Inference-Time Scaling for Flow Models via Stochastic Generation and Rollover Budget Forcing
Authors: Jaihoon Kim
TaeHoon Yoon
Jisung Hwang
Minhyuk Sung
Venue: NeurIPS 2025 poster ()
Keywords: Inference-time scaling
flow model
reward alignment
Primary area: deep_learning
TL;DR: Inference-time scaling for flow models

Abstract: We propose an inference-time scaling approach for pretrained flow models. Recently, inference-time scaling has gained significant attention in LLMs and diffusion models, improving sample quality or better aligning outputs with user preferences by leveraging additional computation. For diffusion models, particle sampling has allowed more efficient scaling due to the stochasticity at intermediate denoising steps. On the contrary, while flow models have gained popularity as an alternative to diffusion models--offering faster generation and high-quality outputs--efficient inference-time scaling methods used for diffusion models cannot be directly applied due to their deterministic generative process. To enable efficient inference-time scaling for flow models, we propose three key ideas: 1) SDE-based generation, enabling particle sampling in flow models, 2) Interpolant conversion, broadening the search space and enhancing sample diversity, and 3) Rollover Budget Forcing (RBF), an adaptive allocation of computational resources across timesteps to maximize budget utilization. Our experiments show that SDE-based generation and variance-preserving (VP) interpolant-based generation, improves the performance of particle sampling methods for inference-time scaling in flow models. Additionally, we demonstrate that RBF with VP-SDE achieves the best performance, outperforming all previous inference-time scaling approaches.

Inference-Time Scaling for Flow Models via Stochastic Generation and Rollover Budget Forcing
Jaihoon Kim
Taehoon Yoon"
Jisung Hwang" KAIST
Minhyuk Sung
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Figure 1: Diverse applications of our inference-time scaling method. Pretrained flow models struggle to generate images that align with complex prompts (left side of each case), whereas our inference-time scaling effectively extends their capabilities to achieve precise alignment (red box).
Abstract
We propose inference-tine scaling approach for pretrained flow models. Re- cently, inference tineencee-imeeeee scaling has gained significant attention in LLMs and diffusion models, improving sample quality or better aligning outputs with preferences by leveraging additional computation. For diffusion models, particle sampling has allowed
efficient scaling due the stochasticity intermediate denoising steps. On the contrary, while flow models have gained popularity-offerine faster generation and high-quality outputs-efficient inference-tine timeenee-immeeeeee scaling methods used for diffusion models cannot be directly applied due to their deterministic generative process. To enable efficient inference-time scaling for flow models, we propose three key ideas: 1) SDE-based generation, enabling particle sampling in flow models, 2) Interpolant conversion, broadening the search space, and 3) Rollover Budget Forcing (RBF), maximizing compute utilization. Our experiments show that SDE-based generation and variance- prraannveneereiinggg.... (VP) interpolant- based genera- tion, improves the performance of particle sampling methods for inference-time scaling flow models. Additionally, we demonstrate that RBF with VP-SDE achieves the best performance, outperforming all previous inference-time scaling approaches. Project page: flow-inferen colo-inffeence time-- camme-scaling scaling, gi thub.
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39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Introduction
Over the past years, scaling laws of Al models have mainly focused on increasing model size and training data. However, recent advancements have shifted attention toward inference-time scaling [57 [9]. leveraging computational resources during inference to enhance model performance. OpenAl ol 42 and DeepSeek R1 [1] exemplify this approach, demonstrating consistent output improvements with increased inference computation. Recent research in LLMs [41] attempting to replicate such improvements has introduced test-time budget forcing, achieving high efficiency with limited token sampling during inference.
For diffusion models (54 [56] which are widely used for generation tasks, research on inference-time
scaling has been growing in the context of swardd-bssed sampling [3 30 53 Given a reward function that measures alignment with user preferences [26] output quality (50 311, the goal is to find the sample from the learned data distribution that best aligns with the reward through repeated sampling. Fig. [ showcases diverse applications of inference-tine scaling using our method, enabling the generation of faithful images that accurately align with complex user descriptions involving objects quantities, logical relationships, and conceptual attributes. Notably, naïve generation from text-to- image models [48] (8 often fails to fully meet user specifications, highlighting the effectiveness of inference-time scaling.
Our goal this work to extend the inference-time scaling capabilities of diffusion models to flow models. Flow models (32 power of image [15] [8] and video generation [ 72), achieving high-quality synthesis with few inference steps, enabled by trajectory stratification techniques during training [35]. Beyond just speed, recent pretrained flow models, equipped with enhanced text-image embeddings [46] and advanced architectures [T], significantly outperform previous pretrained diffusion models in both image and video generation quality.
Despite their advantages in generating high-quality results more efficiently than diffusion models, flow models have an inherent limitation in the context of inference-time scaling. Due to their ODE-based deterministic generative process, they cannot directly incorporate particle sampling at intermediate steps, key mechanism for effective inference-time scaling in diffusion models. Building on the formulation of stochastic interpolant framework (0. we adopt an flow models at inference-ting time,, enabling particle sampling for reward alignment.
sampling method for
To further expand the exploration space, we consider not only stochasticity but also the choice of the interpolant. While typical flow models use a linear interpolant, diffusion models commonly adopt a Sariance-Preservinn
(VP) interpolant [8). Inspired by this, for the first time, we incorporate the VP interpolant into the particle sampling of flow models and demonstrate its effectiveness in increasing sample diversity, enhancing the likelihood of discovering high-reward samples.
We emphasize that while we propose converting the generative process of a pretrained flow model to scaling with
align with that of diffusion models-i.e., VP-See,P----bbaeeee flow models offers significant advantages over diffusion models. Flow models, particularly those with rectification fine-tuning [35 36), produce much clearer expected outputs intermediate steps. enabling more precise future reward estimation and, in turn, more effective particle sampling.
inneraaion-iiffrennee--ee
We additionally explore a strategy for tight budget enforcement in terms of the number of function evaluations (NFEs) of the velocity prediction network. Previous suticll-saapling time scaling approaches for diffusion models (3 53) allocate the NFEs budget uniformly across timesteps in the generative process, which we empirically found to be ineffective in practice. To optimize budget utilization, we propose Rollover Budget Forcing, a method that adaptively reallocates NFES across timesteps. Specifically, we perform a denoising step upon identifying a new particle with a higher expected future reward and allocate the remaining NFEs to subsequent timesteps.
inference-
Experimentally, we demonstrate that our inference-time SDE conversion and VP interpolant conver- sion enable efficient particle sampling in flow models, leading to consistent improvements in reward alignment across two challenging tasks: compositional text-to-image generation and quantity-awan awwae image generation. Additionally, gains, outperforming all previous particle sampling approaches. We also demonstrate that for differen- tiable rewards, such as aesthetic image generation, integrating RBF with gradient-based method creates a synergistic effect, leading to further performance improvements.
Rollover Budget Forcing (RBF) provides further performance
In summary, we introduce inference-time scaling for flow models, analyzing three key factors:

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ODE vs. SDE: We introduce an SDE generative process for flow models to enable particle sampling.
Interpolant: We demonstrate that replacing the linear interpolant of flow models with Variance Pre- serving interpolant expands the search space, facilitating the discovery of higher-reward samples.
NFEs Allocation: We propose Rollover Budget Forcing that adaptively allocates NFEs across timesteps ensure efficient utilization of the available compute budget.
Related Work
2.1 Reward Alignment in Diffusion Models
In the literature of diffusion models, reward alignment approaches can be broadly categorized into
fine-tuning-basse -bsed methods [ 69
67] and tir--eee-it---saiinn bbatn methods [30] While fine-tuning diffusion models enables the generation of samples aligned with
53
64
6
user preferences, requires fine-tuning for each task, potentially limiting scalability. In contrast, inference-tings scaling approaches offer significant advantage as they can be applied to any reward without requiring additional fine-tuning. Moreover, inference-time scaling can also be applied to fine-tuned models to further enhance alignment with the reward. Since proposed approach is an inference-time scaling method, we focus our review on related literature in this domain.
When the reward is differentiable, gradient- based methods g extensively studied. We note that inference-time scaling can be integrated with gradient-basel approaches to achieve synergistic performance improvements.
3
16 17
63
have been
2.2 Particle Sampling with Diffusion Models
The simplest iterative sampling method that can be applied to any generative model is Best-of-N (BoN) (57 599 (88), which generates N samples and selects the one with the highest reward. For diffusion models, however, incorporating particle sampling during the denoising process has been shown to be far more effective than naïve BoN [33 30 This idea has been further developed through
various approaches that sample particles at intermediate steps. For instance, SVDD [30] proposed selecting the particle with the highest reward at every step. CoDe (53) extends this idea by selecting the highest-reward particle only specific intervals. On the other hand, methods based on Sequential employ a probabilistic selection approach, in which particles are sampled from multinomial distribution according their importance weights. Despite the success of particle sampling approaches for diffusion models, they have not been applicable to flow models due to the absence of stochasticity in their generative process. In this work, we present the first inference-time scaling method for flow models based on particle sampling by introducing stochasticity into the generative process and further increasing sampling diversity through trajectory modification.
Monte Carlo (SMC) (64 6 23 13
2.3 Inference-Time Scaling with Flow Models
To our knowledge, Search over Paths (SoP) [3 is the only inference-time scaling method proposed for flow models, which applies forward kernel to sample particles from the deterministic sampling process of flow models. However, SoP does not explore the possibility of modifying the reverse kernel,
which could enable the application of more diverse particlesampling- bamsling-bbsedd methods [30] [33] 23]. To the best of our knowledge, we are the first investigate the application of particle sampling to flow models through the lens of the reverse kernel.
Problem Definition and Background
3.1 Inference-Time Reward Alignment
Given a pretrained flow model that maps the source distribution, a standard Gaussian distribution Pt. into the data distribution Po. our objective is to generate high-reward samples x E Idd from the pretrained flow model without additional training-a task known as inference-time reward alignment. We denote the given reward function as Rd RR which measures text alignment or user preference for a generated sample. Following previous works formulated as finding the following target distribution:
60 61
our objective can be
PG arg max Exg mig [x(x))]
-B Dri. [4llpa)
(I)
Reward
KL. Regularicatinn
3

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which maximizes the expected reward while the KL divergence term prevents (xx) from deviating too far from Pu(X)), with its strength controlled by the hyperparameter 3. As shown in previous work [5), the target distribution pb can be computed as:
1 pā (xx) z (x0) exp
(<(xo)
(2)
where Z is a normalization constant. We present details in Appendix the target distribution is non-trivial.
A.I However, sampling from
A notable approach for sampling from the target distribution is particle sampling, which maintains set of candidate samples-ree rreferred to as particles- and iteratively propagates high-reward samples while discarding lower-reward ones. When combined with the denoising process of diffusion models, particle sampling can improve the efficiency of limited computational resources in inference-time scaling. In the next section, we review particle sampling methods used in diffusion models and, the first time, we explore insights for adapting them to flow models.
3.2 Particle Sampling Using Diffusion Model
A pretrained diffusion model generates data by drawing an initial sample from the standard Gaussian distribution and iteratively sampling from the learned conditional distribution po (x,- |xx). Building on this, previous works [29] [1]] have shown that data from the target distribution in Eq. can be generated by performing the same denoising process while replacing the conditional distribution po(x ar)xx) with the optimal policy:
(x (2x)) exp
(ox) acl
Po (xl=-10)
5
po (x Xx) exp
n(x;
del dx
3
where the details are presented in Appendix A.2
We denote vi()) Rd + R as the optimal value function that estimates the expected future reward of the generated samples at current timestep. Following previous works X8 23
30
3 we approximate the value function using the posterior mean
computed via Tweedie's ' formula 47 given by v(x,) R r(xojt). where Xoje == -xxxxp(xxxxxx) [xo]
Since directly sampling from the optimal policy distribution in Eq. 3 is nontrivial, one can first approximate the distribution using importance sampling while taking Po (x, (xi-At x,, as the proposal distribution:
pà (x Rd
5,00 "
(x/2a
(9 J ~ po(xx-aelx))
(4)
-
where K is the number of particles, wi-At exp exp(1(x (xp(v(((aa1/1) is the weight, and 6 6000 a
is
Dirac distribution. SVDD 330 proposed approximate sampling method for the optimal policy by selecting the sample with the largest weight from Eq. 4
Notably, a key factor in seeking high-reward samples using particle sampling is defining the proposal distribution to sufficiently cover the distribution of high-reward samples. Consider scenario where high-reward samples reside in a low density region of the original data distribution, which is common when generating complex or highly specific samples that deviate from the mode of the pretrained model distribution. In this case, the proposal distribution must have a sufficiently large variance to effectively explore these low density regions. This highlights the importance of the stochasticity of the proposal distribution, which has been instrumental in the successful adoption of particle sampling in diffusion models. In contrast, flow models [32] employ a deterministic sampling process, where all particles Xt- drawn from X, are identical. This restricts the applicability of particle sampling methods in flow models. One of the main contributions is the investigation of how these particle sampling methods can be efficiently applied to flow models.
To this end, we propose an inference- time approach that introduces stochasticity into the generative process of flow models to enable particle sampling. We first transform the deterministic sampling process of flow models into a stochastic process (Sec. 4.2p. p. We further identify a sampling trajectory that expands the search space of the flow models (Sec. 4.3 Note that while stochastic sampling and
trajectory conversion have been studied in prior works, their primary goals have been to improve
sample quality (68) 70 49 24
38 or to accelerate inference [19 52 51 22
To the best of our

