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+# Knowledge Distillation from A Stronger Teacher
+
+Tao Huang1,2 Shan You1∗ Fei Wang3 Chen Qian1 Chang Xu2 1SenseTime Research 2School of Computer Science, Faculty of Engineering, The University of Sydney 3University of Science and Technology of China
+
+# Abstract
+
+Unlike existing knowledge distillation methods focus on the baseline settings, where the teacher models and training strategies are not that strong and competing as state-of-the-art approaches, this paper presents a method dubbed DIST to distill better from a stronger teacher. We empirically find that the discrepancy of predictions between the student and a stronger teacher may tend to be fairly severer. As a result, the exact match of predictions in KL divergence would disturb the training and make existing methods perform poorly. In this paper, we show that simply preserving the relations between the predictions of teacher and student would suffice, and propose a correlation-based loss to capture the intrinsic inter-class relations from the teacher explicitly. Besides, considering that different instances have different semantic similarities to each class, we also extend this relational match to the intra-class level. Our method is simple yet practical, and extensive experiments demonstrate that it adapts well to various architectures, model sizes and training strategies, and can achieve state-of-the-art performance consistently on image classification, object detection, and semantic segmentation tasks. Code is available at: https://github.com/hunto/DIST_KD.
+
+# 1 Introduction
+
+The advent of automatic feature engineering fuels deep neural networks to achieve remarkable success in a plethora of computer vision tasks, such as image classification [17, 19, 38, 48, 53], object detection [2, 23], and semantic segmentation [5, 54]. In the path of pursuing better performance, current deep learning models generally grow deeper and wider [13, 45]. However, such heavy models are clumsy to deploy in practice due to the limitations of computational and memory resources. For an efficient model with competitive performance to those larger models, knowledge distillation (KD) [16] has been proposed to boost the performance of the efficient model (student) by distilling the knowledge of a larger model (teacher) during training.
+
+The essence of knowledge distillation relies on how to formulate and transfer the knowledge from teacher to student. The most intuitive yet effective approach is to match the probabilistic prediction (response) scores between the teacher and student via Kullback–Leibler (KL) divergence [16]. In this way, the student can be guided with more informative signals during training, and is thus expected to have more promising performance than that being trained stand-alone. Besides this vanilla prediction match, other works [11, 14, 34, 41] also investigate the knowledge within intermediate representations to further boost the distillation performance, but this usually induces additional training cost as a consequence. For example, OFD [14] proposes to distill the information via multiple intermediate layers, but requires additional convolutions for feature alignments; CRD [41] introduces a contrastive loss to transfer pair-wise relationships, but it needs to hold a memory bank for all 128-d features of ImageNet images, and produces additional 260M FLOPs of computation cost.
+
+![](images/3c98d691bf1e5ddef9f2ca4076934e2f714fa5c37330374aaa209dc25b59e6bb.jpg)  
+Figure 1: Comparisons of KD and our proposed DIST on ImageNet with different teachers. (a) The ResNet-18 students are trained using baseline strategy with different model sizes of the teacher. (b) The ResNet-18 students are trained using different strategies with ResNet-50 teachers.
+
+Recently, a few studies [8, 29, 39] have been performed to address the poor learning issue of the student network when the student and teacher model sizes significantly differ. For example, TAKD [29] proposes to reduce the discrepancy of teacher and student by resorting to an additional teaching assistant of moderate model size; DGKD [39] further improves TAKD by densely gathering all the assistant models to guide the student. However, increasing the model size is only one of the popular approaches to have a stronger teacher. There lacks a thorough analysis on the training strategies to derive a stronger teacher and their effect on KD. Most importantly, a generic enough solution is preferred to address the difficulty of KD brought by stronger teachers, rather than struggling to deal with different types of stronger teachers (with larger model size or stronger training strategy) individually.
+
+To understand what makes a stronger teacher and their effect on KD, we systematically study the prevalent strategies for designing and training deep neural networks, and show that:
+
+• Beyond scaling up the model size, a stronger teacher can also be derived through advanced training strategies, e.g., label smoothing and data augmentation [51]. However, given a stronger teacher, the student’s performance on the vanilla KD could be dropped, even worse than training from scratch without KD, as shown in Figure 1. • The discrepancy between teacher and student tends to get fairly larger when we switch their training strategy to a stronger one (see Figure 2). In this case, an exact recovery of predictions via KL divergence could be challenging and lead to the failure of vanilla KD. • Preserving the relation of predictions between teacher and student is sufficient and effective. When transferring the knowledge from teacher to student, what we really care about is preserving the preference (relative ranks of predictions) by the teacher, instead of recovering the absolute values accurately. Correlation between teacher and student predictions could be favored to relax the exact match of KL divergence and distill the intrinsic relations.
+
+In this paper, we thus leverage the Pearson correlation coefficient [33] as a new match manner to replace the KL divergence. In addition, besides the inter-class relations in prediction vector (see Figure 3), with the intuition that different instances have different spectrum of similarities with respect to each class, we also propose to distill the intra-class relations for further boosting the performance as Figure 3. Concretely, for each class, we gather its corresponding predicted probabilities of all instances in a batch, then transfer this relation from teacher to student. Our proposed method (dubbed DIST) is super simple, efficient, and practical, which can be implemented with only several lines of code (see Appendix A.1) and has almost the same training cost as the vanilla KD. As a result, the student can be liberated from the burden of matching the exact output of a strong teacher, but only be guided appropriately to distill those truly informative relations.
+
+Extensive experiments are conducted on benchmark datasets to verify our effectiveness on various tasks, including image classification, object detection, and semantic segmentation. Experimental results show that our DIST significantly outperforms vanilla KD and those sophisticatedly-designed state-of-the-art KD methods. For example, with the same baseline settings on ImageNet, our DIST achieves the highest $7 2 . 0 7 \%$ accuracy on ResNet-18. With the stronger strategy, our method obtains $8 2 . 3 \%$ accuracy on the recent transformer Swin-T [27], improving KD by $1 \%$ .
+
+![](images/f3f3dab51630049eab346bb9d566a7028a1d93061af433377c97d78cb92e8c8f.jpg)  
+Figure 2: Discrepancy between the predictions of models trained standalone with different strategies on ImageNet validation set. R18B1 represents ResNet-18 trained with strategy B1 for instance. Details of training strategies B1 and B2 refer to Table 1.
+
+# 2 Revisiting Prediction Match of KD
+
+In vanilla knowledge distillation [16], the knowledge is transferred from a pre-trained teacher model to a student model by minimizing the discrepancy between the prediction scores of the teacher and student models.
+
+Formally, with the logits $Z ^ { ( \mathrm { s } ) } \in \mathbb { R } ^ { B \times C }$ and $Z ^ { ( \mathrm { t } ) } \in \mathbb { R } ^ { B \times C }$ of student and teacher networks, where $B$ and $C$ denote batch size and the number of classes, respectively, the vanilla KD loss [16] is represented as
+
+$$
+\mathcal { L } _ { \mathrm { K D } } : = \frac { \tau ^ { 2 } } { B } \sum _ { i = 1 } ^ { B } \mathrm { K L } ( Y _ { i , : } ^ { ( \mathrm { t } ) } , Y _ { i , : } ^ { ( \mathrm { s } ) } ) = \frac { \tau ^ { 2 } } { B } \sum _ { i = 1 } ^ { B } \sum _ { j = 1 } ^ { C } Y _ { i , j } ^ { ( \mathrm { t } ) } \log \left( \frac { Y _ { i , j } ^ { ( \mathrm { t } ) } } { Y _ { i , j } ^ { ( \mathrm { s } ) } } \right) ,
+$$
+
+where $\mathrm { K L }$ refers to Kullback–Leibler divergence with
+
+$$
+Y _ { i , : } ^ { \mathrm { ( s ) } } = s o f t m a x ( Z _ { i , : } ^ { \mathrm { ( s ) } } / \tau ) , \quad Y _ { i , : } ^ { \mathrm { ( t ) } } = s o f t m a x ( Z _ { i , : } ^ { \mathrm { ( t ) } } / \tau ) ,
+$$
+
+being the probabilistic prediction vectors, and $\tau$ is the temperature factor to control the softness of logits.
+
+In addition to the teacher’s soft targets in Eq.(1), KD [16] stated that it is beneficial to train the student together with ground-truth labels, and the overall training loss is composed of the original classification loss $\mathcal { L } _ { \mathrm { c l s } }$ and KD loss ${ \mathcal { L } } _ { \mathrm { K D } }$ , i.e.,
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { t r } } = \alpha \mathcal { L } _ { \mathrm { c l s } } + \beta \mathcal { L } _ { \mathrm { K D } } , } \end{array}
+$$
+
+where $\mathcal { L } _ { \mathrm { c l s } }$ is usually the cross-entropy loss between the predictions of student network and groundtruth labels, $\alpha$ and $\beta$ are factors for balancing the losses.
+
+# 2.1 Catastrophic discrepancy with a stronger teacher
+
+As illustrated in Section 1, the effect of a teacher on KD has not been sufficiently investigated, especially when the performance of pre-trained teacher grows stronger, such as with larger model size or being trained with more advanced and competing strategies, e.g., label smoothing, mix-up [51], auto augmentations [9], etc. With this regard, as Figure 2, we train ResNet-18 and ResNet-50 standalone with strategy B1 and strategy $\bar { \mathbf { B } \bar { 2 } }$ , and obtain 4 trained models (R18B1, R18B2, R50B1, and R50B2 with accuracies $6 9 . 7 6 \%$ , $7 3 . 4 \%$ , $7 6 . 1 3 \%$ , and $78 . 5 \%$ , respectively), then compare their discrepancy using KL divergence ( $\mathit { \check { \tau } } = 1$ and $\tau = 4$ ) on the predicted probabilities $\mathbf { Y }$ . We have the following observations:
+
+![](images/2bd1eb75068dbf9263a14589f57b9fc951fe07021affa3ae30f02f0e7cd2010b.jpg)  
+Figure 3: Difference between our DIST and existing KD methods. Conventional KD matches the outputs of student $( s \in \mathbb { R } ^ { 5 } )$ ) to teacher $( t \in \mathbb { R } ^ { 5 } )$ ) point-wisely; instance relation methods operate on the feature level and measure the internal correlations (corr.) between instances in student and teacher separately, then transfer the teacher’s correlations to student. Our DIST proposes to maintain the inter-class and intra-class relations between student and teacher. Inter-class relation: correlation between the predicted probabilistic distributions on each instance of teacher and student. Intra-class relation: correlation of the probabilities of all the instances on each class.
+
+• The outputs of ResNet-18 do not change much with the stronger strategy compared to ResNet-50. This implies that the representational capacity limits the student’s performance, and it tends to be fairly challenging for the student to exactly match the teacher’s outputs as their discrepancy becomes larger.   
+When the teacher and student models are trained with a stronger strategy, the discrepancy between teacher and student would be larger. This indicates that when we adopt KD with a stronger training strategy, the misalignment between KD loss and classification loss would be severer, thus disturbing the student’s training.
+
+As a result, the exact match (i.e., the loss reaches the minimal if and only if the teacher and student outputs are exactly identical) with KL divergence seems way too overambitious and demanding since the discrepancy between student and teacher can be considerably huge. Since the exact match can be detrimental with a stronger teacher, our intuition is to develop a relaxed manner for matching the predictions between the teacher and student.
