Stage-wise Dynamics of Classifier-Free Guidance in Diffusion Models
Authors: Cheng Jin
Qitan Shi
Yuantao Gu
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion Models
Classifier-Free Guidance
Primary area: generative models
TL;DR: CFG follows three stages—shift, separation, concentration—explaining why strong guidance reduces diversity and suggesting stage-aware schedules.

Abstract: Classifier-Free Guidance (CFG) is widely used to improve conditional fidelity in diffusion models, but its impact on sampling dynamics remains poorly understood. Prior studies, often restricted to unimodal conditional  distributions or simplified cases, provide only a partial picture.
We analyze CFG under multimodal conditionals and show that the sampling process unfolds in three successive stages. In the Direction Shift stage, guidance accelerates movement toward the weighted mean, introducing initialization bias and norm growth. In the Mode Separation stage, local dynamics remain largely neutral, but the inherited bias suppresses weaker modes, reducing global diversity. In the Concentration stage, guidance amplifies within-mode contraction, diminishing fine-grained variability.
This unified view explains a widely observed phenomenon: stronger guidance improves semantic alignment but inevitably reduces diversity. Experiments support these predictions, showing that early strong guidance erodes global diversity, while late strong guidance suppresses fine-grained variation. Moreover, our theory naturally suggests a time-varying guidance schedule, and empirical results confirm that it consistently improves both quality and diversity.

Published as a conference paper at ICLR 2026
STAGE-WISE DYNAMICS OF CLASSIFIER-FREE GUID- ANCE IN DIFFUSION MODELS
Cheng Jin, Qitan Shi & Yuantao Gu"
Department of Electronic Engineering, Tsinghua University
Beijing, China [jinc21, sqt24) @ma 1ss. edu. gytttssinguu..ddu eewpchuuueedu....
ABSTRACT
Classifier-Fres Guidance (CFG) widely used to improve conditional fidelity diffusion models. but its impact on sampling dynamics remains poorly understood. Prior studies, often restricted to unimodal conditional distributions simplified cases, provide only partial picture. We analyze CFG under multimodal con- ditionals and show that the sampling process unfolds in three successive stages. In the Direction Shift stage, guidance accelerates movement toward the weighted mean, introducing initialization bias and norm growth. In the Mode Separation stage, local dynamics remain largely neutral, but the inherited bias suppresses weaker modes, reducing global diversity. In the Concentration stage, guidance amplifies within-mode contraction, diminishing fine-grained variability. This uni- fied view explains a widely observed phenomenon: stronger guidance improves semantic alignment but inevitably reduces diversity. Experiments support these predictions, showing that early strong guidance crodes global diversity, while late strong guidance suppresses fine-grained variation. Moreover, our theory naturally suggests time-varying guidance schedule, and empirical results confirm that it consistently improves both quality and diversity.
INTRODUCTION
Diffusion models have driven remarkable progress in generative modeling, achieving state-of-the- art results across images, video, audio, and multimodal domains (Rombach et al.] 2022 Podell al. 2023 Cao al. 2024 ZZhan et al., 2025). A central requirement in practice conditional generation, where outputs must follow prompts, class labels, or other structured signals. Among existing strategies, Classifier-Free Guidance (CFG) (Ho & Salimans 202I) has become the de facto standard due to its simplicity and effectiveness, powering nearly all large-scale diffusion pipelines.
Numerous variants further extend its influence (Jin et al., 2025] Gao et al. 2025 Malarz al. 2025
Chung et al., 2025 Sadat et al. 2025 Castillo et al. 2025). Yet despite its widespread adoption, the theoretical underpinnings CFG remain poorly understood.
While recent work has advanced the theory of diffusion sampling (Benton et al. 2023 Chen et al., 2023 Cai Li., 20255 Li et al.) (025a), the analysis of CFG itself is still its infancy, hindered by its heuristic formulation. Prior studies fall into two categories. The first assumes unimodal conditional distributtonnttyicclllll a single Gaussian-yiildding clean derivations but overlooking the multimodal nature of real-world tasks (Chidambaram et al. 2024 Wu et al. 2024 Bradley & Nakkiran 2024; Xia et al.) [2024] Jin et al., 2025 Pavasovic et al. 2025 Li et al. 2025b). The second relaxes these assumptions but imposes only weak conditions, producing broad qualitative insights without sharp predictions (Li & Jiao 2025). Consequently, several well-documented em- pirical phenomena remain theoretically elusive, most notably the collapse of diversity under large guidance weights (Ho & Salimans 2021). Explaining this diversity loss is both key theoretical challenge and a practical necessity for improving conditional generation. A detailed discussion of these related works can be found in Appendix.
In this work, we move beyond the restrictive unimodality assumption by modeling conditional dis- tributions as Gaussian mixtures. This perspective reveals that guided sampling follows natural
"Correponniigg author

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three-stage structure. In the Direction Shift stage (early, high-noise regime), trajectories drawn toward the weighte mean, and guidance amplifies this attraction, inducing initialization bias and norm inflation. In the Mode Separation stage (intenmediate regime), the dynamics remain locally neutral, but the inherited bias suppresses weaker modes, reducing global diversity. In the Concen- tration stage (late regime), contraction within modes is intensified by strong guidance, suppressing local variability and fine-grained diversity.
We empirically validate these predictions on state-of-the-art diffusion models rather than toy set- tings. To test our framework, we first compare early-high, late-high, and all-high schedules, con- firming that strong early guidance reduces global diversity. To probe the late-stage behavior, we initialize trajectories from identical noise and introduce small perturbations during the intermedi- ate stage. We observe that excessively large late-stage guidance forces trajectories to converge toward nearly identical fine details, thereby diminishing local diversity. Beyond these validation studies, we further conduct an additional experiment with a time-varying guidance schedule which can improve performance. Although this is not the central theoretical contribution of this work, underscores the practical relevance of our analysis. Our implementation is publicly available at https://githnn com/ http://t22///////
Contributions. This work makes three contributions: (i) We provide the first theoretical frame- work for analyzing CFG under multimodal conditional distributions, revealing a natural three-stage structure of the sampling dynamics. (ii) We empirically validate the central predictions of this frame- work: strong early guidance reduces global diversity, while strong late guidance suppresses local variability. (iii) As byproduct of our analysis, we observe that varying guidance strength over time trade-off. Together, these results unify theoretical and empirical
can improve the perspectives and establish foundation for the principled design of guided diffusion samplers.
2 PRELIMINARY
2.1 DIFFUSION MODEL
Diffusion models generate data by inverting a forward noising process that progressively transforms samples from the target distribution (a:) into simple Gaussian reference. The forward corrup- tion can be written as
dæ; dt + v 23(
3o
Pdata
t E [0,
where (() and 3()) control the drift and diffusion, and w, is a Wiener process. As grows, the distribution pt converges toward isotropic Gaussian, ensuring that sampling can begin from tractable prior.
By time-reversal theory (Anderson 1982.. sampling is performed through the reverse SDE
dx, [- a(t),, (3( V æ, log (x: dt + 22((t)d
where V... log Pe (xx) is the score function, representing the gradient of the log-density, and n. is backward Wiener process.
Modern implementations typically adopt the probability flow ODE (Song et al. 2021) which elim- inates randomness while preserving the same marginal distributions:
dæ
- aa)) B(t) æ. log (x(t)).
(3)
dt
Its deterministic nature allows the use of high-order ODE solvers (Lu al. 2022 Zhao et al. 2023). enabling efficient generation with few function evaluations and making theoretical analysis more tractable. This ODE view is particularly important for understanding how guidance mechanisms reshape the underlying dynamics.
Conditional generation. When conditioning on external information y (e.g., a class label or a text prompt), one simply replaces the unconditional score with the conditional score:
dx: = (t(()) - B(t) V log Pe (xz [z l)).
(4)
dt
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This conditional probability flow ODE produces samples consistent with while retaining the com- putational benefits of the ODE formulation. However, in practice the learned conditional score may be weak under-confident, leading to poor semantic alignment with the conditioning signal.
2.2 ELASSIFIER-FREE GUIDANCE
While the conditional ODE equation 4 incorporates side information its influence in practice is often insufficient. Classifier-Free Guidance (CFG) addresses this issue by extrapolating between the unconditional score V log po and the conditional score Va, Po (a y) predicted by the same model: (, (1 w) Væ, po w Va, lo v) 1.
(5)
1 reduces to the plain conditional model, whereas larger values enforce stronger alignment with y. While this interpolation is simple and effective, the resulting score no longer corresponds to any valid probabilistic model.
Here the guidance scale controls the trade-off:
Plugging the guided score equation 5 into the probability flow dynamics yields the CFG probability flow ODE:
dæ; dt
att)x; - B(t) (,
(6)
STAGE-WISE BEHAVIOR OF CLASSIFIER FREE GUIDANCE IN SAMPLING
Direction Shift
Mode Separation
Concentration
Cond- ODE
CFG-
ODE
Figure 1: Illustration of the three-stage dynamies of conditional sampling (Cond-ODE, top row) versus Classifier-Free Guidance (CFG-ODE, bottom row) under a multimodal distribution. In the Direction Shift stage (left), CFG trajectories deviate more strongly toward the global weighted mean, introducing initialization bias. In the Mode Separation stage (middle), Cond-ODE trajectories main- tain coverage of multiple modes, while CFG trajectories suppress weaker modes and collapse toward dominant ones. In the Concentration stage (right), CFG trajectories contract excessively within modes, leading to loss of fine- une-grrined diversity. Red dots denote samples, gray arrows connect the start and end points of the same trajectory (indicating their correspondence), and blue crosses mark the weighted mean of conditional modes.
