Title: Tortoise and Hare Guidance: Accelerating Diffusion Model Inference with Multirate Integration
Venue: NeurIPS 2025
Accepted: yes
Keywords: diffusion model solver

diffusion model inference

multirate method

classfier-free guidance
Abstract: In this paper, we propose Tortoise and Hare Guidance (THG), a training-free strategy that accelerates diffusion sampling while maintaining high-fidelity generation. We demonstrate that the noise estimate and the additional guidance term exhibit markedly different sensitivity to numerical error by reformulating the classifier-free guidance (CFG) ODE as a multirate system of ODEs. Our error-bound analysis shows that the additional guidance branch is more robust to approximation, revealing substantial redundancy that conventional solvers fail to exploit. 
Building on this insight, THG significantly reduces the computation of the additional guidance: the noise estimate is integrated with the tortoise equation on the original, fine-grained timestep grid, while the additional guidance is integrated with the hare equation only on a coarse grid. We also introduce (i) an error-bound-aware timestep sampler that adaptively selects step sizes and (ii) a guidance-scale scheduler that stabilizes large extrapolation spans. 
THG reduces the number of function evaluations (NFE) by up to 30% with virtually no loss in generation fidelity (∆ImageReward ≤ 0.032) and outperforms state-of-the-art CFG-based training-free accelerators under identical computation budgets. Our findings highlight the potential of multirate formulations for diffusion solvers, paving the way for real-time high-quality image synthesis without any model retraining. The source code is available at https://github.com/yhlee-add/THG.

Tortoise and Hare Guidance: Accelerating Diffusion Model Inference with Multirate Integration

Abstract

In this paper, we propose Tortoise and Hare Guidance (THG) , a training-free strategy that accelerates diffusion sampling while maintaining high-fidelity generation. We demonstrate that the noise estimate and the additional guidance term exhibit markedly different sensitivity to numerical error by reformulating the classifier-free guidance (CFG) ODE as a multirate system of ODEs . Our error-bound analysis shows that the additional guidance branch is more robust to approximation, revealing substantial redundancy that conventional solvers fail to exploit. Building on this insight, THG significantly reduces the computation of the additional guidance: the noise estimate is integrated with the tortoise equation on the original, fine-grained timestep grid, while the additional guidance is integrated with the hare equation only on a coarse grid. We also introduce (i) an error-boundaware timestep sampler that adaptively selects step sizes and (ii) a guidance-scale scheduler that stabilizes large extrapolation spans. THG reduces the number of function evaluations (NFE) by up to 30% with virtually no loss in generation fidelity ( ∆ ImageReward ≤ 0 . 032 ) and outperforms state-of-the-art CFG-based training-free accelerators under identical computation budgets. Our findings highlight the potential of multirate formulations for diffusion solvers, paving the way for real-time high-quality image synthesis without any model retraining. The source code is available at https://github.com/yhlee-add/THG .

1 Introduction

Diffusion models (DMs) have become the state-of-the-art generative model for images [10, 40, 47] and, more recently, for video [20, 1, 52, 21] and audio-visual content [5, 41]. Despite their impressive quality, sampling is costly: each output is obtained by iteratively denoising a noisy sample, and the latency scales with the total number of function evaluations (NFE) required by the solver.

Many practical scenarios, such as text-to-image synthesis, class-controlled synthesis, or in-context image editing, require conditional generation. The dominant technique for high-quality conditioning is classifier-free guidance (CFG) [18], which improves perceptual quality and controllability. However, CFG runs the denoising network twice per timestep-once conditional and once unconditionalthereby doubling the NFE. For real-time applications, such as interactive editing and large-scale serving, evaluating a deep backbone at every timestep remains a major bottleneck.

A large body of work to accelerate these models has focused on two main approaches. Some approaches reduce the number of steps using higher-order ODE/SDE solvers [45, 46, 30] or distillation [43, 34], while others-such as cache-based strategies like DeepCache [33] and Learning-toCache [32]-lower the cost per step by reusing intermediate features. Nevertheless, both approaches still perform two forward passes whenever CFG is enabled, implicitly assuming that conditional and unconditional calls are equally indispensable.

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Through the lens of numerical analysis, we revisit CFG by reformulating the reverse diffusion process as a two-state multirate system of ODEs whose trajectories are governed by the noise estimate and the additional guidance term. Our error-bound analysis reveals a pronounced asymmetry: the additional guidance term is more robust to approximation than the noise estimate, exposing substantial redundancy that conventional solvers fail to exploit. This finding raises a natural question: Do we need to compute the neural network twice at every fine-grained timestep?

Leveraging this asymmetry, we introduce Tortoise and Hare Guidance (THG) , a training-free sampler that bypasses most additional guidance computation. The noise estimate is integrated with the tortoise equation on the original fine-grained timestep grid. Meanwhile, the additional guidance is integrated with the hare equation only on a coarse grid. We further introduce (i) an error-bound-aware timestep sampler that adaptively determines the coarse grid, and (ii) a guidance-scale scheduler that keeps the trajectory stable over significant gaps.

With these components, THG achieves sampling speeds up to 1 . 43 × faster by reducing the NFE budget from 100 to as low as 70 while maintaining virtually identical generation fidelity ( ∆ ImageReward ≤ 0 . 032 ). Moreover, across Stable Diffusion 1.5 [40], 3.5 Large [47], and AudioLDM 2 [28], our method outperforms state-of-the-art CFG-based training-free accelerators under identical computation budgets. Our study highlights the potential of multirate formulations for accelerating diffusion models and brings us a step closer to achieving real-time performance and high-quality image synthesis without retraining the model.

