SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this section, we show that our UniPC can significantly improve the sampling quality through extensive experiments. Our experiments cover a wide range of datasets, where the image resolution ranges from  $32 \times 32$  to  $256 \times 256$ . Apart from the standard image-space diffusion models [35, 8], we also conduct experiments on the recent prevailing stable-diffusion [29] trained on latent space. We will first present our main results in Section 4.1 and then provide a detailed analysis in Section 4.2.
We start by demonstrating the effectiveness of our UniPC on both unconditional sampling and conditional sampling tasks, with extremely few model evaluations ( $< 10$  NFE). For the sake of clarity, we compare UniPC with the previous state-of-the-art method DPM-Solver++ [26]. We have also conducted experiments with other methods including DDIM [34], DPM-Solver [25], DEIS [40], and PNDM [24]. However, since some of these methods perform very unstable in few-step sampling, we leave their results in Section 4.2 and Appendix G.
Unconditional sampling. We first compare the unconditional sampling quality of different methods on CIFAR10 [21], FFHQ [18], and LSUN Bedroom [39]. The pre-trained diffusion models are from [35] and [29], including both pixel-space and latent-space diffusion models. The results are shown in Figure 2. For DPM-Solver++, we use the multistep 3-order version due to its better performance. For UniPC, we use a combination of UniP-3 and UniC-3, thus the order of accuracy is 4. As shown in Figure 3, we find that our UniPC consistently achieves better sampling quality than DPM-Solver++ on different datasets, especially with fewer NFE. Notably, compared with DPM-Solver++, our UniPC improves the FID by 6.0, 5.9, and 8.5 on CIFAR10, LSUN Bedroom, and FFHQ, respectively. These results clearly demonstrate that our UniPC can effectively improve the unconditional sampling quality with few function evaluations.
Conditional sampling. Conditional sampling is more useful since it allows user-defined input to control the synthesized image. To evaluate the conditional sampling performance of our UniPC, we conduct experiments on two widely used guided sampling settings, including classifier guidance and classifier-free guidance. For classifier guidance, we use the pixel-space diffusion model trained on ImageNet  $256 \times 256$  [7] provided by [8]. Following DPM-Solver++, we use dynamic thresholding [30] to mitigate the gap between training and testing. The results are shown in Figure 3a and 3b, where we compare our UniPC with DPM-Solver++ [26] under different guidance scale ( $s = 8.0 / 4.0$ ). For DPM-Solver++, we use the multistep 2-order version (2M), which achieves the best results
(a) ImageNet (Pixel DPM)
Figure 3: Conditional sampling results. (a)(b) We compare the sample quality measured by FID $\downarrow$  on ImageNet  $256 \times 256$  with guidance scale  $s = 8.0 / 4.0$ ; (c) We adopt the text-to-image model provided by stable-diffusion [29] to compare the convergence error, which is measured by the  $l_{2}$  distance between the results of different methods and 1000-step DDIM. We show that our method outperforms previous ones with various guidance scales and NFE.
(b) ImageNet (Pixel DPM)
(c) MS-COCO2014 (Latent DPM)
Table 1: Ablation on the choice of  $B(h)$ . We consider two implementations of  $B(h)$  and also provide the performance of DPM-Solver++ [26] for comparison. The results are measured by the FID(↓) on CIFAR10 [21] and FFHQ [18]. We show that while the UniPC with both the two forms of  $B(h)$  can outperform DPM-Solver++,  $B_{1}(h)$  performs better at fewer sampling steps.
(a) CIFAR10 (Pixel-space DPM)
Sampling Method	NFE
5	6	8	10
DPM-Solver++ ([26])	29.22	13.28	5.21	4.03
UniPC (B1(h) = h)	23.22	10.33	5.10	3.97
UniPC (B2(h) = e^h - 1)	26.20	11.48	5.11	3.87
(b) FFHQ (Latent-space DPM)
Sampling Method	NFE
5	6	8	10
DPM-Solver++ ([26])	27.94	15.99	9.20	7.36
UniPC (B1(h) = h)	18.66	11.89	8.21	6.99
UniPC (B2(h) = e^h - 1)	21.66	13.21	8.63	7.20
according to the original paper. For our UniPC, we use UniP-2 and UniC-2. It can be seen that our UniPC generates samples with better quality and converges rather faster than other methods. For classifier-free guidance, we adopt the latent-space diffusion model provided by stable-diffusion [29] and set the guidance scale as 1.5 following their original paper. To obtain the input texts, we randomly sample 10K captions from MS-COCO2014 validation dataset [23]. As discussed in [26], the FID of the text-to-image saturates in  $< 10$  steps, possibly because the powerful decoder can generate good image samples from non-converged latent codes. Therefore, to examine how fast a method converges, we follow [26] to compute the  $l_{2}$ -distance between the generated latent code  $\pmb{x}_0$  and the true solution  $\pmb{x}_0^*$  (obtained by running a 999-step DDIM), i.e.,  $\| \pmb{x}_0 - \pmb{x}_0^*\|_2 / \sqrt{D}$ , where  $D$  is the dimension of the latent code. For each text input, we use the same initial value  $\pmb{x}_T^*$  sampled from Gaussian distribution for all the compared methods. It can be seen in Figure 3c that our UniPC consistently has a lower  $l_{2}$ -distance than DPM-Solver++, which indicates that UniPC converges faster in guided sampling.
