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REVIEW 4 major objections 5 minor 2 cited by

CoDe: Blockwise Control for Denoising Diffusion Models

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read CoDe steers a frozen diffusion model toward a downstream reward at inference time by applying best-of-N selection blockwise during denoising, with no gradients or fine-tuning.

desk verdict CoDe is a simple, honest blockwise best-of-N for diffusion with real empirical support; the optimality claim rests on cited work and the Tweedie value approximation at early steps is the main unquantified risk, with a secondary runtime-table contradiction. read the letter →

arxiv 2502.00968 v2 pith:QZ54L2VW submitted 2025-02-03 cs.CV cs.LG

Anuj Singh , Sayak Mukherjee , Ahmad Beirami , Hadi Jamali-Rad This is my paper · ORCID
classification cs.CVcs.LG
keywords diffusionmodelsinference-timeguidancebest-of-NsamplingrewardalignmentKL-regularizedobjectiveblockwisecontrolgradient-freeTweedie'sformula
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 proposes controlled denoising (CoDe), a way to point a pretrained, frozen diffusion model at a downstream reward during generation, without fine-tuning and without needing the reward to be differentiable. The recipe is blockwise best-of-N: every B denoising steps, the sampler draws N short candidate continuations from the base model, scores each partially denoised state with a cheap value estimate (the reward of the Tweedie-predicted clean image), and keeps the best one before resuming. The authors show that this selection rule mirrors the optimal solution of a KL-regularized reward objective, which trades alignment against staying close to the base distribution. In a Gaussian mixture study they report the same win-rate-versus-divergence trade-off as full best-of-N while using roughly 10-15 times fewer samples, and in image experiments they report competitive reward alignment with better prompt fidelity and lower divergence than gradient-based guidance. If correct, CoDe makes inference-time alignment practical for non-differentiable rewards and for users who cannot backpropagate through a reward model.

What carries the argument

The load-bearing object is the blockwise value-guided selection loop built on Theorem 2.1's tilted posterior. The value function $V(x_t;p,c)=E_{x_0\sim p(x_0|x_t,c)}[r(x_0)]$ is approximated by $r(\hat{x}_0)$, with $\hat{x}_0=(x_t-\sqrt{1-\bar\alpha_t}\,\varepsilon_\theta(x_t,c,t))/\sqrt{\bar\alpha_t}$ from Tweedie's formula. Every $B$ steps, CoDe draws $N$ independent $B$-step continuations, scores them with this value estimate, and keeps the argmax; this repeated selection is the Monte Carlo stand-in for sampling from $\pi^*_\lambda$, with the block size $B$ controlling how often selection pressure is applied and therefore how far the output drifts from the base distribution.

What would settle it

Run the Gaussian mixture case study with the exact value $V(x_t)=E[r(x_0)|x_t]$ computed from the known mixture, and compare CoDe's selected branches, win rate, and KL divergence against the Tweedie-approximated version; if exact-value CoDe does not match or beat the claimed best-of-N frontier, the blockwise selection is not doing the work claimed. A sharper version uses a nonlinear reward such as $r(x)=\|x-\mu\|^2$, where $r(E[x_0|x_t])$ and $E[r(x_0)|x_t]$ visibly diverge at large $t$, and checks whether CoDe's early block choices track the exact-value optimum.

Watch

Extended reading notes

Core claim

CoDe's central claim is that the optimal solution of the KL-regularized reward objective, $\pi^*_\lambda(x_{t-1}|x_t) \propto p(x_{t-1}|x_t)\,e^{\lambda V(x_{t-1})}$ (Theorem 2.1), can be approximated at inference time by a blockwise best-of-N procedure on a frozen diffusion model. Every $B$ denoising steps the sampler unrolls $N$ candidate continuations from the base model, scores each with the value estimate $V(x_t)\approx r(\hat{x}_0)$, where $\hat{x}_0$ is the Tweedie clean-image prediction, and keeps the highest-value candidate before continuing. Full best-of-N is the $B=T$ case and per-step selection is the $B=1$ case, so CoDe positions itself as an interpolation that stays close to the optimal reward-versus-divergence frontier: the Gaussian mixture study shows on-par win-rate-versus-KL performance with best-of-N using roughly 10-15 times fewer samples, and the image experiments report competitive reward alignment with better prompt fidelity and lower divergence than gradient-based guidance baselines.

