Alleviating Exposure Bias in Diffusion Models through Sampling with Shifted Time Steps
Authors: Mingxiao Li
Tingyu Qu
Ruicong Yao
Wei Sun
Marie-Francine Moens
Venue: ICLR 2024 poster (2024)
Keywords: diffusion models
sampling methods
exposure bias
Primary area: generative models

Abstract: Diffusion Probabilistic Models (DPM) have shown remarkable efficacy in the synthesis of high-quality images. However, their inference process characteristically requires numerous, potentially hundreds, of iterative steps, which could exaggerate the problem of exposure bias due to the training and inference discrepancy. Previous work has attempted to mitigate this issue by perturbing inputs during training, which consequently mandates the retraining of the DPM. In this work, we conduct a systematic study of exposure bias in DPM and, intriguingly, we find that the exposure bias could be alleviated with a novel sampling method that we propose, without retraining the model. We empirically and theoretically show that, during inference, for each backward time step t and corresponding state ˆxt, there might exist another time step $t_s$ which exhibits superior coupling with $\hat{x}_t$. Based on this finding, we introduce a sampling method
named Time-Shift Sampler. Our framework can be seamlessly integrated to existing sampling algorithms, such as DDPM, DDIM and other high-order solvers, inducing merely minimal additional computations. Experimental results show our method brings significant and consistent improvements in FID scores on different datasets and sampling methods. For example, integrating Time-Shift Sampler to F-PNDM yields a FID=3.88, achieving 44.49% improvements as compared to F-PNDM, on CIFAR-10 with 10 sampling steps, which is more performant than the vanilla DDIM with 100 sampling steps. Our code is available at https://github.com/Mingxiao-Li/TS-DPM.

Published as a conference paper at ICLR 2024
ALLEVIATING EXPOSURE BIAS IN DIFFUSION MOD- ELS THROUGH SAMPLING WITH SHIFTED TIME STEPS
Mingxiao Li Tingyu Qu" Ruicong Yao2, Wei Sun' & Marie-Francinn Moens!
Department of Computer Science, KU Leuven
2 Department of Mathematics, KU Leuven
(mingxiao. 11, tingyu. qu, ruicog. yao, sun si mww.mm.eemmmmeuleeeeeee...
ABSTRACT
Diffusion Probabilistic Models (DPM) have shown remarkable efficacy in the synthesis of high-quality images. However, their inference process characteris- tically requires numerous, potentially hundreds, of iterative steps, which could exaggerate the problem of exposure bias due to the training and inference dis- crepancy. Previous work has attempted to mitigate this issue by perturbing in- puts during training, which consequently mandates the retraining of the DPM. In this work, we conduct a systematic study of exposure bias in DPM and, in- triguingly, find that the exposure bias could be alleviated with novel sam- pling method that we propose, without retraining the model. We empirically and theoretically show that, during inference, for each backward time step and cor- responding state ét, there might exist another time step t, which exhibits supe- rior coupling with ž,. Based on this finding, we introduce sampling method named Time-Shift Sampler. Our framework can be seamlessly integrated to ex- isting sampling algorithms, such as DDPM, DDIM and other high-order solvers, inducing merely minimal additional computations. Experimental results show our method brings significant and consistent improvements in FID scores on differ- ent datasets and sampling methods. For example, integrating Time-Shift Sam- pler to F-PNDM yields a FID=3.88. achieving 44.49% improvements as com- pared F-PNDM, CIFAR-10 10 with 10 sampling steps, which is more perfor- mant than the vanilla DDIM with 100 sampling steps. Our code is available at nttps://githun com /Mi /ww.//iiiniiooo/ Li/TS-
INTRODUCTION
Diffusion Probabilistic Models (DPMs) (Ho et al. 2020 Sohl-Dickstein al.) 2015) class
of generative models that has shown great potential in generating high-quality images. Dhariwal & Nichol 2021 Ramesh al. 2022 Rombach al. 2022a Nichol et al. 2022). DPM consists of a forward and a backward process. In the forward process, images are progressively corrupted with Gaussian noise series of time steps. Conversely, during the backward process, the trained diffusion model generates images by sequentially denoising the white noise.
Despite its success in generating high-quality images, DPMs suffer from the drawbacks of pro- longed inference time. Considerable interest has been expressed in minimizing the number of in- ference steps during the sampling process to speed up generation, while preserving the quality of generated images. Such works include generalizing DDPM (Ho al.) 2020) to non-Markovian processes (Song et al., 2021). deriving optimal variance during sampling (Bao et al. 2002b)), thresh-
olding the pixel values as additional regularization (Saharia al. 2022a). developing pseudo numerical methods for solving differential equations on manifolds (Liu et al. 2022a).
Though showing promising performance, none of them have theoretically and empirically examined the discrepancy between the training and sampling process DPM. If we take a closer look at the training of DPM, at each time step f, ground truth samples zo are given produce corrupted samples H, with noise 6,- The DPM takes both I2 and as input to predict the noise 6,- During
Equal Contribution

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Published as a conference paper at ICLR 2024
X
Xt
X ct
DDPM
Xt
XT-1
XT
(xt,.)
(g)
TS-DDPM
w/2
P
Value
Selection Window
Figure 1: The comparison of 7S-DDDM (ours) and DDPM. The orange and blue arrwws denote the time-state coupling at each denoising step of 7S-DDPM and DDPM, respectively. In 7S-DDPM, we search for coupled time step within the w/2. & w/2) window, until the cutoff time step te.
sampling, one is required to synthesize data samples from white noise without the knowledge of the ground truth distributions. Coupled with the prediction errors of the network, this waining-sampling discrepancy produces errors that could progressively accumulate during inference. This error arising from the difference between training and inference resembles the exposure bias problem as identified in autoregressive generative models (Ranzato al. 2016 Schmidt 2019), given that the network is solely trained with the corrupted ground truth samples, rather than the network predicted samples.
In this work, we focus on the exposure bias problem during sampling. Ning al. (2023) propose to add perturbation to training samples to alleviate the exposure bias problem, which is sub-optimal since the retraining of DPM is computationally expensive. Given that the time step is directly linked to the corruption level of the data samples, we theoretically and empirically show that by adjusting the next time step current generated samples, one effectively alleviate the exposure bias. We search for such time step within a window tu surrounding the current time step to restrict the denoising progress. Furthermore, based the error patterns that the network makes the training samples, propose the use of a cutoff time step tc. For time steps larger than fe, we search for the suitable time step within 4. While for time steps smaller than lc, we keep the original time step. Intuitively, it also suits the nature of a DPM, since the corruption level is smaller for small time steps. We refer to sampling method Time-Shift Sampler. Figure T presents the comparison between DDPM, the stochastic sampling method, and its time-shift variant TS-DDPM. In summary, our contributions are:
during sampling according to the approximated variance of the
We theoretically and empirically study the exposure bias problem of diffusion models, which is often neglected by previous works.
sampling method called Time-Shift Sampler to alleviate the exposure bias problem, which avoids retraining the models. Our method can be seamlessly integrated into existing sampling methods by only introducing minimal computational cost.
We propose
Our Time-Shift Sampler shows consistent and significant improvements pling methods on commonly used image generation benchmarks, indicating the effective- ness of framework. Notably, our method improves the FIDD score on CIFAR-10 from F-PNDM (Liu et al. 2022a) by 44.49% to 3.88 with only 10 sampling steps.
various sam-
2 INVESTIGATING EXPOSURE BIAS IN DIFFUSION PROBABILISTIC MODELS
In this section, we first give a brief introduction of the training and inference procedure for Diffusion Probabilistic Models (DPMs). Then we empirically study the exposure bias problem in DPM by diving deep into the training and inference processes.
2.1 BACKGROUND: DIFFUSION PROBABILISTIC MODELS
DPM encompasses forward process which induces corruption in data sample (e.g., an image) via Gaussian noise, and corresponding inverse process aiming to revert this process in order to generate an image from standard Gaussian noise. Given data distribution g(z)) and forward
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noise schedule Bn 10,1),t 1...T, T, the forward process is structured Markov process, which can be expressed as:
T
( r = II (=) 1=1
(1)
with the transition kernel g( x4-1) ( 3,T), where I denotes the identity matrix, og and Bg are scalars and ag St. With the reparameterization trick, the noisy intermediate state can be computed by the equation below:
Vatoo -
(2)
where a T or and & (0, I). According to the conditional Gaussian distribution, have the transition kernel of backward process as:
1111-1
(2,,
where 3, = - B, and j4: = V /..19 9
+ yorl-a 1) x₁. Considering Equation can be vI-a.
1-o, 1-0 (1). During training, a time-dependent neural network further reformulated jū, /or (#+-
optimized by either learning t4 or <.. Empirically, Ho et al. (2020) observe that predicting e works better. The learning target is to optimize the variational lower bound of the negative log-likelihood, which could also be interpreted as minimizing the KL divergence between the forward and backward process. In practice, Ho et al. (2020) further simplify the loss function as:
siimple E.,. [1), f) <2 <al
We present the training and sampling algorithms of the original Denoising Diffusion Probabilistic Models (DDPM) (Ho et al. 2020) in Algorithm and respectively.
Algorithm Training
Algorithm 2 Sampling
repeat
1: Ty ~ (o, for T. T.... ....... do
2: To ~ g(za)
3: t~ Uniform(1.-.. T)
3: 2 ~ N/.. if > 1, else = 1 4: T:-1 vai (It vi-ai co(xx, t)) +ozz
4:
E~ I), Compute using 2
5:
Take gradient descent step on
5: end for
6:
7:
until converged
6: return 20
2.2
THE EXPOSURE BIAS IN DIFFUSION PROBABILISTIC MODELS
In this section, we empirically demonstrate the phe- nomenon related to the exposure bias problem in DPM using CIFAR-10 10 dataset (Krizhevsky) 2009). We first present the variance distribution of the cor- rupted samples by different time steps in the forward process during training. Models exposed range of inputs during training tend to exhibit greater robustness noise and are consequently less suscep- tible to exposure bias. To further study the behavior of the network during the backward sampling pro- cess, we also examine the evolution of prediction er- rors during sampling. We use DDIM (Song et al.) 2021) sampler to conduct this experiment, it gives a deterministic sampling process.
Var(s.)
wider
0.4
200
400
600
s00
1000
Steps
Figure 2: The density distribution of the vari- ance of 5000 samples from CIFAR-101 by dif- ferent time steps.
Figure 2 presents the changes in variance of sample distributions for different time steps. At each step.
estimate the corrupted image samples with the network predicted noise using Equation 2 We present the details of the figure in Appendix B At time step O, ground truth images serve as the current samples. The distribution of the variance of the ground truth samples spans an approximate
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range of (0, 0.8), showing the diversity of the sample distributions. As the noise is gradually added to the ground truth samples, the span of the variance becomes narrower and narrower. Following 400 steps. the changes in the span range of the variance become stable, and gradually shift towards narrow range surrounding the variance of white noise. The evolution of the sample variance across different time steps indicates that the network exhibits lower sensitivity forward process of DPM, as the variance of the samples can be distributed more sparsely within broader range. Conversely, the network can be more sensitive the later steps (e.g., after 400 steps), as we progressively approach white noise. The constricted variance range during the later stages implies that minor prediction errors during sampling can significantly impact overall performance.
the early steps of the
1.0
0.20
557
(a) 10 steps
(h) 20 steps
(c) 50 steps
(d) 100 steps
Figure 3: CIFAR-10 prediction errors of training samples for different numbers of sampling steps.
In the second experiment, given specific number of sampling steps, compute the mean squared errors between the predicted samples and the ground truth samples at each step, as presented in Fig- ure 3 Details for plotting this figure presented in Appendix [ It can be seen that the evolution of prediction errors adheres to a consistent pattern: initially decreasing before incrementally accu- mulating the sampling process progresses. This phenomenon may be attributed to possible factors: (I) The sampling process originates from white noise, which does not contain any informa- tion about the resultant sample distributions. In early stages, with fewer sampling steps, the error accumulation is less serious. thus the network gradually shapes the predicted distribution into the target distribution. (2) In the later stages, the network more robust the noisy inputs as discussed above. However, due to the exposure bias, the network inevitably makes errors at each step and these errors accumulate along the sequence, resulting in slow but steady progress in error accumulation and larger errors in the end.
In conclusion, the above two experiments together show that during the backward sampling process, the accumulated prediction error, which arises from the exposure bias and the capability of the network, could strongly impact the final results. This demonstrates the importance of alleviating exposure bias in DPM, which could potentially lead to improved results.
3 ALLEVIATING EXPOSURE BIAS VIA TIME STEP SHIFTING
In the backward process of Diffusion Probabilistic Models (DPM), the transition kernel assumed to adhere to Gaussian distribution. To maintain this assumption, the difference between two suc- cessive steps must be sufficiently small, thereby necessitating the extensive training of DPM with hundreds of steps. As previously discussed, the network prediction error coupled with discrepancy between training and inference phases inevitably results in the problem of exposure bias in DPM. We introduce C(ă,, t) referred which can be expressed as:
the input couple for trained DPM-to describe this discrepancy,
Clāe, () - (i,
(5)
where ãt and represent the network input and ground truth states at time step f, respectively, and dis(-.)) denotes the Euclidean distance. Consequently, during the training phase, the relationship C(z,,t) = I holds true for all time steps, as the network always takes ground truth Moreover, a better coupling expressed by C(ă,, tz) reduces the discrepancy between training and inference, thereby alleviating exposure bias. Previous works (Zhang et al. 2023a| Ning al.) 2023) have empirically and statistically affirmed that the network prediction error of DPM follows nor-
input.
mal distribution. In conjunction with Equation 2 during the backward process at time step the
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predicted next state denoted Æt-1 could be represented as:
-1 -t-1 + 11-1
Va-1200 v 02-141-1 9c-10e-1
(6)
V 1:00 + A
In this equation, A²-1 0i-1 + (1 dx-1), 21-1 denotes the ground truth at time step t - 1, 01-16t-1 represents the network prediction errors, and 6t-1, ct-1 and é- conform to normal distribution. Upon observing that Equation O and Equation share a similar structure which the ground truth Zo plus Gaussian noise with variable variance, propose the subsequent assumption.
Assumption 3.1 During inference at time step i, the next state āt-1 predicted by the network, may not optimally align with time step t 1-1 1 within the context of the pretrained diffusion model. In other words, there might be an alternate time step t, that potentially couples better with it-1;
3l. {1 (T...T). s.l.
C(5 C(
900 Step
800 Step
700 Step
600 Step 500 Step
400 Step
300 Step 200 Step
100 Step
Step
+-4
+33
Time Step Shift
+2
+1
-1
-2
5.
01& 0
00A, 0
((A
CEAL 0
CIA U
((R,0)
CtR, 0)
CLA. U
Figure 4: The training and inference discrepancy of DDIM with 10 sampling steps on CIFAR-IO.
The dashed line in each column denotes the couple of predicted â and t. Points on the right side of the dashed line mean that the corresponding time steps couple better with it than time step t.