+
+# 3 DIST: Distillation from A Stronger Teacher
+
+# 3.1 Relaxed match with relations
+
+The prediction scores indicate the teacher’s confidence (or preference) over all classes. For a relaxed match of predictions between the teacher and student, we are motivated to consider what we really care about for the teacher’s output. Instead of the exact probabilistic values, actually, during inference, we are only concerned about their relations, i.e., relative ranks of predictions of teacher.
+
+In this way, for some metric $d ( \cdot , \cdot )$ with $\mathbb { R } ^ { C } \times \mathbb { R } ^ { C }  \mathbb { R } ^ { + }$ , the exact match can be formulated that $d ( { \pmb a } , { \pmb b } ) = 0$ if ${ \pmb a } = { \pmb b }$ for any two prediction vector as $Y _ { i , : } ^ { ( \mathrm { s } ) }$ and $Y _ { i , : } ^ { ( \mathrm { t } ) }$ in the KL divergence of Eq.(1). Then as a relaxed match, we can introduce additional mappings $\phi ( \cdot )$ and $\psi ( \cdot )$ with $\mathbb { R } ^ { C } \to \mathbb { R } ^ { C }$ such that
+
+$$
+d ( \phi ( \mathbf { a } ) , \psi ( \pmb { b } ) ) = d ( \mathbf { a } , \pmb { b } ) , \forall \mathbf { a } , \pmb { b }
+$$
+
+Therefore, $d ( { \pmb a } , { \pmb b } ) = 0$ does not necessarily require $\textbf { \em a }$ and $^ { b }$ should be exactly the same. Nevertheless, since we care about the relation within $\textbf { \em a }$ or $^ { b }$ , the mappings $\phi$ and $\psi$ should be isotone and do not affect the semantic information and inference result of the prediction vector.
+
+With this regard, a simple yet effective choice for the isotone mapping is the positive linear transformation, namely,
+
+$$
+d ( m _ { 1 } { \pmb a } + n _ { 1 } , m _ { 2 } { \pmb b } + n _ { 2 } ) = d ( { \pmb a } , { \pmb b } ) ,
+$$
+
+where $m _ { 1 }$ $_ { \cdot 1 } , m _ { 2 } , n _ { 1 }$ , and $n _ { 2 }$ are constants with $m _ { 1 } \times m _ { 2 } > 0$ . As a result, this match could be invariant under separate changes in scale and shift for the predictions. Actually, to satisfy the property Eq.(5), we can thus adopt the widely-used Pearson’s distance as the metric, i.e.,
+
+$$
+\begin{array} { r } { d _ { \mathrm { p } } ( \pmb { u } , \pmb { v } ) : = 1 - \rho _ { \mathrm { p } } ( \pmb { u } , \pmb { v } ) . } \end{array}
+$$
+
+$\rho _ { \mathrm { p } } ( { \pmb u } , { \pmb v } )$ is the Pearson correlation coefficient between two random variables $\textbf { \em u }$ and $\textbf {  { v } }$
+
+$$
+\rho _ { \mathrm { p } } ( { \pmb u } , { \pmb v } ) : = \frac { \mathrm { C o v } ( { \pmb u } , { \pmb v } ) } { \mathrm { S t d } ( { \pmb u } ) \mathrm { S t d } ( { \pmb v } ) } = \frac { \sum _ { i = 1 } ^ { C } ( u _ { i } - \bar { u } ) ( v _ { i } - \bar { v } ) } { \sqrt { \sum _ { i = 1 } ^ { C } ( u _ { i } - \bar { u } ) ^ { 2 } \sum _ { i = 1 } ^ { C } ( v _ { i } - \bar { v } ) ^ { 2 } } }
+$$
+
+where $\operatorname { C o v } ( u , v )$ is the covariance of $\textbf { \em u }$ and $v , { \bar { u } }$ and $\operatorname { S t d } ( { \pmb u } )$ denote the mean and standard derivation of $\textbf { \em u }$ , respectively.
+
+In this way, we can define the relation as correlation. More specifically, and the original exact match in vanilla KD [16] can thus be relaxed and replaced by maximizing the linear correlation to preserve the relation of teacher and student on the probabilistic distribution of each instance, which we call inter-class relation. Formally, for each pair of prediction vector ${ Y } _ { i , : } ^ { ( \mathrm { s } ) }$ ) and Y (t)i,: , the inter-relation loss can be formulated as
+
+$$
+\mathcal { L } _ { \mathrm { i n t e r } } : = \frac { 1 } { B } \sum _ { i = 1 } ^ { B } d _ { \mathrm { p } } ( Y _ { i , : } ^ { ( \mathrm { s } ) } , Y _ { i , : } ^ { ( \mathrm { t } ) } ) .
+$$
+
+Some isotone mappings or metrics can also be used to relax the match as Eq.(4), such as cosine similarity investigated empirically in Section 4.5; other more advanced and delicate choices could be left as future work.
+
+# 3.2 Better distillation with intra-relations
+
+Besides the inter-class relation, where we transfer the relation of multiple classes in each instance, the prediction scores of multiple instances in each class are also informative and useful. This scores indicate the similarities of multiple instances to one class. For instance, suppose we have three images containing “cat”, “dog”, and “plane”, respectively, and they have three prediction scores on the ‘cat’ class, denoted as $e$ , $f$ , and $g$ . Generally, the picture “cat” should have the largest score to the “cat” class, while the “plane” should have the smallest score since it is inanimate. This relation of $^ { \bullet } e > f > g ^ { , \bullet }$ could also be transferred to the student. Besides, even for the images from the same class, the intrinsic intra-class variance of the semantic similarities is actually also informative. It indicates the prior from the teacher that which one is more reliable to cast in this class.
+
+Therefore, we also encourage to distill this intra-relation for better performance. Actually, define prediction matrix Y (s) and Y (t) with each row as Y (si,: ) and Y (t)i,: , then the above inter-relation is to maximize the correlation row-wisely (see Figure 3). In contrast, for intra-relation, the corresponding loss is thus to maximize the correlation column-wisely, i.e.,
+
+$$
+\mathcal { L } _ { \mathrm { \mathrm { i n t r a } } } : = \frac { 1 } { C } \sum _ { j = 1 } ^ { C } d _ { \mathrm { p } } ( Y _ { : , j } ^ { ( \mathrm { s } ) } , Y _ { : , j } ^ { ( \mathrm { t } ) } ) .
+$$
+
+As a result, the overall training loss $\mathcal { L } _ { \mathrm { t r } }$ can be composed of the classification loss, inter-class KD loss, and intra-class KD loss, i.e.,
+
+$$
+\mathcal { L } _ { \mathrm { t r } } = \alpha \mathcal { L } _ { \mathrm { c l s } } + \beta \mathcal { L } _ { \mathrm { i n t e r } } + \gamma \mathcal { L } _ { \mathrm { i n t r a } } ,
+$$
+
+where $\alpha , \beta$ , and $\gamma$ are factors for balancing the losses. In this way, via the relation loss, we have endowed the student with freedom more or less to match the teacher network’s output adaptively, thus boosting the distillation performance to a great extent.
+
+Table 1: Training strategies on image classification tasks. BS: batch size; $L R$ : learning rate; WD: weight decay; LS: label smoothing; EMA: model exponential moving average; RA: RandAugment [9]; $R E$ : random erasing; CJ: color jitter.   
+
+<table><tr><td>Strategy</td><td>Dataset</td><td>Epochs</td><td>Total Initial BS</td><td>LR</td><td>Optimizer</td><td>WD</td><td></td><td></td><td>LS EMA LR scheduler</td><td></td><td>Data augmentation</td></tr><tr><td>A1</td><td>CIFAR-100</td><td>240</td><td>64</td><td>0.05</td><td>SGD</td><td>5×10-4</td><td></td><td></td><td></td><td>×0.1 at 150,180,210 epochs crop + flip</td><td></td></tr><tr><td>B1</td><td>ImageNet</td><td>100</td><td>256</td><td>0.1</td><td>SGD</td><td>1×10-4</td><td></td><td>1</td><td>-</td><td>×0.1 every 30 epochs</td><td>crop + flip</td></tr><tr><td>B2</td><td>ImageNet</td><td>450</td><td>768</td><td>0.048</td><td>3RMSProp</td><td>1×10-5</td><td></td><td></td><td></td><td>0.10.9999 ×0.97 every 2.4 epochs</td><td>{Bl} +RA + RE</td></tr><tr><td>B3</td><td>ImageNet</td><td>300</td><td>1024</td><td>5e-4</td><td>AdamW</td><td>5×10-²</td><td>0.1</td><td></td><td>1 cosine</td><td></td><td>{B2} +CJ+ Mixup +CutMix</td></tr></table>
+
+Table 2: Evaluation results of baseline settings on ImageNet. We use ResNet-34 and ResNet-50 released by Torchvision [28] as our teacher networks, and follow the standard training strategy (B1). Student (teacher) Teacher Student KD [16] OFD [14] CRD [41] SRRL [47] Review [7] DIST   
+
+<table><tr><td>ResNet-18 (ResNet-34)</td><td>Top-1 Top-5</td><td>73.31 91.42</td><td>69.76 89.08</td><td>70.66 89.88</td><td>71.08 90.07</td><td>71.17 90.13</td><td>71.73 90.60</td><td>71.61 90.51</td><td>72.07 90.42</td></tr><tr><td>MobileNet (ResNet-50)</td><td>Top-1 Top-5</td><td>76.16 92.86</td><td>70.13 89.49</td><td>70.68 90.30</td><td>71.25 90.34</td><td>71.37 90.41</td><td>72.49 90.92</td><td>72.56 91.00</td><td>73.24 91.12</td></tr></table>
+
+# 4 Experiments
+
+# 4.1 Experimental settings
+
+Training strategies. The training strategies of image classification task are summarized in Table 1. CIFAR-100. For fair comparisons, we use the same training strategies (referred to $A l$ in Table 1) and pretrained models following CRD [41]. ImageNet. B1: for comparisons with previous KD methods, we train our baselines with the same simple training strategy as CRD [41]. B2: to validate the effectiveness of KD methods on modern training strategies, we follow EfficientNet [40] and design a training strategy B2, which can significantly improve the performance compared to B1. B3: the strategy B3 is used for training Swin-Transformers [27], and contains even more stronger data augmentations and regularization.
+
+Loss weights. On CIFAR-100 and ImageNet, we set $\alpha = 1$ , $\beta = 2$ , and $\gamma = 2$ in Eq.(10). On object detection and semantic segmentation, these three factors are all equal to 1. For KD [16], we set $\alpha = 0 . 9$ , $\beta = 1$ in Eq.(3), and use a default temperature $\tau = 4$ . Specifically, instead of using $\tau = 1$ on ImageNet, we choose a larger temperature $\tau = 4$ on CIFAR-100, as it is easy to get overfit and the learned probabilistic distribution is sharp on CIFAR-100.
+
+# 4.2 Image Classification
+
+Baseline results on ImageNet. We first compare our method with prior works using the baseline settings. As shown in Table 2, our DIST significantly outperforms prior KD methods. Note that our method is only conducted on the outputs of models, and has a similar computational cost as KD [16]. Nevertheless, it even achieves better performance compared to those sophisticatedlydesigned methods. For example, CRD [41] needs to preserve a memory bank for all 128-d features of ImageNet images, and produces additional 260M FLOPs of computation cost; SRRL [47] and Review [7] require additional convolutions for feature alignments. The implementation of DIST can be found in Appendix A.1, which is quite simple compared to these methods.
