We are grateful for your constructive feedback and your recognition of our efforts. In particular, we appreciate your insightful comments on intelligently selecting the go-back temperature, the potential for continued improvement through small refinements, the intuitive BoN-based illustration of ABCD behavior, the comparison with SoP, and your careful attention to experimental settings. Below, we provide responses to all of the questions and concerns you raised:

[Question 1 & Weakness 3] About the simpler termination method & premature termination issue

Thank you for the insightful question. As you pointed out, there are indeed situations where local exploration continues to improve. To account for this, our ABCD framework always includes the lowest go-back temperature (i.e., 0) across all tasks. In effect, the algorithm terminates only when the best sample is the one that does not require further refinement for some consecutive cycles. Additionally, the issue of early termination can be mitigated by using a stricter $\kappa$ condition as well.

[Question 2] I can imagine that when all initial samples receive very low rewards, adding small noises to them and denoising them again can be a waste of compute. It might be better to just start with fresh samples.

This is another very intuitive question. One of the strengths of ABCD is that it can handle all these cases. This is because it sends the particles across all temperature levels. In the scenario you mentioned, ABCD would likely choose the particle sent to the highest temperature $T$, allowing ABCD to effectively perform a full restart when necessary.

[Question 3] What is the benefit of ABCD over SoP?

We believe that ABCD offers several key advantages over SoP.

1. First of all, ABCD enables unbounded cycling between denoising and re-noising, whereas SoP does not allow further refinement once it reaches $t = 0$. This limits SoP’s ability to scale inference-time computation. In contrast, ABCD introduces an additional axis of scalability by allowing repeated refinement cycles even after reaching $t = 0$, enabling it to flexibly allocate more computation based on task complexity.
2. ABCD adapts the generative process on a per-instance basis. ABCD allows on-line adaptive inference-time compute whereas SoP has constant & freezed inference computation regardless of the difficulty or other requirements such as desired quality-speed tradeoff.
3. As you correctly mentioned, unlike SoP, ABCD does not rely on approximating the reward using $r(\mathbb{E}[x_0 | x_t])$. Instead, it evaluates the actual reward at the generated $x_0$ via fast denoising, thereby avoiding the approximation error inherent in SoP’s approach.

These properties collectively make ABCD a more flexible and potentially more powerful framework for guided generation.

[Weakness 1] As pointed out by another reviewer, large-scale experiments are not strictly necessary, as evaluating the proposed method in a controlled setting—as done in the current work—is sufficient. Therefore, small-scale experimentation is not considered a Weakness.

Thank you for advocating the value of our relatively small-scale experiments. At the same time, several reviewers raised concerns about the scalability of our approach to larger domains such as images. In response, we conducted additional experiments in the text-to-image domain. The results were consistent with our other experiments. The results of which are discussed in [Question 1] of reviewer WZvc.

[Weakness 2] selecting $t'$ more intelligently can be possible

Thank you for pointing out this highly relevant issue—we completely share the motivation behind it. During the development of our method, we explored a more intelligent alternative go-back strategy but found it doesn’t show meaningful improvement.

For instance, we also experimented with a more informed go-back mechanism, where particles were redistributed to different noise levels in proportion to the average rewards observed at those levels across the population. As shown in the results below, however, this approach introduced additional complexity without yielding meaningful performance improvements. Consequently, we chose to adopt a simpler version in the final design of ABCD.

Nonetheless, as you insightfully suggested, incorporating more sophisticated techniques—such as leveraging concepts from SMC within the temperature pool—could further enhance the method. We consider this a compelling avenue for future exploration.

