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REVIEW 5 major objections 5 minor 28 references

Solving Inverse Problems via Diffusion Optimal Control

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Framing the reverse diffusion process as an iLQR optimal-control episode replaces the intractable conditional likelihood and approximate score guidance with a gradient propagated through a full rollout, and reports state-of-the-art…

desk verdict A genuine algorithmic contribution with strong FFHQ numbers, but the proof of the posterior-sampling recovery is internally inconsistent and the theory does not match the implemented method. read the letter →

arxiv 2412.16748 v1 pith:C5AD6MWB submitted 2024-12-21 cs.LG

Henry Li , Marcus Pereira This is my paper · ORCID
classification cs.LG
keywords diffusionmodelsinverseproblemsoptimalcontroliLQRposteriorsamplingimagereconstructionzero-shotrestorationscore-basedgenerative
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 argues that inverse problems—recovering an image from a lossy measurement—can be solved by treating the discretized reverse diffusion process as an optimal control problem instead of a probabilistic sampling problem. With an iLQR controller and a terminal cost equal to the negative log-likelihood of the measurement, the control signal is exactly proportional to the gradient of the measurement likelihood with respect to the noisy state, and with a specific Tikhonov parameter it reproduces the idealized posterior sampling equation. On a 256×256 face dataset, the method reports lower FID and LPIPS than existing diffusion-based inverse solvers across super-resolution, inpainting, and deblurring tasks, and it remains stable with far fewer diffusion steps.

What carries the argument

The carrying object is the discretized reverse diffusion map $x_{t-1} = h(x_t, u_t)$, together with the iterative Linear Quadratic Regulator (iLQR) algorithm. iLQR propagates a quadratic approximation of the value function backwards through the rollout; with zero running costs and a terminal cost given by the negative measurement log-likelihood, the second-order terms vanish and the feedforward gain reduces to $-\frac{1}{\alpha}V_x$, so the control becomes the likelihood gradient. The two ways of injecting control—input perturbation, where the control is added before the diffusion step, and output perturbation, where it is added after—produce the same structure, and the Tikhonov regularizer $\alpha I$ on the $Q_{uu}$ inverse is the knob that matches the idealized posterior sampler. The paper also adds randomized low-rank approximations and matrix-free products to make the high-dimensional Jacobians and Hessians tractable.

What would settle it

Run the method on a low-dimensional linear inverse problem with a known Gaussian prior and known posterior, compute the iLQR control $u_t$ numerically, and compare it with $\alpha \nabla_{x_t} \log p(y|x_0)$ across values of $\alpha$: Theorem 4.1 predicts an exact proportional match, so any mismatch in scaling or sign would falsify the derivation.

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Extended reading notes

Core claim

The central claim is that the drawbacks of diffusion-based inverse solvers—intractable conditional likelihood, dependence on score-network quality, and poor $\mathbf{x}_0$ prediction—are not inherent to diffusion sampling but to the probabilistic framing. By defining the discrete-time reverse diffusion trajectory as the dynamics of an optimal control episode with zero running costs and terminal cost $-\log p(y|x_0)$, the iLQR update with Tikhonov regularization produces controls $u_t = \alpha \nabla_{x_t} \log p(y|x_0)$ (Theorem 4.1); with $\alpha = 1/(g(t)^2 \Delta t)$ and the deterministic sampler this coincides with the idealized posterior sampling equation. The same construction in input-perturbation mode yields a predictor-corrector sampler (Theorem 4.3). Empirically, the paper reports a new baseline in reconstruction performance on FFHQ 256×256 for super-resolution, random and box inpainting, Gaussian deblurring, and motion deblurring, and demonstrates that the method continues to produce feasible solutions even with a randomly initialized score network, where diffusion posterior sampling fails.

Load-bearing premise

The recovery of exact posterior sampling assumes the deterministic ODE sampler and that the noisy state at intermediate times contains no information about the measurement beyond the clean image, and it takes the forward rollout through the learned score network as exact.

Editorial extensions

If this is right

  • Because the controller only needs a differentiable forward operator, the same recipe applies zero-shot to linear and nonlinear measurements, including neural classifiers.
  • The posterior sampling equation used by existing probabilistic solvers becomes a special case, so improvements to the control solver directly improve posterior sampling rather than requiring a new likelihood approximation.
  • The method's stability at small total diffusion steps means inverse-problem solvers built this way can use fewer steps, cutting computation without the usual quality collapse.
  • The reported tolerance to a poorly approximated score suggests the solver can work with approximate or biased priors better than probabilistic samplers can.