+
+Distillation from stronger teacher models. As the stronger teachers come from larger model sizes and stronger strategies, we here first conduct experiments to compare our DIST with the vanilla KD on different scales (model sizes) of ResNets with baseline strategy B1. As shown in Table 3, when the teacher goes larger, the ResNet-18 students perform even worse than that with a medium-sized ResNet-50 teacher. Nevertheless, our DIST shows an upward trend with larger teachers, and the improvements compared to KD also become more significant, indicating that our DIST tackles better on the large discrepancy between the student and larger teacher.
+
+Distillation from stronger training strategies. Recently, the performance of models on ImageNet has been significantly improved by the sophisticated training strategies and strong data augmentations (e.g., TIMM [44] achieves $8 0 . 4 \%$ accuracy on ResNet-50 while the baseline strategy B1 only obtains $7 6 . 1 \%$ . However, most of the KD methods still conduct experiments with simple training settings. It is seldomly investigated whether the KD methods are suitable to the advanced strategies. In this way, we conduct experiments with advanced training strategies and compare our method with vanilla KD, instance relation-based RKD [30], and SRRL [47].
+
+Table 3: Performance of ResNet-18 and ResNet-34 on ImageNet with different sizes of teachers.   
+
+<table><tr><td rowspan="2">Student</td><td rowspan="2">Teacher</td><td colspan="4">Top-1 ACC (%)</td></tr><tr><td> student</td><td>teacher</td><td>KD</td><td>DIST</td></tr><tr><td rowspan="4">ResNet-18</td><td>ResNet-34</td><td rowspan="4">69.76</td><td>73.31</td><td>71.21</td><td>72.07 (+0.86)</td></tr><tr><td>ResNet-50</td><td>76.13</td><td>71.35</td><td>72.12 (+0.77)</td></tr><tr><td>ResNet-101</td><td>77.37</td><td>71.09</td><td>72.08 (+0.99)</td></tr><tr><td>ResNet-152</td><td>78.31</td><td>71.12</td><td>72.24 (+1.12)</td></tr><tr><td rowspan="4">ResNet-34</td><td>ResNet-50</td><td rowspan="2">73.31</td><td>76.13</td><td>74.73</td><td>75.06 (+0.33)</td></tr><tr><td>ResNet-101</td><td>77.37</td><td>74.89</td><td>75.36 (+0.47)</td></tr><tr><td>ResNet-152</td><td>78.31</td><td>74.87</td><td>75.42 (+0.55)</td></tr><tr><td></td><td></td><td></td><td></td></tr></table>
+
+Table 4: Performance of students trained with strong strategies on ImageNet. The Swin- $T$ is trained with strategy B3 in Table 1, others are trained with B2. †: trained by [44]. ‡: Pretrained on ImageNet-22K.   
+
+<table><tr><td rowspan="2">Teacher</td><td rowspan="2">Student</td><td colspan="4">Top-1 ACC (%)</td></tr><tr><td>teacher student</td><td>KD[16] RKD[30]</td><td>SRRL [47]</td><td>DIST</td></tr><tr><td rowspan="4">ResNet-50t</td><td>ResNet-18</td><td rowspan="4">80.1</td><td>73.4</td><td>72.6 72.9 71.2</td><td>74.5</td></tr><tr><td>ResNet-34</td><td>76.8</td><td>77.2 76.6 76.7</td><td> 77.8</td></tr><tr><td>MobileNetV2</td><td>73.6</td><td>71.7 73.1 69.2</td><td> 74.4</td></tr><tr><td>EfficientNet-B0</td><td>78.0 77.4</td><td>77.5 77.3</td><td>78.6</td></tr><tr><td rowspan="2">Swin-L</td><td>ResNet-50</td><td rowspan="2">86.3</td><td>78.5</td><td>80.0 78.9 78.6</td><td>80.2</td></tr><tr><td>Swin-T</td><td>81.3 81.5</td><td>81.2 81.5</td><td> 82.3</td></tr></table>
+
+We first train traditional CNNs with strong strategies, and also use a strong ResNet-50 with $8 0 . 1 \%$ accuracy trained by [44] as the teacher. As results shown in Table 4, on both similar architectures (ResNet-18, ResNet-34) and dissimilar architectures (MobileNetV2, EfficientNet-B0), our DIST can achieve the best performance. Note that RKD and SRRL can perform worse than training from scratch, especially when the students are small (ResNet-18 and MobileNet) or the architectures of teacher and student are fairly different (ResNet-50 and Swin-L), this might be because they focus on the intermediate features, which can be more challenging for the student to recover teacher’s features compared to predictions.
+
+Furthermore, we experiment on the recent state-of-the-art Swin-Transformer [27]. The results show that our DIST gains improvements on even more stronger models and strategies. For example, with Swin-L teacher, our method improves ResNet-50 and Swin-T by $1 . 7 \%$ and $1 . 0 \%$ , respectively.
+
+CIFAR-100. The results on CIFAR-100 dataset in Table 5 show that, by distilling on the predicted logits, our method even outperforms those sophisticatedly-designed feature distillation methods.
+
+# 4.3 Object Detection
+
+We further investigate the effectiveness of DIST on downstream tasks. We conduct experiments on MS COCO object detection dataset [25], and simply leverage our DIST as an additional supervision on the final predictions of classes. Following [37, 52], we use the same standard training strategies and utilize Cascade Mask R-CNN [2] with ResNeXt-101 backbone as the teacher for two-stage student of Faster R-CNN [23] with ResNet-50 backbone; while for one-stage RetinaNet [24] with ResNet-50 backbone, the RetinaNet with ResNeXt-101 backbone is utilized as the teacher.
+
+As shown in Table 6, our DIST achieves competitive results on COCO validation set. For comparisons, we train the vanilla KD under the same settings as our DIST, the results show that our DIST significantly outperforms vanilla KD by simply replacing the loss functions. Moreover, by combining
+
+Table 5: Evaluation results on CIFAR-100 dataset. The upper and lower models denote teacher and student, respectively.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Same architecture style</td><td colspan="3">Different architecture style</td></tr><tr><td>WRN-40-2 WRN-40-1</td><td>ResNet-56 ResNet-20</td><td>ResNet-32x4 ResNet-8x4</td><td>ResNet-50 MobileNetV2 ShuffleNetV1 ShuffleNetV2</td><td>ResNet-32x4 ResNet-32x4</td><td></td></tr><tr><td>Teacher</td><td>75.61</td><td>72.34</td><td>79.42</td><td>79.34</td><td>79.42</td><td>79.42</td></tr><tr><td>Student</td><td>71.98</td><td>69.06</td><td>72.50</td><td>64.6</td><td>70.5</td><td>71.82</td></tr><tr><td colspan="7">Feature-based methods</td></tr><tr><td>FitNet [35]</td><td>72.24±0.24 69.21±0.36</td><td></td><td>73.50±0.28</td><td>63.16±0.47</td><td>73.59±0.15</td><td>73.54±0.22</td></tr><tr><td>VID [1]</td><td>73.30±0.13</td><td>70.38±0.14</td><td>73.09±0.21</td><td>67.57±0.28</td><td>73.38±0.09</td><td>73.40±0.17</td></tr><tr><td>RKD [30]</td><td>72.22±0.20</td><td>69.61±0.06</td><td>71.90±0.11</td><td>64.43±0.42</td><td>72.28±0.39</td><td>73.21±0.28</td></tr><tr><td>PKT [31]</td><td>73.45±0.19</td><td>70.34±0.04</td><td>73.64±0.18</td><td>66.52±0.33</td><td>74.10±0.25</td><td>74.69±0.34</td></tr><tr><td>CRD [41]</td><td>74.14±0.22</td><td>71.16±0.17</td><td>75.51±0.18</td><td>69.11±0.28</td><td>75.11±0.32</td><td>75.65±0.10</td></tr><tr><td colspan="7">Logits-based methods</td></tr><tr><td>KD [16]</td><td>73.54±0.20 70.66±0.24 73.33±0.25</td><td></td><td></td><td>67.35±0.32</td><td>74.07±0.19</td><td>74.45±0.27</td></tr><tr><td> DIST</td><td></td><td>74.73±0.24 71.75±0.30 76.31±0.19</td><td></td><td>68.66±0.23</td><td>76.34±0.18</td><td>77.35±0.25</td></tr></table>
+
+Table 6: Results on COCO validation set. T: teacher; S: student. \*: We implement KD using $\tau = 1$ and other settings are the same as DIST.   
+
+<table><tr><td>Method</td><td colspan="5">AP AP50 AP75 APs APM APL</td></tr><tr><td>T: Cascade Mask RCNN-X101</td><td>Two-stage detectors</td><td>45.6 64.1 49.7</td><td>26.2</td><td>49.6 42.1</td><td>60.0 50.3</td></tr><tr><td>S: Faster RCNN-R50 KD [16]* FKD [52]</td><td>38.4 39.7 41.5</td><td>59.0 61.2 62.2</td><td>42.0 43.0 45.1</td><td>21.5 23.2 23.5</td><td>43.3 51.7 45.0 55.3</td></tr><tr><td>CWD [37] DIST DIST + mimic</td><td>41.7 40.4 61.7</td><td>62.0 45.5 43.8</td><td>23.3 23.9</td><td>45.5 44.6 41.8 62.4 45.6 23.4 46.1</td><td>55.5 52.6 55.0</td></tr><tr><td>T: RetinaNet-X101 S:RetinaNet-R50 KD [16]*</td><td>One-stage detectors 41.0 37.2 56.5</td><td>60.944.0 37.4 56.739.6</td><td>39.3 20.4</td><td>23.9 45.2 20.0 40.7 40.4</td><td>54.0 49.7 49.5</td></tr><tr><td>FKD [52] CWD [37]</td><td>39.6 40.8</td><td>558.8 60.4 59.5</td><td>42.1 43.4</td><td>22.7 43.3 22.7 44.5</td><td>52.5 55.3</td></tr><tr><td>DIST DIST + mimic</td><td>39.8 40.1 59.4</td><td>42.5 43.0</td><td>22.0</td><td>43.7 23.2 44.0</td><td>53.0 53.6</td></tr></table>
+
+Table 7: Results on Cityscapes val dataset. All models are pretrained on ImageNet.   
+
+<table><tr><td rowspan=1 colspan=3>Method</td><td rowspan=1 colspan=1>mIoU (%)</td></tr><tr><td rowspan=1 colspan=3>T: DeepLabV3-R101</td><td rowspan=1 colspan=1>78.07</td></tr><tr><td rowspan=1 colspan=3>S: DeepLabV3-R18</td><td rowspan=1 colspan=1>74.21</td></tr><tr><td rowspan=1 colspan=3>SKD [26]</td><td rowspan=3 colspan=1>75.4275.59</td></tr><tr><td rowspan=2 colspan=2>IFVD [43]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=3>CWD [37]</td></tr><tr><td rowspan=1 colspan=1>75.55</td></tr><tr><td rowspan=1 colspan=3>CIRKD [46]</td><td rowspan=1 colspan=1>76.38</td></tr><tr><td rowspan=1 colspan=3>DIST</td><td rowspan=1 colspan=1>77.10</td></tr><tr><td rowspan=1 colspan=3>S: PSPNet-R18</td><td rowspan=1 colspan=1>72.55</td></tr><tr><td rowspan=1 colspan=3>SKD [26]</td><td rowspan=3 colspan=1>73.2973.7174.36</td></tr><tr><td rowspan=1 colspan=3>IFVD [43]</td></tr><tr><td rowspan=1 colspan=3>CWD [37]</td></tr><tr><td rowspan=1 colspan=3>CIRKD [46]</td><td rowspan=1 colspan=1>74.73</td></tr><tr><td rowspan=1 colspan=3> DIST</td><td rowspan=1 colspan=1>76.31</td></tr></table>
+
+DIST with mimic, which minimizes the mean square error between FPN features of teacher and student, we can even outperform the state-of-the-art KD methods designed for object detection.
