diff --git a/parse/train/-TwO99rbVRu/-TwO99rbVRu.md b/parse/train/-TwO99rbVRu/-TwO99rbVRu.md new file mode 100644 index 0000000000000000000000000000000000000000..15ee7cf6ba04deb63baf4dc268897589b2604495 --- /dev/null +++ b/parse/train/-TwO99rbVRu/-TwO99rbVRu.md @@ -0,0 +1,331 @@ +# PSEUDOSEG: DESIGNING PSEUDO LABELS FOR SEMANTIC SEGMENTATION + +Yuliang $\mathbf { Z o u } ^ { 1 * }$ Zizhao Zhang2 Han Zhang3 Chun-Liang Li2 Xiao Bian2 Jia-Bin Huang1 Tomas Pfister2 1Virginia Tech 2Google Cloud AI 3Google Brain + +# ABSTRACT + +Recent advances in semi-supervised learning (SSL) demonstrate that a combination of consistency regularization and pseudo-labeling can effectively improve image classification accuracy in the low-data regime. Compared to classification, semantic segmentation tasks require much more intensive labeling costs. Thus, these tasks greatly benefit from data-efficient training methods. However, structured outputs in segmentation render particular difficulties (e.g., designing pseudo-labeling and augmentation) to apply existing SSL strategies. To address this problem, we present a simple and novel re-design of pseudo-labeling to generate well-calibrated structured pseudo labels for training with unlabeled or weaklylabeled data. Our proposed pseudo-labeling strategy is network structure agnostic to apply in a one-stage consistency training framework. We demonstrate the effectiveness of the proposed pseudo-labeling strategy in both low-data and highdata regimes. Extensive experiments have validated that pseudo labels generated from wisely fusing diverse sources and strong data augmentation are crucial to consistency training for semantic segmentation. The source code is available at https://github.com/googleinterns/wss. + +# 1 INTRODUCTION + +Image semantic segmentation is a core computer vision task that has been studied for decades. Compared with other vision tasks, such as image classification and object detection, human annotation of pixel-accurate segmentation is dramatically more expensive. Given sufficient pixellevel labeled training data (i.e., high-data regime), the current state-of-the-art segmentation models (e.g., DeepLabv $^ { 3 + }$ (Chen et al., 2018)) produce satisfactory segmentation prediction for common practical usage. Recent exploration demonstrates improvement over high-data regime settings with large-scale data, including self-training (Chen et al., 2020a; Zoph et al., 2020) and backbone pretraining (Zhang et al., 2020a). + +In contrast to the high-data regime, the performance of segmentation models drop significantly, given very limited pixel-labeled data (i.e., low-data regime). Such ineffectiveness at the low-data regime hinders the applicability of segmentation models. Therefore, instead of improving high-data regime segmentation, our work focuses on data-efficient segmentation training that only relies on few pixellabeled data and leverages the availability of extra unlabeled or weakly annotated (e.g., image-level) data to improve performance, with the aim of narrowing the gap to the supervised models trained with fully pixel-labeled data. + +Our work is inspired by the recent success in semi-supervised learning (SSL) for image classification, demonstrating promising performance given very limited labeled data and a sufficient amount of unlabeled data. Successful examples include MeanTeacher (Tarvainen & Valpola, 2017), UDA (Xie et al., 2019), MixMatch (Berthelot et al., 2019b), FeatMatch (Kuo et al., 2020), and FixMatch (Sohn et al., 2020a). One outstanding idea in this type of SSL is consistency training: making predictions consistent among multiple augmented images. FixMatch (Sohn et al., 2020a) shows that using high-confidence one-hot pseudo labels obtained from weakly-augmented unlabeled data to train strongly-augmented counterpart is the key to the success of SSL in image classification. + +However, effective pseudo labels and well-designed data augmentation are non-trivial to satisfy for semantic segmentation. Although we observe that many related works explore the second condition (i.e., augmentation) for image segmentation to enable consistency training framework (French et al., 2020; Ouali et al., 2020), we show that a wise design of pseudo labels for segmentation has great veiled potentials. + +In this paper, we propose PseudoSeg, a one-stage training framework to improve image semantic segmentation by leveraging additional data either with image-level labels (weakly-labeled data) or without any labels. PseudoSeg presents a novel design of pseudo-labeling to infer effective structured pseudo labels of additional data. It then optimizes the prediction of strongly-augmented data to match its corresponding pseudo labels. In summary, we make the following contributions: + +• We propose a simple one-stage framework to improve semantic segmentation by using a limited amount of pixel-labeled data and sufficient unlabeled data or image-level labeled data. Our framework is simple to apply and therefore network architecture agnostic. Directly applying consistency training approaches validated in image classification renders particular challenges in segmentation. We first demonstrate how well-calibrated soft pseudo labels obtained through wise fusion of predictions from diverse sources can greatly improve consistency training for segmentation. We conduct extensive experimental studies on the PASCAL VOC 2012 and COCO datasets. Comprehensive analyses are conducted to validate the effectiveness of this method at not only the low-data regime but also the high-data regime. Our experiments study multiple important open questions about transferring SSL advances to segmentation tasks. + +# 2 RELATED WORK + +Semi-supervised classification. Semi-supervised learning (SSL) aims to improve model performance by incorporating a large amount of unlabeled data during training. Consistency regularization and entropy minimization are two common strategies for SSL. The intuition behind consistencybased approaches (Laine & Aila, 2016; Sajjadi et al., 2016; Miyato et al., 2018; Tarvainen & Valpola, 2017) is that, the model output should remain unchanged when the input is perturbed. On the other hand, the entropy minimization strategy (Grandvalet & Bengio, 2005) argues that the unlabeled data can be used to ensured classes are well-separated, which can be achieved by encouraging the model to output low-entropy predictions. Pseudo-labeling (Lee, 2013) is one of the methods for implicit entropy minimization. Recently, holistic approaches (Berthelot et al., 2019b;a; Sohn et al., 2020a) combining both strategies have been proposed and achieved significant improvement. By redesigning the pseudo label, we propose an efficient one-stage semi-supervised learning framework of semantic segmentation for consistency training. + +Semi-supervised semantic segmentation. Collecting pixel-level annotations for semantic segmentation is costly and prone to error. Hence, leveraging unlabeled data in semantic segmentation is a natural fit. Early methods utilize a GAN-based model either to generate additional training data (Souly et al., 2017) or to learn a discriminator between the prediction and the ground truth mask (Hung et al., 2018; Mittal et al., 2019). Consistency regularization based approaches have also been proposed recently, by enforcing the predictions to be consistent, either from augmented input images (French et al., 2020; Kim et al., 2020), perturbed feature embeddings (Ouali et al., 2020), or different networks (Ke et al., 2020). Recently, Luo & Yang (2020) proposes a dual-branch training network to jointly learn from pixel-accurate and coarse labeled data, achieving good segmentation performance. To push the performance of state of the arts, iterative self-training approaches (Chen et al., 2020a; Zoph et al., 2020; Zhu et al., 2020) have been proposed. These methods usually assume the available labeled data is enough to train a good teacher model, which will be used to generate pseudo labels for the student model. However, this condition might not satisfy in the low-data regime. Our proposed method, on the other hand, realizing the ideas of both consistency regularization and pseudo-labeling in segmentation, consistently improves the supervised baseline in both low-data and high-data regimes. + +Weakly-supervised semantic segmentation. Instead of supervising network training with accurate pixel-level labels, many prior works exploit weaker forms of annotations (e.g., bounding boxes (Dai et al., 2015), scribbles (Lin et al., 2016), image-level labels). Most recent approaches use imagelevel labels as the supervisory signal, which exploits the idea of class activation map (CAM) (Zhou et al., 2016). Since the vanilla CAM only focus on the most discriminative region of objects, different ways to refine CAM have been proposed, including partial image/feature erasing (Hou et al., 2018; Wei et al., 2017; Li et al., 2018), using an additional saliency estimation model (Oh et al., 2017; Huang et al., 2018; Wei et al., 2018), utilizing pixel similarity to propagate the initial score map (Ahn & Kwak, 2018; Wang et al., 2020), or mining and co-segment the same category of objects across images (Sun et al., 2020; Zhang et al., 2020b). While achieving promising results using the approaches mentioned above, most of them require a multi-stage training strategy. The refined score maps are optimized again using a dense-CRF model (Krahenb ¨ uhl & Koltun ¨ , 2011), and then used as the target to train a separate segmentation network. On the other hand, we assume there exists a small number of fully-annotated data, which allows us to learn stronger segmentation models than general methods without needing pixel-labeled data. + +![](images/55439202eaf26b71f27a6328f988f0027a61cd98c089773b1b8032b98ac5ec58.jpg) +Figure 1: Overview of unlabeled data training branch. Given an image, the weakly augmented version is fed into the network to get the decoder prediction and Self-attention Grad-CAM (SGC). The two sources are then combined via a calibrated fusion strategy to form the pseudo label. The network is trained to make its decoder prediction from strongly augmented image to match the pseudo label by a per-pixel cross-entropy loss. + +# 3 THE PROPOSED METHOD + +In analogous to SSL for classification, our training objective in PseudoSeg consists of a supervised loss $\mathcal { L } _ { \mathrm { s } }$ applied to pixel-level labeled data $\mathcal { D } _ { l }$ , and a consistency constraint $\mathcal { L } _ { \mathrm { u } }$ applied to unlabeled data $\mathcal { D } _ { u }$ 1. Specifically, the supervised loss $\mathcal { L } _ { \mathrm { s } }$ is the standard pixel-wise cross-entropy loss on the weakly augmented pixel-level labeled examples: + +$$ +\mathcal { L } _ { \mathrm { s } } = \frac { 1 } { N \times | \mathcal { D } _ { l } | } \sum _ { x \in \mathcal { D } _ { l } } \sum _ { i = 0 } ^ { N - 1 } \mathrm { C r o s s E n t r o p y } \left( y _ { i } , f _ { \theta } ( \omega ( x _ { i } ) ) \right) , +$$ + +where $\theta$ represents the learnable parameters of the network function $f$ and $N$ denotes the number of valid labeled pixels in an image $\boldsymbol { x } \in \mathbb { R } ^ { H \times W \times 3 }$ . $y _ { i } \in \mathbb { R } ^ { C }$ is the ground truth label of a pixel $i$ in $H \times W$ dimensions, and $f _ { \theta } ( \omega ( x _ { i } ) ) \in \mathbb { R } ^ { C }$ is the predicted probability of pixel $i$ , where $C$ is the number of classes to predict and $\omega ( \cdot )$ denotes the weak (common) data augmentation operations used by Chen et al. (2018). + +During training, the proposed PseudoSeg estimates a pseudo label $\widetilde { y } \in \mathbb { R } ^ { H \times W \times C }$ for each stronglyaugmented unlabeled data $x$ in $\mathcal { D } _ { u }$ e, which is then used for computing the cross-entropy loss. The unsupervised objective can then be written as: + +$$ +\mathcal { L } _ { \sf u } = \frac { 1 } { N \times | \mathcal { D } _ { u } | } \sum _ { x \in \mathcal { D } _ { u } } \sum _ { i = 0 } ^ { N - 1 } \mathrm { C r o s s E n t r o p y } \left( \widetilde { y } _ { i } , f _ { \theta } ( \beta \circ \omega ( x _ { i } ) ) \right) , +$$ + +where $\beta ( \cdot )$ denotes a stronger data augmentation operation, which will be described in Section 3.2. +We illustrate the unlabeled data training branch in Figure 1. + +# 3.1 THE DESIGN OF STRUCTURED PSEUDO LABELS + +The next important question is how to generate the desirable pseudo label $\widetilde { y }$ . A straightforward soluetion is directly using the decoder output of a trained segmentation model after confidence thresholding, as suggested by Sohn et al. (2020a); Zoph et al. (2020); Xie et al. (2020); Sohn et al. (2020b). However, as we demonstrate later in the experiments, the generated pseudo hard/soft labels as well as other post-processing of outputs are barely satisfactory in the low-data regime, and thus yield inferior final results. To address this issue, our design of pseudo-labeling has two key insights. First, we seek for a distinct yet efficient decision mechanisms to compensate for the potential errors of decoder outputs. Second, wisely fusing multiple sources of predictions to generate an ensemble and better-calibrated version of pseudo labels. + +Starting with localization. Compared with precise segmentation, learning localization is a simpler task as it only needs to provide coarser-grained outputs than pixel level of objects in images. Based on this motivation, we improve decoder predictions from the localization perspective. Class activation map (CAM) (Zhou et al., 2016) is a popular approach to provide localization for class-specific regions. CAM-based methods (Hou et al., 2018; Wei et al., 2017; Ahn & Kwak, 2018) have been successfully adopted to tackle a different weakly supervised semantic segmentation task from us, where they assume only image-level labels are available. In practice, we adopt a variant of class activation map, Grad-CAM (Selvaraju et al., 2017) in PseudoSeg. + +From localization to segmentation. CAM estimates the strength of classifier responses on local feature maps. Thus, an inherent limitation of CAM-based approaches is that it is prone to attending only to the most discriminative regions. Although many weakly-supervised segmentation approaches (Ahn & Kwak, 2018; Ahn et al., 2019; Sun et al., 2020) aim at refining CAM localization maps to segmentation masks, most of them have complicated post-processing steps, such as dense CRF (Krahenb ¨ uhl & Koltun ¨ , 2011), which increases the model complexity when used for consistency training. Here we present a computationally efficient yet effective refinement alternative, which is learnable using available pixel-labeled data. + +Although CAM only localizes partial regions of interests, if we know the pairwise similarities between regions, we can propagate the CAM scores from the discriminative regions to the rest unattended regions. Actually, it has been shown in many works that the learned high-level deep features are usually good at similarity measurements of visual objects. In this paper, we find hypercolumn (Hariharan et al., 2015) with a learnable similarity measure function works fairly effective. + +Given the vanilla Grad-CAM output for all $C$ classes, which can be viewed as a spatially-flatten 2-D vector of weight $m \in \mathbb { R } ^ { L \times C }$ , where each row $m _ { i }$ is the response weight per class for one region $i$ . Using a kernel function $\mathcal { K } ( \cdot , \cdot ) : \mathbb { R } ^ { H } \times \mathbb { R } ^ { H } \mathbb { R }$ that measures element-wise similarity given feature $h \in { \bar { \mathbb { R } } } ^ { H }$ of two regions, the propagated score $\hat { m } _ { i } \in \mathbb { R } ^ { C }$ can be computed as follows + +$$ +\hat { m } _ { i } = \left( m _ { i } + \sum _ { j = 0 } ^ { L - 1 } \frac { e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { j } ) } } { \sum _ { k = 0 } ^ { L - 1 } e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { k } ) } } m _ { j } \right) \cdot W _ { c } . +$$ + +The goal of this function is to train $\Theta = \{ W _ { k } , W _ { v } \in \mathbb { R } ^ { H \times H } , W _ { c } \in \mathbb { R } ^ { C \times C } \}$ in order to propagate the high value in $m$ to all adjacent elements in the feature space $\mathbb { R } ^ { H }$ (i.e., hypercolumn features) to region $i$ . Adding $m _ { i }$ in equation 3 indicates the skip-connection. To compute propagated score for all regions, the operations in equation 3 can be efficiently implemented with self-attention dotproduct (Vaswani et al., 2017). For brevity, we denote this efficient refinement process output as selfattention Grad-CAM (SGC) maps in $\mathbb { R } ^ { H \times H \times C }$ . Figure 6 in Appendix A specifies the architecture. + +Calibrated prediction fusion. SGC maps are obtained from low-resolution feature maps. It is then resized to the desired output resolution, and thus not sufficient at delineating crisp boundaries. However, compared to the segmentation decoder, SGC is capable of generating more locally-consistent masks. Thus, we propose a novel calibrated fusion strategy to take advantage of both decoder and SCG predictions for better pseudo labels. + +Specifically, given a batch of decoder outputs (pre-softmax logits) $\hat { p } = f _ { \theta } ( \omega ( x ) )$ and SGC maps $\hat { m }$ computed from weakly-augmented data $\omega ( x )$ , we generate the pseudo labels $\widetilde { y }$ by + +$$ +\mathcal { F } ( \hat { p } , \hat { m } ) = \mathrm { S h a r p e n } \left( \gamma \operatorname { S o f t m a x } \left( \frac { \hat { p } } { \operatorname { N o r m } ( \hat { p } , \hat { m } ) } \right) + ( 1 - \gamma ) \operatorname { S o f t m a x } \left( \frac { \hat { m } } { \operatorname { N o r m } ( \hat { p } , \hat { m } ) } \right) , T \right) . +$$ + +Two critical procedures are proposed to use here to make the fusion process successful. First, $\hat { p }$ and $\hat { m }$ are from different decision mechanisms and they could have very different degrees of overconfidence. Therefore, we introduce the operation $\begin{array} { r } { \mathrm { N o r m } ( a , b ) = \sqrt { \sum _ { i } ^ { | a | } ( a _ { i } ^ { 2 } + b _ { i } ^ { 2 } ) } } \end{array}$ as a normalization factor. It alleviates the over-confident probability after softmax, which could unfavorably dominate the resulted $\gamma$ -averaged probability. Second, the distribution sharpening operation Sharpen $\begin{array} { r } { ( a , T ) _ { i } ~ = ~ a _ { i } ^ { 1 / T } / \sum _ { j } ^ { C } a _ { j } ^ { 1 / T } } \end{array}$ adjusts the temperature scalar $T$ of categorical distribution (Berthelot et al., 2019b; Chen et al., 2020b). Figure 2 illustrates the predictions from different sources. More importantly, we investigate the pseudo-labeling from a calibration perspective (Section 4.3), demonstrating that the proposed soft pseudo label $\widetilde { y }$ leads to a better calibration metric comparing to other possible fusion alternatives, and justifying why it benefits the final segmentation performance. + +![](images/da24262064ec7dc7f8d685b2d6566bc0dcda547c410ce0f2f99758c6eebe8bf2.jpg) +Figure 2: Visualization of pseudo labels and other predictions. The generated pseudo label by fusing the predictions from the decoder and SGC map is used to supervise the decoder (strong) predictions of the strongly-augmented counterpart. + +Training. Our final training objective contains two extra losses: a classification loss $\mathcal { L } _ { x }$ , and a segmentation loss $\mathcal { L } _ { s a }$ . First, to compute Grad-CAM, we add a one-layer classification head after the segmentation backbone and a multi-label classification loss $\mathcal { L } _ { x }$ . Second, as specified in Appendix A (Figure 6), SGC maps are scaled as pixel-wise probabilities using one-layer convolution followed by softmax in equation 3. Learning $\Theta$ to predict SGC maps needs pixel-labeled data $D _ { l }$ . It is achieved by an extra segmentation loss $\mathcal { L } _ { s a }$ between SGC maps of pixel-labeled data and corresponding ground truth. All the loss terms are jointly optimized (i.e., $\mathcal { L } _ { u } + \mathcal { L } _ { s } + \mathcal { L } _ { x } + \mathcal { L } _ { s a } )$ , while $\mathcal { L } _ { s a }$ only optimizes $\Theta$ (achieved by stopping gradient). See Figure 7 in the appendix for further details. + +# 3.2 INCORPORATING IMAGE-LEVEL LABELS AND AUGMENTATION + +The proposed PseudoSeg can easily incorporate image-level label information (if available) into our one-stage training framework, which also leads to consistent improvement as we demonstrate in experiments. We utilize the image-level data with two following steps. First, we directly use ground truth image-level labels to generate Grad-CAMs instead of using classifier outputs. Second, they are used to increase classification supervision beyond pixel-level labels for the classifier head. + +For strong data augmentation, we simply follow color jittering operations from SimCLR (Chen et al., 2020b) and remove all geometric transformations. The overall strength of augmentation can be controlled by a scalar (studied in experiments). We also apply once random CutOut (DeVries & Taylor, 2017) with a region of $5 0 \times 5 0$ pixels since we find it gives consistent though minor improvement (pixels inside CutOut regions are ignored in computing losses). + +# 4 EXPERIMENTAL RESULTS + +We start by specifying the experimental details. Then, we evaluate the method in the settings of using pixel-level labeled data and unlabeled data, as well as using pixel-level labeled data and image-level labeled data, respectively. Next, we conduct various ablation studies to justify our design choices. Lastly, we conduct more comparative experiments in specific settings. + +To evaluate the proposed method, we conduct the main experiments and ablation studies on the PASCAL VOC 2012 dataset (VOC12) (Everingham et al., 2015), which contains 21 classes including background. The standard VOC12 dataset has 1,449 images as the training set and 1,456 images as the validation set. We randomly subsample 1/2, 1/4, 1/8, and 1/16 of images in the standard training set to construct the pixel-level labeled data. The remaining images in the standard training set, together with the images in the augmented set (Hariharan et al., 2011) (around $9 \mathrm { k }$ images), are used as unlabeled or image-level labeled data. To further verify the effectiveness of the proposed method, we also conduct experiments on the COCO dataset (Lin et al., 2014). The COCO dataset has 118,287 images as the training set, and 5,000 images as the validation set. We evaluate on the 80 foreground classes and the background, as in the object detection task. As the COCO dataset is larger than VOC12, we randomly subsample smaller ratios, 1/32, 1/64, 1/128, 1/256, 1/512, of images from the training set to construct the pixel-level labeled data. The remaining images in the training set are used as unlabeled data or image-level labeled data. We evaluate the performance using the standard mean intersection-over-union (mIoU) metric. Implementation details can be found in Appendix B. + +![](images/97a7070768c40a2ee90f5b98acf7964c79ffade7edbbd0d82e10f3742a8ab33c.jpg) +Figure 3: Improvement over the strong supervised baseline, in a semi-supervised setting (w/ unlabeled data) on VOC12 val (left) and COCO val (right). + +# 4.1 EXPERIMENTS USING PIXEL-LEVEL LABELED DATA AND UNLABELED DATA + +Improvement over a strong baseline. We first demonstrate the effectiveness of the proposed method by comparing it with the DeepLabv $^ { 3 + }$ model trained with only the pixel-level labeled data. As shown in Figure 3 (a), the proposed method consistently outperforms the supervised training baseline on VOC12, by utilizing the pixel-level labeled data and the unlabeled data. The proposed method not only achieves a large performance boost in the low-data regime (when only $6 . 2 5 \%$ pixellevel labels available), but also improves the performance when the entire training set (1.4k images) is available. In Figure 3 (b), we again observe consistent improvement on the COCO dataset. + +Comparisons with the others. Next, we compare the proposed method with recent state of the arts on both the public $1 . 4 \mathrm { k } / 9 \mathrm { k }$ split (in Table 1) and the created low-data splits (in Table 2), on VOC12. Our method compares favorably with the others. + +Table 1: Comparison with state of the arts on VOC12 val set (w/ pixel-level labeled data and unlabeled data). We use the official training set (1.4k) as labeled data, and the augmented set (9k) as unlabeled data. + +
MethodNetworkmIoU (%)
GANSeg (Souly et al., 2017)VGG1664.10
AdvSemSeg (Hung et al., 2018)ResNet-10168.40
CCT (Ouali et al., 2020)ResNet-5069.40
PseudoSeg (Ours)ResNet-5071.00
PseudoSeg (Ours)ResNet-10173.23
+ +Table 2: Comparison with state of the arts on VOC12 val set (w/ pixel-level labeled data and unlabeled data) using low-data splits. The exact numbers of pixel-labeled images are shown in brackets. All the methods use ResNet-101 as backbone except CCT (Ouali et al., 2020), which uses ResNet-50. \* indicates implementation from Ke et al. (2020), \*\* indicates implementation from French et al. (2020). + +
Method1/2 (732)1/4 (366)1/8 (183)1/16 (92)
AdvSemSeg (Hung et al., 2018)65.2759.9747.5839.69
CCT (Ouali et al., 2020)62.1058.8047.6033.10
*MT (Tarvainen & Valpola, 2017)69.1663.0155.8148.70
GCT (Ke et al., 2020)70.6764.7154.9846.04
**VAT (Miyato et al., 2018)63.3456.8849.3536.92
CutMix (French et al., 2020)69.8468.3663.2055.58
PseudoSeg (Ours)72.4169.1465.5057.60
+ +![](images/e591d5f2246b80ae34e3f3b7d110f34c16cf3c68f8df57d4c593602c1b0a3fc4.jpg) +Figure 4: Improvement over the strong supervised baseline, in a semi-supervised setting (w/ image-level labeled data) on VOC12 val (left) and COCO val (right). + +Table 3: Comparison with state of the arts on VOC12 val set (w/ pixel-level labeled data and image-level labeled data). We use the official training set (1.4k) as labeled data, and the augmented set (9k) as image-level labeled data. + +
MethodModelNetworkmIoU (%)
WSSN (Papandreou et al., 2015)DeepLab-CRFVGG1664.60
GAIN (Li et al., 2018)DeepLab-CRF-LFOVVGG1660.50
MDC (Wei et al.,2018)DeepLab-CRF-LFOVVGG1665.70
DSRG (Huang et al.,2018)DeepLabv2VGG1664.30
GANSeg (Souly et al.,2017)FCNVGG1665.80
FickleNet (Lee et al.,2019)DeepLabv2ResNet-10165.80
CCT (Ouali et al., 2020)PSP-NetResNet-5073.20
PseudoSeg (Ours)DeepLabv3+ResNet-5073.80
+ +Table 4: Comparison with state of the arts on VOC12 val set with pixel-level labeled data and image-level labeled data. Four ratios of pixel-level labeled examples are tested. Both CCT (Ouali et al., 2020) and our method use ResNet-50 as backbone. + +
SplitCCTPseudoSeg
1/266.8073.51
1/467.6071.79
1/862.5069.15
1/1651.8065.44
+ +Similar to semi-supervised learning using pixel-level labeled data and unlabeled data, we first demonstrate the efficacy of our method by comparing it with a strong supervised baseline. As shown in Figure 4, the proposed method consistently improves the strong baseline on both datasets. In Table 3, we evaluate on the public $1 . 4 \mathrm { k } / 9 \mathrm { k }$ split. The proposed method compares favorably with the other methods. Moreover, we further compare to best compared CCT on the created low-data splits (in Table 4). Both experiments show that the proposed PseudoSeg is more robust than the compared method given less data. On all splits on both datasets, using pixel-level labeled data and image-labeled data shows higher mIoU than the setting using pixel-level labeled data and unlabeled data. + +# 4.3 ABLATION STUDY + +In this section, we conduct extensive ablation experiments on VOC12 to validate our design choices. + +How to construct pseudo label? We investigate the effectiveness of the proposed pseudo labeling. Table 5 demonstrates quantitative results, indicating that using either decoder output or SGC alone gives an inferior performance. Naively using decoder output as pseudo labels can hardly work well. The proposed fusion consistently performs better, either with or without additional image-level labels. To further answer why our pseudo labels are effective, we study from the model calibration perspective. We measure the expected calibration error (ECE) (Guo et al., 2017) scores of all the intermediate steps and other fusion variants. As shown in Figure 5 (a), the proposed fusion strategy (denoted as G in the figure) achieves the lowest ECE scores, indicating that the significance of jointly using normalization with sharpening (see equation 4) compared with other fusion alternatives. We hypothesize using well-calibrated soft labels makes model training less affected by label noises. The comprehensive calibration study is left as a future exploration direction. + +Using hypercolumn feature or not? In Figure 5 (b), we study the effectiveness of using hypercolumn features instead of the last feature maps in equation 3. We conduct the experiments on the 1/16 split of VOC12. As we can see, hypercolumn features substantially improve performance. + +Soft or hard pseudo label? How to utilize predictions as pseudo labels remains an active question in SSL. Next, we study whether we should use soft or hard one-hot pseudo labels. We conduct the experiments in the setting where pixel-level labeled data and image-level labeled data are available. As shown in Figure 5 (c), using all predictions as soft pseudo label yields better performance than selecting confident predictions. This suggests that well-calibrated soft pseudo labels might be important in segmentation than over-simplified confidence thresholding. + +Table 5: Comparison to alternative pseudo labeling strategies. We conduct experiments using 1/4, 1/8, 1/16 of the pixel-level labeled data, the exact numbers of images are shown in the brackets. + +
SourceUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Decoder only70.2269.3553.20
SGC only67.0762.6153.42
Calibrated fusion73.7973.1367.06
Decoder only73.9573.0567.54
SGC only71.7367.5764.26
Calibrated fusion75.2974.7071.22
+ +![](images/22e308dca704e89de560810e1f3d6d1ababe29fdb06d4ffaa1dc2fe834ea9bc0.jpg) +Figure 5: Ablation studies on different factors. See Section 4.3 for complete details. + +Temperature sharpening or not? We study the effect of temperature sharpening in equation 4. We conduct the experiments in the setting where pixel-level labeled data and image-level labeled data are available. As shown in Figure 5 (d), temperature sharpening shows consistent and clear improvements. + +Strong augmentation strength. In Figure 5 (e), we study the effects of color jittering in the strong augmentation. The magnitude of jittering strength is controlled by a scalar (Chen et al., 2020b). We conduct the experiments in the setting where pixel-level labeled data and unlabeled data are available. If the magnitude is too small, performance drops significantly, suggesting the importance of strong augmentation. + +Impact of different feature backbones. In Figure 5 (f), we compare the performance of using ResNet-50, ResNet-101, and Xception-65 as backbone architectures, respectively. We conduct the experiments in the setting where pixel-level labeled data and unlabeled data are available. As we can see, the proposed method consistently improves the baseline by a substantial margin across different backbone architectures. + +# 4.4 COMPARISON WITH SELF-TRAINING + +Several recent approaches (Chen et al., 2020a; Zoph et al., 2020) exploit the Student-Teacher selftraining idea to improve the performance with additional unlabeled data. However, these methods only apply self-training in the high-data regime (i.e., sufficient pixel-labeled data to train teachers). + +Table 6: Comparison with self-training. We use our supervised baseline as the teacher to generate one-hot pseudo labels, following Zoph et al. (2020). + +
MethodUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Supervised (Teacher)70.2064.0056.03
Self-training (Student)172.8569.8864.20
PseudoSeg (Ours)-73.7973.1367.06
PseudoSeg (Ours)√75.2974.7071.22
+ +Here we compare these methods in the low-data regimes, where we focus on. To generate offline pseudo labels, we closely follow segmentation experiments in Zoph et al. (2020): pixels with a confidence score higher than 0.5 will be used as one-hot pseudo labels, while the remaining are treated as ignored regions. This step is considered important to suppress noisy labels. A student model is then trained using the combination of unlabeled data in VOC12 train and augmented sets with generated one-hot pseudo labels and all the available pixel-level labeled data. As shown in Table 6, although the self-training pretty well improves over the supervised baseline, it is inferior to the proposed method 2. We conjecture that the teacher model usually produces low confidence scores to pixels around boundaries, so pseudo labels of these pixels are filtered in student training. However, boundary pixels are important for improving the performance of segmentation (Kirillov et al., 2020). On the other hand, the design of our method (online soft pseudo labeling process) bypass this challenge. We will conduct more verification of this hypothesis in future work. + +# 5 IMPROVING THE FULLY-SUPERVISED METHOD WITH ADDITIONAL DAT + +We have validated the effectiveness of the proposed method in the low-data regime. In this section, we want to explore whether the proposed method can further improve supervised training in the full training set using additional data. We use the training set (1.4k) in VOC12 as the pixel-level labeled data. The additional data contains additional VOC 9k $( V _ { 9 k } )$ , COCO training set $( C _ { t r } )$ , and COCO unlabeled data $( C _ { u } )$ . More training details can be found in Appendix D. As shown in Table 7, the proposed PseudoSeg is able to improve upon the supervised baseline even in the high-data regime, using additional unlabeled or image-level labeled data. + +Table 7: Improving fully supervised model with extra data. No test-time augmentation is used. + +
MethodBaseline丨PseudoSeg (w/o image-level labels)丨PseudoSeg (w/image-level labels)
Extra data1Ctr+CuCtr + Cu + V9kCtrCtr +Vgk
mIoU (%)76.9677.40 (+0.44)78.20 (+1.24)77.80 (+0.84)79.28 (+2.32)
+ +# 5 DISCUSSION AND CONCLUSION + +The key to the good performance of our method in the low-data regime is the novel re-design of pseudo-labeling strategy, which pursues a different decision mechanism from weakly-supervised localization to “remedy” weak predictions from segmentation head. Then augmentation consistency training progressively improves segmentation head quality. For the first time, we demonstrate that, with well-calibrated soft pseudo labels, utilizing unlabeled or image-labeled data significantly improves segmentation at low-data regimes. Further exploration of fusing stronger and better-calibrated pseudo labels worth more study as future directions (e.g., multi-scaling). Although color jittering works within our method as strong data augmentation, we have extensively explored geometric augmentations (leveraging STN (Jaderberg et al., 2015) to align pixels in pseudo labels and strongly-augmented predictions) for segmentation but find it not helpful. We believe data augmentation needs re-thinking beyond current success in classification for segmentation usage. + +# ACKNOWLEDGEMENT + +We thank Liang-Chieh Chen and Barret Zoph for their valuable comments. + +# REFERENCES + +Jiwoon Ahn and Suha Kwak. Learning pixel-level semantic affinity with image-level supervision for weakly supervised semantic segmentation. In CVPR, 2018. 3, 4 + +Jiwoon Ahn, Sunghyun Cho, and Suha Kwak. Weakly supervised learning of instance segmentation with inter-pixel relations. In CVPR, 2019. 4, 14 + +David Berthelot, Nicholas Carlini, Ekin D Cubuk, Alex Kurakin, Kihyuk Sohn, Han Zhang, and Colin Raffel. Remixmatch: Semi-supervised learning with distribution matching and augmentation anchoring. In ICLR, 2019a. 2 + +David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel. Mixmatch: A holistic approach to semi-supervised learning. 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Resnest: Split-attention networks. arXiv preprint arXiv:2004.08955, 2020a. 1 +Xiaolin Zhang, Yunchao Wei, and Yi Yang. Inter-image communication for weakly supervised localization. In ECCV, 2020b. 3 +Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discriminative localization. In CVPR, 2016. 2, 4 +Yi Zhu, Zhongyue Zhang, Chongruo Wu, Zhi Zhang, Tong He, Hang Zhang, R Manmatha, Mu Li, and Alexander Smola. Improving semantic segmentation via self-training. arXiv preprint arXiv:2004.14960, 2020. 2 +Barret Zoph, Golnaz Ghiasi, Tsung-Yi Lin, Yin Cui, Hanxiao Liu, Ekin D Cubuk, and Quoc V Le. Rethinking pre-training and self-training. arXiv preprint arXiv:2006.06882, 2020. 1, 2, 4, 8, 9 + +# APPENDIX + +# A SELF-ATTENTION GRAD-CAM + +We elaborate the detailed pipeline of generating Self-attention Grad-CAM (SGC) maps (equation 3) in Figure 6. To construct the hypercolumn feature, we extract the feature maps from the last two convolutional stages of the backbone network and concatenate them together. We then project the hypercolumn feature to two separate low-dimension embedding spaces to construct “key” and “query”, using two $1 \times 1$ convolutional layers. An attention matrix can then be computed via matrix multiplication of “key” and “query”. To construct “value”, we compute Grad-CAM for each foreground class and then concatenate them together. This results in a $H \times W \times ( C - 1 )$ score map, where the maximum score of each category is normalized to one separately. We then use image-level labels (either from classifier prediction or ground truth annotation) to set the score maps of non-existing classes to be zero. For each pixel localization, we use one to subtract the maximum score to construct the background score map, which is then concatenated with the foreground score maps to form “value” $( H \times W \times C )$ . The attention score matrix can then be used to reweight and propagate the scores in “value”. The propagated score is added back to the “value” score map, and the pass through a $1 \times 1$ convolution (w/ batch normalization) to output the SGC map. + +![](images/e601a68b369dd2bea5df204593cc86e5c58bb436da07925d9f96859b296a94ec.jpg) +Figure 6: Diagram of Self-attention Grad-CAM (SGC) . + +# B IMPLEMENTATION DETAILS + +We implement our method on top of the publicly available official DeepLab codebase.3 Unless specified, we adopt the DeepLabv $^ { 3 + }$ model with Xception-65 (Chollet, 2017) as the feature backbone, which is pre-trained on the ImageNet dataset (Russakovsky et al., 2015). We train our model following the default hyper-parameters (e.g., an initial learning rate of 0.007 with a polynomial learning rate decay schedule, a crop size of $5 1 3 \times 5 1 3$ , and an encoder output stride of 16), using 16 GPUs 4. We use a batch size of 4 for each GPU for pixel-level labeled data, and 4 for unlabeled/image-level labeled data. For VOC12, we train the model for 30,000 iterations. For COCO, we train the model for 200,000 iterations. We set $\gamma = 0 . 5$ and $T = 0 . 5$ unless specified. We do not apply any test time augmentations. + +# C LOW-DATA SAMPLING IN PASCAL VOC 2012 + +Unlike random sampling in image classification, it is difficult to sample uniformly in a low-data case for semantic segmentation due to the imbalance of rare classes. To avoid the missing classes at extremely low data regimes, we repeat the random sampling process for 1/16 three times (while ensuring each class has a certain amount) and report the results. We use Split 1 in the main manuscript. All splits will be released to encourage reproducibility. The results of all the three splits are shown as in Table 8. + +![](images/43b7ca0d025624a2b1b4a19d13c29bcfbb2a797f53df2327a7d1cf85d90a40e5.jpg) +Figure 7: Training. For each network component, we show the loss supervision and the corresponding data. + +Table 8: Full results of 1/16 split in VOC12. + +
MethodUsing image-level labelsSplit 1Split 2Split 3
Supervised■56.0356.8755.92
PseudoSeg (Ours)167.0664.1266.09
PseudoSeg (Ours)√71.2268.1169.72
+ +# D HIGH-DATA EXPERIMENTAL SETTINGS + +Here we provide more details about the experiments in Section 4.5. Since we have a lot more unlabeled/image-level labeled data, we adopt a longer training schedule (90,000 iterations) 5. We also adopt a slightly different fusion strategy in this setting by using $T = 0 . 7$ and $\gamma = 0 . 3$ . + +# E COMPARISON WITH WEAKLY-SUPERVISED APPROACHES + +In Table 9, we benchmark recent weakly supervised semantic segmentation performance on PASCAL VOC 2012 val set. Instead of enforcing the consistency between different augmented images as we do, these approaches tackle the semantic segmentation task from a different perspective, by exploiting the weaker annotations (image-level labels). As we can see, by exploiting the imagelevel labels with careful designs, weakly-supervised semantic segmentation methods could achieve reasonably well performance. We believe that both perspectives are feasible and promising for low-data regime semantic segmentation tasks, and complementary to each other. Therefore, these designs could be potentially integrated into our framework to generate better pseudo labels, which leads to improved performance. + +Table 9: Benchmarking state-of-the-art weakly supervised semantic segmentation methods. All the methods use image-level labels from VOC12 training (1.4k) and augmented (9k) sets. + +
MethodPixel-level labeled datamIoU (%)
FickleNet (Lee et al., 2019)64.9
IRNet (Ahn et al., 2019)63.5
OAA+ (Jiang et al., 2019)65.2
SEAM (Wang et al., 2020)64.5
MCIS (Sun et al., 2020)66.2
PseudoSeg (Ours)1/16 (92)71.22
+ +# F PERFORMANCE ANALYSIS FOR TEMPERATURE SHARPENING + +We conduct an additional performance analysis for temporal sharpening. We conduct experiments over T on the 1/16 split of VOC using pixel-level labeled data and image-level labeled data. As shown in Table 10, adopting a $T < 1$ for distribution sharpening generally leads to improved performance. + +Table 10: Performance analysis over T. + +
Temperature (T)mIoU (%)
0.171.11
0.370.11
0.5 (default)71.22
0.772.37
1.0 (no sharpening)68.15
+ +# G EXPERIMENTS ON CITYSCAPES + +In this section, we conduct additional experiments on the Cityscapes dataset (Cordts et al., 2016). The Cityscapes dataset contains 50 real-world driving sequences. Among these video sequences, 2,975 frames are selected as the training set, and 500 frames are selected as the validation set. Following previous common practice, we evaluate on 19 semantic classes. + +Comparison with state of the art. We compare our method with the current state-of-the-art method (French et al., 2020), in the setting of using pixel-level labeled and unlabeled data. We randomly subsample 1/4, 1/8, and 1/30 of the training set to construct the pixel-level labeled data, using the first random seed provided by French et al. (2020). Both French et al. (2020) and our method use ResNet-101 as the feature backbone and DeepLabv $^ { 3 + }$ (Chen et al., 2018) as the segmentation model. As shown in Table 11, the proposed method achieves promising results on all the three label ratios. + +Table 11: Experiments on Cityscapes (w/ pixel-level labeled data and unlabeled data). + +
Method1/4 (744)1/8 (372)1/30 (100)
CutMix (French et al., 2020)68.3365.8255.71
PseudoSeg (Ours)72.3669.8160.96
+ +Per-class performance analysis. Next, we provide per-class performance break down analysis. We compare our method with the supervised baseline on the 1/30 split, using pixel-level labeled data and unlabeled data. As shown in Table 12, the distribution of the labeled pixels is severely imbalanced. Although our method does not in particular address the data imbalance issue, our method improves upon the supervised baseline on most of the classes (except for “Wall” and “Pole”). + +Table 12: Per-class performance analysis on Cityscapes (w/ pixel-level labeled data and unlabeled data). + +
Class Pixel ratio (%)Road 36.36Sidewalk 5.61Building 20.99Wall 0.53Fence 0.98Pole 1.19Traffic light 0.14Traffic sign 0.51Vegetation 19.61Terrain 1.29
Supervised PseudoSeg (Ours)96.0371.2687.5319.7529.1152.1950.1968.0989.9345.79
96.6475.0688.6319.6734.0951.7558.1969.9590.4350.48
ClassSkyPersonRiderCarTruckBusTrainMotorcycleBicycle
Pixel ratio (%)3.701.100.166.490.380.130.230.060.54
Supervised91.0174.1243.9189.917.6814.1917.7825.8669.88
PseudoSeg (Ours)92.9975.1646.0991.6020.3926.3022.1343.9671.30
+ +Discussion. Although the scene layouts are quite similar for all the full images, it is still feasible to generate different image-level labels through a more aggressive geometric data augmentation (e.g., scaling, cropping, translation, etc.). In practice, standard segmentation preprocessing steps only crop a sub-region of the whole training images. It only contains partial images with a certain subset of image labels, making the training batches have diverse image-level labels (converted from pixellevel labels, in the fully-labeled+unlabeled setting). Moreover, in the fully-labeled+weakly-labeled setting, in practice, we can collect diverse Internet images and weakly label them, instead of weakly labeling images from Cityscapes. + +# H QUALITATIVE RESULTS + +We visualize several model prediction results for PASCAL VOC 2012 (Figure 8) and COCO (Figure 9). As we can see, the supervised baseline struggles to segment some of the categories and small objects, when trained in the low-data regime. On the other hand, PseudoSeg utilizes unlabeled or weakly-labeled data to generate more satisfying predictions. + +![](images/135161c6e5516766b8441d8a505a601aedeabfbaf045d2d39d9f58370fe109db.jpg) +Figure 8: Qualitative results of PASCAL VOC 2012. Models are trained with 1/16 pixel-level labeled data in the training set. + +![](images/bb1c575e94a6e95b23b16542a8e7720bafb4f0dec7ccae56a226c86d7f8ac3c3.jpg) +Figure 9: Qualitative results of COCO. Models are trained with 1/512 pixel-level labeled data in the training set. Note that white pixel in the ground truth indicates this pixel is not annotated for evaluation. \ No newline at end of file diff --git a/parse/train/-TwO99rbVRu/-TwO99rbVRu_content_list.json b/parse/train/-TwO99rbVRu/-TwO99rbVRu_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d6fca8ffaa0c447db72a78796463b8a9c0ca7db5 --- /dev/null +++ b/parse/train/-TwO99rbVRu/-TwO99rbVRu_content_list.json @@ -0,0 +1,1615 @@ +[ + { + "type": "text", + "text": "PSEUDOSEG: DESIGNING PSEUDO LABELS FOR SEMANTIC SEGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 745, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yuliang $\\mathbf { Z o u } ^ { 1 * }$ Zizhao Zhang2 Han Zhang3 Chun-Liang Li2 Xiao Bian2 Jia-Bin Huang1 Tomas Pfister2 1Virginia Tech 2Google Cloud AI 3Google Brain ", + "bbox": [ + 186, + 169, + 624, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 250, + 544, + 265 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent advances in semi-supervised learning (SSL) demonstrate that a combination of consistency regularization and pseudo-labeling can effectively improve image classification accuracy in the low-data regime. Compared to classification, semantic segmentation tasks require much more intensive labeling costs. Thus, these tasks greatly benefit from data-efficient training methods. However, structured outputs in segmentation render particular difficulties (e.g., designing pseudo-labeling and augmentation) to apply existing SSL strategies. To address this problem, we present a simple and novel re-design of pseudo-labeling to generate well-calibrated structured pseudo labels for training with unlabeled or weaklylabeled data. Our proposed pseudo-labeling strategy is network structure agnostic to apply in a one-stage consistency training framework. We demonstrate the effectiveness of the proposed pseudo-labeling strategy in both low-data and highdata regimes. Extensive experiments have validated that pseudo labels generated from wisely fusing diverse sources and strong data augmentation are crucial to consistency training for semantic segmentation. The source code is available at https://github.com/googleinterns/wss. ", + "bbox": [ + 233, + 284, + 764, + 503 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 535, + 336, + 550 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Image semantic segmentation is a core computer vision task that has been studied for decades. Compared with other vision tasks, such as image classification and object detection, human annotation of pixel-accurate segmentation is dramatically more expensive. Given sufficient pixellevel labeled training data (i.e., high-data regime), the current state-of-the-art segmentation models (e.g., DeepLabv $^ { 3 + }$ (Chen et al., 2018)) produce satisfactory segmentation prediction for common practical usage. Recent exploration demonstrates improvement over high-data regime settings with large-scale data, including self-training (Chen et al., 2020a; Zoph et al., 2020) and backbone pretraining (Zhang et al., 2020a). ", + "bbox": [ + 174, + 564, + 823, + 675 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In contrast to the high-data regime, the performance of segmentation models drop significantly, given very limited pixel-labeled data (i.e., low-data regime). Such ineffectiveness at the low-data regime hinders the applicability of segmentation models. Therefore, instead of improving high-data regime segmentation, our work focuses on data-efficient segmentation training that only relies on few pixellabeled data and leverages the availability of extra unlabeled or weakly annotated (e.g., image-level) data to improve performance, with the aim of narrowing the gap to the supervised models trained with fully pixel-labeled data. ", + "bbox": [ + 174, + 683, + 825, + 780 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our work is inspired by the recent success in semi-supervised learning (SSL) for image classification, demonstrating promising performance given very limited labeled data and a sufficient amount of unlabeled data. Successful examples include MeanTeacher (Tarvainen & Valpola, 2017), UDA (Xie et al., 2019), MixMatch (Berthelot et al., 2019b), FeatMatch (Kuo et al., 2020), and FixMatch (Sohn et al., 2020a). One outstanding idea in this type of SSL is consistency training: making predictions consistent among multiple augmented images. FixMatch (Sohn et al., 2020a) shows that using high-confidence one-hot pseudo labels obtained from weakly-augmented unlabeled data to train strongly-augmented counterpart is the key to the success of SSL in image classification. ", + "bbox": [ + 174, + 786, + 825, + 898 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, effective pseudo labels and well-designed data augmentation are non-trivial to satisfy for semantic segmentation. Although we observe that many related works explore the second condition (i.e., augmentation) for image segmentation to enable consistency training framework (French et al., 2020; Ouali et al., 2020), we show that a wise design of pseudo labels for segmentation has great veiled potentials. ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we propose PseudoSeg, a one-stage training framework to improve image semantic segmentation by leveraging additional data either with image-level labels (weakly-labeled data) or without any labels. PseudoSeg presents a novel design of pseudo-labeling to infer effective structured pseudo labels of additional data. It then optimizes the prediction of strongly-augmented data to match its corresponding pseudo labels. In summary, we make the following contributions: ", + "bbox": [ + 176, + 180, + 825, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose a simple one-stage framework to improve semantic segmentation by using a limited amount of pixel-labeled data and sufficient unlabeled data or image-level labeled data. Our framework is simple to apply and therefore network architecture agnostic. Directly applying consistency training approaches validated in image classification renders particular challenges in segmentation. We first demonstrate how well-calibrated soft pseudo labels obtained through wise fusion of predictions from diverse sources can greatly improve consistency training for segmentation. We conduct extensive experimental studies on the PASCAL VOC 2012 and COCO datasets. Comprehensive analyses are conducted to validate the effectiveness of this method at not only the low-data regime but also the high-data regime. Our experiments study multiple important open questions about transferring SSL advances to segmentation tasks. ", + "bbox": [ + 215, + 255, + 825, + 410 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 429, + 344, + 446 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Semi-supervised classification. Semi-supervised learning (SSL) aims to improve model performance by incorporating a large amount of unlabeled data during training. Consistency regularization and entropy minimization are two common strategies for SSL. The intuition behind consistencybased approaches (Laine & Aila, 2016; Sajjadi et al., 2016; Miyato et al., 2018; Tarvainen & Valpola, 2017) is that, the model output should remain unchanged when the input is perturbed. On the other hand, the entropy minimization strategy (Grandvalet & Bengio, 2005) argues that the unlabeled data can be used to ensured classes are well-separated, which can be achieved by encouraging the model to output low-entropy predictions. Pseudo-labeling (Lee, 2013) is one of the methods for implicit entropy minimization. Recently, holistic approaches (Berthelot et al., 2019b;a; Sohn et al., 2020a) combining both strategies have been proposed and achieved significant improvement. By redesigning the pseudo label, we propose an efficient one-stage semi-supervised learning framework of semantic segmentation for consistency training. ", + "bbox": [ + 173, + 457, + 825, + 623 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Semi-supervised semantic segmentation. Collecting pixel-level annotations for semantic segmentation is costly and prone to error. Hence, leveraging unlabeled data in semantic segmentation is a natural fit. Early methods utilize a GAN-based model either to generate additional training data (Souly et al., 2017) or to learn a discriminator between the prediction and the ground truth mask (Hung et al., 2018; Mittal et al., 2019). Consistency regularization based approaches have also been proposed recently, by enforcing the predictions to be consistent, either from augmented input images (French et al., 2020; Kim et al., 2020), perturbed feature embeddings (Ouali et al., 2020), or different networks (Ke et al., 2020). Recently, Luo & Yang (2020) proposes a dual-branch training network to jointly learn from pixel-accurate and coarse labeled data, achieving good segmentation performance. To push the performance of state of the arts, iterative self-training approaches (Chen et al., 2020a; Zoph et al., 2020; Zhu et al., 2020) have been proposed. These methods usually assume the available labeled data is enough to train a good teacher model, which will be used to generate pseudo labels for the student model. However, this condition might not satisfy in the low-data regime. Our proposed method, on the other hand, realizing the ideas of both consistency regularization and pseudo-labeling in segmentation, consistently improves the supervised baseline in both low-data and high-data regimes. ", + "bbox": [ + 174, + 628, + 825, + 849 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Weakly-supervised semantic segmentation. Instead of supervising network training with accurate pixel-level labels, many prior works exploit weaker forms of annotations (e.g., bounding boxes (Dai et al., 2015), scribbles (Lin et al., 2016), image-level labels). Most recent approaches use imagelevel labels as the supervisory signal, which exploits the idea of class activation map (CAM) (Zhou et al., 2016). Since the vanilla CAM only focus on the most discriminative region of objects, different ways to refine CAM have been proposed, including partial image/feature erasing (Hou et al., 2018; Wei et al., 2017; Li et al., 2018), using an additional saliency estimation model (Oh et al., 2017; Huang et al., 2018; Wei et al., 2018), utilizing pixel similarity to propagate the initial score map (Ahn & Kwak, 2018; Wang et al., 2020), or mining and co-segment the same category of objects across images (Sun et al., 2020; Zhang et al., 2020b). While achieving promising results using the approaches mentioned above, most of them require a multi-stage training strategy. The refined score maps are optimized again using a dense-CRF model (Krahenb ¨ uhl & Koltun ¨ , 2011), and then used as the target to train a separate segmentation network. On the other hand, we assume there exists a small number of fully-annotated data, which allows us to learn stronger segmentation models than general methods without needing pixel-labeled data. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/55439202eaf26b71f27a6328f988f0027a61cd98c089773b1b8032b98ac5ec58.jpg", + "image_caption": [ + "Figure 1: Overview of unlabeled data training branch. Given an image, the weakly augmented version is fed into the network to get the decoder prediction and Self-attention Grad-CAM (SGC). The two sources are then combined via a calibrated fusion strategy to form the pseudo label. The network is trained to make its decoder prediction from strongly augmented image to match the pseudo label by a per-pixel cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 176, + 104, + 820, + 228 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 320, + 825, + 462 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 THE PROPOSED METHOD ", + "text_level": 1, + "bbox": [ + 176, + 482, + 411, + 498 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In analogous to SSL for classification, our training objective in PseudoSeg consists of a supervised loss $\\mathcal { L } _ { \\mathrm { s } }$ applied to pixel-level labeled data $\\mathcal { D } _ { l }$ , and a consistency constraint $\\mathcal { L } _ { \\mathrm { u } }$ applied to unlabeled data $\\mathcal { D } _ { u }$ 1. Specifically, the supervised loss $\\mathcal { L } _ { \\mathrm { s } }$ is the standard pixel-wise cross-entropy loss on the weakly augmented pixel-level labeled examples: ", + "bbox": [ + 174, + 511, + 825, + 568 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c4e6ef258681327d5a51db4fbe3715327b16f29a336a86c4284b030629a615d4.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { s } } = \\frac { 1 } { N \\times | \\mathcal { D } _ { l } | } \\sum _ { x \\in \\mathcal { D } _ { l } } \\sum _ { i = 0 } ^ { N - 1 } \\mathrm { C r o s s E n t r o p y } \\left( y _ { i } , f _ { \\theta } ( \\omega ( x _ { i } ) ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 312, + 569, + 684, + 616 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\theta$ represents the learnable parameters of the network function $f$ and $N$ denotes the number of valid labeled pixels in an image $\\boldsymbol { x } \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ . $y _ { i } \\in \\mathbb { R } ^ { C }$ is the ground truth label of a pixel $i$ in $H \\times W$ dimensions, and $f _ { \\theta } ( \\omega ( x _ { i } ) ) \\in \\mathbb { R } ^ { C }$ is the predicted probability of pixel $i$ , where $C$ is the number of classes to predict and $\\omega ( \\cdot )$ denotes the weak (common) data augmentation operations used by Chen et al. (2018). ", + "bbox": [ + 174, + 617, + 825, + 688 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "During training, the proposed PseudoSeg estimates a pseudo label $\\widetilde { y } \\in \\mathbb { R } ^ { H \\times W \\times C }$ for each stronglyaugmented unlabeled data $x$ in $\\mathcal { D } _ { u }$ e, which is then used for computing the cross-entropy loss. The unsupervised objective can then be written as: ", + "bbox": [ + 174, + 694, + 825, + 736 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/10f5c9adc85e295e82223bd240ae987ee529c2d193b4b7a1eec4a96192fbde37.jpg", + "text": "$$\n\\mathcal { L } _ { \\sf u } = \\frac { 1 } { N \\times | \\mathcal { D } _ { u } | } \\sum _ { x \\in \\mathcal { D } _ { u } } \\sum _ { i = 0 } ^ { N - 1 } \\mathrm { C r o s s E n t r o p y } \\left( \\widetilde { y } _ { i } , f _ { \\theta } ( \\beta \\circ \\omega ( x _ { i } ) ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 295, + 738, + 700, + 785 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\beta ( \\cdot )$ denotes a stronger data augmentation operation, which will be described in Section 3.2. \nWe illustrate the unlabeled data training branch in Figure 1. ", + "bbox": [ + 173, + 787, + 823, + 816 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 THE DESIGN OF STRUCTURED PSEUDO LABELS ", + "text_level": 1, + "bbox": [ + 174, + 833, + 547, + 848 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The next important question is how to generate the desirable pseudo label $\\widetilde { y }$ . A straightforward soluetion is directly using the decoder output of a trained segmentation model after confidence thresholding, as suggested by Sohn et al. (2020a); Zoph et al. (2020); Xie et al. (2020); Sohn et al. (2020b). However, as we demonstrate later in the experiments, the generated pseudo hard/soft labels as well as other post-processing of outputs are barely satisfactory in the low-data regime, and thus yield inferior final results. To address this issue, our design of pseudo-labeling has two key insights. First, we seek for a distinct yet efficient decision mechanisms to compensate for the potential errors of decoder outputs. Second, wisely fusing multiple sources of predictions to generate an ensemble and better-calibrated version of pseudo labels. ", + "bbox": [ + 174, + 857, + 820, + 887 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Starting with localization. Compared with precise segmentation, learning localization is a simpler task as it only needs to provide coarser-grained outputs than pixel level of objects in images. Based on this motivation, we improve decoder predictions from the localization perspective. Class activation map (CAM) (Zhou et al., 2016) is a popular approach to provide localization for class-specific regions. CAM-based methods (Hou et al., 2018; Wei et al., 2017; Ahn & Kwak, 2018) have been successfully adopted to tackle a different weakly supervised semantic segmentation task from us, where they assume only image-level labels are available. In practice, we adopt a variant of class activation map, Grad-CAM (Selvaraju et al., 2017) in PseudoSeg. ", + "bbox": [ + 173, + 205, + 825, + 318 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "From localization to segmentation. CAM estimates the strength of classifier responses on local feature maps. Thus, an inherent limitation of CAM-based approaches is that it is prone to attending only to the most discriminative regions. Although many weakly-supervised segmentation approaches (Ahn & Kwak, 2018; Ahn et al., 2019; Sun et al., 2020) aim at refining CAM localization maps to segmentation masks, most of them have complicated post-processing steps, such as dense CRF (Krahenb ¨ uhl & Koltun ¨ , 2011), which increases the model complexity when used for consistency training. Here we present a computationally efficient yet effective refinement alternative, which is learnable using available pixel-labeled data. ", + "bbox": [ + 173, + 320, + 825, + 433 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Although CAM only localizes partial regions of interests, if we know the pairwise similarities between regions, we can propagate the CAM scores from the discriminative regions to the rest unattended regions. Actually, it has been shown in many works that the learned high-level deep features are usually good at similarity measurements of visual objects. In this paper, we find hypercolumn (Hariharan et al., 2015) with a learnable similarity measure function works fairly effective. ", + "bbox": [ + 174, + 439, + 823, + 510 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given the vanilla Grad-CAM output for all $C$ classes, which can be viewed as a spatially-flatten 2-D vector of weight $m \\in \\mathbb { R } ^ { L \\times C }$ , where each row $m _ { i }$ is the response weight per class for one region $i$ . Using a kernel function $\\mathcal { K } ( \\cdot , \\cdot ) : \\mathbb { R } ^ { H } \\times \\mathbb { R } ^ { H } \\mathbb { R }$ that measures element-wise similarity given feature $h \\in { \\bar { \\mathbb { R } } } ^ { H }$ of two regions, the propagated score $\\hat { m } _ { i } \\in \\mathbb { R } ^ { C }$ can be computed as follows ", + "bbox": [ + 174, + 515, + 825, + 573 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/315a83a19acb724a2334523b444d0778b85206c5895fe1814d98891b2b465de8.jpg", + "text": "$$\n\\hat { m } _ { i } = \\left( m _ { i } + \\sum _ { j = 0 } ^ { L - 1 } \\frac { e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { j } ) } } { \\sum _ { k = 0 } ^ { L - 1 } e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { k } ) } } m _ { j } \\right) \\cdot W _ { c } .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 575, + 674, + 625 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The goal of this function is to train $\\Theta = \\{ W _ { k } , W _ { v } \\in \\mathbb { R } ^ { H \\times H } , W _ { c } \\in \\mathbb { R } ^ { C \\times C } \\}$ in order to propagate the high value in $m$ to all adjacent elements in the feature space $\\mathbb { R } ^ { H }$ (i.e., hypercolumn features) to region $i$ . Adding $m _ { i }$ in equation 3 indicates the skip-connection. To compute propagated score for all regions, the operations in equation 3 can be efficiently implemented with self-attention dotproduct (Vaswani et al., 2017). For brevity, we denote this efficient refinement process output as selfattention Grad-CAM (SGC) maps in $\\mathbb { R } ^ { H \\times H \\times C }$ . Figure 6 in Appendix A specifies the architecture. ", + "bbox": [ + 173, + 627, + 825, + 713 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Calibrated prediction fusion. SGC maps are obtained from low-resolution feature maps. It is then resized to the desired output resolution, and thus not sufficient at delineating crisp boundaries. However, compared to the segmentation decoder, SGC is capable of generating more locally-consistent masks. Thus, we propose a novel calibrated fusion strategy to take advantage of both decoder and SCG predictions for better pseudo labels. ", + "bbox": [ + 174, + 717, + 825, + 787 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Specifically, given a batch of decoder outputs (pre-softmax logits) $\\hat { p } = f _ { \\theta } ( \\omega ( x ) )$ and SGC maps $\\hat { m }$ computed from weakly-augmented data $\\omega ( x )$ , we generate the pseudo labels $\\widetilde { y }$ by ", + "bbox": [ + 173, + 792, + 825, + 823 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8a2cc9c7dadb07213de1f86511ba990a5e698739952cb66fb24d2207c967f266.jpg", + "text": "$$\n\\mathcal { F } ( \\hat { p } , \\hat { m } ) = \\mathrm { S h a r p e n } \\left( \\gamma \\operatorname { S o f t m a x } \\left( \\frac { \\hat { p } } { \\operatorname { N o r m } ( \\hat { p } , \\hat { m } ) } \\right) + ( 1 - \\gamma ) \\operatorname { S o f t m a x } \\left( \\frac { \\hat { m } } { \\operatorname { N o r m } ( \\hat { p } , \\hat { m } ) } \\right) , T \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 186, + 825, + 790, + 861 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Two critical procedures are proposed to use here to make the fusion process successful. First, $\\hat { p }$ and $\\hat { m }$ are from different decision mechanisms and they could have very different degrees of overconfidence. Therefore, we introduce the operation $\\begin{array} { r } { \\mathrm { N o r m } ( a , b ) = \\sqrt { \\sum _ { i } ^ { | a | } ( a _ { i } ^ { 2 } + b _ { i } ^ { 2 } ) } } \\end{array}$ as a normalization factor. It alleviates the over-confident probability after softmax, which could unfavorably dominate the resulted $\\gamma$ -averaged probability. Second, the distribution sharpening operation Sharpen $\\begin{array} { r } { ( a , T ) _ { i } ~ = ~ a _ { i } ^ { 1 / T } / \\sum _ { j } ^ { C } a _ { j } ^ { 1 / T } } \\end{array}$ adjusts the temperature scalar $T$ of categorical distribution (Berthelot et al., 2019b; Chen et al., 2020b). Figure 2 illustrates the predictions from different sources. More importantly, we investigate the pseudo-labeling from a calibration perspective (Section 4.3), demonstrating that the proposed soft pseudo label $\\widetilde { y }$ leads to a better calibration metric comparing to other possible fusion alternatives, and justifying why it benefits the final segmentation performance. ", + "bbox": [ + 173, + 862, + 825, + 915 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/da24262064ec7dc7f8d685b2d6566bc0dcda547c410ce0f2f99758c6eebe8bf2.jpg", + "image_caption": [ + "Figure 2: Visualization of pseudo labels and other predictions. The generated pseudo label by fusing the predictions from the decoder and SGC map is used to supervise the decoder (strong) predictions of the strongly-augmented counterpart. " + ], + "image_footnote": [], + "bbox": [ + 174, + 101, + 816, + 162 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 224, + 825, + 340 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training. Our final training objective contains two extra losses: a classification loss $\\mathcal { L } _ { x }$ , and a segmentation loss $\\mathcal { L } _ { s a }$ . First, to compute Grad-CAM, we add a one-layer classification head after the segmentation backbone and a multi-label classification loss $\\mathcal { L } _ { x }$ . Second, as specified in Appendix A (Figure 6), SGC maps are scaled as pixel-wise probabilities using one-layer convolution followed by softmax in equation 3. Learning $\\Theta$ to predict SGC maps needs pixel-labeled data $D _ { l }$ . It is achieved by an extra segmentation loss $\\mathcal { L } _ { s a }$ between SGC maps of pixel-labeled data and corresponding ground truth. All the loss terms are jointly optimized (i.e., $\\mathcal { L } _ { u } + \\mathcal { L } _ { s } + \\mathcal { L } _ { x } + \\mathcal { L } _ { s a } )$ , while $\\mathcal { L } _ { s a }$ only optimizes $\\Theta$ (achieved by stopping gradient). See Figure 7 in the appendix for further details. ", + "bbox": [ + 174, + 344, + 825, + 457 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 INCORPORATING IMAGE-LEVEL LABELS AND AUGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 474, + 642, + 488 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The proposed PseudoSeg can easily incorporate image-level label information (if available) into our one-stage training framework, which also leads to consistent improvement as we demonstrate in experiments. We utilize the image-level data with two following steps. First, we directly use ground truth image-level labels to generate Grad-CAMs instead of using classifier outputs. Second, they are used to increase classification supervision beyond pixel-level labels for the classifier head. ", + "bbox": [ + 174, + 497, + 825, + 566 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For strong data augmentation, we simply follow color jittering operations from SimCLR (Chen et al., 2020b) and remove all geometric transformations. The overall strength of augmentation can be controlled by a scalar (studied in experiments). We also apply once random CutOut (DeVries & Taylor, 2017) with a region of $5 0 \\times 5 0$ pixels since we find it gives consistent though minor improvement (pixels inside CutOut regions are ignored in computing losses). ", + "bbox": [ + 174, + 574, + 825, + 643 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 665, + 419, + 680 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We start by specifying the experimental details. Then, we evaluate the method in the settings of using pixel-level labeled data and unlabeled data, as well as using pixel-level labeled data and image-level labeled data, respectively. Next, we conduct various ablation studies to justify our design choices. Lastly, we conduct more comparative experiments in specific settings. ", + "bbox": [ + 174, + 694, + 825, + 750 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To evaluate the proposed method, we conduct the main experiments and ablation studies on the PASCAL VOC 2012 dataset (VOC12) (Everingham et al., 2015), which contains 21 classes including background. The standard VOC12 dataset has 1,449 images as the training set and 1,456 images as the validation set. We randomly subsample 1/2, 1/4, 1/8, and 1/16 of images in the standard training set to construct the pixel-level labeled data. The remaining images in the standard training set, together with the images in the augmented set (Hariharan et al., 2011) (around $9 \\mathrm { k }$ images), are used as unlabeled or image-level labeled data. To further verify the effectiveness of the proposed method, we also conduct experiments on the COCO dataset (Lin et al., 2014). The COCO dataset has 118,287 images as the training set, and 5,000 images as the validation set. We evaluate on the 80 foreground classes and the background, as in the object detection task. As the COCO dataset is larger than VOC12, we randomly subsample smaller ratios, 1/32, 1/64, 1/128, 1/256, 1/512, of images from the training set to construct the pixel-level labeled data. The remaining images in the training set are used as unlabeled data or image-level labeled data. We evaluate the performance using the standard mean intersection-over-union (mIoU) metric. Implementation details can be found in Appendix B. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/97a7070768c40a2ee90f5b98acf7964c79ffade7edbbd0d82e10f3742a8ab33c.jpg", + "image_caption": [ + "Figure 3: Improvement over the strong supervised baseline, in a semi-supervised setting (w/ unlabeled data) on VOC12 val (left) and COCO val (right). " + ], + "image_footnote": [], + "bbox": [ + 204, + 103, + 800, + 256 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 305, + 825, + 347 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 EXPERIMENTS USING PIXEL-LEVEL LABELED DATA AND UNLABELED DATA ", + "text_level": 1, + "bbox": [ + 174, + 369, + 733, + 382 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Improvement over a strong baseline. We first demonstrate the effectiveness of the proposed method by comparing it with the DeepLabv $^ { 3 + }$ model trained with only the pixel-level labeled data. As shown in Figure 3 (a), the proposed method consistently outperforms the supervised training baseline on VOC12, by utilizing the pixel-level labeled data and the unlabeled data. The proposed method not only achieves a large performance boost in the low-data regime (when only $6 . 2 5 \\%$ pixellevel labels available), but also improves the performance when the entire training set (1.4k images) is available. In Figure 3 (b), we again observe consistent improvement on the COCO dataset. ", + "bbox": [ + 174, + 390, + 825, + 488 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Comparisons with the others. Next, we compare the proposed method with recent state of the arts on both the public $1 . 4 \\mathrm { k } / 9 \\mathrm { k }$ split (in Table 1) and the created low-data splits (in Table 2), on VOC12. Our method compares favorably with the others. ", + "bbox": [ + 174, + 492, + 823, + 535 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/7dc01ac8aa35a31e2e6ee4fc30ab2876ef40c10e92c233a07ab5670dfc41cdf0.jpg", + "table_caption": [ + "Table 1: Comparison with state of the arts on VOC12 val set (w/ pixel-level labeled data and unlabeled data). We use the official training set (1.4k) as labeled data, and the augmented set (9k) as unlabeled data. " + ], + "table_footnote": [], + "table_body": "
MethodNetworkmIoU (%)
GANSeg (Souly et al., 2017)VGG1664.10
AdvSemSeg (Hung et al., 2018)ResNet-10168.40
CCT (Ouali et al., 2020)ResNet-5069.40
PseudoSeg (Ours)ResNet-5071.00
PseudoSeg (Ours)ResNet-10173.23
", + "bbox": [ + 290, + 595, + 707, + 703 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/9203a0f288452039135c22a7ed17100e16fb37d08c9e0dc9a4259bcec589a2f7.jpg", + "table_caption": [ + "Table 2: Comparison with state of the arts on VOC12 val set (w/ pixel-level labeled data and unlabeled data) using low-data splits. The exact numbers of pixel-labeled images are shown in brackets. All the methods use ResNet-101 as backbone except CCT (Ouali et al., 2020), which uses ResNet-50. \\* indicates implementation from Ke et al. (2020), \\*\\* indicates implementation from French et al. (2020). " + ], + "table_footnote": [], + "table_body": "
Method1/2 (732)1/4 (366)1/8 (183)1/16 (92)
AdvSemSeg (Hung et al., 2018)65.2759.9747.5839.69
CCT (Ouali et al., 2020)62.1058.8047.6033.10
*MT (Tarvainen & Valpola, 2017)69.1663.0155.8148.70
GCT (Ke et al., 2020)70.6764.7154.9846.04
**VAT (Miyato et al., 2018)63.3456.8849.3536.92
CutMix (French et al., 2020)69.8468.3663.2055.58
PseudoSeg (Ours)72.4169.1465.5057.60
", + "bbox": [ + 235, + 797, + 754, + 921 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/e591d5f2246b80ae34e3f3b7d110f34c16cf3c68f8df57d4c593602c1b0a3fc4.jpg", + "image_caption": [ + "Figure 4: Improvement over the strong supervised baseline, in a semi-supervised setting (w/ image-level labeled data) on VOC12 val (left) and COCO val (right). " + ], + "image_footnote": [], + "bbox": [ + 215, + 103, + 802, + 250 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/eb7b923cd88434032cfbcc8bcc469b576f879830405bd525e8c663d309aec896.jpg", + "table_caption": [ + "Table 3: Comparison with state of the arts on VOC12 val set (w/ pixel-level labeled data and image-level labeled data). We use the official training set (1.4k) as labeled data, and the augmented set (9k) as image-level labeled data. " + ], + "table_footnote": [], + "table_body": "
MethodModelNetworkmIoU (%)
WSSN (Papandreou et al., 2015)DeepLab-CRFVGG1664.60
GAIN (Li et al., 2018)DeepLab-CRF-LFOVVGG1660.50
MDC (Wei et al.,2018)DeepLab-CRF-LFOVVGG1665.70
DSRG (Huang et al.,2018)DeepLabv2VGG1664.30
GANSeg (Souly et al.,2017)FCNVGG1665.80
FickleNet (Lee et al.,2019)DeepLabv2ResNet-10165.80
CCT (Ouali et al., 2020)PSP-NetResNet-5073.20
PseudoSeg (Ours)DeepLabv3+ResNet-5073.80
", + "bbox": [ + 176, + 352, + 550, + 446 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/bed5989fe8e0872d448e030d1d8d57de1db612740b64cbb18930fedff50be1c6.jpg", + "table_caption": [ + "Table 4: Comparison with state of the arts on VOC12 val set with pixel-level labeled data and image-level labeled data. Four ratios of pixel-level labeled examples are tested. Both CCT (Ouali et al., 2020) and our method use ResNet-50 as backbone. " + ], + "table_footnote": [], + "table_body": "
SplitCCTPseudoSeg
1/266.8073.51
1/467.6071.79
1/862.5069.15
1/1651.8065.44
", + "bbox": [ + 604, + 385, + 789, + 463 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Similar to semi-supervised learning using pixel-level labeled data and unlabeled data, we first demonstrate the efficacy of our method by comparing it with a strong supervised baseline. As shown in Figure 4, the proposed method consistently improves the strong baseline on both datasets. In Table 3, we evaluate on the public $1 . 4 \\mathrm { k } / 9 \\mathrm { k }$ split. The proposed method compares favorably with the other methods. Moreover, we further compare to best compared CCT on the created low-data splits (in Table 4). Both experiments show that the proposed PseudoSeg is more robust than the compared method given less data. On all splits on both datasets, using pixel-level labeled data and image-labeled data shows higher mIoU than the setting using pixel-level labeled data and unlabeled data. ", + "bbox": [ + 173, + 505, + 825, + 630 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 176, + 648, + 339, + 662 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we conduct extensive ablation experiments on VOC12 to validate our design choices. ", + "bbox": [ + 173, + 672, + 821, + 688 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "How to construct pseudo label? We investigate the effectiveness of the proposed pseudo labeling. Table 5 demonstrates quantitative results, indicating that using either decoder output or SGC alone gives an inferior performance. Naively using decoder output as pseudo labels can hardly work well. The proposed fusion consistently performs better, either with or without additional image-level labels. To further answer why our pseudo labels are effective, we study from the model calibration perspective. We measure the expected calibration error (ECE) (Guo et al., 2017) scores of all the intermediate steps and other fusion variants. As shown in Figure 5 (a), the proposed fusion strategy (denoted as G in the figure) achieves the lowest ECE scores, indicating that the significance of jointly using normalization with sharpening (see equation 4) compared with other fusion alternatives. We hypothesize using well-calibrated soft labels makes model training less affected by label noises. The comprehensive calibration study is left as a future exploration direction. ", + "bbox": [ + 174, + 691, + 825, + 844 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Using hypercolumn feature or not? In Figure 5 (b), we study the effectiveness of using hypercolumn features instead of the last feature maps in equation 3. We conduct the experiments on the 1/16 split of VOC12. As we can see, hypercolumn features substantially improve performance. ", + "bbox": [ + 174, + 849, + 825, + 891 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Soft or hard pseudo label? How to utilize predictions as pseudo labels remains an active question in SSL. Next, we study whether we should use soft or hard one-hot pseudo labels. We conduct the experiments in the setting where pixel-level labeled data and image-level labeled data are available. As shown in Figure 5 (c), using all predictions as soft pseudo label yields better performance than selecting confident predictions. This suggests that well-calibrated soft pseudo labels might be important in segmentation than over-simplified confidence thresholding. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/1428232eb8af7478e83809145a4cde79932a04e250f1fcd9c83f6c06df1cca51.jpg", + "table_caption": [ + "Table 5: Comparison to alternative pseudo labeling strategies. We conduct experiments using 1/4, 1/8, 1/16 of the pixel-level labeled data, the exact numbers of images are shown in the brackets. " + ], + "table_footnote": [], + "table_body": "
SourceUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Decoder only70.2269.3553.20
SGC only67.0762.6153.42
Calibrated fusion73.7973.1367.06
Decoder only73.9573.0567.54
SGC only71.7367.5764.26
Calibrated fusion75.2974.7071.22
", + "bbox": [ + 235, + 132, + 756, + 244 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/22e308dca704e89de560810e1f3d6d1ababe29fdb06d4ffaa1dc2fe834ea9bc0.jpg", + "image_caption": [ + "Figure 5: Ablation studies on different factors. See Section 4.3 for complete details. " + ], + "image_footnote": [], + "bbox": [ + 174, + 260, + 823, + 535 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 571, + 825, + 628 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Temperature sharpening or not? We study the effect of temperature sharpening in equation 4. We conduct the experiments in the setting where pixel-level labeled data and image-level labeled data are available. As shown in Figure 5 (d), temperature sharpening shows consistent and clear improvements. ", + "bbox": [ + 174, + 632, + 823, + 688 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Strong augmentation strength. In Figure 5 (e), we study the effects of color jittering in the strong augmentation. The magnitude of jittering strength is controlled by a scalar (Chen et al., 2020b). We conduct the experiments in the setting where pixel-level labeled data and unlabeled data are available. If the magnitude is too small, performance drops significantly, suggesting the importance of strong augmentation. ", + "bbox": [ + 174, + 693, + 825, + 762 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Impact of different feature backbones. In Figure 5 (f), we compare the performance of using ResNet-50, ResNet-101, and Xception-65 as backbone architectures, respectively. We conduct the experiments in the setting where pixel-level labeled data and unlabeled data are available. As we can see, the proposed method consistently improves the baseline by a substantial margin across different backbone architectures. ", + "bbox": [ + 174, + 767, + 825, + 837 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 COMPARISON WITH SELF-TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 858, + 462, + 871 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Several recent approaches (Chen et al., 2020a; Zoph et al., 2020) exploit the Student-Teacher selftraining idea to improve the performance with additional unlabeled data. However, these methods only apply self-training in the high-data regime (i.e., sufficient pixel-labeled data to train teachers). ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2715b1a36b360cce6e5e8e954ddad46481951e5efe9bfac4ffd4e9083d94e9a5.jpg", + "table_caption": [ + "Table 6: Comparison with self-training. We use our supervised baseline as the teacher to generate one-hot pseudo labels, following Zoph et al. (2020). " + ], + "table_footnote": [], + "table_body": "
MethodUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Supervised (Teacher)70.2064.0056.03
Self-training (Student)172.8569.8864.20
PseudoSeg (Ours)-73.7973.1367.06
PseudoSeg (Ours)√75.2974.7071.22
", + "bbox": [ + 220, + 132, + 772, + 218 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Here we compare these methods in the low-data regimes, where we focus on. To generate offline pseudo labels, we closely follow segmentation experiments in Zoph et al. (2020): pixels with a confidence score higher than 0.5 will be used as one-hot pseudo labels, while the remaining are treated as ignored regions. This step is considered important to suppress noisy labels. A student model is then trained using the combination of unlabeled data in VOC12 train and augmented sets with generated one-hot pseudo labels and all the available pixel-level labeled data. As shown in Table 6, although the self-training pretty well improves over the supervised baseline, it is inferior to the proposed method 2. We conjecture that the teacher model usually produces low confidence scores to pixels around boundaries, so pseudo labels of these pixels are filtered in student training. However, boundary pixels are important for improving the performance of segmentation (Kirillov et al., 2020). On the other hand, the design of our method (online soft pseudo labeling process) bypass this challenge. We will conduct more verification of this hypothesis in future work. ", + "bbox": [ + 173, + 233, + 825, + 400 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 IMPROVING THE FULLY-SUPERVISED METHOD WITH ADDITIONAL DAT ", + "text_level": 1, + "bbox": [ + 194, + 417, + 691, + 430 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have validated the effectiveness of the proposed method in the low-data regime. In this section, we want to explore whether the proposed method can further improve supervised training in the full training set using additional data. We use the training set (1.4k) in VOC12 as the pixel-level labeled data. The additional data contains additional VOC 9k $( V _ { 9 k } )$ , COCO training set $( C _ { t r } )$ , and COCO unlabeled data $( C _ { u } )$ . More training details can be found in Appendix D. As shown in Table 7, the proposed PseudoSeg is able to improve upon the supervised baseline even in the high-data regime, using additional unlabeled or image-level labeled data. ", + "bbox": [ + 174, + 439, + 825, + 537 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/ff061b56d6ee624a1ae28039431568030c3c487b3a4ade5abd9338e0c6c7e09f.jpg", + "table_caption": [ + "Table 7: Improving fully supervised model with extra data. No test-time augmentation is used. " + ], + "table_footnote": [], + "table_body": "
MethodBaseline丨PseudoSeg (w/o image-level labels)丨PseudoSeg (w/image-level labels)
Extra data1Ctr+CuCtr + Cu + V9kCtrCtr +Vgk
mIoU (%)76.9677.40 (+0.44)78.20 (+1.24)77.80 (+0.84)79.28 (+2.32)
", + "bbox": [ + 187, + 568, + 803, + 627 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 DISCUSSION AND CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 648, + 470, + 665 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The key to the good performance of our method in the low-data regime is the novel re-design of pseudo-labeling strategy, which pursues a different decision mechanism from weakly-supervised localization to “remedy” weak predictions from segmentation head. Then augmentation consistency training progressively improves segmentation head quality. For the first time, we demonstrate that, with well-calibrated soft pseudo labels, utilizing unlabeled or image-labeled data significantly improves segmentation at low-data regimes. Further exploration of fusing stronger and better-calibrated pseudo labels worth more study as future directions (e.g., multi-scaling). Although color jittering works within our method as strong data augmentation, we have extensively explored geometric augmentations (leveraging STN (Jaderberg et al., 2015) to align pixels in pseudo labels and strongly-augmented predictions) for segmentation but find it not helpful. We believe data augmentation needs re-thinking beyond current success in classification for segmentation usage. ", + "bbox": [ + 173, + 670, + 825, + 823 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENT ", + "text_level": 1, + "bbox": [ + 176, + 845, + 357, + 858 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We thank Liang-Chieh Chen and Barret Zoph for their valuable comments. ", + "bbox": [ + 174, + 875, + 663, + 888 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 285, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Jiwoon Ahn and Suha Kwak. 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Rethinking pre-training and self-training. arXiv preprint arXiv:2006.06882, 2020. 1, 2, 4, 8, 9 ", + "bbox": [ + 171, + 87, + 826, + 929 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 93, + 826, + 574 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A SELF-ATTENTION GRAD-CAM ", + "text_level": 1, + "bbox": [ + 178, + 137, + 418, + 151 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We elaborate the detailed pipeline of generating Self-attention Grad-CAM (SGC) maps (equation 3) in Figure 6. To construct the hypercolumn feature, we extract the feature maps from the last two convolutional stages of the backbone network and concatenate them together. We then project the hypercolumn feature to two separate low-dimension embedding spaces to construct “key” and “query”, using two $1 \\times 1$ convolutional layers. An attention matrix can then be computed via matrix multiplication of “key” and “query”. To construct “value”, we compute Grad-CAM for each foreground class and then concatenate them together. This results in a $H \\times W \\times ( C - 1 )$ score map, where the maximum score of each category is normalized to one separately. We then use image-level labels (either from classifier prediction or ground truth annotation) to set the score maps of non-existing classes to be zero. For each pixel localization, we use one to subtract the maximum score to construct the background score map, which is then concatenated with the foreground score maps to form “value” $( H \\times W \\times C )$ . The attention score matrix can then be used to reweight and propagate the scores in “value”. The propagated score is added back to the “value” score map, and the pass through a $1 \\times 1$ convolution (w/ batch normalization) to output the SGC map. ", + "bbox": [ + 173, + 166, + 825, + 359 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/e601a68b369dd2bea5df204593cc86e5c58bb436da07925d9f96859b296a94ec.jpg", + "image_caption": [ + "Figure 6: Diagram of Self-attention Grad-CAM (SGC) . " + ], + "image_footnote": [], + "bbox": [ + 178, + 380, + 825, + 523 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 588, + 395, + 602 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We implement our method on top of the publicly available official DeepLab codebase.3 Unless specified, we adopt the DeepLabv $^ { 3 + }$ model with Xception-65 (Chollet, 2017) as the feature backbone, which is pre-trained on the ImageNet dataset (Russakovsky et al., 2015). We train our model following the default hyper-parameters (e.g., an initial learning rate of 0.007 with a polynomial learning rate decay schedule, a crop size of $5 1 3 \\times 5 1 3$ , and an encoder output stride of 16), using 16 GPUs 4. We use a batch size of 4 for each GPU for pixel-level labeled data, and 4 for unlabeled/image-level labeled data. For VOC12, we train the model for 30,000 iterations. For COCO, we train the model for 200,000 iterations. We set $\\gamma = 0 . 5$ and $T = 0 . 5$ unless specified. We do not apply any test time augmentations. ", + "bbox": [ + 173, + 616, + 825, + 742 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C LOW-DATA SAMPLING IN PASCAL VOC 2012 ", + "text_level": 1, + "bbox": [ + 174, + 765, + 531, + 780 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Unlike random sampling in image classification, it is difficult to sample uniformly in a low-data case for semantic segmentation due to the imbalance of rare classes. To avoid the missing classes at extremely low data regimes, we repeat the random sampling process for 1/16 three times (while ensuring each class has a certain amount) and report the results. We use Split 1 in the main manuscript. All splits will be released to encourage reproducibility. The results of all the three splits are shown as in Table 8. ", + "bbox": [ + 174, + 792, + 825, + 876 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/43b7ca0d025624a2b1b4a19d13c29bcfbb2a797f53df2327a7d1cf85d90a40e5.jpg", + "image_caption": [ + "Figure 7: Training. For each network component, we show the loss supervision and the corresponding data. " + ], + "image_footnote": [], + "bbox": [ + 238, + 111, + 754, + 296 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/e9b82779e99eaed63ad8c809cc6d6dc842d8807f08589ccf327db27ce8cbd91e.jpg", + "table_caption": [ + "Table 8: Full results of 1/16 split in VOC12. " + ], + "table_footnote": [], + "table_body": "
MethodUsing image-level labelsSplit 1Split 2Split 3
Supervised■56.0356.8755.92
PseudoSeg (Ours)167.0664.1266.09
PseudoSeg (Ours)√71.2268.1169.72
", + "bbox": [ + 243, + 373, + 754, + 445 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D HIGH-DATA EXPERIMENTAL SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 472, + 475, + 487 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we provide more details about the experiments in Section 4.5. Since we have a lot more unlabeled/image-level labeled data, we adopt a longer training schedule (90,000 iterations) 5. We also adopt a slightly different fusion strategy in this setting by using $T = 0 . 7$ and $\\gamma = 0 . 3$ . ", + "bbox": [ + 176, + 500, + 825, + 541 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "E COMPARISON WITH WEAKLY-SUPERVISED APPROACHES ", + "text_level": 1, + "bbox": [ + 174, + 563, + 588, + 575 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In Table 9, we benchmark recent weakly supervised semantic segmentation performance on PASCAL VOC 2012 val set. Instead of enforcing the consistency between different augmented images as we do, these approaches tackle the semantic segmentation task from a different perspective, by exploiting the weaker annotations (image-level labels). As we can see, by exploiting the imagelevel labels with careful designs, weakly-supervised semantic segmentation methods could achieve reasonably well performance. We believe that both perspectives are feasible and promising for low-data regime semantic segmentation tasks, and complementary to each other. Therefore, these designs could be potentially integrated into our framework to generate better pseudo labels, which leads to improved performance. ", + "bbox": [ + 173, + 587, + 825, + 713 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/4d0847f24656dd0475d73e7048e0a6968e21e34dda181eb6e7f0646acb151e5a.jpg", + "table_caption": [ + "Table 9: Benchmarking state-of-the-art weakly supervised semantic segmentation methods. All the methods use image-level labels from VOC12 training (1.4k) and augmented (9k) sets. " + ], + "table_footnote": [], + "table_body": "
MethodPixel-level labeled datamIoU (%)
FickleNet (Lee et al., 2019)64.9
IRNet (Ahn et al., 2019)63.5
OAA+ (Jiang et al., 2019)65.2
SEAM (Wang et al., 2020)64.5
MCIS (Sun et al., 2020)66.2
PseudoSeg (Ours)1/16 (92)71.22
", + "bbox": [ + 266, + 758, + 730, + 880 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F PERFORMANCE ANALYSIS FOR TEMPERATURE SHARPENING", + "text_level": 1, + "bbox": [ + 181, + 104, + 614, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We conduct an additional performance analysis for temporal sharpening. We conduct experiments over T on the 1/16 split of VOC using pixel-level labeled data and image-level labeled data. As shown in Table 10, adopting a $T < 1$ for distribution sharpening generally leads to improved performance. ", + "bbox": [ + 173, + 128, + 825, + 185 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/b8c478af059105baa12a29931140f43e29439d24d1c3e40c9ea022b42ed5c14a.jpg", + "table_caption": [ + "Table 10: Performance analysis over T. " + ], + "table_footnote": [], + "table_body": "
Temperature (T)mIoU (%)
0.171.11
0.370.11
0.5 (default)71.22
0.772.37
1.0 (no sharpening)68.15
", + "bbox": [ + 377, + 213, + 616, + 313 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "G EXPERIMENTS ON CITYSCAPES ", + "text_level": 1, + "bbox": [ + 176, + 327, + 423, + 342 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section, we conduct additional experiments on the Cityscapes dataset (Cordts et al., 2016). The Cityscapes dataset contains 50 real-world driving sequences. Among these video sequences, 2,975 frames are selected as the training set, and 500 frames are selected as the validation set. Following previous common practice, we evaluate on 19 semantic classes. ", + "bbox": [ + 173, + 353, + 825, + 410 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Comparison with state of the art. We compare our method with the current state-of-the-art method (French et al., 2020), in the setting of using pixel-level labeled and unlabeled data. We randomly subsample 1/4, 1/8, and 1/30 of the training set to construct the pixel-level labeled data, using the first random seed provided by French et al. (2020). Both French et al. (2020) and our method use ResNet-101 as the feature backbone and DeepLabv $^ { 3 + }$ (Chen et al., 2018) as the segmentation model. As shown in Table 11, the proposed method achieves promising results on all the three label ratios. ", + "bbox": [ + 173, + 414, + 825, + 511 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/614e49f5ab8345f6334ef24d10e80e6811aa3249760ac77af38650c91eb5769a.jpg", + "table_caption": [ + "Table 11: Experiments on Cityscapes (w/ pixel-level labeled data and unlabeled data). " + ], + "table_footnote": [], + "table_body": "
Method1/4 (744)1/8 (372)1/30 (100)
CutMix (French et al., 2020)68.3365.8255.71
PseudoSeg (Ours)72.3669.8160.96
", + "bbox": [ + 263, + 539, + 728, + 597 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Per-class performance analysis. Next, we provide per-class performance break down analysis. We compare our method with the supervised baseline on the 1/30 split, using pixel-level labeled data and unlabeled data. As shown in Table 12, the distribution of the labeled pixels is severely imbalanced. Although our method does not in particular address the data imbalance issue, our method improves upon the supervised baseline on most of the classes (except for “Wall” and “Pole”). ", + "bbox": [ + 173, + 609, + 825, + 681 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/d5a02c39eb789c1b3d3f8abc44df37bc5e9229a425f57e56447e631dfb53e845.jpg", + "table_caption": [ + "Table 12: Per-class performance analysis on Cityscapes (w/ pixel-level labeled data and unlabeled data). " + ], + "table_footnote": [], + "table_body": "
Class Pixel ratio (%)Road 36.36Sidewalk 5.61Building 20.99Wall 0.53Fence 0.98Pole 1.19Traffic light 0.14Traffic sign 0.51Vegetation 19.61Terrain 1.29
Supervised PseudoSeg (Ours)96.0371.2687.5319.7529.1152.1950.1968.0989.9345.79
96.6475.0688.6319.6734.0951.7558.1969.9590.4350.48
ClassSkyPersonRiderCarTruckBusTrainMotorcycleBicycle
Pixel ratio (%)3.701.100.166.490.380.130.230.060.54
Supervised91.0174.1243.9189.917.6814.1917.7825.8669.88
PseudoSeg (Ours)92.9975.1646.0991.6020.3926.3022.1343.9671.30
", + "bbox": [ + 174, + 723, + 821, + 829 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Discussion. Although the scene layouts are quite similar for all the full images, it is still feasible to generate different image-level labels through a more aggressive geometric data augmentation (e.g., scaling, cropping, translation, etc.). In practice, standard segmentation preprocessing steps only crop a sub-region of the whole training images. It only contains partial images with a certain subset of image labels, making the training batches have diverse image-level labels (converted from pixellevel labels, in the fully-labeled+unlabeled setting). Moreover, in the fully-labeled+weakly-labeled setting, in practice, we can collect diverse Internet images and weakly label them, instead of weakly labeling images from Cityscapes. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "H QUALITATIVE RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 148, + 364, + 164 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We visualize several model prediction results for PASCAL VOC 2012 (Figure 8) and COCO (Figure 9). As we can see, the supervised baseline struggles to segment some of the categories and small objects, when trained in the low-data regime. On the other hand, PseudoSeg utilizes unlabeled or weakly-labeled data to generate more satisfying predictions. ", + "bbox": [ + 174, + 174, + 825, + 231 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/135161c6e5516766b8441d8a505a601aedeabfbaf045d2d39d9f58370fe109db.jpg", + "image_caption": [ + "Figure 8: Qualitative results of PASCAL VOC 2012. Models are trained with 1/16 pixel-level labeled data in the training set. " + ], + "image_footnote": [], + "bbox": [ + 174, + 94, + 816, + 906 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/bb1c575e94a6e95b23b16542a8e7720bafb4f0dec7ccae56a226c86d7f8ac3c3.jpg", + "image_caption": [ + "Figure 9: Qualitative results of COCO. Models are trained with 1/512 pixel-level labeled data in the training set. Note that white pixel in the ground truth indicates this pixel is not annotated for evaluation. " + ], + "image_footnote": [], + "bbox": [ + 174, + 101, + 816, + 887 + ], + "page_idx": 17 + } +] \ No newline at end of file diff --git a/parse/train/-TwO99rbVRu/-TwO99rbVRu_middle.json b/parse/train/-TwO99rbVRu/-TwO99rbVRu_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..959c48b2440738370d93503fe949f7508d6fe940 --- /dev/null +++ b/parse/train/-TwO99rbVRu/-TwO99rbVRu_middle.json @@ -0,0 +1,37906 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 456, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 459, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 459, + 97 + ], + "score": 1.0, + "content": "PSEUDOSEG: DESIGNING PSEUDO LABELS FOR", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 97, + 311, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 311, + 118 + ], + "score": 1.0, + "content": "SEMANTIC SEGMENTATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 114, + 134, + 382, + 169 + ], + "lines": [ + { + "bbox": [ + 115, + 134, + 382, + 147 + ], + "spans": [ + { + "bbox": [ + 115, + 134, + 150, + 147 + ], + "score": 1.0, + "content": "Yuliang", + "type": "text" + }, + { + "bbox": [ + 150, + 135, + 178, + 146 + ], + "score": 0.8, + "content": "\\mathbf { Z o u } ^ { 1 * }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 134, + 382, + 147 + ], + "score": 1.0, + "content": "Zizhao Zhang2 Han Zhang3 Chun-Liang Li2", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 114, + 145, + 309, + 159 + ], + "spans": [ + { + "bbox": [ + 114, + 145, + 309, + 159 + ], + "score": 1.0, + "content": "Xiao Bian2 Jia-Bin Huang1 Tomas Pfister2", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 115, + 157, + 318, + 170 + ], + "spans": [ + { + "bbox": [ + 115, + 157, + 318, + 170 + ], + "score": 1.0, + "content": "1Virginia Tech 2Google Cloud AI 3Google Brain", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 210 + ], + "lines": [ + { + "bbox": [ + 276, + 198, + 335, + 211 + ], + "spans": [ + { + "bbox": [ + 276, + 198, + 335, + 211 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 225, + 468, + 399 + ], + "lines": [ + { + "bbox": [ + 142, + 225, + 469, + 237 + ], + "spans": [ + { + "bbox": [ + 142, + 225, + 469, + 237 + ], + "score": 1.0, + "content": "Recent advances in semi-supervised learning (SSL) demonstrate that a combi-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "score": 1.0, + "content": "nation of consistency regularization and pseudo-labeling can effectively improve", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 247, + 469, + 259 + ], + "spans": [ + { + "bbox": [ + 141, + 247, + 469, + 259 + ], + "score": 1.0, + "content": "image classification accuracy in the low-data regime. 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Successful examples include MeanTeacher (Tarvainen & Valpola, 2017), UDA", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "(Xie et al., 2019), MixMatch (Berthelot et al., 2019b), FeatMatch (Kuo et al., 2020), and FixMatch", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "(Sohn et al., 2020a). One outstanding idea in this type of SSL is consistency training: making", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "predictions consistent among multiple augmented images. FixMatch (Sohn et al., 2020a) shows", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 690, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 701 + ], + "score": 1.0, + "content": "that using high-confidence one-hot pseudo labels obtained from weakly-augmented unlabeled data", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "to train strongly-augmented counterpart is the key to the success of SSL in image classification.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 622, + 506, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 93 + ], + "score": 1.0, + "content": "However, effective pseudo labels and well-designed data augmentation are non-trivial to satisfy for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "semantic segmentation. Although we observe that many related works explore the second condition", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "(i.e., augmentation) for image segmentation to enable consistency training framework (French et al.,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "2020; Ouali et al., 2020), we show that a wise design of pseudo labels for segmentation has great", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 177, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 177, + 138 + ], + "score": 1.0, + "content": "veiled potentials.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 108, + 143, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "In this paper, we propose PseudoSeg, a one-stage training framework to improve image semantic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "segmentation by leveraging additional data either with image-level labels (weakly-labeled data) or", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "without any labels. PseudoSeg presents a novel design of pseudo-labeling to infer effective struc-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "tured pseudo labels of additional data. It then optimizes the prediction of strongly-augmented data", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 478, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 478, + 200 + ], + "score": 1.0, + "content": "to match its corresponding pseudo labels. In summary, we make the following contributions:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 132, + 202, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 132, + 202, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 132, + 202, + 505, + 214 + ], + "score": 1.0, + "content": "• We propose a simple one-stage framework to improve semantic segmentation by using a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "limited amount of pixel-labeled data and sufficient unlabeled data or image-level labeled", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 224, + 479, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 224, + 479, + 236 + ], + "score": 1.0, + "content": "data. Our framework is simple to apply and therefore network architecture agnostic.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 139, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 139, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "Directly applying consistency training approaches validated in image classification ren-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 141, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "ders particular challenges in segmentation. We first demonstrate how well-calibrated soft", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "pseudo labels obtained through wise fusion of predictions from diverse sources can greatly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 270, + 332, + 281 + ], + "spans": [ + { + "bbox": [ + 142, + 270, + 332, + 281 + ], + "score": 1.0, + "content": "improve consistency training for segmentation.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "We conduct extensive experimental studies on the PASCAL VOC 2012 and COCO datasets.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "Comprehensive analyses are conducted to validate the effectiveness of this method at not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 142, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "only the low-data regime but also the high-data regime. Our experiments study multiple", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 314, + 469, + 327 + ], + "spans": [ + { + "bbox": [ + 142, + 314, + 469, + 327 + ], + "score": 1.0, + "content": "important open questions about transferring SSL advances to segmentation tasks.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 340, + 211, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 213, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 213, + 356 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "Semi-supervised classification. Semi-supervised learning (SSL) aims to improve model perfor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "mance by incorporating a large amount of unlabeled data during training. Consistency regularization", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "and entropy minimization are two common strategies for SSL. The intuition behind consistency-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "based approaches (Laine & Aila, 2016; Sajjadi et al., 2016; Miyato et al., 2018; Tarvainen & Valpola,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "2017) is that, the model output should remain unchanged when the input is perturbed. On the other", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "hand, the entropy minimization strategy (Grandvalet & Bengio, 2005) argues that the unlabeled", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "data can be used to ensured classes are well-separated, which can be achieved by encouraging the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "model to output low-entropy predictions. Pseudo-labeling (Lee, 2013) is one of the methods for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "score": 1.0, + "content": "implicit entropy minimization. Recently, holistic approaches (Berthelot et al., 2019b;a; Sohn et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 504, + 473 + ], + "score": 1.0, + "content": "2020a) combining both strategies have been proposed and achieved significant improvement. By re-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "designing the pseudo label, we propose an efficient one-stage semi-supervised learning framework", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 482, + 309, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 309, + 496 + ], + "score": 1.0, + "content": "of semantic segmentation for consistency training.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "score": 1.0, + "content": "Semi-supervised semantic segmentation. Collecting pixel-level annotations for semantic seg-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "mentation is costly and prone to error. Hence, leveraging unlabeled data in semantic segmentation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 517, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 506, + 533 + ], + "score": 1.0, + "content": "is a natural fit. Early methods utilize a GAN-based model either to generate additional training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "data (Souly et al., 2017) or to learn a discriminator between the prediction and the ground truth", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "mask (Hung et al., 2018; Mittal et al., 2019). Consistency regularization based approaches have also", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "been proposed recently, by enforcing the predictions to be consistent, either from augmented input", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "images (French et al., 2020; Kim et al., 2020), perturbed feature embeddings (Ouali et al., 2020), or", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "different networks (Ke et al., 2020). Recently, Luo & Yang (2020) proposes a dual-branch training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "network to jointly learn from pixel-accurate and coarse labeled data, achieving good segmentation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "performance. To push the performance of state of the arts, iterative self-training approaches (Chen", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "et al., 2020a; Zoph et al., 2020; Zhu et al., 2020) have been proposed. These methods usually assume", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "the available labeled data is enough to train a good teacher model, which will be used to generate", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "pseudo labels for the student model. However, this condition might not satisfy in the low-data", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "regime. Our proposed method, on the other hand, realizing the ideas of both consistency regular-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "ization and pseudo-labeling in segmentation, consistently improves the supervised baseline in both", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 661, + 237, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 237, + 675 + ], + "score": 1.0, + "content": "low-data and high-data regimes.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "Weakly-supervised semantic segmentation. Instead of supervising network training with accurate", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "pixel-level labels, many prior works exploit weaker forms of annotations (e.g., bounding boxes (Dai", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "score": 1.0, + "content": "et al., 2015), scribbles (Lin et al., 2016), image-level labels). Most recent approaches use image-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "level labels as the supervisory signal, which exploits the idea of class activation map (CAM) (Zhou", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "et al., 2016). Since the vanilla CAM only focus on the most discriminative region of objects, dif-", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 93 + ], + "score": 1.0, + "content": "However, effective pseudo labels and well-designed data augmentation are non-trivial to satisfy for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "semantic segmentation. Although we observe that many related works explore the second condition", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "(i.e., augmentation) for image segmentation to enable consistency training framework (French et al.,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "2020; Ouali et al., 2020), we show that a wise design of pseudo labels for segmentation has great", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 177, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 177, + 138 + ], + "score": 1.0, + "content": "veiled potentials.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 505, + 138 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 143, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "In this paper, we propose PseudoSeg, a one-stage training framework to improve image semantic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "segmentation by leveraging additional data either with image-level labels (weakly-labeled data) or", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "without any labels. PseudoSeg presents a novel design of pseudo-labeling to infer effective struc-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "tured pseudo labels of additional data. It then optimizes the prediction of strongly-augmented data", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 478, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 478, + 200 + ], + "score": 1.0, + "content": "to match its corresponding pseudo labels. In summary, we make the following contributions:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 143, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 202, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 132, + 202, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 132, + 202, + 505, + 214 + ], + "score": 1.0, + "content": "• We propose a simple one-stage framework to improve semantic segmentation by using a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "limited amount of pixel-labeled data and sufficient unlabeled data or image-level labeled", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 224, + 479, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 224, + 479, + 236 + ], + "score": 1.0, + "content": "data. Our framework is simple to apply and therefore network architecture agnostic.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 139, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 139, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "Directly applying consistency training approaches validated in image classification ren-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 141, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "ders particular challenges in segmentation. We first demonstrate how well-calibrated soft", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "pseudo labels obtained through wise fusion of predictions from diverse sources can greatly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 270, + 332, + 281 + ], + "spans": [ + { + "bbox": [ + 142, + 270, + 332, + 281 + ], + "score": 1.0, + "content": "improve consistency training for segmentation.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "We conduct extensive experimental studies on the PASCAL VOC 2012 and COCO datasets.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "Comprehensive analyses are conducted to validate the effectiveness of this method at not", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 142, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "only the low-data regime but also the high-data regime. Our experiments study multiple", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 314, + 469, + 327 + ], + "spans": [ + { + "bbox": [ + 142, + 314, + 469, + 327 + ], + "score": 1.0, + "content": "important open questions about transferring SSL advances to segmentation tasks.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 132, + 202, + 506, + 327 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 340, + 211, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 213, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 213, + 356 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "Semi-supervised classification. Semi-supervised learning (SSL) aims to improve model perfor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "mance by incorporating a large amount of unlabeled data during training. Consistency regularization", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "and entropy minimization are two common strategies for SSL. The intuition behind consistency-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "based approaches (Laine & Aila, 2016; Sajjadi et al., 2016; Miyato et al., 2018; Tarvainen & Valpola,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "2017) is that, the model output should remain unchanged when the input is perturbed. On the other", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "hand, the entropy minimization strategy (Grandvalet & Bengio, 2005) argues that the unlabeled", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "data can be used to ensured classes are well-separated, which can be achieved by encouraging the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "model to output low-entropy predictions. Pseudo-labeling (Lee, 2013) is one of the methods for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "score": 1.0, + "content": "implicit entropy minimization. Recently, holistic approaches (Berthelot et al., 2019b;a; Sohn et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 504, + 473 + ], + "score": 1.0, + "content": "2020a) combining both strategies have been proposed and achieved significant improvement. By re-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "designing the pseudo label, we propose an efficient one-stage semi-supervised learning framework", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 482, + 309, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 309, + 496 + ], + "score": 1.0, + "content": "of semantic segmentation for consistency training.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 362, + 505, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "score": 1.0, + "content": "Semi-supervised semantic segmentation. Collecting pixel-level annotations for semantic seg-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "mentation is costly and prone to error. Hence, leveraging unlabeled data in semantic segmentation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 517, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 506, + 533 + ], + "score": 1.0, + "content": "is a natural fit. Early methods utilize a GAN-based model either to generate additional training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "data (Souly et al., 2017) or to learn a discriminator between the prediction and the ground truth", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "mask (Hung et al., 2018; Mittal et al., 2019). Consistency regularization based approaches have also", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "been proposed recently, by enforcing the predictions to be consistent, either from augmented input", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "images (French et al., 2020; Kim et al., 2020), perturbed feature embeddings (Ouali et al., 2020), or", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "different networks (Ke et al., 2020). Recently, Luo & Yang (2020) proposes a dual-branch training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "network to jointly learn from pixel-accurate and coarse labeled data, achieving good segmentation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "performance. To push the performance of state of the arts, iterative self-training approaches (Chen", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "et al., 2020a; Zoph et al., 2020; Zhu et al., 2020) have been proposed. These methods usually assume", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "the available labeled data is enough to train a good teacher model, which will be used to generate", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "pseudo labels for the student model. However, this condition might not satisfy in the low-data", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "regime. Our proposed method, on the other hand, realizing the ideas of both consistency regular-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "ization and pseudo-labeling in segmentation, consistently improves the supervised baseline in both", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 661, + 237, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 237, + 675 + ], + "score": 1.0, + "content": "low-data and high-data regimes.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 495, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "Weakly-supervised semantic segmentation. Instead of supervising network training with accurate", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "pixel-level labels, many prior works exploit weaker forms of annotations (e.g., bounding boxes (Dai", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "score": 1.0, + "content": "et al., 2015), scribbles (Lin et al., 2016), image-level labels). Most recent approaches use image-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "level labels as the supervisory signal, which exploits the idea of class activation map (CAM) (Zhou", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "et al., 2016). Since the vanilla CAM only focus on the most discriminative region of objects, dif-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 254, + 504, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 504, + 267 + ], + "score": 1.0, + "content": "ferent ways to refine CAM have been proposed, including partial image/feature erasing (Hou et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "2018; Wei et al., 2017; Li et al., 2018), using an additional saliency estimation model (Oh et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "2017; Huang et al., 2018; Wei et al., 2018), utilizing pixel similarity to propagate the initial score", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "map (Ahn & Kwak, 2018; Wang et al., 2020), or mining and co-segment the same category of ob-", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 297, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 506, + 312 + ], + "score": 1.0, + "content": "jects across images (Sun et al., 2020; Zhang et al., 2020b). 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A straightforward solu-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 690, + 504, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 504, + 702 + ], + "score": 1.0, + "content": "etion is directly using the decoder output of a trained segmentation model after confidence threshold-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "score": 1.0, + "content": "ing, as suggested by Sohn et al. (2020a); Zoph et al. (2020); Xie et al. (2020); Sohn et al. 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First,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "we seek for a distinct yet efficient decision mechanisms to compensate for the potential errors of", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "decoder outputs. Second, wisely fusing multiple sources of predictions to generate an ensemble and", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 275, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 275, + 160 + ], + "score": 1.0, + "content": "better-calibrated version of pseudo labels.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 679, + 504, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "score": 1.0, + "content": "ing, as suggested by Sohn et al. (2020a); Zoph et al. (2020); Xie et al. (2020); Sohn et al. (2020b).", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "However, as we demonstrate later in the experiments, the generated pseudo hard/soft labels as well", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "as other post-processing of outputs are barely satisfactory in the low-data regime, and thus yield", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "inferior final results. To address this issue, our design of pseudo-labeling has two key insights. First,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "we seek for a distinct yet efficient decision mechanisms to compensate for the potential errors of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "decoder outputs. Second, wisely fusing multiple sources of predictions to generate an ensemble and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 275, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 275, + 160 + ], + "score": 1.0, + "content": "better-calibrated version of pseudo labels.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 163, + 505, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "Starting with localization. Compared with precise segmentation, learning localization is a simpler", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 174, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 187 + ], + "score": 1.0, + "content": "task as it only needs to provide coarser-grained outputs than pixel level of objects in images. Based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 504, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 504, + 197 + ], + "score": 1.0, + "content": "on this motivation, we improve decoder predictions from the localization perspective. Class activa-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "tion map (CAM) (Zhou et al., 2016) is a popular approach to provide localization for class-specific", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "regions. CAM-based methods (Hou et al., 2018; Wei et al., 2017; Ahn & Kwak, 2018) have been", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "successfully adopted to tackle a different weakly supervised semantic segmentation task from us,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 504, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 504, + 240 + ], + "score": 1.0, + "content": "where they assume only image-level labels are available. In practice, we adopt a variant of class", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 239, + 370, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 370, + 253 + ], + "score": 1.0, + "content": "activation map, Grad-CAM (Selvaraju et al., 2017) in PseudoSeg.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "score": 1.0, + "content": "From localization to segmentation. CAM estimates the strength of classifier responses on lo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "cal feature maps. Thus, an inherent limitation of CAM-based approaches is that it is prone to at-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "score": 1.0, + "content": "tending only to the most discriminative regions. Although many weakly-supervised segmentation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "approaches (Ahn & Kwak, 2018; Ahn et al., 2019; Sun et al., 2020) aim at refining CAM local-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "ization maps to segmentation masks, most of them have complicated post-processing steps, such as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "dense CRF (Krahenb ¨ uhl & Koltun ¨ , 2011), which increases the model complexity when used for con-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "sistency training. Here we present a computationally efficient yet effective refinement alternative,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 332, + 319, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 319, + 343 + ], + "score": 1.0, + "content": "which is learnable using available pixel-labeled data.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 504, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "Although CAM only localizes partial regions of interests, if we know the pairwise similarities be-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "tween regions, we can propagate the CAM scores from the discriminative regions to the rest un-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "attended regions. Actually, it has been shown in many works that the learned high-level deep fea-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "tures are usually good at similarity measurements of visual objects. In this paper, we find hypercol-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 392, + 491, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 491, + 405 + ], + "score": 1.0, + "content": "umn (Hariharan et al., 2015) with a learnable similarity measure function works fairly effective.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 278, + 421 + ], + "score": 1.0, + "content": "Given the vanilla Grad-CAM output for all", + "type": "text" + }, + { + "bbox": [ + 279, + 412, + 287, + 419 + ], + "score": 0.87, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "classes, which can be viewed as a spatially-flatten 2-D", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 417, + 507, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 174, + 434 + ], + "score": 1.0, + "content": "vector of weight", + "type": "text" + }, + { + "bbox": [ + 175, + 420, + 223, + 430 + ], + "score": 0.9, + "content": "m \\in \\mathbb { R } ^ { L \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 417, + 294, + 434 + ], + "score": 1.0, + "content": ", where each row", + "type": "text" + }, + { + "bbox": [ + 294, + 425, + 306, + 431 + ], + "score": 0.78, + "content": "m _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 417, + 497, + 434 + ], + "score": 1.0, + "content": "is the response weight per class for one region", + "type": "text" + }, + { + "bbox": [ + 497, + 422, + 501, + 430 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 417, + 507, + 434 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 201, + 444 + ], + "score": 1.0, + "content": "Using a kernel function", + "type": "text" + }, + { + "bbox": [ + 202, + 432, + 298, + 443 + ], + "score": 0.9, + "content": "\\mathcal { K } ( \\cdot , \\cdot ) : \\mathbb { R } ^ { H } \\times \\mathbb { R } ^ { H } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "that measures element-wise similarity given feature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 441, + 443, + 455 + ], + "spans": [ + { + "bbox": [ + 107, + 442, + 140, + 452 + ], + "score": 0.91, + "content": "h \\in { \\bar { \\mathbb { R } } } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 441, + 289, + 455 + ], + "score": 1.0, + "content": "of two regions, the propagated score", + "type": "text" + }, + { + "bbox": [ + 289, + 442, + 327, + 453 + ], + "score": 0.92, + "content": "\\hat { m } _ { i } \\in \\mathbb { R } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 441, + 443, + 455 + ], + "score": 1.0, + "content": "can be computed as follows", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 456, + 413, + 495 + ], + "lines": [ + { + "bbox": [ + 198, + 456, + 413, + 495 + ], + "spans": [ + { + "bbox": [ + 198, + 456, + 413, + 495 + ], + "score": 0.93, + "content": "\\hat { m } _ { i } = \\left( m _ { i } + \\sum _ { j = 0 } ^ { L - 1 } \\frac { e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { j } ) } } { \\sum _ { k = 0 } ^ { L - 1 } e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { k } ) } } m _ { j } \\right) \\cdot W _ { c } .", + "type": "interline_equation", + "image_path": "315a83a19acb724a2334523b444d0778b85206c5895fe1814d98891b2b465de8.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 198, + 456, + 413, + 469.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 198, + 469.0, + 413, + 482.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 198, + 482.0, + 413, + 495.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 104, + 495, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 251, + 513 + ], + "score": 1.0, + "content": "The goal of this function is to train", + "type": "text" + }, + { + "bbox": [ + 251, + 498, + 416, + 511 + ], + "score": 0.87, + "content": "\\Theta = \\{ W _ { k } , W _ { v } \\in \\mathbb { R } ^ { H \\times H } , W _ { c } \\in \\mathbb { R } ^ { C \\times C } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 495, + 507, + 513 + ], + "score": 1.0, + "content": "in order to propagate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 178, + 523 + ], + "score": 1.0, + "content": "the high value in", + "type": "text" + }, + { + "bbox": [ + 179, + 511, + 189, + 519 + ], + "score": 0.73, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 507, + 372, + 523 + ], + "score": 1.0, + "content": "to all adjacent elements in the feature space", + "type": "text" + }, + { + "bbox": [ + 373, + 510, + 388, + 520 + ], + "score": 0.84, + "content": "\\mathbb { R } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "(i.e., hypercolumn features)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 145, + 533 + ], + "score": 1.0, + "content": "to region", + "type": "text" + }, + { + "bbox": [ + 146, + 522, + 150, + 531 + ], + "score": 0.57, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 521, + 189, + 533 + ], + "score": 1.0, + "content": ". Adding", + "type": "text" + }, + { + "bbox": [ + 189, + 522, + 202, + 532 + ], + "score": 0.87, + "content": "m _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "in equation 3 indicates the skip-connection. To compute propagated score", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "for all regions, the operations in equation 3 can be efficiently implemented with self-attention dot-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "product (Vaswani et al., 2017). For brevity, we denote this efficient refinement process output as self-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 550, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 255, + 567 + ], + "score": 1.0, + "content": "attention Grad-CAM (SGC) maps in", + "type": "text" + }, + { + "bbox": [ + 255, + 552, + 296, + 563 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { H \\times H \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 550, + 504, + 567 + ], + "score": 1.0, + "content": ". Figure 6 in Appendix A specifies the architecture.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "Calibrated prediction fusion. SGC maps are obtained from low-resolution feature maps. It is then", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "resized to the desired output resolution, and thus not sufficient at delineating crisp boundaries. How-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "ever, compared to the segmentation decoder, SGC is capable of generating more locally-consistent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "masks. Thus, we propose a novel calibrated fusion strategy to take advantage of both decoder and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 613, + 272, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 272, + 623 + ], + "score": 1.0, + "content": "SCG predictions for better pseudo labels.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 372, + 642 + ], + "score": 1.0, + "content": "Specifically, given a batch of decoder outputs (pre-softmax logits)", + "type": "text" + }, + { + "bbox": [ + 373, + 629, + 429, + 641 + ], + "score": 0.93, + "content": "\\hat { p } = f _ { \\theta } ( \\omega ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 628, + 494, + 642 + ], + "score": 1.0, + "content": "and SGC maps", + "type": "text" + }, + { + "bbox": [ + 494, + 630, + 504, + 639 + ], + "score": 0.77, + "content": "\\hat { m }", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 434, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 267, + 654 + ], + "score": 1.0, + "content": "computed from weakly-augmented data", + "type": "text" + }, + { + "bbox": [ + 268, + 640, + 289, + 652 + ], + "score": 0.92, + "content": "\\omega ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 640, + 414, + 654 + ], + "score": 1.0, + "content": ", we generate the pseudo labels", + "type": "text" + }, + { + "bbox": [ + 414, + 641, + 421, + 652 + ], + "score": 0.83, + "content": "\\widetilde { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 640, + 434, + 654 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 654, + 484, + 682 + ], + "lines": [ + { + "bbox": [ + 114, + 654, + 484, + 682 + ], + "spans": [ + { + "bbox": [ + 114, + 654, + 484, + 682 + ], + "score": 0.91, + "content": "\\mathcal { F } ( \\hat { p } , \\hat { m } ) = \\mathrm { S h a r p e n } \\left( \\gamma \\operatorname { S o f t m a x } \\left( \\frac { \\hat { p } } { \\operatorname { N o r m } ( \\hat { p } , \\hat { m } ) } \\right) + ( 1 - \\gamma ) \\operatorname { S o f t m a x } \\left( \\frac { \\hat { m } } { \\operatorname { N o r m } ( \\hat { p } , \\hat { m } ) } \\right) , T \\right) .", + "type": "interline_equation", + "image_path": "8a2cc9c7dadb07213de1f86511ba990a5e698739952cb66fb24d2207c967f266.jpg" + } + ] + } + ], + "index": 49, + "virtual_lines": [ + { + "bbox": [ + 114, + 654, + 484, + 663.3333333333334 + ], + "spans": [], + "index": 48 + }, + { + "bbox": [ + 114, + 663.3333333333334, + 484, + 672.6666666666667 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 114, + 672.6666666666667, + 484, + 682.0000000000001 + ], + "spans": [], + "index": 50 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 683, + 505, + 725 + ], + "lines": [ + { + "bbox": [ + 106, + 683, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 696 + ], + "score": 1.0, + "content": "Two critical procedures are proposed to use here to make the fusion process successful. First,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 693, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 113, + 706 + ], + "score": 0.83, + "content": "\\hat { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 693, + 133, + 707 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 133, + 695, + 144, + 704 + ], + "score": 0.82, + "content": "\\hat { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 693, + 505, + 707 + ], + "score": 1.0, + "content": "are from different decision mechanisms and they could have very different degrees of", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 103, + 705, + 507, + 728 + ], + "spans": [ + { + "bbox": [ + 103, + 705, + 333, + 728 + ], + "score": 1.0, + "content": "overconfidence. Therefore, we introduce the operation", + "type": "text" + }, + { + "bbox": [ + 333, + 705, + 465, + 726 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathrm { N o r m } ( a , b ) = \\sqrt { \\sum _ { i } ^ { | a | } ( a _ { i } ^ { 2 } + b _ { i } ^ { 2 } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 705, + 507, + 728 + ], + "score": 1.0, + "content": "as a nor-", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 82, + 505, + 160 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 163, + 505, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "Starting with localization. Compared with precise segmentation, learning localization is a simpler", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 174, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 187 + ], + "score": 1.0, + "content": "task as it only needs to provide coarser-grained outputs than pixel level of objects in images. Based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 504, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 504, + 197 + ], + "score": 1.0, + "content": "on this motivation, we improve decoder predictions from the localization perspective. Class activa-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "tion map (CAM) (Zhou et al., 2016) is a popular approach to provide localization for class-specific", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "regions. CAM-based methods (Hou et al., 2018; Wei et al., 2017; Ahn & Kwak, 2018) have been", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "successfully adopted to tackle a different weakly supervised semantic segmentation task from us,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 504, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 504, + 240 + ], + "score": 1.0, + "content": "where they assume only image-level labels are available. In practice, we adopt a variant of class", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 239, + 370, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 370, + 253 + ], + "score": 1.0, + "content": "activation map, Grad-CAM (Selvaraju et al., 2017) in PseudoSeg.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 162, + 505, + 253 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "score": 1.0, + "content": "From localization to segmentation. CAM estimates the strength of classifier responses on lo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "cal feature maps. Thus, an inherent limitation of CAM-based approaches is that it is prone to at-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 504, + 289 + ], + "score": 1.0, + "content": "tending only to the most discriminative regions. Although many weakly-supervised segmentation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "approaches (Ahn & Kwak, 2018; Ahn et al., 2019; Sun et al., 2020) aim at refining CAM local-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "ization maps to segmentation masks, most of them have complicated post-processing steps, such as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "dense CRF (Krahenb ¨ uhl & Koltun ¨ , 2011), which increases the model complexity when used for con-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "sistency training. Here we present a computationally efficient yet effective refinement alternative,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 332, + 319, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 319, + 343 + ], + "score": 1.0, + "content": "which is learnable using available pixel-labeled data.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 255, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 504, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "Although CAM only localizes partial regions of interests, if we know the pairwise similarities be-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "tween regions, we can propagate the CAM scores from the discriminative regions to the rest un-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "attended regions. Actually, it has been shown in many works that the learned high-level deep fea-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "tures are usually good at similarity measurements of visual objects. In this paper, we find hypercol-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 392, + 491, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 491, + 405 + ], + "score": 1.0, + "content": "umn (Hariharan et al., 2015) with a learnable similarity measure function works fairly effective.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 348, + 506, + 405 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 278, + 421 + ], + "score": 1.0, + "content": "Given the vanilla Grad-CAM output for all", + "type": "text" + }, + { + "bbox": [ + 279, + 412, + 287, + 419 + ], + "score": 0.87, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "classes, which can be viewed as a spatially-flatten 2-D", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 417, + 507, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 174, + 434 + ], + "score": 1.0, + "content": "vector of weight", + "type": "text" + }, + { + "bbox": [ + 175, + 420, + 223, + 430 + ], + "score": 0.9, + "content": "m \\in \\mathbb { R } ^ { L \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 417, + 294, + 434 + ], + "score": 1.0, + "content": ", where each row", + "type": "text" + }, + { + "bbox": [ + 294, + 425, + 306, + 431 + ], + "score": 0.78, + "content": "m _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 417, + 497, + 434 + ], + "score": 1.0, + "content": "is the response weight per class for one region", + "type": "text" + }, + { + "bbox": [ + 497, + 422, + 501, + 430 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 417, + 507, + 434 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 201, + 444 + ], + "score": 1.0, + "content": "Using a kernel function", + "type": "text" + }, + { + "bbox": [ + 202, + 432, + 298, + 443 + ], + "score": 0.9, + "content": "\\mathcal { K } ( \\cdot , \\cdot ) : \\mathbb { R } ^ { H } \\times \\mathbb { R } ^ { H } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "that measures element-wise similarity given feature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 441, + 443, + 455 + ], + "spans": [ + { + "bbox": [ + 107, + 442, + 140, + 452 + ], + "score": 0.91, + "content": "h \\in { \\bar { \\mathbb { R } } } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 441, + 289, + 455 + ], + "score": 1.0, + "content": "of two regions, the propagated score", + "type": "text" + }, + { + "bbox": [ + 289, + 442, + 327, + 453 + ], + "score": 0.92, + "content": "\\hat { m } _ { i } \\in \\mathbb { R } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 441, + 443, + 455 + ], + "score": 1.0, + "content": "can be computed as follows", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 409, + 507, + 455 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 456, + 413, + 495 + ], + "lines": [ + { + "bbox": [ + 198, + 456, + 413, + 495 + ], + "spans": [ + { + "bbox": [ + 198, + 456, + 413, + 495 + ], + "score": 0.93, + "content": "\\hat { m } _ { i } = \\left( m _ { i } + \\sum _ { j = 0 } ^ { L - 1 } \\frac { e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { j } ) } } { \\sum _ { k = 0 } ^ { L - 1 } e ^ { K ( W _ { k } h _ { i } , W _ { v } h _ { k } ) } } m _ { j } \\right) \\cdot W _ { c } .", + "type": "interline_equation", + "image_path": "315a83a19acb724a2334523b444d0778b85206c5895fe1814d98891b2b465de8.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 198, + 456, + 413, + 469.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 198, + 469.0, + 413, + 482.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 198, + 482.0, + 413, + 495.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 104, + 495, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 251, + 513 + ], + "score": 1.0, + "content": "The goal of this function is to train", + "type": "text" + }, + { + "bbox": [ + 251, + 498, + 416, + 511 + ], + "score": 0.87, + "content": "\\Theta = \\{ W _ { k } , W _ { v } \\in \\mathbb { R } ^ { H \\times H } , W _ { c } \\in \\mathbb { R } ^ { C \\times C } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 495, + 507, + 513 + ], + "score": 1.0, + "content": "in order to propagate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 178, + 523 + ], + "score": 1.0, + "content": "the high value in", + "type": "text" + }, + { + "bbox": [ + 179, + 511, + 189, + 519 + ], + "score": 0.73, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 507, + 372, + 523 + ], + "score": 1.0, + "content": "to all adjacent elements in the feature space", + "type": "text" + }, + { + "bbox": [ + 373, + 510, + 388, + 520 + ], + "score": 0.84, + "content": "\\mathbb { R } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "(i.e., hypercolumn features)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 145, + 533 + ], + "score": 1.0, + "content": "to region", + "type": "text" + }, + { + "bbox": [ + 146, + 522, + 150, + 531 + ], + "score": 0.57, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 521, + 189, + 533 + ], + "score": 1.0, + "content": ". Adding", + "type": "text" + }, + { + "bbox": [ + 189, + 522, + 202, + 532 + ], + "score": 0.87, + "content": "m _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "in equation 3 indicates the skip-connection. To compute propagated score", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "for all regions, the operations in equation 3 can be efficiently implemented with self-attention dot-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "product (Vaswani et al., 2017). For brevity, we denote this efficient refinement process output as self-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 550, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 255, + 567 + ], + "score": 1.0, + "content": "attention Grad-CAM (SGC) maps in", + "type": "text" + }, + { + "bbox": [ + 255, + 552, + 296, + 563 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { H \\times H \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 550, + 504, + 567 + ], + "score": 1.0, + "content": ". Figure 6 in Appendix A specifies the architecture.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 495, + 507, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "Calibrated prediction fusion. SGC maps are obtained from low-resolution feature maps. It is then", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "resized to the desired output resolution, and thus not sufficient at delineating crisp boundaries. How-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "ever, compared to the segmentation decoder, SGC is capable of generating more locally-consistent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "masks. Thus, we propose a novel calibrated fusion strategy to take advantage of both decoder and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 613, + 272, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 272, + 623 + ], + "score": 1.0, + "content": "SCG predictions for better pseudo labels.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 568, + 505, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 372, + 642 + ], + "score": 1.0, + "content": "Specifically, given a batch of decoder outputs (pre-softmax logits)", + "type": "text" + }, + { + "bbox": [ + 373, + 629, + 429, + 641 + ], + "score": 0.93, + "content": "\\hat { p } = f _ { \\theta } ( \\omega ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 628, + 494, + 642 + ], + "score": 1.0, + "content": "and SGC maps", + "type": "text" + }, + { + "bbox": [ + 494, + 630, + 504, + 639 + ], + "score": 0.77, + "content": "\\hat { m }", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 434, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 267, + 654 + ], + "score": 1.0, + "content": "computed from weakly-augmented data", + "type": "text" + }, + { + "bbox": [ + 268, + 640, + 289, + 652 + ], + "score": 0.92, + "content": "\\omega ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 640, + 414, + 654 + ], + "score": 1.0, + "content": ", we generate the pseudo labels", + "type": "text" + }, + { + "bbox": [ + 414, + 641, + 421, + 652 + ], + "score": 0.83, + "content": "\\widetilde { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 640, + 434, + 654 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 628, + 504, + 654 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 654, + 484, + 682 + ], + "lines": [ + { + "bbox": [ + 114, + 654, + 484, + 682 + ], + "spans": [ + { + "bbox": [ + 114, + 654, + 484, + 682 + ], + "score": 0.91, + "content": "\\mathcal { F } ( \\hat { p } , \\hat { m } ) = \\mathrm { S h a r p e n } \\left( \\gamma \\operatorname { S o f t m a x } \\left( \\frac { \\hat { p } } { \\operatorname { N o r m } ( \\hat { p } , \\hat { m } ) } \\right) + ( 1 - \\gamma ) \\operatorname { S o f t m a x } \\left( \\frac { \\hat { m } } { \\operatorname { N o r m } ( \\hat { p } , \\hat { m } ) } \\right) , T \\right) .", + "type": "interline_equation", + "image_path": "8a2cc9c7dadb07213de1f86511ba990a5e698739952cb66fb24d2207c967f266.jpg" + } + ] + } + ], + "index": 49, + "virtual_lines": [ + { + "bbox": [ + 114, + 654, + 484, + 663.3333333333334 + ], + "spans": [], + "index": 48 + }, + { + "bbox": [ + 114, + 663.3333333333334, + 484, + 672.6666666666667 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 114, + 672.6666666666667, + 484, + 682.0000000000001 + ], + "spans": [], + "index": 50 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 683, + 505, + 725 + ], + "lines": [ + { + "bbox": [ + 106, + 683, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 696 + ], + "score": 1.0, + "content": "Two critical procedures are proposed to use here to make the fusion process successful. First,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 693, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 113, + 706 + ], + "score": 0.83, + "content": "\\hat { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 693, + 133, + 707 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 133, + 695, + 144, + 704 + ], + "score": 0.82, + "content": "\\hat { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 693, + 505, + 707 + ], + "score": 1.0, + "content": "are from different decision mechanisms and they could have very different degrees of", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 103, + 705, + 507, + 728 + ], + "spans": [ + { + "bbox": [ + 103, + 705, + 333, + 728 + ], + "score": 1.0, + "content": "overconfidence. Therefore, we introduce the operation", + "type": "text" + }, + { + "bbox": [ + 333, + 705, + 465, + 726 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathrm { N o r m } ( a , b ) = \\sqrt { \\sum _ { i } ^ { | a | } ( a _ { i } ^ { 2 } + b _ { i } ^ { 2 } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 705, + 507, + 728 + ], + "score": 1.0, + "content": "as a nor-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 178, + 504, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 504, + 190 + ], + "score": 1.0, + "content": "malization factor. It alleviates the over-confident probability after softmax, which could unfavor-", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 217, + 202 + ], + "score": 1.0, + "content": "ably dominate the resulted", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 218, + 191, + 225, + 200 + ], + "score": 0.79, + "content": "\\gamma", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 225, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "-averaged probability. Second, the distribution sharpening operation", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 200, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 139, + 214 + ], + "score": 1.0, + "content": "Sharpen", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 140, + 200, + 253, + 217 + ], + "score": 0.92, + "content": "\\begin{array} { r } { ( a , T ) _ { i } ~ = ~ a _ { i } ^ { 1 / T } / \\sum _ { j } ^ { C } a _ { j } ^ { 1 / T } } \\end{array}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 255, + 202, + 384, + 216 + ], + "score": 1.0, + "content": "adjusts the temperature scalar", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 385, + 203, + 393, + 213 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 394, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "of categorical distribution", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "(Berthelot et al., 2019b; Chen et al., 2020b). Figure 2 illustrates the predictions from different", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "sources. More importantly, we investigate the pseudo-labeling from a calibration perspective (Sec-", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 504, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 353, + 249 + ], + "score": 1.0, + "content": "tion 4.3), demonstrating that the proposed soft pseudo label", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 353, + 237, + 361, + 248 + ], + "score": 0.84, + "content": "\\widetilde { y }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 361, + 237, + 504, + 249 + ], + "score": 1.0, + "content": "leads to a better calibration metric", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "score": 1.0, + "content": "comparing to other possible fusion alternatives, and justifying why it benefits the final segmentation", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 260, + 162, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 162, + 271 + ], + "score": 1.0, + "content": "performance.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 52, + "bbox_fs": [ + 103, + 683, + 507, + 728 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 80, + 500, + 129 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 80, + 500, + 129 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 80, + 500, + 129 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 500, + 129 + ], + "score": 0.962, + "type": "image", + "image_path": "da24262064ec7dc7f8d685b2d6566bc0dcda547c410ce0f2f99758c6eebe8bf2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 80, + 500, + 96.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 96.33333333333333, + 500, + 112.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 112.66666666666666, + 500, + 129.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 131, + 493, + 165 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 131, + 495, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 495, + 145 + ], + "score": 1.0, + "content": "Figure 2: Visualization of pseudo labels and other predictions. The generated pseudo label by", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 141, + 487, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 487, + 156 + ], + "score": 1.0, + "content": "fusing the predictions from the decoder and SGC map is used to supervise the decoder (strong)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 311, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 311, + 167 + ], + "score": 1.0, + "content": "predictions of the strongly-augmented counterpart.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 504, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 504, + 190 + ], + "score": 1.0, + "content": "malization factor. It alleviates the over-confident probability after softmax, which could unfavor-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 217, + 202 + ], + "score": 1.0, + "content": "ably dominate the resulted", + "type": "text" + }, + { + "bbox": [ + 218, + 191, + 225, + 200 + ], + "score": 0.79, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "-averaged probability. Second, the distribution sharpening operation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 200, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 139, + 214 + ], + "score": 1.0, + "content": "Sharpen", + "type": "text" + }, + { + "bbox": [ + 140, + 200, + 253, + 217 + ], + "score": 0.92, + "content": "\\begin{array} { r } { ( a , T ) _ { i } ~ = ~ a _ { i } ^ { 1 / T } / \\sum _ { j } ^ { C } a _ { j } ^ { 1 / T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 202, + 384, + 216 + ], + "score": 1.0, + "content": "adjusts the temperature scalar", + "type": "text" + }, + { + "bbox": [ + 385, + 203, + 393, + 213 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "of categorical distribution", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "(Berthelot et al., 2019b; Chen et al., 2020b). Figure 2 illustrates the predictions from different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "sources. More importantly, we investigate the pseudo-labeling from a calibration perspective (Sec-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 504, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 353, + 249 + ], + "score": 1.0, + "content": "tion 4.3), demonstrating that the proposed soft pseudo label", + "type": "text" + }, + { + "bbox": [ + 353, + 237, + 361, + 248 + ], + "score": 0.84, + "content": "\\widetilde { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 237, + 504, + 249 + ], + "score": 1.0, + "content": "leads to a better calibration metric", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "score": 1.0, + "content": "comparing to other possible fusion alternatives, and justifying why it benefits the final segmentation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 260, + 162, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 162, + 271 + ], + "score": 1.0, + "content": "performance.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 462, + 286 + ], + "score": 1.0, + "content": "Training. Our final training objective contains two extra losses: a classification loss", + "type": "text" + }, + { + "bbox": [ + 462, + 274, + 475, + 285 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 273, + 506, + 286 + ], + "score": 1.0, + "content": ", and a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 179, + 297 + ], + "score": 1.0, + "content": "segmentation loss", + "type": "text" + }, + { + "bbox": [ + 179, + 285, + 195, + 296 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { s a }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 285, + 505, + 297 + ], + "score": 1.0, + "content": ". First, to compute Grad-CAM, we add a one-layer classification head after the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 344, + 308 + ], + "score": 1.0, + "content": "segmentation backbone and a multi-label classification loss", + "type": "text" + }, + { + "bbox": [ + 344, + 296, + 357, + 307 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 296, + 505, + 308 + ], + "score": 1.0, + "content": ". Second, as specified in Appendix A", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "(Figure 6), SGC maps are scaled as pixel-wise probabilities using one-layer convolution followed by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 237, + 330 + ], + "score": 1.0, + "content": "softmax in equation 3. Learning", + "type": "text" + }, + { + "bbox": [ + 237, + 318, + 246, + 327 + ], + "score": 0.78, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 318, + 432, + 330 + ], + "score": 1.0, + "content": "to predict SGC maps needs pixel-labeled data", + "type": "text" + }, + { + "bbox": [ + 433, + 318, + 445, + 329 + ], + "score": 0.87, + "content": "D _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 318, + 505, + 330 + ], + "score": 1.0, + "content": ". It is achieved", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 234, + 342 + ], + "score": 1.0, + "content": "by an extra segmentation loss", + "type": "text" + }, + { + "bbox": [ + 234, + 329, + 250, + 340 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { s a }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "between SGC maps of pixel-labeled data and corresponding", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 344, + 353 + ], + "score": 1.0, + "content": "ground truth. All the loss terms are jointly optimized (i.e.,", + "type": "text" + }, + { + "bbox": [ + 344, + 339, + 437, + 351 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { u } + \\mathcal { L } _ { s } + \\mathcal { L } _ { x } + \\mathcal { L } _ { s a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 339, + 466, + 353 + ], + "score": 1.0, + "content": ", while", + "type": "text" + }, + { + "bbox": [ + 466, + 340, + 483, + 351 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { s a }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "only", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 481, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 147, + 363 + ], + "score": 1.0, + "content": "optimizes", + "type": "text" + }, + { + "bbox": [ + 148, + 351, + 157, + 361 + ], + "score": 0.79, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 351, + 481, + 363 + ], + "score": 1.0, + "content": "(achieved by stopping gradient). See Figure 7 in the appendix for further details.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 376, + 393, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 394, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 394, + 389 + ], + "score": 1.0, + "content": "3.2 INCORPORATING IMAGE-LEVEL LABELS AND AUGMENTATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 505, + 449 + ], + "lines": [ + { + "bbox": [ + 107, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "The proposed PseudoSeg can easily incorporate image-level label information (if available) into our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "one-stage training framework, which also leads to consistent improvement as we demonstrate in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "experiments. We utilize the image-level data with two following steps. First, we directly use ground", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "truth image-level labels to generate Grad-CAMs instead of using classifier outputs. Second, they are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 468, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 468, + 451 + ], + "score": 1.0, + "content": "used to increase classification supervision beyond pixel-level labels for the classifier head.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "For strong data augmentation, we simply follow color jittering operations from SimCLR (Chen", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "score": 1.0, + "content": "et al., 2020b) and remove all geometric transformations. The overall strength of augmentation can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "be controlled by a scalar (studied in experiments). We also apply once random CutOut (DeVries", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 247, + 501 + ], + "score": 1.0, + "content": "& Taylor, 2017) with a region of", + "type": "text" + }, + { + "bbox": [ + 247, + 488, + 282, + 499 + ], + "score": 0.9, + "content": "5 0 \\times 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 487, + 505, + 501 + ], + "score": 1.0, + "content": "pixels since we find it gives consistent though minor", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 499, + 415, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 415, + 512 + ], + "score": 1.0, + "content": "improvement (pixels inside CutOut regions are ignored in computing losses).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 257, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 258, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 258, + 541 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "We start by specifying the experimental details. Then, we evaluate the method in the settings of using", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "pixel-level labeled data and unlabeled data, as well as using pixel-level labeled data and image-level", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "labeled data, respectively. Next, we conduct various ablation studies to justify our design choices.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 583, + 388, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 388, + 596 + ], + "score": 1.0, + "content": "Lastly, we conduct more comparative experiments in specific settings.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "To evaluate the proposed method, we conduct the main experiments and ablation studies on the PAS-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "CAL VOC 2012 dataset (VOC12) (Everingham et al., 2015), which contains 21 classes including", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "background. The standard VOC12 dataset has 1,449 images as the training set and 1,456 images", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "as the validation set. We randomly subsample 1/2, 1/4, 1/8, and 1/16 of images in the standard", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "training set to construct the pixel-level labeled data. The remaining images in the standard training", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 440, + 667 + ], + "score": 1.0, + "content": "set, together with the images in the augmented set (Hariharan et al., 2011) (around", + "type": "text" + }, + { + "bbox": [ + 441, + 655, + 453, + 666 + ], + "score": 0.35, + "content": "9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "images), are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "used as unlabeled or image-level labeled data. To further verify the effectiveness of the proposed", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "method, we also conduct experiments on the COCO dataset (Lin et al., 2014). The COCO dataset", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "has 118,287 images as the training set, and 5,000 images as the validation set. We evaluate on the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "80 foreground classes and the background, as in the object detection task. As the COCO dataset", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "is larger than VOC12, we randomly subsample smaller ratios, 1/32, 1/64, 1/128, 1/256, 1/512, of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "images from the training set to construct the pixel-level labeled data. 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See Figure 7 in the appendix for further details.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 273, + 506, + 363 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 376, + 393, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 394, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 394, + 389 + ], + "score": 1.0, + "content": "3.2 INCORPORATING IMAGE-LEVEL LABELS AND AUGMENTATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 505, + 449 + ], + "lines": [ + { + "bbox": [ + 107, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "The proposed PseudoSeg can easily incorporate image-level label information (if available) into our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "one-stage training framework, which also leads to consistent improvement as we demonstrate in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "experiments. We utilize the image-level data with two following steps. First, we directly use ground", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "truth image-level labels to generate Grad-CAMs instead of using classifier outputs. Second, they are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 468, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 468, + 451 + ], + "score": 1.0, + "content": "used to increase classification supervision beyond pixel-level labels for the classifier head.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 394, + 505, + 451 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "For strong data augmentation, we simply follow color jittering operations from SimCLR (Chen", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "score": 1.0, + "content": "et al., 2020b) and remove all geometric transformations. The overall strength of augmentation can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "be controlled by a scalar (studied in experiments). We also apply once random CutOut (DeVries", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 247, + 501 + ], + "score": 1.0, + "content": "& Taylor, 2017) with a region of", + "type": "text" + }, + { + "bbox": [ + 247, + 488, + 282, + 499 + ], + "score": 0.9, + "content": "5 0 \\times 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 487, + 505, + 501 + ], + "score": 1.0, + "content": "pixels since we find it gives consistent though minor", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 499, + 415, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 415, + 512 + ], + "score": 1.0, + "content": "improvement (pixels inside CutOut regions are ignored in computing losses).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 455, + 505, + 512 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 257, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 258, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 258, + 541 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "We start by specifying the experimental details. 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MethodNetworkmIoU (%)