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Linear
VP
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tel. Earr
Linor
(a) Lincar-ODE
(b) LinearSSDE
(C) VP-SDE
Figure 3:
Sample diversity test using
Figure 2: Comparison of Linear-ODE, Linear- SDE, and VP-SDE. The visualization shows how trajectories evolve under different dynamics starting from the same noise latent.
FLUX [28] under linear and VP interpolant.
All samples share the same initial latent. Prompt: "A steaming cup of coffee".
knowledge, we are the first to investigate sampling stochasticity and trajectory conversion for efficient particle-based sampling in flow models.
Additionally, previous particle sampling methods in diffusion models allocated a fixed computational budget (i.e., uniform number of particles) across all denoising timesteps, potentially limiting exploration. We explore sampling with the rollover strategy, which adaptively allocates compute across timesteps during the sampling process (Sec. 6).
4 SDE-Based Particle Sampling in Flow Models
In this section, we review flow and diffusion models within the unified stochastic interpolant frame- work (Sec. 4.1 and introduce our inference-time approaches for efficient particle sampling in flow models (Sec. and
and 4.3
4.1 Background: Stochastic Interpolant Framework
At the core of both diffusion and flow models is the construction of probability paths (pelosesis where Pt. serves as a bridge between Pi and Xo Po:
X Xg + 0,X1,
where a and o, are smooth functions satisfying ag = o1 1, 01 = og = 0, and ò₃ < 0, og > 0; we denote the dot as time derivative. This formulation provides flexible choice of interpolant (at, σt) which determines the sampling trajectory.
4.2
Inference-Time SDE Conversion
Flow models 32 35 learn the velocity field u : Rd - Rd, which enables sampling of xo by solving
the Probability Flow-ODE [66] backward in time:
dx; = uy (xxx))
The deterministic process in Eq. accelerates the sampling process enabling few-step generation of high-fidelity samples. However, discussed in Sec. 3.2 the deterministic nature of this sampling process limits the applicability of particle sampling in flow models.
To address this, transform the deterministic sampling process into stochastic process. The reverse-ting time SDE that shares the same marginal densities as the deterministic process in Eq.
of v (log)((x)) 2
dx, fx(x)dt + g,dw, f((x)) u, (x,)
where f, (x;) and represent the drift and diffusion coefficient, respectively, and w is the standard Wiener process. This conversion introduces noise schedule 9t, which can be freely chosen. Although SiT (3 arrives at the same conclusion, we provide more comprehensive proof in Appendix B In
our case, we set gt = (2, scaled by factor of 3. Note that in the case where gr = the process reduces to deterministic sampling in Eq. 6
Using the velocity u, (xx) predicted by a pretrained flow model, the score function v log ( appearing in the drift coefficient (x;:) can be computed as:
1 a,4 (xx) di
V lo
(8)
âi - 04g0
5

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This enables the conversion of the deterministic sampling to stochastic sampling, which we refer to as inference-time SDE conversion. Given the drift coefficient term f, fx,)) and diffusion coefficient gr+ the proposal distribution in the discrete-time domain is derived as follows:
(9) While previous works have proposed converting an SDE to an ODE to improve sampling effi-
po (x = (X - fx(x) 9. I).
ciency 22 55 56]. the reverse approach-traasfommiii an ODE into an SDE-has received
relatively less attention and has primarily focused on improving sample quality 668 38). To the best
of our knowledge, this work is the first to explore SDE conversion in flow models specifically to expand the search space of proposal distribution for efficient particle sampling.
Since flow models utilize the linear interpolant (a,, 1 t, t), refer the generative processes of the flow models using 6q. 66 and Eq. [ as Linear-ODE and Linear-SDE. respectively. In Fig. 2 (left), we visualize the sampling trajectones of Linear-OD -DEE and Linear-SDE. The samples generated using Linear-ODE are identical and collapse to a single point, restricting exploration. In contrast, Linear-SDE introduces sample variance, allowing for broader exploration and increasing the likelihood of discovering high-reward samples.
In Fig. 3 (a-b), visualize images sampled from Linear-ODE and Linear-SDE using FLUX [28] As discussed previously, the particles drawn from the proposal distribution of Linear-ODE are identical. In contrast, Linear-SDE introduces variation across different particles, thereby expanding the search space for identifying high-reward samples. In the next section, we introduce inference-time interpolant conversion, which further increases the search space.
4.3 Inference-Time Interpolant Conversion
To further expand the search space of Linear-SDE, we draw inspiration from the effective use of particle sampling in diffusion models, where we identified a key difference: the interpolant. While the forward process in diffusion models follows the Variance Preserving (VP) interpolant
(a, = exp lo f,ds G = /1- exp SC B,ds), with B. denoting a predefined variance schedule, flow models adopt a linear interpolant.
we note that given a velocity model based an interpolant (a,, σ,) (e.g., linear), one can transform the vector field and generate a sample based on new interpolant (ū,,ö,) (e.g.. VP) at inference-times The two paths x, = 0,X0 + T,Xi and X = 0, + 0,X1, are connected through scale-time transformation:
As shown in the previous works 33 52
x, = Ct t, = (p(s)) C = =0//Ot.
(10)
where p() = ae and p(s) 3, define the signal-to- noise ratio of the original and the new interpolant, respectively. The velocity for the new interpolant is given as:
a. (..x)) - Co
x. + catau,
X5
é or, न. a.
i. o3 (0,, - au, ò.,)"
oi, (ā.â. - ā....)
(1I)
Co
of,
Plugging the transformed velocity into the proposal distribution in Eq. 9 after computing the score using Eq. g giives our efficient proposal distribution.
Pa (x = N x, (x.)) v log log As, o²As
(12)
Since the new trajectory follows the VP interpolant, we refer to this as VP-SDE. We visualize VP-SDE sampling Fig. (right). At inference-time, we query the velocity the new interpolant from the original interpolant (purple arrow). In Fig. (c), we visualize the sample diversity under VP-SDE using FLUX (28) which generates more diverse samples than Linear-SDE. This property of VP-SDE effectively expands the search space, improving particle sampling efficiency in flow models. we provide further analysis on how interpolant conversion contributes to sample diversity.
In Sec. S
12 24 and accelerated inference [5T] 22 52). We utilize interpolant conversion enhance the sample diversity in particle sampling, which has not been unexplored before. Importantly, while we modify the generative process of flow models to align with that of diffusion models, inference-time scaling with flow models still provides distinct advantages. The rectified trajectories of flow models K5 36 28
Previous works focused on interpolant conversion that enables stable training 49
allow for a much clearer posterior mean, leading to more precise future reward estimation and, in turn, more effective particle filtering.

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5 Analysis of the Interpolant Conversion and Sample Diversity
In this section, we analyze how the interpolant conver- sion affects the variance of the proposal distribution and explain why VP-SDE yields higher sample di- versity than Linear-SDE, as illustrated in Fig. To investigate this behavior, Fig- 4 visualizes the log- SNR (aog(aff (a]/ ) of commonly used interpolants
Cosine
Nef SM4
SNI
including linear and VP across timesteps E (0. 1).
Then consider initializing the Linear-SDr timesteps (o using the timestep conversion in Eq. 10
This ensures that the log-SNR of the corresponding latents Xt, matches that of the VP-SDE latents x, each step (see the horizontal dashed line in Fig. 4
Timestep
Under this condition, the proposal distributions of the two processes are expressed as follows:
Figure 4: Interpolant log-SNR. Dashed lines
show a reference SNR and timestep.
Linear-SDB (t&): pe(X), A1, xx.)) = N (xt - fr. (x.,) At g.. At I)
VP-SDE:
Pa(X:-As x,) = N (x,,x,,,...)AA (x g As I) po c,xt,).
Since At, As at early denoising steps, timestep conversion results in smaller variance gř At, than VP-SDE for a fixed diffusion coefficient. Hence, to match the variance of Linear-SDEE (1,) to that of VP-SDE, one can scale the diffusion coefficient to gi. = 9 As/ Atg. Note that this scaling significantly increases the stochasticity at early denoising steps C2 Re 1 and As/Att >> 1. While this scaling can enhance sample diversity, applying it in isolation injects excessive noise, causing samples to deviate from the predefined denoising trajectory and ultimately degrading output quality,
In fact, interpolant conversion counteracts excessive noise injection by pairing diffusion coefficient scaling with timestep conversion. The two mechanisms act synergistically to increase the sample diversity without harming the sample quality. Sec. 7 we validate this analysis with an ablation study that isolates the effect of each factor.
Comparison under Identical Timestep and Diffusion Coefficient. We next analyze the case where both Linear-SDE and VP-SDE operate under identical, fixed timestep schedules and diffusion coefficients. While this setting yields identical proposal distribution variances, Fig. 4 shows that the VP interpolant maintains a consistently lower log- SNR, indicating that at any given timestep, VP-SDE samples contain larger noise component (see the vertical dashed line). Consequently, the VP-SDE proposal distribution effectively samples from a noisier latent at each step. resulting in higher sample diversity. This reflects the observation that noisier latents produce more diverse samples [40). While this work focuses on the interpolant perspective, a systematic exploration of timestep scheduling and diffusion coefficient scaling remains a promising direction for future research.
6 Rollover Budget Forcing
In the previous sections, we have introduced our inference-time approaches expand the search space of proposal distribution. Here, we propose new budget-forcing strategy to maximize the use of limited compute in inference-tine -imeenece-tmeeeeee scaling. To the best of our knowledge, previous particle sampling methods for diffusion models [30] [33] employ a fixed number of particles across all denoising steps. However, our analysis shows that this uniform allocation may lead to inefficiency, where the NFEs required at each denoising step to obtain sample x₁-At with higher reward than the current sample x, significantly varies across different runs. We present the analysis results in Appendix C
This motivates us to adopt a rollover strategy that adaptively allocates NFEs across timesteps. Given a total NFEs budget, the NFEs quota Q is allocated uniformly across timesteps. Then each timestep, if a particle X;-At yields a higher reward than the current sample x; within the quota, we immediately proceed the next timestep from the newly identified high-reward sample. rolling over the remaining NFEs to the next step. If the allocated quota is exhausted without identifying better sample, we select the particle with the highest expected future reward from the current set, following the strategy used in SVDD (3) The pseudocode of RBF is presented in Appendix D In the next section, we demonstrate the effectiveness of RBF, along with SDE conversion and interpolant conversion.
7

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VOAScore '
Inst. BLIP (held-out) f
Aesthetic Score
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0.925
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Figure 5: Quantitative results of compositional text-to-image generation. denotes the given reward used in inference-time scaling (left). Notably, performance consistently improves from Linear-ODE to Linear-SDE and VP-SDE for both given and held-out rewards (left, middle), without
significant quality degradation, evidenced by the comparable aesthetic score (50) (right).
7 Applications
In this section, we present the experimental results of particle sampling methods for inference- time reward alignment. In Appendix, present implementation details of the search algorithms, ii) aesthetic image generation, iii) comparisons between diffusion and flow models, iv) scaling behavior comparison of Best-of-N (BoN) and RBF, and additional qualitative results.
7.1 Experiment Setup
Tasks. this section, we present the results for the following applications: compositional text-to- image generation and quantity-aware image generation, where the rewards are non-differentiable. For the differentiable reward case, we consider aesthetic image generation (Appendix E. In compositional text-to-image generation, use all 121 text prompts from GenAl-Bench |21 that
contain three or more advanced compositional elements. For quantity- aware image generation, we use 100 randomly sampled prompts from T21- Pel--oomBBench++ (20) numeracy category.
For all applications, we use FLUX 28 as the pretrained flow model. We fix the total number of function evaluations (NFEs) to 500 and set the number of denoising steps 10, which allocates 50 NFEs per denoising step. As reference, we also include the results of the base pretrained models without inference-time scaling. Additionally, we present a comparison between flow models and diffusion models in Appendix
E.2
Baselines. We evaluate inference-time search algorithms discussed in Sec. including Best-of-N (BoN), Search over Paths (SoP) [39]. SMC 233, CoDe [53], and SVDD [30 We categorize BoN and SoP Linear-ODE-basee methods, as their generative processes follow the deterministic process in Eq. 6 For SMC, we adopt DAS [23]: however, when the reward is non-differentiabll, we use the reverse transition kernel of the pretrained model the proposal distribution.
7.2 Compositional Text-to-Image Generation
Evaluation Metrics. this work, we refer to the reward used for inference-time scaling as the given reward. Here, the given reward is VQAScore, measured with CLIP-FlanTS [31], which evaluates text-image alignment. For the held-out reward, which is not used during inference, we evaluate the score using different model, InstructBLIP [10]. Additionally, evaluate aesthetic score [50) to assess the quality of the generated images.
Inference-Time SDE and Interpolant Conversion. The quantitative and qualitative results of compositional text-to-image generation are presented in Fig. aand Fig. 6 respectively. As discussed in Sec. 4.2 the deterministic sampling process in flow models limits the effectiveness of particle sampling, whereas introducing stochasticity significantly expands the search space and improves Citormmance-hhhlighhheeee
key contribution of our work: enabling effective particle sampling in flow models. The results in Fig. support this finding, showing that Linear-SDF (yellow) consistently improves the given reward (left in Fig. 5) over the Linear-ODE (green) across particle sampling
methods, even surpassing BoN and SoP [39], which were previously the only available inference-

