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+# COMPLEMENT OBJECTIVE TRAINING
+
+Hao-Yun Chen1, Pei-Hsin Wang1, Chun-Hao Liu1, Shih-Chieh Chang1, 2, Jia-Yu Pan3, Yu-Ting Chen3, Wei Wei3, and Da-Cheng Juan3
+
+1Department of Computer Science, National Tsing-Hua University, Hsinchu, Taiwan   
+2Electronic and Optoelectronic System Research Laboratories, ITRI, Hsinchu, Taiwan 3Google Research, Mountain View, CA, USA {haoyunchen,peihsin,newgod1992}@gapp.nthu.edu.tw scchang@cs.nthu.edu.tw {jypan, yutingchen, wewei, dacheng}@google.com
+
+# ABSTRACT
+
+Learning with a primary objective, such as softmax cross entropy for classification and sequence generation, has been the norm for training deep neural networks for years. Although being a widely-adopted approach, using cross entropy as the primary objective exploits mostly the information from the ground-truth class for maximizing data likelihood, and largely ignores information from the complement (incorrect) classes. We argue that, in addition to the primary objective, training also using a complement objective that leverages information from the complement classes can be effective in improving model performance. This motivates us to study a new training paradigm that maximizes the likelihood of the groundtruth class while neutralizing the probabilities of the complement classes. We conduct extensive experiments on multiple tasks ranging from computer vision to natural language understanding. The experimental results confirm that, compared to the conventional training with just one primary objective, training also with the complement objective further improves the performance of the state-of-the-art models across all tasks. In addition to the accuracy improvement, we also show that models trained with both primary and complement objectives are more robust to single-step adversarial attacks.
+
+# 1 INTRODUCTION
+
+Statistical learning algorithms work by optimizing towards a training objective. A dominant principle for training is to optimize likelihood (Mitchell et al., 1997), which measures the probability of data given the model under a specific set of parameters. The popularity of deep neural networks has given rise to the use of cross entropy (Kullback & Leibler, 1951) as its primary training objective, since minimizing cross entropy is essentially equivalent to maximizing likelihood for disjoint classes. Cross entropy has become the standard training objective for many tasks including classification (Krizhevsky et al., 2012) and sequence generation (Sutskever et al., 2014).
+
+Let $\mathbf { y } _ { i } \in \{ 0 , 1 \} ^ { K }$ be the label of the $i ^ { \mathrm { { t h } } }$ sample in one-hot encoded representation and $\hat { \mathbf { y } } _ { i } \in [ 0 , 1 ] ^ { K }$ be the predicted probabilities, the cross entropy $H ( \mathbf { y } , { \hat { \mathbf { y } } } )$ is defined as:
+
+$$
+\begin{array} { c } { \displaystyle { H ( \mathbf { y } , \hat { \mathbf { y } } ) = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathbf { y } _ { i } ^ { T } \cdot \log ( \hat { \mathbf { y } } _ { i } ) } } \\ { \displaystyle { = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log ( \hat { \mathbf { y } } _ { i g } ) } } \end{array}
+$$
+
+where $\hat { \mathsf { y } } _ { i g }$ represents the predicted probability of the ground-truth class for the $i ^ { \mathrm { { t h } } }$ sample. Training with cross entropy as the primary objective aims at finding $\hat { \pmb { \theta } } \ : = \ : \arg \operatorname* { m i n } _ { \pmb { \theta } } H ( \mathbf { y } , \hat { \mathbf { y } } )$ , where $\hat { \mathbf { y } } = h _ { \pmb { \theta } } ( \mathbf { x } )$ , $h _ { \theta }$ is a neural network and $\mathbf { x }$ is a sample. Although training using the cross entropy as
+
+(a) $\hat { \mathbf { y } }$ from the model trained with cross entropy.
+
+![](images/1dc5977e854e7588d2baf16b17ed3bad98160b360a1c3942ff50f9df7db4a2ba.jpg)  
+Figure 1: Predicted probabilities $\hat { \mathbf { y } }$ from two training paradigms: (a) With cross entropy as the primary objective. (b) COT: with both primary and complement objectives. The model is ResNet110 and the sample image is from CIFAR10 dataset. The ground-truth class is “horse.” Compared to (b), the model in (a) is confused by other classes such as “airplane” and “automobile,” which suggests (a) might be more susceptible for generalization issues and potentially adversarial attacks.
+
+![](images/52fc17efbfc6fc24b30e39d913e4e8ca185c1ea6e1d2913504d5378b452994d8.jpg)  
+(b) $\hat { \mathbf { y } }$ from the model trained with COT.
+
+(a) Embeddings from the model trained with cross entropy.
+
+![](images/2f3ef2de74e9c0a2882a3ca94e68eb6591d5a4b3dc3b475945017e011a63837b.jpg)  
+(b) Embeddings from the model trained with COT.
+
+![](images/8b447a5f9635b9f771ea9ae7c843e9547b2b185e61aef47375a7aaba1879393d.jpg)  
+Figure 2: Embeddings for CIFAR10 test images from two training paradigms: (a) With cross entropy as the primary objective. (b) COT: training with both primary and complement objectives. The model is ResNet-110, and the “embedding” is the vector representation before taking the softmax operation. The embedding representation of each sample is projected to two dimensions using t-SNE for visualization purpose. Compared to (a), the cluster of each class in (b) is “narrower” in terms of intra-cluster distance. Also, the clusters in (b) seem to have clean and separable boundaries, leading to more accurate and robust classification results.
+
+the primary objective has achieved tremendous success, we have observed one limitation: it exploits mostly the information from the ground-truth class as Eq(1) shows; the information from complement classes (i.e., incorrect classes) has been largely ignored, since the predicted probabilities other than $\hat { \mathsf { y } } _ { i g }$ are zeroed out due to the dot product calculation with the one-hot encoded $\mathbf { y } _ { i }$ . Therefore, for classes other than the ground truth, the model behavior is not explicitly optimized — their predicted probabilities are indirectly minimized when $\hat { \mathbf { y } } _ { i g }$ is maximized since the probabilities sum up to 1. One way to utilize the information from the complement classes is to neutralize their predicted probabilities. To this end, we propose Complement Objective Training (COT), a new training paradigm that achieves this optimization goal without compromising the model’s primary objective. Figure 1 illustrates the comparison between Figure 1a: the predicted probability $\hat { \mathbf { y } }$ from the model trained with just cross entropy as the primary objective, and Figure 1b: $\hat { \mathbf { y } }$ from the model trained with both primary and complement objectives. Training with the complement objective finds the parameters $\pmb \theta$ that evenly suppress complement classes without compromising the primary objective (i.e., maximizing $\hat { \mathsf { y } } _ { g } ^ { \phantom { \dagger } } .$ ), making the model more confident of the ground-truth class.
+
+Figure 2 further illustrates the embeddings of CIFAR10 images calculated from ResNet-110 using two training paradigms: cross entropy and COT. An embedding of an image is the vector representation computed by the ResNet-110 model, before taking the softmax operation. Compared to Figure 2a, the clusters in Figure 2b seem to have clean and separable boundaries, leading to more accurate and robust classification results. The experimental results later in Section 3 further confirm this observation.
+
+Complement objective training requires a function that complements the primary objective. In this paper, we propose “complement entropy” (defined in Section 2) to complement the softmax cross entropy for neutralizing the effects of complement classes. The neural net parameters $\pmb { \theta }$ are then updated by alternating iteratively between (a) minimizing cross entropy to increase $\hat { \mathsf { y } } _ { g }$ , and (b) maximizing complement entropy to neutralize $\hat { \mathbf { y } } _ { j \neq g }$ . Experimental results (in Section 3) confirm that COT improves the accuracies of the state-of-the-art methods for both (a) the image classification tasks on ImageNet-2012, Tiny ImageNet, CIFAR-10, CIFAR-100, and SVHN, and (b) language understanding tasks on machine translation and speech recognition. Furthermore, experimental results also show that models trained by COT are more robust to adversarial attacks.
+
+# 2 COMPLEMENT OBJECTIVE TRAINING
+
+In this section, we first define “Complement Entropy” as the complement objective, and then provide a new training algorithm for updating neural network parameters $\pmb { \theta }$ by alternating iteratively between the primary objective and the complement objective.
+
+# 2.1 COMPLEMENT ENTROPY
+
+Conventionally, training with cross entropy as the primary objective aims at maximizing the predicted probability of the ground-truth class $\hat { \mathsf { y } } _ { g }$ in $\operatorname { E q } ( 1 )$ . As mentioned in the introduction, the proposed COT also maximizes the complement objective for neutralizing the predicted probabilities of the complement classes. To achieve this, we propose “complement entropy” as the complement objective; complement entropy $C ( \cdot )$ is defined to be the average of sample-wise entropies over complement classes in a mini-batch:
+
+$$
+\begin{array} { l } { { \displaystyle C ( \hat { \mathbf { y } } _ { \bar { c } } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { H } ( \hat { \mathbf { y } } _ { i \bar { c } } ) } } \\ { { \displaystyle ~ = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 , j \neq g } ^ { K } ( \frac { \hat { \mathbf { y } } _ { i j } } { 1 - \hat { \mathbf { y } } _ { i g } } ) \log ( \frac { \hat { \mathbf { y } } _ { i j } } { 1 - \hat { \mathbf { y } } _ { i g } } ) } } \end{array}
+$$
+
+$\mathcal { H } ( \cdot )$ is the entropy function. All the symbols and notations used in this paper are summarized in Table 1. One thing worth noticing is that this sample-wise entropy is calculated by considering only the complement classes other than the ground-truth class $g$ . The sample-wise predicted probability $\hat { \mathsf { y } } _ { i j }$ is normalized by one minus the ground-truth probability (i.e., $\mathrm { 1 - } \hat { \mathrm { y } } _ { i g } )$ . The term $\hat { \mathsf { y } } _ { i j } / ( 1 - \hat { \mathsf { y } } _ { i g } )$ can be understood as: conditioned on the ground-truth class $g$ not happening, the predicted probability to see the class $j$ for the $i ^ { \mathrm { { t h } } }$ sample. Since the entropy is maximized when the events are equally likely to occur, optimizing on the complement entropy drives $\hat { \mathsf { y } } _ { i j }$ to $( 1 - \hat { \mathbf y } _ { i g } ) / ( K - 1 )$ , which essentially neutralizes the predicted probability of complement classes as $K$ grows large. In other words, maximizing the complement entropy “flattens” the predicted probabilities of complement classes $\hat { \mathsf { y } } _ { j \neq g }$ . We conjecture that, when $\hat { \mathsf { y } } _ { j \neq g }$ are neutralized, the neural net $h _ { \theta }$ generalizes better, since it is less likely to have an incorrect class with a sufficiently high predicted probability to “challenge” the ground-truth class.
+
+# 2.2 TRAINING WITH COMPLEMENT OBJECTIVE
+
+Given a training procedure using a primary objective, such as softmax cross entropy, one can easily adopt the complement entropy to turn the procedure into a Complement Objective Training (COT). Algorithm 1 describes the new training mechanism by alternating iteratively between the primary and complement objectives. At each training step, the cross entropy is first calculated as the loss value to update the model parameters; next, the complement entropy is calculated as the loss value to perform the second update. Therefore, additional forward and backward propagation are required in each iteration when using the complement objective, making the total training time empirically 1.6 times longer.
+
+Table 1: Notations used in this paper.   
+
+<table><tr><td>Symbol</td><td>Meaning</td></tr><tr><td>yi yi</td><td>One-hot vector representing the label of the ith sample. The predicted probability for each class for the ith sample.</td></tr><tr><td>g</td><td>Index of the ground-truth class.</td></tr><tr><td>yij oryij ye</td><td>The jth class (element) of yi oryi.</td></tr><tr><td>H(,)</td><td>Predicted probabilities of of the complement (incorrect) classes.</td></tr><tr><td>H()</td><td>Cross entropy function. Entropy function.</td></tr><tr><td>C(.)</td><td></td></tr><tr><td>N and K</td><td>Complement entropy. Total number of samples and total number of classes.</td></tr></table>
+
+<table><tr><td>Algorithm 1: Training by alternating between primary and complement objectives</td></tr><tr><td>1 for t ← 1 to ntrain_steps do</td></tr><tr><td>1N 2</td></tr><tr><td>1. Update parameters by Primary Objective: -1Σ-1log(𝑦ig)</td></tr></table>
+
+# 3 EXPERIMENTS
+
+We perform extensive experiments to evaluate COT on tasks in domains ranging from computer vision to natural language understanding and compare it with the baseline algorithms that achieve state-of-the-art in the respective domains. We also perform experiments to evaluate the robustness of the model trained by COT when attacked by adversarial examples. For each task, we select a stateof-the-art model that has an open-source implementation (referred to as “baseline”) and reproduce their results with the hyper-parameters reported in the paper or code repository. Our code is available at https://github.com/henry8527/COT.
+
+# 3.1 BALANCING TRAINING OBJECTIVES
+
+In theory, the loss values between the primary and the complement objectives can be in different scales; therefore, additional efforts for tuning learning rates might be required for optimizers to achieve the best performance. Empirically, we find the complement entropy in Eq(2) can be modified as follows to balance the losses between the two objectives:
+
+$$
+\begin{array} { l } { { \displaystyle C ^ { \prime } ( \hat { \mathbf { y } } _ { \bar { c } } ) = \frac { 1 } { K - 1 } \cdot C ( \hat { \mathbf { y } } _ { \bar { c } } ) } } \\ { { \displaystyle ~ = \frac { 1 } { K - 1 } \cdot \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { H } ( \hat { \mathbf { y } } _ { i \bar { c } } ) } } \end{array}
+$$
+
+where $K$ is the number of classes. This modification can be treated as the complement entropy $C ( \cdot )$ being “normalized” by $( K - 1 )$ . For all the experiments conducted in this paper, we use this normalized complement entropy as the complement objective to improve the baselines without further tuning of learning rates.
+
+# 3.2 IMAGE CLASSIFICATION
+
+We consider the following datasets for experiments with image classification: CIFAR-10, CIFAR100, SVHN, Tiny ImageNet and ImageNet-2012. For CIFAR-10, CIFAR-100 and SVHN, we choose the following baseline models: ResNet-110 (He et al., 2016b), PreAct ResNet-18 (He et al., 2016a), ResNeXt-29 $( 2 \times 6 4 \mathrm { d } )$ (Xie et al., 2017), WideResNet-28-10 (Zagoruyko & Komodakis, 2016) and DenseNet-BC-121 (Huang et al., 2017b) with a growth rate of 32. For those five models, we use a consistent set of settings below, which is described in (He et al., 2016b). Specifically, the models are trained using SGD optimizer with momentum of 0.9. Weight decay is set to be 0.0001 and learning rate starts at 0.1, then being divided by 10 at the $1 0 0 ^ { \mathrm { { t h } } }$ and $1 5 \dot { 0 } ^ { \mathrm { t h } }$ epoch. The models are trained for 200 epochs, with mini-batches of size 128. The only exception here is for training WideResNet-28-10, we follow the settings described in (Zagoruyko & Komodakis, 2016), and the learning rate is divided by 10 at the $6 0 ^ { \mathrm { { \bar { t } h } } }$ , $1 2 0 ^ { \mathrm { t h } }$ and $1 8 0 ^ { \mathrm { t h } }$ epoch. In addition, no dropout (Srivastava et al., 2014) is applied to any baseline according to the best practices in (Ioffe & Szegedy, 2015). For Tiny ImageNet and ImageNet-2012, the baseline models are slightly different: we follow the settings from (Zhang et al., 2018), and the details are described in the corresponding paragraphs.
+
+CIFAR-10 and CIFAR-100. CIFAR-10 and CIFAR-100 are datasets (Krizhevsky, 2009) that contain colored natural images of $3 2 \mathrm { x } 3 2 $ pixels, in 10 and 100 classes, respectively. We follow the baseline settings (He et al., 2016b) to pre-process the datasets; both datasets are split into a training set with 50,000 samples and a testing set with 10,000 samples. During training, zero-padding, random cropping, and horizontal mirroring are applied to the images with a probability of 0.5. For the testing images, we use the original images of $3 2 \mathrm { x } 3 2 $ pixels.
+
+A comparison between the models trained using the primary objective and the COT model is illustrated in Figures 3a and 4a for CIFAR-10 and CIFAR-100 respectively. We show that COT consistently outperforms the baseline models. Some of the models, for example, ResNetXt-29, achieves a significant performance boost of $12 . 5 \%$ in terms of classification errors. For some other models such as WideResNet-28-10 and DenseNet-BC-121, the improvements are not as significant but are still large enough to justify the differences. Similar conclusions can be observed from the CIFAR-100 dataset. In addition to the comparisons of the performance, we also present the change of testing errors over the course of the training in Figures 3b and 4b for the ResNet-110 model. Following the standard training practice, learning rates drop after the $1 0 0 ^ { \mathrm { t h } }$ epoch, which corresponds to a drop in testing errors. As we can see from the plot, COT outperforms consistently compared to the baseline models when the models are close to the convergence.
+
+Street View House Numbers (SVHN). The SVHN dataset (Netzer et al., 2011) consists of images extracted from Google Street View. We divide the dataset into a set of 73,257 digits for training and a set of 26,032 digits for testing. When pre-processing the training and validation images, we follow the general practice to normalize pixel values into [-1,1]. Table 2 shows the experimental results and confirms that COT consistently improves the baseline models with the biggest improvement being the ResNet-110 with $1 1 . 7 \%$ reduction on the error rate.
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-110</td><td>7.56</td><td>6.84 (6.99±0.12)</td></tr><tr><td>PreActResNet-18</td><td>5.46</td><td>4.86 (5.08±0.14)</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>5.20</td><td>4.55 (4.69±0.12)</td></tr><tr><td>WideResNet-28-10</td><td>4.40</td><td>4.30 (4.34±0.03)</td></tr><tr><td>DenseNet-BC-121</td><td>4.72</td><td>4.62 (4.67±0.03)</td></tr></table>
+
+(a) Test errors (in $\%$ ) on CIFAR-10. For COT, we repeat 5 runs and report the “best (mean $\pm$ std)” error values.
+
+![](images/b2bf43ea10f0b8c379adb442f9f81f0fe049c4df2f78f096e0127bc9e1ee701d.jpg)  
+(b) Test errors of ResNet-110 on CIFAR-10 over epochs.
+
+Figure 3: Classification errors on CIFAR-10: (a) COT improves all 5 state-of-the-art models. (b) The improvement over epochs. Notice that the performance improvement from COT becomes stable after the $1 0 0 ^ { \mathrm { t h } }$ epoch due to the learning rate decrease.
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-110</td><td>29.22</td><td>27.90</td></tr><tr><td>PreAct ResNet-18</td><td>25.44</td><td>24.73</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>23.45</td><td>21.90</td></tr><tr><td>WideResNet-28-10</td><td>21.91</td><td>20.99</td></tr><tr><td>DenseNet-BC-121</td><td>21.73</td><td>20.54</td></tr></table>
+
+(a) Test errors (in $\%$ ) on CIFAR-100. For COT, we repeat 3 runs and report the mean value.
+
+![](images/bab9830308a6e1257ccc21de57da13917bf8305377892942d5dbb16c5289fe08.jpg)  
+(b) Test errors of ResNet-110 on CIFAR-100 over the epochs.
+
+Figure 4: Classification errors on CIFAR-100: (a) COT improves all 5 state-of-the-art models. (b) The improvement over epochs. Similar to the trend observed in CIFAR-10, the performance improvement from COT becomes stable after the $1 0 0 ^ { \mathrm { t h } }$ epoch due to the learning rate decrease.
+
+Table 2: Test errors $( \mathrm { i n } \% )$ of the baseline models and the COT-trained models on the SVHN dataset. The values presented are the mean values of 3 runs.   
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-110</td><td>4.94</td><td>4.36</td></tr><tr><td>PreAct ResNet-18</td><td>4.31</td><td>3.96</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>4.22</td><td>3.76</td></tr><tr><td>WideResNet-28-10</td><td>3.72</td><td>3.50</td></tr><tr><td>DenseNet-BC-121</td><td>3.52</td><td>3.47</td></tr></table>
+
+Tiny ImageNet. Tiny ImageNet1 dataset is a subset of ImageNet (Deng et al., 2009), which contains 100,000 images for training and 10,000 for testing images across 200 classes. In this dataset, each image is down-sampled to $6 4 \mathrm { x } 6 4$ pixels from the original $2 5 6 \mathrm { x } 2 5 6$ pixels. We consider four state-of-the-art models as baselines: ResNet-50, ResNet-101 (He et al., 2016b), ResNeXt-50 $( 3 2 \times 4 \mathrm { d } )$ and ResNeXt-101 $( 3 2 \times 4 \mathrm { d } )$ (Xie et al., 2017). During training, we follow the standard data-augmentation techniques, such as random cropping, horizontal flipping, and normalization. For each model, the stride of the first convolution layer is modified to adapt images of size $6 4 \mathrm { x } 6 4$ (Huang et al., 2017a). For evaluation, the testing data is only augmented with 56x56 central cropping. The rest of the experimental details are the same as the ones described at the beginning of Section 3.2. Table 3 provides the experimental results, which demonstrate that COT consistently improves the performance of all baseline models.
+
+Table 3: Top-1 Validation errors (in $\%$ ) for the Tiny ImageNet experiments (mean values of 3 runs).   
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-50</td><td>39.39</td><td>39.20</td></tr><tr><td>ResNet-101</td><td>38.23</td><td>37.35</td></tr><tr><td>ResNeXt-50 (32×4d)</td><td>37.36</td><td>36.69</td></tr><tr><td>ResNeXt-101 (32×4d)</td><td>37.02</td><td>36.14</td></tr></table>
+
+ImageNet. ImageNet-2012 dataset (Russakovsky et al., 2015) is one of the largest datasets for image classification, which contains 1.3 million images for training and 50,000 images for testing with 1,000 classes. Random crops and horizontal flips are applied during training (He et al., 2016b), while images in the testing set use $2 2 4 \mathbf { x } 2 2 4$ center crops (1-crop testing) for data augmentation. ResNet-50 is selected as the baseline model, and we follow (Goyal et al., 2017) for the experimental setup: 256 minibatch size, 90 total training epochs, and 0.1 as the initial learning rate starting that is decayed by dividing 10 at the $3 0 ^ { \mathrm { t h } }$ , $6 0 ^ { \mathrm { \tilde { t h } } }$ and $8 0 ^ { \mathrm { t h } }$ epoch. Table 4 shows (a) the error rate2 of baseline reported by (He et al., 2016b) and (b) the error rate of baseline model trained by COT, which confirms COT further improves the baseline performance.
+
+Table 4: Validation errors $( \mathrm { i n } \% )$ ) for the ImageNet-2012 experiments.   
+
+<table><tr><td>Model</td><td></td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-50</td><td>Top-1 Error</td><td>24.7</td><td>24.4</td></tr></table>
+
+# 3.3 NATURAL LANGUAGE UNDERSTANDING
+
+COT is also evaluated on two natural language understanding (NLU) tasks: machine translation and speech recognition. One distinct characteristic of most NLU tasks is a large number of target classes. For example, the machine translation dataset used in this paper, IWSLT 2015 English-Vietnamese (Cettolo et al., 2015), consists of vocabularies of 17,191 English words and 7,709 Vietnamese words. This necessitates the normalized complement entropy in Eq(3).
+
+Machine translation. Neural machine translation (NMT) has popularized the use of neural sequence models (Sutskever et al., 2014; Cho et al., 2014). Specifically, we apply COT on the seq2seq model with Luong attention mechanism (Luong et al., 2015) on the IWSLT 2015 EnglishVietnamese dataset, which contains 133 thousand translation pairs. For validation and testing, we use TED tst2012 and TED tst2013, respectively. For the baseline implementation, we follow the official TensorFlow-NMT implementation3. That is, the number of total training steps is 12,000 and the weight decay starts at the $8 { , } 0 0 0 ^ { \mathrm { t h } }$ step then applied for every 1,000 steps. We experiment models with both greedy decoder and beam search decoder. The model trained by COT gives the best testing results when the beam width is 3, while the baseline uses 10 as the best beam width. Table 5 illustrates the experimental results, showing COT improves testing BLEU scores compared to the baseline NMT model on both greedy decoder and the beam search decoder.
+
+Table 5: Results of IWSLT 2015 English-Vietnamese. The BLEU scores on tst2013 are reported.   
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>NMT (greedy)</td><td>25.5</td><td>25.7</td></tr><tr><td>NMT (beam search)</td><td>26.1</td><td>26.4</td></tr></table>
+
+Speech recognition. For speech recognition, we experiment on Google Commands Dataset (Warden, 2018), which consists of 65,000 one-second utterances of 30 different types such as “Yes,” “No,” “Up,” “Down” and “Stop.” Our baseline model is referenced from (Zhang et al., 2018). We apply the same pre-processing steps as shown in the paper, and perform the short-time Fourier transform on the original waveforms first at a sampling rate of $4 \mathrm { k H z }$ to receive the corresponding spectrograms. We then zero-pad these spectrograms to equalize each sample’s length. For the baseline model, we select VGG-11 (Simonyan & Zisserman, 2014) and train the model for 30 epochs following the steps in (Zhang et al., 2018). We use SGD optimizer with momentum, and weight decay is 0.0001. The learning rate starts at 0.0001 and then is divided by 10 at the $1 0 ^ { \mathrm { t h } }$ and $2 0 ^ { \mathrm { { t h } } }$ epoch. COT improves the baseline by further reducing the error rate by $1 . 5 6 \%$ , as shown in Table 6.
+
+# 3.4 ADVERSARIAL EXAMPLES
+
+An adversarial example is an imperceptibly-perturbed input that results in the model outputting an incorrect answer with high confidence (Szegedy et al., 2014; Goodfellow et al., 2015). Prior literatures have shown that there are several methods to generate effective adversarial examples that greatly mislead the model toward providing wrong predictions.
+
+Table 6: Test errors (in $\%$ ) on Google Commands Dataset (mean values of 3 runs).   
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>VGG-11</td><td>6.06</td><td>4.50</td></tr></table>
+
+As shown in Figure 2, the proposed COT generates embeddings where the class boundaries are clear and well-separated. We believe that the models trained using COT generalize better and are more robust to adversarial attacks. To verify this conjecture, we conduct experiments of white-box attacks to the models trained by COT. We consider a common approach of single-step adversarial attacks: Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2015) that uses the gradient to determine the direction of the perturbation to apply on an input for creating an adversarial example. To set up FGSM white-box attacks on a baseline model, adversarial examples are generated using the gradients calculated based on the primary objective (referred to as the “primary gradient”) of the baseline model. For FGSM white-box attacks on COT, adversarial perturbations are generated based on the sum of the primary gradient and the complement gradient (i.e., the gradient calculated from the complement objective), both gradients from the model trained by COT. In our experiments, the baseline models are the same as in Section 3.2, and the amount of perturbation is limited to a maximum value of 0.1 as described in (Goodfellow et al., 2015) when creating adversarial examples. Furthermore, we also conduct experiments on FGSM transfer attacks, which use the adversarial examples from a baseline model to attack and test the robustness of the model trained by COT.
+
+Table 7: Classification errors $( \mathrm { i n } \% )$ ) on CIFAR-10 under FGSM white-box & transfer attacks.   
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT (White-Box)</td><td>COT (Transfer)</td></tr><tr><td>ResNet-110</td><td>62.23</td><td>52.72</td><td>54.96</td></tr><tr><td>PreAct ResNet-18</td><td>65.60</td><td>56.17</td><td>59.39</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>70.24</td><td>61.55</td><td>65.83</td></tr><tr><td>WideResNet-28-10</td><td>59.39</td><td>55.53</td><td>57.33</td></tr><tr><td>DenseNet-BC-121</td><td>65.97</td><td>55.99</td><td>62.40</td></tr></table>
+
+Table 7 shows the performance of the models on the CIFAR-10 dataset under FGSM white-box and transfer attacks. Generally, the models trained using COT have lower classification error under both FGSM white-box and transfer attacks, which is an indicator that COT models are more robust to both kinds of attacks. We also conduct experiments on the basic iterative attacks using I-FGSM (Kurakin et al., 2017) and the corresponding results can be found in Appendix A.
+
+We conjecture that since the main goal of the complement gradients is to neutralize the probabilities of incorrect classes (instead of maximizing the probability of the correct class), the complement gradients may “push away” primary gradients when forming adversarial perturbations, which might partially answer why COT is more robust to FGSM white-box attacks compared to the baseline. Regarding the transfer attacks, only the primary objective of the baseline model is used to calculate the gradients for generating adversarial examples. In other words, the complement gradients are not considered when generating adversarial examples in the transfer attack, and this might be the reason why models trained by COT are more robust to transfer attacks. Both conjectures leave a large space for future work: using complement objective to defend against more advanced adversarial attacks.
+
+# 4 CONCLUSION AND FUTURE WORK
+
+In this paper, we study Complement Objective Training (COT), a new training paradigm that optimizes the complement objective in addition to the primary objective. We propose complement entropy as the complement objective for neutralizing the effects of complement (incorrect) classes.
+
+Models trained using COT demonstrate superior performance compared to the baseline models. We also find that COT makes the models robust to single-step adversarial attacks.
+
+COT can be extended in several ways: first, in this paper, the complement objective is chosen to be the complement entropy. Non-entropy-based complement objectives should also be considered for future studies, which is left as a straight-line future work. Secondly, the exploration of COT on broader applications remains as an open research question. One example would be applying COT on generative models such as Generative Adversarial Networks (Goodfellow et al., 2014). Another example would be using COT on object detection and segmentation. Finally, in this work, we show using complement objective help defend single-step adversarial attacks; the behavior of COT on more advanced adversarial attacks deserves further investigation and is left as another future work.
+
+# REFERENCES
+
+Mauro Cettolo, Jan Niehues, Sebastian Stu ker, Luisa Bentivogli, Roldano Cattoni, and Marcello Federico. The IWSLT 2015 evaluation campaign. In ICSLP’15, 2015.   
+Kyunghyun Cho, Bart van Merrienboer, C¸ aglar Gulc¸ehre, Fethi Bougares, Holger Schwenk, and ¨ Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. In EMNLP’14, 2014.   
+Jia Deng, Wei Dong, Richard Socher, Li jia Li, Kai Li, and Li Fei-fei. ImageNet: A large-scale hierarchical image database. In CVPR’09, 2009.   
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NIPS’14. 2014.   
+Ian Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. In ICLR’15, 2015.   
+Priya Goyal, Piotr Dollar, Ross B. Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, ´ Andrew Tulloch, Yangqing Jia, and Kaiming He. Accurate, large minibatch SGD: Training ImageNet in 1 hour. arXiv preprint arXiv:1706.02677, 2017.   
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In ECCV’16, 2016a.   
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR’16, 2016b.   
+Gao Huang, Yixuan Li, Geoff Pleiss, Zhuang Liu, John E. Hopcroft, and Kilian Q. Weinberger. Snapshot ensembles: Train 1, get M for free. In ICLR’17, 2017a.   
+Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected convolutional networks. In CVPR’17, 2017b.   
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML’15, 2015.   
+Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, 2009.   
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In NIPS’12, 2012.   
+Solomon Kullback and Richard A. Leibler. On information and sufficiency. Ann. Math. Statist., 1951.   
+Alexey Kurakin, Ian J. Goodfellow, and Samy Bengio. Adversarial examples in the physical world. In ICLR’17 Workshop, 2017.   
+Minh-Thang Luong, Hieu Pham, and Christopher D. Manning. Effective approaches to attentionbased neural machine translation. In EMNLP’15, 2015.   
+Tom M Mitchell et al. Machine learning. Burr Ridge, IL: McGraw Hill, 1997.
+
+Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y. Ng. Reading digits in natural images with unsupervised feature learning. In NIPS’11 Workshop, 2011.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision, 2015.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR’15, 2014.
+
+Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 2014.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V. Le. Sequence to sequence learning with neural networks. In NIPS’14, 2014.
+
+Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In ICLR’14, 2014.
+
+Pete Warden. Speech Commands: A dataset for limited-vocabulary speech recognition. arXiv preprint arXiv:1804.03209, 2018.
+
+Saining Xie, Ross B. Girshick, Piotr Dollar, Zhuowen Tu, and Kaiming He. Aggregated residual ´ transformations for deep neural networks. In CVPR’17, 2017.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. In BMVC’16, 2016.
+
+Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. Mixup: Beyond empirical risk minimization. In ICLR’18, 2018.
+
+# A ITERATIVE FAST GRADIENT SIGN METHOD
+
+Table 8: Classification errors $( \mathrm { i n } \%$ ) on CIFAR-10 under I-FGSM transfer attacks.   
+
+<table><tr><td>Model</td><td>Baseline</td><td>COT (Transfer)</td></tr><tr><td>ResNet-110</td><td>88.00</td><td>84.36</td></tr><tr><td>PreAct ResNet-18</td><td>84.56</td><td>83.77</td></tr><tr><td>ResNeXt-29 (2× 64d)</td><td>87.79</td><td>86.43</td></tr><tr><td>WideResNet-28-10</td><td>81.97</td><td>80.85</td></tr><tr><td>DenseNet-BC-121</td><td>86.95</td><td>83.66</td></tr></table>
+
+Table 8 shows the performance of the models on the CIFAR-10 dataset under I-FGSM transfer attacks. Generally, the models trained using COT have lower classification error under I-FGSM transfer attacks. The number of iteration is set to 10 in the experiment.

parse/train/HyM7AiA5YX/HyM7AiA5YX_content_list.json

@@ -0,0 +1,1056 @@
+[
+    {
+        "type": "text",
+        "text": "COMPLEMENT OBJECTIVE TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hao-Yun Chen1, Pei-Hsin Wang1, Chun-Hao Liu1, Shih-Chieh Chang1, 2, Jia-Yu Pan3, Yu-Ting Chen3, Wei Wei3, and Da-Cheng Juan3 ",
+        "bbox": [
+            199,
+            150,
+            799,
+            180
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Department of Computer Science, National Tsing-Hua University, Hsinchu, Taiwan   \n2Electronic and Optoelectronic System Research Laboratories, ITRI, Hsinchu, Taiwan 3Google Research, Mountain View, CA, USA {haoyunchen,peihsin,newgod1992}@gapp.nthu.edu.tw scchang@cs.nthu.edu.tw {jypan, yutingchen, wewei, dacheng}@google.com ",
+        "bbox": [
+            246,
+            193,
+            753,
+            270
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            311,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning with a primary objective, such as softmax cross entropy for classification and sequence generation, has been the norm for training deep neural networks for years. Although being a widely-adopted approach, using cross entropy as the primary objective exploits mostly the information from the ground-truth class for maximizing data likelihood, and largely ignores information from the complement (incorrect) classes. We argue that, in addition to the primary objective, training also using a complement objective that leverages information from the complement classes can be effective in improving model performance. This motivates us to study a new training paradigm that maximizes the likelihood of the groundtruth class while neutralizing the probabilities of the complement classes. We conduct extensive experiments on multiple tasks ranging from computer vision to natural language understanding. The experimental results confirm that, compared to the conventional training with just one primary objective, training also with the complement objective further improves the performance of the state-of-the-art models across all tasks. In addition to the accuracy improvement, we also show that models trained with both primary and complement objectives are more robust to single-step adversarial attacks. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Statistical learning algorithms work by optimizing towards a training objective. A dominant principle for training is to optimize likelihood (Mitchell et al., 1997), which measures the probability of data given the model under a specific set of parameters. The popularity of deep neural networks has given rise to the use of cross entropy (Kullback & Leibler, 1951) as its primary training objective, since minimizing cross entropy is essentially equivalent to maximizing likelihood for disjoint classes. Cross entropy has become the standard training objective for many tasks including classification (Krizhevsky et al., 2012) and sequence generation (Sutskever et al., 2014). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $\\mathbf { y } _ { i } \\in \\{ 0 , 1 \\} ^ { K }$ be the label of the $i ^ { \\mathrm { { t h } } }$ sample in one-hot encoded representation and $\\hat { \\mathbf { y } } _ { i } \\in [ 0 , 1 ] ^ { K }$ be the predicted probabilities, the cross entropy $H ( \\mathbf { y } , { \\hat { \\mathbf { y } } } )$ is defined as: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a9683b0ec8f753a6a83c46138ad296d517fadfa07f82ff39f04557e636d3cdb.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle { H ( \\mathbf { y } , \\hat { \\mathbf { y } } ) = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathbf { y } _ { i } ^ { T } \\cdot \\log ( \\hat { \\mathbf { y } } _ { i } ) } } \\\\ { \\displaystyle { = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log ( \\hat { \\mathbf { y } } _ { i g } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            781,
+            612,
+            869
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\mathsf { y } } _ { i g }$ represents the predicted probability of the ground-truth class for the $i ^ { \\mathrm { { t h } } }$ sample. Training with cross entropy as the primary objective aims at finding $\\hat { \\pmb { \\theta } } \\ : = \\ : \\arg \\operatorname* { m i n } _ { \\pmb { \\theta } } H ( \\mathbf { y } , \\hat { \\mathbf { y } } )$ , where $\\hat { \\mathbf { y } } = h _ { \\pmb { \\theta } } ( \\mathbf { x } )$ , $h _ { \\theta }$ is a neural network and $\\mathbf { x }$ is a sample. Although training using the cross entropy as ",
+        "bbox": [
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+            876,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "(a) $\\hat { \\mathbf { y } }$ from the model trained with cross entropy. ",
+        "bbox": [
+            181,
+            287,
+            464,
+            301
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/1dc5977e854e7588d2baf16b17ed3bad98160b360a1c3942ff50f9df7db4a2ba.jpg",
+        "image_caption": [
+            "Figure 1: Predicted probabilities $\\hat { \\mathbf { y } }$ from two training paradigms: (a) With cross entropy as the primary objective. (b) COT: with both primary and complement objectives. The model is ResNet110 and the sample image is from CIFAR10 dataset. The ground-truth class is “horse.” Compared to (b), the model in (a) is confused by other classes such as “airplane” and “automobile,” which suggests (a) might be more susceptible for generalization issues and potentially adversarial attacks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            106,
+            462,
+            279
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/52fc17efbfc6fc24b30e39d913e4e8ca185c1ea6e1d2913504d5378b452994d8.jpg",
+        "image_caption": [
+            "(b) $\\hat { \\mathbf { y } }$ from the model trained with COT. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            532,
+            103,
+            818,
+            280
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "(a) Embeddings from the model trained with cross entropy. ",
+        "bbox": [
+            173,
+            551,
+            500,
+            577
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f3ef2de74e9c0a2882a3ca94e68eb6591d5a4b3dc3b475945017e011a63837b.jpg",
+        "image_caption": [
+            "(b) Embeddings from the model trained with COT. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            415,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/8b447a5f9635b9f771ea9ae7c843e9547b2b185e61aef47375a7aaba1879393d.jpg",
+        "image_caption": [
+            "Figure 2: Embeddings for CIFAR10 test images from two training paradigms: (a) With cross entropy as the primary objective. (b) COT: training with both primary and complement objectives. The model is ResNet-110, and the “embedding” is the vector representation before taking the softmax operation. The embedding representation of each sample is projected to two dimensions using t-SNE for visualization purpose. Compared to (a), the cluster of each class in (b) is “narrower” in terms of intra-cluster distance. Also, the clusters in (b) seem to have clean and separable boundaries, leading to more accurate and robust classification results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            501,
+            421,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "the primary objective has achieved tremendous success, we have observed one limitation: it exploits mostly the information from the ground-truth class as Eq(1) shows; the information from complement classes (i.e., incorrect classes) has been largely ignored, since the predicted probabilities other than $\\hat { \\mathsf { y } } _ { i g }$ are zeroed out due to the dot product calculation with the one-hot encoded $\\mathbf { y } _ { i }$ . Therefore, for classes other than the ground truth, the model behavior is not explicitly optimized — their predicted probabilities are indirectly minimized when $\\hat { \\mathbf { y } } _ { i g }$ is maximized since the probabilities sum up to 1. One way to utilize the information from the complement classes is to neutralize their predicted probabilities. To this end, we propose Complement Objective Training (COT), a new training paradigm that achieves this optimization goal without compromising the model’s primary objective. Figure 1 illustrates the comparison between Figure 1a: the predicted probability $\\hat { \\mathbf { y } }$ from the model trained with just cross entropy as the primary objective, and Figure 1b: $\\hat { \\mathbf { y } }$ from the model trained with both primary and complement objectives. Training with the complement objective finds the parameters $\\pmb \\theta$ that evenly suppress complement classes without compromising the primary objective (i.e., maximizing $\\hat { \\mathsf { y } } _ { g } ^ { \\phantom { \\dagger } } .$ ), making the model more confident of the ground-truth class. ",
+        "bbox": [
+            173,
+            729,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 further illustrates the embeddings of CIFAR10 images calculated from ResNet-110 using two training paradigms: cross entropy and COT. An embedding of an image is the vector representation computed by the ResNet-110 model, before taking the softmax operation. Compared to Figure 2a, the clusters in Figure 2b seem to have clean and separable boundaries, leading to more accurate and robust classification results. The experimental results later in Section 3 further confirm this observation. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Complement objective training requires a function that complements the primary objective. In this paper, we propose “complement entropy” (defined in Section 2) to complement the softmax cross entropy for neutralizing the effects of complement classes. The neural net parameters $\\pmb { \\theta }$ are then updated by alternating iteratively between (a) minimizing cross entropy to increase $\\hat { \\mathsf { y } } _ { g }$ , and (b) maximizing complement entropy to neutralize $\\hat { \\mathbf { y } } _ { j \\neq g }$ . Experimental results (in Section 3) confirm that COT improves the accuracies of the state-of-the-art methods for both (a) the image classification tasks on ImageNet-2012, Tiny ImageNet, CIFAR-10, CIFAR-100, and SVHN, and (b) language understanding tasks on machine translation and speech recognition. Furthermore, experimental results also show that models trained by COT are more robust to adversarial attacks. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 COMPLEMENT OBJECTIVE TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            340,
+            516,
+            357
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first define “Complement Entropy” as the complement objective, and then provide a new training algorithm for updating neural network parameters $\\pmb { \\theta }$ by alternating iteratively between the primary objective and the complement objective. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 COMPLEMENT ENTROPY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            431,
+            383,
+            445
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Conventionally, training with cross entropy as the primary objective aims at maximizing the predicted probability of the ground-truth class $\\hat { \\mathsf { y } } _ { g }$ in $\\operatorname { E q } ( 1 )$ . As mentioned in the introduction, the proposed COT also maximizes the complement objective for neutralizing the predicted probabilities of the complement classes. To achieve this, we propose “complement entropy” as the complement objective; complement entropy $C ( \\cdot )$ is defined to be the average of sample-wise entropies over complement classes in a mini-batch: ",
+        "bbox": [
+            174,
+            457,
+            823,
+            541
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/123510907524a0c7aeb4968c5c1cfc35999c5dc9edc001187efa721b3f81d670.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle C ( \\hat { \\mathbf { y } } _ { \\bar { c } } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathcal { H } ( \\hat { \\mathbf { y } } _ { i \\bar { c } } ) } } \\\\ { { \\displaystyle ~ = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 , j \\neq g } ^ { K } ( \\frac { \\hat { \\mathbf { y } } _ { i j } } { 1 - \\hat { \\mathbf { y } } _ { i g } } ) \\log ( \\frac { \\hat { \\mathbf { y } } _ { i j } } { 1 - \\hat { \\mathbf { y } } _ { i g } } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            547,
+            666,
+            638
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { H } ( \\cdot )$ is the entropy function. All the symbols and notations used in this paper are summarized in Table 1. One thing worth noticing is that this sample-wise entropy is calculated by considering only the complement classes other than the ground-truth class $g$ . The sample-wise predicted probability $\\hat { \\mathsf { y } } _ { i j }$ is normalized by one minus the ground-truth probability (i.e., $\\mathrm { 1 - } \\hat { \\mathrm { y } } _ { i g } )$ . The term $\\hat { \\mathsf { y } } _ { i j } / ( 1 - \\hat { \\mathsf { y } } _ { i g } )$ can be understood as: conditioned on the ground-truth class $g$ not happening, the predicted probability to see the class $j$ for the $i ^ { \\mathrm { { t h } } }$ sample. Since the entropy is maximized when the events are equally likely to occur, optimizing on the complement entropy drives $\\hat { \\mathsf { y } } _ { i j }$ to $( 1 - \\hat { \\mathbf y } _ { i g } ) / ( K - 1 )$ , which essentially neutralizes the predicted probability of complement classes as $K$ grows large. In other words, maximizing the complement entropy “flattens” the predicted probabilities of complement classes $\\hat { \\mathsf { y } } _ { j \\neq g }$ . We conjecture that, when $\\hat { \\mathsf { y } } _ { j \\neq g }$ are neutralized, the neural net $h _ { \\theta }$ generalizes better, since it is less likely to have an incorrect class with a sufficiently high predicted probability to “challenge” the ground-truth class. ",
+        "bbox": [
+            173,
+            643,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 TRAINING WITH COMPLEMENT OBJECTIVE ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            828,
+            509,
+            842
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a training procedure using a primary objective, such as softmax cross entropy, one can easily adopt the complement entropy to turn the procedure into a Complement Objective Training (COT). Algorithm 1 describes the new training mechanism by alternating iteratively between the primary and complement objectives. At each training step, the cross entropy is first calculated as the loss value to update the model parameters; next, the complement entropy is calculated as the loss value to perform the second update. Therefore, additional forward and backward propagation are required in each iteration when using the complement objective, making the total training time empirically 1.6 times longer. ",
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+        "img_path": "images/e77f304234323518a5ccd0550ab0f6e7d02c7b29f9b6909cebc2e517bc1fe0c6.jpg",
+        "table_caption": [
+            "Table 1: Notations used in this paper. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Symbol</td><td>Meaning</td></tr><tr><td>yi yi</td><td>One-hot vector representing the label of the ith sample. The predicted probability for each class for the ith sample.</td></tr><tr><td>g</td><td>Index of the ground-truth class.</td></tr><tr><td>yij oryij ye</td><td>The jth class (element) of yi oryi.</td></tr><tr><td>H(,)</td><td>Predicted probabilities of of the complement (incorrect) classes.</td></tr><tr><td>H()</td><td>Cross entropy function. Entropy function.</td></tr><tr><td>C(.)</td><td></td></tr><tr><td>N and K</td><td>Complement entropy. Total number of samples and total number of classes.</td></tr></table>",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm 1: Training by alternating between primary and complement objectives</td></tr><tr><td>1 for t ← 1 to ntrain_steps do</td></tr><tr><td>1N 2</td></tr><tr><td>1. Update parameters by Primary Objective: -1Σ-1log(𝑦ig)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We perform extensive experiments to evaluate COT on tasks in domains ranging from computer vision to natural language understanding and compare it with the baseline algorithms that achieve state-of-the-art in the respective domains. We also perform experiments to evaluate the robustness of the model trained by COT when attacked by adversarial examples. For each task, we select a stateof-the-art model that has an open-source implementation (referred to as “baseline”) and reproduce their results with the hyper-parameters reported in the paper or code repository. Our code is available at https://github.com/henry8527/COT. ",