In summary, our contributions are threefold:

We are the first to cast the reverse diffusion ODE as a two-state multirate system of ODEs and to provide an error-bound analysis showing that the additional guidance term can be safely approximated at a much coarser temporal resolution.

We design Tortoise and Hare Guidance (THG) , a training-free sampler that eliminates the need for a significant amount of additional guidance term evaluation. THG is compatible with any diffusion backbone.

Using image-text pairs from the COCO 2014 dataset, we demonstrate that THG can reduce NFEs up to 30% with virtually no loss in generation fidelity ( ∆ ImageReward ≤ 0 . 032 ). THG outperforms state-of-the-art CFG-based accelerators under identical compute budgets.

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2 Related work

Diffusion models Denoising Diffusion Probabilistic Models (DDPMs) [19] laid the foundation for modern diffusion models by introducing a probabilistic framework. A forward Markov process gradually corrupts a data point x 0 into Gaussian noise. In the reverse process, at each timestep t , a neural network ˆ ϵ θ ( x t , t ) estimates and removes the noise component in x t to recover x t -1 , ultimately reconstructing x 0 . The denoising trajectory can be interpreted either as a stochastic differential equation (SDE) or its deterministic counterpart, the probability flow ODE (PF-ODE) [46]. Denoising Diffusion Implicit Models (DDIMs) [45] drop the strict Markov assumption of DDPMs and apply Tweedie's formula [9] to jump directly from x t to x s , cutting sampling steps from hundreds of steps to as few as 50 and effectively solving the PF-ODE in a single deterministic pass [46].

ODE-based integrators Viewing diffusion sampling as an initial-value ODE problem enables high-order integration techniques. Concretely, DPM-solver [30] observes that the diffusion ODE

has a semi-linear term f ( t ) x t . The need for approximation for the linear term is eliminated by solving the semi-linear ODE using the variation of constants formula. This semi-linear integrator then affords large step sizes with minimal approximation error. Inspired by these semi-linear methods, we introduce a multirate formulation for the classifier-free guidance (CFG) scheme [18] that adjusts the step size of each component of CFG to its own dynamics, achieving further reductions in the number of function evaluations (NFE) without degrading sample quality.

Classifier-free guidance and its variations In real-world applications, diffusion models must produce samples that satisfy a given condition (e.g., class label or text prompt). Classifier Guidance [8] achieves this by incorporating a pre-trained classifier p ϕ ( c | x t ) , effectively sampling from the sharpened density p ( x ) p ( c | x ) ω , where ω controls the strength of the bias towards class c . ClassifierFree Guidance (CFG) [18] eliminates the need for an external classifier by training a single denoising network that gives both conditional and unconditional outputs. Concretely, if ˆ ϵ θ ( x t , c ) and ˆ ϵ θ ( x t , ∅ ) denote the network's noise predictions with and without condition c , respectively, then CFG defines

Subsequent variants focus on finding the optimal strength and timing of guidance for balancing condition fidelity against sample diversity. Guidance Interval [26] restricts the use of CFG to midlevel noise steps, avoiding over-conditioning at the beginning and final stages of the sampling process. CADS and Dynamic-CFG [42] slowly anneal either the conditioning vector or the scale ω during the early denoising steps, preserving diversity in the final samples. PCG [2] reformulates CFG as a predictor-corrector method (with ω ′ = 2 ω -1 ) that alternates between denoising and sharpening phases. CFG++ [7] treats guidance as an explicit loss term rather than a sampling bias, splitting each DDIM iteration into 'denoising' and 'renoising' phases. Unlike these methods, we reformulate the diffusion ODE using a multirate method, integrating the noise estimate on a fine-grained grid and the additional guidance term on a coarse grid, reducing the NFE while preserving sample quality.

Efficient diffusion models Beyond advanced ODE/SDE solvers, various methods have been proposed to speed up pre-trained diffusion models. Distillation methods [43, 34] compress a pretrained 'teacher' model into a 'student' model that can advance multiple timesteps in one forward pass. While these methods reduce the number of sampling steps, they incur substantial retraining costs. Cache-based techniques exploit feature redundancy within the denoising neural network ˆ ϵ θ . DeepCache [33] reuses high-level U-Net activations across adjacent steps. Learning-to-Cache [32] introduces a layer-wise caching mechanism that dynamically reuses transformer activations across timesteps via a timestep-conditioned router. ∆ -Dit [4] leverages stage-adaptive caching of blockspecific feature offsets in DiT models to speed up inference without retraining. These methods deliver inference speedups without retraining but depend heavily on the model's internal architecture. More recently, several works have noted that CFG doubles the NFE per denoising step and have proposed methods to reduce this extra cost. Adaptive Guidance [3] adaptively skips redundant guidance steps based on cosine similarity between conditional and unconditional predictions. FasterCache [31] reuses attention features and conditional-unconditional residuals to mitigate CFG overhead. Although these methods reduce the NFE, they lack a rigorous theoretical foundation and leave further savings on the table. Our approach delivers a more efficient and theoretically grounded method of guided diffusion by directly exploiting the CFG's intrinsic dynamics.

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3 Method

In this section, we introduce Tortoise and Hare Guidance (THG) , which accelerates diffusion model inference by leveraging the asymmetry between the noise estimate and the additional guidance terms. Since the additional guidance term varies more slowly w.r.t. the denoising timestep t than the noise estimate term, we apply a multirate integration scheme that uses a coarser timestep grid for the additional guidance term (Sec. 3.1 and Sec. 3.2). We then perform an approximation error-bound analysis to determine the appropriate grid granularity (Sec. 3.3). Finally, we propose an adaptive guidance scale to compensate for any performance degradation resulting from the reduced number of evaluation points (Sec. 3.4).

