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+# CONTEXT AUTOENCODER FOR SELF-SUPERVISEDREPRESENTATION LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present a novel masked image modeling (MIM) approach, context autoencoder (CAE), for self-supervised representation pretraining. The goal is to pretrain an encoder by solving the pretext task: estimate the masked patches from the visible patches in an image. Our approach first feeds the visible patches into the encoder, extracting the representations. Then, we make predictions from visible patches to masked patches in the encoded representation space. We introduce an alignment constraint, encouraging that the representations for masked patches, predicted from the encoded representations of visible patches, are aligned with the masked patch presentations computed from the encoder. In other words, the predicted representations are expected to lie in the encoded representation space, which empirically shows the benefit to representation learning. Last, the predicted masked patch representations are mapped to the targets of the pretext task through a decoder.
+
+One additional characteristic is that our approach encourages the separation of the representation learning part (encoder), and the pretext task completion part that will be replaced by the downstream task part. In contrast, previous MIM methods (e.g., BEiT and MAE) couple the two parts, potentially limiting the representation learning quality. We demonstrate the effectiveness of our CAE through superior transfer performance in downstream tasks: semantic segmentation, and object detection and instance segmentation.
+
+# 1 INTRODUCTION
+
+We study the masked image modeling task for self-supervised representation learning. Masked image modeling (MIM) is a pretext task of masking some patches of the input image and estimate the masked patches from the visible patches. It is expected that the resulting encoder pretrained through solving the MIM task is able to extract the patch representations taking on semantics that are transferred to solving downstream tasks.
+
+BEiT (Bao et al., 2021) and the method studied in the ViT paper (Dosovitskiy et al., 2021), two MIM methods, learn a ViT to estimate the patch tokens and the pixels, respectively, and use the resulting ViT as the pretrained encoder. They take the visible patches and mask tokens representing the masked patches as input, and make estimations for both the visible and masked patches, where the estimations only for masked patches are evaluated during training. The two methods use the single ViT structure simultaneously for both representation learning and task completion. Thus, only the partial capacity of the ViT is explored for representation learning, limiting the representation quality. Masked autoencoder (MAE) (He et al., 2022) prepends an extra ViT structure that only receives visible patches as the so-called encoder followed by a lightweight decoder taking all the patches as input. Unfortunately, the decoder might play a partial role in representation learning, thus distracting the responsibility of representation learning.
+
+We present a context autoencoder (CAE) approach, illustrated in Figure 1, for improving the encoding quality. We randomly partition the image into two sets of patches: visible patches and masked patches. The architecture contains an encoder, a latent contextual regressor with an alignment constraint, and a decoder, The encoder takes only the visible patches as input and learns the representations only for the visible patches. The latent contextual regressor predicts the masked patch representations according to the visible patch representations, where the predicted masked patch representations are constrained to align with the masked patch representations computed from the encoder. The decoder maps the predicted masked patch representations to the targets for masked patches.
+
+![](images/03141b0b6df68f2bee5709817790b89a14466832cb613c52b920c4796a9a1075.jpg)  
+Figure 1: The pipeline of context autoencoder. Our approach feeds visible patches into the encoder and extracts their representations $\mathbf { Z } _ { v }$ and then completes the pretext task by predicting the representations $\mathbf { Z } _ { m }$ of the masked patches from the visible patches in the encoded representation space with latent contextual regressor and alignment constraint, and mapping predicted representations $\mathbf { Z } _ { m }$ of masked patches to the targets. The pretrained encoder in (a) is applied to downstream tasks by simply replacing the pretext task part (b) with the downstream task part.
+
+The prediction from the visible patches to the masked patches, i.e., generating a plausible semantic guess for the masked patches, is performed on the encoded representation space using latent contextual regressor. The predicted representations for the masked patches are constrained to match with the representations computed from the encoder, rendering that the predicted representations also lie in the encoded representation space. Making predictions in the encoded representation space encourages that the encoded representations take on a larger extent of semantics, empirically validated by the experiments.
+
+In addition, the encoder in the top stream in Figure 1 operates on visible patches, only focusing on learning semantic representations. The CAE design also expects that the responsibility of representation learning is taken by the encoder through two things: The latent representations of visible patches are not updated in the other parts; and the alignment constraint expects that the predicted representations through latent contextual regressor also lie in the encoded representation space. In comparison to BEiT, MAE, and the approach in the ViT paper, our CAE encoder exploits the greater capability for learning the representation, thus improving the representation quality.
+
+We present the empirical performance of our approach on downstream tasks, semantic segmentation, and object detection and instance segmentation. The results show that our approach outperforms supervised pretraining, contrastive pretraining, and other MIM methods.
+
+# 2 RELATED WORK
+
+Self-supervised representation learning has been widely studied in computer vision , including: context prediction (Doersch et al., 2015; Tian et al., 2021), clustering-based methods (Xie et al., 2016; Yang et al., 2016; Caron et al., 2018; Asano et al., 2019; Zhuang et al., 2019; Huang et al., 2019; Caron et al., 2019; Goyal et al., 2021), contrastive learning (Li et al., 2020; Oord et al., 2018; Henaff, 2020; Wang et al., 2022), instance discrimination (Dosovitskiy et al., 2014; 2015), image discretization (Gidaris et al., 2020a;b), masked image modeling (Li et al., 2021; Fang et al., 2022; Tian et al., 2022), and information maximization (Ermolov et al., 2021; Zbontar et al., 2021; Bardes et al., 2021). The following mainly reviews closely-related methods.
+
+Autoencoding. Traditionally, autoencoders were used for dimensionality reduction or feature learning (LeCun, 1987; Gallinari et al., 1987; Hinton & Zemel, 1994; Hinton & Salakhutdinov, 2006; Ranzato et al., 2007; Vincent et al., 2008; Kingma & Welling, 2013). The denoising autoencoder (DAE) is an autoencoder that receives a corrupted data point as input and is trained to estimate the original, uncorrupted data point as its output. The variants or modifications of DAE were adopted for self-supervised representation learning, e.g., corruption by masking pixels (Vincent et al., 2010;
+
+Pathak et al., 2016; Chen et al., 2020a), removing color channels (Zhang et al., 2016), shuffling image patches (Noroozi & Favaro, 2016), denoising pixel-level noise (Atito et al., 2021) and so on.
+
+Contrastive learning. In computer vision, contrastive learning1 has been popular for self-supervised representation learning (Chen et al., 2020b; He et al., 2020; Tian et al., 2020; Chen et al., 2021; Grill et al., 2020; Caron et al., 2021; Chen & He, 2021; Caron et al., 2020; Wu et al., 2018; Peng et al., 2022). The basic idea is to maximize the similarity between the views augmented from the same image and optionally minimize the similarity between the views augmented from different images. Random cropping is an important augmentation scheme, and thus typical contrastive learning methods (e.g., MoCo v3) tend to learn knowledge mainly from the center regions of the original images. Some dense variants (Wang et al., 2021; Xie et al., 2021a) eliminate the tendency in a limited degree by considering an extra contrastive loss with dense patches.
+
+Masked image modeling. Motivated by BERT for masked language modeling (Devlin et al., 2019), the method studied in (Dosovitskiy et al., 2021) and BEiT (Bao et al., 2021) use the ViT structure to solve the masked image modeling task, e.g., estimate the pixels or the discrete tokens. But they do not have explicitly an encoder or a decoder and the ViT structure is essentially a mixture of encoder and decoder, limiting the representation learning quality.
+
+Several subsequent MIM methods are developed to improve the encoder quality, such as designing pretraining architectures: iBOT (Zhou et al., 2021), Masked Autoencoder (MAE) (He et al., 2022), SplitMask (El-Nouby et al., 2021), and Simple MIM (SimMIM) (Xie et al., 2021b); adopting new reconstruction targets: Masked Feature Prediction (MaskFeat) (Wei et al., 2021), Perceptual Codebook for BEiT (PeCo) (Dong et al., 2021), and data2vec (Baevski et al., 2022). Our approach belongs to the former one and propose a new pretraining architecture through separating the representation pretraining module and the MIM task completion module and making predictions in the encoded representation space. More about the concurrently-developed methods are provided in Appendix.
+
+# 3 APPROACH
+
+# 3.1 ARCHITECTURE
+
+Our context autoencoder (CAE) pretrains the encoder by solving the masked image modeling task. We randomly split an image into two sets of patches: visible patches $\mathbf { X } _ { v }$ and masked patches $\mathbf { X } _ { m }$ . The pretext task is to predict the masked patches from visible patches. The key is to make predictions from visible patches to masked patches in the encoded representation space, and then map the predicted representations of masked patches to the targets. The architecture shown in Figure 1 contains: an encoder, a latent contextual regressor with the alignment constraint, and a decoder.
+
+Encoder. The encoder $\mathcal { F }$ maps the visible patches $\mathbf { X } _ { v }$ to the latent representations $\mathbf { Z } _ { v }$ . It only handles the visible patches. We use the ViT to form our encoder. It first embeds the visible patches by linear projection as patch embeddings, and adds the positional embeddings ${ \bf P } _ { v }$ . Then it sends the combined embeddings into a sequence of transformer blocks that are based on self-attention, generating $\mathbf { Z } _ { v }$ .
+
+Latent contextual regressor. The latent contextual regressor $\mathcal { H }$ predicts the latent representations $\mathbf { Z } _ { m }$ for the masked patches from the latent representations $\mathbf { Z } _ { v }$ of the visible patches output from the encoder. We form the latent contextual regressor $\mathcal { H }$ using a series of transformer blocks that are based on cross-attention.
+
+The initial queries $\mathbf { Q } _ { m }$ , called mask queries, are mask tokens that are learned as model parameters and are the same for all the masked patches. The keys and the values are the same and consist of the visible patch representations $\mathbf { Z } _ { v }$ and the output of the previous cross-attention layer (mask queries for the first cross-attention layer). The corresponding positional embeddings are considered when computing the cross-attention weights between the queries and the keys. In this process, the latent representations $\mathbf { Z } _ { v }$ of the visible patches are not updated.
+
+Alignment constraint. The latent representation alignment constraint is imposed on the latent representations $\mathbf { Z } _ { m }$ of the masked patches predicted by the latent contextual regressor. We feed the masked patches $\mathbf { X } _ { m }$ into the encoder, which is the same as the one for encoding visible patches, and generate the representations $\bar { \mathbf Z } _ { m }$ of the masked patches. We then align the two latent representations $\bar { \mathbf { Z } } _ { m }$ and $\bar { \mathbf Z } _ { m }$ for the masked patches.
+
+Decoder. The decoder $\mathcal { G }$ maps the latent representations $\mathbf { Z } _ { m }$ of the masked patches to some forms of the masked patches, $\mathbf { Y } _ { m }$ , with the discrete tokens as the targets as done in BEiT. The decoder, similar to the encoder, is a stack of transformer blocks that are based on self-attention, followed by a linear layer predicting the targets. The decoder only receives the latent representations of the masked patches (the output of the latent contextual regressor), and the positional embeddings of the masked patches as input without directly using the information of the visible patches.
+
+# 3.2 OBJECTIVE FUNCTION
+
+Masking and targets. Following BEiT (Bao et al., 2021), we adopt the random block-wise masking strategy (illustrated in Figure 2) to split the input image into two sets of patches, visible and masked patches. For each image, 98 of 196 $1 4 \times 1 4 )$ patches are masked.
+
+We form the targets using the discrete tokenizer, e.g., the tokenizer trained with d-VAE on ImageNet-1K without using the labels or the DALL-E tokenizer (trained with d-VAE on 400M images) (Ramesh et al., 2021) used in BEiT Bao et al. (2021). The input image is fed into the tokenizer, assigning a discrete token to each patch. The target tokens for the masked patches are denoted as $\bar { \mathbf { Y } } _ { m }$ .
+
+Loss function. The loss function consists of a decoding loss: $\ell _ { y } ( { \mathbf { Y } } _ { m } , \bar { { \mathbf { Y } } } _ { m } )$ , and an alignment loss: $\ell _ { z } ( \mathbf { Z } _ { m } , \bar { \mathbf { Z } } _ { m } )$ . The whole loss is a weighted sum:
+
+![](images/cffb56bbd7ddadf485220b27666f678d66a607d5d4e0ac19e2775777ac43710a.jpg)  
+Figure 2: Illustration of random block-wise sampling and random cropping. Random block-wise sampling is used in our approach. Random cropping is a key data-augmentation scheme for contrastive pretraining.
+
+$$
+\ell _ { y } ( { \mathbf Y } _ { m } , \bar { { \mathbf Y } } _ { m } ) + \lambda \ell _ { z } ( { \mathbf Z } _ { m } , \mathrm { s g } [ \bar { { \mathbf Z } } _ { m } ] ) .
+$$
+
+We use the MSE loss for $\ell _ { z } ( \mathbf { Z } _ { m } , \bar { \mathbf { Z } } _ { m } )$ and the cross-entropy loss for $\ell _ { y } ( { \mathbf { Y } } _ { m } , \bar { { \mathbf { Y } } } _ { m } ) . \mathrm { s g } [ \cdot ]$ stands for stop gradient. $\lambda$ is 2 in our experiments.
+
+# 4 ANALYSIS AND DISCUSSION
+
+Predictions are made in the encoded representation space. Our CAE attempts to make predictions in the encoded representation space: predict the representations for the masked patches from the encoded representations of the visible patches. In other words, it is expected that the output representations of the latent contextual regressor also lie in the encoded representation space, which is ensured by an alignment constraint. The constraint about the predicted representation space encourages the learned representation to take on a large extent of semantics for prediction from visible patches to masked patches, benefiting the representation learning of the encoder.
+
+We empirically verify that the predicted representations lie in the encoded representation space through image reconstruction. We train the CAE using the pixel colors as the prediction targets, for two cases: with and without the alignment constraint. For reconstruction, we feed all the patches (without masking, all the image patches are visible) of an image (from the ImageNet validation set) into the pretrained encoder, then skip the latent contextual regressor and directly send all the encoded patch representations to the pretrained decoder for reconstructing the whole image.
+
+Figure 3 provides reconstruction results for several examples randomly sampled from the ImageNet1K validation set. One can see that our approach can successfully reconstruct the images, implying that the input and output representations of latent contextual regressor are in the same space. In contrast, without the alignment constraint, the reconstructed images are noisy, indicating the input and output representations of latent contextual regressor are in the different spaces. The results suggest that the alignment constraint is critical for ensuring that predictions are made in the encoded representation space.
+
+Representation alignment in CAE and contrastive learning. Representation alignment is also used in contrastive learning methods, such as MoCo, BYOL, SimCLR, and methods mixing contrastive learning and masked image modeling, such as iBOT, and MST, and the alignment loss could be the MSE loss or the contrastive loss that CAE may aslo take advantage of. In the CAE, the alignment is imposed over the representations ${ \bf Z } _ { m } = \mathcal { H } ( \dot { \mathcal { F } } ( { \bf X } _ { v } )$ predicted from the representations $\mathcal { F } ( \mathbf { X } _ { v } )$ of visible patches through the regressor $\mathcal { H }$ , and the representations $\bar { \mathbf Z } _ { m } = \mathcal { F } ( \mathbf { X } _ { m } )$ computed from the encoder $\mathcal { F }$ , both about the masked patches. Differently, the alignment in contrastive learning is imposed over the representations $\{ \mathcal { F } ( \mathbf { V } _ { 1 } ) , \mathcal { F } ( \mathbf { V } _ { 2 } ) , \cdots , \mathcal { F } ( \mathbf { \bar { V } } _ { N } ) \}$ of different views $\{ \mathbf { V } _ { 1 } , \mathbf { V } _ { 2 } , \cdots , \mathbf { V } _ { N } \}$ .
+
+![](images/080ef22866b951d2d997f2618ae6f62963073f082d6d9904cfdf5e456496b30d.jpg)  
+Figure 3: Illustrating that predictions are made in the representation space. We reconstruct the image by feeding the full image (1st, 4th, and 7th) into the pretrained CAE encoder and then the pretrained CAE decoder outputting the reconstructed image (2nd, 5th, and 8th). It can be seen that the image can be constructed with the semantics kept when skipping latent contextual regressor, verifying the input and the predicted representations lie in the same space. We also show the reconstructed images (3rd, 6th, and 9th) from the encoder and the decoder pretrained without the alignment constraint. We can see that those images are meaningless, indicating that the alignment constraint is critical for ensuring that predictions are made in the representation space.
+
+Relation to autoencoder. The original autoencoder (LeCun, 1987; Gallinari et al., 1987; Hinton & Zemel, 1994) consists of an encoder and a decoder. The encoder maps the input into a latent representation, and the decoder reconstructs the input from the latent representation. The denoising autoencoder (DAE) (Vincent et al., 2010), a variant of autoencoder, corrupts the input by adding noises and still reconstructs the non-corrupted input.
+
+Our CAE encoder is similar to the original autoencoder and also contains an encoder and a decoder. Different from the autoencoder where the encoder and the decoder process the whole image, our encoder takes a portion of patches as input and our decoder takes the estimated latent representations of the other portion of patches as input. Importantly, the CAE introduces a latent contextual regressor that makes predictions in the latent space from the visible patches to the masked patches.
+
+Relation to BEiT and MAE. The CAE encoder processes the visible patches, to extract their representations, without making predictions for masked patches. Latent contextual regressor does not update the representations for visible patches: the representations of the visible patches in the regressor are the values and keys for cross-attention; the alignment constraint expects that the output of latent contextual regressor lies in the representation space same with the encoder output. The decoder only processes the predicted representations of masked patches. Therefore, the encoder takes the responsibility of and is only for representation learning.
+
+In contrast, BEiT (Bao et al., 2021) and the MIM part of iBOT do not separate the representation extraction role and the task completion role and uses a single network, with both the visible and masked patches as the input, simultaneously for the two roles. In MAE (He et al., 2022), the so-called decoder may play a partial role for representation learning as the representations of the visible patches are also updated in the MAE decoder. Unlike CAE, MAE does not explicitly regress the representations (that lie in the space the same as the encoded representation space) for masked patches.
+
+When the pretrained encoder is applied to downstream tasks, one often replaces the pretext task completion part using the downstream task layer, e.g., segmentation layer or detection layer. The separation of representation learning (encoding) and pretext task completion helps that downstream task applications take good advantage of representation pretraining.
+
+Comparison to contrastive learning. Typical contrastive learning methods, e.g., SimCLR (Chen et al., 2020b) and MoCo (He et al., 2020; Chen et al., 2021), pretrain the networks by solving the pretext task, maximizing the similarities between augmented views (e.g., random crops) from the same image and minimizing the similarities between augmented views from different images.
+
+It is shown in (Chen et al., 2020b) that random cropping plays an important role in view augmentation for contrastive learning. Through analyzing random crops (illustrated in Figure 2), we observe that the center pixels in the original image space have large chances to belong to random crops. We suspect that the global representation, learned by contrastive learning for a random crop possibly with other augmentation schemes, tends to focus mainly on the center pixels in the original image, so that the representations of different crops from the same image can be possibly similar. Figure 4 (the second row) shows that the center region of the original image for the typical contrastive learning approach, MoCo v3, is highly attended.
+
+![](images/0aab5f83b15db5a95a49f3b5fb2b2186405a600905291ab7e5871af90db93bd6.jpg)  
+Figure 4: Illustrating the attention map averaged over 12 attention heads between the class token and the patch tokens in the last layer of the ViT encoder pretrained on ImageNet-1K. The region inside the blue contour is obtained by thresholding the attention weights to keep $5 0 \%$ of the mass. Top: Input image, Middle: MoCo v3, a typical contrastive learning method, and Bottom: our CAE. One can see that MoCo v3 tends to focus mainly on the centering regions and little on other patches, and our CAE tends to consider almost all the patches.
+
+In contrast, our CAE method (and other MIM methods) randomly samples the patches from the augmented views to form the visible and masked patches. All the patches are possible to be randomly masked for the augmented views and accordingly the original image. Thus, the CAE encoder needs to learn good representations for all the patches, to make good predictions for the masked patches from the visible patches. Figure 4 (the third row) illustrates that almost all the patches in the original images are considered in our CAE encoder.
+
+Considering that the instances of the 1000 categories in ImageNet-1K locate mainly around the center of the original images (Russakovsky et al., 2015), typical contrastive learning methods, e.g., MoCo v3, learn the knowledge mainly about the 1000 categories, which is similar to supervised pretraining. But our CAE and other MIM methods are able to learn more knowledge beyond the 1000 categories from the non-center image regions. This indicates that the CAE has the potential to perform better for downstream tasks.
+
+# 5 EXPERIMENTS
+
+# 5.1 IMPLEMENTATION
+
+We study the standard ViT small, base and large architectures, ViT-S (12 transformer blocks with dimension 384), ViT-B (12 transformer blocks with dimension 768) and ViT-L (24 transformer blocks with dimension 1024). The latent contextual regressor consists of 4 transformer blocks based on cross-attention, and the decoder consists of 4 transformer blocks based on self-attention, and an extra linear projection for making predictions.
+
+We train the CAE on ImageNet-1K. We partition the image of $2 2 4 \times 2 2 4$ into $1 4 \times 1 4$ patches with the patch size being $1 6 \times 1 6$ . We use standard random cropping and horizontal flipping for data augmentation. The pretraining settings are almost the same as BEiT (Bao et al., 2021). (See Appendix for details).
+
+# 5.2 PRETRAINING EVALUATION
+
+Linear probing. Linear probing is widely used as a proxy of pretraining quality evaluation for self-supervised representation learning. It learns a linear classifier over the image-level representation output from the pretrained encoder by using the labels of the images, and then tests the performance on the validation set.
+
+Attentive probing. The output of the encoder pretrained with MIM methods are representations for all the patches. It is not suitable to linearly probe the representation, averagely-pooled from patch representations, because the image label in ImageNet-1K only corresponds to a portion of patches. It is also not suitable to use the default class token within the encoder because the default class token serves as a role of aggregating the patch representations for better patch representation extraction and is not merely for the portion of patches corresponding to the image label.
+
+![](images/41871d8089b5c0018f649251fd2d0f6daafe83110d733ad2087a6315f7c1765e.jpg)  
+Figure 5: Illustrating the cross-attention unit in attentive probing. The attention map is the average of cross-attention maps over 12 heads between the extra class token and the patches. One can see that the attended region lies mainly in the object, which helps image classification.
+
+To use the image-level label as a proxy of evaluating the pretraining quality for the encoder pretrained with MIM methods, we need to attend the patches that are related to the label. We introduce a simple modification by using a cross-attention unit with an extra class token (that is different from the class token in the encoder) as the query and the output patch representations of the encoder as the keys and the values, followed by a linear classifier. The introduced cross-attention unit is able to care mainly about the patches belonging to the 1000 classes in ImageNet-1K and remove the interference of other patches. Figure 5 illustrates the effect of the cross-attention unit, showing that the extra cross-attention unit can to some degree attend the regions that are related to the 1000 ImageNet-1K classes.
+
+Results. Table 1 shows the results with three schemes, linear probing (LIN), attentive probing (ATT), and fine-tuning (FT) for representative contrastive pretraining (MoCo v3 and DINO) and MIM (BEiT and MAE) methods, as well as our approach with the targets formed with the DALL-E tokenizer (trained on 400M images) and the d-VAE tokenizer (trained on ImageNet-1K without using the labels), denoted as $\mathbf { C A E ^ { * } }$ and CAE, respectively. The models of MAE with 300 epochs and BEiT are pretrained by us using the official implementations, and other models are the officially released models.
+
+We highlight a few observations. The fine-tuning performance for these methods are very similar and there is only a minor difference similar to the observation (Zhou et al., 2021). We think that the reason is that self-supervised pretraining and fine-tuning are conducted on the same dataset and no extra knowledge is introduced for image classification. The minor difference might come from the optimization aspect: different initialization (provided by pretrained models) for fine-tuning.
+
+In terms of linear probing, the scores of the contrastive learning methods, MoCo v3 and DINO, are higher than the MIM methods. This is as expected because contrastive learning focuses mainly on learning the representations for 1000 classes (See discussion in Section 4). The pretraining is relatively easier than existing MIM methods as contrastive learning mainly cares about the 1000 classes and MIM methods may care about the classes beyond the 1000 classes.
+
+For the MIM methods, the scores of attentive probing are much larger than linear probing. This validates our analysis: the MIM methods extract the representations for all the patches, and the classification task needs to attend the corresponding portion of patches.
+
+The LIN and ATT scores are similar for contrastive pretraining: e.g., with ViT-B, (76.2 vs 77.0) for MoCo v3, and (77.3 vs 77.8) for DINO. This means that the extra cross-attention in attentive probing does not make a big difference, which is one more evidence for our analysis in Section 4 that they already focus mainly on the region where the instance in the 1000 categories lies.
+
+# 5.3 ABLATION STUDIES
+
+The CAE architecture contains three components for pretraining the encoder: latent contextual regressor, decoder, and alignment constraint. We cannot remove the latent contextual regressor that is the only unit to make predictions for masked patches from visible patches in our architecture. We study the other two components, the decoder (when the decoder is removed, we use a linear layer to predict the targets) and the alignment constraint.
+
+Table 2 shows the ablation results. We report the scores for attentive probing, and downstream tasks: semantic segmentation on ADE20K and object detection on COCO with the DALL-E tokenizer as the
+
+Table 1: Pretraining quality evaluation in terms of fine-tuning (FT), linear probing (LIN), and attentive probing (ATT). ‡ means the number of effective epochs in (Zhou et al., 2021) as they adopt multi-crop augmentation (equivalently take a larger number of epochs compared to one-crop augmentation). We report the top-1 accuracy (in the column ATT) of the supervised training approach DeiT (Touvron et al., 2020) to show how far the ATT score is from supervised training. The scores for other models and our models are based on our implementations if not specified. Except that \* denotes using the DALL-E tokenizer, CAE adopts the d-VAE tokenizer trained on ImageNet-1K only. †: these results are from (He et al., 2022).
+
+Table 3: Semantic segmentation on ADE20K. All the results are based on the same implementation for semantic segmentation. #Epochs refers to the number of pretraining epochs. ‡ means the number of effective epochs in (Zhou et al., 2021) as the method uses multi-crop pretraining augmentation (See Table 1). †: these results are from (He et al., 2022).   
+
+<table><tr><td>Method</td><td>#Epochs</td><td>#Crops</td><td>FT</td><td>LIN</td><td>ATT</td></tr><tr><td colspan="6">Methods using ViT-S:</td></tr><tr><td>DeiT</td><td>300</td><td>=</td><td></td><td></td><td>79.9</td></tr><tr><td>MoCo v3</td><td>600t</td><td>2</td><td>81.7</td><td>73.1</td><td>73.8</td></tr><tr><td>BEiT</td><td>300</td><td>1</td><td>81.7</td><td>15.7</td><td>23.6</td></tr><tr><td>CAE*</td><td>300</td><td>1</td><td>82.0</td><td>51.8</td><td>65.0</td></tr><tr><td colspan="6">Methods using ViT-B:</td></tr><tr><td>DeiT</td><td>300</td><td>=</td><td></td><td></td><td>81.8</td></tr><tr><td>MoCo v3</td><td>600t</td><td>2</td><td>83.0</td><td>76.2</td><td>77.0</td></tr><tr><td>DINO</td><td>1600t</td><td>12</td><td>83.3</td><td>77.3</td><td>77.8</td></tr><tr><td>BEiT</td><td>300</td><td>1</td><td>83.0</td><td>37.6</td><td>49.4</td></tr><tr><td>MAE</td><td>300</td><td>1</td><td>82.9</td><td>61.5</td><td>71.1</td></tr><tr><td>MAE</td><td>1600</td><td>1</td><td>83.6</td><td>67.8</td><td>74.2</td></tr><tr><td>iBOT</td><td>1600*</td><td>12</td><td>83.8</td><td>79.5</td><td>79.8</td></tr><tr><td>CAE*</td><td>300</td><td>1</td><td>83.6</td><td>64.1</td><td>73.8</td></tr><tr><td>CAE*</td><td>800</td><td>1</td><td>83.8</td><td>68.6</td><td>75.9</td></tr><tr><td>CAE*</td><td>1600</td><td>1</td><td>83.9</td><td>70.4</td><td>77.1</td></tr><tr><td>CAE</td><td>1600</td><td>1</td><td>83.9</td><td>71.4</td><td>77.4</td></tr><tr><td colspan="6">Methods using ViT-L:</td></tr><tr><td>MoCo v3t</td><td>600+</td><td>2</td><td>84.1</td><td>=</td><td></td></tr><tr><td>BEiT†</td><td>1600</td><td>1</td><td>85.2</td><td>=</td><td>1</td></tr><tr><td>MAE</td><td>1600</td><td>1</td><td>86.0</td><td>76.0</td><td>78.8</td></tr><tr><td>CAE*</td><td>1600</td><td>1</td><td>86.3</td><td>78.1</td><td>81.2</td></tr><tr><td>CAE</td><td>1600</td><td>1</td><td>86.3</td><td>77.9</td><td>81.2</td></tr></table>
+
+Table 2: Ablation studies for the decoder and the alignment constraint in our CAE. All the models are pretrained on ImageNet-1K with 300 epochs.   
+
+<table><tr><td></td><td>Decoder</td><td>Align</td><td>ATT</td><td>ADE</td><td>CoCO</td></tr><tr><td>CAE*</td><td>×</td><td>×</td><td>71.2</td><td>47.0</td><td>46.9</td></tr><tr><td>CAE*</td><td>√</td><td>×</td><td>72.7</td><td>47.1</td><td>47.2</td></tr><tr><td>CAE*</td><td>√</td><td>√</td><td>73.8</td><td>48.3</td><td>48.4</td></tr></table>
+
+<table><tr><td>Method</td><td>#Epochs</td><td>mIoU</td></tr><tr><td colspan="3">Methodsusing ViT-B:</td></tr><tr><td>DeiT</td><td>300</td><td>47.0</td></tr><tr><td>MoCo v3</td><td>600</td><td>47.2</td></tr><tr><td>DINO</td><td>1600t</td><td>47.2</td></tr><tr><td>BEiT</td><td>300</td><td>45.5</td></tr><tr><td>BEiT</td><td>800</td><td>46.5</td></tr><tr><td>MAE</td><td>300</td><td>45.8</td></tr><tr><td>MAE</td><td>1600</td><td>48.1</td></tr><tr><td>iBOT</td><td>1600*</td><td>50.0</td></tr><tr><td>CAE*</td><td>300</td><td>48.3</td></tr><tr><td>CAE*</td><td>800</td><td>49.7</td></tr><tr><td>CAE*</td><td>1600</td><td>50.2</td></tr><tr><td>CAE</td><td>1600</td><td>50.1</td></tr><tr><td colspan="3">Methodsusing ViT-L:</td></tr><tr><td>MoCo v3t</td><td>600t</td><td>49.1</td></tr><tr><td>BEiT†</td><td>1600</td><td>53.3</td></tr><tr><td>MAE</td><td>1600</td><td>53.6</td></tr><tr><td>CAE*</td><td>1600</td><td>54.7</td></tr><tr><td>CAE</td><td>1600</td><td>54.6</td></tr></table>
+
+targets. One can see that the downstream task performance is almost the same when only the decoder is added and that the performance increases when the decoder and the alignment constraint are both added. This also verifies that the alignment constraint is important for ensuring that the predicted representations of masked patches lie in the encoded representation space and thus the predictions are made in the encoded representation space, and accordingly improving the representation quality.
+
+# 5.4 DOWNSTREAM TASKS
+
+Semantic segmentation on ADE20K (Zhou et al., 2017). We follow the implementation (Bao et al., 2021) to use UperNet (Xiao et al., 2018) (Appendix for training details). In both downstream tasks, the CAE with the tokenizers learned over ImageNet-1K performs almost the same as the tokenizers learned over 400M images provided by DALL-E (CAE\*), implying that the tokenizer trained on ImageNet-1K (without using the labels) or a larger dataset does not affect the pretraining quality and accordingly the downstream task performance.
+
+Table 4: Object detection and instance segmentation on COCO. Mask R-CNN is adopted and trained with the $1 \times$ schedule. All the results are based on the same implementation for object detection and instance segmentation. #Epochs refers to the number of pretraining epochs on ImageNet-1K. ‡ means the number of effective epochs in (Zhou et al., 2021) (See Table 1).   
+Table 3 shows that using the ViT-B, our $\mathrm { C A E ^ { * } }$ with 300 training epochs performs better than DeiT, MoCo v3, DINO, MAE (1600 epochs) and BEiT. Our $\mathrm { C A E ^ { * } }$ (1600 epochs) further improves the segmentation scores and outperforms MAE (1600 epochs), MoCo v3 and DeiT by 2.1, 3.0 and 3.2, respectively. Using ViT-L, our $\mathrm { C A E ^ { * } }$ (1600 epochs) outperforms BEiT (1600 epochs) and MAE (1600 epochs) by 1.4 and 1.1, respectively.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2"></td><td rowspan="2"></td><td rowspan="2">#EpochsSupervisedSelf-supervised</td><td colspan="3">Object detection</td><td colspan="3">Instance segmentation</td></tr><tr><td>AP</td><td>AP0</td><td>AP75</td><td>APm</td><td>AP</td><td>AP75</td></tr><tr><td colspan="10">Methods using ViT-S:</td></tr><tr><td>DeiT</td><td>300</td><td>√</td><td>×</td><td>43.1</td><td>65.2</td><td>46.6</td><td>38.4</td><td>61.8</td><td>40.6</td></tr><tr><td>MoCo v3</td><td>600</td><td>×</td><td>√</td><td>43.3</td><td>64.9</td><td>46.8</td><td>38.8</td><td>61.6</td><td>41.1</td></tr><tr><td>BEiT</td><td>300</td><td>×</td><td>√</td><td>35.6</td><td>56.7</td><td>38.3</td><td>32.6</td><td>53.3</td><td>34.2</td></tr><tr><td>CAE*</td><td>300</td><td>×</td><td>L</td><td>44.1</td><td>64.6</td><td>48.2</td><td>39.2</td><td>61.4</td><td>42.2</td></tr><tr><td colspan="10">Methods using ViT-B:</td></tr><tr><td>DeiT</td><td>300</td><td>√</td><td></td><td>46.9</td><td>68.9</td><td>51.0</td><td>41.5</td><td>65.5</td><td>44.4</td></tr><tr><td>MoCo v3</td><td>600</td><td>×</td><td>×√√√</td><td>45.5</td><td>67.1</td><td>49.4</td><td>40.5</td><td>63.7</td><td>43.4</td></tr><tr><td>DINO</td><td>1600t</td><td>×</td><td></td><td>46.8</td><td>68.6</td><td>50.9</td><td>41.5</td><td>65.3</td><td>44.5</td></tr><tr><td>BEiT</td><td>300</td><td>×</td><td></td><td>39.5</td><td>60.6</td><td>43.0</td><td>35.9</td><td>57.7</td><td>38.5</td></tr><tr><td>BEiT</td><td>800</td><td>×</td><td>√</td><td>42.1</td><td>63.3</td><td>46.0</td><td>37.8</td><td>60.1</td><td>40.6</td></tr><tr><td>MAE</td><td>300</td><td>×</td><td></td><td>45.4</td><td>66.4</td><td>49.6</td><td>40.6</td><td>63.4</td><td>43.7</td></tr><tr><td>MAE</td><td>1600</td><td>×</td><td></td><td>48.4</td><td>69.4</td><td>53.1</td><td>42.6</td><td>66.1</td><td>45.9</td></tr><tr><td>iBOT</td><td>1600*</td><td>×</td><td>√√√√V</td><td>48.2</td><td>69.7</td><td>52.8</td><td>42.7</td><td>66.5</td><td>46.0</td></tr><tr><td>CAE*</td><td>300</td><td>×</td><td></td><td>48.4</td><td>69.2</td><td>52.9</td><td>42.6</td><td>66.1</td><td>45.8</td></tr><tr><td>CAE*</td><td>800</td><td>×</td><td></td><td>49.8</td><td>70.7</td><td>54.6</td><td>43.9</td><td>67.8</td><td>47.4</td></tr><tr><td>CAE*</td><td>1600</td><td>×</td><td></td><td>50.0</td><td>70.9</td><td>54.8</td><td>44.0</td><td>67.9</td><td>47.6</td></tr><tr><td>CAE</td><td>1600</td><td>×</td><td></td><td>50.2</td><td>71.0</td><td>54.9</td><td>44.2</td><td>68.3</td><td>47.9</td></tr><tr><td colspan="10">Methods using ViT-L:</td></tr><tr><td>MAE</td><td>1600</td><td>×</td><td></td><td>54.0</td><td>74.3</td><td>59.5</td><td>47.1</td><td>71.5</td><td>51.0</td></tr><tr><td>CAE*</td><td>1600</td><td>×</td><td></td><td>54.5</td><td>75.2</td><td>60.1</td><td>47.6</td><td>72.2</td><td>51.9</td></tr><tr><td>CAE</td><td>1600</td><td>×</td><td></td><td>54.6</td><td>75.2</td><td>59.9</td><td>47.6</td><td>72.0</td><td>51.9</td></tr></table>
+
+The superior results over supervised and contrastive pretraining methods, DeiT, MoCo v3 and DINO, stem from that our approach captures the knowledge beyond the 1000 classes in ImageNet-1K. The superior results over BEiT and MAE stem from that our CAE makes predictions in the encoded representation space and that representation learning and pretext task completion are separated.
+
+Object detection and instance segmentation on COCO (Lin et al., 2014). We adopt the Mask R-CNN approach (He et al., 2017) that produces bounding boxes and instance masks simultaneously, with the ViT as the backbone (see Appendix for training details). The results are given in Table 4. We report the box AP for object detection and the mask AP for instance segmentation. The observations are consistent with those for semantic segmentation in Table 3. Our CAE\* (300 epochs, ViT-B) is superior to all the other models except that a little lower than MAE (1600 epochs). Our approach (1600 epochs) outperforms MAE (1600 epochs), MoCo v3 and DeiT by 1.6, 4.5 and 3.1, respectively. Using ViT-L, our CAE achieves 54.6 box AP and outperforms MAE by 0.6.
+
+# 6 CONCLUSION
+
+The core design of our CAE architecture for masked image modeling is that predictions are made from visible patches to masked patches in the encoded representation space. Experiments demonstrate the effectiveness of our design. In addition, we also point out that the advantage of MIM methods over typical contrastive pretraining and supervised pretraining on ImageNet-1K is that MIM learns the representations for all the patches, while typical contrastive pretraining (e.g., MoCo and SimCLR) and supervised pretraining tend to learn semantics mainly from center patches of the original images and little from non-center patches.
