Towards a Golden Classifier-Free Guidance Path via Foresight Fixed Point Iterations
Authors: Kaibo Wang
Jianda Mao
Tong Wu
Yang Xiang
Venue: NeurIPS 2025 spotlight ()
Keywords: Classifier-free guidance
Diffusion models
Fixed point iteration
Primary area: deep_learning

Abstract: Classifier-Free Guidance (CFG) is an essential component of text-to-image diffusion models, and understanding and advancing its operational mechanisms remains a central focus of research. Existing approaches stem from divergent theoretical interpretations, thereby limiting the design space and obscuring key design choices. To address this, we propose a unified perspective that reframes conditional guidance as fixed point iterations, seeking to identify a golden path where latents produce consistent outputs under both conditional and unconditional generation. We demonstrate that CFG and its variants constitute a special case of single-step short-interval iteration, which is theoretically proven to exhibit inefficiency. To this end, we introduce Foresight Guidance (FSG), which prioritizes solving longer-interval subproblems in early diffusion stages with increased iterations. Extensive experiments across diverse datasets and model architectures validate the superiority of FSG over state-of-the-art methods in both image quality and computational efficiency. Our work offers novel perspectives for conditional guidance and unlocks the potential of adaptive design.

Towards a Golden Classifier-Free Guidance Path via Foresight Fixed Point Iterations
Kaibo Wang', Jianda Mao', Tong Wu', Yang Xiang
'Department of Mathematics, The Hong Kong University of Science and Technology 2 Shenzhen-Hong Kong Collaborative Innovation Research Institute, HKUST (kwangbi, jmaoao, txubi)@connect. waa.cct.....tt. hk, maxiang@ust. hk
Abstract
Classifier-Free Guidance (CFG) an essential component of text-to-image diffu- sion models, and understanding and advancing its operational mechanisms remains a central focus research. Existing approaches stem from divergent theoretical interpretations, thereby limiting the design space and obscuring key design choices. To address this, we propose a unified perspective that reframes conditional guidance fixed point iterations, seeking identify golden path where latents produce consistent outputs under both conditional and unconditional generation. We demon- strate that CFG and its variants constitute special case of single-step short-interval iteration, which is theoretically proven to exhibit inefficiency. To this end, we introduce Foresight Guidance (FSG), which prioritizes solving longer-interval sub- problems in early diffusion stages with increased iterations. Extensive experiments across diverse datasets and model architectures validate the superiority of FSG over state-of-the of-thh--rtttttt- methods in both image quality and computational efficiency. Our work offers novel perspectives for conditional guidance and unlocks the potential of adaptive design
Introduction
Diffusion models 14 27 16 have emerged as a transformative paradigm for conditional genera- tion, achieving remarkable success in synthesizing lighfiideity images from text prompts 24 22
Classifier-free guidance (CFG) [15] serves key component, steering the generation towards prompt alignment by amplifying the conditional outputs of a model. However, its over-amplified guidance introduces critical trade-offs, including compromised image quality and limited diversity (5). which highlight the necessity for deeper theoretical and practical insights into guidance mechanisms.
Existing efforts to mitigate the limitations of CFG are typically based on distinct conditional sampling perspectives. For example, some methods conceptualize CFG as sampling from sharpened probability distributions {8 [9] while others mitigate off--manfoll deviations through posterior sampling [5] or improve semantic alignment through reflective sampling II 20 Although these approaches are theoretically sound, the absence of unified interpretation narrows the design space of guidance mechanisms, rendering each method a tightly coupled framework where components cannot be independently modified. To integrate the effective components of existing approaches and further unlock the potential of guidance mechanisms, a critical question arises:
How can we systematically explore the design space for conditional guidance and develop more effective algorithms from unified perspective?
"Corresponding author
'The codes are available at http:p////hhhhoooooo/////////ata
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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We address this question by proposing a unified framework based on fixed point iterations. Observing that the latent variable 2, achieves better generation quality and alignment when its unconditional generation matches the prompt-conditioned generation 34 (Figure (a)), we reframe conditional generation as identifying a golden path composed of such points. Our framework comprises two decoupled steps: calibration and denoising. At each timestep t, the calibration step iteratively refines I, to & via fixed point iterations. Following this, the denoising step samples from p(r;-1 &t) with unconditional noise prediction, depicted in Figure (b). We demonstrate that CFG [I5] and its variants [ [00] are special cases of this framework. By disentangling guidance from sampling, approach enables (1) systematic comparison of design choices across methods and (2) upgrades existing algorithms through fixed point iteration design.
The number of iterations and the consistency intervals in fixed point algorithms are critical yet underexplored design dimensions. Current methods typically solve multiple short-interval subprob- lems using single iteration each, suboptimal strategy proven by our theoretical analysis. We first demonstrate that existing methods can be upgraded simply by increasing the iteration count (e.g., CFG x K) Beyond that, we propose Foresight Guidance (FSG), which prioritizes solving longer-interval subproblems during the earlier stages of the diffusion process with more iterations (Figure (c)). FSG enhances alignment by propagating guidance signals over extended time horizons while improving efficiency by reducing the number of subproblems.
Extensive experiments across datasets and models demonstrate that existing guidance methods directly benefit from increased iterations, and FSG further improves generation quality with lower computational overhead. We provide novel insights into advancing the development of efficient and adaptable guidance mechanisms. Our contributions are threefold:
We model conditional guidance calibration task toward golden path, unifying CFG and related methods under fixed point iteration framework.
We upgrade existing methods and propose FSG by designing consistency intervals and iteration schedules, achieving better alignment and efficiency.
We validate the effectiveness of FSG through comprehensive experiments and demonstrate the potential of the framework in guidance design.
2 Preliminaries
2.1 Diffusion Models
Diffusion models [4 27 are generative models that synthesize images by progressively reversing a predefined noising process. The forward diffusion process (t = 0 + 'T) gradually corrupts initial clean image No through a noising function 7 determined by a noise schedule 01:r:
It+1 (z=1+1(21)) vart1 + V ac+1c, e~,,)
(I)
The reverse process reconstructs the data by iteratively denoising to 21-1 p(---1 | ax)) through ft-st-1, which consists of neural network-based noise predictor c(xx)) and reverse sampler:
_t-1 f(-(-1(2)) Sampler(z,, e(z))),
(2)
where the sampler design is flexible, provided that vs, follows the marginal distribution px(x)) of the forward process. Notably, DDIM [27 operates as deterministic sampler, corresponding to a discrete ODE formulation [28). Let a. I Cts, its update rule is defined as:
(xx)) Ft VI @)((x1)
Vār
(3)
xt-1 = v0(=) + /I1 - 1=
To generate an image, we first sample ør ~ N(0,1) and then compute xo = fr-o(IT) fi-10 o... (IT)-
=
2.2 Classifier-Free Guidance
In conditional generation tasks (e.g., generating images for a given prompt c), Classifier-Fres Guidance (CFG) [15] is widely used to improve alignment by extrapolating from the unconditional

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8
Single-step short-sighted calibration
Unconditionl frnciin (K)
X
F (et)
Short-interval subproblems
Multi-steps foresighted calibration
Golden Path
"Mountains" Fr-e (r)
Calibrate to fixed point &
steps
A--0 (enon (2,) fE-0 (3.)
Long interval subprablems
(a) Consistent paths are better (b) Fixed point framework
(c) Foresight fixed point guidance (FSG)
Figure 1: From left to right: (a) Illustration of the golden path. Latents I generating mountains unconditionally produce higher-quality images when guided by mountain-related prompts. (b) Unified framework for fixed point iterations. The state 2; is calibrated toward ž; on the golden path via fixed point iterations. (c) Proposed foresight guidance (FSG). We enhance efficiency and alignment by conducting fixed point iterations over longer intervals with increased iterations.
noise prediction e towards the conditional prediction e", using guidance scale parameter w > 1. The adjusted noise prediction e" computed as:
n (we) = (2x,) w (
For brevity, we denote the denoising processes using e", E and e f f and f respectively. CFG+ [5] enhances CFG for manifold preservation through tunable parameter À E [0,1]: 1]:
Tt- = Vor-
11-
s (x.)
(rt v1 one (5)
7-sampling 11 and Resampling (0 first adjust xı &; through reflection, and then apply update
using < (1x)) The reflection in 7-sampling uses da = f o fi (xt). while Resampling employs & = - ft (x)) Here, fi-111 = va(+1d(z)) + /1 @.+1c(()) denotes
DDIM inversion 27
Methodology
We formalize conditional guidance calibration toward golden paths, then propose unified fixed point iteration framework encompassing existing methods (Section 3.1), thus enabling systematic comparison and extension of different design choices. Next, we introduce foresight guidance (FSG), a strategy to address subproblems across longer intervals through multi-step iterations, improving efficacy and efficiency (Section 3.2
3.1 Towards the Golden Path via Fixed Point Iterations
Consistent denoising paths under conditional and unconditional guidance are golden paths. In diffusion models that generate images zo ~ p(æ c) from initial noise zT ~ ~ I), empirical observations suggest that specific denoising paths yield higher-quality images and better alignment under target conditions c [4 M We term these paths as golden paths. Our objective is to calibrate the denoising process towards these golden paths.
Let (a)) and (xz) denote the conditional and unconditional denoising processes starting from Ty, then the latents 2, golden path satisfy: fr (&,) = (ft) For instance, if â unconditionally generates mountain images, achieves superior conditional generation performance with mountain-related prompts compared to other latents Iz, visualized in Figure aa). The underlying intuition is: when (x+) and f(ro(a))) differ significantly, the model requires sharp turn, often sacrificing texture detail and aesthetic quality to achieve conditional alignment. Conversely,

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when they align, the model balances both conditional alignment and generation quality. Further discussion is provided in Appendix B
We therefore aim to calibrate 2e toward bt by solving
(x=) (zee,
Tr. MM,
(6)
where M, denotes the probability manifold at timestep & 0
A unified framework based on fixed point iteration. We propose a framework based fixed point
iteration to approximately solve (6). and demonstrate that classifier-free guidance [[5] (CFG) and state-of-the-ar variants [5] 20 I are instances of this unified framework. Specifically,
divide
the transition It + xt-1 into three phases: (1) Construct a fixed point operator F F(r) satisfying
F(x.) E Mr and 2, F F
fi==0(a)) (āz) ff-)
(2) Perform K fixed point iterations
from zi
(0) = mς: It = F(z; (a=(-"))), ). ).k 1, 1,,......... K and denote &t = 2t (K), (3) Derive 2t-1 via the
(k)
unconditional noise prediction e" c"(ēt) and reverse sampler. This workflow can be expressed as:
......... F(zz). T--1 Sampler (&,, e" (i,)).
c p(K) = Fo...
4 Calibrate
Denoise
Our framework decouples calibration and denoising, enabling the injection of conditional guidance via the calibration step without binding it to the denoising process. Thus, the key questions of interest for conditional guidance become: (1) What are the design dimensions of fixed point iteration? (2) How can we design along these dimensions for effective guidance? We address the first question below and investigate the second in Section 4.2
Design space. Our fixed point iteration framework identifies four key dimensions: (1) Consistency in- terval. Since the clean semantics pufe are inaccessible during denoising, we consider consistency Longer intervals help to aggregate semantics from more distant timesteps but increase computational cost. (2) Fixed point operator. The fixed point operators typically used include the linear operator 2t = It + (((""n((Z))) f ff and the bockward-forwald operator 2; =["->b o (f)) (3) Guidance strength/sccheduler We can schedule the magnitude of updates for different timesteps. Similar to the learning rate, inappropriate settings can lead to slow convergence or deviation from the data manifold. (4) Number of iterations
over intervals a - b as an alternative, i.c. (z,) = fū=
K. Ideally, the fixed point algorithm converges linearly w.r.t. K, i.e., || - zell O(p²), ). where E (0, 1) is the spectral radius of F"
Surprisingly, we find that CFG and its variants can be incorporated into fixed point framework despite originating from different derivations, which provides us with a unified perspective. We present their design choices in Table [
Sketch of the unification. (detailed in Appendix A We isolate the unconditional noise in the update of DDIM (27 and unify CFG [15 and CFG++ 5 as linear fixed point operators in the interval
[t [t- 1,4,, using the equivalence of (x,,) = e (x,, to 1 = s (xt). Approximating M by fi+e+1 or 71--141. we equate reflective sampling in Z-sampling M and Resampling (0) to backward-forwee operators. Further nested CFG updates yield the results shown in Table
Comparison of design choices. (1) Iteration strength scheduler for CFG (wE+) and CFG++ (AEx): Compared to CFG, CFG++ provides more stable iteration strength during the critical early generation stage, avoiding drastic decay. This explains the improved stability of CFG++ over CFG. (2) Fixed point operators: Compared to the linear operators I2 + WC,Ac(z)) used in CFG and CFG++, we observe that the exhibits lower empirical contraction rates. This facilitates faster convergence, albeit at the cost of increased model evaluations. A more detailed discussion is provided in Appendix A.5
operator -= o (xx) with larger consistency intervals
Extension of design choices. The unified fixed point framework allows us to directly upgrade existing algorithms by increasing iteration counts. We extend CFG and CFG++ to CFG/ / CFG++ x K, where K denotes the number of iterations (Algorithm B However, this approach leads to substantial increase in model evaluations. Therefore, we propose Foresight Guidance, which balances efficiency and effecctivnees by prioritizing longer intervals and more iterations in the early stages.
2This constraint aims to prevent inaccurate score estimation off the manifold. Since satisfying this constraint strictly is not the primary focus of this paper. we simply follow empirical settings, e.g., use smaller step sizes.