+
+# 4.4 Semantic Segmentation
+
+We also perform experiments on semantic segmentation, a challenging dense prediction task. Following [37, 43, 46], we train DeepLabV3 [6] and PSPNet [54] with ResNet-18 backbone on Cityscapes dataset, and adopt our DIST on the predictions of classification head using a teacher with ResNet101 backbone of DeepLabV3. As the results summarized in Table 7, with only the supervision of class predictions, our DIST can significantly outperform existing knowledge distillation methods on semantic segmentation task. For example, our DIST outperforms recent state-of-the-art method CIRKD [46] by $1 . 5 8 \%$ on PSPNet-R18. This demonstrate our effectiveness on relation modeling.
+
+# 4.5 Ablation studies
+
+Effects of inter-class and intra-class correlations. This paper proposes two types of relations: interclass and intra-class relations. To validate the effectiveness of each relation, we conduct experiments to train students with these relations separately. The results on Table 8 verify that, both inter-class and intra-class relations can outperform the vanilla KD; also, the performance could be further boosted by combining them together.
+
+Table 8: Ablation of inter-class and intra-class relations on ImageNet. The student and teacher models are ResNet-18 and ResNet-34, respectively.   
+
+<table><tr><td>Method</td><td>Inter</td><td>Intra</td><td>ACC (%)</td></tr><tr><td>KD</td><td>-</td><td>1</td><td>71.21</td></tr><tr><td>DIST (KL div.) DIST (KL div.)</td><td>× √</td><td>√ &lt;</td><td>70.61 71.62</td></tr><tr><td>DIST</td><td>√</td><td>×</td><td>71.63</td></tr><tr><td>DIST</td><td>×</td><td>√</td><td>71.55</td></tr><tr><td>DIST</td><td>√</td><td>√</td><td></td></tr><tr><td></td><td></td><td></td><td>72.07</td></tr></table>
+
+Effect of intra-class relation in vanilla KD. To investigate the effectiveness of intra-class relation in vanilla KD, we adopt experiments to train our DIST using KL divergence as the relation metric, denoted as $D I S T ( K L { \dot { d } } i \nu . ) ^ { 3 }$ . As the results summarized in Table 8, adding intra-class relation in the vanilla KD can also improve the performance (from $7 1 . 2 1 \%$ to $7 1 . 6 2 \%$ ). However, when the student is trained with intra-class relation only, the improvement of using KL divergence is less significant than using Pearson correlation $7 0 . 6 1 \%$ vs. $7 1 . 5 5 \%$ ), since the means and variances of intra-class distributions could be varied.
+
+Effect of training students with KD loss only. Training student with only the KD loss can better reflect the distillation ability and the information richness of supervision signals. As results in Table 9 show that, when the student is trained with only the KD loss, our DIST significantly outperforms the vanilla KD. Without using the ground-truth labels, it can even outperform the standalone training accuracy, which indicates the effectiveness of our DIST in distilling those truly-beneficial relations.
+
+Table 9: Comparisons of training KD with or without the classification loss on ImageNet. The student and teacher models are ResNet-18 and ResNet-34, respectively. The original accuracy of ResNet-18 without KD is $6 9 . 7 6 \%$ .
+
+<table><tr><td>Method</td><td>w/ cls. loss</td><td>w/o cls. loss</td></tr><tr><td>KD</td><td>71.21</td><td>68.12</td></tr><tr><td>DIST</td><td>72.07</td><td>70.65</td></tr></table>
+
+More ablation studies can be found in Section A.3.
+
+# 5 Conclusion
+
+This paper presents a new knowledge distillation (KD) method named DIST to implement better distillation from a stronger teacher. We empirically study the catastrophic discrepancy problem between the student and a stronger teacher, and propose a relation-based loss to relax the exact match of KL divergence in a linear sense. Our method DIST is simple yet effective in handling strong teachers. Extensive experiments show our superiority in various benchmark tasks. For example, DIST even outperforms state-of-the-art KD methods designed specifically for object detection and semantic segmentation.
+
+# Acknowledgements
+
+This work was supported in part by the Australian Research Council under Project DP210101859 and the University of Sydney Research Accelerator (SOAR) Prize.
+
+# References
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Appendix.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Training details are provided in the paper. Training code and logs are released at GitHub: https://github.com/hunto/DIST_KD.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Standard deviations on CIFAR-100 are reported.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Code and training logs are included.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/3itjR9QxFw/3itjR9QxFw.md

@@ -0,0 +1,433 @@
+# ANALOG BITS: GENERATING DISCRETE DATA USING DIFFUSION MODELS WITH SELF-CONDITIONING
+
+Ting Chen, Ruixiang Zhang†, Geoffrey Hinton
+
+Google Research, Brain Team {iamtingchen,ruixiangz,geoffhinton}@google.com
+
+# ABSTRACT
+
+We present Bit Diffusion: a simple and generic approach for generating discrete data with continuous state and continuous time diffusion models. The main idea behind our approach is to first represent the discrete data as binary bits, and then train a continuous diffusion model to model these bits as real numbers which we call analog bits. To generate samples, the model first generates the analog bits, which are then thresholded to obtain the bits that represent the discrete variables. We further propose two simple techniques, namely Self-Conditioning and Asymmetric Time Intervals, which lead to a significant improvement in sample quality. Despite its simplicity, the proposed approach can achieve strong performance in both discrete image generation and image captioning tasks. For discrete/categorical image generation, we significantly improve previous state-of-the-art on both CIFAR10 (which has $3 K$ discrete 8-bit tokens) and IMAGENET $6 4 \times 6 4$ (which has $1 2 K$ discrete 8-bit tokens), outperforming the best autoregressive model in both sample quality (measured by FID) and efficiency. For image captioning on MS-COCO dataset, our approach achieves competitive results compared to autoregressive models.
+
+# 1 INTRODUCTION
+
+State-of-the-art generative models for discrete data, such as discrete images and text, are based on autoregressive modeling (Van den Oord et al., 2016; Salimans et al., 2017; Parmar et al., 2018; Child et al., 2019; Roy et al., 2021; Jun et al., 2020; Sutskever et al., 2014; Brown et al., 2020; Chowdhery et al., 2022), where the networks, often Transformers (Vaswani et al., 2017), are trained to predict each token given its preceding ones in a sequential manner or with causal attention masks. One major drawback of such approaches is that they typically require computation and memory that is quadratic to the dimension of data (e.g., sequence length or image size), leading to difficulties in modeling large images or sequences. Another drawback is that, during generation, autoregressive models generate one token at a time so the total number of sequential sampling steps is often the same as the dimension of data, making it slow in generating large images or long sequences.
+
+In contrast, diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020), or score-based generative models (Song & Ermon, 2019; 2020; Song et al., 2021), can model much higher dimensional data without running into computation and memory issues. During generation, diffusion models iteratively refine samples with a high degree of parallelism, so the total number of sequential sampling steps can be much less than the dimension of data. However, state-of-the-art diffusion models (Dhariwal & Nichol, 2021; Ho et al., 2022; Nichol et al., 2021; Ramesh et al., 2022; Saharia et al., 2022) can only generate continuous data (mainly real valued pixels), and have not yet achieved results competitive with autoregressive models in generating discrete/categorical data, such as generating discrete/categorical images (Hoogeboom et al., 2021; Austin et al., 2021).
+
+In this work, we propose a simple and generic approach for enabling continuous state diffusion models to generate discrete data. The key ingredient in our approach is analog bits: real numbers used to model the bits that represent the discrete data. Analog bits can be directly modeled by continuous state diffusion models, without requiring a discrete state space or re-formulation of the continuous diffusion process. At sampling time, the generated analog bits can be decoded into discrete variables by a simple thresholding operation. Our approach, as illustrated in Figure 1, is based on the following high-level conjecture. With strong continuous generative models (diffusion models in particular), it should not be too difficult to generate highly concentrated bimodal data where each real-valued analog bit is close to a binary bit. To reduce the prediction loss (such as negative log likelihood), the network has to model structures among analog bits that can actually lead to meaningful discrete variables after thresholding.
+
+![](images/c0ed764f9997aa81e12b49cd8179d3850e34208d49de4be3680434adec455e9c.jpg)  
+Figure 1: Bit Diffusion: modeling discrete data using continuous diffusion models with analog bits.
+
+Besides analog bits, we further propose two simple techniques, namely Self-Conditioning and Asymmetric Time Intervals that greatly improve the sample quality. We evaluate the proposed approach on both discrete image generation, and image-conditional text / caption generation. On discrete CIFAR-10 and IMAGENET $6 4 \times 6 4$ , the proposed Bit Diffusion model significantly improves both existing discrete diffusion models but also the best autoregressive model. For example, on categorical CIFAR-10, the best autoregressive model (Jun et al., 2020) obtains a FID of 12.75, while our model (with $^ 1 / 3$ of the model size of the autoregressive model, using 100 instead of 3072 sequential inference steps) achieves a much better 6.93. For image captioning on MS-COCO dataset, our model achieves a result competitive with a strong autoregressive captioner based on a Transformer.
+
+# 2 METHOD
+
+Preliminaries We start with a short introduction to diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020; 2021). Diffusion models learn a series of state transitions to map noise $\epsilon$ from a known prior distribution to $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ from the data distribution. To learn this (reverse) transition from the noise distribution to the data distribution, a forward transition from $\scriptstyle { \pmb x } _ { 0 }$ to $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ is first defined:
+
+$$
+{ \pmb x } _ { t } = \sqrt { \gamma ( t ) } { \pmb x } _ { 0 } + \sqrt { 1 - \gamma ( t ) } { \pmb \epsilon } ,
+$$
+
+where $\epsilon \sim \mathcal { N } ( 0 , I )$ , $t \sim \mathcal { U } ( 0 , T )$ is a continuous variable, and $\gamma ( t )$ is a monotonically decreasing function from 1 to 0. Instead of directly learning a neural net to model the transition from $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ to $\pmb { x } _ { t - \Delta }$ , one can learn a neural net $f ( \boldsymbol { x } _ { t } , \dot { t } )$ to predict $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ (or $\epsilon$ ) from $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ , and estimate $\pmb { x } _ { t - \Delta }$ from $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ and estimated $\tilde { \mathbf { x } } _ { 0 }$ (or $\tilde { \epsilon }$ ). This training of $f ( \pmb { x } _ { t } , t )$ is based on denoising with a $\ell _ { 2 }$ regression loss:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { { \pmb x } _ { 0 } } = \mathbb { E } _ { t \sim \mathcal { U } ( 0 , T ) , { \epsilon } \sim \mathcal { N } ( { \mathbf 0 } , { \mathbf 1 } ) } \| f ( \sqrt { \gamma ( t ) } { \pmb x } _ { 0 } + \sqrt { 1 - \gamma ( t ) } { \pmb \epsilon } , t ) - { \pmb x } _ { 0 } \| ^ { 2 } . } \end{array}
+$$
+
+To generate samples from a learned model, it follows a series of (reverse) state transition $\mathbf { \delta } _ { \mathbf { x } _ { T } } \to \mathbf { \delta } $ ${ \pmb x } _ { T - \Delta }  \cdot \cdot \cdot  { \pmb x } _ { 0 }$ . This can be achieved by iteratively applying denoising function $f$ on each state $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ to estimate $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ , and then make a transition to $\mathbf { \delta x } _ { t - \Delta }$ with the estimated $\tilde { \mathbf { x } } _ { 0 }$ (using transition rules such as those specified in DDPM (Ho et al., 2020) or DDIM (Song et al., 2020)). Note that state transitions in these diffusion models assume a continuous data space and state space. Therefore, one cannot directly apply it to model and generate discrete/categorical data.