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Linear ODE
Linear SIDE
VP-SD
Linear ODE
Linear SDE
VP. SDE
"Five origami cranes hang from the celling. only one of which is red, and the others are white."
"Three small, non a large
CoDe 5
SMC
"A nouse pad has two pencils on it, shorter pencil green and the longer one not."
"Fuve onts are carrying biscuits, and on ant that is not carrying biscuits standing on green leaf directing them..
R SyDs
BBF
Figure 6: Qualitative results of compositional text-to-image generation. We use VQAScore B1 which measures text-image alignment, as the given reward for inference-time scaling. SDE and interpolant conversion enable more effective exploration during inference, enhancing the performance including RBF.
of all particle sampling methods [3 30
scaling approaches for ODE-based flow models. Additionally, through inference-time interpolant conversion, VP-SDE (red) further improves performance across all particle sampling methods on both given and held-out rewards (left, middle in Fig. (5) by expanding the search space, demonstrating the effectiveness of our proposed distribution. Notably, particle sampling methods with Linear- SDE and VP-SDE generate high-reward samples without significantly compromising image quality, as evidenced by aesthetic scores that remain comparable to the base FLUX model [28] (right in Fig. 5). Qualitatively, SDE conversion and interpolant conversion shown in Fig. 6 bring consistent performance improvements (see Appendix G for additional results).
Rollover Budget Forcing. As discussed in Sec. 6 instead of fixing the number of particles
throughout the denoising process, we explore adaptive budget allocation through RBF. We demonstrate that budget forcing provides additional performance improvements, outperforming the previous particle sampling methods in the given reward (left in Fig. 5) We present qualitative comparisons of inference-tine time scaling methods in Appendix G.2
Ablation study of interpolant conversion. Building the analysis in Sec. 5 we examine how interpolant conversion contributes sample diversity and reward alignment through its two underlying mechanisms, timestep conversion and diffusion coefficient scaling. Tab. extends the results Fig. by isolating the effect of each com- ponent to sample diversity, measured by LPIPS- MPD 23 and reward alignment.
Table 1: Ablation Study of Interpolant Con-
version. denotes the given reward.
Method
LPIPS MPD
VQAScore
Inst, BLIP
Linear-ODE
0.788
0.789
Linear SDE
0.158
0.900
0.813
Adapt. Time.
0,270
0.908
0.813
Adapt. Diff.
0.429
0.702
0.571
VP-SDE
0.509
0.925
We observe that timestep conversion (row 3) yields only modest diversity gains: the benefit of sampling lower log-SNR (Fig. 4 is offset by smaller discretization steps that reduce proposal variance, limiting improvements in reward alignment. On the other hand, applying diffusion coeffi- cient scaling without timestep conversion (row 4) increases sample diversity but simultaneously leads to significant drop in reward alignment indicating excessive noise injection. Lastly, the VP-SDE interpolant conversion (row 5) synergistically combines both components, achieving high sample diversity without sacrificing quality and consequently yielding the highest reward.
7.3 Quantity-Aware Image Generation
Evaluation Metrics. Here, the given reward the negation of the Residual Sum of Squares (RSS) between the target counts and the detected object counts, computed using GroundingDINO (34 and SAM (25) (details in Appendix Fp. Additionally, we report object count accuracy, which evaluates whether all object quantities correctly shown in the image. For the held-out reward, report

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Linea-CDE
Accuracy
VOAScare [held-our]]
0.050
0,750
0.02
0:228
0.76
9.76
0.703
0.6
5,580
3,675
10
1,400
0.4
0.70
5.182
0.2
641
644
0.330
0.40
0.00
Figure 7: Quantitative results of quantity- aware image generation. denotes the given reward, RSS (34, with the y-axis truncated for better visualization (left). We observe consistent performance improvements by converting Linear-ODE to Linear- Linear-DDE, and VP-SDE for most cases.
Linca-ODE
Linear-SDE
VP-SDE
LincarODDE
Lincar-SDE
VP-SDE
" Four balloons,
"Four candles, two balloons, one dog. two tomatoes and three helicopters."
four desks, two dogs and four microwaves."
SNC
CoDe
"light chairs"
"One three camels, four cars and four pillows.
SynD AD
RBF
Figure 8: Qualitative results of quant image generation. At inference-time, we guide generation using the negation of RSS (34] (Residual Sum of Squares) the given reward, which measures the discrepancy between detected and target object counts. SDE and interpolant conversion expands the search space to identify high reward samples.
VQAScore measured with CLIP-FlanT5 [31. As in the previous application, we evaluate the quality of the generated images using the aesthetic score [50).
Results. The quantitative and qualitative results are presented Fig. and Fig. respectively. The trend in Fig. 7 align with those in Sec. 77. demonstrating that SDE conversion and interpolant conversion synergistically enhance the identification of high-reward samples. Notably, particle sampling methods with Linear-SDE. already outperform Linear-ODE-hn
methods (BoN and 6x improvement
SoP (39). while interpolant conversion further improves accuracy, achieving over the base model (28). Our RBF achieves the highest accuracy, outperforming all other particle sampling methods. Qualitatively, Fig. shows that SDE and interpolant conversion effectively identify high-reward samples that accurately match the specified object categories and quantities. Additional qualitative comparisons of the inference-time scaling methods are provided Appendix G.2
8 Conclusion and Limitation
We introduced a novel inference-time scaling method for flow models with three key contributions: (I) ODE-to-SE conversion for particle sampling in flow models, (2) Linear-to- VP interpolant conversion for enhanced diversity and search efficiency, and (3) Rollover Budget Forcing (RBF) VP-SDE-basee generation
for adaptive compute allocation. We demonstrated the effectiveness in applying off-the-she the-f particle sampling to flow models and showed that our RBF combined with VP-SDE generation outperforms previous methods. However, our method introduces additional inference-tine overhead, which could become a bottleneck when the base model prediction is computationally intensive. Also, since the pretrained model may have been trained on uncurated datasets, approach may produce undesirable outputs upon malicious attempts.
10

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Acknowledgments
This work was supported by the NRF of Korea (RS-2023-00209722)) ПIРР grants (RS-2022- II220594, RS-2023-00227592, RS-2024-003998 17, RS-2025-25441313, RS-2025-254433116 RS- 2025-02653113); and the Technology Innovation Program BRS-2025-023173326 all funded by the
Korean government (MSIT and MOTIE), well by the SketchTheFuture Research Center.
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Appendix
A Proofs
A.1 Derivation of the Target Distribution
From Eq. I we obtain the target distribution Po. which maximizes the reward while maintaining proximity to the distribution of the pretrained model po:
(x(x))= = arg max Exo~a [r(x))] 3DKL [gl1po],
r(xo) Blog ((xu)
arg max E Exorg
po (x(x))
log g(x0)
arg min Exong
rx(x)
px(xx)
/ ((x)) log
(xx))
/ ((x))r(x))) (xo)dxn-
= arg min
dxo
(x()
This can be solved via calculus of variation where the functional J is given follows:
I [g(xo)] = ((xx)) [log g(Xo)
3 (x(0))
dxg.
Pu (x0)
Substituting ū(xo,e] <) g(xo) + (x)( gives:
T [[((x, = @(x0, c) (log (xxo,e)
Po(Xo) B r(xx)) dxa.
where (xx) an arbitrary smooth function, and is scalar parameter.
Introducing a Lagrange multiplier À to constraint ((xx))xx))) = 1 gives:
I [ā(xo c)] = f q(xu, c) loog ((xp, 1 po(xo) 3 (((x0)) + >ap(xo, c)dxo
=
f flq:xo
Then the problem boils down to finding a function q(xo, c) satisfying:
0J
0
Ōc
This can be solved using the Euler-Lagrange equation:
Øf
0f
dq
dxp 0uy
where of is a derivative of q with respect to xa and tilde notation is dropped since the condition is to be satisfied at
Note that q' does not appear in f, so the Euler-Lagrange equation simplifies to:
0f θ
((xx)
(oog ((xo) 1
3 3r(80))
>a(xo))
0q
po(xo)
log ((x0)
((xo) +1+
(13)
(x(o
Solving Eq. 13 gives the target distribution pú, which minimizes the objective function in Eq
r(xp)
(x())) = -x((x) exp
1 2
(14)
8
22

---

Lastly, the Lagrangian multiplier À is obtained from the normalization constraint. exp(A) r(xo) - dxg. Plugging this into Eq. 14 gives the target distribution presented
(xo) exp
in Eq. 2
po (x0) exp
r(xo)
pō(xo) =
(15)
p exp exp
r(xo)
dxo
A.2 Derivation of the Optimal Policy
Here, provide the derivations of the optimal policy given in Eq. for completeness, which is proposed in previous works [60 61].
To sample from the target distribution defined in Eq. 15 previous studies utilize an optimal policy ()(x))) The optimal value function v(x,) is defined as the expected future reward at current timestep t:
r(xo) B
]
vxx,) =Blog (xpopo(xxolxs)
exp
(16)
The optimal policy is the policy that maximizes the objective function:
arg max -am-((())))) [x(x (r)- - 33DL (a(- xx)]
o( xx)
po(x-_r( exp (xx(
(17)
I po (x) exp (fu(xe-s)) dxt-Att
po(xi-a1(X) exp s((x--s))
(18)
exp (((x))))
where the last equality follows from the soft-Bellman equations 61 the theorem.
For completeness, we present
Theorem 1. (Theorem of Uehara al. [61]) The induced distribution of the optimal policy in Eg. 77 is the target distribution in Eq. 15
(x(x)) / pi pi(xi)
II pō 4- +1x+) dox 4.1.
However, computing the optimal value function in Eq. 16 is non-trivial. Hence, we follow the
previous works 23 30 and approximate using the posterior mean Xuje == Exo (x.) [xo]:
(fes) exp
r(xp)
vxx:) = 3 log
pe (xo xe )cbxo
B
r((xoe))
))
2s B log
exp
r(Xoje)=
(19)
B
B Choice of Diffusion Coefficient
Ma et al. [88 have shown that the diffusion coefficient can be chosen freely within the stochastic interpolant framework (I] Here, present more comprehensive proof. We use w interchangeably to denote the standard Wiener process for both forward and reverse time flows.
Proposition 1. For a linear stochastic process X, = axX2 + 0,X1 and the Probability-Flon ODE dx, = (x,(xx,) dt that yields the marginal density Pt (xx). the following forward and reverse SDEs
23

---

with an arbitrary diffusion coefficient 2 0 share the same marginal density:
9i v log logg:(x;)
Forward SDE: dx;-"
u (x())
dt g.dw
(20)
2
g log dt g.dw.
Reverse SDE: dx;;
ux(xx)
(21)
2
When velocity field ut generates a probability density path pt, it satisfies the continuity equation:
Proof.
0
Ht Pr(x))
(x:(xx)u,(xx,)
(22)
Similarly, for the SDE dx, = fr (x,)dt + g,dw, the Fokker-Plance equation describes the time evolution of ps:
0 0t (x,) = (xx + 5 Vz,
1
(23)
where 2 denotes the Laplace operator.
To find an SDE that yields the same marginal probability density as the ODE, we equate the probability density functions in Eq. 23] and Eq. 22 resulting in the following equation:
(p. (x) + 506 V px(x)) = - (pe(x:)ux(xi)
(pi( (x)) - un (xx))) = 2 V (x) (x,)
(24)
This implies that any SDE with drift coefficient f, (xx, and diffusion coefficient ge that satisfies Eq. will generate Pt. One particular choice is to set pi (x,, uy (x+) proportional to (p:(X)) (x..))(xx)(f((x))) (x4)) A. (x))).) Then Eq. 24 can be rewritten as:
24
A V p.
2 Px(),,
which leads to the relation A, 2 Similarly, the drift coefficient is given by:
1 Vp((x))
fx(x,) = un (xx) + う (x(x
1
(x,) + 96 V Pxxx)
Thus, a family of SDEs that generate pa takes the following form:
dx,
Ue (x:) + 2 log (x.)) dt + g.dw,
which is the forward SDE presented in Eq. 20 Similarly, the reverse SDE in Eq. 21 can be derived by applying the time reversal formula, following Anderson al. [2].
Corollary 1. If diffusioo coefficient chosen as 9t = /2 (a of dee
then the score function
V loggp(x)) inside the forward SDE vanish and it can be written as:
dx, = dg x,dt + v2 (arr of á² ai
dw
(25)
at
Proof. Velocity field (x.)) for linear stochastie process
X, ax X1
is given as:
un (xx)
x;
0100
of dit
loggp(
(26)
Plugging this equation into forward SDE Eq.
20 we can immediately see that when St
/ (orrr o? 0z the score function term vanishes and the remaining terms constitute Eq. 25
L