• Adaptive ABCD vs. ABCD in Sudoku |Method|Add. Compute|Accuracy|Clock Time (s)| |-|-|-|-| |Adaptive ABCD|perc=0.6,𝜅=1|0.810|0.220| ||perc=0.8,𝜅=1|0.895|0.395| ||perc=1.0,𝜅=3|0.979|0.549| |ABCD|perc=0.6,𝜅=1|0.821|0.240| ||perc=0.8,𝜅=1|0.903|0.350| ||perc=1.0,𝜅=3|0.979|0.557|

[Question 4 & Weakness 5] Does the same number of particles imply the same number of forward passes of the score function? If not, it would be better to report the latter as well, so that readers can clearly compare the computational efficiency in terms of FLOPs. & [Weakness 4] In the experimental results, the performance gap between some existing baselines appears moderate. For instance, in Sudoku, SoP performs on par with the proposed method.

Indeed, the same number of particles does not necessarily imply an equal number of score function evaluations. To clarify this, we explicitly report the Number of Function Evaluations (NFE) per 1 sample, which measures the total number of score function calls during inference. We found that evaluating in NFE provides a more effective way to highlight the advantage of our method over baselines. Specifically, in tasks like Sudoku, the performance and speed gap between ABCD and existing methods becomes more pronounced, effectively addressing your concern in [Weakness 4]. Although we can not include the updated figure here, we provide the updated summary table below and will incorporate the full results in the final camera-ready version to ensure a clear and fair comparison.

• Sudoku |Method|Config|Accuracy|NFEs| |-|-|-|-| |Base Diffusion|No add|0.35|50| |Best-of-N (BoN)|N=480|0.639|24000| ||N=960|0.662|48000| |SMC|p=5|0.643|2000| ||p=1|0.745|2000| |Beam Search|p=2|0.757|1500| ||p=1|0.844|2460| |Search-over-Path (SoP)|Δf=10, Δb=5|0.763|3860| ||Δf=4, Δb=2|0.886|4220| ||Δf=6, Δb=5|0.957|11540| |ABCD (Ours)|perc=0.6, 𝜅=1|0.821|1059| ||perc=1.0, 𝜅=5|0.986|3205| ||perc=1.0, 𝜅=20|0.996|5013|
• Pixel Maze |Method|Config|Success Rate|NFEs| |-|-|-|-| |Base Diffusion|No add|0.02|50| |Best-of-N (BoN)|N=40|0.20|2000| ||N=160|0.34|8000| ||N=200|0.38|10000| |SMC|p=5|0.32|2000| ||p=1|0.36|2000| |Beam Search|p=2|0.39|1500| ||p=1|0.61|2460| |Search-over-Path (SoP)|Δf=10, Δb=5|0.76|3860| ||Δf=8, Δb=4|0.85|3900| ||Δf=4, Δb=2|0.95|4220| |ABCD (Ours)|perc=0.2, 𝜅=1|0.44|562| ||perc=0.3, 𝜅=1|0.96|1226| ||perc=1.0, 𝜅=3|1.00|1854|
• OGBench Maze Giant |Method|Config|Performance|NFEs| |-|-|-|-| |Base Diffusion|No add|12±13|1280| |Best-of-N (BoN)|N=64|54±22|81920| ||N=288|64±17|368640| ||N=544|76±20|696320| |SMC|p=20|20±13|41376| ||p=10|22±11|41792| |Beam Search|p=5|50±16|132960| ||p=1|72±16|618560| |Search-over-Path (SoP)|Δf=100,Δb=50|60±15|59664| ||Δf=50,Δb=25|86±13|71784| ||Δf=6,Δb=3|96±12|98360| |ABCD (Ours)|max_iter=5|94±9|19020| ||max_iter=10|96±8|29720| ||max_iter=35|100±0|83220|
• OGBench Maze Large |Method|Con|Performance|NFEs| |-|-|-|-| |Base Diffusion|No add|38±11|1280| |Best-of-N (BoN)|N=4|70±16|5120| ||N=8|80±9|10240| ||N=16|88±10|20480| |SMC|p=20|82±11|41376| ||p=10|76±15|41792| |Beam Search|p=15|84±12|52720| ||p=5|88±10|132960| |Search-over-Path (SoP)|Δf=100,Δb=50|76±12|58164| ||Δf=50,Δb=25|86±9|71034| ||Δf=6,Δb=3|96±8|98270| |ABCD (Ours)|J=32|100±0|6360|
• Image Generation

Please refer to our response to [Question 1] from Reviewer WZvc.