Preliminaries To accommodate different definitions of the diffusion process [19, 46, 49], we adopt a general notation [30] so that the forward process and the diffusion ODE are described as follows:

where f ( t ) = dlog α t d t , g 2 ( t ) = d σ 2 t d t -2 dlog α t d t σ 2 t , and t ∈ [0 , T ] . ( v -prediction models are covered in Appendix A.) α t and σ t are the predefined noise schedule of the diffusion model. Although modern diffusion models primarily operate in the latent space [40], we adopt x (instead of z ), as our framework is agnostic to this choice. For brevity, we denote the unconditional noise estimate ˆ ϵ ∅ ( x t ) := ˆ ϵ θ ( x t , ∅ ) , the conditional noise estimate ˆ ϵ c ( x t ) = ˆ ϵ θ ( x t , c ) , the difference of the two ∆ˆ ϵ c ( x t ) := ˆ ϵ c ( x t ) -ˆ ϵ ∅ ( x t ) , and the CFG noise estimate ˆ ϵ ω c ( x t ) = ˆ ϵ CFG θ ( x t , c ) following [7].

3.1 A multirate formulation

We propose a multirate formulation [39], in which the reverse diffusion process is decomposed into numerically sensitive and less sensitive components to reduce the number of function evaluations (NFE). We begin by writing the diffusion ODE in Eq. 3 by explicitly separating it into two distinct terms, the noise estimate and the additional guidance term. By the definition of CFG, we have

Substituting Eq. 4 into Eq. 3 yields the following:

We observe a significant difference in temporal sensitivity between the noise estimate term and the additional guidance term. Figure 2 plots the time-derivative norms of ˆ ϵ c ( x t ) and δ ˆ ϵ c ( x t ) , confirming that the noise estimate varies more rapidly than the additional guidance term. This result

Algorithm 1 Tortoise and Hare Guidance Algorithm

clearly demonstrates that the noise estimate exhibits greater numerical sensitivity than the additional guidance.

This motivates the use of a multirate method [44] where the sensitive term is integrated on a finegrained grid, and the less sensitive term is integrated on a coarse grid. We split the diffusion ODE (Eq. 5) into the following system of ODEs:

where x T T = x T , x H T = 0 , and x t := x T t + x H t . The tortoise x T t covers the noise estimate part of the diffusion ODE, while the hare x H t takes care of the additional guidance term. We call the ODE integrated on the fine-grained grid the tortoise equation (Eq. 6), and the ODE integrated on the coarse grid the hare equation (Eq. 7). Intuitively, the hare equation uses coarser timestep intervals-i.e. larger steps-allowing it to skip unnecessary computation and thus significantly improve the efficiency of integrating the diffusion ODE. Moreover, because both equations retain the standard diffusion ODE form, existing solvers such as DDIM [45] can be applied to each equation without modification.

3.2 Tortoise and Hare Guidance

Solving the hare equation (Eq. 7) on the coarse grid is straightforward, since every coarse timestep is also a fine-grained timestep. By contrast, because the tortoise equation (Eq. 6) requires the full state x t = x T t + x H t at every fine-grained timestep, we must infer x H t at those intermediate points [39]. Instead of using generic extrapolation methods [31], we exploit a property of diffusion model solvers: given x t and ˆ ϵ θ ( x t ) , they can deterministically compute x s for any s < t by running the chosen solver from t to s . From each coarse timestep, we run the solver not only to compute x H t for the next coarse timestep but also to populate x H t for all intermediate fine-grained timesteps, thereby constructing the full trajectory of x H t on the fine-grained grid for use in integrating the tortoise equation.

Building on this formulation, we propose an implementation strategy summarized in Algorithm 1. While the standard diffusion solver evaluates both ˆ ϵ c ( x t ) and ∆ˆ ϵ c ( x t ) at every fine-grained timestep, our scheme evaluates ∆ˆ ϵ c ( x t ) only on the coarse grid C ⊂ { t 0 , . . . , t N } , thereby significantly reducing NFE. At each coarse step t i ∈ C , the updated guidance term is used to integrate the hare

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equation across the fine-grained grid until the next coarse step. We then use the resulting x H t values during the subsequent tortoise equation steps. As a result, the NFE is reduced from 2 N to N + | C | -1 while preserving the dynamics of the original diffusion ODE. Moreover, it slots seamlessly into existing diffusion pipelines without any changes to their core logic.

3.3 Approximation error bound analysis

To determine an appropriate coarse grid C for the hare equation, we now turn to an error-based criterion. Our objective is to ensure that the integration error of x H t remains sufficiently small relative to that of x T t . To this end, we adopt a standard multirate strategy [11]. We select coarse step sizes such that the ratio between the hare's approximation error and the tortoise's approximation error does not exceed a user-specified threshold ρ such that ρ ≈ 1 :

Here, x T s and x H s denote the analytical solutions to the tortoise and hare equations at timestep s , while ˆ x T s and ˆ x H s are the corresponding numerical solutions obtained using the diffusion model solver. Given that the solver has order p , the local integration error at a single step scales as [14]:

where ∆ t is the fine-grained step size and c is an unknown constant. Let the coarse step size be m ∆ t , meaning the hare leaps m tortoise steps per update. Then, the local integration error of the hare equation over one coarse step becomes:

In contrast, the tortoise equation accumulates error over m fine-grained steps:

Taking the ratio from Eq. 8 and ignoring higher-order terms, we obtain:

Since m must be a positive integer, we define the maximum allowable value as:

Estimating the error constants To compute m max , we need estimates of ∥ c T ∥ and ∥ c H ∥ without relying on the analytic solution x s . We accomplish this using the Richardson extrapolation method [14] . First, solve the ODE once using step size ∆ t :

Algorithm 2 Look before you leap

Next, solve again using two steps of size ∆ t/ 2 :

Subtracting Eq. 14 and Eq. 15 yields

If we ignore the higher-order terms, the norm of this difference provides a direct estimate proportional to ∥ c ∥ . We apply this procedure independently to both the tortoise and hare equations to estimate ∥ c T ∥ and ∥ c H ∥ , respectively. Empirical results (Fig. 3) on 30,000 prompts from the COCO 2014 dataset [27, 37] show that ∥ c T ∥ is greater than ∥ c H ∥ for most cases, confirming that the tortoise equation is more sensitive to timestep resolution.

Example usage scenario To generate samples using Tortoise and Hare Guidance, we first calculate m max of each fine-grained timestep (Eq. 13) using the average ∥ c T ∥ and ∥ c H ∥ over a batch of inputs. (More details are covered in Appendix C.) Then we build the coarse timestep grid C via the 'look before you leap' strategy (Algorithm 2). Starting at the first fine-grained timestep t 0 , we insert coarse timesteps so that they lie m max ( t i ) steps ahead, keeping the local error ratio below ρ . Finally, we generate samples using Algorithm 1. Note that C could be reused for all subsequent inferences without any additional NFEs.

3.4 Adjusting Guidance Scales

Approximating the hare at fine-grained timesteps can lead to a degradation in output quality. To compensate for this, we propose adjusting the guidance scale whenever the additional guidance term is used more than once per timestep. In particular, we introduce a constant boost factor b and scale the guidance term: ∆ˆ ϵ c ← b · ∆ˆ ϵ c . This simple multiplicative adjustment improves sample quality, especially in cases where the inner loop (which integrates the hare equation) is repeated multiple times for each coarse step. Our method draws inspiration from prior work such as CFG-Cache [31], which amplifies guidance in the frequency domain using FFT. However, unlike FFT-based methods, our approach avoids the overhead of spectral transforms, which can be computationally expensive for highdimensional latent variables. The additional guidance term predominantly contains low-frequency information in the early stages of sampling and vice versa [15]. Therefore, selectively enhancing the frequency components of the additional guidance term per timestep has low significance.

Furthermore, CFG and the additional guidance term are of low significance at the later phase of the reverse diffusion process [26, 3]. We leverage this fact by introducing a threshold timestep index i hi and substituting ∆ˆ ϵ c ← 0 if i ≥ i hi. This simple adjustment helps reduce the NFE even further.

4 Experiments

4.1 Experimental Settings

Compared methods To demonstrate the effectiveness of our approach, we compare against CFGCache [31], a training-free acceleration technique that reuses conditional and unconditional outputs in video diffusion models. Given that CFG-Cache exploits a timestep-adaptive enhancement technique

to mitigate fine-detail degradation, we evaluate both the full CFG-Cache (with enhancement) and a variant without this enhancement (denoted 'CFG-Cache w/o FFT'). All variants are adapted to the diffusion model's modality for a fair comparison. More comparisons with CFG variants are given in Appendix E.

Implementation details We build Tortoise and Hare Guidance with PyTorch [36], Diffusers [48], and Accelerate [13]. We evaluate three pretrained diffusion models-Stable Diffusion 1.5 [40], Stable Diffusion 3.5 Large [47, 10], and AudioLDM 2 [28]. For Stable Diffusion (SD) models, we use prompt-image pairs randomly sampled from COCO 2014 [27, 37]: 30,000 pairs for SD 1.5 and 1,000 pairs for SD 3.5 Large. For AudioLDM 2, we use 2,230 prompt-audio pairs from the validation set of AudioCaps [25]. Experiments are run on a server with an AMD EPYC 74F3 26934-core CPU, 1 TB of RAM, and 8 NVIDIA A100 80GB GPUs. Hyperparameters ( N,ω,ρ,b,i hi ) are set to (50 , 7 . 5 , 1 . 1 , 1 . 1 , 38) for SD 1.5, (28 , 3 . 5 , 1 . 0 , 1 . 2 , 21) for SD 3.5 Large, and (50 , 3 . 5 , 0 . 9 , 1 . 15 , 39) for AudioLDM 2.

4.2 Main Results

Quantitative comparison Table 1 compares our method to the CFG-Cache variants in terms of distributional similarity metrics such as FID [17, 35], CMMD [22], and FAD [24], together with prompt fidelity metrics such as CLIP Score (CS) [16], ImageReward (IR) [51], and CLAP Score [50] under the same number of function evaluations (NFE). Refer to Appendix F for more details on metrics. On SD 1.5, all methods cut NFE from 100 to 70; ours lowers FID (14.165 vs. 14.240),

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matches CMMD, and improves CS and IR over CFG-Cache w/o FFT, and beats full CFG-Cache on CS and IR while keeping FID competitive. On SD 3.5 Large, all cut NFE from 56 to 38; although CFG-Cache slightly leads on FID and CMMD, our method delivers nearly equal FID/CMMD with the highest IR and tied CS. On AudioLDM 2 [28], all cut NFE from 100 to 70; ours lowers FAD (2.764 vs. 2.901) and improves CLAP Score over CFG-Cache w/o FFT. CFG-Cache is excluded since its enhancement is inapplicable to the audio domain. These results show that THG generalizes across solvers and scales, preserving sample distribution and text alignment under aggressive step reduction. The tradeoff of distributional similarity and prompt fidelity is further discussed in Appendix G.