---

Figure 9: Analysis of number of function evaluations (NFEs) across timesteps. The NFEs re- quired to achieve a higher reward for each timestep. The plot il- lustrates the +1 sigma variation band. The blue-dotted line rep- resents the uniform allocation of compute (NFEs) across timesteps. We observe that the NFEs required to identify higher-reward sample may exceed the uniformly allocated budget (blue dotted line).
Unifore
10
20 First occurence of higher reward
C Adaptive Time Scheduling and Rollover Strategy
In this section, provide details of adaptive time scheduling and NFE analysis result which inspired rollover strategy.
Adaptive Time Scheduling. As discussed Sec. 4.3 to maximize the exploration space VP- SDE sampling, we design the time scheduler to take smaller steps during the initial phase phase-when variance is high and gradually increase the step size later stages. Specifically, we define the time scheduler as fnew = /I - (1 1)3. While this approach can be problematic when the number of steps is too low-resulting in excessively large discretization steps in later iterations-we find that using a reasonable number of steps (c.g..10) works well practice, benefiting from the few-step generation capability of flow models. This setup effectively balances broad exploration space with fast inference time, highlighting one of the key advantages of flow models over diffusion models.
NFE Analysis. As discussed in Sec. 6 we analyze the number of function evaluations (NFEs) required to obtain sample with a higher reward than the current one. In Fig. 9 we visualize the variance band of the required NFEs across timesteps, with the blue-dotted line representing the uniform allocation used in previous particle sampling methods 30 53 Notably, uniform compute allocation may constrain exploration and fail to identify high-reward samples, as evidenced by crossings within the variance band. This observation motivates the use of rollover strategy to optimize compute utilization efficiently. As demonstrated Sec. 7 our experiments confirm that
REF provides additional improvements over previous particle sampling methods [30 53
Renard
Reviadd
O
Figure 10: Schematics of inference-time search algorithms. Linear-ODts ODE-- methods, BoN and SoP use deterministic sampling process, whereas particle-sampling- based methods follow stochastic process. Note that RBF adaptively allocates NFEs across denoising timesteps.
D Search Algorithms
In this section, we introduce the inference-time search algorithms discussed in Sec. along with their implementation details. An illustrative figure of the algorithms is provided in Fig. 10 Here,
25

---

define the batch size (N) as the number of initial latent samples and the particle size (K) as the number samples drawn from the proposal distribution (x, A xx) each denoising step.
Best-of-N (BoN) 57 is form of rejection sampling. 1
Given N generated samples
{xo Ti=1-
BoN selects the sample with the highest reward.
arg max r(xb)) (8) 1
As presented in Sec. 7 we fixed the total compute budget to 500 NFEs and the number of denoising steps to 10, which sets the batch size of BoN to N 50.
Search over Paths (SoP) 33 begins by sampling N initial noises and running the ODE
solver to a predefined timestep to. Then the following two operations iterate until reaching t = 0:
1. Applying the forward kernel: For each sample in the batch at time t, K particles are sampled using the forward kernel, which propagates them from L + Ay.
Solving the ODE: The resulting N K particles are then evolved from t + A, to t + A - As by solving the ODE. The top N candidates with the highest rewards selected.
We followed the original implementations 39 for A and Ap. We used 2 and K
Sequential Monte Carlo (SMC) 23
14 extends the idea of importance sampling to time-sequential setting by maintaining samples and updating their importance weights over time:
p5(x1-A1|x))
px xx) exp
[ (x(x(-A)//))
w (1)
(8) w.
wi (8)
q (xx-Ar( exp ((x( )/3)
where ((x;- is proposal distribution and the last equality follows from the optimal policy Eq. [8] We used the reverse process of the pretrained model as the proposal distribution, which leads to the tollowing importance weight equation:
exp 0(x (xxx/11)/B
t Ar exp(v( ((x(())))) (2) wi
(27)
At each step when effective sample size (Σi=1 w()i) )² E (2) is below the threshold, we (1) 1 perform resampling, i.c... indices {a, Ji=ı are first sampled from a multinomial distribution based on the normalized importance weights:
I Multinomial
N,
W
w
These ancestor indices af are then used replicate high-weight particles and discard low-weight ones, yiclding the resampled set {x! (a(*)) \WW If resampling is not performed, the indices are simply
(a) (1) ) li=1:
set as di i. Lastly, one-step denoised samples are obtained from {x,
Ae Pe(x
When resampling is performed, the importance weights are reinitialized to one, i.e.,, = 1. The importance weights for the next step, W) are subsequently computed according to Eq. 27 regardless of whether resampling was applied. We used N 50 for all applications.
26

---

Controlled Denoising (CoDe) step after every L denoising steps.
[3 extends BoN by incorporating an interleaved selection
arg max exp (o(x
X LA
(x) La
We used = 2, K 25, and T. = 2 for all applications.
SVDD
30
approximates the optimal policy in Eq. by leveraging weighted K particles:
K 2
(28)
w L-
1=1
1 poo -=((x:-(((x)
" exp
(1( (xxx--a)/11)
At each timestep, the approximate optimal policy in Eq. 28 is sampled by first drawing an index
dt-Ar Ar from categorical distribution:
w w A-
0t At Categorical
(29)
(5)
E== t- Ar
This index is then used to select the sample from {x (x;-Arliz) i.e.,, X &- (a1-a)) In practice, Xi-A!"
SVDD uses 3 = O, replacing sampling from the categorical distribution with a direct arg max operation, i.e., selecting the particle with the largest importance weight. Following the original implementation [30]. we used N and 25 for all applications.
Rollover Budget Forcing (RBF) adaptively allocates compute across denoising timesteps. At each timestep, when particle with a higher reward than the previous is discovered, it immediately takes a denoising step, and the remaining NFEs are rolled over to the next timestep, ensuring efficient utilization of the available compute. To maintain consistency with SVDD (30), we set N = 2, with the compute initially allocated uniformly across all timesteps. We present the pseudocode for sampling from the stochastic proposal distribution with interpolant conversion in Alg. [ Specifically, the pseudocode for RBF with SDE conversion and interpolant conversion is provided Alg. 2 Here, we denote (S(()) as sequence timesteps descending order, where s(2) = and (())) 0, and M the total number of denoising steps.
E
Additional Results
E.1
Aesthetic Image Generation
In this section, we demonstrate that inference-time scaling can also be applied to gradient-based methods, such as DPS [8] for differentiable rewards. Specifically, we consider aesthetic image generation and show that RBF leads to synergistic performance improvements. We first derive the formulation of the proposal distribution for differentiable rewards and then present qualitative and quantitative results.
E.1.1
Gradient-Based Guidance
Uehara et al. 61 have shown that the marginal distribution P (xx) is computed as follows:
(v(x))
(x())) exp (x(x)j))
pt (xx) x exp
pc px(x)..
B
27

---

Algorithm 1: :eoos_denoise: 1-step
Algorithm 2: Rollover Budget Forcing (RBF)
Inputs:
Number of denoising steps M,
stochastic denoising
timesteps {S(1 1
Ti=1s NFE quota
Inputs: original velocity field u,
{QO) Ji=1
original interpolant (a.
Outputs: Aligned sample X X1 ~ (0,1)) r' < ((oo1)
new interpolant (ã, Ø),
for i {1 (1...., M) do
diffusion coefficient current sample x,, current timestep s, denoising step size As
8 S(1 Ås t S - 5(++) + Q
for jE {I...... o) do
x!6) 3-A= As + stoch stoch_ demoise (x,, As))
// Alg
Outputs: Stochastically denoised
if
<r(0))
())
then
sample x p (p(s)) ca t 0a/or,
0
Q(++1)
Q(+1)) + Q(1 -
// Sec.
úg f- Ex,, + cotolle, & 12
// Eq.
010 (x Re-As
6
3 S, f- 1 1
// Eq.
x 5- break
0, a.d, -0,
= Us,
L. Sa
// Eq.
2
10
j = then
I(,
11
k' - arg max (1.....) r(x0)=_be)
(K)
As 8, G,A. g.v Aoz
12
Xz- t x-
FLUX
DPS
SVDD DPS
RBF (Ours)
DPS
"Bird"
Table 2: Quantitative results of aesthetic image generation. denotess the given re- ward used in inference time. The best result in each row is highlighted in bold.
Model
Aesthetic Score'
ImugeRRward
50
67 (held-out)
5,795
DPS
g
6,438
0.605
SVDD
30 DPS
6.887
1.077
RBF (Ouss)=DPS
7.170
1.152
Figure 11: Qualitative results of aesthetic image guide generate
generation. At inference-tine using the aesthetic score [50] as the given reward, which assesses visual appeal.
where the approximation follows from Eq.
When the reward is differentiable (c.g.. aesthetic score [50)), one simulate samples from pi (x:) by computing its score function:
19
((Xoje)
V log (x,) = V log exp
Pe (xx))
Vr(xo1) + (x:)
(30)
Guidance
Pretrained Score
For differentiable rewards, we incorporate the gradient-based guidance defined Eq. 30 into the SDE sampling process described in Eq. Notably, this approach is orthogonal to inference-time
28

---

Table 3: Comparison of diffusion and flow models.
Type
Model
ImageReward
67
IIPS
CLIP
Steps
SD2
48
0,429
0.280
0,218
0.269
Diffusion
SANA-I.5
66
0.894
0.284
0.222
0.270
0.294
0.277
Flow
FLUX 28
1,054
0.290
0.226
0.275
scaling, and RBF can be additionally utilized to further enhance performance. In the next section, we experimentally demonstrate that RBF can be effectively integrated with gradient-based guidance.
E.1.2 Aesthetic Image Generation Results
The aesthetic image generation task aims to sample images that best capture human preferences, such as visual appeal. We use 45 animal prompts from previous work, DDPO [5]. The aesthetic score [50] serves the given reward, while ImageReward (67 is used the held-out reward.
We present quantitative and qualitative results of aesthetic image generation in Tab. 2ab.2nm. and Fig. Notably, RBF, implemented with DPS [8], achieves significant improvements on both the given and held-out rewards, even surpassing SVDD (3) Qualitatively, RBF effectively adapts the pretrained flow model to better align with human preferences, particularly in terms of visual appeal.
E.2
Comparison of Diffusion and Flow Models
We present quantitative comparisons between text-to-image diffusion and flow models Tab. 3 using compositional text prompts from GenAl-Bench [21]. As shown, flow-based models outper- form diffusion models across all evaluation metrics assessing image quality [6 65 26 and text
alignment [44] 67]. In the flow-based models, FLUX (28 achieves competitive performance while
requiring fewer steps compared to Stable Diffusion 3 [5]
E.3 Scaling Behavior Comparison
As discussed in Sec. expanding the exploration space and applying budget forcing significantly enhance the efficiency RBF, leading to superior performance improvements over BoN, Here, we compare the scaling behavior of BoN, a representative Linear-ODE-baser method, with RBF across different numbers function evaluations (NFEs).
We report qualitative and quantitative scaling results for quantity-awee aware image generation (Fig. 12 Tab. 4 and for compositional text-to-image generation (Fig. [3] Tab. 5). respectively. Our results indicate that allocating more compute leads to performance improvements for both BoN and RBF. However, the performance of BoN plateaus after 300 NFEs, whereas RBF continues to scale and achieves the highest reward in both tasks. Notably, RBF shows similar trend in the held-out reward, outperforming BoN and demonstrating its efficiency.
Time Complexity and Compute Analysis. We present time complexity of scaling methods in Tab. [ Let S as the number of denoising steps, N the NFE budget, and and the costs of the denoising and verification, respectively. Since all methods share the same NFE budget N, the total denoising cost is fixed at N ca. For the verification cost, although BoN has the lowest cost, REF consistently outperforms BoN across all NFE budget regimes in both compositional text-to-image generation and quantity marginal increase in verification overhead.
image generation tasks (Fig. and Fig. 13] while incurring only
Additionally, at inference time, user can specify the compute budget (NFEs), which determines the total runtime our method. We report the runtime of RBF in Tab. 7 Under 500-NFE budget, scaling for compositional text-to-image generation (VQAScore (31) requires 635.01 seconds per image. Runtime can be reduced by lowering the NFE budget-aa at the cost of reward performance-aae and further accelerated by decreasing output resolution or increasing batch size.
29