In this section, we will provide more detailed analyses to further evaluate the effectiveness of UniPC.
Ablation on the choice of  $B(h)$ . In Section 3, we mentioned that  $B(h)$  is set to be any non-zero function of  $h$  that satisfies  $B(h) = \mathcal{O}(h)$ . We now investigate how the choice of  $B(h)$  would affect the performance of our UniPC. Specifically, we test two simple forms:  $B_{1}(h) = h$  and  $B_{2}(h) = e^{h} - 1$  and the results are summarized in Table 1, where we also provide the performance of DPM-Solver++ [26] for reference. We show that UniPC with either implementation of  $B(h)$  can outperform DPM-Solver++. When the NFE is extremely small (5~6), we observe that  $B_{1}(h)$  consistently outperforms  $B_{2}(h)$  by 1~3 in FID. On the other hand, as the NFE increases, the performance of  $B_{2}(h)$  catches up and even surpasses  $B_{1}(h)$  in some experiments (e.g., on CIFAR10 and LSUN Bedroom). As for the guided sampling, we find  $B_{1}(h)$  is worse than  $B_{2}(h)$  consistently (see Appendix G for detailed results and discussions). These results also inspire us that our UniPC can be further improved by designing better  $B(h)$ , which we leave to future work.
Table 2: Applying UniC to any solvers. We show that UniC can be a plug-and-play component to boost the performance of both singlestep/multistep solvers with different orders. The sampling quality is measured by FID $\downarrow$  on the CIFAR10 dataset.
Sampling Method	Order	NFE
5	6	8	10
DDIM ([34])	1	55.04	41.81	27.54	20.02
+ UniC (Ours)	2	47.22	33.70	19.20	12.77
DPM-Solver++(2M) ([26])	2	33.86	21.12	10.24	6.83
+ UniC (Ours)	3	31.23	17.96	8.09	5.51
DPM-Solver++ (3S) ([26])	3	51.49	38.83	11.98	6.46
+ UniC (Ours)	4	50.62	24.59	10.32	5.50
DPM-Solver++(3M) ([26])	3	29.22	13.28	5.21	4.03
+ UniC (Ours)	4	25.50	11.72	5.04	3.90
Table 3: Exploring the upper bound of UniC. We compare the performance of UniC and UniC-oracle by applying them to the DPM Solver++. Note that the NFE of UniC-oracle is twice the number of sampling steps. Our results show that UniC still has room for improvement.
Sampling Method	Sampling Steps
5	6	8	10
LSUN Bedroom, Latent-space DPM
DPM Solver++ ([26])	17.79	8.03	4.04	3.63
+ UniC	13.79	6.53	3.98	3.52
+ UniC-oracle	6.06	4.39	3.46	3.22
FFHQ, Latent-space DPM
DPM Solver++ ([26])	27.15	15.60	8.98	7.39
+ UniC	21.73	13.38	8.67	7.22
+ UniC-oracle	15.29	11.25	8.33	7.03
UniC for any order solver. As shown in Algorithm 1, our UniC- $p$  can be applied after any  $p$ -order solver to increase the order of accuracy. To verify this, we perform experiments on a wide range of solvers. The existing solvers for DPM can be roughly categorized by the orders or the updating method (i.e., singlestep or multistep). Since DPM-Solver++ [26] by design has both singlestep and multistep variants of  $1 \sim 3$  orders, we apply our UniC to different versions of DPM-Solver++ to see whether UniC can bring improvements. The results are reported in Table 2, where the sampling quality is measured by FID $\downarrow$  on CIFAR10 by sampling from a pixel-space DPM [35]. We also provide the order of accuracy of each baseline method without/with our UniC. Apart from the DDIM [34], which can be also viewed as 1-order singlestep DPM-Solver++, we consider another 3 variants of DPM-Solver++ including 2-order multistep (2M), 3-order singlestep (3S) and 3-order multistep (3M). It can be found that our UniC can increase the order of accuracy of the baseline methods by 1 and consistently improve the sampling quality for the solvers with different updating methods and orders.