To verify our assumption, we initially conduct a statistical examination of the discrepancy between training and inference pretrained diffusion model. A sample size of 5000 instances was ran- domly selected from the CIFAR-10 10 training set and Equation 2 was utilized to generate ground truth states 24 for sequence of time steps. Subsequently, we compare the C(.,, t) with C(+,, t.). Details of plotting this figure are presented in Appendix B Here we show the results of 10 inference steps in the backward process and only consider time step l, within
range of Figure 4 for certain backward steps, there alternate time steps t, that display stronger correla- tion with the predicted next state é compared to time step t. We also observe that when approaching the zero time step, all nearby time steps converge to the same distribution. Similar findings observed for other pretrained DPMs on different datasets including CelebA with varying settings, such different numbers of inference steps and ranges of f.. See Appendix E for details.
to L + 4. As depicted in
Our empirical findings lend substantial support to Assumption 3.1 This naturally prompts the ques- tion: How can we identify the time step that best couples with the predicted â.-1? By optimizing the KL divergence between the predicted Ω and Ht, at time step lg. we arrive at the following Theorem 3.1 with the complete derivation provided Appendix J.1
Theorem 3.1 Let in represent given state and ê,-1 represent the predicted subsequent state. We assume is sufficiently large such that the distribution of it-1 is still close to the initialized normal distribution with diagonal covariance matrix. In addition, the selected time step F, to couple with - is among those time steps closely surrounding have the following variance:
Then the optimal tg should
||e|
T₁, o ddd -1
(8)
where d is the dimension of the input, e represents the network prediction error, and 0:-1 is the variance of the predicted 8T-1-
If is close to then vōr-1 Voe, so O. See details in Appendix J.1
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The derivation of Theorem mainly follows two steps: Firstly, we optimize the KL divergence between 2t, and 8 to obtain the variance of zt,, Secondly, we establish the relationship between the variance within a single sample of - and the variance of 2 The results articulated in Theorem 3.1] could be further simplified to ot, x 0t-1, when t is large and given the assumption that the network prediction error the current time step is minimal. This assumption has been found to hold well in practice.
Based on the findings of Theorem 3.1 we pro- pose the Time-Shift Sampler, a method that can be seamlessly incorporated into existing sampling algorithms, such as Denoising Dif- fusion Implicit Models (DDIM) (Song et al.] 2021). Denoising Diffusion Probabilistic Mod- els (DDPM) (Ho al. 2020) or sampling meth- ods based on high-order numerical solvers (Liu ct al. 2022aj. Moreover, in light of the ear- lier discussion that the model's predictions of nearby time steps tend to converge to the same distribution during the later stage of the in- ference process and the condition of large L in the derivation in Appendix J.I we remove the time-shift operation when the time step is smaller then predefined cutoff time step Our
Algorithm 3 Time-Shin Sampler
1: Input : Trained diffusion model cØ: Win-
dow size w; Reverse Time series {T,T
1.... 0): Cutoff threshold Lc
2: Initialize: IT ~ N 1) :4====
=-
for
1,, do
4:
If * -1 then Enext = t, else tnext = t 4 = co (Z+, (neert)
6:
take a sampling step with knert to get 12-1
7:
if > te then
8:
Get variance for time steps within
the window: Σ = {1 Tz-w/22 1
Ck 0t-w/2+1,'' ..
(6tw/2)
to arg min, ||var (Ix-1) orll for
algorithm is detailed in Algorithm 3] Specifi- cally, given trained DPM co. we sample for 10:
o, E Σ and r E (t - w/2, t + 2/2]
else
Fr, ( = 1, 2, ..., T using arbitrarily any sam- 11:
t, -
pling method. For each time step >> les where 12: end if
te is the cutoff threshold, we replace the next 13:
end for
time step fnzt with the time step to that best 14: return x0 couples with the variance of 2 within win-
dow of size w. In the search of such f.,, we first take a sampling step with the current Ineat to get Tt-1- 1- which is used to compute the variance of æ;-1 as var(x,-1)- Then get the variance E of the time steps within the window. The optimal 6, can be obtained via arg min, [|var(z, for a, E E. Finally, the obtained is passed to the next sampling iteration as bneant. We repeat this process until < fe, after which we perform the conventional sampling steps from the sampling method of choice.
or ||.
4 EXPERIMENTAL SETUP
We integrate our Time-Shift Samplers to various sampling methods including DDPM (Ho et al. 2020): the stochastic sampling method: DDIM (Song et al.) (02T): the deterministic version of DDPM: S-PNDM (Liu et al. 2022a):: the sampling method based on second-order ODE solver; and F-PNDM (Liu et al.) 2002aa: the sampling method based on fourth-order ODE solver. We term our Time-Shift Sampler as .. with respect to the bascline sampling methods. For example, the time shift variant of DDIM is referred to TS-DDIM. Following DDIM, consider two types of time step selection procedures during sampling, namely uniform and quadratic. For t4 < T: (1) uniform: select time steps such that f; [ci], for a constant value c. (2) quadratic: we select time steps such that &4- = [ci²] for a constant value c.
We report main results using pre-trained DDPM on CIFAR- 10 (Krizhvsky 2009 and CelebA 64 (Liu et al.] 2015). Moreover, based on DDIM sampler, a comparison to ADM-IP (Ning et al. (023) is made, which uses ADM (Dhariwal & Nichol 202T) the backbone model. More experiments can be found in the appendix. We conduct experiments for varying sampling time steps, namely 5, 10, 20, 50, and 100. We use the Frechet Inception Distance (FID) (Heusel et al., (017) for evaluating the quality of the generated images. We further discuss the influence of window sizes and cutoff values in Sec. 5.4 More details can be found in Appendix C
Thhe latter refers to the variance computed by considering corresponding elements across samples of ft-1-
We discuss the influence of the window size and the cutoff value in Sec. 5.4

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5 RESULTS
5.1 MAIN RESULTS
Dataset
Sampliny Method
steps
deps
20 steps
50
100 sleps
DDIM (quadranic)
41.37
13:70
6.91
4.71
4.23
IS-DDIMiquodrutic
38.09 (+8.37%)
11.93 (+12299%)
(+11.43))%
4.16 (+11.68%)
3.81 (+9.93%)
(InM(u)foom)
44.60
18.71
11:05
7.09
5.06
TS DDIManiform)
35213.221
12.21 (+34.74%)
8.03 (+27333))
5.56 5
4.56 (+19.43%)
DDPM (uniform)
83.90
4204
24.00
14.76
10.66
CIFAR-10
TS DDDM (smifoorm)
67.06 (+20.07%)
33.36 (+21.65%)
22.21 (+9.72%)
13.64 (+7.59% )
9.69 (+9.10%
S-PNIDM (uniform)
9.49
5:37
3.74
3.71
TS S. PNDM (unifrarm)
3.14
4.42 (+17.69%%
3.71 (+0.80%)
3.60 (+2.96%
PENDMM (anijorm)
31.30
6.99
4,34
3.71
4.03
TS-F.PNDI (unifrorm)
31.11 (+4.07%)
3.88 (+44.49%)
3.60 (+17.05%)
3.56 (+4.04%)
3,86 (+4.22%)
DDIM (quadratic)
27.28
10.33
6.34
5.20
4,96
TS IS-DDIM (quadratic)
24.24 (+11.14%)
9.36 (+14.36
5.08 (+22.325%)
4.20 (+1..23%)
4.18 (+15.73%)
DDIM tuniform))
24.69
17.18
13:50
9.12
6.00
IS-DDIM (uniforne)
21.32(+133655)
10.61 (+38.24%)
7.01 (+48.30%%
5.29 (+42.000%
6.50 (+1.52%)
CelehA
DDPM (uniform)
42,83
34.12
26.02
18.49
13.90
TS DDPM (snifform)
33.87 (+20.92%)
27.17 (+20.37%)
20.42 (+21.52%)
13.54 (+26.77%)
12.83 (+7.70%)
PNDM (uniform)
3K.67
11.36
7.51
5.24
4.74
78.5.PDDN (unifrrmm)
29.77 (+23.02%)
10.50 (+7.57%)
7.34 (+2.26%)
5.03 (+4.01%)
4.40 (+7.17%)
P.NDMM (aniform)
94.94
9.23
5:91
4.61
4.62
FS.F.PNDM (uniffarm)
94.26 (+0.72%)
6.96
(+24.59%%
5.84 (+1.18)
4.50 (+2.39%%
4.42 (+4.33%)
Table I: Quality of the image generation measured with FID 1 on CIFAR-10 (32x32) and CelebA (64x64) with varying time steps for different sampling algorithms.
