation backward in time, so the computational complexity scales with the number of data points rather than with the dimension of the state space.
  • The estimator includes nonparametric estimators of uncontrolled infinitesimal generators as the special case $\pi=0$, so the same construction yields a data-driven method for approximating Fokker-Planck-Kolmogorov operators.
  • Because the representation is in continuous time, the method does not depend on a chosen time lag of the data, in contrast to discrete-time Koopman and transfer-operator methods, and it returns explicit value and policy functions.
  • In the reported experiments, the learned kernel policy achieves lower accumulated stage costs than a receding-horizon nonlinear MPC solver on the same pendulum and cartpole dynamics, and the method needs fewer data points and is less sensitive to the kernel lengthscale than the discrete-time kernel HJB baseline.

Reading between the lines

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

  • Beyond the paper: the same kernel generator matrices could be used for data-driven certification, since the matrices represent $L^*$ on the sample points and therefore let one evaluate the sign of the generator applied to a candidate Lyapunov or barrier function.
  • Beyond the paper: the unquantified finite-difference bias under Assumption 2 could be characterized by comparing exact and approximated kernel matrices on a system with a known generator; the paper does not attempt this comparison.
  • Beyond the paper: the final-value ODE can be re-solved over a moving horizon, yielding an online data-driven model-predictive controller with a kernel terminal cost, an extension the paper leaves implicit.
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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

3 major / 5 minor

Summary. The paper proposes a data-driven method for stochastic optimal control by learning the infinitesimal generator of a controlled diffusion in an RKHS. The key theoretical elements are Lemma 1, which expresses the empirical target kernel matrices K_pi in terms of drift derivatives and the kernel's gradient and Hessian, and Proposition 2, which reduces an infinite-dimensional HJB final-value problem to an N-dimensional ODE system, the "HJB-FVP" recursion. Algorithm 1, named IG-KHJB, then outputs a feedback policy from the solution of this finite-dimensional recursion. The paper reports experiments on a Van der Pol oscillator, an inverted pendulum, and a cartpole, comparing against a discrete-time kernel HJB method and against Altro-NMPC.

Significance. The paper's main contribution is a clean operator-theoretic derivation that avoids explicit grid-based PDE discretization: the derivative-reproducing property turns generator regression into kernel matrix computations, and the control-affine structure yields a finite-dimensional HJB recursion. If the infinite-horizon and approximation issues are resolved, the approach would be a practical data-driven alternative for nonlinear stochastic optimal control. The paper is explicit about the need for derivative data in Assumption 2 and uses finite differences in the simulator experiments, but the consequences of that approximation are not analyzed. The central claims are plausible, but they currently lack the error analysis needed to support the phrase "approximately globally optimal".