Exploring the upper bound of UniC. According to Algorithm 1, our UniC works by leveraging the rough prediction  $\tilde{\pmb{x}}_{t_i}$  as another data point to perform correction and increase the order of accuracy. Note that to make sure there is no extra NFE, we directly feed  $\epsilon_{\theta}(\tilde{\pmb{x}}_{t_i}, t_i)$  to the next updating step instead of re-computing a  $\epsilon_{\theta}(\tilde{\pmb{x}}_{t_i}^c, t_i)$  for the corrected  $\tilde{\pmb{x}}_{t_i}^c$ . Although the error caused by the misalignment between  $\epsilon_{\theta}(\tilde{\pmb{x}}_{t_i}, t_i)$  and  $\epsilon_{\theta}(\tilde{\pmb{x}}_{t_i}^c, t_i)$  has no influence on the order of accuracy (as proved in Appendix E.7), we are still interested in how this error will affect the performance. Therefore, we conduct experiments where we re-compute the  $\epsilon_{\theta}(\tilde{\pmb{x}}_{t_i}^c, t_i)$  as the input for the next sampling step, which we name as "UniC-oracle". Due to the multiple function evaluations on each  $t_i$  for both  $\tilde{\pmb{x}}_{t_i}$  and  $\tilde{\pmb{x}}_{t_i}^c$ , the real NFE for UniC-oracle is twice as the standard UniC for the same sampling steps. However, UniC-oracle is very helpful to explore the upper bound of UniC, and thus can be used in pre-experiments to examine whether the corrector is potentially effective. We compare the performance of UniC and UniC-oracle in Table 3, where we apply them to the DPM Solver++ [26] on LSUN Bedroom [39] and FFHQ [18] datasets. We observe that the UniC-oracle can significantly improve the sampling quality over the baseline methods. Although the approximation error caused by the misalignment makes UniC worse than UniC-oracle, we find that UniC can still remarkably increases the sampling quality over the baselines, especially with few sampling steps.
Customizing order schedule via UniPC. Thanks to the unified analytical form of UniPC, we are able to investigate the performance of arbitrary-order solvers and customize the order schedule freely. As a first attempt, we conduct experiments on CIFAR10 with our UniPC, varying the order schedule (the order at each sampling step). Some results are listed in Table 4, where we test different order schedules with NFE=6/7 because the search space is not too big. Note that the order schedule in Table 4 represents the order of accuracy of the UniP, while the actual order is increased by 1 because of UniC. Our default order schedule follows the implementation of DPM-Solver++ [26], where lower-order solvers are used in the final few steps. Interestingly, we find some customized order schedules can yield better results, such as 123432 for NFE=6 and 1223334 for NFE=7. We also show that simply increasing the order as much as possible is harmful to the sampling quality.
Sampling diversity. Apart from the sampling quality, we are also interested in the diversity of the images generated by UniPC. In Table 6, we compare the sampling diversity of UniPC and
Table 4: Customizing order schedule via UniPC. We investigate different order schedules with UniPC and find that some customized order schedules behave better than the default settings, while simply increasing the order as much as possible is harmful to the sampling quality.
CIFAR10, NFE = 6
Order Schedule	123321	123432	123443	123456
FID↓	10.33	9.03	11.23	22.98
CIFAR10, NFE = 7
Order Schedule	1233321	1223334	1234321	1234567
FID↓	6.41	6.29	7.24	60.99
Table 6: Comparisons of sampling diversity. We compute the Inception Score (IS) on CIFAR10 [21] and find UniPC can generate more diverse samples than DPM-Solver++ [26].
Inception Score (IS↑)	NFE
5	6	8	10
DPM-Solver++ [26]	7.27	8.62	9.52	9.69
UniPC	7.55	8.81	9.59	9.83
Table 5: Comparisons with more NFE. We compare the sampling quality between UniPC and previous methods with 10-25 NFE on ImageNet  $256\times 256$  and show our UniPC still outperforms previous methods by a large margin.