+
+Analog Bits A discrete data variable from an alphabet of size $K$ can be represented using $n =$ $\lceil \log _ { 2 } K \rceil$ bits, as $\{ 0 , 1 \} ^ { n }$ . Due to the discreteness, existing work has to re-formulate continuous diffusion models by adopting a discrete data space and state space (Sohl-Dickstein et al., 2015; Hoogeboom et al., 2021; Austin et al., 2021). In contrast, we propose to simply cast the binary bits $\{ 0 , \bar  1 \} ^ { n }$ into real numbers $\mathbb { R } ^ { n }$ for the continuous diffusion models 1. We term these real numbers analog bits since they learn to share the same bimodal values as binary bits but are modeled as real numbers. To draw samples, we follow the same procedure as sampling in a continuous diffusion model, except that we apply a quantization operation at the end by simply thresholding the generated analog bits. This yields binary bits which can be then converted into original discrete/categorical variables. Notably, there is no hard constraint to force the model to generate exact binary bits, but we expect a strong continuous generative model to generate real numbers that exhibit very clear bimodal concentrations and this is what happens in our experiments.
+
+![](images/6db7425edf9e8249ebde61d2284a0730646c07a01b44aa9f2ca88cdf548a8357.jpg)  
+(a) Standard reverse diffusion steps.
+
+![](images/f737574d4fc79cdf05a4a018341a712d2e29771633a9a7a0ad5b245f095afdb3.jpg)  
+(b) Self-Conditioning on the previous $\scriptstyle \mathbf { { \vec { x } } } 0$ estimate.   
+Figure 2: An illustration of reverse diffusion sampling steps (a) without or (b) with Self-Conditioning. $\tilde { \mathbf { x } } _ { 0 }$ denotes the estimation of data sample by the denoising network $f$ at a sampling step. We propose to condition the network directly on its previously generated/estimated samples.
+
+For simplicity, we use the same regression loss function (Eq. 2) for modeling analog bits. However, it is possible to use other loss functions such as the cross entropy loss. We also note that the binary encoding mechanism for constructing analog bits is extensible as well (e.g., one-hot encoding). Extensions of loss functions and binary encoding are described in the appendix B.
+
+Self-Conditioning Conditioning is a useful technique for improving diffusion models (Nichol & Dhariwal, 2021; Ho et al., 2022). However, a typical conditioning variable is either from some external sources, such as class labels (Nichol & Dhariwal, 2021) or low-resolution images from another network (Nichol & Dhariwal, 2021; Saharia et al., 2021; Ho et al., 2022). Here we propose a technique for the model to directly condition on previously generated samples of its own during the iterative sampling process, which can significantly improve the sample quality of diffusion models.
+
+In a typical diffusion sampling process, the model iteratively predicts $\scriptstyle { \mathbf { { \vec { x } } } } _ { 0 }$ (or $\epsilon$ ) in order to progress the chain of mapping noise into data. However, as shown in Figure 2a, the previously estimated $\tilde { \mathbf { x } } _ { 0 }$ is simply discarded when estimating $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ from a new time step, i.e. the denoising function $f ( \pmb { x } _ { t } , t )$ does not directly depend on a previously estimated $\tilde { \mathbf { x } } _ { 0 }$ . Here we consider a slightly different denoising function of $f ( \pmb { x } _ { t } , \tilde { \pmb { x } } _ { 0 } , t )$ that also takes previous generated samples as its input, illustrated in Figure 2b. A simple implementation of Self-Conditioning is to concatenate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ with previously estimated $\tilde { \mathbf { x } } _ { 0 }$ . Given that $\tilde { \mathbf { x } } _ { 0 }$ is from the earlier prediction of the model in the sampling chain, this comes at a negligible extra cost during sampling. In order to train the denoising function $f ( \pmb { x } _ { t } , \tilde { \pmb { x } } _ { 0 } , t )$ , we make some small changes to the training. With some probability (e.g., $5 0 \%$ ), we set $\tilde { \pmb { x } } _ { 0 } = { \bf 0 }$ which falls back to modeling without Self-Conditioning. At other times, we first estimate $\tilde { \pmb { x } } _ { 0 } = f ( \pmb { x } _ { t } , \mathbf { 0 } , t )$ and then use it for Self-Conditioning. Note that we do not backpropagate through the estimated $\tilde { \mathbf { x } } _ { 0 }$ so the overall increase of training time is small (e.g., less than $2 5 \%$ ).
+
+Asymmetric Time Intervals Besides Self-Conditioning, we identify another factor, time step $t$ , that can also impact Bit Diffusion models. Time step $t$ is an integral part of both denoising network $f ( \pmb { x } _ { t } , t )$ as well as the state transitions. During a typical reverse diffusion process, the model takes symmetric time intervals (i.e., $\Delta$ as in $t \to t - \Delta _ { * }$ ) for both the state transition and time reduction itself, resulting in the same/shared $t$ for both arguments of $f ( \pmb { x } _ { t } , t )$ . However, we find that, when taking large reverse steps, using asymmetric time intervals, implemented via a simple manipulation of time scheduling at generation, can lead to improved sampling quality for Bit Diffusion models.
+
+More specially, with asymmetric time intervals during the sampling process, we have $f ( \pmb { x } _ { t } , t ^ { \prime } )$ , where $t ^ { \prime } = t + \xi$ and $\xi$ is a small non-negative time difference parameter. Note that training remains unchanged, and the same/shared $t$ is used for both arguments of the $f ( \pmb { x } _ { t } , t )$ . Figure 3 illustrates the effect with a trained Bit Diffusion model, where it is asked to take two reversing steps from a state $x _ { t }$ constructed using the forward diffusion, and it shows that asymmetric time intervals reduce the number of noisy pixels (after thresholding and converting back to discrete variables).
+
+![](images/f72c43281e8ed1f0ec8e1e837f38cc9b0d83da14b82d3a69ed816260e93c9399.jpg)  
+Figure 3: When taking a large reverse step from ${ \bf { x } } _ { t = 0 . 6 }$ to ${ \bf { x } } _ { t = 0 . 1 }$ in Bit Diffusion with maximum time $T = 1 . 0$ , we see that asymmetric time intervals with a positive time difference $\xi$ improve the denoising quality of ${ \mathbf { \mathcal { x } } } _ { t = 0 . 1 }$ (by reducing the number of noisy pixels).
+
+Putting it together Algorithm 1 and 2 summarize the training and sampling algorithms for the proposed Bit Diffusion model with Analog Bits, Self-Conditioning, and Asymmetric Time Intervals (via the td parameter). The proposed changes to the existing diffusion models are highlighted in blue. Note that unlike standard diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Nichol & Dhariwal, 2021), we use a continuous time parameterization between 0 and 1 instead of a fixed discrete time for maximal flexibility but they perform similarly. More details of the algorithm (including some important functions) can be found in Appendix A.
+
+# Algorithm 1 Bit Diffusion training algorithm.
+
+# Algorithm 2 Bit Diffusion sampling algorithm.
+
+def train_loss(x): # Binary encoding: discrete to analog bits. x_bits $=$ int2bit(x).astype(float) x_bits $=$ (x_bits \* 2 - 1) \* scale # Corrupt data. t $=$ uniform(0, 1) eps $=$ normal(mean ${ } = 0$ , std $= 1$ ) x_crpt $=$ sqrt(gamma(t)) \* x_bits + sqrt(1 - gamma(t)) \* eps # Compute self-cond estimate. x_pred $=$ zeros_like(x_crpt) if self_cond and uniform(0, 1) > 0.5: x_pred $=$ net(cat([x_crpt, x_pred], -1), t) x_pred $=$ stop_gradient(x_pred) # Predict and compute loss. x_pred $=$ net(cat([x_crpt, x_pred], -1), t) loss $=$ (x_pred - x_bits)\*\*2 return loss.mean()
+
+def generate(steps, td=0): x_t $=$ normal(mean=0, std=1) x_pred $=$ zeros_like(x_t) for step in range(steps): # Get time for current and next states. t_now $\ c = ~ 1$ - step / steps t_next $=$ max(1 - (step+1+td) / steps, 0) # Predict x_0. if not self_cond: x_pred $=$ zeros_like $( \mathrm { x \_ t } )$ ) x_pred $=$ net(cat([x_t,x_pred],-1), t_now) # Estimate x at t_next. x_t $=$ ddim_or_ddpm_step( x_t, x_pred, t_now, t_next) # Binary decoding to discrete data. return bit2int(x_pred $> 0$ )
+
+# 3 EXPERIMENTS
+
+We experiment with two different discrete data generation tasks, namely discrete/categorical image generation, and image captioning (image-conditional text generation).
+
+# 3.1 EXPERIMENTAL SETUP AND IMPLEMENTATION DETAILS
+
+Datasets We use CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET $6 4 \times 6 4$ (Deng et al., 2009) 2 for image generation experiments. We adopt widely used FID (Heusel et al., 2017) as the main evaluation metric, and it is computed between 50K generated samples and the whole training set. For image captioning, following (Chen et al., 2022), we use MS-COCO 2017 captioning dataset (Lin et al., 2014).
+
+Binary encoding Each pixel consists of 3 sub-pixels (RGB channels), and each sub-pixel is an integer in [0, 256) representing the intensity. Standard continuous generative models cast RGB channels as real numbers and normalize them in $[ - 1 , 1 ]$ . For discrete image generation, we consider three discrete encoding for sub-pixels, namely UINT8, GRAY CODE, and UINT8 (RAND). In UINT8, we use 8-bit binary codes converted from the corresponding sub-pixel integer in [0, 256). In GRAY CODE, we assign 8-bit binary codes uniquely to each sub-pixel integer such that two adjacent integers only differ by 1 bit. And in UINT8 (RAND), we assign 8-bit binary codes to every sub-pixel integer by randomly shuffling the integer-to-bits mapping in UINT8. The binary codes in UINT8 and GRAY CODE are loosely correlated with its original sub-pixel intensities, while UINT8 (RAND) has no correlation so each sub-pixel is a categorical variable. The details of the binary codes and their correlations with sub-pixel intensity can be found in the appendix C. We shift and scale the binary bits from $0 , 1$ to $- 1 , 1$ for the analog bits.
+
+For image captioning, we follow (Chen et al., 2022), and use sentencepiece (Kudo & Richardson, 2018) with a vocabulary of size 32K to tokenize the captions. After tokenization, we encode each token into 15 analog bits using the binary codes converted from the corresponding integer. We set the maximum number of tokens to 64 so the total sequence length is 960 bits. Since we directly model bits, it is also possible to directly work with their byte representations without a tokenizer, but we leave this for future work.
+
+Table 1: Comparison of FIDs on unconditional and class-conditional CIFAR-10. Note that both UINT8 and GRAY CODE are only partial/weakly ordinal (see Appendix C). Our Bit Diffusion achieves state-of-the-art FIDs in generating discrete images, beating the best autoregressive model.   
