Inference-Time Scaling for Flow Models via Stochastic Generation and Rollover Budget Forcing

[W1: Choice of hyperparameters]

We 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.

Tab. R1: Number of denoising step

Tab. R2: Diffusion coefficient

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.

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.

[W2: Clarification of contributions]

We 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.

Due 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.

From 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.

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[W3: Non-uniform timestep schedule]

Thank 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.

Instead, 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.

Tab. R3: Time scheduler comparison on Linear SDE

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[W4: Inference-time cost]

Here, we provide a more comprehensive analysis in Tab. R4, where we evaluate runtime and performance across varying NFE budgets using NVIDIA L40S.

Tab. R4: Runtime and reward alignment under varying NFE budgets

As 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.

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[W5: Component ablations and hyperparameter rationale]

We 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).

To 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.

Tab. R5: Ablation study across particle sampling variants

We 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.

Lastly, 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.

[Q1: Non-uniform scheduler]

Yes, 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.

[Q2: Applicability to other domains]

We 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.

[W1: Clarification of our contributions]

We 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.

To 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.

Additionally, 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.

Our contributions span three key aspects:

• 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.

• 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.

• We introduce Rollover Budget Forcing (RBF), a novel strategy that adaptively reallocates compute across timesteps to maximize utilization of compute budget.

We appreciate the reviewer’s feedback and will revise the main paper to clearly highlight our novel contributions and better distinguish them from prior work.

[W2: How interpolant conversion enhances sample diversity?]

We 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.

In 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.

First, 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.

In 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.

These 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:

• It introduces stronger stochasticity via norm scaling,
• While stabilizing the sampling trajectory through an annealed timestep schedule.

To further validate this, we present an ablation study by isolating and evaluating each component separately.

[Ablation study]

Tab. R1: Ablation on interpolant conversion strategies

Tab. R1 isolates the effects of each component (stochasticity and interpolant conversion) and evaluates their impact on:

• Sample diversity (LPIPS-MPD [23]),
• Sample quality (Aesthetic Score),
• Reward alignment (using the compositional text-to-image task in the main experiments).

All 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.

In 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.

In 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.

[L1: Inference-time overhead]

Thank 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.

Tab. R2: Runtime and known reward under different NFE budgets

As 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.

[W1: Clarification of our contributions]

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.

To 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.

Additionally, 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.

Our contributions span three key aspects:

• We reinterpret SDE conversion as a tool for constructing an effective proposal distribution in particle sampling, enabling efficient inference-time scaling in flow models.

• 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.

• Additionally, we introduce Rollover Budget Forcing (RBF), a novel particle sampling strategy that adaptively reallocates compute across timesteps to maximize utilization of compute budget.

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[W2: Ablation study of SDE and interpolant conversion in sample diversity and reward alignment]

Tab. R1: Ablation study of SDE and interpolant conversion

We 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.

We 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.

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[Q1: Time complexity and RBF scaling]

We 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.

Tab. R2: Inference cost

• $S$: Denoising steps
• $N$: NFE budget
• $c_d$, $c_v$: Cost of denoising and verification (reward)

As shown in Tab. R2, since all methods share the same NFE budget, the cost for denoising is set to $N * c_d$. The 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.

Tab. R3: Scaling comparison

The 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.

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[Q2: Ambiguity of “much clearer posterior mean”]

We 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.

As 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.

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[Q3: Hyperparameter choice and rationale]

We 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.

Tab. R4: Number of denoising step

Tab. R5: Diffusion coefficient

As 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.

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.

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[Q4: Particle size and RBF schedule]

We clarify that varying the particle size is equivalent to scaling the NFE budget. Please see our response to Q1 for details.

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[Q5: Other flow models and domains]

We 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.

[W1: Comparison with Diffusion Models under Particle Sampling]

We sincerely thank the reviewer for this insightful point regarding comparisons with diffusion models.

While 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.

To 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:

Tab. R1: Comparison of particle sampling between SANA (diffusion) and FLUX (flow)

This 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.

In 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.

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[W2: Clarification on "making flow-models as diffusion model like as possible"]

Thank 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:

1. 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.

2. 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.

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[Q1: RBF reduces to SVDD without rollover strategy]

If 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.

We would like to clarify two key points regarding the behavior of RBF.

First, 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.

Second, 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.

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[L1: Aesthetic Score Drops]

We 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).

In 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.

Additionally, 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.