Sampling Method \ NFE	10	15	20	25
DDIM ([34])	13.04	11.27	10.21	9.87
DPM-Solver ([25])	114.62	44.05	20.33	9.84
PNDM ([24])	99.80	37.59	15.50	11.54
DEIS ([40])	19.12	11.37	10.08	9.75
DPM-Solver++ ([26])	9.56	8.64	8.50	8.39
UniPC (Ours)	7.51	6.76	6.65	6.58
Table 7: Comparisons of inference time. We compare the inference time of sampling from a stable-diffusion [29] and find UniPC achieves a similar speed to DPM-Solver++ [26].
Inference Time (s)	NFE
5	10	15
DPM-Solver++ [26]	0.48	0.77	1.07
UniPC	0.49	0.78	1.07
DPM-Solver++ [26], measured by the inception score (IS) on CIFAR10 dataset. We find that UniPC consistently generates more diverse samples in a variant number of function evaluations.
Comparisons with more NFE. To further evaluate the effectiveness of our UniPC, we also perform experiments with  $10\sim 25$  NFE. Specifically, we perform guided sampling on ImageNet  $256\times 256$  [7] with guidance scale 8.0 and compare our UniPC with more existing methods including DDIM, DPM-Solver, PNDM, DEIS, and DPM-Solver++. We summarize the results in Table 5, where some results of the previous methods are from [26]. The results clearly demonstrate that our UniPC surpasses previous methods by a large margin.
Inference speed and memory. We test the wall-clock time of UniPC by sampling from a stable-diffusion model [29] using a single NVIDIA RTX 3090 GPU and the results are shown in Table 7. We find the actual inference time of UniPC is very similar to DPM-Solver++ [26]. As for memory usage, it is only related to how many previous model outputs are stored. Therefore, our UniPC also costs similar memory to DPM-Solver++. For example, both UniPC-2 and DPM-Solver++(2M) cost about 6.3GB of memory when sampling from a stable-diffusion model.
Visualizations. We provide a qualitative comparison between our UniPC and previous methods with only 7 NFE, as shown in Figure 1. We use different methods to perform guided sampling with a guidance scale of 8.0 from a DPM trained on ImageNet  $256 \times 256$ . We find DEIS [40] tends to crash with extremely few steps, while the sample generated by DDIM [34] is relatively blurry. Compared with DPM-Solver++, the state-of-the-art method in guided sampling, UniPC can generate more plausible samples with better visual details. We further compare the sampling quality of our method UniPC and DPM-Solver++ using Stable-Diffusion-XL, a newly released model that can generate  $1024 \times 1024$  images. The results in Figure 4 show that our method consistently generates more realistic images with fewer visual flaws.
Limitations and broader impact. Despite the effectiveness of UniPC, it still lags behind training-based methods such as [31]. How to further close the gap between training-free methods and training-based methods requires future efforts.
In this paper, we have proposed a new unified predictor-corrector framework named UniPC for the fast sampling of DPMs. Unlike previous methods, UniPC has a unified formulation for its two components (UniP and UniC) for any order. The universality of UniPC makes it possible to customize arbitrary order schedules and to improve the order of accuracy of off-the-shelf fast sampler via UniC.
DPM-Solver++ [26]	UniPC (Ours)	DPM-Solver++ [26]	UniPC (Ours)
“black and white cat is sitting on top of a wooden bench”	“vase filled with flowers sitting on a table”
“little girl sitting on top of a teddy bear”	“cat is sitting on top of a bed in front of a living room”
“baseball player swinging a bat at a game”	“bathroom with a sink and a mirror”
“bowl of fruits and vegetables on it”	“man standing next to a dog”
Figure 4: Comparisons of text-to-image results between UniPC and DPM-Solver++[26]. Images are sampled from the newly released Stable-Diffusion-XL (1024×1024) using 15 NFE. We show that the images generated by DPM-Solver++ contain visible artifacts while UniPC consistently produces images with better quality. Please view the images in color and zoom in for easier comparison.
Extensive experiments have demonstrated the effectiveness of UniPC on unconditional/conditional sampling tasks with pixel-space/latent-space pre-trained DPMs. We have also discovered several directions where UniPC can be further improved, such as choosing a better  $B(h)$ , estimating a more accurate  $\epsilon_{\theta}(\tilde{\boldsymbol{x}}_{t_i}^c,t_i)$ , and designing a better order schedule. We hope our attempt can inspire future work to further explore the fast sampling of DPMs in very few steps.
This work was supported in part by the National Key Research and Development Program of China under Grant 2022ZD0160102, and in part by the National Natural Science Foundation of China under Grant 62321005, Grant 62336004, Grant 12271287 and Grant 62125603.