[Question 5] report the number of forward passes instead of “Mean Cyc” in Table 2?

The purpose of Table 2 is to highlight the necessity of employing multiple go-back step sizes. However, we agree that including the number of forward passes would provide a clearer picture of computational costs. We add the NFEs to Table 2 and observe the same trend, further reinforce our conclusion. We will update the table accordingly to reflect this clearly.

[Question 6] In Table 1, what does “success” mean? The distance to the goal is not 0. Do you use a threshold value and consider a particle successful when the distance is below that threshold? This part is not clearly explained.

This MoG task was framed as a mode search problem, meaning that success corresponds to reaching the mode that contains the goal point. In the implementation, we compute the closest mode for each particle and considering it successful if that mode matches the goal mode.

Thanks for the detailed response. However, my two concerns still remain.

This is another very intuitive question. One of the strengths of ABCD is that it can handle all these cases. This is because it sends the particles across all temperature levels. In the scenario you mentioned, ABCD would likely choose the particle sent to the highest temperature $T$, allowing ABCD to effectively perform a full restart when necessary.

But doing so requires J times more compute, no? In your experiment, in Maze Large, this would mean ABCD is 32x more expensive than BoN. In real-world cases where we can't indefinitely increase the parallel compute, it could mean ABCD is at most 32x slower.

Overall, the core idea of this paper is that we can mitigate the exploration-exploitation dilemma by just doing everything simultaneously. This is only possible if we can afford the additional computational cost to do so. This is the most important limitation and should be clarified explicitly.

To account for this, our ABCD framework always includes the lowest go-back temperature (i.e., 0) across all tasks.

With that choice, your adaptive rule is equivalent to the naive algorithm I mentioned above, which simply "checks whether the reward has stagnated." Terminating a search algorithm when a criterion stops improving is a general idea and not particularly new. The current explanation seems to be an overly complicated way to describe this simple algorithm. As this is presented as a core contribution of the paper, upon the author's clarification, I'm lowering my score, but I'm still inclined toward acceptance and happy to be corrected.

Thank you for your constructive feedback! We find it helpful in clarifying our paper. Please feel free to let us know if you have more questions after reading our answer in the following.

But doing so requires J times more compute, no? In your experiment, in Maze Large, this would mean ABCD is 32x more expensive than BoN.

No, having large J (e.g., 32) in ABCD doesn’t mean it requires 32 times more compute than BoN. It’s important to note that ABCD’s computation depends both J and K. In our Large Maze experiment, to consider the fact that J is large, e.g., 32, we set K=1 when J=32. Thus, only a maximum of N=KxJ=32 particles are used in ABCD’s noising and denoising cycles. Therefore, a fair comparison to BoN in terms of particle count is to set N=32 for BoN.

Given this, the most accurate way to compare computational efficiency is through NFE and wall-clock time rather than particle count. For both, our experiment results clearly demonstrate that ABCD delivers substantially better performance per unit of computation. Specifically, as shown in the table above, ABCD with N=32 requires only 6360 NFEs to achieve a 100% success rate. In contrast, BoN with N=32 uses 6.4 times more NFEs (40960) yet achieves only a 94% success rate. Similarly, BoN with N=4 requires comparable NFEs (5120) to ABCD (N=32) but achieves only a 70% success rate. This same trend appears in the wall-clock time experiment which, unlike NFE, can also consider the efficiency by parallelism. ABCD with N=32 (K=1,J=32) requires only 10.5 seconds to achieve a 100% success rate, while BoN (N=32) achieves just 94% success despite taking 4 times more time (41.7 seconds).