+
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+
+# APPENDIX
+
+# A TRAINING DETAILS
+
+Pretraining. The settings are almost the same as BEiT (Bao et al., 2021). We use AdamW (Loshchilov & Hutter, 2017) for optimization and train the CAE for 300/800/1600 epochs with the batch size being 2048. We set the learning rate as 1.5e-3 with cosine learning rate decay. The weight decay is set as 0.05. The warmup epochs for 300/800/1600 epochs pre-training are 10/20/40, respectively. We employ drop path (Huang et al., 2016) rate 0.1 and dropout rate 0.
+
+Fine-tuning on ImageNet. We follow the fine-tuning protocol in BEiT to use layer-wise learning rate decay, weight decay and AdamW. The batch size is 4096, the warmup epoch is 5 and the weight decay is 0.05. For ViT-S, we train 200 epochs with learning rate 1.6e-2 and layer-wise decay rate 0.75. For ViT-B, we train 100 epochs with learning rate 8e-3 and layer-wise decay rate 0.65. For ViT-L, we train 50 epochs with learning rate 2e-3 and layer-wise decay rate 0.75.
+
+Linear probing. We use the LARS (You et al., 2017) optimizer with momentum 0.9. The model is trained for 90 epochs. The batch size is 16384, the warmup epoch is 10 and the learning rate is 6.4. Following (He et al., 2022), we adopt an extra BatchNorm layer (Ioffe & Szegedy, 2015) without affine transformation (affine $=$ False) before the linear classifier. We do not use mixup (Zhang et al., 2017), cutmix (Yun et al., 2019), drop path (Huang et al., 2016), or color jittering, and we set weight decay as zero.
+
+Attentive probing. The parameters of the encoder are fixed during attentive probing. A crossattention module, a BatchNorm layer (affine $=$ False), and a linear classifier are appended after the encoder. The extra class token representation in cross-attention is learned as model parameters. The keys and the values are the patch representations output from the encoder. There is no MLP or skip connection operation in the extra cross-attention module. We use the SGD optimizer with momentum 0.9 and train the model for 90 epochs. The batch size is 8192, the warmup epoch is 10 and the learning rate is 0.4. Same as linear probing, we do not use mixup (Zhang et al., 2017), cutmix (Yun et al., 2019), drop path, or color jittering, and we set weight decay as zero.
+
+Hyperparameter choice. There is a tradeoff variable $\lambda$ in the loss function given in Equation 1 in the main paper. We did not do an extensive study and only tried three choices, $\lambda = 1$ , $\lambda = 1 . 5$ and $\lambda = 2$ . The linear probing results are 63.4, 63.7 and 64.1, respectively. The choice $\lambda = 1$ works also well, slightly worse than $\lambda = 2$ that is adopted in our experiment.
+
+We also conduct experiments with different mask ratios including $4 0 \%$ , $5 0 \%$ , and $6 0 \%$ . Results are listed in Table 5. We find that ratio $5 0 \%$ gets better results than ratio $\cdot$ . Adopting a higher mask ratio $( 6 0 \% )$ could further improve the performance of linear probing and attentive probing, while the semantic segmentation performance is reduced by $0 . 2 \%$ . We choose $\cdot$ in our work unless specified.
+
+Table 5: Ablation study on different mask ratios. ViT-B is used here.   
+
+<table><tr><td>Mask Ratio</td><td>LIN</td><td>ATT</td><td>ADE Seg</td></tr><tr><td>40%</td><td>63.1</td><td>73.0</td><td>47.2</td></tr><tr><td>50%</td><td>64.1</td><td>73.8</td><td>48.3</td></tr><tr><td>60%</td><td>64.8</td><td>74.2</td><td>48.1</td></tr></table>
+
+Table 6: Ablation study on number of layers in the latent contextual regressor and decoder. 4 layers means we use 4 layers in latent contextual regressor and 4 layers in decoder. ViT-B is used here.   
+
+<table><tr><td>Layer Num</td><td>LIN</td></tr><tr><td>1</td><td>58.7 67.5</td></tr><tr><td>2</td><td>62.1 71.7</td></tr><tr><td>4</td><td>73.8</td></tr><tr><td>5</td><td>73.7</td></tr></table>
+
+![](images/c0fd9905c8c72742aec5c13af69859231aff6ebc77d27c77b0db7bbbc6267953.jpg)  
+Figure 6: t-SNE visualization (one color for one category) of representations extracted from the images in ADE20K. Left: ViT pretrained with our CAE; Right: ViT with random weights.
+
+For the number of layers in the latent contextual regressor and decoder, we tried four choices: 1-layer, 2-layers, 4-layer and 5-layer. Results are listed in Table 6. We empirically observed that 4-layer outperforms other choices overall.
+
+Object detection and instance segmentation on COCO. We utilize multi-scale training and resize the image with the size of the short side between 480 and 800 and the longe side no larger than 1333. The batch size is 32. For the ViT-S, the learning rate is 3e-4, the layer-wise decay rate is 0.75, and the drop path rate is 0.1. For the ViT-B, the learning rate is 3e-4, the layer-wise decay rate is 0.75, and the drop path rate is 0.2. For the ViT-L, the learning rate is 2e-4, the layer-wise decay rate is 0.8, and the drop path rate is 0.2. We train the network with the $1 \times$ schedule: 12 epochs with the learning rate decayed by $1 0 \times$ at epochs 9 and 11. We do not use multi-scale testing. The Mask R-CNN implementation follows MMDetection (Chen et al., 2019).
+
+Semantic segmentation on ADE20K. We use AdamW as the optimizer. The input resolution is $5 1 2 \times 5 1 2$ . The batch size is 16. For the ViT-B, the layer-wise decay rate is 0.65 and the drop path rate is 0.1. We search from four learning rates, 1e-4, 2e-4, 3e-4 and 4e-4, for all the results in Table 3 in the main paper. For the ViT-L, the layer-wise decay rate is 0.95 and drop path rate is 0.15. We search from three learning rates for all the methods, 3e-5, 4e-5 and 5e-5, We conduct fine-tuning for 160K steps. We do not use multi-scale testing.
+
+# B INTERPRETATION
+
+Intuitive Interpretation for CAE. Humans are able to hallucinate what appears in the masked regions and how they appear according to the visible regions. We speculate that humans do this possibly in a way similar as the following example: given that only the region of the dog’s head is visible and the remaining parts are missing, one can (a) recognize the visible region to be about a dog, (b) predict the regions where the other parts of the dog appear, and (c) guess what the other parts look like.
+
+Our CAE encoder is in some sense like the human recognition step (a). It understands the content by mapping the visual patches into latent representations that lie in the subspace that corresponds to the category $\deg$ . The latent contextual regressor is like step (b). It produces a plausible hypothesis for the masked patches, and describes the regions corresponding to the other parts of the dog using latent representations. The CAE decoder is like step (c), mapping the latent representations to the targets. It should be noted that the latent representations might contain other information besides the semantic information, e.g., the part information and the information for making predictions.
+
+We adopt t-SNE (Van der Maaten & Hinton, 2008) to visualize the high-dimensional patch representations output from our CAE encoder on ADE20K (Zhou et al., 2017) in Figure 6. ADE20K has a total of 150 categories. For each patch in the image, we set its label to be the category that more than half of the pixels belong to. We collect up to 1000 patches for each category from sampled 500 images.
+
+![](images/bc1ac2add4b102e741ba2329874c7f06332d0d41d67322a81503824458633eaf.jpg)  
+Figure 7: The computational graphs for (a) a context autoencoder (CAE), (b) BEiT (Bao et al., 2021), (c) a denoising autoencoder (DAE), and (d) MAE (He et al., 2022) and the one stream in SplitMask (El-Nouby et al., 2021). The parts in cornflower blue are for loss function. (a) The encoder $\mathcal { F }$ receives visible patches $\mathbf { X } _ { v }$ and outputs their latent representations $\mathbf { Z } _ { v }$ . The latent contextual regressor $\mathcal { H }$ predicts the latent representations $\mathbf { Z } _ { m }$ for masked patches from $\mathbf { Z } _ { v }$ . The decoder predicts the targets ${ \bf Y } _ { m }$ for masked patches from $\mathbf { Z } _ { m }$ . $\ell _ { z }$ and $\ell _ { y }$ are the loss functions. During training, the gradient is stopped for $\bar { \mathbf { Z } } _ { m }$ . See the detail in Section 3 in the main paper. (b) The input includes both visible patches $\mathbf { X } _ { v }$ and mask queries $\mathbf { Q } _ { m }$ representing masked patches, and the representations for them are updated within the function $\mathcal { R }$ . (c) The function $\mathcal { N }$ is a noising function generating the noisy version $\tilde { \mathbf { X } }$ from the input $\mathbf { X }$ . $\mathcal { F }$ and $\mathcal { G }$ are the normal encoder and decoder, respectively. (d) The two functions, ${ \mathcal { F } } ^ { \prime }$ and $\mathcal { R }$ , are both based on self-attention. ${ \mathcal { F } } ^ { \prime }$ (called encoder in MAE) only processes the visible patches $\mathbf { X } _ { v }$ , and $\mathcal { R }$ (called decoder in MAE) processes both the latent representations $\mathbf { Z } _ { v }$ of the visible patches and the mask queries $( \mathbf { Q } _ { m } )$ and updates them simultaneously. For simplicity, the positional embeddings are not included in computational graphs. (a) CAE and (c) $D A E$ perform the encoding and MIM task completion roles explicitly and separately, (b) BEiT and (d) MAE perform the encoding and MIM task completion roles implicitly and simultaneously.
+
+As shown in the figure, the latent representations of CAE are clustered to some degree for different categories (though not perfect as our CAE is pretrained on ImageNet-1K). Similar observations could be found for other MIM methods.
+
+Probabilistic interpretation for CAE. The MIM problem can be formulated in the probabilistic form, maximizing the probability of the predictions $\mathbf { Y } _ { m }$ of the masked patches given the conditions, the visible patches $\mathbf { X } _ { v }$ , the positions $\mathbf { P } _ { v }$ of the visible patches, and the positions $\mathbf { P } _ { m }$ of the masked patches: $P ( \mathbf { Y } _ { m } | \mathbf { X } _ { v } , \mathbf { P } _ { v } , \mathbf { P } _ { m } )$ . It can be solved by introducing latent representations $\mathbf { Z } _ { m }$ and $\mathbf { Z } _ { v }$ , with the assumption that $\mathbf { Z } _ { v }$ and $\mathbf { P } _ { m }$ ( $\mathbf { Y } _ { m }$ and $\mathbf { P } _ { v }$ ) are conditionally independent:
+
+$$
+P ( \mathbf { Y } _ { m } | \mathbf { X } _ { v } , \mathbf { P } _ { v } , \mathbf { P } _ { m } ) = P ( \mathbf { Z } _ { v } | \mathbf { X } _ { v } , \mathbf { P } _ { v } ) P ( \mathbf { Z } _ { m } | \mathbf { Z } _ { v } , \mathbf { P } _ { v } , \mathbf { P } _ { m } ) P ( \mathbf { Y } _ { m } | \mathbf { Z } _ { m } , \mathbf { P } _ { m } ) ,
+$$
+
+where the three terms on the right side correspond to three parts of our CAE: the encoder, the latent contextual regressor, and the decoder, respectively.
+
+The latent representation alignment constraint can be written as a conditional probability, $P ( \mathbf { Z } _ { m } | \bar { \mathbf { Z } } _ { m } )$ where $\bar { \mathbf Z } _ { m }$ is the masked patch representations computed from the encoder.
+
+Intuitive interpretation for contrastive learning. We consider the case in ImageNet-1K that the object mainly lies in the center of an image3. There are $N$ randomly sampled crops from an image, and each crop ${ \mathbf I } _ { n }$ contains a part of the center object, ${ \bf O } _ { n }$ . To maximize the similarity between two crops ${ \mathbf I } _ { m }$ and ${ \mathbf I } _ { n }$ , the pretraining might contain the processes: Select the regions $\mathbf { O } _ { m }$ and ${ \bf O } _ { n }$ from the two crops ${ \mathbf I } _ { m }$ and ${ \mathbf I } _ { n }$ , extract their features $\mathbf { f } _ { o m }$ and $\mathbf { f } _ { o n }$ , and predict the feature of the object, $\mathbf { f } _ { o }$ , from the part features $\mathbf { f } _ { o m }$ and $\mathbf { f } _ { o n }$ . In this way, the features of the crops from the same image could be similar. Among the $N$ random crops, most crops contain a part of the object in the center, and a few crops that do not contain a part of the center object could be viewed as noises when optimizing the contrastive loss.
+
+![](images/863fa591103dd42790ec3b00d8199da4de08dabdc78564e02f2a818cf4d1b291.jpg)  
+Figure 8: The computational graph for MAE (He et al., 2022) and the one stream in SplitMask (ElNouby et al., 2021). The two functions, ${ \mathcal { F } } ^ { \prime }$ and $\mathcal { R }$ , are both based on self-attention. ${ \mathcal { F } } ^ { \prime }$ (called encoder in MAE) only processes the visible patches $\mathbf { X } _ { v }$ , and $\mathcal { R }$ (called decoder in MAE) processes both the latent representations $\mathbf { Z } _ { v }$ of the visible patches and the mask queries $( \mathbf { Q } _ { m } )$ and updates them simultaneously.
+
+![](images/cfa7023d1ef7e9fd6de32e6016a965df619b32f401fd80eb6c1f321a8c903cf3.jpg)  
+Figure 9: The attention maps over two sets of randomly cropped images (the 1st the 5th rows) for MoCo v3 (the 2nd the 6th rows), MAE (the 3rd the 7th rows), and our CAE (the 4th the 8th rows) pretrained on ImageNet-1K. The contrastive learning method, MoCo v3, tends to focus mainly on the object region and little on other regions. However, MIM-based models, CAE and MAE, tend to consider almost all the patches. The attention maps over the original images are shown in Figure 4 in the main paper.
+
+After pretrained on ImageNet-1K (where the object mainly lies in the center) the encoder is able to learn the knowledge of the 1000 classes and localize the region containing the object belonging to the 1000 classes. It is not necessary that the object lies in the center for the testing image. We show the attention maps of MoCo v3 and our CAE for random crops in Figure 9. This further verifies that MoCo v3 (contrastive pretraining) pretrained on ImageNet-1K tends to attend to the object region, corresponding to the center region of the original image as shown in Figure 4 in the main paper.
+
+![](images/809cb5fede0e879c0dee572c3793c8961e35e3306d21874f07cb52155ec504d6.jpg)  
+Figure 10: The architecture for image reconstruction by skipping the latent regressor. After we pretrain the CAE model, we feed the full image (including visible patches and masked patches) to the pretrained CAE encoder and then the decoder by skipping the latent regressor.
+
+Computational graph. We provide the computational graph for CAE, BEiT (Bao et al., 2021), denoising autoencoder, Masked autoencoder (He et al., 2022) and SplitMask (El-Nouby et al., 2021) (one stream) in Figure 7. Compared to our CAE, the main issue of MAE is that the so-called decoder $\mathcal { R }$ might have also the encoding role, i.e., learning semantic representations of the visible patches.
+
+Image reconstruction by skipping the latent regressor. To verify that the input and output representation of latent contextual regressor lie in the same space, we conduct experiments on image reconstruction (e.g., the 2nd, 5th, and 8th columns of Figure 3 in the main paper) by using only the CAE encoder and decoder with the latent regressor skipped. After we pretrain the CAE model under two settings: w/ the alignment constraint and w/o the alignment constraint, we feed the full image (including visible patches and masked patches) to the pretrained CAE encoder and then the decoder by skipping the latent regressor, as shown in Figure 10. The pretrained CAE decoder outputs the reconstructed image. It can be seen that the image can be reconstructed with the semantics kept when training with the alignment constraint, verifying the input and the predicted representations lie in the same space. Otherwise, the reconstructed images are meaningless.
+
+# C CONCURRENT WORK AND MORE RESULTS
+
+Concurrent work. There are concurrently-developed MIM methods, e.g., extending MIM to frequency domain (Xie et al., 2022a; Liu et al., 2022), studying the scalability of MIM (Xie et al., 2022c), combining MIM with contrastive learning (Tao et al., 2022; Jing et al., 2022; Yi et al., 2022; Huang et al., 2022b), efficient pretraining (Zhang et al., 2022; Huang et al., 2022a; Chen et al., 2022), designing mask strategy Kakogeorgiou et al. (2022); Li et al. (2022a;c), studying how MIM works (Xie et al., 2022b; Li et al., 2022b; Kong & Zhang, 2022). Other variants (Wei et al., 2022; Li et al., 2022d) extend MIM through forming the targets using semantic encoders which is essentially a supervised learning method other than self-supervised learning.
+
+Table 7: The results of some concurrently-developed self-supervised MIM methods for semantic segmentation on ADE20K, and object detection and instance segmentation on COCO with the Cascaded Mask-RCNN framework $1 \times$ schedule). ViT-B is used for all experiments. The segmentation results of other methods are from the corresponding paper, and all the detection results are from our implementation.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Pretraining Dataset</td><td rowspan="2">#Epochs</td><td colspan="2">ADE</td><td colspan="2">COCO</td></tr><tr><td>mIoU</td><td></td><td>AP</td><td>APm</td></tr><tr><td>SplitMask (El-Nouby et al., 2021)</td><td>ADE20K</td><td>21000</td><td>45.7</td><td></td><td>=</td><td></td></tr><tr><td>Ge²-AE (Liu et al., 2022)</td><td>ImageNet-1K</td><td>800</td><td>48.9</td><td></td><td></td><td></td></tr><tr><td>A²MIM (Li et al., 2022b)</td><td>ImageNet-1K</td><td>800</td><td>49.0</td><td></td><td>-</td><td>-</td></tr><tr><td>MAE (He et al., 2022)</td><td>ImageNet-1K</td><td>1600</td><td>48.1</td><td></td><td>51.3</td><td>44.3</td></tr><tr><td>iBOT (Zhou et al., 2021)</td><td>ImageNet-1K</td><td>1600</td><td>50.0</td><td></td><td>51.2</td><td>44.2</td></tr><tr><td>CAE*</td><td>ImageNet-1K</td><td>300</td><td>48.3</td><td></td><td>51.6</td><td>44.6</td></tr><tr><td>CAE*</td><td>ImageNet-1K</td><td>800</td><td>49.7</td><td></td><td>52.8</td><td>45.5</td></tr><tr><td>CAE*</td><td>ImageNet-1K</td><td>1600</td><td>50.2</td><td></td><td>52.9</td><td>45.5</td></tr></table>
+
+Segmentation and detection. Table 7 reports the results of semantic segmentation on ADE20K for some concurrent papers. We also report the results of object detection and instance segmentation under the Cascaded Mask R-CNN framework (Cai & Vasconcelos, 2021).
+
+Downstream classification. We conduct fine-tuning experiments on three datasets: Food101 (Bossard et al., 2014), Clipart (Castrejon et al., 2016) and Sketch (Castrejon et al., 2016). Results in Table 8 shows that the proposed method outperforms previous supervised method (DeiT) and self-supervised methods (DINO, MAE).
+
+Table 8: Top-1 classification accuracy on Food-101, Clipart and Sketch. ViT-B is used here.   
+
+<table><tr><td>Method</td><td>Supervised</td><td>Self-supervised</td><td>Food-101</td><td>Clipart</td><td>Sketch</td></tr><tr><td>Random Init.</td><td>×</td><td>×</td><td>82.77</td><td>52.90</td><td>46.42</td></tr><tr><td>DeiT</td><td>√</td><td>×</td><td>91.81</td><td>81.18</td><td>73.45</td></tr><tr><td>DINO</td><td>×</td><td>√</td><td>91.67</td><td>80.72</td><td>73.13</td></tr><tr><td>MAE</td><td>×</td><td>√</td><td>93.19</td><td>80.63</td><td>73.87</td></tr><tr><td>CAE*</td><td>×</td><td>√</td><td>93.32</td><td>81.84</td><td>74.65</td></tr></table>
+
+Impact of pretraining targets. To study the impact of different pretraining targets on model performance, we conduct additional experiments on the RGB pixel value target. Comparing the results with DALL-E tokenizer and d-VAE tokenizer trained on ImageNet-1K, the model shows better linear probe and segmentation results but inferior in attentive probe, as shown in Table 9. Pretraining with these three targets obtains similar performance, illustrating that CAE does not rely on specific pretraining targets.
+
+Table 9: Evaluation of different pretraining targets on the performance of CAE. ViT-B is used here. Models are trained for 1600 epochs.   
+
+<table><tr><td>Targets</td><td>LIN</td><td>ATT</td><td>ADE Seg</td></tr><tr><td>DALL-E tokenizer</td><td>70.4</td><td>77.1</td><td>50.2</td></tr><tr><td>d-VAE tokenizer</td><td>71.4</td><td>77.4</td><td>50.1</td></tr><tr><td>RGB pixel value</td><td>72.4</td><td>77.0</td><td>50.4</td></tr></table>
+
+Training costs analysis. We report the runtime cost for the training process for ViT-B under the number of 1600 epochs in Table 10. The time cost is got with 4 machines with 8-GPU A100 and batch size is 2048. One can see that the time costs for different models are similar ( $\_$ BEiT slightly).
+
+Table 10: Training cost analysis. ViT-B is used here. Models are trained for 1600 epochs.   
+
+<table><tr><td>Method</td><td>Epochs</td><td>Total Hours</td><td>Memory/GPU</td></tr><tr><td>BEiT</td><td>1600</td><td>115 hours</td><td>14330MiB</td></tr><tr><td>MAE</td><td>1600</td><td>85 hours</td><td>11222 MiB</td></tr><tr><td>CAE</td><td>1600</td><td>109 hours</td><td>13730 MiB</td></tr></table>

parse/dev/QDdJhACYrlX/QDdJhACYrlX.md

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+# THOMAS: TRAJECTORY HEATMAP OUTPUT WITH LEARNED MULTI-AGENT SAMPLING
+
+Thomas Gilles1,2, Stefano Sabatini1, Dzmitry Tsishkou1, Bogdan Stanciulescu2, Fabien Moutarde2
+
+1IoV team, Paris Research Center, Huawei Technologies France 2Center for robotics, MINES ParisTech {stefano.sabatini, dzmitry.tsishkou}@huawei.com {thomas.gilles, bogdan.stanciulescu, fabien.moutarde}@mines-paristech.fr
+
+# ABSTRACT
+
+In this paper, we propose THOMAS, a joint multi-agent trajectory prediction framework allowing for an efficient and consistent prediction of multi-agent multimodal trajectories. We present a unified model architecture for simultaneous agent future heatmap estimation, in which we leverage hierarchical and sparse image generation for fast and memory-efficient inference. We propose a learnable trajectory recombination model that takes as input a set of predicted trajectories for each agent and outputs its consistent reordered recombination. This recombination module is able to realign the initially independent modalities so that they do no collide and are coherent with each other. We report our results on the Interaction multi-agent prediction challenge and rank $1 ^ { s t }$ on the online test leaderboard.
+
+# 1 INTRODUCTION
+
+Motion forecasting is an essential step in the pipeline of an autonomous driving vehicle, transforming perception data into future prediction which are then leveraged to plan the future moves of the autonomous cars. The self-driving stacks needs to predict the future trajectories for all the neighbor agents, in a fast and coherent way.
+
+The interactivity between agents plays an important role for accurate trajectory prediction. Agents need to be aware of their neighbors in order to adapt their speed, yield right of way and merge in neighbor lanes. To do so, different interaction mechanisms have been developed, such as social pooling (Alahi et al., 2016; Lee et al., 2017; Deo & Trivedi, 2018), graphs (Salzmann et al., 2020; Zeng et al., 2021) or attention (Mercat et al., 2020; Messaoud et al., 2020; Luo et al., 2020; Gao et al., 2020; Liang et al., 2020), benefiting from the progress of powerful transformer architectures (Li et al., 2020; Yuan et al., 2021; Girgis et al., 2021; Ngiam et al., 2021) . These mechanisms allow agents to look at and share features with neighbors and to take them into account in their own predictions.
+
+Multi-modality is another important aspect of the possible future trajectories. A car can indeed chose to turn right or left, or decide to realise a certain maneuver in various ways. Uncertainty modeled as variance of Gaussians is insufficient to model these multiple cases, as it can only represent a continuous spread and cannot show multiple discrete possibilities. Therefore, current state-of-the-art produces not one but $K$ possible trajectories for each agent predicted, and most recent benchmarks (Caesar et al., 2020; Chang et al., 2019; Zhan et al., 2019; Ettinger et al., 2021) include multimodality in their metrics, taking only the minimum error over a predicted set of $K$ trajectories.
+
+However, up until very recently and the opening of multi-agent joint interaction challenges (Ettinger et al., 2021; Zhan et al., 2021), no motion forecasting prediction datasets were taking into account the coherence of modalities between different agents predicted at the same time. As a result, the most probable predicted modality of a given agent could crash with the most probable modality of another agent.
+
+Our THOMAS model encodes the past trajectories of all the agents present in the scene, as well as the HD-Map lanelet graph, and then predicts for each agent a sparse heatmap representing the future probability distribution at a fixed timestep in the future, using hierarchical refinement for very efficient decoding. A deterministic sampling algorithm then iteratively selects the best K trajectory endpoints according to the heatmap for each agent, in order to represent a wide and diverse spectrum of modalities. Given this wide spectrum of endpoints, a recombination module takes care of addressing consistency in the scene among agents.
+
+Our contributions are summarized as follow:
+
+• We propose a hierarchical heatmap decoder allowing for unconstrained heatmap generation with optimized computational costs, enabling efficient simultaneous multi-agent prediction.   
+• We design a novel recombination model able to recombine the sampled endpoints to obtain scene-consistent trajectories across the agents.
+
+# 2 RELATED WORK
+
+Learning-based models have quickly overtaken physics-based methods for trajectory prediction for several reasons. First, the sequential nature of trajectories is a logical application for recurrent architectures (Alahi et al., 2016; Altche & de La Fortelle, 2017; Lee et al., 2017; Mercat et al., ´ 2020; Khandelwal et al., 2020). Then, benefiting from the latest progresses in computer vision, convolutional layers can easily be applied to bird-view rasters of the map context (Lee et al., 2017; Tang & Salakhutdinov, 2019; Cui et al., 2019; Hong et al., 2019; Salzmann et al., 2020; Chai et al., 2020; Gilles et al., 2021b). A more compact representation closer to the trajectory space can encode surrounding HD-Maps (usually formalized as connected lanelets), using Graph Neural Networks (Gao et al., 2020; Liang et al., 2020; Zeng et al., 2021; Gilles et al., 2021a). Finally, some pointbased approaches (Ye et al., 2021) can be applied in a broader way to trajectory prediction, as both lanes and trajectories can be considered as ordered set of points.
+
+Multi-modality in prediction can be obtained through a multiple prediction head in the model (Cui et al., 2019; Liang et al., 2020; Ngiam et al., 2021; Deo et al., 2021). However, some methods rather adopt a candidate-based approach where potential endpoints are obtained either from anchor trajectories through clustering (Chai et al., 2020; Phan-Minh et al., 2020) or from a model-based generator (Song et al., 2021). Other approaches use a broader set of candidates from the context graph (Zhang et al., 2020; Zhao et al., 2020; Zeng et al., 2021; Kim et al., 2021) or a dense grid around the target agent (Deo & Trivedi, 2020; Gu et al., 2021; Gilles et al., 2021b;a). Another family of approaches use variational inference to generate diverse predictions through latent variables (Lee et al., 2017; Rhinehart et al., 2018; Tang & Salakhutdinov, 2019; Casas et al., 2020) or GAN (Gupta et al., 2018; Rhinehart et al., 2018; Sadeghian et al., 2019) but the sampling of these trajectories is stochastic and does not provide any probability value for each sample.
+
+While very little work has directly tackled multi-agent prediction and evaluation so far, multiple methods hint at the ability to predict multiple agents at the same time (Liang et al., 2020; Ivanovic et al., 2020; Zeng et al., 2021) even if they then focus on a more single-agent oriented framework. Other works (Alahi et al., 2016; Tang & Salakhutdinov, 2019; Rhinehart et al., 2019; Girgis et al., 2021) use autoregressive roll-outs to condition the future step of an agent on the previous steps of all the other agents. SceneTransformer (Ngiam et al., 2021) repeats each agent features across possible modalities, and performs self-attention operations inside each modality before using a loss computed jointly among agents to train a model and evaluate on the WOMD dataset (Ettinger et al., 2021) interaction track. ILVM (Casas et al., 2020) uses scene latent representations conditioned on all agents to generate scene-consistent samples, but its variational inference does not provide a confidence score for each modality, hence LookOut (Cui et al., 2021) proposes a scenario scoring function and a diverse sampler to improve sample efficiency. AIR2 (Wu & Wu, 2021) extends Multipath (Chai et al., 2020) and produces a cross-distribution for two agents along all possible trajectory anchors, but it scales exponentially with the number of agents, making impractical for a real-time implementation that could encounter more than 10 agents at the same time.
+
+The approach most related to this paper is GOHOME (Gilles et al., 2021a), which uses a similar graph encoder and then leverage lane rasters to generate a probability heatmap in a sparse manner. However, this lane-generated heatmap remains constrained to the drivable area and to fixed lanewidths. Another close approach is the one of DenseTNT (Gu et al., 2021), which also uses attention to a grid of points in order to obtain a dense prediction, but their grid also remains constrained to a neighborhood of the drivable area. Finally, none of these previous methods considers the problems of scene consistency for multi-agent prediction.
+
+![](images/5a3ba25a987b6d34843c249d488e98067ab815e58842c8473070164fbc0fe1ae.jpg)  
+Figure 1: Illustration of the THOMAS multi-agent prediction pipeline
+
+# 3 METHOD
+
+Our goal is to predict the future $T$ timesteps of $A$ agents using their past history made of $H$ timesteps and the HD-Map context. Similar to recent works (Zhao et al., 2020; Zeng et al., 2021; Gu et al., 2021), we will divide the problem into goal-based prediction followed by full trajectory reconstruction. Our prediction pipeline is displayed in Fig. 1. We first encode each agent trajectory and the HD-Map context graph into a common representation. We then decode a future probability heatmap for each agent in the scene, which we sample heuristically to maximize coverage. Finally, we recombine the sampled endpoints into scene-consistent modalities across agents and build the full trajectories for each agent.
+
+Our pipeline shares the same graph encoder, sampling algorithm and full trajectory generation as GOHOME (Gilles et al., 2021a), but uses a novel efficient hierarchical heatmap process that enables to scale to simultaneous multi-agent prediction. Furthermore, we add a novel scene-consistency module that recombines the marginal outputs into a joint prediction.
+
+# 3.1 MODEL BACKBONE
+
+# 3.1.1 GRAPH ENCODER
+
+We use the same encoder as the GOHOME model (Gilles et al., 2021a). The agent trajectories are encoded though T rajEncoder using a 1D CNN followed by a UGRU recurrent layer (Rozenberg et al., 2021), and the HD-Map is encoded as a lanelet graph using a GNN GraphEncoder made of graph convolutions. We then run cross-attention Lanes2Agents to add context information to the agent features, followed by self-attention Agents2Agents to observe interaction between agents. The final result is an encoding $F _ { a }$ for each agent, where history, context and interactions have been summarized. This encoding $F _ { a }$ is used in the next decoder operations and is also stored to be potentially used in modality recombination described in Sec. 3.2. The resulting architecture of these encoding operations is illustrated in the first half of Fig. 2.
+
+![](images/0703cba6cd80ceb93683e3249927f2f8c77652689a2df7e6e0a92b83f65aafd0.jpg)  
+Figure 2: Model architecture for multi-agent prediction with shared backbone
+
+![](images/cb2812d32407c26900f1d01873383bc5ca02a8bbe3edf9f8734a6647821212af.jpg)  
+Figure 3: Hierarchical iterative refinement of the grid probabilities. First, the full grid is evaluated at a very low resolution, then the highest cells are up-sampled and evaluated at a higher resolution, until final resolution is reached. We highlight in grey the restricted area considered for refinement at each step.
+
+# 3.1.2 HIERARCHICAL GRID DECODER
+
+Our aim here is to decode each agent encoding into a heatmap representing its future probability distribution at prediction horizon $T$ . Since we create this heatmap for each agent in the scene, the decoding process has to be fast so that it can be applied to a great number of agents in parallel. We use hierarchical predictions at various levels of resolutions so that the decoder has the possibility of predicting over the full surroundings of the agent but learns to refine with more precision only in places where the agent will end up with high probability. This hierarchical process is illustrated in Fig. 3.
+
+Starting from an initial full dense grid probability at low resolution $R _ { 0 } \times R _ { 0 }$ by pixels, we iteratively refine the resolution by a fixed factor $f$ until we reach the desired final resolution $R _ { f i n a l } \times R _ { f i n a l }$ . At each iteration $i$ , we select only the $N _ { i }$ highest ranking grid points from the previous iteration, and upsample only these points to the $R _ { i } \times R _ { i } = R _ { i - 1 } / f \times R _ { i - 1 } / f$ . At each step, the grid points features are computed by a 2-layer MLP applied on the point coordinates, these features are then concatenated to the agent encoding and passed through a linear layer, and finally enriched by a 2- layer cross-attention on the graph lane features, before applying a linear layer with sigmoid to get the probability.
+
+For a given $W$ output range, this hierarchical process allows the model to only operate on $W / _ { R _ { 0 } } \times$ $\textstyle W / R _ { 0 } { \dot { + } } \sum _ { i } N _ { i } \times { \bar { f } } ^ { 2 }$ grid points instead of the $W \big / R _ { f i n a l } \ \times \ W \big / R _ { f i n a l }$ available. In practice, for a final output range of 192 meters with desired $R _ { f i n a l } = 0 . 5 m$ resolution, we start with an initial resolution of $R _ { 0 } = 8 m$ and use two iterations of $\left( N _ { 1 } , N _ { 2 } \right) = \left( 1 6 , 6 4 \right)$ points each and an upscaling factor $f = 4$ . This way, we compute only 1856 grid points from the 147 456 available, with no performance loss.
+
+The heatmap is trained on each resolution level using as pixel-wise focal loss as in Gilles et al. (2021b), detailed in Appendix A.3, with the ground truth being a Gaussian centered at the target agent future position.
+
+# 3.1.3 FULL TRAJECTORY GENERATION
+
+Having a heatmap as output grants us the good property of letting us decide to privilege coverage in order to give a wide spectrum of candidates to the scene recombination process. From each heatmap, we therefore decode $K$ end points using the same MissRate optimization algorithm as Gilles et al. (2021b). We then generate the full trajectories for each end point using also the same model, a fully-connected MLP. The MLP takes the car history and predicted endpoint as input and produces $T$ 2D-coordinates representing the full future trajectory. At training time, this model is trained using ground-truth endpoints.
+
+![](images/167fcee0a0b323d60a446df059598a89bce034f2549248e69ff9cb8b16df5573.jpg)  
+Figure 4: Illustration of THOMAS methods for generation of scene-consistent agent modalities
+
+# 3.2 MODALITY RECOMBINATION FOR MULTI-AGENT CONSISTENT PREDICTION
+
+The difficulty in multimodal multi-agent prediction resides in having coherent modalities between each agent. Since the modalities are considered scene-wise, each agent’s most probable modality has to match with the most probable prediction of the other agents, and so on. Moreover, these modalities must not collide with each other, as they should represent realistic scenarios.
+
+Initially, the prediction output of the model can be defined as marginal, as all $A$ agents have been predicted and sampled independently in order to get $K$ possible endpoint each. Our goal is to output a set of scene predictions $\bar { \mathcal { I } } = ( \bar { L } , A )$ from the marginal prediction $\mathcal { M } = ( A , K )$ , where each scene modality $l$ belonging to $\mathcal { I }$ is an association of endpoints $p _ { l } ^ { a }$ for each agent $a$ . To achieve this, our main hypothesis lays in the fact that good trajectory proposals are already present in the marginal predictions, but they need to be coherently aligned among agents to achieve a set of overall consistent scene predictions. For a given agent $a$ , the modality selected for the scene $l$ would be a combination of this agent available marginal modalities $p _ { k } ^ { a } \in \mathcal { M }$ . Each scene modality $l$ would select a different association between the agents.
+
+We illustrate our scene modality generation process in Fig. 4. The model learns $L$ scene modality embeddings $S _ { l }$ of $D$ features each. $K \times A$ agent modality vectors $A _ { k } ^ { a }$ are also derived from each agent modality endpoint. These vectors are obtained through a 2-layer MLP applied on the agent modality coordinates $p _ { k } ^ { a }$ , to which the stored agent encoding $F _ { a }$ (previously described in Sec. 3.1.1) is concatenated in order to help the model recognise modalities from the same agent. The scene modality vectors $S _ { l }$ are ’specialized’ by querying the available modality proposals $A _ { k } ^ { a }$ of each agent through cross-attention layers. Subsequently, a matching score $c _ { l } ^ { k , a }$ between each scene modality $S _ { l }$ and each agent modality $A _ { k } ^ { a }$ is computed. This matching score can be intuitively interpreted as the selection of the role (maneuver $k$ ) that the agent $a$ would play in the overall traffic scene $l$ :
+
+$$
+c _ { l } ^ { k , a } = S _ { l } . A _ { k } ^ { a T }
+$$
+
+Since argmax is non-differentiable, we employ a soft argmax as a weighted linear combination of the agent modalities $p _ { k } ^ { a }$ using a softmax on the $c _ { l } ^ { k , a }$ scores:
+
+$$
+p _ { l } ^ { a } = \sum _ { k } s o f t m a x ( c _ { l } ^ { k , a } ) p _ { k } ^ { a }
+$$
+
+The recombination module is trained with the scene-level minimum displacement error, where for each modality inside a single scene all predicted agent displacement losses are averaged before taking the minimum (winner-takes-all loss) over the $L$ scene modalities. A formal definiton of this loss, also used as a metric, is given in Eq. 2 in Sec. 4.2 $\mathrm { \ m i n S F D E } _ { k }$ ).
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASET
+
+We use the Interaction v1.2 dataset that has recently opened a new multi-agent track in the context of its Interpret challenge. It contains 47 584 training cases, 11 794 validation cases and 2 644 testing cases, with each case containing between 1 and 40 agents to predict simultaneously.