GANSeg (Souly et al., 2017)VGG1664.10
AdvSemSeg (Hung et al., 2018)ResNet-10168.40
CCT (Ouali et al., 2020)ResNet-5069.40
PseudoSeg (Ours)ResNet-5071.00
PseudoSeg (Ours)ResNet-10173.23
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Method1/2 (732)1/4 (366)1/8 (183)1/16 (92)
AdvSemSeg (Hung et al., 2018)65.2759.9747.5839.69
CCT (Ouali et al., 2020)62.1058.8047.6033.10
*MT (Tarvainen & Valpola, 2017)69.1663.0155.8148.70
GCT (Ke et al., 2020)70.6764.7154.9846.04
**VAT (Miyato et al., 2018)63.3456.8849.3536.92
CutMix (French et al., 2020)69.8468.3663.2055.58
PseudoSeg (Ours)72.4169.1465.5057.60
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MethodNetworkmIoU (%)
GANSeg (Souly et al., 2017)VGG1664.10
AdvSemSeg (Hung et al., 2018)ResNet-10168.40
CCT (Ouali et al., 2020)ResNet-5069.40
PseudoSeg (Ours)ResNet-5071.00
PseudoSeg (Ours)ResNet-10173.23
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Method1/2 (732)1/4 (366)1/8 (183)1/16 (92)
AdvSemSeg (Hung et al., 2018)65.2759.9747.5839.69
CCT (Ouali et al., 2020)62.1058.8047.6033.10
*MT (Tarvainen & Valpola, 2017)69.1663.0155.8148.70
GCT (Ke et al., 2020)70.6764.7154.9846.04
**VAT (Miyato et al., 2018)63.3456.8849.3536.92
CutMix (French et al., 2020)69.8468.3663.2055.58
PseudoSeg (Ours)72.4169.1465.5057.60
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MethodModelNetworkmIoU (%)
WSSN (Papandreou et al., 2015)DeepLab-CRFVGG1664.60
GAIN (Li et al., 2018)DeepLab-CRF-LFOVVGG1660.50
MDC (Wei et al.,2018)DeepLab-CRF-LFOVVGG1665.70
DSRG (Huang et al.,2018)DeepLabv2VGG1664.30
GANSeg (Souly et al.,2017)FCNVGG1665.80
FickleNet (Lee et al.,2019)DeepLabv2ResNet-10165.80
CCT (Ouali et al., 2020)PSP-NetResNet-5073.20
PseudoSeg (Ours)DeepLabv3+ResNet-5073.80
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SplitCCTPseudoSeg
1/266.8073.51
1/467.6071.79
1/862.5069.15
1/1651.8065.44
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As", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "shown in Figure 4, the proposed method consistently improves the strong baseline on both datasets.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 257, + 446 + ], + "score": 1.0, + "content": "In Table 3, we evaluate on the public", + "type": "text" + }, + { + "bbox": [ + 257, + 433, + 289, + 444 + ], + "score": 0.64, + "content": "1 . 4 \\mathrm { k } / 9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "split. The proposed method compares favorably with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "the other methods. Moreover, we further compare to best compared CCT on the created low-data", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "splits (in Table 4). Both experiments show that the proposed PseudoSeg is more robust than the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "compared method given less data. On all splits on both datasets, using pixel-level labeled data and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "image-labeled data shows higher mIoU than the setting using pixel-level labeled data and unlabeled", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 128, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 128, + 500 + ], + "score": 1.0, + "content": "data.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 514, + 208, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 209, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 209, + 527 + ], + "score": 1.0, + "content": "4.3 ABLATION STUDY", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 503, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "In this section, we conduct extensive ablation experiments on VOC12 to validate our design choices.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 548, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 562 + ], + "score": 1.0, + "content": "How to construct pseudo label? We investigate the effectiveness of the proposed pseudo labeling.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Table 5 demonstrates quantitative results, indicating that using either decoder output or SGC alone", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "gives an inferior performance. Naively using decoder output as pseudo labels can hardly work", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "well. The proposed fusion consistently performs better, either with or without additional image-level", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "labels. To further answer why our pseudo labels are effective, we study from the model calibration", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "perspective. We measure the expected calibration error (ECE) (Guo et al., 2017) scores of all the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 628 + ], + "score": 1.0, + "content": "intermediate steps and other fusion variants. As shown in Figure 5 (a), the proposed fusion strategy", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 637 + ], + "score": 1.0, + "content": "(denoted as G in the figure) achieves the lowest ECE scores, indicating that the significance of jointly", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "using normalization with sharpening (see equation 4) compared with other fusion alternatives. We", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 659 + ], + "score": 1.0, + "content": "hypothesize using well-calibrated soft labels makes model training less affected by label noises. The", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 658, + 395, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 395, + 671 + ], + "score": 1.0, + "content": "comprehensive calibration study is left as a future exploration direction.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 673, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 504, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 504, + 685 + ], + "score": 1.0, + "content": "Using hypercolumn feature or not? In Figure 5 (b), we study the effectiveness of using hypercol-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "umn features instead of the last feature maps in equation 3. We conduct the experiments on the 1/16", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 694, + 468, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 468, + 708 + ], + "score": 1.0, + "content": "split of VOC12. As we can see, hypercolumn features substantially improve performance.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Soft or hard pseudo label? How to utilize predictions as pseudo labels remains an active question", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "in SSL. Next, we study whether we should use soft or hard one-hot pseudo labels. 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We use the official training set (1.4k) as labeled data,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 266, + 330, + 279 + ], + "spans": [ + { + "bbox": [ + 107, + 266, + 330, + 279 + ], + "score": 1.0, + "content": "and the augmented set (9k) as image-level labeled data.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 279, + 337, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 279, + 337, + 354 + ], + "spans": [ + { + "bbox": [ + 108, + 279, + 337, + 354 + ], + "score": 0.974, + "html": "
MethodModelNetworkmIoU (%)
WSSN (Papandreou et al., 2015)DeepLab-CRFVGG1664.60
GAIN (Li et al., 2018)DeepLab-CRF-LFOVVGG1660.50
MDC (Wei et al.,2018)DeepLab-CRF-LFOVVGG1665.70
DSRG (Huang et al.,2018)DeepLabv2VGG1664.30
GANSeg (Souly et al.,2017)FCNVGG1665.80
FickleNet (Lee et al.,2019)DeepLabv2ResNet-10165.80
CCT (Ouali et al., 2020)PSP-NetResNet-5073.20
PseudoSeg (Ours)DeepLabv3+ResNet-5073.80
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SplitCCTPseudoSeg
1/266.8073.51
1/467.6071.79
1/862.5069.15
1/1651.8065.44
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As", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "shown in Figure 4, the proposed method consistently improves the strong baseline on both datasets.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 257, + 446 + ], + "score": 1.0, + "content": "In Table 3, we evaluate on the public", + "type": "text" + }, + { + "bbox": [ + 257, + 433, + 289, + 444 + ], + "score": 0.64, + "content": "1 . 4 \\mathrm { k } / 9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "split. 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We investigate the effectiveness of the proposed pseudo labeling.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Table 5 demonstrates quantitative results, indicating that using either decoder output or SGC alone", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "gives an inferior performance. Naively using decoder output as pseudo labels can hardly work", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "well. The proposed fusion consistently performs better, either with or without additional image-level", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "labels. To further answer why our pseudo labels are effective, we study from the model calibration", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "perspective. We measure the expected calibration error (ECE) (Guo et al., 2017) scores of all the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 628 + ], + "score": 1.0, + "content": "intermediate steps and other fusion variants. As shown in Figure 5 (a), the proposed fusion strategy", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 637 + ], + "score": 1.0, + "content": "(denoted as G in the figure) achieves the lowest ECE scores, indicating that the significance of jointly", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "using normalization with sharpening (see equation 4) compared with other fusion alternatives. We", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 659 + ], + "score": 1.0, + "content": "hypothesize using well-calibrated soft labels makes model training less affected by label noises. The", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 658, + 395, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 395, + 671 + ], + "score": 1.0, + "content": "comprehensive calibration study is left as a future exploration direction.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 547, + 505, + 671 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 673, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 504, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 504, + 685 + ], + "score": 1.0, + "content": "Using hypercolumn feature or not? In Figure 5 (b), we study the effectiveness of using hypercol-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "umn features instead of the last feature maps in equation 3. We conduct the experiments on the 1/16", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 694, + 468, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 468, + 708 + ], + "score": 1.0, + "content": "split of VOC12. As we can see, hypercolumn features substantially improve performance.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 673, + 505, + 708 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Soft or hard pseudo label? How to utilize predictions as pseudo labels remains an active question", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "in SSL. Next, we study whether we should use soft or hard one-hot pseudo labels. We conduct", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "the experiments in the setting where pixel-level labeled data and image-level labeled data are avail-", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "able. As shown in Figure 5 (c), using all predictions as soft pseudo label yields better performance", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "than selecting confident predictions. This suggests that well-calibrated soft pseudo labels might be", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 486, + 396, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 396, + 500 + ], + "score": 1.0, + "content": "important in segmentation than over-simplified confidence thresholding.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 144, + 105, + 463, + 194 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 502, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 495, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 495, + 94 + ], + "score": 1.0, + "content": "Table 5: Comparison to alternative pseudo labeling strategies. We conduct experiments using", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 504, + 102 + ], + "spans": [ + { + "bbox": [ + 107, + 91, + 504, + 102 + ], + "score": 1.0, + "content": "1/4, 1/8, 1/16 of the pixel-level labeled data, the exact numbers of images are shown in the brackets.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 144, + 105, + 463, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 105, + 463, + 194 + ], + "spans": [ + { + "bbox": [ + 144, + 105, + 463, + 194 + ], + "score": 0.941, + "html": "
SourceUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Decoder only70.2269.3553.20
SGC only67.0762.6153.42
Calibrated fusion73.7973.1367.06
Decoder only73.9573.0567.54
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SourceUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Decoder only70.2269.3553.20
SGC only67.0762.6153.42
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A student", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 229, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 240 + ], + "score": 1.0, + "content": "model is then trained using the combination of unlabeled data in VOC12 train and augmented sets", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "with generated one-hot pseudo labels and all the available pixel-level labeled data. As shown in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "Table 6, although the self-training pretty well improves over the supervised baseline, it is inferior", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "to the proposed method 2. We conjecture that the teacher model usually produces low confidence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 505, + 286 + ], + "score": 1.0, + "content": "scores to pixels around boundaries, so pseudo labels of these pixels are filtered in student training.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "However, boundary pixels are important for improving the performance of segmentation (Kirillov", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "et al., 2020). On the other hand, the design of our method (online soft pseudo labeling process)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 306, + 470, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 470, + 318 + ], + "score": 1.0, + "content": "bypass this challenge. We will conduct more verification of this hypothesis in future work.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 119, + 331, + 423, + 341 + ], + "lines": [ + { + "bbox": [ + 116, + 330, + 425, + 342 + ], + "spans": [ + { + "bbox": [ + 116, + 330, + 425, + 342 + ], + "score": 1.0, + "content": "5 IMPROVING THE FULLY-SUPERVISED METHOD WITH ADDITIONAL DAT", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "We have validated the effectiveness of the proposed method in the low-data regime. 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The additional data contains additional VOC 9k", + "type": "text" + }, + { + "bbox": [ + 326, + 382, + 348, + 393 + ], + "score": 0.83, + "content": "( V _ { 9 k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 381, + 432, + 394 + ], + "score": 1.0, + "content": ", COCO training set", + "type": "text" + }, + { + "bbox": [ + 432, + 382, + 453, + 393 + ], + "score": 0.85, + "content": "( C _ { t r } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 381, + 505, + 394 + ], + "score": 1.0, + "content": ", and COCO", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 168, + 405 + ], + "score": 1.0, + "content": "unlabeled data", + "type": "text" + }, + { + "bbox": [ + 169, + 393, + 187, + 404 + ], + "score": 0.79, + "content": "( C _ { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 393, + 505, + 405 + ], + "score": 1.0, + "content": ". More training details can be found in Appendix D. As shown in Table 7, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "proposed PseudoSeg is able to improve upon the supervised baseline even in the high-data regime,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 415, + 327, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 327, + 428 + ], + "score": 1.0, + "content": "using additional unlabeled or image-level labeled data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "table", + "bbox": [ + 115, + 450, + 492, + 497 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 436, + 502, + 448 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 435, + 503, + 449 + ], + "spans": [ + { + "bbox": [ + 108, + 435, + 503, + 449 + ], + "score": 1.0, + "content": "Table 7: Improving fully supervised model with extra data. No test-time augmentation is used.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "table_body", + "bbox": [ + 115, + 450, + 492, + 497 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 115, + 450, + 492, + 497 + ], + "spans": [ + { + "bbox": [ + 115, + 450, + 492, + 497 + ], + "score": 0.973, + "html": "
MethodBaseline丨PseudoSeg (w/o image-level labels)丨PseudoSeg (w/image-level labels)
Extra data1Ctr+CuCtr + Cu + V9kCtrCtr +Vgk
mIoU (%)76.9677.40 (+0.44)78.20 (+1.24)77.80 (+0.84)79.28 (+2.32)
", + "type": "table", + "image_path": "ff061b56d6ee624a1ae28039431568030c3c487b3a4ade5abd9338e0c6c7e09f.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 115, + 450, + 492, + 465.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 115, + 465.6666666666667, + 492, + 481.33333333333337 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 115, + 481.33333333333337, + 492, + 497.00000000000006 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 26.0 + }, + { + "type": "title", + "bbox": [ + 108, + 514, + 288, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 290, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 290, + 529 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "The key to the good performance of our method in the low-data regime is the novel re-design of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "pseudo-labeling strategy, which pursues a different decision mechanism from weakly-supervised", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "localization to “remedy” weak predictions from segmentation head. Then augmentation consis-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "tency training progressively improves segmentation head quality. For the first time, we demon-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 573, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 505, + 590 + ], + "score": 1.0, + "content": "strate that, with well-calibrated soft pseudo labels, utilizing unlabeled or image-labeled data sig-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "nificantly improves segmentation at low-data regimes. 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Although", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 609, + 504, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 504, + 620 + ], + "score": 1.0, + "content": "color jittering works within our method as strong data augmentation, we have extensively explored", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "geometric augmentations (leveraging STN (Jaderberg et al., 2015) to align pixels in pseudo labels", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 644 + ], + "score": 1.0, + "content": "and strongly-augmented predictions) for segmentation but find it not helpful. We believe data aug-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 640, + 477, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 477, + 654 + ], + "score": 1.0, + "content": "mentation needs re-thinking beyond current success in classification for segmentation usage.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 670, + 219, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 221, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 221, + 684 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENT", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 693, + 406, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 692, + 408, + 706 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 408, + 706 + ], + "score": 1.0, + "content": "We thank Liang-Chieh Chen and Barret Zoph for their valuable comments.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 712, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 117, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "2It is difficult to directly compare to Zoph et al. (2020) in their setting because of enormous parallel training,", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 329, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 329, + 732 + ], + "score": 1.0, + "content": "uncommon backbones, and inaccessible pre-training datasets.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 135, + 105, + 473, + 173 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 503, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 6: Comparison with self-training. 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MethodUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Supervised (Teacher)70.2064.0056.03
Self-training (Student)172.8569.8864.20
PseudoSeg (Ours)-73.7973.1367.06
PseudoSeg (Ours)√75.2974.7071.22
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To generate offline", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "score": 1.0, + "content": "pseudo labels, we closely follow segmentation experiments in Zoph et al. (2020): pixels with a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 205, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 220 + ], + "score": 1.0, + "content": "confidence score higher than 0.5 will be used as one-hot pseudo labels, while the remaining are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "treated as ignored regions. This step is considered important to suppress noisy labels. A student", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 229, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 240 + ], + "score": 1.0, + "content": "model is then trained using the combination of unlabeled data in VOC12 train and augmented sets", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "with generated one-hot pseudo labels and all the available pixel-level labeled data. As shown in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "Table 6, although the self-training pretty well improves over the supervised baseline, it is inferior", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "to the proposed method 2. 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More training details can be found in Appendix D. As shown in Table 7, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "proposed PseudoSeg is able to improve upon the supervised baseline even in the high-data regime,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 415, + 327, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 327, + 428 + ], + "score": 1.0, + "content": "using additional unlabeled or image-level labeled data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 348, + 506, + 428 + ] + }, + { + "type": "table", + "bbox": [ + 115, + 450, + 492, + 497 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 436, + 502, + 448 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 435, + 503, + 449 + ], + "spans": [ + { + "bbox": [ + 108, + 435, + 503, + 449 + ], + "score": 1.0, + "content": "Table 7: Improving fully supervised model with extra data. 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MethodBaseline丨PseudoSeg (w/o image-level labels)丨PseudoSeg (w/image-level labels)
Extra data1Ctr+CuCtr + Cu + V9kCtrCtr +Vgk
mIoU (%)76.9677.40 (+0.44)78.20 (+1.24)77.80 (+0.84)79.28 (+2.32)
", + "type": "table", + "image_path": "ff061b56d6ee624a1ae28039431568030c3c487b3a4ade5abd9338e0c6c7e09f.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 115, + 450, + 492, + 465.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 115, + 465.6666666666667, + 492, + 481.33333333333337 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 115, + 481.33333333333337, + 492, + 497.00000000000006 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 26.0 + }, + { + "type": "title", + "bbox": [ + 108, + 514, + 288, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 290, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 290, + 529 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "The key to the good performance of our method in the low-data regime is the novel re-design of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "pseudo-labeling strategy, which pursues a different decision mechanism from weakly-supervised", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "localization to “remedy” weak predictions from segmentation head. Then augmentation consis-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "tency training progressively improves segmentation head quality. For the first time, we demon-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 573, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 505, + 590 + ], + "score": 1.0, + "content": "strate that, with well-calibrated soft pseudo labels, utilizing unlabeled or image-labeled data sig-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "nificantly improves segmentation at low-data regimes. 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Although", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 609, + 504, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 504, + 620 + ], + "score": 1.0, + "content": "color jittering works within our method as strong data augmentation, we have extensively explored", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "geometric augmentations (leveraging STN (Jaderberg et al., 2015) to align pixels in pseudo labels", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 644 + ], + "score": 1.0, + "content": "and strongly-augmented predictions) for segmentation but find it not helpful. We believe data aug-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 640, + 477, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 477, + 654 + ], + "score": 1.0, + "content": "mentation needs re-thinking beyond current success in classification for segmentation usage.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 531, + 506, + 654 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 670, + 219, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 221, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 221, + 684 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENT", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 693, + 406, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 692, + 408, + 706 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 408, + 706 + ], + "score": 1.0, + "content": "We thank Liang-Chieh Chen and Barret Zoph for their valuable comments.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 106, + 692, + 408, + 706 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 175, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 99, + 503, + 120 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 505, + 111 + ], + "score": 1.0, + "content": "Jiwoon Ahn and Suha Kwak. 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To avoid the missing classes at", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "extremely low data regimes, we repeat the random sampling process for 1/16 three times (while en-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "suring each class has a certain amount) and report the results. We use Split 1 in the main manuscript.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "All splits will be released to encourage reproducibility. 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To construct the hypercolumn feature, we extract the feature maps from the last two con-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "volutional stages of the backbone network and concatenate them together. 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MethodUsing image-level labelsSplit 1Split 2Split 3
Supervised■56.0356.8755.92
PseudoSeg (Ours)167.0664.1266.09
PseudoSeg (Ours)√71.2268.1169.72
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MethodPixel-level labeled datamIoU (%)
FickleNet (Lee et al., 2019)64.9
IRNet (Ahn et al., 2019)63.5
OAA+ (Jiang et al., 2019)65.2
SEAM (Wang et al., 2020)64.5
MCIS (Sun et al., 2020)66.2
PseudoSeg (Ours)1/16 (92)71.22
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MethodUsing image-level labelsSplit 1Split 2Split 3
Supervised■56.0356.8755.92
PseudoSeg (Ours)167.0664.1266.09
PseudoSeg (Ours)√71.2268.1169.72
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Instead of enforcing the consistency between different augmented images", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 487, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 487, + 505, + 502 + ], + "score": 1.0, + "content": "as we do, these approaches tackle the semantic segmentation task from a different perspective, by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 498, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 513 + ], + "score": 1.0, + "content": "exploiting the weaker annotations (image-level labels). 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Therefore, these", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "designs could be potentially integrated into our framework to generate better pseudo labels, which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 554, + 235, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 235, + 566 + ], + "score": 1.0, + "content": "leads to improved performance.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 466, + 506, + 566 + ] + }, + { + "type": "table", + "bbox": [ + 163, + 601, + 447, + 697 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 577, + 491, + 600 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 577, + 493, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 493, + 590 + ], + "score": 1.0, + "content": "Table 9: Benchmarking state-of-the-art weakly supervised semantic segmentation methods.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 587, + 480, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 480, + 602 + ], + "score": 1.0, + "content": "All the methods use image-level labels from VOC12 training (1.4k) and augmented (9k) sets.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "table_body", + "bbox": [ + 163, + 601, + 447, + 697 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 163, + 601, + 447, + 697 + ], + "spans": [ + { + "bbox": [ + 163, + 601, + 447, + 697 + ], + "score": 0.979, + "html": "
MethodPixel-level labeled datamIoU (%)
FickleNet (Lee et al., 2019)64.9
IRNet (Ahn et al., 2019)63.5
OAA+ (Jiang et al., 2019)65.2
SEAM (Wang et al., 2020)64.5
MCIS (Sun et al., 2020)66.2
PseudoSeg (Ours)1/16 (92)71.22
", + "type": "table", + "image_path": "4d0847f24656dd0475d73e7048e0a6968e21e34dda181eb6e7f0646acb151e5a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 163, + 601, + 447, + 633.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 163, + 633.0, + 447, + 665.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 163, + 665.0, + 447, + 697.0 + ], + "spans": [], + "index": 27 + } + ] + } + ], + "index": 24.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 111, + 83, + 376, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 82, + 378, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 378, + 95 + ], + "score": 1.0, + "content": "F PERFORMANCE ANALYSIS FOR TEMPERATURE SHARPENING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "We conduct an additional performance analysis for temporal sharpening. We conduct experiments", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "over T on the 1/16 split of VOC using pixel-level labeled data and image-level labeled data. As", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 124, + 231, + 138 + ], + "score": 1.0, + "content": "shown in Table 10, adopting a", + "type": "text" + }, + { + "bbox": [ + 231, + 125, + 260, + 135 + ], + "score": 0.9, + "content": "T < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "for distribution sharpening generally leads to improved per-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 150, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 150, + 148 + ], + "score": 1.0, + "content": "formance.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 231, + 169, + 377, + 248 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 223, + 155, + 387, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 222, + 153, + 389, + 168 + ], + "spans": [ + { + "bbox": [ + 222, + 153, + 389, + 168 + ], + "score": 1.0, + "content": "Table 10: Performance analysis over T.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 231, + 169, + 377, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 231, + 169, + 377, + 248 + ], + "spans": [ + { + "bbox": [ + 231, + 169, + 377, + 248 + ], + "score": 0.968, + "html": "
Temperature (T)mIoU (%)
0.171.11
0.370.11
0.5 (default)71.22
0.772.37
1.0 (no sharpening)68.15
", + "type": "table", + "image_path": "b8c478af059105baa12a29931140f43e29439d24d1c3e40c9ea022b42ed5c14a.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 169, + 377, + 208.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 231, + 208.5, + 377, + 248.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 5.75 + }, + { + "type": "title", + "bbox": [ + 108, + 259, + 259, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 259, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 259, + 272 + ], + "score": 1.0, + "content": "G EXPERIMENTS ON CITYSCAPES", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "In this section, we conduct additional experiments on the Cityscapes dataset (Cordts et al., 2016).", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "The Cityscapes dataset contains 50 real-world driving sequences. Among these video sequences,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "2,975 frames are selected as the training set, and 500 frames are selected as the validation set.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 312, + 405, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 405, + 326 + ], + "score": 1.0, + "content": "Following previous common practice, we evaluate on 19 semantic classes.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "Comparison with state of the art. We compare our method with the current state-of-the-art", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "method (French et al., 2020), in the setting of using pixel-level labeled and unlabeled data. We ran-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "domly subsample 1/4, 1/8, and 1/30 of the training set to construct the pixel-level labeled data, using", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 361, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 372 + ], + "score": 1.0, + "content": "the first random seed provided by French et al. (2020). Both French et al. (2020) and our method", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 330, + 384 + ], + "score": 1.0, + "content": "use ResNet-101 as the feature backbone and DeepLabv", + "type": "text" + }, + { + "bbox": [ + 331, + 373, + 343, + 383 + ], + "score": 0.7, + "content": "^ { 3 + }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "(Chen et al., 2018) as the segmentation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "model. As shown in Table 11, the proposed method achieves promising results on all the three label", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 394, + 135, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 135, + 406 + ], + "score": 1.0, + "content": "ratios.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "table", + "bbox": [ + 161, + 427, + 446, + 473 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 123, + 413, + 486, + 425 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 123, + 413, + 488, + 426 + ], + "spans": [ + { + "bbox": [ + 123, + 413, + 488, + 426 + ], + "score": 1.0, + "content": "Table 11: Experiments on Cityscapes (w/ pixel-level labeled data and unlabeled data).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 161, + 427, + 446, + 473 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 161, + 427, + 446, + 473 + ], + "spans": [ + { + "bbox": [ + 161, + 427, + 446, + 473 + ], + "score": 0.969, + "html": "
Method1/4 (744)1/8 (372)1/30 (100)
CutMix (French et al., 2020)68.3365.8255.71
PseudoSeg (Ours)72.3669.8160.96
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Class Pixel ratio (%)Road 36.36Sidewalk 5.61Building 20.99Wall 0.53Fence 0.98Pole 1.19Traffic light 0.14Traffic sign 0.51Vegetation 19.61Terrain 1.29
Supervised PseudoSeg (Ours)96.0371.2687.5319.7529.1152.1950.1968.0989.9345.79
96.6475.0688.6319.6734.0951.7558.1969.9590.4350.48
ClassSkyPersonRiderCarTruckBusTrainMotorcycleBicycle
Pixel ratio (%)3.701.100.166.490.380.130.230.060.54
Supervised91.0174.1243.9189.917.6814.1917.7825.8669.88
PseudoSeg (Ours)92.9975.1646.0991.6020.3926.3022.1343.9671.30
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Although the scene layouts are quite similar for all the full images, it is still feasible to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "generate different image-level labels through a more aggressive geometric data augmentation (e.g.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "scaling, cropping, translation, etc.). In practice, standard segmentation preprocessing steps only", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "crop a sub-region of the whole training images. It only contains partial images with a certain subset", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "of image labels, making the training batches have diverse image-level labels (converted from pixel-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "level labels, in the fully-labeled+unlabeled setting). Moreover, in the fully-labeled+weakly-labeled", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 111, + 83, + 376, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 82, + 378, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 378, + 95 + ], + "score": 1.0, + "content": "F PERFORMANCE ANALYSIS FOR TEMPERATURE SHARPENING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "We conduct an additional performance analysis for temporal sharpening. We conduct experiments", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "over T on the 1/16 split of VOC using pixel-level labeled data and image-level labeled data. As", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 124, + 231, + 138 + ], + "score": 1.0, + "content": "shown in Table 10, adopting a", + "type": "text" + }, + { + "bbox": [ + 231, + 125, + 260, + 135 + ], + "score": 0.9, + "content": "T < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "for distribution sharpening generally leads to improved per-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 150, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 150, + 148 + ], + "score": 1.0, + "content": "formance.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 102, + 505, + 148 + ] + }, + { + "type": "table", + "bbox": [ + 231, + 169, + 377, + 248 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 223, + 155, + 387, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 222, + 153, + 389, + 168 + ], + "spans": [ + { + "bbox": [ + 222, + 153, + 389, + 168 + ], + "score": 1.0, + "content": "Table 10: Performance analysis over T.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 231, + 169, + 377, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 231, + 169, + 377, + 248 + ], + "spans": [ + { + "bbox": [ + 231, + 169, + 377, + 248 + ], + "score": 0.968, + "html": "
Temperature (T)mIoU (%)
0.171.11
0.370.11
0.5 (default)71.22
0.772.37
1.0 (no sharpening)68.15
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Method1/4 (744)1/8 (372)1/30 (100)
CutMix (French et al., 2020)68.3365.8255.71
PseudoSeg (Ours)72.3669.8160.96
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Class Pixel ratio (%)Road 36.36Sidewalk 5.61Building 20.99Wall 0.53Fence 0.98Pole 1.19Traffic light 0.14Traffic sign 0.51Vegetation 19.61Terrain 1.29
Supervised PseudoSeg (Ours)96.0371.2687.5319.7529.1152.1950.1968.0989.9345.79
96.6475.0688.6319.6734.0951.7558.1969.9590.4350.48
ClassSkyPersonRiderCarTruckBusTrainMotorcycleBicycle
Pixel ratio (%)3.701.100.166.490.380.130.230.060.54
Supervised91.0174.1243.9189.917.6814.1917.7825.8669.88
PseudoSeg (Ours)92.9975.1646.0991.6020.3926.3022.1343.9671.30
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Although the scene layouts are quite similar for all the full images, it is still feasible to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "generate different image-level labels through a more aggressive geometric data augmentation (e.g.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "scaling, cropping, translation, etc.). In practice, standard segmentation preprocessing steps only", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "crop a sub-region of the whole training images. It only contains partial images with a certain subset", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "of image labels, making the training batches have diverse image-level labels (converted from pixel-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "level labels, in the fully-labeled+unlabeled setting). 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Method1/2 (732)1/4 (366)1/8 (183)1/16 (92)
AdvSemSeg (Hung et al., 2018)65.2759.9747.5839.69
CCT (Ouali et al., 2020)62.1058.8047.6033.10
*MT (Tarvainen & Valpola, 2017)69.1663.0155.8148.70
GCT (Ke et al., 2020)70.6764.7154.9846.04
**VAT (Miyato et al., 2018)63.3456.8849.3536.92
CutMix (French et al., 2020)69.8468.3663.2055.58
PseudoSeg (Ours)72.4169.1465.5057.60
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MethodNetworkmIoU (%)
GANSeg (Souly et al., 2017)VGG1664.10
AdvSemSeg (Hung et al., 2018)ResNet-10168.40
CCT (Ouali et al., 2020)ResNet-5069.40
PseudoSeg (Ours)ResNet-5071.00
PseudoSeg (Ours)ResNet-10173.23
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MethodModelNetworkmIoU (%)
WSSN (Papandreou et al., 2015)DeepLab-CRFVGG1664.60
GAIN (Li et al., 2018)DeepLab-CRF-LFOVVGG1660.50
MDC (Wei et al.,2018)DeepLab-CRF-LFOVVGG1665.70
DSRG (Huang et al.,2018)DeepLabv2VGG1664.30
GANSeg (Souly et al.,2017)FCNVGG1665.80
FickleNet (Lee et al.,2019)DeepLabv2ResNet-10165.80
CCT (Ouali et al., 2020)PSP-NetResNet-5073.20
PseudoSeg (Ours)DeepLabv3+ResNet-5073.80
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SplitCCTPseudoSeg
1/266.8073.51
1/467.6071.79
1/862.5069.15
1/1651.8065.44
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Decoder only70.2269.3553.20
SGC only67.0762.6153.42
Calibrated fusion73.7973.1367.06
Decoder only73.9573.0567.54
SGC only71.7367.5764.26
Calibrated fusion75.2974.7071.22
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MethodUsing image-level labels1/4 (366)1/8 (183)1/16 (92)
Supervised (Teacher)70.2064.0056.03
Self-training (Student)172.8569.8864.20
PseudoSeg (Ours)-73.7973.1367.06
PseudoSeg (Ours)√75.2974.7071.22
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MethodBaseline丨PseudoSeg (w/o image-level labels)丨PseudoSeg (w/image-level labels)
Extra data1Ctr+CuCtr + Cu + V9kCtrCtr +Vgk
mIoU (%)76.9677.40 (+0.44)78.20 (+1.24)77.80 (+0.84)79.28 (+2.32)
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MethodPixel-level labeled datamIoU (%)
FickleNet (Lee et al., 2019)64.9
IRNet (Ahn et al., 2019)63.5
OAA+ (Jiang et al., 2019)65.2
SEAM (Wang et al., 2020)64.5
MCIS (Sun et al., 2020)66.2
PseudoSeg (Ours)1/16 (92)71.22
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MethodUsing image-level labelsSplit 1Split 2Split 3
Supervised■56.0356.8755.92
PseudoSeg (Ours)167.0664.1266.09
PseudoSeg (Ours)√71.2268.1169.72
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Class Pixel ratio (%)Road 36.36Sidewalk 5.61Building 20.99Wall 0.53Fence 0.98Pole 1.19Traffic light 0.14Traffic sign 0.51Vegetation 19.61Terrain 1.29
Supervised PseudoSeg (Ours)96.0371.2687.5319.7529.1152.1950.1968.0989.9345.79
96.6475.0688.6319.6734.0951.7558.1969.9590.4350.48
ClassSkyPersonRiderCarTruckBusTrainMotorcycleBicycle
Pixel ratio (%)3.701.100.166.490.380.130.230.060.54
Supervised91.0174.1243.9189.917.6814.1917.7825.8669.88
PseudoSeg (Ours)92.9975.1646.0991.6020.3926.3022.1343.9671.30
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Method1/4 (744)1/8 (372)1/30 (100)
CutMix (French et al., 2020)68.3365.8255.71
PseudoSeg (Ours)72.3669.8160.96
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Temperature (T)mIoU (%)
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0.370.11
0.5 (default)71.22
0.772.37
1.0 (no sharpening)68.15
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a/parse/train/E4PK0rg2eP/E4PK0rg2eP.md b/parse/train/E4PK0rg2eP/E4PK0rg2eP.md new file mode 100644 index 0000000000000000000000000000000000000000..6c2858be4cbdd9f1fd0ea45ddf7b00f0121f7b1e --- /dev/null +++ b/parse/train/E4PK0rg2eP/E4PK0rg2eP.md @@ -0,0 +1,328 @@ +# PARAMETER-EFFICIENT TRANSFER LEARNING WITH DIFF PRUNING + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +While task-specific finetuning of deep networks pretrained with self-supervision has led to significant empirical advances in NLP, their large size makes the standard finetuning approach difficult to apply to multi-task, memory-constrained settings, as storing the full model parameters for each task become prohibitively expensive. We propose diff pruning as a simple approach to enable parameterefficient transfer learning within the pretrain-finetune framework. This approach views finetuning as learning a task-specific “diff” vector that is applied on top of the pretrained parameter vector, which remains fixed and is shared across different tasks. The diff vector is adaptively pruned during training with a differentiable approximation to the $L _ { 0 }$ -norm penalty to encourage sparsity. Diff pruning becomes parameter-efficient as the number of tasks increases, as it requires storing only the nonzero positions and weights of the diff vector for each task, while the cost of storing the shared pretrained model remains constant. We find that models finetuned with diff pruning can match the performance of fully finetuned baselines on the GLUE benchmark while only modifying $0 . 5 \%$ of the pretrained model’s parameters per task. + +# 1 INTRODUCTION + +Task-specific finetuning of pretrained deep networks has become the dominant paradigm in contemporary NLP, achieving state-of-the-art results across a suite of natural language understanding tasks (Devlin et al., 2019; Liu et al., 2019c; Yang et al., 2019; Lan et al., 2020). While straightforward and empirically effective, this approach is difficult to scale to multi-task, memory-constrained settings (e.g. for on-device applications), as it requires shipping and storing a full set of model parameters for each task. Inasmuch as these models are learning generalizable, task-agnostic language representations through self-supervised pretraining, finetuning the entire model for each task is an especially inefficient use of model parameters. + +A popular approach to parameter-efficiency is to learn sparse models for each task where a subset of the final model parameters are exactly zero (Gordon et al., 2020; Sajjad et al., 2020; Zhao et al., 2020; Sanh et al., 2020). Such approaches often face a steep sparsity/performance tradeoff, and a substantial portion of nonzero parameters (e.g. $10 \% { - } 3 0 \% )$ ) are still typically required to match the performance of the dense counterparts. An alternative is to use multi-task learning or feature-based transfer for more parameter-efficient transfer learning with pretrained models (Liu et al., 2019b; Clark et al., 2019; Stickland & Murray, 2019; Reimers & Gurevych, 2019; Feng et al., 2020). These methods learn only a small number of additional parameters (e.g. a linear layer) on top of a shared model. However, multi-task learning generally requires access to all tasks during training to prevent catastrophic forgetting (French, 1999), while feature-based transfer learning (e.g. based on taskagnostic sentence representations) is typically outperformed by full finetuning (Howard & Ruder, 2018). + +Adapters (Rebuffi et al., 2018) have recently emerged as a promising approach to parameterefficient transfer learning within the pretrain-finetune paradigm (Houlsby et al., 2019; Pfeiffer et al., 2020a;b;c). Adapter layers are smaller, task-specific modules that are inserted between layers of a pretrained model, which remains fixed and is shared across tasks. These approaches do not require access to all tasks during training making them attractive in settings where one hopes to obtain and share performant models as new tasks arrive in stream. Houlsby et al. (2019) find that adapter layers trained on BERT can match the performance of fully finetuned BERT on the GLUE benchmark (Wang et al., 2019a) while only requiring $3 . 6 \%$ additional parameters (on average) per task. + +In this work, we consider a similar setting as adapters but propose a new diff pruning approach with the goal of even more parameter-efficient transfer learning. Diff pruning views finetuning as learning a task-specific difference vector that is applied on top of the pretrained parameter vector, which remains fixed and is shared across different tasks. In order to learn this vector, we reparameterize the task-specific model parameters as $\theta _ { \mathrm { t a s k } } = \theta _ { \mathrm { p r e t r a i n e d } } + \delta _ { \mathrm { t a s k } }$ , where the pretrained parameter vector $\theta _ { \mathrm { p r e t r a i n e d } }$ is fixed and the task-specific diff vector $\delta _ { \mathrm { t a s k } }$ is finetuned. The diff vector is regularized with a differentiable approximation to the $L _ { 0 }$ -norm penalty (Louizos et al., 2018) to encourage sparsity. This approach can become parameter-efficient as the number of tasks increases as it only requires storing the nonzero positions and weights of the diff vector for each task. The cost of storing the shared pretrained model remains constant and is amortized across multiple tasks. On the GLUE benchmark (Wang et al., 2019a), diff pruning can match the performance of the fully finetuned BERT baselines while finetuning only $0 . 5 \%$ of the pretrained parameters per task, making it a potential alternative to adapters for parameter-efficient transfer learning. + +# 2 BACKGROUND: TRANSFER LEARNING FOR NLP + +The field of NLP has recently seen remarkable progress through transfer learning with a pretrainand-finetune paradigm, which initializes a subset of the model parameters for all tasks from a pretrained model and then finetunes on a task specific objective. Pretraining objectives include context prediction (Mikolov et al., 2013), autoencoding (Dai & Le, 2015), machine translation (McCann et al., 2017), and more recently, variants of language modeling (Peters et al., 2018; Radford et al., 2018; Devlin et al., 2019) objectives. + +Here we consider applying transfer learning to multiple tasks. We consider a setting with a potentially unknown set of tasks, where each $\tau \in \mathcal { T }$ has an associated training set $\{ x _ { \tau } ^ { ( n ) } , y _ { \tau } ^ { ( n ) } \} _ { n = 1 } ^ { N }$ 1 For . all tasks, the goal is to produce (possibly tied) model parameters to minimize the empirical risk, + +$$ +\operatorname* { m i n } _ { \pmb { \theta } _ { \tau } } \ \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \mathcal { L } \left( f ( x _ { \tau } ^ { ( n ) } ; \pmb { \theta } _ { \tau } ) , y _ { \tau } ^ { ( n ) } \right) + \lambda R ( \pmb { \theta } _ { \tau } ) +$$ + +where $f ( \cdot ; \pmb \theta )$ is a parameterized function over the input (e.g. a neural network), $\mathcal L ( \cdot , \cdot )$ is a loss function (e.g. cross-entropy), and $R ( \cdot )$ is an optional regularizer with hyperparameter $\lambda$ . + +This multi-task setting can use the pretrain-then-finetune approach by simply learning independent parameters for each task; however the large size of pretrained models makes this approach exceedingly parameter inefficient. For example, widely-adopted models such as BERTBASE and BERTLARGE have 110M and 340M parameters respectively, while their contemporaries such as T5 (Raffel et al., 2020), Megatron-LM (Shoeybi et al., 2019), and Turing-NLG (Rajbhandari et al., 2019) have parameter counts in the billions. Storing the fully finetuned models becomes difficult even for a moderate number of tasks.2 A classic approach to tackling this parameterinefficiency (Caruana, 1997) is to train a single shared model (along with a task-specific output layer) against multiple tasks through joint training. However, the usual formulation of multi-task learning requires the set of tasks $\tau$ to be known in advance in order to prevent catastrophic forgetting (French, 1999),3 making it unsuitable for applications in which the set of tasks is unknown (e.g. when tasks arrive in stream). + +# 3 DIFF PRUNING + +Diff pruning formulates task-specific finetuning as learning a diff vector $\delta _ { \tau }$ that is added to the pretrained model parameters $\theta _ { \mathrm { p r e t r a i n e d } }$ . We first reparameterize the task-specific model parameters, + +$$ +\begin{array} { r } { \pmb { \theta } _ { \tau } = \pmb { \theta } _ { \mathrm { p r e t r a i n e d } } + \delta _ { \tau } , } \end{array} +$$ + +which results in the following empirical risk minimization problem, + +$$ +\operatorname* { m i n } _ { \delta _ { \tau } } \ \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \mathcal { L } \left( f ( x _ { \tau } ^ { ( n ) } ; \theta _ { \mathrm { p r e t r a i n e d } } + \delta _ { \tau } ) , y _ { \tau } ^ { ( n ) } \right) + \lambda R ( \theta _ { \mathrm { p r e t r a i n e d } } + \delta _ { \tau } ) . +$$ + +This trivial reparameterization is equivalent to the original formulation. Its benefit comes in the multi-task setting where the cost of storing the pretrained parameters $\theta _ { \mathrm { p r e t r a i n e d } }$ is amortized across tasks, and the only marginal cost for new tasks is the diff vector. If we can regularize $\delta _ { \tau }$ to be sparse such that $\lVert \delta _ { \tau } \rVert _ { 0 } \ll \lVert \bar { \pmb { \theta } } _ { \mathrm { p r e t r a i n e d } } \rVert _ { 0 }$ , then this approach can become more parameter-efficient as the number of tasks increases. We can specify this goal with an $L _ { 0 }$ -norm penalty on the diff vector, + +$$ +R ( \theta _ { \mathrm { p r e t r a i n e d } } + \delta _ { \tau } ) = \| \pmb { \delta } _ { \tau } \| _ { 0 } = \sum _ { i = 1 } ^ { d } \mathbb { 1 } \{ \pmb { \delta } _ { \tau , i } \neq 0 \} . +$$ + +# 3.1 DIFFERENTIABLE APPROXIMATION TO THE $L _ { 0 }$ -NORM + +This regularizer is difficult to directly optimize as it is non-differentiable. In order to approximate this $L _ { 0 }$ objective, we follow the standard approach for gradient-based learning with $L _ { 0 }$ sparsity using a relaxed mask vector (Louizos et al., 2018). This approach involves relaxing a binary vector into continuous space, and then multiplying it with a dense weight vector to determine how much of the weight vector is applied during training. After training, the mask is deterministic and a large portion of the diff vector is true zero. + +To apply this method we first decompose $\delta _ { \tau }$ into a binary mask vector multiplied with a dense vector, + +$$ +\begin{array} { r } { \delta _ { \tau } = \mathbf { z } _ { \tau } \odot \mathbf { w } _ { \tau } , \qquad \mathbf { z } _ { \tau } \in \{ 0 , 1 \} ^ { d } , \mathbf { w } _ { \tau } \in \mathbb { R } ^ { d } } \end{array} +$$ + +We can now instead optimize an expectation with respect to ${ \bf z } _ { \tau }$ , whose distribution $p ( \mathbf { z } _ { \tau } ; \pmb { \alpha } _ { \tau } )$ is initially Bernoulli with parameters $\pmb { \alpha } _ { \tau }$ , + +$$ +\operatorname* { m i n } _ { \alpha _ { \tau } , \mathbf { w } _ { \tau } } \mathbb { E } _ { \mathbf { z } _ { \tau } \sim p ( \mathbf { z } _ { \tau } ; \alpha _ { \tau } ) } \left[ \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \mathcal { L } \left( f ( x _ { \tau } ^ { ( n ) } ; \boldsymbol { \theta } _ { \mathrm { p r e t r a i n e d } } + \mathbf { z } _ { \tau } \odot \mathbf { w } _ { \tau } , ) , y _ { \tau } ^ { ( n ) } \right) + \lambda \lVert \delta _ { \tau } \rVert _ { 0 } \right] . +$$ + +This objective is still difficult in practice due to ${ \bf z } _ { \tau }$ ’s being discrete (which requires the score function gradient estimator), but the expectation provides some guidance for empirically effective relaxations. We follow prior work (Louizos et al., 2018; Wang et al., 2019b) and relax ${ \bf z } _ { \tau }$ into continuous space $[ 0 , 1 ] ^ { d }$ with a stretched Hard-Concrete distribution (Jang et al., 2017; Maddison et al., 2017), which allows for the use of pathwise gradient estimators. Specifically, ${ \bf z } _ { \tau }$ is now defined to be a deterministic and (sub)differentiable function of a sample $\mathbf { u }$ from a uniform distribution, + +$$ +\begin{array} { r } { \mathbf { u } \sim U ( \mathbf { 0 } , \mathbf { 1 } ) , \qquad \mathbf { s } _ { \tau } = \sigma \left( \log \mathbf { u } - \log ( 1 - \mathbf { u } ) + \alpha _ { \tau } \right) , } \\ { \bar { \mathbf { s } } _ { \tau } = \mathbf { s } _ { \tau } \times ( r - l ) + l , \qquad \mathbf { z } _ { \tau } = \operatorname* { m i n } ( \mathbf { 1 } , \operatorname* { m a x } ( \mathbf { 0 } , \bar { \mathbf { s } } _ { \tau } ) ) . } \end{array} +$$ + +Here $l < 0$ and $r > 1$ are two constants used to stretch ${ \bf s } _ { \tau }$ into the interval $( l , r ) ^ { d }$ before it is clamped to $[ 0 , 1 ] ^ { d }$ with the $\operatorname* { m i n } ( \mathbf { 1 } , \operatorname* { m a x } ( \mathbf { 0 } , \cdot ) )$ operation. In this case we have a differentiable closed-form expression for the expected $L _ { 0 }$ -norm, + +$$ +\mathbb { E } \left[ \left. \pmb { \delta } _ { \tau } \right. _ { 0 } \right] = \sum _ { i = 1 } ^ { d } \mathbb { E } \left[ \mathbb { 1 } \left\{ \mathbf { z } _ { \tau , i } > 0 \right\} \right] = \sum _ { i = 1 } ^ { d } \sigma \left( \alpha _ { \tau , i } - \log \frac { - l } { r } \right) . +$$ + +Thus the final optimization problem is given by, + +$\operatorname* { m i n } _ { \tau _ { \tau } , \mathbf { w } _ { \tau } } \mathbb { E } _ { \mathbf { u } \sim U [ \mathbf { 0 } , 1 ] } \left[ \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \mathcal { L } \left( f ( x _ { \tau } ^ { ( n ) } ; \boldsymbol { \theta } _ { \mathrm { p r e r a i n e d } } + \mathbf { z } _ { \tau } \odot \mathbf { w } _ { \tau } , ) , y _ { \tau } ^ { ( n ) } \right) \right] + \lambda \sum _ { i = 1 } ^ { d } \sigma \left( \alpha _ { \tau , i } - \log \frac { - l } { r } \right) ,$ and we can now utilize pathwise gradient estimators to optimize the first term with respect to $\pmb { \alpha } _ { \tau }$ since the expectation no longer depends on it.4 After training we obtain the final diff vector $\delta _ { \tau }$ by sampling $\mathbf { u }$ once to obtain ${ \bf z } _ { \tau }$ (which is not necessarily a binary vector but has a significant number of dimensions equal to exactly zero due to the clamping function), then setting $\pmb { \delta } _ { \tau } = \mathbf { z } _ { \tau } \odot \mathbf { w } _ { \tau }$ . 5 + +3.2 $L _ { 0 }$ -BALL PROJECTION WITH MAGNITUDE PRUNING FOR SPARSITY CONTROL + +Differentiable $L _ { 0 }$ regularization provides a strong way to achieve high sparsity rate. However, it would be ideal to have more fine-grained control into the exact sparsity rate in the diff vector, especially considering applications which require specific parameter budgets. As $\lambda$ is just the Lagrangian multiplier for the constraint $\mathbb { E } \left[ \lVert \pmb { \delta } _ { \tau } \rVert _ { 0 } \right] < \eta$ for some $\eta$ , this could be achieved in principle by searching over different values of $\lambda$ . However we found it more efficient and empirically effective to achieve an exact sparsity rate by simply projecting onto the $L _ { 0 }$ -ball after training. + +Specifically we use magnitude pruning on the diff vector $\delta _ { \tau }$ and target a sparsity rate $t \%$ by only keeping the top $t \% \times \bar { d }$ values in $\delta _ { \tau }$ .6 Note that unlike standard magnitude pruning, this is based on the magnitude of the diff vector values and not the model parameters. As is usual in magnitude pruning, we found it important to further finetune $\delta _ { \tau }$ with the nonzero masks fixed to maintain good performance (Han et al., 2016). Since this type of parameter-efficiency through projection onto the $L _ { 0 }$ -ball can be applied without adaptive diff pruning,7 such an approach will serve as one of our baselines in the empirical study. + +# 3.3 STRUCTURED DIFF PRUNING + +Diff pruning, as presented above, is architecture-agnostic and does not exploit the underlying model structure—each dimension of ${ \bf z } _ { \tau }$ is independent from one another. While this makes the approach potentially more flexible, we might expect to achieve better sparsity/performance tradeoff through a structured formulation which encourages active parameters to group together and other areas to be fully sparse. Motivated by this intuition, we first partition the parameter indices into $G$ groups $\{ g ( 1 ) , \ldots , g ( G ) \}$ where $g ( j )$ is a subset of parameter indices governed by group $g ( j )$ .8 We then introduce a scalar $\mathbf { z } _ { \tau } ^ { j }$ (with the associated parameter $\alpha _ { \tau } ^ { j }$ ) for each group $g ( j )$ , and decompose the task-specific parameter for index $i \in g ( j )$ as $\begin{array} { r } { \delta _ { \tau , i } ^ { j } = \mathbf { z } _ { \tau , i } \times \mathbf { z } _ { \tau } ^ { j } \times \mathbf { w } _ { \tau , i } . } \end{array}$ The expected $L _ { 0 }$ -norm is then given by, + +$$ +\Sigma \left[ \left. \delta _ { \tau } \right. _ { 0 } \right] = \sum _ { j = 1 } ^ { G } \sum _ { i \in g ( j ) } \mathbb { E } \left[ \mathbb { I } \left\{ \mathbf { z } _ { \tau , i } \cdot \mathbf { z } _ { \tau } ^ { g } > 0 \right\} \right] = \sum _ { j = 1 } ^ { G } \sum _ { i \in g ( j ) } \sigma \left( \alpha _ { \tau , i } - \log { \frac { - l } { r } } \right) \times \sigma \left( \alpha _ { \tau } ^ { j } - \log { \frac { - l } { r } } \right) , +$$ + +and we can train with gradient-based optimization as before. + +# 4 EXPERIMENTS + +# 4.1 MODEL AND DATASETS + +For evaluation we use the GLUE benchmark (Wang et al., 2019b), a popular finetuning dataset. Following adapters (Houlsby et al., 2019), we test our approach on the following subset of the GLUE tasks: Multi-Genre Natural Language Inference (MNLI), where the goal is two predict whether the relationship between two sentences is entailment, contradiction, or neutral (we test on both $\mathrm { M N L L } \mathrm { I } _ { m }$ and $\mathrm { M N L I } _ { m m }$ which respectively tests on matched/mismatched domains); Quora Question Pairs (QQP), a classification task to predict whether two question are semantically equivalent; Question Natural Language Inference (QNLI), which must predict whether a sentence is a correct answer to the question; Stanford Sentiment Treebank (SST-2), a sentence classification task to predict the sentiment of movie reviews; Corpus of Linguistic Acceptability (CoLA), where the goal is predict whether a sentence is linguistically acceptable or not; Semantic Textual Similarity Benchmark (STS$\mathbf { B }$ ), which must predict a similarity rating between two sentences; Microsoft Research Paraphrase Corpus (MRPC), where the goal is to predict whether two sentences are semantically equivalent; Recognizing Textual Entailment (RTE), which must predict whether a second sentence is entailed by the first. For evaluation, the benchmark uses Matthew’s correlation for CoLA, Spearman for STS-B, $\mathrm { F _ { 1 } }$ score for MRPC/QQC, and accuracy for MNLI/QNLI/SST-2/RTE. + +
Total paramsNew params per taskQNLI*SST-2 MNLImMNLImmCoLA MRPC STS-B RTEQQPAvg
Full finetuning9.00×100%91.194.986.785.960.589.387.670.172.180.9
Adapters (8-256)1.32×3.6%90.794.084.985.159.589.586.971.571.880.4
Adapters (64)1.19×2.1%91.494.285.384.656.989.687.368.671.879.8
Full finetuning9.00×100%93.494.186.786.059.688.986.671.271.780.6
Last layer1.34×3.8%79.891.671.472.940.280.167.358.663.368.2
Non-adap. diff pruning1.05×0.5%89.793.684.984.851.281.578.261.568.675.5
Diff pruning1.05×0.5%92.993.885.785.660.587.083.568.170.679.4
Diff pruning (struct.)1.05×0.5%93.394.186.486.061.189.786.070.671.180.6
+ +Table 1: GLUE benchmark test server results with BERTLARGE models. (Top) Results with adapter bottleneck layers (brackets indicate the size of bottlenecks), taken from from Houlsby et al. (2019). (Bottom) Results from this work. ${ } ^ { * } \mathrm { Q N L I }$ results are not directly comparable across the two works as the GLUE benchmark has updated the test set since then. To make our results comparable the average column is calculated without QNLI. + +For all experiments, we use the BERTLARGE model from Devlin et al. (2019), which has 24 layers, 1024 hidden size, 16 attention heads, and 340M parameters. We use the Huggingface Transformer library (Wolf et al., 2019) to conduct our experiments. + +# 4.2 BASELINES + +We compare both structured and non-structured variants of diff pruning against the following baselines: Full finetuning, which fully finetunes $\mathrm { B E R T _ { L } }$ ARGE as usual; Last layer finetuning, which only finetunes the penultimate layer (along with the final output layer)9; Adapters from Houlsby et al. (2019), which train task-specific bottleneck layers between between each layer of a pretrained model, where parameter-efficiency can be controlled by varying the size of the bottleneck layers; and Non-adaptive diff pruning, which performs diff pruning just based on magnitude pruning (i.e., we obtain $\pmb { \theta } _ { \tau }$ through usual finetuning, set $\delta _ { \tau } = \pmb { \theta } _ { \tau } - \pmb { \theta } _ { \mathrm { p r e t r a i n e d } }$ , and then apply magnitude pruning followed by additional finetuning on $\delta _ { \tau }$ ). For diff pruning we set our target sparsity rate to $0 . 5 \%$ and investigate the effect of different target sparsity rates in section 5.1. + +# 4.3 IMPLEMENTATION DETAILS AND HYPERPARAMETERS + +Diff pruning introduces additional hyperparameters $l , r$ (for stretching the Hard-Concrete distribution) and $\lambda$ (for weighting the approximate $L _ { 0 }$ -norm penalty). We found $l = - 1 . 5 , r = 1 . 5 , \lambda =$ $1 . 2 5 \times 1 0 ^ { - 7 }$ to work well across all tasks. We also initialize the weight vector ${ \bf w } _ { \tau }$ to 0, and $\pmb { \alpha } _ { \tau }$ to a positive vector (we use 5) to encourage ${ \bf z } _ { \tau }$ to be close to 1 at the start of training. While we mainly experiment with BERTLARGE to compare against prior work with adapters (Houlsby et al., 2019), in preliminary experiments we found these hyperparameters to work for finetuning RoBERTa (Liu et al., $2 0 1 9 \mathrm { c }$ ) and XLNet (Yang et al., 2019) models as well. + +For all tasks we use a learning rate of $1 \times 1 0 ^ { - 5 }$ and perform a hyperparameter search over batch size $\in \{ 4 , 6 , 8 , 1 0 \}$ and the number of epochs $\in \{ 2 , 3 , 4 , 5 \}$ .10 However we found the default settings used for regular finetuning as suggested in the original BERT paper to work well for most tasks. Finetuning with the fixed mask after projecting onto the $L _ { 0 }$ -ball with magnitude pruning is done with a learning rate of $5 \times 1 0 ^ { - 5 }$ for 3 or 5 epochs (3 epochs for QNLI, SST-2, MNLI-m, MNLI-mm, CoLA, QQP, 5 epochs for MRPC, STS-B, RTE). Grouping for the structured version of diff pruning is based on the matrix/bias vectors (i.e. parameters that belong to the same matrix or bias vector are assumed to be in the same group), which results in 393 groups.1 + +# 5 RESULTS AND ANALYSIS + +Our main results on the GLUE benchmark are shown in Table 1. Structured diff pruning can match the performance of a fully finetuned BERTLARGE model while only requiring $0 . 5 \%$ additional parameters per task. Diff pruning without structured sparsity also performs well, though slightly worse than the structured approach. Non-adaptive diff pruning, which magnitude prunes the diff vector without learning the binary mask $\mathbf { z } _ { \tau }$ , performs significantly worse, indicating the importance of learning the masking vector. Compared to adapters, diff pruning obtains similar performance while requiring fewer parameters per task, making it a potential alternative for parameter-efficient transfer learning.12 We now perform a series of analysis experiments on the validation set. + +
Non-structuredPruned Diff GroupsStructured
#%#%
MRPC246.15213.2
STS-B256.44812.2
RTE287.15012.7
Avg25.76.550.012.7
+ +![](images/595b6e69a0b27f0e5a01c20b6b20af16c35c98e03db9d2bd1a18a59ee3317e0f.jpg) +Figure 1: (Left) Average performance on the GLUE validation set across different target sparsity rates for the different methods. (Right) Number of groups where all of the parameters in the group are fully zero for structured vs. non-structured diff pruning at $0 . 5 \%$ target sparsity. We group based on each matrix/bias vector, resulting in 393 groups in total. + +
Diff vector target sparsityQNLISST-2MNLImMNLImmCoLAMRPCSTS-BRTEQQPAvg
0.10%92.793.385.685.958.087.486.368.685.282.5
0.25%93.294.286.286.563.390.988.471.586.184.5
0.50%93.494.286.486.963.591.389.571.586.684.8
1.00%93.394.286.487.066.391.489.971.186.685.1
100%93.594.186.587.162.891.989.871.887.685.0
+ +Table 2: Structured diff pruning results on the validation set with different target sparsity rates. Average performance includes all 9 tasks. + +# 5.1 VARYING THE TARGET SPARSITY + +In Figure 1 (left), we plot results on the GLUE validation set averaged across all tasks at target sparsity rates of $0 . 1 \%$ , $0 . 2 5 \%$ , $0 . 5 \%$ , $1 . 0 \%$ for the different baselines. Structured diff pruning consistently outperforms non-structured and and non-adaptive variants across different sparsity rates. The advantage of adaptive methods becomes more pronounced at extreme sparsity rates. In Table 2, we report the breakdown of accuracy of structured diff pruning across different tasks and sparsity rates, where we observe that different tasks have different sensitivity to target sparsity rates. This suggests that we can obtain even greater parameter-efficiency through targeting task-specific sparsity rates in the diff vector. + +# 5.2 STRUCTURED VS. NON-STRUCTURED DIFF PRUNING + +Structured diff pruning introduces an additional mask per group, which encourages pruning of entire groups. This is less restrictive than traditional group sparsity techniques that have been used with $L _ { 0 }$ -norm relaxations which force all parameters in a group to share the same mask (Louizos et al., 2018; Wang et al., 2019b). However we still expect entire groups to be pruned out more often in the structured case, which might bias the learning process towards either eliminating completely or clustering together nonzero diffs. In Figure 1 (right), we indeed find that structured diff pruning leads to finetuned models that are much more likely to leave entire groups unchanged from their pretrained values (zero diffs). + +# 5.3 TASK-SPECIFIC SPARSITY + +Different layers of pretrained models have argued to encode different information (Liu et al., 2019a; Tenney et al., 2019). Given that each task will likely recruit different kinds of language phenomena embedded in the hidden layers, we hypothesize that diff pruning will modify different parts of the pretrained model through task-specific finetuning. Figure 2 shows the percentage of nonzero diff parameters attributable to the different layers for each task. We find that different tasks indeed modify different parts of the network, although there are some qualitative similarities between some tasks, for example between QNLI & QQP (both must encode questions), and MRPC & STS-B (both must predict similarity between sentences). The embedding layer is very sparsely modified for all tasks. While some of the variations in the sparsity distributions is due to simple randomness, we do observe some level of consistency over multiple runs of the same task, as shown in Figure 3 of the appendix. + +![](images/a2618320f9f0e0193cd71e3737c981cd5b834522d27110d75f63705a8284b8ae.jpg) + +Figure 2: Percentage of modified parameters attributable to each layer for different tasks at $0 . 5 \%$ target sparsity. The layers are ordered from earlier to later (i.e. the embedding layer is shown at the top). The $\mathbf { X }$ -axis for each plot goes from $0 \%$ to $20 \%$ . + +
QNLISST-2MNLImMNLImmCoLAMRPCSTS-BRTEQQPAvg
Sparsity1.5%0.6%0.8%0.8%1.6%2.4%3.3%0.7%0.6%1.4%
Performance93.894.086.286.863.191.989.771.886.584.9
With 0.5% sparsity93.494.286.486.963.591.389.571.586.684.8
+ +Table 3: (Top) Sparsity and performance before magnitude pruning on the validation set with structured diff pruning. (Bottom) Performance with $0 . 5 \%$ target sparsity. + +The ability to modify different parts of the pretrained model for each task could explain the improved parameter-efficiency of our approach compared to Houlsby et al. (2019)’s adapter layers, which can only read/write to the pretrained model at certain points of the computational graph.13 This potentially suggests that adapter layers with more fine-grained access into model internals (e.g. adapters for key/value/query transformations) might result in even greater parameter-efficiency. While left as future work, we also note that diff pruning can be applied in conjunction with adapters, which might further improve results. + +# 5.4 EFFECT OF $\mathrm { L } _ { 0 }$ -BALL PROJECTION VIA MAGNITUDE PRUNING + +Applying magnitude pruning to project onto the $\mathrm { L } _ { 0 }$ -ball was crucial in achieving exact sparsity targets. As shown in Table 3, we observed little loss in performance through magnitude pruning. We re-iterate that it was crucial to finetune with the fixed mask in order to maintain good performance.14 + +# 5.5 SQUAD EXTRACTIVE QUESTION ANSWERING + +To demonstrate the effectiveness of our approach beyond classification, we additionally experiment on the extractive question answering task SQuAD, which asks model to select the answer span to a question given a Wikipedia paragraph. To make direct comparisons with Houlsby et al. (2019), we run all experiments on SQuAD v1.1. For diff pruning, we use the same general hyper-parameters as our full finetuning baseline.15 Results are shown in Table 4. Diff pruning is able achieve comparable or better performance with only $1 \%$ additional parameters. Notably, we see that our method can improve the F1 score of full finetuning baseline by a significant margin (e.g. $9 0 . 8 \% \Rightarrow 9 3 . 2 \% )$ ) + +
SparsityF1
Full finetuning Adapters100% 2%90.7% 90.4%
Full finetuning100%90.8%
Diff pruning1%92.1%
Diff pruning (struct.)1%93.2%
+ +Table 4: SQuAD validation results with BERTLARGE model. + +while modifying many fewer parameters (e.g., $1 0 0 \% \Rightarrow 1 \%$ ), which potentially implies that diff pruning can have a useful regularization effect. + +# 6 DISCUSSION + +# 6.1 MEMORY REQUIREMENTS + +For training, our approach requires more memory than usual finetuning due to additionally optimizing $\pmb { \alpha } _ { \tau }$ and ${ \bf w } _ { \tau }$ . This did not present a significant challenge for pretrained models that we experimented with in this study, since majority of GPU memory was utilized by the minibatch’s activation layers. However, this could present an issue as model sizes get larger and larger. While training efficiency was not a primary concern of this work, diff pruning takes approxiamtely $1 . 5 \times$ to $2 \times$ more time per batch, which results in slower training. + +After training, storing the task-specific diff vector requires storing a compressed version with both the nonzero positions and weights, which incurs additional storage requirements. + +# 6.2 INFORMATION-EFFICIENT TRANSFER LEARNING + +Efficiently representing pretrained models adapted to new tasks is becoming an increasingly important problem in contemporary NLP. This paper focuses on a rather narrow definition of efficiency— parameter-efficiency. An interesting direction might be to target generalizations of parameterefficiency, for example, information-efficiency, which aims to minimize the number of bits required to represent the task-specific model when given the pretrained model for free. This view can suggest other avenues for achieving information-efficient transfer learning: for example, “what is the minimum number of (potentially synthetic) datapoints that we can finetune BERT on to obtain a good task-specific model?”,16 or “what is the shortest prefix string that we can condition GPT3 on for it to become a good task-specific model”? + +# 7 RELATED WORK + +Multi-task learning Multi-task learning (Caruana, 1997), broadly construed, aims to learn models and representations that can be utilized across a diverse range of tasks, and offers a natural approach to training parameter-efficient deep models. Several works have shown that a single BERT model can obtain good performance across multiple tasks when jointly trained (Liu et al., 2019b; Clark et al., 2019; Stickland & Murray, 2019). Adapter layers, which are task-specific layers that read and write to layers of a shared model (Rebuffi et al., 2018), offer an alternative approach to multi-task learning that does not require access to all tasks during training, and have also been applied to obtain parameter-efficient BERT models (Houlsby et al., 2019; Pfeiffer et al., 2020a;b;c). A related line of work targets extreme parameter-efficiency through task-agnostic sentence representations that can be used without finetuning for downstream tasks (Le & Mikolov, 2014; Kiros et al., 2015; Wieting et al., 2016; Hill et al., 2016; Arora et al., 2017; Conneau et al., 2017; Cer et al., 2018; Zhang et al., 2018; Subramanian et al., 2018; Zhang et al., 2020). Reimers & Gurevych (2019), building on the earlier work of Conneau et al. (2017), show that BERT finetuned on natural language inference obtains sentence representations that perform well across multiple sentence-level tasks. These feature-based transfer learning methods are however generally outperformed by fully finetuned models (Howard & Ruder, 2018). + +Model compression There has been much recent work on compressing pretrained trained with self-supervision (see Ganesh et al. (2020) for a recent survey). A particularly promising line of work focuses on obtaining smaller pretrained models (for subsequent finetuning) through weight pruning (Gordon et al., 2020; Sajjad et al., 2020; Chen et al., 2020) and/or knowledge distillation (Sanh et al., 2019; Sun et al., 2019; Turc et al., 2019; Jiao et al., 2019; Sun et al., 2020). It would be interesting to see whether our approach can be applied on top of these smaller pretrained models to for even greater parameter-efficiency. + +Learning to prune Our work is closely related to the line of work on learning to prune pretrained models with differentiable relaxations of binary masks (Wang et al., 2019b; Zhao et al., 2020; Sanh et al., 2020; Radiya-Dixit & Wang, 2020). While these works also enable parameter-efficient transfer learning, they generally apply the masks directly on the pretrained parameters instead of on the difference vector as in the present work. + +Regularization towards pretrained models Finally, diff pruning is also related to works which regularize the learning process towards pretrained models for continual learning (Kirkpatrick et al., 2017; Schwarz et al., 2018), domain adaptation (Wiese et al., 2017; Miceli Barone et al., 2017), and stable finetuning (Lee et al., 2020). These works typically do not utilize sparse regularizers and target a different goal than parameter-efficiency. + +# 8 CONCLUSION + +We propose diff pruning as a simple approach for parameter-efficient transfer learning with pretrained models. Experiments on standard NLP benchmarks and models show that diff pruning can match the performance of fully finetuned baselines while requiring only a few additional parameters per task. We also propose a structured variant of diff pruning which provides further improvements. Future work will consider (i) applying this approach to other architectures (e.g. ConvNets for vision applications), (ii) injecting parameter-efficiency objectives directly into the pretraining process (to pretrain models that are better suited towards sparse transfer learning), and (iii) combining diff pruning with other techniques (e.g. adapters) to achieve even greater parameter-efficiency. + +# REFERENCES + +Sanjeev Arora, Yingyu Liang, and Tengyu Ma. 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Towards Universal Paraphrastic Sentence Embeddings. In Proceedings of ICLR, 2016. + +Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick ´ von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. Huggingface’s transformers: Stateof-the-art natural language processing. ArXiv, abs/1910.03771, 2019. + +John M. Wu, Yonatan Belinkov, Hassan Sajjad, Nadir Durrani, Fahim Dalvi, and James Glass. Similarity Analysis of Contextual Word Representation Models. In Proceedings of ACL, 2020. + +![](images/8c95c374d96e969461975571abc9c78de78634e6f294904982d24204df94d38e.jpg) +Figure 3: Percentage of modified parameters attributable to each layer for 5 different runs of SST-2 at $0 . 5 \%$ target sparsity. The layers are ordered from earlier to later (i.e. the embedding layer is shown at the top). The $\mathbf { X }$ -axis for each plot goes from $0 \%$ to $20 \%$ . + +Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Russ R Salakhutdinov, and Quoc V Le. XLNet: Generalized Autoregressive Pretraining for Language Understanding. In Proceedings of NeurIPS, 2019. + +Minghua Zhang, Yunfang Wu, Weikang Li, and Wei Li. Learning universal sentence representations with mean-max attention autoencoder. In Proceedings of EMNLP, 2018. + +Yan Zhang, Ruidan He, Zuozhu Liu, Kwan Hui Lim, and Lidong Bing. An Unsupervised Sentence Embedding Method byMutual Information Maximization. In Proceedings of EMNLP, 2020. + +Mengjie Zhao, Tao Lin, Martin Jaggi, and Hinrich Schutze. Masking as an Efficient Alternative to Finetuning for Pretrained Language Models. arXiv:2004.12406, 2020. + +# A APPENDIX + +# A.1 CONSISTENCY OF NONZERO PARAMETERS + +Figure 3 shows the percentage of modified parameters attributable to each layer across 5 runs of SST2. We find that there is nonotrivial variation in sparsity across runs, but also a degree of consistency. For example, the first layer is modified considerably more than other layers across all runs. \ No newline at end of file diff --git a/parse/train/E4PK0rg2eP/E4PK0rg2eP_content_list.json b/parse/train/E4PK0rg2eP/E4PK0rg2eP_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..425105ea0f4ec08fa0ddbcbf0c87bd5ad1bced88 --- /dev/null +++ b/parse/train/E4PK0rg2eP/E4PK0rg2eP_content_list.json @@ -0,0 +1,1651 @@ +[ + { + "type": "text", + "text": "PARAMETER-EFFICIENT TRANSFER LEARNING WITH DIFF PRUNING ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 250 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While task-specific finetuning of deep networks pretrained with self-supervision has led to significant empirical advances in NLP, their large size makes the standard finetuning approach difficult to apply to multi-task, memory-constrained settings, as storing the full model parameters for each task become prohibitively expensive. We propose diff pruning as a simple approach to enable parameterefficient transfer learning within the pretrain-finetune framework. This approach views finetuning as learning a task-specific “diff” vector that is applied on top of the pretrained parameter vector, which remains fixed and is shared across different tasks. The diff vector is adaptively pruned during training with a differentiable approximation to the $L _ { 0 }$ -norm penalty to encourage sparsity. Diff pruning becomes parameter-efficient as the number of tasks increases, as it requires storing only the nonzero positions and weights of the diff vector for each task, while the cost of storing the shared pretrained model remains constant. We find that models finetuned with diff pruning can match the performance of fully finetuned baselines on the GLUE benchmark while only modifying $0 . 5 \\%$ of the pretrained model’s parameters per task. ", + "bbox": [ + 233, + 263, + 764, + 484 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 497, + 336, + 513 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Task-specific finetuning of pretrained deep networks has become the dominant paradigm in contemporary NLP, achieving state-of-the-art results across a suite of natural language understanding tasks (Devlin et al., 2019; Liu et al., 2019c; Yang et al., 2019; Lan et al., 2020). While straightforward and empirically effective, this approach is difficult to scale to multi-task, memory-constrained settings (e.g. for on-device applications), as it requires shipping and storing a full set of model parameters for each task. Inasmuch as these models are learning generalizable, task-agnostic language representations through self-supervised pretraining, finetuning the entire model for each task is an especially inefficient use of model parameters. ", + "bbox": [ + 174, + 520, + 825, + 631 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A popular approach to parameter-efficiency is to learn sparse models for each task where a subset of the final model parameters are exactly zero (Gordon et al., 2020; Sajjad et al., 2020; Zhao et al., 2020; Sanh et al., 2020). Such approaches often face a steep sparsity/performance tradeoff, and a substantial portion of nonzero parameters (e.g. $10 \\% { - } 3 0 \\% )$ ) are still typically required to match the performance of the dense counterparts. An alternative is to use multi-task learning or feature-based transfer for more parameter-efficient transfer learning with pretrained models (Liu et al., 2019b; Clark et al., 2019; Stickland & Murray, 2019; Reimers & Gurevych, 2019; Feng et al., 2020). These methods learn only a small number of additional parameters (e.g. a linear layer) on top of a shared model. However, multi-task learning generally requires access to all tasks during training to prevent catastrophic forgetting (French, 1999), while feature-based transfer learning (e.g. based on taskagnostic sentence representations) is typically outperformed by full finetuning (Howard & Ruder, 2018). ", + "bbox": [ + 174, + 638, + 825, + 805 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Adapters (Rebuffi et al., 2018) have recently emerged as a promising approach to parameterefficient transfer learning within the pretrain-finetune paradigm (Houlsby et al., 2019; Pfeiffer et al., 2020a;b;c). Adapter layers are smaller, task-specific modules that are inserted between layers of a pretrained model, which remains fixed and is shared across tasks. These approaches do not require access to all tasks during training making them attractive in settings where one hopes to obtain and share performant models as new tasks arrive in stream. Houlsby et al. (2019) find that adapter layers trained on BERT can match the performance of fully finetuned BERT on the GLUE benchmark (Wang et al., 2019a) while only requiring $3 . 6 \\%$ additional parameters (on average) per task. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we consider a similar setting as adapters but propose a new diff pruning approach with the goal of even more parameter-efficient transfer learning. Diff pruning views finetuning as learning a task-specific difference vector that is applied on top of the pretrained parameter vector, which remains fixed and is shared across different tasks. In order to learn this vector, we reparameterize the task-specific model parameters as $\\theta _ { \\mathrm { t a s k } } = \\theta _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\mathrm { t a s k } }$ , where the pretrained parameter vector $\\theta _ { \\mathrm { p r e t r a i n e d } }$ is fixed and the task-specific diff vector $\\delta _ { \\mathrm { t a s k } }$ is finetuned. The diff vector is regularized with a differentiable approximation to the $L _ { 0 }$ -norm penalty (Louizos et al., 2018) to encourage sparsity. This approach can become parameter-efficient as the number of tasks increases as it only requires storing the nonzero positions and weights of the diff vector for each task. The cost of storing the shared pretrained model remains constant and is amortized across multiple tasks. On the GLUE benchmark (Wang et al., 2019a), diff pruning can match the performance of the fully finetuned BERT baselines while finetuning only $0 . 5 \\%$ of the pretrained parameters per task, making it a potential alternative to adapters for parameter-efficient transfer learning. ", + "bbox": [ + 173, + 103, + 825, + 284 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND: TRANSFER LEARNING FOR NLP", + "text_level": 1, + "bbox": [ + 174, + 297, + 607, + 314 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The field of NLP has recently seen remarkable progress through transfer learning with a pretrainand-finetune paradigm, which initializes a subset of the model parameters for all tasks from a pretrained model and then finetunes on a task specific objective. Pretraining objectives include context prediction (Mikolov et al., 2013), autoencoding (Dai & Le, 2015), machine translation (McCann et al., 2017), and more recently, variants of language modeling (Peters et al., 2018; Radford et al., 2018; Devlin et al., 2019) objectives. ", + "bbox": [ + 174, + 321, + 825, + 406 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Here we consider applying transfer learning to multiple tasks. We consider a setting with a potentially unknown set of tasks, where each $\\tau \\in \\mathcal { T }$ has an associated training set $\\{ x _ { \\tau } ^ { ( n ) } , y _ { \\tau } ^ { ( n ) } \\} _ { n = 1 } ^ { N }$ 1 For . all tasks, the goal is to produce (possibly tied) model parameters to minimize the empirical risk, ", + "bbox": [ + 174, + 412, + 825, + 459 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/209034c1e384d6fcc31ab4692e6f4c361c0f837f239fb577916b24303febed90.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\pmb { \\theta } _ { \\tau } } \\ \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathcal { L } \\left( f ( x _ { \\tau } ^ { ( n ) } ; \\pmb { \\theta } _ { \\tau } ) , y _ { \\tau } ^ { ( n ) } \\right) + \\lambda R ( \\pmb { \\theta } _ { \\tau } )\n$$", + "text_format": "latex", + "bbox": [ + 346, + 465, + 650, + 508 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $f ( \\cdot ; \\pmb \\theta )$ is a parameterized function over the input (e.g. a neural network), $\\mathcal L ( \\cdot , \\cdot )$ is a loss function (e.g. cross-entropy), and $R ( \\cdot )$ is an optional regularizer with hyperparameter $\\lambda$ . ", + "bbox": [ + 173, + 515, + 821, + 545 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This multi-task setting can use the pretrain-then-finetune approach by simply learning independent parameters for each task; however the large size of pretrained models makes this approach exceedingly parameter inefficient. For example, widely-adopted models such as BERTBASE and BERTLARGE have 110M and 340M parameters respectively, while their contemporaries such as T5 (Raffel et al., 2020), Megatron-LM (Shoeybi et al., 2019), and Turing-NLG (Rajbhandari et al., 2019) have parameter counts in the billions. Storing the fully finetuned models becomes difficult even for a moderate number of tasks.2 A classic approach to tackling this parameterinefficiency (Caruana, 1997) is to train a single shared model (along with a task-specific output layer) against multiple tasks through joint training. However, the usual formulation of multi-task learning requires the set of tasks $\\tau$ to be known in advance in order to prevent catastrophic forgetting (French, 1999),3 making it unsuitable for applications in which the set of tasks is unknown (e.g. when tasks arrive in stream). ", + "bbox": [ + 173, + 550, + 825, + 717 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 DIFF PRUNING ", + "text_level": 1, + "bbox": [ + 176, + 731, + 328, + 747 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Diff pruning formulates task-specific finetuning as learning a diff vector $\\delta _ { \\tau }$ that is added to the pretrained model parameters $\\theta _ { \\mathrm { p r e t r a i n e d } }$ . We first reparameterize the task-specific model parameters, ", + "bbox": [ + 173, + 756, + 823, + 785 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/641357162c8cf5fd67763b84ed9ce3a1b4889bb6c238750d670e7c517ca4be3b.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { \\theta } _ { \\tau } = \\pmb { \\theta } _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\tau } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 428, + 791, + 568, + 809 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "which results in the following empirical risk minimization problem, ", + "bbox": [ + 173, + 103, + 619, + 118 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/67153bebf4dae0992cb925cc59d54228833fd661c3be6236e95797eb4ef05e58.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\delta _ { \\tau } } \\ \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathcal { L } \\left( f ( x _ { \\tau } ^ { ( n ) } ; \\theta _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\tau } ) , y _ { \\tau } ^ { ( n ) } \\right) + \\lambda R ( \\theta _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\tau } ) .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 121, + 728, + 164 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This trivial reparameterization is equivalent to the original formulation. Its benefit comes in the multi-task setting where the cost of storing the pretrained parameters $\\theta _ { \\mathrm { p r e t r a i n e d } }$ is amortized across tasks, and the only marginal cost for new tasks is the diff vector. If we can regularize $\\delta _ { \\tau }$ to be sparse such that $\\lVert \\delta _ { \\tau } \\rVert _ { 0 } \\ll \\lVert \\bar { \\pmb { \\theta } } _ { \\mathrm { p r e t r a i n e d } } \\rVert _ { 0 }$ , then this approach can become more parameter-efficient as the number of tasks increases. We can specify this goal with an $L _ { 0 }$ -norm penalty on the diff vector, ", + "bbox": [ + 173, + 172, + 826, + 243 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/db87601c916bc51947dfd29fe65493f44e30dcb3ea1509908209393cfeaa86fe.jpg", + "text": "$$\nR ( \\theta _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\tau } ) = \\| \\pmb { \\delta } _ { \\tau } \\| _ { 0 } = \\sum _ { i = 1 } ^ { d } \\mathbb { 1 } \\{ \\pmb { \\delta } _ { \\tau , i } \\neq 0 \\} .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 244, + 656, + 287 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 DIFFERENTIABLE APPROXIMATION TO THE $L _ { 0 }$ -NORM", + "text_level": 1, + "bbox": [ + 173, + 308, + 583, + 323 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This regularizer is difficult to directly optimize as it is non-differentiable. In order to approximate this $L _ { 0 }$ objective, we follow the standard approach for gradient-based learning with $L _ { 0 }$ sparsity using a relaxed mask vector (Louizos et al., 2018). This approach involves relaxing a binary vector into continuous space, and then multiplying it with a dense weight vector to determine how much of the weight vector is applied during training. After training, the mask is deterministic and a large portion of the diff vector is true zero. ", + "bbox": [ + 173, + 329, + 825, + 415 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To apply this method we first decompose $\\delta _ { \\tau }$ into a binary mask vector multiplied with a dense vector, ", + "bbox": [ + 173, + 420, + 823, + 449 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/50baad9ae5d43e25ead61a531e7ee5bd6f11e2d43dad20fb8b2093d5a5ab6bab.jpg", + "text": "$$\n\\begin{array} { r } { \\delta _ { \\tau } = \\mathbf { z } _ { \\tau } \\odot \\mathbf { w } _ { \\tau } , \\qquad \\mathbf { z } _ { \\tau } \\in \\{ 0 , 1 \\} ^ { d } , \\mathbf { w } _ { \\tau } \\in \\mathbb { R } ^ { d } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 303, + 449, + 694, + 469 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can now instead optimize an expectation with respect to ${ \\bf z } _ { \\tau }$ , whose distribution $p ( \\mathbf { z } _ { \\tau } ; \\pmb { \\alpha } _ { \\tau } )$ is initially Bernoulli with parameters $\\pmb { \\alpha } _ { \\tau }$ , ", + "bbox": [ + 173, + 472, + 825, + 500 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3f81f16930f14f81b0b5df85f8e68d18a457c5c95c09d04e34b40d88c3eccece.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\alpha _ { \\tau } , \\mathbf { w } _ { \\tau } } \\mathbb { E } _ { \\mathbf { z } _ { \\tau } \\sim p ( \\mathbf { z } _ { \\tau } ; \\alpha _ { \\tau } ) } \\left[ \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathcal { L } \\left( f ( x _ { \\tau } ^ { ( n ) } ; \\boldsymbol { \\theta } _ { \\mathrm { p r e t r a i n e d } } + \\mathbf { z } _ { \\tau } \\odot \\mathbf { w } _ { \\tau } , ) , y _ { \\tau } ^ { ( n ) } \\right) + \\lambda \\lVert \\delta _ { \\tau } \\rVert _ { 0 } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 222, + 502, + 776, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This objective is still difficult in practice due to ${ \\bf z } _ { \\tau }$ ’s being discrete (which requires the score function gradient estimator), but the expectation provides some guidance for empirically effective relaxations. We follow prior work (Louizos et al., 2018; Wang et al., 2019b) and relax ${ \\bf z } _ { \\tau }$ into continuous space $[ 0 , 1 ] ^ { d }$ with a stretched Hard-Concrete distribution (Jang et al., 2017; Maddison et al., 2017), which allows for the use of pathwise gradient estimators. Specifically, ${ \\bf z } _ { \\tau }$ is now defined to be a deterministic and (sub)differentiable function of a sample $\\mathbf { u }$ from a uniform distribution, ", + "bbox": [ + 173, + 546, + 825, + 631 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2a6654c74a3353f08edc978223c41515541788a721f3d19b6042113955bb2af0.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { u } \\sim U ( \\mathbf { 0 } , \\mathbf { 1 } ) , \\qquad \\mathbf { s } _ { \\tau } = \\sigma \\left( \\log \\mathbf { u } - \\log ( 1 - \\mathbf { u } ) + \\alpha _ { \\tau } \\right) , } \\\\ { \\bar { \\mathbf { s } } _ { \\tau } = \\mathbf { s } _ { \\tau } \\times ( r - l ) + l , \\qquad \\mathbf { z } _ { \\tau } = \\operatorname* { m i n } ( \\mathbf { 1 } , \\operatorname* { m a x } ( \\mathbf { 0 } , \\bar { \\mathbf { s } } _ { \\tau } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 632, + 736, + 670 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here $l < 0$ and $r > 1$ are two constants used to stretch ${ \\bf s } _ { \\tau }$ into the interval $( l , r ) ^ { d }$ before it is clamped to $[ 0 , 1 ] ^ { d }$ with the $\\operatorname* { m i n } ( \\mathbf { 1 } , \\operatorname* { m a x } ( \\mathbf { 0 } , \\cdot ) )$ operation. In this case we have a differentiable closed-form expression for the expected $L _ { 0 }$ -norm, ", + "bbox": [ + 174, + 671, + 825, + 714 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/edb3942bcbccde41788b4dc919f478a284c87096f975cee4e8d84e9352a91af3.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\left. \\pmb { \\delta } _ { \\tau } \\right. _ { 0 } \\right] = \\sum _ { i = 1 } ^ { d } \\mathbb { E } \\left[ \\mathbb { 1 } \\left\\{ \\mathbf { z } _ { \\tau , i } > 0 \\right\\} \\right] = \\sum _ { i = 1 } ^ { d } \\sigma \\left( \\alpha _ { \\tau , i } - \\log \\frac { - l } { r } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 715, + 700, + 760 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Thus the final optimization problem is given by, ", + "bbox": [ + 174, + 761, + 488, + 776 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "$\\operatorname* { m i n } _ { \\tau _ { \\tau } , \\mathbf { w } _ { \\tau } } \\mathbb { E } _ { \\mathbf { u } \\sim U [ \\mathbf { 0 } , 1 ] } \\left[ \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathcal { L } \\left( f ( x _ { \\tau } ^ { ( n ) } ; \\boldsymbol { \\theta } _ { \\mathrm { p r e r a i n e d } } + \\mathbf { z } _ { \\tau } \\odot \\mathbf { w } _ { \\tau } , ) , y _ { \\tau } ^ { ( n ) } \\right) \\right] + \\lambda \\sum _ { i = 1 } ^ { d } \\sigma \\left( \\alpha _ { \\tau , i } - \\log \\frac { - l } { r } \\right) ,$ and we can now utilize pathwise gradient estimators to optimize the first term with respect to $\\pmb { \\alpha } _ { \\tau }$ since the expectation no longer depends on it.4 After training we obtain the final diff vector $\\delta _ { \\tau }$ by sampling $\\mathbf { u }$ once to obtain ${ \\bf z } _ { \\tau }$ (which is not necessarily a binary vector but has a significant number of dimensions equal to exactly zero due to the clamping function), then setting $\\pmb { \\delta } _ { \\tau } = \\mathbf { z } _ { \\tau } \\odot \\mathbf { w } _ { \\tau }$ . 5 ", + "bbox": [ + 173, + 777, + 826, + 880 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 $L _ { 0 }$ -BALL PROJECTION WITH MAGNITUDE PRUNING FOR SPARSITY CONTROL ", + "bbox": [ + 176, + 103, + 746, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Differentiable $L _ { 0 }$ regularization provides a strong way to achieve high sparsity rate. However, it would be ideal to have more fine-grained control into the exact sparsity rate in the diff vector, especially considering applications which require specific parameter budgets. As $\\lambda$ is just the Lagrangian multiplier for the constraint $\\mathbb { E } \\left[ \\lVert \\pmb { \\delta } _ { \\tau } \\rVert _ { 0 } \\right] < \\eta$ for some $\\eta$ , this could be achieved in principle by searching over different values of $\\lambda$ . However we found it more efficient and empirically effective to achieve an exact sparsity rate by simply projecting onto the $L _ { 0 }$ -ball after training. ", + "bbox": [ + 174, + 126, + 825, + 210 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Specifically we use magnitude pruning on the diff vector $\\delta _ { \\tau }$ and target a sparsity rate $t \\%$ by only keeping the top $t \\% \\times \\bar { d }$ values in $\\delta _ { \\tau }$ .6 Note that unlike standard magnitude pruning, this is based on the magnitude of the diff vector values and not the model parameters. As is usual in magnitude pruning, we found it important to further finetune $\\delta _ { \\tau }$ with the nonzero masks fixed to maintain good performance (Han et al., 2016). Since this type of parameter-efficiency through projection onto the $L _ { 0 }$ -ball can be applied without adaptive diff pruning,7 such an approach will serve as one of our baselines in the empirical study. ", + "bbox": [ + 173, + 217, + 825, + 314 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 STRUCTURED DIFF PRUNING ", + "text_level": 1, + "bbox": [ + 176, + 328, + 416, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Diff pruning, as presented above, is architecture-agnostic and does not exploit the underlying model structure—each dimension of ${ \\bf z } _ { \\tau }$ is independent from one another. While this makes the approach potentially more flexible, we might expect to achieve better sparsity/performance tradeoff through a structured formulation which encourages active parameters to group together and other areas to be fully sparse. Motivated by this intuition, we first partition the parameter indices into $G$ groups $\\{ g ( 1 ) , \\ldots , g ( G ) \\}$ where $g ( j )$ is a subset of parameter indices governed by group $g ( j )$ .8 We then introduce a scalar $\\mathbf { z } _ { \\tau } ^ { j }$ (with the associated parameter $\\alpha _ { \\tau } ^ { j }$ ) for each group $g ( j )$ , and decompose the task-specific parameter for index $i \\in g ( j )$ as $\\begin{array} { r } { \\delta _ { \\tau , i } ^ { j } = \\mathbf { z } _ { \\tau , i } \\times \\mathbf { z } _ { \\tau } ^ { j } \\times \\mathbf { w } _ { \\tau , i } . } \\end{array}$ The expected $L _ { 0 }$ -norm is then given by, ", + "bbox": [ + 173, + 349, + 825, + 479 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6dd9f1cd7a0ba610a4afd20bc615d654b40294d36a6f77a7d0098203f1f2eddd.jpg", + "text": "$$\n\\Sigma \\left[ \\left. \\delta _ { \\tau } \\right. _ { 0 } \\right] = \\sum _ { j = 1 } ^ { G } \\sum _ { i \\in g ( j ) } \\mathbb { E } \\left[ \\mathbb { I } \\left\\{ \\mathbf { z } _ { \\tau , i } \\cdot \\mathbf { z } _ { \\tau } ^ { g } > 0 \\right\\} \\right] = \\sum _ { j = 1 } ^ { G } \\sum _ { i \\in g ( j ) } \\sigma \\left( \\alpha _ { \\tau , i } - \\log { \\frac { - l } { r } } \\right) \\times \\sigma \\left( \\alpha _ { \\tau } ^ { j } - \\log { \\frac { - l } { r } } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 484, + 825, + 531 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "and we can train with gradient-based optimization as before. ", + "bbox": [ + 173, + 537, + 570, + 553 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 565, + 328, + 582 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 MODEL AND DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 590, + 380, + 604 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For evaluation we use the GLUE benchmark (Wang et al., 2019b), a popular finetuning dataset. Following adapters (Houlsby et al., 2019), we test our approach on the following subset of the GLUE tasks: Multi-Genre Natural Language Inference (MNLI), where the goal is two predict whether the relationship between two sentences is entailment, contradiction, or neutral (we test on both $\\mathrm { M N L L } \\mathrm { I } _ { m }$ and $\\mathrm { M N L I } _ { m m }$ which respectively tests on matched/mismatched domains); Quora Question Pairs (QQP), a classification task to predict whether two question are semantically equivalent; Question Natural Language Inference (QNLI), which must predict whether a sentence is a correct answer to the question; Stanford Sentiment Treebank (SST-2), a sentence classification task to predict the sentiment of movie reviews; Corpus of Linguistic Acceptability (CoLA), where the goal is predict whether a sentence is linguistically acceptable or not; Semantic Textual Similarity Benchmark (STS$\\mathbf { B }$ ), which must predict a similarity rating between two sentences; Microsoft Research Paraphrase Corpus (MRPC), where the goal is to predict whether two sentences are semantically equivalent; Recognizing Textual Entailment (RTE), which must predict whether a second sentence is entailed by the first. For evaluation, the benchmark uses Matthew’s correlation for CoLA, Spearman for STS-B, $\\mathrm { F _ { 1 } }$ score for MRPC/QQC, and accuracy for MNLI/QNLI/SST-2/RTE. ", + "bbox": [ + 173, + 613, + 825, + 820 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/7e11a32275eb2c11b417cefece91c401cc0ca72c9474a3ea349c1c085c7a67e7.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Total paramsNew params per taskQNLI*SST-2 MNLImMNLImmCoLA MRPC STS-B RTEQQPAvg
Full finetuning9.00×100%91.194.986.785.960.589.387.670.172.180.9
Adapters (8-256)1.32×3.6%90.794.084.985.159.589.586.971.571.880.4
Adapters (64)1.19×2.1%91.494.285.384.656.989.687.368.671.879.8
Full finetuning9.00×100%93.494.186.786.059.688.986.671.271.780.6
Last layer1.34×3.8%79.891.671.472.940.280.167.358.663.368.2
Non-adap. diff pruning1.05×0.5%89.793.684.984.851.281.578.261.568.675.5
Diff pruning1.05×0.5%92.993.885.785.660.587.083.568.170.679.4
Diff pruning (struct.)1.05×0.5%93.394.186.486.061.189.786.070.671.180.6
", + "bbox": [ + 176, + 101, + 825, + 222 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 1: GLUE benchmark test server results with BERTLARGE models. (Top) Results with adapter bottleneck layers (brackets indicate the size of bottlenecks), taken from from Houlsby et al. (2019). (Bottom) Results from this work. ${ } ^ { * } \\mathrm { Q N L I }$ results are not directly comparable across the two works as the GLUE benchmark has updated the test set since then. To make our results comparable the average column is calculated without QNLI. ", + "bbox": [ + 174, + 223, + 825, + 273 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For all experiments, we use the BERTLARGE model from Devlin et al. (2019), which has 24 layers, 1024 hidden size, 16 attention heads, and 340M parameters. We use the Huggingface Transformer library (Wolf et al., 2019) to conduct our experiments. ", + "bbox": [ + 174, + 284, + 825, + 325 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 BASELINES", + "text_level": 1, + "bbox": [ + 174, + 335, + 294, + 349 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We compare both structured and non-structured variants of diff pruning against the following baselines: Full finetuning, which fully finetunes $\\mathrm { B E R T _ { L } }$ ARGE as usual; Last layer finetuning, which only finetunes the penultimate layer (along with the final output layer)9; Adapters from Houlsby et al. (2019), which train task-specific bottleneck layers between between each layer of a pretrained model, where parameter-efficiency can be controlled by varying the size of the bottleneck layers; and Non-adaptive diff pruning, which performs diff pruning just based on magnitude pruning (i.e., we obtain $\\pmb { \\theta } _ { \\tau }$ through usual finetuning, set $\\delta _ { \\tau } = \\pmb { \\theta } _ { \\tau } - \\pmb { \\theta } _ { \\mathrm { p r e t r a i n e d } }$ , and then apply magnitude pruning followed by additional finetuning on $\\delta _ { \\tau }$ ). For diff pruning we set our target sparsity rate to $0 . 5 \\%$ and investigate the effect of different target sparsity rates in section 5.1. ", + "bbox": [ + 174, + 357, + 825, + 483 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.3 IMPLEMENTATION DETAILS AND HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 173, + 492, + 588, + 507 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Diff pruning introduces additional hyperparameters $l , r$ (for stretching the Hard-Concrete distribution) and $\\lambda$ (for weighting the approximate $L _ { 0 }$ -norm penalty). We found $l = - 1 . 5 , r = 1 . 5 , \\lambda =$ $1 . 2 5 \\times 1 0 ^ { - 7 }$ to work well across all tasks. We also initialize the weight vector ${ \\bf w } _ { \\tau }$ to 0, and $\\pmb { \\alpha } _ { \\tau }$ to a positive vector (we use 5) to encourage ${ \\bf z } _ { \\tau }$ to be close to 1 at the start of training. While we mainly experiment with BERTLARGE to compare against prior work with adapters (Houlsby et al., 2019), in preliminary experiments we found these hyperparameters to work for finetuning RoBERTa (Liu et al., $2 0 1 9 \\mathrm { c }$ ) and XLNet (Yang et al., 2019) models as well. ", + "bbox": [ + 174, + 511, + 825, + 609 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For all tasks we use a learning rate of $1 \\times 1 0 ^ { - 5 }$ and perform a hyperparameter search over batch size $\\in \\{ 4 , 6 , 8 , 1 0 \\}$ and the number of epochs $\\in \\{ 2 , 3 , 4 , 5 \\}$ .10 However we found the default settings used for regular finetuning as suggested in the original BERT paper to work well for most tasks. Finetuning with the fixed mask after projecting onto the $L _ { 0 }$ -ball with magnitude pruning is done with a learning rate of $5 \\times 1 0 ^ { - 5 }$ for 3 or 5 epochs (3 epochs for QNLI, SST-2, MNLI-m, MNLI-mm, CoLA, QQP, 5 epochs for MRPC, STS-B, RTE). Grouping for the structured version of diff pruning is based on the matrix/bias vectors (i.e. parameters that belong to the same matrix or bias vector are assumed to be in the same group), which results in 393 groups.1 ", + "bbox": [ + 173, + 616, + 825, + 728 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 RESULTS AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 741, + 413, + 757 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our main results on the GLUE benchmark are shown in Table 1. Structured diff pruning can match the performance of a fully finetuned BERTLARGE model while only requiring $0 . 5 \\%$ additional parameters per task. Diff pruning without structured sparsity also performs well, though slightly worse than the structured approach. Non-adaptive diff pruning, which magnitude prunes the diff vector without learning the binary mask $\\mathbf { z } _ { \\tau }$ , performs significantly worse, indicating the importance of learning the masking vector. Compared to adapters, diff pruning obtains similar performance while requiring fewer parameters per task, making it a potential alternative for parameter-efficient transfer learning.12 We now perform a series of analysis experiments on the validation set. ", + "bbox": [ + 174, + 768, + 823, + 797 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/51e4628c9fdcedc7b5148c6713e9133f068ca6743dacfae1df379a77b1f70a0f.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Non-structuredPruned Diff GroupsStructured
#%#%
MRPC246.15213.2
STS-B256.44812.2
RTE287.15012.7
Avg25.76.550.012.7
", + "bbox": [ + 531, + 136, + 789, + 242 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/595b6e69a0b27f0e5a01c20b6b20af16c35c98e03db9d2bd1a18a59ee3317e0f.jpg", + "image_caption": [ + "Figure 1: (Left) Average performance on the GLUE validation set across different target sparsity rates for the different methods. (Right) Number of groups where all of the parameters in the group are fully zero for structured vs. non-structured diff pruning at $0 . 5 \\%$ target sparsity. We group based on each matrix/bias vector, resulting in 393 groups in total. " + ], + "image_footnote": [], + "bbox": [ + 205, + 116, + 446, + 262 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/cf93b9d4d94aac37759a017bb2787dcedf4a0259386fb92fc7fa433b7d3a4591.jpg", + "table_caption": [], + "table_footnote": [ + "Table 2: Structured diff pruning results on the validation set with different target sparsity rates. Average performance includes all 9 tasks. " + ], + "table_body": "
Diff vector target sparsityQNLISST-2MNLImMNLImmCoLAMRPCSTS-BRTEQQPAvg
0.10%92.793.385.685.958.087.486.368.685.282.5
0.25%93.294.286.286.563.390.988.471.586.184.5
0.50%93.494.286.486.963.591.389.571.586.684.8
1.00%93.394.286.487.066.391.489.971.186.685.1
100%93.594.186.587.162.891.989.871.887.685.0
", + "bbox": [ + 236, + 325, + 756, + 407 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 445, + 825, + 529 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 VARYING THE TARGET SPARSITY ", + "text_level": 1, + "bbox": [ + 174, + 541, + 444, + 556 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Figure 1 (left), we plot results on the GLUE validation set averaged across all tasks at target sparsity rates of $0 . 1 \\%$ , $0 . 2 5 \\%$ , $0 . 5 \\%$ , $1 . 0 \\%$ for the different baselines. Structured diff pruning consistently outperforms non-structured and and non-adaptive variants across different sparsity rates. The advantage of adaptive methods becomes more pronounced at extreme sparsity rates. In Table 2, we report the breakdown of accuracy of structured diff pruning across different tasks and sparsity rates, where we observe that different tasks have different sensitivity to target sparsity rates. This suggests that we can obtain even greater parameter-efficiency through targeting task-specific sparsity rates in the diff vector. ", + "bbox": [ + 174, + 564, + 825, + 675 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 STRUCTURED VS. NON-STRUCTURED DIFF PRUNING ", + "text_level": 1, + "bbox": [ + 174, + 689, + 581, + 702 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Structured diff pruning introduces an additional mask per group, which encourages pruning of entire groups. This is less restrictive than traditional group sparsity techniques that have been used with $L _ { 0 }$ -norm relaxations which force all parameters in a group to share the same mask (Louizos et al., 2018; Wang et al., 2019b). However we still expect entire groups to be pruned out more often in the structured case, which might bias the learning process towards either eliminating completely or clustering together nonzero diffs. In Figure 1 (right), we indeed find that structured diff pruning leads to finetuned models that are much more likely to leave entire groups unchanged from their pretrained values (zero diffs). ", + "bbox": [ + 174, + 714, + 825, + 825 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.3 TASK-SPECIFIC SPARSITY ", + "text_level": 1, + "bbox": [ + 174, + 835, + 393, + 849 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Different layers of pretrained models have argued to encode different information (Liu et al., 2019a; Tenney et al., 2019). Given that each task will likely recruit different kinds of language phenomena embedded in the hidden layers, we hypothesize that diff pruning will modify different parts of the pretrained model through task-specific finetuning. Figure 2 shows the percentage of nonzero diff parameters attributable to the different layers for each task. We find that different tasks indeed modify different parts of the network, although there are some qualitative similarities between some tasks, for example between QNLI & QQP (both must encode questions), and MRPC & STS-B (both must predict similarity between sentences). The embedding layer is very sparsely modified for all tasks. While some of the variations in the sparsity distributions is due to simple randomness, we do observe some level of consistency over multiple runs of the same task, as shown in Figure 3 of the appendix. ", + "bbox": [ + 176, + 858, + 823, + 901 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/a2618320f9f0e0193cd71e3737c981cd5b834522d27110d75f63705a8284b8ae.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 179, + 103, + 818, + 251 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f1421724012c23f06006dcf661bb337cb0a8a1761b9ef97a21766239f94a5d77.jpg", + "table_caption": [ + "Figure 2: Percentage of modified parameters attributable to each layer for different tasks at $0 . 5 \\%$ target sparsity. The layers are ordered from earlier to later (i.e. the embedding layer is shown at the top). The $\\mathbf { X }$ -axis for each plot goes from $0 \\%$ to $20 \\%$ . " + ], + "table_footnote": [ + "Table 3: (Top) Sparsity and performance before magnitude pruning on the validation set with structured diff pruning. (Bottom) Performance with $0 . 5 \\%$ target sparsity. " + ], + "table_body": "
QNLISST-2MNLImMNLImmCoLAMRPCSTS-BRTEQQPAvg
Sparsity1.5%0.6%0.8%0.8%1.6%2.4%3.3%0.7%0.6%1.4%
Performance93.894.086.286.863.191.989.771.886.584.9
With 0.5% sparsity93.494.286.486.963.591.389.571.586.684.8
", + "bbox": [ + 186, + 304, + 805, + 364 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 414, + 825, + 525 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The ability to modify different parts of the pretrained model for each task could explain the improved parameter-efficiency of our approach compared to Houlsby et al. (2019)’s adapter layers, which can only read/write to the pretrained model at certain points of the computational graph.13 This potentially suggests that adapter layers with more fine-grained access into model internals (e.g. adapters for key/value/query transformations) might result in even greater parameter-efficiency. While left as future work, we also note that diff pruning can be applied in conjunction with adapters, which might further improve results. ", + "bbox": [ + 174, + 532, + 825, + 628 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.4 EFFECT OF $\\mathrm { L } _ { 0 }$ -BALL PROJECTION VIA MAGNITUDE PRUNING ", + "text_level": 1, + "bbox": [ + 174, + 640, + 637, + 654 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Applying magnitude pruning to project onto the $\\mathrm { L } _ { 0 }$ -ball was crucial in achieving exact sparsity targets. As shown in Table 3, we observed little loss in performance through magnitude pruning. We re-iterate that it was crucial to finetune with the fixed mask in order to maintain good performance.14 ", + "bbox": [ + 176, + 659, + 825, + 700 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.5 SQUAD EXTRACTIVE QUESTION ANSWERING ", + "text_level": 1, + "bbox": [ + 174, + 723, + 537, + 738 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To demonstrate the effectiveness of our approach beyond classification, we additionally experiment on the extractive question answering task SQuAD, which asks model to select the answer span to a question given a Wikipedia paragraph. To make direct comparisons with Houlsby et al. (2019), we run all experiments on SQuAD v1.1. For diff pruning, we use the same general hyper-parameters as our full finetuning baseline.15 Results are shown in Table 4. Diff pruning is able achieve comparable or better performance with only $1 \\%$ additional parameters. Notably, we see that our method can improve the F1 score of full finetuning baseline by a significant margin (e.g. $9 0 . 8 \\% \\Rightarrow 9 3 . 2 \\% )$ ) ", + "bbox": [ + 173, + 750, + 825, + 847 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/4f39fcc11bdeb82064d1bfc004bd6998c756639a075f86e91f7cfa4d23221bd5.jpg", + "table_caption": [], + "table_footnote": [ + "Table 4: SQuAD validation results with BERTLARGE model. " + ], + "table_body": "
SparsityF1
Full finetuning Adapters100% 2%90.7% 90.4%
Full finetuning100%90.8%
Diff pruning1%92.1%
Diff pruning (struct.)1%93.2%
", + "bbox": [ + 352, + 101, + 640, + 207 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "while modifying many fewer parameters (e.g., $1 0 0 \\% \\Rightarrow 1 \\%$ ), which potentially implies that diff pruning can have a useful regularization effect. ", + "bbox": [ + 174, + 252, + 825, + 280 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 297, + 310, + 314 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.1 MEMORY REQUIREMENTS ", + "text_level": 1, + "bbox": [ + 174, + 324, + 397, + 338 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For training, our approach requires more memory than usual finetuning due to additionally optimizing $\\pmb { \\alpha } _ { \\tau }$ and ${ \\bf w } _ { \\tau }$ . This did not present a significant challenge for pretrained models that we experimented with in this study, since majority of GPU memory was utilized by the minibatch’s activation layers. However, this could present an issue as model sizes get larger and larger. While training efficiency was not a primary concern of this work, diff pruning takes approxiamtely $1 . 5 \\times$ to $2 \\times$ more time per batch, which results in slower training. ", + "bbox": [ + 174, + 344, + 825, + 428 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "After training, storing the task-specific diff vector requires storing a compressed version with both the nonzero positions and weights, which incurs additional storage requirements. ", + "bbox": [ + 174, + 434, + 823, + 463 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 INFORMATION-EFFICIENT TRANSFER LEARNING", + "text_level": 1, + "bbox": [ + 174, + 477, + 550, + 491 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Efficiently representing pretrained models adapted to new tasks is becoming an increasingly important problem in contemporary NLP. This paper focuses on a rather narrow definition of efficiency— parameter-efficiency. An interesting direction might be to target generalizations of parameterefficiency, for example, information-efficiency, which aims to minimize the number of bits required to represent the task-specific model when given the pretrained model for free. This view can suggest other avenues for achieving information-efficient transfer learning: for example, “what is the minimum number of (potentially synthetic) datapoints that we can finetune BERT on to obtain a good task-specific model?”,16 or “what is the shortest prefix string that we can condition GPT3 on for it to become a good task-specific model”? ", + "bbox": [ + 173, + 496, + 825, + 622 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 645, + 343, + 661 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Multi-task learning Multi-task learning (Caruana, 1997), broadly construed, aims to learn models and representations that can be utilized across a diverse range of tasks, and offers a natural approach to training parameter-efficient deep models. Several works have shown that a single BERT model can obtain good performance across multiple tasks when jointly trained (Liu et al., 2019b; Clark et al., 2019; Stickland & Murray, 2019). Adapter layers, which are task-specific layers that read and write to layers of a shared model (Rebuffi et al., 2018), offer an alternative approach to multi-task learning that does not require access to all tasks during training, and have also been applied to obtain parameter-efficient BERT models (Houlsby et al., 2019; Pfeiffer et al., 2020a;b;c). A related line of work targets extreme parameter-efficiency through task-agnostic sentence representations that can be used without finetuning for downstream tasks (Le & Mikolov, 2014; Kiros et al., 2015; Wieting et al., 2016; Hill et al., 2016; Arora et al., 2017; Conneau et al., 2017; Cer et al., 2018; Zhang et al., 2018; Subramanian et al., 2018; Zhang et al., 2020). Reimers & Gurevych (2019), building on the earlier work of Conneau et al. (2017), show that BERT finetuned on natural language inference obtains sentence representations that perform well across multiple sentence-level tasks. These feature-based transfer learning methods are however generally outperformed by fully finetuned models (Howard & Ruder, 2018). ", + "bbox": [ + 173, + 680, + 825, + 900 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Model compression There has been much recent work on compressing pretrained trained with self-supervision (see Ganesh et al. (2020) for a recent survey). A particularly promising line of work focuses on obtaining smaller pretrained models (for subsequent finetuning) through weight pruning (Gordon et al., 2020; Sajjad et al., 2020; Chen et al., 2020) and/or knowledge distillation (Sanh et al., 2019; Sun et al., 2019; Turc et al., 2019; Jiao et al., 2019; Sun et al., 2020). It would be interesting to see whether our approach can be applied on top of these smaller pretrained models to for even greater parameter-efficiency. ", + "bbox": [ + 174, + 103, + 825, + 200 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Learning to prune Our work is closely related to the line of work on learning to prune pretrained models with differentiable relaxations of binary masks (Wang et al., 2019b; Zhao et al., 2020; Sanh et al., 2020; Radiya-Dixit & Wang, 2020). While these works also enable parameter-efficient transfer learning, they generally apply the masks directly on the pretrained parameters instead of on the difference vector as in the present work. ", + "bbox": [ + 174, + 210, + 825, + 279 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Regularization towards pretrained models Finally, diff pruning is also related to works which regularize the learning process towards pretrained models for continual learning (Kirkpatrick et al., 2017; Schwarz et al., 2018), domain adaptation (Wiese et al., 2017; Miceli Barone et al., 2017), and stable finetuning (Lee et al., 2020). These works typically do not utilize sparse regularizers and target a different goal than parameter-efficiency. ", + "bbox": [ + 174, + 287, + 825, + 358 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 378, + 318, + 395 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose diff pruning as a simple approach for parameter-efficient transfer learning with pretrained models. Experiments on standard NLP benchmarks and models show that diff pruning can match the performance of fully finetuned baselines while requiring only a few additional parameters per task. We also propose a structured variant of diff pruning which provides further improvements. Future work will consider (i) applying this approach to other architectures (e.g. ConvNets for vision applications), (ii) injecting parameter-efficiency objectives directly into the pretraining process (to pretrain models that are better suited towards sparse transfer learning), and (iii) combining diff pruning with other techniques (e.g. adapters) to achieve even greater parameter-efficiency. ", + "bbox": [ + 174, + 410, + 825, + 522 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 544, + 285, + 559 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sanjeev Arora, Yingyu Liang, and Tengyu Ma. A Simple but Tough-to-Beat Baseline for Sentence Embeddings . In Proceedings of ICLR, 2017. ", + "bbox": [ + 174, + 568, + 823, + 595 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Tom B. 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Neural domain adaptation for biomedical question answering. In Proceedings of CoNLL, August 2017. ", + "bbox": [ + 171, + 742, + 823, + 771 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "John Wieting, Mohit Bansal, Kevin Gimpel, and Karen Livescu. Towards Universal Paraphrastic Sentence Embeddings. In Proceedings of ICLR, 2016. ", + "bbox": [ + 173, + 779, + 823, + 809 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick ´ von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. Huggingface’s transformers: Stateof-the-art natural language processing. ArXiv, abs/1910.03771, 2019. ", + "bbox": [ + 181, + 816, + 823, + 887 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "John M. Wu, Yonatan Belinkov, Hassan Sajjad, Nadir Durrani, Fahim Dalvi, and James Glass. Similarity Analysis of Contextual Word Representation Models. In Proceedings of ACL, 2020. ", + "bbox": [ + 176, + 895, + 818, + 924 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/8c95c374d96e969461975571abc9c78de78634e6f294904982d24204df94d38e.jpg", + "image_caption": [ + "Figure 3: Percentage of modified parameters attributable to each layer for 5 different runs of SST-2 at $0 . 5 \\%$ target sparsity. The layers are ordered from earlier to later (i.e. the embedding layer is shown at the top). The $\\mathbf { X }$ -axis for each plot goes from $0 \\%$ to $20 \\%$ . " + ], + "image_footnote": [], + "bbox": [ + 197, + 104, + 799, + 301 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Russ R Salakhutdinov, and Quoc V Le. XLNet: Generalized Autoregressive Pretraining for Language Understanding. In Proceedings of NeurIPS, 2019. ", + "bbox": [ + 174, + 382, + 826, + 424 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Minghua Zhang, Yunfang Wu, Weikang Li, and Wei Li. Learning universal sentence representations with mean-max attention autoencoder. In Proceedings of EMNLP, 2018. ", + "bbox": [ + 174, + 433, + 823, + 463 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Yan Zhang, Ruidan He, Zuozhu Liu, Kwan Hui Lim, and Lidong Bing. An Unsupervised Sentence Embedding Method byMutual Information Maximization. In Proceedings of EMNLP, 2020. ", + "bbox": [ + 176, + 470, + 823, + 501 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Mengjie Zhao, Tao Lin, Martin Jaggi, and Hinrich Schutze. Masking as an Efficient Alternative to Finetuning for Pretrained Language Models. arXiv:2004.12406, 2020. ", + "bbox": [ + 178, + 508, + 825, + 537 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 565, + 299, + 580 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 CONSISTENCY OF NONZERO PARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 595, + 514, + 611 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Figure 3 shows the percentage of modified parameters attributable to each layer across 5 runs of SST2. We find that there is nonotrivial variation in sparsity across runs, but also a degree of consistency. For example, the first layer is modified considerably more than other layers across all runs. ", + "bbox": [ + 173, + 622, + 825, + 664 + ], + "page_idx": 12 + } +] \ No newline at end of file diff --git a/parse/train/E4PK0rg2eP/E4PK0rg2eP_middle.json b/parse/train/E4PK0rg2eP/E4PK0rg2eP_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..619fcf5137fea29e257733bd3ae8f2d55456359b --- /dev/null +++ b/parse/train/E4PK0rg2eP/E4PK0rg2eP_middle.json @@ -0,0 +1,35670 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 97 + ], + "score": 1.0, + "content": "PARAMETER-EFFICIENT TRANSFER LEARNING WITH", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 217, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 217, + 117 + ], + "score": 1.0, + "content": "DIFF PRUNING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 198 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 209, + 468, + 384 + ], + "lines": [ + { + "bbox": [ + 142, + 208, + 469, + 222 + ], + "spans": [ + { + "bbox": [ + 142, + 208, + 469, + 222 + ], + "score": 1.0, + "content": "While task-specific finetuning of deep networks pretrained with self-supervision", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 232 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 232 + ], + "score": 1.0, + "content": "has led to significant empirical advances in NLP, their large size makes the stan-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 230, + 469, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 230, + 469, + 244 + ], + "score": 1.0, + "content": "dard finetuning approach difficult to apply to multi-task, memory-constrained set-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 242, + 469, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 242, + 469, + 255 + ], + "score": 1.0, + "content": "tings, as storing the full model parameters for each task become prohibitively", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 254, + 469, + 265 + ], + "spans": [ + { + "bbox": [ + 142, + 254, + 469, + 265 + ], + "score": 1.0, + "content": "expensive. 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We find that models fine-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 352, + 469, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 352, + 469, + 363 + ], + "score": 1.0, + "content": "tuned with diff pruning can match the performance of fully finetuned baselines", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 339, + 375 + ], + "score": 1.0, + "content": "on the GLUE benchmark while only modifying", + "type": "text" + }, + { + "bbox": [ + 340, + 362, + 362, + 374 + ], + "score": 0.87, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 362, + 469, + 375 + ], + "score": 1.0, + "content": "of the pretrained model’s", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 225, + 385 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 225, + 385 + ], + "score": 1.0, + "content": "parameters per task.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 394, + 206, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 208, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 208, + 409 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "Task-specific finetuning of pretrained deep networks has become the dominant paradigm in contem-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "porary NLP, achieving state-of-the-art results across a suite of natural language understanding tasks", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "(Devlin et al., 2019; Liu et al., 2019c; Yang et al., 2019; Lan et al., 2020). While straightforward and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "empirically effective, this approach is difficult to scale to multi-task, memory-constrained settings", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "(e.g. for on-device applications), as it requires shipping and storing a full set of model parameters for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "each task. Inasmuch as these models are learning generalizable, task-agnostic language representa-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "score": 1.0, + "content": "tions through self-supervised pretraining, finetuning the entire model for each task is an especially", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 252, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 252, + 501 + ], + "score": 1.0, + "content": "inefficient use of model parameters.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "A popular approach to parameter-efficiency is to learn sparse models for each task where a subset", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "of the final model parameters are exactly zero (Gordon et al., 2020; Sajjad et al., 2020; Zhao et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "2020; Sanh et al., 2020). Such approaches often face a steep sparsity/performance tradeoff, and a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 539, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 299, + 553 + ], + "score": 1.0, + "content": "substantial portion of nonzero parameters (e.g.", + "type": "text" + }, + { + "bbox": [ + 299, + 539, + 342, + 550 + ], + "score": 0.82, + "content": "10 \\% { - } 3 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 539, + 506, + 553 + ], + "score": 1.0, + "content": ") are still typically required to match the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "performance of the dense counterparts. 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These", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "methods learn only a small number of additional parameters (e.g. a linear layer) on top of a shared", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "model. 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This approach", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 275, + 470, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 287 + ], + "score": 1.0, + "content": "views finetuning as learning a task-specific “diff” vector that is applied on top of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 286, + 470, + 298 + ], + "spans": [ + { + "bbox": [ + 142, + 286, + 470, + 298 + ], + "score": 1.0, + "content": "the pretrained parameter vector, which remains fixed and is shared across different", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "score": 1.0, + "content": "tasks. The diff vector is adaptively pruned during training with a differentiable ap-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 470, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 218, + 320 + ], + "score": 1.0, + "content": "proximation to the", + "type": "text" + }, + { + "bbox": [ + 219, + 308, + 231, + 319 + ], + "score": 0.89, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 308, + 470, + 320 + ], + "score": 1.0, + "content": "-norm penalty to encourage sparsity. Diff pruning becomes", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 319, + 470, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 470, + 331 + ], + "score": 1.0, + "content": "parameter-efficient as the number of tasks increases, as it requires storing only the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 470, + 341 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 470, + 341 + ], + "score": 1.0, + "content": "nonzero positions and weights of the diff vector for each task, while the cost of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 341, + 469, + 352 + ], + "spans": [ + { + "bbox": [ + 141, + 341, + 469, + 352 + ], + "score": 1.0, + "content": "storing the shared pretrained model remains constant. We find that models fine-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 352, + 469, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 352, + 469, + 363 + ], + "score": 1.0, + "content": "tuned with diff pruning can match the performance of fully finetuned baselines", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 339, + 375 + ], + "score": 1.0, + "content": "on the GLUE benchmark while only modifying", + "type": "text" + }, + { + "bbox": [ + 340, + 362, + 362, + 374 + ], + "score": 0.87, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 362, + 469, + 375 + ], + "score": 1.0, + "content": "of the pretrained model’s", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 225, + 385 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 225, + 385 + ], + "score": 1.0, + "content": "parameters per task.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 141, + 208, + 470, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 394, + 206, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 208, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 208, + 409 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "Task-specific finetuning of pretrained deep networks has become the dominant paradigm in contem-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "porary NLP, achieving state-of-the-art results across a suite of natural language understanding tasks", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "(Devlin et al., 2019; Liu et al., 2019c; Yang et al., 2019; Lan et al., 2020). While straightforward and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "empirically effective, this approach is difficult to scale to multi-task, memory-constrained settings", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "(e.g. for on-device applications), as it requires shipping and storing a full set of model parameters for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "each task. Inasmuch as these models are learning generalizable, task-agnostic language representa-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "score": 1.0, + "content": "tions through self-supervised pretraining, finetuning the entire model for each task is an especially", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 252, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 252, + 501 + ], + "score": 1.0, + "content": "inefficient use of model parameters.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 412, + 505, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "A popular approach to parameter-efficiency is to learn sparse models for each task where a subset", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "of the final model parameters are exactly zero (Gordon et al., 2020; Sajjad et al., 2020; Zhao et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "2020; Sanh et al., 2020). Such approaches often face a steep sparsity/performance tradeoff, and a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 539, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 299, + 553 + ], + "score": 1.0, + "content": "substantial portion of nonzero parameters (e.g.", + "type": "text" + }, + { + "bbox": [ + 299, + 539, + 342, + 550 + ], + "score": 0.82, + "content": "10 \\% { - } 3 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 539, + 506, + 553 + ], + "score": 1.0, + "content": ") are still typically required to match the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "performance of the dense counterparts. 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These", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "methods learn only a small number of additional parameters (e.g. a linear layer) on top of a shared", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "model. 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(2019) find that adapter lay-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "ers trained on BERT can match the performance of fully finetuned BERT on the GLUE benchmark", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 474, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 273, + 733 + ], + "score": 1.0, + "content": "(Wang et al., 2019a) while only requiring", + "type": "text" + }, + { + "bbox": [ + 273, + 721, + 296, + 731 + ], + "score": 0.84, + "content": "3 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 721, + 474, + 733 + ], + "score": 1.0, + "content": "additional parameters (on average) per task.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In this work, we consider a similar setting as adapters but propose a new diff pruning approach with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "the goal of even more parameter-efficient transfer learning. Diff pruning views finetuning as learning", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "a task-specific difference vector that is applied on top of the pretrained parameter vector, which", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "remains fixed and is shared across different tasks. 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While interesting,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "these models generally underperform task-specific models and require billions of parameters, though recent", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 413, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 413, + 712 + ], + "score": 1.0, + "content": "work suggests that they can be made substantially smaller (Schick & Schutze, 2020).", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "3However, work on continual learning mitigates these issues to an extent (Shin et al., 2017; Lopez-Paz &", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 381, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 381, + 732 + ], + "score": 1.0, + "content": "Ranzato, 2017; Lee et al., 2017; Kirkpatrick et al., 2017; Parisi et al., 2018).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In this work, we consider a similar setting as adapters but propose a new diff pruning approach with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "the goal of even more parameter-efficient transfer learning. 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For example, widely-adopted models such as BERTBASE and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "BERTLARGE have 110M and 340M parameters respectively, while their contemporaries such as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "T5 (Raffel et al., 2020), Megatron-LM (Shoeybi et al., 2019), and Turing-NLG (Rajbhandari", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "et al., 2019) have parameter counts in the billions. Storing the fully finetuned models becomes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "difficult even for a moderate number of tasks.2 A classic approach to tackling this parameter-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "inefficiency (Caruana, 1997) is to train a single shared model (along with a task-specific output", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "layer) against multiple tasks through joint training. However, the usual formulation of multi-task", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 239, + 548 + ], + "score": 1.0, + "content": "learning requires the set of tasks", + "type": "text" + }, + { + "bbox": [ + 239, + 536, + 249, + 546 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "to be known in advance in order to prevent catastrophic forget-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "ting (French, 1999),3 making it unsuitable for applications in which the set of tasks is unknown (e.g.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 558, + 223, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 223, + 569 + ], + "score": 1.0, + "content": "when tasks arrive in stream).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 436, + 506, + 569 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 579, + 201, + 592 + ], + "lines": [ + { + "bbox": [ + 104, + 578, + 203, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 203, + 595 + ], + "score": 1.0, + "content": "3 DIFF PRUNING", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 408, + 612 + ], + "score": 1.0, + "content": "Diff pruning formulates task-specific finetuning as learning a diff vector", + "type": "text" + }, + { + "bbox": [ + 408, + 600, + 420, + 610 + ], + "score": 0.87, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "that is added to the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 502, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 223, + 624 + ], + "score": 1.0, + "content": "pretrained model parameters", + "type": "text" + }, + { + "bbox": [ + 223, + 610, + 258, + 623 + ], + "score": 0.83, + "content": "\\theta _ { \\mathrm { p r e t r a i n e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 610, + 502, + 624 + ], + "score": 1.0, + "content": ". We first reparameterize the task-specific model parameters,", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 599, + 505, + 624 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 627, + 348, + 641 + ], + "lines": [ + { + "bbox": [ + 262, + 627, + 348, + 641 + ], + "spans": [ + { + "bbox": [ + 262, + 627, + 348, + 641 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\pmb { \\theta } _ { \\tau } = \\pmb { \\theta } _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\tau } , } \\end{array}", + "type": "interline_equation", + "image_path": "641357162c8cf5fd67763b84ed9ce3a1b4889bb6c238750d670e7c517ca4be3b.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 262, + 627, + 348, + 641 + ], + "spans": [], + "index": 43 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 379, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 380, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 380, + 97 + ], + "score": 1.0, + "content": "which results in the following empirical risk minimization problem,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 96, + 446, + 130 + ], + "lines": [ + { + "bbox": [ + 164, + 96, + 446, + 130 + ], + "spans": [ + { + "bbox": [ + 164, + 96, + 446, + 130 + ], + "score": 0.95, + "content": "\\operatorname* { m i n } _ { \\delta _ { \\tau } } \\ \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\mathcal { L } \\left( f ( x _ { \\tau } ^ { ( n ) } ; \\theta _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\tau } ) , y _ { \\tau } ^ { ( n ) } \\right) + \\lambda R ( \\theta _ { \\mathrm { p r e t r a i n e d } } + \\delta _ { \\tau } ) .", + "type": "interline_equation", + "image_path": "67153bebf4dae0992cb925cc59d54228833fd661c3be6236e95797eb4ef05e58.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 164, + 96, + 446, + 107.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 164, + 107.33333333333333, + 446, + 118.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 164, + 118.66666666666666, + 446, + 130.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 137, + 506, + 193 + ], + "lines": [ + { + "bbox": [ + 106, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "This trivial reparameterization is equivalent to the original formulation. Its benefit comes in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 388, + 161 + ], + "score": 1.0, + "content": "multi-task setting where the cost of storing the pretrained parameters", + "type": "text" + }, + { + "bbox": [ + 388, + 148, + 424, + 160 + ], + "score": 0.9, + "content": "\\theta _ { \\mathrm { p r e t r a i n e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "is amortized across", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 443, + 172 + ], + "score": 1.0, + "content": "tasks, and the only marginal cost for new tasks is the diff vector. If we can regularize", + "type": "text" + }, + { + "bbox": [ + 443, + 159, + 455, + 170 + ], + "score": 0.88, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "to be sparse", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 147, + 183 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 147, + 170, + 240, + 182 + ], + "score": 0.93, + "content": "\\lVert \\delta _ { \\tau } \\rVert _ { 0 } \\ll \\lVert \\bar { \\pmb { \\theta } } _ { \\mathrm { p r e t r a i n e d } } \\rVert _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 170, + 505, + 183 + ], + "score": 1.0, + "content": ", then this approach can become more parameter-efficient as the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 489, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 346, + 195 + ], + "score": 1.0, + "content": "number of tasks increases. 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This approach involves relaxing a binary vector", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "into continuous space, and then multiplying it with a dense weight vector to determine how much", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "of the weight vector is applied during training. After training, the mask is deterministic and a large", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 317, + 257, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 257, + 330 + ], + "score": 1.0, + "content": "portion of the diff vector is true zero.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 504, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 281, + 347 + ], + "score": 1.0, + "content": "To apply this method we first decompose", + "type": "text" + }, + { + "bbox": [ + 281, + 334, + 293, + 345 + ], + "score": 0.88, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "into a binary mask vector multiplied with a dense", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 343, + 137, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 137, + 359 + ], + "score": 1.0, + "content": "vector,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 356, + 425, + 372 + ], + "lines": [ + { + "bbox": [ + 186, + 356, + 425, + 372 + ], + "spans": [ + { + "bbox": [ + 186, + 356, + 425, + 372 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\delta _ { \\tau } = \\mathbf { z } _ { \\tau } \\odot \\mathbf { w } _ { \\tau } , \\qquad \\mathbf { z } _ { \\tau } \\in \\{ 0 , 1 \\} ^ { d } , \\mathbf { w } _ { \\tau } \\in \\mathbb { R } ^ { d } } \\end{array}", + "type": "interline_equation", + "image_path": "50baad9ae5d43e25ead61a531e7ee5bd6f11e2d43dad20fb8b2093d5a5ab6bab.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 186, + 356, + 425, + 372 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 357, + 387 + ], + "score": 1.0, + "content": "We can now instead optimize an expectation with respect to", + "type": "text" + }, + { + "bbox": [ + 357, + 376, + 369, + 385 + ], + "score": 0.87, + "content": "{ \\bf z } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 372, + 452, + 387 + ], + "score": 1.0, + "content": ", whose distribution", + "type": "text" + }, + { + "bbox": [ + 453, + 374, + 493, + 386 + ], + "score": 0.91, + "content": "p ( \\mathbf { z } _ { \\tau } ; 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In order to approximate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 504, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 124, + 286 + ], + "score": 1.0, + "content": "this", + "type": "text" + }, + { + "bbox": [ + 124, + 273, + 137, + 284 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 272, + 456, + 286 + ], + "score": 1.0, + "content": "objective, we follow the standard approach for gradient-based learning with", + "type": "text" + }, + { + "bbox": [ + 456, + 274, + 469, + 284 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 272, + 504, + 286 + ], + "score": 1.0, + "content": "sparsity", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "using a relaxed mask vector (Louizos et al., 2018). This approach involves relaxing a binary vector", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "into continuous space, and then multiplying it with a dense weight vector to determine how much", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "of the weight vector is applied during training. After training, the mask is deterministic and a large", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 317, + 257, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 257, + 330 + ], + "score": 1.0, + "content": "portion of the diff vector is true zero.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 261, + 506, + 330 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 504, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 281, + 347 + ], + "score": 1.0, + "content": "To apply this method we first decompose", + "type": "text" + }, + { + "bbox": [ + 281, + 334, + 293, + 345 + ], + "score": 0.88, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "into a binary mask vector multiplied with a dense", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 343, + 137, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 137, + 359 + ], + "score": 1.0, + "content": "vector,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 333, + 506, + 359 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 356, + 425, + 372 + ], + "lines": [ + { + "bbox": [ + 186, + 356, + 425, + 372 + ], + "spans": [ + { + "bbox": [ + 186, + 356, + 425, + 372 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\delta _ { \\tau } = \\mathbf { z } _ { \\tau } \\odot \\mathbf { w } _ { \\tau } , \\qquad \\mathbf { z } _ { \\tau } \\in \\{ 0 , 1 \\} ^ { d } , \\mathbf { w } _ { \\tau } \\in \\mathbb { R } ^ { d } } \\end{array}", + "type": "interline_equation", + "image_path": "50baad9ae5d43e25ead61a531e7ee5bd6f11e2d43dad20fb8b2093d5a5ab6bab.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 186, + 356, + 425, + 372 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 357, + 387 + ], + "score": 1.0, + "content": "We can now instead optimize an expectation with respect to", + "type": "text" + }, + { + "bbox": [ + 357, + 376, + 369, + 385 + ], + "score": 0.87, + "content": "{ \\bf z } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 372, + 452, + 387 + ], + "score": 1.0, + "content": ", whose distribution", + "type": "text" + }, + { + "bbox": [ + 453, + 374, + 493, + 386 + ], + "score": 0.91, + "content": "p ( \\mathbf { z } _ { \\tau } ; 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However, it", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "would be ideal to have more fine-grained control into the exact sparsity rate in the diff vector, espe-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 410, + 135 + ], + "score": 1.0, + "content": "cially considering applications which require specific parameter budgets. As", + "type": "text" + }, + { + "bbox": [ + 410, + 123, + 417, + 132 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "is just the Lagrangian", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 218, + 147 + ], + "score": 1.0, + "content": "multiplier for the constraint", + "type": "text" + }, + { + "bbox": [ + 218, + 133, + 276, + 146 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ \\lVert \\pmb { \\delta } _ { \\tau } \\rVert _ { 0 } \\right] < \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 132, + 314, + 147 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 314, + 135, + 321, + 145 + ], + "score": 0.76, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 132, + 505, + 147 + ], + "score": 1.0, + "content": ", this could be achieved in principle by search-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 222, + 157 + ], + "score": 1.0, + "content": "ing over different values of", + "type": "text" + }, + { + "bbox": [ + 223, + 145, + 230, + 154 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 144, + 505, + 157 + ], + "score": 1.0, + "content": ". However we found it more efficient and empirically effective to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 434, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 345, + 168 + ], + "score": 1.0, + "content": "achieve an exact sparsity rate by simply projecting onto the", + "type": "text" + }, + { + "bbox": [ + 345, + 156, + 357, + 166 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 155, + 434, + 168 + ], + "score": 1.0, + "content": "-ball after training.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 340, + 185 + ], + "score": 1.0, + "content": "Specifically we use magnitude pruning on the diff vector", + "type": "text" + }, + { + "bbox": [ + 340, + 172, + 352, + 183 + ], + "score": 0.88, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 172, + 456, + 185 + ], + "score": 1.0, + "content": "and target a sparsity rate", + "type": "text" + }, + { + "bbox": [ + 457, + 173, + 470, + 182 + ], + "score": 0.85, + "content": "t \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "by only", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 172, + 196 + ], + "score": 1.0, + "content": "keeping the top", + "type": "text" + }, + { + "bbox": [ + 172, + 183, + 204, + 194 + ], + "score": 0.92, + "content": "t \\% \\times \\bar { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 182, + 244, + 196 + ], + "score": 1.0, + "content": "values in", + "type": "text" + }, + { + "bbox": [ + 244, + 183, + 255, + 194 + ], + "score": 0.87, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 182, + 506, + 196 + ], + "score": 1.0, + "content": ".6 Note that unlike standard magnitude pruning, this is based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "on the magnitude of the diff vector values and not the model parameters. As is usual in magnitude", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 104, + 204, + 304, + 218 + ], + "score": 1.0, + "content": "pruning, we found it important to further finetune", + "type": "text" + }, + { + "bbox": [ + 305, + 205, + 316, + 216 + ], + "score": 0.89, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 204, + 506, + 218 + ], + "score": 1.0, + "content": "with the nonzero masks fixed to maintain good", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "performance (Han et al., 2016). Since this type of parameter-efficiency through projection onto the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 119, + 238 + ], + "score": 0.87, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "-ball can be applied without adaptive diff pruning,7 such an approach will serve as one of our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 236, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 236, + 251 + ], + "score": 1.0, + "content": "baselines in the empirical study.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 260, + 255, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 256, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 256, + 273 + ], + "score": 1.0, + "content": "3.3 STRUCTURED DIFF PRUNING", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "Diff pruning, as presented above, is architecture-agnostic and does not exploit the underlying model", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 228, + 302 + ], + "score": 1.0, + "content": "structure—each dimension of", + "type": "text" + }, + { + "bbox": [ + 228, + 290, + 239, + 300 + ], + "score": 0.86, + "content": "{ \\bf z } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "is independent from one another. While this makes the approach", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "potentially more flexible, we might expect to achieve better sparsity/performance tradeoff through", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "a structured formulation which encourages active parameters to group together and other areas to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 464, + 335 + ], + "score": 1.0, + "content": "be fully sparse. 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"The expected", + "type": "text" + }, + { + "bbox": [ + 438, + 357, + 451, + 368 + ], + "score": 0.89, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 355, + 506, + 371 + ], + "score": 1.0, + "content": "-norm is then", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 367, + 146, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 146, + 381 + ], + "score": 1.0, + "content": "given by,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 384, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 111, + 384, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 111, + 384, + 505, + 421 + ], + "score": 0.92, + "content": "\\Sigma \\left[ \\left. \\delta _ { \\tau } \\right. _ { 0 } \\right] = \\sum _ { j = 1 } ^ { G } \\sum _ { i \\in g ( j ) } \\mathbb { E } \\left[ \\mathbb { I } \\left\\{ \\mathbf { z } _ { \\tau , i } \\cdot \\mathbf { z } _ { \\tau } ^ { g } > 0 \\right\\} \\right] = \\sum _ { j = 1 } ^ { G } \\sum _ { i \\in g ( j ) } \\sigma \\left( \\alpha _ { \\tau , i } - \\log { \\frac { - l } { r } } \\right) \\times \\sigma \\left( \\alpha _ { \\tau } ^ { j } - \\log { \\frac { - l } { r } } \\right) ,", + "type": "interline_equation", + "image_path": "6dd9f1cd7a0ba610a4afd20bc615d654b40294d36a6f77a7d0098203f1f2eddd.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 111, + 384, + 505, + 396.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 396.3333333333333, + 505, + 408.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 111, + 408.66666666666663, + 505, + 420.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 349, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 350, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 350, + 440 + ], + "score": 1.0, + "content": "and we can train with gradient-based optimization as before.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 201, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 201, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 201, + 463 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 468, + 233, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 234, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 234, + 480 + ], + "score": 1.0, + "content": "4.1 MODEL AND DATASETS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 104, + 484, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 505, + 499 + ], + "score": 1.0, + "content": "For evaluation we use the GLUE benchmark (Wang et al., 2019b), a popular finetuning dataset.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "Following adapters (Houlsby et al., 2019), we test our approach on the following subset of the GLUE", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "tasks: Multi-Genre Natural Language Inference (MNLI), where the goal is two predict whether the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 518, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 469, + 533 + ], + "score": 1.0, + "content": "relationship between two sentences is entailment, contradiction, or neutral (we test on both", + "type": "text" + }, + { + "bbox": [ + 469, + 519, + 504, + 530 + ], + "score": 0.87, + "content": "\\mathrm { M 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However we found it more efficient and empirically effective to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 434, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 345, + 168 + ], + "score": 1.0, + "content": "achieve an exact sparsity rate by simply projecting onto the", + "type": "text" + }, + { + "bbox": [ + 345, + 156, + 357, + 166 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 155, + 434, + 168 + ], + "score": 1.0, + "content": "-ball after training.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 100, + 506, + 168 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 340, + 185 + ], + "score": 1.0, + "content": "Specifically we use magnitude pruning on the diff vector", + "type": "text" + }, + { + "bbox": [ + 340, + 172, + 352, + 183 + ], + "score": 0.88, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 172, + 456, + 185 + ], + "score": 1.0, + "content": "and target a sparsity rate", + "type": "text" + }, + { + "bbox": [ + 457, + 173, + 470, + 182 + ], + "score": 0.85, + "content": "t \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "by only", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 172, + 196 + ], + "score": 1.0, + "content": "keeping the top", + "type": "text" + }, + { + "bbox": [ + 172, + 183, + 204, + 194 + ], + "score": 0.92, + "content": "t \\% \\times \\bar { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 182, + 244, + 196 + ], + "score": 1.0, + "content": "values in", + "type": "text" + }, + { + "bbox": [ + 244, + 183, + 255, + 194 + ], + "score": 0.87, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 182, + 506, + 196 + ], + "score": 1.0, + "content": ".6 Note that unlike standard magnitude pruning, this is based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "on the magnitude of the diff vector values and not the model parameters. 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"The expected", + "type": "text" + }, + { + "bbox": [ + 438, + 357, + 451, + 368 + ], + "score": 0.89, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 355, + 506, + 371 + ], + "score": 1.0, + "content": "-norm is then", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 367, + 146, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 146, + 381 + ], + "score": 1.0, + "content": "given by,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 277, + 506, + 381 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 384, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 111, + 384, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 111, + 384, + 505, + 421 + ], + "score": 0.92, + "content": "\\Sigma \\left[ \\left. \\delta _ { \\tau } \\right. _ { 0 } \\right] = \\sum _ { j = 1 } ^ { G } \\sum _ { i \\in g ( j ) } \\mathbb { E } \\left[ \\mathbb { I } \\left\\{ \\mathbf { z } _ { \\tau , i } \\cdot \\mathbf { z } _ { \\tau } ^ { g } > 0 \\right\\} \\right] = \\sum _ { j = 1 } ^ { G } \\sum _ { i \\in g ( j ) } \\sigma \\left( \\alpha _ { \\tau , i } - \\log { \\frac { - l } { r } } \\right) \\times \\sigma \\left( \\alpha _ { \\tau } ^ { j } - \\log { \\frac { - l } { r } } \\right) ,", + "type": "interline_equation", + "image_path": "6dd9f1cd7a0ba610a4afd20bc615d654b40294d36a6f77a7d0098203f1f2eddd.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 111, + 384, + 505, + 396.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 396.3333333333333, + 505, + 408.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 111, + 408.66666666666663, + 505, + 420.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 349, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 350, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 350, + 440 + ], + 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Total paramsNew params per taskQNLI*SST-2 MNLImMNLImmCoLA MRPC STS-B RTEQQPAvg
