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+# SHAPE-TEXTURE DEBIASED NEURAL NETWORK TRAINING
+
+Yingwei $\mathbf { L i } ^ { 1 }$ , Qihang $\mathbf { Y u } ^ { 1 }$ , Mingxing $\mathbf { T a n } ^ { 2 }$ , Jieru Mei1, Peng Tang1, Wei Shen3   
+Alan Yuille1 & Cihang Xie4   
+1Johns Hopkins University 2Google Brain 3Shanghai Jiaotong University   
+4University of California, Santa Cruz
+
+# ABSTRACT
+
+Shape and texture are two prominent and complementary cues for recognizing objects. Nonetheless, Convolutional Neural Networks are often biased towards either texture or shape, depending on the training dataset. Our ablation shows that such bias degenerates model performance. Motivated by this observation, we develop a simple algorithm for shape-texture debiased learning. To prevent models from exclusively attending on a single cue in representation learning, we augment training data with images with conflicting shape and texture information (e.g., an image of chimpanzee shape but with lemon texture) and, most importantly, provide the corresponding supervisions from shape and texture simultaneously.
+
+Experiments show that our method successfully improves model performance on several image recognition benchmarks and adversarial robustness. For example, by training on ImageNet, it helps ResNet-152 achieve substantial improvements on ImageNet $( + 1 . 2 \% )$ , ImageNet-A $( + 5 . 2 \% )$ , ImageNet-C $( + 8 . 3 \% )$ and Stylized-ImageNet $( + 1 1 . 1 \% )$ , and on defending against FGSM adversarial attacker on ImageNet $( + 1 4 . 4 \% )$ . Our method also claims to be compatible to other advanced data augmentation strategies, e.g., Mixup and CutMix. The code is available here: https://github.com/LiYingwei/ ShapeTextureDebiasedTraining.
+
+# 1 INTRODUCTION
+
+It is known that both shape and texture serve as essential cues for object recognition. A decade ago, computer vision researchers had explicitly designed a variety of hand-crafted features, either based on shape (e.g., shape context (Belongie et al., 2002) and inner distance shape context (Ling & Jacobs, 2007)) or texture (e.g., textons (Malik et al., 2001)), for object recognition. Moreover, researchers found that properly combining shape and texture can further recognition performance (Shotton et al., 2009; Zheng et al., 2007), demonstrating the superiority of possessing both features.
+
+Nowadays, as popularized by Convolutional Neural Networks (CNNs) (Krizhevsky et al., 2012), the features used for object recognition are automatically learned, rather than manually designed. This change not only eases human efforts on feature engineering, but also yields much better performance on a wide range of visual benchmarks (Simonyan & Zisserman, 2015; He et al., 2016; Girshick et al., 2014; Girshick, 2015; Ren et al., 2015; Long et al., 2015; Chen et al., 2015). But interestingly, as pointed by Geirhos et al. (2019), the features learned by CNNs tend to bias toward either shape or texture, depending on the training dataset.
+
+We verify that such biased representation learning (towards either shape or texture) weakens CNNs’ performance.1 Nonetheless, surprisingly, we also find (1) the model with shape-biased representations and the model with texture-biased representations are highly complementary to each other, e.g., they focus on completely different cues for predictions (an example is provided in Figure 1); and (2) being biased towards either cue may inevitably limit model performance, e.g., models may not be able to tell the difference between a lemon and an orange without texture information. These observations altogether deliver a promising message—biased models (e.g., ImageNet trained (texturebiased) CNNs (Geirhos et al., 2019) or (shape-biased) CNNs (Shi et al., 2020)) are improvable.
+
+![](images/83235d307c0b6ec9fed007c2a255dfacb36f496a9e7ac9b8cefa396660526811.jpg)  
+Figure 1: Both shape and texture are essential cues for object recognition, and biasing towards either one degenerates model performance. As shown above, when classifying this fur coat image, the shape-biased model is confounded by the cloth-like shape therefore predict it as a poncho, and the texture-biased model confuses it as an Egyptian cat because of the misleading texture. Nonetheless, our debiased model can successfully recognize it as a fur coat by leveraging both shape and texture.
+
+To this end, we hereby develop a shape-texture debiased neural network training framework to guide CNNs for learning better representations. Our method is a data-driven approach, which let CNNs automatically figure out how to avoid being biased towards either shape or texture from their training samples. Specifically, we apply style transfer to generate cue conflict images, which breaks the correlation between shape and texture, for augmenting the original training data. The most important recipe of training a successful shape-texture debiased model is that we need to provide supervision from both shape and texture on these generated cue conflict images, otherwise models will remain being biased.
+
+Experiments show that our proposed shape-texture debiased neural network training significantly improves recognition models. For example, on the challenging ImageNet dataset (Russakovsky et al., 2015), our method helps ResNet-152 gain an absolute improvement of $1 . 2 \%$ , achieving $7 9 . 8 \%$ top-1 accuracy. Additionally, compared to its vanilla counterpart, this debiased ResNet-152 shows better generalization on ImageNet-A (Hendrycks et al., 2019) $( + 5 . 2 \% )$ , ImageNet-C (Hendrycks & Dietterich, 2019) $( + 8 . 3 \% )$ and Stylized ImageNet (Geirhos et al., 2019) $( + 1 1 . 1 \% )$ , and stronger robustness on defending against FGSM adversarial attacker on ImageNet $( + 1 4 . 4 \% )$ . Our shape-texture debiased neural network training is orthogonal to other advanced data augmentation strategies, e.g., it further boosts CutMix-ResNeXt-101 (Yun et al., 2019) by $0 . 7 \%$ on ImageNet, achieving $8 1 . 2 \%$ top-1 accuracy.
+
+# 2 SHAPE/TEXTURE BIASED NEURAL NETWORKS
+
+The biased feature representation of CNNs mainly stems from the training dataset, e.g., Geirhos et al. (2019) point out that models will be biased towards shape if trained on Stylized-ImageNet dataset. Following Geirhos et al. (2019), we hereby present a similar training pipeline to acquire shapebiased models or texture-biased models. By evaluating these two kinds of models, we observe the necessity of possessing both shape and texture representations for CNNs to better recognize objects.
+
+# 2.1 MODEL ACQUISITION
+
+Data generation. Similar to Geirhos et al. (2019), we apply images with conflicting shape and texture information as training samples to obtain shape-biased or texture-biased models. But different from Geirhos et al. (2019), an important change in our cue conflict image generation procedure is that we override the original texture information with the informative texture patterns from another randomly selected image, rather than with the uninformative style of randomly selected artistic paintings. That being said, to create a new training sample, we need to first select a pair of images from the training set uniformly at random, and then apply style transfer to blend their shape and texture information. Such a generated example is shown in Figure 2, i.e., the image of chimpanzee shape but with lemon texture.
+
+![](images/dec6f3cfa7570413769bd477e908d876c8ce5e7c56a24a017462103df59a15ae.jpg)  
+Figure 2: Illustration of the our training pipeline for acquiring (a) a shape-biased model, (b) a texture-biased model, and (c) a shape-texture debiased model. Specifically, these models share the same training samples, i.e. images with conflicting texture and shape information, generated by style transfer between two randomly selected images; but apply distinct labelling strategies: in (a) & (b), labels are determined by the images that provides shape (or texture) information in style transfer, for guiding models to learn more shape (or texture) representations; in (c), labels are jointly determined by the pair of images in style transfer, for avoiding bias in representation learning.
+
+Label assignment. The way of assigning labels to cue conflict images controls the bias of learned models. Without loss of generality, we show the case of learning a texture-biased model. To guide the model to attend more on texture, the labels assigned to the cue conflict images here will be exclusively based on the texture information, e.g., the image of chimpanzee shape but with lemon texture will be labelled as lemon, shown in Figure 2(b). By this way, the texture information is highly related to the “ground-truth” while the shape information only serves as a nuisance factor during learning. Similarly, to learn a shape-biased model, the label assignment of cue conflict images will be based on shape only, e.g., the image of chimpanzee shape but with lemon texture now will be labelled as chimpanzee, shown in Figure 2(a).
+
+# 2.2 EVALUATION AND OBSERVATION
+
+To reduce the computational overhead in this ablation, all models are trained and evaluated on ImageNet-200, which is a 200 classes subset of the original ImageNet, including 100,000 images (500 images per class) for training and 10,000 images (50 images per class) for validation. Akin to Geirhos et al. (2019), we observe that the models with biased feature representations tend to have inferior accuracy than their vanilla counterparts. For example, our shape-biased ResNet-18 only achieves $7 3 . 9 \%$ top-5 ImageNet-200 accuracy, which is much lower than the vanilla ResNet-18 with $8 8 . 2 \%$ top-5 ImageNet-200 accuracy.
+
+Though biased representations weaken the overall classification accuracy, surprisingly, we find they are highly complementary to each other. We first visualize the attended image regions of biased models, via Class Activation Mapping (Zhou et al., 2016), in Figure 3. As we can see here, the shape-biased model and the texture-biased model concentrate on different cues for predictions. For instance, on the leftmost tabby cat image, the shape-biased model mainly focuses on the cat head, while the texture-biased model mainly focuses on the lower body and the front legs of the cat. Such attention mechanisms are correlated to their learned representations—the shape-biased model extracts the shape of the cat head as an important signal for predictions, while the texture-biased model relies on the texture information of cat fur for predictions.
+
+![](images/46445119eee72fa07255beb2dc4195191637244b96b8d02509c096f6a55e6f06.jpg)  
+Figure 3: The shape-biased model and the texture-biased model attend on complementary cues for predictions. We use Class Activation Mapping to visualize which image regions are attended by models. Redder regions indicates more attentions are paid by models.
+
+![](images/896f20f3dc475d08249a0debfd12f702f8e65a4a9ce6afce1bc6ef0e1d66f59a.jpg)  
+Figure 4: The shape-biased model and the texture-biased model are good/bad at classifying different object categories. We sort these object categories according to the model’s corresponding top-1 accuracy, where the righter one indicates a lower accuracy achieved by the model.
+
+As distinct cues are picked by shape-biased/texture-biased models, a more concrete observation is they are good/bad at classifying quite different object categories. As showed in Figure 4, the shapebiased model is good at recognizing objects with representative shape structure like obelisk, but is bad at recognizing objects whose shape is uninformative or almost indistinguishable from others like fur coat. Similarly, the texture-biased model can effectively recognize objects with unique texture patterns like brain coral but may fail to recognize objects with unpredictable texture like trolleybus (as its side body can be painted with different advertisements). Besides, biased models may inevitably perform poorly on certain categories as insufficient cues are applied. For examples, it is challenging to distinguish between a lemon and an orange if texture information cannot be utilized, or to distinguish between an lion and a tabby cat without shape information.
+
+Given the analysis above, we can conclude that biased representations limit models’ recognition ability. But meanwhile, our ablation delivers a promising message—the features learned by biased models are highly complementary to each other. This observation indicates the current training framework is improvable (as the resulted models are biased towards texture (Geirhos et al., 2019) or shape (Shi et al., 2020)), and offers a potential direction for building a stronger one—we should train models to properly acquire both shape and texture feature representations. We will introduce a simple method for doing so next.
+
+# 3 SHAPE-TEXTURE DEBIASED NEURAL NETWORK TRAINING
+
+Recall that when obtaining a biased model, the strategy of label assignment is pivot—when the labels are exclusively determined by the images that provide shape (or texture) information in style transfer, we will obtain a shape-biased (or texture-biased) model. Therefore, to guide models for leveraging both shape and texture for predictions, we hereby propose a simple way, which is inspired by Mixup (Zhang et al., 2018), to softly construct labels during training. In other words, given the one-hot label of the shape-source image $y _ { s }$ and the one-hot label of the texture-source image $y _ { t }$ , the new label that we assigned to the cue conflict image is
+
+$$
+\widetilde { y } = \gamma * y _ { s } + ( 1 - \gamma ) * y _ { t } ,
+$$
+
+where $\gamma \in [ 0 , 1 ]$ is a manually selected hyperparameter to control the relative importance between shape and texture. By ranging the shape-texture coefficient $\gamma$ from 0 to 1, we obtain a path to evolve the model from being a texture-biased one (i.e., $\gamma = 0$ ) to being a shape-biased one (i.e., $\gamma = 1$ ). Although the two extreme ends lead to biased models with inferior performance, we empirically show that there exist a sweet point along this interpolation path, i.e., the learned models can properly acquires both shape and texture feature representations and achieve superior performance on a wide range of image recognition benchmarks.
+
+We name this simple method as shape-texture debiased neural network training, and illustrate the training pipeline in Figure 2(c). It is worth to mention that, although Figure 2 only shows the procedure of applying our method to the image classification task, this training framework is general and has the potential to be extended to other computer vision tasks, e.g., a simple showcase on semantic segmentation is presented in Section 4.4.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTS SETUP
+
+Datasets. We evaluate models on ImageNet classification and PASCAL VOC semantic segmentation. ImageNet dataset (Russakovsky et al., 2015) consists of 1.2 million images for training, and 50,000 for validation, from 1,000 classes. PASCAL VOC 2012 segmentation dataset (Everingham et al., 2012) with extra annotated images from (Hariharan et al., 2011) involves 20 foreground object classes and one background class, including 10,582 training images and 1,449 validation images.
+
+Going beyond the standard benchmarks, we further evaluate models’ generalization on ImageNetA, ImageNet-C and Stylized-ImageNet, and robustness by defending against FGSM adversarial attacker on ImageNet. ImageNet- $C$ (Hendrycks & Dietterich, 2019) is a benckmark dataset that measures models’ corruption robustness. It is constructed by applying 75 common visual corruptions to the ImageNet validation set. ImageNet-A (Hendrycks et al., 2019) includes 7,500 natural adversarial examples that successfully attacks unseen classifiers. These examples are much harder than original ImageNet validation images due to scene complications encountered in the long tail of scene configurations and by exploiting classifier blind spots (Hendrycks et al., 2019). StylizedImageNet (Geirhos et al., 2019) is a stylized version of ImageNet that constructed by re-rendering the original images by AdaIN stylizer (Huang & Belongie, 2017). The generated images keep the original global shape information but removes the local texture information. FGSM (Goodfellow et al., 2015) is a widely used adversarial attacker to evaluate model robustness. We set the maximum perturbation change per pixel $\epsilon = 1 6 / 2 5 5$ for FGSM.
+
+Implementation details. We choose ResNet (He et al., 2016) as the default architecture. For image classification tasks, our implementation is based on the publicly available framework in PyTorch2. To generate cue conflict images, we follow Geirhos et al. (2019) to use Adaptive Instance Normalization (Huang & Belongie, 2017) in style transfer, and set stylization coefficient $\alpha = 0 . 5$ . Importantly, to increase the diversity of training samples, we generate these cue conflict images on-the-fly during training. We choose the shape-texture coefficient $\gamma = 0 . 8$ when assigning labels.
+
+When training shape-biased, texture-biased and our shape-texture debiased models, we always apply the auxiliary batch normalization (BN) design (Xie et al., 2020; Xie & Yuille, 2020; Chen et al.,
+
+Table 1: The performance of the vanilla training, the shape-biased (S-biased) training, the texturebiased (T-biased) training, and our shape-texture debiased training on ImageNet. For all ResNet models, our debiased training shows the best performance among others.   
+
+<table><tr><td></td><td>VANILLA</td><td>2×EPOCHS</td><td>S-BIASED</td><td>T-BIASED</td><td>DEBIASED</td></tr><tr><td>ResNet-50</td><td>76.4</td><td>76.4 (+0.0)</td><td>76.2 (-0.2)</td><td>75.3 (-1.1)</td><td>76.9 (+0.5)</td></tr><tr><td>ResNet-101</td><td>78.0</td><td>78.0 (+0.0)</td><td>78.0 (-0.0)</td><td>77.4 (-0.6)</td><td>78.9 (+0.9)</td></tr><tr><td>ResNet-152</td><td>78.6</td><td>79.1 (+0.5)</td><td>78.6 (-0.0)</td><td>78.1 (-0.5)</td><td>79.8 3(+1.2)</td></tr></table>
+
+<table><tr><td></td><td>IN-A Acc. ↑</td><td>IN-C mCE↓</td><td>S-IN Acc. ↑</td><td>FGSM Acc.↑</td></tr><tr><td>ResNet-50 +Debiased</td><td>2.0 3.5 (+1.5)</td><td>75.0 67.5 (-7.5)</td><td>7.4 17.4 (+10.0)</td><td>17.1 27.4 (+10.3)</td></tr><tr><td>ResNet-101 +Debiased</td><td>5.6 9.1 1 (+3.5)</td><td>69.8 62.2 (-7.6)</td><td>9.9 22.0 (+12.1)</td><td>23.1 34.4 (+11.3)</td></tr><tr><td>ResNet-152 +Debiased</td><td>7.4 12.6 (+5.2)</td><td>67.2 58.9 (-8.3)</td><td>11.3 22.4 (+11.1)</td><td>25.2 39.6 (+14.4)</td></tr></table>
+
+Table 2: The model robustness on ImageNet-A (IN-A), ImageNet-C (IN-C), Stylized-ImageNet (SIN), and on defending against FGSM adversarial attacker on ImageNet. Our shape-texture debiased neural network training significantly boosts the model robustness over the vanilla training baseline.
+
+2021) to bridge the domain gap between the original data and the augmented data, i.e., the main BN is exclusively running on original ImageNet images and the auxiliary BN is exclusively running on cue conflict images. We follow Xie et al. (2020) to always apply the main BN for performance evaluation. Besides, since our biased models and debiased models are all trained with both the original data and the augmented data (i.e., $2 \times$ data are used in training), we also consider a stronger baseline (i.e., $2 \times$ epochs training) which doubles the schedule of the vanilla training baseline, for the purpose of matching the total training cost.
+
+# 4.2 RESULTS
+
+Model accuracy. Table 1 shows the results on ImageNet. For all ResNet models, the proposed shape-texture debiased neural network training consistently outperforms the vanilla training baseline. For example, it helps ResNet-50 achieve $7 6 . 9 \%$ top-1 accuracy, beating its vanilla counterpart by $0 . 5 \%$ . Our method works better for larger models, e.g., it further improves the vanilla ResNet-152 by $1 . 2 \%$ , achieving $7 9 . 8 \%$ top-1 accuracy.
+
+We then compare our shape-texture debiased training to the $2 \times$ epochs training baseline. We find that simply doubling the schedule of the vanilla training baseline cannot effectively lead to improvements like ours. For examples, compared to the vanilla ResNet-101, this $2 \times$ epochs training fails to provide additional improvements, while ours furthers the top-1 accuracy by $1 . 0 \%$ . This result suggests that it is non-trivial to improve performance even if more computational budgets are given.
+
+Lastly, we compare ours to the biased training methods. Though the only difference between our method and the biased training methods is the strategy of label assignment (as shown in Figure 2), it imperatively affects model performance. For example, compared to the vanilla baseline, both the shape-biased training and the texture-biased training fail to improve (sometimes even slightly hurt) the model accuracy, while our shape-texture debiased neural network training successfully leads to consistent and substantial accuracy improvements.
+
+Model robustness. Next, we evaluate models’ generalization on ImageNet-A, ImageNet-C and Stylized-ImageNet, and robustness on defending against FGSM on ImageNet. We note these tasks are much more challenging than the original ImageNet classification, e.g., the ImageNet trained ResNet-50 only achieves $2 . 0 \%$ accuracy on ImageNet-A, $7 5 . 0 \%$ mCE on ImageNet-C, $7 . 4 \%$ accuracy on Stylized-ImageNet, and $1 7 . 1 \%$ accuracy on defending against FGSM adversarial attacker. As shown in Table 2, our shape-texture debiased neural network training beats the vanilla training baseline by a large margin on all tasks for all ResNet models. For example, it substantially boosts ResNet-152’s performance on ImageNet-A $( + 5 . 2 \%$ , from $7 . 4 \%$ to $1 2 . 6 \%$ ), ImageNet-C $( - 8 . 3 \%$ , from $6 7 . 2 \%$ to $5 8 . 9 \%$ , the lower the better) and Stylized-ImageNet $( + 1 1 . 1 \%$ , from $1 1 . 3 \%$ to $2 2 . 4 \%$ ), and on defending against FGSM on ImageNet $+ 1 4 . 4 \%$ , from $2 5 . 2 \%$ to $3 9 . 6 \%$ ). These results altogether suggest that our shape-texture debiased neural network training is an effective way to mitigate the issue of shortcut learning (Geirhos et al., 2020).
+
+<table><tr><td></td><td>IN Acc. 个</td><td>IN-A Acc.个</td><td>IN-C mCE↓</td><td>S-IN Acc. 个</td><td>FGSM Acc.个</td></tr><tr><td>ResNet-50</td><td>76.4</td><td>2.0</td><td>75.0</td><td>7.4</td><td>17.1</td></tr><tr><td>CutMix + MoEx (Li et al., 2021)</td><td>79.0</td><td>8.0</td><td>74.8</td><td>5.0</td><td>41.0</td></tr><tr><td>DeepAugment + AugMix (Hendrycks et al., 2020)</td><td>75.8</td><td>3.9</td><td>53.6</td><td>21.2</td><td>18.8</td></tr><tr><td>SIN (Geirhos et al., 2019)</td><td>60.2</td><td>2.4</td><td>77.3</td><td>56.2</td><td>5.6</td></tr><tr><td>Shape-Texture Debiased Training (ours)</td><td>76.9</td><td>3.5</td><td>67.5</td><td>17.4</td><td>27.4</td></tr></table>
+
+Table 3: Compare with state-of-the-art methods using ResNet-50 on ImageNet (IN), ImageNet-A (IN-A), ImageNet-C (IN-C), Stylized-ImageNet (S-IN), and on defending against FGSM on ImageNet. We use green to denote significant improvement, red to denote performance drop, and gray to denote similar performance. We observe our shape-texture debiased training is the only method that successfully leads to improvements over the vanilla baseline on all benchmarks.
+
+<table><tr><td>Datasets</td><td>VANILLA</td><td>S-BIASED</td><td>T-BIASED</td><td>DEBIASED</td></tr><tr><td>ImageNet-Sketch</td><td>23.8</td><td>27.9</td><td>24.3</td><td>28.4</td></tr><tr><td>ImageNet-R</td><td>36.2</td><td>40.6</td><td>36.7</td><td>40.8</td></tr><tr><td>Kylberg Texture</td><td>99.5</td><td>99.1</td><td>99.6</td><td>99.5</td></tr><tr><td>FlickerMaterial</td><td>74.6</td><td>73.3</td><td>79.2</td><td>75.8</td></tr></table>
+
+Table 4: The performance comparison between Vanilla, Shape-biased, Texture-biased, and ShapeTexture Debiased models on ImageNet-Sketch, ImageNet-R, Kylberg Texture, and Flicker Material datasets. We note the shape-biased and the shape-texture debiased models perform better on shape datasets (ImageNet-Sketch and ImageNet-R); the texture-biased and the shape-texture debiased models perform better on texture datasets (Kylberg Texture and Flicker Material).
+
+Comparing to SoTAs. We further compare our shape-texture debiased model with the SoTA on ImageNet and ImageNet-A (CutMix $^ +$ MoEx (Li et al., 2021)), the SoTA on ImageNet-C (DeepAugment $^ +$ AugMix (Hendrycks et al., 2020)), and the SoTA on Stylized-ImageNet (SIN (Geirhos et al., 2019)). Interestingly, we note the improvements of all these SoTAs are not consistent across different benchmarks. For example, as shown in Table 3, SIN significantly improves the results on Stylized-ImageNet, but at the cost of huge performance drop on ImageNet $( - 1 6 . 2 \% )$ and ImageNetC $( - 2 . 3 \% )$ . Our shape-texture debiased training stands as the only method that can improve the vanilla training baseline holistically.
+
+# 4.3 ABLATIONS
+
+Comparing to model ensembles. An alternative but na¨ıve way for obtaining the model with both shape and texture information is to ensemble a shape-biased model and a texture-biased model. We note this ensemble strategy yields a model of on-par performance with our shape-texture debiased model on ImageNet $7 7 . 2 \%$ vs. $7 6 . 9 \%$ ). Nonetheless, interestingly, when measuring model robustness, such model ensemble strategy is inferior than ours. For example, compared to our proposed debiased training, this ensemble strategy is $1 . 5 \%$ worse on ImageNet-A ( $2 . 0 \%$ vs. $3 . 5 \%$ ), $1 . 1 \%$ worse on ImageNet-C $6 8 . 6 ~ \mathrm { m C E }$ vs. $6 7 . 5 ~ \mathrm { m C E }$ ), $1 . 1 \%$ worse on Stylized-ImageNet ( $1 6 . 3 \%$ vs. $1 7 . 4 \%$ ), and $7 . 0 \%$ worse on defending against FGSM $2 0 . 4 \%$ vs. $2 7 . 4 \%$ ). Moreover, due to model ensemble, this strategy is $2 \times$ expensive at the inference stage. These evidences clearly demonstrate the effectiveness and efficiency of the proposed shape-texture debiased training.
+
+Does our method help models to learn debiased shape-texture representations? Here we take a close look at whether our method indeed prevents models from being biased toward shape or texture during learning. We evaluate models in Section 4.2 on two kinds of datasets: (1) ImageNetSketch dataset (Wang et al., 2019) and ImageNet-R (Hendrycks et al., 2020) for examining how well models can capture shape; and (2) Kylberg Texture dataset (Kylberg, 2011) and Flicker Material dataset (Sharan et al., 2014) for examining how well models can capture texture. Specifically, since object categories from two texture datasets are not compatible to that from ImageNet dataset, we retrain the last fc-layer (while keeping all other layers untouched) of all models on Kylberg Texture dataset or Flicker Material dataset for 5 epochs. The results are shown in Table 4.
+
+We first analyze results on ImageNet-Sketch dataset. We observe our shape-texture debiased models are as good as the shape-biased models, and significantly outperforms the texture-biased models and the vanilla training models. For instance, using ResNet-50, our shape-texture debiased training and shape-biased training achieve $2 8 . 4 \%$ top-1 accuracy and $2 7 . 9 \%$ top-1 accuracy, while texture-biased training and vanilla training only get $2 4 . 3 \%$ top-1 accuracy and $2 3 . 8 \%$ top-1 accuracy. A similar observation can be seen from ImageNet-R. These results support that our method helps models acquire stronger shape representations than the vanilla training.
+
+![](images/3b8f58c07b58608ea11407af951dddb69b2f963cbff7c4a52d31732da1be47c0.jpg)  
+Figure 5: Illustration of the data preparation pipeline of our shape-texture debiased neural network training on the semantic segmentation task.
+
+We next analyze results on Kylberg Texture dataset. Similarly, we observe that our debiased model are comparable to the texture-biased model and the vanilla training model, and get better performance than the shape-biased model. On Flicker Material dataset, we observe that our debiased models are better than the vanilla training model and the shape-biased model. This phenomenon suggests texture information is effectively caught by our shape-texture debiased training. As a side note, it is expected that vanilla training are better than shape-biased training on these texture datasets, as Geirhos et al. (2019) point out that ImageNet trained models (i.e., vanilla training) also tend to be biased towards texture.
+
+With the analysis above, we conclude that, compared to vanilla training, our shape-texture debiased training successfully helps networks effectively acquire both shape and texture representations.
+
+Combining with other data augmentation methods. Our shape-texture debiased neural network training can be viewed as a data augmentation method, which trains models on cue conflict images. Nonetheless, our method specifically guides the model to learn debiased shape and texture representations, which could potentially serve as a complementary feature to other data augmentation methods. To validate this argument, we train models using a combination of our method and an existing data augmentation method (i.e., Mixup (Zhang et al., 2018) or CutMix (Yun et al., 2019)).
+
+We choose ResNeXt-101 (Xie et al., 2017) as the backbone network, which reports the best top-1 ImageNet accuracy in both the Mixup paper, i.e., $7 9 . 9 \%$ , and the CutMix paper, i.e., $8 0 . 5 \%$ . Though building upon very strong baselines, our shape-texture debiased neural network training still leads to substantial improvements, e.g., it furthers ResNeXt-101-Mixup’s accuracy to $8 0 . 5 \%$ $( + 0 . 6 \% )$ , and ResNeXt-101-CutMix’s accuracy to $8 1 . 2 \%$ $( + 0 . 7 \% )$ . Meanwhile, models’ generalization also get greatly improved. For example, by combining CutMix and our method, ResNeXt-101 gets additional improvements on ImageNet-A $( + 1 . 4 \% )$ , ImageNet-C $( - 5 . 9 \%$ , the lower the better) and Stylized ImageNet $( + 7 . 5 \% )$ . These results support that our shape-texture debiased neural network training is compatible to existing data augmentation methods.
+
+Shape-texture coefficient $\gamma$ . We set $\gamma = 0 . 8$ in our shape-texture debiased training. This value is found via the grid search over ImageNet-200 using ResNet-18. We now ablate its sensitivity on ImageNet using ResNet-50, where $\gamma$ is linearly interpolated between 0.0 and 1.0. By increasing the value of $\gamma$ , we observe that the corresponding accuracy on ImageNet first monotonically goes up, and then monotonically goes down. The sweet point can be reached by setting $\gamma = 0 . 7$ , where ResNet-50 achieves $7 7 . 0 \%$ top-1 ImageNet accuracy. Besides, we note that by setting $\gamma \in [ 0 . 5 , 0 . 9 ]$ can always lead to performance improvements over the vanilla baseline. These results demonstrate the robustness of our shape-texture debiased neural network training w.r.t. the coefficient $\gamma$ .
+
+# 4.4 SEMANTIC SEGMENTATION RESULTS
+
+We extend our shape-texture debiased neural network training to the segmentation task. We select DeepLabv3-ResNet-101 (Chen et al., 2017) as our backbone. To better incorporate our method with the segmentation task, the following changes are made when generating cue conflict images: (1) unlike in the classification task where the whole image is used as the texture source, we use a specific object (which can cropped from the background using the segmentation ground-truth) to provide texture information in style transfer; (2) when composing the soft label for the cue conflict image, we set the label mask from texture source as the full image (since the pattern from the texture source will fill the whole image after style transfer); and (3) we set stylization coefficient $\alpha = 0 . 2$ and shape-texture coefficient $\gamma = 0 . 9 5$ to prevent object boundaries from being overly blurred in style transfer. Figure 5 shows an illustration of our data preparation pipeline.
+
+Results. Our shape-texture debiased training can also effectively improve segmentation models. For example, our method helps DeepLabv3-ResNet-101 achieve $7 7 . 6 \%$ mIOU, significantly beating its vanilla counterpart by $1 . 1 \%$ . Our method still shows advantages when compared to the $2 \times$ epochs training baseline. Doubling the learning schedule of the vanilla training can only lead to an improvement of $0 . 2 \%$ , which is still $0 . 9 \%$ worse than our shape-texture debiased training. These results demonstrate the potential of our methods in helping recognition tasks in general.
+
+# 5 RELATED WORK
+
+Data augmentation. Data augmentation is essential for the success of deep learning (LeCun et al., 1998; Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; Zhong et al., 2020; Cubuk et al., 2019; Lim et al., 2019; Cubuk et al., 2020). Our shape-texture debiased neural network training is related to a specific family of data augmentation, called Mixup (Zhang et al., 2018), which blends pairs of images and their labels in a convex manner, either at pixel-level (Zhang et al., 2018; Yun et al., 2019) or feature-level (Verma et al., 2019; Li et al., 2021). Our method can be interpreted as a special instantiation of Mixup which blends pairs of images at the abstraction level—images’ texture information and shape information are mixed. Our method successfully guides CNNs to learn better shape and texture representations, which is an important but missing piece in existing data argumentation methods.
+
+Style transfer. Style transfer, closely related to texture synthesis and transfer, means generating a stylized image by combining a shape-source image and a texture-source image (Efros & Leung, 1999; Efros & Freeman, 2001; Elad & Milanfar, 2017). The seminal work (Gatys et al., 2016) demonstrate impressive style transfer results by matching feature statistics in convolutional layers of a CNN. Later follow-ups further improve the generation quality and speed (Huang & Belongie, 2017; Chen & Schmidt, 2016; Ghiasi et al., 2017; Li et al., 2017). In this work, we follow Geirhos et al. (2019) to use AdaIN (Huang & Belongie, 2017) to generate stylized images. Nonetheless, instead of applying style transfer between an image and an artistic paintings as in Geirhos et al. (2019), we directly apply style transfer on a pair of images to generate cue conflict images. This change is vital as it enables us to provide supervisions from both shape and texture during training.
+
+# 6 CONCLUSION
+
+There is a long-time debate about which cue dominates the object recognition. By carefully ablate the shape-biased model and the texture-biased model, we found though biased feature representations lead to performance degradation, they are complementary to each other and are both necessary for image recognition. To this end, we propose shape-texture debiased neural network training for guiding CNNs to learn better feature representations. The key in our method is that we should not only augment training set with cue conflict images, but also provide supervisions from both shape and texture. We empirically demonstrate the advantages of our shape-texture debiased neural network training on boosting both accuracy and robustness. Our method is conceptually simple and is generalizable to different image recognition tasks. We hope our work will shed light on understanding and improving convolutional neural networks.
+
+# ACKNOWLEDGEMENT
+
+This project is partially supported by ONR N00014-18-1-2119 and ONR N00014-20-1-2206. Cihang Xie is supported by the Facebook PhD Fellowship and a gift grant from Open Philanthropy. Yingwei Li thanks Zhiwen Wang for suggestions on figures.