+
+# 4.2 METRICS
+
+For a set of $k$ predictions $p _ { k } ^ { a }$ for each agent $a$ in a scene, we report the minimum Final Displacement Error $( \mathrm { m i n F D E } _ { k } ,$ ) to the ground truth $\hat { p } ^ { a }$ and the MissRate $( \mathrm { M R } _ { k } )$ , whose definition of a miss is specified according to the evaluated dataset in Appendix A.1. In their usual marginal definition, these metrics are averaged over agents after the minimum operation, which means that the best modality of each agent is selected independently for each and then averaged:
+
+$$
+m i n F D E _ { k } = \frac { 1 } { A } \sum _ { a } m i n _ { k } \| p _ { k } ^ { a } - \hat { p } ^ { a } \| _ { 2 } , M R _ { k } = \frac { 1 } { A } \sum _ { a } m i n _ { k } \mathbb { 1 } _ { m i s s k } ^ { a }
+$$
+
+For consistent scene multi-agent prediction, we report the joint scene-level metrics, where the average operation over the agents is done before the minimum operator. In this formulation, the minimum is taken over scene modalities, meaning that only the best scene (joint over agents) modality is taken into account:
+
+$$
+m i n S F D E _ { k } = m i n _ { k } \frac { 1 } { A } \sum _ { a } \| p _ { k } ^ { a } - \hat { p } ^ { a } \| _ { 2 } , S M R _ { k } = m i n _ { k } \frac { 1 } { A } \sum _ { a } \mathbb { 1 } _ { m i s s k } ^ { a }
+$$
+
+In other words, the marginal metrics pick their optimal solutions in a pool of $k$ to the power of $a$ predicted solutions, while the joint metrics restrict this pool to only $k$ possibilities, making it a much more complex problem.
+
+We also report the scene collision rate (SCR) which is the percentage of modalities where two or more agents collide together, and the consistent collision-free joint Miss Rate (cSMR), which is $S M R$ where colliding modalities are also counted as misses.
+
+# 4.3 COMPARISON WITH STATE-OF-THE-ART
+
+We compare our THOMAS model performance with other joint predictions methods ILVM (Casas et al., 2020) and SceneTransformer (Ngiam et al., 2021). For fair comparison, we use a GOHOME encoder for each of the method, and adapt them accordingly so that they predict only endpoints similar to our method. For each method, we focus on implementing the key idea meant to solve scene consistency and keep the remaining part of the model as close as possible to our approach for fair comparison:
+
+ILVM (Casas et al., 2020) uses variational inference to learn a latent representation of the scene conditioned of each agent with a Scene Interaction Module, and decodes it with a similar Scene Interaction Module. The required number of modalities is obtained by sampling the latent space as many times as require. Even though the sampling is independent for each agent, the latent space is generated conditionally on all the agents.
+
+SceneTransformer (Ngiam et al., 2021) duplicates the agent encoding with the number of modalities required and adds a one-hot encoding specific to each modality. They then apply a shared transformer architecture on all modalities to encode intra-modality interactions between agents and generate the final trajectories.
+
+More details on the state-of-the-art method and the re-implementations we used are given in Appendix A.4. The results are reported in Tab. 1. While having comparable marginal distance performance (demonstrating that our model is not inherently more powerful or leveraging more information), THOMAS significantly outperforms other methods on every joint metric. SMR is improved by about $2 5 \%$ and SCR by almost $3 0 \%$ , leading to a combined cSMR decreased by also more than $2 \dot { 5 } \%$ . We also report our numbers from the Interpret multi-agent online test set leaderboard in Tab.
+
+Table 1: Comparison of consistent solutions on Interpret multi-agent validation track   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Marginal metrics</td><td colspan="3">Joint metrics</td></tr><tr><td>mADE mFDE</td><td>MR</td><td>mFDE</td><td>MR</td><td>SCR</td><td>cMR</td></tr><tr><td>ILVM (Casas et al., 2020)</td><td>0.30 0.62</td><td>10.8</td><td>0.84</td><td>19.8</td><td>5.7</td><td>21.3</td></tr><tr><td>SceneTranformer (Ngiam et al., 2021)</td><td>0.29 0.59</td><td>10.5</td><td>0.84</td><td>15.7</td><td>3.4</td><td>17.3</td></tr><tr><td>THOMAS</td><td>0.31 0.60</td><td>8.2</td><td>0.76</td><td>11.8</td><td>2.4</td><td>12.7</td></tr></table>
+
+5 of Appendix A.5. For better comparison with existing single-agent trajectory prediction state-ofthe-art, we provide evaluation numbers on the single-agents benchmarks Argoverse (Chang et al., 2019) and NuScenes (Caesar et al., 2020) in Appendix A.7.
+
+# 4.4 ABLATION STUDIES
+
+# 4.4.1 RECOMBINATION MODULE
+
+We establish the following baselines to assess the effects our THOMAS recombination.
+
+Scalar output: we train a model with the GOHOME graph encoder and a multimodal scalar regression head similar to Liang et al. (2020). We optimize it with either marginal and joint loss formulation.
+
+Heatmap output with deterministic sampling: we try various sampling methods applied on the heatmap, with either the deterministic sampling as described in Gilles et al. (2021a) or a learned sampling trained to directly regress the sampled modalities from the input heatmap.
+
+We report the comparison between the aforementioned baselines and THOMAS in Tab. 2. Compared to these baselines, THOMAS can be seen as an hybrid sampling method that takes the result of deterministic sampling as input and learns to recombine it into a more coherent solution. With regard to the joint algorithmic sampling that only tackled collisions but has little to no effect on actual consistency, as highlighted by the big drop in collision rate from $7 . 2 \%$ to $2 . 6 \%$ but a similar joint SMR, THOMAS actually brings a lot of consistency in the multi-agent prediction and drops the joint SMR from $1 4 . 8 \%$ to $\mathrm { { \dot { 1 } 1 . 8 \% } }$ .
+
+Table 2: Comparison of consistent solutions on Interpret multi-agent validation track   
+
+<table><tr><td></td><td></td><td></td><td colspan="3">Marginal metrics</td><td colspan="4">Jointmetrics</td></tr><tr><td>Output</td><td>Sampling</td><td>Objective</td><td>mADE</td><td>mFDE</td><td>MR</td><td>mFDE</td><td>MR</td><td>Col</td><td>cMR</td></tr><tr><td>Scalar</td><td></td><td>Marg</td><td>0.28</td><td>0.59</td><td>10.4</td><td>1.04</td><td>23.7</td><td>6.4</td><td>24.9</td></tr><tr><td>Scalar</td><td></td><td>Joint</td><td>0.34</td><td>0.77</td><td>16.2</td><td>0.90</td><td>19.9</td><td>49</td><td>21.7</td></tr><tr><td>Heat</td><td>Learned</td><td>Marg</td><td>0.26</td><td>0.46</td><td>4.9</td><td>0.98</td><td>20.9</td><td>4.1</td><td>21.9</td></tr><tr><td>Heat</td><td>Learned</td><td>Joint</td><td>0.29</td><td>0.58</td><td>9.8</td><td>0.88</td><td>15.2</td><td>3.0</td><td>16.4</td></tr><tr><td>Heat</td><td>Algo</td><td>Marg</td><td>0.29</td><td>0.54</td><td>3.8</td><td>0.83</td><td>14.8</td><td>7.2</td><td>15.9</td></tr><tr><td>Heat</td><td>Algo</td><td>Joint</td><td>0.29</td><td>0.54</td><td>3.8</td><td>0.83</td><td>14.8</td><td>2.6</td><td>15.6</td></tr><tr><td>Heat</td><td>Combi</td><td>Joint</td><td>0.31</td><td>0.60</td><td>8.2</td><td>0.76</td><td>11.8</td><td>2.4</td><td>12.7</td></tr></table>
+
+Usually, scalar marginal models already suffer from learning difficulties as only one output modality, the closest one to ground, can be trained at a time. Some modalities may therefore converge faster to acceptable solutions, and benefit from a much increased number of training samples compared to the others. This problem is aggravated in the joint training case, since the modality selected is the same for all agents in a training sample. The joint scalar model therefore actually fails to learn multi-modality as illustrated by a higher marginal $\mathrm { m i n F D E _ { 6 } }$ than any other model, and an extremely high crossCollisionRate since some modalities never train and always point to the same coordinates regardless of the queried agent. Note that, despite a similar training loss, SceneTransformer doesn’t suffer of the same pitfalls in Tab. 1 as is shares the same weights between all modalities and only differentiates them in the initialization of the features.
+
+# 4.5 HIERARCHICAL HEATMAP REFINEMENT
+
+We assess the speed gain of our proposed hierarchical decoder compared to the lane rasters of GOHOME Gilles et al. (2021a). In Tab. 3. We report training time for 16 epochs with a batchsize of 16, and inference time for 32 and 128 simultaneous agents.
+
+Table 3: Comparison of consistent solutions on Interpret multi-agent validation track   
+
+<table><tr><td></td><td>Training</td><td>Inference 32 agents</td><td>Inference 128 agents</td></tr><tr><td>GOHOME</td><td>12.8 hours</td><td>36 ms</td><td>90 ms</td></tr><tr><td>THOMAS</td><td>7.5 hours</td><td>20 ms</td><td>31 ms</td></tr></table>
+
+For additional comparison, the other existing dense prediction method DenseTNT (Gu et al., 2021) reports an inference speed of $1 0 0 \mathrm { m s }$ per sample for their model.
+
+# 4.6 QUALITATIVE EXAMPLES
+
+In this section, we will mostly compare the model before recombination, which we will refer by the Before model, to the model after recombination, referenced as the Af ter model. We display four qualitative examples in Fig. 5 with colliding modalities in the Before model (in dashed orange) and the solved modality (in full line orange) after recombination. For each model (Before-dashed or Af ter-full), the highlighted modality in orange is the best modality according to the $S M R _ { 6 }$ metric among the 6 available modalities. We also display in dashed grey the other 5 predicted Before modalities, and highlight that the recombination model indeed selects modalities already available in the vanilla set and reorders them so that non-colliding modalities are aligned together.
+
+![](images/2ab60ac29fca607437fdd3b01a0d78606f2110dc5906eba1c06697aeea3f8509.jpg)  
+Figure 5: Qualitative examples of recombination model assembling collision-free modalities together compared to initial colliding modalities. For each example we display the general context with highlighted agents and area of interest, then two zooms in on the agents, one displaying the initial best modality before recombination in dashed orange and all the other available modalities in grey. The second zooms shows the best modality after recombination in full line orange.
+
+We also show more qualitative examples in Fig. 6, where we highlight the comparison in modality diversity between the Before model (in dashed lines) and the Af ter model (in full lines). While the Bef ore model tries to spread the modalities for all agents to minimize marginal miss-rate, the recombined model presents much less spread compared to the original model, maintaining a multimodal behavior only in presence of very different possible agent intentions such as different possible exits or turn choices. For most other agents, almost all modalities are located at the same position, that is, the one deemed the most likely by the model. Thus, if the truly uncertain agents have to select the second or third most likely modality, the other agents still have their own most likely modality
+
+![](images/f2aa394fced168b31ab355c9972d010e64ac3fc2ef104a28f4c3825000c2ecf9.jpg)  
+Figure 6: Qualitative examples of recombination model selecting fewer but more pronounced and impactful agent modalities compared to initial colliding modalities. For each example we display on the left the vanilla model modalities with dashed lines, with the initial best modality in dashed orange and all the other available modalities in grey. On the right we display the selected modalities after recombination, where the model focuses on the most likely occurrence in most agents.
+
+# 5 CONCLUSION
+
+We have presented THOMAS, a recombination module that can be added after any trajectory prediction module outputing multi-modal predictions. By design, THOMAS allows to generate sceneconsistent modalities across all agents by making the scene modalities select coherent agent modalities and restricting the modality budget on the agents that truly need it. We show significant performance increase when adding the THOMAS module compared to the vanilla model and achieve state-of-the-art results compared to already existing methods tackling scene-consistent predictions.
+
+# REPRODUCIBILITY STATEMENT
+
+We use the publicly available Interaction 1.2 dataset (Zhan et al., 2019) available at http:// challenge.interaction-dataset.com/dataset/download. We detail dataset preprocessing in Appendix A.2, training process in Appendix A.3 and give a precise illustration of our model architecture with each layer size in A.8.
+
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+
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+
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+
+# A APPENDIX
+
+# A.1 METRIC DETAILS
+
+We adopt for each dataset evaluation the miss definition as specified by the dataset benchmark. Argoverse (Chang et al., 2019) and NuScenes (Caesar et al., 2020) define a miss as the prediction being further than 2 meters to the ground truth, while Waymo (Ettinger et al., 2021) and Interaction Zhan et al. (2019)count a miss when the prediction is closer than a lateral (1m) and an longitudinal threshold with regard to speed:
+
+$$
+\mathrm { T h r e s h o l d } _ { \mathrm { \scriptsize ~ l o n } } = \left\{ \begin{array} { c c } { 1 } & { v < 1 . 4 m / s } \\ { 1 + \frac { v - 1 . 4 } { 1 1 - 1 . 4 } } & { 1 . 4 m / s \le v \le 1 1 m / s } \\ { 2 } & { v \ge 1 1 m / s } \end{array} \right.
+$$
+
+# A.2 DATASET PROCESSING
+
+We use the training/validation split provided in Interaction 1.2. We select for each scene a reference agent on which the scene will be centered and oriented according to its heading at prediction time. At training time, this reference agent is chosen randomly across all agents. At validation and testing time, we compute the barycenter of all agents and select the closest agent as the scene reference point. We use for training all agents that have ground-truth available at prediction horizon $\mathrm { T } { = } 3 \mathrm { s }$ .
+
+We use the provided dataset HD-Maps to create the context graphs. As in Liang et al. (2020), we use four relations:{predecessor, successor, $l e f t , r i g h t \}$ obtained from the lane graph connectivity. We upsample long lanelets to have a maximum of 10 points in each lanelet.
+
+For each agent, we use their trajectory history made of position, yaw and speed in the past second sampled at $1 0 \mathrm { H z }$ . We also provide the model a mask indicating if the agent was present in a given time-frame, and pad with zeros when the agent is not tracked at a specific timestep.
+
+# A.3 TRAINING DETAILS
+
+We train all models with Adam optimizer and batchsize 32. We initialize the learning rate at $1 e ^ { - 3 }$ and divide it by 2 at epochs 3, 6, 9 and 13, before stopping the training at epoch 16. We use ReLU activation after every linear layer unless specified otherwise, and LayerNormalization after every attention and graph convolution layer.
+
+During training, since a scene can contain up to 40 agents, which is too much for gradient computation in batches of 32, we restrict the number of predicted agents to 8 by randomly sampling them across available agents. We do not use other data augmentation.
+
+The final heatmap $Y$ predicted cover a range of 192 meters at resolution $0 . 5 \mathrm { m }$ , hence a (384, 384) image. For heatmap generation, we use the same pixel-wise focal loss over the pixels $p$ as in Gilles et al. (2021b):
+
+$$
+L = - \frac { 1 } { P } \sum _ { p } ( Y _ { p } - \hat { Y } _ { p } ) ^ { 2 } f ( Y _ { p } , \hat { Y } _ { p } ) \mathrm { w i t h } f ( Y _ { p } , \hat { Y } _ { p } ) = \left\{ \begin{array} { l l } { \log ( \hat { Y } _ { p } ) } & { \mathrm { i f ~ } Y _ { p } = 1 } \\ { ( 1 - Y _ { p } ) ^ { 4 } \log ( 1 - \hat { Y } _ { p } ) } & { \mathrm { e l s e } } \end{array} \right.
+$$
+
+where $\hat { Y }$ is defined as a Gaussian centered on the agent future position at prediction horizon $T$ with a standard deviation of 4 pixels, equivalent to $2 \mathrm { m }$ .
+
+# A.4 BASELINES IMPLEMENTATIONS
+
+# A.4.1 IMPLICIT LATENT VARIABLE MODEL
+
+We use a GOHOME encoder for the prior, posterior and decoder Scene Interaction Modules. We weight the KL term with $\beta = 1$ which worked best according to our experiments.
+
+# A.4.2 SCENE TRANSFORMER
+
+The initial paper applies a transformer architecture on a $[ F , A , T , D ]$ tensor where $F$ is the potential modality dimension, $A$ the agent dimension and $T$ the time dimension, with $D$ the feature embedding, with factorized self-attention to the agent and time dimensions separately, so that agents can look at each-other inside a specific scene modality. The resulting output is optimized using a jointly formalized loss. For our implementation, we get rid of the $T$ dimension as we focus on endpoint prediction and coherence between the $A$ agents. The initial encoded $[ A , D ]$ tensor is obtained with a GOHOME encoder, multiplied across the $F$ futures and concatenated with a modality-specific one-hot encoding as in Ngiam et al. (2021) to obtain the $[ F , A , D ]$ tensor. We then apply two layers of agent self-attention similar to the original paper, before decoding the endpoints through a MLP.
+
+# A.4.3 COLLISION-FREE ENDPOINT SAMPLING BASELINE
+
+We designed a deterministic sampling algorithm based on the heatmaps generated in previous section in order to sample endpoints for each agent in a collision aware manner. We use the same sampling algorithm as Gilles et al. (2021a) based on MR optimization, but add a sequential iteration over the agents for each modalities.
+
+For a single modality $k$ , we predict the possible endpoint of a first agent $a$ by taking the maximum accumulated predicted probability under an area of radius $r$ . We then not only set to zero the heatmap values of this agent heatmap $\mathcal { T } _ { k ^ { \prime } } ^ { a }$ around the sampled location so not to sample it in the next modalities $k ^ { \prime }$ , but we also set to zero the same area on the heatmaps $\mathcal { T } _ { k } ^ { a ^ { \prime } }$ of the other agents $a ^ { \prime }$ on the same modality $k$ , so that these other agents cannot be sampled at the same position for this modality. This way, we try to enforce collision-free endpoints, and expect that considering collisions brings logic to improve the overall consistency of the predictions. However, as will be highlighted in Sec. 4.4, this methods significantly improves the collision rate without the need for any additional learned model but it does barely improve the multi agent consistency.
+
+# A.5 RESULTS ON INTERPRET CHALLENGE
+
+We also report our numbers from the Interpret multi-agent track challenge online leaderboard in Table 5. It is to be noted that the DenseTNT solution explicitely checks for collisions in the search for its proposed predictions, which we don’t, hence their $0 \%$ collision rate (SCR) and its direct impact on consistent collision-free joint Miss Rate (cSMR).
+
+Table 4: Results on Interpret multi-agent regular scene leaderboard (test set)   
+
+<table><tr><td></td><td>minSADE</td><td>minSFDE</td><td>SMR</td><td>SCR</td><td>cSMR</td></tr><tr><td>MoliNet</td><td>0.73</td><td>2.55</td><td>44.4</td><td>7.5</td><td>47.4</td></tr><tr><td>ReCoG2 (Mo et al., 2020)</td><td>0.47</td><td>1.16</td><td>23.8</td><td>6.9</td><td>26.8</td></tr><tr><td>DenseTNT (Gu et al., 2021)</td><td>0.42</td><td>1.13</td><td>22.4</td><td>0.0</td><td>22.4</td></tr><tr><td>THOMAS</td><td>0.42</td><td>0.97</td><td>17.9</td><td>12.8</td><td>25.2</td></tr></table>
+
+Table 5: Results on Interpret multi-agent conditional scene leaderboard (test set)   
+
+<table><tr><td></td><td>minSADE</td><td>minSFDE</td><td>SMR</td><td>SCR</td><td>cSMR</td></tr><tr><td>ReCoG2 (Mo et al., 2020)</td><td>0.33</td><td>0.87</td><td>14.98</td><td>0.09</td><td>15.12</td></tr><tr><td>DenseTNT (Gu et al., 2021)</td><td>0.28</td><td>0.89</td><td>15.02</td><td>0.0</td><td>15.02</td></tr><tr><td>THOMAS</td><td>0.31</td><td>0.72</td><td>10.67</td><td>0.84</td><td>11.63</td></tr></table>
+
+# A.6 SPEED / PERFORMANCE TRADE-OFF WITH HIERARCHICAL REFINEMENT
+
+We also display the trade-off between inference speed and coverage from hierarchical refinement in Fig. 7, evaluated on the Interpret multi-agent dataset with marginal MissRate6. The curve is obtained setting the number $N$ of upsampled points at the last refinement iteration from 2 to 128.
+
+From $N = 1 6$ and lower, coverage performance starts to diminish while little speed gains are made. We still kept a relatively high $_ { \mathrm { N = 6 4 } }$ in our model as we wanted to insure a wide coverage, and the time loss between $4 1 ~ \mathrm { m s }$ and $4 6 \mathrm { m s }$ remains acceptable.
+
+![](images/4f12d90162723a8705e4a9c69025eb31c341b546352eed37d0aaf56dddc7c749.jpg)  
+Figure 7: Curve of MissRate $\mathrm { { \dot { 6 } } }$ with regard to inference time with varying number of points upsampled at the last hierarchical refinement iteration.
+
+A.7 ADDITIONAL RESULTS ON SINGLE-AGENT DATASETS
+
+Table 6: Argoverse Leaderboard (Chang et al.)   
+
+<table><tr><td rowspan="2"></td><td colspan="2">K=1</td><td colspan="3">K=6</td></tr><tr><td>minFDE</td><td>MR</td><td>minADE</td><td>minFDE</td><td>MR</td></tr><tr><td>LaneGCN (Liang et al., 2020)</td><td>3.78</td><td>59.1</td><td>0.87</td><td>1.36</td><td>16.3</td></tr><tr><td>Autobot (Girgis et al., 2021)</td><td></td><td></td><td>0.89</td><td>1.41</td><td>16</td></tr><tr><td>TPCN (Ye et al., 2021)</td><td>3.64</td><td>58.6</td><td>0.85</td><td>1.35</td><td>15.9</td></tr><tr><td>Jean (Mercat et al., 2020)</td><td>4.24</td><td>68.6</td><td>1.00</td><td>1.42</td><td>13.1</td></tr><tr><td>SceneTrans (Ngiam et al.,2021)</td><td>4.06</td><td>59.2</td><td>0.80</td><td>1.23</td><td>12.6</td></tr><tr><td>LaneRCNN (Zeng et al., 2021)</td><td>3.69</td><td>56.9</td><td>0.90</td><td>1.45</td><td>12.3</td></tr><tr><td>PRIME (Song et al., 2021)</td><td>3.82</td><td>58.7</td><td>1.22</td><td>1.56</td><td>11.5</td></tr><tr><td>DenseTNT (Gu et al., 2021)</td><td>3.70</td><td>59.9</td><td>0.94</td><td>1.49</td><td>10.5</td></tr><tr><td>GOHOME (Gilles et al., 2021a)</td><td>3.65</td><td>57.2</td><td>0.94</td><td>1.45</td><td>10.5</td></tr><tr><td>HOME (Gilles et al., 2021b)</td><td>3.65</td><td>57.1</td><td>0.93</td><td>1.44</td><td>9.8</td></tr><tr><td>THOMAS</td><td>3.59</td><td>56.1</td><td>0.94</td><td>1.44</td><td>10.4</td></tr></table>
+
+Table 7: NuScenes Leaderboard (Caesar et al.)   
+
+<table><tr><td rowspan="2"></td><td colspan="2">K=5</td><td colspan="2">K=10</td><td rowspan="2">k=1 minFDE</td></tr><tr><td>minADE</td><td>MR</td><td>minADE</td><td>MR</td></tr><tr><td>CoverNet (Phan-Minh etal., 2020) Trajectron++ (Salzmann et al.,2020)</td><td>1.96</td><td>67</td><td>1.48</td><td>57</td><td></td></tr><tr><td></td><td>1.88</td><td>70</td><td>1.51</td><td></td><td>9.52</td></tr><tr><td>ALAN (Narayanan et al., 2021)</td><td>1.87</td><td>60</td><td>1.22</td><td>49</td><td>9.98</td></tr><tr><td>SG-Net (Wang et al., 2021)</td><td>1.86</td><td>67</td><td>1.40</td><td>52</td><td>9.25</td></tr><tr><td>WIMP (Khandelwal et al., 2020)</td><td>1.84</td><td>55</td><td>1.11</td><td>43</td><td>8.49</td></tr><tr><td>MHA-JAM (Messaoud et al., 2020)</td><td>1.81</td><td>59</td><td>1.24</td><td>46</td><td>8.57</td></tr><tr><td>CXX (Luo et al., 2020)</td><td>1.63</td><td>69</td><td>1.29</td><td>60</td><td>8.86</td></tr><tr><td>LaPred (Kim et al.,2021)</td><td>1.53</td><td></td><td>1.12</td><td></td><td>8.12</td></tr><tr><td>P2T (Deo &amp; Trivedi,2020)</td><td>1.45</td><td>64</td><td>1.16</td><td>46</td><td>10.50</td></tr><tr><td>GOHOME (Gilles et al., 2021a)</td><td>1.42</td><td>57</td><td>1.15</td><td>47</td><td>6.99</td></tr><tr><td>Autobot (Girgis et al., 2021)</td><td>1.37</td><td>62</td><td>1.03</td><td>44</td><td>8.19</td></tr><tr><td>PGP (Deo &amp; Trivedi, 2020)</td><td>1.30</td><td>57</td><td>0.98</td><td>37</td><td>7.72</td></tr><tr><td>THOMAS</td><td>1.33</td><td>55</td><td>1.04</td><td>1.04</td><td>6.71</td></tr></table>
+
+![](images/95909725efdeccdf0707b864994c727b2d2d3bc667aa020ba734d223f2e82f52.jpg)  
+Figure 8: Detailed illustration of our heatmap generator model
+
+![](images/9becef61539f1118df436f5341192c61d1cb7f755f868cd727b3bf797788ee40.jpg)  
+Figure 9: Qualitative examples of heatmap output from our multi-agent model. All the heatmaps from one scene are the results from one single forward pass in our model predicting all agents at once. We use matching colors for the agent history, current location and their future predicted heatmap (best viewed in color).

parse/dev/QDdJhACYrlX/QDdJhACYrlX_content_list.json

@@ -0,0 +1,1986 @@
+[
+    {
+        "type": "text",
+        "text": "THOMAS: TRAJECTORY HEATMAP OUTPUT WITH LEARNED MULTI-AGENT SAMPLING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Thomas Gilles1,2, Stefano Sabatini1, Dzmitry Tsishkou1, Bogdan Stanciulescu2, Fabien Moutarde2 ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1IoV team, Paris Research Center, Huawei Technologies France 2Center for robotics, MINES ParisTech {stefano.sabatini, dzmitry.tsishkou}@huawei.com {thomas.gilles, bogdan.stanciulescu, fabien.moutarde}@mines-paristech.fr ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            263,
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+            279
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose THOMAS, a joint multi-agent trajectory prediction framework allowing for an efficient and consistent prediction of multi-agent multimodal trajectories. We present a unified model architecture for simultaneous agent future heatmap estimation, in which we leverage hierarchical and sparse image generation for fast and memory-efficient inference. We propose a learnable trajectory recombination model that takes as input a set of predicted trajectories for each agent and outputs its consistent reordered recombination. This recombination module is able to realign the initially independent modalities so that they do no collide and are coherent with each other. We report our results on the Interaction multi-agent prediction challenge and rank $1 ^ { s t }$ on the online test leaderboard. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            336,
+            476
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Motion forecasting is an essential step in the pipeline of an autonomous driving vehicle, transforming perception data into future prediction which are then leveraged to plan the future moves of the autonomous cars. The self-driving stacks needs to predict the future trajectories for all the neighbor agents, in a fast and coherent way. ",
+        "bbox": [
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+            492,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The interactivity between agents plays an important role for accurate trajectory prediction. Agents need to be aware of their neighbors in order to adapt their speed, yield right of way and merge in neighbor lanes. To do so, different interaction mechanisms have been developed, such as social pooling (Alahi et al., 2016; Lee et al., 2017; Deo & Trivedi, 2018), graphs (Salzmann et al., 2020; Zeng et al., 2021) or attention (Mercat et al., 2020; Messaoud et al., 2020; Luo et al., 2020; Gao et al., 2020; Liang et al., 2020), benefiting from the progress of powerful transformer architectures (Li et al., 2020; Yuan et al., 2021; Girgis et al., 2021; Ngiam et al., 2021) . These mechanisms allow agents to look at and share features with neighbors and to take them into account in their own predictions. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multi-modality is another important aspect of the possible future trajectories. A car can indeed chose to turn right or left, or decide to realise a certain maneuver in various ways. Uncertainty modeled as variance of Gaussians is insufficient to model these multiple cases, as it can only represent a continuous spread and cannot show multiple discrete possibilities. Therefore, current state-of-the-art produces not one but $K$ possible trajectories for each agent predicted, and most recent benchmarks (Caesar et al., 2020; Chang et al., 2019; Zhan et al., 2019; Ettinger et al., 2021) include multimodality in their metrics, taking only the minimum error over a predicted set of $K$ trajectories. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, up until very recently and the opening of multi-agent joint interaction challenges (Ettinger et al., 2021; Zhan et al., 2021), no motion forecasting prediction datasets were taking into account the coherence of modalities between different agents predicted at the same time. As a result, the most probable predicted modality of a given agent could crash with the most probable modality of another agent. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our THOMAS model encodes the past trajectories of all the agents present in the scene, as well as the HD-Map lanelet graph, and then predicts for each agent a sparse heatmap representing the future probability distribution at a fixed timestep in the future, using hierarchical refinement for very efficient decoding. A deterministic sampling algorithm then iteratively selects the best K trajectory endpoints according to the heatmap for each agent, in order to represent a wide and diverse spectrum of modalities. Given this wide spectrum of endpoints, a recombination module takes care of addressing consistency in the scene among agents. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are summarized as follow: ",
+        "bbox": [
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+            152,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a hierarchical heatmap decoder allowing for unconstrained heatmap generation with optimized computational costs, enabling efficient simultaneous multi-agent prediction.   \n• We design a novel recombination model able to recombine the sampled endpoints to obtain scene-consistent trajectories across the agents. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            285,
+            341,
+            301
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning-based models have quickly overtaken physics-based methods for trajectory prediction for several reasons. First, the sequential nature of trajectories is a logical application for recurrent architectures (Alahi et al., 2016; Altche & de La Fortelle, 2017; Lee et al., 2017; Mercat et al., ´ 2020; Khandelwal et al., 2020). Then, benefiting from the latest progresses in computer vision, convolutional layers can easily be applied to bird-view rasters of the map context (Lee et al., 2017; Tang & Salakhutdinov, 2019; Cui et al., 2019; Hong et al., 2019; Salzmann et al., 2020; Chai et al., 2020; Gilles et al., 2021b). A more compact representation closer to the trajectory space can encode surrounding HD-Maps (usually formalized as connected lanelets), using Graph Neural Networks (Gao et al., 2020; Liang et al., 2020; Zeng et al., 2021; Gilles et al., 2021a). Finally, some pointbased approaches (Ye et al., 2021) can be applied in a broader way to trajectory prediction, as both lanes and trajectories can be considered as ordered set of points. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Multi-modality in prediction can be obtained through a multiple prediction head in the model (Cui et al., 2019; Liang et al., 2020; Ngiam et al., 2021; Deo et al., 2021). However, some methods rather adopt a candidate-based approach where potential endpoints are obtained either from anchor trajectories through clustering (Chai et al., 2020; Phan-Minh et al., 2020) or from a model-based generator (Song et al., 2021). Other approaches use a broader set of candidates from the context graph (Zhang et al., 2020; Zhao et al., 2020; Zeng et al., 2021; Kim et al., 2021) or a dense grid around the target agent (Deo & Trivedi, 2020; Gu et al., 2021; Gilles et al., 2021b;a). Another family of approaches use variational inference to generate diverse predictions through latent variables (Lee et al., 2017; Rhinehart et al., 2018; Tang & Salakhutdinov, 2019; Casas et al., 2020) or GAN (Gupta et al., 2018; Rhinehart et al., 2018; Sadeghian et al., 2019) but the sampling of these trajectories is stochastic and does not provide any probability value for each sample. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While very little work has directly tackled multi-agent prediction and evaluation so far, multiple methods hint at the ability to predict multiple agents at the same time (Liang et al., 2020; Ivanovic et al., 2020; Zeng et al., 2021) even if they then focus on a more single-agent oriented framework. Other works (Alahi et al., 2016; Tang & Salakhutdinov, 2019; Rhinehart et al., 2019; Girgis et al., 2021) use autoregressive roll-outs to condition the future step of an agent on the previous steps of all the other agents. SceneTransformer (Ngiam et al., 2021) repeats each agent features across possible modalities, and performs self-attention operations inside each modality before using a loss computed jointly among agents to train a model and evaluate on the WOMD dataset (Ettinger et al., 2021) interaction track. ILVM (Casas et al., 2020) uses scene latent representations conditioned on all agents to generate scene-consistent samples, but its variational inference does not provide a confidence score for each modality, hence LookOut (Cui et al., 2021) proposes a scenario scoring function and a diverse sampler to improve sample efficiency. AIR2 (Wu & Wu, 2021) extends Multipath (Chai et al., 2020) and produces a cross-distribution for two agents along all possible trajectory anchors, but it scales exponentially with the number of agents, making impractical for a real-time implementation that could encounter more than 10 agents at the same time. ",
+        "bbox": [
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+            638,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The approach most related to this paper is GOHOME (Gilles et al., 2021a), which uses a similar graph encoder and then leverage lane rasters to generate a probability heatmap in a sparse manner. However, this lane-generated heatmap remains constrained to the drivable area and to fixed lanewidths. Another close approach is the one of DenseTNT (Gu et al., 2021), which also uses attention to a grid of points in order to obtain a dense prediction, but their grid also remains constrained to a neighborhood of the drivable area. Finally, none of these previous methods considers the problems of scene consistency for multi-agent prediction. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a3ba25a987b6d34843c249d488e98067ab815e58842c8473070164fbc0fe1ae.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the THOMAS multi-agent prediction pipeline "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            314,
+            282,
+            330
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to predict the future $T$ timesteps of $A$ agents using their past history made of $H$ timesteps and the HD-Map context. Similar to recent works (Zhao et al., 2020; Zeng et al., 2021; Gu et al., 2021), we will divide the problem into goal-based prediction followed by full trajectory reconstruction. Our prediction pipeline is displayed in Fig. 1. We first encode each agent trajectory and the HD-Map context graph into a common representation. We then decode a future probability heatmap for each agent in the scene, which we sample heuristically to maximize coverage. Finally, we recombine the sampled endpoints into scene-consistent modalities across agents and build the full trajectories for each agent. ",
+        "bbox": [
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+            347,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our pipeline shares the same graph encoder, sampling algorithm and full trajectory generation as GOHOME (Gilles et al., 2021a), but uses a novel efficient hierarchical heatmap process that enables to scale to simultaneous multi-agent prediction. Furthermore, we add a novel scene-consistency module that recombines the marginal outputs into a joint prediction. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 MODEL BACKBONE",
+        "text_level": 1,
+        "bbox": [
+            176,
+            540,
+            349,
+            554
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 GRAPH ENCODER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            566,
+            349,
+            580
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We use the same encoder as the GOHOME model (Gilles et al., 2021a). The agent trajectories are encoded though T rajEncoder using a 1D CNN followed by a UGRU recurrent layer (Rozenberg et al., 2021), and the HD-Map is encoded as a lanelet graph using a GNN GraphEncoder made of graph convolutions. We then run cross-attention Lanes2Agents to add context information to the agent features, followed by self-attention Agents2Agents to observe interaction between agents. The final result is an encoding $F _ { a }$ for each agent, where history, context and interactions have been summarized. This encoding $F _ { a }$ is used in the next decoder operations and is also stored to be potentially used in modality recombination described in Sec. 3.2. The resulting architecture of these encoding operations is illustrated in the first half of Fig. 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/0703cba6cd80ceb93683e3249927f2f8c77652689a2df7e6e0a92b83f65aafd0.jpg",
+        "image_caption": [
+            "Figure 2: Model architecture for multi-agent prediction with shared backbone "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            746,
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+            887
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb2812d32407c26900f1d01873383bc5ca02a8bbe3edf9f8734a6647821212af.jpg",
+        "image_caption": [
+            "Figure 3: Hierarchical iterative refinement of the grid probabilities. First, the full grid is evaluated at a very low resolution, then the highest cells are up-sampled and evaluated at a higher resolution, until final resolution is reached. We highlight in grey the restricted area considered for refinement at each step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.2 HIERARCHICAL GRID DECODER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            375,
+            446,
+            388
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our aim here is to decode each agent encoding into a heatmap representing its future probability distribution at prediction horizon $T$ . Since we create this heatmap for each agent in the scene, the decoding process has to be fast so that it can be applied to a great number of agents in parallel. We use hierarchical predictions at various levels of resolutions so that the decoder has the possibility of predicting over the full surroundings of the agent but learns to refine with more precision only in places where the agent will end up with high probability. This hierarchical process is illustrated in Fig. 3. ",
+        "bbox": [
+            173,
+            406,
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+            503
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Starting from an initial full dense grid probability at low resolution $R _ { 0 } \\times R _ { 0 }$ by pixels, we iteratively refine the resolution by a fixed factor $f$ until we reach the desired final resolution $R _ { f i n a l } \\times R _ { f i n a l }$ . At each iteration $i$ , we select only the $N _ { i }$ highest ranking grid points from the previous iteration, and upsample only these points to the $R _ { i } \\times R _ { i } = R _ { i - 1 } / f \\times R _ { i - 1 } / f$ . At each step, the grid points features are computed by a 2-layer MLP applied on the point coordinates, these features are then concatenated to the agent encoding and passed through a linear layer, and finally enriched by a 2- layer cross-attention on the graph lane features, before applying a linear layer with sigmoid to get the probability. ",
+        "bbox": [
+            174,
+            510,
+            825,
+            622
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For a given $W$ output range, this hierarchical process allows the model to only operate on $W / _ { R _ { 0 } } \\times$ $\\textstyle W / R _ { 0 } { \\dot { + } } \\sum _ { i } N _ { i } \\times { \\bar { f } } ^ { 2 }$ grid points instead of the $W \\big / R _ { f i n a l } \\ \\times \\ W \\big / R _ { f i n a l }$ available. In practice, for a final output range of 192 meters with desired $R _ { f i n a l } = 0 . 5 m$ resolution, we start with an initial resolution of $R _ { 0 } = 8 m$ and use two iterations of $\\left( N _ { 1 } , N _ { 2 } \\right) = \\left( 1 6 , 6 4 \\right)$ points each and an upscaling factor $f = 4$ . This way, we compute only 1856 grid points from the 147 456 available, with no performance loss. ",
+        "bbox": [
+            174,
+            628,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The heatmap is trained on each resolution level using as pixel-wise focal loss as in Gilles et al. (2021b), detailed in Appendix A.3, with the ground truth being a Gaussian centered at the target agent future position. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.3 FULL TRAJECTORY GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            795,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Having a heatmap as output grants us the good property of letting us decide to privilege coverage in order to give a wide spectrum of candidates to the scene recombination process. From each heatmap, we therefore decode $K$ end points using the same MissRate optimization algorithm as Gilles et al. (2021b). We then generate the full trajectories for each end point using also the same model, a fully-connected MLP. The MLP takes the car history and predicted endpoint as input and produces $T$ 2D-coordinates representing the full future trajectory. At training time, this model is trained using ground-truth endpoints. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/167fcee0a0b323d60a446df059598a89bce034f2549248e69ff9cb8b16df5573.jpg",
+        "image_caption": [
+            "Figure 4: Illustration of THOMAS methods for generation of scene-consistent agent modalities "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            95,
+            820,
+            366