Full finetuning9.00×100%91.194.986.785.960.589.387.670.172.180.9
Adapters (8-256)1.32×3.6%90.794.084.985.159.589.586.971.571.880.4
Adapters (64)1.19×2.1%91.494.285.384.656.989.687.368.671.879.8
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Last layer1.34×3.8%79.891.671.472.940.280.167.358.663.368.2
Non-adap. diff pruning1.05×0.5%89.793.684.984.851.281.578.261.568.675.5
Diff pruning1.05×0.5%92.993.885.785.660.587.083.568.170.679.4
Diff pruning (struct.)1.05×0.5%93.394.186.486.061.189.786.070.671.180.6
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(Top) Results with adapter bottleneck", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 187, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 198 + ], + "score": 1.0, + "content": "layers (brackets indicate the size of bottlenecks), taken from from Houlsby et al. (2019). 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To make our results comparable the average column is calculated without QNLI.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 238 + ], + "score": 1.0, + "content": "For all experiments, we use the BERTLARGE model from Devlin et al. (2019), which has 24 layers,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "1024 hidden size, 16 attention heads, and 340M parameters. We use the Huggingface Transformer", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 246, + 325, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 325, + 259 + ], + "score": 1.0, + "content": "library (Wolf et al., 2019) to conduct our experiments.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 266, + 180, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 181, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 181, + 279 + ], + "score": 1.0, + "content": "4.2 BASELINES", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 283, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 504, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 504, + 296 + ], + "score": 1.0, + "content": "We compare both structured and non-structured variants of diff pruning against the following base-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 290, + 307 + ], + "score": 1.0, + "content": "lines: Full finetuning, which fully finetunes", + "type": "text" + }, + { + "bbox": [ + 290, + 295, + 321, + 306 + ], + "score": 0.25, + "content": "\\mathrm { B E R T _ { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "ARGE as usual; Last layer finetuning, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 304, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 320 + ], + "score": 1.0, + "content": "only finetunes the penultimate layer (along with the final output layer)9; Adapters from Houlsby", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "et al. (2019), which train task-specific bottleneck layers between between each layer of a pretrained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "model, where parameter-efficiency can be controlled by varying the size of the bottleneck layers;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "and Non-adaptive diff pruning, which performs diff pruning just based on magnitude pruning (i.e.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 348, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 148, + 364 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + }, + { + "bbox": [ + 148, + 350, + 160, + 361 + ], + "score": 0.87, + "content": "\\pmb { \\theta } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 348, + 278, + 364 + ], + "score": 1.0, + "content": "through usual finetuning, set", + "type": "text" + }, + { + "bbox": [ + 279, + 349, + 361, + 362 + ], + "score": 0.91, + "content": "\\delta _ { \\tau } = \\pmb { \\theta } _ { \\tau } - \\pmb { \\theta } _ { \\mathrm { p r e t r a i n e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 348, + 505, + 364 + ], + "score": 1.0, + "content": ", and then apply magnitude pruning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 253, + 374 + ], + "score": 1.0, + "content": "followed by additional finetuning on", + "type": "text" + }, + { + "bbox": [ + 253, + 361, + 264, + 372 + ], + "score": 0.85, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 360, + 465, + 374 + ], + "score": 1.0, + "content": "). For diff pruning we set our target sparsity rate to", + "type": "text" + }, + { + "bbox": [ + 465, + 361, + 487, + 371 + ], + "score": 0.86, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 377, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 377, + 384 + ], + "score": 1.0, + "content": "investigate the effect of different target sparsity rates in section 5.1.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 106, + 390, + 360, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 361, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 361, + 403 + ], + "score": 1.0, + "content": "4.3 IMPLEMENTATION DETAILS AND HYPERPARAMETERS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 316, + 418 + ], + "score": 1.0, + "content": "Diff pruning introduces additional hyperparameters", + "type": "text" + }, + { + "bbox": [ + 316, + 406, + 330, + 417 + ], + "score": 0.88, + "content": "l , r", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 404, + 505, + 418 + ], + "score": 1.0, + "content": "(for stretching the Hard-Concrete distribu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 145, + 429 + ], + "score": 1.0, + "content": "tion) and", + "type": "text" + }, + { + "bbox": [ + 146, + 417, + 153, + 426 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 416, + 283, + 429 + ], + "score": 1.0, + "content": "(for weighting the approximate", + "type": "text" + }, + { + "bbox": [ + 284, + 417, + 296, + 428 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 416, + 405, + 429 + ], + "score": 1.0, + "content": "-norm penalty). We found", + "type": "text" + }, + { + "bbox": [ + 405, + 417, + 505, + 428 + ], + "score": 0.89, + "content": "l = - 1 . 5 , r = 1 . 5 , \\lambda =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 426, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 427, + 158, + 438 + ], + "score": 0.92, + "content": "1 . 2 5 \\times 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 426, + 419, + 440 + ], + "score": 1.0, + "content": "to work well across all tasks. We also initialize the weight vector", + "type": "text" + }, + { + "bbox": [ + 420, + 429, + 434, + 438 + ], + "score": 0.88, + "content": "{ \\bf w } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 426, + 473, + 440 + ], + "score": 1.0, + "content": "to 0, and", + "type": "text" + }, + { + "bbox": [ + 473, + 429, + 487, + 438 + ], + "score": 0.86, + "content": "\\pmb { \\alpha } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 426, + 506, + 440 + ], + "score": 1.0, + "content": "to a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 265, + 451 + ], + "score": 1.0, + "content": "positive vector (we use 5) to encourage", + "type": "text" + }, + { + "bbox": [ + 266, + 441, + 277, + 450 + ], + "score": 0.88, + "content": "{ \\bf z } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "to be close to 1 at the start of training. While we mainly", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "experiment with BERTLARGE to compare against prior work with adapters (Houlsby et al., 2019),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "in preliminary experiments we found these hyperparameters to work for finetuning RoBERTa (Liu", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 349, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 130, + 484 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 131, + 471, + 157, + 482 + ], + "score": 0.3, + "content": "2 0 1 9 \\mathrm { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 471, + 349, + 484 + ], + "score": 1.0, + "content": ") and XLNet (Yang et al., 2019) models as well.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 255, + 500 + ], + "score": 1.0, + "content": "For all tasks we use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 255, + 487, + 292, + 498 + ], + "score": 0.92, + "content": "1 \\times 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "and perform a hyperparameter search over batch size", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 497, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 167, + 511 + ], + "score": 0.91, + "content": "\\in \\{ 4 , 6 , 8 , 1 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 497, + 277, + 513 + ], + "score": 1.0, + "content": "and the number of epochs", + "type": "text" + }, + { + "bbox": [ + 277, + 499, + 332, + 511 + ], + "score": 0.9, + "content": "\\in \\{ 2 , 3 , 4 , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 497, + 506, + 513 + ], + "score": 1.0, + "content": ".10 However we found the default settings", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "used for regular finetuning as suggested in the original BERT paper to work well for most tasks.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 339, + 533 + ], + "score": 1.0, + "content": "Finetuning with the fixed mask after projecting onto the", + "type": "text" + }, + { + "bbox": [ + 339, + 522, + 352, + 532 + ], + "score": 0.89, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "-ball with magnitude pruning is done", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 196, + 545 + ], + "score": 1.0, + "content": "with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 196, + 532, + 232, + 543 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "for 3 or 5 epochs (3 epochs for QNLI, SST-2, MNLI-m, MNLI-mm,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "CoLA, QQP, 5 epochs for MRPC, STS-B, RTE). Grouping for the structured version of diff pruning", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "is based on the matrix/bias vectors (i.e. parameters that belong to the same matrix or bias vector are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 565, + 361, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 361, + 577 + ], + "score": 1.0, + "content": "assumed to be in the same group), which results in 393 groups.1", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 587, + 253, + 600 + ], + "lines": [ + { + "bbox": [ + 104, + 585, + 255, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 255, + 602 + ], + "score": 1.0, + "content": "5 RESULTS AND ANALYSIS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "Our main results on the GLUE benchmark are shown in Table 1. Structured diff pruning can match", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 423, + 633 + ], + "score": 1.0, + "content": "the performance of a fully finetuned BERTLARGE model while only requiring", + "type": "text" + }, + { + "bbox": [ + 423, + 620, + 445, + 631 + ], + "score": 0.84, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "additional pa-", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 672, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 115, + 668, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 115, + 668, + 506, + 686 + ], + "score": 1.0, + "content": "11This definition of groups is implementation-specific since it depends on how one concatenates the", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 681, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 692 + ], + "score": 1.0, + "content": "input vector before each affine layer. Our grouping is based on Huggingface’s BERT implementation at", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 692, + 504, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 504, + 703 + ], + "score": 1.0, + "content": "commit 656e1386a296d696327a9db37de2ccccc79e2cc7 (available at https://github.com/", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 702, + 489, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 489, + 712 + ], + "score": 1.0, + "content": "huggingface/transformers/blob/656e1386a296d696327a9db37de2ccccc79e2cc7/", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 712, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 505, + 723 + ], + "score": 1.0, + "content": "src/transformers/modeling_bert.py). In preliminary experiments we found this simple definition", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 722, + 430, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 430, + 732 + ], + "score": 1.0, + "content": "to work well compared to alternative group definitions (e.g. based on individual neurons).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 109, + 640, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 117, + 637, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 117, + 637, + 506, + 654 + ], + "score": 1.0, + "content": "9Wu et al. (2020) observe that finetuning later layers generally performs better than finetuning earlier layers", + "type": "text" + } + ] + }, + { + "bbox": [ + 114, + 648, + 507, + 664 + ], + "spans": [ + { + "bbox": [ + 114, + 648, + 507, + 664 + ], + "score": 1.0, + "content": "10For the larger QNLI, SST-2, MNLI-m, MNLI-mm, CoLA, QQP datasets, we use batch size of 8 over 3", + "type": "text" + } + ] + }, + { + "bbox": [ + 107, + 661, + 438, + 672 + ], + "spans": [ + { + "bbox": [ + 107, + 661, + 438, + 672 + ], + "score": 1.0, + "content": "epochs. For the smaller MRPC, STS-B, RTE datasets, we use batch size of 8 over 3 epochs.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 80, + 505, + 176 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 80, + 505, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 505, + 176 + ], + "score": 0.981, + "html": "
Total paramsNew params per taskQNLI*SST-2 MNLImMNLImmCoLA MRPC STS-B RTEQQPAvg
Full finetuning9.00×100%91.194.986.785.960.589.387.670.172.180.9
Adapters (8-256)1.32×3.6%90.794.084.985.159.589.586.971.571.880.4
Adapters (64)1.19×2.1%91.494.285.384.656.989.687.368.671.879.8
Full finetuning9.00×100%93.494.186.786.059.688.986.671.271.780.6
Last layer1.34×3.8%79.891.671.472.940.280.167.358.663.368.2
Non-adap. diff pruning1.05×0.5%89.793.684.984.851.281.578.261.568.675.5
Diff pruning1.05×0.5%92.993.885.785.660.587.083.568.170.679.4
Diff pruning (struct.)1.05×0.5%93.394.186.486.061.189.786.070.671.180.6
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(Top) Results with adapter bottleneck", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 187, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 198 + ], + "score": 1.0, + "content": "layers (brackets indicate the size of bottlenecks), taken from from Houlsby et al. (2019). (Bottom) Results", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 165, + 208 + ], + "score": 1.0, + "content": "from this work.", + "type": "text" + }, + { + "bbox": [ + 165, + 197, + 193, + 207 + ], + "score": 0.68, + "content": "{ } ^ { * } \\mathrm { Q N L I }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "results are not directly comparable across the two works as the GLUE benchmark has", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "score": 1.0, + "content": "updated the test set since then. To make our results comparable the average column is calculated without QNLI.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 106, + 176, + 505, + 218 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 238 + ], + "score": 1.0, + "content": "For all experiments, we use the BERTLARGE model from Devlin et al. (2019), which has 24 layers,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "1024 hidden size, 16 attention heads, and 340M parameters. We use the Huggingface Transformer", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 246, + 325, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 325, + 259 + ], + "score": 1.0, + "content": "library (Wolf et al., 2019) to conduct our experiments.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 224, + 505, + 259 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 266, + 180, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 181, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 181, + 279 + ], + "score": 1.0, + "content": "4.2 BASELINES", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 283, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 504, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 504, + 296 + ], + "score": 1.0, + "content": "We compare both structured and non-structured variants of diff pruning against the following base-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 290, + 307 + ], + "score": 1.0, + "content": "lines: Full finetuning, which fully finetunes", + "type": "text" + }, + { + "bbox": [ + 290, + 295, + 321, + 306 + ], + "score": 0.25, + "content": "\\mathrm { B E R T _ { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "ARGE as usual; Last layer finetuning, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 304, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 320 + ], + "score": 1.0, + "content": "only finetunes the penultimate layer (along with the final output layer)9; Adapters from Houlsby", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "et al. (2019), which train task-specific bottleneck layers between between each layer of a pretrained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "model, where parameter-efficiency can be controlled by varying the size of the bottleneck layers;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "and Non-adaptive diff pruning, which performs diff pruning just based on magnitude pruning (i.e.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 348, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 148, + 364 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + }, + { + "bbox": [ + 148, + 350, + 160, + 361 + ], + "score": 0.87, + "content": "\\pmb { \\theta } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 348, + 278, + 364 + ], + "score": 1.0, + "content": "through usual finetuning, set", + "type": "text" + }, + { + "bbox": [ + 279, + 349, + 361, + 362 + ], + "score": 0.91, + "content": "\\delta _ { \\tau } = \\pmb { \\theta } _ { \\tau } - \\pmb { \\theta } _ { \\mathrm { p r e t r a i n e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 348, + 505, + 364 + ], + "score": 1.0, + "content": ", and then apply magnitude pruning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 253, + 374 + ], + "score": 1.0, + "content": "followed by additional finetuning on", + "type": "text" + }, + { + "bbox": [ + 253, + 361, + 264, + 372 + ], + "score": 0.85, + "content": "\\delta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 360, + 465, + 374 + ], + "score": 1.0, + "content": "). For diff pruning we set our target sparsity rate to", + "type": "text" + }, + { + "bbox": [ + 465, + 361, + 487, + 371 + ], + "score": 0.86, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 377, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 377, + 384 + ], + "score": 1.0, + "content": "investigate the effect of different target sparsity rates in section 5.1.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 284, + 506, + 384 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 390, + 360, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 361, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 361, + 403 + ], + "score": 1.0, + "content": "4.3 IMPLEMENTATION DETAILS AND HYPERPARAMETERS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 316, + 418 + ], + "score": 1.0, + "content": "Diff pruning introduces additional hyperparameters", + "type": "text" + }, + { + "bbox": [ + 316, + 406, + 330, + 417 + ], + "score": 0.88, + "content": "l , r", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 404, + 505, + 418 + ], + "score": 1.0, + "content": "(for stretching the Hard-Concrete distribu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 145, + 429 + ], + "score": 1.0, + "content": "tion) and", + "type": "text" + }, + { + "bbox": [ + 146, + 417, + 153, + 426 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 416, + 283, + 429 + ], + "score": 1.0, + "content": "(for weighting the approximate", + "type": "text" + }, + { + "bbox": [ + 284, + 417, + 296, + 428 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 416, + 405, + 429 + ], + "score": 1.0, + "content": "-norm penalty). We found", + "type": "text" + }, + { + "bbox": [ + 405, + 417, + 505, + 428 + ], + "score": 0.89, + "content": "l = - 1 . 5 , r = 1 . 5 , \\lambda =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 426, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 427, + 158, + 438 + ], + "score": 0.92, + "content": "1 . 2 5 \\times 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 426, + 419, + 440 + ], + "score": 1.0, + "content": "to work well across all tasks. We also initialize the weight vector", + "type": "text" + }, + { + "bbox": [ + 420, + 429, + 434, + 438 + ], + "score": 0.88, + "content": "{ \\bf w } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 426, + 473, + 440 + ], + "score": 1.0, + "content": "to 0, and", + "type": "text" + }, + { + "bbox": [ + 473, + 429, + 487, + 438 + ], + "score": 0.86, + "content": "\\pmb { \\alpha } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 426, + 506, + 440 + ], + "score": 1.0, + "content": "to a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 265, + 451 + ], + "score": 1.0, + "content": "positive vector (we use 5) to encourage", + "type": "text" + }, + { + "bbox": [ + 266, + 441, + 277, + 450 + ], + "score": 0.88, + "content": "{ \\bf z } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "to be close to 1 at the start of training. While we mainly", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "experiment with BERTLARGE to compare against prior work with adapters (Houlsby et al., 2019),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "in preliminary experiments we found these hyperparameters to work for finetuning RoBERTa (Liu", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 349, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 130, + 484 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 131, + 471, + 157, + 482 + ], + "score": 0.3, + "content": "2 0 1 9 \\mathrm { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 471, + 349, + 484 + ], + "score": 1.0, + "content": ") and XLNet (Yang et al., 2019) models as well.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 404, + 506, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 255, + 500 + ], + "score": 1.0, + "content": "For all tasks we use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 255, + 487, + 292, + 498 + ], + "score": 0.92, + "content": "1 \\times 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "and perform a hyperparameter search over batch size", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 497, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 167, + 511 + ], + "score": 0.91, + "content": "\\in \\{ 4 , 6 , 8 , 1 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 497, + 277, + 513 + ], + "score": 1.0, + "content": "and the number of epochs", + "type": "text" + }, + { + "bbox": [ + 277, + 499, + 332, + 511 + ], + "score": 0.9, + "content": "\\in \\{ 2 , 3 , 4 , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 497, + 506, + 513 + ], + "score": 1.0, + "content": ".10 However we found the default settings", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "used for regular finetuning as suggested in the original BERT paper to work well for most tasks.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 339, + 533 + ], + "score": 1.0, + "content": "Finetuning with the fixed mask after projecting onto the", + "type": "text" + }, + { + "bbox": [ + 339, + 522, + 352, + 532 + ], + "score": 0.89, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "-ball with magnitude pruning is done", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 196, + 545 + ], + "score": 1.0, + "content": "with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 196, + 532, + 232, + 543 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "for 3 or 5 epochs (3 epochs for QNLI, SST-2, MNLI-m, MNLI-mm,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "CoLA, QQP, 5 epochs for MRPC, STS-B, RTE). Grouping for the structured version of diff pruning", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "is based on the matrix/bias vectors (i.e. parameters that belong to the same matrix or bias vector are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 565, + 361, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 361, + 577 + ], + "score": 1.0, + "content": "assumed to be in the same group), which results in 393 groups.1", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 487, + 506, + 577 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 587, + 253, + 600 + ], + "lines": [ + { + "bbox": [ + 104, + 585, + 255, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 255, + 602 + ], + "score": 1.0, + "content": "5 RESULTS AND ANALYSIS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "Our main results on the GLUE benchmark are shown in Table 1. Structured diff pruning can match", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 423, + 633 + ], + "score": 1.0, + "content": "the performance of a fully finetuned BERTLARGE model while only requiring", + "type": "text" + }, + { + "bbox": [ + 423, + 620, + 445, + 631 + ], + "score": 0.84, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "additional pa-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "rameters per task. Diff pruning without structured sparsity also performs well, though slightly worse", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "than the structured approach. Non-adaptive diff pruning, which magnitude prunes the diff vector", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 246, + 387 + ], + "score": 1.0, + "content": "without learning the binary mask", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 246, + 376, + 258, + 386 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { \\tau }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 258, + 375, + 505, + 387 + ], + "score": 1.0, + "content": ", performs significantly worse, indicating the importance of", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "learning the masking vector. Compared to adapters, diff pruning obtains similar performance while", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 409 + ], + "score": 1.0, + "content": "requiring fewer parameters per task, making it a potential alternative for parameter-efficient transfer", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 406, + 436, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 436, + 421 + ], + "score": 1.0, + "content": "learning.12 We now perform a series of analysis experiments on the validation set.", + "type": "text", + "cross_page": true + } + ], + "index": 22 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 608, + 506, + 633 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 325, + 108, + 483, + 192 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 325, + 108, + 483, + 192 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 325, + 108, + 483, + 192 + ], + "spans": [ + { + "bbox": [ + 325, + 108, + 483, + 192 + ], + "score": 0.95, + "html": "
Non-structuredPruned Diff GroupsStructured
#%#%
MRPC246.15213.2
STS-B256.44812.2
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Average", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 336, + 227, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 227, + 346 + ], + "score": 1.0, + "content": "performance includes all 9 tasks.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + } + ], + "index": 14.25 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "rameters per task. Diff pruning without structured sparsity also performs well, though slightly worse", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "than the structured approach. Non-adaptive diff pruning, which magnitude prunes the diff vector", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 246, + 387 + ], + "score": 1.0, + "content": "without learning the binary mask", + "type": "text" + }, + { + "bbox": [ + 246, + 376, + 258, + 386 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 375, + 505, + 387 + ], + "score": 1.0, + "content": ", performs significantly worse, indicating the importance of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "learning the masking vector. Compared to adapters, diff pruning obtains similar performance while", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 409 + ], + "score": 1.0, + "content": "requiring fewer parameters per task, making it a potential alternative for parameter-efficient transfer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 406, + 436, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 436, + 421 + ], + "score": 1.0, + "content": "learning.12 We now perform a series of analysis experiments on the validation set.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 272, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 274, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 274, + 442 + ], + "score": 1.0, + "content": "5.1 VARYING THE TARGET SPARSITY", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "In Figure 1 (left), we plot results on the GLUE validation set averaged across all tasks at target", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 173, + 471 + ], + "score": 1.0, + "content": "sparsity rates of", + "type": "text" + }, + { + "bbox": [ + 174, + 458, + 195, + 469 + ], + "score": 0.82, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 458, + 199, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 199, + 458, + 226, + 470 + ], + "score": 0.8, + "content": "0 . 2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 458, + 229, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 458, + 252, + 469 + ], + "score": 0.8, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 458, + 254, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 458, + 277, + 469 + ], + "score": 0.71, + "content": "1 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "for the different baselines. Structured diff pruning con-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "sistently outperforms non-structured and and non-adaptive variants across different sparsity rates.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "The advantage of adaptive methods becomes more pronounced at extreme sparsity rates. In Table 2,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "we report the breakdown of accuracy of structured diff pruning across different tasks and sparsity", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "rates, where we observe that different tasks have different sensitivity to target sparsity rates. This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "suggests that we can obtain even greater parameter-efficiency through targeting task-specific sparsity", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 198, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 198, + 536 + ], + "score": 1.0, + "content": "rates in the diff vector.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 107, + 546, + 356, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 357, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 357, + 558 + ], + "score": 1.0, + "content": "5.2 STRUCTURED VS. NON-STRUCTURED DIFF PRUNING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "Structured diff pruning introduces an additional mask per group, which encourages pruning of entire", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "groups. This is less restrictive than traditional group sparsity techniques that have been used with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 107, + 589, + 119, + 599 + ], + "score": 0.87, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "-norm relaxations which force all parameters in a group to share the same mask (Louizos et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "2018; Wang et al., 2019b). However we still expect entire groups to be pruned out more often in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "the structured case, which might bias the learning process towards either eliminating completely or", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 619, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 505, + 635 + ], + "score": 1.0, + "content": "clustering together nonzero diffs. In Figure 1 (right), we indeed find that structured diff pruning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "leads to finetuned models that are much more likely to leave entire groups unchanged from their", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 226, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 226, + 655 + ], + "score": 1.0, + "content": "pretrained values (zero diffs).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 241, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 662, + 243, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 243, + 675 + ], + "score": 1.0, + "content": "5.3 TASK-SPECIFIC SPARSITY", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "Different layers of pretrained models have argued to encode different information (Liu et al., 2019a;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "Tenney et al., 2019). Given that each task will likely recruit different kinds of language phenomena", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 701, + 506, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 506, + 715 + ], + "score": 1.0, + "content": "embedded in the hidden layers, we hypothesize that diff pruning will modify different parts of the", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 112, + 722, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 115, + 718, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 115, + 718, + 505, + 734 + ], + "score": 1.0, + "content": "12However diff pruning incurs additional storage cost due to storing the nonzero positions of the diff vector.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 325, + 108, + 483, + 192 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 325, + 108, + 483, + 192 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 325, + 108, + 483, + 192 + ], + "spans": [ + { + "bbox": [ + 325, + 108, + 483, + 192 + ], + "score": 0.95, + "html": "
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Average", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 336, + 227, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 227, + 346 + ], + "score": 1.0, + "content": "performance includes all 9 tasks.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + } + ], + "index": 14.25 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 419 + ], + "lines": [], + "index": 19.5, + "bbox_fs": [ + 105, + 352, + 506, + 421 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 272, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 274, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 274, + 442 + ], + "score": 1.0, + "content": "5.1 VARYING THE TARGET SPARSITY", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "In Figure 1 (left), we plot results on the GLUE validation set averaged across all tasks at target", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 173, + 471 + ], + "score": 1.0, + "content": "sparsity rates of", + "type": "text" + }, + { + "bbox": [ + 174, + 458, + 195, + 469 + ], + "score": 0.82, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 458, + 199, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 199, + 458, + 226, + 470 + ], + "score": 0.8, + "content": "0 . 2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 458, + 229, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 458, + 252, + 469 + ], + "score": 0.8, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 458, + 254, + 471 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 458, + 277, + 469 + ], + "score": 0.71, + "content": "1 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "for the different baselines. Structured diff pruning con-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "sistently outperforms non-structured and and non-adaptive variants across different sparsity rates.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "The advantage of adaptive methods becomes more pronounced at extreme sparsity rates. In Table 2,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "we report the breakdown of accuracy of structured diff pruning across different tasks and sparsity", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "rates, where we observe that different tasks have different sensitivity to target sparsity rates. This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "suggests that we can obtain even greater parameter-efficiency through targeting task-specific sparsity", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 198, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 198, + 536 + ], + "score": 1.0, + "content": "rates in the diff vector.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 446, + 506, + 536 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 546, + 356, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 357, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 357, + 558 + ], + "score": 1.0, + "content": "5.2 STRUCTURED VS. NON-STRUCTURED DIFF PRUNING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "Structured diff pruning introduces an additional mask per group, which encourages pruning of entire", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "groups. This is less restrictive than traditional group sparsity techniques that have been used with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 107, + 589, + 119, + 599 + ], + "score": 0.87, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "-norm relaxations which force all parameters in a group to share the same mask (Louizos et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "2018; Wang et al., 2019b). However we still expect entire groups to be pruned out more often in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "the structured case, which might bias the learning process towards either eliminating completely or", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 619, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 505, + 635 + ], + "score": 1.0, + "content": "clustering together nonzero diffs. In Figure 1 (right), we indeed find that structured diff pruning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "leads to finetuned models that are much more likely to leave entire groups unchanged from their", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 226, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 226, + 655 + ], + "score": 1.0, + "content": "pretrained values (zero diffs).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 566, + 506, + 655 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 241, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 662, + 243, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 243, + 675 + ], + "score": 1.0, + "content": "5.3 TASK-SPECIFIC SPARSITY", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "Different layers of pretrained models have argued to encode different information (Liu et al., 2019a;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "Tenney et al., 2019). Given that each task will likely recruit different kinds of language phenomena", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 701, + 506, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 506, + 715 + ], + "score": 1.0, + "content": "embedded in the hidden layers, we hypothesize that diff pruning will modify different parts of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 328, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 506, + 341 + ], + "score": 1.0, + "content": "pretrained model through task-specific finetuning. Figure 2 shows the percentage of nonzero diff", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "parameters attributable to the different layers for each task. 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QNLISST-2MNLImMNLImmCoLAMRPCSTS-BRTEQQPAvg
Sparsity1.5%0.6%0.8%0.8%1.6%2.4%3.3%0.7%0.6%1.4%
Performance93.894.086.286.863.191.989.771.886.584.9
With 0.5% sparsity93.494.286.486.963.591.389.571.586.684.8
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Sparsity1.5%0.6%0.8%0.8%1.6%2.4%3.3%0.7%0.6%1.4%
Performance93.894.086.286.863.191.989.771.886.584.9
With 0.5% sparsity93.494.286.486.963.591.389.571.586.684.8
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This did not present a significant challenge for pretrained models that we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "experimented with in this study, since majority of GPU memory was utilized by the minibatch’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "activation layers. However, this could present an issue as model sizes get larger and larger. While", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 482, + 329 + ], + "score": 1.0, + "content": "training efficiency was not a primary concern of this work, diff pruning takes approxiamtely", + "type": "text" + }, + { + "bbox": [ + 482, + 317, + 504, + 328 + ], + "score": 0.85, + "content": "1 . 5 \\times", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 327, + 347, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 116, + 341 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 328, + 131, + 338 + ], + "score": 0.86, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 327, + 347, + 341 + ], + "score": 1.0, + "content": "more time per batch, which results in slower training.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 504, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "After training, storing the task-specific diff vector requires storing a compressed version with both", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 431, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 431, + 369 + ], + "score": 1.0, + "content": "the nonzero positions and weights, which incurs additional storage requirements.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 378, + 337, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 377, + 338, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 338, + 390 + ], + "score": 1.0, + "content": "6.2 INFORMATION-EFFICIENT TRANSFER LEARNING", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "Efficiently representing pretrained models adapted to new tasks is becoming an increasingly impor-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 405, + 501, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 501, + 418 + ], + "score": 1.0, + "content": "tant problem in contemporary NLP. This paper focuses on a rather narrow definition of efficiency—", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "parameter-efficiency. An interesting direction might be to target generalizations of parameter-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "efficiency, for example, information-efficiency, which aims to minimize the number of bits required", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "to represent the task-specific model when given the pretrained model for free. This view can suggest", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "other avenues for achieving information-efficient transfer learning: for example, “what is the min-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "imum number of (potentially synthetic) datapoints that we can finetune BERT on to obtain a good", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 467, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 507, + 484 + ], + "score": 1.0, + "content": "task-specific model?”,16 or “what is the shortest prefix string that we can condition GPT3 on for it", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 268, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 268, + 494 + ], + "score": 1.0, + "content": "to become a good task-specific model”?", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 511, + 210, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 213, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 213, + 526 + ], + "score": 1.0, + "content": "7 RELATED WORK", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "Multi-task learning Multi-task learning (Caruana, 1997), broadly construed, aims to learn models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "and representations that can be utilized across a diverse range of tasks, and offers a natural approach", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "to training parameter-efficient deep models. 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Adapter layers, which are task-specific layers that read and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "write to layers of a shared model (Rebuffi et al., 2018), offer an alternative approach to multi-task", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "score": 1.0, + "content": "learning that does not require access to all tasks during training, and have also been applied to obtain", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "parameter-efficient BERT models (Houlsby et al., 2019; Pfeiffer et al., 2020a;b;c). 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This did not present a significant challenge for pretrained models that we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "experimented with in this study, since majority of GPU memory was utilized by the minibatch’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "activation layers. However, this could present an issue as model sizes get larger and larger. While", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 482, + 329 + ], + "score": 1.0, + "content": "training efficiency was not a primary concern of this work, diff pruning takes approxiamtely", + "type": "text" + }, + { + "bbox": [ + 482, + 317, + 504, + 328 + ], + "score": 0.85, + "content": "1 . 5 \\times", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 327, + 347, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 116, + 341 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 328, + 131, + 338 + ], + "score": 0.86, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 327, + 347, + 341 + ], + "score": 1.0, + "content": "more time per batch, which results in slower training.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 273, + 505, + 341 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 504, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "After training, storing the task-specific diff vector requires storing a compressed version with both", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 431, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 431, + 369 + ], + "score": 1.0, + "content": "the nonzero positions and weights, which incurs additional storage requirements.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 344, + 505, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 378, + 337, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 377, + 338, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 338, + 390 + ], + "score": 1.0, + "content": "6.2 INFORMATION-EFFICIENT TRANSFER LEARNING", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "Efficiently representing pretrained models adapted to new tasks is becoming an increasingly impor-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 405, + 501, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 501, + 418 + ], + "score": 1.0, + "content": "tant problem in contemporary NLP. This paper focuses on a rather narrow definition of efficiency—", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "parameter-efficiency. An interesting direction might be to target generalizations of parameter-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "efficiency, for example, information-efficiency, which aims to minimize the number of bits required", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "to represent the task-specific model when given the pretrained model for free. This view can suggest", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "other avenues for achieving information-efficient transfer learning: for example, “what is the min-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "imum number of (potentially synthetic) datapoints that we can finetune BERT on to obtain a good", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 467, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 507, + 484 + ], + "score": 1.0, + "content": "task-specific model?”,16 or “what is the shortest prefix string that we can condition GPT3 on for it", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 268, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 268, + 494 + ], + "score": 1.0, + "content": "to become a good task-specific model”?", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 393, + 507, + 494 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 511, + 210, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 213, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 213, + 526 + ], + "score": 1.0, + "content": "7 RELATED WORK", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "Multi-task learning Multi-task learning (Caruana, 1997), broadly construed, aims to learn models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "and representations that can be utilized across a diverse range of tasks, and offers a natural approach", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "to training parameter-efficient deep models. Several works have shown that a single BERT model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "can obtain good performance across multiple tasks when jointly trained (Liu et al., 2019b; Clark", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "et al., 2019; Stickland & Murray, 2019). Adapter layers, which are task-specific layers that read and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "write to layers of a shared model (Rebuffi et al., 2018), offer an alternative approach to multi-task", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "score": 1.0, + "content": "learning that does not require access to all tasks during training, and have also been applied to obtain", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "parameter-efficient BERT models (Houlsby et al., 2019; Pfeiffer et al., 2020a;b;c). A related line of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "work targets extreme parameter-efficiency through task-agnostic sentence representations that can be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "used without finetuning for downstream tasks (Le & Mikolov, 2014; Kiros et al., 2015; Wieting et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 647, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 661 + ], + "score": 1.0, + "content": "2016; Hill et al., 2016; Arora et al., 2017; Conneau et al., 2017; Cer et al., 2018; Zhang et al., 2018;", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "Subramanian et al., 2018; Zhang et al., 2020). Reimers & Gurevych (2019), building on the earlier", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "work of Conneau et al. (2017), show that BERT finetuned on natural language inference obtains", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "sentence representations that perform well across multiple sentence-level tasks. These feature-based", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "transfer learning methods are however generally outperformed by fully finetuned models (Howard", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 702, + 174, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 174, + 715 + ], + "score": 1.0, + "content": "& Ruder, 2018).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 537, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Model compression There has been much recent work on compressing pretrained trained with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "self-supervision (see Ganesh et al. (2020) for a recent survey). A particularly promising line of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "work focuses on obtaining smaller pretrained models (for subsequent finetuning) through weight", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "pruning (Gordon et al., 2020; Sajjad et al., 2020; Chen et al., 2020) and/or knowledge distillation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "(Sanh et al., 2019; Sun et al., 2019; Turc et al., 2019; Jiao et al., 2019; Sun et al., 2020). It would be", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "interesting to see whether our approach can be applied on top of these smaller pretrained models to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 258, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 258, + 162 + ], + "score": 1.0, + "content": "for even greater parameter-efficiency.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "Learning to prune Our work is closely related to the line of work on learning to prune pretrained", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "models with differentiable relaxations of binary masks (Wang et al., 2019b; Zhao et al., 2020; Sanh", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 200 + ], + "score": 1.0, + "content": "et al., 2020; Radiya-Dixit & Wang, 2020). While these works also enable parameter-efficient transfer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "learning, they generally apply the masks directly on the pretrained parameters instead of on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 211, + 268, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 268, + 223 + ], + "score": 1.0, + "content": "difference vector as in the present work.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "Regularization towards pretrained models Finally, diff pruning is also related to works which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "regularize the learning process towards pretrained models for continual learning (Kirkpatrick et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 264 + ], + "score": 1.0, + "content": "2017; Schwarz et al., 2018), domain adaptation (Wiese et al., 2017; Miceli Barone et al., 2017),", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "and stable finetuning (Lee et al., 2020). 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Total paramsNew params per taskQNLI*SST-2 MNLImMNLImmCoLA MRPC STS-B RTEQQPAvg
Full finetuning9.00×100%91.194.986.785.960.589.387.670.172.180.9
Adapters (8-256)1.32×3.6%90.794.084.985.159.589.586.971.571.880.4
Adapters (64)1.19×2.1%91.494.285.384.656.989.687.368.671.879.8
Full finetuning9.00×100%93.494.186.786.059.688.986.671.271.780.6
Last layer1.34×3.8%79.891.671.472.940.280.167.358.663.368.2
Non-adap. diff pruning1.05×0.5%89.793.684.984.851.281.578.261.568.675.5
Diff pruning1.05×0.5%92.993.885.785.660.587.083.568.170.679.4
Diff pruning (struct.)1.05×0.5%93.394.186.486.061.189.786.070.671.180.6
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Sparsity1.5%0.6%0.8%0.8%1.6%2.4%3.3%0.7%0.6%1.4%
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With 0.5% sparsity93.494.286.486.963.591.389.571.586.684.8
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+sunyj@math.ucla.edu + +Wotao Yin UCLA wotaoyin@math.ucla.edu + +# Abstract + +Stochastic nested optimization, including stochastic bilevel, min-max, and compositional optimization, is gaining popularity in many machine learning applications. While the three problems share a nested structure, existing works often treat them separately, thus developing problem-specific algorithms and analyses. Among various exciting developments, simple SGD-type updates (potentially on multiple variables) are still prevalent in solving this class of nested problems, but they are believed to have a slower convergence rate than non-nested problems. This paper unifies several SGD-type updates for stochastic nested problems into a single SGD approach that we term ALternating Stochastic gradient dEscenT (ALSET) method. By leveraging the hidden smoothness of the problem, this paper presents a tighter analysis of ALSET for stochastic nested problems. Under the new analysis, to achieve an $\epsilon$ -stationary point of the nested problem, it requires $\mathcal { O } ( \epsilon ^ { - 2 } )$ samples in total. Under certain regularity conditions, applying our results to stochastic compositional, min-max, and reinforcement learning problems either improves or matches the best-known sample complexity in the respective cases. Our results explain why simple SGD-type algorithms in stochastic nested problems all work very well in practice without the need for further modifications. + +# 1 Introduction + +Stochastic gradient descent (SGD) methods [1] are prevalent in solving large-scale machine learning problems. Often, SGD is applied to solve stochastic problems with a relatively simple structure. Specifically, applying SGD to minimize the function $\mathbb { E } _ { \xi } ^ { - } \left[ f ( x ; \xi ) \right]$ over the variable $x \in \mathbb { R } ^ { d }$ , we have the iterative update $\boldsymbol { x } ^ { k + 1 } = \boldsymbol { x } ^ { k } - \alpha \nabla f ( x ^ { k } ; \xi ^ { k } )$ , where $\alpha > 0$ is the stepsize and $\nabla f ( x ^ { k } ; \xi ^ { k } )$ is the stochastic gradient at the iterate $x ^ { k }$ and the sample $\xi ^ { k }$ . However, many problems in machine learning today, such as meta learning, deep learning, hyper-parameter optimization, and reinforcement learning, go beyond the above simple minimization structure (termed the non-nested problem thereafter). For example, the objective function may be the compositions of multiple functions, where each composition may introduce an additional expectation [2]; and, the objective function may depend on the solution of another optimization problem [3]. In these problems, how to apply SGD and the efficiency of running SGD are not fully understood. + +To answer these questions, in this paper, we consider the following form of stochastic nested optimization problems, which is a generalization of the non-nested problems, given by + +$$ +\begin{array} { r l } { \underset { x \in \mathbb { R } ^ { d } } { \operatorname* { m i n } } } & { F ( x ) : = \mathbb { E } _ { \xi } \left[ f \left( x , y ^ { * } ( x ) ; \xi \right) \right] } \\ { \mathrm { s . t . ~ } } & { y ^ { * } ( x ) = \underset { y \in \mathbb { R } ^ { d ^ { \prime } } } { \operatorname { a r g m i n } } \ \mathbb { E } _ { \phi } [ g ( x , y ; \phi ) ] } \end{array} +$$ + +where $f$ and $g$ are differentiable functions; and, $\xi$ and $\phi$ are random variables. In the optimization literature [4–6], the problem (1) is referred to as the stochastic bilevel problem, where the upper-level optimization problem depends on the solution of the lower-level optimization over $y \in \mathbb { R } ^ { d ^ { \prime } }$ , denoted as $y ^ { \ast } ( x )$ , which depends on the value of upper-level variable $\boldsymbol { x } \in \mathbb { R } ^ { d }$ . + +The stochastic bilevel nested problem (1) encompasses two popular formulations with the nested structure: stochastic min-max problems and stochastic compositional problems. Therefore, results on the general nested problem (1) will also imply the results in the special cases. For example, if the lower-level objective $g$ is the negative of the upper-level objective $f$ , i.e., $g ( x , y ; \phi ) : = - \bar { f } ( x , y ; \xi )$ , the stochastic bilevel problem (1) reduces to the stochastic min-max problem + +$$ +\operatorname { I f } g ( x , y ; \phi ) : = - f ( x , y ; \xi ) \quad \Rightarrow \quad \operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } F ( x ) : = \operatorname* { m a x } _ { y \in \mathbb { R } ^ { d ^ { \prime } } } \mathbb { E } _ { \xi } \left[ f ( x , y ; \xi ) \right] . +$$ + +Motivated by applications in zero-sum games, adversarial learning and training GANs, significant efforts have been recently made for solving the stochastic min-max problem; see e.g., [7–11]. + +For example, if the upper-level objective $f$ is only a function of $y$ , i.e., $f ( x , y ; \xi ) : = f ( y ; \xi )$ , and the lower-level objective $g$ is a quadratic function of $y$ , i.e., $g ( x , y ; \phi ) : = \| y - h ( x ; \phi ) \| ^ { 2 }$ with a smooth function $h$ of $x$ , then the variable $y ^ { * } ( x )$ admits a closed-form solution, and thus the stochastic bilevel problem (1) reduces to the stochastic compositional problem [12–14] + +$$ +\mathrm { I f } \ g ( x , y ; \phi ) : = \| y - h ( x ; \phi ) \| ^ { 2 } \quad \Rightarrow \quad \operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } \ F ( x ) : = \mathbb { E } _ { \xi } \left[ f \big ( \mathbb { E } _ { \phi } [ h ( x ; \phi ) ] ; \xi \big ) \right] . +$$ + +Stochastic compositional problems in the form of (3) have been studied in the applications in model-agnostic meta learning and policy evaluation in reinforcement learning; see e.g., [2, 15]. + +To solve the nested problem (1) by SGD, one natural solution is to apply alternating SGD updates on $x$ and $y$ based on their stochastic gradients + +$$ +y ^ { k + 1 } = y ^ { k } - \beta _ { k } h _ { g } ^ { k } ~ \mathrm { a n d } ~ x ^ { k + 1 } = x ^ { k } - \alpha _ { k } h _ { f } ^ { k } +$$ + +where $h _ { g } ^ { k }$ is the unbiased stochastic gradient of $\mathbb { E } _ { \phi } [ g ( x ^ { k } , y ^ { k } ; \phi ) ]$ and $h _ { f } ^ { k }$ is the (possibly biased) stochastic gradient of $F ( x ^ { k } )$ ; and, $\beta _ { k }$ and $\alpha _ { k }$ are the stepsizes. A key challenge of running (4) for the nested problem is that (stochastic) gradient of the upper-level variable $x$ is prohibitively expensive to compute. As we will show later, computing an unbiased stochastic gradient of $F ( x )$ requires solving the lower-level problem exactly to obtain $y ^ { * } ( x )$ . + +An accurate stochastic gradient $h _ { f } ^ { k }$ can be obtained in roughly three ways. One way is to run SGD updates on $y ^ { k }$ multiple times before updating $x ^ { k }$ , which yields a double-loop algorithm. To guarantee convergence, it typically requires either the increasing number of lower-level $y$ -update or the growing number of batch size to estimate $h _ { g } ^ { k }$ ; see e.g., [16, 17]. The second way is to update $y ^ { k }$ in a timescale faster than that of $x ^ { k }$ so that $x ^ { k }$ is relatively static with respect to $y ^ { k }$ ; i.e., $\scriptstyle \operatorname* { l i m } _ { k \to \infty } \alpha _ { k } / \beta _ { k } = 0$ ; see e.g., [18]. The third way is to modify the direction $h _ { g } ^ { k }$ of $y ^ { k }$ by incorporating additional correction term, which adds extra computation burden; see e.g., [19]. At a high level, these modifications either deviate from the lightweight implementation of SGD or sacrifice the sample complexity of SGD. + +To this end, the main goal of this paper is to study the efficiency of running the vanilla alternating SGD (4) for the nested problem (1) and its implications on the special problem classes (2)-(3). + +# 1.1 Main results + +This paper analyzes a unifying algorithm for the stochastic bilevel problems that runs SGD on each variable alternatingly. We provide sample complexity that matches the complexity of SGD for single-level stochastic problems. Our results explain why SGD-type algorithms in stochastic bilevel, min-max, and compositional problems work very well in practice without modifications, including correction, increasing batch size, and two-timescale stepsizes. + +In the context of existing methods, our contributions can be summarized as follows. + +C1) We connect three different classes of stochastic nested optimization problems (stochastic compositional, min-max, and bilevel optimization), and unify three popular SGD-type updates for the respective problems into a single SGD-type method. We call it the ALternating Stochastic gradient dEscenT (ALSET) method. + +
ALSETBSATTSAstocBiOSTABLESUSTAIN/RSVRB
batch size0(1)0(1)0(1)0(c-1）0(1)0(1)
y-updateSGDO（c-）)SGD stepsSGDSGDcorrectionmomentum
samples in gsamples in（5∈-2）(-2)（k-2）（-3)O(KPe-（P-（5∈-2）(-2)O(KPe-2）O(KPe-2)O(KP-2）（P
+ +Table 1: Sample complexity of stochastic bilevel algorithms (BSA in [16], TTSA in [18], stocBiO in [17], STABLE in [19], SUSTAIN in [25], RSVRB in [26]) to achieve an $\epsilon$ -stationary point of $F ( x )$ ; the notation $\widetilde { \mathcal { O } } ( \cdot )$ hides the terms of $\log \epsilon ^ { - 1 }$ ; the notation $\kappa ^ { p }$ denotes a polynomial function of $\kappa$ since the dependence on $\kappa$ is not explicit in [18, 19, 25, 26]. + +C2) Under the same assumptions made in most of the previous work, we discover that the solution of the lower-level problem is smooth – a property that is overlooked by the previous analyses. By leveraging the hidden smoothness, we present a tighter analysis of ALSET for the stochastic bilevel problems. Under the new analysis, to achieve an $\epsilon$ -stationary point of the nested problem, ALSET requires $\mathcal { O } ( \epsilon ^ { - 2 } )$ samples in total, rather than the $\mathcal { O } \dot { ( \epsilon ^ { - 5 / 2 } ) }$ sample complexity in the existing literature. +C3) We further customize the analysis to the two special cases – the compositional and min-max problems, and establish the improved sample complexity relative to that in the literature. We apply a new analysis to the celebrated actor-critic method for reinforcement learning problems. Under some regularity conditions, we show that, to achieve an $\epsilon$ -stationary point, the single-loop actor-critic method requires $\mathcal { O } ( \epsilon ^ { - 2 } )$ samples with i.i.d. sampling, which improves the best-known result of $\mathcal { O } ( \epsilon ^ { - 5 / 2 } )$ in the literature. + +# 1.2 Other related works + +To put our work in context, we review prior art that we group in the following three categories. + +Stochastic bilevel optimization. We can trace the study of bilevel optimization to the 1950s [20]. Many recent efforts have been made to solve the bilevel problems. One successful approach is to reformulate the bilevel problem as a single-level problem by replacing the lower-level problem by its optimality conditions [4, 5]. Recently, gradient-based methods for bilevel optimization have gained popularity. They iteratively approximate the (stochastic) gradient of the upper-level problem either in a forward or backward manner [21, 3, 22, 23]. Recent work has also studied the case where the lower-level problem does not have a unique solution [24]. + +The non-asymptotic analysis of bilevel optimization algorithms has been recently studied in some pioneering works, e.g., [16, 18, 17], just to name a few. In both [16, 17], bilevel stochastic optimization algorithms have been developed that run in a double-loop manner. To achieve an $\epsilon$ -stationary point, they only need the sample complexities $\mathcal { O } ( \epsilon ^ { - 3 } )$ and $\mathcal { O } ( \epsilon ^ { - 2 } )$ , respectively, comparable to that of SGD for the single-level case. Recently, a single-loop two-timescale stochastic approximation algorithm has been developed in [18] for the bilevel problem (1). Due to the nature of the two-timescale update, it incurs the sub-optimal sample complexity $\mathcal { O } ( \epsilon ^ { - 5 / 2 } )$ . A single-loop single-timescale stochastic bilevel optimization method has been recently developed in [19]. While the method can achieve the sample complexity $\mathcal { O } ( \epsilon ^ { - 2 } )$ , the resultant update on $y$ needs extra matrix projection, which can be costly. Very recently, the momentum-based acceleration has been incorporated into both the $x$ - and $y$ -updates in [25, 26] and also in [27] after our submission to the conference, where the new algorithms therein enjoy an improved sample complexity $\mathcal { O } ( \epsilon ^ { - 3 / 2 } )$ . However, these results cannot imply the $\mathcal { O } ( \epsilon ^ { - 2 } )$ sample complexity of the alternating SGD update (4), and are orthogonal to our results. A comparison of our results with prior work can be found in Table 1. + +Stochastic min-max optimization. In the context of min-max problems, the alternating version of the stochastic gradient descent ascent (GDA) method can be viewed as the alternating SGD updates (4) for the special nested problem (2). To mitigate the cycling behavior of GDA for convex-concave min-max problems, several variants have been developed by incorporating the idea of optimism; see e.g., [7, 8, 11, 29]. The analysis of stochastic GDA in the nonconvex-strongly concave setting is closely related to this paper; e.g., [9, 10, 30, 28]. Specifically, for stochastic GDA (SGDA), the $\mathcal { O } ( \epsilon ^ { - 2 } )$ sample complexity has been established in [28] under an increasing batch size $\mathcal { O } ( \epsilon ^ { - 1 } )$ . As highlighted in [28], how to achieve the $\mathcal { O } ( \epsilon ^ { - 2 } )$ sample complexity under an $\mathcal { O } ( 1 )$ constant batch size remains open. The reduction of our results to the min-max setting will provide an answer to this open question. In the same setting, accelerated GDA algorithms have been developed in [31–33]. Going beyond the one-side concave settings, algorithms and their convergence analysis have been studied for nonconvex-nonconcave min-max problems with certain benign structure; see e.g., [8, 34–36]. A comparison of our results with prior work can be found in Table 2. + +Stochastic compositional optimization. Stochastic compositional gradient algorithms developed in [12, 37] can be viewed as the alternating SGD updates (4) for the special compositional problem (3). However, to ensure convergence, the algorithms [12, 37] use two sequences of variables being updated in two different time scales, and thus the complexity of [12] and [37] is worse than $\mathcal { O } ( \epsilon ^ { - 2 } )$ of SGD for the non-compositional case. While most of existing algorithms rely on either two-timescale updates, the single-timescale single-loop approaches have been recently developed in [14, 38, 39], which achieve the sample complexity $\bar { \mathcal { O } } \bar { ( } \epsilon ^ { - 2 } \bar { ) }$ , same as SGD for the non-nested problems. However, the algorithms proposed therein are not the vanilla alternating SGD update in the sense of (4). Other related compositional algorithms also include [40–42]. A comparison can be found in Table 3. + +Organization. The basic background of bilevel optimization is reviewed, and the tighter analysis of the unifying ALSET method is presented in Section 2. The reduction of the main results to the special stochastic nested problems is provided in Section 3, and its applications to the actor-critic method are discussed in Section 4, followed by the conclusions in Section 5. + +# 2 Improved Analysis of Alternating Stochastic Gradient Method + +In this section, we will first provide background of bilevel problems and then introduce ALSET for stochastic nested problems. + +# 2.1 Preliminaries + +We use $\| \cdot \|$ to denote the $\ell _ { 2 }$ norm for vectors and Frobenius norm for matrices. For convenience, we define the deterministic functions as $g ( x , y ) : = \mathbb { E } _ { \phi } [ g ( x , y ; \phi ) ]$ and $f ( x , y ) : = \mathbb { E } _ { \xi } [ f ( x , y ; \xi ) ]$ . + +We also define $\nabla _ { y y } ^ { 2 } g \big ( x , y \big )$ as the Hessian matrix of $g$ with respect to $y$ and define $\nabla _ { x y } ^ { 2 } g \left( x , y \right)$ as + +$$ +\nabla _ { x y } ^ { 2 } g ( x , y ) : = \left[ \begin{array} { l l l } { \frac { \partial ^ { 2 } } { \partial x _ { 1 } \partial y _ { 1 } } g ( x , y ) } & { \cdot \cdot \cdot } & { \frac { \partial ^ { 2 } } { \partial x _ { 1 } \partial y _ { d ^ { \prime } } } g ( x , y ) } \\ & { \cdot \cdot \cdot } \\ { \frac { \partial ^ { 2 } } { \partial x _ { d } \partial y _ { 1 } } g ( x , y ) } & { \cdot \cdot \cdot } & { \frac { \partial ^ { 2 } } { \partial x _ { d } \partial y _ { d ^ { \prime } } } g ( x , y ) } \end{array} \right] . +$$ + +We make the following assumptions, which are common in the bilevel optimization literature [16– 18, 26]. + +Assumption 1 (Lipschitz continuity). Assume that $f , \nabla f , \nabla g , \nabla ^ { 2 } g$ are respectively $\ell _ { f , 0 } .$ , $\ell _ { f , 1 } , \ell _ { g , 1 } , \ell _ { g , 2 }$ -Lipschitz continuous; that is, for $z _ { 1 } : = [ x _ { 1 } ; y _ { 1 } ]$ , $z _ { 2 } : = [ x _ { 2 } ; y _ { 2 } ]$ , we have $\parallel f ( x _ { 1 } , y _ { 1 } ) -$ $\begin{array} { r c l } { f ( x _ { 2 } , y _ { 2 } ) \| ^ { - } \le } & { \ell _ { f , 0 } \| z _ { 1 } - z _ { 2 } \| , \| \nabla f ( x _ { 1 } , y _ { 1 } ) - \nabla f ( x _ { 2 } , y _ { 2 } ) \| } & { \le } & { \ell _ { f , 1 } \| z _ { 1 } - x _ { 2 } \| , } \end{array}$ $\nabla g ( x _ { 2 } , y _ { 2 } ) \| \leq \ell _ { g , 1 } \| z _ { 1 } - z _ { 2 } \|$ , $\begin{array} { r } { \| \nabla ^ { 2 } g ( x _ { 1 } , y _ { 1 } ) - \nabla ^ { 2 } g ( x _ { 2 } , y _ { 2 } ) \| \le \ell _ { g , 2 } \| z _ { 1 } - z _ { 2 } \| . } \end{array}$ . + +Assumption 2 (Strong convexity of $g$ in $y$ ). For any fixed $x$ , $g ( x , y )$ is $\mu _ { g }$ -strongly convex in $y$ + +Assumptions 1 and 2 together ensure that the first- and second-order derivations of $f ( x , y ) , g ( x , y )$ as well as the solution mapping $y ^ { \ast } ( x )$ , are well-behaved. Define the condition number $\kappa : = \ell _ { g , 1 } / \mu _ { g }$ + +Assumption 3 (Stochastic derivatives). The stochastic derivatives $\nabla f ( x , y ; \xi )$ , $\nabla g ( x , y ; \phi )$ , $\nabla ^ { 2 } g ( x , \mathbf { \bar { y } } , \phi )$ are unbiased estimators of $\nabla f ( x , y )$ , $\nabla g ( x , y )$ , $\nabla ^ { 2 } g ( x , y )$ , respectively; and their variances are bounded by $\sigma _ { f } ^ { 2 } , \sigma _ { g , 1 } ^ { 2 }$ , $\sigma _ { g , 2 } ^ { 2 }$ , respectively. + +
ALSETSCGDNASA
batch size0(1)0(1)0(1)
y-updateSGDSGDcorrection
samplesO(c-²）0(c-4）O(c-2)
+ +Table 2: Sample complexity of stochastic minmax algorithms (BSA in [16], GDA in [28], SMD in [9]) to achieve an $\epsilon$ -stationary point of $F ( x )$ . + +
ALSETSGDASMD
batch size0(1)0(e-1)1
y-updateSGDSGDsubproblem
samplesO（kc-2）O（κ³-²）O（k³-2）
+ +Table 3: Sample complexity of stochastic compositional algorithms (SCGD in [12], NASA in [14]) to achieve an $\epsilon$ -stationary point of $F ( x )$ . + +Assumptions 2 and 3 together imply that the second moments are bounded by + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \xi } [ \| \nabla f ( x , y ; \xi ) \| ^ { 2 } ] \le \ell _ { f , 0 } ^ { 2 } + \sigma _ { f } ^ { 2 } : = C _ { f } ^ { 2 } } \\ & { \mathbb { E } _ { \phi } [ \| \nabla ^ { 2 } g ( x , y ; \phi ) \| ^ { 2 } ] \le \ell _ { g , 1 } ^ { 2 } + \sigma _ { g , 2 } ^ { 2 } : = C _ { g } ^ { 2 } . } \end{array} +$$ + +Assumption 3 is the counterpart of the unbiasedness and bounded variance assumption in the singlelevel stochastic optimization. In addition, the bounded moments in Assumption 3 ensure the Lipschitz continuity of the upper-level gradient $\nabla F ( x )$ . + +We first highlight the inherent challenge of directly applying the alternating SGD method to the bilevel problem (1). To illustrate this point, we derive the gradient of the upper-level function $F ( x )$ in the next proposition; see the proof in the supplementary document. + +Proposition 1. Under Assumptions $_ { I - 3 }$ , we have the gradients + +$$ +\begin{array} { r } { \nabla F ( x ) = \nabla _ { x } f ( x , y ^ { * } ( x ) ) - \nabla _ { x y } ^ { 2 } g ( x , y ^ { * } ( x ) ) \left[ \nabla _ { y y } ^ { 2 } g ( x , y ^ { * } ( x ) ) \right] ^ { - 1 } \nabla _ { y } f ( x , y ^ { * } ( x ) ) . } \end{array} +$$ + +urthermore, $\nabla F ( x )$ and $y ^ { * } ( x )$ are Lipschitz continuous with constants $L _ { F } , L _ { y }$ , respectively. + +Notice that obtaining an unbiased stochastic estimate of $\nabla F ( x )$ and applying SGD on $x$ face two main difficulties: i) the gradient $\nabla F ( x )$ at $x$ depends on the minimizer of the lower-level problem $y ^ { \ast } ( x )$ ; ii) even if $y ^ { * } ( x )$ is known, it is hard to apply the stochastic approximation to obtain an unbiased estimate of $\dot { \nabla } F ( { \boldsymbol { x } } )$ since $\nabla F ( x )$ is nonlinear in $\nabla _ { y y } ^ { 2 } g ( x , y ^ { * } ( x ) )$ . + +Similar to some existing stochastic bilevel algorithms [16, 18, 17], we evaluate $\nabla F ( x )$ on a certain vector $y$ in place of $y ^ { * } ( x )$ . Replacing the $y ^ { * } ( x )$ in definition (6) by $y$ , we define + +$$ +\overline { { \nabla } } _ { x } f \big ( x , y \big ) : = \nabla _ { x } f \big ( x , y \big ) - \nabla _ { x y } ^ { 2 } g \big ( x , y \big ) \left[ \nabla _ { y y } ^ { 2 } g \big ( x , y \big ) \right] ^ { - 1 } \nabla _ { y } f \big ( x , y \big ) . +$$ + +And to reduce the bias in (7), we estimate $\left[ \nabla _ { y y } ^ { 2 } g ( x , y ) \right] ^ { - 1 }$ via + +$$ +\left[ \nabla _ { y y } ^ { 2 } g ( x , y ) \right] ^ { - 1 } \approx \Big [ \frac { N } { \ell _ { g , 1 } } \prod _ { n = 1 } ^ { N ^ { \prime } } \Big ( I - \frac { 1 } { \ell _ { g , 1 } } \nabla _ { y y } ^ { 2 } g ( x , y ; \phi _ { ( n ) } ) \Big ) \Big ] +$$ + +where $N ^ { \prime }$ is drawn from $\{ 1 , 2 , \ldots , N \}$ uniformly at random and $\{ \phi ^ { ( 1 ) } , \dots , \phi ^ { ( N ^ { \prime } ) } \}$ are i.i.d. samples. It has been shown in [16] that using (8), the estimation bias of $\left[ \nabla _ { y y } ^ { 2 } g ( x , y ) \right] ^ { - 1 }$ exponentially decreases with the number of samples $N$ . + +# 2.2 Main results: Tighter analysis of ALSET + +In this subsection, we first describe the general ALSET algorithm for the stochastic bilevel problem, and then present its new convergence result. + +This algorithm is very simple to implement. At each iteration $k$ , ALSET alternates between the stochastic gradient update on $y ^ { k }$ and that on $x ^ { k }$ . Although it is possible that $T = 1$ , for generality, we run $T$ steps of SGD on + +# Algorithm 1 ALSET for the stochastic bilevel problem (1) + +1: initialize: $x ^ { 0 } , y ^ { 0 }$ , stepsizes $\{ \alpha _ { k } , \beta _ { k } \}$ . +2: for $k = 0 , 1 , \ldots , K - 1$ do +3: for $t = 0 , 1 , \dots , T - 1$ do +4: update $y ^ { k , t + 1 } = y ^ { k , t } - \beta _ { k } h _ { g } ^ { k , t }$ . set $y ^ { k , 0 } = y ^ { k }$ +5: end for +6: update $x ^ { k + 1 } = x ^ { k } - \alpha _ { k } h _ { f } ^ { k } \qquad \Join$ +7: end for + +the lower-level variable $y ^ { k }$ before updating upper-level variable $x ^ { k }$ . With $\alpha _ { k }$ and $\beta _ { k }$ denoting the stepsizes of $x ^ { k }$ and $y ^ { k }$ that decrease at the same rate as SGD, the ALSET update is + +$$ +\begin{array} { r l } & { y ^ { k , t + 1 } = y ^ { k , t } - \beta _ { k } h _ { g } ^ { k , t } , t = 0 , \ldots , T \quad \mathrm { w i t h } y ^ { k , 0 } : = y ^ { k } ; y ^ { k + 1 } : = y ^ { k , T } } \\ & { x ^ { k + 1 } = x ^ { k } - \alpha _ { k } h _ { f } ^ { k } } \end{array} +$$ + +where the update direction of $y$ is the stochastic gradient $h _ { g } ^ { k , t } : = \nabla _ { y } g ( x ^ { k } , y ^ { k , t } ; \phi ^ { k , t } )$ ; and, with the Hessian inverse estimator (8), the update direction of $x$ is the slightly biased gradient + +$$ +\begin{array} { r l } & { \boldsymbol { h } _ { f } ^ { k } : = \nabla _ { \boldsymbol { x } } f ( \boldsymbol { x } ^ { k } , \boldsymbol { y } ^ { k + 1 } ; \boldsymbol { \xi } ^ { k } ) } \\ & { \qquad - \nabla _ { \boldsymbol { x } \boldsymbol { y } } ^ { 2 } g ( \boldsymbol { x } ^ { k } , \boldsymbol { y } ; \phi _ { ( 0 ) } ^ { k } ) \Bigl [ \frac { N } { \ell _ { g , 1 } } \displaystyle \prod _ { n = 1 } ^ { N ^ { \prime } } \left( I - \frac { 1 } { \ell _ { g , 1 } } \nabla _ { \boldsymbol { y } \boldsymbol { y } } ^ { 2 } g ( \boldsymbol { x } ^ { k } , \boldsymbol { y } ^ { k + 1 } ; \phi _ { ( n ) } ^ { k } ) \right) \Bigr ] \nabla _ { \boldsymbol { y } } f ( \boldsymbol { x } ^ { k } , \boldsymbol { y } ^ { k + 1 } ; \boldsymbol { \xi } ^ { k } ) . } \end{array} +$$ + +The alternating update (9) serves as a template for running SGD on stochastic nested problems. As we will show in the subsequent sections, we can generate stochastic algorithms for min-max, compositional, and even reinforcement learning problems following (9) as a template, but they differ in the particular forms of the stochastic gradients $h _ { g } ^ { k } , h _ { f } ^ { k }$ for the specific upper- and lower-level objective functions. See Algorithm 1 for a summary of ALSET for the bilevel problem. + +Comparison between ALSET with existing works. Readers who are familiar with recent developments on stochastic optimization for bilevel problems may readily recognize the similarities between the general ALSET update (1) that we will analyze and the SGD-based updates in BSA [16], TTSA [18] and stocBiO [17]. However, the update (1) is different from BSA in that the number of $y$ -update, denoted as $T$ , is a constant in (1) that does not grow with the accuracy $\epsilon ^ { - 1 }$ ; the update (1) is different from stocBiO in that the stochastic gradient $h _ { g } ^ { k , \bar { t } }$ used in the $y$ -update (9a) is obtained by a fixed batch size that does not depend on the accuracy $\epsilon ^ { - 1 }$ ; and, the update (1) is different from TTSA in that the stepsizes $\alpha _ { k }$ and $\beta _ { k }$ in (9) decrease at the same timescale. + +We next present the convergence result of ALSET. + +Theorem 1 (Bilevel problems). Suppose Assumptions $_ { I - 3 }$ hold. Define the constants as + +$$ +\bar { \alpha } _ { 1 } = \frac { 1 } { 2 L _ { F } + 4 L _ { f } L _ { y } + \frac { 2 L _ { f } L _ { y x } } { L _ { y } \eta } } , \bar { \alpha } _ { 2 } = \frac { 1 6 T \mu _ { g } \ell _ { g , 1 } } { ( \mu _ { g } + \ell _ { g , 1 } ) ^ { 2 } ( 8 L _ { f } L _ { y } + 2 \eta L _ { y x } \tilde { C } _ { f } ^ { 2 } \bar { \alpha } _ { 1 } ) } +$$ + +where $\eta > 0$ is a control constant that will be specified in each special case to achieve the best sample complexity. With $\alpha > 0$ being a control constant that will be specified later, choose the stepsizes as + +$$ +\alpha _ { k } = \operatorname* { m i n } \left\{ \bar { \alpha } _ { 1 } , \bar { \alpha } _ { 2 } , \frac { \alpha } { \sqrt { K } } \right\} \mathrm { a n d } \beta _ { k } = \frac { 8 L _ { f } L _ { y } + 2 \eta L _ { y x } \tilde { C } _ { f } ^ { 2 } \bar { \alpha } _ { 1 } } { 4 T \mu _ { g } } \alpha _ { k } . +$$ + +For any $T \geq 1$ and $N = \mathcal { O } ( \log K )$ , the iterates $\{ x ^ { k } , y ^ { k } \}$ generated by Algorithm $I$ satisfy + +$$ +\frac { 1 } { K } \sum _ { k = 1 } ^ { K } \mathbb { E } \left[ \left. \nabla F ( x ^ { k } ) \right. ^ { 2 } \right] = \mathcal { O } \Big ( \frac { 1 } { \sqrt { K } } \Big ) \ \mathrm { ~ a n d ~ } \ \mathbb { E } \left[ \left. y ^ { K } - y ^ { * } ( x ^ { K } ) \right. ^ { 2 } \right] = \mathcal { O } \Big ( \frac { 1 } { \sqrt { K } } \Big ) +$$ + +where $y ^ { * } ( x ^ { K } )$ is the minimizer of the lower-level problem in (1b). + +Proposition 2. Under the same assumptions and the choice of parameters of Theorem $^ { l }$ , with $\begin{array} { r } { \kappa : = { \frac { \ell _ { g , 1 } } { \mu _ { g } } } } \end{array}$ \`g,1µg being the condition number, select α = Θ(κ−5/2), T = Θ(κ4), η = O(κ) in (12), and then + +$$ +\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } [ \| \nabla F ( x ^ { k } ) \| ^ { 2 } ] = \mathcal { O } \left( \frac { \kappa ^ { 3 } } { K } + \frac { \kappa ^ { \frac { 5 } { 2 } } } { \sqrt { K } } \right) . +$$ + +Discussion of Theorem 1. To achieve $\epsilon$ -stationary point, we need $K = \mathcal { O } ( \kappa ^ { 5 } \epsilon ^ { - 2 } )$ , and the number of evaluations of $h _ { f } ^ { k } , h _ { g } ^ { k , t }$ are $\mathcal { O } ( \kappa ^ { 5 } \epsilon ^ { - 2 } )$ and $\mathcal { O } ( \kappa ^ { 9 } \epsilon ^ { - 2 } )$ , respectively. Therefore, the sample complexity is on the same order of SGD’s sample complexity for the single-level nonconvex problems [43], and improves the state-of-the-art single-loop TTSA’s sample complexity $\mathcal { O } ( \epsilon ^ { - 5 / 2 } )$ [18]. Compared to [17], ALSET achieves the same sample complexity in terms of both $\epsilon$ and $\kappa$ , without using a growing batch size. Importantly, we obtain this tighter bound without introducing additional assumptions. + +# 2.3 Proof sketch + +In this subsection, we highlight the key steps of the proof towards Theorem 1, and highlight the differences between our analysis and the existing ones. + +For simplicity, we define the following Lyapunov function as $\begin{array} { r } { \mathbb { V } ^ { k } : = F ( x ^ { k } ) + \frac { L _ { f } } { L _ { y } } \| y ^ { k } - y ^ { * } ( x ^ { k } ) \| ^ { 2 } . } \end{array}$ We first quantify the difference between two Lyapunov functions as + +$$ +\mathbb { V } ^ { k + 1 } - \mathbb { V } ^ { k } = \underbrace { F ( x ^ { k + 1 } ) - F ( x ^ { k } ) } _ { \mathrm { L e m m a ~ 1 } } + \ \frac { L _ { f } } { L _ { y } } ( \| y ^ { k + 1 } - y ^ { * } ( x ^ { k + 1 } ) \| ^ { 2 } - \| y ^ { k } - y ^ { * } ( x ^ { k } ) \| ^ { 2 } ) . +$$ + +The difference in (15) consists of two difference terms: the first term quantifies the descent of the overall objective functions; the second term characterizes the descent of the lower-level errors. + +We will first analyze the descent of the upper-level objective in the next lemma. + +Lemma 1 (Descent of upper level). Suppose Assumptions $_ { I - 3 }$ hold. Define $\bar { h } _ { f } ^ { k } : = \mathbb { E } [ h _ { f } ^ { k } | x ^ { k } , y ^ { k + 1 } ]$ and $\| \bar { h } _ { f } ^ { k } - \overline { { \nabla } } f ( x ^ { k } , y ^ { k + 1 } ) \| \leq b _ { k }$ . The sequence of $x ^ { k }$ generated by Algorithm $I$ satisfies + +$$ +\begin{array} { r } { \mathbb { E } [ F ( { x } ^ { k + 1 } ) ] - \mathbb { E } [ F ( { x } ^ { k } ) ] \le - \frac { \alpha _ { k } } { 2 } \mathbb { E } [ \| \nabla F ( { x } ^ { k } ) \| ^ { 2 } ] - \left( \frac { \alpha _ { k } } { 2 } - \frac { L _ { F } \alpha _ { k } ^ { 2 } } { 2 } \right) \mathbb { E } [ \| \bar { h } _ { f } ^ { k } \| ^ { 2 } ] } \\ { + L _ { f } ^ { 2 } \alpha _ { k } \mathbb { E } [ \| y ^ { k + 1 } - y ^ { * } ( { x } ^ { k } ) \| ^ { 2 } ] + \alpha _ { k } b _ { k } ^ { 2 } + \frac { L _ { F } \alpha _ { k } ^ { 2 } } { 2 } \tilde { \sigma } _ { f } ^ { 2 } } \end{array} +$$ + +where constants $L _ { f } , L _ { F } , \sigma _ { f } ^ { 2 }$ are defined in Lemma 4 of the supplementary document. + +Lemma 1 implies that the descent of the upper-level objective functions depends on the error of the lower-level variable $y ^ { k }$ . We will next analyze the error of the lower-level variable, which is the key step to improving the existing results. + +Before we analyze the error of $y ^ { k }$ , we introduce a lemma that characterizes the smoothness of $y ^ { \ast } ( x )$ and the bounded moments of $h _ { f } ^ { k }$ . The smoothness and the bounded moments have not been explored by previous analysis such as [16–18], and they play an essential role in our improved analysis of $y ^ { k }$ . + +Lemma 2 (Smoothness and boundedness). Under Assumptions $^ { l }$ and 2, we have + +$$ +\begin{array} { r } { \| \nabla y ^ { * } ( x _ { 1 } ) - \nabla y ^ { * } ( x _ { 2 } ) \| \leq L _ { y x } \| x _ { 1 } - x _ { 2 } \| ; \quad \mathbb { E } [ \| h _ { f } ^ { k } \| ^ { 2 } | x ^ { k } , y ^ { k + 1 } ] \leq \tilde { C } _ { f } ^ { 2 } } \end{array} +$$ + +where $L _ { y x }$ and $\tilde { C } _ { f } ^ { 2 }$ depend on the constants defined in Assumptions 1-2. + +Building upon Lemma 2, we establish the progress of the lower-level update. + +Lemma 3 (Error of lower level). Suppose that Assumptions 1–3 hold, and $y ^ { k + 1 }$ is generated by running iteration (9) given $x ^ { k }$ . If we choose $\begin{array} { r } { \beta _ { k } \le \frac { 2 ^ { \binom { - } { q } } } { \mu _ { g } + \ell _ { g , 1 } } } \end{array}$ , then $y ^ { k + 1 }$ satisfies + +$$ +\begin{array} { r l } & { \mathbb { E } [ \| y ^ { k + 1 } - y ^ { * } ( x ^ { k } ) \| ^ { 2 } ] \leq ( 1 - \mu _ { g } \beta _ { k } ) ^ { T } \mathbb { E } [ \| y ^ { k } - y ^ { * } ( x ^ { k } ) \| ^ { 2 } ] + T \beta _ { k } ^ { 2 } \sigma _ { g , 1 } ^ { 2 } } \\ & { \mathbb { E } [ \| y ^ { k + 1 } - y ^ { * } ( x ^ { k + 1 } ) \| ^ { 2 } ] \leq \Big ( 1 + 4 L _ { f } L _ { y } \alpha _ { k } + \frac { \eta L _ { y x } \tilde { C } _ { f } ^ { 2 } } { 2 } \alpha _ { k } ^ { 2 } \Big ) \mathbb { E } [ \| y ^ { k + 1 } - y ^ { * } ( x ^ { k } ) \| ^ { 2 } ] } \\ & { \qquad + \Big ( L _ { y } ^ { 2 } + \frac { L _ { y } } { 4 L _ { f } \alpha _ { k } } + \frac { L _ { y x } } { 2 \eta } \Big ) \alpha _ { k } ^ { 2 } \mathbb { E } [ \| \bar { h } _ { f } ^ { k } \| ^ { 2 } ] + \Big ( L _ { y } ^ { 2 } + \frac { L _ { y x } } { 2 \eta } \Big ) \alpha _ { k } ^ { 2 } \tilde { \sigma } _ { f } ^ { 2 } } \end{array} +$$ + +where $\eta > 0$ is a fixed constant that will be chosen to obtain the tighter complexity bound. + +The improved analysis of the lower-level problem. Next we explain where we can obtain improved analysis. Plugging (18a) into (18b), and selecting stepsizes $\alpha _ { k } , \beta _ { k }$ properly, we can show that + +$$ +\begin{array} { r } { \mathbb { E } [ \| y ^ { k + 1 } - y ^ { * } ( x ^ { k + 1 } ) \| ^ { 2 } ] \leq ( 1 - \delta _ { 1 } ) \mathbb { E } [ \| y ^ { k } - y ^ { * } ( x ^ { k } ) \| ^ { 2 } ] + \delta _ { 2 } \mathbb { E } [ \| \bar { h } _ { f } ^ { k } \| ^ { 2 } ] + \delta _ { 3 } T \sigma _ { g , 1 } ^ { 2 } + \delta _ { 4 } \tilde { \sigma } _ { f } ^ { 2 } } \end{array} +$$ + +where the constants are $\delta _ { 1 } \in [ 0 , 1 ) , \delta _ { 2 } = \mathcal { O } ( \alpha _ { k } ) , \delta _ { 3 } = \mathcal { O } ( \beta _ { k } ^ { 2 } ) , \delta _ { 4 } = \mathcal { O } ( \alpha _ { k } ^ { 2 } )$ . As we will show in our supplementary material, the term $\mathbb { E } [ \| \bar { h } _ { f } ^ { k } \| ^ { 2 } ]$ will be canceled when combined with (16) in our analysis. Hence, choosing $\alpha _ { k } = \mathcal { O } ( k ^ { - 1 / 2 } )$ and $\beta _ { k } = \mathcal { O } ( k ^ { - 1 / 2 } )$ makes the variance terms in (19) decrease at the same $\mathcal { O } ( k ^ { - 1 / 2 } )$ rate as the vanilla SGD for stochastic non-nested problems. + +As a comparison, the progress of the lower-level problem in [18, 17] can be summarized as + +$$ +\mathbb { E } [ \| y ^ { k + 1 } - y ^ { * } ( x ^ { k + 1 } ) \| ^ { 2 } ] \leq ( 1 - \delta _ { 1 } ) \mathbb { E } [ \| y ^ { k } - y ^ { * } ( x ^ { k } ) \| ^ { 2 } ] + \delta _ { 5 } \sigma ^ { 2 } +$$ + +where $\sigma ^ { 2 }$ is some variance term, and the constant is $\delta _ { 5 } = \mathcal { O } ( \beta _ { k } ^ { 2 } + \alpha _ { k } ^ { 2 } / \beta _ { k } )$ or $\mathcal { O } ( 1 / B _ { k } )$ with $B _ { k }$ being the batch size at iteration $k$ . To balance the two terms in $\delta _ { 5 } = \mathcal { O } ( \beta _ { k } ^ { 2 } + \alpha _ { k } ^ { 2 } / \beta _ { k } )$ , two timescales of stepsizes $\begin{array} { r } { \operatorname* { l i m } _ { k \to \infty } \alpha _ { k } / \beta _ { k } = 0 } \end{array}$ are needed, which will make the variance term of the $y$ -update in (20) and that of the $x$ -update in (16) decrease at two different rates, slower than that of SGD; and to reduce $\delta _ { 5 } = \mathcal { O } ( 1 / B _ { k } )$ , a growing batch size $B _ { k } = \mathcal { O } ( k )$ is needed for the $y$ -update. + +# 3 Applications to Stochastic Min-Max and Compositional Problems + +Building upon the general results for the bilevel problems in Section 2, this section will identify special features of the stochastic min-max and stochastic compositional problems, and customize the general results to yield state-of-the-art convergence results for two special nested problems. + +# 3.1 Stochastic min-max problems + +We first apply our results to the stochastic min-max problem (2). In this special case, the lower-level function is $\bar { g ( x , y ; \phi ) } = - f ( x , y ; \xi )$ , and the bilevel gradient in (6) reduces to + +$$ +\nabla F ( x ) : = \nabla _ { x } f { \big ( } x , y ^ { * } ( x ) { \big ) } + \nabla _ { x } y ^ { * } ( x ) ^ { \top } \nabla _ { y } f { \big ( } x , y ^ { * } ( x ) { \big ) } = \nabla _ { x } f { \big ( } x , y ^ { * } ( x ) { \big ) } +$$ + +where the second equality follows from the optimality condition of the lower-level problem, i.e., $\nabla _ { y } f ( x , y ^ { * } ( x ) ) = 0$ . Similar to Section 2, we again approximate $\nabla F ( x )$ on a certain vector $y$ in place of $y ^ { * } ( x )$ . Therefore, the alternating stochastic gradients for this special case are given by + +$$ +\begin{array} { r } { h _ { g } ^ { k , t } = - \nabla _ { y } f ( x ^ { k } , y ^ { k , t } ; \xi _ { 1 } ^ { k , t } ) ~ \mathrm { a n d } ~ h _ { f } ^ { k } = \nabla _ { x } f ( x ^ { k } , y ^ { k + 1 } ; \xi _ { 2 } ^ { k } ) . } \end{array} +$$ + +Plugging the stochastic gradient into the general update (9), we summarize the update in Algorithm 2. +When the number of $y$ -update is $T = 1$ , the ALSET algorithm reduces to the SGDA method in [28]. + +Proposition 3 (Min-max problems). Choose the same choice of parameters as those in Theorem $I$ , and follow the same assumption as those in Theorem $^ { l }$ except that $f ( \cdot , y )$ is only Lipchitz over $x \in \mathbb { R } ^ { d }$ but not that $f ( x , \cdot )$ is Lipschitz continuous over $y \in \mathbb { R } ^ { d ^ { \prime } }$ . If we select $\alpha = \Theta ( \kappa ^ { - 1 } )$ , $T = \Theta ( \kappa )$ , $\eta = 1$ in (12), the iterates generated by Algorithm 2 satisfy + +$$ +\frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left[ \left. \nabla F ( x ^ { k } ) \right. ^ { 2 } \right] = \mathcal { O } \left( \frac { \kappa ^ { 2 } } { K } + \frac { \kappa } { \sqrt { K } } \right) . +$$ + +Proposition 3 implies that for the minmax problem, the convergence rate of ALSET to the stationary point of $\begin{array} { r } { F ( x ) \ \mathrel { \mathop : } = \ \operatorname* { m a x } _ { y \in \mathbb { R } ^ { d ^ { \prime } } } \mathbb { E } _ { \xi } \left[ f ( x , y ; \xi ) \right] } \end{array}$ is $\mathcal { O } ( K ^ { - 1 / 2 } )$ . To achieve $\epsilon$ -stationary point, we need $K = \mathcal { O } ( \kappa ^ { 2 } \epsilon ^ { - 2 } )$ . And the number of gradient evaluations for $h _ { f } ^ { k } , h _ { g } ^ { k , t }$ are $\mathcal { O } \bar { ( \kappa ^ { 2 } \epsilon ^ { - 2 } ) }$ and $\mathcal { O } ( \kappa ^ { 3 } \epsilon ^ { - 2 } )$ , respectively. Comparing with the results in [28], we achieve the same sample complexity without an increasing batch size $\mathsf { \bar { \mathcal { O } } } ( \epsilon ^ { - 1 } )$ , and improve their sample complexity $\mathcal { O } ( \epsilon ^ { - 5 / 2 } )$ under a fixed batch size. + +Algorithm 2 ALSET for the min-max problem (2) + +1: initialize: $x ^ { 0 } , y ^ { 0 }$ , stepsizes $\{ \alpha _ { k } , \beta _ { k } \}$ . +2: for $k = 0 , 1 , \ldots , K - 1$ do +3: set $y ^ { k , 0 } = y ^ { k }$ +4: for $t = 0 , 1 , \dots , T - 1$ do +5: update $y ^ { k , t + 1 } = y ^ { k , t } - \beta _ { k } \nabla _ { y } f ( x ^ { k } , y ^ { k , t } ; \xi _ { 1 } ^ { k , t } )$ +6: end for +7: set $y ^ { k + 1 } = y ^ { k , T }$ +8: update $\boldsymbol { x } ^ { k + \mathrm { i } } = x ^ { k } - \alpha _ { k } \nabla _ { x } f ( x ^ { k } , y ^ { k + 1 } ; \xi _ { 2 } ^ { k } )$ +9: end for + +However, it is also worth mentioning that compared with [28], our analysis requires the additional Lipschitz continuity assumption of $f ( \cdot , y )$ over $x \in \mathbb { R } ^ { d }$ , which inherits from the analysis for the general bilevel problem. Therefore, our result complements, rather than improves, the analysis in [28]. We view our contribution in min-max problems as a supplementary of existing results. + +# 3.2 Stochastic compositional problems + +In this section, we apply our results to the stochastic compositional problem (3). In this special case, the upper-level function is $f ( x , y ; \xi ) : = f ( y ; \xi )$ , and the lower-level function is $g ( x , y ; \phi ) =$ $\| y - h ( x ; \phi ) \| ^ { 2 }$ , and the bilevel gradient in (6) reduces to + +$$ +\begin{array} { r l } & { \nabla F ( x ) : = \nabla _ { x } f \big ( x , y ^ { * } ( x ) \big ) - \nabla _ { x y } ^ { 2 } g ( x , y ^ { * } ( x ) ) \big [ \nabla _ { y y } ^ { 2 } g ( x , y ^ { * } ( x ) ) \big ] ^ { - 1 } \nabla _ { y } f ( x , y ^ { * } ( x ) ) } \\ & { \qquad = \nabla h ( x ; \phi ) ^ { \top } \nabla _ { y } f ( y ^ { * } ( x ) ) } \end{array} +$$ + +where we use the fact that $\nabla _ { y y } ^ { 2 } g ( x , y ; \phi ) = \mathbf { I } _ { d ^ { \prime } \times d ^ { \prime } } , \nabla _ { x y } ^ { 2 } g ( x , y ; \phi ) = - \nabla h ( x ; \phi ) ^ { \top }$ . Similar to Section 2, we again evaluate $\nabla F ( x )$ on a certain vector $y$ in place of $y ^ { \ast } ( x )$ . Therefore, by choosing $T = 1$ , the alternating stochastic gradients $h _ { f } ^ { k } , h _ { g } ^ { k , t }$ for this special case are much simpler, given by + +$$ +h _ { g } ^ { k , t } = h _ { g } ^ { k } = y ^ { k } - h ( x ^ { k } ; \phi ^ { k } ) ~ \mathrm { a n d } ~ h _ { f } ^ { k } = \nabla h ( x ^ { k } ; \phi ^ { k } ) \nabla f ( y ^ { k + 1 } ; \xi ^ { k } ) . +$$ + +Plugging the stochastic gradient into the general update (9), we summarize the update in Algorithm 3. +When $T = 1$ , the ALSET algorithm reduces to SCGD proposed in [12]. + +In the supplementary document, we have verified that the standard assumptions of stochastic compositional optimization in [12, 37, 14, 41, 38] are sufficient for Assumptions 1–3 to hold. + +Proposition 4 (Compositional problems). Under the same assumptions and the parameters as those in Theorem $I$ , if we select $\begin{array} { r } { T ^ { } = 1 , \alpha = 1 , \eta = \frac { 1 } { L _ { y x } } } \end{array}$ in (12), the iterates of Algorithm 3 satisfy + +$$ +\frac { 1 } { K } \sum _ { k = 1 } ^ { K } \mathbb { E } \left[ \left\| \nabla F ( x ^ { k } ) \right\| ^ { 2 } \right] = \mathcal { O } \Big ( \frac { 1 } { \sqrt { K } } \Big ) . +$$ + +Since each iteration of ALSET only uses $\mathcal { O } ( 1 )$ samples (see Algorithm 3), Proposition 4 implies that the sample complexity to achieve an $\epsilon$ -stationary point of (3) is $\mathcal { O } ( \epsilon ^ { - 2 } )$ . Comparing with the results + +Algorithm 3 ALSET for the compositional problem (3) + +1: initialize: $x ^ { 0 } , y ^ { 0 }$ , stepsizes $\{ \alpha _ { k } , \beta _ { k } \}$ . +2: for $k = 0 , 1 , \ldots , K - 1$ do +3: update $y ^ { k + 1 } = y ^ { k } - \beta _ { k } ( y ^ { k } - h ( x ^ { k } ; \phi ^ { k } ) )$ +4: update $x ^ { k + 1 } = x ^ { k } - \alpha _ { k } \nabla f ( y ^ { k + 1 } ; \xi ^ { k } ) \nabla h ( x ^ { k } ; \phi ^ { k } )$ +5: end for + +of the SCGD method in [12], our result improves the sample complexity $\mathcal { O } ( \epsilon ^ { - 4 } )$ under a fixed batch size. Importantly, our analysis does not introduce additional assumption compared to [12]. + +# 4 Applications to Actor-Critic Methods + +In this section, we apply our tighter analysis to the actor-critic (AC) method with linear value function approximation [44], which can be viewed as a special case of the stochastic bilevel algorithm [45, 46]. + +Consider a Markov decision process described by $\mathcal { M } = \{ { \cal S } , \mathcal { A } , \mathcal { P } , { \cal R } , \gamma \}$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $\mathcal { P } ( s ^ { \prime } | s , a )$ is the probability of transitioning to $s ^ { \prime } \in \mathcal { S }$ given state $s \in S$ and action $a \in { \mathcal { A } }$ , and $R ( s , a , s ^ { \prime } )$ is the reward associated with $( s , a , s ^ { \prime } )$ , and $\gamma \in [ 0 , 1 )$ is a discount factor. For a policy $\pi _ { \theta }$ , define the value function $V _ { \pi _ { \theta } } ( s )$ that satisfies the Bellman equation [47] + +$$ +\begin{array} { r } { V _ { \pi _ { \theta } } ( s ) = \mathbb { E } _ { a \sim \pi _ { \theta } ( . | s ) , s ^ { \prime } \sim \mathcal { P } ( \cdot | s , a ) } \left[ r ( s , a , s ^ { \prime } ) + \gamma V _ { \pi _ { \theta } } ( s ^ { \prime } ) \right] . } \end{array} +$$ + +Given the state feature mapping $\phi ( \cdot ) : \mathcal { S } \mathbb { R } ^ { d _ { y } }$ , we approximate the value function linearly as $V _ { \pi _ { \boldsymbol { \theta } } } ( s ) \approx \hat { V } _ { y } ( s ) : = \boldsymbol { \phi } ( s ) ^ { \top } \boldsymbol { y }$ , where $\boldsymbol { y } \in \mathbb { R } ^ { d _ { y } }$ is the critic parameter. The task of finding the best $y$ such that $V _ { \pi _ { \theta } } ( s ) \approx \hat { V } _ { y } ( s )$ is usually addressed by TD learning [48]. + +Defining the stationary distribution induced by the policy parameter $\theta _ { k }$ as $\mu _ { \theta _ { k } }$ and the $k$ th transition as $\xi _ { k } : = ( s _ { k } , a _ { k } , s _ { k + 1 } )$ , which is sampled from $s _ { k } \sim \mu _ { \theta _ { k } } , a \sim \pi _ { \theta _ { k } } , s _ { k + 1 } \sim \mathcal { P }$ , the TD-error is + +$$ +\hat { \delta } ( \xi _ { k } , y _ { k } ) : = r ( s _ { k } , a _ { k } , s _ { k + 1 } ) + \gamma \phi ( s _ { k + 1 } ) ^ { \top } y _ { k } - \phi ( s _ { k } ) ^ { \top } y _ { k } +$$ + +and the critic gradient $h _ { g } ( \xi _ { k } , y _ { k } ) : = \hat { \delta } ( \xi _ { k } , y _ { k } ) \nabla \hat { V } _ { y _ { k } } ( s _ { k } )$ . We update the parameter $y$ via + +$$ +y _ { k + 1 } = \Pi _ { R _ { y } } \big ( y _ { k } + \beta _ { k } h _ { g } ( \xi _ { k } , y _ { k } ) \big ) , +$$ + +where $\beta _ { k }$ is the critic stepsize, and $\Pi _ { R _ { y } }$ is the projection to control the norm of the gradient. A pre-defined constant $R _ { y }$ will be specified in the supplementary document. + +The goal of policy optimization is to solve ${ \mathrm { m a x } } _ { \theta \in \mathbb { R } ^ { d } } F ( \theta )$ with $F ( \theta ) : = \mathbb { E } _ { s \sim \eta } [ V _ { \pi _ { \theta } } ( s ) ]$ , where $\eta$ is the initial distribution. Leveraging the value function approximation and the policy gradient theorem [49], we have the policy gradient $h _ { f } ( \xi , \theta , y ) : = \hat { \delta } ( \xi , \bar { y ) \psi } _ { \theta } ( s , a )$ , which gives the policy update + +$$ +\theta _ { k + 1 } = \theta _ { k } + \alpha _ { k } h _ { f } ( \xi _ { k } ^ { \prime } , \theta _ { k } , y _ { k + 1 } ) , +$$ + +where $\alpha _ { k }$ is the stepsize and $\psi _ { \boldsymbol \theta } ( s , a ) : = \nabla \log \pi _ { \boldsymbol \theta } ( a | s )$ . Note that the sample $\xi _ { k } ^ { \prime } : = ( s _ { k } ^ { \prime } , a _ { k } ^ { \prime } , s _ { k + 1 } ^ { \prime } )$ used in (30) is independent from $\xi _ { k }$ in (29). Specifically, $\xi _ { k } ^ { \prime }$ is sampled from $s _ { k } ^ { \prime } \sim d _ { \theta _ { k } } , a _ { k } ^ { \prime } \sim$ $\pi _ { \boldsymbol { \theta } _ { k } } , s _ { k + 1 } ^ { \prime } \sim \mathcal { P }$ with $d _ { \theta _ { k } }$ being the discounted state action visitation measure under $\theta _ { k }$ . + +The alternating AC update (29)-(30) is a special case of ALSET, where the critic update is the lower-level update, and the actor update is the upper-level update. + +Due to space limitation, we will directly present the results of the alternating AC next, and defer presentation of the proof and the corresponding assumptions, which are the counterparts of Assumptions 1–3 in the context of AC, to the supplementary document. + +Theorem 2 (Actor-critic). Under the some regularity conditions that are specified in the supplementary document, selecting step size $\begin{array} { r } { \alpha _ { k } = \alpha = \overset { \cdot } { \mathcal { O } } ( \frac { 1 } { \sqrt { K } } ) } \end{array}$ , $\begin{array} { r } { \beta _ { k } = \beta = \mathcal { O } ( \frac { 1 } { \sqrt { K } } ) } \end{array}$ , it holds + +$$ +\frac { 1 } { K } \sum _ { k = 1 } ^ { K } \mathbb { E } \left[ \| \nabla F ( \theta _ { k } ) \| ^ { 2 } \right] = \mathcal { O } \left( \frac { 1 } { \sqrt { K } } \right) + \epsilon _ { \mathrm { a p p } } +$$ + +where $\epsilon _ { \mathrm { a p p } }$ , defined in the supplementary document, captures the richness of the linear function class. + +Both sides of Theorem 2. As an application of our tighter analysis, Theorem 2 establishes for the first time that the sample complexity of the single-loop alternating actor-critic method is $\mathcal { O } ( \epsilon ^ { - 2 } )$ . On the positive side, this new result improves the previous complexity $\mathcal { O } ( \epsilon ^ { - 5 / 2 } )$ for the single-loop AC [50], and $\mathcal { O } ( \epsilon ^ { - 2 } \log \epsilon ^ { - 1 } )$ for the nested-loop AC [51], and matches $\mathcal { O } ( \epsilon ^ { - 2 } )$ for AC with an exact critic oracle [52]. In addition to using two independent samples, one limitation of our result is that inheriting from the analysis for the general bilevel case, our analysis of AC requires the smoothness of the critic fixed-point $y ^ { * } ( \theta )$ . As shown in the supplementary document, this implicitly requires the additional bounded and Lipschitz continuity assumption on the stationary distribution $\mu _ { \theta }$ . The removal of this assumption and the extension to Markovian sampling are left for future research. + +# 5 Preliminary Experiments + +To validate our new theoretical results, we have conducted the simple experiment using the riskaverse portfolio management task on a benchmark dataset - 100 Book-to-Market. This is a typical application of stochastic compositional optimization (3) that is used in [40, 41]. We compared the popular two-timescale SCGD approach [12] with our single-timescale ALSET approach. + +We use the same initialization of $x ^ { 0 } , y ^ { 0 }$ for both SCGD and ALSET, and tune the stepsizes $\alpha _ { k } , \beta _ { k }$ by following the suggested order in the original SCGD paper and then using a grid search for the multiplicative constant $c$ , that is + +
Iter kln kSCGDALSETALSET-const
102.305.325.315.63
1004.613.783.493.63
2005.303.402.943.06
4005.993.042.402.55
10006.912.571.652.06
+ +The constant $c$ is chosen from the searching + +Table 4: Comparison of $\begin{array} { r l } { { \ln ( \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \| \nabla F ( x ^ { k } ) \| ^ { 2 } ) } \quad } & { { } } \end{array}$ among the two-timescale and single-timescale algorithms. + +grid $\{ 1 0 ^ { - 3 } , 5 \times 1 0 ^ { - 4 } , 1 0 ^ { - 4 } \}$ and is optimized for each algorithm in terms of ergodic average gradient norm versus the number of iterations. In Table 4, we report the logarithmic value of the average gradient norm performance of SCGD, ALSET with both the above decreasing stepsizes and ALSETconst with the constant stepsizes (replacing $k$ with $K = 1 0 0 0$ ). Since SCGD and ALSET use the same number of samples and gradient evaluations per iteration, we report the progress in terms of iterations. By calculating the decay rate, we can observe that the empirical convergence rate of ALSET is no worse than the theoretical rate $\mathcal { O } ( k ^ { - 1 / 2 } )$ , and ALSET outperforms SCGD thanks to its single-timescale stepsizes. We will pursue more comprehensive experiments in our future work. + +# 6 Conclusions + +This paper unifies several SGD-type updates for stochastic nested problems into a single nested SGD approach that we term ALternating Stochastic gradient dEscenT (ALSET) method. ALSET runs in the single-timescale and uses a fixed batch size. This paper presents a tighter analysis for using ALSET to solve stochastic nested problems. Under the new analysis, to achieve an $\epsilon$ -stationary point of the nested problem, ALSET requires $\mathcal { O } ( \epsilon ^ { - 2 } )$ samples in total. As a by-product, this general result also improves the existing sample complexity of the min-max and compositional cases. It matches the sample complexity of SGD for single-level stochastic problems. Applying our analysis to an alternating version of the actor-critic algorithm also yields a state-of-the-art sample complexity. + +Potential limitations of our results include additional assumptions in the min-max and actor-critic cases, which inherit from the assumptions of general bilevel problems. Nevertheless, our work can also lead to promising future research in understanding the theoretical performance of many successful empirical nested optimization algorithms. To this end, our future work consists of relaxing the regularity conditions needed to achieve our theoretical results and Possible extensions include applying our the tighter analysis in this paper to the existing two-timescale Hessian-free bilevel optimization algorithms and decentralized stochastic nested optimization algorithms. + +# Acknowledgements + +The work of T. 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SEGMENTING NATURAL LANGUAGE SENTENCES VIA LEXICAL UNIT ANALYSIS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +In this work, we present Lexical Unit Analysis (LUA), a framework for general sequence segmentation tasks. Given a natural language sentence, LUA scores all the valid segmentation candidates and utilizes dynamic programming (DP) to extract the maximum scoring one. LUA enjoys a number of appealing properties such as inherently guaranteeing the predicted segmentation to be valid and facilitating globally optimal training and inference. Besides, the practical time complexity of LUA can be reduced to linear time, which is very efficient. We have conducted extensive experiments on 5 tasks, including syntactic chunking, named entity recognition (NER), slot filling, Chinese word segmentation, and Chinese part-of-speech (POS) tagging, across 15 datasets. Our models have achieved the state-of-the-art performances on 13 of them. The results also show that the F1 score of identifying long-length segments is notably improved. + +# 1 INTRODUCTION + +Sequence segmentation is essentially the process of partitioning a sequence of fine-grained lexical units into a sequence of coarse-grained ones. In some scenarios, each composed unit is assigned a categorical label. For example, Chinese word segmentation splits a character sequence into a word sequence (Xue, 2003). Syntactic chunking segments a word sequence into a sequence of labeled groups of words (i.e., constituents) (Sang & Buchholz, 2000). + +There are currently two mainstream approaches to sequence segmentation. The most common is to regard it as a sequence labeling problem by using IOB tagging scheme (Mesnil et al., 2014; Ma & Hovy, 2016; Liu et al., 2019b; Chen et al., 2019a; Luo et al., 2020). A representative work is Bidirectional LSTM-CRF (Huang et al., 2015), which adopts LSTM (Hochreiter & Schmidhuber, 1997) to read an input sentence and CRF (Lafferty et al., 2001) to decode the label sequence. This type of method is very effective, providing tons of state-of-the-art performances. However, it is vulnerable to producing invalid labels, for instance, “O, I-tag, I-tag”. This problem is very severe in low resource settings (Peng et al., 2017). In experiments (see section 4.6), we also find that it performs poorly in recognizing long-length segments. + +Recently, there is a growing interest in span-based models (Zhai et al., 2017; Li et al., 2019; Yu et al., 2020). They treat a span rather than a token as the basic unit for labeling. Li et al. (2019) cast named entity recognition (NER) to a machine reading comprehension (MRC) task, where entities are extracted as retrieving answer spans. Yu et al. (2020) rank all the spans in terms of the scores predicted by a bi-affine model (Dozat & Manning, 2016). In NER, span-based models have significantly outperformed their sequence labeling based counterparts. While these methods circumvent the use of IOB tagging scheme, they still rely on post-processing rules to guarantee the extracted span set to be valid. Moreover, since span-based models are locally normalized at span level, they potentially suffer from the label bias problem (Lafferty et al., 2001). + +This paper seeks to provide a new framework which infers the segmentation of a unit sequence by directly selecting from all valid segmentation candidates, instead of manipulating tokens or spans. To this end, we propose Lexical Unit Analysis (LUA) in this paper. LUA assigns a score to every valid segmentation candidate and leverages dynamic programming (DP) (Bellman, 1966) to search for the maximum scoring one. The score of a segmentation is computed by using the scores of its all segments. Besides, we adopt neural networks to score every segment of the input sentence. + +![](images/b34a0a4b29cfa62fc1064e82fa52cff24949a657d0e6f47cf445c7abc4ae1d90.jpg) +Figure 1: A toy example to show LUA and how it differs from prior methods. The items in blue and red respectively denote valid and invalid predictions. + +The purpose of using DP is to solve the intractability of extracting the maximum scoring segmentation candidate by brute-force search. The time complexity of LUA is quadratic time, yet it can be optimized to linear time in practice by performing parallel matrix computations. For training criterion, we incur a hinge loss between the ground truth and the predictions. We also extend LUA to unlabeled segmentation and capturing label correlations. + +Figure 1 illustrates the comparison between previous methods and the proposed LUA. Prior models at token level and span level are vulnerable to generating invalid predictions, and hence rely on heuristic rules to fix them. For example, in the middle part of Figure 1, the spans of two inferred named entities, [Word $\mathrm { C u p } \mathrm { _ { M I S C } }$ and $[ \mathrm { C u p } ] _ { \mathrm { M I S C } }$ , conflicts, which is mitigated by comparing the predicted scores. LUA scores all possible segmentation candidates and uses DP to extract the maximum scoring one. In this way, our models guarantee the predictions to be valid. Moreover, the globality of DP addresses the label bias problem. + +Extensive experiments are conducted on syntactic chunking, NER, slot filling, Chinese word segmentation, and Chinese part-of-speech (POS) tagging across 15 tasks. We have obtained new stateof-the-art results on 13 of them and performed competitively on the others. In particular, we observe that LUA is expert at identifying long-length segments. + +# 2 METHODOLOGY + +We denote an input sequence (i.e., fine-grained lexical units) as $\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { n } ]$ , where $n$ is the sequence length. An output sequence (i.e., coarse-grained lexical units) is represented as the segmentation $\mathbf { y } = [ y _ { 1 } , y _ { 2 } , \cdots , y _ { m } ]$ with each segment $y _ { k }$ being a triple $( i _ { k } , j _ { k } , t _ { k } )$ . $m$ denotes its length. $( i _ { k } , j _ { k } )$ specifies a span that corresponds to the phrase $\mathbf { x } _ { i _ { k } , j _ { k } } = [ x _ { i _ { k } } , x _ { i _ { k } + 1 } , \cdot \cdot \cdot , x _ { j _ { k } } ]$ . $t _ { k }$ is a label from the label space $\mathcal { L }$ . We define a valid segmentation candidate as its segments are non-overlapping and fully cover the input sequence. + +A case extracted from CoNLL-2003 dataset (Sang & De Meulder, 2003): + +$$ +\begin{array} { r } { \mathbf { x } = [ [ \mathrm { S O S } ] , \mathrm { S a n g t h a i } , \mathrm { G l o r y } , 2 2 / 1 1 / 9 6 , 3 0 0 0 , \mathrm { S i n g a p o r e } ] } \\ { \mathbf { y } = [ ( 1 , 1 , 0 ) , ( 2 , 3 , \mathrm { M I S C } ) , ( 4 , 4 , 0 ) , ( 5 , 5 , 0 ) , ( 6 , 6 , \mathrm { L O C } ) ] } \end{array} . +$$ + +Start-of-sentence symbol [SOS] is added in the pre-processing stage. + +# 2.1 MODEL: SCORING SEGMENTATION CANDIDATES + +We denote $\mathcal { V }$ as the universal set that contains all valid segmentation candidates. Given one of its members $\mathbf { y } \in \mathcal { V }$ , we compute the score $f ( \mathbf { y } )$ as + +$$ +f ( \mathbf { y } ) = \sum _ { ( i , j , t ) \in \mathbf { y } } \Big ( s _ { i , j } ^ { c } + s _ { i , j , t } ^ { l } \Big ) , +$$ + +Input: Composition score $s _ { i , j } ^ { c }$ and label score $s _ { i , j , t } ^ { l }$ for every possible segment $( i , j , t )$ . + +utput: The maximum segmentation scoring candidate $\hat { \mathbf { y } }$ and its score $f ( \hat { \mathbf { y } } )$ . + +1 Set two $n \times n$ shaped matrices, $\mathbf { c } ^ { L }$ and ${ \bf b } ^ { c }$ , for computing maximum scoring labels. + +2 Set two $n$ -length vectors, $\mathbf { g }$ and $\mathbf { b } ^ { g }$ , for computing maximum scoring segmentation. + +4 Compute the maximum label score for each span $( i , j ) \colon s _ { i , j } ^ { L } = \operatorname* { m a x } _ { t \in \mathcal { L } } s _ { i , j , t } ^ { l }$ +5 Record the backtracking index: $b _ { i , j } ^ { c } = \arg \operatorname* { m a x } _ { t \in \mathcal { L } } s _ { i , j , t } ^ { l }$ . + +6 Initialize the value of the base case $\mathbf { x } _ { 1 , 1 } \colon g _ { 1 } = s _ { 1 , 1 } ^ { c } + s _ { 1 , 1 } ^ { L }$ + +8 Compute the value of the prefix $\begin{array} { r } { \mathbf { x } _ { 1 , i } { \mathrm { : ~ } } g _ { i } = \operatorname* { m a x } _ { 1 \leq j \leq i - 1 } \left( g _ { i - j } + \left( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } \right) \right) } \end{array}$ +9 Record the backtracking index: $b _ { i } ^ { g } = \arg \operatorname* { m a x } _ { 1 \leq j \leq i - 1 } \left( g _ { i - j } + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } ) \right) .$ . +10 Get the maximum scoring candidate $\hat { \mathbf { y } }$ by back tracing the tables $\mathbf { b } ^ { g }$ and $\mathbf { b } ^ { c }$ . +11 Get the maximum segmentation score: $f ( \hat { \mathbf { y } } ) = g _ { n }$ . + +where $s _ { i , j } ^ { c }$ is the composition score to estimate the feasibility of merging several fine-grained units $[ x _ { i } , x _ { i + 1 } , \cdot \cdot \cdot , x _ { j } ]$ into a coarse-grained unit and $s _ { i , j , t } ^ { l }$ is the label score to measure how likely the label of this segment is $t$ . Both scores are obtained by a scoring model. + +Scoring Model. a scoring model scores all possible segments $( i , j , t )$ for an input sentence $\mathbf { x }$ . Firstly, we get the representation for each fine-grained unit. Following prior works (Li et al., 2019; Luo et al., 2020; Yu et al., 2020), we adopt BERT (Devlin et al., 2018), a powerful pre-trained language model, as the sentence encoder. Specifically, we have + +$$ +[ \mathbf { h } _ { 1 } ^ { w } , \mathbf { h } _ { 2 } ^ { w } \cdot \cdot \cdot \mathbf { \epsilon } , \mathbf { h } _ { n } ^ { w } ] = \mathrm { B E R T } ( \mathbf { x } ) , +$$ + +Then, we compute the representation for a coarse-grained unit $\mathbf { x } _ { i , j } , 1 \leq i \leq j \leq n$ as + +$$ +\mathbf { h } _ { i , j } ^ { p } = \mathbf { h } _ { i } ^ { w } \oplus \mathbf { h } _ { j } ^ { w } \oplus \left( \mathbf { h } _ { i } ^ { w } - \mathbf { h } _ { j } ^ { w } \right) \oplus \big ( \mathbf { h } _ { i } ^ { w } \odot \mathbf { h } _ { j } ^ { w } \big ) , +$$ + +where $\oplus$ is vector concatenation and $\odot$ is element-wise product. + +Eventually, we employ two non-linear feedforward networks to score a segment $( i , j , t )$ : + +$$ +\begin{array} { r } { \boldsymbol { s } _ { i , j } ^ { c } = \left( \mathbf { v } ^ { c } \right) ^ { T } \operatorname { t a n h } ( \mathbf { W } ^ { c } \mathbf { h } _ { i , j } ^ { p } ) , \boldsymbol { s } _ { i , j , t } ^ { l } = \left( \mathbf { v } _ { t } ^ { l } \right) ^ { T } \operatorname { t a n h } ( \mathbf { W } ^ { l } \mathbf { h } _ { i , j } ^ { p } ) , } \end{array} +$$ + +where $\mathbf { v } _ { } ^ { c }$ , $\mathbf { W } ^ { c }$ , $\mathbf { v } _ { t } ^ { l } , t \in \mathcal { L }$ , and $\mathbf { W } ^ { l }$ are all learnable parameters. Besides, the scoring model used here can be flexibly replaced by any regression method. + +# 2.2 INFERENCE VIA DYNAMIC PROGRAMMING + +The prediction of the maximum scoring segmentation candidate can be formulated as + +$$ +{ \hat { \mathbf { y } } } = \operatorname * { a r g m a x } _ { \mathbf { y } \in \mathcal { Y } } f ( \mathbf { y } ) . +$$ + +Because the size of search space $| \mathcal { V } |$ increases exponentially with respect to the sequence length $n$ , brute-force search to solve Equation 5 is computationally infeasible. LUA uses DP to address this issue, which is facilitated by the decomposable nature of Equation 1. + +DP is a well-known optimization method which solves a complicated problem by breaking it down into simpler sub-problems in a recursive manner. The relation between the value of the larger problem and the values of its sub-problems is called the Bellman equation. + +Sub-problem. In the context of LUA, the sub-problem of segmenting an input unit sequence $\mathbf { x }$ is segmenting its prefixes $\mathbf { x } _ { 1 , i } , 1 \leq i \leq n$ . We define $g _ { i }$ as the maximum segmentation score of the prefix $\mathbf { x } _ { 1 , i }$ . Under this scheme, we have $\textstyle \operatorname* { m a x } _ { \mathbf { y } \in { \mathcal { y } } } f ( \mathbf { y } ) = g _ { n }$ . + +The Bellman Equation. The relatinship between segmenting a sequence $\mathbf { x } _ { 1 , i } , i > 1$ and segmenting its prefixes $x _ { 1 , i - j } , 1 \leq j \leq i - 1$ is built by the last segments $( i - j + 1 , i , t )$ : + +$$ +g _ { i } = \operatorname* { m a x } _ { 1 \leq j \leq i - 1 } \big ( g _ { i - j } + \big ( s _ { i - j + 1 , i } ^ { c } + \operatorname* { m a x } _ { t \in \mathcal { L } } s _ { i - j + 1 , i , t } ^ { l } \big ) \big ) . +$$ + +In practice, to reduce the time complexity of above equation, the last term is computed beforehand as $\begin{array} { r } { \dot { s } _ { i , j } ^ { L } = \operatorname* { m a x } _ { t \in \mathcal { L } } s _ { i , j , t } ^ { l } , 1 \leq i \leq j \dot { \leq } n } \end{array}$ . Hence, Equation 6 is reformulated as + +$$ +g _ { i } = \operatorname* { m a x } _ { 1 \leq j \leq i - 1 } \big ( g _ { i - j } + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } ) \big ) . +$$ + +The base case is the first token $\mathbf { x } _ { 1 , 1 } = [ [ \mathrm { S O S } ] ]$ . We get its score $g _ { 1 }$ as $s _ { 1 , 1 } ^ { c } + s _ { 1 , 1 } ^ { L }$ + +Algorithm 1 shows how DP is applied in inference. Firstly, we set two matrices and two vectors to store the solutions to the sub-problems (1-st to 2-nd lines). Secondly, we get the maximum label scores for all the spans (3-rd to 5-th lines). Then, we initialize the trivial case $g _ { 1 }$ and recursively calculate the values for prefixes $\mathbf { x } _ { 1 , i } , i > 1$ (6-th to 9-th lines). Finally, we get the predicted segmentation $\hat { \mathbf { y } }$ and its score $f ( \hat { \mathbf { y } } )$ (10-th to 11-th lines). + +The time complexity of Algorithm 1 is $\mathcal { O } ( n ^ { 2 } )$ . By performing the max operation of Equation 7 in parallel on GPU, it can be optimized to only ${ \mathcal { O } } ( n )$ , which is highly efficient. Besides, DP, as the backbone of the proposed model, is non-parametric. The trainable parameters only exist in the scoring model part. These show LUA is a very light-weight algorithm. + +# 2.3 TRAINING CRITERION + +We adopt max-margin penalty as the loss function for training. Given the predicted segmentation $\hat { \mathbf { y } }$ and the ground truth segmentation $\mathbf { y } ^ { * }$ , we have + +$$ +\mathcal { T } = \operatorname* { m a x } \big ( 0 , 1 - f ( \mathbf { y } ^ { * } ) + f ( \hat { \mathbf { y } } ) \big ) . +$$ + +# 3 EXTENSIONS OF LUA + +We propose two extensions of LUA for generalizing it to different scenarios. + +Unlabeled Segmentation. In some tasks (e.g., Chinese word segmentation), the segments are unlabeled. Under this scheme, the Equation 1 and Equation 7 are reformulated as + +$$ +f ( \mathbf { y } ) = \sum _ { ( i , j ) \in \mathbf { y } } s _ { i , j } ^ { c } , ~ g _ { i } = \operatorname* { m a x } _ { 1 \leq j \leq i - 1 } ( g _ { i - j } + s _ { i - j + 1 , i } ^ { c } ) . +$$ + +Capturing Label Correlations. In some tasks (e.g., syntactic chunking), the labels of segments are strongly correlated. To incorporate this information, we redefine $f ( \mathbf { y } )$ as + +$$ +f ( { \bf { y } } ) = \sum _ { 1 \le k \le m } \left( s _ { i _ { k } , j _ { k } } ^ { c } + s _ { i _ { k } , j _ { k } , t _ { k } } ^ { l } \right) + \sum _ { 1 \le k \le m } s _ { t _ { k - q + 1 } , t _ { k - q + 2 } , \cdots , t _ { k } } ^ { d } . +$$ + +Score In pra $s _ { t _ { k - q + 1 } , t _ { k - q + 2 } , \cdots , t _ { k } } ^ { d }$ models the label dependencies among balances the efficiency and the effecti $q$ successive segments, ness well, and thus pa $_ { \mathbf { y } _ { k - q + 1 , k } }$ $q = 2$ +a learnable matrix $\mathbf { W } ^ { d } \in \mathbb { R } ^ { | \nu | \times | \nu | }$ to implement it. + +The corresponding Bellman equation to above scoring function is + +$$ +g _ { i , t } = \underset { 1 \leq j \leq i - 1 } { \operatorname* { m a x } } \big ( \underset { t ^ { \prime } \in \mathcal { L } } { \operatorname* { m a x } } ( g _ { i - j , t ^ { \prime } } + s _ { t ^ { \prime } , t } ^ { d } ) + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i , t } ^ { l } ) \big ) , +$$ + +where $g _ { i , t }$ is the maximum score of labeling the last segment of the prefix $\mathbf { x } _ { 1 , i }$ with $t$ . For initialization, we set the value of $g _ { . 1 , \mathrm { O } } ^ { d }$ as 0 and the others as $- \infty$ . By performing the inner loops of two max operations in parallel, the practical time complexity for computing $g _ { i , t } , 1 \leq i \leq n , t \in \mathcal { L }$ is also ${ \mathcal { O } } ( n )$ . Ultimately, the segmentation score $f ( \hat { \mathbf { y } } )$ is obtained by $\operatorname* { m a x } _ { t \in \mathcal { L } } g _ { n , t }$ . + +This extension further improves the results on syntactic chunking and Chinese POS tagging, as both tasks have rich sequential features among the labels of segments. + +Table 1: Experiment results on Chinese word segmentation. + +
ModelASMSRCITYUPKUCTB6
Rich Pretraining(Yang et al.,2017)95.797.596.996.396.2
Bi-LSTM(Ma et al.,2018)96.298.197.296.196.7
Multi-Criteria_Learning +_BERT_(Huang etal., 2019)96.697.997.696.697.6
BERT (Meng et al.,2019)96.598.197.696.5=
Glyce + BERT (Meng et al.,2019)96.798.397.996.7-
Unlabeled LUA96.9498.2798.2196.8898.13
+ +Table 2: Experiment results on the four datasets of Chinese POS tagging. + +
ModelCTB5CTB6CTB9UD1
Bi-RNN+ CRF (Single) (Shao et al., 2017) Bi-RNN + CRF (Ensemble) (Shao et al.,2017)94.0790.8191.8989.41
Lattice-LSTM(Meng et al.,2019)94.38-92.3489.75
Glyce + Lattice-LSTM (Meng et al., 2019)95.1491.4392.1390.09
BERT (Meng et al.,2019)95.6191.9292.3890.87
Glyce +BERT (Meng et al., 2019)96.0694.7792.2994.79
96.6195.4193.1596.14
This WorkLUA96.7995.3993.2296.01
LUA w/Label Correlations97.9696.6393.9597.08
+ +# 4 EXPERIMENTS + +We have conducted extensive studies on 5 tasks, including Chinese word segmentation, Chinese POS tagging, syntactic chunking, NER, and slot filling, across 15 datasets. Firstly, Our models have achieved new state-of-the-art performances on 13 of them. Secondly, the results demonstrate that the F1 score of identifying long-length segments has been notably improved. Lastly, we show that LUA is a very efficient algorithm concerning the running time. + +# 4.1 SETTINGS + +We use the same configurations for all 15 datasets. L2 regularization and dropout ratio are respectively set as $1 \times 1 0 ^ { - 6 }$ and 0.2 for reducing overfit. We use Adam (Kingma & Ba, 2014) to optimize our model. Following prior works, BERTBASE is adopted as the sentence encoder. We use uncased BERTBASE for slot filling, Chinese BERTBASE for Chinese tasks (e.g., Chinese POS tagging), and cased BERTBASE for others (e.g., syntactic chunking). In addition, the improvements of our model over baselines are statistically significant with $p < 0 . 0 5$ under t-test. + +# 4.2 CHINESE WORD SEGMENTATION + +Chinese word segmentation splits a Chinese character sequence into a sequence of Chinese words. We use SIGHAN 2005 bake-off (Emerson, 2005) and Chinese Treebank 6.0 (CTB6) (Xue et al., 2005). SIGHAN 2005 back-off consists of 5 datasets, namely AS, MSR, CITYU, and PKU. Following Ma et al. (2018), we randomly select $1 0 \%$ training data as development set. We convert all digits, punctuation, and Latin letters to half-width for handling full/half-width mismatch between training and test set. We also convert AS and CITYU to simplified Chinese. For CTB6, we follow the same format and partition as in Yang et al. (2017); Ma et al. (2018). + +Table 1 depicts the experiment results. All the results of baselines are from Yang et al. (2017); Ma et al. (2018); Huang et al. (2019); Meng et al. (2019). We have achieved new state-of-the-art performance on all datasets except MSR. Our model improves the F1 score by $0 . 2 5 \%$ on AS, $0 . 3 2 \%$ on CITYU, $0 . 1 9 \%$ on PKU, and $0 . 5 4 \%$ on CTB6. Note that our model doesn’t use any external resources, such as glyph information (Meng et al., 2019) or POS tags (Yang et al., 2017). Despite this, our model is still competitive with Glyce $^ +$ BERT on MSR. + +# 4.3 CHINESE POS TAGGING + +Chinese POS tagging jointly segments a Chinese character sequence and assigns a POS tag to each segmented unit. We use Chinese Treebank 5.0 (CTB5), CTB6, Chinese Treebank 9.0 (CTB9) (Xue et al., 2005), and the Chinese section of Universal Dependencies 1.4 (UD1) (Nivre et al., 2016). CTB5 is comprised of newswire data. CTB9 consists of source texts in various genres, which cover CTB5. we convert the texts in UD1 from traditional Chinese into simplified Chinese. We follow the same train/dev/test split for above datasets as in Shao et al. (2017). + +Table 3: Experiment results on syntactic chunking and NER. + +
ModelChunkingNER
CoNLL-2000CoNLL-2003OntoNotes5.0
Bi-LSTM + CRF (Huang et al., 2015)94.4690.101
Flair Embeddings (Akbik et al., 2018)96.7293.0989.3
GCDT w/BERT (Liu et al., 2019b)96.8193.231
BERT-MRC (Li et al., 2019)-93.0491.11
HCR w/BERT (Luo et al., 2020)=93.3790.30
BERT-Biaffine Model (Yu et al., 2020) LUA-93.591.3
This Work96.9593.4692.09
LUA w/Label Correlations97.23--
+ +Table 2 shows the experiment results. The performances of all baselines are reported from Meng et al. (2019). Our model LUA w/ Label Correlations has yielded new state-of-the-art results on all the datasets: it improves the F1 scores by $1 . 3 5 \%$ on CTB5, $1 . 2 2 \%$ on CTB6, $0 . 8 \%$ on CTB9, and $0 . 9 4 \%$ on UD1. Moreover, the basic LUA without capturing the label correlations also outperforms the strongest baseline, Glyce $^ +$ BERT, by $0 . 1 8 \%$ on CTB5 and $0 . 0 7 \%$ on CTB9. All these facts further verify the effectiveness of LUA and its extension. + +# 4.4 SYNTACTIC CHUNKING AND NER + +Syntactic chunking aims to find phrases related to syntatic category for a sentence. We use CoNLL2000 dataset (Sang & Buchholz, 2000), which defines 11 syntactic chunk types (NP, VP, PP, etc.) and follow the standard splittings of training and test datasets as previous work. NER locates the named entities mentioned in unstructured text and meanwhile classifies them into predefined categories. We use CoNLL-2003 dataset (Sang & De Meulder, 2003) and OntoNotes 5.0 dataset (Pradhan et al., 2013). CoNLL-2003 dataset consists of 22137 sentences totally and is split into 14987, 3466, and 3684 sentences for the training set, development set, and test set, respectively. It is tagged with four linguistic entity types (PER, LOC, ORG, MISC). OntoNotes 5.0 dataset contains 76714 sentences from a wide variety of sources (e.g., magazine and newswire). It includes 18 types of named entity, which consists of 11 types (Person, Organization, etc.) and 7 values (Date, Percent, etc.). We follow the same format and partition as in Li et al. (2019); Luo et al. (2020); Yu et al. (2020). In order to fairly compare with previous reported results, we convert the predicted segments into IOB format and utilize conlleval script1 to compute the F1 score at test time. + +Table 3 shows the results. Most of baselines are directly taken from Akbik et al. (2018); Li et al. (2019); Luo et al. (2020); Yu et al. (2020). Besides, following Luo et al. (2020), we rerun the source code2 of GCDT and report its result on CoNLL-2000 with standard evaluation method. Generally, our proposed models LUA w/o Label Correlations yield competitive performance over state-of-theart models on both Chunking and NER tasks. Specifically, regarding to the NER task, on CoNLL2003 dataset our model LUA outperforms several strong baselines including Flair Embedding, and it is comparable to the state-of-the-art model (i.e., BERT-Biaffine Model). In particular, on OntoNotes dataset, LUA outperforms it by $0 . 7 9 \%$ points and establishes a new state-of-the-art result. Regarding to the Chunking task, LUA advances the best model (GCDT) and the improvements are further enlarged to $0 . 4 \hat { 2 } \%$ points by LUA w/ Label Correlations. + +# 4.5 SLOT FILLING + +Slot filling, as an important task in spoken language understanding (SLU), extracts semantic constituents from an utterance. We use ATIS dataset (Hemphill et al., 1990), SNIPS dataset (Coucke et al., 2018), and MTOD dataset (Schuster et al., 2018). ATIS dataset consists of audio recordings of + +Table 4: Experiment results on the three datasets of slot filling. + +
ModelATISSNIPSMTOD
Slot-Gated SLU (Goo et al.,2018) Bi-LSTM + EMLo (Siddhant et al., 2019)95.2088.3095.12
95.4293.90-
Joint BERT (Chen et al., 2019b) CM-Net (Liu et al.,2019c)96.10 96.2097.0096.48
96.1597.15-
This WorkLUA LUA w/ Intent Detection96.2797.1097.53
97.2097.55
+ +
Model1-3(8695)4-7(2380)8-11(151)12-24(31)Overall
HCRw/BERT91.1585.2250.4320.6790.27
BERT-Biaffine Model91.6787.2370.2440.5591.26
LUA92.3188.5277.3457.2792.09
+ +Table 5: The F1 scores for NER models on different segment lengths. $A - B ( N )$ denotes that there are $N$ entities whose span lengths are between $A$ and $B$ . + +people making flight reservations. The training set contains 4478 utterances and the test set contains 893 utterances. SNIPS dataset is collected by Snips personal voice assistant. The training set contains 13084 utterances and the test set contains 700 utterances. MTOD dataset has three domains, including Alarm, Reminder, and Weather. We use the English part of MTOD dataset, where training set, dev set, and test set respectively contain 30521, 4181, and 8621 utterances. We follow the same partition of above datasets as in Goo et al. (2018); Schuster et al. (2018). + +Table 4 summarizes the experiment results for slot filling. On ATIS and SNIPS, we take the results of all baselines as reported in Liu et al. (2019c) for comparison. On MTOD, we rerun the open source toolkits, Slot-gated $\mathrm { S L U } ^ { 3 }$ and Joint BERT4. As all previous approaches jointly model slot filling and intent detection (a classification task in SLU), we follow them to augment LUA with intent detection for a fair comparison. As shown in Table 4, the augmented LUA has surpassed all baselines and obtained state-of-the-art results on the three datasets: it increases the F1 scores by around $0 . 0 5 \%$ on ATIS and SNIPS, and delivers a substantial gain of $1 . 1 1 \%$ on MTOD. It’s worth mentioning that LUA even outperforms the strong baseline Joint BERT with a margin of $0 . 1 8 \%$ and $0 . 2 1 \%$ on ATIS and SNIPS without modeling intent detection. + +# 4.6 LONG-LENGTH SEGMENT IDENTIFICATION + +Since LUA doesn’t resort to IOB tagging scheme, it should be more accurate in recognizing longlength segments than prior methods. To verify this intuition, we evaluate different models on the segments of different lengths. This study is investigated on OntoNotes 5.0 dataset. Two strong models are adopted as the baselines: one is the best sequence labeling model (i.e., HCR) and the other is the best span-based model (i.e., BERT-Biaffine Model). Both baselines are reproduced by rerunning their open source codes, biaffine-ner5 and Hire-NER6. + +The results are shown in Table 5. On the one hand, both LUA and Biaffine Model obtain much higher scores of extracting long-length entities than HCR. For example, LUA outperforms HCR w/ BERT by almost twofold on range $1 2 - 2 4$ . On the other hand, LUA achieves even better results than BERT-Biaffine Model. For instance, the F1 score improvements of LUA over it are $1 0 . 1 1 \%$ on range $8 - 1 1$ and $4 1 . 2 3 \%$ on range $1 2 - 2 4$ . + +# 4.7 RUNNING TIME ANALYSIS + +Table 6 shows the running time comparison among different models. The middle two columns are the time complexity of decoding a label sequence. The last column is the time cost of one epoch in training. We set the batch size as 16 and run all the models on 1 GPU. The results indicate that + +Table 6: Running time comparison on the syntactic chunking dataset. + +
ModelTheoretical ComplexityPractical ComplexityRunning Time
BERT BERT+CRFO(n) O(n|C|2)0(1)5m11s 7m33s
LUAO(n²)O(n) O(n)6m25s
LUA w/Label CorrelationsO(n²|C1²)0(n)7m09s