+
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+        "type": "text",
+        "text": "SHAPE-TEXTURE DEBIASED NEURAL NETWORK TRAINING ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Yingwei $\\mathbf { L i } ^ { 1 }$ , Qihang $\\mathbf { Y u } ^ { 1 }$ , Mingxing $\\mathbf { T a n } ^ { 2 }$ , Jieru Mei1, Peng Tang1, Wei Shen3   \nAlan Yuille1 & Cihang Xie4   \n1Johns Hopkins University 2Google Brain 3Shanghai Jiaotong University   \n4University of California, Santa Cruz ",
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+        ],
+        "page_idx": 0
+    },
+    {
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+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Shape and texture are two prominent and complementary cues for recognizing objects. Nonetheless, Convolutional Neural Networks are often biased towards either texture or shape, depending on the training dataset. Our ablation shows that such bias degenerates model performance. Motivated by this observation, we develop a simple algorithm for shape-texture debiased learning. To prevent models from exclusively attending on a single cue in representation learning, we augment training data with images with conflicting shape and texture information (e.g., an image of chimpanzee shape but with lemon texture) and, most importantly, provide the corresponding supervisions from shape and texture simultaneously. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experiments show that our method successfully improves model performance on several image recognition benchmarks and adversarial robustness. For example, by training on ImageNet, it helps ResNet-152 achieve substantial improvements on ImageNet $( + 1 . 2 \\% )$ , ImageNet-A $( + 5 . 2 \\% )$ , ImageNet-C $( + 8 . 3 \\% )$ and Stylized-ImageNet $( + 1 1 . 1 \\% )$ , and on defending against FGSM adversarial attacker on ImageNet $( + 1 4 . 4 \\% )$ . Our method also claims to be compatible to other advanced data augmentation strategies, e.g., Mixup and CutMix. The code is available here: https://github.com/LiYingwei/ ShapeTextureDebiasedTraining. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "It is known that both shape and texture serve as essential cues for object recognition. A decade ago, computer vision researchers had explicitly designed a variety of hand-crafted features, either based on shape (e.g., shape context (Belongie et al., 2002) and inner distance shape context (Ling & Jacobs, 2007)) or texture (e.g., textons (Malik et al., 2001)), for object recognition. Moreover, researchers found that properly combining shape and texture can further recognition performance (Shotton et al., 2009; Zheng et al., 2007), demonstrating the superiority of possessing both features. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Nowadays, as popularized by Convolutional Neural Networks (CNNs) (Krizhevsky et al., 2012), the features used for object recognition are automatically learned, rather than manually designed. This change not only eases human efforts on feature engineering, but also yields much better performance on a wide range of visual benchmarks (Simonyan & Zisserman, 2015; He et al., 2016; Girshick et al., 2014; Girshick, 2015; Ren et al., 2015; Long et al., 2015; Chen et al., 2015). But interestingly, as pointed by Geirhos et al. (2019), the features learned by CNNs tend to bias toward either shape or texture, depending on the training dataset. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We verify that such biased representation learning (towards either shape or texture) weakens CNNs’ performance.1 Nonetheless, surprisingly, we also find (1) the model with shape-biased representations and the model with texture-biased representations are highly complementary to each other, e.g., they focus on completely different cues for predictions (an example is provided in Figure 1); and (2) being biased towards either cue may inevitably limit model performance, e.g., models may not be able to tell the difference between a lemon and an orange without texture information. These observations altogether deliver a promising message—biased models (e.g., ImageNet trained (texturebiased) CNNs (Geirhos et al., 2019) or (shape-biased) CNNs (Shi et al., 2020)) are improvable. ",
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+    {
+        "type": "image",
+        "img_path": "images/83235d307c0b6ec9fed007c2a255dfacb36f496a9e7ac9b8cefa396660526811.jpg",
+        "image_caption": [
+            "Figure 1: Both shape and texture are essential cues for object recognition, and biasing towards either one degenerates model performance. As shown above, when classifying this fur coat image, the shape-biased model is confounded by the cloth-like shape therefore predict it as a poncho, and the texture-biased model confuses it as an Egyptian cat because of the misleading texture. Nonetheless, our debiased model can successfully recognize it as a fur coat by leveraging both shape and texture. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "To this end, we hereby develop a shape-texture debiased neural network training framework to guide CNNs for learning better representations. Our method is a data-driven approach, which let CNNs automatically figure out how to avoid being biased towards either shape or texture from their training samples. Specifically, we apply style transfer to generate cue conflict images, which breaks the correlation between shape and texture, for augmenting the original training data. The most important recipe of training a successful shape-texture debiased model is that we need to provide supervision from both shape and texture on these generated cue conflict images, otherwise models will remain being biased. ",
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+    {
+        "type": "text",
+        "text": "Experiments show that our proposed shape-texture debiased neural network training significantly improves recognition models. For example, on the challenging ImageNet dataset (Russakovsky et al., 2015), our method helps ResNet-152 gain an absolute improvement of $1 . 2 \\%$ , achieving $7 9 . 8 \\%$ top-1 accuracy. Additionally, compared to its vanilla counterpart, this debiased ResNet-152 shows better generalization on ImageNet-A (Hendrycks et al., 2019) $( + 5 . 2 \\% )$ , ImageNet-C (Hendrycks & Dietterich, 2019) $( + 8 . 3 \\% )$ and Stylized ImageNet (Geirhos et al., 2019) $( + 1 1 . 1 \\% )$ , and stronger robustness on defending against FGSM adversarial attacker on ImageNet $( + 1 4 . 4 \\% )$ . Our shape-texture debiased neural network training is orthogonal to other advanced data augmentation strategies, e.g., it further boosts CutMix-ResNeXt-101 (Yun et al., 2019) by $0 . 7 \\%$ on ImageNet, achieving $8 1 . 2 \\%$ top-1 accuracy. ",
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+    {
+        "type": "text",
+        "text": "2 SHAPE/TEXTURE BIASED NEURAL NETWORKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The biased feature representation of CNNs mainly stems from the training dataset, e.g., Geirhos et al. (2019) point out that models will be biased towards shape if trained on Stylized-ImageNet dataset. Following Geirhos et al. (2019), we hereby present a similar training pipeline to acquire shapebiased models or texture-biased models. By evaluating these two kinds of models, we observe the necessity of possessing both shape and texture representations for CNNs to better recognize objects. ",
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+    {
+        "type": "text",
+        "text": "2.1 MODEL ACQUISITION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Data generation. Similar to Geirhos et al. (2019), we apply images with conflicting shape and texture information as training samples to obtain shape-biased or texture-biased models. But different from Geirhos et al. (2019), an important change in our cue conflict image generation procedure is that we override the original texture information with the informative texture patterns from another randomly selected image, rather than with the uninformative style of randomly selected artistic paintings. That being said, to create a new training sample, we need to first select a pair of images from the training set uniformly at random, and then apply style transfer to blend their shape and texture information. Such a generated example is shown in Figure 2, i.e., the image of chimpanzee shape but with lemon texture. ",
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+    {
+        "type": "image",
+        "img_path": "images/dec6f3cfa7570413769bd477e908d876c8ce5e7c56a24a017462103df59a15ae.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the our training pipeline for acquiring (a) a shape-biased model, (b) a texture-biased model, and (c) a shape-texture debiased model. Specifically, these models share the same training samples, i.e. images with conflicting texture and shape information, generated by style transfer between two randomly selected images; but apply distinct labelling strategies: in (a) & (b), labels are determined by the images that provides shape (or texture) information in style transfer, for guiding models to learn more shape (or texture) representations; in (c), labels are jointly determined by the pair of images in style transfer, for avoiding bias in representation learning. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Label assignment. The way of assigning labels to cue conflict images controls the bias of learned models. Without loss of generality, we show the case of learning a texture-biased model. To guide the model to attend more on texture, the labels assigned to the cue conflict images here will be exclusively based on the texture information, e.g., the image of chimpanzee shape but with lemon texture will be labelled as lemon, shown in Figure 2(b). By this way, the texture information is highly related to the “ground-truth” while the shape information only serves as a nuisance factor during learning. Similarly, to learn a shape-biased model, the label assignment of cue conflict images will be based on shape only, e.g., the image of chimpanzee shape but with lemon texture now will be labelled as chimpanzee, shown in Figure 2(a). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 EVALUATION AND OBSERVATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To reduce the computational overhead in this ablation, all models are trained and evaluated on ImageNet-200, which is a 200 classes subset of the original ImageNet, including 100,000 images (500 images per class) for training and 10,000 images (50 images per class) for validation. Akin to Geirhos et al. (2019), we observe that the models with biased feature representations tend to have inferior accuracy than their vanilla counterparts. For example, our shape-biased ResNet-18 only achieves $7 3 . 9 \\%$ top-5 ImageNet-200 accuracy, which is much lower than the vanilla ResNet-18 with $8 8 . 2 \\%$ top-5 ImageNet-200 accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Though biased representations weaken the overall classification accuracy, surprisingly, we find they are highly complementary to each other. We first visualize the attended image regions of biased models, via Class Activation Mapping (Zhou et al., 2016), in Figure 3. As we can see here, the shape-biased model and the texture-biased model concentrate on different cues for predictions. For instance, on the leftmost tabby cat image, the shape-biased model mainly focuses on the cat head, while the texture-biased model mainly focuses on the lower body and the front legs of the cat. Such attention mechanisms are correlated to their learned representations—the shape-biased model extracts the shape of the cat head as an important signal for predictions, while the texture-biased model relies on the texture information of cat fur for predictions. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/46445119eee72fa07255beb2dc4195191637244b96b8d02509c096f6a55e6f06.jpg",
+        "image_caption": [
+            "Figure 3: The shape-biased model and the texture-biased model attend on complementary cues for predictions. We use Class Activation Mapping to visualize which image regions are attended by models. Redder regions indicates more attentions are paid by models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/896f20f3dc475d08249a0debfd12f702f8e65a4a9ce6afce1bc6ef0e1d66f59a.jpg",
+        "image_caption": [
+            "Figure 4: The shape-biased model and the texture-biased model are good/bad at classifying different object categories. We sort these object categories according to the model’s corresponding top-1 accuracy, where the righter one indicates a lower accuracy achieved by the model. "
+        ],
+        "image_footnote": [],
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+    {
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "As distinct cues are picked by shape-biased/texture-biased models, a more concrete observation is they are good/bad at classifying quite different object categories. As showed in Figure 4, the shapebiased model is good at recognizing objects with representative shape structure like obelisk, but is bad at recognizing objects whose shape is uninformative or almost indistinguishable from others like fur coat. Similarly, the texture-biased model can effectively recognize objects with unique texture patterns like brain coral but may fail to recognize objects with unpredictable texture like trolleybus (as its side body can be painted with different advertisements). Besides, biased models may inevitably perform poorly on certain categories as insufficient cues are applied. For examples, it is challenging to distinguish between a lemon and an orange if texture information cannot be utilized, or to distinguish between an lion and a tabby cat without shape information. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given the analysis above, we can conclude that biased representations limit models’ recognition ability. But meanwhile, our ablation delivers a promising message—the features learned by biased models are highly complementary to each other. This observation indicates the current training framework is improvable (as the resulted models are biased towards texture (Geirhos et al., 2019) or shape (Shi et al., 2020)), and offers a potential direction for building a stronger one—we should train models to properly acquire both shape and texture feature representations. We will introduce a simple method for doing so next. ",
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+    {
+        "type": "text",
+        "text": "3 SHAPE-TEXTURE DEBIASED NEURAL NETWORK TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Recall that when obtaining a biased model, the strategy of label assignment is pivot—when the labels are exclusively determined by the images that provide shape (or texture) information in style transfer, we will obtain a shape-biased (or texture-biased) model. Therefore, to guide models for leveraging both shape and texture for predictions, we hereby propose a simple way, which is inspired by Mixup (Zhang et al., 2018), to softly construct labels during training. In other words, given the one-hot label of the shape-source image $y _ { s }$ and the one-hot label of the texture-source image $y _ { t }$ , the new label that we assigned to the cue conflict image is ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/4f950a9bd266422054993735d61ea755c4b53244435cd8b186407cb8d19f45aa.jpg",
+        "text": "$$\n\\widetilde { y } = \\gamma * y _ { s } + ( 1 - \\gamma ) * y _ { t } ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma \\in [ 0 , 1 ]$ is a manually selected hyperparameter to control the relative importance between shape and texture. By ranging the shape-texture coefficient $\\gamma$ from 0 to 1, we obtain a path to evolve the model from being a texture-biased one (i.e., $\\gamma = 0$ ) to being a shape-biased one (i.e., $\\gamma = 1$ ). Although the two extreme ends lead to biased models with inferior performance, we empirically show that there exist a sweet point along this interpolation path, i.e., the learned models can properly acquires both shape and texture feature representations and achieve superior performance on a wide range of image recognition benchmarks. ",
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+        "type": "text",
+        "text": "We name this simple method as shape-texture debiased neural network training, and illustrate the training pipeline in Figure 2(c). It is worth to mention that, although Figure 2 only shows the procedure of applying our method to the image classification task, this training framework is general and has the potential to be extended to other computer vision tasks, e.g., a simple showcase on semantic segmentation is presented in Section 4.4. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTS SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Datasets. We evaluate models on ImageNet classification and PASCAL VOC semantic segmentation. ImageNet dataset (Russakovsky et al., 2015) consists of 1.2 million images for training, and 50,000 for validation, from 1,000 classes. PASCAL VOC 2012 segmentation dataset (Everingham et al., 2012) with extra annotated images from (Hariharan et al., 2011) involves 20 foreground object classes and one background class, including 10,582 training images and 1,449 validation images. ",
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+    {
+        "type": "text",
+        "text": "Going beyond the standard benchmarks, we further evaluate models’ generalization on ImageNetA, ImageNet-C and Stylized-ImageNet, and robustness by defending against FGSM adversarial attacker on ImageNet. ImageNet- $C$ (Hendrycks & Dietterich, 2019) is a benckmark dataset that measures models’ corruption robustness. It is constructed by applying 75 common visual corruptions to the ImageNet validation set. ImageNet-A (Hendrycks et al., 2019) includes 7,500 natural adversarial examples that successfully attacks unseen classifiers. These examples are much harder than original ImageNet validation images due to scene complications encountered in the long tail of scene configurations and by exploiting classifier blind spots (Hendrycks et al., 2019). StylizedImageNet (Geirhos et al., 2019) is a stylized version of ImageNet that constructed by re-rendering the original images by AdaIN stylizer (Huang & Belongie, 2017). The generated images keep the original global shape information but removes the local texture information. FGSM (Goodfellow et al., 2015) is a widely used adversarial attacker to evaluate model robustness. We set the maximum perturbation change per pixel $\\epsilon = 1 6 / 2 5 5$ for FGSM. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation details. We choose ResNet (He et al., 2016) as the default architecture. For image classification tasks, our implementation is based on the publicly available framework in PyTorch2. To generate cue conflict images, we follow Geirhos et al. (2019) to use Adaptive Instance Normalization (Huang & Belongie, 2017) in style transfer, and set stylization coefficient $\\alpha = 0 . 5$ . Importantly, to increase the diversity of training samples, we generate these cue conflict images on-the-fly during training. We choose the shape-texture coefficient $\\gamma = 0 . 8$ when assigning labels. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "When training shape-biased, texture-biased and our shape-texture debiased models, we always apply the auxiliary batch normalization (BN) design (Xie et al., 2020; Xie & Yuille, 2020; Chen et al., ",
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+    {
+        "type": "table",
+        "img_path": "images/3226623f79a7419f1e3a7fdfa5004b54ed0d97ac5b7253f036406bddf835c523.jpg",
+        "table_caption": [
+            "Table 1: The performance of the vanilla training, the shape-biased (S-biased) training, the texturebiased (T-biased) training, and our shape-texture debiased training on ImageNet. For all ResNet models, our debiased training shows the best performance among others. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>VANILLA</td><td>2×EPOCHS</td><td>S-BIASED</td><td>T-BIASED</td><td>DEBIASED</td></tr><tr><td>ResNet-50</td><td>76.4</td><td>76.4 (+0.0)</td><td>76.2 (-0.2)</td><td>75.3 (-1.1)</td><td>76.9 (+0.5)</td></tr><tr><td>ResNet-101</td><td>78.0</td><td>78.0 (+0.0)</td><td>78.0 (-0.0)</td><td>77.4 (-0.6)</td><td>78.9 (+0.9)</td></tr><tr><td>ResNet-152</td><td>78.6</td><td>79.1 (+0.5)</td><td>78.6 (-0.0)</td><td>78.1 (-0.5)</td><td>79.8 3(+1.2)</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/31d7e7a38b998909962c1c56056015789cfa4a28acdb15d94b62a2d167874304.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: The model robustness on ImageNet-A (IN-A), ImageNet-C (IN-C), Stylized-ImageNet (SIN), and on defending against FGSM adversarial attacker on ImageNet. Our shape-texture debiased neural network training significantly boosts the model robustness over the vanilla training baseline. "
+        ],
+        "table_body": "<table><tr><td></td><td>IN-A Acc. ↑</td><td>IN-C mCE↓</td><td>S-IN Acc. ↑</td><td>FGSM Acc.↑</td></tr><tr><td>ResNet-50 +Debiased</td><td>2.0 3.5 (+1.5)</td><td>75.0 67.5 (-7.5)</td><td>7.4 17.4 (+10.0)</td><td>17.1 27.4 (+10.3)</td></tr><tr><td>ResNet-101 +Debiased</td><td>5.6 9.1 1 (+3.5)</td><td>69.8 62.2 (-7.6)</td><td>9.9 22.0 (+12.1)</td><td>23.1 34.4 (+11.3)</td></tr><tr><td>ResNet-152 +Debiased</td><td>7.4 12.6 (+5.2)</td><td>67.2 58.9 (-8.3)</td><td>11.3 22.4 (+11.1)</td><td>25.2 39.6 (+14.4)</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "2021) to bridge the domain gap between the original data and the augmented data, i.e., the main BN is exclusively running on original ImageNet images and the auxiliary BN is exclusively running on cue conflict images. We follow Xie et al. (2020) to always apply the main BN for performance evaluation. Besides, since our biased models and debiased models are all trained with both the original data and the augmented data (i.e., $2 \\times$ data are used in training), we also consider a stronger baseline (i.e., $2 \\times$ epochs training) which doubles the schedule of the vanilla training baseline, for the purpose of matching the total training cost. ",
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+    {
+        "type": "text",
+        "text": "4.2 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Model accuracy. Table 1 shows the results on ImageNet. For all ResNet models, the proposed shape-texture debiased neural network training consistently outperforms the vanilla training baseline. For example, it helps ResNet-50 achieve $7 6 . 9 \\%$ top-1 accuracy, beating its vanilla counterpart by $0 . 5 \\%$ . Our method works better for larger models, e.g., it further improves the vanilla ResNet-152 by $1 . 2 \\%$ , achieving $7 9 . 8 \\%$ top-1 accuracy. ",
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+    {
+        "type": "text",
+        "text": "We then compare our shape-texture debiased training to the $2 \\times$ epochs training baseline. We find that simply doubling the schedule of the vanilla training baseline cannot effectively lead to improvements like ours. For examples, compared to the vanilla ResNet-101, this $2 \\times$ epochs training fails to provide additional improvements, while ours furthers the top-1 accuracy by $1 . 0 \\%$ . This result suggests that it is non-trivial to improve performance even if more computational budgets are given. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lastly, we compare ours to the biased training methods. Though the only difference between our method and the biased training methods is the strategy of label assignment (as shown in Figure 2), it imperatively affects model performance. For example, compared to the vanilla baseline, both the shape-biased training and the texture-biased training fail to improve (sometimes even slightly hurt) the model accuracy, while our shape-texture debiased neural network training successfully leads to consistent and substantial accuracy improvements. ",
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+    {
+        "type": "text",
+        "text": "Model robustness. Next, we evaluate models’ generalization on ImageNet-A, ImageNet-C and Stylized-ImageNet, and robustness on defending against FGSM on ImageNet. We note these tasks are much more challenging than the original ImageNet classification, e.g., the ImageNet trained ResNet-50 only achieves $2 . 0 \\%$ accuracy on ImageNet-A, $7 5 . 0 \\%$ mCE on ImageNet-C, $7 . 4 \\%$ accuracy on Stylized-ImageNet, and $1 7 . 1 \\%$ accuracy on defending against FGSM adversarial attacker. As shown in Table 2, our shape-texture debiased neural network training beats the vanilla training baseline by a large margin on all tasks for all ResNet models. For example, it substantially boosts ResNet-152’s performance on ImageNet-A $( + 5 . 2 \\%$ , from $7 . 4 \\%$ to $1 2 . 6 \\%$ ), ImageNet-C $( - 8 . 3 \\%$ , from $6 7 . 2 \\%$ to $5 8 . 9 \\%$ , the lower the better) and Stylized-ImageNet $( + 1 1 . 1 \\%$ , from $1 1 . 3 \\%$ to $2 2 . 4 \\%$ ), and on defending against FGSM on ImageNet $+ 1 4 . 4 \\%$ , from $2 5 . 2 \\%$ to $3 9 . 6 \\%$ ). These results altogether suggest that our shape-texture debiased neural network training is an effective way to mitigate the issue of shortcut learning (Geirhos et al., 2020). ",
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+    {
+        "type": "table",
+        "img_path": "images/7aa013da64fb4640b9132834a6a5044fae31fc5ba20684619a16b822e91ff8f8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>IN Acc. 个</td><td>IN-A Acc.个</td><td>IN-C mCE↓</td><td>S-IN Acc. 个</td><td>FGSM Acc.个</td></tr><tr><td>ResNet-50</td><td>76.4</td><td>2.0</td><td>75.0</td><td>7.4</td><td>17.1</td></tr><tr><td>CutMix + MoEx (Li et al., 2021)</td><td>79.0</td><td>8.0</td><td>74.8</td><td>5.0</td><td>41.0</td></tr><tr><td>DeepAugment + AugMix (Hendrycks et al., 2020)</td><td>75.8</td><td>3.9</td><td>53.6</td><td>21.2</td><td>18.8</td></tr><tr><td>SIN (Geirhos et al., 2019)</td><td>60.2</td><td>2.4</td><td>77.3</td><td>56.2</td><td>5.6</td></tr><tr><td>Shape-Texture Debiased Training (ours)</td><td>76.9</td><td>3.5</td><td>67.5</td><td>17.4</td><td>27.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Compare with state-of-the-art methods using ResNet-50 on ImageNet (IN), ImageNet-A (IN-A), ImageNet-C (IN-C), Stylized-ImageNet (S-IN), and on defending against FGSM on ImageNet. We use green to denote significant improvement, red to denote performance drop, and gray to denote similar performance. We observe our shape-texture debiased training is the only method that successfully leads to improvements over the vanilla baseline on all benchmarks. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8118b5e0cdeca5554779ff8abaa53a7a54eed6b3de3728e2e155e98987a9e1a2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>VANILLA</td><td>S-BIASED</td><td>T-BIASED</td><td>DEBIASED</td></tr><tr><td>ImageNet-Sketch</td><td>23.8</td><td>27.9</td><td>24.3</td><td>28.4</td></tr><tr><td>ImageNet-R</td><td>36.2</td><td>40.6</td><td>36.7</td><td>40.8</td></tr><tr><td>Kylberg Texture</td><td>99.5</td><td>99.1</td><td>99.6</td><td>99.5</td></tr><tr><td>FlickerMaterial</td><td>74.6</td><td>73.3</td><td>79.2</td><td>75.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4: The performance comparison between Vanilla, Shape-biased, Texture-biased, and ShapeTexture Debiased models on ImageNet-Sketch, ImageNet-R, Kylberg Texture, and Flicker Material datasets. We note the shape-biased and the shape-texture debiased models perform better on shape datasets (ImageNet-Sketch and ImageNet-R); the texture-biased and the shape-texture debiased models perform better on texture datasets (Kylberg Texture and Flicker Material). ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Comparing to SoTAs. We further compare our shape-texture debiased model with the SoTA on ImageNet and ImageNet-A (CutMix $^ +$ MoEx (Li et al., 2021)), the SoTA on ImageNet-C (DeepAugment $^ +$ AugMix (Hendrycks et al., 2020)), and the SoTA on Stylized-ImageNet (SIN (Geirhos et al., 2019)). Interestingly, we note the improvements of all these SoTAs are not consistent across different benchmarks. For example, as shown in Table 3, SIN significantly improves the results on Stylized-ImageNet, but at the cost of huge performance drop on ImageNet $( - 1 6 . 2 \\% )$ and ImageNetC $( - 2 . 3 \\% )$ . Our shape-texture debiased training stands as the only method that can improve the vanilla training baseline holistically. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.3 ABLATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparing to model ensembles. An alternative but na¨ıve way for obtaining the model with both shape and texture information is to ensemble a shape-biased model and a texture-biased model. We note this ensemble strategy yields a model of on-par performance with our shape-texture debiased model on ImageNet $7 7 . 2 \\%$ vs. $7 6 . 9 \\%$ ). Nonetheless, interestingly, when measuring model robustness, such model ensemble strategy is inferior than ours. For example, compared to our proposed debiased training, this ensemble strategy is $1 . 5 \\%$ worse on ImageNet-A ( $2 . 0 \\%$ vs. $3 . 5 \\%$ ), $1 . 1 \\%$ worse on ImageNet-C $6 8 . 6 ~ \\mathrm { m C E }$ vs. $6 7 . 5 ~ \\mathrm { m C E }$ ), $1 . 1 \\%$ worse on Stylized-ImageNet ( $1 6 . 3 \\%$ vs. $1 7 . 4 \\%$ ), and $7 . 0 \\%$ worse on defending against FGSM $2 0 . 4 \\%$ vs. $2 7 . 4 \\%$ ). Moreover, due to model ensemble, this strategy is $2 \\times$ expensive at the inference stage. These evidences clearly demonstrate the effectiveness and efficiency of the proposed shape-texture debiased training. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Does our method help models to learn debiased shape-texture representations? Here we take a close look at whether our method indeed prevents models from being biased toward shape or texture during learning. We evaluate models in Section 4.2 on two kinds of datasets: (1) ImageNetSketch dataset (Wang et al., 2019) and ImageNet-R (Hendrycks et al., 2020) for examining how well models can capture shape; and (2) Kylberg Texture dataset (Kylberg, 2011) and Flicker Material dataset (Sharan et al., 2014) for examining how well models can capture texture. Specifically, since object categories from two texture datasets are not compatible to that from ImageNet dataset, we retrain the last fc-layer (while keeping all other layers untouched) of all models on Kylberg Texture dataset or Flicker Material dataset for 5 epochs. The results are shown in Table 4. ",
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+    },
+    {
+        "type": "text",
+        "text": "We first analyze results on ImageNet-Sketch dataset. We observe our shape-texture debiased models are as good as the shape-biased models, and significantly outperforms the texture-biased models and the vanilla training models. For instance, using ResNet-50, our shape-texture debiased training and shape-biased training achieve $2 8 . 4 \\%$ top-1 accuracy and $2 7 . 9 \\%$ top-1 accuracy, while texture-biased training and vanilla training only get $2 4 . 3 \\%$ top-1 accuracy and $2 3 . 8 \\%$ top-1 accuracy. A similar observation can be seen from ImageNet-R. These results support that our method helps models acquire stronger shape representations than the vanilla training. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b8f58c07b58608ea11407af951dddb69b2f963cbff7c4a52d31732da1be47c0.jpg",
+        "image_caption": [
+            "Figure 5: Illustration of the data preparation pipeline of our shape-texture debiased neural network training on the semantic segmentation task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We next analyze results on Kylberg Texture dataset. Similarly, we observe that our debiased model are comparable to the texture-biased model and the vanilla training model, and get better performance than the shape-biased model. On Flicker Material dataset, we observe that our debiased models are better than the vanilla training model and the shape-biased model. This phenomenon suggests texture information is effectively caught by our shape-texture debiased training. As a side note, it is expected that vanilla training are better than shape-biased training on these texture datasets, as Geirhos et al. (2019) point out that ImageNet trained models (i.e., vanilla training) also tend to be biased towards texture. ",
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+    {
+        "type": "text",
+        "text": "With the analysis above, we conclude that, compared to vanilla training, our shape-texture debiased training successfully helps networks effectively acquire both shape and texture representations. ",
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+        "type": "text",
+        "text": "Combining with other data augmentation methods. Our shape-texture debiased neural network training can be viewed as a data augmentation method, which trains models on cue conflict images. Nonetheless, our method specifically guides the model to learn debiased shape and texture representations, which could potentially serve as a complementary feature to other data augmentation methods. To validate this argument, we train models using a combination of our method and an existing data augmentation method (i.e., Mixup (Zhang et al., 2018) or CutMix (Yun et al., 2019)). ",
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+        "type": "text",
+        "text": "We choose ResNeXt-101 (Xie et al., 2017) as the backbone network, which reports the best top-1 ImageNet accuracy in both the Mixup paper, i.e., $7 9 . 9 \\%$ , and the CutMix paper, i.e., $8 0 . 5 \\%$ . Though building upon very strong baselines, our shape-texture debiased neural network training still leads to substantial improvements, e.g., it furthers ResNeXt-101-Mixup’s accuracy to $8 0 . 5 \\%$ $( + 0 . 6 \\% )$ , and ResNeXt-101-CutMix’s accuracy to $8 1 . 2 \\%$ $( + 0 . 7 \\% )$ . Meanwhile, models’ generalization also get greatly improved. For example, by combining CutMix and our method, ResNeXt-101 gets additional improvements on ImageNet-A $( + 1 . 4 \\% )$ , ImageNet-C $( - 5 . 9 \\%$ , the lower the better) and Stylized ImageNet $( + 7 . 5 \\% )$ . These results support that our shape-texture debiased neural network training is compatible to existing data augmentation methods. ",
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+    {
+        "type": "text",
+        "text": "Shape-texture coefficient $\\gamma$ . We set $\\gamma = 0 . 8$ in our shape-texture debiased training. This value is found via the grid search over ImageNet-200 using ResNet-18. We now ablate its sensitivity on ImageNet using ResNet-50, where $\\gamma$ is linearly interpolated between 0.0 and 1.0. By increasing the value of $\\gamma$ , we observe that the corresponding accuracy on ImageNet first monotonically goes up, and then monotonically goes down. The sweet point can be reached by setting $\\gamma = 0 . 7$ , where ResNet-50 achieves $7 7 . 0 \\%$ top-1 ImageNet accuracy. Besides, we note that by setting $\\gamma \\in [ 0 . 5 , 0 . 9 ]$ can always lead to performance improvements over the vanilla baseline. These results demonstrate the robustness of our shape-texture debiased neural network training w.r.t. the coefficient $\\gamma$ . ",
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+    {
+        "type": "text",
+        "text": "4.4 SEMANTIC SEGMENTATION RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We extend our shape-texture debiased neural network training to the segmentation task. We select DeepLabv3-ResNet-101 (Chen et al., 2017) as our backbone. To better incorporate our method with the segmentation task, the following changes are made when generating cue conflict images: (1) unlike in the classification task where the whole image is used as the texture source, we use a specific object (which can cropped from the background using the segmentation ground-truth) to provide texture information in style transfer; (2) when composing the soft label for the cue conflict image, we set the label mask from texture source as the full image (since the pattern from the texture source will fill the whole image after style transfer); and (3) we set stylization coefficient $\\alpha = 0 . 2$ and shape-texture coefficient $\\gamma = 0 . 9 5$ to prevent object boundaries from being overly blurred in style transfer. Figure 5 shows an illustration of our data preparation pipeline. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Results. Our shape-texture debiased training can also effectively improve segmentation models. For example, our method helps DeepLabv3-ResNet-101 achieve $7 7 . 6 \\%$ mIOU, significantly beating its vanilla counterpart by $1 . 1 \\%$ . Our method still shows advantages when compared to the $2 \\times$ epochs training baseline. Doubling the learning schedule of the vanilla training can only lead to an improvement of $0 . 2 \\%$ , which is still $0 . 9 \\%$ worse than our shape-texture debiased training. These results demonstrate the potential of our methods in helping recognition tasks in general. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Data augmentation. Data augmentation is essential for the success of deep learning (LeCun et al., 1998; Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; Zhong et al., 2020; Cubuk et al., 2019; Lim et al., 2019; Cubuk et al., 2020). Our shape-texture debiased neural network training is related to a specific family of data augmentation, called Mixup (Zhang et al., 2018), which blends pairs of images and their labels in a convex manner, either at pixel-level (Zhang et al., 2018; Yun et al., 2019) or feature-level (Verma et al., 2019; Li et al., 2021). Our method can be interpreted as a special instantiation of Mixup which blends pairs of images at the abstraction level—images’ texture information and shape information are mixed. Our method successfully guides CNNs to learn better shape and texture representations, which is an important but missing piece in existing data argumentation methods. ",
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+    {
+        "type": "text",
+        "text": "Style transfer. Style transfer, closely related to texture synthesis and transfer, means generating a stylized image by combining a shape-source image and a texture-source image (Efros & Leung, 1999; Efros & Freeman, 2001; Elad & Milanfar, 2017). The seminal work (Gatys et al., 2016) demonstrate impressive style transfer results by matching feature statistics in convolutional layers of a CNN. Later follow-ups further improve the generation quality and speed (Huang & Belongie, 2017; Chen & Schmidt, 2016; Ghiasi et al., 2017; Li et al., 2017). In this work, we follow Geirhos et al. (2019) to use AdaIN (Huang & Belongie, 2017) to generate stylized images. Nonetheless, instead of applying style transfer between an image and an artistic paintings as in Geirhos et al. (2019), we directly apply style transfer on a pair of images to generate cue conflict images. This change is vital as it enables us to provide supervisions from both shape and texture during training. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "There is a long-time debate about which cue dominates the object recognition. By carefully ablate the shape-biased model and the texture-biased model, we found though biased feature representations lead to performance degradation, they are complementary to each other and are both necessary for image recognition. To this end, we propose shape-texture debiased neural network training for guiding CNNs to learn better feature representations. The key in our method is that we should not only augment training set with cue conflict images, but also provide supervisions from both shape and texture. We empirically demonstrate the advantages of our shape-texture debiased neural network training on boosting both accuracy and robustness. Our method is conceptually simple and is generalizable to different image recognition tasks. We hope our work will shed light on understanding and improving convolutional neural networks. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This project is partially supported by ONR N00014-18-1-2119 and ONR N00014-20-1-2206. Cihang Xie is supported by the Facebook PhD Fellowship and a gift grant from Open Philanthropy. Yingwei Li thanks Zhiwen Wang for suggestions on figures. ",
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+        "text": "Cihang Xie and Alan Yuille. Intriguing properties of adversarial training at scale. In ICLR, 2020.   \nCihang Xie, Mingxing Tan, Boqing Gong, Jiang Wang, Alan Yuille, and Quoc V Le. Adversarial examples improve image recognition. In CVPR, 2020.   \nSaining Xie, Ross Girshick, Piotr Dollar, Zhuowen Tu, and Kaiming He. Aggregated residual trans- ´ formations for deep neural networks. In CVPR, 2017.   \nSangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In ICCV, 2019.   \nHongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In ICLR, 2018.   \nSongfeng Zheng, Zhuowen Tu, and Alan L. Yuille. Detecting object boundaries using low-, mid-, and high-level information. In CVPR, 2007.   \nZhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang. Random erasing data augmentation. In AAAI, 2020.   \nBolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discriminative localization. In CVPR, 2016. ",
+        "bbox": [
+            171,
+            99,
+            826,
+            400
+        ],
+        "page_idx": 11
+    }
+]

parse/train/HklUCCVKDB/HklUCCVKDB.md

@@ -0,0 +1,361 @@
+# UNCERTAINTY-GUIDED CONTINUAL LEARNING WITH BAYESIAN NEURAL NETWORKS
+
+Sayna Ebrahimi∗ Mohamed Elhoseiny† UC Berkeley KAUST, Stanford University
+
+Trevor Darrell UC Berkeley
+
+Marcus Rohrbach Facebook AI Research
+
+# ABSTRACT
+
+Continual learning aims to learn new tasks without forgetting previously learned ones. This is especially challenging when one cannot access data from previous tasks and when the model has a fixed capacity. Current regularization-based continual learning algorithms need an external representation and extra computation to measure the parameters’ importance. In contrast, we propose Uncertaintyguided Continual Bayesian Neural Networks (UCB), where the learning rate adapts according to the uncertainty defined in the probability distribution of the weights in networks. Uncertainty is a natural way to identify what to remember and what to change as we continually learn, and thus mitigate catastrophic forgetting. We also show a variant of our model, which uses uncertainty for weight pruning and retains task performance after pruning by saving binary masks per tasks. We evaluate our UCB approach extensively on diverse object classification datasets with short and long sequences of tasks and report superior or on-par performance compared to existing approaches. Additionally, we show that our model does not necessarily need task information at test time, i.e. it does not presume knowledge of which task a sample belongs to.
+
+# 1 INTRODUCTION
+
+Humans can easily accumulate and maintain knowledge gained from previously observed tasks, and continuously learn to solve new problems or tasks. Artificial learning systems typically forget prior tasks when they cannot access all training data at once but are presented with task data in sequence.
+
+Overcoming these challenges is the focus of continual learning, sometimes also referred to as lifelong learning or sequential learning. Catastrophic forgetting (McCloskey & Cohen, 1989; McClelland et al., 1995) refers to the significant drop in the performance of a learner when switching from a trained task to a new one. This phenomenon occurs because trained parameters on the initial task change in favor of learning new objectives.
+
+Given a network of limited capacity, one way to address this problem is to identify the importance of each parameter and penalize further changes to those parameters that were deemed to be important for the previous tasks (Kirkpatrick et al., 2017; Aljundi et al., 2018; Zenke et al., 2017). An alternative is to freeze the most important parameters and allow future tasks to only adapt the remaining parameters to new tasks (Mallya & Lazebnik, 2018). Such models rely on the explicit parametrization of importance. We propose here implicit uncertainty-guided importance representation.
+
+Bayesian approaches to neural networks (MacKay, 1992b) can potentially avoid some of the pitfalls of explicit parameterization of importance in regular neural networks. Bayesian techniques, naturally account for uncertainty in parameters estimates. These networks represent each parameter with a distribution defined by a mean and variance over possible values drawn from a shared latent probability distribution (Blundell et al., 2015). Variational inference can approximate posterior distributions using Monte Carlo sampling for gradient estimation. These networks act like ensemble methods in that they reduce the prediction variance but only use twice the number of parameters present in a regular neural network. We propose to use the predicted mean and variance of the latent distributions to characterize the importance of each parameter. We perform continual learning with
+
+![](images/e2a287656723b19af987cc33a73abe212da1921569874429b9a7713c410eca4d.jpg)  
+Illustration of evolution of weight distributions through learning two tasks. (a) circles represent Figure 1: Illustration of the evolution of weight distributions – uncertain weights adapt more quickly – from Ɲ(0,0.1). As an example we show five color-coded and plot their distributions. (b) Shows when learning two tasks using UCB. (a) weight parameter initialized by distributions initialized with posterior distribution after learning Task 1. Whicontributions in learning Task 1), W3, W4, and mean and variance values randomly sampled from $\mathcal { N } ( 0 , 0 . 1 )$ xhibit lower uncertainties (more ve larger uncertainties, with the . (b) posterior distribution after learning task one; while $\theta _ { 1 }$ higand $\theta _ { 2 }$ STD in W5, making them available to learn more tasks. (c) Task 2 is learned using highexhibit lower uncertainties after learning the first task, $\theta _ { 3 } , \theta _ { 4 }$ , and $\theta _ { 5 }$ have learning rates for previously uncertain parameters (W3 and W4, W5) while learning rates for W1 and W2 are moderated according to their predicted low uncertainty after finishing task 1. larger uncertainties, making them available to learn more tasks. (c) a second task is learned using 1 2 higher learning rates for previously uncertain parameters $( \theta _ { 1 } , \theta _ { 2 } , \theta _ { 3 }$ , and $\theta _ { 4 }$ ) while learning rates for $\theta _ { 1 }$ and $\theta _ { 2 }$ 3 4 5 are reduced. Size of the arrows indicate the magnitude of the change of the distribution mean upon gradient update.
+
+Bayesian neural networks by controlling the learning rate of each parameter as a function of its uncertainty. Figure 1 illustrates how posterior distributions evolve for certain and uncertain weight distributions while learning two consecutive tasks. Intuitively, the more uncertain a parameter is, the more learnable it can be and therefore, larger gradient steps can be taken for it to learn the current task. As a hard version of this regularization technique, we also show that pruning, i.e., preventing the most important model parameters from any change and learning new tasks with the remaining parameters, can be also integrated into UCB. We refer to this method as UCB-P.
+
+Contributions: We propose to perform continual learning with Bayesian neural networks and develop a new method which exploits the inherent measure of uncertainty therein to adapt the learning rate of individual parameters (Sec. 4). Second, we introduce a hard-threshold variant of our method that decides which parameters to freeze (Sec. 4.2). Third, in Sec. 5, we extensively validate our approach experimentally, comparing it to prior art both on single datasets split into different tasks, as well as for the more difficult scenario of learning a sequence of different datasets. Forth, in contrast to most prior work, our approach does not rely on knowledge about task boundaries at inference time, which humans do not need and might not be always available. We show in Sec. 6 that our approach naturally supports this scenario and does not require task information at test time, sometimes also referred to as a “single head” scenario for all tasks. We refer to evaluation metric of a “single head” model without task information at test time as “generalized accuracy”. Our code is available at https://github.com/SaynaEbrahimi/UCB.
+
+# 2 RELATED WORK
+
+Conceptually, approaches to continual learning can be divided into the following categories: dynamic architectural methods, memory-based methods, and regularization methods.
+
+Dynamic architectural methods: In this setting, the architecture grows while keeping past knowledge fixed and storing new knowledge in different forms such as additional layers, nodes, or modules. In this approach, the objective function remains fixed whereas the model capacity grows –often exponentially– with the number of tasks. Progressive networks (Rusu et al., 2016; Schwarz et al., 2018) was one of the earliest works in this direction and was successfully applied to reinforcement learning problems; the base architecture was duplicated and lateral connections added in response to new tasks. Dynamically Expandable Network (DEN) (Yoon et al., 2018) also expands its network by selecting drifting units and retraining them on new tasks. In contrast to our method, these approaches require the architecture grow with each new task.
+
+Memory-based methods: In this regime, previous information is partially stored to be used later as a form of rehearsal (Robins, 1995). Gradient episodic memory (GEM) (Lopez-Paz et al., 2017) uses this idea to store the data at the end of each episode to be used later to prevent gradient updates from deviating from their previous values. GEM also allows for positive backward knowledge transfer, i.e, an improvement on previously learned tasks, and it was the first method capable of learning using a single training example. Recent approaches in this category have mitigated forgetting by using external data combined with distillation loss and/or confidence-based sampling strategies to select the most representative samples. (Castro et al., 2018; Wu et al., 2019; Lee et al., 2019)
+
+Regularization methods: In these approaches, significant changes to the representation learned for previous tasks are prevented. This can be performed through regularizing the objective function or directly enforced on weight parameters. Typically, this importance measure is engineered to represent the importance of each parameter. Inspired by Bayesian learning, in elastic weight consolidation (EWC) method (Kirkpatrick et al., 2017) important parameters are those to have the highest in terms of the Fisher information matrix. In Synaptic Intelligence (SI) (Zenke et al., 2017) this parameter importance notion is engineered to correlate with the loss function: parameters that contribute more to the loss are more important. Similar to SI, Memory-aware Synapses (MAS) (Aljundi et al., 2018) proposed an online way of computing importance adaptive to the test set using the change in the model outputs w.r.t the inputs. While all the above algorithms are task-dependent, in parallel development to this work, (Aljundi et al., 2019) has recently investigated task-free continual learning by building upon MAS and using a protocol to update the weights instead of waiting until the tasks are finished. PackNet (Mallya & Lazebnik, 2018) used iterative pruning to fully restrict gradient updates on important weights via binary masks. This method requires knowing which task is being tested to use the appropriate mask. PackNet also ranks the weight importance by their magnitude which is not guaranteed to be a proper importance indicative. HAT (Serra et al., 2018) identifies important neurons by learning an attention vector to the task embedding to control the gradient propagation. It maintains the information learned on previous tasks using an almost-binary mask per previous tasks.
+
+Bayesian approaches: Using Bayesian approach in learning neural networks has been studied for few decades (MacKay, 1992b;a). Several approaches have been proposed for Bayesian neural networks, based on, e.g., the Laplace approximation (MacKay, 1992a), Hamiltonian Monte Carlo (Neal, 2012), variational inference (Hinton & Van Camp, 1993; Graves, 2011), and probabilistic backpropagation (Hernandez-Lobato & Adams, 2015). Variational continual learning (Nguyen et al., ´ 2018) uses Bayesian inference to perform continual learning where new posterior distribution is simply obtained by multiplying the previous posterior by the likelihood of the dataset belonging to the new task. They also showed that by using a core-set, a small representative set of data from previous tasks, VCL can experience less forgetting. In contrast, we rely on Bayesian neural networks to use their predictive uncertainty to perform continual learning. Moreover, we do not use episodic memory or any other way to access or store previous data in our approach.
+
+Natural gradient descent methods: A fast natural gradient descent method for variational inference was introduced in (Khan & Nielsen, 2018) in which, the Fisher Information matrix is approximated using the generalized Gauss-Newton method. In contrast, in our work, we use classic gradient descent. Although second order optimization algorithms are proven to be more accurate than the first order methods, they add considerable computational cost. Tseran et al. (2018); Chen et al. (2019) both investigate the effect of natural gradient descent methods as an alternative to classic gradient descent used in VCL and EWC methods. GNG (Chen et al., 2019) uses Gaussian natural gradients in the Adam optimizer (Kingma & Ba, 2014) in the framework of VCL because as opposed to conventional gradient methods which perform in Euclidian space, natural gradients cause a small difference in terms of distributions following the changes in parameters in the Riemannian space. Similar to VCL, they obtained their best performance by adding a coreset of previous examples. Tseran et al. (2018) introduce two modifications to VCL called Natural-VCL (N-VCL) and VCL-Vadam. N-VCL (Tseran et al., 2018) uses a Gauss-Newton approximation introduced by (Schraudolph, 2002; Graves, 2011) to estimate the VCL objective function and used natural gradient method proposed in (Khan et al., 2018) to exploit the Riemannian geometry of the variational posterior by scaling the gradient with an adaptive learning rate equal to $\bar { \sigma } ^ { - 2 }$ obtained by approximating the Fisher Information matrix in an online fashion. VCL-Vadam (Tseran et al., 2018) is a simpler version of N-VCL to trade-off accuracy for simplicity which uses Vadam (Khan et al., 2018) to update the gradients by perturbing the weights with a Gaussian noise using a reparameterization trick and scaling by $\sigma ^ { - 1 }$ instead of its squared. N-VCL/VCL-Vadam both use variational inference to adapt the learning rate within Adam optimizer at every time step, whereas in our method below, gradient decent is used with constant learning rate during each task where learning rate scales with uncertainty only after finishing a task. We show extensive comparison with state-of-the-art results on short and relatively long sequence of vision datasets with Bayesian convolutional neural networks, whereas VCL-Vadam only rely on multi-layer perceptron networks. We also like to highlight that this is the first work which evaluates and shows the working of convolutional Bayesian Neural Networks rather than only fully connected MLP models for continual learning.
+
+# 3 BACKGROUND: VARIATIONAL BAYES-BY-BACKPROP
+
+In this section, we review the Bayes-by-Backprop (BBB) framework which was introduced by (Blundell et al., 2015); to learn a probability distribution over network parameters. (Blundell et al., 2015) showed a back-propagation-compatible algorithm which acts as a regularizer and yields comparable performance to dropout on the MNIST dataset. In Bayesian models, latent variables are drawn from a prior density $p ( \mathbf { w } )$ which are related to the observations through the likelihood $p ( \mathbf { x } | \mathbf { w } )$ During inference, the posterior distribution $p ( \mathbf { w } | \mathbf { x } )$ is computed conditioned on the given input data. However, in practice, this probability distribution is intractable and is often estimated through approximate inference. Markov Chain Monte Carlo (MCMC) sampling (Hastings, 1970) has been widely used and explored for this purpose, see (Robert & Casella, 2013) for different methods under this category. However, MCMC algorithms, despite providing guarantees for finding asymptotically exact samples from the target distribution, are not suitable for large datasets and/or large models as they are bounded by speed and scalability issues. Alternatively, variational inference provides a faster solution to the same problem in which the posterior is approximated using optimization rather than being sampled from a chain (Hinton & Van Camp, 1993).Variational inference methods always take advantage of fast optimization techniques such as stochastic methods or distributed methods, which allow them to explore data models quickly. See (Blei et al., 2017) for a complete review of the theory and (Shridhar et al., 2018) for more discussion on how to use Bayes by Backprop (BBB) in convolutioal neural networks.
+
+# 3.1 BAYES BY BACKPROP (BBB)
+
+Let $\mathbf { x } \in \mathbb { R } ^ { n }$ be a set of observed variables and w be a set of latent variables. A neural network, as a probabilistic model $P ( \mathbf { y } | \mathbf { x } , \mathbf { w } )$ , given a set of training examples $\mathcal { D } = ( \mathbf { x } , \mathbf { y } )$ can output y which belongs to a set of classes by using the set of weight parameters w. Variational inference aims to calculate this conditional probability distribution over the latent variables by finding the closest proxy to the exact posterior by solving an optimization problem.