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 MODALITY RECOMBINATION FOR MULTI-AGENT CONSISTENT PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            416,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The difficulty in multimodal multi-agent prediction resides in having coherent modalities between each agent. Since the modalities are considered scene-wise, each agent’s most probable modality has to match with the most probable prediction of the other agents, and so on. Moreover, these modalities must not collide with each other, as they should represent realistic scenarios. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Initially, the prediction output of the model can be defined as marginal, as all $A$ agents have been predicted and sampled independently in order to get $K$ possible endpoint each. Our goal is to output a set of scene predictions $\\bar { \\mathcal { I } } = ( \\bar { L } , A )$ from the marginal prediction $\\mathcal { M } = ( A , K )$ , where each scene modality $l$ belonging to $\\mathcal { I }$ is an association of endpoints $p _ { l } ^ { a }$ for each agent $a$ . To achieve this, our main hypothesis lays in the fact that good trajectory proposals are already present in the marginal predictions, but they need to be coherently aligned among agents to achieve a set of overall consistent scene predictions. For a given agent $a$ , the modality selected for the scene $l$ would be a combination of this agent available marginal modalities $p _ { k } ^ { a } \\in \\mathcal { M }$ . Each scene modality $l$ would select a different association between the agents. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We illustrate our scene modality generation process in Fig. 4. The model learns $L$ scene modality embeddings $S _ { l }$ of $D$ features each. $K \\times A$ agent modality vectors $A _ { k } ^ { a }$ are also derived from each agent modality endpoint. These vectors are obtained through a 2-layer MLP applied on the agent modality coordinates $p _ { k } ^ { a }$ , to which the stored agent encoding $F _ { a }$ (previously described in Sec. 3.1.1) is concatenated in order to help the model recognise modalities from the same agent. The scene modality vectors $S _ { l }$ are ’specialized’ by querying the available modality proposals $A _ { k } ^ { a }$ of each agent through cross-attention layers. Subsequently, a matching score $c _ { l } ^ { k , a }$ between each scene modality $S _ { l }$ and each agent modality $A _ { k } ^ { a }$ is computed. This matching score can be intuitively interpreted as the selection of the role (maneuver $k$ ) that the agent $a$ would play in the overall traffic scene $l$ : ",
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/6f10896928ecdcb564bf6b0ae0cc8855193544b8d66f00fc70668f183db29218.jpg",
+        "text": "$$\nc _ { l } ^ { k , a } = S _ { l } . A _ { k } ^ { a T }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Since argmax is non-differentiable, we employ a soft argmax as a weighted linear combination of the agent modalities $p _ { k } ^ { a }$ using a softmax on the $c _ { l } ^ { k , a }$ scores: ",
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/a3cb1246375e83911e1c3dd3a3d07c8fd688315b5853000c0482ebd0667a70c6.jpg",
+        "text": "$$\np _ { l } ^ { a } = \\sum _ { k } s o f t m a x ( c _ { l } ^ { k , a } ) p _ { k } ^ { a }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The recombination module is trained with the scene-level minimum displacement error, where for each modality inside a single scene all predicted agent displacement losses are averaged before taking the minimum (winner-takes-all loss) over the $L$ scene modalities. A formal definiton of this loss, also used as a metric, is given in Eq. 2 in Sec. 4.2 $\\mathrm { \\ m i n S F D E } _ { k }$ ). ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 DATASET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use the Interaction v1.2 dataset that has recently opened a new multi-agent track in the context of its Interpret challenge. It contains 47 584 training cases, 11 794 validation cases and 2 644 testing cases, with each case containing between 1 and 40 agents to predict simultaneously. ",
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+    {
+        "type": "text",
+        "text": "4.2 METRICS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "For a set of $k$ predictions $p _ { k } ^ { a }$ for each agent $a$ in a scene, we report the minimum Final Displacement Error $( \\mathrm { m i n F D E } _ { k } ,$ ) to the ground truth $\\hat { p } ^ { a }$ and the MissRate $( \\mathrm { M R } _ { k } )$ , whose definition of a miss is specified according to the evaluated dataset in Appendix A.1. In their usual marginal definition, these metrics are averaged over agents after the minimum operation, which means that the best modality of each agent is selected independently for each and then averaged: ",
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+    {
+        "type": "equation",
+        "img_path": "images/b9989602e5f081dc7ac4ea8a3040de0d3256ce416ffb3534d6e4f937f7ccba15.jpg",
+        "text": "$$\nm i n F D E _ { k } = \\frac { 1 } { A } \\sum _ { a } m i n _ { k } \\| p _ { k } ^ { a } - \\hat { p } ^ { a } \\| _ { 2 } , M R _ { k } = \\frac { 1 } { A } \\sum _ { a } m i n _ { k } \\mathbb { 1 } _ { m i s s k } ^ { a }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "For consistent scene multi-agent prediction, we report the joint scene-level metrics, where the average operation over the agents is done before the minimum operator. In this formulation, the minimum is taken over scene modalities, meaning that only the best scene (joint over agents) modality is taken into account: ",
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+    {
+        "type": "equation",
+        "img_path": "images/1b94d148c583832f28d75ad3381567a69292fee542c6dfe4df891608ddf87383.jpg",
+        "text": "$$\nm i n S F D E _ { k } = m i n _ { k } \\frac { 1 } { A } \\sum _ { a } \\| p _ { k } ^ { a } - \\hat { p } ^ { a } \\| _ { 2 } , S M R _ { k } = m i n _ { k } \\frac { 1 } { A } \\sum _ { a } \\mathbb { 1 } _ { m i s s k } ^ { a }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In other words, the marginal metrics pick their optimal solutions in a pool of $k$ to the power of $a$ predicted solutions, while the joint metrics restrict this pool to only $k$ possibilities, making it a much more complex problem. ",
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+    {
+        "type": "text",
+        "text": "We also report the scene collision rate (SCR) which is the percentage of modalities where two or more agents collide together, and the consistent collision-free joint Miss Rate (cSMR), which is $S M R$ where colliding modalities are also counted as misses. ",
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+    {
+        "type": "text",
+        "text": "4.3 COMPARISON WITH STATE-OF-THE-ART ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare our THOMAS model performance with other joint predictions methods ILVM (Casas et al., 2020) and SceneTransformer (Ngiam et al., 2021). For fair comparison, we use a GOHOME encoder for each of the method, and adapt them accordingly so that they predict only endpoints similar to our method. For each method, we focus on implementing the key idea meant to solve scene consistency and keep the remaining part of the model as close as possible to our approach for fair comparison: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ILVM (Casas et al., 2020) uses variational inference to learn a latent representation of the scene conditioned of each agent with a Scene Interaction Module, and decodes it with a similar Scene Interaction Module. The required number of modalities is obtained by sampling the latent space as many times as require. Even though the sampling is independent for each agent, the latent space is generated conditionally on all the agents. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "SceneTransformer (Ngiam et al., 2021) duplicates the agent encoding with the number of modalities required and adds a one-hot encoding specific to each modality. They then apply a shared transformer architecture on all modalities to encode intra-modality interactions between agents and generate the final trajectories. ",
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+    {
+        "type": "text",
+        "text": "More details on the state-of-the-art method and the re-implementations we used are given in Appendix A.4. The results are reported in Tab. 1. While having comparable marginal distance performance (demonstrating that our model is not inherently more powerful or leveraging more information), THOMAS significantly outperforms other methods on every joint metric. SMR is improved by about $2 5 \\%$ and SCR by almost $3 0 \\%$ , leading to a combined cSMR decreased by also more than $2 \\dot { 5 } \\%$ . We also report our numbers from the Interpret multi-agent online test set leaderboard in Tab. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/78c5707c37e62673482e90589f137cb427dad00250dc496c025197526a81676f.jpg",
+        "table_caption": [
+            "Table 1: Comparison of consistent solutions on Interpret multi-agent validation track "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Marginal metrics</td><td colspan=\"3\">Joint metrics</td></tr><tr><td>mADE mFDE</td><td>MR</td><td>mFDE</td><td>MR</td><td>SCR</td><td>cMR</td></tr><tr><td>ILVM (Casas et al., 2020)</td><td>0.30 0.62</td><td>10.8</td><td>0.84</td><td>19.8</td><td>5.7</td><td>21.3</td></tr><tr><td>SceneTranformer (Ngiam et al., 2021)</td><td>0.29 0.59</td><td>10.5</td><td>0.84</td><td>15.7</td><td>3.4</td><td>17.3</td></tr><tr><td>THOMAS</td><td>0.31 0.60</td><td>8.2</td><td>0.76</td><td>11.8</td><td>2.4</td><td>12.7</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 of Appendix A.5. For better comparison with existing single-agent trajectory prediction state-ofthe-art, we provide evaluation numbers on the single-agents benchmarks Argoverse (Chang et al., 2019) and NuScenes (Caesar et al., 2020) in Appendix A.7. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.4.1 RECOMBINATION MODULE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We establish the following baselines to assess the effects our THOMAS recombination. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Scalar output: we train a model with the GOHOME graph encoder and a multimodal scalar regression head similar to Liang et al. (2020). We optimize it with either marginal and joint loss formulation. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Heatmap output with deterministic sampling: we try various sampling methods applied on the heatmap, with either the deterministic sampling as described in Gilles et al. (2021a) or a learned sampling trained to directly regress the sampled modalities from the input heatmap. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We report the comparison between the aforementioned baselines and THOMAS in Tab. 2. Compared to these baselines, THOMAS can be seen as an hybrid sampling method that takes the result of deterministic sampling as input and learns to recombine it into a more coherent solution. With regard to the joint algorithmic sampling that only tackled collisions but has little to no effect on actual consistency, as highlighted by the big drop in collision rate from $7 . 2 \\%$ to $2 . 6 \\%$ but a similar joint SMR, THOMAS actually brings a lot of consistency in the multi-agent prediction and drops the joint SMR from $1 4 . 8 \\%$ to $\\mathrm { { \\dot { 1 } 1 . 8 \\% } }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/29f4a090994a20fbb44a7af0746accfcb9463b582e153b1faa386f86a93d3f6b.jpg",
+        "table_caption": [
+            "Table 2: Comparison of consistent solutions on Interpret multi-agent validation track "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td colspan=\"3\">Marginal metrics</td><td colspan=\"4\">Jointmetrics</td></tr><tr><td>Output</td><td>Sampling</td><td>Objective</td><td>mADE</td><td>mFDE</td><td>MR</td><td>mFDE</td><td>MR</td><td>Col</td><td>cMR</td></tr><tr><td>Scalar</td><td></td><td>Marg</td><td>0.28</td><td>0.59</td><td>10.4</td><td>1.04</td><td>23.7</td><td>6.4</td><td>24.9</td></tr><tr><td>Scalar</td><td></td><td>Joint</td><td>0.34</td><td>0.77</td><td>16.2</td><td>0.90</td><td>19.9</td><td>49</td><td>21.7</td></tr><tr><td>Heat</td><td>Learned</td><td>Marg</td><td>0.26</td><td>0.46</td><td>4.9</td><td>0.98</td><td>20.9</td><td>4.1</td><td>21.9</td></tr><tr><td>Heat</td><td>Learned</td><td>Joint</td><td>0.29</td><td>0.58</td><td>9.8</td><td>0.88</td><td>15.2</td><td>3.0</td><td>16.4</td></tr><tr><td>Heat</td><td>Algo</td><td>Marg</td><td>0.29</td><td>0.54</td><td>3.8</td><td>0.83</td><td>14.8</td><td>7.2</td><td>15.9</td></tr><tr><td>Heat</td><td>Algo</td><td>Joint</td><td>0.29</td><td>0.54</td><td>3.8</td><td>0.83</td><td>14.8</td><td>2.6</td><td>15.6</td></tr><tr><td>Heat</td><td>Combi</td><td>Joint</td><td>0.31</td><td>0.60</td><td>8.2</td><td>0.76</td><td>11.8</td><td>2.4</td><td>12.7</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Usually, scalar marginal models already suffer from learning difficulties as only one output modality, the closest one to ground, can be trained at a time. Some modalities may therefore converge faster to acceptable solutions, and benefit from a much increased number of training samples compared to the others. This problem is aggravated in the joint training case, since the modality selected is the same for all agents in a training sample. The joint scalar model therefore actually fails to learn multi-modality as illustrated by a higher marginal $\\mathrm { m i n F D E _ { 6 } }$ than any other model, and an extremely high crossCollisionRate since some modalities never train and always point to the same coordinates regardless of the queried agent. Note that, despite a similar training loss, SceneTransformer doesn’t suffer of the same pitfalls in Tab. 1 as is shares the same weights between all modalities and only differentiates them in the initialization of the features. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 HIERARCHICAL HEATMAP REFINEMENT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We assess the speed gain of our proposed hierarchical decoder compared to the lane rasters of GOHOME Gilles et al. (2021a). In Tab. 3. We report training time for 16 epochs with a batchsize of 16, and inference time for 32 and 128 simultaneous agents. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/054d25d7647464cf61e8e14f7f060605ed87a8902916f25638d32a25704ecbe6.jpg",
+        "table_caption": [
+            "Table 3: Comparison of consistent solutions on Interpret multi-agent validation track "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Training</td><td>Inference 32 agents</td><td>Inference 128 agents</td></tr><tr><td>GOHOME</td><td>12.8 hours</td><td>36 ms</td><td>90 ms</td></tr><tr><td>THOMAS</td><td>7.5 hours</td><td>20 ms</td><td>31 ms</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For additional comparison, the other existing dense prediction method DenseTNT (Gu et al., 2021) reports an inference speed of $1 0 0 \\mathrm { m s }$ per sample for their model. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.6 QUALITATIVE EXAMPLES",
+        "text_level": 1,
+        "bbox": [
+            174,
+            343,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will mostly compare the model before recombination, which we will refer by the Before model, to the model after recombination, referenced as the Af ter model. We display four qualitative examples in Fig. 5 with colliding modalities in the Before model (in dashed orange) and the solved modality (in full line orange) after recombination. For each model (Before-dashed or Af ter-full), the highlighted modality in orange is the best modality according to the $S M R _ { 6 }$ metric among the 6 available modalities. We also display in dashed grey the other 5 predicted Before modalities, and highlight that the recombination model indeed selects modalities already available in the vanilla set and reorders them so that non-colliding modalities are aligned together. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2ab60ac29fca607437fdd3b01a0d78606f2110dc5906eba1c06697aeea3f8509.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative examples of recombination model assembling collision-free modalities together compared to initial colliding modalities. For each example we display the general context with highlighted agents and area of interest, then two zooms in on the agents, one displaying the initial best modality before recombination in dashed orange and all the other available modalities in grey. The second zooms shows the best modality after recombination in full line orange. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also show more qualitative examples in Fig. 6, where we highlight the comparison in modality diversity between the Before model (in dashed lines) and the Af ter model (in full lines). While the Bef ore model tries to spread the modalities for all agents to minimize marginal miss-rate, the recombined model presents much less spread compared to the original model, maintaining a multimodal behavior only in presence of very different possible agent intentions such as different possible exits or turn choices. For most other agents, almost all modalities are located at the same position, that is, the one deemed the most likely by the model. Thus, if the truly uncertain agents have to select the second or third most likely modality, the other agents still have their own most likely modality ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f2aa394fced168b31ab355c9972d010e64ac3fc2ef104a28f4c3825000c2ecf9.jpg",
+        "image_caption": [
+            "Figure 6: Qualitative examples of recombination model selecting fewer but more pronounced and impactful agent modalities compared to initial colliding modalities. For each example we display on the left the vanilla model modalities with dashed lines, with the initial best modality in dashed orange and all the other available modalities in grey. On the right we display the selected modalities after recombination, where the model focuses on the most likely occurrence in most agents. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            611
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            631,
+            318,
+            647
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented THOMAS, a recombination module that can be added after any trajectory prediction module outputing multi-modal predictions. By design, THOMAS allows to generate sceneconsistent modalities across all agents by making the scene modalities select coherent agent modalities and restricting the modality budget on the agents that truly need it. We show significant performance increase when adding the THOMAS module compared to the vanilla model and achieve state-of-the-art results compared to already existing methods tackling scene-consistent predictions. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY STATEMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            393,
+            118
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We use the publicly available Interaction 1.2 dataset (Zhan et al., 2019) available at http:// challenge.interaction-dataset.com/dataset/download. We detail dataset preprocessing in Appendix A.2, training process in Appendix A.3 and give a precise illustration of our model architecture with each layer size in A.8. ",
+        "bbox": [
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+            128,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            205,
+            285,
+            220
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 METRIC DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            136,
+            338,
+            151
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We adopt for each dataset evaluation the miss definition as specified by the dataset benchmark. Argoverse (Chang et al., 2019) and NuScenes (Caesar et al., 2020) define a miss as the prediction being further than 2 meters to the ground truth, while Waymo (Ettinger et al., 2021) and Interaction Zhan et al. (2019)count a miss when the prediction is closer than a lateral (1m) and an longitudinal threshold with regard to speed: ",
+        "bbox": [
+            173,
+            164,
+            826,
+            234
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3ef5b43ccb2e661888ac50707a19658f8b2699c2c0f09ad2fd9022345745223.jpg",
+        "text": "$$\n\\mathrm { T h r e s h o l d } _ { \\mathrm { \\scriptsize ~ l o n } } = \\left\\{ \\begin{array} { c c } { 1 } & { v < 1 . 4 m / s } \\\\ { 1 + \\frac { v - 1 . 4 } { 1 1 - 1 . 4 } } & { 1 . 4 m / s \\le v \\le 1 1 m / s } \\\\ { 2 } & { v \\ge 1 1 m / s } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            256,
+            684,
+            306
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 DATASET PROCESSING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            325,
+            375,
+            339
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use the training/validation split provided in Interaction 1.2. We select for each scene a reference agent on which the scene will be centered and oriented according to its heading at prediction time. At training time, this reference agent is chosen randomly across all agents. At validation and testing time, we compute the barycenter of all agents and select the closest agent as the scene reference point. We use for training all agents that have ground-truth available at prediction horizon $\\mathrm { T } { = } 3 \\mathrm { s }$ . ",
+        "bbox": [
+            173,
+            352,
+            825,
+            424
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use the provided dataset HD-Maps to create the context graphs. As in Liang et al. (2020), we use four relations:{predecessor, successor, $l e f t , r i g h t \\}$ obtained from the lane graph connectivity. We upsample long lanelets to have a maximum of 10 points in each lanelet. ",
+        "bbox": [
+            174,
+            429,
+            821,
+            472
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For each agent, we use their trajectory history made of position, yaw and speed in the past second sampled at $1 0 \\mathrm { H z }$ . We also provide the model a mask indicating if the agent was present in a given time-frame, and pad with zeros when the agent is not tracked at a specific timestep. ",
+        "bbox": [
+            174,
+            478,
+            823,
+            521
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            542,
+            352,
+            556
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We train all models with Adam optimizer and batchsize 32. We initialize the learning rate at $1 e ^ { - 3 }$ and divide it by 2 at epochs 3, 6, 9 and 13, before stopping the training at epoch 16. We use ReLU activation after every linear layer unless specified otherwise, and LayerNormalization after every attention and graph convolution layer. ",
+        "bbox": [
+            174,
+            570,
+            825,
+            627
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "During training, since a scene can contain up to 40 agents, which is too much for gradient computation in batches of 32, we restrict the number of predicted agents to 8 by randomly sampling them across available agents. We do not use other data augmentation. ",
+        "bbox": [
+            174,
+            632,
+            825,
+            676
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The final heatmap $Y$ predicted cover a range of 192 meters at resolution $0 . 5 \\mathrm { m }$ , hence a (384, 384) image. For heatmap generation, we use the same pixel-wise focal loss over the pixels $p$ as in Gilles et al. (2021b): ",
+        "bbox": [
+            174,
+            681,
+            825,
+            724
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af8722ba66d345311be1ae680a9516837a89a87fe3cd2b4f0a2fd97d4e38d10f.jpg",
+        "text": "$$\nL = - \\frac { 1 } { P } \\sum _ { p } ( Y _ { p } - \\hat { Y } _ { p } ) ^ { 2 } f ( Y _ { p } , \\hat { Y } _ { p } ) \\mathrm { w i t h } f ( Y _ { p } , \\hat { Y } _ { p } ) = \\left\\{ \\begin{array} { l l } { \\log ( \\hat { Y } _ { p } ) } & { \\mathrm { i f ~ } Y _ { p } = 1 } \\\\ { ( 1 - Y _ { p } ) ^ { 4 } \\log ( 1 - \\hat { Y } _ { p } ) } & { \\mathrm { e l s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            734,
+            787,
+            779
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { Y }$ is defined as a Gaussian centered on the agent future position at prediction horizon $T$ with a standard deviation of 4 pixels, equivalent to $2 \\mathrm { m }$ . ",
+        "bbox": [
+            171,
+            791,
+            825,
+            820
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 BASELINES IMPLEMENTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
+            436,
+            856
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4.1 IMPLICIT LATENT VARIABLE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            868,
+            488,
+            885
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use a GOHOME encoder for the prior, posterior and decoder Scene Interaction Modules. We weight the KL term with $\\beta = 1$ which worked best according to our experiments. ",
+        "bbox": [
+            174,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4.2 SCENE TRANSFORMER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            390,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The initial paper applies a transformer architecture on a $[ F , A , T , D ]$ tensor where $F$ is the potential modality dimension, $A$ the agent dimension and $T$ the time dimension, with $D$ the feature embedding, with factorized self-attention to the agent and time dimensions separately, so that agents can look at each-other inside a specific scene modality. The resulting output is optimized using a jointly formalized loss. For our implementation, we get rid of the $T$ dimension as we focus on endpoint prediction and coherence between the $A$ agents. The initial encoded $[ A , D ]$ tensor is obtained with a GOHOME encoder, multiplied across the $F$ futures and concatenated with a modality-specific one-hot encoding as in Ngiam et al. (2021) to obtain the $[ F , A , D ]$ tensor. We then apply two layers of agent self-attention similar to the original paper, before decoding the endpoints through a MLP. ",
+        "bbox": [
+            174,
+            127,
+            825,
+            253
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4.3 COLLISION-FREE ENDPOINT SAMPLING BASELINE",
+        "text_level": 1,
+        "bbox": [
+            174,
+            270,
+            576,
+            282
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We designed a deterministic sampling algorithm based on the heatmaps generated in previous section in order to sample endpoints for each agent in a collision aware manner. We use the same sampling algorithm as Gilles et al. (2021a) based on MR optimization, but add a sequential iteration over the agents for each modalities. ",
+        "bbox": [
+            174,
+            294,
+            825,
+            349
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For a single modality $k$ , we predict the possible endpoint of a first agent $a$ by taking the maximum accumulated predicted probability under an area of radius $r$ . We then not only set to zero the heatmap values of this agent heatmap $\\mathcal { T } _ { k ^ { \\prime } } ^ { a }$ around the sampled location so not to sample it in the next modalities $k ^ { \\prime }$ , but we also set to zero the same area on the heatmaps $\\mathcal { T } _ { k } ^ { a ^ { \\prime } }$ of the other agents $a ^ { \\prime }$ on the same modality $k$ , so that these other agents cannot be sampled at the same position for this modality. This way, we try to enforce collision-free endpoints, and expect that considering collisions brings logic to improve the overall consistency of the predictions. However, as will be highlighted in Sec. 4.4, this methods significantly improves the collision rate without the need for any additional learned model but it does barely improve the multi agent consistency. ",
+        "bbox": [
+            174,
+            356,
+            825,
+            484
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 RESULTS ON INTERPRET CHALLENGE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            502,
+            478,
+            516
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We also report our numbers from the Interpret multi-agent track challenge online leaderboard in Table 5. It is to be noted that the DenseTNT solution explicitely checks for collisions in the search for its proposed predictions, which we don’t, hence their $0 \\%$ collision rate (SCR) and its direct impact on consistent collision-free joint Miss Rate (cSMR). ",
+        "bbox": [
+            176,
+            529,
+            825,
+            584
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/e4ef50aaf114a39c69ff6f5ee7abc0787937d0c7df0978812d6ad66c180576a1.jpg",
+        "table_caption": [
+            "Table 4: Results on Interpret multi-agent regular scene leaderboard (test set) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>minSADE</td><td>minSFDE</td><td>SMR</td><td>SCR</td><td>cSMR</td></tr><tr><td>MoliNet</td><td>0.73</td><td>2.55</td><td>44.4</td><td>7.5</td><td>47.4</td></tr><tr><td>ReCoG2 (Mo et al., 2020)</td><td>0.47</td><td>1.16</td><td>23.8</td><td>6.9</td><td>26.8</td></tr><tr><td>DenseTNT (Gu et al., 2021)</td><td>0.42</td><td>1.13</td><td>22.4</td><td>0.0</td><td>22.4</td></tr><tr><td>THOMAS</td><td>0.42</td><td>0.97</td><td>17.9</td><td>12.8</td><td>25.2</td></tr></table>",
+        "bbox": [
+            227,
+            630,
+            772,
+            704
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/e85b0ac8caee07f38ee0a088bcfdbb93f03571dff59a21651d8a8b1ff2345225.jpg",
+        "table_caption": [
+            "Table 5: Results on Interpret multi-agent conditional scene leaderboard (test set) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>minSADE</td><td>minSFDE</td><td>SMR</td><td>SCR</td><td>cSMR</td></tr><tr><td>ReCoG2 (Mo et al., 2020)</td><td>0.33</td><td>0.87</td><td>14.98</td><td>0.09</td><td>15.12</td></tr><tr><td>DenseTNT (Gu et al., 2021)</td><td>0.28</td><td>0.89</td><td>15.02</td><td>0.0</td><td>15.02</td></tr><tr><td>THOMAS</td><td>0.31</td><td>0.72</td><td>10.67</td><td>0.84</td><td>11.63</td></tr></table>",
+        "bbox": [
+            225,
+            763,
+            772,
+            824
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6 SPEED / PERFORMANCE TRADE-OFF WITH HIERARCHICAL REFINEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            856,
+            714,
+            869
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We also display the trade-off between inference speed and coverage from hierarchical refinement in Fig. 7, evaluated on the Interpret multi-agent dataset with marginal MissRate6. The curve is obtained setting the number $N$ of upsampled points at the last refinement iteration from 2 to 128. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "From $N = 1 6$ and lower, coverage performance starts to diminish while little speed gains are made. We still kept a relatively high $_ { \\mathrm { N = 6 4 } }$ in our model as we wanted to insure a wide coverage, and the time loss between $4 1 ~ \\mathrm { m s }$ and $4 6 \\mathrm { m s }$ remains acceptable. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/4f12d90162723a8705e4a9c69025eb31c341b546352eed37d0aaf56dddc7c749.jpg",
+        "image_caption": [
+            "Figure 7: Curve of MissRate $\\mathrm { { \\dot { 6 } } }$ with regard to inference time with varying number of points upsampled at the last hierarchical refinement iteration. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            351,
+            180,
+            633,
+            340
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.7 ADDITIONAL RESULTS ON SINGLE-AGENT DATASETS",
+        "bbox": [
+            173,
+            416,
+            583,
+            430
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/37dc479a7781e802dbc17207592d70ffd4ec4225875d7a9b2f6714da5804f816.jpg",
+        "table_caption": [
+            "Table 6: Argoverse Leaderboard (Chang et al.) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">K=1</td><td colspan=\"3\">K=6</td></tr><tr><td>minFDE</td><td>MR</td><td>minADE</td><td>minFDE</td><td>MR</td></tr><tr><td>LaneGCN (Liang et al., 2020)</td><td>3.78</td><td>59.1</td><td>0.87</td><td>1.36</td><td>16.3</td></tr><tr><td>Autobot (Girgis et al., 2021)</td><td></td><td></td><td>0.89</td><td>1.41</td><td>16</td></tr><tr><td>TPCN (Ye et al., 2021)</td><td>3.64</td><td>58.6</td><td>0.85</td><td>1.35</td><td>15.9</td></tr><tr><td>Jean (Mercat et al., 2020)</td><td>4.24</td><td>68.6</td><td>1.00</td><td>1.42</td><td>13.1</td></tr><tr><td>SceneTrans (Ngiam et al.,2021)</td><td>4.06</td><td>59.2</td><td>0.80</td><td>1.23</td><td>12.6</td></tr><tr><td>LaneRCNN (Zeng et al., 2021)</td><td>3.69</td><td>56.9</td><td>0.90</td><td>1.45</td><td>12.3</td></tr><tr><td>PRIME (Song et al., 2021)</td><td>3.82</td><td>58.7</td><td>1.22</td><td>1.56</td><td>11.5</td></tr><tr><td>DenseTNT (Gu et al., 2021)</td><td>3.70</td><td>59.9</td><td>0.94</td><td>1.49</td><td>10.5</td></tr><tr><td>GOHOME (Gilles et al., 2021a)</td><td>3.65</td><td>57.2</td><td>0.94</td><td>1.45</td><td>10.5</td></tr><tr><td>HOME (Gilles et al., 2021b)</td><td>3.65</td><td>57.1</td><td>0.93</td><td>1.44</td><td>9.8</td></tr><tr><td>THOMAS</td><td>3.59</td><td>56.1</td><td>0.94</td><td>1.44</td><td>10.4</td></tr></table>",
+        "bbox": [
+            218,
+            478,
+            781,
+            662
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/0e16590bc5e21889da989e0f21b74c83807d9e8ed62d7139efd320ed44558b0e.jpg",
+        "table_caption": [
+            "Table 7: NuScenes Leaderboard (Caesar et al.) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">K=5</td><td colspan=\"2\">K=10</td><td rowspan=\"2\">k=1 minFDE</td></tr><tr><td>minADE</td><td>MR</td><td>minADE</td><td>MR</td></tr><tr><td>CoverNet (Phan-Minh etal., 2020) Trajectron++ (Salzmann et al.,2020)</td><td>1.96</td><td>67</td><td>1.48</td><td>57</td><td></td></tr><tr><td></td><td>1.88</td><td>70</td><td>1.51</td><td></td><td>9.52</td></tr><tr><td>ALAN (Narayanan et al., 2021)</td><td>1.87</td><td>60</td><td>1.22</td><td>49</td><td>9.98</td></tr><tr><td>SG-Net (Wang et al., 2021)</td><td>1.86</td><td>67</td><td>1.40</td><td>52</td><td>9.25</td></tr><tr><td>WIMP (Khandelwal et al., 2020)</td><td>1.84</td><td>55</td><td>1.11</td><td>43</td><td>8.49</td></tr><tr><td>MHA-JAM (Messaoud et al., 2020)</td><td>1.81</td><td>59</td><td>1.24</td><td>46</td><td>8.57</td></tr><tr><td>CXX (Luo et al., 2020)</td><td>1.63</td><td>69</td><td>1.29</td><td>60</td><td>8.86</td></tr><tr><td>LaPred (Kim et al.,2021)</td><td>1.53</td><td></td><td>1.12</td><td></td><td>8.12</td></tr><tr><td>P2T (Deo &amp; Trivedi,2020)</td><td>1.45</td><td>64</td><td>1.16</td><td>46</td><td>10.50</td></tr><tr><td>GOHOME (Gilles et al., 2021a)</td><td>1.42</td><td>57</td><td>1.15</td><td>47</td><td>6.99</td></tr><tr><td>Autobot (Girgis et al., 2021)</td><td>1.37</td><td>62</td><td>1.03</td><td>44</td><td>8.19</td></tr><tr><td>PGP (Deo &amp; Trivedi, 2020)</td><td>1.30</td><td>57</td><td>0.98</td><td>37</td><td>7.72</td></tr><tr><td>THOMAS</td><td>1.33</td><td>55</td><td>1.04</td><td>1.04</td><td>6.71</td></tr></table>",
+        "bbox": [
+            202,
+            417,
+            795,
+            631
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/95909725efdeccdf0707b864994c727b2d2d3bc667aa020ba734d223f2e82f52.jpg",
+        "image_caption": [
+            "Figure 8: Detailed illustration of our heatmap generator model "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            137,
+            789,
+            792
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/9becef61539f1118df436f5341192c61d1cb7f755f868cd727b3bf797788ee40.jpg",
+        "image_caption": [
+            "Figure 9: Qualitative examples of heatmap output from our multi-agent model. All the heatmaps from one scene are the results from one single forward pass in our model predicting all agents at once. We use matching colors for the agent history, current location and their future predicted heatmap (best viewed in color). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            131,
+            825,
+            821
+        ],
+        "page_idx": 17
+    }
+]

parse/dev/QkLpGxUboF/QkLpGxUboF.md

@@ -0,0 +1,260 @@
+# ProPILE: Probing Privacy Leakage in Large Language Models
+
+# Siwon Kim1,∗
+
+Sangdoo Yun3 Hwaran Lee3 Martin Gubri4,5 Sungroh Yoon1,2,† Seong Joon Oh5,6,†
+
+1 Department of Electrical and Computer Engineering, Seoul National University   
+2 Interdisciplinary Program in Artificial Intelligence, Seoul National University 3 NAVER AI Lab 4 University of Luxembourg 5 Parameter Lab 6 Tübingen AI Center, University of Tübingen
+
+# Abstract
+
+The rapid advancement and widespread use of large language models (LLMs) have raised significant concerns regarding the potential leakage of personally identifiable information (PII). These models are often trained on vast quantities of web-collected data, which may inadvertently include sensitive personal data. This paper presents ProPILE, a novel probing tool designed to empower data subjects, or the owners of the PII, with awareness of potential PII leakage in LLM-based services. ProPILE lets data subjects formulate prompts based on their own PII to evaluate the level of privacy intrusion in LLMs. We demonstrate its application on the OPT-1.3B model trained on the publicly available Pile dataset. We show how hypothetical data subjects may assess the likelihood of their PII being included in the Pile dataset being revealed. ProPILE can also be leveraged by LLM service providers to effectively evaluate their own levels of PII leakage with more powerful prompts specifically tuned for their in-house models. This tool represents a pioneering step towards empowering the data subjects for their awareness and control over their own data on the web. The demo can be found here: https://parameterlab.de/research/propile
+
+# 1 Introduction
+
+Recent years have seen staggering advances in large language models (LLMs) [25, 3, 31, 6, 28, 32, 22]. The remarkable improvement is commonly attributed to the massive scale of training data crawled indiscriminately from the web. The web-collected data is likely to contain sensitive personal information crawled from personal web pages, social media, personal profiles on online forums, and online databases such as collections of in-house emails [13]. They include various types of personally identifiable information (PII) for the data subjects [8], including their names, phone numbers, addresses, education, career, family members, and religion, to name a few.
+
+This poses an unprecedented level of privacy concern not matched by prior web-based products like social media. In social media, the affected data subjects were precisely the users who have consciously shared their private data with the awareness of associated risks. In contrast, products based on LLMs trained on uncontrolled, web-scaled data have quickly expanded the scope of the affected data subjects far beyond the actual users of the LLM products. Virtually anyone who has left some form of PII on the world-wide-web is now relevant to the question of PII leakage.
+
+![](images/456a5e39bb231f292ba0a24eb6235fd762f98588e191ffacb0088f8fdc945d24.jpg)  
+Figure 1: ProPILE. Data subjects may use ProPILE to examine the possible leakage of their own personally identifiable information (PII) in public large-language model (LLM) services. ProPILE helps data subjects formulate an LLM prompt based on $M - 1$ of their PII items to task the LLM to output the $M ^ { \mathrm { { t h } } }$ PII not given in the prompt. If the generated responses include similar strings to the true PII, this can be considered as a privacy threat to the data subject.
+
+Currently, there is no assurance that adequate safeguards are in place to prevent the inadvertent disclosure of PII. Understanding of the probability and mechanisms through which PII could leak under specific prompt conditions remains insufficient. This knowledge gap highlights the ongoing need for comprehensive research and implementation of robust leakage measurement tools.
+
+In this regard, we introduce ProPILE, a tool to let the data subjects examine the possible inclusion and subsequent leakage of their own PII in LLM products in deployment. The data subject has only black-box access to LLM products; they can only send prompts and receive the generated sentences or likelihoods. Nevertheless, since the data subject possesses complete access to their own PII, ProPILE leverages this to generate effective prompts aimed at assessing the potential PII leakage in LLMs. See Figure 1 for an overview of the ProPILE framework. Importantly, this tool holds considerable value not only for data subjects but also for LLM service providers. ProPILE provides the service providers with a tool to effectively assess their own levels of PII leakage with more powerful prompts specifically tuned for their in-house models. Through this, the service providers can proactively address potential privacy vulnerabilities and enhance the overall robustness of their LLMs.
+
+Our experiments on the Open Pre-trained Transformers (OPT) [35] trained on the Pile dataset [10] confirm the following. 1) A significant portion of the diverse types of PII included in the training data can be disclosed through strategically crafted prompts. 2) By refining the prompt, having access to model parameters, and utilizing a few hundred training data points for the LLM, the degree of PII leakage can be significantly magnified. We envision our proposition and the insights gathered through ProPILE as the initial step towards enhancing the awareness of data subjects and LLM service providers regarding potential PII leakage.
+
+# 2 Related Works
+
+# 2.1 Privacy Leakage in Learned Models: Pre-LLM Era
+
+The successful development of machine learning (ML) technologies and related web products led to privacy concerns. ML models may unintentionally include PII of certain data subjects in ML training data. As those models become publicly available, concerns have been raised that such PIIs may be accessed by millions of users using the ML service. Researchers have assessed the possibility of reconstructing PII-relevant training data from a learned model [9, 11, 34, 37, 36]. The task is referred to as training data reconstruction or model inversion. Previous work has shown that it is often possible to reconstruct training data well enough to reveal sensitive attributes (e.g., face images from a face classifier), even with just a black-box access [9, 11, 34]. Researchers have also designed a more evaluation-friendly surrogate task, membership inference attack [29], that tests whether each of the given samples has been included in the training data of the learned model. Subsequent work has shown that this is indeed possible for a wide range of models, including text-generation models [12, 30] and image-generation models [5]. For a comprehensive review of the field up to 2020, refer to the overview by Rigaki & Garcia [27].