+ +the success of our models in performances does not lead to serious side-effects on efficiency. For example, with the same practical time complexity, BERT $^ +$ CRF is slower than the proposed LUA by $1 5 . 0 1 \%$ and LUA w/ Label Correlations by $5 . \dot { 3 } 0 \%$ . + +# 5 RELATED WORK + +Sequence segmentation aims to partition a fine-grained unit sequence into multiple labeled coarsegrained units. Traditionally, there are two types of methods. The most common is to cast it into a sequence labeling task (Mesnil et al., 2014; Ma & Hovy, 2016; Chen et al., 2019a) by using IOB tagging scheme. This method is simple and effective, providing a number of state-of-the-art results. Akbik et al. (2018) present Flair Embeddings that pretrain character embedding in a large corpus and directly use it, instead of word representation, to encode a sentence. Liu et al. (2019b) introduce GCDT that deepens the state transition path at each position in a sentence, and further assigns each word with global representation. Luo et al. (2020) use hierarchical contextualized representations to incorporate both sentence-level and document-level information. Nevertheless, these models are vulnerable to producing invalid labels and perform poorly in identifying longlength segments. This problem is very severe in low-resource setting. Ye & Ling (2018); Liu et al. (2019a) adopt Semi-Markov CRF (Sarawagi & Cohen, 2005) that improves CRF at phrase level. However, the computation of CRF loss is costly in practice and the potential to model the label dependencies among segments is limited. An alternative approach that is less studied uses a transition-based system to incrementally segment and label an input sequence (Zhang et al., 2016; Lample et al., 2016). For instance, Qian et al. (2015) present a transition-based model for joint word segmentation, POS tagging, and text normalization. Wang et al. (2017) employ a transitionbased model to disfluency detection task, which helps capture non-local chunk-level features. These models have many advantages like theoretically lower time complexity and labeling the extracted mentions at span level. However, to our best knowledge, no recent transition-based models surpass their sequence labeling based counterparts. + +More recently, there is a surge of interests in span-based models. They treat a segment, instead of a fine-grained token, as the basic unit for labeling. For example, Li et al. (2019) regard NER as a MRC task, where entities are recognized as retrieving answer spans. Since these methods are locally normalized at span level rather than sequence level, they potentially suffer from the label bias problem. Additionally, they rely on rules to ensure the extracted span set to be valid. Spanbased methods also emerge in other fields of NLP. In dependency parsing, Wang & Chang (2016) propose a LSTM-based sentence segment embedding method named LSTM-Minus. Stern et al. (2017) integrate LSTM-minus feature into constituent parsing models. In coreference resolution, Lee et al. (2018) consider all spans in a document as the potential mentions and learn distributions over all the possible antecedents for each other. + +# 6 CONCLUSION + +This work proposes a novel LUA for general sequence segmentation tasks. LUA directly scores all the valid segmentation candidates and uses dynamic programming to extract the maximum scoring one. Compared with previous models, LUA naturally guarantees the predicted segmentation to be valid and circumvents the label bias problem. Extensive studies are conducted on 5 tasks across 15 datasets. We have achieved the state-of-the-art performances on 13 of them. 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In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 421–431, 2016. \ No newline at end of file diff --git a/parse/train/PQlC91XxqK5/PQlC91XxqK5_content_list.json b/parse/train/PQlC91XxqK5/PQlC91XxqK5_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..517f46f950a7b41fd1ce37cfa08b72a402562fad --- /dev/null +++ b/parse/train/PQlC91XxqK5/PQlC91XxqK5_content_list.json @@ -0,0 +1,1705 @@ +[ + { + "type": "text", + "text": "SEGMENTING NATURAL LANGUAGE SENTENCES VIA LEXICAL UNIT ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we present Lexical Unit Analysis (LUA), a framework for general sequence segmentation tasks. Given a natural language sentence, LUA scores all the valid segmentation candidates and utilizes dynamic programming (DP) to extract the maximum scoring one. LUA enjoys a number of appealing properties such as inherently guaranteeing the predicted segmentation to be valid and facilitating globally optimal training and inference. Besides, the practical time complexity of LUA can be reduced to linear time, which is very efficient. We have conducted extensive experiments on 5 tasks, including syntactic chunking, named entity recognition (NER), slot filling, Chinese word segmentation, and Chinese part-of-speech (POS) tagging, across 15 datasets. Our models have achieved the state-of-the-art performances on 13 of them. The results also show that the F1 score of identifying long-length segments is notably improved. ", + "bbox": [ + 233, + 268, + 764, + 435 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 465, + 336, + 482 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sequence segmentation is essentially the process of partitioning a sequence of fine-grained lexical units into a sequence of coarse-grained ones. In some scenarios, each composed unit is assigned a categorical label. For example, Chinese word segmentation splits a character sequence into a word sequence (Xue, 2003). Syntactic chunking segments a word sequence into a sequence of labeled groups of words (i.e., constituents) (Sang & Buchholz, 2000). ", + "bbox": [ + 174, + 500, + 825, + 569 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There are currently two mainstream approaches to sequence segmentation. The most common is to regard it as a sequence labeling problem by using IOB tagging scheme (Mesnil et al., 2014; Ma & Hovy, 2016; Liu et al., 2019b; Chen et al., 2019a; Luo et al., 2020). A representative work is Bidirectional LSTM-CRF (Huang et al., 2015), which adopts LSTM (Hochreiter & Schmidhuber, 1997) to read an input sentence and CRF (Lafferty et al., 2001) to decode the label sequence. This type of method is very effective, providing tons of state-of-the-art performances. However, it is vulnerable to producing invalid labels, for instance, “O, I-tag, I-tag”. This problem is very severe in low resource settings (Peng et al., 2017). In experiments (see section 4.6), we also find that it performs poorly in recognizing long-length segments. ", + "bbox": [ + 174, + 575, + 825, + 702 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, there is a growing interest in span-based models (Zhai et al., 2017; Li et al., 2019; Yu et al., 2020). They treat a span rather than a token as the basic unit for labeling. Li et al. (2019) cast named entity recognition (NER) to a machine reading comprehension (MRC) task, where entities are extracted as retrieving answer spans. Yu et al. (2020) rank all the spans in terms of the scores predicted by a bi-affine model (Dozat & Manning, 2016). In NER, span-based models have significantly outperformed their sequence labeling based counterparts. While these methods circumvent the use of IOB tagging scheme, they still rely on post-processing rules to guarantee the extracted span set to be valid. Moreover, since span-based models are locally normalized at span level, they potentially suffer from the label bias problem (Lafferty et al., 2001). ", + "bbox": [ + 174, + 708, + 825, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper seeks to provide a new framework which infers the segmentation of a unit sequence by directly selecting from all valid segmentation candidates, instead of manipulating tokens or spans. To this end, we propose Lexical Unit Analysis (LUA) in this paper. LUA assigns a score to every valid segmentation candidate and leverages dynamic programming (DP) (Bellman, 1966) to search for the maximum scoring one. The score of a segmentation is computed by using the scores of its all segments. Besides, we adopt neural networks to score every segment of the input sentence. ", + "bbox": [ + 174, + 840, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/b34a0a4b29cfa62fc1064e82fa52cff24949a657d0e6f47cf445c7abc4ae1d90.jpg", + "image_caption": [ + "Figure 1: A toy example to show LUA and how it differs from prior methods. The items in blue and red respectively denote valid and invalid predictions. " + ], + "image_footnote": [], + "bbox": [ + 207, + 99, + 790, + 287 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The purpose of using DP is to solve the intractability of extracting the maximum scoring segmentation candidate by brute-force search. The time complexity of LUA is quadratic time, yet it can be optimized to linear time in practice by performing parallel matrix computations. For training criterion, we incur a hinge loss between the ground truth and the predictions. We also extend LUA to unlabeled segmentation and capturing label correlations. ", + "bbox": [ + 174, + 357, + 825, + 428 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Figure 1 illustrates the comparison between previous methods and the proposed LUA. Prior models at token level and span level are vulnerable to generating invalid predictions, and hence rely on heuristic rules to fix them. For example, in the middle part of Figure 1, the spans of two inferred named entities, [Word $\\mathrm { C u p } \\mathrm { _ { M I S C } }$ and $[ \\mathrm { C u p } ] _ { \\mathrm { M I S C } }$ , conflicts, which is mitigated by comparing the predicted scores. LUA scores all possible segmentation candidates and uses DP to extract the maximum scoring one. In this way, our models guarantee the predictions to be valid. Moreover, the globality of DP addresses the label bias problem. ", + "bbox": [ + 173, + 434, + 825, + 532 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Extensive experiments are conducted on syntactic chunking, NER, slot filling, Chinese word segmentation, and Chinese part-of-speech (POS) tagging across 15 tasks. We have obtained new stateof-the-art results on 13 of them and performed competitively on the others. In particular, we observe that LUA is expert at identifying long-length segments. ", + "bbox": [ + 174, + 539, + 825, + 594 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 614, + 341, + 631 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We denote an input sequence (i.e., fine-grained lexical units) as $\\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \\cdot \\cdot \\cdot , x _ { n } ]$ , where $n$ is the sequence length. An output sequence (i.e., coarse-grained lexical units) is represented as the segmentation $\\mathbf { y } = [ y _ { 1 } , y _ { 2 } , \\cdots , y _ { m } ]$ with each segment $y _ { k }$ being a triple $( i _ { k } , j _ { k } , t _ { k } )$ . $m$ denotes its length. $( i _ { k } , j _ { k } )$ specifies a span that corresponds to the phrase $\\mathbf { x } _ { i _ { k } , j _ { k } } = [ x _ { i _ { k } } , x _ { i _ { k } + 1 } , \\cdot \\cdot \\cdot , x _ { j _ { k } } ]$ . $t _ { k }$ is a label from the label space $\\mathcal { L }$ . We define a valid segmentation candidate as its segments are non-overlapping and fully cover the input sequence. ", + "bbox": [ + 173, + 646, + 826, + 731 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A case extracted from CoNLL-2003 dataset (Sang & De Meulder, 2003): ", + "bbox": [ + 176, + 737, + 653, + 752 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f88d55dbc3255902eb4f47a9e662f9a54902436565787429bd2615935f3acfc4.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { x } = [ [ \\mathrm { S O S } ] , \\mathrm { S a n g t h a i } , \\mathrm { G l o r y } , 2 2 / 1 1 / 9 6 , 3 0 0 0 , \\mathrm { S i n g a p o r e } ] } \\\\ { \\mathbf { y } = [ ( 1 , 1 , 0 ) , ( 2 , 3 , \\mathrm { M I S C } ) , ( 4 , 4 , 0 ) , ( 5 , 5 , 0 ) , ( 6 , 6 , \\mathrm { L O C } ) ] } \\end{array} .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 758, + 709, + 790 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Start-of-sentence symbol [SOS] is added in the pre-processing stage. ", + "bbox": [ + 176, + 796, + 622, + 813 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 MODEL: SCORING SEGMENTATION CANDIDATES ", + "text_level": 1, + "bbox": [ + 173, + 828, + 555, + 843 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We denote $\\mathcal { V }$ as the universal set that contains all valid segmentation candidates. Given one of its members $\\mathbf { y } \\in \\mathcal { V }$ , we compute the score $f ( \\mathbf { y } )$ as ", + "bbox": [ + 173, + 854, + 823, + 883 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/b4eeb1215c28a8c7d8a70ce88bcebb7c8716dddcb809969942d46b5d4d1b1a17.jpg", + "text": "$$\nf ( \\mathbf { y } ) = \\sum _ { ( i , j , t ) \\in \\mathbf { y } } \\Big ( s _ { i , j } ^ { c } + s _ { i , j , t } ^ { l } \\Big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 393, + 890, + 602, + 928 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Input: Composition score $s _ { i , j } ^ { c }$ and label score $s _ { i , j , t } ^ { l }$ for every possible segment $( i , j , t )$ . ", + "bbox": [ + 169, + 126, + 746, + 142 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "utput: The maximum segmentation scoring candidate $\\hat { \\mathbf { y } }$ and its score $f ( \\hat { \\mathbf { y } } )$ . ", + "bbox": [ + 187, + 143, + 714, + 156 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1 Set two $n \\times n$ shaped matrices, $\\mathbf { c } ^ { L }$ and ${ \\bf b } ^ { c }$ , for computing maximum scoring labels. ", + "bbox": [ + 160, + 157, + 725, + 170 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Set two $n$ -length vectors, $\\mathbf { g }$ and $\\mathbf { b } ^ { g }$ , for computing maximum scoring segmentation. ", + "bbox": [ + 160, + 171, + 718, + 185 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 Compute the maximum label score for each span $( i , j ) \\colon s _ { i , j } ^ { L } = \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i , j , t } ^ { l }$ \n5 Record the backtracking index: $b _ { i , j } ^ { c } = \\arg \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i , j , t } ^ { l }$ . ", + "bbox": [ + 165, + 199, + 704, + 233 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "6 Initialize the value of the base case $\\mathbf { x } _ { 1 , 1 } \\colon g _ { 1 } = s _ { 1 , 1 } ^ { c } + s _ { 1 , 1 } ^ { L }$ ", + "bbox": [ + 160, + 236, + 555, + 252 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "8 Compute the value of the prefix $\\begin{array} { r } { \\mathbf { x } _ { 1 , i } { \\mathrm { : ~ } } g _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\left( g _ { i - j } + \\left( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } \\right) \\right) } \\end{array}$ \n9 Record the backtracking index: $b _ { i } ^ { g } = \\arg \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\left( g _ { i - j } + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } ) \\right) .$ . \n10 Get the maximum scoring candidate $\\hat { \\mathbf { y } }$ by back tracing the tables $\\mathbf { b } ^ { g }$ and $\\mathbf { b } ^ { c }$ . \n11 Get the maximum segmentation score: $f ( \\hat { \\mathbf { y } } ) = g _ { n }$ . ", + "bbox": [ + 165, + 267, + 779, + 300 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 156, + 303, + 671, + 333 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $s _ { i , j } ^ { c }$ is the composition score to estimate the feasibility of merging several fine-grained units $[ x _ { i } , x _ { i + 1 } , \\cdot \\cdot \\cdot , x _ { j } ]$ into a coarse-grained unit and $s _ { i , j , t } ^ { l }$ is the label score to measure how likely the label of this segment is $t$ . Both scores are obtained by a scoring model. ", + "bbox": [ + 174, + 367, + 825, + 412 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Scoring Model. a scoring model scores all possible segments $( i , j , t )$ for an input sentence $\\mathbf { x }$ . Firstly, we get the representation for each fine-grained unit. Following prior works (Li et al., 2019; Luo et al., 2020; Yu et al., 2020), we adopt BERT (Devlin et al., 2018), a powerful pre-trained language model, as the sentence encoder. Specifically, we have ", + "bbox": [ + 173, + 429, + 825, + 486 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d9e94df8f089510bd74aa3185e0dcd1b7e94b88c40f2f670c76dd9fd397c10d4.jpg", + "text": "$$\n[ \\mathbf { h } _ { 1 } ^ { w } , \\mathbf { h } _ { 2 } ^ { w } \\cdot \\cdot \\cdot \\mathbf { \\epsilon } , \\mathbf { h } _ { n } ^ { w } ] = \\mathrm { B E R T } ( \\mathbf { x } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 392, + 493, + 604, + 511 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Then, we compute the representation for a coarse-grained unit $\\mathbf { x } _ { i , j } , 1 \\leq i \\leq j \\leq n$ as ", + "bbox": [ + 173, + 517, + 733, + 534 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/27e2d0c5384994b44168713ea50683326c64817a294acd44ec9c0adea92468ca.jpg", + "text": "$$\n\\mathbf { h } _ { i , j } ^ { p } = \\mathbf { h } _ { i } ^ { w } \\oplus \\mathbf { h } _ { j } ^ { w } \\oplus \\left( \\mathbf { h } _ { i } ^ { w } - \\mathbf { h } _ { j } ^ { w } \\right) \\oplus \\big ( \\mathbf { h } _ { i } ^ { w } \\odot \\mathbf { h } _ { j } ^ { w } \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 344, + 540, + 651, + 560 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\oplus$ is vector concatenation and $\\odot$ is element-wise product. ", + "bbox": [ + 171, + 565, + 594, + 580 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Eventually, we employ two non-linear feedforward networks to score a segment $( i , j , t )$ : ", + "bbox": [ + 179, + 587, + 751, + 602 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/bec80a12be79ba41c21d919bcb58d452ae9381b46291688e763333d89b2a2884.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { s } _ { i , j } ^ { c } = \\left( \\mathbf { v } ^ { c } \\right) ^ { T } \\operatorname { t a n h } ( \\mathbf { W } ^ { c } \\mathbf { h } _ { i , j } ^ { p } ) , \\boldsymbol { s } _ { i , j , t } ^ { l } = \\left( \\mathbf { v } _ { t } ^ { l } \\right) ^ { T } \\operatorname { t a n h } ( \\mathbf { W } ^ { l } \\mathbf { h } _ { i , j } ^ { p } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 609, + 700, + 633 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { v } _ { } ^ { c }$ , $\\mathbf { W } ^ { c }$ , $\\mathbf { v } _ { t } ^ { l } , t \\in \\mathcal { L }$ , and $\\mathbf { W } ^ { l }$ are all learnable parameters. Besides, the scoring model used here can be flexibly replaced by any regression method. ", + "bbox": [ + 173, + 642, + 825, + 671 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 INFERENCE VIA DYNAMIC PROGRAMMING ", + "text_level": 1, + "bbox": [ + 173, + 689, + 513, + 704 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The prediction of the maximum scoring segmentation candidate can be formulated as ", + "bbox": [ + 174, + 714, + 732, + 731 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/caa4c2a21b43a8e4714ab0d3f8d55af123600d3527111674492360b846f69e31.jpg", + "text": "$$\n{ \\hat { \\mathbf { y } } } = \\operatorname * { a r g m a x } _ { \\mathbf { y } \\in \\mathcal { Y } } f ( \\mathbf { y } ) .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 738, + 565, + 765 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Because the size of search space $| \\mathcal { V } |$ increases exponentially with respect to the sequence length $n$ , brute-force search to solve Equation 5 is computationally infeasible. LUA uses DP to address this issue, which is facilitated by the decomposable nature of Equation 1. ", + "bbox": [ + 174, + 773, + 823, + 816 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "DP is a well-known optimization method which solves a complicated problem by breaking it down into simpler sub-problems in a recursive manner. The relation between the value of the larger problem and the values of its sub-problems is called the Bellman equation. ", + "bbox": [ + 174, + 821, + 825, + 866 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Sub-problem. In the context of LUA, the sub-problem of segmenting an input unit sequence $\\mathbf { x }$ is segmenting its prefixes $\\mathbf { x } _ { 1 , i } , 1 \\leq i \\leq n$ . We define $g _ { i }$ as the maximum segmentation score of the prefix $\\mathbf { x } _ { 1 , i }$ . Under this scheme, we have $\\textstyle \\operatorname* { m a x } _ { \\mathbf { y } \\in { \\mathcal { y } } } f ( \\mathbf { y } ) = g _ { n }$ . ", + "bbox": [ + 174, + 881, + 823, + 926 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The Bellman Equation. The relatinship between segmenting a sequence $\\mathbf { x } _ { 1 , i } , i > 1$ and segmenting its prefixes $x _ { 1 , i - j } , 1 \\leq j \\leq i - 1$ is built by the last segments $( i - j + 1 , i , t )$ : ", + "bbox": [ + 171, + 103, + 821, + 133 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4ea631d18eb1fb1f52c566c91fa1d6d6420b272c87e744cc707644d49966264d.jpg", + "text": "$$\ng _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\big ( g _ { i - j } + \\big ( s _ { i - j + 1 , i } ^ { c } + \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i - j + 1 , i , t } ^ { l } \\big ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 137, + 673, + 165 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In practice, to reduce the time complexity of above equation, the last term is computed beforehand as $\\begin{array} { r } { \\dot { s } _ { i , j } ^ { L } = \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i , j , t } ^ { l } , 1 \\leq i \\leq j \\dot { \\leq } n } \\end{array}$ . Hence, Equation 6 is reformulated as ", + "bbox": [ + 173, + 169, + 828, + 199 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/34d8e85a05ba47dd37e6f441f4752efb18981451822aae6ba358f4e3517ee83e.jpg", + "text": "$$\ng _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\big ( g _ { i - j } + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 205, + 651, + 232 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The base case is the first token $\\mathbf { x } _ { 1 , 1 } = [ [ \\mathrm { S O S } ] ]$ . We get its score $g _ { 1 }$ as $s _ { 1 , 1 } ^ { c } + s _ { 1 , 1 } ^ { L }$ ", + "bbox": [ + 176, + 238, + 705, + 256 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 shows how DP is applied in inference. Firstly, we set two matrices and two vectors to store the solutions to the sub-problems (1-st to 2-nd lines). Secondly, we get the maximum label scores for all the spans (3-rd to 5-th lines). Then, we initialize the trivial case $g _ { 1 }$ and recursively calculate the values for prefixes $\\mathbf { x } _ { 1 , i } , i > 1$ (6-th to 9-th lines). Finally, we get the predicted segmentation $\\hat { \\mathbf { y } }$ and its score $f ( \\hat { \\mathbf { y } } )$ (10-th to 11-th lines). ", + "bbox": [ + 174, + 260, + 825, + 332 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The time complexity of Algorithm 1 is $\\mathcal { O } ( n ^ { 2 } )$ . By performing the max operation of Equation 7 in parallel on GPU, it can be optimized to only ${ \\mathcal { O } } ( n )$ , which is highly efficient. Besides, DP, as the backbone of the proposed model, is non-parametric. The trainable parameters only exist in the scoring model part. These show LUA is a very light-weight algorithm. ", + "bbox": [ + 174, + 335, + 825, + 393 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 TRAINING CRITERION ", + "text_level": 1, + "bbox": [ + 174, + 410, + 369, + 424 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We adopt max-margin penalty as the loss function for training. Given the predicted segmentation $\\hat { \\mathbf { y } }$ and the ground truth segmentation $\\mathbf { y } ^ { * }$ , we have ", + "bbox": [ + 173, + 435, + 821, + 464 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/457d011c2cb217443885823246c257c541dd779d0f588fa29aaf2bef5c6a620f.jpg", + "text": "$$\n\\mathcal { T } = \\operatorname* { m a x } \\big ( 0 , 1 - f ( \\mathbf { y } ^ { * } ) + f ( \\hat { \\mathbf { y } } ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 469, + 614, + 488 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 EXTENSIONS OF LUA ", + "text_level": 1, + "bbox": [ + 176, + 506, + 388, + 522 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We propose two extensions of LUA for generalizing it to different scenarios. ", + "bbox": [ + 173, + 536, + 674, + 553 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Unlabeled Segmentation. In some tasks (e.g., Chinese word segmentation), the segments are unlabeled. Under this scheme, the Equation 1 and Equation 7 are reformulated as ", + "bbox": [ + 171, + 565, + 823, + 595 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e031642fd8af3f46620e1ea6c4a50cf093b131fe03f0a02674e2595726f5da4a.jpg", + "text": "$$\nf ( \\mathbf { y } ) = \\sum _ { ( i , j ) \\in \\mathbf { y } } s _ { i , j } ^ { c } , ~ g _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } ( g _ { i - j } + s _ { i - j + 1 , i } ^ { c } ) .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 601, + 676, + 637 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Capturing Label Correlations. In some tasks (e.g., syntactic chunking), the labels of segments are strongly correlated. To incorporate this information, we redefine $f ( \\mathbf { y } )$ as ", + "bbox": [ + 173, + 650, + 825, + 679 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e0811db3c3ddd656234cd29dda83006120b3b4bd9af090d84908e46367e14c5f.jpg", + "text": "$$\nf ( { \\bf { y } } ) = \\sum _ { 1 \\le k \\le m } \\left( s _ { i _ { k } , j _ { k } } ^ { c } + s _ { i _ { k } , j _ { k } , t _ { k } } ^ { l } \\right) + \\sum _ { 1 \\le k \\le m } s _ { t _ { k - q + 1 } , t _ { k - q + 2 } , \\cdots , t _ { k } } ^ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 683, + 717, + 720 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Score In pra $s _ { t _ { k - q + 1 } , t _ { k - q + 2 } , \\cdots , t _ { k } } ^ { d }$ models the label dependencies among balances the efficiency and the effecti $q$ successive segments, ness well, and thus pa $_ { \\mathbf { y } _ { k - q + 1 , k } }$ $q = 2$ \na learnable matrix $\\mathbf { W } ^ { d } \\in \\mathbb { R } ^ { | \\nu | \\times | \\nu | }$ to implement it. ", + "bbox": [ + 173, + 727, + 825, + 773 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The corresponding Bellman equation to above scoring function is ", + "bbox": [ + 174, + 780, + 602, + 795 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9cae8d71dd0da00eaa135294c9af6a6cbb256d6c4f0214b771b20ea37e723112.jpg", + "text": "$$\ng _ { i , t } = \\underset { 1 \\leq j \\leq i - 1 } { \\operatorname* { m a x } } \\big ( \\underset { t ^ { \\prime } \\in \\mathcal { L } } { \\operatorname* { m a x } } ( g _ { i - j , t ^ { \\prime } } + s _ { t ^ { \\prime } , t } ^ { d } ) + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i , t } ^ { l } ) \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 284, + 799, + 712, + 825 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $g _ { i , t }$ is the maximum score of labeling the last segment of the prefix $\\mathbf { x } _ { 1 , i }$ with $t$ . For initialization, we set the value of $g _ { . 1 , \\mathrm { O } } ^ { d }$ as 0 and the others as $- \\infty$ . By performing the inner loops of two max operations in parallel, the practical time complexity for computing $g _ { i , t } , 1 \\leq i \\leq n , t \\in \\mathcal { L }$ is also ${ \\mathcal { O } } ( n )$ . Ultimately, the segmentation score $f ( \\hat { \\mathbf { y } } )$ is obtained by $\\operatorname* { m a x } _ { t \\in \\mathcal { L } } g _ { n , t }$ . ", + "bbox": [ + 174, + 830, + 825, + 890 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This extension further improves the results on syntactic chunking and Chinese POS tagging, as both tasks have rich sequential features among the labels of segments. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/01ef1500ea87c73cf80341c505cfc841fc0043a8912d34c933cff91bf688af24.jpg", + "table_caption": [ + "Table 1: Experiment results on Chinese word segmentation. " + ], + "table_footnote": [], + "table_body": "
ModelASMSRCITYUPKUCTB6
Rich Pretraining(Yang et al.,2017)95.797.596.996.396.2
Bi-LSTM(Ma et al.,2018)96.298.197.296.196.7
Multi-Criteria_Learning +_BERT_(Huang etal., 2019)96.697.997.696.697.6
BERT (Meng et al.,2019)96.598.197.696.5=
Glyce + BERT (Meng et al.,2019)96.798.397.996.7-
Unlabeled LUA96.9498.2798.2196.8898.13
", + "bbox": [ + 200, + 101, + 797, + 194 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/81b0f9609d55192d669a0737e0bf384714999e6f84337f7f6ad042fe99b86ced.jpg", + "table_caption": [ + "Table 2: Experiment results on the four datasets of Chinese POS tagging. " + ], + "table_footnote": [], + "table_body": "
ModelCTB5CTB6CTB9UD1
Bi-RNN+ CRF (Single) (Shao et al., 2017) Bi-RNN + CRF (Ensemble) (Shao et al.,2017)94.0790.8191.8989.41
Lattice-LSTM(Meng et al.,2019)94.38-92.3489.75
Glyce + Lattice-LSTM (Meng et al., 2019)95.1491.4392.1390.09
BERT (Meng et al.,2019)95.6191.9292.3890.87
Glyce +BERT (Meng et al., 2019)96.0694.7792.2994.79
96.6195.4193.1596.14
This WorkLUA96.7995.3993.2296.01
LUA w/Label Correlations97.9696.6393.9597.08
", + "bbox": [ + 232, + 229, + 766, + 348 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 395, + 328, + 411 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We have conducted extensive studies on 5 tasks, including Chinese word segmentation, Chinese POS tagging, syntactic chunking, NER, and slot filling, across 15 datasets. Firstly, Our models have achieved new state-of-the-art performances on 13 of them. Secondly, the results demonstrate that the F1 score of identifying long-length segments has been notably improved. Lastly, we show that LUA is a very efficient algorithm concerning the running time. ", + "bbox": [ + 173, + 425, + 825, + 496 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 SETTINGS ", + "text_level": 1, + "bbox": [ + 174, + 512, + 284, + 526 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use the same configurations for all 15 datasets. L2 regularization and dropout ratio are respectively set as $1 \\times 1 0 ^ { - 6 }$ and 0.2 for reducing overfit. We use Adam (Kingma & Ba, 2014) to optimize our model. Following prior works, BERTBASE is adopted as the sentence encoder. We use uncased BERTBASE for slot filling, Chinese BERTBASE for Chinese tasks (e.g., Chinese POS tagging), and cased BERTBASE for others (e.g., syntactic chunking). In addition, the improvements of our model over baselines are statistically significant with $p < 0 . 0 5$ under t-test. ", + "bbox": [ + 174, + 537, + 825, + 622 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 CHINESE WORD SEGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 638, + 442, + 654 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Chinese word segmentation splits a Chinese character sequence into a sequence of Chinese words. We use SIGHAN 2005 bake-off (Emerson, 2005) and Chinese Treebank 6.0 (CTB6) (Xue et al., 2005). SIGHAN 2005 back-off consists of 5 datasets, namely AS, MSR, CITYU, and PKU. Following Ma et al. (2018), we randomly select $1 0 \\%$ training data as development set. We convert all digits, punctuation, and Latin letters to half-width for handling full/half-width mismatch between training and test set. We also convert AS and CITYU to simplified Chinese. For CTB6, we follow the same format and partition as in Yang et al. (2017); Ma et al. (2018). ", + "bbox": [ + 174, + 665, + 825, + 762 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 1 depicts the experiment results. All the results of baselines are from Yang et al. (2017); Ma et al. (2018); Huang et al. (2019); Meng et al. (2019). We have achieved new state-of-the-art performance on all datasets except MSR. Our model improves the F1 score by $0 . 2 5 \\%$ on AS, $0 . 3 2 \\%$ on CITYU, $0 . 1 9 \\%$ on PKU, and $0 . 5 4 \\%$ on CTB6. Note that our model doesn’t use any external resources, such as glyph information (Meng et al., 2019) or POS tags (Yang et al., 2017). Despite this, our model is still competitive with Glyce $^ +$ BERT on MSR. ", + "bbox": [ + 174, + 768, + 825, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.3 CHINESE POS TAGGING ", + "text_level": 1, + "bbox": [ + 174, + 869, + 383, + 883 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Chinese POS tagging jointly segments a Chinese character sequence and assigns a POS tag to each segmented unit. We use Chinese Treebank 5.0 (CTB5), CTB6, Chinese Treebank 9.0 (CTB9) (Xue et al., 2005), and the Chinese section of Universal Dependencies 1.4 (UD1) (Nivre et al., 2016). CTB5 is comprised of newswire data. CTB9 consists of source texts in various genres, which cover CTB5. we convert the texts in UD1 from traditional Chinese into simplified Chinese. We follow the same train/dev/test split for above datasets as in Shao et al. (2017). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/046cdaf77be2ed2e24da69ff4d35d36af359dc163ceb6f8adf35a5565af6b32d.jpg", + "table_caption": [ + "Table 3: Experiment results on syntactic chunking and NER. " + ], + "table_footnote": [], + "table_body": "
ModelChunkingNER
CoNLL-2000CoNLL-2003OntoNotes5.0
Bi-LSTM + CRF (Huang et al., 2015)94.4690.101
Flair Embeddings (Akbik et al., 2018)96.7293.0989.3
GCDT w/BERT (Liu et al., 2019b)96.8193.231
BERT-MRC (Li et al., 2019)-93.0491.11
HCR w/BERT (Luo et al., 2020)=93.3790.30
BERT-Biaffine Model (Yu et al., 2020) LUA-93.591.3
This Work96.9593.4692.09
LUA w/Label Correlations97.23--
", + "bbox": [ + 214, + 101, + 784, + 232 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 281, + 823, + 338 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 2 shows the experiment results. The performances of all baselines are reported from Meng et al. (2019). Our model LUA w/ Label Correlations has yielded new state-of-the-art results on all the datasets: it improves the F1 scores by $1 . 3 5 \\%$ on CTB5, $1 . 2 2 \\%$ on CTB6, $0 . 8 \\%$ on CTB9, and $0 . 9 4 \\%$ on UD1. Moreover, the basic LUA without capturing the label correlations also outperforms the strongest baseline, Glyce $^ +$ BERT, by $0 . 1 8 \\%$ on CTB5 and $0 . 0 7 \\%$ on CTB9. All these facts further verify the effectiveness of LUA and its extension. ", + "bbox": [ + 173, + 344, + 825, + 429 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.4 SYNTACTIC CHUNKING AND NER ", + "text_level": 1, + "bbox": [ + 176, + 445, + 450, + 460 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Syntactic chunking aims to find phrases related to syntatic category for a sentence. We use CoNLL2000 dataset (Sang & Buchholz, 2000), which defines 11 syntactic chunk types (NP, VP, PP, etc.) and follow the standard splittings of training and test datasets as previous work. NER locates the named entities mentioned in unstructured text and meanwhile classifies them into predefined categories. We use CoNLL-2003 dataset (Sang & De Meulder, 2003) and OntoNotes 5.0 dataset (Pradhan et al., 2013). CoNLL-2003 dataset consists of 22137 sentences totally and is split into 14987, 3466, and 3684 sentences for the training set, development set, and test set, respectively. It is tagged with four linguistic entity types (PER, LOC, ORG, MISC). OntoNotes 5.0 dataset contains 76714 sentences from a wide variety of sources (e.g., magazine and newswire). It includes 18 types of named entity, which consists of 11 types (Person, Organization, etc.) and 7 values (Date, Percent, etc.). We follow the same format and partition as in Li et al. (2019); Luo et al. (2020); Yu et al. (2020). In order to fairly compare with previous reported results, we convert the predicted segments into IOB format and utilize conlleval script1 to compute the F1 score at test time. ", + "bbox": [ + 174, + 473, + 825, + 652 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 3 shows the results. Most of baselines are directly taken from Akbik et al. (2018); Li et al. (2019); Luo et al. (2020); Yu et al. (2020). Besides, following Luo et al. (2020), we rerun the source code2 of GCDT and report its result on CoNLL-2000 with standard evaluation method. Generally, our proposed models LUA w/o Label Correlations yield competitive performance over state-of-theart models on both Chunking and NER tasks. Specifically, regarding to the NER task, on CoNLL2003 dataset our model LUA outperforms several strong baselines including Flair Embedding, and it is comparable to the state-of-the-art model (i.e., BERT-Biaffine Model). In particular, on OntoNotes dataset, LUA outperforms it by $0 . 7 9 \\%$ points and establishes a new state-of-the-art result. Regarding to the Chunking task, LUA advances the best model (GCDT) and the improvements are further enlarged to $0 . 4 \\hat { 2 } \\%$ points by LUA w/ Label Correlations. ", + "bbox": [ + 173, + 660, + 825, + 799 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.5 SLOT FILLING ", + "text_level": 1, + "bbox": [ + 174, + 816, + 313, + 830 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Slot filling, as an important task in spoken language understanding (SLU), extracts semantic constituents from an utterance. We use ATIS dataset (Hemphill et al., 1990), SNIPS dataset (Coucke et al., 2018), and MTOD dataset (Schuster et al., 2018). ATIS dataset consists of audio recordings of ", + "bbox": [ + 178, + 843, + 825, + 885 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/b85c1a025f9b33999faf6903084d89eb9f902b629107d69eea5992dff1e779bb.jpg", + "table_caption": [ + "Table 4: Experiment results on the three datasets of slot filling. " + ], + "table_footnote": [], + "table_body": "
ModelATISSNIPSMTOD
Slot-Gated SLU (Goo et al.,2018) Bi-LSTM + EMLo (Siddhant et al., 2019)95.2088.3095.12
95.4293.90-
Joint BERT (Chen et al., 2019b) CM-Net (Liu et al.,2019c)96.10 96.2097.0096.48
96.1597.15-
This WorkLUA LUA w/ Intent Detection96.2797.1097.53
97.2097.55
", + "bbox": [ + 274, + 101, + 723, + 194 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/0d8a988459cd38745b0972009b489a7a6c36555d632195fbb5dfc48458492265.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Model1-3(8695)4-7(2380)8-11(151)12-24(31)Overall
HCRw/BERT91.1585.2250.4320.6790.27
BERT-Biaffine Model91.6787.2370.2440.5591.26
LUA92.3188.5277.3457.2792.09
", + "bbox": [ + 217, + 229, + 781, + 286 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 5: The F1 scores for NER models on different segment lengths. $A - B ( N )$ denotes that there are $N$ entities whose span lengths are between $A$ and $B$ . ", + "bbox": [ + 173, + 297, + 825, + 325 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "people making flight reservations. The training set contains 4478 utterances and the test set contains 893 utterances. SNIPS dataset is collected by Snips personal voice assistant. The training set contains 13084 utterances and the test set contains 700 utterances. MTOD dataset has three domains, including Alarm, Reminder, and Weather. We use the English part of MTOD dataset, where training set, dev set, and test set respectively contain 30521, 4181, and 8621 utterances. We follow the same partition of above datasets as in Goo et al. (2018); Schuster et al. (2018). ", + "bbox": [ + 174, + 352, + 825, + 435 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 4 summarizes the experiment results for slot filling. On ATIS and SNIPS, we take the results of all baselines as reported in Liu et al. (2019c) for comparison. On MTOD, we rerun the open source toolkits, Slot-gated $\\mathrm { S L U } ^ { 3 }$ and Joint BERT4. As all previous approaches jointly model slot filling and intent detection (a classification task in SLU), we follow them to augment LUA with intent detection for a fair comparison. As shown in Table 4, the augmented LUA has surpassed all baselines and obtained state-of-the-art results on the three datasets: it increases the F1 scores by around $0 . 0 5 \\%$ on ATIS and SNIPS, and delivers a substantial gain of $1 . 1 1 \\%$ on MTOD. It’s worth mentioning that LUA even outperforms the strong baseline Joint BERT with a margin of $0 . 1 8 \\%$ and $0 . 2 1 \\%$ on ATIS and SNIPS without modeling intent detection. ", + "bbox": [ + 174, + 443, + 825, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.6 LONG-LENGTH SEGMENT IDENTIFICATION", + "text_level": 1, + "bbox": [ + 174, + 585, + 511, + 599 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Since LUA doesn’t resort to IOB tagging scheme, it should be more accurate in recognizing longlength segments than prior methods. To verify this intuition, we evaluate different models on the segments of different lengths. This study is investigated on OntoNotes 5.0 dataset. Two strong models are adopted as the baselines: one is the best sequence labeling model (i.e., HCR) and the other is the best span-based model (i.e., BERT-Biaffine Model). Both baselines are reproduced by rerunning their open source codes, biaffine-ner5 and Hire-NER6. ", + "bbox": [ + 174, + 611, + 823, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The results are shown in Table 5. On the one hand, both LUA and Biaffine Model obtain much higher scores of extracting long-length entities than HCR. For example, LUA outperforms HCR w/ BERT by almost twofold on range $1 2 - 2 4$ . On the other hand, LUA achieves even better results than BERT-Biaffine Model. For instance, the F1 score improvements of LUA over it are $1 0 . 1 1 \\%$ on range $8 - 1 1$ and $4 1 . 2 3 \\%$ on range $1 2 - 2 4$ . ", + "bbox": [ + 174, + 702, + 825, + 772 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.7 RUNNING TIME ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 790, + 398, + 804 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 6 shows the running time comparison among different models. The middle two columns are the time complexity of decoding a label sequence. The last column is the time cost of one epoch in training. We set the batch size as 16 and run all the models on 1 GPU. The results indicate that ", + "bbox": [ + 176, + 815, + 823, + 857 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/3639b599177115723ea18d0f6cf9562ca42b4a8d5ff61ee459784a3e91af33fd.jpg", + "table_caption": [ + "Table 6: Running time comparison on the syntactic chunking dataset. " + ], + "table_footnote": [], + "table_body": "
ModelTheoretical ComplexityPractical ComplexityRunning Time
BERT BERT+CRFO(n) O(n|C|2)0(1)5m11s 7m33s
LUAO(n²)O(n) O(n)6m25s
LUA w/Label CorrelationsO(n²|C1²)0(n)7m09s
", + "bbox": [ + 186, + 101, + 812, + 176 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "the success of our models in performances does not lead to serious side-effects on efficiency. For example, with the same practical time complexity, BERT $^ +$ CRF is slower than the proposed LUA by $1 5 . 0 1 \\%$ and LUA w/ Label Correlations by $5 . \\dot { 3 } 0 \\%$ . ", + "bbox": [ + 176, + 233, + 825, + 275 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 303, + 343, + 319 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Sequence segmentation aims to partition a fine-grained unit sequence into multiple labeled coarsegrained units. Traditionally, there are two types of methods. The most common is to cast it into a sequence labeling task (Mesnil et al., 2014; Ma & Hovy, 2016; Chen et al., 2019a) by using IOB tagging scheme. This method is simple and effective, providing a number of state-of-the-art results. Akbik et al. (2018) present Flair Embeddings that pretrain character embedding in a large corpus and directly use it, instead of word representation, to encode a sentence. Liu et al. (2019b) introduce GCDT that deepens the state transition path at each position in a sentence, and further assigns each word with global representation. Luo et al. (2020) use hierarchical contextualized representations to incorporate both sentence-level and document-level information. Nevertheless, these models are vulnerable to producing invalid labels and perform poorly in identifying longlength segments. This problem is very severe in low-resource setting. Ye & Ling (2018); Liu et al. (2019a) adopt Semi-Markov CRF (Sarawagi & Cohen, 2005) that improves CRF at phrase level. However, the computation of CRF loss is costly in practice and the potential to model the label dependencies among segments is limited. An alternative approach that is less studied uses a transition-based system to incrementally segment and label an input sequence (Zhang et al., 2016; Lample et al., 2016). For instance, Qian et al. (2015) present a transition-based model for joint word segmentation, POS tagging, and text normalization. Wang et al. (2017) employ a transitionbased model to disfluency detection task, which helps capture non-local chunk-level features. These models have many advantages like theoretically lower time complexity and labeling the extracted mentions at span level. However, to our best knowledge, no recent transition-based models surpass their sequence labeling based counterparts. ", + "bbox": [ + 173, + 338, + 825, + 630 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "More recently, there is a surge of interests in span-based models. They treat a segment, instead of a fine-grained token, as the basic unit for labeling. For example, Li et al. (2019) regard NER as a MRC task, where entities are recognized as retrieving answer spans. Since these methods are locally normalized at span level rather than sequence level, they potentially suffer from the label bias problem. Additionally, they rely on rules to ensure the extracted span set to be valid. Spanbased methods also emerge in other fields of NLP. In dependency parsing, Wang & Chang (2016) propose a LSTM-based sentence segment embedding method named LSTM-Minus. Stern et al. (2017) integrate LSTM-minus feature into constituent parsing models. In coreference resolution, Lee et al. (2018) consider all spans in a document as the potential mentions and learn distributions over all the possible antecedents for each other. ", + "bbox": [ + 173, + 637, + 825, + 776 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 804, + 318, + 820 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "This work proposes a novel LUA for general sequence segmentation tasks. LUA directly scores all the valid segmentation candidates and uses dynamic programming to extract the maximum scoring one. Compared with previous models, LUA naturally guarantees the predicted segmentation to be valid and circumvents the label bias problem. Extensive studies are conducted on 5 tasks across 15 datasets. We have achieved the state-of-the-art performances on 13 of them. Importantly, the F1 score of identifying long-length segments is significantly improved. 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", + "bbox": [ + 174, + 536, + 825, + 579 + ], + "page_idx": 10 + } +] \ No newline at end of file diff --git a/parse/train/PQlC91XxqK5/PQlC91XxqK5_middle.json b/parse/train/PQlC91XxqK5/PQlC91XxqK5_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..2de5457def6d62cd5e2c0f0998fa663b79ac17b7 --- /dev/null +++ b/parse/train/PQlC91XxqK5/PQlC91XxqK5_middle.json @@ -0,0 +1,30295 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "score": 1.0, + "content": "SEGMENTING NATURAL LANGUAGE SENTENCES VIA", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 299, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 299, + 118 + ], + "score": 1.0, + "content": "LEXICAL UNIT ANALYSIS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 213, + 468, + 345 + ], + "lines": [ + { + "bbox": [ + 141, + 213, + 469, + 226 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 469, + 226 + ], + "score": 1.0, + "content": "In this work, we present Lexical Unit Analysis (LUA), a framework for general", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 224, + 470, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 224, + 470, + 236 + ], + "score": 1.0, + "content": "sequence segmentation tasks. Given a natural language sentence, LUA scores all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 235, + 469, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 469, + 247 + ], + "score": 1.0, + "content": "the valid segmentation candidates and utilizes dynamic programming (DP) to ex-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 246, + 469, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 469, + 258 + ], + "score": 1.0, + "content": "tract the maximum scoring one. LUA enjoys a number of appealing properties", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 257, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 469, + 268 + ], + "score": 1.0, + "content": "such as inherently guaranteeing the predicted segmentation to be valid and facil-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 470, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 470, + 280 + ], + "score": 1.0, + "content": "itating globally optimal training and inference. Besides, the practical time com-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "score": 1.0, + "content": "plexity of LUA can be reduced to linear time, which is very efficient. We have", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "score": 1.0, + "content": "conducted extensive experiments on 5 tasks, including syntactic chunking, named", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "score": 1.0, + "content": "entity recognition (NER), slot filling, Chinese word segmentation, and Chinese", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "score": 1.0, + "content": "part-of-speech (POS) tagging, across 15 datasets. Our models have achieved the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 469, + 334 + ], + "score": 1.0, + "content": "state-of-the-art performances on 13 of them. The results also show that the F1", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 395, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 395, + 346 + ], + "score": 1.0, + "content": "score of identifying long-length segments is notably improved.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 369, + 206, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 208, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 208, + 385 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "Sequence segmentation is essentially the process of partitioning a sequence of fine-grained lexical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "units into a sequence of coarse-grained ones. In some scenarios, each composed unit is assigned a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "score": 1.0, + "content": "categorical label. For example, Chinese word segmentation splits a character sequence into a word", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "sequence (Xue, 2003). Syntactic chunking segments a word sequence into a sequence of labeled", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 354, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 354, + 452 + ], + "score": 1.0, + "content": "groups of words (i.e., constituents) (Sang & Buchholz, 2000).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "There are currently two mainstream approaches to sequence segmentation. The most common is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "to regard it as a sequence labeling problem by using IOB tagging scheme (Mesnil et al., 2014; Ma", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "& Hovy, 2016; Liu et al., 2019b; Chen et al., 2019a; Luo et al., 2020). A representative work is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "score": 1.0, + "content": "Bidirectional LSTM-CRF (Huang et al., 2015), which adopts LSTM (Hochreiter & Schmidhuber,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "1997) to read an input sentence and CRF (Lafferty et al., 2001) to decode the label sequence. This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "type of method is very effective, providing tons of state-of-the-art performances. However, it is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "vulnerable to producing invalid labels, for instance, “O, I-tag, I-tag”. This problem is very severe", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "in low resource settings (Peng et al., 2017). In experiments (see section 4.6), we also find that it", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 324, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 324, + 558 + ], + "score": 1.0, + "content": "performs poorly in recognizing long-length segments.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "Recently, there is a growing interest in span-based models (Zhai et al., 2017; Li et al., 2019; Yu", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "et al., 2020). They treat a span rather than a token as the basic unit for labeling. Li et al. (2019) cast", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "named entity recognition (NER) to a machine reading comprehension (MRC) task, where entities", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "are extracted as retrieving answer spans. Yu et al. (2020) rank all the spans in terms of the scores", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "predicted by a bi-affine model (Dozat & Manning, 2016). In NER, span-based models have signif-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "icantly outperformed their sequence labeling based counterparts. While these methods circumvent", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "the use of IOB tagging scheme, they still rely on post-processing rules to guarantee the extracted", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "score": 1.0, + "content": "span set to be valid. Moreover, since span-based models are locally normalized at span level, they", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 380, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 380, + 661 + ], + "score": 1.0, + "content": "potentially suffer from the label bias problem (Lafferty et al., 2001).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "This paper seeks to provide a new framework which infers the segmentation of a unit sequence by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "directly selecting from all valid segmentation candidates, instead of manipulating tokens or spans.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "To this end, we propose Lexical Unit Analysis (LUA) in this paper. LUA assigns a score to every", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "valid segmentation candidate and leverages dynamic programming (DP) (Bellman, 1966) to search", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "for the maximum scoring one. The score of a segmentation is computed by using the scores of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "its all segments. Besides, we adopt neural networks to score every segment of the input sentence.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "score": 1.0, + "content": "SEGMENTING NATURAL LANGUAGE SENTENCES VIA", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 299, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 299, + 118 + ], + "score": 1.0, + "content": "LEXICAL UNIT ANALYSIS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 112, + 136, + 245, + 159 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 213, + 468, + 345 + ], + "lines": [ + { + "bbox": [ + 141, + 213, + 469, + 226 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 469, + 226 + ], + "score": 1.0, + "content": "In this work, we present Lexical Unit Analysis (LUA), a framework for general", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 224, + 470, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 224, + 470, + 236 + ], + "score": 1.0, + "content": "sequence segmentation tasks. Given a natural language sentence, LUA scores all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 235, + 469, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 469, + 247 + ], + "score": 1.0, + "content": "the valid segmentation candidates and utilizes dynamic programming (DP) to ex-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 246, + 469, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 469, + 258 + ], + "score": 1.0, + "content": "tract the maximum scoring one. LUA enjoys a number of appealing properties", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 257, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 469, + 268 + ], + "score": 1.0, + "content": "such as inherently guaranteeing the predicted segmentation to be valid and facil-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 470, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 470, + 280 + ], + "score": 1.0, + "content": "itating globally optimal training and inference. Besides, the practical time com-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "score": 1.0, + "content": "plexity of LUA can be reduced to linear time, which is very efficient. We have", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "score": 1.0, + "content": "conducted extensive experiments on 5 tasks, including syntactic chunking, named", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "score": 1.0, + "content": "entity recognition (NER), slot filling, Chinese word segmentation, and Chinese", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 469, + 324 + ], + "score": 1.0, + "content": "part-of-speech (POS) tagging, across 15 datasets. Our models have achieved the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 469, + 334 + ], + "score": 1.0, + "content": "state-of-the-art performances on 13 of them. The results also show that the F1", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 395, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 395, + 346 + ], + "score": 1.0, + "content": "score of identifying long-length segments is notably improved.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10.5, + "bbox_fs": [ + 141, + 213, + 470, + 346 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 369, + 206, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 208, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 208, + 385 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "Sequence segmentation is essentially the process of partitioning a sequence of fine-grained lexical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "units into a sequence of coarse-grained ones. In some scenarios, each composed unit is assigned a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "score": 1.0, + "content": "categorical label. For example, Chinese word segmentation splits a character sequence into a word", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "sequence (Xue, 2003). Syntactic chunking segments a word sequence into a sequence of labeled", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 354, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 354, + 452 + ], + "score": 1.0, + "content": "groups of words (i.e., constituents) (Sang & Buchholz, 2000).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 396, + 505, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "There are currently two mainstream approaches to sequence segmentation. The most common is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "to regard it as a sequence labeling problem by using IOB tagging scheme (Mesnil et al., 2014; Ma", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "& Hovy, 2016; Liu et al., 2019b; Chen et al., 2019a; Luo et al., 2020). A representative work is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "score": 1.0, + "content": "Bidirectional LSTM-CRF (Huang et al., 2015), which adopts LSTM (Hochreiter & Schmidhuber,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "1997) to read an input sentence and CRF (Lafferty et al., 2001) to decode the label sequence. This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "type of method is very effective, providing tons of state-of-the-art performances. However, it is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "vulnerable to producing invalid labels, for instance, “O, I-tag, I-tag”. This problem is very severe", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "in low resource settings (Peng et al., 2017). In experiments (see section 4.6), we also find that it", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 324, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 324, + 558 + ], + "score": 1.0, + "content": "performs poorly in recognizing long-length segments.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 456, + 506, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "Recently, there is a growing interest in span-based models (Zhai et al., 2017; Li et al., 2019; Yu", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "et al., 2020). They treat a span rather than a token as the basic unit for labeling. Li et al. (2019) cast", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "named entity recognition (NER) to a machine reading comprehension (MRC) task, where entities", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "are extracted as retrieving answer spans. Yu et al. (2020) rank all the spans in terms of the scores", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "predicted by a bi-affine model (Dozat & Manning, 2016). In NER, span-based models have signif-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "icantly outperformed their sequence labeling based counterparts. While these methods circumvent", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "the use of IOB tagging scheme, they still rely on post-processing rules to guarantee the extracted", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "score": 1.0, + "content": "span set to be valid. 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The items in blue and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 251, + 319, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 319, + 264 + ], + "score": 1.0, + "content": "red respectively denote valid and invalid predictions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 283, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 297 + ], + "score": 1.0, + "content": "The purpose of using DP is to solve the intractability of extracting the maximum scoring segmen-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "tation candidate by brute-force search. The time complexity of LUA is quadratic time, yet it can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 304, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 320 + ], + "score": 1.0, + "content": "be optimized to linear time in practice by performing parallel matrix computations. For training", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "criterion, we incur a hinge loss between the ground truth and the predictions. We also extend LUA", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 326, + 344, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 344, + 342 + ], + "score": 1.0, + "content": "to unlabeled segmentation and capturing label correlations.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "Figure 1 illustrates the comparison between previous methods and the proposed LUA. 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Besides, the scoring model used", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 519, + 330, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 330, + 533 + ], + "score": 1.0, + "content": "here can be flexibly replaced by any regression method.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 507, + 505, + 533 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 546, + 314, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 314, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 314, + 559 + ], + "score": 1.0, + "content": "2.2 INFERENCE VIA DYNAMIC PROGRAMMING", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 448, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 450, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 450, + 581 + ], + "score": 1.0, + "content": "The prediction of the maximum scoring segmentation candidate can be formulated as", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 566, + 450, + 581 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 585, + 346, + 606 + ], + "lines": [ + { + "bbox": [ + 264, + 585, + 346, + 606 + ], + "spans": [ + { + "bbox": [ + 264, + 585, + 346, + 606 + ], + "score": 0.93, + "content": "{ \\hat { \\mathbf { y } } } = \\operatorname * { a r g m a x } _ { \\mathbf { y } \\in \\mathcal { Y } } f ( \\mathbf { y } ) .", + "type": "interline_equation", + "image_path": "caa4c2a21b43a8e4714ab0d3f8d55af123600d3527111674492360b846f69e31.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 264, + 585, + 346, + 606 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 613, + 504, + 647 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 239, + 626 + ], + "score": 1.0, + "content": "Because the size of search space", + "type": "text" + }, + { + "bbox": [ + 239, + 613, + 254, + 625 + ], + "score": 0.89, + "content": "| \\mathcal { V } |", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 613, + 494, + 626 + ], + "score": 1.0, + "content": "increases exponentially with respect to the sequence length", + "type": "text" + }, + { + "bbox": [ + 494, + 616, + 501, + 623 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 613, + 505, + 626 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "brute-force search to solve Equation 5 is computationally infeasible. LUA uses DP to address this", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 635, + 383, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 383, + 647 + ], + "score": 1.0, + "content": "issue, which is facilitated by the decomposable nature of Equation 1.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 613, + 505, + 647 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 651, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 106, + 652, + 504, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 504, + 664 + ], + "score": 1.0, + "content": "DP is a well-known optimization method which solves a complicated problem by breaking it down", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 662, + 504, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 504, + 675 + ], + "score": 1.0, + "content": "into simpler sub-problems in a recursive manner. The relation between the value of the larger prob-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 674, + 388, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 388, + 686 + ], + "score": 1.0, + "content": "lem and the values of its sub-problems is called the Bellman equation.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 652, + 504, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 487, + 712 + ], + "score": 1.0, + "content": "Sub-problem. In the context of LUA, the sub-problem of segmenting an input unit sequence", + "type": "text" + }, + { + "bbox": [ + 487, + 701, + 495, + 709 + ], + "score": 0.48, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 201, + 723 + ], + "score": 1.0, + "content": "segmenting its prefixes", + "type": "text" + }, + { + "bbox": [ + 202, + 710, + 268, + 722 + ], + "score": 0.91, + "content": "\\mathbf { x } _ { 1 , i } , 1 \\leq i \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 709, + 317, + 723 + ], + "score": 1.0, + "content": ". We define", + "type": "text" + }, + { + "bbox": [ + 318, + 711, + 327, + 721 + ], + "score": 0.83, + "content": "g _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "as the maximum segmentation score of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 719, + 353, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 132, + 735 + ], + "score": 1.0, + "content": "prefix", + "type": "text" + }, + { + "bbox": [ + 132, + 722, + 149, + 733 + ], + "score": 0.89, + "content": "\\mathbf { x } _ { 1 , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 719, + 268, + 735 + ], + "score": 1.0, + "content": ". Under this scheme, we have", + "type": "text" + }, + { + "bbox": [ + 268, + 721, + 348, + 733 + ], + "score": 0.9, + "content": "\\textstyle \\operatorname* { m a x } _ { \\mathbf { y } \\in { \\mathcal { y } } } f ( \\mathbf { y } ) = g _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 719, + 353, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 698, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 503, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 405, + 96 + ], + "score": 1.0, + "content": "The Bellman Equation. The relatinship between segmenting a sequence", + "type": "text" + }, + { + "bbox": [ + 406, + 83, + 448, + 94 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 1 , i } , i > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "and segment-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 436, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 167, + 107 + ], + "score": 1.0, + "content": "ing its prefixes", + "type": "text" + }, + { + "bbox": [ + 168, + 94, + 255, + 106 + ], + "score": 0.9, + "content": "x _ { 1 , i - j } , 1 \\leq j \\leq i - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 93, + 369, + 107 + ], + "score": 1.0, + "content": "is built by the last segments", + "type": "text" + }, + { + "bbox": [ + 370, + 94, + 431, + 106 + ], + "score": 0.91, + "content": "( i - j + 1 , i , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 93, + 436, + 107 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 109, + 412, + 131 + ], + "lines": [ + { + "bbox": [ + 199, + 109, + 412, + 131 + ], + "spans": [ + { + "bbox": [ + 199, + 109, + 412, + 131 + ], + "score": 0.9, + "content": "g _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\big ( g _ { i - j } + \\big ( s _ { i - j + 1 , i } ^ { c } + \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i - j + 1 , i , t } ^ { l } \\big ) \\big ) .", + "type": "interline_equation", + "image_path": "4ea631d18eb1fb1f52c566c91fa1d6d6420b272c87e744cc707644d49966264d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 199, + 109, + 412, + 131 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 134, + 507, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 134, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 505, + 147 + ], + "score": 1.0, + "content": "In practice, to reduce the time complexity of above equation, the last term is computed beforehand", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 144, + 420, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 118, + 161 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 145, + 263, + 159 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\dot { s } _ { i , j } ^ { L } = \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i , j , t } ^ { l } , 1 \\leq i \\leq j \\dot { \\leq } n } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 144, + 420, + 161 + ], + "score": 1.0, + "content": ". Hence, Equation 6 is reformulated as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 163, + 399, + 184 + ], + "lines": [ + { + "bbox": [ + 212, + 163, + 399, + 184 + ], + "spans": [ + { + "bbox": [ + 212, + 163, + 399, + 184 + ], + "score": 0.93, + "content": "g _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\big ( g _ { i - j } + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } ) \\big ) .", + "type": "interline_equation", + "image_path": "34d8e85a05ba47dd37e6f441f4752efb18981451822aae6ba358f4e3517ee83e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 212, + 163, + 399, + 184 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 189, + 432, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 428, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 231, + 206 + ], + "score": 1.0, + "content": "The base case is the first token", + "type": "text" + }, + { + "bbox": [ + 231, + 189, + 291, + 202 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { 1 , 1 } = [ [ \\mathrm { S O S } ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 185, + 361, + 206 + ], + "score": 1.0, + "content": ". We get its score", + "type": "text" + }, + { + "bbox": [ + 362, + 191, + 372, + 201 + ], + "score": 0.83, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 185, + 384, + 206 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 385, + 189, + 428, + 203 + ], + "score": 0.92, + "content": "s _ { 1 , 1 } ^ { c } + s _ { 1 , 1 } ^ { L }", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 206, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Algorithm 1 shows how DP is applied in inference. Firstly, we set two matrices and two vectors to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "store the solutions to the sub-problems (1-st to 2-nd lines). Secondly, we get the maximum label", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 504, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 428, + 241 + ], + "score": 1.0, + "content": "scores for all the spans (3-rd to 5-th lines). Then, we initialize the trivial case", + "type": "text" + }, + { + "bbox": [ + 428, + 230, + 439, + 240 + ], + "score": 0.84, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 229, + 504, + 241 + ], + "score": 1.0, + "content": "and recursively", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 237, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 237, + 254 + ], + "score": 1.0, + "content": "calculate the values for prefixes", + "type": "text" + }, + { + "bbox": [ + 238, + 240, + 284, + 252 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 1 , i } , i > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 237, + 505, + 254 + ], + "score": 1.0, + "content": "(6-th to 9-th lines). Finally, we get the predicted seg-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 251, + 320, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 148, + 263 + ], + "score": 1.0, + "content": "mentation", + "type": "text" + }, + { + "bbox": [ + 149, + 251, + 156, + 262 + ], + "score": 0.84, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 251, + 210, + 263 + ], + "score": 1.0, + "content": "and its score", + "type": "text" + }, + { + "bbox": [ + 210, + 251, + 231, + 263 + ], + "score": 0.91, + "content": "f ( \\hat { \\mathbf { y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 251, + 320, + 263 + ], + "score": 1.0, + "content": "(10-th to 11-th lines).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 270, + 280 + ], + "score": 1.0, + "content": "The time complexity of Algorithm 1 is", + "type": "text" + }, + { + "bbox": [ + 270, + 267, + 297, + 279 + ], + "score": 0.92, + "content": "\\mathcal { O } ( n ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 267, + 506, + 280 + ], + "score": 1.0, + "content": ". By performing the max operation of Equation 7", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 304, + 291 + ], + "score": 1.0, + "content": "in parallel on GPU, it can be optimized to only", + "type": "text" + }, + { + "bbox": [ + 304, + 279, + 327, + 290 + ], + "score": 0.91, + "content": "{ \\mathcal { O } } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 278, + 505, + 291 + ], + "score": 1.0, + "content": ", which is highly efficient. Besides, DP, as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "the backbone of the proposed model, is non-parametric. The trainable parameters only exist in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 300, + 389, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 389, + 313 + ], + "score": 1.0, + "content": "scoring model part. These show LUA is a very light-weight algorithm.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 325, + 226, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 227, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 227, + 338 + ], + "score": 1.0, + "content": "2.3 TRAINING CRITERION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 503, + 368 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 504, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 496, + 359 + ], + "score": 1.0, + "content": "We adopt max-margin penalty as the loss function for training. Given the predicted segmentation", + "type": "text" + }, + { + "bbox": [ + 496, + 346, + 504, + 357 + ], + "score": 0.7, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 357, + 297, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 245, + 370 + ], + "score": 1.0, + "content": "and the ground truth segmentation", + "type": "text" + }, + { + "bbox": [ + 245, + 357, + 257, + 368 + ], + "score": 0.88, + "content": "\\mathbf { y } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 357, + 297, + 370 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 372, + 376, + 387 + ], + "lines": [ + { + "bbox": [ + 235, + 372, + 376, + 387 + ], + "spans": [ + { + "bbox": [ + 235, + 372, + 376, + 387 + ], + "score": 0.92, + "content": "\\mathcal { T } = \\operatorname* { m a x } \\big ( 0 , 1 - f ( \\mathbf { y } ^ { * } ) + f ( \\hat { \\mathbf { y } } ) \\big ) .", + "type": "interline_equation", + "image_path": "457d011c2cb217443885823246c257c541dd779d0f588fa29aaf2bef5c6a620f.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 235, + 372, + 376, + 387 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 238, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 400, + 239, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 239, + 416 + ], + "score": 1.0, + "content": "3 EXTENSIONS OF LUA", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 413, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 414, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 414, + 439 + ], + "score": 1.0, + "content": "We propose two extensions of LUA for generalizing it to different scenarios.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 105, + 448, + 504, + 472 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "Unlabeled Segmentation. In some tasks (e.g., Chinese word segmentation), the segments are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 433, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 433, + 473 + ], + "score": 1.0, + "content": "unlabeled. Under this scheme, the Equation 1 and Equation 7 are reformulated as", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 476, + 414, + 505 + ], + "lines": [ + { + "bbox": [ + 196, + 476, + 414, + 505 + ], + "spans": [ + { + "bbox": [ + 196, + 476, + 414, + 505 + ], + "score": 0.93, + "content": "f ( \\mathbf { y } ) = \\sum _ { ( i , j ) \\in \\mathbf { y } } s _ { i , j } ^ { c } , ~ g _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } ( g _ { i - j } + s _ { i - j + 1 , i } ^ { c } ) .", + "type": "interline_equation", + "image_path": "e031642fd8af3f46620e1ea6c4a50cf093b131fe03f0a02674e2595726f5da4a.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 476, + 414, + 490.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 196, + 490.5, + 414, + 505.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 528 + ], + "score": 1.0, + "content": "Capturing Label Correlations. In some tasks (e.g., syntactic chunking), the labels of segments", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 526, + 414, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 380, + 539 + ], + "score": 1.0, + "content": "are strongly correlated. To incorporate this information, we redefine", + "type": "text" + }, + { + "bbox": [ + 380, + 526, + 401, + 538 + ], + "score": 0.92, + "content": "f ( \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 526, + 414, + 539 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 541, + 439, + 571 + ], + "lines": [ + { + "bbox": [ + 172, + 541, + 439, + 571 + ], + "spans": [ + { + "bbox": [ + 172, + 541, + 439, + 571 + ], + "score": 0.94, + "content": "f ( { \\bf { y } } ) = \\sum _ { 1 \\le k \\le m } \\left( s _ { i _ { k } , j _ { k } } ^ { c } + s _ { i _ { k } , j _ { k } , t _ { k } } ^ { l } \\right) + \\sum _ { 1 \\le k \\le m } s _ { t _ { k - q + 1 } , t _ { k - q + 2 } , \\cdots , t _ { k } } ^ { d } .", + "type": "interline_equation", + "image_path": "e0811db3c3ddd656234cd29dda83006120b3b4bd9af090d84908e46367e14c5f.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 172, + 541, + 439, + 551.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 172, + 551.0, + 439, + 561.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 172, + 561.0, + 439, + 571.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 100, + 568, + 501, + 602 + ], + "spans": [ + { + "bbox": [ + 100, + 568, + 132, + 602 + ], + "score": 1.0, + "content": "Score In pra", + "type": "text" + }, + { + "bbox": [ + 132, + 575, + 210, + 590 + ], + "score": 0.91, + "content": "s _ { t _ { k - q + 1 } , t _ { k - q + 2 } , \\cdots , t _ { k } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 568, + 367, + 602 + ], + "score": 1.0, + "content": "models the label dependencies among balances the efficiency and the effecti", + "type": "text" + }, + { + "bbox": [ + 368, + 579, + 374, + 588 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 568, + 462, + 602 + ], + "score": 1.0, + "content": "successive segments, ness well, and thus pa", + "type": "text" + }, + { + "bbox": [ + 462, + 579, + 501, + 589 + ], + "score": 0.89, + "content": "_ { \\mathbf { y } _ { k - q + 1 , k } }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 187, + 590, + 212, + 600 + ], + "spans": [ + { + "bbox": [ + 187, + 590, + 212, + 600 + ], + "score": 0.86, + "content": "q = 2", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 599, + 313, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 181, + 614 + ], + "score": 1.0, + "content": "a learnable matrix", + "type": "text" + }, + { + "bbox": [ + 181, + 600, + 245, + 611 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { d } \\in \\mathbb { R } ^ { | \\nu | \\times | \\nu | }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 599, + 313, + 614 + ], + "score": 1.0, + "content": "to implement it.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 369, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 371, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 371, + 632 + ], + "score": 1.0, + "content": "The corresponding Bellman equation to above scoring function is", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 633, + 436, + 654 + ], + "lines": [ + { + "bbox": [ + 174, + 633, + 436, + 654 + ], + "spans": [ + { + "bbox": [ + 174, + 633, + 436, + 654 + ], + "score": 0.92, + "content": "g _ { i , t } = \\underset { 1 \\leq j \\leq i - 1 } { \\operatorname* { m a x } } \\big ( \\underset { t ^ { \\prime } \\in \\mathcal { L } } { \\operatorname* { m a x } } ( g _ { i - j , t ^ { \\prime } } + s _ { t ^ { \\prime } , t } ^ { d } ) + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i , t } ^ { l } ) \\big ) ,", + "type": "interline_equation", + "image_path": "9cae8d71dd0da00eaa135294c9af6a6cbb256d6c4f0214b771b20ea37e723112.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 174, + 633, + 436, + 654 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 133, + 670 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 660, + 149, + 671 + ], + "score": 0.87, + "content": "g _ { i , t }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 658, + 409, + 670 + ], + "score": 1.0, + "content": "is the maximum score of labeling the last segment of the prefix", + "type": "text" + }, + { + "bbox": [ + 409, + 660, + 426, + 671 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { 1 , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 658, + 448, + 670 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 449, + 660, + 453, + 668 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 658, + 505, + 670 + ], + "score": 1.0, + "content": ". For initial-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 670, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 218, + 685 + ], + "score": 1.0, + "content": "ization, we set the value of", + "type": "text" + }, + { + "bbox": [ + 218, + 670, + 236, + 685 + ], + "score": 0.92, + "content": "g _ { . 1 , \\mathrm { O } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 670, + 327, + 685 + ], + "score": 1.0, + "content": "as 0 and the others as", + "type": "text" + }, + { + "bbox": [ + 328, + 672, + 347, + 681 + ], + "score": 0.85, + "content": "- \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 670, + 506, + 685 + ], + "score": 1.0, + "content": ". By performing the inner loops of two", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 400, + 695 + ], + "score": 1.0, + "content": "max operations in parallel, the practical time complexity for computing", + "type": "text" + }, + { + "bbox": [ + 401, + 682, + 494, + 694 + ], + "score": 0.91, + "content": "g _ { i , t } , 1 \\leq i \\leq n , t \\in \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 691, + 430, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 125, + 708 + ], + "score": 1.0, + "content": "also", + "type": "text" + }, + { + "bbox": [ + 126, + 693, + 148, + 705 + ], + "score": 0.92, + "content": "{ \\mathcal { O } } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 691, + 293, + 708 + ], + "score": 1.0, + "content": ". Ultimately, the segmentation score", + "type": "text" + }, + { + "bbox": [ + 294, + 693, + 314, + 705 + ], + "score": 0.92, + "content": "f ( \\hat { \\mathbf { y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 691, + 374, + 708 + ], + "score": 1.0, + "content": "is obtained by", + "type": "text" + }, + { + "bbox": [ + 374, + 694, + 425, + 705 + ], + "score": 0.84, + "content": "\\operatorname* { m a x } _ { t \\in \\mathcal { L } } g _ { n , t }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 691, + 430, + 708 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "This extension further improves the results on syntactic chunking and Chinese POS tagging, as both", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 368, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 368, + 733 + ], + "score": 1.0, + "content": "tasks have rich sequential features among the labels of segments.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 307, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 307, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 307, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 503, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 405, + 96 + ], + "score": 1.0, + "content": "The Bellman Equation. The relatinship between segmenting a sequence", + "type": "text" + }, + { + "bbox": [ + 406, + 83, + 448, + 94 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 1 , i } , i > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "and segment-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 436, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 167, + 107 + ], + "score": 1.0, + "content": "ing its prefixes", + "type": "text" + }, + { + "bbox": [ + 168, + 94, + 255, + 106 + ], + "score": 0.9, + "content": "x _ { 1 , i - j } , 1 \\leq j \\leq i - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 93, + 369, + 107 + ], + "score": 1.0, + "content": "is built by the last segments", + "type": "text" + }, + { + "bbox": [ + 370, + 94, + 431, + 106 + ], + "score": 0.91, + "content": "( i - j + 1 , i , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 93, + 436, + 107 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 107 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 109, + 412, + 131 + ], + "lines": [ + { + "bbox": [ + 199, + 109, + 412, + 131 + ], + "spans": [ + { + "bbox": [ + 199, + 109, + 412, + 131 + ], + "score": 0.9, + "content": "g _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\big ( g _ { i - j } + \\big ( s _ { i - j + 1 , i } ^ { c } + \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i - j + 1 , i , t } ^ { l } \\big ) \\big ) .", + "type": "interline_equation", + "image_path": "4ea631d18eb1fb1f52c566c91fa1d6d6420b272c87e744cc707644d49966264d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 199, + 109, + 412, + 131 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 134, + 507, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 134, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 505, + 147 + ], + "score": 1.0, + "content": "In practice, to reduce the time complexity of above equation, the last term is computed beforehand", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 144, + 420, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 118, + 161 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 145, + 263, + 159 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\dot { s } _ { i , j } ^ { L } = \\operatorname* { m a x } _ { t \\in \\mathcal { L } } s _ { i , j , t } ^ { l } , 1 \\leq i \\leq j \\dot { \\leq } n } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 144, + 420, + 161 + ], + "score": 1.0, + "content": ". Hence, Equation 6 is reformulated as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 134, + 505, + 161 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 163, + 399, + 184 + ], + "lines": [ + { + "bbox": [ + 212, + 163, + 399, + 184 + ], + "spans": [ + { + "bbox": [ + 212, + 163, + 399, + 184 + ], + "score": 0.93, + "content": "g _ { i } = \\operatorname* { m a x } _ { 1 \\leq j \\leq i - 1 } \\big ( g _ { i - j } + ( s _ { i - j + 1 , i } ^ { c } + s _ { i - j + 1 , i } ^ { L } ) \\big ) .", + "type": "interline_equation", + "image_path": "34d8e85a05ba47dd37e6f441f4752efb18981451822aae6ba358f4e3517ee83e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 212, + 163, + 399, + 184 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 189, + 432, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 428, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 231, + 206 + ], + "score": 1.0, + "content": "The base case is the first token", + "type": "text" + }, + { + "bbox": [ + 231, + 189, + 291, + 202 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { 1 , 1 } = [ [ \\mathrm { S O S } ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 185, + 361, + 206 + ], + "score": 1.0, + "content": ". We get its score", + "type": "text" + }, + { + "bbox": [ + 362, + 191, + 372, + 201 + ], + "score": 0.83, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 185, + 384, + 206 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 385, + 189, + 428, + 203 + ], + "score": 0.92, + "content": "s _ { 1 , 1 } ^ { c } + s _ { 1 , 1 } ^ { L }", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 185, + 428, + 206 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 206, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Algorithm 1 shows how DP is applied in inference. Firstly, we set two matrices and two vectors to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "store the solutions to the sub-problems (1-st to 2-nd lines). Secondly, we get the maximum label", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 504, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 428, + 241 + ], + "score": 1.0, + "content": "scores for all the spans (3-rd to 5-th lines). Then, we initialize the trivial case", + "type": "text" + }, + { + "bbox": [ + 428, + 230, + 439, + 240 + ], + "score": 0.84, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 229, + 504, + 241 + ], + "score": 1.0, + "content": "and recursively", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 237, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 237, + 254 + ], + "score": 1.0, + "content": "calculate the values for prefixes", + "type": "text" + }, + { + "bbox": [ + 238, + 240, + 284, + 252 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 1 , i } , i > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 237, + 505, + 254 + ], + "score": 1.0, + "content": "(6-th to 9-th lines). Finally, we get the predicted seg-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 251, + 320, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 148, + 263 + ], + "score": 1.0, + "content": "mentation", + "type": "text" + }, + { + "bbox": [ + 149, + 251, + 156, + 262 + ], + "score": 0.84, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 251, + 210, + 263 + ], + "score": 1.0, + "content": "and its score", + "type": "text" + }, + { + "bbox": [ + 210, + 251, + 231, + 263 + ], + "score": 0.91, + "content": "f ( \\hat { \\mathbf { y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 251, + 320, + 263 + ], + "score": 1.0, + "content": "(10-th to 11-th lines).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 207, + 505, + 263 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 270, + 280 + ], + "score": 1.0, + "content": "The time complexity of Algorithm 1 is", + "type": "text" + }, + { + "bbox": [ + 270, + 267, + 297, + 279 + ], + "score": 0.92, + "content": "\\mathcal { O } ( n ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 267, + 506, + 280 + ], + "score": 1.0, + "content": ". By performing the max operation of Equation 7", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 304, + 291 + ], + "score": 1.0, + "content": "in parallel on GPU, it can be optimized to only", + "type": "text" + }, + { + "bbox": [ + 304, + 279, + 327, + 290 + ], + "score": 0.91, + "content": "{ \\mathcal { O } } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 278, + 505, + 291 + ], + "score": 1.0, + "content": ", which is highly efficient. Besides, DP, as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "the backbone of the proposed model, is non-parametric. The trainable parameters only exist in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 300, + 389, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 389, + 313 + ], + "score": 1.0, + "content": "scoring model part. 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In some tasks (e.g., Chinese word segmentation), the segments are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 433, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 433, + 473 + ], + "score": 1.0, + "content": "unlabeled. 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We use uncased", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "BERTBASE for slot filling, Chinese BERTBASE for Chinese tasks (e.g., Chinese POS tagging), and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "cased BERTBASE for others (e.g., syntactic chunking). In addition, the improvements of our model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 482, + 380, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 292, + 494 + ], + "score": 1.0, + "content": "over baselines are statistically significant with", + "type": "text" + }, + { + "bbox": [ + 292, + 482, + 330, + 493 + ], + "score": 0.9, + "content": "p < 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 482, + 380, + 494 + ], + "score": 1.0, + "content": "under t-test.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 271, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 273, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 273, + 519 + ], + "score": 1.0, + "content": "4.2 CHINESE WORD SEGMENTATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "Chinese word segmentation splits a Chinese character sequence into a sequence of Chinese words.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "score": 1.0, + "content": "We use SIGHAN 2005 bake-off (Emerson, 2005) and Chinese Treebank 6.0 (CTB6) (Xue et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "2005). 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Our model improves the F1 score by", + "type": "text" + }, + { + "bbox": [ + 419, + 631, + 446, + 642 + ], + "score": 0.87, + "content": "0 . 2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 631, + 476, + 644 + ], + "score": 1.0, + "content": "on AS,", + "type": "text" + }, + { + "bbox": [ + 477, + 631, + 504, + 642 + ], + "score": 0.86, + "content": "0 . 3 2 \\%", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 156, + 655 + ], + "score": 1.0, + "content": "on CITYU,", + "type": "text" + }, + { + "bbox": [ + 156, + 642, + 184, + 654 + ], + "score": 0.93, + "content": "0 . 1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 642, + 242, + 655 + ], + "score": 1.0, + "content": "on PKU, and", + "type": "text" + }, + { + "bbox": [ + 243, + 642, + 270, + 653 + ], + "score": 0.89, + "content": "0 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "on CTB6. 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ModelCTB5CTB6CTB9UD1
Bi-RNN+ CRF (Single) (Shao et al., 2017) Bi-RNN + CRF (Ensemble) (Shao et al.,2017)94.0790.8191.8989.41
Lattice-LSTM(Meng et al.,2019)94.38-92.3489.75
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BERT (Meng et al.,2019)95.6191.9292.3890.87
Glyce +BERT (Meng et al., 2019)96.0694.7792.2994.79
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In addition, the improvements of our model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 482, + 380, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 292, + 494 + ], + "score": 1.0, + "content": "over baselines are statistically significant with", + "type": "text" + }, + { + "bbox": [ + 292, + 482, + 330, + 493 + ], + "score": 0.9, + "content": "p < 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 482, + 380, + 494 + ], + "score": 1.0, + "content": "under t-test.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 426, + 506, + 494 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 271, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 273, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 273, + 519 + ], + "score": 1.0, + "content": "4.2 CHINESE WORD SEGMENTATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "Chinese word segmentation splits a Chinese character sequence into a sequence of Chinese words.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 551 + ], + "score": 1.0, + "content": "We use SIGHAN 2005 bake-off (Emerson, 2005) and Chinese Treebank 6.0 (CTB6) (Xue et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "2005). SIGHAN 2005 back-off consists of 5 datasets, namely AS, MSR, CITYU, and PKU. Fol-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 288, + 572 + ], + "score": 1.0, + "content": "lowing Ma et al. (2018), we randomly select", + "type": "text" + }, + { + "bbox": [ + 288, + 560, + 308, + 570 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "training data as development set. We convert all", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "digits, punctuation, and Latin letters to half-width for handling full/half-width mismatch between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "training and test set. We also convert AS and CITYU to simplified Chinese. For CTB6, we follow", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 593, + 393, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 393, + 605 + ], + "score": 1.0, + "content": "the same format and partition as in Yang et al. (2017); Ma et al. (2018).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 527, + 505, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "Table 1 depicts the experiment results. All the results of baselines are from Yang et al. (2017);", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "Ma et al. (2018); Huang et al. (2019); Meng et al. (2019). We have achieved new state-of-the-art", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 631, + 504, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 418, + 644 + ], + "score": 1.0, + "content": "performance on all datasets except MSR. Our model improves the F1 score by", + "type": "text" + }, + { + "bbox": [ + 419, + 631, + 446, + 642 + ], + "score": 0.87, + "content": "0 . 2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 631, + 476, + 644 + ], + "score": 1.0, + "content": "on AS,", + "type": "text" + }, + { + "bbox": [ + 477, + 631, + 504, + 642 + ], + "score": 0.86, + "content": "0 . 3 2 \\%", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 156, + 655 + ], + "score": 1.0, + "content": "on CITYU,", + "type": "text" + }, + { + "bbox": [ + 156, + 642, + 184, + 654 + ], + "score": 0.93, + "content": "0 . 1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 642, + 242, + 655 + ], + "score": 1.0, + "content": "on PKU, and", + "type": "text" + }, + { + "bbox": [ + 243, + 642, + 270, + 653 + ], + "score": 0.89, + "content": "0 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "on CTB6. Note that our model doesn’t use any external", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 654, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 504, + 666 + ], + "score": 1.0, + "content": "resources, such as glyph information (Meng et al., 2019) or POS tags (Yang et al., 2017). Despite", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 664, + 365, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 290, + 676 + ], + "score": 1.0, + "content": "this, our model is still competitive with Glyce", + "type": "text" + }, + { + "bbox": [ + 291, + 665, + 299, + 675 + ], + "score": 0.76, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 664, + 365, + 676 + ], + "score": 1.0, + "content": "BERT on MSR.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 609, + 506, + 676 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 689, + 235, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 236, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 236, + 702 + ], + "score": 1.0, + "content": "4.3 CHINESE POS TAGGING", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Chinese POS tagging jointly segments a Chinese character sequence and assigns a POS tag to each", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "score": 1.0, + "content": "segmented unit. We use Chinese Treebank 5.0 (CTB5), CTB6, Chinese Treebank 9.0 (CTB9) (Xue", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "et al., 2005), and the Chinese section of Universal Dependencies 1.4 (UD1) (Nivre et al., 2016).", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "CTB5 is comprised of newswire data. CTB9 consists of source texts in various genres, which cover", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "CTB5. we convert the texts in UD1 from traditional Chinese into simplified Chinese. We follow the", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 257, + 374, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 374, + 269 + ], + "score": 1.0, + "content": "same train/dev/test split for above datasets as in Shao et al. (2017).", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 131, + 80, + 480, + 184 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 131, + 80, + 480, + 184 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 131, + 80, + 480, + 184 + ], + "spans": [ + { + "bbox": [ + 131, + 80, + 480, + 184 + ], + "score": 0.981, + "html": "
ModelChunkingNER
CoNLL-2000CoNLL-2003OntoNotes5.0
Bi-LSTM + CRF (Huang et al., 2015)94.4690.101
Flair Embeddings (Akbik et al., 2018)96.7293.0989.3
GCDT w/BERT (Liu et al., 2019b)96.8193.231
BERT-MRC (Li et al., 2019)-93.0491.11
HCR w/BERT (Luo et al., 2020)=93.3790.30
BERT-Biaffine Model (Yu et al., 2020) LUA-93.591.3
This Work96.9593.4692.09
LUA w/Label Correlations97.23--
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CTB9 consists of source texts in various genres, which cover", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "CTB5. we convert the texts in UD1 from traditional Chinese into simplified Chinese. We follow the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 257, + 374, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 374, + 269 + ], + "score": 1.0, + "content": "same train/dev/test split for above datasets as in Shao et al. (2017).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "score": 1.0, + "content": "Table 2 shows the experiment results. The performances of all baselines are reported from Meng", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "et al. (2019). Our model LUA w/ Label Correlations has yielded new state-of-the-art results on all", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 277, + 308 + ], + "score": 1.0, + "content": "the datasets: it improves the F1 scores by", + "type": "text" + }, + { + "bbox": [ + 277, + 295, + 305, + 307 + ], + "score": 0.87, + "content": "1 . 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 295, + 349, + 308 + ], + "score": 1.0, + "content": "on CTB5,", + "type": "text" + }, + { + "bbox": [ + 349, + 295, + 377, + 306 + ], + "score": 0.89, + "content": "1 . 2 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 295, + 421, + 308 + ], + "score": 1.0, + "content": "on CTB6,", + "type": "text" + }, + { + "bbox": [ + 421, + 295, + 444, + 307 + ], + "score": 0.88, + "content": "0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "on CTB9, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 134, + 318 + ], + "score": 0.88, + "content": "0 . 9 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "on UD1. Moreover, the basic LUA without capturing the label correlations also outperforms", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 226, + 330 + ], + "score": 1.0, + "content": "the strongest baseline, Glyce", + "type": "text" + }, + { + "bbox": [ + 226, + 318, + 236, + 328 + ], + "score": 0.76, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 317, + 280, + 330 + ], + "score": 1.0, + "content": "BERT, by", + "type": "text" + }, + { + "bbox": [ + 280, + 317, + 307, + 329 + ], + "score": 0.89, + "content": "0 . 1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 317, + 368, + 330 + ], + "score": 1.0, + "content": "on CTB5 and", + "type": "text" + }, + { + "bbox": [ + 368, + 317, + 396, + 329 + ], + "score": 0.87, + "content": "0 . 0 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "on CTB9. All these facts", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 329, + 336, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 336, + 341 + ], + "score": 1.0, + "content": "further verify the effectiveness of LUA and its extension.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 353, + 276, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 277, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 277, + 366 + ], + "score": 1.0, + "content": "4.4 SYNTACTIC CHUNKING AND NER", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "Syntactic chunking aims to find phrases related to syntatic category for a sentence. We use CoNLL-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 385, + 504, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 504, + 397 + ], + "score": 1.0, + "content": "2000 dataset (Sang & Buchholz, 2000), which defines 11 syntactic chunk types (NP, VP, PP, etc.) and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "follow the standard splittings of training and test datasets as previous work. NER locates the named", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "entities mentioned in unstructured text and meanwhile classifies them into predefined categories.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 418, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 432 + ], + "score": 1.0, + "content": "We use CoNLL-2003 dataset (Sang & De Meulder, 2003) and OntoNotes 5.0 dataset (Pradhan et al.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "2013). CoNLL-2003 dataset consists of 22137 sentences totally and is split into 14987, 3466, and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "3684 sentences for the training set, development set, and test set, respectively. It is tagged with four", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "linguistic entity types (PER, LOC, ORG, MISC). OntoNotes 5.0 dataset contains 76714 sentences", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "from a wide variety of sources (e.g., magazine and newswire). It includes 18 types of named entity,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "which consists of 11 types (Person, Organization, etc.) and 7 values (Date, Percent, etc.). We follow", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "the same format and partition as in Li et al. (2019); Luo et al. (2020); Yu et al. (2020). 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(2018); Li et al.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "(2019); Luo et al. (2020); Yu et al. (2020). Besides, following Luo et al. (2020), we rerun the source", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "code2 of GCDT and report its result on CoNLL-2000 with standard evaluation method. 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ModelATISSNIPSMTOD
Slot-Gated SLU (Goo et al.,2018) Bi-LSTM + EMLo (Siddhant et al., 2019)95.2088.3095.12
95.4293.90-
Joint BERT (Chen et al., 2019b) CM-Net (Liu et al.,2019c)96.10 96.2097.0096.48
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Model1-3(8695)4-7(2380)8-11(151)12-24(31)Overall
HCRw/BERT91.1585.2250.4320.6790.27
BERT-Biaffine Model91.6787.2370.2440.5591.26
LUA92.3188.5277.3457.2792.09
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Both baselines are reproduced by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 540, + 366, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 366, + 551 + ], + "score": 1.0, + "content": "rerunning their open source codes, biaffine-ner5 and Hire-NER6.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "score": 1.0, + "content": "The results are shown in Table 5. On the one hand, both LUA and Biaffine Model obtain much", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "score": 1.0, + "content": "higher scores of extracting long-length entities than HCR. 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The middle two columns are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "the time complexity of decoding a label sequence. The last column is the time cost of one epoch", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "in training. We set the batch size as 16 and run all the models on 1 GPU. 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ModelATISSNIPSMTOD
Slot-Gated SLU (Goo et al.,2018) Bi-LSTM + EMLo (Siddhant et al., 2019)95.2088.3095.12
95.4293.90-
Joint BERT (Chen et al., 2019b) CM-Net (Liu et al.,2019c)96.10 96.2097.0096.48
96.1597.15-
This WorkLUA LUA w/ Intent Detection96.2797.1097.53
97.2097.55
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Model1-3(8695)4-7(2380)8-11(151)12-24(31)Overall
HCRw/BERT91.1585.2250.4320.6790.27
BERT-Biaffine Model91.6787.2370.2440.5591.26
LUA92.3188.5277.3457.2792.09
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The training set contains 4478 utterances and the test set contains", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "893 utterances. SNIPS dataset is collected by Snips personal voice assistant. The training set con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "tains 13084 utterances and the test set contains 700 utterances. MTOD dataset has three domains,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "score": 1.0, + "content": "including Alarm, Reminder, and Weather. We use the English part of MTOD dataset, where training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "set, dev set, and test set respectively contain 30521, 4181, and 8621 utterances. We follow the same", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 334, + 398, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 398, + 346 + ], + "score": 1.0, + "content": "partition of above datasets as in Goo et al. (2018); Schuster et al. (2018).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 279, + 506, + 346 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "Table 4 summarizes the experiment results for slot filling. On ATIS and SNIPS, we take the results", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "of all baselines as reported in Liu et al. (2019c) for comparison. On MTOD, we rerun the open", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 371, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 216, + 387 + ], + "score": 1.0, + "content": "source toolkits, Slot-gated", + "type": "text" + }, + { + "bbox": [ + 216, + 372, + 240, + 384 + ], + "score": 0.66, + "content": "\\mathrm { S L U } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 371, + 506, + 387 + ], + "score": 1.0, + "content": "and Joint BERT4. As all previous approaches jointly model slot", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 384, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 504, + 396 + ], + "score": 1.0, + "content": "filling and intent detection (a classification task in SLU), we follow them to augment LUA with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "intent detection for a fair comparison. As shown in Table 4, the augmented LUA has surpassed all", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "baselines and obtained state-of-the-art results on the three datasets: it increases the F1 scores by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 137, + 429 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 137, + 417, + 165, + 428 + ], + "score": 0.88, + "content": "0 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 416, + 387, + 429 + ], + "score": 1.0, + "content": "on ATIS and SNIPS, and delivers a substantial gain of", + "type": "text" + }, + { + "bbox": [ + 388, + 417, + 415, + 428 + ], + "score": 0.88, + "content": "1 . 1 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "on MTOD. It’s worth", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 460, + 440 + ], + "score": 1.0, + "content": "mentioning that LUA even outperforms the strong baseline Joint BERT with a margin of", + "type": "text" + }, + { + "bbox": [ + 460, + 428, + 487, + 439 + ], + "score": 0.89, + "content": "0 . 1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 438, + 357, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 134, + 450 + ], + "score": 0.87, + "content": "0 . 2 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 438, + 357, + 452 + ], + "score": 1.0, + "content": "on ATIS and SNIPS without modeling intent detection.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 351, + 506, + 452 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 464, + 313, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 314, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 314, + 477 + ], + "score": 1.0, + "content": "4.6 LONG-LENGTH SEGMENT IDENTIFICATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 504, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "score": 1.0, + "content": "Since LUA doesn’t resort to IOB tagging scheme, it should be more accurate in recognizing long-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "length segments than prior methods. To verify this intuition, we evaluate different models on the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 521 + ], + "score": 1.0, + "content": "segments of different lengths. This study is investigated on OntoNotes 5.0 dataset. Two strong", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "models are adopted as the baselines: one is the best sequence labeling model (i.e., HCR) and the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "other is the best span-based model (i.e., BERT-Biaffine Model). Both baselines are reproduced by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 540, + 366, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 366, + 551 + ], + "score": 1.0, + "content": "rerunning their open source codes, biaffine-ner5 and Hire-NER6.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 483, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "score": 1.0, + "content": "The results are shown in Table 5. On the one hand, both LUA and Biaffine Model obtain much", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "score": 1.0, + "content": "higher scores of extracting long-length entities than HCR. For example, LUA outperforms HCR w/", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 249, + 591 + ], + "score": 1.0, + "content": "BERT by almost twofold on range", + "type": "text" + }, + { + "bbox": [ + 249, + 578, + 284, + 589 + ], + "score": 0.53, + "content": "1 2 - 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 578, + 505, + 591 + ], + "score": 1.0, + "content": ". On the other hand, LUA achieves even better results", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 459, + 603 + ], + "score": 1.0, + "content": "than BERT-Biaffine Model. For instance, the F1 score improvements of LUA over it are", + "type": "text" + }, + { + "bbox": [ + 459, + 589, + 492, + 600 + ], + "score": 0.89, + "content": "1 0 . 1 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 285, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 131, + 612 + ], + "score": 1.0, + "content": "range", + "type": "text" + }, + { + "bbox": [ + 131, + 600, + 159, + 611 + ], + "score": 0.39, + "content": "8 - 1 1", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 599, + 177, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 600, + 210, + 611 + ], + "score": 0.89, + "content": "4 1 . 2 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 599, + 248, + 612 + ], + "score": 1.0, + "content": "on range", + "type": "text" + }, + { + "bbox": [ + 248, + 600, + 281, + 611 + ], + "score": 0.57, + "content": "1 2 - 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 599, + 285, + 612 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 556, + 505, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 626, + 244, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 246, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 246, + 639 + ], + "score": 1.0, + "content": "4.7 RUNNING TIME ANALYSIS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 646, + 504, + 679 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "score": 1.0, + "content": "Table 6 shows the running time comparison among different models. The middle two columns are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "the time complexity of decoding a label sequence. The last column is the time cost of one epoch", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "in training. We set the batch size as 16 and run all the models on 1 GPU. The results indicate that", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 645, + 505, + 681 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 114, + 80, + 497, + 140 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 114, + 80, + 497, + 140 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 80, + 497, + 140 + ], + "spans": [ + { + "bbox": [ + 114, + 80, + 497, + 140 + ], + "score": 0.981, + "html": "
ModelTheoretical ComplexityPractical ComplexityRunning Time
BERT BERT+CRFO(n) O(n|C|2)0(1)5m11s 7m33s
LUAO(n²)O(n) O(n)6m25s
LUA w/Label CorrelationsO(n²|C1²)0(n)7m09s
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For", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 339, + 209 + ], + "score": 1.0, + "content": "example, with the same practical time complexity, BERT", + "type": "text" + }, + { + "bbox": [ + 339, + 197, + 347, + 206 + ], + "score": 0.58, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "CRF is slower than the proposed LUA", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 207, + 324, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 119, + 219 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 207, + 151, + 218 + ], + "score": 0.88, + "content": "1 5 . 0 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 207, + 293, + 219 + ], + "score": 1.0, + "content": "and LUA w/ Label Correlations by", + "type": "text" + }, + { + "bbox": [ + 294, + 207, + 321, + 218 + ], + "score": 0.88, + "content": "5 . \\dot { 3 } 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 207, + 324, + 219 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 240, + 210, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 239, + 213, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 213, + 255 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Sequence segmentation aims to partition a fine-grained unit sequence into multiple labeled coarse-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "grained units. 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ModelTheoretical ComplexityPractical ComplexityRunning Time
BERT BERT+CRFO(n) O(n|C|2)0(1)5m11s 7m33s
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LUA w/Label CorrelationsO(n²|C1²)0(n)7m09s
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ModelCTB5CTB6CTB9UD1
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Lattice-LSTM(Meng et al.,2019)94.38-92.3489.75
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BERT (Meng et al.,2019)95.6191.9292.3890.87
Glyce +BERT (Meng et al., 2019)96.0694.7792.2994.79
96.6195.4193.1596.14
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LUA w/Label Correlations97.9696.6393.9597.08
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ModelASMSRCITYUPKUCTB6
Rich Pretraining(Yang et al.,2017)95.797.596.996.396.2
Bi-LSTM(Ma et al.,2018)96.298.197.296.196.7
Multi-Criteria_Learning +_BERT_(Huang etal., 2019)96.697.997.696.697.6
BERT (Meng et al.,2019)96.598.197.696.5=
Glyce + BERT (Meng et al.,2019)96.798.397.996.7-
Unlabeled LUA96.9498.2798.2196.8898.13
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ModelChunkingNER
CoNLL-2000CoNLL-2003OntoNotes5.0
Bi-LSTM + CRF (Huang et al., 2015)94.4690.101
Flair Embeddings (Akbik et al., 2018)96.7293.0989.3
GCDT w/BERT (Liu et al., 2019b)96.8193.231
BERT-MRC (Li et al., 2019)-93.0491.11
HCR w/BERT (Luo et al., 2020)=93.3790.30
BERT-Biaffine Model (Yu et al., 2020) LUA-93.591.3
This Work96.9593.4692.09
LUA w/Label Correlations97.23--
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95.4293.90-
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96.1597.15-
This WorkLUA LUA w/ Intent Detection96.2797.1097.53
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ModelTheoretical ComplexityPractical ComplexityRunning Time
BERT BERT+CRFO(n) O(n|C|2)0(1)5m11s 7m33s
LUAO(n²)O(n) O(n)6m25s
LUA w/Label CorrelationsO(n²|C1²)0(n)7m09s
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{ + "page_no": 10, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/SkmiegW0b/SkmiegW0b.md b/parse/train/SkmiegW0b/SkmiegW0b.md new file mode 100644 index 0000000000000000000000000000000000000000..b8680fe089b9ad62aaacbadc0a348a7290c24ad5 --- /dev/null +++ b/parse/train/SkmiegW0b/SkmiegW0b.md @@ -0,0 +1,250 @@ +# CHALLENGES IN DISENTANGLING INDEPENDENT FAC-TORS OF VARIATION + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +We study the problem of building models that disentangle independent factors of variation. Such models encode features that can efficiently be used for classification and to transfer attributes between different images in image synthesis. As data we use a weakly labeled training set, where labels indicate what single factor has changed between two data samples, although the relative value of the change is unknown. This labeling is of particular interest as it may be readily available without annotation costs. We introduce an autoencoder model and train it through constraints on image pairs and triplets. We show the role of feature dimensionality and adversarial training theoretically and experimentally. We formally prove the existence of the reference ambiguity, which is inherently present in the disentangling task when weakly labeled data is used. The numerical value of a factor has different meaning in different reference frames. When the reference depends on other factors, transferring that factor becomes ambiguous. We demonstrate experimentally that the proposed model can successfully transfer attributes on several datasets, but show also cases when the reference ambiguity occurs. + +# 1 INTRODUCTION + +One way to simplify the problem of classifying or regressing attributes of interest from data is to build an intermediate representation, a feature, where the information about the attributes is better separated than in the input data. Better separation means that some entries of the feature vary only with respect to one and only one attribute. In this way, classifiers and regressors would not need to build invariance to many nuisance attributes. Instead, they could devote more capacity to discriminating the attributes of interest, and possibly achieve better performance. We call this task disentangling factors of variation, and we identify attributes with the factors. In addition to facilitating classification and regression, this task is beneficial to image synthesis. One could build a model to render images, where each input varies only one attribute of the output, and to transfer attributes between images. + +When labeling is possible and available, supervised learning can be used to solve this task. In general, however, some attributes may not be easily quantifiable (e.g., style). Therefore, we consider using weak labeling, where we only know what attribute has changed between two images, although we do not know by how much. This type of labeling may be readily available in many cases without manual annotation. For example, image pairs from a stereo system are automatically labeled with a viewpoint change, albeit unknown. A practical model that can learn from these labels is an encoder-decoder pair subject to a reconstruction constraint. In this model the weak labels can be used to define similarities between subsets of the feature obtained from two input images. + +We introduce a novel adversarial training of autoencoders to solve the disentangling task when only weak labels are available. Compared to previous methods, our discriminator is not conditioned on class labels, but takes image pairs as inputs. This way the number of parameters can be kept constant. + +We describe the shortcut problem, where all the the information is encoded only in one part of the feature, while other part is completely ignored, as fig. 1 illustrates. We prove our method solves this problem and demonstrate it experimentally. + +We formally prove existence of the reference ambiguity, that is inherently present in the disentangling task when weak labels are used. Thus no algorithm can provably learn disentangling. As fig. 1 shows, the reference ambiguity means that a factor (for example viewpoint) can have different meaning when using a different reference frame that depends on another factor (for example car type). We show experimentally that this ambiguity rarely arise, we can observe it only when the data is complex. + +![](images/fded6d23f694cb78ea35bd328451d6c6df4a13df4ab823b7f12b98e6e7a12b42.jpg) +Figure 1: Challenges of disentangling. We disentangle the feature into two parts, one representing the viewpoint, the other the car type. We use the features for attribute transfer. For all subfigures the viewpoint feature is taken from the leftmost column and the car type feature is taken from the topmost row. (a) ideal solution: the viewpoint and the car type are transferred correctly. (b) shortcut problem: the car type is not transferred. (c) reference ambiguity: compared to the others the viewpoint orientation is flipped for the blue car. + +# 2 RELATED WORK + +Autoencoders. Autoencoders in Bourlard & Kamp (1988), Hinton & Salakhutdinov (2006), Bengio et al. (2013) learn to reconstruct the input data as $\mathbf { x } = \mathrm { D e c } ( \mathrm { E n c } ( \mathbf { x } ) )$ , where $\operatorname { E n c } ( \mathbf { x } )$ is the internal image representation (the encoder) and Dec (the decoder) reconstructs the input of the encoder. Variational autoencoders in Kingma & Welling (2014) use a generative model; $p ( \bar { \mathbf { x } } , \mathbf { z } ) = p ( \mathbf { x } | \mathbf { z } ) p ( \mathbf { z } )$ , where $\mathbf { x }$ is the observed data (images), and $\mathbf { z }$ are latent variables. The encoder estimates the parameters of the posterior, $\operatorname { E n c } ( \mathbf { x } ) = p ( \mathbf { z } | \mathbf { x } )$ , and the decoder estimates the conditional likelihood, $\mathrm { D e c } ( \mathbf { z } ) = p ( \mathbf { x } | \mathbf { z } )$ . In Hinton et al. (2011) autoencoders are trained with transformed image input pairs. The relative transformation parameters are also fed to the network. Because the internal representation explicitly represents the objects presence and location, the network can learn their absolute position. One important aspect of the autoencoders is that they encourage latent representations to keep as much information about the input as possible. + +GAN. Generative Adversarial Nets Goodfellow et al. (2014) learn to sample realistic images with two competing neural networks. The generator Dec creates images $\mathbf { x } = \mathrm { D e c } ( \mathbf { z } )$ from a random noise sample $\mathbf { z }$ and tries to fool a discriminator Dsc, which has to decide whether the image is sampled from the generator $p _ { g }$ or from real images $p _ { r e a l }$ . After a successful training the discriminator cannot distinguish the real from the generated samples. Adversarial training is often used to enforce constraints on random variables. BIGAN, Donahue et al. (2016) learns a feature representation with adversarial nets by training an encoder Enc, such that $\operatorname { E n c } ( \mathbf { x } )$ is Gaussian, when $\mathbf { x } \sim p _ { r e a l }$ . CoGAN, Liu & Tuzel (2016) learns the joint distribution of multi-domain images by having generators and discriminators in each domain, and sharing their weights. They can transform images between domains without being given correspondences. InfoGan, Chen et al. (2016) learns a subset of factors of variation by reproducing parts of the input vector with the discriminator. + +Disentangling and independence. Many recent methods use neural networks for disentangling features, with various degrees of supervision. In Xi Peng (2017) multi-task learning is used with full supervision for pose invariant face recognition. Using both identity and pose labels Tran et al. (2017) can learn pose invariant features and synthesize frontalized faces from any pose. In Yang et al. (2015) autoencoders are used to generate novel viewpoints of objects. They disentangle the object category factor from the viewpoint factor by using as explicit supervision signals: the relative viewpoint transformations between image pairs. In Cheung et al. (2014) the output of the encoder is split in two parts: one represents the class label and the other represents the nuisance factors. Their objective function has a penalty term for misclassification and a cross-covariance cost to disentangle class from nuisance factors. Hierarchical Boltzmann Machines are used in Reed et al. (2014) for disentangling. A subset of hidden units are trained to be sensitive to a specific factor of variation, while being invariant to others. Variational Fair Autoencoders Louizos et al. (2016) learn a representation that is invariant to specific nuisance factors, while retaining as much information as possible. Autoencoders can also be used for visual analogy Reed et al. (2015). GAN is used for disentangling intrinsic image factors (albedo and normal map) in Shu et al. (2017) without using ground truth labeling. They achieve this by explicitly modeling the physics of the image formation in their network. + +The work most related to ours is Mathieu et al. (2016), where an autoencoder restores an image from another by swapping parts of the internal image representation. Their main improvement over Reed et al. (2015) is the use of adversarial training, which allows for learning with image pairs instead of image triplets. Therefore, expensive labels like viewpoint alignment between different car types are no longer needed. One of the differences between this method and ours is that it trains a discriminator for each of the given labels. A benefit of this approach is the higher selectivity of the discriminator, but a drawback is that the number of model parameters grows linearly with the number of labels. In contrast, we work with image pairs and use a single discriminator so that our method is uninfluenced by the number of labels. Moreover, we show formally and experimentally the difficulties of disentangling factors of variation. + +# 3 DISENTANGLING FACTORS OF VARIATION + +We are interested in the design and training of two models. One should map a data sample (e.g., an image) to a feature that is explicitly partitioned into subvectors, each associated to a specific factor of variation. The other model should map this feature back to an image. We call the first model the encoder and the second model the decoder. For example, given the image of a car we would like the encoder to yield a feature with two subvectors: one related to the car viewpoint, and the other related to the car type. The subvectors of the feature obtained from the encoder should be useful for classification or regression of the corresponding factor that they depend on (the car viewpoint and type in the example). This separation would also be very useful to the decoder. It would enable advanced editing of images, for example, the transfer of the viewpoint or car types from an image to another, by swapping the corresponding subvectors. Next, we introduce our model of the data and formal definitions of our encoder and decoder. + +Data model. We assume that our observed data $\mathbf { x }$ is generated through some unknown deterministic invertible and smooth process $f$ that depends on the factors $\mathbf { v }$ and $\mathbf { c }$ , so that $\mathbf { x } = f ( \mathbf { v } , \mathbf { c } )$ . In our earlier example, $\mathbf { x }$ is an image, $\mathbf { v }$ is a viewpoint, c is a car type, and $f$ is the rendering engine. It is reasonable to assume that $f$ is invertible, as for most cases the factors are readily apparent form the image. We assume $f$ is smooth, because a small change in the factors should only result in a small change in the image and vice versa. We denote the inverse of the rendering engine as $f ^ { - 1 } = [ f _ { \mathbf { v } } ^ { - 1 } , f _ { \mathbf { c } } ^ { - 1 } ]$ , where the subscript refers to the recovered factor. + +Weak labeling. In the training we are given pairs of images $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ , where they differ in $\mathbf { v }$ (varying factor), but they have the same c (common factor). We also assume that the two varying factors and the common factor are sampled independently, $\mathbf { v } _ { 1 } \sim p _ { \mathbf { v } }$ , $\mathbf { v } _ { 2 } \sim p _ { \mathbf { v } }$ and $\mathbf { c } \sim p _ { \mathbf { c } }$ . The images are generated as $\mathbf { x } _ { 1 } = f ( \mathbf { v } _ { 1 } , \mathbf { c } )$ and $\mathbf { x } _ { 1 } = f ( \mathbf { v } _ { 2 } , \mathbf { c } )$ . We call this labeling weak, because we do not know the absolute values of either the $\mathbf { v }$ or c factors or even relative changes between $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ . All we know is that the image pairs share the same common factor. + +The encoder. Let Enc be the encoder mapping images to features. For simplicity, we consider features split into only two column subvectors, $N _ { \mathbf { v } }$ and $N _ { \mathbf { c } }$ , one associated to the varying factor $\mathbf { v }$ and the other associated to the common factor c. Then, we have that $\mathrm { E n c } ( \mathbf { x } ) = [ N _ { \mathbf { v } } ( \mathbf { x } ) , \bar { N _ { \mathbf { c } } } ( \mathbf { x } ) ]$ . Ideally, we would like to find the inverse of the image formation function, $[ N _ { \bf v } , N _ { \bf c } ] = f ^ { - 1 }$ , which separates and recovers the factors $\mathbf { v }$ and $\mathbf { c }$ from data samples $\mathbf { x }$ , i.e., + +$$ +N _ { \mathbf { v } } ( f ( \mathbf { v } , \mathbf { c } ) ) = \mathbf { v } \qquad N _ { \mathbf { c } } ( f ( \mathbf { v } , \mathbf { c } ) ) = \mathbf { c } . +$$ + +In practice, this is not possible because any bijective transformation of $\mathbf { v }$ and c could be undone by $f$ and produce the same output $\mathbf { x }$ . Therefore, we aim for $N _ { \mathbf { v } }$ and $N _ { \mathbf { c } }$ that satisfy the following feature disentangling properties + +$$ +R _ { \mathbf { v } } ( N _ { \mathbf { v } } ( f ( \mathbf { v } , \mathbf { c } ) ) ) = \mathbf { v } \qquad R _ { \mathbf { c } } ( N _ { \mathbf { c } } ( f ( \mathbf { v } , \mathbf { c } ) ) ) = \mathbf { c } +$$ + +for all $\mathbf { v } , \mathbf { c }$ , and for some bijective functions $R _ { \mathbf { v } }$ and $R _ { \mathbf { c } }$ , so that $N _ { \mathbf { v } }$ is invariant to $\mathbf { c }$ and $N _ { \mathbf { c } }$ is invariant to $\mathbf { v }$ . + +The decoder. Let Dec be the decoder mapping features to images. The sequence encoder-decoder is constrained to form an autoencoder, so + +$$ +\mathrm { D e c } ( N _ { \mathbf { v } } ( { \mathbf { x } } ) , N _ { \mathbf { c } } ( { \mathbf { x } } ) ) = { \mathbf { x } } , \qquad \forall { \mathbf { x } } . +$$ + +To use the decoder for image synthesis, so that each input subvector affects only one factor in the rendered image, the ideal decoder should satisfy the data disentangling property + +$$ +\mathrm { D e c } ( N _ { \mathbf { v } } ( f ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 1 } ) ) , N _ { \mathbf { c } } ( f ( \mathbf { v } _ { 2 } , \mathbf { c } _ { 2 } ) ) ) = f ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 2 } ) +$$ + +for any $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { c } _ { 1 }$ , and $\mathbf { c } _ { 2 }$ . The equation above describes the transfer of the varying factor $\mathbf { v } _ { 1 }$ of $\mathbf { x } _ { 1 }$ and the common factor $\mathbf { c } _ { 2 }$ of $\mathbf { x } _ { 2 }$ to a new image ${ \bf x } _ { 1 \oplus 2 } = f ( { \bf v } _ { 1 } , { \bf c } _ { 2 } )$ . + +In the next section we describe our training method for disentangling. We introduce a novel adversarial term, that does not need to be conditioned on the common factor, rather it uses only image pairs, that keeps the model parameters constant. Then we address the two main challenges of disentangling, the shortcut problem and the reference ambiguity. We discuss which disentanglement properties can be (provably) achieved by our (or any) method. + +# 3.1 MODEL TRAINING + +In our training procedure we use two terms in the objective function: an autoencoder loss and an adversarial loss. We describe these losses in functional form, however the components are implemented using neural networks. In all our terms we use the following sampling of independent factors + +$$ +\mathbf { c } _ { 1 } , \mathbf { c } _ { 3 } \sim p _ { \mathbf { c } } , \quad \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \sim p _ { \mathbf { v } } . +$$ + +The images are formed as $\mathbf { x } _ { 1 } = f ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 1 } )$ , $\mathbf { x } _ { 2 } = f ( \mathbf { v } _ { 2 } , \mathbf { c } _ { 1 } )$ and $\mathbf { x } _ { 3 } = f ( \mathbf { v } _ { 3 } , \mathbf { c } _ { 3 } )$ . The images $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ share the same common factor, and $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 3 }$ are independent. In our objective functions, we use either pairs or triplets of the above images. + +Autoencoder loss. In this term, we use images $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ with the same common factor $\mathbf { c } _ { 1 }$ . We feed both images to the encoder. Since both images share the same $\mathbf { c } _ { 1 }$ , we impose that the decoder should reconstruct $\mathbf { x } _ { 1 }$ from the encoder subvector $N _ { \mathbf { v } } ( \mathbf { x } _ { 1 } )$ and the encoder subvector $N _ { \mathbf { c } } ( \mathbf { x } _ { 2 } )$ , and similarly for the reconstruction of $\mathbf { x } _ { 2 }$ . The autoencoder objective is thus defined as + +$$ +\mathcal { L } _ { A E } \doteq E _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } } \left[ \left| \mathbf { x } _ { 1 } - \mathrm { D e c } ( N _ { \mathbf { v } } ( \mathbf { x } _ { 1 } ) , N _ { \mathbf { c } } ( \mathbf { x } _ { 2 } ) ) \right| ^ { 2 } + \left| \mathbf { x } _ { 2 } - \mathrm { D e c } ( N _ { \mathbf { v } } ( \mathbf { x } _ { 2 } ) , N _ { \mathbf { c } } ( \mathbf { x } _ { 1 } ) ) \right| ^ { 2 } \right] . +$$ + +Adversarial loss. We introduce an adversarial training where the generator is our encoder-decoder pair and the discriminator Dsc is a neural network, which takes image pairs as input. The discriminator learns to distinguish between real image pairs $[ \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } ]$ and fake ones $[ \mathbf { x } _ { 1 } , \mathbf { x } _ { 3 \oplus 1 } ]$ , where $\mathbf { x } _ { 3 \oplus 1 } \doteq$ $\mathrm { D e c } ( N _ { \mathbf { v } } ( \mathbf { x } _ { 3 } ) , N _ { \mathbf { c } } ( \mathbf { x } _ { 1 } ) )$ . If the encoder were ideal, the image $\mathbf { x } _ { \mathrm { 3 \oplus 1 } }$ would be the result of taking the common factor from $\mathbf { x } _ { 1 }$ and the varying factor from $\mathbf { x } _ { 3 }$ . The generator learns to fool the discriminator, so that $\mathbf { x } _ { 3 \oplus 1 }$ looks like the random variable $\mathbf { x } _ { 2 }$ (the common factor is $\mathbf { c } _ { 1 }$ and the varying factor is independent of $\mathbf { v } _ { 1 }$ ). To this purpose, the decoder must make use of $N _ { \mathbf { c } } ( \mathbf { x } _ { 1 } )$ , since $\mathbf { x } _ { 3 }$ does not carry any information about $\mathbf { c } _ { 1 }$ . The objective function is thus defined as + +$$ +\mathcal { L } _ { G A N } \doteq E _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } } \Big [ \log ( \mathrm { D s c } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } ) ) \Big ] + E _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 3 } } \Big [ \log ( 1 - \mathrm { D s c } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 3 \oplus 1 } ) ) \Big ] . +$$ + +Composite loss. Finally, we optimize the weighted sum of the two losses $\mathcal { L } = \mathcal { L } _ { A E } + \lambda \mathcal { L } _ { G A N }$ + +$$ +\operatorname* { m i n } _ { \mathrm { D e c , E n c } } \operatorname* { m a x } _ { \mathrm { D s c } } \mathcal { L } _ { A E } ( \mathrm { D e c , E n c } ) + \lambda \mathcal { L } _ { G A N } ( \mathrm { D e c , E n c , D s c } ) +$$ + +where $\lambda$ regulates the relative importance of the two losses. + +# 3.2 SHORTCUT PROBLEM. + +Ideally, at the global minimum of $\mathcal { L } _ { A E }$ , $N _ { \mathbf { v } }$ relates only to the factor $\mathbf { v }$ and $N _ { \mathbf { c } }$ only to c. However, the encoder may map a complete description of its input into $N _ { \mathbf { v } }$ and the decoder may completely ignore $N _ { \mathbf { c } }$ . We call this challenge the shortcut problem. When the shortcut problem occurs, the decoder is invariant to its second input, so it does not transfer the $\mathbf { c }$ factor correctly, + +$$ +\mathrm { D e c } ( N _ { \mathbf { v } } ( { \mathbf { x } } _ { 3 } ) , N _ { \mathbf { c } } ( { \mathbf { x } } _ { 1 } ) ) = { \mathbf { x } } _ { 3 } . +$$ + +The shortcut problem can be addressed by reducing the dimensionality of $N _ { \mathbf { v } }$ , so it cannot build a complete representation of all input images. This also forces the encoder and decoder to make use of $N _ { \mathbf { c } }$ for the common factor. However, this strategy may not be convenient as it leads to a time consuming trial-and-error procedure to find the correct dimensionality. A better way to address the shortcut problem is to use adversarial training (7) (8). + +Proposition 1. Let $\mathbf { x } _ { 1 }$ , $\mathbf { x } _ { 2 }$ and $\mathbf { x } _ { 3 }$ data samples generated according to (5), where the factors $\mathbf { c } _ { 1 } , \mathbf { c } _ { 3 } , \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 }$ are jointly independent, and $\mathbf { x } _ { 3 \oplus 1 } \doteq D e c ( N _ { \mathbf { v } } ( \mathbf { x } _ { 3 } ) , N _ { \mathbf { c } } ( \mathbf { x } _ { 1 } ) )$ . When the global optimum of the composite loss (8) is reached, the c factor is transferred to $\mathbf { x } _ { \mathrm { 3 \oplus 1 } }$ , i.e. $f _ { \mathbf { c } } ^ { - 1 } ( \mathbf { x } _ { 3 \oplus 1 } ) = \mathbf { c } _ { 1 }$ + +Proof. When the global optimum of (8) is reached, the distribution of real $\left[ \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } \right]$ and fake $[ \mathbf { x } _ { 1 } , \mathbf { x } _ { 3 \oplus 1 } ]$ image pairs are identical. We compute statistics of the inverse of the rendering engine of the common factor $\bar { f } _ { \mathbf { c } } ^ { - 1 }$ on the data. For the images $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ we obtain + +$$ +\begin{array} { r } { { E } _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } } \Big [ | f _ { \mathbf { c } } ^ { - 1 } ( \mathbf { x } _ { 1 } ) - f _ { \mathbf { c } } ^ { - 1 } ( \mathbf { x } _ { 2 } ) | ^ { 2 } \Big ] = { E } _ { \mathbf { c } _ { 1 } } \Big [ | \mathbf { c } _ { 1 } - \mathbf { c } _ { 1 } | ^ { 2 } \Big ] = 0 } \end{array} +$$ + +by construction (of $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ ). For the images $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { \mathrm { 3 \oplus 1 } }$ we obtain + +$$ +\begin{array} { r } { E _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 3 } } \Big [ | f _ { \mathbf { c } } ^ { - 1 } ( \mathbf { x } _ { 1 } ) - f _ { \mathbf { c } } ^ { - 1 } ( \mathbf { x } _ { 3 \oplus 1 } ) | ^ { 2 } \Big ] = E _ { \mathbf { v } _ { 1 } , \mathbf { c } _ { 1 } , \mathbf { v } _ { 3 } , \mathbf { c } _ { 3 } } \Big [ | \mathbf { c } _ { 1 } - \mathbf { c } _ { 3 \oplus 1 } | ^ { 2 } \Big ] \geq 0 , } \end{array} +$$ + +where ${ \bf c } _ { 3 \oplus 1 } = f _ { \bf c } ^ { - 1 } ( { \bf x } _ { 3 \oplus 1 } )$ . We achieve equality if and only if $\mathbf { c } _ { 1 } = \mathbf { c } _ { 3 \oplus 1 }$ everywhere. + +# 3.3 REFERENCE AMBIGUITY + +Let us consider the ideal case where we observe the space of all images. When weak labels are made available to us, we also know what images $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ share the same c factor (for example, which images have the same car). This labeling is equivalent to defining the probability density function $p _ { \mathbf { c } }$ and the joint conditional $\displaystyle p _ { { \mathbf { x } } _ { 1 } , { \mathbf { x } } _ { 2 } | { \mathbf { c } } }$ , where + +$$ +p _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } | \mathbf { c } } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } | \mathbf { c } ) = \int \delta ( \mathbf { x } _ { 1 } - f ( \mathbf { v } _ { 1 } , \mathbf { c } ) ) \delta ( \mathbf { x } _ { 2 } - f ( \mathbf { v } _ { 2 } , \mathbf { c } ) ) p ( \mathbf { v } _ { 1 } ) p ( \mathbf { v } _ { 2 } ) d \mathbf { v } _ { 1 } d \mathbf { v } _ { 2 } . +$$ + +Firstly, we show that the labeling allows us to satisfy the feature disentangling property for $\mathbf { c }$ (2). For any $[ \mathbf { \dot { x } } _ { 1 } , \mathbf { x } _ { 2 } ] \sim p _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } | \mathbf { c } }$ we impose $N _ { \mathbf { c } } ( \mathbf { x } _ { 1 } ) = N _ { \mathbf { c } } ( \mathbf { x } _ { 2 } )$ . In particular, this equation is true for pairs when one of the two images is held fixed. Thus, a function $C ( \mathbf { c } ) = N _ { \mathbf { c } } ( \mathbf { x } _ { 1 } )$ can be defined, where the $C$ only depends on c, because $N _ { \mathbf { c } }$ is invariant to $\mathbf { v }$ . Lastly, images with the same $\mathbf { v }$ , but different c must also result in different features, $C ( \mathbf { c } _ { 1 } ) = N _ { \mathbf { v } } ( f ( \mathbf { v } , \mathbf { c } _ { 1 } ) ) \neq N _ { \mathbf { v } } ( \mathbf { v } , \mathbf { c } _ { 2 } ) = C ( \mathbf { c } _ { 2 } )$ , otherwise the autoencoder constraint (3) cannot be satisfied. Then, there exists a bijective function $\dot { R } _ { \bf c } = { C } ^ { - 1 }$ such that property (2) is satisfied for c. Unfortunately the other disentangling properties can not provably be satisfied. + +Definition 1. A function $g$ reproduces the data distribution, when it generates samples $\mathbf { y } _ { 1 } = g ( \mathbf { v } _ { 1 } , \mathbf { c } )$ and $\mathbf { y } _ { 2 } = g ( \mathbf { v } _ { 2 } , \mathbf { c } )$ that have the same distribution as the data. Formally, $[ \mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } ] \sim p _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } }$ , where the latent factors are independent, $\mathbf { v } _ { 1 } \sim p _ { \mathbf { v } }$ , $\mathbf { v } _ { 2 } \sim p _ { \mathbf { v } }$ and $\mathbf { c } \sim p _ { \mathbf { c } }$ . + +The reference ambiguity occurs, when a decoder reproduces the data without satisfying the disentangling properties. + +Proposition 2. Let $p _ { \mathbf { v } }$ assign the same probability value to at least two different instances of v. Then, we can find encoders that reproduce the data distribution, but do not satisfy the disentangling properties for v in (2) and (4). + +Proof. We already saw that $N _ { \mathbf { c } }$ satisfies (2), so we can choose $N _ { \mathbf { c } } = f _ { \mathbf { c } } ^ { - 1 }$ , the inverse of the rendering engine. Now we look at defining $N _ { \mathbf { v } }$ and the decoder. The iso-probability property of $p _ { \mathbf { v } }$ implies that there exists a mapping $T ( \mathbf { v } , \mathbf { c } )$ , such that $T ( \mathbf { v } , \mathbf { c } ) \sim p _ { \mathbf { v } }$ and $T ( \mathbf { v } , \bar { \mathbf { c } _ { 1 } } ) \bar { \neq } T ( \bar { \mathbf { v } } , \bar { \mathbf { c } _ { 2 } } )$ for some $\mathbf { v }$ and $\mathbf { c } _ { 1 } \neq \mathbf { c } _ { 2 }$ . For example, let us denote with $\mathbf { v } _ { 1 } \neq \mathbf { v } _ { 2 }$ two varying components such that $p _ { \mathbf { v } } ( \mathbf { v } _ { 1 } ) = p _ { \mathbf { v } } ( \mathbf { v } _ { 2 } )$ . Then, let + +$$ +T ( \mathbf { v } , \mathbf { c } ) \dot { = } \left\{ \begin{array} { l l } { \mathbf { v } } & { \mathrm { i f } \ \mathbf { v } \neq \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } } \\ { \mathbf { v } _ { 1 } } & { \mathrm { i f } \ \mathbf { v } = \mathbf { v } _ { 1 } \lor \mathbf { v } _ { 2 } \mathrm { a n d } \mathbf { c } \in \mathcal { C } } \\ { \mathbf { v } _ { 2 } } & { \mathrm { i f } \ \mathbf { v } = \mathbf { v } _ { 1 } \lor \mathbf { v } _ { 2 } \mathrm { a n d } \mathbf { c } \not \in \mathcal { C } } \end{array} \right. +$$ + +and $\mathcal { C }$ is a subset of the domain of $\mathbf { c }$ , where $\begin{array} { r } { \int _ { \mathcal { C } } p _ { \mathbf { c } } ( \mathbf { c } ) d \mathbf { c } = 1 / 2 } \end{array}$ . Now, let us define the encoder as $N _ { \mathbf { v } } ( f ( \mathbf { v } , \mathbf { c } ) ) = T ( \mathbf { v } , \mathbf { c } )$ . By using the autoencoder constraint, the decoder satisfies + +$$ +\operatorname { D e c } ( N _ { \mathbf { v } } ( f ( \mathbf { v } , \mathbf { c } ) ) , N _ { \mathbf { c } } ( f ( \mathbf { v } , \mathbf { c } ) ) ) = \operatorname { D e c } ( T ( \mathbf { v } , \mathbf { c } ) , \mathbf { c } ) = f ( \mathbf { v } , \mathbf { c } ) . +$$ + +Even though $T ( \mathbf { v } , \mathbf { c } )$ depends on $\mathbf { c }$ functionally, they are statistically independent. Because $T ( \mathbf { v } , \mathbf { c } ) \sim$ $p _ { \mathbf { v } }$ and $\mathbf { c } \sim p _ { \mathbf { c } }$ by construction, our encoder-decoder pair defines a data distribution identical to that given as training set + +$$ +\left[ \operatorname { D e c } ( T ( \mathbf { v } _ { 1 } , \mathbf { c } ) , \mathbf { c } ) , \operatorname { D e c } ( T ( \mathbf { v } _ { 2 } , \mathbf { c } ) , \mathbf { c } ) \right] \sim p _ { \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } } . +$$ + +The feature disentanglement property is not satisfied because $N _ { \mathbf { v } } ( f ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 1 } ) ) ~ = ~ T ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 1 } ) ~ \neq$ $T ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 2 } ) \ = \ N _ { \mathbf { v } } ( f ( { \bar { \mathbf { v } } } _ { 1 } , \mathbf { c } _ { 2 } ) )$ , when $\mathbf { c } _ { 1 } ~ \in ~ { \mathcal { C } }$ and $\mathbf { c } _ { 2 } \notin \mathcal { C }$ . Similarly, the data disentanglement property does not hold, because $\mathrm { D e c } ( T ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 1 } ) , \mathbf { c } _ { 1 } ) \neq \mathrm { D e c } ( T ( \mathbf { v } _ { 1 } , \mathbf { c } _ { 2 } ) , \mathbf { c } _ { 2 } )$ . □ + +The above proposition implies that we cannot provably disentangle all the factors of variation from weakly labeled data, even if we had access to all the data and knew the distributions $p _ { \mathbf { v } }$ and $p _ { \mathbf { c } }$ . + +To better understand it, let us consider a practical example. Let $\mathbf { v } \sim \mathcal { U } [ - \pi , \pi ]$ be the (continuous) viewpoint (the azimuth angle) and $\mathbf { c } \sim B ( 0 . 5 )$ the car type, where $\mathcal { U }$ denotes the uniform distribution and $B ( 0 . 5 )$ the Bernoulli distribution with probability $p _ { \mathbf { c } } ( \mathbf { c } = 0 ) = p _ { \mathbf { c } } ( \mathbf { c } = 1 ) = 0 . 5$ (i.e., there are only 2 car types). In this case, every instance of $\mathbf { v }$ is iso-probable in $p _ { \mathbf { v } }$ so we have the worst scenario for the reference ambiguity. We can define the function $T ( \mathbf { v } , \mathbf { c } ) = \mathbf { v } ( 2 \mathbf { c } - 1 )$ so that the mapping of $\mathbf { v }$ is mirrored as we change the car type. By construction $T ( \mathbf { v } , \mathbf { c } ) \sim \mathcal { U } [ - \pi , \pi ]$ for any $\mathbf { c }$ and $T ( \mathbf { v } , \mathbf { c } _ { 1 } ) \neq T ( \mathbf { v } , \mathbf { c } _ { 2 } )$ for $\mathbf { v } \neq 0$ and $\mathbf { c } _ { 1 } \neq \mathbf { c } _ { 2 }$ . So we cannot tell the difference between $T$ and the ideal correct mapping to the viewpoint factor. This is equivalent to an encoder $N _ { \mathbf { v } } ( f ( \mathbf { v } , \mathbf { c } ) ) = T ( \mathbf { v } , \mathbf { c } )$ that reverses the ordering of the azimuth of car 1 with respect to car 0. Each car has its own reference system, and thus it is not possible to transfer the viewpoint from one system to the other, as it is illustrated in fig. 1. + +# 3.4 IMPLEMENTATION + +In our implementation we use convolutional neural networks for all the models. We denote with $\theta$ the parameters associated to each network. Then, the optimization of the composite loss can be written as + +$$ +\hat { \theta } _ { \mathrm { { D e c } } } , \hat { \theta } _ { \mathrm { { E n c } } } , \hat { \theta } _ { \mathrm { { D s c } } } = \arg \operatorname* { m i n } _ { \theta _ { \mathrm { { D e c } } } , \theta _ { \mathrm { { E n c } } } } \operatorname* { m a x } _ { \theta _ { \mathrm { { D s c } } } } \mathcal { L } ( \theta _ { \mathrm { { D e c } } } , \theta _ { \mathrm { { E n c } } } , \theta _ { \mathrm { { D s c } } } ) . +$$ + +We choose $\lambda = 1$ and also add regularization to the adversarial loss so that each logarithm has a minimum value. We define $\log _ { \epsilon } \bar { \mathrm { D s c } } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } ) = \log ( \epsilon + \mathrm { D s c } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } ) )$ (and similarly for the other logarithmic term) and use $\epsilon = 1 0 ^ { - 1 2 }$ . The main components of our neural network are shown in Fig. 2. The architecture of the encoder and the decoder were taken from DCGAN Radford et al. (2015), with slight modifications. We added fully connected layers at the output of the encoder and to the input of the decoder. For the discriminator we used a simplified version of the VGG Simonyan & Zisserman (2014) network. As the input to the discriminator is an image pair, we concatenate them along the color channels. + +Normalization. In our architecture both the encoder and the decoder networks use blocks with a convolutional layer, a nonlinear activation function (ReLU/leaky ReLU) and a normalization layer, typically, batch normalization (BN). As an alternative to BN we consider the recently introduced instance normalization (IN) Ulyanov et al. (2017). The main difference between BN and IN is that the latter just computes the mean and standard deviation across the spatial domain of the input and not along the batch dimension. Thus, the shift and scaling for the output of each layer is the same at every iteration for the same input image. In practice, we find that IN improves the performance. + +# 4 EXPERIMENTS + +We tested our method on the MNIST, Sprites and ShapeNet datasets. We performed ablation studies on the shortcut problem using ShapeNet cars. We focused on the effect of the feature dimensionality and having the adversarial term $( \mathcal { L } _ { A E } + \mathcal { L } _ { G A N } )$ or not $( \mathcal { L } _ { A E } )$ . We also show that in most cases the reference ambiguity does not arise in practice (MNIST, Sprites, ShapeNet cars), we can only observe it when the data is more complex (ShapeNet chairs). + +![](images/90a0f2e4ff16668ce677bfd81f19be4a9e730377fbd5bbdf49085baa9daeb981.jpg) +Figure 2: Learning to disentangle factors of variation. The scheme above shows how the encoder (Enc), the decoder (Dec) and the discriminator (Dsc) are trained with input triplets. The components with the same name share weights. + +# 4.1 SHORTCUT PROBLEM + +ShapeNet cars. The ShapeNet dataset Chang et al. (2015) contains 3D objects than we can render from different viewpoints. We consider only one category (cars) for a set of fixed viewpoints. Cars have high intraclass variability and they do not have rotational symmetries. We used approximately 3K car types for training and 300 for testing. We rendered 24 possible viewpoints around each object in a full circle, resulting in 80K images in total. The elevation was fixed to 15 degrees and azimuth angles were spaced 15 degrees apart. We normalized the size of the objects to fit in a $1 0 0 \times 1 0 0$ pixel bounding box, and placed it in the middle of a $1 2 8 \times 1 2 8$ pixel image. + +Fig. 3 shows the attribute transfer on the Shapenet cars. We compare the methods $\mathcal { L } _ { A E }$ and $\mathcal { L } _ { A E } +$ $\mathcal { L } _ { G A N }$ with different feature dimension of $N _ { \mathbf { v } }$ . The size of the common feature $N _ { \mathbf { c } }$ was fixed to 1024 dimensions. We can observe that the transferring performance degrades for $\mathcal { L } _ { A E }$ , when we increase the feature size of $N _ { \mathbf { v } }$ . As expected, the autoencoder takes the shortcut and tries to store all information into $N _ { \mathbf { v } }$ . The model $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ instead renders images without loss of quality, independently of the feature dimension. + +![](images/5c12b702322a9c6bbc47ab8de3e5581a0bb893cfe305a9a1e478823279c0c10d.jpg) +Figure 3: Feature transfer on Shapenet. (a) synthesized images with $\mathcal { L } _ { A E }$ , where the top row shows images from which the car type is taken. The second, third and fourth row show the decoder renderings using 2, 16 and 128 dimensions for the feature $N _ { \mathbf { v } }$ . (b) images synthesized with $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ . The setting for the inputs and feature dimensions are the same as in (a). + +In Fig. 4 we visualize the t-SNE embeddings of the $N _ { \mathbf { v } }$ features for several models using different feature sizes. For the $2 D$ case, we do not modify the data. We can see that both $\mathcal { L } _ { A E }$ with 2 dimensions and $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ with 128 separate the viewpoints well, but $\mathcal { L } _ { A E }$ with 128 dimensions does not due to the shortcut problem. We investigate the effect of dimensionality of the $N _ { \mathbf { v } }$ features on the nearest neighbor classification task. The performance is measured by the mean average precision. For $N _ { \mathbf { v } }$ we use the viewpoint as ground truth. Fig. 4 also shows the results on $\mathcal { L } _ { A E }$ and $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ models with different $N _ { \mathbf { v } }$ feature dimensions. The dimension of $N _ { \mathbf { c } }$ was fixed to 1024 for this experiment. One can now see quantitatively that $\mathcal { L } _ { A E }$ is sensitive to the size of $N _ { \mathbf { v } }$ , while $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ is not. $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ also achieves a better performance. + +![](images/682c15c0ae852d517b1ac69b302654a800de4fd0f02c77d85fbeaeb17fe5a1ca.jpg) +Figure 4: The effect of dimensions and objective function on $N _ { v }$ features. (a), (b), (c) t-SNE embeddings on $N _ { \mathbf { v } }$ features. Colors correspond to the ground truth viewpoint. The objective functions and the $N _ { \mathbf { v } }$ dimensions are: (a) $\mathcal { L } _ { A E }$ 2 dim, (b) $\mathcal { L } _ { A E }$ 128 dim, (c) $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N } \mathrm { ~ 1 ~ }$ 128 dim. (d) Mean average precision curves for the viewpoint prediction from the viewpoint feature using different models and dimensions for $N _ { \mathbf { v } }$ . + +Table 1: Nearest neighbor classification on $N _ { \mathbf { v } }$ and $N _ { \mathbf { c } }$ features using different normalization techniques on ShapeNet cars. + +
NormalizationNv mAPNc mAP
None0.470.13
Batch0.500.08
Instance0.500.20