+
+We first assume a family of probability densities over the latent variables w parametrized by $\theta$ , i.e., $q ( \mathbf { w } | \theta )$ . We then find the closest member of this family to the true conditional probability of interest $P ( \mathbf { w } | \mathcal { D } )$ by minimizing the Kullback-Leibler (KL) divergence between $q$ and $P$ which is equivalent to minimizing variational free energy or maximizing the expected lower bound:
+
+$$
+\begin{array} { r } { \theta ^ { * } = \arg \operatorname* { m i n } _ { \theta } { \mathrm { K L } } \big ( q ( \mathbf { w } | \theta ) \| P ( \mathbf { w } | \mathcal { D } ) \big ) } \end{array}
+$$
+
+The objective function can be written as:
+
+$$
+\mathcal { L } _ { B B B } ( \theta , \mathcal { D } ) = \mathrm { K L } \big [ q ( \mathbf { w } | \theta ) \| P ( \mathbf { w } ) \big ] - \mathbb { E } _ { q ( \mathbf { w } | \theta ) } \big [ \log ( P ( \mathcal { D } | \mathbf { w } ) ) \big ]
+$$
+
+Eq. 2 can be approximated using $N$ Monte Carlo samples $\mathbf { w } _ { i }$ from the variational posterior (Blundell et al., 2015):
+
+$$
+\mathcal { L } _ { B B B } ( \boldsymbol { \theta } , \mathcal { D } ) \approx \sum _ { i = 1 } ^ { N } \log q ( \mathbf { w } _ { i } | \boldsymbol { \theta } ) - \log P ( \mathbf { w } _ { i } ) - \log ( P ( \mathcal { D } | \mathbf { w } _ { i } ) )
+$$
+
+We assume $q ( \mathbf { w } | \theta )$ to have a Gaussian pdf with diagonal covariance and parametrized by $\theta = \left( \mu , \rho \right)$ . A sample weight of the variational posterior can be obtained by sampling from a unit Gaussian and reparametrized by $\mathbf { w } = \mu + \sigma \circ \epsilon$ where $\epsilon$ is the noise drawn from unit Gaussian, and $\circ$ is a pointwise multipliation. Standard deviation is parametrized as $\sigma = \log ( 1 + \exp ( \rho ) )$ and thus is always positive. For the prior, as suggested by Blundell et al. (2015), a scale mixture of two Gaussian pdfs are chosen which are zero-centered while having different variances of $\sigma _ { 1 } ^ { 2 }$ and $\sigma _ { 2 } ^ { 2 }$ . The uncertainty obtained for every parameter has been successfully used in model compression (Han et al., 2015) and uncertainty-based exploration in reinforcement learning (Blundell et al., 2015). In this work we propose to use this framework to learn sequential tasks without forgetting using per-weight uncertainties.
+
+# 4 UNCERTAINTY-GUIDED CONTINUAL LEARNING IN BAYESIAN NEURAL NETWORKS
+
+In this section, we introduce Uncertainty-guided Continual learning approach with Bayesian neural networks (UCB), which exploits the estimated uncertainty of the parameters’ posterior distribution to regulate the change in “important” parameters both in a soft way (Section 4.1) or setting a hard threshold (Section 4.2).
+
+# 4.1 UCB WITH LEARNING RATE REGULARIZATION
+
+A common strategy to perform continual learning is to reduce forgetting by regularizing further changes in the model representation based on parameters’ importance. In UCB the regularization is performed with the learning rate such that the learning rate of each parameter and hence its gradient update becomes a function of its importance. As shown in the following equations, in particular, we scale the learning rate of $\mu$ and $\rho$ for each parameter distribution inversely proportional to its importance $\Omega$ to reduce changes in important parameters while allowing less important parameters to alter more in favor of learning new tasks.
+
+$$
+\begin{array} { r } { \alpha _ { \mu }  \alpha _ { \mu } / \Omega _ { \mu } } \\ { \alpha _ { \rho }  \alpha _ { \rho } / \Omega _ { \rho } } \end{array}
+$$
+
+The core idea of this work is to base the definition of importance on the well-defined uncertainty in parameters distribution of Bayesian neural networks, i.e., setting the importance to be inversely proportional to the standard deviation $\sigma$ which represents the parameter uncertainty in the Baysian neural network:
+
+$$
+\Omega \propto { } ^ { 1 } / \sigma
+$$
+
+We explore different options to set $\Omega$ in our ablation study presented in Section A.2 of the appendix, Table 1. We empirically found that $\Omega _ { \mu } = 1 / \sigma$ and not adapting the learning rate for $\rho$ (i.e. $\Omega _ { \rho } = 1 $ ) yields the highest accuracy and the least forgetting.
+
+The key benefit of UCB with learning rate as the regularizer is that it neither requires additional memory, as opposed to pruning technique nor tracking the change in parameters with respect to the previously learned task, as needed in common weight regularization methods.
+
+More importantly, this method does not need to be aware of task switching as it only needs to adjust the learning rates of the means in the posterior distribution based on their current uncertainty. The complete algorithm for UCB is shown in Algorithm 1 with parameter update function given in Algorithm 2.
+
+# 4.2 UCB USING WEIGHT PRUNING (UCB-P)
+
+In this section, we introduce a variant of our method, UCB-P, which is related to recent efforts in weight pruning in the context of reducing inference computation and network compression (Liu et al., 2017; Molchanov et al., 2016). More specifically, weight pruning has been recently used in continual learning (Mallya & Lazebnik, 2018), where the goal is to continue learning multiple tasks using a single network’s capacity. (Mallya & Lazebnik, 2018) accomplished this by freeing up parameters deemed to be unimportant to the current task according to their magnitude. Forgetting is prevented in pruning by saving a task-specific binary mask of important vs. unimportant parameters. Here, we adapt pruning to Bayesian neural networks. Specifically, we propose a different criterion for measuring importance: the statistically-grounded uncertainty defined in Bayesian neural networks.
+
+Unlike regular deep neural networks, in a BBB model weight parameters are represented by probability distributions parametrized by their mean and standard deviation. Similar to (Blundell et al., 2015), in order to take into account both mean and standard deviation, we use the signal-to-noise ratio (SNR) for each parameter defined as
+
+$$
+\Omega = \mathrm { S N R } = | \mu | / \sigma
+$$
+
+1: Require Training data for all tasks $\mathcal { D } = ( \mathbf { x } , \mathbf { y } )$ , $\mu$ (mean of posterior), $\rho , \sigma _ { 1 }$ and $\sigma _ { 2 }$ (std for the scaled   
+mixture Gaussian pdf of prior), $\pi$ (weighting factor for prior), $N$ (number of samples in a mini-batch), $M$   
+(Number of minibatches per epoch), initial learning rate $( \alpha _ { 0 } )$   
+2: $\alpha _ { \mu } = \alpha _ { \rho } = \alpha _ { 0 }$   
+3: for every task do   
+4: repeat   
+5: $\begin{array} { r l r } & { \mathbf { \Lambda } _ { \epsilon \sim \mathcal { N } ( 0 , I ) } ^ { \mathbf { \epsilon } } } & { \sim \mathcal { N } ( 0 , I ) } \\ & { \boldsymbol { \epsilon } \sim \mathrm { l o g } ( 1 + \exp ( \rho ) ) } & { \triangleright \mathrm { E n s u r e s ~ } \sigma \mathrm { ~ i s ~ a l w a y s ~ p o s i t i v e } } \\ & { \mathbf { w } = \mu + \sigma \circ \boldsymbol { \epsilon } } & { \triangleright \mathbf { w } = \{ \mathbf { w } _ { 1 } , \dots , \mathbf { w } _ { i } , \dots , \mathbf { w } _ { N } \} \mathrm { ~ p o s t e r i o r ~ s a m p l e s ~ o f ~ w e i g h t } } \\ & { l _ { 1 } = \sum _ { i = 1 } ^ { N } \log N ( \mathbf { w } _ { i } | \mu , \sigma ^ { 2 } ) } & { \triangleright \mathbf { \epsilon } [ \mathbf { u } _ { 1 } , \dots , \mathbf { u } _ { 0 } , \dots , \mathbf { u } _ { 0 } , \dots , \mathbf { u } _ { 0 } ] } \\ & { l _ { 2 } = \sum _ { i = 1 } ^ { N } \log \big ( \pi N ( \mathbf { \epsilon } _ { i } | 0 , \sigma _ { 1 } ^ { 2 } ) + ( 1 - \pi ) \mathcal { N } ( \mathbf { w } _ { i } | 0 , \sigma _ { 2 } ^ { 2 } ) \big ) } & { \qquad v _ { l } : = \mathrm { L o g - p o s t e r i o r } } \\ & { l _ { 3 } = \sum _ { i = 1 } ^ { N } \log ( p ( \mathcal { D } | \mathbf { w } _ { i } ) ) } & { \qquad v _ { l } : = \mathrm { L o g - h i k e l i h o o d ~ o f ~ d a t a } } \\ & { \mathcal { L } _ { B B B } = \frac { 1 } { M } ( l _ { 1 } - l _ { 2 } - l _ { 3 } ) } & \\ & { \mu  \mu - \alpha _ { \mu } \nabla \mathcal { L } _ { B B B \mu } } \\ & { \boldsymbol { \rho }  \boldsymbol { \rho } - \alpha _ { \rho } \nabla \mathcal { L } _ { B B B \rho } } & \end{array}$   
+6:   
+10:   
+11:   
+12:   
+13:   
+14: until loss plateaus   
+15: αµ, $\alpha _ { \rho } \gets$ LearningRateUpdate $( \alpha _ { \mu } , \alpha _ { \rho } , \sigma , \mu )$ $\vartriangleright$ See Algorithm 2 for UCB and 3 for UCB-P   
+16: end for
+
+Algorithm 1 Uncertainty-guided Continual Learning with Bayesian Neural Networks UCB   
+
+<table><tr><td>Algorithm2LearningRateUpdate in UCB</td><td>Algorithm 3LearningRateUpdate in UCB-P</td></tr><tr><td>1:function LearningRateUpdate(αμ, α,)</td><td>1:function LearningRateUpdate(αμ, @p,0, μ)</td></tr><tr><td>2: for each parameter do</td><td>2: for each parameter j in each layer l do</td></tr><tr><td>3: Ωμ←1/σ</td><td>3: Ω←lμ//g Signal to noise ratio</td></tr><tr><td>4: ←1</td><td>4: if 2[j] ∈ top p% of SΩs in l then</td></tr><tr><td>5: aμ←aμ/Ωμ</td><td>5: αμ=αp=0</td></tr><tr><td>6: ap←αp/p</td><td>6: end if</td></tr><tr><td>7: end for</td><td>7: end for</td></tr><tr><td>8:end function</td><td>8:end function</td></tr></table>
+
+SNR is a commonly used measure in signal processing to distinguish between “useful” information from unwanted noise contained in a signal. In the context of neural models, the SNR can be thought as an indicative of parameter importance; the higher the SNR, the more effective or important the parameter is to the model predictions for a given task.
+
+UCB-P, as shown in Algorithms 1 and 3, is performed as follows: for every layer, convolutional or fully-connected, the parameters are ordered by their SNR value and those with the lowest importance are pruned (set to zero). The pruned parameters are marked using a binary mask so that they can be used later in learning new tasks whereas the important parameters remain fixed throughout training on future tasks. Once a task is learned, an associated binary mask is saved which will be used during inference to recover key parameters and hence the exact performance to the desired task.
+
+The overhead memory per parameter in encoding the mask as well as saving it on the disk is as follows. Assuming we have $n$ tasks to learn using a single network, the total number of required bits to encode an accumulated mask for a parameter is at max $\log _ { 2 } n$ bits assuming a parameter deemed to be important from task 1 and kept being encoded in the mask.
+
+# 5 RESULTS
+
+# 5.1 EXPERIMENTAL SETUP
+
+Datasets: We evaluate our approach in two common scenarios for continual learning: 1) classincremental learning of a single or two randomly alternating datasets, where each task covers only a subset of the classes in a dataset, and 2) continual learning of multiple datasets, where each task is a dataset. We use Split MNIST with 5 tasks (5-Split MNIST) similar to (Nguyen et al., 2018; Chen et al., 2019; Tseran et al., 2018) and permuted MNIST (Srivastava et al., 2013) for class incremental learning with similar experimental settings as used in (Serra et al., 2018; Tseran et al., 2018). Furthermore, to have a better understanding of our method, we evaluate our approach on continually learning a sequence of 8 datasets with different distributions using the identical sequence as in (Serra et al., 2018), which includes FaceScrub ( $\mathrm { N g }$ & Winkler, 2014), MNIST, CIFAR100, NotMNIST (Bulatov, 2011), SVHN (Netzer et al., 2011), CIFAR10, TrafficSigns (Stallkamp et al., 2011), and FashionMNIST (Xiao et al., 2017). Details of each are summarized in Table 4 in appendix. No data augmentation of any kind has been used in our analysis.
+
+Baselines: Within the Bayesian framework, we compare to three models which do not incorporate the importance of parameters, namely fine-tuning, feature extraction, and joint training. In fine-tuning (BBB-FT), training continues upon arrival of new tasks without any forgetting avoidance strategy. Feature extraction, denoted as (BBB-FE), refers to freezing all layers in the network after training the first task and training only the last layer for the remaining tasks. In joint training (BBB-JT) we learn all the tasks jointly in a multitask learning fashion which serves as the upper bound for average accuracy on all tasks, as it does not adhere to the continual learning scenario. We also perform the counterparts for FT, FE, and JT using ordinary neural networks and denote them as ORD-FT, ORDFE, and ORD-JT. From the prior work, we compare with state-of-the-art approaches including Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017), Incremental Moment Matching (IMM) (Lee et al., 2017), Learning Without Forgetting (LWF) (Li & Hoiem, 2016), Less-Forgetting Learning (LFL) (Jung et al., 2016), PathNet (Fernando et al., 2017), Progressive neural networks (PNNs) (Rusu et al., 2016), and Hard Attention Mask (HAT) (Serra et al., 2018) using implementations provided by (Serra et al., 2018). On Permuted MNIST results for SI (Zenke et al., 2017) are reported from (Serra et al., 2018). On Split and Permuted MNIST, results for VCL (Nguyen et al., 2018) are obtained using their original provided code whereas for VCL-GNG (Chen et al., 2019) and VCL-Vadam (Tseran et al., 2018) results are reported from the original work without re-implementation. Because our method lies into the regularization-based regime, we only compare against baselines which do not benefit from episodic or coreset memory.
+
+Hyperparameter tuning: Unlike commonly used tuning techniques which use a validation set composed of all classes in the dataset, we only rely on the first two task and their validations set, similar to the setup in (Chaudhry et al., 2019). In all our experiments we consider a 0.15 split for the validation set on the first two tasks. After tuning, training starts from the beginning of the sequence. Our scheme is different from (Chaudhry et al., 2019), where the models are trained on the first (e.g. three) tasks for validation and then training is restarted for the remaining ones and the reported performance is only on the remaining tasks.
+
+Training details: It is important to note that in all our experiments, no pre-trained model is used. We used stochastic gradient descent with a batch size of 64 and a learning rate of 0.01, decaying it by a factor of 0.3 once the loss plateaued. Dataset splits and batch shuffle are identically in all UCB experiments and all baselines.
+
+Pruning procedure and mask size: Once a task is learned, we compute the performance drop for a set of arbitrary pruning percentages from the maximum training accuracy achieved when no pruning is applied. The pruning portion is then chosen using a threshold beyond which the performance drop is not accepted. Mask size is chosen without having the knowledge of how many tasks to learn in the future. Upon learning each task we used a uniform distribution of pruning ratios $( 5 0 - 1 0 0 \% )$ ) and picked the ratio resulted in at most $1 \%$ , $2 \%$ , and $3 \%$ forgetting for MNIST, CIFAR, and 8tasks experiments, respectively. We did not tune this parameter because in our hyperparameter tuning, we only assume we have validation sets of the first two tasks.
+
+Parameter regularization and importance measurement: Table 1 ablates different ways to compute the importance $\Omega$ of an parameter in Eq. 4 and 5. As shown in Table 1 the configuration that yields the highest accuracy and the least forgetting (maximum BWT) occurs when the learning rate regularization is performed only on $\mu$ of the posteriors using $\Omega _ { \mu } = 1 / \sigma$ as the importance and $\Omega _ { \rho } = 1$ .
+
+Performance measurement: Let $n$ be the total number of tasks. Once all are learned, we evaluate our model on all $n$ tasks. ACC is the average test classification accuracy across all tasks. To measure forgetting we report backward transfer, BWT, which indicates how much learning new tasks has influenced the performance on previous tasks. While BWT $< 0$ directly reports catastrophic forgetting, $\mathrm { B W T } > 0$ indicates that learning new tasks has helped with the preceding tasks. Formally, BWT and ACC are as follows:
+
+$$
+\mathrm { B W T } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } R _ { i , n } - R _ { i , i } , \quad \mathrm { A C C } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } R _ { i , n }
+$$
+
+Table 1: Variants of learning rate regularization and importance measurement on 2-Split MNIST   
+Table 2: Continually learning on different datasets. BWT and ACC in $\%$ . $( ^ { * } )$ denotes that methods do not adhere to the continual learning setup: BBB-JT and ORD-JT serve as the upper bound for ACC for BBB/ORD networks, respectively. $^ \ddag$ denotes results reported by (Serra et al., 2018). $^ \dagger$ denotes the result reported from original work. BWT was not reported in $^ \ddag$ and $\dagger$ . All others results are (re)produced by us and are averaged over 3 runs with standard deviations given in Section A.3 of the appendix.   
+
+<table><tr><td>Method</td><td>μ</td><td>p</td><td>Importance Ω</td><td>BWT (%)</td><td>ACC (%)</td></tr><tr><td>UCB</td><td>X</td><td>1</td><td>1/g</td><td>0.00</td><td>99.2</td></tr><tr><td>UCB</td><td>1</td><td>X</td><td>1/g</td><td>-0.04</td><td>98.7</td></tr><tr><td>UCB</td><td>X</td><td>X</td><td>1/g</td><td>-0.02</td><td>98.0</td></tr><tr><td>UCB</td><td>X</td><td>-</td><td>|μ//σ</td><td>-0.03</td><td>98.4</td></tr><tr><td>UCB</td><td>-</td><td>X</td><td>/g</td><td>-0.52</td><td>98.7</td></tr><tr><td>UCB</td><td>X</td><td>X</td><td>lμ//σ</td><td>-0.32</td><td>98.8</td></tr><tr><td>UCB-P</td><td>X</td><td>X</td><td>μ/σ</td><td>-0.01</td><td>99.0</td></tr><tr><td>UCB-P</td><td>X</td><td>X</td><td>1/g</td><td>-0.01</td><td>98.9</td></tr></table>
+
+(a) 5-Split MNIST, 5 tasks.   
+
+<table><tr><td>Method</td><td>BWT</td><td>ACC</td></tr><tr><td>VCL-Vadam†</td><td></td><td>99.17</td></tr><tr><td>VCL-GNG†</td><td></td><td>96.50</td></tr><tr><td>VCL</td><td>-0.56</td><td>98.20</td></tr><tr><td>IMM</td><td>-11.20</td><td>88.54</td></tr><tr><td>EWC</td><td>-4.20</td><td>95.78</td></tr><tr><td>HAT</td><td>0.00</td><td>99.59</td></tr><tr><td>ORD-FT</td><td>-9.18</td><td>90.60</td></tr><tr><td>ORD-FE</td><td>0.00</td><td>98.54</td></tr><tr><td>BBB-FT</td><td>-6.45</td><td>93.42</td></tr><tr><td>BBB-FE</td><td>0.00</td><td>98.76</td></tr><tr><td>UCB-P (Ours)</td><td>-0.72</td><td>99.32</td></tr><tr><td>UCB (Ours)</td><td></td><td>0.00 99.63</td></tr><tr><td>ORD-JT*</td><td>0.00</td><td>99.78</td></tr><tr><td>BBB-JT*</td><td>0.00</td><td>99.87</td></tr></table>
+
+(b) Permuted MNIST, 10 permutations. (c) Alternating CIFAR10/100   
+
+<table><tr><td>Method</td><td>#Params</td><td>BWTACC</td></tr><tr><td>SI</td><td>0.1M</td><td>- 86.0</td></tr><tr><td>EWC #</td><td>0.1M</td><td>- 88.2</td></tr><tr><td>HAT t</td><td>0.1M</td><td>- 91.6</td></tr><tr><td>VCL-Vadam†</td><td>0.1M</td><td>-86.34</td></tr><tr><td>VCL-GNG†</td><td>0.1M</td><td>-90.50</td></tr><tr><td>VCL</td><td>0.1M</td><td>-7.90 88.80</td></tr><tr><td>UCB (Ours)</td><td>0.1M</td><td>-0.38 91.44</td></tr><tr><td>LWF</td><td>1.9M</td><td>-31.17 65.65</td></tr><tr><td>IMM</td><td>1.9M</td><td>-7.14 90.51</td></tr><tr><td>HAT</td><td>1.9M</td><td>0.03 97.34</td></tr><tr><td>BBB-FT</td><td>1.9M</td><td>-0.58 90.01</td></tr><tr><td>BBB-FE</td><td>1.9M</td><td>0.02 93.54</td></tr><tr><td>UCB-P(Ours)1.9M</td><td></td><td>-0.95 97.24</td></tr><tr><td>UCB (Ours)</td><td>1.9M</td><td>0.03 97.42</td></tr><tr><td></td><td></td><td></td></tr><tr><td>BBB-JT*</td><td>1.9M</td><td>0.00 98.12</td></tr></table>
+
+(d) Sequence of 8 tasks   
+
+<table><tr><td>Method</td><td>BWT</td><td>ACC</td></tr><tr><td>LFL</td><td>-10.0</td><td>8.61</td></tr><tr><td>PathNet</td><td>0.00</td><td>20.22</td></tr><tr><td>LWF</td><td>-54.3</td><td>28.22</td></tr><tr><td>IMM</td><td>-38.5</td><td>43.93</td></tr><tr><td>EWC</td><td>-18.04</td><td>50.68</td></tr><tr><td>PNN</td><td>0.00</td><td>76.78</td></tr><tr><td>HAT</td><td>-0.14</td><td>81.59</td></tr><tr><td>BBB-FT</td><td>-23.1</td><td>43.09</td></tr><tr><td>BBB-FE</td><td>-0.01</td><td>58.07</td></tr><tr><td>UCB-P (Ours)</td><td>-2.54</td><td>80.38</td></tr><tr><td>UCB (Ours) BBB-JT*</td><td>-0.84 84.04 -1.2</td><td>84.1</td></tr></table>
+
+<table><tr><td>Method</td><td>BWT</td><td>ACC</td></tr><tr><td>PathNet</td><td>0.00</td><td>28.94</td></tr><tr><td>LWF</td><td>-37.9</td><td>42.93</td></tr><tr><td>LFL</td><td>-24.22</td><td>47.67</td></tr><tr><td>IMM</td><td>-12.23</td><td>69.37</td></tr><tr><td>PNN</td><td>0.00</td><td>70.73</td></tr><tr><td>EWC</td><td>-1.53</td><td>72.46</td></tr><tr><td>HAT</td><td>-0.04</td><td>78.32</td></tr><tr><td>BBB-FE</td><td>-0.04</td><td>51.04</td></tr><tr><td>BBB-FT</td><td>-7.43</td><td>68.89</td></tr><tr><td>UCB-P (Ours)</td><td>-1.89</td><td>77.32</td></tr><tr><td>UCB (Ours)</td><td></td><td>-0.72 79.44</td></tr><tr><td>BBB-JT*</td><td>1.52</td><td>83.93</td></tr></table>
+
+where $R _ { i , n }$ is the test classification accuracy on task $i$ after sequentially finishing learning the $n ^ { \mathrm { t h } }$ task. Note that in UCB-P, $R _ { i , i }$ refers the test accuracy on task $i$ before pruning and $R _ { i , n }$ after pruning which is equivalent to the end of sequence performance. In Section 6, we show that our UCB model can be used when tasks labels are not available at inference time by training it with a “single head” architecture with a sum of number of classes for all tasks. We refer to the ACC measured for this scenario as “Generalized Accuracy”.
+
+# 5.2 5-SPLIT MNIST
+
+We first present our results for class incremental learning of MNIST (5-Split MNIST) in which we learn the digits $0 - 9$ in five tasks with 2 classes at a time in 5 pairs of $0 / 1 , 2 / 3 , 4 / 5 , 6 / 7$ , and $8 / 9$ . Table 2a shows the results for reference baselines in Bayesian and non-Bayesian neural networks including fine-tuning (BBB-FT, ORD-FT), feature extraction (BBB-FE, ORD-FE) and, joint training (BBB-JT, ORD-JT) averaged over 3 runs and standard deviations are given in Table 9 in the appendix. Although the MNIST dataset is an “easy” dataset, we observe throughout all experiments that Bayesian fine-tuning and joint training perform significantly better than their counterparts, ORD-FT and ORD-JT. For Bayesian methods, we compare against VCL and its variations named as VCL with Variational Adam (VCL-Vadam), VCL with Adam and Gaussian natural gradients (VCL-GNG). For non-Bayesian methods, we compare against HAT, IMM, and EWC (EWC can be regarded as Bayesian-inspired). VCL-Vadam $\mathrm { \Delta A C C = } 9 9 . 1 7 \%$ ) appears to be outperforming VCL $\mathrm { \Delta A C C { = } 9 8 . 2 0 \% }$ ) and VCL-GNG ( $\mathrm { A C C } { = } 9 6 . 5 0 \%$ ) in average accuracy. However, full comparison is not possible because forgetting was not reported for Vadam and GNG. Nevertheless, UCB $\mathrm { \ A C C { = } 9 9 . 6 3 \% }$ ) is able to surpass all the baselines including VCL-Vadam in average accuracy while in zero forgetting it is on par with HAT $( \mathrm { A C C { = } 9 9 . 5 9 \% }$ ). We also report results on incrementally learning MNIST in two tasks (2-Split MNIST) in Table 8 in the appendix, where we compare it against PackNet, HAT, and LWF where PackNet, HAT, UCB-P, and UCB have zero forgetting while UCB has marginally higher accuracy than all others.
+
+# 5.3 PERMUTED MNIST
+
+Permuted MNIST is a popular variant of the MNIST dataset to evaluate continual learning approaches in which each task is considered as a random permutation of the original MNIST pixels. Following the literature, we learn a sequence of 10 random permutations and report average accuracy at the end. Table 2b shows ACC and BWT of UCB and UCB-P in comparison to state-of-the-art models using a small and a large network with 0.1M and $1 . 9 \mathrm { M }$ parameters, respectively (architecture details are given in Section A.2 of the appendix). The accuracy achieved by UCB $( \mathrm { A C C = 9 1 . 4 4 \pm 0 . 0 4 \% } )$ using the small network outperforms the ACC reported by Serra et al. (2018) for SI $\mathrm { \Delta A C C { = } 8 6 . 0 \% }$ ), EWC $( \mathrm { A C C { = } 8 8 . 2 \% }$ ), while HAT attains a slightly better performance $( \mathrm { { A C C = 9 1 . 6 \% } }$ ). Comparing the average accuracy reported in VCL-Vadam $\mathrm { A C C } { = } 8 6 . 3 4 \%$ ) and VCL-GNG $( \mathrm { A C C } { = } 9 0 . 5 0 \%$ ) as well as obtained results for VCL $\mathrm { \Delta A C C { = } 8 8 . 8 0 \% }$ ) shows UCB with $\mathbf { B W T } { = } ( 0 . 0 3 \% \pm 0 . 0 0 \% )$ is able to outperform other Bayesian approaches in accuracy while forgetting significantly less compared to VCL with $\mathrm { B W T = - 7 . 9 \% }$ . While we do not experiment with memory in this work, not surprisingly adding memory to most approaches will improve their performance significantly as it allows looking into past tasks. E.g. Chen et al. (2019) report $\mathrm { A C C { = } 9 4 . 3 7 \% }$ for VCL-GNC when adding a memory of size 200.
+
+Next, we compare the results for the larger network (1.9M). While HAT and UCB have zero forgetting, UCB, reaching $\mathrm { A C C = 9 7 . 4 2 \pm 0 . 0 1 \% }$ , performs better than all baselines including HAT which obtains $\mathrm { A C C = 9 7 . 3 4 \pm 0 . 0 5 \% }$ using 1.9M parameters. We also observe again that BBB-FT, despite being not specifically penalized to prevent forgetting, exhibits reasonable negative BWT values, performing better than IMM and LWF baselines. It is close to joint training, BBB-JT, with $\mathrm { A C C { = } 9 8 . \bar { 1 } \% }$ , which can be seen as an upper bound.
+
+# 5.4 ALTERNATING CIFAR10 AND CIFAR100
+
+In this experiment, we randomly alternate between class incremental learning of CIFAR10 and CIFAR100. Both datasets are divided into 5 tasks each with 2 and 20 classes per task, respectively. Table 2c presents ACC and BWT obtained with UCB-P, UCB, and three BBB reference methods compared against various continual learning baselines. Among the baselines presented in Table 2c, PNN and PathNet are the only zero-forgetting-guaranteed approaches. It is interesting to note that in this setup, some baselines (PathNet, LWF, and LFL) do not perform better than the naive accuracy achieved by feature extraction. PathNet suffers from bad pre-assignment of the network’s capacity per task which causes poor performance on the initial task from which it never recovers. IMM performs almost similar to fine-tuning in ACC, yet forgets more. PNN, EWC, and HAT are the only baselines that perform better than BBB-FE and BBB-FT. EWC and HAT are both allowed to forget by construction, however, HAT shows zero forgetting behavior. While EWC is outperformed by both of our UCB variants, HAT exhibits $1 \%$ better ACC over UCB-P. Despite having a slightly higher forgetting, the overall accuracy of UCB is higher, reaching $7 9 . 4 \%$ . BBB-JT in this experiment achieves a positive BWT which shows that learning the entire sequence improves the performance on earlier tasks.
+
+# 5.5 MULTIPLE DATASETS LEARNING
+
+Finally, we present our results for continual learning of 8 tasks using UCB-P and UCB in Table 2d. Similar to the previous experiments we look at both ACC and BWT obtained for UCB-P, UCB, BBB references (FT, FE, JT) as well as various baselines. Considering the ACC achieved by BBB-FE or BBB-FT $( 5 8 . 1 \% )$ as a lower bound we observe again that some baselines are not able to do better than BBB-FT including LFL, PathNet, LWF, IMM, and EWC while PNN and HAT remain the only strong baselines for our UCB-P and UCB approaches. UCB-P again outperforms PNN by $3 . 6 \%$ in ACC. HAT exhibits only $- 0 . 1 \%$ BWT, but our UCB achieves $2 . 4 \%$ higher ACC.
+
+# 6 SINGLE HEAD AND GENERALIZED ACCURACY OF UCB
+
+UCB can be used even if the task information is not given at test time. For this purpose, at training time, instead of using a separate fully connected classification head for each task, we use a single head with the total number of outputs for all tasks. For example in the 8-dataset experiment we only use one head with 293 number of output classes, rather than using 8 separate heads, during training and inference time.
+
+Table 3: Single Head vs. Multi-Head architecture and Generalized vs. Standard Accuracy. Generalized accuracy means that task information is not available at test time. SM, PM, CF, and 8T denote the 5-Split MNIST, Permuted MNIST, Alternating CIFAR10/100, and sequence of 8 tasks, respectively.   
+
+<table><tr><td rowspan="3"></td><td rowspan="3" colspan="2">Generalized ACC Single Head</td><td colspan="4">ACC</td></tr><tr><td colspan="2">Single Head</td><td colspan="2">Multi Head</td></tr><tr><td>BBB-FT</td><td>UCB BBB-FT</td><td>UCB</td><td>BBB-FT</td></tr><tr><td>Exp SM</td><td>UCB 98.7</td><td>98.1</td><td>98.9</td><td>98.7</td><td>99.2</td><td>98.4</td></tr><tr><td>PM</td><td>92.5</td><td>86.1</td><td>95.1</td><td>88.3</td><td>97.7</td><td>90.0</td></tr><tr><td>CF</td><td>71.2</td><td>65.2</td><td>74.3</td><td>67.8</td><td>79.4</td><td>68.9</td></tr><tr><td>8T</td><td>76.8</td><td>47.6</td><td>79.9</td><td>53.2</td><td>84.0</td><td>43.1</td></tr></table>
+
+Table 3 presents our results for UCB and BBB-FT trained with a single head against having a multi-head architecture, in columns 4-7. Interestingly, we see only a small performance degrade for UCB from training with multi-head to a single head. The ACC reduction is $0 . 3 \%$ , $2 . 6 \%$ , ${ \bar { 5 } } . 1 \%$ , and $4 . 1 \%$ for 2-Split MNIST, Permuted MNIST, Alternating CIFAR10/100, and sequence of 8 tasks experiments, respectively.
+
+We evaluated UCB and BBB-FT with a more challenging metric where the prediction space covers the classes across all the tasks. Hence, confusion of similar class labels across tasks can be measured. Performance for this condition is reported as Generalized ACC in Table 3 in columns 2-3. We observe a small performance reduction in going from ACC to Generalized ACC, suggesting non-significant confusion caused by the presence of more number of classes at test time. The performance degradation from ACC to Generalized ACC is $0 . 2 \%$ , $2 . 6 \%$ , $3 . 1 \%$ , and $3 . 1 \%$ for 2-Split MNIST, Permuted MNIST, Alternating CIFAR10/100, and sequence of 8 tasks, respectively. This shows that UCB can perform competitively in more realistic conditions such as unavailability of task information at test time. We believe the main insight of our approach is that instead of computing additional measurements of importance, which are often task, input or output dependent, we directly use predicted weight uncertainty to find important parameters. We can freeze them using a binary mask, as in UCB-P, or regularize changes conditioned on current uncertainty, as in UCB.
+
+# 7 CONCLUSION
+
+In this work, we propose a continual learning formulation with Bayesian neural networks, called UCB, that uses uncertainty predictions to perform continual learning: important parameters can be either fully preserved through a saved binary mask (UCB-P) or allowed to change conditioned on their uncertainty for learning new tasks (UCB). We demonstrated how the probabilistic uncertainty distributions per weight are helpful to continually learning short and long sequences of benchmark datasets compared against baselines and prior work. We show that UCB performs superior or on par with state-of-the-art models such as HAT (Serra et al., 2018) across all the experiments. Choosing between the two UCB variants depends on the application scenario: While UCB-P enforces no forgetting after the initial pruning stage by saving a small binary mask per task, UCB does not require additional memory and allows for more learning flexibility in the network by allowing small forgetting to occur. UCB can also be used in a single head setting where the right subset of classes belonging to the task is not known during inference leading to a competitive model that can be deployed where it is not possible to distinguish tasks in a continuous stream of the data at test time. UCB can also be deployed in a single head scenario and where tasks information is not available at test time.
+
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+
+# A APPENDIX
+
+# A.1 DATASETS
+
+Table 4 shows a summary of the datasets utilized in our work along with their size and number of classes. In all the experiments we resized images to $3 2 \times 3 2 \times 3$ if necessary. For datasets with monochromatic images, we replicate the image across all RGB channels.
+
+Table 4: Utilized datasets summary   
+
+<table><tr><td>Names</td><td>#Classes</td><td>Train</td><td>Test</td></tr><tr><td>FaceScrub (Ng &amp; Winkler,2014)</td><td>100</td><td>20.600</td><td>2,289</td></tr><tr><td>MNIST (LeCun et al.,1998)</td><td>10</td><td>60,000</td><td>10,000</td></tr><tr><td>CIFAR100 (Krizhevsky &amp; Hinton,2009)</td><td>100</td><td>50.000</td><td>10,000</td></tr><tr><td>NotMNIST (Bulatov,2011)</td><td>10</td><td>16,853</td><td>1,873</td></tr><tr><td>SVHN (Netzer et al., 2011)</td><td>10</td><td>73,257</td><td>26.032</td></tr><tr><td>CIFAR10 (Krizhevsky &amp; Hinton,2009)</td><td>10</td><td>39,209</td><td>12.630</td></tr><tr><td>TrafficSigns (Stallkamp et al., 2011)</td><td>43</td><td>39,209</td><td>12,630</td></tr><tr><td>FashionMNIST (Xiao et al.,2017)</td><td>10</td><td>60,000</td><td>10,000</td></tr></table>
+
+# A.2 IMPLEMENTATION DETAILS
+
+In this section we take a closer look at elements of our UCB model on MNIST and evaluate variants of parameter regularization, importance measurement, as well as the effect of the number of samples drawn from the posited posterior.
+
+Bayes-by-backprop (BBB) Hyperparamters: Table 5 shows the search space for hyperparamters in the BBB algorithm Blundell et al. (2015) which we used for tuning on the validation set of the first two tasks.
+
+Table 5: Search space for hyperparamters in BBB given by Blundell et al. (2015)   
+
+<table><tr><td rowspan=1 colspan=1>BBB hyperparamters</td><td rowspan=1 colspan=1>-logσ1</td><td rowspan=1 colspan=1>-logσ2</td><td rowspan=1 colspan=1>T</td></tr><tr><td rowspan=1 colspan=1>Search space</td><td rowspan=1 colspan=1>{0,1,2}</td><td rowspan=1 colspan=1>{6,7,8}</td><td rowspan=1 colspan=1>{0.25,0.5,0.75}</td></tr></table>
+
+Network architecture: For Split MNIST and Permuted MNIST experiments, we have used a twolayer perceptron which has 1200 units. Because there is more number of parameters in our Bayesian neural network compared to its equivalent regular neural net, we ensured fair comparison by matching the total number of parameters between the two to be $1 . 9 \mathrm { M }$ unless otherwise is stated. For the multiple datasets learning scenario, as well as alternating incremental CIFAR10/100 datasets, we have used a ResNet18 Bayesian neural network with 7.1-11.3M parameters depending on the experiment. However, the majority of the baselines provided in this work are originally developed using some variants of AlexNet structure and altering that, e.g. to ResNet18, resulted in degrading in their reported and experimented performance as shown in Table 6. Therefore, we kept the architecture for baselines as AlexNet and ours as ResNet18 and only matched their number of parameters to ensure having equal capacity across different approaches.
+
+Table 6: Continually learning on CIFAR10/100 using AlexNet and ResNet18 for UCB (our method) and HAT (Serra et al., 2018). BWT and ACC in $\%$ . All results are (re)produced by us.   
+
+<table><tr><td>Method</td><td>BWT</td><td>ACC</td></tr><tr><td>HAT (AlexNet) HAT (ResNet18)</td><td>0.0</td><td>78.3</td></tr><tr><td>UCB (AlexNet)</td><td>-9.0 -0.7</td><td>56.8 79.44</td></tr><tr><td>UCB (ResNet18)</td><td>-0.7</td><td>79.70</td></tr></table>
+
+Number of Monte Carlo samples: UCB is ensured to be robust to random noise using multiple samples drawn from posteriors. Here we explore different number of samples and the effect on final performance for ACC and BWT. We have used $\Omega _ { \mu } = 1 / \sigma$ as importance and regularization has been performed on mean values only. Following the result in Table 7 we chose the number of samples to be 10 for all experiments.
+
+Table 7: Number of Monte Carlo samples $( \mathrm { N } )$ in 2-Split MNIST   
+
+<table><tr><td>Method</td><td>N</td><td>BWT (%)</td><td>ACC (%)</td></tr><tr><td>UCB</td><td>1</td><td>0.00</td><td>98.0</td></tr><tr><td>UCB</td><td>2</td><td>0.00</td><td>98.3</td></tr><tr><td>UCB</td><td>5</td><td>-0.15</td><td>99.0</td></tr><tr><td>UCB</td><td>10</td><td>0.00</td><td>99.2</td></tr><tr><td>UCB</td><td>15</td><td>-0.01</td><td>98.3</td></tr></table>
+
+# A.3 ADDITIONAL RESULTS
+
+Here we include some additional results such as Table 8 for 2-split MNIST and some complementary results for tables in the main text as follows: 9, 10, and 11 include standard deviation for results shown in Table 2a, 2b, 2c, respectively.
+
+Table 8: Continually learning on 2-Split MNIST. BWT and ACC in $\%$ . $( ^ { * } )$ denotes that methods do not adhere to the continual learning setup: BBB-JT and ORD-JT serve as the upper bound for ACC for BBB/ORD networks, respectively. All results are (re)produced by us.   
+
+<table><tr><td>Method</td><td>BWT</td><td>ACC</td></tr><tr><td>PackNet (Mallya&amp;Lazebnik,2018)</td><td>0.04±0.01</td><td>98.91±0.03</td></tr><tr><td>LWF (Li&amp; Hoiem,2016)</td><td>-0.22 ± 0.04</td><td>99.12 ±0.03</td></tr><tr><td>HAT (Serra et al., 2018)</td><td>0.01 ±0.00</td><td>99.02 ±0.00</td></tr><tr><td>ORD-FT</td><td>-6.81± 0.03</td><td>92.42 ±0.02</td></tr><tr><td>ORD-FE</td><td>0.04±0.04</td><td>97.90 ±0.04</td></tr><tr><td>BBB-FT</td><td>-0.61±0.03</td><td>98.44±0.03</td></tr><tr><td>BBB-FE</td><td>0.02 ±0.05</td><td>98.03±0.05</td></tr><tr><td>UCB-P (Ours)</td><td>0.03 ±0.04</td><td>99.02 ±0.01</td></tr><tr><td>UCB (Ours)</td><td>0.01±0.00</td><td>99.18 ±0.01</td></tr><tr><td>ORD-JT*</td><td>0.02±0.03</td><td>99.13±0.03</td></tr><tr><td>BBB-JT*</td><td>0.03 ±0.02</td><td>99.51 ±0.02</td></tr></table>
+
+Table 9: Continually learning on 5-Split MNIST. BWT and ACC in $\%$ . $( ^ { \ast } )$ denotes that methods do not adhere to the continual learning setup: BBB-JT and ORD-JT serve as the upper bound for ACC for BBB/ORD networks, respectively. All results are (re)produced by us.   