Load-bearing premise

The method's value estimate treats the reward of the average predicted clean image as the average reward; early in denoising that equality can fail badly, and wrong branch choices would defeat the alignment even though each candidate is a valid base-model sample.

Editorial extensions

If this is right

  • A frozen diffusion model can be aligned to differentiable or non-differentiable rewards without fine-tuning and without gradient access, using only base-model samples and reward queries.
  • CoDe interpolates between best-of-N and per-step selection, so $N$, $B$, and $\eta$ give the user a tunable trade-off among reward, divergence from the base model, and compute.
  • In the Gaussian mixture study, CoDe reaches the same win-rate-versus-KL frontier as best-of-N with $N\in[2,10]$ rather than $N\in[30,500]$, a roughly 10-15x sample-efficiency gain.
  • The KL divergence from the base model is bounded by $(\log N - (N-1)/N)\,\eta T / B$, so divergence grows logarithmically in $N$, shrinks with larger blocks, and is further controlled by the noise-conditioning ratio $\eta$.
  • On image tasks, CoDe and its noise-conditioned variant CoDe($\eta$) report competitive reward while keeping better text alignment and lower divergence than gradient-based guidance methods, at lower runtime than per-step and universal guidance.

Reading between the lines

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

  • The efficiency gain over best-of-N plausibly comes from applying selection pressure before the final image exists, so one surviving trajectory amortizes the cost of many rejected continuations; the paper demonstrates the gain but does not decompose it.
  • If the Tweedie value approximation is the active weakness, a learned or ensemble value function accurate at high noise levels would likely push CoDe's frontier further; this is a natural next step the paper does not take.
  • The adaptive-control ablation suggests the optimal block schedule is task-dependent: compression benefits from late-block control, while tasks that fix global layout early would likely benefit from early-block control.
  • Since CoDe only needs samples and a scalar reward, it transfers to any generative diffusion setting, such as molecules or audio, where the reward is non-differentiable or black-box.
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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

4 major / 5 minor

Summary. The paper proposes CoDe (Controlled Denoising), an inference-time, gradient-free guidance method for frozen diffusion models. The method performs blockwise best-of-N sampling: every B denoising steps it draws N candidate continuations from the base model, scores them with a value function approximated by r(E[x0|xt]) via Tweedie's formula, and keeps only the highest-scoring candidate. The authors claim that this procedure approximately samples from the optimal KL-regularized reward-tilted posterior of Theorem 2.1, and they evaluate it on a 2D Gaussian mixture model and on Stable Diffusion across style, face, stroke, compression, and aesthetic guidance tasks, comparing against DPS, UG, MPGD, Freedom, BoN, and SVDD-PM. The headline empirical claims are that CoDe matches the reward-versus-divergence trade-off of BoN with roughly 10-15x fewer samples in the GMM study, and that it offers competitive image/text alignment with lower divergence than several baselines.

Significance. If the central claims hold, CoDe is a practically attractive contribution: it requires no gradient access, no fine-tuning, and no trained value network, and it naturally handles non-differentiable rewards. The paper also ships code, which is a concrete strength. However, the most load-bearing claims are not fully supported. The approximation in Eq. (12) replaces E[r(x0)|xt] with r(E[x0|xt]), which is uncontrolled at early denoising steps, and the paper does not prove that blockwise argmax selection (beam width 1) actually samples from the optimal KL-regularized objective. A separate internal inconsistency in Section 6.4, where CoDe is called faster than BoN while Table 5 shows the opposite, weakens the efficiency narrative. The empirical work is extensive and suggests the method works reasonably in practice, but the theoretical positioning needs substantial revision.

major comments (4)
  1. [Section 3, Eq. (12)] The value function approximation V(xt) ≈ r(E[x0|xt]) is not justified with any error bound. V(xt) is defined in Eq. (6) as E_{x0∼p(x0|xt)}[r(x0)], and replacing the expectation inside r is exact only for linear rewards or point-mass posteriors. At early denoising steps p(x0|xt) is broad, so the surrogate can select branches that do not maximize true expected reward. Because CoDe keeps a single survivor per block (beam width 1), a wrong early selection cannot be repaired later. This directly affects Contribution I, which claims CoDe samples from the optimal KL-regularized objective of Theorem 2.1, since that theorem requires the true value function. The paper's own caveat that the approximation is 'expected to be more accurate towards the end of denoising' is not a substitute for a bound or correction. Please either provide an analysis of the approximation gap, or revise the optimality claim to clearly state that CoDe optimizes a surrogate objective.
  2. [Appendix A, Eq. (18)] The proof of Theorem 2.1 contains a mathematical error in the definition of the normalizing constant. Eq. (18) defines Zλ(xt,c) = E_{xt−1∼p}[p(xt−1|xt,c)e^{λV(xt−1;p,c)}], but the density p(xt−1|xt,c) should not appear inside the expectation; the correct normalizer is Zλ(xt,c) = E_{xt−1∼p(·|xt,c)}[e^{λV(xt−1;p,c)}]. As written, the derivation of Eq. (19)-(23) is internally inconsistent, even though the theorem statement itself is a standard variational result. This needs to be corrected.
  3. [Section 6.4, Table 5] The text in Section 6.4 states that 'CoDe, CoDe(η) with its blockwise guidance remains to be faster and more efficient than BoN', but Table 5 reports CoDe at 489.00 s/img versus BoN at 266.77 s/img, and Table 9 shows CoDe slower than BoN in all three scenarios (e.g., style: 441.81 vs 266.02; face: 583.12 vs 268.43; stroke: 442.08 vs 265.86). Moreover, CoDe has NT/B reward queries versus BoN's N, so it is not more query-efficient either. The efficiency claim is therefore contradicted by the paper's own tables and must be corrected. If the intent is to claim efficiency per unit of achieved reward at a lower N, that should be stated explicitly and supported with the trade-off curves in Fig. 10.
  4. [Section 3, 'Why blockwise BoN is almost optimal'] The paper asserts that blockwise argmax selection with beam width 1 approximates the optimal reward-tilted posterior of Theorem 2.1, but no proof is given for this blockwise algorithm with a single survivor. The cited results (Beirami et al. 2024; Yang et al. 2024; Mudgal et al. 2024) concern best-of-N sampling and controlled decoding in particular settings; they do not directly imply that CoDe's resampling every B steps with one survivor samples from π*_λ in Eq. (10). The connection in Appendix C (argmax as a zero-temperature categorical) is a selection rule, not a statement about the final sample distribution. Please either provide a theoretical argument specific to CoDe, or explicitly reclassify it as a heuristic approximation to the optimal sampler.
minor comments (5)
  1. [Section 5] The claim that CoDe is 'roughly 10-15× more efficient than BoN' should be qualified as 'requires 10-15× fewer samples N' rather than 'more efficient' without qualification, since CoDe incurs additional per-block reward queries and inference steps as shown in Table 5.
  2. [Figure 5 caption / Appendix F] Figure 5's caption says 'Reward vs. KL-divergence', but for image experiments Appendix F only provides an upper bound on KL (Lemma F.1) and does not describe an empirical estimate for images. Please state clearly whether the plotted quantity is the theoretical upper bound or an empirical estimate.
  3. [Section 6.4] The text in Section 6.4 says CoDe offers a '4× faster runtime than UG', but Table 5 gives 489.00 vs 1241.47 s/img, which is approximately 2.5×. Please check this number.
  4. [Throughout] There are several typos, including 'comptetitive' in Section 4, 'scneario' near Table 1, and 'SVPP-PM' in Appendix E. A light proofreading pass is needed.
  5. [Section 2.2, Eq. (10)] In Theorem 2.1 and the surrounding text, the theorem is stated for the unconditional setting, while the algorithm conditions on c. The conditional extension is deferred to Appendix A, which is fine, but the main text should flag this distinction earlier to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2.1 is proved directly and the empirical comparisons are independent; the value approximation and runtime-table issues are correctness risks, not tautologies.