Your adaptive rule is equivalent to the naive algorithm which simply "checks whether the reward has stagnated." Terminating a search algorithm when a criterion stops improving is a general idea and not particularly new.

We believe that the presented Adaptive Thinking Time section presents a nontrivial contribution for the following reasons.

First, as we introduce the new concept of cyclic iterative refinement, we identify that determining when to stop the inference process emerges as a new important research challenge in diffusion-based inference-time scaling. This identification not only clarifies a new axis of difficulty in this domain but also lays the groundwork for future research to develop improved solutions. Note that, to our best knowledge, our paper is the first in this line of research to address this aspect.

Second, from a technical perspective, we emphasize that we are in an interesting and distinctive setting where we can find a stopping criteria w.r.t. a distribution of particles. We introduce the new perspective that we can see from which temperature the particles came from as the uncertainty-aware criterion. This probabilistic, uncertainty-aware perspective offers a more general framework than conventional heuristics such as “stop if improvement stagnates” because this framework provides the opportunity to define different stoping criterion (e.g., measuring KL divergence between a target distribution and the current particle distribution.). In addition, in the rebuttal to Reviewer WZvc (Weakness 1), we also provide a new theoretic analysis result to show this process’s convergence property.

Then, as a practical implementation of this general principle, we propose the specific stopping rule counting temperature=0 particles. While this final implementation appears similar to a familiar heuristic, it should not obscure the broader conceptual contribution: (i) identifying a new challenge, (ii) proposing a general probabilistic perspective grounded in particle uncertainty, and (iii) demonstrating a concrete method that implements this principle. We argue that these complete intellectual trajectory constitutes a meaningful contribution.

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About Lowering the Score

We thank the reviewer for the constructive feedback, which greatly helped us identify key areas where our manuscript could be improved further. In addition to the clarifications above, we would like to highlight that this rebuttal also addresses two primary concerns raised in the review. Specifically, we have (i) added additional experimental results on high-dimensional image generation tasks, where our method consistently outperforms the baselines, and (ii) included evaluations in terms of NFE alongside wall-clock time, both of which consistently demonstrate the superiority of ABCD.

We would like to ask respectfully the reviewer to consider these improvements as well when reconsidering the score.

We sincerely appreciate your thoughtful feedback, including your comments on applying our method to more complex problems, handling poor initialization, and experimental details. Below, we provide responses to all of the questions and concerns you raised:

[Question 1 & Weakness 3] How does ABCD perform when DDIM initialization produces only poor samples? Are there any other possible failure modes?

Thank you for highlighting this important possible failure case. Rather than being a weakness, we believe that possible poor initialization is actually a setting in which ABCD demonstrates its strengths. Specifically, ABCD is particularly advantageous in scenarios with bad initializations (i.e., an initial low-quality generation), as it does not terminate after a single pass but instead enables adaptive, iterative refinement for each sample.

Additionally, the key hypothesis behind ABCD's design is that reaching the correct mode doesn't require high-quality fine-grained moves initially. Instead, it's more efficient to quickly approach the good mode (by DDIM) by trading off some quality. Once we're near the appropriate mode, we can then focus on taking fine-grained moves (low-temperature moves) to refine the result.

[Question 2 & Weakness 1] Have you tested ABCD on more complex problems (such as small scale image generation)?

We appreciate your point and, in response, have conducted additional experiments in more complex problems such as image generation. The results were consistent with our other experiments. Please refer to our response to [Question 1] from Reviewer WZvc above for more details.

[Question 3] What time-step schedulers and sampling algorithms (e.g. DDIM, DDPM) were used for each baseline method?

Within each task, we used the same pretrained model during inference and applied the same DDIM framework across all baseline methods. However, ABCD uniquely employed a jumpy denoising schedule, where the model skips several diffusion steps during each iteration to enable faster exploration. Specifically, we used 10-step jumpy denoising over $T=50$ for Sudoku, 5-step jumpy denoising over $T=50$ for Pixel Maze, 10-step jumpy denoising over $T=256$ for OGBench, and 10-step jumpy denoising over $T=1000$ for Molecular generation.