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Table I: Unification of different guidance methods as fixed point iterations via xh+1 = F(a aiming for (1:) (1a). Multi-iter. indicates whether the original algorithm supports multi-step iterations. Abbreviations: (I) w > 1, = vI- ā. var at for CFG; (2) A E
[0, 11. vI a2 for CFG++; (3) fr denotes denoising with noise c" + + - -"),, Ae(+) denotes e() e (.), and id denotes the identity mapping.
Methods
Fixed point operator F(x))
Interval (a - b)
Multi-iter.
CFG 15
2t w (
+ 1
CFG++
St (xe)
I 1 - 1
X
Z-sampling
(id wf fi+1- o St (It)
t + tt --1
X
Resampling 20
(id wC s M (It)
++ 1
FSG (ours)
(id )f,De) fi- At (2x)
t At
3.2
Foresight Guidance
We aim to minimize the gap between condi- tional and unconditional paths. Existing meth- ods typically divide the problem into T sub- problems with intervals of t/t + 1 + f 1, each solved with one step of fixed point it- eration (illustrated each subproblem is relatively simple, the large number of subproblems limits efficiency, espe- cially for high-overhead fixed point operators such as werkwwdddfowwrr operators. More- over, small intervals limit the benefit of cali- bration, as only the semantics of neighboring timesteps can be obtained for guidance.
Algorithm 1: Foresight Guidance (FSG)
Input IIiiiill noise ær, Condition c. Timestops T. Iteration set S = {(t; Ks, At:)
Strengths y 1, X E [0, 1].
Output::ennnran :Generated image 270
Figure [ (b)). While
for 4- to do
if [t, At) E S then Foresight Fixed Point Calibration; for x- to do
at- = 8 ((+))) 3t = Si- (TC-
end end
CFG++ Calibration; 10 it = xz Xxx(c((x)) e" (xx))):
To improve efficiency, instead of allocating one iteration for each small interval, allocate multiple iterations for fewer long in- tervals. We theoretically demonstrate that the single-step short-interval strategy is typically suboptimal. Given the total iteration budget N and timesteps T. we uniformly divide fr fo into M subproblems ((T/M-((-1)/A f E [M], solving each only at timesteps iT M with N/M fixed point iterations (assuming T/M e Z). The M T case represents the short-interval single-step strategy.
11 Denoising Step;
12 11-1 Sampler(fr, ((x)))
13
end
14 return 20
Theorem 1. (Detailed in Appendix Ch Let C -
(i.) denote the average gap c" (",
over calibrated trajectories án E R.d. with B as the Fuclidean norm bound for in and (f.(). T. as the smoothness constant of (-), and r E (0, 1) as the upper bound of the contraction rate of F,, i [M). Under mild assumptions (Appendix Cj there exists constant C > 0 such that
L= T Σ
($x)) (2.)) < B² (cr
(Cr
2N +
2L
(8)
M
Setting the derivative of the right-hand side to zero yields the optimal M that minimizes the upper bound. The optimal M* is typically not T. indicating that performing fixed point iterations at every timestep is unnecessary. Key insights include: (1) Smaller L (smoother noise predictors) reduces M* favoring fewer, longer-interval subproblems; (2) Sufficient computational resources (N - oo) drive M + T', recovering the short-interval subproblems. In practice, limited resources suggest using moderate interval sizes to enhance fixed point solving efficiency. Intuitively, longer intervals provide stronger guidance during early denoising stages, where short-term approximations (f""/_,) yield limited benefits due to insufficient estimates of the clean image. Thus, generating prototypes through
5

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Table 2: The quantitative results on the SDXL model with NFE 50, 100, 150 (Time denotes seconds per image. denotes higher is better, The best results under the same NFE are bolded).
Datasets
DrawBench
25
Pick-a-Pic
[7
Method
NFE
Time
IR
HPSv21
AEST
CLIP
IRt
HPPvv1
AES
CLIP
CFG
50
6.71
59.02
28.73
6.07
32.29
82.14
28.46
6.73
33.53
CFG++
50
6.82
65.21
28.98
6.08
32.60
89.75
28.72
6.67
33.86
7-Sampling
50
6.66
72.75
29.08
6.00
32.59
96.77
28.68
6.59
33.97
Resampling
50
6.48
59.99
28.80
5.99
32.21
82.65
28.46
6.61
33,46
FSG (ours)
50
6.77
82.81
29.42
6.01
32.65
98.59
28.89
6.60
34.32
CFGx2
100
12.51
77.71
29.36
6.06
32.44
96.06
28.84
6.64
34.13
CFG++X2
100
12.60
79.42
29.42
6.01
32.61
99.90
29.00
6.61
34.18
Z-Sampling
100
12.61
77.46
29.26
6.03
32.39
94.98
28.79
6.61
34.01
Resampling
100
12.49
77.26
29.12
6.00
32.46
79.36
28.61
6.02
33.61
FSG (ours)
12.56 100
84.12
29.54
6.02
32.76
102.82
29.05
6.66
34.30
CFGx3
150
18.47
83.56
29.51
5.95
32.66
102.13
29.04
6.61
34,28
CFG++X3
150
18.47
82.58
29.45
5.93
32.66
103.32
29.05
6.57
34.20
Z-Sampling
150
18.26
78.35
29.40
6.06
32.43
97.25
28.90
6.67
34.20
Resampling
150
18.26
79.98
29.23
6.05
32.32
87.48
28.70
6.59
33.49
FSG (ours)
150
18.49
88.18
29.44
5.96
32.70
104.86
29.04
6.65
34.28
extended conditional denoising processes fi+t-Ai and preserving information into d. via the inverse ODE A appears more effective.
Practical design of foresight guidance. We integrate these design choices into Foresight Guidance (FSG), as outlined in Algorithm [ We perform multi-step lixed-point iterations with long time intervals at specific timesteps, parameterized by a set S = (4, K,, A where each tuple denotes the starting timestep, number of iterations, and interval length, respectively. To reduce computational overhead, we employ forward-hackward operator with a single-step ODE solver (DDIM) for both fi and fi At- As most image semantics are determined in the early stages of diffusion [3]] we allocate more iterations (K;) and longer intervals (At,) during these phases, following ratio of approximately 3:2:1 across early, middle, and late stages. Outside these scheduled iterations, we apply CFG++ to maintain stable guidance strength and avoid oscillations. This allows early generation stages benefit from the semantics of future states, motivating the term foresight guidance.
4 Experiments
4.1 Experimental Setup
Datasets. We assess generation performance across four benchmark datasets: DrawBench (25]. Pick-a-Pic [17), Geneval [00, and PartiPrompts (32]. Detailed experimental setups are provided in
Appendix I.I and results for PartiPrompts are included in Appendix [.2]
Metrics. To evaluate results, we employ IR (30 and HPSv2 29 as human preference metrics,
ClipScore 12 for prompt alignment assessment, and AES 26 for aesthetic quality analysis.
Baselines. Under our unified fixed point iteration framework, we systematically analyze five methods: CFG [15], CFG++ [5]. Z-sampling [ Resampling [20], and our proposed FSG. Leveraging the fixed point perspective, we enhance CFG and CFG++ by increasing iterations (denoted as x x2/x3), Experiments primarily adopt configurations from original studies. The fixed point interval FSG set within (0.02T, 0. 0217..), with larger intervals and iterations allocated to early steps. Implementation details provided in Appendix D.
novel extension beyond prior literature 15
5

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Z-Sampling
8.5%
13.5%
20.5%
23.550
(x)]ll
3.0
2.8
5.8910.9%
2.6
ASE
2.4
CFG
Z ZaSmllnn
FSG
CFG++
Resampling
CLIP
24.5%
18.5%
18.0%
1000
800
600
400
200
1.0
Denosing steps:
(a) Top-1 Rate.
(b) Prediction gap after calibration.
Figure 2: Left: Proportion of samples that outperform the other four methods (Top-1 rate). Right: Prediction gap during denoising, indicating the effectiveness of the fixed point iteration. Dataset: DrawBench, NFE: 50, see Appendix D.3 and D.4 for more results.
Table Quantitative results on Geneval dataset. Model: SDXL; NFE: 50 (CFG), 150 (others).
Method
Overallf
Single Objectt
Two Objectt
Countingt
Colorst
Position
Color Attribution
CFG
48.39 %
97.50 %
61.62 %
22.50 %
78.72 %
14.00 %
16.00
CFGx3
55.94 %
98.75 %
75.76
40.00 %
85.11
8.00 %
28.00 %
CFG++x3
56.03 %
97.50
78.79
45.00
81.91
10.00
23.00
Z sampling
56.70 %
100.00
75.76
46.25
86.17
20.00
Resampling
56.65 %
100.00 %
84.85 %
40.00
84.04
7.00 %
24.00
FSG (ours)
57.95 %
100.00 %
79.80 %
43.75 %
86.17 %
12.00 %
28.00 %
4.2 Experimental Results
Quantitative analysis. We evaluate the performance of different methods NFE (number of function evaluations) of 50, 100, and 150 using SDXL with the DDIM sampler, as shown in Table 2 The following discussion analyzes design choices within the fixed point iteration framework:
Extended intervals and increased iterations enhance efficiency and alignment of FSG. FSG demonstrates superior performance across datasets and NFEs, with particularly notable improvements lower NFEs (50, 100). As shown Table and the Top-1 rate in Figure FSG improves IR by 10.06 and achieves Top-1 rate of 34%, while improving HPSv2 by 0.34 with a Top-1 rate of 46%. This improvement attributed to more effective fixed point solving. The reduced fixed point error in Figure confirms that balanced subproblem decomposition enables better convergence and generation quality, consistent with Theorem Additionally, longer intervals enhance semantic guidance and prompt-alignment in FSG, as evidenced by consistent CLIPScore and improvements across all NFEs.
2. Existing methods benefit from increased iterations. Within the fixed point framework, CFG and CFG++ outperform vanilla CFG when the number of iterations is increased (denoted as x/)x3). This demonstrates the potential of our framework to enable test-time scaling for improved performance through additional inference resources, preferable to simply increasing inference steps (see Appendix D.5).
3. Strength schedulers and fixed point operators. CFG++ achieves marginal gains over CFG through more stable intensity scheduling. Backward- -orwardd operators with small intervals show limited improvement compared to linear operators at higher NFEs due to insufficient guidance, necessitating longer intervals for adequate semantic guidance.
Qualitative analysis. We present qualitative results in Figure comparing the performance of FSG with baseline methods. FSG achieves both high image aesthetics and strong adherence to prompt requirements. By leveraging longer intervals, FSG achieves more precise guidance for generating fine

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Table Quantitative results on different models and samplers. Dataset:Pick-a-pii: NFE: 50 (CFG). 150 (others); see Appendix D.6 for more results.
Models Method
SD-2.1 (24], DDIM
Hlunyuan-DiT [18].
DDIM
SDXL. DDPM
[4
IR1
HPSv2f AEST CLIP
IRt
HPSv21
AEST CLIP1
IRt
HPSv21
AEST
CLJP
CFG
62.83
25.51
5.88
30.38
115.63
29.00
6.82
33.09
73.57
28.42
6.73
33.62
CFG x3
1.08
27.25
5.92
32.50
115.32
28.98
6.50
32.88
91.37
28.69
6.62
33.73
CPG+ x3
3.98
27.24
5.96
32.51
115.63
29.03
6.63
32.97
89.17
28.79
6,60
33.60
Z-sampling
3.65
27.24
6.07
32.60
128.72
29.23
6.72
33.42
90.35
28.63
6.65
33.58
Resampling
8.07
27.03
5.85
32.31
117.65
29.28
6.73
33.18
90.92
28.64
6.65
33.73
FSG (ours)
16.26
27.60
6.10
32.80
132.88
29.37
6.68
33.48
91.53
28.56
6.65
33.79
Prompts:
FSG (ours)
CFG
CFG++
Z-Sampling
Resampling
bowl of soup looks a monster with tofu soys deep learning Prompt details and text
Five frosted glass bottles t Counting
A cat in space suit walking on the moon
1 age details
storefront with Gooole
Google
Research Pizza Cafe' written on it.
t Long text
One computer technical sketch white background
Prompt pt alignment
Figure 3: Results of qualitative analysis. Our proposed FSG demonstrates effectiveness in several dimensions including text, details, and counting. We present more examples in Appendix B
details (e.g., cat faces, text). Furthermore, FSG-generated images exhibit fewer structural or visual artifacts and align more closely with human preferences. These results highlight the robustness and versatility of FSG in addressing complex image generation tasks. We present additional visualizations and discuss failure cases in Appendix
Object-focused evaluation. We assess fine-grained instruction compliance on Geneval [0], as shown in Table FSG significantly addresses the deficiencies of vanilla CFG in counting accuracy (+23.75%) and two-object generation (+23.23% achieving wate-of-the-an overall performance. Notably, CFG/CFG+ with increased fixed point iterations also demonstrate enhanced alignment, underscoring the benefits of fixed point itcrations.
Class conditional generation. To investigate whether enhanced fixed point iterations reduce diversity, we conduct experiments on the ImageNet (6] 256 x 256 conditional generation task using DiT [21 models, generating 1K images per class (totaling 50K images). We report FID [ and Vendi scores 7 at NFE=25/50. As shown in Table 5 FSG and CFG/CFG++ at NFE=50 both improve generation quality and diversity, indicating that fixed point iterations do not cause mode collapse. hypothesize this occurs because the iterations are performed locally in the neighborhood of st thereby improving quality while preserving randomness.

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Table 5: Quantitative results on ImageNet
Table 6: Synergistic effects of FSG and noise opti- mization model (NPNet) Dataset: Pick-a-Pic.
256 x 256. Model: DIT.
NFE Methods
25
50
Methods
IR1
HPSv21
AES1
CLIP
FID 4 Vendi f
FID
Vendi t
CFG50
82.14
28.46
6.73
33.53
CFG (x2)
17.81
3.44
14.69
3.79
FSG50
98.59
28.89
6.60
34.32
CFG++ (x2)
13.27
3.91
8.85
4.43
FSG100
102.82
29.05
6.66
34.30
Resample
17.54
3.50
9.05
4.47
NPNet
91.66
28.60
6.70
33.57
Z-sampling
19.89
3.40
8.62
4.64
+FSG50
112.64
29.04
6.54
34.09
FSG (ours)
10.56
4.73
7.91
5.79
+FSG100
111.83
29.15
6.57
34.13
Table 7: Synergistic effects of FSG and preference
Prompt:
alignment model (SPO). Dataset: Pick-a-Pe pic.
3D Pac Man real life
Method
IRf
HPSv21
AEST
CLIP
SDXL CFG50
SOKL F5C100
CFG50
82.14
28.46
6.73
33.53
FSG100
102.82
29.05
6.66
34.30
SPO
111.86
29.08
6.91
33.22
+FSG50
115,86
29.16
6.91
33.12
+FSG100
117.93
29.20
6.93
33.24
SPO CR6S0
SPO F5G50
SPO FSG100
+FSG150 116.49
28.74
6,85
33.30
Figure 4: FSG provides better guidance on SPO
and progressively improves generated images.
Different models and samplers. Results across models (SD2.1 (24 and Hunyuan-DiT (8)) and
samplers (DDPM (4) are presented in Table Compared to other methods, FSG demonstrates
greater improvement on the weaker SD2.1 baseline (IR: 18:19; HPSv2: +0.35) while still providing
additional quality gains for the ttte-offthe-aat art Hunyuan-DiT model (IR: +4.16; HPSv2: +0.09). As
sample-agnostic framework, our method naturally extends to stochastic samplers without additional derivations, where fixed point iteration similarly improves prompt alignment.
Synergy with orthogonal methods. As fixed point algorithms are sensitive initial values, explore the synergistic effects between FSG and orthogonal approaches, including preference-aligned models [19] and noise optimization methods (34). SPO [19] improves the noise predictions of the diffusion model via preference fine-tuning, while NPNet [4] employs lightweight networks to tailor initial noise to given prompt. Both strategies supply better initial values to the fixed-point iteration, facilitating boarse-to-fine calibration. Experiments conducted with NPNet [34] (Table and SPO 19 ( demonstrate that FSG alone outperforms noise optimization, but falls short of preference-aligned models. When FSG is integrated with either approach, synergistic improvements are observed, leading to further gains in aesthetic quality and prompt alignment (IR: 112.64 with NPNet; 11 117.93 with SPO). Figure illustrates the progressive quality enhancement when combining FSG with SPO, confirming the compatibility of our framework with orthogonal approaches.
Figure/[illustratee
Ablation studies. Table summarizes the impactful design choices at NFE = 50/150. Key findings: (I) Consistency intervals (x2/42): low NFEs, appropriate interval selection provides more effective guidance than short-interval strategies. Overly large intervals introduce distant guidance that exceeds the current optimization stage, hindering fixed point iteration convergence.
(2) Guidance strength: As shown in Figure 5] FSG adapts to À E [0.4. 1.3] (beyond the [0.5. 1] range in CFG++). Fixed point iteration narrows the conditionalluncondttooaa gap while suppressing quality degradation from excessive guidance. The scheduler of CFG++ prevents premature intensity decay, making it suitable for our approach. (3) Iterations (x 2/12);: While 2-3 iterations suffice for high-quality results, excessive iterations introduce computational overhead and risk divergence from the data manifold. (4) Timestep prioritization: Early-stage ((KT,T)) foresighted guidance drives quality gains, while later timestep calibrations refine details.

---

Table 8:
Ablation study on differ- choices (metries differences shown). Dataset: Pick-a
106
102
29.0
28.8
Method
IRT
HPSv21
AEST CLIP
R
28.6
HPSV2
28.4
FSG50
98.59
28.89
6.60
34.32
0.2
1.0
6.85
34.6
Interval x Va
-8.20
-0.04
0.01
-0.13
AES
Intervalx2
-2,40
-0.12
-0.01
-0.23
6.75
CLIPSCore
34.2 W
FSG150
104.86
29.04
6.65
34.28
6.65
Iterations x V22
-6,16
-0.21
+0.03
-0.23
34.0
Iterations x2
-2,41
-0.50
-0.13
-0.07
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Early
-4.82
-0.19
-0.08
-0.14
Guidance strength
Early+Mid
-2.33
-0.12
-0.07
-0.14
Figure 5: Guidance strength analysis. FSG shows robust-
w.o. CFG++
-4.78
-0.08
+0.03
-0.03
ness on À E [0.4, 1.3]. Dataset: Pick-a-Pic; NFE:
Related Works
Classifier-Free Guidance 11 (CFG) is crucial for modern text-to-image diffusion models, yet its mechanisms remain unclear. While initially framed as sampling from p(zcc) & p(z)p(((z>)" this interpretation has proven inconsistent with implementation behaviors 4 Recent works pursue distinct objectives: sampling from p(xrcc) x p(r|c) R(x, c) [9], minimizing score distillation sampling (2). and satisfying Fokker-Planck equations [33].
loss [5), functioning as a predictor-correate However, these approaches' entanglement with diffusion sampling dynamics impedes both theoretical progress and unified method comparison. Our fixed point iteration framework circumvents this coupling through a sampling-independer interpretation.
The golden path phenomenon states that specific noise yields superior generation outcomes 23
Empirical observations demonstrate that noise capable of unconditionally generating image matching condition achieves enhanced performance when guided by [3 This observation has motivated various approaches, including optimizing initial noise [11
23 neural network- noise
prediction 34. and progressive refinement during inference [1. While effective, these techniques incur significant computational overhead. To the best of our knowledge, our work presents the first
established connection between this phenomenon and CFG.
6 Conclusion
In this work, we introduce unified framework for classifier-free guidance (CFG) in text-to-image -maie diffusion models by reinterpreting conditional generation calibration process toward a golden path. First, propose fixed point iteration methodological tool to enforce latent consistency between conditional and unconditional outputs, establishing a sanpling-independent framework with broader design applicability. We unify existing CFG and its variants as short-interval single-step approaches under this framework, which are theoretically inefficient. Second, we present Foresight Guidance (FSG), multi-step iteration paradigm that reduces subproblems while enabling long- horizon guidance, achieving an enhanced alignment-quality balance. Comprehensive experiments demonstrate the superiority of FSG over state-of-the-art methods. We anticipate that our unified perspective will catalyze advancements in adaptive, efficient diffusion guidance strategies.
Acknowledgements
We would like to thank the anonymous reviewers for their valuable suggestions theory and experi- ments. This work is supported by the Project of Hetao Shenzhen-HKLU HKST Innovation Cooperation Zone HZQBKCZYB-202000
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The paper does not involve crowdsourcing nor research with human subjects.
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18

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For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
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Does the paper describe the usage of LLMs if is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Question:
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19

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A Unified Framework
In this section, establish theoretical framework that unifies guidance methods under the perspec- tive fixed-point iterations. Through systematic examination of Classifier- Guidance (CFG), its variant CFG++, and reflective sampling techniques including Z-Sampling and Resampling, demonstrate how these approaches achieve conditional generation through distinct yet analogous fixed point operators. Our framework decouples guidance denoising-agnostic calibration steps, facilitating the design of guidance algorithms.
A.1 Classifier-Free Guidance (CFG) 15
Classifier-Free Guidance (CFG) enhances conditional generation by interpolating conditional and unconditional noise predictions during the diffusion process. Our analysis reveals that CFG's forward process corresponds to linear fixed-point operator with consistency intervals spanning - t 1, requiring NFE per iteration.
To formalize this, we begin with the DDIM update step governed by the predicted noise <(r)):
T
04-1
I - 11- ā,c(z)) + v 1 ā)-1c(z(),
(9)
/
where a IT:=1 0; representation that better elucidates the noise dependency:
a Through algebraic rearrangement, we derive an equivalent
00--
va-
It-1
2t +
(V--ōr-1
āc-1
vI ās c(x))
Vā
Vā=
0
vI-
Dt
+
(1-
-i-1
E(xxt).
(10)
Var
vai
CFG introduces rescaled noise prediction to amplify conditional guidance:
(x,) c((x)) + w(<((x,) - <(()),
(11)
where w modulates guidance strength. Substituting e" (xx) into the update rule yields:
xt-1 va-i
vār xe + \ a (VI-0--1-
Vai
x (x)( + w(e - (xx)))
(12)
This formulation admits insightful reinterpretation two-step process. First, the latent variable & is calibrated via:
f = xt w (V1- - a; - Vas - ac) (xx)
((((x))-e - - (xx)),
(13)
followed by an unconditional denoise:
v
(VI- VI- ar
Ft-1
it +
(x)))
(14)
Vav
Vai
The iterative variant CFGx K is detailed in Algorithm
2
A.2 CFG++
CFG++ derives conditional guidance through manifold constraints formulated from an inverse problem perspective, closely aligned with CFG in implementation. Our analysis demonstrates that CFG++ similarly corresponds linear fixed-point operator iteration but diverges from CFG solely in its strength scheduling strategy, retaining the consistency interval / + l at a computational cost of NFE per iteration.
CFG++ adjust the guidance strength through parameter applied to the noise difference term:
var-1
Tt-1
{ot
v1
(x)) + X(e - (zx)))
Vas
+ /1 (xx))
(15)
20

---

This admits similar calibration calibration-interprttaii interpretatirnn, where the latent calibration becomes:
f It Av1 >vI-aa(<((a)) e" (xx)),
(16)
followed by the same unconditional update:
It-1
vau-l ae + Vāt
(V1- at-1
vI aee
E (Iz))
(17)
Vave
Both CFG and CFG++ exhibit fixed-point behavior through linear operators:
FCFG( = + e FC (x1) = It (=()
(18)
(19)
where & (VI- 0i- Va a4) E = =I= a respectively. Note that fixed-point F(x,) 2,
implies consistency (x²) E" (a1)),
enforcing alignment between conditional and unconditional
update paths: f 1
(x+) = fi- (xT). The iterative variant CFG++X K is detailed in Algorithm
Algorithm 2: CFGXK
Algorithm 3: CFG++X
Input :Initial noise TT, Condition c, Timesteps
Input IIitial noise ær. Coudition c, Timesteps
T. Iterations K, Strength w >> 1.
T. Iterations K, Strength E [0, 1].
:Genprated image To
image for ( <- to
for
< T do
(0) 21'
2; (0)
CFG Fixed Point Calibration; for 4- K do
3
CFG++ Fixed Point Calibration;
4
for k + to do
/1 Ot
Vat- āi:
& v (k-
(k-1)
we (t=
z It
end
7
end
Denoising Step;
Denoising Step;
T1-1 Sampler(1; , _e"(x) ,x"(x( (1(
(K)
2t-1 Sampler(x; (8) , exx((( (x((K-1)))
10 end
end
return
return 200
A.3 Z-Sampling
Z-sampling augments conditional guidance by reflective sampling steps: inversion using unconditional noise followed by forward using high-strength guided noise. This procedure is mathematically equivalent to a fixed-point iteration using an hackward-forwant operator. When integrated with CFG after reflective sampling updates, the process achieves consistency interval of t + 1 at cost of 3 NFE per iteration.
Z-S Z-Sampling enforces bookward-foraatt consistency via the relation:
(z.)) (2141(21)
2t f2 (2))
(20)
Here, f denotes denoising with noise c" c"+7(c-e²) )²). In fact, the equality ft fi ( ff (2)) can be viewed
generalization of fi (2x) fi (xt). This stems from the relationship:
e" (((x)) = e (x, implies e ("xz) = =6" (((x) + (1) - ((x)), which establishes that (1+))1((1))
fi (xx) can be extended to li = ( (x.4
Approximating the inverse process
fi fiti- we derive: ã = fiti-e o (xt).
(21)
Subsequent application of CFG's calibration step yields:
t. H. - w (vI ai - Vos ās) -("())),,
(22)
1 v0-- t1 + (VI- VI-
(zt).
(23)
Var
Vai
This constructs composite fixed- fixed-point iteration over the extended interval i + - t 1-1.
21

---

A.4
Resampling [00]
Resampling also utilizes backward- forward fixed-point iterations but differs by incorporating a model- free stochastic noise function for the inversion step. Consequently, it maintains the + 1 + update interval while reducing the computational cost to NFE per iteration.
The stochastic backward step is defined as:
T (z+) = Vo + -1- ast16, E~ (0,1),
(24)
which leads to the complete update sequence:
Tt = f o u (11441(001));
(25)
f. = 3t - w (V1 Õr -Vo - az) (< - <" ("())),
(26)
-4-1 vāc-L is + ac-1 vI āt
(Zt).
(27)
vā+
vaz
Although computationally efficient, the stochastic approximation n4-+1+1 introduces semantic distor- tion compared to deterministic counterparts.
We demonstrate that Resampling be viewed approximation of Z-Sampling when the step size sufficiently small and y is sufficiently large. First, we express the DDIM forward process 1i explicitly:
x++1 + (V1- a₁ VI- VI-ai āg var+1
(41-(1441
(x2+1))
(28)
vou+1
For sufficiently small step size, approximate e( (1rx) and substitute 2441 VI Qc+16 obtain the fixed-point update form:
v0++112
8. vot+1 (Vart101 + v a1116)+ + (v- v1 ā vatt
E (I++1)
(29)
= σ4 + m4E m4(e" (xt) + -
-
where m. -I- aauti and = /1 ã₁ -1- ā,
/ /auti are time-dependent con- stants. Compared to the fixed-point objective (xx) - (xx)) = other operators, the objective for Resampling's fixed-point operator becomes:
1 Im,e -m n,e (x;)].
(x.) e
(30)
Theoretically, the objective of Resampling aligns with other fixed-point operators when y is suf- ficiently large. In practice, however, is not set to an extremely large value to maintain sample quality, which causes Resampling underperform compared other operators. Nevertheless, Table 2 shows that Resampling still benefits from increased iteration counts. This empirically confirms that Resampling's 's fixed-point operator a functional, albeit suboptimal, choice.
This systematic analysis demonstrates how various guidance methods can be unified under our fixed point framework, as summarized in Table of the main text.
A.5 Comparison of Design Choices
This section provides comprehensive analysis of the design choices within our unified fixed point iteration framework, examining both iteration strength schedulers and fixed point operators.
Iteration strength schedulers. The schedulers for CFG and CFG++ given by & -I- C7: Va, Or and & vT a,, respectively. Using the a, setting in DDIM example, Figure shows the & and & at different timesteps.
22

---

It can be observed that, in the early stages where guidance has a greater impact on generation (t [0.67, 1). the guid- ance strength provided by & decays rapidly, which limits generation quality. In contrast, & provides more stable iteration strength during these critical early stages, ex- plaining the improved stability of CFG++ over CFG. The
0.175
Schoduler stren gtteggtth
0.150
0.325
mathematical derivation these schedulers reveals that
& contains additional term va de that causes the
9.075
rapid decay, while maintains more gradual decrease throughout the diffusion process.
0,000
CFS
0+++
Fixed point operators. In FSG, a forward-backwarr operator with larger interval A/ selected, as ex- hibits smaller contraction rate, thereby accelerating the convergence of the sxcd-ppoint algorithm.
Figure 6: Comparison of iteration strength schedulers for CFG (Ex) and CFG+ (E:) across different timesteps.
Due to the complexity of neural networks, direct compu- tation of the model's contraction rate infeasible. Instead, empirical estimation of the local contraction rate within a neighborhood of It. defined
we employ as set points generated by mixing the same zo with slightly different noises. The local contraction rate for an operator I under the E2 norm is measured as:
[( F(x1)
(31)
x 2 12
where ne, o are obtained by mixing different noises with Z0o-
Table 9: Empirical contraction rates i for different operators across denoising timesteps.
Operators
=02
00.4
=066 t =0.8
id wi Ac (CFG, CFG++)
1.00
1.00
1.00
1.00
f dt f (Z-sampling)
1.04
0.99
0.97
0.99
17 di o o (Resampling)
0.89
0.97
1.03
1.07
11/2-st fi-1/2
1.03
0.95
0.96
0.98
fu f1/4-1t fi-e/4
0.61
0.91
0.88
0.91
18 Si-00
0.62
0.70
0.75
0.79
Notably, long-interval operators (e.g., SöLzt f demonstrate lower contraction rates < 1) compared to short-interval operators such as 7-sampling or Resampling. To balance convergence speed and operator complexity, we select an intermediate interval At (dt At t) for the operator fi- fint-A
The contraction rate of operator depends on the choice of metric. While operators in CFG/CFG+ are non-contracting under the l norm, they may exhibit contraction in other carefully designed metrics. Empirically, as shown in Table 2, these methods also benefit from additional iterations, supporting the ixee-poiin framework.
B Discussion of Golden Path
The concept of the golden path, wherein latent representations yield consistent outputs under both conditional and unconditional generation, is central to our framework. This section provides empirical evidence supporting this theoretical foundation and demonstrates its practical implications. We construct matched and unmatched samples (on and off the golden path, respectively) by design.
Match case: An image zf is generated conditioned on a prompt c. We then obtain æt via DDIM inversion 277 using unconditional noise c(xx), i.e., Ty. (2%). These T, exhibit consistency between conditional and unconditional generation paths, satisfying
x x fz;) fz)))
23

---

Mismatch case: For each matched sample, we assign a different prompt c' + to create mismatch, where x xi f (24) * fi
Using these as starting points, we evaluate generation performance under matched and mis- matched prompt conditions using CFG [15]. As
Table 10: Performance from iy under matched (c)
and mismatched (c' + c) conditions.
shown in Table 10 when /e/u| - 0(2x)) is mis- matched, both the alignment and quality of the generated images decrease significantly. Intu-
IR Match Mismatch Match Mismatch
HPSv2
itively. when f + 0(2;) aligns poorly with
0.8 97.09
-69.96
28.85
25.02
condition c, a larger proportion of the compo- nents in Iy become unsuitable for generating content consistent with c. This forces the diffu- sion model make more substantial corrections within the limited t + O interval, ultimately compromising generation quality and prompt adherence. These results empirically validate our theo- retical framework and motivate the use of fixed-point iterations to achieve golden path consistency.
0.6 89.45
-171.16
28.73
22.84
0.4
84.42
-196.38
28.53
21.86
0.2 81.43
-201.60
28.47
21.53
C Proofs
We now present the theoretical foundations for FSG. summarized in Theorem I We first clarify the notations and state the key assumptions. Under mild conditions, provide theoretical guidance for allocating the total number of fixed-point iterations N. Recall that we partition the diffusion process into M intervals 4 t; At; for 1, ....... M, and perform K; fixed-point iterations each interval using j ..... K2 for FSG:
operator F, == Ft, Ar,- Specifically, for each iteration
1. Starting from 24, (5-1), ()-1) "4--Ar
apply the conditionallly-guieed ODE solver fi,+te-bt, to obtain
2. Then, use the unconditionalll-guude. reverse ODE solver fi,-st--t, to map back and update =!(
This -arkaadrd procedure iteratively calibrates It, by leveraging both conditional and unconditional guidance over the interval, promoting consistency between the two trajectories. For other methods such as CFG and CFG++, the operator F, can be defined described in Section A (Appendix A). Our analysis remains applicable under these definitions.
For the clarity of theoretical analysis, we consider a fixed-length interval partition, setting each interval size W so that the total number of timesteps T is divided into M M= T/W intervals. We assume M divides N, and allocate K = N/M fixed-point iterations to calibrate each 2t,, where t; iW for I...,.M M. This leads the following procedure to produce the trajectory d; from given
FP w (xt), if mod W = Dt.
ft
24-1 ft t.....T
(32)
otherwise
Let e (xr, t) and 8 (xr,t) denote the unconditional and conditional noise predictions, respectively. The domain regarding æ for all the functions, including e*/e (z, t), f (z) and Fz-)((z), etc., is assumed to be Rd (unless otherwise specified). And we use || to denote the Euclidean norm in IR. Key assumptions are follows:
1. Boundedness:
The latent variables and noise predictions are bounded, i.e., [ ||e (a.)) < for some constant B dassumption 1..
2. Smoothness:
The noise functions <//(()) are Lipschitz continuous with smoothness constant (assumption 2). Additionally, the ODE dynamics derived from these noise functions are smooth (assumption 3).
24

---

3. Contraction: The fixed-pont soiddppoint operator F; is contraction with rate bounded by r E (0, 1)
|assmmption 4).
Building on the assumptions, we establish framework for optimizing the allocation of fixed- pointt iterations. For the procedure described in (32 we analyze the allocation of iterations across intervals [ii - 1)W, iW] for i E [.]. where the fixed point operator F:w 1iW is applied. The sampling ratio is defined as:
K
W T
B
(33)
M
where M is the number of intervals, K is the number of iterations per interval, N is the total number of iterations, and T is the total number of timesteps. We derive upper bound to guide the optimal choice of J.
Theorem (detailed version of heorem p. Assume that Assumptions 1 to 4 hold. Let in denote the 1 M (0, 1). There exists an upper bound function:
trajectory produced as in (22) with 3
9(3, L,r) (0(1) 0 222 + +2.22
(34)
such that:
T
L | (dt, - E" (8r, (3T
(35)
The proof is deferred to the end of this section.
Here, (I.) denotes an upper bound up to constant, i.e., S CLL) for some C > 0; (f)))) denotes both upper and lower bounds up to constants, i.e., cf(L) S. S Cf(L) for some c, C' >
The assumption provides trivial bound L < 4B while large L may lead to g(3, L,r) r) > so
our bound more meaningful for smooth noise function with smaller L. For L = O(), the bound simplifies to:
g L.r = 0(, (3N) 2L...²
(36)
Optimal 8 for minimizing
g L,)) For fixed N. L. and the optimal 3 is obtained by solving:
d d (B, L,r 4T.² e (2N (n(1))(((1/)) 22N) O,
(37)
L.3 ONNI((1))(11))) -29N)
Here, L increases linearly with 3, while the term N (1)1 2AN decays exponentially, guaranteeing unique minimizer 3 Importantly, the noise smoothness L decreases, the optimal 3 increases, suggesting that smoother noise allows for fewer, longer intervals (i.e., larger W and K). Conversely, the total iteration budget N grows, 3r approaches zero, recovering the short-interval setting (M = T) as a limiting case.
Next, we formalize the boundedness assumptions for images and noise, which are standard in diffusion model analysis:
Assumption (Boundedness). There exists B such that for all t and all x E M
||||| S B, ||E" (I.t)ll
(38)
For images with entries in [0, 1] and size di x d2, we may take d = d, d2 and B = vd
We next formalize the smoothness properties of the noise function, introducing constant I. 0(1) that is independent of the data dimension d and the number of diffusion steps T.
Assumption 2 (Smoothness of Noise). There exists a smoothness constant E (0, such that the following properties hold for any initial point and time interval (a. b):
1. Smoothness at fixed time: For any t E [a, b] and any pair of points I T2, the noise function satisfies:
4// (21,t)
(x2, t) || (x2,
L|| z2|1-
(39)
25

---

2. Smoothness over varying time: Let u/c = saye (x) for E [a, b). Then, for any 11, l2 [a, bl. the noise function satisfies:
t t2
|| (1% - ///C (I) (2) || S LB
T
11 12
|/ t1) - su/e (x 12) - LB
T
Building on the smoothness of noise (assumption 2), we analyze the properties of fa (I)). Recall that fu/e (a) represents an ODE evolving from time to b, initialized at point z. The ODE is defined
u/c d d 1
(V-==a
az E0 w/c (2 (1)
(40)
dt
Vol
dt
where 0 denotes the parameters of the neural network, which we omit for simplicity in subsequent discussions. This ODE yields the following explicit form for the time derivative of s.N/c.
dx, u/c
pext u/c + AxEa/o (2//o,t) ,t),
(41)
dt
where
Aeg Vās d
(V (VT=0a
at āi
Nt. == =
(42)
dt Assumption Let be the smoothness constant in assumption 2 E 10 such that:
t=to"
dt Vou t=t
There exist constants C 1 and
1. Smooth Dependence on Initial Error:
I (x)) -Sale (v) || S Cla,,,1111r-- yll:
(43)
where
Lab.L = (.5.==(2al | 1b -a| 1
(44)
2. Noise Control:
Ilfa (z) - Sa (x) 2 challe - E" (I, a) |1|b al.
(45)
Motivation:
This assumption is motivated by a first-order Taylor expansion of fale (x/v).
For smoothness intial error:
Sale (x) Satt (v) 20 di (Sala / (x) - Sayl (1)) (b a) + (x - 4)
1=a
=Mo(z y) Ao "//c (z, - ea (y, a)) (6 1 a) (x -v),
(ewee( - (y. a) || s L| - y/l from assumption 2
11fo (x) fuue (v) S Class || v/1.
where la,6,I == (1A,a + [maa))( al + captures the effect of the smoothness of noise, the length of time interval (a, b) the and dynamics of ā, in it. C > 1 is constant that compensates for the error in the Taylor expansion.
For noise control:
S0 (xt) - f (x) R d
di (Su (I) 1o (z)) (b-a)
= Xa (e (x, a) - E" (a, a)) (b a):
Ilfa-sh (x) f (v)
2 = (x, - (z,a) a) b||b al,
where E (0, 1) compensates for the error in the Taylor expansion.
26

---

Assumption 4 (Contraction and Fixed Point Existence). Letr E (0, 1). For any time interval [a, b), the fixed point operator Fo-sh satisfies the following:
1. Fixed point equivalence: There exists a fixed point x' of Fa-tb such that:
Fa-b(z²)) = xx" + fa (x") = fa- (x*).
(46)
2. Contraction: For all I, y:
[{Fa)s())) (a)i S <r|bz - y/1-
(47)
As example, the main text considers the fixed point operator Fa-zb(z)= fa-tb (z). For brevity, we write F(x) when the interval (a, b] is clear from context.
Lemma 1. Assume that Assumptions Tro 4 hold. For any given x, the following inequality holds:
Isa- (F(R) (z)) - faa (F(H (x)) |I s 2C - -x'1l 4Clo,b,L B. (48)
Proof. We start by expanding the difference:
Ilā-- (F(A) (x)) - -/ (F()) (x)) ||
= =I1sa=b (F((A) (x)) - ga+ (K()) (1) - (Ja-b (F(R) (x') - -Sa-b (F(A) (x*) I|
S &11fa (E((k) (x)) fart (E((A) (I")) + |1 (F(R) (x)) - sa-t (pltt) (x*)) ||.
Using the smoothness property of lä-s (-) and pa (-) from assumption 3 we have:
I (F(t) fa-b (p()) (x*)) S Cla (F(() (x) F(() (x*) ||.
(49)
115 (E(R) (x)) fa+bb (F(N) (x*)) || s Class |F(M) (x) - F(h) (x*) ||.
(50)
Combining these, we get:
Ilfā (p()) (x)) fä-b (F(*) (x) S 2C (x) - F(N) (x*)
(51)
Next, applying the contraction property of F' from assumption 4 we have:
|F()) (x) ((() (x*) || S r² x-l--
(52)
Substituting this into the inequality, we obtain:
1e (F(()) (a)) ميتك (1))) (p()(()) (x)) || s 2h r. 12 a=1l
(53)
Finally, using the boundedness assumption from assumption 1 || x= $2B, conclude:
|| (F(t) (x)) Sā- (F()) (m:) || S 4la Al B.
(54)
With these assumptions and lemmas established, we are now ready to prove the main theorem.
Proof. We analyze the error in the first interval, as the analysis for other intervals is similar. Let F
bes the simplified notation for Fw-so. The error can be bounded as follows:
W
s (it, 1) - E" 1) 1|2
W-1
=||5" (āw, W) 5n (ow, W) 112 +
Σ
e (fe, - 5" (E., 112
t=l
(55)
(0)
(1 3(W - 1))]le (ow, W) - E" (øw, W) 112
W-1
+3
(Ile" (Gr, t) - 5 (ow, W) 112 + l|e" (Ee, 1) - E" (ow, W) 11?)
27

---

where (a) follows from ||a + + elI < 3(1a||2 + ||b|| + [|cll?).
Using the bounds in assumption and assumption 3] we have:
w
Σ 1ee tb), - -0"(0e,0 (8t, 1|?
s (3W - fW (F.1) (xw) - Sw (F(K) 11
x W
(56)
+ 6 B2 W-1
I
(W i)²
T
Lemma
C3W 2)
B²
(W-
12W 1)
(A6.,o.r K B)²
+ -6
L
2 W2
T
6
Now, summing over all intervals:
Σ E (o., - =" = (&,,t)
M W
| (BW- - W +1) - ($W iW - W + +0)
(APA,,--Iw. B) C9(W -2)
exGg W22
(57)
B²
L2 (W- - -1)
+ 6
T
T/W
C3W - 2)
MW=T
B²
P2 16 Σ
xw W22 r 22
1..(i-1)W.L
2N2w
(.2 1)(2W - 1)
+ 6
6T
Expanding (w. we obtain:
c (B,, t) =" t)),
T/W
((Aw L + |Ww +12 C3W - 2)
2
< 16B 1=1
2 2x2 W2
LW 1)(2) -1)
+ 6B²
6T
(W-1) -1)2222)
T/W
([AwlL++ +1) C3 -2) 22 +
122
16B
Σ
+BB
222
6T
T/W
((Aw |+ + |aw DW + 1)2 C3 2) 22 2),2 + B2T
< 2 x 16
Σ
222 W2
(58)
Observe that (ww h.llwwlwi E [M] are constants, and we define
T/W
((owLL + (www//)l 1)W + +12 C (3W -2) - 0(L2)++0)
C = 16
2 W2
28

---

Finally, grouping terms:
T C=
(2z, - ( & B2T
(59)
t=1
where
( 22N + 32.
D Experimental Details
D.1 Experimental Setup
In this section, we demonstrate the specific setup the experiment, including the dataset, metries, and hyperparameter settings for different methods.
Datasets. The evaluation leverages four datasets assess text-to-image models.
1. Pick-a-Pic [17 collects real-world user preferences from text-to-image web app. For each
prompt, users compare two generated images and select their preferred option (or mark tie). We use the first 100 prompts from this dataset to test model performance.
2. DrawBench 25 is comprehensive and challenging benchmark with ~-20 prompts spanning 11 categories like color, counting, and text rendering. These prompts test how well models handle complex or ambiguous descriptions.
3. GenEval [0] evaluates whether generated images correctly follow object- instruc- tions. Its 553 prompts cover object placement, quantity, color, leveraging object detection models to evaluate text-to-image models on a variety of generation tasks.
4. PartiPrompts [32 contains 1,600+ diverse prompts covering creative, technical, and abstract concepts. We randomly pick 100 prompts to assess how models balance language understanding and visual creativity.
Metrics.
Four metrics are employed to quantify image quality and alignment.
1. Aesthetic Score (AES)
26 quantifies assigning scores (often 1-10) of visual quality by analyzing contrast, composition, color harmony, and detail richness. AES reflects human aesthetic preferences to help refine image generation.
2. ImageReward (IR) 30 a reward model trained on 137,000 expert comparisons using rating/ranking methods, which can be integrated with reinforcement learning enhance output alignment with human preferences.
3. Human Preference Score v2 (HPSvv2 [2] fine-tuned from CLIP based on Human Preference Dataset v2 with 798,090 human preference choices. HPSv2 can predicts authentic human perceptions of beauty and style.
4. CLIPScore 12 measures text-image alignment via CLIP embedding cosine similarity. This metric is essential in text-to-image synthesis, ensuring that generated visuals maintain semantic alignment.
Hyper-parameters. For Classifier-Free Guidance (CFG), adopted guidance strength 5.5. while CFG++ used = 0.6. Both methods utilized 50 inference steps. In Z-Sampling. forward guidance strength was set to 5.5, and reverse guidance strength to 0. Following the setting in ( reflective sampling was applied during first 12/25 steps for NFE=50, 25/50 steps for NFE=100, and all 50 steps for NFE=150. For Resampling, configurations varied by NFE: NFE=50, 25 inference steps with one resample per step; for NFE=100/150, 50 steps with or resamples per step, respectively.
For our Foresight Guidance (FSG) method, we set X = 1.0 for NFE=50/100 and X 0.7 for NFF=150.. We allocate fixed-point iterations (4,, Al, K;) using stage- wase strategy that prioritizes early timesteps:
29

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Table 11: Performance under different hyperparameter settings. Dataset: Pick-a-Pic; NFE: 100.
Setting
IRt
HPSv21
AESt
CLIP1
3:2:1 (Default)
102.82
29.05
6.66
34.30
1:1:1
99.13
28.95
6.67
34.14
1:2:3
93.23
28.70
6.63
33.75
Intra-stage perturbation
102.29 (+: 1.01)
28.72 (+ 0.22)
6.59 (+: 0.04)
34.33 (# 0.17)
Interval perturbation
102.17 (# 0.79) 29.04 (# 0.02)
6.67 (# 0.01)
34.28 (= 0.04)
Table 12:
The quantitative results on PartiPrompts Dataset with the SDXL model and NFE 50, 100, 150 (f denotes higher is better, best results under the same NFE are bolded).
Method
NFE
IRT
HPSv21
AES
CLIP
CFG
50
99.13
28.72
6,41
33.01
CFG++
50
106.80
28.96
6.42
33.20
Z-Sampling
50
114.74
29.04
6.32
33.41
Resampling
50
96.73
28.73
6.17
33.35
FSG (ours)
50
117.65
29.20
6.30
33.53
CFGx2
100
117.84
29.25
6.33
33.42
CFG++X2
100
115.35
29.33
6.26
33.46
Z-Sampling
100
118.39
29.22
6.34
33.47
Resampling
100
112.72
29.23
6.31
33.20
FSG (ours)
100
121.11
29.38
6.29
33.46
CFGx3
150
117.00
29.37
6.22
33.40
CFG++X3
150
117.02
29.29
6.22
33.41
7-Sampling
150
120.20
29.35
6.35
33.34
Resampling
150
113.62
29.16
6.24
33.16
FSG (ours)
150
123.28
29.40
6.34
33.41
NFE=50 (limited budget): We employ 40 inference steps and concentrate iterations early stages. Fixed-point iterations are performed at timesteps E (1.0, 0.875, 0.625] with interval size At 0.125, executing K = 2, iterations respectively.
NFE=100 (moderate budget): We adopt 50 inference steps with 3:2:1 allocation ratio across early, middle, and late stages. Specifically: (i) Early stage (t {0.68, 0.72, ... 1.0)): 24 NFEs with At 0.06, applying iterations at t E {1.0, 0.92, 0.8) and iteration at remaining timesteps; (ii) Middle stage (t E (0.36, 0.40, ... 0..44)): 16 NFEs with At 0.04; (iii) Late stage (t f0.08, 0.14. 0.32))) 10 NFEs with At 0.02.
NFE=150 (ample budget): With sufficient computational resources, NFE=100 configuration by adding supplementary fixed-point iterations at intermediate timesteps {0.02, 0.06. ... 0.98} with interval At 0.02 to enhance image detail.
augment the
Despite the apparent complexity of the hyperparameters (44, At, K;)). they are relatively straight- forward to configure and not require extensive fine-tuning. As demonstrated in Table [T FSG maintains stable performance under random perturbations timestep positions (#0.02) interval lengths (x0.7-1.3), provided the overall stage-wise allocation ratio (c.g.. 3:2:1) is preserved.
D.2 Experiment Results PartiPrompts Dataset
In this section, present comparative experiments between our method and various baselines on the PartiPrompts dataset, aiming to mitigate dataset-induced bias and further validate the effectiveness of approach.
30

---

PSG
CFG
CPG
Z-Samping
Rosa Rosamping
F3G
CPG
CPG
2-Sampl
Rezammp
18
5%
13.5%
20.9%
23.5% 99
1R
24 035
24.0%
22
KP3+2
17.0%
18.8%
HP552
13.5%
10.0%
19
AE5
21.2%
km
10.7%
15.7%
22.
AES
18.5%
25.0%
19.5%
27.0%
CLIP
24.5%
18.0%
18.0%
23.0%
OUI
37.3
19.0%
36.0%
20.0%
04
(a) DrawBench, NFE:
Pick-a-Pia NFE: 50
Resa
FSG
CFG
Z-Gamal
Resamp
18
23.0%
12.5%
12.5%
18.0%
34 34.2%
IR
27.0%
14:8 0%
19.0%
15.0%
25. 076
1P5+2
34.8%
22%
21.0%
12.0%
26.0%
HP542
36.3%
11.0%
25.3%
7.8%
10.3%
A25
20.4%
26.2%
12.7%
15.5%
25. 25.2%
AP5
22.0%
23.0%
11.5%
11.0%
25.5%
C1.P
59
17.5%
20.0%
13.0%
30.0%
CUI
28 0%
15.0%
21.0%
14.0%
22.0%
(c) Draw Bench, NFE: 100
(d) Pick-a- P-c,, NFE: 100
F5G
Samal
Resampir
72% 5m
12 12.5%
16.5%
37.5%
31.01
22.0% 2%
17.0%
25 3%
15.0%
21.0%
195v2
20.7%
40%
19.0%
22.0%
24.8%
19562
31.0%
9.0%
26.5%
11.0%
22.5%
455
12.5%
30.2%
10.0%
20.0%
27.35
AES
21. 21.0%
20.0%
14.3%
13.3%
21.5%
CLI
19.5%
24.0%
25.0%
13.0%
19.5%
CUP
20.0%
18.0%
29.0%
20.0%
13.0%
(c) DrawBench, NFE: 150
(f) Pick-a-Pic, NFE: 150
Figure 7: Proportion samples that outperform the other four methods (Top-1 rate).
As shown in Table 12
our method FSG consistently achieves state-of-the-an performance across IR and HPSv2 metrics. For the CLIP metric, FSG outperforms other baselines NFE 50 and NFE 150, with only marginal difference (-0.01) against the best performance at NFE 100. Notably. FSG demonstrates substantial improvements over CFG in the metric across all NFE levels (50, 100, and 150), with performance gains of +18.52, +3.27, and +6.28 respectively. Compared to the second-best method, we achieve consistent improvements of +2.91, +2.72, and +3.08, demonstrating superior image generation quality. The strong performance on both HPSv2 and CLIP metrics further confirms that our method maintains excellent alignment while producing high-quality outputs.
These results collectively demonstrate that our fixed-point iteration strategy enables more balanced sub-pue brob-probem decoupling, leading to better convergence behavior and improved performance both image quality and text alignment. These findings align with the results obtained on the DreamBench and Pick-a-P a-Pic datasets, as presented in Ta of the main text.
D.3 Additional Results Top-1 Rate
In this section, we present the Top-1 rate of different methods on SDXL. As evidenced by the Top-1 rate Figure 7 FSG exhibits a significant advantage NFE=50, particularly in HPSv2 (46% on DrawBench, 40.5% on Pick-a-Pic) and IR (34% on DrawBench, 29% Pick-a-Pic). Even as NFE increases, FSG retains superiority HPSv2, suggesting that long-interval guidance effectively strengthens alignment with human preferences.
31

---

3.2
CFG
2-5amping
FSG
CFG
Z-Sampling Resampling
FSG
Resampling
CFG-
3.0-
3.0-
2.8
2.6
2.0-
2.4
2.2
1000
Denosing steps:
Denasing steps: r
(a) DrawBench, NFE: 50
(h) Pick-a-Pic, NFE: 50
CFG
2 ZaSampiing Resampling
FSG
CFG
Z-Sampling Resampling
FS5
CFG+
4.0
3.5
3.0
2.5-
2.0
1000
800
6:0
400
2000
1000
Denosing steps:
Denosing steps:
(c) DrawBench, NFE: 100
(d) Pick-a-Pic, NFE: 100
CFG
2:5ampiing Resampling
FSG
5.5
CFG
Sammlinn Resampling
FS5
GFG-+
5.0-
CrG++
0.5
8.5
4.0
3.5
3.0
3.0
1000
600
400
Denosing steps:
Denasing steps:
(c) DrawBench, NFE: 150
(f) Pick-a-Pic, NFE: 150
Figure 8: Prediction gap during denoising. indicating the effectiveness of the fixed point iteration.
notable observation is the divergent performance characteristics of fixed-point iteration variants. For example, NFE=150, CFG enhances AES metrics, while Resampling improves IR of specific samples, likely attributable to its stochastic nature. These findings highlight the potential for adaptive frameworks that exploit the unique properties of fixed-point operators address diverse objectives, presenting promising direction for future research.
D.4 Additional Results on Prediction Gap
In this section, we analyze the noise prediction gap (z,) - (xx) 2 during the denoising process of different methods on SDXL. We select the prediction gap as key metric since it provides insight into score estimation accuracy along the generation trajectory. We hypothesize that models achieve optimal score 28 (noise) estimation when ('())) (xx) + 0, and accumulated score estimation errors Σ, (x)) (xx) - along the trajectory are known to correlate with generation quality [3]. Since conditional and unconditional diffusion models share most parameters and differ only in cross- attention layers, larger discrepancies between ((,)) and e (xr,) present greater learning challenges and potentially induce larger prediction enrors. Through examination of the prediction gap along trajectories, we demonstrate that FSG may benefit from more accurate score estimation, which translates to improved generation performance.
32

---

Table 13: Comparison of fixed point iterations with increasing inference steps on the SDXL model with NFE 50, 100, 150 (1 denotes higher is better, best results under the same NFE are bolded).
Datasets
DrawBench
25
Pick-a-Pie 17
Method
NFE
IRt
HPSv21
AEST
CLIP
IRt
HPSv21
AESt
CLIP
CFG
50
59.02
28.73
6.07
32.29
82.14
28.46
6.73
33.53
FSG (ours)
50
82.81
29.42
6.01
32.65
98.59
28.89
6.60
34.32
CFGx2
100
77.71
29.36
6.06
32.44
96.06
28.84
6.64
34.13
CFG-100
100
67.87
28.85
6.12
32.36
85.11
28.55
6.73
33.51
FSG (ours)
100
84.12
29.54
6.02
32.76
102.82
29.05
6.66
34.30
CFGx3
150
83.56
29.51
5,95
32.66
102.13
29.04
6.61
34.28
CFG-150
150
29.37
28.12
6.05
32.43
62.50
27.91
6.65
32.98
FSG (ours)
150
88.18
29.44
5.96
32.70
104.86
29.04
6.65
34.28
Table 14: Quantitative results SD2 model. Dataset: Pick-a-pic; NFE: 50 and 100.
NFE
100
Method
IRt
HPSv21
AEST
CLIP
IRt
HPSv2f
AESt
CLIP
CFG
-92.00
25.16
5.63
29.76
-30.30
26.51
5.97
31.70
CFG x1/2
-92.80
25.16
5.63
29.76
-9.55
26.80
5.88
32.05
CFG++x1/2
-71.92
25.49
5.76
30.23
-2.15
26.97
5.92
32.34
7-sampling
-6.22
26.97
6.06
32.53
0.80
27.21
6.06
32.58
Resampling
-17.30
26.54
5.83
32.01
-10.29
26.87
5.91
31.94
FSG (ours)
3.30
27.21
5.93
32.59
12.94
27.37
5.97
32.77
As illustrated in Figure 8 FSG accelerates fixed-point convergence at NFE=50 by integrating long- interval guidance and prioritizing early-stage iterations, accounting for its superior performance under low computational budgets. At higher NFE levels, however, the benefits of long-interval guidance diminish, resulting in comparable convergence speeds across fixed-point operators. Notably, Resampling exhibits slower mid-term convergence due to its stochastic nature, whereas CFG++ achieves greater stability through smoother strength scheduler.
D.5 Comparison Fixed Point Iterations with Increasing Inference Steps
In this section, compare the performance difference between extending the fixed point iterations and inference steps. As shown in Table [3] allocating inference resources to increase fixed soint iterations yields greater performance improvements compared to increasing the number of inference steps. Gains from additional inference steps saturate rapidly and exhibit instability when the step count not evenly divisible by T. In contrast, extending fixed-point iterations aligns the denoising trajectory closer to the golden path, producing outputs more consistent with human preferences. For instance, at NFE=100, CFGx2 achieves approximate IR improvement of 10. Our proposed FSG enhances this advantage by optimizing subproblem-solving strategies.
D.6 Additional Results on Different Models and Samplers
In this section, we present experimental results for the DDPM sampler 14,, SD2 24), and Hunyun- DiT [18] models under NFE 50 and NFE 100 settings, complementing the NFE 150 case discussed in the main text.
As shown in Table and Table 15 FSG maintains superior performance across IR, HPSv2, and
CLIP metries at NFE = 50 and NFE 100 for both Hunyun-Dis and SD2 models, consistent with the NFE= 150 results. With Hunyun-DiT as the base model, FSG improves the IR metrie by +11.39
33

---

Table 15: Quantitative results on Hunyuan-Din DiT model. Dataset: Pick-a-pic; NFE: 50 and 100.
NFE
50
100
Method
IR1
HPSv21
AEST
CLIP
IRt
HPSv21
AEST
CLIP1
CFG
116.82
29.09
6.59
33.00
120.50
29.12
6.74
33.20
CFG x1/2
116.82
29.09
6.59
33.00
121.05
29.15
6.54
32.96
CFG++x1/2 1/2
115.63
29.03
6,63
32.97
119.71
29.19
6.64
33.14
Z-sampling
127.82
29.21
6.69
33.56
128.45
29.37
6.71
33.40
Resampling
116.62
29.23
6.65
33.33
121.77
29.32
6.69
33.27
FSG (ours)
128.21
29.40
6.68
33.56
129.70
29.38
6.68
33.48
Table 16: Quantitative results on DDPM sampler. Dataset: Pick-a-pic; NFE: 50 and 100.
NFE Method
50
100
IRT
HPSv2f
AES1
CLIP
IRT
HPSv21
AEST
CLIP
CFG
61.85
27.90
6.72
32.88
81.52
28.61
6.67
33.42
CFG x1/2
61.85
27.90
6.72
32.88
83.56
28.56
6.70
33.61
CFG++x1/3
67.19
28.00
6.68
33.19
81,56
28.60
6.69
33.72
Z-sampling
83.99
28.54
6.60
33.78
87.26
28.60
6.66
33.56
Resampling
73.58
28.23
6.61
33.34
84.04
28.51
6.67
33.55
FSG (ours)
90.95
28.80
6.67
33.85
91.82
28.66
6.63
33.81
(NfE 50) and 9.20 (NFE= 100) over the CFG baseline. For the SD2.0 base model, FSG achieves more pronounced improvements in IR, with gains of +96.10 10 (NFE = 50) and +43.24 (NFE 100). These results confirm the model-agnostic characteristic FSG, demonstrating the plug-and-play compatibility of FSG across different diffusion models and inference steps. Notably, weaker base models exhibit greater benefits from the improved convergence properties FSG.
The DDPM sampler results under NFE 50 and NFE = 100 are presented in Table 16 Aligning with the main text's NFE=150 findings. FSG outperforms other methods in all metrics. It shows significant quality improvements at NFE 50 (IR: +29.10; HPSv2: +0.90) and NFE = 100 (IR: + +10.30; HPSv2: +0.39), verifying that FSG's fixed-point iteration strategy effectively enhances both generation quality and text alignment for stochastic samplers under varying computational budgets.
These supplementary experiments collectively demonstrate the robust performance of FSG across different base models, samplers, and computational budgets. The model/sampl cadde//sampee--agnoose design of FSG and efficient resource allocation consistently improve generation quality and text alignment.
E
Visualization
E.1
Additional Visual Results
In this section, showcase more images generated by different methods. As visualized in Figure FSG demonstrates superior alignment with target prompts compared to baseline methods, For instance:
9
In case (a), FSG generates a single blue pizza, whereas other methods produce multiple red pizzas;
In (b), FSG accurately renders snowboard on the bench;
In (c), FSG achieves precise text rendering;
(d), FSG strictly adheres to the "in nature" requirement.
Notably, FSG mitigates semantic interference common in generative tasks. For example:
34

---

FSG (ours)
CFG
C
Resampling
Z-Sampling
(a)
blue coloured pizza.
FSG (ours)
CFG
CFG++
Resampling
Z-Sampling
(b)
photo of bench and a snowboard
FSG (ours)
CFG
CFG++
Resampling
Z-Sampling
DON'T FART
DONT HARIT FAiiT
DONT FART
DOW IT FART
DONT FART
(c)
Sign that says DON'T FART
FSG (ours)
CFG
CFG++
Resampling
Z-Sampling
(d)
mechanical bee flying in nature, electronics, motors, wires, buttons, led
FSG (ours)
CFG
CFG++
Resampling
Z-Sampling
(e)
an evil spirit holding an orange cat in a dark room
FSG (ours)
CFG
CFG++
Resampling
Z-Sampling
(f)
blue bird and a brown bear.
Figure 9: Additional results of qualitative analysis.
35

---

Photo dog animal Viadimirr volegov beautiful blonde frecklas dining table food cake desert background glass translucent view Fiffel tower warm glow neon frreorks night
sensual
A black apple and green backpack.
A chess queen to the right of chess knight
Abadd
yellow wall with the word KA-BOOM on
laptop on top of teddy bear
A fish eating pelican.
RKA0OM K3OONI
Figure 10: Representative failure cases of FSG.
In (e), baselines erroneously depict an "evil spirit" cat due the "cat" in prompt, while
FSG avoids this bias;
In (f), baselines incorrectly blend "brown" semantics into "blue bird" generation, whereas FSG almost preserves color fidelity.
These results highlight the robustness of FSG in aligning with complex prompts, attributed to the long-interval guidance strategy FSG that stabilizes semantic coherence during generation.
E.2 Failure Case Analysis
Although FSG demonstrates superior performance across diverse benchmarks, we acknowledge its limitations through failure case analysis. Figure I0 presents representative failure modes encountered by FSG. Most failures originate from inherent deficiencies in the underlying diffusion models. As shown in Figure T0 FSG inherits the base model's difficulties with complex text rendering and counter-intuitive scene compositions.
A notable failure mode specific to FSG stems from its design philosophy of prioritizing strong guidance during early diffusion stages. When the base model exhibits semantic misinterpretations in early timesteps, such conflating chess piece with royalty for the term "queen," or overemphasizing individual tokens like "dog". FSG's intensive early-stage calibration can inadvertently reinforce these misconeeptions. This amplification effect propagates through subsequent denoising steps, resulting in misaligned generation.
We hypothesize that prompt-aware adaptive guidance, which dynamically adjusts calibration intensity based on semantic understanding, could mitigate this issue. We leave the exploration of such adaptive strategies to future work.
36

---

Border Impact and Limitation
Border Impact. The advancement of text-to-image diffusion models carries significant commercial potential for industries such as digital art, advertising, and content creation. A critical challenge in these applications lies in ensuring precise alignment between user prompts and generated outputs, which hinges on effective guidance mechanisms. Our work addresses this challenge by introducing a unified framework of fixed point iterations that reinterprets conditional guidance as a calibration process toward an idealized golden path, thereby decoupling guidance design from sampling dynamics. By demonstrating that prevalent approaches like classifier-free guidance (CFG) and variants correspond to short-interval iterations within our framework, reveal their inherent inefficiencies and motivate the proposed foresight guidance (FSG). FSG addresses longer-interval interval subproblems during early diffusion stages, optimizing computational resource allocation.
The potential impact of our work is threefold. First, it enables practitioners flexibly integrate diverse guidance mechanisms without theoretical constraints. Second, the framework exhibits extensibility to other domains of conditional generation, such as 3D or video synthesis. Third, it remains compatible with existing techniques like noise search and preference alignment, facilitating test-time scaling (TTS) to enhance image quality by leveraging increased computational resources during inference. Collectively, this work advances adaptive design and offers novel perspectives for unlocking the potential of conditional guidance in diffusion models.
Limitations In this work, we propose unified framework grounded in fixed point iteration and introduce foresight guidance (FSG) to enhance the alignment and efficiency conditional guidance. While empirically effective, the framework presents limitations that merit further investigation. First, the concept of the golden path remains empirically observed but not fully theoretically character- ized. Second, hyperparameters such as consistency intervals and iteration schedules are determined heuristically rather than through principled optimization. Despite these challenges, our framework provides foundational step toward systematizing guidance mechanisms, offering flexible platform for further innovation in efficient and controllable generative modeling.
37


Reviews:
[{"review_id": "NcuLsGzccq", "reviewer": "Reviewer_eeiK", "summary": "This paper introduces a unified framework for understanding Classifier-Free Guidance (CFG) in text-to-image diffusion models by reframing conditional generation as a fixed-point iteration problem. The objective is to identify a \"golden path\" where conditional and unconditional generation processes are consistent. The authors demonstrate that existing methods like CFG are special cases of inefficient, single-step \"short-sighted\" iterations. To address this, they propose Foresight Guidance (FSG), a novel method that solves longer-interval subproblems with multiple iterations, prioritizing the early stages of diffusion. Extensive experiments across multiple datasets and models are presented to show that FSG outperforms state-of-the-art methods in image quality and prompt alignment.", "questions": "- **On Computational Overhead:** Could the authors provide a more direct analysis of the computational costs? A comparison of wall-clock time and total UNet evaluations per generated image would be crucial for a complete assessment of the method's practical efficiency.\n- **On the Golden Path Proxy:** Figure 2b provides valuable empirical evidence, showing that FSG reduces the single-step _prediction gap_ ($||ε_c(x_t) - ε_u(x_t)||$). However, the paper defines the ideal golden path based on the consistency of the _full remaining trajectory_ ($f^c_{t→0}(x_t) = f^u_{t→0}(x_t)$). Could the authors comment on how well minimizing the single-step prediction gap serves as a proxy for achieving full-path consistency? Is it possible for this proxy to be a poor indicator, for instance, in cases where small, consistent errors in prediction accumulate to cause a large divergence in the final generated images?\n- **On the Contraction Assumption:** The theoretical analysis relies on the assumption that the fixed-point operator `F` is a contraction (Assumption 4). Could the authors provide further justification or empirical evidence for this assumption? This appears to be a very strong condition for an operator whose application involves composing two ODE solves, each making repeated calls to a deep neural network.\n- **On Hyperparameter Selection:** The appendix notes that the FSG schedule is chosen heuristically. How sensitive is the method's performance to these choices?", "limitations": "The authors appropriately acknowledge key limitations in Appendix D.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths:**\n\n- **A Novel and Unifying Conceptual Framework (Originality/Quality):** A key strength of the paper is its novel reframing of conditional guidance. The appendix provides a detailed mathematical unification of several guidance methods (CFG, CFG++, etc.), offering a new theoretical perspective. This presents a useful and systematic framework for analyzing and designing guidance mechanisms.\n- **Comprehensive Empirical Validation (Significance):** The proposed FSG method demonstrates strong performance, outperforming baselines across a range of settings. The consistent improvements across these settings suggest the method's robustness and general applicability.\n- **Theoretical Grounding (Quality/Originality):** The paper's claims are supported by a theoretical analysis in Appendix B. Theorem 1 provides a motivation, grounded in optimization theory, for why the proposed FSG strategy could be more effective than existing approaches under a finite computational budget.\n\n**Weaknesses:**\n\n- **Insufficient Analysis of Computational Efficiency (Clarity/Significance):** The paper's efficiency claims are based on the Number of Function Evaluations (NFE). This metric is insufficient for a fair comparison, as the computational work encapsulated by a single \"function evaluation\" differs substantially between FSG and the baselines. For instance, an FSG iteration involves expensive ODE solves, which is not directly comparable to a single UNet pass in CFG. A more direct comparison of wall-clock time or total UNet calls is necessary to substantiate the claims of improved efficiency.\n- **High Hyperparameter Complexity (Clarity):** The proposed method introduces considerable hyperparameter complexity. As detailed in Appendix C.1, FSG relies on a manually-designed, heuristic schedule of intervals, interval lengths, and iteration counts. This may hinder practical adoption and reproducibility compared to the relative simplicity of standard CFG.\n- **The \"Golden Path\" Concept Lacks Rigorous Formulation (Significance):** While motivationally effective, the conceptual foundation of the \"golden path\" remains more metaphorical than rigorously defined. The paper does not fully characterize the properties of this path (e.g., existence, uniqueness), which limits its utility as a formal theoretical tool. (As mentioned in Appendix D)", "was_revised": true, "final_justification": "After careful consideration of the authors' rebuttal and follow-up responses, I maintain my score of 4: Borderline accept.\n\n**Issues Successfully Resolved:**\n\n- Computational efficiency concerns were addressed with convincing wall-clock timing data showing FSG has equivalent runtime to baselines\n- The contraction assumption was supported with empirical evidence across different operators and timesteps\n- Hyperparameter sensitivity was demonstrated to be reasonable with robustness analysis\n\n**Areas for Improvement:**\n\n- Algorithm clarity: The core FSG algorithm and its hybrid implementation (selective application at specific timesteps) could be clearer in the main text, as readers currently need to piece together details across sections\n\n- Theoretical consistency: The relationship between the \"golden path\" motivation (full trajectory consistency), the actual optimization target (interval consistency), and the visualization method (single-step gaps) would benefit from better integration in the main paper\nPresentation structure: Some key implementation details in the appendix would enhance reader comprehension if moved to the main text\n\n**Overall Assessment:**\n\nThe paper presents solid technical contributions with strong empirical results and novel theoretical insights. The authors have demonstrated good faith in addressing reviewer concerns and provided valuable additional experiments. Importantly, the authors have acknowledged these presentation concerns and committed to specific revisions that would address them, including adding detailed algorithm explanations and relocating key content to the main text.\n\nThe technical merit and empirical strength support acceptance, and the authors' commitment to improving clarity suggests the final version will be significantly more accessible.", "rebuttal": "Thank you for the valuable feedback, here's our response to the concerned problems.\n___\n**Q1: Computational Overhead**  \n**A1:** Our method (FSG) has **identical wall-clock time** to baselines (Tab. 1).\n\n**Tab. 1**: Running time (seconds per image) for different methods at NFE=50/100/150.\n|NFE|CFG|CFG++|Resample|Z-Sampling|FSG\n|-|-|-|-|-|-\n|50|6.71|6.82|6.48|6.66|6.77\n|100|12.51|12.60|12.49|12.61|12.56\n|150|18.47|18.47|18.26|18.26|18.49\n\nFor a fair comparison, all NFEs in the main text correspond to a single backbone call. \nAlthough FSG employs longer-interval ODE operators, we reduce its computational time by using single-step ODE solver. \nAs fixed-point iterations occur mainly in the early denoising phase, the potential discretization error is acceptable. As a result, FSG delivers superior performance while maintaining computational efficiency equivalent to the baselines.\n___\n**Q2: Golden Path Proxy** \n\n**A2:** Thank you for your questions, which help improve the clarity of our work. We clarify that while we use $\\lVert \\epsilon^c(x_t) - \\epsilon^u(x_t) \\rVert_2^2$ for visualization in Fig. 2 (main text), the **golden path proxy actually minimized in FSG is $f^c _ {t-\\Delta t}(x_t) = f^u _ {t-\\Delta t}(x_t)$** (see line 180 in the main text).\n\nThe rationale for this proxy is as follows: (1) directly targeting $f^c _ {t\\to 0}(x_t) = f^u _ {t \\to 0}(x_t)$ is suboptimal for optimizing $x_t$, as the long-range generation process causes the dependence of $f^{c/u} _ {t\\to 0}(x_t)$ on $x_t$ to diminish. (2) Minimizing $\\lVert \\epsilon^c(x_t) - \\epsilon^u(x_t) \\rVert_2^2$ only regularizes the trajectory locally around $x_t$. Therefore, FSG minimizes $f^c _ {t-\\Delta t}(x_t) = f^u _ {t-\\Delta t}(x_t)$, which balances these two extremes by selecting an appropriate interval length $\\Delta t$.\n\nFor visualization, we use $\\lVert \\epsilon^c(x_t) - \\epsilon^u(x_t) \\rVert_2^2$ because it is equivalent to the **expectation of clean image gap** $\\lVert \\mathbb{E}^c (x_0 \\mid x_t) - \\mathbb{E}^u(x_0 \\mid x_t) \\rVert_2^2$, where $\\mathbb{E}^{c/u} (x_0 \\mid x_t)$ denotes the conditional/unconditional expectation of the clean image $x_0$ given $x_t$. This expectation approximates $f^{c/u} _ {t\\to 0}(x_t)$, with the equivalence following from Tweedie’s formula: $  \\mathbb{E}^{c/u} (x_0 \\mid x_t) = [x_t - \\sqrt{1-\\bar{\\alpha} _ t} \\epsilon^{c/u}(x_t)] / \\sqrt{\\bar{\\alpha} _ t}$. \n\nEmpirically, FSG-guided $f_{t\\to 0}^{c/u}(x_t)$ also exhibits stronger semantic similarity, indicating improved full-path consistency. To further support this, we will provide additional visualization examples in the Appendix and revise our paper to enhance clarity.\n___\n**Q3: Contraction Assumption**\n\n**A3:** We empirically measure local contraction rates for operators $F$ under $\\ell_2$ norm: $\\hat{r}=\\mathbb{E}_{x_0\\sim p(x)} \\frac{\\lVert F(x_t)-F(x_t') \\rVert_2^2}{\\lVert x_t-x_t' \\rVert_2^2}$, where $x_t,x_t'$ are obtained by mixing different noises with $x_0$.\n\n**Tab.2** Empirical contraction rates $\\hat{r}$ for different operators across denoising timesteps ($t\\in[0,1],dt=0.05$)\nOperators|$t=$ 0.2|0.4|0.6|0.8\n-|-|-|-|-\n$\\text{id}-w_t\\Delta\\epsilon$ (CFG, CFG++)|1.00|1.00|1.00|1.00\n $f^{\\gamma} _ {t+dt\\to t}\\circ f^{u} _ {t\\to t+dt}$ (Z-sampling)|1.04|0.99|0.97|0.99\n$f^{\\gamma} _ {t+dt\\to t}\\circ n^{u} _ {t\\to t+dt}$ (Resample)|0.89|0.97|1.03|1.07\n$f^u _ {t/2\\to t}\\circ f^{\\gamma}_{t\\to t/2}$|1.03|0.95|0.96|0.98\n$f^u _ {t/4\\to t}\\circ f^{\\gamma} _ {t\\to t/4}$|0.61|0.91|0.88|0.91\n$f^u _ {0\\to t}\\circ f^{\\gamma} _ {t\\to 0}$|0.62|0.70|0.75|0.79\n\n- Notably, long-interval operators (e.g., $f^u _ {0\\to t}\\circ f^{\\gamma} _ {t\\to 0}$) demonstrate **lower contraction rates ($\\hat{r} < 1$)** compared to short-interval operators such as Z-sampling or Resample. To balance convergence speed and operator complexity, we select an intermediate interval $\\Delta t$ ($dt < \\Delta t < t$) for the operator $f^u _ {t-\\Delta t\\to t}\\circ f^{\\gamma} _ {t\\to t-\\Delta t}$.\n\n- The contraction rate of an operator depends on the choice of metric. While operators in CFG/CFG++ are non-contracting under the $\\ell_2$ norm, they may exhibit contraction in other carefully designed metrics. Empirically, as shown in Tab. 2 (main text), these methods also benefit from additional iterations, supporting the fixed-point framework.\n\n___\n**Q4: Hyperparameter Selection**\n\n**A4:** Our scheduler assigns more fixed-point iterations to the early denoising stage. Specifically, we distribute NFE in a **3:2:1** ratio across the stages $1 \\to 0.67$ (early), $0.67 \\to 0.33$ (mid), and $0.33 \\to 0$ (late), with iterations uniformly distributed within each stage.\n\nAlternative ratios (such as 1:1:1 or 1:2:3) lead to degraded performance (see Tab. 3). Under the 3:2:1 setting, FSG demonstrates **robustness to hyperparameters**: perturbing the intra-stage distribution or operator intervals yields stable results. \nWe conducted 5 trials, randomly perturbing the position of points intra-stage ($\\pm 1$) or the interval length ($\\times 70\\\\%-130\\\\%$) on our chosen hyperparameters.\n\n**Tab. 3**: Performance under different hyperparameter settings  \nSetting|IR|HPSv2|AES|CLIPScore\n-|-|-|-|-\n3:2:1 (Default)|102.82|29.05|6.66|34.30\n1:1:1|99.13|28.95|6.67|34.14       \n1:2:3|93.23|28.70|6.63|33.75\nIntra-stage perturbation|102.29 ($\\pm$ 1.01)|28.72 ($\\pm$ 0.22)|6.59 ($\\pm$ 0.04)|34.33 ($\\pm$ 0.17)\nInterval perturbation|102.17 ($\\pm$ 0.79)|29.04 ($\\pm$ 0.02)|6.67 ($\\pm$ 0.01)|34.28 ($\\pm$ 0.04)"}, {"review_id": "lYEAA7M4yn", "reviewer": "Reviewer_5edQ", "summary": "The paper proposed a unified framework based on fixed-point iteration that includes variants of classifier-free guidance as special cases, and outlines the design choices under this framework. The author also provides theoretical studies on how to balance the trade-off between the effectiveness and efficiency of iterative denoising image generation. Based on the observation, the author propose the foresight guidance, a new design combination under the framework that decomposes the iterative generation into longer but fewer subproblems. Various ablation and experiments on multiple benchmarks validate the effectiveness of the proposed method.", "questions": "1. It would be good to desribe the reasoning behind specific design choices such as the fixed point operaters for the foresight guidance.\n2. It would be good to make Table 1 easier to understand.", "limitations": "No limitation in the main section.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. The paper is generally well-written.\n2. The fixed-point iteration view allows clean decomposition of the design spaces for methods based on classifier-free guidance.\n3. A theoretically grounded decomposition of iterative denoising generation into subproblems with longer intervals.\n4. The experiments are thorough and convincing, investigating multiple controlled scenarios on different benchmarks.\n\nWeakness:\n1. Table 1 lacks necessary details in the main text to be fully understood. On top of this, while the proposed method fits into the fixed-point framework, the reasoning behind choosing the specific fixed point operator and scheduler is unclear.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable suggestions regarding our work. Please find our responses below.\n\n___\n**Q1: Specific design choices for the foresight guidance: fixed point operator and scheduler.**\n\n**A1:**\n\n1. In FSG, we select the forward-backward operator with a larger interval $\\Delta t$ because it exhibits a **smaller contraction rate**, thereby accelerating the convergence of the fixed-point algorithm.\nWe empirically measure local contraction rates for operators $F$ under $\\ell_2$ norm: $\\hat{r}=\\mathbb{E}_{x_0\\sim p(x)} \\frac{\\lVert F(x_t)-F(x_t') \\rVert_2^2}{\\lVert x_t-x_t' \\rVert_2^2}$, where $x_t,x_t'$ are obtained by mixing different noises with $x_0$ (See Tab. 1).\n\n**Tab. 1**:  Empirical contraction rates $\\hat{r}$ for different operators across denoising timesteps ($t\\in[0,1],dt=0.05$)\nOperators|$t=$ 0.2|0.4|0.6|0.8\n-|-|-|-|-\n$\\text{id}-w_t\\Delta\\epsilon$ (CFG, CFG++)|1.00|1.00|1.00|1.00\n $f^{\\gamma} _ {t+dt\\to t}\\circ f^{u} _ {t\\to t+dt}$ (Z-sampling)|1.04|0.99|0.97|0.99\n$f^{\\gamma} _ {t+dt\\to t}\\circ n^{u} _ {t\\to t+dt}$ (Resample)|0.89|0.97|1.03|1.07\n$f^u _ {t/2\\to t}\\circ f^{\\gamma}_{t\\to t/2}$|1.03|0.95|0.96|0.98\n$f^u _ {t/4\\to t}\\circ f^{\\gamma} _ {t\\to t/4}$|0.61|0.91|0.88|0.91\n$f^u _ {0\\to t}\\circ f^{\\gamma} _ {t\\to 0}$|0.62|0.70|0.75|0.79\n\nWe observe that, compared to the linear fixed-point operator ($\\text{id}-w_t\\Delta\\epsilon$), the contraction rate of the forward-backward operator decreases as the interval $\\Delta t$ increases. Therefore, to balance operator complexity and convergence speed, we choose the forward-backward operator $f^u _ {t - \\Delta t \\to t}\\circ f^{\\gamma} _ {t\\to t - \\Delta t}$ with a larger $\\Delta t$.\n\n\n2. In FSG, we adopt the CFG++ scheduler because it provides **more stable guidance strength** during the early stages.\n\nThe schedulers for CFG and CFG++ are given by $\\xi_t = \\sqrt{1-\\bar{\\alpha} _ t} - \\sqrt{\\alpha_t - \\bar{\\alpha} _ t}$ and $\\hat{\\xi}_t = \\sqrt{1-\\bar{\\alpha} _ t}$, respectively. Using the $\\alpha_t$ setting in DDIM as an example, the derivatives of $\\xi_t$ and $\\tilde{\\xi}_t$ at different timesteps are shown in Tab. 2:\n\n**Tab. 2:** Derivatives of $\\xi_t$ and $\\tilde{\\xi}_t$ at different timesteps\n$t$|0.9|0.8|0.7|0.6|0.5|0.4|0.3|0.2|0.1\n-|-|-|-|-|-|-|-|-|-\n$\\xi'_t$ (CFG)|-0.78|-0.71|-0.61|-0.51|-0.41|-0.32|-0.23|-0.09|0.37\n$\\tilde{\\xi}'_t$ (CFG++)|-0.03|-0.08|-0.15|-0.25|-0.38|-0.51|-0.65|-0.76|-0.93\n\nIt can be observed that, in the early stages where the guidance has a greater impact on generation ($t \\in [0.67, 1]$), the guidance strength provided by $\\xi_t$ decays rapidly, which limits the generation quality.\n\n\n___\n\n**Q2: Enhancing the comprehensibility of Tab.1 (main text)**  \n\n**A2:** Thank you for your valuable suggestions. To improve clarity, we will expand the description of Tab.1 in the main text. This will include complete definitions of all abbreviations in the caption (e.g., $\\text{id}$: identity mapping, $\\Delta\\epsilon(x_t):=\\epsilon^c(x_t)-\\epsilon^u(x_t)$), as well as clear explanations for \"Interval\" and \"Iter.\" \n\nSpecifically, the Interval $a\\to b$ indicates that the method aims to satisfy $f^c_{a\\to b} = f^u_{a\\to b}$, while Iter describes whether the original version of the method supports increasing the number of fixed-point iterations (Iter=1: not supported; Iter=N: supported). The default versions of CFG, CFG++, and Z-sampling do not mention increasing iterations to improve performance. The author of Resampling noted that iterations can be increased, but did not provide an explanation from the fixed-point perspective. \n\nIn contrast, our fixed-point iteration framework naturally enables this enhancement, and we validate the performance of CFG/CFG++ with additional iterations as CFG/CFG++ ($\\times 2/3$) in Tab.2 (main text). These clarifications will be incorporated into the main text as well."}, {"review_id": "mwPZ1uutMq", "reviewer": "Reviewer_QUUy", "summary": "This paper reframes classifier-free guidance (CFG) for text-to-image diffusion as a fixed-point calibration problem. at each timestep, the latent should lie on a “golden path” where unconditional and conditional denoising trajectories coincide. Existing guidance methods are shown to be single-step, short-sighted fixed-point iterations.  Based on this unified fixed-point view, the authors introduce Foresight Guidance (FSG), which 1- decomposes the diffusion trajectory into fewer, longer intervals, and 2- allocates multiple forward–backward iterations early in the process.  A theoretical bound predicts that such balanced sub-problem sizing minimizes the conditional–unconditional score gap under a fixed compute budget. Empirically, the authors show FSG outperforms several baselines in prompt fidelity, Human Preference Score, ImageReward and Aesthetic score.", "questions": "1. What is the exact wall clock time of FSG compared to the baseline in different NFE settings?\n2. Can we learn the interval schedule?\n3. Can you provide FID values for the experiments? I'm worried about mode collapse.\n4. Have you tried empirically verifying the convergence rate in synthetic settings?\n\nOverall I think this paper makes meaningful contributions to controlled generation and will increase my score if the authors provide answers to the questions above.", "limitations": "Even if diversity decreases (if that’s why FID was omitted) acknowledging this limitation and discussing its impact would significantly improve the paper’s honesty and transparency.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths:\n1. Casting diverse guidance methods as fixed-point iterations. This clarifies design choices.\n2.  FSG is derived from a clear theoretical analysis predicting why larger intervals help under limited compute.\n3. Extensive evaluation on prompt fidelity and image rewards. \n4. FSG outperforms the baselines in almost all considered metrics and settings.\n\n---\n### Weaknesses:\n1. Image diversity metrics are not reported. The focus of the paper is on alignment and image quality and loss of sample diversity from stronger early calibration is not measured.\n2. Wall clock time are not reported. Wall-clock time analysis is paramount to understanding the overhead of extra forward backward passes, and gives reader a clear picture of the paper's practical contributions.\n3. (Minor) Several hyper-parameters are introduced which require tuning.\n4. (Minor) Qualitative failure cases could be added for when too many iterations are performed, with a discussion on limitations.", "was_revised": true, "final_justification": "My main concerns were addressed by authors. I've revised my rating accordingly.", "rebuttal": "Thank you for your valuable questions. Please find our responses below.\n___\n**Q1:  Wall clock time of FSG**\n\n**A1:** Our method (FSG) has **identical wall-clock time** to baselines (Tab. 1).\n\n**Tab. 1**: Running time (seconds per image) for different methods at NFE=50/100/150.\n|NFE|CFG|CFG++|Resample|Z-Sampling|FSG\n|-|-|-|-|-|-\n|50|6.71|6.82|6.48|6.66|6.77\n|100|12.51|12.60|12.49|12.61|12.56\n|150|18.47|18.47|18.26|18.26|18.49\n\nFor a fair comparison, all NFEs in the main text count a single backbone call. \nAlthough FSG employs longer-interval ODE operators, we reduce its computational time by using single-step ODE solver. \nThe potential discretization error is acceptable, as fixed-point iterations occur mainly in the early denoising phase. \nAs a result, FSG delivers superior performance while maintaining computational efficiency equivalent to the baselines.\n___\n\n**Q2: Can we learn the interval schedule?**\n\n**A2: Our interval scheduler can be set relatively easily and does not require extensive fine-tuning or learning.**\n\nFor example, when NFE=100, we allocate the NFEs in a **3:2:1** ratio across three stages: $1 \\rightarrow 0.67$ (early), $0.67 \\rightarrow 0.33$ (mid), and $0.33 \\rightarrow 0$ (late), with iterations uniformly distributed within each stage.\n\nAs shown in Tab. 2, as long as the overall ratio is not violated (e.g., 1:1:1 or 3:2:1), FSG is **robust to hyperparameter choices**: perturbing the intra-stage distribution or operator intervals leads to stable results. We conducted 5 trials with random perturbations to the position of points intra-stage ($\\pm 1$) or interval lengths ($\\times 70\\\\%-130\\\\%$), and observed stable performance.\n\n\n**Tab. 2**: Performance under hyperparameter settings \nSetting|IR|HPSv2|AES|CLIPScore\n-|-|-|-|-\n3:2:1 (Default)|102.82|29.05|6.66|34.30\n1:1:1|99.13|28.95|6.67|34.14       \n1:2:3|93.23|28.70|6.63|33.75\nIntra-stage perturbation|102.29 ($\\pm$ 1.01)|28.72 ($\\pm$ 0.22)|6.59 ($\\pm$ 0.04)|34.33 ($\\pm$ 0.17)\nInterval perturbation|102.17 ($\\pm$ 0.79)|29.04 ($\\pm$ 0.02)|6.67 ($\\pm$ 0.01)|34.28 ($\\pm$ 0.04)\n___\n**Q3: FID Metric and Mode Collapse**\n\n**A3:** FSG **improves sample quality and diversity**, as demonstrated by the FID and Vendi scores in Tab. 3 below.\n\nSince FID is challenging to use for large-scale text-to-image models, we conducted experiments on the ImageNet 256×256 conditional generation task, generating 1K images per class (totaling 50K images). \nWe report FID and Vendi (diversity metric)[1] at NFE=25/50. The results show: \n1. FSG achieves better FID than baselines in class-conditional generation, and even slightly improves diversity. \n2. CFG/CFG++$\\times 2$ (NFE=50) also benefit from increased fixed-point iteration steps compared to CFG/CFG++ (NFE), improving both generation quality and diversity, indicating that fixed-point iterations do not cause mode collapse.\n\nWe hypothesize that this is because **fixed-point iterations are performed locally** in the neighborhood of $x_t$, preserving randomness. Intuitively, the initial noise for $x_t$ is $\\epsilon$, and fixed-point iteration finds a better $\\epsilon'$ in the vicinity of $\\epsilon$, enhancing generation quality without sacrificing the inherent randomness of $\\epsilon$.\n\nWe will provide detailed experimental settings and results in the Appendix.\n\n**Tab. 3**: FID and Vendi scores for different methods on ImageNet 256×256 conditional generation (lower FID and higher Vendi are better).\nMethods (NFE=25)|FID $\\downarrow$ |Vendi $\\uparrow$|Methods (NFE=50)|FID $\\downarrow$|Vendi $\\uparrow$\n-|-|-|-|-|-\nCFG|17.81|3.44|CFG ($\\times 2$)|14.69|3.79\nCFG++|13.27|3.91|CFG++ ($\\times 2$)|8.85|4.43\nResample|17.54|3.50|Resample|9.05|4.47\nZ-sampling|19.89|3.40|Z-sampling|8.62|4.64\nFSG|**10.56**|**4.73**|FSG|**7.91**|**5.79**\n___\n**Q4: Convergence rate in synthetic settings**\n\n**A4: FSG accelerates convergence under the Gaussian mixture setting.**\n\nWe simulated the convergence rates of FSG and CFG under a Gaussian mixture setting. Following [2], the target distribution is a Gaussian mixture $\\mathcal{N}(\\pm \\mathbf{m}, \\sigma^2I)$ with two equally weighted components ($\\mathbf{m}\\in \\mathbb{R}^d, \\lVert \\mathbf{m} \\rVert_2^2=d$). The conditional sampling target is the component with mean $+\\mathbf{m}$. Assuming the score is estimated accurately, [2] shows that CFG achieves faster convergence than using only the conditional prediction $\\epsilon^c(x_t)$, as quantified by $\\mathbb{E}(x_t)\\cdot\\mathbf{m}/\\sqrt{d}$. \n\nBuilding on this, we simulated FSG with an interval of 0.1 at $t=0,0.1,...,0.9$, and observed significantly faster convergence. The results indicate that FSG provides stronger and more effective guidance, especially in the early stages.\n\n**Tab. 4**: Convergence speed under the Gaussian mixture setting ($\\times 10^{-2}$), using the same metric as in [1] (higher values indicate better performance).\n$t$|$1-e^{-2}$|$1-e^{-1.5}$|$1-e^{-1}$|$1-e^{-0.5}$|0\n-|-|-|-|-|-\nConditional|0.45|1.09|2.02|4.08|5.77\nCFG ($w=4$)|0.98|2.41|4.12|6.62|11.50\nFSG ($w=4$)|3.47|4.59|6.32|8.89|14.70\n\nWe will provide additional details and visualizations of the simulation experiments in the Appendix.\n\n___\n**Q5: Qualitative failure cases with a discussion on limitations**\n\n**A5:** When excessive iterations are performed, sample quality may degrade due to violation of the manifold constraint $x_t \\in \\mathcal{M}_t$. We will include qualitative failure cases and discuss these limitations in the Appendix.\n___\n**Reference**\n\n[1] The Vendi Score: A Diversity Evaluation Metric for Machine Learning\n\n[2] Classifier-Free Guidance: From High-Dimensional Analysis to Generalized Guidance Forms"}, {"review_id": "COGrJjbEDx", "reviewer": "Reviewer_Wo6Z", "summary": "This paper proposes improve diffusion CFG sample quality by using fixed point iteration to align unconditional and conditional score outputs during generation. The authors identify four design spaces, consistency interval, fixed point operator, guidance strength, and number of iterations. In Section 3.2, the authors propose Foresight Guidance (FSG), which is a collection of choices for each design space. It is shown that FSG outperforms CFG, CFG++, Z-Sampling, and Resampling on a variety of benchmark tasks.", "questions": "I am willing to raise the score to *borderline accept* if the authors provide appropriate explanation for the concerns in **Quality** and **Clarity**.", "limitations": "The authors discuss the limitations of the work in Appendix D.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths and weaknesses are denoted [S#] and [W#], respectively.\n\n**Quality**\n- [S1] Overall, the paper is well-written with clear notations, comprehensive baselines, and thorough ablations. I had no problem understanding the paper.\n- [W1] The paper lacks experiments on canonical tasks such as ImageNet generation. I believe results which demonstrate that FSG works on e.g., CIFAR10 or ImageNet class-conditional generation will make the paper much stronger.\n- [W2] The paper also lacks diversity metrics such as the Vendi score [1]. I believe the authors should compare quality-diversity trade-off compared to other CFG-based methods to verify whether FSG truly improves the pareto frontier.\n\n**Clarity**\n\nI found some parts of the paper to be written imprecisely, or stated without reference, proof, or analysis. See below.\n\n- [W1] Lines 96 - 98, \"when $f^c_{t \\rightarrow 0}(x_t)$ and $f^u_{t \\rightarrow 0}(x_t)$ differ significantly, the model tends to generate conditional-aligned structures at the expense of texture detail and aesthetic quality. Conversely, when they align, the model balances both conditional alignment and generation quality\". I believe supporting this claim with appropriate evidence is crucial for this paper, since this intuition serves as the basis of the proposed method.\n- [W2] Lines 99 - 100, \"we therefore aim to calibrate $x\\_t$ toward $\\hat{x}\\_t$ by solving $f^u_{t \\rightarrow 0}(x_t) = f^c_{t \\rightarrow 0}(x_t), x_t \\in \\mathcal{M}\\_t$ where $\\mathcal{M}\\_t$ denotes the data manifold at time $t$.\" To my understanding, this paper only solves $f^u\\_{t \\rightarrow 0}(x_t) = f^c_{t \\rightarrow 0}(x_t)$ without the constraint $x_t \\in \\mathcal{M}\\_t$. Indeed, as written at line 251, the authors admit that \"excessive iterations risk divergence from the data manifold\". I believe that the authors should clarify the proposed method does not solve $f^u\\_{t \\rightarrow 0}(x_t) = f^c\\_{t \\rightarrow 0}(x_t), x\\_t \\in \\mathcal{M}\\_t$ but solves $f^u_{t \\rightarrow 0}(x_t) = f^c_{t \\rightarrow 0}(x_t)$.\n- [W3] Lines 127 - 128, \"approximating $f^{u_{(-1)}}\\_{t+1 \\rightarrow t}$ by $f^u_{t \\rightarrow t+1}$ or $n_{t \\rightarrow t+1}$ ...\". Is $n_{t \\rightarrow t+1}$ a valid approximation of $f^{u_{(-1)}}\\_{t+1 \\rightarrow t}$? One is stochastic and the other is deterministic.\n\n**Significance**\n- [S1] FSG outperforms CFG, CFG++, Z-Sampling, and Resampling on a variety of benchmark tasks on popular models such as SDXL.\n\n**Originality**\n- [S1] Interpretation of CFG as a specific instance of FPI is novel and interesting.\n- [W1] The proposed method, FSG, is a collection of rather incremental engineering techniques. Specifically, it is a collection of reverse diffusion interval adjustment, FPI, and CFG++ scheduler.\n\n[1] The Vendi Score: A Diversity Evaluation Metric for Machine Learning, TMLR, 2023", "was_revised": true, "final_justification": "While the authors' rebuttal has addressed my concerns regarding\n\n- Quality [W1], [W2],\n- Clarity [W1], [W2],\n\nthere are still remaining concerns with whether $n_{t \\rightarrow t+1}$ a valid approximation of $f^{u_{(-1)}}\\_{t+1 \\rightarrow t}$, and I am still ambivalent about the originality of this work. Hence, I have raised the score to **borderline accept**.", "rebuttal": "Thank you for the valuable feedback, here's our response to the concerned problems.\n___\n**Quality W1/W2: FID, Diversity Metrics, and Class-Conditional Generation**\n\n**A1: FSG improves sample quality and diversity**, as demonstrated by the FID and Vendi scores in Tab. 1 below.\n\nWe conducted experiments on the ImageNet 256×256 conditional generation task, generating 1K images per class (totaling 50K images). \nWe report FID and Vendi at NFE=25/50. The results show: \n1. FSG achieves better FID than baselines in class-conditional generation, and even slightly improves diversity. \n2. CFG/CFG++$\\times 2$ (NFE=50) also benefit from increased fixed-point iteration steps compared to CFG/CFG++ (NFE), improving both generation quality and diversity, indicating that fixed-point iterations do not cause mode collapse.\n\nWe hypothesize that this is because **fixed-point iterations are performed locally** in the neighborhood of $x_t$, preserving randomness. Intuitively, the initial noise for $x_t$ is $\\epsilon$, and fixed-point iteration finds a better $\\epsilon'$ in the vicinity of $\\epsilon$, enhancing generation quality without sacrificing the inherent randomness of $\\epsilon$.\n\nWe will provide detailed experimental settings and results in the Appendix.\n**Tab. 1**: FID and Vendi scores for different methods on ImageNet 256×256 conditional generation (lower FID and higher Vendi are better).\nMethods (NFE=25)|FID $\\downarrow$ |Vendi $\\uparrow$|Methods (NFE=50)|FID $\\downarrow$|Vendi $\\uparrow$\n-|-|-|-|-|-\nCFG|17.81|3.44|CFG ($\\times 2$)|14.69|3.79\nCFG++|13.27|3.91|CFG++ ($\\times 2$)|8.85|4.43\nResample|17.54|3.50|Resample|9.05|4.47\nZ-sampling|19.89|3.40|Z-sampling|8.62|4.64\nFSG|**10.56**|**4.73**|FSG|**7.91**|**5.79**\n___\n**Clarity W1: Improved Generation Performance When $f^c _ {t\\to 0}(x_t)$ Aligns with $f^u _ {t\\to 0}(x_t)$**\n\n**A2:** Thank you for your constructive feedback. We provide further evidence to support this statement.\n\nFirst, we generate data points $x_t$ such that $f^u _ {t\\to 0}(x_t) \\approx f^c _ {t\\to 0}(x_t)$. Specifically, we generate images $x^c$ conditioned on $c$, and obtain $x_t$ via DDIM inversion using unconditional noise $\\epsilon^u(x_t)$, i.e., $x_t = f _ {0\\to t}^u (x^c)$. These $x_t$ can be regarded as consistent conditionally / unconditionally generation on $c$, i.e., $x^c \\approx f^u _ {t\\to 0}(x_t) \\approx f^c _ {t\\to 0}(x_t)$. We then randomly sample a $c' \\ne c$ from prompts except $c$, so $x^c \\approx f^u _ {t\\to 0}(x_t) \\ne f^{c'} _ {t\\to 0}(x_t)$.\n\n**Tab. 2**: Generation performance of FSG from $x_t$ under matched ($c$) and mismatched ($c' \\ne c$) conditions.\n\n$t$|IR (match)|IR (mismatch)|HPSv2 (match)|HPSv2 (mismatch)\n-|-|-|-|-\n0.8|97.09|-69.96|28.85|25.02\n0.6|89.45|-171.16|28.73|22.84\n0.4|84.42|-196.38|28.53|21.86\n0.2|81.43|-201.60|28.47|21.53\n___\nIt can be seen that when $f^{c/u} _ {t\\to 0}(x_t)$ is mismatched, both the alignment and quality of the generated images decrease significantly. Intuitively, the less $f^u _ {t\\to 0}(x_t)$ matches the condition $c$, the more components in $x_t$ are unsuitable for condition $c$, making it harder for the diffusion model to correct these components within the $t\\to 0$ interval.\n\nWe will add a section in the Appendix to introduce the golden path, along with detailed experimental settings and visualization results.\n\n___\n**Clarity W2: Satisfaction of Manifold Constraints**\n\n**A3:** Here we clarify that satisfying the manifold constraint is meaningful; otherwise, score estimation outside the probability manifold may be inaccurate. In practice, we avoid violating the manifold assumption via empirical settings. \n\n- As shown in Fig. 2(b) (main text), the norm of $\\Delta\\epsilon (x_t):=\\epsilon^c(x_t)-\\epsilon^u(x_t)$ in CFG and CFG++ updates is relatively small. Therefore, as long as the guidance strength $w$ is not set too large, $x_t+w\\xi_t\\Delta\\epsilon (x_t)$ can still be regarded as remaining on the probability manifold $P_t(x_t)$. \n\n- For forward/backward operators, we consider that a suitable ODE/SDE sampler $f_{a\\to b}$ can sample $x_b\\in \\mathcal{M}_b$ from $x_a\\in \\mathcal{M}_a$, thus satisfying the manifold constraint.\n\nOur paper follows these empirical settings and does not focus on strict satisfaction of the manifold constraint. For rigor, we will remove the strong constraint $s.t. x_t \\in \\mathcal{M}_t$ from the formulas in the main text and instead discuss it in the text.\n\n___\n\n\n**Clarity W3: $n_{t\\to t+1}(x_t)$ as an approximation of $f^u_{t\\to t+1}(x_t)$**\n\n**A4:** The use of noise addition as a backward operator was proposed in [1], where the authors argue that $x_t$ is obtained from $x_{t+1}$ via conditional denoising, and that the injected conditional signals are not completely removed by the noise process $n _ {t\\to t+1}$. Therefore, $f^{\\gamma} _ {t+1 \\to t} \\circ n_{t\\to t+1}(x_t)$ contains more conditional semantics compared to $x_t$.\n\nIn our view, $n _ {t\\to t+1}$, like $f^u _ {t\\to t+1}$, samples $x _ {t+1} \\in \\mathcal{M}_{t+1}$ from $x_t$ (differing only in whether the sampler is stochastic or deterministic). However, there is still a gap between $n _ {t\\to t+1}$ and $f^u _ {t\\to t+1}$, which causes Resample requires more iterations to improve the generation quality.\n\n___\n**Originality W1: FSG as a collection of engineering techniques**\n\n**A5:** Within our fixed-point iteration framework, FSG can be viewed as a composition of settings like the forward-backward operator and the CFG++ scheduler. However, the novelty of our paper lies in proposing a **unified fixed-point iteration framework** that enables the engineering-composable design of guidance algorithms. The effectiveness of the CFG/CFG++$\\times N$ instances constructed under our fixed-point iteration framework has also been experimentally validated. We anticipate that the fixed-point iteration framework will provide deeper insights for future guidance design.\n___\n**Reference**\n\n[1] Repaint: Inpainting using denoising diffusion probabilistic models."}]

Author rebuttal:

Meta-review: All reviewers recommend acceptance of this submission, and the AC concurs. The final version should include all
reviewer feedback and suggestions from the discussion.
Decision: Accept (spotlight)