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+    {
+        "type": "text",
+        "text": "3.1 BALANCING TRAINING OBJECTIVES ",
+        "text_level": 1,
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+        "text": "In theory, the loss values between the primary and the complement objectives can be in different scales; therefore, additional efforts for tuning learning rates might be required for optimizers to achieve the best performance. Empirically, we find the complement entropy in Eq(2) can be modified as follows to balance the losses between the two objectives: ",
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+    {
+        "type": "equation",
+        "img_path": "images/531d60f0397032c1ba60ccb7fda1e376ea5d086062531b169dce7208a69bc1b3.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle C ^ { \\prime } ( \\hat { \\mathbf { y } } _ { \\bar { c } } ) = \\frac { 1 } { K - 1 } \\cdot C ( \\hat { \\mathbf { y } } _ { \\bar { c } } ) } } \\\\ { { \\displaystyle ~ = \\frac { 1 } { K - 1 } \\cdot \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathcal { H } ( \\hat { \\mathbf { y } } _ { i \\bar { c } } ) } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $K$ is the number of classes. This modification can be treated as the complement entropy $C ( \\cdot )$ being “normalized” by $( K - 1 )$ . For all the experiments conducted in this paper, we use this normalized complement entropy as the complement objective to improve the baselines without further tuning of learning rates. ",
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+    {
+        "type": "text",
+        "text": "3.2 IMAGE CLASSIFICATION ",
+        "text_level": 1,
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+        "text": "We consider the following datasets for experiments with image classification: CIFAR-10, CIFAR100, SVHN, Tiny ImageNet and ImageNet-2012. For CIFAR-10, CIFAR-100 and SVHN, we choose the following baseline models: ResNet-110 (He et al., 2016b), PreAct ResNet-18 (He et al., 2016a), ResNeXt-29 $( 2 \\times 6 4 \\mathrm { d } )$ (Xie et al., 2017), WideResNet-28-10 (Zagoruyko & Komodakis, 2016) and DenseNet-BC-121 (Huang et al., 2017b) with a growth rate of 32. For those five models, we use a consistent set of settings below, which is described in (He et al., 2016b). Specifically, the models are trained using SGD optimizer with momentum of 0.9. Weight decay is set to be 0.0001 and learning rate starts at 0.1, then being divided by 10 at the $1 0 0 ^ { \\mathrm { { t h } } }$ and $1 5 \\dot { 0 } ^ { \\mathrm { t h } }$ epoch. The models are trained for 200 epochs, with mini-batches of size 128. The only exception here is for training WideResNet-28-10, we follow the settings described in (Zagoruyko & Komodakis, 2016), and the learning rate is divided by 10 at the $6 0 ^ { \\mathrm { { \\bar { t } h } } }$ , $1 2 0 ^ { \\mathrm { t h } }$ and $1 8 0 ^ { \\mathrm { t h } }$ epoch. In addition, no dropout (Srivastava et al., 2014) is applied to any baseline according to the best practices in (Ioffe & Szegedy, 2015). For Tiny ImageNet and ImageNet-2012, the baseline models are slightly different: we follow the settings from (Zhang et al., 2018), and the details are described in the corresponding paragraphs. ",
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+    {
+        "type": "text",
+        "text": "CIFAR-10 and CIFAR-100. CIFAR-10 and CIFAR-100 are datasets (Krizhevsky, 2009) that contain colored natural images of $3 2 \\mathrm { x } 3 2 $ pixels, in 10 and 100 classes, respectively. We follow the baseline settings (He et al., 2016b) to pre-process the datasets; both datasets are split into a training set with 50,000 samples and a testing set with 10,000 samples. During training, zero-padding, random cropping, and horizontal mirroring are applied to the images with a probability of 0.5. For the testing images, we use the original images of $3 2 \\mathrm { x } 3 2 $ pixels. ",
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+        "type": "text",
+        "text": "A comparison between the models trained using the primary objective and the COT model is illustrated in Figures 3a and 4a for CIFAR-10 and CIFAR-100 respectively. We show that COT consistently outperforms the baseline models. Some of the models, for example, ResNetXt-29, achieves a significant performance boost of $12 . 5 \\%$ in terms of classification errors. For some other models such as WideResNet-28-10 and DenseNet-BC-121, the improvements are not as significant but are still large enough to justify the differences. Similar conclusions can be observed from the CIFAR-100 dataset. In addition to the comparisons of the performance, we also present the change of testing errors over the course of the training in Figures 3b and 4b for the ResNet-110 model. Following the standard training practice, learning rates drop after the $1 0 0 ^ { \\mathrm { t h } }$ epoch, which corresponds to a drop in testing errors. As we can see from the plot, COT outperforms consistently compared to the baseline models when the models are close to the convergence. ",
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+    {
+        "type": "text",
+        "text": "Street View House Numbers (SVHN). The SVHN dataset (Netzer et al., 2011) consists of images extracted from Google Street View. We divide the dataset into a set of 73,257 digits for training and a set of 26,032 digits for testing. When pre-processing the training and validation images, we follow the general practice to normalize pixel values into [-1,1]. Table 2 shows the experimental results and confirms that COT consistently improves the baseline models with the biggest improvement being the ResNet-110 with $1 1 . 7 \\%$ reduction on the error rate. ",
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+        "table_footnote": [
+            "(a) Test errors (in $\\%$ ) on CIFAR-10. For COT, we repeat 5 runs and report the “best (mean $\\pm$ std)” error values. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-110</td><td>7.56</td><td>6.84 (6.99±0.12)</td></tr><tr><td>PreActResNet-18</td><td>5.46</td><td>4.86 (5.08±0.14)</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>5.20</td><td>4.55 (4.69±0.12)</td></tr><tr><td>WideResNet-28-10</td><td>4.40</td><td>4.30 (4.34±0.03)</td></tr><tr><td>DenseNet-BC-121</td><td>4.72</td><td>4.62 (4.67±0.03)</td></tr></table>",
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+    {
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+        "img_path": "images/b2bf43ea10f0b8c379adb442f9f81f0fe049c4df2f78f096e0127bc9e1ee701d.jpg",
+        "image_caption": [
+            "(b) Test errors of ResNet-110 on CIFAR-10 over epochs. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Figure 3: Classification errors on CIFAR-10: (a) COT improves all 5 state-of-the-art models. (b) The improvement over epochs. Notice that the performance improvement from COT becomes stable after the $1 0 0 ^ { \\mathrm { t h } }$ epoch due to the learning rate decrease. ",
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+        "table_caption": [],
+        "table_footnote": [
+            "(a) Test errors (in $\\%$ ) on CIFAR-100. For COT, we repeat 3 runs and report the mean value. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-110</td><td>29.22</td><td>27.90</td></tr><tr><td>PreAct ResNet-18</td><td>25.44</td><td>24.73</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>23.45</td><td>21.90</td></tr><tr><td>WideResNet-28-10</td><td>21.91</td><td>20.99</td></tr><tr><td>DenseNet-BC-121</td><td>21.73</td><td>20.54</td></tr></table>",
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+    {
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+        "img_path": "images/bab9830308a6e1257ccc21de57da13917bf8305377892942d5dbb16c5289fe08.jpg",
+        "image_caption": [
+            "(b) Test errors of ResNet-110 on CIFAR-100 over the epochs. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Figure 4: Classification errors on CIFAR-100: (a) COT improves all 5 state-of-the-art models. (b) The improvement over epochs. Similar to the trend observed in CIFAR-10, the performance improvement from COT becomes stable after the $1 0 0 ^ { \\mathrm { t h } }$ epoch due to the learning rate decrease. ",
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+        "table_caption": [
+            "Table 2: Test errors $( \\mathrm { i n } \\% )$ of the baseline models and the COT-trained models on the SVHN dataset. The values presented are the mean values of 3 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-110</td><td>4.94</td><td>4.36</td></tr><tr><td>PreAct ResNet-18</td><td>4.31</td><td>3.96</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>4.22</td><td>3.76</td></tr><tr><td>WideResNet-28-10</td><td>3.72</td><td>3.50</td></tr><tr><td>DenseNet-BC-121</td><td>3.52</td><td>3.47</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Tiny ImageNet. Tiny ImageNet1 dataset is a subset of ImageNet (Deng et al., 2009), which contains 100,000 images for training and 10,000 for testing images across 200 classes. In this dataset, each image is down-sampled to $6 4 \\mathrm { x } 6 4$ pixels from the original $2 5 6 \\mathrm { x } 2 5 6$ pixels. We consider four state-of-the-art models as baselines: ResNet-50, ResNet-101 (He et al., 2016b), ResNeXt-50 $( 3 2 \\times 4 \\mathrm { d } )$ and ResNeXt-101 $( 3 2 \\times 4 \\mathrm { d } )$ (Xie et al., 2017). During training, we follow the standard data-augmentation techniques, such as random cropping, horizontal flipping, and normalization. For each model, the stride of the first convolution layer is modified to adapt images of size $6 4 \\mathrm { x } 6 4$ (Huang et al., 2017a). For evaluation, the testing data is only augmented with 56x56 central cropping. The rest of the experimental details are the same as the ones described at the beginning of Section 3.2. Table 3 provides the experimental results, which demonstrate that COT consistently improves the performance of all baseline models. ",
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+    {
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+        "img_path": "images/9c6e6bc365c3f07c6dfc2c814ef60810abd585e3cfc7a590494f8f4d6d72fe15.jpg",
+        "table_caption": [
+            "Table 3: Top-1 Validation errors (in $\\%$ ) for the Tiny ImageNet experiments (mean values of 3 runs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-50</td><td>39.39</td><td>39.20</td></tr><tr><td>ResNet-101</td><td>38.23</td><td>37.35</td></tr><tr><td>ResNeXt-50 (32×4d)</td><td>37.36</td><td>36.69</td></tr><tr><td>ResNeXt-101 (32×4d)</td><td>37.02</td><td>36.14</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "ImageNet. ImageNet-2012 dataset (Russakovsky et al., 2015) is one of the largest datasets for image classification, which contains 1.3 million images for training and 50,000 images for testing with 1,000 classes. Random crops and horizontal flips are applied during training (He et al., 2016b), while images in the testing set use $2 2 4 \\mathbf { x } 2 2 4$ center crops (1-crop testing) for data augmentation. ResNet-50 is selected as the baseline model, and we follow (Goyal et al., 2017) for the experimental setup: 256 minibatch size, 90 total training epochs, and 0.1 as the initial learning rate starting that is decayed by dividing 10 at the $3 0 ^ { \\mathrm { t h } }$ , $6 0 ^ { \\mathrm { \\tilde { t h } } }$ and $8 0 ^ { \\mathrm { t h } }$ epoch. Table 4 shows (a) the error rate2 of baseline reported by (He et al., 2016b) and (b) the error rate of baseline model trained by COT, which confirms COT further improves the baseline performance. ",
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+        "text": "",
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+        "table_caption": [
+            "Table 4: Validation errors $( \\mathrm { i n } \\% )$ ) for the ImageNet-2012 experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td></td><td>Baseline</td><td>COT</td></tr><tr><td>ResNet-50</td><td>Top-1 Error</td><td>24.7</td><td>24.4</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.3 NATURAL LANGUAGE UNDERSTANDING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "COT is also evaluated on two natural language understanding (NLU) tasks: machine translation and speech recognition. One distinct characteristic of most NLU tasks is a large number of target classes. For example, the machine translation dataset used in this paper, IWSLT 2015 English-Vietnamese (Cettolo et al., 2015), consists of vocabularies of 17,191 English words and 7,709 Vietnamese words. This necessitates the normalized complement entropy in Eq(3). ",
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+    {
+        "type": "text",
+        "text": "Machine translation. Neural machine translation (NMT) has popularized the use of neural sequence models (Sutskever et al., 2014; Cho et al., 2014). Specifically, we apply COT on the seq2seq model with Luong attention mechanism (Luong et al., 2015) on the IWSLT 2015 EnglishVietnamese dataset, which contains 133 thousand translation pairs. For validation and testing, we use TED tst2012 and TED tst2013, respectively. For the baseline implementation, we follow the official TensorFlow-NMT implementation3. That is, the number of total training steps is 12,000 and the weight decay starts at the $8 { , } 0 0 0 ^ { \\mathrm { t h } }$ step then applied for every 1,000 steps. We experiment models with both greedy decoder and beam search decoder. The model trained by COT gives the best testing results when the beam width is 3, while the baseline uses 10 as the best beam width. Table 5 illustrates the experimental results, showing COT improves testing BLEU scores compared to the baseline NMT model on both greedy decoder and the beam search decoder. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f46b31ceda5b282b42e2693f44d7e79c182cc8914d0e884dd91edfe0320ed8e7.jpg",
+        "table_caption": [
+            "Table 5: Results of IWSLT 2015 English-Vietnamese. The BLEU scores on tst2013 are reported. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>NMT (greedy)</td><td>25.5</td><td>25.7</td></tr><tr><td>NMT (beam search)</td><td>26.1</td><td>26.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Speech recognition. For speech recognition, we experiment on Google Commands Dataset (Warden, 2018), which consists of 65,000 one-second utterances of 30 different types such as “Yes,” “No,” “Up,” “Down” and “Stop.” Our baseline model is referenced from (Zhang et al., 2018). We apply the same pre-processing steps as shown in the paper, and perform the short-time Fourier transform on the original waveforms first at a sampling rate of $4 \\mathrm { k H z }$ to receive the corresponding spectrograms. We then zero-pad these spectrograms to equalize each sample’s length. For the baseline model, we select VGG-11 (Simonyan & Zisserman, 2014) and train the model for 30 epochs following the steps in (Zhang et al., 2018). We use SGD optimizer with momentum, and weight decay is 0.0001. The learning rate starts at 0.0001 and then is divided by 10 at the $1 0 ^ { \\mathrm { t h } }$ and $2 0 ^ { \\mathrm { { t h } } }$ epoch. COT improves the baseline by further reducing the error rate by $1 . 5 6 \\%$ , as shown in Table 6. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.4 ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "An adversarial example is an imperceptibly-perturbed input that results in the model outputting an incorrect answer with high confidence (Szegedy et al., 2014; Goodfellow et al., 2015). Prior literatures have shown that there are several methods to generate effective adversarial examples that greatly mislead the model toward providing wrong predictions. ",
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+    {
+        "type": "table",
+        "img_path": "images/cae281f1aa45910bc2b649a1212ae164e4e960972c647fd8b4b5f5c5e104283a.jpg",
+        "table_caption": [
+            "Table 6: Test errors (in $\\%$ ) on Google Commands Dataset (mean values of 3 runs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT</td></tr><tr><td>VGG-11</td><td>6.06</td><td>4.50</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "As shown in Figure 2, the proposed COT generates embeddings where the class boundaries are clear and well-separated. We believe that the models trained using COT generalize better and are more robust to adversarial attacks. To verify this conjecture, we conduct experiments of white-box attacks to the models trained by COT. We consider a common approach of single-step adversarial attacks: Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2015) that uses the gradient to determine the direction of the perturbation to apply on an input for creating an adversarial example. To set up FGSM white-box attacks on a baseline model, adversarial examples are generated using the gradients calculated based on the primary objective (referred to as the “primary gradient”) of the baseline model. For FGSM white-box attacks on COT, adversarial perturbations are generated based on the sum of the primary gradient and the complement gradient (i.e., the gradient calculated from the complement objective), both gradients from the model trained by COT. In our experiments, the baseline models are the same as in Section 3.2, and the amount of perturbation is limited to a maximum value of 0.1 as described in (Goodfellow et al., 2015) when creating adversarial examples. Furthermore, we also conduct experiments on FGSM transfer attacks, which use the adversarial examples from a baseline model to attack and test the robustness of the model trained by COT. ",
+        "bbox": [
+            174,
+            246,
+            825,
+            454
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/0693c6e8411b4204a486fa60d4638a9c4a9ee882bdb5a2d6379294d5c92677ff.jpg",
+        "table_caption": [
+            "Table 7: Classification errors $( \\mathrm { i n } \\% )$ ) on CIFAR-10 under FGSM white-box & transfer attacks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT (White-Box)</td><td>COT (Transfer)</td></tr><tr><td>ResNet-110</td><td>62.23</td><td>52.72</td><td>54.96</td></tr><tr><td>PreAct ResNet-18</td><td>65.60</td><td>56.17</td><td>59.39</td></tr><tr><td>ResNeXt-29 (2×64d)</td><td>70.24</td><td>61.55</td><td>65.83</td></tr><tr><td>WideResNet-28-10</td><td>59.39</td><td>55.53</td><td>57.33</td></tr><tr><td>DenseNet-BC-121</td><td>65.97</td><td>55.99</td><td>62.40</td></tr></table>",
+        "bbox": [
+            240,
+            502,
+            756,
+            602
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 7 shows the performance of the models on the CIFAR-10 dataset under FGSM white-box and transfer attacks. Generally, the models trained using COT have lower classification error under both FGSM white-box and transfer attacks, which is an indicator that COT models are more robust to both kinds of attacks. We also conduct experiments on the basic iterative attacks using I-FGSM (Kurakin et al., 2017) and the corresponding results can be found in Appendix A. ",
+        "bbox": [
+            173,
+            625,
+            825,
+            694
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We conjecture that since the main goal of the complement gradients is to neutralize the probabilities of incorrect classes (instead of maximizing the probability of the correct class), the complement gradients may “push away” primary gradients when forming adversarial perturbations, which might partially answer why COT is more robust to FGSM white-box attacks compared to the baseline. Regarding the transfer attacks, only the primary objective of the baseline model is used to calculate the gradients for generating adversarial examples. In other words, the complement gradients are not considered when generating adversarial examples in the transfer attack, and this might be the reason why models trained by COT are more robust to transfer attacks. Both conjectures leave a large space for future work: using complement objective to defend against more advanced adversarial attacks. ",
+        "bbox": [
+            174,
+            700,
+            825,
+            827
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            848,
+            495,
+            864
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we study Complement Objective Training (COT), a new training paradigm that optimizes the complement objective in addition to the primary objective. We propose complement entropy as the complement objective for neutralizing the effects of complement (incorrect) classes. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Models trained using COT demonstrate superior performance compared to the baseline models. We also find that COT makes the models robust to single-step adversarial attacks. ",
+        "bbox": [
+            173,
+            103,
+            821,
+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "COT can be extended in several ways: first, in this paper, the complement objective is chosen to be the complement entropy. Non-entropy-based complement objectives should also be considered for future studies, which is left as a straight-line future work. Secondly, the exploration of COT on broader applications remains as an open research question. One example would be applying COT on generative models such as Generative Adversarial Networks (Goodfellow et al., 2014). Another example would be using COT on object detection and segmentation. Finally, in this work, we show using complement objective help defend single-step adversarial attacks; the behavior of COT on more advanced adversarial attacks deserves further investigation and is left as another future work. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            251
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A ITERATIVE FAST GRADIENT SIGN METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+            573,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/bfe592ab0c9e451276bceae9db649b5b1e2026a3dc05b8c7e1679dc582dec09f.jpg",
+        "table_caption": [
+            "Table 8: Classification errors $( \\mathrm { i n } \\%$ ) on CIFAR-10 under I-FGSM transfer attacks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Baseline</td><td>COT (Transfer)</td></tr><tr><td>ResNet-110</td><td>88.00</td><td>84.36</td></tr><tr><td>PreAct ResNet-18</td><td>84.56</td><td>83.77</td></tr><tr><td>ResNeXt-29 (2× 64d)</td><td>87.79</td><td>86.43</td></tr><tr><td>WideResNet-28-10</td><td>81.97</td><td>80.85</td></tr><tr><td>DenseNet-BC-121</td><td>86.95</td><td>83.66</td></tr></table>",
+        "bbox": [
+            313,
+            169,
+            684,
+            270
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 8 shows the performance of the models on the CIFAR-10 dataset under I-FGSM transfer attacks. Generally, the models trained using COT have lower classification error under I-FGSM transfer attacks. The number of iteration is set to 10 in the experiment. ",
+        "bbox": [
+            173,
+            291,
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+        ],
+        "page_idx": 10
+    }
+]

parse/train/Q-PA3D1OsDz/Q-PA3D1OsDz.md

@@ -0,0 +1,285 @@
+# MetaAvatar: Learning Animatable Clothed Human Models from Few Depth Images
+
+Shaofei Wang1 shaofei.wang@inf.ethz.ch
+
+Marko Mihajlovic1 marko.mihajlovic@inf.ethz.ch
+
+Qianli Ma1,2 qianli.ma@tue.mpg.de
+
+Andreas Geiger2,3 a.geiger@uni-tuebingen.de
+
+# Siyu Tang1
+
+siyu.tang@inf.ethz.ch
+
+1ETH Zürich 2Max Planck Institute for Intelligent Systems, Tübingen 3University of Tübingen
+
+# Abstract
+
+In this paper, we aim to create generalizable and controllable neural signed distance fields (SDFs) that represent clothed humans from monocular depth observations. Recent advances in deep learning, especially neural implicit representations, have enabled human shape reconstruction and controllable avatar generation from different sensor inputs. However, to generate realistic cloth deformations from novel input poses, watertight meshes or dense full-body scans are usually needed as inputs. Furthermore, due to the difficulty of effectively modeling pose-dependent cloth deformations for diverse body shapes and cloth types, existing approaches resort to per-subject/cloth-type optimization from scratch, which is computationally expensive. In contrast, we propose an approach that can quickly generate realistic clothed human avatars, represented as controllable neural SDFs, given only monocular depth images. We achieve this by using meta-learning to learn an initialization of a hypernetwork that predicts the parameters of neural SDFs. The hypernetwork is conditioned on human poses and represents a clothed neural avatar that deforms non-rigidly according to the input poses. Meanwhile, it is metalearned to effectively incorporate priors of diverse body shapes and cloth types and thus can be much faster to fine-tune, compared to models trained from scratch. We qualitatively and quantitatively show that our approach outperforms state-of-the-art approaches that require complete meshes as inputs while our approach requires only depth frames as inputs and runs orders of magnitudes faster. Furthermore, we demonstrate that our meta-learned hypernetwork is very robust, being the first to generate avatars with realistic dynamic cloth deformations given as few as 8 monocular depth frames.
+
+# 1 Introduction
+
+Representing clothed humans as neural implicit functions is a rising research topic in the computer vision community. Earlier works in this direction address geometric reconstruction of clothed humans from static monocular images [35, 36, 63, 64], RGBD videos [37, 38, 71, 78, 80] or sparse point clouds [12] as direct extensions of neural implicit functions for rigid objects [11, 45, 46, 52]. More recent works advocate to learn shapes in a canonical pose [7, 27, 75] in order to not only handle reconstruction, but also build controllable neural avatars from sensor inputs. However, these works do not model pose-dependent cloth deformation, limiting their realism.
+
+![](images/cd969b2fa547db4de07e2ca7b836bfb967ef08f426e63bcefc978f760839e9c7.jpg)  
+Figure 1: Given as few as 8 monocular depth images and their SMPL fittings, our meta-learned model yields a controllable neural SDF in 2 minutes which synthesizes realistic cloth deformations for unseen body poses. Here we show results of two different subjects wearing different clothes.
+
+On the other hand, traditional parametric human body models [41, 50, 54, 77] can represent pose-dependent soft tissue deformations of minimally-clothed human bodies. Several recent methods [13, 48] proposed to learn neural implicit functions to approximate such parametric models from watertight meshes. However, they cannot be straightforwardly extended to model clothed humans. SCANimate [65] proposed to learn canonicalized dynamic neural Signed Distance Fields (SDFs) controlled by human pose inputs and trained with Implicit Geometric Regularization (IGR [21]), thus circumventing the requirement of watertight meshes. However, SCANimate works only on dense full-body scans with accurate surface normals and further requires expensive per-subject/cloth-type training. These factors limit the applicability of SCANimate for building personalized human avatars from commodity RGBD sensors.
+
+Contrary to all the aforementioned works, we propose to use meta-learning to effectively incorporate priors of dynamic neural SDFs of clothed humans, thus enabling fast fine-tuning (few minutes) for generating new avatars given only a few monocular depth images of unseen clothed humans as inputs. More specifically, we build upon recently proposed ideas of meta-learned initialization for implicit representations [67, 72] to enable fast fine-tuning. Similar to [67], we represent a specific category of objects (in our case, clothed human bodies in the canonical pose) with a neural implicit function and use meta-learning algorithms such as [16, 49] to learn a meta-model. However, unlike [67, 72], where the implicit functions are designed for static reconstruction, we target the generation of dynamic neural SDFs that are controllable by user-specified body poses. We observe that directly conditioning neural implicit functions (represented as a multi-layer perceptron) on body poses lacks the expressiveness to capture high-frequency details of diverse cloth types, and hence propose to meta-learn a hypernetwork [25] that predicts the parameters of the neural implicit function. Overall, the proposed approach, which we name MetaAvatar, yields controllable neural SDFs with dynamic surfaces in minutes via fast fine-tuning, given only a few depth observations of an unseen clothed human and the underlying SMPL [41] fittings (Fig. 1) as inputs. Code and data are public at https://neuralbodies.github.io/metavatar/.
+
+# 2 Related Work
+
+Our approach lies at the intersection of clothed human body modeling, neural implicit representations, and meta-learning. We review related works in the following.
+
+Clothed Human Body Modeling: Earlier works for clothed human body modeling utilize parametric human body models [5, 26, 28, 41, 50, 54, 77] combined with deformation layers [2, 3, 7, 8] to model cloth deformations. However, these approaches cannot model fine clothing details due to their fixed topology, and they cannot handle pose-dependent cloth deformations. Mesh-based approaches that handle articulated deformations of clothes either require accurate surface registration [33, 43, 79, 83] or synthetic data [22, 24, 53] for training. Such requirement for data can be freed by using neural implicit surfaces [10, 51, 65, 73]. For example, SCANimate [65] proposed a weakly supervised approach to learn dynamic clothed human body models from 3D full-body scans which only requires minimally-clothed body registration. However, its training process usually takes one day for each subject/cloth-type combination and requires accurate surface normal information extracted from dense scans. Recent explicit clothed human models [9, 42, 44, 81] can also be learned from unregistered data. Like our method, concurrent work [44] also models pose-dependent shapes across different subjects/cloth-types, but it requires full-body scans for training. In contrast, our approach enables learning of clothed body models in minutes from as few as 8 depth images.
+
+Neural Implicit Representations: Neural implicit representations [11, 45, 46, 52, 55] have been used to tackle both image-based [27, 35, 36, 57, 59, 63, 64, 87] and point cloud-based [7, 12] clothed human reconstruction. Among these works, ARCH [27] was the first one to represent clothed human bodies as a neural implicit function in a canonical pose. However, ARCH does not handle posedependent cloth deformations. Most recently, SCANimate [65] proposed to condition neural implicit functions on joint-rotation vectors (in the form of unit quaternions), such that the canonicalized shapes of the neural avatars change according to the joint angles of the human body, thus representing pose-dependent cloth deformations. However, diverse and complex cloth deformations make it hard to learn a unified prior from different body shapes and cloth types, thus SCANimate resorts to per-subject/cloth-type training which is computationally expensive.
+
+Meta-Learning: Meta-learning is typically used to address few-shot learning, where a few training examples of a new task are given, and the model is required to learn from these examples to achieve good performance on the new task [1, 14, 15, 17–19, 23, 29, 32, 58, 60, 62, 66, 70, 74, 76, 82, 86]. We focus on optimization-based meta-learning, where Model-Agnostic Meta Learning (MAML [16]), Reptile [49] and related alternatives are typically used to learn such models [4, 6, 20, 34, 39, 61]. In general, this line of algorithms tries to learn a "meta-model" that can be updated quickly from new observations with only few gradient steps. Recently, meta-learning has been used to learn a universal initialization of implicit representations for static neural SDFs [67] and radiance fields [72]. MetaSDF [67] demonstrates that only a few gradient update steps are needed to achieve comparable or better results than slower auto-decoder-based approaches [52]. However, [67, 72] only meta-learn static representations, whereas we are interested in dynamic representations conditioned on human body poses. To our best knowledge, we are the first to meta-learn the hypernetwork to generate the parameters of neural SDF networks.
+
+# 3 Fundamentals
+
+We start by briefly reviewing the linear blend skinning (LBS) method [41] and the recent implicit skinning networks [48, 65] that learn to predict skinning weights of cloth surfaces in a weakly supervised manner. Using the learned implicit skinning networks allows us to canonicalize meshes or depth observations of clothed humans, given only minimally-clothed human body model registrations to the meshes. Canonicalization of meshes or points is a necessary step as the dynamic neural SDFs introduced in Section 4 are modeled in canonical space.
+
+# 3.1 Linear Blend Skinning
+
+Linear blend skinning (LBS) is a commonly used technique to deform parametric human body models [5, 26, 41, 50, 54, 77] according to user-specified rigid bone transformations. Given a set of $N$ points in a canonical space, $\hat { \mathbf { X } } = \{ \hat { \mathbf { x } } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ , LBS takes a set of rigid bone transformations (in our case we use 23 local transformations plus one global transformation, assuming an underlying SMPL model) $\{ \mathbf { B } _ { b } \} _ { b = 1 } ^ { 2 4 }$ as inputs, each $\mathbf { B } _ { b }$ being a $4 \times 4$ rotation-translation matrix. For a 3D point xˆ(i) ∈ Xˆ 1, a skinning weight vector is a probability simplex w(i) ∈ [0, 1]24, s.t. P24b=1 $\begin{array} { r } { \sum _ { b = 1 } ^ { 2 4 } \mathbf { w } _ { b } ^ { ( i ) } = 1 } \end{array}$ , that defines the affinity of the point $\hat { \mathbf { x } } ^ { ( i ) }$ to each of the bone transformations $\{ \mathbf { B } _ { b } \} _ { b = 1 } ^ { 2 4 }$ . The set of transformed points $\mathbf { X } = \{ \mathbf { x } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ of the clothed human is related to $\hat { \mathbf X }$ via:
+
+$$
+\begin{array} { l } { { \displaystyle { \bf x } ^ { ( i ) } = L B S \left( \hat { { \bf x } } ^ { ( i ) } , \{ { \bf B } _ { b } \} , { \bf w } ^ { ( i ) } \right) = \left( \sum _ { b = 1 } ^ { 2 4 } { \bf w } _ { b } ^ { ( i ) } { \bf B } _ { b } \right) \hat { \bf x } ^ { ( i ) } } , ~ \forall i = 1 , \ldots , N }  \\ { { \displaystyle \hat { \bf x } ^ { ( i ) } = L B S ^ { - 1 } \left( { \bf x } ^ { ( i ) } , \{ { \bf B } _ { b } \} , { \bf w } ^ { ( i ) } \right) = \left( \sum _ { b = 1 } ^ { 2 4 } { \bf w } _ { b } ^ { ( i ) } { \bf B } _ { b } \right) ^ { - 1 } { \bf x } ^ { ( i ) } } , ~ \forall i = 1 , \ldots , N } \end{array}
+$$
+
+where Eq. (1) is referred to as the LBS function and Eq. (2) is referred to as the inverse-LBS function. The process of applying Eq. (1) to all points in $\hat { \bf X }$ is often referred to as forward skinning while the process of applying Eq. (2) is referred to as inverse skinning.
+
+# 3.2 Implicit Skinning Networks
+
+Recent articulated implicit representations [48, 65] have proposed to learn functions that predict the forward/inverse skinning weights for arbitrary points in $\bar { \mathbb { R } ^ { 3 } }$ . We follow this approach, but take advantage of a convolutional point-cloud encoder [56] for improved generalization. Formally, we define the implicit forward and inverse skinning networks as $h _ { \mathrm { f w d } } ( \cdot , \cdot ) : ( \mathbb { R } ^ { 3 \times K } , \mathbb { R } ^ { 3 } ) \mapsto \mathbb { R } ^ { 2 \bar { 4 } }$ and $h _ { \mathrm { i n v } } ( \cdot , \cdot ) : ( \mathbb { R } ^ { 5 \times K } , \mathbb { R } ^ { 3 } ) \mapsto \mathbb { R } ^ { 2 4 }$ , respectively. Both networks take as input a point cloud with $K$ points and a query point for which they predict skinning weights. Therefore, we can analogously re-define Eq. $\displaystyle ( \begin{array} { l l } \end{array} , \begin{array} { \ l } \end{array} )$ respectively as:
+
+$$
+\begin{array} { l } { { \displaystyle { \bf x } ^ { ( i ) } = \left( \sum _ { b = 1 } ^ { 2 4 } h _ { \mathrm { f w d } } ( \hat { { \bf X } } , \hat { { \bf x } } ^ { ( i ) } ) _ { b } { \bf B } _ { b } \right) \hat { { \bf x } } ^ { ( i ) } } , ~ \forall i = 1 , \ldots , N }  \\ { { \displaystyle { \hat { \bf x } ^ { ( i ) } = \left( \sum _ { b = 1 } ^ { 2 4 } h _ { \mathrm { i n v } } ( { \bf X } , { \bf x } ^ { ( i ) } ) _ { b } { \bf B } _ { b } \right) ^ { - 1 } { \bf x } ^ { ( i ) } } , ~ \forall i = 1 , \ldots , N } } \end{array}
+$$
+
+Training the Skinning Network: We follow the setting of SCANimate [65], where a dataset of
+
+observed point clouds $\{ { \bf X } \}$ and their underlying SMPL registration are known. For a sample $\mathbf { X }$ in the dataset, we first define the re-projected points $\bar { \mathbf { X } } = \{ \bar { \mathbf { x } } \} _ { i = 1 } ^ { N }$ as $\mathbf { X }$ mapped to canonical space via Eq. (4) and then mapped back to transformed space via Eq. (3). We then define the training loss:
+
+$$
+\mathcal { L } ( \mathbf { X } ) = \lambda _ { r } \mathcal { L } _ { r } + \lambda _ { s } \mathcal { L } _ { s } + \lambda _ { s k i n } \mathcal { L } _ { s k i n } ,
+$$
+
+where $\mathcal { L } _ { r }$ represents a re-projection loss that penalizes the L2 distance between an input point $\mathbf { x }$ and the re-projected point $\bar { \bf x }$ , $\mathcal { L } _ { s }$ represents L1 distances between the predicted forward skinning weights and inverse skinning weights, and $\mathcal { L } _ { s k i n }$ represents the L1 distances between the predicted (forward and inverse) skinning weights and the barycentrically interpolated skinning weights $\mathbf { w } ^ { ( i ) }$ on the registered SMPL shape that is closest to point $\mathbf { x } ^ { ( i ) }$ ; please refer to the Supp. Mat. for hyperparameters and details.
+
+We train two skinning network types, the first one takes a partial point cloud extracted from a depth image as input and performs the inverse skinning, while the second one takes a full point cloud sampled from iso-surface points generated from the dynamic neural SDF in the canonical space and performs forward skinning.
+
+Canonicalization: We use the learned inverse skinning network to canonicalize complete or partial point clouds $\{ \hat { \bf X } \}$ via Eq. (4) which are further used to learn the canonicalized dynamic neural SDFs.
+
+# 4 MetaAvatar
+
+Our approach meta-learns a unified clothing deformation prior from the training set that consists of different subjects wearing different clothes. This meta-learned model is further efficiently fine-tuned to produce a dynamic neural SDF from an arbitrary amount of fine-tuning data of unseen subjects. In extreme cases, MetaAvatar requires as few as 8 depth frames and takes only 2 minutes for fine-tuning to yield a subject/cloth-type-specific dynamic neural SDF (Fig. 1).
+
+We assume that each subject/cloth-type combination in the training set has a set of registered bone transformations and canonicalized points, denoted as $\{ \{ \mathbf { B } _ { b } \} _ { b = 1 } ^ { 2 4 } , \hat { \mathbf { X } } \}$ . Points in $\hat { \bf X }$ are normalized to the range $[ - 1 , 1 ] ^ { 3 }$ according to their corresponding registered SMPL shape. With slight abuse of notation, we also define $\mathbf { X }$ as all possible points in $[ - \bar { 1 } , 1 ] ^ { 3 }$ . Our goal is to meta-learn a hypernetwork [25, 69] which takes $\{ \mathbf { B } _ { b } \} _ { b = 1 } ^ { 2 4 }$ ( $\{ { \bf B } _ { b } \}$ for shorthand) as inputs and predicts parameters of the neural SDFs in the canonical space. Denoting the hypernetwork as $g _ { \psi } ( \left\{ \bar { \mathbf { B } } _ { b } \right\} )$ and the predicted neural SDF as $f _ { \phi } ( \mathbf { x } ) \vert _ { \phi = g _ { \psi } ( \{ \mathbf { B } _ { b } \} ) }$ , we use the following IGR [21] loss to supervise the learning of $g$ :
+
+$$
+\begin{array} { l } { { \displaystyle \dot { \mathrm {  ~ \ l ~ } } _ { { \mathrm { L G R } } } \big ( f _ { \phi } \big ( \hat { \bf X } \big ) \big | _ { \phi = g _ { \psi } ( { \{ { \bf B } _ { b } \} } ) } \big ) = \sum _ { { \bf x } \in \hat { \bf X } } \lambda _ { s d f } \Big | f _ { \phi } \big ( { \bf x } \big ) \big | _ { \phi = g _ { \psi } ( { \{ { \bf B } _ { b } \} } ) } \Big | + \lambda _ { \bf n } \left( 1 - \langle { \bf n } ( { \bf x } ) , \nabla _ { { \bf x } } f _ { \phi } ( { \bf x } ) \big | _ { \phi = g _ { \psi } ( { \{ { \bf B } _ { b } \} } ) } \rangle \right) } } \\ { { \displaystyle \qquad + \lambda _ { E } \Big | \big | \nabla _ { { \bf x } } f _ { \phi } \big ( { \bf x } \big ) \big | _ { \phi = g _ { \psi } ( \{ { { \bf B } _ { b } \} } ) } \big | \big | _ { 2 } - 1 \Big | \qquad \quad \mathrm { ( o n \mathrm { - s u r f a c e ~ l o s s } ) } } } \\ { { \displaystyle \qquad + \sum _ { { \bf x } \sim { \bf X } \backslash \hat { \bf X } } \qquad \quad - \alpha \cdot \Big | f _ { \phi } \big ( { \bf x } \big ) \big | _ { \phi = g _ { \psi } ( \{ { { \bf B } _ { b } \} } ) } \Big | \Big ) } } \\ { { \displaystyle \qquad + \lambda _ { E } \Big | \big | \nabla _ { { \bf x } } f _ { \phi } \big ( { \bf x } \big ) \big | _ { \phi = g _ { \psi } ( \{ { { \bf B } _ { b } \} } ) } \big | \big | _ { 2 } - 1 \Big | \qquad \mathrm { ( o f f \mathrm { - s u r f a c e ~ l o s s } ) } } } \end{array}
+$$
+
+where $\mathbf { n _ { x } }$ is the surface normal of point $\mathbf { x }$ . We assume that this information, along with the groundtruth correspondences from transformed space to canonical space is available when learning the meta-model on the training set, but do not require this for fine-tuning $g$ on unseen subjects.
+
+In practice, we found that directly learning the hypernetwork $g _ { \psi }$ via Eq. (6) does not converge, and thus we decompose the meta-learning of $g _ { \psi }$ into two steps. First, we learn a meta-SDF [67] (without conditioning on $\{ \mathbf { B } _ { b } \}$ , Sec. 4.1), and then we meta-learn a hypernetwork that takes $\{ { \bf B } _ { b } \}$ as input and predicts the residuals to the parameters of the previously learned meta-SDF (Sec. 4.2).
+
+# 4.1 Meta-learned Initialization of Static Neural SDFs
+
+To effectively learn a statistical prior of clothed human bodies, we ignore the input bone transformations $\{ { \bf B } _ { b } \}$ and meta-learn the static neural SDF $f _ { \phi } ( \mathbf { \dot { x } } ) : [ - 1 , 1 ] ^ { 3 } \mapsto \mathbb { R }$ , parameterized by $\phi$ , from all canonicalized points of subjects with different genders, body shapes, cloth types, and poses. Furthermore, for faster and more stable convergence, the neural SDF $f _ { \phi }$ function additionally leverages the periodic activation functions [68].
+
+![](images/d5ef2bd65a2017e3a496dc05e47de251369d4edd7223f0cb3c23cfcc5cd6c2c0.jpg)  
+Figure 2: Overview of the meta-SDF network. We use a 5-layer SIREN [67] network with 256 neurons for each layer.
+
+The full meta-learning algorithm for the static neural SDFs is described in Alg. 1.
+
+# Algorithm 1 Meta-learning SDF with Reptile [49]
+
+Initialize: meta-network parameters $\phi$ , meta learning rate $\beta$ , inner learning rate $\alpha$ , max training   
+iteration $N$ , inner-loop iteration $m$ , batch size $M$   
+1: for $i = 1 , \ldots , N$ do   
+2: Sample a batch of $M$ training samples $\{ \hat { \mathbf { X } } ^ { ( j ) } \} _ { j = 1 } ^ { M }$   
+3: for $\begin{array} { r l } & { \mathbf { \Lambda } _ { [ = 1 , \dots , M \mathbf { \delta } ] \mathbf { \delta } \mathbf { 1 } } ^ { [ = 1 , \dots , M \mathbf { \delta } ] \mathbf { \delta } \mathbf { 0 } } } \\ & { \mathbf { \Phi } _ { \mathbf { 0 } } ^ { ( j ) } = \phi } \\ & { \mathbf { \Phi } \mathbf { 0 } \mathbf { r } \ k = 1 , \dots , m \mathbf { \delta } \mathbf { d } \mathbf { \mathbf { 0 } } } \\ & { \quad \phi _ { k } ^ { ( j ) } = \phi _ { k - 1 } ^ { ( j ) } - \alpha \nabla _ { \phi } \mathcal { L } _ { \mathrm { I G R } } \big ( f _ { \phi } ( \hat { \mathbf { X } } ^ { ( j ) } ) \big | _ { \phi = \phi _ { k - 1 } ^ { ( j ) } } \big ) } \end{array}$   
+4: φ(j) =   
+5: f   
+6:   
+7: end for   
+8: 9: $\begin{array} { r l } & { \mathrel { \phantom { = } } \phi  \phi + \beta \frac { 1 } { M } \sum _ { j = 1 } ^ { M } ( \phi _ { m } ^ { ( j ) } - \phi ) } \end{array}$   
+10: end for
+
+# 4.2 Meta-learned Initialization of HyperNetwork for Dynamic Neural SDFs
+
+The meta-learned static neural SDF explained in the previous section can efficiently adapt to new observations, however it is not controllable by user-specified bone transformations $\{ \bar { \mathbf { B } } _ { b } \}$ . Therefore, to enable non-rigid pose-dependent cloth deformations, we further meta-learn a hypernetwork [25] to predict residuals to the learned parameters of the meta-SDF in Alg. 1.
+
+The key motivation for meta-learning the hypernetwork is to build an effective unified prior for articulated clothed humans, which enables the recovery of the non-rigid clothing deformations at test time via the efficient fine-tuning process from several depth images of unseen subjects.
+
+# Algorithm 2 Meta-learning hypernetwork with Modified Reptile
+
+Initialize: meta-hypernetwork parameters $\psi$ , pre-trained meta-SDF parameters $\phi ^ { * }$ , meta learning rate $\beta$ , inner learning rate $\alpha$ , max training iteration $N$ , inner-loop iteration $m$ . 1: for $i = 1 , \ldots , N$ do 2: $\psi _ { 0 } = \psi$ 3: Randomly choose a subject/cloth-type combination $n$ 4: Uniformly sample $M \sim \{ 1 , \dots , D ^ { ( n ) } \}$ where $D ^ { ( n ) }$ is the number of datapoints of subject/cloth-type combination $n$ 5: Sample $M$ datapoints from subject/cloth-type combination $n$ , denoting these datapoints as $\mathcal { S } = \{ \{ \bar { \bf B } _ { b } \} ^ { ( j ) } , \hat { \bf X } ^ { ( j ) } \} _ { j = 1 } ^ { M }$ 6: for $k = 1 , \ldots , m$ do 7: L = 1M P({Bb},Xˆ )∈S LIGR(fφ(Xˆ )|φ=gψk−1 ({Bb})+φ∗ ) 8: ψk = ψk−1 − α∇ψk−1L 9: end for 10: $\psi  \psi + \beta ( \psi _ { m } - \psi _ { 0 } )$ 11: end for
+
+Denoting the meta-SDF learned by Alg. 1 as $\phi ^ { * }$ and our hypernetwork as $g _ { \psi } ( \{ \mathbf { B } _ { b } \} )$ , we implement Alg. 2. This algorithm differs from the original Reptile [49] algorithm in that it tries to optimize the inner-loop on arbitrary amount of data. Note that for brevity the loss in the innerloop (line 7-line 8) is computed over the whole batch $s$ , whereas in practice we used stochastic gradient descent (SGD) with fixed mini-batch size over $s$ since $s$ can contain hundreds of samples; SGD is used with the mini-batch size of 12 for the inner-loop.
+
+![](images/7e4a5f9cf29a9498401db73082b325c23e30988ddf4b5de04970a3d0ded562d1.jpg)  
+Figure 3: Overview of the meta-hypernetwork. It predicts residuals to $\phi ^ { * }$ which is learned in Sec. 4.1
+
+Inference: At test-time, we are given a small fine-tuning set {{Bb}f ine,(j), Xˆ f ine,(j)}Mj= and the validation set {{Bb}val,(j)}Kj= The fine-tuning set is used to optimize the hypernetwork parameters $\psi$ ( $m = 2 5 6$ SGD epochs) that
+
+are then used to generate neural SDFs from bone transformations available in the validation set. The overall inference pipeline including the inverse and the forward LBS stages is shown in Fig. 4.
+
+Bone Transformation Encoding: We found that a small hierarchical MLP proposed in LEAP [48] for encoding bone transformations works slightly better than the encoding of unit quaternions used in SCANimate [65]. Thus, we employ the hierarchical MLP encoder to encode $\{ \mathbf { B } _ { b } \}$ for $g$ unless specified otherwise; we ablate different encoding types in the experiment section.
+
+# 5 Experiments
+
+We validate the proposed MetaAvatar model for learning meta-models and controllable dynamic neural SDFs of clothed humans by first comparing our MetaAvatar to the established approaches [13, 48, 65]. Then, we ablate the modeling choices for the proposed controllable neural SDFs. And lastly, we demonstrate MetaAvatar’s capability to tackle the challenging task of learning animatable clothed human models from reduced data, to the point that only 8 depth images are available as input.
+
+Datasets: We use the CAPE dataset [43] as the major test bed for our experiments. This dataset consists of 148584 pairs of clothed meshes, capturing 15 human subjects wearing different clothes while performing different actions. We use 10 subjects for meta-learning, which we denote as the training set. We use four unseen subjects (00122, 00134, 00215, 03375)2 for fine-tuning and validation; for each of these four subjects, the corresponding action sequences are split into finetuning set and validation set. The fine-tuning set is used for fine-tuning the MetaAvatar models, it is also used to evaluate pose interpolation task. The validation set is used for evaluating novel pose extrapolation. Among the four unseen subjects, two of them (00122, 00215) perform actions that are present in the training set for the meta-learning; we randomly split actions of these two subjects with $70 \%$ fine-tuning and $30 \%$ validation. Subject 00134 and 03375 perform two trials of actions unseen in the training set for meta-learning. We use the first trial as the fine-tuning set and the second trial as the validation set. Subject 03375 also has one cloth type (blazer) that is unseen during meta-learning.
+
+![](images/499f555f0f574cbcc116e67f0ea4e1494f130391d8a750d3dc7d9a366cd3ba36.jpg)  
+Figure 4: Overview of our inference pipeline. The inverse LBS net (Sec. 3.2) takes a small set of input depth frames together with their underlying SMPL registrations to canonicalize the depth points; then the meta-learned hypernetwork (Sec. 4.2) is fine-tuned to represent the instance specific dynamic SDF; given novel poses, the updated hypernetwork generates pose-dependent cloth-deformations in canonical space, and the animated meshes are obtained via the forward LBS network (Sec. 3.2).
+
+Baselines: We use NASA [13], LEAP [48], and SCANimate [65] as our baselines. NASA and SCANimate cannot handle multi-subject-cloth with a single model so we train per-subject/cloth-type models from scratch for each of them on the fine-tuning set. LEAP is a generalizable neural-implicit human body model that has shown to work on minimally-clothed bodies. We extend LEAP by adding a one-hot encoding to represent different cloth types (similarly to [43]) and train it jointly on the full training and the fine-tuning set.
+
+As for the input format, we use depth frames rendered from CAPE meshes for our MetaAvatar. To render the depth frames, we fixed the camera and rotate the CAPE meshes around the y-axis (in SMPL space) at different angles with an interval of 45 degrees; note that for each mesh we only render it on one angle, simulating a monocular camera taking a round-view of a moving person. For the baselines, we use watertight meshes and provide the occupancy [45] loss to supervise the training of NASA and LEAP, while sampling surface points and normals on watertight meshes to provide the IGR loss supervision for SCANimate. Note that our model is at great disadvantage, as for fine-tuning we only use discrete monocular depth observations without accurate surface normal information.
+
+Tasks and Evaluation: Our goal is to generate realistic clothing deformations from arbitrary input human pose parameters. To systematically understand the generalization capability of the MetaAvatar representation, we validate the baselines and MetaAvatar on two tasks, pose interpolation and extrapolation. For interpolation, we sample every 10th frame on the fine-tuning set for training/finetuning, and sample every 5th frame (excluding the training frames) also on the fine-tuning set for validation. For extrapolation, we sample every 5th frame on the fine-tuning set for training, and sample every 5th frame on the validation set for validation.