+ +We compare the different normalization choices in Table 1. We evaluate the case when batch, instance and no normalization are used and compute the performance on the nearest neighbor classification task. We fixed the feature dimensions at 1024 for both $N _ { \mathbf { v } }$ and $N _ { \mathbf { c } }$ features in all normalization cases. We can see that both batch and instance normalization perform equally well on viewpoint classification and no normalization is slightly worse. For the car type classification instance normalization is clearly better. + +# 4.2 REFERENCE AMBIGUITY + +MNIST. The MNIST dataset LeCun et al. (1998) contains handwritten grayscale digits of size $2 8 \times 2 8$ pixel. There are 60K images of 10 classes for training and 10K for testing. The common factor is the digit class and the varying factor is the intraclass variation. We take image pairs that have the same digit for training, and use our full model $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ with dimensions 64 for $N _ { \mathbf { v } }$ and 64 for $N _ { \mathbf { c } }$ . In Fig. 5 (a) and (b) we show the transfer of varying factors. Qualitatively, both our method and Mathieu et al. (2016) perform well. We observe neither the reference ambiguity nor the shortcut problem in this case. + +![](images/19dbb062afd2ee7e2e211be55c604266d4e81e0c4b40072158decf54298db406.jpg) +Figure 5: Renderings of transferred features. In all figures the variable factor is transferred from the left column and the common factor from the top row. (a) MNIST Mathieu et al. (2016); (b) MNIST (ours); (c) Sprites Mathieu et al. (2016); (d) Sprites (ours). + +![](images/55049684dd4908d747a5bd58b28114a25a2f6b44c1aa8542f673f3beeb04f571.jpg) +Figure 6: Attribute transfer on ShapeNet. For both subfigures the viewpoint is taken from the leftmost column and the car/chair type is taken from the first row. (a) Cars: the factors are transferred correctly. (b) Chairs: in the bottom three rows the viewpoint is not transferred correctly due to the reference ambiguity. + +Sprites. The Sprites dataset Reed et al. (2015) contains 60 pixel color images of animated characters (sprites). There are 672 sprites, 500 for training, 100 for testing and 72 for validation. Each sprite has 20 animations and 178 images, so the full dataset has 120K images in total. There are many changes in the appearance of the sprites, they differ in their body shape, gender, hair, armour, arm type, greaves, and weapon. We consider character identity as the common factor and the pose as the varying factor. We train our system using image pairs of the same sprite and do not exploit labels on their pose. We train the $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ model with dimensions 64 for $N _ { \mathbf { v } }$ and 448 for $N _ { \mathbf { c } }$ . Fig. 5 (c) and (d) show results on the attribute transfer task. Both our method and Mathieu et al. (2016)’s transfer the identity of the sprites correctly, the reference ambiguity does not arise. + +ShapeNet chairs. We render the ShapeNet chairs with the same settings (viewpoints, image size) as the cars. There are 3500 chair types for training and 3200 for testing, so the dataset contains 160K images. We trained $\mathcal { L } _ { A E } + \mathcal { L } _ { G A N }$ , and set the feature dimensions to 1024 for both $N _ { \mathbf { v } }$ and $N _ { \mathbf { c } }$ . In Fig. 6 we show results on attribute transfer and compare it with ShapeNet cars. We found that the reference ambiguity does not emerge for cars, but it does for chairs, possibly due to the higher complexity, as cars have much less variability than chairs. + +# 5 CONCLUSIONS + +In this paper we studied the challenges of disentangling factors of variation, mainly the shortcut problem and the reference ambiguity. The shortcut problem occurs when all information is stored in only one feature chunk, while the other is ignored. The reference ambiguity means that the reference in which a factor is interpreted, may depend on other factors. This makes the attribute transfer ambiguous. We introduced a novel training of autoencoders to solve disentangling using image triplets. We showed theoretically and experimentally how to keep the shortcut problem under control through adversarial training, and enable to use large feature dimensions. We proved that the reference ambiguity is inherently present in the disentangling task when weak labels are used. Most importantly this can be stated independently of the learning algorithm. We demonstrated that training and transfer of factors of variation may not be guaranteed. However, in practice we observe that our trained model works well on many datasets and exhibits good generalization capabilities. + +# REFERENCES + +Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE transactions on pattern analysis and machine intelligence, 35(8):1798–1828, 2013. + +Hervé Bourlard and Yves Kamp. Auto-association by multilayer perceptrons and singular value decomposition. Biological cybernetics, 59(4):291–294, 1988. +Angel X. Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, Jianxiong Xiao, Li Yi, and Fisher Yu. ShapeNet: An Information-Rich 3D Model Repository. Technical Report arXiv:1512.03012 [cs.GR], 2015. +Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In NIPS, 2016. +Brian Cheung, Jesse A Livezey, Arjun K Bansal, and Bruno A Olshausen. Discovering hidden factors of variation in deep networks. arXiv:1412.6583, 2014. +Jeff Donahue, Philipp Krähenbühl, and Trevor Darrell. Adversarial feature learning. arXiv:1605.09782, 2016. +Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NIPS, 2014. +Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006. +Geoffrey E Hinton, Alex Krizhevsky, and Sida D Wang. Transforming auto-encoders. In International Conference on Artificial Neural Networks, pp. 44–51. Springer, 2011. +Diederik P Kingma and Max Welling. Auto-encoding variational bayes. In ICLR, 2014. +Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. +Ming-Yu Liu and Oncel Tuzel. Coupled generative adversarial networks. In Advances in Neural Information Processing Systems, pp. 469–477, 2016. +Christos Louizos, Kevin Swersky, Yujia Li, Max Welling, and Richard Zemel. The variational fair autoencoder. In ICLR, 2016. +Michael F Mathieu, Junbo Jake Zhao, Junbo Zhao, Aditya Ramesh, Pablo Sprechmann, and Yann LeCun. Disentangling factors of variation in deep representation using adversarial training. In Advances in Neural Information Processing Systems, pp. 5041–5049, 2016. +Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv:1511.06434, 2015. +Scott Reed, Kihyuk Sohn, Yuting Zhang, and Honglak Lee. Learning to disentangle factors of variation with manifold interaction. In Proceedings of the 31st International Conference on Machine Learning (ICML-14), pp. 1431–1439, 2014. +Scott E Reed, Yi Zhang, Yuting Zhang, and Honglak Lee. Deep visual analogy-making. In Advances in Neural Information Processing Systems, pp. 1252–1260, 2015. +Zhixin Shu, Ersin Yumer, Sunil Hadap, Kalyan Sunkavalli, Eli Shechtman, and Dimitris Samaras. Neural face editing with intrinsic image disentangling. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. +Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. +Luan Tran, Xi Yin, and Xiaoming Liu. Disentangled representation learning gan for pose-invariant face recognition. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. +Dmitry Ulyanov, Andrea Vedaldi, and Victor S. Lempitsky. Improved texture networks: Maximizing quality and diversity in feed-forward stylization and texture synthesis. In CVPR, 2017. +Kihyuk Sohn Dimitris Metaxas Manmohan Chandraker Xi Peng, Xiang Yu. Reconstruction for feature disentanglement in pose-invariant face recognition. arXiv:1702.03041, 2017. +Jimei Yang, Scott E Reed, Ming-Hsuan Yang, and Honglak Lee. Weakly-supervised disentangling with recurrent transformations for 3d view synthesis. In NIPS, 2015. \ No newline at end of file diff --git a/parse/train/SkmiegW0b/SkmiegW0b_content_list.json b/parse/train/SkmiegW0b/SkmiegW0b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..1f4759ffc2946d0881b20f83032ec28820a1d5f1 --- /dev/null +++ b/parse/train/SkmiegW0b/SkmiegW0b_content_list.json @@ -0,0 +1,1155 @@ +[ + { + "type": "text", + "text": "CHALLENGES IN DISENTANGLING INDEPENDENT FAC-TORS OF VARIATION", + "text_level": 1, + "bbox": [ + 176, + 98, + 826, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We study the problem of building models that disentangle independent factors of variation. Such models encode features that can efficiently be used for classification and to transfer attributes between different images in image synthesis. As data we use a weakly labeled training set, where labels indicate what single factor has changed between two data samples, although the relative value of the change is unknown. This labeling is of particular interest as it may be readily available without annotation costs. We introduce an autoencoder model and train it through constraints on image pairs and triplets. We show the role of feature dimensionality and adversarial training theoretically and experimentally. We formally prove the existence of the reference ambiguity, which is inherently present in the disentangling task when weakly labeled data is used. The numerical value of a factor has different meaning in different reference frames. When the reference depends on other factors, transferring that factor becomes ambiguous. We demonstrate experimentally that the proposed model can successfully transfer attributes on several datasets, but show also cases when the reference ambiguity occurs. ", + "bbox": [ + 233, + 266, + 766, + 474 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 502, + 336, + 517 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One way to simplify the problem of classifying or regressing attributes of interest from data is to build an intermediate representation, a feature, where the information about the attributes is better separated than in the input data. Better separation means that some entries of the feature vary only with respect to one and only one attribute. In this way, classifiers and regressors would not need to build invariance to many nuisance attributes. Instead, they could devote more capacity to discriminating the attributes of interest, and possibly achieve better performance. We call this task disentangling factors of variation, and we identify attributes with the factors. In addition to facilitating classification and regression, this task is beneficial to image synthesis. One could build a model to render images, where each input varies only one attribute of the output, and to transfer attributes between images. ", + "bbox": [ + 174, + 532, + 825, + 659 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "When labeling is possible and available, supervised learning can be used to solve this task. In general, however, some attributes may not be easily quantifiable (e.g., style). Therefore, we consider using weak labeling, where we only know what attribute has changed between two images, although we do not know by how much. This type of labeling may be readily available in many cases without manual annotation. For example, image pairs from a stereo system are automatically labeled with a viewpoint change, albeit unknown. A practical model that can learn from these labels is an encoder-decoder pair subject to a reconstruction constraint. In this model the weak labels can be used to define similarities between subsets of the feature obtained from two input images. ", + "bbox": [ + 174, + 665, + 825, + 776 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We introduce a novel adversarial training of autoencoders to solve the disentangling task when only weak labels are available. Compared to previous methods, our discriminator is not conditioned on class labels, but takes image pairs as inputs. This way the number of parameters can be kept constant. ", + "bbox": [ + 176, + 784, + 825, + 825 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We describe the shortcut problem, where all the the information is encoded only in one part of the feature, while other part is completely ignored, as fig. 1 illustrates. We prove our method solves this problem and demonstrate it experimentally. ", + "bbox": [ + 176, + 833, + 825, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We formally prove existence of the reference ambiguity, that is inherently present in the disentangling task when weak labels are used. Thus no algorithm can provably learn disentangling. As fig. 1 shows, the reference ambiguity means that a factor (for example viewpoint) can have different meaning when using a different reference frame that depends on another factor (for example car type). We show experimentally that this ambiguity rarely arise, we can observe it only when the data is complex. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/fded6d23f694cb78ea35bd328451d6c6df4a13df4ab823b7f12b98e6e7a12b42.jpg", + "image_caption": [ + "Figure 1: Challenges of disentangling. We disentangle the feature into two parts, one representing the viewpoint, the other the car type. We use the features for attribute transfer. For all subfigures the viewpoint feature is taken from the leftmost column and the car type feature is taken from the topmost row. (a) ideal solution: the viewpoint and the car type are transferred correctly. (b) shortcut problem: the car type is not transferred. (c) reference ambiguity: compared to the others the viewpoint orientation is flipped for the blue car. " + ], + "image_footnote": [], + "bbox": [ + 223, + 148, + 774, + 292 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 419, + 341, + 435 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Autoencoders. Autoencoders in Bourlard & Kamp (1988), Hinton & Salakhutdinov (2006), Bengio et al. (2013) learn to reconstruct the input data as $\\mathbf { x } = \\mathrm { D e c } ( \\mathrm { E n c } ( \\mathbf { x } ) )$ , where $\\operatorname { E n c } ( \\mathbf { x } )$ is the internal image representation (the encoder) and Dec (the decoder) reconstructs the input of the encoder. Variational autoencoders in Kingma & Welling (2014) use a generative model; $p ( \\bar { \\mathbf { x } } , \\mathbf { z } ) = p ( \\mathbf { x } | \\mathbf { z } ) p ( \\mathbf { z } )$ , where $\\mathbf { x }$ is the observed data (images), and $\\mathbf { z }$ are latent variables. The encoder estimates the parameters of the posterior, $\\operatorname { E n c } ( \\mathbf { x } ) = p ( \\mathbf { z } | \\mathbf { x } )$ , and the decoder estimates the conditional likelihood, $\\mathrm { D e c } ( \\mathbf { z } ) = p ( \\mathbf { x } | \\mathbf { z } )$ . In Hinton et al. (2011) autoencoders are trained with transformed image input pairs. The relative transformation parameters are also fed to the network. Because the internal representation explicitly represents the objects presence and location, the network can learn their absolute position. One important aspect of the autoencoders is that they encourage latent representations to keep as much information about the input as possible. ", + "bbox": [ + 173, + 450, + 826, + 604 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GAN. Generative Adversarial Nets Goodfellow et al. (2014) learn to sample realistic images with two competing neural networks. The generator Dec creates images $\\mathbf { x } = \\mathrm { D e c } ( \\mathbf { z } )$ from a random noise sample $\\mathbf { z }$ and tries to fool a discriminator Dsc, which has to decide whether the image is sampled from the generator $p _ { g }$ or from real images $p _ { r e a l }$ . After a successful training the discriminator cannot distinguish the real from the generated samples. Adversarial training is often used to enforce constraints on random variables. BIGAN, Donahue et al. (2016) learns a feature representation with adversarial nets by training an encoder Enc, such that $\\operatorname { E n c } ( \\mathbf { x } )$ is Gaussian, when $\\mathbf { x } \\sim p _ { r e a l }$ . CoGAN, Liu & Tuzel (2016) learns the joint distribution of multi-domain images by having generators and discriminators in each domain, and sharing their weights. They can transform images between domains without being given correspondences. InfoGan, Chen et al. (2016) learns a subset of factors of variation by reproducing parts of the input vector with the discriminator. ", + "bbox": [ + 173, + 611, + 825, + 763 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Disentangling and independence. Many recent methods use neural networks for disentangling features, with various degrees of supervision. In Xi Peng (2017) multi-task learning is used with full supervision for pose invariant face recognition. Using both identity and pose labels Tran et al. (2017) can learn pose invariant features and synthesize frontalized faces from any pose. In Yang et al. (2015) autoencoders are used to generate novel viewpoints of objects. They disentangle the object category factor from the viewpoint factor by using as explicit supervision signals: the relative viewpoint transformations between image pairs. In Cheung et al. (2014) the output of the encoder is split in two parts: one represents the class label and the other represents the nuisance factors. Their objective function has a penalty term for misclassification and a cross-covariance cost to disentangle class from nuisance factors. Hierarchical Boltzmann Machines are used in Reed et al. (2014) for disentangling. A subset of hidden units are trained to be sensitive to a specific factor of variation, while being invariant to others. Variational Fair Autoencoders Louizos et al. (2016) learn a representation that is invariant to specific nuisance factors, while retaining as much information as possible. Autoencoders can also be used for visual analogy Reed et al. (2015). GAN is used for disentangling intrinsic image factors (albedo and normal map) in Shu et al. (2017) without using ground truth labeling. They achieve this by explicitly modeling the physics of the image formation in their network. ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The work most related to ours is Mathieu et al. (2016), where an autoencoder restores an image from another by swapping parts of the internal image representation. Their main improvement over Reed et al. (2015) is the use of adversarial training, which allows for learning with image pairs instead of image triplets. Therefore, expensive labels like viewpoint alignment between different car types are no longer needed. One of the differences between this method and ours is that it trains a discriminator for each of the given labels. A benefit of this approach is the higher selectivity of the discriminator, but a drawback is that the number of model parameters grows linearly with the number of labels. In contrast, we work with image pairs and use a single discriminator so that our method is uninfluenced by the number of labels. Moreover, we show formally and experimentally the difficulties of disentangling factors of variation. ", + "bbox": [ + 174, + 180, + 825, + 319 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 DISENTANGLING FACTORS OF VARIATION ", + "text_level": 1, + "bbox": [ + 174, + 340, + 549, + 356 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We are interested in the design and training of two models. One should map a data sample (e.g., an image) to a feature that is explicitly partitioned into subvectors, each associated to a specific factor of variation. The other model should map this feature back to an image. We call the first model the encoder and the second model the decoder. For example, given the image of a car we would like the encoder to yield a feature with two subvectors: one related to the car viewpoint, and the other related to the car type. The subvectors of the feature obtained from the encoder should be useful for classification or regression of the corresponding factor that they depend on (the car viewpoint and type in the example). This separation would also be very useful to the decoder. It would enable advanced editing of images, for example, the transfer of the viewpoint or car types from an image to another, by swapping the corresponding subvectors. Next, we introduce our model of the data and formal definitions of our encoder and decoder. ", + "bbox": [ + 174, + 369, + 825, + 523 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Data model. We assume that our observed data $\\mathbf { x }$ is generated through some unknown deterministic invertible and smooth process $f$ that depends on the factors $\\mathbf { v }$ and $\\mathbf { c }$ , so that $\\mathbf { x } = f ( \\mathbf { v } , \\mathbf { c } )$ . In our earlier example, $\\mathbf { x }$ is an image, $\\mathbf { v }$ is a viewpoint, c is a car type, and $f$ is the rendering engine. It is reasonable to assume that $f$ is invertible, as for most cases the factors are readily apparent form the image. We assume $f$ is smooth, because a small change in the factors should only result in a small change in the image and vice versa. We denote the inverse of the rendering engine as $f ^ { - 1 } = [ f _ { \\mathbf { v } } ^ { - 1 } , f _ { \\mathbf { c } } ^ { - 1 } ]$ , where the subscript refers to the recovered factor. ", + "bbox": [ + 173, + 531, + 825, + 628 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Weak labeling. In the training we are given pairs of images $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ , where they differ in $\\mathbf { v }$ (varying factor), but they have the same c (common factor). We also assume that the two varying factors and the common factor are sampled independently, $\\mathbf { v } _ { 1 } \\sim p _ { \\mathbf { v } }$ , $\\mathbf { v } _ { 2 } \\sim p _ { \\mathbf { v } }$ and $\\mathbf { c } \\sim p _ { \\mathbf { c } }$ . The images are generated as $\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 1 } , \\mathbf { c } )$ and $\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 2 } , \\mathbf { c } )$ . We call this labeling weak, because we do not know the absolute values of either the $\\mathbf { v }$ or c factors or even relative changes between $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ . All we know is that the image pairs share the same common factor. ", + "bbox": [ + 173, + 635, + 825, + 719 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The encoder. Let Enc be the encoder mapping images to features. For simplicity, we consider features split into only two column subvectors, $N _ { \\mathbf { v } }$ and $N _ { \\mathbf { c } }$ , one associated to the varying factor $\\mathbf { v }$ and the other associated to the common factor c. Then, we have that $\\mathrm { E n c } ( \\mathbf { x } ) = [ N _ { \\mathbf { v } } ( \\mathbf { x } ) , \\bar { N _ { \\mathbf { c } } } ( \\mathbf { x } ) ]$ . Ideally, we would like to find the inverse of the image formation function, $[ N _ { \\bf v } , N _ { \\bf c } ] = f ^ { - 1 }$ , which separates and recovers the factors $\\mathbf { v }$ and $\\mathbf { c }$ from data samples $\\mathbf { x }$ , i.e., ", + "bbox": [ + 173, + 724, + 825, + 795 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f05d5a8c8d2a1d069976f7a15f58c01103a8388f9b9b200d56c1872194d14df2.jpg", + "text": "$$\nN _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = \\mathbf { v } \\qquad N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = \\mathbf { c } .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 801, + 632, + 819 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In practice, this is not possible because any bijective transformation of $\\mathbf { v }$ and c could be undone by $f$ and produce the same output $\\mathbf { x }$ . Therefore, we aim for $N _ { \\mathbf { v } }$ and $N _ { \\mathbf { c } }$ that satisfy the following feature disentangling properties ", + "bbox": [ + 174, + 824, + 825, + 867 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/229e09169a1617310ada564c46542252b122f24e4909b120d447fffe62bf8b90.jpg", + "text": "$$\nR _ { \\mathbf { v } } ( N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) ) = \\mathbf { v } \\qquad R _ { \\mathbf { c } } ( N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) ) = \\mathbf { c }\n$$", + "text_format": "latex", + "bbox": [ + 334, + 872, + 663, + 890 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "for all $\\mathbf { v } , \\mathbf { c }$ , and for some bijective functions $R _ { \\mathbf { v } }$ and $R _ { \\mathbf { c } }$ , so that $N _ { \\mathbf { v } }$ is invariant to $\\mathbf { c }$ and $N _ { \\mathbf { c } }$ is invariant to $\\mathbf { v }$ . ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The decoder. Let Dec be the decoder mapping features to images. The sequence encoder-decoder is constrained to form an autoencoder, so ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/bf06ce5db3f7455eea29b0e5dcee4161bdc1360ef3b9becbb352a1a666eb8a8d.jpg", + "text": "$$\n\\mathrm { D e c } ( N _ { \\mathbf { v } } ( { \\mathbf { x } } ) , N _ { \\mathbf { c } } ( { \\mathbf { x } } ) ) = { \\mathbf { x } } , \\qquad \\forall { \\mathbf { x } } .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 136, + 616, + 154 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To use the decoder for image synthesis, so that each input subvector affects only one factor in the rendered image, the ideal decoder should satisfy the data disentangling property ", + "bbox": [ + 171, + 159, + 823, + 186 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/917cc8fd290a28a204e30435cdee2a4200eedc267530b0945e3be26d05c615eb.jpg", + "text": "$$\n\\mathrm { D e c } ( N _ { \\mathbf { v } } ( f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ) , N _ { \\mathbf { c } } ( f ( \\mathbf { v } _ { 2 } , \\mathbf { c } _ { 2 } ) ) ) = f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 334, + 190, + 663, + 208 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "for any $\\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { c } _ { 1 }$ , and $\\mathbf { c } _ { 2 }$ . The equation above describes the transfer of the varying factor $\\mathbf { v } _ { 1 }$ of $\\mathbf { x } _ { 1 }$ and the common factor $\\mathbf { c } _ { 2 }$ of $\\mathbf { x } _ { 2 }$ to a new image ${ \\bf x } _ { 1 \\oplus 2 } = f ( { \\bf v } _ { 1 } , { \\bf c } _ { 2 } )$ . ", + "bbox": [ + 176, + 212, + 821, + 242 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the next section we describe our training method for disentangling. We introduce a novel adversarial term, that does not need to be conditioned on the common factor, rather it uses only image pairs, that keeps the model parameters constant. Then we address the two main challenges of disentangling, the shortcut problem and the reference ambiguity. We discuss which disentanglement properties can be (provably) achieved by our (or any) method. ", + "bbox": [ + 173, + 247, + 825, + 318 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 MODEL TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 334, + 341, + 348 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In our training procedure we use two terms in the objective function: an autoencoder loss and an adversarial loss. We describe these losses in functional form, however the components are implemented using neural networks. In all our terms we use the following sampling of independent factors ", + "bbox": [ + 173, + 359, + 825, + 415 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8cf7d20c2f999ca2f57e5412d13d36f48a19c78074f065d60b49938e9aa5cb05.jpg", + "text": "$$\n\\mathbf { c } _ { 1 } , \\mathbf { c } _ { 3 } \\sim p _ { \\mathbf { c } } , \\quad \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { v } _ { 3 } \\sim p _ { \\mathbf { v } } .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 424, + 606, + 438 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The images are formed as $\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } )$ , $\\mathbf { x } _ { 2 } = f ( \\mathbf { v } _ { 2 } , \\mathbf { c } _ { 1 } )$ and $\\mathbf { x } _ { 3 } = f ( \\mathbf { v } _ { 3 } , \\mathbf { c } _ { 3 } )$ . The images $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ share the same common factor, and $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 3 }$ are independent. In our objective functions, we use either pairs or triplets of the above images. ", + "bbox": [ + 176, + 441, + 826, + 484 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Autoencoder loss. In this term, we use images $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ with the same common factor $\\mathbf { c } _ { 1 }$ . We feed both images to the encoder. Since both images share the same $\\mathbf { c } _ { 1 }$ , we impose that the decoder should reconstruct $\\mathbf { x } _ { 1 }$ from the encoder subvector $N _ { \\mathbf { v } } ( \\mathbf { x } _ { 1 } )$ and the encoder subvector $N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } )$ , and similarly for the reconstruction of $\\mathbf { x } _ { 2 }$ . The autoencoder objective is thus defined as ", + "bbox": [ + 173, + 491, + 825, + 546 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d198eac559418e416a222cd5c4ae34b398fd38aa7b97356827fd1a0c9091aea2.jpg", + "text": "$$\n\\mathcal { L } _ { A E } \\doteq E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\left[ \\left| \\mathbf { x } _ { 1 } - \\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 1 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } ) ) \\right| ^ { 2 } + \\left| \\mathbf { x } _ { 2 } - \\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 2 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) ) \\right| ^ { 2 } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 223, + 551, + 774, + 578 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Adversarial loss. We introduce an adversarial training where the generator is our encoder-decoder pair and the discriminator Dsc is a neural network, which takes image pairs as input. The discriminator learns to distinguish between real image pairs $[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ]$ and fake ones $[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 \\oplus 1 } ]$ , where $\\mathbf { x } _ { 3 \\oplus 1 } \\doteq$ $\\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 3 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) )$ . If the encoder were ideal, the image $\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }$ would be the result of taking the common factor from $\\mathbf { x } _ { 1 }$ and the varying factor from $\\mathbf { x } _ { 3 }$ . The generator learns to fool the discriminator, so that $\\mathbf { x } _ { 3 \\oplus 1 }$ looks like the random variable $\\mathbf { x } _ { 2 }$ (the common factor is $\\mathbf { c } _ { 1 }$ and the varying factor is independent of $\\mathbf { v } _ { 1 }$ ). To this purpose, the decoder must make use of $N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } )$ , since $\\mathbf { x } _ { 3 }$ does not carry any information about $\\mathbf { c } _ { 1 }$ . The objective function is thus defined as ", + "bbox": [ + 173, + 588, + 825, + 700 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0254827e2494ad8529a8f04663895c6a092faf37fe1509920e8ab46b652ebb25.jpg", + "text": "$$\n\\mathcal { L } _ { G A N } \\doteq E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\Big [ \\log ( \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) ) \\Big ] + E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 } } \\Big [ \\log ( 1 - \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 \\oplus 1 } ) ) \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 253, + 704, + 745, + 732 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Composite loss. Finally, we optimize the weighted sum of the two losses $\\mathcal { L } = \\mathcal { L } _ { A E } + \\lambda \\mathcal { L } _ { G A N }$ ", + "bbox": [ + 176, + 742, + 803, + 758 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/522327b33c38fe4c3ca69b99c16c2158b35bd3c55f4562d00489d3bde72d2042.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathrm { D e c , E n c } } \\operatorname* { m a x } _ { \\mathrm { D s c } } \\mathcal { L } _ { A E } ( \\mathrm { D e c , E n c } ) + \\lambda \\mathcal { L } _ { G A N } ( \\mathrm { D e c , E n c , D s c } )\n$$", + "text_format": "latex", + "bbox": [ + 318, + 762, + 679, + 785 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\lambda$ regulates the relative importance of the two losses. ", + "bbox": [ + 171, + 790, + 562, + 805 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 SHORTCUT PROBLEM. ", + "text_level": 1, + "bbox": [ + 176, + 821, + 366, + 837 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Ideally, at the global minimum of $\\mathcal { L } _ { A E }$ , $N _ { \\mathbf { v } }$ relates only to the factor $\\mathbf { v }$ and $N _ { \\mathbf { c } }$ only to c. However, the encoder may map a complete description of its input into $N _ { \\mathbf { v } }$ and the decoder may completely ignore $N _ { \\mathbf { c } }$ . We call this challenge the shortcut problem. When the shortcut problem occurs, the decoder is invariant to its second input, so it does not transfer the $\\mathbf { c }$ factor correctly, ", + "bbox": [ + 173, + 847, + 825, + 904 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7d0078287b4a15239b3b5a8df7d67055394d89138ef884e274c71c9292c0de05.jpg", + "text": "$$\n\\mathrm { D e c } ( N _ { \\mathbf { v } } ( { \\mathbf { x } } _ { 3 } ) , N _ { \\mathbf { c } } ( { \\mathbf { x } } _ { 1 } ) ) = { \\mathbf { x } } _ { 3 } .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 907, + 580, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The shortcut problem can be addressed by reducing the dimensionality of $N _ { \\mathbf { v } }$ , so it cannot build a complete representation of all input images. This also forces the encoder and decoder to make use of $N _ { \\mathbf { c } }$ for the common factor. However, this strategy may not be convenient as it leads to a time consuming trial-and-error procedure to find the correct dimensionality. A better way to address the shortcut problem is to use adversarial training (7) (8). ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 1. Let $\\mathbf { x } _ { 1 }$ , $\\mathbf { x } _ { 2 }$ and $\\mathbf { x } _ { 3 }$ data samples generated according to (5), where the factors $\\mathbf { c } _ { 1 } , \\mathbf { c } _ { 3 } , \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { v } _ { 3 }$ are jointly independent, and $\\mathbf { x } _ { 3 \\oplus 1 } \\doteq D e c ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 3 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) )$ . When the global optimum of the composite loss (8) is reached, the c factor is transferred to $\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }$ , i.e. $f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 3 \\oplus 1 } ) = \\mathbf { c } _ { 1 }$ ", + "bbox": [ + 173, + 178, + 823, + 220 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. When the global optimum of (8) is reached, the distribution of real $\\left[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } \\right]$ and fake $[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 \\oplus 1 } ]$ image pairs are identical. We compute statistics of the inverse of the rendering engine of the common factor $\\bar { f } _ { \\mathbf { c } } ^ { - 1 }$ on the data. For the images $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ we obtain ", + "bbox": [ + 173, + 234, + 825, + 279 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d63503b72bdcbc84bb8b5313db9996fceb9c0fa3e4cf33136f2453fd2527a5bc.jpg", + "text": "$$\n\\begin{array} { r } { { E } _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\Big [ | f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 1 } ) - f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 2 } ) | ^ { 2 } \\Big ] = { E } _ { \\mathbf { c } _ { 1 } } \\Big [ | \\mathbf { c } _ { 1 } - \\mathbf { c } _ { 1 } | ^ { 2 } \\Big ] = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 284, + 687, + 311 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "by construction (of $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ ). For the images $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }$ we obtain ", + "bbox": [ + 171, + 315, + 637, + 332 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8ef50f6da86620ea7ff42242fee4dd9d0a38fbc043dc1de97b3292a81bcda76b.jpg", + "text": "$$\n\\begin{array} { r } { E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 } } \\Big [ | f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 1 } ) - f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 3 \\oplus 1 } ) | ^ { 2 } \\Big ] = E _ { \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } , \\mathbf { v } _ { 3 } , \\mathbf { c } _ { 3 } } \\Big [ | \\mathbf { c } _ { 1 } - \\mathbf { c } _ { 3 \\oplus 1 } | ^ { 2 } \\Big ] \\geq 0 , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 337, + 730, + 364 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where ${ \\bf c } _ { 3 \\oplus 1 } = f _ { \\bf c } ^ { - 1 } ( { \\bf x } _ { 3 \\oplus 1 } )$ . We achieve equality if and only if $\\mathbf { c } _ { 1 } = \\mathbf { c } _ { 3 \\oplus 1 }$ everywhere. ", + "bbox": [ + 174, + 371, + 732, + 387 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 REFERENCE AMBIGUITY ", + "text_level": 1, + "bbox": [ + 174, + 402, + 383, + 416 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let us consider the ideal case where we observe the space of all images. When weak labels are made available to us, we also know what images $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ share the same c factor (for example, which images have the same car). This labeling is equivalent to defining the probability density function $p _ { \\mathbf { c } }$ and the joint conditional $\\displaystyle p _ { { \\mathbf { x } } _ { 1 } , { \\mathbf { x } } _ { 2 } | { \\mathbf { c } } }$ , where ", + "bbox": [ + 174, + 428, + 823, + 486 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c5c10d21f164dc958f0ae014970750e9436e47bd027d67fbf3d27aa4f8cc5a22.jpg", + "text": "$$\np _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } ) = \\int \\delta ( \\mathbf { x } _ { 1 } - f ( \\mathbf { v } _ { 1 } , \\mathbf { c } ) ) \\delta ( \\mathbf { x } _ { 2 } - f ( \\mathbf { v } _ { 2 } , \\mathbf { c } ) ) p ( \\mathbf { v } _ { 1 } ) p ( \\mathbf { v } _ { 2 } ) d \\mathbf { v } _ { 1 } d \\mathbf { v } _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 238, + 492, + 759, + 525 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Firstly, we show that the labeling allows us to satisfy the feature disentangling property for $\\mathbf { c }$ (2). For any $[ \\mathbf { \\dot { x } } _ { 1 } , \\mathbf { x } _ { 2 } ] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } }$ we impose $N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) = N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } )$ . In particular, this equation is true for pairs when one of the two images is held fixed. Thus, a function $C ( \\mathbf { c } ) = N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } )$ can be defined, where the $C$ only depends on c, because $N _ { \\mathbf { c } }$ is invariant to $\\mathbf { v }$ . Lastly, images with the same $\\mathbf { v }$ , but different c must also result in different features, $C ( \\mathbf { c } _ { 1 } ) = N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } _ { 1 } ) ) \\neq N _ { \\mathbf { v } } ( \\mathbf { v } , \\mathbf { c } _ { 2 } ) = C ( \\mathbf { c } _ { 2 } )$ , otherwise the autoencoder constraint (3) cannot be satisfied. Then, there exists a bijective function $\\dot { R } _ { \\bf c } = { C } ^ { - 1 }$ such that property (2) is satisfied for c. Unfortunately the other disentangling properties can not provably be satisfied. ", + "bbox": [ + 173, + 530, + 825, + 643 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 1. A function $g$ reproduces the data distribution, when it generates samples $\\mathbf { y } _ { 1 } = g ( \\mathbf { v } _ { 1 } , \\mathbf { c } )$ and $\\mathbf { y } _ { 2 } = g ( \\mathbf { v } _ { 2 } , \\mathbf { c } )$ that have the same distribution as the data. Formally, $[ \\mathbf { y } _ { 1 } , \\mathbf { y } _ { 2 } ] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } }$ , where the latent factors are independent, $\\mathbf { v } _ { 1 } \\sim p _ { \\mathbf { v } }$ , $\\mathbf { v } _ { 2 } \\sim p _ { \\mathbf { v } }$ and $\\mathbf { c } \\sim p _ { \\mathbf { c } }$ . ", + "bbox": [ + 174, + 647, + 825, + 690 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The reference ambiguity occurs, when a decoder reproduces the data without satisfying the disentangling properties. ", + "bbox": [ + 174, + 700, + 823, + 729 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 2. Let $p _ { \\mathbf { v } }$ assign the same probability value to at least two different instances of v. Then, we can find encoders that reproduce the data distribution, but do not satisfy the disentangling properties for v in (2) and (4). ", + "bbox": [ + 173, + 733, + 825, + 776 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. We already saw that $N _ { \\mathbf { c } }$ satisfies (2), so we can choose $N _ { \\mathbf { c } } = f _ { \\mathbf { c } } ^ { - 1 }$ , the inverse of the rendering engine. Now we look at defining $N _ { \\mathbf { v } }$ and the decoder. The iso-probability property of $p _ { \\mathbf { v } }$ implies that there exists a mapping $T ( \\mathbf { v } , \\mathbf { c } )$ , such that $T ( \\mathbf { v } , \\mathbf { c } ) \\sim p _ { \\mathbf { v } }$ and $T ( \\mathbf { v } , \\bar { \\mathbf { c } _ { 1 } } ) \\bar { \\neq } T ( \\bar { \\mathbf { v } } , \\bar { \\mathbf { c } _ { 2 } } )$ for some $\\mathbf { v }$ and $\\mathbf { c } _ { 1 } \\neq \\mathbf { c } _ { 2 }$ . For example, let us denote with $\\mathbf { v } _ { 1 } \\neq \\mathbf { v } _ { 2 }$ two varying components such that $p _ { \\mathbf { v } } ( \\mathbf { v } _ { 1 } ) = p _ { \\mathbf { v } } ( \\mathbf { v } _ { 2 } )$ . Then, let ", + "bbox": [ + 173, + 790, + 825, + 862 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/bb4ae1e763a78a1dae1156e86bc9cd8f148c155f09fc90d3857031b049127b49.jpg", + "text": "$$\nT ( \\mathbf { v } , \\mathbf { c } ) \\dot { = } \\left\\{ \\begin{array} { l l } { \\mathbf { v } } & { \\mathrm { i f } \\ \\mathbf { v } \\neq \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } } \\\\ { \\mathbf { v } _ { 1 } } & { \\mathrm { i f } \\ \\mathbf { v } = \\mathbf { v } _ { 1 } \\lor \\mathbf { v } _ { 2 } \\mathrm { a n d } \\mathbf { c } \\in \\mathcal { C } } \\\\ { \\mathbf { v } _ { 2 } } & { \\mathrm { i f } \\ \\mathbf { v } = \\mathbf { v } _ { 1 } \\lor \\mathbf { v } _ { 2 } \\mathrm { a n d } \\mathbf { c } \\not \\in \\mathcal { C } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 351, + 869, + 647, + 921 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "and $\\mathcal { C }$ is a subset of the domain of $\\mathbf { c }$ , where $\\begin{array} { r } { \\int _ { \\mathcal { C } } p _ { \\mathbf { c } } ( \\mathbf { c } ) d \\mathbf { c } = 1 / 2 } \\end{array}$ . Now, let us define the encoder as $N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = T ( \\mathbf { v } , \\mathbf { c } )$ . By using the autoencoder constraint, the decoder satisfies ", + "bbox": [ + 169, + 102, + 823, + 133 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/54a73ddb2a2bc6d4404212dd98ca2cbb4232243f51a7f880456ef26805957772.jpg", + "text": "$$\n\\operatorname { D e c } ( N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) , N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) ) = \\operatorname { D e c } ( T ( \\mathbf { v } , \\mathbf { c } ) , \\mathbf { c } ) = f ( \\mathbf { v } , \\mathbf { c } ) .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 138, + 705, + 156 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Even though $T ( \\mathbf { v } , \\mathbf { c } )$ depends on $\\mathbf { c }$ functionally, they are statistically independent. Because $T ( \\mathbf { v } , \\mathbf { c } ) \\sim$ $p _ { \\mathbf { v } }$ and $\\mathbf { c } \\sim p _ { \\mathbf { c } }$ by construction, our encoder-decoder pair defines a data distribution identical to that given as training set ", + "bbox": [ + 173, + 161, + 825, + 204 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f6bcda88538e042aaccb0e9ecdcfead5aed1204c01d813be25799d45cf17aefe.jpg", + "text": "$$\n\\left[ \\operatorname { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } ) , \\mathbf { c } ) , \\operatorname { D e c } ( T ( \\mathbf { v } _ { 2 } , \\mathbf { c } ) , \\mathbf { c } ) \\right] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 343, + 209, + 653, + 227 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The feature disentanglement property is not satisfied because $N _ { \\mathbf { v } } ( f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ) ~ = ~ T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ~ \\neq$ $T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } ) \\ = \\ N _ { \\mathbf { v } } ( f ( { \\bar { \\mathbf { v } } } _ { 1 } , \\mathbf { c } _ { 2 } ) )$ , when $\\mathbf { c } _ { 1 } ~ \\in ~ { \\mathcal { C } }$ and $\\mathbf { c } _ { 2 } \\notin \\mathcal { C }$ . Similarly, the data disentanglement property does not hold, because $\\mathrm { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) , \\mathbf { c } _ { 1 } ) \\neq \\mathrm { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } ) , \\mathbf { c } _ { 2 } )$ . □ ", + "bbox": [ + 174, + 231, + 825, + 275 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The above proposition implies that we cannot provably disentangle all the factors of variation from weakly labeled data, even if we had access to all the data and knew the distributions $p _ { \\mathbf { v } }$ and $p _ { \\mathbf { c } }$ . ", + "bbox": [ + 173, + 289, + 823, + 318 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To better understand it, let us consider a practical example. Let $\\mathbf { v } \\sim \\mathcal { U } [ - \\pi , \\pi ]$ be the (continuous) viewpoint (the azimuth angle) and $\\mathbf { c } \\sim B ( 0 . 5 )$ the car type, where $\\mathcal { U }$ denotes the uniform distribution and $B ( 0 . 5 )$ the Bernoulli distribution with probability $p _ { \\mathbf { c } } ( \\mathbf { c } = 0 ) = p _ { \\mathbf { c } } ( \\mathbf { c } = 1 ) = 0 . 5$ (i.e., there are only 2 car types). In this case, every instance of $\\mathbf { v }$ is iso-probable in $p _ { \\mathbf { v } }$ so we have the worst scenario for the reference ambiguity. We can define the function $T ( \\mathbf { v } , \\mathbf { c } ) = \\mathbf { v } ( 2 \\mathbf { c } - 1 )$ so that the mapping of $\\mathbf { v }$ is mirrored as we change the car type. By construction $T ( \\mathbf { v } , \\mathbf { c } ) \\sim \\mathcal { U } [ - \\pi , \\pi ]$ for any $\\mathbf { c }$ and $T ( \\mathbf { v } , \\mathbf { c } _ { 1 } ) \\neq T ( \\mathbf { v } , \\mathbf { c } _ { 2 } )$ for $\\mathbf { v } \\neq 0$ and $\\mathbf { c } _ { 1 } \\neq \\mathbf { c } _ { 2 }$ . So we cannot tell the difference between $T$ and the ideal correct mapping to the viewpoint factor. This is equivalent to an encoder $N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = T ( \\mathbf { v } , \\mathbf { c } )$ that reverses the ordering of the azimuth of car 1 with respect to car 0. Each car has its own reference system, and thus it is not possible to transfer the viewpoint from one system to the other, as it is illustrated in fig. 1. ", + "bbox": [ + 173, + 324, + 825, + 478 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 494, + 341, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In our implementation we use convolutional neural networks for all the models. We denote with $\\theta$ the parameters associated to each network. Then, the optimization of the composite loss can be written as ", + "bbox": [ + 174, + 520, + 823, + 549 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ad8a7a239b6e53b3ca59924edbecb1c1d2d6c239f4721fbd445add5e1ca1471a.jpg", + "text": "$$\n\\hat { \\theta } _ { \\mathrm { { D e c } } } , \\hat { \\theta } _ { \\mathrm { { E n c } } } , \\hat { \\theta } _ { \\mathrm { { D s c } } } = \\arg \\operatorname* { m i n } _ { \\theta _ { \\mathrm { { D e c } } } , \\theta _ { \\mathrm { { E n c } } } } \\operatorname* { m a x } _ { \\theta _ { \\mathrm { { D s c } } } } \\mathcal { L } ( \\theta _ { \\mathrm { { D e c } } } , \\theta _ { \\mathrm { { E n c } } } , \\theta _ { \\mathrm { { D s c } } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 554, + 673, + 582 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We choose $\\lambda = 1$ and also add regularization to the adversarial loss so that each logarithm has a minimum value. We define $\\log _ { \\epsilon } \\bar { \\mathrm { D s c } } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) = \\log ( \\epsilon + \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) )$ (and similarly for the other logarithmic term) and use $\\epsilon = 1 0 ^ { - 1 2 }$ . The main components of our neural network are shown in Fig. 2. The architecture of the encoder and the decoder were taken from DCGAN Radford et al. (2015), with slight modifications. We added fully connected layers at the output of the encoder and to the input of the decoder. For the discriminator we used a simplified version of the VGG Simonyan & Zisserman (2014) network. As the input to the discriminator is an image pair, we concatenate them along the color channels. ", + "bbox": [ + 173, + 585, + 825, + 698 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Normalization. In our architecture both the encoder and the decoder networks use blocks with a convolutional layer, a nonlinear activation function (ReLU/leaky ReLU) and a normalization layer, typically, batch normalization (BN). As an alternative to BN we consider the recently introduced instance normalization (IN) Ulyanov et al. (2017). The main difference between BN and IN is that the latter just computes the mean and standard deviation across the spatial domain of the input and not along the batch dimension. Thus, the shift and scaling for the output of each layer is the same at every iteration for the same input image. In practice, we find that IN improves the performance. ", + "bbox": [ + 173, + 704, + 825, + 803 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 821, + 326, + 838 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We tested our method on the MNIST, Sprites and ShapeNet datasets. We performed ablation studies on the shortcut problem using ShapeNet cars. We focused on the effect of the feature dimensionality and having the adversarial term $( \\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N } )$ or not $( \\mathcal { L } _ { A E } )$ . We also show that in most cases the reference ambiguity does not arise in practice (MNIST, Sprites, ShapeNet cars), we can only observe it when the data is more complex (ShapeNet chairs). ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/90a0f2e4ff16668ce677bfd81f19be4a9e730377fbd5bbdf49085baa9daeb981.jpg", + "image_caption": [ + "Figure 2: Learning to disentangle factors of variation. The scheme above shows how the encoder (Enc), the decoder (Dec) and the discriminator (Dsc) are trained with input triplets. The components with the same name share weights. " + ], + "image_footnote": [], + "bbox": [ + 354, + 101, + 643, + 285 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 SHORTCUT PROBLEM ", + "text_level": 1, + "bbox": [ + 176, + 376, + 361, + 390 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "ShapeNet cars. The ShapeNet dataset Chang et al. (2015) contains 3D objects than we can render from different viewpoints. We consider only one category (cars) for a set of fixed viewpoints. Cars have high intraclass variability and they do not have rotational symmetries. We used approximately 3K car types for training and 300 for testing. We rendered 24 possible viewpoints around each object in a full circle, resulting in 80K images in total. The elevation was fixed to 15 degrees and azimuth angles were spaced 15 degrees apart. We normalized the size of the objects to fit in a $1 0 0 \\times 1 0 0$ pixel bounding box, and placed it in the middle of a $1 2 8 \\times 1 2 8$ pixel image. ", + "bbox": [ + 173, + 404, + 825, + 502 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Fig. 3 shows the attribute transfer on the Shapenet cars. We compare the methods $\\mathcal { L } _ { A E }$ and $\\mathcal { L } _ { A E } +$ $\\mathcal { L } _ { G A N }$ with different feature dimension of $N _ { \\mathbf { v } }$ . The size of the common feature $N _ { \\mathbf { c } }$ was fixed to 1024 dimensions. We can observe that the transferring performance degrades for $\\mathcal { L } _ { A E }$ , when we increase the feature size of $N _ { \\mathbf { v } }$ . As expected, the autoencoder takes the shortcut and tries to store all information into $N _ { \\mathbf { v } }$ . The model $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ instead renders images without loss of quality, independently of the feature dimension. ", + "bbox": [ + 173, + 508, + 826, + 593 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/5c12b702322a9c6bbc47ab8de3e5581a0bb893cfe305a9a1e478823279c0c10d.jpg", + "image_caption": [ + "Figure 3: Feature transfer on Shapenet. (a) synthesized images with $\\mathcal { L } _ { A E }$ , where the top row shows images from which the car type is taken. The second, third and fourth row show the decoder renderings using 2, 16 and 128 dimensions for the feature $N _ { \\mathbf { v } }$ . (b) images synthesized with $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ . The setting for the inputs and feature dimensions are the same as in (a). " + ], + "image_footnote": [], + "bbox": [ + 186, + 609, + 812, + 738 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Fig. 4 we visualize the t-SNE embeddings of the $N _ { \\mathbf { v } }$ features for several models using different feature sizes. For the $2 D$ case, we do not modify the data. We can see that both $\\mathcal { L } _ { A E }$ with 2 dimensions and $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ with 128 separate the viewpoints well, but $\\mathcal { L } _ { A E }$ with 128 dimensions does not due to the shortcut problem. We investigate the effect of dimensionality of the $N _ { \\mathbf { v } }$ features on the nearest neighbor classification task. The performance is measured by the mean average precision. For $N _ { \\mathbf { v } }$ we use the viewpoint as ground truth. Fig. 4 also shows the results on $\\mathcal { L } _ { A E }$ and $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ models with different $N _ { \\mathbf { v } }$ feature dimensions. The dimension of $N _ { \\mathbf { c } }$ was fixed to 1024 for this experiment. One can now see quantitatively that $\\mathcal { L } _ { A E }$ is sensitive to the size of $N _ { \\mathbf { v } }$ , while $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ is not. $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ also achieves a better performance. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/682c15c0ae852d517b1ac69b302654a800de4fd0f02c77d85fbeaeb17fe5a1ca.jpg", + "image_caption": [ + "Figure 4: The effect of dimensions and objective function on $N _ { v }$ features. (a), (b), (c) t-SNE embeddings on $N _ { \\mathbf { v } }$ features. Colors correspond to the ground truth viewpoint. The objective functions and the $N _ { \\mathbf { v } }$ dimensions are: (a) $\\mathcal { L } _ { A E }$ 2 dim, (b) $\\mathcal { L } _ { A E }$ 128 dim, (c) $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N } \\mathrm { ~ 1 ~ }$ 128 dim. (d) Mean average precision curves for the viewpoint prediction from the viewpoint feature using different models and dimensions for $N _ { \\mathbf { v } }$ . " + ], + "image_footnote": [], + "bbox": [ + 240, + 99, + 758, + 214 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/b952c7556c42a601464b737e324c1e90893cbadb4fec0f246e68d9f3dde1e3eb.jpg", + "table_caption": [ + "Table 1: Nearest neighbor classification on $N _ { \\mathbf { v } }$ and $N _ { \\mathbf { c } }$ features using different normalization techniques on ShapeNet cars. " + ], + "table_footnote": [], + "table_body": "
NormalizationNv mAPNc mAP
None0.470.13
Batch0.500.08
Instance0.500.20
", + "bbox": [ + 361, + 353, + 635, + 425 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 452, + 823, + 481 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare the different normalization choices in Table 1. We evaluate the case when batch, instance and no normalization are used and compute the performance on the nearest neighbor classification task. We fixed the feature dimensions at 1024 for both $N _ { \\mathbf { v } }$ and $N _ { \\mathbf { c } }$ features in all normalization cases. We can see that both batch and instance normalization perform equally well on viewpoint classification and no normalization is slightly worse. For the car type classification instance normalization is clearly better. ", + "bbox": [ + 173, + 487, + 826, + 571 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 REFERENCE AMBIGUITY ", + "text_level": 1, + "bbox": [ + 174, + 590, + 383, + 604 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "MNIST. The MNIST dataset LeCun et al. (1998) contains handwritten grayscale digits of size $2 8 \\times 2 8$ pixel. There are 60K images of 10 classes for training and 10K for testing. The common factor is the digit class and the varying factor is the intraclass variation. We take image pairs that have the same digit for training, and use our full model $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ with dimensions 64 for $N _ { \\mathbf { v } }$ and 64 for $N _ { \\mathbf { c } }$ . In Fig. 5 (a) and (b) we show the transfer of varying factors. Qualitatively, both our method and Mathieu et al. (2016) perform well. We observe neither the reference ambiguity nor the shortcut problem in this case. ", + "bbox": [ + 173, + 617, + 825, + 715 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/19dbb062afd2ee7e2e211be55c604266d4e81e0c4b40072158decf54298db406.jpg", + "image_caption": [ + "Figure 5: Renderings of transferred features. In all figures the variable factor is transferred from the left column and the common factor from the top row. (a) MNIST Mathieu et al. (2016); (b) MNIST (ours); (c) Sprites Mathieu et al. (2016); (d) Sprites (ours). " + ], + "image_footnote": [], + "bbox": [ + 181, + 728, + 816, + 866 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/55049684dd4908d747a5bd58b28114a25a2f6b44c1aa8542f673f3beeb04f571.jpg", + "image_caption": [ + "Figure 6: Attribute transfer on ShapeNet. For both subfigures the viewpoint is taken from the leftmost column and the car/chair type is taken from the first row. (a) Cars: the factors are transferred correctly. (b) Chairs: in the bottom three rows the viewpoint is not transferred correctly due to the reference ambiguity. " + ], + "image_footnote": [], + "bbox": [ + 192, + 98, + 803, + 321 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sprites. The Sprites dataset Reed et al. (2015) contains 60 pixel color images of animated characters (sprites). There are 672 sprites, 500 for training, 100 for testing and 72 for validation. Each sprite has 20 animations and 178 images, so the full dataset has 120K images in total. There are many changes in the appearance of the sprites, they differ in their body shape, gender, hair, armour, arm type, greaves, and weapon. We consider character identity as the common factor and the pose as the varying factor. We train our system using image pairs of the same sprite and do not exploit labels on their pose. We train the $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ model with dimensions 64 for $N _ { \\mathbf { v } }$ and 448 for $N _ { \\mathbf { c } }$ . Fig. 5 (c) and (d) show results on the attribute transfer task. Both our method and Mathieu et al. (2016)’s transfer the identity of the sprites correctly, the reference ambiguity does not arise. ", + "bbox": [ + 173, + 420, + 825, + 546 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ShapeNet chairs. We render the ShapeNet chairs with the same settings (viewpoints, image size) as the cars. There are 3500 chair types for training and 3200 for testing, so the dataset contains 160K images. We trained $\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }$ , and set the feature dimensions to 1024 for both $N _ { \\mathbf { v } }$ and $N _ { \\mathbf { c } }$ . In Fig. 6 we show results on attribute transfer and compare it with ShapeNet cars. We found that the reference ambiguity does not emerge for cars, but it does for chairs, possibly due to the higher complexity, as cars have much less variability than chairs. ", + "bbox": [ + 174, + 553, + 825, + 637 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 661, + 328, + 678 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper we studied the challenges of disentangling factors of variation, mainly the shortcut problem and the reference ambiguity. The shortcut problem occurs when all information is stored in only one feature chunk, while the other is ignored. The reference ambiguity means that the reference in which a factor is interpreted, may depend on other factors. This makes the attribute transfer ambiguous. We introduced a novel training of autoencoders to solve disentangling using image triplets. We showed theoretically and experimentally how to keep the shortcut problem under control through adversarial training, and enable to use large feature dimensions. We proved that the reference ambiguity is inherently present in the disentangling task when weak labels are used. Most importantly this can be stated independently of the learning algorithm. We demonstrated that training and transfer of factors of variation may not be guaranteed. However, in practice we observe that our trained model works well on many datasets and exhibits good generalization capabilities. ", + "bbox": [ + 174, + 695, + 825, + 849 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 873, + 285, + 888 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE transactions on pattern analysis and machine intelligence, 35(8):1798–1828, 2013. ", + "bbox": [ + 176, + 897, + 823, + 922 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Hervé Bourlard and Yves Kamp. Auto-association by multilayer perceptrons and singular value decomposition. Biological cybernetics, 59(4):291–294, 1988. \nAngel X. Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, Jianxiong Xiao, Li Yi, and Fisher Yu. ShapeNet: An Information-Rich 3D Model Repository. Technical Report arXiv:1512.03012 [cs.GR], 2015. \nXi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In NIPS, 2016. \nBrian Cheung, Jesse A Livezey, Arjun K Bansal, and Bruno A Olshausen. 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Coupled generative adversarial networks. In Advances in Neural Information Processing Systems, pp. 469–477, 2016. \nChristos Louizos, Kevin Swersky, Yujia Li, Max Welling, and Richard Zemel. The variational fair autoencoder. In ICLR, 2016. \nMichael F Mathieu, Junbo Jake Zhao, Junbo Zhao, Aditya Ramesh, Pablo Sprechmann, and Yann LeCun. Disentangling factors of variation in deep representation using adversarial training. In Advances in Neural Information Processing Systems, pp. 5041–5049, 2016. \nAlec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv:1511.06434, 2015. \nScott Reed, Kihyuk Sohn, Yuting Zhang, and Honglak Lee. Learning to disentangle factors of variation with manifold interaction. In Proceedings of the 31st International Conference on Machine Learning (ICML-14), pp. 1431–1439, 2014. \nScott E Reed, Yi Zhang, Yuting Zhang, and Honglak Lee. Deep visual analogy-making. In Advances in Neural Information Processing Systems, pp. 1252–1260, 2015. \nZhixin Shu, Ersin Yumer, Sunil Hadap, Kalyan Sunkavalli, Eli Shechtman, and Dimitris Samaras. Neural face editing with intrinsic image disentangling. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. \nKaren Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. \nLuan Tran, Xi Yin, and Xiaoming Liu. Disentangled representation learning gan for pose-invariant face recognition. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. \nDmitry Ulyanov, Andrea Vedaldi, and Victor S. Lempitsky. Improved texture networks: Maximizing quality and diversity in feed-forward stylization and texture synthesis. In CVPR, 2017. \nKihyuk Sohn Dimitris Metaxas Manmohan Chandraker Xi Peng, Xiang Yu. Reconstruction for feature disentanglement in pose-invariant face recognition. arXiv:1702.03041, 2017. \nJimei Yang, Scott E Reed, Ming-Hsuan Yang, and Honglak Lee. Weakly-supervised disentangling with recurrent transformations for 3d view synthesis. In NIPS, 2015. ", + "bbox": [ + 169, + 32, + 828, + 897 + ], + "page_idx": 9 + } +] \ No newline at end of file diff --git a/parse/train/SkmiegW0b/SkmiegW0b_middle.json b/parse/train/SkmiegW0b/SkmiegW0b_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..468137e4333bc971585d998494032f4a31c4e8f7 --- /dev/null +++ b/parse/train/SkmiegW0b/SkmiegW0b_middle.json @@ -0,0 +1,33307 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 508, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 508, + 98 + ], + "score": 1.0, + "content": "CHALLENGES IN DISENTANGLING INDEPENDENT FAC-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 258, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 258, + 118 + ], + "score": 1.0, + "content": "TORS OF VARIATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 211, + 469, + 376 + ], + "lines": [ + { + "bbox": [ + 142, + 212, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 142, + 212, + 469, + 223 + ], + "score": 1.0, + "content": "We study the problem of building models that disentangle independent factors of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 222, + 469, + 235 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 469, + 235 + ], + "score": 1.0, + "content": "variation. 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One could build a model to render images,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 499, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 499, + 524 + ], + "score": 1.0, + "content": "where each input varies only one attribute of the output, and to transfer attributes between images.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "When labeling is possible and available, supervised learning can be used to solve this task. In general,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "however, some attributes may not be easily quantifiable (e.g., style). 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Such models encode features that can efficiently be used for classification", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 233, + 469, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 246 + ], + "score": 1.0, + "content": "and to transfer attributes between different images in image synthesis. As data", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 469, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 469, + 256 + ], + "score": 1.0, + "content": "we use a weakly labeled training set, where labels indicate what single factor has", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 256, + 469, + 267 + ], + "spans": [ + { + "bbox": [ + 142, + 256, + 469, + 267 + ], + "score": 1.0, + "content": "changed between two data samples, although the relative value of the change is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "unknown. This labeling is of particular interest as it may be readily available", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 277, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 142, + 277, + 469, + 289 + ], + "score": 1.0, + "content": "without annotation costs. We introduce an autoencoder model and train it through", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "constraints on image pairs and triplets. We show the role of feature dimensionality", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 300, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 142, + 300, + 469, + 311 + ], + "score": 1.0, + "content": "and adversarial training theoretically and experimentally. We formally prove the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 470, + 322 + ], + "score": 1.0, + "content": "existence of the reference ambiguity, which is inherently present in the disentan-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 321, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 142, + 321, + 470, + 333 + ], + "score": 1.0, + "content": "gling task when weakly labeled data is used. The numerical value of a factor", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "score": 1.0, + "content": "has different meaning in different reference frames. When the reference depends", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 469, + 356 + ], + "score": 1.0, + "content": "on other factors, transferring that factor becomes ambiguous. We demonstrate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "score": 1.0, + "content": "experimentally that the proposed model can successfully transfer attributes on", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 364, + 442, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 442, + 378 + ], + "score": 1.0, + "content": "several datasets, but show also cases when the reference ambiguity occurs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 212, + 470, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 208, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 208, + 413 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "One way to simplify the problem of classifying or regressing attributes of interest from data is to build", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "an intermediate representation, a feature, where the information about the attributes is better separated", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "than in the input data. Better separation means that some entries of the feature vary only with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "respect to one and only one attribute. In this way, classifiers and regressors would not need to build", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "invariance to many nuisance attributes. Instead, they could devote more capacity to discriminating the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "attributes of interest, and possibly achieve better performance. We call this task disentangling factors", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "of variation, and we identify attributes with the factors. In addition to facilitating classification and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 498, + 507, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 507, + 514 + ], + "score": 1.0, + "content": "regression, this task is beneficial to image synthesis. One could build a model to render images,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 499, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 499, + 524 + ], + "score": 1.0, + "content": "where each input varies only one attribute of the output, and to transfer attributes between images.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 423, + 507, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "When labeling is possible and available, supervised learning can be used to solve this task. In general,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "however, some attributes may not be easily quantifiable (e.g., style). Therefore, we consider using", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "weak labeling, where we only know what attribute has changed between two images, although we do", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "not know by how much. This type of labeling may be readily available in many cases without manual", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "annotation. For example, image pairs from a stereo system are automatically labeled with a viewpoint", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "change, albeit unknown. A practical model that can learn from these labels is an encoder-decoder pair", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 594, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 605 + ], + "score": 1.0, + "content": "subject to a reconstruction constraint. In this model the weak labels can be used to define similarities", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 603, + 361, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 361, + 617 + ], + "score": 1.0, + "content": "between subsets of the feature obtained from two input images.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 527, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 621, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "We introduce a novel adversarial training of autoencoders to solve the disentangling task when only", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "weak labels are available. Compared to previous methods, our discriminator is not conditioned on", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 642, + 507, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 507, + 656 + ], + "score": 1.0, + "content": "class labels, but takes image pairs as inputs. This way the number of parameters can be kept constant.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 620, + 507, + 656 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "We describe the shortcut problem, where all the the information is encoded only in one part of the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "feature, while other part is completely ignored, as fig. 1 illustrates. We prove our method solves this", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 682, + 282, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 282, + 694 + ], + "score": 1.0, + "content": "problem and demonstrate it experimentally.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 659, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "We formally prove existence of the reference ambiguity, that is inherently present in the disentangling", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "task when weak labels are used. Thus no algorithm can provably learn disentangling. As fig. 1 shows,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "the reference ambiguity means that a factor (for example viewpoint) can have different meaning when", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "using a different reference frame that depends on another factor (for example car type). We show", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 493, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 493, + 107 + ], + "score": 1.0, + "content": "experimentally that this ambiguity rarely arise, we can observe it only when the data is complex.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "using a different reference frame that depends on another factor (for example car type). We show", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 493, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 493, + 107 + ], + "score": 1.0, + "content": "experimentally that this ambiguity rarely arise, we can observe it only when the data is complex.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "image", + "bbox": [ + 137, + 118, + 474, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 137, + 118, + 474, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 118, + 474, + 232 + ], + "spans": [ + { + "bbox": [ + 137, + 118, + 474, + 232 + ], + "score": 0.973, + "type": "image", + "image_path": "fded6d23f694cb78ea35bd328451d6c6df4a13df4ab823b7f12b98e6e7a12b42.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 137, + 118, + 474, + 156.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 137, + 156.0, + 474, + 194.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 137, + 194.0, + 474, + 232.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 240, + 505, + 308 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "score": 1.0, + "content": "Figure 1: Challenges of disentangling. We disentangle the feature into two parts, one representing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "the viewpoint, the other the car type. We use the features for attribute transfer. For all subfigures", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "the viewpoint feature is taken from the leftmost column and the car type feature is taken from the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "topmost row. (a) ideal solution: the viewpoint and the car type are transferred correctly. (b) shortcut", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "problem: the car type is not transferred. (c) reference ambiguity: compared to the others the viewpoint", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 297, + 256, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 256, + 308 + ], + "score": 1.0, + "content": "orientation is flipped for the blue car.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + } + ], + "index": 5.25 + }, + { + "type": "title", + "bbox": [ + 108, + 332, + 209, + 345 + ], + "lines": [ + { + "bbox": [ + 104, + 331, + 210, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 331, + 210, + 347 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 506, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "Autoencoders. Autoencoders in Bourlard & Kamp (1988), Hinton & Salakhutdinov (2006), Bengio", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 309, + 381 + ], + "score": 1.0, + "content": "et al. (2013) learn to reconstruct the input data as", + "type": "text" + }, + { + "bbox": [ + 309, + 369, + 383, + 381 + ], + "score": 0.9, + "content": "\\mathbf { x } = \\mathrm { D e c } ( \\mathrm { E n c } ( \\mathbf { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 369, + 415, + 381 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 415, + 369, + 446, + 381 + ], + "score": 0.87, + "content": "\\operatorname { E n c } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "is the internal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "image representation (the encoder) and Dec (the decoder) reconstructs the input of the encoder.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 415, + 403 + ], + "score": 1.0, + "content": "Variational autoencoders in Kingma & Welling (2014) use a generative model;", + "type": "text" + }, + { + "bbox": [ + 415, + 390, + 502, + 403 + ], + "score": 0.93, + "content": "p ( \\bar { \\mathbf { x } } , \\mathbf { z } ) = p ( \\mathbf { x } | \\mathbf { z } ) p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 390, + 506, + 403 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 402, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 134, + 414 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 403, + 143, + 412 + ], + "score": 0.53, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 402, + 294, + 414 + ], + "score": 1.0, + "content": "is the observed data (images), and", + "type": "text" + }, + { + "bbox": [ + 294, + 404, + 301, + 412 + ], + "score": 0.61, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 402, + 506, + 414 + ], + "score": 1.0, + "content": "are latent variables. The encoder estimates the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 218, + 425 + ], + "score": 1.0, + "content": "parameters of the posterior,", + "type": "text" + }, + { + "bbox": [ + 219, + 413, + 289, + 424 + ], + "score": 0.93, + "content": "\\operatorname { E n c } ( \\mathbf { x } ) = p ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 412, + 506, + 425 + ], + "score": 1.0, + "content": ", and the decoder estimates the conditional likelihood,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 423, + 507, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 177, + 436 + ], + "score": 0.92, + "content": "\\mathrm { D e c } ( \\mathbf { z } ) = p ( \\mathbf { x } | \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 423, + 507, + 437 + ], + "score": 1.0, + "content": ". In Hinton et al. (2011) autoencoders are trained with transformed image input pairs.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "The relative transformation parameters are also fed to the network. 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CoGAN,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "Liu & Tuzel (2016) learns the joint distribution of multi-domain images by having generators and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "discriminators in each domain, and sharing their weights. They can transform images between", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "domains without being given correspondences. InfoGan, Chen et al. (2016) learns a subset of factors", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 594, + 408, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 408, + 607 + ], + "score": 1.0, + "content": "of variation by reproducing parts of the input vector with the discriminator.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 484, + 506, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 504, + 623 + ], + "score": 1.0, + "content": "Disentangling and independence. Many recent methods use neural networks for disentangling", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "features, with various degrees of supervision. In Xi Peng (2017) multi-task learning is used with full", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "supervision for pose invariant face recognition. Using both identity and pose labels Tran et al. (2017)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "can learn pose invariant features and synthesize frontalized faces from any pose. In Yang et al. (2015)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "score": 1.0, + "content": "autoencoders are used to generate novel viewpoints of objects. They disentangle the object category", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "factor from the viewpoint factor by using as explicit supervision signals: the relative viewpoint", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "transformations between image pairs. In Cheung et al. (2014) the output of the encoder is split in two", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "parts: one represents the class label and the other represents the nuisance factors. Their objective", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "function has a penalty term for misclassification and a cross-covariance cost to disentangle class from", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 708, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 507, + 723 + ], + "score": 1.0, + "content": "nuisance factors. Hierarchical Boltzmann Machines are used in Reed et al. (2014) for disentangling.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "A subset of hidden units are trained to be sensitive to a specific factor of variation, while being", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "invariant to others. Variational Fair Autoencoders Louizos et al. (2016) learn a representation that is", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "invariant to specific nuisance factors, while retaining as much information as possible. Autoencoders", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "can also be used for visual analogy Reed et al. (2015). GAN is used for disentangling intrinsic image", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 506, + 129 + ], + "score": 1.0, + "content": "factors (albedo and normal map) in Shu et al. (2017) without using ground truth labeling. They", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 458, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 458, + 140 + ], + "score": 1.0, + "content": "achieve this by explicitly modeling the physics of the image formation in their network.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 611, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "invariant to others. Variational Fair Autoencoders Louizos et al. (2016) learn a representation that is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "invariant to specific nuisance factors, while retaining as much information as possible. Autoencoders", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "can also be used for visual analogy Reed et al. (2015). GAN is used for disentangling intrinsic image", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 506, + 129 + ], + "score": 1.0, + "content": "factors (albedo and normal map) in Shu et al. (2017) without using ground truth labeling. They", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 458, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 458, + 140 + ], + "score": 1.0, + "content": "achieve this by explicitly modeling the physics of the image formation in their network.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "The work most related to ours is Mathieu et al. (2016), where an autoencoder restores an image", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "from another by swapping parts of the internal image representation. Their main improvement over", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "Reed et al. (2015) is the use of adversarial training, which allows for learning with image pairs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "instead of image triplets. Therefore, expensive labels like viewpoint alignment between different", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "car types are no longer needed. One of the differences between this method and ours is that it trains", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "a discriminator for each of the given labels. A benefit of this approach is the higher selectivity of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "the discriminator, but a drawback is that the number of model parameters grows linearly with the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "number of labels. In contrast, we work with image pairs and use a single discriminator so that our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "method is uninfluenced by the number of labels. Moreover, we show formally and experimentally the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 299, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 299, + 254 + ], + "score": 1.0, + "content": "difficulties of disentangling factors of variation.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 270, + 336, + 282 + ], + "lines": [ + { + "bbox": [ + 104, + 268, + 338, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 338, + 284 + ], + "score": 1.0, + "content": "3 DISENTANGLING FACTORS OF VARIATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "We are interested in the design and training of two models. One should map a data sample (e.g., an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "image) to a feature that is explicitly partitioned into subvectors, each associated to a specific factor", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "of variation. The other model should map this feature back to an image. We call the first model the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "encoder and the second model the decoder. For example, given the image of a car we would like", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "the encoder to yield a feature with two subvectors: one related to the car viewpoint, and the other", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "related to the car type. The subvectors of the feature obtained from the encoder should be useful", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "for classification or regression of the corresponding factor that they depend on (the car viewpoint", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "and type in the example). This separation would also be very useful to the decoder. It would enable", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "score": 1.0, + "content": "advanced editing of images, for example, the transfer of the viewpoint or car types from an image to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "another, by swapping the corresponding subvectors. Next, we introduce our model of the data and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 293, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 293, + 415 + ], + "score": 1.0, + "content": "formal definitions of our encoder and decoder.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 299, + 433 + ], + "score": 1.0, + "content": "Data model. We assume that our observed data", + "type": "text" + }, + { + "bbox": [ + 299, + 423, + 307, + 431 + ], + "score": 0.59, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "is generated through some unknown deterministic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 231, + 444 + ], + "score": 1.0, + "content": "invertible and smooth process", + "type": "text" + }, + { + "bbox": [ + 231, + 432, + 239, + 443 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 432, + 353, + 444 + ], + "score": 1.0, + "content": "that depends on the factors", + "type": "text" + }, + { + "bbox": [ + 353, + 434, + 361, + 442 + ], + "score": 0.51, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 432, + 380, + 444 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 434, + 387, + 442 + ], + "score": 0.29, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 432, + 421, + 444 + ], + "score": 1.0, + "content": ", so that", + "type": "text" + }, + { + "bbox": [ + 421, + 432, + 472, + 444 + ], + "score": 0.93, + "content": "\\mathbf { x } = f ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 432, + 506, + 444 + ], + "score": 1.0, + "content": ". In our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 176, + 455 + ], + "score": 1.0, + "content": "earlier example,", + "type": "text" + }, + { + "bbox": [ + 176, + 444, + 184, + 453 + ], + "score": 0.63, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 442, + 239, + 455 + ], + "score": 1.0, + "content": "is an image,", + "type": "text" + }, + { + "bbox": [ + 240, + 444, + 247, + 452 + ], + "score": 0.55, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 442, + 397, + 455 + ], + "score": 1.0, + "content": "is a viewpoint, c is a car type, and", + "type": "text" + }, + { + "bbox": [ + 397, + 443, + 405, + 454 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "is the rendering engine.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 235, + 466 + ], + "score": 1.0, + "content": "It is reasonable to assume that", + "type": "text" + }, + { + "bbox": [ + 236, + 454, + 243, + 465 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "is invertible, as for most cases the factors are readily apparent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 223, + 477 + ], + "score": 1.0, + "content": "form the image. We assume", + "type": "text" + }, + { + "bbox": [ + 223, + 465, + 231, + 476 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "is smooth, because a small change in the factors should only result", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "in a small change in the image and vice versa. We denote the inverse of the rendering engine as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 485, + 385, + 501 + ], + "spans": [ + { + "bbox": [ + 107, + 486, + 181, + 499 + ], + "score": 0.93, + "content": "f ^ { - 1 } = [ f _ { \\mathbf { v } } ^ { - 1 } , f _ { \\mathbf { c } } ^ { - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 485, + 385, + 501 + ], + "score": 1.0, + "content": ", where the subscript refers to the recovered factor.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 363, + 516 + ], + "score": 1.0, + "content": "Weak labeling. In the training we are given pairs of images", + "type": "text" + }, + { + "bbox": [ + 363, + 505, + 375, + 514 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 502, + 395, + 516 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 505, + 407, + 514 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 502, + 496, + 516 + ], + "score": 1.0, + "content": ", where they differ in", + "type": "text" + }, + { + "bbox": [ + 496, + 505, + 504, + 514 + ], + "score": 0.32, + "content": "\\mathbf { v }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 513, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 506, + 528 + ], + "score": 1.0, + "content": "(varying factor), but they have the same c (common factor). We also assume that the two varying", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 524, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 350, + 539 + ], + "score": 1.0, + "content": "factors and the common factor are sampled independently,", + "type": "text" + }, + { + "bbox": [ + 350, + 527, + 388, + 537 + ], + "score": 0.73, + "content": "\\mathbf { v } _ { 1 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 524, + 392, + 539 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 392, + 527, + 430, + 537 + ], + "score": 0.75, + "content": "\\mathbf { v } _ { 2 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 524, + 449, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 527, + 480, + 537 + ], + "score": 0.89, + "content": "\\mathbf { c } \\sim p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 524, + 506, + 539 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 205, + 550 + ], + "score": 1.0, + "content": "images are generated as", + "type": "text" + }, + { + "bbox": [ + 205, + 536, + 263, + 548 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 1 } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 536, + 282, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 536, + 340, + 549 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 2 } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 536, + 506, + 550 + ], + "score": 1.0, + "content": ". We call this labeling weak, because we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 286, + 559 + ], + "score": 1.0, + "content": "do not know the absolute values of either the", + "type": "text" + }, + { + "bbox": [ + 286, + 550, + 294, + 558 + ], + "score": 0.51, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 547, + 475, + 559 + ], + "score": 1.0, + "content": "or c factors or even relative changes between", + "type": "text" + }, + { + "bbox": [ + 475, + 549, + 487, + 558 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 558, + 392, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 118, + 569 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 558, + 392, + 570 + ], + "score": 1.0, + "content": ". All we know is that the image pairs share the same common factor.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "The encoder. Let Enc be the encoder mapping images to features. For simplicity, we consider", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 290, + 599 + ], + "score": 1.0, + "content": "features split into only two column subvectors,", + "type": "text" + }, + { + "bbox": [ + 291, + 587, + 305, + 597 + ], + "score": 0.89, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 586, + 322, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 323, + 586, + 336, + 597 + ], + "score": 0.89, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 586, + 480, + 599 + ], + "score": 1.0, + "content": ", one associated to the varying factor", + "type": "text" + }, + { + "bbox": [ + 480, + 588, + 487, + 596 + ], + "score": 0.48, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 363, + 610 + ], + "score": 1.0, + "content": "the other associated to the common factor c. Then, we have that", + "type": "text" + }, + { + "bbox": [ + 363, + 597, + 470, + 609 + ], + "score": 0.9, + "content": "\\mathrm { E n c } ( \\mathbf { x } ) = [ N _ { \\mathbf { v } } ( \\mathbf { x } ) , \\bar { N _ { \\mathbf { c } } } ( \\mathbf { x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 596, + 506, + 610 + ], + "score": 1.0, + "content": ". Ideally,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 606, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 369, + 622 + ], + "score": 1.0, + "content": "we would like to find the inverse of the image formation function,", + "type": "text" + }, + { + "bbox": [ + 370, + 609, + 435, + 620 + ], + "score": 0.92, + "content": "[ N _ { \\bf v } , N _ { \\bf c } ] = f ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 606, + 506, + 622 + ], + "score": 1.0, + "content": ", which separates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 617, + 340, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 203, + 632 + ], + "score": 1.0, + "content": "and recovers the factors", + "type": "text" + }, + { + "bbox": [ + 204, + 621, + 211, + 629 + ], + "score": 0.7, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 617, + 229, + 632 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 229, + 621, + 236, + 629 + ], + "score": 0.4, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 617, + 312, + 632 + ], + "score": 1.0, + "content": "from data samples", + "type": "text" + }, + { + "bbox": [ + 312, + 621, + 319, + 629 + ], + "score": 0.66, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 617, + 340, + 632 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 635, + 387, + 649 + ], + "lines": [ + { + "bbox": [ + 224, + 635, + 387, + 649 + ], + "spans": [ + { + "bbox": [ + 224, + 635, + 387, + 649 + ], + "score": 0.91, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = \\mathbf { v } \\qquad N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = \\mathbf { c } .", + "type": "interline_equation", + "image_path": "f05d5a8c8d2a1d069976f7a15f58c01103a8388f9b9b200d56c1872194d14df2.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 224, + 635, + 387, + 649 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 384, + 667 + ], + "score": 1.0, + "content": "In practice, this is not possible because any bijective transformation of", + "type": "text" + }, + { + "bbox": [ + 384, + 655, + 392, + 663 + ], + "score": 0.71, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 652, + 497, + 667 + ], + "score": 1.0, + "content": "and c could be undone by", + "type": "text" + }, + { + "bbox": [ + 497, + 654, + 504, + 665 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 223, + 677 + ], + "score": 1.0, + "content": "and produce the same output", + "type": "text" + }, + { + "bbox": [ + 224, + 666, + 231, + 674 + ], + "score": 0.68, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 663, + 325, + 677 + ], + "score": 1.0, + "content": ". Therefore, we aim for", + "type": "text" + }, + { + "bbox": [ + 325, + 665, + 340, + 675 + ], + "score": 0.9, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 663, + 358, + 677 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 358, + 665, + 372, + 675 + ], + "score": 0.89, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "that satisfy the following feature", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 674, + 206, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 206, + 689 + ], + "score": 1.0, + "content": "disentangling properties", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 691, + 406, + 705 + ], + "lines": [ + { + "bbox": [ + 205, + 691, + 406, + 705 + ], + "spans": [ + { + "bbox": [ + 205, + 691, + 406, + 705 + ], + "score": 0.89, + "content": "R _ { \\mathbf { v } } ( N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) ) = \\mathbf { v } \\qquad R _ { \\mathbf { c } } ( N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) ) = \\mathbf { c }", + "type": "interline_equation", + "image_path": "229e09169a1617310ada564c46542252b122f24e4909b120d447fffe62bf8b90.jpg" + } + ] + } + ], + "index": 49, + "virtual_lines": [ + { + "bbox": [ + 205, + 691, + 406, + 705 + ], + "spans": [], + "index": 49 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 135, + 722 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 135, + 711, + 154, + 720 + ], + "score": 0.33, + "content": "\\mathbf { v } , \\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 709, + 294, + 722 + ], + "score": 1.0, + "content": ", and for some bijective functions", + "type": "text" + }, + { + "bbox": [ + 295, + 710, + 309, + 721 + ], + "score": 0.89, + "content": "R _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 709, + 329, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 329, + 710, + 342, + 721 + ], + "score": 0.89, + "content": "R _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 709, + 378, + 722 + ], + "score": 1.0, + "content": ", so that", + "type": "text" + }, + { + "bbox": [ + 378, + 710, + 392, + 721 + ], + "score": 0.9, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 709, + 453, + 722 + ], + "score": 1.0, + "content": "is invariant to", + "type": "text" + }, + { + "bbox": [ + 454, + 712, + 460, + 720 + ], + "score": 0.6, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 709, + 480, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 480, + 710, + 494, + 721 + ], + "score": 0.88, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 165, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 154, + 732 + ], + "score": 1.0, + "content": "invariant to", + "type": "text" + }, + { + "bbox": [ + 154, + 722, + 161, + 730 + ], + "score": 0.69, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 720, + 165, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 104, + 83, + 506, + 140 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "The work most related to ours is Mathieu et al. (2016), where an autoencoder restores an image", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "from another by swapping parts of the internal image representation. Their main improvement over", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "Reed et al. (2015) is the use of adversarial training, which allows for learning with image pairs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "instead of image triplets. Therefore, expensive labels like viewpoint alignment between different", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "car types are no longer needed. One of the differences between this method and ours is that it trains", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "a discriminator for each of the given labels. A benefit of this approach is the higher selectivity of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "the discriminator, but a drawback is that the number of model parameters grows linearly with the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "number of labels. In contrast, we work with image pairs and use a single discriminator so that our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "method is uninfluenced by the number of labels. Moreover, we show formally and experimentally the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 299, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 299, + 254 + ], + "score": 1.0, + "content": "difficulties of disentangling factors of variation.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 142, + 506, + 254 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 270, + 336, + 282 + ], + "lines": [ + { + "bbox": [ + 104, + 268, + 338, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 338, + 284 + ], + "score": 1.0, + "content": "3 DISENTANGLING FACTORS OF VARIATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "We are interested in the design and training of two models. One should map a data sample (e.g., an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "image) to a feature that is explicitly partitioned into subvectors, each associated to a specific factor", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "of variation. The other model should map this feature back to an image. We call the first model the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "encoder and the second model the decoder. For example, given the image of a car we would like", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "the encoder to yield a feature with two subvectors: one related to the car viewpoint, and the other", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "related to the car type. The subvectors of the feature obtained from the encoder should be useful", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "for classification or regression of the corresponding factor that they depend on (the car viewpoint", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "and type in the example). This separation would also be very useful to the decoder. It would enable", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "score": 1.0, + "content": "advanced editing of images, for example, the transfer of the viewpoint or car types from an image to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "another, by swapping the corresponding subvectors. Next, we introduce our model of the data and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 293, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 293, + 415 + ], + "score": 1.0, + "content": "formal definitions of our encoder and decoder.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 293, + 506, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 299, + 433 + ], + "score": 1.0, + "content": "Data model. We assume that our observed data", + "type": "text" + }, + { + "bbox": [ + 299, + 423, + 307, + 431 + ], + "score": 0.59, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "is generated through some unknown deterministic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 231, + 444 + ], + "score": 1.0, + "content": "invertible and smooth process", + "type": "text" + }, + { + "bbox": [ + 231, + 432, + 239, + 443 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 432, + 353, + 444 + ], + "score": 1.0, + "content": "that depends on the factors", + "type": "text" + }, + { + "bbox": [ + 353, + 434, + 361, + 442 + ], + "score": 0.51, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 432, + 380, + 444 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 434, + 387, + 442 + ], + "score": 0.29, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 432, + 421, + 444 + ], + "score": 1.0, + "content": ", so that", + "type": "text" + }, + { + "bbox": [ + 421, + 432, + 472, + 444 + ], + "score": 0.93, + "content": "\\mathbf { x } = f ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 432, + 506, + 444 + ], + "score": 1.0, + "content": ". In our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 176, + 455 + ], + "score": 1.0, + "content": "earlier example,", + "type": "text" + }, + { + "bbox": [ + 176, + 444, + 184, + 453 + ], + "score": 0.63, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 442, + 239, + 455 + ], + "score": 1.0, + "content": "is an image,", + "type": "text" + }, + { + "bbox": [ + 240, + 444, + 247, + 452 + ], + "score": 0.55, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 442, + 397, + 455 + ], + "score": 1.0, + "content": "is a viewpoint, c is a car type, and", + "type": "text" + }, + { + "bbox": [ + 397, + 443, + 405, + 454 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "is the rendering engine.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 235, + 466 + ], + "score": 1.0, + "content": "It is reasonable to assume that", + "type": "text" + }, + { + "bbox": [ + 236, + 454, + 243, + 465 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "is invertible, as for most cases the factors are readily apparent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 223, + 477 + ], + "score": 1.0, + "content": "form the image. We assume", + "type": "text" + }, + { + "bbox": [ + 223, + 465, + 231, + 476 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "is smooth, because a small change in the factors should only result", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "in a small change in the image and vice versa. We denote the inverse of the rendering engine as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 485, + 385, + 501 + ], + "spans": [ + { + "bbox": [ + 107, + 486, + 181, + 499 + ], + "score": 0.93, + "content": "f ^ { - 1 } = [ f _ { \\mathbf { v } } ^ { - 1 } , f _ { \\mathbf { c } } ^ { - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 485, + 385, + 501 + ], + "score": 1.0, + "content": ", where the subscript refers to the recovered factor.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 420, + 506, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 363, + 516 + ], + "score": 1.0, + "content": "Weak labeling. In the training we are given pairs of images", + "type": "text" + }, + { + "bbox": [ + 363, + 505, + 375, + 514 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 502, + 395, + 516 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 505, + 407, + 514 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 502, + 496, + 516 + ], + "score": 1.0, + "content": ", where they differ in", + "type": "text" + }, + { + "bbox": [ + 496, + 505, + 504, + 514 + ], + "score": 0.32, + "content": "\\mathbf { v }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 513, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 506, + 528 + ], + "score": 1.0, + "content": "(varying factor), but they have the same c (common factor). We also assume that the two varying", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 524, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 350, + 539 + ], + "score": 1.0, + "content": "factors and the common factor are sampled independently,", + "type": "text" + }, + { + "bbox": [ + 350, + 527, + 388, + 537 + ], + "score": 0.73, + "content": "\\mathbf { v } _ { 1 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 524, + 392, + 539 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 392, + 527, + 430, + 537 + ], + "score": 0.75, + "content": "\\mathbf { v } _ { 2 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 524, + 449, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 527, + 480, + 537 + ], + "score": 0.89, + "content": "\\mathbf { c } \\sim p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 524, + 506, + 539 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 205, + 550 + ], + "score": 1.0, + "content": "images are generated as", + "type": "text" + }, + { + "bbox": [ + 205, + 536, + 263, + 548 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 1 } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 536, + 282, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 536, + 340, + 549 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 2 } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 536, + 506, + 550 + ], + "score": 1.0, + "content": ". We call this labeling weak, because we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 286, + 559 + ], + "score": 1.0, + "content": "do not know the absolute values of either the", + "type": "text" + }, + { + "bbox": [ + 286, + 550, + 294, + 558 + ], + "score": 0.51, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 547, + 475, + 559 + ], + "score": 1.0, + "content": "or c factors or even relative changes between", + "type": "text" + }, + { + "bbox": [ + 475, + 549, + 487, + 558 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 558, + 392, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 118, + 569 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 558, + 392, + 570 + ], + "score": 1.0, + "content": ". All we know is that the image pairs share the same common factor.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 502, + 506, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "The encoder. Let Enc be the encoder mapping images to features. For simplicity, we consider", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 290, + 599 + ], + "score": 1.0, + "content": "features split into only two column subvectors,", + "type": "text" + }, + { + "bbox": [ + 291, + 587, + 305, + 597 + ], + "score": 0.89, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 586, + 322, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 323, + 586, + 336, + 597 + ], + "score": 0.89, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 586, + 480, + 599 + ], + "score": 1.0, + "content": ", one associated to the varying factor", + "type": "text" + }, + { + "bbox": [ + 480, + 588, + 487, + 596 + ], + "score": 0.48, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 363, + 610 + ], + "score": 1.0, + "content": "the other associated to the common factor c. Then, we have that", + "type": "text" + }, + { + "bbox": [ + 363, + 597, + 470, + 609 + ], + "score": 0.9, + "content": "\\mathrm { E n c } ( \\mathbf { x } ) = [ N _ { \\mathbf { v } } ( \\mathbf { x } ) , \\bar { N _ { \\mathbf { c } } } ( \\mathbf { x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 596, + 506, + 610 + ], + "score": 1.0, + "content": ". Ideally,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 606, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 369, + 622 + ], + "score": 1.0, + "content": "we would like to find the inverse of the image formation function,", + "type": "text" + }, + { + "bbox": [ + 370, + 609, + 435, + 620 + ], + "score": 0.92, + "content": "[ N _ { \\bf v } , N _ { \\bf c } ] = f ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 606, + 506, + 622 + ], + "score": 1.0, + "content": ", which separates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 617, + 340, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 203, + 632 + ], + "score": 1.0, + "content": "and recovers the factors", + "type": "text" + }, + { + "bbox": [ + 204, + 621, + 211, + 629 + ], + "score": 0.7, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 617, + 229, + 632 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 229, + 621, + 236, + 629 + ], + "score": 0.4, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 617, + 312, + 632 + ], + "score": 1.0, + "content": "from data samples", + "type": "text" + }, + { + "bbox": [ + 312, + 621, + 319, + 629 + ], + "score": 0.66, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 617, + 340, + 632 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 575, + 506, + 632 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 635, + 387, + 649 + ], + "lines": [ + { + "bbox": [ + 224, + 635, + 387, + 649 + ], + "spans": [ + { + "bbox": [ + 224, + 635, + 387, + 649 + ], + "score": 0.91, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = \\mathbf { v } \\qquad N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = \\mathbf { c } .", + "type": "interline_equation", + "image_path": "f05d5a8c8d2a1d069976f7a15f58c01103a8388f9b9b200d56c1872194d14df2.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 224, + 635, + 387, + 649 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 384, + 667 + ], + "score": 1.0, + "content": "In practice, this is not possible because any bijective transformation of", + "type": "text" + }, + { + "bbox": [ + 384, + 655, + 392, + 663 + ], + "score": 0.71, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 652, + 497, + 667 + ], + "score": 1.0, + "content": "and c could be undone by", + "type": "text" + }, + { + "bbox": [ + 497, + 654, + 504, + 665 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 223, + 677 + ], + "score": 1.0, + "content": "and produce the same output", + "type": "text" + }, + { + "bbox": [ + 224, + 666, + 231, + 674 + ], + "score": 0.68, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 663, + 325, + 677 + ], + "score": 1.0, + "content": ". 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Let Dec be the decoder mapping features to images. 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We introduce a novel adversarial", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "score": 1.0, + "content": "term, that does not need to be conditioned on the common factor, rather it uses only image pairs, that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "keeps the model parameters constant. Then we address the two main challenges of disentangling, the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "shortcut problem and the reference ambiguity. We discuss which disentanglement properties can be", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 285, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 285, + 253 + ], + "score": 1.0, + "content": "(provably) achieved by our (or any) method.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 265, + 209, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 210, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 210, + 278 + ], + "score": 1.0, + "content": "3.1 MODEL TRAINING", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "In our training procedure we use two terms in the objective function: an autoencoder loss and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "an adversarial loss. We describe these losses in functional form, however the components are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "implemented using neural networks. In all our terms we use the following sampling of independent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 137, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 137, + 330 + ], + "score": 1.0, + "content": "factors", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 336, + 371, + 347 + ], + "lines": [ + { + "bbox": [ + 240, + 336, + 371, + 347 + ], + "spans": [ + { + "bbox": [ + 240, + 336, + 371, + 347 + ], + "score": 0.88, + "content": "\\mathbf { c } _ { 1 } , \\mathbf { c } _ { 3 } \\sim p _ { \\mathbf { c } } , \\quad \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { v } _ { 3 } \\sim p _ { \\mathbf { v } } .", + "type": "interline_equation", + "image_path": "8cf7d20c2f999ca2f57e5412d13d36f48a19c78074f065d60b49938e9aa5cb05.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 240, + 336, + 371, + 347 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 350, + 506, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 211, + 363 + ], + "score": 1.0, + "content": "The images are formed as", + "type": "text" + }, + { + "bbox": [ + 211, + 350, + 274, + 362 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { 1 } = f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 349, + 278, + 363 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 278, + 350, + 341, + 362 + ], + "score": 0.91, + "content": "\\mathbf { x } _ { 2 } = f ( \\mathbf { v } _ { 2 } , \\mathbf { c } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 349, + 359, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 360, + 350, + 422, + 362 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { 3 } = f ( \\mathbf { v } _ { 3 } , \\mathbf { c } _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 349, + 475, + 363 + ], + "score": 1.0, + "content": ". The images", + "type": "text" + }, + { + "bbox": [ + 475, + 352, + 487, + 361 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 118, + 372 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 361, + 259, + 374 + ], + "score": 1.0, + "content": "share the same common factor, and", + "type": "text" + }, + { + "bbox": [ + 260, + 363, + 272, + 372 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 361, + 289, + 374 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 363, + 301, + 372 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "are independent. In our objective functions, we use", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 371, + 280, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 280, + 386 + ], + "score": 1.0, + "content": "either pairs or triplets of the above images.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 304, + 402 + ], + "score": 1.0, + "content": "Autoencoder loss. 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We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 378, + 412 + ], + "score": 1.0, + "content": "feed both images to the encoder. Since both images share the same", + "type": "text" + }, + { + "bbox": [ + 378, + 402, + 388, + 411 + ], + "score": 0.86, + "content": "\\mathbf { c } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 399, + 506, + 412 + ], + "score": 1.0, + "content": ", we impose that the decoder", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 183, + 423 + ], + "score": 1.0, + "content": "should reconstruct", + "type": "text" + }, + { + "bbox": [ + 183, + 412, + 195, + 422 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 411, + 309, + 423 + ], + "score": 1.0, + "content": "from the encoder subvector", + "type": "text" + }, + { + "bbox": [ + 310, + 411, + 342, + 423 + ], + "score": 0.93, + "content": "N _ { \\mathbf { v } } ( \\mathbf { x } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 411, + 452, + 423 + ], + "score": 1.0, + "content": "and the encoder subvector", + "type": "text" + }, + { + "bbox": [ + 452, + 411, + 484, + 423 + ], + "score": 0.92, + "content": "N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 411, + 505, + 423 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 423, + 438, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 243, + 434 + ], + "score": 1.0, + "content": "similarly for the reconstruction of", + "type": "text" + }, + { + "bbox": [ + 244, + 423, + 255, + 433 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 423, + 438, + 434 + ], + "score": 1.0, + "content": ". The autoencoder objective is thus defined as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 437, + 474, + 458 + ], + "lines": [ + { + "bbox": [ + 137, + 437, + 474, + 458 + ], + "spans": [ + { + "bbox": [ + 137, + 437, + 474, + 458 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { A E } \\doteq E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\left[ \\left| \\mathbf { x } _ { 1 } - \\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 1 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } ) ) \\right| ^ { 2 } + \\left| \\mathbf { x } _ { 2 } - \\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 2 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) ) \\right| ^ { 2 } \\right] .", + "type": "interline_equation", + "image_path": "d198eac559418e416a222cd5c4ae34b398fd38aa7b97356827fd1a0c9091aea2.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 137, + 437, + 474, + 458 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "Adversarial loss. We introduce an adversarial training where the generator is our encoder-decoder", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "pair and the discriminator Dsc is a neural network, which takes image pairs as input. The discriminator", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 300, + 502 + ], + "score": 1.0, + "content": "learns to distinguish between real image pairs", + "type": "text" + }, + { + "bbox": [ + 300, + 489, + 332, + 501 + ], + "score": 0.93, + "content": "[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 488, + 395, + 502 + ], + "score": 1.0, + "content": "and fake ones", + "type": "text" + }, + { + "bbox": [ + 395, + 489, + 437, + 501 + ], + "score": 0.93, + "content": "[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 \\oplus 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 488, + 469, + 502 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 469, + 489, + 505, + 501 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { 3 \\oplus 1 } \\doteq", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 198, + 512 + ], + "score": 0.9, + "content": "\\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 3 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 500, + 349, + 513 + ], + "score": 1.0, + "content": ". If the encoder were ideal, the image", + "type": "text" + }, + { + "bbox": [ + 349, + 502, + 371, + 512 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "would be the result of taking the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 189, + 523 + ], + "score": 1.0, + "content": "common factor from", + "type": "text" + }, + { + "bbox": [ + 189, + 513, + 200, + 522 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 511, + 309, + 523 + ], + "score": 1.0, + "content": "and the varying factor from", + "type": "text" + }, + { + "bbox": [ + 310, + 512, + 321, + 522 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 511, + 506, + 523 + ], + "score": 1.0, + "content": ". The generator learns to fool the discriminator,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 136, + 534 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 136, + 523, + 158, + 534 + ], + "score": 0.89, + "content": "\\mathbf { x } _ { 3 \\oplus 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 522, + 285, + 534 + ], + "score": 1.0, + "content": "looks like the random variable", + "type": "text" + }, + { + "bbox": [ + 285, + 523, + 298, + 533 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 522, + 391, + 534 + ], + "score": 1.0, + "content": "(the common factor is", + "type": "text" + }, + { + "bbox": [ + 391, + 523, + 402, + 533 + ], + "score": 0.85, + "content": "\\mathbf { c } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "and the varying factor is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 168, + 546 + ], + "score": 1.0, + "content": "independent of", + "type": "text" + }, + { + "bbox": [ + 168, + 534, + 180, + 544 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 532, + 375, + 546 + ], + "score": 1.0, + "content": "). To this purpose, the decoder must make use of", + "type": "text" + }, + { + "bbox": [ + 375, + 533, + 406, + 545 + ], + "score": 0.92, + "content": "N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 532, + 433, + 546 + ], + "score": 1.0, + "content": ", since", + "type": "text" + }, + { + "bbox": [ + 433, + 534, + 445, + 544 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "does not carry", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 544, + 376, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 197, + 555 + ], + "score": 1.0, + "content": "any information about", + "type": "text" + }, + { + "bbox": [ + 197, + 545, + 208, + 555 + ], + "score": 0.86, + "content": "\\mathbf { c } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 544, + 376, + 555 + ], + "score": 1.0, + "content": ". The objective function is thus defined as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 558, + 456, + 580 + ], + "lines": [ + { + "bbox": [ + 155, + 558, + 456, + 580 + ], + "spans": [ + { + "bbox": [ + 155, + 558, + 456, + 580 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { G A N } \\doteq E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\Big [ \\log ( \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) ) \\Big ] + E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 } } \\Big [ \\log ( 1 - \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 \\oplus 1 } ) ) \\Big ] .", + "type": "interline_equation", + "image_path": "0254827e2494ad8529a8f04663895c6a092faf37fe1509920e8ab46b652ebb25.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 155, + 558, + 456, + 580 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 588, + 492, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 489, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 404, + 604 + ], + "score": 1.0, + "content": "Composite loss. Finally, we optimize the weighted sum of the two losses", + "type": "text" + }, + { + "bbox": [ + 405, + 589, + 489, + 600 + ], + "score": 0.89, + "content": "\\mathcal { L } = \\mathcal { L } _ { A E } + \\lambda \\mathcal { L } _ { G A N }", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 604, + 416, + 622 + ], + "lines": [ + { + "bbox": [ + 195, + 604, + 416, + 622 + ], + "spans": [ + { + "bbox": [ + 195, + 604, + 416, + 622 + ], + "score": 0.87, + "content": "\\operatorname* { m i n } _ { \\mathrm { D e c , E n c } } \\operatorname* { m a x } _ { \\mathrm { D s c } } \\mathcal { L } _ { A E } ( \\mathrm { D e c , E n c } ) + \\lambda \\mathcal { L } _ { G A N } ( \\mathrm { D e c , E n c , D s c } )", + "type": "interline_equation", + "image_path": "522327b33c38fe4c3ca69b99c16c2158b35bd3c55f4562d00489d3bde72d2042.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 195, + 604, + 416, + 622 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 626, + 344, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 345, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 133, + 639 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 627, + 140, + 636 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 626, + 345, + 639 + ], + "score": 1.0, + "content": "regulates the relative importance of the two losses.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 108, + 651, + 224, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 226, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 226, + 664 + ], + "score": 1.0, + "content": "3.2 SHORTCUT PROBLEM.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 243, + 685 + ], + "score": 1.0, + "content": "Ideally, at the global minimum of", + "type": "text" + }, + { + "bbox": [ + 243, + 672, + 264, + 683 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { A E }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 671, + 267, + 685 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 268, + 672, + 282, + 683 + ], + "score": 0.84, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 671, + 383, + 685 + ], + "score": 1.0, + "content": "relates only to the factor", + "type": "text" + }, + { + "bbox": [ + 384, + 674, + 391, + 682 + ], + "score": 0.62, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 671, + 409, + 685 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 409, + 672, + 423, + 683 + ], + "score": 0.88, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "only to c. However,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 355, + 695 + ], + "score": 1.0, + "content": "the encoder may map a complete description of its input into", + "type": "text" + }, + { + "bbox": [ + 355, + 683, + 370, + 694 + ], + "score": 0.89, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "and the decoder may completely", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 694, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 135, + 707 + ], + "score": 1.0, + "content": "ignore", + "type": "text" + }, + { + "bbox": [ + 136, + 694, + 149, + 705 + ], + "score": 0.88, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 694, + 506, + 707 + ], + "score": 1.0, + "content": ". We call this challenge the shortcut problem. When the shortcut problem occurs, the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 704, + 441, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 367, + 717 + ], + "score": 1.0, + "content": "decoder is invariant to its second input, so it does not transfer the", + "type": "text" + }, + { + "bbox": [ + 368, + 707, + 374, + 714 + ], + "score": 0.41, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 704, + 441, + 717 + ], + "score": 1.0, + "content": "factor correctly,", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 719, + 355, + 733 + ], + "lines": [ + { + "bbox": [ + 235, + 719, + 355, + 733 + ], + "spans": [ + { + "bbox": [ + 235, + 719, + 355, + 733 + ], + "score": 0.92, + "content": "\\mathrm { D e c } ( N _ { \\mathbf { v } } ( { \\mathbf { x } } _ { 3 } ) , N _ { \\mathbf { c } } ( { \\mathbf { x } } _ { 1 } ) ) = { \\mathbf { x } } _ { 3 } .", + "type": "interline_equation", + "image_path": "7d0078287b4a15239b3b5a8df7d67055394d89138ef884e274c71c9292c0de05.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 235, + 719, + 355, + 733 + ], + "spans": [], + "index": 44 + } + ] + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "The decoder. Let Dec be the decoder mapping features to images. The sequence encoder-decoder is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 265, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 265, + 106 + ], + "score": 1.0, + "content": "constrained to form an autoencoder, so", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 108, + 377, + 122 + ], + "lines": [ + { + "bbox": [ + 234, + 108, + 377, + 122 + ], + "spans": [ + { + "bbox": [ + 234, + 108, + 377, + 122 + ], + "score": 0.92, + "content": "\\mathrm { D e c } ( N _ { \\mathbf { v } } ( { \\mathbf { x } } ) , N _ { \\mathbf { c } } ( { \\mathbf { x } } ) ) = { \\mathbf { x } } , \\qquad \\forall { \\mathbf { x } } .", + "type": "interline_equation", + "image_path": "bf06ce5db3f7455eea29b0e5dcee4161bdc1360ef3b9becbb352a1a666eb8a8d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 234, + 108, + 377, + 122 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 126, + 504, + 148 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "To use the decoder for image synthesis, so that each input subvector affects only one factor in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 429, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 429, + 150 + ], + "score": 1.0, + "content": "rendered image, the ideal decoder should satisfy the data disentangling property", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 125, + 505, + 150 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 151, + 406, + 165 + ], + "lines": [ + { + "bbox": [ + 205, + 151, + 406, + 165 + ], + "spans": [ + { + "bbox": [ + 205, + 151, + 406, + 165 + ], + "score": 0.89, + "content": "\\mathrm { D e c } ( N _ { \\mathbf { v } } ( f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ) , N _ { \\mathbf { c } } ( f ( \\mathbf { v } _ { 2 } , \\mathbf { c } _ { 2 } ) ) ) = f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } )", + "type": "interline_equation", + "image_path": "917cc8fd290a28a204e30435cdee2a4200eedc267530b0945e3be26d05c615eb.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 205, + 151, + 406, + 165 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 168, + 503, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 504, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 137, + 182 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 138, + 171, + 179, + 180 + ], + "score": 0.28, + "content": "\\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { c } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 167, + 200, + 182 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 200, + 170, + 211, + 180 + ], + "score": 0.84, + "content": "\\mathbf { c } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 167, + 469, + 182 + ], + "score": 1.0, + "content": ". 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We introduce a novel adversarial", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "score": 1.0, + "content": "term, that does not need to be conditioned on the common factor, rather it uses only image pairs, that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "keeps the model parameters constant. Then we address the two main challenges of disentangling, the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "shortcut problem and the reference ambiguity. 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We describe these losses in functional form, however the components are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "implemented using neural networks. 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The images", + "type": "text" + }, + { + "bbox": [ + 475, + 352, + 487, + 361 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 118, + 372 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 361, + 259, + 374 + ], + "score": 1.0, + "content": "share the same common factor, and", + "type": "text" + }, + { + "bbox": [ + 260, + 363, + 272, + 372 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 361, + 289, + 374 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 363, + 301, + 372 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "are independent. 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We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 378, + 412 + ], + "score": 1.0, + "content": "feed both images to the encoder. 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The autoencoder objective is thus defined as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 389, + 506, + 434 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 437, + 474, + 458 + ], + "lines": [ + { + "bbox": [ + 137, + 437, + 474, + 458 + ], + "spans": [ + { + "bbox": [ + 137, + 437, + 474, + 458 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { A E } \\doteq E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\left[ \\left| \\mathbf { x } _ { 1 } - \\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 1 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } ) ) \\right| ^ { 2 } + \\left| \\mathbf { x } _ { 2 } - \\mathrm { D e c } ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 2 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) ) \\right| ^ { 2 } \\right] .", + "type": "interline_equation", + "image_path": "d198eac559418e416a222cd5c4ae34b398fd38aa7b97356827fd1a0c9091aea2.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 137, + 437, + 474, + 458 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "Adversarial loss. We introduce an adversarial training where the generator is our encoder-decoder", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "pair and the discriminator Dsc is a neural network, which takes image pairs as input. 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If the encoder were ideal, the image", + "type": "text" + }, + { + "bbox": [ + 349, + 502, + 371, + 512 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "would be the result of taking the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 189, + 523 + ], + "score": 1.0, + "content": "common factor from", + "type": "text" + }, + { + "bbox": [ + 189, + 513, + 200, + 522 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 511, + 309, + 523 + ], + "score": 1.0, + "content": "and the varying factor from", + "type": "text" + }, + { + "bbox": [ + 310, + 512, + 321, + 522 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 511, + 506, + 523 + ], + "score": 1.0, + "content": ". The generator learns to fool the discriminator,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 136, + 534 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 136, + 523, + 158, + 534 + ], + "score": 0.89, + "content": "\\mathbf { x } _ { 3 \\oplus 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 522, + 285, + 534 + ], + "score": 1.0, + "content": "looks like the random variable", + "type": "text" + }, + { + "bbox": [ + 285, + 523, + 298, + 533 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 522, + 391, + 534 + ], + "score": 1.0, + "content": "(the common factor is", + "type": "text" + }, + { + "bbox": [ + 391, + 523, + 402, + 533 + ], + "score": 0.85, + "content": "\\mathbf { c } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "and the varying factor is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 168, + 546 + ], + "score": 1.0, + "content": "independent of", + "type": "text" + }, + { + "bbox": [ + 168, + 534, + 180, + 544 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 532, + 375, + 546 + ], + "score": 1.0, + "content": "). 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The objective function is thus defined as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 466, + 506, + 555 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 558, + 456, + 580 + ], + "lines": [ + { + "bbox": [ + 155, + 558, + 456, + 580 + ], + "spans": [ + { + "bbox": [ + 155, + 558, + 456, + 580 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { G A N } \\doteq E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\Big [ \\log ( \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) ) \\Big ] + E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 } } \\Big [ \\log ( 1 - \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 \\oplus 1 } ) ) \\Big ] .", + "type": "interline_equation", + "image_path": "0254827e2494ad8529a8f04663895c6a092faf37fe1509920e8ab46b652ebb25.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 155, + 558, + 456, + 580 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 588, + 492, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 489, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 404, + 604 + ], + "score": 1.0, + "content": "Composite loss. Finally, we optimize the weighted sum of the two losses", + "type": "text" + }, + { + "bbox": [ + 405, + 589, + 489, + 600 + ], + "score": 0.89, + "content": "\\mathcal { L } = \\mathcal { L } _ { A E } + \\lambda \\mathcal { L } _ { G A N }", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 586, + 489, + 604 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 604, + 416, + 622 + ], + "lines": [ + { + "bbox": [ + 195, + 604, + 416, + 622 + ], + "spans": [ + { + "bbox": [ + 195, + 604, + 416, + 622 + ], + "score": 0.87, + "content": "\\operatorname* { m i n } _ { \\mathrm { D e c , E n c } } \\operatorname* { m a x } _ { \\mathrm { D s c } } \\mathcal { L } _ { A E } ( \\mathrm { D e c , E n c } ) + \\lambda \\mathcal { L } _ { G A N } ( \\mathrm { D e c , E n c , D s c } )", + "type": "interline_equation", + "image_path": "522327b33c38fe4c3ca69b99c16c2158b35bd3c55f4562d00489d3bde72d2042.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 195, + 604, + 416, + 622 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 626, + 344, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 345, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 133, + 639 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 627, + 140, + 636 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 626, + 345, + 639 + ], + "score": 1.0, + "content": "regulates the relative importance of the two losses.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 626, + 345, + 639 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 651, + 224, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 226, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 226, + 664 + ], + "score": 1.0, + "content": "3.2 SHORTCUT PROBLEM.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 243, + 685 + ], + "score": 1.0, + "content": "Ideally, at the global minimum of", + "type": "text" + }, + { + "bbox": [ + 243, + 672, + 264, + 683 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { A E }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 671, + 267, + 685 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 268, + 672, + 282, + 683 + ], + "score": 0.84, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 671, + 383, + 685 + ], + "score": 1.0, + "content": "relates only to the factor", + "type": "text" + }, + { + "bbox": [ + 384, + 674, + 391, + 682 + ], + "score": 0.62, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 671, + 409, + 685 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 409, + 672, + 423, + 683 + ], + "score": 0.88, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "only to c. However,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 355, + 695 + ], + "score": 1.0, + "content": "the encoder may map a complete description of its input into", + "type": "text" + }, + { + "bbox": [ + 355, + 683, + 370, + 694 + ], + "score": 0.89, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "and the decoder may completely", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 694, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 135, + 707 + ], + "score": 1.0, + "content": "ignore", + "type": "text" + }, + { + "bbox": [ + 136, + 694, + 149, + 705 + ], + "score": 0.88, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 694, + 506, + 707 + ], + "score": 1.0, + "content": ". We call this challenge the shortcut problem. When the shortcut problem occurs, the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 704, + 441, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 367, + 717 + ], + "score": 1.0, + "content": "decoder is invariant to its second input, so it does not transfer the", + "type": "text" + }, + { + "bbox": [ + 368, + 707, + 374, + 714 + ], + "score": 0.41, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 704, + 441, + 717 + ], + "score": 1.0, + "content": "factor correctly,", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 671, + 506, + 717 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 719, + 355, + 733 + ], + "lines": [ + { + "bbox": [ + 235, + 719, + 355, + 733 + ], + "spans": [ + { + "bbox": [ + 235, + 719, + 355, + 733 + ], + "score": 0.92, + "content": "\\mathrm { D e c } ( N _ { \\mathbf { v } } ( { \\mathbf { x } } _ { 3 } ) , N _ { \\mathbf { c } } ( { \\mathbf { x } } _ { 1 } ) ) = { \\mathbf { x } } _ { 3 } .", + "type": "interline_equation", + "image_path": "7d0078287b4a15239b3b5a8df7d67055394d89138ef884e274c71c9292c0de05.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 235, + 719, + 355, + 733 + ], + "spans": [], + "index": 44 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 407, + 95 + ], + "score": 1.0, + "content": "The shortcut problem can be addressed by reducing the dimensionality of", + "type": "text" + }, + { + "bbox": [ + 407, + 83, + 421, + 93 + ], + "score": 0.89, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 82, + 506, + 95 + ], + "score": 1.0, + "content": ", so it cannot build a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "complete representation of all input images. This also forces the encoder and decoder to make use", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 118, + 118 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 105, + 132, + 115 + ], + "score": 0.89, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "for the common factor. However, this strategy may not be convenient as it leads to a time", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "consuming trial-and-error procedure to find the correct dimensionality. A better way to address the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 322, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 322, + 140 + ], + "score": 1.0, + "content": "shortcut problem is to use adversarial training (7) (8).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 504, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 189, + 154 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 189, + 143, + 200, + 152 + ], + "score": 0.8, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 141, + 205, + 154 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 143, + 218, + 152 + ], + "score": 0.7, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 141, + 239, + 154 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 142, + 251, + 152 + ], + "score": 0.74, + "content": "\\mathbf { x } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 141, + 506, + 154 + ], + "score": 1.0, + "content": "data samples generated according to (5), where the factors", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 152, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 107, + 154, + 176, + 164 + ], + "score": 0.88, + "content": "\\mathbf { c } _ { 1 } , \\mathbf { c } _ { 3 } , \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { v } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 152, + 298, + 165 + ], + "score": 1.0, + "content": "are jointly independent, and", + "type": "text" + }, + { + "bbox": [ + 299, + 152, + 426, + 164 + ], + "score": 0.91, + "content": "\\mathbf { x } _ { 3 \\oplus 1 } \\doteq D e c ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 3 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 152, + 506, + 165 + ], + "score": 1.0, + "content": ". When the global", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 502, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 393, + 177 + ], + "score": 1.0, + "content": "optimum of the composite loss (8) is reached, the c factor is transferred to", + "type": "text" + }, + { + "bbox": [ + 394, + 165, + 416, + 175 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 163, + 434, + 177 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 434, + 163, + 502, + 175 + ], + "score": 0.92, + "content": "f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 3 \\oplus 1 } ) = \\mathbf { c } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 504, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 394, + 201 + ], + "score": 1.0, + "content": "Proof. When the global optimum of (8) is reached, the distribution of real", + "type": "text" + }, + { + "bbox": [ + 394, + 187, + 426, + 199 + ], + "score": 0.93, + "content": "\\left[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 184, + 461, + 201 + ], + "score": 1.0, + "content": "and fake", + "type": "text" + }, + { + "bbox": [ + 462, + 187, + 504, + 199 + ], + "score": 0.92, + "content": "[ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 \\oplus 1 } ]", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "image pairs are identical. We compute statistics of the inverse of the rendering engine of the common", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 347, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 132, + 222 + ], + "score": 1.0, + "content": "factor", + "type": "text" + }, + { + "bbox": [ + 133, + 209, + 150, + 221 + ], + "score": 0.91, + "content": "\\bar { f } _ { \\mathbf { c } } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 208, + 262, + 222 + ], + "score": 1.0, + "content": "on the data. For the images", + "type": "text" + }, + { + "bbox": [ + 262, + 210, + 273, + 220 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 208, + 291, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 292, + 210, + 304, + 220 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 208, + 347, + 222 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 225, + 421, + 247 + ], + "lines": [ + { + "bbox": [ + 190, + 225, + 421, + 247 + ], + "spans": [ + { + "bbox": [ + 190, + 225, + 421, + 247 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { E } _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } \\Big [ | f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 1 } ) - f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 2 } ) | ^ { 2 } \\Big ] = { E } _ { \\mathbf { c } _ { 1 } } \\Big [ | \\mathbf { c } _ { 1 } - \\mathbf { c } _ { 1 } | ^ { 2 } \\Big ] = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "d63503b72bdcbc84bb8b5313db9996fceb9c0fa3e4cf33136f2453fd2527a5bc.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 190, + 225, + 421, + 247 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 250, + 390, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 390, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 185, + 265 + ], + "score": 1.0, + "content": "by construction (of", + "type": "text" + }, + { + "bbox": [ + 185, + 253, + 196, + 262 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 249, + 214, + 265 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 215, + 253, + 226, + 262 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 249, + 295, + 265 + ], + "score": 1.0, + "content": "). For the images", + "type": "text" + }, + { + "bbox": [ + 295, + 253, + 307, + 262 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 249, + 325, + 265 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 253, + 347, + 263 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 249, + 390, + 265 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 267, + 447, + 289 + ], + "lines": [ + { + "bbox": [ + 163, + 267, + 447, + 289 + ], + "spans": [ + { + "bbox": [ + 163, + 267, + 447, + 289 + ], + "score": 0.92, + "content": "\\begin{array} { r } { E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 } } \\Big [ | f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 1 } ) - f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 3 \\oplus 1 } ) | ^ { 2 } \\Big ] = E _ { \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } , \\mathbf { v } _ { 3 } , \\mathbf { c } _ { 3 } } \\Big [ | \\mathbf { c } _ { 1 } - \\mathbf { c } _ { 3 \\oplus 1 } | ^ { 2 } \\Big ] \\geq 0 , } \\end{array}", + "type": "interline_equation", + "image_path": "8ef50f6da86620ea7ff42242fee4dd9d0a38fbc043dc1de97b3292a81bcda76b.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 163, + 267, + 447, + 289 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 448, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 451, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 133, + 309 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 294, + 212, + 307 + ], + "score": 0.93, + "content": "{ \\bf c } _ { 3 \\oplus 1 } = f _ { \\bf c } ^ { - 1 } ( { \\bf x } _ { 3 \\oplus 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 293, + 353, + 309 + ], + "score": 1.0, + "content": ". We achieve equality if and only if", + "type": "text" + }, + { + "bbox": [ + 354, + 296, + 397, + 306 + ], + "score": 0.89, + "content": "\\mathbf { c } _ { 1 } = \\mathbf { c } _ { 3 \\oplus 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 293, + 451, + 309 + ], + "score": 1.0, + "content": "everywhere.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 319, + 235, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 236, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 236, + 331 + ], + "score": 1.0, + "content": "3.3 REFERENCE AMBIGUITY", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 504, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "Let us consider the ideal case where we observe the space of all images. When weak labels are made", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 281, + 363 + ], + "score": 1.0, + "content": "available to us, we also know what images", + "type": "text" + }, + { + "bbox": [ + 281, + 353, + 293, + 362 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 351, + 311, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 352, + 323, + 362 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "share the same c factor (for example, which", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 361, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 493, + 376 + ], + "score": 1.0, + "content": "images have the same car). This labeling is equivalent to defining the probability density function", + "type": "text" + }, + { + "bbox": [ + 493, + 364, + 504, + 374 + ], + "score": 0.79, + "content": "p _ { \\mathbf { c } }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 373, + 270, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 206, + 387 + ], + "score": 1.0, + "content": "and the joint conditional", + "type": "text" + }, + { + "bbox": [ + 206, + 374, + 239, + 386 + ], + "score": 0.91, + "content": "\\displaystyle p _ { { \\mathbf { x } } _ { 1 } , { \\mathbf { x } } _ { 2 } | { \\mathbf { c } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 373, + 270, + 387 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 390, + 465, + 416 + ], + "lines": [ + { + "bbox": [ + 146, + 390, + 465, + 416 + ], + "spans": [ + { + "bbox": [ + 146, + 390, + 465, + 416 + ], + "score": 0.91, + "content": "p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } ) = \\int \\delta ( \\mathbf { x } _ { 1 } - f ( \\mathbf { v } _ { 1 } , \\mathbf { c } ) ) \\delta ( \\mathbf { x } _ { 2 } - f ( \\mathbf { v } _ { 2 } , \\mathbf { c } ) ) p ( \\mathbf { v } _ { 1 } ) p ( \\mathbf { v } _ { 2 } ) d \\mathbf { v } _ { 1 } d \\mathbf { v } _ { 2 } .", + "type": "interline_equation", + "image_path": "c5c10d21f164dc958f0ae014970750e9436e47bd027d67fbf3d27aa4f8cc5a22.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 146, + 390, + 465, + 416 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 464, + 434 + ], + "score": 1.0, + "content": "Firstly, we show that the labeling allows us to satisfy the feature disentangling property for", + "type": "text" + }, + { + "bbox": [ + 465, + 423, + 471, + 431 + ], + "score": 0.39, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 419, + 506, + 434 + ], + "score": 1.0, + "content": "(2). For", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 431, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 124, + 446 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 124, + 431, + 201, + 444 + ], + "score": 0.92, + "content": "[ \\mathbf { \\dot { x } } _ { 1 } , \\mathbf { x } _ { 2 } ] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 431, + 249, + 446 + ], + "score": 1.0, + "content": "we impose", + "type": "text" + }, + { + "bbox": [ + 249, + 431, + 326, + 443 + ], + "score": 0.93, + "content": "N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) = N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 431, + 506, + 446 + ], + "score": 1.0, + "content": ". In particular, this equation is true for pairs", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 347, + 456 + ], + "score": 1.0, + "content": "when one of the two images is held fixed. Thus, a function", + "type": "text" + }, + { + "bbox": [ + 347, + 443, + 412, + 456 + ], + "score": 0.92, + "content": "C ( \\mathbf { c } ) = N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "can be defined, where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 121, + 466 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 455, + 130, + 464 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 454, + 242, + 466 + ], + "score": 1.0, + "content": "only depends on c, because", + "type": "text" + }, + { + "bbox": [ + 243, + 455, + 256, + 465 + ], + "score": 0.89, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 454, + 313, + 466 + ], + "score": 1.0, + "content": "is invariant to", + "type": "text" + }, + { + "bbox": [ + 314, + 456, + 321, + 464 + ], + "score": 0.45, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 454, + 442, + 466 + ], + "score": 1.0, + "content": ". Lastly, images with the same", + "type": "text" + }, + { + "bbox": [ + 443, + 456, + 450, + 464 + ], + "score": 0.4, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 454, + 505, + 466 + ], + "score": 1.0, + "content": ", but different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 258, + 478 + ], + "score": 1.0, + "content": "c must also result in different features,", + "type": "text" + }, + { + "bbox": [ + 258, + 465, + 446, + 478 + ], + "score": 0.91, + "content": "C ( \\mathbf { c } _ { 1 } ) = N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } _ { 1 } ) ) \\neq N _ { \\mathbf { v } } ( \\mathbf { v } , \\mathbf { c } _ { 2 } ) = C ( \\mathbf { c } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 465, + 505, + 478 + ], + "score": 1.0, + "content": ", otherwise the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 438, + 488 + ], + "score": 1.0, + "content": "autoencoder constraint (3) cannot be satisfied. Then, there exists a bijective function", + "type": "text" + }, + { + "bbox": [ + 439, + 476, + 483, + 487 + ], + "score": 0.92, + "content": "\\dot { R } _ { \\bf c } = { C } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "that property (2) is satisfied for c. Unfortunately the other disentangling properties can not provably", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 499, + 156, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 156, + 509 + ], + "score": 1.0, + "content": "be satisfied.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 205, + 526 + ], + "score": 1.0, + "content": "Definition 1. A function", + "type": "text" + }, + { + "bbox": [ + 206, + 515, + 212, + 525 + ], + "score": 0.43, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 511, + 447, + 526 + ], + "score": 1.0, + "content": "reproduces the data distribution, when it generates samples", + "type": "text" + }, + { + "bbox": [ + 447, + 513, + 505, + 525 + ], + "score": 0.93, + "content": "\\mathbf { y } _ { 1 } = g ( \\mathbf { v } _ { 1 } , \\mathbf { c } )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 124, + 538 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 524, + 182, + 536 + ], + "score": 0.93, + "content": "\\mathbf { y } _ { 2 } = g ( \\mathbf { v } _ { 2 } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 523, + 403, + 538 + ], + "score": 1.0, + "content": "that have the same distribution as the data. Formally,", + "type": "text" + }, + { + "bbox": [ + 404, + 524, + 474, + 536 + ], + "score": 0.91, + "content": "[ \\mathbf { y } _ { 1 } , \\mathbf { y } _ { 2 } ] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 523, + 506, + 538 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 372, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 245, + 549 + ], + "score": 1.0, + "content": "the latent factors are independent,", + "type": "text" + }, + { + "bbox": [ + 245, + 537, + 281, + 547 + ], + "score": 0.78, + "content": "\\mathbf { v } _ { 1 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 534, + 284, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 285, + 537, + 320, + 547 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { 2 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 534, + 339, + 549 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 536, + 367, + 547 + ], + "score": 0.88, + "content": "\\mathbf { c } \\sim p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 534, + 372, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 504, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "score": 1.0, + "content": "The reference ambiguity occurs, when a decoder reproduces the data without satisfying the disentan-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 567, + 174, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 174, + 579 + ], + "score": 1.0, + "content": "gling properties.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 187, + 594 + ], + "score": 1.0, + "content": "Proposition 2. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 583, + 200, + 593 + ], + "score": 0.82, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "assign the same probability value to at least two different instances of v.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "Then, we can find encoders that reproduce the data distribution, but do not satisfy the disentangling", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 230, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 230, + 615 + ], + "score": 1.0, + "content": "properties for v in (2) and (4).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 104, + 624, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 624, + 216, + 641 + ], + "score": 1.0, + "content": "Proof. We already saw that", + "type": "text" + }, + { + "bbox": [ + 216, + 628, + 230, + 638 + ], + "score": 0.9, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 624, + 349, + 641 + ], + "score": 1.0, + "content": "satisfies (2), so we can choose", + "type": "text" + }, + { + "bbox": [ + 350, + 627, + 393, + 639 + ], + "score": 0.93, + "content": "N _ { \\mathbf { c } } = f _ { \\mathbf { c } } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 624, + 506, + 641 + ], + "score": 1.0, + "content": ", the inverse of the rendering", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 243, + 651 + ], + "score": 1.0, + "content": "engine. Now we look at defining", + "type": "text" + }, + { + "bbox": [ + 244, + 639, + 258, + 649 + ], + "score": 0.88, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 637, + 459, + 651 + ], + "score": 1.0, + "content": "and the decoder. The iso-probability property of", + "type": "text" + }, + { + "bbox": [ + 460, + 640, + 471, + 650 + ], + "score": 0.84, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "implies", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 221, + 662 + ], + "score": 1.0, + "content": "that there exists a mapping", + "type": "text" + }, + { + "bbox": [ + 221, + 649, + 253, + 661 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 649, + 298, + 662 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 299, + 649, + 356, + 661 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 649, + 376, + 662 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 649, + 464, + 661 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\bar { \\mathbf { c } _ { 1 } } ) \\bar { \\neq } T ( \\bar { \\mathbf { v } } , \\bar { \\mathbf { c } _ { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 114, + 670 + ], + "score": 0.54, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 659, + 134, + 673 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 134, + 660, + 172, + 671 + ], + "score": 0.9, + "content": "\\mathbf { c } _ { 1 } \\neq \\mathbf { c } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 659, + 316, + 673 + ], + "score": 1.0, + "content": ". For example, let us denote with", + "type": "text" + }, + { + "bbox": [ + 317, + 661, + 357, + 672 + ], + "score": 0.89, + "content": "\\mathbf { v } _ { 1 } \\neq \\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "two varying components such that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 671, + 222, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 178, + 683 + ], + "score": 0.92, + "content": "p _ { \\mathbf { v } } ( \\mathbf { v } _ { 1 } ) = p _ { \\mathbf { v } } ( \\mathbf { v } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 671, + 222, + 684 + ], + "score": 1.0, + "content": ". Then, let", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 689, + 396, + 730 + ], + "lines": [ + { + "bbox": [ + 215, + 689, + 396, + 730 + ], + "spans": [ + { + "bbox": [ + 215, + 689, + 396, + 730 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\dot { = } \\left\\{ \\begin{array} { l l } { \\mathbf { v } } & { \\mathrm { i f } \\ \\mathbf { v } \\neq \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } } \\\\ { \\mathbf { v } _ { 1 } } & { \\mathrm { i f } \\ \\mathbf { v } = \\mathbf { v } _ { 1 } \\lor \\mathbf { v } _ { 2 } \\mathrm { a n d } \\mathbf { c } \\in \\mathcal { C } } \\\\ { \\mathbf { v } _ { 2 } } & { \\mathrm { i f } \\ \\mathbf { v } = \\mathbf { v } _ { 1 } \\lor \\mathbf { v } _ { 2 } \\mathrm { a n d } \\mathbf { c } \\not \\in \\mathcal { C } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "bb4ae1e763a78a1dae1156e86bc9cd8f148c155f09fc90d3857031b049127b49.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 689, + 396, + 709.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 215, + 709.5, + 396, + 730.0 + ], + "spans": [], + "index": 43 + } + ] + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 295, + 504, + 305 + ], + "lines": [ + { + "bbox": [ + 496, + 297, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 496, + 297, + 504, + 304 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 407, + 95 + ], + "score": 1.0, + "content": "The shortcut problem can be addressed by reducing the dimensionality of", + "type": "text" + }, + { + "bbox": [ + 407, + 83, + 421, + 93 + ], + "score": 0.89, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 82, + 506, + 95 + ], + "score": 1.0, + "content": ", so it cannot build a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "complete representation of all input images. This also forces the encoder and decoder to make use", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 118, + 118 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 105, + 132, + 115 + ], + "score": 0.89, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "for the common factor. However, this strategy may not be convenient as it leads to a time", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "consuming trial-and-error procedure to find the correct dimensionality. A better way to address the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 322, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 322, + 140 + ], + "score": 1.0, + "content": "shortcut problem is to use adversarial training (7) (8).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 504, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 189, + 154 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 189, + 143, + 200, + 152 + ], + "score": 0.8, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 141, + 205, + 154 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 143, + 218, + 152 + ], + "score": 0.7, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 141, + 239, + 154 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 142, + 251, + 152 + ], + "score": 0.74, + "content": "\\mathbf { x } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 141, + 506, + 154 + ], + "score": 1.0, + "content": "data samples generated according to (5), where the factors", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 152, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 107, + 154, + 176, + 164 + ], + "score": 0.88, + "content": "\\mathbf { c } _ { 1 } , \\mathbf { c } _ { 3 } , \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { v } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 152, + 298, + 165 + ], + "score": 1.0, + "content": "are jointly independent, and", + "type": "text" + }, + { + "bbox": [ + 299, + 152, + 426, + 164 + ], + "score": 0.91, + "content": "\\mathbf { x } _ { 3 \\oplus 1 } \\doteq D e c ( N _ { \\mathbf { v } } ( \\mathbf { x } _ { 3 } ) , N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 152, + 506, + 165 + ], + "score": 1.0, + "content": ". When the global", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 502, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 393, + 177 + ], + "score": 1.0, + "content": "optimum of the composite loss (8) is reached, the c factor is transferred to", + "type": "text" + }, + { + "bbox": [ + 394, + 165, + 416, + 175 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 163, + 434, + 177 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 434, + 163, + 502, + 175 + ], + "score": 0.92, + "content": "f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 3 \\oplus 1 } ) = \\mathbf { c } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 141, + 506, + 177 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 504, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 394, + 201 + ], + "score": 1.0, + "content": "Proof. 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We compute statistics of the inverse of the rendering engine of the common", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 347, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 132, + 222 + ], + "score": 1.0, + "content": "factor", + "type": "text" + }, + { + "bbox": [ + 133, + 209, + 150, + 221 + ], + "score": 0.91, + "content": "\\bar { f } _ { \\mathbf { c } } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 208, + 262, + 222 + ], + "score": 1.0, + "content": "on the data. 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For the images", + "type": "text" + }, + { + "bbox": [ + 295, + 253, + 307, + 262 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 249, + 325, + 265 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 253, + 347, + 263 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { \\mathrm { 3 \\oplus 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 249, + 390, + 265 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 249, + 390, + 265 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 267, + 447, + 289 + ], + "lines": [ + { + "bbox": [ + 163, + 267, + 447, + 289 + ], + "spans": [ + { + "bbox": [ + 163, + 267, + 447, + 289 + ], + "score": 0.92, + "content": "\\begin{array} { r } { E _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 3 } } \\Big [ | f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 1 } ) - f _ { \\mathbf { c } } ^ { - 1 } ( \\mathbf { x } _ { 3 \\oplus 1 } ) | ^ { 2 } \\Big ] = E _ { \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } , \\mathbf { v } _ { 3 } , \\mathbf { c } _ { 3 } } \\Big [ | \\mathbf { c } _ { 1 } - \\mathbf { c } _ { 3 \\oplus 1 } | ^ { 2 } \\Big ] \\geq 0 , } \\end{array}", + "type": "interline_equation", + "image_path": "8ef50f6da86620ea7ff42242fee4dd9d0a38fbc043dc1de97b3292a81bcda76b.