Reading between the lines

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

  • Beyond the paper: the same iLQR controller could steer toward differentiable objectives other than measurement likelihood, such as perceptual or semantic scoring functions, inside a single rollout.
  • Beyond the paper: applying this construction to distilled or quantized diffusion models would test whether the reported tolerance to score error survives when the prior itself is only approximately preserved.
  • The paper leaves open whether rank-one Hessian approximations preserve the posterior-sampling identity for strongly non-Gaussian terminal costs; a direct numerical check in a low-dimensional setting would settle that.
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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 proposes 'diffusion optimal control', a method for solving inverse problems by treating the discretized reverse diffusion process as an iLQR optimal control problem, with a pretrained unconditional diffusion model as the dynamics and a task-dependent terminal cost. The authors claim that Tikhonov-regularized iLQR yields controls equal to the conditional score (Theorem 4.1), that an appropriate choice of the regularization constant recovers posterior sampling (Lemma 4.2), that an analogous input-perturbation result holds (Theorem 4.3), and that the resulting solver achieves state-of-the-art image reconstruction on FFHQ 256x256 across super-resolution, inpainting, and deblurring, with additional class-conditional MNIST experiments.

Significance. If correct, this would be a valuable zero-shot inverse-problem solver: it avoids training conditional score models, works with any differentiable forward operator, and uses an off-the-shelf unconditional diffusion prior. The paper also ships code, includes ablations over rank, alpha, and T, and evaluates against many baselines. I do not see a circularity problem: the method is an optimization-based solver with a data-fidelity terminal cost, and the posterior-recovery statement is a claimed derived property rather than an input assumption. However, the central theoretical derivation has several load-bearing algebraic inconsistencies, and the implemented algorithm is not the one analyzed in the theorems; the empirical results are interesting but do not by themselves establish the advertised theoretical bridge.

major comments (5)
  1. [Appendix C, Theorem 4.1] The statement of Theorem 4.1 (Eq. 29) asserts u_t = alpha * grad_{x_t} log p(y|x0), but the proof concludes in Eq. (64) with u_t = (1/alpha) * grad_{x_t} log p(y|x0). This is not a notational slip: with the value of alpha chosen in Lemma C.1, alpha = 1/(g(t)^2 Delta t), the two expressions differ by a factor (g(t)^2 Delta t)^2. Since Lemma C.1 substitutes the theorem's control into Eq. (18), the claimed recovery of posterior sampling does not follow from the proof as written.
  2. [Appendix C, Eqs. (51)-(58)] The Riccati step conflates the input- and output-perturbation formulas. For output-perturbation dynamics (Eq. 18) h_u = I, so Q_uu = l_uu + V'_xx, not h_x^T V'_xx h_x + alpha I as used in Eq. (58). In addition, at the first backward sweep V'_xx is the terminal Hessian d^2 l_0/dx_0^2, generically nonzero, so the cancellation in Eqs. (52)-(53) is invalid. Eq. (56) is also arithmetically wrong: h_x^T V'_x + h_x^T V'_xx (V'_xx)^-1 V'_x equals 2 h_x^T V'_x, not h_x^T V'_x. The induction therefore does not establish that V_xx vanishes or that the control law holds.
  3. [Appendix C, Lemma C.1 and Eq. (65)] Equation (65) is dimensionally inconsistent: the left side is a state x_{t-1}, while the right side is a velocity-like expression without the state x_t and without the structure of Eq. (18). Even setting that aside, substituting alpha = 1/(g(t)^2 Delta t) into Eq. (18) yields a control contribution proportional to (g(t)^2 Delta t) grad log p(y|x0), not the -1/2 g(t)^2 Delta t grad log p(y|x0) term needed to turn Eq. (3) into the posterior update of Eq. (9). The sign inconsistency between Eq. (3), which has a minus before the bracket, and Eq. (9), which has a plus, compounds the problem. Lemma C.1 therefore does not demonstrate the claimed recovery of posterior sampling.
  4. [Section 4, Algorithm 1, Table 2] The reported experiments do not instantiate the analyzed controller. The theorems assume output- or input-perturbation dynamics with zero running cost and a log-likelihood terminal cost, while Algorithm 1 and Table 2 use input-perturbation mode, running cost alpha ||u_t||, terminal cost ||A(x0)-y||, alpha = 1e-4, rank-1 projected second-order terms, and Adam-preconditioned updates. Theorem 4.3 is the only input-mode result, but it still assumes zero running cost and terminal cost +log p(y|x0), and its proof has the same 1/alpha and sign issues. Thus the theoretical results do not cover the method whose performance is reported, and no separate analysis of the implemented cost is given.
  5. [Section 4, Figure 2] The claim that x0 is computed exactly on every step is overstated. The forward rollout uses the learned score network s_theta(x_t,t) inside h (Eq. (3)), so the computed x0 is exact only if the score model is exact; otherwise it inherits score-approximation error. The subsequent claim that the method is robust to score error is an empirical observation, not a consequence of exact x0 computation, and the abstract's statement that the method sidesteps dependence on score approximation needs to be qualified.
minor comments (5)
  1. [Section 2.1, Eq. (2)] Equation (2) contains the typo 'dxt = dxt + ...'; the first term should be the drift, not dxt.
  2. [Section 2.3, Eqs. (10) and (34)] The notation 'JT = 1X t=T ...' is confusing; the summation should be written explicitly as a sum over t = T, ..., 1.
  3. [Section 2.1, Eqs. (3) and (9)] The sign structure of Eq. (9) is inconsistent with Eq. (3): if Eq. (3) is the Euler discretization of the PF-ODE, the conditional Euler step should have the same minus-before-bracket form, but Eq. (9) has a plus. This inconsistency affects the reading of Lemma C.1.
  4. [Table 1] The row 'Ours (NFE = 2500)' uses 2.5 times the NFE of most baselines; presenting a matched-NFE comparison in the main table, or referring more prominently to the budget analysis in the appendix, would make the performance claims easier to interpret fairly.
  5. [Appendix D.1] The statement that vector-Jacobian products 'have runtime complexity O(1)' is imprecise; the intended point is that full Jacobians need not be materialized, but the actual arithmetic cost is not O(1) in dimension.