+
+# 2.2 Privacy Leakage in Learned Models: Post-LLM Era
+
+The appearance of billion-scale large-language models (LLMs) and the highly successful products including ChatGPT [22], leads to an even higher level of privacy concerns. Their training data includes not only the data consciously or voluntarily provided by the data subjects [8], but also a massive crawl of the entire web such as personal web pages, social media accounts, personal profiles on online forums, and databases of in-house emails [13]. Building a model-based service on such a web-crawled dataset and making it available to millions of users worldwide poses a novel, serious threat to the data rights of the data subjects. Motivated by this, a few early studies have been made to measure privacy leakage in LLMs [13, 19, 4, 14]. However, although [13] initiated the discussion on PII leakage in LLMs, it was limited to the preliminary analysis of only email addresses. [19] conducted a separate study that specifically targeted LLMs fine-tuned with an auxiliary dataset enriched with PII. Furthermore, their study specifically concentrated on scenarios where the prefix or suffix associated with the PII was known. In contrast, ProPILE aims to provide a more comprehensive tool for probing LLMs already in deployment without LLM fine-tuning or prefix retrieval.
+
+# 2.3 Prompt Tuning
+
+Prompt engineering [26, 18] improves downstream task performance of LLMs by well-designing prompts without further LLM fine-tuning. In soft prompt tuning [15, 16], a few learnable soft token embeddings concatenated to the original prompts are trained while LLM is frozen, so that more optimal prompts for the downstream task can be obtained. The white-box approach of ProPILE leverages soft prompt tuning to further refine the black-box approach’s hand-crafted prompts.
+
+# 3 ProPILE: Probing PII Leakage of Large Language Models
+
+In this section, we propose ProPILE, a probing tool to profile the PII leakage of LLMs. We first introduce the two attributes of PII, namely linkability and structurality, which are important for the subsequent analysis. We also describe our threat model and eventually introduce probing methods of ProPILE. Finally, we discuss the quantification of the degrees of privacy leakage.
+
+# 3.1 Formulation of PII
+
+# 3.1.1 Linkability
+
+From a privacy standpoint, the random disclosure of PII may not necessarily pose a substantial risk. For instance, when a phone number is generated in an unrelated context, there are no identifiable markers linking the number to its owner. However, if targeted PII is presented within a context directly tied to the owner, it could pose a severe privacy risk as it unequivocally associates the number with its owner. In light of this, the linkability of PII items has been considered critical for the study of privacy leakage [24]. We formalize the linkability of PII in the definition below.
+
+Definition 1 (Linkable PII leakage). Let $\mathcal { A } : = \{ a _ { 1 } , . . . , a _ { M } \}$ be $M$ PII items relevant to a data subject $S$ . Each element $a _ { m }$ denotes a PII item of a specific PII type. Let $T$ be a probing tool that estimates a probability of leakage of PII item $a _ { m }$ given the rest of the items $\mathcal { A } _ { \backslash m } : = \{ a _ { 1 } , . . . , a _ { m - 1 } , a _ { m + 1 } , . . . , a _ { M } \}$ . We say that $T$ exposes the linkability of PII items for the data subject $S$ when the likelihood of reconstructing the true PII, $\operatorname* { P r } ( a _ { m } | \mathcal { A } _ { \setminus m } , T )$ , is greater than the unconditional, context-free likelihood $\Pr ( a _ { m } )$ .
+
+# 3.1.2 Structurality
+
+We consider PII in LLM training data in a string format. Certain types of PII tend to be more structured than others. The structurality of PII has significant implications for practical countermeasures against privacy leakage. We discuss them below.
+
+Structured PII refers to the PII type that often appears in a structured pattern. For example, phone numbers and social security numbers are written down in a recognizable pattern like (xxx) xxx-xxxx that is often consistent within each country. Email addresses also follow a distinct pattern id@domain and are considered structured. Though less intuitive, we also consider physical addresses structured: [building, street, state, country, postal code].
+
+We expect structured PII to be easily detectable with simple regular expressions [1]. This implies apparently simple remedies against privacy leakage. Structured PII may easily be purged out from training data through regular expression detection. Moreover, leakage of such PII may be controlled through detection and redaction in the LLM outputs. However, in practice, the complete removal of structured PII in training data and LLM-generated content is difficult. Regulating the generation of useful public information, such as the phone number and address of the emergency clinic, will significantly limit the utility of LLM services. It is often difficult to distinguish PII and public information that fall within the same pattern category. As such, it is not impossible to find structured PII in the actual LLM training data, such as the Pile dataset (section 4.1) [10], and the leakage of PII in actual LLM outputs [17]. We thus study the leakage of structured PII in this work.
+
+Unstructured PII refers to the PII type that does not follow an easy regular expression pattern. For example, information about a data subject’s family members is sensitive PII that does not follow a designated pattern in text. One could write "{name1}’s father is {name2}", but this is not the only way to convey this information. Other examples include the affiliation, employer, and educational background of data subjects. Unstructured PII indeed poses greater threats of unrecognized privacy leakage than structured PII. In this work, we consider family relationships and affiliation as representative cases of unstructured PII (section 4.3).
+
+# 3.2 Threat Model
+
+Our goal is to enable data subjects to probe how likely LLMs are to leak their PII. We organize the relevant actors surrounding our PII probing tool and the resources they have access to.
+
+Actors in the threat model. First of all, there are data subjects whose PII is included in the training data for LLMs. They have their ownership, or the data rights [8], over the PII. LLM providers train LLMs using web-crawled data that may potentially include PII from corresponding data subjects. Finally, LLM users have access to the LLM-based services to send prompts and receive text responses.
+
+Available resources. LLM-based services, especially proprietary ones, are often available as APIs, allowing only black-box access to LLM users. They formulate the inputs within the boundary of rate limit policy and inappropriate-content regulations and receive outputs from the models. On the other hand, LLM providers have white-box access to the LLM training data, LLM training algorithm, and hyperparameters, as well as LLM model parameters and gradients. Data subjects may easily acquire black-box access to the LLMs by registering themselves as LLM users, but it is unlikely that they will get white-box access. Importantly, data subjects have rightful access to their own PII. We show how they can utilize their own PII to effectively probe the privacy leakage in LLMs.
+
+# 3.3 Probing Methods
+
+We present two probing methods, one designed for data subjects with only black-box access to LLMs and the other for model providers with white-box access.
+
+# 3.3.1 Black-box Probing
+
+Actor’s goal. In a black-box probing scenario, an actor with black-box access aims to probe whether there is a possibility that the LLM leaks one of their PII. Particularly, an actor has a list of their own PII $\mathcal { A }$ with $M$ PII items and aims to check if the target PII $a _ { m } \in { \mathcal { A } }$ leaks from an LLM.
+
+Probing strategy. For a target PII $a _ { m }$ , a set of query prompts $\tau$ is created by associating the remaining PII $\mathcal { A } _ { \backslash m }$ . Particularly, $\mathcal { A } _ { \backslash m }$ is prompted with $K$ different templates $t _ { k }$ as $\mathcal { T } = \{ t _ { 1 } ( \mathcal { A } _ { \backslash m } ) , . . . , t _ { K } ( \dot { \mathcal { A } _ { \backslash m } } ) \}$ . Then, the user sends the set of probing prompts $\tau$ to the target LLM for as much as $N$ times. Assuming the target LLM performs sampling, the user will receive $N \times K$ responses along with the likelihood scores $\mathbf { \bar { \boldsymbol { \mathcal { L } } } } \in \mathbb { R } ^ { K \times L \times V }$ , where $L$ and $V$ denote the length of the response and the vocabulary size of the target LLM, respectively. Please note that the likelihood is identical for the same query regardless of repeated queries. Example prompts are shown in Figure 2.
+
+# 3.3.2 White-box Probing
+
+Actor’s goal. In the white-box probing scenario, the goal of the actor is to find a tighter worst-case leakage (lower bound on the likelihood) of specific types of PII $( a _ { m } )$ . The actor is given additional resources beyond the black-box case. They have access to the training dataset, model parameters, and model gradients.
+
+Probing strategy. We use soft prompt tuning to achieve the goal, of finding a prompt that induces more leakage than the handcrafted prompts in the black-box case. First, we denote a set of PII lists included in the training dataset of target LLM as $\mathcal { D } = \{ \mathcal { A } ^ { i } \} _ { i = 1 } ^ { N }$ . White-box approach assumes that an actor has access to a subset of training data $\tilde { \mathcal { D } } \subset \mathcal { D }$ , where $| \tilde { D } | = n$ for $n \ll N$ . Let us denote a query prompt as $X$ that is created by one of the templates used in the black-box probing $X = t _ { n } ( \hat { \mathcal { A } } _ { \backslash m } ^ { i } )$ . Then $X$ is tokenized and embedded into $X _ { e } \in \mathbb { R } ^ { L _ { X } \times d }$ , where $L _ { X }$ denotes the length of the query sequence and $d$ denotes the embedding dimension of the target LLM. The soft prompt $\theta _ { s } \in \mathbb { R } ^ { \hat { L } _ { s } \times \check { d } }$ , technically learnable parameters, are appended ahead of $X _ { e }$ making $[ \theta _ { s } ; X _ { e } ] \in \mathbb { R } ^ { ( L _ { s } + L _ { X } ) \times d }$ , where $L _ { s }$ denotes the number of soft prompt tokens to be prepended. The soft embedding is trained to maximize the expected reconstruction likelihood of the target PII over $\tilde { \mathcal { D } }$ . Therefore, the training is conducted to minimize negative log-likelihood defined as below:
+
+$$
+\begin{array} { r } { \theta _ { s } ^ { * } = \underset { \theta _ { s } } { \mathrm { a r g m i n } } \underset { A \sim \tilde { \mathcal { D } } } { \mathbb { E } } \left[ - \log ( \operatorname* { P r } ( a _ { m } | [ \theta _ { s } ; X _ { e } ] ) ) \right] . } \end{array}
+$$
+
+After the training, the learned soft embedding $\theta _ { s } ^ { * }$ is prepended to prompts $t _ { n } ( \mathcal { A } _ { \backslash m } )$ made of unseen data subject’s PII to measure the leakage of $a _ { m }$ of the subject.
+
+# 3.4 Quantifying PII leakage
+
+For both black-box and white-box probing, the risk of PII leakage is quantified using two types of metrics depending on the output that the users receive.
+
+Quantification based on string match. Users receive generated text from the LLMs. Naturally, the string match between the generated text and the target PII serves as a primary metric to quantify the leakage. Exact match represents a verbatim reconstruction of a PII; the generated string is identical to the ground truth PII.
+
+Quantification based on likelihood. We consider the scenario that black-box LLMs can provide likelihood scores for candidate text outputs. The availability of likelihood scores enables a more precise assessment of the level of privacy leakage. It also lets one simulate the chance of LLMs revealing the PII when it is deployed at a massive scale. Reconstruction likelihood implies the probability of the target PII being reconstructed given the query prompt. Therefore, the likelihood defined as follows is used to quantify the leakage:
+
+$$
+\operatorname* { P r } ( a _ { m } | \mathcal { A } _ { \backslash m } ) = \prod _ { r = 1 } ^ { L _ { r } } p ( a _ { m , r } | x _ { 1 } , x _ { 2 } , . . . , x _ { L _ { q } + r - 1 } ) .
+$$
+
+In this equation, $a _ { m }$ represents the target PII and the product is taken over the range from $r = 1$ to $L _ { r }$ , where $L _ { r }$ represents the length of the target PII $( a _ { m } )$ . $x _ { 1 } , x _ { 2 } , . . . , x _ { L _ { q } + r - 1 }$ correspond to the tokens or words comprising the query prompt of length $L _ { q }$ followed by the response.
+
+Even a low level of likelihood has critical implications for privacy leakage, particularly for systems deployed at scale. For example, ChatGPT has been deployed to more than 100 million users worldwide [23]. The likelihood of $0 . 0 1 \%$ of reconstructing the PII implies 100 cases of PII reconstruction if only $0 . 0 1 \%$ of the 100 million users attempt the reconstruction 10 times each.1 The inverse of the likelihood indicates the expected number of sampling or queries needed to generate the exact PII.
+
+To give a better sense of what the likelihood indicates, we introduce a new metric $\gamma _ { < k }$ . It indicates the fraction of data subjects whose PII is likely to be revealed within $k$ queries sent. For example, $\gamma _ { < 1 0 0 , m } = 0 . 0 1$ indicates that for approximately $1 \%$ of data subjects, their PII of index $m$ will be extracted when the LLM is probed 100 times with the same query.
+
+![](images/bfcbf72e687bb63aec701d41c28c2048c9d605cc0bb282758687812eb3c4c062.jpg)  
+Figure 2: Probing prompts. (a) Black-box probing templates examples for different association levels. Blue text denotes the associated PII to be included in the prompt, and Red text indicates the target PII and the type of it. (b) Examples from the evaluation dataset. Text in Pile dataset is converted to dictionary.
+
+$$
+\gamma _ { < k , m } = { \frac { \# \left\{ { \mathrm { P I I } } { \mathrm { ~ } } { \mathrm { ~ } } { \mathrm { ~ } } A { \mathrm { ~ f o r ~ d a t a ~ s u b j e c t s ~ i n ~ } } { \mathcal { D } } \mid { \mathrm { P r } } ( a _ { m } | A _ { \left. m \right. } > { \frac { 1 } { k } } \right\} } { \# { \mathrm { ~ o f ~ d a t a ~ s u b j e c t s ~ i n ~ } } { \mathcal { D } } } }
+$$
+
+# 4 Probing Existing LLMs
+
+# 4.1 Experimental Setup
+
+Target LLM to be probed. In our experiments, the selection of the target LLM was guided by two specific requirements. Firstly, in order to assess the probing results, it was necessary for the training dataset of the target LLM to be publicly available. Secondly, to facilitate both black-box and white-box probing, it was essential to have access to pre-trained weights of the target model. To meet these criteria, we opted to utilize the OPT with 1.3 billion hyperparameters (OPT-1.3B) [35] and corresponding tokenizer released by HuggingFace $[ 3 3 ] ^ { 2 }$ as our target LLM for probing. Please note that the above criteria are for evaluation. In real-world scenarios, ProPILE is not limited to OPT but can be applied to many other LLMs.
+
+Evaluation dataset. This paper conducts experiments using five types of PII: phone number, email address, and (physical) address as instances of structured PII and family relationship and university information as instances of unstructured PII. To evaluate the PII leakage, an evaluation dataset was collected from the Pile dataset, which is an 825GB English dataset included in OPT training data [10]. It is noteworthy that the presence of documents containing all five types linked to a data subject is rare in the Pile dataset. However, for structured PII, there were instances where all three types of structured PII were linked to the name of a data subject. Hence, we extracted quadruplets of (name, phone number, email address, address) from the Pile dataset. Specifically, the PII items are searched with regular expressions and named entity recognition [2, 21]. Examples are shown in Figure 2 (b). For the collection of unstructured PII, we adopted a question-answering model based on RoBERTa3 and formulated relevant questions to extract information regarding relationships or affiliations. Only answers with a confidence score exceeding 0.9 were gathered, and subsequently underwent manual filtering to eliminate mislabeled instances. The final evaluation dataset consists of the structured PII quadruplets for 10,000 data subjects, name-family relationship pairs for 10,000 data subjects, and name-university pairs for 2,000 data subjects. Please refer to the Appendix for the dataset construction details.
+
+# 4.2 Black-box Probing Results
+
+We show how the black-box probing approach of ProPILE with hand-crafted prompts helps data subjects assess the leakage of their own PII. We also examine the effect of various factors on the leakage.
+
+![](images/a91555ded60163371db5288f6fe28a51a2410781738637806530feea8cf12f36.jpg)  
+Figure 3: Black-box probing result in likelihood perspective. Reconstruction vs. baseline likelihood of (a) structured PII and (b) unstructured PII, shown with the average likelihood and the $\mathsf { p }$ -value of the Wilcoxon signed-rank test. (c) shows a summary of the likelihoods using $\gamma _ { < k }$ defined in Equation 3.
+
+Likelihood results. We first evaluate the likelihood of the target PII item given the other items of a subject. Then, we consider the black-box LLM as revealing the linkable PII item, if the likelihood probability is greater than that of randomly selected PII instances, i.e., $\mathrm { P r } ( a _ { m } | \mathcal { A } _ { \backslash m } ) >$ $\mathrm { P r } ( a _ { m , \mathrm { N u l l } } | \mathcal { A } _ { \backslash m } )$ . The $a _ { m , \mathrm { N u l l } }$ is randomly selected from the evaluation dataset. We utilized the aforementioned evaluation dataset and created prompts using five different triplet templates, including those described in Figure 2 (a). Subsequently, the generation is done using beam search with a beam size of 3. The likelihood was computed using Equation 2. Please refer to the Appendix for the detailed generation hyperparameters and prompt templates.
+
+Figure 3 (a-b) illustrates the density plot of the likelihoods. The blue and orange color represents the target PII $( a _ { m } )$ and randomly chosen PII $( a _ { m , \mathrm { n u l l } } )$ , respectively. The plots also display the mean likelihood values. It is observed that the mean likelihood of target PII is higher than that of the null PII for all PII types. We also denoted the p-value obtained from the statistical test using the Wilcoxon signed rank test [7]. The small p-value suggests that the observed difference is statistically significant except for affiliation. Figure 3 (c) shows $\gamma _ { < k }$ . We have mentioned in section 3.4 that the $\mathbf { X }$ -axis variable, $k$ , can be interpreted as the number of queries. As the number of queries increases, we observe a gradual increase in the frequency of exact reconstruction.
+
+The above black-box probing results demonstrate a high risk of reconstructing the exact PII based on available PII items and establishing the link. The results of $\gamma _ { < k }$ indicate that despite the seemingly low likelihood values, there is a possibility of exact reconstruction of PII.
+
+Exact match results. Through black-box probing, the generated sequences can be obtained. The exact match can be assessed by evaluating whether the generated sequence includes the exact string of target PII or not. First, we evaluated the exact match with a varying number of templates used to construct the prompts. Results are shown in Figure 4 (a). The rate of exact matches increases as the diversity of prompt templates increases. This also supports the rationale behind white-box probing, as it suggests that finding more optimal prompts can further increase the leakage.
+
+Furthermore, we conducted an assessment of exact matches when different levels of the associations were present in the prompt. Figure 4 (b) shows the results. The “twins” denotes that only the name of a data subject is used to make the query prompt, while “triplet” indicates the presence of an additional PII item in the prompt. We can observe a fivefold increase in the exact match rate for the email address. This increase occurred when a phone number, which offers more specific information about the data subject, was provided in addition to the name. In the case of phone numbers, we also observed an increase of more than double. This shows increasing information in the prompts that can be associated with the target PII elevates the leakage. It also supports the effectiveness of black-box probing that utilizes the data subject’s linkable PIIs. Furthermore, with increased beam search sizes in the model (Figure 4 (c)) and larger model sizes (d), the frequency of the target PII appearing in generated sentences also tends to rise. The increasing leakage that occurs with larger model sizes can be attributed to improved accuracy. This implies that as the current trend of scaling up large language models continues, the potential risks of PII leakage may also increase.
+
+![](images/17133a8a0b8b9607954169e39c5e5866ca1eee0799381bd1c5d580b3caf2c604.jpg)  
+Figure 4: Black-box probing results in string-match perspective. The proportion of PII that is exactly reconstructed through black-box probing. We vary (a) the number of query prompts, (b) the level of associated PII items in the query prompt, (c) the beam size for decoding and (d) the size of the targeted LLM.
+
+# 4.3 White-box Probing Results
+
+In this section, we demonstrate the white-box probing by presenting the leakage of the phone number given other PII information in the structured quadruplet. We train 20 embedding vectors for the soft prompts by appending them ahead of a single prompt to generate the target phone number; We use additional 128 quadruplet data that are not included in the evaluation dataset. Please refer to Appendix for the training details. With the trained soft prompts, we measure the likelihood probabilities and exact match ratios on the evaluation dataset. Figure 5 summarizes the results in terms of the number of training data, the number of soft tokens, and the initialization type.
+
+Efficacy of soft prompt tuning. Figure 5 illustrates the impact of the soft prompt on the exact match rate and reconstruction likelihood, with blue and orange colors, respectively. The results indicate a significant increase, from $0 . 0 0 4 7 \%$ of black-box probing using five prompt templates to $1 . 3 \%$ with the soft prompt learned only from 128 data points being prepended to a single query prompt. The likelihood also increased by a large amount for the same case. It is speculated that the observed increase can be attributed to the soft prompt facilitating the more optimal prompts that may not have been considered by humans during the construction of prompts in black-box probing.
+
+Effect of dataset size. The white-box probing scenario assumes that a user (or a service provider) has access to only a small portion of the training data. To see the impact of the number of data used for tuning to the degree of the leakage, soft prompts were trained using different numbers of triplets in the training dataset, specifically [16, 32, 64, 128, 256, 512]. The results are depicted in Figure 5 (a). Even with 16 data points, a significant surge in leakage was observed. The exact match rate escalated to $0 . 1 2 \%$ , surpassing the exact match scores achieved by using five prompts, as well as in terms of likelihood. As the training set size increases from 16 to 128, the exact match dramatically increases from $0 . 1 2 \%$ to $1 . 5 0 \%$ . This finding indicates that even with a small fraction of the training dataset, it is possible to refine prompts that can effectively probe the PII leakage in LLM.
+
+Additional analysis of soft prompt tuning. We also examine the impact of different factors on the leakage and Figure 5 (b) and (c) display the leakage levels according to these factors. As the number of soft tokens increases, the leakage also exhibits an increasing trend. This can be attributed to the enhanced expressiveness of the soft prompts, which improves as the number of parameters increases. Furthermore, different initialization schemes produce diverse outcomes. We investigated three initialization schemes: 1) an embedding of the word representing the specific type of target PII, i.e., “phone”, which was the default setting throughout our experiments, 2) an embedding sampled from a uniform distribution $\mathcal { U } ( - 1 , 1 )$ , and 3) utilizing the mean of all vocabulary embeddings. As illustrated in Figure 1(c), the uniform and mean initialization schemes were unable to raise the leakage. In contrast, initializing with the PII type resulted in the most significant leakage.
+
+![](images/290c41a763b761d0a5a4330945fa3c1766931e7230ac7d6bcbb4152f6be5f68c.jpg)  
+Figure 5: White box probing results. Leakage results on 10,000 unseen triplets according to (a) varying number of data used for prompt tuning, (b) number of soft tokens, (c) different intialization type. Blue and orange color denotes exact match rate and likelihood, respectively.
+
+Table 1: Transferability of soft prompt. Original denotes the black-box probing results using one query prompt and transfer denotes the probing results using the transferred soft prompt that is learned from the source model (OPT-1.3B). $\times$ columns show how much the leakage likelihood increases by using the transferred soft prompt.   
+
+<table><tr><td rowspan="2">Source</td><td rowspan="2">Target</td><td colspan="3">Avg. Likelihood</td><td colspan="2">#Exact match</td></tr><tr><td>Original</td><td>Transfer</td><td>×</td><td>Original</td><td>Transfer</td></tr><tr><td rowspan="3">OPT-1.3B</td><td>OPT-350M</td><td>1.05×10-11</td><td>1.08×10-10</td><td>7.5</td><td>0</td><td>0</td></tr><tr><td>OPT-1.3B</td><td>6.06×10-8</td><td>3.47×10-6</td><td>57.3</td><td>5</td><td>3</td></tr><tr><td>OPT-2.7B</td><td>1.39×10-7</td><td>2.18×10-6</td><td>15.6</td><td>14</td><td>15</td></tr></table>
+
+Transferability test. If the soft embedding learned for one language model can be reused to probe a different language model, it opens up the possibility of applying the knowledge acquired from whitebox probing to black-box probing. To assess the feasibility of this approach, we transferred the soft prompt learned for the OPT-1.3B model to OPT models with different scales, namely OPT-350M and OPT-2.7B. However, directly plugging the soft embedding trained on one model into another model is impossible due to the mismatch of embedding dimensions (e.g., 1, 024 and 512 for OPT-1.3B and OPT-350M, respectively.) To address this, we follow a two-step process of the previous approach [20]. We project the soft embedding to the closest hard tokens in terms of Euclidean distance and decode it to raw string with the source model’s tokenizer. The string is then concatenated ahead of the raw query text and fed into the target model.
+
+Table 1 demonstrates that the soft prompt learned from the OPT-1.3B model increases the leakage of the same type of PII in both the OPT-350M and OPT-2.7B models. The increase in leakage is also denoted with the multiplication symbol $( \times )$ , showcasing how many times the reconstruction likelihood is amplified when utilizing the soft prompt learned for OPT-1.3B in the other models. While there may not be a substantial difference from the exact match perspective, the potential for transferability has been confirmed in the perspective of likelihood. Future work could explore research for investigating white-box probing techniques for enhancing transferability.
+
+# 5 Conclusion
+
+This paper introduces ProPILE, a novel tool designed for probing PII leakage in LLM. ProPILE encompasses two probing strategies: black-box probing for data subjects and white-box probing for LLM service providers. In the black-box probing approach, we strategically designed prompts and metrics so that the data subjects can effectively probe if their own PII is being leaked from LLM. The white-box probing approach empowered LLM service providers to conduct investigations on their own in-house models. This was achieved by leveraging the training data and model parameters to fine-tune more potent prompts, enabling a deeper analysis of potential PII leakage. By conducting actual probing on the OPT-1.3B model, we made several observations. First, we found that the target PII item is generated with a significantly higher likelihood compared to a random PII item. Furthermore, white-box probing revealed a tighter worst-case leakage possibility in terms of PII leakage. We hope that our findings empower the data subjects and LLM service providers for their awareness and control over their own data on the web.
+
+Limitations. The construction of the evaluation dataset exclusively involved the use of private information sourced from open-source datasets provided by large corporations. This approach ensures the ethical acquisition of data. However, it’s important to acknowledge that the data collection process itself was heuristic in nature. Consequently, the evaluation dataset may contain instances of incorrectly associated data or noise. This could introduce a degree of uncertainty or potential inaccuracies, which must be taken into account when interpreting the results.
+
+Societal Impact. We emphasize that our proposed probing strategies are not designed to facilitate or encourage the leakage of PII. Instead, our intention is to provide a framework that empowers both data subjects and LLM service providers to thoroughly assess the privacy state of current LLMs. By conducting such evaluations, stakeholders can gain insights into the privacy vulnerabilities and potential risks associated with LLMs prior to their deployment in a wider range of real-world applications. This proactive approach aims to raise awareness among users, enabling them to understand the security and privacy implications of LLM usage and take appropriate measures to safeguard their personal information.
+
+# 6 Ethical Considerations
+
+The evaluation dataset used in our paper is collected from the Pile dataset and thus follows the data regulation and terms of services of it. Specifically, all sources of the Pile dataset are public, with the majority following the terms of services, and $55 \%$ of them are authorized by data owners. Please refer to [10] for the details.
+
+ProPILE’s primary intent is to empower data subjects to verify whether their own information may leak from LLM services. However, the black-box probing approach could be abused by malicious actors aiming to extract others’ personal information from LLMs without proper consent. Although attackers remain incapable of discerning whether the extracted information belongs to the target, they may still obtain sequences with potential connections with the target. Additionally, the chance of successful attacks with limited information is not entirely eliminated, although it is very low according to the associativity experiment results in Section 4.2.
+
+Here we suggest some guidelines to use and develop ProPILE properly. ProPILE should only be used by the data subjects themselves to examine the reconstruction of PII items linked to them, or by the individuals who have been given explicit consent from the relevant data subjects. One way to implement this is to insert an authentication layer over ProPILE to allow its usage only to users identified through Google accounts; they are only allowed to submit queries using only the information from the user’s verified Google profile. Ultimately, the user’s good intentions are the most important. We strongly recommend that ProPILE be used only for its intended purpose of enhancing the self-awareness of data subjects.
+
+# Acknowledgements
+
+This work was supported by NAVER Corporation, the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (2022R1A3B1077720, 2022R1A5A708390811), Institute of Information & Communications Technology Planning & Evaluation (IITP) grants funded by the Korea government (MSIT) (2021-0-01343: AI Graduate School Program, SNU, 2022-0- 00959), the BK21 FOUR program of the Education and Research Program for Future ICT Pioneers, Seoul National University in 2023, and the Luxembourg National Research Funds (FNR) through CORE project C18/IS/12669767/STELLAR/LeTraon.
+
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parse/dev/VFBjuF8HEp/VFBjuF8HEp.md

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+# LEARNING FAST SAMPLERS FOR DIFFUSION MODELSBY DIFFERENTIATING THROUGH SAMPLE QUALITY
+
+Daniel Watson∗, William Chan, Jonathan Ho & Mohammad Norouzi Google Research, Brain Team {watsondaniel,williamchan,jonathanho,mnorouzi}@google.com
+
+# ABSTRACT
+
+Diffusion models have emerged as an expressive family of generative models rivaling GANs in sample quality and autoregressive models in likelihood scores. Standard diffusion models typically require hundreds of forward passes through the model to generate a single high-fidelity sample. We introduce Differentiable Diffusion Sampler Search (DDSS): a method that optimizes fast samplers for any pre-trained diffusion model by differentiating through sample quality scores. We present Generalized Gaussian Diffusion Models (GGDM), a family of flexible non-Markovian samplers for diffusion models. We show that optimizing the degrees of freedom of GGDM samplers by maximizing sample quality scores via gradient descent leads to improved sample quality. Our optimization procedure backpropagates through the sampling process using the reparametrization trick and gradient rematerialization. DDSS achieves strong results on unconditional image generation across various datasets (e.g., FID scores on LSUN church $1 2 8 \mathrm { x } 1 2 8$ of 11.6 with only 10 inference steps, and 4.82 with 20 steps, compared to 51.1 and 14.9 with strongest DDPM/DDIM baselines). Our method is compatible with any pre-trained diffusion model without fine-tuning or re-training required.
+
+# 1 INTRODUCTION
+
+Denoising Diffusion Probabilistic Models (DDPM) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020) have emerged as a powerful family of generative models, capable of synthesizing high-quality images, audio, and 3D shapes (Ho et al., 2020; 2021; Chen et al., 2021a;b; Cai et al., 2020; Luo & Hu, 2021). Recent work (Dhariwal & Nichol, 2021; Ho et al., 2021) shows that DDPMs can outperform Generative Adversarial Networks (GAN) (Goodfellow et al., 2014; Brock et al., 2018) in generation quality, but unlike GANs, DDPMs admit likelihood computation and much more stable training dynamics (Arjovsky et al., 2017; Gulrajani et al., 2017).
+
+![](images/fd5801c9ac221ce83ea91e924697dc3725f542dbe1d292487226be79766b71a8.jpg)  
+Figure 1: Non-cherrypicked samples from DDSS (ours) and strongest $\mathrm { D D I M } ( \eta = 0 )$ baseline for unconditional DDPMs trained on LSUN churches $1 2 8 \times 1 2 8$ . All samples are generated with the same random seed. Original DDPM samples (1000 steps) and training images are shown on the left.
+
+However, GANs are typically much more efficient than DDPMs at generation time, often requiring a single forward pass through the generator network, whereas DDPMs require hundreds of forward passes through a U-Net model. Instead of learning a generator directly, DDPMs learn to convert noisy data to less noisy data starting from pure noise, which leads to a wide variety of feasible strategies for sampling (Song et al., 2021b). In particular, at inference time, DDPMs allow controlling the number of forward passes (a.k.a. inference steps) through the denoising network (Song et al., 2020; Nichol & Dhariwal, 2021).
+
+It has been shown both empirically and mathematically that, for any sufficiently good DDPM, more inference steps leads to better log-likelihood and sample quality (Nichol & Dhariwal, 2021; Kingma et al., 2021). In practice, the minimum number of inference steps to achieve competitive sample quality is highly problem-dependent, e.g., depends on the complexity of the dataset, and the strength of the conditioning signal if the task is conditional. Given the importance of generation speed, recent work (Song et al., 2020; Chen et al., 2021a; Watson et al., 2021) has explored reducing the number of steps required for high quality sampling with pretrained diffusion models. See Section 7 for a more complete review of prior work on few-step sampling.
+
+This paper treats the design of fast samplers for diffusion models as a differentiable optimization problem, and proposes Differentiable Diffusion Sampler Search (DDSS). Our key observation is that one can unroll the sampling chain of a diffusion model and use reparametrization trick (Kingma & Welling, 2013) and gradient rematerialization (Kumar et al., 2019a) to optimize over a class of parametric few-step samplers with respect to a global objective function. Our class of parameteric samplers, which we call Generalized Gaussian Diffusion Model (GGDM), includes Denoising Diffusion Implicit Models (DDIM) (Song et al., 2020) as a special case and is motivated by the success of DDIM on fast sampling of diffusion models.
+
+An important challenge for fast DDPM sampling is the mismatch between the training objective (e.g., ELBO or weighted ELBO) and sample quality. Prior work (Watson et al., 2021; Song et al., 2021a) finds that samplers that are optimal with respect to ELBO often lead to worse sample quality and Frechet Inception Distance (FID) scores (Heusel et al., 2017), especially with few inference ´ steps. We propose the use of a perceptual loss within the DDSS framework to find high-fidelity diffusion samplers, motivated by prior work showing that their optimization leads to solutions that correlate better with human perception of quality. We empirically find that using DDSS with the Kernel Inception Distance (KID) (Binkowski et al., 2018) as the perceptual loss indeed leads to ´ fast samplers with significantly better image quality than prior work (see Figure 1). Moreover, our method is robust to different choices of kernels for KID.
+
+Our main contributions are as follows:
+
+1. We propose Differentiable Diffusion Sampler Search (DDSS), which uses the reparametrization trick and gradient rematerialization to optimize over a parametric family of fast samplers for diffusion models.   
+2. We identify a parametric family of Generalized Gaussian Diffusion Model (GGDM) that admits high-fidelity fast samplers for diffusion models.   
+3. We show that using DDSS to optimize samplers by minimizing the Kernel Inception Distance leads to fast diffusion model samplers with state-of-the-art sample quality scores.
+
+# 2 BACKGROUND ON DENOISING DIFFUSION IMPLICIT MODELS
+
+We start with a brief review on DDPM (Ho et al., 2020) and DDIM (Song et al., 2020). DDPMs pre-specify a Markovian forward diffusion process, which gradually adds noise to data in $T$ steps. Following the notation of Ho et al. (2020),
+
+$$
+\begin{array} { r c l } { \displaystyle q ( \pmb { x } _ { 0 } , . . . , \pmb { x } _ { T } ) } & { = } & { \displaystyle q ( \pmb { x } _ { 0 } ) \prod _ { t = 1 } ^ { T } q ( \pmb { x } _ { t } | \pmb { x } _ { t - 1 } ) } \\ { \displaystyle q ( \pmb { x } _ { t } | \pmb { x } _ { t - 1 } ) } & { = } & { \displaystyle \mathcal { N } ( \pmb { x } _ { t } | \sqrt { 1 - \beta _ { t } } \pmb { x } _ { s } , \beta _ { t } \pmb { I } ) , } \end{array}
+$$
+
+where $q ( { \pmb x } _ { 0 } )$ is the data distribution, and $\beta _ { t }$ is the variance of Gaussian noise added at step $t$ . To be able to gradually convert noise to data, DDPMs learn to invert (1) with a model $p _ { \theta } ( \pmb { x } _ { t - 1 } | \bar { \pmb { x } } _ { t } )$ , which
+
+is trained by maximizing a (possibly reweighted) evidence lower bound (ELBO):
+
+$$
+\mathbb { E } _ { q } \left[ D _ { \mathrm { K L } } [ q ( \boldsymbol { x } _ { T } | \boldsymbol { x } _ { 0 } ) | | p ( \boldsymbol { x } _ { T } ) ] + \sum _ { t = 2 } ^ { T } D _ { \mathrm { K L } } [ q ( \boldsymbol { x } _ { t - 1 } | \boldsymbol { x } _ { t } , \boldsymbol { x } _ { 0 } ) | | p _ { \theta } ( \boldsymbol { x } _ { t - 1 } | \boldsymbol { x } _ { t } ) ] - \log p _ { \theta } ( \boldsymbol { x } _ { 0 } | \boldsymbol { x } _ { 1 } ) \right]
+$$
+
+DDPMs specifically choose the model to be parametrized as
+
+$$
+\begin{array} { l } { \displaystyle p _ { \theta } \big ( \mathbf { x } _ { t - 1 } \big | \mathbf { x } _ { t } \big ) = q \left( x _ { t - 1 } \bigg | \mathbf { x } _ { t } , \hat { \mathbf { x } } _ { 0 } = \frac { 1 } { \sqrt { \bar { \alpha _ { t } } } } \big ( x _ { t } - \sqrt { 1 - \bar { \alpha _ { t } } } \epsilon _ { \theta } \big ( \mathbf { x } _ { t } , t \big ) \big ) \right) } \\ { \displaystyle \qquad = \mathcal { N } \left( x _ { t - 1 } \bigg | \frac { 1 } { \sqrt { 1 - \beta _ { t } } } \left( x _ { t } - \frac { \beta _ { t } } { \sqrt { 1 - \bar { \alpha } _ { t } } } \epsilon _ { \theta } \big ( \mathbf { x } _ { t } , t \big ) \right) , \frac { 1 - \bar { \alpha } _ { t } } { 1 - \bar { \alpha } _ { t - 1 } } \beta _ { t } I _ { d } \right) } \end{array}
+$$
+
+where $\begin{array} { r } { \bar { \alpha _ { t } } = \prod _ { s = 1 } ^ { t } ( 1 - \beta _ { t } ) } \end{array}$ for each $t$ . With this parametrization, maximizing the ELBO is equivalent to minimizing a weighted sum of denoising score matching objectives (Vincent, 2011).