In this section, we systematically investigate the effect of Classifier-Fres Guidance on sampling when the conditional distribution multi-modal. We identify three-stage progression of the dy-
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namies: in the Direction Shift stage (early, high-noise regime), CFG accelerates convergence toward the class-weigs weighte mean and induces early norm inflation; in the Mode Separation stage (intermedi- ate regime). CFG primarily accelerates the pre-existing mode-attraction dynamics while remaining relatively neutral with respect to the geometry of attraction-nontrvvaal regions that flow to weaker modes persist and independent of the guidance scale. Nevertheless, the initialization bias inher- ited from the first stage causes far fewer trajectories to enter these regions, so that the interaction effective loss of diversity. Finally, the Concentration stage
of the first two stages leads (late regime), CFG amplifies the restoring force toward mode centers, causing trajectories within the same mode to contract more tightly. As result, intra-class variability is suppressed, leading to a loss of fine-grained diversity despite the samples appearing sharper and more aligned.
We begin by stating the distributional and noise-schedule assumptions that ground our analysis. These provide a simplified yet representative setting for characterizing the three stages and their interactions.
Assumption 3.1. Let a E IRd the data variable and y E y the condition.
Ia). This aligns with the standard Gaussian prior used in latent diffusion models, making it both realistic and analytically convenient.
(A1) Unconditional Distribution. po(x:) = (O,
(A2) Conditional Distribution. For each y, model the conditional distribution Gaussian mixture:
k(y) Σ
N(x;; ue (v) La)a) T
(w)
p v)
0, 0, E. = and oy < This multimodal formulation makes theoretically possible to analyze the impact of CFG sampling on diversity. For notational simplicity, we omit subscripts and superscripts associated with y when no confusion arises.
where &
(A3) Noise Schedule. In the forward SDE equation I
we set
1 (l1) 1-t B(1)
t
E
1-
giving
p( 20) - -N((1 ())oo 1I))
This choice corresponds to flow-matching model with linear mean decay. We emphasize that this assumption made for technical convenience, as different diffusion formulations can be trans- formed into another through appropriate reparameterizatiios (Karras al. 2022
3.1 FIRST STAGE: ACCELERATION AND DIRECTION SHIFT
In the early stage (high-noise), fine-grained multimodal information is largely suppressed, so the score function reflects only global statisties of the conditional distribution rather than detailed mode structure. In this stage, Classifier- Free Guidance strengthens the global attraction and thereby alters both the speed and the direction of early trajectories by amplifying the conditional score. prac- tice, CFG accelerates motion toward the cligh-weighted mean of the mixture, while also pushing trajectories outward and inflating their norms. We make this precise in the following theorem.
Theorem 3.2 (In-expectation early-stage proximity under CFG), Let j -EK,, Thd denote the class-weighter mean of the Gaussian mixture prior, and assume the same initialization ~ (O, I) for both trajectories. Then for any w there exists time point t.1 < such that, for all 1 E [1t, 1),
E [|11:| wHlll]]
< E 11 -wp-lppl 2
where w is the guidance weight, at is the solution to the conditional probability flow ODE and (CPG) the solution the CFG-driven ODE, both starting from Tı. In expectation, the Criver trajectory remains closer to the reference point wp than the purely conditional one in the early stage of sampling.
xt
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Interpretation.
Theorem 3.2 reveals two characteristic consequences of CFG in the high-noise
regime.
First, because strong noise suppresses fine-graines structure, the score is dominated by global aver- ages. When CFG amplifies this score, it effectively increases the pull toward the scaled mean wp, leading to acceleration effect: trajectories approach the global mean faster than they would under the conditional flow alone.
Second, since w enlarges the mean by factor, the target point wp lies farther from the origin. Trajectories that are pulled toward this more distant point inevitably attain larger Euclidean norms, producing what term norm inflation.
Together, these two effects introduce a structural bias at the very beginning of sampling. By steer- ing trajectories rapidly toward a magnified global mean, CFG suppresses the natural exploration of multimodal variability and predisposes samples to collapse into the dominant mode once the multi- modal structure becomes relevant. Thus, the first stage not only governs the pace and scale of early dynamics but also seeds the mode-selection mechanism that shapes the second stage.
3.2 SECOND STAGE: INTRA-CLASS MODE SEPARATION
As noise decays and the multimodal structure becomes dominant, trajectories cease to move toward the global mean and instead diverge into distinct attraction basins. This marks the onset of mode separation: the state space partitioned, and each basin deterninistically leads to a different mode.
In this regime, CFG plays a neutral role. Amplifying the conditional score merely accelerates convergence within whichever basin a trajectory already occupies, but does not reshape the basin geometry redirect trajectories across basins. Hence, once trajectory falls into the attraction region of particular mode, its final outcome is determined regardless of the guidance weight.
Theorem 3 formalizes this neutrality in the two-component case. It establishes that the weaker mode retains a genuine basin of attraction Us, that is invariant to w: any trajectory entering this region will remain aligned with the weaker mode throughout its evolution.
Theorem 3.3 (Persistence of the weaker mode under CFG). Consider a two-component Gaussian mixture
po( v) лI 11 Wxx; 11, + NT M2, III,, || || T1 TT2-
Under some mild assumptions, there exist taz E (0, 1) and w-independent region Usz C Rd depending only on (Ty, M4x o). such that for any w Z if Øt., E Um and æs follows the CFG probability flow ODE equation Oon [0, teyl. then
Ti Max (1 t)pm, (t2 + +1-1))0011
12 t)p2, (1 +1 -1)2021)) VI txu)
Interpretation. Theorem 3.3 shows that the weaker mode μt possesses a bona fide basin of attrac- tion Us, that is independent of w. This result highlights the neutrality of CFG in the mode-separation regime: amplifying the conditional score does not distort the geometry of attraction basins. Once a trajectory falls into U.,, will remain aligned with μm throughout its evolution, no matter how large the guidance weight is chosen. Thus, the disappearance of weaker modes observed in practice cannot be ascribed the intrinsic geometry of second-stage dynamics alone.
Interaction with the first stage. The missing element is the initialization bias accumulated during the high-noise regime. While CFG is neutral once modes separate, it rarely supplies trajectories that actually enter U.y. In the early stage, the amplified score consistently drags samples toward the scaled mean wj.. By the time noise decays enough for the multimodal structure to emerge, most trajectories have already been displaced into regions where the drift field unequivocally favors the dominant mode μ22 In other words, the second stage does not destroy weaker modes, but the first stage strongly reduces the likelihood that any trajectory remains close enough them for mode separation to take effect. This mechanism is formalized in Proposition 3.4
Proposition 3.4 (Initialization bias from the first stage). Let æ evolve under the CFG probability flow ODE equation 6 with w 2 1. Then there exists 0 < ls < such that, for any l > 1, one can find a radius r r(k) satisfying: if |
kjj r(k),
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then for all t S tsg it holds that
Th Ni (1 - t)pi, +1++(1 (1-1))00))1 < T2 Nan;; (1 1),,, (P + (1 (1)
Moreover, r(k) grows monotonically with k.
Interpretation. Proposition makes explicit how the first stage biases the sampling distribution. Once trajectory is drawn sufficiently close to the scaled mean ki,, the surrounding drift field guarantees that the posterior likelihood of μl will dominate that of μ4 at all carlier times. According Theorem 3.2, larger w causes early-stage trajectories be closer to wp,, corresponding larger effective k and thus expanding the region dominated by the stronger mode. Consequently, although the basin of μ4 mathematically persists, becomes increasingly difficult for trajectories to reach it under strong guidance. The scarcity of weaker-mode samples in practice therefore reflects not the elimination of Usg. but the fact that most trajectories are already preconditioned by the first stage to fall outside it.
Stage summary. The second stage reveals subtle but important contrast. On theoretical level, the dynamics preserve all modes, including weaker components, whose attraction basins remain stable and w-independent. On a practical level, however, the initialization bias carried over from the first stage severely limits access these basins. Thus, the loss of weaker-mode diversity is not due to second-stage suppression, but to the compounded effect of early-stage displacement and reduced basin occupancy. This interplay explains why weaker modes remain viable theory yet vanish in empirical samples.
3.3 THIRD STAGE: CONCENTRATION
In the final stage, when the noise level has decayed to a certain level, the fine-grained structure of the conditional distribution takes full control of the dynamics. Unlike the first stage, where global statistics dominate, or the second stage, where trajectories are separated into different basins, the third stage is governed by local geometry around each mode μ4g. In this regime, trajectories evolve almost exclusively under the influence of the nearest mode, and the role of CFG becomes entirely within-mode.
This yields a concentration effect: relative to standard conditional sampling (w=1), CFG-guided trajectories contract more aggressively within basin, thereby reducing pairwise separation and diminishing within-class dispersion (fine-grained variability conditioned on a fixed semantic label). The following theorem formalizes this effect.
Theorem 3.5 (CFG yields stronger wide contraction). Under some mild assumptions, there exist time two (0, 1) and radius >
O such that the following holds uniformly. For any
l E (1,2} and any initial pair
Di Ft.g [B((1- teg) pkx r),
(v) be the solutions initialized at the same pair (Zt.. zt.n)) ) at time (CFG) = (CFG) (resp. a!!)) 2, let st
te under the CFG flow with weight w (resp. the standard conditional flow with w 1). Then for all t [0, tzg).