In [ compare four sampling methods, namely DDIM, DDPM. S-PNDM and F-PNDM, and their Time-Shift variants on two datasets, where we vary the time steps and the time step selection procedures. We take larger window sizes for fewer sampling steps. The cutoff value is within the range of [200, 400], and is dependent to the number of time steps we take. As expected, our Timer-Shift Sampler consistently improves the quality of the generated images with respect that of the baseline sampling methods. We observe that for time steps less than 20 with uniform time step selection, the performance improvements are extremely significant as compared the original sampling methods. To our surprise, even for very strong high-order sampling methods, our Time- Shift Sampler can still bring significant improvements. Notably, we obtain FID=3.88 with TS-F- PNDM CIFAR-I0 using 10 sampling steps, which is better than DDIM CIFAR-10 for 100 sampling steps. Our method also manages to improve the performance of the baseline sampling methods for both uniform and quadratic time selection, showing the versatility of our method.
5.2 DISCUSSION ON EFFICIENCY AND PERFORMANCE
Our Time-Shift Sampler involves searching for suitable time steps using computed variance of the generated samples, which inevitably brings additional computation during sampling. To compare the efficiency of different sam- pling methods, methods, we present the average sampling time" in Figure 5 by running each sampling method 300 times CIFAR-10 using DDPM as backbone model. For all three sampling methods, the additional computation time dur- ing sampling with 5.10 and 20 sampling steps negligible. The additional computation time is visually larger for 50 sampling steps. Yet, the actual additional computation time remains ac- ceptable for small backbone like DDPM. For example, 7S-S-PNDM requires 9.71% more sampling time average than S-PNDM for 5,10 and steps. For 50 steps, 7S-S-PNDM
DDIM
40
TS-DDIM
S-PNDM
TS-S PNDS
Score
30-
F-PNDM
TS-F-PNDM
FDD
20
50
100
150
200
Sampling Time (ms)
Figure 5: Sampling time VS FID on CIFAR-10 10 using DDPM as backbone with various sampling methods. We report the results of (5.10,20,50) sampling steps from left right for each sampler, denoted with "x" symbol.
"Test conducted AMD EPYC7502 CPU and RTX 3090 GPU using Pytorch time estimation API.

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requires 9.86% more sampling time than S-PNDM. We report the detailed sampling time in Ap- pendix F The additional computation time is also dependant to the choice of the backbone, which further elaborate in Sec. 5.3
5.3
COMPARISON WITH
METHOD ADM-IP
Model ADM
Sampling
steps
steps
steps
steps
DDIM
28,98
12.11
7.14
4.45
ADM-IP ADM
DDIM
50.58 (-74.53%)
20.95 (-73.00%)
7.01 (+1.82%)
2.86 (+35.73%)
7S-DDIM
(+7.04%)
10.73 (+11.40%
5.35 (+25.07%)
3.52 (+20.90%%)
Table Performance comparison on CIFAR-I0 with ADM and ADM-IP as backbone models.
Our method is also proven to be effective al- ternative model architectures than DDPM. In this section present the results ADM (Dhariwal & Nichol 2021), and the variant of ADM named ADM-IP (Ning al. 2023), which tries to allevi- ate exposure bias in DPM by retraining ADM with added input perturbation. We conduct our Time- Shift Sampler using ADM as backbone model to show the merits of our method compared to ADM-IP. We present the comparison on FID) scores in Table 2 and the comparison on sampling time Figure 6 The results indicate that when employ- ing a small sampling step, which is favored in prac- tice, ADM-IP performs much worse than the original ADM. As the number of sampling steps increases, the performance of ADM-IP improves. We manage to improve the performance of the ADM model with our method by a large margin. Note that we achieve these significant improvements without retraining the model as is required by ADM-IP. We also obtain roughly the same sampling time for DDIM and TS-DDIM with ADM as the backbone model. makes our method more favorable in practice given merely zero additional cost for com- putation and significant performance improvements.
00
ADM DRIM
ADM w/ TS DS-DDIM
FID Score
Sampling Time (ms)
Figure 6: Sampling time VS FID on CIFAR- 10 with ADM as backbone using DDIM and TS-DDIM for sampling.
5.4 INFLUENCE OF WINDOW SIZES AND CUTOFF VALUES
13 Steps
Stops
20 Stess
255
Stago
120
WindOW
WincER:
200
CLEaiT Nats
212 210
CutrfT
Figure 7: FID of generated CIFAR-10 images using TS-DDIM (uniform) with (a) various win- dow sizes using cutoff value=300; (b) various cutoff values using window size= (40:30:8:2) for (10:20;50;100} steps.
We first conduct study on the effect of window sizes. The results are presented in Figure 7 (a), where we fix the cutoff value=300, and vary the window sizes for different numbers of sampling steps. In the assessment of 10 and 20 sampling steps, larger window sizes evaluated to ensure
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adequate search space for suitable time step. And adopt smaller window sizes for more sam- pling steps, given the limitation of step sizes. Figure 7 (a) illustrates that the Time-Shift algorithm is not sensitive to the selection of window size when using 10, 20 and 50 sampling steps. While when the number of sampling step is increased, such as 100 steps, the algorithm become more sen- sitive to the window size, which could be attributed to the enhanced accuracy of per-step predictions achievable with smaller step sizes.
The influence of different cutoff' values on the image generation performance on CIFAR-10 are presented in Figure (b). From the density distribution plot of the sample variance as discussed in Sec. 2.2 we can have a good estimation of the range of the cutoff values to be within [200, 400). While sampling with 10 steps, a smaller cutoff value (200) is preferred as compared to the scenarios with more sampling steps. One possible reason is that fewer sampling steps lead to larger step size, which makes more time-shift operations beneficial.
6 RELATED WORK
Denoising Diffusion Probabilistic Model. The denoising diffusion probabilistic model (DDPM) was first introduced by [Sohl-Dickstein et al. (2015) and further advanced by Nichol & Dhariwal (2021), where they include variance learning in the model and optimize it with new weighted vari- ational bound. Song et al. (2020) further connect DDPMs with stochastic differential equations by considering DDPMs with infinitssimal timesteps. They also find that both score-based generative models (Song & Ermon 2019) and that DDPMs can be formulated by stochastic differential equa- tions with different discretization. Some variants of DDPMs are recently introduced, including the variational diffusion model (VDM) Kingma et al. (202T) that also learns the forward process, and consistency model (Song al.) 2023) and rectified flow model (Liu et al. (022b), both of which aim to learn to generate natural images in one inference step.