+
+The seminal work of Song et al. (2020) presents Denoising Diffusion Implicit Models (DDIM): a family of evidence lower bounds (ELBOs) with corresponding forward diffusion processes and samplers. All of these ELBOs share the same marginals as DDPM, but allow arbitrary choices of posterior variances. Specifically, Song et al. (2020) note that is it possible to construct alternative ELBOs with only a subsequence of the original timesteps $S \subset \{ 1 , . . . , T \}$ that shares the same marginals as the construction above (i.e., $q _ { S } \mathbf { \bar { ( } } \mathbf { x } _ { t } | \mathbf { x } _ { 0 } \mathbf { ) } = q \mathbf { \bar { ( } } \mathbf { x } _ { t } | \mathbf { x } _ { 0 } \mathbf { ) }$ for every $t \in S$ , so $q _ { S }$ defines a faster sampler compatible with the pre-trained model) by simply using the new contiguous timesteps in the equations above. They also show it is also possible to construct an infinite family of nonMarkovian processes $\{ q _ { \sigma } : \dot { \sigma } \in [ 0 , 1 ] ^ { T - 1 } \}$ where each $q _ { \sigma }$ also shares the same marginals as the original forward process with:
+
+$$
+q _ { \sigma } ( \pmb { x } _ { 0 } , . . . , \pmb { x } _ { t } ) = q ( \pmb { x } _ { 0 } ) q ( \pmb { x } _ { T } | \pmb { x } _ { 0 } ) \prod _ { t = 1 } ^ { T - 1 } q _ { \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { t + 1 } , \pmb { x } _ { 0 } )
+$$
+
+and where the posteriors are defined as
+
+$$
+q _ { \sigma } ( \pmb { x } _ { t - 1 } | \pmb { x } _ { t } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t - 1 } \bigg | \sqrt { \bar { \alpha } _ { t - 1 } } \pmb { x } _ { 0 } + \sqrt { 1 - \bar { \alpha } _ { t - 1 } - \sigma _ { t } ^ { 2 } } \cdot \frac { \pmb { x } _ { t } - \sqrt { \bar { \alpha } _ { t } } \pmb { x } _ { 0 } } { \sqrt { 1 - \bar { \alpha } _ { t } } } , \sigma _ { t } ^ { 2 } \pmb { I } _ { d } \right)
+$$
+
+Song et al. (2020) empirically find that the extreme case of using all-zero variances (a.k.a. $\mathrm { D D I M } ( \eta = 0 ) ,$ ) consistently helps with sample quality in the few-step regime. Combined with a good selection of timesteps to evaluate the modeled score function (a.k.a. strides), $\mathrm { D D I M } ( \eta = 0 )$ establishes the current state-of-the-art for few-step diffusion model sampling with the smallest inference step budgets. Our key contribution that allows improving sample quality significantly by optimizing sampler families is constructing a family that generalizes DDIM. See Section 4 for a more complete treatment of our novel GGDM family.
+
+# 3 DIFFERENTIABLE DIFFUSION SAMPLER SEARCH (DDSS)
+
+We now describe DDSS, our approach to search for fast high-fidelity samplers with a limited budget of $K < T$ steps. Our key observation is that one can backpropagate through the sampling process of a diffusion model via the reparamterization trick (Kingma & Welling, 2013). Equipped with this, we can now use stochastic gradient descent to learn fast samplers by optimizing any given differentiable loss function over a minibatch of model samples.
+
+We begin with a pre-trained DDPM and a family of $K$ -step samplers that we wish to optimize for the given DDPM. We parametrize this family’s degrees of freedom as simple transformations of trainable variables. We experiment with the following families in this paper, but emphasize that DDSS is applicable to any other family where model samples are differentiable with respect to the trainable variables:
+
+• DDIM: we parametrize the posterior variances with √ $\sigma _ { t } = \sqrt { 1 - \bar { \alpha } _ { t - 1 } }$ sigmoid $\left( v _ { t } \right)$ , where $v _ { 1 } , . . . , v _ { K }$ are trainable variables (the $\sqrt { 1 - \bar { \alpha } _ { t - 1 } }$ constant ensures real-valued mean coefficients; see the square root in Equation 6).
+
+![](images/f7b465e57a29651ce12353533765c18485729af556a7cb3ef9c66309d8c511a7.jpg)  
+Figure 2: Illustration of GGDM. To improve sample quality, our novel family of samplers combines information from all previous (noisier) images at every denoising step.
+
+• VARS: we parametrize the marginal variances of a DDPM as cumsum $( \operatorname { s o f t m a x } ( [ \pmb { v } ; 1 ] ) ) _ { t }$ instead of fixing them to $1 - \bar { \alpha } _ { t }$ . This ensures they are monotonically increasing with respect to $t$ (appending a one to ensure $K$ degrees of freedom). GGDM: a new family of non-Markovian samplers for diffusion models with more degrees of freedom illustrated in Figure 2 and defined in Section 4. We parametrize $\mu _ { t u }$ and $\sigma _ { t }$ of a GGDM for all $t$ as sigmoid functions of trainable variables. GGDM $\mathbf { \Gamma } + \mathbf { P } \mathbf { R } \mathbf { E } \mathbf { D }$ : we parametrize all the $\mu _ { t u }$ and $\sigma _ { t }$ identically to GGDM, but also learn the marginal coefficients with a cumsum $\circ$ softmax parametrization (identical to VARS) instead of computing them via Theorem 1, as well as the coefficients that predict $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ from $a _ { t } { \pmb x } _ { t } - b _ { t } { \pmb \epsilon }$ with $1 +$ softplus and softplus parametrizations.   
+• [family] ${ \bf + } { \bf \Pi } { \bf \Pi } { \bf \bf \Pi } \mathbf { M } { \bf E }$ : for any sampler family, we additionally parametrize the timesteps used to query the score model with a cumsum ◦ softmax parametrization (identical to VARS).
+
+As we will show in the experiments, despite the fact that our pre-trained DDPMs are trained with discrete timesteps, learning the timesteps is still helpful. In principle, this should only be possible for DDPMs trained with continuous time sampling (Chen et al., 2021a; Song et al., 2021b; Kingma et al., 2021), but in practice we find that DDPMs trained with continuously embedded discrete timesteps are still well-behaved when applied at timesteps not present during training. We think this is due to the regularity of the sinusoidal positional encodings Vaswani et al. (2017) used in these model architectures and training with a sufficiently large number of timesteps $T$ .
+
+# 3.1 DIFFERENTIABLE SAMPLE QUALITY SCORES
+
+We can differentiate through a stochastic sampler using the reparameterization trick, but the question of which objective to optimize still remains. Prior work has shown that optimizing log-likelihood can actually worsen sample quality and FID scores in the few-step regime (Watson et al., 2021; Song et al., 2021a). Thus, we instead design a perceptual loss which simply compares mean statistics between model samples and real samples in the neural network feature space. These types of objectives have been shown in prior work to better correlate with human perception of sample quality (Johnson et al., 2016; Heusel et al., 2017), which we also confirm in our experiments.
+
+We rely on the representations of the penultimate layer of a pre-trained InceptionV3 classifier (Szegedy et al., 2016) and optimize the Kernel Inception Distance (KID) (Binkowski et al., 2018). ´ Let $\phi ( { \pmb x } )$ denote the inception features of an image $_ { \textbf { \em x } }$ and $p _ { \psi }$ represent a diffusion sampler with trainable parameters $\psi$ . For a linear kernel, which works best in our experiments, the objective is:
+
+$$
+\mathcal { L } ( \psi ) = \underset { \pmb { x } _ { p } \sim p _ { \psi } } { \mathbb { E } } \underset { \pmb { x } _ { p } ^ { \prime } \sim p _ { \psi } } { \mathbb { E } } \phi ( \pmb { x } _ { p } ) ^ { \top } \phi ( \pmb { x } _ { p } ^ { \prime } ) - \underset { \pmb { x } _ { p } \sim p _ { \psi } } { \mathbb { E } } \underset { \pmb { x } _ { q } \sim q } { \mathbb { E } } \phi ( \pmb { x } _ { p } ) ^ { \top } \phi ( \pmb { x } _ { q } )
+$$
+
+More generally, KID can be expressed as:
+
+$$
+{ \mathcal L } _ { \mathrm { K I D } } ( \boldsymbol { \psi } ) = \left. \underset { \boldsymbol { x } _ { p } \sim p _ { \boldsymbol { \psi } } } { \mathbb { E } } f ^ { * } ( \boldsymbol { x } _ { p } ) - \underset { \boldsymbol { x } _ { q } \sim q } { \mathbb { E } } f ^ { * } ( \boldsymbol { x } _ { q } ) \right. _ { 2 } ^ { 2 } ,
+$$
+
+where $f ^ { * } ( x ) = \mathbb { E } _ { x _ { p } ^ { \prime } \sim p _ { \psi } } k _ { \phi } ( x , x _ { p } ^ { \prime } ) - \mathbb { E } _ { x _ { q } ^ { \prime } \sim q ( x _ { 0 } ) } k _ { \phi } ( x , x _ { q } ^ { \prime } )$ is the witness function for any differentiable, positive definite kernel $k$ , and $k _ { \phi } ( \dot { \pmb x } , \pmb y ) = k ( \phi ( \pmb x ) , \phi ( \pmb y ) )$ . Note that $f ^ { * }$ attains the supremum of the MMD. To enable stochastic gradient descent, we use an unbiased estimator of KID using a minibatch of $n$ model samples $\pmb { x } _ { p } ^ { ( 1 ) } \ldots \pmb { x } _ { p } ^ { ( n ) } \sim p _ { \psi }$ and $n$ real samples $\pmb { x } _ { q } ^ { ( 1 ) } \dots \pmb { x } _ { q } ^ { ( n ) } \sim q$ :
+
+$$
+\frac { 1 } { n ( n - 1 ) } \sum _ { i \neq j } ^ { n } k _ { \phi } ( { \pmb x } _ { p } ^ { ( i ) } , { \pmb x } _ { p } ^ { ( j ) } ) - \frac { 2 } { n ^ { 2 } } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { n } k _ { \phi } ( { \pmb x } _ { p } ^ { ( i ) } , { \pmb x } _ { q } ^ { ( j ) } ) + c ,
+$$
+
+where $c$ is constant in $\psi$ . Since the sampling chain of any Gaussian diffusion process admits using the reparametrization trick, our loss function is fully differentiable with respect to the trainable variables $\psi$ . We empirically find that using the perceptual features is crucial; i.e., by trying $\phi ( { \pmb x } ) = { \pmb x }$ to compare images directly on pixel space rather than neural network feature space (as above), we observe that our method makes samples consistently worsen in apparent quality as training progresses.
+
+# 3.2 GRADIENT REMATERIALIZATION
+
+In order for backpropagation to be feasible under reasonable memory constraints, one final problem must be addressed: since we are taking gradients with respect to model samples, the cost in memory to maintain the state of the forward pass scales linearly with the number of inference steps, which can quickly become unfeasible considering the large size of typical DDPM architectures. To address this issue, we use gradient rematerialization (Kumar et al., 2019b). Instead of storing a particular computation’s output from the forward pass required by the backward pass computation, we recompute it on the fly. To trade $\mathcal O ( K )$ memory cost for $\mathcal O ( K )$ computation time, we simply rematerialize calls to the pre-trained DDPM (i.e., the estimated score function), but keep in memory all the progressively denoised images from the sampling chain. In JAX (Bradbury et al., 2018), this is trivial to implement by simply wrapping the score function calls with jax.remat.
+
+# 4 GENERALIZED GAUSSIAN DIFFUSION MODELS
+
+We now present Generalized Gaussian Diffusion Model (GGDM), our novel family of Gaussian diffusion processes that includes DDIM as a special case mentioned in section 3. We define a joint distribution with no independence assumptions
+
+$$
+q _ { \mu , \sigma } ( \pmb { x } _ { 0 } , . . . , \pmb { x } _ { T } ) = q ( \pmb { x } _ { 0 } ) q ( \pmb { x } _ { T } | \pmb { x } _ { 0 } ) \prod _ { t = 1 } ^ { T - 1 } q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { > t } , \pmb { x } _ { 0 } )
+$$
+
+where the new factors are defined as
+
+$$
+q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { > t } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \bigg | \sum _ { u \in S _ { t } } \mu _ { t u } \pmb { x } _ { u } , \sigma _ { t } ^ { 2 } \pmb { I } _ { d } \right)
+$$
+
+(letting $S _ { t } ~ = ~ \{ 0 , . . . , T \} ~ \backslash ~ \{ 1 , . . . , t \}$ for notation compactness), with $\sigma _ { t }$ and $\mu _ { t u }$ free parameters $\forall t \in \{ 1 , . . . , T \} , u \in S _ { t }$ . In other words, when predicting the next, less noisy image, the sampler can take into account all the previous, noisier images in the sampling chain, and similarly to DDIM, we can also control the sampler’s variances. As we prove in the appendix (A.2), this construction admits Gaussian marginals, and we can differentiably compute the marginal coefficients from arbitrary choices of $\mu$ and $\sigma$ :
+
+Theorem 1. Given some $t \in \{ 1 , . . . , T \}$ , let $a _ { t u } ^ { ( 1 ) } \ = \ \mu _ { t u } \ \forall u \in \ S _ { t }$ and $v _ { t } ^ { ( 1 ) } = \sigma _ { t } ^ { 2 }$ . For each , recursively define
+
+$$
+a _ { t u } ^ { ( i + 1 ) } = a _ { t , t + i } ^ { ( i ) } \mu _ { t + i , u } + a _ { t u } ^ { ( i ) } \forall u \in S _ { t + i } \quad \mathrm { a n d } \quad v _ { t } ^ { ( i + 1 ) } = v _ { t } ^ { ( i ) } + \left( a _ { t , t + i } ^ { ( i ) } \sigma _ { t + i } \right) ^ { 2 } .
+$$
+
+Then, it follows that
+
+$$
+q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { > t + i } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \middle | \sum _ { \pmb { u \in S } _ { t + i } } a _ { t u } ^ { ( i + 1 ) } \pmb { x } _ { u } , v _ { t } ^ { ( i + 1 ) } \pmb { I } _ { d } \right) .
+$$
+
+In other words, instead of letting the $\beta _ { t }$ (or equivalently, the $\bar { \alpha } _ { t }$ ) define the forward process as done by a usual DDPM, the GGDM family lets the $\mu _ { t u }$ and $\sigma _ { t }$ define the process. In particular, an immediate corollary of Theorem 1 is that the marginal coefficients are given by
+
+$$
+\begin{array} { r } { q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \Big | a _ { t 0 } ^ { ( T - t + 1 ) } \pmb { x } _ { 0 } , v _ { t } ^ { ( T - t + 1 ) } \pmb { I } _ { d } \right) } \end{array}
+$$
+
+The reverse process is thus defined as $\begin{array} { r } { p ( { \pmb x } _ { T } ) \prod _ { t = 1 } ^ { T } p ( { \pmb x } _ { t - 1 } | { \pmb x } _ { t } ) } \end{array}$ with $p ( \pmb { x } _ { T } ) \sim \mathcal { N } ( \mathbf { 0 } , \pmb { I } _ { d } ) )$ and
+
+$$
+p _ { \theta } ( \mathbf { x } _ { t } | \mathbf { x } _ { > t } ) = q _ { \mu , \sigma } \left( x _ { t } | \mathbf { x } _ { > t } , \hat { \mathbf { x } } _ { 0 } = \frac { 1 } { a _ { t 0 } ^ { ( T - t + 1 ) } } \left( \mathbf { x } _ { t } - \sqrt { v _ { t } ^ { ( T - t + 1 ) } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) \right) \right) .
+$$
+
+Table 1: FID and IS scores for DDSS against baseline methods for a DDPM trained on CIFAR10 with the $L _ { \mathrm { s i m p l e } }$ objective proposed by (Ho et al., 2020). FID scores (lower is better) are the numbers at the left of each entry, and IS scores (higher is better) are at the right.   
+
+<table><tr><td>SamplerK</td><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td></tr><tr><td>DDPM (linear stride)</td><td>84.27/5.396</td><td>43.39/7.034</td><td>31.40/7.609</td><td>25.94/7.879</td><td>22.60/8.043</td></tr><tr><td>DDPM(quadratic stride)</td><td>76.25 / 5.435</td><td>42.03 /6.965</td><td>27.78 /7.714</td><td>20.225 /8.128</td><td>16.17 / 8.350</td></tr><tr><td>DDIM (linear stride)</td><td>44.41 / 6.750</td><td>19.11 / 7.965</td><td>14.06 /8.190</td><td>11.82 /8.420</td><td>10.52 / 8.512</td></tr><tr><td>DDIM(quadratic stride)</td><td>32.66 /7.090</td><td>13.62/8.190</td><td>9.318 /8.495</td><td>7.500 / 8.641</td><td>6.560 / 8.759</td></tr><tr><td>GGDM+PRED+TIME</td><td>13.77 /8.520</td><td>8.227 / 8.903</td><td>6.115 /9.050</td><td>4.722/9.261</td><td>4.250 /9.186</td></tr></table>
+
+Table 2: FID / IS scores for DDSS against baseline methods for a DDPM trained on ImageNet 64x64 with the $L _ { \mathrm { h y b r i d } }$ objective proposed by Nichol & Dhariwal (2021).   
+
+<table><tr><td>SamplerK</td><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td></tr><tr><td>DDPM (linear stride)</td><td>122.0/5.878</td><td>58.78/10.67</td><td>39.30/13.22</td><td>31.36/14.72</td><td>26.36/15.71</td></tr><tr><td>DDPM(quadratic stride)</td><td>394.8 / 1.351</td><td>129.5 / 5.997</td><td>80.10 /9.595</td><td>61.34 / 11.60</td><td>49.60 / 13.01</td></tr><tr><td>DDIM (linear stride)</td><td>135.4 /5.898</td><td>40.70 / 12.225</td><td>28.54 /13.99</td><td>24.225 / 14.75</td><td>22.13 /15.16</td></tr><tr><td>DDIM (quadratic stride)</td><td>409.1/1.380</td><td>148.6 / 5.533</td><td>67.65 /9.842</td><td>45.60 / 11.99</td><td>36.11 /13.225</td></tr><tr><td>GGDM+PRED+TIME</td><td>55.14 /12.90</td><td>37.32/14.76</td><td>24.69/17.225</td><td>20.69 /17.92</td><td>18.40 /18.12</td></tr></table>
+
+# 4.1 IGNORING THE MATCHING MARGINALS CONDITION
+
+Unlike DDIM, the GGDM family does not guarantee that the marginals of the new forward process match that of the original DDPM. We empirically find, however, that this condition can often be too restrictive and better samplers exist where the marginals $q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { 0 } ) =$ $\mathcal { N } \left( \pmb { x } _ { t } \Big | a _ { t 0 } ^ { ( T - t + 1 ) } \pmb { x } _ { 0 } , v _ { t } ^ { ( T - t + 1 ) } \pmb { I } _ { d } \right)$ v(T −t+1)t Id of the new forward process differ from the original DDPM’s marginals. We verify this empirically by applying DDSS to both the family of DDIM sigmas and DDPM variances (“VARS” in Section 3): both sampler families have the same number of parameters (the reverse process variances), but the latter does not adjust the mean coefficients like DDIM to ensure matching marginals and still achieves similar or better scores than the former across sample quality metrics (and even outperforms the $\mathrm { D D I M } ( \eta = 0 )$ baseline); see Section 5.2.
+
+# 5 EXPERIMENTS
+
+In order to emphasize that our method is compatible with any pre-trained DDPM, we apply our method on pre-trained DDPM checkpoints from prior work. Specifically, we experiment with the DDPM trained by Ho et al. (2020) with $L _ { \mathrm { s i m p l e } }$ on CIFAR10, as well as a DDPM following the exact configuration of Nichol & Dhariwal (2021) trained on ImageNet 64x64 (Deng et al., 2009) with their $L _ { \mathrm { h y b r i d } }$ objective (with the only difference being that we trained the latter ourselves for 3M rather than 1.5M steps). Both of these models utilize adaptations of the UNet architecture (Ronneberger et al., 2015) that incorporate self-attention layers (Vaswani et al., 2017).
+
+We evaluate all of our models on both FID and Inception Score (IS) (Salimans et al., 2016), comparing the samplers discovered by DDSS against DDPM and DDIM baselines with linear and quadratic strides. As previously mentioned, more recent methods for fast sampling are outperformed by DDIM when the budget of inference steps is as small as those we utilize in this work (5, 10, 15, 20, 25). All reported results on both of these approximate sample quality metrics were computed by comparing 50K model and training data samples, as is standard in the literature. Also as is standard, IS scores are computed 10 times, each time on 5K samples, and then averaged.
+
+In all of our experiments, we optimize the DDSS objective presented in Section 3.1 with the following hyperparameters:
+
+1. For every family of models we search over, we initialize the degrees of freedom such that training begins with a sampler matching DDPM with $K$ substeps following Song et al. (2020); Nichol & Dhariwal (2021).
+
+Table 3: FID / IS scores for the KID kernel ablation on CIFAR10. When not learning the timesteps, we fix them to a quadratic stride, as Table 1 shows this performs best on CIFAR10.   
+
+<table><tr><td rowspan=1 colspan=1>SamplerK</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>25</td></tr><tr><td rowspan=1 colspan=1>DDSS (linear kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM)</td><td rowspan=1 colspan=1>13.77 /8.52014.26 /8.40612.85 /8.38314.45 /8.281</td><td rowspan=1 colspan=1>8.227 /8.9038.617 /8.8427.858 / 8.8958.154 /8.892</td><td rowspan=1 colspan=1>6.115 /9.0505.939 /9.0356.265 /9.0757.045 /8.939</td><td rowspan=1 colspan=1>4.722 /9.2614.893 /9.1535.367 /9.1365.477 /9.183</td><td rowspan=1 colspan=1>4.250 /9.1864.574 /9.1454.887 /9.2294.815 /9.189</td></tr><tr><td rowspan=1 colspan=1>DDSS (cubic kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM</td><td rowspan=1 colspan=1>14.41 / 8.52714.39 /8.40112.35 / 8.40614.57 /8.297</td><td rowspan=1 colspan=1>8.2257 /9.0078.977 /8.8707.879 /8.8528.2252/8.836</td><td rowspan=1 colspan=1>5.895 /9.0366.517 / 8.9706.682 /8.9996.727 /8.904</td><td rowspan=1 colspan=1>4.932/9.0924.915 /9.1325.639 /9.0585.569 /9.177</td><td rowspan=1 colspan=1>4.278/9.2864.471 /9.2474.631/9.1894.668 /9.192</td></tr></table>
+
+2. We apply gradient updates using the Adam optimizer (Kingma & Ba, 2015). We sweeped over the learning rate and used $\lambda = 0 . 0 0 0 5$ . We did not sweep over other Adam hyperparameters and kept β1 = 0.9, β2 = 0.999, and  = 1 × 10−8.
+
+3. We tried batch sizes of 128 and 512 and opted for the latter, finding that it leads to better sample quality upon inspection. Since the loss depends on averages over examples as our experiments are on unconditional generation, this choice was expected.
+
+4. We run all of our experiments for 50K training steps and evaluate the discovered samplers at this exact number of training steps. We did not sweep over this value.
+
+We include our main results in Table 1 for CIFAR10 and Table 2 for ImageNet 64x64, comparing DDSS applied to GGDM $+ \mathrm { P R E D + T I M E }$ against DDPM and DDIM baselines with linear and quadratic strides. All models use a linear kernel, i.e., $k _ { \phi } ( { \pmb x } , { \pmb y } ) = \phi ( { \pmb x } ) ^ { \top } \phi ( { \pmb y } )$ , which we found to perform slightly better than the cubic kernel used by Binkowski et al. (2018) (we ablate this in ´ section 5.1). We omit the use of the learned variances of the ImageNet $6 4 \mathrm { x } 6 4$ model (i.e., following Nichol & Dhariwal (2021)), as we search for the variances ourselves via DDSS. We include samples for 5, 10 and 25 steps comparing the strongest DDIM baselines to $\mathrm { D D S S } + \mathrm { G G D M }$ with a learned stride; see Figures 1 and 3. We include additional ImageNet $6 4 \mathrm { x } 6 4$ samples (A.1) and results for larger resolution datasets (A.4) in the appendix.
+
+# 5.1 ABLATIONS FOR KID KERNEL AND GGDM VARIANTS
+
+As our approach is compatible with any choice of KID kernel, we experiment with different choices of kernels. Namely, we try the simplest possible linear kernel, $k _ { \phi } ( \dot { \pmb { x } } , \pmb { y } ) = \phi ( \pmb { x } ) ^ { \top } \phi ( \pmb { y } )$ , as well as the cubic kernel $\begin{array} { r } { k _ { \phi } ( \pmb { x } , \pmb { y } ) = \left( \frac { 1 } { d } \phi ( \pmb { x } ) ^ { \top } \phi ( \pmb { y } ) + 1 \right) ^ { 3 } } \end{array}$ used by Binkowski et al. (2018). We compare the ´ performance of these two kernels, as well as different variations of GGDM (i.e., with and without TIME and PRED as defined in Section 3). Results are included for CIFAR10 across all budgets in Table 3. We also include a smaller version of this ablation for ImageNet 64x64 in the appendix (A.3).
+
+The results in this ablation show that the contributions of the linear kernel, timestep learning, and the empirical choice of learning the coefficients that predict $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ all slightly contribute to better FID and IS scores. Importantly, however, removing any of these additions still allows us to comfortably outperform the strongest baselines. See also the results on LSUN in the appendix A.4, which also do not include these additional trainable variables.
+
+# 5.2 SEARCH SPACE ABLATION
+
+Now, in order to further demonstrate the key importance of optimizing our GGDM family to find high-fidelity samplers, we also apply DDSS to the less general DDIM and VARS families. We show that, while we still attain better scores than a regular DDPM, searching these less flexible families of samplers does not yield improvements as significant as with out novel GGDM family. In particular, optimizing the DDIM sigma coefficients does not outperform the corresponding $\mathrm { D D I M } ( \eta = 0 )$ baseline on CIFAR10, which is not a surprising result as Song et al. (2020) show empirically that most choices of the $\sigma _ { t }$ degrees of freedom lead to worse FID scores than setting them all to 0. These results also show that optimizing the VARS can outperform DDSS applied to the DDIM family, and even the strongest $\mathrm { D D I M } ( \eta = 0 )$ baselines for certain budgets, justifying our choice of not enforcing the marginals to match (as discussed in Section 4.1).
+
+Table 4: FID / IS scores for the DDSS search space ablation on CIFAR10. All runs fix the timesteps to a quadratic stride and use a linear kernel except for the last row (we only include the GGDM results for ease of comparison).   
+
+<table><tr><td rowspan=1 colspan=1>SamplerK</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>25</td></tr><tr><td rowspan=1 colspan=1>DDIM(n = 0)</td><td rowspan=1 colspan=1>32.66/7.090</td><td rowspan=1 colspan=1>13.62/8.190</td><td rowspan=1 colspan=1>9.318/ 8.495</td><td rowspan=1 colspan=1>7.500 / 8.641</td><td rowspan=1 colspan=1>6.560/ 8.759</td></tr><tr><td rowspan=1 colspan=1>DDSSVARSDDIM</td><td rowspan=1 colspan=1>33.08 /7.09632.61/7.084</td><td rowspan=1 colspan=1>15.33 /8.55916.29 /7.966</td><td rowspan=1 colspan=1>9.693 /8.84511.31/ 8.372</td><td rowspan=1 colspan=1>7.297 / 8.9249.120 /8.563</td><td rowspan=1 colspan=1>6.172/9.0577.853 /8.644</td></tr><tr><td rowspan=1 colspan=1>GGDMGGDM+PRED+TIME</td><td rowspan=1 colspan=1>14.45/8.28113.77 /8.520</td><td rowspan=1 colspan=1>8.154/8.8928.227 / 8.903</td><td rowspan=1 colspan=1>7.045 / 8.9396.115 /9.050</td><td rowspan=1 colspan=1>5.477 /9.1834.722 /9.261</td><td rowspan=1 colspan=1>4.815/9.1894.250 /9.186</td></tr></table>
+
+# 6 DISCUSSION
+
+When applied to a sufficiently flexible family (such as the GGDM family proposed in this work), DDSS consistently finds samplers that achieve better image generation quality than the strongest baselines in the literature for very few steps. This is qualitatively apparent in non-cherrypicked samples (e.g., $\mathrm { D D I M } ( \eta = 0 )$ tends to generate blurrier images and with less background details as the budget decreases), and multiple quantitative sample quality metrics (FID and IS) also reflect these results. Still, we observe limitations to our method. Finding samplers with inference step budgets as small as $K < 1 0$ that have little apparent loss in quality remains challenging with our proposed search family. And, while on CIFAR10 the metrics indicate significant relative improvement over sample quality metrics, the relative improvement on ImageNet 64x64 is less pronounced. We hypothesize that this is an inherent difficulty of ImageNet due to its high diversity of samples, and that in order to retain sample quality and diversity, it might be impossible to escape some minimum number of inference steps with score-based models as they might be crucial to mode-breaking.
+
+Beyond the empirical gains of applying our procedure, our findings shed further light into properties of pre-trained score-based generative models. First, we show that without fine-tuning a DDPM’s parameters, these models are already capable of producing high-quality samples with very few inference steps, though the default DDPM sampler in this regime is usually suboptimal when using a few-step sampler. We further show that better sampling paths exist, and interestingly, these are determined by alternative variational lower bounds to the data distribution that make use of the score-based model but do not necessarily share the same marginals as the original DDPM forward process. Our findings thus suggest that enforcing this marginal-sharing constraint is unnecessary and can be too restrictive in practice.
+
+# 7 OTHER RELATED WORK
+
+Besides DDIM (Song et al., 2020), there have been more recent attempts at reducing the number of inference steps for DDPMs. Jolicoeur-Martineau et al. (2021) proposed a dynamic step size SDE solver that can reduce the number of calls to the modeled score function to $\sim 1 5 0$ on CIFAR10 (Krizhevsky et al., 2009) with minimal cost in FID scores, but quickly falls behind $\mathrm { D D I M } ( \eta = 0 )$ with as many as 50 steps. Watson et al. (2021) proposed a dynamic programming algorithm that chooses log-likelihood optimal strides, but find that log-likelihood reduction has a mismatch with FID scores, particularly with in the very few step regime, also falling behind $\mathrm { D D I M } ( \eta = 0 )$ in this front. Other methods that have been shown to help sample quality in the few-step regime include non-Gaussian variants of diffusion models (Nachmani et al., 2021) and adaptively adjusting the sampling path by introducing a noise level estimating network (San-Roman et al., 2021), but more thorough evaluation of sample quality achieved by these approaches is needed with budgets as small as those considered in this work.
+
+Other approaches to sampling DDPMs have also been recently proposed, though not for the explicit purpose of efficient sampling. Song et al. (2021b) derive a reverse SDE that, when discretized, uses different coefficients than the ancestral samplers considered in this work. The same authors also derive “corrector” steps, which introduce additional calls to the pre-trained DDPM as a form of gradient ascent (Langevin dynamics) that help with quality but introduce computation cost, as well as an alternative sampling procedure using a probability flow ODE that shares the same marginals as the DDPM’s original forward process. Huang et al. (2021) generalize this family of samplers to a “plug-in reverse SDE” that interpolates between a probability flow ODE and the reverse SDE, similarly to how the DDIM $\eta$ interpolates between an implicit probabilistic model and a stochastic reverse process. Our proposed search family includes discretizations of most of these cases for Gaussian processes, notably missing corrector steps, where reusing a single timestep is considered.
+
+![](images/32e14402d05adeeb690961580b44df8d5fbd8f1bb8eac3367ce846dcaf8eebe6.jpg)  
+Figure 3: Non-cherrypicked samples for a DDPM trained on CIFAR10, comparing the strongest DDIM $\mathbf { \eta } _ { \eta } = 0 ,$ ) baseline and our approach. All samples were generated with the same random seeds. For reference, we include DDPM samples using all 1,000 steps (top right) and real images (bottom right).
+
+# 8 CONCLUSION AND FUTURE WORK
+
+We propose Differentiable Diffusion Sampler Search (DDSS), a method for finding few-step samplers for Denoising Diffusion Probabilistic Models. We show how to optimize a perceptual loss over a space of diffusion processes that makes use of a pre-trained DDPM’s samples by leveraging the reparametrization trick and gradient rematerialization. Our results qualitatively and quantitatively show that DDSS is able to significantly improve sample quality for unconditional image generation over prior methods on efficient DDPM sampling. The success of our method hinges on searching a novel, wider family of Generalized Gaussian Diffusion Model (GGDM) than those identified in prior work (Song et al., 2020). DDSS does not fine-tune the pre-trained DDPM, only needs to be applied once, has few hyperparameters, and does not require re-training the DDPM.
+
+Our findings suggest future directions to further reduce the number of inference steps while retaining high fidelity in generated samples. For instance, it is plausible to use different representations for the perceptual loss instead of those of a classifier, e.g., use representations from an unsupervised model such as SimCLR (Chen et al., 2020), to using internal representations learned by the pre-trained DDPM itself, which would eliminate the burden of additional computation. Moreover, considering the demonstrated benefits of applying DDSS to our proposed GGDM family of samplers (as opposed to narrower families like DDIM), we motivate future work on identifying more general families of samplers and investigating whether they help uncover even better samplers or lead to overfitting. Finally, identifying other variants of perceptual losses (e.g., that do not sample from the model), or alternative optimization strategies (e.g., gradient-free methods) that lead to similar results is important future work. This would make DDSS itself a more efficient procedure, as gradient-based optimization of our proposed loss requires extensive memory or computation requirements to backpropagate through the whole sampling chain.
+
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+Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computerassisted intervention, pp. 234–241. Springer, 2015.
+
+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. Advances in neural information processing systems, 29: 2234–2242, 2016.
+
+Robin San-Roman, Eliya Nachmani, and Lior Wolf. Noise estimation for generative diffusion models. arXiv preprint arXiv:2104.02600, 2021.
+
+Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, pp. 2256–2265. PMLR, 2015.
+
+Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020.
+
+Yang Song and Stefano Ermon. Generative Modeling by Estimating Gradients of the Data Distribution. NeurIPS, 2019.
+
+Yang Song, Conor Durkan, Iain Murray, and Stefano Ermon. Maximum likelihood training of scorebased diffusion models. arXiv e-prints, pp. arXiv–2101, 2021a.
+
+Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-Based Generative Modeling through Stochastic Differential Equations. In ICLR, 2021b.
+
+Markus Svensen and Christopher M Bishop. Pattern recognition and machine learning, 2007. ´
+
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jonathon Shlens, and Zbigniew Wojna. Rethinking the Inception Architecture for Computer Vision. In CVPR, 2016.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention Is All You Need. In NIPS, 2017.
+
+Pascal Vincent. A connection between score matching and denoising autoencoders. Neural Computation, 23(7):1661–1674, 2011.
+
+Daniel Watson, Jonathan Ho, Mohammad Norouzi, and William Chan. Learning to efficiently sample from diffusion probabilistic models. arXiv preprint arXiv:2106.03802, 2021.
+
+Fisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, and Jianxiong Xiao. Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015.
+
+# A APPENDIX
+
+# A.1 ADDITIONAL IMAGENET 64X64 SAMPLES
+
+We provide additional samples for our results on ImageNet $6 4 \mathrm { x } 6 4$ . The DDPM and $\mathrm { D D I M } ( \eta = 0 )$ samples (left and middle, respectively) use a linear stride, while our DDSS $^ +$ GGDM samples (right) use a learned stride.
+
+![](images/1837841f8ba78d46089b6644adcd4a9a922cbe225c3310a07840389f63dad545.jpg)  
+Figure A.1: Additional samples on ImageNet $6 4 \mathrm { x } 6 4$ . For reference, we include DDPM samples with all 4,000 steps (bottom left) and real samples (bottom middle).