full rationale

The derivation chain is self-contained at its core. Theorem 2.1 is proved in Appendix A by standard variational calculus: the KL-regularized objective is rewritten as -KL(pi || p_lambda) plus a constant, so the optimum is p_lambda. Algorithm 1 is an explicit blockwise selector, not a quantity derived from the objective; its reward/divergence behavior is measured empirically against BoN, SVDD-PM, DPS, and UG in the GMM and image experiments. The near-optimality of blockwise BoN is imported from Beirami/Yang/Mudgal; although these works share an author with the present paper, they are external theorems about BoN asymptotics that do not incorporate this paper's fitted values, so they are independent support under the stated rules. The Tweedie surrogate V(x_t) approximately r(E[x_0|x_t]) (Eq. 12) is a stated approximation rather than a tautology: it can diverge from E[r(x_0)] at early steps, which is a correctness risk the paper concedes ('expected to be more accurate towards the end of denoising'), but it is not an input-output equivalence. The runtime claim in Section 6.4 contradicts Table 5 (CoDe at 489.00 s/img versus BoN at 266.77 s/img) and is a consistency/correctness issue, not circularity. Therefore no circular step is exhibited.

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

The paper introduces no new physical or architectural entities. The central claim rests on four tuned hyperparameters (N, B, eta, and the nominal lambda), the Tweedie value approximation, and imported near-optimality results for blockwise best-of-N from prior work, including work co-authored by one of the present authors. The empirical comparisons give the main independent grounding, while the theoretical framing is largely borrowed.

free parameters (4)
  • lambda (KL-regularization coefficient)
    Introduced in Eq. (9) as the reward-divergence trade-off weight, but Algorithm 1 sweeps N, B, and eta instead of setting lambda. The connection between lambda and the implemented sampling procedure is never made explicit.
  • N (candidate count per block) = swept from 2 to 500; best settings typically 40 to 100
    N controls the number of trajectories sampled at each block intervention. The paper sweeps N to produce trade-off curves, so the reported performance depends on a tuned hyperparameter.
  • B (block size) = swept from 1 to 320; best settings typically 5 to 80
    B controls how often blockwise best-of-N interventions occur. The trade-off results are parameterized by B, so the central empirical claims depend on this tuned value.
  • eta (noise-conditioning ratio) = best values: 0.6 for style/stroke, 0.7 for face, 0.8 for compression
    eta controls how much of the reference image is injected as initial noise in the CoDe(eta) variant. It is an extra design knob selected per scenario and materially affects the reported I-Gram and T-CLIP results.
assumptions (5)
  • domain assumption Tweedie's formula gives an accurate estimate of E[r(x0)] via r(E[x0|xt]) (Eq. 12).
    Used in Algorithm 1 line 6 and Algorithm 2 line 8. The approximation is unbiased only for a perfectly trained model and approximately linear rewards, and it is known to degrade at early denoising steps.
  • domain assumption Blockwise best-of-N sampling is almost optimal for the KL-regularized reward objective (Eq. 9).
    The paper's Section 3 'Why blockwise BoN is almost optimal' cites Mudgal et al., Beirami et al., and Yang et al. rather than proving the blockwise diffusion case. The claim is load-bearing for the 'optimal KL-regularized objective' framing.
  • domain assumption The KL divergence bound in Lemma F.1, KL(CoDe) <= (log N minus (N-1)/N) times T/B, holds for the implemented algorithm.
    The bound is imported from Beirami et al. and Mroueh's theorems. The paper does not verify the bound's tightness in the diffusion setting and notes in Appendix F that the bounds are loose near degeneracy.
  • domain assumption Reward functions used in evaluation (CLIP cosine similarity, JPEG size, FaceNet negative L1, aesthetic predictor) are valid proxies for the stated alignment goals.
    The paper explicitly uses I-Gram as an unseen alignment metric because the guidance reward is 'seen' by the model, acknowledging that reward metrics do not fully capture alignment.
  • domain assumption In Case Study I, KL divergence between guided and base distributions can be estimated by fitting a Gaussian to 1000 samples.
    Appendix F.1 describes this procedure even though the base prior is a multimodal Gaussian mixture; the Gaussian fit is an approximation and may understate multimodality in guided samples.