DDPMs are initially applied in the pixel space achieving impressive success in generating images of high quality, however, they suffer from the significant drawbacks such as prolonged inference time and substantial training cost. Rombach et al. (2022b) propose to use a diffusion model in the latent image space, which drastically reduces the computational time and training cost. DDPMs have been widely used in different fields, for example controllable image generation (Rombach al. 20226 Choi et al., 2021 /Zhang & Agrawala, 2023 Mou et al. (203)). language generation (Zhang et al., 2023b Ye et al. 2023 Lin et al. 2022). image super-resolution (Saharia et al., 2022b) and video
generation (Hu et al. 2023 Karras et al. 2023 He et al. 2022).
DDPM Accelerator. Though diffusion models have shown powerful generation performance, normally takes hundreds of steps to generate high-quality images (Ho et al., (020). To accelerate the generation speed, [Song et al. (2021) propose the denoising diffusion implicit model (DDIM) which derives ordinary differential equations for the diffusion model showing the possibility of generating
high-quality images in much less steps. Liu et al. (202aa) introduce numerical solvers to further reduce the number of generation steps. [Bao et al. 2022b) find that the variance in the backward process can be analytically computed and improve the quality of the generated images. [ao et al. (202a)) further remove the diagonal variance matrix assumption and boost the image generation performance. Lu et al. (2022) leverage a high-order ordinary differential equations solver to reduce the generation steps of a diffusion model to 10, while maintaining the image quality. We follow this line of research and propose method to better select time steps.
7 CONCLUSION
In this paper, we have proposed novel method to alleviate the exposure bias problem in diffusion probabilistic models. Different from previous work, which tries to reduce exposure bias by retrain- ing the diffusion model with extra noisy inputs, we demonstrate that this bias can be mitigated by identifying the timestep that most aligns with the subsequent predicted step. Furthermore, we theo- retically derive the variance of this optimal next timestep. Leveraging these insights, we introduce novel and training-frr free inference algorithm, the Time-Shifted Sampler, to alleviate exposure bias in diffusion models. Our algorithm can be seamlessly incorporated into existing sampling algorithms for diffusion models including DDPM, DDIM, S-PNDM and F-PNDM. Extensive experimental re- sults prove the effectiveness of our method.

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8 ACKNOWLEDGEMENTS
This work funded by the CALCULUS project (European Research Council Advanced Grant H2020-ERCC2017-ADD 788506) the Flanders AI Research Program, and the China Scholarship Council.
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A LIMITATION AND FUTURE WORK
In this work, we introduce the Time-Shiff Sampler, a training-free mechanism devised to mitigate the exposure bias problem inherent to the diffusion model, thereby enhancing the quality of gen- erated images. Despite its efficacy, our sampler suffers from the limitation that it introduces two parameters, i.e., window size and cutoff value. We estimate their values based on the statistical anal- ysis of training data. However, more advanced methods to analytically derive the optimal value of these two parameters might be possible since both the cutoff value and the window size are related to the noise level of each step. We leave this further exploration to future work. We also foresee possibilities to use the concept of the Time-Shift Sampler in other Markovian processes where a reduction in processing steps is desired.
B FIGURE DETAILS
In this section, we present the detailed procedures to plot Figures
2
3 and
To plot Figure 2 we compute the variance at the instance level so that we can measure how much the variance of each sample varies as time progresses. Specifically, suppose we obtain a sample
(x, of size 3 x 32 x 32, we first flatten it to 3072. Then we compute the variance of x as n-1 where 0 is the mean æ. Thus, for each sample we obtain a variance of its own, which leads to the density plot in Figure
Figure shows the mean square error (MSE) between the prediction and ground truth at each step in the backward process. Given image denoted as zo. by applying Equation 2 we could obtain sequence of æ,, = 1... T 1. Taking each xt and the paired time step to run the backward process, would obtain sequence of predicted a t == 1,2,... T 1. Ideally,
would expect all these predicted $6. = 1, .... T 1, to be exactly equal to the ground truth To. as they are generated using the given T5. However, this is not the case in practice. In our experiments, we found that only when t < t. (around 650 steps in our experiment using DDIM) we could obtain the original zo by running the backward process with paired (1,, For t > t.. the image created using (I,, t) differs from the original øo. This observation also reveals that the image generation process of diffusion models basically contains two stages. the first stage, the model moves the Gaussian distribution towards the image distribution and no modes are presented at this stage, which means we can not know which images will be generated. In the second stage, the prediction shows that clear patterns and modes are presented. We can predict which images will be generated following the backward process. This observation led us to divide the error computation into two stages. The full explanation, including the equations, is shown in Figure g
To plot Figure | we follow the method used in plotting Figure generate the ground truth example for each step and compute the MSE between predicted it, and the ground truth 2; and Tt,,
C ADDITIONAL EXPERIMENTAL SETUP
Instead of generating many samples of 2;-1 to estimate var(x;-1). which brings additional compu- tational workload, we find that under some assumption (see derivation of Theorem 3.1 (T.) the var(s;-1) could be estimated using the inter variance of a single æt-1, Thus during sampling, we compute the variance within each Ft-1-
Following DDIM (Song et al. 2021), we use two types of time step selection procedures, namely uniform and quadratic. For comparing different sampling methods, the architecture of our models follows the one of DDPM (Ho et al. 2020). For all datasets, we use the same pretrained models for evaluating the sampling methods. For CIFAR-10 and LSUN- bedroom, we obtain the checkpoints from the original DDPM implementation; for CelebA, adopt the DDIM checkpoints. To carry out the comparison between ADM (Dhariwal & Nichol, (0ch),,002I), ADM-IP (Ning et al. 2023) and our method, we choose the ADM architecture and the checkpoints from ADM-IP.
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Given data sample Xo
Applying Equation (2) to generate a sequence of states X
XT XT-1 XT-2
Xts Xts-
X3
X2
X1
X
Stage generate same Xn
Stage I: unable to know which image will be generated
The Computation of the MSE Stage 1:
Applying Equation (2) to obtain the ground truth sequence: XT- ...,,, Xt, Using (Xy, 7) as input to the model to generate a sequence predicted x till time t.
(xdel(x, T) 87-1.8--2 8t,
Mx 11& xx11
MSE;
R
The Computation of the MSE at Stage
Applying Equation (2) to obtain Xt,
Using (xy, ts) as input the model to generate sequence predicted & at each time step.
MSE, Σ -xo12
Figure The detailed procedure for computing the MSE.
D ERROR ANALYSIS
We provide here more examples of prediction errors on training samples for different sampling steps for the different datasets. Figure presents the prediction errors obtained in the CelebA dataset, which show similar patterns as those of the CIFAR- 10 dataset as dipicted in Figure
580
90p
(a) 10 steps
(b) 20 steps
(c) 50 steps
(d) 100 stops
Figure 9: CelebA prediction emrors of training samples for different numbers of sampling steps.
E TRAININ-INFERRENE DISCREPANCY
We provide more examples on the training-inference discrepancy for the different datasets in Figure 10 19 and 20 For varying numbers of time steps, the same training-inference discrepancy pattern can be observed as in Figure 4 Specifically, for a certain backward step t, given a time step window w, f w| surrounding t. there exists time step t, that might couple better with the predicted next state 2;-1- And as sampling progresses, the coupling effects for á become identical for surrounding time steps.
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Time Step ShR
Figure 10: The training and inference discrepancy of DDIM with 10 sampling steps for window
sizc=20: Left: CelebA dataset; Right: CIFAR-10 dataset.