+
+# A.2 PROOF FOR THEOREM 1
+
+Theorem 1. Given some $t \in \{ 1 , . . . , T \}$ , let $a _ { t u } ^ { ( 1 ) } \ = \ \mu _ { t u } \ \forall u \in \ S _ { t }$ and $v _ { t } ^ { ( 1 ) } = \sigma _ { t } ^ { 2 }$ . For each , recursively define
+
+$$
+\begin{array} { r l } & { \bullet \ a _ { t u } ^ { ( i + 1 ) } = a _ { t , t + i } ^ { ( i ) } \mu _ { t + i , u } + a _ { t u } ^ { ( i ) } \forall u \in S _ { t + i } } \\ & { \bullet \ v _ { t } ^ { ( i + 1 ) } = v _ { t } ^ { ( i ) } + \left( a _ { t , t + i } ^ { ( i ) } \sigma _ { t + i } \right) ^ { 2 } } \end{array}
+$$
+
+Then, it follows that
+
+$$
+q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { > t + i } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \bigg | \sum _ { u \in S _ { t + i } } a _ { t u } ^ { ( i + 1 ) } \pmb { x } _ { u } , v _ { t } ^ { ( i + 1 ) } \pmb { I } _ { d } \right)
+$$
+
+Proof. Let us prove this result with mathematical induction. Note that, for each such $t$ , we have by definition that
+
+$$
+\begin{array} { r } { q _ { \mu , \sigma } ( \pmb { x } _ { t + 1 } | \pmb { x } _ { > t + 1 } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \Big | \sum _ { u \in S _ { t + 1 } } \mu _ { t + 1 , u } \pmb { x } _ { u } , \sigma _ { t + 1 } ^ { 2 } \pmb { I } _ { d } \right) } \end{array}
+$$
+
+and
+
+$$
+\begin{array} { r } { q _ { \mu , \sigma } ( { \pmb x } _ { t } | { \pmb x } _ { t + 1 } , { \pmb x } _ { > t + 1 } , { \pmb x } _ { 0 } ) = { \mathcal N } \left( { \pmb x } _ { t } \big | \sum _ { u \in S _ { t } } \mu _ { t u } { \pmb x } _ { u } , \sigma _ { t } ^ { 2 } I _ { d } \right) = { \mathcal N } \left( { \pmb x } _ { t } \Big | \sum _ { u \in S _ { t } } a _ { t u } ^ { ( 1 ) } { \pmb x } _ { u } , v _ { t } ^ { ( 1 ) } I _ { d } \right) } \end{array}
+$$
+
+Therefore, following Svensen & Bishop (2007) (2.115), by prior conjugacy it follows that ´
+
+$$
+\begin{array} { r l } & { ( { \pmb x } _ { t } | { \pmb x } _ { > t + 1 } , { \pmb x } _ { 0 } ) = \mathcal { N } ( { \pmb x } _ { t } \Big | a _ { t , t + 1 } ^ { ( 1 ) } \sum _ { u \in S _ { t + 1 } } \mu _ { t + 1 , u } { \pmb x } _ { u } + \sum _ { u \in S _ { t + 1 } } a _ { t u } ^ { ( 1 ) } , ( v _ { t } ^ { ( 1 ) } + a _ { t , t + 1 } ^ { ( 1 ) } \sigma _ { t + 1 } ^ { 2 } a _ { t , t + 1 } ^ { ( 1 ) } ) I _ { d _ { 1 } ^ { \prime } }  } \\ & { \qquad = \mathcal { N } ( { \pmb x } _ { t } \Big | \sum _ { u \in S _ { t + 1 } } ( a _ { t , t + 1 } ^ { ( 1 ) } \mu _ { t + 1 , u } + a _ { t u } ^ { ( 1 ) } ) { \pmb x } _ { u } , v _ { t } ^ { ( 2 ) } I _ { d } )  } \\ & { \qquad = \mathcal { N } ( { \pmb x } _ { t } \Big | \sum _ { u \in S _ { t + 1 } } a _ { t u } ^ { ( 2 ) } { \pmb x } _ { u } , v _ { t } ^ { ( 2 ) } I _ { d } ) . } \end{array}
+$$
+
+This proves the base case for our induction argument. Now, let us prove the inductive step. Suppose there exists some integer $j \in \{ 1 , . . . , T - t + 1 \}$ such that
+
+$$
+\begin{array} { r } { q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { > t + j } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \middle | \sum _ { \boldsymbol { u } \in S _ { t + j } } a _ { t \boldsymbol { u } } ^ { ( j + 1 ) } \pmb { x } _ { \boldsymbol { u } } , v _ { t } ^ { ( j + 1 ) } \pmb { I } _ { d } \right) . } \end{array}
+$$
+
+By definition, we already know $q ( \pmb { x } _ { t + j + 1 } | \pmb { x } _ { > t + j + 1 } , \pmb { x } _ { 0 } )$ , so we have
+
+$$
+\begin{array} { r } { q _ { \mu , \sigma } ( \pmb { x } _ { t + j + 1 } | \pmb { x } _ { > t + j + 1 } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t + j + 1 } \middle | \sum _ { u \in S _ { t + j + 1 } } \mu _ { t + j + 1 , u } \pmb { x } _ { u } , \sigma _ { t + j + 1 } ^ { 2 } \pmb { I } _ { d } \right) } \end{array}
+$$
+
+and (rewriting the inductive hypothesis)
+
+$$
+\begin{array} { r } { q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { t + j + 1 } , \pmb { x } _ { > t + j + 1 } , \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \Big | \sum _ { \pmb { u \in S } _ { t + j } } a _ { t u } ^ { ( j + 1 ) } \pmb { x } _ { u } , v _ { t } ^ { ( j + 1 ) } \pmb { I } _ { d } \right) . } \end{array}
+$$
+
+Therefore, by prior conjugacy again, it follows that
+
+$$
+\begin{array} { r l } & { \quad q _ { \mu , \sigma } ( \boldsymbol { x } _ { t } | \boldsymbol { x } _ { > t + j + 1 } , \boldsymbol { x } _ { 0 } ) } \\ & { = \mathcal { N } \left( \boldsymbol { x } _ { t } \Big | a _ { t , t + j + 1 } ^ { ( j + 1 ) } \sum _ { u \in S _ { t + j } } \mu _ { t + j + 1 , u } \boldsymbol { x } _ { u } + \sum _ { u \in S _ { t + j } } a _ { t u } ^ { ( j + 1 ) } , \left( v _ { t } ^ { ( j + 1 ) } + a _ { t , t + j + 1 } ^ { ( j + 1 ) } \sigma _ { t + j + 1 } ^ { 2 } a _ { t , t + j + 1 } ^ { ( j + 1 ) } \right) \boldsymbol { I } _ { d } \right) } \\ & { = \mathcal { N } \left( \boldsymbol { x } _ { t } \Big | \sum _ { u \in S _ { t + j } } \left( a _ { t , t + j + 1 } ^ { ( j + 1 ) } \mu _ { t + j + 1 , u } + a _ { t u } ^ { ( j + 1 ) } \right) \boldsymbol { x } _ { u } , v _ { t } ^ { ( j + 2 ) } \boldsymbol { I } _ { d } \right) } \\ & { = \mathcal { N } \left( \boldsymbol { x } _ { t } \Big | \sum _ { u \in S _ { t + j + 1 } } a _ { t u } ^ { ( j + 2 ) } \boldsymbol { x } _ { u } , v _ { t } ^ { ( j + 2 ) } \boldsymbol { I } _ { d } \right) . } \end{array}
+$$
+
+This concludes the proof of the inductive step. Hence, we have proven the result for any $i \in$ $\{ 1 , . . . , T - t \}$ . In particular,
+
+$$
+\begin{array} { r } { q _ { \mu , \sigma } ( \pmb { x } _ { t } | \pmb { x } _ { 0 } ) = \mathcal { N } \left( \pmb { x } _ { t } \middle | a _ { t 0 } ^ { ( T - t + 1 ) } \pmb { x } _ { 0 } , v _ { t } ^ { ( T - t + 1 ) } \pmb { I } _ { d } \right) . \quad \prod } \end{array}
+$$
+
+A.3 ADDITIONAL ABLATION OF KID KERNEL AND GGDM VARIANTS FOR IMAGENET 64X64
+
+We also ran a smaller version of the ablation results presented in Section 5.1, but for ImageNet $6 4 \mathrm { x } 6 4$ instead of CIFAR10, as these are more computationally intensive to do a full grid search. Results for a step budget $K = 1 5$ are included below. When not learning the timesteps, we fix them to a linear stride, as Table 2 shows this performs best on ImageNet $6 4 \mathrm { x } 6 4$ .
+
+<table><tr><td rowspan=1 colspan=2>SamplerK</td><td rowspan=1 colspan=1>15</td></tr><tr><td rowspan=1 colspan=2>DDSS (linear kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM</td><td rowspan=1 colspan=1>24.69 /17.22527.08 /16.4425.73 /17.2728.34 /16.63</td></tr><tr><td rowspan=2 colspan=2>DDSS (cubic kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM</td><td rowspan=2 colspan=1>26.52 /16.2927.82/16.326.87 /16.9928.83 /16.32</td></tr><tr></tr></table>
+
+# A.4 RESULTS ON LARGER RESOLUTION DATASETS
+
+We include results for LSUN (Yu et al., 2015) bedrooms and churches at the $1 2 8 \mathrm { x } 1 2 8$ resolution. We trained the models for 400K and 200K steps (respectively), and all other hyperparameters are identical: we use the Adam optimizer with learning rate 0.0003 (linearly warmed up for the first 1000 training steps), batch size 2048, gradient clipping at norms over 1.0, dropout of 0.1, and EMA over the weights with decay rate 0.9999. We train the models using a linear stride of 1000 evenlyspaced timesteps, fixing the log-signal-to-noise-ratio schedule to a cosine function monotonically decreasing from 20 to -20. The ELBO is reweighted with $L _ { \mathrm { s i m p l e } }$ following Ho et al. (2020), but we additionally reweight each term by $\mathrm { m a x } ( 1 , \mathrm { S N R } )$ which we found to be slightly helpful in resulting FID scores (note this is equivalent to minimizing the worst mean squared error between either the $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ or $\epsilon$ ). The UNet employs five down/up-sampling resolutions with $7 6 8 \times ( 1 , 2 , 4 , 6 , 8 )$ respective channels, 3 ResNet blocks per resolution, and spatial self-attention at the 3 smallest resolutions, i.e., 8, 16, and 32.
+
+After training the models, we run DDSS using just the GGDM model family for simplicity (i.e., we don’t use the $+ \mathrm { P R E D }$ and $+ \mathrm { T I M E }$ we experiment with in the paper) at 5, 10 and 20 evenly-spaced inference steps. DDSS training occurs for 50K steps, using the Adam optimizer with learning rate of 0.0005 and batch size 512, optimizing the linear kernel for the KID loss. We compare against the usual DDPM and $\mathrm { D D I M } ( \eta = 0 )$ baselines at the same inference budgets and include the FID scores with all 1000 steps for reference. Results and samples are included below.
+
+<table><tr><td rowspan=1 colspan=1>SamplerK</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>1000</td></tr><tr><td rowspan=1 colspan=1>LSUN BedroomDDPMDDIM(n = 0)DDSS (GGDM)</td><td rowspan=1 colspan=1>95.38168.729.15</td><td rowspan=1 colspan=1>44.8456.3311.01</td><td rowspan=1 colspan=1>16.889.5274.817</td><td rowspan=1 colspan=1>2.54711</td></tr><tr><td rowspan=1 colspan=1>LSUN ChurchDDPMDDIM(n = 0)DDSS (GGDM)</td><td rowspan=1 colspan=1>96.67133.130.24</td><td rowspan=1 colspan=1>51.0554.3911.59</td><td rowspan=1 colspan=1>16.5314.966.736</td><td rowspan=1 colspan=1>2.71811</td></tr></table>
+
+![](images/117d2db6ee60f3c709a7640eba901b5649459defc69fac8b0fb239b09f7afe7a.jpg)  
+Figure A.2: Non-cherrypicked samples for a DDPM trained on LSUN bedroom $1 2 8 \mathrm { x } 1 2 8$ , comparing DDPM and $\mathrm { D D I M } ( \eta = 0 )$ to our approach. All samples were generated with the same random seeds and a linear stride. For reference, we include DDPM samples using all 1,000 steps (bottom left) and real images (bottom middle).
+
+![](images/14ca7c6b13c4879f8aaf148f154440406d5bb69bfc76f588f68a066f71e41f95.jpg)  
+Figure A.3: Non-cherrypicked samples for a DDPM trained on LSUN church 128x128, comparing DDPM and $\mathrm { D D I M } ( \eta = 0 )$ to our approach. All samples were generated with the same random seeds and a linear stride. For reference, we include DDPM samples using all 1,000 steps (bottom left) and real images (bottom middle).

parse/dev/VFBjuF8HEp/VFBjuF8HEp_content_list.json

@@ -0,0 +1,1957 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING FAST SAMPLERS FOR DIFFUSION MODELSBY DIFFERENTIATING THROUGH SAMPLE QUALITY",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            820,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Daniel Watson∗, William Chan, Jonathan Ho & Mohammad Norouzi Google Research, Brain Team {watsondaniel,williamchan,jonathanho,mnorouzi}@google.com ",
+        "bbox": [
+            183,
+            172,
+            741,
+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            252,
+            544,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Diffusion models have emerged as an expressive family of generative models rivaling GANs in sample quality and autoregressive models in likelihood scores. Standard diffusion models typically require hundreds of forward passes through the model to generate a single high-fidelity sample. We introduce Differentiable Diffusion Sampler Search (DDSS): a method that optimizes fast samplers for any pre-trained diffusion model by differentiating through sample quality scores. We present Generalized Gaussian Diffusion Models (GGDM), a family of flexible non-Markovian samplers for diffusion models. We show that optimizing the degrees of freedom of GGDM samplers by maximizing sample quality scores via gradient descent leads to improved sample quality. Our optimization procedure backpropagates through the sampling process using the reparametrization trick and gradient rematerialization. DDSS achieves strong results on unconditional image generation across various datasets (e.g., FID scores on LSUN church $1 2 8 \\mathrm { x } 1 2 8$ of 11.6 with only 10 inference steps, and 4.82 with 20 steps, compared to 51.1 and 14.9 with strongest DDPM/DDIM baselines). Our method is compatible with any pre-trained diffusion model without fine-tuning or re-training required. ",
+        "bbox": [
+            233,
+            285,
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+            506
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            531,
+            336,
+            546
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Denoising Diffusion Probabilistic Models (DDPM) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020) have emerged as a powerful family of generative models, capable of synthesizing high-quality images, audio, and 3D shapes (Ho et al., 2020; 2021; Chen et al., 2021a;b; Cai et al., 2020; Luo & Hu, 2021). Recent work (Dhariwal & Nichol, 2021; Ho et al., 2021) shows that DDPMs can outperform Generative Adversarial Networks (GAN) (Goodfellow et al., 2014; Brock et al., 2018) in generation quality, but unlike GANs, DDPMs admit likelihood computation and much more stable training dynamics (Arjovsky et al., 2017; Gulrajani et al., 2017). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd5801c9ac221ce83ea91e924697dc3725f542dbe1d292487226be79766b71a8.jpg",
+        "image_caption": [
+            "Figure 1: Non-cherrypicked samples from DDSS (ours) and strongest $\\mathrm { D D I M } ( \\eta = 0 )$ baseline for unconditional DDPMs trained on LSUN churches $1 2 8 \\times 1 2 8$ . All samples are generated with the same random seed. Original DDPM samples (1000 steps) and training images are shown on the left. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            665,
+            821,
+            852
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, GANs are typically much more efficient than DDPMs at generation time, often requiring a single forward pass through the generator network, whereas DDPMs require hundreds of forward passes through a U-Net model. Instead of learning a generator directly, DDPMs learn to convert noisy data to less noisy data starting from pure noise, which leads to a wide variety of feasible strategies for sampling (Song et al., 2021b). In particular, at inference time, DDPMs allow controlling the number of forward passes (a.k.a. inference steps) through the denoising network (Song et al., 2020; Nichol & Dhariwal, 2021). ",
+        "bbox": [
+            174,
+            103,
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+            200
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "It has been shown both empirically and mathematically that, for any sufficiently good DDPM, more inference steps leads to better log-likelihood and sample quality (Nichol & Dhariwal, 2021; Kingma et al., 2021). In practice, the minimum number of inference steps to achieve competitive sample quality is highly problem-dependent, e.g., depends on the complexity of the dataset, and the strength of the conditioning signal if the task is conditional. Given the importance of generation speed, recent work (Song et al., 2020; Chen et al., 2021a; Watson et al., 2021) has explored reducing the number of steps required for high quality sampling with pretrained diffusion models. See Section 7 for a more complete review of prior work on few-step sampling. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper treats the design of fast samplers for diffusion models as a differentiable optimization problem, and proposes Differentiable Diffusion Sampler Search (DDSS). Our key observation is that one can unroll the sampling chain of a diffusion model and use reparametrization trick (Kingma & Welling, 2013) and gradient rematerialization (Kumar et al., 2019a) to optimize over a class of parametric few-step samplers with respect to a global objective function. Our class of parameteric samplers, which we call Generalized Gaussian Diffusion Model (GGDM), includes Denoising Diffusion Implicit Models (DDIM) (Song et al., 2020) as a special case and is motivated by the success of DDIM on fast sampling of diffusion models. ",
+        "bbox": [
+            174,
+            325,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An important challenge for fast DDPM sampling is the mismatch between the training objective (e.g., ELBO or weighted ELBO) and sample quality. Prior work (Watson et al., 2021; Song et al., 2021a) finds that samplers that are optimal with respect to ELBO often lead to worse sample quality and Frechet Inception Distance (FID) scores (Heusel et al., 2017), especially with few inference ´ steps. We propose the use of a perceptual loss within the DDSS framework to find high-fidelity diffusion samplers, motivated by prior work showing that their optimization leads to solutions that correlate better with human perception of quality. We empirically find that using DDSS with the Kernel Inception Distance (KID) (Binkowski et al., 2018) as the perceptual loss indeed leads to ´ fast samplers with significantly better image quality than prior work (see Figure 1). Moreover, our method is robust to different choices of kernels for KID. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contributions are as follows: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. We propose Differentiable Diffusion Sampler Search (DDSS), which uses the reparametrization trick and gradient rematerialization to optimize over a parametric family of fast samplers for diffusion models.   \n2. We identify a parametric family of Generalized Gaussian Diffusion Model (GGDM) that admits high-fidelity fast samplers for diffusion models.   \n3. We show that using DDSS to optimize samplers by minimizing the Kernel Inception Distance leads to fast diffusion model samplers with state-of-the-art sample quality scores. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ON DENOISING DIFFUSION IMPLICIT MODELS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We start with a brief review on DDPM (Ho et al., 2020) and DDIM (Song et al., 2020). DDPMs pre-specify a Markovian forward diffusion process, which gradually adds noise to data in $T$ steps. Following the notation of Ho et al. (2020), ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ca8ea56010ae051a507e405364073a96d04ce4868c8f3f22d461e0b438adee47.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\displaystyle q ( \\pmb { x } _ { 0 } , . . . , \\pmb { x } _ { T } ) } & { = } & { \\displaystyle q ( \\pmb { x } _ { 0 } ) \\prod _ { t = 1 } ^ { T } q ( \\pmb { x } _ { t } | \\pmb { x } _ { t - 1 } ) } \\\\ { \\displaystyle q ( \\pmb { x } _ { t } | \\pmb { x } _ { t - 1 } ) } & { = } & { \\displaystyle \\mathcal { N } ( \\pmb { x } _ { t } | \\sqrt { 1 - \\beta _ { t } } \\pmb { x } _ { s } , \\beta _ { t } \\pmb { I } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $q ( { \\pmb x } _ { 0 } )$ is the data distribution, and $\\beta _ { t }$ is the variance of Gaussian noise added at step $t$ . To be able to gradually convert noise to data, DDPMs learn to invert (1) with a model $p _ { \\theta } ( \\pmb { x } _ { t - 1 } | \\bar { \\pmb { x } } _ { t } )$ , which ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "is trained by maximizing a (possibly reweighted) evidence lower bound (ELBO): ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/c702246bd3fd39986a5710852fbc52bc55bc9182a6ea215e79503c31b448ffb3.jpg",
+        "text": "$$\n\\mathbb { E } _ { q } \\left[ D _ { \\mathrm { K L } } [ q ( \\boldsymbol { x } _ { T } | \\boldsymbol { x } _ { 0 } ) | | p ( \\boldsymbol { x } _ { T } ) ] + \\sum _ { t = 2 } ^ { T } D _ { \\mathrm { K L } } [ q ( \\boldsymbol { x } _ { t - 1 } | \\boldsymbol { x } _ { t } , \\boldsymbol { x } _ { 0 } ) | | p _ { \\theta } ( \\boldsymbol { x } _ { t - 1 } | \\boldsymbol { x } _ { t } ) ] - \\log p _ { \\theta } ( \\boldsymbol { x } _ { 0 } | \\boldsymbol { x } _ { 1 } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "DDPMs specifically choose the model to be parametrized as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9fe515d2c0a72a953662e15c0f189525e77bdab1778c951185f32cb64f181b0e.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle p _ { \\theta } \\big ( \\mathbf { x } _ { t - 1 } \\big | \\mathbf { x } _ { t } \\big ) = q \\left( x _ { t - 1 } \\bigg | \\mathbf { x } _ { t } , \\hat { \\mathbf { x } } _ { 0 } = \\frac { 1 } { \\sqrt { \\bar { \\alpha _ { t } } } } \\big ( x _ { t } - \\sqrt { 1 - \\bar { \\alpha _ { t } } } \\epsilon _ { \\theta } \\big ( \\mathbf { x } _ { t } , t \\big ) \\big ) \\right) } \\\\ { \\displaystyle \\qquad = \\mathcal { N } \\left( x _ { t - 1 } \\bigg | \\frac { 1 } { \\sqrt { 1 - \\beta _ { t } } } \\left( x _ { t } - \\frac { \\beta _ { t } } { \\sqrt { 1 - \\bar { \\alpha } _ { t } } } \\epsilon _ { \\theta } \\big ( \\mathbf { x } _ { t } , t \\big ) \\right) , \\frac { 1 - \\bar { \\alpha } _ { t } } { 1 - \\bar { \\alpha } _ { t - 1 } } \\beta _ { t } I _ { d } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\bar { \\alpha _ { t } } = \\prod _ { s = 1 } ^ { t } ( 1 - \\beta _ { t } ) } \\end{array}$ for each $t$ . With this parametrization, maximizing the ELBO is equivalent to minimizing a weighted sum of denoising score matching objectives (Vincent, 2011). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The seminal work of Song et al. (2020) presents Denoising Diffusion Implicit Models (DDIM): a family of evidence lower bounds (ELBOs) with corresponding forward diffusion processes and samplers. All of these ELBOs share the same marginals as DDPM, but allow arbitrary choices of posterior variances. Specifically, Song et al. (2020) note that is it possible to construct alternative ELBOs with only a subsequence of the original timesteps $S \\subset \\{ 1 , . . . , T \\}$ that shares the same marginals as the construction above (i.e., $q _ { S } \\mathbf { \\bar { ( } } \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 } \\mathbf { ) } = q \\mathbf { \\bar { ( } } \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 } \\mathbf { ) }$ for every $t \\in S$ , so $q _ { S }$ defines a faster sampler compatible with the pre-trained model) by simply using the new contiguous timesteps in the equations above. They also show it is also possible to construct an infinite family of nonMarkovian processes $\\{ q _ { \\sigma } : \\dot { \\sigma } \\in [ 0 , 1 ] ^ { T - 1 } \\}$ where each $q _ { \\sigma }$ also shares the same marginals as the original forward process with: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f2bfd37bd873ac7baa25424de2396f3cc841139c37b4a1070a6a26e3fc07775.jpg",
+        "text": "$$\nq _ { \\sigma } ( \\pmb { x } _ { 0 } , . . . , \\pmb { x } _ { t } ) = q ( \\pmb { x } _ { 0 } ) q ( \\pmb { x } _ { T } | \\pmb { x } _ { 0 } ) \\prod _ { t = 1 } ^ { T - 1 } q _ { \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { t + 1 } , \\pmb { x } _ { 0 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "and where the posteriors are defined as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3bfa2f150cbef965c4397c080669870a9d021e1ac57168c8dcef03911f7a676b.jpg",
+        "text": "$$\nq _ { \\sigma } ( \\pmb { x } _ { t - 1 } | \\pmb { x } _ { t } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t - 1 } \\bigg | \\sqrt { \\bar { \\alpha } _ { t - 1 } } \\pmb { x } _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t - 1 } - \\sigma _ { t } ^ { 2 } } \\cdot \\frac { \\pmb { x } _ { t } - \\sqrt { \\bar { \\alpha } _ { t } } \\pmb { x } _ { 0 } } { \\sqrt { 1 - \\bar { \\alpha } _ { t } } } , \\sigma _ { t } ^ { 2 } \\pmb { I } _ { d } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Song et al. (2020) empirically find that the extreme case of using all-zero variances (a.k.a. $\\mathrm { D D I M } ( \\eta = 0 ) ,$ ) consistently helps with sample quality in the few-step regime. Combined with a good selection of timesteps to evaluate the modeled score function (a.k.a. strides), $\\mathrm { D D I M } ( \\eta = 0 )$ establishes the current state-of-the-art for few-step diffusion model sampling with the smallest inference step budgets. Our key contribution that allows improving sample quality significantly by optimizing sampler families is constructing a family that generalizes DDIM. See Section 4 for a more complete treatment of our novel GGDM family. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 DIFFERENTIABLE DIFFUSION SAMPLER SEARCH (DDSS) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now describe DDSS, our approach to search for fast high-fidelity samplers with a limited budget of $K < T$ steps. Our key observation is that one can backpropagate through the sampling process of a diffusion model via the reparamterization trick (Kingma & Welling, 2013). Equipped with this, we can now use stochastic gradient descent to learn fast samplers by optimizing any given differentiable loss function over a minibatch of model samples. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We begin with a pre-trained DDPM and a family of $K$ -step samplers that we wish to optimize for the given DDPM. We parametrize this family’s degrees of freedom as simple transformations of trainable variables. We experiment with the following families in this paper, but emphasize that DDSS is applicable to any other family where model samples are differentiable with respect to the trainable variables: ",
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+        "type": "text",
+        "text": "• DDIM: we parametrize the posterior variances with √ $\\sigma _ { t } = \\sqrt { 1 - \\bar { \\alpha } _ { t - 1 } }$ sigmoid $\\left( v _ { t } \\right)$ , where $v _ { 1 } , . . . , v _ { K }$ are trainable variables (the $\\sqrt { 1 - \\bar { \\alpha } _ { t - 1 } }$ constant ensures real-valued mean coefficients; see the square root in Equation 6). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/f7b465e57a29651ce12353533765c18485729af556a7cb3ef9c66309d8c511a7.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of GGDM. To improve sample quality, our novel family of samplers combines information from all previous (noisier) images at every denoising step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• VARS: we parametrize the marginal variances of a DDPM as cumsum $( \\operatorname { s o f t m a x } ( [ \\pmb { v } ; 1 ] ) ) _ { t }$ instead of fixing them to $1 - \\bar { \\alpha } _ { t }$ . This ensures they are monotonically increasing with respect to $t$ (appending a one to ensure $K$ degrees of freedom). GGDM: a new family of non-Markovian samplers for diffusion models with more degrees of freedom illustrated in Figure 2 and defined in Section 4. We parametrize $\\mu _ { t u }$ and $\\sigma _ { t }$ of a GGDM for all $t$ as sigmoid functions of trainable variables. GGDM $\\mathbf { \\Gamma } + \\mathbf { P } \\mathbf { R } \\mathbf { E } \\mathbf { D }$ : we parametrize all the $\\mu _ { t u }$ and $\\sigma _ { t }$ identically to GGDM, but also learn the marginal coefficients with a cumsum $\\circ$ softmax parametrization (identical to VARS) instead of computing them via Theorem 1, as well as the coefficients that predict $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ from $a _ { t } { \\pmb x } _ { t } - b _ { t } { \\pmb \\epsilon }$ with $1 +$ softplus and softplus parametrizations.   \n• [family] ${ \\bf + } { \\bf \\Pi } { \\bf \\Pi } { \\bf \\bf \\Pi } \\mathbf { M } { \\bf E }$ : for any sampler family, we additionally parametrize the timesteps used to query the score model with a cumsum ◦ softmax parametrization (identical to VARS). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As we will show in the experiments, despite the fact that our pre-trained DDPMs are trained with discrete timesteps, learning the timesteps is still helpful. In principle, this should only be possible for DDPMs trained with continuous time sampling (Chen et al., 2021a; Song et al., 2021b; Kingma et al., 2021), but in practice we find that DDPMs trained with continuously embedded discrete timesteps are still well-behaved when applied at timesteps not present during training. We think this is due to the regularity of the sinusoidal positional encodings Vaswani et al. (2017) used in these model architectures and training with a sufficiently large number of timesteps $T$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 DIFFERENTIABLE SAMPLE QUALITY SCORES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We can differentiate through a stochastic sampler using the reparameterization trick, but the question of which objective to optimize still remains. Prior work has shown that optimizing log-likelihood can actually worsen sample quality and FID scores in the few-step regime (Watson et al., 2021; Song et al., 2021a). Thus, we instead design a perceptual loss which simply compares mean statistics between model samples and real samples in the neural network feature space. These types of objectives have been shown in prior work to better correlate with human perception of sample quality (Johnson et al., 2016; Heusel et al., 2017), which we also confirm in our experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We rely on the representations of the penultimate layer of a pre-trained InceptionV3 classifier (Szegedy et al., 2016) and optimize the Kernel Inception Distance (KID) (Binkowski et al., 2018). ´ Let $\\phi ( { \\pmb x } )$ denote the inception features of an image $_ { \\textbf { \\em x } }$ and $p _ { \\psi }$ represent a diffusion sampler with trainable parameters $\\psi$ . For a linear kernel, which works best in our experiments, the objective is: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e8bc2152fb9dc4c76fa96a5a968679039e6918a61accb5eb567c87c8b575c076.jpg",
+        "text": "$$\n\\mathcal { L } ( \\psi ) = \\underset { \\pmb { x } _ { p } \\sim p _ { \\psi } } { \\mathbb { E } } \\underset { \\pmb { x } _ { p } ^ { \\prime } \\sim p _ { \\psi } } { \\mathbb { E } } \\phi ( \\pmb { x } _ { p } ) ^ { \\top } \\phi ( \\pmb { x } _ { p } ^ { \\prime } ) - \\underset { \\pmb { x } _ { p } \\sim p _ { \\psi } } { \\mathbb { E } } \\underset { \\pmb { x } _ { q } \\sim q } { \\mathbb { E } } \\phi ( \\pmb { x } _ { p } ) ^ { \\top } \\phi ( \\pmb { x } _ { q } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "More generally, KID can be expressed as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a90d412372f4568bf2d4a5dffe2b62c79045c5af3cd67802e3e533927fb7b23.jpg",
+        "text": "$$\n{ \\mathcal L } _ { \\mathrm { K I D } } ( \\boldsymbol { \\psi } ) = \\left. \\underset { \\boldsymbol { x } _ { p } \\sim p _ { \\boldsymbol { \\psi } } } { \\mathbb { E } } f ^ { * } ( \\boldsymbol { x } _ { p } ) - \\underset { \\boldsymbol { x } _ { q } \\sim q } { \\mathbb { E } } f ^ { * } ( \\boldsymbol { x } _ { q } ) \\right. _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $f ^ { * } ( x ) = \\mathbb { E } _ { x _ { p } ^ { \\prime } \\sim p _ { \\psi } } k _ { \\phi } ( x , x _ { p } ^ { \\prime } ) - \\mathbb { E } _ { x _ { q } ^ { \\prime } \\sim q ( x _ { 0 } ) } k _ { \\phi } ( x , x _ { q } ^ { \\prime } )$ is the witness function for any differentiable, positive definite kernel $k$ , and $k _ { \\phi } ( \\dot { \\pmb x } , \\pmb y ) = k ( \\phi ( \\pmb x ) , \\phi ( \\pmb y ) )$ . Note that $f ^ { * }$ attains the supremum of the MMD. To enable stochastic gradient descent, we use an unbiased estimator of KID using a minibatch of $n$ model samples $\\pmb { x } _ { p } ^ { ( 1 ) } \\ldots \\pmb { x } _ { p } ^ { ( n ) } \\sim p _ { \\psi }$ and $n$ real samples $\\pmb { x } _ { q } ^ { ( 1 ) } \\dots \\pmb { x } _ { q } ^ { ( n ) } \\sim q$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5974c80fe4a40addfe8b6ea62dfcc3e9b6e41333ea387afe9001c10fbfe535df.jpg",
+        "text": "$$\n\\frac { 1 } { n ( n - 1 ) } \\sum _ { i \\neq j } ^ { n } k _ { \\phi } ( { \\pmb x } _ { p } ^ { ( i ) } , { \\pmb x } _ { p } ^ { ( j ) } ) - \\frac { 2 } { n ^ { 2 } } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } k _ { \\phi } ( { \\pmb x } _ { p } ^ { ( i ) } , { \\pmb x } _ { q } ^ { ( j ) } ) + c ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $c$ is constant in $\\psi$ . Since the sampling chain of any Gaussian diffusion process admits using the reparametrization trick, our loss function is fully differentiable with respect to the trainable variables $\\psi$ . We empirically find that using the perceptual features is crucial; i.e., by trying $\\phi ( { \\pmb x } ) = { \\pmb x }$ to compare images directly on pixel space rather than neural network feature space (as above), we observe that our method makes samples consistently worsen in apparent quality as training progresses. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 GRADIENT REMATERIALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In order for backpropagation to be feasible under reasonable memory constraints, one final problem must be addressed: since we are taking gradients with respect to model samples, the cost in memory to maintain the state of the forward pass scales linearly with the number of inference steps, which can quickly become unfeasible considering the large size of typical DDPM architectures. To address this issue, we use gradient rematerialization (Kumar et al., 2019b). Instead of storing a particular computation’s output from the forward pass required by the backward pass computation, we recompute it on the fly. To trade $\\mathcal O ( K )$ memory cost for $\\mathcal O ( K )$ computation time, we simply rematerialize calls to the pre-trained DDPM (i.e., the estimated score function), but keep in memory all the progressively denoised images from the sampling chain. In JAX (Bradbury et al., 2018), this is trivial to implement by simply wrapping the score function calls with jax.remat. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 GENERALIZED GAUSSIAN DIFFUSION MODELS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We now present Generalized Gaussian Diffusion Model (GGDM), our novel family of Gaussian diffusion processes that includes DDIM as a special case mentioned in section 3. We define a joint distribution with no independence assumptions ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/573967dde19f9f874da0c994bbdf32e6718672b7cb8d462f5c467f655c59ce4d.jpg",
+        "text": "$$\nq _ { \\mu , \\sigma } ( \\pmb { x } _ { 0 } , . . . , \\pmb { x } _ { T } ) = q ( \\pmb { x } _ { 0 } ) q ( \\pmb { x } _ { T } | \\pmb { x } _ { 0 } ) \\prod _ { t = 1 } ^ { T - 1 } q _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { > t } , \\pmb { x } _ { 0 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where the new factors are defined as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/42160437fd50763c1745fbebf5056ff42cd23f96c6339df0411367a6937340e8.jpg",
+        "text": "$$\nq _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { > t } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\bigg | \\sum _ { u \\in S _ { t } } \\mu _ { t u } \\pmb { x } _ { u } , \\sigma _ { t } ^ { 2 } \\pmb { I } _ { d } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(letting $S _ { t } ~ = ~ \\{ 0 , . . . , T \\} ~ \\backslash ~ \\{ 1 , . . . , t \\}$ for notation compactness), with $\\sigma _ { t }$ and $\\mu _ { t u }$ free parameters $\\forall t \\in \\{ 1 , . . . , T \\} , u \\in S _ { t }$ . In other words, when predicting the next, less noisy image, the sampler can take into account all the previous, noisier images in the sampling chain, and similarly to DDIM, we can also control the sampler’s variances. As we prove in the appendix (A.2), this construction admits Gaussian marginals, and we can differentiably compute the marginal coefficients from arbitrary choices of $\\mu$ and $\\sigma$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Given some $t \\in \\{ 1 , . . . , T \\}$ , let $a _ { t u } ^ { ( 1 ) } \\ = \\ \\mu _ { t u } \\ \\forall u \\in \\ S _ { t }$ and $v _ { t } ^ { ( 1 ) } = \\sigma _ { t } ^ { 2 }$ . For each , recursively define ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/235fc0a859660837f93b44fa63fbcbddc6e03ed958904366b298fe2fb1b80831.jpg",
+        "text": "$$\na _ { t u } ^ { ( i + 1 ) } = a _ { t , t + i } ^ { ( i ) } \\mu _ { t + i , u } + a _ { t u } ^ { ( i ) } \\forall u \\in S _ { t + i } \\quad \\mathrm { a n d } \\quad v _ { t } ^ { ( i + 1 ) } = v _ { t } ^ { ( i ) } + \\left( a _ { t , t + i } ^ { ( i ) } \\sigma _ { t + i } \\right) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Then, it follows that ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8185d39c066238d2d7f38fe1c7c073d4ac012d6a88ba6a6696f62584ca0a0763.jpg",
+        "text": "$$\nq _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { > t + i } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\middle | \\sum _ { \\pmb { u \\in S } _ { t + i } } a _ { t u } ^ { ( i + 1 ) } \\pmb { x } _ { u } , v _ { t } ^ { ( i + 1 ) } \\pmb { I } _ { d } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In other words, instead of letting the $\\beta _ { t }$ (or equivalently, the $\\bar { \\alpha } _ { t }$ ) define the forward process as done by a usual DDPM, the GGDM family lets the $\\mu _ { t u }$ and $\\sigma _ { t }$ define the process. In particular, an immediate corollary of Theorem 1 is that the marginal coefficients are given by ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/115d065b8b640800dccea91bf1c522055e6d095eab19d7f54140179b87b3c0d7.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\Big | a _ { t 0 } ^ { ( T - t + 1 ) } \\pmb { x } _ { 0 } , v _ { t } ^ { ( T - t + 1 ) } \\pmb { I } _ { d } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The reverse process is thus defined as $\\begin{array} { r } { p ( { \\pmb x } _ { T } ) \\prod _ { t = 1 } ^ { T } p ( { \\pmb x } _ { t - 1 } | { \\pmb x } _ { t } ) } \\end{array}$ with $p ( \\pmb { x } _ { T } ) \\sim \\mathcal { N } ( \\mathbf { 0 } , \\pmb { I } _ { d } ) )$ and ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff958d13aab89336c07f53644913922a5204232953eccc5fd0b572e4e9b1dd6e.jpg",
+        "text": "$$\np _ { \\theta } ( \\mathbf { x } _ { t } | \\mathbf { x } _ { > t } ) = q _ { \\mu , \\sigma } \\left( x _ { t } | \\mathbf { x } _ { > t } , \\hat { \\mathbf { x } } _ { 0 } = \\frac { 1 } { a _ { t 0 } ^ { ( T - t + 1 ) } } \\left( \\mathbf { x } _ { t } - \\sqrt { v _ { t } ^ { ( T - t + 1 ) } } \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , t ) \\right) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f5252ac0b3b35058e4d25fc2797ea22da0c360c8701670587f913f1d13bab59e.jpg",
+        "table_caption": [
+            "Table 1: FID and IS scores for DDSS against baseline methods for a DDPM trained on CIFAR10 with the $L _ { \\mathrm { s i m p l e } }$ objective proposed by (Ho et al., 2020). FID scores (lower is better) are the numbers at the left of each entry, and IS scores (higher is better) are at the right. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>SamplerK</td><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td></tr><tr><td>DDPM (linear stride)</td><td>84.27/5.396</td><td>43.39/7.034</td><td>31.40/7.609</td><td>25.94/7.879</td><td>22.60/8.043</td></tr><tr><td>DDPM(quadratic stride)</td><td>76.25 / 5.435</td><td>42.03 /6.965</td><td>27.78 /7.714</td><td>20.225 /8.128</td><td>16.17 / 8.350</td></tr><tr><td>DDIM (linear stride)</td><td>44.41 / 6.750</td><td>19.11 / 7.965</td><td>14.06 /8.190</td><td>11.82 /8.420</td><td>10.52 / 8.512</td></tr><tr><td>DDIM(quadratic stride)</td><td>32.66 /7.090</td><td>13.62/8.190</td><td>9.318 /8.495</td><td>7.500 / 8.641</td><td>6.560 / 8.759</td></tr><tr><td>GGDM+PRED+TIME</td><td>13.77 /8.520</td><td>8.227 / 8.903</td><td>6.115 /9.050</td><td>4.722/9.261</td><td>4.250 /9.186</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/f034ddcf8593b9daa07f390fc77a4069e1be03f4862dee267cd680337b0295e7.jpg",
+        "table_caption": [