---

Table 4: Quantitative results of quantity- aware image generation in NFE scaling ex- priment. We use the same 100 prompts from
RBF (Ours)
500
BoN
0.-
T21-CompBench (20 denotes the given re-
Accuracy
ward.
0.6
300
500
NHEs
RSS A4
Acc.
VQAScure (held-out)
B1]
Aestbetic Score 50
100
0.4
50
50
4.360
0.400
0.758
5.408
100
3,280
0,510
0.750
5.522
R55 (Log Scale)
BoN
5.463
1,760
0.580
0.756
5,420
Figure 12: Quantity-aware image generation scaling behavior comparison of BoN and RBF, We plot the known reward (RSS) [4] against accuracy for different numbers of function eval- uations: {50. 100, 300, 500, 1, 000}. Note that the horizontal axis displayed on logarithmic scale.
1000
1,340
0.590
0.759
5.466
50
3,250
0.410
0.756
5,560
RBF (Ous)
100
1.860
0.590
0.760
5.627
300
0.690
0.720
0.779
5.503
0.540
0,800
0.769
5.581
0.777
5.526
Table 5: Quantitative results of compositional text-to-image generation in NFE scaling ex- Priment. We use the 121 prompts from GenAI- Bench (21) denotes the given reward.
0.88
RBF (Ours)
0.86
BoN
300
500
Inst. BLIP
1000
NFTs
VQAScore
Inst BLIP 10
Aesthetic
1000
(held-out)
f
500
0,8310
0.8011
5.2246
50
100
0.8459
0.7959
5,2594
BoN
300
0.8775
0.8250
5.1414
500
0,8790
0.8200
5,1620
VQAScore
0.88
1000
0,8886
0.8269
5,2055
Figure 13: Compositional text-to-image gen- eration scaling behavior comparison of BoN and RBF. We plot the known reward (VQAS-
0.8577
0.8253
5,2704
RBP (Ours)
100
0.8824
0.8212
5.3213
300
0.9146
0.8387
5,2837
0.9250
0.8430
5.2370
core) 31 against the held-out reward [10]
1000
0.9283
0.8369
5.2593
for different numbers function evaluations:
{50, 100, 300, 500, 1, 0003.
Table 6: Time complexity of scaling methods.
Table 7: Runtime of RBF.
SVDD BU,
50
300
500
1000
Base
BoN
SMC
RBP
Runtime (sec)
84.00
140.11
383.89
635.01
1243.68
S-ea N oo
1op N vog + N.c.
N +
VQAScore 33
0.858
0.882
0.915
0.925
0.928
For all experiments, we use FLUX 28 which requires approximately 32GB of GPU memory, accounting for the majority of overall memory usage. All evaluations are performed on an NVIDIA RTX A6000 GPU.
F
Implementation Details
F1
Choice of Hyperparameters
We report quantitative results on aesthetic score [50] and diversity (23 for images generated under different settings of the number of denoising steps and the diffusion coefficient. As shown in Tab. 8 a), the number of denoising steps beyond 10 gives marginal gains. Hence, we fixed the number of denoising steps 10 to ensure fair and efficient evaluation across all methods, Note that once the
30

---

Table 8: Choice of hyperparameters. Evaluation of the images generated with different (a) number of denoising steps and (b) diffusion coefficient.
Norm
Aesthetic (50) g(t)
Diversity 23
Aesthetic
50
Diversity g(t)
A1
Steps
Aesthetic
50
Diversity
g(t)
g(t)
10
5.635
0.084
5.635
20
5,680
0.103
(a) Number of denoising steps
4.608
0.223
4.838
(b) Diffusion coefficient
number of denoising steps is fixed, the total partiele count per step is automatically determined by dividing the total NFE budget by the number of steps. Additionally, Tab, ((b) reports results obtained under varying diffusion coefficients scaled by different norms. We found that using g(t) 3t²
consistently offered the best trade-off between sample diversity and output fidelity, so we adopt this setting for all SDE sampling.
F.2 Compositional Text-to-Image Generation
In the compositional text-to-image generation task, we use the VQAScore as the reward, which evaluates image-text alignment using visual question-answerte and InstructBLIP [TO). Specifically, VQAScore measures the probability that given attribute or object is present in the generated image. To compute the reward, we scale the probability value by setting 3 = 0.1 in Eq.
(VQA) model (CLIP-Flan'SS [31
F.3 Quantity-Aware Image Generation
In quantity-aware image generation, text prompts specify objects along with their respective quantities. To generate images that accurately match the specified object counts, we use the negation of the Residual Sum of Squares (RSS) as the given reward. Here, RSS is computed to measure the discrepancy between the detected object count Ó; and the target object count C₁ in the text prompt:
RSS Σi= (c. ( CC) where is the total number object categories in the prompt. We additionally report accuracy, which is defined as 1 when RSS = 0 and 0 otherwise. For the held-out reward, we report VQAScore measured with CLIP-FlanTS
model.
Object Detection Implementation Details. To compute the given reward, RSS, it is necessary to detect the number of objects category, C. Here, we leverage the state-of-the-art object detection model, GroundingDINO 34] and the object segmentation model SAM 25 which specifically used to filter out duplicate detections.
We observe that naively using the detection model (34] compute RSS leads to poor detection accuracy due to two key issues: inner-class duplication and cross-class duplication. Inner-class duplication occurs when multiple detections are assigned to the same object within category, leading to overcounting. This often happens when an object is detected both individually and as part of larger group. Cross-class duplication arises when an object is assigned to multiple categories due to shared characteristics (e.g... toy airplane being classified as both a toy and an airplane), making it difficult to assign a single category.
To address inner-class duplication, we refine the object bounding boxes detected by Ground- ingDINO (34] using SAM (25] and filter out overlapping detections. Smaller bounding boxes are prioritized, and larger ones that significantly overlap with existing detections are discarded. This ensures that each object is counted only once within its category. To resolve cross-class duplication, we assign each object to the category with the highest GroundingDINO 34 confidence score which prevents duplicate counting across multiple classes.
More qualitative results are presented in the following pages.
31

---

Additional Qualitative Results
G.1
Comparisons of Inference-Time SDE Conversion and Interpolant Conversion
Lincar-ODE
Lincar-SDe SDE
VP-SDE
LincarODE
Lincar-SDE
VP-SDE
Sii people guthered for picatie."
"Four condles, two bulloons, une dlgg Prio Anmaturs and liree helivopterss"
SMC 2
Quanity
"Kour rubbits, three apples, two mice and jour televisions."
"evvn pigs smorted and played in mud."
SMC 23 Quanity
"Tho frogs competing high Jamp.
purpic genstones pink genutonn, with the pink gonsione hoving the smoothen looking surfoce:"
The frog blue tacksuit jumps highter than the frog not blue tracknoi."
S.C 23
Compostion
Linear-ODE
Lineur-SDi
VP-SDE
Linear-ODE ODE
Linear Linear-SDE SDE
VP. SDD
"Seven helmery"
"Four concher, three condies, tvie firh, one frog and three plates."
CoDe 53 Quanity
"Seven lampy."
"Seven
CoDe 133 Quanity
"A with busebvll rap ronuching unuther fing without cap crowling on ligger loaf"
hats, plain, feathers.
CoDe 133
Compositon
Figure 14: Additional qualitative results of inference-time SDE conversion and interpolant conversion.
32

---

Lincar-ODE
Lincar-SDE
VP-SDE
Lincar-ODE
Lincar-SDE
VP-SDE
"Sta boitles"
"Fue kumbargers siculed on the piill
SVDD Quaniity
chickons ships."
SVDD 30 Quanity
"Two and bicycles the bicyclee the lorger belongs
"There cuger the table without coffee the other coffee."
SVDD 30
Compos potiiit
Lincar-ODE
Lincar-DEE
VP-SDE
Lincar-ODE
Lincar-SDE
VP-SDE
"Two piraffes. three brends. and
pears. desky, thmee
RBF (Ours) Quanity
"Feur batloons, four desks, four vakceewavve.."
"Sevon
RBF (Ours)
Quanity
"Two binds are chasing each other the alc, with the higher having other bird having toil."
"Three satiioute the cach with sails differenn color"
RBF (Ours)
Campostion
Figure 15: Additional qualitative results of inference-time SDE conversion and interpolant conversion.
33

---

G.2
Comparisons of Inference- Time Scaling
BoN
SoP
39
SMC
CoDe
SVDD
30
RBF (Ours)
"In room, all the chairs are occupied except one."
"Three mugs are placed side by side;
the two closest to the faucet each contain toothbrush, while the one furthest away is empty.
"Three flowers in the meadow. with only the red rose blooming: the others are not open."
"In pack of wolves, each one howls at the moon, but one remains silent."
"An open biscuit contains three biscuits, one without sultanas is square-shaped and the other two are round-shaped""
"A rose that is not fully bloomed is higher than a rose that is already in bloom."
"There are two colors of pots in the flower garden; all pots have tulips in them and all yellow pots have no flowers in them."
Figure 16: Additional qualitative results of compositional text-to-image generation task.
34

---

BoN
SoP 39
SMC 223
CoDe 53
SVDD
30
RBP (Ours)
"Seven balloons, four bears and four swans.
"Six horses and six deer and four balloons." 1
"Eight apples, three bicycles and five rabbits."
"Six helicopters buzzed over eight pillows."
"Five swans and seven ducks swam in the pond." "
Four drums, seven tomatoes, and five candles."
"Three chickens, four birds, and eight pears."
"Six airplanes flying over a desert with seven camels walking below."
oe
Aeft
NARNWEN
Figure 17: Additional qualitative results of quantity-aware image generation task.
35