F
SAMPLING TIME COMPARISON
We report a detailed comparison of sampling time using different sampling methods in Table B and
4
Sampling Method
5 steps
10 steps
20 steps
50 steps
DDIM
19.57
43.46
90.90 ms
232.47
TS-DDIM
22.30 (+13.9%)
47.53 ms (+9.4%)
101.01 (+11.1%)
271.25 ms (+16.7%)
S-PNDM
24.35 ms
48.32 ms
93.69 ms
235.93 ms
7S-S-PNDM
26.44 ms (+8.5%)
52.08 ms (+7.78%)
105.72 ms (+12.84%)
259.19 ms (+9.86%)
F-PNDM
ms
ms
132.03 ms
268 ms
7S-F-PNDM
64.39 ms (+0.7%)
92.61 ms (+7.39%)
141.38 ms (+7.08%)
302.88 (+ (+12.74%)
Table 3: Sampling time comparison for different sampling methods on CIFAR-10 10 using DDPM as backbone.
Sampling Method
5 steps
steps 79.30 ms
steps 157.43 ms
steps 393.77 ms
ADM w/ DDIM
39,26 ms
ADM w/ TS-DDIM
40.09 ms (+2. 10%)
82.26 ms (+3.70%)
160.45 ms (+1.92%)
394.95 ms (+0.29%)
Table 4: Sampling time comparison for DDIM and TS-DDIM on 10 with ADM as backbone.
G
CASE STUDY
VateffValue
100
200
300
500
500
600
700
800
900
100
200
320
420
480
580
680
800
900
Figure 11: Example of generation process of 7S-DDIM and DDIM on CIFAR-10, --. We use the horizontal black arrow represent the time line, with the DDIM generation chain above it, TS- DDIM generation chain underneath it. We sample steps with window [t 20, t + 20) and cutoff value=200.
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The proposed novel sampling method is motivated by the theoretical and empirical analysis of the exposure bias problem in DDPM. Experimental results show that our sampling method can effec- tively improve the quality of the generated images compared to images generated with the original DDIM/DDPM models quantatively measured with the FID score. We present the comparison of the generation chain of 7S-DDIM (our method) and DDIM in Figure [T]] It can be seen that 7/SDDDIM shifted most of the time steps before reaching the cutoff value, which yields much better generation quality the image of horse. Additional qualitative examples will be presented in the Appendix
We also visualize the selected timestep trajectory in Figure 12 Before the cutoff value 200, time shift happens to all the timesteps. This is especially visible for the time steps within the range of (300, 600), where more time shifts happen, and time steps can shift to both larger or smaller time steps within the window. Figure 12 demonstrates that most of the time shifting happens in the intermediate range of the backward process. This phenomenon might be due to the fact that at the initial stage (when is large), the samples are predominantly influenced by the initialized normal distribution, leading to minimal variance change. Conversely, in the later stage (when t is small), the samples are predominantly composed of image data, containing most of the image information. This makes it easier for the model to make accurate predictions.
Tr Trajectory of Selected Timesteps Original trajectory by DDIM Trajectory TS-DDIM
Timestep Trajectory Zoomed In (300 (300-600)
Original trajectory by DDIM
Trajectory TS-DDIM
Selected Timesteps
Selected Timestops
$500
450
Cutoff yalue
400
600
800
30000
350
400
450
550
600
Original Timesteps
Original Timesteps
Figure 12: Comparison of timestep trajectory for TS-DDIM and DDIM CIFAR-10 with DDPM as backbone using 10 sampling steps and uniform time selection. We use the red dashed line and blue line present the time step trajectories for the original DDIM and our TS-DDIM, respectively. To improve visibility, we also zoom in for the time steps between 300 to 600.
H QUALITATIVE EXAMPLES
In this section present the example images generated using different sampling methods with uniform time selection procedure for varying sampling
time steps. Generated examples can found and [8] It can be seen that we can generate images with good quality
in Figure I3
14
15
16 17
for less than 10 sampling steps.
I ADDITIONAL RESULTS
We present more results obtained with the LSUN-bedroom dataset (Yu et al. 2016) in Table 5 Limited by computational resources, do properly tune the parameters. i.e., window size and cutoff values, for the -SUN-bedroom. We leave the exploration of a more efficient tuning strategy on high-resolution images for future work.
In Table we compare our method with Analytic-DPM (Bao et al. 2022b). The results of Analytic- DPM are directly taken from the paper.
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Sampling Time Steps
10
20
50
100
TS-DDPM
Figure 13: CIFAR-10 10 samples for varying time steps using TS-DDPM.
5
Sampling Time Steps
10
20
50
100
TS-DDIM
Figure 14: CIFAR-10 samples for varying time steps using TS-DDIM.
Sampling Time Steps
10
20
50
100
TS-DDPM
Figure 15: CelebA samples for varying time steps using TS-DDPM.
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5
Sampling Time Steps
10
20
50
100
TS-DDIM
Figure 16: CelebA samples for varying time steps using TS-DDIM.
Sampling Time Steps
10
20
50
TS-DDPM
Figure 17: LSUN-bedroom samples for varying time steps using TS-DDPM.
5
Sampling Time Steps
10
20
50
TS-DDIM
Figure 18: LSUN-bedroom samples for varying time steps using TS-DDIM.
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Dataset
Sampling Method
steps
10 steps
steps
50 steps
DIIM(uniform)
52.29
16.90
8.78
6.74
TS- T'S-DDMMuniform
51.57
16.66
8.29
6,90
LSUNN-bedrom
DDPM (uniform)
85.60
42.82
22.66
10.79
TS-DDPM (uniform)
79.05
32.47
15.40
10.20
Table 5: Quality of the image generation measured with FID 4 on LSUN-bedroom (256x256) with varying time steps for different sampling algorithms.
Dataset
Sampling Method
10 steps
50 steps
100 steps
Analytic-DPM(DDIM)
14.00
4.04
3.55
CIFAR-10
7S-DDIM
11.93
4.16
3.81
InalyiccDPM((DDI))
15.62
6.13
4.29
CelebA
TS-DDIM
9.36
4.20
4.18
Table Comparison of TS-DDIM with Analytic-DPM on CIFAR-1O (32x32) and CelebA (64x64)
Following (Xiao et al. 2022 DDIM with ADM as backbone on CIFARIO. As shown in Table our TS-DDIM tends to achieve much higher recall while maintaining the level of the precision obtained by DDIM.
present the precision and recall results from TS-DDIM and
5 steps
10 steps
20 steps
50 steps
Sampling Method
Precision
Recall
Precision
Recall
Precision
Recall
Precision
Recall
ADM w/ DDIM
0.59
0.47
0.62
0.52
0.64
0.57
0.66
0.60
ADM w/ TS-DDIM
0.57
0.46
0.62
0.55
0.64
0.60
0.65
0.62
Table 7: Comparison of precision and recall for DDIM and 7'-DDDIM CIFAR-10 with ADM as backbone.
We also evaluate our method on the ImageNet dataset as presented in Table We adopt ADM as the backbone model with classifier guidance.
Sampling Method 5 steps
10 steps
20 steps
DDIM
67.63
13.74
6.83
TS-DDIM
39.47(441.64%%
13.45(+2.11%)
6.57(43.81 %)
Table 8: Performance comparison with DDIM on ImageNet (64x64) with classifier guidance.
We further integrate our method into the DPM-solver (Lu et al. 2022) and DEIS (Zhang & Chen 2023 showcase its versatility. The results presented Table Our model can still improve the performance of both the DPM-solver and DEIS samplers. However, in comparison to DDIM, our method again provides substantial improvements although the improvements for these samplers are a bit smaller than for the other tested samplers. This could be attributed to the fact that both the DPM-solver and DEIS utilize the particular structure the semi-linear ODE, which already largely reduces the error of estimating Zt.