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 163, + 267, + 447, + 289 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 448, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 451, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 133, + 309 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 294, + 212, + 307 + ], + "score": 0.93, + "content": "{ \\bf c } _ { 3 \\oplus 1 } = f _ { \\bf c } ^ { - 1 } ( { \\bf x } _ { 3 \\oplus 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 293, + 353, + 309 + ], + "score": 1.0, + "content": ". We achieve equality if and only if", + "type": "text" + }, + { + "bbox": [ + 354, + 296, + 397, + 306 + ], + "score": 0.89, + "content": "\\mathbf { c } _ { 1 } = \\mathbf { c } _ { 3 \\oplus 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 293, + 451, + 309 + ], + "score": 1.0, + "content": "everywhere.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 293, + 451, + 309 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 319, + 235, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 236, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 236, + 331 + ], + "score": 1.0, + "content": "3.3 REFERENCE AMBIGUITY", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 504, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "Let us consider the ideal case where we observe the space of all images. When weak labels are made", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 281, + 363 + ], + "score": 1.0, + "content": "available to us, we also know what images", + "type": "text" + }, + { + "bbox": [ + 281, + 353, + 293, + 362 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 351, + 311, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 352, + 323, + 362 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "share the same c factor (for example, which", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 361, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 493, + 376 + ], + "score": 1.0, + "content": "images have the same car). This labeling is equivalent to defining the probability density function", + "type": "text" + }, + { + "bbox": [ + 493, + 364, + 504, + 374 + ], + "score": 0.79, + "content": "p _ { \\mathbf { c } }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 373, + 270, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 206, + 387 + ], + "score": 1.0, + "content": "and the joint conditional", + "type": "text" + }, + { + "bbox": [ + 206, + 374, + 239, + 386 + ], + "score": 0.91, + "content": "\\displaystyle p _ { { \\mathbf { x } } _ { 1 } , { \\mathbf { x } } _ { 2 } | { \\mathbf { c } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 373, + 270, + 387 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 340, + 506, + 387 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 390, + 465, + 416 + ], + "lines": [ + { + "bbox": [ + 146, + 390, + 465, + 416 + ], + "spans": [ + { + "bbox": [ + 146, + 390, + 465, + 416 + ], + "score": 0.91, + "content": "p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } ) = \\int \\delta ( \\mathbf { x } _ { 1 } - f ( \\mathbf { v } _ { 1 } , \\mathbf { c } ) ) \\delta ( \\mathbf { x } _ { 2 } - f ( \\mathbf { v } _ { 2 } , \\mathbf { c } ) ) p ( \\mathbf { v } _ { 1 } ) p ( \\mathbf { v } _ { 2 } ) d \\mathbf { v } _ { 1 } d \\mathbf { v } _ { 2 } .", + "type": "interline_equation", + "image_path": "c5c10d21f164dc958f0ae014970750e9436e47bd027d67fbf3d27aa4f8cc5a22.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 146, + 390, + 465, + 416 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 464, + 434 + ], + "score": 1.0, + "content": "Firstly, we show that the labeling allows us to satisfy the feature disentangling property for", + "type": "text" + }, + { + "bbox": [ + 465, + 423, + 471, + 431 + ], + "score": 0.39, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 419, + 506, + 434 + ], + "score": 1.0, + "content": "(2). For", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 431, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 124, + 446 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 124, + 431, + 201, + 444 + ], + "score": 0.92, + "content": "[ \\mathbf { \\dot { x } } _ { 1 } , \\mathbf { x } _ { 2 } ] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } | \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 431, + 249, + 446 + ], + "score": 1.0, + "content": "we impose", + "type": "text" + }, + { + "bbox": [ + 249, + 431, + 326, + 443 + ], + "score": 0.93, + "content": "N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } ) = N _ { \\mathbf { c } } ( \\mathbf { x } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 431, + 506, + 446 + ], + "score": 1.0, + "content": ". In particular, this equation is true for pairs", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 347, + 456 + ], + "score": 1.0, + "content": "when one of the two images is held fixed. Thus, a function", + "type": "text" + }, + { + "bbox": [ + 347, + 443, + 412, + 456 + ], + "score": 0.92, + "content": "C ( \\mathbf { c } ) = N _ { \\mathbf { c } } ( \\mathbf { x } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "can be defined, where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 121, + 466 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 455, + 130, + 464 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 454, + 242, + 466 + ], + "score": 1.0, + "content": "only depends on c, because", + "type": "text" + }, + { + "bbox": [ + 243, + 455, + 256, + 465 + ], + "score": 0.89, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 454, + 313, + 466 + ], + "score": 1.0, + "content": "is invariant to", + "type": "text" + }, + { + "bbox": [ + 314, + 456, + 321, + 464 + ], + "score": 0.45, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 454, + 442, + 466 + ], + "score": 1.0, + "content": ". Lastly, images with the same", + "type": "text" + }, + { + "bbox": [ + 443, + 456, + 450, + 464 + ], + "score": 0.4, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 454, + 505, + 466 + ], + "score": 1.0, + "content": ", but different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 258, + 478 + ], + "score": 1.0, + "content": "c must also result in different features,", + "type": "text" + }, + { + "bbox": [ + 258, + 465, + 446, + 478 + ], + "score": 0.91, + "content": "C ( \\mathbf { c } _ { 1 } ) = N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } _ { 1 } ) ) \\neq N _ { \\mathbf { v } } ( \\mathbf { v } , \\mathbf { c } _ { 2 } ) = C ( \\mathbf { c } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 465, + 505, + 478 + ], + "score": 1.0, + "content": ", otherwise the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 438, + 488 + ], + "score": 1.0, + "content": "autoencoder constraint (3) cannot be satisfied. Then, there exists a bijective function", + "type": "text" + }, + { + "bbox": [ + 439, + 476, + 483, + 487 + ], + "score": 0.92, + "content": "\\dot { R } _ { \\bf c } = { C } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "that property (2) is satisfied for c. Unfortunately the other disentangling properties can not provably", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 499, + 156, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 156, + 509 + ], + "score": 1.0, + "content": "be satisfied.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 419, + 506, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 205, + 526 + ], + "score": 1.0, + "content": "Definition 1. A function", + "type": "text" + }, + { + "bbox": [ + 206, + 515, + 212, + 525 + ], + "score": 0.43, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 511, + 447, + 526 + ], + "score": 1.0, + "content": "reproduces the data distribution, when it generates samples", + "type": "text" + }, + { + "bbox": [ + 447, + 513, + 505, + 525 + ], + "score": 0.93, + "content": "\\mathbf { y } _ { 1 } = g ( \\mathbf { v } _ { 1 } , \\mathbf { c } )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 124, + 538 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 524, + 182, + 536 + ], + "score": 0.93, + "content": "\\mathbf { y } _ { 2 } = g ( \\mathbf { v } _ { 2 } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 523, + 403, + 538 + ], + "score": 1.0, + "content": "that have the same distribution as the data. Formally,", + "type": "text" + }, + { + "bbox": [ + 404, + 524, + 474, + 536 + ], + "score": 0.91, + "content": "[ \\mathbf { y } _ { 1 } , \\mathbf { y } _ { 2 } ] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 523, + 506, + 538 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 372, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 245, + 549 + ], + "score": 1.0, + "content": "the latent factors are independent,", + "type": "text" + }, + { + "bbox": [ + 245, + 537, + 281, + 547 + ], + "score": 0.78, + "content": "\\mathbf { v } _ { 1 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 534, + 284, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 285, + 537, + 320, + 547 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { 2 } \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 534, + 339, + 549 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 536, + 367, + 547 + ], + "score": 0.88, + "content": "\\mathbf { c } \\sim p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 534, + 372, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 511, + 506, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 504, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 569 + ], + "score": 1.0, + "content": "The reference ambiguity occurs, when a decoder reproduces the data without satisfying the disentan-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 567, + 174, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 174, + 579 + ], + "score": 1.0, + "content": "gling properties.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 554, + 505, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 187, + 594 + ], + "score": 1.0, + "content": "Proposition 2. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 583, + 200, + 593 + ], + "score": 0.82, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "assign the same probability value to at least two different instances of v.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "Then, we can find encoders that reproduce the data distribution, but do not satisfy the disentangling", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 230, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 230, + 615 + ], + "score": 1.0, + "content": "properties for v in (2) and (4).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 581, + 506, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 104, + 624, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 624, + 216, + 641 + ], + "score": 1.0, + "content": "Proof. We already saw that", + "type": "text" + }, + { + "bbox": [ + 216, + 628, + 230, + 638 + ], + "score": 0.9, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 624, + 349, + 641 + ], + "score": 1.0, + "content": "satisfies (2), so we can choose", + "type": "text" + }, + { + "bbox": [ + 350, + 627, + 393, + 639 + ], + "score": 0.93, + "content": "N _ { \\mathbf { c } } = f _ { \\mathbf { c } } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 624, + 506, + 641 + ], + "score": 1.0, + "content": ", the inverse of the rendering", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 243, + 651 + ], + "score": 1.0, + "content": "engine. Now we look at defining", + "type": "text" + }, + { + "bbox": [ + 244, + 639, + 258, + 649 + ], + "score": 0.88, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 637, + 459, + 651 + ], + "score": 1.0, + "content": "and the decoder. The iso-probability property of", + "type": "text" + }, + { + "bbox": [ + 460, + 640, + 471, + 650 + ], + "score": 0.84, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "implies", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 221, + 662 + ], + "score": 1.0, + "content": "that there exists a mapping", + "type": "text" + }, + { + "bbox": [ + 221, + 649, + 253, + 661 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 649, + 298, + 662 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 299, + 649, + 356, + 661 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\sim p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 649, + 376, + 662 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 649, + 464, + 661 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\bar { \\mathbf { c } _ { 1 } } ) \\bar { \\neq } T ( \\bar { \\mathbf { v } } , \\bar { \\mathbf { c } _ { 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 114, + 670 + ], + "score": 0.54, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 659, + 134, + 673 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 134, + 660, + 172, + 671 + ], + "score": 0.9, + "content": "\\mathbf { c } _ { 1 } \\neq \\mathbf { c } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 659, + 316, + 673 + ], + "score": 1.0, + "content": ". For example, let us denote with", + "type": "text" + }, + { + "bbox": [ + 317, + 661, + 357, + 672 + ], + "score": 0.89, + "content": "\\mathbf { v } _ { 1 } \\neq \\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "two varying components such that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 671, + 222, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 178, + 683 + ], + "score": 0.92, + "content": "p _ { \\mathbf { v } } ( \\mathbf { v } _ { 1 } ) = p _ { \\mathbf { v } } ( \\mathbf { v } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 671, + 222, + 684 + ], + "score": 1.0, + "content": ". Then, let", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 624, + 506, + 684 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 689, + 396, + 730 + ], + "lines": [ + { + "bbox": [ + 215, + 689, + 396, + 730 + ], + "spans": [ + { + "bbox": [ + 215, + 689, + 396, + 730 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\dot { = } \\left\\{ \\begin{array} { l l } { \\mathbf { v } } & { \\mathrm { i f } \\ \\mathbf { v } \\neq \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } } \\\\ { \\mathbf { v } _ { 1 } } & { \\mathrm { i f } \\ \\mathbf { v } = \\mathbf { v } _ { 1 } \\lor \\mathbf { v } _ { 2 } \\mathrm { a n d } \\mathbf { c } \\in \\mathcal { C } } \\\\ { \\mathbf { v } _ { 2 } } & { \\mathrm { i f } \\ \\mathbf { v } = \\mathbf { v } _ { 1 } \\lor \\mathbf { v } _ { 2 } \\mathrm { a n d } \\mathbf { c } \\not \\in \\mathcal { C } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "bb4ae1e763a78a1dae1156e86bc9cd8f148c155f09fc90d3857031b049127b49.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 689, + 396, + 709.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 215, + 709.5, + 396, + 730.0 + ], + "spans": [], + "index": 43 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 124, + 96 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 83, + 131, + 92 + ], + "score": 0.8, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 81, + 248, + 96 + ], + "score": 1.0, + "content": "is a subset of the domain of", + "type": "text" + }, + { + "bbox": [ + 249, + 85, + 255, + 92 + ], + "score": 0.43, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 81, + 287, + 96 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 287, + 82, + 362, + 96 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\int _ { \\mathcal { C } } p _ { \\mathbf { c } } ( \\mathbf { c } ) d \\mathbf { c } = 1 / 2 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 81, + 506, + 96 + ], + "score": 1.0, + "content": ". Now, let us define the encoder as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 438, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 202, + 106 + ], + "score": 0.92, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 94, + 438, + 107 + ], + "score": 1.0, + "content": ". By using the autoencoder constraint, the decoder satisfies", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 110, + 432, + 124 + ], + "lines": [ + { + "bbox": [ + 178, + 110, + 432, + 124 + ], + "spans": [ + { + "bbox": [ + 178, + 110, + 432, + 124 + ], + "score": 0.89, + "content": "\\operatorname { D e c } ( N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) , N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) ) = \\operatorname { D e c } ( T ( \\mathbf { v } , \\mathbf { c } ) , \\mathbf { c } ) = f ( \\mathbf { v } , \\mathbf { c } ) .", + "type": "interline_equation", + "image_path": "54a73ddb2a2bc6d4404212dd98ca2cbb4232243f51a7f880456ef26805957772.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 178, + 110, + 432, + 124 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 128, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 158, + 141 + ], + "score": 1.0, + "content": "Even though", + "type": "text" + }, + { + "bbox": [ + 158, + 128, + 189, + 140 + ], + "score": 0.93, + "content": "T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 128, + 236, + 141 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 236, + 131, + 243, + 138 + ], + "score": 0.31, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 128, + 461, + 141 + ], + "score": 1.0, + "content": "functionally, they are statistically independent. Because", + "type": "text" + }, + { + "bbox": [ + 461, + 128, + 505, + 141 + ], + "score": 0.91, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\sim", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 118, + 151 + ], + "score": 0.84, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 140, + 136, + 151 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 136, + 141, + 165, + 151 + ], + "score": 0.89, + "content": "\\mathbf { c } \\sim p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 140, + 506, + 151 + ], + "score": 1.0, + "content": "by construction, our encoder-decoder pair defines a data distribution identical to that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 189, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 189, + 163 + ], + "score": 1.0, + "content": "given as training set", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 166, + 400, + 180 + ], + "lines": [ + { + "bbox": [ + 210, + 166, + 400, + 180 + ], + "spans": [ + { + "bbox": [ + 210, + 166, + 400, + 180 + ], + "score": 0.89, + "content": "\\left[ \\operatorname { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } ) , \\mathbf { c } ) , \\operatorname { D e c } ( T ( \\mathbf { v } _ { 2 } , \\mathbf { c } ) , \\mathbf { c } ) \\right] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } .", + "type": "interline_equation", + "image_path": "f6bcda88538e042aaccb0e9ecdcfead5aed1204c01d813be25799d45cf17aefe.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 210, + 166, + 400, + 180 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 183, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 370, + 198 + ], + "score": 1.0, + "content": "The feature disentanglement property is not satisfied because", + "type": "text" + }, + { + "bbox": [ + 371, + 183, + 505, + 196 + ], + "score": 0.92, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ) ~ = ~ T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ~ \\neq", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 225, + 207 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } ) \\ = \\ N _ { \\mathbf { v } } ( f ( { \\bar { \\mathbf { v } } } _ { 1 } , \\mathbf { c } _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 194, + 256, + 208 + ], + "score": 1.0, + "content": ", when", + "type": "text" + }, + { + "bbox": [ + 256, + 195, + 291, + 206 + ], + "score": 0.89, + "content": "\\mathbf { c } _ { 1 } ~ \\in ~ { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 194, + 311, + 208 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 195, + 346, + 207 + ], + "score": 0.89, + "content": "\\mathbf { c } _ { 2 } \\notin \\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 194, + 506, + 208 + ], + "score": 1.0, + "content": ". Similarly, the data disentanglement", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 235, + 219 + ], + "score": 1.0, + "content": "property does not hold, because", + "type": "text" + }, + { + "bbox": [ + 235, + 207, + 404, + 218 + ], + "score": 0.91, + "content": "\\mathrm { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) , \\mathbf { c } _ { 1 } ) \\neq \\mathrm { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } ) , \\mathbf { c } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 205, + 409, + 219 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 207, + 505, + 217 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 504, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "The above proposition implies that we cannot provably disentangle all the factors of variation from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 487, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 442, + 254 + ], + "score": 1.0, + "content": "weakly labeled data, even if we had access to all the data and knew the distributions", + "type": "text" + }, + { + "bbox": [ + 443, + 243, + 454, + 253 + ], + "score": 0.86, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 240, + 472, + 254 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 472, + 243, + 483, + 253 + ], + "score": 0.84, + "content": "p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 240, + 487, + 254 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 365, + 270 + ], + "score": 1.0, + "content": "To better understand it, let us consider a practical example. Let", + "type": "text" + }, + { + "bbox": [ + 365, + 258, + 423, + 270 + ], + "score": 0.93, + "content": "\\mathbf { v } \\sim \\mathcal { U } [ - \\pi , \\pi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "be the (continuous)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 242, + 281 + ], + "score": 1.0, + "content": "viewpoint (the azimuth angle) and", + "type": "text" + }, + { + "bbox": [ + 242, + 269, + 289, + 280 + ], + "score": 0.91, + "content": "\\mathbf { c } \\sim B ( 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 268, + 367, + 281 + ], + "score": 1.0, + "content": "the car type, where", + "type": "text" + }, + { + "bbox": [ + 367, + 270, + 376, + 279 + ], + "score": 0.74, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "denotes the uniform distribution", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 124, + 293 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 280, + 152, + 291 + ], + "score": 0.91, + "content": "B ( 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 279, + 330, + 293 + ], + "score": 1.0, + "content": "the Bernoulli distribution with probability", + "type": "text" + }, + { + "bbox": [ + 330, + 280, + 460, + 292 + ], + "score": 0.91, + "content": "p _ { \\mathbf { c } } ( \\mathbf { c } = 0 ) = p _ { \\mathbf { c } } ( \\mathbf { c } = 1 ) = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "(i.e., there", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 321, + 303 + ], + "score": 1.0, + "content": "are only 2 car types). In this case, every instance of", + "type": "text" + }, + { + "bbox": [ + 321, + 293, + 329, + 300 + ], + "score": 0.41, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 291, + 403, + 303 + ], + "score": 1.0, + "content": "is iso-probable in", + "type": "text" + }, + { + "bbox": [ + 403, + 293, + 415, + 302 + ], + "score": 0.83, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "so we have the worst", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 372, + 315 + ], + "score": 1.0, + "content": "scenario for the reference ambiguity. We can define the function", + "type": "text" + }, + { + "bbox": [ + 372, + 302, + 459, + 313 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) = \\mathbf { v } ( 2 \\mathbf { c } - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 300, + 505, + 315 + ], + "score": 1.0, + "content": "so that the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 312, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 154, + 326 + ], + "score": 1.0, + "content": "mapping of", + "type": "text" + }, + { + "bbox": [ + 155, + 314, + 162, + 322 + ], + "score": 0.63, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 312, + 383, + 326 + ], + "score": 1.0, + "content": "is mirrored as we change the car type. By construction", + "type": "text" + }, + { + "bbox": [ + 384, + 313, + 465, + 324 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\sim \\mathcal { U } [ - \\pi , \\pi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 312, + 497, + 326 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 497, + 315, + 504, + 323 + ], + "score": 0.28, + "content": "\\mathbf { c }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 123, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 324, + 208, + 336 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\mathbf { c } _ { 1 } ) \\neq T ( \\mathbf { v } , \\mathbf { c } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 323, + 222, + 337 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 223, + 324, + 249, + 335 + ], + "score": 0.92, + "content": "\\mathbf { v } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 323, + 266, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 324, + 299, + 335 + ], + "score": 0.91, + "content": "\\mathbf { c } _ { 1 } \\neq \\mathbf { c } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 323, + 464, + 337 + ], + "score": 1.0, + "content": ". So we cannot tell the difference between", + "type": "text" + }, + { + "bbox": [ + 465, + 324, + 473, + 334 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 409, + 348 + ], + "score": 1.0, + "content": "ideal correct mapping to the viewpoint factor. This is equivalent to an encoder", + "type": "text" + }, + { + "bbox": [ + 409, + 334, + 505, + 347 + ], + "score": 0.9, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "that reverses the ordering of the azimuth of car 1 with respect to car 0. Each car has its own reference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "system, and thus it is not possible to transfer the viewpoint from one system to the other, as it is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 185, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 185, + 381 + ], + "score": 1.0, + "content": "illustrated in fig. 1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 392, + 209, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 210, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 210, + 405 + ], + "score": 1.0, + "content": "3.4 IMPLEMENTATION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 504, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 483, + 425 + ], + "score": 1.0, + "content": "In our implementation we use convolutional neural networks for all the models. We denote with", + "type": "text" + }, + { + "bbox": [ + 483, + 413, + 489, + 423 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "parameters associated to each network. Then, the optimization of the composite loss can be written as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 439, + 412, + 461 + ], + "lines": [ + { + "bbox": [ + 199, + 439, + 412, + 461 + ], + "spans": [ + { + "bbox": [ + 199, + 439, + 412, + 461 + ], + "score": 0.92, + "content": "\\hat { \\theta } _ { \\mathrm { { D e c } } } , \\hat { \\theta } _ { \\mathrm { { E n c } } } , \\hat { \\theta } _ { \\mathrm { { D s c } } } = \\arg \\operatorname* { m i n } _ { \\theta _ { \\mathrm { { D e c } } } , \\theta _ { \\mathrm { { E n c } } } } \\operatorname* { m a x } _ { \\theta _ { \\mathrm { { D s c } } } } \\mathcal { L } ( \\theta _ { \\mathrm { { D e c } } } , \\theta _ { \\mathrm { { E n c } } } , \\theta _ { \\mathrm { { D s c } } } ) .", + "type": "interline_equation", + "image_path": "ad8a7a239b6e53b3ca59924edbecb1c1d2d6c239f4721fbd445add5e1ca1471a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 199, + 439, + 412, + 461 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 104, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 153, + 478 + ], + "score": 1.0, + "content": "We choose", + "type": "text" + }, + { + "bbox": [ + 154, + 466, + 180, + 475 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "and also add regularization to the adversarial loss so that each logarithm has a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 221, + 488 + ], + "score": 1.0, + "content": "minimum value. We define", + "type": "text" + }, + { + "bbox": [ + 221, + 476, + 389, + 488 + ], + "score": 0.91, + "content": "\\log _ { \\epsilon } \\bar { \\mathrm { D s c } } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) = \\log ( \\epsilon + \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "(and similarly for the other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 485, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 214, + 500 + ], + "score": 1.0, + "content": "logarithmic term) and use", + "type": "text" + }, + { + "bbox": [ + 215, + 487, + 259, + 497 + ], + "score": 0.89, + "content": "\\epsilon = 1 0 ^ { - 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 485, + 506, + 500 + ], + "score": 1.0, + "content": ". The main components of our neural network are shown in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "Fig. 2. The architecture of the encoder and the decoder were taken from DCGAN Radford et al.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "(2015), with slight modifications. We added fully connected layers at the output of the encoder and to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "the input of the decoder. For the discriminator we used a simplified version of the VGG Simonyan &", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "Zisserman (2014) network. As the input to the discriminator is an image pair, we concatenate them", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 542, + 209, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 209, + 554 + ], + "score": 1.0, + "content": "along the color channels.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "Normalization. In our architecture both the encoder and the decoder networks use blocks with a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "convolutional layer, a nonlinear activation function (ReLU/leaky ReLU) and a normalization layer,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "typically, batch normalization (BN). As an alternative to BN we consider the recently introduced", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "instance normalization (IN) Ulyanov et al. (2017). The main difference between BN and IN is that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "the latter just computes the mean and standard deviation across the spatial domain of the input and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "score": 1.0, + "content": "not along the batch dimension. Thus, the shift and scaling for the output of each layer is the same at", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 624, + 491, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 491, + 638 + ], + "score": 1.0, + "content": "every iteration for the same input image. In practice, we find that IN improves the performance.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 108, + 651, + 200, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 201, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 201, + 666 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "We tested our method on the MNIST, Sprites and ShapeNet datasets. We performed ablation studies", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "on the shortcut problem using ShapeNet cars. We focused on the effect of the feature dimensionality", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 235, + 712 + ], + "score": 1.0, + "content": "and having the adversarial term", + "type": "text" + }, + { + "bbox": [ + 235, + 699, + 300, + 710 + ], + "score": 0.89, + "content": "( \\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 699, + 328, + 712 + ], + "score": 1.0, + "content": "or not", + "type": "text" + }, + { + "bbox": [ + 329, + 699, + 355, + 710 + ], + "score": 0.88, + "content": "( \\mathcal { L } _ { A E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 699, + 505, + 712 + ], + "score": 1.0, + "content": ". We also show that in most cases the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "reference ambiguity does not arise in practice (MNIST, Sprites, ShapeNet cars), we can only observe", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 317, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 317, + 733 + ], + "score": 1.0, + "content": "it when the data is more complex (ShapeNet chairs).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 124, + 96 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 83, + 131, + 92 + ], + "score": 0.8, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 81, + 248, + 96 + ], + "score": 1.0, + "content": "is a subset of the domain of", + "type": "text" + }, + { + "bbox": [ + 249, + 85, + 255, + 92 + ], + "score": 0.43, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 81, + 287, + 96 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 287, + 82, + 362, + 96 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\int _ { \\mathcal { C } } p _ { \\mathbf { c } } ( \\mathbf { c } ) d \\mathbf { c } = 1 / 2 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 81, + 506, + 96 + ], + "score": 1.0, + "content": ". Now, let us define the encoder as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 438, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 202, + 106 + ], + "score": 0.92, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 94, + 438, + 107 + ], + "score": 1.0, + "content": ". By using the autoencoder constraint, the decoder satisfies", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 107 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 110, + 432, + 124 + ], + "lines": [ + { + "bbox": [ + 178, + 110, + 432, + 124 + ], + "spans": [ + { + "bbox": [ + 178, + 110, + 432, + 124 + ], + "score": 0.89, + "content": "\\operatorname { D e c } ( N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) , N _ { \\mathbf { c } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) ) = \\operatorname { D e c } ( T ( \\mathbf { v } , \\mathbf { c } ) , \\mathbf { c } ) = f ( \\mathbf { v } , \\mathbf { c } ) .", + "type": "interline_equation", + "image_path": "54a73ddb2a2bc6d4404212dd98ca2cbb4232243f51a7f880456ef26805957772.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 178, + 110, + 432, + 124 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 128, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 158, + 141 + ], + "score": 1.0, + "content": "Even though", + "type": "text" + }, + { + "bbox": [ + 158, + 128, + 189, + 140 + ], + "score": 0.93, + "content": "T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 128, + 236, + 141 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 236, + 131, + 243, + 138 + ], + "score": 0.31, + "content": "\\mathbf { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 128, + 461, + 141 + ], + "score": 1.0, + "content": "functionally, they are statistically independent. Because", + "type": "text" + }, + { + "bbox": [ + 461, + 128, + 505, + 141 + ], + "score": 0.91, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\sim", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 118, + 151 + ], + "score": 0.84, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 140, + 136, + 151 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 136, + 141, + 165, + 151 + ], + "score": 0.89, + "content": "\\mathbf { c } \\sim p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 140, + 506, + 151 + ], + "score": 1.0, + "content": "by construction, our encoder-decoder pair defines a data distribution identical to that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 189, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 189, + 163 + ], + "score": 1.0, + "content": "given as training set", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 128, + 506, + 163 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 166, + 400, + 180 + ], + "lines": [ + { + "bbox": [ + 210, + 166, + 400, + 180 + ], + "spans": [ + { + "bbox": [ + 210, + 166, + 400, + 180 + ], + "score": 0.89, + "content": "\\left[ \\operatorname { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } ) , \\mathbf { c } ) , \\operatorname { D e c } ( T ( \\mathbf { v } _ { 2 } , \\mathbf { c } ) , \\mathbf { c } ) \\right] \\sim p _ { \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } } .", + "type": "interline_equation", + "image_path": "f6bcda88538e042aaccb0e9ecdcfead5aed1204c01d813be25799d45cf17aefe.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 210, + 166, + 400, + 180 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 183, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 370, + 198 + ], + "score": 1.0, + "content": "The feature disentanglement property is not satisfied because", + "type": "text" + }, + { + "bbox": [ + 371, + 183, + 505, + 196 + ], + "score": 0.92, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ) ~ = ~ T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) ~ \\neq", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 225, + 207 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } ) \\ = \\ N _ { \\mathbf { v } } ( f ( { \\bar { \\mathbf { v } } } _ { 1 } , \\mathbf { c } _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 194, + 256, + 208 + ], + "score": 1.0, + "content": ", when", + "type": "text" + }, + { + "bbox": [ + 256, + 195, + 291, + 206 + ], + "score": 0.89, + "content": "\\mathbf { c } _ { 1 } ~ \\in ~ { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 194, + 311, + 208 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 195, + 346, + 207 + ], + "score": 0.89, + "content": "\\mathbf { c } _ { 2 } \\notin \\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 194, + 506, + 208 + ], + "score": 1.0, + "content": ". Similarly, the data disentanglement", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 235, + 219 + ], + "score": 1.0, + "content": "property does not hold, because", + "type": "text" + }, + { + "bbox": [ + 235, + 207, + 404, + 218 + ], + "score": 0.91, + "content": "\\mathrm { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 1 } ) , \\mathbf { c } _ { 1 } ) \\neq \\mathrm { D e c } ( T ( \\mathbf { v } _ { 1 } , \\mathbf { c } _ { 2 } ) , \\mathbf { c } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 205, + 409, + 219 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 207, + 505, + 217 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 182, + 506, + 219 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 504, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "The above proposition implies that we cannot provably disentangle all the factors of variation from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 487, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 442, + 254 + ], + "score": 1.0, + "content": "weakly labeled data, even if we had access to all the data and knew the distributions", + "type": "text" + }, + { + "bbox": [ + 443, + 243, + 454, + 253 + ], + "score": 0.86, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 240, + 472, + 254 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 472, + 243, + 483, + 253 + ], + "score": 0.84, + "content": "p _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 240, + 487, + 254 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 229, + 505, + 254 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 365, + 270 + ], + "score": 1.0, + "content": "To better understand it, let us consider a practical example. Let", + "type": "text" + }, + { + "bbox": [ + 365, + 258, + 423, + 270 + ], + "score": 0.93, + "content": "\\mathbf { v } \\sim \\mathcal { U } [ - \\pi , \\pi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "be the (continuous)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 242, + 281 + ], + "score": 1.0, + "content": "viewpoint (the azimuth angle) and", + "type": "text" + }, + { + "bbox": [ + 242, + 269, + 289, + 280 + ], + "score": 0.91, + "content": "\\mathbf { c } \\sim B ( 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 268, + 367, + 281 + ], + "score": 1.0, + "content": "the car type, where", + "type": "text" + }, + { + "bbox": [ + 367, + 270, + 376, + 279 + ], + "score": 0.74, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "denotes the uniform distribution", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 124, + 293 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 280, + 152, + 291 + ], + "score": 0.91, + "content": "B ( 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 279, + 330, + 293 + ], + "score": 1.0, + "content": "the Bernoulli distribution with probability", + "type": "text" + }, + { + "bbox": [ + 330, + 280, + 460, + 292 + ], + "score": 0.91, + "content": "p _ { \\mathbf { c } } ( \\mathbf { c } = 0 ) = p _ { \\mathbf { c } } ( \\mathbf { c } = 1 ) = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "(i.e., there", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 321, + 303 + ], + "score": 1.0, + "content": "are only 2 car types). In this case, every instance of", + "type": "text" + }, + { + "bbox": [ + 321, + 293, + 329, + 300 + ], + "score": 0.41, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 291, + 403, + 303 + ], + "score": 1.0, + "content": "is iso-probable in", + "type": "text" + }, + { + "bbox": [ + 403, + 293, + 415, + 302 + ], + "score": 0.83, + "content": "p _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "so we have the worst", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 372, + 315 + ], + "score": 1.0, + "content": "scenario for the reference ambiguity. We can define the function", + "type": "text" + }, + { + "bbox": [ + 372, + 302, + 459, + 313 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) = \\mathbf { v } ( 2 \\mathbf { c } - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 300, + 505, + 315 + ], + "score": 1.0, + "content": "so that the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 312, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 154, + 326 + ], + "score": 1.0, + "content": "mapping of", + "type": "text" + }, + { + "bbox": [ + 155, + 314, + 162, + 322 + ], + "score": 0.63, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 312, + 383, + 326 + ], + "score": 1.0, + "content": "is mirrored as we change the car type. By construction", + "type": "text" + }, + { + "bbox": [ + 384, + 313, + 465, + 324 + ], + "score": 0.9, + "content": "T ( \\mathbf { v } , \\mathbf { c } ) \\sim \\mathcal { U } [ - \\pi , \\pi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 312, + 497, + 326 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 497, + 315, + 504, + 323 + ], + "score": 0.28, + "content": "\\mathbf { c }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 123, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 324, + 208, + 336 + ], + "score": 0.92, + "content": "T ( \\mathbf { v } , \\mathbf { c } _ { 1 } ) \\neq T ( \\mathbf { v } , \\mathbf { c } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 323, + 222, + 337 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 223, + 324, + 249, + 335 + ], + "score": 0.92, + "content": "\\mathbf { v } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 323, + 266, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 324, + 299, + 335 + ], + "score": 0.91, + "content": "\\mathbf { c } _ { 1 } \\neq \\mathbf { c } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 323, + 464, + 337 + ], + "score": 1.0, + "content": ". So we cannot tell the difference between", + "type": "text" + }, + { + "bbox": [ + 465, + 324, + 473, + 334 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 409, + 348 + ], + "score": 1.0, + "content": "ideal correct mapping to the viewpoint factor. This is equivalent to an encoder", + "type": "text" + }, + { + "bbox": [ + 409, + 334, + 505, + 347 + ], + "score": 0.9, + "content": "N _ { \\mathbf { v } } ( f ( \\mathbf { v } , \\mathbf { c } ) ) = T ( \\mathbf { v } , \\mathbf { c } )", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "that reverses the ordering of the azimuth of car 1 with respect to car 0. Each car has its own reference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "system, and thus it is not possible to transfer the viewpoint from one system to the other, as it is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 185, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 185, + 381 + ], + "score": 1.0, + "content": "illustrated in fig. 1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 257, + 506, + 381 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 392, + 209, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 210, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 210, + 405 + ], + "score": 1.0, + "content": "3.4 IMPLEMENTATION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 504, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 483, + 425 + ], + "score": 1.0, + "content": "In our implementation we use convolutional neural networks for all the models. We denote with", + "type": "text" + }, + { + "bbox": [ + 483, + 413, + 489, + 423 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "parameters associated to each network. Then, the optimization of the composite loss can be written as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 412, + 506, + 437 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 439, + 412, + 461 + ], + "lines": [ + { + "bbox": [ + 199, + 439, + 412, + 461 + ], + "spans": [ + { + "bbox": [ + 199, + 439, + 412, + 461 + ], + "score": 0.92, + "content": "\\hat { \\theta } _ { \\mathrm { { D e c } } } , \\hat { \\theta } _ { \\mathrm { { E n c } } } , \\hat { \\theta } _ { \\mathrm { { D s c } } } = \\arg \\operatorname* { m i n } _ { \\theta _ { \\mathrm { { D e c } } } , \\theta _ { \\mathrm { { E n c } } } } \\operatorname* { m a x } _ { \\theta _ { \\mathrm { { D s c } } } } \\mathcal { L } ( \\theta _ { \\mathrm { { D e c } } } , \\theta _ { \\mathrm { { E n c } } } , \\theta _ { \\mathrm { { D s c } } } ) .", + "type": "interline_equation", + "image_path": "ad8a7a239b6e53b3ca59924edbecb1c1d2d6c239f4721fbd445add5e1ca1471a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 199, + 439, + 412, + 461 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 104, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 153, + 478 + ], + "score": 1.0, + "content": "We choose", + "type": "text" + }, + { + "bbox": [ + 154, + 466, + 180, + 475 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "and also add regularization to the adversarial loss so that each logarithm has a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 221, + 488 + ], + "score": 1.0, + "content": "minimum value. We define", + "type": "text" + }, + { + "bbox": [ + 221, + 476, + 389, + 488 + ], + "score": 0.91, + "content": "\\log _ { \\epsilon } \\bar { \\mathrm { D s c } } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) = \\log ( \\epsilon + \\mathrm { D s c } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "(and similarly for the other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 485, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 214, + 500 + ], + "score": 1.0, + "content": "logarithmic term) and use", + "type": "text" + }, + { + "bbox": [ + 215, + 487, + 259, + 497 + ], + "score": 0.89, + "content": "\\epsilon = 1 0 ^ { - 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 485, + 506, + 500 + ], + "score": 1.0, + "content": ". The main components of our neural network are shown in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "Fig. 2. The architecture of the encoder and the decoder were taken from DCGAN Radford et al.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "(2015), with slight modifications. We added fully connected layers at the output of the encoder and to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "the input of the decoder. For the discriminator we used a simplified version of the VGG Simonyan &", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "Zisserman (2014) network. As the input to the discriminator is an image pair, we concatenate them", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 542, + 209, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 209, + 554 + ], + "score": 1.0, + "content": "along the color channels.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 464, + 506, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "Normalization. In our architecture both the encoder and the decoder networks use blocks with a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "convolutional layer, a nonlinear activation function (ReLU/leaky ReLU) and a normalization layer,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "typically, batch normalization (BN). As an alternative to BN we consider the recently introduced", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "instance normalization (IN) Ulyanov et al. (2017). The main difference between BN and IN is that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "the latter just computes the mean and standard deviation across the spatial domain of the input and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "score": 1.0, + "content": "not along the batch dimension. Thus, the shift and scaling for the output of each layer is the same at", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 624, + 491, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 491, + 638 + ], + "score": 1.0, + "content": "every iteration for the same input image. In practice, we find that IN improves the performance.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 559, + 506, + 638 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 651, + 200, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 201, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 201, + 666 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "We tested our method on the MNIST, Sprites and ShapeNet datasets. We performed ablation studies", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "on the shortcut problem using ShapeNet cars. We focused on the effect of the feature dimensionality", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 235, + 712 + ], + "score": 1.0, + "content": "and having the adversarial term", + "type": "text" + }, + { + "bbox": [ + 235, + 699, + 300, + 710 + ], + "score": 0.89, + "content": "( \\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 699, + 328, + 712 + ], + "score": 1.0, + "content": "or not", + "type": "text" + }, + { + "bbox": [ + 329, + 699, + 355, + 710 + ], + "score": 0.88, + "content": "( \\mathcal { L } _ { A E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 699, + 505, + 712 + ], + "score": 1.0, + "content": ". We also show that in most cases the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "reference ambiguity does not arise in practice (MNIST, Sprites, ShapeNet cars), we can only observe", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 317, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 317, + 733 + ], + "score": 1.0, + "content": "it when the data is more complex (ShapeNet chairs).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 677, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 217, + 80, + 394, + 226 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 217, + 80, + 394, + 226 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 217, + 80, + 394, + 226 + ], + "spans": [ + { + "bbox": [ + 217, + 80, + 394, + 226 + ], + "score": 0.968, + "type": "image", + "image_path": "90a0f2e4ff16668ce677bfd81f19be4a9e730377fbd5bbdf49085baa9daeb981.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 217, + 80, + 394, + 93.27272727272728 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 217, + 93.27272727272728, + 394, + 106.54545454545456 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 217, + 106.54545454545456, + 394, + 119.81818181818184 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 217, + 119.81818181818184, + 394, + 133.09090909090912 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 217, + 133.09090909090912, + 394, + 146.3636363636364 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 217, + 146.3636363636364, + 394, + 159.63636363636368 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 217, + 159.63636363636368, + 394, + 172.90909090909096 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 217, + 172.90909090909096, + 394, + 186.18181818181824 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 217, + 186.18181818181824, + 394, + 199.45454545454552 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 217, + 199.45454545454552, + 394, + 212.7272727272728 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 217, + 212.7272727272728, + 394, + 226.00000000000009 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 239, + 506, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "Figure 2: Learning to disentangle factors of variation. The scheme above shows how the encoder", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "(Enc), the decoder (Dec) and the discriminator (Dsc) are trained with input triplets. The components", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 260, + 247, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 247, + 273 + ], + "score": 1.0, + "content": "with the same name share weights.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 108, + 298, + 221, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 223, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 223, + 310 + ], + "score": 1.0, + "content": "4.1 SHORTCUT PROBLEM", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "ShapeNet cars. The ShapeNet dataset Chang et al. (2015) contains 3D objects than we can render", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "from different viewpoints. We consider only one category (cars) for a set of fixed viewpoints. Cars", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "have high intraclass variability and they do not have rotational symmetries. We used approximately", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "score": 1.0, + "content": "3K car types for training and 300 for testing. We rendered 24 possible viewpoints around each object", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 377 + ], + "score": 1.0, + "content": "in a full circle, resulting in 80K images in total. The elevation was fixed to 15 degrees and azimuth", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 439, + 388 + ], + "score": 1.0, + "content": "angles were spaced 15 degrees apart. 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Fig. 5", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "(c) and (d) show results on the attribute transfer task. Both our method and Mathieu et al. (2016)’s", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 422, + 437, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 437, + 434 + ], + "score": 1.0, + "content": "transfer the identity of the sprites correctly, the reference ambiguity does not arise.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "ShapeNet chairs. 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We found that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "score": 1.0, + "content": "the reference ambiguity does not emerge for cars, but it does for chairs, possibly due to the higher", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 493, + 340, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 340, + 506 + ], + "score": 1.0, + "content": "complexity, as cars have much less variability than chairs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 201, + 537 + ], + "lines": [ + { + "bbox": [ + 104, + 522, + 203, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 203, + 540 + ], + "score": 1.0, + "content": "5 CONCLUSIONS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "In this paper we studied the challenges of disentangling factors of variation, mainly the shortcut", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "problem and the reference ambiguity. The shortcut problem occurs when all information is stored in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "only one feature chunk, while the other is ignored. The reference ambiguity means that the reference", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "in which a factor is interpreted, may depend on other factors. This makes the attribute transfer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "ambiguous. 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However, in practice we observe that our trained model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 662, + 406, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 406, + 674 + ], + "score": 1.0, + "content": "works well on many datasets and exhibits good generalization capabilities.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 692, + 175, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 691, + 176, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 176, + 705 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 108, + 711, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 115, + 722, + 439, + 732 + ], + "spans": [ + { + "bbox": [ + 115, + 722, + 439, + 732 + ], + "score": 1.0, + "content": "IEEE transactions on pattern analysis and machine intelligence, 35(8):1798–1828, 2013.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 118, + 78, + 492, + 255 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 118, + 78, + 492, + 255 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 78, + 492, + 255 + ], + "spans": [ + { + "bbox": [ + 118, + 78, + 492, + 255 + ], + "score": 0.969, + "type": "image", + "image_path": "55049684dd4908d747a5bd58b28114a25a2f6b44c1aa8542f673f3beeb04f571.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 118, + 78, + 492, + 137.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 118, + 137.0, + 492, + 196.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 118, + 196.0, + 492, + 255.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 264, + 506, + 308 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 265, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 276 + ], + "score": 1.0, + "content": "Figure 6: Attribute transfer on ShapeNet. For both subfigures the viewpoint is taken from the leftmost", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 276, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 506, + 287 + ], + "score": 1.0, + "content": "column and the car/chair type is taken from the first row. 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We trained", + "type": "text" + }, + { + "bbox": [ + 213, + 461, + 272, + 472 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { A E } + \\mathcal { L } _ { G A N }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 460, + 471, + 474 + ], + "score": 1.0, + "content": ", and set the feature dimensions to 1024 for both", + "type": "text" + }, + { + "bbox": [ + 472, + 461, + 487, + 471 + ], + "score": 0.89, + "content": "N _ { \\mathbf { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 107, + 472, + 120, + 482 + ], + "score": 0.85, + "content": "N _ { \\mathbf { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 471, + 505, + 484 + ], + "score": 1.0, + "content": ". In Fig. 6 we show results on attribute transfer and compare it with ShapeNet cars. We found that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 496 + ], + "score": 1.0, + "content": "the reference ambiguity does not emerge for cars, but it does for chairs, possibly due to the higher", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 493, + 340, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 340, + 506 + ], + "score": 1.0, + "content": "complexity, as cars have much less variability than chairs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 437, + 506, + 506 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 201, + 537 + ], + "lines": [ + { + "bbox": [ + 104, + 522, + 203, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 203, + 540 + ], + "score": 1.0, + "content": "5 CONCLUSIONS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "In this paper we studied the challenges of disentangling factors of variation, mainly the shortcut", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "problem and the reference ambiguity. The shortcut problem occurs when all information is stored in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "only one feature chunk, while the other is ignored. The reference ambiguity means that the reference", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "in which a factor is interpreted, may depend on other factors. This makes the attribute transfer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "ambiguous. We introduced a novel training of autoencoders to solve disentangling using image", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 607, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 619 + ], + "score": 1.0, + "content": "triplets. We showed theoretically and experimentally how to keep the shortcut problem under control", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "through adversarial training, and enable to use large feature dimensions. We proved that the reference", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "ambiguity is inherently present in the disentangling task when weak labels are used. Most importantly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "this can be stated independently of the learning algorithm. We demonstrated that training and transfer", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "of factors of variation may not be guaranteed. 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a/parse/train/rklEj2EFvB/rklEj2EFvB_content_list.json b/parse/train/rklEj2EFvB/rklEj2EFvB_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..1d782052bfbb5ad7896479637adb13aa9975dec7 --- /dev/null +++ b/parse/train/rklEj2EFvB/rklEj2EFvB_content_list.json @@ -0,0 +1,3670 @@ +[ + { + "type": "text", + "text": "ESTIMATING GRADIENTS FOR DISCRETE RANDOM VARIABLES BY SAMPLING WITHOUT REPLACEMENT ", + "text_level": 1, + "bbox": [ + 176, + 99, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Herke van Hoof University of Amsterdam h.c.vanhoof@uva.nl ", + "bbox": [ + 397, + 170, + 575, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Wouter Kool \nUniversity of Amsterdam ORTEC \nw.w.m.kool@uva.nl Max Welling \nUniversity of Amsterdam CIFAR \nm.welling@uva.nl ", + "bbox": [ + 183, + 170, + 352, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 617, + 170, + 787, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 262, + 544, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We derive an unbiased estimator for expectations over discrete random variables based on sampling without replacement, which reduces variance as it avoids duplicate samples. We show that our estimator can be derived as the RaoBlackwellization of three different estimators. Combining our estimator with REINFORCE, we obtain a policy gradient estimator and we reduce its variance using a built-in control variate which is obtained without additional model evaluations. The resulting estimator is closely related to other gradient estimators. Experiments with a toy problem, a categorical Variational Auto-Encoder and a structured prediction problem show that our estimator is the only estimator that is consistently among the best estimators in both high and low entropy settings. ", + "bbox": [ + 233, + 295, + 764, + 434 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 460, + 336, + 477 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Put replacement in your basement! We derive the unordered set estimator1: an unbiased (gradient) estimator for expectations over discrete random variables based on (unordered sets of) samples without replacement. In particular, we consider the problem of estimating (the gradient of) the expectation of $f ( { \\pmb x } )$ where $_ { \\textbf { \\em x } }$ has a discrete distribution $p$ over the domain $D$ , i.e. ", + "bbox": [ + 174, + 491, + 825, + 547 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/a18d756698f674fda23751c9deccb14a866693f84ffad8812a7db5d0aa84f303.jpg", + "text": "$$\n\\mathbb { E } _ { { \\pmb x } \\sim p ( { \\pmb x } ) } [ f ( { \\pmb x } ) ] = \\sum _ { { \\pmb x } \\in D } p ( { \\pmb x } ) f ( { \\pmb x } ) .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 555, + 622, + 582 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This expectation comes up in reinforcement learning, discrete latent variable modelling (e.g. for compression), structured prediction (e.g. for translation), hard attention and many other tasks that use models with discrete operations in their computational graphs (see e.g. Jang et al. (2016)). In general, $_ { \\textbf { \\em x } }$ has structure (such as a sequence), but we can treat it as a ‘flat’ distribution, omitting the bold notation, so $x$ has a categorical distribution over $D$ given by $p ( x ) , x \\in D$ . Typically, the distribution has parameters $\\pmb { \\theta }$ , which are learnt through gradient descent. This requires estimating the gradient $\\nabla _ { \\pmb { \\theta } } \\bar { \\mathbb { E } } _ { \\boldsymbol { x } \\sim p _ { \\pmb { \\theta } } ( \\boldsymbol { x } ) } [ f ( \\boldsymbol { x } ) ]$ , using a set of samples $S$ . A gradient estimate $e ( S )$ is unbiased if ", + "bbox": [ + 173, + 587, + 825, + 685 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/f611a4818678ca6d6a94189e9ceb1ebc2fee17ab410928ec5188501d7f982d21.jpg", + "text": "$$\n\\mathbb { E } _ { S } [ e ( S ) ] = \\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { \\pmb { x } \\sim p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ f ( \\pmb { x } ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 693, + 606, + 712 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The samples $S$ can be sampled independently or using alternatives such as stratified sampling which reduce variance to increase the speed of learning. In this paper, we derive an unbiased gradient estimator that reduces variance by avoiding duplicate samples, i.e. by sampling $S$ without replacement. This is challenging as samples without replacement are dependent and have marginal distributions that are different from $p ( x )$ . We further reduce the variance by deriving a built-in control variate, which maintains the unbiasedness and does not require additional samples. ", + "bbox": [ + 173, + 717, + 825, + 801 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Related work. Many algorithms for estimating gradients for discrete distributions have been proposed. A general and widely used estimator is REINFORCE (Williams, 1992). Biased gradients based on a continuous relaxations of the discrete distribution (known as Gumbel-Softmax or Concrete) were jointly introduced by Jang et al. (2016) and Maddison et al. (2016). These can be combined with the straight through estimator (Bengio et al., 2013) if the model requires discrete samples or be used to construct control variates for REINFORCE, as in REBAR (Tucker et al., 2017) or ", + "bbox": [ + 174, + 815, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "RELAX (Grathwohl et al., 2018). Many other methods use control variates and other techniques to reduce the variance of REINFORCE (Paisley et al., 2012; Ranganath et al., 2014; Gregor et al., 2014; Mnih & Gregor, 2014; Gu et al., 2016; Mnih & Rezende, 2016). ", + "bbox": [ + 173, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Some works rely on explicit summation of the expectation, either for the marginal distribution (Titsias & Lazaro-Gredilla, 2015) or globally summing some categories while sampling from the re- ´ mainder (Liang et al., 2018; Liu et al., 2019). Other approaches use a finite difference approximation to the gradient (Lorberbom et al., 2018; 2019). Yin et al. (2019) introduced ARSM, which uses multiple model evaluations where the number adapts automatically to the uncertainty. ", + "bbox": [ + 173, + 152, + 825, + 223 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the structured prediction setting, there are many algorithms for optimizing a quantity under a sequence of discrete decisions, using (weak) supervision, multiple samples (or deterministic model evaluations), or a combination both (Ranzato et al., 2016; Shen et al., 2016; He et al., 2016; Norouzi et al., 2016; Bahdanau et al., 2017; Edunov et al., 2018; Leblond et al., 2018; Negrinho et al., 2018). Most of these algorithms are biased and rely on pretraining using maximum likelihood or gradually transitioning from supervised to reinforcement learning. Using Gumbel-Softmax based approaches in a sequential setting is difficult as the bias accumulates because of mixing errors (Gu et al., 2018). ", + "bbox": [ + 173, + 229, + 825, + 328 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 349, + 339, + 364 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Throughout this paper, we will denote with $B ^ { k }$ an ordered sample without replacement of size $k$ and with $S ^ { k }$ an unordered sample (of size $k$ ) from the categorical distribution $p$ . ", + "bbox": [ + 174, + 381, + 823, + 411 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Restricted distribution. When sampling without replacement, we remove the set $C \\subset D$ already sampled from the domain and we denote with $p ^ { D \\setminus C }$ the distribution restricted to the domain $D \\backslash C$ : ", + "bbox": [ + 171, + 426, + 823, + 458 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/13de092ebbad144ed1582389f19c0e3ba4185aa7ab5fb3003afb94aebee46b00.jpg", + "text": "$$\np ^ { D \\setminus C } ( x ) = \\frac { p ( x ) } { 1 - \\sum _ { c \\in C } p ( c ) } , \\quad x \\in D \\setminus C .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 465, + 643, + 501 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Ordered sample without replacement $B ^ { k }$ . Let $B ^ { k } = ( b _ { 1 } , . . . , b _ { k } ) , b _ { i } \\in D$ be an ordered sample without replacement, which is generated from the distribution $p$ as follows: first, sample $b _ { 1 } \\sim p$ , then sample $b _ { 2 } \\sim p ^ { D \\setminus \\{ b _ { 1 } \\} }$ , $b _ { 3 } ^ { ^ { - } } \\sim p ^ { D \\setminus \\{ b _ { 1 } , b _ { 2 } \\} }$ , etc. i.e. elements are sampled one by one without replacement. Using this procedure, $B ^ { k }$ can be seen as a (partial) ranking according to the PlackettLuce model (Plackett, 1975; Luce, 1959) and the probability of obtaining the vector $B ^ { k }$ is ", + "bbox": [ + 174, + 517, + 825, + 590 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/86b02d654fa38adab9eb042c932a2ca5c18e66b0e7b09ae5c06e7b3c1f7804c2.jpg", + "text": "$$\np ( B ^ { k } ) = \\prod _ { i = 1 } ^ { k } p ^ { D \\setminus B ^ { i - 1 } } ( b _ { i } ) = \\prod _ { i = 1 } ^ { k } \\frac { p ( b _ { i } ) } { 1 - \\sum _ { j < i } p ( b _ { j } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 598, + 656, + 647 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We can also restrict $B ^ { k }$ to the domain $D \\backslash C$ , which means that $b _ { i } \\notin C$ for $i = 1 , . . . , k$ : ", + "bbox": [ + 173, + 656, + 746, + 674 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/7d2dbcdc06bd8030e00f817db260a946e3aa85c9999d074ee060a76a475ee636.jpg", + "text": "$$\np ^ { D \\setminus C } ( B ^ { k } ) = \\prod _ { i = 1 } ^ { k } { \\frac { p ^ { D \\setminus C } ( b _ { i } ) } { 1 - \\sum _ { j < i } p ^ { D \\setminus C } ( b _ { j } ) } } = \\prod _ { i = 1 } ^ { k } { \\frac { p ( b _ { i } ) } { 1 - \\sum _ { c \\in C } p ( c ) - \\sum _ { j < i } p ( b _ { j } ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 271, + 681, + 727, + 731 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Unordered sample without replacement. Let $S ^ { k } \\subseteq D$ be an unordered sample without replacement from the distribution $p$ , which can be generated simply by generating an ordered sample and discarding the order. We denote elements in the sample with $s \\in \\breve { S } ^ { k }$ (so without index) and we write $B ( S ^ { k } )$ as the set of all $k !$ ! permutations (orderings) $\\bar { B ^ { k } }$ that correspond to (could have generated) $S ^ { k }$ . It follows that the probability for sampling $S ^ { k }$ is given by: ", + "bbox": [ + 173, + 747, + 825, + 819 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/6d296d81c7b9c0171b827bfda9cc4f8fde6f7b94ec7b11437c2b6839d948c0c7.jpg", + "text": "$$\np ( S ^ { k } ) = \\sum _ { B ^ { k } \\in B ( S ^ { k } ) } p ( B ^ { k } ) = \\sum _ { B ^ { k } \\in B ( S ^ { k } ) } \\prod _ { i = 1 } ^ { k } \\frac { p ( b _ { i } ) } { 1 - \\sum _ { j < i } p ( b _ { j } ) } = \\left( \\prod _ { s \\in S ^ { k } } p ( s ) \\right) \\cdot \\sum _ { B ^ { k } \\in B ( S ^ { k } ) } \\prod _ { i = 1 } ^ { k } \\frac { 1 } { 1 - \\sum _ { j < i } p ( b _ { j } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 184, + 825, + 812, + 880 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The last step follows since $B ^ { k } \\in B ( S ^ { k } )$ is an ordering of $S ^ { k }$ , such that $\\begin{array} { r } { \\prod _ { i = 1 } ^ { k } p ( b _ { i } ) = \\prod _ { s \\in S } p ( s ) } \\end{array}$ . \nNaive computation of $p ( S ^ { k } )$ is $O ( k ! )$ , but in Appendix $\\mathbf { B }$ we show how to compute it efficiently. ", + "bbox": [ + 174, + 890, + 825, + 925 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "When sampling from the distribution restricted to $D \\backslash C$ , we sample $S ^ { k } \\subseteq D \\setminus C$ with probability: ", + "bbox": [ + 171, + 102, + 823, + 119 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f1416a090c7000a0924172bf9118f60f0a3e8afda4a60413536211213eafb11e.jpg", + "text": "$$\np ^ { D \\setminus C } ( S ^ { k } ) = \\left( \\prod _ { s \\in S ^ { k } } p ( s ) \\right) \\cdot \\sum _ { B ^ { k } \\in \\mathcal { B } ( S ^ { k } ) } \\prod _ { i = 1 } ^ { k } \\frac { 1 } { 1 - \\sum _ { c \\in C } p ( c ) - \\sum _ { j < i } p ( b _ { j } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 123, + 725, + 176 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The Gumbel-Top- $k$ trick. As an alternative to sequential sampling, we can also sample $B ^ { k }$ and $S ^ { k }$ by taking the top $k$ of Gumbel variables (Yellott, 1977; Vieira, 2014; Kim et al., 2016). Following notation from Kool et al. $( 2 0 1 9 \\mathrm { c } )$ , we define the perturbed log-probability $g _ { \\phi _ { i } } = \\phi _ { i } + g _ { i }$ , where $\\phi _ { i } = \\log p ( i )$ and $g _ { i } \\sim \\mathrm { G u m b e l } ( 0 )$ . Then let $b _ { 1 } = \\arg \\operatorname* { m a x } _ { i \\in D } g _ { \\phi _ { i } }$ , $b _ { 2 } = \\arg \\operatorname* { m a x } _ { i \\in D \\backslash \\{ b _ { 1 } \\} } g _ { \\phi _ { i } }$ , etc., so $B ^ { k }$ is the top $k$ of the perturbed log-probabilities in decreasing order. The probability of obtaining $B _ { k }$ using this procedure is given by equation 4, so this provides an alternative sampling method which is effectively a (non-differentiable) reparameterization of sampling without replacement. For a differentiable reparameterization, see Grover et al. (2019). ", + "bbox": [ + 173, + 189, + 826, + 306 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "It follows that taking the top $k$ perturbed log-probabilities without order, we obtain the unordered sample set $S ^ { k }$ . This way of sampling underlies the efficient computation of $p ( S ^ { k } )$ in Appendix B. ", + "bbox": [ + 173, + 311, + 825, + 340 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 359, + 341, + 376 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we derive the unordered set policy gradient estimator: a low-variance, unbiased estimator of $\\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ f ( \\pmb { x } ) ]$ based on an unordered sample without replacement $S ^ { k }$ . First, we derive the generic (non-gradient) estimator for $\\mathbb { E } [ f ( x ) ]$ as the Rao-Blackwellized version of a single sample Monte Carlo estimator (and two other estimators!). Then we combine this estimator with REINFORCE (Williams, 1992) and we show how to reduce its variance using a built-in baseline. ", + "bbox": [ + 173, + 391, + 826, + 463 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 RAO-BLACKWELLIZATION OF THE SINGLE SAMPLE ESTIMATOR", + "text_level": 1, + "bbox": [ + 174, + 479, + 653, + 494 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A very crude but simple estimator for $\\mathbb { E } [ f ( x ) ]$ based on the ordered sample $B ^ { k }$ is to only use the first element $b _ { 1 }$ , which by definition is a sample from the distribution $p$ . We define this estimator as the single sample estimator, which is unbiased, since ", + "bbox": [ + 173, + 503, + 825, + 547 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/02f2b3f7b50f0fc7fea7eafa76908daeb1becae1ddbe2c49d64505835578994c.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { x \\sim p ( x ) } [ f ( x ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 553, + 686, + 571 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Discarding all but one sample, the single sample estimator is inefficient, but we can use RaoBlackwellization (Casella $\\&$ Robert, 1996) to signficantly improve it. To this end, we consider the distribution $B ^ { k } | S ^ { k }$ , which is, knowing the unordered sample $S ^ { k }$ , the conditional distribution over ordered samples $B ^ { k } \\in B ( S ^ { k } )$ that could have generated $S ^ { \\hat { k } }$ .2 Using $B ^ { k } | S ^ { k }$ , we rewrite $\\mathbb { E } [ f ( b _ { 1 } ) ]$ as ", + "bbox": [ + 174, + 583, + 825, + 640 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/266311ced4be4415de977a8b098b3fa79a51a05a8a5ba8855a81cddf903a16de.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { S ^ { k } \\sim p ( S ^ { k } ) } \\left[ \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } \\mid S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] \\right] = \\mathbb { E } _ { S ^ { k } \\sim p ( S ^ { k } ) } \\left[ \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } \\mid S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 645, + 808, + 665 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The Rao-Blackwellized version of the single sample estimator computes the inner conditional expectation exactly. Since $B ^ { k }$ is an ordering of $S ^ { k }$ , we have $b _ { 1 } \\in S ^ { k }$ and we can compute this as ", + "bbox": [ + 173, + 676, + 821, + 704 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/596870a40feaefe9b1a493c1183747aa6ee26d2fce368c656b98b1f97b2dccf6.jpg", + "text": "$$\n\\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s )\n$$", + "text_format": "latex", + "bbox": [ + 341, + 709, + 656, + 744 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where, in a slight abuse of notation, $P ( b _ { 1 } = s | S ^ { k } )$ is the probability that the first sampled element $b _ { 1 }$ takes the value $s$ , given that the complete set of $k$ samples is $S ^ { k }$ . Using Bayes’ Theorem we find ", + "bbox": [ + 174, + 751, + 825, + 781 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1f9be78c95a7fbbee7be9b15c59a94a262dff1f16e5d8e1f33172e666357253e.jpg", + "text": "$$\nP ( b _ { 1 } = s | S ^ { k } ) = \\frac { p ( S ^ { k } | b _ { 1 } = s ) P ( b _ { 1 } = s ) } { p ( S ^ { k } ) } = \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s ) } { p ( S ^ { k } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 271, + 785, + 725, + 821 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The step $p ( S ^ { k } | b _ { 1 } = s ) = p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} )$ comes from analyzing sequential sampling without replacement: given that the first element sampled is $s$ , the remaining elements have a distribution restricted to $D \\backslash \\{ s \\}$ , so sampling $S ^ { k }$ (including $s$ ) given the first element $s$ is equivalent to sampling the remainder $S ^ { k } \\setminus \\{ s \\}$ from the restricted distribution, which has probability $p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} )$ (see equation 7). ", + "bbox": [ + 173, + 827, + 825, + 901 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The unordered set estimator. For notational convenience, we introduce the leave-one-out ratio. \nDefinition 1. The leave-one-out ratio of $s$ w.r.t. the set $S$ is given by $\\begin{array} { r } { R ( S ^ { k } , s ) = \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } \\end{array}$ . ", + "bbox": [ + 171, + 102, + 820, + 146 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Rewriting equation 10 as $P ( b _ { 1 } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )$ shows that the probability of sampling $s$ first, given $S ^ { k }$ , is simply the unconditional probability multiplied by the leave-one-out ratio. We now define the unordered set estimator as the Rao-Blackwellized version of the single-sample estimator. ", + "bbox": [ + 173, + 156, + 825, + 200 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1. The unordered set estimator, given by ", + "bbox": [ + 176, + 204, + 509, + 218 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/03ee959b283d9cfc0e39f81176d5ac72d29d6fce2c91e22ff52fc67c6b0db17c.jpg", + "text": "$$\ne ^ { U S } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s )\n$$", + "text_format": "latex", + "bbox": [ + 383, + 224, + 614, + 261 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "is the Rao-Blackwellized version of the (unbiased!) single sample estimator. ", + "bbox": [ + 173, + 266, + 669, + 282 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proof. Using $P ( b _ { 1 } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )$ in equation 9 we have ", + "bbox": [ + 173, + 295, + 619, + 313 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/51b7f2077f8163e0ce349e2825cb373b1056003fe7e01d54c2ffc9c3c267dffe.jpg", + "text": "$$\n\\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s ) = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 253, + 318, + 743, + 353 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The implication of this theorem is that the unordered set estimator, in explicit form given by equation 11, is an unbiased estimator of $\\mathbb { E } [ f ( x ) ]$ since it is the Rao-Blackwellized version of the unbiased single sample estimator. Also, as expected by taking multiple samples, it has variance equal or lower than the single sample estimator by the Rao-Blackwell Theorem (Lehmann & Scheffe, 1950). ´ ", + "bbox": [ + 173, + 388, + 825, + 446 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 RAO-BLACKWELLIZATION OF OTHER ESTIMATORS ", + "text_level": 1, + "bbox": [ + 174, + 463, + 565, + 478 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The unordered set estimator is also the result of Rao-Blackwellizing two other unbiased estimators: the stochastic sum-and-sample estimator and the importance-weighted estimator. ", + "bbox": [ + 169, + 488, + 823, + 518 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The sum-and-sample estimator. We define as sum-and-sample estimator any estimator that relies on the identity that for any $C \\subset D$ ", + "bbox": [ + 173, + 531, + 823, + 561 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/71c31bf9e4547b1b6d796ed465395787980be036d959b15057290c3d36ca596b.jpg", + "text": "$$\n\\mathbb { E } _ { x \\sim p ( x ) } [ f ( x ) ] = \\mathbb { E } _ { x \\sim p ^ { D \\setminus C } ( x ) } \\left[ \\sum _ { c \\in C } p ( c ) f ( c ) + \\left( 1 - \\sum _ { c \\in C } p ( c ) \\right) f ( x ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 259, + 568, + 736, + 611 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For the derivation, see Appendix C.1 or Liang et al. (2018); Liu et al. (2019). In general, a sum-andsample estimator with a budget of $k > 1$ evaluations sums expectation terms for a set of categories $C$ (s.t. $| C | < k )$ explicitly (e.g. selected by their value $f$ (Liang et al., 2018) or probability $p$ (Liu et al., 2019)), and uses $k - | C |$ (down-weighted) samples from $D \\backslash C$ to estimate the remaining terms. As is noted by Liu et al. (2019), selecting $C$ such that $\\frac { 1 - \\sum _ { c \\in C } p ( c ) } { k - | C | }$ is minimized guarantees to reduce variance compared to a standard minibatch of $k$ samples (which is equivalent to setting $C = \\varnothing$ ). See also Fearnhead $\\&$ Clifford (2003) for a discussion on selecting $C$ optimally. The ability to optimize $C$ depends on whether $p ( c )$ can be computed efficiently a-priori (before sampling). This is difficult in high-dimensional settings, e.g. sequence models which compute the probability incrementally while ancestral sampling. An alternative is to select $C$ stochastically (as equation 13 holds for any $C$ ), and we choose $C = B ^ { k - 1 }$ to define the stochastic sum-and-sample estimator: ", + "bbox": [ + 173, + 616, + 825, + 779 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9ed8f24def9f93383a37e69286f2f5833e1c4dc8a9d8a9cf54ab1cede0ce92e5.jpg", + "text": "$$\ne ^ { { \\mathrm { S S A S } } } ( B ^ { k } ) = \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) \\right) f ( b _ { k } ) .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 785, + 687, + 835 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For simplicity, we consider the version that sums $k - 1$ terms here, but the following results also hold for a version that sums $k - m$ terms and uses $m$ samples (without replacement) (see Appendix C.3). Sampling without replacement, it holds that bk|Bk−1 ∼ pD\\Bk−1, so the unbiasedness follows from equation 13 by separating the expectation over $B ^ { k }$ into expectations over $B ^ { k - 1 }$ and $b _ { k } | B ^ { k - 1 }$ : ", + "bbox": [ + 173, + 839, + 826, + 900 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1619865ccf403398a0293e9ef050ffb2ee63a39836b62b1c47416adfc582475b.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { B ^ { k - 1 } \\sim p ( B ^ { k - 1 } ) } \\left[ \\mathbb { E } _ { b _ { k } \\sim p ( b _ { k } | B ^ { k - 1 } ) } \\left[ e ^ { S S A S } ( B ^ { k } ) \\right] \\right] = \\mathbb { E } _ { B ^ { k - 1 } \\sim p ( B ^ { k - 1 } ) } \\left[ \\mathbb { E } [ f ( x ) ] \\right] = \\mathbb { E } [ f ( x ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 209, + 906, + 787, + 926 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In general, a sum-and-sample estimator reduces variance if the probability mass is concentrated on the summed categories. As typically high probability categories are sampled first, the stochastic sum-and-sample estimator sums high probability categories, similar to the estimator by Liu et al. (2019) which we refer to as the deterministic sum-and-sample estimator. As we show in Appendix C.2, Rao-Blackwellizing the stochastic sum-and-sample estimator also results in the unordered set estimator. This even holds for a version that uses $m$ samples and $k - m$ summed terms (see Appendix C.3), which means that the unordered set estimator has equal or lower variance than the optimal (in terms of $m$ ) stochastic sum-and-sample estimator, but conveniently does not need to choose $m$ . ", + "bbox": [ + 173, + 102, + 825, + 215 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The importance-weighted estimator. The importance-weighted estimator (Vieira, 2017) is ", + "bbox": [ + 174, + 229, + 787, + 246 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8b61001740dc7d42db3c377579f08d9f93cdff9e7e3528a276595849d54f819e.jpg", + "text": "$$\ne ^ { \\mathrm { I W } } ( S ^ { k } , \\kappa ) = \\sum _ { s \\in S ^ { k } } { \\frac { p ( s ) } { q ( s , \\kappa ) } } f ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 252, + 606, + 291 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This estimator is based on the idea of priority sampling (Duffield et al., 2007). It does not use the order of the sample, but assumes sampling using the Gumbel-Top- $k$ trick and requires access to $\\kappa$ , the $( k + 1 )$ -th largest perturbed log-probability, which can be seen as the ‘threshold’ since $g _ { \\phi _ { s } } ~ > ~ \\kappa ~ \\forall s ~ \\in ~ S ^ { k }$ . $\\bar { q ( s , a ) } = P ( g _ { \\phi _ { s } } > a )$ can be interpreted as the inclusion probability of $s \\in S ^ { k }$ (assuming a fixed threshold $a$ instead of a fixed sample size $k$ ). For details and a proof of unbiasedness, see Vieira (2017) or Kool et al. (2019c). As the estimator has high variance, Kool et al. (2019c) resort to normalizing the importance weights, resulting in biased estimates. Instead, we use Rao-Blackwellization to eliminate stochasticity by $\\kappa$ . Again, the result is the unordered set estimator (see Appendix D.1), which thus has equal or lower variance. ", + "bbox": [ + 173, + 297, + 825, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 THE UNORDERED SET POLICY GRADIENT ESTIMATOR ", + "text_level": 1, + "bbox": [ + 174, + 443, + 584, + 457 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Writing $p _ { \\pmb { \\theta } }$ to indicate the dependency on the model parameters $\\pmb { \\theta }$ , we can combine the unordered set estimator with REINFORCE (Williams, 1992) to obtain the unordered set policy gradient estimator. ", + "bbox": [ + 174, + 468, + 823, + 497 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Corollary 1. The unordered set policy gradient estimator, given by ", + "bbox": [ + 173, + 501, + 617, + 516 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1dbc150d5df5f608b513f6396d6dc46e4c366286427d7ad9da89ab67b5e56ddd.jpg", + "text": "$$\ne ^ { U S P G } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } \\log p _ { \\theta } ( s ) f ( s ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) f ( s ) ,\n$$", + "text_format": "latex", + "bbox": [ + 212, + 522, + 758, + 558 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "is an unbiased estimate of the policy gradient. ", + "bbox": [ + 174, + 564, + 475, + 579 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. Using REINFORCE (Williams, 1992) combined with the unordered set estimator we find: ", + "bbox": [ + 171, + 594, + 812, + 609 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/aab23b5a4539c051761f39cd7a8e17024373c54bc3fc29e240863c4159a74352.jpg", + "text": "$$\n\\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } ( x ) } [ f ( x ) ] = \\mathbb { E } _ { p _ { \\theta } ( x ) } [ \\nabla _ { \\theta } \\log p _ { \\theta } ( x ) f ( x ) ] = \\mathbb { E } _ { S ^ { k } \\sim p _ { \\theta } ( S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } \\log p _ { \\theta } ( s ) f ( s ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 616, + 826, + 666 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Variance reduction using a built-in control variate. The variance of REINFORCE can be reduced by subtracting a baseline from $f$ . When taking multiple samples (with replacement), a simple and effective baseline is to take the mean of other (independent!) samples (Mnih & Rezende, 2016). Sampling without replacement, we can use the same idea to construct a baseline based on the other samples, but we have to correct for the fact that the samples are not independent. ", + "bbox": [ + 173, + 702, + 825, + 773 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 2. The unordered set policy gradient estimator with baseline, given by ", + "bbox": [ + 173, + 776, + 702, + 792 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2bbfebe328fefece3095174414fb002d79ec5114db4b9f927fd8cd5b4c3d72fe.jpg", + "text": "$$\ne ^ { U S P G B L } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\left( f ( s ) - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 204, + 797, + 766, + 848 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 854, + 215, + 868 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6c8b0893165f90a463454da2a941978b46de124e3b5b3602947a4a6ac32e8a96.jpg", + "text": "$$\nR ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) = { \\frac { p _ { \\theta } ^ { D \\setminus \\{ s , s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s , s ^ { \\prime } \\} ) } { p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } }\n$$", + "text_format": "latex", + "bbox": [ + 357, + 864, + 640, + 906 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "is the second order leave-one-out ratio, is an unbiased estimate of the policy gradient. ", + "bbox": [ + 171, + 910, + 733, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. See Appendix E.1. ", + "bbox": [ + 174, + 104, + 344, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This theorem shows how to include a built-in baseline based on dependent samples (without replacement), without introducing bias. By having a built-in baseline, the value $f ( s )$ for sample $s$ is compared against an estimate of its expectation $\\mathbb { E } [ f ( s ) ]$ , based on the other samples. The difference is an estimate of the advantage (Sutton & Barto, 2018), which is positive if the sample $s$ is ‘better’ than average, causing $p _ { \\pmb { \\theta } } ( s )$ to be increased (reinforced) through the sign of the gradient, and vice versa. By sampling without replacement, the unordered set estimator forces the estimator to compare different alternatives, and reinforces the best among them. ", + "bbox": [ + 173, + 142, + 825, + 241 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Including the pathwise derivative. So far, we have only considered the scenario where $f$ does not depend on $\\pmb \\theta$ . If $f$ does depend on $\\pmb \\theta$ , for example in a VAE (Kingma & Welling, 2014; Rezende et al., 2014), then we use the notation $f _ { \\theta }$ and we can write the gradient (Schulman et al., 2015) as ", + "bbox": [ + 174, + 257, + 825, + 301 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6d9200624e6d99ee5fa00ffa9450b95f696e999e1febff3fc26aa6f635fccce8.jpg", + "text": "$$\n\\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ f _ { \\pmb { \\theta } } ( \\pmb { x } ) ] = \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ \\nabla _ { \\pmb { \\theta } } \\log p _ { \\pmb { \\theta } } ( \\pmb { x } ) f _ { \\pmb { \\theta } } ( \\pmb { x } ) + \\nabla _ { \\pmb { \\theta } } f _ { \\pmb { \\theta } } ( \\pmb { x } ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 299, + 309, + 699, + 328 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The additional second (‘pathwise’) term can be estimated (using the same samples) with the standard unordered set estimator. This results in the full unordered set policy gradient estimator: ", + "bbox": [ + 171, + 337, + 830, + 364 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/4cab27528948a20ca5fd07651bbd08bcb77a11aa80de85ca79d21a7721f0cde6.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle e ^ { \\mathrm { F U S P G } } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) f _ { \\theta } ( s ) + \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } f _ { \\theta } ( s ) } } \\\\ { { \\displaystyle \\quad = \\sum _ { s \\in S ^ { k } } R ( S ^ { k } , s ) \\nabla _ { \\theta } \\left( p _ { \\theta } ( s ) f _ { \\theta } ( s ) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 372, + 746, + 445 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Equation 20 is straightforward to implement using an automatic differentiation library. We can also include the baseline (as in equation 17) but we must make sure to call STOP GRADIENT (DETACH in PyTorch) on the baseline (but not on $f _ { \\pmb \\theta } ( s ) ! )$ . Importantly, we should never track gradients through the leave-one-out ratio $R ( S ^ { k } , s )$ which means it can be efficiently computed in pure inference mode. ", + "bbox": [ + 174, + 453, + 821, + 511 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Scope & limitations. We can use the unordered set estimator for any discrete distribution from which we can sample without replacement, by treating it as a univariate categorical distribution over its domain. This includes sequence models, from which we can sample using Stochastic Beam Search (Kool et al., 2019c), as well as multivariate categorical distributions which can also be treated as sequence models (see Section 4.2). In the presence of continuous variables or a stochastic function $f$ , we may separate this stochasticity from the stochasticity over the discrete distribution, as in Lorberbom et al. (2019). The computation of the leave-one-out ratios adds some overhead, although they can be computed efficiently, even for large $k$ (see Appendix B). For a moderately sized model, the costs of model evaluation and backpropagation dominate the cost of computing the estimator. ", + "bbox": [ + 173, + 527, + 825, + 654 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 RELATION TO OTHER MULTI-SAMPLE ESTIMATORS ", + "text_level": 1, + "bbox": [ + 174, + 674, + 563, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Relation to Murthy’s estimator. We found out that the ‘vanilla’ unordered set estimator (equation 11) is actually a special case of the estimator by Murthy (1957), known in statistics literature for estimation of a population total $\\begin{array} { r } { \\Theta = \\sum _ { i \\in D } y _ { i } } \\end{array}$ . Using $y _ { i } = p ( i ) f ( i )$ , we have $\\Theta = \\mathbb { E } [ f ( i ) ]$ , so Murthy’s estimator can be used to estimate expectations (see equation 11). Murthy derives the estimator by ‘unordering’ a convex combination of Raj (1956) estimators, which, using $y _ { i } = p ( i ) f ( i )$ , are stochastic sum-and-sample estimators in our analogy. ", + "bbox": [ + 173, + 700, + 825, + 785 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Murthy (1957) also provides an unbiased estimator of the variance, which may be interesting for future applications. Since Murthy’s estimator can be used with arbitrary sampling distribution, it is straightforward to derive importance-sampling versions of our estimators. In particular, we can sample $S$ without replacement using $q ( x ) > 0 , x \\in D$ , and use equations 11, 16, 17 and 20, as long as we compute the leave-one-out ratio $R ( S ^ { k } , s )$ using $q$ . ", + "bbox": [ + 174, + 790, + 825, + 862 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "While part of our derivation coincides with Murthy (1957), we are not aware of previous work using this estimator to estimate expectations. Additionally, we discuss practical computation of $p ( S )$ (Appendix B), we show the relation to the importance-weighted estimator, and we provide the extension to estimating policy gradients, especially including a built-in baseline without adding bias. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Relation to the empirical risk estimator. The empirical risk loss (Edunov et al., 2018) estimates the expectation in equation 1 by summing only a subset $S$ of the domain, using normalized probabilities $\\begin{array} { r } { \\hat { p } _ { \\pmb { \\theta } } ( s ) = \\frac { \\bar { p } _ { \\pmb { \\theta } } ( s ) } { \\sum _ { s ^ { \\prime } \\in S } p _ { \\pmb { \\theta } } ( s ) } } \\end{array}$ . Using this loss, the (biased) estimate of the gradient is given by ", + "bbox": [ + 173, + 103, + 823, + 155 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/16c4a246849e072f4944d39d2f161557419ea23787c012dd3296694e026e574b.jpg", + "text": "$$\ne ^ { \\mathrm { R I S K } } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\pmb { \\theta } } \\left( \\frac { p _ { \\pmb { \\theta } } ( s ) } { \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\pmb { \\theta } } ( s ^ { \\prime } ) } \\right) f ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 343, + 155, + 655, + 196 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The risk estimator is similar to the unordered set policy gradient estimator, with two important differences: 1) the individual terms are normalized by the total probability mass rather than the leave-one-out ratio and 2) the gradient w.r.t. the normalization factor is taken into account. As a result, samples ‘compete’ for probability mass and only the best can be reinforced. This has the same effect as using a built-in baseline, which we prove in the following theorem. ", + "bbox": [ + 174, + 198, + 825, + 268 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 3. By taking the gradient w.r.t. the normalization factor into account, the risk estimator has a built-in baseline, which means it can be written as ", + "bbox": [ + 173, + 271, + 823, + 299 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/904b5d4d9996c49ac35bd50c3cc3eaa34cee54f06b1fb5ec574ba84ccd5a5819.jpg", + "text": "$$\ne ^ { R I S K } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) \\frac { 1 } { \\sum _ { s ^ { \\prime \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime \\prime } ) } \\left( f ( s ) - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) \\frac { 1 } { \\sum _ { s ^ { \\prime \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime \\prime } ) } f ( s ^ { \\prime } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 184, + 300, + 787, + 351 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof. See Appendix F.1 ", + "bbox": [ + 174, + 363, + 339, + 377 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This theorem highlights the similarity between the biased risk estimator and our unbiased estimator (equation 17), and suggests that their only difference is the weighting of terms. Unfortunately, the implementation by Edunov et al. (2018) has more sources of bias (e.g. length normalization), which are not compatible with our estimator. However, we believe that our analysis helps analyze the bias of the risk estimator and is a step towards developing unbiased estimators for structured prediction. ", + "bbox": [ + 173, + 392, + 825, + 463 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Relation to VIMCO. VIMCO (Mnih & Rezende, 2016) is an estimator that uses $k$ samples (with replacement) to optimize an objective of the form log $\\textstyle { \\frac { 1 } { k } } \\sum _ { i } f ( x _ { i } )$ , which is a multi-sample stochastic lower bound in the context of variational inference. VIMCO reduces the variance by using a local baseline for each of the $k$ samples, based on the other $k - 1$ samples. While we do not have a log term, as our goal is to optimize general $\\mathbb { E } [ f ( x ) ]$ , we adopt the idea of forming a baseline based on the other samples, and we define REINFORCE with replacement (with built-in baseline) as the estimator that computes the gradient estimate using samples with replacement $X ^ { k } = ( x _ { 1 } , . . . , x _ { k } )$ as ", + "bbox": [ + 173, + 476, + 825, + 575 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/351a340e6952afd897fb774a727f1e14b17f5a4500b3ff32102096cd6837dcf3.jpg", + "text": "$$\ne ^ { \\mathrm { R F W R } } ( X ^ { k } ) = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\nabla _ { \\theta } \\log p _ { \\theta } ( x _ { i } ) \\left( f ( x _ { i } ) - \\frac { 1 } { k - 1 } \\sum _ { j \\ne i } f ( x _ { j } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 577, + 720, + 627 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This estimator is unbiased, as $\\mathbb { E } _ { x _ { i } , x _ { j } } [ \\nabla _ { \\pmb { \\theta } } \\log p _ { \\pmb { \\theta } } ( x _ { i } ) f ( x _ { j } ) ] = 0$ for $i \\neq j$ (see also Kool et al. (2019b)). We think of the unordered set estimator as the without-replacement version of this estimator, which weights terms by $p _ { \\pmb { \\theta } } ( s ) R ( S ^ { k } , s )$ instead of $\\frac { 1 } { k }$ . This puts more weight on higher probability elements to compensate for sampling without replacement. If probabilities are small and (close to) uniform, there are (almost) no duplicate samples and the weights will be close to $\\frac { 1 } { k }$ , so the gradient estimate of the with- and without-replacement versions are similar. ", + "bbox": [ + 173, + 628, + 825, + 713 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Relation to ARSM. ARSM (Yin et al., 2019) also uses multiple evaluations (‘pseudo-samples’) of $p _ { \\theta }$ and $f$ . This can be seen as similar to sampling without replacement, and the estimator also has a built-in control variate. Compared to ARSM, our estimator allows direct control over the computational cost (through the sample size $k$ ) and has wider applicability, for example it also applies to multivariate categorical variables with different numbers of categories per dimension. ", + "bbox": [ + 173, + 727, + 825, + 797 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Relation to stratified/systematic sampling. Our estimator aims to reduce variance by changing the sampling distribution for multiple samples by sampling without replacement. There are alternatives, such as using stratified or systematic sampling (see, e.g. Douc & Cappe (2005)). Both partition ´ the domain $D$ into $k$ strata and take a single sample from each stratum, where systematic sampling uses common random numbers for each stratum. In applications involving high-dimensional or structured domains, it is unclear how to partition the domain and how to sample from each partition. Additionally, as samples are not independent, it is non-trivial to include a built-in baseline, which we find is a key component that makes our estimator perform well. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1827e94fdb564b2088e10a208547d35fffe208d74d3041a8d194d3868ada9f35.jpg", + "image_caption": [ + "Figure 1: Bernoulli gradient variance (on log scale) as a function of the number of model evaluations (including baseline evaluations, so the sum-and-sample estimators with sampled baselines use twice as many evaluations). Note that for some estimators, the variance is 0 (log variance $- \\infty )$ for $k = 8$ . " + ], + "image_footnote": [], + "bbox": [ + 187, + 89, + 807, + 219 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 285, + 326, + 301 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.1 BERNOULLI TOY EXPERIMENT", + "text_level": 1, + "bbox": [ + 176, + 316, + 424, + 332 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use the code by Liu et al. (2019) to reproduce their Bernoulli toy experiment. Given a vector $\\mathbf { p } =$ (0.6, 0.51, 0.48) the goal is to minimize the loss $\\begin{array} { r } { \\mathcal { L } ( \\eta ) \\ : = \\ : \\mathbb { E } _ { x _ { 1 } , x _ { 2 } , x _ { 3 } \\sim \\mathrm { B e r n } ( \\sigma ( \\eta ) ) } \\left[ \\sum _ { i = 1 } ^ { 3 } ( x _ { i } - p _ { i } ) ^ { 2 } \\right] } \\end{array}$ . Here $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are i.i.d. from the Bernoulli $( \\sigma ( \\eta ) )$ distribution, parameterized by a scalar $\\eta \\in \\mathbb { R }$ , where $\\sigma ( \\eta ) = ( 1 + \\exp ( - \\eta ) ) ^ { - 1 }$ is the sigmoid function. We compare different estimators, with and without baseline (either ‘built-in’ or using additional samples, referred to as REINFORCE+ in Liu et al. (2019)). We report the (log-)variance of the scalar gradient $\\frac { \\partial \\mathcal { L } } { \\partial \\eta }$ as a function of the number of model evaluations, which is twice as high when using a sampled baseline (for each term). ", + "bbox": [ + 173, + 342, + 825, + 453 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As can be seen in Figure 1, the unordered set estimator is the only estimator that has consistently the lowest (or comparable) variance in both the high $\\langle \\eta = 0 \\rangle$ ) and low entropy $\\hphantom { - } \\eta = - 4 )$ regimes and for different number of samples/model evaluations. This suggests that it combines the advantages of the other estimators. We also ran the actual optimization experiment, where with as few as $k = 3$ samples the trajectory was indistinguishable from using the exact gradient (see Liu et al. (2019)). ", + "bbox": [ + 174, + 459, + 825, + 529 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 CATEGORICAL VARIATIONAL AUTO-ENCODER ", + "text_level": 1, + "bbox": [ + 174, + 545, + 537, + 560 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use the code from Yin et al. (2019) to train a categorical Variational Auto-Encoder (VAE) with 20 dimensional latent space, with 10 categories per dimension (details in Appendix G.1). To use our estimator, we treat this as a single factorized distribution with $1 0 ^ { 2 0 }$ categories from which we can sample without replacement using Stochastic Beam Search (Kool et al., 2019c), sequentially sampling each dimension as if it were a sequence model. We also perform experiments with $1 0 ^ { \\dot { 2 } }$ latent space, which provides a lower entropy setting, to highlight the advantage of our estimator. ", + "bbox": [ + 174, + 571, + 825, + 656 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Measuring the variance. In Table 1, we report the variance of different gradient estimators with $k = 4$ samples, evaluated on a trained model. The unordered set estimator has the lowest variance in both the small and large domain (low and high entropy) setting, being on-par with the best of the (stochastic3) sum-and-sample estimator and REINFORCE with replacement4. This confirms the toy experiment, suggesting that the unordered set estimator provides the best of both estimators. In Appendix G.2 we repeat the same experiment at different stages of training, with similar results. ", + "bbox": [ + 173, + 670, + 825, + 755 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/38fff474333f745080d8627c5fb4566e88f353059b7a66955110f7f26afb9b13.jpg", + "image_caption": [ + "Figure 2: VAE smoothed training curves (-ELBO) of two independent runs when training with different estimators with $k = 1$ , 4 or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value. " + ], + "image_footnote": [], + "bbox": [ + 191, + 104, + 808, + 272 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ELBO optimization. We use different estimators to optimize the ELBO (details in Appendix G.1). Additionally to the baselines by Yin et al. (2019) we compare against REINFORCE with replacement and the stochastic sum-and-sample estimator. In Figure 2 we observe that our estimator performs on par with REINFORCE with replacement (and built-in baseline, equation 23) and outperforms other estimators in at least one of the settings. There are a lot of other factors, e.g. exploration that may explain why we do not get a strictly better result despite the lower variance. We note some overfitting (see validation curves in Appendix G.2), but since our goal is to show improved optimization, and to keep results directly comparable to Yin et al. (2019), we consider regularization a separate issue outside the scope of this paper. These results are using MNIST binarized by a threshold of 0.5. In Appendix G.2 we report results using the standard binarized MNIST dataset from Salakhutdinov & Murray (2008). ", + "bbox": [ + 173, + 354, + 825, + 506 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 STRUCTURED PREDICTION FOR THE TRAVELLING SALESMAN PROBLEM ", + "text_level": 1, + "bbox": [ + 176, + 523, + 714, + 537 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To show the wide applicability of our estimator, we consider the structured prediction task of predicting routes (sequences) for the Travelling Salesman Problem (TSP) (Vinyals et al., 2015; Bello et al., 2016; Kool et al., 2019a). We use the code by Kool et al. (2019a)5 to reproduce their TSP experiment with 20 nodes. For details, see Appendix H. ", + "bbox": [ + 174, + 549, + 823, + 606 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We implement REINFORCE with replacement (and built-in baseline) as well as the stochastic sumand-sample estimator and our estimator, using Stochastic Beam Search (Kool et al., 2019c) for sampling. Also, we include results using the biased normalized importance-weighted policy gradient estimator with built-in baseline (derived in Kool et al. (2019b), see Appendix D.2). Additionally, we compare against REINFORCE with greedy rollout baseline (Rennie et al., 2017) used by Kool et al. (2019c) and a batch-average baseline. For reference, we also include the biased risk estimator, either ‘sampling’ using stochastic or deterministic beam search (as in Edunov et al. (2018)). ", + "bbox": [ + 174, + 613, + 823, + 709 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Figure 3a, we compare training progress (measured on the validation set) as a function of the number of training steps, where we divide the batch size by $k$ to keep the total number of samples equal. Our estimator outperforms REINFORCE with replacement, the stochastic sum-and-sample estimator and the strong greedy rollout baseline (which uses additional baseline model evaluations) and performs on-par with the biased risk estimator. In Figure 3b, we plot the same results against the number of instances, which shows that, compared to the single sample estimators, we can train with less data and less computational cost (as we only need to run the encoder once for each instance). ", + "bbox": [ + 174, + 718, + 825, + 814 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/3cf7aa8c67e64f59fc3b77ff931ff72bff898a29559db5fa6f7eddaa573687f1.jpg", + "image_caption": [ + "Figure 3: TSP validation set optimality gap measured during training. Raw results are light, smoothed results are darker (2 random seeds). We compare our estimator against different unbiased and biased (dotted) multi-sample estimators and against single-sample REINFORCE, with batch-average or greedy rollout baseline. " + ], + "image_footnote": [], + "bbox": [ + 205, + 97, + 794, + 247 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 327, + 310, + 343 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We introduced the unordered set estimator, a low-variance, unbiased gradient estimator based on sampling without replacement, which can be used as an alternative to the popular biased GumbelSoftmax estimator (Jang et al., 2016; Maddison et al., 2016). Our estimator is the result of RaoBlackwellizing three existing estimators, which guarantees equal or lower variance, and is closely related to a number of other estimators. It has wide applicability, is parameter free (except for the sample size $k$ ) and has competitive performance to the best of alternatives in both high and low entropy regimes. ", + "bbox": [ + 174, + 359, + 825, + 457 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In our experiments, we found that REINFORCE with replacement, with multiple samples and a built-in baseline as inspired by VIMCO (Mnih & Rezende, 2016), is a simple yet strong estimator which has performance similar to our estimator in the high entropy setting. We are not aware of any recent work on gradient estimators for discrete distributions that has considered this estimator as baseline, while it may be often preferred given its simplicity. In future work, we want to investigate if we can apply our estimator to estimate gradients ‘locally’ (Titsias & Lazaro-Gredilla, 2015), as ´ locally we have a smaller domain and expect more duplicate samples. ", + "bbox": [ + 174, + 463, + 825, + 561 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 578, + 326, + 592 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This research was funded by ORTEC. We would like to thank anonymous reviewers for their feedback that helped improve the paper. ", + "bbox": [ + 174, + 602, + 821, + 630 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 651, + 285, + 666 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Dzmitry Bahdanau, Philemon Brakel, Kelvin Xu, Anirudh Goyal, Ryan Lowe, Joelle Pineau, Aaron Courville, and Yoshua Bengio. An actor-critic algorithm for sequence prediction. In International Conference on Learning Representations, 2017. 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The relationship between Luce’s choice axiom, Thurstone’s theory of comparative judgment, and the double exponential distribution. Journal of Mathematical Psychology, 15(2): 109–144, 1977. ", + "bbox": [ + 171, + 602, + 823, + 643 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Mingzhang Yin, Yuguang Yue, and Mingyuan Zhou. Arsm: Augment-reinforce-swap-merge estimator for gradient backpropagation through categorical variables. In International Conference on Machine Learning, pp. 7095–7104, 2019. ", + "bbox": [ + 174, + 652, + 823, + 695 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A NOTATION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 299, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Throughout this appendix we will use the following notation from Maddison et al. (2014): ", + "bbox": [ + 173, + 132, + 764, + 148 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/821cbe0666f811f0f17fdbd24322d0cfd6934666b30ce92d9f2fdccfc37edb08.jpg", + "text": "$$\n\\begin{array} { l } { { e _ { \\phi } ( g ) = \\exp ( - g + \\phi ) } } \\\\ { { F _ { \\phi } ( g ) = \\exp ( - \\exp ( - g + \\phi ) ) } } \\\\ { { f _ { \\phi } ( g ) = e _ { \\phi } ( g ) F _ { \\phi } ( g ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 395, + 154, + 602, + 209 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This means that $F _ { \\phi } ( g )$ is the CDF and $f _ { \\phi } ( g )$ the PDF of the Gumbel $\\left( \\phi \\right)$ distribution. Additionally we will use the identities by Maddison et al. (2014): ", + "bbox": [ + 173, + 212, + 823, + 242 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e3101e89cd302781233ce9a26c02ed7b482a567d9cb9a5e50a37ae7ff37a8a08.jpg", + "text": "$$\n\\begin{array} { c } { { F _ { \\phi } ( g ) F _ { \\gamma } ( g ) = F _ { \\log ( \\exp ( \\phi ) + \\exp ( \\gamma ) ) } ( g ) } } \\\\ { { \\displaystyle \\int _ { g = a } ^ { b } e _ { \\gamma } ( g ) F _ { \\phi } ( g ) \\partial g = ( F _ { \\phi } ( b ) - F _ { \\phi } ( a ) ) \\frac { \\exp ( \\gamma ) } { \\exp ( \\phi ) } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 246, + 661, + 306 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Also, we will use the following notation, definitions and identities (see Kool et al. (2019c)): ", + "bbox": [ + 169, + 318, + 772, + 333 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/74b04d313c6fa9193ba14f401130ebf9ca9cdbf24b7db98e9027266e865313ae.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\phi _ { i } = \\log p ( i ) } \\\\ & { \\displaystyle \\phi _ { S } = \\log \\displaystyle \\sum _ { i \\in S } p ( i ) = \\log \\sum _ { i \\in S } \\exp \\phi _ { i } } \\\\ & { \\displaystyle \\phi _ { D \\setminus S } = \\log \\sum _ { i \\in D \\setminus S } p ( i ) = \\log \\left( 1 - \\sum _ { i \\in S } p ( i ) \\right) = \\log ( 1 - \\exp ( \\phi _ { S } ) ) } \\\\ & { \\displaystyle { G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { i } ) } } \\\\ & { \\displaystyle { G _ { \\phi _ { S } } = \\sum _ { i \\in S } G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { S } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 338, + 722, + 481 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For a proof of equation 30, see Maddison et al. (2014). ", + "bbox": [ + 174, + 486, + 532, + 501 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ebdb2bbb39718391052a7e6fd23cc99c971cdbc1773a9535484decfa338d1f69.jpg", + "text": "$$\np ( S ^ { k } ) , p ^ { D \\setminus C } ( S \\setminus C ) \\ \\mathrm { A N D } \\ R ( S ^ { k } , s )\n$$", + "text_format": "latex", + "bbox": [ + 364, + 518, + 640, + 540 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We can sample the set $S ^ { k }$ from the Plackett-Luce distribution using the Gumbel-Top- $k$ trick by drawing Gumbel variables $G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { i } )$ for each element and returning the indices of the $k$ largest Gumbels. If we ignore the ordering, this means we will obtain the set $S ^ { k }$ if $\\mathrm { m i n } _ { i \\in S ^ { k } } G _ { \\phi _ { i } } >$ $\\operatorname* { m a x } _ { i \\in D \\setminus S ^ { k } } G _ { \\phi _ { i } }$ . Omitting the superscript $k$ for clarity, we can use the Gumbel-Max trick, i.e. that $G _ { \\phi _ { D \\setminus S } } = \\operatorname* { m a x } _ { i \\notin S } G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { D \\setminus S } )$ (equation 30) and marginalize over $G _ { \\phi _ { D \\setminus S } }$ : ", + "bbox": [ + 173, + 553, + 825, + 630 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/806f97cd9da1c0ad1baa8ecdc50026ff7932d71042f7f521a043bd018768c996.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle p ( S ) = P ( \\underset { i \\in S } { \\operatorname* { m i n } } G _ { \\phi _ { i } \\setminus } \\ G _ { \\phi _ { D \\setminus S } } ) } \\\\ { \\displaystyle } & { = P ( G _ { \\phi _ { i } } > G _ { \\phi _ { D \\setminus S } } , i \\in S ) } \\\\ { \\displaystyle } & { = \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } ) P ( G _ { \\phi _ { i } } > g _ { \\phi _ { D \\setminus S } } , i \\in S ) \\partial g _ { \\phi _ { D \\setminus S } } } \\\\ { \\displaystyle } & { = \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } ) \\prod _ { i \\in S } \\left( 1 - F _ { \\phi _ { \\star } } ( g _ { \\phi _ { D \\setminus S } } ) \\right) \\partial g _ { \\phi _ { D \\setminus S } } } \\\\ { \\displaystyle } & { = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S } \\left( 1 - F _ { \\phi _ { i } } \\left( F _ { \\phi _ { D \\setminus S } } ^ { - 1 } ( u ) \\right) \\right) \\partial u } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 636, + 717, + 801 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we have used a change of variables $u = F _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } )$ . This expression can be efficiently numerically integrated (although another change of variables may be required for numerical stability depending on the values of $\\phi$ ). ", + "bbox": [ + 173, + 808, + 823, + 851 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Exact computation in $O ( 2 ^ { k } )$ . The integral in equation 31 can be computed exactly using the identity ", + "bbox": [ + 173, + 864, + 823, + 892 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/dd04b747902262790d7c58d260268b6984c64f5106491fbb6d615230c69419e4.jpg", + "text": "$$\n\\prod _ { i \\in S } ( a _ { i } - b _ { i } ) = \\sum _ { C \\subseteq S } ( - 1 ) ^ { | C | } \\prod _ { i \\in C } b _ { i } \\prod _ { i \\in S \\setminus C } a _ { i }\n$$", + "text_format": "latex", + "bbox": [ + 352, + 891, + 642, + 928 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "which gives ", + "bbox": [ + 173, + 103, + 254, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/7ba5abea6b75d40021fc45a9d7e85ccd6775706b57325b46c30787c210537343.jpg", + "text": "$$\n\\begin{array} { r l } { { \\mu ( S ) = \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } \\longrightarrow \\int _ { \\partial \\sigma _ { \\phi ( S ) } } \\bigl ( g _ { \\phi ( S ) , s } \\bigr ) \\prod ( 1 - F _ { \\phi _ { \\phi } } ( g _ { \\phi ( S ) , s } ) ) \\partial g _ { \\phi _ { \\phi ( S ) , s } } } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } \\big | \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } \\int _ { \\Phi _ { \\phi ( S ) - } ( \\partial g _ { \\phi ( S ) - } ) } \\prod \\int _ { s _ { \\phi } } F _ { \\phi _ { \\phi } } ( g _ { \\phi ( S ) - s } ) \\prod \\big | \\partial g _ { \\phi _ { \\phi ( S ) - } s } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } \\big | \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } e _ { \\phi _ { \\phi ( S ) - } ( \\partial g _ { \\phi ( S ) - } ) } F _ { \\phi _ { \\phi _ { \\phi ( S ) - } s } } ( g _ { \\phi _ { S } , s } ) F _ { \\phi _ { \\phi ( S ) - } } ( g _ { \\phi _ { S } , s } ) \\big | \\mathcal { P } _ { \\phi _ { \\phi ( S ) - } } \\big | \\mathcal { P } _ { \\phi _ { \\phi ( S ) - } } \\big | \\mathcal { P } _ { \\phi _ { \\phi ( S ) - } } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } \\big | \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } e _ { \\phi _ { \\phi ( S ) - } ( \\partial g _ { \\phi ( S ) - } ) } F _ { \\phi _ { \\phi ( S ) - } , \\phi _ { \\phi ( S ) - } } ( g _ { \\phi _ { S } , s } ) \\big | \\partial g _ { \\phi _ { \\phi ( S ) - } } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } ( 1 - 0 ) \\frac { \\exp ( \\phi _ { \\phi ( S ) - } ) } { \\exp ( \\phi _ { \\phi ( S ) - } ) } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } ( 1 - 0 ) \\exp ( \\phi _ { \\phi ( S ) - } ) } \\\\ & = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } ( 1 - \\sum \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 125, + 769, + 373 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Computation of $p ^ { D \\setminus C } ( S \\setminus C )$ . When using the Gumbel-Top- $k$ trick over the restricted domain $D \\backslash C$ , we do not need to renormalize the log-probabilities $\\phi _ { s } , s \\in D \\setminus C$ since the Gumbel-Top- $k$ trick applies to unnormalized log-probabilities. Also, assuming $C \\subseteq S ^ { k }$ , it holds that $\\left( D \\backslash C \\right) \\backslash ( \\mathbf { \\bar { S } } )$ $C ) = D \\backslash S$ . This means that we can compute $p ^ { D \\setminus C } ( S \\setminus C )$ similar to equation 31: ", + "bbox": [ + 173, + 385, + 825, + 445 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/71b1ede5eb7d716390ae712d93a77350d15fd8d99080e5f5d9d65b83e209d14e.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle p ^ { D \\setminus C } ( S \\setminus C ) = P ( \\underset { i \\in S \\backslash C } { \\mathrm { m i n } } G _ { \\phi _ { i } } > G _ { \\phi _ { ( D \\setminus C ) \\setminus ( S \\setminus C ) } } ) } & { } \\\\ { = P ( \\underset { i \\in S \\backslash C } { \\mathrm { m i n } } G _ { \\phi _ { i } } > G _ { \\phi _ { D \\setminus S } } ) } & { } \\\\ { = \\displaystyle \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } \\big ( g _ { \\phi _ { D \\setminus S } } \\big ) \\prod _ { i \\in S \\backslash C } \\left( 1 - F _ { \\phi _ { i } } \\big ( g _ { \\phi _ { D \\setminus S } } \\big ) \\right) \\partial g _ { \\phi _ { D \\setminus S } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 452, + 756, + 549 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Computation of $R ( S ^ { k } , s )$ . Note that, using equation 10, it holds that ", + "bbox": [ + 173, + 563, + 640, + 580 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/37fa10b9c1fd720eb745b828e611bb6706d79858280070ca02012d93d4503d8a.jpg", + "text": "$$\n\\sum _ { s \\in S ^ { k } } { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s ) } { p ( S ^ { k } ) } } = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) = 1\n$$", + "text_format": "latex", + "bbox": [ + 316, + 585, + 681, + 627 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "from which it follows that ", + "bbox": [ + 174, + 633, + 346, + 647 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2a4ec5a0625a14701521c45a48439b3f0ca24500a7cee376c7f650e285dca087.jpg", + "text": "$$\np ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s )\n$$", + "text_format": "latex", + "bbox": [ + 375, + 642, + 622, + 678 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "such that ", + "bbox": [ + 173, + 681, + 235, + 694 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/410624f2557ad5303a82af31b26e72b713b414348fbc1f7ef00af886da23f8f8.jpg", + "text": "$$\nR ( S ^ { k } , s ) = { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } = { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ^ { D \\setminus \\{ s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s ^ { \\prime } \\} ) p ( s ^ { \\prime } ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 689, + 723, + 728 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This means that, to compute the leave-one-out ratio for all $s ~ \\in ~ S ^ { k }$ , we only need to compute $p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} )$ for $s \\in S ^ { k }$ . When using the numerical integration or summation in $O ( 2 ^ { k } )$ , we can reuse computation, whereas using the naive method, the cost is $O ( k \\cdot ( k - 1 ) ! ) = O ( k ! )$ , making the total computational cost comparable to computing just $p ( S ^ { k } )$ , and the same holds when computing the ‘second-order’ leave one out ratios for the built-in baseline (equation 17). ", + "bbox": [ + 173, + 731, + 825, + 804 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Details of numerical integration. For computation of the leave-one-out ratio (equation 35) for large $k$ we can use the numerical integration, where we need to compute equation 34 with $C = \\{ s \\}$ . For this purpose, we rewrite the integral as ", + "bbox": [ + 171, + 103, + 825, + 146 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c310d2db9b1fa7110a785947e4cf56a00a645a76a168ecfb1126319fea07ae03.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle p ^ { D \\setminus C } ( S \\setminus C ) = \\int _ { \\phi _ { \\partial \\setminus \\mathcal { D } } \\setminus \\phi = - \\phi } ^ { \\phi _ { \\partial \\setminus \\mathcal { D } } } f _ { \\phi _ { D \\setminus \\mathcal { D } } } ( g _ { \\phi \\phi _ { D \\setminus \\mathcal { D } } } , s ) \\prod _ { i \\in S \\setminus \\mathcal { C } } \\left( 1 - F _ { \\phi _ { i } } ( g _ { \\phi _ { D \\setminus i } } , s ) \\right) \\partial g _ { \\phi _ { D \\setminus i } } } \\\\ { \\displaystyle \\qquad = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - F _ { \\phi _ { i } } \\Big ( F _ { \\phi _ { D \\setminus i } } ^ { - 1 } \\Big ( u \\Big ) \\Big ) \\Big ) \\partial u } \\\\ { \\displaystyle \\qquad = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - u ^ { \\alpha \\setminus \\phi _ { i } ( \\phi _ { i } - \\phi _ { D \\setminus i } ) s } \\Big ) \\partial u } \\\\ { \\displaystyle \\qquad = \\exp ( b ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\phi _ { i } ( b ) - 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } - \\phi _ { D \\setminus i } - \\phi _ { i } ) } \\Big ) \\partial v } \\\\ { \\displaystyle \\qquad = \\exp ( a + \\phi _ { D \\setminus S } ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + \\phi _ { D \\setminus i } - s ) - 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + s ) } \\Big ) \\partial v } \\\\ { \\displaystyle \\qquad = \\exp ( a + \\phi _ { D \\setminus \\mathcal { D } } ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + \\phi _ { D \\setminus i } - s ) - 1 } v \\in \\mathrm { S } ^ { \\setminus \\Phi _ { i } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 218, + 155, + 779, + 372 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Here we have used change of variables $v = u ^ { e x p ( - b ) }$ and $a = b - \\phi _ { D \\backslash S }$ . This form allows to compute the integrands efficiently, as ", + "bbox": [ + 174, + 381, + 823, + 411 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/3ab65bbdffb221dc2e3ddf5ae942c5101495262c958abebcbf1015b13a972aee.jpg", + "text": "$$\n\\prod _ { i \\in S \\setminus C } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) = \\frac { \\prod _ { i \\in S } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) } { \\prod _ { i \\in C } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) }\n$$", + "text_format": "latex", + "bbox": [ + 325, + 420, + 669, + 464 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the numerator only needs to computed once, and, since $C = \\{ s \\}$ when computing equation 35, the denominator only consists of a single term. ", + "bbox": [ + 173, + 474, + 823, + 503 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The choice of $a$ may depend on the setting, but we found that $a \\ : = \\ : 5$ is a good default option which leads to an integral that is generally smooth and can be accurately approximated using the trapezoid rule. We compute the integrands in logarithmic space and sum the terms using the stable LOGSUMEXP trick. In our code we provide an implementation which also computes all second-order leave-one-out ratios efficiently. ", + "bbox": [ + 173, + 510, + 825, + 580 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C THE SUM-AND-SAMPLE ESTIMATOR ", + "text_level": 1, + "bbox": [ + 174, + 603, + 504, + 619 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.1 UNBIASEDNESS OF THE SUM-AND-SAMPLE ESTIMATOR ", + "text_level": 1, + "bbox": [ + 174, + 637, + 601, + 651 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We show that the sum-and-sample estimator is unbiased for any set $C \\subset D$ (see also Liang et al. (2018); Liu et al. (2019)): ", + "bbox": [ + 174, + 662, + 825, + 693 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/b9e074f65ae7cf0608fff4cbda5585c6ae47c099e3db5c45711392f7bb8c436a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\mathrm { s e } \\sim p ^ { N } \\times ( \\pi ) } \\left[ \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) f ( c ) + \\left( 1 - \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) \\right) f ( s ) \\right] } \\\\ & { = \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) f ( e ) + \\left( 1 - \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) \\right) \\mathbb { E } _ { \\alpha \\sim p ^ { N } \\times ( \\pi ) } [ f ( x ) ] } \\\\ & { = \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) f ( e ) + \\left( 1 - \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) \\right) \\underset { s \\in \\mathcal { C } } { \\sum } \\frac { p ( x ) } { 1 - \\sum _ { s \\in \\mathcal { C } } p ( c ) } f ( x ) } \\\\ & { = \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) f ( e ) + \\underset { s \\in \\mathcal { D } } { \\sum } p ( x ) f ( x ) } \\\\ & { = \\underset { s \\in \\mathcal { C } } { \\sum } p ( c ) f ( x ) + \\underset { s \\in \\mathcal { D } \\backslash \\mathcal { C } } { \\sum } } \\\\ & { = \\underset { s \\in \\mathcal { D } } { \\sum } p ( x ) f ( x ) } \\\\ & { = \\underset { c \\in \\mathcal { C } \\sim ( \\mathcal { C } ) } { \\sum } [ f ( x ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 700, + 715, + 931 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this section we give the proof that Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered set estimator. ", + "bbox": [ + 174, + 135, + 823, + 165 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Theorem 4. Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered set estimator, i.e. ", + "bbox": [ + 171, + 174, + 825, + 204 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/0b9d7c9d1ed7944469fbeac8c9a5bc6f5f685b09515131a1595aefaf726d7cb7.jpg", + "text": "$$\n\\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } | S ^ { k } ) } \\left[ \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) \\right) f ( b _ { k } ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 196, + 222, + 772, + 273 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. To give the proof, we first prove three Lemmas. ", + "bbox": [ + 173, + 330, + 534, + 347 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma 1. ", + "bbox": [ + 173, + 393, + 246, + 409 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/317c6fc716831ad249f80ed035ca691c5c122a60da3a87b0f11283a54e6d79c1.jpg", + "text": "$$\nP ( b _ { k } = s | S ^ { k } ) = { \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } { \\frac { p ( s ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ s \\} } p ( s ^ { \\prime } ) } }\n$$", + "text_format": "latex", + "bbox": [ + 325, + 424, + 673, + 462 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. Similar to the derivation of $P ( b _ { 1 } = s | S ^ { k } )$ (equation 10 in the main paper), we can write: ", + "bbox": [ + 169, + 520, + 802, + 537 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/6c885fc524dc9e072ef6b44df93fc4809a19f07582e36c8423a970c39b38f389.jpg", + "text": "$$\n\\begin{array} { r l } & { P ( b _ { k } = s | S ^ { k } ) = \\displaystyle \\frac { P ( S ^ { k } \\cap b _ { k } = s ) } { p ( S ^ { k } ) } } \\\\ & { \\qquad = \\displaystyle \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) p ^ { D \\setminus ( S ^ { k } \\setminus \\{ s \\} ) } ( s ) } { p ( S ^ { k } ) } } \\\\ & { \\qquad = \\displaystyle \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } \\displaystyle \\frac { p ( s ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ s \\} } p ( s ^ { \\prime } ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 558, + 676, + 671 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The step from the first to the second row comes from analyzing the event $S ^ { k } \\cap b _ { k } = s$ using sequential sampling: to sample $S ^ { k }$ (including $s$ ) with $s$ being the $k$ -th element means that we should first sample $S ^ { k } \\setminus \\{ s \\bar { \\} }$ (in any order), and then sample $s$ from the distribution restricted to $D \\setminus \\left( S ^ { k } \\setminus \\left\\{ s \\right\\} \\right)$ . □ ", + "bbox": [ + 174, + 690, + 823, + 736 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma 2. ", + "bbox": [ + 173, + 786, + 246, + 801 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/7c0852d8179f1a62f8ea46378e393a74a336a7f30eadf02092aad28ae432b7d6.jpg", + "text": "$$\np ( S ) + p ( S \\setminus \\{ s \\} ) { \\frac { 1 - \\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } { 1 - \\sum _ { s ^ { \\prime } \\in S \\setminus \\{ s \\} } p ( s ^ { \\prime } ) } } = p ^ { D \\setminus \\{ s \\} } ( S \\setminus \\{ s \\} )\n$$", + "text_format": "latex", + "bbox": [ + 299, + 815, + 697, + 854 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Dividing equation 33 by $\\begin{array} { r } { 1 - \\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } \\end{array}$ on both sides, we obtain ", + "bbox": [ + 171, + 907, + 611, + 926 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 173, + 103, + 217, + 118 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/debf08f1ef48f69c0813d75fce15229f44e5af54cbf61079be4a12af114655f3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad _ { 1 } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } \\\\ & { = \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } } \\\\ & { \\quad _ { 1 } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\Big ( \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } + \\mathsf { E } \\eta ^ { \\mathsf { C P S } _ { k } } \\Big ) + \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } } \\\\ & = \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\frac { \\mathsf { E } ^ { ( k ) } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } \\\\ & \\quad _ { 1 } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } 1 - \\sum _ k \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 174, + 125, + 816, + 500 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Multiplying by $\\begin{array} { r } { 1 - \\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } \\end{array}$ and rearranging terms proves Lemma 2. ", + "bbox": [ + 174, + 526, + 651, + 544 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma 3. ", + "bbox": [ + 173, + 570, + 246, + 584 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/f849db37933e0df0ed77d77694a6da529c83d635bb55db7375ebdcf9171da09e.jpg", + "text": "$$\np ( s ) + \\left( 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) P ( b _ { k } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )\n$$", + "text_format": "latex", + "bbox": [ + 305, + 588, + 691, + 638 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof. First using Lemma 1 and then Lemma 2 we find ", + "bbox": [ + 173, + 671, + 540, + 688 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/377d01fdf87f7d91e3098db3d580ff2258aaa4b7ed1d2cd4efd8f12a2bc3fc82.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad p ( s ) + \\left( 1 - \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) P ( b _ { k } = s | S ^ { k } ) } \\\\ & { = p ( s ) + \\left( 1 - \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) \\frac { p ( S ^ { k } \\setminus \\{ S \\} ) } { p ( S ^ { k } ) } \\frac { p ( s ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ S \\} } p ( s ^ { \\prime } ) } } \\\\ & { = \\frac { p ( S ) } { p ( S ^ { k } ) } \\left( p ( S ^ { k } ) + \\displaystyle \\frac { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ P ( s ^ { \\prime } ) \\} } p ( s ^ { \\prime } ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ S \\} } p ( s ^ { \\prime } ) } p ( S ^ { k } \\setminus \\{ s \\} ) \\right) } \\\\ & { = \\frac { p ( s ) } { p ( S ^ { k } ) } p ^ { D \\setminus \\{ S \\} } ( S ^ { k } \\setminus \\{ s \\} ) } \\\\ & { \\quad - p ( s ) R ( S ^ { k } , s ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 699, + 710, + 898 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Now we can complete the proof of Theorem 4 by adding $p ( b _ { k } ) f ( b _ { k } ) - p ( b _ { k } ) f ( b _ { k } ) = 0$ to the estimator, moving the terms independent of $B ^ { k }$ outside the expectation and using Lemma 3: ", + "bbox": [ + 169, + 102, + 823, + 133 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/30c1840633db154981388efc9babecf7dc93fdfb0811c0fef42d725b51213f11.jpg", + "text": "$$\n\\begin{array} { r l } & \\quad \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } \\\\ & \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } \\\\ & \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } \\\\ & { \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\infty } [ ( 1 - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } ) f ^ { ( i i i i ) } } ] f ^ { ( i i i i ) } } \\\\ & \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } \\\\ & { \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } [ f ^ { ( i i ) } + \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ] f ^ { ( i i i i ) } } \\\\ & { \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ( f ^ { ( i i i ) } + \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ) f ^ { ( i i i i ) } } \\\\ & { \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ( f ^ { ( i i i ) } + \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ) f ^ { ( i i i ) } } \\\\ & \\quad \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ( f ^ { ( i i ) } + \\sum _ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 140, + 707, + 428 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C.3 THE STOCHASTIC SUM-AND-SAMPLE ESTIMATOR WITH MULTIPLE SAMPLES ", + "text_level": 1, + "bbox": [ + 173, + 462, + 740, + 477 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "As was discussed in Liu et al. (2019), one can trade off the number of summed terms and number of sampled terms to maximize the achieved variance reduction. As a generalization of Theorem 4 (the stochastic sum-and-sample estimator with $k - 1$ summed terms), we introduce here the stochastic sum-and-sample estimator that sums $k - m$ terms and samples $m > 1$ terms without replacement. To estimate the sampled term, we use the unordered set estimator on the $m$ samples without replacement, on the domain restricted to $D \\setminus B ^ { k - m }$ . In general, we denote the unordered set estimator restricted to the domain $D \\backslash C$ by ", + "bbox": [ + 173, + 487, + 826, + 587 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/ae6f53cdd3e12b7696af2cbaed29929c0d66349ef59ae9486808cda4504d3d23.jpg", + "text": "$$\ne ^ { { \\bf U S } , D \\setminus C } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } \\setminus C } p ( s ) R ^ { D \\setminus C } ( S ^ { k } , s ) f ( s )\n$$", + "text_format": "latex", + "bbox": [ + 343, + 592, + 655, + 628 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $R ^ { D \\setminus C } ( S ^ { k } , s )$ is the leave-one-out ratio restricted to the domain $D \\backslash C$ , similar to the second order leave-one-out ratio in equation 18: ", + "bbox": [ + 174, + 636, + 823, + 665 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/262dc87328c00006d729bdd5cefd83d20437f1eb35b0ef68dbc8d80f134f5054.jpg", + "text": "$$\nR ^ { D \\setminus C } ( S ^ { k } , s ) = { \\frac { p _ { \\theta } ^ { ( D \\setminus C ) \\setminus \\{ s \\} } ( ( S ^ { k } \\setminus C ) \\setminus \\{ s \\} ) } { p _ { \\theta } ^ { D \\setminus C } ( S ^ { k } \\setminus C ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 670, + 651, + 713 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "While we can also constrain $S ^ { k } \\subseteq ( D \\backslash C )$ , this definition is consistent with equation 18 and allows simplified notation. ", + "bbox": [ + 176, + 719, + 823, + 750 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Theorem 5. Rao-Blackwellizing the stochastic sum-and-sample estimator with $m \\ > \\ 1$ samples results in the unordered set estimator, i.e. ", + "bbox": [ + 173, + 752, + 823, + 781 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/c9ac6892079fc79013e0e1be3f725409a4351778d410ea7fdeda86d9dc2a5412.jpg", + "text": "$$\n\\ ? \\ ? \\complement _ { B ^ { k } \\sim p ( B ^ { k } \\mid S ^ { k } ) } \\left[ \\sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) \\right) e ^ { U S , D \\setminus B ^ { k - m } } ( S ^ { k } ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 785, + 833, + 837 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. Recall that for the unordered set estimator, it holds that ", + "bbox": [ + 173, + 863, + 584, + 878 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/530122c7b7780623682b4f30cd05eb78970872a06f15a7c9c139ea6d2478afee.jpg", + "text": "$$\ne ^ { \\mathrm { U S } } ( S ^ { k } ) = \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\mathbb { E } _ { x \\sim p ( x ) } \\left[ f ( x ) \\middle | x \\in S ^ { k } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 303, + 883, + 692, + 905 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "which for the restricted equivalent (with restricted distribution $p ^ { D \\setminus C }$ ) translates into ", + "bbox": [ + 171, + 102, + 723, + 118 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/95f74612504c9115d909cb70e4eade4f448adb087befe3d0cd39383613e5771d.jpg", + "text": "$$\ne ^ { \\mathrm { U S } , D \\setminus C } ( S ^ { k } ) = \\mathbb { E } _ { x \\sim p ^ { D \\setminus C } ( x ) } \\left[ f ( x ) \\big | x \\in S ^ { k } \\right] = \\mathbb { E } _ { x \\sim p ( x ) } \\left[ f ( x ) \\big | x \\in S ^ { k } , x \\not \\in C \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 236, + 123, + 761, + 143 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now we consider the distribution $b _ { k - m + 1 } | S ^ { k } , B ^ { k - m }$ : the distribution of the first element sampled (without replacement) after sampling $B ^ { k - m }$ , given (conditionally on the event) that the set of $k$ samples is $S ^ { k }$ , so we have $b _ { k - m + 1 } \\stackrel { - } { \\in } S ^ { k }$ and $\\mathsf { \\bar { b } } _ { k - m + 1 } \\notin B ^ { k - m }$ . This means that its conditional expectation of $f ( b _ { k - m + 1 } )$ is the restricted unordered set estimator for $C = B ^ { k - m }$ since ", + "bbox": [ + 173, + 148, + 825, + 207 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/07a2fbd8fe56d665e43e09c3001882623419cefdd4f57e0ab0f3785a4c9020f6.jpg", + "text": "$$\n\\begin{array} { r l } { { e ^ { \\mathrm { U S } , D \\setminus B ^ { k - m } } ( S ^ { k } ) = \\mathbb { E } _ { x \\sim p ( x ) } [ f ( x ) \\vert x \\in S ^ { k } , x \\not \\in B ^ { k - m } ] } \\quad } & { } \\\\ & { = \\mathbb { E } _ { b _ { k - m + 1 } \\sim p ( b _ { k - m + 1 } \\mid S ^ { k } , B ^ { k - m } ) } [ f ( b _ { k - m + 1 } ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 210, + 710, + 255 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Observing that the definition (equation 42) of the stochastic sum-and-sample estimator does not depend on the actual order of the $m$ samples, and using equation 45, we can reduce the multisample estimator to the stochastic sum-and-sample estimator with $k ^ { \\prime } = k - m + 1$ , such that the result follows from equation 36. ", + "bbox": [ + 173, + 256, + 825, + 313 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/d8526bc56964422d83f6abb56b1004632519ad61650cc88894ec9d0df48c8e72.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\begin{array} { l } { { { \\cal L } _ { \\mathrm { e x c } , \\mathrm { t r i g h } , \\mathrm { \\scriptsize ~ 1 . . } } \\sum _ { j = 0 } ^ { N } \\phi _ { j } \\Big | Z _ { j } ^ { ( k ) } \\phi _ { j } \\Big | + ( \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\phi _ { j } } ) Z _ { j } ^ { ( k ) } z ^ { ( k ) } z ^ { ( k ) } - ( \\beta _ { j } - \\gamma ) \\delta _ { j } ^ { \\prime } \\Big | } } } \\\\ { { \\displaystyle - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\mathrm { \\cdots ~ 1 } } } \\end{array} \\} } \\\\ \\displaystyle - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } ( \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\phi _ { j } } ) \\Bigg | \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { k = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } } \\\\ \\displaystyle - \\sum _ j = \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 318, + 816, + 655 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D THE IMPORTANCE-WEIGHTED ESTIMATOR ", + "text_level": 1, + "bbox": [ + 173, + 670, + 560, + 686 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D.1 RAO-BLACKWELLIZATION OF THE IMPORTANCE-WEIGHTED ESTIMATOR", + "text_level": 1, + "bbox": [ + 174, + 700, + 718, + 715 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this section we give the proof that Rao-Blackwellizing the importance-weighted estimator results in the unordered set estimator. ", + "bbox": [ + 171, + 726, + 825, + 755 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Theorem 6. Rao-Blackwellizing the importance-weighted estimator results in the unordered set estimator, i.e.: ", + "bbox": [ + 173, + 758, + 823, + 786 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/01e8ad49e2b31698b00a425d91b8cf16f8e6028f37590b04c330c6d3a3bf6f9c.jpg", + "text": "$$\n\\mathbb { E } _ { \\kappa \\sim p ( \\kappa | S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } \\frac { p ( s ) } { 1 - F _ { \\phi _ { s } } ( \\kappa ) } f ( s ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .\n$$", + "text_format": "latex", + "bbox": [ + 285, + 790, + 710, + 840 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Here we have slightly rewritten the definition of the importance-weighted estimator, using that $q ( s , a ) = P ( g _ { \\phi _ { s } } ^ { \\mathrm { ~ ~ } } > \\stackrel { \\cdot } { a } ) = 1 - F _ { \\phi _ { s } } ( a )$ , where $F _ { \\phi _ { s } }$ is the CDF of the Gumbel distribution (see Appendix A). ", + "bbox": [ + 173, + 852, + 826, + 895 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. We first prove the following Lemma: ", + "bbox": [ + 174, + 909, + 465, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma 4. ", + "bbox": [ + 173, + 103, + 246, + 118 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/ea6ebad1366a3fc44c7c07e5829e5eaef7cab655a4b439ea9150102a75887e5b.jpg", + "text": "$$\n\\mathbb { E } _ { \\kappa \\sim p ( \\kappa | S ^ { k } ) } \\left[ \\frac { 1 } { 1 - F _ { \\phi _ { s } } ( \\kappa ) } \\right] = R ( S ^ { k } , s )\n$$", + "text_format": "latex", + "bbox": [ + 369, + 113, + 629, + 150 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. Conditioning on $S ^ { k }$ , we know that the elements in $S ^ { k }$ have the $k$ largest perturbed logprobabilities, so $\\kappa$ , the $( k + 1 )$ -th largest perturbed log-probability is the largest perturbed logprobability in $D \\backslash S ^ { k }$ , and satisfies $\\begin{array} { r } { \\kappa = \\operatorname* { m a x } _ { s \\in D \\backslash S ^ { k } } g _ { \\phi _ { s } } = g _ { \\phi _ { D \\backslash S ^ { k } } } \\sim \\mathrm { G u m b e l } ( \\phi _ { D \\backslash S ^ { k } } ) } \\end{array}$ . Computing $p ( \\kappa | S ^ { k } )$ using Bayes’ Theorem, we have ", + "bbox": [ + 173, + 165, + 825, + 227 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/72d64bf455a71d7bfe77493ed53bfb56de65d59129300cc1d8bd30091b9c3013.jpg", + "text": "$$\np ( \\kappa | S ^ { k } ) = \\frac { p ( S ^ { k } | \\kappa ) p ( \\kappa ) } { p ( S ^ { k } ) } = \\frac { \\prod _ { s \\in S ^ { k } } ( 1 - F _ { \\phi _ { s } } ( \\kappa ) ) f _ { \\phi _ { D \\setminus S ^ { k } } } ( \\kappa ) } { p ( S ^ { k } ) }\n$$", + "text_format": "latex", + "bbox": [ + 299, + 233, + 697, + 272 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "which allows us to compute (using equation 34 with $C = \\{ s \\}$ and $g _ { \\phi _ { D \\setminus S } } = \\kappa$ ", + "bbox": [ + 173, + 285, + 687, + 303 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/270db507263957445b50afe3dffcba35712863dac88728966242bbb04b5dec1b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { n \\sim p ( n | S ^ { k } ) } \\left[ \\frac { 1 } { 1 - F _ { \\phi _ { n } } ( \\kappa ) } \\right] } \\\\ & { = \\int _ { n = - \\infty } ^ { \\infty } p ( \\kappa | S ^ { k } ) \\frac { 1 } { 1 - F _ { \\phi _ { n } } ( \\kappa ) } \\partial \\kappa } \\\\ & { = \\int _ { \\kappa = - \\infty } ^ { \\infty } \\frac { \\prod _ { s \\in S ^ { k } } ( 1 - F _ { \\phi _ { s } } ( \\kappa ) ) f _ { \\phi _ { n ; s } } ( \\kappa ) } { p ( S ^ { k } ) } \\frac { 1 } { 1 - F _ { \\phi _ { n } } ( \\kappa ) } \\partial \\kappa } \\\\ & { = \\frac { 1 } { p ( S ^ { k } ) } \\int _ { \\kappa = - \\infty } ^ { \\infty } \\underset { s \\in S ^ { k } \\backslash \\{ s \\} } { \\prod } \\ ( 1 - F _ { \\phi _ { s } } ( \\kappa ) ) f _ { \\phi _ { n ; s } \\mu \\kappa } ( \\kappa ) \\partial \\kappa } \\\\ & { = \\frac { 1 } { p ( S ^ { k } ) } p ^ { D \\cup S } ( S \\setminus \\{ s \\} ) } \\\\ & { = R ( S ^ { k } , s ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 309, + 689, + 517 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Using Lemma 4 we find ", + "bbox": [ + 174, + 553, + 334, + 568 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/189a05ec559dd1c3ad78a3bb4788e233438705067db6a1ef44e13d2ded16f5f7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\kappa \\sim p ( \\kappa | S ^ { k } ) } \\left[ \\displaystyle \\sum _ { s \\in S ^ { k } } \\frac { p ( s ) } { 1 - F _ { \\phi _ { s } } ( \\kappa ) } f ( s ) \\right] } \\\\ & { = \\displaystyle \\sum _ { s \\in S ^ { k } } p ( s ) \\mathbb { E } _ { \\kappa \\sim p ( \\kappa | S ^ { k } ) } \\left[ \\frac { 1 } { 1 - F _ { \\phi _ { s } } ( \\kappa ) } \\right] f ( s ) } \\\\ & { = \\displaystyle \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 575, + 642, + 702 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "D.2 THE IMPORTANCE-WEIGHTED POLICY GRADIENT ESTIMATOR WITH BUILT-IN BASELINE", + "text_level": 1, + "bbox": [ + 171, + 739, + 823, + 755 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For self-containment we include this section, which is adapted from our unpublished workshop paper (Kool et al., 2019b). The importance-weighted policy gradient estimator combines REINFORCE (Williams, 1992) with the importance-weighted estimator (Duffield et al., 2007; Vieira, 2017) in equation 15 which results in an unbiased estimator of the policy gradient $\\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ f _ { \\pmb { \\theta } } ( \\pmb { x } ) ]$ : ", + "bbox": [ + 173, + 765, + 825, + 823 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/15056798f35b0a4a5f225aa5c7bdb4f9651cc7e2dc85aed31fd13666948ec7ab.jpg", + "text": "$$\ne ^ { \\mathrm { I W P G } } ( S ^ { k } , \\kappa ) = \\sum _ { s \\in S ^ { k } } \\frac { p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } \\nabla _ { \\theta } \\log p _ { \\theta } ( s ) f ( s ) = \\sum _ { s \\in S ^ { k } } \\frac { \\nabla _ { \\theta } p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } f ( s )\n$$", + "text_format": "latex", + "bbox": [ + 267, + 830, + 730, + 871 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Recall that $\\kappa$ is the $( k + 1 )$ -th largest perturbed log-probability (see Section 3.2). We compute a lower variance but biased variant by normalizing the importance weights using the normalization W (Sk) = Ps∈Sk pθ(s)qθ,κ(s) . ", + "bbox": [ + 174, + 877, + 825, + 929 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "As we show in Kool et al. (2019b), we can include a ‘baseline’ B(Sk) = Ps∈Sk pθ(s)qθ,κ(s) f and correct for the bias (since it depends on the complete sample $S ^ { k }$ ) by weighting individual terms of the estimator by $\\begin{array} { r } { 1 - p _ { \\pmb { \\theta } } ( s ) + \\frac { p _ { \\pmb { \\theta } } ( s ) } { q _ { \\pmb { \\theta } , \\kappa } ( s ) } } \\end{array}$ : ", + "bbox": [ + 173, + 101, + 826, + 159 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/ed94fc41ed206e40c574476efbaf81fc4a9410e07b249158520c562ae93e9b64.jpg", + "text": "$$\ne ^ { \\mathrm { I W P G B L } } ( S ^ { k } , \\kappa ) = \\sum _ { s \\in S ^ { k } } { \\frac { \\nabla _ { \\theta } p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } } \\left( f ( s ) \\left( 1 - p _ { \\theta } ( s ) + { \\frac { p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } } \\right) - B ( S ^ { k } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 245, + 164, + 751, + 204 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For the normalized version, we use the normalization W (Sk) = Ps∈Sk q for the baseline, and $\\begin{array} { r } { W _ { i } ( S ^ { k } ) = W ( S ^ { k } ) - \\frac { p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } + p _ { \\theta } ( s ) } \\end{array}$ to normalize the individual terms: ", + "bbox": [ + 173, + 219, + 825, + 261 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/428c5674023653acf72976a9a4a14f8ea48e5542ff29a4b3bc54d6b3079b8a91.jpg", + "text": "$$\n\\nabla _ { \\pmb \\theta } \\mathbb { E } _ { \\pmb \\theta \\sim p _ { \\pmb \\theta } ( \\pmb y ) } \\left[ f ( \\pmb y ) \\right] \\approx \\sum _ { s \\in S ^ { k } } \\frac { 1 } { W _ { i } ( S ^ { k } ) } \\cdot \\frac { \\nabla _ { \\pmb \\theta } p _ { \\pmb \\theta } ( s ) } { q _ { \\pmb \\theta , \\kappa } ( s ) } \\left( f ( s ) - \\frac { B ( S ^ { k } ) } { W ( S ^ { k } ) } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 276, + 267, + 722, + 309 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "It seems odd to normalize the terms in the outer sum by $\\frac { 1 } { W _ { i } ( S ^ { k } ) }$ instead of 1W (Sk) , but equation 52 can be rewritten into a form similar to equation 17, i.e. with a different baseline for each sample, but this form is more convenient for implementation (Kool et al., 2019b). ", + "bbox": [ + 173, + 315, + 826, + 363 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "E THE UNORDERED SET POLICY GRADIENT ESTIMATOR ", + "text_level": 1, + "bbox": [ + 173, + 382, + 651, + 398 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "E.1 PROOF OF UNBIASEDNESS OF THE UNORDERED SET POLICY GRADIENT ESTIMATOR WITH BASELINE ", + "bbox": [ + 176, + 412, + 790, + 441 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To prove the unbiasedness of result we need to prove that the control variate has expectation 0: Lemma 5. ", + "bbox": [ + 171, + 452, + 795, + 486 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/7dcde4fb0e8b1866026dbef50261833923c9fad3f33392df0058daff30d20272.jpg", + "text": "$$\n\\mathbb { E } _ { S ^ { k } \\sim p _ { \\theta } ( S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right] = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 489, + 754, + 540 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. Similar to equation 10, we apply Bayes’ Theorem conditionally on $b _ { 1 } ~ = ~ s$ to derive for $s ^ { \\prime } \\neq s$ ", + "bbox": [ + 169, + 553, + 825, + 583 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/b7826c18422de652fa96e5046dce9e2b17066522210c25cb22c5f01a87170f56.jpg", + "text": "$$\n\\begin{array} { c } { P ( b _ { 2 } = s ^ { \\prime } | S ^ { k } , b _ { 1 } = s ) = \\displaystyle { \\frac { P ( S ^ { k } | b _ { 2 } = s ^ { \\prime } , b _ { 1 } = s ) P ( b _ { 2 } = s ^ { \\prime } | b _ { 1 } = s ^ { \\prime } ) } { P ( S ^ { k } | b _ { 1 } = s ) } } } \\\\ { = \\displaystyle { \\frac { p _ { \\theta } ^ { D \\setminus \\{ s , s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s , s ^ { \\prime } \\} ) p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( s ^ { \\prime } ) } { p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } } } \\\\ { { = \\displaystyle { \\frac { p _ { \\theta } ( s ^ { \\prime } ) } { 1 - p _ { \\theta } ( s ) } R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 587, + 725, + 703 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For $s ^ { \\prime } = s$ we have $R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) = 1$ by definition, so using equation 54 we can show that ", + "bbox": [ + 169, + 707, + 782, + 724 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/941373c9e6ed240fe6a0ad7ccfcb8bc26b621b25ffdc7ff97d9ddb312f31505b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D N \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } \\frac { p _ { \\theta } ( s ^ { \\prime } ) } { 1 - p _ { \\theta } ( s ) } R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } P ( b _ { 2 } = s ^ { \\prime } | S ^ { k } , b _ { 1 } = s ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ { s ^ { \\prime } \\circ p _ { \\theta } ( b _ { 2 } \\mid S ^ { k } , b _ { 1 } = s ) } [ f ( b _ { 2 } ) ] } \\\\ & { = \\mathbb { E } _ { \\delta \\to p _ { \\theta } ( b _ { 2 } \\mid S ^ { k } , b _ { 1 } = s ) } [ p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 729, + 730, + 925 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Now we can show that the control variate is actually the result of Rao-Blackwellization: ", + "bbox": [ + 174, + 102, + 750, + 119 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/5aa17a281b84d6ea5ad10f6f5e8779f8f9604e2a43778445838faba1d23ee89e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\nabla \\theta ^ { ( k ) } ( \\theta ^ { k ) } H ^ { ( S ^ { k } , \\ell ) } \\cdot \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } \\varphi ( \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ( \\theta ^ { ( k ) } , s ^ { \\prime } ) \\nabla \\theta ( \\theta , \\theta ^ { \\prime } ) \\wedge \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } ) \\cdot \\xi ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } - s | S ^ { k } | \\nabla \\theta ) \\cdot \\Big ( \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } ) \\Big ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\hat { b } _ { 1 } - s | S ^ { k } | \\nabla \\theta ) \\cdot \\Big ( \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { ( k ) } , \\theta ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\operatorname* { m i n } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\nabla \\theta ^ { ( k ) } \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ] \\xi ^ { \\prime } s ^ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 126, + 830, + 388 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "This expression depends only on $b _ { 1 }$ and $b _ { 2 }$ and we recognize the stochastic sum-and-sample estimator for $k = 2$ used as ‘baseline’. As a special case of equation 13 for $C = \\{ b _ { 1 } \\}$ , we have ", + "bbox": [ + 173, + 398, + 825, + 428 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/2cafa0492e5f1eacfe47000c2ad3ffce6075884ea511cb796118f38c1fbd87c7.jpg", + "text": "$$\n\\mathbb { E } _ { b _ { 2 } \\sim p _ { \\theta } ( b _ { 2 } \\mid b _ { 1 } ) } \\left[ \\left( p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) \\right) \\right] = \\mathbb { E } _ { i \\sim p _ { \\theta } ( i ) } \\left[ f ( i ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 433, + 728, + 452 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Using this, and the fact that $\\begin{array} { r } { \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\right] = \\nabla _ { \\theta } \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ 1 \\right] = \\nabla _ { \\theta } 1 = 0 } \\end{array}$ we find ", + "bbox": [ + 173, + 458, + 802, + 477 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/bdf452d0c81a2a447c9b897bf8923e610b1dd3efe14bea58ceb1b7bbda495407.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { S ^ { k } \\sim p _ { \\theta } ( S ^ { k } ) } \\left[ \\displaystyle \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right] } \\\\ & { = \\mathbb { E } _ { B ^ { k } \\sim p _ { \\theta } ( B ^ { k } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\left( p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) \\right) \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\mathbb { E } _ { b _ { 2 } \\sim p _ { \\theta } ( b _ { 2 } | b _ { 1 } ) } \\left[ ( p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ) \\right] \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\right] \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] } \\\\ & { = 0 \\cdot \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] } \\\\ & { = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 498, + 764, + 666 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "F THE RISK ESTIMATOR ", + "text_level": 1, + "bbox": [ + 174, + 102, + 397, + 118 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "F.1 PROOF OF BUILT-IN BASELINE", + "text_level": 1, + "bbox": [ + 176, + 133, + 423, + 147 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We show that the RISK estimator, taking gradients through the normalization factor actually has a built-in baseline. We first use the log-derivative trick to rewrite the gradient of the ratio as the ratio times the logarithm of the gradient, and then swap the summation variables in the double sum that arises: ", + "bbox": [ + 174, + 159, + 825, + 214 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/d4fc2e2e5933c5709e0115282aca9abd4d4a5eeb1046ee13e85dc7bc3a3c9cbf.jpg", + "text": "$$\n\\begin{array} { r l } \\varepsilon ^ { \\mathrm { s c } } > 5 1 - \\sum _ { j \\in \\mathcal { K } } \\frac { \\partial } { \\partial x _ { j } } \\varepsilon ( \\frac \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 256, + 219, + 743, + 553 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G CATEGORICAL VARIATIONAL AUTO-ENCODER ", + "text_level": 1, + "bbox": [ + 173, + 564, + 598, + 580 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G.1 EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 595, + 392, + 609 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We use the code6 by Yin et al. (2019) to reproduce their categorical VAE experiment, of which we include details here for self-containment. The dataset is MNIST, statically binarized by thresholding at 0.5 (although we include results using the standard binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray (2011) in Section G.2). The latent representation $_ { z }$ is $K = 2 0$ dimensional with $C = 1 0$ categories per dimension with a uniform prior $p ( z _ { k } = c ) = 1 / C , k = 1 , . . . , K$ . The encoder is parameterized by $\\phi$ as $\\begin{array} { r } { q _ { \\phi } ( z | \\pmb { x } ) = \\prod _ { k } q _ { \\phi } ( z _ { k } | \\pmb { \\bar { x } } ) } \\end{array}$ and has two fully connected hidden layers with 512 and 256 hidden nodes respectively, with LeakyReLU $\\mathit { \\Phi } _ { \\mathrm { ( \\alpha ) } } = 0 . 1$ ) activations. The decoder, parameterized by $\\pmb \\theta$ , is given by $\\begin{array} { r } { \\bar { p _ { \\pmb \\theta } } ( \\pmb x | z ) \\dot { = } \\prod _ { i } p _ { \\pmb \\theta } ( x _ { i } | \\pmb z ) } \\end{array}$ , where $x _ { i } \\in \\{ 0 , 1 \\}$ are the pixel values, and has fully connected hidden layers with 256 and 512 nodes and LeakyReLU activation. ", + "bbox": [ + 173, + 621, + 825, + 747 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "ELBO optimization. The evidence lower bound (ELBO) that we optimize is given by ", + "bbox": [ + 166, + 761, + 753, + 776 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/11fc4483bac1bab571c6eb2989f4310edc1f93d523bcb801442db4bd3f334b47.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\phi , \\pmb { \\theta } ) = \\mathbb { E } _ { z \\sim q _ { \\phi } ( z \\vert x ) } \\left[ \\ln p _ { \\pmb { \\theta } } ( \\pmb { x } \\vert z ) + \\ln p ( z ) - \\ln q _ { \\phi } ( z \\vert x ) \\right] } \\\\ { = \\mathbb { E } _ { z \\sim q _ { \\phi } ( z \\vert x ) } \\left[ \\ln p _ { \\pmb { \\theta } } ( \\pmb { x } \\vert z ) \\right] - K L ( q _ { \\phi } ( z \\vert x ) \\vert \\vert p ( z ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 780, + 697, + 820 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For the decoder parameters $\\pmb \\theta$ , since $q _ { \\phi } ( z | \\boldsymbol { x } )$ does not depend on $\\pmb \\theta$ , it follows that ", + "bbox": [ + 173, + 821, + 712, + 838 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/141335888e50bf6a027bede7f5a9b8c0923d73d3b68ac93b2a5cde2fa4e49d8e.jpg", + "text": "$$\n\\nabla _ { \\pmb \\theta } \\mathcal L ( \\phi , \\pmb \\theta ) = \\mathbb E _ { z \\sim q _ { \\phi } ( z | x ) } \\left[ \\nabla _ { \\pmb \\theta } \\ln p _ { \\pmb \\theta } ( \\pmb x | z ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 842, + 642, + 861 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "or the encoder parameters $\\phi$ , we can write $\\nabla _ { \\phi } \\mathcal { L } ( \\phi , \\theta )$ using equation 57 and equation 19 as ", + "bbox": [ + 192, + 863, + 787, + 880 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/278208fac71600b9253b8dca8238bbf449eaac59045f527748b682df7a879ecb.jpg", + "text": "$$\n\\nabla _ { \\phi } \\mathcal { L } ( \\phi , \\pmb { \\theta } ) = \\mathbb { E } _ { z \\sim q _ { \\phi } ( z | \\pmb { x } ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | \\pmb { x } ) \\ln p _ { \\theta } ( \\pmb { x } | z ) \\right] - \\nabla _ { \\phi } K L ( q _ { \\phi } ( z | \\pmb { x } ) | | p ( z ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 212, + 882, + 758, + 901 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "This assumes we can compute the $\\mathrm { K L }$ divergence analytically. Alternatively, we can use a sample estimate for the KL divergence, and use equation 56 with equation 19 to obtain ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/9047984934ee20c3367dfc54f7d948b00ec5fc53723581b47394e47e41f9a1e7.jpg", + "text": "$$\n\\begin{array} { l } { \\nabla _ { \\phi } \\mathcal { L } ( \\phi , \\theta ) = \\mathbb { E } _ { z \\sim q _ { \\phi } ( z | x ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) ( \\ln p _ { \\theta } ( x | z ) + \\ln p ( z ) - \\ln q _ { \\phi } ( z | x ) ) + \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) \\right] , } \\\\ { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { = \\mathbb { E } _ { z \\sim q _ { \\phi } ( z | x ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) ( \\ln p _ { \\theta } ( x | z ) - \\ln q _ { \\phi } ( z | x ) ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 176, + 137, + 821, + 193 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Here we have left out the term $\\begin{array} { r } { \\mathbb { E } _ { z \\sim q _ { \\phi } ( z | \\pmb { x } ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | \\pmb { x } ) \\right] = 0 } \\end{array}$ , similar to Roeder et al. (2017), and, assuming a uniform (i.e. constant) prior $\\ln p ( z )$ , the term ${ { \\mathbb E } } _ { z \\sim q _ { \\phi } ( z | x ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) \\ln p ( z ) \\right] = 0$ . With a built-in baseline, this second term cancels out automatically, even if it is implemented. Despite the similarity of the equation 56 and equation 57, their gradient estimates (equation 60 and equation 59) are structurally dissimilar and care should be taken to implement the REINFORCE estimator (or related estimators such as ARSM and the unordered set estimator) correctly using automatic differentiation software. Using Gumbel-Softmax and RELAX, we take gradients ‘directly’ through the objective in equation 57. ", + "bbox": [ + 174, + 198, + 825, + 313 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We optimize the ELBO using the analytic KL for 1000 epochs using the Adam (Kingma & Ba, 2015) optimizer. We use a learning rate of $\\mathrm { i 0 ^ { - 3 } }$ for all estimators except Gumbel-Softmax and RELAX, which use a learning rate of $\\bar { 1 } 0 ^ { - 4 }$ as we found they diverged with a higher learning rate. For ARSM, as an exception we use the sample KL, and a learning rate of $3 \\cdot 1 0 ^ { - \\overline { { 4 } } }$ , as suggested by the authors. All reported ELBO values are computed using the analytic KL. Our code is publicly available7. ", + "bbox": [ + 174, + 319, + 825, + 388 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "G.2 ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 406, + 374, + 420 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Gradient variance during training. We also evaluate gradient variance of different estimators during different stages of training. We measure the variance of different estimators with $k = 4$ samples during training with REINFORCE with replacement, such that all estimators are computed for the same model parameters. The results during training, given in Figure 4, are similar to the results for the trained model in Table 1, except for at the beginning of training, although the rankings of different estimator are mostly the same. ", + "bbox": [ + 174, + 431, + 825, + 515 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Negative ELBO on validation set. Figure 5 shows the -ELBO evaluated during training on the validation set. For the large latent space, we see validation error quickly increase (after reaching a minimum) which is likely because of overfitting (due to improved optimization), a phenomenon observed before (Tucker et al., 2017; Grathwohl et al., 2018). Note that before the overfitting starts, both REINFORCE without replacement and the unordered set estimator achieve a validation error similar to the other estimators, such that in a practical setting, one can use early stopping. ", + "bbox": [ + 174, + 530, + 825, + 614 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Results using standard binarized MNIST dataset. Instead of using the MNIST dataset binarized by thresholding values at 0.5 (as in the code and paper by Yin et al. (2019)) we also experiment with the standard (fixed) binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray (2011), for which we plot train and validation curves for two runs on the small and large domain in Figure 6. This gives more realistic (higher) -ELBO scores, although we still observe the effect of overfitting. As this is a bit more unstable setting, one of the runs using REINFORCE with replacement diverged, but in general the relative performance of estimators is similar to using the dataset with 0.5 threshold. ", + "bbox": [ + 173, + 630, + 825, + 741 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/3cf64085ef97333430dfc6ae343dba9cc3bb81a28a8d5d46e4e908f59c58c425.jpg", + "image_caption": [ + "Training log variance (10 2 latent space), $k = 4$ samples ", + "Figure 4: Gradient log variance of different unbiased estimators with $k = 4$ samples, estimated every 100 (out of 1000) epochs while training using REINFORCE with replacement. Each estimator is computed 1000 times with different latent samples for a fixed minibatch (the first 100 records of training data). We report (the logarithm of) the sum of the variances per parameter (trace of the covariance matrix). Some lines coincide, so we sort the legend by the last measurement and report its value. " + ], + "image_footnote": [], + "bbox": [ + 189, + 193, + 808, + 348 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/9db0e1232d97d8b19cc6e8b5122295f21e9ef07cda2dbb6e7624da7dadc43d3b.jpg", + "image_caption": [ + "Figure 5: Smoothed validation -ELBO curves during training of two independent runs when with different estimators with $k = 1$ , 4 or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value. " + ], + "image_footnote": [], + "bbox": [ + 191, + 617, + 808, + 785 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/d1fca44c1c3c18bdebf0fcdc5ea9a18b01b73cd2389f61f27b68f5e6d89e5bae.jpg", + "image_caption": [ + "Figure 6: Smoothed training and validation -ELBO curves during training on the standard binarized MNIST dataset (Salakhutdinov & Murray, 2008; Larochelle & Murray, 2011) of two independent runs when with different estimators with $k = 1$ , 4 or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value. " + ], + "image_footnote": [], + "bbox": [ + 187, + 303, + 808, + 641 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "H TRAVELLING SALESMAN PROBLEM ", + "text_level": 1, + "bbox": [ + 174, + 102, + 504, + 118 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "The Travelling Salesman Problem (TSP) is a discrete optimization problem that consists of finding the order in which to visit a set of locations, given as $x , y$ coordinates, to minimize the total length of the tour, starting and ending at the same location. As a tour can be considered a sequence of locations, this problem can be set up as a sequence modelling problem, that can be either addressed using supervised (Vinyals et al., 2015) or reinforcement learning (Bello et al., 2016; Kool et al., 2019a). ", + "bbox": [ + 174, + 133, + 825, + 217 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Kool et al. (2019a) introduced the Attention Model, which is an encoder-decoder model which considers a TSP instances as a fully connected graph. The encoder computes embeddings for all nodes (locations) and the decoder produces a tour, which is sequence of nodes, selecting one note at the time using an attention mechanism, and uses this autoregressively as input to select the next node. In Kool et al. (2019a), this model is trained using REINFORCE, with a greedy rollout used as baseline to reduce variance. ", + "bbox": [ + 174, + 223, + 825, + 308 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We use the code by Kool et al. (2019a) to train the exact same Attention Model (for details we refer to Kool et al. (2019a)), and minimize the expected length of a tour predicted by the model, using different gradient estimators. We did not do any hyperparameter optimization and used the exact same training details, using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of $1 0 ^ { - 4 }$ (no decay) for 100 epochs for all estimators. For the baselines, we used the same batch size of 512, but for estimators that use $k = 4$ samples, we used a batch size of $\\frac { 5 1 2 } { 4 } = 1 2 8$ to compensate for the additional samples (this makes multi-sample methods actually faster since the encoder still needs to be evaluated only once). 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We further reduce the variance by deriving a built-in control variate,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 407, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 407, + 636 + ], + "score": 1.0, + "content": "which maintains the unbiasedness and does not require additional samples.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "Related work. Many algorithms for estimating gradients for discrete distributions have been pro-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "posed. A general and widely used estimator is REINFORCE (Williams, 1992). 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We further reduce the variance by deriving a built-in control variate,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 407, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 407, + 636 + ], + "score": 1.0, + "content": "which maintains the unbiasedness and does not require additional samples.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 569, + 506, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "Related work. Many algorithms for estimating gradients for discrete distributions have been pro-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "posed. A general and widely used estimator is REINFORCE (Williams, 1992). Biased gradients", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "based on a continuous relaxations of the discrete distribution (known as Gumbel-Softmax or Con-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 681, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 692 + ], + "score": 1.0, + "content": "crete) were jointly introduced by Jang et al. (2016) and Maddison et al. (2016). These can be com-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "bined with the straight through estimator (Bengio et al., 2013) if the model requires discrete samples", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "score": 1.0, + "content": "or be used to construct control variates for REINFORCE, as in REBAR (Tucker et al., 2017) or", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 647, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "RELAX (Grathwohl et al., 2018). Many other methods use control variates and other techniques", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "to reduce the variance of REINFORCE (Paisley et al., 2012; Ranganath et al., 2014; Gregor et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 390, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 390, + 117 + ], + "score": 1.0, + "content": "2014; Mnih & Gregor, 2014; Gu et al., 2016; Mnih & Rezende, 2016).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "Some works rely on explicit summation of the expectation, either for the marginal distribution (Tit-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "sias & Lazaro-Gredilla, 2015) or globally summing some categories while sampling from the re- ´", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "mainder (Liang et al., 2018; Liu et al., 2019). Other approaches use a finite difference approxima-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "tion to the gradient (Lorberbom et al., 2018; 2019). Yin et al. (2019) introduced ARSM, which uses", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 450, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 450, + 179 + ], + "score": 1.0, + "content": "multiple model evaluations where the number adapts automatically to the uncertainty.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "In the structured prediction setting, there are many algorithms for optimizing a quantity under a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "sequence of discrete decisions, using (weak) supervision, multiple samples (or deterministic model", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "score": 1.0, + "content": "evaluations), or a combination both (Ranzato et al., 2016; Shen et al., 2016; He et al., 2016; Norouzi", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "et al., 2016; Bahdanau et al., 2017; Edunov et al., 2018; Leblond et al., 2018; Negrinho et al., 2018).", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "Most of these algorithms are biased and rely on pretraining using maximum likelihood or gradually", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "transitioning from supervised to reinforcement learning. 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Many other methods use control variates and other techniques", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "to reduce the variance of REINFORCE (Paisley et al., 2012; Ranganath et al., 2014; Gregor et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 390, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 390, + 117 + ], + "score": 1.0, + "content": "2014; Mnih & Gregor, 2014; Gu et al., 2016; Mnih & Rezende, 2016).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "Some works rely on explicit summation of the expectation, either for the marginal distribution (Tit-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "sias & Lazaro-Gredilla, 2015) or globally summing some categories while sampling from the re- ´", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "mainder (Liang et al., 2018; Liu et al., 2019). Other approaches use a finite difference approxima-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "tion to the gradient (Lorberbom et al., 2018; 2019). Yin et al. (2019) introduced ARSM, which uses", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 450, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 450, + 179 + ], + "score": 1.0, + "content": "multiple model evaluations where the number adapts automatically to the uncertainty.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 122, + 505, + 179 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "In the structured prediction setting, there are many algorithms for optimizing a quantity under a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "sequence of discrete decisions, using (weak) supervision, multiple samples (or deterministic model", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "score": 1.0, + "content": "evaluations), or a combination both (Ranzato et al., 2016; Shen et al., 2016; He et al., 2016; Norouzi", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "et al., 2016; Bahdanau et al., 2017; Edunov et al., 2018; Leblond et al., 2018; Negrinho et al., 2018).", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "Most of these algorithms are biased and rely on pretraining using maximum likelihood or gradually", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "transitioning from supervised to reinforcement learning. Using Gumbel-Softmax based approaches", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "in a sequential setting is difficult as the bias accumulates because of mixing errors (Gu et al., 2018).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 182, + 506, + 261 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 277, + 208, + 289 + ], + "lines": [ + { + "bbox": [ + 104, + 275, + 209, + 292 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 209, + 292 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 504, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 286, + 315 + ], + "score": 1.0, + "content": "Throughout this paper, we will denote with", + "type": "text" + }, + { + "bbox": [ + 287, + 302, + 300, + 313 + ], + "score": 0.87, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 302, + 497, + 315 + ], + "score": 1.0, + "content": "an ordered sample without replacement of size", + "type": "text" + }, + { + "bbox": [ + 497, + 303, + 504, + 313 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 429, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 144, + 328 + ], + "score": 1.0, + "content": "and with", + "type": "text" + }, + { + "bbox": [ + 144, + 313, + 156, + 324 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 312, + 276, + 328 + ], + "score": 1.0, + "content": "an unordered sample (of size", + "type": "text" + }, + { + "bbox": [ + 276, + 315, + 282, + 324 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 312, + 418, + 328 + ], + "score": 1.0, + "content": ") from the categorical distribution", + "type": "text" + }, + { + "bbox": [ + 418, + 316, + 424, + 326 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 312, + 429, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 302, + 504, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 338, + 504, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 441, + 352 + ], + "score": 1.0, + "content": "Restricted distribution. 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Then let", + "type": "text" + }, + { + "bbox": [ + 284, + 185, + 372, + 197 + ], + "score": 0.86, + "content": "b _ { 1 } = \\arg \\operatorname* { m a x } _ { i \\in D } g _ { \\phi _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 183, + 375, + 199 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 375, + 186, + 483, + 198 + ], + "score": 0.83, + "content": "b _ { 2 } = \\arg \\operatorname* { m a x } _ { i \\in D \\backslash \\{ b _ { 1 } \\} } g _ { \\phi _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 183, + 506, + 199 + ], + "score": 1.0, + "content": ", etc.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 196, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 196, + 117, + 211 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 118, + 197, + 131, + 208 + ], + "score": 0.88, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 196, + 169, + 211 + ], + "score": 1.0, + "content": "is the top", + "type": "text" + }, + { + "bbox": [ + 170, + 198, + 176, + 208 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 196, + 506, + 211 + ], + "score": 1.0, + "content": "of the perturbed log-probabilities in decreasing order. The probability of obtaining", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 107, + 209, + 120, + 219 + ], + "score": 0.89, + "content": "B _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "using this procedure is given by equation 4, so this provides an alternative sampling method", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "which is effectively a (non-differentiable) reparameterization of sampling without replacement. For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 348, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 348, + 243 + ], + "score": 1.0, + "content": "a differentiable reparameterization, see Grover et al. 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This way of sampling underlies the efficient computation of", + "type": "text" + }, + { + "bbox": [ + 409, + 258, + 434, + 270 + ], + "score": 0.93, + "content": "p ( S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 256, + 500, + 272 + ], + "score": 1.0, + "content": "in Appendix B.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 285, + 209, + 298 + ], + "lines": [ + { + "bbox": [ + 104, + 284, + 211, + 302 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 211, + 302 + ], + "score": 1.0, + "content": "3 METHODOLOGY", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "In this section, we derive the unordered set policy gradient estimator: a low-variance, unbiased", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 160, + 335 + ], + "score": 1.0, + "content": "estimator of", + "type": "text" + }, + { + "bbox": [ + 160, + 321, + 226, + 334 + ], + "score": 0.93, + "content": "\\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ f ( \\pmb { x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 320, + 444, + 335 + ], + "score": 1.0, + "content": "based on an unordered sample without replacement", + "type": "text" + }, + { + "bbox": [ + 445, + 321, + 457, + 332 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 320, + 506, + 335 + ], + "score": 1.0, + "content": ". First, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 292, + 346 + ], + "score": 1.0, + "content": "derive the generic (non-gradient) estimator for", + "type": "text" + }, + { + "bbox": [ + 293, + 333, + 325, + 345 + ], + "score": 0.93, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "as the Rao-Blackwellized version of a single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "sample Monte Carlo estimator (and two other estimators!). Then we combine this estimator with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 354, + 496, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 496, + 369 + ], + "score": 1.0, + "content": "REINFORCE (Williams, 1992) and we show how to reduce its variance using a built-in baseline.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 400, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 401, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 401, + 393 + ], + "score": 1.0, + "content": "3.1 RAO-BLACKWELLIZATION OF THE SINGLE SAMPLE ESTIMATOR", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 263, + 413 + ], + "score": 1.0, + "content": "A very crude but simple estimator for", + "type": "text" + }, + { + "bbox": [ + 263, + 401, + 295, + 413 + ], + "score": 0.93, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 399, + 417, + 413 + ], + "score": 1.0, + "content": "based on the ordered sample", + "type": "text" + }, + { + "bbox": [ + 417, + 400, + 431, + 411 + ], + "score": 0.89, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "is to only use the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 158, + 424 + ], + "score": 1.0, + "content": "first element", + "type": "text" + }, + { + "bbox": [ + 159, + 412, + 168, + 423 + ], + "score": 0.87, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 411, + 384, + 424 + ], + "score": 1.0, + "content": ", which by definition is a sample from the distribution", + "type": "text" + }, + { + "bbox": [ + 384, + 414, + 390, + 423 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 411, + 505, + 424 + ], + "score": 1.0, + "content": ". We define this estimator as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 423, + 320, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 320, + 434 + ], + "score": 1.0, + "content": "the single sample estimator, which is unbiased, since", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 438, + 420, + 453 + ], + "lines": [ + { + "bbox": [ + 190, + 438, + 420, + 453 + ], + "spans": [ + { + "bbox": [ + 190, + 438, + 420, + 453 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { x \\sim p ( x ) } [ f ( x ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "02f2b3f7b50f0fc7fea7eafa76908daeb1becae1ddbe2c49d64505835578994c.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 190, + 438, + 420, + 453 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Discarding all but one sample, the single sample estimator is inefficient, but we can use Rao-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 212, + 485 + ], + "score": 1.0, + "content": "Blackwellization (Casella", + "type": "text" + }, + { + "bbox": [ + 213, + 474, + 221, + 483 + ], + "score": 0.4, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "Robert, 1996) to signficantly improve it. To this end, we consider the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 155, + 498 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 156, + 484, + 183, + 496 + ], + "score": 0.91, + "content": "B ^ { k } | S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 482, + 356, + 498 + ], + "score": 1.0, + "content": ", which is, knowing the unordered sample", + "type": "text" + }, + { + "bbox": [ + 356, + 484, + 368, + 494 + ], + "score": 0.89, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 482, + 506, + 498 + ], + "score": 1.0, + "content": ", the conditional distribution over", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 174, + 509 + ], + "score": 1.0, + "content": "ordered samples", + "type": "text" + }, + { + "bbox": [ + 174, + 495, + 226, + 507 + ], + "score": 0.93, + "content": "B ^ { k } \\in B ( S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 493, + 332, + 509 + ], + "score": 1.0, + "content": "that could have generated", + "type": "text" + }, + { + "bbox": [ + 332, + 495, + 344, + 505 + ], + "score": 0.85, + "content": "S ^ { \\hat { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 493, + 379, + 509 + ], + "score": 1.0, + "content": ".2 Using", + "type": "text" + }, + { + "bbox": [ + 380, + 495, + 407, + 507 + ], + "score": 0.92, + "content": "B ^ { k } | S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 493, + 456, + 509 + ], + "score": 1.0, + "content": ", we rewrite", + "type": "text" + }, + { + "bbox": [ + 457, + 495, + 492, + 507 + ], + "score": 0.92, + "content": "\\mathbb { E } [ f ( b _ { 1 } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 493, + 505, + 509 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 511, + 495, + 527 + ], + "lines": [ + { + "bbox": [ + 115, + 511, + 495, + 527 + ], + "spans": [ + { + "bbox": [ + 115, + 511, + 495, + 527 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { S ^ { k } \\sim p ( S ^ { k } ) } \\left[ \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } \\mid S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] \\right] = \\mathbb { E } _ { S ^ { k } \\sim p ( S ^ { k } ) } \\left[ \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } \\mid S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "266311ced4be4415de977a8b098b3fa79a51a05a8a5ba8855a81cddf903a16de.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 115, + 511, + 495, + 527 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 503, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "The Rao-Blackwellized version of the single sample estimator computes the inner conditional ex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 546, + 487, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 204, + 560 + ], + "score": 1.0, + "content": "pectation exactly. Since", + "type": "text" + }, + { + "bbox": [ + 204, + 546, + 218, + 557 + ], + "score": 0.89, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 546, + 287, + 560 + ], + "score": 1.0, + "content": "is an ordering of", + "type": "text" + }, + { + "bbox": [ + 288, + 546, + 299, + 557 + ], + "score": 0.89, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 546, + 339, + 560 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 339, + 546, + 372, + 558 + ], + "score": 0.93, + "content": "b _ { 1 } \\in S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 546, + 487, + 560 + ], + "score": 1.0, + "content": "and we can compute this as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 562, + 402, + 590 + ], + "lines": [ + { + "bbox": [ + 209, + 562, + 402, + 590 + ], + "spans": [ + { + "bbox": [ + 209, + 562, + 402, + 590 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s )", + "type": "interline_equation", + "image_path": "596870a40feaefe9b1a493c1183747aa6ee26d2fce368c656b98b1f97b2dccf6.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 209, + 562, + 402, + 590 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 253, + 609 + ], + "score": 1.0, + "content": "where, in a slight abuse of notation,", + "type": "text" + }, + { + "bbox": [ + 253, + 595, + 312, + 608 + ], + "score": 0.94, + "content": "P ( b _ { 1 } = s | S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "is the probability that the first sampled element", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 116, + 618 + ], + "score": 0.87, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 606, + 179, + 619 + ], + "score": 1.0, + "content": "takes the value", + "type": "text" + }, + { + "bbox": [ + 180, + 610, + 185, + 617 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 606, + 308, + 619 + ], + "score": 1.0, + "content": ", given that the complete set of", + "type": "text" + }, + { + "bbox": [ + 309, + 608, + 315, + 617 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 606, + 360, + 619 + ], + "score": 1.0, + "content": "samples is", + "type": "text" + }, + { + "bbox": [ + 361, + 606, + 373, + 617 + ], + "score": 0.89, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 606, + 505, + 619 + ], + "score": 1.0, + "content": ". 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\\setminus C } ( S ^ { k } ) = \\left( \\prod _ { s \\in S ^ { k } } p ( s ) \\right) \\cdot \\sum _ { B ^ { k } \\in \\mathcal { B } ( S ^ { k } ) } \\prod _ { i = 1 } ^ { k } \\frac { 1 } { 1 - \\sum _ { c \\in C } p ( c ) - \\sum _ { j < i } p ( b _ { j } ) } .", + "type": "interline_equation", + "image_path": "f1416a090c7000a0924172bf9118f60f0a3e8afda4a60413536211213eafb11e.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 165, + 98, + 444, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 165, + 112.0, + 444, + 126.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 165, + 126.0, + 444, + 140.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 506, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 183, + 164 + ], + "score": 1.0, + "content": "The Gumbel-Top-", + "type": "text" + }, + { + "bbox": [ + 183, + 152, + 190, + 162 + 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As an alternative to sequential sampling, we can also sample", + "type": "text" + }, + { + "bbox": [ + 473, + 151, + 486, + 162 + ], + "score": 0.89, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 150, + 506, + 164 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 161, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 107, + 162, + 119, + 173 + ], + "score": 0.87, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 161, + 187, + 176 + ], + "score": 1.0, + "content": "by taking the top", + "type": "text" + }, + { + "bbox": [ + 188, + 163, + 195, + 173 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 161, + 506, + 176 + ], + "score": 1.0, + "content": "of Gumbel variables (Yellott, 1977; Vieira, 2014; Kim et al., 2016). Following", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 172, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 104, + 172, + 212, + 187 + ], + "score": 1.0, + "content": "notation from Kool et al.", + "type": "text" + }, + { + "bbox": [ + 212, + 174, + 243, + 185 + ], + "score": 0.27, + "content": "( 2 0 1 9 \\mathrm { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 172, + 412, + 187 + ], + "score": 1.0, + "content": ", we define the perturbed log-probability", + "type": "text" + }, + { + "bbox": [ + 412, + 174, + 473, + 186 + ], + "score": 0.91, + "content": "g _ { \\phi _ { i } } = \\phi _ { i } + g _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 172, + 506, + 187 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 183, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 107, + 184, + 161, + 196 + ], + "score": 0.92, + "content": "\\phi _ { i } = \\log p ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 183, + 178, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 185, + 245, + 196 + ], + "score": 0.79, + "content": "g _ { i } \\sim \\mathrm { G u m b e l } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 183, + 284, + 199 + ], + "score": 1.0, + "content": ". Then let", + "type": "text" + }, + { + "bbox": [ + 284, + 185, + 372, + 197 + ], + "score": 0.86, + "content": "b _ { 1 } = \\arg \\operatorname* { m a x } _ { i \\in D } g _ { \\phi _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 183, + 375, + 199 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 375, + 186, + 483, + 198 + ], + "score": 0.83, + "content": "b _ { 2 } = \\arg \\operatorname* { m a x } _ { i \\in D \\backslash \\{ b _ { 1 } \\} } g _ { \\phi _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 183, + 506, + 199 + ], + "score": 1.0, + "content": ", etc.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 196, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 196, + 117, + 211 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 118, + 197, + 131, + 208 + ], + "score": 0.88, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 196, + 169, + 211 + ], + "score": 1.0, + "content": "is the top", + "type": "text" + }, + { + "bbox": [ + 170, + 198, + 176, + 208 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 196, + 506, + 211 + ], + "score": 1.0, + "content": "of the perturbed log-probabilities in decreasing order. The probability of obtaining", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 107, + 209, + 120, + 219 + ], + "score": 0.89, + "content": "B _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "using this procedure is given by equation 4, so this provides an alternative sampling method", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "which is effectively a (non-differentiable) reparameterization of sampling without replacement. For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 348, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 348, + 243 + ], + "score": 1.0, + "content": "a differentiable reparameterization, see Grover et al. (2019).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 150, + 506, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 225, + 260 + ], + "score": 1.0, + "content": "It follows that taking the top", + "type": "text" + }, + { + "bbox": [ + 225, + 248, + 232, + 257 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "perturbed log-probabilities without order, we obtain the unordered", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 500, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 151, + 272 + ], + "score": 1.0, + "content": "sample set", + "type": "text" + }, + { + "bbox": [ + 151, + 258, + 163, + 268 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 256, + 408, + 272 + ], + "score": 1.0, + "content": ". This way of sampling underlies the efficient computation of", + "type": "text" + }, + { + "bbox": [ + 409, + 258, + 434, + 270 + ], + "score": 0.93, + "content": "p ( S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 256, + 500, + 272 + ], + "score": 1.0, + "content": "in Appendix B.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 247, + 505, + 272 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 285, + 209, + 298 + ], + "lines": [ + { + "bbox": [ + 104, + 284, + 211, + 302 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 211, + 302 + ], + "score": 1.0, + "content": "3 METHODOLOGY", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "In this section, we derive the unordered set policy gradient estimator: a low-variance, unbiased", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 160, + 335 + ], + "score": 1.0, + "content": "estimator of", + "type": "text" + }, + { + "bbox": [ + 160, + 321, + 226, + 334 + ], + "score": 0.93, + "content": "\\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ f ( \\pmb { x } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 320, + 444, + 335 + ], + "score": 1.0, + "content": "based on an unordered sample without replacement", + "type": "text" + }, + { + "bbox": [ + 445, + 321, + 457, + 332 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 320, + 506, + 335 + ], + "score": 1.0, + "content": ". First, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 292, + 346 + ], + "score": 1.0, + "content": "derive the generic (non-gradient) estimator for", + "type": "text" + }, + { + "bbox": [ + 293, + 333, + 325, + 345 + ], + "score": 0.93, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "as the Rao-Blackwellized version of a single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "sample Monte Carlo estimator (and two other estimators!). Then we combine this estimator with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 354, + 496, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 496, + 369 + ], + "score": 1.0, + "content": "REINFORCE (Williams, 1992) and we show how to reduce its variance using a built-in baseline.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 310, + 506, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 400, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 401, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 401, + 393 + ], + "score": 1.0, + "content": "3.1 RAO-BLACKWELLIZATION OF THE SINGLE SAMPLE ESTIMATOR", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 263, + 413 + ], + "score": 1.0, + "content": "A very crude but simple estimator for", + "type": "text" + }, + { + "bbox": [ + 263, + 401, + 295, + 413 + ], + "score": 0.93, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 399, + 417, + 413 + ], + "score": 1.0, + "content": "based on the ordered sample", + "type": "text" + }, + { + "bbox": [ + 417, + 400, + 431, + 411 + ], + "score": 0.89, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "is to only use the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 158, + 424 + ], + "score": 1.0, + "content": "first element", + "type": "text" + }, + { + "bbox": [ + 159, + 412, + 168, + 423 + ], + "score": 0.87, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 411, + 384, + 424 + ], + "score": 1.0, + "content": ", which by definition is a sample from the distribution", + "type": "text" + }, + { + "bbox": [ + 384, + 414, + 390, + 423 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 411, + 505, + 424 + ], + "score": 1.0, + "content": ". We define this estimator as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 423, + 320, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 320, + 434 + ], + "score": 1.0, + "content": "the single sample estimator, which is unbiased, since", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 399, + 505, + 434 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 438, + 420, + 453 + ], + "lines": [ + { + "bbox": [ + 190, + 438, + 420, + 453 + ], + "spans": [ + { + "bbox": [ + 190, + 438, + 420, + 453 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { x \\sim p ( x ) } [ f ( x ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "02f2b3f7b50f0fc7fea7eafa76908daeb1becae1ddbe2c49d64505835578994c.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 190, + 438, + 420, + 453 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Discarding all but one sample, the single sample estimator is inefficient, but we can use Rao-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 212, + 485 + ], + "score": 1.0, + "content": "Blackwellization (Casella", + "type": "text" + }, + { + "bbox": [ + 213, + 474, + 221, + 483 + ], + "score": 0.4, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "Robert, 1996) to signficantly improve it. To this end, we consider the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 155, + 498 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 156, + 484, + 183, + 496 + ], + "score": 0.91, + "content": "B ^ { k } | S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 482, + 356, + 498 + ], + "score": 1.0, + "content": ", which is, knowing the unordered sample", + "type": "text" + }, + { + "bbox": [ + 356, + 484, + 368, + 494 + ], + "score": 0.89, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 482, + 506, + 498 + ], + "score": 1.0, + "content": ", the conditional distribution over", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 174, + 509 + ], + "score": 1.0, + "content": "ordered samples", + "type": "text" + }, + { + "bbox": [ + 174, + 495, + 226, + 507 + ], + "score": 0.93, + "content": "B ^ { k } \\in B ( S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 493, + 332, + 509 + ], + "score": 1.0, + "content": "that could have generated", + "type": "text" + }, + { + "bbox": [ + 332, + 495, + 344, + 505 + ], + "score": 0.85, + "content": "S ^ { \\hat { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 493, + 379, + 509 + ], + "score": 1.0, + "content": ".2 Using", + "type": "text" + }, + { + "bbox": [ + 380, + 495, + 407, + 507 + ], + "score": 0.92, + "content": "B ^ { k } | S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 493, + 456, + 509 + ], + "score": 1.0, + "content": ", we rewrite", + "type": "text" + }, + { + "bbox": [ + 457, + 495, + 492, + 507 + ], + "score": 0.92, + "content": "\\mathbb { E } [ f ( b _ { 1 } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 493, + 505, + 509 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 461, + 506, + 509 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 511, + 495, + 527 + ], + "lines": [ + { + "bbox": [ + 115, + 511, + 495, + 527 + ], + "spans": [ + { + "bbox": [ + 115, + 511, + 495, + 527 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \\mathbb { E } _ { S ^ { k } \\sim p ( S ^ { k } ) } \\left[ \\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } \\mid S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] \\right] = \\mathbb { E } _ { S ^ { k } \\sim p ( S ^ { k } ) } \\left[ \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } \\mid S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "266311ced4be4415de977a8b098b3fa79a51a05a8a5ba8855a81cddf903a16de.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 115, + 511, + 495, + 527 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 503, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "The Rao-Blackwellized version of the single sample estimator computes the inner conditional ex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 546, + 487, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 204, + 560 + ], + "score": 1.0, + "content": "pectation exactly. Since", + "type": "text" + }, + { + "bbox": [ + 204, + 546, + 218, + 557 + ], + "score": 0.89, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 546, + 287, + 560 + ], + "score": 1.0, + "content": "is an ordering of", + "type": "text" + }, + { + "bbox": [ + 288, + 546, + 299, + 557 + ], + "score": 0.89, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 546, + 339, + 560 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 339, + 546, + 372, + 558 + ], + "score": 0.93, + "content": "b _ { 1 } \\in S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 546, + 487, + 560 + ], + "score": 1.0, + "content": "and we can compute this as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 536, + 505, + 560 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 562, + 402, + 590 + ], + "lines": [ + { + "bbox": [ + 209, + 562, + 402, + 590 + ], + "spans": [ + { + "bbox": [ + 209, + 562, + 402, + 590 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s )", + "type": "interline_equation", + "image_path": "596870a40feaefe9b1a493c1183747aa6ee26d2fce368c656b98b1f97b2dccf6.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 209, + 562, + 402, + 590 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 253, + 609 + ], + "score": 1.0, + "content": "where, in a slight abuse of notation,", + "type": "text" + }, + { + "bbox": [ + 253, + 595, + 312, + 608 + ], + "score": 0.94, + "content": "P ( b _ { 1 } = s | S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "is the probability that the first sampled element", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 116, + 618 + ], + "score": 0.87, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 606, + 179, + 619 + ], + "score": 1.0, + "content": "takes the value", + "type": "text" + }, + { + "bbox": [ + 180, + 610, + 185, + 617 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 606, + 308, + 619 + ], + "score": 1.0, + "content": ", given that the complete set of", + "type": "text" + }, + { + "bbox": [ + 309, + 608, + 315, + 617 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 606, + 360, + 619 + ], + "score": 1.0, + "content": "samples is", + "type": "text" + }, + { + "bbox": [ + 361, + 606, + 373, + 617 + ], + "score": 0.89, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 606, + 505, + 619 + ], + "score": 1.0, + "content": ". Using Bayes’ Theorem we find", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 106, + 595, + 505, + 619 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 622, + 444, + 651 + ], + "lines": [ + { + "bbox": [ + 166, + 622, + 444, + 651 + ], + "spans": [ + { + "bbox": [ + 166, + 622, + 444, + 651 + ], + "score": 0.94, + "content": "P ( b _ { 1 } = s | S ^ { k } ) = \\frac { p ( S ^ { k } | b _ { 1 } = s ) P ( b _ { 1 } = s ) } { p ( S ^ { k } ) } = \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s ) } { p ( S ^ { k } ) } .", + "type": "interline_equation", + "image_path": "1f9be78c95a7fbbee7be9b15c59a94a262dff1f16e5d8e1f33172e666357253e.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 166, + 622, + 444, + 631.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 166, + 631.6666666666666, + 444, + 641.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 166, + 641.3333333333333, + 444, + 650.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 104, + 654, + 507, + 670 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 145, + 670 + ], + "score": 1.0, + "content": "The step", + "type": "text" + }, + { + "bbox": [ + 145, + 655, + 293, + 669 + ], + "score": 0.92, + "content": "p ( S ^ { k } | b _ { 1 } = s ) = p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 654, + 507, + 670 + ], + "score": 1.0, + "content": "comes from analyzing sequential sampling without", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 320, + 680 + ], + "score": 1.0, + "content": "replacement: given that the first element sampled is", + "type": "text" + }, + { + "bbox": [ + 320, + 670, + 326, + 677 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 667, + 505, + 680 + ], + "score": 1.0, + "content": ", the remaining elements have a distribution", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 677, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 155, + 692 + ], + "score": 1.0, + "content": "restricted to", + "type": "text" + }, + { + "bbox": [ + 156, + 678, + 187, + 691 + ], + "score": 0.92, + "content": "D \\backslash \\{ s \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 677, + 240, + 692 + ], + "score": 1.0, + "content": ", so sampling", + "type": "text" + }, + { + "bbox": [ + 241, + 678, + 253, + 689 + ], + "score": 0.86, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 677, + 297, + 692 + ], + "score": 1.0, + "content": "(including", + "type": "text" + }, + { + "bbox": [ + 298, + 681, + 303, + 689 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 677, + 397, + 692 + ], + "score": 1.0, + "content": ") given the first element", + "type": "text" + }, + { + "bbox": [ + 397, + 681, + 403, + 689 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 677, + 505, + 692 + ], + "score": 1.0, + "content": "is equivalent to sampling", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 689, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 104, + 689, + 164, + 704 + ], + "score": 1.0, + "content": "the remainder", + "type": "text" + }, + { + "bbox": [ + 164, + 691, + 199, + 703 + ], + "score": 0.92, + "content": "S ^ { k } \\setminus \\{ s \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 689, + 414, + 704 + ], + "score": 1.0, + "content": "from the restricted distribution, which has probability", + "type": "text" + }, + { + "bbox": [ + 414, + 690, + 485, + 703 + ], + "score": 0.93, + "content": "p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 689, + 505, + 704 + ], + "score": 1.0, + "content": "(see", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 702, + 157, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 157, + 714 + ], + "score": 1.0, + "content": "equation 7).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 654, + 507, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 502, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 502, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 502, + 95 + ], + "score": 1.0, + "content": "The unordered set estimator. For notational convenience, we introduce the leave-one-out ratio.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 497, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 271, + 117 + ], + "score": 1.0, + "content": "Definition 1. 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We now", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 504, + 160 + ], + "score": 1.0, + "content": "define the unordered set estimator as the Rao-Blackwellized version of the single-sample estimator.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 162, + 312, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 312, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 312, + 176 + ], + "score": 1.0, + "content": "Theorem 1. The unordered set estimator, given by", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 178, + 376, + 207 + ], + "lines": [ + { + "bbox": [ + 235, + 178, + 376, + 207 + ], + "spans": [ + { + "bbox": [ + 235, + 178, + 376, + 207 + ], + "score": 0.95, + "content": "e ^ { U S } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s )", + "type": "interline_equation", + "image_path": "03ee959b283d9cfc0e39f81176d5ac72d29d6fce2c91e22ff52fc67c6b0db17c.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 178, + 376, + 192.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 235, + 192.5, + 376, + 207.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 410, + 224 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 411, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 411, + 225 + ], + "score": 1.0, + "content": "is the Rao-Blackwellized version of the (unbiased!) single sample estimator.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 379, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 379, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 162, + 249 + ], + "score": 1.0, + "content": "Proof. Using", + "type": "text" + }, + { + "bbox": [ + 163, + 234, + 287, + 248 + ], + "score": 0.92, + "content": "P ( b _ { 1 } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 234, + 379, + 249 + ], + "score": 1.0, + "content": "in equation 9 we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 252, + 455, + 280 + ], + "lines": [ + { + "bbox": [ + 155, + 252, + 455, + 280 + ], + "spans": [ + { + "bbox": [ + 155, + 252, + 455, + 280 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s ) = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "51b7f2077f8163e0ce349e2825cb373b1056003fe7e01d54c2ffc9c3c267dffe.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 155, + 252, + 455, + 280 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "The implication of this theorem is that the unordered set estimator, in explicit form given by equa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 246, + 333 + ], + "score": 1.0, + "content": "tion 11, is an unbiased estimator of", + "type": "text" + }, + { + "bbox": [ + 246, + 320, + 279, + 332 + ], + "score": 0.93, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "since it is the Rao-Blackwellized version of the unbiased", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "single sample estimator. Also, as expected by taking multiple samples, it has variance equal or lower", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 343, + 482, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 482, + 355 + ], + "score": 1.0, + "content": "than the single sample estimator by the Rao-Blackwell Theorem (Lehmann & Scheffe, 1950). ´", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 367, + 346, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 347, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 347, + 380 + ], + "score": 1.0, + "content": "3.2 RAO-BLACKWELLIZATION OF OTHER ESTIMATORS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 104, + 387, + 504, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "The unordered set estimator is also the result of Rao-Blackwellizing two other unbiased estimators:", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 398, + 433, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 433, + 412 + ], + "score": 1.0, + "content": "the stochastic sum-and-sample estimator and the importance-weighted estimator.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 504, + 445 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "score": 1.0, + "content": "The sum-and-sample estimator. We define as sum-and-sample estimator any estimator that relies", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 434, + 245, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 215, + 445 + ], + "score": 1.0, + "content": "on the identity that for any", + "type": "text" + }, + { + "bbox": [ + 215, + 434, + 245, + 444 + ], + "score": 0.88, + "content": "C \\subset D", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 450, + 451, + 484 + ], + "lines": [ + { + "bbox": [ + 159, + 450, + 451, + 484 + ], + "spans": [ + { + "bbox": [ + 159, + 450, + 451, + 484 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { x \\sim p ( x ) } [ f ( x ) ] = \\mathbb { E } _ { x \\sim p ^ { D \\setminus C } ( x ) } \\left[ \\sum _ { c \\in C } p ( c ) f ( c ) + \\left( 1 - \\sum _ { c \\in C } p ( c ) \\right) f ( x ) \\right] .", + "type": "interline_equation", + "image_path": "71c31bf9e4547b1b6d796ed465395787980be036d959b15057290c3d36ca596b.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 159, + 450, + 451, + 461.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 159, + 461.3333333333333, + 451, + 472.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 159, + 472.66666666666663, + 451, + 483.99999999999994 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "score": 1.0, + "content": "For the derivation, see Appendix C.1 or Liang et al. (2018); Liu et al. (2019). In general, a sum-and-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 500, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 241, + 513 + ], + "score": 1.0, + "content": "sample estimator with a budget of", + "type": "text" + }, + { + "bbox": [ + 242, + 501, + 266, + 510 + ], + "score": 0.89, + "content": "k > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 500, + 495, + 513 + ], + "score": 1.0, + "content": "evaluations sums expectation terms for a set of categories", + "type": "text" + }, + { + "bbox": [ + 495, + 501, + 504, + 510 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 124, + 524 + ], + "score": 1.0, + "content": "(s.t.", + "type": "text" + }, + { + "bbox": [ + 125, + 511, + 161, + 523 + ], + "score": 0.89, + "content": "| C | < k )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 510, + 312, + 524 + ], + "score": 1.0, + "content": "explicitly (e.g. selected by their value", + "type": "text" + }, + { + "bbox": [ + 312, + 513, + 319, + 523 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 510, + 454, + 524 + ], + "score": 1.0, + "content": "(Liang et al., 2018) or probability", + "type": "text" + }, + { + "bbox": [ + 455, + 513, + 461, + 523 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "(Liu et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 174, + 535 + ], + "score": 1.0, + "content": "2019)), and uses", + "type": "text" + }, + { + "bbox": [ + 174, + 523, + 205, + 534 + ], + "score": 0.91, + "content": "k - | C |", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 521, + 334, + 535 + ], + "score": 1.0, + "content": "(down-weighted) samples from", + "type": "text" + }, + { + "bbox": [ + 334, + 523, + 359, + 534 + ], + "score": 0.89, + "content": "D \\backslash C", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "to estimate the remaining terms. As", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 263, + 552 + ], + "score": 1.0, + "content": "is noted by Liu et al. (2019), selecting", + "type": "text" + }, + { + "bbox": [ + 263, + 537, + 272, + 547 + ], + "score": 0.78, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 531, + 313, + 552 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 313, + 534, + 363, + 551 + ], + "score": 0.95, + "content": "\\frac { 1 - \\sum _ { c \\in C } p ( c ) } { k - | C | }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 531, + 506, + 552 + ], + "score": 1.0, + "content": "is minimized guarantees to reduce", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 290, + 563 + ], + "score": 1.0, + "content": "variance compared to a standard minibatch of", + "type": "text" + }, + { + "bbox": [ + 290, + 552, + 296, + 560 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 550, + 453, + 563 + ], + "score": 1.0, + "content": "samples (which is equivalent to setting", + "type": "text" + }, + { + "bbox": [ + 453, + 550, + 480, + 561 + ], + "score": 0.87, + "content": "C = \\varnothing", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "). See", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 169, + 574 + ], + "score": 1.0, + "content": "also Fearnhead", + "type": "text" + }, + { + "bbox": [ + 169, + 562, + 178, + 572 + ], + "score": 0.36, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 561, + 358, + 574 + ], + "score": 1.0, + "content": "Clifford (2003) for a discussion on selecting", + "type": "text" + }, + { + "bbox": [ + 358, + 563, + 367, + 571 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "optimally. The ability to optimize", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 107, + 573, + 115, + 582 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 571, + 199, + 585 + ], + "score": 1.0, + "content": "depends on whether", + "type": "text" + }, + { + "bbox": [ + 200, + 573, + 217, + 584 + ], + "score": 0.92, + "content": "p ( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "can be computed efficiently a-priori (before sampling). This is difficult", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "score": 1.0, + "content": "in high-dimensional settings, e.g. sequence models which compute the probability incrementally", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 317, + 608 + ], + "score": 1.0, + "content": "while ancestral sampling. 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For notational convenience, we introduce the leave-one-out ratio.", + "type": "text" + } + ], + "index": 0, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 93, + 497, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 271, + 117 + ], + "score": 1.0, + "content": "Definition 1. The leave-one-out ratio of", + "type": "text" + }, + { + "bbox": [ + 271, + 103, + 277, + 110 + ], + "score": 0.73, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 93, + 329, + 117 + ], + "score": 1.0, + "content": "w.r.t. the set", + "type": "text" + }, + { + "bbox": [ + 329, + 101, + 337, + 110 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 93, + 384, + 117 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 384, + 96, + 493, + 115 + ], + "score": 0.93, + "content": "\\begin{array} { r } { R ( S ^ { k } , s ) = \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 93, + 497, + 117 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 502, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 104, + 122, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 122, + 210, + 140 + ], + "score": 1.0, + "content": "Rewriting equation 10 as", + "type": "text" + }, + { + "bbox": [ + 211, + 124, + 338, + 137 + ], + "score": 0.94, + "content": "P ( b _ { 1 } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 122, + 498, + 140 + ], + "score": 1.0, + "content": "shows that the probability of sampling", + "type": "text" + }, + { + "bbox": [ + 498, + 128, + 504, + 135 + ], + "score": 0.53, + "content": "s", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 150, + 149 + ], + "score": 1.0, + "content": "first, given", + "type": "text" + }, + { + "bbox": [ + 150, + 136, + 163, + 146 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 135, + 505, + 149 + ], + "score": 1.0, + "content": ", is simply the unconditional probability multiplied by the leave-one-out ratio. We now", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 504, + 160 + ], + "score": 1.0, + "content": "define the unordered set estimator as the Rao-Blackwellized version of the single-sample estimator.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 122, + 505, + 160 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 162, + 312, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 312, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 312, + 176 + ], + "score": 1.0, + "content": "Theorem 1. The unordered set estimator, given by", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 159, + 312, + 176 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 178, + 376, + 207 + ], + "lines": [ + { + "bbox": [ + 235, + 178, + 376, + 207 + ], + "spans": [ + { + "bbox": [ + 235, + 178, + 376, + 207 + ], + "score": 0.95, + "content": "e ^ { U S } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s )", + "type": "interline_equation", + "image_path": "03ee959b283d9cfc0e39f81176d5ac72d29d6fce2c91e22ff52fc67c6b0db17c.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 178, + 376, + 192.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 235, + 192.5, + 376, + 207.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 410, + 224 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 411, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 411, + 225 + ], + "score": 1.0, + "content": "is the Rao-Blackwellized version of the (unbiased!) single sample estimator.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 210, + 411, + 225 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 379, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 379, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 162, + 249 + ], + "score": 1.0, + "content": "Proof. Using", + "type": "text" + }, + { + "bbox": [ + 163, + 234, + 287, + 248 + ], + "score": 0.92, + "content": "P ( b _ { 1 } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 234, + 379, + 249 + ], + "score": 1.0, + "content": "in equation 9 we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 234, + 379, + 249 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 252, + 455, + 280 + ], + "lines": [ + { + "bbox": [ + 155, + 252, + 455, + 280 + ], + "spans": [ + { + "bbox": [ + 155, + 252, + 455, + 280 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s ) = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "51b7f2077f8163e0ce349e2825cb373b1056003fe7e01d54c2ffc9c3c267dffe.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 155, + 252, + 455, + 280 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "The implication of this theorem is that the unordered set estimator, in explicit form given by equa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 246, + 333 + ], + "score": 1.0, + "content": "tion 11, is an unbiased estimator of", + "type": "text" + }, + { + "bbox": [ + 246, + 320, + 279, + 332 + ], + "score": 0.93, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "since it is the Rao-Blackwellized version of the unbiased", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "single sample estimator. Also, as expected by taking multiple samples, it has variance equal or lower", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 343, + 482, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 482, + 355 + ], + "score": 1.0, + "content": "than the single sample estimator by the Rao-Blackwell Theorem (Lehmann & Scheffe, 1950). ´", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 308, + 506, + 355 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 367, + 346, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 347, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 347, + 380 + ], + "score": 1.0, + "content": "3.2 RAO-BLACKWELLIZATION OF OTHER ESTIMATORS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 104, + 387, + 504, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "The unordered set estimator is also the result of Rao-Blackwellizing two other unbiased estimators:", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 398, + 433, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 433, + 412 + ], + "score": 1.0, + "content": "the stochastic sum-and-sample estimator and the importance-weighted estimator.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 387, + 505, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 504, + 445 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "score": 1.0, + "content": "The sum-and-sample estimator. 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(2018); Liu et al. (2019). 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This is difficult", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "score": 1.0, + "content": "in high-dimensional settings, e.g. sequence models which compute the probability incrementally", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 317, + 608 + ], + "score": 1.0, + "content": "while ancestral sampling. An alternative is to select", + "type": "text" + }, + { + "bbox": [ + 318, + 595, + 326, + 604 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 592, + 505, + 608 + ], + "score": 1.0, + "content": "stochastically (as equation 13 holds for any", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 603, + 436, + 619 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 115, + 615 + ], + "score": 0.74, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 603, + 184, + 619 + ], + "score": 1.0, + "content": "), and we choose", + "type": "text" + }, + { + "bbox": [ + 184, + 605, + 229, + 615 + ], + "score": 0.91, + "content": "C = B ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 603, + 436, + 619 + ], + "score": 1.0, + "content": "to define the stochastic sum-and-sample estimator:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 489, + 506, + 619 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 622, + 421, + 662 + ], + "lines": [ + { + "bbox": [ + 190, + 622, + 421, + 662 + ], + "spans": [ + { + "bbox": [ + 190, + 622, + 421, + 662 + ], + "score": 0.95, + "content": "e ^ { { \\mathrm { S S A S } } } ( B ^ { k } ) = \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) \\right) f ( b _ { k } ) .", + "type": "interline_equation", + "image_path": "9ed8f24def9f93383a37e69286f2f5833e1c4dc8a9d8a9cf54ab1cede0ce92e5.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 622, + 421, + 642.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 190, + 642.0, + 421, + 662.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 506, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 309, + 678 + ], + "score": 1.0, + "content": "For simplicity, we consider the version that sums", + "type": "text" + }, + { + "bbox": [ + 310, + 666, + 334, + 677 + ], + "score": 0.88, + "content": "k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "terms here, but the following results also", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 219, + 690 + ], + "score": 1.0, + "content": "hold for a version that sums", + "type": "text" + }, + { + "bbox": [ + 220, + 678, + 246, + 687 + ], + "score": 0.89, + "content": "k - m", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 676, + 309, + 690 + ], + "score": 1.0, + "content": "terms and uses", + "type": "text" + }, + { + "bbox": [ + 309, + 681, + 318, + 686 + ], + "score": 0.78, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "samples (without replacement) (see Appendix", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 103, + 686, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 103, + 686, + 393, + 705 + ], + "score": 1.0, + "content": "C.3). Sampling without replacement, it holds that bk|Bk−1 ∼ pD\\Bk−1,", + "type": "text" + }, + { + "bbox": [ + 389, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "so the unbiasedness follows", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 103, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 103, + 699, + 317, + 714 + ], + "score": 1.0, + "content": "from equation 13 by separating the expectation over", + "type": "text" + }, + { + "bbox": [ + 317, + 702, + 330, + 711 + ], + "score": 0.87, + "content": "B ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 699, + 422, + 714 + ], + "score": 1.0, + "content": "into expectations over", + "type": "text" + }, + { + "bbox": [ + 422, + 700, + 446, + 711 + ], + "score": 0.91, + "content": "B ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 699, + 464, + 714 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 700, + 500, + 713 + ], + "score": 0.93, + "content": "b _ { k } | B ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 699, + 506, + 714 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 103, + 666, + 506, + 714 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 718, + 482, + 734 + ], + "lines": [ + { + "bbox": [ + 128, + 718, + 482, + 734 + ], + "spans": [ + { + "bbox": [ + 128, + 718, + 482, + 734 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } _ { B ^ { k - 1 } \\sim p ( B ^ { k - 1 } ) } \\left[ \\mathbb { E } _ { b _ { k } \\sim p ( b _ { k } | B ^ { k - 1 } ) } \\left[ e ^ { S S A S } ( B ^ { k } ) \\right] \\right] = \\mathbb { E } _ { B ^ { k - 1 } \\sim p ( B ^ { k - 1 } ) } \\left[ \\mathbb { E } [ f ( x ) ] \\right] = \\mathbb { E } [ f ( x ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "1619865ccf403398a0293e9ef050ffb2ee63a39836b62b1c47416adfc582475b.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 128, + 718, + 482, + 734 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "In general, a sum-and-sample estimator reduces variance if the probability mass is concentrated on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "the summed categories. As typically high probability categories are sampled first, the stochastic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "sum-and-sample estimator sums high probability categories, similar to the estimator by Liu et al.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "(2019) which we refer to as the deterministic sum-and-sample estimator. As we show in Appendix", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "C.2, Rao-Blackwellizing the stochastic sum-and-sample estimator also results in the unordered set", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 299, + 149 + ], + "score": 1.0, + "content": "estimator. This even holds for a version that uses", + "type": "text" + }, + { + "bbox": [ + 299, + 139, + 309, + 147 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 137, + 360, + 149 + ], + "score": 1.0, + "content": "samples and", + "type": "text" + }, + { + "bbox": [ + 361, + 138, + 385, + 148 + ], + "score": 0.91, + "content": "k - m", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "summed terms (see Appendix", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "C.3), which means that the unordered set estimator has equal or lower variance than the optimal (in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 488, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 142, + 172 + ], + "score": 1.0, + "content": "terms of", + "type": "text" + }, + { + "bbox": [ + 142, + 161, + 152, + 169 + ], + "score": 0.75, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 159, + 474, + 172 + ], + "score": 1.0, + "content": ") stochastic sum-and-sample estimator, but conveniently does not need to choose", + "type": "text" + }, + { + "bbox": [ + 475, + 161, + 484, + 169 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 159, + 488, + 172 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 482, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 484, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 484, + 197 + ], + "score": 1.0, + "content": "The importance-weighted estimator. The importance-weighted estimator (Vieira, 2017) is", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 200, + 371, + 231 + ], + "lines": [ + { + "bbox": [ + 240, + 200, + 371, + 231 + ], + "spans": [ + { + "bbox": [ + 240, + 200, + 371, + 231 + ], + "score": 0.95, + "content": "e ^ { \\mathrm { I W } } ( S ^ { k } , \\kappa ) = \\sum _ { s \\in S ^ { k } } { \\frac { p ( s ) } { q ( s , \\kappa ) } } f ( s ) .", + "type": "interline_equation", + "image_path": "8b61001740dc7d42db3c377579f08d9f93cdff9e7e3528a276595849d54f819e.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 200, + 371, + 215.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 240, + 215.5, + 371, + 231.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "This estimator is based on the idea of priority sampling (Duffield et al., 2007). It does not use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 394, + 260 + ], + "score": 1.0, + "content": "the order of the sample, but assumes sampling using the Gumbel-Top-", + "type": "text" + }, + { + "bbox": [ + 394, + 248, + 401, + 258 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "trick and requires access", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 117, + 271 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 261, + 125, + 268 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 259, + 145, + 271 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 146, + 258, + 178, + 270 + ], + "score": 0.89, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "-th largest perturbed log-probability, which can be seen as the ‘threshold’ since", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 268, + 507, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 189, + 281 + ], + "score": 0.75, + "content": "g _ { \\phi _ { s } } ~ > ~ \\kappa ~ \\forall s ~ \\in ~ S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 268, + 198, + 283 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 198, + 269, + 298, + 282 + ], + "score": 0.88, + "content": "\\bar { q ( s , a ) } = P ( g _ { \\phi _ { s } } > a )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 268, + 507, + 283 + ], + "score": 1.0, + "content": "can be interpreted as the inclusion probability of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 281, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 137, + 293 + ], + "score": 0.89, + "content": "s \\in S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 281, + 253, + 294 + ], + "score": 1.0, + "content": "(assuming a fixed threshold", + "type": "text" + }, + { + "bbox": [ + 254, + 284, + 260, + 292 + ], + "score": 0.73, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 281, + 384, + 294 + ], + "score": 1.0, + "content": "instead of a fixed sample size", + "type": "text" + }, + { + "bbox": [ + 384, + 282, + 390, + 292 + ], + "score": 0.7, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 281, + 507, + 294 + ], + "score": 1.0, + "content": "). For details and a proof of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "unbiasedness, see Vieira (2017) or Kool et al. (2019c). As the estimator has high variance, Kool", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "et al. (2019c) resort to normalizing the importance weights, resulting in biased estimates. Instead,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 342, + 327 + ], + "score": 1.0, + "content": "we use Rao-Blackwellization to eliminate stochasticity by", + "type": "text" + }, + { + "bbox": [ + 342, + 317, + 349, + 325 + ], + "score": 0.69, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 315, + 506, + 327 + ], + "score": 1.0, + "content": ". Again, the result is the unordered set", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 389, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 389, + 338 + ], + "score": 1.0, + "content": "estimator (see Appendix D.1), which thus has equal or lower variance.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 351, + 358, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 350, + 358, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 358, + 363 + ], + "score": 1.0, + "content": "3.3 THE UNORDERED SET POLICY GRADIENT ESTIMATOR", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 504, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 139, + 384 + ], + "score": 1.0, + "content": "Writing", + "type": "text" + }, + { + "bbox": [ + 140, + 373, + 151, + 383 + ], + "score": 0.84, + "content": "p _ { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 371, + 358, + 384 + ], + "score": 1.0, + "content": "to indicate the dependency on the model parameters", + "type": "text" + }, + { + "bbox": [ + 358, + 372, + 365, + 381 + ], + "score": 0.77, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 371, + 505, + 384 + ], + "score": 1.0, + "content": ", we can combine the unordered set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "score": 1.0, + "content": "estimator with REINFORCE (Williams, 1992) to obtain the unordered set policy gradient estimator.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 378, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 378, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 378, + 412 + ], + "score": 1.0, + "content": "Corollary 1. The unordered set policy gradient estimator, given by", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 130, + 414, + 464, + 442 + ], + "lines": [ + { + "bbox": [ + 130, + 414, + 464, + 442 + ], + "spans": [ + { + "bbox": [ + 130, + 414, + 464, + 442 + ], + "score": 0.92, + "content": "e ^ { U S P G } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } \\log p _ { \\theta } ( s ) f ( s ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) f ( s ) ,", + "type": "interline_equation", + "image_path": "1dbc150d5df5f608b513f6396d6dc46e4c366286427d7ad9da89ab67b5e56ddd.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 130, + 414, + 464, + 423.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 130, + 423.3333333333333, + 464, + 432.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 130, + 432.66666666666663, + 464, + 441.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 291, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 292, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 292, + 460 + ], + "score": 1.0, + "content": "is an unbiased estimate of the policy gradient.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 471, + 497, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 498, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 498, + 485 + ], + "score": 1.0, + "content": "Proof. Using REINFORCE (Williams, 1992) combined with the unordered set estimator we find:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 488, + 506, + 528 + ], + "lines": [ + { + "bbox": [ + 111, + 488, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 111, + 488, + 506, + 528 + ], + "score": 0.93, + "content": "\\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } ( x ) } [ f ( x ) ] = \\mathbb { E } _ { p _ { \\theta } ( x ) } [ \\nabla _ { \\theta } \\log p _ { \\theta } ( x ) f ( x ) ] = \\mathbb { E } _ { S ^ { k } \\sim p _ { \\theta } ( S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } \\log p _ { \\theta } ( s ) f ( s ) \\right] .", + "type": "interline_equation", + "image_path": "aab23b5a4539c051761f39cd7a8e17024373c54bc3fc29e240863c4159a74352.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 111, + 488, + 506, + 501.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 111, + 501.3333333333333, + 506, + 514.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 111, + 514.6666666666666, + 506, + 528.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "Variance reduction using a built-in control variate. The variance of REINFORCE can be re-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 254, + 581 + ], + "score": 1.0, + "content": "duced by subtracting a baseline from", + "type": "text" + }, + { + "bbox": [ + 255, + 569, + 262, + 580 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 567, + 505, + 581 + ], + "score": 1.0, + "content": ". When taking multiple samples (with replacement), a simple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "and effective baseline is to take the mean of other (independent!) samples (Mnih & Rezende, 2016).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "Sampling without replacement, we can use the same idea to construct a baseline based on the other", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 601, + 430, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 430, + 613 + ], + "score": 1.0, + "content": "samples, but we have to correct for the fact that the samples are not independent.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 430, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 430, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 430, + 630 + ], + "score": 1.0, + "content": "Theorem 2. The unordered set policy gradient estimator with baseline, given by", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 632, + 469, + 672 + ], + "lines": [ + { + "bbox": [ + 125, + 632, + 469, + 672 + ], + "spans": [ + { + "bbox": [ + 125, + 632, + 469, + 672 + ], + "score": 0.94, + "content": "e ^ { U S P G B L } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\left( f ( s ) - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right) ,", + "type": "interline_equation", + "image_path": "2bbfebe328fefece3095174414fb002d79ec5114db4b9f927fd8cd5b4c3d72fe.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 125, + 632, + 469, + 645.3333333333334 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 125, + 645.3333333333334, + 469, + 658.6666666666667 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 125, + 658.6666666666667, + 469, + 672.0000000000001 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 132, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 135, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 135, + 690 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 685, + 392, + 718 + ], + "lines": [ + { + "bbox": [ + 219, + 685, + 392, + 718 + ], + "spans": [ + { + "bbox": [ + 219, + 685, + 392, + 718 + ], + "score": 0.94, + "content": "R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) = { \\frac { p _ { \\theta } ^ { D \\setminus \\{ s , s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s , s ^ { \\prime } \\} ) } { p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } }", + "type": "interline_equation", + "image_path": "6c8b0893165f90a463454da2a941978b46de124e3b5b3602947a4a6ac32e8a96.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 685, + 392, + 701.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 219, + 701.5, + 392, + 718.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 721, + 449, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 720, + 450, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 450, + 733 + ], + "score": 1.0, + "content": "is the second order leave-one-out ratio, is an unbiased estimate of the policy gradient.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 533, + 504, + 543 + ], + "lines": [ + { + "bbox": [ + 496, + 534, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 496, + 534, + 504, + 543 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "In general, a sum-and-sample estimator reduces variance if the probability mass is concentrated on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "the summed categories. As typically high probability categories are sampled first, the stochastic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "sum-and-sample estimator sums high probability categories, similar to the estimator by Liu et al.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "(2019) which we refer to as the deterministic sum-and-sample estimator. As we show in Appendix", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "C.2, Rao-Blackwellizing the stochastic sum-and-sample estimator also results in the unordered set", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 299, + 149 + ], + "score": 1.0, + "content": "estimator. This even holds for a version that uses", + "type": "text" + }, + { + "bbox": [ + 299, + 139, + 309, + 147 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 137, + 360, + 149 + ], + "score": 1.0, + "content": "samples and", + "type": "text" + }, + { + "bbox": [ + 361, + 138, + 385, + 148 + ], + "score": 0.91, + "content": "k - m", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "summed terms (see Appendix", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "C.3), which means that the unordered set estimator has equal or lower variance than the optimal (in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 488, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 142, + 172 + ], + "score": 1.0, + "content": "terms of", + "type": "text" + }, + { + "bbox": [ + 142, + 161, + 152, + 169 + ], + "score": 0.75, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 159, + 474, + 172 + ], + "score": 1.0, + "content": ") stochastic sum-and-sample estimator, but conveniently does not need to choose", + "type": "text" + }, + { + "bbox": [ + 475, + 161, + 484, + 169 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 159, + 488, + 172 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 83, + 506, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 482, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 484, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 484, + 197 + ], + "score": 1.0, + "content": "The importance-weighted estimator. The importance-weighted estimator (Vieira, 2017) is", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 181, + 484, + 197 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 200, + 371, + 231 + ], + "lines": [ + { + "bbox": [ + 240, + 200, + 371, + 231 + ], + "spans": [ + { + "bbox": [ + 240, + 200, + 371, + 231 + ], + "score": 0.95, + "content": "e ^ { \\mathrm { I W } } ( S ^ { k } , \\kappa ) = \\sum _ { s \\in S ^ { k } } { \\frac { p ( s ) } { q ( s , \\kappa ) } } f ( s ) .", + "type": "interline_equation", + "image_path": "8b61001740dc7d42db3c377579f08d9f93cdff9e7e3528a276595849d54f819e.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 200, + 371, + 215.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 240, + 215.5, + 371, + 231.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "This estimator is based on the idea of priority sampling (Duffield et al., 2007). It does not use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 394, + 260 + ], + "score": 1.0, + "content": "the order of the sample, but assumes sampling using the Gumbel-Top-", + "type": "text" + }, + { + "bbox": [ + 394, + 248, + 401, + 258 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "trick and requires access", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 117, + 271 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 261, + 125, + 268 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 259, + 145, + 271 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 146, + 258, + 178, + 270 + ], + "score": 0.89, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "-th largest perturbed log-probability, which can be seen as the ‘threshold’ since", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 268, + 507, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 189, + 281 + ], + "score": 0.75, + "content": "g _ { \\phi _ { s } } ~ > ~ \\kappa ~ \\forall s ~ \\in ~ S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 268, + 198, + 283 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 198, + 269, + 298, + 282 + ], + "score": 0.88, + "content": "\\bar { q ( s , a ) } = P ( g _ { \\phi _ { s } } > a )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 268, + 507, + 283 + ], + "score": 1.0, + "content": "can be interpreted as the inclusion probability of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 281, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 137, + 293 + ], + "score": 0.89, + "content": "s \\in S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 281, + 253, + 294 + ], + "score": 1.0, + "content": "(assuming a fixed threshold", + "type": "text" + }, + { + "bbox": [ + 254, + 284, + 260, + 292 + ], + "score": 0.73, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 281, + 384, + 294 + ], + "score": 1.0, + "content": "instead of a fixed sample size", + "type": "text" + }, + { + "bbox": [ + 384, + 282, + 390, + 292 + ], + "score": 0.7, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 281, + 507, + 294 + ], + "score": 1.0, + "content": "). For details and a proof of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "unbiasedness, see Vieira (2017) or Kool et al. (2019c). As the estimator has high variance, Kool", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "et al. (2019c) resort to normalizing the importance weights, resulting in biased estimates. 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Again, the result is the unordered set", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 389, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 389, + 338 + ], + "score": 1.0, + "content": "estimator (see Appendix D.1), which thus has equal or lower variance.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 236, + 507, + 338 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 351, + 358, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 350, + 358, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 358, + 363 + ], + "score": 1.0, + "content": "3.3 THE UNORDERED SET POLICY GRADIENT ESTIMATOR", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 504, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 139, + 384 + ], + "score": 1.0, + "content": "Writing", + "type": "text" + }, + { + "bbox": [ + 140, + 373, + 151, + 383 + ], + "score": 0.84, + "content": "p _ { \\pmb { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 371, + 358, + 384 + ], + "score": 1.0, + "content": "to indicate the dependency on the model parameters", + "type": "text" + }, + { + "bbox": [ + 358, + 372, + 365, + 381 + ], + "score": 0.77, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 371, + 505, + 384 + ], + "score": 1.0, + "content": ", we can combine the unordered set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "score": 1.0, + "content": "estimator with REINFORCE (Williams, 1992) to obtain the unordered set policy gradient estimator.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 371, + 505, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 378, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 378, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 378, + 412 + ], + "score": 1.0, + "content": "Corollary 1. The unordered set policy gradient estimator, given by", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 396, + 378, + 412 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 130, + 414, + 464, + 442 + ], + "lines": [ + { + "bbox": [ + 130, + 414, + 464, + 442 + ], + "spans": [ + { + "bbox": [ + 130, + 414, + 464, + 442 + ], + "score": 0.92, + "content": "e ^ { U S P G } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } \\log p _ { \\theta } ( s ) f ( s ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) f ( s ) ,", + "type": "interline_equation", + "image_path": "1dbc150d5df5f608b513f6396d6dc46e4c366286427d7ad9da89ab67b5e56ddd.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 130, + 414, + 464, + 423.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 130, + 423.3333333333333, + 464, + 432.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 130, + 432.66666666666663, + 464, + 441.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 291, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 292, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 292, + 460 + ], + "score": 1.0, + "content": "is an unbiased estimate of the policy gradient.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 445, + 292, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 471, + 497, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 498, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 498, + 485 + ], + "score": 1.0, + "content": "Proof. Using REINFORCE (Williams, 1992) combined with the unordered set estimator we find:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 470, + 498, + 485 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 488, + 506, + 528 + ], + "lines": [ + { + "bbox": [ + 111, + 488, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 111, + 488, + 506, + 528 + ], + "score": 0.93, + "content": "\\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } ( x ) } [ f ( x ) ] = \\mathbb { E } _ { p _ { \\theta } ( x ) } [ \\nabla _ { \\theta } \\log p _ { \\theta } ( x ) f ( x ) ] = \\mathbb { E } _ { S ^ { k } \\sim p _ { \\theta } ( S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } \\log p _ { \\theta } ( s ) f ( s ) \\right] .", + "type": "interline_equation", + "image_path": "aab23b5a4539c051761f39cd7a8e17024373c54bc3fc29e240863c4159a74352.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 111, + 488, + 506, + 501.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 111, + 501.3333333333333, + 506, + 514.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 111, + 514.6666666666666, + 506, + 528.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "Variance reduction using a built-in control variate. The variance of REINFORCE can be re-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 254, + 581 + ], + "score": 1.0, + "content": "duced by subtracting a baseline from", + "type": "text" + }, + { + "bbox": [ + 255, + 569, + 262, + 580 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 567, + 505, + 581 + ], + "score": 1.0, + "content": ". When taking multiple samples (with replacement), a simple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "and effective baseline is to take the mean of other (independent!) samples (Mnih & Rezende, 2016).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "Sampling without replacement, we can use the same idea to construct a baseline based on the other", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 601, + 430, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 430, + 613 + ], + "score": 1.0, + "content": "samples, but we have to correct for the fact that the samples are not independent.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 557, + 506, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 430, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 430, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 430, + 630 + ], + "score": 1.0, + "content": "Theorem 2. The unordered set policy gradient estimator with baseline, given by", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 614, + 430, + 630 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 632, + 469, + 672 + ], + "lines": [ + { + "bbox": [ + 125, + 632, + 469, + 672 + ], + "spans": [ + { + "bbox": [ + 125, + 632, + 469, + 672 + ], + "score": 0.94, + "content": "e ^ { U S P G B L } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\left( f ( s ) - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right) ,", + "type": "interline_equation", + "image_path": "2bbfebe328fefece3095174414fb002d79ec5114db4b9f927fd8cd5b4c3d72fe.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 125, + 632, + 469, + 645.3333333333334 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 125, + 645.3333333333334, + 469, + 658.6666666666667 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 125, + 658.6666666666667, + 469, + 672.0000000000001 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 132, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 135, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 135, + 690 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 675, + 135, + 690 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 685, + 392, + 718 + ], + "lines": [ + { + "bbox": [ + 219, + 685, + 392, + 718 + ], + "spans": [ + { + "bbox": [ + 219, + 685, + 392, + 718 + ], + "score": 0.94, + "content": "R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) = { \\frac { p _ { \\theta } ^ { D \\setminus \\{ s , s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s , s ^ { \\prime } \\} ) } { p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } }", + "type": "interline_equation", + "image_path": "6c8b0893165f90a463454da2a941978b46de124e3b5b3602947a4a6ac32e8a96.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 685, + 392, + 701.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 219, + 701.5, + 392, + 718.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 721, + 449, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 720, + 450, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 450, + 733 + ], + "score": 1.0, + "content": "is the second order leave-one-out ratio, is an unbiased estimate of the policy gradient.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 720, + 450, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 211, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 213, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 213, + 96 + ], + "score": 1.0, + "content": "Proof. See Appendix E.1.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 113, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "This theorem shows how to include a built-in baseline based on dependent samples (without re-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 124, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 421, + 137 + ], + "score": 1.0, + "content": "placement), without introducing bias. 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The difference", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 465, + 158 + ], + "score": 1.0, + "content": "is an estimate of the advantage (Sutton & Barto, 2018), which is positive if the sample", + "type": "text" + }, + { + "bbox": [ + 466, + 148, + 472, + 156 + ], + "score": 0.7, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "is ‘bet-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 213, + 171 + ], + "score": 1.0, + "content": "ter’ than average, causing", + "type": "text" + }, + { + "bbox": [ + 213, + 157, + 237, + 169 + ], + "score": 0.91, + "content": "p _ { \\pmb { \\theta } } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 156, + 506, + 171 + ], + "score": 1.0, + "content": "to be increased (reinforced) through the sign of the gradient, and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 180 + ], + "score": 1.0, + "content": "vice versa. 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This results in the full unordered set policy gradient estimator:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 295, + 457, + 353 + ], + "lines": [ + { + "bbox": [ + 153, + 295, + 457, + 353 + ], + "spans": [ + { + "bbox": [ + 153, + 295, + 457, + 353 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle e ^ { \\mathrm { F U S P G } } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) f _ { \\theta } ( s ) + \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } f _ { \\theta } ( s ) } } \\\\ { { \\displaystyle \\quad = \\sum _ { s \\in S ^ { k } } R ( S ^ { k } , s ) \\nabla _ { \\theta } \\left( p _ { \\theta } ( s ) f _ { \\theta } ( s ) \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "4cab27528948a20ca5fd07651bbd08bcb77a11aa80de85ca79d21a7721f0cde6.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 153, + 295, + 457, + 314.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 153, + 314.3333333333333, + 457, + 333.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 153, + 333.66666666666663, + 457, + 352.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 359, + 503, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "Equation 20 is straightforward to implement using an automatic differentiation library. We can also", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "include the baseline (as in equation 17) but we must make sure to call STOP GRADIENT (DETACH in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 255, + 394 + ], + "score": 1.0, + "content": "PyTorch) on the baseline (but not on", + "type": "text" + }, + { + "bbox": [ + 255, + 382, + 285, + 394 + ], + "score": 0.91, + "content": "f _ { \\pmb \\theta } ( s ) ! )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 381, + 505, + 394 + ], + "score": 1.0, + "content": ". Importantly, we should never track gradients through", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 198, + 406 + ], + "score": 1.0, + "content": "the leave-one-out ratio", + "type": "text" + }, + { + "bbox": [ + 198, + 392, + 235, + 405 + ], + "score": 0.93, + "content": "R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "which means it can be efficiently computed in pure inference mode.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 504, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 504, + 432 + ], + "score": 1.0, + "content": "Scope & limitations. We can use the unordered set estimator for any discrete distribution from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "which we can sample without replacement, by treating it as a univariate categorical distribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "over its domain. This includes sequence models, from which we can sample using Stochastic Beam", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Search (Kool et al., 2019c), as well as multivariate categorical distributions which can also be treated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "as sequence models (see Section 4.2). In the presence of continuous variables or a stochastic function", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 114, + 485 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 474, + 505, + 486 + ], + "score": 1.0, + "content": ", we may separate this stochasticity from the stochasticity over the discrete distribution, as in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "Lorberbom et al. (2019). The computation of the leave-one-out ratios adds some overhead, although", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 298, + 507 + ], + "score": 1.0, + "content": "they can be computed efficiently, even for large", + "type": "text" + }, + { + "bbox": [ + 298, + 496, + 305, + 505 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "(see Appendix B). For a moderately sized model,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 496, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 496, + 520 + ], + "score": 1.0, + "content": "the costs of model evaluation and backpropagation dominate the cost of computing the estimator.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 534, + 345, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 347, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 347, + 546 + ], + "score": 1.0, + "content": "3.4 RELATION TO OTHER MULTI-SAMPLE ESTIMATORS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Relation to Murthy’s estimator. We found out that the ‘vanilla’ unordered set estimator (equa-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "tion 11) is actually a special case of the estimator by Murthy (1957), known in statistics literature", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 575, + 507, + 593 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 249, + 593 + ], + "score": 1.0, + "content": "for estimation of a population total", + "type": "text" + }, + { + "bbox": [ + 250, + 577, + 308, + 590 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Theta = \\sum _ { i \\in D } y _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 575, + 340, + 593 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + }, + { + "bbox": [ + 340, + 577, + 397, + 589 + ], + "score": 0.93, + "content": "y _ { i } = p ( i ) f ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 575, + 437, + 593 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 437, + 577, + 489, + 589 + ], + "score": 0.92, + "content": "\\Theta = \\mathbb { E } [ f ( i ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 575, + 507, + 593 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "Murthy’s estimator can be used to estimate expectations (see equation 11). Murthy derives the esti-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 598, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 598, + 444, + 613 + ], + "score": 1.0, + "content": "mator by ‘unordering’ a convex combination of Raj (1956) estimators, which, using", + "type": "text" + }, + { + "bbox": [ + 445, + 599, + 501, + 611 + ], + "score": 0.93, + "content": "y _ { i } = p ( i ) f ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 598, + 506, + 613 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 336, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 336, + 623 + ], + "score": 1.0, + "content": "are stochastic sum-and-sample estimators in our analogy.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "Murthy (1957) also provides an unbiased estimator of the variance, which may be interesting for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "future applications. Since Murthy’s estimator can be used with arbitrary sampling distribution, it", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "is straightforward to derive importance-sampling versions of our estimators. In particular, we can", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 137, + 674 + ], + "score": 1.0, + "content": "sample", + "type": "text" + }, + { + "bbox": [ + 137, + 660, + 145, + 670 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 659, + 254, + 674 + ], + "score": 1.0, + "content": "without replacement using", + "type": "text" + }, + { + "bbox": [ + 255, + 660, + 323, + 671 + ], + "score": 0.9, + "content": "q ( x ) > 0 , x \\in D", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 659, + 505, + 674 + ], + "score": 1.0, + "content": ", and use equations 11, 16, 17 and 20, as long", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 668, + 334, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 261, + 686 + ], + "score": 1.0, + "content": "as we compute the leave-one-out ratio", + "type": "text" + }, + { + "bbox": [ + 261, + 671, + 298, + 683 + ], + "score": 0.92, + "content": "R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 668, + 323, + 686 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 324, + 673, + 329, + 682 + ], + "score": 0.67, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 668, + 334, + 686 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "While part of our derivation coincides with Murthy (1957), we are not aware of previous work", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "using this estimator to estimate expectations. Additionally, we discuss practical computation of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 127, + 722 + ], + "score": 0.9, + "content": "p ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(Appendix B), we show the relation to the importance-weighted estimator, and we provide the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "extension to estimating policy gradients, especially including a built-in baseline without adding bias.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 83, + 504, + 93 + ], + "lines": [ + { + "bbox": [ + 496, + 84, + 504, + 93 + ], + "spans": [ + { + "bbox": [ + 496, + 84, + 504, + 93 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 211, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 213, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 213, + 96 + ], + "score": 1.0, + "content": "Proof. See Appendix E.1.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 213, + 96 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 113, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "This theorem shows how to include a built-in baseline based on dependent samples (without re-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 124, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 421, + 137 + ], + "score": 1.0, + "content": "placement), without introducing bias. By having a built-in baseline, the value", + "type": "text" + }, + { + "bbox": [ + 422, + 124, + 441, + 136 + ], + "score": 0.91, + "content": "f ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 124, + 488, + 137 + ], + "score": 1.0, + "content": "for sample", + "type": "text" + }, + { + "bbox": [ + 488, + 126, + 495, + 134 + ], + "score": 0.7, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 124, + 505, + 137 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 297, + 149 + ], + "score": 1.0, + "content": "compared against an estimate of its expectation", + "type": "text" + }, + { + "bbox": [ + 297, + 135, + 328, + 147 + ], + "score": 0.92, + "content": "\\mathbb { E } [ f ( s ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 135, + 506, + 149 + ], + "score": 1.0, + "content": ", based on the other samples. The difference", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 465, + 158 + ], + "score": 1.0, + "content": "is an estimate of the advantage (Sutton & Barto, 2018), which is positive if the sample", + "type": "text" + }, + { + "bbox": [ + 466, + 148, + 472, + 156 + ], + "score": 0.7, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "is ‘bet-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 213, + 171 + ], + "score": 1.0, + "content": "ter’ than average, causing", + "type": "text" + }, + { + "bbox": [ + 213, + 157, + 237, + 169 + ], + "score": 0.91, + "content": "p _ { \\pmb { \\theta } } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 156, + 506, + 171 + ], + "score": 1.0, + "content": "to be increased (reinforced) through the sign of the gradient, and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 180 + ], + "score": 1.0, + "content": "vice versa. By sampling without replacement, the unordered set estimator forces the estimator to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 376, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 376, + 191 + ], + "score": 1.0, + "content": "compare different alternatives, and reinforces the best among them.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 113, + 506, + 191 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 475, + 217 + ], + "score": 1.0, + "content": "Including the pathwise derivative. So far, we have only considered the scenario where", + "type": "text" + }, + { + "bbox": [ + 475, + 205, + 483, + 217 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "does", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 165, + 228 + ], + "score": 1.0, + "content": "not depend on", + "type": "text" + }, + { + "bbox": [ + 166, + 216, + 172, + 226 + ], + "score": 0.74, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 214, + 186, + 228 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 186, + 216, + 193, + 227 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 214, + 259, + 228 + ], + "score": 1.0, + "content": "does depend on", + "type": "text" + }, + { + "bbox": [ + 259, + 216, + 266, + 226 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 214, + 505, + 228 + ], + "score": 1.0, + "content": ", for example in a VAE (Kingma & Welling, 2014; Rezende", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 227, + 498, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 258, + 239 + ], + "score": 1.0, + "content": "et al., 2014), then we use the notation", + "type": "text" + }, + { + "bbox": [ + 259, + 227, + 270, + 238 + ], + "score": 0.88, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 227, + 498, + 239 + ], + "score": 1.0, + "content": "and we can write the gradient (Schulman et al., 2015) as", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 205, + 505, + 239 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 245, + 428, + 260 + ], + "lines": [ + { + "bbox": [ + 183, + 245, + 428, + 260 + ], + "spans": [ + { + "bbox": [ + 183, + 245, + 428, + 260 + ], + "score": 0.89, + "content": "\\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ f _ { \\pmb { \\theta } } ( \\pmb { x } ) ] = \\mathbb { E } _ { p _ { \\pmb { \\theta } } ( \\pmb { x } ) } [ \\nabla _ { \\pmb { \\theta } } \\log p _ { \\pmb { \\theta } } ( \\pmb { x } ) f _ { \\pmb { \\theta } } ( \\pmb { x } ) + \\nabla _ { \\pmb { \\theta } } f _ { \\pmb { \\theta } } ( \\pmb { x } ) ] .", + "type": "interline_equation", + "image_path": "6d9200624e6d99ee5fa00ffa9450b95f696e999e1febff3fc26aa6f635fccce8.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 183, + 245, + 428, + 260 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 267, + 508, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 267, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 506, + 279 + ], + "score": 1.0, + "content": "The additional second (‘pathwise’) term can be estimated (using the same samples) with the standard", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 277, + 456, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 456, + 290 + ], + "score": 1.0, + "content": "unordered set estimator. This results in the full unordered set policy gradient estimator:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 267, + 506, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 295, + 457, + 353 + ], + "lines": [ + { + "bbox": [ + 153, + 295, + 457, + 353 + ], + "spans": [ + { + "bbox": [ + 153, + 295, + 457, + 353 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle e ^ { \\mathrm { F U S P G } } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) f _ { \\theta } ( s ) + \\sum _ { s \\in S ^ { k } } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\nabla _ { \\theta } f _ { \\theta } ( s ) } } \\\\ { { \\displaystyle \\quad = \\sum _ { s \\in S ^ { k } } R ( S ^ { k } , s ) \\nabla _ { \\theta } \\left( p _ { \\theta } ( s ) f _ { \\theta } ( s ) \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "4cab27528948a20ca5fd07651bbd08bcb77a11aa80de85ca79d21a7721f0cde6.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 153, + 295, + 457, + 314.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 153, + 314.3333333333333, + 457, + 333.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 153, + 333.66666666666663, + 457, + 352.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 359, + 503, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "Equation 20 is straightforward to implement using an automatic differentiation library. We can also", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "include the baseline (as in equation 17) but we must make sure to call STOP GRADIENT (DETACH in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 255, + 394 + ], + "score": 1.0, + "content": "PyTorch) on the baseline (but not on", + "type": "text" + }, + { + "bbox": [ + 255, + 382, + 285, + 394 + ], + "score": 0.91, + "content": "f _ { \\pmb \\theta } ( s ) ! )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 381, + 505, + 394 + ], + "score": 1.0, + "content": ". Importantly, we should never track gradients through", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 198, + 406 + ], + "score": 1.0, + "content": "the leave-one-out ratio", + "type": "text" + }, + { + "bbox": [ + 198, + 392, + 235, + 405 + ], + "score": 0.93, + "content": "R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "which means it can be efficiently computed in pure inference mode.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 360, + 505, + 406 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 504, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 504, + 432 + ], + "score": 1.0, + "content": "Scope & limitations. We can use the unordered set estimator for any discrete distribution from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "which we can sample without replacement, by treating it as a univariate categorical distribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "over its domain. This includes sequence models, from which we can sample using Stochastic Beam", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Search (Kool et al., 2019c), as well as multivariate categorical distributions which can also be treated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "as sequence models (see Section 4.2). In the presence of continuous variables or a stochastic function", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 114, + 485 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 474, + 505, + 486 + ], + "score": 1.0, + "content": ", we may separate this stochasticity from the stochasticity over the discrete distribution, as in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "Lorberbom et al. (2019). The computation of the leave-one-out ratios adds some overhead, although", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 298, + 507 + ], + "score": 1.0, + "content": "they can be computed efficiently, even for large", + "type": "text" + }, + { + "bbox": [ + 298, + 496, + 305, + 505 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "(see Appendix B). For a moderately sized model,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 496, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 496, + 520 + ], + "score": 1.0, + "content": "the costs of model evaluation and backpropagation dominate the cost of computing the estimator.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 418, + 506, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 534, + 345, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 347, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 347, + 546 + ], + "score": 1.0, + "content": "3.4 RELATION TO OTHER MULTI-SAMPLE ESTIMATORS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Relation to Murthy’s estimator. We found out that the ‘vanilla’ unordered set estimator (equa-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "tion 11) is actually a special case of the estimator by Murthy (1957), known in statistics literature", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 575, + 507, + 593 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 249, + 593 + ], + "score": 1.0, + "content": "for estimation of a population total", + "type": "text" + }, + { + "bbox": [ + 250, + 577, + 308, + 590 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Theta = \\sum _ { i \\in D } y _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 575, + 340, + 593 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + }, + { + "bbox": [ + 340, + 577, + 397, + 589 + ], + "score": 0.93, + "content": "y _ { i } = p ( i ) f ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 575, + 437, + 593 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 437, + 577, + 489, + 589 + ], + "score": 0.92, + "content": "\\Theta = \\mathbb { E } [ f ( i ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 575, + 507, + 593 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "Murthy’s estimator can be used to estimate expectations (see equation 11). Murthy derives the esti-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 598, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 598, + 444, + 613 + ], + "score": 1.0, + "content": "mator by ‘unordering’ a convex combination of Raj (1956) estimators, which, using", + "type": "text" + }, + { + "bbox": [ + 445, + 599, + 501, + 611 + ], + "score": 0.93, + "content": "y _ { i } = p ( i ) f ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 598, + 506, + 613 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 336, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 336, + 623 + ], + "score": 1.0, + "content": "are stochastic sum-and-sample estimators in our analogy.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 554, + 507, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "Murthy (1957) also provides an unbiased estimator of the variance, which may be interesting for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "future applications. Since Murthy’s estimator can be used with arbitrary sampling distribution, it", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "is straightforward to derive importance-sampling versions of our estimators. In particular, we can", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 137, + 674 + ], + "score": 1.0, + "content": "sample", + "type": "text" + }, + { + "bbox": [ + 137, + 660, + 145, + 670 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 659, + 254, + 674 + ], + "score": 1.0, + "content": "without replacement using", + "type": "text" + }, + { + "bbox": [ + 255, + 660, + 323, + 671 + ], + "score": 0.9, + "content": "q ( x ) > 0 , x \\in D", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 659, + 505, + 674 + ], + "score": 1.0, + "content": ", and use equations 11, 16, 17 and 20, as long", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 668, + 334, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 261, + 686 + ], + "score": 1.0, + "content": "as we compute the leave-one-out ratio", + "type": "text" + }, + { + "bbox": [ + 261, + 671, + 298, + 683 + ], + "score": 0.92, + "content": "R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 668, + 323, + 686 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 324, + 673, + 329, + 682 + ], + "score": 0.67, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 668, + 334, + 686 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 626, + 506, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "While part of our derivation coincides with Murthy (1957), we are not aware of previous work", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "using this estimator to estimate expectations. Additionally, we discuss practical computation of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 127, + 722 + ], + "score": 0.9, + "content": "p ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(Appendix B), we show the relation to the importance-weighted estimator, and we provide the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "extension to estimating policy gradients, especially including a built-in baseline without adding bias.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 686, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Relation to the empirical risk estimator. The empirical risk loss (Edunov et al., 2018) estimates", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 333, + 106 + ], + "score": 1.0, + "content": "the expectation in equation 1 by summing only a subset", + "type": "text" + }, + { + "bbox": [ + 334, + 94, + 342, + 104 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "of the domain, using normalized proba-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 478, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 136, + 125 + ], + "score": 1.0, + "content": "bilities", + "type": "text" + }, + { + "bbox": [ + 136, + 104, + 219, + 122 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { p } _ { \\pmb { \\theta } } ( s ) = \\frac { \\bar { p } _ { \\pmb { \\theta } } ( s ) } { \\sum _ { s ^ { \\prime } \\in S } p _ { \\pmb { \\theta } } ( s ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 104, + 478, + 125 + ], + "score": 1.0, + "content": ". Using this loss, the (biased) estimate of the gradient is given by", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 123, + 401, + 156 + ], + "lines": [ + { + "bbox": [ + 210, + 123, + 401, + 156 + ], + "spans": [ + { + "bbox": [ + 210, + 123, + 401, + 156 + ], + "score": 0.95, + "content": "e ^ { \\mathrm { R I S K } } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\pmb { \\theta } } \\left( \\frac { p _ { \\pmb { \\theta } } ( s ) } { \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\pmb { \\theta } } ( s ^ { \\prime } ) } \\right) f ( s ) .", + "type": "interline_equation", + "image_path": "16c4a246849e072f4944d39d2f161557419ea23787c012dd3296694e026e574b.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 123, + 401, + 139.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 210, + 139.5, + 401, + 156.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 157, + 505, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "The risk estimator is similar to the unordered set policy gradient estimator, with two important", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "differences: 1) the individual terms are normalized by the total probability mass rather than the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "leave-one-out ratio and 2) the gradient w.r.t. the normalization factor is taken into account. As a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "result, samples ‘compete’ for probability mass and only the best can be reinforced. This has the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 435, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 435, + 214 + ], + "score": 1.0, + "content": "same effect as using a built-in baseline, which we prove in the following theorem.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "Theorem 3. By taking the gradient w.r.t. the normalization factor into account, the risk estimator", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 333, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 333, + 237 + ], + "score": 1.0, + "content": "has a built-in baseline, which means it can be written as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 238, + 482, + 278 + ], + "lines": [ + { + "bbox": [ + 113, + 238, + 482, + 278 + ], + "spans": [ + { + "bbox": [ + 113, + 238, + 482, + 278 + ], + "score": 0.93, + "content": "e ^ { R I S K } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) \\frac { 1 } { \\sum _ { s ^ { \\prime \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime \\prime } ) } \\left( f ( s ) - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) \\frac { 1 } { \\sum _ { s ^ { \\prime \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime \\prime } ) } f ( s ^ { \\prime } ) \\right) .", + "type": "interline_equation", + "image_path": "904b5d4d9996c49ac35bd50c3cc3eaa34cee54f06b1fb5ec574ba84ccd5a5819.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 113, + 238, + 482, + 251.33333333333334 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 113, + 251.33333333333334, + 482, + 264.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 113, + 264.6666666666667, + 482, + 278.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 208, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 209, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 209, + 300 + ], + "score": 1.0, + "content": "Proof. See Appendix F.1", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "This theorem highlights the similarity between the biased risk estimator and our unbiased estimator", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "(equation 17), and suggests that their only difference is the weighting of terms. Unfortunately, the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "implementation by Edunov et al. (2018) has more sources of bias (e.g. length normalization), which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "are not compatible with our estimator. However, we believe that our analysis helps analyze the bias", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 356, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 504, + 368 + ], + "score": 1.0, + "content": "of the risk estimator and is a step towards developing unbiased estimators for structured prediction.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 438, + 390 + ], + "score": 1.0, + "content": "Relation to VIMCO. VIMCO (Mnih & Rezende, 2016) is an estimator that uses", + "type": "text" + }, + { + "bbox": [ + 439, + 379, + 446, + 388 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "samples (with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 388, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 388, + 316, + 404 + ], + "score": 1.0, + "content": "replacement) to optimize an objective of the form log", + "type": "text" + }, + { + "bbox": [ + 316, + 389, + 364, + 402 + ], + "score": 0.88, + "content": "\\textstyle { \\frac { 1 } { k } } \\sum _ { i } f ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 388, + 506, + 404 + ], + "score": 1.0, + "content": ", which is a multi-sample stochastic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "lower bound in the context of variational inference. 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While we do not have a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 281, + 435 + ], + "score": 1.0, + "content": "log term, as our goal is to optimize general", + "type": "text" + }, + { + "bbox": [ + 282, + 422, + 314, + 434 + ], + "score": 0.92, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 421, + 506, + 435 + ], + "score": 1.0, + "content": ", we adopt the idea of forming a baseline based", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "on the other samples, and we define REINFORCE with replacement (with built-in baseline) as the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 419, + 457 + ], + "score": 1.0, + "content": "estimator that computes the gradient estimate using samples with replacement", + "type": "text" + }, + { + "bbox": [ + 419, + 444, + 493, + 456 + ], + "score": 0.93, + "content": "X ^ { k } = ( x _ { 1 } , . . . , x _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 457, + 441, + 497 + ], + "lines": [ + { + "bbox": [ + 171, + 457, + 441, + 497 + ], + "spans": [ + { + "bbox": [ + 171, + 457, + 441, + 497 + ], + "score": 0.94, + "content": "e ^ { \\mathrm { R F W R } } ( X ^ { k } ) = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\nabla _ { \\theta } \\log p _ { \\theta } ( x _ { i } ) \\left( f ( x _ { i } ) - \\frac { 1 } { k - 1 } \\sum _ { j \\ne i } f ( x _ { j } ) \\right) .", + "type": "interline_equation", + "image_path": "351a340e6952afd897fb774a727f1e14b17f5a4500b3ff32102096cd6837dcf3.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 171, + 457, + 441, + 470.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 171, + 470.3333333333333, + 441, + 483.66666666666663 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 171, + 483.66666666666663, + 441, + 496.99999999999994 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 234, + 512 + ], + "score": 1.0, + "content": "This estimator is unbiased, as", + "type": "text" + }, + { + "bbox": [ + 234, + 498, + 369, + 512 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { x _ { i } , x _ { j } } [ \\nabla _ { \\pmb { \\theta } } \\log p _ { \\pmb { \\theta } } ( x _ { i } ) f ( x _ { j } ) ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 498, + 388, + 512 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 388, + 499, + 416, + 511 + ], + "score": 0.91, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "(see also Kool et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "(2019b)). 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This puts more weight on higher", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 532, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 544 + ], + "score": 1.0, + "content": "probability elements to compensate for sampling without replacement. If probabilities are small and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 103, + 539, + 508, + 559 + ], + "spans": [ + { + "bbox": [ + 103, + 539, + 468, + 559 + ], + "score": 1.0, + "content": "(close to) uniform, there are (almost) no duplicate samples and the weights will be close to", + "type": "text" + }, + { + "bbox": [ + 468, + 542, + 475, + 556 + ], + "score": 0.85, + "content": "\\frac { 1 } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 539, + 508, + 559 + ], + "score": 1.0, + "content": ", so the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 411, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 411, + 566 + ], + "score": 1.0, + "content": "gradient estimate of the with- and without-replacement versions are similar.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "Relation to ARSM. 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This can be seen as similar to sampling without replacement, and the estimator also", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 599, + 504, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 504, + 610 + ], + "score": 1.0, + "content": "has a built-in control variate. 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Both partition ´", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 154, + 691 + ], + "score": 1.0, + "content": "the domain", + "type": "text" + }, + { + "bbox": [ + 154, + 677, + 164, + 687 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 675, + 183, + 691 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 183, + 677, + 190, + 687 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 675, + 505, + 691 + ], + "score": 1.0, + "content": "strata and take a single sample from each stratum, where systematic sampling", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "uses common random numbers for each stratum. 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The empirical risk loss (Edunov et al., 2018) estimates", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 333, + 106 + ], + "score": 1.0, + "content": "the expectation in equation 1 by summing only a subset", + "type": "text" + }, + { + "bbox": [ + 334, + 94, + 342, + 104 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "of the domain, using normalized proba-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 478, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 136, + 125 + ], + "score": 1.0, + "content": "bilities", + "type": "text" + }, + { + "bbox": [ + 136, + 104, + 219, + 122 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { p } _ { \\pmb { \\theta } } ( s ) = \\frac { \\bar { p } _ { \\pmb { \\theta } } ( s ) } { \\sum _ { s ^ { \\prime } \\in S } p _ { \\pmb { \\theta } } ( s ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 104, + 478, + 125 + ], + "score": 1.0, + "content": ". Using this loss, the (biased) estimate of the gradient is given by", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 125 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 123, + 401, + 156 + ], + "lines": [ + { + "bbox": [ + 210, + 123, + 401, + 156 + ], + "spans": [ + { + "bbox": [ + 210, + 123, + 401, + 156 + ], + "score": 0.95, + "content": "e ^ { \\mathrm { R I S K } } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\pmb { \\theta } } \\left( \\frac { p _ { \\pmb { \\theta } } ( s ) } { \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\pmb { \\theta } } ( s ^ { \\prime } ) } \\right) f ( s ) .", + "type": "interline_equation", + "image_path": "16c4a246849e072f4944d39d2f161557419ea23787c012dd3296694e026e574b.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 123, + 401, + 139.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 210, + 139.5, + 401, + 156.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 157, + 505, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "The risk estimator is similar to the unordered set policy gradient estimator, with two important", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "differences: 1) the individual terms are normalized by the total probability mass rather than the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "leave-one-out ratio and 2) the gradient w.r.t. the normalization factor is taken into account. As a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "result, samples ‘compete’ for probability mass and only the best can be reinforced. This has the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 435, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 435, + 214 + ], + "score": 1.0, + "content": "same effect as using a built-in baseline, which we prove in the following theorem.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 156, + 506, + 214 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "Theorem 3. By taking the gradient w.r.t. the normalization factor into account, the risk estimator", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 333, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 333, + 237 + ], + "score": 1.0, + "content": "has a built-in baseline, which means it can be written as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 106, + 214, + 505, + 237 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 238, + 482, + 278 + ], + "lines": [ + { + "bbox": [ + 113, + 238, + 482, + 278 + ], + "spans": [ + { + "bbox": [ + 113, + 238, + 482, + 278 + ], + "score": 0.93, + "content": "e ^ { R I S K } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) \\frac { 1 } { \\sum _ { s ^ { \\prime \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime \\prime } ) } \\left( f ( s ) - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) \\frac { 1 } { \\sum _ { s ^ { \\prime \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime \\prime } ) } f ( s ^ { \\prime } ) \\right) .", + "type": "interline_equation", + "image_path": "904b5d4d9996c49ac35bd50c3cc3eaa34cee54f06b1fb5ec574ba84ccd5a5819.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 113, + 238, + 482, + 251.33333333333334 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 113, + 251.33333333333334, + 482, + 264.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 113, + 264.6666666666667, + 482, + 278.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 208, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 209, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 209, + 300 + ], + "score": 1.0, + "content": "Proof. See Appendix F.1", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 288, + 209, + 300 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "This theorem highlights the similarity between the biased risk estimator and our unbiased estimator", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "(equation 17), and suggests that their only difference is the weighting of terms. Unfortunately, the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "implementation by Edunov et al. (2018) has more sources of bias (e.g. length normalization), which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "are not compatible with our estimator. However, we believe that our analysis helps analyze the bias", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 356, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 504, + 368 + ], + "score": 1.0, + "content": "of the risk estimator and is a step towards developing unbiased estimators for structured prediction.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 311, + 505, + 368 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 438, + 390 + ], + "score": 1.0, + "content": "Relation to VIMCO. VIMCO (Mnih & Rezende, 2016) is an estimator that uses", + "type": "text" + }, + { + "bbox": [ + 439, + 379, + 446, + 388 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "samples (with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 388, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 388, + 316, + 404 + ], + "score": 1.0, + "content": "replacement) to optimize an objective of the form log", + "type": "text" + }, + { + "bbox": [ + 316, + 389, + 364, + 402 + ], + "score": 0.88, + "content": "\\textstyle { \\frac { 1 } { k } } \\sum _ { i } f ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 388, + 506, + 404 + ], + "score": 1.0, + "content": ", which is a multi-sample stochastic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "lower bound in the context of variational inference. VIMCO reduces the variance by using a local", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 207, + 423 + ], + "score": 1.0, + "content": "baseline for each of the", + "type": "text" + }, + { + "bbox": [ + 207, + 412, + 214, + 421 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 411, + 334, + 423 + ], + "score": 1.0, + "content": "samples, based on the other", + "type": "text" + }, + { + "bbox": [ + 334, + 411, + 360, + 422 + ], + "score": 0.88, + "content": "k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 411, + 506, + 423 + ], + "score": 1.0, + "content": "samples. While we do not have a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 281, + 435 + ], + "score": 1.0, + "content": "log term, as our goal is to optimize general", + "type": "text" + }, + { + "bbox": [ + 282, + 422, + 314, + 434 + ], + "score": 0.92, + "content": "\\mathbb { E } [ f ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 421, + 506, + 435 + ], + "score": 1.0, + "content": ", we adopt the idea of forming a baseline based", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "on the other samples, and we define REINFORCE with replacement (with built-in baseline) as the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 419, + 457 + ], + "score": 1.0, + "content": "estimator that computes the gradient estimate using samples with replacement", + "type": "text" + }, + { + "bbox": [ + 419, + 444, + 493, + 456 + ], + "score": 0.93, + "content": "X ^ { k } = ( x _ { 1 } , . . . , x _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 379, + 506, + 457 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 457, + 441, + 497 + ], + "lines": [ + { + "bbox": [ + 171, + 457, + 441, + 497 + ], + "spans": [ + { + "bbox": [ + 171, + 457, + 441, + 497 + ], + "score": 0.94, + "content": "e ^ { \\mathrm { R F W R } } ( X ^ { k } ) = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\nabla _ { \\theta } \\log p _ { \\theta } ( x _ { i } ) \\left( f ( x _ { i } ) - \\frac { 1 } { k - 1 } \\sum _ { j \\ne i } f ( x _ { j } ) \\right) .", + "type": "interline_equation", + "image_path": "351a340e6952afd897fb774a727f1e14b17f5a4500b3ff32102096cd6837dcf3.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 171, + 457, + 441, + 470.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 171, + 470.3333333333333, + 441, + 483.66666666666663 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 171, + 483.66666666666663, + 441, + 496.99999999999994 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 234, + 512 + ], + "score": 1.0, + "content": "This estimator is unbiased, as", + "type": "text" + }, + { + "bbox": [ + 234, + 498, + 369, + 512 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { x _ { i } , x _ { j } } [ \\nabla _ { \\pmb { \\theta } } \\log p _ { \\pmb { \\theta } } ( x _ { i } ) f ( x _ { j } ) ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 498, + 388, + 512 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 388, + 499, + 416, + 511 + ], + "score": 0.91, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "(see also Kool et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "(2019b)). We think of the unordered set estimator as the without-replacement version of this es-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 518, + 507, + 536 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 244, + 536 + ], + "score": 1.0, + "content": "timator, which weights terms by", + "type": "text" + }, + { + "bbox": [ + 244, + 520, + 304, + 533 + ], + "score": 0.93, + "content": "p _ { \\pmb { \\theta } } ( s ) R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 518, + 350, + 536 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 351, + 520, + 358, + 534 + ], + "score": 0.86, + "content": "\\frac { 1 } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 518, + 507, + 536 + ], + "score": 1.0, + "content": ". This puts more weight on higher", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 532, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 544 + ], + "score": 1.0, + "content": "probability elements to compensate for sampling without replacement. If probabilities are small and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 103, + 539, + 508, + 559 + ], + "spans": [ + { + "bbox": [ + 103, + 539, + 468, + 559 + ], + "score": 1.0, + "content": "(close to) uniform, there are (almost) no duplicate samples and the weights will be close to", + "type": "text" + }, + { + "bbox": [ + 468, + 542, + 475, + 556 + ], + "score": 0.85, + "content": "\\frac { 1 } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 539, + 508, + 559 + ], + "score": 1.0, + "content": ", so the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 411, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 411, + 566 + ], + "score": 1.0, + "content": "gradient estimate of the with- and without-replacement versions are similar.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 103, + 498, + 508, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "Relation to ARSM. ARSM (Yin et al., 2019) also uses multiple evaluations (‘pseudo-samples’)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 117, + 600 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 590, + 129, + 599 + ], + "score": 0.83, + "content": "p _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 587, + 148, + 600 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 149, + 588, + 156, + 600 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 587, + 505, + 600 + ], + "score": 1.0, + "content": ". This can be seen as similar to sampling without replacement, and the estimator also", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 599, + 504, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 504, + 610 + ], + "score": 1.0, + "content": "has a built-in control variate. Compared to ARSM, our estimator allows direct control over the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 293, + 623 + ], + "score": 1.0, + "content": "computational cost (through the sample size", + "type": "text" + }, + { + "bbox": [ + 293, + 610, + 300, + 620 + ], + "score": 0.66, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 610, + 505, + 623 + ], + "score": 1.0, + "content": ") and has wider applicability, for example it also", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 621, + 490, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 490, + 634 + ], + "score": 1.0, + "content": "applies to multivariate categorical variables with different numbers of categories per dimension.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 576, + 505, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "score": 1.0, + "content": "Relation to stratified/systematic sampling. Our estimator aims to reduce variance by changing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 656, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 504, + 667 + ], + "score": 1.0, + "content": "the sampling distribution for multiple samples by sampling without replacement. There are alterna-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "tives, such as using stratified or systematic sampling (see, e.g. Douc & Cappe (2005)). Both partition ´", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 154, + 691 + ], + "score": 1.0, + "content": "the domain", + "type": "text" + }, + { + "bbox": [ + 154, + 677, + 164, + 687 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 675, + 183, + 691 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 183, + 677, + 190, + 687 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 675, + 505, + 691 + ], + "score": 1.0, + "content": "strata and take a single sample from each stratum, where systematic sampling", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "uses common random numbers for each stratum. In applications involving high-dimensional or", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 699, + 504, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 710 + ], + "score": 1.0, + "content": "structured domains, it is unclear how to partition the domain and how to sample from each partition.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "Additionally, as samples are not independent, it is non-trivial to include a built-in baseline, which", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 375, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 375, + 732 + ], + "score": 1.0, + "content": "we find is a key component that makes our estimator perform well.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 642, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 71, + 494, + 174 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 71, + 494, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 71, + 494, + 174 + ], + "spans": [ + { + "bbox": [ + 115, + 71, + 494, + 174 + ], + "score": 0.971, + "type": "image", + "image_path": "1827e94fdb564b2088e10a208547d35fffe208d74d3041a8d194d3868ada9f35.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 71, + 494, + 105.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 105.33333333333334, + 494, + 139.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 139.66666666666669, + 494, + 174.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 179, + 505, + 212 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "Figure 1: Bernoulli gradient variance (on log scale) as a function of the number of model evaluations", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 203 + ], + "score": 1.0, + "content": "(including baseline evaluations, so the sum-and-sample estimators with sampled baselines use twice", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 439, + 213 + ], + "score": 1.0, + "content": "as many evaluations). Note that for some estimators, the variance is 0 (log variance", + "type": "text" + }, + { + "bbox": [ + 439, + 201, + 460, + 212 + ], + "score": 0.8, + "content": "- \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 201, + 476, + 213 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 477, + 201, + 501, + 210 + ], + "score": 0.89, + "content": "k = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 201, + 505, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 200, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 201, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 201, + 241 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 251, + 260, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 262, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 262, + 263 + ], + "score": 1.0, + "content": "4.1 BERNOULLI TOY EXPERIMENT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 359 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 486, + 286 + ], + "score": 1.0, + "content": "We use the code by Liu et al. (2019) to reproduce their Bernoulli toy experiment. Given a vector", + "type": "text" + }, + { + "bbox": [ + 486, + 274, + 505, + 284 + ], + "score": 0.83, + "content": "\\mathbf { p } =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 279, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 307, + 305 + ], + "score": 1.0, + "content": "(0.6, 0.51, 0.48) the goal is to minimize the loss", + "type": "text" + }, + { + "bbox": [ + 308, + 283, + 501, + 303 + ], + "score": 0.82, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\eta ) \\ : = \\ : \\mathbb { E } _ { x _ { 1 } , x _ { 2 } , x _ { 3 } \\sim \\mathrm { B e r n } ( \\sigma ( \\eta ) ) } \\left[ \\sum _ { i = 1 } ^ { 3 } ( x _ { i } - p _ { i } ) ^ { 2 } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 279, + 506, + 305 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 128, + 314 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 304, + 169, + 313 + ], + "score": 0.87, + "content": "x _ { 1 } , x _ { 2 } , x _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 300, + 286, + 314 + ], + "score": 1.0, + "content": "are i.i.d. from the Bernoulli", + "type": "text" + }, + { + "bbox": [ + 286, + 301, + 313, + 314 + ], + "score": 0.87, + "content": "( \\sigma ( \\eta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 300, + 473, + 314 + ], + "score": 1.0, + "content": "distribution, parameterized by a scalar", + "type": "text" + }, + { + "bbox": [ + 473, + 303, + 501, + 313 + ], + "score": 0.82, + "content": "\\eta \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 300, + 505, + 314 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 133, + 325 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 312, + 236, + 325 + ], + "score": 0.93, + "content": "\\sigma ( \\eta ) = ( 1 + \\exp ( - \\eta ) ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "is the sigmoid function. We compare different estimators, with and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "without baseline (either ‘built-in’ or using additional samples, referred to as REINFORCE+ in Liu", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 333, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 367, + 349 + ], + "score": 1.0, + "content": "et al. (2019)). We report the (log-)variance of the scalar gradient", + "type": "text" + }, + { + "bbox": [ + 367, + 333, + 380, + 349 + ], + "score": 0.9, + "content": "\\frac { \\partial \\mathcal { L } } { \\partial \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 333, + 506, + 349 + ], + "score": 1.0, + "content": "as a function of the number of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 346, + 465, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 465, + 360 + ], + "score": 1.0, + "content": "model evaluations, which is twice as high when using a sampled baseline (for each term).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 377 + ], + "score": 1.0, + "content": "As can be seen in Figure 1, the unordered set estimator is the only estimator that has consistently", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 318, + 388 + ], + "score": 1.0, + "content": "the lowest (or comparable) variance in both the high", + "type": "text" + }, + { + "bbox": [ + 318, + 375, + 344, + 387 + ], + "score": 0.87, + "content": "\\langle \\eta = 0 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 374, + 416, + 388 + ], + "score": 1.0, + "content": ") and low entropy", + "type": "text" + }, + { + "bbox": [ + 417, + 376, + 452, + 387 + ], + "score": 0.87, + "content": "\\hphantom { - } \\eta = - 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "regimes and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "for different number of samples/model evaluations. This suggests that it combines the advantages", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 396, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 478, + 410 + ], + "score": 1.0, + "content": "of the other estimators. We also ran the actual optimization experiment, where with as few as", + "type": "text" + }, + { + "bbox": [ + 479, + 397, + 504, + 407 + ], + "score": 0.88, + "content": "k = 3", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 495, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 495, + 421 + ], + "score": 1.0, + "content": "samples the trajectory was indistinguishable from using the exact gradient (see Liu et al. (2019)).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 432, + 329, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 330, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 330, + 445 + ], + "score": 1.0, + "content": "4.2 CATEGORICAL VARIATIONAL AUTO-ENCODER", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "We use the code from Yin et al. (2019) to train a categorical Variational Auto-Encoder (VAE) with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 463, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 478 + ], + "score": 1.0, + "content": "20 dimensional latent space, with 10 categories per dimension (details in Appendix G.1). To use", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 376, + 488 + ], + "score": 1.0, + "content": "our estimator, we treat this as a single factorized distribution with", + "type": "text" + }, + { + "bbox": [ + 376, + 475, + 396, + 486 + ], + "score": 0.88, + "content": "1 0 ^ { 2 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "categories from which we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "can sample without replacement using Stochastic Beam Search (Kool et al., 2019c), sequentially", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 496, + 504, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 488, + 510 + ], + "score": 1.0, + "content": "sampling each dimension as if it were a sequence model. We also perform experiments with", + "type": "text" + }, + { + "bbox": [ + 488, + 497, + 504, + 507 + ], + "score": 0.85, + "content": "1 0 ^ { \\dot { 2 } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 507, + 492, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 492, + 522 + ], + "score": 1.0, + "content": "latent space, which provides a lower entropy setting, to highlight the advantage of our estimator.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "score": 1.0, + "content": "Measuring the variance. In Table 1, we report the variance of different gradient estimators with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 107, + 543, + 133, + 553 + ], + "score": 0.89, + "content": "k = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "samples, evaluated on a trained model. The unordered set estimator has the lowest variance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "in both the small and large domain (low and high entropy) setting, being on-par with the best of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "the (stochastic3) sum-and-sample estimator and REINFORCE with replacement4. This confirms the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "toy experiment, suggesting that the unordered set estimator provides the best of both estimators. In", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 587, + 492, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 492, + 599 + ], + "score": 1.0, + "content": "Appendix G.2 we repeat the same experiment at different stages of training, with similar results.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 117, + 648, + 496, + 669 + ], + "lines": [ + { + "bbox": [ + 118, + 645, + 497, + 660 + ], + "spans": [ + { + "bbox": [ + 118, + 645, + 448, + 660 + ], + "score": 1.0, + "content": "3We cannot use the deterministic version by Liu et al. (2019) since we cannot select the top", + "type": "text" + }, + { + "bbox": [ + 449, + 648, + 455, + 657 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 645, + 497, + 660 + ], + "score": 1.0, + "content": "categories.