+
+<table><tr><td>Method</td><td>BWT</td><td>ACC</td></tr><tr><td>VCL-Vadam (Tseran et al., 2018)</td><td></td><td>99.17 ± 0.05</td></tr><tr><td>VCL-GNG (Chen et al., 2019)</td><td>=</td><td>96.50± 0.07</td></tr><tr><td>VCL (Nguyen et al., 2018)</td><td>-0.56 ±0.03</td><td>98.20±0.03</td></tr><tr><td>IMM (Lee et al., 2017)</td><td>-11.20 ± 1.57</td><td>88.54 ± 1.56</td></tr><tr><td>EWC (Kirkpatrick et al., 2017)</td><td>-4.20 ±1.08</td><td>95.78 ±1.08</td></tr><tr><td>HAT (Serra et al., 2018)</td><td>0.00±0.02</td><td>99.59 ±0.02</td></tr><tr><td>ORD-FT*</td><td>-9.18 ± 1.12</td><td>90.60 ±1.12</td></tr><tr><td>ORD-FE*</td><td>0.00 ±1.56</td><td>98.54 ± 1.57</td></tr><tr><td>BBB-FT*</td><td>-6.45 ± 1.99</td><td>93.42 ±1.98</td></tr><tr><td>BBB-FE*</td><td>0.00 ± 2.23</td><td>98.76 ± 2.23</td></tr><tr><td>UCB-P (Ours)</td><td>-0.72 ± 0.04</td><td>99.32 ±0.04</td></tr><tr><td>UCB (Ours)</td><td>0.00 ±0.04</td><td>99.63 ± 0.03</td></tr><tr><td>ORD-JT*</td><td>0.00±0.02</td><td>99.78±0.02</td></tr><tr><td>BBB-JT*</td><td>0.00 ± 0.01</td><td>99.87 ± 0.01</td></tr></table>
+
+Table 10: Continually learning on Permuted MNIST. BWT and ACC in $\%$ . $( ^ { * } )$ denotes that method does not adhere to the continual learning setup: BBB-JT serves as the upper bound for ACC for BBB network. $^ \ddag$ denotes results reported by (Serra et al., 2018). $\dagger$ denotes the result reported from original work. BWT was not reported in $^ \ddag$ and $\dagger$ . All others results are (re)produced by us.   
+
+<table><tr><td>Method</td><td>#Params</td><td>BWT</td><td>ACC</td></tr><tr><td>SI (Zenke et al., 2017)‡</td><td>0.1M</td><td>=</td><td>86.0</td></tr><tr><td>EWC (Kirkpatrick et al., 2017)‡</td><td>0.1M</td><td>=</td><td>88.2</td></tr><tr><td>HAT (Serra et al., 2018)‡</td><td>0.1M</td><td></td><td>91.6</td></tr><tr><td>VCL-Vadamt</td><td>0.1M</td><td></td><td>93.34</td></tr><tr><td>VCL-GNGt</td><td>0.1M</td><td></td><td>94.62</td></tr><tr><td>VCL</td><td>0.1M</td><td>-7.90 ±0.23</td><td>88.80±0.23</td></tr><tr><td>UCB (Ours)</td><td>0.1M</td><td>-0.38 ± 0.02</td><td>91.44±0.04</td></tr><tr><td>LWF (Li&amp; Hoiem,2016)</td><td>1.9M</td><td>-31.17 ± 0.05</td><td>65.65 ± 0.05</td></tr><tr><td>IMM (Lee et al., 2017)</td><td>1.9M</td><td>-7.14±0.07</td><td>90.51 ± 0.08</td></tr><tr><td>HAT (Serra et al., 2018)</td><td>1.9M</td><td>0.03 ±0.05</td><td>97.34 ± 0.05</td></tr><tr><td>BBB-FT</td><td>1.9M</td><td>-0.58 ± 0.05</td><td>90.01 ±0.05</td></tr><tr><td>BBB-FE</td><td>1.9M</td><td>0.02 ±0.03</td><td>93.54± 0.04</td></tr><tr><td>UCB-P (Ours)</td><td>1.9M</td><td>-0.95 ± 0.06</td><td>97.24± 0.06</td></tr><tr><td>UCB (Ours)</td><td>1.9M</td><td>0.03 ±0.00</td><td>97.42 ± 0.01</td></tr><tr><td>BBB-JT*</td><td>1.9M</td><td>0.00±0.00</td><td>98.12±0.01</td></tr></table>
+
+Table 11: Continually learning on CIFAR10/100. BWT and ACC in $\%$ . $( ^ { * } )$ denotes that method does not adhere to the continual learning setup: BBB-JT serves as the upper bound for ACC for BBB network. All results are (re)produced by us.   
+
+<table><tr><td>Method</td><td>BWT</td><td>ACC</td></tr><tr><td>PathNet (Fernando et al., 2017)</td><td>0.00 ±0.00</td><td>28.94±0.03</td></tr><tr><td>LWF (Li&amp; Hoiem,2016)</td><td>-37.9 ± 0.32</td><td>42.93 ± 0.30</td></tr><tr><td>LFL (Jung et al., 2016)</td><td>-24.22 ± 0.21</td><td>47.67 ± 0.22</td></tr><tr><td>IMM (Lee et al., 2017)</td><td>-12.23 ±0.06</td><td>69.37 ±0.06</td></tr><tr><td>PNN (Rusu et al., 2016)</td><td>0.00±0.00</td><td>70.73 ±0.08</td></tr><tr><td>EWC (Kirkpatrick et al., 2017)</td><td>-1.53 ± 0.07</td><td>72.46 ± 0.06</td></tr><tr><td>HAT (Serra et al., 2018)</td><td>0.04±0.06</td><td>78.32 ±0.06</td></tr><tr><td>BBB-FE</td><td>0.04±0.02</td><td>51.04 ± 0.03</td></tr><tr><td>BBB-FT</td><td>-7.43 ± 0.07</td><td>68.89 ± 0.07</td></tr><tr><td>UCB-P (Ours)</td><td>-1.89±0.03</td><td>77.32 ± 0.03</td></tr><tr><td>UCB (Ours)</td><td>-0.72 ±0.02</td><td>79.44± 0.02</td></tr><tr><td>BBB-JT*</td><td>1.52 ± 0.04</td><td>83.93±0.04</td></tr></table>

parse/train/HyEl3o05Fm/HyEl3o05Fm.md

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+# STOCHASTIC ADVERSARIAL VIDEO PREDICTION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Being able to predict what may happen in the future requires an in-depth understanding of the physical and causal rules that govern the world. A model that is able to do so has a number of appealing applications, from robotic planning to representation learning. However, learning to predict raw future observations, such as frames in a video, is exceedingly challenging—the ambiguous nature of the problem can cause a naively designed model to average together possible futures into a single, blurry prediction. Recently, this has been addressed by two distinct approaches: (a) latent variational variable models that explicitly model underlying stochasticity and (b) adversarially-trained models that aim to produce naturalistic images. However, a standard latent variable model can struggle to produce realistic results, and a standard adversarially-trained model underutilizes latent variables and fails to produce diverse predictions. We show that these distinct methods are in fact complementary. Combining the two produces predictions that look more realistic to human raters and better cover the range of possible futures. Our method outperforms prior works in these aspects.
+
+# 1 INTRODUCTION
+
+When we interact with objects in our environment, we can easily imagine the consequences of our actions: push a ball and it will roll; drop a vase and it will break. The ability to imagine future outcomes provides an appealing avenue for learning about the world. Unlabeled video sequences can be gathered autonomously with minimal human intervention, and a machine that learns to predict future events will gain an in-depth and functional understanding of its environment. This leads naturally to the problem of video prediction—given a sequence of context frames, and optionally a proposed action sequence, generate the pixels of the future frames. Once trained, such a model could be used to determine which actions can bring about desired outcomes (Finn et al., 2016; Ebert et al., 2017). Unfortunately, accurate and naturalistic video prediction remains an open problem.
+
+One major challenge in video prediction is the ambiguous nature of the problem. While frames in the immediate future can be extrapolated with high precision, the space of possibilities diverges beyond a few frames, and the problem becomes multimodal by nature. Methods that use deterministic models and loss functions unequipped to handle this inherent uncertainty, such as mean-squared error (MSE), will average together possible futures, producing blurry predictions. Prior works have explored stochastic models for video prediction (Babaeizadeh et al., 2018; Denton & Fergus, 2018), using the framework of variational autoencoders (VAEs) (Kingma & Welling, 2014). These models predict possible futures by sampling latent variables. During training, they optimize a variational lower bound on the likelihood of the data in a latent variable model. However, the posterior is still a pixel-wise MSE loss, corresponding to the log-likelihood under a fully factorized Gaussian distribution. This makes training tractable, but causes them to still make blurry and unrealistic predictions when the latent variables alone do not adequately capture the uncertainty.
+
+Another relevant branch of recent work has been generative adversarial networks (GANs) (Goodfellow et al., 2014) for image generation. Here, a generator network is trained to produce images that are indistinguishable from real images, under the guidance of a learned discriminator network trained to classify images as real or generated. The discriminator operates on patches or entire images, and is thus capable of modeling the joint distribution of pixels. Although this overcomes the limitations of pixel-wise losses, GANs are notoriously susceptible to mode collapse, where latent random variables are often ignored by the model, especially in the conditional setting. This makes them difficult to apply to generation of diverse and plausible futures, conditioned on context frames.
+
+![](images/bf524931faf800d7e2fd4dfc6d66c43592606f2192036461c7837785ec6f3929.jpg)  
+Figure 1: Example results. While the SV2P method (Babaeizadeh et al., 2018) produces blurry images, our method maintains sharpness and realism through time. The prior SVG-LP method (Denton & Fergus, 2018) produces sharper predictions, but still blurs out objects in the background (left) or objects that interact with the robot such as the baseball (right).
+
+To address these challenges, we propose a model that combines both adversarial losses and latent variables to enable realistic stochastic video prediction. Our model consists of a video prediction network that can sample multiple plausible futures by sampling time-varying stochastic latent variables and decoding them into multiple frames. At training time, an inference network estimates the distribution of these latent variables, and video discriminator networks classify generated videos from real. The full training objective is the variational lower bound used in VAEs combined with the adversarial loss used in GANs. This enables us to capture stochastic posterior distributions of videos while also modeling the spatiotemporal joint distribution of pixels. VAEs with no adversarial losses are also capable of modeling joint distributions, provided that the generator model doesn’t assume any factorization of the pixels. This is the case of pixel-autoregressive models, such as Pixel Video Networks (Kalchbrenner et al., 2017), though training and inference with these models are impractically slow. In this work, we take a different approach and we instead focus on the losses.
+
+The primary contribution of our work is an stochastic video prediction model based on VAE-GANs. To our knowledge, this is the first stochastic video prediction model that combines an adversarial loss with a latent variable model trained via the variational lower bound. Our experiments show that the VAE component greatly improves the diversity of the generated images, while the adversarial loss attains prediction results that are substantially more realistic than state-of-the-art methods, as shown in Fig. 1. We further present a comparison of various types of prediction models and losses, including VAE, GAN, and VAE-GAN models, and analyze the impact of these choices on prediction realism, diversity, and accuracy.
+
+# 2 RELATED WORK
+
+Recent developments in expressive generative models based on deep networks has led to impressive developments in video generation and prediction. Earlier approaches to prediction focused on models that generate pixels directly from the latent state of the model using both feed-forward (Ranzato et al., 2014; Mathieu et al., 2016) and recurrent (Oh et al., 2015; Xingjian et al., 2015) architectures. An alternative to generating pixels is transforming them by applying a constrained geometric distortion to a previous frame (Finn et al., 2016; De Brabandere et al., 2016; Xue et al., 2016; Byravan & Fox, 2016; Vondrick & Torralba, 2017; van Amersfoort et al., 2017; Liu et al., 2017; Chen et al., 2017; Lu et al., 2017; Walker et al., 2015; 2016; Liang et al., 2017). Aside from the design of the generator architecture, performance is strongly affected by the training objective. Simply minimizing MSE loss can lead to strong results for deterministic synthetic videos (Oh et al., 2015; Chiappa et al., 2017). However, on real-world videos that contain uncertainty, this loss can result in blurry predictions, as the model averages futures to avoid incurring a large MSE loss (Mathieu et al., 2016).
+
+Incorporating uncertainty is critical for addressing this issue. One approach is to model the full joint distribution using pixel-autoregressive models (van den Oord et al., 2016; Kalchbrenner et al., 2017; Reed et al., 2017), though training and inference are impractically slow. Another approach is to train a latent variable model, such as in variational autoencoders (VAEs) (Kingma & Welling, 2014). Conditional VAEs have been used for prediction of optical flow trajectories (Walker et al.,
+
+2016), single-frame prediction (Xue et al., 2016), and recently for stochastic multi-frame video prediction (Babaeizadeh et al., 2018; Denton & Fergus, 2018). While these models can model distributions over possible futures, the prediction distribution is still fully factorized over pixels, which still tends to produce blurry predictions.
+
+Adversarial losses (Goodfellow et al., 2014) for image generation can produce substantially improved realism. However, these networks tend to be difficult to train and are susceptible to mode collapse. A number of prior works have used adversarial losses for deterministic video prediction (Mathieu et al., 2016; Vondrick & Torralba, 2017; Villegas et al., 2017; Lu et al., 2017; Zhou & Berg, 2016; Bhattacharjee & Das, 2017). Several prior works have also sought to produce unconditioned video generations (Vondrick et al., 2016; Saito et al., 2017; Tulyakov et al., 2018) and conditional generation with input noise (Chen et al., 2017; Tulyakov et al., 2018; Wang et al., 2018). We show that a GAN model with input noise can indeed generate realistic videos, but fails to adequately cover the space of possible futures. In contrast, our method, which combines latent variable models with an adversarial loss, produces videos that are both visually plausible and diverse.
+
+Prior works have combined VAEs and GANs to produce stochastic and realistic predictions. Walker et al. (2017) predicts videos of humans by decomposing the problem into a VAE that predicts stochastic future poses and a GAN that generates videos conditioned on those poses and an image. VAE-GANs, which jointly optimize the VAE and GAN losses, have shown promising results for unconditional and conditional image generation (Larsen et al., 2016; Bao et al., 2017; Zhu et al., 2017), but have not been applied to video prediction. The video prediction setting presents two important challenges. First, conditional image generation can handle large appearance changes between the input and output, but suffer when attempting to produce large spatial changes. The video prediction setting is precisely the opposite—the appearance remains largely the same from frame to frame, but the most important changes are spatial. Secondly, video prediction involves sequential prediction, where it’s increasingly difficult to predict farther into the future. Our approach is the first to use VAE-GANs in a recurrent setting for stochastic video prediction.
+
+# 3 VIDEO PREDICTION WITH STOCHASTIC ADVERSARIAL MODELS
+
+Our goal is to learn a stochastic video prediction model that can predict videos that are diverse and perceptually realistic, and where all predictions are plausible futures for the given initial image. In practice, we use a short initial sequence of images (typically two frames), though we will omit this in our derivation for ease of notation. Our model consists of a recurrent generator network $G$ , which is a deterministic video prediction model that maps an initial image $\mathbf { x } _ { \mathrm { 0 } }$ and a sequence of latent random codes ${ \bf z } _ { 0 : T - 1 }$ , to the predicted sequence of future images $\hat { \mathbf { x } } _ { 1 : T }$ . Intuitively, the latent codes encapsulate any ambiguous or stochastic events that might affect the future. At test time, we sample videos by first sampling the latent codes from a prior distribution $p ( \mathbf { z } _ { t } )$ , and then passing them to the generator. We use a fixed unit Gaussian prior, $\mathcal { N } ( 0 , 1 )$ . The training procedure for this includes elements of variational inference and generative adversarial networks. Before describing the training procedure, we formulate the problem in the context of VAEs and GANs.
+
+# 3.1 VARIATIONAL AUTOENCODERS
+
+Our recurrent generator predicts each frame given the previous frame and a random latent code. The previous frame passed to the generator is denoted as $\tilde { \mathbf { x } } _ { t - 1 }$ to indicate that it could be a ground truth frame $\mathbf { x } _ { t - 1 }$ (for the initial frames) or the last prediction $\hat { \mathbf { x } } _ { t - 1 }$ . The generator specifies a distribution $p ( \mathbf { x } _ { t } | \mathbf { x } _ { t - 1 } , \mathbf { z } _ { t - 1 } )$ , parametrized as a fixed-variance Laplacian distribution with mean $\hat { \mathbf { x } } _ { t } = G ( \mathbf { x } _ { t - 1 } , \mathbf { z } _ { t - 1 } )$ . The likelihood of the data $p ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } )$ cannot be directly maximized, since it involves marginalizing over the latent variables, which is intractable in general. Thus, we instead maximize the variational lower bound of the log-likelihood. We approximate the posterior with a recognition model $q \big ( \mathbf { z } _ { t } \big | \mathbf { x } _ { t : t + 1 } \big )$ , which is parametrized as a conditionally Gaussian distribution $\sqrt { ( \mu _ { \mathbf { z } _ { t } } ) } , \sigma _ { \mathbf { z } _ { t } } ^ { 2 } )$ , represented by a network $E ( \mathbf { x } _ { t : t + 1 } )$ . The encoder $E$ is conditioned on adjacent frames $\mathbf { x } _ { t }$ and $\mathbf { x } _ { t + 1 }$ in order to have temporally local latent variables $\mathbf { z } _ { t }$ that capture the ambiguity for only that transition, a sensible choice when using independent and identically distributed Gaussian priors. Another choice is to use temporally correlated latent variables, which would require a stronger prior (e.g. as in Denton & Fergus (2018)). For simplicity, we opted for the former.
+
+![](images/20ea8a442331722f73f03624a9d9ec5efdd27b5ded042fd6787239574618bd4c.jpg)  
+Figure 2: Our proposed video prediction model. (a) During testing, we synthesize new frames by sampling random latent codes $\mathbf { z }$ from a prior distribution $p ( \mathbf { z } )$ independently at each time step. The generator $G$ takes a previous frame and a latent code to synthesize a new frame. (b) During training, the generator is optimized to predict videos that match the distribution of real videos, using learned discriminators. The discriminators operate on entire sequences. We sample latent codes from two distributions: (1) the prior distribution, and (2) a posterior distribution approximated by a learned encoder $E$ . For the latter, the regression $\mathcal { L } _ { 1 }$ loss is used. Separate discriminators $D$ and $D ^ { \mathrm { V A E } }$ are used depending on the distribution used to sample the latent code.
+
+During training, the latent code is sampled from $q \big ( \mathbf { z } _ { t } | \mathbf { x } _ { t : t + 1 } \big )$ . The generation of each frame can be thought of as the reconstruction of frame $\hat { \mathbf { x } } _ { t + 1 }$ , where the ground truth frame $\mathbf { x } _ { t + 1 }$ (along with $\mathbf { x } _ { t }$ ) is encoded into a latent code $\mathbf { z } _ { t }$ , and then it (along with the last frame) is mapped back to $\hat { \mathbf { x } } _ { t + 1 }$ . Since the latent code has ground truth information about the frame being reconstructed, the model is encouraged to use it during training. This is a conditional version of VAEs, where the encoder and decoder are conditioned on the previous frame $\mathbf { \check { x } } _ { t }$ or $\hat { \mathbf { x } } _ { t } .$ ). To allow back-propagation through the encoder, the reconstruction term is rewritten using the re-parametrization trick,
+
+$$
+\mathcal { L } _ { 1 } ( G , E ) = \mathbb { E } _ { \mathbf { x } _ { 0 : T } , \mathbf { z } _ { t } \sim E \left( \mathbf { x } _ { t : t + 1 } \right) | _ { t = 0 } ^ { T - 1 } } \left[ \sum _ { t = 1 } ^ { T } | | \mathbf { x } _ { t } - G ( \mathbf { x } _ { t - 1 } , \mathbf { z } _ { t - 1 } ) | | _ { 1 } \right] .
+$$
+
+To enable sampling from the prior at test time, a regularization term encourages the approximate posterior to be close to the prior distribution,
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { { K L } } } ( E ) = \mathbb { E } _ { { \mathbf { x } _ { 0 : T } } } \left[ \sum _ { t = 1 } ^ { T } \mathcal { D } _ { \mathrm { { K L } } } ( E ( { \mathbf { x } _ { t - 1 : t } } ) | | p ( \mathbf { z } _ { t - 1 } ) ) \right] . } \end{array}
+$$
+
+The VAE optimization involves minimizing the following objective, where the relative weighting of $\lambda _ { 1 }$ and $\lambda _ { \mathrm { K L } }$ is determined by the (fixed) variance of conditional likelihood $p ( \mathbf { x } _ { t } | \mathbf { x } _ { t - 1 } , \mathbf { z } _ { t - 1 } )$ ,
+
+$$
+G ^ { * } , E ^ { * } = \arg \operatorname* { m i n } _ { G , E } \lambda _ { 1 } \mathcal { L } _ { 1 } ( G , E ) + \lambda _ { \mathrm { K L } } \mathcal { L } _ { \mathrm { K L } } ( E ) .
+$$
+
+# 3.2 GENERATIVE ADVERSARIAL NETWORKS
+
+Without overcoming the problem of modeling pixel covariances, it is likely not possible to produce sharp and clean predictions. Indeed, as shown in our experiments, the pure VAE model tends to produce blurry futures. We can force the predictions to stay on the video manifold by matching the distributions of predicted and real videos. Given a classifier $D$ that is capable of distinguishing generated videos $\hat { \mathbf { x } } _ { 1 : T }$ from real videos $\mathbf { x } _ { \mathrm { 1 : } T }$ , the generator can be trained to match the statistics of the real data distribution using the binary cross-entropy loss,
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { G a N } } ( G , D ) = \mathbb { E } _ { \mathbf { x } _ { 1 : T } } [ \log D ( \mathbf { x } _ { 1 : T } ) ] + \mathbb { E } _ { \mathbf { x } _ { 1 : T } , \mathbf { z } _ { t } \sim p ( \mathbf { z } _ { t } ) | _ { t = 0 } ^ { T - 1 } } [ \log ( 1 - D ( G ( \mathbf { x } _ { 0 } , \mathbf { z } _ { 0 : T - 1 } ) ) ) ] . } \end{array}
+$$
+
+The overloaded notation $G ( { \bf x } _ { 0 } , { \bf z } _ { 0 : T - 1 } )$ indicates the generated sequence $\hat { \mathbf { x } } _ { 1 : T }$ . The classifier, which is not known a priori and is problem-specific, can be realized as a deep discriminator network that can be adversarially learned,
+
+$$
+G ^ { * } = \arg \operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \mathcal { L } _ { \mathrm { G A N } } ( G , D ) .
+$$
+
+This is the setting of GANs. In the conditional case, a per-pixel reconstruction term $\mathcal { L } _ { 1 } ^ { \mathrm { G A N } }$ is added to the objective, which is analogous to $\mathcal { L } _ { 1 }$ , except that the latent codes are sampled from the prior.
+
+# 3.3 STOCHASTIC ADVERSARIAL VIDEO PREDICTION
+
+The VAE and GAN models provide complementary strengths. GANs use a learned loss function through the discriminator, which learns the statistics of natural videos. However, GANs can suffer from the problem of mode collapse, especially in the conditional setting (Pathak et al., 2016; Isola et al., 2017; Zhu et al., 2017). VAEs explicitly encourage the latent code to be more expressive and meaningful, since the learned encoder produces codes that are useful for making accurate predictions at training time. However, during training, VAEs only observe latent codes that are encodings of ground truth images, and never train on completely randomly drawn latent codes, leading to a potential train and test mismatch. GANs, however, are trained with randomly drawn codes.
+
+Our stochastic adversarial video prediction (SAVP) model combines both approaches, shown in Fig. 2. Another term $\mathcal { L } _ { \mathrm { G A N } } ^ { \mathrm { V A E } }$ is introduced, which is analogous to $\mathcal { L } _ { \mathrm { G A N } }$ except that it uses latent codes sampled from $q \big ( \mathbf { z } _ { t } | \mathbf { x } _ { t : t + 1 } \big )$ and a video discriminator . The objective of our SAVP model is
+
+$$
+G ^ { * } , E ^ { * } = \operatorname * { a r g m i n } _ { G , E } \operatorname* { m a x } _ { D , D ^ { \mathrm { v a g } } } \lambda _ { 1 } \mathcal { L } _ { 1 } ( G , E ) + \lambda _ { \mathrm { { K L } } } \mathcal { L } _ { \mathrm { { K L } } } ( E ) + \mathcal { L } _ { \mathrm { G a N } } ( G , D ) + \mathcal { L } _ { \mathrm { G A N } } ^ { \mathrm { v a g } } ( G , E , D ^ { \mathrm { v a E } } ) .
+$$
+
+# 3.4 NETWORK ARCHITECTURES
+
+The generator is a convolutional LSTM (Xingjian et al., 2015) that predicts pixel-space transformations between the current and next frame, with additional skip connections with the first frame as done in SNA (Ebert et al., 2017). At every time step, the network is conditioned on the current frame and latent code. After the initial frames, the network is conditioned on its own predictions. The conditioning on the latent codes is realized by concatenating them along the channel dimension to the inputs of all the convolutional layers of the convolutional LSTM. We note that the warping component of this generator assumes that the frames in the videos can be described as transformations of pixels, which is the case for the datasets that we consider. And although the generator used in this work is based on SNA, any video generator (including the one from Denton & Fergus (2018)) could be used with our losses. The encoder is a feed-forward convolutional network that, at every time step, encodes a pair of images $\mathbf { x } _ { t }$ and $\mathbf { x } _ { t + 1 }$ into $\mu _ { \mathbf { z } _ { t } }$ and $\log \sigma _ { \mathbf { z } _ { t } }$ .
+
+The video discriminator is a feed-forward convolutional network with 3D filters, based on SNGAN (Miyato et al., 2018) but with the filters “inflated” from 2D to 3D. The network takes in a spatiotemporal cube of all the predicted pixels and outputs a single logit. The ground-truth context frames are not provided to the network. We found that spectral normalization in the discriminator and conditioning only on the predicted frames were important for a stable training. We also found that image discriminators that operate on single frames were not necessary. See Fig. 8 and Appendix A for additional details.
+
+# 3.5 DISCUSSION OF RELATED VAE MODELS
+
+Aside from the adversarial losses, the VAE component of our model is related to prior work on stochastic video prediction. Although the variational losses are the same, there are differences on encoding the posterior distribution and sampling the latent variables at training and test time.
+
+The inference network of Babaeizadeh et al. (2018) estimates a single distribution $q ( \mathbf { z } | \mathbf { x } _ { 1 : T } )$ by using a feed-forward network that encodes the entire video sequence at once. At test time, the latent variable is sampled from a unit Gaussian prior. They propose two variants for sampling. The latent is sampled once for the entire sequence in the time-invariant case or at every time step in the time-variant case.
+
+On the other hand, the inference network of Denton & Fergus (2018) estimates a time-varying distribution $q \big ( \mathbf { z } _ { t } | \mathbf { x } _ { 1 : t + 1 } \big )$ by using a recurrent network that encodes all the frames up to the next frame. They propose two versions for the prior. The prior is either a fixed unit Gaussian distribution or a time-varying distribution $p ( \mathbf { z } _ { t } | \mathbf { x } _ { 1 : t } )$ , which is learned and estimated by a recurrent network. In both cases, the latent variable is sampled at every time step.
+
+In contrast, our inference network estimates a single distribution $q \big ( \mathbf { z } _ { t } | \mathbf { x } _ { t : t + 1 } \big )$ by using a feedforward network that encodes the current and next frames. Unlike both prior works, the posterior distribution is temporally local and is conditioned on only two adjacent frames. The latent variables are sampled at every time step and, like the time-variant SV2P and fixed-prior SVG, the prior is a fixed unit Gaussian distribution for every time step.
+
+# 4 EXPERIMENTS
+
+Our experimental evaluation studies the realism, diversity, and accuracy of the videos generated by our approach and prior methods, and evaluates the importance of various design decisions, including the form of the reconstruction loss and the presence of the variational and adversarial objectives. Evaluating the performance of stochastic video prediction models is exceedingly challenging: not only should the samples from the model be physically realistic and visually plausible given the context frames, but the model should also be able to produce diverse samples that match the conditional distribution in the data. This is difficult to evaluate precisely: realism is not accurately reflected with simple metrics of reconstruction accuracy, and the true conditional distribution in the data is unknown, since real-world datasets only have a single future for each initial sequence. Below, we discuss the metrics that we use to evaluate realism, diversity, and accuracy. No single metric alone provides a clear answer as to which model is better, but considering multiple metrics can provide us with a more complete understanding of the performance and trade-offs of each approach.
+
+# 4.1 EVALUATION METRICS
+
+Realism: comparisons to real videos using human judges. The realism of the predicted videos is evaluated based on a real vs. fake two-alternative forced choice (2AFC) test. Human judges on Amazon Mechanical Turk (AMT) are presented with a pair of videos—one generated and one real—and asked to identify the generated, or “fake” video. We use the implementation from (Zhang et al., 2016), modified for videos. Each video is 10 frames long and shown over 2.5 seconds. For each method, we gather 1000 judgments from 25 human judges. Each human evaluator is provided with 10 training trials followed by 40 test trials. A method that produces perfectly realistic videos would achieve a fooling rate of $5 \dot { 0 } \%$ .
+
+Diversity: distance between samples. Realism is not the only factor in determining the performance of a video prediction model: aside from generating predictions that look physically plausible and realistic, a successful model must also adequately cover the range of possible futures in an uncertain environment. We compute diversity as the average distance between randomly sampled video predictions, similar to Zhu et al. (2017). Distance is measured in the VGG feature space (pretrained on ImageNet classification), averaged across five layers, which has been shown to correlate well with human perception (Dosovitskiy & Brox, 2016; Johnson et al., 2016; Zhang et al., 2018).
+
+Accuracy: similarity of the best sample. One weakness of the above metric is that the samples may be diverse but still not cover the feasible output space. Though we do not have the true output distribution, we can still leverage the single ground truth instance. This can be done by sampling the model a finite number of times, and evaluating the similarity between the best sample and the ground truth. This has been explored in prior work on stochastic video prediction (Babaeizadeh et al., 2018; Denton & Fergus, 2018), using PSNR or SSIM as the evaluation metric. In addition to these, we use cosine similarity in the pretrained VGG feature space.
+
+# 4.2 DATASETS
+
+We evaluate on two real-world datasets: the BAIR action-free robot pushing dataset (Ebert et al., 2017) and the KTH human actions dataset (Schuldt et al., 2004). See Appendix B.2 for additional results on the action-conditioned version of the robot pushing dataset.
+
+BAIR action-free. This dataset consists of a randomly moving robotic arm that pushes objects on a table. This dataset is particularly challenging since (a) it contains large amounts of stochasticity due to random arm motion, and (b) it is a real-world application, with a diverse set of objects and large cluttered scene (rather than a single frame-centered object with a neutral background). The frame resolution is $6 4 \times 6 4$ . We condition on 2 frames and train to predict the next 10 frames. We predict 10 future frames for the 2AFC experiments and 28 future frames for the other experiments.
+
+KTH. This dataset consists of a human subject doing one of six activities: walking, jogging, running, boxing, hand waving, and hand clapping. For the first three activities, the human enters and leaves the frame multiple times, leaving the frame empty with a mostly static background for multiple frames at a time. The sequences are particularly stochastic when the initial frames are all empty since the human can enter the frame at any point in the future. As a preprocessing step, we center-crop each frame to a $1 2 0 \times 1 2 0$ square and then resize to a spatial resolution of $6 4 \times 6 4$ . We condition on the first 10 frames and train to predict the next 10 frames. We predict 10 future frames for the 2AFC experiments and 30 future frames for the other experiments. For each sequence, subclips of the desired length are randomly sampled at training and test time.
+
+![](images/722111e2f6b14ee02e57d554b8d72aa9dbc686b46155961b48cd10606613a4cb.jpg)  
+Figure 3: Qualitative Results (BAIR action-free dataset). Unless labeled otherwise, we show the closest generated sample to ground truth using VGG cosine similarity. SV2P immediately produces blurry results. Our GAN and VAE-based variants, as well as SVG-LP produce sharper results. However, SVG-LP still blurs out the jar on the right side of the image when it is touched by the robot, while our GAN-based models keep the jar sharp. We show three results for our SAVP model: using the closest, furthest, and random samples. There is large variation between the three samples in the arm motion, and even the furthest sample from the ground truth looks realistic. (bottom) We show a failure case where the arm disappears.
+
+# 4.3 METHODS: ABLATIONS AND COMPARISONS
+
+We compare the following variants of our method, in our effort to evaluate the effect of each loss term. Videos, code, and models are available at our website1.
+
+Ours, SAVP. Our stochastic adversarial video prediction model, with the VAE and GAN objectives.
+
+Ours, GAN-only. An ablation of our model with only a conditional GAN, without the variational autoencoder. This model still takes a noise sample as input, but the noise is sampled from the prior during training. This model is broadly representative of prior stochastic GAN-based methods.
+
+Ours, VAE-only. An ablation of our model with only a conditional VAE, with the reconstruction $\mathcal { L } _ { 1 }$ loss but without the adversarial loss. This model is broadly representative of prior stochastic VAE-based methods.
+
+Ours, deterministic. A deterministic ablation of our model with the reconstruction $\mathcal { L } _ { 1 }$ loss but without the VAE nor the GAN objectives. The model uses the same generator architecture but without the latent variables.
+
+![](images/aac14dd29f2c0325f5cb11aeadddf63dc2697e0ee140a4398b8ff417f9ff8a5a.jpg)  
+Figure 4: Qualitative Results (KTH dataset). We show qualitative comparisons to ablations of our model. Models are conditioned on 10 frames and trained to predict 10 future frames (vertical dashed line). For stochastic models, we show the closest generated sample to ground truth using VGG cosine similarity. We hypothesize that this dataset has much less stochasticity; even our deterministic model produces reasonable predictions. (top) Both the deterministic and VAE models generate images that are slightly blurry, but that do not degrade over time. The GAN-based methods produce sharper predictions. (middle) Our VAE model generates images where small limbs disappear further into the future, whereas our SAVP method preserves them. (bottom) All conditioning frames are empty except for a shadow on the left. All our variants are able to use this cue to predict that a person is coming from the left, although our SAVP model generates the most realistic sequence.
+
+We also compare to prior stochastic VAE-based methods Stochastic Variational Video Prediction (SV2P) (Babaeizadeh et al., 2018) and Stochastic Video Generation (SVG) (Denton & Fergus, 2018), both of which use the reconstruction $\mathcal { L } _ { 2 }$ loss and no adversarial loss.
+
+# 4.4 EXPERIMENTAL RESULTS
+
+We show qualitative results on the BAIR and KTH datasets in Fig. 3 and Fig. 4, respectively. For the quantitative results, we evaluate the realism, diversity, and accuracy of the predicted videos.
+
+Does our method produce realistic results? In Fig. 5, variants of our method are compared to prior work. On the BAIR action-free dataset, our GAN variant achieves the highest fooling rate, whereas our proposed SAVP model, a VAE-GAN-based method, achieves a fooling rate that is roughly halfway between the GAN and VAE models alone. The SV2P method (Babaeizadeh et al., 2018) does not achieve realistic results. The VAE-based SVG-LP method (Denton & Fergus, 2018) achieves high realism, similar to our VAE variant, but substantially below our GAN-based variants. On the KTH dataset, our SAVP model achieves the highest realism score, substantially above our GAN variant. Among the VAE-based methods without adversarial losses, our VAE-only model outperforms SV2P and SVG-FP (Denton & Fergus, 2018) in terms of realism.
+
+Does our method generate diverse results? We measure diversity by taking the distance between random samples. Diversity results are also shown in Fig. 5. For a qualitative visualization of diversity, see Appendix B.4. While the GAN-only approach achieves realistic results, it shows lower diversity than the VAE-based methods. This is an example of the commonly known phenomenon of mode-collapse, where multiple latent codes produce the same or similar images on the output (Goodfellow, 2016). Intuitively, the VAE-based methods explicitly encourage the latent code to be more expressive by using an encoder from the output space into the latent space during training. This is verified in our experiments, as the VAE-based variants, including our SAVP model, achieve higher diversity than our GAN-only models on both datasets. On the KTH dataset, our VAE variant and VAE-based SVG-FP method (Denton & Fergus, 2018) both achieve significantly higher diversity than all the other methods. Although the VAE-based SV2P methods (Babaeizadeh et al., 2018) mode-collapse on the KTH dataset, we note that they did not evaluate on this dataset, and as such, their method could benefit from different hyperparameters that are better suited for this dataset.
+
+![](images/7ba923b43a6fb87dddf9124260096ba043a47a530fe496348b12c92fdea85e51.jpg)  
+Figure 5: Realism vs Diversity. We measure realism using a real vs fake Amazon Mechanical Turk (AMT) test, and diversity using average VGG cosine distance. Higher is better on both metrics. Our VAE variant achieves higher realism and diversity than the SV2P (Babaeizadeh et al., 2018) and SVG (Denton & Fergus, 2018) methods based on VAEs. Our GAN variant achieves higher realism than the pure VAE methods, at the expense of significantly lower diversity. Our SAVP model, based on VAE-GANs, improves along the realism axis compared to a pure VAE method, and improves along the diversity axis compared to a pure GAN method. Although the SV2P methods mode-collapse on the KTH dataset, we note that they did not evaluate on this dataset, and their method could benefit from hyperparameters that are better suited for this dataset.
+
+Does our method generate accurate results? Following recent work on VAE-based video prediction (Babaeizadeh et al., 2018; Denton & Fergus, 2018), we evaluate on full-reference metrics by sampling multiple predictions from the model. We draw 100 samples for each video, find the “best” sample by computing similarity to the ground truth video, and show the average similarity across the test set as a function of time. The results on the BAIR and KTH datasets are shown in Fig. 14 and Fig. 15, respectively. We test generalization ability by running the model for more time steps than it was trained for. Even though the model is only trained to predict 10 future frames, we observe graceful degradation over time.
+
+While PSNR and SSIM (Wang et al., 2004) are commonly used for video prediction, these metrics are not necessarily indicative of prediction quality. In video prediction, structural ambiguities and geometric deformations are a dominant factor, and SSIM is not an appropriate metric in such situations (Sampat et al., 2009; Zhang et al., 2018). This is particularly noticeable with the SV2P method, which achieves high PSNR and SSIM scores, but produces blurry and unrealistic images. Furthermore, we additionally trained our VAE and deterministic variants using the standard MSE loss $\mathcal { L } _ { 2 }$ to understand the relationship between the form of the reconstruction loss and the metrics. The general trend is that models trained with $\mathcal { L } _ { 2 }$ , which favors blurry predictions, are better on PSNR and SSIM, but models trained with $\mathcal { L } _ { 1 }$ are better on VGG cosine similarity. See Appendix B.1 for quantitative results comparing models trained with $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ . In addition, we expect for our GAN-based variants to underperform on PSNR and SSIM since GANs prioritize matching joint distributions of pixels over per-pixel reconstruction accuracy.
+
+To partially overcome the limitations of these metrics, we also evaluate using distances in a deep feature space (Dosovitskiy & Brox, 2016; Johnson et al., 2016), which have been shown to correspond better with human perceptual judgments (Zhang et al., 2018). We use cosine similarity between VGG features averaged across five layers. Otherwise, a model trained for it would unfairly and artificially achieve better similarities by exploiting potential flaws on that metric. Our VAE variant, along with SVG (Denton & Fergus, 2018), performs best on this metric. Although our SAVP model improves on diversity and realism, it also performs worse in accuracy compared to pure VAE models (both our own ablation and SVG). This is to be expected, since accuracy and realism are at odds with each other. This tradeoff has recently been proved and it holds even for similarity distances based on VGG features (Blau & Michaeli; Blau et al., 2018). Among the VAE-based methods, SV2P (Babaeizadeh et al., 2018) achieves the lowest VGG similarity.
+
+![](images/94b60e66972d804ca57bac66a9ddd03db9c4795a3ecd886101306ee63a1a3610.jpg)  
+Figure 6: Similarity of the best sample (BAIR action-free dataset). We show the similarity (higher is better) between the best sample (of 100) as a function of prediction time step across different methods and evaluation metrics. (top) Although SV2P produces blurry and unrealistic images, it achieves the highest PSNR. Both SAVP and SVG-LP outperform SV2P on VGG similarity. We expect our GAN-based variants to underperform on PSNR and SSIM since GANs prioritize matching joint distributions of pixels over per-pixel reconstruction accuracy. (bottom) We compare to ablated versions of our model. Our VAE variant achieves higher scores than our SAVP model, which in turn achieves significantly higher VGG similarities compared to our GAN-only model. Note that the models were only trained to predict 10 future frames (indicated by the vertical line), but is being tested on generalization to longer sequences.
+
+On stochastic environments, such as in the BAIR action-free dataset, there is correlation between diversity and accuracy of the best sample: a model with diverse predictions is more likely to sample a video that is close to the ground truth. This relation can be seen in Fig. 5 and Fig. 14 for the robot dataset, e.g. our SAVP model is both more diverse and achieves higher similarity than our GAN-only variant. This is not true on less stochastic environments. We hypothesize that the KTH dataset is not as stochastic when conditioning on 10 frames, as evidenced the similarities between the predictions from the deterministic and stochastic models. This would explain why our GAN variant and SV2P achieve modest similarities despite achieving low diversity on the KTH dataset.
+
+Does combining the VAE and GAN produce better predictions? The GAN alone achieves high realism but low diversity. The VAE alone achieves lower realism but increased diversity. Adding the GAN to the VAE model increases the realism without sacrificing diversity, at only a small or no cost in realism on stochastic datasets. This is consistent with Zhu et al. (2017), which showed that combining GAN and VAE-based models provides benefits in the case of image generation. To our knowledge, our method is the first to extend this class of models to the video prediction setting, and the first to illustrate that this leads to improved realism with a degree of diversity comparable to the best VAE models in stochastic environments. The results show that this combination of losses is the best choice for realistic coverage of diverse stochastic futures.
+
+![](images/258c8108dc999f452806f8a47b01e93d0700b6378a94dbe37583176f878b4549.jpg)  
+Figure 7: Similarity of the best sample (KTH dataset). We evaluate the similarity between the best predicted sample (out of a 100 samples) and the ground truth video. (top) As in the case of the robot dataset, SV2P achieves high PSNR values, even though it produces blurry and unrealistic images. Although all three methods achieve comparable VGG similarities for the first 10 future frames (which is what the models were trained for, and indicated by the vertical line), our SAVP model predicts videos that are substantially more realistic, as shown in our subjective human evaluation, thus achieving a desirable balance between realism and accuracy. (bottom) We compare to ablated versions of our model. Our VAE-only method outperforms all our other variants on the three metrics. In addition, our deterministic model is not that far behind in terms of similarity, leading us to believe that the KTH dataset is not as stochastic when conditioning on the past 10 frames.