+
+Interpolation is evaluated using three metrics: point-based ground-truth-to-mesh distance $( D _ { p } \downarrow ,$ in cm), face-based ground-truth-to-mesh distance $( D _ { f } \downarrow ,$ , in cm), and point-based ground-truth-to-mesh normal consistency $( N C \uparrow$ , in range $[ - 1 , 1 ] )$ . For computing these interpolation metrics we ignore non-clothed body parts such as hands, feet, head, and neck. For extrapolation, we note that clothdeformation are often stochastic; in such a case, predicting overly smooth surfaces can result in lower distances and higher normal consistency. Thus, we also conduct a large-scale perceptual study using
+
+![](images/04778bfeba6e8531b15fb31b8bf10ba2173d2807c481cc0b0182997a03765d4c.jpg)  
+Figure 5: Qualitative comparison on extrapolation results with blazer outfit. NASA shows consistent blocky artifacts. LEAP predicts overly smooth surfaces missing the tails of the blazer outfit. SCANimate does not generalize as this specific pose has not been seen during training. Directly meta-learning a SIREN [68] network that conditions on input poses produces a smooth surface that does not capture the blazer tails well.
+
+![](images/c48f179a578faed555e41852a3e3ab47ab69bbf01a2b531186854ba0fb96fb79.jpg)  
+Figure 6: Qualitative comparison on extrapolation results when reducing fine-tuning data on subject 00215 wearing poloshirt. The caption indicates the amount of fine-tuning data used to fine-tune the meta-hypernetwork on this unseen subject. Our meta-learned model captures for this unseen subject the sliding effect of the poloshirt at this pose in which the person raising arms, even fine-tuned with just 8 depth images.
+
+Amazon Mechanical Turk, and report the perceptual scores $( \mathrm { P S } \uparrow )$ which reflects the percentage of users who favor the outputs of baselines over MetaAvatar. Details about user study design can be found in the Supp. Mat.
+
+# 5.1 Evaluation Against Baselines
+
+In this section, we report results on both interpolation and extrapolation tasks against various baselines described above. For NASA and SCANimate, we train one model for each subject/clothtype combination on the fine-tuning set. For them it usually takes several thousand of epochs to converge for each subject/cloth-type combination, which roughly equals to 10-24 hours of training. For LEAP, we train a single model on both the training and the fine-tuning set using two days. For the MetaAvatar, we meta-learn a single model on the training set, and for each subject/cloth-type combination we fine-tune the model for 256 epochs to produce subject/cloth-type specific models. The exact fine-tuning time ranges from 40 minutes to 3 hours depending on the amount of available data since we are running a fixed number of epochs; see the Supp. Mat. for detailed runtime comparison on each subject/cloth-type combination. Note that MetaAvatar uses partial depth observations while the other baselines are trained on complete meshes. The results are reported in Table 1.
+
+Importantly, models of NASA and SCANimate are over-fitted to each subject/cloth-type combination as they cannot straightforwardly leverage prior knowledge from multiple training subjects. LEAP is trained on all training and fine-tuning data with input encodings to distinguish different body shapes and cloth types, but it fails to capture high-frequency details of clothes, often predicting smooth surfaces (Fig. 5); this is evidenced by its lower perceptual scores (PS) compared to SCANimate and our MetaAvatar. In contrast to these baselines, MetaAvatar successfully captures a unified clothing deformation prior of diverse body shapes and cloth types, which generalizes well to unseen body shapes (00122, 00215), unseen poses (00134, 03375), and unseen cloth types (03375 with blazer outfit); although we did not outperform LEAP on the interpolation task for subject 00134 and 03375, we note that 1) our method uses only 2.5D input for fine-tuning, while LEAP has access to groundtruth canonical meshes during training; 2) subject 00134 and 03375 comprise much more missing frames than subject 00122 and 00215, resulting in higher stochasticity and thus predicting smooth surfaces (such as LEAP) may yield better performance; this is also evidenced by LEAP’s much lower perceptual scores on subject 00134 and 03375, although obtaining the best performance for pose interpolation. We encourage the readers to watch the side-by-side comparison videos available on our project page: https://neuralbodies.github.io/metavatar/.
+
+<table><tr><td></td><td></td><td colspan="2">3DInput</td><td>2.5D Input</td></tr><tr><td></td><td>NASA</td><td>LEAP</td><td>SCANimate</td><td>Ours</td></tr><tr><td colspan="5">Subj 100122，00215</td></tr><tr><td>Ex.</td><td>PS↑</td><td>0.078</td><td>0.314 0.333</td><td>0.5</td></tr><tr><td rowspan="3">Int.</td><td>Dp↓</td><td>0.484</td><td>0.454 0.586</td><td>0.450</td></tr><tr><td>Df↓</td><td>0.327 0.293</td><td>0.489</td><td>0.273</td></tr><tr><td>NC↑</td><td>0.752</td><td>0.807 0.793</td><td>0.821</td></tr><tr><td colspan="5">Subj 00134，03375</td></tr><tr><td>Ex.</td><td>PS↑</td><td>0.182 0.224</td><td>0.481</td><td>0.5</td></tr><tr><td rowspan="3">Int.</td><td>Dp↓</td><td>0.595</td><td>0.483</td><td>0.518</td></tr><tr><td>Df↓</td><td>0.469 0.340</td><td>0.542</td><td>0.367</td></tr><tr><td>NC↑</td><td>0.693 0.780</td><td>0.755</td><td>0.773</td></tr><tr><td colspan="5">Averge per-model training/fine-tuning time (hours)</td></tr><tr><td colspan="2">&gt;10</td><td>1</td><td>&gt;10</td><td>1.60</td></tr></table>
+
+Table 1: Comparison to baselines. $D _ { p } , D _ { f }$ and $N C$ are reported for interpolation (Int.) while PS is reported for extrapolation $\left( \mathrm { E x . } \right)$ . Note that MetaAvatar is fine-tuned on depth images while all other baselines are trained on complete meshes. The training/fine-tuning times are just rough estimates, as ours does not include the time for meta-learning, while many factors, including varying training schedules, disk-IOs and hardware setups, can affect the final speed.
+
+<table><tr><td></td><td></td><td>MLP</td><td>PosEnc</td><td>SIREN</td><td>Hyper Quat</td><td>Hyper BoneEnc</td></tr><tr><td colspan="7">Subj 00122,00215</td></tr><tr><td rowspan="3">Int.</td><td>Dp↓</td><td>3.278</td><td>1.806</td><td>0.472</td><td>0.460</td><td>0.461</td></tr><tr><td>D↓</td><td>2.201</td><td>0.998</td><td>0.301</td><td>0.288</td><td>0.288</td></tr><tr><td>NC↑</td><td>-0.279</td><td>-0.045</td><td>0.815</td><td>0.818</td><td>0.820</td></tr><tr><td colspan="7">Subj 100134,03375</td></tr><tr><td rowspan="3">Int.</td><td>Dp↓</td><td>3.320</td><td>1.498</td><td>0.532</td><td>0.526</td><td>0.523</td></tr><tr><td>Df↓</td><td>2.190</td><td>0.772</td><td>0.385</td><td>0.378</td><td>0.374</td></tr><tr><td>NC↑</td><td>-0.300</td><td>-0.099</td><td>0.773</td><td>0.772</td><td>0.772</td></tr></table>
+
+Table 2: Ablation for different architectures on the interpolation task. Hyper-Quat is our model that takes the relative joint-rotations (in the form of unit quaternions) as inputs. HyperBoneEnc is our full model with hierarchical bone encoding MLP of LEAP [48]. Models in the table are fine-tuned for 128 epochs.
+
+# 5.2 Ablation Study on Model Architectures
+
+We further ablate model architecture choices for MetaAvatar. We compare against (1) a plain MLP that takes the concatenation of the relative joint-rotations (in the form of unit quaternions) and query points as input (MLP), (2) a MLP that takes the concatenation of the relative joint-rotations and the positional encodings of query point coordinates as input (PosEnc), and (3) a SIREN network that takes the concatenation of the relative joint-rotations and query points as input (SIREN). The evaluation task is interpolation; results are reported in Table 2. For the baselines (MLP, PosEnc and SIREN), we directly use Alg. 2 to meta-learn the corresponding models with $\phi ^ { * } = 0$ . For MLP and PosEnc, the corresponding models fail to produce reasonable shapes. For SIREN, it produces unnaturally smooth surfaces which cannot capture fine clothing details such as wrinkles (Fig. 5).
+
+# 5.3 Few-shot learning of MetaAvatar
+
+In this section, we evaluate the few-shot learning capabilities of MetaAvatar. As shown in Table 3, we reduce the amount of data on the fine-tuning set, and report the performance of models finetuned on reduced amount of data. Note that with $< 1 \%$ data, we require only one frame from each action sequence available for a subject/cloth-type combination, this roughly equals to 8-20 depth frames depending on the amount of data for that subject/cloth-type combination. For interpolation, the performance drops because the stochastic nature of cloth deformation becomes dominant when the amount of fine-tuning data decreases. On the other hand, the perceptual scores (PS) are better than NASA and LEAP even with ${ \tt c 1 \% }$ data in the form of partial depth observations, and better
+
+Table 3: Ablation for few-shot learning. We report performance of MetaAvatar on reduced amount of fine-tuning data. Fine-tuning time scales linearly with the amount of data, since we run for a fixed number of epochs.   
+
+<table><tr><td>Fine-tune data (%)</td><td></td><td>100</td><td>50</td><td>20</td><td>10</td><td>5</td><td>&lt;1</td></tr><tr><td colspan="8">Subj 00122,00215</td></tr><tr><td>Ex.</td><td>PS个</td><td>0.5</td><td>0.471</td><td>0.509</td><td>0.473</td><td>0.373</td><td>0.510</td></tr><tr><td rowspan="3">Int.</td><td>Dp↓</td><td>-</td><td>0.450</td><td>0.480</td><td>0.512</td><td>0.543</td><td>0.592</td></tr><tr><td>D↓</td><td>=</td><td>0.273</td><td>0.310</td><td>0.353</td><td>0.391</td><td>0.450</td></tr><tr><td>NC↑</td><td>-</td><td>0.821</td><td>0.808</td><td>0.795</td><td>0.785</td><td>0.768</td></tr><tr><td colspan="8">Subj 00134， ，03375</td></tr><tr><td>Ex.</td><td>PS个</td><td>0.5</td><td>0.476</td><td>0.424</td><td>0.463</td><td>0.439</td><td>0.387</td></tr><tr><td rowspan="3">Int.</td><td>Dp↓</td><td>1</td><td>0.518</td><td>0.545</td><td>0.576</td><td>0.603</td><td>0.619</td></tr><tr><td>Df↓</td><td></td><td>0.367</td><td>0.400</td><td>0.438</td><td>0.471</td><td>0.489</td></tr><tr><td>NC↑</td><td>-</td><td>0.773</td><td>0.762</td><td>0.753</td><td>0.745</td><td>0.737</td></tr><tr><td colspan="8">Average per-model training/fine-tuning time ( (hours) 0.08 0.02</td></tr><tr><td colspan="8">1.60 0.8 0.32 0.16</td></tr></table>
+
+than or comparable to SCANimate in most cases. The qualitative comparison on extrapolation results of reduced fine-tuning data is shown in Fig. 6. Please see the Supp. Mat. for more qualitative results on few-shot learning, including results on depth from raw scans, results on real depth images and comparison with pre-trained SCANimate model.
+
+# 6 Conclusion
+
+We introduced MetaAvatar, a meta-learned hypernetwork that represents controllable dynamic neural SDFs applicable for generating clothed human avatars. Compared to existing methods, MetaAvatar learns from less data (temporally discrete monocular depth frames) and requires less time to represent novel unseen clothed humans. We demonstrated that the meta-learned deformation prior is robust and can be used to effectively generate realistic clothed human avatars in 2 minutes from as few as 8 depth observations.
+
+MetaAvatar is compatible with automatic registration methods [8, 75], human motion models [84, 85], implicit hand models [30, 31] and rendering primitives [40, 47] that could jointly enable an efficient end-to-end photo-realistic digitization of humans from commodity RGBD sensors, which has broad applicability in movies, games, and telepresence applications. However, this digitization may raise privacy concerns that need to be addressed carefully before deploying the introduced technology.
+
+# 7 Acknowledgment
+
+Siyu Tang acknowledges funding by the Swiss National Science Foundation under project 200021_204840. Andreas Geiger was supported by the ERC Starting Grant LEGO-3D (850533) and DFG EXC number 2064/1 - project number 390727645. We thank Jinlong Yang for providing results of SCANimate on the CAPE dataset. We thank Yebin Liu for sharing POSEFusion [38] data and results during the rebuttal period. We also thank Yan Zhang, Siwei Zhang and Korrawe Karunratanakul for proof reading the paper.
+
+# References
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+        "text": "MetaAvatar: Learning Animatable Clothed Human Models from Few Depth Images ",
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+        "text": "1ETH Zürich 2Max Planck Institute for Intelligent Systems, Tübingen 3University of Tübingen ",
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+        "text": "Abstract ",
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+        "text": "In this paper, we aim to create generalizable and controllable neural signed distance fields (SDFs) that represent clothed humans from monocular depth observations. Recent advances in deep learning, especially neural implicit representations, have enabled human shape reconstruction and controllable avatar generation from different sensor inputs. However, to generate realistic cloth deformations from novel input poses, watertight meshes or dense full-body scans are usually needed as inputs. Furthermore, due to the difficulty of effectively modeling pose-dependent cloth deformations for diverse body shapes and cloth types, existing approaches resort to per-subject/cloth-type optimization from scratch, which is computationally expensive. In contrast, we propose an approach that can quickly generate realistic clothed human avatars, represented as controllable neural SDFs, given only monocular depth images. We achieve this by using meta-learning to learn an initialization of a hypernetwork that predicts the parameters of neural SDFs. The hypernetwork is conditioned on human poses and represents a clothed neural avatar that deforms non-rigidly according to the input poses. Meanwhile, it is metalearned to effectively incorporate priors of diverse body shapes and cloth types and thus can be much faster to fine-tune, compared to models trained from scratch. We qualitatively and quantitatively show that our approach outperforms state-of-the-art approaches that require complete meshes as inputs while our approach requires only depth frames as inputs and runs orders of magnitudes faster. Furthermore, we demonstrate that our meta-learned hypernetwork is very robust, being the first to generate avatars with realistic dynamic cloth deformations given as few as 8 monocular depth frames. ",
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+        "text": "1 Introduction ",
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+        "text": "Representing clothed humans as neural implicit functions is a rising research topic in the computer vision community. Earlier works in this direction address geometric reconstruction of clothed humans from static monocular images [35, 36, 63, 64], RGBD videos [37, 38, 71, 78, 80] or sparse point clouds [12] as direct extensions of neural implicit functions for rigid objects [11, 45, 46, 52]. More recent works advocate to learn shapes in a canonical pose [7, 27, 75] in order to not only handle reconstruction, but also build controllable neural avatars from sensor inputs. However, these works do not model pose-dependent cloth deformation, limiting their realism. ",
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+            "Figure 1: Given as few as 8 monocular depth images and their SMPL fittings, our meta-learned model yields a controllable neural SDF in 2 minutes which synthesizes realistic cloth deformations for unseen body poses. Here we show results of two different subjects wearing different clothes. "
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+        "text": "On the other hand, traditional parametric human body models [41, 50, 54, 77] can represent pose-dependent soft tissue deformations of minimally-clothed human bodies. Several recent methods [13, 48] proposed to learn neural implicit functions to approximate such parametric models from watertight meshes. However, they cannot be straightforwardly extended to model clothed humans. SCANimate [65] proposed to learn canonicalized dynamic neural Signed Distance Fields (SDFs) controlled by human pose inputs and trained with Implicit Geometric Regularization (IGR [21]), thus circumventing the requirement of watertight meshes. However, SCANimate works only on dense full-body scans with accurate surface normals and further requires expensive per-subject/cloth-type training. These factors limit the applicability of SCANimate for building personalized human avatars from commodity RGBD sensors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Contrary to all the aforementioned works, we propose to use meta-learning to effectively incorporate priors of dynamic neural SDFs of clothed humans, thus enabling fast fine-tuning (few minutes) for generating new avatars given only a few monocular depth images of unseen clothed humans as inputs. More specifically, we build upon recently proposed ideas of meta-learned initialization for implicit representations [67, 72] to enable fast fine-tuning. Similar to [67], we represent a specific category of objects (in our case, clothed human bodies in the canonical pose) with a neural implicit function and use meta-learning algorithms such as [16, 49] to learn a meta-model. However, unlike [67, 72], where the implicit functions are designed for static reconstruction, we target the generation of dynamic neural SDFs that are controllable by user-specified body poses. We observe that directly conditioning neural implicit functions (represented as a multi-layer perceptron) on body poses lacks the expressiveness to capture high-frequency details of diverse cloth types, and hence propose to meta-learn a hypernetwork [25] that predicts the parameters of the neural implicit function. Overall, the proposed approach, which we name MetaAvatar, yields controllable neural SDFs with dynamic surfaces in minutes via fast fine-tuning, given only a few depth observations of an unseen clothed human and the underlying SMPL [41] fittings (Fig. 1) as inputs. Code and data are public at https://neuralbodies.github.io/metavatar/. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Our approach lies at the intersection of clothed human body modeling, neural implicit representations, and meta-learning. We review related works in the following. ",
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+    {
+        "type": "text",
+        "text": "Clothed Human Body Modeling: Earlier works for clothed human body modeling utilize parametric human body models [5, 26, 28, 41, 50, 54, 77] combined with deformation layers [2, 3, 7, 8] to model cloth deformations. However, these approaches cannot model fine clothing details due to their fixed topology, and they cannot handle pose-dependent cloth deformations. Mesh-based approaches that handle articulated deformations of clothes either require accurate surface registration [33, 43, 79, 83] or synthetic data [22, 24, 53] for training. Such requirement for data can be freed by using neural implicit surfaces [10, 51, 65, 73]. For example, SCANimate [65] proposed a weakly supervised approach to learn dynamic clothed human body models from 3D full-body scans which only requires minimally-clothed body registration. However, its training process usually takes one day for each subject/cloth-type combination and requires accurate surface normal information extracted from dense scans. Recent explicit clothed human models [9, 42, 44, 81] can also be learned from unregistered data. Like our method, concurrent work [44] also models pose-dependent shapes across different subjects/cloth-types, but it requires full-body scans for training. In contrast, our approach enables learning of clothed body models in minutes from as few as 8 depth images. ",
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+        "text": "",
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+        "type": "text",
+        "text": "Neural Implicit Representations: Neural implicit representations [11, 45, 46, 52, 55] have been used to tackle both image-based [27, 35, 36, 57, 59, 63, 64, 87] and point cloud-based [7, 12] clothed human reconstruction. Among these works, ARCH [27] was the first one to represent clothed human bodies as a neural implicit function in a canonical pose. However, ARCH does not handle posedependent cloth deformations. Most recently, SCANimate [65] proposed to condition neural implicit functions on joint-rotation vectors (in the form of unit quaternions), such that the canonicalized shapes of the neural avatars change according to the joint angles of the human body, thus representing pose-dependent cloth deformations. However, diverse and complex cloth deformations make it hard to learn a unified prior from different body shapes and cloth types, thus SCANimate resorts to per-subject/cloth-type training which is computationally expensive. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Meta-Learning: Meta-learning is typically used to address few-shot learning, where a few training examples of a new task are given, and the model is required to learn from these examples to achieve good performance on the new task [1, 14, 15, 17–19, 23, 29, 32, 58, 60, 62, 66, 70, 74, 76, 82, 86]. We focus on optimization-based meta-learning, where Model-Agnostic Meta Learning (MAML [16]), Reptile [49] and related alternatives are typically used to learn such models [4, 6, 20, 34, 39, 61]. In general, this line of algorithms tries to learn a \"meta-model\" that can be updated quickly from new observations with only few gradient steps. Recently, meta-learning has been used to learn a universal initialization of implicit representations for static neural SDFs [67] and radiance fields [72]. MetaSDF [67] demonstrates that only a few gradient update steps are needed to achieve comparable or better results than slower auto-decoder-based approaches [52]. However, [67, 72] only meta-learn static representations, whereas we are interested in dynamic representations conditioned on human body poses. To our best knowledge, we are the first to meta-learn the hypernetwork to generate the parameters of neural SDF networks. ",
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+    {
+        "type": "text",
+        "text": "3 Fundamentals ",
+        "text_level": 1,
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+        "text": "We start by briefly reviewing the linear blend skinning (LBS) method [41] and the recent implicit skinning networks [48, 65] that learn to predict skinning weights of cloth surfaces in a weakly supervised manner. Using the learned implicit skinning networks allows us to canonicalize meshes or depth observations of clothed humans, given only minimally-clothed human body model registrations to the meshes. Canonicalization of meshes or points is a necessary step as the dynamic neural SDFs introduced in Section 4 are modeled in canonical space. ",
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+    {
+        "type": "text",
+        "text": "3.1 Linear Blend Skinning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Linear blend skinning (LBS) is a commonly used technique to deform parametric human body models [5, 26, 41, 50, 54, 77] according to user-specified rigid bone transformations. Given a set of $N$ points in a canonical space, $\\hat { \\mathbf { X } } = \\{ \\hat { \\mathbf { x } } ^ { ( i ) } \\} _ { i = 1 } ^ { N }$ , LBS takes a set of rigid bone transformations (in our case we use 23 local transformations plus one global transformation, assuming an underlying SMPL model) $\\{ \\mathbf { B } _ { b } \\} _ { b = 1 } ^ { 2 4 }$ as inputs, each $\\mathbf { B } _ { b }$ being a $4 \\times 4$ rotation-translation matrix. For a 3D point xˆ(i) ∈ Xˆ 1, a skinning weight vector is a probability simplex w(i) ∈ [0, 1]24, s.t. P24b=1 $\\begin{array} { r } { \\sum _ { b = 1 } ^ { 2 4 } \\mathbf { w } _ { b } ^ { ( i ) } = 1 } \\end{array}$ , that defines the affinity of the point $\\hat { \\mathbf { x } } ^ { ( i ) }$ to each of the bone transformations $\\{ \\mathbf { B } _ { b } \\} _ { b = 1 } ^ { 2 4 }$ . The set of transformed points $\\mathbf { X } = \\{ \\mathbf { x } ^ { ( i ) } \\} _ { i = 1 } ^ { N }$ of the clothed human is related to $\\hat { \\mathbf X }$ via: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/efb2e2897c41c3f9b5050a44e2deff85e41ee3f4bffae0d82043ed12e60d1bc7.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\bf x } ^ { ( i ) } = L B S \\left( \\hat { { \\bf x } } ^ { ( i ) } , \\{ { \\bf B } _ { b } \\} , { \\bf w } ^ { ( i ) } \\right) = \\left( \\sum _ { b = 1 } ^ { 2 4 } { \\bf w } _ { b } ^ { ( i ) } { \\bf B } _ { b } \\right) \\hat { \\bf x } ^ { ( i ) } } , ~ \\forall i = 1 , \\ldots , N }  \\\\ { { \\displaystyle \\hat { \\bf x } ^ { ( i ) } = L B S ^ { - 1 } \\left( { \\bf x } ^ { ( i ) } , \\{ { \\bf B } _ { b } \\} , { \\bf w } ^ { ( i ) } \\right) = \\left( \\sum _ { b = 1 } ^ { 2 4 } { \\bf w } _ { b } ^ { ( i ) } { \\bf B } _ { b } \\right) ^ { - 1 } { \\bf x } ^ { ( i ) } } , ~ \\forall i = 1 , \\ldots , N } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where Eq. (1) is referred to as the LBS function and Eq. (2) is referred to as the inverse-LBS function. The process of applying Eq. (1) to all points in $\\hat { \\bf X }$ is often referred to as forward skinning while the process of applying Eq. (2) is referred to as inverse skinning. ",
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+    {
+        "type": "text",
+        "text": "3.2 Implicit Skinning Networks ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recent articulated implicit representations [48, 65] have proposed to learn functions that predict the forward/inverse skinning weights for arbitrary points in $\\bar { \\mathbb { R } ^ { 3 } }$ . We follow this approach, but take advantage of a convolutional point-cloud encoder [56] for improved generalization. Formally, we define the implicit forward and inverse skinning networks as $h _ { \\mathrm { f w d } } ( \\cdot , \\cdot ) : ( \\mathbb { R } ^ { 3 \\times K } , \\mathbb { R } ^ { 3 } ) \\mapsto \\mathbb { R } ^ { 2 \\bar { 4 } }$ and $h _ { \\mathrm { i n v } } ( \\cdot , \\cdot ) : ( \\mathbb { R } ^ { 5 \\times K } , \\mathbb { R } ^ { 3 } ) \\mapsto \\mathbb { R } ^ { 2 4 }$ , respectively. Both networks take as input a point cloud with $K$ points and a query point for which they predict skinning weights. Therefore, we can analogously re-define Eq. $\\displaystyle ( \\begin{array} { l l } \\end{array} , \\begin{array} { \\ l } \\end{array} )$ respectively as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f7f741dfe65ce74bd391b5afb49ed5b3245aa384f07350ec90a977a5c0fc8f22.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\bf x } ^ { ( i ) } = \\left( \\sum _ { b = 1 } ^ { 2 4 } h _ { \\mathrm { f w d } } ( \\hat { { \\bf X } } , \\hat { { \\bf x } } ^ { ( i ) } ) _ { b } { \\bf B } _ { b } \\right) \\hat { { \\bf x } } ^ { ( i ) } } , ~ \\forall i = 1 , \\ldots , N }  \\\\ { { \\displaystyle { \\hat { \\bf x } ^ { ( i ) } = \\left( \\sum _ { b = 1 } ^ { 2 4 } h _ { \\mathrm { i n v } } ( { \\bf X } , { \\bf x } ^ { ( i ) } ) _ { b } { \\bf B } _ { b } \\right) ^ { - 1 } { \\bf x } ^ { ( i ) } } , ~ \\forall i = 1 , \\ldots , N } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Training the Skinning Network: We follow the setting of SCANimate [65], where a dataset of ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "observed point clouds $\\{ { \\bf X } \\}$ and their underlying SMPL registration are known. For a sample $\\mathbf { X }$ in the dataset, we first define the re-projected points $\\bar { \\mathbf { X } } = \\{ \\bar { \\mathbf { x } } \\} _ { i = 1 } ^ { N }$ as $\\mathbf { X }$ mapped to canonical space via Eq. (4) and then mapped back to transformed space via Eq. (3). We then define the training loss: ",
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+    {
+        "type": "equation",
+        "img_path": "images/2a9893a5d8fb4b44a64650dcd2a309f04980d854bdc87c351fdd40ed43ba2123.jpg",
+        "text": "$$\n\\mathcal { L } ( \\mathbf { X } ) = \\lambda _ { r } \\mathcal { L } _ { r } + \\lambda _ { s } \\mathcal { L } _ { s } + \\lambda _ { s k i n } \\mathcal { L } _ { s k i n } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { r }$ represents a re-projection loss that penalizes the L2 distance between an input point $\\mathbf { x }$ and the re-projected point $\\bar { \\bf x }$ , $\\mathcal { L } _ { s }$ represents L1 distances between the predicted forward skinning weights and inverse skinning weights, and $\\mathcal { L } _ { s k i n }$ represents the L1 distances between the predicted (forward and inverse) skinning weights and the barycentrically interpolated skinning weights $\\mathbf { w } ^ { ( i ) }$ on the registered SMPL shape that is closest to point $\\mathbf { x } ^ { ( i ) }$ ; please refer to the Supp. Mat. for hyperparameters and details. ",
+        "bbox": [
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+        "page_idx": 3
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+        "type": "text",
+        "text": "We train two skinning network types, the first one takes a partial point cloud extracted from a depth image as input and performs the inverse skinning, while the second one takes a full point cloud sampled from iso-surface points generated from the dynamic neural SDF in the canonical space and performs forward skinning. ",
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+        "type": "text",
+        "text": "Canonicalization: We use the learned inverse skinning network to canonicalize complete or partial point clouds $\\{ \\hat { \\bf X } \\}$ via Eq. (4) which are further used to learn the canonicalized dynamic neural SDFs. ",
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+    {
+        "type": "text",
+        "text": "4 MetaAvatar ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our approach meta-learns a unified clothing deformation prior from the training set that consists of different subjects wearing different clothes. This meta-learned model is further efficiently fine-tuned to produce a dynamic neural SDF from an arbitrary amount of fine-tuning data of unseen subjects. In extreme cases, MetaAvatar requires as few as 8 depth frames and takes only 2 minutes for fine-tuning to yield a subject/cloth-type-specific dynamic neural SDF (Fig. 1). ",
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+    {
+        "type": "text",
+        "text": "We assume that each subject/cloth-type combination in the training set has a set of registered bone transformations and canonicalized points, denoted as $\\{ \\{ \\mathbf { B } _ { b } \\} _ { b = 1 } ^ { 2 4 } , \\hat { \\mathbf { X } } \\}$ . Points in $\\hat { \\bf X }$ are normalized to the range $[ - 1 , 1 ] ^ { 3 }$ according to their corresponding registered SMPL shape. With slight abuse of notation, we also define $\\mathbf { X }$ as all possible points in $[ - \\bar { 1 } , 1 ] ^ { 3 }$ . Our goal is to meta-learn a hypernetwork [25, 69] which takes $\\{ \\mathbf { B } _ { b } \\} _ { b = 1 } ^ { 2 4 }$ ( $\\{ { \\bf B } _ { b } \\}$ for shorthand) as inputs and predicts parameters of the neural SDFs in the canonical space. Denoting the hypernetwork as $g _ { \\psi } ( \\left\\{ \\bar { \\mathbf { B } } _ { b } \\right\\} )$ and the predicted neural SDF as $f _ { \\phi } ( \\mathbf { x } ) \\vert _ { \\phi = g _ { \\psi } ( \\{ \\mathbf { B } _ { b } \\} ) }$ , we use the following IGR [21] loss to supervise the learning of $g$ : ",
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+        "type": "text",
+        "text": "",
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+        "img_path": "images/38f57af7590cca1f4be01cc92cc21bcf0e51dd8c4fd0e61be16b7100183cdaed.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\dot { \\mathrm {  ~ \\ l ~ } } _ { { \\mathrm { L G R } } } \\big ( f _ { \\phi } \\big ( \\hat { \\bf X } \\big ) \\big | _ { \\phi = g _ { \\psi } ( { \\{ { \\bf B } _ { b } \\} } ) } \\big ) = \\sum _ { { \\bf x } \\in \\hat { \\bf X } } \\lambda _ { s d f } \\Big | f _ { \\phi } \\big ( { \\bf x } \\big ) \\big | _ { \\phi = g _ { \\psi } ( { \\{ { \\bf B } _ { b } \\} } ) } \\Big | + \\lambda _ { \\bf n } \\left( 1 - \\langle { \\bf n } ( { \\bf x } ) , \\nabla _ { { \\bf x } } f _ { \\phi } ( { \\bf x } ) \\big | _ { \\phi = g _ { \\psi } ( { \\{ { \\bf B } _ { b } \\} } ) } \\rangle \\right) } } \\\\ { { \\displaystyle \\qquad + \\lambda _ { E } \\Big | \\big | \\nabla _ { { \\bf x } } f _ { \\phi } \\big ( { \\bf x } \\big ) \\big | _ { \\phi = g _ { \\psi } ( \\{ { { \\bf B } _ { b } \\} } ) } \\big | \\big | _ { 2 } - 1 \\Big | \\qquad \\quad \\mathrm { ( o n \\mathrm { - s u r f a c e ~ l o s s } ) } } } \\\\ { { \\displaystyle \\qquad + \\sum _ { { \\bf x } \\sim { \\bf X } \\backslash \\hat { \\bf X } } \\qquad \\quad - \\alpha \\cdot \\Big | f _ { \\phi } \\big ( { \\bf x } \\big ) \\big | _ { \\phi = g _ { \\psi } ( \\{ { { \\bf B } _ { b } \\} } ) } \\Big | \\Big ) } } \\\\ { { \\displaystyle \\qquad + \\lambda _ { E } \\Big | \\big | \\nabla _ { { \\bf x } } f _ { \\phi } \\big ( { \\bf x } \\big ) \\big | _ { \\phi = g _ { \\psi } ( \\{ { { \\bf B } _ { b } \\} } ) } \\big | \\big | _ { 2 } - 1 \\Big | \\qquad \\mathrm { ( o f f \\mathrm { - s u r f a c e ~ l o s s } ) } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { n _ { x } }$ is the surface normal of point $\\mathbf { x }$ . We assume that this information, along with the groundtruth correspondences from transformed space to canonical space is available when learning the meta-model on the training set, but do not require this for fine-tuning $g$ on unseen subjects. ",
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+    {
+        "type": "text",
+        "text": "In practice, we found that directly learning the hypernetwork $g _ { \\psi }$ via Eq. (6) does not converge, and thus we decompose the meta-learning of $g _ { \\psi }$ into two steps. First, we learn a meta-SDF [67] (without conditioning on $\\{ \\mathbf { B } _ { b } \\}$ , Sec. 4.1), and then we meta-learn a hypernetwork that takes $\\{ { \\bf B } _ { b } \\}$ as input and predicts the residuals to the parameters of the previously learned meta-SDF (Sec. 4.2). ",
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+    {
+        "type": "text",
+        "text": "4.1 Meta-learned Initialization of Static Neural SDFs ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To effectively learn a statistical prior of clothed human bodies, we ignore the input bone transformations $\\{ { \\bf B } _ { b } \\}$ and meta-learn the static neural SDF $f _ { \\phi } ( \\mathbf { \\dot { x } } ) : [ - 1 , 1 ] ^ { 3 } \\mapsto \\mathbb { R }$ , parameterized by $\\phi$ , from all canonicalized points of subjects with different genders, body shapes, cloth types, and poses. Furthermore, for faster and more stable convergence, the neural SDF $f _ { \\phi }$ function additionally leverages the periodic activation functions [68]. ",
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+    {
+        "type": "image",
+        "img_path": "images/d5ef2bd65a2017e3a496dc05e47de251369d4edd7223f0cb3c23cfcc5cd6c2c0.jpg",
+        "image_caption": [
+            "Figure 2: Overview of the meta-SDF network. We use a 5-layer SIREN [67] network with 256 neurons for each layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The full meta-learning algorithm for the static neural SDFs is described in Alg. 1. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Meta-learning SDF with Reptile [49] ",
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+        "type": "text",
+        "text": "Initialize: meta-network parameters $\\phi$ , meta learning rate $\\beta$ , inner learning rate $\\alpha$ , max training   \niteration $N$ , inner-loop iteration $m$ , batch size $M$   \n1: for $i = 1 , \\ldots , N$ do   \n2: Sample a batch of $M$ training samples $\\{ \\hat { \\mathbf { X } } ^ { ( j ) } \\} _ { j = 1 } ^ { M }$   \n3: for $\\begin{array} { r l } & { \\mathbf { \\Lambda } _ { [ = 1 , \\dots , M \\mathbf { \\delta } ] \\mathbf { \\delta } \\mathbf { 1 } } ^ { [ = 1 , \\dots , M \\mathbf { \\delta } ] \\mathbf { \\delta } \\mathbf { 0 } } } \\\\ & { \\mathbf { \\Phi } _ { \\mathbf { 0 } } ^ { ( j ) } = \\phi } \\\\ & { \\mathbf { \\Phi } \\mathbf { 0 } \\mathbf { r } \\ k = 1 , \\dots , m \\mathbf { \\delta } \\mathbf { d } \\mathbf { \\mathbf { 0 } } } \\\\ & { \\quad \\phi _ { k } ^ { ( j ) } = \\phi _ { k - 1 } ^ { ( j ) } - \\alpha \\nabla _ { \\phi } \\mathcal { L } _ { \\mathrm { I G R } } \\big ( f _ { \\phi } ( \\hat { \\mathbf { X } } ^ { ( j ) } ) \\big | _ { \\phi = \\phi _ { k - 1 } ^ { ( j ) } } \\big ) } \\end{array}$   \n4: φ(j) =   \n5: f   \n6:   \n7: end for   \n8: 9: $\\begin{array} { r l } & { \\mathrel { \\phantom { = } } \\phi  \\phi + \\beta \\frac { 1 } { M } \\sum _ { j = 1 } ^ { M } ( \\phi _ { m } ^ { ( j ) } - \\phi ) } \\end{array}$   \n10: end for ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Meta-learned Initialization of HyperNetwork for Dynamic Neural SDFs ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The meta-learned static neural SDF explained in the previous section can efficiently adapt to new observations, however it is not controllable by user-specified bone transformations $\\{ \\bar { \\mathbf { B } } _ { b } \\}$ . Therefore, to enable non-rigid pose-dependent cloth deformations, we further meta-learn a hypernetwork [25] to predict residuals to the learned parameters of the meta-SDF in Alg. 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The key motivation for meta-learning the hypernetwork is to build an effective unified prior for articulated clothed humans, which enables the recovery of the non-rigid clothing deformations at test time via the efficient fine-tuning process from several depth images of unseen subjects. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Meta-learning hypernetwork with Modified Reptile ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            147,
+            599,
+            162
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Initialize: meta-hypernetwork parameters $\\psi$ , pre-trained meta-SDF parameters $\\phi ^ { * }$ , meta learning rate $\\beta$ , inner learning rate $\\alpha$ , max training iteration $N$ , inner-loop iteration $m$ . 1: for $i = 1 , \\ldots , N$ do 2: $\\psi _ { 0 } = \\psi$ 3: Randomly choose a subject/cloth-type combination $n$ 4: Uniformly sample $M \\sim \\{ 1 , \\dots , D ^ { ( n ) } \\}$ where $D ^ { ( n ) }$ is the number of datapoints of subject/cloth-type combination $n$ 5: Sample $M$ datapoints from subject/cloth-type combination $n$ , denoting these datapoints as $\\mathcal { S } = \\{ \\{ \\bar { \\bf B } _ { b } \\} ^ { ( j ) } , \\hat { \\bf X } ^ { ( j ) } \\} _ { j = 1 } ^ { M }$ 6: for $k = 1 , \\ldots , m$ do 7: L = 1M P({Bb},Xˆ )∈S LIGR(fφ(Xˆ )|φ=gψk−1 ({Bb})+φ∗ ) 8: ψk = ψk−1 − α∇ψk−1L 9: end for 10: $\\psi  \\psi + \\beta ( \\psi _ { m } - \\psi _ { 0 } )$ 11: end for ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Denoting the meta-SDF learned by Alg. 1 as $\\phi ^ { * }$ and our hypernetwork as $g _ { \\psi } ( \\{ \\mathbf { B } _ { b } \\} )$ , we implement Alg. 2. This algorithm differs from the original Reptile [49] algorithm in that it tries to optimize the inner-loop on arbitrary amount of data. Note that for brevity the loss in the innerloop (line 7-line 8) is computed over the whole batch $s$ , whereas in practice we used stochastic gradient descent (SGD) with fixed mini-batch size over $s$ since $s$ can contain hundreds of samples; SGD is used with the mini-batch size of 12 for the inner-loop. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7e4a5f9cf29a9498401db73082b325c23e30988ddf4b5de04970a3d0ded562d1.jpg",
+        "image_caption": [
+            "Figure 3: Overview of the meta-hypernetwork. It predicts residuals to $\\phi ^ { * }$ which is learned in Sec. 4.1 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Inference: At test-time, we are given a small fine-tuning set {{Bb}f ine,(j), Xˆ f ine,(j)}Mj= and the validation set {{Bb}val,(j)}Kj= The fine-tuning set is used to optimize the hypernetwork parameters $\\psi$ ( $m = 2 5 6$ SGD epochs) that ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "are then used to generate neural SDFs from bone transformations available in the validation set. The overall inference pipeline including the inverse and the forward LBS stages is shown in Fig. 4. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Bone Transformation Encoding: We found that a small hierarchical MLP proposed in LEAP [48] for encoding bone transformations works slightly better than the encoding of unit quaternions used in SCANimate [65]. Thus, we employ the hierarchical MLP encoder to encode $\\{ \\mathbf { B } _ { b } \\}$ for $g$ unless specified otherwise; we ablate different encoding types in the experiment section. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We validate the proposed MetaAvatar model for learning meta-models and controllable dynamic neural SDFs of clothed humans by first comparing our MetaAvatar to the established approaches [13, 48, 65]. Then, we ablate the modeling choices for the proposed controllable neural SDFs. And lastly, we demonstrate MetaAvatar’s capability to tackle the challenging task of learning animatable clothed human models from reduced data, to the point that only 8 depth images are available as input. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets: We use the CAPE dataset [43] as the major test bed for our experiments. This dataset consists of 148584 pairs of clothed meshes, capturing 15 human subjects wearing different clothes while performing different actions. We use 10 subjects for meta-learning, which we denote as the training set. We use four unseen subjects (00122, 00134, 00215, 03375)2 for fine-tuning and validation; for each of these four subjects, the corresponding action sequences are split into finetuning set and validation set. The fine-tuning set is used for fine-tuning the MetaAvatar models, it is also used to evaluate pose interpolation task. The validation set is used for evaluating novel pose extrapolation. Among the four unseen subjects, two of them (00122, 00215) perform actions that are present in the training set for the meta-learning; we randomly split actions of these two subjects with $70 \\%$ fine-tuning and $30 \\%$ validation. Subject 00134 and 03375 perform two trials of actions unseen in the training set for meta-learning. We use the first trial as the fine-tuning set and the second trial as the validation set. Subject 03375 also has one cloth type (blazer) that is unseen during meta-learning. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/499f555f0f574cbcc116e67f0ea4e1494f130391d8a750d3dc7d9a366cd3ba36.jpg",
+        "image_caption": [
+            "Figure 4: Overview of our inference pipeline. The inverse LBS net (Sec. 3.2) takes a small set of input depth frames together with their underlying SMPL registrations to canonicalize the depth points; then the meta-learned hypernetwork (Sec. 4.2) is fine-tuned to represent the instance specific dynamic SDF; given novel poses, the updated hypernetwork generates pose-dependent cloth-deformations in canonical space, and the animated meshes are obtained via the forward LBS network (Sec. 3.2). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Baselines: We use NASA [13], LEAP [48], and SCANimate [65] as our baselines. NASA and SCANimate cannot handle multi-subject-cloth with a single model so we train per-subject/cloth-type models from scratch for each of them on the fine-tuning set. LEAP is a generalizable neural-implicit human body model that has shown to work on minimally-clothed bodies. We extend LEAP by adding a one-hot encoding to represent different cloth types (similarly to [43]) and train it jointly on the full training and the fine-tuning set. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As for the input format, we use depth frames rendered from CAPE meshes for our MetaAvatar. To render the depth frames, we fixed the camera and rotate the CAPE meshes around the y-axis (in SMPL space) at different angles with an interval of 45 degrees; note that for each mesh we only render it on one angle, simulating a monocular camera taking a round-view of a moving person. For the baselines, we use watertight meshes and provide the occupancy [45] loss to supervise the training of NASA and LEAP, while sampling surface points and normals on watertight meshes to provide the IGR loss supervision for SCANimate. Note that our model is at great disadvantage, as for fine-tuning we only use discrete monocular depth observations without accurate surface normal information. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Tasks and Evaluation: Our goal is to generate realistic clothing deformations from arbitrary input human pose parameters. To systematically understand the generalization capability of the MetaAvatar representation, we validate the baselines and MetaAvatar on two tasks, pose interpolation and extrapolation. For interpolation, we sample every 10th frame on the fine-tuning set for training/finetuning, and sample every 5th frame (excluding the training frames) also on the fine-tuning set for validation. For extrapolation, we sample every 5th frame on the fine-tuning set for training, and sample every 5th frame on the validation set for validation. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Interpolation is evaluated using three metrics: point-based ground-truth-to-mesh distance $( D _ { p } \\downarrow ,$ in cm), face-based ground-truth-to-mesh distance $( D _ { f } \\downarrow ,$ , in cm), and point-based ground-truth-to-mesh normal consistency $( N C \\uparrow$ , in range $[ - 1 , 1 ] )$ . For computing these interpolation metrics we ignore non-clothed body parts such as hands, feet, head, and neck. For extrapolation, we note that clothdeformation are often stochastic; in such a case, predicting overly smooth surfaces can result in lower distances and higher normal consistency. Thus, we also conduct a large-scale perceptual study using ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/04778bfeba6e8531b15fb31b8bf10ba2173d2807c481cc0b0182997a03765d4c.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative comparison on extrapolation results with blazer outfit. NASA shows consistent blocky artifacts. LEAP predicts overly smooth surfaces missing the tails of the blazer outfit. SCANimate does not generalize as this specific pose has not been seen during training. Directly meta-learning a SIREN [68] network that conditions on input poses produces a smooth surface that does not capture the blazer tails well. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c48f179a578faed555e41852a3e3ab47ab69bbf01a2b531186854ba0fb96fb79.jpg",
+        "image_caption": [
+            "Figure 6: Qualitative comparison on extrapolation results when reducing fine-tuning data on subject 00215 wearing poloshirt. The caption indicates the amount of fine-tuning data used to fine-tune the meta-hypernetwork on this unseen subject. Our meta-learned model captures for this unseen subject the sliding effect of the poloshirt at this pose in which the person raising arms, even fine-tuned with just 8 depth images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Amazon Mechanical Turk, and report the perceptual scores $( \\mathrm { P S } \\uparrow )$ which reflects the percentage of users who favor the outputs of baselines over MetaAvatar. Details about user study design can be found in the Supp. Mat. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.1 Evaluation Against Baselines ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we report results on both interpolation and extrapolation tasks against various baselines described above. For NASA and SCANimate, we train one model for each subject/clothtype combination on the fine-tuning set. For them it usually takes several thousand of epochs to converge for each subject/cloth-type combination, which roughly equals to 10-24 hours of training. For LEAP, we train a single model on both the training and the fine-tuning set using two days. For the MetaAvatar, we meta-learn a single model on the training set, and for each subject/cloth-type combination we fine-tune the model for 256 epochs to produce subject/cloth-type specific models. The exact fine-tuning time ranges from 40 minutes to 3 hours depending on the amount of available data since we are running a fixed number of epochs; see the Supp. Mat. for detailed runtime comparison on each subject/cloth-type combination. Note that MetaAvatar uses partial depth observations while the other baselines are trained on complete meshes. The results are reported in Table 1. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Importantly, models of NASA and SCANimate are over-fitted to each subject/cloth-type combination as they cannot straightforwardly leverage prior knowledge from multiple training subjects. LEAP is trained on all training and fine-tuning data with input encodings to distinguish different body shapes and cloth types, but it fails to capture high-frequency details of clothes, often predicting smooth surfaces (Fig. 5); this is evidenced by its lower perceptual scores (PS) compared to SCANimate and our MetaAvatar. In contrast to these baselines, MetaAvatar successfully captures a unified clothing deformation prior of diverse body shapes and cloth types, which generalizes well to unseen body shapes (00122, 00215), unseen poses (00134, 03375), and unseen cloth types (03375 with blazer outfit); although we did not outperform LEAP on the interpolation task for subject 00134 and 03375, we note that 1) our method uses only 2.5D input for fine-tuning, while LEAP has access to groundtruth canonical meshes during training; 2) subject 00134 and 03375 comprise much more missing frames than subject 00122 and 00215, resulting in higher stochasticity and thus predicting smooth surfaces (such as LEAP) may yield better performance; this is also evidenced by LEAP’s much lower perceptual scores on subject 00134 and 03375, although obtaining the best performance for pose interpolation. We encourage the readers to watch the side-by-side comparison videos available on our project page: https://neuralbodies.github.io/metavatar/. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f00b42d53f7f835ffea83b8ada755d8dca0810240d7a099e7094fa3d3c14cd90.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"2\">3DInput</td><td>2.5D Input</td></tr><tr><td></td><td>NASA</td><td>LEAP</td><td>SCANimate</td><td>Ours</td></tr><tr><td colspan=\"5\">Subj 100122，00215</td></tr><tr><td>Ex.</td><td>PS↑</td><td>0.078</td><td>0.314 0.333</td><td>0.5</td></tr><tr><td rowspan=\"3\">Int.</td><td>Dp↓</td><td>0.484</td><td>0.454 0.586</td><td>0.450</td></tr><tr><td>Df↓</td><td>0.327 0.293</td><td>0.489</td><td>0.273</td></tr><tr><td>NC↑</td><td>0.752</td><td>0.807 0.793</td><td>0.821</td></tr><tr><td colspan=\"5\">Subj 00134，03375</td></tr><tr><td>Ex.</td><td>PS↑</td><td>0.182 0.224</td><td>0.481</td><td>0.5</td></tr><tr><td rowspan=\"3\">Int.</td><td>Dp↓</td><td>0.595</td><td>0.483</td><td>0.518</td></tr><tr><td>Df↓</td><td>0.469 0.340</td><td>0.542</td><td>0.367</td></tr><tr><td>NC↑</td><td>0.693 0.780</td><td>0.755</td><td>0.773</td></tr><tr><td colspan=\"5\">Averge per-model training/fine-tuning time (hours)</td></tr><tr><td colspan=\"2\">&gt;10</td><td>1</td><td>&gt;10</td><td>1.60</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Comparison to baselines. $D _ { p } , D _ { f }$ and $N C$ are reported for interpolation (Int.) while PS is reported for extrapolation $\\left( \\mathrm { E x . } \\right)$ . Note that MetaAvatar is fine-tuned on depth images while all other baselines are trained on complete meshes. The training/fine-tuning times are just rough estimates, as ours does not include the time for meta-learning, while many factors, including varying training schedules, disk-IOs and hardware setups, can affect the final speed. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/6042ca69cef7218b3ddc873e059b4c9a4b481e09ecfeb2cc3a0d6ef758d306cb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td>MLP</td><td>PosEnc</td><td>SIREN</td><td>Hyper Quat</td><td>Hyper BoneEnc</td></tr><tr><td colspan=\"7\">Subj 00122,00215</td></tr><tr><td rowspan=\"3\">Int.</td><td>Dp↓</td><td>3.278</td><td>1.806</td><td>0.472</td><td>0.460</td><td>0.461</td></tr><tr><td>D↓</td><td>2.201</td><td>0.998</td><td>0.301</td><td>0.288</td><td>0.288</td></tr><tr><td>NC↑</td><td>-0.279</td><td>-0.045</td><td>0.815</td><td>0.818</td><td>0.820</td></tr><tr><td colspan=\"7\">Subj 100134,03375</td></tr><tr><td rowspan=\"3\">Int.</td><td>Dp↓</td><td>3.320</td><td>1.498</td><td>0.532</td><td>0.526</td><td>0.523</td></tr><tr><td>Df↓</td><td>2.190</td><td>0.772</td><td>0.385</td><td>0.378</td><td>0.374</td></tr><tr><td>NC↑</td><td>-0.300</td><td>-0.099</td><td>0.773</td><td>0.772</td><td>0.772</td></tr></table>",
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Ablation for different architectures on the interpolation task. Hyper-Quat is our model that takes the relative joint-rotations (in the form of unit quaternions) as inputs. HyperBoneEnc is our full model with hierarchical bone encoding MLP of LEAP [48]. Models in the table are fine-tuned for 128 epochs. ",
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Ablation Study on Model Architectures ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We further ablate model architecture choices for MetaAvatar. We compare against (1) a plain MLP that takes the concatenation of the relative joint-rotations (in the form of unit quaternions) and query points as input (MLP), (2) a MLP that takes the concatenation of the relative joint-rotations and the positional encodings of query point coordinates as input (PosEnc), and (3) a SIREN network that takes the concatenation of the relative joint-rotations and query points as input (SIREN). The evaluation task is interpolation; results are reported in Table 2. For the baselines (MLP, PosEnc and SIREN), we directly use Alg. 2 to meta-learn the corresponding models with $\\phi ^ { * } = 0$ . For MLP and PosEnc, the corresponding models fail to produce reasonable shapes. For SIREN, it produces unnaturally smooth surfaces which cannot capture fine clothing details such as wrinkles (Fig. 5). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3 Few-shot learning of MetaAvatar ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "In this section, we evaluate the few-shot learning capabilities of MetaAvatar. As shown in Table 3, we reduce the amount of data on the fine-tuning set, and report the performance of models finetuned on reduced amount of data. Note that with $< 1 \\%$ data, we require only one frame from each action sequence available for a subject/cloth-type combination, this roughly equals to 8-20 depth frames depending on the amount of data for that subject/cloth-type combination. For interpolation, the performance drops because the stochastic nature of cloth deformation becomes dominant when the amount of fine-tuning data decreases. On the other hand, the perceptual scores (PS) are better than NASA and LEAP even with ${ \\tt c 1 \\% }$ data in the form of partial depth observations, and better ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e2f34edcfbf9ef14db48617cb7333275649a65faa44a4001ec9938212893a8df.jpg",
+        "table_caption": [
+            "Table 3: Ablation for few-shot learning. We report performance of MetaAvatar on reduced amount of fine-tuning data. Fine-tuning time scales linearly with the amount of data, since we run for a fixed number of epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Fine-tune data (%)</td><td></td><td>100</td><td>50</td><td>20</td><td>10</td><td>5</td><td>&lt;1</td></tr><tr><td colspan=\"8\">Subj 00122,00215</td></tr><tr><td>Ex.</td><td>PS个</td><td>0.5</td><td>0.471</td><td>0.509</td><td>0.473</td><td>0.373</td><td>0.510</td></tr><tr><td rowspan=\"3\">Int.</td><td>Dp↓</td><td>-</td><td>0.450</td><td>0.480</td><td>0.512</td><td>0.543</td><td>0.592</td></tr><tr><td>D↓</td><td>=</td><td>0.273</td><td>0.310</td><td>0.353</td><td>0.391</td><td>0.450</td></tr><tr><td>NC↑</td><td>-</td><td>0.821</td><td>0.808</td><td>0.795</td><td>0.785</td><td>0.768</td></tr><tr><td colspan=\"8\">Subj 00134， ，03375</td></tr><tr><td>Ex.</td><td>PS个</td><td>0.5</td><td>0.476</td><td>0.424</td><td>0.463</td><td>0.439</td><td>0.387</td></tr><tr><td rowspan=\"3\">Int.</td><td>Dp↓</td><td>1</td><td>0.518</td><td>0.545</td><td>0.576</td><td>0.603</td><td>0.619</td></tr><tr><td>Df↓</td><td></td><td>0.367</td><td>0.400</td><td>0.438</td><td>0.471</td><td>0.489</td></tr><tr><td>NC↑</td><td>-</td><td>0.773</td><td>0.762</td><td>0.753</td><td>0.745</td><td>0.737</td></tr><tr><td colspan=\"8\">Average per-model training/fine-tuning time ( (hours) 0.08 0.02</td></tr><tr><td colspan=\"8\">1.60 0.8 0.32 0.16</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "than or comparable to SCANimate in most cases. The qualitative comparison on extrapolation results of reduced fine-tuning data is shown in Fig. 6. Please see the Supp. Mat. for more qualitative results on few-shot learning, including results on depth from raw scans, results on real depth images and comparison with pre-trained SCANimate model. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We introduced MetaAvatar, a meta-learned hypernetwork that represents controllable dynamic neural SDFs applicable for generating clothed human avatars. Compared to existing methods, MetaAvatar learns from less data (temporally discrete monocular depth frames) and requires less time to represent novel unseen clothed humans. We demonstrated that the meta-learned deformation prior is robust and can be used to effectively generate realistic clothed human avatars in 2 minutes from as few as 8 depth observations. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "MetaAvatar is compatible with automatic registration methods [8, 75], human motion models [84, 85], implicit hand models [30, 31] and rendering primitives [40, 47] that could jointly enable an efficient end-to-end photo-realistic digitization of humans from commodity RGBD sensors, which has broad applicability in movies, games, and telepresence applications. However, this digitization may raise privacy concerns that need to be addressed carefully before deploying the introduced technology. ",
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+    },
+    {
+        "type": "text",
+        "text": "7 Acknowledgment ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Siyu Tang acknowledges funding by the Swiss National Science Foundation under project 200021_204840. Andreas Geiger was supported by the ERC Starting Grant LEGO-3D (850533) and DFG EXC number 2064/1 - project number 390727645. We thank Jinlong Yang for providing results of SCANimate on the CAPE dataset. We thank Yebin Liu for sharing POSEFusion [38] data and results during the rebuttal period. We also thank Yan Zhang, Siwei Zhang and Korrawe Karunratanakul for proof reading the paper. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            450,
+            266,
+            467
+        ],
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+    },
+    {
+        "type": "text",
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parse/train/SJBr9Mcxl/SJBr9Mcxl.md

@@ -0,0 +1,228 @@
+# UNDERSTANDING TRAINED CNNS BY INDEXING NEURON SELECTIVITY
+
+Ivet Rafegas & Maria Vanrell
+
+Computer Vision Center   
+Universitat Autonoma de Barcelona \`   
+Bellaterra, Barcelona (Spain)   
+{ivet.rafegas, maria.vanrell}@uab.cat   
+Lu´ıs A. Alexandre   
+Department of Computer Science   
+Universidade da Beira Interior   
+Covilha , Portugal ˜   
+lfbaa@ubi.pt
+
+# ABSTRACT
+
+The impressive performance and plasticity of convolutional neural networks to solve different vision problems are shadowed by their black-box nature and its consequent lack of full understanding. To reduce this gap we propose to describe the activity of individual neurons by quantifying their inherent selectivity to specific properties. Our approach is based on the definition of feature selectivity indexes that allow the ranking of neurons according to specific properties. Here we report the results of exploring selectivity indexes for: (a) an image feature (color); and (b) an image label (class membership). Our contribution is a framework to seek or classify neurons by indexing on these selectivity properties. It helps to find color selective neurons, such as a red-mushroom neuron in layer conv4 or class selective neurons such as dog-face neurons in layer conv5, and establishes a methodology to derive other selectivity properties. Indexing on neuron selectivity can statistically draw how features and classes are represented through layers at a moment when the size of trained nets is growing and automatic tools to index can be helpful.