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A UniPC for Data Prediction Model
A.1 Comparison of data prediction and noise prediction
The data prediction model is a simple linear transformation of the noise prediction model, namely  $\pmb{x}_{\theta} = (\pmb{x}_t - \sigma_t\pmb{\epsilon}_{\theta}) / \alpha_t$  [19]. However, high-order solvers based on  $\pmb{x}_{\theta}$  and  $\pmb{\epsilon}_{\theta}$  are essentially different [30, 26]. As we shall see, the formulae of UniPC for the data prediction model differ from those for the noise prediction model. On the other hand, for image data,  $\pmb{x}_{\theta}$  is bounded in  $[-1,1]$ , while  $\pmb{\epsilon}_{\theta}$  is generally unbounded and thus can push the sample out of the bound. Therefore, solvers for data prediction models are preferred, since thresholding method [30] can be directly applied and alleviate the "train-test mismatch" problem.
A.2 Adapting UniPC to Data Prediction Model
As shown in the Proposition 4.1 of [26], for an initial value  $x_{s}$  at time  $s > 0$ , the solution at time  $t \in [0, s]$  of diffusion ODEs is
$$
\boldsymbol {x} _ {t} = \frac {\sigma_ {t}}{\sigma_ {s}} \boldsymbol {x} _ {s} + \sigma_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\theta} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) \mathrm {d} \lambda , \tag {9}
$$
where we use the notation  $\hat{\pmb{x}}_{\theta}$  and  $\hat{\pmb{x}}_{\lambda}$  to denote changing from the domain of time  $(t)$  to the domain of half log-SNR  $(\lambda)$ , i.e.,  $\hat{\pmb{x}}_{\lambda} \coloneqq \pmb{x}_{t_{\lambda}(\lambda)}$  and  $\hat{\pmb{x}}_{\theta}(\cdot, \lambda) \coloneqq \pmb{x}_{\theta}(\cdot, t_{\lambda}(\lambda))$ . We are also able to adapt our UniPC to the data prediction model and utilize more information from previous data points. Recall that  $h_i = \lambda_{t_i} - \lambda_{t_{i-1}}$ . For any nonzero increasing sequence  $r_1 < r_2 < \dots < r_p = 1$ ,  $\lambda_{s_m} = r_m h_i + \lambda_{t_{i-1}}$ ,  $s_m = t_{\lambda}(\lambda_{s_m})$ ,  $m = 1, \dots, p$ . The UniPC- $p$  is given by
$$
\tilde {\boldsymbol {x}} _ {t _ {i}} = \frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} \tilde {\boldsymbol {x}} _ {t _ {i - 1}} + \alpha_ {t _ {i}} \left(1 - e ^ {- h _ {i}}\right) \boldsymbol {x} _ {\theta} \left(\tilde {\boldsymbol {x}} _ {t _ {i - 1}}, t _ {i - 1}\right) + \alpha_ {t _ {i}} B \left(h _ {i}\right) \sum_ {m = 1} ^ {p - 1} \frac {a _ {m}}{r _ {m}} D _ {m} ^ {x}, \quad (\text {P r e d i c t o r}) \tag {10}
$$
$$
\tilde {\boldsymbol {x}} _ {t _ {i}} ^ {c} = \frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} \tilde {\boldsymbol {x}} _ {t _ {i - 1}} + \alpha_ {t _ {i}} \left(1 - e ^ {- h _ {i}}\right) \boldsymbol {x} _ {\theta} \left(\tilde {\boldsymbol {x}} _ {t _ {i - 1}}, t _ {i - 1}\right) + \alpha_ {t _ {i}} B \left(h _ {i}\right) \sum_ {m = 1} ^ {p} \frac {c _ {m}}{r _ {m}} D _ {m} ^ {x}, \quad (\text {C o r r e c t o r}) \tag {11}
$$
where  $D_{m}^{x} = \pmb{x}_{\theta}(\tilde{\pmb{x}}_{s_{m}}, s_{m}) - \pmb{x}_{\theta}(\tilde{\pmb{x}}_{t_{i-1}}, t_{i-1})$ . Importantly, the corrector (UniC) can be also applied to any solver for the data prediction model that outputs  $\tilde{\pmb{x}}_{t_{i}}$ . Denote  $\pmb{a}_{p} = (a_{1}, \dots, a_{p-1})^{\top}$ ,  $\pmb{c}_{p} = (c_{1}, \dots, c_{p})^{\top}$ . Let
$$
\boldsymbol {g} _ {p} (h) = \left(g _ {1} (h), \dots , g _ {p} (h)\right) ^ {\top}, \quad g _ {n} (h) = h ^ {n} n! \psi_ {n + 1} (h), \tag {12}
$$