(CFG)
(CFG) < ||25.00 2, -.00
x
Here B(c,r) = (x gd || - cll < r denotes the open Euclidean ball.
Interprefation. Theorem 3.5 demonstrates that, under continuous dynamics, CFG strengthens the contraction of trajectories inside each mode. The intuition is straightforward: when noise is neg- ligible, the conditional score essentially acts as a linear restoring force that points toward the local mean. Scaling this score by w > increases the strength of this force, causing pairs of nearby trajectories to converge faster than they would under standard conditional sampling. Consequently, their separation shrinks more rapidly, reducing local variability.
From a generative perspective, this contraction has a dual effect. On one hand, explains why large guidance weights often produce samples that appear sharper, cleaner, and more faithfully aligned with the conditioning signal because trajectories are pulled tightly into the most semantically rep- resentative regions of each mode. On the other hand, it also clarifies why intra-class diversity suffers:

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by collapsing trajectories together, CFG suppresses natural variations such pose, texture, fine- grained stylistic features, which would otherwise emerge from looser sampling around the same mode.
(a) Constant high weight
(b) Early low weight
(c) Early high weight
Figure 2: Comparison of guidance schedules on the prompt "A view of a bathroom that is clean." The (a) Constant schedule and (c) Early-high schedule both collapse diversity, with most samples converging to layouts dominated by large windows and uniform cool tones. The (b) Early-low schedule mitigates this effect, producing more varied spatial structures and color palettes.
3.4 SUMMARY ACROSS STAGES
The analysis above reveals that the effect of Classifier-Free Guidance (CFG) cannot be understood isolation at any single point in the sampling process. Instead, its influence unfolds sequentially across three distinct stages, each of which leaves lasting imprint the final distribution of samples.
Stage I: Acceleration and Direction Shift. Under strong noise, fine-grained multimodal informa- tion suppressed, and the score reflects only global statistics. CFG amplifies this global signal, accelerating convergence toward the class-weighted mean and inflating trajectory norms. This early bias seeds initialization effect: samples are drawn disproportionately toward regions aligned with the global mean.
Stage II: Mode Separation. As noise decreases, trajectories diverge into distinct basins associated with different modes. CFG itself does not alter the geometry of catraction-baaat remain intact and weaker modes preserve nontrivial regions of influence. However, because most trajectories were preconditioned by the first stage, they rarely enter the weaker basins. The disappearance of weaker- mode samples thus arises not from second-stage geometry but from the initialization bias inherited from Stage I.
Stage I: Concentration. Once noise becomes negligible, dynamics are dominated by local con- traction within each mode. By scaling the conditional score, CFG sharpens this contraction, sup- pressing intra-class variability. Samples therefore appear sharper and more semantically faithful, but diversity within each mode is eroded.
Taken together, these stages explain the dual empirical effects of CFG: it improves conditional fi- delity and visual sharpness, yet simultaneously diminishes both global coverage (loss of weaker modes) and local diversity (within-mode contraction). Interestingly, this resonates with prior work (Balaji al.) 2022 Li al.) 2024) that empirically found diffusion models determine the overall shape early in the process and fine details late, which corresponds to our distinction between global diversity and local diversity. This stage-wise perspective provides unified theoretical frame- work for understanding diversity loss under CFG and points to principled strategies for designing time-van varying guidance schedules.
EXPERIMENTS
We now turn to empirical validation of our theoretical analysis. We begin by testing two key pre- dictions: (1) strong early guidance erodes global diversity, and (2) strong late guidance reduces

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Published as a conference paper at ICLR 2026
(a) Late high schedule, MSE=0.0025
(b) Constant schedule. MSE=0.0103
Futuristic city at sunset, glass towers with neon skybridges, flying cars leaving light trails, woman in silver robe holding holographic tablet balcony, cinematic lighting, ultra- detailed, 8K render. Under the late-high schedule (a), the highlighted regions reveal cars that are nearly identical across samples, indicating reduced diversity. In contrast, the constant schedule (b) preserves greater variability in both car shapes and positions, which is also reflected by a larger mean squared error (MSE).
Figure 3: Prompt:
fine-grained diversity. We then evaluate the effectiveness of the time-varying schedule derived from theory, which weakens guidance in the early and late stages while strengthening it in the middle. Although the main contribution of this work is explain the mechanism behind diversity loss un- der CFG, we also show that the proposed schedule yields tangible improvements, thereby bridging theory and practice. Detailed experimental settings are provided in the Appendix.
Table 1: Results with NFE fixed at 10 across guidance scales. Bold indicates the best value within each method at the same w (higher is better for CLIP/IR; lower better for FID).
CFG
APG
Metric
Variant
w=3
w=5
w=7
w=9
w=3
w=5
w=7
w=9
vanilla
0.320
0.322
0.319
0.313
0.322
0.322
0.321
0.317
interval
0.315
0.317
0.318
0.319
0.315
0.317
0.318
0.319
CLIP 1
TV(Ours)
0.319
0.319
0.319
0.319
0.319
0.320
0.320
0.319
0.316
0.318
0.318
0.316
0.317
0.318
0.318
0.317
vanilla
0.894
0.806
0.553
0.223
0.909
0.890
0.712
0.434
interval
0.602
0.694
0.727
0.723
0.602
0.693
0.724
0.725
IR f
TV(Ours)
0.859
0.935
0,950
0.932
0.860
0.937
0.958
0.941
B
0.705
0.822
0.831
0.790
0.737
0.846
0.867
0.833
vanilla
28.305
29.275
32.859
38.988
28.028
28.208
30.268
34,950
interval
30.485
28.167
27.884
28.413
30.461
28.155
28.000
28.427
FID +
TV(Ours)
27.898
27.722
28.547
30.259
27.917
27.935
28.503
29.838
B
28.679
28.571
29.774
32.184 184
28. 163
28.082
29.148
31.615
vanilla
0.283
0.408
0.513
0.574
0.266
0.385
0.500
0.576
interval
0.173
0.178
0.185
0.193
0.172
0.177
0.183
0.192
Saturation
TV(Ours)
0.197
0.229
0.263
0.298
0.193
0.221
0.253
0.288
B
0.184
0.219
0.259
0.305
0.186
0.218
0.257
0,304
vamilla
1.066
1.101
1.105
1.081
1.059
1.115
1.136
1.128
interval
1.013
1.073
1.122
1.160
1.013
1.072
1.121
1.159
Diversity 1
TV(Ours) B
1.092
1.158
1.196
1.223
1.088
1.154
1.194
1.222
1.123
1.181
1.217
1.242
1.121
1.178
1.214
1.241
4.1 VALIDATION OF THEORY.
We first test whether the predictions of our theoretical analysis manifest in practice. In addition to vanilla CFG with constant high weight, evaluate two time-varying variants: that applies high weight early and low weight late, and the other with the opposite schedule. Figure[ compares generated samples across these three strategies. Relative vanilla CPG and the early-high variant,
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(a) Vanilla-CFG
(b) interval-CFG
(c) TV-CFG
(d) 8-CFG
Generated samples with the prompt "A view of bathroom that is clean" at high sam- pling budget (NFE=50). While constant schedules yield semantically consistent but overly uniform results, methods adhering to the low-high-low scheduling principle exhibit significantly higher di- versity.
Figure 4:
the early-low variant achieves substantially higher diversity, directly supporting our first theoretical claim: excessive early guidance induces mean-shift bias, suppressing weaker modes and thereby reducing multimodal coverage.
We next evaluate our second claim: that strong late-stage guidance enforces excessive similarity in fine details. Starting from identical noise, we integrate to an intermediate step, inject small Gaussian perturbations, and then continue sampling. One schedule keeps a low weight throughout, while the other switches that are more alike in local structure, whereas the constant-low schedule preserves greater variabil- ity. This confirms that strong late guidance diminishes fine-gra fine-grained diversity: once trajectories confined to basins, large late weights amplify mode contraction and suppress injected pertur- bations, leading to nearly indistinguishable outcomes.
higher weight late. As shown in Fig. the late-high schedule yields outputs
Table 2: Results with guidance weight fixed at = 9 while varying the NFE budget. Bold indicates best value within each method at the same NFE (higher better for CLIP/IR; lower better for FID).