+
+<table><tr><td>Method</td><td>State space</td><td>FID (Unconditional)</td><td>FID (Conditional)</td></tr><tr><td>On continuous pixels (as reference):</td><td></td><td></td><td></td></tr><tr><td>DDPM (Ho et al., 2020)</td><td>Continuous</td><td>3.17</td><td></td></tr><tr><td>DDPM (our reproduction)</td><td>Continuous</td><td>3.14</td><td>2.95</td></tr><tr><td>On discrete (partial) ordinal pixels:</td><td></td><td></td><td></td></tr><tr><td>D3PM Gauss+Logistic (Austin et al., 2021)</td><td>Discrete</td><td>7.34</td><td></td></tr><tr><td>TLDR-10 (Campbell et al., 2022)</td><td>Discrete</td><td>3.74</td><td></td></tr><tr><td>Bit Diffusion on UINT8</td><td>Continuous</td><td>3.48</td><td>2.72</td></tr><tr><td>Bit Diffusion on GRAY CODE</td><td>Continuous</td><td>3.86</td><td>2.94</td></tr><tr><td>On categorical pixels:</td><td></td><td></td><td></td></tr><tr><td>D3PM uniform (Austin et al., 2021)</td><td>Discrete</td><td>51.27</td><td></td></tr><tr><td>D3PM absorbing (Austin et al.,2021)</td><td>Discrete</td><td>30.97</td><td></td></tr><tr><td>Autoregressive Transformer (Jun et al.,2020)</td><td>Discrete</td><td>12.75</td><td></td></tr><tr><td>Bit Diffusion on UINT8 (RAND)</td><td>Continuous</td><td>6.93</td><td>6.43</td></tr></table>
+
+Table 2: Comparison of FIDs on class-conditional IMAGENET $6 4 \times 6 4$ . The corresponding samples can be found in Figure 4 and 11.
+
+<table><tr><td>DDPM (our repo.) on continuous pixels</td><td>Bit Diffusion on UINT8</td><td>Bit Diffusion On GRAY CODE</td><td>Bit Diffusion On UINT8 (RAND)</td></tr><tr><td>3.43</td><td>4.84</td><td>5.14</td><td>8.76</td></tr></table>
+
+Architecture We use the U-Net architecture (Ho et al., 2020; Nichol & Dhariwal, 2021; Ronneberger et al., 2015) for image generation. For CIFAR-10, we use a single channel dimension of 256, 3 stages and 3 residual blocks (He et al., 2016) per stage, with a total of 51M parameters. We only use dropout (Srivastava et al., 2014) of 0.3 for continuous diffusion models on CIFAR-10. For IMAGENET $6 4 \times 6 4$ , following (Nichol & Dhariwal, 2021), we use a base channel dimension of 192, multiplied by 1,2,3,4 in 4 stages and 3 residual blocks per stage, which account for a total of 240M parameters 3. For UINT8 (RAND) encoding, we find the following “softmax factorization” architectural tweak on the final output layer can lead to a better performance. Instead of using a linear output layer to predict analog bits directly, we first predict a probability distribution over 256 classes per sub-pixel (with each class corresponds to one of the 256 different 8-bit codes), and then map class distribution into analog bits by taking weighted average over all 256 different 8-bit codes.
+
+For image captioning, we follow the architecture used in (Chen et al., 2021; 2022), with a pre-trained image encoder using the object detection task, for both autoregressive baseline as well as the proposed method. Both decoders are randomly initialized 6-layer Transformer (Vaswani et al., 2017) decoder with 512 dimension per layer. For the autoregressive decoder, the token attention matrix is offset by the causal masks, but it is non-masked all-to-all attention for our Bit Diffusion.
+
+Other settings We train our models with the Adam optimizer (Kingma & Ba, 2014). For CIFAR-10, we train the model for $1 . 5 { \bf M }$ steps with a constant learning rate of 0.0001 and batch size of 128. For IMAGENET $6 4 \times 6 4$ , we train the model for 500K steps with a constant learning rate of 0.0002 4 and batch size of 1024. For Bit Diffusion, we use Self-Conditioning by default, unless otherwise specified. We use an exponential moving average of the weights during training with a decay factor of 0.9999. For our best image generation results, we sweep over a few sampling hyper-parameters, such as sampler (DDIM vs DDPM), sampling steps in $\{ 1 0 0 , 2 5 0 , 4 0 0 , 1 0 0 0 \}$ , and time difference in $\{ 0 . , 0 . 0 1 , 0 . { \overset { \cdot } { 1 } } , 0 . 2 , 0 . 5 \}$ .
+
+# 3.2 DISCRETE IMAGE GENERATION
+
+We compare our model against state-of-the-art generative models (Ho et al., 2020; Austin et al., 2021; Campbell et al., 2022; Jun et al., 2020) on generating discrete CIFAR-10 images in Table 1. Our model achieves better results compared to both existing discrete diffusion models and the best autoregressive model. When compared to continuous diffusion models (i.e., DDPM), our Bit Diffusion models on UINT8 and GRAY CODE can achieve similar performance.
+
+![](images/07c841d2d466ccb4fe24e93ec5f8d90ad34328248073fe60f875375244969167.jpg)  
+Figure 4: Class-conditional generations on continuous v.s. discrete ImageNet $6 4 \times 6 4$ . Each row represents random samples conditioned on a class, and the classes are adopted from (Nichol & Dhariwal, 2021), namely, 9: ostrich, 11: goldfinch, 130: flamingo, 141: redshank, 154: pekinese, 157: papillon, 97: drake and 28: spotted salamander. More samples from random classes are shown in Figure 11.
+
+Discrete generation of IMAGENET $6 4 \times 6 4$ is significantly harder than CIFAR-10, and we have not found other competing methods that report FIDs, so we only compare the proposed method against DDPM on continuous pixels. Results are shown in Table 2. We find that the diffusion model on continuous pixels has the best FID while the diffusion model on UINT8 (RAND), i.e., categorical data, has the worst FID, indicating the increase of hardness when removing intensity/order information in sub-pixels. Note that, in these experiments, there is no extra model capacity to compensate for the loss of intensity/order information since the model sizes are the same. Figure 4 shows generated images of different diffusion models on continuous and discrete IMAGENET $6 4 \times 6 4$ . Despite the differences in FIDs, visually these samples look similar.
+
+Ablation of Self-Conditioning Figure 5 shows the effectiveness of the Self-Conditioning technique in both Bit Diffusion and continuous diffusion models. Note that the experiments are performed in three settings, namely CIFAR-10 with UINT8, CIFAR-10 with UINT8 (RAND), and IMAGENET
+
+![](images/f1829098c0e53d5b75d339e11962393f4d8d156eddb8d59297c587f22cdabe85.jpg)  
+Figure 5: Self-conditioning is a generic technique that not only greatly improves Bit Diffusion but also leads to improved results for continuous diffusion models.
+
+![](images/a222e73539e23289a138d993c86b82a68f7a1323b7a8eec48d0c2856682e5f18.jpg)  
+Figure 6: Effect of time difference in class-conditional IMAGENET $6 4 \times 6 4$ . Optimal time difference shrinks to zero as the number of sampling steps increases. For 100 sampling steps, non-zero time difference leads to improved FIDs.
+
+$6 4 \times 6 4$ with continuous pixels, where the only difference for pairs in each setting is whether the Self-Conditioning is used. For CIFAR-10, we find that Self-Conditioning greatly improves the performance across different binary encodings. We also notice that for Bit Diffusion, predicting $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ is much more effective than predicting . For IMAGENET $6 4 \times 6 4$ , we find that the proposed Self-Conditioning also leads to improved FIDs for continuous diffusion (i.e., DDPM). Therefore, we conclude that Self-Conditioning by itself is a generic technique that can benefit diffusion models on both continuous and discrete data.
+
+Ablation of asymmetric time intervals Figure 6 shows the FID on generated IMAGENET $6 4 \times 6 4$ samples as we vary the time difference parameter during the sampling process. We find that as the number of steps increases (from 100 to 400), the optimal time difference shrinks to 0. For 100 steps, a non-zero time difference leads to a significant improvement of FID. We also note that for Bit Diffusion on UINT8 (RAND), using 400 sampling steps actually leads to a drastically worse sample quality than using 100 steps. This is related to how the Self-Conditioning is applied and we present alternative Self-Conditioning sampling strategies in the Appendix G, some of which lead to improved FIDs at a cost of longer sampling time.
+
+Concentration of generated analog bits Figure 7 visualizes the distribution of generated analog bits from 64 generated images on IMAGENET $6 4 \times 6 4$ . Although there is no hard constraint on the analog bits being binary / bimodal, the generated ones are highly concentrated on two modes, which makes the thresholding / quantization easy and robust.
+
+![](images/2711cf1eb2ef77e5962d7c44640a23d9e8daf8fe2b1b075ac8e049e8405a0470.jpg)  
+Figure 7: Histogram distribution (50 bins) of the analog bits from 64 randomly generated IMAGENET $6 4 \times 6 4$ samples at $\tilde { { \boldsymbol { x } } } _ { 0 }$ , with 100 DDIM steps. Most of the generated analog bits are very concentrated.
+
+# 3.3 IMAGE CAPTIONING
+
+We compare our Bit Diffusion model with an autoregressive Transformer baseline (Chen et al., 2022). As mentioned, both models have similar architectures, with an object detection pretrained (Chen et al., 2021) image encoder, and a randomly initialized Transformer (Vaswani et al., 2017) decoder. Table 3 presents the main comparison. Overall, our model achieves similar performance as the autoregressive model. We find that generally it only needs about 10 steps for the model to achieve good results, despite that there are a total of maximum 960 bits for caption that the model has to model. We find that the asymmetric time intervals play an important role in the final performance of our model, as demonstrated in Table 4, especially when sampling steps are fewer.
+
+Table 3: Image captioning results on MS-COCO dataset with a randomly initialized text decoder.   
+
+<table><tr><td>Method</td><td>BLEU-4</td><td>CIDEr</td><td>ROUGE-L</td></tr><tr><td>Autoregressive Transformer</td><td>33.9</td><td>1.18</td><td>0.57</td></tr><tr><td>Bit Diffusion (5 steps)</td><td>31.5</td><td>1.00</td><td>0.55</td></tr><tr><td>Bit Diffusion (10 steps)</td><td>34.5</td><td>1.13</td><td>0.57</td></tr><tr><td>Bit Diffusion (20 steps)</td><td>34.7</td><td>1.15</td><td>0.58</td></tr><tr><td>Bit Diffusion (40 steps)</td><td>34.4</td><td>1.15</td><td>0.57</td></tr></table>
+
+Table 4: Asymmetric time intervals significantly improves the performance of Bit Diffusion.   
+Table 5: Generated image captions under different number of sampling steps.   
+
+<table><tr><td rowspan="2"></td><td colspan="9">Time difference</td></tr><tr><td>0.0</td><td>1.0</td><td>2.0</td><td>3.0</td><td>4.0</td><td>5.0</td><td>6.0</td><td>7.0</td><td>8.0</td></tr><tr><td>5 steps</td><td>17.1</td><td>27.8</td><td>30.8</td><td>31.5</td><td>31.6</td><td>31.5</td><td>31.5</td><td>31.5</td><td>31.6</td></tr><tr><td>10 steps</td><td>17.6</td><td>26.3</td><td>30.7</td><td>32.6</td><td>33.4</td><td>34.0</td><td>34.3</td><td>34.5</td><td>34.6</td></tr><tr><td>20 steps</td><td>20.0</td><td>27.9</td><td>30.6</td><td>32.0</td><td>32.3</td><td>33.9</td><td>34.4</td><td>34.7</td><td>34.5</td></tr><tr><td>40 steps</td><td>20.7</td><td>27.5</td><td>30.7</td><td>32.2</td><td>32.9</td><td>33.2</td><td>33.8</td><td>34.4</td><td>34.4</td></tr></table>
+
+Table 5 provides some generated samples of our model when different inference steps are used. The model makes mistakes when the sampling steps are too few, and the mistakes may not always be interpretable due to that the model directly predicts the bits behind the tokenized word pieces and a small difference in bits can lead to total different words.