Qualitative comparison Figure 4 compares images generated by our method and the two CFGCache variants. The results demonstrate that THG effectively preserves image fidelity and fine details. More visual comparisons are shown in Appendix I.

4.3 Ablation Studies

Boost factor b We perform ablation studies on hyperparameters using SD 1.5 with DDIM. Table 2 shows how varying the boost factor b affects inference quality at 70 NFE budget with the same latents x T . As b increases from 1.00 to 1.20, we observe a steady rise in IR from 0.09395 up to 0.13478, indicating stronger image-text alignment, and a modest gain in CS. However, this comes at the cost of higher FID and CMMD values, reflecting a gradual drop in distributional similarity. We select b = 1 . 10 as our default because it strikes the best balance: it substantially boosts IR (0.11576) with only a moderate increase in FID (14.232) and CMMD (0.59354) relative to lower b values.

Error-ratio threshold ρ Table 3 summarizes the effect of varying ρ with the same latents x T . Lowering ρ from 1.1 to 0.9 results in more conservative hare leaps-NFE rise from 70 to 75-and yields slightly better FID (14.128 vs. 14.232) and CMMD (0.59044 vs. 0.59354), at the expense of marginally lower IR (0.11942 vs. 0.11576). Increasing ρ to 1.3 reduces NFE to 67 but degrades FID (14.280) and IR (0.10849). We choose ρ = 1 . 1 as our default since it achieves the best trade-off: a 30% NFE reduction (70 NFE) while maintaining competitive fidelity and alignment metrics.

5 Conclusion

We present Tortoise and Hare Guidance, a training-free acceleration framework for diffusion sampling that leverages a multirate reformulation of classifier-free guidance (CFG). Exploiting the asymmetric sensitivity of the noise estimate and the additional guidance term to numerical error, Tortoise and Hare Guidance integrates the noise estimate on a fine-grained grid while integrating the additional guidance term on a coarse grid. This approach allows for a substantial reduction in the number of function evaluations (NFE) without sacrificing generation quality. With an error-bound-aware timestep sampler and a guidance scale adjustment, our method achieves up to 30% faster sampling while preserving fidelity across models like Stable Diffusion 1.5, 3.5 Large, and AudioLDM 2, demonstrating the effectiveness of multirate integration for real-time high-quality generation.

Limitations Our experiments are currently limited to latent diffusion models and a few benchmark datasets such as COCO 2014 and AudioCaps. Extending the evaluation to a wider range of architectures, modalities, and downstream tasks will help assess the generality and robustness of our method.

Broader Impact By reducing sampling cost without retraining, Tortoise and Hare Guidance lowers the barrier to deploying diffusion models in real-time applications such as creative tools, accessibility services, and mobile environments. This could result in accelerating the production of synthetic media, including deepfakes and misleading content. Nonetheless, the capabilities of Tortoise and Hare Guidance remain bounded by those of the underlying diffusion model, introducing a limited impact to the quality of such synthetic media.

Acknowledgments and Disclosure of Funding

We sincerely thank Byeongju Woo, Chanyong Lee, and Eunjin Koh for their constructive discussions and support. We also appreciate Minseo Kim and Minkyu Song for providing insightful feedback. This work was supported by the Agency For Defense Development Grant Funded by the Korean Government (912A45701).

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A v -prediction models

Recent models such as Stable Diffusion 3.5 [47] directly infer v , or the velocity field of the reverse diffusion process. The diffusion ODE is then defined as

By the definition of CFG [18], we have

where ∆ˆ v c ( x t ) := ˆ v c ( x t ) -ˆ v ∅ ( x t ) . Substituting Eq. 18 into Eq. 17 yields the following:

We split this diffusion ODE into a multirate system of ODEs similar to Section 3.1.

Both equations retain the form of Eq. 17 so that existing solvers as the Euler method can be applied to each equation without modification. Furthermore, Algorithm 1 could be utilized unchanged since it is agnostic to the form of equation or the type of the diffusion model solver.

B Proof for approximation error bound analysis

We provide a proof for error accumulation presented in Section 3.3. More rigourous analysis of error bounds could be found in Section II. 3. of [14].

Theorem 1. Assume the local integration error of an ODE using a solver of order p and timestep size ∆ t is given by:

for sufficiently small ∆ t . Then the error of using the same solver repeatedly for m steps is given by

Proof. We use mathematical induction. ( Base step ) For m = 1 , Eq. 22 reduces to the assumption. ( Inductive step ) Assume the error of using the solver m times is given by Eq. 22. We proceed to the next iteration to obtain ˆ x t -( m +1)∆ t . Let ˜ x t -( m +1)∆ t be the exact solution given by solving the ODE from t -m ∆ t to t -( m +1)∆ t using ˆ x t -m ∆ t . The error in Eq. 22 is transported to the next timestep as

On the other hand, the local error of the next iteration is also given by Eq. 21:

The error of using the solver m +1 times is thus

Therefore the error of using the ODE solver m times is given by Eq. 22 for all positive integer m .

C More details for Richardson Extrapolation

We specify further details about the computation of the coarse timestep grid C . We calculate ∥ ˆ x T (1) s -ˆ x T (2) s ∥ and ∥ ˆ x H (1) s -ˆ x H (2) s ∥ by solving both the tortoise and hare equations on the finegrained timestep grid using Algorithm 3. In particular, for each denoising step t i , we first find ˆ x (1) t i +1 by using the diffusion model solver once from t i to t i +1 . Then we find ˆ x (2) t i +1 by using the diffusion model solver twice, from t i to ( t i + t i +1 ) / 2 and from ( t i + t i +1 ) / 2 to t i +1 . We use ˆ x (1) t i +1 for the next denoising step to ensure that we follow the reference trajectory of CFG [18]. Together with Algorithm 2, we obtain the coarse timestep grid C specified in Table 4.