major comments (3)
  1. [Section 3.1, Eq. (HJB-FVP), and Algorithm 1] The paper formulates an infinite-horizon average-cost problem (2) but computes the policy from the solution of a finite-horizon FVP with zero terminal condition, V(T)=0. For the exact infinite-horizon HJB, V_T(0,x) = lambda*T + w(x) + o(1) as T tends to infinity, and the T-dependent term cancels in the policy because B^* 1 = 0. In the finite-rank approximation, the constant function 1 is not in the RKHS for the Gaussian or Laplace kernels used in the experiments, so no vector e in R^N satisfies the approximate identity exactly; the solution v0 therefore contains a component proportional to T, and the policy contains the unexamined term lambda*T*<B e, k(x)>. The manuscript provides no theorem on the T-to-infinity limit, no bound on ||B e||, and no numerical sweep over the horizon H (Table 1 fixes H to 1000, 500, and 3000). As a result, the claim that Algorithm 1 solves the infinite-horizon problem (2) is not supported. The authors should either prove that the T-dependence cancels in the finite-dimensional recursion, add a T-sensitivity analysis, or reframe the contribution as finite-horizon optimal control.
  2. [Assumption 2 and Section 5 (Implementation details)] Assumption 2 requires exact instantaneous derivative measurements at the data points. In the Dojo experiments these derivatives are approximated by finite differences, as stated in Section 5, but no bias or error bound is given. Lemma 1 uses the derivative-reproducing property on the exact drift values; a finite-difference approximation biases the target matrices K_pi and hence the estimated operators in Eq. (12) in a way that is not captured by the RKHS regression framework. Since this assumption is the main practical limitation of the method, the paper should quantify the resulting error, for example in terms of the finite-difference step and the smoothness of f and G, or provide empirical validation of the finite-difference approximation.
  3. [Section 4, Proposition 2] Proposition 2 establishes an algebraic equivalence between the infinite-dimensional estimated HJB and the finite-dimensional recursion, but it is not an approximation result: it does not state how the solution v0 and the resulting policy relate to the true value function or optimal policy as N tends to infinity, gamma tends to zero, or as the empirical estimator converges. The abstract and Section 6 use the phrase "approximately globally optimal" without a quantitative error bound. At minimum, a consistency statement, such as convergence of the reconstructed value function to the true relative value function in an appropriate norm under Assumptions 1 and 2, is needed to support the central claim.
minor comments (5)
  1. [Section 4, Proposition 2] In the definition of the matrix B_i, the index m should be i; the current subscript is inconsistent with the surrounding notation.
  2. [Section 1] The phrase "Structure1." contains a missing space and should read "Structure. 1." or similar.
  3. [Sections 5.2 and 5.3] The text refers to Figure 4(a) and Figure 4(b), but the displayed figure is captioned Figure 3; the figure numbering should be corrected.
  4. [Table 1] The column headers and entries in Table 1 are difficult to parse because the delimiters are not aligned; the authors should reformat the table so that each column is unambiguous.
  5. [Section 4, Proposition 2 and Algorithm 1] The notation D_r(Bv) is overloaded: in the proposition it denotes a vector in R^N, while in Algorithm 1 it is defined through the regularized inverse K_gamma^{-1}. Please use distinct symbols for the pointwise function D_r and its coefficient vector.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the kernel generator estimates are derived from operator identities and the policy is obtained by solving the projected HJB equation, not by fitting it to observed optimal behavior.

full rationale

The core derivation chain is not circular. Lemma 1 computes the kernel matrix entries (K_pi)_ij from the Itô formula and the derivative reproducing property, which is an exact evaluation of the adjoint infinitesimal generator on kernel sections using the derivative measurements in Assumption 2. The matrices A and B in (12) are assembled from these evaluations, and the N-dimensional FVP in Proposition 2 is obtained by projecting the estimated HJB equation onto the kernel sections. The value coefficients v0 and the policy pi*(x)=u*(<Bv0,k(x)>) are outputs of solving that FVP, not parameters fitted to observed optimal trajectories or to stage-cost samples beyond the q term. There is therefore no fitted-input-called-prediction or self-definitional reduction. The citations to Houska (2025) and Bevanda et al. (2024b) supply the convex FPK/HJB duality and strong-duality conditions; those are prior stated theorems whose assumptions do not include the kernel construction of this paper, so they count as independent support even though the author lists overlap. The finite-horizon truncation and the finite-difference approximation of derivatives in the Dojo experiments are correctness or robustness concerns, as the paper provides no T-to-infinity bound, but they do not make the derivation circular. Overall score 1: no significant circularity, with only non-load-bearing self-citations.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The approach uses the standard infinitesimal generator of a diffusion and RKHS machinery.