+
+# 5 CONCLUSION
+
+We develop a video prediction model that combines latent variables trained via a variational lower bound with an adversarial loss to produce a high degree of visual and physical realism. VAE-style training enables our method to make diverse stochastic predictions, and our experiments show that the adversarial loss is effective at producing predictions that are more visually realistic according to human raters. Evaluation of video prediction models is a major challenge, and we evaluate our method, as well as ablated variants that consist of only the VAE or only the GAN loss, in terms of a variety of quantitative and qualitative measures, including human ratings, diversity, and accuracy of the predicted samples. Our results demonstrate that our approach produces more realistic predictions than prior methods, while preserving the sample diversity of VAE-based methods.
+
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+
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+
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+
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+
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+
+Yochai Blau, Roey Mechrez, Radu Timofte, Tomer Michaeli, and Lihi Zelnik-Manor. 2018 pirm challenge on perceptual image super-resolution. arXiv preprint arXiv:1809.07517, 2018.
+
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+
+# A NETWORKS AND TRAINING DETAILS
+
+# A.1 NETWORK DETAILS
+
+# A.1.1 GENERATOR.
+
+Our generator network, shown in Fig. 8, is inspired by the convolutional dynamic neural advection (CDNA) model proposed by Finn et al. (2016). The video prediction setting is a sequential prediction problem, so we use a convolutional LSTM (Hochreiter & Schmidhuber, 1997; Xingjian et al., 2015) to predict future frames. We initialize the prediction on the initial sequence of ground truth frames (2 or 10 frames for the BAIR and KTH datasets, respectively), and predict 10 future frames. The model predicts a sequence of future frames by repeatedly making next-frame predictions and feeding those predictions back to itself. For each one-step prediction, the predicted frame is given by a compositing layer, which composes intermediate frames with predicted compositing masks. The intermediate frames include the previous frame, transformed versions of the previous frame, and a frame with pixels directly synthesized by the network. The transformed versions of the frame are produced by convolving in the input image with predicted convolutional kernels, allowing for different shifted versions of the input. In more recent work, the first frame of the sequence is also given as one of the intermediate frames (Ebert et al., 2017).
+
+To enable stochastic sampling, the generator is also conditioned on time-varying latent codes, which are sampled at training and test time. Each latent code $\mathbf { z } _ { t }$ is an 8-dimensional vector. At each prediction step, the latent code is passed through a fully-connected LSTM to facilitate correlations in time of the latent variables. The encoded latent code is then passed to all the convolutional layers of the main network, by concatenating it along the channel dimension to the inputs of these layers. Since they are vectors with no spatial dimensions, they are replicated spatially to match the spatial dimensions of the inputs.
+
+We made a variety of architectural improvements to the original CDNA (Finn et al., 2016) and SNA (Ebert et al., 2017) models, which overall produced better results on the per-pixel loss and similarity metrics. See Fig. 11 for a quantitative comparison of our deterministic variant (without
+
+![](images/c695f84530310c5e0b041dd245d60f70e2cf318bbdb4d54957e0604cbf590118.jpg)  
+Figure 8: Architecture of our generator network. Our network uses a convolutional LSTM (Hochreiter & Schmidhuber, 1997; Xingjian et al., 2015) with skip-connection between internal layers. As proposed by Finn et al. (2016), the network predicts (1) a set of convolution kernels to produce a set of transformed input images (2) synthesized pixels at the input resolution and (3) a compositing mask. Using the mask, the network can choose how to composite together the set of warped pixels, the first frame, previous frame, and synthesized pixels. One of the internal feature maps is given to a fully-connected layer to compute the kernels that specify pixel flow. The output of the main network is passed to two separate heads, each with two convolutional layers, to predict the synthesized frame and the composite mask. These two outputs use sigmoid and softmax non-linearities, respectively, to ensure proper normalization. We enable stochastic sampling of the model by conditioning the generator network on latent codes. These are first passed through a fully-connected LSTM, and then given to all the convolutional layers of the the convolutional LSTM.
+
+VAE or GAN losses) to the SNA model on the BAIR action-conditioned dataset. Each convolutional layer is followed by instance normalization (Ulyanov et al., 2016) and ReLU activations. We also use instance normalization on the LSTM pre-activations (i.e., the input, forget, and output gates, as well as the transformed and next cell of the LSTM). In addition, we modify the spatial downsampling and upsampling mechanisms. Standard subsampling and upsampling between convolutions is known to produce artifacts for dense image generation tasks (Odena et al., 2016; Zhao et al., 2017; Niklaus et al., 2017). In the encoding layers, we reduce the spatial resolution of the feature maps by average pooling, and in the decoding layers, we increase the resolution by using bilinear interpolation. All convolutions in the generator use a stride of 1. In the case of the action-conditioned dataset, actions are concatenated to the inputs of all the convolutional layers of the main network, as opposed to only the bottleneck.
+
+# A.1.2 ENCODER.
+
+The encoder is a standard convolutional network that, at every time step, encodes a pair of images $\mathbf { x } _ { t }$ and $\mathbf { x } _ { t + 1 }$ into $\mu _ { \mathbf { z } _ { t } }$ and $\log \sigma _ { \mathbf { z } _ { t } }$ . The latent variable $\mathbf { z } _ { t }$ is sampled at every time step and the same encoder network with shared weights is used at every step. The encoder architecture consists of three convolutional layers, followed by average pooling of all the spatial dimensions. Two separate fullyconnected layers are then used to estimate $\mu _ { \mathbf { z } _ { t } }$ and $\log \sigma _ { \mathbf { z } _ { t } }$ , respectively. The convolutional layers use instance normalization, leaky ReLU non-linearities, and stride 2. This encoder architecture is the same one used in BicyleGAN (Zhu et al., 2017) except that the inputs are pair of images, concatenated along the channel dimension.
+
+# A.1.3 DISCRIMINATOR.
+
+The discriminator is a 3D convolutional neural network that takes in all the images of the video at once. We use spectral normalization and the SNGAN discriminator architecture (Miyato et al., 2018), except that we “inflate” the convolution filters from 2D to 3D. The two video discriminators, $D$ and $D _ { \mathrm { v A E } }$ , share the same architecture, but not the weights, as done in BicycleGAN (Zhu et al., 2017).
+
+# A.2 TRAINING DETAILS
+
+Our generator network uses scheduled sampling during training as in Finn et al. (2016), such that at the beginning the model is trained for one-step predictions, while by the end of training the model is fully autoregressive. We trained all models with Adam (Kingma & Ba, 2015) for 300000 iterations, linearly decaying the learning rate to 0 for the last 100000 iterations. The same training schedule was used for all the models, except for SVG, which was trained by its author. Our GAN-based variants used an optimizer with $\beta _ { 1 } = 0 . 5$ , $\beta _ { 2 } = 0 . 9 9 9$ , learning rate of 0.0002, and a batch size of 16. Our deterministic and VAE models (including SNA and SV2P from prior work) used an optimizer with $\beta _ { 1 } = 0 . 9$ , $\beta _ { 2 } = 0 . 9 9 9$ , learning rate of 0.001, and a batch size of 32.
+
+We used $\lambda _ { 1 } = 1 0 0$ for our GAN-based variants, and $\lambda _ { 1 } = 1$ for all the other models. For our VAEbased variants, we linearly anneal the weight on the KL divergence term from 0 to the final value $\lambda _ { \mathrm { K L } }$ during training, as proposed by Bowman et al. (2016), from iterations 50000 to 100000. We used a relative weighting of $\lambda _ { \mathrm { K L } } / \lambda _ { 1 } = 0 . 0 0 1$ for the BAIR robot pushing datasets, and $\lambda _ { \mathrm { K L } } / \lambda _ { 1 } = 0 . 0 0 0 0 1$ for the KTH dataset. This hyperparameter was empirically chosen by computing similarity metrics on the validation set.
+
+# B ADDITIONAL EXPERIMENTS
+
+# B.1 COMPARISON OF PIXEL-WISE $\mathcal { L } _ { 1 }$ AND $\mathcal { L } _ { 2 }$ LOSSES
+
+We train our deterministic and VAE variants with the $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ losses to compare the effects of these reconstruction losses on the full-reference metrics used in this work. The pixel-wise $\mathcal { L } _ { 1 }$ loss assumes that pixels are generated according to a fully factorized Laplacian distribution, whereas the $\mathcal { L } _ { 2 }$ loss corresponds to a fully factorized Gaussian distribution. See Fig. 9 and Fig. 11 for quantitative results on the action-free and action-conditioned BAIR datasets, respectively.
+
+![](images/2779b0e1924ab3428fb17dbc07843bc42099346afe112c023d80b14743242b06.jpg)  
+Figure 9: Similarity of the best sample (BAIR action-conditioned dataset). We show the similarity between the predicted video and the ground truth, using the same evaluation as in Fig. 14. We compare our deterministic and VAE variants when trained with $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ losses, and observe that they have a significant impact on the quality of our predictions. The models trained with $\mathcal { L } _ { 1 }$ produce videos that are qualitatively better and achieve higher VGG similarity than the equivalent models trained with $\mathcal { L } _ { 2 }$ .
+
+![](images/2fa7428aabdf8d48de0f454a5a67e95dcf02e55fc6f04f09b3d8c5aee81ffc84.jpg)  
+Figure 10: Realism vs Diversity (BAIR action-conditioned dataset). The SV2P method (Babaeizadeh et al., 2018) from prior work produces images with low realism, whereas our GAN, VAE, and SAVP models fool the human judges at a rate of around $3 5 \mathrm { - } 4 0 \%$ . Our VAE-based models also produce videos with higher diversity, though lower diversity than other datasets, as this task involves much less stochasticity. The trend is the same as in the other datasets. Our SAVP model improves the realism of the predictions compared to our VAE-only model, and improves the diversity compared to our GAN-only model.
+
+The general trend is that models trained with the $\mathcal { L } _ { 2 }$ loss tend to generate blurry predictions but achieve higher PSNR scores than equivalent models trained with the $\mathcal { L } _ { 1 }$ loss. This is because PSNR and $\mathcal { L } _ { 2 }$ are closely related, the former being a logarithmic function of the latter. The opposite is true for the VGG cosine similarity metric, which has been shown to correspond better with human perceptual judgments (Zhang et al., 2018). Models trained with $\mathcal { L } _ { 1 }$ significantly outperforms equivalent models trained with $\mathcal { L } _ { 2 }$ . On the SSIM metric, models trained with $\mathcal { L } _ { 1 }$ achieve roughly the same or better similarities than models trained with $\mathcal { L } _ { 2 }$ . Although both losses are pixel-wise losses, the choice between $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ have a significant impact on the quality of our predicted videos.
+
+# B.2 RESULTS ON ACTION-CONDITIONED BAIR ROBOT PUSHING DATASET
+
+We use the same dataset as the one in the main paper, except that we use the robot actions. Each action is a 4-dimensional vector corresponding to Cartesian translations and a value indicating if the gripper has been closed or opened. As in the action-free dataset, we condition on the first 2 frames of the sequence and train to predict the next 10 frames. In this dataset, the video prediction model is now also conditioned on a sequence of actions ${ \bf a } _ { 0 : T - 1 }$ , in addition to the initial frames. The generator network is modified to take an action $\mathbf { a } _ { t }$ at each time step, by concatenating the action to the inputs of all the convolutional layers of the main network, similar to how the latent code $\mathbf { z } _ { t }$ is passed in (but without the additional fully-connected LSTM).
+
+![](images/b8c728d3fa845c27311e80de8513d50c454e74e4b589963d63085208644a3cfb.jpg)  
+Figure 11: Similarity of the best sample (BAIR action-conditioned dataset). We show the similarity between the predicted video and the ground truth, using the same evaluation as in Fig. 14, except that we condition on robot actions. (top) We compare to prior SV2P (Babaeizadeh et al., 2018) and ours ablations. Our VAE and deterministic models both outperform SV2P, even though it is VAE-based. However, notice that the gap in performance between our VAE and deterministic models is small, as the dataset is less stochastic when conditioning on actions. Our SAVP model achieves much lower scores on all three metrics. We hypothesize that our SAVP model, as well as SV2P, is underutilizing the provided actions and thus achieving more stochasticity at the expense of accuracy. (bottom) We compare deterministic models—SNA (Ebert et al., 2017) and ours— and our VAE model when trained with $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ losses. As in the action-free case, we observe that the choice of the pixel-wise reconstruction loss significantly affects prediction accuracy. Models trained with $\mathcal { L } _ { 1 }$ are substantially better in SSIM and VGG cosine similarity compared to equivalent models trained with $\mathcal { L } _ { 2 }$ . Surprisingly, the VAE model trained with $\mathcal { L } _ { 1 }$ outperforms the other models even on the PSNR metric. We hypothesize that VAE models trained with $\mathcal { L } _ { 1 }$ are better equipped to separate multiple modes of futures, whereas the ones trained with $\mathcal { L } _ { 2 }$ might still average some of the modes. In fact, we evidenced this in preliminary experiments on the toy shapes dataset used by Babaeizadeh et al. (2018). Among the deterministic models, ours improves upon SNA (Ebert et al., 2017), which is currently the best deterministic action-conditioned model on this dataset.
+
+We show the realism and diversity results in Fig. 10 and the accuracy results in Fig. 11. In addition to the methods compared in the action-free dataset, we also compare to SNA (Ebert et al., 2017), an action-conditioned deterministic video prediction model. The results indicate that our VAE model significantly outperforms prior methods on the full-reference metrics, and that our models significantly outperforms the model by Babaeizadeh et al. (2018) both in terms of diversity of predictions and realism.
+
+![](images/5c57ff346756a1b3af02827c95369a2178bf367eb1b95ccc7c085acdc583990d.jpg)  
+Figure 12: Example generations of MoCoGAN. We use the unconditional version of MoCoGAN (Tulyakov et al., 2018) to generate videos from the BAIR robot pushing dataset. We chose this model as a representative recent example of purely GAN-based unconditioned video generation. MoCoGAN produces impressive results on various applications related to human action, which are focused on a actor in the middle of the frame. However, this model struggles in the robot dataset where multiple entities are moving at a time. Note that since the patch-based discriminator has a limited receptive field of the image, the model can produce videos with two robot arms (last row) even though this is not in the dataset. We did not observe this behavior with their image-based discriminator.
+
+# B.3 EXAMPLE GENERATIONS OF AN UNCONDITIONAL GAN
+
+We consider the motion and content decomposed GAN (MoCoGAN) model (Tulyakov et al., 2018) as a representative unconditional GAN method from prior work, and use it to generate videos from the BAIR robot pushing dataset. We use their publicly available code and show qualitative results in Fig. 12. The results show the variant that uses patch-based discriminators, since this one achieved higher realism than the variant that uses image-based discriminators. Since this prior work demonstrates competitive results in comparison to other prior unconditional GAN methods, we chose it as the most representative recent example of purely GAN-based video generation for this comparison. MoCoGAN produces impressive results on various applications related to human action, which are focused on a single actor in the middle of the frame. However, it struggles on videos in the robot pushing domain where multiple entities are moving at a time, i.e. the robot arm and the objects it interacts with.
+
+# B.4 QUALITATIVE VISUALIZATION OF DIVERSITY
+
+We show a qualitative visualization of diversity in Fig. 13 by averaging multiple samples.
+
+![](images/d7f750510b3de7003f9cbd832aae82bdb2eefcf8ee6efc702a4aae10946532e0.jpg)  
+Figure 13: Qualitative visualization of diversity. We show predictions of our models, averaged over 100 samples. A model that produces diverse outputs should predict that the robot arm moves in random directions at each time step, and thus the arm should “disappear” over time in these averaged predictions. Consistent with our quantitative evaluation of diversity, we see that both our SAVP model and our VAE variant produces diverse samples, whereas the GAN-only method is prone to mode-collapse.
+
+![](images/40131816f41ce4a6b3eb3f7a4419598ff993197699853f521de25e7afa7980c4.jpg)  
+Figure 14: Similarity of the best sample (BAIR action-free dataset), updated plot. We show the similarity (higher is better) between the best sample (of 100) as a function of prediction time step across different methods and evaluation metrics. The three leftmost plots show the similarity (higher is better) between the best sample (of 100) as a function of prediction time step across different methods and evaluation metrics. Besides the standard metrics, we also use the Learned Perceptual Image Patch Similarity (LPIPS) metric (Zhang et al., 2018), which has been shown to correlate well with human perception. This distance is measured in the AlexNet feature space (pretrained on ImageNet classification) with linear weights calibrated to match human judgements. The plot on the right shows the diversity (higher is better) as a function of prediction time step, computed as the LPIPS distance between pairs of samples. Aside for the first two predicted frames, our SAVP model achieves similar LPIPS distances as the VAE models, both our VAE ablation and the SVG model from Denton & Fergus (2018). In addition, not only our SAVP method substantially improve sample diversity compared to the GAN-only model, but it also produces more diverse samples than both of the VAE models.
+
+![](images/4abc85c32e9970656c0c3b68fd6ffb68f99a463748593eeda9e0341cb31001f1.jpg)  
+Figure 15: Similarity of the best sample and diversity (KTH dataset), updated plot. The three leftmost plots show the similarity (higher is better) between the best sample (of 100) as a function of prediction time step. The plot on the right shows the diversity (higher is better) as a function of prediction time step, computed as the LPIPS distance between pairs of samples. The top and bottom plots show results when conditioning on 10 and 2 frames, respectively. Among the VAE methods, our VAE-only model achieves substantially higher similarities and diversities than the SVG model from prior work (Denton & Fergus, 2018). The GAN-only model mode-collapses and generates samples that lack diversity. Our SAVP method, which incorporates the variational loss, improves both sample diversity and similarities, compared to the GAN-only model. Our SAVP model also achieves higher accuracy than SVG.

parse/train/KYPz4YsCPj/KYPz4YsCPj.md

@@ -0,0 +1,537 @@
+# INDUCTIVE REPRESENTATION LEARNING IN TEMPORAL NETWORKS VIA CAUSAL ANONYMOUS WALKS
+
+Yanbang Wang1∗, Yen-Yu Chang2, Yunyu $\mathbf { L i u ^ { 3 } }$ , Jure Leskovec1, Pan $\mathbf { L i ^ { 1 , 3 } }$
+
+1Department of Computer Science, 2Electrical Engineering, Stanford Univers 3Department of Computer Science, Purdue University   
+{ywangdr,jure}@cs.stanford.edu,yenyu@stanford.edu {liu3154,panli}@purdue.edu
+
+# ABSTRACT
+
+Temporal networks serve as abstractions of many real-world dynamic systems. These networks typically evolve according to certain laws, such as the law of triadic closure, which is universal in social networks. Inductive representation learning of temporal networks should be able to capture such laws and further be applied to systems that follow the same laws but have not been unseen during the training stage. Previous works in this area depend on either network node identities or rich edge attributes and typically fail to extract these laws. Here, we propose Causal Anonymous Walks (CAWs) to inductively represent a temporal network. CAWs are extracted by temporal random walks and work as automatic retrieval of temporal network motifs to represent network dynamics while avoiding the time-consuming selection and counting of those motifs. CAWs adopt a novel anonymization strategy that replaces node identities with the hitting counts of the nodes based on a set of sampled walks to keep the method inductive, and simultaneously establish the correlation between motifs. We further propose a neural-network model CAW-N to encode CAWs, and pair it with a CAW sampling strategy with constant memory and time cost to support online training and inference. CAW-N is evaluated to predict links over 6 real temporal networks and uniformly outperforms previous SOTA methods by averaged $15 \%$ AUC gain in the inductive setting. CAW-N also outperforms previous methods in 5 out of the 6 networks in the transductive setting.
+
+# 1 INTRODUCTION
+
+Temporal networks consider dynamically interacting elements as nodes, interactions as temporal links, with labels of when those interactions happen. Such temporal networks provide abstractions to study many real-world dynamic systems (Holme & Saramäki, 2012). Researchers have investigated temporal networks in recent several decades and concluded many insightful laws that essentially reflect how these real-world systems evolve over time (Kovanen et al., 2011; Benson et al., 2016; Paranjape et al., 2017; Zitnik et al., 2019). For example, the law of triadic closure in social networks, describing that two nodes with common neighbors tend to have a mutual interaction later, reflects how people establish social connections (Simmel, 1950). Later, a more elaborate law on the correlation between the interaction frequency between two individuals and the degree that they share social connections, further got demonstrated (Granovetter, 1973; Toivonen et al., 2007). Feedforward control loops that consist of a direct interaction (from node $w$ to node $u$ ) and an indirect interaction (from $w$ through another node $v$ to $u$ ), also work as a law in the modulation of gene regulatory systems (Mangan & Alon, 2003) and also as the control principles of many engineering systems (Gorochowski et al., 2018). Although research on temporal networks has achieved the above success, it can hardly be generalized to study more complicated laws: Researchers have to investigate an exponentially increasing number of patterns when incorporating more interacting elements let alone their time-evolving aspects.
+
+Recently, representation learning, via learning vector representations of data based on neural networks, has offered unprecedented possibilities to extract, albeit implicitly, more complex structural patterns (Hamilton et al., 2017b; Battaglia et al., 2018). However, as opposed to the study on static networks, representation learning of temporal networks is far from mature. Two challenges on temporal networks have been frequently discussed. First, the entanglement of structural and temporal
+
+![](images/40467c5e271e5773d80798469e21f9352137a12f2468d6a9cc93669eeb3ca6ab.jpg)  
+Figure 1: Triadic closure and feed-forward loops: Causal anonymous walks (CAW) capture the laws.
+
+Example: three 3-step walks $( t _ { x } , X$ are the default timestamp and the default node when no historical links can be found)
+
+![](images/7540d82ab67e2037f4b9150644f1d5f6696f950c9818baa6667bec84f0ec6546.jpg)
+
+Count number of $b$ ’s in different positions:
+
+![](images/e040abd14669a9fa48b1000380074bf83b5459d0cae4973f7127d0f34a681fdb.jpg)  
+Figure 2: Causal anonymous walks (CAW): causality extraction and set-based anonymization.
+
+![](images/859b7ac2a5c3f2a0ec2f95acc989adbe9316089986c8c2ebf0405488bc3b4d7b.jpg)
+
+patterns required an elegant model to digest the two-side information. Second, the model scalability becomes more crucial over temporal networks as new arriving links need to be processed timely while a huge link set due to the repetitive links between two nodes needs to be digested simultaneously.
+
+In contrast to the above two challenges, another challenge, the inductive capability of the temporalnetwork representation, is often ignored. However, it is equally important if not more, as the inductive capability indicates whether the models indeed capture the dynamic laws of the systems and can be further generalized to the system that share the same laws but have not been used to train these models. These laws may only depend on structures such as the triadic closure or feed-forward control loops as aforementioned. These laws may also correlate with node attributes, such as interactions between people affected by their gender and age (Kovanen et al., 2013). But in both cases, the laws should be independent from network node identities. Although previous works tend to learn inductive models by removing node identities (Trivedi et al., 2019; Xu et al., 2020), they run into other issues to inductively represent the dynamic laws, for which we leave more detailed discussion in Sec. 2.
+
+Here we propose Causal Anonymous Walks (CAW) for modeling temporal networks. Our idea for inductive learning is inspired by the recent investigation on temporal network motifs that correspond to connected subgraphs with links that appear within a restricted time range (Kovanen et al., 2011; Paranjape et al., 2017). Temporal network motifs essentially reflect network dynamics: Both triadic closure and feed-forward control can be viewed as temporal network motifs evolving (Fig. 1); An inductive model should predict the 3rd link in both cases when it captures the correlation of these two links as they share a common node, while the model is agnostic to the node identities of these motifs.
+
+Our CAW model has two important properties (Fig. 2): (1) Causality extraction — a CAW starts from a link of interest and backtracks several adjacent links over time to encode the underlying causality of network dynamics. Each walk essentially gives a temporal network motif; (2) Set-based anonymization — CAWs remove the node identities over the walks to guarantee inductive learning while encoding relative node identities based on the counts that they appear at a certain position according to a set of sampled walks. Relative node identities guarantee that the structures of motifs and their correlations are still kept after removing node identities. To predict temporal links between two nodes of interest, we propose a model CAW-Network (CAW-N) that samples a few CAWs related to the two nodes of interest, encodes and aggregates these CAWs via RNNs (Rumelhart et al., 1986) and set-pooling respectively to make the prediction.
+
+Experiments show that CAW-N is extremely effective. CAW-N does not need to enumerate the types of motifs and count their numbers that have been used as features to predict network dynamics (Lahiri & Berger-Wolf, 2007; Rahman & Al Hasan, 2016; Rossi et al., 2019; AbuOda et al., 2019; Li & Milenkovic, 2017), which significantly saves feature-engineering effort. CAW-N also keeps all fine-grained temporal information along the walks that may be removed by directly counting motifs (Ahmed et al., 2015; Paranjape et al., 2017). CAWs share a similar idea as anonymous walks (AW) (Micali & Zhu, 2016) to remove node identities. However, AWs have only been used for entire static graph embedding (Ivanov & Burnaev, 2018) and are not directedly applied to represent temporal networks: AWs cannot capture causality; AWs get anonymized based on each single walk and hence lose the correlation between network motifs. In contrast, CAWs capture all the information, temporal, structural, motif-correlation that are needed, to represent temporal networks.
+
+We conclude our contributions in three-folds: (1) A novel approach to represent temporal network CAW-N is proposed, which leverages CAWs to encode temporal network motifs to capture network dynamics while keeping fully inductive. CAW-N is evaluated to predict links over 6 real-world temporal networks. CAW-N outperforms all SOTA methods by about $15 \%$ averaged over 6 networks in the inductive setting and also significantly beat all SOTA methods over 5 networks in the transductive setting; (2) CAW-N significantly decreases the feature-engineering effort in traditional motif selection and counting approaches and keeps fine-grained temporal information; (3) CAW-N is paired with a CAW sampling method with constant memory and time cost, which conduces to online learning.
+
+# 2 RELATED WORK
+
+Prior work on representation learning of temporal networks preprocesses the networks by simply aggregating the sequence of links within consecutive time windows into network snapshots, and use graph neural networks (GNN) (Scarselli et al., 2008; Kipf & Welling, 2017) and RNNs or transformer networks (Vaswani et al., 2017) to encode structural patterns and temporal patterns respectively (Pareja et al., 2020; Manessi et al., 2020; Goyal et al., 2020; Hajiramezanali et al., 2019; Sankar et al., 2020). The main drawback of these approaches is that they need to predetermine a time granularity for link aggregation, which is hard to learn structural dynamics in different time scales. Therefore, approaches that work on link streams directly have been recently proposed (Trivedi et al., 2017; 2019; Kumar et al., 2019; Xu et al., 2020). Know-E (Trivedi et al., 2017), DyRep (Trivedi et al., 2019) and JODIE (Kumar et al., 2019) use RNNs to propagate messages across interactions to update node representations. Know-E, JODIE consider message exchanges between two directly interacted nodes while DyRep considers an additional hop of interactions. Therefore, DyRep gives a more expressive model at a cost of high complexity. TGAT (Xu et al., 2020) in contrast mimics GraphSAGE (Hamilton et al., 2017a) and GAT (Velickovi ˇ c et al., 2018) to propagate messages in a ´ GNN-like way from sampled historical neighbors of a node of interest. TGAT’s sampling strategy requires to store all historical neighbors, which is unscalable for online learning. Our CAW-N directly works on link streams and only requires to memorize constant many most recent links for each node.
+
+Most of the above models are not inductive because they associate each node with an onehot identity (or the corresponding row of the adjacency matrix, or a free-trained vector) (Li et al., 2018; Chen et al., 2019; Kumar et al., 2019; Hajiramezanali et al., 2019; Sankar et al., 2020; Manessi et al., 2020; Goyal et al., 2020).
+
+![](images/d2549a96a4e14bdf943d40de6abf627c66f6937ae8a05294220ece0616340abb.jpg)  
+Figure 3: Ambiguity due to removing node identities in TGAT (Xu et al., 2020) $( t _ { 1 } < t _ { 2 } < t _ { 3 } )$ ).
+
+TGAT (Xu et al., 2020) claimed to be inductive by removing node identities and just encoding link timestamps and attributes. However, TGAT was only evaluated over networks with rich link attributes, where the structural dynamics is not captured essentially: If we focus on structural dynamics only, it is easy to show a case when TGAT confuses node representations and will fail: Suppose in the history, two node pairs $\{ a , b \}$ and $\{ a ^ { \prime } , b ^ { \prime } \}$ only interact within each pair but share the timestamps (Fig. 3). Intuitively, a proper model should predict that future links still appear within each pair. However, TGAT cannot distinguish $a$ v.s. $a ^ { \prime }$ , and $b$ v.s. $b ^ { \prime }$ , which leads to incorrect prediction. Note that GraphSAGE (Hamilton et al., 2017a) and GAT (Velickovi ˇ c et al., 2018) also share the similar ´ issue when representing static networks for link prediction (Zhang et al., 2020; Srinivasan & Ribeiro, 2019). DyRep (Trivedi et al., 2019) is able to relieve such ambiguity by merging node representations with their neighbors’ via RNNs. However, when DyRep runs over a new network, it frequently encounters node representations unseen during its training and will fail to make correct prediction.
+
+# Algorithm 1: Temporal Walk Extraction $( \mathcal { E } , \alpha , M , m , w _ { 0 } , t _ { 0 } )$
+
+Initialize $M$ walks: $W _ { i } \gets ( ( w _ { 0 } , t _ { 0 } ) )$ , $1 \leq i \leq M$ ;
+
+The rule: “one node (e.g. u) interacts with other nodes only if another node interacts with this node at least twice.”
+
+3 for $i$ from 1 to $M$ do 4 $( w _ { \mathsf { p } } , t _ { \mathsf { p } } ) \gets$ the last (node, time) pair in $W _ { i }$ ; 5 Sample one $( e , t ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ with prob. $\propto \exp ( \alpha ( t - t _ { \mathrm { p } } ) )$ Denote $\boldsymbol { e } = \{ w ^ { \prime } , w \}$ and then $W _ { i } \gets W _ { i } \oplus ( w ^ { \prime } , t )$ ;
+
+6 Return $\{ W _ { i } | 1 \leq i \leq M \}$ ;
+
+![](images/48cffeb4acfb34c6a6577ae037e112f82ed82578a1cad56743fe7d423e49c370.jpg)  
+?? takes action $u \mathrm { N O T }$ takes action Figure 4: The correlation between walks needs to be captured to learn this law.
+
+Our CAW-N removes node identities and leverages relative node identities to avoid the issue in Fig. 3.   
+Detailed explanations are given in Sec.4.2.
+
+Network-embedding approaches may also be applied to temporal networks (Zhou et al., 2018; Du et al., 2018; Mahdavi et al., 2018; Singer et al., 2019; Nguyen et al., 2018). However, they directly assign each node with a learnable vector. Therefore, they are not inductive and cannot digest attributes.
+
+# 3 PROBLEM FORMULATION AND NOTATIONS
+
+Problem Formulation. A temporal network can be represented as a sequence of links that come in over time, i.e. $\mathcal { E } = \{ ( e _ { 1 } , t _ { 1 } ) , ( \bar { e } _ { 2 } , t _ { 2 } ) , . . . \}$ where $e _ { i }$ is a link and $t _ { i }$ is the timestamp showing when $e _ { i }$ arrives. Each link $e _ { i }$ corresponds to a dyadic event between two nodes $\{ v _ { i } , u _ { i } \}$ . For simplicity, we first assume those links to be undirected and without attributes while later we discuss how to generalized our method to directed attributed networks. The sequence of links encodes network dynamics. Therefore, the capability of a model for representation learning of temporal networks is typically evaluated by how accurately it may predict future links based on the historical links (Sarkar et al., 2012). In this work, we also use link prediction as the metric. Note that we care not only the link prediction between the nodes that have been seen during the training. We also expect the models to predict links between the nodes that has never been seen as the inductive evaluation.
+
+Notations. We define $\mathcal { E } _ { v , t } = \{ ( e , t ^ { \prime } ) \in \mathcal { E } | t ^ { \prime } < t , v \in e \}$ to include the links attached to a node $v$ before certain time $t$ . A walk $W$ (reverse over time) on temporal networks can be represented as
+
+$$
+W = ( ( w _ { 0 } , t _ { 0 } ) , ( w _ { 1 } , t _ { 1 } ) , . . . , ( w _ { m } , t _ { m } ) ) , \ t _ { 0 } > t _ { 1 } > \cdot \cdot \cdot > t _ { m } , \ ( \{ w _ { i - 1 } , w _ { i } \} , t _ { i } ) \in \mathcal { E } \mathrm { ~ f o r ~ a l l ~ } \ t _ { 0 } < \mathbb { E } \mathrm { ~ o r ~ } \ t _ { 1 } < \mathbb { E } ,
+$$
+
+We use $W [ i ]$ to denote the $i$ th node-time pair, and $W [ i ] [ 0 ]$ and $W [ i ] [ 1 ]$ to denote the corresponding node and time in $W [ i ]$ correspondingly. Later, we also use $\oplus$ as vector concatenation.
+
+Temporal network motifs are defined as connected subgraphs that consist of links appearing within a restricted time range (Kovanen et al., 2011). Based this definition, each walk defined in Eq. 1 naturally corresponds to a temporal network motif as long as $\left( t _ { 1 } - t _ { m } \right)$ is in the time range.
+
+# 4 PROPOSED METHOD: CAUSAL ANONYMOUS WALK-NETWORK
+
+# 4.1 PRELIMINARIES: ANONYMOUS WALK AND TEMPORAL NETWORK MOTIF
+
+Anonymous walks were first considered by Micali & Zhu (2016) to study the reconstruction of a Markov process from the records without sharing a common “name space”. AWs can be directly rephrased in the network context. Specifically, an AW starts from a node, performs random walks over the graph to collect a walk of nodes, e.g. $( v _ { 1 } , v _ { 2 } , . . . , v _ { m } )$ . AW has an important anonymization step to replace the node identities by the orders of their appearance in each walk, which we term relative node identities in AW and define it as
+
+$$
+I _ { A W } ( w ; W ) \triangleq | \{ v _ { 0 } , v _ { 1 } , . . . , v _ { k ^ { * } } \} |
+$$
+
+Note that the set operation above removes duplicated elements. Although AW removes node identities, those nodes are still distinguishable within this walk. Therefore, an AW can be viewed as a network motif while the information on which specific nodes form this motif is removed. Examples of AWs are shown as follows. While these are two different walks, they may be mapped to the same AW when node identities get removed.
+
+![](images/d6e9fb23b0ffa2f5f256b5887a40dd252ed31e92613a66243d5005342b3a9e6c.jpg)
+
+# 4.2 CAUSAL ANONYMOUS WALK
+
+We propose CAW that shares the high-level concept with AW to remove the original node identities. However, CAW has a different causal sampling strategy and a novel set-based approach for node anonymization, which are specifically designed to encode temporal network dynamics (Fig. 2).
+
+Causality Extraction. Alg. 1 shows our causal sampling: We sample connected links by backtracking over time to extract the underlying causality of network dynamics. More recent links may be more informative and thus we introduce a non-negative hyper-parameter $\alpha$ to sample a link with a probability proportional to $\exp ( \alpha ( t - t _ { \mathrm { p } } ) )$ where $t , t _ { \mathrm { p } }$ are the timestamps of this link and the link previously sampled respectively. A large $\alpha$ can emphasize more on recent links while zero $\alpha$ leads to uniform sampling. In Sec.4.4, we will discuss an efficient sampling strategy for the step 5 in Alg.1, which avoids computing those probabilities by visiting the entire historical links.
+
+Then, given a link $\{ u _ { 0 } , v _ { 0 } \}$ and a time $t _ { 0 }$ , we use Alg. 1 to collect $M$ many $m$ -step walks starting from both $u _ { 0 }$ and $v _ { 0 }$ , and record them in $S _ { u }$ and $S _ { v }$ respectively. For convenience, a walk $W$ from a starting node $w _ { 0 } \in \{ u , v \}$ can be represented as Eq.1.
+
+Set-based Anonymization. Based on $S _ { u }$ and $S _ { v }$ , we may anonymize each node identity $w$ that appears on at least one walk in $S _ { u } \cup S _ { v }$ and design relative node identity $I _ { C A W } ( w ; \{ S _ { u } , \overrightharpoonup { S } _ { v } \} )$ for $w$ . Our design has the following consideration. $I _ { A W }$ (Eq.2) only depends on a single path, which results from the original assumption that any two AWs do not even share the name space (i.e., node identities) (Micali & Zhu, 2016). However, in our case, node identities are actually accessible, though an inductive model is not allowed to use them directly. Instead, correlation across different walks could be a key to reflect laws of network dynamics: Consider the case when the link $\{ u , v \}$ happens only if there is another node appearing in multiple links connected to $u$ (Fig. 4). Therefore, we propose to use node identities to first establish such correlation and then remove the original identities.
+
+Specifically, we define $I _ { C A W } ( w ; \{ S _ { u } , S _ { v } \} )$ as follows: For $w _ { 0 } \in \{ u , v \}$ , let $g ( w , S _ { w _ { 0 } } ) \in \mathbb { Z } ^ { m + 1 }$ count the times in $S _ { w _ { 0 } }$ node $w$ appears at certain positions, i.e., $g ( w , S _ { w _ { 0 } } ) [ i ] \triangleq | \{ W | W \in S _ { w _ { 0 } } , w =$ $W [ i ] [ 0 ] \}$ for $i \in \{ 0 , 1 , . . . , m \}$ . Further, define
+
+$$
+I _ { C A W } ( w ; \{ S _ { u } , S _ { v } \} ) \triangleq \{ g ( w , S _ { u } ) , g ( w , S _ { v } ) \} .
+$$
+
+Essentially, $g ( w , S _ { u } )$ and $g ( w , S _ { v } )$ encode the correlation between walks within $S _ { u }$ and $S _ { v }$ respectively and the set operation in Eq.3 establishes the correlation across $S _ { u }$ and $S _ { v }$ . For the case in Fig.3, suppose $S _ { a } , S _ { b } , S _ { a ^ { \prime } } , S _ { b ^ { \prime } }$ include all historical one-step walks (before $t _ { 3 }$ ) starting from $a , b , a ^ { \prime } , b ^ { \prime }$ respectively. Then, it is easy to show that $I _ { C A W } ( a ; \{ { \bar { S } } _ { a } , S _ { b } \} ) \neq I _ { C A W } ( a ^ { \prime } ; \{ S _ { a ^ { \prime } } , { \bar { S } } _ { b } \} )$ that allows differentiating $a$ and $a ^ { \prime }$ , while TGAT ( $\mathrm { X u }$ et al., 2020) as discussed in Sec.2 fails. From the networkmotif point of view, $I _ { C A W }$ not only encodes each network motif that corresponds to one single walk as $I _ { A W }$ does but also establish the correlation among these network motifs. $I _ { A W }$ cannot establish the correlation between motifs and will also fail to distinguish $a$ and $a ^ { \prime }$ in Fig.3. We see it as a significant breakthrough as such correlation often gets neglected in previous works that directly count motifs or adopt AW-type anonymization $I _ { A W }$ .
+
+Later, we use $I _ { C A W } ( w )$ for simplicity when the reference set $\{ S _ { u } , S _ { v } \}$ can be inferred from the context. Then, each walk $W$ (Eq.1) can be anonymized as
+
+$$
+\hat { W } = ( ( I _ { C A W } ( w _ { 0 } ) , t _ { 0 } ) , ( I _ { C A W } ( w _ { 1 } ) , t _ { 1 } ) , . . . , ( I _ { C A W } ( w _ { m } ) , t _ { m } ) ) .
+$$
+
+The following theorem indicates that $I _ { C A W }$ does not depend on node identities to guarantee the inductive property of the models, which can be easily justified.
+
+Theorem 4.1. For two pairs of walk sets $\{ S _ { u } , S _ { v } \}$ and $\{ S _ { u ^ { \prime } } , S _ { v ^ { \prime } } \}$ , if there exists a bijective mapping $\pi$ between node identities such that each walk $W$ in $S _ { u } \cup S _ { v }$ can be bijectively mapped to one walk $W ^ { \prime }$ in $S _ { u ^ { \prime } } \cup S _ { v ^ { \prime } }$ according to $\pi ( W [ i ] [ 0 ] ) = W ^ { \prime } [ i ] [ 0 ]$ for all $i \in [ 0 , m ]$ . Then $I _ { C A W } ( w | \{ S _ { u } , S _ { v } \} ) =$ $I _ { C A W } ( \pi ( w ) | \{ S _ { u ^ { \prime } } , S _ { v ^ { \prime } } \} )$ for all nodes $w$ that appear in at least one walk in $S _ { u } \cup S _ { v }$ .
+
+# 4.3 NEURAL ENCODING FOR CAUSAL ANONYMOUS WALKS
+
+After we collect CAWs, neural networks can be conveniently leveraged to extract their structural (in $I _ { C A W } ( \cdot ) )$ and fine-grained temporal information by encoding CAWs: We will propose the model CAW-N to first encode each walk $\hat { W }$ (Eq.4) and then aggregate all encoded walks in $S _ { u } \cup S _ { v }$ .
+
+<table><tr><td>Measurement</td><td>Reddit</td><td>Wikipedia</td><td>MOOC</td><td>Social Evo.</td><td>Enron</td><td>UCI</td></tr><tr><td>nodes&amp; temporal links attributes for nodes &amp; links</td><td>10,985&amp;672,447 172&amp;172</td><td>9,227&amp;157,474 172 &amp;172</td><td>7145&amp;411,749</td><td>184&amp;125,235</td><td>74&amp;2,099,520</td><td>1,899&amp;59,835 0&amp;0</td></tr><tr><td>avg. link stream intensity T</td><td>4.57×10-5</td><td>1.27 × 10-5</td><td>0&amp;4 4.48×10-5</td><td>0&amp;0 6.50 × 10-5</td><td>0&amp;0 4.98 × 10-3</td><td>3.59×10-5</td></tr></table>
+
+Table 1: Summary of dataset statistics. Average link stream intensity is calculated by $2 | E | / ( | V | T )$ , where $T$ is the total time range of all edges in unit of seconds, $| V |$ and $| E |$ are number of nodes and temporal links.