D Sensitivity of C

Guidance weight ω The coarse timestep grid C depends on guidance weight ω , since the error bounds computed by Algorithm 3 depend on ω . As ω increases, both the hare term and its approximation error grow, resulting in a denser coarse grid C . Table 5 shows C evaluated under different ω values. While a bigger guidance scale results in a denser C , the overall trend is consistent; one can obtain the same C by adjusting ρ .

Batch size While we computed C with sample trajectories on 30,000 prompts from the COCO 2014 dataset, it is possible to compute C using fewer sample trajectories. Table 6 shows C computed on a batch of 1,000 prompts, compared to C computed on 30,000 prompts. The results closely matched our original, large-scale estimate. Compared to the original estimate, the 1,000 sample estimate demonstrated 95.25% IoU (i.e., Jaccard index) in average over 30 trials.

E More comparisons with CFG variants

Table 7 extends our comparison to include additonal baselines [12, 26] across a wider range of NFEs. Note that we used a different number of fine-grained timesteps N to control the NFE of CFG. The results demonstrate that our approach achieves superior image quality with the same NFE budget. While Selective Guidance [12] shows lower FID and CMMD values, they generate images with low prompt fidelity and degraded details indicated by lower CS and IR values as noted in Appendix G. Aside from this phenomenon, THG achieves superior overall performance.

Comparison with vanilla CFG Although vanilla CFG with 70 NFE yields a FID of 13.342 (better than THG's 14.232 with 70 NFE), its PSNR drops substantially to 19.42 dB compared to 24.16 dB for THG. This aligns with observations in [6], where reducing N sometimes lowers FID. Moreover, [35, 23] show that FID may not reliably reflect perceptual quality when the image structures diverge, so we focus on comparisons at the same N .

F More details for metrics

FID, FAD Fréchet Inception Distance [17] (FID) is a ubiquitously used metric for developing and adopting image generative models. It measures the distance between real and generated images in a deep feature space to capture relevant features of the two distributions [35]. Therefore, a lower FID value indicates realistic generation. To measure FID, one first uses an InceptionV3 feature extractor model to compute features from real and generated images. Under the assumption that the resulting feature sets follows a multidimensional Gaussian distribution, the distance of the two distributions N ( µ r , Σ r ) and N ( µ g , Σ g ) is given by the Fréchet distance

Similarly, Fréchet Audio Distance [24] (FAD) measures the distance between real and generated audio by using a VGGish embedding model to extract features from audio clips.

CMMD CMMD[22] is a recently proposed alternative for FID. It uses CLIP [38] embeddings and the Maximum Mean Discrepancy (MMD) distance instead of InceptionV3 features and the Fréchet distance. CLIP is trained on 400 million images with corresponding text descriptions and therefore

much more suitable for capturing rich and diverse content. MMD does not make any assumptions about the underlying distributions, unlike the Fréchet distance which assumes multidimensional Gaussian distributions. By combining CLIP and MMD, CMMD avoids the drawbacks of FID.

CLIP Score, CLAP Score CLIP Score [16] uses CLIP [38] to assess image-caption compatibility. CLIP learns a multimodal embedding space by jointly traning an image encoder and a text encoder, while the cosine similarity of matching image and text embeddings are maximized. Leveraging this design, CLIP score is defined by the cosine similarity of the image embedding E I and text prompt embedding E C as

We report the average CLIP score over the generated image set. Similarly, CLAP Score [50] uses the CLAP model to assess audio-caption compatibility.

ImageReward ImageReward [51] is a general-purpose text-to-image human preference reward model, effectively encoding human preferences by traning on actual human feedback. Experiments shows that ImageReward aligns with human ranking better than zero-shot FID and CLIP Score. Also, ImageReward values have larger quantile value range than that of CLIP Score, demonstrating that it can well distinguish the quality of images from each other. We report the average ImageReward value over the generated image set.

G Tradeoff of distributional similarity and prompt fidelity

Tables 1 and 2 demonstrate a tradeoff between distributional similarity metrics (FID, CMMD) and prompt fidelity metrics (CS, IR). When the prompt fidelity metrics improve so that each image matches better with the given prompt, the distributional similarity metrics worsen so that the distribution of the images is further from that of real images.

We further investigate this phenomenon by conducting an additional ablation study for the guidance scale ω using Stable Diffusion 1.5 and CFG. Table 8 shows how the metrics change as ω is changed. The minimum FID is achieved at ω = 2 . 5 and the minimum CMMD is achieved at ω = 3 . 5 . However,

they suffer from low CS and IR. Generated images using ω = 2 . 5 are visualized in Fig. 5, showing degraded details or insufficient text alignment. This suggests that lower FID or CMMD does not always indicate better generation quality. While these distributional similarity metrics measure both image plausibility and diversity, they can possibly fail to report high-quality details of the images with lower values.

Since the global structure of each image is determined by the initial few steps of the reverse diffusion process [26, 3], the images generated by the methods in Table 1 have mostly shared global structures and differ on delicate details. Given that, we suggest that the human-perceived quality of generated samples could be better explained by the prompt fidelity metrics compared to the distributional similarity metrics. Our results in Table 1 with slightly higher FID or CMMD therefore do not indicate a significant degradation of generation quality.