Circularity Check

0 steps flagged · score 2.0 of 10

No meaningful circularity: the iLQR derivation and empirical evaluation are self-contained; only non-load-bearing self-citations appear.

full rationale

The paper's central derivation (Theorem 4.1, Lemma 4.2, Theorem 4.3) is an optimal-control argument. The control law is obtained from Bellman recursion and the terminal cost -log p(y|x0), not by postulating the posterior sampler as an input. The claimed controls are backpropagated gradients of that terminal cost, and recovering Eq. (9) is an attempted equivalence, not an assumed conclusion. The forward rollout and x0 computation are algorithmic claims, not fits to the target measurement. Empirically, the method is evaluated on held-out FFHQ and MNIST with pretrained weights, so no prediction is statistically forced. The self-citations present (Li et al. 2024 in Section 2.1 for likelihood computation; Pereira et al. 2020 in Related Work) are background citations and are not load-bearing; neither is invoked to exclude alternatives. The paper does contain serious proof inconsistencies: Eq. (29) states u_t = alpha * grad log p(y|x0), while Eq. (64) concludes u_t = (1/alpha) * grad log p(y|x0); Lemma 4.2's Eq. (65) is dimensionally and sign inconsistent with Eq. (9); and the Riccati cancellation at Eq. (52) omits the terminal Hessian and a factor of 2. These are correctness risks, not circularity, because the target result is not assumed at the start of the derivation. Hence a low circularity score of 2 is appropriate.

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

The method does not introduce new physical entities. Its free parameters are standard optimization hyperparameters (regularizer, step size, rank, step count). The key assumptions are the accuracy of the pretrained score network and the idealized zero-running-cost setup used in the proofs, plus the log p(y|x_t) = log p(y|x_0) condition for the posterior-sampling recovery.