We used the jumpy denoising only in our method because that is the main inference-scaling mechanism implementing the main hypothesis of our method (described in the above answer to [Question 1 & Weakness 3]). For the baselines, this coarse jumpy denoising is actually harmful because, unlike ABCD, they do not allow additional chance to improve once it reaches at $x_0$.

[Weakness 2] A more detailed analysis of computational overhead—including how varying the budget allocated to each ABCD component affects final quality—would be helpful for understanding the trade-offs.

Thank you for raising this important point regarding computational trade-offs. To better understand how different computational budgets affect final performance, we added ablation study analyzing the two core components of ABCD—Adaptive Thinking Time and Automatic Exploration–Exploitation Balancing—under varying configurations. Specifically, we conducted experiments that vary the level of computation allocated to each mechanism to examine how efficiency and accuracy are impacted in both the Sudoku and text-to-image generation domains.

In the Sudoku task, we focused on the Adaptive Thinking Time mechanism by varying the termination percentage and the value of $\kappa$, two key hyperparameters that define ABCD’s adaptive terminal condition. As shown in the two tables, increasing $\kappa$ generally improves accuracy but also leads to longer inference time, and similarly, raising the termination percentage improves performance at the cost of additional computation. These results show that by adjusting the strictness of the termination criterion, we can effectively control the trade-off between accuracy and efficiency.

• ABCD – Sudoku (Changing 𝜅) |Metric|perc=1.0,𝜅=1|perc=1.0,𝜅=5|perc=1.0,𝜅=10| |------|-------------|-------------|--------------| |Performance|0.958|0.986|0.994| |Time (s)|0.443|0.608|0.754| |NFEs|2296|3205|3942|
• ABCD – Sudoku (Changing percentage) |Metric|perc=0.4,𝜅=1|perc=0.6,𝜅=1|perc=0.8,𝜅=1| |------|-------------|-------------|-------------| |Performance|0.698|0.821|0.903| |Time (s)|0.112|0.210|0.329| |NFEs|520|1059|1667|

In the text-to-image generation task, we analyzed the Automatic Exploration–Exploitation Balancing mechanism by varying the granularity of the temperature pool—specifically, the number of go-back temperatures included in the pool. We fixed the number of particles and terminal conditions (max_iter=50, $\kappa=30$) and varied the temperature pool size. As shown in the table, using a finer pool (i.e., more diverse temperature values) improves the final compressibility and aesthetic score but also increases both inference time and NFE. This experiment highlights the impact of temperature diversity on controllability and performance.

• Compressibility |Temperature pool size|Compressibility|Time (s)|NFEs| |---|---|---|---| |2|-55.19±4.93|233.33|8188| |3|-48.84±2.14|285.83|10500| |6|-46.14±2.16|356.5|12800|
• Image Aesthetic |Temperature pool size|Aesthetic|Time(s)|NFEs| |---|---|---|---| |2|6.2345±0.0535|237.92|8200| |3|6.3179±0.0488|286.58|10369| |6|6.3624±0.1419|354.67|12578|

As you suggested, we will include a more comprehensive analysis of these trade-offs in the final camera-ready version. We also plan to report results using Number of Function Evaluations (NFE) as the x-axis instead of inference wall-clock time, as this metric provides a more standardized basis for comparing computational efficiency. A summary table is already available in Reviewer jBRb’s [Question 4 & Weakness 5].

Thank you for your constructive feedback, including your comments on applying our method to high-dimensional outputs, concerns regarding unnecessary computation, and detailed questions about our experimental setup. We sincerely appreciate your thoughtful observations, and we have provided responses to all of your questions and concerns below:

[Weakness 1] It is unclear how well your method would scale to high-dimensional outputs, such as high-resolution text-to-image generation?

We appreciate your point and, in response, have conducted additional experiments in a high-dimensional domain. The results were consistent with other experiments, outperforming the rest of the baselines. For more details, please refer to our response to [Question 1] from Reviewer WZvc below.

[Weakness 2] Some amount of unnecessary compute is being performed by distributing each sample uniformly along the noise levels:

Thank you for highlighting this insightful point—it aligns closely with our own considerations during the development of ABCD. Indeed, we explored a more intelligent go-back strategy but found it doesn’t show meaningful improvement.

Specifically, we tested strategy where particles were redistributed to different noise levels in proportion to the average reward observed at those levels across the population. However, as shown in the table below, despite its intuitive appeal, the performance improvement was marginal compared to the added complexity. Therefore, we ultimately chose a simpler and more efficient formulation for the current version of ABCD. Nevertheless, as you rightly noted, we believe that incorporating smarter strategies—such as applying ideas from SMC to the temperature pool—could be a promising direction.

• Adaptive ABCD vs. ABCD in Sudoku |Method|Add. Compute|Accuracy|Wall Clock Time (sec.)| |-------|----------|----------|---------------------| |Adaptive ABCD|perc=0.6,𝜅=1|0.810|0.220| ||perc=0.8,𝜅=1|0.895|0.395| ||perc=1.0,𝜅=1|0.951|0.489| ||perc=1.0,𝜅=3|0.979|0.549| |ABCD|perc=0.6,𝜅=1|0.821|0.240| ||perc=0.8,𝜅=1|0.903|0.350| ||perc=1.0,𝜅=1|0.958|0.450| ||perc=1.0,𝜅=3|0.979|0.557|

[Question 1] How many denoising steps are used at each of the noise levels ${t_1, t_2, ..., t_M}$? Is this a constant fixed amount or does it change based on the noise level?

The denoising (DDIM) step size is fixed to the same constant across all temperatures (noise levels). For example, for sudoku task, the step size is set to 10. So, the actual number of denoising steps are set to remaining_total_steps(t_k)/ddim_step_size.

[Question 2] Does the algorithm utilize any form of "history" to save the best top-K during the different cycles? This question mainly comes from the fact that the curves in Figures 6,7 are not always increasing w.r.t. the inference time which seems counter-intuitive:

Yes, ABCD utilizes a form of history to save the best top-K during the cycles. More specifically, we included temperature=0 in the go-back temperature pool, allowing the algorithm to preserve the best particles from the previous cycle unless further improvements are made. This allows the algorithm to keep the best case and thus improve monotonically.

Due to this mechanism, ABCD's performance in terms of verifier score consistently improves over time. In contrast, some baselines in Figure 6 lack this property and occasionally show performance decreases.

However, in Figure 7, we observe some performance drops of ABCD mainly in panels (a), (b), and (c). For panels (a) and (b), this occurs because we intentionally excluded the memory feature that maintains the temperature=0 configuration as part of our ablation study. For panel (c), the slight drop in success rate results from a mismatch between the verifier score (which determines our top-K selection) and the actual success rate, which aren't perfectly correlated. We will clarify this in the revision.

Thank you for your valuable comments, including your concerns regarding the lack of theoretical justification and questions about the applicability of our method to domains such as text-to-image or text-to-video generation. We sincerely appreciate your thoughtful feedback, and we have addressed all of your questions in detail below:

[Weakness 1] the lack of theoretical justification:

We thank the reviewer for comment regarding the theoretical justification of our method. We agree that a formal analysis complements our empirical results and strengthens the paper's contributions.

Motivated by the reviewer's valuable feedback, we have developed a theoretical analysis for the convergence of our automatic termination condition. Due to the space limit, we provide a brief sketch below but the full proof will be included in the revised manuscript.

I. Formal statement

Let T = {t₁, …, tₘ} be the “temperature pool” with lowest temperature t₁ = 0. At cycle c, let Dₖ(c) be the multiset of origin‐temperatures of the top‑k particles after denoising.
Stopping rule: terminate at the first cycle N such that for all i in {N − κ + 1, …, N − 1, N}, Dₖ(i) = {0, 0, …, 0}, i.e., all top‑k particles originate from t₁ = 0 for κ consecutive cycles.

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Assumptions

1. Bounded reward. There is a reward function r: ℝᵈ → ℝ with finite supremum r* < ∞.
2. Monotonic selection. Let rₖ(c) = min{rewards of top‑k at cycle c}. Then rₖ(c) is non‐decreasing in c.
3. Attainment. There exists at least one state x* such that r(x*) = r*.

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Theorem (Termination guarantee). Under these assumptions, ABCD’s Adaptive Thinking Time criterion triggers termination in finite time.

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The proof of this theorem is built upon several key lemmas, which we now introduce. We will include the proof for the lemmas in the revised manuscript.

Lemma 1 (Uniform Hit‐Chance). Let x* be any maximizer of the reward, r(x*) = r*. Under the ABCD dynamics (Algorithm 1), there is a constant p > 0 such that for each cycle c, Pr(E_c | 𝓕_{c−1}) ≥ p, where E_c = “x* appears among the selected top‑K at cycle c” and 𝓕_{c−1} is the sigma‑algebra of all randomness up to (but not including) cycle c.

Lemma 2 (Optimality). Under the assumptions:

1. Monotonicity and Boundedness: rₖ(c) is non‑decreasing and rₖ(c) ≤ r* < ∞, so rₖ(c) → r_{∞} ≤ r*.
2. Supremum Attained: there exists x* with r(x*) = r*.
3. Uniform Hit‑Chance: at each cycle c, Pr(E_c | past) ≥ p > 0.
   Then Pr[r_{∞} = r*] = 1.

Lemma 3 (Finite‐Time Hitting). Under the Uniform Hit‐Chance lemma, define T = inf{ c ≥ 1 | x* appears among the top‑K at cycle c }. Then Pr(T < ∞) = 1, Pr(T > n) ≤ (1 − p)ⁿ for all n ∈ ℕ, where p > 0 is the per‑cycle success lower bound.

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II. Proof of Main Theorem

We now assemble the results from the lemmas to prove the termination guarantee theorem.

1. Monotone convergence of rₖ(c). The sequence rₖ(0), rₖ(1), rₖ(2), … is non‑decreasing and bounded above by r*, hence it converges to some limit r_{∞} ≤ r*.
2. Limit must equal r*. According to the Optimality Lemma, r_{∞} = r*.
3. Optimal is reached in finite time. By the Finite‑Time Hitting Lemma, rₖ(c) reaches r* at some finite cycle C.
4. Once optimal is reached, only t₁‑origins survive. At cycle C where rₖ(C) = r*, any particle noised at temperatures t > 0 and then denoised back cannot exceed r*. Thus, the only way to preserve the top‑k set of reward‑r* particles is via replicas from t₁ = 0, which leave them unchanged. Consequently, for all c ≥ C, Dₖ(c) = {0, 0, …, 0}.
5. κ‑persistence triggers termination. Therefore, once cycle C is reached, the condition “all top‑k from temperature 0 for κ consecutive cycles” is satisfied by cycle C + κ − 1, and ABCD terminates.

[Question 1] Would it work well for tasks such as text-to-image or text-to-video that diffusion models are widely applied to?

As suggested by multiple reviewers, we evaluated ABCD on the text-to-image generation tasks using Stable Diffusion v1.5, a latent diffusion model, to generate 512×512 images $x \sim p_\theta(x|y)$ conditioned on textual prompts $y$. The results confirm ABCD consistently outperforms key baselines (BoN and SoP), exhibiting performance trends aligned with our previous experiments.

For example, with around 10k NFEs, ABCD achieves compressibility of −49.17, while SoP and BoN achieves −72.64 and −81.89, respectively. The same trend is observed w.r.t. aesthetic and human preference score. This demonstrates ABCD’s capability to efficiently navigate the high-dimensional image space.

Specifically, we comprehensively assessed ABCD’s performance using three metrics: compressibility, measured as negative JPEG file size (in kilobytes) after compression following [3]; aesthetic evaluation, computed using LAION’s V2 Aesthetic Predictor [1], a linear MLP built upon CLIP embeddings, trained on over 400,000 human ratings; and human preference evaluation, based on the HPSv2 scorer [2], a CLIP model fine-tuned on 798,090 human-selected rankings across 433,760 image pairs. For compressibility and aesthetic evaluations, we used animal category prompts from [3] (e.g., Dog, Cat, Panda), whereas for human preference evaluation we used human instruction prompts from [2]. Detailed quantitative results are provided below, and we will include visualizations of generated images in the camera-ready version.

• Compressibility |Inference Method|Add.Compute|Compressibility|Clock Time(s)|NFEs| |----------------|:---------:|:-------------:|------------:|---:| |Base Diffusion|No add|−100.07±2.88|58.42|2000| |Best-of-N(BoN)|N=5|−81.89±5.71|257.58|10000| ||N=10|−77.86±8.42|494.33|20000| |Search-over-Path(SoP)|Δf=510,Δb=255|−88.36±4.88|188.75|7548| ||Δf=280,Δb=140|−72.64±5.61|257.67|10266| ||Δf=180,Δb=90|−64.59±3.43|280.25|10974| |ABCD|max_iter=30|−62.33±4.33|103.67|3340| ||max_iter=80|−51.49±4.34|254.50|8240| ||max_iter=100|−49.17±4.29|306.08|10176|
• Image Aesthetic |Inference Method|Add.Compute|Image Aesthetic|Clock time (sec.)|NFEs| |----------------|-----------|---------------|-----------------|----| |Base Diffusion|No add|5.57 ± 0.03|58.83|2000| |Best-of-N (BoN)|N=5|5.79 ± 0.05|258.33|10000| ||N=10|5.86 ± 0.05|494.25|20000| |Search-over-Path (SoP)|Δf=510,Δb=255|5.68 ± 0.03|189.42|7548| ||Δf=280,Δb=140|5.84 ± 0.08|258.67|10266| ||Δf=170,Δb=85|5.87 ± 0.04|285.08|10704| |ABCD|max_iter=30|6.13 ± 0.09|105.42|3340| ||max_iter=80|6.23 ± 0.07|233.83|7677| ||max_iter=100|6.25 ± 0.09|288.42|9147|
• Human Preference Score |Inference Method|Add.Compute|Human Preference|Clock time (sec.)|NFEs| |----------------|-----------|----------------|-----------------|----| |Base Diffusion|No add|0.2710 ± 0.0028|61.00|2000| |Best-of-N (BoN)|N=5|0.2753 ± 0.0017|260.92|10000| |Search-over-Path (SoP)|Δf=510,Δb=255|0.2725 ± 0.0031|192.83|7548| ||Δf=230,Δb=115|0.2751 ± 0.0029|272.25|10218| ||Δf=130,Δb=65|0.2765 ± 0.0046|811.25|11232| |ABCD|max_iter=30|0.2797 ± 0.0023|115.92|3340| ||max_iter=80|0.2815 ± 0.0030|272.08|8060| ||max_iter=100|0.2819 ± 0.0032|322.08|9718|

[1] Christoph Schuhmann and Romain Beaumont. Laion-aesthetics. LAION. AI, 2022.

[2] Wu, Xiaoshi, et al. Human preference score v2: A solid benchmark for evaluating human preferences of text-to-image synthesis. (2023).

[3] Black, Kevin, et al. Training diffusion models with reinforcement learning.(2023).