Sampling Method DPM-solver-2 TS-DPM-solvve--2
steps
10 steps
20 steps
32.30
10.92
4.30
31.02(+3.96%% %)
9.82(+10.07%
4.11(+4.42%)
DEIS-order-2-IAB
24.64
5,88
4.13
TS-DEIS --AB
22.57(+8.40%)
5.41(+8.00%)
3.62(+12.30%)
Table 9: Performance comparison with DPM-solver and DEIS obtained on CIFAR-10 (32x32).
Finally, in Table 10 we report the performance of our method performed on text-to-image generation on MSCOCO 10
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Sampling Method
10 steps
steps
DDIM
27.80
25.47
7S-DDIM
26.32 (+5.32%)
24.80 (+2.63%)
Table 10: Performance comparison with DDIM for text-to-image generation obtained MSCOCO
val2017.
J DERIVATION
J.1 DERIVATION OF THEOREM 3.1
In this section, we prove Theorem 3.1
proof. We first find the optimal time step la by minimizing the KL divergence between á. and Zt,--
As g |zr) = N ( CyTo, (1 āe II) and assume p(o 1 is a probability density function of the distribution for d with mean úz-1 and covariance matrix Et- then according to Lemma 2. of Bao et al. (2022b), we have:
D Dxulp(te-122))llcau. (xo))
DRI(N 11 22,))
H (z 2.11) - H(p(2r-112.))
2 (log( (I,,)) Tr TTT (E-1E (((1)) + (f4-1 - (4,,) ET (f4-1 - 11,)7) 11011-1,)))
(9)
2 (dlog(1- - -a,)) + 1- a. Tr(Ei-1) + Tt.
1i - 1t,11 +
where C 1/d
1 - log( sو | and d is the dimension of
Tp. Denoted P as the ground truth of the mean of the distribution of g(x+-1) and according to Equation] 2 we have Mr-1
0 1J0. fi-1 can be rewritten as;
p 1r- + - 10 +
(10)
Here, e is the network prediction error. Since pr. = Va, Zo, Equation 9 can rewritten as:
D DKr(p( 1 (zo))
1 (d (4.,) + 1-as, at.
11( 0t-1 v Va,, 100 + +ell3)+0 (11)
Tr((S-1) + 1 at,
if t. is close to t 1, then vā-i Vār, x O. We have
1 (d - Th,)) + 1-
D (mo))
-T((S, +
|ell 11+++
Tt.
(12)
further calculate the derivative of DKL with respect to Ee, = =1 We know that DEL gets its minimum at
Σ, = Tr({{{, |lee|1
(13)
We next estimate the Σt-1 in Equation distribution N N(uss o) with a being the variance, and p; 1 py if o j. then the covariance P oI and we have:
13
Assuming each pixel of image P E R follows
(p - 2)
d d-1
Σn(g + 4 2p.p)
(14)
d 1
E,(?3 dp2
20

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Published as a conference paper at ICLR 2024
Taking expectation on both sides, we achieve
E B Σ, (E[p?))
dE p
d-1
E,(o + +n4)
d
E PiPija1
E[(
(15)
d d-1
d 3-1
d
do
Σ
d
E, Pip
181(
d-
The last term on the RHS of Equation [5|an be rewritten as
d
E
Pi
)9
d
E(p))
E(p;))((p)))
d
1 d2
GS
(E(pi))) + +0) + Σ E(p:)E(()))
d
1
d
ifj
(16)
0
+
(B(pi))2 +
Σ [p(;) B(p))
d-1
d(d 1)
d d(d- Σ Hi)
By combining Equation 15 and Equation
16 and denoting fr = E, Mi
we have
M. do
1,) dld 1)
Elor 1]
d
d 1
d-
Σ 13
( M,)2
1
d(d- 1))
o + (n dF
(17)
d-1
E, 2/1 + nT9)
d-1
- E1 -T7)
d-1
Here is the mean of μ4; and μ4 is the mean of the distribution of each pixel in âz-1 at time step t 1. The ground truth 2t-1 ~ M(Va,_1o0 (1
[I), thus u - In practice, the wr is normalized to stay the range of -1 to 1, and va, is close to zero when is large. Define Ç as the difference between j and T and denote that ç.e = u F and Ç² = 44 P., then when t is large we have 2 ~ O. Considering the network prediction error, we reach
S5" +e
(18)
Thus Equation 17 can be rewritten as
E[a.- + Eee))
le|
+ d
(19)
d d-1
Multiplying I on both sides and taking trace
dlle|
da,-1 R do +
(20)
d-1
Here we assume the sample variance is approximately equal to its expected value because the di-
mension of the image is usually large. Bring Equation 20 to Equation [3
||e| 2
ot = 0 d(d (dd-1) 1)
(21)
In the above derivation, assume that when t is large Equation 21 holds. This assumption cor- responds to the cutoff mechanism our proposed algorithm, where we stop conducting time shift
when t is small, as the assumption does not hold and we are not able to estimate the Σt,, in
Equation 13
21

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Published as a conference paper at ICLR 2024
J.2 ANALYTICAL ESTIMATION OF WINDOW SIZE
In this section, we derive the bounds of window size w with optimal time step 4, E [[- 11-1-wah w/2, 2, 1 + tr/2.. In Algorithm 3, we predefine window size and search the optimal time step l,2 around time step / - 1 within w. In the above derivation in Section J.T after Equation [T we assume I, is thus we can omit the term vā Vāt,, and the following derivations (Equations t2 give us the estimated variance of the optimal time step tg. In order to estimate the window size w. we first relax our assumption. Instead of directly assuming ty close
close
1, in the last term of Equation 11 we assume that the norm of (vot-1 vor, Jap is sufficiently smaller than the norm ye, where 0 < 7 < 1. Thus we have:
l|vār-1 V 0r
||00|||
$ >|le||
(llār-1 var,1l) 70 ||e
(22)
|| 1
04-1 ai, 2 201-110 7² ||e|12
S 0
(23)
|
Solving Equation 23
we obtain:
2 _ā, 4(ār-1 72
2
Var 2
(24)
2/ā + v 4dt-1 - -((ār-1 - 72
||c/13 |
Vār, S
2
In Equation 24 is the network prediction eπro and r is predefined value. could be estimated using a small amount of data or a trained error prediction network similar to the work of Bao et al. (2022a). 0₂ on the LHS is the predefined noise schedule. Since ã₂ is monotonic function with respect to f, denoted as a, = ft)), for a given a₁, the corresponding time step t could be obtained through the inverse function of f. that is f (ā,) Thus, the bounds of window size w could be estimated from the bounds of ā, through Equation 24 To obtain the bounds of w, one could first compute the right hand sides of Equation 24 for the predefined noise schedule of given diffusion model. Then for time step t, one could find the largest and smallest f., denoted as tmar and emin respectively, satisfying Equation 24 The w is then bounded by the 2 x min(Imas 1 (min). ). If the above condition holds, then the last term of Equation is dominated by e, and (Va--1 Vor, 200
can be ignored in above derivations (Equations 12
22

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Published as a conference paper at ICLR 2024
Higure 19: The training and inference discrepancy of DDIM with 50 sampling steps CIFAR 10 for windoe size= 10.
Figure 20: The training and inference discrepancy of DDIM with 50 sampling steps on CelebA for windoe size= 10.