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Cite this review

Pith. "Pith review of CoDe: Blockwise Control for Denoising Diffusion Models." pith.science (2026). https://pith.science/paper/QZ54L2VW

@misc{pith2026250200968,
  author       = {Pith},
  title        = {Pith review of: CoDe: Blockwise Control for Denoising Diffusion Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZ54L2VW}},
  note         = {Machine review of arXiv:2502.00968}
}
read the original abstract

Aligning diffusion models to downstream tasks often requires finetuning new models or gradient-based guidance at inference time to enable sampling from the reward-tilted posterior. In this work, we explore a simple inference-time gradient-free guidance approach, called controlled denoising (CoDe), that circumvents the need for differentiable guidance functions and model finetuning. CoDe is a blockwise sampling method applied during intermediate denoising steps, allowing for alignment with downstream rewards. Our experiments demonstrate that, despite its simplicity, CoDe offers a favorable trade-off between reward alignment, prompt instruction following, and inference cost, achieving a competitive performance against the state-of-the-art baselines. Our code is available at: https://github.com/anujinho/code.

Figures

Figures reproduced from arXiv: 2502.00968 by the authors.

Figure 1
Figure 1. CoDe generates high quality compression (non￾differentiable reward), style, face and stroke (differentiable re￾wards) guided images. Diffusion models have emerged as a pow￾erful tool for generating high-fidelity real￾istic images, videos, natural language con￾tent and even molecular data (Ho et al., 2020; Song et al., 2020; Bar-Tal et al., 2024; Wu et al., 2022). While diffusion models have proved to be effective at… view at source ↗
Figure 2
Figure 2. Setup (left, middle) and reward vs. divergence trade-off (right) for Case Study I. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. In contrast to BoN, SVDD-PM, CoDe with and without noise-conditioning (η = 0.6, η = 1, resp.) are robust against increased distance between reward and prior distributions. SVDD-PM’s generated samples offer almost zero variance indicating reward over-optimization. In contrast, UG and DPS tend to exhibit higher KL divergence, as they often collapse to the mode of the reward distribution when the guidance scale is incr… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: CoDe(η) demonstrates a superior trade-off between compressibility, image and text alignment as compared to other baselines on the (T+I)2I settings. 6.1 Non-Differentiable Reward: Compression [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: CoDe(η) offers a better reward vs. KL￾divergence trade-off as compared to BoN(η) for all N values. SVDD-PM(η) demonstrates a higher reward beyond N = 7, but at the cost of a much higher KL￾divergence. First, we consider a scenario with non-differentiable reward, where …
Figure 6
Figure 6. Figure 6: The style alignment offered by CoDe(η) stands on par or outperforms other baselines in terms of quality and preserving nuances of the reference image, while adhering to the text-prompt [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Same narrative as in Fig [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Qualitative evaluation across methods for aesthetic guidance. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Ablation on the block size (B) and the noise ratio (η). the right, changing the noise ratio η toward higher values, reduces the conditioning strength (as indi￾cated also in (Meng et al., 2021; Koohpayegani et al., 2023)) resulting in lower image alignment capac￾ity (I-…
Figure 10
Figure 10. Figure 10: Reward vs. Compute and FID vs Compute trade-off curves for Style guidance. [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Setup (top row) and reward vs. divergence [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Quality evaluation across methods for style guidance without noise-conditioning. [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: Reward vs. divergence trade-off curves for style guidance. [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: Reward vs. divergence trade-off curves for face guidance. [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]
Figure 15
Figure 15. Figure 15: Quality evaluation across methods for face guidance. [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: Quality evaluation across methods for face guidance without noise-conditioning. [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: Quality evaluation across methods for stroke guidance without noise-conditioning. [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: Reward vs. divergence trade-off curves for aesthetic guidance. [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]
Figure 19
Figure 19. Figure 19: Qualitative examples of reward over-optimized images from SVDD-PM( [PITH_FULL_IMAGE:figures/full_fig_p035_19.png]
Figure 20
Figure 20. Figure 20: Qualitative examples of low text-alignment (T-CLIP) for UG with higher guidance scales, in the [PITH_FULL_IMAGE:figures/full_fig_p036_20.png]
Figure 21
Figure 21. Figure 21: Qualitative demonstration of Reward vs I-GRAM for different style guidance generated images. [PITH_FULL_IMAGE:figures/full_fig_p037_21.png]
Figure 22
Figure 22. Figure 22: Quality evaluation across methods for style guidance on additional settings without noise [PITH_FULL_IMAGE:figures/full_fig_p038_22.png]
Figure 23
Figure 23. Figure 23: Quality evaluation of CoDe for style guidance on additional settings. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_23.png]
Figure 24
Figure 24. Figure 24: Quality evaluation of CoDe for face guidance on additional settings. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_24.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.