+
+# 1 INTRODUCTION
+
+In parallel with the success of CNNs to solve vision problems, there is a growing interest in developing methodologies to understand and visualize the internal representations of these networks. How the responses of a trained CNN encode the visual information is a fundamental question for computer and eventually for human vision.
+
+Several works have proposed different methodologies to address the understanding problem. Recently, in Li et al. (2016) two main groups of works are mentioned. On one side those works that deal with the problem from a theoretical point of view. These are works such as Montavon et al. (2011) where kernel sequences are used to conclude that deep networks create increasingly better representations as the number of layer increases, Paul & Venkatasubramanian (2014) which explains why a deep learning network learns simple features first and that the representation complexity increases as the layers get deeper, Goodfellow et al. (2014) where an explanation for why an adversarial example created for one network is still valid in many others and they usually assign it the same (wrong) class, or Arora et al. (2014) that presents algorithms for training certain deep generative models with provable polynomial running time. On the other side, an empirical point of view, which comprises approaches that pursuit methodologies to visualize intermediate features in the image space, or approaches that analyze the effect of modifying a given feature map in a neuron activation. Our work is framed in the first subset of empirical approaches.
+
+Visualizing intermediate features seeks to describe the activity of individual neurons. This description is the basis of this work hypothesis that is based on the idea that a proper understanding of the activity of the individual neurons allow us to draw a map of the CNN behavior. This behavior can be understood either in terms of relevant image features or in terms of the discriminative power of the neurons across the full architecture.
+
+The first and most obvious way to describe the activity of a single neuron is given by the inherent set of weights of the learned filters. These weights can be used to compare neurons between them, either within the same layer or versus neurons in similar CNNs which have been trained under different initialization conditions, as it is proposed by Li et al. (2016). A direct visualization of these weights is intuitive when they belong to neurons of a first convolutional layer. However, when layers are stacked, that intuition disappears and the capability to understand the neuron activity is lost.
+
+A second method to describe neuron activity is projecting the filter weights into the image space, trying to get the inherent feature that maximally activates the filter. The projection can be computed by composing the inversion of the layer operators under a specific neuron towards the image space: this was called a Decoded Filter (DF) in Rafegas & Vanrell (2016). The resulting image represents an estimation of the feature that should highly activate such neuron. The disentangling algorithm that inverts the filter would give a good estimation of the feature image if most of the layer operators were invertible. However, when the number of non-invertible operators increases, the estimation becomes unintelligible. The appearance of the DFs can be seen in Fig. 1 of Rafegas & Vanrell (2016). They have also been explored by Springenberg et al. (2015) for architectures with no pooling layers since pooling is the less invertible operator. They point out the interest of obtaining such a representation, since it would allow the understanding of neuron activity independently of the input image. However, the majority of proficient CNNs contain pooling layers.
+
+A third way to describe neuron activity is by exploring the images that maximally activate the neuron. One of the most relevant works pursuing the visualization of intermediate features, is the one proposed by Zeiler & Fergus (2014), where they project intrinsic features of neurons from the image that have provoked a maximum spike to a certain neuron, the network representation is projected into the image space by isolating them in the deconvolution approach Zeiler et al. (2010). By observing different projections that maximally activate a certain neuron they get the intuition about the main features learned on the network. Later on, in Springenberg et al. (2015) the guided backpropagation improves the deconvolution approach by a new way of inverting rectified linear (ReLu) nonlinearities, achieving better visualizations of the activations. These approaches present a main drawback, their feature visualization is image-specific, since the maximum activation of a neuron not always generalize the intrinsic feature of the neuron. To solve this problem, in some works instead of using the image that provokes the maximum activation, they use optimization techniques to generate an image that maximizes the activation. The key point of these works is using an appropriate regularization in the generation process, otherwise, the resulting image appearance is unrealistic and difficult to understand. Simonyan et al. (2014) propose a method to generate an image which is representative of a certain class by maximizing the score of this image to be classified in a certain class (or highly activates the specified neuron) with an $L _ { 2 }$ -regularization. A similar work was performed afterwards in Yosinski et al. (2015) but taking advantage of combining three different regularizations to achieve more recognizable images. Although they have explored different regularizations to achieve more realistic intrinsic feature representations, their visualizations present important artifacts that complicate the understanding of the intrinsic property.
+
+Finally, other works focus on proposing approaches able to reconstruct the input image given a feature map, going further of analyzing the individual neuron activity. Mahendran & Vedaldi (2015) make use of optimization algorithms to search for an image whose feature map best matches a given feature map by incorporating natural image priors. Contrary, in Dosovitskiy & Brox (2015), the authors propose to reconstruct the input image from its feature maps of a given convolutional network by training a new deconvolutional network to learn filter weights that minimize the image reconstruction error when these filters are applied to the image feature maps. With this approach they are also able to get an image reconstruction with natural priors.
+
+In the second subset of empirical approaches, Alexey Dosovitskiy (2015) train a generative deconvolutional network to create images from neuron activations. With this methodology, the variation of the activations enables the visualization of the differences in the generated images. A similar analysis is done by Aubry & Russell (2015), but instead of forward-propagate different activations to the image space and comparing them, they observe the changes on neuron activations when similar computer-generated images with different scene factors are introduced into a CNN. These works contribute in giving a deeper understanding on the internal CNN behavior. Both works conclude that there are specific neurons which are sensitive to color changes, point of views, scale or lighting configurations.
+
+Likewise, in Zeiler & Fergus (2014) in this work we pursuit visualizing the intrinsic feature of a neuron by analyzing the images that maximally activates a specific neuron. However, to avoid the lack of generality of this approach, we define the Neuron Feature which is not based on a single maximum activation. The Neuron Feature is a weighted average version of a set of maximum activation images that capture the essential properties shared by the most important activations and makes it not to be image-specific. Additionally, our Neuron Feature overcomes the problem of unrealistic representation we metnioned earlier, by directly averaging on the image space. In this way we achieve two main advantages: (a) keeping the properties of the natural images, and (b) providing a very straightforward approach to compute it.
+
+![](images/0204076260902bf8f42015443e877c4cbca53d9cfd28aecfc5a3a6d3427e5f70.jpg)  
+Figure 1: Normalized activations of a subset of neurons for the first 400 ranked images through all convolutional layers. For each layer we plot the normalized activation for the neurons with highest and smallest AUC (Area Under Curve), and some other examples in between these extremes. For all neurons the highest normalized activations is 1, and the percentage of AUC is computed with respect to the neuron AUC achieving the biggest area in the entire network.
+
+Afterwards, we introduce the concept of neuron selectivity index, that is used in human vision research to characterize the response of specific cells to specific stimuli (Shapley & Hawken (2011)). This concept allows to achieve a higher level of abstraction in the understanding of a single neuron. In this work we provide two selectivity indexes which are different in their essence: a color selectivity index that quantifies the degree of response of a neuron to a specific color; a class selectivity index that quantifies the degree of response of a neuron to a specific class label. Indexes are derived from the neuron feature or directly from the set of images with maximum activations. We analyze both indexes on a VGG-M network (Chatfield et al. (2014)) trained on ImageNet (Deng et al. (2009)) and we confirm their flexibility to cluster neurons according to their index values and extract conclusions in terms of their task in the net. By selecting color selective neurons we are able to outline how color is represented by the network. Curiously we found some parallelism between color representation in the first convolutional layer and known evidences about the representation in the human visual system. Color selective-neurons also show some preferences towards specific colors which coincide with ImageNet color biases. Indexing on class selectivity neurons we found highly class selective neurons like digital-clock at conv2, cardoon at conv3 and ladybug at conv5, much before the fully connected layers.
+
+# 2 NEURON FEATURE
+
+As we mentioned in the previous section we propose to visualize the image feature that activates a neuron, whenever is possible, by directly computing a weighted average of the $N$ -th first images that maximally activate this neuron. We will refer to it as the Neuron Feature (NF).
+
+In order to build the NF we need to calculate the activations associated to each individual neuron. They need to be accordingly ranked with the rest of activations of the layer. For each neuron we select the set of images that achieve a minimum normalized activation value but constrained to a maximum number of images for practical reasons. By normalized activation we mean the value of the maximum activation of a neuron for a specific input image, which is normalized by the maximum of these values achieved by the same neuron over all the images in the dataset.
+
+In Fig. 1 we can see the behavior of the ranked normalized responses of a subset of neurons for every convolution layer of the VGG-M CNN trained on ImageNet by Chatfield et al. (2014). The y-axis represents the normalized activation value of a single neuron to an image of the dataset. Images are ranked on the $\mathbf { X }$ -axis according with their activation value, from highest to lowest activation (we just plot the first 400 images for each neuron). Therefore, the first relative activation value is always 1 for all neurons and then the normalized activation values decrease monotonically. This normalization allows to compare different neuron behaviors, from neurons which are activated by most of the images (flatter behavior), to neurons that highly activates only for a subset of images and have very little activation for the rest (steeper behavior). In this figure we also provide the percentage of area for each plotted curve. This percentage is computed over the area of the neuron that presents the maximum AUC in the entire architecture. We can observe different behaviors in all layers. In general, we can state that in deeper layers the behavior of the neurons is steeper (lower AUC), i.e. neurons highly spike for a small number of images. However, in shallower layers the behavior is flatter, i.e. neurons highly spike for a lot of images. This is an expected behavior, since the image features spiking neurons in first layers (e.g. oriented edges) are shared by almost all the images, while the features spiking shallow neurons are more selective features (e.g. faces) that only spike for specific images. The observation of the responses confirms the adequacy of our assumption to fix a minimum value for the activation and a maximum number of images to capture the most important activations for all the neurons. Similar observations have been made for other networks like VGG-S and VGG-F Chatfield et al. (2014) 1.
+
+![](images/bc333c47e3a2156f0c6e04657a0f50c1d14adaa6eef8573e29aa44452330362e.jpg)  
+Figure 2: Neuron Feature (NF) visualizations (top) for 5 neuronsof the different convolutional layers of VGG-M with their corresponding 100 cropped images (bottom). We scale all layers to the same size due to space constraints.
+
+![](images/eabb814d949626d99a3357fde25e7c632a6ca4811368de4893acef50f1fbddd9.jpg)  
+Figure 3: Examples of NFs for each convolutional layer of the network VGG-M (see section 4.1. (a) 20 examples of structured NF, (b), blurred NF. Although sizes of NF increments through layers, we scale them into the same size. Original sizes are: 7x7x3 , 27x27x3, 75x75x3, $1 0 7 \mathrm { x } 1 0 7 \mathrm { x } 3$ and $1 3 9 \mathrm { x } 1 3 9 \mathrm { x } 3$ for conv1, conv2, conv3, conv4 and conv5, respectively.
+
+Thus, the NF is computed as:
+
+$$
+N F ( n ^ { L , i } ) = \frac { 1 } { N _ { m a x } } \sum _ { j = 1 } ^ { N _ { m a x } } w _ { j , i , L } I _ { j }
+$$
+
+where $w _ { j , i , L }$ is the relative activation of the $j$ -th cropped image, denoted as $I _ { j }$ , of the $i$ -th neuron $n ^ { L , i }$ at layer $L$ . The relative activation is the activation $a _ { j , i }$ of a neuron, given a input image, with respect to its maximum activation obtained for any image, $\begin{array} { r } { \dot { w } _ { j , i , L } = \frac { a _ { j , i } } { a _ { m a x , i } } } \end{array}$ where $a _ { m a x , i } = \operatorname* { m a x } a _ { k , i } , \forall k$ .
+
+In Fig. 2 we can see some NFs and their corresponding set of first 100 maximum activations, and in Fig. 3 (a) we can see a selected subset of $2 0 \mathrm { N F }$ per layer. In this image we can identify specific shapes that display the intrinsic property that fires a single neuron. At first glance, we can see how in this particular network the first two layers are devoted to basic properties. Oriented edges of different frequencies and in different colors in the first layer; textures, blobs, bars and more specific curves in the second layer. The rest of the layers seem to be devoted to more complex objects. We can see that dog and human faces, cars and flowers are detected at different scales in different layers, since the size of the NF and their corresponding cropped images increase with depth. This visualization of the neuron activity can be seen as a way to visualize a trained vocabulary of the CNN that opens multiple ways to analyze the global behavior of the network from its single units. However, not all neurons present such a clear tuning to an identifiable shape. Some neurons present a blurred version of NF, such as, those in Fig. 3(b). The level of blurring is directly related to a high variability between the maximally activated images for a neuron.
+
+At this point, we want to make a short parenthesis to relate the previous representational observations with the scientific problem about neural coding that is focus of attention in visual brain research (Kriegeskorte & Kreiman (2011)). We are referring to the hypothesis about distributed representations that encode object information in neuron population codes, that co-exist with strong evidences of neurons which are only activated by a very specific object. In line with this idea, we invite to speculate about neurons presenting a highly structured NF could be closer to localist code neurons while neurons with a blurred NF as closer to a distributed code. We return on this discussion later on at sections 4.3 and 5.
+
+Finally, we want to add a further analysis about how neuron feature is related to the neuron activity is representing. In Fig. 4 we plot the level of the neuron responses when the input image is its own NF. We can observe a high degree of activation (in green) between the NF and the response of the net to this feature. However we have some disagreements between the NF and the neuron activations: an important example is shown in layer 2, that is curiously bigger than in layer 3 and 4. This is explained by the high number of dead neurons2 and also by a higher presence of texture selective neurons, that is observed in Fig. 3. Another example, which is more understandable, is the clear increase of disagreement that happens through layers 3, 4 and 5, that seems to be explained by an increase in invariance that is obvious when the size of the image increases.
+
+![](images/3e15e431a10f442e23ee018e2a9d7bb95132b38afc8fc4f4276f0cba02508f52.jpg)  
+Figure 4: Number of neurons and degree of activation as a response to their own NF. Activations values are normalized to a specific range within each layer.
+
+![](images/269bd2c285bed5348fb4bd4742f3382a7c289711514d4782091a8ffba0e55a6e.jpg)  
+Figure 5: Conv1 NFs sorted by their color selectivity index.
+
+# 3 NEURON SELECTIVITY INDEX
+
+In this section we propose to describe neurons by their inherent response to a specific property, using an index. The index has to allow to rank them in a proportional order between their response and the existence of the property in the input image. Therefore, we translate the problem of describing neuron activity to the problem of proposing methods which are able to quantify specific image facets that correlate with the degree of activation of the neuron holding such a property. A selectivity index of a single unit is a flexible an independent method for discriminating or clustering between neurons inside the same network. Selectivity indexes can be defined either for image features or for image labels. In what follows, we propose two selectivity indexes one on each group.
+
+# 3.1 COLOR SELECTIVITY INDEX
+
+Color selectivity is a property that can be proved in specific neurons of the human brain. The level of activation of the neuron when the observer is exposed to a stimulus with a strong color bias, and its corresponding low activation when the color is not present, is the object of attention in vision research that pursuits the understanding of how color is coded in the human visual system (Shapley & Hawken (2011),Conway & Tsao (2009)).
+
+Here we propose a method to compute a color selectivity index for neurons in artificial neural networks. We propose to base it directly on the image properties of the NF we have defined above. We quantify the selectivity to a specific chromaticity directly from the color distribution of the NF. We define this index as the angle between the first principal component (v) of the color distribution of the NF and the intensity axis (b) of the Opponent Color Space (OPP). To compute (v) we use a weighted Principal Component Analysis Delchambre (2014) that allows to strengthen the selectivity of small color areas. Weights are applied to each pixel in order to reinforce those pixels that are shared by most cropped images and that highly contribute to the NF. Therefore, the weights are the inverse of the standard deviation. In this way, a NF defined by cropped images with different colors will tend to be represented by a grayish image and its principal component will be close to the intensity axis in the OPP color space and it will receive a low selectivity index. We formulate this index (in degrees) as follows:
+
+$$
+\alpha ( n ^ { L , i } ) = { \frac { 1 } { 9 0 } } \operatorname { a r c c o s } \left( { \frac { \mathbf { b } \cdot \mathbf { v } } { \| \mathbf { b } \| \| \mathbf { v } \| } } \right)
+$$
+
+Other selectivity indexes that can be derived from this, are those related to color attributes. We can easily extract color name labels using a color naming approach such as Benavente et al. (2008) and directly define color selectivity to basic names such as red, or green, among others.
+
+# 3.2 CLASS SELECTIVITY INDEX
+
+Class selectivity is a property of a neuron that can help to establish its discriminative power for one specific class or can allow to cluster neurons accordingly with the ontological properties of their class labels.
+
+We propose a method to compute a class selectivity index for individual neurons by compiling the class labels of the images that maximally activates this neuron in a single descriptor. We define class selectivity from the set of class labels of the $N$ images used to build the NF. To quantify this index we build the class label distribution of the full set of images. As in the color selectivity index, we weight the significance of a class label by the relative activation of its image. Thus, the relative frequency of each class $c$ for a certain neuron is defined as:
+
+$$
+f _ { c } ( n ^ { i , L } ) = \frac { \sum _ { j } ^ { N _ { c } } w _ { j , i , L } } { \sum _ { l } ^ { N } w _ { l , i , L } }
+$$
+
+where $N _ { c }$ refers to the number of images, among the $N$ cropped images activating this neuron, that belong to class $c$ .
+
+Given the densities for all the classes. Finally, our class selectivity index is defined as follows:
+
+$$
+\gamma ( n ^ { L , i } ) = \frac { N - M } { N - 1 }
+$$
+
+where $M$ is the minimum number of classes that covers a pre-fixed ratio, $^ { t h }$ , of the neuron activation, this can be denoted as $\begin{array} { r } { \sum _ { c } ^ { M } f _ { c } \geq t h } \end{array}$ . This threshold allow to avoid considering class labels with very small activation weight. Jointly with the index value the selectivity provides the set of $M$ classes that describe the neuron selectivity and their corresponding relative frequency values.
+
+Therefore, a low class selectivity index indicates a poor contribution of this neuron to a single class (minimum is 0 when $M = N$ ), while a high value (maximum is 1) indicates a strong contribution of this neuron to a single class. In between we can have different degrees of selectivity to different number of classes. Obviously, this index is irrelevant for the last fully connected layers in a CNN, but it allows to group related neurons across different convolutional layers.
+
+Here we want to point out, that this index can also contribute to give some insights about the problem of how information is coded through layers, in the debate of localist and distributed neural codes we mentioned before (Kriegeskorte & Kreiman (2011)). Neurons with high class selectivity index should be in line with a localist code, while neurons with low class selectivity index should be part of a distributed code. This way the index is defined allow a large range of interpretations in between these two kinds of coding as it has been outlined in the visual coding literature.
+
+# 4 RESULTS
+
+In this section we report some empirical results to show how the proposed selectivity indexes perform and what representational conclusions we can extract from the subsets of neurons sharing indexed properties.
+
+# 4.1 EXPERIMENTAL SETUP
+
+In this paper we analyze the neurons of a CNN architecture trained on ImageNet ILSVRC dataset Deng et al. (2009) (using a subset of 1.2M images classified in 1.000 categories). We report the results for the VGG-M CNN that was trained by Chatfield et al. (2014) for a generic visual task of object recognition. The details of the CNN architecture are given in table 1. We selected this network since it has a similar structure to those which have been reported as having a representational performance that competes with human performance (as was proved in Cadieu et al. (2014)). Nevertheless, we have obtained similar results for VGG-F and VGG-S that are provided in Chatfield et al. (2014). We used the Matconvnet library provided by Vedaldi & Lenc (2015) for all the experiments.
+
+<table><tr><td rowspan=1 colspan=1>conv1</td><td rowspan=1 colspan=1>conv2</td><td rowspan=1 colspan=1>conv3</td><td rowspan=1 colspan=1>conv4</td><td rowspan=1 colspan=1>conv5</td><td rowspan=1 colspan=1>full6</td><td rowspan=1 colspan=1>ful17</td><td rowspan=1 colspan=1>ful18</td></tr><tr><td rowspan=1 colspan=1>96x7x7st.2, pad.0LRN, x2 pool</td><td rowspan=1 colspan=1>256x5x5st.2, pad.1LRN, x2 pool</td><td rowspan=1 colspan=1>512x3x3st.1, pad.1</td><td rowspan=1 colspan=1>512x3x3st.1, pad.1</td><td rowspan=1 colspan=1>512x3x3st.1, pad.1x2 pool</td><td rowspan=1 colspan=1>4096dropout</td><td rowspan=1 colspan=1>4096dropout</td><td rowspan=1 colspan=1>1000softmax</td></tr></table>
+
+Table 1: VGG-M architecture designed by Chatfield et al. (2014), where $M \times N \times P$ corresponds to number of filters, number of rows and columns of the filters respectively. $S t .$ . and pad. refers to stride and padding respectively; LRN is a ReLU and the corresponding pooling $( p o o l )$ if applied.
+
+![](images/ec52b975e9b7673884b7f4c98981a113f82f63806d12b5422c37a054c3332daf.jpg)  
+Figure 6: Neurons with different color selectivity indexes. Images in 4 rows (1st and 3rd row are NFs, 2nd and 4th rows are sets of cropped images that maximally activates the neuron).
+
+# 4.2 COLOR SELECTIVITY
+
+General purpose CNN architectures are usually trained on RGB color images. However there is a strong belief in the computer vision community that color is a dispensable property. The results we obtain by indexing color selective neurons make us conclude that there is no basis for such a belief. Results show that color is strongly entangled at all levels of the CNN representation. In a preliminary experiment we have tested a subset of ImageNet images with VGG-M in their original color and the same subset in a gray scale representation. Classification results show a considerable decrease: while original RGB images are classified with a $2 7 . 5 0 \%$ top-1 error and $1 0 . 1 4 \%$ top-5 error, gray scale image versions present $5 1 . 1 2 \%$ and $2 6 . 3 7 \%$ errors, top-1 and top-5 errors respectively.
+
+In a first experiment we extract how many NFs are related to color in each convolutional layer using the proposed color selectivity index. The bars in Fig. 8 plot the relative quantity of neurons that are color selective compared to those that are not. Grey represents the ratio of neurons that do not spike for the presence of a color and reddish represent neurons that are highly activated by the presence of a color. In the graphic we can observe that shallow layers are the main responsible for the color representation on the images: $5 0 \%$ and $4 0 \%$ of neurons are color selective in layers conv1 and conv2, respectively. Nevertheless, we also still found around $2 5 \%$ of color selective neurons in in deeper layers. Therefore, although neurons in deeper layers tend to be color invariant, an important part of the representation is devoted to color, that reinforces the discriminative power of color in object recognition. In Fig. 6 we show some examples of NFs with different degrees of color selectivity at different layers of the network and showing the corresponding cropped images.
+
+Regarding color representation in layer 1 we want to point out two more observations derived from the NFs (see Fig. 5): (a) selectivity to different spatial-frequencies is only tackled by gray-level neurons; and (b) four main color axis emerge (black-white, blue-yellow, orange-cyan and cyanmagenta). Curiously, these two observations correlate with evidences in the human visual system (Shapley & Hawken (2011)).
+
+![](images/38ae245bdac305005a49af38201628f343495291f93b9e70566d7c4ddf6b71b4.jpg)  
+Figure 7: Distribution of color selective neurons on a hue color space through layers. Maximum activation images for 4 top color selective neurons for each layer. Dashed rings connect NFs of color selective neurons through layers, from inner ring (conv1) to outer ring (conv5).
+
+![](images/389eb3416b3fe5532429c5273e7b5230f16c9613bad875e027ff7dc80f590d3a.jpg)  
+Figure 8: Number of neurons and degree of color selectivity through layers. Grayish bars are for low index values and reddish for high index values.
+
+![](images/edafef2e65b9e7d6de0863b0a3d4db8d04862593e8b1d81481eafa9c927daf8c.jpg)  
+Figure 9: Number of neurons and degree of class selectivity through layers. Grayish bars are for low index values and bluish for high index values.
+
+In a second experiment, we analyze how color selective neurons from all layers cover the color space. Figure 7 displays the distribution of color selective neurons with $\alpha \ge 0 . 4 0$ . Each NF is plotted on the hue angle that represents the projection of its first principal component on the OPP chromaticity plane (red-green and blue-yellow components). Dashed rings identify different convolutional layers from conv1 (inner ring) to conv5 (outer ring) linking the NFs that belong to the same layer. We can appreciate the emergence of an axis (from orange to cyan) that connects a crowded area of color selective neurons. We can add a low population of NFs in the magenta area, that becomes more crowded on the opposite side where green and yellow selectivity has several neurons. The interest of this explanation relies on the fact that a similar distribution appears in the ImageNet color distribution, that is plotted at the bottom of the same images, where a similar interpretation in terms of emergent axes can be done. A more in depth study is required to prove this correlation, but we illustrate how neuron selectivity helps in the understanding of how a specific property is represented by the CNN.
+
+# 4.3 CLASS SELECTIVITY
+
+Following with the analysis of ranking neurons by their response to a certain property, here we focus on the proposed selectivity index that relates to image labels instead of to an image property, is the class selectivity index, which only applies for classification networks. We report the results of different experiments where we have fixed $t h = 1$ , which means we consider all the class labels for the $N = 1 0 0$ images that maximally activates the neuron. As we mentioned before, this index can enlighten how classes are encoded through the net layers, that again it can be related to the scientific problem of how general object recognition is encoded in the human brain. Here we hypothesize that the difference between localist or distributed codes could correlate with the idea of neurons highly selective to a single class and neurons highly selective to several classes, we resume on this later at section 5.
+
+In a first experiment we analyze how many neurons present different degrees of class selectivity through layers. The bars in Fig. 9 plot the relative quantity of neurons that are class selective compared to those that are not. Grey represents the ratio of neurons that are not activated by a single class and bluish represent neurons that are highly activated by a single class. Opposite to what we showed about color selectivity, we found most of class selective neurons in deeper layers, and no class selectivity in shallow layers, as expected. We have moved from a very basic image property, color, to a very high level property, class label. This fact corroborates the idea that CNNs start by defining basic feature detectors that are share by most of the classes, and the neurons become more specialized when they belong to deeper layers representing larger areas in the image space and therefore more complex shapes. We start to have neurons with relevant class selectivity in layer conv3, where a $5 \%$ of neurons is quite class selective and we found some neurons with a degree of selective close to 1. These ratios progressively increase up to layer conv5 where we have more than a $5 0 \%$ of neurons with a class selectivity index greater than 0.6, that means that we have less than 40 different classes activating this neuron, which is a very selective ratio considering the number of classes of the ImageNet dataset. In the same layer a $2 0 \%$ of neurons present a high class selectivity index, than means less than 20 different classes. Further experiments should explore how this graphic evolves by moving from current class labels which are on the leaves of the ImageNet ontology towards higher nodes with more generic classes.
+
+Secondly, we have visualized the properties of a set of images presenting different degrees of class selectivity in Fig. 10 for different levels of depth. We visualize each neuron with their NF visualization and the corresponding cropped images. We also show two tag clouds of each neuron. They visualize the importance of each class label. With an orange frame we plot the leave classes of the ImageNet ontology, while in the green frame we plot generic classes. This second analysis could help finding neurons that are specialized to a general semantic concept that different final classes share. Note that neurons with high class selectivity index have a set of cropped images that we can identify as belonging to the same class.
+
+Finally we stress the utility of ranking images by selectivity indexes in Fig. 11, where we show interesting neurons in different convolutional layers that present high values for both selectivity indexes, neurons which are both, color and class selective.
+
+# 5 CONCLUSIONS
+
+In this paper we propose a framework to analyze a trained CNN by dissecting individual neurons using their indexes of selectivity to specific properties. We have proposed two properties of different nature: (a) color, that is a low-level image property that we have shown to be entangled in all the representations levels of the net; (b) class label, that is a high-level image property that can be analyzed at different levels of abstraction. We have shown that while the number of color selective neurons decreases with depth, the number of class selective neuron increases. In this line of describing the activity of individual images, we have also proposed to visualize the activity with what we have called the neuron feature (NF), that allows to arise interesting structures that are shared by the images that highly activate a neuron.
+
+The proposed work have made us to speculate about two different ways to address the coding properties of individual neurons (localist versus distributed). Firstly, we have mentioned the possibility that a blurred NF, i.e. without a clear structure, belongs to a neuron that can be part of a distributed code where the neuron does not represent a selectivity to a single shape, maybe to diverse shapes than can be part of a code in deeper neurons. Secondly, we speculate about the possibility that neurons with high class selective index can represent a localist code, and part of a distributed when is low. In parallel, the analysis of the color selective neurons have made to arise some parallelism between color representation in the 1st convolutional layer and known evidences about the representation in the human visual system.
+
+![](images/826a14425cb5a5a953fbdc7fb8a24c699475862a82ae5cded3c360c4c2e80e81.jpg)  
+Figure 10: Neurons with different class selectivity indexes. For each neuron two images (top: NF, bottom: cropped images) and two tag clouds (top: leave classes, bottom: all classes in the ontology).
+
+![](images/6b3f1a583f754b14d49b9250a1f19ec877fac1f49023d6121dede424b497c131.jpg)  
+Figure 11: Examples of neurons with high color and class selectivity indexes.
+
+As further work we need to fully exploit the potential of the indexes in different CNN architectures, and defining new selectivity indexes like shape or texture, that could be a perfect complement to current ones.
+
+# ACKNOWLEDGMENTS
+
+Project funded by MINECO Ref. TIN (TIN2014-61068-R)
+
+# REFERENCES
+
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+
+Sanjeev Arora, Aditya Bhaskara, Rong Ge, and Tengyu Ma. Provable bounds for learning some deep representations. In ICML, pp. 584–592, 2014.
+
+Mathieu Aubry and Bryan C. Russell. Understanding deep features with computer-generated imagery. In ICCV, 2015.
+
+Robert Benavente, Maria Vanrell, and Ramon Baldrich. Parametric fuzzy sets for automatic color naming. JOSA, 25(10):2582–2593, Oct 2008.
+
+Charles F Cadieu, Ha Hong, Daniel L K Yamins, Nicolas Pinto, Diego Ardila, Ethan A Solomon, Najib J Majaj, and James J DiCarlo. Deep neural networks rival the representation of primate it cortex for core visual object recognition. PLoS computational biology, 10, 2014 Dec 2014.
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+Bevil R. Conway and Doris Y. Tsao. Color-tuned neurons are spatially clustered according to color preference within alert macaque posterior inferior temporal cortex. Proc Natl Acad Sci U S A., 42 (106):18034–18039, 2009.