CFG
APG
Metric
Variant
NFE 5
NFE 10
NFE
15
NFE 20
NFE
NFE
10
NFE 15
NFE 20
vanilla
0.275
0,313
0.319
0.320
0.282
0.317
0.320
0.320
CLIP
interval
0.314
0.319
0.318
0.317
0.314
0.319
0.319
0.317
TV(Ours)
0.312
0.319
0.319
0.319
0.312
0.319
0.319
0.319
8
0.302
0.316
0.318
0.318
0.304
0.317
0.318
0.318
vanilla
-1.137
0.223
0.616
0.820
-0,926
0.434
0.735
0.860
interval
0.145
0.723
0.827
0.863
0.145
0.725
0.825
0.865
TV(Ours)
0.176
0.932
1.016
1.049
0.174
0.941
1,029
1,061
B
0..28
0.790
0.971
1.024
-0,202
0.833
0,984
1,036
vanilla
988
31.863
29.059
73.751
34,950
30.122
interval
47.239
28.413
25,747
25,385
47.735
28.427
25.691
25,365
TV(Ours)
46.895
30.259
29.256
28.458
46.878
29.838
28.834
27 969955
62.979
32.184
29.246
28.348
$8.398
31.615
28.836
27.968
vanilla
0.589
0.574
0.526
0.476
0.610
0.576
0.521
0.472
interval
0.197
0.193
0.212
0.233
0.194
0.192
0,209
0,229
Saturation
TV(Our))
8
0.236
0.305
0.316
0.330
0.243
0.304
0.307
0.315
vanilla
0.867
1.081
1.171
1.201
0.917
1.128
1.186
1,200
Diversity t
interval
1.140
1.160
1.178
1.202
1.140
1.159
1.176
1,200
TV(Our)) 3
1.191
1.223
1.233
1.231
1.191
1.222
1.232
1.231
1.196
1.242
1.241
1.239
1.198
1.241
1.239
1.238
4.2 METHODOLOGICAL IMPLICATIONS.
Guided by the above analysis, we propose linear time- varying schedule where the guidance weight rises early, peaks at the intermediate stage, and decreases late (TV-CFG). We also considered Interval-CFF (Kynkääänieemi et al.) (024) and 3-CFG methods (Malarz et al.) (025). Both adhere the time-varying scheduling principle of being weak in the early stages, strong in the middle, and

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weak in the late stages, which theoretically contributes sampling diversity. The specific schedul- ing principles are detailed in the Appendix. Though not our main focus, this yields empirical gains and highlights how stage-wise insights translate into practical improvements in robustness and diversity.
design
Figure shows that schedule preserves diversity while reducing over-saturation. Table further reports consistent gains in ImageReward (IR) (Xu et al. 2024), which better reflects overall quality than CLIP. Notably, all three methods that follow our design principle effectively improve generation diversity while maintaining good generation quality, At low weights our method matches vanilla CFG, but high weights-vv wwhere vanilla CFG degrades it achieves clear advantages and stronger best-case performance. The same principle applies to APG (Sadat et al. 2025), where TV-APG likewise improves
Table 2 summarizes results across NFEs. At low NFEs, vanilla CFG nearly collapses, producing poor IR and FID; at higher NFEs it partially recovers but still suffers from over-saturation, with high saturation values and limited diversity gains. We attribute this to norm inflation from strong early guidance under coarse discretization, which destabilizes dynamics and cannot be fully corrected by increasing NFEs. Thus, theory-guide schedules are particularly effective in low- and medium-NFE regimes, where budgets are limited and naive CFG most fragile.
CONCLUSION
We present the first systematic analysis of Classifier-Free Guidance (CFG) under multimodal con- ditional distribution assumption, characterizing its dynamics in three stages: early Direction Shift, mid Mode Separation, and late Concentration. Our theory explains how diversity deteriorates: early mean-shift bias suppresses weaker modes, while late-stage contraction reduces intra-class variabil- ity. This stage-wist wise perspective clarifies long-standing empirical observations and provides concrete predictions consistent with behaviors observed in modern diffusion models. Beyond theoretical analysis, we further show that attenuating guidance the early and late stages while emphasizing the mid stage offers simple yet effective strategy for mitigating diversity loss. Finally, extending the stage-wise framework to stochastic differential equation (SDE) sampling and rigorously charac- terizing the effects of numerical discretization remain important directions for future research.
ACKNOWLEDGEMENTS
The authors are with the Department of Electronic Engineering, Beijing National Research Cen- ter for Information Science and Technology, Tsinghua University, Beijing 100084, China. This work was supported by the National Key Research and Development Program of China (Grant No. 2025YFF05 16601) and the National Natural Science Foundation of China (NSAF U2230201).
REFERENCES
Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Ap- plications, 12(3):313-326, 1982.
Yogesh Balaji, Seungjun Nah, Xun Huang, Arash Vahdat, Jiaming Song, Qinsheng Zhang, Karsten Kreis, Miika Aittala, Timo Aila, Samuli Laine, et al. ediff-i: Text-to-image diffusion models with an ensemble of expert denoisers. arXiv preprint arXiv:2211.01324, 2022.
Joe Benton, Valentin De Bortoli, Arnaud Doucet, and George Deligiannidis. Nearly d-linear conver- gence bounds for diffusion models via stochastic localization. arXiv preprint arXiv:2308.030m 03686, 2023.
Arwen Bradley and Preetum Nakkiran. Classifier-free guidance is predictor-corrector. arXiv preprint w7X00:2408.09000, 2024.
Changxiao Cai and Gen Li. Minimax optimality of the probability flow ode for diffusion models. arXiv preprint arXiv:2503.09583, 2025.
10

---

Published as a conference paper at ICLR 2026
Boyuan Cao, Jiaxin Ye, Yujie Wei, and Hongming Shan. Ap-Idm: Attentive and progres- sive latent diffusion model for training-fin free high-resolution image generation. arXiv preprint arXiv:2410.06055, 2024.
Angela Castillo, Jonas Kohler, Juan Perez, Juan Pablo Perez, Albert Pumarola, Bernard Ghanem, Pablo Arbelaez, and Ali Thabet. Adaptive guidance: Training- free acceleration of conditional diffusion models. In Proceedings of the AAAI Conference on Artificial Intelligence, 2025.
Hongrui Chen, Holden Lee, and Jianfeng Lu. Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learning, pp. 4735-4763. PMLR, 2023.
Muthu Chidambaram, Khashayar Gatmiry, Sitan Chen, Holden Lee, and Jianfeng Lu. What does guidance do? fine-grained analysis in simple setting. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
Hyungjin Chung, Jeongsol Kim, Geon Yeong Park, Hyelin Nam, and Jong Chul Ye. CFG++: Manifold-constrinned classifier free guidance for diffusion models. In The Thirteenth Interna- tional Conference on Learning Representations, 2025.
Zhengqi Gao, Kaiwen Zha, Tianyuan Zhang, Zihui Xue, and Duane S Boning. Reg: Rectified gradient guidance for conditional diffusion models. arXiv preprint arXiv:2501.1888 2025.
Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance. In NewrIPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021.
Cheng Jin, Zhenyu Xiao, Chutao Liu, and Yuantao Gu. Angle domain guidance: Latent diffusion
requires rotation rather than extrapolation. In Forty-second International Conference on Machine Learning. 2025.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. Advances in neural information processing systems, 35:26565-26557 2022.
Tuomas Kynkäänniemi, Mila Aittala, Tero Karras, Samuli Laine, Timo Aila, and Jaakko Lehtinen. Applying guidance in limited interval improves sample and distribution quality in diffusion models. arXiv preprint arXiv:2404.0777 2024. doi: 10.48550/aXx..22440.7772
Gen and Yuchen Jiao. Provable efficiency of guidance in diffusion models for general data distribution. In Forty-second International Conference Machine Learning, 2025.
Gen Li, Changxiao Cai, and Yuting Wei. Dimension-free convergence of diffusion models for approximate gaussian mixtures. arXiv preprint arXiv:2504.053000 2025a.
Senmao Li, Taihang Hu, Joost van de Weijer, Fahad Khan, Tao Liu, Linxuan Li, Shiqi Yang. Yaxing Wang, Ming-Ming Cheng, and Jian Yang. Faster diffusion: Rethinking the role of the encoder for diffusion model inference. Advances in Neural Information Processing Systems, 37: 12203-85240, 2024.
Xiang Li, Rongrong Wang, and Qing Qu. Towards understanding the mechanisms of classifier-free guidance. arXiv preprint
2025b.
Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: fast ode solver for diffusion probabilistic model sampling in around 10 steps. In Advances in Neural Information Processing Systems (NeurIPS), volume 35, pp. 5775-5787, 2022.
Dawid Malarz, Artur Kasymov, Maciej Zieba, Jacek Tabor, and Przemystaw Spurek. Classifier-free guidance with adaptive scaling. arXiv e-prints, pp. arXiv-2502. 2025.
Krunoslav Lehman Pavasovic, Jakob Verbeek, Giulio Biroli, and Mare Mezard. Classifier-fres guidance: From high-dimensional analysis to generalized guidance forms. arXiv preprint arXiv:2502.07884, 2025.
11

---

Published as a conference paper at ICLR 2026
Dustin Podell, Zion English, Kyle Lacey, Andreas Blattmann, Tim Dockhom, Jonas Müller, Joe Penna, and Robin Rombach. Sdxl: Improving latent diffusion models for high-resolution image synthesis. arXiv preprint arXtv:2307.01952, 2023.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjöm Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF confer- ence on computer vision and pattern recognition, pp. 10684-10695, 2022.
Seyedmorteza Sadat, Otmar Hilliges, and Romann M. Weber. Eliminating oversaturation and arti- facts of high guidance scales in diffusion models. In The Thirleenth International Conference Learning Reprexentations, 2025.
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- tional Conference on Learning Representations, 2021.
Yuchen Wu, Minshuo Chen, Zihao Li, Mengdi Wang, and Yuting Wei. Theoretical insights for diffu- sion guidance: case study for gaussian mixture models. In Forty-first International Conference on Machine Learning. 2024.
Mengfei Xia, Nan Xue, Yujun Shen, Ran Yi, Tieliang Gong, and Yong- Jin Liu. Rectified diffusion guidance for conditional generation. arXiv preprint arX1v:2410.18737, 2024.
Jiazheng Xu, Xiao Liu, Yuchen Wu, Yuxuan Tong, Qinkai Li, Ming Ding, Jie Tang, and Yuxiao Dong. ImageReward: Learning and evaluating human preferences for text-to-image generation. Advances in Neural Information Processing Systems, 36, 2024.
Jinjin Zhang, Qiuyu Huang, Junjie Liu, Xiefan Guo, and Di Huang. Diffusion-4k: Ultra- resolution image synthesis with latent diffusion models. In Proceedings of the Computer Vision and Pattern Recognition Conference, pp.