free parameters (5)
  • Tikhonov regularization / running cost weight α = 1e-4 (Table 2)
    Chosen to prevent singular matrix inversions and as the control cost weight; ablations show robustness but there is no theoretical guide for the value.
  • iLQR step size (λ in Algorithm 1, 'step_size' in Table 2) = 1e-3
    Controls the magnitude of feedforward control updates; chosen empirically.
  • Low-rank approximation rank k = 1
    Selected for computational efficiency; ablations show diminishing returns beyond k=1.
  • Number of diffusion steps T = 50
    A user-selected balance between sample quality and compute (Table 6).
  • Number of iLQR iterations num_iters = 50-100
    Chosen to ensure convergence; not derived from theory.
assumptions (5)
  • domain assumption The learned score network s_θ(x_t, t) accurately approximates the true data score ∇ log p_t(x_t).
    The paper relies on a pretrained diffusion model for the dynamics and for theoretical recovery of posterior sampling (Sections 2.1, 4).
  • domain assumption The forward measurement operator A is differentiable and the noise η is Gaussian (Eq. 5).
    Required for computing ∇ log p(y|x_0) and for the terminal cost to be differentiable.
  • domain assumption The reverse diffusion process can be discretized with Euler steps and treated as a deterministic controlled dynamical system (Eq. 3, Eq. 18).
    The iLQR framework assumes a deterministic transition function h(x_t, u_t).
  • ad hoc to paper For the theoretical results, running costs ℓ_t are zero and the terminal cost is twice-differentiable (Theorem 4.1).
    This is an idealization that simplifies the iLQR derivation; in practice ℓ_t = α||u_t|| is used (Table 2).
  • ad hoc to paper Under the deterministic ODE sampler, log p(y|x_t) = log p(y|x_0) (Lemma C.1).
    The paper states this assumption to recover posterior sampling, but it is only valid if each x_t has a unique path to x_0, which is not strictly true for the discretized process.

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Cite this review

Pith. "Pith review of Solving Inverse Problems via Diffusion Optimal Control." pith.science (2026). https://pith.science/paper/C5AD6MWB

@misc{pith2026241216748,
  author       = {Pith},
  title        = {Pith review of: Solving Inverse Problems via Diffusion Optimal Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C5AD6MWB}},
  note         = {Machine review of arXiv:2412.16748}
}
abstract

Existing approaches to diffusion-based inverse problem solvers frame the signal recovery task as a probabilistic sampling episode, where the solution is drawn from the desired posterior distribution. This framework suffers from several critical drawbacks, including the intractability of the conditional likelihood function, strict dependence on the score network approximation, and poor $\mathbf{x}_0$ prediction quality. We demonstrate that these limitations can be sidestepped by reframing the generative process as a discrete optimal control episode. We derive a diffusion-based optimal controller inspired by the iterative Linear Quadratic Regulator (iLQR) algorithm. This framework is fully general and able to handle any differentiable forward measurement operator, including super-resolution, inpainting, Gaussian deblurring, nonlinear deblurring, and even highly nonlinear neural classifiers. Furthermore, we show that the idealized posterior sampling equation can be recovered as a special case of our algorithm. We then evaluate our method against a selection of neural inverse problem solvers, and establish a new baseline in image reconstruction with inverse problems.

Figures

Figures reproduced from arXiv: 2412.16748 by the authors.

Figure 1
Figure 1. Conceptual illustration comparing a probabilistic posterior sampler to our proposed optimal control-based sampler. In a probabilistic sampler, the model relies on an approximation x˜0 ≈ x0 to guide each step (left). We are able to compute x0 exactly on each step, resulting in much higher quality gradients ∇ log p(y|x˜0) and an improved trajectory update (right). To address these issues, we propose a novel framework … view at source ↗
Figure 2
Figure 2. Predicted x0 used in a probabilistic framework (above) compared to ours (below) for a general diffusion trajectory. The full forward rollout in our proposed framework allows for the predicted x0 (and therefore ∇xt log p(y|x0)) to be efficiently computed for all t = 0, . . . , T. probability-flow (PF) ODE given by dxt = dxt + [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Inverse problem solution as a function of total diffusion timesteps T for the 4× super-resolution task. Compared to DPS (top row), our method (bottom row) produces solutions that are higher quality, in greater agreement with the inverse problem contraint Ax = y, and more stable across T. 2.3 Optimal Control Optimal control is the structured and principled approach to the guidance of dynamical systems over time. Many… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Examples from inverse problem tasks on FFHQ 256 × 256. From left to right each column contains ground truth, measurement, Diffusion Posterior Sampling (DPS), and ours. In Appendix D.1 we propose and analyze three modifications to the standard iLQR algorithm: randomized…
Figure 6
Figure 6. Figure 6: Robustness to approximation quality of the score function. We consider the 4× super-resolution task with a randomly initialized diffusion model. Since the reverse diffusion process is no longer well approxi￾mated, DPS cannot produce a feasible solu￾tion, while our meth…

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Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper

    Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We demonstrate our results through rigorous analysis of our algorithm and extensive experiments on multiple inverse problem settings over several datasets. Guidelines: • The answer NA means that the...

    work page

  2. [2]

    Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: Yes, the paper discusses the runtime cost of the work, and provides an equivalent budget analysis, where it still demonstrates competitive performance on each benchmark

    work page

  3. [3]