H Licenses

Stable Diffusion 1.5 - weights released under the CreativeML Open RAIL-M license (v1.0; https://github.com/CompVis/stable-diffusion/blob/main/LICENSE )

Stable Diffusion 3.5 Large - weights released under the Stability AI Community Licence v3 (research & commercial use for organizations or individuals with < USD 1 M annual revenue; https://stability.ai/license )

FID - clean-FID implementation by Parmar et al., released under the MIT License (v1.0; https://github.com/GaParmar/clean-fid/blob/main/LICENSE )

CMMD - PyTorch implementation of CLIP Maximum Mean Discrepancy by Sayak Paul, released under the Apache License 2.0 (v2.0; https://github.com/sayakpaul/ cmmd-pytorch/blob/main/LICENSE )

CLIP Score - TorchMetrics' CLIPScore module released under the Apache License 2.0 (v2.0; https://github.com/Lightning-AI/metrics/blob/master/LICENSE ; Lightning-AI)

ImageReward - model and evaluation code released under the Apache License 2.0 (v2.0; https://github.com/THUDM/ImageReward/blob/main/LICENSE ; Xu et al., 2023)

· MS COCO 2014 :

-Annotations released under the Creative Commons Attribution 4.0 International license (CC BY 4.0; https://creativecommons.org/licenses/by/4.0/ )

-Underlying images governed by Flickr Terms of Use; users must comply with Flickr's rules when reusing or redistributing any COCO images.

AudioLDM 2 -weights released under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License ( https://github.com/ haoheliu/AudioLDM2/blob/main/LICENSE )

AudioCaps - dataset released under the MIT License (v1.0; https://github.com/ cdjkim/audiocaps/blob/master/LICENSE )

FAD - PyTorch implementation of Frechet Audio Distance by Hao Hao Tan, released under the MIT License (v1.0; https://github.com/gudgud96/frechet-audio-distance/ blob/main/LICENSE )

CLAP Score - model and evaluation code released under the Creative Commons CC0 1.0 Universal License; public domain dedication ( https://github.com/LAION-AI/CLAP/ blob/main/LICENSE )

I More qualitative results

Figure 6 shows more qualitative results for Stable Diffusion 1.5. Figures 7 and 8 show more qualitative results for Stable Diffusion 3.5 Large.

Wi

Wi

Ml

STOP

(a) CFG (baseline) NFE = 56

STOP

(b) CFG-Cache w/o FFT NFE = 38

STOP

(c) CFG-Cache NFE = 38

STOP

(d) Ours NFE = 38

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Tortoise and Hare Guidance (THG), a training-free sampling acceleration method for diffusion model. THG reformulates classifier-free guidance (CFG) by utilizing the asymmetric sensitivity of the noise estimate (tortoise branch) and the additional guidance term (hare branch) to numerical error, reducing the number of function evaluations (NFE) by up to 30% with small loss in sample quality. THG achieves good performance on Stable Diffusion 1.5 and 3.5 Large.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is well-organized and the story is interesting. The proposed method is built upon multirate formulation with error bound analyzed for choosing proper coarse time schedule, which is novel.\n- THG achieves considerable acceleration without training, making it applicable to existing diffusion pipelines.\n- Experiments on Stable Diffusion models demonstrate competitive performance in terms of FID, CLIP Score, and ImageReward compared to the baseline.\n\nWeaknesses:\n- This paper only includes CFG-cache for comparison. However, there are a series of works [1, 2, 3], which are also mentioned in Section 2, focusing on accelerating CFG sampling but are not quantitatively compared. \n- Gradually increasing the guidance scale by $b$ is kind of wired, as it breaks the fair comparison with baseline where the guidance scale is fixed.\n- The experiment could be conducted more thoroughly. Currently, each model only has one choice of NFE. Including more choices like 20 or 10 steps could further enrich the results and demonstrate the superiority of THG.\n\nReferences\n\n[1] Angela Castillo, Jonas Kohler, Juan C Pérez, Juan Pablo Pérez, Albert Pumarola, Bernard Ghanem, Pablo Arbeláez, and Ali Thabet. Adaptive guidance: Training-free acceleration of conditional diffusion models. arXiv preprint arXiv:2312.12487, 2023.\n\n[2] Pareesa Ameneh Golnari, Zhewei Yao, and Yuxiong He. Selective guidance: Are all the denoising steps of guided diffusion important? arXiv preprint arXiv:2305.09847, 2023.\n\n[3] T. Kynkäänniemi, M. Aittala, T. Karras, S. Laine, T. Aila, and J. Lehtinen. Applying guidance in a limited interval improves sample and distribution quality in diffusion models. In Proc. NeurIPS, 2024.\n\nQUESTIONS:\n- Though choosing the coarse time schedule based on error bound analysis is convincing, it takes considerable efforts to estimate the error constants. How is the performance gain of this carefully designed schedule? Could a naive choice also provide good performance? A naive choice of the coarse time schedule can also make the use of higher-order solvers reasonable.\n- It could be better to include the latency comparison with existing method to show the superiority of the proposed method.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Tortoise and Hare Guidance (THG), a training-free method to accelerate CFG sampling by reformulating it as a multirate system of ODEs. The key insight is that the noise estimate and additional guidance term have different sensitivities to numerical error. This allows the guidance term to be computed on a coarser timestep grid while maintaining the noise estimate on a fine grid. The method includes an error-bound analysis to determine appropriate coarse grid spacing and achieves up to 30% reduction in NFEs while preserving generation quality on text-to-image generation.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n\nThe ODE formulation of CFG and decomposition of CFG into separate tortoise and hare equations are quite novel. Especially, the decomposition of CFG based on their different sensitivities seems well-supported by the empirical evidence shown in Figure 2. The idea of treating these components differently is intuitive and their algorithms are mathematically sound. The error-bound analysis makes their approach more grounded approach.\n\n**Weakness**\n1. If I understand correctly, Algorithm 3 is used to obtain the coarse grid before running Algorithm 1. This preprocessing step likely requires additional NFEs that should be included when comparing computational efficiency. It would be more fair to account for these extra NFEs in the overall NFE comparison.\n\n2. The paper only evaluates performance at specific NFE values. It would be valuable to see performance curves across a wider range of NFE values, particularly at more limited NFE settings, to better understand when and how much the method helps.\n\n3. The experiments are limited to first-order solvers (DDIM, Euler). Since higher-order solvers like DPM-Solver are commonly used in practice, demonstrating the effectiveness on these methods would strengthen the paper's impact.\n\n4. Given that four different evaluation metrics are used (FID, CMMD, CS, IR), it would be helpful to include brief descriptions of these metrics in the experimental section or appendix for readers who may not be familiar with all of them.\n\n\nminor:\nThe notation for the difference estimation could be clearer to avoid confusion, as the current notation might be interpreted as a product of delta and epsilon.\n\nQUESTIONS:\nAlgorithm 3 appears to compute the coarse grid for each example individually, which seems correct from an accuracy perspective. However, I'm curious about the variance in coarse time grids across different initial noise samples and text prompts. How much do these grids differ? For practical efficiency, would it be possible to compute coarse grids in a batch-wise manner across multiple text prompts?\n\n\nAs I found this work interesting, If the authors could provide sufficiently convincing explanations or additional experiments that address the experimental concerns I raised, I would be willing to increase my score.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper presents a method called "Tortoise and Hare Guidance" which accelerates the diffusion sampling under CFG with a coarser timestep grid. It took a principled approach by analyzing the ODEs and developed a method based on theoretical analysis. The proposed method gave a 30% reduction in NFE while maintaining sample quality with SD on Coco dataset.

STRENGTHS AND WEAKNESSES:
(+) The paper took a first-principled numerical-analysis-inspired approach to improving the CFG schedule. The novelty is good in my opinion.

(+) The paper is training free. It can be plugged into existing trained models with minimal changes. 

(+) The experimental results are relatively strong in my opinion for the method it tested and the dataset it tested on

(+) The writing and the flow of the paper is good

(-) Only SD series of models are tested, and it only works for 1st order solvers?

(-) Only tested on a limited datasets such as COCO. Not sure how well it generalizes to other datasets / models, and how sensitive the hyperparameters are across datasets

QUESTIONS:
See the (-) points in the strength and weakness section. It also contains questions

LIMITATIONS:
n/a
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose Tortoise and Hare guidance,  a training free framework for reducing the number of function evaluations during guided sampling in diffusion model. This is achieved by splitting the evaluations into a branch that is sensitive to change in time step (noise estimate) and a second branch which is relatively less sensitive to change in time step (guidance term). The signal at each time step is then calculated as a sum of the noise estimate (tortoise branch) and the guidance term (hare branch). The noise estimated is updated on a finer time scale whereas the guidance term is only update on a coarse time grid. The qualitative performance is shown to be close to standard diffusion and comparable quantitative performance is reported with respect to relevant baselines.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. **Writing quality**: The paper is well written with attention to detail. The core motivation for the approach and the central idea of the paper is presented in a manner that is easy to understand and appreciate. \n\n2. **Simple and elegant**: The proposed solution is relatively straightforward and explained in detail aiding in the ease of reproduction.  \n\n3. **Theoretically Grounded**: The motivation for the multi rate approach is theoretically grounded and error bound analysis has been provided to support the choice and design of certain hyper-parametes .  \n\n4. **Adaptive guidance scale**: This is an interesting trick to help maintain quality of samples depending on the resolution of coarse grid.  \n\n5. **Generality**: The proposed approach is a fundamental change to denoising process, allowing it to be used in any kind of diffusion model. Additionally, since it is a training free approach, it can be swapped into existing inference pipelines easily. \n\n## Weaknesses\n1. **Limited Qualitative results**: The main paper shows only 2 qualitative result for the difference in quality between the proposed approach and other baselines. Although the appendix includes some more. Demonstrating this on a wide variety of examples would be beneficial to understand and appreciate the effectiveness of the approach.  \n\n2. **Effect of schedule**: No ablation or analysis has been provided to indicate the effect of the timestep schedule during inference. Are the error bounds and performance guarantees the same for all time step schedules? Some discussion about this would be insightful   \n\n3. **Calculation of boosting term**: The authors ablate this term as well as motivate the need for it. But since the sensitivity changes across time steps, should the $b$ term be time dependent? Some discussion on this would be helpful.  \n\n4. **Evaluations**: Although the authors acknowledge this in the limitation section, the approach has been evaluated only on the COCO dataset. Since the algorithm is general enough, checking the performance on a wide variety of diffusion models would be beneficial. Although not a strict requirement, this could potentially also be evaluate on other modalities like video and audio.   \n\n5. **Coarse grid calculation**: The coarse grid is calculated based on the error bounds. Is this calculation independent of starting noise and the guidance weight?\n\nQUESTIONS:\n1. What is the result if $x_t^H$ is only updates at the coarse time steps and its approximation for intermediate fine time steps not calculated ? In particular, instead of calculating $x_s^H$ for $s<t$, would it be possible to use $x_t^H$ for every $s$ between $x_t$ and $x_{t+1}$.   \n  \n2. The authors mainly mention the use of DDIM and Euler method for evaluation. Is this approach also applicable to higher order methods for step reduction, and if not, what are the major considerations?\n\nLIMITATIONS:\nAn adequate treatment of the limitations of the approach has been provided. In particular the authors acknowledge that the approach is only tested on first order solvers and on limited evaluation datasets.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}