free parameters (6)
  • Diffusion coefficient epsilon = 0.01 (oscillator, cartpole), 0.02 (pendulum)
    Chosen per experiment as a proxy for process noise; not estimated from data. The method solves the stochastic control problem with this epsilon, so the resulting policy depends on it (Table 1).
  • Kernel lengthscale sigma = 43 for IG-KHJB in oscillator comparison; 3.0, 25, etc. per Table 1
    Hyperparameter of the RBF/Laplace kernel; selected via sweeps or hand-tuning. In Figure 1, the 'optimal' sigma is chosen from the sweep to report best RMSE.
  • Regularization gamma = 10^-8 to 10^-12 depending on experiment
    Tikhonov regularization in KRR; chosen to stabilize the Gram matrix inversion (Table 1).
  • Time horizon H and time step Delta t = e.g., H=1000, Delta t=0.01 for oscillator
    Numerical integration parameters for the finite-horizon HJB displayed in Table 1; the ergodic limit is approximated by a finite T.
  • Smoothing parameter for Laplace kernel derivative at 0 = sigma/100
    Footnote 5: the derivative at 0 is replaced by that of an RBF kernel with sigma/100 to stabilize computations. This is a manual hack.
  • Finite-difference step for derivative and kernel partial derivative approximations = Not specified
    In Section 5, dynamics and kernel partial derivatives are approximated by finite differences; the step size is not reported.
assumptions (6)
  • domain assumption Assumption 1: Existence of a stabilizing feedback and Lyapunov function, strongly convex control penalty r, and a feasible interior control set U.
    Imported from Bevanda et al. 2024b and Houska 2025 to guarantee well-posedness and existence of the ergodic optimal control problem (Section 2).
  • domain assumption Assumption 2: The dataset D_N contains state observations and instantaneous derivative measurements of f(x) and G(x) at the same states under u=0 and one-hot inputs.
    This is the load-bearing data assumption; it requires direct access to derivative labels, not just state trajectories (Section 2, Eq. 3).
  • domain assumption Assumption 3: RKHS H is norm-equivalent to H^s(X) with s > n_x/2 + 1.
    Used to ensure the inclusion H -> H^1(X) is Hilbert-Schmidt via Maurin's theorem, making the regression well-defined (Section 4). Fulfilled by Matern and Gaussian kernels.
  • standard math Ito's formula and derivative reproducing property (Zhou 2008, Theorem 1) for the generator acting on kernel sections.
    Used in Lemma 1 proof to derive the kernel target matrix formula (11).
  • domain assumption Strong duality and equivalence between the stochastic control problem and the convex FPK primal and HJB dual (Houska 2025, Thm. 1; Bevanda et al. 2024b, Thm. 2).
    The entire HJB approach relies on this prior duality result, which is imported rather than reproved.
  • standard math Fokker-Planck-Kolmogorov operator theory: the transition semigroup and its generator are well defined on H^1(X).
    Standard background for diffusion processes, referenced to Bogachev et al. 2015 and Engel 2000.

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Cite this review

Pith. "Pith review of Kernel-Based Optimal Control: An Infinitesimal Generator Approach." pith.science (2026). https://pith.science/paper/NUQCJSJV

@misc{pith2026241201591,
  author       = {Pith},
  title        = {Pith review of: Kernel-Based Optimal Control: An Infinitesimal Generator Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUQCJSJV}},
  note         = {Machine review of arXiv:2412.01591}
}
read the original abstract

This paper presents a novel operator-theoretic approach for optimal control of nonlinear stochastic systems within reproducing kernel Hilbert spaces. Our learning framework leverages data samples of system dynamics and stage cost functions, with only control penalties and constraints provided. The proposed method directly learns the infinitesimal generator of a controlled stochastic diffusion in an infinite-dimensional hypothesis space. We demonstrate that our approach seamlessly integrates with modern convex operator-theoretic Hamilton-Jacobi-Bellman recursions, enabling a data-driven solution to the optimal control problems. Furthermore, our learning framework includes nonparametric estimators for uncontrolled infinitesimal generators as a special case. Numerical experiments, ranging from synthetic differential equations to simulated robotic systems, showcase the advantages of our approach compared to both modern data-driven and classical nonlinear programming methods for optimal control.

Figures

Figures reproduced from arXiv: 2412.01591 by the authors.

Figure 1
Figure 1. Comparison of RMSE to the known optimal policy π ⋆ ∞(x) between KHJB ( Bevanda et al. (2024b)) and our IG-KHJB approach for the Van der Pol Oscillator. performance of KHJB (Bevanda et al., 2024b) against our proposed IG-KHJB for a varying length￾scale ( [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Contour plots of the value and controller func￾tions learned from 2.5 · 103 samples using our IG-KHJB approach (upright at ( ˙θ, θ)=(0, 0)). We evaluate our method on a swing￾up and stabilization task at the up￾right equilibrium θ = 0 for the in￾verted pendulum. The value func￾tion and policy shown in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Accumulated stage costs using our learned policy [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.