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parse/train/SJBr9Mcxl/SJBr9Mcxl_content_list.json

@@ -0,0 +1,1237 @@
+[
+    {
+        "type": "text",
+        "text": "UNDERSTANDING TRAINED CNNS BY INDEXING NEURON SELECTIVITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ivet Rafegas & Maria Vanrell ",
+        "bbox": [
+            184,
+            170,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Computer Vision Center   \nUniversitat Autonoma de Barcelona \\`   \nBellaterra, Barcelona (Spain)   \n{ivet.rafegas, maria.vanrell}@uab.cat   \nLu´ıs A. Alexandre   \nDepartment of Computer Science   \nUniversidade da Beira Interior   \nCovilha , Portugal ˜   \nlfbaa@ubi.pt ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            276,
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+            291
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The impressive performance and plasticity of convolutional neural networks to solve different vision problems are shadowed by their black-box nature and its consequent lack of full understanding. To reduce this gap we propose to describe the activity of individual neurons by quantifying their inherent selectivity to specific properties. Our approach is based on the definition of feature selectivity indexes that allow the ranking of neurons according to specific properties. Here we report the results of exploring selectivity indexes for: (a) an image feature (color); and (b) an image label (class membership). Our contribution is a framework to seek or classify neurons by indexing on these selectivity properties. It helps to find color selective neurons, such as a red-mushroom neuron in layer conv4 or class selective neurons such as dog-face neurons in layer conv5, and establishes a methodology to derive other selectivity properties. Indexing on neuron selectivity can statistically draw how features and classes are represented through layers at a moment when the size of trained nets is growing and automatic tools to index can be helpful. ",
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In parallel with the success of CNNs to solve vision problems, there is a growing interest in developing methodologies to understand and visualize the internal representations of these networks. How the responses of a trained CNN encode the visual information is a fundamental question for computer and eventually for human vision. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Several works have proposed different methodologies to address the understanding problem. Recently, in Li et al. (2016) two main groups of works are mentioned. On one side those works that deal with the problem from a theoretical point of view. These are works such as Montavon et al. (2011) where kernel sequences are used to conclude that deep networks create increasingly better representations as the number of layer increases, Paul & Venkatasubramanian (2014) which explains why a deep learning network learns simple features first and that the representation complexity increases as the layers get deeper, Goodfellow et al. (2014) where an explanation for why an adversarial example created for one network is still valid in many others and they usually assign it the same (wrong) class, or Arora et al. (2014) that presents algorithms for training certain deep generative models with provable polynomial running time. On the other side, an empirical point of view, which comprises approaches that pursuit methodologies to visualize intermediate features in the image space, or approaches that analyze the effect of modifying a given feature map in a neuron activation. Our work is framed in the first subset of empirical approaches. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Visualizing intermediate features seeks to describe the activity of individual neurons. This description is the basis of this work hypothesis that is based on the idea that a proper understanding of the activity of the individual neurons allow us to draw a map of the CNN behavior. This behavior can be understood either in terms of relevant image features or in terms of the discriminative power of the neurons across the full architecture. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The first and most obvious way to describe the activity of a single neuron is given by the inherent set of weights of the learned filters. These weights can be used to compare neurons between them, either within the same layer or versus neurons in similar CNNs which have been trained under different initialization conditions, as it is proposed by Li et al. (2016). A direct visualization of these weights is intuitive when they belong to neurons of a first convolutional layer. However, when layers are stacked, that intuition disappears and the capability to understand the neuron activity is lost. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A second method to describe neuron activity is projecting the filter weights into the image space, trying to get the inherent feature that maximally activates the filter. The projection can be computed by composing the inversion of the layer operators under a specific neuron towards the image space: this was called a Decoded Filter (DF) in Rafegas & Vanrell (2016). The resulting image represents an estimation of the feature that should highly activate such neuron. The disentangling algorithm that inverts the filter would give a good estimation of the feature image if most of the layer operators were invertible. However, when the number of non-invertible operators increases, the estimation becomes unintelligible. The appearance of the DFs can be seen in Fig. 1 of Rafegas & Vanrell (2016). They have also been explored by Springenberg et al. (2015) for architectures with no pooling layers since pooling is the less invertible operator. They point out the interest of obtaining such a representation, since it would allow the understanding of neuron activity independently of the input image. However, the majority of proficient CNNs contain pooling layers. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A third way to describe neuron activity is by exploring the images that maximally activate the neuron. One of the most relevant works pursuing the visualization of intermediate features, is the one proposed by Zeiler & Fergus (2014), where they project intrinsic features of neurons from the image that have provoked a maximum spike to a certain neuron, the network representation is projected into the image space by isolating them in the deconvolution approach Zeiler et al. (2010). By observing different projections that maximally activate a certain neuron they get the intuition about the main features learned on the network. Later on, in Springenberg et al. (2015) the guided backpropagation improves the deconvolution approach by a new way of inverting rectified linear (ReLu) nonlinearities, achieving better visualizations of the activations. These approaches present a main drawback, their feature visualization is image-specific, since the maximum activation of a neuron not always generalize the intrinsic feature of the neuron. To solve this problem, in some works instead of using the image that provokes the maximum activation, they use optimization techniques to generate an image that maximizes the activation. The key point of these works is using an appropriate regularization in the generation process, otherwise, the resulting image appearance is unrealistic and difficult to understand. Simonyan et al. (2014) propose a method to generate an image which is representative of a certain class by maximizing the score of this image to be classified in a certain class (or highly activates the specified neuron) with an $L _ { 2 }$ -regularization. A similar work was performed afterwards in Yosinski et al. (2015) but taking advantage of combining three different regularizations to achieve more recognizable images. Although they have explored different regularizations to achieve more realistic intrinsic feature representations, their visualizations present important artifacts that complicate the understanding of the intrinsic property. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Finally, other works focus on proposing approaches able to reconstruct the input image given a feature map, going further of analyzing the individual neuron activity. Mahendran & Vedaldi (2015) make use of optimization algorithms to search for an image whose feature map best matches a given feature map by incorporating natural image priors. Contrary, in Dosovitskiy & Brox (2015), the authors propose to reconstruct the input image from its feature maps of a given convolutional network by training a new deconvolutional network to learn filter weights that minimize the image reconstruction error when these filters are applied to the image feature maps. With this approach they are also able to get an image reconstruction with natural priors. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the second subset of empirical approaches, Alexey Dosovitskiy (2015) train a generative deconvolutional network to create images from neuron activations. With this methodology, the variation of the activations enables the visualization of the differences in the generated images. A similar analysis is done by Aubry & Russell (2015), but instead of forward-propagate different activations to the image space and comparing them, they observe the changes on neuron activations when similar computer-generated images with different scene factors are introduced into a CNN. These works contribute in giving a deeper understanding on the internal CNN behavior. Both works conclude that there are specific neurons which are sensitive to color changes, point of views, scale or lighting configurations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Likewise, in Zeiler & Fergus (2014) in this work we pursuit visualizing the intrinsic feature of a neuron by analyzing the images that maximally activates a specific neuron. However, to avoid the lack of generality of this approach, we define the Neuron Feature which is not based on a single maximum activation. The Neuron Feature is a weighted average version of a set of maximum activation images that capture the essential properties shared by the most important activations and makes it not to be image-specific. Additionally, our Neuron Feature overcomes the problem of unrealistic representation we metnioned earlier, by directly averaging on the image space. In this way we achieve two main advantages: (a) keeping the properties of the natural images, and (b) providing a very straightforward approach to compute it. ",
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+    {
+        "type": "image",
+        "img_path": "images/0204076260902bf8f42015443e877c4cbca53d9cfd28aecfc5a3a6d3427e5f70.jpg",
+        "image_caption": [
+            "Figure 1: Normalized activations of a subset of neurons for the first 400 ranked images through all convolutional layers. For each layer we plot the normalized activation for the neurons with highest and smallest AUC (Area Under Curve), and some other examples in between these extremes. For all neurons the highest normalized activations is 1, and the percentage of AUC is computed with respect to the neuron AUC achieving the biggest area in the entire network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Afterwards, we introduce the concept of neuron selectivity index, that is used in human vision research to characterize the response of specific cells to specific stimuli (Shapley & Hawken (2011)). This concept allows to achieve a higher level of abstraction in the understanding of a single neuron. In this work we provide two selectivity indexes which are different in their essence: a color selectivity index that quantifies the degree of response of a neuron to a specific color; a class selectivity index that quantifies the degree of response of a neuron to a specific class label. Indexes are derived from the neuron feature or directly from the set of images with maximum activations. We analyze both indexes on a VGG-M network (Chatfield et al. (2014)) trained on ImageNet (Deng et al. (2009)) and we confirm their flexibility to cluster neurons according to their index values and extract conclusions in terms of their task in the net. By selecting color selective neurons we are able to outline how color is represented by the network. Curiously we found some parallelism between color representation in the first convolutional layer and known evidences about the representation in the human visual system. Color selective-neurons also show some preferences towards specific colors which coincide with ImageNet color biases. Indexing on class selectivity neurons we found highly class selective neurons like digital-clock at conv2, cardoon at conv3 and ladybug at conv5, much before the fully connected layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 NEURON FEATURE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As we mentioned in the previous section we propose to visualize the image feature that activates a neuron, whenever is possible, by directly computing a weighted average of the $N$ -th first images that maximally activate this neuron. We will refer to it as the Neuron Feature (NF). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In order to build the NF we need to calculate the activations associated to each individual neuron. They need to be accordingly ranked with the rest of activations of the layer. For each neuron we select the set of images that achieve a minimum normalized activation value but constrained to a maximum number of images for practical reasons. By normalized activation we mean the value of the maximum activation of a neuron for a specific input image, which is normalized by the maximum of these values achieved by the same neuron over all the images in the dataset. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 1 we can see the behavior of the ranked normalized responses of a subset of neurons for every convolution layer of the VGG-M CNN trained on ImageNet by Chatfield et al. (2014). The y-axis represents the normalized activation value of a single neuron to an image of the dataset. Images are ranked on the $\\mathbf { X }$ -axis according with their activation value, from highest to lowest activation (we just plot the first 400 images for each neuron). Therefore, the first relative activation value is always 1 for all neurons and then the normalized activation values decrease monotonically. This normalization allows to compare different neuron behaviors, from neurons which are activated by most of the images (flatter behavior), to neurons that highly activates only for a subset of images and have very little activation for the rest (steeper behavior). In this figure we also provide the percentage of area for each plotted curve. This percentage is computed over the area of the neuron that presents the maximum AUC in the entire architecture. We can observe different behaviors in all layers. In general, we can state that in deeper layers the behavior of the neurons is steeper (lower AUC), i.e. neurons highly spike for a small number of images. However, in shallower layers the behavior is flatter, i.e. neurons highly spike for a lot of images. This is an expected behavior, since the image features spiking neurons in first layers (e.g. oriented edges) are shared by almost all the images, while the features spiking shallow neurons are more selective features (e.g. faces) that only spike for specific images. The observation of the responses confirms the adequacy of our assumption to fix a minimum value for the activation and a maximum number of images to capture the most important activations for all the neurons. Similar observations have been made for other networks like VGG-S and VGG-F Chatfield et al. (2014) 1. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/bc333c47e3a2156f0c6e04657a0f50c1d14adaa6eef8573e29aa44452330362e.jpg",
+        "image_caption": [
+            "Figure 2: Neuron Feature (NF) visualizations (top) for 5 neuronsof the different convolutional layers of VGG-M with their corresponding 100 cropped images (bottom). We scale all layers to the same size due to space constraints. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/eabb814d949626d99a3357fde25e7c632a6ca4811368de4893acef50f1fbddd9.jpg",
+        "image_caption": [
+            "Figure 3: Examples of NFs for each convolutional layer of the network VGG-M (see section 4.1. (a) 20 examples of structured NF, (b), blurred NF. Although sizes of NF increments through layers, we scale them into the same size. Original sizes are: 7x7x3 , 27x27x3, 75x75x3, $1 0 7 \\mathrm { x } 1 0 7 \\mathrm { x } 3$ and $1 3 9 \\mathrm { x } 1 3 9 \\mathrm { x } 3$ for conv1, conv2, conv3, conv4 and conv5, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, the NF is computed as: ",
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35a37e2ddf7581485acc6c739d1222ed28051cb2535e64042b5de4408294dc80.jpg",
+        "text": "$$\nN F ( n ^ { L , i } ) = \\frac { 1 } { N _ { m a x } } \\sum _ { j = 1 } ^ { N _ { m a x } } w _ { j , i , L } I _ { j }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { j , i , L }$ is the relative activation of the $j$ -th cropped image, denoted as $I _ { j }$ , of the $i$ -th neuron $n ^ { L , i }$ at layer $L$ . The relative activation is the activation $a _ { j , i }$ of a neuron, given a input image, with respect to its maximum activation obtained for any image, $\\begin{array} { r } { \\dot { w } _ { j , i , L } = \\frac { a _ { j , i } } { a _ { m a x , i } } } \\end{array}$ where $a _ { m a x , i } = \\operatorname* { m a x } a _ { k , i } , \\forall k$ . ",
+        "bbox": [
+            174,
+            377,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 2 we can see some NFs and their corresponding set of first 100 maximum activations, and in Fig. 3 (a) we can see a selected subset of $2 0 \\mathrm { N F }$ per layer. In this image we can identify specific shapes that display the intrinsic property that fires a single neuron. At first glance, we can see how in this particular network the first two layers are devoted to basic properties. Oriented edges of different frequencies and in different colors in the first layer; textures, blobs, bars and more specific curves in the second layer. The rest of the layers seem to be devoted to more complex objects. We can see that dog and human faces, cars and flowers are detected at different scales in different layers, since the size of the NF and their corresponding cropped images increase with depth. This visualization of the neuron activity can be seen as a way to visualize a trained vocabulary of the CNN that opens multiple ways to analyze the global behavior of the network from its single units. However, not all neurons present such a clear tuning to an identifiable shape. Some neurons present a blurred version of NF, such as, those in Fig. 3(b). The level of blurring is directly related to a high variability between the maximally activated images for a neuron. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At this point, we want to make a short parenthesis to relate the previous representational observations with the scientific problem about neural coding that is focus of attention in visual brain research (Kriegeskorte & Kreiman (2011)). We are referring to the hypothesis about distributed representations that encode object information in neuron population codes, that co-exist with strong evidences of neurons which are only activated by a very specific object. In line with this idea, we invite to speculate about neurons presenting a highly structured NF could be closer to localist code neurons while neurons with a blurred NF as closer to a distributed code. We return on this discussion later on at sections 4.3 and 5. ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, we want to add a further analysis about how neuron feature is related to the neuron activity is representing. In Fig. 4 we plot the level of the neuron responses when the input image is its own NF. We can observe a high degree of activation (in green) between the NF and the response of the net to this feature. However we have some disagreements between the NF and the neuron activations: an important example is shown in layer 2, that is curiously bigger than in layer 3 and 4. This is explained by the high number of dead neurons2 and also by a higher presence of texture selective neurons, that is observed in Fig. 3. Another example, which is more understandable, is the clear increase of disagreement that happens through layers 3, 4 and 5, that seems to be explained by an increase in invariance that is obvious when the size of the image increases. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/3e15e431a10f442e23ee018e2a9d7bb95132b38afc8fc4f4276f0cba02508f52.jpg",
+        "image_caption": [
+            "Figure 4: Number of neurons and degree of activation as a response to their own NF. Activations values are normalized to a specific range within each layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/269bd2c285bed5348fb4bd4742f3382a7c289711514d4782091a8ffba0e55a6e.jpg",
+        "image_caption": [
+            "Figure 5: Conv1 NFs sorted by their color selectivity index. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 NEURON SELECTIVITY INDEX ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we propose to describe neurons by their inherent response to a specific property, using an index. The index has to allow to rank them in a proportional order between their response and the existence of the property in the input image. Therefore, we translate the problem of describing neuron activity to the problem of proposing methods which are able to quantify specific image facets that correlate with the degree of activation of the neuron holding such a property. A selectivity index of a single unit is a flexible an independent method for discriminating or clustering between neurons inside the same network. Selectivity indexes can be defined either for image features or for image labels. In what follows, we propose two selectivity indexes one on each group. ",
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+    {
+        "type": "text",
+        "text": "3.1 COLOR SELECTIVITY INDEX ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Color selectivity is a property that can be proved in specific neurons of the human brain. The level of activation of the neuron when the observer is exposed to a stimulus with a strong color bias, and its corresponding low activation when the color is not present, is the object of attention in vision research that pursuits the understanding of how color is coded in the human visual system (Shapley & Hawken (2011),Conway & Tsao (2009)). ",
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+    {
+        "type": "text",
+        "text": "Here we propose a method to compute a color selectivity index for neurons in artificial neural networks. We propose to base it directly on the image properties of the NF we have defined above. We quantify the selectivity to a specific chromaticity directly from the color distribution of the NF. We define this index as the angle between the first principal component (v) of the color distribution of the NF and the intensity axis (b) of the Opponent Color Space (OPP). To compute (v) we use a weighted Principal Component Analysis Delchambre (2014) that allows to strengthen the selectivity of small color areas. Weights are applied to each pixel in order to reinforce those pixels that are shared by most cropped images and that highly contribute to the NF. Therefore, the weights are the inverse of the standard deviation. In this way, a NF defined by cropped images with different colors will tend to be represented by a grayish image and its principal component will be close to the intensity axis in the OPP color space and it will receive a low selectivity index. We formulate this index (in degrees) as follows: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/433b1e1a5b424dc9c27e9c99499be1d0d4147d84fdca50e10a8abca6c59280d7.jpg",
+        "text": "$$\n\\alpha ( n ^ { L , i } ) = { \\frac { 1 } { 9 0 } } \\operatorname { a r c c o s } \\left( { \\frac { \\mathbf { b } \\cdot \\mathbf { v } } { \\| \\mathbf { b } \\| \\| \\mathbf { v } \\| } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Other selectivity indexes that can be derived from this, are those related to color attributes. We can easily extract color name labels using a color naming approach such as Benavente et al. (2008) and directly define color selectivity to basic names such as red, or green, among others. ",
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+    {
+        "type": "text",
+        "text": "3.2 CLASS SELECTIVITY INDEX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Class selectivity is a property of a neuron that can help to establish its discriminative power for one specific class or can allow to cluster neurons accordingly with the ontological properties of their class labels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose a method to compute a class selectivity index for individual neurons by compiling the class labels of the images that maximally activates this neuron in a single descriptor. We define class selectivity from the set of class labels of the $N$ images used to build the NF. To quantify this index we build the class label distribution of the full set of images. As in the color selectivity index, we weight the significance of a class label by the relative activation of its image. Thus, the relative frequency of each class $c$ for a certain neuron is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/822f84a66b6daa3f06b4604051be83fd43a820f3306095c31924f7e673a83782.jpg",
+        "text": "$$\nf _ { c } ( n ^ { i , L } ) = \\frac { \\sum _ { j } ^ { N _ { c } } w _ { j , i , L } } { \\sum _ { l } ^ { N } w _ { l , i , L } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $N _ { c }$ refers to the number of images, among the $N$ cropped images activating this neuron, that belong to class $c$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given the densities for all the classes. Finally, our class selectivity index is defined as follows: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4dcc7d1808724fad5c43b02618529e770298efeebcad97f6df40d8eec512371d.jpg",
+        "text": "$$\n\\gamma ( n ^ { L , i } ) = \\frac { N - M } { N - 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $M$ is the minimum number of classes that covers a pre-fixed ratio, $^ { t h }$ , of the neuron activation, this can be denoted as $\\begin{array} { r } { \\sum _ { c } ^ { M } f _ { c } \\geq t h } \\end{array}$ . This threshold allow to avoid considering class labels with very small activation weight. Jointly with the index value the selectivity provides the set of $M$ classes that describe the neuron selectivity and their corresponding relative frequency values. ",
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+    {
+        "type": "text",
+        "text": "Therefore, a low class selectivity index indicates a poor contribution of this neuron to a single class (minimum is 0 when $M = N$ ), while a high value (maximum is 1) indicates a strong contribution of this neuron to a single class. In between we can have different degrees of selectivity to different number of classes. Obviously, this index is irrelevant for the last fully connected layers in a CNN, but it allows to group related neurons across different convolutional layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here we want to point out, that this index can also contribute to give some insights about the problem of how information is coded through layers, in the debate of localist and distributed neural codes we mentioned before (Kriegeskorte & Kreiman (2011)). Neurons with high class selectivity index should be in line with a localist code, while neurons with low class selectivity index should be part of a distributed code. This way the index is defined allow a large range of interpretations in between these two kinds of coding as it has been outlined in the visual coding literature. ",
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section we report some empirical results to show how the proposed selectivity indexes perform and what representational conclusions we can extract from the subsets of neurons sharing indexed properties. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper we analyze the neurons of a CNN architecture trained on ImageNet ILSVRC dataset Deng et al. (2009) (using a subset of 1.2M images classified in 1.000 categories). We report the results for the VGG-M CNN that was trained by Chatfield et al. (2014) for a generic visual task of object recognition. The details of the CNN architecture are given in table 1. We selected this network since it has a similar structure to those which have been reported as having a representational performance that competes with human performance (as was proved in Cadieu et al. (2014)). Nevertheless, we have obtained similar results for VGG-F and VGG-S that are provided in Chatfield et al. (2014). We used the Matconvnet library provided by Vedaldi & Lenc (2015) for all the experiments. ",
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+    {
+        "type": "table",
+        "img_path": "images/3bc62250481e242926d7f55f9fbc5dfd081225e41647bd3f446c6cc39b4b1b9d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>conv1</td><td rowspan=1 colspan=1>conv2</td><td rowspan=1 colspan=1>conv3</td><td rowspan=1 colspan=1>conv4</td><td rowspan=1 colspan=1>conv5</td><td rowspan=1 colspan=1>full6</td><td rowspan=1 colspan=1>ful17</td><td rowspan=1 colspan=1>ful18</td></tr><tr><td rowspan=1 colspan=1>96x7x7st.2, pad.0LRN, x2 pool</td><td rowspan=1 colspan=1>256x5x5st.2, pad.1LRN, x2 pool</td><td rowspan=1 colspan=1>512x3x3st.1, pad.1</td><td rowspan=1 colspan=1>512x3x3st.1, pad.1</td><td rowspan=1 colspan=1>512x3x3st.1, pad.1x2 pool</td><td rowspan=1 colspan=1>4096dropout</td><td rowspan=1 colspan=1>4096dropout</td><td rowspan=1 colspan=1>1000softmax</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Table 1: VGG-M architecture designed by Chatfield et al. (2014), where $M \\times N \\times P$ corresponds to number of filters, number of rows and columns of the filters respectively. $S t .$ . and pad. refers to stride and padding respectively; LRN is a ReLU and the corresponding pooling $( p o o l )$ if applied. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/ec52b975e9b7673884b7f4c98981a113f82f63806d12b5422c37a054c3332daf.jpg",
+        "image_caption": [
+            "Figure 6: Neurons with different color selectivity indexes. Images in 4 rows (1st and 3rd row are NFs, 2nd and 4th rows are sets of cropped images that maximally activates the neuron). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 COLOR SELECTIVITY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "General purpose CNN architectures are usually trained on RGB color images. However there is a strong belief in the computer vision community that color is a dispensable property. The results we obtain by indexing color selective neurons make us conclude that there is no basis for such a belief. Results show that color is strongly entangled at all levels of the CNN representation. In a preliminary experiment we have tested a subset of ImageNet images with VGG-M in their original color and the same subset in a gray scale representation. Classification results show a considerable decrease: while original RGB images are classified with a $2 7 . 5 0 \\%$ top-1 error and $1 0 . 1 4 \\%$ top-5 error, gray scale image versions present $5 1 . 1 2 \\%$ and $2 6 . 3 7 \\%$ errors, top-1 and top-5 errors respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In a first experiment we extract how many NFs are related to color in each convolutional layer using the proposed color selectivity index. The bars in Fig. 8 plot the relative quantity of neurons that are color selective compared to those that are not. Grey represents the ratio of neurons that do not spike for the presence of a color and reddish represent neurons that are highly activated by the presence of a color. In the graphic we can observe that shallow layers are the main responsible for the color representation on the images: $5 0 \\%$ and $4 0 \\%$ of neurons are color selective in layers conv1 and conv2, respectively. Nevertheless, we also still found around $2 5 \\%$ of color selective neurons in in deeper layers. Therefore, although neurons in deeper layers tend to be color invariant, an important part of the representation is devoted to color, that reinforces the discriminative power of color in object recognition. In Fig. 6 we show some examples of NFs with different degrees of color selectivity at different layers of the network and showing the corresponding cropped images. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Regarding color representation in layer 1 we want to point out two more observations derived from the NFs (see Fig. 5): (a) selectivity to different spatial-frequencies is only tackled by gray-level neurons; and (b) four main color axis emerge (black-white, blue-yellow, orange-cyan and cyanmagenta). Curiously, these two observations correlate with evidences in the human visual system (Shapley & Hawken (2011)). ",
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+    {
+        "type": "image",
+        "img_path": "images/38ae245bdac305005a49af38201628f343495291f93b9e70566d7c4ddf6b71b4.jpg",
+        "image_caption": [
+            "Figure 7: Distribution of color selective neurons on a hue color space through layers. Maximum activation images for 4 top color selective neurons for each layer. Dashed rings connect NFs of color selective neurons through layers, from inner ring (conv1) to outer ring (conv5). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/389eb3416b3fe5532429c5273e7b5230f16c9613bad875e027ff7dc80f590d3a.jpg",
+        "image_caption": [
+            "Figure 8: Number of neurons and degree of color selectivity through layers. Grayish bars are for low index values and reddish for high index values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/edafef2e65b9e7d6de0863b0a3d4db8d04862593e8b1d81481eafa9c927daf8c.jpg",
+        "image_caption": [
+            "Figure 9: Number of neurons and degree of class selectivity through layers. Grayish bars are for low index values and bluish for high index values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In a second experiment, we analyze how color selective neurons from all layers cover the color space. Figure 7 displays the distribution of color selective neurons with $\\alpha \\ge 0 . 4 0$ . Each NF is plotted on the hue angle that represents the projection of its first principal component on the OPP chromaticity plane (red-green and blue-yellow components). Dashed rings identify different convolutional layers from conv1 (inner ring) to conv5 (outer ring) linking the NFs that belong to the same layer. We can appreciate the emergence of an axis (from orange to cyan) that connects a crowded area of color selective neurons. We can add a low population of NFs in the magenta area, that becomes more crowded on the opposite side where green and yellow selectivity has several neurons. The interest of this explanation relies on the fact that a similar distribution appears in the ImageNet color distribution, that is plotted at the bottom of the same images, where a similar interpretation in terms of emergent axes can be done. A more in depth study is required to prove this correlation, but we illustrate how neuron selectivity helps in the understanding of how a specific property is represented by the CNN. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "4.3 CLASS SELECTIVITY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Following with the analysis of ranking neurons by their response to a certain property, here we focus on the proposed selectivity index that relates to image labels instead of to an image property, is the class selectivity index, which only applies for classification networks. We report the results of different experiments where we have fixed $t h = 1$ , which means we consider all the class labels for the $N = 1 0 0$ images that maximally activates the neuron. As we mentioned before, this index can enlighten how classes are encoded through the net layers, that again it can be related to the scientific problem of how general object recognition is encoded in the human brain. Here we hypothesize that the difference between localist or distributed codes could correlate with the idea of neurons highly selective to a single class and neurons highly selective to several classes, we resume on this later at section 5. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In a first experiment we analyze how many neurons present different degrees of class selectivity through layers. The bars in Fig. 9 plot the relative quantity of neurons that are class selective compared to those that are not. Grey represents the ratio of neurons that are not activated by a single class and bluish represent neurons that are highly activated by a single class. Opposite to what we showed about color selectivity, we found most of class selective neurons in deeper layers, and no class selectivity in shallow layers, as expected. We have moved from a very basic image property, color, to a very high level property, class label. This fact corroborates the idea that CNNs start by defining basic feature detectors that are share by most of the classes, and the neurons become more specialized when they belong to deeper layers representing larger areas in the image space and therefore more complex shapes. We start to have neurons with relevant class selectivity in layer conv3, where a $5 \\%$ of neurons is quite class selective and we found some neurons with a degree of selective close to 1. These ratios progressively increase up to layer conv5 where we have more than a $5 0 \\%$ of neurons with a class selectivity index greater than 0.6, that means that we have less than 40 different classes activating this neuron, which is a very selective ratio considering the number of classes of the ImageNet dataset. In the same layer a $2 0 \\%$ of neurons present a high class selectivity index, than means less than 20 different classes. Further experiments should explore how this graphic evolves by moving from current class labels which are on the leaves of the ImageNet ontology towards higher nodes with more generic classes. ",
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+    {
+        "type": "text",
+        "text": "Secondly, we have visualized the properties of a set of images presenting different degrees of class selectivity in Fig. 10 for different levels of depth. We visualize each neuron with their NF visualization and the corresponding cropped images. We also show two tag clouds of each neuron. They visualize the importance of each class label. With an orange frame we plot the leave classes of the ImageNet ontology, while in the green frame we plot generic classes. This second analysis could help finding neurons that are specialized to a general semantic concept that different final classes share. Note that neurons with high class selectivity index have a set of cropped images that we can identify as belonging to the same class. ",
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "Finally we stress the utility of ranking images by selectivity indexes in Fig. 11, where we show interesting neurons in different convolutional layers that present high values for both selectivity indexes, neurons which are both, color and class selective. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "In this paper we propose a framework to analyze a trained CNN by dissecting individual neurons using their indexes of selectivity to specific properties. We have proposed two properties of different nature: (a) color, that is a low-level image property that we have shown to be entangled in all the representations levels of the net; (b) class label, that is a high-level image property that can be analyzed at different levels of abstraction. We have shown that while the number of color selective neurons decreases with depth, the number of class selective neuron increases. In this line of describing the activity of individual images, we have also proposed to visualize the activity with what we have called the neuron feature (NF), that allows to arise interesting structures that are shared by the images that highly activate a neuron. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The proposed work have made us to speculate about two different ways to address the coding properties of individual neurons (localist versus distributed). Firstly, we have mentioned the possibility that a blurred NF, i.e. without a clear structure, belongs to a neuron that can be part of a distributed code where the neuron does not represent a selectivity to a single shape, maybe to diverse shapes than can be part of a code in deeper neurons. Secondly, we speculate about the possibility that neurons with high class selective index can represent a localist code, and part of a distributed when is low. In parallel, the analysis of the color selective neurons have made to arise some parallelism between color representation in the 1st convolutional layer and known evidences about the representation in the human visual system. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/826a14425cb5a5a953fbdc7fb8a24c699475862a82ae5cded3c360c4c2e80e81.jpg",
+        "image_caption": [
+            "Figure 10: Neurons with different class selectivity indexes. For each neuron two images (top: NF, bottom: cropped images) and two tag clouds (top: leave classes, bottom: all classes in the ontology). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b3f1a583f754b14d49b9250a1f19ec877fac1f49023d6121dede424b497c131.jpg",
+        "image_caption": [
+            "Figure 11: Examples of neurons with high color and class selectivity indexes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As further work we need to fully exploit the potential of the indexes in different CNN architectures, and defining new selectivity indexes like shape or texture, that could be a perfect complement to current ones. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Project funded by MINECO Ref. TIN (TIN2014-61068-R) ",
+        "bbox": [
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+        "page_idx": 11
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+    {
+        "type": "text",
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parse/train/fV4vvs1J5iM/fV4vvs1J5iM.md

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+# A REDUCTION APPROACH TOCONSTRAINED REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Many applications of reinforcement learning (RL) optimize a long-term reward subject to risk, safety, budget, diversity or other constraints. Though constrained RL problem has been studied to incorporate various constraints, existing methods either tie to specific families of RL algorithms or require storing infinitely many individual policies found by an RL oracle to approach a feasible solution. In this paper, we present a novel reduction approach for constrained RL problem that ensures convergence when using any off-the-shelf RL algorithm to construct an RL oracle yet requires storing at most constantly many policies. The key idea is to reduce the constrained RL problem to a distance minimization problem, and a novel variant of Frank-Wolfe algorithm is proposed for this task. Throughout the learning process, our method maintains at most constantly many individual policies, where the constant is shown to be worst-case optimal to ensure convergence of any RL oracle. Our method comes with rigorous convergence and complexity analysis, and does not introduce any extra hyper-parameter. Experiments on a grid-world navigation task demonstrate the efficiency of our method.
+
+# 1 INTRODUCTION
+
+Contemporary approaches in reinforcement learning (RL) largely focus on optimizing the behavior of an agent against a single reward function. RL algorithms like value function methods (Zou et al., 2019; Zheng et al., 2018) or policy optimization methods (Chen et al., 2019; Zhao et al., 2017) are widely used in real-world tasks. This can be sufficient for simple tasks. However, for complicated applications, designing a reward function that implicitly defines the desired behavior can be challenging. For instance, applications concerning risk (Geibel & Wysotzki, 2005; Chow & Ghavamzadeh, 2014; Chow et al., 2017), safety (Chow et al., 2018) or budget (Boutilier & Lu, 2016; Xiao et al., 2019) are naturally modelled by augmenting the RL problem with orthant constraints. Exploration suggestions, such as to visit all states as evenly as possible, can be modelled by using a vector to measure the behavior of the agent, and to find a policy whose measurement vector lies in a convex set (Miryoosefi et al., 2019).
+
+To solve RL problem under constraints, existing methods either ensure convergence only on a specific family of RL algorithms, or treat the underlying RL algorithms as a black box oracle to find individual policy, and look for mixed policy that randomizes among these individual policies. Though the second group of methods has the advantage of working with arbitrary RL algorithms that best suit the underlying problem, existing methods have practically infeasible memory requirement. To get an $\epsilon$ -approximate solution, they require storing $\bar { O } ( 1 / \epsilon )$ individual policies, and an exact solution requires storing infinitely many policies. This limits the prevalence of such methods, especially when the individual policy uses deep neural networks.
+
+In this paper, we propose a novel reduction approach for the general convex constrained RL (C2RL) problem. Our approach has the advantage of the second group of methods, yet requires storing at most constantly many policies. For a vector-valued Markov Decision Process (MDP) and any given target convex set, our method finds a mixed policy whose measurement vector lies in the target convex set, using any off-the-shelf RL algorithm that optimizes a scalar reward as a RL oracle. To do so, the C2RL problem is reduced to a distance minimization problem between a polytope and a convex set, and a novel variant of Frank-Wolfe type algorithm is proposed to solve this distance minimization problem. To find an $\epsilon$ -approximate solution in an $m$ -dimensional vector-valued MDP, our method only stores at most $m + 1$ policies, which improves from infinitely many $O ( 1 / \epsilon )$ (Le et al., 2019; Miryoosefi et al., 2019) to a constant. We also show this $m + 1$ constant is worstcase optimal to ensure convergence of RL algorithms using deterministic policies. Moreover, our method introduces no extra hyper-parameter, which is favorable for practical usage. A preliminary experimental comparison demonstrates the performance of the proposed method and the sparsity of the policy found.
+
+Table 1: Comparison with previous approaches. To find an $\epsilon$ -approximate solution, time complexity under orthant or convex constraints is compared using the numbers of RL oracle calls. The memory requirement is measured by the number of individual policies stored for an $\epsilon \cdot$ -approximate solution.   
+
+<table><tr><td>Method</td><td>Orthant constraint</td><td>Convex constraint</td><td>Converge for any</td><td>No extra hyper-</td><td>Memory requirement</td></tr><tr><td></td><td>To a fixed point</td><td>X</td><td>RL algo.</td><td>parameter X</td><td>1</td></tr><tr><td>Tessler et al. (2018) Le et al. (2019)</td><td>0(1/e)</td><td>×</td><td>X</td><td>X</td><td>0(1/e)</td></tr><tr><td>Miryoosefi et al. (2019)</td><td>0(1/e)</td><td>0(1/e)</td><td>√</td><td>X</td><td>0(1/e)</td></tr><tr><td>C2RL (this paper)</td><td>0(1/e)</td><td>0(1/e)</td><td>√</td><td>√</td><td>≤m+1</td></tr></table>
+
+# 2 RELATED WORK
+
+For high dimensional constrained RL, one line of approaches incorporates the constraint as a penalty signal into the reward function, and makes updates in a multiple time-scale scheme (Tessler et al., 2018; Chow & Ghavamzadeh, 2014). When used with policy gradient or actor-critic algorithms (Sutton & Barto, 2018), this penalty signal guides the policy to converge to a constraint satisfying one (Paternain et al., 2019; Chow et al., 2017). However, the convergence guarantee requires the RL algorithm can find a single policy that satisfies the constraint, hence ruling out methods that search for deterministic policies, such as Deep Q-Networks (DQN) (Mnih et al., 2013), Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2015) and their variants (Van Hasselt et al., 2015; Wang et al., 2016; Fujimoto et al., 2018; Barth-Maron et al., 2018).
+
+Another line of approaches uses a game-theoretic framework, and does not tie to specific families of RL algorithm. The constrained problem is relaxed to a zero-sum game, whose equilibrium is solved by online learning (Agarwal et al., 2018). The game is played repeatedly, each time any RL algorithm can be used to find a best response policy to play against a no-regret online learner. The mixed policy that uniformly distributed among all played policies can be shown to converge to an optimal policy of the constrained problem (Freund & Schapire, 1999; Abernethy et al., 2011). Taking this approach, Le et al. (2019) uses Lagrangian relaxation to solve the orthant constraint case, and Miryoosefi et al. (2019) uses conic duality to solve the convex constraint case. However, since the convergence is established by the no-regret property, the policy found by these methods requires randomization among policies found during the learning process, which limits their prevalence.
+
+Different from the game-theoretic approaches, we reduce the C2RL to a distance minimization problem and propose a novel variant of Frank-Wolfe (FW) algorithm to solve it. Our result builds on recent finding that the standard FW algorithm emerges as computing the equilibrium of a special convex-convave zero sum game (Abernethy & Wang, 2017). This connects our approach with previous approaches from game-theoretic framework (Agarwal et al., 2018; Le et al., 2019; Miryoosefi et al., 2019). The main advantage of our reduction approach is that the convergence of FW algorithm does not rely on the no-regret property of an online learner. Hence there is no need to introduce extra hyper-parameters, such as learning rate of the online learner, and intuitively, we can eliminate unnecessary policies to achieve better sparsity. To do so, we extend Wolfe’s method for minimum norm point problem (Wolfe, 1976) to solve our distance minimization problem. Throughout the learning process, we maintain an active policy set, and constantly eliminate policies whose measurement vector are affinely dependent of others. Unlike norm function in Wolfe’s method, our objective function is not strongly convex. Hence we cannot achieve the linear convergence of Wolfe’s method as shown in Lacoste-Julien & Jaggi (2015). Instead, we analyze the complexity of our method based on techniques from Chakrabarty et al. (2014). A theoretical comparison between our method and various approaches in constrained RL is provided in Table 1.
+
+# 3 PRELIMINARIES
+
+A vector-valued Markov decision process can be identified by a tuple $\{ \boldsymbol { S } , \mathcal { A } , \beta , P , c \}$ , where $s$ is a set of states, $\mathcal { A }$ is the set of actions and $\beta$ is the initial state distribution. At the start of each episode, an initial state $s _ { 0 }$ is drawn following the distribution $\beta$ . Then, at each step $t = 0 , 1 , \ldots$ , the agent observes a state $s _ { t } \in S$ and makes a decision to take an action $a _ { t }$ . After $a _ { t }$ is chosen, at the next observation the state evolves to state $s _ { t + 1 } \in S$ with probability $\textstyle P ( s _ { t + 1 } | s _ { t } , a _ { t } )$ . However, instead of a scalar reward, in our setting, the agent receives an $m$ -dimensional vector $\boldsymbol { c } _ { t } \in \mathbb { R } ^ { m }$ that may implicitly contain measurements of reward, risk or violation of other constraints. The episode ends after a certain number of steps, called the horizon, or when a terminate state is reached.
+
+Actions are typically selected according to a policy $\pi$ , where $\pi ( s )$ is a distribution over actions for any $s \in { \mathcal { S } }$ . Policies that take a single action for any state are deterministic policies, and can be identified by the mapping $\pi : { \mathcal { S } } \mapsto A$ . The set of all deterministic policies is denoted by $\Pi$ . For a discount factor $\gamma \in [ 0 , 1 )$ , the discounted long-term measurement vector of a policy $\pi \in \Pi$ is defined as
+
+$$
+\pmb { c } ( \pi ) : = \mathbb { E } ( \sum _ { t = 0 } ^ { T } \gamma ^ { t } \pmb { c } _ { t } ( s _ { t } , \pi ( s _ { t } ) ) ) ,
+$$
+
+where the expectation is over trajectories generated by the described random process.
+
+Unlike unconstrained setting, for a constrained RL problem, it is possible that all feasible policies are non-deterministic (see Appendix $\mathbf { D }$ for an example). This limits the usage of RL algorithms that search for deterministic policies in the setting of constrained RL problem.
+
+One workaround is to use mixed policies. For a set of policies $\mathcal { U }$ , a mixed policy is a distribution over $\mathcal { U }$ , and the set of all mixed policies over $\mathcal { U }$ is denoted by $\Delta ( \mathcal { U } )$ . To execute a mixed policy $\mu \in \Delta ( \mathcal { U } )$ , we first select a policy $\pi \in { \mathcal { U } }$ according to $\pi \sim \mu ( \pi )$ , and then execute $\pi$ for the entire episode. Altman (1999) shows that any $c ( \cdot )$ achievable can be achieved by some mixed deterministic policies $\mu \in \Delta ( \Pi )$ . Therefore, though an off-shelves RL algorithm may not converge to any constraint-satisfying policy, it can be used as a subroutine to find individual policies (possibly deterministic), and a randomization among these policies can converge to a feasible policy. The discounted long-term measurement vector of a mixed policy $\mu \in \Delta ( \Pi )$ is defined similarly
+
+$$
+\pmb { c } ( \mu ) : = \mathbb { E } _ { \pi \sim \mu } ( \pmb { c } ( \pi ) ) = \sum _ { \pi \in \Pi } \mu ( \pi ) \pmb { c } ( \pi ) .
+$$
+
+For a mixed policy $\mu \in \Delta ( \mathcal { U } )$ , its active set is defined to be the set of policies with non-zero weights $\mathcal { A } : = \{ \pi \in \mathcal { U } | \mu ( \pi ) > 0 \}$ . The memory requirement of storing $\mu$ , is then proportional to the size of its active set. Since a mixed policy can be interpreted as a convex combination of policies in its active set, in the following, the term sparsity of a mixed policy refers to the sparsity of this combination.
+
+Our learning problem, the convex constrained reinforcement learning (C2RL), is to find a policy whose expected long-term measurement vector lies in a given convex set; i.e., for a given convex target set ${ \mathcal { C } } \subset \mathbb { R } ^ { m }$ , our target is to
+
+Any policy $\mu ^ { * }$ that satisfies $\pmb { c } ( \mu ^ { * } ) \in \Omega$ is called a feasible policy, and a C2RL problem is feasible if there exists some feasible policies. In the following, we assume the C2RL problem is feasible.
+
+# 4 APPROACH, ALGORITHM AND ANALYSIS
+
+We now show how the C2RL (3) can be reduced to a distance minimization problem (7) between a polytope and a convex set. A novel variant of Frank-Wolfe-type algorithm is then proposed to solve the distance minimization problem, followed by theoretic analysis about convergence and sparsity of the proposed method.
+
+# 4.1 REDUCE C2RL TO A DISTANCE MINIMIZATION PROBLEM
+
+Let $| | \cdot | |$ denote the Euclidean norm. For a convex set $\Omega \in \mathbb { R } ^ { m }$ , let $\begin{array} { r } { \operatorname* { P r o j } _ { \Omega } ( \pmb { x } ) \in \arg \operatorname* { m i n } _ { \pmb { y } \in \Omega } | | \pmb { x } - \pmb { y } | | } \end{array}$ be the projection operator, and $\begin{array} { r } { \mathtt { d i s t } ^ { 2 } ( { \pmb x } , \Omega ) : = \frac { 1 } { 2 } | | { \pmb x } - \mathtt { P r o j } _ { \Omega } ( { \pmb x } ) | | ^ { 2 } } \end{array}$ be half of the squared Euclidean
+
+distance function. Then we consider the problem to find a policy whose measurement vector is closest to the target convex set,
+
+$$
+\operatorname * { a r g m i n } _ { \mu \in \Delta ( \Pi ) } \mathtt { d i s t } ^ { 2 } ( { \pmb { c } } ( \mu ) , \Omega ) .
+$$
+
+A policy $\mu ^ { * } \in \Delta ( \Pi )$ is defined to be an optimal solution if it minimizes (4). Otherwise, the approximation error of $\dot { \mu } \in \Delta ( \Pi )$ is defined as
+
+$$
+\mathsf { e r r } ( \mu ) : = \mathsf { d i s t } ^ { 2 } ( { \pmb { c } } ( \mu ) , \Omega ) - \mathsf { d i s t } ^ { 2 } ( { \pmb { c } } ( { \mu } ^ { * } ) , \Omega )
+$$
+
+(Approximation Error)
+
+where $\mu ^ { * }$ is an optimal solution, and a policy is defined to be an $\epsilon$ -approximate solution if its approximation error is no larger than $\epsilon$ .
+
+When C2RL (3) is feasible, the equivalence of being optimal to (4) and being feasible to C2RL can be easily established. Since a feasible policy of C2RL problem lies inside $\Omega$ , it minimizes the non-negative dist2 function, and hence is optimal to (4). Vice versa, any optimal solution to (4) lies inside $\Omega$ and is a feasible solution to C2RL.
+
+From a geometric perspective, let $\pmb { c } ( \Pi ) : = \{ \pmb { c } ( \pi ) | \pi \in \Pi \}$ be the set of all values achievable by deterministic policies. If the MDP has finite states and actions (though may be extremely large), then $\Pi$ is finite as well, and hence $c ( \Pi )$ contains finitely many points in $\mathbb { R } ^ { m }$ . Then the set of values achievable by mixed deterministic policies
+
+$$
+c ( \Delta ( \Pi ) ) : = \{ c ( \mu ) | \mu \in \Delta ( \Pi ) \} = \{ \sum _ { \pi } \mu ( \pi ) c ( \pi ) | \sum _ { \pi } \mu ( \pi ) = 1 , \mu ( \pi ) \geq 0 \} \subset \mathbb { R } ^ { m }
+$$
+
+is the convex hull of $c ( \Pi )$ ; i.e., $c ( \Delta ( \Pi ) )$ is a $m$ -dimension polytope whose vertices are $c ( \Pi )$ . Therefore finding a policy whose value is closest to the target convex set (4) is equivalent to find a point in the polytope $c ( \dot { \Delta ( \Pi ) } )$ that is closest to the convex set $\Omega$
+
+$$
+\operatorname* { a r g m i n } _ { \mathbf { c } ( \mu ) \in c ( \Delta ( \Pi ) ) } \mathtt { d i s t } ^ { 2 } ( { \pmb { c } } ( { \mu } ) , \Omega ) \qquad \mathtt { ( D i s t a n c e m i n i m i z a t i o n p r o b l e m ) . }
+$$
+
+To solve this constrained optimization problem, it might be tempting to consider projection methods. However, constructing a projection operator for $c ( \Delta ( \Pi ) )$ is non-trivial. For any given measurement vector, it is obscure how to modify a general RL algorithm to update the parameters such that the discounted expected measurement vector is closest to the given value. Therefore, projection-free methods are preferable for this task.
+
+Frank-Wolfe (FW) algorithm does not require any projection operation, instead it uses a linear minimizer oracle. Intuitively, finding a linear minimizer is similar to the reward maximization process of what a general RL algorithm does. In section 4.3, we formalize this idea. We show that after simple modifications, any RL algorithm that maximizes a scalar reward can be used to construct such a linear minimizer oracle. Before getting into details of the construction process, we discuss FW-type algorithms over polytope and its applications in the distance minimization problem (7).
+
+# 4.2 DISTANCE MINIMIZATION BY FRANK-WOLFE-TYPE ALGORITHMS
+
+The Frank-Wolfe algorithm (FW) is a first-order method to minimize a convex function $f : \mathcal { P } \mapsto \mathbb { R }$ over a compact and convex set $\mathcal { P }$ , with only access to a linear minimizer oracle. When the feasible set is a polytope $\mathcal { P } : = \mathsf { c o n v } ( \{ s _ { 1 } , s _ { 2 } , \hdots , s _ { n } \} ) \subset \mathbb { R } ^ { m }$ defined as the convex hull of finitely many points, FW-type algorithms are discussed by Lacoste-Julien & Jaggi (2015) to optimize
+
+$$
+\operatorname* { m i n } _ { x \in \mathcal { P } } f ( x ) \quad { \mathrm { ~ u s i n g ~ } } \quad 0 \mathrm { r a c 1 } \mathbf { e } ( \mathrm { v } ) : = \operatorname * { a r g m i n } _ { s \in \{ s _ { 1 } , \ldots , s _ { n } \} } s ^ { T } v .
+$$
+
+The standard FW (Algorithm 2 in Appendix A.1) consists of making repeated calls to the linear minimizer oracle to find an improving point $\pmb { s }$ , followed by a convex averaging step of the current iterate ${ \mathbf { \mathcal { x } } } _ { t - 1 }$ and the oracle’s output $\pmb { s }$ .
+
+If we have already constructed a RL oracl $\mathfrak { a } ( \lambda )$ that outputs a policy $\pi \in \arg \operatorname* { m i n } _ { \pi \in \Pi } \lambda ^ { T } { \pmb { c } } ( \pi )$ together with its measurement vector $\pmb { c } ( \pi )$ , then the distance minimizing problem (7) can be solved with standard FW by using
+
+$$
+\pi , c ( \pi ) \gets \mathrm { R L . o r a c l e } ( \boldsymbol { \nabla } \mathrm { d i s t } ^ { 2 } ( \boldsymbol { x } _ { t - 1 } , \Omega ) ) = \mathrm { R L . o r a c l e } ( \boldsymbol { x } _ { t - 1 } - \boldsymbol { \mathrm { P r o j } } _ { \Omega } ( \boldsymbol { x } _ { t - 1 } ) )
+$$
+
+# Algorithm 1 Convex Constrained Reinforcement Learning (C2RL)
+
+Input. RL Oracle constructed by any RL algorithm, projection operator to target set $\mathtt { P r o j } _ { \Omega }$ .   
+Initialize. Random policy $\pi$ , value $\pmb { x } = \pmb { c } ( \pi )$ , active sets $S _ { p } : = [ \pi ] , S _ { c } : = [ { \pmb x } ]$ and weight $\bar { \lambda ( \cdot ) } = [ 1 ]$ .   
+Output. Mixed policy $\mu$ and its value $\mathbf { \boldsymbol { c } } ( \mu )$ s.t. $\mathbf { \boldsymbol { c } } ( \mu )$ minimizes the distance to the target set $\Omega$ .
+
+// Major cycle
+
+$$
+{ \pmb y } , { \pmb \alpha } \gets \mathsf { A f f i n e M i n i m i z e r } ( S _ { c } , { \pmb \omega } )
+$$
+
+16: return $u , c ( { \boldsymbol { \mu } } ) \gets \mathbf { { x } }$
+
+to find an improving policy and its measurement vector. For $\begin{array} { r } { \eta _ { t } : = \frac { 2 } { t + 2 } } \end{array}$ , the convex averaging steps
+
+$$
+\mu _ { t }  ( 1 - \eta _ { t } ) \mu _ { t - 1 } + \eta _ { t } \pi , \quad x _ { t }  ( 1 - \eta _ { t } ) x _ { t - 1 } + \eta _ { t } c ( \pi ) ,
+$$
+
+then maintain the mixed policy, and the corresponding measurement vector, respectivel
+
+However, after $T$ rounds of iteration, the $\mu _ { t }$ found has an active set containing up to $T$ individual polices, and is not sparse enough. If neural networks are used to parameterize the policy, that requires storing $T$ copies of parameters for the individual network, which is unaffordable for largescale usage.
+
+To find even more sparse policies, we turn to variants of FW-type algorithms. In particular, Wolfe’s method for minimum norm point in a polytope (Wolfe, 1976; De Loera et al., 2018). In Wolfe’s method (Algorithm 3 in Appendix A.2), the loop in FW is called a major cycle, and the convex averaging step is replaced by a weight optimization process, called minor cycle. Wolfe’s method maintains an active set $s$ , and the current point can be represented by a sparse combination of points in the active set. The minor cycles maintain $s$ to be an affinely independent set such that the affine minimizer is inside $S ^ { t }$ , which Wolfe calls corrals. Recall an affine minimizer is defined as $\begin{array} { r } { \arg \operatorname* { m i n } _ { \pmb { \mathscr { s } } \in \mathsf { a f f } ( \mathscr { S } ) } | | \pmb { \mathscr { s } } | | _ { 2 } } \end{array}$ , where $\begin{array} { r } { \mathtt { a f f } ( \mathscr { S } ) : = \{ y | y = \sum _ { z \in S } \alpha _ { z } ^ { T } \pmb { x } , \sum _ { z \in S } \alpha _ { z } = 1 \} } \end{array}$ is the affine hull formed by $s$ . Since the active set is affinely independent, the number of active atoms is at most $m + 1$ at any time. Wolfe’s method is shown to strictly decrease the approximation error between two major cycles.
+
+# 4.3 OUR MAIN ALGORITHM
+
+The main obstacle to apply Wolfe’s method to our distance minimization problem (7) is that the objective function in Wolfe’s method is the norm function. However, in our problem, the objective function is the distance function to a convex set. Unlike the norm function, the distance function to a convex set is not strongly convex and affine minimizer is ill-defined with respect to a convex set. To tackle these problems, we modify the Wolfe’s method. At the core of our new variant of FW algorithm, we add a projection step to Wolfe’s method.
+
+Projection Step In each major cycle, we minimize the distance to a projected point $\omega : = \mathrm { P r o j } _ { \Omega } ( \pmb { x } )$ . Intuitively, since the distance to the convex set is upper bounded by the distance to this projected point $\omega$ , if the distance to $\omega$ converges, so does the distance to the target convex set.
+
+Formally, for a set of points $S \subset \mathbb { R } ^ { m }$ , and a point $\pmb { x } \in \mathbb { R } ^ { m }$ , we extend the definition of an affine minimizer to define affine minimizer with respect to $_ { \textbf { \em x } }$ as $\begin{array} { r } { \arg \operatorname* { m i n } _ { { s } \in \mathrm { a f f } ( S ) } \left| \left| \pmb { \mathscr { s } } - \pmb { \mathscr { x } } \right| \right| _ { 2 } } \end{array}$ . For $_ { \textbf { \em x } }$ being the affine minimizer of $s$ with respect to $\omega$ , the extended affine minimizer property gives （ $\mathbf { \mathrm { \textbf { j } } } \mathbf { \mathrm { { v e n } } } \omega , \forall \pmb { v } \in \mathbf { \mathrm { a f f } } ( S ) , ( \pmb { v } - \pmb { x } ) ^ { T } ( \pmb { x } - \pmb { \omega } ) = 0$ (Extended affine minimizer property)
+
+Similar to Wolfe’s method, our C2RL method (Algo. 1) contains an outer loop (called major cycle) to find improving policies and their measurement vectors, and an inner loop (called minor cycle) to maintain the affinely independent property of the active set $ { \boldsymbol { S } } _ { c }$ . At the start of each major cycle step, the $ { \boldsymbol { S } } _ { c }$ is an affinely independent set. Then, the RL oracle (defined in (15)) finds a potential improving policy $\pi \in { \mathcal { U } }$ , and its long-term measurement vector $\pmb { c } ( \pi )$ . If the $\pmb { c } ( \pi )$ does not get strictly closer to the $\omega : = \mathtt { P r o j } ( \pmb { x } )$ , then we are done, and $_ { \textbf { \em x } }$ is the optimal value. Otherwise, the $\pmb { c } ( \pi )$ is added into the active set, and the minor cycle is run to eliminate policies whose measurement vectors are affinely dependent.