+            "Table 2: FID / IS scores for DDSS against baseline methods for a DDPM trained on ImageNet 64x64 with the $L _ { \\mathrm { h y b r i d } }$ objective proposed by Nichol & Dhariwal (2021). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>SamplerK</td><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td></tr><tr><td>DDPM (linear stride)</td><td>122.0/5.878</td><td>58.78/10.67</td><td>39.30/13.22</td><td>31.36/14.72</td><td>26.36/15.71</td></tr><tr><td>DDPM(quadratic stride)</td><td>394.8 / 1.351</td><td>129.5 / 5.997</td><td>80.10 /9.595</td><td>61.34 / 11.60</td><td>49.60 / 13.01</td></tr><tr><td>DDIM (linear stride)</td><td>135.4 /5.898</td><td>40.70 / 12.225</td><td>28.54 /13.99</td><td>24.225 / 14.75</td><td>22.13 /15.16</td></tr><tr><td>DDIM (quadratic stride)</td><td>409.1/1.380</td><td>148.6 / 5.533</td><td>67.65 /9.842</td><td>45.60 / 11.99</td><td>36.11 /13.225</td></tr><tr><td>GGDM+PRED+TIME</td><td>55.14 /12.90</td><td>37.32/14.76</td><td>24.69/17.225</td><td>20.69 /17.92</td><td>18.40 /18.12</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 IGNORING THE MATCHING MARGINALS CONDITION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Unlike DDIM, the GGDM family does not guarantee that the marginals of the new forward process match that of the original DDPM. We empirically find, however, that this condition can often be too restrictive and better samplers exist where the marginals $q _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } ) =$ $\\mathcal { N } \\left( \\pmb { x } _ { t } \\Big | a _ { t 0 } ^ { ( T - t + 1 ) } \\pmb { x } _ { 0 } , v _ { t } ^ { ( T - t + 1 ) } \\pmb { I } _ { d } \\right)$ v(T −t+1)t Id\u0011 of the new forward process differ from the original DDPM’s marginals. We verify this empirically by applying DDSS to both the family of DDIM sigmas and DDPM variances (“VARS” in Section 3): both sampler families have the same number of parameters (the reverse process variances), but the latter does not adjust the mean coefficients like DDIM to ensure matching marginals and still achieves similar or better scores than the former across sample quality metrics (and even outperforms the $\\mathrm { D D I M } ( \\eta = 0 )$ baseline); see Section 5.2. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In order to emphasize that our method is compatible with any pre-trained DDPM, we apply our method on pre-trained DDPM checkpoints from prior work. Specifically, we experiment with the DDPM trained by Ho et al. (2020) with $L _ { \\mathrm { s i m p l e } }$ on CIFAR10, as well as a DDPM following the exact configuration of Nichol & Dhariwal (2021) trained on ImageNet 64x64 (Deng et al., 2009) with their $L _ { \\mathrm { h y b r i d } }$ objective (with the only difference being that we trained the latter ourselves for 3M rather than 1.5M steps). Both of these models utilize adaptations of the UNet architecture (Ronneberger et al., 2015) that incorporate self-attention layers (Vaswani et al., 2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate all of our models on both FID and Inception Score (IS) (Salimans et al., 2016), comparing the samplers discovered by DDSS against DDPM and DDIM baselines with linear and quadratic strides. As previously mentioned, more recent methods for fast sampling are outperformed by DDIM when the budget of inference steps is as small as those we utilize in this work (5, 10, 15, 20, 25). All reported results on both of these approximate sample quality metrics were computed by comparing 50K model and training data samples, as is standard in the literature. Also as is standard, IS scores are computed 10 times, each time on 5K samples, and then averaged. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In all of our experiments, we optimize the DDSS objective presented in Section 3.1 with the following hyperparameters: ",
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+    },
+    {
+        "type": "text",
+        "text": "1. For every family of models we search over, we initialize the degrees of freedom such that training begins with a sampler matching DDPM with $K$ substeps following Song et al. (2020); Nichol & Dhariwal (2021). ",
+        "bbox": [
+            214,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/7fcb584c4f595cd06c0beac00e70900ed752f78dd02d4a171700a53959364c8b.jpg",
+        "table_caption": [
+            "Table 3: FID / IS scores for the KID kernel ablation on CIFAR10. When not learning the timesteps, we fix them to a quadratic stride, as Table 1 shows this performs best on CIFAR10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>SamplerK</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>25</td></tr><tr><td rowspan=1 colspan=1>DDSS (linear kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM)</td><td rowspan=1 colspan=1>13.77 /8.52014.26 /8.40612.85 /8.38314.45 /8.281</td><td rowspan=1 colspan=1>8.227 /8.9038.617 /8.8427.858 / 8.8958.154 /8.892</td><td rowspan=1 colspan=1>6.115 /9.0505.939 /9.0356.265 /9.0757.045 /8.939</td><td rowspan=1 colspan=1>4.722 /9.2614.893 /9.1535.367 /9.1365.477 /9.183</td><td rowspan=1 colspan=1>4.250 /9.1864.574 /9.1454.887 /9.2294.815 /9.189</td></tr><tr><td rowspan=1 colspan=1>DDSS (cubic kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM</td><td rowspan=1 colspan=1>14.41 / 8.52714.39 /8.40112.35 / 8.40614.57 /8.297</td><td rowspan=1 colspan=1>8.2257 /9.0078.977 /8.8707.879 /8.8528.2252/8.836</td><td rowspan=1 colspan=1>5.895 /9.0366.517 / 8.9706.682 /8.9996.727 /8.904</td><td rowspan=1 colspan=1>4.932/9.0924.915 /9.1325.639 /9.0585.569 /9.177</td><td rowspan=1 colspan=1>4.278/9.2864.471 /9.2474.631/9.1894.668 /9.192</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2. We apply gradient updates using the Adam optimizer (Kingma & Ba, 2015). We sweeped over the learning rate and used $\\lambda = 0 . 0 0 0 5$ . We did not sweep over other Adam hyperparameters and kept β1 = 0.9, β2 = 0.999, and \u000f = 1 × 10−8. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3. We tried batch sizes of 128 and 512 and opted for the latter, finding that it leads to better sample quality upon inspection. Since the loss depends on averages over examples as our experiments are on unconditional generation, this choice was expected. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4. We run all of our experiments for 50K training steps and evaluate the discovered samplers at this exact number of training steps. We did not sweep over this value. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We include our main results in Table 1 for CIFAR10 and Table 2 for ImageNet 64x64, comparing DDSS applied to GGDM $+ \\mathrm { P R E D + T I M E }$ against DDPM and DDIM baselines with linear and quadratic strides. All models use a linear kernel, i.e., $k _ { \\phi } ( { \\pmb x } , { \\pmb y } ) = \\phi ( { \\pmb x } ) ^ { \\top } \\phi ( { \\pmb y } )$ , which we found to perform slightly better than the cubic kernel used by Binkowski et al. (2018) (we ablate this in ´ section 5.1). We omit the use of the learned variances of the ImageNet $6 4 \\mathrm { x } 6 4$ model (i.e., following Nichol & Dhariwal (2021)), as we search for the variances ourselves via DDSS. We include samples for 5, 10 and 25 steps comparing the strongest DDIM baselines to $\\mathrm { D D S S } + \\mathrm { G G D M }$ with a learned stride; see Figures 1 and 3. We include additional ImageNet $6 4 \\mathrm { x } 6 4$ samples (A.1) and results for larger resolution datasets (A.4) in the appendix. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 ABLATIONS FOR KID KERNEL AND GGDM VARIANTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As our approach is compatible with any choice of KID kernel, we experiment with different choices of kernels. Namely, we try the simplest possible linear kernel, $k _ { \\phi } ( \\dot { \\pmb { x } } , \\pmb { y } ) = \\phi ( \\pmb { x } ) ^ { \\top } \\phi ( \\pmb { y } )$ , as well as the cubic kernel $\\begin{array} { r } { k _ { \\phi } ( \\pmb { x } , \\pmb { y } ) = \\left( \\frac { 1 } { d } \\phi ( \\pmb { x } ) ^ { \\top } \\phi ( \\pmb { y } ) + 1 \\right) ^ { 3 } } \\end{array}$ used by Binkowski et al. (2018). We compare the ´ performance of these two kernels, as well as different variations of GGDM (i.e., with and without TIME and PRED as defined in Section 3). Results are included for CIFAR10 across all budgets in Table 3. We also include a smaller version of this ablation for ImageNet 64x64 in the appendix (A.3). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results in this ablation show that the contributions of the linear kernel, timestep learning, and the empirical choice of learning the coefficients that predict $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ all slightly contribute to better FID and IS scores. Importantly, however, removing any of these additions still allows us to comfortably outperform the strongest baselines. See also the results on LSUN in the appendix A.4, which also do not include these additional trainable variables. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 SEARCH SPACE ABLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            800,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Now, in order to further demonstrate the key importance of optimizing our GGDM family to find high-fidelity samplers, we also apply DDSS to the less general DDIM and VARS families. We show that, while we still attain better scores than a regular DDPM, searching these less flexible families of samplers does not yield improvements as significant as with out novel GGDM family. In particular, optimizing the DDIM sigma coefficients does not outperform the corresponding $\\mathrm { D D I M } ( \\eta = 0 )$ baseline on CIFAR10, which is not a surprising result as Song et al. (2020) show empirically that most choices of the $\\sigma _ { t }$ degrees of freedom lead to worse FID scores than setting them all to 0. These results also show that optimizing the VARS can outperform DDSS applied to the DDIM family, and even the strongest $\\mathrm { D D I M } ( \\eta = 0 )$ baselines for certain budgets, justifying our choice of not enforcing the marginals to match (as discussed in Section 4.1). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4a8a7d5257cee010c0bfdad5ea933f271f2808140610c2bbc64badf7b82fe1f2.jpg",
+        "table_caption": [
+            "Table 4: FID / IS scores for the DDSS search space ablation on CIFAR10. All runs fix the timesteps to a quadratic stride and use a linear kernel except for the last row (we only include the GGDM results for ease of comparison). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>SamplerK</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>25</td></tr><tr><td rowspan=1 colspan=1>DDIM(n = 0)</td><td rowspan=1 colspan=1>32.66/7.090</td><td rowspan=1 colspan=1>13.62/8.190</td><td rowspan=1 colspan=1>9.318/ 8.495</td><td rowspan=1 colspan=1>7.500 / 8.641</td><td rowspan=1 colspan=1>6.560/ 8.759</td></tr><tr><td rowspan=1 colspan=1>DDSSVARSDDIM</td><td rowspan=1 colspan=1>33.08 /7.09632.61/7.084</td><td rowspan=1 colspan=1>15.33 /8.55916.29 /7.966</td><td rowspan=1 colspan=1>9.693 /8.84511.31/ 8.372</td><td rowspan=1 colspan=1>7.297 / 8.9249.120 /8.563</td><td rowspan=1 colspan=1>6.172/9.0577.853 /8.644</td></tr><tr><td rowspan=1 colspan=1>GGDMGGDM+PRED+TIME</td><td rowspan=1 colspan=1>14.45/8.28113.77 /8.520</td><td rowspan=1 colspan=1>8.154/8.8928.227 / 8.903</td><td rowspan=1 colspan=1>7.045 / 8.9396.115 /9.050</td><td rowspan=1 colspan=1>5.477 /9.1834.722 /9.261</td><td rowspan=1 colspan=1>4.815/9.1894.250 /9.186</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "When applied to a sufficiently flexible family (such as the GGDM family proposed in this work), DDSS consistently finds samplers that achieve better image generation quality than the strongest baselines in the literature for very few steps. This is qualitatively apparent in non-cherrypicked samples (e.g., $\\mathrm { D D I M } ( \\eta = 0 )$ tends to generate blurrier images and with less background details as the budget decreases), and multiple quantitative sample quality metrics (FID and IS) also reflect these results. Still, we observe limitations to our method. Finding samplers with inference step budgets as small as $K < 1 0$ that have little apparent loss in quality remains challenging with our proposed search family. And, while on CIFAR10 the metrics indicate significant relative improvement over sample quality metrics, the relative improvement on ImageNet 64x64 is less pronounced. We hypothesize that this is an inherent difficulty of ImageNet due to its high diversity of samples, and that in order to retain sample quality and diversity, it might be impossible to escape some minimum number of inference steps with score-based models as they might be crucial to mode-breaking. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Beyond the empirical gains of applying our procedure, our findings shed further light into properties of pre-trained score-based generative models. First, we show that without fine-tuning a DDPM’s parameters, these models are already capable of producing high-quality samples with very few inference steps, though the default DDPM sampler in this regime is usually suboptimal when using a few-step sampler. We further show that better sampling paths exist, and interestingly, these are determined by alternative variational lower bounds to the data distribution that make use of the score-based model but do not necessarily share the same marginals as the original DDPM forward process. Our findings thus suggest that enforcing this marginal-sharing constraint is unnecessary and can be too restrictive in practice. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 OTHER RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            690,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Besides DDIM (Song et al., 2020), there have been more recent attempts at reducing the number of inference steps for DDPMs. Jolicoeur-Martineau et al. (2021) proposed a dynamic step size SDE solver that can reduce the number of calls to the modeled score function to $\\sim 1 5 0$ on CIFAR10 (Krizhevsky et al., 2009) with minimal cost in FID scores, but quickly falls behind $\\mathrm { D D I M } ( \\eta = 0 )$ with as many as 50 steps. Watson et al. (2021) proposed a dynamic programming algorithm that chooses log-likelihood optimal strides, but find that log-likelihood reduction has a mismatch with FID scores, particularly with in the very few step regime, also falling behind $\\mathrm { D D I M } ( \\eta = 0 )$ in this front. Other methods that have been shown to help sample quality in the few-step regime include non-Gaussian variants of diffusion models (Nachmani et al., 2021) and adaptively adjusting the sampling path by introducing a noise level estimating network (San-Roman et al., 2021), but more thorough evaluation of sample quality achieved by these approaches is needed with budgets as small as those considered in this work. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Other approaches to sampling DDPMs have also been recently proposed, though not for the explicit purpose of efficient sampling. Song et al. (2021b) derive a reverse SDE that, when discretized, uses different coefficients than the ancestral samplers considered in this work. The same authors also derive “corrector” steps, which introduce additional calls to the pre-trained DDPM as a form of gradient ascent (Langevin dynamics) that help with quality but introduce computation cost, as well as an alternative sampling procedure using a probability flow ODE that shares the same marginals as the DDPM’s original forward process. Huang et al. (2021) generalize this family of samplers to a “plug-in reverse SDE” that interpolates between a probability flow ODE and the reverse SDE, similarly to how the DDIM $\\eta$ interpolates between an implicit probabilistic model and a stochastic reverse process. Our proposed search family includes discretizations of most of these cases for Gaussian processes, notably missing corrector steps, where reusing a single timestep is considered. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/32e14402d05adeeb690961580b44df8d5fbd8f1bb8eac3367ce846dcaf8eebe6.jpg",
+        "image_caption": [
+            "Figure 3: Non-cherrypicked samples for a DDPM trained on CIFAR10, comparing the strongest DDIM $\\mathbf { \\eta } _ { \\eta } = 0 ,$ ) baseline and our approach. All samples were generated with the same random seeds. For reference, we include DDPM samples using all 1,000 steps (top right) and real images (bottom right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "8 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We propose Differentiable Diffusion Sampler Search (DDSS), a method for finding few-step samplers for Denoising Diffusion Probabilistic Models. We show how to optimize a perceptual loss over a space of diffusion processes that makes use of a pre-trained DDPM’s samples by leveraging the reparametrization trick and gradient rematerialization. Our results qualitatively and quantitatively show that DDSS is able to significantly improve sample quality for unconditional image generation over prior methods on efficient DDPM sampling. The success of our method hinges on searching a novel, wider family of Generalized Gaussian Diffusion Model (GGDM) than those identified in prior work (Song et al., 2020). DDSS does not fine-tune the pre-trained DDPM, only needs to be applied once, has few hyperparameters, and does not require re-training the DDPM. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our findings suggest future directions to further reduce the number of inference steps while retaining high fidelity in generated samples. For instance, it is plausible to use different representations for the perceptual loss instead of those of a classifier, e.g., use representations from an unsupervised model such as SimCLR (Chen et al., 2020), to using internal representations learned by the pre-trained DDPM itself, which would eliminate the burden of additional computation. Moreover, considering the demonstrated benefits of applying DDSS to our proposed GGDM family of samplers (as opposed to narrower families like DDIM), we motivate future work on identifying more general families of samplers and investigating whether they help uncover even better samplers or lead to overfitting. Finally, identifying other variants of perceptual losses (e.g., that do not sample from the model), or alternative optimization strategies (e.g., gradient-free methods) that lead to similar results is important future work. This would make DDSS itself a more efficient procedure, as gradient-based optimization of our proposed loss requires extensive memory or computation requirements to backpropagate through the whole sampling chain. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
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+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 12
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+    {
+        "type": "text",
+        "text": "A.1 ADDITIONAL IMAGENET 64X64 SAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            509,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We provide additional samples for our results on ImageNet $6 4 \\mathrm { x } 6 4$ . The DDPM and $\\mathrm { D D I M } ( \\eta = 0 )$ samples (left and middle, respectively) use a linear stride, while our DDSS $^ +$ GGDM samples (right) use a learned stride. ",
+        "bbox": [
+            174,
+            159,
+            825,
+            202
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/1837841f8ba78d46089b6644adcd4a9a922cbe225c3310a07840389f63dad545.jpg",
+        "image_caption": [
+            "Figure A.1: Additional samples on ImageNet $6 4 \\mathrm { x } 6 4$ . For reference, we include DDPM samples with all 4,000 steps (bottom left) and real samples (bottom middle). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            209,
+            805,
+            847
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 PROOF FOR THEOREM 1 ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Given some $t \\in \\{ 1 , . . . , T \\}$ , let $a _ { t u } ^ { ( 1 ) } \\ = \\ \\mu _ { t u } \\ \\forall u \\in \\ S _ { t }$ and $v _ { t } ^ { ( 1 ) } = \\sigma _ { t } ^ { 2 }$ . For each , recursively define ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/614aef9881d501c3d331c019e3592d77819accf9ce0f2fc619b180efb38f9cae.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\bullet \\ a _ { t u } ^ { ( i + 1 ) } = a _ { t , t + i } ^ { ( i ) } \\mu _ { t + i , u } + a _ { t u } ^ { ( i ) } \\forall u \\in S _ { t + i } } \\\\ & { \\bullet \\ v _ { t } ^ { ( i + 1 ) } = v _ { t } ^ { ( i ) } + \\left( a _ { t , t + i } ^ { ( i ) } \\sigma _ { t + i } \\right) ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            167,
+            495,
+            223
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then, it follows that ",
+        "bbox": [
+            173,
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+            310,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39af935af7295c62a1cf6cfb57c99ca9e9c6cb5ff7f8ffb3c8be48768c06cf2c.jpg",
+        "text": "$$\nq _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { > t + i } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\bigg | \\sum _ { u \\in S _ { t + i } } a _ { t u } ^ { ( i + 1 ) } \\pmb { x } _ { u } , v _ { t } ^ { ( i + 1 ) } \\pmb { I } _ { d } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            251,
+            694,
+            303
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let us prove this result with mathematical induction. Note that, for each such $t$ , we have by definition that ",
+        "bbox": [
+            171,
+            313,
+            821,
+            342
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f88e9dbb5d3f2e5e36357a1a63cc97874d57b8f6588fd865b2753c8b05c4c723.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\mu , \\sigma } ( \\pmb { x } _ { t + 1 } | \\pmb { x } _ { > t + 1 } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\Big | \\sum _ { u \\in S _ { t + 1 } } \\mu _ { t + 1 , u } \\pmb { x } _ { u } , \\sigma _ { t + 1 } ^ { 2 } \\pmb { I } _ { d } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            345,
+            632,
+            372
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "and ",
+        "bbox": [
+            173,
+            382,
+            202,
+            395
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8c8a56618f7377286971055fbd2aa174ccba213e054b054df291f650858e4da1.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\mu , \\sigma } ( { \\pmb x } _ { t } | { \\pmb x } _ { t + 1 } , { \\pmb x } _ { > t + 1 } , { \\pmb x } _ { 0 } ) = { \\mathcal N } \\left( { \\pmb x } _ { t } \\big | \\sum _ { u \\in S _ { t } } \\mu _ { t u } { \\pmb x } _ { u } , \\sigma _ { t } ^ { 2 } I _ { d } \\right) = { \\mathcal N } \\left( { \\pmb x } _ { t } \\Big | \\sum _ { u \\in S _ { t } } a _ { t u } ^ { ( 1 ) } { \\pmb x } _ { u } , v _ { t } ^ { ( 1 ) } I _ { d } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            402,
+            805,
+            430
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Therefore, following Svensen & Bishop (2007) (2.115), by prior conjugacy it follows that ´ ",
+        "bbox": [
+            173,
+            441,
+            763,
+            458
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/543fae8ff59668a86dae9ba72ebb35e5aa8af0f0c65fbea837651a27a061898d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { ( { \\pmb x } _ { t } | { \\pmb x } _ { > t + 1 } , { \\pmb x } _ { 0 } ) = \\mathcal { N } ( { \\pmb x } _ { t } \\Big | a _ { t , t + 1 } ^ { ( 1 ) } \\sum _ { u \\in S _ { t + 1 } } \\mu _ { t + 1 , u } { \\pmb x } _ { u } + \\sum _ { u \\in S _ { t + 1 } } a _ { t u } ^ { ( 1 ) } , ( v _ { t } ^ { ( 1 ) } + a _ { t , t + 1 } ^ { ( 1 ) } \\sigma _ { t + 1 } ^ { 2 } a _ { t , t + 1 } ^ { ( 1 ) } ) I _ { d _ { 1 } ^ { \\prime } }  } \\\\ & { \\qquad = \\mathcal { N } ( { \\pmb x } _ { t } \\Big | \\sum _ { u \\in S _ { t + 1 } } ( a _ { t , t + 1 } ^ { ( 1 ) } \\mu _ { t + 1 , u } + a _ { t u } ^ { ( 1 ) } ) { \\pmb x } _ { u } , v _ { t } ^ { ( 2 ) } I _ { d } )  } \\\\ & { \\qquad = \\mathcal { N } ( { \\pmb x } _ { t } \\Big | \\sum _ { u \\in S _ { t + 1 } } a _ { t u } ^ { ( 2 ) } { \\pmb x } _ { u } , v _ { t } ^ { ( 2 ) } I _ { d } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            202,
+            463,
+            861,
+            547
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This proves the base case for our induction argument. Now, let us prove the inductive step. Suppose there exists some integer $j \\in \\{ 1 , . . . , T - t + 1 \\}$ such that ",
+        "bbox": [
+            173,
+            558,
+            823,
+            588
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6bc385bbba9eb4539578d54b6e5880d320a7c91860ddc71e934527f37b27571a.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { > t + j } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\middle | \\sum _ { \\boldsymbol { u } \\in S _ { t + j } } a _ { t \\boldsymbol { u } } ^ { ( j + 1 ) } \\pmb { x } _ { \\boldsymbol { u } } , v _ { t } ^ { ( j + 1 ) } \\pmb { I } _ { d } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            593,
+            705,
+            621
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "By definition, we already know $q ( \\pmb { x } _ { t + j + 1 } | \\pmb { x } _ { > t + j + 1 } , \\pmb { x } _ { 0 } )$ , so we have ",
+        "bbox": [
+            176,
+            626,
+            625,
+            642
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3561f87e2232c9748ad1639896c8b2e3311be47d8e502a3e8031492ad07f5c3.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\mu , \\sigma } ( \\pmb { x } _ { t + j + 1 } | \\pmb { x } _ { > t + j + 1 } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t + j + 1 } \\middle | \\sum _ { u \\in S _ { t + j + 1 } } \\mu _ { t + j + 1 , u } \\pmb { x } _ { u } , \\sigma _ { t + j + 1 } ^ { 2 } \\pmb { I } _ { d } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            248,
+            647,
+            772,
+            675
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "and (rewriting the inductive hypothesis) ",
+        "bbox": [
+            173,
+            683,
+            436,
+            698
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f66e8fd094e4c040bcb89f855e0a3052242d68e14b5077e3bb7b0239bcfacfb.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { t + j + 1 } , \\pmb { x } _ { > t + j + 1 } , \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\Big | \\sum _ { \\pmb { u \\in S } _ { t + j } } a _ { t u } ^ { ( j + 1 ) } \\pmb { x } _ { u } , v _ { t } ^ { ( j + 1 ) } \\pmb { I } _ { d } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            707,
+            710,
+            734
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Therefore, by prior conjugacy again, it follows that ",
+        "bbox": [
+            173,
+            739,
+            509,
+            755
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c7f5dcb62110321491ec0d33a01f51312dd8769a86fb3bfa575e141e2fda6346.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad q _ { \\mu , \\sigma } ( \\boldsymbol { x } _ { t } | \\boldsymbol { x } _ { > t + j + 1 } , \\boldsymbol { x } _ { 0 } ) } \\\\ & { = \\mathcal { N } \\left( \\boldsymbol { x } _ { t } \\Big | a _ { t , t + j + 1 } ^ { ( j + 1 ) } \\sum _ { u \\in S _ { t + j } } \\mu _ { t + j + 1 , u } \\boldsymbol { x } _ { u } + \\sum _ { u \\in S _ { t + j } } a _ { t u } ^ { ( j + 1 ) } , \\left( v _ { t } ^ { ( j + 1 ) } + a _ { t , t + j + 1 } ^ { ( j + 1 ) } \\sigma _ { t + j + 1 } ^ { 2 } a _ { t , t + j + 1 } ^ { ( j + 1 ) } \\right) \\boldsymbol { I } _ { d } \\right) } \\\\ & { = \\mathcal { N } \\left( \\boldsymbol { x } _ { t } \\Big | \\sum _ { u \\in S _ { t + j } } \\left( a _ { t , t + j + 1 } ^ { ( j + 1 ) } \\mu _ { t + j + 1 , u } + a _ { t u } ^ { ( j + 1 ) } \\right) \\boldsymbol { x } _ { u } , v _ { t } ^ { ( j + 2 ) } \\boldsymbol { I } _ { d } \\right) } \\\\ & { = \\mathcal { N } \\left( \\boldsymbol { x } _ { t } \\Big | \\sum _ { u \\in S _ { t + j + 1 } } a _ { t u } ^ { ( j + 2 ) } \\boldsymbol { x } _ { u } , v _ { t } ^ { ( j + 2 ) } \\boldsymbol { I } _ { d } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            760,
+            839,
+            863
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This concludes the proof of the inductive step. Hence, we have proven the result for any $i \\in$ $\\{ 1 , . . . , T - t \\}$ . In particular, ",
+        "bbox": [
+            169,
+            866,
+            825,
+            896
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc528ce694a815ada4f6e8d5315520c51e0c0fb6b8beae9a90775e39feb52ba1.jpg",
+        "text": "$$\n\\begin{array} { r } { q _ { \\mu , \\sigma } ( \\pmb { x } _ { t } | \\pmb { x } _ { 0 } ) = \\mathcal { N } \\left( \\pmb { x } _ { t } \\middle | a _ { t 0 } ^ { ( T - t + 1 ) } \\pmb { x } _ { 0 } , v _ { t } ^ { ( T - t + 1 ) } \\pmb { I } _ { d } \\right) . \\quad \\prod } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            901,
+            684,
+            929
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3 ADDITIONAL ABLATION OF KID KERNEL AND GGDM VARIANTS FOR IMAGENET 64X64 ",
+        "bbox": [
+            169,
+            103,
+            782,
+            132
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We also ran a smaller version of the ablation results presented in Section 5.1, but for ImageNet $6 4 \\mathrm { x } 6 4$ instead of CIFAR10, as these are more computationally intensive to do a full grid search. Results for a step budget $K = 1 5$ are included below. When not learning the timesteps, we fix them to a linear stride, as Table 2 shows this performs best on ImageNet $6 4 \\mathrm { x } 6 4$ . ",
+        "bbox": [
+            174,
+            154,
+            825,
+            210
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/4b3078e48185bc7d8f84a15aee10167ad5deebbd730a8a14ed3b2bd0bbef10b1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>SamplerK</td><td rowspan=1 colspan=1>15</td></tr><tr><td rowspan=1 colspan=2>DDSS (linear kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM</td><td rowspan=1 colspan=1>24.69 /17.22527.08 /16.4425.73 /17.2728.34 /16.63</td></tr><tr><td rowspan=2 colspan=2>DDSS (cubic kernel)GGDM+PRED+TIMEGGDM+PREDGGDM+TIMEGGDM</td><td rowspan=2 colspan=1>26.52 /16.2927.82/16.326.87 /16.9928.83 /16.32</td></tr><tr></tr></table>",
+        "bbox": [
+            357,
+            247,
+            638,
+            392
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.4 RESULTS ON LARGER RESOLUTION DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            469,
+            535,
+            483
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We include results for LSUN (Yu et al., 2015) bedrooms and churches at the $1 2 8 \\mathrm { x } 1 2 8$ resolution. We trained the models for 400K and 200K steps (respectively), and all other hyperparameters are identical: we use the Adam optimizer with learning rate 0.0003 (linearly warmed up for the first 1000 training steps), batch size 2048, gradient clipping at norms over 1.0, dropout of 0.1, and EMA over the weights with decay rate 0.9999. We train the models using a linear stride of 1000 evenlyspaced timesteps, fixing the log-signal-to-noise-ratio schedule to a cosine function monotonically decreasing from 20 to -20. The ELBO is reweighted with $L _ { \\mathrm { s i m p l e } }$ following Ho et al. (2020), but we additionally reweight each term by $\\mathrm { m a x } ( 1 , \\mathrm { S N R } )$ which we found to be slightly helpful in resulting FID scores (note this is equivalent to minimizing the worst mean squared error between either the $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ or $\\epsilon$ ). The UNet employs five down/up-sampling resolutions with $7 6 8 \\times ( 1 , 2 , 4 , 6 , 8 )$ respective channels, 3 ResNet blocks per resolution, and spatial self-attention at the 3 smallest resolutions, i.e., 8, 16, and 32. ",
+        "bbox": [
+            173,
+            505,
+            825,
+            671
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "After training the models, we run DDSS using just the GGDM model family for simplicity (i.e., we don’t use the $+ \\mathrm { P R E D }$ and $+ \\mathrm { T I M E }$ we experiment with in the paper) at 5, 10 and 20 evenly-spaced inference steps. DDSS training occurs for 50K steps, using the Adam optimizer with learning rate of 0.0005 and batch size 512, optimizing the linear kernel for the KID loss. We compare against the usual DDPM and $\\mathrm { D D I M } ( \\eta = 0 )$ baselines at the same inference budgets and include the FID scores with all 1000 steps for reference. Results and samples are included below. ",
+        "bbox": [
+            174,
+            679,
+            825,
+            762
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/a035e70f7d1d72942648af82275919d957d2e2e29016a2c9ec711307d6605fd1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>SamplerK</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>1000</td></tr><tr><td rowspan=1 colspan=1>LSUN BedroomDDPMDDIM(n = 0)DDSS (GGDM)</td><td rowspan=1 colspan=1>95.38168.729.15</td><td rowspan=1 colspan=1>44.8456.3311.01</td><td rowspan=1 colspan=1>16.889.5274.817</td><td rowspan=1 colspan=1>2.54711</td></tr><tr><td rowspan=1 colspan=1>LSUN ChurchDDPMDDIM(n = 0)DDSS (GGDM)</td><td rowspan=1 colspan=1>96.67133.130.24</td><td rowspan=1 colspan=1>51.0554.3911.59</td><td rowspan=1 colspan=1>16.5314.966.736</td><td rowspan=1 colspan=1>2.71811</td></tr></table>",
+        "bbox": [
+            326,
+            801,
+            671,
+            921
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/117d2db6ee60f3c709a7640eba901b5649459defc69fac8b0fb239b09f7afe7a.jpg",
+        "image_caption": [
+            "Figure A.2: Non-cherrypicked samples for a DDPM trained on LSUN bedroom $1 2 8 \\mathrm { x } 1 2 8$ , comparing DDPM and $\\mathrm { D D I M } ( \\eta = 0 )$ to our approach. All samples were generated with the same random seeds and a linear stride. For reference, we include DDPM samples using all 1,000 steps (bottom left) and real images (bottom middle). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            87,
+            803,
+            729
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/14ca7c6b13c4879f8aaf148f154440406d5bb69bfc76f588f68a066f71e41f95.jpg",
+        "image_caption": [
+            "Figure A.3: Non-cherrypicked samples for a DDPM trained on LSUN church 128x128, comparing DDPM and $\\mathrm { D D I M } ( \\eta = 0 )$ to our approach. All samples were generated with the same random seeds and a linear stride. For reference, we include DDPM samples using all 1,000 steps (bottom left) and real images (bottom middle). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            87,
+            803,
+            729
+        ],
+        "page_idx": 16
+    }
+]

parse/dev/pOEN7dDC0d/pOEN7dDC0d.md

@@ -0,0 +1,558 @@
+# On the Word Boundaries of Emergent Languages Based on Harris’s Articulation Scheme
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 The purpose of this paper is to investigate whether Harris’s articulation scheme   
+2 (HAS) also holds in emergent languages. HAS is thought to be a universal property   
+3 in natural languages that articulatory boundaries can be obtained from statistical   
+4 information of phonems alone, without referring to word meanings. Emergent   
+5 languages are artificial communication protocols that arise between agents in a   
+6 simulated environment and have been attracting attention in recent years. It is   
+7 considerd important to study the structure of emergent languages and the simi  
+8 larity to natural languages. In this paper, we employ HAS as an unsupervised   
+9 word segmentation method and verify whether emergent languages arising from   
+10 signaling games have meaningful segments. Our experiments showed that the   
+11 emergent languages arising from signaling games satisfy some preconditions for   
+12 HAS. However, it was also suggested that the HAS-based segmentation boundaries   
+13 are not necessarily semantically valid.
+
+# 14 1 Introduction
+
+15 Communication protocols emerging among artificial agents in a simulated environment are called   
+16 emergent languages [Lazaridou and Baroni, 2020]. It is important to investigate their structure to   
+17 recognize and bridge the gap between natural and emergent languages, as several structural gaps have   
+18 been reported [Kottur et al., 2017, Chaabouni et al., 2019]. For instance, Kottur et al. [2017] pointed   
+19 out that emergent languages are not necessarily compositional. Such gaps are undesirable because   
+20 major motivations in this area are to develop interactive AI [Foerster et al., 2016, Mordatch and   
+21 Abbeel, 2018, Lazaridou et al., 2020] and to simulate the evolution of human language [Kirby, 2001,   
+22 Graesser et al., 2019, Dagan et al., 2021]. Previous work examined whether emergent languages   
+23 have the same properties as natural languages, such as compositionality [e.g., Kottur et al., 2017],   
+24 grammar [van der Wal et al., 2020], entropy minimization [Kharitonov et al., 2020], and Zipf’s law of   
+25 abbreviation (ZLA) [e.g., Chaabouni et al., 2019]. 1 Word segmentation would be another direction to   
+26 understand the structure of emergent languages because natural languages not only have construction   
+27 from word to sentence but also from phoneme to word [Martinet, 1960]. However, previous studies   
+28 have not gone so far as to address word segmentation, as they treat each symbol in emergent messages   
+29 as if it were a “word” [Kottur et al., 2017, van der Wal et al., 2020], or ensure that a whole message   
+30 constructs just one “word” [Chaabouni et al., 2019, Kharitonov et al., 2020].   
+31 The purpose of this paper is to study whether Harris’s articulation scheme (HAS) [Harris, 1955,   
+32 Tanaka-Ishii, 2021] also holds in emergent languages. HAS is a statistical universal in natural   
+33 languages. Its basic idea is that we can obtain word segments from the statistical information of   
+34 phonemes, but without referring to word meanings. 2 HAS can be used for unsupervised word   
+35 segmentation [Tanaka-Ishii, 2005] to allow us to study the structure of emergent languages. In   
+36 addition, it should be promising to apply such unsupervised methods, since word segments and   
+37 meanings are not available beforehand in emergent languages.   
+38 The problem is whether emergent languages have meaningful segments. If not, then it means that we   
+39 find another gap between emergent and natural languages. In this paper, we pose several verifiable   
+40 questions to answer whether their segments are meaningful.   
+41 To simulate the emergence of language, we adopt Lewis’s signaling game [Lewis, 1969]. This   
+42 game involves two agents called sender $S$ and receiver $R$ , and allows only one-way communication   
+43 from $S$ to $R$ . In each play, $S$ obtains an input $i \in \mathcal { Z }$ and converts $i$ into a sequential message   
+44 $m = S ( i ) \in \mathcal { M }$ . Then, $R$ receives $m \in \mathcal { M }$ and predicts the original input. The goal of the game is   
+45 the correct prediction $R ( m ) = i$ . Figure 1 illustrates the signaling game. Here, we consider the set   
+46 $\{ m \in \mathcal { M } \mid m = S ( i ) \} _ { i \in \mathcal { I } }$ as the dataset of an emergent language, to which the HAS-based boundary   
+47 detection [Tanaka-Ishii, 2005] is applicable. The algorithm yields the segments of messages.   
+48 Our experimental results showed that emergent languages arising from signaling games satisfy   
+49 two preconditions for HAS: (i) the conditional entropy (Eq. 2) decreases monotonically and (ii)   
+50 the branching entropy (Eq. 1) repeatedly falls and rises. However, it was also suggested that the   
+51 HAS-based boundaries are not necessarily meaningful. Segments divided by the boundaries may not   
+52 serve as meaning units, while words in natural languages do [Martinet, 1960]. It is left for future   
+53 work to bridge the gap between emergent and natural languages in terms of HAS, by giving rise to   
+54 meaningful word boundaries.
+
+![](images/28ebbd4cfe074135aef10d67a815231894d2067e731cc7fe0d0f5d3b062d35be.jpg)  
+Figure 1: Illustration of a signaling game. Section 3.1 gives its formal definition. In each play, a sender agent obtains an input and converts it to a sequential message. A receiver agent receives the messsage and converts it to an output. Each agent is represented as an encoder-decoder model.
+
+# 55 2 Harris’s Articulation Scheme
+
+56 In the paper “From phoneme to morpheme” [Harris, 1955], Harris hypothesized that word boundaries   
+57 tend to occur at points where the number of possible successive phonemes reaches a local peak in a   
+58 given context. Harris [1955] exemplifies the utterance “He’s clever” that has the phoneme sequence   
+59 /hiyzclev@r/.3 The number of possible successors after the first phoneme $/ \mathrm { h } I$ is 9: /w,y,i,e,æ,a,@,o,u/.   
+60 Next, the number of possible successors after /hi/ increases to 14. Likewise, the number of possible   
+61 phonemes increases to 29 after /hiy/, stays at 29 after /hiyz/, decreases to 11 after /hiyzk/, decreases   
+62 to 7 after /hiyzkl/, and so on. Peak numbers are found at $/ \mathrm { y } / , / \mathrm { z } /$ , and $/ \mathrm { r } /$ , which divides the phoneme   
+63 sequence into /hiy/+/z/+/klev@r/. Thus, the utterance is divided into “He”, “s”, and “clever”.   
+64 Harris’s hypothesis can be reformulated from an information-theoretic point of view by replacing   
+65 the number of successors with entropy. In the following sections, we review the mathematical   
+66 formulation of the hypothesis as Harris’s articulation scheme (HAS) and the HAS-based boundary   
+67 detection [Tanaka-Ishii, 2005]. HAS does involve statistical information of phonemes but does not   
+68 involve word meanings. This is important because it gives a natural explanation for a well-known   
+69 linguistic concept called double articulation [Martinet, 1960]. Martinet [1960] pointed out that   
+70 languages have two structures: phonemes (irrelevant to meanings) and meaning units (i.e., words and   
+71 morphemes). HAS can construct meaning units without referring to meanings.
+
+# 2.1 Mathematical Formulation of Harris’s Hypothesis
+
+While Harris [1955] focuses on phonemes for word boundary detection, Tanaka-Ishii [2021] suggests that the hypothesis is also applicable to units other than phonemes. Therefore, in this section, a set
+
+75 of units is called an alphabet $\mathcal { X }$ as a purely mathematical notion that is not restricted to phonemes.   
+76 Tanaka-Ishii [2005] uses characters for the same purpose. Moreover, