Adaptive Inference-Time Scaling via Cyclic Diffusion Search
Authors: Gyubin Lee
Bao N Nguyen Truong
Jaesik Yoon
Dongwoo Lee
Minsu Kim
Yoshua Bengio
Sungjin Ahn
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion model
Inference-time scaling
Primary area: deep_learning
TL;DR: We introduce the challenge of adaptive inference-time scaling—dynamically adjusting computational effort during inference—and propose Adaptive Bi-directional Cyclic Diffusion (ABCD), a flexible, search-based inference framework.

Abstract: Diffusion models have demonstrated strong generative capabilities across domains ranging from image synthesis to complex reasoning tasks. However, most inference-time scaling methods rely on fixed denoising schedules, limiting their ability to allocate computation based on instance difficulty or task-specific demands adaptively. We introduce the challenge of adaptive inference-time scaling—dynamically adjusting computational effort during inference—and propose Adaptive Bi-directional Cyclic Diffusion (ABCD), a flexible, search-based inference framework. ABCD refines outputs through bi-directional diffusion cycles while adaptively controlling exploration depth and termination. It comprises three components: Cyclic Diffusion Search, Automatic Exploration-Exploitation Balancing, and Adaptive Thinking Time. Experiments show that ABCD improves performance across diverse tasks while maintaining computational efficiency.

Adaptive Inference-Time Scaling via Cyclic Diffusion Search
Gyubin Lee" KAIST
Truong Nhat Nguyen Bao" KAIST
dlrbgls06210Kaistt.ee
axoongahattngyyyobbookkkat....tt
Jaesik Yoon
Dongwoo Lee KAIST
Minsu Kim
KAIST & SAP
Mila Quebec Al Institute KAIST
jaos
dongwoolee@kaist.acc ac.kr
minsu kim@mi 1a quebec
Yoshua Bengio
Sungjin Ahn KAIST & NYU
Mila - Quebec Al Institute
Université de Montréal
sungjin in ahn@kaist ac. kc.kr
yoshua. bengio@mila, quebe
Abstract
Diffusion models have demonstrated strong generative capabilities across domains ranging from image synthesis to complex reasoning tasks. However, most inference- time scaling methods rely on fixed denoising schedules, limiting their ability to al- locate computation based on instance difficulty or task-specific demands adaptively. We introduce the challenge of adaptive inference-time scaling-dynamicciln dynamically adjust- ing computational effort during inference-ant propose Adaptive Bi-directional Cyclic Diffusion (ABCD), flexible, search-based inference framework. ABCD refines outputs through bi-directional diffusion cycles while adaptively control- ling exploration depth and termination. It comprises three components: Cyclic Diffusion Search, Automatic Thinking Time. Experiments show that ABCD improves performance across diverse tasks while maintaining computational efficiency.
Balancing, and Adaptive
Introduction
Diffusion models have become a leading class of generative models, achieving state-of-the-ar performance across diverse domains, from high-fidelity image synthesis complex language gen- eration 221. A core strength of diffusion models is their ability to represent complex, multimodal distributions through hierarchical, multi-step denoising process 27 This iterative refinement makes them well-suited not only for content generation but also for challenging search and reasoning tasks, such planning controls 4. maze navigation 34 35), and puzzle solving 87.
Despite their success, a key open challenge is how to effectively realize inference-time scaling the ability improve model performance by allocating additional computation during inference, either on per-task basis or, ideally, adaptively per instance. While several approaches have recently explored inference-time scaling for diffusion models [5] 17), they still face notable limitations.
"gqual contribution. Correspondence to Gyubin Lee and Sungjin Ahn <dIrbqls0521 kaist.ac.kr and sungjin. ahn @ kk.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Among the limitations, the most critical is the prevalent reliance on fixed inference procedures. Most existing methods such as Best-of-N [36]. Sequential Monte Carlo 24 31], Beam Search [T7]. and Search-over-Path [6 execute predetermined number of denoising steps along trajectory. Once this trajectory completes, the inference is terminated, regardless of the complexity or specific requirements of the input instance. This rigidity fundamentally restricts the model's ability to adapt computation based on instance difficulty, ambiguity, or the need for higher confidence or output quality. As a result, performance could suffer more challenging inputs, and computation may be either underutilized or inefficiently spent.
In this paper, we propose novel method method-Adapiin Adaptive Bi-directional Cyclic Diffusion (ABCD) to address the challenge of adaptive inference-time scaling. This approach reframes diffusion model inference as a flexible and efficient search process, enabling adaptive computation and instance- refinement based on task difficulty and evolving solution quality. ABCD consists of three components. Cyclic Diffusion Search (CDS) enables iterative refinement by cycling bi-directionally through the diffusion process-altereee alternating between denoising and re-noising steps-alee allowing the model to escape local minima and explore alternative generative trajectories. Automatic Exploration-EEplooitttioo Balancing (AEEB) introduces mechanism for adaptively controlling the depth of exploration by distributing particles across multiple re-noising levels, allowing the model to implicitly identify the appropriate amount of computation for each instance. Finally, Adaptive Thinking Time (ATT) provides a principled stopping criterion by monitoring the evolution of solution quality over inference cycles, ensuring computation is allocated efficiently and terminated when additional refinement yields diminishing returns. We demonstrate the effectiveness of ABCD across range of wosks-includdig control planning, maze solving, Sudoku, and molecule generation-showee inference-time flexibility, computational efficiency, and solution quality.
significant gains in
Our main contributions of this paper are as follows: (1) We formalize the problem of adaptive inference-time scaling in diffusion models and identify the limitations of fixed-step inference. (2) We propose Adaptive Bi-directional Cyclic Diffusion, a novel framework that frames inference as flexible, search-based process with dynamic compute allocation. (3) We introduce three compo- Balance, and Adaptive Thinking
nents Cyclic Diffusion Search, Automatic peploration-Exploitteee Time-that enable iterative refinement, adaptive search control, and automatic stopping. (4) We empirically validate ABCD across diverse tasks, including planning, maze solving, Sudoku, and molecule generation, showing improvements in flexibility, accuracy, and compute efficiency.
2 Preliminaries
Diffusion model. Diffusion models 9 25 26 27 have recently emerged as a powerful generative framework that formulates sample generation as gradual denoising process. This framework consists of two complementary components: fixed forward process, which incrementally injects noise into data to transform structured inputs into nearly pure noise, and a learned reverse process, which aims to recover clean data from noisy observations. Formally, the forward process defines a sequence of latent variables X X2, Xy conditioned on the original sample X, via a Markov chain: q(x++1 xxx) (x / /ax,, (1 oz) where o, E (0, 1) denotes a pre-defined noise schedule. This process ultimately transforms Xy into Gaussian noise ør ~ NO, T). The forward transition an arbitrary step f is given by:
VI oze, 6 (,,
where ã; =I ==1 Og denotes the cumulative product of the noise schedule. This expression provides the marginal distribution of X, conditioned on Xp-
The reverse process uses neural network to predict the noise component cx l), which is then used to estimate the clean sample: (x(x)) Va, (x, vI āi (x(x,, Using this estimate, the previous latent state is computed via update:
X var- + / ā, oo co + ozz z N~ ~(011,
where a, controls the level of stochasticity. The reverse process can be parameterized in various equivalent forms, including noise prediction co(x,, (), posterior mean prediction po (x(, t), or direct clean-sample prediction {x,x.

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Cycle
Converge
Fast Denoising
Temperature pool
ATT
Top- K Selection
Select the Best
Automatic Exploration Balancing
Figure Overview of our method. (1) Start with particles, use jumpy denoising øT Φт-} Zo. (2) Replicate each particles times and send each particles to multiple noise levels (AEEB). (3) Re-denoise particles. (4) Select Top K particles. (5) We repeat steps (2)-(4) until the adaptive terminal condition is satisfied (CDS). The numher of cycles executed in this way defines the Adaptive Thinking Time (ATT).
Reward-guided generation. In many practical scenarios, is desirable only generate realistic samples from diffusion model, but also to align them with a predefined reward function r(x) E IR. common objective is sample from distribution that biases toward high-reward samples while staying close to the original model distribution (28). This can be formalized as:
pla) exp(r(x)/o) Ppee(x),
where > 0 is temperature parameter controlling the trade-off between reward optimization and fidelity the pre-trained distribution Ppre- As a - 0, the distribution becomes increasingly
concentrated on reward hoauddmaximizing samples [1
ABCD: Adaptive Bi-directional Cyclic Diffusion
We propose Adaptive Bi-directional Cyclic Diffusion (ABCD), a novel inference framework for diffusion models that enables adaptive and efficient scaling with respect the available computation budget. ABCD is designed to flexibly trade off exploration and exploitation during inference and to dynamically determine when terminate. The framework is composed of three core components: (i) Cyclic Diffusion Search, (ii) Automatic Exploration, and (iii) Adaptive Thinking Time.
3.1
Cyclie Diffusion Search
The core idea of Cyclic Diffusion Search is enable iterative inference by cycling bi-directionally through the diffusion timeline, alternating between the denoising (reverse) and noising (forward) processes. This approach allows the model refine its predictions over time while adaptively revisiting earlier stages of the generative process to explore alternative solution paths. Each iteration of the cycle consists of three main stages: Denoising, Selection-ande cattiiinnnnn-ooppppp Copy, and Noising.
Denoising. As particle- based method, the inference process begins by sampling a set of particles from the standard Gaussian prior distribution (0, I). Leveraging the pretrained diffusion model, 7 = using a coarse, accelerated Denoising Diffusion Implicit Models (DDIM) [26] trajectory that skips intermediate steps. This allows to quickly obtain initial sample estimates X, from the noisy latents X7, significantly reducing the time to acquire preliminary solution candidate. While this fast DDIM process may yield suboptimal initial outputs, the key idea of our framework to mitigate this through subsequent cyclic refinement. Unlike some prior approaches [16] that interleave intermediate noising steps during the denoising pass, prioritize reaching = -0 quickly to establish a diverse set of starting points for the cyclic search.
each particle is then denoised from
Selection and copy. Once all particles reach = O, we evaluate them using task-specifi reward function, yielding scores r', f2...., Tv. To guide the subsequent search towards promising regions, we select the top-K particles with the highest reward scores. These K selected particles serve as "anchor" points the next phase of exploration. Each of these particles is then replicated times, producing total of K particles.
3

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Algorithm ABCD: Adaptive Bi-directional Cyclic Diffusion
1: procedure ABCD(verifier, T. T, max_maiiitt max_iter, K, N, K, J)
2: Initialize N particles {z/]² } from (0,
3: Denoise to obtain {ag (0) via Fast denoising
4: while less than max iter do
5: Selection: Select top- particles using verifier
6: If Selected top-K particles come from smallest &g for times consecutively break
7:
Copy: Replicate each K particles J time Noising: Send each K particles to each ty ET Denoising: Denoise from each tg go-back temperature 11: end while
8:
9:
return Select best particle from top- K particles
13: end procedure
Noising, The replicated particles are then sent back to an earlier diffusion step S < T referreed to as the go-back timestep-ve via the noising process of the diffusion model: ((xy XX)) This operation reintroduces stochasticity and enables the particle to re-enter the denoising trajectory from a different point in the diffusion space. The full denissnn,,,, selection-copy, and noising then repeated iteratively until a predefined stopping criterion is met.
3.2 Automatic Exploration-EExlooittaio Balancing
key limitation of the basic Cyclic Diffusion Search, as described above, lies its reliance on fixed go-back timestep, treated as manually specified hyperparameter. In principle, there may exist go-back step that optimally balances exploration and exploitation at each stage of inference. However, identifying this optimal step is generally difficult and cone-consuming. Moreover, show that the optimal go-back step not only varies across different instances but could also evolve dynamically throughout the inference trajectory of single instance. Consequently, relying fixed go-back step is both hard to tune and fundamentally suboptimal for adaptive inference.
experiments
Illustrative example. Consider solving Sudoku puzzle. In the early stages, when only few digits are provided, global exploration beneficial to fill in plausible candidates and explore a wide solution space. In contrast, in the later stages-when most of the board is correctly filled aeged-sgeeee global changes can corrupt nearly-complete solutions. At that point, cautious local refinement is preferable. This illustrates the need to dynamically adjust the pegloration-exploitttiii
balance during inference.
Implementation. To implement automatic exploration mechanism, we modify the noising step of the previously described cyclic diffusion search as follows. We begin by defining set of predefined go-back timesteps, referred as the "temperature pool", denoted by T (1,,...... (M)), where each t; corresponds to distinct level of exploration. A simple and effective way to construct T to uniformly partition the diffusion timeline into M segments. After the select-and-copy step. yielding K x J particles, the replicas are distributed across the predefined temperatures in T via the forward (noising) process, rather than being sent to a single go-back timestep. This allows each anchor particle to probe multiple exploration depths in parallel, enabling the system implicitly identify effective go-back steps through iterative feedback. After completing the re- re-noising, all K J particles denoised back to t O, and a top-K set is selected. This way, particles from suboptimal temperatures will automatically be discarded. This cycle continues until a stopping criterion is satisfied. See Figure for illustration of the algorithm.
3.3 Adaptive Thinking Time
When should we stop this eyclic inference computation? Unlike single-pass. inference methods that terminate once reaching 0, ABCD's cyelic structure raises nontrivial question of when to conclude the iterative search. A naive solution is to impose fixed computation budget (e.g.. a maximum number of cycles), but this risks either underutilizing available compute or wasting resources when further refinement yields no benefit.

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To address this, ABCD incorporates an implicit uncertainty measure derived from the temperature distribution of the top-K particles. After each cycle, we record the re-noising temperatures from which the current top-K particles originated. If high-temperature particles are present in the top- K, this suggests that exploration is still yielding valuable improvements. Conversely, if all top- K particles originate from the lowest temperature, it implies that global exploration has little impact and the search is now confined to local refinement-Signallee
that the process may have converged.
As practical implementation of this idea, we adopt simple yet effective stopping rule: terminate the inference when all top-k K particles originate from the lowest temperature for x consecutive cycles. The hyperparameter к controls the sensitivity of the termination criterion, balancing early stopping against the opportunity for further refinement.
Related works
Inference-time scaling with pre-trained diffusion models. There is growing interest in lever- aging pre-trained diffusion models more effectively during inference, by adaptively steering the generation process during sampling. A unified perspective (3]28) frames this as sampling from augmented distribution of the form pxx) exp(Ar(x)). A common approach is classifier guid- ance [O), which assumes access to a differentiable value functioo.. For non-differentiabll rewards, derivative-free free strategies such as SVDD [3]. Beam Search [7]. DSearch [14, and MCTS [35] have been proposed. Alternatively, sampling- based based approaches like Importance Sampling and Sequen- tial Monte Carlo (e.g.. TDS [1I, FKD (24) offer principled swward-weighter trajectory selection. However, most methods operate under uni-directional denoising and rely static search schedules.
Planning and reasoning with diffusion models. Diffusion models have recently gained traction for reinterprets planning as conditional extend this idea by using diffusion models world models. On the reasoning side, recent studies explore diffusion models for structured inference. Energy-based diffusion models have been applied to symbolic reasoning tasks 7 8 35]. and SRM 30 applies diffusion to visual spatial reasoning.
long-horizon planning and complex reasoning tasks. Diffuser 13
generation over offline trajectories. Subsequent works [
36
5
Experiments
We evaluate our proposed method on a diverse suite of challenging tasks that require generating sam- ples in sparse previously unexplored regions at training phase: (1) toy Mixture of Gaussian(MoG))
(2) point-mass maze navigation [19], (3) Sudoku puzzle completion [8, 29 (4) path generation
on unseen pixel maze image (12). (5) molecular structure prediction [I]. and (6) text-to-image
generation. These tasks present distinct challenges: MoG shows the importance of multiple go-back temperature pool as the proof-of-concept; point-mass maze requires long-horizon planning over 1000 steps; Sudoku demands logical consistency across row, column, and block constraints; path generation on unseen maze image tests generalization novel environmental structures; molecular structure prediction requires generating valid 3D conformations under chemical and physical constraints; and image generation illustrates that our approach also operates effectively in the high-dimensional setting, demonstrating its scalability beyond structured low-dimensional tasks. Detailed task configurations are discussed Appendix B.2
5.1 Baselines
We compare ABCD against several strong representative baselines: Base Diffusion: The standard diffusion model without additional computational scaling, serving as our primary baseline. Best-of-N (BoN): A computationally scaled variant that generates N independent samples and selects the best according task-specific verifier used consistently across all models. Diffusion Beam Search (BS) 1 Combines diffusion with beam search to scale the denoising process. pruning unpromising candidates during generation. Sequential Monte Carlo Diffusion (SMC) (2 B1: Employs resampling during the denoising process to focus computational resources on more promising trajectories. Implementation details were adapted from (24] to suit our experimental setting. Search over Paths (SoP) [16]: Enhances generation by allowing limited backward steps (adding noise) to promising samples, enabling exploration of wider solution spaces compared to strictly unidirectional approaches. Detailed Baseline configurations discussed in Appendix B.3
5

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dog with plate food on the ground
Dataset
17 Provided
Maze Size 15
PointMaae Giant
HH
(a) MoG
(b) Sudoku
(c) Pixel Maze
(d) Molecule
OGBench
(f) Text-to-Image
Figure 2: Evaluated Tasks. (a) Mixture Gaussian for Proof-of-coneept that we need multiple go back temper-
ature. (b) Sudoku puzzle completion demanding logical consistency across multiple constraints simultaneously:
(c) Pixel Maze Path Finding testing generalization to novel environmental structures not encountered during training: (d) Molecular structure prediction requiring physically and chemically valid 3D conformations. (e) OGBench PointMaze navigation requiring complex planning over 1000+ steps (f) Text-to-image generation requiring complex, high-dimensional outputs that are well aligned with the given textual descriptions;
Table 1: Left: Success rate (%) and distance the goal on MoG. We compare multi ga-back (ABCD) and fixed single go-back (GB 20, 40, 60, 80).
Table 2: Right: Success rate (%) and mean itera- tion reach the goal on MoG. Every method termi- nates when reaches the goal mode.
Method
Dataset 1
Dataset 2
Method
Datasel
Dafaset 2 Mean Cyc
Success
Distance
Success
Distance
Success
Mean Cyc
Success
ABCD (Ours)
100
0.08 a: 0.0
100
0,03 a
ABCD (Ours)
100
3.36
100
7.02
GB 20
0.17 + 0.0
26
3.97 6.0
GB 20
100
2,06
79
16.14
GB
2.90 0.6
59
1.73
GB
46.38
100
6.77
GB 60
0
7,80 0.8
7
2.94 1.1
GB 60
0
100
15.05
GB
42.81 6.6
36.46
GB 80
6
43.33
All baselines used the same pre-trained diffusion model per tasks, and inference-ting budgets were expanded much as possible while preserving their core operational characteristics. In BS and SoP, inference-time compute was increased by more frequent expansion and selection, while SMC scaled compute via higher resampling frequency, and BoN by increasing the number of particles N. For fair comparison, the total number of particles was set equal across all methods (except BoN). While we primarily report wall-clock time in the main paper analyze inference-time scaling behavior, additionally provide a analysis of the Number of Function Evaluations (NFEs) in Appendix B.16
Additional experimental details for each tasks are provided in the Appendix B.6 B.8 B.10
B.12
5.2 Mixture of Gaussian
Setup. We construct two toy datasets to illustrate the need for multiple go-back noise levels and terminal condition during inference refinement (Figure aa)). so small-step refinement suffices; Dataset 2's modes lie in distant regions, requiring occasional large "jumps" to escape poor initial prediction. We train separate diffusion models on each and condition sampling on distance to target (red star). Full details appear in Appendix B.4
())g Dataset 1 has modes clustered locally,
Result. We evaluate ABCD against four fixed single go-back temperatures (GB 20, 40, 60, 80) under two inference regimes: a fixed 25-cycle budget and an open-ended budget of up to 100 cycles with early stopping upon goal attainment. As shown in comparison with fixed 25 cycles [T): only ABCD attains 100 % success and minimal final distance on both datasets. Fixed small go-back steps work Dataset 1 but cannot escape poor initializations in Dataset 2, whereas fixed large steps succeed escaping from poor initializations on Dataset 2 but over-cycle on Dataset I and 2 (e.g.. reaching the goal cycle 20 but diverging by cycle 25 after losing useful information). As shown in comparison with up to 100 cycles with early stopping (Table 2 some fixed levels solve one dataset, but only ABCD consistently achieves both 100 % success with comparable average cycles. These findings underscore that exploration (coarse jumps) and exploitation (fine steps) require different go-back levels a balance that ABCD's 's adaptive termination naturally realizes.
5.3 Sudoku puzzle completion
Setup. We evaluated ABCD against baseline methods on the Sudoku puzzle completion task 18 29 a canonical logical-reasont benchmark. In Sudoku, 9x9 grid must be filled under row, column, and 3x3-block constraints, with each new grid filled based the current grid state. An adaptive

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Base Diflusion
BoN
SMC
85
ABCD (Ours)
Sudoku Harderset
Pixel Maze Size 15
QM9
10
1.0
Molecule Stalaliity
Ratt
0.93
Accuracy
0.90
0.6
0.6
0.85
04
0.00
02
0.2
0.700
Inforence Wall Clock Time
Inferince Wall Clock Time
Inference Wall Clock Time (6)
Figure 4: Left: Sudoku Puzzle Completion result. Mean accuracy on Harder dataset (17-28 entities provided) by giving more computational budget (sec.). Middle: Pixel maze path finding result. Success rate on the OOD Pixel Maze size-15 test Right: Molecular 3D structure prediction task result. Molecular Stahility on QM9 dataset. For each method, inference-time computation was scaled as much as possible along its core controllable axis for inference time scaling to ensure a fair comparison under expanded budgets.
Size 11
Maze Size 12
Size
Size 14
Maze Size
Success Rate
67
03
Inference Time
Inference Time (s)
Inference Time (s)
Inference Time (s)
Inference Time (s)
Figure 5: Pixel Maze path finding results on multiple maze size. Success rate comparison across maze sizes in the Pixel Maze task. All settings involve OOD evaluation, where models are trained smaller mazes (sizes 4-v6) and tested larger ones. As size increases, the search space grows and the performance gap between other methods widens, highlighting importance adaptive inference-tinn exploration complex scenarios.
search strategy- exploring when few clues are given and then locally refining once most entries are correct-proves particularly effecttve. We trained all models on puzzles with 31-42 given digits (29) and tested them on more challenging instances containing only 17-28 givens [8] thereby
measuring each method's ability produce logically consistent solutions. See Appendix B.8 for additional experimental details.
Result. Figure Leftt) shows that ABCD consistently outperforms all baselines, achieving higher accuracy in less wall-clock time and displaying superior inference- time scaling. Notably, SoP [16] never achieves perfect accuracy (95.5 % accuracy), whereas ABCD attains 100 % accuracy in fewer cycles. Moreover, as illustrated in Figure 3 ABCD's accuracy remains stable across all difficulty levels, while baseline methods exhibit marked performance degradation on the hardest instances. We attribute this robustness to ABCD's dynamic compu- tation allocation: it adaptively concentrates additional inference effort on more challenging puzzles, in contrast baselines that expend a fixed, uniform budget.
SMC
ABCD(Ours)
1.0
0.8
Accuracy
0.0
0.4
Very Hard
Hard Task Difficulties
Medium
Easy
Figure 3: Mean accuracy across four dif- ficulty levels- -Very Hard (17-19 provided), Hard (20-21 provided), Medium (22-24 pro- vided), Easy (25-27 provided).
5.4 Path generation on unseen pixel maze image
Setup. We evaluated ABCD's adaptability path-generation task using mazes of previously unseen sizes [12. Every method leveraged the same diffusion model pretrained on small mazes (sizes 4, 5, and 6) and then tested larger, structurally distinct mazes (35) This setting rigorously probes each algorithm's capacity to generalize to novel spatial configurations at inference time. Additional experimental details are provided in Appendix [B.
Result. As shown in Figure Middle), ABCD reaches near-perfect success rates significantly faster than all basclines, evidencing superior time-accuracy trade-off. In contrast, most competing

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Base Diffusion + BoN
SMC
-
BS
SoP
ABCD (Ours)
PointMaze Large
PointMaze Giant
1.0
1.0
Succoss Rats
0.8
Successs Rate
0.8
0.0
0.5
64
0.2
0.0
60 Inference Wall Clock
60
Inference
(s)
Figure 6: OGBench PointMaze task results. Success rates across large and giant mazes. Our ABCD approach consistently outperforms baselines, rapidly achieving higher success rates with fewer inference time and attaining perfect performance, particularly in the Giant maze, emphasizing its superior efficiency.
cerchods-nncludee Base Diffusion, BoN, BS, and SMC never exceed a 60 % success rate even under generous computational budgets. ABCD, by comparison, attains 100 % success in substantially less time. Figure [5 further examines performance across maze sizes 1 11-15 in the Pixel Maze task. Here again, ABCD consistently dominates both success rate and inference time at every size, and the margin over baselines widens as search space grows. These findings underscore the value of adaptive, instance-specific exploration strategies especially in complex search demanding tasks.
5.5 Molecule generation
Setup. We evaluate ABCD on the 3D molecule generation task using the QM9 dataset [20), which contains 130k small molecules with annotated properties and atomic coordinates. For our evaluation, we utilized the Equivariant Diffusion Model (EDM) [1]] as the pre-trained model. We then applied various inference-time methods to generate valid 3D conformations.
Result. As shown in the Figure Right,, ABCD significantly outperforms baselines by achieving higher molecular stability with less inference wall-clock time. For instance, ABCD reaches molecule stability of 0.97 within approximately 20 seconds, a threshold that other baselines either fail to attain or require considerably more computational time to approach. Notably, our method achieved a peak molecule stability of approximately 0.99, which is about 5.32% higher than the peak stability of 0.94 achieved by SoP [16), the next best-performing baseline. Furthermore, the performance gap between ABCD and the baselines widens at higher stability levels, emphasizing the robustness of our method in complex molecular generation tasks. See Appendix B.12 for more details.
5.6 OGBench Point Maze
Setup. We further evaluate the ABCD framework using OGBench's PointMaze environments 19 In this task, agents must navigate specified goal region. The base diffusion model follows Diffuser [13. and is trained using the offline dataset provided by OGBench [19], containing trajectories collected by a noisy expert policy. Each environment includes 5 distinct tasks with different start-goal pairs. At inference time, the goal information is provided to the verifier for scoring the particles. To let the model efficiently plan over long horizon while reaching the goal as quickly as possible, we employed value guidance function
5 to minimize the distance from the final state to the goal.
Result. As shown in Figure 6 ABCD consistently surpasses other baselines. Moreover, only ABCD achieves perfect performance on both mazes. It reflects the advantages of our method: by quickly denoising to
locates valid trajectories almost immediately, rather than spending compute on uniformly long reverse sampling path. In maze giant, the gap widens because larger mazes demand more on both global exploration (to reach the goal) and fine refinement (to thread tight corridors). Further results analysis and visualization of the trajectories can be founded in the Appendix B.7
5.7 Text-to-image generation
Setup. Finally, we evaluate ABCD on the image generation tasks. adopt Stable Diffusion v1.5, a latent diffusion model, to serve the image prior for generating 512 x 512 samples (o(l))) conditioned on the textual prompt y. For downstream reward function, we utilize metrics including compressibility, aesthetic evaluation (using LAION's V2 Aesthetic Predictor (22), and human
8

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Base Diffusion
BoN
--- SoP
ABCD
(Ours)
-10
Humen Preference Score
0.278
Compressibillty
-70
Aesthetic
-3
-90
0.274
0.272
150
550
210 Inference Wall Clock Time [s]
Inference Wall Clock Time
Inference Wall Clack Time
Figure 7: Text-to-image generation result. Average reward with respect to compressibility, aesthetic score and human preference score.
14 provided Instance
44 provided Instance
Pixel Maze Size 15
Average Thinking Cycles
0.5
1.0
Number of Cycles
0.
Success Rate
-= 10
Accurac
Accuroo
20
16
9
0.00
0.
0.4
-
ABCD w/o Adletive temmaal
ABCD Aduptive terinnl
8
0.3
10
20
30
42
50
10
17 Number of Provided Numbers
19 23 23 25 20
Cycle
Cycle
Inference Wall Clock Time (9)
(a)
(b)
(c)
(d)
Figure (a-b) Per-instance analysis inference- scaling behavior across different go-back tempera- tures. (a) Results for Sudoku instances with 14 provided entities. (b) Results for instances with 44 provided
entities. (c) Performance comparison with vs. without adaptive terminal condition in Pixel Maze Size 15.
x-axis is the average inference time per sample. (d) Different thinking assignment per hardness in Sudoku (provided number from to 33).
preference evaluation (HPSv2 from 332 for comprehensive evaluation. Further details regarding the experiments can be found Appendix B.14
Result, Figure 7 shows that ABCD consistently outperforms two strong baselines, BoN and SoP, in generating high-reward samples. efficiently navigates the higg--iimessionll image space identify high-reward samples. As computational budget increases, all methods see reward improvements, but ABCD benefits the most. For example, in all metrics, we double or triple the inference time, BoN barely climbs, while SoP achieves moderate gains. By contrast, ABCD can hit the same compression level that SoP reaches at 272s in less than a quarter of the time. In addition, to reach human preference score of 0.2764, SoP needs over 300s, while ABCD does it in under 81s. These highlights that its instance-specific exploration strategy effectively leverages additional computation to better align generated samples. Visualizations of the generated images are provided in Appendix 15.
5.8 Do we need to care about go-back temperature level?
Instance-level optimal temperature. Figure a-bb plots cycle-wise accuracy for two Sudoku instances (14 44 givens) under fixed go-back noise levels. Certain noise settings plateau after only a few cycles and the plateau point differs between puzzles, indicating that each instance possesses its own optimal noise level. (sce Appendix C.I.
Temperature selection per cycle. Figure tracks the provenance selected particles source across cycles for three instance types in Sudoku (17, 19, and 32 givens). Harder puzzles continue sampling from a broad range of noise levels, while easier ones converge rapidly to low-variance perturbations. Early iterations emphasize high-variance (global) moves; later iterations favor low-variance (local) refinements. These patterns demonstrate ABCD's dynamic, per-instance generative process through adaptive exploration vs. exploitation.
5.9
Do we need adaptive terminal condition?
Efficiency adaptive terminal condition. An ablation study on Pixel maze size (Figure K(c) compares fixed cycle length inference (blue) our adaptive terminal criterion (red) with the same temperature pool. The adaptive policy stops early on simple avoiddng wasted compute-yer allocates extra iterations to hard ones, yielding consistently superior time-success trade-offs.

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from go-back
30
40
17 Provided
19 Provided
32 Provided
Count Particies O-Nu www.auose.
15
30
55
Cycle
Cycle
Cycle
Figure 9: The origin of the top-10 selected particles at each cycle. The above plot shows, for different Sudoku difficulty cases, the origin of the Top- selected particles cach cycle. We observe that the behavior varies significantly across cases, and harder instances tend to require longer search. Additionally, early iterations tend select samples with large modifications, while later cycles focus on smaller refinements. This indicates that our model automatically expands and adjusts the diffusion generation process, resulting in different generation graph for each final Xo prediction.
ABCD automatically assigns proper thinking time. Figure dd) demonstrates that ABCD allocates more thinking cycles to harder puzzles (fewer givens) and terminates earlier on simpler ones, vali- dating that our adaptive terminal criterion scales compute with problem difficulty. Furthermore, the stop-flag hyperparameter к. directly controls exploration depth: larger values enforce additional cycles before stopping, which boosts success on challenging instances. Identical trends hold on Pixel Maze task. (see Appendix B.17
C.2
for more detailed analysis).
6 Limitations and discussion
While ABCD performs well on moderately sized planning and reasoning tasks, its scalability to high- dimensional output spaces sraces-such&ae such as high-resolution image video generation rereraiionremmminns unexplored. Extending ABCD to such domains may require structural priors or hicrarchical variants to maintain tractability and efficiency. In addition, although ABCD demonstrates strong empirical performance through its search-based inference strategy, its theoretical properties remain largely unexamined. A deeper theoretical understanding could be gained by connecting ABCD principled frameworks such as Bayesian optimization or approximate inference.
A particularly promising future direction lies in amortizing the search process. The optimal re-noising steps and decisions discovered during ABCD's iterative inference could treated as pseudo-labels for training context-aware policy that predicts adaptive search strategies conditioned on the input. Such a learned policy could retain the adaptivity of ABCD while significantly reducing inference- cost by eliminating online search. We leave the development of such amortized inference mechanisms to future work.
Conclusion
We introduced Adaptive Bi-directional Cyclic Diffusion (ABCD), novel inference framework that enables instance-aware, compute-adaptioe generation with diffusion models. By framing inference as flexible search process, ABCD dynamically allocates computation through bi-directional dif- fusion cycles, adaptive exploration, and principled stopping. Across diverse domains-including planning, reasoning, and molecule generation-ABCI improves performance and efficiency over fixed-schedule baselines. This work underscores the promise of adaptive inference-time scaling.
Acknowledgments and Disclosure of Funding
We thank Junyeob Baek, Doojin Baek, and Hyeonseo Cho for insightful discussions and as- sistance with this project. This research was supported by GRDC (Global Research Develop- ment Center) Cooperative Hub Program (R5-2024-004361655) and Brain Pool Plus Program (No. 2021HID3A2AA330333
through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT. The authors of Mila acknowledge the support from CIFAR, NSERC, the Future of Life Institute.
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A Ethic statement
Our method is general-purpose inference-time optimization framework designed to maximize task-specific reward functions. The ethical implications of its use are inherently tied to how the reward function is defined. While this flexibility allows for beneficial applications enabling faster and more targeted generation aligned with desirable outcomes- it also raises the potential for misuse the reward function encodes harmful or biased objectives. Therefore, the responsibility for ethical deployment lies in the careful design and oversight of the reward specification.
B
Experiment details
B.1
Computational resource
Our implementation is built using the PyTorch framework. The experiments are conducted on 2 machines: Ubuntu 20.04 machine equipped with an Intel(R) Xeon(R) Gold 6348 CPU @ 2.60GHz with 1 12 cores, 384 GB RAM, and NVIDIA GeForce RTX 4090 GPUs; Ubuntu 22.04 machine equipped with an Intel(R) (R) Gold 6230R CPU @ 2.10HHz with 104 cores, 256 GB RAM, and NVIDIA GeForce RTX 4090 GPUs. Each experiment is run individually on NVIDIA GeForce RTX 4090 GPU.
B.2
Task detail
We introduce the setting of experiments and diffusion models we used. For Mixture of Gaussian, we constructed the two datasets generated from the mixture of gaussians. Dataset is generated from region of Gaussians and Dataset 2 is generated from 2 region of the islands as you can on the Figure a). After pre-training two diffusion models on each dataset, we guide sampling at inference time by providing the distance to target goal toward the desired goal state. For OGBench Maze, we adopted the task setups from [9], running sparse reward navigation tasks in each environment, each take hundreds of steps accomplish. The model is trained on standard dataset collected by noisy expert policy that repeatedly reaching randomly sampled goals. At test time, the model is required generate proper plan using guidance from value function. For Sudoku, the basic setting is adopted from IRED 8. where the diffusion harder RRN dataset [18 with 17 to 28 provided samples. For Pixel Maze, the basic setting is adopted from T-SCEND [35], where the dataset are generated by maze- dataset [12), and the diffusion model is trained with Maze sizes of 4 x 4 to 6 X 6 and tested with size of 11 X 11 to 15 x 15. For Molecule Generation, the basic setting is adopted from EDM I1]. The diffusion model is trained to generate 3D molecular geometries using the QM9 dataset 200 which contains approximately 130k small molecules, each with up to heavy atoms. We utilized the standard splits of this dataset for training, validation, and testing. For Text- generation, we use Stable Diffusion v1.5-a widely adopted text-to-image diffusion model-as our pretrained backbone.
reward, encouraging generation
model is trained SAT-Net dataset with 31 42 provided samples (29] and tested
Verifiers. For OGBench Maze, we employ a soft verifier that scores each generated plan based on the proportion of collision-free states throughout the trajectory and the proximity the goal the end of the trajectory. For Sudoku, we use the ratio of correct answer that matches the ground truth label. For Pixel Maze, we use weighted sum of Precision, Recall, Fl score with rate of success score which indicates whether the path reaches the goal without passing the wall. For Molecule Generation, we use the Negative Likeliieldoood (NLL) to evaluate the generated molecular structures. For Text-to-Image generation, we use three verifier scores:
compressibility 2, aesthetic [22. and
human preference score 33
Evaluation metrics. For OGBench Maze, we used the success rate (whether the planned trajectory reaches the goal not) of the actual rollouts from the generated trajectory plans. For Sudoku, we use the ratio of correct answer that matches the ground truth label. For Pixel Maze, we use success score (0 or 1) which indicates whether the path reaches the goal without passing the wall. For Molecule Generation, we use molecule stability (0 or 1), the proportion of generated molecules for which all atoms are stable, For Text-to-Iman Image generation, we use three evaluation metries: compressibility [2), aesthetic (22, and human preference score 33 which correspond to specific verifiers, respectively.
20

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B.3 Baseline detail
We compare ABCD (ours) to several representative methods capable of inference-tine scaling.
Base Diffusion is the most basic baseline, which generates samples without any additional com- putation during inference. When a differentiable value function is available, we include Classifier Guidance [5] base diffusion baseline.
Best-of-N(BoN) generates N samples from the pre-trained diffusion model and returns the sample with the highest reward score according to the verifier.
Search over Paths (SoP) [16] is recently proposed method that performs inference- time scaling by alternating (noiiiiingg and (envaiiddnoiiiiggg transitions in the denoising space. Starting from M particles, the method perturbs each particle with K different noise moving to Af steps backward, resulting in M x K expanded candidates (expansion). Each of these is then denoised forward for Ab steps. After reaching the new states, each particles are scored using verifier, and the top M are selected to continue (selection). This process is repeated until the particles reach 0, with the constraint that Af Ab to ensure overall forward progression.
Diffusion Beam Search (BS) [7 is another baseline that performs inference-time search over denoising trajectories. Starting from M initial particles, the method branches each into K candidate continuations every denoising steps, and retains the top M candidates based on reward estimates. This beam-style selection continues until the denoising process reaches zo. Our implementation is inspired by 17,, with necessary modifications for our setting.
Sequential Monte Carlo Diffusion (SMC) [4] is sampling- sampling-based method that employs Sequential Monte Carlo (SMC) to steer diffusion trajectories toward high-reward regions(31]. Inspired by the Feynman-Kac Steering Diffusion framework (FKD) [24], it begins with an initial set of N proposal particles and performs iterative resampling every p steps during the denoising process, using importance weights defined by tailored potentials. These potentials are designed to approximate the weightee posterior distribution p(x)p(c x) (where the reward encodes the target condition), effectively guiding particle evolution toward desirable outputs. Our implementation follows the general structure of FKD, with modifications to accommodate our conditional generation setting. Specifically, we define p(c x) using the task-specific verifier score guide the sampling process. As this is a sampling based approach, we select the highest-scoring particle from the final set of N samples drawn from the guided distribution.
ABCD's inference-time termination is controlled by three hyperparameters: the percentage threshold, the maximum iteration bound (max_ iter), and the persistence parameter The percentage threshold defines the minimum fraction of top- K particles that must originate from the zero go-back noise level in order to raise termination flag. Inference is terminated once this condition holds for к consccutive iterations. We scale inference- titre computation by varying both the percentage threshold and k, adjust max max_iter iter
which jointly determine the strictness of the stopping condition. Additionally, to cap the total number of iterations. As these hyperparameters increase, the termination criterion becomes more conservative, typically leading to longer inference time and improved performance through extended refinement.
Unless otherwise specified, in both SoP, BS and SMC, since we have reward function defined on Xo space, the reward evaluations at each step x, are performed using predicted clean sample & (x;) (approximation of ExorPRee lxo xx)xx.). which is then passed into the verifier 15| 24]. With this choice, we get the intermediate reward r( r(x,, c) r(xo- = &,, c). This ensures that reward signals are consistent across trajectories and noise levels.
B.4 MoG experiment detail
We constructed two Mixture of Gaussians datasets: Dataset 1 has 36 modes concentrated in a single region, while Dataset distributes 36 modes across two regions (18 modes each). Two separate diffusion models were pretrained on each datasets using 100 denoising steps. In each experiment, we initialize with M = 2 particles sampled from the prior, and use K distinct noise samples to guide the particles backward.
21

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B.5
MoG result detail
B.5.1 Small go-back levels vs. large go-back levels
As shown in Figure 10 small go-back levels effectively guide the particle toward the goal when navigating within single region landscape. However, in the two-regions scenario (Figure LPp small go-back levels struggle to move particles across regions to reach the goal. By contrast, large go-back levels significantly improve cross-region transitions.
go-back step size 20 Cycle 5
Initial particles
Cycle
Cycle 9
Final samples
go-back step size 80 Cycle
Initial particles
Cycle
Cycle 19
Final samples
o
Figure Compare different go-back step size on task 1.
go back step size 20
Cycle
Cycle
Cycle
samples
go-back step size Cycle
particles
Cycle
Cycle
samples
go-back step [0,
Initial particles
Cycle
Cycle
Cycle
Final samples
Figure 11: Compare single and multiple go ga-back step size task 2.
22

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B.5.2 Single go-back levels vs. multiple go-back levels
After leveraging the large go-back to traverse between regions, switching to small go-back becomes advantageous for shifting from exploration to exploitation. This transition enables fine-grained goal-oriented refinement, as demonstrated in Figure II All instances begin from initial incorrect prediction, requiring additional correction during inference. go-buck = 20 fails to escape the initial cluster of modes, whereas go-buck = 80 enables transitions across regions but refines inefficiently. Our method combines coarse transitions to explore distant regions and fine adjustments for local refinement, enabling faster and more effective inference time adjustment.
B.6 OGBench Point Maze experiment detail
All methods were evaluated under the same setup, using 32 particles to ensure a fair comparison. For SoP, we used M = 4 and K = 8 in Maze Giant, and M and K" = 32 in Maze Large. BS used M = 8 and = 4 in both mazes, with lookahead estimator [17] to have better predicted & (x,) with value guidance. SMC (24] was implemented with POTENTIAL TYPE = "sum", X 0.1 and used N 32 particles in both mazes. ABCD was configured with 32,
and Maze Giant and N 32, K 1 and . 32 in Maze Large. The adaptive terminal condition was set once more than 90% of top-K particles consistently originated from the zero noise level over x consecutive steps. We used 30 in Maze Giant and k = 5 in Maze Large. All methods used base diffusion model trained with 256 denoising steps. ABCD employed jumpy denoising with a jump length of 10.
in
To scale inference compute, we increased the branching and-selection frequency p for BS and SMC. For BoN, we scaled by increasing N. For SoP, we varied the step size for back-and-forth moves, as smaller steps incur higher compute due finer-grained trajectory updates. For ABCD, we scaled by increasing the number of max itte. Inference time was measured as the average time required to generate and execute a single plan in the environment.
23

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B.7 OGBench Point Maze result
F compares the performance using an adaptive guidance scheme that increases guidance as the diffusion step approaches 0. For reference, the performance of all methods under the standard guidance scheme reported Table 3 and Table I ABCD consistently outperforms baselines, and is the only method to achieve perfect success rate within the given compute budget.
Table 3: Comparison of timereece--imee strategies in OGBench Point Maze Giant.
Inference Method
Add. Compute
Performance
Inference wall clock time (sec.)
Base Diffusion
No add
12+13
41.02
N 32
36+15
43.12
N 64
54+22
45.34
N 96
48+18
45.38
BoN
N 128
60+24
48.64
= 192
64+15
53.78
288
64+17
87.94
544
76+20
169.98
A f=200, Ab=100
40+9
59.12
A f==100, Ab=50
60+15
76.56
f-00, Ab=25
86213
88.06
A/=40, Ab=20
92+10
99.08
A A/=30, Ab=15
9449
108.84
SoP
f=200, At=10
90+13
131.90
f==10, Al=5
94+13
181.94
f Ab=3
96+12
250.06
A4=,, Ab=2
8649
342.94
Af=2, A/=1
80+0
588.98
p 50
28+20
52.52
p 25
20+13
62.48
20
20113
67.20
SMC
15
22+14
73.82
= 10
22+11
90.82
5
18+19
142.48
p 50
38+11
65.40
p 20
30+10
92.30
15
52+18
74.84
BS
10
46+16
122.08
5
50+16
239.12
=
72+16
1014.30
max iteiter-
82+11 1
12.70
max iter =
94+9
34.02
max_ iter 10
96+8
65.52
max_ite _ex_iter_ = 15
96+8
85.98
max_ iteriter = 20
9648
117.54
ABCD
max_iter = 25
9648
138.04
max_ jex_iter- = 30
98+6
184.40
max_ iter 35
100+0
202.58
max_ iter 40
100=0
214.08
max_iter -xx_tter 45
100+0
271.68
hax_iter 50
100+0
295.12
24

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Table 4: Comparison of inference-time strategies in OGBench Point Maze Large.
Inference Method
Add. Compute
Performance
Inference wall clock time (sec.)
Base Diffusion
No add
38+11
25.10
N
62+11
28.88
N
70+16
34.82
N =
8049
41.58
BoN
N = 16
88+10
39,66
N = 32
9449
41.74
N 64
98+6
40.12
A/=200, Ab=100
64+15
58.22
Af=100, Ab=50
76+12
78.36
Af=50, 2b=25
86+9
92.52
Af=40, Ab=20
80+15
94.28
Af=30, Ab=15
92+10
94.32
SoP
Af=20, Ab=10
98+6
121.12
Af=10, Ab=5
96+8
158,48
A f= At=3
96+8
214.18
f= At=2
98+6
285.60
A A!=1
94+9
323.36
p 50
8448
44.82
p
84+8
51,26
SMC
=
82+11
50.92
= 15
82+
53.08
p 10
76+15
61.40
p
74+9
81.90
p 50
88+10
58.44
P =
88+10
60.50
= 15
84+12
89.16
BS
= 10
90+13
115.64
p 5
88+10
158.30
p
9449
801.42
ABCD
32
100+0
10.52
25

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B.7.1 OGBench Point Maze visualization
Planning Trajectory
Initial particles
Cycle
Cycle 2
Figure 12: Visualization of the generated trajectories through cycle with ABCD. The figure highlighted with red border indicates the selected plan for executing the environment.
As shown in Figure 12 the initial trajectories proposed by jumpy denoising effectively capture rough path from the start to the goal, yet contain noticeable invalid segments. Across subsequent cycles, Cyclic Diffusion Search progressively refines these trajectories by focusing on problematic regions and improves them toward viable plans. The highlighted plan selected for execution (red border) showcases ABCD's capability preserve the nearly-complete solution by leveraging Automatic Degloration-EExloiaatiioo Balancing mechanism.
B.8 Sudoku experiment detail
Experiments were conducted 335 Sudoku test instances, each containing between 17 and 28 provided digits. Reported accuracy reflects the average performance across the full test set. Inference time measured the average time required generate single sample, computed by dividing the total time taken to generate all 335 outputs by 335. The base diffusion model was trained using 50 denoising steps. We follow the training configuration described in (35). Each digit from to 9 is represented as one-hot vector, normalized, and passed through the diffusion model. During inference, the model outputs continuous predictions, which are then discretized by selecting the index with the highest predicted value as the final label.
To evaluate inference-timee scaling strategies, each baseline was scaled along its primary computational axis while preserving its core algorithmic structure. Base Diffusion employed a single particle (N 1). BoN increased inference cost by evaluating models with N 40, 480, and 960 particles. BS and SoP were implemented with M and K = 10. In BS, inference-time compute was scaled by increasing the wranching-and-selertiee frequency p. SoP additionally scaled computation by reducing the step sizes for forward and backward transitions, (Ab, A SMC used N = 40 particles and scaled inference-tine tiie compute by adjusting the resampling frequency p across timesteps. Following the FKD framework (24), we set the POTENTIAL TYPE to "max" and À 100 to enforce an optimization-style behavior. ABCD (ours) was configured with N 10. K 10, and / corresponding to four non-zero go-back noise levels (10, 20, 30, 40) and one zero level. The adaptive terminal condition terminated inference when specified percentage top-N particles consistently originated from the zero noise level for к consecutive steps. We fixed max max_iter to 100 and scaled ABCD's s inference time by varying both the persistence parameter x E 21, 5, 10, 15, 20) and percentage thresholds E (0.6, 0.8, 1.0). For fast denoising, ABCD employed five total inference steps, each performing a 10-step jump in the diffusion space. Please refer Table for detailed results.
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B.9 Sudoku result
We provide detailed results Table corresponding the main Figure (idddle) presented in the main paper in this section. Additionally, Section B.9.1 visualizes the evolution of Xy predictions across refinement cycles in the Sudoku task, illustrating how different go-back temperature step sizes affect the iterative denoising trajectory.
Table 5: Comparison of various inference- strategies across compute and performance the Sudoku harder test set setting (from 17 to 28 provided entities). For each method, inference-tine compute was scaled by adjusting its primary control axis such as branching frequency, resampling rate, or particle count- coult-while preserving its core algorithmic behavior.
Inference Method
Add. Compute
Accuracy
Clock Time (sec.)
NFE
Base Diffusion
No add
0.35 + 0.1
0.28
50
N 40
0.55 0.49
0.29
2000
BoN
N 480
0.63 0.48
0.68
24000
N 960
0.66 + 0.09
1.32
48000
p
0.64 + 0.1
0.3
2000
SMC
0.70 + 0.1
0.31
2000
0.74 + 0.1
0.32
2000
5 p
0.65 + 0.46
0.32
900
BS
0.75 0.48
0.40
1500
p p=
0.84 0.08
0.54
2460
A 40, Ab 20
0.61 0.09
0.41
2940
Aʃ = 20, A
0.67 + 0.1
0.46
3420
A = 10, A = 5
0.76 + 0.1
0.51
3860
A 6, Ab = 3
0.83 + 0.11
0.52
3940
SoP
A 4, Ab 2
0.88 0.09
0.55
4220
A 2, Ab 1
0.06
0.65
4980
A = 4, Ab 3
0.955 + 0.06
0.99
8260
f 5, Ab = 4
0.958 + 0.06
1.14
9900
A 6, A6 5
0.957 = 0.06
1.33
11540
percentage 0.6,
0.82 + 0.14
0.24
1059
percentage 0.8, 1
0.90 + 0.1
0.35
1667
percentage 1.0, ==
0.95 0.06
0.45
2296
percentage 1.0, == 2
0.97 0.05
0.49
2607
ABCD
percentage 1.0. == 3
0.979 + 0.04
0.55
2827
percentage 1.0, ==
0.986 + 0.03
0.60
3205
percentage 1.0, == 10
0.75
3942
percentage 1.0, == 15
0.995
0.85
4499
percentage 1.0, == 20
0.99
0.92
5013
27

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B.9.1 Sudoku visualization showing that proper go-back is necessary
P
1
L
B
1
2
S
B
P
L
2
9
6
1
7
E
9
2
2
E
2
22
E
E
E
6
S 3
2
Answer
2 E
G
9
2
L 9
9
2
E
T
5
R
2
R
6
9
S
S
9
6
9
E
6
E
6
R
R
9
P
LE
6
2
T
9
6
S
2
9
9
Z
9
E
2
6
2
6
6
II
L
5
6
5
3
P
R
5
3
ACC-
Cycle=0 Acc: 27/59
6
2
2
9
S
2
T
C
3
2
5
7
9/55
6510
5
5
9
9
5
G
E
W
9
5
4 X
6 1
T
8
5
6
5 8
6
5
5
2
G
5
5
6
2
4
Acc: 32/59
Cycle=10
2
3
L
3
5
65/00
8
5
8
-
S
9
0
6 7
6 2
9
-
5
9
9
6
5
2
ACC-
2
4
3
Acc: 36/59
Cycle=22 20
2
3
5
6
5
5
3
4
2
5
6
5
3
7
9
6
5
3
9
2
6
2
5
6
5
9
2
P
2
5
5
6
5
A20
2
5
ACC-
Acc: 36/59
Cycle=30
2
2
5
9
9
2
3
2
5
58
6
5
5
5
65
5
4
2
9
9
5
5
5
3
1
9
R
5
S
6
S
5
4
C
ACCC
2
Acc: 36/59
Cycle=40
2
9
2
S
6
9
6
3
37//
S
50/5
8
S
R
6
Figure 13: Visualization of predictions across cycles for four go go-back step sizes. The rightmost column corresponds go-back 10, followed by 20, 30, and 40 in ascending order from right to left.
28

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As shown in the visualization results for Sudoku in Figure 13 when the go-back noise level is 10, the zo prediction remains unchanged despite repeated iterations failing to fix incorrect predictions. In contrast, with noise level of 40, the zo prediction appears to update randomly at each iteration altering even the correct ones, as if generating entirely new guesses without accumulating information from previous predictions. For noise levels 20 and 30 e0-especially 20-we observed meaningful performance improvement across iterations, where incorrect predictions were corrected while correct ones were unchanged. These findings confirm that appropriately selecting the go-back noise letcl-both per task and per instance-is is crucial for maximizing the benefits of inference-time iterative refinement.
B.10 Pixel Maze experiment details
Experiments were conducted on 100 Pixel Maze test instances for each maze size (from 11 15). The reported success rate reflects the proportion of cases in which the generated trajectory successfully reaches the goal without crossing any walls. Inference time was measured as the average time required to generate single sample, computed by dividing the total generation time for 100 samples by 100. The base diffusion model was pretrained using = 50 denoising steps, We follow the training configuration described in [35]. Both the maze layout (with walls encoded I and empty spaces as 0) and the start/goal positions are represented as one-hot vectors and normalized before being passed into the diffusion model. During inference, the model produces continuous predictions, which are discretized by selecting the index with the highest value at each position, yielding the final binary path representation.
To assess inference- time scaling strategies, each baseline was sealed along its primary computational axis while preserving its core algorithmic behavior. Base Diffusion employed single particle (N 1). BoN increased inference cost by using a larger number of particles N. BS and SoP were implemented with M = 4 and K 10. In BS, inference-time compute was scaled by increasing the branching and selection frequency p. In SoP, additional scaling was achieved by reducing the step sizes for forward and backward transitions, (Ab, f SMC used N 40 particles and scaled inference-time time compute by varying the resampling frequency over steps. Following FKD [24), we used POTENTIAL TYPE "max" and set X 100 to emphasize optimization behavior. ABCD (ours) was configured with N = 10, = 10, and J = 5, corresponding to four non-zero go-back noise levels (10, 20, 30, 40) and zero level. The adaptive terminal condition terminated inference once a specified proportion of top-N particles consistently originated from the zero noise level for k consecutive iterations. We scaled ABCD's inference time by varying K E 2,, 2,3 and percentage thresholds E {0.1, 0.2, 0.3, 0.6, 1.0). To ensure computational efficiency, ABCD employed ten denoising steps, each executing 5-step jump in the diffusion process. Please refer to Table 6 for detailed results.
29

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B.11 Pixel Maze result
We provide detailed results corresponding to the main Figure Left) presented in the main paper in Section 11.11.I. and further demonstrate that the varying inference-time behavior-ppeviione previously observed in the Sudoku setting also holds the Pixel Maze setting, as discussed in Section 1.11.2
B.11.1 Result table for Pixel Maze size = 15
This section provides breakdown of the scaling factors we used, the performances and corresponding inference times for each method in Table
Table 6: Comparison of various inference-time strategies across compute and performance in the Pixel Maze setting (size 15). For method, inference-time compute was scaled by adjusting primary control axis such as branching frequency, resampling rate, or particle count- count-while preserving its core algorithmic behavior.
Inference Method
Add. Compute
Success rate
clock Time (sec.)
NFE
Base Diffusion
No add
0.02 + 0.13
0.2
50
N 20
0.17 + 0.37
1.87
1000
N 40
0.2 + 0.4
2.67
2000
BoN
N 80
0.24 + 0.42
5.46
4000
N = 160
0.34 + 0.47
10.52
8000
N = 200
0.38 # 0.48
13.0
10000
25
0.2 # 0.4
3.21
2000
10
0.28 = 0.44
3.61
2000
SMC
p
0.32 # 0.46
4.58
2000
p
0.39 + 0.48
6.58
2000
0.36 # 0.48
10.31
2000
p =
0.25 + 0.43
2.69
1020
BS
p= 3
0.32 + 0,46
4.11
1180
2 p ==
0.39 # 0.48
5.78
1500
0.61 + 0.48
10.79
2460
A 40, Ab 20
0.25 + 0.43
4.05
2940
A 20, Ab
0.54 0.49
4.98
3420
A = 10, Ab =
0.76
6.35
3860
SoP
A 8, Ab 4
0.35
6.69
3900
A 6, Ab 3
0.91
7.08
3940
A 4, A6 2
0.21
8.43
4220
A = 2, Ab 1
0.98 0.13
12.42
4980
percentage 0.1, K
0.17 + 0.37
0.65
310
percentage 0.2, ==
0.44 # 0.49
1.22
562
ABCD
percentage 0.3, == I
0.96 0.19
2.75
1226
percentage 1.0, k=
0.99 0.09
3.11
1400
percentage 1.0, 3=3
1.0
3.75
1854
30

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B.11.2 Investigation on automatic thinking assignment across different difficulty levels
The Figure 15 visualize how our method allocates varying amounts of inference iterations and time depending on task difficulty for Pixel Maze. It demonstrates that, similar to Sudoku, the model adaptively assigns greater computational effort to larger and more complex mazes, confirming the generality adaptive inference-time allocation across domains.
Thinking iteration
Thinking time
Thinking Iterations
Thinking Time
Fg 20
Tim Sh
mop 20
1ng. 3eg
802 -3
Provided Numbers
Provided Numbers
Maze
Size
Figure Thinking iterations and inference time in Sudoku. Left: Mean and standard deviation of iter- ation counts across varying numbers of provided digits. Right: Mean and standard deviation of per- instance inference time. The model adapts computation dy- namically, allocating more iterations harder puzzles with fewer initial clues.
Figure 15: Thinking itcrations and inference time in Pixel Maze. Left: Mean and standard deviation of iteration counts across different maze sizes. Right: Mean and standard deviation of per-sample inference time. Larger mazes require more search, the model allocates compute accordingly, demonstrating instance-aware inference adaptation.
B.12 Molecular 3D structure prediction task experiment detail
Experiments were conducted 100 molecules randomly sampled from the QM9 test dataset. For all methods, we utilized the same checkpoint of diffusion model from the official EDM repository, where T' = 1000. Following [II] we compared molecule stability (the proportion of generated molecules for which all atoms are stable). We also measured the average inference time required for a single sample.
To compare inference-time scaling strategies, each baseline was scaled along its primary compu- tational axis while preserving its core algorithmic behavior. Base Diffusion employed a single particle (N = 1). BoN was evaluated with using more to increase inference cost. BS and SoP were implemented with M 4 and K = 10, and inference-time compute was scaled by increasing the branching and selection frequency. Specifically, for BS, we scale inference- compute by adjusting the branching and-selection frequency p. SoP additionally scaled computation by reducing the step sizes for Back-and-fortd moves A and Ab. SMC used N 40 particles and increased inference-time compute by adjusting the resampling frequency p over different steps. Following the FKD (2), we use POTENTIAL TYPE = "max" and = 100 to make it as optimization. ABCD (ours) was configured with N 10, K 10, and J 5, corresponding to four non-zero go-back noise levels (50, 100, 200, 400) and a zero level. The adaptive terminal condition halted inference specified fraction of top-N particles repeatedly originated from the zero noise level for consecutive steps. We scaled ABCD's inference time by varying both E {1,2) and percentage thresholds (0.4,0.5), To ensure computational efficiency, ABCD employed 100 denoising steps, each executing a 10-step jump in the diffusion process.
31

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B.13 Molecular 3D structure prediction task experiment result
We provide detailed results corresponding the main Figure (Right), with breakdown of the scaling factors we used in Table 7
Table 7: Comparison of various inference-time time strategies across compute and performance in the QM9 setting.
Inference Method
Add. Compute
Success rate
Clock Time (sec.)
NFE
Base Diffusion
No add
0.74
4.07
1001
N = 40
0.86
17.33
40040
BoN
N 80
0.86
35.58
80080
N 160
0.89
70.28
160160
p
0.88
24.17
48040
SMC
p
0.85
35.18
60040
==
0.9
52.29
80010
p =
0.82
24.17
24010
BS
== 2
0.83
37.95
45010
5
0.91
69.95
80010
A 50, A 25
0.9
32.21
73690
SoP
f 100, Ab 50
0.92
40.06
74410
f 200, Ab 100
94
49.84
75850
percentage 0.4, K
0.93
4.35
7482
ABCD
percentage 0.4, K = 2
0.97
23.09
43009
percentage 0.5, k ==
0.99
46.45
82010
B.14
Text-to-image generation experiment detail
We adopt Stable Diffusion v1.5, a widely used text-to-image diffusion model, as our pre-trained model. we comprehensively assessed ABCD's performance using three metrics: compressibility, measured as negative JPEG file size (in kilobytes) after compression following [2]: aesthetic evaluation, computed using LAION's V2 Aesthetic Predictor [22]. linear MLP built upon CLIP embeddings, trained on over 400,000 human ratings; and human preference evaluation, based on the HPSv2 scorer [33], CLIP model fine-tuned on 798,090 compressibility and aesthetic evaluations, we used animal category prompts from 2 (e.g., Dog, Cat, Panda), whereas for human preference evaluation we used human instruction prompts from (33.
selected rankings across 433, 760 image pairs. For
Each inference method is evaluated by generating 12 images. Inference time is calculated as the average duration produce one image. To analyze inference-time scaling, each baseline method was scaled by adjusting its primary computational parameters without altering its fundamental algorithmic design. Base Diffusion utilized single particle
1). BoN scaled inference by incrementing particle count from N = up to N 12. SoP increased computational cost through smaller incremental steps during both forward and backward transitions, denoted by (Af, Ab). Our proposed ABCD method employed N = 4, K 1, and J = 4, translating to three non-zero noise levels (1, 301, 621) for backward transitions and one zero-noise level. The adaptive stopping criterion halted inference when the top-K particles persistently emerged from the zero noise level for x = 30 consecutive steps. ABCD's inference scaling was achieved by varying the number of max_iter. For fast denoising, ABCD employed 50 inference steps with each step covering 20-step jump in the diffusion space. Detailed results are provided in the next section B.15
32

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B.15 Text-to-image generation experiment result
We present detailed results related Figure along with a breakdown the scaling factors listed in Table Table 9 and Table For each method, inference- compute was scaled by adjusting its primary control axis while preserving its core algorithmic behavior.
Table 8: Comparison of various inference-ting strategics across compute and image compressibility in the Text-to-image generation task.
Inference Method
Add. Compute
Compressibility
Clock Time (sec.)
NFEs
Base Diffusion
No add
-100.07 + 2.88
58.42
2000
=
-91.59 + 4.43
109.25
4000
N
-83.70 + 7.60
150.17
6000
=
-81.89 + 5.71
200.00
8000
=
-81.89 + 5.71
257.58
10000
= 6
-81.64 + 5.92
292.75
12000
BoN
N N
-78.21 + 8.68
348.33
14000
=
-78.22 + 8.69
403.00
16000
N N
-78.00 8.40
444.58
18000
N = 10
-77.86 + 8.42
494.33
20000
N 11
77.53 8.77
556.33
22000
N
77.02 8.88
583.25
24000
AF 510, Ab 255
88.36 + 4.88
188.75
7548
A 410, Ab 205
85.32 3.21
215.50
8490
SoP
AF 350, Ao 175
80.82 + 4.14
227.92
9192
A 280, Ab 140
-22.64 + 5.61
257.67
10266
A 220, Ab 110
-67.79 + 4.65
271.25
10710
A 180, Ab 90
64.59 3.43
280.25
10974
max_ited
74.91 + 5.58
41.83
1380
max_ites iiter = 20
667.19 + 5.84
73.67
2360
max_iter = 30
-6233 + 4.33
103.67
3340
ABCD
max_iter- 40
-59.01 + 4.34
130.50
4320
max_ite 50
-56.08 + 4.02
163.67
5300
bax_ited = 80
-51.49 + 4.34
254.50
8240
max_iters = 100
-49.17 + 4.29
306.08
10176
33

---

Table 9: Comparison of various inference-time strategies compule and aesthetic the Text-to-image generation task.
Inference Method
Add. Compute
Image Aesthetic
Clock time (sec.)
NFEs
Base Diffusion
No add
5.57 = 0.03
58.83
2000
N =
5.63 0.08
109.75
4000
N =
5.72 = 0.09
150.92
6000
N 4
5.75 0.07
200.58
8000
N
5.79 0.05
258.33
10000
N =
5.81 0.02
293.42
12000
BoN
N =
5.81 0.02
349.00
14000
N =
5.82 = 0.03
403.50
16000
N
5.86 0.05
445.25
18000
N 10
5.86 0.05
494.25
20000
N = 11
5.87 = 0.04
556.75
22000
N = 12
5.90
583.17
24000
A = 510, A 255
5.68 = 0.03
189.42
7548
A f 400, A 200
5.78 0.03
206.83
9636
A f 280, Ab 140
5.84 0.08
258.67
10266
SoP
AS 170, A 85
5.87 0.04
285.08
10704
Á = 130, A 65
5.93 - 0.08
301.42
11232
A 110, A 55
5.98 0.02
313.67
11478
max_ eax_iter= = 10
5.95 + 0.06
40.00
1380
max_iter 20
6.08 0.06
72.17
2360
max_iter: 30
6.13 0.09
105,42
3340
ABCD
max max_ites iter
6.15 0.08
132.00
4254
max_ited ittr = 60
6.18 + 0.05
178.75
6043
max max_iter iter 80
6.23 # 0.07
233.83
7677
max_ites 100
6.25 0.09
288.42
9147
34

---

Table 10: Comparison of various inference-time stralegies compute and human preference in the Text-to-image generation task.
Inference Method
Add. Compute
Human Preference
Clock time (sec.)
NFEs
Base Diffusion
No add
0.2710 + 0.0028
61.00
2000
=
0.2749 + 0.0013
112.25
4000
N = 3
0.2752 = 0.0017
153.00
6000
BoN
N = 4
0.2752 + 0.0018
203.08
8000
5
0.2753 + 0.0017
260.92
10000
A 510, Ab 255
0.2725 + 0.0031
192.83
7548
A 410, Ab 205
0.2728 + 0.0030
221.00
8490
A 350, Ab 175
0,2739 + 0.0044
233.00
9192
SoP
A f 230, A6 115
0.2751 + 0.0029
272.25
10218
f 130, Δb 65
0.2765 + 0.0046
811.25
11232
A 70, A6 35
0.2777 = 0.0039
335,42
11778
A 20, A
0.2787 + 0.0027
431.42
13056
max_ iter =
0.2776 + 0.0027
46.83
1380
max iter = 20
0,2791 + 0.0021
82.83
2360
max_iter = 30
0.2797 = 0.0023
115.92
3340
max_iter- 40
0.2804 + 0.0027
142.17
4320
ABCD
max_ites 50
0.2808 = 0.0028
183.42
5300
max_ _ex_iter= = 80
0,2815 = 0.0030
272.08
8060
max_iter = 100
0.2819 = 0.0032
322.08
9718
max_iter = 200
0.2823 + 0.0033
432.92
12716
35

---

B.15.1
Visualization of generated images
In this section, we show more generated samples. Figures T6 17 and [8 compare outputs from baseline approaches and ABCD with respect to compressibility, aesthetic score, and human preference score, respectively.
Butterfly
Cat
Monkey
Horse
Rabbit
Dog
Base Diffusion
-95.34
-124.01
-86.57
-99.71
-95.85
-115.38
BoN
SoP
ABCD
66.25
-55.66
-6140
-55.55
-64.77
-5.000
Figure 16: Visual comparison images produced by different inference-time strategies optimizing compressibility reward. Numbers at the bottom of each image show the corresponding compressibility scores. Base Diffusion produces photorealistic textures and complex hackgrounds, but at the cost of very large file sizes. BoN occasionally yields simpler backgrounds but still leaves lot of high-frequency detail. SoP further flatten color gradients, shaving further and smoother regions- e.g. the Cat is rendered with minimal fur grain, the Horse with a crisp silhouette against a solid tone.
size. Under ABCD, each image sports a clean, uncluttered background
Butterfly
Cat
Monkey
Horse
Rabbit
Dog
Diffusion
6.22
BoN
5.74
5,79
6.21
6.22
5.77
5.82
SoP
5,76
5,60
5.96
6
6.01
5.95
ABCD
6.57
5.21
6.23
5.85
6.04
Figure Visual comparison images produced by different inference-time strategies optimizing aesthetic score reward. Numbers at the bottom of each image show the corresponding aesthetic scores.
36

---

poir planes parkad small airfildd
cars city street with people
dog plate on the ground
young man riding through top skateboard.
person stringg motorcycle grass.
a generic public bathroam
dog
Base Diffusion
0.2567
0.2626
0,2612
0:2584
0.2891
0.3009
BoN
0.2677
SoP
0,2648
0.2813
0.2755
0.2855
ABCD
0.2755
0,2650
0.2889
0,2700
0.2999
0.5040
Figure 18: Visual comparison images produced by different inference-time strategies optimizing human preference score reward. Numbers at the bottom of each image show the corresponding human preference scores.
B.16 Comparison with respect to number of function evaluations of the diffusion model
In this section, we present a more detailed and fair comparison with baselines by evaluating not only the wall-clock time but also the Number of Function Evaluations (NFEs) during inference across tasks. The NFE measures how many times the pretrained diffusion function is called, and to isolate pure computational cost from parallelization effects, count one function evaluation per particle ABCD demonstrates superior inference-time scaling compared to other baselines, consistent with wall-clock time results.
per step. As shown in Figure 19 20
Base Diffusion
BoN
SMC
85
SeP
ABCD (Ours)
Sudoku Harder set
Pixel Maze Size 15
QM9
1.00
Melecule Stability
08
0.8
0.55
Rocuracy
06-
0.4
02
0.75
0.0
5c
126
13k
20c
25k
0.0
22
35
3x
0.70
25k
50k
75x.
100k
123k
250k
Inference Compure (NFF)
Inference Compune (NFF)
Inference Compute (NFF)
Figure 19: Comparison wart Left: Mean accuracy on Sudoku Harder dataset (17-28 entities provided). Middle: Pixel maze path finding result. (Size 15). Success rate on the OOD size-15 15 pixel maze test set. Right: Molecular 3D structure prediction task result. Molecular Stability rate on the QM9 dataset.
Base Diffusion
BoN
SoP
(Ours)
Preference Score:
0.202
Compressibilit
Aesthetic
8.0
8.78
6374
-90
-100
5000
10000
15000
20000
25000
5000
10000
15000
20000
25050
2000
4000
6000
8020
10500 12000
Inference Compute (NFE)
Inferance Compute (NFE)
Inferance Compute (NFEI
Figure Comparison w.nt NFES on Text-to-Image generation task. Text-to-image generation result. Average reward with respect to compressibility, aesthetic score and human preference score.
37

---

C Ablation studies
C.1
Investigate on existance of inference time scalable go-back noise level per task
OGBench
Sudoku
Pixel Maze
1.0
1.0
1.0
Performance
0.8
0.9
0.8
10
0.6
0B-b4
0.8
0.6
6B-128
0.7
0.4
0B-163
0B-192
0.2
0.2
0B-224
5
10
25
30
15 20 25 30 35 40 45 Iteration Number
55
10
15
20
25
Iteration Number
Iteration Number
Figure 21: Existence Inference- Time Scalable go-back Noise Level per Task. Left: OGBench Point Maze Giant. Middle: Sudoku harder dataset(17 to 34 provided). Right: Pixel Maze (Size 15). Performance across iterations for different go-buck noise levels. Each task exhibits distinct optimal go-buck level, demonstrating the necessity of spec-specific noise-level selection for effective inference-time scaling.
Existence of inference time scalable go-back noise level per tasks. investigate the impact of the go-back noise level on inference-time scaling, we conducted an additional study across OGBench Point Maze, Sudoku, and Pixel Maze. As shown in Figure [T] the optimal go-back noise civeel-expresscee as fraction of the total denoising steps-varies significantly across tasks: approximately 3/4 for OGBench, 2/5 for Sudoku, and 4/5 for Pixel Maze. These results highlight that selecting appropriate go-back step size is critical for effective iterative refinement during inference time.
C.2 Investigate on different configuration of Adaptive Thinking Time
As the terminal condition becomes more stringent, both performance and inference cost increase. To better understand how different computational budgets affect final performance, we added ablation study analyzing the core components of ABCD ABCDDAdaptive 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.
-under
In the Sudoku task, We focused on the Adaptive Thinking Time mechanism by varying the termination percentage and the value of K, two key hyperparameters that define ABCD's adaptive terminal condition. As shown in the two tables [2 increasing K 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.
Table 11: ABCD ABCD-Sudon Sudoku (Changing s). We report
Table 12: ABCD Sudoku (Changing percentage).
accuracy, wall-clock time, and NFEs under different We report accuracy, wall-clock time, and NFEs under values.
different percentage values.
Metric
1
R
5
10
Metric
perc=0.4
perc=0.6
perc=0.8
Accuracy 0.958
0,986
0.994
Accuracy
0.698
0.821
0.903
Time 0.443
0.608
0.754
Time (s)
0.112
0.210
0.329
NFEs
2296
3205
3942
NFEs
520
1059
1667
In the text-to-image generation task, we analyzed the Automatic Exploration-Explooiaatoon Balancing mechanism by varying the granularity of the temperature pool-speciffiallyy the number of go- back temperatures included in the pool. We fixed the number of particles and terminal conditions (max_iter=50) x = 30) and varied the temperature pool size. As shown in the two tables [3 14 using 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.
38

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Table More diverse temperature values on Compressibility score. We report Compressibility score, wall-clock time, and NFEs under different tem- perature pool size.
Table diverse temperature values on Aes-
thetic score. We report Aesthetic score, wall-clock
time. and NFEs under different lemperature pool size.
T size
Aesth
Time (s)
NFEs
T size
Comp
Time (s)
NFEs
6,23=0.05
237.9
8200
15.1
233.3
8188
6.31=0.04
286.5
10369
-48.8+2.1
285.8
10500
4
6.3620.14
354.6
12578
-46. 142.1
356.5
12800
C.3
Comparison with more intelligent go-back strategy
Since amount of unnecessary compute is being performed by distributing each sample uniformly along the noise levels, we explored a more intelligent go-back doesn't show meaningful improvement. Specifically, redistributed to different noise levels in proportion to the average reward observed at those levels across the population. However, as shown in the table despite its intuitive appeal, the performance improvement was marginal compared to the added complexity. Therefore, we ultimately chose simpler and more efficient formulation for the current version of ABCD. Nevertheless, believe that incorporating smarter stategies-sun such as applying ideas from SMC to the temperature pool could be a promising direction.
ABCI> found tested strategy where particles were
Table 15: Comparison between Adaptive ABCD and ABCD. We report accuracy and wall-clod time under different compute configurations.
Method
Add. Compute
Accuracy
Wall Clock Time (8)
Adaptive ABCD
pere=0.6, x=1
0.810
0.220
perc=0.8, x=1
0.895
0.395
perc=1.0,
0.951
0.489
pere=1.0, x=3
0.979
0.549
ABCD
perc=0.6, k=1
0.821
0.240
pere=0,8, x=1
0.903
0.350
perc= x=1
0.958
0.450
perc=1.0, x=3
0.979
0.557
39

---

C.4 Investigate on the diversity preservation of ABCD
We measured diversity using the average pairwise cosine similarity CLIP embeddings, where lower score reflects greater variability in outputs and broader exploration of the data space. While optimization naturally reduces some diversity, our method successfully preserves the underlying multi-modality of the pretrained model while steering toward high-reward regions.
Table 16: Comparison of diversity scores across Compressibility, Aesthetic, and HPS tasks. We report the diversity mean + std under different compute configurations.
Method
Add compute
Compressibility
Acsthetic
HPS
Base Diffusion
0.2957=000481
0.2957=#0.0481
0,4559=0.0215
BoN
N
0.2899=000010
0.2870=0.0002
0,4424=0.0095
N=10
0.27811000184
0.291910.0285
SoP
Af -50510, Ab=255
0.2795+000375
0.2912+t0.030S
0.4677=000227
= A4=140
0.297510.0512
0.300910..019
Af=230, Ab=115
0.4508=0.0152
A 18=180 Ab=90
0.279810.0154
17=1 Ab=85
0.2672++0.0223
A/=130, Ab=65
0.441710.0054
ABCD
iter=30
0.312810.0084
0..04810.0221
0.467211.0160
max_iter=80
0.3162++.0028
0.2082+0.0222
0.4543=0.0143
max_iter=100
0.3037++00011
0.2880+0.0022
0.4555=0.0167
40

---

D Prediction visualization
In this section, we provide illustrative visualizations instance-level predictions generated by our model to facilitate intuitive understanding. Figure 22 displays representative outputs from both the Pixel Maze and Sudoku tasks.
Pixel Maze
Maze size = 15
Maze size 13
Maze size = 11
Sudoku
32 Provided
17 Provided
20 Provided
3
7
4
8
1
2
1 2
9 6 4
8
3
9
6
4 2 8
1
2
5
3
9
6
4
5 4
3
1
9
7
2
5
3
9
2
8
9
5
3
8
1
1
6
2
9
3
1
7
5
1
6
5
7
7
5
4
9
2
8
6
6 8
2
5
3
9
4
3
2
9
1
4
6
3
7
1
2
6
4
8
2 6
7
5
3
6
7
3
9
8
9
2
4
7
5
6
8
2
9
4
9
5
1
7
2
3
1
8
5
Figure 22: This figure presents the final outputs obtained by our ABCD method on both the Pixel Maze and Sudoku tasks. The visualizations demonstrate that, across range of maze sizes and Sudoku difficulty levels, ABCD produces accurate predictions, particularly in instances that demand greater computational effort.
41

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E
Termination guarantee of the Adaptive Thinking Time
In this section, we present detailed proof establishing the termination guarantee the Adaptive Thinking Time algorithm introduced in Section 3.3
E.1
Formal statement
Let
T= (t.....,tt
be the "temperature pool" with lowest temperature O. At cycle let D&(c) be the multiset of of the top-k particles after denoising. Define the stopping rule: terminate the first cycle N such that
Vi (N - x N N-1,NH, (a(1) =
ie. all top-k particles originate from 1 for к consecutive cycles.
Assumptions
1. Bounded reward. There is reward function r: R + R with finite supremum r* <
2. Monotonic selection. Let
(c) = min (rewards of top-k at cycle c).
Then rkk (c) is non-decreasing in c.
3. Attainment. There exists at least one state x" such that r(=")(())))))) =
Theorem 1. Under these assumptions, ABCD's 's Adaptive Thinking Time criterion triggers termination in finite time.
The proof of this theorem is built upon several key lemmas, which we will now introduce and prove.
E.2
Uniform Hit-Chance
Lemma E.1. Let x² be any maximizer of the reward, r(x*) = r² Under the ABCD dynamics (Algorithm 7) there is constant >> such that for each cycle c,
Pr(Ee Fo-1) 2 P.
where Eg "x" appears among the selected top-K at cycle c" and Fo- is the d-algebra of all randomness up to (but not including) cycle
Proof.
1. At the start of cycle we have anchor particles at 0. We replicate each J times and send each of those KJ copies through the forward noising process g( ap) at every i' T. Exactly K of those replicas go the highest temperature t'm
2. Conditioned on F. 1, the noising and denoising of these KJ particles is independent of the past and identically distributed across cycles. In particular, the K samples at Im are fresh each time. Let
= Pr (noising at tm) then (denoise) yields æ²].
By full support of the forward kernel and nonzero recovery probability of the denoiser, q 0.
3. Those K draws at tm are independent, so the chance none equals x² is (1 g)x Hence
p = 1 (1 -0- 0.
4. If any replicate equals x", its reward is r", guaranteeing xt is in the top- K. Thus for every history,
Pr Fa-1) = Pr(some -eplliia = FF-11) 2 p.
This completes the proof of the uniform hit-chance lemma.
42

---

E.3
Optimality
Lemma E.2. Under the assumptions
1. Monotonicity and Boundedness: Ti (e) is non-decreasing and rx(c) S < 0o, so rc(c)
Teo rr"
2. Supremum Attained: 3x* with r(x") =r"
3. Uniform Hit-Chance: at each cycle e, Pr(Ee past) 2 p > 0.
Then
(e[Too=11 = 1.
Proof.
1. Define the "hit" event Eg as above. By assumption 3 E Pr(E,
past) >
E. p = co.
2. By the conditional Borel-Cantelli lemma 22 Pr(E. io.) 1, so infinitely many hits occur almost surely.
3. Let C be the first cycle where Ec occurs. Then
rx(C) 2 r(x*)= r rx(C) Sr", = rx(CC) = r"
4. Monotonicity then implies ri((c) = r* for all 2 C, Too =
Hence Pr(m.0 = r"] =1
E.4 Finite-To Vinite-Time Hitting
Lemma E.3, Under the Uniform Hit- Chance lemma, define
= -nf c 2 x appears among the top- K at cycle c)-
Then
Pr(T < oo) =1, Pr(T > S p)" VnEN,
where >> O is the per-cycle success lower bound.
Proof.
1. Fallure-to-hit bound.
Let
F
E
E. {cycle hits x*}.
c=1
By the Uniform Hit-Chance lemma,
Pr(Ee Fo-1) 22
Pr(Eg Fo-1) < p.
Hence, by the tower property,
Pr(F.) Pr(Efnn--. E Er Fイ 1 S (1 p) Pr(Fu-1).
and by induction
Pr(F,) < ".
2. Hitting in finite time w.p.1. Since (T > n) = Fn. we get
Pr(T> > n) S (1 - p)" n-000 0,
so Pr(TT = xx) = and hence Pr(T < x:) = 1.
43

---

E,5 Proof of Main Theorem
We now assemble the results from the lemmas to prove Theorem
1. Monotone convergence of ri (c). The sequence r& (l) ... is non-decreasing and bounded above by r", hence it converges to some limit Toc S r".
2. Limit must equal r". According to Lemma E.2 Too
r"
3. Optimal is reached in finite time. By Lemma E.3
(c(C) hits r" some finite cycle
4. Once optimal is reached, only ti-rrggins survive. At cycle C where ra (C((C) r", any particle noised at temperatures > and then denoised back cannot exceed r" Thus the only way preserve the top-k set of reward-r* particles is via replicas from leave them unchanged. Consequently, for all e 2 C, Dx(c) = (0.0..... (0.0.....,);
which
5. k-persistence triggers termination. Therefore, once cycle C' is reached, the condition "all top-k from temperature 0 for consecutive eycles" is satisfied by cycle C + and ABCD terminates.
44


Reviews:
[{"review_id": "0HOPlk8csY", "reviewer": "Reviewer_D4Gw", "summary": "The paper tackles the problem of inference-time optimization in diffusion models when a predefined reward function is available. The authors propose Adaptive Bi-directional Cyclic Diffusion (ABCD), a search-based inference framework composed of three main steps.\nFirst is the Cyclic Diffusion Search: $N$ particles are denoised from pure Gaussian noise, evaluated using the reward function, and the top-$K$ are selected. These are then re-noised, and the denoise-rank-renoise cycle is repeated. To adaptively balance exploration and exploitation, the method re-noises to multiple noise levels rather than a fixed one, with higher noise levels corresponding to more exploratory search and the lower noise levels corresponding to local refinement. The cycle terminates once all top-$K$ samples are obtained from low renoised samples, indicating local convergence.\nThe authors evaluate ABCD on several reasoning and planning tasks, including Sudoku and maze solving, showing impressive  improvements over previous methods.", "questions": "* How many denoising steps are used at each of the noise levels $\\{t_1, t_2, \\dots, t_M\\}$? Is this a constant fixed amount or does it change based on the noise level?\n* 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.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n* The paper is very well written and easy to follow. \n* The proposed approach is simple, intuitive, and quite elegant. \n* The experimental improvements are impressive, with the proposed approach improving on both speed and quality (success rate). \n\n**Weaknesses:**\nThe paper is an extremely good one, and the following weaknesses are more nitpicks rather than actual weaknesses. \n* All experiments are done on relatively low-dimensional data. As pointed out by the authors, it remains unclear how well the approach would scale to high-dimensional outputs, such as high-resolution text-to-image generation. \n* One of the main concepts of the paper is using multiple noise levels during the renoising step, discussed in 3.2. However, denoising higher noise levels requires more function evaluations of the base model than lower noise levels. As a result, during the end of the search process, where only local refinement is needed, some amount of unnecessary compute is being performed by distributing each sample uniformly along the $M$ noise levels.", "was_revised": true, "final_justification": "The paper proposes an elegant test-time scaling approach for improving diffusion model sampling and evaluate their approach on several puzzle/reasoning/maze benchmarks with impressive results. I recommend accepting this paper.", "rebuttal": "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: \n\n### **[Weakness 1] It is unclear how well your method would scale to high-dimensional outputs, such as high-resolution text-to-image generation?**\nWe 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.\n### **[Weakness 2] Some amount of unnecessary compute is being performed by distributing each sample uniformly along the noise levels:**\nThank 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.**\n\nSpecifically, 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. \n\n- **Adaptive ABCD vs. ABCD in Sudoku** \n|Method|Add. Compute|Accuracy|Wall Clock Time (sec.)|\n|-------|----------|----------|---------------------|\n|**Adaptive ABCD**|perc=0.6,𝜅=1|0.810|0.220|\n||perc=0.8,𝜅=1|0.895|0.395|\n||perc=1.0,𝜅=1|0.951|0.489|\n||perc=1.0,𝜅=3|0.979|0.549|\n|**ABCD**|perc=0.6,𝜅=1|0.821|0.240|\n||perc=0.8,𝜅=1|0.903|0.350|\n||perc=1.0,𝜅=1|0.958|0.450|\n||perc=1.0,𝜅=3|0.979|0.557|\n### **[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?**\n**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.\n\n### **[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:**\n**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.\n\nDue 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.**\n\nHowever, 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."}, {"review_id": "rcm6MxnUbx", "reviewer": "Reviewer_WZvc", "summary": "The paper proposes adaptive bidirectional Cyclic Diffusion (ABCD), a search-based inference framework for adaptive inference-time scaling for diffusion models with respect to the available computation budget. The authors identified fixed denoising schedules as a bottleneck for inference time scaling, and proposed a renoising and denoising schedule. Experiment results show good performance on a variety of tasks including Sudoku, Pixel Maze, Molecule Generation and OGBench Point Maze", "questions": "Would the method work well for tasks such as text-to-image or text-to-video that diffusion models are widely applied to?", "limitations": "n/a", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "(+) the idea of Adaptive Inference-Time Scaling and the ABCD frame, to my knowledge, is novel and good first explorations to this direction and can inspire future work\n\n(+) Experiments were conducted in a comprehensive set of tasks including Sudoku, Pixel Maze, Molecule Generation and OGBench Point Maze and all show promising results\n\n(+) ablations seems good to me \n\n(-) The experiments and proposed method are mostly experimental instead of theoretical. Not much theoretical justification", "was_revised": false, "final_justification": "", "rebuttal": "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:\n### **[Weakness 1] the lack of theoretical justification:**\nWe 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.\n\nMotivated 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.\n\n### **I. Formal statement**\nLet T = {t₁, …, tₘ}  be the “temperature pool” with lowest temperature t₁ = 0.\nAt cycle c, let Dₖ(c) be the multiset of origin‐temperatures of the top‑k particles after denoising.  \n**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.\n\n---\n**Assumptions**\n\n1. **Bounded reward.** There is a reward function r: ℝᵈ → ℝ with finite supremum r* < ∞.\n2. **Monotonic selection.** Let rₖ(c) = min{rewards of top‑k at cycle c}. Then rₖ(c) is non‐decreasing in c.\n3. **Attainment.** There exists at least one state x* such that r(x*) = r*.\n\n---\n**Theorem (Termination guarantee).** Under these assumptions, ABCD’s Adaptive Thinking Time criterion triggers termination in finite time.\n\n---\nThe 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.\n\n**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.\n\n**Lemma 2 (Optimality).**  Under the assumptions:  \n\n1. *Monotonicity and Boundedness:* rₖ(c) is non‑decreasing and rₖ(c) ≤ r* < ∞, so rₖ(c) → r_{∞} ≤ r*.  \n2. *Supremum Attained:* there exists x* with r(x*) = r*.  \n3. *Uniform Hit‑Chance:* at each cycle c, Pr(E_c | past) ≥ p > 0.  \nThen Pr[r_{∞} = r*] = 1.\n\n**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.\n\n---\n### **II. Proof of Main Theorem**\n\nWe now assemble the results from the lemmas to prove the termination guarantee theorem.\n\n1. **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*.\n2. **Limit must equal** r*. According to the Optimality Lemma, r_{∞} = r*.\n3. **Optimal is reached in finite time.** By the Finite‑Time Hitting Lemma, rₖ(c) reaches r* at some finite cycle C.\n4. **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}.\n5. **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.\n\n### **[Question 1] Would it work well for tasks such as text-to-image or text-to-video that diffusion models are widely applied to?**\nAs 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.\n\nFor 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.\n\nSpecifically, 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.\n- **Compressibility**\n|Inference Method|Add.Compute|Compressibility|Clock Time(s)|NFEs|\n|----------------|:---------:|:-------------:|------------:|---:|\n|**Base Diffusion**|No add|−100.07±2.88|58.42|2000|\n|**Best-of-N(BoN)**|N=5|−81.89±5.71|257.58|10000|\n||N=10|−77.86±8.42|494.33|20000|\n|**Search-over-Path(SoP)**|Δf=510,Δb=255|−88.36±4.88|188.75|7548|\n||Δf=280,Δb=140|−72.64±5.61|257.67|10266|\n||Δf=180,Δb=90|−64.59±3.43|280.25|10974|\n|**ABCD**|max_iter=30|−62.33±4.33|103.67|3340|\n||max_iter=80|−51.49±4.34|254.50|8240|\n||max_iter=100|**−49.17±4.29**|306.08|10176|\n- **Image Aesthetic**\n|Inference Method|Add.Compute|Image Aesthetic|Clock time (sec.)|NFEs|\n|----------------|-----------|---------------|-----------------|----|\n|**Base Diffusion**|No add|5.57 ± 0.03|58.83|2000|\n|**Best-of-N (BoN)**|N=5|5.79 ± 0.05|258.33|10000|\n||N=10|5.86 ± 0.05|494.25|20000|\n|**Search-over-Path (SoP)**|Δf=510,Δb=255|5.68 ± 0.03|189.42|7548|\n||Δf=280,Δb=140|5.84 ± 0.08|258.67|10266|\n||Δf=170,Δb=85|5.87 ± 0.04|285.08|10704|\n|**ABCD**|max_iter=30|6.13 ± 0.09|105.42|3340|\n||max_iter=80|6.23 ± 0.07|233.83|7677|\n||max_iter=100|**6.25 ± 0.09**|288.42|9147|\n- **Human Preference Score**\n|Inference Method|Add.Compute|Human Preference|Clock time (sec.)|NFEs|\n|----------------|-----------|----------------|-----------------|----|\n|**Base Diffusion**|No add|0.2710 ± 0.0028|61.00|2000|\n|**Best-of-N (BoN)**|N=5|0.2753 ± 0.0017|260.92|10000|\n|**Search-over-Path (SoP)**|Δf=510,Δb=255|0.2725 ± 0.0031|192.83|7548|\n||Δf=230,Δb=115|0.2751 ± 0.0029|272.25|10218|\n||Δf=130,Δb=65|0.2765 ± 0.0046|811.25|11232|\n|**ABCD**|max_iter=30|0.2797 ± 0.0023|115.92|3340|\n||max_iter=80|0.2815 ± 0.0030|272.08|8060|\n||max_iter=100|**0.2819 ± 0.0032**|322.08|9718|\n\n[1] Christoph Schuhmann and Romain Beaumont. Laion-aesthetics. LAION. AI, 2022.\n\n[2] Wu, Xiaoshi, et al. Human preference score v2: A solid benchmark for evaluating human preferences of text-to-image synthesis. (2023).\n\n[3] Black, Kevin, et al. Training diffusion models with reinforcement learning.(2023)."}, {"review_id": "o2PczQ1bXs", "reviewer": "Reviewer_Zz6X", "summary": "This work introduces Adaptive Bi-directional Cyclic Diffusion (ABCD), a novel inference framework for diffusion models that automatically discovers improved denoising trajectories based on a given reward function. The core innovation lies in replacing fixed denoising schedules with a flexible, search-based approach that cycles bidirectionally through the diffusion process. ABCD has three main components: (1) Cyclic Diffusion Search, which alternates between denoising and re-noising steps; (2) Automatic Exploration-Exploitation Balancing, which distributes particles across multiple noise levels; and (3) Adaptive Thinking Time, which provides a stopping criterion. The authors evaluate their method on a range of tasks, including maze navigation, Sudoku solving, molecular generation, and planning, demonstrating improved performance and computational efficiency compared to baselines.", "questions": "- How does ABCD perform when DDIM initialization produces only poor samples? Are there any other possible failure modes?\n\n - Have you tested ABCD on more complex problems (such as small scale image generation)? Even modest increases in performance would provide valuable insights into scalability toward real world applications.\n\n- What time-step schedulers and sampling algorithms (e.g. DDIM, DDPM) were used for each baseline method?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### **Strengths**\n\n* The paper addresses an important limitation in current diffusion model inference—namely, the rigidity of fixed denoising schedules.\n* The bi-directional cycling approach and adaptive termination are novel contributions.\n* The adaptive termination criterion based on the temperature distribution is conceptually interesting.\n* The writing is clear, with effective use of figures, and the method is presented with sufficient detail for reproducibility.\n* The experimental evaluation spans multiple domains, demonstrating consistent improvements.\n* The method is compatible with both differentiable and non-differentiable reward functions.\n\n\n### **Weaknesses**\n\n* The main weakness of the paper is the lack of experiments on more complex, real-world problems such as image generation or language modeling. Including such experiments would significantly strengthen the contribution by demonstrating ABCD’s applicability to practical scenarios. At present, it remains unclear how well the method translates to real-world tasks like text-to-image generation.\n\n* The highest-performing configurations of ABCD still require significantly more inference time than baseline sampling. 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.\n\n* The method’s reliance on DDIM initialization may introduce failure modes that are not fully addressed. An ablation study evaluating the impact of different initialization strategies would help assess the robustness of ABCD.", "was_revised": true, "final_justification": "This paper presents an interesting and novel inference framework for diffusion models that automatically discovers improved denoising trajectories based on a specified reward function. Considering the thorough evaluation and the additional experiments provided by the authors in the rebuttal, I would like to recommend acceptance of this work.", "rebuttal": "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:\n\n### **[Question 1 & Weakness 3] How does ABCD perform when DDIM initialization produces only poor samples? Are there any other possible failure modes?**\n\nThank 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**. \n\nAdditionally, **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.\n\n### **[Question 2 & Weakness 1] Have you tested ABCD on more complex problems (such as small scale image generation)?**\n\nWe 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. \n\n### **[Question 3] What time-step schedulers and sampling algorithms (e.g. DDIM, DDPM) were used for each baseline method?**\n\nWithin 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.  \n\nWe 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 $x_0$.\n\n### **[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.**\n\nThank 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.\n\nIn 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.\n\n- **ABCD – Sudoku (Changing 𝜅)**\n|Metric|perc=1.0,𝜅=1|perc=1.0,𝜅=5|perc=1.0,𝜅=10|\n|------|-------------|-------------|--------------|\n|Performance|0.958|0.986|0.994|\n|Time (s)|0.443|0.608|0.754|\n|NFEs|2296|3205|3942|\n- **ABCD – Sudoku (Changing percentage)**\n|Metric|perc=0.4,𝜅=1|perc=0.6,𝜅=1|perc=0.8,𝜅=1|\n|------|-------------|-------------|-------------|\n|Performance|0.698|0.821|0.903|\n|Time (s)|0.112|0.210|0.329|\n|NFEs|520|1059|1667|\n\nIn 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.\n\n- **Compressibility**\n|Temperature pool size|Compressibility|Time (s)|NFEs|\n|---|---|---|---|\n|2|-55.19±4.93|233.33|8188|\n|3|-48.84±2.14|285.83|10500|\n|6|-46.14±2.16|356.5|12800|\n- **Image Aesthetic**\n|Temperature pool size|Aesthetic|Time(s)|NFEs|\n|---|---|---|---|\n|2|6.2345±0.0535|237.92|8200|\n|3|6.3179±0.0488|286.58|10369|\n|6|6.3624±0.1419|354.67|12578|\n\nAs 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].**"}, {"review_id": "9EYgKiDLMK", "reviewer": "Reviewer_jBRb", "summary": "This paper proposes an idea to use a pre-trained diffusion model to generate $\\arg\\max_x r(x)$ for some black-box reward function $r$. To simultaneously handle exploitation and exploration, they map the initially predicted data into multiple noise levels and re-denoise them in parallel. They also propose a way to adaptively terminate the sampling process.", "questions": "- Have the authors tried a simpler adaptive termination method that only checks whether the reward is improving or not? Does the proposed method offer any advantage over this simple baseline? The proposed method terminates when high-reward samples are mostly from small $t'$, but this discards the case where local exploration keeps improving, in which case it may be too early to terminate the algorithm. \n- Intuitively, low $t'$ encourages exploitation, whereas a large $t'$ promotes exploration. Does this mean that in environments where exploration is more important (e.g., when the reward support is far from the pre-trained model's support), BoN will outperform the proposed method? 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.\n- As mentioned in the paper, SoP uses a similar back-and-forth denoising->re-noising process. What is the benefit of ABCD over SoP? One disadvantage of SoP (and all other baselines compared) is that it uses $r(\\mathbb{E}[x_0 \\mid x_t])$ as an approximation of the true reward, which is not very reliable--correct? Is there any advantage of SoP over ABCD? A more detailed comparison would be helpful.\n\n\n>For fair comparison, the total number of particles was set equal across all methods (except BoN).\n\nDoes 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.\n\nFor the same reason, wouldn’t it be better to report the number of forward passes instead of “Mean Cyc” in Table 2?\n\nIn 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.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 2, "strengths_and_weaknesses": "# Strengths:\n\n- The idea is simple, and the motivation is sound. The concept of performing both exploitation and exploration in a parallel manner is interesting.\n- The topic--learning to optimize test-time reward using a pre-trained generative prior--is timely and of interest to the community.\n\n# Weaknesses:\n\n- Experiments are done on a relatively small scale. However, this is sufficient to analyze the proposed method in a controlled way, so this is not considered a significant weakness in the reviewer's opinion.\n- Always denoising all samples from all $t'$ seems inefficient. This could be made more efficient by selecting $t'$ more intelligently.\n- The adaptive termination criterion could have been compared to a simpler stopping criterion that checks whether the reward has stagnated.\n- 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.\n- Comparing wall-clock time does not seem to provide the full picture. See questions.", "was_revised": true, "final_justification": "The proposed adaptive rule is equivalent to the naive algorithm that 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.", "rebuttal": "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:\n### **[Question 1 & Weakness 3] About the simpler termination method & premature termination issue**\nThank 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.**\n### **[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.**\nThis 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.\n### **[Question 3] What is the benefit of ABCD over SoP?**\nWe believe that ABCD offers several key advantages over SoP.\n\n1. **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. \n2. **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.\n3. 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.\n\nThese properties collectively make ABCD a more flexible and potentially more powerful framework for guided generation.\n### **[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.**\n**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**.\n### **[Weakness 2] selecting $t' $ more intelligently can be possible**\nThank 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.**\n\nFor 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.\n\nNonetheless, 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.\n- **Adaptive ABCD vs. ABCD in Sudoku**\n|Method|Add. Compute|Accuracy|Clock Time (s)|\n|-|-|-|-|\n|**Adaptive ABCD**|perc=0.6,𝜅=1|0.810|0.220|\n||perc=0.8,𝜅=1|0.895|0.395|\n||perc=1.0,𝜅=3|0.979|0.549|\n|**ABCD**|perc=0.6,𝜅=1|0.821|0.240|\n||perc=0.8,𝜅=1|0.903|0.350|\n||perc=1.0,𝜅=3|0.979|0.557|\n### **[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.**\nIndeed, 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.\n- **Sudoku**\n|Method|Config|Accuracy|NFEs|\n|-|-|-|-|\n|**Base Diffusion**|No add|0.35|50|\n|**Best-of-N (BoN)**|N=480|0.639|24000|\n||N=960|0.662|48000|\n|**SMC**|p=5|0.643|2000|\n||p=1|0.745|2000|\n|**Beam Search**|p=2|0.757|1500|\n||p=1|0.844|2460|\n|**Search-over-Path (SoP)**|Δf=10, Δb=5|0.763|3860|\n||Δf=4, Δb=2|0.886|4220|\n||Δf=6, Δb=5|**0.957**|**11540**|\n|**ABCD (Ours)**|perc=0.6, 𝜅=1|0.821|1059|\n||perc=1.0, 𝜅=5|0.986|3205|\n||perc=1.0, 𝜅=20|**0.996**|**5013**|\n- **Pixel Maze**\n|Method|Config|Success Rate|NFEs|\n|-|-|-|-|\n|**Base Diffusion**|No add|0.02|50|\n|**Best-of-N (BoN)**|N=40|0.20|2000|\n||N=160|0.34|8000|\n||N=200|0.38|10000|\n|**SMC**|p=5|0.32|2000|\n||p=1|0.36|2000|\n|**Beam Search**|p=2|0.39|1500|\n||p=1|0.61|2460|\n|**Search-over-Path (SoP)**|Δf=10, Δb=5|0.76|3860|\n||Δf=8, Δb=4|0.85|3900|\n||Δf=4, Δb=2|**0.95**|**4220**|\n|**ABCD (Ours)**|perc=0.2, 𝜅=1|0.44|562|\n||perc=0.3, 𝜅=1|0.96|1226|\n||perc=1.0, 𝜅=3|**1.00**|**1854**|\n- **OGBench Maze Giant**\n|Method|Config|Performance|NFEs|\n|-|-|-|-|\n|**Base Diffusion**|No add|12±13|1280|\n|**Best-of-N (BoN)**|N=64|54±22|81920|\n||N=288|64±17|368640|\n||N=544|76±20|696320|\n|**SMC**|p=20|20±13|41376|\n||p=10|22±11|41792|\n|**Beam Search**|p=5|50±16|132960|\n||p=1|72±16|618560|\n|**Search-over-Path (SoP)**|Δf=100,Δb=50|60±15|59664|\n||Δf=50,Δb=25|86±13|71784|\n||Δf=6,Δb=3|**96±12**|**98360**|\n|**ABCD (Ours)**|max_iter=5|94±9|19020|\n||max_iter=10|96±8|29720|\n||max_iter=35|**100±0**|**83220**|\n- **OGBench Maze Large**\n|Method|Con|Performance|NFEs|\n|-|-|-|-|\n|**Base Diffusion**|No add|38±11|1280|\n|**Best-of-N (BoN)**|N=4|70±16|5120|\n||N=8|80±9|10240|\n||N=16|88±10|20480|\n|**SMC**|p=20|82±11|41376|\n||p=10|76±15|41792|\n|**Beam Search**|p=15|84±12|52720|\n||p=5|88±10|132960|\n|**Search-over-Path (SoP)**|Δf=100,Δb=50|76±12|58164|\n||Δf=50,Δb=25|86±9|71034|\n||Δf=6,Δb=3|**96±8**|**98270**|\n|**ABCD (Ours)**|J=32|**100±0**|**6360**|\n- **Image Generation**\n\nPlease refer to our response to **[Question 1] from Reviewer WZvc**.\n### **[Question 5] report the number of forward passes instead of “Mean Cyc” in Table 2?**\nThe 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. \n|Method|Success rate|Mean Cyc|NFEs|Success rate|Mean Cyc|NFEs|\n|-|-|-|-|-|-|-|\n|**ABCD (Ours)**|**100**|**3.36**|**308.32**|**100**|**7.02**|**535.24**|\n|GB 21|**100**|**2.06**|**182.4**|79|16.14|745.6|\n|GB 41|8|46.38|3439.36|**100**|**6.77**|**587.44**|\n|GB 61|0|–|–|100|15.05|1665.2|\n|GB 81|0|–|–|6|43.33|5992.88|\n### **[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.**\nThis 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."}]

Author rebuttal:

Meta-review: This paper introduces Adaptive Bi-directional Cyclic Diffusion to address the challenge of adaptive inference-time scaling. Reviewers are satisfied with the writing of this paper and impressive experimental results of the proposed method. The main concerns, including low-dimensional data and detailed comparison with other methods, are addressed with further experimental results provided by the authors. In the end, the reviewers all recommend accept. With these new results added into the revised version of this submission, I think this work meets the quality threshold of NeurIPS conference and recommend acceptance.
Decision: Accept (poster)