+
+Line 6 to line 13 contains the minor cycle, which is the same as the original Wolfe’s method (except in line 6, we find affine minimizer with respect to $\omega$ ). The elimination is executed as a series of affine projections. The minor cycle terminates if active set $ { \boldsymbol { S } } _ { c }$ is affinely independent. Though the interleaving of major and minor cycles oscillate the size of active set $S _ { c }$ , the minor cycles keep $| S _ { c } |$ an affinely independent set, and is terminated whenever $S _ { c }$ contains a single element. Therefore at the start of any major cycle, the size of the active set satisfies $| S _ { c } | \in [ 0 , \bar { m } + 1 ]$ . More background about the minor cycle in Wolfe’s method is provided in Appendix A.2.
+
+Construction of RL Oracle The construction of our RL oracle can use any off-the-shelf RL algorithm that maximizes a scalar reward. For any given $\pmb { \lambda } \in \mathbb { R } ^ { m }$ , we define any algorithm that finds a policy minimizing the linear function $\lambda ^ { T } c ( \cdot )$ as a $R L$ oracle, that is
+
+$$
+\mathtt { R L \_ o r a c l e } _ { \mathtt { p } } ( \lambda ) \in \underset { \pi \in \Pi } { \arg \operatorname* { m i n } } \lambda ^ { T } \pmb { c } ( \pi ) .
+$$
+
+Recall that standard RL algorithm receives a scalar reward after each state transition, instead of the long-term measurement vector $\pmb { c } ( \pi ) \in \mathbb { R } ^ { m }$ . We then use the following linear property to reformulate the right hand side of (12) to a standard RL problem
+
+$$
+\underset { \pi \in \Pi } { \arg \operatorname* { m i n } } \ : \lambda ^ { T } c ( \pi ) = \underset { \pi \in \Pi } { \arg \operatorname* { m i n } } \ : \lambda ^ { T } \mathbb { E } ( \sum _ { t = 0 } ^ { T } \gamma ^ { t } c _ { t } ) = - \underset { \pi \in \Pi } { \arg \operatorname* { m a x } } \ : \mathbb { E } ( \sum _ { t = 0 } ^ { T } \gamma ^ { t } ( - \lambda ^ { T } c _ { t } ) ) .
+$$
+
+This shows that if we consider the Markov decision process with the same state, action, and transition probability, and construct a scalar reward $r : = \overline { { ( - \lambda ^ { T } } } c _ { t } )$ , then any policy that maximizes the expected $r$ is a linear minimizer of (12). Therefore any RL algorithm that best suits the underlying problems can be used to construct a RL oracle.
+
+Certifying constraint satisfaction amounts to evaluate the measurement vector of the current policy. This is handy in online settings, where simulations can be used to evaluate the measurement vector of the policy directly. Otherwise, in batch settings, various off-policy evaluation methods, such as importance sampling (Precup, 2000; Precup et al., 2001) or doubly robust (Jiang & Li, 2016; Dud´ık et al., 2011), can be used to evaluate the policy.
+
+$$
+{ \mathrm { R L } } _ { \mathrm { - } } { \mathrm { o r a c l e } } _ { \mathrm { c } } ( \lambda ) : = c ( \operatorname* { a r g m i n } _ { \pi \in \Pi } \lambda ^ { T } c ( \pi ) ) = \operatorname * { a r g m i n } _ { c ( \pi ) , \pi \in \Pi } \lambda ^ { T } c ( \pi ) .
+$$
+
+To simplify notation, we assume a RL Oracle returns a policy as well as its measurement vector
+
+$$
+\mathtt { R L \_ O r a c l e } ( \lambda ) : = \pi , \pm ( \pi ) = \mathtt { R L \_ o r a c l e } _ { \mathtt { p } } ( \lambda ) , \mathtt { R L \_ o r a c l e } _ { \mathtt { c } } ( \lambda )
+$$
+
+Finding Extended Affine Minimizer The process AffineMinimizer $( S , { \pmb x } )$ returns the $( y , \alpha )$ the affine minimizer of $S$ with respect to $_ { \textbf { \em x } }$ where $\textbf {  { y } }$ is the affine minimizer and $\alpha : = \{ \alpha _ { s } | \forall s \in$ $ { \boldsymbol { S } } _ { c }   \boldsymbol { \} }$ is the set of coefficient expressing $\textbf {  { y } }$ as an affine combination of points in $S$ , that is ${ \textbf { 3 } } =$ $\sum _ { s \in S _ { c } } \alpha _ { s } s$ , where $\alpha _ { s }$ is the weight associated with $\pmb { s }$ . The process AffineMinimizer $( S , { \pmb x } )$ can be straightforwardly implemented using linear algebra. Wolfe (1976) also provides a more efficient implementation that uses a triangular array representation of the active set.
+
+# 4.4 CONVERGENCE AND SPARSITY
+
+In this section, we analyze the convergence and complexity of the proposed C2RL method (Algo. 1). We first show that approximation error of C2RL strictly decreases between any two major cycle steps and it converges in $O ( 1 / t )$ rate. Then we show our method ensures convergence of arbitrary RL algorithm, including those searching for deterministic policies. Moreover, concerning the memory complexity, we show that maintaining an active policy set of $m { + 1 }$ is worst case optimal to ensure the convergence of arbitrary RL algorithm. Therefore, the proposed C2RL indeed achieves the optimal sparsity for the found policy, making it favorable for large-scale usage.
+
+The main difference between the convergence analysis of C2RL and Wolfe’s method is the addition of the projection step. Intuitively, at each major step, if we are making a significant progress toward the projected point, then the distance to the convex set is decreased by at least the same amount.
+
+Time Complexity. In our analysis, we consider the approximation error as defined in (5). We use superscript $t$ to denote the variable in $t$ -th major cycle before executing any minor cycle. To simplify notions, we let $x ^ { t } : = c ( \mu ^ { t } )$ and $s ^ { t } : = c ( \pi ^ { t } )$ . When discussing one step with $t$ fixed, let $y ^ { i }$ denote the affine minimizer found in $i$ -th minor cycle (line 6 of Algo. 1). We first show that the C2RL method strictly reduces approximation error between two calls of the RL oracle.
+
+Theorem 4.1 (Approximation Error Strictly Decreases). For any non-terminal step $t ,$ , we have $\mathsf { e r r } ( \mu ^ { t + 1 } ) \ < \ \mathsf { e r r } ( \mu ^ { t } )$ . That is, the measurement vector of $\mu ^ { t }$ found by the C2RL method gets strictly closer to the convex set $\Omega$ after major cycle step.
+
+The proof is provided in Appendix B. The idea is to consider the distance between $\mathbf { \boldsymbol { x } } ^ { t }$ and $\omega ^ { t }$ . When the major cycle has no minor cycle, the non-terminal condition and the affine minimizer property implies $\mathrm { d i } \mathbf { s } \ t ^ { 2 } ( \underline { { { \boldsymbol { x } } } } ^ { t + 1 } , \omega ^ { t } ) < \mathrm { d i } \mathbf { s } \ t ^ { 2 } ( \boldsymbol { x } ^ { t } , \omega ^ { t } )$ . Otherwise we show that the first minor cycle strictly reduces the $\mathsf { d i s t ^ { 2 } } ( \boldsymbol { x } ^ { t } , \omega ^ { t } )$ by moving along the segment joining $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ , and the subsequent minor cycle cannot increase it. Since $\omega ^ { t } \in \Omega$ , we conclude $\mathrm { e r r } ( \bar { { \bf x } ^ { t + 1 } } ) \le \mathrm { d i s t ^ { 2 } } ( { \bf x } ^ { t + 1 } , \omega ^ { \hat { t } } ) <$ $\mathrm { d i s t ^ { 2 } } ( \dot { { \bf x } } ^ { t } , \omega ^ { t } ) = \mathrm { e r r } ( { \bf x } ^ { t } )$ , and the approximation error strictly decreases.
+
+Given the approximation error strictly decreases, Wolfe’s method for minimum norm point can be shown to terminate finitely (Wolfe, 1976). However, this finitely terminating property does not hold for our algorithm. Since a changed $\omega ^ { t }$ may yield a lower distance to the same active set $S _ { c } ^ { t }$ , the active set may stay unchanged across major cycles (see Figure 2 Middle for an example). Therefore we establish the convergence of the C2RL method by the following theorem.
+
+Theorem 4.2 (Convergence in Approximation Error). For $t \geq 1$ , the mixed policy $\mu ^ { t }$ found by the C2RL method satisfies
+
+$$
+\mathsf { e r r } ( \mu ^ { t } ) \leq 1 6 Q ^ { 2 } / ( t + 2 ) ,
+$$
+
+where $Q : = \operatorname* { m a x } _ { \mu \in \Delta ( \mathcal { U } ) } | | c ( \mu ) | |$ is the maximum norm of a measurement vector.
+
+The proof is provided in Appendix C, which relies on the following two lemmas. We briefly discuss the main idea here. Define major cycle steps with at most one minor cycle as ”non-drop step” and major cycle steps with more than one minor cycles as ”drop steps”. We show that in each non-drop step, Algorithm 1 is guaranteed to make enough progress in the following lemma.
+
+Lemma 4.3. For a non-drop step in C2RL method, we have $\mathsf { e r r } ( \mu ^ { t } ) - \mathsf { e r r } ( \mu ^ { t + 1 } ) \geq \mathsf { e r r } ^ { 2 } ( \mu ^ { t } ) / 8 Q ^ { 2 } .$
+
+Though this does not hold for drop steps, we can bound the frequency of drop steps by the following.
+
+Lemma 4.4. After t major cycle steps of C2RL method, the number of drop steps is less than $t / 2$ .
+
+Since the approximation error strictly decreases (Thm. 4.1), and in more than half of the major cycles steps, the C2RL method makes significantly progress. The Thm. (4.2) can then be proved using an induction argument (Appendix C).
+
+Convergence with Arbitrary RL Algo. The convergence of the C2RL method when used with RL algorithms that search for deterministic policies, such as DQN, DDPG and variants, is indeed straightforward. In (8), though each time the oracle yields a vertex, the FW-type algorithms indeed optimize over the polytope formed by these vertices. Then since citetaltman1999constrained shows that any $c ( \cdot )$ achievable can be achieved by some mixed deterministic policies, we conclude that if the underlying problem is feasible, then our C2RL method is able to find a feasible policy.
+
+Memory Complexity We then discuss the sparsity of mixed policy for constrained RL problem. We give a constructive proof in Appendix $\mathrm { D }$ to show that to ensure convergence for RL algorithms that search for deterministic policies, storing $m + 1$ policies is required in the worst case.
+
+![](images/1740e8b4ed7f6e9b1be1310e621d034c19a3b19de3e313a3da3466bbf086957e.jpg)  
+Figure 1: Left: The Risky Mars Rover environment. The agent is required to navigate from the starting point to reach the goal point without staying long (0.5 steps in expectation) in the risky area (cross-hatching region). Middle, Right. Example of an optimal mixed policy found by C2RL in a single run. After 10k samples, C2RL finds a mixed policy that randomizes among two policies with weight 0.49 and 0.51. The visitation probabilities of the two policies are plotted.
+
+Theorem 4.5 (Memory Complexity Bound). For an constrained RL problem with $m$ -dimensional measurement vector, in the worst case, a mixed policy needs to randomize among $m + 1$ individual policies to ensure convergence of RL oracles that search for deterministic policies.
+
+Since the minor cycles in the C2RL method eliminate policies with affinely dependent measurement vectors, after the termination of minor cycles, the size of the active set is at most $m + 1$ . That is, the policy found by the C2RL method requires randomization among no more than $m + 1$ individual policies. Therefore the proposed C2RL indeed achieves the optimal sparsity in the worst case, making it favorable for large-scale usage.
+
+Corollary 4.5.1. The C2RL method that randomizes among at most $m + 1$ policies is worst-case optimal to ensure convergence of any RL oracle.
+
+# 5 EXPERIMENTS
+
+We evaluate the performance of C2RL in a grid-world navigation task (Fig. 1), and demonstrate its ability to efficiently find sparse policy. In this Risky Mars Rover environment, the agent is required to navigate from the starting point to the goal point, by moving to one of the four neighborhood cells at each step. The episodes terminate when the goal point is reached or after 300 steps. To enforce robustness, we add a risky area to indicate the dangerous states. The agent receives a measurement vector to indicate the steps it takes (0.1 for every step), and whether it stays in the risky area (0.1 for every risky step, and 0 otherwise), with discount factor $\gamma = 0 . 9 9$ . We constrain the agent to reach the goal point with expected cumulative steps measure within 1.1 and the expected cumulative risky steps within 0.05. Note that by design, the shortest path from the starting point to the goal point does not satisfy the constraint. This is common in practice, as robustness typically evolves trade-off between the reward and the constraint satisfaction.
+
+The proposed C2RL method is compared with approachability-based policy optimization (ApproPO) (Miryoosefi et al., 2019) and with reward constrained policy optimization (RCPO) (Tessler et al., 2018). ApproPO solves the same convex constrained RL problem by using an RL oracle to play against a no-regret online learner (Hazan et al., 2008; Zinkevich, 2003). Since ApproPO and C2RL both use a RL oracle, ApproPO is a natural baseline to be compared with our method. Besides, we also compare with RCPO, which takes a Lagrangian approach to incorporate the constraints as a penalty signal into the reward. Using an advantage actor critic (A2C) Mnih et al. (2016), RCPO has been shown to converge to a fixed point. For a fair comparison, C2RL and ApproPO uses an A2C agent as the RL oracle, with the same hyperparameter as used in RCPO. The approximation errors are compared after training for the same number of samples.
+
+Note that the C2RL method does not introduce any extra hyper-parameter. For ApproPO and RCPO, they require extra hyper-parameter for the initialization and learning rate of a variable equivalent to our $\boldsymbol { \lambda }$ in the outer loop. This is because our approach does not rely on the online learning framework, and therefore there is no need to tune the initialization and learning rate for our $\boldsymbol { \lambda }$ and ease the usage.
+
+We first showcase the consequences of our theoretical results using an optimal RL oracle. For any $\pmb { x } \in \mathbb { R } ^ { m }$ , an optimal policy can be easily found via Dijkstra’s algorithm. If multiple optimal paths exist, one is randomly picked to form a deterministic policy.
+
+![](images/389628de1f75b64669d9098cb4d5f09762538a3a7d4405553c5ff04232cfe97e.jpg)  
+Figure 2: Left: Visualization of the distance minimization problem (7) in $\mathbb { R } ^ { 2 }$ , where the number of steps and the number of steps in risky zone are measured. The green hatched region is the polytope formed by values achievable by mixed deterministic policies $c \bar { ( \Delta ( \Pi ) ) }$ , and the red hatched region is the target set. Middle: Using an optimal RL oracle, 10 paths are sampled to showcase the convergence property of C2RL and ApproPO, where each cross on the dashed line corresponds to a call to the oracle. Right: If we zoom in, ApproPO suffers from the zig-zagging problem.
+
+![](images/3a7be78b8557e141d82da1b543f857a22bcca7efa9c5e96bc6f94f86cad03f20.jpg)  
+Figure 3: Left: Time complexity measured by number of calling an optimal RL oracle. Middle, Right: Using A2C to approximate an RL oracle, time complexity measured by thousands of samples and memory complexity measured by the number of policies stored are compared.
+
+Using this as an optimal RL oracle, the convergence property of C2RL and ApproPo are compared. Figure 2 Middle shows the value of policies $c ( \mu ^ { t } )$ found after each call to the oracle. In Figure 2 Right, when approaching the boundary of the feasible set, the iterations of approachability-based methods start to zigzag. Since C2RL contains a minor cycle to re-optimize the weights among the active set, C2RL progresses quickly to reach the exact optimal solution. In Figure 3 Left, the approximation error is shown for 300 calls of the optimal RL oracle.
+
+We then compare C2RL, ApproPO and RCPO using the same A2C agent (details of the model structures and hyper-parameters are provided in Appendix E). We run each algorithm for 50 times, and each run for a maximum of 100 thousands of samples. The mean and standard deviation of the results are presented in Figure 3. The original paper of ApproPO suggests using a cache to save memory, and the memory requirement of this variant is also presented. Figure 3 demonstrates that C2RL converges to an optimal policy faster than previous methods, and a sparse combination of individual policies is maintained throughout the iteration process.
+
+# 6 CONCLUSION
+
+In this paper, we introduce C2RL, an algorithm to solve RL problems under orthant or convex constraints. Our method reduces the constrained RL problem to a distance minimization problem, and a novel variant of Frank-Wolfe type algorithm is proposed to solve this. Our method comes with rigorous theoretical guarantees and does not introduce any extra hyper-parameter. To find an $\epsilon$ -approximation solution, C2RL takes $O ( 1 / \epsilon )$ calls of any RL oracle and ensures convergence to work with arbitrary RL algorithm. Moreover, C2RL strictly reduces the approximation error between consecutive calls of RL oracle, and for $m$ -dimensional constraints, the memory requirement is reduced from storing infinitely many policies $( O ( 1 / \epsilon ) )$ ) to storing at most constantly many $( m + 1 )$ polices. We further show that the constant is worst-case optimal to ensure the convergence for RL algorithms that search for deterministic policies. Experimentally, we demonstrate that the proposed C2RL method finds sparse solution efficiently, and outperforms previous methods.
+
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+Hado Van Hasselt, Arthur Guez, and David Silver. Deep reinforcement learning with double qlearning. arXiv preprint arXiv:1509.06461, 2015.
+
+Ziyu Wang, Tom Schaul, Matteo Hessel, Hado Hasselt, Marc Lanctot, and Nando Freitas. Dueling network architectures for deep reinforcement learning. In International conference on machine learning, pp. 1995–2003, 2016.
+
+Philip Wolfe. Convergence theory in nonlinear programming. Integer and nonlinear programming, pp. 1–36, 1970.
+
+Philip Wolfe. Finding the nearest point in a polytope. Mathematical Programming, 11(1):128–149, 1976.
+
+Shuai Xiao, Le Guo, Zaifan Jiang, Lei Lv, Yuanbo Chen, Jun Zhu, and Shuang Yang. Model-based constrained mdp for budget allocation in sequential incentive marketing. In Proceedings of the 28th ACM International Conference on Information and Knowledge Management, pp. 971–980, 2019.
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+Guanjie Zheng, Fuzheng Zhang, Zihan Zheng, Yang Xiang, Nicholas Jing Yuan, Xing Xie, and Zhenhui Li. Drn: A deep reinforcement learning framework for news recommendation. In Proceedings of the 2018 World Wide Web Conference, pp. 167–176, 2018.
+
+Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th international conference on machine learning (icml-03), pp. 928–936, 2003.
+
+Lixin Zou, Long Xia, Zhuoye Ding, Jiaxing Song, Weidong Liu, and Dawei Yin. Reinforcement learning to optimize long-term user engagement in recommender systems. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 2810–2818, 2019.
+
+A MORE ON FRANK-WOLFE-TYPE ALGORITHMS
+
+A.1 STANDARD FRANK-WOLFE ALGORITHM
+
+A.2 WOLFE’S METHOD FOR MINIMUM NORM POINT   
+
+<table><tr><td>Algorithm 2 Frank-Wolfe algorithm (Frank et al.,1956)</td></tr><tr><td>Input: obj. f : V -→ R,oracle O(·), init. xo ∈ Y</td></tr><tr><td>1: for t=1,2,3...,Tdo</td></tr><tr><td> 2:s←0racle(Vf(xt-1))= argming∈{s1,.,sn) sTVf(xt-1)</td></tr><tr><td>:xt←(1-nt)xt-1+nts,fornt:= t²2 3: 2</td></tr><tr><td></td></tr><tr><td>4: end for 5: return xT</td></tr></table>
+
+For a convex function $f : \mathcal { X } \mapsto \mathbb { R }$ the Frank-Wolfe algorithm (FW) solves the constrained optimization problem over a compact and convex set $\mathcal { X }$ . The standard FW is known to have a sublinear convergence rate, and various methods are proposed to improve the performance. For example, when the underlying feasible set is a polytope, and the objective function is strongly convex, multiple variants, such as away-step FW (Wolfe, 1970; Jaggi, 2013), pairwise FW (Mitchell et al., 1974), and Wolfe’s method (Wolfe, 1976) are shown to enjoy linear convergence rate. Linear convergence under other conditions is also studied (Beck & Shtern, 2017; Garber & Hazan, 2013a;b).
+
+Algorithm 3 Wolfe’s Method for Minimum Norm Point   
+Initialize $\boldsymbol { \mathscr { x } } \in \mathcal { P }$ , active set $s = [ { \pmb x } ]$ and weight $\lambda = [ 1 ]$ .   
+Output: $\boldsymbol { \mathscr { x } } \in \mathcal { P }$ that has the minimum Euclidean norm.   
+1: while true do // Major cycle 2: $\pmb { s } \gets 0 \mathtt { r a c l e } ( \pmb { x } )$ // Potential improving point 3: if $| | \pmb { x } | | ^ { 2 } \leq \pmb { x } ^ { T } s ^ { ' } + \epsilon$ then break   
+4: $S \gets S \cup \{ \mathbf { s } \}$   
+5: while true do // Minor cycle 6: ${ \pmb y } , { \pmb \alpha } \gets \mathsf { A f f i n e M i n i m i z e r } ( { \pmb S } )$ $/ / \pmb { y } = \arg \operatorname* { m i n } _ { \pmb { s } \in \mathrm { a f f } ( \pmb { S } ) } \left| \left| \pmb { s } \right| \right| _ { 2 }$ 7: if $\alpha _ { s } > 0$ for all $\pmb { s }$ then break // $\pmb { y } \in \mathsf { c o n v } ( S )$ 8: $/ /$ If $\pmb { \mathscr { z } } \cos \tau ( S )$ , then update $\textbf {  { y } }$ to the intersection of $\mathsf { c o n v } ( S )$ and segment joining $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ . Then remove points in $s$ unnecessary for describing $\textbf {  { y } }$ .   
+9: $\begin{array} { r l } & { \overset { \vartriangle } { \theta }  \operatorname* { m i n } _ { i : \boldsymbol { \alpha } _ { i } \leq 0 } \frac { \dot { \lambda } _ { i } } { \lambda _ { i } - \alpha _ { i } } } \\ & { \overset { \vartriangle } { \boldsymbol { y } }  \theta \pmb { y } + ( 1 - \theta ) \pmb { x } , \lambda _ { i } = \theta \alpha _ { i } + ( 1 - \theta ) \lambda _ { i } } \\ & { \overset { \boldsymbol { S } } { \epsilon }  \{ \mathbf { s } _ { i } | \mathbf { s } _ { i } \in \mathcal { S } \mathrm { a n d } \lambda _ { i } > 0 \} } \end{array}$ // Recall $\boldsymbol { \lambda }$ satisfies $\begin{array} { r } { \pmb { x } = \sum _ { s \in \mathcal { S } } \lambda _ { s } \pmb { s } } \end{array}$ 10:   
+11:   
+12: end while   
+13: Update ${ \mathbf { \mathscr { x } } } = { \mathbf { \mathscr { y } } }$ and $\lambda = \alpha$ .   
+14: end while   
+15: return $_ { \textbf { \em x } }$
+
+Wolfe’s method is an iterative algorithm for finding the point with minimum Euclidean norm in a polytope, which is defined as the convex hull of a set of finitely many points.
+
+The Wolfe’s method consists of a finite number of major cycles, each of which consists of a finite number of minor cycles. At the start of each major cycle, let $H ( \pmb { x } ) : = \{ \pmb { y } ^ { T } \pmb { x } = \pmb { x } ^ { \pmb { x } } \}$ be the hyperplane defined by $_ { \textbf { \em x } }$ . If $H ( { \pmb x } )$ separates the polytope from the origin, then the major cycle is terminated. Otherwise, it invokes an oracle to find any point on the near side of the hyperplane. The point is then added into the active set $s$ , and starts a minor cycle.
+
+In a minor cycle, let $\textbf {  { y } }$ be the point of smallest norm in of the affine hull aff $( S )$ . If $\textbf {  { y } }$ is in the relative interior of the convex hull $\mathsf { c o n v } ( S )$ , then $_ { \textbf { \em x } }$ is updated to $\textbf {  { y } }$ and the minor cycle is terminated. Otherwise, $\textbf {  { y } }$ is updated to the nearest point to $\textbf {  { y } }$ on the line segment $\mathsf { c o n v } ( S ) \cap [ \pmb { x } , \pmb { y } ]$ . Thus $\textbf {  { y } }$ is updated to a boundary point of $\mathsf { c o n v } ( S )$ , and any point that is not on the face of $\mathsf { c o n v } ( S )$ in which $\textbf {  { y } }$ lies is deleted. The minor cycles are executed repeatedly until $s$ becomes a corral, that is, a set whose affine minimizer lies inside its convex hull. Since a set of one point is always a corral, the minor cycles is terminated after a finite number of runs.
+
+# B PROOF OF THEOREM 4.1
+
+Theorem 4.1 (Approximation Error Strictly Decreases). For any non-terminal step $t ,$ , we have $\mathsf { e r r } ( \mu ^ { t + 1 } ) \ < \ \mathsf { e r r } ( \mu ^ { t } )$ . That is, the measurement vector of $\mu ^ { t }$ found by the C2RL method gets strictly closer to the convex set $\Omega$ after major cycle step.
+
+Proof. If the current step is a major cycle with no minor cycle, then $\boldsymbol { x } ^ { t + 1 }$ is the affine minimizer of aff $( S \cup \{ s ^ { t } \} )$ with respect to $\omega ^ { \bar { t } }$ . Then the affine minimizer property implies $( \pmb { s } ^ { t } - \pmb { x } ^ { t + 1 } ) ( \pmb { x } ^ { t + 1 } -$ $\omega ^ { t } ) \ = \ 0$ . Since iteration does not terminate at step $t$ , we have $( { \pmb x } ^ { t } - { \bar { \pmb \omega } } ^ { t } ) ^ { T } ( { \pmb x } ^ { t } - { \pmb s } ^ { t } ) > 0$ , and therefore $\pmb { x } ^ { t + 1 }$ not equal to $\mathbf { \boldsymbol { x } } ^ { t }$ . Then $\boldsymbol { x } ^ { t + 1 }$ is the unique affine minimizer implies $f _ { \Omega } ( { \pmb x } ^ { t + 1 } ) =$ $\begin{array} { r } { \operatorname* { m i n } _ { \omega \in \Omega } | | { \pmb x } ^ { t + 1 } - \omega | | ^ { \bar { 2 } } \leq | | { \pmb x } ^ { t + 1 } - \omega ^ { t } | | ^ { 2 } < | | { \pmb x } ^ { t } - \omega ^ { t } | | ^ { 2 } = f _ { \Omega } ( { \pmb x } ^ { t } ) } \end{array}$ .
+
+Otherwise the current step contains one or more minor cycles. In this case, we show that the first minor cycle strictly reduces the approximation error, and the (possibly) following minor cycles cannot increase it. For the first minor cycle, the affine minimizer $\bar { \boldsymbol { y } } ^ { 0 }$ of $\mathsf { a f f } ( S \cup \{ s ^ { t } \} )$ with respect to $\omega ^ { t }$ is outside c $\operatorname { m v } ( S \cup \{ s ^ { t } \} )$ . Let $\dot { z } = \theta { \bf y } ^ { 0 } + ( 1 - \theta ) { \bf x } ^ { t }$ be the intersection of $\mathsf { c o n v } ( \boldsymbol { S } \cup \{ \bar { \boldsymbol { s } } ^ { t } \} )$ and segment joining $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ . Let $\mathcal { V } ^ { 0 } : = \mathcal { S } ^ { t }$ and $\mathcal { V } ^ { i }$ denote the active set after the $i$ -th minor cycle. Then since $y ^ { \bar { 1 } }$ is the affine minimizer of $\mathcal { V } ^ { 1 }$ with respect to $\omega ^ { t }$ , we have
+
+$$
+| | z - \omega ^ { t } | | = | | \theta y ^ { 0 } + ( 1 - \theta ) x ^ { t } - \omega ^ { t } | | \leq \theta | | y ^ { 0 } - \omega ^ { t } | | + ( 1 - \theta ) | | x ^ { t } - \omega ^ { t } | | < | | x ^ { t } - \omega ^ { t } | | ,
+$$
+
+where the second step uses the triangle inequality and the last step follows since the segment $\boldsymbol { x } ^ { t } \boldsymbol { y } ^ { 0 }$ intersects the interior of $\mathsf { c o n v } ( S \cup \{ s ^ { \bar { t } } \} )$ , and the distance to $\omega ^ { t }$ strictly decreases along this segment. Therefore the point $_ { z }$ found by first minor cycle satisfies
+
+$$
+f _ { \Omega } ( z ) = \operatorname* { m i n } _ { \omega \in \Omega } | | z - \omega | | ^ { 2 } \leq | | z - \omega ^ { t } | | ^ { 2 } < | | x ^ { t } - \omega ^ { t } | | = f _ { \Omega } ( x ^ { t } ) .
+$$
+
+Hence $h ( \pmb { y } ^ { 1 } ) < h ( \pmb { x } ^ { t } )$ , and the first minor cycle strictly decreases the approximation error. By a similar argument, in subsequent minor cycles the approximation error cannot be increased. However, after the first minor cycle, the iterating point may already at the intersection point and the strict inequality in last step of Eq. 17 need to be replaced by non-strict inequality.
+
+Therefore any major cycle either finds an improving point and continue, or enters minor cycles where the first minor cycle finds an improving point, and the subsequent minor cycles does not increase the distance. Adding both side of $f _ { \Omega } ( \hat { \mathbf { x } ^ { t + 1 } } ) < f _ { \Omega } ( \mathbf { x } ^ { t } )$ by $f _ { \Omega } \bar { ( } x ^ { * } )$ and we have the approximation error $h ( \mathbf { x } ^ { t + 1 } ) < h ( \bar { \mathbf { x } } ^ { t } )$ strictly decreases. □
+
+# C PROOF OF THEOREM 4.2
+
+We first prove the Theorem 4.2, using Lemma 4.3 and Lemma 4.4. Then we present the proof of the lemmas.
+
+Theorem 4.2 (Convergence in Approximation Error). For $t \geq 1$ , the mixed policy $\mu ^ { t }$ found by the C2RL method satisfies
+
+$$
+\mathsf { e r r } ( \mu ^ { t } ) \leq 1 6 Q ^ { 2 } / ( t + 2 ) .
+$$
+
+where $Q : = \operatorname* { m a x } _ { \mu \in \Delta ( \mathcal { U } ) } | | c ( \mu ) | |$ is the maximum norm of a measurement vector.
+
+Proof. Since Lemma 4.4 shows that drop steps are no more than half of total major cycle steps, and Theorem 4.1 guarantees these drop steps reducing the approximation error, we can safely skip these step, and re-index the step numbers to include non-drop steps only using $k$ .
+
+For these non-drop steps, we claim that $\mathsf { e r r } ( \mu ^ { k } ) \leq 8 Q ^ { 2 } / ( k + 1 )$ . Using Lemma 4.3, we prove the convergence rate using induction. We first bound the error of any $\mathsf { e r r } ( \bar { \mu } ^ { \bar { k } } )$ . For any $k \geq 1$
+
+$$
+\begin{array} { r l } & { \mathsf { e r r } ( \mu ^ { k } ) = \mathtt { d i s t ^ { 2 } } ( c ( \mu ^ { k } ) , \Omega ) - \mathtt { d i s t ^ { 2 } } ( c ( \mu ^ { * } ) , \Omega ) } \\ & { \quad \quad = 1 / 2 | | c ( \mu ^ { k } ) - \mathtt { P r o j } _ { \Omega } ( c ( \mu ^ { k } ) ) | | ^ { 2 } - 1 / 2 | | c ( \mu ^ { * } ) - \mathtt { P r o j } _ { \Omega } ( c ( \mu ^ { * } ) ) | | ^ { 2 } } \\ & { \quad \le 1 / 2 ( | | c ( \mu ^ { k } ) | | ^ { 2 } + | | \mathsf { P r o j } _ { \Omega } ( c ( \mu ^ { k } ) ) | | ^ { 2 } - | | c ( \mu ^ { * } ) | | ^ { 2 } - | | \mathsf { P r o j } _ { \Omega } ( c ( \mu ^ { * } ) ) | | ^ { 2 } ) } \\ & { \quad \le | | c ( \mu ^ { k } ) | | ^ { 2 } - | | c ( \mu ^ { * } ) | | ^ { 2 } } \\ & { \quad \le | | c ( \mu ^ { k } ) | | ^ { 2 } } \\ & { \quad \le Q ^ { 2 } , } \end{array}
+$$
+
+where Eq. 21 uses the definition of our squared Euclidean distance function. Eq. 22 follows from triangle inequality, and Eq. 23 is by the contractive property of the Euclidean distance.
+
+When $k = 1$ , the Eq. 25 established the based case. Now for $k \geq 1$ , assume that $\mathtt { e r r } ( \mu ^ { k } ) \le$ $8 Q ^ { 2 } / ( k + 1 )$ for $k \geq 1$ , then Lemma 4.3 gives $\mathsf { e r r } ( \mu ^ { k + 1 } ) \le \mathsf { e r r } ( \mu ^ { k } ) - \mathsf { e r r } ^ { 2 } ( \mu ^ { k } ) / 8 Q ^ { 2 }$ . Since the quadratic function of the right hand side is monotonically increasing on $\left( - \infty , 4 Q ^ { 2 } \right]$ , using the inductive hypothesis
+
+$$
+\mathbf { e r r } ( \mu ^ { k + 1 } ) \le \mathbf { e r r } ( \mu ^ { k } ) - \mathbf { e r r } ^ { 2 } ( \mu ^ { k } ) / 8 Q ^ { 2 } \le 8 Q ^ { 2 } / ( k + 1 ) - 8 Q ^ { 2 } / ( k + 1 ) ^ { 2 } \le Q ^ { 2 } / ( k + 2 )
+$$
+
+Since for $t$ steps of major cycle steps, the number of non-drop steps $k > t / 2$ , we conclude that $\mathsf { e r r } ( \mu ^ { t } ) \leq 1 6 \bar { Q } ^ { 2 } / ( t + \bar { 2 } )$ .
+
+Then we prove the lemmas.
+
+Lemma 4.3. For a non-drop step, we have $\mathbf { e r r } ( \mu ^ { t } ) - \mathbf { e r r } ( \mu ^ { t + 1 } ) \geq \mathbf { e r r } ^ { 2 } ( \mu ^ { t } ) / 8 Q ^ { 2 } .$
+
+Proof. The non-drop step contains either no minor cycle or one minor cycle. We first consider the no minor cycle case.
+
+If a major cycle contains no minor cycle, then $\pmb { x } ^ { t + 1 }$ is the affine minimizer of the $S \cup \{ s ^ { t } \}$
+
+$$
+\begin{array} { r l } { \mathrm { e r r } ( \mu ^ { t } ) - \mathrm { e r r } ( \mu ^ { t + 1 } ) = \mathrm { d i s t } ^ { 2 } ( x ^ { t } , \Omega ) - \mathrm { d i s t } ^ { 2 } ( x ^ { t + 1 } , \Omega ) } \\ & { = 1 / 2 ( \| x ^ { t } - \omega ^ { t } \| ^ { 2 } - \underset { \omega \in \Omega } { \operatorname* { m i n } } \| x ^ { t + 1 } - \omega \| ^ { 2 } ) } \\ & { \geq 1 / 2 ( \| x ^ { t } - \omega ^ { t } \| ^ { 2 } - \| x ^ { t + 1 } - \omega ^ { t } \| ^ { 2 } ) } \\ & { = 1 / 2 ( \| x ^ { t } - \omega ^ { t } \| ^ { 2 } + \| x ^ { t + 1 } - \omega ^ { t } \| ^ { 2 } - 2 \| x ^ { t + 1 } - \omega ^ { t } \| ^ { 2 } ) } \\ & { = 1 / 2 ( \| x ^ { t } - \omega ^ { t } \| ^ { 2 } + \| x ^ { t + 1 } - \omega ^ { t } \| ^ { 2 } - 2 ( x ^ { t } - \omega ^ { t } ) ^ { T } ( x ^ { t + 1 } - \omega ^ { t } ) ) } \\ & { = 1 / 2 ( \| x ^ { t } - x ^ { t + 1 } \| ^ { 2 } ) , } \end{array}
+$$
+
+where the equation (31) follows from the affine minimizer property Eq. (11). For $\lVert \mathbf { x } ^ { t } - \mathbf { x } ^ { t + 1 } \rVert$ in the last equation, and $\forall \mathbf { \pmb { q } } \in \mathsf { a f f } ( S \cup \{ \pmb { \mathscr { s } } ^ { t } \} ) ,$ , we have
+
+$$
+\begin{array} { r l r } { \| \pmb { x } ^ { t } - \pmb { x } ^ { t + 1 } \| \geq \| \pmb { x } ^ { t } - \pmb { x } ^ { t + 1 } \| \frac { \| \pmb { x } ^ { t } \| + \| \pmb { q } \| } { 2 Q } } & { \qquad \mathrm { ( D e f i n i t i o n ~ o f ~ } Q \mathrm { ) } } & \\ { \geq \| \pmb { x } ^ { t } - \pmb { x } ^ { t + 1 } \| \frac { \| \pmb { x } ^ { t } - \pmb { q } \| } { 2 Q } } & { \qquad \mathrm { ( ~ T r i a n g l e ~ i n e q u a l i t y ) } } & \\ { \geq \displaystyle \frac { 1 } { 2 Q } ( \pmb { x } ^ { t } - \pmb { x } ^ { t + 1 } ) ( \pmb { x } ^ { t } - \pmb { q } ) } & { \qquad \mathrm { ( ~ C a u c h y - S c h w a r z ~ i n e q u a l i t y ) } } & \\ { = \displaystyle \frac { 1 } { 2 Q } ( \pmb { x } ^ { t } - \omega ^ { t } ) ( \pmb { x } ^ { t } - \pmb { q } ) } & { \qquad \mathrm { ( ~ A f f i n e ~ m i n i m i z e r ~ p r o p e r t y ) . } } & \end{array}
+$$
+
+Then it suffices to show that $( { \pmb x } ^ { t } - { \pmb \omega } ^ { t } ) ( { \pmb x } ^ { t } - { \pmb q } ) \geq \mathrm { e r r } ( \mu ^ { t } )$ .
+
+Since $\Omega$ is a convex set, the squared Euclidean distance function dist ${ \mathbf \xi } ^ { 2 } ( { \pmb x } , { \pmb \Omega } )$ is convex for $_ { \textbf { \em x } }$ , which implies
+
+$$
+\mathrm { d i s t } ^ { 2 } ( { \pmb x } ^ { t } , \Omega ) + ( { \pmb q } - { \pmb x } ^ { t } ) \nabla \mathrm { d i s t } ^ { 2 } ( { \pmb x } ^ { t } , \Omega ) \leq \mathrm { d i s t } ^ { 2 } ( { \pmb q } , \Omega ) .
+$$
+
+Putting in $\nabla \mathrm { d i s t } ^ { 2 } ( \boldsymbol { x } ^ { t } , \Omega ) = \left( \boldsymbol { x } ^ { t } - \mathrm { P r o j } _ { \Omega } ( \boldsymbol { x } ^ { t } ) \right) = \left( \boldsymbol { x } ^ { t } - \omega ^ { t } \right)$ , we get $( { \pmb x } ^ { t } - { \pmb \omega } ^ { t } ) ( { \pmb x } ^ { t } - { \pmb q } ) \geq \mathtt { e r r } ( { \pmb \mu } ^ { t } )$ , which together with Eq. 32 and Eq. 36 concludes that for non-drop step with no minor cycles, we have $\mathsf { e r r } ( \mu ^ { t } ) - \mathsf { e r r } ( \mu ^ { \bar { t } + 1 } ) \geq \mathsf { e r r } ^ { \bar { 2 } } ( \mu ^ { t } ) / 8 Q ^ { 2 }$ .
+
+For non-drop step with one minor cycle, we use the Theorem 6 of (Chakrabarty et al., 2014). By a linear translation of adding all points with $- \omega ^ { t }$ , it gives
+
+$$
+| | \pmb { x } ^ { t } - \omega ^ { t } | | ^ { 2 } - | | \pmb { x } ^ { t + 1 } - \omega ^ { t } | | ^ { 2 } \geq ( ( \pmb { x } ^ { t } - \omega ^ { t } ) ( \pmb { x } ^ { t } - \pmb { q } ) ) ^ { 2 } / 8 Q ^ { 2 } .
+$$
+
+Then applying the same argument as Eq. 37, and we finished our proof.
+
+Lemma 4.4. After t major cycle steps of C2RL method, the number of drop steps is less than $t / 2$
+
+Proof. Recall that at the termination of a minor cycle, the size of the active set $| S _ { c } | \in [ 1 , m ]$ . Since in each major cycle steps, the size of active set $S _ { t }$ increases by one, and each drop step reduces the size of $S _ { t }$ by at least one, the number of drop steps is always less than half of total number of the major cycle steps. □
+
+# D PROOF OF THEOREM 4.5
+
+Theorem 4.5 (Memory Complexity Bound). For an constrained RL problem with $m$ -dimensional measurement vector, in the worst case, a mixed policy needs to randomize among $m + 1$ individual policies to ensure convergence of RL oracles that search for deterministic policies.
+
+Proof. We give a constructive proof. Consider a $m$ -dimensional vector-valued MDP with a single state, $m + 1$ actions, and $\pmb { c } ( a _ { i } ) : = \pmb { e } _ { i }$ is the unit vector of $i$ -th dimension for $i \in [ 1 , m ]$ , and $\pmb { c } ( a _ { m + 1 } ) : = \mathbf { 0 }$ , and the episode terminates after 1 steps. The constrained RL problem is to find a policy whose measurement vector lies in the convex set of a single point $\{ \mathbf { 1 } / 2 m \}$ . By linear programming, it is clear that the only feasible mixed deterministic policy is to select $a _ { m + 1 }$ with $1 / 2$ probability, and the rest $m$ actions with $1 / 2 m$ probability; i.e. the unique feasible policy to this problem has an active set containing $m + 1$ deterministic policies. Therefore any method randomize among less than $m + 1$ individual policies does not ensure convergence when used with RL algorithms searching for deterministic policies. □
+
+# E ADDITIONAL EXPERIMENT DETAILS
+
+All the methods use the same A2C agent. The input is the one-hot encoded current position index. The A2C is the standard fully connected multi-layer perceptron with ReLU activation function. The actor and critic share the internal representation and have their only final layer. Both actor and critic networks use Adam optimizer with learning rate set to $1 e ^ { - 2 }$ . The network is as follows
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Actor</td><td rowspan=1 colspan=1>Critic</td></tr><tr><td rowspan=1 colspan=1>Input layer</td><td rowspan=1 colspan=2>One-hot encoded state index (dim=54)</td></tr><tr><td rowspan=1 colspan=1>Hidden layer</td><td rowspan=1 colspan=2>Linear(in=54, out=128,act=&quot;relu&quot;)</td></tr><tr><td rowspan=1 colspan=1>Output layer</td><td rowspan=1 colspan=1>Linear(in=128,out=4,act=&quot;relu&quot;)</td><td rowspan=1 colspan=1>Linear(in=128,out=1,act=&quot;relu&quot;)</td></tr><tr><td rowspan=1 colspan=1>Output name</td><td rowspan=1 colspan=1>Action score</td><td rowspan=1 colspan=1>State value</td></tr></table>
+
+For ApproPO, the constant $\kappa$ for projection convex set to convex cone is set to be 20. The $\pmb \theta$ is initialized to 0. Following the original paper.
+
+For RCPO, the learning rate of its $\boldsymbol { \lambda }$ is set to $2 . 5 e ^ { - 5 }$ , and its $\lambda$ is initialized to 0 and updated by online gradient descent with learning rate set to 1, as used by the original paper.
+
+The proposed C2RL introduces no extra hyper-parameters, and has nothing to report.

parse/train/qcjOWDHAc4J/qcjOWDHAc4J.md

@@ -0,0 +1,561 @@
+# Social Processes: Self-Supervised Forecasting of Nonverbal Cues in Social Conversations
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+The default paradigm for the forecasting of human behavior in social conversations is characterized by top-down approaches. These involve identifying predictive relationships between low level nonverbal cues and future semantic events of interest (e.g. turn changes, group leaving). A common hurdle however, is the limited availability of labeled data for supervised learning. In this work, we take the first step in the direction of a bottom-up self-supervised approach in the domain. We formulate the task of Social Cue Forecasting to leverage the larger amount of unlabeled low-level behavior cues, and characterize the modeling challenges involved. To address these, we take a meta-learning approach and propose the Social Process (SP) models—socially aware sequence-to-sequence (Seq2Seq) models within the Neural Process (NP) family. SP models learn extractable representations of non-semantic future cues for each participant, while capturing global uncertainty by jointly reasoning about the future for all members of the group. Evaluation on synthesized and real-world behavior data shows that our SP models achieve higher log-likelihood than the NP baselines, and also highlights important considerations for applying such techniques within the domain of social human interactions.
+
+# 17 1 Introduction
+
+18 Picture a situated interactive agent such as a social robot conversing with a group of people. How   
+19 can agents act in such a setting? We sustain conversations spatially and temporally through explicit   
+20 behavioral cues—examples include locations of partners, their orientation, gestures, gaze, and floor   
+21 control actions [1–3]. Evidence suggests that we employ an anticipation of these and other cues to   
+22 navigate daily social interactions [1, 4–8]. Consequently, the ability to forecast the future constitutes   
+23 a natural objective towards the realization of machines with social skills. As such, interactive agents   
+24 typically contend with uncertainties in inferences surrounding cues [3]. So beyond making real-time   
+25 inferences, such systems may achieve more fluid interactions by leveraging the ability to forecast   
+26 future states of the conversation [9].   
+27 In addition to the development of social agents, behavior forecasting is also of significance in social   
+28 psychology, where the focus is on gaining insight into human behavior. Since human-interpretability   
+29 is of essence, top-down approaches largely constitute the default paradigm, where specific events of   
+30 semantic interest are selected first for consideration and their relationship to potentially predictive cues   
+31 are studied in isolation—either in controlled interactions in lab settings, or in subsequent statistical   
+32 analyses [10, 11]. Examples of such semantic events include speaker turn transitions [5, 12, 13],   
+33 mimicry episodes [14], or the termination of an interaction [9, 15]. However, one hurdle in the   
+34 top-down paradigm is limited data. The events (that constitute the labels or the dependent variables)   
+35 often occur infrequently over a longer interaction, reducing the effective amount of labeled data. This   
+36 precludes the use of neural supervised learning techniques that tend to be data intensive.   
+37 In this work, we take an initial step towards a bottom-up approach to forecasting human behavior for   
+38 free standing conversational groups. Our guiding motivation is to learn predictive representations of   
+39 general future social behavior by utilizing unlabeled streams of low-level behavioral features. We do   
+40 this by regressing future sequences of these features from observed sequences of the same features in   
+41 a self-supervised manner. We term this task of non-semantic future behavior forecasting as Social   
+42 Cue Forecasting (SCF).   
+43 Our approach is built on the observation that the social signal [17]—the high-level attitudes and social   
+44 meaning transferred in interactions—is already embedded in the low-level cues [18]. To conceptually   
+45 illustrate the contrasting top-down and bottom-up approaches on an example task, Figure 1 depicts   
+46 an instance of a group leaving event in a naturalistic social conversation. Evidence suggests that   
+47 such events can be anticipated from certain preceding rituals [15] reflected in the postural changes of   
+48 conversing members [1]. van Doorn [15] built a predictor using 200 instances of group leaving found   
+49 in over 90 minutes of mingling interaction and hand-crafted features. In contrast, our bottom-up   
+50 approach would entail learning task agnostic representations of future behavior using the entire 90   
+51 minutes of data, and then training simpler predictors for group leaving using the learnt representations   
+52 as input. The figure also illustrates the complexity of naturalistic interactions where cross-group   
+53 social influence exists. In this work we focus on the simpler setting of a single group in a scene.   
+54 There are several challenges intrinsic to computationally modeling future behavior in social conversa  
+55 tions. The future is intrinsically uncertain, the forecasts for interaction partners are inter-dependent,   
+56 and the social dynamics is unique for each grouping of individuals. We address these through the   
+57 following contributions:
+
+![](images/075804c0c5de43a9d7c73e8bb27136d7c99fd35d0e60c15f3fcb826c5068b063.jpg)  
+Figure 1: Conceptual illustration of forecasting approaches on an in-the-wild conversation from the MatchNMingle dataset [16]. a. The top-down approach entails predicting a semantic event or action of interest for the observed window $\mathbf { \mathit { t } _ { \mathrm { o b s } } } : = [ o \bar { 1 } \dots o T ]$ . Here we illustrate group leaving [15]; the circled individual in the center leaves a group in the future. b. In contrast, we propose a bottom-up approach in the social conversation forecasting domain through the task of Social Cue Forecasting. This entails using the non-semantic low-level cues over $\mathbf { \Delta } \mathbf { \mathcal { t } } _ { \mathrm { o b s } }$ to regress the same cues over the future window $\pmb { t } _ { \mathrm { f u t } } : = [ f 1 \ldots f { \bar { T } } ]$ . In this example we depict the cues of head pose (solid normal), body pose (hollow normal), and speaking status (speaker in orange). The hypothetical uncertainty estimates over $\scriptstyle t _ { \mathrm { f u t } }$ are also depicted as shaded spreads.
+
+• We formalize the task of SCF. We characertize the modeling challenges involved, and cast the problem into the meta-learning paradigm, allowing for data-efficient generalization to unseen groups at evaluation without learning group-specific models. We propose and evaluate two socially aware Sequence-to-Sequence (Seq2Seq) models within the Neural Process (NP) family [19] for SCF in social conversations. Our method encodes complex social dynamics informative of future group behavior into extractable representations for each individual.
+
+65 This paper is organized as follows. In Section 2 we formally define and characterize the task of   
+66 SCF. We situate this work within broader literature in Section 3, and review background concepts   
+67 in Section 4. We propose the Social Process models in Section 5 and describe our experiments in   
+68 Section 6, concluding with a discussion of our findings in Section 7.
+
+# 2 Social Cue Forecasting
+
+70 The objective of SCF is to predict future behavioral cues of all people involved in a social encounter   
+71 given an observed sequence of their behavioral features. More formally, let us denote a window   
+72 of observed timesteps as $\begin{array} { r c l } { t _ { \mathrm { o b s } } } & { : = } & { [ o 1 , o 2 , . . . , o T ] } \end{array}$ , and an unobserved future time window as   
+73 $\pmb { t } _ { \mathrm { f u t } } : = [ f 1 , f 2 , . . . , \bar { f } T ]$ , $f 1 > o T$ . Note that $\mathbf { \Delta } t _ { \mathrm { f u t } }$ and $\pmb { t } _ { \mathrm { o b s } }$ are typically non-overlapping, can be of   
+74 different lengths, and $\mathbf { \Delta } \mathbf { \mathbf { t } } _ { \mathrm { f u t } }$ need not immediately follow $\mathbf { \Delta } \mathbf { t } _ { \mathrm { o b s } }$ . Given a set of $n$ interacting participants,   
+75 let us denote their social cues over a $t _ { \mathrm { o b s } }$ and $\mathbf { \Delta } \mathbf { \mathbf { t } } _ { \mathrm { f u t } }$ respectively as
+
+$$
+X : = [ b _ { t } ^ { i } ; t \in t _ { \mathrm { o b s } } ] _ { i = 1 } ^ { n } , \quad Y : = [ b _ { t } ^ { i } ; t \in t _ { \mathrm { f u t } } ] _ { i = 1 } ^ { n } .
+$$
+
+The vector 76 $ { b _ { t } ^ { i } }$ encapsulates the multimodal cues of interest from participant $i$ at time $t$ . These can 77 include head and body pose, speaking status, facial expressions, gestures, and verbal content—any 78 information stream that combine to transfer social meaning.
+
+79 In its simplest form, given an $\boldsymbol { X }$ , the objective of SCF is to learn a single function $f$ such that   
+80 $Y = f ( X )$ . However, an inherent challenge in forecasting behavior is that an observed sequence   
+81 of interaction does not have a deterministic future and can result in multiple socially valid ones—a   
+82 window of overlapping speech between people both may and may not result in a change of speaker   
+83 [12, 20], a change in head orientation may continue into a sweeping glance across the room or a darting   
+84 glance stopping at a recipient of interest [21]. In some cases certain observed behaviors—intonation   
+85 and gaze cues [5, 13] or synchronization in speaker-listener speech [22] for turn-taking—might   
+86 make some outcomes more likely than others. Given that there are both supporting and challenging   
+87 arguments for how these observations influence subsequent behaviors [22, p. 5; 13, p. 22], it would   
+88 be beneficial if a data-driven model expresses a measure of uncertainty in its forecasts. We do this by   
+89 modeling the distribution over possible futures $p ( \mathbf { { Y } } | \boldsymbol { { X } } )$ rather than forecasting a single future.   
+90 Another design consideration arises from a defining characteristic of focused interactions—the   
+91 participants’ behaviors are interdependent. Participants in a group sustain equal access to the shared   
+92 interaction space through cooperative maneuvering [1, p. 220]. Moreover, when multiple groups   
+93 are co-located, outsiders unengaged in these intra-group maneuvers may also influence the behavior   
+94 of those within the group [23, p. 91;1, p. 233], sometimes causing them to leave (see Figure 1). It   
+95 is therefore essential to capture uncertainty in forecasts at the global level—jointly forecasting one   
+96 future for all participants at a time, rather than at a local output level—one future for each individual   
+97 independent of the remaining participants’ futures.   
+98 How participants coordinate their behaviors is a function of several individual factors [24, Chap. 1; 1,   
+99 p. 237]. Consequently, the social dynamics guiding an interaction also has unique attributes for every   
+100 unique grouping of individuals. Rather than learning group-specific models to capture these unique   
+101 dynamics, we formulate the forecasting problem in terms of meta-learning, or few-shot function   
+102 estimation. We interpret each unique group of individuals as the meta-learning notion of a task. The   
+103 core idea is that we can learn to predict a distribution over futures for a target sequence $\boldsymbol { X }$ having   
+104 captured the group’s unique behavioral tendencies from a context set $C$ of their observed-future   
+105 sequences. We can then generalize to unseen groups at evaluation by conditioning on a short observed   
+106 slice of their interaction. We believe that this approach is especially suitable for social conversation   
+107 forecasting—a setting that involves a limited data regime where good uncertainty estimates are   
+108 desirable. Note that when conditioning on context is removed $C = \varnothing$ ), we simply revert to the   
+109 formulation $p ( \mathbf { { Y } } | \boldsymbol { { X } } )$ .
+
+# 110 3 Related Work
+
+111 Free-standing conversations are an example of what social scientists call focused interactions, said to   
+112 arise when a “group of persons gather close together and openly cooperate to sustain a single focus   
+113 of attention, typically by taking turns at talking” [23, p. 24]. A long-standing topic of study has been   
+114 the systematic organization of turn-taking [25–27], with a particular interest in the event of upcoming   
+115 speaking turns [5–8]. There has also been some interest in the forecasting task itself, to anticipate   
+116 disengagement from an interaction [9, 15], the splitting or merging of groups [28], the time-evolving   
+117 size of a group [29] or semantic social action labels [30, 31]. Most of these works use heuristics,   
+118 either to generate semantic labels [9], model the dynamics itself [29], or hand-craft features [15].   
+119 Although not a forecasting task, the closest work that shares our motivation in predicting non-semantic   
+120 low-level features is the recently introduced task of Social Signal Prediction (SSP) [32]. The objective   
+121 is to predict the social cues1 of a target person using cues from the communication partners as   
+122 input (Joo et al. focus on predictions within the same time window [32, Eq. 6]). While the most   
+123 general formulation of SSP involves forecasting a single timestep for a target person given the   
+124 entire group’s past behavior [32, Eq. 3], generalizing this formulation runs into an inherent problem;   
+125 applying the definition to forecasting entails iteratively treating each individual as target, learning   
+126 separate functions for every person. However, as we discuss in Section 2, these futures of interacting   
+127 individuals are not independent given observed group behavior. Furthermore, a constrained definition   
+128 of forecasting that predicts an immediate step into the future is limiting, since forecasting an event   
+129 that occurs after a delay (e.g. a time lagged synchrony [33] or mimicry [14] episode) might be of   
+130 interest. Operationalizing this definition would entail a sliding window iteratively using predictions   
+131 over the offset between $t _ { \mathrm { o b s } }$ and $\pmb { t } _ { \mathrm { f u t } }$ as input, which would cascade prediction errors.   
+132 A related social setting where forecasting has been of interest is that of unfocused interactions. These   
+133 occur when individuals find themselves by circumstance in the immediate presence of each other,   
+134 such as pedestrians walking in proximity. Early approaches for forecasting pedestrian trajectories   
+135 were heuristic based, involving hand-crafted energy potentials to describe the influence pedestrians   
+136 have on each other [34–41]. More recent approaches encode the relative positional information   
+137 directly into a neural architecture [42–46].   
+138 In a broad sense, the self-supervised learning aspects of this work has some overlap with recent   
+139 approaches focusing on the non-interaction task of visual forecasting. These works have taken a   
+140 non-semantic approach to predict low level pixel-based features or intermediate representations   
+141 [38, 47–52], and demonstrated a utility of the learned representation for other tasks like semi  
+142 supervised classification [53], or training agents in immersive environments [54].
+
+# 143 4 Preliminaries
+
+144 Meta-learning. A supervised learning algorithm can be viewed as a function mapping a dataset   
+145 $C : = ( X _ { C } , \bar { Y _ { C } } ) : = \{ ( \bar { { \bf x } } _ { i } , { \bf y } _ { i } ) \} _ { i \in [ N _ { C } ] }$ to a predictor $f ( { \pmb x } )$ . Here $N _ { C }$ is the number of datapoints in   
+146 $C$ , and $[ N _ { C } ] : = \{ 1 , \dots , N _ { C } \}$ . The key idea of meta-learning is to learn the learning process itself,   
+147 modeling this function representing the initial algorithm using another supervised learning algorithm;   
+148 hence the name meta-learning. In meta-learning literature, a task refers to each dataset in a collection   
+149 $\mathcal { M } : = \{ \mathcal { T } _ { i } \} _ { i = 1 } ^ { N _ { \mathrm { t a s k s } } }$ of related datasets [55]. For each task $\tau$ , a meta-learner is episodically trained   
+150 to fit a subset of target points $D : = ( X , Y ) : = \{ ( { \pmb x } _ { i } , { \pmb y } _ { i } ) \} _ { i \in [ N _ { D } ] }$ given another subset of context   
+151 observations $C$ . At meta-test time, the resulting predictor $f ( \pmb { x } , C )$ uses the information obtained   
+152 during meta-learning to make predictions for unseen target points conditioned on context sets unseen   
+153 at meta-training.   
+154 Neural Processes Sharing the same core motivations, NPs are a family of latent variable models   
+155 that extend the idea of meta-learning to situations where uncertainty in the predictions $f ( \pmb { x } , C )$ are   
+156 desirable. They do this by meta-learning a map from datasets to stochastic processes, estimating a   
+157 distribution over the predictions $p ( { \boldsymbol { Y } } | { \boldsymbol { X } } , { \boldsymbol { C } } )$ . To capture this distribution, NPs model the conditional   
+158 latent distribution $p ( z | C )$ from which a task representation $z \in \mathbb { R } ^ { d }$ is sampled. This constitutes   
+159 the model’s latent path. The context can also be incorporated through a deterministic path, via a   
+160 representation $r _ { C } \in \mathbb { R } ^ { d }$ aggregated over $C$ . An observation model $p ( \pmb { y } _ { i } | \pmb { x } _ { i } , \pmb { r } _ { C } , \pmb { z } )$ then fits the target   
+161 observations in $D$ . The generative process for the NP is written as
+
+$$
+p ( { \pmb Y } | { \pmb X } , { \pmb C } ) : = \int p ( { \pmb Y } | { \pmb X } , { \pmb C } , { \pmb z } ) p ( { \pmb z } | { \pmb C } ) d { \pmb z } = \int p ( { \pmb Y } | { \pmb X } , { \pmb r } _ { C } , { \pmb z } ) q ( { \pmb z } | s _ { C } ) d { \pmb z } ,
+$$
+
+162 where $\begin{array} { r } { p ( { \pmb Y } | { \pmb X } , { \pmb r } _ { C } , z ) : = \prod _ { i \in [ N _ { D } ] } p ( { \pmb y } _ { i } | { \pmb x } _ { i } , { \pmb r } _ { C } , z ) } \end{array}$ . The latent $_ z$ is modeled by a factorized Gaussian   
+163 parameterized by $\bullet _ { C } : = f _ { s } ( C )$ , with $f _ { s }$ being a deterministic function invariant to order permutation   
+164 over $C$ . When the conditioning on context is removed $C = \varnothing$ ), we have $q ( z | s _ { \emptyset } ) : \bar { = } p ( z )$ , the   
+165 zero-information prior on $_ { z }$ . $C$ is encoded on the deterministic path using a function $f _ { r }$ similar to   
+166 $f _ { s }$ , so that $\pmb { r } _ { C } : = f _ { r } ( C )$ . In practice this is implemented as $\textstyle { \pmb r } _ { C } \doteq \sum _ { i \in [ N _ { C } ] } \operatorname { M L P } ( { \pmb x } _ { i } , { \pmb y } _ { i } ) / N _ { C }$ . The   
+167 observation model is referred to as the decoder, and $q , f _ { r } , f _ { s }$ comprise the encoders. The parameters   
+168 of the NP are learned for random subsets $C$ and $D$ by maximizing the evidence lower bound (ELBO)
+
+![](images/7cda7fc64afbe0b23d6bdadb0c6190cf34a8132631c02b3936d08c58e0a213a6.jpg)  
+Figure 2: Architecture of the SP and ASP family.
+
+# 169 5 Social Processes
+
+170 In this section we present our socially aware Seq2Seq models within the NP family that is agnostic to   
+171 group member identities and group size. To setup the task, we split the contextual interaction on which   
+172 we condition into pairs of observed and future sequences, writing the context as $C : = ( X _ { C } , Y _ { C } ) : =$   
+173 $( X _ { j } , Y _ { k } ) _ { ( j , k ) \in [ N _ { C } ] \times [ N _ { C } ] }$ , where every $X _ { j }$ occurs before the corresponding $Y _ { k }$ . As discussed in   
+174 Section 3, domain experts focusing on behavior analysis might be interested in settings where $t _ { \mathrm { o b s } }$   
+175 and $\mathbf { \Delta } t _ { \mathrm { f u t } }$ are offset by an arbitrary delay. Consequently, the $j$ th $t _ { \mathrm { o b s } }$ can have multiple associated $\mathbf { \Delta } \mathbf { \mathbf { { t } } } _ { \mathrm { { f u t } } }$   
+176 windows. Denoting the set of target window pairs as $D : = ( X , Y ) : = ( X _ { j } , Y _ { k } ) _ { ( j , k ) \in [ N _ { D } ] \times [ N _ { D } ] }$ , our   
+177 focus in the rest of this work is to model the distribution $p ( { \boldsymbol { Y } } | { \boldsymbol { X } } , { \boldsymbol { C } } )$ .   
+178 The generative process for our model we call the Social Process (SP) follows Eq. 2, which we   
+179 extend to social forecasting in two ways. We embed an observed sequence $_ { \textbf { \em x } }$ for an individual   
+180 into a condensed encoding $\mathbf { \bar { \boldsymbol { e } } } \in \mathbb { R } ^ { d }$ that is then decoded into the future sequence using a Seq2Seq   
+181 architecture [56, 57]. Our intuition is that this would cause the representation to encode temporal   
+182 information about the future. Further, for every individual we model this $e$ as a function of their own   
+183 behavior, and that of their partners as viewed by them. The intuition is that this captures the spatial   
+184 influence partners have on the participant over the $t _ { \mathrm { o b s } }$ . Using notation we established in Section 2,   
+185 we define the observation model for the SP for a single participant $\mathrm { p } _ { i }$ as
+
+$$
+p ( \pmb { y } ^ { i } | \pmb { x } ^ { i } , C , z ) : = p ( \pmb { b } _ { f 1 } ^ { i } , \ldots , \pmb { b } _ { f T } ^ { i } | \pmb { b } _ { o 1 } ^ { i } , \ldots , \pmb { b } _ { o T } ^ { i } , C , z ) = p ( \pmb { b } _ { f 1 } ^ { i } , \ldots , \pmb { b } _ { f T } ^ { i } | \pmb { e } ^ { i } , \pmb { r } _ { C } , z ) .
+$$
+
+186 If decoding is carried out in an auto-regressive manner, we can further write the right hand side of   
+187 Eq. 4 as $\begin{array} { r } { \dot { \prod _ { t = f 1 } ^ { f T } } p ( b _ { t } ^ { i } | b _ { t - 1 } ^ { i } , \dots , b _ { f 1 } ^ { i } , e ^ { i } , r _ { C } , z ) } \end{array}$ . Following the standard NP setting, we implement the   
+188 observation model as a set of Gaussian distributions factorized over time and feature dimensions.   
+189 We also incorporate the cross-attention mechanism from the Attentive Neural Process (ANP) [58] to   
+190 define the variant Attentive Social Process (ASP). Following Eq. 4 and the definition of the ANP, the   
+191 corresponding observation model of the ASP for a single participant is defined as
+
+$$
+p ( \pmb { y } ^ { i } | \pmb { x } ^ { i } , C , z ) = p ( b _ { f 1 } ^ { i } , \ldots , b _ { f T } ^ { i } | \pmb { e } ^ { i } , r ^ { * } ( C , \pmb { x } ^ { i } ) , z ) .
+$$
+
+92 Here each target query sequence $\pmb { x } _ { \ast } ^ { i }$ attends to the context sequences $X _ { C }$ to produce a query-specific   
+93 representation $r _ { * } : = r ^ { * } ( C , \pmb { x } _ { * } ^ { i } ) \in \mathbb { R } ^ { d }$ . The model architectures are illustrated in Figure 2.   
+194 Encoding Partner Behavior. While a typical Seq2Seq setup conditions the sequence decoder on   
+195 solely a compact representation of the observed sequence, we’d like to condition an individual’s   
+196 forecast on the observed behavior of both, themselves and their partners. We do this using a pair   
+197 of sequence encoders: one to encode the temporal dynamics of participant $\mathrm { p } _ { i }$ ’s features, $\bar { e } _ { \mathrm { s e l f } } ^ { i } =$   
+198 $f _ { \mathrm { s e l f } } ( \mathbf { \bar { x } } _ { i } )$ , and another to encode the dynamics of a transformed representation of the features of $\mathrm { p } _ { i }$ ’s   
+199 partners, $e _ { \mathrm { p a r t n e r } } ^ { i } = f _ { \mathrm { p a r t n e r } } ( \psi ( \pmb { x } _ { j , ( j \neq i ) } ) )$ . Using a separate network to encode partner behavior   
+200 grants the practical advantage of being able to sample an individual’s and partners’ features at different   
+201 sampling rates.   
+202 How do we model $\psi ( \pmb { x } _ { j } )$ ? We want the partners’ representation to possess two properties: per  
+203 mutation invariance—changing the order of the partners should not affect the representation; and   
+204 group size independence—we want to compactly represent all partners independent of the group size.   
+205 Beyond coordinate space invariance, we wish to intuitively capture a view of the interaction from   
+206 $\mathrm { p } _ { i }$ ’s perspective. We extend the approach Qi et al. [59] applied to point clouds to focused interactions   
+207 by computing pooled embeddings of relative behavioral features. Since most commonly considered   
+208 nonverbal cues in literature (see Section 6.3) include the attributes of orientation or location (e.g.   
+209 head/body pose or keypoints) or a binary indicator (such as speaking status), we specify how we   
+210 transform these. The 3D pose (orientation, location) of every partner $\mathrm { p } _ { j }$ is transformed to a frame of   
+211 reference defined by $\mathrm { p } _ { i }$ ’s pose. At timestep $t$ , denoting orientation, location, and binary speaking   
+212 status for $\mathrm { p } _ { i }$ as $b _ { t } ^ { i } = [ \mathbf { q } ^ { i } ; \mathbf { l } ^ { i } ; \mathbf { s } ^ { i } ]$ , and those for $\mathrm { p } _ { j }$ as $\mathbf { } b _ { t } ^ { j } = [ \mathbf { q } ^ { j } ; \mathbf { l } ^ { j } ; \mathbf { s } ^ { j } ]$ , we have
+
+$$
+\mathbf { q } ^ { r e l } = \mathbf { q } ^ { i } * ( \mathbf { q } ^ { j } ) ^ { - 1 } , \quad \mathbf { l } ^ { r e l } = \mathbf { l } ^ { j } - \mathbf { l } ^ { i } , \quad \mathbf { s } ^ { r e l } = \mathbf { s } ^ { j } - \mathbf { s } ^ { i } .
+$$
+
+213 Note that we use unit quaternions (denoted $\mathbf { q }$ ) for representing orientation due their various benefits   
+214 over other representations of rotation [60, Sec. 3.2]. The operator $^ *$ denotes the Hamilton product of   
+215 the quaternions. These transformed features for each $\mathrm { p } _ { j }$ are encoded using an embedder MLP. The   
+216 outputs are concatenated with $e _ { \mathrm { s e l f } } ^ { j }$ and processed by a pre-pooler MLP, which is followed by the   
+217 symmetric element-wise Max-pooling function to obtain $\psi ( \pmb { x } ^ { j } )$ at each timestep. We capture the   
+218 dynamics in the pooled representation over $t _ { \mathrm { o b s } }$ using $f _ { \mathrm { p a r t n e r } }$ . Finally, we combine $e _ { \mathrm { s e l f } } ^ { i }$ and $e _ { \mathrm { p a r t n e r } } ^ { i }$   
+219 for $\mathrm { p } _ { i }$ through a linear projection (defined by a weight matrix $W$ ) to obtain the individual’s embedding   
+220 $e _ { \mathrm { i n d } } ^ { i } = W . [ \bar { e } _ { \mathrm { s e l f } } ^ { i } ; e _ { \mathrm { p a r t n e r } } ^ { i } ]$ . Our intuition is that with information about both $\mathrm { p } _ { i }$ themselves, and of   
+221 $\mathrm { p } _ { i }$ ’s partners from $\mathrm { p } _ { i }$ ’s point-of-view, $e _ { \mathrm { i n d } } ^ { i }$ now contains the information required to predict $\mathrm { p } _ { i }$ ’s   
+222 future behavior.   
+223 Encoding Future Window Offset. As we’ve discussed at the start of this section, a single $t _ { \mathrm { o b s } }$   
+224 might have multiple associated $\mathbf { \Delta } \mathbf { \mathbf { \mathit { t } } _ { \mathrm { f u t } } }$ windows at different offsets. Our intuition is that training a   
+225 sequence decoder to decode the same $e _ { \mathrm { i n d } } ^ { i }$ into multiple sequences (corresponding to the multiple   
+226 $\pmb { t } _ { \mathrm { f u t } } )$ ) in the absence of any timing information might cause an averaging effect in either the decoder   
+227 or the information encoded in $e _ { \mathrm { i n d } } ^ { \bar { i } }$ . One way around this would be to start decoding one timestep   
+228 following the end of $t _ { \mathrm { o b s } }$ and discard the predictions in the gap between $t _ { \mathrm { o b s } }$ and $\mathbf { \Delta } t _ { \mathrm { f u t } }$ . However,   
+229 if decoding is done auto-regressively this might lead to cascading errors over the gap. Instead, we   
+230 address this one-to-many issue by injecting the offset information into $e _ { \mathrm { i n d } } ^ { i }$ so that the decoder   
+231 receives a unique encoded representation for every $\mathbf { \Delta } \mathbf { \mathbf { t } } _ { \mathrm { f u t } }$ to decode over. We do this by repurposing   
+232 the idea of sinusoidal positional encodings [61] to encode offsets rather than relative positions in   
+233 sequences. For a given $\mathbf { \Delta } \mathbf { t } _ { \mathrm { o b s } }$ and $\mathbf { \Delta } \mathbf { \mathbf { \mathit { t } } _ { \mathrm { f u t } } }$ , and $d _ { e }$ -dimensional $e _ { \mathrm { i n d } } ^ { i }$ we define the offset as $\Delta t = f 1 - o T$   
+234 and the corresponding offset encoding $O E _ { \Delta t }$ as
+
+$$
+O E _ { ( \Delta t , 2 m ) } = \sin ( \Delta t / 1 0 0 0 0 ^ { 2 m / d _ { c } } ) , \quad O E _ { ( \Delta t , 2 m + 1 ) } = \cos ( \Delta t / 1 0 0 0 0 ^ { 2 m / d _ { c } } ) .
+$$
+
+Here 235 $m$ refers to the dimension index in the encoding. We finally compute the representation $e ^ { i }$ for 236 Eqs. 4 and 5 as
+
+$$
+e ^ { i } = e _ { \mathrm { i n d } } ^ { i } + O E _ { \Delta t } .
+$$
+
+237 Auxiliary Loss Functions. We incorporate a geometric loss function that improves performance in   
+238 pose regression tasks. For $\mathrm { p } _ { i }$ at time $t$ , given the ground truth $b _ { t } ^ { i } = [ \mathbf { q } ; 1 ; \mathrm { s } ]$ , and the predicted mean   
+239 $\hat { b } _ { t } ^ { i } = [ \hat { \mathbf { q } } ; \hat { \mathbf { l } } ; \hat { \mathbf { s } } ]$ , we denote the tuple $( b _ { t } ^ { i } , b _ { t } ^ { i } )$ as $B _ { t } ^ { i }$ . We then have the location loss in Eucliden space   
+240 $\mathcal { L } _ { 1 } ( B _ { t } ^ { i } ) = \left. \mathbf { l } - \hat { \mathbf { l } } \right.$ , and we can regress the quaternion values using
+
+$$
+\mathcal { L } _ { \mathrm { q } } ( B _ { t } ^ { i } ) = \left\| \mathbf { q } - \frac { \hat { \mathbf { q } } } { \| \hat { \mathbf { q } } \| } \right\| .
+$$
+
+241 Kendall and Cipolla [60] show how these losses can be combined using the homoscedastic uncertainties in position and orientation, 242 $\hat { \sigma } _ { 1 } ^ { 2 }$ and $\hat { \sigma } _ { \mathrm { q } } ^ { 2 }$ :
+
+$$
+\mathcal { L } _ { \sigma } ( B _ { t } ^ { i } ) = \mathcal { L } _ { 1 } ( B _ { t } ^ { i } ) \exp ( - \hat { s } _ { 1 } ) + \hat { s } _ { 1 } + \mathcal { L } _ { \boldsymbol { \mathrm { q } } } ( B _ { t } ^ { i } ) \exp ( - \hat { s } _ { \mathrm { q } } ) + \hat { s } _ { \boldsymbol { \mathrm { q } } } ,
+$$
+
+where 243 $\hat { s } : = \log \hat { \sigma } ^ { 2 }$ . Using the binary cross-entropy loss for speaking status $\mathcal { L } _ { \mathrm { s } } ( B _ { t } ^ { i } )$ , we have the 244 overall auxiliary loss over $t \in { \pmb t } _ { \mathrm { f u t } }$ :
+
+$$
+\mathcal { L } _ { \mathrm { a u x } } ( \boldsymbol { Y } , \hat { \boldsymbol { Y } } ) = \sum _ { i } \sum _ { t } \mathcal { L } _ { \sigma } ( B _ { t } ^ { i } ) + \mathcal { L } _ { \mathrm { s } } ( B _ { t } ^ { i } ) .
+$$
+
+245 The parameters of the SP and ASP are trained by maximizing the ELBO in Eq. 3 and minimizing this   
+246 auxiliary loss function for each of our sequence decoders.
+
+![](images/42651c2b36add882c5b51ca88501608167a5fe63ce5ae59c255468d4faac0dd6.jpg)  
+Figure 3: Ground truths and model predictions for the toy task simulating the forecasting of glancing behavior.
+
+Table 1: Mean (Std.) Negative LogLikelihood (NLL) on the Haggling Test Sets. The reported mean and std. are over individual sequences in the test sets. Lower is better. The superscript ⇤ indicates best NLL within family, boldface best overall.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">Context</td></tr><tr><td>Random</td><td>Fixed-Initial</td></tr><tr><td>Baselines</td><td></td><td></td></tr><tr><td>NP-latent</td><td>38.34 (19.1)</td><td>37.64 (18.1)</td></tr><tr><td>NP-latent+det</td><td>40.41 (23.9)</td><td>40.15 (23.0)</td></tr><tr><td>ANP-dot</td><td>35.66* (20.8)</td><td>38.06* (20.6)</td></tr><tr><td>ANP-multihead</td><td>40.60 (19.2)</td><td>41.11 (19.2)</td></tr><tr><td>Ours (MLP)</td><td></td><td></td></tr><tr><td>SP-latent</td><td>-74.06 (6.0)</td><td>-74.19 (5.9)</td></tr><tr><td>SP-latent+det</td><td>-77.49 (7.8)</td><td>-76.90 (8.4)</td></tr><tr><td>ASP-dot</td><td>-76.33 (6.5)</td><td>-75.15 (6.5)</td></tr><tr><td>ASP-multihead</td><td>-83.77* (10.3)</td><td>-83.43* (9.7)</td></tr><tr><td>Ours (GRU)</td><td></td><td></td></tr><tr><td>SP-latent</td><td>-4.23 (27.4)</td><td>-3.72 (30.7)</td></tr><tr><td>SP-latent+det</td><td>-17.38* (50.5)</td><td>-16.08* (52.2)</td></tr><tr><td>ASP-dot</td><td>19.91 (46.7)</td><td>31.39 (77.0)</td></tr><tr><td>ASP-multihead</td><td>-7.11 (26.9)</td><td>-0.51 (28.8)</td></tr></table>
+
+# 247 6 Experiments and Results
+
+# 6.1 Models and Baselines
+
+Our modeling assumption is that the underlying stochastic process generating the behaviors does not evolve over time. Stated differently, we assume that the individual factors determining how participants coordinate behaviors—age, cultural background, personality variables [24, Chap. 1; 1, p. 237]—are likely to remain the same over the short duration of a single interaction. This is in contrast to a related line of work that deals with meta-transfer learning, where the stochastic process itself changes over time [62–65]. We therefore compare against the NP and ANP family which share our model assumptions and meta-learning attributes. Note that in contrast to our methods, these baselines have direct access to the future sequences in the context, and therefore constitute a strong baseline. We consider two variants: -latent denoting only the latent path; and -latent+det, containing both deterministic and stochastic paths. We further consider two attention mechanisms for the cross-attention module: -dot with dot attention, and -multihead with wide multi-head attention [58]. We operationalize the original definitions of the baseline models to sequences by collapsing the timestep and feature dimensions. While the ANP-RNN model [66] shares our model assumptions, it is defined for a task analogous to SSP for concurrent car locations, and cannot be operationalized to forecasting in any simple way (see Section 3 discussing the distinction). We experiment with two choices of architectures for the sequence encoders and decoders in our proposed models: multi-layer perceptrons (MLP), and Gated Recurrent Units (GRU). Implementation and training details for our experiments can be found in Appendix C.
+
+# 6.2 Evaluation on Synthesized Behavior: Forecasting Glancing Behavior
+
+With limited behavioral data availability, a common practice in the domain is to train and evaluate methods on synthesized behavior dynamics [31, 67]. In keeping with this practice, we construct a synthesized dataset simulating two glancing behaviors in social settings [21]. We use a 1D sinusoid to represent horizontal head rotation over 20 timesteps. The sweeping Type I glance is represented by a pristine sinusoid, while the gaze fixating Type III glance is denoted by clipping the amplitude for the last six timesteps. The task is to forecast the signal over the last 10 timesteps $( \pmb { t } _ { \mathrm { f u t } } )$ by observing the first 10 $( \pmb { t } _ { \mathrm { o b s } } )$ . Consequently, the first half of $\mathbf { \delta } _ { t _ { \mathrm { f u t } } }$ is certain, while uncertainty over the last half results from every observed sinusoid having two ground-truths. It is impossible to infer from an observed sequence alone if the head rotation will stop partway through the future. We describe additional data setup, model details, and quantitative results for this setting in Appendices A.1, C and D.1, respectively. Figure 3 illustrates the ground truths, predicted means and std. deviations for a sequence within and outside the context set. We observe that all models estimate the mean reasonably well, although our proposed SP models learn a slightly better fit. More crucially, the SP models—especially the SP-GRU—learn much better uncertainty estimates over the certain and uncertain parts of the future compared to the NP baseline.
+
+# 6.3 Real-World Behavior: The Haggling Dataset
+
+We also evaluate our models on real-world behavior data, using the Haggling dataset of triadic interactions [32]. Participants are engaged in an unscripted game where two sellers compete to sell a fictional product to a buyer who has to choose between the two. We use the same split of 79 training sets (groups) and 28 test sets used by Joo et al. [32]. In our experiments we consider the following social cues: head pose described by the 3D location of the nose keypoint and a face normal; body pose described by the location of the mid-point of the shoulders and a body normal; and binary speaking status. Apart from being the most commonly considered cues in computational analyses of such conversations [68–70], pose and turn taking are found to be crucial in the sustaining of conversation [1, 12, 18]. We specify the dataset preprocessing details in Appendix D.2.
+
+# 6.4 Evaluation
+
+Context Regimes. We evaluate all models on two context regimes: random, and fixed-initial. The random regime follows the standard NP setting that the models are trained in. Context samples (sequence-pairs) are selected as a random subset of target samples, so the model is exposed to behaviors from any phase of the interaction lifecycle. Here we ensure that batches contain unique $t _ { \mathrm { o b s } }$ to prevent any single observed sequence from dominating the aggregation of representations over the context split. At evaluation, we take $5 0 \%$ of the batch as context. In the fixed-initial context regime, we investigate how the model can generalize knowledge of group specific characteristics from observing the initial dynamics of an interaction where certain gestures and patterns are more distinctive [1, Chap. 6]. This matches what a social agent might face in a real-world scenario. Here we treat the first $2 0 \%$ of the entire interaction as context, treating sequences from the rest as target.
+
+304 Evaluation Metrics. We report the negative log-likelihood (NLL) $- \log p ( \boldsymbol { Y } | \boldsymbol { X } , \boldsymbol { C } )$ in Table 1   
+305 (computed by summing over feature dimensions and people, and averaging over timesteps). Beyond   
+306 the NLL, we also report the error in the predicted means over test sequences in Table 2: mean-squared   
+307 error (MSE) for the head and body keypoint locations; mean absolute error (MAE) in orientation   
+308 in degrees; and speaking status accuracy. Note that while the ground truth orientation normals are   
+309 constrained in the horizontal plane, we don’t constrain our predicted quaternions. We therefore report   
+310 the absolute error in rotation in 3D. The reported mean and std. deviation of all metrics are over   
+311 sequences in the test sets. We further report the metrics for every timestep over $\mathbf { \Delta } \mathbf { \mathbf { \mathit { t } } _ { \mathrm { f u t } } }$ in Appendix A.2,   
+312 and qualitative visualizations of the forecasts in Appendix B.
+
+# 6.5 Ablations
+
+Encoding Partner Behavior. Modeling the interaction from the perspective of each individual is a central idea in our apindividual representations $\bar { r } _ { \mathrm { i n d } } ^ { i }$ ch. We investigate the influence of encoding partner behavior intoon the performance. We train the SP-latent+det GRU variant in two configurations: no-pool, where we do not encode any partner behavior; and pool-oT where we pool over partner representations only at the last timestep (similar to [44]). We choose the SP-GRU model since it achieves the best trade-off between minimizing NLL and forecasting cues consistent with human behavior. Both configurations lead to worse NLL and location errors (Appendix A.3).
+
+Deterministic Decoding and Social Encoder Sharing. Error gradients can flow back into our sequence encoders through two paths: from the final stochastic sequence decoder, as well as the deterministic decoders on the latent and deterministic paths. We investigate the effect of the deterministic decoders by training the SP-latent+det GRU model without them. We also investigate sharing a single social encoder between the Process Encoder and Process Decoder in Figure 2. We find that removing the decoders only improves log-likelihood if the encoders are shared, and at the cost of head orientation errors (Appendix A.3).
+
+Table 2: Mean (Std.) Errors in Predicted Means over Sequences in the Haggling Test Sets. Lower is better for all metrics except for speaking status accuracy. ⇤ indicates best measure within family, boldface best overall.   
+
+<table><tr><td rowspan="2"></td><td colspan="5">Random Context</td><td colspan="5">Fixed-Initial Context</td></tr><tr><td>Head Loc. MSE (cm)</td><td>Body Loc. MSE (cm)</td><td>Head Ori. MAE()</td><td>Body Ori. MAE()</td><td>Speaking Accuracy</td><td>Head Loc. MSE (cm)</td><td>Body Loc. MSE (cm)</td><td>Head Ori. MAE(°)</td><td>Body Ori. MAE(°)</td><td>Speaking Accuracy</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>NP-latent</td><td>14.21 (6.5)</td><td>15.06 (6.1)</td><td>16.29 (13.8)</td><td>12.82 (13.7)</td><td>0.787 (0.23)</td><td>13.85 (6.1)</td><td>14.71 (5.7)</td><td>16.22 (14.1)</td><td>12.69* (13.9)</td><td>0.774* (0.24)</td></tr><tr><td>NP-latent+det</td><td>15.01 (7.3)</td><td>15.97 (7.2)</td><td>17.45 (18.3)</td><td>14.65 (20.0)</td><td>0.715 (0.24)</td><td>15.01 (7.5)</td><td>15.95 (7.5)</td><td>17.26 (15.9)</td><td>14.68 (18.7)</td><td>0.701 (0.24)</td></tr><tr><td>ANP-dot</td><td>11.86* (5.4)</td><td>12.22* (5.5)</td><td>15.44* (13.3)</td><td>12.56* (18.0)</td><td>0.806* (0.23)</td><td>12.83* (5.9)</td><td>13.26* (6.0)</td><td>16.19* (13.7)</td><td>13.56 (17.8)</td><td>0.717 (0.23)</td></tr><tr><td>ANP-multihead</td><td>16.36 (7.4)</td><td>17.17 (7.2)</td><td>19.41 (20.4)</td><td>16.02 (22.1)</td><td>0.692 (0.21)</td><td>16.68 (7.9)</td><td>17.43 (7.7)</td><td>19.78 (21.2)</td><td>15.57 (20.3)</td><td>0.682 (0.21)</td></tr><tr><td>Ours (MLP)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SP-latent</td><td>25.58 (10.1)</td><td>26.57* (9.0)</td><td>91.07 (23.9)</td><td>97.09 (22.5)</td><td>0.638 (0.08)</td><td>25.27 (10.0)</td><td>26.33* (8.9)</td><td>91.14 (23.8)</td><td>97.09 (22.5)</td><td>0.640 (0.09)</td></tr><tr><td>SP-latent+det</td><td>31.99 (8.2)</td><td>36.33 (7.3)</td><td>91.08 (23.9)</td><td>91.36 (23.9)</td><td>0.629 (0.18)</td><td>32.93 (9.4)</td><td>37.16 (8.5)</td><td>91.15 (23.9)</td><td>91.36 (23.9)</td><td>0.633 (0.18)</td></tr><tr><td>ASP-dot</td><td>27.16 (7.7)</td><td>31.19 (7.1)</td><td>90.88 (23.9)</td><td>91.43 (23.8)</td><td>0.704 (0.19)</td><td>27.94 (7.8)</td><td>31.83 (7.1)</td><td>90.93 (23.9)</td><td>91.43 (23.8)</td><td>0.628 (0.20)</td></tr><tr><td>ASP-multihead</td><td>23.88* (7.8)</td><td>27.13 (7.7)</td><td>90.50* (23.9)</td><td>91.04* (24.1)</td><td>0.792* (0.24)</td><td>24.07* (8.1)</td><td>27.35 (8.3)</td><td>90.53* (23.9)</td><td>91.07* (24.1)</td><td>0.770* (0.25)</td></tr><tr><td>Ours (GRU)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SP-latent</td><td>17.18 (6.5)</td><td>17.41 (6.2)</td><td>17.76* (15.8)</td><td>14.78* (20.7)</td><td>0.713 (0.23)</td><td>16.66 (6.2)</td><td>17.17 (6.0)</td><td>17.67* (16.0)</td><td>14.64* (20.3)</td><td>0.705 (0.23)</td></tr><tr><td>SP-latent+det</td><td>15.84 (5.5)</td><td>17.76 (7.5)</td><td>20.65 (19.9)</td><td>21.73 (29.5)</td><td>0.671 (0.22)</td><td>16.53* (6.0)</td><td>18.20 (8.0)</td><td>20.74 (19.5)</td><td>21.31 (28.9)</td><td>0.674 (0.22)</td></tr><tr><td>ASP-dot</td><td>22.49 (8.7)</td><td>22.64 (11.1)</td><td>17.99 (12.8)</td><td>15.58 (19.6)</td><td>0.722 (0.25)</td><td>23.66 (8.7)</td><td>24.50 (11.7)</td><td>19.22 (14.8)</td><td>16.82 (19.4)</td><td>0.620 (0.27)</td></tr><tr><td>ASP-multihead</td><td>15.18* (6.7)</td><td>15.01* (6.0)</td><td>24.26 (21.3)</td><td>35.06 (38.5)</td><td>0.778* (0.23)</td><td>16.84 (6.9)</td><td>16.80* (6.3)</td><td>25.37 (21.3)</td><td>35.44 (38.0)</td><td>0.725*(0.23)</td></tr></table>
+
+# 328 7 Discussion and Conclusion
+
+What qualifies as the best performing model for SCF? Our SP-GRU learns the best fit for synthesized behavior. On the commonly used metric of NLL [19, 58, 62], our SP-MLP models perform the best for real-world data. However, they fare the worst at estimating the mean. On the other hand, the SP-GRU models estimate a better likelihood than the NP baselines with comparable errors in mean forecast. While the NP baselines attain the lowest errors in predicted means, they also achieve the worst NLL. From the qualitative visualizations and ablations, it seems that the models minimize NLL at the cost of orientation errors; in the case of SP-MLP seemingly by predicting the majority orientation of the two sellers who face the same direction. Also, the NP models forecast largely static futures. In contrast, while being more dynamic, the SP-GRU forecasts also contain some smoothing.
+
+338 Our synthesized glancing behavior is grounded in social literature, and matches the head pose features   
+339 in the real-world data (horizontal orientation). Why do we see a large discrepancy in qualitative   
+340 forecasts? One crucial distinction between the synthetic and real data is the subtlety and sparsity   
+341 of motion. Our synthesized data makes the common implicit assumption that head pose is a proxy   
+342 for gaze [31, 67, 68, 70–72]. In real-world data, attention shifts through changes in gaze are not   
+343 always accompanied by similar head rotations [73, Fig. 5], and gaze is harder to record non-invasively   
+344 in-the-wild with reasonable accuracy. The consequence of this approximation is exacerbated in the   
+345 triadic Haggling setting where people are arranged roughly in a triangle and within each other’s   
+346 field of vision, making head movements even more subtle. In natural settings, groups occupy varied   
+347 formations such as side-by-side, or $L$ -arrangement [60, p. 213]. Here the more accentuated pose   
+348 changes could aid in anticipating behavior. From this perspective, the combination of limited data and   
+349 our simplifying assumption of a single group in a scene is a primary limitation of this work. The only   
+350 publicly available dataset meeting our assumptions is the Haggling dataset, where all interactions   
+351 follow similar patterns. As targeted development of techniques for recording such datasets in-the  
+352 wild gain momentum [74], evaluating these models in the different interaction settings would yield   
+353 increased insight. Nevertheless, our aim in evaluating on synthesized as well as real-world data   
+354 was to highlight the influence that such common implicit assumptions can have on performance   
+355 when applying methods. As an aside, we believe that this subtlety and sparsity of motion is also an   
+356 important distinction between forecasting in focused and unfocused interactions. While the same   
+357 techniques can be applied in both scenarios, pedestrian location is a perpetually changing data stream.   
+358 The broader goal of this paper is to take a step towards bridging a gap we perceive between research   
+359 domains; on one hand, we notice that there is a growing trend of applying deep learning techniques   
+360 in the small data regime that is social behavior data [30, 75]. Without citing specific works as   
+361 negative exemplars, this is occasionally accompanied by surface treatment of social science literature.   
+362 On the other hand, in our conversations we have also perceived a preemptive resistance to deep   
+363 learning methods precisely due to limited data. We believe that our work here—specifically our   
+364 conceptualization of conversations groups as meta-learning tasks grounded in extensive considerations   
+365 from social literature; our approach of learning extractable task-agnostic representations of predictive   
+366 behavior; and the distinction between real-world and synthesized dynamics commonly used for   
+367 evaluation—is of value in stimulating a broader community discussion about the considerations when   
+368 applying machine learning approaches within the domain of free-standing social conversations.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We mention the contributions of the task formulation and method in Section 1, along with our simplifying assumption for the setting, and circle back to discussing its implications in Section 7.   
+(b) Did you describe the limitations of your work? [Yes] Please refer to the discussion in Section 7.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code, processed data, splits, and test batches are provided for reproduction in the Supplementary material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Please refer to Appendices C and D.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We have reported mean and std. for the metrics over individual sequences in the test sets.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Please refer to Appendix C.2. We specify the specific GPUs used for our experiments and their corresponding memory capacities.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We have cited and discussed the original work proposing the Haggling dataset.   
+(b) Did you mention the license of the assets? [No] The Haggling dataset is freely available for non-commercial and research purpose only.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] The Haggling dataset is freely available for non-commercial and research purpose only.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] However, we visualized the features in Blender3D from scratch to not display original videos with the subjects.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "Social Processes: Self-Supervised Forecasting of Nonverbal Cues in Social Conversations ",
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+        "text": "The default paradigm for the forecasting of human behavior in social conversations is characterized by top-down approaches. These involve identifying predictive relationships between low level nonverbal cues and future semantic events of interest (e.g. turn changes, group leaving). A common hurdle however, is the limited availability of labeled data for supervised learning. In this work, we take the first step in the direction of a bottom-up self-supervised approach in the domain. We formulate the task of Social Cue Forecasting to leverage the larger amount of unlabeled low-level behavior cues, and characterize the modeling challenges involved. To address these, we take a meta-learning approach and propose the Social Process (SP) models—socially aware sequence-to-sequence (Seq2Seq) models within the Neural Process (NP) family. SP models learn extractable representations of non-semantic future cues for each participant, while capturing global uncertainty by jointly reasoning about the future for all members of the group. Evaluation on synthesized and real-world behavior data shows that our SP models achieve higher log-likelihood than the NP baselines, and also highlights important considerations for applying such techniques within the domain of social human interactions. ",
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+        "text": "17 1 Introduction ",
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+        "text": "18 Picture a situated interactive agent such as a social robot conversing with a group of people. How   \n19 can agents act in such a setting? We sustain conversations spatially and temporally through explicit   \n20 behavioral cues—examples include locations of partners, their orientation, gestures, gaze, and floor   \n21 control actions [1–3]. Evidence suggests that we employ an anticipation of these and other cues to   \n22 navigate daily social interactions [1, 4–8]. Consequently, the ability to forecast the future constitutes   \n23 a natural objective towards the realization of machines with social skills. As such, interactive agents   \n24 typically contend with uncertainties in inferences surrounding cues [3]. So beyond making real-time   \n25 inferences, such systems may achieve more fluid interactions by leveraging the ability to forecast   \n26 future states of the conversation [9].   \n27 In addition to the development of social agents, behavior forecasting is also of significance in social   \n28 psychology, where the focus is on gaining insight into human behavior. Since human-interpretability   \n29 is of essence, top-down approaches largely constitute the default paradigm, where specific events of   \n30 semantic interest are selected first for consideration and their relationship to potentially predictive cues   \n31 are studied in isolation—either in controlled interactions in lab settings, or in subsequent statistical   \n32 analyses [10, 11]. Examples of such semantic events include speaker turn transitions [5, 12, 13],   \n33 mimicry episodes [14], or the termination of an interaction [9, 15]. However, one hurdle in the   \n34 top-down paradigm is limited data. The events (that constitute the labels or the dependent variables)   \n35 often occur infrequently over a longer interaction, reducing the effective amount of labeled data. This   \n36 precludes the use of neural supervised learning techniques that tend to be data intensive.   \n37 In this work, we take an initial step towards a bottom-up approach to forecasting human behavior for   \n38 free standing conversational groups. Our guiding motivation is to learn predictive representations of   \n39 general future social behavior by utilizing unlabeled streams of low-level behavioral features. We do   \n40 this by regressing future sequences of these features from observed sequences of the same features in   \n41 a self-supervised manner. We term this task of non-semantic future behavior forecasting as Social   \n42 Cue Forecasting (SCF).   \n43 Our approach is built on the observation that the social signal [17]—the high-level attitudes and social   \n44 meaning transferred in interactions—is already embedded in the low-level cues [18]. To conceptually   \n45 illustrate the contrasting top-down and bottom-up approaches on an example task, Figure 1 depicts   \n46 an instance of a group leaving event in a naturalistic social conversation. Evidence suggests that   \n47 such events can be anticipated from certain preceding rituals [15] reflected in the postural changes of   \n48 conversing members [1]. van Doorn [15] built a predictor using 200 instances of group leaving found   \n49 in over 90 minutes of mingling interaction and hand-crafted features. In contrast, our bottom-up   \n50 approach would entail learning task agnostic representations of future behavior using the entire 90   \n51 minutes of data, and then training simpler predictors for group leaving using the learnt representations   \n52 as input. The figure also illustrates the complexity of naturalistic interactions where cross-group   \n53 social influence exists. In this work we focus on the simpler setting of a single group in a scene.   \n54 There are several challenges intrinsic to computationally modeling future behavior in social conversa  \n55 tions. The future is intrinsically uncertain, the forecasts for interaction partners are inter-dependent,   \n56 and the social dynamics is unique for each grouping of individuals. We address these through the   \n57 following contributions: ",
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+            "Figure 1: Conceptual illustration of forecasting approaches on an in-the-wild conversation from the MatchNMingle dataset [16]. a. The top-down approach entails predicting a semantic event or action of interest for the observed window $\\mathbf { \\mathit { t } _ { \\mathrm { o b s } } } : = [ o \\bar { 1 } \\dots o T ]$ . Here we illustrate group leaving [15]; the circled individual in the center leaves a group in the future. b. In contrast, we propose a bottom-up approach in the social conversation forecasting domain through the task of Social Cue Forecasting. This entails using the non-semantic low-level cues over $\\mathbf { \\Delta } \\mathbf { \\mathcal { t } } _ { \\mathrm { o b s } }$ to regress the same cues over the future window $\\pmb { t } _ { \\mathrm { f u t } } : = [ f 1 \\ldots f { \\bar { T } } ]$ . In this example we depict the cues of head pose (solid normal), body pose (hollow normal), and speaking status (speaker in orange). The hypothetical uncertainty estimates over $\\scriptstyle t _ { \\mathrm { f u t } }$ are also depicted as shaded spreads. "
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+        "text": "• We formalize the task of SCF. We characertize the modeling challenges involved, and cast the problem into the meta-learning paradigm, allowing for data-efficient generalization to unseen groups at evaluation without learning group-specific models. We propose and evaluate two socially aware Sequence-to-Sequence (Seq2Seq) models within the Neural Process (NP) family [19] for SCF in social conversations. Our method encodes complex social dynamics informative of future group behavior into extractable representations for each individual. ",
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+        "text": "65 This paper is organized as follows. In Section 2 we formally define and characterize the task of   \n66 SCF. We situate this work within broader literature in Section 3, and review background concepts   \n67 in Section 4. We propose the Social Process models in Section 5 and describe our experiments in   \n68 Section 6, concluding with a discussion of our findings in Section 7. ",
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+        "type": "text",
+        "text": "2 Social Cue Forecasting ",
+        "text_level": 1,
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+        "text": "70 The objective of SCF is to predict future behavioral cues of all people involved in a social encounter   \n71 given an observed sequence of their behavioral features. More formally, let us denote a window   \n72 of observed timesteps as $\\begin{array} { r c l } { t _ { \\mathrm { o b s } } } & { : = } & { [ o 1 , o 2 , . . . , o T ] } \\end{array}$ , and an unobserved future time window as   \n73 $\\pmb { t } _ { \\mathrm { f u t } } : = [ f 1 , f 2 , . . . , \\bar { f } T ]$ , $f 1 > o T$ . Note that $\\mathbf { \\Delta } t _ { \\mathrm { f u t } }$ and $\\pmb { t } _ { \\mathrm { o b s } }$ are typically non-overlapping, can be of   \n74 different lengths, and $\\mathbf { \\Delta } \\mathbf { \\mathbf { t } } _ { \\mathrm { f u t } }$ need not immediately follow $\\mathbf { \\Delta } \\mathbf { t } _ { \\mathrm { o b s } }$ . Given a set of $n$ interacting participants,   \n75 let us denote their social cues over a $t _ { \\mathrm { o b s } }$ and $\\mathbf { \\Delta } \\mathbf { \\mathbf { t } } _ { \\mathrm { f u t } }$ respectively as ",
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+        "img_path": "images/029dce264f5981a194ed2b6893a4727efec475bc408df8af2d567019b5974f66.jpg",
+        "text": "$$\nX : = [ b _ { t } ^ { i } ; t \\in t _ { \\mathrm { o b s } } ] _ { i = 1 } ^ { n } , \\quad Y : = [ b _ { t } ^ { i } ; t \\in t _ { \\mathrm { f u t } } ] _ { i = 1 } ^ { n } .\n$$",
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+        "text": "The vector 76 $ { b _ { t } ^ { i } }$ encapsulates the multimodal cues of interest from participant $i$ at time $t$ . These can 77 include head and body pose, speaking status, facial expressions, gestures, and verbal content—any 78 information stream that combine to transfer social meaning. ",
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+        "text": "79 In its simplest form, given an $\\boldsymbol { X }$ , the objective of SCF is to learn a single function $f$ such that   \n80 $Y = f ( X )$ . However, an inherent challenge in forecasting behavior is that an observed sequence   \n81 of interaction does not have a deterministic future and can result in multiple socially valid ones—a   \n82 window of overlapping speech between people both may and may not result in a change of speaker   \n83 [12, 20], a change in head orientation may continue into a sweeping glance across the room or a darting   \n84 glance stopping at a recipient of interest [21]. In some cases certain observed behaviors—intonation   \n85 and gaze cues [5, 13] or synchronization in speaker-listener speech [22] for turn-taking—might   \n86 make some outcomes more likely than others. Given that there are both supporting and challenging   \n87 arguments for how these observations influence subsequent behaviors [22, p. 5; 13, p. 22], it would   \n88 be beneficial if a data-driven model expresses a measure of uncertainty in its forecasts. We do this by   \n89 modeling the distribution over possible futures $p ( \\mathbf { { Y } } | \\boldsymbol { { X } } )$ rather than forecasting a single future.   \n90 Another design consideration arises from a defining characteristic of focused interactions—the   \n91 participants’ behaviors are interdependent. Participants in a group sustain equal access to the shared   \n92 interaction space through cooperative maneuvering [1, p. 220]. Moreover, when multiple groups   \n93 are co-located, outsiders unengaged in these intra-group maneuvers may also influence the behavior   \n94 of those within the group [23, p. 91;1, p. 233], sometimes causing them to leave (see Figure 1). It   \n95 is therefore essential to capture uncertainty in forecasts at the global level—jointly forecasting one   \n96 future for all participants at a time, rather than at a local output level—one future for each individual   \n97 independent of the remaining participants’ futures.   \n98 How participants coordinate their behaviors is a function of several individual factors [24, Chap. 1; 1,   \n99 p. 237]. Consequently, the social dynamics guiding an interaction also has unique attributes for every   \n100 unique grouping of individuals. Rather than learning group-specific models to capture these unique   \n101 dynamics, we formulate the forecasting problem in terms of meta-learning, or few-shot function   \n102 estimation. We interpret each unique group of individuals as the meta-learning notion of a task. The   \n103 core idea is that we can learn to predict a distribution over futures for a target sequence $\\boldsymbol { X }$ having   \n104 captured the group’s unique behavioral tendencies from a context set $C$ of their observed-future   \n105 sequences. We can then generalize to unseen groups at evaluation by conditioning on a short observed   \n106 slice of their interaction. We believe that this approach is especially suitable for social conversation   \n107 forecasting—a setting that involves a limited data regime where good uncertainty estimates are   \n108 desirable. Note that when conditioning on context is removed $C = \\varnothing$ ), we simply revert to the   \n109 formulation $p ( \\mathbf { { Y } } | \\boldsymbol { { X } } )$ . ",
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+        "type": "text",
+        "text": "110 3 Related Work ",
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+        "text": "111 Free-standing conversations are an example of what social scientists call focused interactions, said to   \n112 arise when a “group of persons gather close together and openly cooperate to sustain a single focus   \n113 of attention, typically by taking turns at talking” [23, p. 24]. A long-standing topic of study has been   \n114 the systematic organization of turn-taking [25–27], with a particular interest in the event of upcoming   \n115 speaking turns [5–8]. There has also been some interest in the forecasting task itself, to anticipate   \n116 disengagement from an interaction [9, 15], the splitting or merging of groups [28], the time-evolving   \n117 size of a group [29] or semantic social action labels [30, 31]. Most of these works use heuristics,   \n118 either to generate semantic labels [9], model the dynamics itself [29], or hand-craft features [15].   \n119 Although not a forecasting task, the closest work that shares our motivation in predicting non-semantic   \n120 low-level features is the recently introduced task of Social Signal Prediction (SSP) [32]. The objective   \n121 is to predict the social cues1 of a target person using cues from the communication partners as   \n122 input (Joo et al. focus on predictions within the same time window [32, Eq. 6]). While the most   \n123 general formulation of SSP involves forecasting a single timestep for a target person given the   \n124 entire group’s past behavior [32, Eq. 3], generalizing this formulation runs into an inherent problem;   \n125 applying the definition to forecasting entails iteratively treating each individual as target, learning   \n126 separate functions for every person. However, as we discuss in Section 2, these futures of interacting   \n127 individuals are not independent given observed group behavior. Furthermore, a constrained definition   \n128 of forecasting that predicts an immediate step into the future is limiting, since forecasting an event   \n129 that occurs after a delay (e.g. a time lagged synchrony [33] or mimicry [14] episode) might be of   \n130 interest. Operationalizing this definition would entail a sliding window iteratively using predictions   \n131 over the offset between $t _ { \\mathrm { o b s } }$ and $\\pmb { t } _ { \\mathrm { f u t } }$ as input, which would cascade prediction errors.   \n132 A related social setting where forecasting has been of interest is that of unfocused interactions. These   \n133 occur when individuals find themselves by circumstance in the immediate presence of each other,   \n134 such as pedestrians walking in proximity. Early approaches for forecasting pedestrian trajectories   \n135 were heuristic based, involving hand-crafted energy potentials to describe the influence pedestrians   \n136 have on each other [34–41]. More recent approaches encode the relative positional information   \n137 directly into a neural architecture [42–46].   \n138 In a broad sense, the self-supervised learning aspects of this work has some overlap with recent   \n139 approaches focusing on the non-interaction task of visual forecasting. These works have taken a   \n140 non-semantic approach to predict low level pixel-based features or intermediate representations   \n141 [38, 47–52], and demonstrated a utility of the learned representation for other tasks like semi  \n142 supervised classification [53], or training agents in immersive environments [54]. ",
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+        "text": "143 4 Preliminaries ",
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+        "text": "144 Meta-learning. A supervised learning algorithm can be viewed as a function mapping a dataset   \n145 $C : = ( X _ { C } , \\bar { Y _ { C } } ) : = \\{ ( \\bar { { \\bf x } } _ { i } , { \\bf y } _ { i } ) \\} _ { i \\in [ N _ { C } ] }$ to a predictor $f ( { \\pmb x } )$ . Here $N _ { C }$ is the number of datapoints in   \n146 $C$ , and $[ N _ { C } ] : = \\{ 1 , \\dots , N _ { C } \\}$ . The key idea of meta-learning is to learn the learning process itself,   \n147 modeling this function representing the initial algorithm using another supervised learning algorithm;   \n148 hence the name meta-learning. In meta-learning literature, a task refers to each dataset in a collection   \n149 $\\mathcal { M } : = \\{ \\mathcal { T } _ { i } \\} _ { i = 1 } ^ { N _ { \\mathrm { t a s k s } } }$ of related datasets [55]. For each task $\\tau$ , a meta-learner is episodically trained   \n150 to fit a subset of target points $D : = ( X , Y ) : = \\{ ( { \\pmb x } _ { i } , { \\pmb y } _ { i } ) \\} _ { i \\in [ N _ { D } ] }$ given another subset of context   \n151 observations $C$ . At meta-test time, the resulting predictor $f ( \\pmb { x } , C )$ uses the information obtained   \n152 during meta-learning to make predictions for unseen target points conditioned on context sets unseen   \n153 at meta-training.   \n154 Neural Processes Sharing the same core motivations, NPs are a family of latent variable models   \n155 that extend the idea of meta-learning to situations where uncertainty in the predictions $f ( \\pmb { x } , C )$ are   \n156 desirable. They do this by meta-learning a map from datasets to stochastic processes, estimating a   \n157 distribution over the predictions $p ( { \\boldsymbol { Y } } | { \\boldsymbol { X } } , { \\boldsymbol { C } } )$ . To capture this distribution, NPs model the conditional   \n158 latent distribution $p ( z | C )$ from which a task representation $z \\in \\mathbb { R } ^ { d }$ is sampled. This constitutes   \n159 the model’s latent path. The context can also be incorporated through a deterministic path, via a   \n160 representation $r _ { C } \\in \\mathbb { R } ^ { d }$ aggregated over $C$ . An observation model $p ( \\pmb { y } _ { i } | \\pmb { x } _ { i } , \\pmb { r } _ { C } , \\pmb { z } )$ then fits the target   \n161 observations in $D$ . The generative process for the NP is written as ",
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+        "img_path": "images/d84f7e3c6432e79fa8e4435631182ad50a5f55a377b3dc94b12b60535252273f.jpg",
+        "text": "$$\np ( { \\pmb Y } | { \\pmb X } , { \\pmb C } ) : = \\int p ( { \\pmb Y } | { \\pmb X } , { \\pmb C } , { \\pmb z } ) p ( { \\pmb z } | { \\pmb C } ) d { \\pmb z } = \\int p ( { \\pmb Y } | { \\pmb X } , { \\pmb r } _ { C } , { \\pmb z } ) q ( { \\pmb z } | s _ { C } ) d { \\pmb z } ,\n$$",
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+        "text": "162 where $\\begin{array} { r } { p ( { \\pmb Y } | { \\pmb X } , { \\pmb r } _ { C } , z ) : = \\prod _ { i \\in [ N _ { D } ] } p ( { \\pmb y } _ { i } | { \\pmb x } _ { i } , { \\pmb r } _ { C } , z ) } \\end{array}$ . The latent $_ z$ is modeled by a factorized Gaussian   \n163 parameterized by $\\bullet _ { C } : = f _ { s } ( C )$ , with $f _ { s }$ being a deterministic function invariant to order permutation   \n164 over $C$ . When the conditioning on context is removed $C = \\varnothing$ ), we have $q ( z | s _ { \\emptyset } ) : \\bar { = } p ( z )$ , the   \n165 zero-information prior on $_ { z }$ . $C$ is encoded on the deterministic path using a function $f _ { r }$ similar to   \n166 $f _ { s }$ , so that $\\pmb { r } _ { C } : = f _ { r } ( C )$ . In practice this is implemented as $\\textstyle { \\pmb r } _ { C } \\doteq \\sum _ { i \\in [ N _ { C } ] } \\operatorname { M L P } ( { \\pmb x } _ { i } , { \\pmb y } _ { i } ) / N _ { C }$ . The   \n167 observation model is referred to as the decoder, and $q , f _ { r } , f _ { s }$ comprise the encoders. The parameters   \n168 of the NP are learned for random subsets $C$ and $D$ by maximizing the evidence lower bound (ELBO) ",
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+            "Figure 2: Architecture of the SP and ASP family. "
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+        "text": "169 5 Social Processes ",
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+        "text": "170 In this section we present our socially aware Seq2Seq models within the NP family that is agnostic to   \n171 group member identities and group size. To setup the task, we split the contextual interaction on which   \n172 we condition into pairs of observed and future sequences, writing the context as $C : = ( X _ { C } , Y _ { C } ) : =$   \n173 $( X _ { j } , Y _ { k } ) _ { ( j , k ) \\in [ N _ { C } ] \\times [ N _ { C } ] }$ , where every $X _ { j }$ occurs before the corresponding $Y _ { k }$ . As discussed in   \n174 Section 3, domain experts focusing on behavior analysis might be interested in settings where $t _ { \\mathrm { o b s } }$   \n175 and $\\mathbf { \\Delta } t _ { \\mathrm { f u t } }$ are offset by an arbitrary delay. Consequently, the $j$ th $t _ { \\mathrm { o b s } }$ can have multiple associated $\\mathbf { \\Delta } \\mathbf { \\mathbf { { t } } } _ { \\mathrm { { f u t } } }$   \n176 windows. Denoting the set of target window pairs as $D : = ( X , Y ) : = ( X _ { j } , Y _ { k } ) _ { ( j , k ) \\in [ N _ { D } ] \\times [ N _ { D } ] }$ , our   \n177 focus in the rest of this work is to model the distribution $p ( { \\boldsymbol { Y } } | { \\boldsymbol { X } } , { \\boldsymbol { C } } )$ .   \n178 The generative process for our model we call the Social Process (SP) follows Eq. 2, which we   \n179 extend to social forecasting in two ways. We embed an observed sequence $_ { \\textbf { \\em x } }$ for an individual   \n180 into a condensed encoding $\\mathbf { \\bar { \\boldsymbol { e } } } \\in \\mathbb { R } ^ { d }$ that is then decoded into the future sequence using a Seq2Seq   \n181 architecture [56, 57]. Our intuition is that this would cause the representation to encode temporal   \n182 information about the future. Further, for every individual we model this $e$ as a function of their own   \n183 behavior, and that of their partners as viewed by them. The intuition is that this captures the spatial   \n184 influence partners have on the participant over the $t _ { \\mathrm { o b s } }$ . Using notation we established in Section 2,   \n185 we define the observation model for the SP for a single participant $\\mathrm { p } _ { i }$ as ",
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+        "img_path": "images/223d2bb647db808d0ab35622f95633fb70c8118da9ace3c1febf1c7cb7d99a84.jpg",
+        "text": "$$\np ( \\pmb { y } ^ { i } | \\pmb { x } ^ { i } , C , z ) : = p ( \\pmb { b } _ { f 1 } ^ { i } , \\ldots , \\pmb { b } _ { f T } ^ { i } | \\pmb { b } _ { o 1 } ^ { i } , \\ldots , \\pmb { b } _ { o T } ^ { i } , C , z ) = p ( \\pmb { b } _ { f 1 } ^ { i } , \\ldots , \\pmb { b } _ { f T } ^ { i } | \\pmb { e } ^ { i } , \\pmb { r } _ { C } , z ) .\n$$",
+        "text_format": "latex",
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+        "text": "186 If decoding is carried out in an auto-regressive manner, we can further write the right hand side of   \n187 Eq. 4 as $\\begin{array} { r } { \\dot { \\prod _ { t = f 1 } ^ { f T } } p ( b _ { t } ^ { i } | b _ { t - 1 } ^ { i } , \\dots , b _ { f 1 } ^ { i } , e ^ { i } , r _ { C } , z ) } \\end{array}$ . Following the standard NP setting, we implement the   \n188 observation model as a set of Gaussian distributions factorized over time and feature dimensions.   \n189 We also incorporate the cross-attention mechanism from the Attentive Neural Process (ANP) [58] to   \n190 define the variant Attentive Social Process (ASP). Following Eq. 4 and the definition of the ANP, the   \n191 corresponding observation model of the ASP for a single participant is defined as ",
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+        "text": "$$\np ( \\pmb { y } ^ { i } | \\pmb { x } ^ { i } , C , z ) = p ( b _ { f 1 } ^ { i } , \\ldots , b _ { f T } ^ { i } | \\pmb { e } ^ { i } , r ^ { * } ( C , \\pmb { x } ^ { i } ) , z ) .\n$$",
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+        "text": "92 Here each target query sequence $\\pmb { x } _ { \\ast } ^ { i }$ attends to the context sequences $X _ { C }$ to produce a query-specific   \n93 representation $r _ { * } : = r ^ { * } ( C , \\pmb { x } _ { * } ^ { i } ) \\in \\mathbb { R } ^ { d }$ . The model architectures are illustrated in Figure 2.   \n194 Encoding Partner Behavior. While a typical Seq2Seq setup conditions the sequence decoder on   \n195 solely a compact representation of the observed sequence, we’d like to condition an individual’s   \n196 forecast on the observed behavior of both, themselves and their partners. We do this using a pair   \n197 of sequence encoders: one to encode the temporal dynamics of participant $\\mathrm { p } _ { i }$ ’s features, $\\bar { e } _ { \\mathrm { s e l f } } ^ { i } =$   \n198 $f _ { \\mathrm { s e l f } } ( \\mathbf { \\bar { x } } _ { i } )$ , and another to encode the dynamics of a transformed representation of the features of $\\mathrm { p } _ { i }$ ’s   \n199 partners, $e _ { \\mathrm { p a r t n e r } } ^ { i } = f _ { \\mathrm { p a r t n e r } } ( \\psi ( \\pmb { x } _ { j , ( j \\neq i ) } ) )$ . Using a separate network to encode partner behavior   \n200 grants the practical advantage of being able to sample an individual’s and partners’ features at different   \n201 sampling rates.   \n202 How do we model $\\psi ( \\pmb { x } _ { j } )$ ? We want the partners’ representation to possess two properties: per  \n203 mutation invariance—changing the order of the partners should not affect the representation; and   \n204 group size independence—we want to compactly represent all partners independent of the group size.   \n205 Beyond coordinate space invariance, we wish to intuitively capture a view of the interaction from   \n206 $\\mathrm { p } _ { i }$ ’s perspective. We extend the approach Qi et al. [59] applied to point clouds to focused interactions   \n207 by computing pooled embeddings of relative behavioral features. Since most commonly considered   \n208 nonverbal cues in literature (see Section 6.3) include the attributes of orientation or location (e.g.   \n209 head/body pose or keypoints) or a binary indicator (such as speaking status), we specify how we   \n210 transform these. The 3D pose (orientation, location) of every partner $\\mathrm { p } _ { j }$ is transformed to a frame of   \n211 reference defined by $\\mathrm { p } _ { i }$ ’s pose. At timestep $t$ , denoting orientation, location, and binary speaking   \n212 status for $\\mathrm { p } _ { i }$ as $b _ { t } ^ { i } = [ \\mathbf { q } ^ { i } ; \\mathbf { l } ^ { i } ; \\mathbf { s } ^ { i } ]$ , and those for $\\mathrm { p } _ { j }$ as $\\mathbf { } b _ { t } ^ { j } = [ \\mathbf { q } ^ { j } ; \\mathbf { l } ^ { j } ; \\mathbf { s } ^ { j } ]$ , we have ",
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+        "text": "213 Note that we use unit quaternions (denoted $\\mathbf { q }$ ) for representing orientation due their various benefits   \n214 over other representations of rotation [60, Sec. 3.2]. The operator $^ *$ denotes the Hamilton product of   \n215 the quaternions. These transformed features for each $\\mathrm { p } _ { j }$ are encoded using an embedder MLP. The   \n216 outputs are concatenated with $e _ { \\mathrm { s e l f } } ^ { j }$ and processed by a pre-pooler MLP, which is followed by the   \n217 symmetric element-wise Max-pooling function to obtain $\\psi ( \\pmb { x } ^ { j } )$ at each timestep. We capture the   \n218 dynamics in the pooled representation over $t _ { \\mathrm { o b s } }$ using $f _ { \\mathrm { p a r t n e r } }$ . Finally, we combine $e _ { \\mathrm { s e l f } } ^ { i }$ and $e _ { \\mathrm { p a r t n e r } } ^ { i }$   \n219 for $\\mathrm { p } _ { i }$ through a linear projection (defined by a weight matrix $W$ ) to obtain the individual’s embedding   \n220 $e _ { \\mathrm { i n d } } ^ { i } = W . [ \\bar { e } _ { \\mathrm { s e l f } } ^ { i } ; e _ { \\mathrm { p a r t n e r } } ^ { i } ]$ . Our intuition is that with information about both $\\mathrm { p } _ { i }$ themselves, and of   \n221 $\\mathrm { p } _ { i }$ ’s partners from $\\mathrm { p } _ { i }$ ’s point-of-view, $e _ { \\mathrm { i n d } } ^ { i }$ now contains the information required to predict $\\mathrm { p } _ { i }$ ’s   \n222 future behavior.   \n223 Encoding Future Window Offset. As we’ve discussed at the start of this section, a single $t _ { \\mathrm { o b s } }$   \n224 might have multiple associated $\\mathbf { \\Delta } \\mathbf { \\mathbf { \\mathit { t } } _ { \\mathrm { f u t } } }$ windows at different offsets. Our intuition is that training a   \n225 sequence decoder to decode the same $e _ { \\mathrm { i n d } } ^ { i }$ into multiple sequences (corresponding to the multiple   \n226 $\\pmb { t } _ { \\mathrm { f u t } } )$ ) in the absence of any timing information might cause an averaging effect in either the decoder   \n227 or the information encoded in $e _ { \\mathrm { i n d } } ^ { \\bar { i } }$ . One way around this would be to start decoding one timestep   \n228 following the end of $t _ { \\mathrm { o b s } }$ and discard the predictions in the gap between $t _ { \\mathrm { o b s } }$ and $\\mathbf { \\Delta } t _ { \\mathrm { f u t } }$ . However,   \n229 if decoding is done auto-regressively this might lead to cascading errors over the gap. Instead, we   \n230 address this one-to-many issue by injecting the offset information into $e _ { \\mathrm { i n d } } ^ { i }$ so that the decoder   \n231 receives a unique encoded representation for every $\\mathbf { \\Delta } \\mathbf { \\mathbf { t } } _ { \\mathrm { f u t } }$ to decode over. We do this by repurposing   \n232 the idea of sinusoidal positional encodings [61] to encode offsets rather than relative positions in   \n233 sequences. For a given $\\mathbf { \\Delta } \\mathbf { t } _ { \\mathrm { o b s } }$ and $\\mathbf { \\Delta } \\mathbf { \\mathbf { \\mathit { t } } _ { \\mathrm { f u t } } }$ , and $d _ { e }$ -dimensional $e _ { \\mathrm { i n d } } ^ { i }$ we define the offset as $\\Delta t = f 1 - o T$   \n234 and the corresponding offset encoding $O E _ { \\Delta t }$ as ",
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+        "type": "equation",
+        "img_path": "images/2c93382fc306bd6ab7ad882a8fa0f06f6ac2fb2b53a2e31577765348c8981849.jpg",
+        "text": "$$\nO E _ { ( \\Delta t , 2 m ) } = \\sin ( \\Delta t / 1 0 0 0 0 ^ { 2 m / d _ { c } } ) , \\quad O E _ { ( \\Delta t , 2 m + 1 ) } = \\cos ( \\Delta t / 1 0 0 0 0 ^ { 2 m / d _ { c } } ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here 235 $m$ refers to the dimension index in the encoding. We finally compute the representation $e ^ { i }$ for 236 Eqs. 4 and 5 as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d46e11bd1c887c68dbf1afa6f4da9008d3c5d822d4baf121868f79fdf8a4a75.jpg",
+        "text": "$$\ne ^ { i } = e _ { \\mathrm { i n d } } ^ { i } + O E _ { \\Delta t } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "237 Auxiliary Loss Functions. We incorporate a geometric loss function that improves performance in   \n238 pose regression tasks. For $\\mathrm { p } _ { i }$ at time $t$ , given the ground truth $b _ { t } ^ { i } = [ \\mathbf { q } ; 1 ; \\mathrm { s } ]$ , and the predicted mean   \n239 $\\hat { b } _ { t } ^ { i } = [ \\hat { \\mathbf { q } } ; \\hat { \\mathbf { l } } ; \\hat { \\mathbf { s } } ]$ , we denote the tuple $( b _ { t } ^ { i } , b _ { t } ^ { i } )$ as $B _ { t } ^ { i }$ . We then have the location loss in Eucliden space   \n240 $\\mathcal { L } _ { 1 } ( B _ { t } ^ { i } ) = \\left. \\mathbf { l } - \\hat { \\mathbf { l } } \\right.$ , and we can regress the quaternion values using ",
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+    {
+        "type": "equation",
+        "img_path": "images/cfacc2a1e892d60a11b919ecd2cc60858ba9d9bc19567dab34febe3a7fc45111.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { q } } ( B _ { t } ^ { i } ) = \\left\\| \\mathbf { q } - \\frac { \\hat { \\mathbf { q } } } { \\| \\hat { \\mathbf { q } } \\| } \\right\\| .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "241 Kendall and Cipolla [60] show how these losses can be combined using the homoscedastic uncertainties in position and orientation, 242 $\\hat { \\sigma } _ { 1 } ^ { 2 }$ and $\\hat { \\sigma } _ { \\mathrm { q } } ^ { 2 }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/24e56766d43f607c3286c4cff13446dba7875a79e71b9eefbaed4d1cd25676e4.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\sigma } ( B _ { t } ^ { i } ) = \\mathcal { L } _ { 1 } ( B _ { t } ^ { i } ) \\exp ( - \\hat { s } _ { 1 } ) + \\hat { s } _ { 1 } + \\mathcal { L } _ { \\boldsymbol { \\mathrm { q } } } ( B _ { t } ^ { i } ) \\exp ( - \\hat { s } _ { \\mathrm { q } } ) + \\hat { s } _ { \\boldsymbol { \\mathrm { q } } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where 243 $\\hat { s } : = \\log \\hat { \\sigma } ^ { 2 }$ . Using the binary cross-entropy loss for speaking status $\\mathcal { L } _ { \\mathrm { s } } ( B _ { t } ^ { i } )$ , we have the 244 overall auxiliary loss over $t \\in { \\pmb t } _ { \\mathrm { f u t } }$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/71a8159a4f0391f9d15a4c9b12c1fa22c66b3c86affed64255468d87d9ffdb9c.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { a u x } } ( \\boldsymbol { Y } , \\hat { \\boldsymbol { Y } } ) = \\sum _ { i } \\sum _ { t } \\mathcal { L } _ { \\sigma } ( B _ { t } ^ { i } ) + \\mathcal { L } _ { \\mathrm { s } } ( B _ { t } ^ { i } ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "245 The parameters of the SP and ASP are trained by maximizing the ELBO in Eq. 3 and minimizing this   \n246 auxiliary loss function for each of our sequence decoders. ",
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+    {
+        "type": "image",
+        "img_path": "images/42651c2b36add882c5b51ca88501608167a5fe63ce5ae59c255468d4faac0dd6.jpg",
+        "image_caption": [
+            "Figure 3: Ground truths and model predictions for the toy task simulating the forecasting of glancing behavior. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1dcd98ddd96a830df6688dbabda8f1455cfee50df5a755420d981e5b6a050217.jpg",
+        "table_caption": [
+            "Table 1: Mean (Std.) Negative LogLikelihood (NLL) on the Haggling Test Sets. The reported mean and std. are over individual sequences in the test sets. Lower is better. The superscript ⇤ indicates best NLL within family, boldface best overall. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Context</td></tr><tr><td>Random</td><td>Fixed-Initial</td></tr><tr><td>Baselines</td><td></td><td></td></tr><tr><td>NP-latent</td><td>38.34 (19.1)</td><td>37.64 (18.1)</td></tr><tr><td>NP-latent+det</td><td>40.41 (23.9)</td><td>40.15 (23.0)</td></tr><tr><td>ANP-dot</td><td>35.66* (20.8)</td><td>38.06* (20.6)</td></tr><tr><td>ANP-multihead</td><td>40.60 (19.2)</td><td>41.11 (19.2)</td></tr><tr><td>Ours (MLP)</td><td></td><td></td></tr><tr><td>SP-latent</td><td>-74.06 (6.0)</td><td>-74.19 (5.9)</td></tr><tr><td>SP-latent+det</td><td>-77.49 (7.8)</td><td>-76.90 (8.4)</td></tr><tr><td>ASP-dot</td><td>-76.33 (6.5)</td><td>-75.15 (6.5)</td></tr><tr><td>ASP-multihead</td><td>-83.77* (10.3)</td><td>-83.43* (9.7)</td></tr><tr><td>Ours (GRU)</td><td></td><td></td></tr><tr><td>SP-latent</td><td>-4.23 (27.4)</td><td>-3.72 (30.7)</td></tr><tr><td>SP-latent+det</td><td>-17.38* (50.5)</td><td>-16.08* (52.2)</td></tr><tr><td>ASP-dot</td><td>19.91 (46.7)</td><td>31.39 (77.0)</td></tr><tr><td>ASP-multihead</td><td>-7.11 (26.9)</td><td>-0.51 (28.8)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "247 6 Experiments and Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "6.1 Models and Baselines ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our modeling assumption is that the underlying stochastic process generating the behaviors does not evolve over time. Stated differently, we assume that the individual factors determining how participants coordinate behaviors—age, cultural background, personality variables [24, Chap. 1; 1, p. 237]—are likely to remain the same over the short duration of a single interaction. This is in contrast to a related line of work that deals with meta-transfer learning, where the stochastic process itself changes over time [62–65]. We therefore compare against the NP and ANP family which share our model assumptions and meta-learning attributes. Note that in contrast to our methods, these baselines have direct access to the future sequences in the context, and therefore constitute a strong baseline. We consider two variants: -latent denoting only the latent path; and -latent+det, containing both deterministic and stochastic paths. We further consider two attention mechanisms for the cross-attention module: -dot with dot attention, and -multihead with wide multi-head attention [58]. We operationalize the original definitions of the baseline models to sequences by collapsing the timestep and feature dimensions. While the ANP-RNN model [66] shares our model assumptions, it is defined for a task analogous to SSP for concurrent car locations, and cannot be operationalized to forecasting in any simple way (see Section 3 discussing the distinction). We experiment with two choices of architectures for the sequence encoders and decoders in our proposed models: multi-layer perceptrons (MLP), and Gated Recurrent Units (GRU). Implementation and training details for our experiments can be found in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "6.2 Evaluation on Synthesized Behavior: Forecasting Glancing Behavior ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With limited behavioral data availability, a common practice in the domain is to train and evaluate methods on synthesized behavior dynamics [31, 67]. In keeping with this practice, we construct a synthesized dataset simulating two glancing behaviors in social settings [21]. We use a 1D sinusoid to represent horizontal head rotation over 20 timesteps. The sweeping Type I glance is represented by a pristine sinusoid, while the gaze fixating Type III glance is denoted by clipping the amplitude for the last six timesteps. The task is to forecast the signal over the last 10 timesteps $( \\pmb { t } _ { \\mathrm { f u t } } )$ by observing the first 10 $( \\pmb { t } _ { \\mathrm { o b s } } )$ . Consequently, the first half of $\\mathbf { \\delta } _ { t _ { \\mathrm { f u t } } }$ is certain, while uncertainty over the last half results from every observed sinusoid having two ground-truths. It is impossible to infer from an observed sequence alone if the head rotation will stop partway through the future. We describe additional data setup, model details, and quantitative results for this setting in Appendices A.1, C and D.1, respectively. Figure 3 illustrates the ground truths, predicted means and std. deviations for a sequence within and outside the context set. We observe that all models estimate the mean reasonably well, although our proposed SP models learn a slightly better fit. More crucially, the SP models—especially the SP-GRU—learn much better uncertainty estimates over the certain and uncertain parts of the future compared to the NP baseline. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "6.3 Real-World Behavior: The Haggling Dataset ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We also evaluate our models on real-world behavior data, using the Haggling dataset of triadic interactions [32]. Participants are engaged in an unscripted game where two sellers compete to sell a fictional product to a buyer who has to choose between the two. We use the same split of 79 training sets (groups) and 28 test sets used by Joo et al. [32]. In our experiments we consider the following social cues: head pose described by the 3D location of the nose keypoint and a face normal; body pose described by the location of the mid-point of the shoulders and a body normal; and binary speaking status. Apart from being the most commonly considered cues in computational analyses of such conversations [68–70], pose and turn taking are found to be crucial in the sustaining of conversation [1, 12, 18]. We specify the dataset preprocessing details in Appendix D.2. ",
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+    {
+        "type": "text",
+        "text": "6.4 Evaluation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Context Regimes. We evaluate all models on two context regimes: random, and fixed-initial. The random regime follows the standard NP setting that the models are trained in. Context samples (sequence-pairs) are selected as a random subset of target samples, so the model is exposed to behaviors from any phase of the interaction lifecycle. Here we ensure that batches contain unique $t _ { \\mathrm { o b s } }$ to prevent any single observed sequence from dominating the aggregation of representations over the context split. At evaluation, we take $5 0 \\%$ of the batch as context. In the fixed-initial context regime, we investigate how the model can generalize knowledge of group specific characteristics from observing the initial dynamics of an interaction where certain gestures and patterns are more distinctive [1, Chap. 6]. This matches what a social agent might face in a real-world scenario. Here we treat the first $2 0 \\%$ of the entire interaction as context, treating sequences from the rest as target. ",
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+    {
+        "type": "text",
+        "text": "304 Evaluation Metrics. We report the negative log-likelihood (NLL) $- \\log p ( \\boldsymbol { Y } | \\boldsymbol { X } , \\boldsymbol { C } )$ in Table 1   \n305 (computed by summing over feature dimensions and people, and averaging over timesteps). Beyond   \n306 the NLL, we also report the error in the predicted means over test sequences in Table 2: mean-squared   \n307 error (MSE) for the head and body keypoint locations; mean absolute error (MAE) in orientation   \n308 in degrees; and speaking status accuracy. Note that while the ground truth orientation normals are   \n309 constrained in the horizontal plane, we don’t constrain our predicted quaternions. We therefore report   \n310 the absolute error in rotation in 3D. The reported mean and std. deviation of all metrics are over   \n311 sequences in the test sets. We further report the metrics for every timestep over $\\mathbf { \\Delta } \\mathbf { \\mathbf { \\mathit { t } } _ { \\mathrm { f u t } } }$ in Appendix A.2,   \n312 and qualitative visualizations of the forecasts in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "6.5 Ablations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Encoding Partner Behavior. Modeling the interaction from the perspective of each individual is a central idea in our apindividual representations $\\bar { r } _ { \\mathrm { i n d } } ^ { i }$ ch. We investigate the influence of encoding partner behavior intoon the performance. We train the SP-latent+det GRU variant in two configurations: no-pool, where we do not encode any partner behavior; and pool-oT where we pool over partner representations only at the last timestep (similar to [44]). We choose the SP-GRU model since it achieves the best trade-off between minimizing NLL and forecasting cues consistent with human behavior. Both configurations lead to worse NLL and location errors (Appendix A.3). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Deterministic Decoding and Social Encoder Sharing. Error gradients can flow back into our sequence encoders through two paths: from the final stochastic sequence decoder, as well as the deterministic decoders on the latent and deterministic paths. We investigate the effect of the deterministic decoders by training the SP-latent+det GRU model without them. We also investigate sharing a single social encoder between the Process Encoder and Process Decoder in Figure 2. We find that removing the decoders only improves log-likelihood if the encoders are shared, and at the cost of head orientation errors (Appendix A.3). ",
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+        "img_path": "images/4751f6ed151e375140d276bb813210a0d5794709136e6578725bfe5e8d6495c7.jpg",
+        "table_caption": [
+            "Table 2: Mean (Std.) Errors in Predicted Means over Sequences in the Haggling Test Sets. Lower is better for all metrics except for speaking status accuracy. ⇤ indicates best measure within family, boldface best overall. "
+        ],
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+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"5\">Random Context</td><td colspan=\"5\">Fixed-Initial Context</td></tr><tr><td>Head Loc. MSE (cm)</td><td>Body Loc. MSE (cm)</td><td>Head Ori. MAE()</td><td>Body Ori. MAE()</td><td>Speaking Accuracy</td><td>Head Loc. MSE (cm)</td><td>Body Loc. MSE (cm)</td><td>Head Ori. MAE(°)</td><td>Body Ori. MAE(°)</td><td>Speaking Accuracy</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>NP-latent</td><td>14.21 (6.5)</td><td>15.06 (6.1)</td><td>16.29 (13.8)</td><td>12.82 (13.7)</td><td>0.787 (0.23)</td><td>13.85 (6.1)</td><td>14.71 (5.7)</td><td>16.22 (14.1)</td><td>12.69* (13.9)</td><td>0.774* (0.24)</td></tr><tr><td>NP-latent+det</td><td>15.01 (7.3)</td><td>15.97 (7.2)</td><td>17.45 (18.3)</td><td>14.65 (20.0)</td><td>0.715 (0.24)</td><td>15.01 (7.5)</td><td>15.95 (7.5)</td><td>17.26 (15.9)</td><td>14.68 (18.7)</td><td>0.701 (0.24)</td></tr><tr><td>ANP-dot</td><td>11.86* (5.4)</td><td>12.22* (5.5)</td><td>15.44* (13.3)</td><td>12.56* (18.0)</td><td>0.806* (0.23)</td><td>12.83* (5.9)</td><td>13.26* (6.0)</td><td>16.19* (13.7)</td><td>13.56 (17.8)</td><td>0.717 (0.23)</td></tr><tr><td>ANP-multihead</td><td>16.36 (7.4)</td><td>17.17 (7.2)</td><td>19.41 (20.4)</td><td>16.02 (22.1)</td><td>0.692 (0.21)</td><td>16.68 (7.9)</td><td>17.43 (7.7)</td><td>19.78 (21.2)</td><td>15.57 (20.3)</td><td>0.682 (0.21)</td></tr><tr><td>Ours (MLP)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SP-latent</td><td>25.58 (10.1)</td><td>26.57* (9.0)</td><td>91.07 (23.9)</td><td>97.09 (22.5)</td><td>0.638 (0.08)</td><td>25.27 (10.0)</td><td>26.33* (8.9)</td><td>91.14 (23.8)</td><td>97.09 (22.5)</td><td>0.640 (0.09)</td></tr><tr><td>SP-latent+det</td><td>31.99 (8.2)</td><td>36.33 (7.3)</td><td>91.08 (23.9)</td><td>91.36 (23.9)</td><td>0.629 (0.18)</td><td>32.93 (9.4)</td><td>37.16 (8.5)</td><td>91.15 (23.9)</td><td>91.36 (23.9)</td><td>0.633 (0.18)</td></tr><tr><td>ASP-dot</td><td>27.16 (7.7)</td><td>31.19 (7.1)</td><td>90.88 (23.9)</td><td>91.43 (23.8)</td><td>0.704 (0.19)</td><td>27.94 (7.8)</td><td>31.83 (7.1)</td><td>90.93 (23.9)</td><td>91.43 (23.8)</td><td>0.628 (0.20)</td></tr><tr><td>ASP-multihead</td><td>23.88* (7.8)</td><td>27.13 (7.7)</td><td>90.50* (23.9)</td><td>91.04* (24.1)</td><td>0.792* (0.24)</td><td>24.07* (8.1)</td><td>27.35 (8.3)</td><td>90.53* (23.9)</td><td>91.07* (24.1)</td><td>0.770* (0.25)</td></tr><tr><td>Ours (GRU)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SP-latent</td><td>17.18 (6.5)</td><td>17.41 (6.2)</td><td>17.76* (15.8)</td><td>14.78* (20.7)</td><td>0.713 (0.23)</td><td>16.66 (6.2)</td><td>17.17 (6.0)</td><td>17.67* (16.0)</td><td>14.64* (20.3)</td><td>0.705 (0.23)</td></tr><tr><td>SP-latent+det</td><td>15.84 (5.5)</td><td>17.76 (7.5)</td><td>20.65 (19.9)</td><td>21.73 (29.5)</td><td>0.671 (0.22)</td><td>16.53* (6.0)</td><td>18.20 (8.0)</td><td>20.74 (19.5)</td><td>21.31 (28.9)</td><td>0.674 (0.22)</td></tr><tr><td>ASP-dot</td><td>22.49 (8.7)</td><td>22.64 (11.1)</td><td>17.99 (12.8)</td><td>15.58 (19.6)</td><td>0.722 (0.25)</td><td>23.66 (8.7)</td><td>24.50 (11.7)</td><td>19.22 (14.8)</td><td>16.82 (19.4)</td><td>0.620 (0.27)</td></tr><tr><td>ASP-multihead</td><td>15.18* (6.7)</td><td>15.01* (6.0)</td><td>24.26 (21.3)</td><td>35.06 (38.5)</td><td>0.778* (0.23)</td><td>16.84 (6.9)</td><td>16.80* (6.3)</td><td>25.37 (21.3)</td><td>35.44 (38.0)</td><td>0.725*(0.23)</td></tr></table>",
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+        "text": "328 7 Discussion and Conclusion ",
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+        "text": "What qualifies as the best performing model for SCF? Our SP-GRU learns the best fit for synthesized behavior. On the commonly used metric of NLL [19, 58, 62], our SP-MLP models perform the best for real-world data. However, they fare the worst at estimating the mean. On the other hand, the SP-GRU models estimate a better likelihood than the NP baselines with comparable errors in mean forecast. While the NP baselines attain the lowest errors in predicted means, they also achieve the worst NLL. From the qualitative visualizations and ablations, it seems that the models minimize NLL at the cost of orientation errors; in the case of SP-MLP seemingly by predicting the majority orientation of the two sellers who face the same direction. Also, the NP models forecast largely static futures. In contrast, while being more dynamic, the SP-GRU forecasts also contain some smoothing. ",
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+        "text": "338 Our synthesized glancing behavior is grounded in social literature, and matches the head pose features   \n339 in the real-world data (horizontal orientation). Why do we see a large discrepancy in qualitative   \n340 forecasts? One crucial distinction between the synthetic and real data is the subtlety and sparsity   \n341 of motion. Our synthesized data makes the common implicit assumption that head pose is a proxy   \n342 for gaze [31, 67, 68, 70–72]. In real-world data, attention shifts through changes in gaze are not   \n343 always accompanied by similar head rotations [73, Fig. 5], and gaze is harder to record non-invasively   \n344 in-the-wild with reasonable accuracy. The consequence of this approximation is exacerbated in the   \n345 triadic Haggling setting where people are arranged roughly in a triangle and within each other’s   \n346 field of vision, making head movements even more subtle. In natural settings, groups occupy varied   \n347 formations such as side-by-side, or $L$ -arrangement [60, p. 213]. Here the more accentuated pose   \n348 changes could aid in anticipating behavior. From this perspective, the combination of limited data and   \n349 our simplifying assumption of a single group in a scene is a primary limitation of this work. The only   \n350 publicly available dataset meeting our assumptions is the Haggling dataset, where all interactions   \n351 follow similar patterns. As targeted development of techniques for recording such datasets in-the  \n352 wild gain momentum [74], evaluating these models in the different interaction settings would yield   \n353 increased insight. Nevertheless, our aim in evaluating on synthesized as well as real-world data   \n354 was to highlight the influence that such common implicit assumptions can have on performance   \n355 when applying methods. As an aside, we believe that this subtlety and sparsity of motion is also an   \n356 important distinction between forecasting in focused and unfocused interactions. While the same   \n357 techniques can be applied in both scenarios, pedestrian location is a perpetually changing data stream.   \n358 The broader goal of this paper is to take a step towards bridging a gap we perceive between research   \n359 domains; on one hand, we notice that there is a growing trend of applying deep learning techniques   \n360 in the small data regime that is social behavior data [30, 75]. Without citing specific works as   \n361 negative exemplars, this is occasionally accompanied by surface treatment of social science literature.   \n362 On the other hand, in our conversations we have also perceived a preemptive resistance to deep   \n363 learning methods precisely due to limited data. We believe that our work here—specifically our   \n364 conceptualization of conversations groups as meta-learning tasks grounded in extensive considerations   \n365 from social literature; our approach of learning extractable task-agnostic representations of predictive   \n366 behavior; and the distinction between real-world and synthesized dynamics commonly used for   \n367 evaluation—is of value in stimulating a broader community discussion about the considerations when   \n368 applying machine learning approaches within the domain of free-standing social conversations. ",
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