+
+![](images/9aac52c53817dd45cbdf8027351f9eb9775cd9bd356b23f8455de14c97f024e6.jpg)
+
+# 4 RELATED WORK
+
+Autoregressive models for discrete data Autoregressive models have demonstrated state-of-the-art results when it comes to generating discrete data. In particular, text generation, or language modeling, is dominated by autoregressive approaches (Sutskever et al., 2014; Brown et al., 2020; Chowdhery et al., 2022). Autoregressive models are also applied to discrete/categorical image generation (Van den Oord et al., 2016; Salimans et al., 2017; Parmar et al., 2018; Child et al., 2019; Roy et al., 2021; Jun et al., 2020; Chen et al., 2020a), where they work well on small image resolutions. However, the computation cost and memory requirement increase drastically (typically in a quadratic relation) as the size of sequence or the image resolution increase, so it becomes very challenging to scale these approaches to data with large dimensions.
+
+Diffusion models for discrete data State-of-the-art diffusion models (Dhariwal & Nichol, 2021; Ho et al., 2022; Nichol et al., 2021; Ramesh et al., 2022; Saharia et al., 2022) cannot generate discrete or categorical data. Existing extensions of these continuous diffusion models to discrete data are based on both discrete data space and state space (Sohl-Dickstein et al., 2015; Hoogeboom et al., 2021; Austin et al., 2021; Campbell et al., 2022). Compared to discrete state space, continuous state space is more flexible and potentially more efficient. Our approach is also compatible with both discrete and continuous time, and does not require re-formulation of existing continuous models, thus it is simpler and can potentially be plugged into a broader family of generative models.
+
+Another line of discrete diffusion models is based on the embedding of discrete data (Li et al., 2022). One can also consider our binary encoding with analog bits as a simple fixed encoder, and the decoding / quantization of bimodal analog bits is easy and robust via a simple thresholding operation. In contrast, the quantization of real numbers in generated continuous embedding vectors may contain multiple modes per dimension, leading to potential difficulty in thresholding/quantization.
+
+Normalizing Flows for discrete data Normalizing Flows (Rezende & Mohamed, 2015; Dinh et al., 2017; Kingma & Dhariwal, 2018) are a powerful family of generative models for high-dimensional continuous distributions based on some invertible mapping. However, straightforward application of flow-based models on categorical data is limited due to the inherent challenges on discrete support. Discrete flows (Tran et al., 2019; Hoogeboom et al., 2019; Lindt & Hoogeboom, 2021) introduce invertible transformations of random variables in discrete space without the need of computing the log-determinant of Jacobian. Other works (Lippe & Gavves, 2021; Hoogeboom et al., 2021; Tan et al., 2021) introduce various embedding methods for transforming discrete data into continuous space with disjoint support, which can be interpreted as a variational inference problem (Theis et al., 2015) with different dequantization distribution families. Several works (Kingma et al., 2016; Ziegler & Rush, 2019; Zhang et al., 2020) also explore normalizing flows on discrete data under the Variational Autoencoders (Kingma & Welling, 2013) framework by enriching the prior. Compared to our diffusion-based approach, these models suffer from strict invertible restrictions on network architecture, thus limiting their capacity.
+
+Other generative models for discrete data Other generative models, such as Varational Autoencoders (VAE) (Kingma & Welling, 2013), Generateive Adversarial Networks (GAN) (Goodfellow et al., 2014; Yu et al., 2017; Che et al., 2017; Hjelm et al., 2017; Fedus et al., 2018) have also been applied to generate discrete data. These methods have not yet achieved the level of performance as autoregressive models on tasks such as discrete image generation or text generation, in terms of sample quality or data likelihood. Potentially, the proposed analog bits can also be applied to these continuous generative models, by having the networks directly model and generate analog bits, but it is not explored in this work.
+
+Other related work The proposed Self-Conditioning technique shares some similarities with selfmodulation in GANs (Chen et al., 2018a) (where the earlier latent state can directly modulate the later latent states) and SUNDAE (Savinov et al., 2021) (where an inference step is incorporated for denoising).
+
+# 5 CONCLUSION
+
+We introduce a simple and generic technique that enables continuous state diffusion models to generate discrete data. The main idea is to encode discrete or categorical data into bits and then model these bits as real numbers that we call analog bits. We also propose two simple techniques, namely Self-Conditioning (i.e., condition the diffusion models directly on their previously generated samples) and Asymmetric Time Intervals, that lead to improved sample quality. We demonstrate that our approach leads to state-of-the-art results in discrete / categorical image generation, beating the best autoregressive model. In an image-conditional text generation task on MS-COCO dataset, we also achieve competitive results compared to autoregressive models. One limitation of our approach, similar to other existing diffusion models, is that they still require a significant number of inference steps for generating good (image) samples. However, we expect that future improvements from diffusion models for continuous data can also transfer to discrete data using analog bits.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Priyank Jaini, Kevin Swersky for providing helpful feedback to our draft. Our implementation is partially based on the Pix2Seq codebase, and we thank Lala Li, Saurabh Saxena, for their contributions to the Pix2Seq codebase.
+
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+
+# A MORE DETAILS OF ALGORITHM 1 AND 2
+
+Algorithm 3 and 4 provide more detailed implementations of functions in Algorithm 1 and 2.
+
+Algorithm 3 Binary encoding and decoding algorithms (in Tensorflow).
+
+import tensorflow as tf
+
+def int2bit(x, $\mathrm { n } { = } 8$ ): # Convert integers into the corresponding binary bits. $\textrm { \textbf { x } } =$ tf.bitwise.right_shift(tf.expand_dims(x, -1), tf.range(n)) $\textrm { \textbf { x } } =$ tf.math.mod(x, 2) return x
+
+def bit2int(x): # Convert binary bits into the corresponding integers. $\textrm { \textbf { x } } =$ tf.cast(x, tf.int32) n $=$ x.shape[-1] $\textrm { \textbf { x } } =$ tf.math.reduce_sum(x \* (2 \*\* tf.range(n)), -1) return x
+
+# Algorithm 4 $x _ { t }$ estimation with DDIM / DDPM updating rules.
+
+def gamma(t, $\scriptstyle \mathrm { n s = 0 } . 0 0 0 2$ , ds=0.00025): # A scheduling function based on cosine function. return numpy.cos(((t + ns) / (1 + ds)) $\star$ numpy.pi / 2)\*\*2 def ddim_step(x_t, x_pred, t_now, t_next): # Estimate x at t_next with DDIM updating rule. γnow $=$ gamma(t_now) γnext $=$ gamma(t_next) x_pred $=$ clip(x_pred, -scale, scale) eps = √ 11−γnow (x_t - γnow \* x_pred) x_next $\mathbf { \Sigma } = \mathbf { \Sigma } \sqrt { \gamma _ { \mathrm { n e x t } } }$ \* x_pred $^ +$ $\sqrt { 1 - \gamma _ { \mathrm { n e x t } } }$ \* eps return x_next
+
+def ddpm_step(x_t, x_pred, t_now, t_next): # Estimate x at t_next with DDPM updating rule. γnow $=$ gamma(t_now) $\alpha _ { \Omega \circ w } =$ gamma(t_now) / gamma(t_next) $\sigma _ { \mathrm { n o w } } =$ sqrt $\mathrm { ~ ( ~ 1 ~ ~ - ~ } ~ \alpha _ { \mathrm { n o w } }$ ) $z =$ normal(mean ${ } = 0$ , std $^ { = 1 }$ ) x_pred $=$ clip(x_pred, -scale, scale) eps $= { \frac { 1 } { \sqrt { 1 - \gamma _ { \mathrm { n o w } } } } }$ \* (x_t - γnow \* x_pred) x_next = √ 1αnow \* (x_t - 1−γnow √1−αnow \* eps) + σnow \* z return x_next
+
+# B ALTERNATIVE BINARY ENCODING AND LOSS FUNCTIONS
+
+B.1 ANALOG BITS BASED ON ONE-HOT ENCODING
+
+An alternative binary encoding to the base-2 encoding of the discrete data used in the main paper, is the one-hot encoding, where a discrete variable is represented as a vector whose length is the same as the vocabulary size $K$ , with a single slot being 1 and the rest being 0. The resulting one-hot vector can be similarly treated as analog bits and modeled by continuous state diffusion models. To obtain discrete variables corresponding to the generated analog bits, we use an arg max operation over all candidate categories, instead of the thresholding operation in base-2 analog bits. Note that the one-hot encoding requires $K$ bits, which is less efficient compared to base-2 encoding that only requires $\lceil \log _ { 2 } K \rceil$ bits, especially for large $K$ . 5
+
+# B.2 SIGMOID CROSS ENTROPY LOSS
+
+As we use $\ell _ { 2 }$ loss by default for its simplicity and compatibility with continuous diffusion models. The proposed Bit Diffusion models can work with other loss functions too. Since the analog bits are bimodal, we can use the following sigmoid cross entropy loss:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \pmb { x } _ { 0 } , \pmb { x } _ { t } , t } = \log \sigma ( \pmb { x } _ { 0 } f ( \pmb { x } _ { t } , t ) ) , } \end{array}
+$$
+
+where we assume $\pmb { x } _ { 0 } \in \{ - 1 , 1 \} ^ { n }$ , and $\sigma$ is a sigmoid function. During the sampling process, we use $2 \sigma ( f ( \pmb { x } _ { t } , t ) ) - 1$ as the output of denoising network.
+
+# B.3 SOFTMAX CROSS ENTROPY LOSS
+
+For one-hot analog bits, one could also add a softmax activation function for the output of denosing network $f$ , and use the following softmax cross entropy loss:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \pmb { x } _ { 0 } , \pmb { x } _ { t } , t } = \pmb { x } _ { 0 } \log \mathrm { s o f t m a x } ( f ( \pmb { x } _ { t } , t ) ) , } \end{array}
+$$
+
+where we assume $\pmb { x } _ { 0 } \in \{ 0 , 1 \} ^ { n }$ which is the one-hot representation.
+
+# B.4 PRELIMINARY EXPERIMENTS
+
+Table 6 presents FIDs of Bit Diffusion models with different types of analog bits and loss functions on unconditional CIFAR-10. Note that it is possible some of these results can be improved by more tuning of hyper-parameters or tweaks of the network, but we do not focus on them in this work.
+
+Table 6: FIDs of Bit Diffusion models with different types of analog bits and loss functions on unconditional CIFAR-10.   
+
+<table><tr><td></td><td>l2 loss</td><td>Logistic loss</td><td>Softmax loss</td></tr><tr><td>ONE HOT</td><td>46.32</td><td>26.82</td><td>29.49</td></tr><tr><td>UINT8</td><td>3.48</td><td>3.53</td><td>1</td></tr><tr><td>GRAY CODE</td><td>3.86</td><td>3.71</td><td>=</td></tr><tr><td>UINT8 (RAND)</td><td>6.93</td><td>49.29</td><td>-</td></tr></table>
+
+C ON BINARY ENCODING OF PIXELS: UINT8, GRAY CODE, UINT8 (RAND)
+
+In the main paper, we describe three different types of binary encodings of pixels. Here we provide additional detail on how we generate UINT8 (RAND): we first apply a random permutation to 256 sub-pixel values, and then assign the binary binary bits of permuted integers to the non-permuted integers. For example, assume 0 is mapped to 228 after the permutation, the analog bits of 0 would be the binary bits of 228. The random permutation is generated by numpy.random.seed(42); numpy.random.shuffle(numpy.arange(256)).
+
+![](images/5c5048a370cbdae3ad4f4add0ee59c4dc8d0e03265bf05ba9e1544806945fd40.jpg)  
+Figure 8: Correlation between (absolute) difference in subpixel intensity and the Hamming distance of the corresponding binary bits.
+
+Figure 8 show the correlation between Hamming distance of three different binary encodings we use and the (absolute) difference of sub-pixel intensity. This is done by taking every pair of subpixel integers (in [0, 256)), compute their absolute difference, as well as the Hamming distance between the corresponding binary bits. We find that both UINT8 and GRAY CODE exhibit partial correlation between the two quantities (with different correlation patterns), meaning that these codes partially contain the order information about the original sub-pixel intensity. However, UINT8 (RAND) exhibits no correlation between hamming distance and sub-pixel intensity, indicating the order information is fully removed, thus can be considered as categorical data.
+
+# D A TOY EXAMPLE ON CONTINUOUS MODELING OF DISCRETE VARIABLES
+
+An intuitive toy example of how a continuous generative model can generate binary data is given in Figure 9, where a mapping from prior distribution at $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } T }$ to data distribution at $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ is shown. With a deterministic sampler (such as DDIM), it is straight-forward how they can represent any Bernoulli distribution by dividing the prior into two regions of probability densities corresponding to the Bernoulli distribution. For stochastic samplers, they can achieve a similar effect but the mapping from noise to data is stochastic. For an arbitrary discrete variable, represented as m-dimensional Bernoulli distribution, the mapping from continuous noise distribution to the target Bernoulli distribution also exists but it is more complicated (and difficult to visualize).
+
+![](images/5f6d559af6c6f590dc65c06be016cdac1595cb78b27323a2db579ba392e43fe2.jpg)  
+Figure 9: A toy example on continuous modeling of discrete variables.
+
+# E ON OTHER SAMPLERS FOR CONTINUOUS DIFFUSION MODELS
+
+As our models are based on continuous diffusion models, in theory our models are able to incorporate faster samplers. To this end, we conduct preliminary exploration of using DPM-Solver (Lu et al., 2022) for sampling some of our models.
+
+We find that DPM-Solver provides a boost to diffusion models based on analog bits, similar to what it is able to do for continuous data. This shows a potential of our model enjoying faster continuous sampler while other baselines (e.g., D3PM) may not be able to do due to their use of discrete states. Table 7 below shows the FID scores of bit diffusion models on ImageNet-64x64 under different binary encoding schemes. We find that the DPM-Solver is able to provide a significant reduction in function evaluations for bit diffusion on discrete/categorical data (with 30 NFEs it gets comparable FIDs as 100 NFEs of DDIM), similar to that in continuous diffusion models.
+
+Furthermore, we also find that self-conditioning continues to provide a boost with DPM-solver. For example, the table 8 shows FID scores of diffusion models on ImageNet 64x64 (continuous rgb values). And we find that the self-conditioning consistently improves the performance of DPM-Solver with fixed number of function evaluations.
+
+# F EXTRA RANDOM SAMPLES ON CIFAR-10 AND IMAGENET $6 4 \times 6 4$
+
+Figure 10 shows random samples (non cherry-picked) from unconditional diffusion models on CIFAR-10 with continuous pixels and analog bits.
+
+Figure 11 shows random samples (non cherry-picked) from class-conditional diffusion models on IMAGENET $6 4 \times 6 4$ with continuous pixels and analog bits.
+
+Table 7: Comparison of Continuous Diffusion Samplers. FIDs on ImageNet $6 4 \times 6 4$ shown below.   
+
+<table><tr><td>Samplers</td><td>UINT8</td><td>Gray Code</td><td>UINT8 (RAND)</td></tr><tr><td>DDIM @ 1000 NFE</td><td>5.51</td><td>5.14</td><td>58.45</td></tr><tr><td>DDPM @ 1000 NFE</td><td>7.71</td><td>6.91</td><td>64.08</td></tr><tr><td>DDIM @ 400 NFE</td><td>5.00</td><td>5.52</td><td>38.44</td></tr><tr><td>DDPM @ 400 NFE</td><td>4.84</td><td>5.37</td><td>40.91</td></tr><tr><td>DDIM @ 100 NFE</td><td>8.80</td><td>11.31</td><td>8.76</td></tr><tr><td>DDPM @ 100 NFE</td><td>13.04</td><td>12.77</td><td>9.25</td></tr><tr><td>DPM-Solver @ 30 NFE</td><td>7.85</td><td>9.64</td><td>10.39</td></tr><tr><td>DPM-Solver @ 50 NFE</td><td>6.46</td><td>7.61</td><td>10.96</td></tr></table>
+
+Table 8: The effect of Self-Conditioning for sampling with DPM-Solver. FIDs on ImageNet $6 4 \times 6 4$ shown below.   
+
+<table><tr><td>Model</td><td>DPM-Solver @ 20 NFE|DPM-Solver @ 30 NFE</td><td></td></tr><tr><td>E prediction, w/o self-conditioning e prediction, w/ self-conditioning</td><td>6.10 4.24</td><td>5.58 4.15</td></tr><tr><td>xo prediction, w/o self-conditioning xo prediction, w/ self-conditioning</td><td>12.13 6.94</td><td>11.05 6.43</td></tr></table>
+
+![](images/39950aa93042e67fdc28e63a17ee0ba480d0808b29d949a36ad47b1fd1319276.jpg)  
+Figure 10: Random samples from unconditional models trained on CIFAR-10. (a) is for continuous image generation, (b), (c), and (d) are for discrete/categorial image generation.
+
+![](images/0087973f3f7fbbac5ed2501a0fe621a7bbfa949861342d891ce84a7ba0cba029.jpg)  
+Figure 11: Random samples from class-conditional models trained on IMAGENET $6 4 \times 6 4$ . (a) is for continuous image generation, (b), (c), and (d) are for discrete/categorial image generation.
+
+# G ON SAMPLING STRATEGIES WITH SELF-CONDITIONING
+
+# G.1 METHOD
+
+In this section, we present extensions to the default sampling strategy with Self-Conditioning in Algorithm 2. The default sampling strategy utilizes data estimate from the previous step as the conditional input to the denoising network for producing data estimate at the current step. While this is both simple and effective, we observe that, for UINT8 (RAND) encoding of pixels, as the number of sampling steps increases (with both DDIM or DDPM samplers), the generated samples tend to be over-smoothed. We propose the following two extensions of the default sampling strategy to mitigate the issues and provide improvements when using larger sampling steps.
+
+Self-Conditioning based on Momentum Estimate The first extension to the default sampling strategy is to adopt an exponential moving average over the previous data estimate to provide a more reliable conditioning input, similar to a momentum optimizer. The detailed procedure is shown in algorithm 5, where the differences from the default sampling strategy are highlighted in blue. Note that the default sampling strategy can also be considered as a special case of this generalized form in that the momentum is set to zero.
+
+Self-Conditioning based on Self-Guidance One potential issue with the default sampling strategy is the slight discrepancy of the Self-Conditioning signal during training and inference/sampling. Specifically, during training, the Self-Conditioning signal is the data estimate from the same time step, while, during sampling, it is from the past time step(s). Therefore, here we propose an approach that also use the same step data estimate for self-conditioning, which comes at the cost of extra forward pass over the denoising network at sampling time. Specifically, we conduct two forward passes of denoising network per sampling step, one with zero data estimate and the other with current data step estimate, and then we use a weighted combination, similar to (Ho & Salimans, 2021), of both prediction to form the final prediction at the current step. The detailed procedure is given in algorithm 6 with differences to the default sampling strategy highlighted.
+
+# Algorithm 5 Sampling with Self-Conditioning based on Momentum Estimate.
+
+# Algorithm 6 Sampling with Self-Conditioning based on Self-Guidance.
+
+![](images/0056fbc804ba1f5cfe054d28ced48060002aa3630bff0e17c22d7352b912d3ce.jpg)
+
+# G.2 EXPERIMENTS
+
+Table 9 reports the best FID scores across various sampling strategies discussed here (as well as samplers, sampling steps, time difference in asymmetric time intervals).
+
+Table 9: Best FIDs of Bit Diffusion models with different Self-Conditioning sampling strategies on conditional IMAGENET $6 4 \times 6 4$ .   
+
+<table><tr><td></td><td>UINT8</td><td>GRAY CODE</td><td>UINT8 (RAND)</td></tr><tr><td>Default sampling (momentum= 0)</td><td>4.84</td><td>5.14</td><td>8.76</td></tr><tr><td>Momentum Estimate</td><td>4.85</td><td>5.14</td><td>8.51</td></tr><tr><td>Self-Guidance</td><td>5.15</td><td>5.65</td><td>7.87</td></tr></table>
+
+Figure 12 shows FIDs on conditional IMAGENET $6 4 \times 6 4$ with UINT8 encoding, using Momentum Estimate with different sampling steps. We find that the momentum on the data estimate is only helpful when sampling steps are larger.
+
+Figure 13 shows FIDs on conditional IMAGENET $6 4 \times 6 4$ with UINT8 encoding, using Self-Guidance with different sampling steps. We find that a guidance weight between 3.0 and 5.0 is generally preferable and robust to other hyper-parameters (such as sampler choice, sampling steps, and time difference).
+
+# G.3 SAMPLES
+
+Figure 14 and 15 provide generated samples from different sampling strategies with 100 and 1000 DDIM sampling steps, respectively.
+
+![](images/e0cc712c3974b0d7aa91d39f7f4ca1200feea15ccb966b114d05e815e0f4bdfa.jpg)
+
+![](images/8d8810ce1d18fb7fcccc7187ed9ecd89bfb6e4e3edc67cf6004e1873fe0127b4.jpg)  
+Figure 12: FID on conditional IMAGENET $6 4 \times 6 4$ with UINT8 (RAND) encoding using selfcondition sampling based on momentum estimate. The statistics of FID scores in each group are aggregated over the number of sampling steps in $\{ 1 0 0 , 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 , 1 0 0 0 \}$ , time difference in $\{ 0 . 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 \}$ .   
+Figure 13: FID on conditional IMAGENET $6 4 \times 6 4$ with UINT8 (RAND) encoding using self-condition sampling based on self-guidance. The statistics of FID scores in each group are aggregated over the number of sampling steps in $\{ 1 0 0 , 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 , 1 0 0 0 \}$ , time difference in $\{ 0 . 0 , 0 . 1 \}$ .
+
+![](images/98c330bd3cfa80d1b5cfa628e7d92a67251242f56ef91dce4e362169079ed40f.jpg)  
+Figure 14: Random samples of Bit Diffusion with UINT8 (RAND) on categorical IMAGENET $6 4 \times 6 4$ using various Self-Conditioning sampling strategies. Different plots share the same set of $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } \mathcal { T } }$ . Sampling with 100 steps of DDIM without asymmetric time intervals.
+
+![](images/b0406388784e46f6aaa31871b6b6b44bf9fb3267ea5db5e28d46a84a39b8b3f1.jpg)  
+Figure 15: Random samples of Bit Diffusion with UINT8 (RAND) on categorical IMAGENET $6 4 \times 6 4$ using various Self-Conditioning sampling strategies. Different plots share the same set of $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } \mathcal { T } }$ . Sampling with 1000 steps of DDIM without asymmetric time intervals.