2025.
Wenliang Zhao, Lujia Bai, Yongming Rao, Jie Zhou, and Jiwen Lu. Unipe: A unified predic- tor-corrector framework for fast sampling of diffusion models. In NeurlPS 2023, 2023.
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A LLM USAGE
We used large language models (LLMs) solely for language polishing. All research ideas, theoretical analyses, experiment designs, and results were developed entirely by the authors. The LLMs did not contribute ideation or scientific content.
B RELATED WORK
Although Classifier-Free Guidance (CFG) was introduced early and has since become standard sampling strategy for diffusion models, theoretical analyses of CFG only began to emerge around 2024. Most existing studies focus on simplified distributional settings, such as one-dimensional or Gaussian cases.
Chidambaram et al. (2024) show that in one-dimensional setting where the unconditional distri- bution is bimodal and the conditional distribution is unimodal, CFG-ODE sampling tends to con- centrate the distribution's edges the guidance weight increases. Moreover, estimation errors can cause severe deviations from the target support under large guidance weights. Extending higher dimensions, Pavasovic Pavvssovi al. (2025) analyze two-component Gaussian mixture and demonstrate that such edge-concentration effects vanish in high dimensions.
small score
Bradley & Nakkiran (2024) prove that when both conditional and unconditional distributions one-dimensional zero-mean Gaussians, the closed-form solution of CFG deviates from the ex- pected samma-weightee distribution. Xia al. (2024) further extend this result to nigh-dimensional isotropic Gaussians with differing conditional and unconditional parameters, deriving closed-form output that again diverges from the hamma--eighted distribution. Li al. (2025b) consider the effect of the covariance structure of Gaussian distributions on the CFG sampling process, and point out that CFG enhances generation quality by amplifying class-specific features while suppressing generic ones.
Building on these findings. Wu et al. (2024) show that when unconditional distribution a mixture of approximately orthogonal Gaussians and the conditional distribution is Gaussian, CFG sampling theoretically increases classification confidence while simultancously decreasing the en- tropy of the output distribution. The entropy reduction phenomenon is consistent with analysis of the third-stage dynamics of CFG sampling, where strong guidance induces distributional con- traction and suppresses inter-modal variability. Jin al. (2025) further relax these assumptions to more general Gaussian mixture settings, and reveal that CFG induces norm inflation and anomalous diffusion effects, thereby providing theoretical explanation for the over-saturation phenomena ob- served in practice. Finally, Li & Jiao (2025) analyze CFG under considerably weaker assumptions without restricting the distributional form, and demonstrate that the improvement in classification confidence holds only in an average sense, rather than uniformly across all initial conditions.
In summary, with the exception of Li & Jiao (2025), existing works primarily focus on settings where the conditional distribution is unimodal. This restriction limits their ability to explain crit- ical phenomena such as the loss of diversity under large guidance weights. Although [i & Jiao 2025) relax the distributional assumptions considerably, their overly general setting similarly fails to capture the underlying mechanism the phenomenon.
C PROOF OF THEOREM AND PROPOSITION
We begin by stating key lemma, which serves as a fundamental tool throughout the subsequent analysis.
Lemma C.1. For the probability flow ODE associated with the forward noising process, the dy- namics of x satisfy
dæ: Elxoo x
Et t
dt
In the unconditional setling where the prior is isotropic Gaussian, the posterior mean admits the closed-form expression
1-t
Zojz == Blz0
1)22 Zt. [t]]
(8)
82
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In the conditional case where the data distribution is modeled Gaussian mixture, the posterior mean can be expressed as a convex combination
K Σ
a(v) Ojr
= E|zy
Ze. vl
(x(
m& (xi);
(9)
km
where the component-wise posterior mean and responsibilities are given respectively by 1² p. + (1 - ()-t)o
mg(xx+)
(10)
(1 1)20² 1²
T& TxNi xii (1- (1-t)pp t)pp: (11-t)o0++ +
T& (xx))
(11)
Ty xe; (1- (1-t) ((1 t) + 1))1)
The first identity follows directly from Tweedie's formula, while the latter two expressions are ob- tained via straightforward posterior computations for Gaussian and Gaussian mixture distributions.
C.1 PROOF OF THEOREM 3.2
Theorem C.2 (Theoren 3.2). Assume the class-conditional prior is a K K-oomponent Gaussian mix-
ture with shared covariance I and weights {= and let μ Σx T&p Let the reverse probability flow ODE and its CFG-driven variant be driven by the estimators from Lemma 1:
(v) (1))
at (2)
Olt (xt
(v)
(12)
(v) (CHO))
+ w Toje (x; ( - Loje
(xF(G) (CPO))
(CFG)
oofe (at
at
(CFG)
(13)
with a shared random initialization (v) = there exists le (0,1) such that for all / (111,
x 1
((CFG)) N((0,) independent of y. Then for any
E [| wHll 1
|| (v) wā [?]]
Proof Sketch. By Lemma C.I. as t + 1 we have the uniform early-time approximations fo (ar) fa and (x) 2 0. Hence the drifts point toward j (no-CFG) and wp (CFG), respectively, Using the Lyapunov functional V(x) - w the initial gap derivative satisfies (17) 0, so by continuity there exists t.1 E (0, 1) with (tt) for all [te1, 1). Thus, in the early regime, the CFG trajectory is (in expectation) closer to wit than the unguided one.
Proof. Define the quadratic functional V(x) = | - wjall3 and the gap
D(t) = ((x (CFO)) - v(x.
Clearly D(1) O since the two processes share the same random initial condition æ1. We first compute the time derivative of D. Under the standard local-Lipschitz and linear- grearr conditions satisfied by the fields in Lemma (see below), both trajectories have uniformly bounded second mo- ments on E [to. 1] and differentiation under the expectation is justified by dominated convergence:
D'(() = E
V(xi (CFG)) - E
= Ef V (1, - F(C,, (1,
dt
de
where (V = 2(x: wpt) and
(v) $ (x) x
(t, = 2((x w
a!/? (x) + w( (x) (x))
Letg (1, x==
2(x - wp)),
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Early-time expansions. By the closed forms in Lemma 1, for any bounded set there exist and to < such that uniformly for ||ж||| bounded and t [to, 1).
for (z)) = p (1, x),
(o)((a) ro(l,
(1, a) [|a(,,2)11 S d(-1) (1+ [++||=]])
(14)
Hence for such (t,, x)).
a (
# O(1 - -t,
(CFG)
(1 +1+ w)
+0(1 -t),
(15)
t
where the 0(1 -)) terms are uniform on bounded sets as / - 1.
The limiting derivative at t 1. Using equation 15 and the fact that z!! 21 in L as t 1, we obtain
D
= wp). -((1 + w) x1)) B[(2((1 - wji), -(p xz)}
Expanding the inner products and using [[x,]]
get
lim (1) 2w E (z) [3] 2w 1 2w 2 >
1-1-
Strict negativity on a full interval. By continuity of all terms in equation 14 lequatttoo 3 and uniform integrability (bounded second moments), there exists tel (0, 1) and constant C, such that
D'(t) 2 c. O. Vt E [). Since D(1) = and D' strictly positive on [le, 1), we integrate backward from conclude that
D(t) - -f.
ds < 0, Vt [ter,
This is exactly the desired inequality in expectation:
E ||,,
(CFG)
wpll ||=)(0) - w = D(I) <
Remark C.3. The proof only uses that ((()) (z) + f and (0)((x)) > uniformly on compact sets, together with bounded second moments of xy, Hence the statement extends to other conditional models admitting the same carly-time limits.
C.2 PROOF THEOREM 3.3
Theorem C.4 (Theorem 3.3 Consider a tvo-component Gaussian mixture
(a( y) = ni (1; o II) + T2 (Z; O'),
μ2,
|| = || 1. T2-
Under mild assumptions, there exist lig E (0, 1) and an w-indeppndent region Usg C R.d, depending only on (Tk: pk. such that for any wi if æt., E Us, and x, follows the CFG probability flow ODE equation 6 on [0, tag), then
TT M(x (1 t)p (62 (1- (1-1))=0011) > n Naxi t)p2, (12 (1+ Vt [0, toy].
Proof Sketch.
To eliminate the effect of scaling, we map the sampling trajectory
which corresponds to the VP-SDE parameterization of the diffusion model. Consider a separating hyperplane
S, = (æc Rdd (x Ce. l41 22) = 0),
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lying between 141 and J42, orthogonal to Au M1 M2z and positioned closer to the weaker class f1. By construction, the conditional ODE vector field on St is orthogonal to Áμ..
This hyperplane Sy has three key properties: 1. On St, the CFG-ODE drift points toward the weaker class p1. As t decreases, the location C₁ of St moves toward the stronger class μ2z- 3. The two regions divided by St differ in posterior mass: the side containing μ4 always assigns higher responsibility to the weaker component.
Combining these properties, conclude that if point Ft, lies in the region of S, containing μa some time lo. then the trajectory æ, will remain in that region for all S lo.
Proof. Let Ap:= := μ1 u2 and of = 12 + (1 t)²0². Introduce the rescaled state 2, := x0:((111) (1-1) and the scalar coordinate
u<() = (z, Ap)..