    Theory Assumptions and Proofs Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [Yes] Justification: The paper provides full proofs for all theory in the appendix

    work page

  4. [4]

    Experimental Result Reproducibility Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] Justification: The paper discloses all hy...

    work page

  5. [5]

    The authors will release code upon acceptance

    Open access to data and code Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [Yes] Justification: The paper provides open access to the data, which is publicly available. The authors will release code upon...

    work page

  6. [6]

    Experimental Setting/Details Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? Answer: [Yes] Justification: The paper provides all details in the appendix

    work page

  7. [7]

    Experiment Statistical Significance 13 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [NA] Justification: Experiments for other works do not provide error bars, therefore error bars would not benefit the analysis in this paper

    work page

  8. [8]

    Experiments Compute Resources Question: For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justification: Experiments can be run on any GPU A4000 or later

    work page

Show all 28 references
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    Code Of Ethics Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? Answer: [Yes] Justification: We confirm to the NeurIPS Code of Ethics in every respect

  2. [10]

    Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: The paper discusses this in the appendix

  3. [11]

    Safeguards Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: The results in ...

  4. [12]

    Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: We credit all cr...

  5. [13]

    New Assets Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] Justification: No new assets are introduced

  6. [14]

    Crowdsourcing and Research with Human Subjects Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? A...

  7. [15]

    Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects 14 Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approv...

    1968

  8. [16]

    We start with an initial guess of the the nominal control trajectory {¯ut}1 t=T and generate the corresponding nominal state trajectory {¯xt}0 t=T using xt = h(xt+1, ut+1)

  9. [17]

    (35) that V (x0, 0) = ℓ(x0) we can obtain expressions for Vx and Vxx evaluated at ¯x0

    By noticing from Eq. (35) that V (x0, 0) = ℓ(x0) we can obtain expressions for Vx and Vxx evaluated at ¯x0

  10. [18]

    (39)-(44) using {¯ut}1 t=T and {¯xt}1 t=T

    Next, we compute the derivatives of Q given by equations. (39)-(44) using {¯ut}1 t=T and {¯xt}1 t=T

  11. [19]

    Using the derivatives of Q, we can compute the feedforward and feedback gains using equations (46)-(47)

  12. [20]

    Finally, using the Riccati equations (48)-(49), we can propagate both Vx and Vxx one step backwards in time

  13. [21]

    We then repeat the steps 3, 4 and 5 until we backpropagate the derivatives of V to time step t = T

  14. [22]

    This completes one iteration of iLQR. At the end of each iteration the gains are used to produce the updated nominal control trajectory as follows: ¯u∗ t = ¯ut + αk + K( ¯xt − xt) (50) where, xt is the state obtained by unrolling the dynamics subject to the updated controls: x...

  15. [23]

    17 C Proofs Theorem 4.1

    The new nominal control trajectory ¯u∗ t is used to produce a new nominal state trajectory ¯x∗ t and the algorithm is repeated from step 2 onwards until convergence or a fixed number of iterations. 17 C Proofs Theorem 4.1. Let Eq. 3 be the discretized sampling equation for the...

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    2) We obtain AΩ = Y ∈ Rm×k

    We sample a random matrix Ω ∼ N(0, In×k). 2) We obtain AΩ = Y ∈ Rm×k. 3) We form a basis over the columns of Y, e.g. by taking the Q matrix in a QR factorization QR = Y. 4) We approximate A ≈ QT QA. Notably, we observe that when A is a Jacobian (or Hessian) matrix, it can be a...

    2011

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    the rank only affects the quadratic approximation of the iLQR algorithm (and does not affect our theoretical results in Theorems 4.1-4.3)

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    k = 1 already allows second-order propagation of the quadratic trust-region regularization, and

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    Tikhonov Regularizer Table 5 demonstrates that our algorithm is relatively robust to the Tikhonov regularization parameter, except when α = 0

    subsequent increases in k have a minimal effect on the performance of the algorithm. Tikhonov Regularizer Table 5 demonstrates that our algorithm is relatively robust to the Tikhonov regularization parameter, except when α = 0. Under this condition, any ill-conditioning of Quu...

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    [2014], resulting in the action update ut = Pkt + Kt(xt − x′ t), (80) where P is the preconditioning matrix produced by the Adam optimizer

    before applying the feedback gains, rather than applying a backtracking line search Tassa et al. [2014], resulting in the action update ut = Pkt + Kt(xt − x′ t), (80) where P is the preconditioning matrix produced by the Adam optimizer. This reduces the overall run- time of th...

    2014

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