Reviews:
[{"review_id": "bB5uxVu2pY", "reviewer": "Reviewer_tyvx", "summary": "This paper adapts particle sampling to flow-matching models. Particle sampling consists of generating several candidates along the denoising process and picking the best according to a defined reward function. Flow-matching models are deterministic, and thus cannot generate several candidates to enable particle sampling. To address this, the paper proposes two changes to the sampling process of flow-matching models: it derives the stochastic path corresponding to the flow-matching ODE (this introduces the stochasticity needed to generate several candidates) and converts the uniform interpolant to the VP interpolant commonly used in diffusion models (this increases the variance of the generated candidates). On top of these changes, the authors propose a greedy budget-allocation strategy that greedily distributes the number of particles generated at each denoising time-step. These 3 changes are evaluated and perform well on compositional text-to-image generation and quantity-aware generation.", "questions": "1. The greedy budget re-allocation scheme is interesting. However, from Figure 9, it seems that it is \"rare\" for the `time of occurrence of higher reward` to exceed the uniform budget. If the authors were to remove the budget reallocation strategy but simply cut the uniform budget early (i.e removing line 7 in alg.2) does RBF simply become CODE? Please correct my understanding if not. Moreover, it appears that RBF is harder to parallelize compared with other search schemes. Could the authors discuss this some more?", "limitations": "A limitation the authors should discuss is the aesthetics score, which appears to be harmed by most of the inference time modifications.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "## Strengths\n- The paper is very well written and builds up its ideas in a clear and understandable manner. Synthetic examples provide good illustrations.\n- The proposed ideas, namely VP-SDE sampling of flow models performs well, compared to the baseline, non-stochastic or linear variants, in some cases.\n\n## Weaknesses\n1. The comparison with diffusion models needs more discussion: the authors effectively propose a method diffusion-ify a pre-trained flow model. The comparisons should be between a particle sampling method applied to diffusion models vs the authors' VP-SDE flow models. The authors provide a table in the appendix comparing base diffusion and base flow models but the more interesting comparison is one where both are using particle-sampling. Does the NFE advantage of flow models persist in that setting as well? Is the paper's message that if one wants to do inference-time scaling, then convert the flow-model to a diffusion model at inference time? Moreover, some care is necessary when comparing more recent, larger flow models with older diffusion models.\n2. The two main ingredients of the paper which are:  1)changing between deterministic ODEs and stochastic paths and 2) interpolant conversion of a pre-trained model are known results. The combination and the end-goals are different, which is why I still recommend acceptance. However, the method is not adapting particle sampling to flow-models, but rather making flow-models as diffusion model like as possible to apply existing inference time-scaling methods.\n---\nminor point: The Laplacian symbol $\\Delta$ might be better to avoid confusion with the Hessian symbol in the proof of Proposition 1.", "was_revised": false, "final_justification": "", "rebuttal": "**[W1: Comparison with Diffusion Models under Particle Sampling]**\n\nWe sincerely thank the reviewer for this insightful point regarding comparisons with diffusion models.\n\nWhile an ideal comparison would involve models trained under identical architecture and dataset conditions, such training is resource-intensive. Instead, we chose the most competitive available models: **FLUX** (flow model, Aug. 2024) and **SANA** (diffusion model, Jan. 2025), both trained on high-quality datasets. We acknowledge that this may appear to be an unfair comparison, which is why we omitted it from the main paper. However, we include it here as a reference point to illustrate the distinct advantages that flow models can offer over diffusion models.\n\nTo address the reviewer’s concern, we apply **particle sampling to both models**, where ×2 and ×3 denote increased compute budgets for the denoising steps. As shown in **Tab. R1**, flow model consistently outperforms SANA across all settings:\n\n**Tab. R1**: *Comparison of particle sampling between SANA (diffusion) and FLUX (flow)*  \n| NFE         | SANA (1×) | SANA (2×) | SANA (3×) | FLUX (Ours) |\n|-------------|-----------|-----------|-----------|----------------------|\n| VQAScore ↑  | 0.682     | 0.695     | 0.706     | **0.925**            |\n| InstBLIP ↑  | 0.749       | 0.737      | 0.745    | **0.843**            |\n| Aesthetic ↑ | 5.288     | 5.338     | 5.354     | **5.237**            |\n\nThis confirms that our approach enables effective inference-time scaling in flow models, not just mimicking diffusion, but actually leveraging the structural advantages of flow models for superior performance. Notably, reflow training in flow models reduces trajectory entanglement, ensuring that each sampling path remains well-separated. When trajectories remain disentangled, the velocity at each intermediate state is predicted based solely on the local sample path, rather than being implicitly averaged across nearby trajectories. This leads to more accurate predictions that point toward the target data sample, rather than toward a smoothed or interpolated region. As a result, the posterior mean estimates are more accurate and less prone to blurring, improving the reliability of reward evaluation in particle sampling. While we briefly mention this in L239–L242, we will revise the final version to ensure that this distinction is better presented. \n\nIn short, our work does not simply convert flow models into diffusion models, but instead expands their inference-time capabilities while preserving their core strengths—offering distinct advantages for efficient scaling. \n\n---\n\n**[W2: Clarification on \"making flow-models as diffusion model like as possible\"]**\n\nThank you for recognizing that the combination and end goals of our approach differ from those of previous works. In response to your comment, we would like to highlight the following:\n\n1. Applying particle sampling to a flow model critically requires the ODE-to-SDE conversion, as stochastic sampling at intermediate steps is essential for particle sampling. More importantly, we have shown in this work that **inference-time particle sampling—traditionally exclusive to diffusion models—is both feasible and effective in flow models**. \n\n2. Our use of interpolant conversion from Linear-SDE to VP-SDE was not merely intended to “diffusion-ify” a flow model. Rather, it was the result of empirically identifying an interpolant that provides sufficient diversity during intermediate sampling without sacrificing output quality. To our knowledge, we are also the **first to demonstrate that the diversity of samples can be controlled by navigating the interpolant space**. We emphasize that our work offers a **principled analysis of how interpolant design governs the variance of the proposal distribution** (**Appendix C**), a connection which was not explored before. Additionally, we present an ablation study in the **response to W2 of Reviewer kdqx**, where the impact of each component of interpolant conversion on sample diversity is systematically evaluated. \n\n---\n\n**[Q1: RBF reduces to SVDD without rollover strategy]**\n\nIf we remove the budget reallocation mechanism (e.g., **Line 7 in Alg. 2**), RBF reduces to SVDD. CoDe runs verification (reward) only at periodic timesteps, whereas SVDD performs candidate selection at every step. \n\nWe would like to clarify two key points regarding the behavior of RBF. \n\nFirst, **Fig. 9** illustrates the standard deviation ($\\sigma$) of the required NFEs to discover a higher-reward sample. Importantly, as more prompts are evaluated, this leads to an increasing number of cases where the optimal timestep requires more NFEs than the uniform allocation. In such cases, RBF’s dynamic reallocation becomes critical for efficiently identifying higher-reward particles. \n\nSecond, while RBF may seem less parallelizable as it involves a sequential component due to its greedy budget reallocation, in practice, RBF can be parallelized efficiently by increasing the batch size (the number of initial samples). In contrast, methods such as SoP and SMC are inherently more difficult to parallelize, as they require correction or resampling steps that involve cross-sample communication, necessitating complex engineering for synchronization. \n\n---\n\n**[L1: Aesthetic Score Drops]**\n\nWe appreciate the reviewer’s thoughtful observation. Indeed, reward over-optimization is a well-known issue in reward alignment tasks, where aggressively optimizing toward a given reward can lead to a significant drop in other metrics (e.g., aesthetic quality). \n\nIn our experiments, while we observe a slight drop in aesthetic score, we emphasize that the **degradation is minor and does not result in noticeable artifacts or perceptual quality loss**. As illustrated in Fig. 5 and 7, the generated images remain sharp and visually compelling—even as alignment performance improves significantly (x6 accuracy improvement in quantity-aware image generation)—demonstrating the robustness of our approach. \n\nAdditionally, we clarify that this trade-off is a **controllable design choice**: users can moderate the level of reward optimization by reducing the NFE budget to suit downstream needs. Overall, our results show that inference-time scaling enables high-impact improvements in alignment while preserving the expressive priors of flow models."}, {"review_id": "cr24rP4uOb", "reviewer": "Reviewer_P9ao", "summary": "The paper introduces VP‑SDE, an inference‑time scaling framework for pretrained flow models that brings the benefits of particle sampling—previously exploited in diffusion models—to the deterministic flow setting. First, it casts flow generation as an SDE, enabling Monte Carlo sampling of multiple trajectories. Second, it enlarges the candidate output space via a Variance‑Preserving (VP) interpolant conversion. Third, it allocates computational budget adaptively across timesteps through Rollover Budget Forcing (RBF), dynamically adjusting the number of function evaluations (NFEs). Empirical results on Compositional Text‑to‑Image Generation and Quantity‑Aware Image Generation demonstrate that VP‑SDE with RBF outperforms prior inference‑time scaling methods for flow and diffusion models.", "questions": "1. Can you provide a detailed time‑ and space‑complexity analysis of each sampling strategy (e.g. vanilla flow, SDE‑based, VP interpolant, RBF) under a fixed compute budget? In particular, is Rollover Budget Forcing (RBF) truly optimal in all regimes, or only once you scale up the model?\n\n2. The phrase “much clearer posterior mean” on line 241 is vague—what metric or visualization supports this description?\n\n3. Could you supply a comprehensive list of all critical hyperparameters (e.g. number of particles, RBF schedule parameters, step size for SDE integration) and the rationale behind their chosen values?\n\n4. How sensitive are your results to the number of particles and the specific RBF allocation schedule?\n\n5. Have you tested VP‑SDE and RBF on other flow model variants (e.g. neural ODE flows, continuous normalizing flows) or non‑image tasks (e.g. audio, tabular data)?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n- The paper pioneers the integration of particle sampling into flow matching models by reformulating the deterministic probability path evolution as a stochastic process, achieving a more effective exploration–exploitation trade‑off.\n\n-  The manuscript is well organized and reads smoothly; theoretical derivations are detailed and easy to follow.\n\n-  The supplementary material is thorough and supports the main text with additional proofs and experimental details.\n\n## Weaknesses\n\n-  Prior works (e.g., SiT) have already applied very similar techniques to improve the sampling process; this paper adopts those ideas almost unchanged, resulting in limited novelty.\n\n-  While the authors claim that stochastic, variance‑preserving sampling strategies help discover high‑reward, low‑density regions, they provide neither rigorous theoretical justification nor dedicated ablation studies to substantiate this claim, leaving the exploration benefit insufficiently supported.", "was_revised": true, "final_justification": "The authors clarified their contributions and conducted a comprehensive ablation study to address my concerns about weaknesses; they also answered my questions regarding confusion and paper details. I think this is a good work on exploring the integration of variance into flow models, only with some ambiguity and lack of evidence for specific claims, so I recommend Borderline accept.", "rebuttal": "**[W1: Clarification of our contributions]**\n\nWe would like to emphasize that our core contribution does not lie in re-deriving techniques such as SDE conversion, but rather in repurposing and integrating them through the lens of particle sampling to **enable efficient inference-time scaling for flow models**.\n\nTo the best of our knowledge, this work is the first to show that **inference-time particle sampling—traditionally exclusive to diffusion models—is both feasible and effective in flow models**. This is a nontrivial insight: when using ODE-based sampling, all particles deterministically collapse into a single trajectory, rendering exploration effectively impossible. The introduction of an SDE injects variance into the proposal distribution, forming the basis for reward-guided sampling in flow models. \n\nAdditionally, we observe that SDE conversion alone is insufficient for achieving effective inference-time scaling. To address this, we establish— **for the first time— a connection between interpolant design and sample diversity**. Our work further provides a **principled analysis of how interpolant design governs the variance of the proposal distribution** (see **Appendix C**). This analysis is supported by an ablation study included in the **response to W2 of Reviewer kdqx**, where the impact of each component of interpolant conversion on sample diversity is analyzed. \n\nOur contributions span three key aspects:\n- We reinterpret SDE conversion as a tool for constructing an effective proposal distribution in particle sampling, **enabling efficient inference-time scaling in flow models**.\n\n- To our knowledge, this is the **first work to view interpolant conversion from the perspective of sample diversity** and apply it for improving the efficiency of inference-time scaling. We present a detailed analysis that reveals how interpolant conversion influences the sample diversity.\n\n- Additionally, we introduce Rollover Budget Forcing (RBF), a novel particle sampling strategy that **adaptively reallocates compute** across timesteps to maximize utilization of compute budget. \n\n---\n\n**[W2: Ablation study of SDE and interpolant conversion in sample diversity and reward alignment]**\n\n**Tab. R1**: *Ablation study of SDE and interpolant conversion* \n\n| Method                      | Diversity ↑ | Aesthetic ↑ | VQAScore ↑ | Inst. BLIP (Held-out) ↑ |\n|----------------------------|-------------|--------------|-----------------|--------------------|\n| Linear-ODE                 | –           | 5.624        | 0.788           | 0.789              |\n| Linear-SDE                 | 0.158       | 5.436        | 0.900           | 0.813              |\n| + Adapt. Diffusion         | 0.429       | 3.986        | 0.702           | 0.571              |\n| + Adapt. Timestep          | 0.270       | 5.487        | 0.908           | 0.813              |\n| VP-SDE (Ours)              | **0.509**   | **5.830**    | **0.925**       | **0.843**          |\n\nWe thank the reviewer for raising this important point. We present empirical results on how increased diversity can enhance reward alignment in **Fig. 4 and 6**. To further support these findings, we include additional quantitative results in **Tab. R1**, which substantiate how both SDE and interpolant conversion contribute to improved sample diversity and reward alignment under a controlled setup. We measure diversity using LPIPS-MPD [23], sample quality using the Aesthetic Score on 1,000 generated images. \n\nWe first highlight that Linear-SDE significantly increases sample diversity over Linear-ODE, thereby expanding the exploration space and improve reward alignment. Building on this, we observe that VP-SDE further enhances sample diversity and expands the exploration space, leading to improved reward alignment. This improvement stems directly from interpolant conversion, which integrates two key mechanisms analyzed in **Appendix C**: diffusion norm scaling and timestep conversion. Due to space constraints, we refer the reviewer to our **response to W2 of Reviewer kdqx** for a more detailed analysis of how each component contributes to sample diversity. \n\n---\n\n**[Q1: Time complexity and RBF scaling]**\n\nWe clarify that the RBF scaling in quantity-aware image generation is presented in **Fig. 12**. Here, we include time complexity analysis and RBF scaling in compositional text-to-image generation. \n\n**Tab. R2**: *Inference cost*\n\n| Method         | Inference Cost                                   |\n|----------------|--------------------------------------------------|\n| Base           | $S \\cdot c_d$                                    |\n| BoN            | $N \\cdot c_d + \\frac{N}{S} \\cdot c_v$            |\n| SMC            | $N \\cdot c_d + N \\cdot c_v$                      |\n| RBF, SVDD     | $N \\cdot c_d + N \\cdot c_v$                      |\n\n- $S$: Denoising steps\n- $N$: NFE budget\n- $c_d$, $c_v$: Cost of denoising and verification (reward)\n\nAs shown in **Tab. R2**, since all methods share the same NFE budget, the cost for denoising is set to $N * c_d$. \nThe verification cost varies across methods. Notably, both SVDD and RBF consistently outperform BoN across all NFE budget regimes (**Fig. 12 and Tab. R3**), despite incurring only marginal overhead in verification cost. In terms of space complexity, all methods have similar memory usage, with the flow model accounting the most (32GB). We appreciate the comment and will revise the paper to include the analysis of time complexity. \n\n**Tab. R3**: *Scaling comparison*\n\n| Method           | NFE | VQAScore | Inst.Blip (Held-out) | Aesthetic |\n|------------------|-----|-----------|------------------------|------------------|\n| **BoN**     | 100 | 0.846     | 0.796                  | 5.259            |\n|                  | 500 | 0.879     | 0.820                  | 5.162            |\n|                  | 1000| 0.889     | 0.827                  | 5.205            |\n| **SVDD (Linear-SDE)** | 100 | 0.861     | 0.791                  | 0.809            |\n|                       | 500 | 0.893     | 0.813                  | 5.052            |\n|                       | 1000| 0.892     | 0.820                  | 0.851            |\n| **SVDD (VP-SDE)**     | 100 | 0.874     | 0.809                  | 5.286            |\n|                       | 500 | 0.915     | 0.847                  | 5.249            |\n|                       | 1000| 0.921     | 0.851                  | 5.281            |\n| **RBF (VP-SDE)**     | 100 | 0.882     | 0.821                  | 5.321            |\n|                  | 500 | 0.925     | 0.843                  | 5.237            |\n|                  | 1000| 0.928     | 0.837                  | 5.259            |\n\nThe results in **Fig. 12 and Tab. R3** show that RBF maintains strong performance across all NFE regimes. Additionally, we note that **the total number of particles per step is automatically determined for all baselines as NFE budget divided by denoising steps**, as described in **Appendix. E**. We further emphasize that RBF introduces no additional hyperparameters. The initial NFE quota is uniformly allocated across all steps from the total budget, and RBF dynamically reallocates any unused quota based on the discovery of higher-reward samples. \n\n---\n\n**[Q2: Ambiguity of “much clearer posterior mean”]**\n\nWe agree that the phrase could be more precise. Flow models trained with rectification exhibit more **disentangled trajectories**, allowing their velocity predictions to be computed with minimal entanglement across samples. This reduces averaging artifacts and blurry posterior means often seen in diffusion models with overlapping trajectories.\n\nAs a result, the posterior mean $\\mathbb{E}[x_0 \\mid x_t]$ (Tweedie’s formula) is more accurate estimate of the final sample, improving reward prediction during inference. We will clarify this point and include supporting visuals in the final version.\n\n---\n\n**[Q3: Hyperparameter choice and rationale]**\n\nWe clarify our choice of hyperparameters. We set denoising steps to 10 for all experiments and diffusion coefficient $g(t) = 3t^2$. Below we provide rationale of this choice.\n\n**Tab. R4**: *Number of denoising step*\n| Steps | Aesthetic | Diversity |\n|-------|------------------|-----------|\n| 10    | **5.635**            | **0.084**     |\n| 20    | 5.680            | 0.103     |\n\n**Tab. R5**: *Diffusion coefficient*\n| Norm | Aesthetic $g(t)=t$ | Diversity $g(t)=t$ | Aesthetic $g(t)=t^2$ | Diversity $g(t)=t^2$ |\n|------|---------------------------|----------------------------|---------------------|----------------------|\n| 1    | 5.635                     | 0.084                      | 5.652               | 0.083                |\n| 3    | 5.168                     | 0.153                      | **5.436**               | **0.158**                |\n| 5    | 4.608                     | 0.223                      | 4.838               | 0.187                |\n\nAs shown in **Tab R4**, the number of denoising steps beyond 10 gives marginal gains. Hence, we fixed the number of denoising steps to 10 to ensure fair and consistent evaluation across all methods. Note that once the number of denoising steps is fixed, the total particle count per step is automatically determined by dividing the total NFE budget by the number of steps.\n\n**Tab. R5** shows sample diversity and sample quality with different diffusion coefficient. We found that using $ g(t) = 3t^2 $ consistently yielded the best balance between sample diversity and output fidelity across tasks. \n\n---\n\n**[Q4: Particle size and RBF schedule]**\n\nWe clarify that varying the particle size is equivalent to scaling the NFE budget. Please see our **response to Q1** for details. \n\n---\n\n**[Q5: Other flow models and domains]**\n\nWe agree that the proposed method is broadly applicable to other modalities—such as molecular generation, audio, or tabular data—and we are planning to extend our framework to these domains."