+
+Encode $\hat { W }$ . Note that each walk is a sequence of node-time pairs. If we encode each node-time pair and plug those pairs in a sequence encoder, e.g., RNNs, we obtain the encoding of $\hat { W }$ :
+
+$$
+\begin{array} { r } { \mathrm { e n c } ( \hat { W } ) = \mathrm { R N N } ( \{ f _ { 1 } \big ( I _ { C A W } ( w _ { i } ) \big ) \oplus f _ { 2 } ( t _ { i - 1 } - t _ { i } ) \} _ { i = 0 , 1 , \dots , m } ) , \mathrm { ~ w h e r e ~ } t _ { - 1 } = t _ { 0 } , } \end{array}
+$$
+
+where $f _ { 1 } , f _ { 2 }$ are two encoding function on $I _ { C A W } ( w _ { i } )$ and $t _ { i - 1 } - t _ { i }$ respectively. One may use transformer networks instead of RNNs to encode the sequences but as the sequences in our case are not long $( 1 \sim 5 )$ , RNNs have achieved good enough performance. Now, we specify the two encoding functions $f _ { 1 } ( I _ { C A W } ( w _ { i } ) )$ and $f _ { 2 } ( t _ { i - 1 } - t _ { i } )$ as follows. Recall the definition of $I _ { C A W } ( w _ { i } )$ (Eq.3).
+
+$\begin{array} { r } { f _ { 1 } \big ( I _ { C A W } \big ( w _ { i } \big ) \big ) = \mathbf { M L P } \big ( g \big ( w , S _ { u } \big ) \big ) + \mathbf { M L P } \big ( g \big ( w , S _ { v } \big ) \big ) . } \end{array}$ , where two MLPs share parameters.
+
+Here the encoding of $I _ { C A W } ( w _ { i } )$ adopts the sum-pooling as the order of $u , v$ is not relevant. For $f _ { 2 } ( t )$ , we adopt random Fourier features to encode time ( $\mathrm { { X u } }$ et al., 2019; Kazemi et al., 2019) which may approach any positive definite kernels according to the Bochner’s theorem (Bochner, 1992).
+
+$f _ { 2 } ( t ) = [ \cos ( \omega _ { 1 } t ) , \sin ( \omega _ { 1 } t ) , . . . , \cos ( \omega _ { d } t ) , \sin ( \omega _ { d } t ) ] .$ , where $\omega _ { i }$ ’s are learnable parameters.
+
+Encode $S _ { u } \cup S _ { v }$ . After encoding each walk in $S _ { u } \cup S _ { v }$ , we aggregate all these walks to obtain the final representation enc $( S _ { u } \cup S _ { v } )$ for prediction. We suggest to use either mean-pooling for algorithmic efficiency or self-attention (Vaswani et al., 2017) followed by mean-pooling to further capture subtle interactions between different walks. Specifically, suppose $\{ \hat { W } _ { i } \} _ { 1 \leq i \leq 2 M }$ are the $2 M$ CAWs in $S _ { u } \cup S _ { v }$ and each enc $( \hat { W } _ { i } ) \in \mathbb { R } ^ { d \times 1 }$ . We set enc $( S _ { u } \cup S _ { v } )$ as
+
+• $\begin{array} { r } { \mathbf { M e a n - A G G } ( S _ { u } \cup S _ { v } ) \colon \frac { 1 } { 2 M } \sum _ { i = 1 } ^ { 2 M } \operatorname { e n c } ( \hat { W } _ { i } ) . } \end{array}$   
+• Self-Att- $\operatorname { A G G } ( S _ { u } \cup S _ { v } )$ $\begin{array} { r l } { \colon } & { { } \frac { 1 } { 2 M } \sum _ { i = 1 } ^ { 2 M } \mathrm { s o f t m a x } ( \{ \mathrm { e n c } ( \hat { W } _ { i } ) ^ { T } Q _ { 1 } \mathrm { e n c } ( \hat { W } _ { j } ) \} _ { 1 \leq j \leq n } ) \mathrm { e n c } ( \hat { W } _ { i } ) Q _ { 2 } } \end{array}$ where $Q _ { 1 } , Q _ { 2 } \in \mathbb { R } ^ { d \times d }$ are two learnable parameter matrices.
+
+We add 2-layer perceptron over enc $( S _ { u } \cup S _ { v } )$ to make the final link prediction.
+
+# 4.4 EXTENSION AND DISCUSSION
+
+Attributed nodes/links and directed links. In some real networks, nodes or links may have attributes available, e.g., the message content in the case of SMS networks. In this case, the walk in Eq.1 can be associated with node/link attributes $X _ { 0 } , X _ { 1 } , . . . , X _ { m }$ where $X _ { i }$ refers to the attributes on link $( \{ w _ { i - 1 } , w _ { i } \} , t _ { i } )$ or on the node $w _ { i }$ or a concatenation of these two parts. Note that the direction of a link can also be viewed as a binary link attribute, where a 2-dimensional one-hot encoding can be used. To incorporate such information, we only need to change enc $( \hat { W } )$ (Eq.5) as
+
+Since $f _ { 1 } ( I _ { C A W } ( w _ { i } ) )$ is a strong signal, in practice it is optional to use another RNN to encode its own dynamics. The derived encoding is then concatenated with enc $( \hat { W } )$ to obtain the enhanced final encoding of $\hat { W }$ .
+
+Efficient link sampling. A naive implementation of the link sampling in step 5 in Alg.1 is to compute and normalize the sampling probabilities of all links in $\mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ , which requires to memorize all historical links and costs much time and memory. To solve this problem, we propose a sampling strategy (Appendix A) with expected time and memory complexity $\begin{array} { r } { \operatorname* { m i n } \{ \frac { 2 \tau } { \alpha } + 1 , \top E _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } } | \} } \end{array}$ if links in Ewp,tp come in by following a Poisson process with intensity $\tau$ (Last & Penrose, 2017). This means that our sampling strategy with a positive $\alpha$ allows the model only recording $\textstyle O ( { \frac { \tau } { \alpha } } )$ recent links for each node instead of the entire history. Our experiments in Sec.5.3 show that $\alpha$ that achieves the best prediction performance makes $\textstyle { \frac { \tau } { \alpha } } \approx 5$ in different datasets. Since the time and memory complexity do not increase with respect to the number of links, our model can be used for online training and inference. Note that the $\alpha = 0$ case reduces to uniform sampling, mostly adopted by previous methods ( $\mathrm { \Delta X u }$ et al., 2020), which requires to record the entire history and thus is not scalable.
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETUP
+
+CAW-N Variants. We test CAW-N-mean and CAW-N-attn which uses mean and attention pooling respectively to encode $S _ { u } \cup S _ { v }$ (Sec. 4.3). Their code is provided in the supplement.
+
+Baselines. Our method is compared with six previous state-of-the-art baselines on representation learning of temporal networks. They can be grouped into two categories based on their input data structure: (1) Snapshot-based methods, including DynAERNN (Goyal et al., 2020), VGRNN (Hajiramezanali et al., 2019) and EvolveGCN (Pareja et al., 2020); (2) Stream-based methods, including TGAT (Xu et al., 2020), JODIE (Kumar et al., 2019) and DyRep (Trivedi et al., 2019). We give their detailed introduction in Appendix C.2.2. For the snapshot-based methods, we view the link aggregation as a way to preprocess historical links. We adopt the aggregation ways suggested in their papers. These models are trained and evaluated over the same link sets as the stream-based methods.
+
+Dataset. We use six real-world public datasets: Wikipedia is a network between wiki pages and human editors. Reddit is a network between posts and users on subreddits. MOOC is a network of students and online course content units. Social Evolution is a network recording the physical proximity between students. Enron is a email communication network. UCI is a network between online posts made by students. We summarize their statistics in Tab.1 and give their detailed description and access in Appendix C.1.
+
+Evaluation Tasks. Two types of tasks are for evaluation: transductive and inductive link prediction.
+
+Transductive link prediction task allows temporal links between all nodes to be observed up to a time point during the training phase, and uses all the remaining links after that time point for testing. In our implementation, we split the total time range $[ 0 , T ]$ into three intervals: [0, $T _ { t r a i n } )$ , $[ T _ { t r a i n }$ , $T _ { v a l }$ ), $[ T _ { v a l } , T ]$ . links occurring within each interval are dedicated to training, validation, and testing set, respectively. For all datasets, we fix $T _ { t r a i n } / T { = } 0 . 7$ , and ${ T _ { v a l } } / T \mathrm { = } 0 . 8 5$ .
+
+Inductive link prediction task predicts links associated with nodes that are not observed in the training set. There are two types of such links: 1) "old vs. new" links, which are links between an observed node and an unobserved node; 2) "new vs. new" links, which are links between two unobserved nodes. Since these two types of links suggest different types of inductiveness, we distinguish them by reporting their performance metrics separately. In practice, we follow two steps to split the data: 1) we use the same setting of the transductive task to first split the links chronologically into training / validation / testing sets; 2) we randomly select $10 \%$ nodes, remove any links associated with them from the training set, and remove any links not associated with them in the validation and testing sets.
+
+Following most baselines, we randomly sample an equal amount of negative links and consider link prediction as a binary classification problem. For fair comparison, we use the same evaluation procedures for all baselines, including the snapshot-based methods.
+
+Training configuration. We use binary cross entropy loss and Adam optimizer to train all the models, and early stopping strategy to select the best epoch to stop training. For hyperparameters, we primarily tune those that control the CAW sampling scheme including the number $M$ , the length $m$ of CAWs and the time decay $\alpha$ . We will investigate their sensitivity in Sec.5.3. For all baselines, we adapt their implemented models into our evaluation pipeline and extensively tune them. Detailed description of all models’ tuning can be found in Appendix C. Finally, we adopt two metrics to evaluate the models’ performance: Area Under the ROC curve (AUC) and Average Precision (AP).
+
+# 5.2 RESULTS AND DISCUSSION
+
+We report AUC scores in Tab.2, and report AP scores in Tab.6 of Appendix D.1. In the inductive setting and especially with "new vs new" links, our models significantly outperform all baselines on all datasets. On average, our best method improves over the strongest baseline by $1 4 . 4 6 \%$ (new vs. new) and $3 . 4 9 \%$ (old vs. new) in relative, or reduces the error $( = 1 - \mathsf { A U C } )$ by $6 9 . 7 3 \%$ (new vs. new) and $5 8 . 6 3 \%$ (old vs. new). Noticeably, out method achieves almost perfect AUCs on UCI’s "new vs. new" edges, when all baselines’ performance is below 0.8.
+
+Even in transductive setting where the baselines claim their primary contribution, our approaches still significantly outperform them on five out of six datasets. Note that our models achieve almost perfect scores on Reddit and Wikpedia when the baselines are far from perfect. Meanwhile, the strongest baseline on these two attributed datasets, TGAT ( $\mathrm { X u }$ et al., 2020), suffers a lot on all the other datasets where informative node / link attributes become unavailable.
+
+Table 2: Performance in AUC (mean in percentage $\pm 9 5 \%$ confidence level.) $\dagger$ highlights the best baselines. ∗, bold font, bold font∗ respectively highlights the case where our models’ performance exceeds the best baseline on average, by $7 0 \%$ confidence, by $9 5 \%$ confidence.   
+
+<table><tr><td colspan="2">Task</td><td>Methods</td><td>Reddit</td><td>Wikipedia</td><td>MOOC</td><td>Social Evo.</td><td>Enron</td><td>UCI</td></tr><tr><td rowspan="7">Trgeecer</td><td rowspan="7">Mwass aaa</td><td>DynAERNN JODIE DyRep</td><td>57.51 ± 2.54 72.49 ± 0.38</td><td>55.16 ± 1.15 70.78 ± 0.75</td><td>60.85 ± 1.61 80.04±0.28†</td><td>52.00±0.16 87.66 ±0.12†</td><td>51.57 ± 2.63 73.99 ± 2.54†</td><td>50.20± 2.78 64.77 ± 0.75</td></tr><tr><td></td><td>62.37 ± 1.49 61.93 ± 0.72</td><td>67.07 ± 1.26</td><td>74.07 ± 1.88</td><td>83.92 ±0.02</td><td>69.74 ± 0.44</td><td>63.76± 4.67</td></tr><tr><td>VGRNN EvolveGCN</td><td>63.31 ±0.53</td><td>60.64 ±0.68 58.01 ± 0.16</td><td>63.01 ± 0.29 52.31 ± 4.14</td><td>66.30 ± 0.84</td><td>61.35 ± 1.10</td><td>61.35 ± 1.10</td></tr><tr><td>TGAT</td><td>94.96 ± 0.88†</td><td>93.53 ± 0.84†</td><td>70.10 ± 0.35</td><td>46.95 ± 0.85</td><td>42.53 ± 2.12</td><td>76.65 ± 0.63†</td></tr><tr><td>CAW-N-mean</td><td>98.30 ± 0.71*</td><td></td><td></td><td>53.27 ± 1.16</td><td>63.34 ± 2.95</td><td>76.36 ± 1.48</td></tr><tr><td>CAW-N-attn</td><td>98.11 ± 0.58*</td><td>96.36±0.48* 97.83 ± 0.67*</td><td>90.29±0.82* 90.40 ± 0.75*</td><td>93.81 ± 0.69* 94.55 ± 0.81*</td><td>94.26±0.62* 93.53 ± 0.63*</td><td>99.62 ± 0.34*</td></tr><tr><td>DynAERNN</td><td>58.79 ±3.01 57.97 ± 2.38</td><td>80.99 ±1.35</td><td>52.31 ±0.59</td><td></td><td></td><td>100.00 ±0.00* 52.26 ± 1.36</td></tr><tr><td rowspan="7">PlO &#x27;s&#x27;A Mau</td><td>JODIE</td><td>76.33 ± 0.03</td><td>74.65 ± 0.06</td><td>87.40 ± 1.71</td><td>91.80 ± 0.01†</td><td>54.36± 1.48 85.24 ± 0.08</td><td>69.95 ± 0.11</td></tr><tr><td>DyRep</td><td>66.13 ± 1.07</td><td>76.72 ± 0.19</td><td>88.23 ± 1.20†</td><td>87.98 ± 0.45</td><td>94.39 ± 0.32†</td><td>93.28 ± 0.96†</td></tr><tr><td>VGRNN</td><td>54.11 ± 0.74</td><td>62.93 ± 0.69</td><td>60.10 ± 0.88</td><td>64.66 ± 0.41</td><td>68.71 ± 0.92</td><td></td></tr><tr><td>EvolveGCN</td><td>65.61 ± 0.37</td><td>56.29 ± 2.17</td><td>50.20 ± 1.92</td><td>50.73 ±1.36</td><td>42.53 ± 2.13</td><td>62.39 ± 1.08</td></tr><tr><td>TGAT</td><td>97.25 ± 0.18†</td><td>95.47 ± 0.17†</td><td>69.30 ± 0.08</td><td></td><td></td><td>70.78 ± 0.22</td></tr><tr><td>CAW-N-mean</td><td>99.88 ± 0.04*</td><td>98.94 ± 0.05*</td><td>90.88 ± 0.54*</td><td>54.22 ± 1.28 95.15 ± 0.40*</td><td>58.76 ± 1.18 94.76 ± 1.05*</td><td>74.19 ± 0.88</td></tr><tr><td></td><td>99.93 ±0.03*</td><td>99.61± 0.25*</td><td>90.89 ±0.56*</td><td>95.74 ± 0.68*</td><td>93.43 ± 1.41</td><td>99.04± 0.34*</td></tr><tr><td rowspan="10">Trarrsreeea</td><td>CAW-N-attn DynAERNN</td><td>83.37 ±1.48</td><td>71.00±1.10</td><td>89.34± 0.24</td><td>67.78±0.80</td><td>63.11 ± 1.13</td><td>98.99 ± 0.44*</td></tr><tr><td>JODIE</td><td>87.71 ± 0.02</td><td>88.43 ± 0.02</td><td>90.50 ± 0.01†</td><td>89.78 ± 0.04</td><td>89.36 ± 0.06</td><td>83.72± 1.79</td></tr><tr><td>DyRep</td><td>67.36 ± 1.23</td><td></td><td></td><td></td><td></td><td>74.63 ± 0.11</td></tr><tr><td>VGRNN</td><td>51.89 ±0.92</td><td>77.40 ± 0.13 71.20 ± 0.65</td><td>90.49 ± 0.03 90.03 ±0.32</td><td>90.85 ± 0.01† 78.28 ± 0.69</td><td>96.71 ± 0.04† 93.84 ±0.58</td><td>95.23 ± 0.25†</td></tr><tr><td>EvolveGCN</td><td>58.42 ±0.52</td><td>60.48 ± 0.47</td><td>50.36 ± 0.85</td><td>60.36 ± 0.65</td><td>74.02 ± 0.31</td><td>89.43 ± 0.27</td></tr><tr><td>TGAT</td><td>96.65 ± 0.06†</td><td>96.36 ± 0.05†</td><td>72.09 ± 0.29</td><td>56.63 ± 0.55</td><td>60.88 ±0.37</td><td>78.30 ± 0.22</td></tr><tr><td>CAW-N-mean</td><td>99.97 ± 0.01*</td><td>99.91 ± 0.04*</td><td>91.99 ± 0.72*</td><td>94.12 ± 0.15*</td><td>93.53 ± 0.73</td><td>77.67 ± 0.27</td></tr><tr><td>CAW-N-attn</td><td>99.98 ±0.01*</td><td>99.89±0.03*</td><td>92.38 ±0.58*</td><td>94.79 ±0.16*</td><td>95.93 ± 0.39</td><td>95.90 ± 0.71</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>98.45±0.49*</td></tr></table>
+
+<table><tr><td>No.</td><td>Ablation</td><td>Wikipedia</td><td>UCI</td><td>Social Evo.</td></tr><tr><td>1.</td><td>original method (CAW-N-mean)</td><td>98.49 ± 0.38</td><td>99.12± 0.33</td><td>94.54± 0.69</td></tr><tr><td>2.</td><td>remove fi(IcAw)</td><td>96.28 ± 0.66</td><td>79.45 ± 0.71</td><td>53.69 ± 0.21</td></tr><tr><td>3.</td><td>remove f2(t)</td><td>97.99 ± 0.13</td><td>95.00 ±0.42</td><td>71.93 ± 2.32</td></tr><tr><td>4.</td><td>remove fi(IcAw),f2(t)</td><td>88.94 ± 0.85</td><td>50.01 ± 0.02</td><td>50.00 ±0.00</td></tr><tr><td>5.</td><td>replace fi(IcAw) byone-hot(IAw)</td><td>96.55 ± 0.21</td><td>85.59 ± 0.34</td><td>68.47 ± 0.86</td></tr><tr><td>6.</td><td>fixα=0</td><td>75.10 ± 3.12</td><td>87.13 ± 0.49</td><td>78.01 ± 0.69</td></tr></table>
+
+Table 3: Ablation study with CAW-N-mean. AUC scores on all inductive test links are reported.
+
+Comparing the two training settings, we observe that the performance of four baselines (JODIE, VGRNN, DynAERNN,DyRep) drop significantly when transiting from the transductive setting to the inductive one, as they mostly record node identities either explicitly (JODIE, VGRNN, DynAERNN) or implicitly (DyRep). TGAT and EvolveGCN do not use node identities and thus their performance gaps between the two settings are small, while they sometimes do not perform well in the transductive setting, as they encounter the ambiguity issue in Fig. 3. In contrast, our methods perform well in both the transductive and inductive settings. We attribute this superiority to the anonymization procedure: the set-based relative node identities well capture the correlation between walks to make good prediction while removing the original node identities to keep entirely inductive. Even when the network structures greatly change and new nodes come in as long as the network evolves according to the same law as the network used for training, CAW-N will always work. Comparing CAW-N-mean and CAW-N-attn, we see that our attention-based variant outperforms the mean-pooling variant, albeit at the cost of high computation complexity. Also note that the strongest baselines on all datasets are stream-based methods, which indicates that the aggregation of links into network snapshots may remove some useful time information (see more discussion in Appendix D.2).
+
+We further conduct ablation studies on Wikipedia (attributed), UCI (non-attributed), and Social Evolution (non-attributed), to validate effectiveness of critical components of our model. Tab. 3 shows the results. By comparing Ab.1 with Ab.2, 3 and 4 respectively, we observe that our proposed node anonymization and encoding, $f _ { 1 } ( I _ { C A W } )$ , contributes most to the performance, though the time encoding ${ \dot { f } } _ { 2 } ( t )$ also helps. Comparing performance across different datasets, we see that the impact of ablation is more prominent when informative node/link attributes are unavailable such as with UCI and Social Evolution. Therefore, in such scenarios our CAWs are highly crucial. In Ab.5, we replace our proposed $I _ { C A W }$ with $I _ { A W }$ (Eq.2), which is used in standard AWs, and we use one-hot encoding of node new identities $I _ { A W }$ . We see by comparing Ab.1, 2, and 5 that such anonymization process is significantly less effective than our $I _ { C A W }$ , though it helps to some extent. Finally, Ab.6 suggests that entirely uniform sampling of the history may hurt performance.
+
+![](images/9ec0d59d700c9619967deb9fcfa86cbc9899ba2e1df8ce4606fe42d07ab3800c.jpg)  
+Figure 5: Hyperparameter sensitivity in CAW sampling. AUC on all inductive test links are reported.
+
+![](images/72705a5154c4f49127ce3c0370e2c6003416fbe4886d79f7e26cb9c346c78665.jpg)  
+Figure 6: Complexity evaluation: The accumulated runtime of (a) temporal random walk extraction (Alg.1) and (b) the entire CAW-N training, timed over one epoch on Wikipedia (using different $| \mathcal { E } |$ for training).
+
+# 5.3 HYPERPARAMETER INVESTIGATION OF CAW SAMPLING
+
+We systematically analyze the effect of hyperparameters used in CAW sampling schemes, including sampling number $M$ , temporal decay coefficient $\alpha$ and walk length $m$ . The experiments are conducted on UCI and Wikipedia datasets using CAW-N-mean. When investigating each hyperparameter, we set the rest two to an optimal value found by grid search, and report the mean AUC performance on all inductive test links (i.e. old vs. new, new vs. new) and their $9 5 \%$ confidence intervals.
+
+The results are summarized in Fig.5. From (a), we observe that only a small number of sampled CAWs are needed to achieve a competitive performance. Meanwhile, the performance gain is saturated as the sampling number increases. We analyze the temporal decay $\alpha$ in (b): $\alpha$ usually has an optimal interval, whose values and length also vary with different datasets to capture the different levels of temporal dynamics; a small $\alpha$ suggests an almost uniform sampling of interaction history, which hurts the performance; an overly large $\alpha$ also damages the model, since it makes the model only sample the most recent few interactions for computation and blind to the rest. Based on our efficient sampling strategy (Sec.4.4), we may combine the optimal $\alpha$ with the average link intensity $\tau$ (Tab.1), and concludes that CAW-N only needs to online record and sample from about a constant times about 5 $\begin{array} { r } { ( \approx \frac { \tau } { \alpha } ) } \end{array}$ most recent links for each node. Plot (c) suggests that the performance may peak at a certain CAW length, while the exact value may vary with datasets. Longer CAWs indicate that the corresponding networks evolve according to more complicated laws encoded in higher-order motifs.
+
+# 5.4 COMPLEXITY EVALUATION
+
+We examine how the runtime of CAW-N depends on the number of edges $| \mathcal { E } |$ used for training. We record the runtimes of CAW-N for training one epoch on the Wikipedia datasets using $M = 3 2$ , $m = 2$ with batch-size $^ { = 3 2 }$ . Specifics of the computing infrastructure are given in Appendix C.5. Fig. 6 (a) shows the accumulated runtime of executing the random walk extraction i.e. Alg.1 only. It well aligns with our theoretical analysis (Thm. A.2) that each step of the random walk extraction has constant complexity (i.e. accumulated runtime linear with $| \mathcal { E } | )$ . Plot (b) shows the entire runtime for one-epoch training, which is also linear with $| \mathcal { E } |$ . Note that $\dot { O ( | \mathcal { E } | ) }$ is the time complexity that one at least needs to pay. The study demonstrates our method is scalable to long edge streams.
+
+# 6 CONCLUSION
+
+We proposed CAW-N to inductively represent the dynamics of temporal networks. CAW-N uses CAWs to implicitly extract network motifs via temporal random walks and adopts novel set-based anonymization to establish the correlation between network motifs. The success of CAW-N points out many promising future research directions on temporal networks: Pairing CAW-N with neural network interpretation techniques (Montavon et al., 2018) may give a chance to automatically discover larger and meaningful motifs/patterns of temporal networks; CAW-N may also be generalized to predict high-order structures (e.g., triangles) that correspond to some function units of temporal networks from different domains (Benson et al., 2016; 2018; Zitnik et al., 2019).
+
+# ACKNOWLEDGMENTS
+
+We thank Jiaxuan You and Rex Ying for their helpful discussion on the idea of Causal Anonymous Walks. We also thank Rok Sosic, Camilo Andres Ruiz and Maria Brbi ˇ c for providing insightful ´ feedback on the abstract. We also gratefully acknowledge the support of DARPA under Nos. FA865018C7880 (ASED), N660011924033 (MCS); ARO under Nos. W911NF-16-1-0342 (MURI), W911NF-16-1-0171 (DURIP); NSF under Nos. OAC-1835598 (CINES), OAC-1934578 (HDR), CCF-1918940 (Expeditions), IIS-2030477 (RAPID); Stanford Data Science Initiative, Wu Tsai Neurosciences Institute, Chan Zuckerberg Biohub, Amazon, Boeing, JPMorgan Chase, Docomo, Hitachi, JD.com, KDDI, NVIDIA, Dell. J. L. is a Chan Zuckerberg Biohub investigator.
+
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+
+# A EFFICIENT LINK SAMPLING
+
+Our efficient link sampling strategy contains two subroutines – Online probability computation (Alg.2) and Iterative sampling (Alg.3). The Online probability computation subroutine Alg.2 essentially works online to assign each new incoming link $( \{ u , v \} , t )$ with a pair of probabilities $\{ p _ { u , t } , p _ { v , t } \}$ such that
+
+$$
+p _ { u , t } = \frac { \exp ( \alpha t ) } { \sum _ { ( e , t ^ { \prime } ) \in E _ { u , t } } \exp ( \alpha t ^ { \prime } ) } , \quad p _ { v , t } = \frac { \exp ( \alpha t ) } { \sum _ { ( e , t ^ { \prime } ) \in E _ { v , t } } \exp ( \alpha t ^ { \prime } ) } .
+$$
+
+These probabilities will be used later in sampling (Alg.3) and do not need to be updated any more.
+
+# Algorithm 2: Online probability computation $( G , \alpha )$
+
+1 Initialize $V  \emptyset$ , $\Omega  \emptyset$ ;   
+2 for $( \{ u , v \} , t ) \in \mathcal { E }$ do   
+3 for $w \in \{ u , v \}$ do   
+4 if $w \not \in V$ then   
+5 V ← V ∪ {w};   
+6 Pw ← exp(αt);   
+7 else   
+8 Find $P _ { w } \in \Omega$ ;   
+9 $P _ { w } \gets P _ { w } + \exp ( \alpha t ) ;$   
+10 p ← exp(αt) ; Pw   
+11 Ω ← Ω ∪ {Pw};
+
+12 Assign two probability scores: $( \{ ( u , p _ { u , t } ) , ( v , p _ { v , t } ) \} , t , ) \gets ( \{ u , v \} , t ) ;$
+
+The Iterative sampling subroutine $\mathrm { A l g } . 3$ is an efficient implementation of step 5 in Alg.1. We may first show that the sampling probability of a link $( e , t )$ in $E _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ is proportional to $\exp ( \alpha ( t - t _ { \mathrm { p } } ) )$ in Prop.A.1.
+
+# Algorithm 3: Iterative Sampling $( \mathcal { E } , \alpha , w _ { \mathsf { p } } , t _ { \mathsf { p } } )$
+
+1 Initialize $V  \emptyset$ , $\Omega  \emptyset$ ;   
+2 for $( e , t ) \in \mathcal { E } _ { w _ { p } , t _ { p } }$ with an decreasing order of $t$ do   
+3 Sample $a \sim { \mathrm { U n i f } } [ 0 , 1 ]$ ;   
+4 $p _ { w _ { \mathrm { p } } , t }$ is the score of this link related to $w _ { \mathsf { p } }$ obtained from Alg.2;   
+5 if $a < p _ { w _ { p } , t }$ then   
+6 Return $( e , t )$ ;
+
+7 Return $( \{ w _ { \mathrm { p } } , X \} , t _ { X } )$ ;
+
+Proposition A.1. Based on the probabilities (Eq.9) pre-computed by Alg.2, Alg.3 will sample a link $( e , t )$ in $\mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ with probability proportional to $\mathrm { e x p } ( \bar { \alpha } ( t - t _ { \mathrm { p } } ) )$ .
+
+Proof. To show this, we first order the timestamps of links in $\mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ between $[ t , t _ { \mathsf { p } } )$ as $t = t _ { 0 } ^ { \prime } < t _ { 1 } ^ { \prime } <$ $t _ { 2 } ^ { \prime } < \cdots < t _ { k } ^ { \prime } < t _ { \mathrm { p } }$ where there exists an link $( e ^ { \prime } , t _ { i } ^ { \prime } ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ for $t _ { i } ^ { \prime }$ . Then, the probability to sample a link $( e , t ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ satisfies
+
+$$
+\begin{array} { l } { p = p _ { w _ { \mathrm { p } } , t } \times \displaystyle \prod _ { i = 1 } ^ { k } \left( 1 - p _ { w _ { \mathrm { p } } , t _ { i } ^ { \prime } } \right) } \\ { = \displaystyle \frac { \exp ( \alpha t ) } { \sum _ { ( e ^ { \prime } , t ^ { \prime } ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t } } \exp { ( \alpha t ^ { \prime } ) } } \times \displaystyle \prod _ { i = 1 } ^ { k } \frac { \sum _ { ( e ^ { \prime } , t ^ { \prime } ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t _ { i - 1 } ^ { \prime } } } \exp ( \alpha t ^ { \prime } ) } { \sum _ { ( e ^ { \prime } , t ^ { \prime } ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t _ { i } ^ { \prime } } } \exp ( \alpha t ^ { \prime } ) } } \\ { = \displaystyle \frac { \exp ( \alpha t ) } { \sum _ { ( e ^ { \prime } , t ^ { \prime } ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t _ { 0 } } } \exp { ( \alpha t ^ { \prime } ) } } = \frac { \exp \left( \alpha \left( t - t _ { \mathrm { p } } \right) \right) } { \sum _ { ( e ^ { \prime } , t ^ { \prime } ) \in \mathcal { E } _ { w _ { \mathrm { p } } , t _ { 0 } } } \exp { \left( \alpha \left( t ^ { \prime } - t _ { \mathrm { p } } \right) \right) } } , } \end{array}
+$$
+
+which is exactly the probability that we need.
+
+We have the following Thm.A.2 with a weak assumption that evaluates the complexity of Alg.3. We assume that links come in by following a Poisson point process with intensity $\tau$ , which is a frequently used assumption to model communication networks (Lavenberg, 1983). This result indicates that if $\alpha > 0$ , for each node, we only need to record the most recent $\textstyle O ( { \frac { \tau } { \alpha } } )$ links to sample. This result is important as it means our method can do online training and inference with time and memory complexity that are not related to the total number of links.
+
+Theorem A.2. If the links that are connected to $w _ { \mathsf { p } }$ appear by following a Poisson point process with intensity $\tau$ . Then, the expected number of iterations of Alg.3 is bounded by $\begin{array} { r } { \operatorname* { m i n } \{ \frac { 2 \tau } { \alpha } + 1 , | \mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } } | \} } \end{array}$ .
+
+Proof. The number of iterations of Alg.3 is always bounded by $| \mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } } |$ . So we only need to prove that the expected number of interactions of Alg.3 is bounded by $\begin{array} { r } { \frac { 2 \tau } { \alpha } + 1 } \end{array}$ .
+
+To show this, we order the timestamps of links in $\mathcal { E } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ between $[ 0 , t _ { \mathrm { p } } )$ as $0 = t _ { 0 } < t _ { 1 } < t _ { 2 } <$ $\cdot \cdot \cdot < t _ { k } < t _ { \mathrm { p } }$ where there exists an link $( e , t _ { i } ) \in \dot { \mathcal { E } } _ { w _ { \mathrm { p } } , t _ { \mathrm { p } } }$ for $t _ { i }$ . Further define we define $Z _ { i } =$ $\exp ( \alpha ( t _ { i } - t _ { i - 1 } ) )$ for $i \in [ 1 , k ]$ .
+
+Due to the definition of Poisson process, we know that each time difference in $\{ t _ { i } - t _ { i - 1 } \} _ { 1 \leq i \leq k }$ follows i.i.d. exponential distribution with parameter $\tau$ . Therefore, $\{ Z _ { i } \} _ { 1 \le i \le k }$ are also i.i.d.. Let $\begin{array} { r } { \psi = \mathbb { E } ( Z _ { i } ^ { - 1 } ) = \frac { \tau } { \alpha + \tau } } \end{array}$ .
+
+The probability that Alg.3 runs $j$ iterations is equal to the probability that the link with timestamp $t _ { k + 1 - j } )$ gets sampled. That is
+
+$$
+\mathbb { P } ( \mathrm { i t e r } = j ) = \frac { \prod _ { i = 1 } ^ { k + 1 - j } Z _ { i } } { \sum _ { h = 1 } ^ { k } \prod _ { i = 1 } ^ { k + 1 - h } Z _ { i } } .
+$$
+
+Therefore, the expected number of iterations is
+
+$$
+\mathbb { E } ( \mathrm { i t e r } ) = \mathbb { E } \left[ \frac { \sum _ { j = 1 } ^ { k } j \prod _ { i = 1 } ^ { k + 1 - j } Z _ { i } } { \sum _ { h = 1 } ^ { k } \prod _ { i = 1 } ^ { k + 1 - h } Z _ { i } } \right] = \sum _ { j = 1 } ^ { k } j \mathbb { E } \left[ \frac { \prod _ { i = 1 } ^ { j - 1 } Z _ { i } ^ { \prime } } { \sum _ { h = 1 } ^ { k } \prod _ { i = 1 } ^ { h - 1 } Z _ { i } ^ { \prime } } \right] .
+$$
+
+where $Z _ { i } ^ { \prime } = Z _ { k + 1 - i } ^ { - 1 }$ and $\textstyle \prod _ { i = 1 } ^ { 0 } Z _ { i } ^ { \prime } = 1$ . Next we will prove that each item in right-hand-side of Eq.10 satisfies
+
+$$
+j \mathbb { E } \left[ \frac { \prod _ { i = 1 } ^ { j - 1 } Z _ { i } ^ { \prime } } { \sum _ { h = 1 } ^ { k } \prod _ { i = 1 } ^ { h - 1 } Z _ { i } ^ { \prime } } \right] \leq [ 1 + ( j - 1 ) ( 1 - \psi ) ] \psi ^ { j - 1 } .
+$$
+
+If this is true, then
+
+$$
+\begin{array} { l } { { \mathbb { E } ( \mathrm { i t e r } ) \leq \displaystyle \sum _ { j = 1 } ^ { k } [ 1 + ( j - 1 ) ( 1 - \psi ) ] \psi ^ { j - 1 } = \sum _ { j = 1 } ^ { k } \psi ^ { j - 1 } + \sum _ { j = 1 } ^ { k } ( j - 1 ) ( 1 - \psi ) \psi ^ { j - 1 } } } \\ { { \leq \displaystyle \frac { 1 } { 1 - \psi } + \frac { \psi } { 1 - \psi } = \frac { 2 \tau } { \alpha } + 1 . } } \end{array}
+$$
+
+Now, let us prove Eq.11. For $j = 1$ , Eq.11 is trivial. For $j > 1$ ,
+
+$$
+\mathbb { E } \left[ \frac { \prod _ { i = 1 } ^ { j - 1 } Z _ { i } ^ { \prime } } { \sum _ { h = 1 } ^ { k } \prod _ { i = 1 } ^ { h - 1 } Z _ { i } ^ { \prime } } \right] \leq \mathbb { E } \left[ \frac { \prod _ { i = 1 } ^ { j - 1 } Z _ { i } ^ { \prime } } { 1 + \sum _ { h = 2 } ^ { j } \prod _ { i = 1 } ^ { h - 1 } Z _ { i } ^ { \prime } } \right] \leq \frac { \prod _ { i = 1 } ^ { j - 1 } \mathbb { E } ( Z _ { i } ^ { \prime } ) } { 1 + \sum _ { h = 2 } ^ { j } \prod _ { i = 1 } ^ { h - 1 } \mathbb { E } ( Z _ { i } ^ { \prime } ) } = \frac { \psi ^ { j - 1 } } { \sum _ { i = 0 } ^ { j - 1 } \psi ^ { i } } ,
+$$
+
+where the second inequality is due to the Jensen’s inequality and the fact that for any positive $c _ { 1 } , c _ { 2 }$ , $\frac { x } { c _ { 1 } + c _ { 2 } x }$ is concave with respect to $x$ . Moreover, we also have
+
+$$
+[ 1 + ( j - 1 ) ( 1 - \psi ) ] \sum _ { i = 0 } ^ { j - 1 } \psi ^ { i } = j + \sum _ { i = 1 } ^ { j - 1 } \psi ^ { i } - ( j - 1 ) \psi ^ { j } \ge j .
+$$
+
+Combining Eq.12 and Eq.13, we prove Eq.11, which concludes the proof.
+
+# B TREE-STRUCTURED SAMPLING
+
+We may further decrease the sampling complexity by revising Alg. 1 into tree-structured sampling. Alg. 1 originally requires to sample link $M m$ times because we need to search $M$ links that connected to $w _ { 0 }$ in the first step and then sample one link for each of the $M$ nodes in each following step. A tree-structured sampling strategy may reduce this number: Specifically, we sample $k _ { i }$ links for each node in step $i$ but we make sure $\textstyle \prod _ { i = 1 } ^ { m } k _ { i } = M$ , which does not change the total number of walks. In this way, the times of link search decrease to $\textstyle \sum _ { i = 1 } ^ { m } k _ { 1 } k _ { 2 } \ldots k _ { i }$ . Suppose $M = 6 4$ , $m = 3$ , and $k _ { 1 } = 4 , k _ { 2 } = 4 , k _ { 3 } = 4$ , then the times of link search decrease to about $0 . 4 4 M m$ . Though empirical results below show that tree-structured sampling achieves slightly worse performance, it provides an opportunity to tradeoff between prediction performance and time complexity.
+
+![](images/cd5d1123535379ab95d15e0c2679d92067a8f48de186d184852ce7ba1c17a25b.jpg)  
+Figure 7: Effect of sampling of different tree structures on inductive performance.
+
+We conduct more experiment to investigate this topic with Wikipedia and UCI datasets. The setup is as follows: first, we fix CAW sampling number $\dot { M } = 6 4 = 2 ^ { 6 }$ and length $m = 2$ , so that we always have $k _ { 1 } k _ { 2 } = M = 2 ^ { 6 }$ ; next, we assign different values to $k _ { 1 }$ , so that the shape of the tree changes accordingly; controlling other hyperparameters to be the optimal combination found by grid search, we plot the corresponding inductive AUC scores of CAW-N-mean on all testing edges in Fig. 7. It is observed that while tree-structured sampling may affect the performance to some extent, its negative impact is less prominent when the first-step sampling number $k _ { 1 }$ is relatively large, and our model still achieves state-of-the-art performance compared to our baselines. That makes the tree-structured sampling a reasonable strategy that can further reduce time complexity.
+
+# C ADDITIONAL EXPERIMENTAL SETUP DETAILS
+
+# C.1 DATASET INTRODUCTION AND ACCESS
+
+We list the introduction of the six datasets as follows.
+
+• Reddit1 is a dataset of posts made by users on subredditts over a month. Its nodes are users and posts, and its links are the timestamped posting requests. Wikipedia2 is a dataset of edits over wiki pages over a month, whose nodes represent human editors and wiki pages and whose links represent timestamped edits.   
+• Social Evolution3 is a dataset recording the detailed evolving physical proximity between students in a dormitory over a year, deterimined from wireless signals of their mobile devices. Enron4 is a communication network whose links are email communication between core employees of a cooperation over several years.   
+• $\mathrm { U C I } ^ { 5 }$ is a dataset recording online posts made by university students on a forum, but is non-attributed.   
+• $\mathbf { M O O C } ^ { 6 }$ is a dataset of online courses where nodes represent students and course content units such as videos and problem sets, and links represent student’s access behavior to a particular unit.
+
+# C.2 BASELINES, IMPLEMENTATION AND TRAINING DETAILS
+
+# C.2.1 CAW-N-MEAN AND CAW-N-ATTN
+
+We first report the general training hyperparameters of our models in addition to those mentioned in the main text: on all datasets, we train both variants with mini-batch size 32 and set learning rate $=$ $1 . 0 \times 1 0 ^ { - 4 }$ ; the maximum training epoch number is 50 though in practice we observe that with early stopping we usually find the optimal epoch in fewer than 10 epochs; our early stopping strategy is that if the validation performance does not increase for more than 3 epoch then we stop and use the third previous epoch for testing; dropout layers with dropout probability $= 0 . 1$ are added to the RNN module, the MLP modules, and the self-attention pooling layer. Please refer to our code for more details.
+
+In terms of the three hyperparameters controlling CAW sampling, we discussed them in Sec 5.3. For all datasets, they are systematically tuned with grid search, whose ranges are reported in Tab.4.   
+Table 4: Hyperparameter search range of CAW sampling.   
+
+<table><tr><td>Dataset</td><td>Sampling number M</td><td>Time decay α</td><td>Walk length m</td></tr><tr><td>Reddit</td><td>32,64,128</td><td>{0.25, 0.5, 1.0,2.0,4.0}×10-5</td><td>1,2,3,4</td></tr><tr><td>Wikipedia</td><td>32,64,128</td><td>{0.25,0.5,1.0,2.0,4.0}×10-6</td><td>2,3,4</td></tr><tr><td>MOOC</td><td>32,64,128</td><td>{0.25,0.5,1.0,2.0,4.0}×10-6</td><td>2,3,4,5</td></tr><tr><td>Social Evo.</td><td>32,64,128</td><td>{0.25,0.5,1.0,2.0,4.0,8.0}x10-6</td><td>1,2,3</td></tr><tr><td>Enron</td><td>32,64,128</td><td>{0.25,0.5,1.0,2.0,4.0}×10-7</td><td>1,2, 3,4</td></tr><tr><td>UCI</td><td>32,64,128</td><td>{0.6,0.8,1.0, 1.2, 1.4}×10-5</td><td>1,2,3</td></tr></table>
+
+Apart from the hyperparameters controlling CAW sampling, hidden dimensions of CAW-N mentioned in Sec.4.3, including that of the various encodings, MLPs, RNN, and attention projection matrices, are relatively less tuned. We select them based on two principles: 1) when node & link attributes are available, dimension of all these modules are set to have the same dimensions as baselines; 2) when node & link attributes are unavailable, the dimensions are picked from 32, 64, 128, whichever leads to a better performance.
+
+# C.2.2 BASELINES
+
+We list the introduction of the six baselines as follows:
+
+• DynAERNN (Goyal et al., 2020) uses a fully connected encoder to acquire network representations, passes them into LSTM and uses a fully connected network to decode the future network structures. JODIE (Kumar et al., 2019) applies RNNs to estimate the future embedding of nodes. The model was proposed for bipartite graphs while we properly modify it for standard graphs if the input graphs are non-bipartite. DyRep (Trivedi et al., 2019) also uses RNNs to learn node embedding while its loss function is built upon temporal point process.   
+• VGRNN (Hajiramezanali et al., 2019) generalizes the variational GAE (Kipf & Welling (2016)) to temporal graphs, which makes the prior depend on the historical dynamics and captures those dynamics with RNNs. EvolveGCN (Pareja et al., 2020) uses a RNN to estimate the GCN parameters for the future snapshots.   
+• TGAT (Xu et al., 2020) leverages GAT to extract node representations where the nodes’ neighbors are sampled from the history and encodes temporal information via Eq.7.
+
+We introduce how we tune these baselines as follows.
+
+DynAERNN. The model with code provided here is adapted into our evaluation pipeline. We follow most of the settings in the code. We tune the embedding size in {32, 64} and lookback in {2, 3, 5} to report the best performance.
+
+JODIE. The model with code provided here is adapted into our evaluation pipeline. JODIE calculates the $L _ { 2 }$ distances between the predicted item embedding to other items and uses the rankings to evaluate their performance. Here, we consider the negative distances as the prediction score. Based on the prediction score, we calculate mAP and AUC. We split the data according to the setting in section 5.1. The model is trained for 50 epoches. The dimensions of the dynamic embedding is searched in{64, 128} and the best performance is reported.
+
+DyRep. The model with code provided here is adapted into our evaluation pipeline. We follow most of the settings in the paper. That is, we set the number of samples for survival to 5, gradient clipping to 100. And we tune the hidden unit size and embedding size in {32, 64} to report the best performance. The model uses likelihood based on point process to predict links and therefore we use these likelihood scores to compute AUC and AP.
+
+VGRNN. The model with code provided here is adapted into our evaluation pipeline. We use several of its default settings: one-hot node features as input when node attributes are unavailable as suggested by the original paper (Hajiramezanali et al., 2019), one layer of GRU network as the history tracking backbone, a learning rate of 1e-2, and training for 1000 epochs. Its hidden dimension is searched in {32, 64} and the best performance is reported.
+
+EvolveGCN. The model with code provided here is adapted into our evaluation pipeline. We utilize EvolveGCN-O version since it can capture more graph structural information. For most hyperparameters, we follow the default values. According to our setting, we sample an equal amount of negative links, which means we set negative_mult_training and negative_mult_test to 1. One central hyperparameter needs to be further tuned is number of previous snapshots used for training and testing. We search its optimal value in {3, 4, 6, 8, 10} when tuning the model for most datasets. Since Enron only contains 11 snapshots, we search its optimal value in {3,4,5,6}.
+
+TGAT. The model with code provided here is adapted into our evaluation pipeline. We use several of their default settings. That is, we use product attention, set the number of attention heads to 2, set the number of graph attention layers to 2, and use 100 as their default hidden dimension. One central hyperparameter that needs to be further tuned is the degree of their neighbor sampling. We search its optimal value in {10, 20, 30} when tuning the model.
+
+C.3 EVALUATION OF SNAPSHOT-BASED BASELINES   
+Table 5: Snapshot split for evaluating snapshot-based baselines.   
+
+<table><tr><td></td><td>Reddit</td><td>Wikipedia</td><td>MOOC</td><td>Social Evo.</td><td>Enron</td><td>UCI</td></tr><tr><td>total snapshots</td><td>174</td><td>20</td><td>20</td><td>27</td><td>11</td><td>88</td></tr><tr><td>exact split</td><td>122 /26 /26</td><td>14/3/3</td><td>14/3/3</td><td>19/4/4</td><td>7/2/2</td><td>62/13/13</td></tr><tr><td>referenced baseline</td><td>EvolveGCN</td><td>-</td><td>-</td><td>VGRNN</td><td>VGRNN</td><td>EvolveGCN</td></tr></table>
+
+We make the following decisions to evaluate snapshot-based baselines in a fair manner, so that their performances are comparable to those derived from the stream-based evaluation procedure. The first step we do is to evenly split the whole dataset chronologically into a number of snapshots. We determine the exact number of snapshots by referring the three snapshot-based baselines we use. For Wikipedia and MOOC dataset which are not used by any snapshot-based baseline, we split them into a total of 20 snapshots. Next, we need to determine the proportions of these snapshots assigned each to training, validation, and testing set. In doing this, our principle is that the proportions of these three sets should be close to 70:15:15 as much as possible, since that ratio is what we use for evaluating stream-based baselines and our proposed method. These decisions lead to our final splitting scheme summarized in Tab. 5.
+
+Extra care should also be taken when testing snapshot-based methods. For a queried link in a snapshot, usually snapshot-based methods only make a binary prediction whether or not that link may exist at any time in that snapshot. They do not, however, take care of the case that the link may appear multiple times at different time points within that snapshot’s time range. This lead to a different evaluation scheme than stream-based methods, which do consider the multiplicity of links. Therefore, when testing snapshot-based methods, if a link appear in a certain snapshot for multiple times, we record the model’s prediction the same number of times before computing its performance metrics.
+
+# C.4 CHOICE OF EVALUATION METRIC
+
+When considering link prediction as a binary classification problem, the existing literature usually choose metrics from the following: Area Under the ROC Curve (AUC), Average Precision (AP), and Accuracy (ACC). The reason we do not use ACC is that a proper confidence threshold of decision is ill-defined in literature, which leads to unfair comparison across different works.
+
+# C.5 COMPUTING INFRASTRUCTURE
+
+All the experiments were carried out on a Ubuntu 16.04 server with Xeon Gold 6148 2.4 GHz 40-core CPU, Nvidia 2080 Ti RTX 11GB GPU, and 768 GB memory.
+
+# D ADDITIONAL EXPERIMENTAL RESULTS
+
+# D.1 PERFORMANCE IN AVERAGE PRECISION
+
+<table><tr><td>Task</td><td>Methods</td><td>Reddit</td><td>Wikipedia</td><td>MOOC</td><td>Social Evo.</td><td>Enron</td><td>UCI</td></tr><tr><td rowspan="7">Trngeeier</td><td>DynAERNN JODIE maas mna</td><td>58.63 ±5.42 80.03 ± 0.13</td><td>54.94±2.29 76.90 ± 0.49</td><td>59.84±1.26 82.27 ± 0.46†</td><td>54.76±1.33 87.96 ± 0.12†</td><td>54.89±3.79 79.80 ± 1.48†</td><td>51.59 ± 3.92 71.64 ± 0.62</td></tr><tr><td>DyRep VGRNN</td><td>61.28 ± 1.89</td><td>57.57 ± 2.56</td><td>62.29 ± 2.09</td><td>75.42 ± 0.32</td><td>69.97 ± 0.92</td><td>63.08 ± 7.40</td></tr><tr><td></td><td>60.64 ± 0.68</td><td>52.55 ± 0.82</td><td>65.44 ± 0.82</td><td>67.83 ± 0.53</td><td>67.93 ± 0.88</td><td>67.50 ± 0.92</td></tr><tr><td>EvolveGCN</td><td>62.99 ± 0.17</td><td>55.64 ±1.03</td><td>52.28 ±1.80</td><td>52.26 ± 1.16</td><td>47.36 ± 1.24</td><td>80.98 ± 1.09†</td></tr><tr><td>TGAT</td><td>95.17 ± 0.91†</td><td>93.18 ± 0.73†</td><td>72.91 ± 0.92</td><td>52.17± 1.94</td><td>63.83 ± 3.70</td><td>75.27 ± 2.34</td></tr><tr><td>CAW-N-mean</td><td>98.05 ± 0.87*</td><td>96.01±0.25*</td><td>90.36 ±0.80*</td><td>92.16 ± 1.03*</td><td>93.93 ± 0.66*</td><td>99.63 ± 0.34*</td></tr><tr><td>CAW-N-attn DynAERNN</td><td>98.08 ±0.66*</td><td>97.86 ±0.63*</td><td>90.35 ±0.81*</td><td>93.29 ±1.90*</td><td>92.73±0.76*</td><td>100.00 ±0.00*</td></tr><tr><td rowspan="8">PO ‘S&#x27;A Mou</td><td></td><td>66.59 ± 2.90</td><td>63.76±2.82</td><td>82.02 ±1.59</td><td>52.54±0.22</td><td>55.50± 2.07</td><td>57.29±2.52</td></tr><tr><td>JODIE</td><td>83.15 ± 0.03</td><td>80.54 ± 0.06</td><td>87.95 ± 0.08</td><td>91.40 ± 0.04†</td><td>89.57± 0.30</td><td>76.34 ± 0.17</td></tr><tr><td>DyRep</td><td>66.73 ± 1.99</td><td>76.89 ± 0.31</td><td>88.25 ± 1.20†</td><td>89.41 ± 0.29</td><td>95.97 ± 0.28†</td><td>93.60 ± 1.47†</td></tr><tr><td>VGRNN</td><td>52.84 ±0.66</td><td>60.99 ± 0.55</td><td>62.95 ± 0.58</td><td>69.20 ± 0.52</td><td>67.93 ± 0.88</td><td>67.50 ± 0.92</td></tr><tr><td>EvolveGCN</td><td>66.29 ± 0.52</td><td>53.82 ± 1.64</td><td>51.53 ± 0.92</td><td>52.01 ± 0.67</td><td>46.56 ± 1.89</td><td>76.30 ± 0.33</td></tr><tr><td>TGAT</td><td>97.09 ± 0.18†</td><td>95.17 ± 0.15†</td><td>71.77 ± 0.23</td><td>52.48 ± 0.52</td><td>59.70 ± 1.49</td><td></td></tr><tr><td>CAW-N-mean</td><td>98.89 ±0.04*</td><td>99.04 ± 0.04*</td><td>90.99 ± 0.96*</td><td>93.71± 0.75*</td><td>92.93 ± 0.94</td><td>75.01 ± 0.72 98.86 ± 0.95*</td></tr><tr><td>CAW-N-attn</td><td>99.90 ± 0.05*</td><td>99.55 ± 0.30*</td><td>91.24 ± 0.93*</td><td>94.17 ± 0.86*</td><td>92.38 ± 1.36</td><td>98.53±0.64*</td></tr><tr><td rowspan="11">Trlrrseeera</td><td>DynAERNN</td><td>85.58 ± 2.12</td><td>76.58 ± 1.41</td><td>89.29± 0.49</td><td>66.58 ± 1.84</td><td>60.90±2.70</td><td>84.95± 2.13</td></tr><tr><td>JODIE</td><td>91.14 ± 0.01</td><td>91.39 ± 0.04</td><td>91.19 ± 0.03†</td><td>89.22 ± 0.01</td><td>91.94 ± 0.01</td><td>80.27 ± 0.08</td></tr><tr><td>DyRep</td><td>67.54 ± 2.02</td><td>77.36 ± 0.25</td><td>90.49 ± 0.03</td><td>94.48 ± 0.01†</td><td>97.14 ± 0.07†</td><td></td></tr><tr><td>VGRNN</td><td>50.87 ± 0.81</td><td>67.66 ± 0.89</td><td>83.70 ± 0.56</td><td>78.66 ± 0.67</td><td>94.02 ± 0.52</td><td>95.29 ± 0.13†</td></tr><tr><td>EvolveGCN</td><td>54.49 ± 0.73</td><td>55.84 ± 0.37</td><td>51.80 ± 0.46</td><td>56.90 ± 0.54</td><td>69.72 ± 0.49</td><td>82.23 ± 0.56</td></tr><tr><td>TGAT</td><td>98.38 ± 0.01†</td><td>96.65 ± 0.06†</td><td>69.75 ± 0.23</td><td>57.37 ± 1.18</td><td>57.37 ± 0.18</td><td>81.63 ± 0.23</td></tr><tr><td>CAW-N-mean</td><td>99.96 ± 0.01*</td><td>99.91±0.03*</td><td>92.05 ± 0.88</td><td>95.90 ± 0.09*</td><td></td><td>60.25 ± 0.31</td></tr><tr><td>CAW-N-attn</td><td>99.99 ± 0.01*</td><td></td><td>92.23±0.76*</td><td></td><td>94.93 ± 0.39</td><td>95.89 ± 0.87</td></tr><tr><td></td><td></td><td>99.89±0.03*</td><td></td><td>96.37 ± 0.09*</td><td>96.13 ± 0.37</td><td>98.86±0.38*</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 6: Performance in Average Precision (AP) (mean in percentage $\pm 9 5 \%$ confidence level.) $\dagger$ highlights the best baselines. ∗, bold font, bold font∗ respectively highlights the case where our models’ performance exceeds the best baseline on average, by $7 0 \%$ confidence, by $9 5 \%$ confidence.
+
+# D.2 MORE DISCUSSION ON STREAM-BASED VS. SNAPSHOT-BASED METHODS
+
+Stream-based methods usually treat each temporal link as an individual training instance. In contrast, snapshot-based methods stack all the temporal links within a time slice into one static graph snapshot and do not further distinguish temporal order of links within that snapshot. In Tab. 2 and 6 we saw that stream-based methods generally exhibit better performance than snapshot-based methods. An important reason is that stream-based methods are able to access the few most recent interactions previous to the target link to predict. This makes them especially advantageous when used to model many common temporal networks whose dynamics are governed by some short-term laws. In contrast, snapshot-based methods are less able to access such immediate history, because they make prediction on all links within one future snapshot all at once. In principle they could alleviate this problem by making very fine-grained snapshots so that less immediate history is missed. However, this is not practical with real-word large temporal graphs whose temporal links come in millions, which leads to snapshot sequences of extreme length intractable to their recurrent neural structure. This issue was also observed by the recent work to model social interacting behaviors (Wang et al., 2020), although temporal convolutional networks may alleviate this issue to some extent. That said, snapshot-based methods usually has the advantage that they usually consume less memory and computation time. Stream-based methods on the other hand need to manage how they sample history very carefully to balance the efficiency and effectiveness. Our proposed algorithm on CAW sampling comes into the place in light of this to solve the problem.
+
+![](images/e3d29759c34f7a2389e6c5903fe4ace1e3257b5998f42cc15a5fc1b8e5c3477e.jpg)  
+Figure 8: Visualizing most discriminatory CAWs, and their occurrence ratios with positive / negative samples.
+
+# E VISUALIZING CAWS AND AWS
+
+Here, we introduce one way to visualize and interpret CAWs. Our interpretation can also illustrate the importance to capture the correlation between walks to represent the dynamics of temporal networks, where the set-based anonymization of CAWs can work while AWs will fail. The basic idea of our intrepretation is to identify different shapes of CAWs via their patterns encoded in $I _ { C A W }$ , and compare their contributions to the link-prediction confidence. The idea will be also used to interpret AWs so that we can compare CAWs with AWs.
+
+First, we define the shapes of walks based on $I _ { C A W }$ . Recall from Eq. 3 that $g ( w , S _ { u } )$ encodes the number of times node $w$ that appears in different walk positions w.r.t source node $u$ . This encoding induces a temporal shortest-path distance $d _ { u w }$ between node $w$ and $u$ : $d _ { u w } \triangleq \operatorname* { m i n } \{ i | g ( w , S _ { u } ) [ i ] >$ $0 \}$ . Note that in temporal networks, there is not a canonical way to define shortest-path distance between two nodes as there is no static structures. So our definition $d _ { u w }$ can be viewed the shortestpath distance between u and w over the subgraph that consists of walks in $S _ { u }$ . Based on the way to define $( d _ { u w } , d _ { v w } )$ , we introduce the mapping from $I _ { C A W }$ of the node $w$ to a coordinate of this node in the subgraph that consists of walks in $S _ { u } \cap S _ { v }$ : $g ( w , S _ { u } ) , g ( w , S _ { v } ) ) \to \mathrm { c o o r } ( w ; u , v ) = ( d _ { u w } , d _ { v u }$ ). This cooredinate can be viewed as a relative coordinate of node $w$ w.r.t. the source nodes $u$ , $v$ . Each walk $W \in S _ { u } \cup S _ { v }$ can then represented as a sequence of such coordinates by mapping each node’s $I _ { C A W }$ to a coordinate. The obtained sequence can be viewed as a shape of $W$ and we denote the obtained shape as $s _ { C A W } ( W )$ . For instance, in the toy example shown by Fig. 2 right, the first CAW in $S _ { u }$ , $u  b  a  c$ is mapped to a new coordinate sequence $( 0 , 2 ) \overset { \cdot } {  } ( \bar { 1 } , 2 ) \overset { - } {  } ( 2 , \infty )  ( 3 , \infty )$ . The $\infty$ marks the setting that node $a , c$ do not appear in $S _ { v }$ .
+
+Next, we score the contributions of CAWs with different shapes. We use CAW-N-Mean as enc $( S _ { u } \cup S _ { v } )$ is simply mean over the encodings of sampled CAWs and further use linear projection $\beta ^ { T } \mathrm { e n c } ( S _ { u } \cup S _ { v } )$ to compute the final scalar logit for prediction. As the two operations mean and projection are commutative, the above setting allows each CAW $\hat { W _ { i } }$ contributing a scalar score $l o g i t ( \hat { W } _ { i } ) = \beta ^ { T } \mathrm { e n c } ( \hat { W } _ { i } )$ to the final logit.
+
+Fig. 8 lists the 3 highest-scored and 3 lowest-scored shapes of CAW, which are extracted from the Wikipedia dataset with $M = 3 2$ and $m = 3$ . A law of general motif closure can be observed from the highest-scored CAWs: two nodes that commonly appear in some types of motif are more inclined to have a link in between. For example, the highest-scored shape of CAW, $( 0 , \infty )  ( 1 , 2 )  ( 2 , 3 ) $ $( 1 , 2 )$ , implies that the nodes except the first in this CAW appear in the sampled common 3-hop neighborhood around the two nodes between which the link is to be predicted. Therefore, CAW-N essentially adaptively samples a temporal motif closure pattern that is very informative to predict this link. CAW-N does not explicitly enumerating or counting these motif patterns. In contrast, when CAWs do not bridge the two nodes, as shown in top-2 lowest-scored CAWs, very unlikely there will exist a link. Fig. 8 also displays each of the 6 CAW’s occurrence ratio with positive and negative links. The difference within each pair of ratios is an indicator of the corresponding CAW’s discriminatory power. We also see that the discriminatory power of CAWs is very strong: highest-scored CAWs almost never occur with negative links, and lowest-scored CAWs also seldom occur with positive links.
+
+![](images/af8de1ab6b3f16fbe94a970a87c2cb8b1159fd24bdc8c1ca711ffa7c020a3c6c.jpg)  
+Figure 9: Visualizing all AWs, and their occurrence ratios with positive / negative samples.
+
+We further apply the similar procedure to analyze the AWs introduced in Sec. 4.1 and the model Ab.5 of Tab. 3 used for ablation study. Note that AW cannot establish the correlation between walks, each single AW itself, say $W = ( v _ { 0 } , v _ { 1 } , . . . , v _ { m } )$ , decides its own shape $s _ { A W } ( W )$ . We directly set $s _ { A W } ( W ) = I _ { A W } ( v _ { 0 } ; W ) \to I _ { A W } ( v _ { 2 } ; W ) \to \cdots \to I _ { A W } ( v _ { m } ; W )$ with $I _ { A W } ( w ; W )$ defined in Eq. 2. As the Wikipedia dataset is a bipartite graph, there are in total four different shapes when $m = 3$ as listed align with the $\mathbf { X }$ -axis of Fig. 9. For illustration, we explain one shape of AW as an example, say $0  1  2  1$ : The corresponding walks have the second and the fourth nodes correspond to the same node, which the first, second and third nodes are different. As shown in Fig. 9, we can see that AW’s occurrence with positive versus negative links are highly mixed-up, compared to CAW’s. That suggests that AWs possess significantly less discriminatory power than CAWs. The main reason is that AWs do not have set-based anonymization, so they cannot capture the correlation between walks/motifs but CAWs can do that. This observation further gives a reason on why the model Ab.5 of Tab. 3 only achieves some performance on-par with the model Ab.2 where we totally remove the anonymization procedure: the anonymization adopted by AWs loses too much information of the structure and cannot benefit the prediction much. However, the original CAW-N well captures such information via the set-based anonymization.

parse/train/rkesVkHtDr/rkesVkHtDr.md

@@ -0,0 +1,470 @@
+# META-LEARNING RUNGE-KUTTA
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Initial value problems, i.e. differential equations with specific, initial conditions, represent a classic problem within the field of ordinary differential equations (ODEs). While the simplest types of ODEs may have closed-form solutions, most interesting cases typically rely on iterative schemes for numerical integration such as the family of Runge-Kutta methods. They are, however, sensitive to the strategy the step size is adapted during integration, which has to be chosen by the experimenter. Here, we show how the design of a step size controller can be cast as a learning problem, allowing deep networks to learn to exploit structure in the initial value problem at hand in an automatic way. The key ingredients for the resulting Meta-Learning Runge-Kutta (MLRK) are the development of a good performance measure and the identification of suitable input features. Traditional approaches suggest the local error estimates as input to the controller. However, by studying the characteristics of the local error function we show that including the partial derivatives of the initial value problem is favorable. Our experiments demonstrate considerable benefits over traditional approaches. In particular, MLRK is able to mitigate sudden spikes in the local error function by a faster adaptation of the step size. More importantly, the additional information in the form of partial derivatives and function values leads to a substantial improvement in performance. The source code can be found at https://www.dropbox.com/sh/ rkctdfhkosywnnx/AABKadysCR8-aHW_0kb6vCtSa?dl $= 0$
+
+# 1 INTRODUCTION
+
+Differential equations in their general form cover an extremely wide variety of disciplines: While many applications are rather intuitive, as for instance simple Newtonian physics and engineering, other more exotic use cases include the governing of price evolution in economics (Black & Scholes, 1973), the study of rates in chemical reactions (Scholz & Scholz, 2014), and the modeling of population growths in biology (Lotka, 1925; Volterra, 1926). In medicine, differential equations may be used to model cancer growth (Ilea et al., 2013), diabetes and the glucose metabolism (Esna-Ashari et al., 2017) as well as for pharmaceutical drug design (Deuflhard, 2000). Recently, differential equations have also been used as a way to design neural networks (Chen et al., 2018). Unfortunately, finding an analytical solution in closed form is in many cases very difficult, if not impossible. Therefore, a variety of numerical integration methods have been developed to obtain accurate, but approximate solutions. Arguably, the most prominent ones are Runge-Kutta methods, a family of integration methods for initial value problems. However, setting up Runge-Kutta involves several design choices, one of which is the step size controller. Using an adaptive step size strategy instead of a constant step size can often increase efficiency by several orders of magnitude, $c . f$ . (Söderlind, 2006). Their performance is hampered by the fact that they only make use of hand-designed features.
+
+Contribution. We show how to cast the design of a step size controller for Runge-Kutta as a learning problem, $c . f$ . Fig. 1. The key ingredients of the resulting Meta-Learning Runge-Kutta (MLRK) are the identification of a good performance measure and appropriate inputs.
+
+Related Work. Various approaches to control the step sizes in Runge-Kutta methods have been proposed. While some rely on signal processing principles where the goal is to produce a smooth step size sequence in conjunction with acceptable local errors, others are based on the assumption that step sizes should be adapted to a prescribed function of the solution, e.g. to preserve structure in geometric integration (Söderlind, 2006). In this work we will focus on control theoretic approaches which aim to keep the local error associated with a single step close to a tolerance parameter.
+
+![](images/ed560a630f9d0f17b08690c45f047927431bcaa3feb93ba76c4a0095a9d6f2fc.jpg)  
+Figure 1: Meta-Learning Runge-Kutta: The model represented by the blue block determines the step size adjustment $\log r _ { n - 1 }$ using a LSTM and a linear layer, $c . f$ . Eq. (10). The Runge-Kutta update then determines the new step size $\log h _ { n } = \log h _ { n - 1 } + \log r _ { n - 1 }$ . The new step size is used to perform the next Runge-Kutta step which computes the approximation $y _ { n + 1 }$ at time step $t _ { n + 1 }$ and an error estimate $\mathrm { e r r } _ { n + 1 }$ according to (12). Then the next input $x _ { n + 1 }$ for the model is computed.
+
+Specifically, the design of the step size control algorithm is learned within MLRK. Indeed, casting algorithm design as a learning problem is not new. Andrychowicz et al. (2016) learned a gradientbased optimizer for nonlinear unconstrained optimization problems. Wichrowska et al. (2017) extended this work by introducing a hierarchical RNN architecture which improves the generalization ability of the optimizer. Schramowski et al. (2018) demonstrated the benefit of a learned projectionfree convex optimization algorithm, which relies on conditional gradients. Finally, Chen et al. (2017) cast the design of gradient-free black-bock optimization as a learning problem.
+
+Furthermore, other approaches to solve differential equations using neural networks have been proposed, too. Lagaris et al. (1998) suggested to use a feed-forward neural network to approximate the solution of an ordinary or partial differential equation with initial or boundary conditions. Inspired by the Galerkin method, Sirignano & Spiliopoulos (2018) used a deep neural network to directly approximate the solution of a high-dimensional partial differential equation. E & Yu (2018) proposed the Deep Ritz method as a means to solve variational problems that arise from partial differential equations. Han et al. (2018) cast the problem of solving semilinear parabolic partial differential equations as a learning problem by using the reformulation of these differential equations as backward stochastic differential equations. In all of the above approaches, a neural network is used to help approximate the solution of the differential equation. The learning process corresponds to the numerical computation of a solution of the differential equation. In contrast, MLRK learns to improve the numerical integration process itself.
+
+We proceed as follows. As the first step, i.e. identifying a good performance measure, we consider the general objective of step size control as well as the simplified and more practical objective that many controllers are based on. Then, we identify useful inputs by analyzing existing step size control algorithms. Furthermore, we show how local information about the ODE can be used as additional favorable inputs. Before concluding, we demonstrate empirically how our proposed controller can be used to improve step size control and investigate the benefit of different inputs.
+
+# 2 INITIAL VALUE PROBLEMS AND RUNGE-KUTTA METHODS
+
+We give a brief introduction to initial value problems and Runge-Kutta methods, a numerical method for solving these problems. Furthermore, we will recap standard step size controllers and point out their underlying assumptions.
+
+Basics of Initial Value Problems and Runge-Kutta Methods. An ordinary differential equation describes a system that depends on one variable, often referred to as time. An initial value problem additionally provides initial values: $y ^ { \prime } = g ( t , y ) , \quad y ( t _ { 0 } ) = y _ { 0 }$ , with a function $g : [ a , b ] \times \mathbb { R } ^ { m } $ $\mathbb { R } ^ { m }$ , $m \in \mathbb { N }$ . As closed-form solutions can be very hard to find, numerical methods such as RungeKutta methods have been developed. Their mathematical underpinning is quite evolved (Butcher, 2008; Hairer et al., 2000; Hairer & Wanner, 2002). A brief recap can be found in the appendix.
+
+For iterative solvers, the local truncation error—the error committed in a single step using step size $h$ —constitutes an important tool to control the step size as it allows to access the fitness of a step size.
+
+Definition 2.1. The local truncation error in step $n + 1$ is defined as
+
+$$
+\mathrm { e r r } _ { n + 1 } ( h ) = \| y ( t _ { n } + h ) - y _ { n + 1 } \| = C ( t _ { n } ) h ^ { p + 1 } + \mathcal { O } ( h ^ { p + 2 } ) ,
+$$
+
+where $y _ { n + 1 } = \Phi ( t _ { n } , y _ { n } , h )$ is a single Runge-Kutta step of a $p$ -th order method using step size $h$ . We call $C ( t _ { n } )$ the principal error term.
+
+We will also denote the local truncation error as local error and refer to the appendix for simple ways to estimate it. We denote the error estimates as $\operatorname { e r r } _ { n }$ .
+
+The Objective of Step Size Control. As numerical integration is used to approximate a solution, high accuracy is a desired property. However, in practice, computational resources might be limited and efficient use of these resources can be crucial in some applications. Hence, an efficient step size controller should maximize the accuracy of the approximate solution while minimizing the computational cost. These are competing goals and require a trade-off, which is achieved by considering the Lagrangian of the error $E ( H )$ and the work cost $W ( H )$ produced by a sequence of step sizes $H \ = \ ( h _ { n } ) _ { n = 0 } ^ { N - 1 }$ , $E ( H ) + \lambda \cdot W ( H )$ . Butcher (2008) uses two integrals describing the error $E ( H )$ and the work cost $W ( H )$ . Under some assumptions one can show that optimal step sizes w.r.t. this objective causes local errors equal to the Lagrange multiplier $\lambda$ , which is often referred to as the tolerance parameter. Namely, we need to assume that the global error is equal to the sum of local errors and all step sizes are small enough so that the loss can be approximated by the integrals suggested by Butcher. However, these are quite strong assumptions. The optimal step size depends on the tolerance parameter, and hence, might be too large. Furthermore, the assumption that the global error is equal to the sum of the local errors is not true in general.
+
+Classical Step Size Control. Most common step size control mechanisms aim to keep the local errors equal to a tolerance parameter so that a standard controller is based on the following idea: For a RungeKutta method of order $p$ , the local error for step size $h$ at time $t$ is approximately err $\approx C ( t ) h ^ { p + 1 }$ , and, accordingly, the desired optimal step size $h _ { \mathrm { o p t } }$ is given by $h _ { \mathrm { o p t } } \approx h ( \mathrm { t o l / e r r } ) ^ { \frac { 1 } { p + 1 } }$ . This formula is then used to adapt the step size as
+
+$$
+h _ { n + 1 } = r _ { n } h _ { n } , \qquad r _ { n }  &  = \operatorname* { m a x } \left( \alpha , \operatorname* { m i n } \left( \beta , \gamma \left( { \frac { \mathrm { t o l } } { \mathrm { e r r } _ { n + 1 } } } \right) ^ { \frac { 1 } { p + 1 } } \right) \right) ,
+$$
+
+where a safety factor $0 < \gamma < 1$ and a minimal and maximal factor $\alpha , \beta \in \mathbb { R } _ { \ge 0 }$ , $\alpha \leq \beta$ are included to avoid excessive step rejection and ensure a smooth step size control. Söderlind (2002) pointed out that the underlying assumptions of Eq. (2) are rather strong. It assumes slow variations in $C$ and requires $h$ to be sufficiently small so that it exhibits its theoretical asymptotic behavior. Both are not true in general.
+
+Control Theory on Step Size Control in Runge-Kutta Methods. Gustafsson et al. (1988) discussed step size control in the context of a proportional-integral-derivative (PID) controller. The classical step size control mechanism in Eq. (2) can be regarded as an I-controller. They then demonstrated the oscillatory behavior of this controller when applied to certain problems. They report the poor stabilizing capability of an I-controller as the origin of these oscillations, which are further accentuated by a large integration gain. To overcome this, they Gustafsson et al. (1988) suggested a PI-controller, which in addition to the integral term used in a standard step size control, also includes a proportional term and can be expressed as
+
+$$
+h _ { n + 1 } = r _ { n } h _ { n } , \qquad r _ { n }  &  = \operatorname* { m a x } \left( \alpha , \operatorname* { m i n } \left( \beta , \gamma \left( { \frac { { \mathrm { t o l ~ } } } { { \mathrm { e r r } } _ { n + 1 } } } \right) ^ { n _ { 1 } } \left( { \frac { { \mathrm { t o l } } } { { \mathrm { e r r } } _ { n } } } \right) ^ { n _ { 2 } } \right) \right) .
+$$
+
+A Predictive Controller. Both the classical step size controller, see Eq. (2), and the PI-controller in Eq. (3) rely on the assumption that the local truncation error can be described as a function of $h$ that remains independent of $t$ . This is certainly not true for all cases. Therefore, Gustafsson (1994), e.g., discussed a controller based on a prediction of the principal error term. A simple model that assumes a constant linear trend in $\log C ( t )$ was suggested. Implementing the predicted principal error leads to
+
+$$
+h _ { n + 1 } = { \frac { h _ { n } } { h _ { n - 1 } } } \left( { \frac { \mathrm { t o l } } { \mathrm { e r r } _ { n + 1 } } } \right) ^ { n _ { 1 } } \left( { \frac { \mathrm { e r r } _ { n } } { \mathrm { e r r } _ { n + 1 } } } \right) ^ { n _ { 2 } } h _ { n } .
+$$
+
+# 3 META-LEARNING RUNGE-KUTTA: NEURAL STEP-SIZE CONTROLLER
+
+Step size controllers for Runge-Kutta are typically still designed by hand. Formulas such as Eq. (2), (3) and (4) require parameter fine-tuning. In this work, we take a different tack and instead suggest to replace these hand-designed update rules with a learned controller, hereinafter referred to as optimizer. The key steps to cast the design of a controller as a learning problem is to determine a good performance measure and appropriate inputs.
+
+Performance measure. As the first step we need to identify an appropriate performance measure. As discussed, a trade-off between numerical accuracy and computational cost of the solution must be made, e.g. by considering the Lagrangian of error and work cost. The work cost is generally correlated with the number of integration steps. It is crucial to take the length of the integration interval $t _ { n } - t _ { 0 }$ into account as well, which leads to the following performance measure:
+
+$$
+l ^ { \mathrm { L a g r a n g e } } ( t _ { n } , y _ { n } , \mathrm { e r r } _ { n } ) = { \frac { \| y ( t _ { n } ) - y _ { n } \| + \mathrm { t o l } \cdot n } { t _ { n } - t _ { 0 } } } .
+$$
+
+In contrast, if we follow the concept of most step size controllers, an optimal step size is achieved if the local error is equal to the tolerance, which suggests the performance measure
+
+$$
+l ^ { \mathrm { l o c a l } } ( t _ { n } , y _ { n } , \mathbf { e r r } _ { n } ) = \left\| \mathbf { e r r } _ { n } - \mathbf { t o l } \right\| .
+$$
+
+Both types of loss functions offer various strengths and weaknesses over one another.
+
+The loss Eq. (5) incorporates the essential quantities that an efficient algorithm should minimize. However, the true global error of numerical integration is generally unknown and can only be approximated. In general this can be computationally demanding and as a result make the training process very expensive. Furthermore, when the global error is employed, optimal step sizes may depend on the integration interval as well, e.g. the choice of step size in the first few steps may have different effects on the global error at different points in the integration interval.
+
+Classic step size controllers are typically based on local errors alone without regards to global strategies. The objective of these controllers is reflected by the loss Eq. (6). Although this objective minimizes the Lagrangian of the global error and the number of steps only under strong assumptions it does qualify as an appropriate practical objective for step size controllers.The advantage here is that estimates of the local error associated with a step size are available and no additional computation to determine the loss of a step size is required. This allows efficient training even when the analytical solution of an initial value problem is unknown.
+
+Input Features. Existing step size controllers rely on error estimates to update the step size. Specifically, $\frac { \mathrm { t o l } } { \mathrm { e r r } _ { n } }$ is important, as evident in Eq. (2), (3) and (4). We intuitively expect a data driven approach to be able to utilize these features. This gives rise to the first set of input features,
+
+$$
+\psi \left( \mathrm { e r r } _ { n + 1 } , \cdot \right) = \log \left( \frac { \mathrm { t o l } } { \mathrm { e r r } _ { n + 1 } } \right) .
+$$
+
+Here, $\psi \left( \mathrm { e r r } _ { n + 1 } , \cdot \right)$ denotes the input for the optimizer with optional variables. Next, we want to investigate additional input features that may be beneficial. The principal error term as well as higher order terms of the local errors can be expressed with elementary differentials of the ODE, $c . f$ . (Hairer et al., 2000, Section II.3). These elementary differentials are formed of function values and partial derivatives of $g$ and allow a Taylor approximation of the local error function $\operatorname { e r r } _ { n + 1 } ( h ) =$ $\phi ( t _ { n } , h ) h ^ { k + 1 } + \mathcal { O } ( h ^ { k + 2 } )$ , where $\phi ( t _ { n } , h )$ is a polynomial in $h$ with coefficients formed from partial derivatives of $g$ up to order $k > p$ . The roots of the polynomial $\phi ( t _ { n } , h ) h ^ { k + 1 } \ - $ tol approximate the optimal step size. Moreover, it is reasonable to assume that the local error takes values greater tol for some $h > 0$ , implying the existence of real roots. Since the complex roots of polynomials are known to depend continuously on the coefficients of the polynomial, $c . f$ . (Rahman & Schmeisser, 2002, Theorem 1.3.1), it is clear that the optimal step size, a real root of $p ( h )$ , depends continuously on the partial derivatives and function values of $g$ as well as the tolerance tol. From the Universal Approximation Theorem (Hornik, 1991), it then follows that the optimal step size function can be approximated by a neural network with input $\cdot \left( \partial ^ { \alpha } g ( t _ { n } , y _ { n } ) \right) _ { | \alpha | = k } , \ldots , \partial g ( t _ { n } , y _ { n } ) , g ( t _ { n } , y _ { n } ) , \mathrm { t o l }$ 
 . This suggests that the partial derivatives of $g$ can also be an appropriate input for our controller,
+
+$$
+\psi \left( { \mathrm { e r r } } _ { n + 1 } , \cdot \right) = \left( \log \left( { \frac { \mathrm { t o l } } { { \mathrm { e r r } } _ { n + 1 } } } \right) , ( \partial ^ { \alpha } g ( t _ { n } , y _ { n } ) ) _ { | \alpha | = p } , \cdot \cdot . . , \partial g ( t _ { n } , y _ { n } ) , g ( t _ { n } , y _ { n } ) \right) .
+$$
+
+However, providing partial derivatives requires additional computation whereas the $p$ function values that were computed during the Runge-Kutta step are conveniently available. As a trade-off between “perfect” information in the form of higher order partial derivatives and the computational effort to compute them, we also propose to use the following input:
+
+$$
+\psi \left( \mathrm { e r r } _ { n + 1 } , \cdot \right) = \left( \log \left( \frac { \mathrm { t o l } } { \mathrm { e r r } _ { n + 1 } } \right) , g _ { n } ^ { ( 1 ) } , \dots , g _ { n } ^ { ( p ) } \right) ,
+$$
+
+$g _ { n } ^ { ( 1 ) } , \ldots , g _ { n } ^ { ( p ) }$ $p$
+
+In contrast to the traditional input (7), our novel inputs (8) and (9) both provide local information about the structure of the problem at hand, which allows one to keep the local error close to the tolerance parameter. Moreover, hand-designed controller and existing Runge-Kutta methods do not make use of these information, and it is difficult—if not impossible—to do so. Training an optimizer with input (8) or (9), as shown next, does so automatically and designs novel Runge-Kutta methods.
+
+Meta-Learner. With a good performance measure and appropriate inputs at hand, we can now solve the step-size control problem as a meta-learning problem as sketched already in Fig. 1. Let the input for the optimizer be $x _ { n } = \psi \left( \mathrm { e r r } _ { n } , \cdot \right)$ . We parameterize the optimizer $c$ using $\phi$ and update the step size as follows, $\log h _ { n + 1 } = \log h _ { n } + c _ { n } ( \psi ( \mathbf { e r r } _ { n + 1 } , \cdot ) , \phi )$ . An approach in log space has the advantage that that we do not need to constrain the output of $c$ to be positive. Since $c _ { n }$ corresponds to $\log r _ { n }$ , we will adjust the notation in a similar fashion. Due to their natural ability to handle sequential tasks, an LSTM was chosen in our experiments as optimizer $c$ , and its prediction determines the step size which will be used in the next Runge-Kutta iteration. We denote the model of the optimizer, represented by the blue block in Fig. 1, by $m$ , its parameters by $\phi$ and the hidden state by $\hat { h } _ { n }$ . It can be expressed by
+
+$$
+\binom { \log r _ { n - 1 } } { \hat { h } _ { n + 1 } } = m \left( x _ { n } , \hat { h } _ { n } , \phi \right) .
+$$
+
+The subsequent Runge-Kutta update in the yellow block of Fig. 1 first updates the step size
+
+$$
+\log h _ { n } = \log h _ { n - 1 } + \log r _ { n - 1 } ,
+$$
+
+and then uses it to execute the next Runge-Kutta step,
+
+$$
+t _ { n + 1 } = t _ { n } + h _ { n } , \left( { \begin{array} { c } { y _ { n + 1 } } \\ { \operatorname { e r r } _ { n + 1 } } \end{array} } \right) = \Phi ( t _ { n } , y _ { n } , h _ { n } ) .
+$$
+
+Here, $\Phi$ denotes the Runge-Kutta step from $y _ { n }$ to $y _ { n + 1 }$ as well as the error estimate $\mathrm { e r r } _ { n + 1 }$ that is computed during that step. Afterwards, the next input for the model will be computed.
+
+Learning Objective. Different initial value problems can require very different step sizes. Using an optimizer that is specialized for a certain class of problems allows to exploit the structure of these problems. The behavior of an initial value problem is described by a function $g$ , therefore we can represent a class of initial value problems with a distribution over the functions $g$ . Thus, an optimizer can be considered optimal for a class of problems, if it minimizes the expected loss:
+
+$$
+L ( \phi ) = \mathbb { E } _ { g } \left( \sum _ { n = 1 } ^ { N } l ( t _ { n } , y _ { n } , \mathrm { e r r } _ { n } ) \right) ,
+$$
+
+where $t _ { n } , y _ { n } , \mathbf { e r r } _ { n }$ were computed according to Eqs. (10), (11), (12). Here, $l$ denotes the loss function defined for an approximation $y _ { n }$ at time point $t _ { n }$ and the local error estimate $\operatorname { e r r } _ { n }$ . Note that our training loss directly corresponds to the performance measure we are interested in. The model $m$ can then learn the behavior of the given class of problems and use this knowledge to generalize to new examples of the same class and new, unseen classes. Specifically, since the provided performance measure $l$ is differentiable a.e., we can optimize the learning objective Eq. (13) using gradient descent on the parameters $\phi$ . An estimate of the gradient $\frac { \partial L ( \phi ) } { \partial \phi }$ can be computed by sampling a function $g$ from the distribution of the class of initial value problems and applying backpropagation, $c . f$ . (Rumelhart et al., 1986).
+
+# 4 EXPERIMENTAL EVIDENCE
+
+Our intention here is to investigate the benefits of Meta-Learning Runge-Kutta (MLRK). To this end we conducted experiments with different classes of initial value problems. Specifically, we
+
+Table 1: On the test set of class (low-freq), MLRK was considerably faster (less many steps) than the baseline controller on average, while causing only a mildly larger mean global error at the end of the integration interval. Both the baseline controller and MLRK showed a gradually increasing mean global error at the end of the integration interval over the test set consisting of 1500 harmonic oscillators of class (med-freq).   
+
+<table><tr><td></td><td colspan="4">(low-freq)</td><td colspan="4">(med-freq)</td></tr><tr><td rowspan="2">interval</td><td colspan="2">steps</td><td colspan="2">error</td><td colspan="2">steps</td><td colspan="2">error</td></tr><tr><td>Baseline</td><td>MLRK</td><td>Baseline</td><td>MLRK</td><td>Baseline</td><td>MLRK</td><td>Baseline</td><td>MLRK</td></tr><tr><td></td><td>3.42</td><td>3.15</td><td>0.000004</td><td>0.000009</td><td>26.24</td><td>7.95</td><td>0.001588</td><td>0.008253</td></tr><tr><td></td><td>7.59</td><td>6.05</td><td>0.000017</td><td>0.000119</td><td>87.16</td><td>29.63</td><td>0.001686</td><td>0.011236</td></tr><tr><td></td><td>11.76</td><td>8.22</td><td>0.000032</td><td>0.000366</td><td>148.05</td><td>53.44</td><td>0.001739</td><td>0.012904</td></tr><tr><td>１３57</td><td>15.80</td><td>10.23</td><td>0.000048</td><td>0.000668</td><td>208.95</td><td>77.54</td><td>0.001735</td><td>0.014222</td></tr><tr><td>10</td><td>21.92</td><td>13.15</td><td>0.000073</td><td>0.001171</td><td>300.32</td><td>113.82</td><td>0.001816</td><td>0.016546</td></tr></table>
+
+(high-freq)   
+
+<table><tr><td rowspan="2">interval</td><td colspan="2">steps</td><td colspan="2">error</td></tr><tr><td>Baseline</td><td>MLRK</td><td>Baseline</td><td>MLRK</td></tr><tr><td>13</td><td>47.15</td><td>12.08</td><td>0.026415</td><td>0.085082</td></tr><tr><td></td><td>157.58</td><td>53.42</td><td>0.023223</td><td>0.081219</td></tr><tr><td>5</td><td>268.03</td><td>96.48</td><td>0.025230</td><td>0.091109</td></tr><tr><td>7</td><td>378.42</td><td>139.69</td><td>0.026177</td><td>0.094129</td></tr><tr><td>10</td><td>544.05</td><td>204.57</td><td>0.024858</td><td>0.094562</td></tr></table>
+
+Table 2: The mean global errors of the baseline stay approximately the same. The optimizer trained on problem instances of class (low-freq) shows only a very gradual increment in the global error on instances of class (high-freq).
+
+investigated our suggested loss functions and our theory that suggests that providing higher order partial derivatives of $g$ can be used as a means to anticipate the evolution of the local error and, hence, may lead to an improved performance of the optimizer. For a class of initial value problems we assume a parametric form of $g$ with a distribution over the parameters; details can be found in the appendix. Our datasets are obtained by sampling from these distributions. Finally, we compare our method to the baseline controller given by Eq. (2).
+
+Harmonic Oscillator. First, we started with a simple class of ODEs. We trained MLRK on low varied harmonic oscillators. The corresponding results for training on high frequencies are similar and can be found in the appendix. As inputs we only used the error estimates, Eq. (7). We started with the dataset of harmonic oscillators with low frequencies (low-freq): the training set contained 30,000 instances, the validation set and test set each 1,500 instances. To measure the performance we considered the $L _ { 1 }$ -Lagrangian loss Eq. (5) as well as the number of steps needed for an integration interval and the corresponding error separately. Since the Lagrangian represents a trade-off between the two it is a particularly good measure.
+
+Fig. 2(a) shows the mean loss during the integration of the test set, which consists of 1500 harmonic oscillators of class (low-freq). The solid lines show the mean loss in step $n$ , the colored areas mark the standard deviation. MLRK achieves a smaller mean loss than the baseline, which confirms its ability to generalize to new problem instances of the same class of problem. When we compare the mean number of steps and the mean global error of MLRK and the baseline controller in Tab. 1, the necessity for a trade-off between the two objectives becomes clear. While the baseline needs on average more steps to complete the integration, it achieves a smaller error, the contrary is true for our optimizer. This makes it hard to compare the two methods based on just these values.
+
+Generalization Capability of Optimizer based on Low Frequency Harmonic Oscillators. By evaluating the controller designed by MLRK on different test sets, we investigated its ability to transfer knowledge to problem instances of other classes. Fig. 2(b) shows that MLRK maintains a smaller mean loss on 1500 harmonic oscillators of class (med-freq). These oscillators are both higher in frequency and amplitude than the problem instances contained in the training set. This indicates that MLRK is able to generalize to these problems. Tab. 1 reveals that MLRK uses on average about a third of the integration steps the baseline needs for the tested integration intervals. The resulting mean global errors are indeed larger than those of the baseline controller, but still very reasonable. Moreover, evaluating MLRK on 1500 harmonic oscillators of class (high-freq) showed a mean loss similar to that of the baseline controller in Fig. 2(c). This suggests that MLRK generalizes to problem instances of this class as well. In Tab. 2 we observe a similar situation as in the previous experiment. The global error of the baseline controller stays approximately the same for different interval lengths, while the global error of MLRK increases very gradually, while using only a fraction of iterations.
+
+![](images/8fb302fef67595b67a0a345f7d0de78e1679bb2d3408dd66a2399a0a042beb83.jpg)  
+Figure 2: The mean $L _ { 1 }$ -Lagrangian loss (5) of the approximation $( t _ { n } , y _ { n } )$ in step $n$ over three different test sets. (a) MLRK achieved a smaller mean loss, thus, it generalizes well to new problem instances of the same class of problems (low-freq) it was trained on. (b) MLRK achieves a lower loss on instances with both higher amplitude and frequency than the problem instances used during training. Thus, MLRK is able to generalize to different problems. (c) Even on higher frequencies than the ones in (med-freq), MLRK shows a mean loss similar to that of the baseline controller.
+
+![](images/dd80c719431d50131a556ec6d2b4b11a1c8053da5a7a64a8594621926ae0bcef.jpg)  
+Figure 3: The mean local error over the test set of van der Pol ODEs for different methods is shown. The optimizer err was trained with input (7), optimizer partial was trained with input (8) and grad was trained with input (9). Providing local information such as those in input (8) and (9) results in increased responsiveness of the optimizer.
+
+van der Pol. "I have a theory that whenever you want to get in trouble with a method, look for the van der Pol equation" – P. Zadunaisky, 1982, $c . f$ . (Hairer et al., 2000).
+
+Next we conducted an experiment regarding the local $L _ { 1 }$ loss (6). Experiments with harmonic oscillators revealed the optimizers ability to keep the local errors close to the tolerance parameter $( c . f .$ . appendix). Keeping the local error equal to a constant tolerance is not a hard problem for simple harmonic oscillators, therefore we turned towards van der Pol oscillators. We considered the following dataset of van der Pol oscillators of class $\mathbf { v d P } ( 0 , 1 )$ : The training set contained 50,000 instances, the validation set and the test set each 1,500 instances. The local errors of van der Pol oscillators vary drastically $\cdot c . f$ . Fig. 5, appendix). High spikes occur when the solution changes from being driven to being damped. As the behavior of van der Pol equations are more complex compared to harmonic oscillators, we expect that providing additional information about the problem such as partial derivatives or function values of $g$ can be beneficial. To investigate this, we considered MLRK with different inputs. One optimizer is only provided with the error estimate Eq. (7), another is provided with the error estimate and all partial derivatives Eq. (8) and the last is provided with the error estimate and the function evaluations that were computed during the Runge-Kutta step, Eq. (9).
+
+The results are summarized in Fig. 3. Using err shows reoccurring high spikes similar to the baseline, although the spikes of the optimizer are lower on average. In contrast, MLRK with additional information on the partial derivatives (partial) shows a much smoother sequence of local errors. This demonstrates that the additional information in the partial derivatives of $g$ improves the ability of MLRK to anticipate the variations of the local errors and respond with adequate step sizes. MLRK with error estimates as well as function values of $g$ (grad) produces a smooth step size sequence similar to the optimizer partial. This shows that the optimizer with additional input-information from $g$ are actually outperforming the baseline. The reoccurring spikes in the local error seem to decrease in amplitude with time. This is largely due to an averaging effect – for different values of $\sigma$ these spikes occur at different integration steps. All three methods reveal a reduction in both the mean number of steps and the mean local errors, $c . f$ . Tab. 3, appendix.
+
+![](images/688744afe207389f2cea3441a67d96af19f602953a92c3e17d919be6f72ee02f.jpg)  
+Figure 4: The mean local error over the test set of double pendulums for different methods is shown. The optimizer err was trained with input (7), optimizer partial was trained with input (8) and grad was trained with input (9). In all three cases MLRK is able to reduce the spikes in the local errors drastically, the use of local information reduces the variance of the local errors even further.
+
+Double Pendulum. To demonstrate the benefit of MLRK in an application we conducted experiments with double pendulum ODEs $_ { c , f }$ . appendix). The chaotic behaviour of the double pendulum leads to sudden spikes in the local error, a particularly challenging problem for step size controllers. To investigate the ability of MLRK to avoid these spikes by adjusting the step size appropriately, we consider the $L _ { 1 }$ local loss as in the previous experiment and a dataset of double pendulums of class (pendulum). The training set contained 50,000 instances, the validation set and the test set each 1,500 instances. Again, the different inputs Eqs. (7), (8) and (9) were examined.
+
+The results are shown in Fig. 4 and reveal the baseline controllers inability to produce steady local errors. Here, the benefit of a learned update rule over a static one becomes evident. The optimizer err that is only provided with the local error estimates is able to reduce the spikes in the local errors to a remarkable degree. Moreover, local information about the problem allows the optimizers partial and grad to reduce the variance in the local errors even further.
+
+# 5 CONCLUSION
+
+We have shown how to cast the design Runge-Kutta methods for solving initial value problems as a learning problem. We established appropriate performance measures and useful inputs for the controller, which constitute the key ingredients of the resulting Meta-learning Runge-Kutta (MLRK), which learns step size controllers that are specialized to a particular class of initial value problems. Our experimental results demonstrate that MLRK can indeed learn to design novel Runge-Kutta methods that perform better than a hand-designed Runge-Kutta approach. Furthermore, we observed a remarkable degree of generalization to other classes of initial value problems. More importantly, examining the effect of different inputs demonstrated that the additional information contained in the function values and partial derivatives of $g$ leads to a substantial improvement in performance of the automatically designed solvers.
+
+There are several interesting avenues for future work. While MLRK generalizes well to problem instances of the same class and even to problems of similar classes, when confronted with a very different type of problem, MLRK does not generalize well yet. Wichrowska et al. (2017) showed how a carefully chosen network architecture and a diverse training set improves generalization of their optimizer to many different classes of optimization problems. A similar approach may lead to improved generalization of MLRK. Another common aspect of Runge-Kutta is the need for step rejection in case the local error exceeds the tolerance. The problem here is that a single large error can in general not be compensated for, even if subsequent step sizes are chosen very small. In this case, a step is usually rejected and repeated using a smaller step size. Here the challenge is to choose a step size small enough to meet the accuracy requirements but at the same time not too small, since the choice of step size influences the entire step size sequence to come.
+
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+
+# A APPENDIX
+
+RUNGE-KUTTA METHODS
+
+The mathematical theory of Runge-Kutta methods is quite evolved. We will only recap a few basics here, for further details see for example (Butcher, 2008; Hairer et al., 2000; Hairer & Wanner, 2002).
+
+Definition A.1. Let $s \in \mathbb { N }$ , $a _ { i j } \in \mathbb { R }$ for $i , j \in \{ 1 , \ldots , s \}$ , $c _ { i } \in \mathbb { R }$ for $i \in \{ 1 , \ldots , s \}$ and $b _ { i } \in \mathbb { R }$ for $i \in \{ 1 , \ldots , s \}$ . Let $\boldsymbol { g } : \mathbb { R } \times \mathbb { R } ^ { m } \to \mathbb { R } ^ { m }$ be a function that describes an initial value problem. The method defined by
+
+$$
+\begin{array} { c } { { y _ { n + 1 } = y _ { n } + h \displaystyle \sum _ { i = 1 } ^ { s } b _ { i } k _ { i } \nonumber , } } \\ { { { } } } \\ { { k _ { i } = g ( t _ { n } + h c _ { i } , y _ { n } + h \displaystyle \sum _ { j = 1 } ^ { s } a _ { i j } k _ { j } ) \nonumber , } } \end{array}
+$$
+
+for step size $h$ is called an $s$ -stage Runge-Kutta method. If $a _ { i j } = 0$ for $i \leq j$ , the method is called explicit, otherwise it is called implicit.
+
+Definition A.2. A Runge-Kutta method is of order $p$ if for sufficiently smooth initial value problems
+
+$$
+\| y ( t _ { 0 } + h ) - y _ { 1 } \| \leq K h ^ { p + 1 }
+$$
+
+holds.
+
+Note that a higher order method yields more accurate results, consequently a high order is a desired characteristic of a Runge-Kutta method. However, higher orders can only be achieved by the use of more stages. For an $s \mathrm { . }$ -stage Runge-Kutta method the order $p$ is bounded by the number of stages, $p \leq s$ . Up to order 4 there exist methods with $p = s$ , for order 5 and higher $p$ is strictly smaller than $s , c . f$ . (Butcher, 2008, Theorem 324B). The order of an $s$ -stage Runge-Kutta method depends on the coefficients $a _ { i j }$ , $b _ { i }$ , $c _ { i }$ . Order conditions for these coefficients have been developed, which ensure a certain order of the method.
+
+The problem of step size control can be described in the following way: Based on some input values, e.g. $t _ { n } , y _ { n } , \mathbf { e r r } _ { n }$ and possibly additional characteristics of $g$ , we must choose a step size $h _ { n }$ ,which is then used to evaluate $g$ at $p$ different points. These points are determined by $h _ { n }$ , see Definition A.1. The resulting stages in Eq. (15) are then used to compute the approximation $y _ { n + 1 }$ of $y ( t _ { n } + h _ { n } )$ , $c . f$ . Eq. (14). The problem is to choose $h _ { n }$ in a way such that the approximation $y _ { n + 1 }$ fulfills some desired properties. For example we may require that the local error of $y _ { n + 1 }$ is close to some tolerance parameter. In fact, this is the standard objective of most common step size controllers.
+
+# ERROR ESTIMATION
+
+The local errors are of particular importance for step size control and can fortunately be estimated very efficiently. To that end, we take a look at a simple idea as described in (Butcher, 2008, p. 198): Suppose we have two approximations for $y ( t _ { n } )$ of order $\hat { p }$ and $\tilde { p }$ respectively, that is
+
+$$
+\begin{array} { r } { \hat { y } _ { n } = y ( t _ { n } ) + \mathcal { O } ( h ^ { \hat { p } + 1 } ) , } \\ { \tilde { y } _ { n } = y ( t _ { n } ) + \mathcal { O } ( h ^ { \hat { p } + 1 } ) , } \end{array}
+$$
+
+where $y ( t )$ is the solution of
+
+$$
+y ^ { \prime } ( t ) = g ( t , y ( t ) ) , \quad y ( t _ { n - 1 } ) = y _ { n - 1 } ,
+$$
+
+and the step size $h$ that was used to obtain both approximations is given by
+
+$$
+h = t _ { n } - t _ { n - 1 } .
+$$
+
+If $\hat { p } > \tilde { p }$
+
+$$
+\hat { y } _ { n } - \tilde { y } _ { n } = y ( t _ { n } ) - \tilde { y } _ { n } + \mathcal { O } ( h ^ { \tilde { p } + 2 } )
+$$
+
+can be used as an approximation for the local truncation error of the $\tilde { p }$ -th order Runge-Kutta method:
+
+$$
+\mathrm { e r r } _ { n } ( h ) \approx \| \hat { y } _ { n } - \tilde { y } _ { n } \| .
+$$
+
+To obtain error estimates each step has to be carried out with two methods of different orders. Here, the easiest approach is to use two separate Runge-Kutta methods to compute $\hat { y }$ and $\tilde { y }$ , however this is quite costly, especially for implicit methods. A more efficient approach to obtain error estimates are the embedded Runge-Kutta methods. Here both approximation use the same stages $k _ { i }$ in Eq. (15) but two different pairs of coefficients $b _ { i }$ , $\hat { b } _ { i }$ in Eq. (14). By doing so, the stages can be reused and an error estimate can be obtained with a cheap extra linear combination of the $k _ { i }$ ’s. We denote the error estimates by
+
+$$
+\mathrm { e r r } _ { n } = \| \hat { y } _ { n } - \tilde { y } _ { n } \| .
+$$
+
+# CLASSES OF INITIAL VALUE PROBLEMS
+
+For loss functions such as the one in Eq. (5) it is useful to consider initial value problems with known solutions so that we can compare the numerical approximation to the true function values of the solution. For this reason we considere simple harmonic oscillators and linear differential equations with constant coefficients. If we employ loss functions as Eq. (6), local rather than global errors are of interest. For these loss functions it is interesting to consider initial value problems for which the local errors vary drastically during the integration such as van der Pol oscillators.
+
+# SIMPLE HARMONIC OSCILLATOR
+
+A simple harmonic oscillator is a harmonic oscillator that is neither driven nor damped and can be characterized by
+
+$$
+m x ^ { \prime \prime } ( t ) = - k x ( t ) ,
+$$
+
+where $m$ is the mass of the oscillator, $x$ the position and $k$ describes the restoring force that is applied, when displaced from its equilibrium position. It can be transformed to
+
+$$
+\begin{array} { l } { { \displaystyle y _ { 1 } ^ { \prime } ( t ) = y _ { 2 } ( t ) \ : , } } \\ { { \displaystyle y _ { 2 } ^ { \prime } ( t ) = - \frac { m } { k } y _ { 1 } ( t ) \ : . } } \end{array}
+$$
+
+The initial values provide the initial position $x ( t _ { 0 } )$ of $x$ and the initial velocity $x ^ { \prime } ( t _ { 0 } )$
+
+$$
+\begin{array} { l } { { y _ { 1 } ( t _ { 0 } ) = x ( t _ { 0 } ) , } } \\ { { y _ { 2 } ( t _ { 0 } ) = x ^ { \prime } ( t _ { 0 } ) . } } \end{array}
+$$
+
+The solution of the simple harmonic oscillator takes the form
+
+$$
+x ( t ) = A \cos ( \omega t + \varphi ) , \quad \omega = \sqrt { \frac { m } { k } } ,
+$$
+
+where $A$ and $\varphi$ are uniquely determined by the initial values. The frequency of the oscillations is determined by $\omega$ , $A$ is the magnitude of the oscillations and $\varphi$ is a phase-shift.
+
+A class of this kind of initial value problvalues or alternatively a distribution over , n beand escribed by a distribution over . The following classes of initia $\frac { m } { k }$ and the initialalue problems $A$ $\omega$ $\varphi$   
+were used in the experiments
+
+$$
+\begin{array} { l c r } { { A \sim U ( 0 , 5 ) , } } \\ { { A \sim U ( 0 , 5 ) , } } \\ { { A \sim U ( 0 , 1 ) , } } \\ { { A \sim U ( 1 , 5 ) , } } \end{array}
+$$
+
+$$
+\begin{array} { r l r } & { } & { \omega \sim U ( 0 , 1 0 ) , \quad \varphi \sim U ( 0 , 1 ) , } \\ & { } & { \omega \sim U ( 0 , 2 0 ) , \quad \varphi \sim U ( 0 , 1 ) , } \\ & { } & { \omega \sim U ( 0 , 1 ) , \quad \varphi \sim U ( 0 , 1 ) , } \\ & { } & { \omega \sim U ( 1 , 5 ) , \quad \varphi \sim U ( 0 , 1 ) , } \end{array}
+$$
+
+where $U ( a , b )$ denotes the uniform distribution over the interval $[ a , b ]$ .
+
+# LINEAR CONSTANT COEFFICIENT
+
+A linear differential equation with constant coefficients can be described by
+
+$$
+y ^ { \prime } ( t ) = A y ( t ) , \qquad y ( t _ { 0 } ) = y _ { 0 } ,
+$$
+
+where $A \in \mathbb { R } ^ { m \times m }$ . The solution of this differential equation is known and can be expressed in terms of the eigenvalues $\lambda _ { 1 } , \ldots , \lambda _ { m }$ with corresponding eigenvectors $v _ { 1 } , \ldots , v _ { m }$ of the matrix $A$ :
+
+$$
+y ( t ) = \sum _ { j = 1 } ^ { m } c _ { j } e ^ { \lambda _ { j } t } v _ { j } ,
+$$
+
+with coefficients $c _ { j } \in \mathbb { R }$ that are uniquely determined by the initial values $y ( t _ { 0 } ) = y _ { 0 }$ . The initial value problem is stable, if $\mathrm { R e } ( \lambda _ { j } ) < 0$ for all $j \in \{ 1 , \dots , m \}$ and unstable if $\operatorname { R e } ( \lambda _ { j } ) > 0$ for some $j ~ \in ~ \{ 1 , \dots , m \}$ . The oscillatory behavior of the problem is determined by $\mathrm { I m } ( \lambda _ { j } )$ . Note that the simple harmonic oscillators are linear differential equations with constant coefficients, where $A \in \mathbb { R } ^ { 2 \times 2 }$ , $\mathbf { R e } ( \lambda _ { j } ) = 0$ for $j \in \{ 1 , 2 \}$ and $\begin{array} { r } { \mathrm { I m } ( \lambda _ { 1 } ) = - \mathrm { I m } ( \lambda _ { 2 } ) = \sqrt { \frac { m } { k } } i } \end{array}$ .
+
+For the experiments we used differential equations with $A \in \mathbb { R } ^ { 3 \times 3 }$ . The matrix $A$ either has three real eigenvalues $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \ \in \ \mathbb { R }$ or one real eigenvalue $\lambda _ { 1 } \in \mathbb { R }$ and two complex eigenvalues $\lambda _ { 2 } = a + b i , \lambda _ { 3 } = a - b i$ , with $a , b \in \mathbb { R }$ . In this case we choose $c _ { 2 } = c _ { 3 }$ to obtain a real solution. We can specify a class of linear constant coefficient differential equations by a distribution on the eigenvalues of $A$ and the coefficients $c _ { j }$ . The classes that were used for the experiments were described by the distributions
+
+$$
+\begin{array} { l } { \lambda _ { 1 } \sim U ( - 2 . 5 , - 0 . 1 ) , } \\ { \lambda _ { 1 } \sim U ( 0 , 1 ) , } \\ { \lambda _ { 1 } \sim U ( - 1 , 0 ) , } \end{array}
+$$
+
+$$
+\begin{array} { r } { a \sim U ( - 2 . 5 , - 0 . 1 ) , ~ b \sim U ( 0 , 5 ) , } \\ { a \sim U ( - 2 . 5 , - 0 . 1 ) , ~ b \sim U ( 0 , 5 ) , } \\ { a \sim U ( 0 , 1 ) , ~ b \sim U ( 1 , 2 ) , } \end{array}
+$$
+
+where $U ( a , b )$ denotes the uniform distribution on the interval $[ a , b ]$ .
+
+![](images/79a23670bf80941cad59ce436ee56a6428b332fbcee2e05c4ef832814c658452.jpg)  
+Figure 5: The upper plot shows the numerically approximated solution $y _ { 1 } ( t )$ of a van der Pol oscillator with $\sigma = 2$ . The lower plot displays the local error estimates for $h \ : = \ : 0 . 1$ . The error estimates show high spikes whenever the solution switches from being driven to being damped.
+
+VAN DER POL OSCILLATOR
+
+Van der Pol oscillators arise in the study of limit cycles. A van der Pol oscillator can be described by
+
+$$
+\begin{array} { l } { { y _ { 1 } ^ { \prime } ( t ) = y _ { 2 } , } } \\ { { y _ { 2 } ^ { \prime } ( t ) = \sigma ( 1 - y _ { 1 } ^ { 2 } ) y _ { 2 } - y _ { 1 } , } } \end{array}
+$$
+
+$$
+\begin{array} { r } { y _ { 1 } ( 0 ) = 2 , } \\ { y _ { 2 } ( 0 ) = 0 , } \end{array}
+$$
+
+with $\sigma > 0$ . For $\sigma = 2$ an approximate solution is displayed in Figure 5. Small oscillations are amplifies and large oscillations damped, $c . f$ . (Hairer et al., 2000, pp. 111). The local errors of a van der Pol oscillator show high variations. For the experiments we used van der Pol oscillators with the following distributions
+
+$$
+\begin{array} { r } { \sigma \sim U ( 0 , 1 ) , } \\ { \sigma \sim U ( 1 , 2 ) . } \end{array}
+$$
+
+$$
+\begin{array} { r } { ( \mathrm { v d P } ( 0 , 1 ) ) } \\ { ( \mathrm { v d P } ( 1 , 2 ) ) } \end{array}
+$$
+
+# DOUBLE PENDULUM
+
+A double pendulum is a pendulum attached to another pendulum, $c . f$ . Fig. 6. The length of the first and second pendulum are given by $L _ { 1 }$ and $L _ { 2 }$ , respectively, their masses by $M _ { 1 }$ and $M _ { 2 }$ and the angle by $\theta _ { 1 }$ and $\theta _ { 2 }$ . The equations of motion for the double pendulum can be written in different forms, we chose the following formulation in terms of the angular acceleration:
+
+$$
+\begin{array} { r l } & { { \ddot { \theta } _ { 1 } } = \frac { { M _ { 2 } L _ { 1 } { { \dot { \theta } } _ { 1 } } ^ { 2 } \sin ( { \theta _ { 2 } } - { \theta _ { 1 } } ) \cos ( { \theta _ { 2 } } - { \theta _ { 1 } } ) + { M _ { 2 } } g \sin ( { \theta _ { 2 } } ) \cos ( { \theta _ { 2 } } - { \theta _ { 1 } } ) } } { { L _ { 1 } { { \left( { M _ { 1 } } + { M _ { 2 } } - { M _ { 2 } } \cos ^ { 2 } ( { \theta _ { 2 } } - { \theta _ { 1 } } ) \right) } } } } } \\ & { \qquad + \frac { { M _ { 2 } } { L _ { 2 } { { \dot { \theta } } _ { 2 } } ^ { 2 } \sin ( { \theta _ { 2 } } - { \theta _ { 1 } } ) } - { \left( { M _ { 1 } } + { M _ { 2 } } \right) g \sin ( { \theta _ { 1 } } ) } } { { L _ { 1 } { { \left( { M _ { 1 } } + { M _ { 2 } } - { M _ { 2 } } \cos ^ { 2 } ( { \theta _ { 2 } } - { \theta _ { 1 } } ) \right) } } } } } \\ & { { \ddot { \theta } _ { 2 } } = \frac { { \left( { M _ { 1 } } + { M _ { 2 } } \right) \left( g \sin ( { \theta _ { 1 } } ) \cos ( { \theta _ { 2 } } - { \theta _ { 1 } } ) - { L _ { 1 } } { { \dot { \theta } } _ { 1 } } ^ { 2 } \sin ( { \theta _ { 2 } } - { \theta _ { 1 } } ) - g \sin ( { \theta _ { 2 } } ) \right) } } { { L _ { 2 } { { \left( { M _ { 1 } } + { M _ { 2 } } - { M _ { 2 } } \cos ^ { 2 } ( { \theta _ { 2 } } - { \theta _ { 1 } } ) \right) } } } } } \\ &  \qquad - \frac { { M _ { 2 } } { L _ { 2 } { \dot { \theta } } _ { 2 } ^ { 2 } \sin ( { \theta _ { 2 } } - { \theta _ { 1 } } ) \cos ( { \theta _ { 2 } } - { \theta _ { 1 } } ) } } { { L _ { 2 } { \left( { M _ { 1 } } + { M _ { 2 } } - { M _ { 2 } } \cos ^ { 2 } ( { \theta _ { 2 } } - { \theta _ { 1 } } ) \right) } } } \end{array}
+$$
+
+The initial angular position and velocity $\theta _ { 1 } , \theta _ { 2 } , { \dot { \theta } } _ { 1 } , { \dot { \theta } } _ { 2 }$ determine the trajectory of the double pendulum. For the experiments we used double pendulums with the following distribution
+
+$$
+M _ { 1 } \sim U ( 0 . 5 , 1 ) , M _ { 2 } \sim U ( 0 . 5 , 2 ) , L _ { 1 } \sim U ( 0 . 5 , 1 ) , L _ { 2 } \sim U ( 0 . 5 , 1 ) .
+$$
+
+# ADDITIONAL EXPERIMENTS
+
+ON VAN DER POL
+
+Table 3 shows the mean number of steps, local error and wall clock time over 1500 van der Pol equations of class $( \mathrm { v d P } ( 0 , 1 ) )$ .
+
+![](images/389adf363f00be8443851980f8b81d3ec4fabac4dfbec03b85a8403edc9754e9.jpg)  
+Figure 6: A double pendulum is one pendulum attached to another pendulum. The masses $M _ { 1 }$ and $M _ { 2 }$ of these pendulums may vary as do the lengths $L _ { 1 }$ and $L _ { 2 }$ . The initial configuration of angular position and velocity determines the trajectory of the double pendulum.
+
+HIGH FREQUENCY HARMONIC OSCILLATORS
+
+In this experiment, we trained a controller specialized for simple harmonic oscillators of class (high-freq) which were described in Sec. A. Thus we chose the following datasets:
+
+Training set: 30000 Harmonic oscillators of class (high-freq). Validation set: 1000 Harmonic oscillators of class (high-freq). Test set: 1000 Harmonic oscillators of class (high-freq).
+
+As loss we chose the $L _ { 1 }$ -Lagrangian Eq. (5). As inputs we used only the error estimates, Eq. (7). The loss of the learned controller and the baseline controller, averaged over the test dataset, is shown in Fig. 7(a). Both controllers are used to execute 100 integration steps for each instance of the test dataset. Subsequently, the loss Eq. (5) is computed for each approximation $( t _ { n } , y _ { n } )$ . The solid lines show the mean loss in step $n$ , the colored areas mark the standard deviation. Here, the loss of our learned controller is on average smaller than the loss of the baseline controller, which demonstrates the controllers generalization capability to new problems of the same class.
+
+Tab. 4 summaizres the global error at the end of the integration interval and the number of steps needed during the integration for different interval lengths in range 1 to 10. All values are averaged over the test dataset. Here, the baseline controller needs much more steps than the learned controller, but maintains a smaller error. In fact, the error caused by the baseline is approximately the same for all shown interval lengths, while the error caused by the trained model increases with interval length. This is due to the assumption that the global error is the sum of the local errors. In particular, the loss Eq. (5) rates step size sequences with a higher global error better, if they can balance the higher error by achieving a longer integration interval.
+
+While the general assumption that the global error increases with the length of the integration interval is often fulfilled, the specific assumption of a linear accumulation of the global error, as in our loss, is usually not fulfilled. This is a good example for why the assumptions leading to the widely used objective to keep the local errors constant are too strong.
+
+Table 3: The mean number of steps, local error and wall clock time over 1500 van der Pol equations are shown for different lengths of the integration interval. Our method err is slightly faster and uses less steps than the baseline while producing smaller local errors during the integration, see also Figure 3. While partial and grad reduce the number of steps even further, wall time is increased. However one can clearly see that partial and grad outperform both the baseline and err regarding the local error.   
+
+<table><tr><td>int</td><td colspan="4">steps</td></tr><tr><td></td><td>baseline</td><td>err</td><td>partial</td><td>grad</td></tr><tr><td>1</td><td>21.59</td><td>16.42</td><td>12.40</td><td>12.09</td></tr><tr><td>3</td><td>33.74</td><td>29.12</td><td>25.16</td><td>24.74</td></tr><tr><td>5</td><td>45.43</td><td>41.07</td><td>36.42</td><td>36.02</td></tr><tr><td>7</td><td>56.84</td><td>52.57</td><td>48.34</td><td>47.97</td></tr><tr><td>10</td><td>73.46</td><td>69.41</td><td>65.40</td><td>65.04</td></tr><tr><td colspan="5"></td></tr><tr><td></td><td>baseline</td><td>err</td><td>partial</td><td>grad</td></tr><tr><td>1</td><td>7.17e-4</td><td>6.58e-4</td><td>4.01e-4</td><td>3.74e-4</td></tr><tr><td>3</td><td>6.40e-4</td><td>5.45e-4</td><td>2.49e-4</td><td>2.28e-4</td></tr><tr><td>5</td><td>5.18e-4</td><td>4.47e-4</td><td>1.95e-4</td><td>1.75e-4</td></tr><tr><td>7</td><td>4.97e-4</td><td>4.16e-4</td><td>1.57e-4</td><td>1.39e-4</td></tr><tr><td>10</td><td>4.59e-4</td><td>3.82e-4</td><td>1.32e-4</td><td>1.18e-4</td></tr><tr><td colspan="5">int</td></tr><tr><td></td><td>baseline</td><td>err</td><td>partial</td><td>grad</td></tr><tr><td>1</td><td>0.0255</td><td>0.0263</td><td>0.0254</td><td>0.0221</td></tr><tr><td>3</td><td>0.0405</td><td>0.0375</td><td>0.0460</td><td>0.0403</td></tr><tr><td>5</td><td>0.0591</td><td>0.0517</td><td>0.0681</td><td>0.0596</td></tr><tr><td>7</td><td>0.0858</td><td>0.0742</td><td>0.1036</td><td>0.0897</td></tr><tr><td>10</td><td>0.0971</td><td>0.0825</td><td>0.1201</td><td>0.1065</td></tr></table>
+
+<table><tr><td rowspan="2">interval</td><td colspan="2">steps</td><td colspan="2">error</td></tr><tr><td>Baseline</td><td>Optimizer</td><td>Baseline</td><td>Optimizer</td></tr><tr><td>1</td><td>45.50</td><td>13.73</td><td>0.023092</td><td>0.030499</td></tr><tr><td>3</td><td>152.11</td><td>43.12</td><td>0.022266</td><td>0.040429</td></tr><tr><td>5</td><td>258.73</td><td>74.40</td><td>0.022731</td><td>0.049529</td></tr><tr><td>7</td><td>365.30</td><td>106.31</td><td>0.022011</td><td>0.055821</td></tr><tr><td>10</td><td>525.18</td><td>154.25</td><td>0.023353</td><td>0.070451</td></tr></table>
+
+Table 4: The mean values over the test set consisting of 1000 harmonic oscillators of class (high-freq) are shown. While the baseline needs much more steps than our trained model, it achieves a smaller global error. The global error of the baseline stays approximately the same for all integration intervals.
+
+It is also interesting to evaluate the learned controller on other classes of initial value problems. Fig. 7(b) shows the mean loss of the learned controller and the baseline controller over a test set consisting of 1500 harmonic oscillators of class (higher-freq). This class contains oscillators of class (high-freq) but also includes higher frequency ones. The learned controller has a lower mean loss than the baseline controller which indicates that it is able to transfer knowledge from problems of class (high-freq) to problems of class (higher-freq).
+
+When we compare the number of steps and the global error at the end of the integration interval, see Tab. 5, we can observe a similar situation as in Tab. 4. The learned controller needs less steps than the baseline controller. The global error however stays approximately the same for the baseline controller while it increases with interval length for the learned controller.
+
+![](images/01b9a89d3e853d783345b125cbc781f04029af741c31aa8bb58f7fd6ebed9798.jpg)  
+Figure 7: The red and blue line show the mean loss Eq. (5) of the approximation $( t _ { n } , y _ { n } )$ in step $n$ over the test set. The red and blue area indicate the standard deviation. (a) The learned model LSTM attains a smaller mean loss on new instances of the same class of problems (high-freq) it was trained on, this is especially pronounced in the steps $n \geq 5$ . (b) The test set consists of 1500 harmonic oscillators of class (higher-freq), which means that it also contains higher frequency oscillators. The mean loss of our learned controller is lower than the mean loss of the baseline controller which indicates that our method generalizes to these different problem instances.
+
+<table><tr><td rowspan="2">interval</td><td colspan="2">steps</td><td colspan="2">error</td></tr><tr><td>Baseline</td><td>Optimizer</td><td>Baseline</td><td>Optimizer</td></tr><tr><td>1</td><td>116.76</td><td>22.26</td><td>0.453698</td><td>0.511174</td></tr><tr><td>3</td><td>386.40</td><td>75.64</td><td>0.442521</td><td>0.620779</td></tr><tr><td>5</td><td>656.06</td><td>141.29</td><td>0.426773</td><td>0.770457</td></tr><tr><td>7</td><td>925.80</td><td>212.19</td><td>0.418467</td><td>0.925463</td></tr><tr><td>10</td><td>1330.31</td><td>319.85</td><td>0.413902</td><td>1.427778</td></tr></table>
+
+Table 5: The mean number of steps and the mean global error at the end of the integration interval was computed over a test set of 1500 harmonic oscillators of class (higher-freq). While the baseline controller needs much more steps than our learned controller, it has on average a lower global error, which stays approximately the same for different interval lengths.
+
+# HARMONIC OSCILLATOR
+
+We learn a controller optimized for simple harmonic oscillators. We choose the following data sets:
+
+Training set: 30000 Harmonic oscillators of class (low-freq).   
+Validation set: 1500 Harmonic oscillators of class (low-freq). Test set: 1500 Harmonic oscillators of class (low-freq).
+
+First, we evaluate our method on a test set consisting of the same class of initial value problems the controller was trained on. Figure 8(a) shows the mean local error of the learned controller and the baseline on this test set. Both controllers are capable to keep the local errors close to the tolerance. Our method is slightly closer to the tolerance. Next, we evaluate our learned controller on different test sets. When we test the learned controller on harmonic oscillators with higher frequencies and higher amplitude, we find that our method generalizes to these problem instances as well. In fact, both the learned and the baseline controller are able to keep the local errors close to the tolerance as displayed in Figure 8(b). Similar results were found for harmonic oscillators of the class (med-freq). Hence, our learned controller is able to transfer knowledge from oscillators with low frequencies to oscillators with higher frequencies.
+
+![](images/00d42f6fa8372bdd03d979a357f4f6350723ca77bad35f127146478f9ec4a7ad.jpg)  
+Figure 8: The mean local error of our learned controller and the baseline controller is shown. (a) Testing on 1500 harmonic oscillators of class (low-freq) shows that both the baseline and the learned controller are able to keep the local errors close to the tolerance. The mean local error of the learned controller is slightly closer to the tolerance. (b) For a test set consisting of 1500 harmonic oscillators of class (high-freq) the two controllers show a similar performance, both are able to keep the local error close to the tolerance. Testing the controllers on problem instances of class (med-freq) yields a similar results.
+
+![](images/81a12dc6aeb497fca2e990f721a990764c5b28cd677f2e2a0cbf3dba85bf4373.jpg)  
+Figure 9: The mean local error over different test sets is shown. (a) The test set consists of 1500 linear differential equations with constant coefficients of class (osc-increasing). For these problem instances the learned controller demonstrates its ability to keep the mean local error close to the tolerance. In fact, the local errors of our controller appear to be closer to the tolerance than the local errors of the baseline for some part of the integration. (b) Testing our controller on a set of 1500 van der Pol oscillators of type $( \mathrm { v d P } ( 0 , 1 ) )$ ) reveals a similar performance to that of the baseline controller. Both controllers show high spikes in the local errors. These occur whenever the solution changes from being driven to being damped, see Figure 5.
+
+We also want to evaluate the generalization capability of our method to problem instances of different classes of initial value problems. Our method shows an acceptable performance on linear differential equations with constant coefficients of the class (osc-increasing), see Figure 9(a). Here, the performance of our controller is similar to the baseline controller. The learned controller obtains local errors which are closer to the tolerance in some steps. Figure 9(b) shows a similar performance of our controller to the baseline controller on van der Pol oscillators. Both methods show high spikes in the local errors. These occur whenever the solution of a van der Pol oscillator changes from being driven to being damped as demonstrated in Figure 5. Our controller does not react quickly to the sudden changes in the local error of van der Pol equations. This behavior is very different from harmonic oscillators, thus generalization can not be expected.