Then we prove that for
Dynamics along Ap. Under the CFG probability-foon ODE,
a olc (z,) zx),
1 (CFG)
(v) (1 w) Zoj+ + w 011"
(1))
For isotropic Gaussians, Tweedie's identity yields the shared affine form
b = d1(1)t) a
(1)) Zoje (x) = at (x)(x) + be x,
oi
where (x) rt, x) μr + (t, a)) u2 and rx x) x Th Nax (1 t) II 0 + 12 1.
Setting 2, = (1 - z; and simplifying gives
2
(v)
a₁
(zt p)(2))))
t(1
1)
tz
-: (1- () 2 + (1 (1))
Taking the inner product with Au and using the identity
#1t 2
Au = O, we define
A
(1-1)2
(u - c(t)) ) 20?
gr(u)= u
tanh
2
so that
(2/0) A14) = s(((2z)))
(16)
where
o²
c(I) = (1() log
v(() = (1 t)2
Zero-thrust hyperplane and U.. The zero-thrust hyperplane at time /. is
S (=: (=()))) 0, u(=) 0, 141, -122 < 0).
or equivalently
S. = {z u(z)) c
e, > O,
for some e, In other words, S, lies between μl and f41 positioned closer to f1 and it is precisely on this hyperplane that the conditional velocity field has zero projection onto the direction Au.
U. == 2 [(())) > et
ie.. the side of S, that contains fA.
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Claim A (direction of the CFG field St). On St we have
(CFG)
.Au (cnn) ( ES, t(1 t) 9r (u(z, ((()((()))))))))))))))))))))))) (w 1) (1 1)(12 + -1-1) (12) u(z;
u(z!crc) (CRG)
(
2
t
u(2, (cFC)) 0.
=0 - (w 1) (12 (1 - 1)2)
(17)
where u(z, (CFG) is because (z, Ap) 0 for 2 E St.
In words, on St the CFG vector field points strictly toward the weaker mode μ4 (in backward time).
Claim B (motion of St). We define
(e((()) = 0, (z, Ap) 00, (z f1, -22 < 0).
St =
or cquivalently
S: {z u(z) et), Ex > 0.
It follows that satisfies
p tanh
π2 0.
et
2(0² + (1-)) 44 - log
Since x² + +(I-g) is strictly increasing in 4, we conclude that
de dt > 0.
In other words, as decreases (i.e., in backward time), the hyperplane S, moves closer to the stronger mode /42-
Backward invariance and conclusion. Thus, by the moving-set viability (Nagumo) criterion applied to the time-reversed system, (Ut (Ee,,.)) is backward invariant: if 2t., E Ut, then 2, E U for all E t0, Finally, 2, E U, implies un c(t), hence
Ti N[;;; () o
म2 Naa; (1 11),,, oi1),
which proves the theorem.
Remark C.5. The mild condition mentioned in the theorem refers to the existence of Sg. In fact, for sufficiently small and w, such a set always exists. To see this, consider the regime where - 0 and a - 0, while the distance between 141 and μ2 remains non-negligible but not too large. this case we have the approximation
(c) Ro u AAul
for all u > 0.
Evaluating u(fi) gives
go (u(())) x (Mi Au) |لuا 0, while for sufficiently small we have ge( < O. Hence, by continuity, there exists some e such that
u(z;)= Ct. 0 < (mn 14) This ensures the non-emptiness of S, in the considered regime.
C.3 PROOF OF PROPOSITION
Proposition C.6 (Proposition 3.4 3.4) Let æ, evolve according to the CFG probability flow ODE equa-
tion O with guidance weight w 1. Then there exists a time 0 < tz such that, for any 1, can find radius r(k) with the following property: if
| - -kpl < r(k),
then for all < ts, holds that
Ti NNNs; (1-t)11y Moreover, the radius r(k) grows monotonically with k.
< Nax t)pu, (t2 (1-1))))0) t)30311).
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Proof Sketch. Consider the hyperplane
: far E Rd (x. p4 (22) 0}.
which is orthogonal to the vector Ap := p p22 One can show that along H, the probability-foow vector field always points toward the stronger mode μ2. Moreover, H partitions the space into two half-spaces, and the one containing μ2 consistently assigns larger posterior weight to the stronger component. Therefore, any trajectory initialized near ku22 that remains on the μ2l side of H can never cross into the weaker side. Equivalently, as long as a point does not exceed this separating boundary, it will remain in the region dominated by μ2 for all future times.
Proof. Let Ap = p22 P1 and j = T1 + T2f42 with m2 > TT. Fix & > 1 and define
(kk) = k Aμ
Ap
Consider any with || kull rk)). Then
x AJA 2 1: j Au || |اpl > ls j - -( 0.
Thus Br (kp) c {x : x Ap > 0}. i.c., the entire ball lies strictly inside the half-space where the second Gaussian component dominates.
Next, note that on the boundary (x : x' Ap = 0}, the CFG dynamics satisfy
a Ap 0,
which means the flow points strictly inward relative to the half-space {x x =TAu > 0}. By Nagumo's ' condition, this half-space positively invariant under the dynamics. Therefore, if æt., e (y (kj)) for some la close to 1, then remains in the region {æ x Au > 0} for all S lar
Finally, since r(k) grows lincarly in k the monotonicity claim follows immediately. This proves the proposition.
C.4 PROOF OF THEOREM 3.5
Theorem C.7 (CFG yields stronger within-mode contraction for small σ). Consider class- conditional Gaussian mixture with component k having covariance I²I,. Assume o² < 1, and the mixture sufficiently well-separated so that on ball Bg(t, r) == B (1 r) one has wr (1, x)) 1- and Σ₁ WWw,(t 2:]l < Cresp uniformly for all
(l, with 0 arbitrarily small by laking the separation large enough and tsg small enough. Let (CPG) (CFG) and x; 2
(e,
(1)) be the solutions of the CFG and conditional flows (same initial pair there exist Lag (0, 1) and 0 (depending on the mixture separation and σ) such that for all E (0,tes).
ar 2
in B& (lag: r) time lzg). Then for any guidance
(CFG)
(CFG) z,
< a (v)
2 (8)
x
Proof Sketch. When the dynamics are predominantly driven by a single mode 14,, the uncondi- tional probability-fon flow ODE reads
d. ----700 ( az.),
where (x)) is an affine combination of and Φt. For the difference between two trajectories, we have
dt (x
d (FF)) 2
(CFO))
d (z. (0)0)) (=(1)(xx)) (xx) (x.)
2(1) 2ofe (2x) + 201.(Zz))
dt
Since the samples are almost entirely governed by the same mode pss we can approximate
1 (())(z))) (xx) a.44: B
(of((z1) B.
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with 3r < 3. because the conditional prior is stronger (o Juncond). Subtracting the two estimators thus leaves a residual term -C with o 0. Hence,
(0 -ze) (CFG)
d (ate =i) (v)
7 (-c((21 - 21)).
Consequently,
(CHO)
_ | -0.l) (0)
d - zell
which establishes that CFG induces stronger contraction of pairwise distances within the same mode.
Proof. Fix k E {1,2} and work inside B& = B(( Du,,)) for & E [0, (g). For any two solutions of the same flow, let D; == ot 2 ( E (CFG))
(A) Explicit posterior differences. For single Gaussian N (Ju,, 2 we have the closed form
o²(1 ()
1
ofcc (x) c2 + Bipss at (1-11 t) Be
12 +1 2)202'
and for the unconditional prior ./O0, I), (z) و with 7/- =++1111p 1jz
1-t
In the mixture model,
ofjl (x) = 2)(t, [) = are + 3e Σ wj (t(t,
Hence, for any x, 2 and A == x-
G.( Σ us 1/ (Vu;)(=+ (A) ds. (18) Br G.(a, (.)
a (v) (((())=/((()) = a.
(B) Explicit ODE for A Using (())) (1, =
(1 - ( (a) x and a (CFO) - (CFG) (x)
z)/t with a 0,t (1 w Boj + we (v)
(CFG)
combining with equation 18 gives
0/2
A (v) I+ & 4.) A (CFG)
(1 (1-w)rtwar-- I with G. A (19)
(C) Two-sided differential inequalities for log | Let M = 2.
sup
[G,(,,,) =)[l
Then from equation
19 and
(t)-1b (t) M - dt
log |,بب| I (.()) + (t) M,
(20)
0a -
Br
((v)
b(w) t
(21)
w) + way
3z.
((CFG)
((CFG) = w t
(D) Pointwise gap. Subtract the upper bound for (CFG) from the lower bound for (v):
(d(y) M) (a(cra) + b(crG) (w 1) 2e ai
(1 ww+
81
M.
Using the small-t expansions
B. = o² O
" a = -(0-2-1)+((t))
t
we find
(a(w) - b()) - (a(cra) + M) ( 1w(o 2 1) - +0
Therefore, if 0² < 1 and M small enough so that
(w -
2 (1 W,
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then there exist Exs (0, 1) and > 0 with
-600-b6-0) ((CFG) (t) + (t) & - (6)
-ot,
AIE (0, tss)
(22)
(E) Concluding the comparison. Combining equation 21 and equation 22
d | log 1aa 2 for all / a0,
di log [1
Since the two distances agree at tes (same initialization), integrating backward in time yields | ||| (CFG)
|| for all t [0. t2y).
D DETAILMENT OF EXPERIMENT
In the experiments evaluating the impact early high guidance (with N 50 NFEs), we set low guidance weight 3 and the high guidance weight 9. In the early-high strategy, the weight switched from low to high after 20% of the iterations, whereas the late-high strategy, the adjustment made in the opposite direction.
In the experiments evaluating the impact of late high guidance (also with = 50 NFEs), we start from the same noise initialization and inject additional perturbation drawn from (00, 0.042) 20% of the iterations. For the constant-low schedule, the guidance weight fixed at 3 throughout. For the late-high schedule, ensure fairness, the weight is set to during the first 20% of the iterations, reduced to between 20% and 60%, and increased to from 60% to 100%.
conducted with Stable Diffusion v3.5 on the COCO 2017 validation set (5,000 captioned images). For each configuration, we generated 5,000 images at 1024 x 1024 resolution and evaluated them using FID, CLIP, ImageReward. and saturation, covering both quality and diversity, All runs used NVIDIA A L100-XXM4400GB GPUs with PyTorch and Hugging Face Di f fusers.
The experiments
We compared four settings: (i) vanilla-CFG with constant guidance w; (ii) time- varying CFG (TV- CFG), which weakens guidance in early/late steps and strengthens it mid-way; (iii) B-CFG (Malarz 2025), which utilizes the probability density function (PDF) of Beta distribution as the weight schedule; and (iv) interval-CFG (Kynkäänniemi et al., 2024), which executes CFG only within a specific interval.
1-11 the timesteps, and {wi, } the scales. The For TV-CFG, let N be the number of NFEs, [tm) schedule is
A
[N/2/ n w w-
n S [N/2],
Wta
A (1722) (N - n) + - n [N/2],
with s w 1. The factor A normalizes the average scale to match the baseline:
N Wen (tn tx+1) = w, ty = 0.
analogous to f dt For interval-CFG and 3-CFG, the values of the guidance scale within the CFG interval are also determined by normalization, and the relevant hyperparameters are set according to the recommendations in the original paper. Figure visualizes the schedules.
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25
vanilla-CFG interval-CFG
beta-CFG
TV-CFG
wl0
1.0
0.8
0.6
0.4
0.2
0.0
Figure 5: Vanilla-CFG, interval-CFG, 3-CFG and TV-CFG guidance scale settings at w 9.
Diversity evaluated by sampling 1,000 prompts from the COCO dataset. For each prompt, 16 images generated using different random seeds. LPIPS features extracted for all generated images, and diversity quantified as the mean squared pairwise distance between feature represen- tations. Larger values correspond higher sample diversity.
E ABLATION STUDY ON PEAK TIMING
In this section, we investigate the sensitivity of our method the peak location of the guidance schedule. Our theoretical framework describes three-stage dynamic: early direction shift, inter- mediate mode separation, and late contraction. The symmetric TV-CFG schedule used in our main experiments serves a minimal, theory-aligned instantiation. To demonstrate that the effective- ness relies on the general stage-wise mechanism rather than fine-grained hyperparameter tuning, we conduct an ablation study varying the peak timing.
We adjust the peak position the schedule to occur at 20%, 40%, 60%, and 80% of total sampling steps, fixing the total steps at NFE 10 and the guidance scale 7. We evaluate the performance across CLIP, IR, FID, saturation, and Diversity metrics.
Table Ablation study on the sensitivity of the peak timing location. Experiments were conducted with NFE = 10 and w = 7. The results demonstrate robustness when the peak lies within the intermediate regime.
Peak Timings
CLIP
IR f
FID
Saturation
Diversity
20%
0.317
0.752
31.669
0.401
1.2214
40%
0.319
0.940
28.770
0.279
1.2309
60%
0.319
0,935
27.722
0.229
1.1963
80%
0.319
0.896
27.092
0.201
1.1622
The quantitative results are presented in Table
We observe the following:
Stability in the Intermediate Regime: When the peak is located within the intermediate stage (40%-60%) the performance remains stable and optimal across most metrics. No- tably, CLIP scores saturate at their highest value (0.319), and IR reaches its peak at 40%.
Impact of Extreme Timings: Setting the peak too early (20%) or too late (80%) leads to expected degradation in specific areas. An early peak (20%) results in lower IR and higher FID, suggesting that applying strong guidance too soon hinders the quality of the samples. Conversely, late peak (80%) yields the lowest Diversity (1.1622) and Saturation, consis- tent with the theoretical prediction that delayed guidance restricts the available generation space.
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These findings confirm that the primary factor for success is the "low - high - low" structural shape predicted by our three-stage theory. The method is robust to the exact peak location, pro- vided it resides within the mode-separation stage. This supports conclusion that the stage-wise mechanism, rather than precise hyperparameter selection, governs the behavior of guided sampling.
F ADDITIONAL VISULIZATION RESULTS
G and present results under different prompts and scheduling strategies, further illustrating how strong early guidance undermines global diversity (NFE=50, w 9). Figures X 9 TO and
11 demonstrate the effect of the proposed time-varying schedule on the generated images (NFE=20,
w 9).
(a) Constant high weight
(b) Early low weight
(c) Early high weight
Figure 6: Comparison of guidance schedules on the prompt bike parked on side walk.". (a) Constant high weight and (c) Early high weight both exhibit directional collapse, with nearly all bicycles oriented in the same side-facing position.
(a) Constant high weight
(b) Early low weight
(c) Early high weight
Figure 7: Comparison of guidance schedules on the prompt "A man standing in front of a mirror in a room.". " (a) Constant high weight and (c) Early high weight both reduce diversity, with most samples converging nearly identical settings: man in dark shirt facing a tall rectangular mirror.
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Figure 8: Generated samples using CFG.
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Figure 9: Generated samples using TV-CFG.
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Figure 10: Generated samples using APG.
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Published as a conference paper at ICLR 2026
Figure 11: Generated samples using TV-APG.
26


Reviews:
[{"review_id": "q1mO1xziKK", "reviewer": "Reviewer_ksW2", "summary": "The paper presents a stage-wise theoretical analysis of Classifier-Free Guidance (CFG) in diffusion models under multimodal conditionals, identifying three distinct phases: Acceleration and Direction Shift (early), Intra-class Mode Separation (mid), and Concentration (late). It explains how CFG causes diversity loss via early-stage bias toward dominant modes and late-stage intra-mode contraction. The authors propose a time-varying guidance schedule that reduces guidance strength in early/late stages while emphasizing the mid stage, showing empirical improvements in diversity without sacrificing fidelity.", "questions": "Address the weakness, especially the novelty issue.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1.\tThe paper provides a principled theoretical foundation for time-varying guidance schedules—specifically, using low guidance strength in the early and late stages of sampling while applying stronger guidance during the intermediate phase—to simultaneously preserve diversity and maintain semantic fidelity.\n2.\tIt reveals when and how Classifier-Free Guidance compromises sample diversity by analyzing the interplay between multimodal conditional structure and temporal dynamics: early-stage bias toward the global mean suppresses weaker modes before they can emerge, while late-stage over-contraction erodes intra-mode variation. This offers a unified explanation for the well-known trade-off between alignment and diversity under CFG.", "weaknesses": "1. The proposed guidance schedule—weak guidance in early/late stages and strong guidance in the middle—is not new. Prior work such as β-CFG[1] has already identified a similar three-stage behavior of CFG and explicitly designed non-uniform guidance schedules based on this insight.\n2. The experimental comparison omits key adaptive guidance methods that also modulate ω over time. In particular, the paper should include comparisons against:Linear ramp-up followed by ramp-down schedules, β-CFG or other non-linear schedules.\n3. The paper relies heavily on ImageReward and saturation as proxies for diversity. However, ImageReward primarily measures aesthetic or semantic alignment rather than sample variance under the same conditioning, and lower saturation does not necessarily correlate with higher diversity (e.g., outputs could be desaturated yet nearly identical). The absence of explicit, widely accepted diversity metrics—such as LPIPS variance across multiple generations per prompt—weakens the empirical support for the claim that the proposed method better preserves diversity.\n4. Although the paper motivates TV-CFG through an asymmetric three-stage dynamic—comprising early direction shift, mid-stage mode separation, and late-stage concentration—the actual guidance schedule is symmetric with respect to the sampling step index and peaks precisely at the midpoint. The paper lacks ablation studies on the sensitivity of performance to the peak location (e.g., shifting the maximum guidance strength earlier or later).\n\n[1] Malarz D, Kasymov A, Zięba M, et al. Classifier-free Guidance with Adaptive Scaling[J]. arXiv preprint arXiv:2502.10574, 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VDevrZoREc", "reviewer": "Reviewer_2UBn", "summary": "This paper theoretically analyzes the effect of CFG on sampling dynamics and shows that its influence can be divided into three stages. In the Direction Shift stage (early in sampling), guidance pushes samples toward a weighted average of class means, potentially biasing the initialization toward specific generations. In the Mode Separation stage, samples are driven toward stronger modes, reducing the diversity of generated examples. In the Concentration stage, strong guidance further decreases diversity while improving alignment. Overall, the authors conclude that early strong guidance reduces global diversity, while late strong guidance suppresses fine-grained variations. This motivates the use of a time-varying guidance schedule (TV-CFG) that peaks in the middle, and the experiments demonstrate that TV-CFG improves the quality–diversity trade-off compared to standard CFG.", "questions": "1. Have the authors considered extending their analysis to other guidance methods, such as autoguidance [1]?\n\n2. How would the analysis differ if an SDE-based sampler were used instead of the ODE solver?\n\nI am open to reconsidering my rating after the discussion with the authors.\n\n[1] Karras T, Aittala M, Kynkäänniemi T, Lehtinen J, Aila T, Laine S. Guiding a diffusion model with a bad version of itself. Advances in Neural Information Processing Systems. 2024 Dec 16;37:52996-3021.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The paper studies the influence of guidance on sampling dynamics, which is an important aspect of modern diffusion models. Given the widespread use of CFG, further intuition in this domain is essential to better understand and improve its behavior. The work aims to bridge the gap between current empirical practices using CFG and the theoretical understanding of its underlying mechanisms.\n\n* The authors validate their findings not only on toy datasets but also on real-world text-to-image models.\n\n* The paper is well-written and easy to follow. The examples effectively illustrate each aspect of the discussion.", "weaknesses": "In my opinion, the main weakness of the paper is that its findings may not fully justify the novelty claim. Specifically, [1] studied the dynamics of CFG in Gaussian mixture models, showing that strong guidance improves classifier confidence (i.e., better alignment) while reducing the final entropy of the generated distribution (i.e., lower diversity). They also analyzed the influence of guidance across different stages of sampling (see Section 6 and Proposition 6.1). However, this reference is described by the authors as focusing only on unimodal distributions and “overlooking the multimodal nature of real-world tasks” (L42).\n\nMoreover, the benefits of time-varying guidance scales have already been explored in prior work from a practical standpoint, as acknowledged by the authors. Therefore, I believe the novelty claims require stronger justification to warrant a separate publication on this topic. While I appreciate the idea and analysis presented, the noted similarities with existing work affect my judgment unless the authors provide clearer differentiation. \n\n[1] Wu Y, Chen M, Li Z, Wang M, Wei Y. Theoretical insights for diffusion guidance: A case study for Gaussian mixture models. arXiv preprint arXiv:2403.01639. 2024 Mar 3.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LeykYbFuOT", "reviewer": "Reviewer_v1Re", "summary": "This paper presents a theoretical analysis of Classifier-Free Guidance (CFG) in diffusion models under multimodal conditional distributions. The authors identify three distinct stages in the sampling dynamics: Direction Shift (early), Mode Separation (middle), and Concentration (late), and explain how CFG leads to diversity loss through these stages. They validate their theory empirically and propose a time-varying guidance schedule.", "questions": "refer to weaknesses.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.  The paper provides a systematic analysis of CFG under multimodal conditional distributions, moving beyond restrictive unimodal assumptions prevalent in prior work.\n\n2. The paper successfully explains the long-standing empirical observation that strong CFG improves semantic alignment but reduces diversity.\n\n3. The experiments on Stable Diffusion v3.5 with COCO validation set appropriately test the theoretical assessments.", "weaknesses": "1.  In Related Work, the treatment of closely related concurrent papers, say, both the  theoretical or algorithmic perspectives on conditional sampling and CFG in diffusion is incomplete. \n    \n2. While experiments are run on a modern diffusion model and COCO captions, the diversity of datasets and generative tasks is limited. For example, all primary experiments use a single text-to-image model/configuration (Stable Diffusion v3.5 with 5,000 COCO images).  Discussion on how these findings generalize to other modalities (e.g., audio, video) or other conditional types (e.g., class or style transfer) are missing. \n    \n3. Although time-varying and interval guidance competitors are implemented (and TV-CFG is compared), the 'interval' baseline is only briefly described, and no discussion is given to recently proposed, possibly stronger or more customized adaptive schedules. Moreover, the ablation or sensitivity analysis on schedule shapes, peak timings, or trade-offs beyond the one setting is insufficient. \n    \n4. The central theoretical perspective is attributed as the 'first systematic analysis' of CFG under multimodal conditionals (see Abstract, Page 2, and Conclusion). However, several recent works—some cited, some missing[1][2][3], have begun to analyze guidance dynamics in GMMs or with alternative assumptions (Li et al. ICML25/ arxiv 2025a,  Jin et al. ICML25, Wu et al. ICML24). The authors should more carefully disentangle which aspects are genuinely new (e.g., the particular stage-wise decomposition or explicit formalism) as opposed to extended or sharpened results over antecedents. \n    \n5. The theory and results emphasize the effect of guidance at various sampling budgets (notably for low NFE), yet the paper lacks comprehensive ablation over NFE and guidance range. Only a select number of NFE and $\\omega$ values are tested, how does the approach performs at extremely low (NFE < 5) or much higher NFE values?\n    \n\n[1] W. Deng et al., *Reflected Schrödinger Bridge for Constrained Generative Modeling. 2024.* \n\n[2] X. R. Li. et al., *A Simple Approach to Unifying Diffusion-based Conditional Generation. 2024.* \n\n[3]  G. M. Favero, et al., *Conditional Diffusion Models are Medical Image Classifiers that Provide Explainability and Uncertainty for Free. 2025.*", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZYZsVWrNyO", "reviewer": "Reviewer_gN79", "summary": "The main contribution of this paper is the description of three distinct phases in the application of Classifier-Free Guidance (CFG). First, in a \"strong noise\" phase, CFG causes an *acceleration and direction shift*, pushing the distribution toward the global, class-weighted mean of the dataset. This essentially acts as an amplification process that inflates the norms of the trajectories. The authors claim this creates an initialization effect that biases the process toward lower-diversity samples.\n\nThe second phase is *mode separation*, where trajectories begin fanning out toward the different modes in the data distribution as noise decreases, settling into their basins of attraction. The issue here, as the authors argue, is that the initial stage has already biased the results, positioning trajectories to favor certain modes over others.\n\nFinally, the third phase is *concentration*, which occurs in the low-noise regime. Here, the dynamics are dominated by contraction, which suppresses intra-class variability. This results in samples that are sharper but less diverse.\n\nAdditional contributions of the paper include empirical validation of the theory and a time-varying guidance protocol for diffusion sampling.", "questions": "* Given the use of $\\omega > 1$ in Theorem 3.2, am I correct in the interpretation that trajectories are directed toward the class-weighted mean scaled *exactly* by the CFG scale? If that's the case, it seems like a result that should be more explicitly called out, especially given the large CFG scales often used in practice. (And if that's not the conclusion here, it would be a confusing reuse of notation.)\n* The authors’ theoretical analysis relies on the countable, discrete components of a GMM. This setting is well-suited to class-conditional models but less obviously applies to uncountable, continuous conditions (e.g. CLIP embeddings). What is the authors' perspective on how to extend their analysis to the continuous-condition domain?\n* Please also address the issues described in the \"Weaknesses\" section.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Overall, my impression of the paper is quite positive.\n\n* The paper is well-written and well-presented. The illustrations, especially in Figure 1, are very helpful for understanding the different stages the authors discuss.\n* A significant contribution of this work is that it lends some theoretical muscle to analyzing CFG. CFG is a technique we all use, but it's essentially a hack that works without a tremendous amount of theory to motivate it. It is valuable to see an analysis from a dynamical perspective, especially as the field moves more toward ODE representations of the diffusion process. This is a genuine contribution to the literature and provides food for thought for additional theoretical development.\n* The theoretical analysis itself, which models the data as a Gaussian mixture, is general enough to provide useful insight, seems sound, and provides a good foundation for the claims. However, see the caveats introduced below.", "weaknesses": "* I occasionally struggled to keep track of some of the theoretical results in the theorems. In some cases, the meaning or interpretation of certain variables wasn't entirely clear. For instance, in Proposition 3.4, a parameter $k$ is introduced that determines a radius. It's later stated that $k$ increases with the CFG weight $\\omega$ (line 273), but it was not completely clear to me why that is the case. It was also not clear to me what \"mild assumptions\" were being made in some of the theorems. To the extent possible without introducing too much clutter, making the theoretical statements more self-contained would be helpful.\n* In the experimental results, the authors' proposed method is referred to as \"TV.\" While it can be inferred that TV means “time-varying,” the paper does not explicitly define what \"TV\" stands for immediately before abbreviating it (unless I missed it). So when looking at the experimental tables, I wasn't sure at first if the \"TV\" results belonged to the authors or someone else.\n* The authors refer to the difference between \"global diversity,\" which is negatively affected by strong early guidance, and \"fine-grained diversity,\" which is reduced by strong late guidance. While I have an intuitive sense of what the authors mean by this terminology, I would like to see a more precise definition of these two concepts.\n* The paper would benefit from discussing its findings in the context of some other relevant work:\n1) The CADS method [1] uses a condition-annealing strategy during inference, where noise is added to the conditioning signal early on in sampling and is then gradually reduced to reveal an unperturbed condition. It would be interesting to see the authors reconcile their theoretical results with that technique. I would also refer to the \"fine-grained detail\" argument the authors make to see how it reconciles with the results reported in the CADS paper.\n2) Two recent papers [2, 3] describe the *biphasic* nature of diffusion sampling, which are distinguished by the relative influence of the conditioning signal. It would be interesting to have the authors discuss their \"triphasic\" findings in the context of this other work.\n* While I think the authors' modeling choices allow for insight into the more general conditional diffusion sampling problem, the theoretical results still rely on what is essentially a toy setting. To the authors' credit, their empirical results operate on real in-the-wild diffusion models. However, a more frank acknowledgement and discussion of the limitations of the theoretical results is needed.\n* I noticed very few (but nonzero) typos in the paper (e.g. an extra period in the caption for Figure 2, using *assumption* where *assumptions* is intended in one theorem statement, etc.). One more reading pass, especially from a fresh-eyed reader, should help eliminate them. \n\n[1] Sadat et al. (2023). CADS: Unleashing the Diversity of Diffusion Models through Condition-Annealed Sampling\n\n[2] Balaji et al. (2022). eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers\n\n[3] Liu et al. (2024). Faster Diffusion: Rethinking the Role of Temporal Attention in Diffusion Models", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)