23


Reviews:
[{"review_id": "SQyP2daZFP", "reviewer": "Reviewer_8DgG", "summary": "The paper proposes to select a suitable timestep for next step instead decreasing it as in most diffusion model sampling algorithms.", "questions": "as above", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and organized.\n\n2. The analysis are sufficient and the findings in section 3. are interesting. \n\n3. According to the tables, the FIDs are improved with different baseline samplers. It is also good to see that the sampling time is not increased significantly as shown in figure 6.", "weaknesses": "1.  Other metric should also be included, such as precision and recall. Is there any reason why text-to-image performances are not added? It would be better if authors could include such results as the task is one of most important tasks of diffusion model. \n\n2. Could you also visualize the selecte timestep trajectory?  It also would better to have more analysis on the selected timesteps and around which timesteps are mostly important.", "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": "txAyq64SVK", "reviewer": "Reviewer_YoEn", "summary": "This work studies the problem of exposure bias in diffusion models and proposes a training-free method to mitigate exposure bias during sampling. The paper first reasons why mitigating exposure bias in diffusion models could result in improved sampling by empirically demonstrating the accumulation of error at different time steps. To this end, a quantity $C(x_t, t)$, called input couple for trained DPM, is used to capture the discrepancy between ground truth and network predictions. The core idea is that there might be an alternate time step $t_s$ that might align better with the next state $\\hat{x}_{t-1}$  predicted by the network. This assumption is empirically demonstrated for different datasets and for different choices of time steps. \n\nIn order to find this alternate optimal time step $t_s$, this work derives the variance of this optimal tilmestep. This can then be used to empirically determine $t_s$ on-the-fly during sampling. This step can be seamlessly integrated into existing sampling methods like DDIM, DDPM, and PNDM. Further, this does not require any fine-tuning or training of diffusion models. However, this comes at the cost of some minor additional overhead in terms of sampling time. Overall, this method gives consistent improvements over baselines of DDIM, DDPM and PNDM in terms of FID scores.", "questions": "1. The explanation of details for plotting Figure 3 is unclear from the description given in Appendix B. Perhaps writing mathematical equations might make the idea more concise and clear. Also, for the purpose of rebuttal, could the authors include this expression here, or alternately provide an explanation of what the figure indicates? It is unclear to me why the error computation for this figure is split into two different stages. Any intuition/reasoning behind choosing the methodology of computing errors in Figure 3 is appreciated.\n2. In DDPM sampling, we use line 4 in algorithm 2 to get the next sample $x_{t-1}$.  It is unclear to me how this $x_{t-1}$ is used to compute $var(x_{t-1})$ in line 9 of Algorithm 3. We cannot use the analytic closed form of variance as it won’t have errors/exposure bias from prediction in network. Thus it needs to be sample variance. In that case, to compute $var(x_{t-1})$, is $z \\sim N(0, I)$ sampled multiple times and then sample $var(x_{t-1})$ computed? As other terms for sampling $x_{t-1}$ are fixed for a given $x_t$ (Line 4 in Algorithm 2), isn’t $z$ the only source of variance in this case? How many samples of $x_{t-1}$ are needed to get a reasonable estimate of this sample variance? Is it possible to add these details in the text that explains the algorithm?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The proposed method for alleviating exposure bias is training-free unlike previously proposed methods.\n2. The proposed method results in consistent improved performance in terms of FID score compared to baselines of DDIM, DDPM, and PNDM sampling (See Table 1), as well as prior works like ADM-IP.\n3. The primary contribution of this work which is empirical demonstration of the fact that correction of exposure bias can be done without retraining diffusion models is valuable.", "weaknesses": "1. The proposed method is training-free but introduces minor overhead in sampling time of diffusion models. Further, as the number of sampling steps increases, the method seems to be sensitive to the choice of hyperparameters like window size. At smaller number of sampling steps, the method is also a bit sensitive to cutoff time. (See Section 5.4)\n2. The writing needs improvement as it is currently a bit ambiguous at certain places. (See 1. and 2. In questions below for further details). For instance, plotting details for Figure 3 are unclear from the appendix. Similarly, it is unclear to me how $var(x_{t-1})$ is computed in line 9 of Algorithm 3. The overall clarity of this paper will greatly improve if these paragraphs are rewritten by adding additional details. Similarly, mathematical expressions should be added at multiple places along with text for improved clarity. For instance, Figure 2, writing $var(x_t)$ is more informative than simply writing $x_t$ for the label of y-axis. In Figure 3, the mathematical form of error can be included instead of labelling y-axis as error.  \n3. Certain parts of derivation of proof of Theorem 3.1 need further explanation. The proof assumes that for an image P, and pixels $p_i, p_j \\in P$, $p_i \\perp p_j$ if $i \\neq j$ which is usually not true in practice as neighboring pixels in image usually have high correlation. It also assumes that each pixel in image $P$ follows distribution $\\mathcal{N}(\\mu_i, \\sigma)$, but later it claims that $\\mu_i$ is the mean of the distribution of $\\hat{x}_{t-1}$ which looks incorrect as the mean of the latter distribution is a vector/tensor (as it is an image) while $\\mu_i$ is a scalar as it is mean of pixel values.", "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": "u2iGZZCzTC", "reviewer": "Reviewer_rQd1", "summary": "This paper examines the exposure bias of diffusion models and proposes a straightforward and efficient method to address it. The study presents clear experiments and motivation to elucidate both the exposure bias phenomenon and its solution.", "questions": "How can we design a more effective strategy for selecting window sizes and cutoff values when dealing with random datasets and image sizes?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper analyzes the exposure bias of diffusion models, presenting clear visual results and detailed analysis.\n2. This paper presents a simple, effective, and training-free solution for exposure bias.\n3. The proposed solution can be applied to both DDPM-like and DDIM-like methods, providing potential benefits for future acceleration work.", "weaknesses": "The selection of window sizes and cutoff values is primarily based on limited experience.", "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": "LNiYzPz6Ub", "reviewer": "Reviewer_zG28", "summary": "To address the issue of exposure bias, this paper proposes a sampling method called the Time-Shift Sampler. Specifically, this method is based on the observation that within a window of size w around the time step t, there exists a better time step ts that corresponds to a closer match between the variance of the true samples and the predicted samples xt. Moreover, the feasibility of this method is theoretically validated by the authors. To validate the effectiveness of the proposed method, extensive experiments are conducted on multiple models and datasets, and the results strongly demonstrate its efficacy. Importantly, compared to other relevant papers addressing exposure bias, this article offers the advantage of not requiring retraining and incurring minimal additional costs.", "questions": "1. Can this method combine with other training-free sampling methods, such as DPM-sovler, DEIS?\n2. How is the performance on ImageNet?\n3. Is there exits a analytical solution for the window size?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper investigate an important problem which ignored by previous works, dubbed exposure bias, and propose an interesting method to remedy it.\n2. The analysis for the exposure bias problem helps to understand the proposed method, Time-Shift Sampler.\n3. Time-Shift Sampler does not require fine-tune the pre-trained diffusion models and incurs minimal additional costs, while effectively mitigating exposure bias. \n4. Moreover, Time-Shift Sampler enables to combine with various diffusion model, which demonstrates a good scalability.", "weaknesses": "1. Due to the conditions imposed by the theoretical derivation, amost 10% sampling time can not be simply ignored.\n2. As mentioned 'seamlessly integrated to existing sampling algorithms', how about the performance combined with the DPM-solver [1], DEIS [2].\n3. In my humble opinion, the analysis of the exposure bias problem is empirically not theoretically as mentioned in the 'contribution'.\n4. There is no obvious advantage over the recent training-free sampling method.\n\n[1] C. Lu, Y. Zhou, F. Bao, J. Chen, C. Li, and J. Zhu. DPM-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems, 35:5775–5787, 2022.\n[2]  Q. Zhang and Y. Chen. Fast sampling of diffusion models with exponential integrator. International Conference on Learning Representations, 2023.", "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)