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 656, + 479, + 672 + ], + "spans": [ + { + "bbox": [ + 118, + 656, + 479, + 672 + ], + "score": 1.0, + "content": "4We cannot compare against VIMCO (Mnih & Rezende, 2016) as it optimizes a different objective.", + "type": "text" + } + ] + } + ] + }, + { + "type": "table_body", + "bbox": [ + 125, + 696, + 486, + 742 + ], + "group_id": 0, + "lines": [] + }, + { + "type": "table_caption", + "bbox": [ + 127, + 676, + 483, + 689 + ], + "group_id": 0, + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 71, + 494, + 174 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 71, + 494, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 71, + 494, + 174 + ], + "spans": [ + { + "bbox": [ + 115, + 71, + 494, + 174 + ], + "score": 0.971, + "type": "image", + "image_path": "1827e94fdb564b2088e10a208547d35fffe208d74d3041a8d194d3868ada9f35.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 71, + 494, + 105.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 105.33333333333334, + 494, + 139.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 139.66666666666669, + 494, + 174.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 179, + 505, + 212 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "Figure 1: Bernoulli gradient variance (on log scale) as a function of the number of model evaluations", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 203 + ], + "score": 1.0, + "content": "(including baseline evaluations, so the sum-and-sample estimators with sampled baselines use twice", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 439, + 213 + ], + "score": 1.0, + "content": "as many evaluations). Note that for some estimators, the variance is 0 (log variance", + "type": "text" + }, + { + "bbox": [ + 439, + 201, + 460, + 212 + ], + "score": 0.8, + "content": "- \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 201, + 476, + 213 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 477, + 201, + 501, + 210 + ], + "score": 0.89, + "content": "k = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 201, + 505, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 200, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 201, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 201, + 241 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 251, + 260, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 262, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 262, + 263 + ], + "score": 1.0, + "content": "4.1 BERNOULLI TOY EXPERIMENT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 359 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 486, + 286 + ], + "score": 1.0, + "content": "We use the code by Liu et al. (2019) to reproduce their Bernoulli toy experiment. Given a vector", + "type": "text" + }, + { + "bbox": [ + 486, + 274, + 505, + 284 + ], + "score": 0.83, + "content": "\\mathbf { p } =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 279, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 307, + 305 + ], + "score": 1.0, + "content": "(0.6, 0.51, 0.48) the goal is to minimize the loss", + "type": "text" + }, + { + "bbox": [ + 308, + 283, + 501, + 303 + ], + "score": 0.82, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\eta ) \\ : = \\ : \\mathbb { E } _ { x _ { 1 } , x _ { 2 } , x _ { 3 } \\sim \\mathrm { B e r n } ( \\sigma ( \\eta ) ) } \\left[ \\sum _ { i = 1 } ^ { 3 } ( x _ { i } - p _ { i } ) ^ { 2 } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 279, + 506, + 305 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 128, + 314 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 304, + 169, + 313 + ], + "score": 0.87, + "content": "x _ { 1 } , x _ { 2 } , x _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 300, + 286, + 314 + ], + "score": 1.0, + "content": "are i.i.d. from the Bernoulli", + "type": "text" + }, + { + "bbox": [ + 286, + 301, + 313, + 314 + ], + "score": 0.87, + "content": "( \\sigma ( \\eta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 300, + 473, + 314 + ], + "score": 1.0, + "content": "distribution, parameterized by a scalar", + "type": "text" + }, + { + "bbox": [ + 473, + 303, + 501, + 313 + ], + "score": 0.82, + "content": "\\eta \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 300, + 505, + 314 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 133, + 325 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 312, + 236, + 325 + ], + "score": 0.93, + "content": "\\sigma ( \\eta ) = ( 1 + \\exp ( - \\eta ) ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "is the sigmoid function. We compare different estimators, with and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "without baseline (either ‘built-in’ or using additional samples, referred to as REINFORCE+ in Liu", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 333, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 367, + 349 + ], + "score": 1.0, + "content": "et al. (2019)). We report the (log-)variance of the scalar gradient", + "type": "text" + }, + { + "bbox": [ + 367, + 333, + 380, + 349 + ], + "score": 0.9, + "content": "\\frac { \\partial \\mathcal { L } } { \\partial \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 333, + 506, + 349 + ], + "score": 1.0, + "content": "as a function of the number of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 346, + 465, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 465, + 360 + ], + "score": 1.0, + "content": "model evaluations, which is twice as high when using a sampled baseline (for each term).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 271, + 506, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 377 + ], + "score": 1.0, + "content": "As can be seen in Figure 1, the unordered set estimator is the only estimator that has consistently", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 318, + 388 + ], + "score": 1.0, + "content": "the lowest (or comparable) variance in both the high", + "type": "text" + }, + { + "bbox": [ + 318, + 375, + 344, + 387 + ], + "score": 0.87, + "content": "\\langle \\eta = 0 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 374, + 416, + 388 + ], + "score": 1.0, + "content": ") and low entropy", + "type": "text" + }, + { + "bbox": [ + 417, + 376, + 452, + 387 + ], + "score": 0.87, + "content": "\\hphantom { - } \\eta = - 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "regimes and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "for different number of samples/model evaluations. This suggests that it combines the advantages", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 396, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 478, + 410 + ], + "score": 1.0, + "content": "of the other estimators. We also ran the actual optimization experiment, where with as few as", + "type": "text" + }, + { + "bbox": [ + 479, + 397, + 504, + 407 + ], + "score": 0.88, + "content": "k = 3", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 495, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 495, + 421 + ], + "score": 1.0, + "content": "samples the trajectory was indistinguishable from using the exact gradient (see Liu et al. (2019)).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 363, + 505, + 421 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 432, + 329, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 330, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 330, + 445 + ], + "score": 1.0, + "content": "4.2 CATEGORICAL VARIATIONAL AUTO-ENCODER", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "We use the code from Yin et al. (2019) to train a categorical Variational Auto-Encoder (VAE) with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 463, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 478 + ], + "score": 1.0, + "content": "20 dimensional latent space, with 10 categories per dimension (details in Appendix G.1). To use", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 376, + 488 + ], + "score": 1.0, + "content": "our estimator, we treat this as a single factorized distribution with", + "type": "text" + }, + { + "bbox": [ + 376, + 475, + 396, + 486 + ], + "score": 0.88, + "content": "1 0 ^ { 2 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "categories from which we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "can sample without replacement using Stochastic Beam Search (Kool et al., 2019c), sequentially", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 496, + 504, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 488, + 510 + ], + "score": 1.0, + "content": "sampling each dimension as if it were a sequence model. We also perform experiments with", + "type": "text" + }, + { + "bbox": [ + 488, + 497, + 504, + 507 + ], + "score": 0.85, + "content": "1 0 ^ { \\dot { 2 } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 507, + 492, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 492, + 522 + ], + "score": 1.0, + "content": "latent space, which provides a lower entropy setting, to highlight the advantage of our estimator.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 453, + 506, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "score": 1.0, + "content": "Measuring the variance. In Table 1, we report the variance of different gradient estimators with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 107, + 543, + 133, + 553 + ], + "score": 0.89, + "content": "k = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "samples, evaluated on a trained model. The unordered set estimator has the lowest variance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "in both the small and large domain (low and high entropy) setting, being on-par with the best of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "the (stochastic3) sum-and-sample estimator and REINFORCE with replacement4. This confirms the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "toy experiment, suggesting that the unordered set estimator provides the best of both estimators. In", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 587, + 492, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 492, + 599 + ], + "score": 1.0, + "content": "Appendix G.2 we repeat the same experiment at different stages of training, with similar results.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 532, + 506, + 599 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 83, + 495, + 216 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 83, + 495, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 83, + 495, + 216 + ], + "spans": [ + { + "bbox": [ + 117, + 83, + 495, + 216 + ], + "score": 0.972, + "type": "image", + "image_path": "38fff474333f745080d8627c5fb4566e88f353059b7a66955110f7f26afb9b13.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 83, + 495, + 127.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 127.33333333333334, + 495, + 171.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 171.66666666666669, + 495, + 216.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 226, + 505, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Figure 2: VAE smoothed training curves (-ELBO) of two independent runs when training with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 210, + 249 + ], + "score": 1.0, + "content": "different estimators with", + "type": "text" + }, + { + "bbox": [ + 210, + 237, + 240, + 247 + ], + "score": 0.86, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 236, + 505, + 249 + ], + "score": 1.0, + "content": ", 4 or 8 (thicker lines) samples (ARSM has a variable number).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 248, + 489, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 489, + 260 + ], + "score": 1.0, + "content": "Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "ELBO optimization. We use different estimators to optimize the ELBO (details in Appendix", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "G.1). Additionally to the baselines by Yin et al. (2019) we compare against REINFORCE with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "replacement and the stochastic sum-and-sample estimator. In Figure 2 we observe that our estima-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "tor performs on par with REINFORCE with replacement (and built-in baseline, equation 23) and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 323, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 104, + 323, + 506, + 339 + ], + "score": 1.0, + "content": "outperforms other estimators in at least one of the settings. There are a lot of other factors, e.g.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "exploration that may explain why we do not get a strictly better result despite the lower variance.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 360 + ], + "score": 1.0, + "content": "We note some overfitting (see validation curves in Appendix G.2), but since our goal is to show", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "improved optimization, and to keep results directly comparable to Yin et al. (2019), we consider", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "regularization a separate issue outside the scope of this paper. These results are using MNIST bina-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 378, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 393 + ], + "score": 1.0, + "content": "rized by a threshold of 0.5. 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In Figure 3b, we plot the same results against the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "number of instances, which shows that, compared to the single sample estimators, we can train with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 634, + 497, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 497, + 647 + ], + "score": 1.0, + "content": "less data and less computational cost (as we only need to run the encoder once for each instance).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 722, + 416, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 720, + 417, + 733 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 417, + 733 + ], + "score": 1.0, + "content": "5https://github.com/wouterkool/attention-learn-to-route", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 83, + 495, + 216 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 83, + 495, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 83, + 495, + 216 + ], + "spans": [ + { + "bbox": [ + 117, + 83, + 495, + 216 + ], + "score": 0.972, + "type": "image", + "image_path": "38fff474333f745080d8627c5fb4566e88f353059b7a66955110f7f26afb9b13.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 83, + 495, + 127.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 127.33333333333334, + 495, + 171.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 171.66666666666669, + 495, + 216.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 226, + 505, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Figure 2: VAE smoothed training curves (-ELBO) of two independent runs when training with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 210, + 249 + ], + "score": 1.0, + "content": "different estimators with", + "type": "text" + }, + { + "bbox": [ + 210, + 237, + 240, + 247 + ], + "score": 0.86, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 236, + 505, + 249 + ], + "score": 1.0, + "content": ", 4 or 8 (thicker lines) samples (ARSM has a variable number).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 248, + 489, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 489, + 260 + ], + "score": 1.0, + "content": "Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "ELBO optimization. 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In Figure 2 we observe that our estima-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "tor performs on par with REINFORCE with replacement (and built-in baseline, equation 23) and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 323, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 104, + 323, + 506, + 339 + ], + "score": 1.0, + "content": "outperforms other estimators in at least one of the settings. There are a lot of other factors, e.g.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "exploration that may explain why we do not get a strictly better result despite the lower variance.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 360 + ], + "score": 1.0, + "content": "We note some overfitting (see validation curves in Appendix G.2), but since our goal is to show", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "improved optimization, and to keep results directly comparable to Yin et al. 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We use the code by Kool et al. (2019a)5 to reproduce their TSP", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 469, + 332, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 332, + 481 + ], + "score": 1.0, + "content": "experiment with 20 nodes. 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We would like to thank anonymous reviewers for their feed-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 487, + 251, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 251, + 501 + ], + "score": 1.0, + "content": "back that helped improve the paper.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 516, + 175, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 176, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 176, + 530 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 506, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "Dzmitry Bahdanau, Philemon Brakel, Kelvin Xu, Anirudh Goyal, Ryan Lowe, Joelle Pineau, Aaron", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 117, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 117, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "Courville, and Yoshua Bengio. 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Our estimator is the result of Rao-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "Blackwellizing three existing estimators, which guarantees equal or lower variance, and is closely", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "related to a number of other estimators. 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In future work, we want to investigate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "if we can apply our estimator to estimate gradients ‘locally’ (Titsias & Lazaro-Gredilla, 2015), as ´", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 433, + 386, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 386, + 447 + ], + "score": 1.0, + "content": "locally we have a smaller domain and expect more duplicate samples.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 367, + 506, + 447 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 458, + 200, + 469 + ], + "lines": [ + { + "bbox": [ + 107, + 459, + 200, + 469 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 200, + 469 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 503, + 499 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "This research was funded by ORTEC. 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Additionally", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 180, + 316, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 316, + 193 + ], + "score": 1.0, + "content": "we will use the identities by Maddison et al. 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(2019c)):", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 268, + 442, + 381 + ], + "lines": [ + { + "bbox": [ + 168, + 268, + 442, + 381 + ], + "spans": [ + { + "bbox": [ + 168, + 268, + 442, + 381 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\phi _ { i } = \\log p ( i ) } \\\\ & { \\displaystyle \\phi _ { S } = \\log \\displaystyle \\sum _ { i \\in S } p ( i ) = \\log \\sum _ { i \\in S } \\exp \\phi _ { i } } \\\\ & { \\displaystyle \\phi _ { D \\setminus S } = \\log \\sum _ { i \\in D \\setminus S } p ( i ) = \\log \\left( 1 - \\sum _ { i \\in S } p ( i ) \\right) = \\log ( 1 - \\exp ( \\phi _ { S } ) ) } \\\\ & { \\displaystyle { G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { i } ) } } \\\\ & { \\displaystyle { G _ { \\phi _ { S } } = \\sum _ { i \\in S } G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { S } ) } } \\end{array}", + "type": "interline_equation", + "image_path": "74b04d313c6fa9193ba14f401130ebf9ca9cdbf24b7db98e9027266e865313ae.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 168, + 268, + 442, + 305.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 168, + 305.6666666666667, + 442, + 343.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 168, + 343.33333333333337, + 442, + 381.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 326, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 327, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 327, + 399 + ], + "score": 1.0, + "content": "For a proof of equation 30, see Maddison et al. 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Omitting the superscript", + "type": "text" + }, + { + "bbox": [ + 279, + 474, + 285, + 484 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "for clarity, we can use the Gumbel-Max trick, i.e. that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 482, + 452, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 275, + 500 + ], + "score": 0.89, + "content": "G _ { \\phi _ { D \\setminus S } } = \\operatorname* { m a x } _ { i \\notin S } G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { D \\setminus S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 482, + 417, + 503 + ], + "score": 1.0, + "content": "(equation 30) and marginalize over", + "type": "text" + }, + { + "bbox": [ + 418, + 486, + 446, + 500 + ], + "score": 0.93, + "content": "G _ { \\phi _ { D \\setminus S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 482, + 452, + 503 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 504, + 439, + 635 + ], + "lines": [ + { + "bbox": [ + 169, + 504, + 439, + 635 + ], + "spans": [ + { + "bbox": [ + 169, + 504, + 439, + 635 + ], + "score": 0.97, + "content": "\\begin{array} { r l } { \\displaystyle p ( S ) = P ( \\underset { i \\in S } { \\operatorname* { m i n } } G _ { \\phi _ { i } \\setminus } \\ G _ { \\phi _ { D \\setminus S } } ) } \\\\ { \\displaystyle } & { = P ( G _ { \\phi _ { i } } > G _ { \\phi _ { D \\setminus S } } , i \\in S ) } \\\\ { \\displaystyle } & { = \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } ) P ( G _ { \\phi _ { i } } > g _ { \\phi _ { D \\setminus S } } , i \\in S ) \\partial g _ { \\phi _ { D \\setminus S } } } \\\\ { \\displaystyle } & { = \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } ) \\prod _ { i \\in S } \\left( 1 - F _ { \\phi _ { \\star } } ( g _ { \\phi _ { D \\setminus S } } ) \\right) \\partial g _ { \\phi _ { D \\setminus S } } } \\\\ { \\displaystyle } & { = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S } \\left( 1 - F _ { \\phi _ { i } } \\left( F _ { \\phi _ { D \\setminus S } } ^ { - 1 } ( u ) \\right) \\right) \\partial u } \\end{array}", + "type": "interline_equation", + "image_path": "806f97cd9da1c0ad1baa8ecdc50026ff7932d71042f7f521a043bd018768c996.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 169, + 504, + 439, + 547.6666666666666 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 169, + 547.6666666666666, + 439, + 591.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 169, + 591.3333333333333, + 439, + 634.9999999999999 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 504, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 277, + 655 + ], + "score": 1.0, + "content": "Here we have used a change of variables", + "type": "text" + }, + { + "bbox": [ + 278, + 640, + 358, + 654 + ], + "score": 0.93, + "content": "u = F _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 639, + 506, + 655 + ], + "score": 1.0, + "content": ". 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(2014):", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 105, + 469, + 120 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 122, + 369, + 166 + ], + "lines": [ + { + "bbox": [ + 242, + 122, + 369, + 166 + ], + "spans": [ + { + "bbox": [ + 242, + 122, + 369, + 166 + ], + "score": 0.93, + "content": "\\begin{array} { l } { { e _ { \\phi } ( g ) = \\exp ( - g + \\phi ) } } \\\\ { { F _ { \\phi } ( g ) = \\exp ( - \\exp ( - g + \\phi ) ) } } \\\\ { { f _ { \\phi } ( g ) = e _ { \\phi } ( g ) F _ { \\phi } ( g ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "821cbe0666f811f0f17fdbd24322d0cfd6934666b30ce92d9f2fdccfc37edb08.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 242, + 122, + 369, + 144.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 242, + 144.0, + 369, + 166.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 504, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 173, + 182 + ], + "score": 1.0, + "content": "This means that", + "type": "text" + }, + { + "bbox": [ + 173, + 169, + 198, + 182 + ], + "score": 0.93, + "content": "F _ { \\phi } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 168, + 264, + 182 + ], + "score": 1.0, + "content": "is the CDF and", + "type": "text" + }, + { + "bbox": [ + 264, + 169, + 288, + 182 + ], + "score": 0.93, + "content": "f _ { \\phi } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 168, + 384, + 182 + ], + "score": 1.0, + "content": "the PDF of the Gumbel", + "type": "text" + }, + { + "bbox": [ + 384, + 169, + 398, + 181 + ], + "score": 0.8, + "content": "\\left( \\phi \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 168, + 505, + 182 + ], + "score": 1.0, + "content": "distribution. Additionally", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 180, + 316, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 316, + 193 + ], + "score": 1.0, + "content": "we will use the identities by Maddison et al. 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(2019c)):", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 251, + 474, + 266 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 268, + 442, + 381 + ], + "lines": [ + { + "bbox": [ + 168, + 268, + 442, + 381 + ], + "spans": [ + { + "bbox": [ + 168, + 268, + 442, + 381 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\phi _ { i } = \\log p ( i ) } \\\\ & { \\displaystyle \\phi _ { S } = \\log \\displaystyle \\sum _ { i \\in S } p ( i ) = \\log \\sum _ { i \\in S } \\exp \\phi _ { i } } \\\\ & { \\displaystyle \\phi _ { D \\setminus S } = \\log \\sum _ { i \\in D \\setminus S } p ( i ) = \\log \\left( 1 - \\sum _ { i \\in S } p ( i ) \\right) = \\log ( 1 - \\exp ( \\phi _ { S } ) ) } \\\\ & { \\displaystyle { G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { i } ) } } \\\\ & { \\displaystyle { G _ { \\phi _ { S } } = \\sum _ { i \\in S } G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { S } ) } } \\end{array}", + "type": "interline_equation", + "image_path": "74b04d313c6fa9193ba14f401130ebf9ca9cdbf24b7db98e9027266e865313ae.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 168, + 268, + 442, + 305.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 168, + 305.6666666666667, + 442, + 343.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 168, + 343.33333333333337, + 442, + 381.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 326, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 327, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 327, + 399 + ], + "score": 1.0, + "content": "For a proof of equation 30, see Maddison et al. (2014).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 385, + 327, + 399 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 411, + 392, + 428 + ], + "lines": [ + { + "bbox": [ + 223, + 411, + 392, + 428 + ], + "spans": [ + { + "bbox": [ + 223, + 411, + 392, + 428 + ], + "score": 0.55, + "content": "p ( S ^ { k } ) , p ^ { D \\setminus C } ( S \\setminus C ) \\ \\mathrm { A N D } \\ R ( S ^ { k } , s )", + "type": "interline_equation", + "image_path": "ebdb2bbb39718391052a7e6fd23cc99c971cdbc1773a9535484decfa338d1f69.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 223, + 411, + 392, + 428 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 203, + 452 + ], + "score": 1.0, + "content": "We can sample the set", + "type": "text" + }, + { + "bbox": [ + 204, + 439, + 216, + 450 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 438, + 461, + 452 + ], + "score": 1.0, + "content": "from the Plackett-Luce distribution using the Gumbel-Top-", + "type": "text" + }, + { + "bbox": [ + 462, + 440, + 468, + 450 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "trick by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 450, + 504, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 215, + 464 + ], + "score": 1.0, + "content": "drawing Gumbel variables", + "type": "text" + }, + { + "bbox": [ + 216, + 451, + 297, + 463 + ], + "score": 0.89, + "content": "G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 450, + 497, + 464 + ], + "score": 1.0, + "content": "for each element and returning the indices of the", + "type": "text" + }, + { + "bbox": [ + 497, + 451, + 504, + 460 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 416, + 476 + ], + "score": 1.0, + "content": "largest Gumbels. If we ignore the ordering, this means we will obtain the set", + "type": "text" + }, + { + "bbox": [ + 416, + 462, + 429, + 473 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 461, + 439, + 476 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 439, + 462, + 505, + 475 + ], + "score": 0.72, + "content": "\\mathrm { m i n } _ { i \\in S ^ { k } } G _ { \\phi _ { i } } >", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 107, + 474, + 173, + 487 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } _ { i \\in D \\setminus S ^ { k } } G _ { \\phi _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 474, + 278, + 486 + ], + "score": 1.0, + "content": ". Omitting the superscript", + "type": "text" + }, + { + "bbox": [ + 279, + 474, + 285, + 484 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "for clarity, we can use the Gumbel-Max trick, i.e. that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 482, + 452, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 275, + 500 + ], + "score": 0.89, + "content": "G _ { \\phi _ { D \\setminus S } } = \\operatorname* { m a x } _ { i \\notin S } G _ { \\phi _ { i } } \\sim \\mathrm { G u m b e l } ( \\phi _ { D \\setminus S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 482, + 417, + 503 + ], + "score": 1.0, + "content": "(equation 30) and marginalize over", + "type": "text" + }, + { + "bbox": [ + 418, + 486, + 446, + 500 + ], + "score": 0.93, + "content": "G _ { \\phi _ { D \\setminus S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 482, + 452, + 503 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 438, + 506, + 503 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 504, + 439, + 635 + ], + "lines": [ + { + "bbox": [ + 169, + 504, + 439, + 635 + ], + "spans": [ + { + "bbox": [ + 169, + 504, + 439, + 635 + ], + "score": 0.97, + "content": "\\begin{array} { r l } { \\displaystyle p ( S ) = P ( \\underset { i \\in S } { \\operatorname* { m i n } } G _ { \\phi _ { i } \\setminus } \\ G _ { \\phi _ { D \\setminus S } } ) } \\\\ { \\displaystyle } & { = P ( G _ { \\phi _ { i } } > G _ { \\phi _ { D \\setminus S } } , i \\in S ) } \\\\ { \\displaystyle } & { = \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } ) P ( G _ { \\phi _ { i } } > g _ { \\phi _ { D \\setminus S } } , i \\in S ) \\partial g _ { \\phi _ { D \\setminus S } } } \\\\ { \\displaystyle } & { = \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } ) \\prod _ { i \\in S } \\left( 1 - F _ { \\phi _ { \\star } } ( g _ { \\phi _ { D \\setminus S } } ) \\right) \\partial g _ { \\phi _ { D \\setminus S } } } \\\\ { \\displaystyle } & { = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S } \\left( 1 - F _ { \\phi _ { i } } \\left( F _ { \\phi _ { D \\setminus S } } ^ { - 1 } ( u ) \\right) \\right) \\partial u } \\end{array}", + "type": "interline_equation", + "image_path": "806f97cd9da1c0ad1baa8ecdc50026ff7932d71042f7f521a043bd018768c996.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 169, + 504, + 439, + 547.6666666666666 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 169, + 547.6666666666666, + 439, + 591.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 169, + 591.3333333333333, + 439, + 634.9999999999999 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 504, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 277, + 655 + ], + "score": 1.0, + "content": "Here we have used a change of variables", + "type": "text" + }, + { + "bbox": [ + 278, + 640, + 358, + 654 + ], + "score": 0.93, + "content": "u = F _ { \\phi _ { D \\setminus S } } ( g _ { \\phi _ { D \\setminus S } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 639, + 506, + 655 + ], + "score": 1.0, + "content": ". 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This means that we can compute", + "type": "text" + }, + { + "bbox": [ + 295, + 340, + 350, + 354 + ], + "score": 0.94, + "content": "p ^ { D \\setminus C } ( S \\setminus C )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 339, + 445, + 354 + ], + "score": 1.0, + "content": "similar to equation 31:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 358, + 463, + 435 + ], + "lines": [ + { + "bbox": [ + 148, + 358, + 463, + 435 + ], + "spans": [ + { + "bbox": [ + 148, + 358, + 463, + 435 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\displaystyle p ^ { D \\setminus C } ( S \\setminus C ) = P ( \\underset { i \\in S \\backslash C } { \\mathrm { m i n } } G _ { \\phi _ { i } } > G _ { \\phi _ { ( D \\setminus C ) \\setminus ( S \\setminus C ) } } ) } & { } \\\\ { = P ( \\underset { i \\in S \\backslash C } { \\mathrm { m i n } } G _ { \\phi _ { i } } > G _ { \\phi _ { D \\setminus S } } ) } & { } \\\\ { = \\displaystyle \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } \\big ( g _ { \\phi _ { D \\setminus S } } \\big ) \\prod _ { i \\in S \\backslash C } \\left( 1 - F _ { \\phi _ { i } } \\big ( g _ { \\phi _ { D \\setminus S } } \\big ) \\right) \\partial g _ { \\phi _ { D \\setminus S } } . } \\end{array}", + "type": "interline_equation", + "image_path": "71b1ede5eb7d716390ae712d93a77350d15fd8d99080e5f5d9d65b83e209d14e.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 148, + 358, + 463, + 383.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 148, + 383.6666666666667, + 463, + 409.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 148, + 409.33333333333337, + 463, + 435.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 392, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 393, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 176, + 461 + ], + "score": 1.0, + "content": "Computation of", + "type": "text" + }, + { + "bbox": [ + 177, + 447, + 213, + 460 + ], + "score": 0.94, + "content": "R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 445, + 393, + 461 + ], + "score": 1.0, + "content": ". Note that, using equation 10, it holds that", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 464, + 417, + 497 + ], + "lines": [ + { + "bbox": [ + 194, + 464, + 417, + 497 + ], + "spans": [ + { + "bbox": [ + 194, + 464, + 417, + 497 + ], + "score": 0.93, + "content": "\\sum _ { s \\in S ^ { k } } { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s ) } { p ( S ^ { k } ) } } = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) = 1", + "type": "interline_equation", + "image_path": "37fa10b9c1fd720eb745b828e611bb6706d79858280070ca02012d93d4503d8a.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 464, + 417, + 480.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 194, + 480.5, + 417, + 497.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 212, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 213, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 213, + 514 + ], + "score": 1.0, + "content": "from which it follows that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 509, + 381, + 537 + ], + "lines": [ + { + "bbox": [ + 230, + 509, + 381, + 537 + ], + "spans": [ + { + "bbox": [ + 230, + 509, + 381, + 537 + ], + "score": 0.91, + "content": "p ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s )", + "type": "interline_equation", + "image_path": "2a4ec5a0625a14701521c45a48439b3f0ca24500a7cee376c7f650e285dca087.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 230, + 509, + 381, + 537 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 144, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 146, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 146, + 551 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 546, + 443, + 577 + ], + "lines": [ + { + "bbox": [ + 168, + 546, + 443, + 577 + ], + "spans": [ + { + "bbox": [ + 168, + 546, + 443, + 577 + ], + "score": 0.93, + "content": "R ( S ^ { k } , s ) = { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } = { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ^ { D \\setminus \\{ s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s ^ { \\prime } \\} ) p ( s ^ { \\prime } ) } } .", + "type": "interline_equation", + "image_path": "410624f2557ad5303a82af31b26e72b713b414348fbc1f7ef00af886da23f8f8.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 168, + 546, + 443, + 556.3333333333334 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 168, + 556.3333333333334, + 443, + 566.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 168, + 566.6666666666667, + 443, + 577.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 355, + 592 + ], + "score": 1.0, + "content": "This means that, to compute the leave-one-out ratio for all", + "type": "text" + }, + { + "bbox": [ + 356, + 579, + 390, + 591 + ], + "score": 0.91, + "content": "s ~ \\in ~ S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 578, + 505, + 592 + ], + "score": 1.0, + "content": ", we only need to compute", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 589, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 107, + 591, + 177, + 604 + ], + "score": 0.92, + "content": "p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 589, + 192, + 605 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 192, + 591, + 221, + 602 + ], + "score": 0.93, + "content": "s \\in S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 589, + 444, + 605 + ], + "score": 1.0, + "content": ". When using the numerical integration or summation in", + "type": "text" + }, + { + "bbox": [ + 445, + 591, + 471, + 604 + ], + "score": 0.92, + "content": "O ( 2 ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 589, + 506, + 605 + ], + "score": 1.0, + "content": ", we can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 358, + 616 + ], + "score": 1.0, + "content": "reuse computation, whereas using the naive method, the cost is", + "type": "text" + }, + { + "bbox": [ + 358, + 603, + 455, + 615 + ], + "score": 0.91, + "content": "O ( k \\cdot ( k - 1 ) ! ) = O ( k ! )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 602, + 505, + 616 + ], + "score": 1.0, + "content": ", making the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 610, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 327, + 629 + ], + "score": 1.0, + "content": "total computational cost comparable to computing just", + "type": "text" + }, + { + "bbox": [ + 327, + 613, + 352, + 626 + ], + "score": 0.92, + "content": "p ( S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 610, + 506, + 629 + ], + "score": 1.0, + "content": ", and the same holds when computing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 624, + 415, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 415, + 638 + ], + "score": 1.0, + "content": "the ‘second-order’ leave one out ratios for the built-in baseline (equation 17).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 156, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 158, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 158, + 96 + ], + "score": 1.0, + "content": "which gives", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 158, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 99, + 471, + 296 + ], + "lines": [ + { + "bbox": [ + 138, + 99, + 471, + 296 + ], + "spans": [ + { + "bbox": [ + 138, + 99, + 471, + 296 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { { \\mu ( S ) = \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } \\longrightarrow \\int _ { \\partial \\sigma _ { \\phi ( S ) } } \\bigl ( g _ { \\phi ( S ) , s } \\bigr ) \\prod ( 1 - F _ { \\phi _ { \\phi } } ( g _ { \\phi ( S ) , s } ) ) \\partial g _ { \\phi _ { \\phi ( S ) , s } } } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } \\big | \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } \\int _ { \\Phi _ { \\phi ( S ) - } ( \\partial g _ { \\phi ( S ) - } ) } \\prod \\int _ { s _ { \\phi } } F _ { \\phi _ { \\phi } } ( g _ { \\phi ( S ) - s } ) \\prod \\big | \\partial g _ { \\phi _ { \\phi ( S ) - } s } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } \\big | \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } e _ { \\phi _ { \\phi ( S ) - } ( \\partial g _ { \\phi ( S ) - } ) } F _ { \\phi _ { \\phi _ { \\phi ( S ) - } s } } ( g _ { \\phi _ { S } , s } ) F _ { \\phi _ { \\phi ( S ) - } } ( g _ { \\phi _ { S } , s } ) \\big | \\mathcal { P } _ { \\phi _ { \\phi ( S ) - } } \\big | \\mathcal { P } _ { \\phi _ { \\phi ( S ) - } } \\big | \\mathcal { P } _ { \\phi _ { \\phi ( S ) - } } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } \\big | \\int _ { s _ { \\phi ( S ) - } - \\infty } ^ { \\infty } e _ { \\phi _ { \\phi ( S ) - } ( \\partial g _ { \\phi ( S ) - } ) } F _ { \\phi _ { \\phi ( S ) - } , \\phi _ { \\phi ( S ) - } } ( g _ { \\phi _ { S } , s } ) \\big | \\partial g _ { \\phi _ { \\phi ( S ) - } } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } ( 1 - 0 ) \\frac { \\exp ( \\phi _ { \\phi ( S ) - } ) } { \\exp ( \\phi _ { \\phi ( S ) - } ) } } \\\\ & { = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } ( 1 - 0 ) \\exp ( \\phi _ { \\phi ( S ) - } ) } \\\\ & = \\sum _ { c \\in S } ^ { \\infty } ( - 1 ) ^ { C } ( 1 - \\sum \\end{array}", + "type": "interline_equation", + "image_path": "7ba5abea6b75d40021fc45a9d7e85ccd6775706b57325b46c30787c210537343.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 138, + 99, + 471, + 164.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 138, + 164.66666666666669, + 471, + 230.33333333333337 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 138, + 230.33333333333337, + 471, + 296.00000000000006 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 305, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 178, + 321 + ], + "score": 1.0, + "content": "Computation of", + "type": "text" + }, + { + "bbox": [ + 178, + 305, + 234, + 319 + ], + "score": 0.93, + "content": "p ^ { D \\setminus C } ( S \\setminus C )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 304, + 366, + 321 + ], + "score": 1.0, + "content": ". When using the Gumbel-Top-", + "type": "text" + }, + { + "bbox": [ + 367, + 307, + 373, + 317 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 304, + 506, + 321 + ], + "score": 1.0, + "content": "trick over the restricted domain", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 316, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 318, + 133, + 330 + ], + "score": 0.91, + "content": "D \\backslash C", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 316, + 344, + 331 + ], + "score": 1.0, + "content": ", we do not need to renormalize the log-probabilities", + "type": "text" + }, + { + "bbox": [ + 345, + 318, + 404, + 329 + ], + "score": 0.87, + "content": "\\phi _ { s } , s \\in D \\setminus C", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 316, + 497, + 331 + ], + "score": 1.0, + "content": "since the Gumbel-Top-", + "type": "text" + }, + { + "bbox": [ + 497, + 318, + 504, + 328 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 327, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 360, + 343 + ], + "score": 1.0, + "content": "trick applies to unnormalized log-probabilities. Also, assuming", + "type": "text" + }, + { + "bbox": [ + 360, + 329, + 393, + 340 + ], + "score": 0.88, + "content": "C \\subseteq S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 327, + 446, + 343 + ], + "score": 1.0, + "content": ", it holds that", + "type": "text" + }, + { + "bbox": [ + 447, + 329, + 505, + 341 + ], + "score": 0.89, + "content": "\\left( D \\backslash C \\right) \\backslash ( \\mathbf { \\bar { S } } )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 339, + 445, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 158, + 353 + ], + "score": 0.92, + "content": "C ) = D \\backslash S", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 339, + 294, + 354 + ], + "score": 1.0, + "content": ". This means that we can compute", + "type": "text" + }, + { + "bbox": [ + 295, + 340, + 350, + 354 + ], + "score": 0.94, + "content": "p ^ { D \\setminus C } ( S \\setminus C )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 339, + 445, + 354 + ], + "score": 1.0, + "content": "similar to equation 31:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 304, + 506, + 354 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 358, + 463, + 435 + ], + "lines": [ + { + "bbox": [ + 148, + 358, + 463, + 435 + ], + "spans": [ + { + "bbox": [ + 148, + 358, + 463, + 435 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\displaystyle p ^ { D \\setminus C } ( S \\setminus C ) = P ( \\underset { i \\in S \\backslash C } { \\mathrm { m i n } } G _ { \\phi _ { i } } > G _ { \\phi _ { ( D \\setminus C ) \\setminus ( S \\setminus C ) } } ) } & { } \\\\ { = P ( \\underset { i \\in S \\backslash C } { \\mathrm { m i n } } G _ { \\phi _ { i } } > G _ { \\phi _ { D \\setminus S } } ) } & { } \\\\ { = \\displaystyle \\int _ { g _ { \\phi _ { D \\setminus S } } = - \\infty } ^ { \\infty } f _ { \\phi _ { D \\setminus S } } \\big ( g _ { \\phi _ { D \\setminus S } } \\big ) \\prod _ { i \\in S \\backslash C } \\left( 1 - F _ { \\phi _ { i } } \\big ( g _ { \\phi _ { D \\setminus S } } \\big ) \\right) \\partial g _ { \\phi _ { D \\setminus S } } . } \\end{array}", + "type": "interline_equation", + "image_path": "71b1ede5eb7d716390ae712d93a77350d15fd8d99080e5f5d9d65b83e209d14e.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 148, + 358, + 463, + 383.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 148, + 383.6666666666667, + 463, + 409.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 148, + 409.33333333333337, + 463, + 435.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 392, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 393, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 176, + 461 + ], + "score": 1.0, + "content": "Computation of", + "type": "text" + }, + { + "bbox": [ + 177, + 447, + 213, + 460 + ], + "score": 0.94, + "content": "R ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 445, + 393, + 461 + ], + "score": 1.0, + "content": ". Note that, using equation 10, it holds that", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 445, + 393, + 461 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 464, + 417, + 497 + ], + "lines": [ + { + "bbox": [ + 194, + 464, + 417, + 497 + ], + "spans": [ + { + "bbox": [ + 194, + 464, + 417, + 497 + ], + "score": 0.93, + "content": "\\sum _ { s \\in S ^ { k } } { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s ) } { p ( S ^ { k } ) } } = \\sum _ { s \\in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) = 1", + "type": "interline_equation", + "image_path": "37fa10b9c1fd720eb745b828e611bb6706d79858280070ca02012d93d4503d8a.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 464, + 417, + 480.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 194, + 480.5, + 417, + 497.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 212, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 213, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 213, + 514 + ], + "score": 1.0, + "content": "from which it follows that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 501, + 213, + 514 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 509, + 381, + 537 + ], + "lines": [ + { + "bbox": [ + 230, + 509, + 381, + 537 + ], + "spans": [ + { + "bbox": [ + 230, + 509, + 381, + 537 + ], + "score": 0.91, + "content": "p ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } } p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) p ( s )", + "type": "interline_equation", + "image_path": "2a4ec5a0625a14701521c45a48439b3f0ca24500a7cee376c7f650e285dca087.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 230, + 509, + 381, + 537 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 144, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 146, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 146, + 551 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 538, + 146, + 551 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 546, + 443, + 577 + ], + "lines": [ + { + "bbox": [ + 168, + 546, + 443, + 577 + ], + "spans": [ + { + "bbox": [ + 168, + 546, + 443, + 577 + ], + "score": 0.93, + "content": "R ( S ^ { k } , s ) = { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } = { \\frac { p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } { \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ^ { D \\setminus \\{ s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s ^ { \\prime } \\} ) p ( s ^ { \\prime } ) } } .", + "type": "interline_equation", + "image_path": "410624f2557ad5303a82af31b26e72b713b414348fbc1f7ef00af886da23f8f8.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 168, + 546, + 443, + 556.3333333333334 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 168, + 556.3333333333334, + 443, + 566.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 168, + 566.6666666666667, + 443, + 577.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 355, + 592 + ], + "score": 1.0, + "content": "This means that, to compute the leave-one-out ratio for all", + "type": "text" + }, + { + "bbox": [ + 356, + 579, + 390, + 591 + ], + "score": 0.91, + "content": "s ~ \\in ~ S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 578, + 505, + 592 + ], + "score": 1.0, + "content": ", we only need to compute", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 589, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 107, + 591, + 177, + 604 + ], + "score": 0.92, + "content": "p ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 589, + 192, + 605 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 192, + 591, + 221, + 602 + ], + "score": 0.93, + "content": "s \\in S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 589, + 444, + 605 + ], + "score": 1.0, + "content": ". 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For computation of the leave-one-out ratio (equation 35) for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 128, + 107 + ], + "score": 1.0, + "content": "large", + "type": "text" + }, + { + "bbox": [ + 129, + 94, + 135, + 104 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 93, + 464, + 107 + ], + "score": 1.0, + "content": "we can use the numerical integration, where we need to compute equation 34 with", + "type": "text" + }, + { + "bbox": [ + 464, + 93, + 501, + 106 + ], + "score": 0.93, + "content": "C = \\{ s \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 93, + 505, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 280, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 280, + 118 + ], + "score": 1.0, + "content": "For this purpose, we rewrite the integral as", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 123, + 477, + 295 + ], + "lines": [ + { + "bbox": [ + 134, + 123, + 477, + 295 + ], + "spans": [ + { + "bbox": [ + 134, + 123, + 477, + 295 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle p ^ { D \\setminus C } ( S \\setminus C ) = \\int _ { \\phi _ { \\partial \\setminus \\mathcal { D } } \\setminus \\phi = - \\phi } ^ { \\phi _ { \\partial \\setminus \\mathcal { D } } } f _ { \\phi _ { D \\setminus \\mathcal { D } } } ( g _ { \\phi \\phi _ { D \\setminus \\mathcal { D } } } , s ) \\prod _ { i \\in S \\setminus \\mathcal { C } } \\left( 1 - F _ { \\phi _ { i } } ( g _ { \\phi _ { D \\setminus i } } , s ) \\right) \\partial g _ { \\phi _ { D \\setminus i } } } \\\\ { \\displaystyle \\qquad = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - F _ { \\phi _ { i } } \\Big ( F _ { \\phi _ { D \\setminus i } } ^ { - 1 } \\Big ( u \\Big ) \\Big ) \\Big ) \\partial u } \\\\ { \\displaystyle \\qquad = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - u ^ { \\alpha \\setminus \\phi _ { i } ( \\phi _ { i } - \\phi _ { D \\setminus i } ) s } \\Big ) \\partial u } \\\\ { \\displaystyle \\qquad = \\exp ( b ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\phi _ { i } ( b ) - 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } - \\phi _ { D \\setminus i } - \\phi _ { i } ) } \\Big ) \\partial v } \\\\ { \\displaystyle \\qquad = \\exp ( a + \\phi _ { D \\setminus S } ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + \\phi _ { D \\setminus i } - s ) - 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + s ) } \\Big ) \\partial v } \\\\ { \\displaystyle \\qquad = \\exp ( a + \\phi _ { D \\setminus \\mathcal { D } } ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + \\phi _ { D \\setminus i } - s ) - 1 } v \\in \\mathrm { S } ^ { \\setminus \\Phi _ { i } } } \\end{array}", + "type": "interline_equation", + "image_path": "c310d2db9b1fa7110a785947e4cf56a00a645a76a168ecfb1126319fea07ae03.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 134, + 123, + 477, + 180.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 134, + 180.33333333333334, + 477, + 237.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 134, + 237.66666666666669, + 477, + 295.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 504, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 271, + 318 + ], + "score": 1.0, + "content": "Here we have used change of variables", + "type": "text" + }, + { + "bbox": [ + 271, + 303, + 329, + 314 + ], + "score": 0.91, + "content": "v = u ^ { e x p ( - b ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 301, + 349, + 318 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 304, + 414, + 317 + ], + "score": 0.92, + "content": "a = b - \\phi _ { D \\backslash S }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 301, + 506, + 318 + ], + "score": 1.0, + "content": ". This form allows to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 315, + 258, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 258, + 327 + ], + "score": 1.0, + "content": "compute the integrands efficiently, as", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 333, + 410, + 368 + ], + "lines": [ + { + "bbox": [ + 199, + 333, + 410, + 368 + ], + "spans": [ + { + "bbox": [ + 199, + 333, + 410, + 368 + ], + "score": 0.93, + "content": "\\prod _ { i \\in S \\setminus C } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) = \\frac { \\prod _ { i \\in S } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) } { \\prod _ { i \\in C } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) }", + "type": "interline_equation", + "image_path": "3ab65bbdffb221dc2e3ddf5ae942c5101495262c958abebcbf1015b13a972aee.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 333, + 410, + 350.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 199, + 350.5, + 410, + 368.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 504, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 366, + 390 + ], + "score": 1.0, + "content": "where the numerator only needs to computed once, and, since", + "type": "text" + }, + { + "bbox": [ + 366, + 376, + 407, + 388 + ], + "score": 0.93, + "content": "C = \\{ s \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 374, + 504, + 390 + ], + "score": 1.0, + "content": "when computing equa-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 387, + 328, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 328, + 400 + ], + "score": 1.0, + "content": "tion 35, the denominator only consists of a single term.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 167, + 417 + ], + "score": 1.0, + "content": "The choice of", + "type": "text" + }, + { + "bbox": [ + 168, + 407, + 174, + 414 + ], + "score": 0.65, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 403, + 371, + 417 + ], + "score": 1.0, + "content": "may depend on the setting, but we found that", + "type": "text" + }, + { + "bbox": [ + 371, + 405, + 401, + 415 + ], + "score": 0.9, + "content": "a \\ : = \\ : 5", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "is a good default option", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "which leads to an integral that is generally smooth and can be accurately approximated using the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "trapezoid rule. We compute the integrands in logarithmic space and sum the terms using the stable", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "LOGSUMEXP trick. In our code we provide an implementation which also computes all second-order", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 448, + 232, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 232, + 460 + ], + "score": 1.0, + "content": "leave-one-out ratios efficiently.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 478, + 309, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 312, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 312, + 493 + ], + "score": 1.0, + "content": "C THE SUM-AND-SAMPLE ESTIMATOR", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 505, + 368, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 369, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 369, + 517 + ], + "score": 1.0, + "content": "C.1 UNBIASEDNESS OF THE SUM-AND-SAMPLE ESTIMATOR", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 383, + 540 + ], + "score": 1.0, + "content": "We show that the sum-and-sample estimator is unbiased for any set", + "type": "text" + }, + { + "bbox": [ + 383, + 527, + 416, + 537 + ], + "score": 0.89, + "content": "C \\subset D", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 525, + 505, + 540 + ], + "score": 1.0, + "content": "(see also Liang et al.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 537, + 212, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 212, + 550 + ], + "score": 1.0, + "content": "(2018); Liu et al. 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For computation of the leave-one-out ratio (equation 35) for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 128, + 107 + ], + "score": 1.0, + "content": "large", + "type": "text" + }, + { + "bbox": [ + 129, + 94, + 135, + 104 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 93, + 464, + 107 + ], + "score": 1.0, + "content": "we can use the numerical integration, where we need to compute equation 34 with", + "type": "text" + }, + { + "bbox": [ + 464, + 93, + 501, + 106 + ], + "score": 0.93, + "content": "C = \\{ s \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 93, + 505, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 280, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 280, + 118 + ], + "score": 1.0, + "content": "For this purpose, we rewrite the integral as", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 118 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 123, + 477, + 295 + ], + "lines": [ + { + "bbox": [ + 134, + 123, + 477, + 295 + ], + "spans": [ + { + "bbox": [ + 134, + 123, + 477, + 295 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle p ^ { D \\setminus C } ( S \\setminus C ) = \\int _ { \\phi _ { \\partial \\setminus \\mathcal { D } } \\setminus \\phi = - \\phi } ^ { \\phi _ { \\partial \\setminus \\mathcal { D } } } f _ { \\phi _ { D \\setminus \\mathcal { D } } } ( g _ { \\phi \\phi _ { D \\setminus \\mathcal { D } } } , s ) \\prod _ { i \\in S \\setminus \\mathcal { C } } \\left( 1 - F _ { \\phi _ { i } } ( g _ { \\phi _ { D \\setminus i } } , s ) \\right) \\partial g _ { \\phi _ { D \\setminus i } } } \\\\ { \\displaystyle \\qquad = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - F _ { \\phi _ { i } } \\Big ( F _ { \\phi _ { D \\setminus i } } ^ { - 1 } \\Big ( u \\Big ) \\Big ) \\Big ) \\partial u } \\\\ { \\displaystyle \\qquad = \\int _ { u = 0 } ^ { 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - u ^ { \\alpha \\setminus \\phi _ { i } ( \\phi _ { i } - \\phi _ { D \\setminus i } ) s } \\Big ) \\partial u } \\\\ { \\displaystyle \\qquad = \\exp ( b ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\phi _ { i } ( b ) - 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } - \\phi _ { D \\setminus i } - \\phi _ { i } ) } \\Big ) \\partial v } \\\\ { \\displaystyle \\qquad = \\exp ( a + \\phi _ { D \\setminus S } ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + \\phi _ { D \\setminus i } - s ) - 1 } \\prod _ { i \\in S \\setminus \\mathcal { C } } \\Big ( 1 - v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + s ) } \\Big ) \\partial v } \\\\ { \\displaystyle \\qquad = \\exp ( a + \\phi _ { D \\setminus \\mathcal { D } } ) \\cdot \\int _ { v = 0 } ^ { 1 } v ^ { \\alpha \\setminus \\Phi _ { i } ( \\phi _ { i } + \\phi _ { D \\setminus i } - s ) - 1 } v \\in \\mathrm { S } ^ { \\setminus \\Phi _ { i } } } \\end{array}", + "type": "interline_equation", + "image_path": "c310d2db9b1fa7110a785947e4cf56a00a645a76a168ecfb1126319fea07ae03.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 134, + 123, + 477, + 180.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 134, + 180.33333333333334, + 477, + 237.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 134, + 237.66666666666669, + 477, + 295.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 504, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 271, + 318 + ], + "score": 1.0, + "content": "Here we have used change of variables", + "type": "text" + }, + { + "bbox": [ + 271, + 303, + 329, + 314 + ], + "score": 0.91, + "content": "v = u ^ { e x p ( - b ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 301, + 349, + 318 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 304, + 414, + 317 + ], + "score": 0.92, + "content": "a = b - \\phi _ { D \\backslash S }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 301, + 506, + 318 + ], + "score": 1.0, + "content": ". This form allows to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 315, + 258, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 258, + 327 + ], + "score": 1.0, + "content": "compute the integrands efficiently, as", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 301, + 506, + 327 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 333, + 410, + 368 + ], + "lines": [ + { + "bbox": [ + 199, + 333, + 410, + 368 + ], + "spans": [ + { + "bbox": [ + 199, + 333, + 410, + 368 + ], + "score": 0.93, + "content": "\\prod _ { i \\in S \\setminus C } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) = \\frac { \\prod _ { i \\in S } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) } { \\prod _ { i \\in C } \\left( 1 - v ^ { \\exp ( \\phi _ { i } + a ) } \\right) }", + "type": "interline_equation", + "image_path": "3ab65bbdffb221dc2e3ddf5ae942c5101495262c958abebcbf1015b13a972aee.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 333, + 410, + 350.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 199, + 350.5, + 410, + 368.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 504, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 366, + 390 + ], + "score": 1.0, + "content": "where the numerator only needs to computed once, and, since", + "type": "text" + }, + { + "bbox": [ + 366, + 376, + 407, + 388 + ], + "score": 0.93, + "content": "C = \\{ s \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 374, + 504, + 390 + ], + "score": 1.0, + "content": "when computing equa-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 387, + 328, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 328, + 400 + ], + "score": 1.0, + "content": "tion 35, the denominator only consists of a single term.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 374, + 504, + 400 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 167, + 417 + ], + "score": 1.0, + "content": "The choice of", + "type": "text" + }, + { + "bbox": [ + 168, + 407, + 174, + 414 + ], + "score": 0.65, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 403, + 371, + 417 + ], + "score": 1.0, + "content": "may depend on the setting, but we found that", + "type": "text" + }, + { + "bbox": [ + 371, + 405, + 401, + 415 + ], + "score": 0.9, + "content": "a \\ : = \\ : 5", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "is a good default option", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "which leads to an integral that is generally smooth and can be accurately approximated using the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "trapezoid rule. We compute the integrands in logarithmic space and sum the terms using the stable", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "LOGSUMEXP trick. In our code we provide an implementation which also computes all second-order", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 448, + 232, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 232, + 460 + ], + "score": 1.0, + "content": "leave-one-out ratios efficiently.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 403, + 505, + 460 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 478, + 309, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 312, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 312, + 493 + ], + "score": 1.0, + "content": "C THE SUM-AND-SAMPLE ESTIMATOR", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 505, + 368, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 369, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 369, + 517 + ], + "score": 1.0, + "content": "C.1 UNBIASEDNESS OF THE SUM-AND-SAMPLE ESTIMATOR", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 383, + 540 + ], + "score": 1.0, + "content": "We show that the sum-and-sample estimator is unbiased for any set", + "type": "text" + }, + { + "bbox": [ + 383, + 527, + 416, + 537 + ], + "score": 0.89, + "content": "C \\subset D", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 525, + 505, + 540 + ], + "score": 1.0, + "content": "(see also Liang et al.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 537, + 212, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 212, + 550 + ], + "score": 1.0, + "content": "(2018); Liu et al. 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Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 151, + 176, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 176, + 162 + ], + "score": 1.0, + "content": "set estimator, i.e.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 176, + 473, + 217 + ], + "lines": [ + { + "bbox": [ + 120, + 176, + 473, + 217 + ], + "spans": [ + { + "bbox": [ + 120, + 176, + 473, + 217 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } | S ^ { k } ) } \\left[ \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) \\right) f ( b _ { k } ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "0b9d7c9d1ed7944469fbeac8c9a5bc6f5f685b09515131a1595aefaf726d7cb7.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 120, + 176, + 473, + 189.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 120, + 189.66666666666666, + 473, + 203.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 120, + 203.33333333333331, + 473, + 216.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 327, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 328, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 328, + 276 + ], + "score": 1.0, + "content": "Proof. To give the proof, we first prove three Lemmas.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 151, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 152, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 152, + 324 + ], + "score": 1.0, + "content": "Lemma 1.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 336, + 412, + 366 + ], + "lines": [ + { + "bbox": [ + 199, + 336, + 412, + 366 + ], + "spans": [ + { + "bbox": [ + 199, + 336, + 412, + 366 + ], + "score": 0.92, + "content": "P ( b _ { k } = s | S ^ { k } ) = { \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } { \\frac { p ( s ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ s \\} } p ( s ^ { \\prime } ) } }", + "type": "interline_equation", + "image_path": "317c6fc716831ad249f80ed035ca691c5c122a60da3a87b0f11283a54e6d79c1.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 336, + 412, + 351.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 199, + 351.0, + 412, + 366.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 412, + 491, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 492, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 246, + 427 + ], + "score": 1.0, + "content": "Proof. Similar to the derivation of", + "type": "text" + }, + { + "bbox": [ + 246, + 413, + 304, + 426 + ], + "score": 0.92, + "content": "P ( b _ { 1 } = s | S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 412, + 492, + 427 + ], + "score": 1.0, + "content": "(equation 10 in the main paper), we can write:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 442, + 414, + 532 + ], + "lines": [ + { + "bbox": [ + 196, + 442, + 414, + 532 + ], + "spans": [ + { + "bbox": [ + 196, + 442, + 414, + 532 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { P ( b _ { k } = s | S ^ { k } ) = \\displaystyle \\frac { P ( S ^ { k } \\cap b _ { k } = s ) } { p ( S ^ { k } ) } } \\\\ & { \\qquad = \\displaystyle \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) p ^ { D \\setminus ( S ^ { k } \\setminus \\{ s \\} ) } ( s ) } { p ( S ^ { k } ) } } \\\\ & { \\qquad = \\displaystyle \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } \\displaystyle \\frac { p ( s ) } { 1 - 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\\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } { 1 - \\sum _ { s ^ { \\prime } \\in S \\setminus \\{ s \\} } p ( s ^ { \\prime } ) } } = p ^ { D \\setminus \\{ s \\} } ( S \\setminus \\{ s \\} )", + "type": "interline_equation", + "image_path": "7c0852d8179f1a62f8ea46378e393a74a336a7f30eadf02092aad28ae432b7d6.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 646, + 427, + 661.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 183, + 661.5, + 427, + 677.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 719, + 374, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 718, + 375, + 736 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 206, + 736 + ], + "score": 1.0, + "content": "Dividing equation 33 by", + "type": "text" + }, + { + "bbox": [ + 206, + 720, + 274, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 1 - \\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 718, + 375, + 736 + ], + "score": 1.0, + "content": "on both sides, we obtain", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 104, + 82, + 470, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 470, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 470, + 95 + ], + "score": 1.0, + "content": "C.2 RAO-BLACKWELLIZATION OF THE STOCHASTIC SUM-AND-SAMPLE ESTIMATOR", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 107, + 504, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 107, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 505, + 121 + ], + "score": 1.0, + "content": "In this section we give the proof that Rao-Blackwellizing the stochastic sum-and-sample estimator", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 119, + 257, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 257, + 131 + ], + "score": 1.0, + "content": "results in the unordered set estimator.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 107, + 505, + 131 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 138, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "Theorem 4. Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 151, + 176, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 176, + 162 + ], + "score": 1.0, + "content": "set estimator, i.e.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 138, + 506, + 162 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 176, + 473, + 217 + ], + "lines": [ + { + "bbox": [ + 120, + 176, + 473, + 217 + ], + "spans": [ + { + "bbox": [ + 120, + 176, + 473, + 217 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { B ^ { k } \\sim p ( B ^ { k } | S ^ { k } ) } \\left[ \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) \\right) f ( b _ { k } ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "0b9d7c9d1ed7944469fbeac8c9a5bc6f5f685b09515131a1595aefaf726d7cb7.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 120, + 176, + 473, + 189.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 120, + 189.66666666666666, + 473, + 203.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 120, + 203.33333333333331, + 473, + 216.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 327, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 328, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 328, + 276 + ], + "score": 1.0, + "content": "Proof. To give the proof, we first prove three Lemmas.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 262, + 328, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 151, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 152, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 152, + 324 + ], + "score": 1.0, + "content": "Lemma 1.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 312, + 152, + 324 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 336, + 412, + 366 + ], + "lines": [ + { + "bbox": [ + 199, + 336, + 412, + 366 + ], + "spans": [ + { + "bbox": [ + 199, + 336, + 412, + 366 + ], + "score": 0.92, + "content": "P ( b _ { k } = s | S ^ { k } ) = { \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } } { \\frac { p ( s ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ s \\} } p ( s ^ { \\prime } ) } }", + "type": "interline_equation", + "image_path": "317c6fc716831ad249f80ed035ca691c5c122a60da3a87b0f11283a54e6d79c1.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 336, + 412, + 351.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 199, + 351.0, + 412, + 366.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 412, + 491, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 492, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 246, + 427 + ], + "score": 1.0, + "content": "Proof. Similar to the derivation of", + "type": "text" + }, + { + "bbox": [ + 246, + 413, + 304, + 426 + ], + "score": 0.92, + "content": "P ( b _ { 1 } = s | S ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 412, + 492, + 427 + ], + "score": 1.0, + "content": "(equation 10 in the main paper), we can write:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 412, + 492, + 427 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 442, + 414, + 532 + ], + "lines": [ + { + "bbox": [ + 196, + 442, + 414, + 532 + ], + "spans": [ + { + "bbox": [ + 196, + 442, + 414, + 532 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { P ( b _ { k } = s | S ^ { k } ) = \\displaystyle \\frac { P ( S ^ { k } \\cap b _ { k } = s ) } { p ( S ^ { k } ) } } \\\\ & { \\qquad = \\displaystyle \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) p ^ { D \\setminus ( S ^ { k } \\setminus \\{ s \\} ) } ( s ) } { p ( S ^ { k } ) } } \\\\ & { \\qquad = \\displaystyle \\frac { p ( S ^ { k } \\setminus \\{ s \\} ) } { p ( S ^ { k } ) } \\displaystyle \\frac { p ( s ) } { 1 - 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\\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } { 1 - \\sum _ { s ^ { \\prime } \\in S \\setminus \\{ s \\} } p ( s ^ { \\prime } ) } } = p ^ { D \\setminus \\{ s \\} } ( S \\setminus \\{ s \\} )", + "type": "interline_equation", + "image_path": "7c0852d8179f1a62f8ea46378e393a74a336a7f30eadf02092aad28ae432b7d6.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 646, + 427, + 661.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 183, + 661.5, + 427, + 677.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 719, + 374, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 718, + 375, + 736 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 206, + 736 + ], + "score": 1.0, + "content": "Dividing equation 33 by", + "type": "text" + }, + { + "bbox": [ + 206, + 720, + 274, + 734 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 1 - \\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 718, + 375, + 736 + ], + "score": 1.0, + "content": "on both sides, we obtain", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 718, + 375, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 133, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 135, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 135, + 97 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 107, + 99, + 500, + 396 + ], + "lines": [ + { + "bbox": [ + 107, + 99, + 500, + 396 + ], + "spans": [ + { + "bbox": [ + 107, + 99, + 500, + 396 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\quad _ { 1 } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } \\\\ & { = \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } } \\\\ & { \\quad _ { 1 } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\Big ( \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } + \\mathsf { E } \\eta ^ { \\mathsf { C P S } _ { k } } \\Big ) + \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } } \\\\ & = \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } - 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\\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 416, + 401, + 433 + ], + "score": 1.0, + "content": "and rearranging terms proves Lemma 2.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 151, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 152, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 152, + 464 + ], + "score": 1.0, + "content": "Lemma 3.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 466, + 423, + 506 + ], + "lines": [ + { + "bbox": [ + 187, + 466, + 423, + 506 + ], + "spans": [ + { + "bbox": [ + 187, + 466, + 423, + 506 + ], + "score": 0.94, + "content": "p ( s ) + \\left( 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) P ( b _ { k } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )", + "type": "interline_equation", + "image_path": "f849db37933e0df0ed77d77694a6da529c83d635bb55db7375ebdcf9171da09e.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 466, + 423, + 486.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 187, + 486.0, + 423, + 506.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 331, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 331, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 331, + 545 + ], + "score": 1.0, + "content": "Proof. First using Lemma 1 and then Lemma 2 we find", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 554, + 435, + 712 + ], + "lines": [ + { + "bbox": [ + 173, + 554, + 435, + 712 + ], + "spans": [ + { + "bbox": [ + 173, + 554, + 435, + 712 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad p ( s ) + \\left( 1 - \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) P ( b _ { k } = s | S ^ { k } ) } \\\\ & { = p ( s ) + \\left( 1 - \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) \\frac { p ( S ^ { k } \\setminus \\{ S \\} ) } { p ( S ^ { k } ) } \\frac { p ( s ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ S \\} } p ( s ^ { \\prime } ) } } \\\\ & { = \\frac { p ( S ) } { p ( S ^ { k } ) } \\left( p ( S ^ { k } ) + \\displaystyle \\frac { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ P ( s ^ { \\prime } ) \\} } p ( s ^ { \\prime } ) } { 1 - 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\\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } \\\\ & { = \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } } \\\\ & { \\quad _ { 1 } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\Big ( \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } + \\mathsf { E } \\eta ^ { \\mathsf { C P S } _ { k } } \\Big ) + \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } } \\\\ & = \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\le \\epsilon _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } \\frac { \\mathsf { E } ^ { ( k ) } } { 1 - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } , \\mathsf { \\Lambda } _ { k } ^ { 2 } ) } \\\\ & \\quad _ { 1 } - \\sum _ { k \\in \\mathcal { N } _ { k } } ^ { \\mathsf { C P S } _ { k } } ( \\mathsf { E } ^ { ( k ) } \\frac { \\epsilon _ { k \\in \\mathcal { N } _ { k } } } 1 - \\sum _ k \\ \\end{array}", + "type": "interline_equation", + "image_path": "debf08f1ef48f69c0813d75fce15229f44e5af54cbf61079be4a12af114655f3.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 107, + 99, + 500, + 198.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 198.0, + 500, + 297.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 107, + 297.0, + 500, + 396.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 399, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 401, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 169, + 433 + ], + "score": 1.0, + "content": "Multiplying by", + "type": "text" + }, + { + "bbox": [ + 169, + 418, + 237, + 431 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 1 - \\sum _ { s ^ { \\prime } \\in S } p ( s ^ { \\prime } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 416, + 401, + 433 + ], + "score": 1.0, + "content": "and rearranging terms proves Lemma 2.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 416, + 401, + 433 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 151, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 152, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 152, + 464 + ], + "score": 1.0, + "content": "Lemma 3.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 451, + 152, + 464 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 466, + 423, + 506 + ], + "lines": [ + { + "bbox": [ + 187, + 466, + 423, + 506 + ], + "spans": [ + { + "bbox": [ + 187, + 466, + 423, + 506 + ], + "score": 0.94, + "content": "p ( s ) + \\left( 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) P ( b _ { k } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )", + "type": "interline_equation", + "image_path": "f849db37933e0df0ed77d77694a6da529c83d635bb55db7375ebdcf9171da09e.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 466, + 423, + 486.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 187, + 486.0, + 423, + 506.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 331, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 331, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 331, + 545 + ], + "score": 1.0, + "content": "Proof. First using Lemma 1 and then Lemma 2 we find", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 532, + 331, + 545 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 554, + 435, + 712 + ], + "lines": [ + { + "bbox": [ + 173, + 554, + 435, + 712 + ], + "spans": [ + { + "bbox": [ + 173, + 554, + 435, + 712 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad p ( s ) + \\left( 1 - \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) P ( b _ { k } = s | S ^ { k } ) } \\\\ & { = p ( s ) + \\left( 1 - \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p ( s ^ { \\prime } ) \\right) \\frac { p ( S ^ { k } \\setminus \\{ S \\} ) } { p ( S ^ { k } ) } \\frac { p ( s ) } { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ S \\} } p ( s ^ { \\prime } ) } } \\\\ & { = \\frac { p ( S ) } { p ( S ^ { k } ) } \\left( p ( S ^ { k } ) + \\displaystyle \\frac { 1 - \\sum _ { s ^ { \\prime } \\in S ^ { k } \\setminus \\{ P ( s ^ { \\prime } ) \\} } p ( s ^ { \\prime } ) } { 1 - 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(2019), one can trade off the number of summed terms and number of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "sampled terms to maximize the achieved variance reduction. 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In general, we denote the unordered set estimator", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 244, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 203, + 466 + ], + "score": 1.0, + "content": "restricted to the domain", + "type": "text" + }, + { + "bbox": [ + 204, + 452, + 230, + 465 + ], + "score": 0.92, + "content": "D \\backslash C", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 451, + 244, + 466 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 469, + 401, + 498 + ], + "lines": [ + { + "bbox": [ + 210, + 469, + 401, + 498 + ], + "spans": [ + { + "bbox": [ + 210, + 469, + 401, + 498 + ], + "score": 0.93, + "content": "e ^ { { \\bf U S } , D \\setminus C } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } \\setminus C } p ( s ) R ^ { D \\setminus C } ( S ^ { k } , s ) f ( s )", + "type": "interline_equation", + "image_path": "ae6f53cdd3e12b7696af2cbaed29929c0d66349ef59ae9486808cda4504d3d23.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 469, + 401, + 483.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 210, + 483.5, + 401, + 498.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 504, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 133, + 519 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 504, + 188, + 517 + ], + "score": 0.93, + "content": "R ^ { D \\setminus C } ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 502, + 389, + 519 + ], + "score": 1.0, + "content": "is the leave-one-out ratio restricted to the domain", + "type": "text" + }, + { + "bbox": [ + 389, + 505, + 416, + 517 + ], + "score": 0.92, + "content": "D \\backslash C", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 502, + 506, + 519 + ], + "score": 1.0, + "content": ", similar to the second", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 270, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 270, + 528 + ], + "score": 1.0, + "content": "order leave-one-out ratio in equation 18:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 531, + 399, + 565 + ], + "lines": [ + { + "bbox": [ + 211, + 531, + 399, + 565 + ], + "spans": [ + { + "bbox": [ + 211, + 531, + 399, + 565 + ], + "score": 0.94, + "content": "R ^ { D \\setminus C } ( S ^ { k } , s ) = { \\frac { p _ { \\theta } ^ { ( D \\setminus C ) \\setminus \\{ s \\} } ( ( S ^ { k } \\setminus C ) \\setminus \\{ s \\} ) } { p _ { \\theta } ^ { D \\setminus C } ( S ^ { k } \\setminus C ) } } .", + "type": "interline_equation", + "image_path": "262dc87328c00006d729bdd5cefd83d20437f1eb35b0ef68dbc8d80f134f5054.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 531, + 399, + 548.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 211, + 548.0, + 399, + 565.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 570, + 504, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 221, + 585 + ], + "score": 1.0, + "content": "While we can also constrain", + "type": "text" + }, + { + "bbox": [ + 221, + 570, + 279, + 583 + ], + "score": 0.93, + "content": "S ^ { k } \\subseteq ( D \\backslash C )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 569, + 505, + 585 + ], + "score": 1.0, + "content": ", this definition is consistent with equation 18 and allows", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 186, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 186, + 594 + ], + "score": 1.0, + "content": "simplified notation.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 504, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 435, + 609 + ], + "score": 1.0, + "content": "Theorem 5. Rao-Blackwellizing the stochastic sum-and-sample estimator with", + "type": "text" + }, + { + "bbox": [ + 435, + 597, + 468, + 607 + ], + "score": 0.89, + "content": "m \\ > \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "samples", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 272, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 272, + 621 + ], + "score": 1.0, + "content": "results in the unordered set estimator, i.e.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 622, + 510, + 663 + ], + "lines": [ + { + "bbox": [ + 111, + 622, + 510, + 663 + ], + "spans": [ + { + "bbox": [ + 111, + 622, + 510, + 663 + ], + "score": 0.94, + "content": "\\ ? \\ ? \\complement _ { B ^ { k } \\sim p ( B ^ { k } \\mid S ^ { k } ) } \\left[ \\sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) \\right) e ^ { U S , D \\setminus B ^ { k - m } } ( S ^ { k } ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "c9ac6892079fc79013e0e1be3f725409a4351778d410ea7fdeda86d9dc2a5412.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 111, + 622, + 510, + 635.6666666666666 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 111, + 635.6666666666666, + 510, + 649.3333333333333 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 111, + 649.3333333333333, + 510, + 662.9999999999999 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 684, + 358, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 358, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 358, + 696 + ], + "score": 1.0, + "content": "Proof. 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(2019), one can trade off the number of summed terms and number of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "sampled terms to maximize the achieved variance reduction. 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In general, we denote the unordered set estimator", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 244, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 203, + 466 + ], + "score": 1.0, + "content": "restricted to the domain", + "type": "text" + }, + { + "bbox": [ + 204, + 452, + 230, + 465 + ], + "score": 0.92, + "content": "D \\backslash C", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 451, + 244, + 466 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 386, + 506, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 469, + 401, + 498 + ], + "lines": [ + { + "bbox": [ + 210, + 469, + 401, + 498 + ], + "spans": [ + { + "bbox": [ + 210, + 469, + 401, + 498 + ], + "score": 0.93, + "content": "e ^ { { \\bf U S } , D \\setminus C } ( S ^ { k } ) = \\sum _ { s \\in S ^ { k } \\setminus C } p ( s ) R ^ { D \\setminus C } ( S ^ { k } , s ) f ( s )", + "type": "interline_equation", + "image_path": "ae6f53cdd3e12b7696af2cbaed29929c0d66349ef59ae9486808cda4504d3d23.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 469, + 401, + 483.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 210, + 483.5, + 401, + 498.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 504, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 133, + 519 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 504, + 188, + 517 + ], + "score": 0.93, + "content": "R ^ { D \\setminus C } ( S ^ { k } , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 502, + 389, + 519 + ], + "score": 1.0, + "content": "is the leave-one-out ratio restricted to the domain", + "type": "text" + }, + { + "bbox": [ + 389, + 505, + 416, + 517 + ], + "score": 0.92, + "content": "D \\backslash C", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 502, + 506, + 519 + ], + "score": 1.0, + "content": ", similar to the second", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 270, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 270, + 528 + ], + "score": 1.0, + "content": "order leave-one-out ratio in equation 18:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 502, + 506, + 528 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 531, + 399, + 565 + ], + "lines": [ + { + "bbox": [ + 211, + 531, + 399, + 565 + ], + "spans": [ + { + "bbox": [ + 211, + 531, + 399, + 565 + ], + "score": 0.94, + "content": "R ^ { D \\setminus C } ( S ^ { k } , s ) = { \\frac { p _ { \\theta } ^ { ( D \\setminus C ) \\setminus \\{ s \\} } ( ( S ^ { k } \\setminus C ) \\setminus \\{ s \\} ) } { p _ { \\theta } ^ { D \\setminus C } ( S ^ { k } \\setminus C ) } } .", + "type": "interline_equation", + "image_path": "262dc87328c00006d729bdd5cefd83d20437f1eb35b0ef68dbc8d80f134f5054.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 531, + 399, + 548.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 211, + 548.0, + 399, + 565.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 570, + 504, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 221, + 585 + ], + "score": 1.0, + "content": "While we can also constrain", + "type": "text" + }, + { + "bbox": [ + 221, + 570, + 279, + 583 + ], + "score": 0.93, + "content": "S ^ { k } \\subseteq ( D \\backslash C )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 569, + 505, + 585 + ], + "score": 1.0, + "content": ", this definition is consistent with equation 18 and allows", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 186, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 186, + 594 + ], + "score": 1.0, + "content": "simplified notation.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 569, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 504, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 435, + 609 + ], + "score": 1.0, + "content": "Theorem 5. Rao-Blackwellizing the stochastic sum-and-sample estimator with", + "type": "text" + }, + { + "bbox": [ + 435, + 597, + 468, + 607 + ], + "score": 0.89, + "content": "m \\ > \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "samples", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 272, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 272, + 621 + ], + "score": 1.0, + "content": "results in the unordered set estimator, i.e.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 596, + 505, + 621 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 622, + 510, + 663 + ], + "lines": [ + { + "bbox": [ + 111, + 622, + 510, + 663 + ], + "spans": [ + { + "bbox": [ + 111, + 622, + 510, + 663 + ], + "score": 0.94, + "content": "\\ ? \\ ? \\complement _ { B ^ { k } \\sim p ( B ^ { k } \\mid S ^ { k } ) } \\left[ \\sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) f ( b _ { j } ) + \\left( 1 - \\sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) \\right) e ^ { U S , D \\setminus B ^ { k - m } } ( S ^ { k } ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "c9ac6892079fc79013e0e1be3f725409a4351778d410ea7fdeda86d9dc2a5412.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 111, + 622, + 510, + 635.6666666666666 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 111, + 635.6666666666666, + 510, + 649.3333333333333 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 111, + 649.3333333333333, + 510, + 662.9999999999999 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 684, + 358, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 358, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 358, + 696 + ], + "score": 1.0, + "content": "Proof. Recall that for the unordered set estimator, it holds that", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 106, + 684, + 358, + 696 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 700, + 424, + 717 + ], + "lines": [ + { + "bbox": [ + 186, + 700, + 424, + 717 + ], + "spans": [ + { + "bbox": [ + 186, + 700, + 424, + 717 + ], + "score": 0.9, + "content": "e ^ { \\mathrm { U S } } ( S ^ { k } ) = \\mathbb { E } _ { b _ { 1 } \\sim p ( b _ { 1 } | S ^ { k } ) } \\left[ f ( b _ { 1 } ) \\right] = \\mathbb { E } _ { x \\sim p ( x ) } \\left[ f ( x ) \\middle | x \\in S ^ { k } \\right]", + "type": "interline_equation", + "image_path": "530122c7b7780623682b4f30cd05eb78970872a06f15a7c9c139ea6d2478afee.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 186, + 700, + 424, + 717 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 443, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 444, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 356, + 97 + ], + "score": 1.0, + "content": "which for the restricted equivalent (with restricted distribution", + "type": "text" + }, + { + "bbox": [ + 357, + 81, + 380, + 94 + ], + "score": 0.9, + "content": "p ^ { D \\setminus C }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 79, + 444, + 97 + ], + "score": 1.0, + "content": ") translates into", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 98, + 466, + 114 + ], + "lines": [ + { + "bbox": [ + 145, + 98, + 466, + 114 + ], + "spans": [ + { + "bbox": [ + 145, + 98, + 466, + 114 + ], + "score": 0.89, + "content": "e ^ { \\mathrm { U S } , D \\setminus C } ( S ^ { k } ) = \\mathbb { E } _ { x \\sim p ^ { D \\setminus C } ( x ) } \\left[ f ( x ) \\big | x \\in S ^ { k } \\right] = \\mathbb { E } _ { x \\sim p ( x ) } \\left[ f ( x ) \\big | x \\in S ^ { k } , x \\not \\in C \\right] .", + "type": "interline_equation", + "image_path": "95f74612504c9115d909cb70e4eade4f448adb087befe3d0cd39383613e5771d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 145, + 98, + 466, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 104, + 117, + 507, + 132 + ], + "spans": [ + { + "bbox": [ + 104, + 117, + 243, + 132 + ], + "score": 1.0, + "content": "Now we consider the distribution", + "type": "text" + }, + { + "bbox": [ + 243, + 118, + 322, + 131 + ], + "score": 0.9, + "content": "b _ { k - m + 1 } | S ^ { k } , B ^ { k - m }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 117, + 507, + 132 + ], + "score": 1.0, + "content": ": the distribution of the first element sampled", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 127, + 504, + 143 + ], + "spans": [ + { + "bbox": [ + 104, + 127, + 261, + 143 + ], + "score": 1.0, + "content": "(without replacement) after sampling", + "type": "text" + }, + { + "bbox": [ + 262, + 131, + 289, + 140 + ], + "score": 0.87, + "content": "B ^ { k - m }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 127, + 497, + 143 + ], + "score": 1.0, + "content": ", given (conditionally on the event) that the set of", + "type": "text" + }, + { + "bbox": [ + 497, + 131, + 504, + 140 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 151, + 153 + ], + "score": 1.0, + "content": "samples is", + "type": "text" + }, + { + "bbox": [ + 152, + 141, + 164, + 151 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 140, + 217, + 153 + ], + "score": 1.0, + "content": ", so we have", + "type": "text" + }, + { + "bbox": [ + 218, + 141, + 277, + 153 + ], + "score": 0.92, + "content": "b _ { k - m + 1 } \\stackrel { - } { \\in } S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 140, + 297, + 153 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 141, + 371, + 153 + ], + "score": 0.92, + "content": "\\mathsf { \\bar { b } } _ { k - m + 1 } \\notin B ^ { k - m }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 140, + 506, + 153 + ], + "score": 1.0, + "content": ". This means that its conditional", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 462, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 166, + 165 + ], + "score": 1.0, + "content": "expectation of", + "type": "text" + }, + { + "bbox": [ + 166, + 152, + 213, + 164 + ], + "score": 0.92, + "content": "f ( b _ { k - m + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 151, + 388, + 165 + ], + "score": 1.0, + "content": "is the restricted unordered set estimator for", + "type": "text" + }, + { + "bbox": [ + 388, + 152, + 436, + 162 + ], + "score": 0.91, + "content": "C = B ^ { k - m }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 151, + 462, + 165 + ], + "score": 1.0, + "content": "since", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 167, + 435, + 202 + ], + "lines": [ + { + "bbox": [ + 176, + 167, + 435, + 202 + ], + "spans": [ + { + "bbox": [ + 176, + 167, + 435, + 202 + ], + "score": 0.93, + "content": "\\begin{array} { r l } { { e ^ { \\mathrm { U S } , D \\setminus B ^ { k - m } } ( S ^ { k } ) = \\mathbb { E } _ { x \\sim p ( x ) } [ f ( x ) \\vert x \\in S ^ { k } , x \\not \\in B ^ { k - m } ] } \\quad } & { } \\\\ & { = \\mathbb { E } _ { b _ { k - m + 1 } \\sim p ( b _ { k - m + 1 } \\mid S ^ { k } , B ^ { k - m } ) } [ f ( b _ { k - m + 1 } ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "07a2fbd8fe56d665e43e09c3001882623419cefdd4f57e0ab0f3785a4c9020f6.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 176, + 167, + 435, + 178.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 176, + 178.66666666666666, + 435, + 190.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 176, + 190.33333333333331, + 435, + 201.99999999999997 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "Observing that the definition (equation 42) of the stochastic sum-and-sample estimator does not", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 249, + 226 + ], + "score": 1.0, + "content": "depend on the actual order of the", + "type": "text" + }, + { + "bbox": [ + 249, + 216, + 259, + 224 + ], + "score": 0.77, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "samples, and using equation 45, we can reduce the multi-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 376, + 237 + ], + "score": 1.0, + "content": "sample estimator to the stochastic sum-and-sample estimator with", + "type": "text" + }, + { + "bbox": [ + 376, + 225, + 447, + 236 + ], + "score": 0.91, + "content": "k ^ { \\prime } = k - m + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 225, + 505, + 237 + ], + "score": 1.0, + "content": ", such that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 236, + 236, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 236, + 248 + ], + "score": 1.0, + "content": "result follows from equation 36.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 252, + 500, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 500, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 500, + 519 + ], + "score": 0.96, + "content": "\\begin{array} { c } { { \\displaystyle \\begin{array} { l } { { { \\cal L } _ { \\mathrm { e x c } , \\mathrm { t r i g h } , \\mathrm { \\scriptsize ~ 1 . . } } \\sum _ { j = 0 } ^ { N } \\phi _ { j } \\Big | Z _ { j } ^ { ( k ) } \\phi _ { j } \\Big | + ( \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\phi _ { j } } ) Z _ { j } ^ { ( k ) } z ^ { ( k ) } z ^ { ( k ) } - ( \\beta _ { j } - \\gamma ) \\delta _ { j } ^ { \\prime } \\Big | } } } \\\\ { { \\displaystyle - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\mathrm { \\cdots ~ 1 } } } \\end{array} \\} } \\\\ \\displaystyle - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } ( \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\phi _ { j } } ) \\Bigg | \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { k = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } } \\\\ \\displaystyle - \\sum _ j = \\end{array}", + "type": "interline_equation", + "image_path": "d8526bc56964422d83f6abb56b1004632519ad61650cc88894ec9d0df48c8e72.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 105, + 252, + 500, + 341.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 105, + 341.0, + 500, + 430.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 105, + 430.0, + 500, + 519.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 531, + 343, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 344, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 344, + 547 + ], + "score": 1.0, + "content": "D THE IMPORTANCE-WEIGHTED ESTIMATOR", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 555, + 440, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 441, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 441, + 568 + ], + "score": 1.0, + "content": "D.1 RAO-BLACKWELLIZATION OF THE IMPORTANCE-WEIGHTED ESTIMATOR", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 105, + 575, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 504, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 504, + 588 + ], + "score": 1.0, + "content": "In this section we give the proof that Rao-Blackwellizing the importance-weighted estimator results", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 587, + 230, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 230, + 599 + ], + "score": 1.0, + "content": "in the unordered set estimator.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 504, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "Theorem 6. Rao-Blackwellizing the importance-weighted estimator results in the unordered set", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 612, + 166, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 166, + 624 + ], + "score": 1.0, + "content": "estimator, i.e.:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 626, + 435, + 666 + ], + "lines": [ + { + "bbox": [ + 175, + 626, + 435, + 666 + ], + "spans": [ + { + "bbox": [ + 175, + 626, + 435, + 666 + ], + "score": 0.96, + "content": "\\mathbb { E } _ { \\kappa \\sim p ( \\kappa | S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } \\frac { p ( s ) } { 1 - F _ { \\phi _ { s } } ( \\kappa ) } f ( s ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "01e8ad49e2b31698b00a425d91b8cf16f8e6028f37590b04c330c6d3a3bf6f9c.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 175, + 626, + 435, + 639.3333333333334 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 175, + 639.3333333333334, + 435, + 652.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 175, + 652.6666666666667, + 435, + 666.0000000000001 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 675, + 506, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 673, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 688 + ], + "score": 1.0, + "content": "Here we have slightly rewritten the definition of the importance-weighted estimator, using that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 107, + 686, + 271, + 698 + ], + "score": 0.91, + "content": "q ( s , a ) = P ( g _ { \\phi _ { s } } ^ { \\mathrm { ~ ~ } } > \\stackrel { \\cdot } { a } ) = 1 - F _ { \\phi _ { s } } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 685, + 304, + 699 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 304, + 686, + 321, + 698 + ], + "score": 0.9, + "content": "F _ { \\phi _ { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 685, + 506, + 699 + ], + "score": 1.0, + "content": "is the CDF of the Gumbel distribution (see", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 696, + 164, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 164, + 710 + ], + "score": 1.0, + "content": "Appendix A).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 285, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 286, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 286, + 734 + ], + "score": 1.0, + "content": "Proof. We first prove the following Lemma:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 443, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 444, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 356, + 97 + ], + "score": 1.0, + "content": "which for the restricted equivalent (with restricted distribution", + "type": "text" + }, + { + "bbox": [ + 357, + 81, + 380, + 94 + ], + "score": 0.9, + "content": "p ^ { D \\setminus C }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 79, + 444, + 97 + ], + "score": 1.0, + "content": ") translates into", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 79, + 444, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 98, + 466, + 114 + ], + "lines": [ + { + "bbox": [ + 145, + 98, + 466, + 114 + ], + "spans": [ + { + "bbox": [ + 145, + 98, + 466, + 114 + ], + "score": 0.89, + "content": "e ^ { \\mathrm { U S } , D \\setminus C } ( S ^ { k } ) = \\mathbb { E } _ { x \\sim p ^ { D \\setminus C } ( x ) } \\left[ f ( x ) \\big | x \\in S ^ { k } \\right] = \\mathbb { E } _ { x \\sim p ( x ) } \\left[ f ( x ) \\big | x \\in S ^ { k } , x \\not \\in C \\right] .", + "type": "interline_equation", + "image_path": "95f74612504c9115d909cb70e4eade4f448adb087befe3d0cd39383613e5771d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 145, + 98, + 466, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 104, + 117, + 507, + 132 + ], + "spans": [ + { + "bbox": [ + 104, + 117, + 243, + 132 + ], + "score": 1.0, + "content": "Now we consider the distribution", + "type": "text" + }, + { + "bbox": [ + 243, + 118, + 322, + 131 + ], + "score": 0.9, + "content": "b _ { k - m + 1 } | S ^ { k } , B ^ { k - m }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 117, + 507, + 132 + ], + "score": 1.0, + "content": ": the distribution of the first element sampled", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 127, + 504, + 143 + ], + "spans": [ + { + "bbox": [ + 104, + 127, + 261, + 143 + ], + "score": 1.0, + "content": "(without replacement) after sampling", + "type": "text" + }, + { + "bbox": [ + 262, + 131, + 289, + 140 + ], + "score": 0.87, + "content": "B ^ { k - m }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 127, + 497, + 143 + ], + "score": 1.0, + "content": ", given (conditionally on the event) that the set of", + "type": "text" + }, + { + "bbox": [ + 497, + 131, + 504, + 140 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 151, + 153 + ], + "score": 1.0, + "content": "samples is", + "type": "text" + }, + { + "bbox": [ + 152, + 141, + 164, + 151 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 140, + 217, + 153 + ], + "score": 1.0, + "content": ", so we have", + "type": "text" + }, + { + "bbox": [ + 218, + 141, + 277, + 153 + ], + "score": 0.92, + "content": "b _ { k - m + 1 } \\stackrel { - } { \\in } S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 140, + 297, + 153 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 141, + 371, + 153 + ], + "score": 0.92, + "content": "\\mathsf { \\bar { b } } _ { k - m + 1 } \\notin B ^ { k - m }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 140, + 506, + 153 + ], + "score": 1.0, + "content": ". 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( \\beta _ { j } - \\gamma ) \\delta _ { j } ^ { \\prime } \\Big | } } } \\\\ { { \\displaystyle - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\mathrm { \\cdots ~ 1 } } } \\end{array} \\} } \\\\ \\displaystyle - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } ( \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\phi _ { j } } ) \\Bigg | \\displaystyle { 1 - \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { k = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { j = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } \\sum _ { l = 0 } ^ { N } \\mathrm { \\scriptsize ~ 1 . } } \\\\ \\displaystyle - \\sum _ j = \\end{array}", + "type": "interline_equation", + "image_path": "d8526bc56964422d83f6abb56b1004632519ad61650cc88894ec9d0df48c8e72.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 105, + 252, + 500, + 341.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 105, + 341.0, + 500, + 430.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 105, + 430.0, + 500, + 519.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 531, + 343, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 344, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 344, + 547 + ], + "score": 1.0, + "content": "D THE IMPORTANCE-WEIGHTED ESTIMATOR", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 555, + 440, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 441, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 441, + 568 + ], + "score": 1.0, + "content": "D.1 RAO-BLACKWELLIZATION OF THE IMPORTANCE-WEIGHTED ESTIMATOR", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 105, + 575, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 504, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 504, + 588 + ], + "score": 1.0, + "content": "In this section we give the proof that Rao-Blackwellizing the importance-weighted estimator results", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 587, + 230, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 230, + 599 + ], + "score": 1.0, + "content": "in the unordered set estimator.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 576, + 504, + 599 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 504, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "Theorem 6. Rao-Blackwellizing the importance-weighted estimator results in the unordered set", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 612, + 166, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 166, + 624 + ], + "score": 1.0, + "content": "estimator, i.e.:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 600, + 505, + 624 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 626, + 435, + 666 + ], + "lines": [ + { + "bbox": [ + 175, + 626, + 435, + 666 + ], + "spans": [ + { + "bbox": [ + 175, + 626, + 435, + 666 + ], + "score": 0.96, + "content": "\\mathbb { E } _ { \\kappa \\sim p ( \\kappa | S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } \\frac { p ( s ) } { 1 - F _ { \\phi _ { s } } ( \\kappa ) } f ( s ) \\right] = \\sum _ { s \\in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) .", + "type": "interline_equation", + "image_path": "01e8ad49e2b31698b00a425d91b8cf16f8e6028f37590b04c330c6d3a3bf6f9c.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 175, + 626, + 435, + 639.3333333333334 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 175, + 639.3333333333334, + 435, + 652.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 175, + 652.6666666666667, + 435, + 666.0000000000001 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 675, + 506, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 673, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 688 + ], + "score": 1.0, + "content": "Here we have slightly rewritten the definition of the importance-weighted estimator, using that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 107, + 686, + 271, + 698 + ], + "score": 0.91, + "content": "q ( s , a ) = P ( g _ { \\phi _ { s } } ^ { \\mathrm { ~ ~ } } > \\stackrel { \\cdot } { a } ) = 1 - F _ { \\phi _ { s } } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 685, + 304, + 699 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 304, + 686, + 321, + 698 + ], + "score": 0.9, + "content": "F _ { \\phi _ { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 685, + 506, + 699 + ], + "score": 1.0, + "content": "is the CDF of the Gumbel distribution (see", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 696, + 164, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 164, + 710 + ], + "score": 1.0, + "content": "Appendix A).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 673, + 506, + 710 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 285, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 286, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 286, + 734 + ], + "score": 1.0, + "content": "Proof. We first prove the following Lemma:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 719, + 286, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 151, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 152, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 152, + 94 + ], + "score": 1.0, + "content": "Lemma 4.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 90, + 385, + 119 + ], + "lines": [ + { + "bbox": [ + 226, + 90, + 385, + 119 + ], + "spans": [ + { + "bbox": [ + 226, + 90, + 385, + 119 + ], + "score": 0.95, + "content": "\\mathbb { E } _ { \\kappa \\sim p ( \\kappa | S ^ { k } ) } \\left[ \\frac { 1 } { 1 - F _ { \\phi _ { s } } ( \\kappa ) } \\right] = R ( S ^ { k } , s )", + "type": "interline_equation", + "image_path": "ea6ebad1366a3fc44c7c07e5829e5eaef7cab655a4b439ea9150102a75887e5b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 226, + 90, + 385, + 119 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 104, + 129, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 104, + 129, + 206, + 146 + ], + "score": 1.0, + "content": "Proof. Conditioning on", + "type": "text" + }, + { + "bbox": [ + 207, + 131, + 219, + 142 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 129, + 351, + 146 + ], + "score": 1.0, + "content": ", we know that the elements in", + "type": "text" + }, + { + "bbox": [ + 351, + 131, + 364, + 142 + ], + "score": 0.88, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 129, + 405, + 146 + ], + "score": 1.0, + "content": "have the", + "type": "text" + }, + { + "bbox": [ + 405, + 132, + 412, + 142 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 129, + 505, + 146 + ], + "score": 1.0, + "content": "largest perturbed log-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 175, + 156 + ], + "score": 1.0, + "content": "probabilities, so", + "type": "text" + }, + { + "bbox": [ + 176, + 145, + 183, + 153 + ], + "score": 0.71, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 142, + 205, + 156 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 205, + 143, + 237, + 155 + ], + "score": 0.89, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "-th largest perturbed log-probability is the largest perturbed log-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 151, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 151, + 162, + 171 + ], + "score": 1.0, + "content": "probability in", + "type": "text" + }, + { + "bbox": [ + 163, + 154, + 192, + 166 + ], + "score": 0.92, + "content": "D \\backslash S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 151, + 246, + 171 + ], + "score": 1.0, + "content": ", and satisfies", + "type": "text" + }, + { + "bbox": [ + 247, + 154, + 453, + 169 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\kappa = \\operatorname* { m a x } _ { s \\in D \\backslash S ^ { k } } g _ { \\phi _ { s } } = g _ { \\phi _ { D \\backslash S ^ { k } } } \\sim \\mathrm { G u m b e l } ( \\phi _ { D \\backslash S ^ { k } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 151, + 506, + 171 + ], + "score": 1.0, + "content": ". 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1.0, + "content": "To prove the unbiasedness of result we need to prove that the control variate has expectation 0:", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 372, + 153, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 153, + 387 + ], + "score": 1.0, + "content": "Lemma 5.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 388, + 462, + 428 + ], + "lines": [ + { + "bbox": [ + 149, + 388, + 462, + 428 + ], + "spans": [ + { + "bbox": [ + 149, + 388, + 462, + 428 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { S ^ { k } \\sim p _ { \\theta } ( S ^ { k } ) } \\left[ \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right] = 0 .", + "type": "interline_equation", + 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Similar to equation 10, we apply Bayes’ Theorem conditionally on", + "type": "text" + }, + { + "bbox": [ + 417, + 439, + 449, + 450 + ], + "score": 0.92, + "content": "b _ { 1 } ~ = ~ s", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 436, + 506, + 452 + ], + "score": 1.0, + "content": "to derive for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 450, + 133, + 461 + ], + "spans": [ + { + "bbox": [ + 107, + 450, + 133, + 461 + ], + "score": 0.88, + "content": "s ^ { \\prime } \\neq s", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 465, + 444, + 557 + ], + "lines": [ + { + "bbox": [ + 166, + 465, + 444, + 557 + ], + "spans": [ + { + "bbox": [ + 166, + 465, + 444, + 557 + ], + "score": 0.95, + "content": "\\begin{array} { c } { P ( b _ { 2 } = s ^ { \\prime } | S ^ { k } , b _ { 1 } = s ) = \\displaystyle { \\frac { P ( S ^ { k } | b _ { 2 } = s ^ { \\prime } , b _ { 1 } = s ) P ( b _ { 2 } = s ^ { \\prime } | b _ { 1 } = s ^ { \\prime } ) } { P ( S ^ { k } | b _ { 1 } = s ) } } } \\\\ { = \\displaystyle { \\frac { p _ { \\theta } ^ { D \\setminus \\{ s , s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s , s ^ { \\prime } \\} ) p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( s ^ { \\prime } ) } { p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } } } \\\\ { { = \\displaystyle { \\frac { p _ { \\theta } ( s ^ { \\prime } ) } { 1 - p _ { \\theta } ( s ) } R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) } . } } \\end{array}", + "type": "interline_equation", + "image_path": "b7826c18422de652fa96e5046dce9e2b17066522210c25cb22c5f01a87170f56.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 166, + 465, + 444, + 495.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 166, + 495.6666666666667, + 444, + 526.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 166, + 526.3333333333334, + 444, + 557.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 560, + 479, + 574 + ], + "lines": [ + { + "bbox": [ + 104, + 559, + 482, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 123, + 576 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 562, + 149, + 572 + ], + "score": 0.91, + "content": "s ^ { \\prime } = s", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 559, + 186, + 576 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 186, + 560, + 267, + 574 + ], + "score": 0.93, + "content": "R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 559, + 482, + 576 + ], + "score": 1.0, + "content": "by definition, so using equation 54 we can show that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 578, + 447, + 733 + ], + "lines": [ + { + "bbox": [ + 165, + 578, + 447, + 733 + ], + "spans": [ + { + "bbox": [ + 165, + 578, + 447, + 733 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D N \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } \\frac { p _ { \\theta } ( s ^ { \\prime } ) } { 1 - p _ { \\theta } ( s ) } R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } P ( b _ { 2 } = s ^ { \\prime } | S ^ { k } , b _ { 1 } = s ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ { s ^ { \\prime } \\circ p _ { \\theta } ( b _ { 2 } \\mid S ^ { k } , b _ { 1 } = s ) } [ f ( b _ { 2 } ) ] } \\\\ & { = \\mathbb { E } _ { \\delta \\to p _ { \\theta } ( b _ { 2 } \\mid S ^ { k } , b _ { 1 } = s ) } [ p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "941373c9e6ed240fe6a0ad7ccfcb8bc26b621b25ffdc7ff97d9ddb312f31505b.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 165, + 578, + 447, + 629.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 165, + 629.6666666666666, + 447, + 681.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 165, + 681.3333333333333, + 447, + 732.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "22", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 126 + ], + "lines": [ + { + "bbox": [ + 102, + 77, + 506, + 100 + ], + "spans": [ + { + "bbox": [ + 102, + 77, + 474, + 100 + ], + "score": 1.0, + "content": "As we show in Kool et al. (2019b), we can include a ‘baseline’ B(Sk) = Ps∈Sk pθ(s)qθ,κ(s) f", + "type": "text" + }, + { + "bbox": [ + 487, + 79, + 506, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 96, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 353, + 109 + ], + "score": 1.0, + "content": "correct for the bias (since it depends on the complete sample", + "type": "text" + }, + { + "bbox": [ + 353, + 97, + 366, + 108 + ], + "score": 0.86, + "content": "S ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 96, + 505, + 109 + ], + "score": 1.0, + "content": ") by weighting individual terms of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 109, + 257, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 173, + 126 + ], + "score": 1.0, + "content": "the estimator by", + "type": "text" + }, + { + "bbox": [ + 173, + 109, + 253, + 126 + ], + "score": 0.92, + "content": "\\begin{array} { r } { 1 - p _ { \\pmb { \\theta } } ( s ) + \\frac { p _ { \\pmb { \\theta } } ( s ) } { q _ { \\pmb { \\theta } , \\kappa } ( s ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 109, + 257, + 126 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 102, + 77, + 506, + 126 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 130, + 460, + 162 + ], + "lines": [ + { + "bbox": [ + 150, + 130, + 460, + 162 + ], + "spans": [ + { + "bbox": [ + 150, + 130, + 460, + 162 + ], + "score": 0.93, + "content": "e ^ { \\mathrm { I W P G B L } } ( S ^ { k } , \\kappa ) = \\sum _ { s \\in S ^ { k } } { \\frac { \\nabla _ { \\theta } p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } } \\left( f ( s ) \\left( 1 - p _ { \\theta } ( s ) + { \\frac { p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } } \\right) - B ( S ^ { k } ) \\right)", + "type": "interline_equation", + "image_path": 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257, + 208 + ], + "score": 0.92, + "content": "\\begin{array} { r } { W _ { i } ( S ^ { k } ) = W ( S ^ { k } ) - \\frac { p _ { \\theta } ( s ) } { q _ { \\theta , \\kappa } ( s ) } + p _ { \\theta } ( s ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 189, + 397, + 209 + ], + "score": 1.0, + "content": "to normalize the individual terms:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 172, + 506, + 209 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 212, + 442, + 245 + ], + "lines": [ + { + "bbox": [ + 169, + 212, + 442, + 245 + ], + "spans": [ + { + "bbox": [ + 169, + 212, + 442, + 245 + ], + "score": 0.93, + "content": "\\nabla _ { \\pmb \\theta } \\mathbb { E } _ { \\pmb \\theta \\sim p _ { \\pmb \\theta } ( \\pmb y ) } \\left[ f ( \\pmb y ) \\right] \\approx \\sum _ { s \\in S ^ { k } } \\frac { 1 } { W _ { i } ( S ^ { k } ) } \\cdot \\frac { \\nabla _ { \\pmb \\theta } p _ { \\pmb \\theta } ( s ) } { q _ { \\pmb \\theta , \\kappa } ( s ) } \\left( f ( s ) - \\frac { B ( S ^ { k } ) } { W ( S ^ { k } ) } \\right)", + "type": "interline_equation", + "image_path": "428c5674023653acf72976a9a4a14f8ea48e5542ff29a4b3bc54d6b3079b8a91.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 169, + 212, + 442, + 223.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 169, + 223.0, + 442, + 234.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 169, + 234.0, + 442, + 245.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 506, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 507, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 334, + 268 + ], + "score": 1.0, + "content": "It seems odd to normalize the terms in the outer sum by", + "type": "text" + }, + { + "bbox": [ + 335, + 250, + 364, + 266 + ], + "score": 0.93, + "content": "\\frac { 1 } { W _ { i } ( S ^ { k } ) }", + "type": 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\\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right] = 0 .", + "type": "interline_equation", + "image_path": "7dcde4fb0e8b1866026dbef50261833923c9fad3f33392df0058daff30d20272.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 149, + 388, + 462, + 401.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 149, + 401.3333333333333, + 462, + 414.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 149, + 414.66666666666663, + 462, + 427.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 438, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 416, + 452 + ], + "score": 1.0, + "content": "Proof. Similar to equation 10, we apply Bayes’ Theorem conditionally on", + "type": "text" + }, + { + "bbox": [ + 417, + 439, + 449, + 450 + ], + "score": 0.92, + "content": "b _ { 1 } ~ = ~ s", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 436, + 506, + 452 + ], + "score": 1.0, + "content": "to derive for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 450, + 133, + 461 + ], + "spans": [ + { + "bbox": [ + 107, + 450, + 133, + 461 + ], + "score": 0.88, + "content": "s ^ { \\prime } \\neq s", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 436, + 506, + 461 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 465, + 444, + 557 + ], + "lines": [ + { + "bbox": [ + 166, + 465, + 444, + 557 + ], + "spans": [ + { + "bbox": [ + 166, + 465, + 444, + 557 + ], + "score": 0.95, + "content": "\\begin{array} { c } { P ( b _ { 2 } = s ^ { \\prime } | S ^ { k } , b _ { 1 } = s ) = \\displaystyle { \\frac { P ( S ^ { k } | b _ { 2 } = s ^ { \\prime } , b _ { 1 } = s ) P ( b _ { 2 } = s ^ { \\prime } | b _ { 1 } = s ^ { \\prime } ) } { P ( S ^ { k } | b _ { 1 } = s ) } } } \\\\ { = \\displaystyle { \\frac { p _ { \\theta } ^ { D \\setminus \\{ s , s ^ { \\prime } \\} } ( S ^ { k } \\setminus \\{ s , s ^ { \\prime } \\} ) p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( s ^ { \\prime } ) } { p _ { \\theta } ^ { D \\setminus \\{ s \\} } ( S ^ { k } \\setminus \\{ s \\} ) } } } \\\\ { { = \\displaystyle { \\frac { p _ { \\theta } ( s ^ { \\prime } ) } { 1 - p _ { \\theta } ( s ) } R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) } . } } \\end{array}", + "type": "interline_equation", + "image_path": "b7826c18422de652fa96e5046dce9e2b17066522210c25cb22c5f01a87170f56.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 166, + 465, + 444, + 495.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 166, + 495.6666666666667, + 444, + 526.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 166, + 526.3333333333334, + 444, + 557.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 560, + 479, + 574 + ], + "lines": [ + { + "bbox": [ + 104, + 559, + 482, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 123, + 576 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 562, + 149, + 572 + ], + "score": 0.91, + "content": "s ^ { \\prime } = s", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 559, + 186, + 576 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 186, + 560, + 267, + 574 + ], + "score": 0.93, + "content": "R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 559, + 482, + 576 + ], + "score": 1.0, + "content": "by definition, so using equation 54 we can show that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 559, + 482, + 576 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 578, + 447, + 733 + ], + "lines": [ + { + "bbox": [ + 165, + 578, + 447, + 733 + ], + "spans": [ + { + "bbox": [ + 165, + 578, + 447, + 733 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D N \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } \\frac { p _ { \\theta } ( s ^ { \\prime } ) } { 1 - p _ { \\theta } ( s ) } R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } P ( b _ { 2 } = s ^ { \\prime } | S ^ { k } , b _ { 1 } = s ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ { s ^ { \\prime } \\circ p _ { \\theta } ( b _ { 2 } \\mid S ^ { k } , b _ { 1 } = s ) } [ f ( b _ { 2 } ) ] } \\\\ & { = \\mathbb { E } _ { \\delta \\to p _ { \\theta } ( b _ { 2 } \\mid S ^ { k } , b _ { 1 } = s ) } [ p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "941373c9e6ed240fe6a0ad7ccfcb8bc26b621b25ffdc7ff97d9ddb312f31505b.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 165, + 578, + 447, + 629.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 165, + 629.6666666666666, + 447, + 681.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 165, + 681.3333333333333, + 447, + 732.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 459, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 460, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 460, + 95 + ], + "score": 1.0, + "content": "Now we can show that the control variate is actually the result of Rao-Blackwellization:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 100, + 508, + 308 + ], + "lines": [ + { + "bbox": [ + 111, + 100, + 508, + 308 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 508, + 308 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\nabla \\theta ^ { ( k ) } ( \\theta ^ { k ) } H ^ { ( S ^ { k } , \\ell ) } \\cdot \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } \\varphi ( \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ( \\theta ^ { ( k ) } , s ^ { \\prime } ) \\nabla \\theta ( \\theta , \\theta ^ { \\prime } ) \\wedge \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } ) \\cdot \\xi ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } - s | S ^ { k } | \\nabla \\theta ) \\cdot \\Big ( \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } ) \\Big ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\hat { b } _ { 1 } - s | S ^ { k } | \\nabla \\theta ) \\cdot \\Big ( \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { ( k ) } , \\theta ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\operatorname* { m i n } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\nabla \\theta ^ { ( k ) } \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ] \\xi ^ { \\prime } s ^ \\end{array}", + "type": "interline_equation", + "image_path": "5aa17a281b84d6ea5ad10f6f5e8779f8f9604e2a43778445838faba1d23ee89e.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 100, + 508, + 169.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 169.33333333333331, + 508, + 238.66666666666663 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 238.66666666666663, + 508, + 307.99999999999994 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 238, + 329 + ], + "score": 1.0, + "content": "This expression depends only on", + "type": "text" + }, + { + "bbox": [ + 239, + 317, + 248, + 327 + ], + "score": 0.88, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 316, + 266, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 317, + 276, + 327 + ], + "score": 0.88, + "content": "b _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "and we recognize the stochastic sum-and-sample estima-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 326, + 461, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 134, + 339 + ], + "score": 1.0, + "content": "tor for", + "type": "text" + }, + { + "bbox": [ + 135, + 327, + 159, + 337 + ], + "score": 0.9, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 326, + 381, + 339 + ], + "score": 1.0, + "content": "used as ‘baseline’. As a special case of equation 13 for", + "type": "text" + }, + { + "bbox": [ + 381, + 327, + 422, + 339 + ], + "score": 0.93, + "content": "C = \\{ b _ { 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 326, + 461, + 339 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 343, + 446, + 358 + ], + "lines": [ + { + "bbox": [ + 164, + 343, + 446, + 358 + ], + "spans": [ + { + "bbox": [ + 164, + 343, + 446, + 358 + ], + "score": 0.88, + "content": "\\mathbb { E } _ { b _ { 2 } \\sim p _ { \\theta } ( b _ { 2 } \\mid b _ { 1 } ) } \\left[ \\left( p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) \\right) \\right] = \\mathbb { E } _ { i \\sim p _ { \\theta } ( i ) } \\left[ f ( i ) \\right] .", + "type": "interline_equation", + "image_path": "2cafa0492e5f1eacfe47000c2ad3ffce6075884ea511cb796118f38c1fbd87c7.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 164, + 343, + 446, + 358 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 363, + 491, + 378 + ], + "lines": [ + { + "bbox": [ + 104, + 361, + 493, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 361, + 217, + 379 + ], + "score": 1.0, + "content": "Using this, and the fact that", + "type": "text" + }, + { + "bbox": [ + 217, + 363, + 457, + 377 + ], + "score": 0.78, + "content": "\\begin{array} { r } { \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\right] = \\nabla _ { \\theta } \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ 1 \\right] = \\nabla _ { \\theta } 1 = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 361, + 493, + 379 + ], + "score": 1.0, + "content": "we find", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 395, + 468, + 528 + ], + "lines": [ + { + "bbox": [ + 142, + 395, + 468, + 528 + ], + "spans": [ + { + "bbox": [ + 142, + 395, + 468, + 528 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { S ^ { k } \\sim p _ { \\theta } ( S ^ { k } ) } \\left[ \\displaystyle \\sum _ { s \\in S ^ { k } } \\nabla _ { \\theta } p _ { \\theta } ( s ) R ( S ^ { k } , s ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) \\right] } \\\\ & { = \\mathbb { E } _ { B ^ { k } \\sim p _ { \\theta } ( B ^ { k } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\left( p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) \\right) \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\mathbb { E } _ { b _ { 2 } \\sim p _ { \\theta } ( b _ { 2 } | b _ { 1 } ) } \\left[ ( p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ) \\right] \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\right] \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] } \\\\ & { = 0 \\cdot \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] } \\\\ & { = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "bdf452d0c81a2a447c9b897bf8923e610b1dd3efe14bea58ceb1b7bbda495407.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 142, + 395, + 468, + 439.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 142, + 439.3333333333333, + 468, + 483.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 142, + 483.66666666666663, + 468, + 528.0 + ], + "spans": [], + "index": 10 + } + ] + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 538, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 496, + 540, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 496, + 540, + 505, + 550 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 459, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 460, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 460, + 95 + ], + "score": 1.0, + "content": "Now we can show that the control variate is actually the result of Rao-Blackwellization:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 82, + 460, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 100, + 508, + 308 + ], + "lines": [ + { + "bbox": [ + 111, + 100, + 508, + 308 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 508, + 308 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\nabla \\theta ^ { ( k ) } ( \\theta ^ { k ) } H ^ { ( S ^ { k } , \\ell ) } \\cdot \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } \\varphi ( \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ( \\theta ^ { ( k ) } , s ^ { \\prime } ) \\nabla \\theta ( \\theta , \\theta ^ { \\prime } ) \\wedge \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } ) \\cdot \\xi ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } - s | S ^ { k } | \\nabla \\theta ) \\cdot \\Big ( \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { k } ) \\Big ) ] } \\\\ & { = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\hat { b } _ { 1 } - s | S ^ { k } | \\nabla \\theta ) \\cdot \\Big ( \\displaystyle \\sum _ { \\ell \\in \\mathbb { N } ^ { k } } \\mathcal { P } ( \\theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \\ell ) } ( \\theta ^ { ( k ) } , \\theta ^ { \\prime } ) f ( s ^ { \\prime } ) ] } \\\\ & = \\mathbb { E } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\displaystyle \\operatorname* { m i n } _ { \\theta ^ { \\star } \\sim \\Theta ( \\theta ^ { ( k ) } ) } [ \\nabla \\theta ^ { ( k ) } \\theta ^ { ( k ) } H ^ { ( S ^ { k } , \\ell ) } ] \\xi ^ { \\prime } s ^ \\end{array}", + "type": "interline_equation", + "image_path": "5aa17a281b84d6ea5ad10f6f5e8779f8f9604e2a43778445838faba1d23ee89e.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 100, + 508, + 169.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 169.33333333333331, + 508, + 238.66666666666663 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 238.66666666666663, + 508, + 307.99999999999994 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 238, + 329 + ], + "score": 1.0, + "content": "This expression depends only on", + "type": "text" + }, + { + "bbox": [ + 239, + 317, + 248, + 327 + ], + "score": 0.88, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 316, + 266, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 317, + 276, + 327 + ], + "score": 0.88, + "content": "b _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "and we recognize the stochastic sum-and-sample estima-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 326, + 461, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 134, + 339 + ], + "score": 1.0, + "content": "tor for", + "type": "text" + }, + { + "bbox": [ + 135, + 327, + 159, + 337 + ], + "score": 0.9, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 326, + 381, + 339 + ], + "score": 1.0, + "content": "used as ‘baseline’. 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p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) \\right) \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\mathbb { E } _ { b _ { 2 } \\sim p _ { \\theta } ( b _ { 2 } | b _ { 1 } ) } \\left[ ( p _ { \\theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \\theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ) \\right] \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] \\right] } \\\\ & { = \\mathbb { E } _ { b _ { 1 } \\sim p _ { \\theta } ( b _ { 1 } ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( b _ { 1 } ) \\right] \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] } \\\\ & { = 0 \\cdot \\mathbb { E } _ { x \\sim p _ { \\theta } ( x ) } \\left[ f ( x ) \\right] } \\\\ & { = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "bdf452d0c81a2a447c9b897bf8923e610b1dd3efe14bea58ceb1b7bbda495407.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 142, + 395, + 468, + 439.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 142, + 439.3333333333333, + 468, + 483.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 142, + 483.66666666666663, + 468, + 528.0 + ], + "spans": [], + "index": 10 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 243, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 245, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 245, + 96 + ], + "score": 1.0, + "content": "F THE RISK ESTIMATOR", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 259, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 259, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 259, + 118 + ], + "score": 1.0, + "content": "F.1 PROOF OF BUILT-IN BASELINE", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "We show that the RISK estimator, taking gradients through the normalization factor actually has a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "built-in baseline. 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\\partial x _ { j } } \\varepsilon ( \\frac \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial \\cdot \\partial 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De-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "spite the similarity of the equation 56 and equation 57, their gradient estimates (equation 60 and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "equation 59) are structurally dissimilar and care should be taken to implement the REINFORCE", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "estimator (or related estimators such as ARSM and the unordered set estimator) correctly using au-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 504, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 504, + 237 + ], + "score": 1.0, + "content": "tomatic differentiation software. Using Gumbel-Softmax and RELAX, we take gradients ‘directly’", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 256, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 256, + 248 + ], + "score": 1.0, + "content": "through the objective in equation 57.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 107, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 107, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "We optimize the ELBO using the analytic KL for 1000 epochs using the Adam (Kingma & Ba, 2015)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 254, + 277 + ], + "score": 1.0, + "content": "optimizer. We use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 255, + 263, + 276, + 274 + ], + "score": 0.9, + "content": "\\mathrm { i 0 ^ { - 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "for all estimators except Gumbel-Softmax and RELAX,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 218, + 288 + ], + "score": 1.0, + "content": "which use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 218, + 275, + 240, + 285 + ], + "score": 0.9, + "content": "\\bar { 1 } 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "as we found they diverged with a higher learning rate. For ARSM,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 352, + 300 + ], + "score": 1.0, + "content": "as an exception we use the sample KL, and a learning rate of", + "type": "text" + }, + { + "bbox": [ + 353, + 285, + 386, + 297 + ], + "score": 0.9, + "content": "3 \\cdot 1 0 ^ { - \\overline { { 4 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 284, + 506, + 300 + ], + "score": 1.0, + "content": ", as suggested by the authors.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 489, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 489, + 311 + ], + "score": 1.0, + "content": "All reported ELBO values are computed using the analytic KL. Our code is publicly available7.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 322, + 229, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 230, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 230, + 334 + ], + "score": 1.0, + "content": "G.2 ADDITIONAL RESULTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "Gradient variance during training. We also evaluate gradient variance of different estimators", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 474, + 366 + ], + "score": 1.0, + "content": "during different stages of training. We measure the variance of different estimators with", + "type": "text" + }, + { + "bbox": [ + 475, + 354, + 504, + 364 + ], + "score": 0.88, + "content": "k = 4", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "samples during training with REINFORCE with replacement, such that all estimators are computed", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "for the same model parameters. The results during training, given in Figure 4, are similar to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "results for the trained model in Table 1, except for at the beginning of training, although the rankings", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 277, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 277, + 410 + ], + "score": 1.0, + "content": "of different estimator are mostly the same.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 504, + 433 + ], + "score": 1.0, + "content": "Negative ELBO on validation set. Figure 5 shows the -ELBO evaluated during training on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 430, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 446 + ], + "score": 1.0, + "content": "validation set. For the large latent space, we see validation error quickly increase (after reaching", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "a minimum) which is likely because of overfitting (due to improved optimization), a phenomenon", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "observed before (Tucker et al., 2017; Grathwohl et al., 2018). Note that before the overfitting starts,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "both REINFORCE without replacement and the unordered set estimator achieve a validation error", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 464, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 464, + 490 + ], + "score": 1.0, + "content": "similar to the other estimators, such that in a practical setting, one can use early stopping.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "Results using standard binarized MNIST dataset. Instead of using the MNIST dataset binarized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "by thresholding values at 0.5 (as in the code and paper by Yin et al. (2019)) we also experiment with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "the standard (fixed) binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "(2011), for which we plot train and validation curves for two runs on the small and large domain", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "in Figure 6. This gives more realistic (higher) -ELBO scores, although we still observe the effect", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "of overfitting. As this is a bit more unstable setting, one of the runs using REINFORCE with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "replacement diverged, but in general the relative performance of estimators is similar to using the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 577, + 214, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 214, + 587 + ], + "score": 1.0, + "content": "dataset with 0.5 threshold.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 113, + 722, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 503, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 503, + 733 + ], + "score": 1.0, + "content": "7https://github.com/wouterkool/estimating-gradients-without-replacement", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "25", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 247, + 96 + ], + "score": 1.0, + "content": "This assumes we can compute the", + "type": "text" + }, + { + "bbox": [ + 247, + 83, + 262, + 93 + ], + "score": 0.37, + "content": "\\mathrm { K L }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "divergence analytically. Alternatively, we can use a sample", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 424, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 424, + 106 + ], + "score": 1.0, + "content": "estimate for the KL divergence, and use equation 56 with equation 19 to obtain", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 108, + 109, + 503, + 153 + ], + "lines": [ + { + "bbox": [ + 108, + 109, + 487, + 153 + ], + "spans": [ + { + "bbox": [ + 108, + 109, + 487, + 153 + ], + "score": 0.64, + "content": "\\begin{array} { l } { \\nabla _ { \\phi } \\mathcal { L } ( \\phi , \\theta ) = \\mathbb { E } _ { z \\sim q _ { \\phi } ( z | x ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) ( \\ln p _ { \\theta } ( x | z ) + \\ln p ( z ) - \\ln q _ { \\phi } ( z | x ) ) + \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) \\right] , } \\\\ { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { = \\mathbb { E } _ { z \\sim q _ { \\phi } ( z | x ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) ( \\ln p _ { \\theta } ( x | z ) - \\ln q _ { \\phi } ( z | x ) ) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "9047984934ee20c3367dfc54f7d948b00ec5fc53723581b47394e47e41f9a1e7.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 108, + 109, + 503, + 123.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 108, + 123.66666666666667, + 503, + 138.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 138.33333333333334, + 503, + 153.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 157, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 104, + 156, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 227, + 173 + ], + "score": 1.0, + "content": "Here we have left out the term", + "type": "text" + }, + { + "bbox": [ + 228, + 158, + 359, + 171 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbb { E } _ { z \\sim q _ { \\phi } ( z | \\pmb { x } ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | \\pmb { x } ) \\right] = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 156, + 506, + 173 + ], + "score": 1.0, + "content": ", similar to Roeder et al. (2017), and,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 169, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 268, + 184 + ], + "score": 1.0, + "content": "assuming a uniform (i.e. constant) prior", + "type": "text" + }, + { + "bbox": [ + 268, + 171, + 298, + 182 + ], + "score": 0.91, + "content": "\\ln p ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 169, + 338, + 184 + ], + "score": 1.0, + "content": ", the term", + "type": "text" + }, + { + "bbox": [ + 339, + 170, + 501, + 183 + ], + "score": 0.9, + "content": "{ { \\mathbb E } } _ { z \\sim q _ { \\phi } ( z | x ) } \\left[ \\nabla _ { \\phi } \\ln q _ { \\phi } ( z | x ) \\ln p ( z ) \\right] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 169, + 506, + 184 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "With a built-in baseline, this second term cancels out automatically, even if it is implemented. De-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "spite the similarity of the equation 56 and equation 57, their gradient estimates (equation 60 and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "equation 59) are structurally dissimilar and care should be taken to implement the REINFORCE", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "estimator (or related estimators such as ARSM and the unordered set estimator) correctly using au-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 504, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 504, + 237 + ], + "score": 1.0, + "content": "tomatic differentiation software. Using Gumbel-Softmax and RELAX, we take gradients ‘directly’", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 256, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 256, + 248 + ], + "score": 1.0, + "content": "through the objective in equation 57.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 156, + 506, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 107, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 107, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "We optimize the ELBO using the analytic KL for 1000 epochs using the Adam (Kingma & Ba, 2015)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 254, + 277 + ], + "score": 1.0, + "content": "optimizer. We use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 255, + 263, + 276, + 274 + ], + "score": 0.9, + "content": "\\mathrm { i 0 ^ { - 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "for all estimators except Gumbel-Softmax and RELAX,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 218, + 288 + ], + "score": 1.0, + "content": "which use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 218, + 275, + 240, + 285 + ], + "score": 0.9, + "content": "\\bar { 1 } 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "as we found they diverged with a higher learning rate. For ARSM,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 352, + 300 + ], + "score": 1.0, + "content": "as an exception we use the sample KL, and a learning rate of", + "type": "text" + }, + { + "bbox": [ + 353, + 285, + 386, + 297 + ], + "score": 0.9, + "content": "3 \\cdot 1 0 ^ { - \\overline { { 4 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 284, + 506, + 300 + ], + "score": 1.0, + "content": ", as suggested by the authors.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 489, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 489, + 311 + ], + "score": 1.0, + "content": "All reported ELBO values are computed using the analytic KL. Our code is publicly available7.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 253, + 506, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 322, + 229, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 230, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 230, + 334 + ], + "score": 1.0, + "content": "G.2 ADDITIONAL RESULTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "Gradient variance during training. We also evaluate gradient variance of different estimators", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 474, + 366 + ], + "score": 1.0, + "content": "during different stages of training. We measure the variance of different estimators with", + "type": "text" + }, + { + "bbox": [ + 475, + 354, + 504, + 364 + ], + "score": 0.88, + "content": "k = 4", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "samples during training with REINFORCE with replacement, such that all estimators are computed", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "for the same model parameters. The results during training, given in Figure 4, are similar to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "results for the trained model in Table 1, except for at the beginning of training, although the rankings", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 277, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 277, + 410 + ], + "score": 1.0, + "content": "of different estimator are mostly the same.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 341, + 506, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 504, + 433 + ], + "score": 1.0, + "content": "Negative ELBO on validation set. Figure 5 shows the -ELBO evaluated during training on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 430, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 446 + ], + "score": 1.0, + "content": "validation set. For the large latent space, we see validation error quickly increase (after reaching", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "a minimum) which is likely because of overfitting (due to improved optimization), a phenomenon", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "observed before (Tucker et al., 2017; Grathwohl et al., 2018). Note that before the overfitting starts,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "both REINFORCE without replacement and the unordered set estimator achieve a validation error", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 464, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 464, + 490 + ], + "score": 1.0, + "content": "similar to the other estimators, such that in a practical setting, one can use early stopping.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 420, + 506, + 490 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "Results using standard binarized MNIST dataset. Instead of using the MNIST dataset binarized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "by thresholding values at 0.5 (as in the code and paper by Yin et al. (2019)) we also experiment with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "the standard (fixed) binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "(2011), for which we plot train and validation curves for two runs on the small and large domain", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "in Figure 6. This gives more realistic (higher) -ELBO scores, although we still observe the effect", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "of overfitting. 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Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 559, + 134, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 134, + 575 + ], + "score": 1.0, + "content": "value.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 309, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 311, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 311, + 95 + ], + "score": 1.0, + "content": "H TRAVELLING SALESMAN PROBLEM", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 120 + ], + "score": 1.0, + "content": "The Travelling Salesman Problem (TSP) is a discrete optimization problem that consists of finding", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 321, + 130 + ], + "score": 1.0, + "content": "the order in which to visit a set of locations, given as", + "type": "text" + }, + { + "bbox": [ + 322, + 119, + 339, + 129 + ], + "score": 0.86, + "content": "x , y", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "coordinates, to minimize the total length", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "of the tour, starting and ending at the same location. As a tour can be considered a sequence of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "score": 1.0, + "content": "locations, this problem can be set up as a sequence modelling problem, that can be either addressed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "using supervised (Vinyals et al., 2015) or reinforcement learning (Bello et al., 2016; Kool et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 140, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 140, + 173 + ], + "score": 1.0, + "content": "2019a).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "Kool et al. 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The encoder computes embeddings for all", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "nodes (locations) and the decoder produces a tour, which is sequence of nodes, selecting one note", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "at the time using an attention mechanism, and uses this autoregressively as input to select the next", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "node. In Kool et al. (2019a), this model is trained using REINFORCE, with a greedy rollout used as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 232, + 220, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 220, + 245 + ], + "score": 1.0, + "content": "baseline to reduce variance.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "We use the code by Kool et al. (2019a) to train the exact same Attention Model (for details we refer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 274 + ], + "score": 1.0, + "content": "to Kool et al. 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We did not do any hyperparameter optimization and used the exact", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 504, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 482, + 295 + ], + "score": 1.0, + "content": "same training details, using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 482, + 282, + 504, + 293 + ], + "score": 0.89, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "(no decay) for 100 epochs for all estimators. 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As a tour can be considered a sequence of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "score": 1.0, + "content": "locations, this problem can be set up as a sequence modelling problem, that can be either addressed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "using supervised (Vinyals et al., 2015) or reinforcement learning (Bello et al., 2016; Kool et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 140, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 140, + 173 + ], + "score": 1.0, + "content": "2019a).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 105, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "Kool et al. (2019a) introduced the Attention Model, which is an encoder-decoder model which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 200 + ], + "score": 1.0, + "content": "considers a TSP instances as a fully connected graph. The encoder computes embeddings for all", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "nodes (locations) and the decoder produces a tour, which is sequence of nodes, selecting one note", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "at the time using an attention mechanism, and uses this autoregressively as input to select the next", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "node. In Kool et al. (2019a), this model is trained using REINFORCE, with a greedy rollout used as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 232, + 220, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 220, + 245 + ], + "score": 1.0, + "content": "baseline to reduce variance.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 178, + 506, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "We use the code by Kool et al. (2019a) to train the exact same Attention Model (for details we refer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 274 + ], + "score": 1.0, + "content": "to Kool et al. (2019a)), and minimize the expected length of a tour predicted by the model, using", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "different gradient estimators. We did not do any hyperparameter optimization and used the exact", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 504, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 482, + 295 + ], + "score": 1.0, + "content": "same training details, using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 482, + 282, + 504, + 293 + ], + "score": 0.89, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "(no decay) for 100 epochs for all estimators. 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