}, {"review_id": "yMmaJ59Y0Z", "reviewer": "Reviewer_kdqx", "summary": "This paper considers inference time scaling for generating high quality images using flow models by leveraging the existing inference time scaling methods. The paper proposes three techniques to achieve the above purpose, which are (1) extending the deterministic ODE sampling to stochastic sampling, (2) transforming the stochastic sampling to VP-based stochastic sampling, and (3) Rollover Budget Forcing to dynamically allocate the computational resources over timesteps.", "questions": "See my comments above.", "limitations": "The authors mentioned that their method introduces additional inference-time overhead. But it is not clear how much additional overhead is introduced by the method.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths: \n(1) The main contribution of the paper is to convert the deterministic ODE sampling of flow models to VP-based stochastic sampling to be able to use existing inference time scaling methods developed for diffusion models. \n(2) Extensive experimental results confirms the effectiveness of the proposed techniques.   \n\nWeaknesses: \n(1) As noted in the paper, the processes for convert the deterministic ODE sampling of flow models to stochastic sampling is already covered in the SiT [38]. Even though the SiT paper focus on improving sampling quality of flow models, the mathematical derivations for converting deterministic sampling of flow models to stochastic sampling are well explained. This makes Subsection 4.3 on Inference-Time SDE Conversion prior work instead of new contribution.  \n\n(2)  It is also known from literature [33] and [52] how to convert a stochastic sampling to a VP-based stochastic sampling.   I would suggest the authors to rewrite Subsection 4.3 on Inference-Time Interpolant Conversion in order to better explain why the conversion makes the search space larger.  In the paper, the authors simply state that \"In Fig.3(c), we visualize the sample diversity under VP-SDE using FLUX which generates more diverse samples than Linear-SDE.\" My question is what the reason behind it. Can the authors provide some intuitive argument?", "was_revised": true, "final_justification": "After checking the rebuttal responses from the authors, I decide to raise the score to bordeline accept. I hope the authors rewrite the paper to clarify their contributions as they promised.", "rebuttal": "**[W1: Clarification of our contributions]**\n\nWe appreciate the reviewer for raising this important point. We would like to emphasize that our core contribution does not lie in re-deriving techniques such as SDE conversion, but rather in repurposing and integrating them through the lens of particle sampling to **enable efficient inference-time scaling for flow models**.\n\nTo the best of our knowledge, this work is the first to show that inference-time particle sampling—traditionally exclusive to diffusion models—is both feasible and effective in flow models. This is a nontrivial insight: when using ODE-based sampling, all particles deterministically collapse into a single trajectory, rendering exploration effectively impossible. The introduction of an SDE injects variance into the proposal distribution, forming the basis for reward-guided sampling in flow models. \n\nAdditionally, we show that SDE conversion alone is insufficient for achieving effective inference-time scaling. To address this, we introduce interpolant conversion as a key mechanism for increasing sample diversity—an insight presented for the first time in this work. In **Appendix C**, we provide a detailed analysis showing how interpolant design governs the variance of the proposal distribution, a factor central to exploration efficiency. Additionally, in **our response to W2**, we present an ablation study that empirically validates these findings. Together, these results establish a previously unexplored but critical connection between interpolant choice and particle sampling effectiveness in flow models. \n\nOur contributions span three key aspects:\n- We reinterpret SDE conversion as a tool for constructing an effective proposal distribution, **enabling efficient particle sampling in flow models**—a capability that was not previously possible.\n\n- To our knowledge, this is the **first work to view interpolant conversion from the perspective of sample diversity** and apply it for improving the efficiency of inference-time scaling. We present a detailed analysis that reveals how interpolant choice influences the sample diversity.\n\n- We introduce Rollover Budget Forcing (RBF), a novel strategy that **adaptively reallocates compute** across timesteps to maximize utilization of compute budget.\n\nWe appreciate the reviewer’s feedback and will revise the main paper to clearly highlight our novel contributions and better distinguish them from prior work. \n\n**[W2: How interpolant conversion enhances sample diversity?]**\n\nWe appreciate the reviewer’s insightful suggestion. Due to space constraints, our analysis of the relationship between interpolants and sample diveristy was provided in **Appendix. C**. We promise the reviewer that we will revise the main paper to incorporate both the analytical discussion from **Appendix C** and the supporting ablation study presented in this response. \n\nIn **Appendix C**, we dissect interpolant conversion into its two core components—diffusion coefficient scaling and timestep conversion—and **analyze their impact on the variance of the proposal distribution**. Here, in this response, we present an ablation study that **explains how each component of interpolant conversion affects sample diversity**, validating our analysis in **Appendix C**. \n\nFirst, to deepen our understanding of why interpolant conversion is effective, we analyze the impact of **diffusion norm scaling** $g_{t_s}' = \\frac{g_s}{c_s} \\cdot \\sqrt{\\frac{\\Delta s}{\\Delta t_s}}$, which increases the stochasticity of the generative process since $c_s \\approx 1$ and $\\sqrt{\\frac{\\Delta s}{\\Delta t_s}} \\gg 1$. While this scaling increases the variance of the proposal distribution and enhances sample diversity, it can also inject excessive noise relative to the current timestep. This may result in detached samples from well-supported regions of the sample manifold—especially for sharp distributions (e.g., small timesteps)—ultimately **harming sampling performance**. \n\nIn contrast, **timestep conversion** shifts sampling toward earlier timesteps, ($s < t_s$) where the noise level is higher and the distribution is more diffuse. In principle, this could promote diversity; however, it also reduces the integration step size $\\Delta t_s$ (see Fig 8, $\\Delta s$ corresponds to smaller $\\Delta t_s$ at early steps), thereby lowering the actual variance of the proposal distribution, given by $g_{t_s}^2 \\Delta t_s$. This diminished variance counteracts the intended benefit of enhanced exploration, leading to only minor improvements in reward alignment and diversity.\n\nThese findings suggest that diffusion norm scaling and timestep conversion are not independently sufficient, but rather act as complementary mechanisms. Interpolant conversion—as implemented in VP-SDE—combines both:\n- It introduces stronger stochasticity via norm scaling,\n- While stabilizing the sampling trajectory through an annealed timestep schedule. \n\nTo further validate this, we present an ablation study by isolating and evaluating each component separately. \n\n**[Ablation study]**\n\n**Tab. R1**: *Ablation on interpolant conversion strategies*  \n| Method                      | Diversity ↑ | Aesthetic ↑ | VQAScore ↑ | Inst. Blip (Held-out) ↑ |\n|----------------------------|-------------|--------------|-----------------|--------------------|\n| Linear-SDE                 | 0.158       | 5.436        | 0.900           | 0.813              |\n| + Adapt. Diffusion         | 0.429       | 3.986        | 0.702           | 0.571              |\n| + Adapt. Timestep          | 0.270       | 5.487        | 0.908           | 0.813              |\n| VP-SDE (Ours)              | **0.509**   | **5.830**    | **0.925**       | **0.843**          |\n\n**Tab. R1** isolates the effects of each component (stochasticity and interpolant conversion) and evaluates their impact on:\n- Sample diversity (LPIPS-MPD [23]),\n- Sample quality (Aesthetic Score),\n- Reward alignment (using the compositional text-to-image task in the main experiments).\n\nAll metrics are computed using 1,000 generated images. To the best of our knowledge, this is the first work to explicitly analyze how these components interact and influence the sample diversity and exploration capability of flow-based generative models. \n\nIn **Tab. R1**, we see that applying diffusion norm scaling alone (“+ Adapt. Diffusion”) results in a noticeable increase in sample diversity, but also leads to a significant drop in both reward alignment and sample quality—likely due to the injection of excessive noise. In contrast, timestep conversion (“+ Adapt. Timestep”) avoids such degradation but yields only modest improvements in diversity. The VP-SDE variant demonstrates that the coordinated use of both components can expand the exploration space without deviating from the underlying data manifold, achieving both high diversity and high fidelity.\n\nIn this work, we primarily focused on the interpolant side and analyzed its connection to timestep conversion and diffusion norm scaling. As noted in **Appendix C**, exploring the reverse direction—designing new interpolants based on desired variance properties of the proposal distribution—would also be an interesting future direction, and we plan to take concrete steps toward this in follow-up work. \n\n**[L1: Inference-time overhead]**\n\nThank you for pointing this out. While we report the runtime overhead of RBF under a 500-NFE budget in **Appendix F.3**, we provide a more comprehensive analysis in **Tab. R2**, where we evaluate runtime using NVIDIA L40S GPU.\n\n**Tab. R2**: *Runtime and known reward under different NFE budgets*\n\n| NFE Budget | Runtime (Sec) | VQAScore ↑ |\n|------------|----------------|----------------|\n| 50         | 84.00          | 0.858        |\n| 100        | 140.11         | 0.882        |\n| 300        | 383.89         | 0.915        |\n| 500        | 635.01         | 0.925        |\n| 1000       | 1243.68        | 0.928        |\n\nAs shown in **Tab. R2**, increasing the NFE budget leads to consistent improvements in reward alignment, demonstrating that RBF effectively scales with more compute. Importantly, this setup enables a **flexible trade-off between runtime and performance**—users can adjust the NFE budget according to their computational constraints or desired fidelity level."}, {"review_id": "4ojedOOabo", "reviewer": "Reviewer_Hs6F", "summary": "The paper proposes a new method for inference-time scaling in pretrained flow models, making them competitive with diffusion models for tasks requiring reward alignment (e.g., compositional or quantity-aware image generation). The key contributions are:  introducing SDE-based stochasticity into flow models for effective particle sampling, interpolant conversion (linear to variance-preserving) to broaden the search space/diversity, and Rollover Budget Forcing (RBF) for adaptive allocation of computational budgets during inference.", "questions": "- In Section 5, the authors suggest that an uneven allocation may be more effective than a uniform one. This idea has been previously explored in several works on flow models [1], [2] and diffusion models [3]. In particular, [2] shows that the trajectory of flow models tends to be more curved at the beginning and end of the denoising process, implying that more NFEs should be allocated to these stages. Does this observation support the design presented in Appendix D, or does it contradict it? \n\n- From my perspective, the inference time scaling for the diffusion model is especially valuable in tasks such as molecular generation where the runtime does not matter. It would be great to see how this work is applied to this domain.", "limitations": "yes\n\n**References**\n\n[1] Shaul, N., Perez, J., Chen, R. T., Thabet, A., Pumarola, A., & Lipman, Y. (2023). Bespoke solvers for generative flow models. arXiv preprint arXiv:2310.19075.\n\n[2] Bao Nguyen, Binh Nguyen, and Viet Anh Nguyen. \"Bellman Optimal Step-size Straightening of Flow-Matching Models.\" arXiv preprint arXiv:2312.16414 (2023).\n\n[3] Lu, Cheng, et al. \"Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps.\" Advances in Neural Information Processing Systems 35 (2022).", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**:\n\n- Includes diverse test cases, comparisons to multiple baselines, and ablation on key techniques (SDE, interpolant conversion, RBF).\n- Outperforms prior inference-time scaling baselines on compositional and quantity-aware image generation tasks. Demonstrates both quantitative and qualitative improvements.\n- Clear and easy-to-follow presentation.\n\n**Weaknesses**: \n\n- It is ambiguous why authors choose this setup heuristically: “In our case, we set $g(t) = t^2$, scaled by a factor of 3.”\n- After all the transformations, the paper finally uses the VP-SDE. I questioned the novelty of the paper. It appears that all transformations can be eliminated by utilizing direct paths in the VP-SDE diffusion model. This point should be discussed, or at least an experiment around this selection should be included.\n- Some papers show that the denoising steps of flow models should not be uniform [1], [2]. A comparison between your time schedule design in Appendix D and the time schedule in these papers is recommended.\n- The paper introduces methods that add computational cost at inference. There’s limited discussion or quantitative analysis of this trade-off.\n- The paper consists of several modules that are built upon each other. However, the paper lacks some ablation studies on them and important hyperparameters.", "was_revised": true, "final_justification": "The authors have fully addressed my concerns, so I decided to raise the score. However, after reading the rebuttal, I realized that some points are outside my expertise, so I lowered my confidence score.", "rebuttal": "**[W1: Choice of hyperparameters]**\n\nWe clarify our choice of hyperparameters. We set denoising steps to 10 for all experiments and diffusion coefficient $g(t) = 3t^2$. Below we provide rationale of this choice.\n\n**Tab. R1**: *Number of denoising step*\n| Steps | Aesthetic | Diversity |\n|-------|------------------|-----------|\n| 10    | **5.635**            | **0.084**     |\n| 20    | 5.680            | 0.103     |\n\n**Tab. R2**: *Diffusion coefficient*\n| Norm | Aesthetic $g(t)=t$ | Diversity $g(t)=t$ | Aesthetic $g(t)=t^2$ | Diversity $g(t)=t^2$ |\n|------|---------------------------|----------------------------|---------------------|----------------------|\n| 1    | 5.635                     | 0.084                      | 5.652               | 0.083                |\n| 3    | 5.168                     | 0.153                     | **5.436**               | **0.158**                |\n| 5    | 4.608                     | 0.223                      | 4.838               | 0.187                |\n\n\n**Number of denoising steps and NFE configuration.** As shown in **Tab R1**, the number of denoising steps beyond 10 gives marginal gains as also observed in prior work on inference-time scaling in diffusion models [39]. Hence, we fixed the number of denoising steps to 10 to ensure fair and consistent evaluation across all methods. Note that once the number of denoising steps is fixed, the total particle count per step is automatically determined by dividing the total NFE budget by the number of steps.\n\n**Diffusion coefficient design and norm.** From the empirical experiments shown in **Tab R2**, $g(t)=3t^2$ yielded the most favorable balance between sample diversity and output fidelity. We observe that using too small norms failed to sufficiently diversify the particle set. Conversely, larger norms induced excessive noise, degrading image quality and alignment. \n\n**[W2: Clarification of contributions]** \n\nWe appreciate the reviewer’s comment. We clarify that our contribution goes beyond simply adopting the VP-SDE interpolant, and instead centers on **analyzing and leveraging interpolant conversion to enable inference-time scaling in flow models**. To the best of our knowledge, this is the **first work to establish a connection between interpolant design and sample diversity**, and to provide both theoretical and empirical analysis supporting this link.\n\nDue to page limit, we provided a detailed analysis in **Appendix. C**, where we establish a formal connection between **interpolant choice and the variance of the proposal distribution**. In **Fig. 4 and 6**, we empirically showed that this variance directly affects the diversity of generated samples and thus the overall effectiveness of inference-time scaling. Additionally, in **our response to W2 of Reviewer kdqx**, we present an ablation study that isolates the impact of each component of interpolant conversion—timestep conversion and diffusion norm scaling on sample diversity. This study empirically confirms that both components synergistically increase sample diversity and sample quality. \n\nFrom our findings, we believe that taking the reverse approach—i.e., using a VP-SDE diffusion model but reverting to a Linear-SDE interpolant at inference—would **decrease sample diversity**, shrink the exploration space, and ultimately degrade the performance of inference-time scaling. \n\n---\n\n**[W3: Non-uniform timestep schedule]**\n\nThank you for highlighting this point—this direction indeed helps enrich our analysis. While the referenced works also employ non-uniform timestep scheduling, they require training for each model type.  This constraint prevented us from directly applying their methods in our setting. Also, it is important to note that their focus is primarily on ODE-based sampling, whereas our goal is to explore SDE-based inference to balance the trade-off between sample diversity and quality for efficient inference-time scaling. \n\nInstead, we include a quantitative comparison in **Tab. R3** using four different timestep schedulers including linear [32], ours, polynomial [32], and EDM [22]. Here, linear and polynomial are special case of setting $n=1$ and $n=2$ in $1-(1-t)^n$, respectively. Our results show that timestep schedules emphasizing finer integration in the early stages—specifically ours and polynomial—consistently yield better performance. \n\n**Tab. R3**: *Time scheduler comparison on Linear SDE*\n\n| Timestep Scheduler     | Diversity | Aesthetic Score | VQAScore | Inst. BLIP (Held-out) |\n|------------------------|-----------|------------------|---------------|------------------|\n| Linear ($n=1$) [32]               | 0.078   | 5.302          | 0.790       | 0.758          |\n| Ours     | **0.158**   | **5.436**          | **0.900**       | **0.813**          |\n| Polynomial ($n=2$) [32]     | 0.134   | 5.429          | 0.890       | 0.807          |\n| EDM  [22]     | 0.079   | 5.237          | 0.834       | 0.768         |\n\n---\n\n**[W4: Inference-time cost]**\n\nHere, we provide a more comprehensive analysis in **Tab. R4**, where we evaluate runtime and performance across varying NFE budgets using NVIDIA L40S.\n\n**Tab. R4**: *Runtime and reward alignment under varying NFE budgets*\n\n| NFE  | Runtime (Sec) | VQAScore |\n|------|--------------|---------------|\n| 50   | 84.00        | 0.858       |\n| 100  | 140.11       | 0.882       |\n| 300  | 383.89       | 0.915       |\n| 500  | 635.01       | 0.925       |\n| 1000 | 1243.68      | 0.928       |\n\nAs shown in **Tab. R4**, increasing the NFE budget leads to consistent performance improvements, demonstrating that RBF effectively scales with more compute. Importantly, this setup enables flexible trade-offs between runtime and reward alignment—users can adjust the NFE budget to meet their specific computational or fidelity requirements.\n\n---\n\n**[W5: Component ablations and hyperparameter rationale]**\n\nWe respectfully clarify that the main paper presents key ablations in **Fig. 4 and 6**, which demonstrate performance improvements from Linear-ODE to Linear-SDE, and further to VP-SDE across all particle-sampling-based methods. These figures also show gains from incorporating the rollover strategy (SVDD to RBF). \n\nTo complement these results, we additionally present a more comprehensive ablation study in **Tab. R5**, where we vary the total NFE budget and isolate the impact of each core component—SDE conversion, interpolant conversion, and rollover strategy. For this experiment, we use the SVDD search strategy as the base, as RBF reduces to SVDD when the rollover mechanism is disabled. \n\n**Tab. R5**: *Ablation study across particle sampling variants*\n\n| Method            | NFE  | VQAScore | Inst.BLIP (Held-out) | Aesthetic |\n|-------------------|------|----------|------------------------|-----------|\n| BoN           | 100  | 0.846    | 0.796                  | 5.259     |\n|                   | 500  | 0.879    | 0.820                  | 5.162     |\n|                   | 1000 | 0.889    | 0.827                  | 5.205     |\n| SVDD (Linear-SDE) | 100  | 0.861    | 0.791                  | 5.045     |\n|                   | 500  | 0.893    | 0.813                  | 5.052     |\n|                   | 1000 | 0.892    | 0.820                  | 4.994     |\n| SVDD (VP-SDE)     | 100  | 0.874    | 0.809                  | 5.286     |\n|                   | 500  | 0.915    | 0.847                  | 5.249     |\n|                   | 1000 | 0.921    | 0.851                  | 5.281     |\n| RBF (VP-SDE)       | 100  | 0.882    | 0.821                  | 5.321     |\n|                   | 500  | 0.925    | 0.843                  | 5.237     |\n|                   | 1000 | 0.928    | 0.837                  | 5.259     |\n\nWe conduct these experiments on the compositional text-to-image generation task, following the setup in the main paper. As shown in **Tab. R5**, we observe consistent improvements in reward alignment across all NFE regimes when moving from Linear-SDE to VP-SDE, and further with the addition of the rollover strategy (RBF). Notably, both VP-SDE SVDD and RBF outperform BoN across all metrics. We observe a similar trend in **Fig. 12** in Appendix for quantity-aware image generation, further supporting the generality of our findings. These results strongly validate the benefit of each proposed component and the modular design of our framework.\n\nLastly, we kindly remind the reviewer that the rationale for key hyperparameter choices is presented in our previous responses to W1 and W3, including: the number of denoising steps (**Tab. R1**), diffusion coefficient (**Tab. R2**), and timestep scheduler (**Tab. R3**). We will incorporate these into the main paper to clearly justify our design choices.\n\n**[Q1: Non-uniform scheduler]**\n\nYes, the observation aligns with our timestep scheduling described in Appendix D. The curved trajectories at the beginning would lead to greater diversity at the early stage, so we allocate more budget at the beginning accordingly.\n\n**[Q2: Applicability to other domains]**\n\nWe thank the reviewer for this thoughtful suggestion. We fully agree that molecule and protein generation are promising domains for applying inference-time scaling. We will explore this direction and include additional results in the revision."}]

Author rebuttal:

Meta-review: (a) The paper proposes VP-SDE, a framework to adapt particle sampling to pre-trained flow models. This allows inference-time scaling previously limited to diffusion models. Contributions include: i) converting deterministic flow ODEs to SDEs, ii) replacing uniform with variance-preserving (VP) interpolants for diversity, and iii) Rollover Budget Forcing (RBF) for adaptive budget allocation. Experiments on compositional and text-to-image tasks show consistent improvements.

(b) It has a clear and well-structured presentation, it extends inference-time scaling to flow models, its performance shows improvements.

(c) Reviewers found the paper to have a limited novelty: they point out that SDE conversion and VP interpolants are established ideas; the contribution seems to lie in combining them for flows. There are baseline gaps: Missing direct comparison with diffusion models using particle sampling. In terms of efficiency, it is not clear how scalable the method is (the added inference-time cost should be better quantified). Some hyperparameter choices lack justification.

(d) Nonetheless, the paper is found overall technically sound, clearly presented, and practically useful. The introduction of particle sampling to flow models is demonstrated satisfactorily. While the novelty is incremental and the evaluation incomplete, the contribution is meaningful for the generative modeling community.

e) Reviewers initially questioned originality, fairness of baselines, and clarity. The rebuttal improved the presentation, added ablations, and resolved many concerns, leading to several score increases. The remaining issues include novelty framing and missing efficiency analysis.

Thus, overall the paper has been found to be suitable for NeurIPS, as its practical contributions outweigh its incremental novelty.
Decision: Accept (poster)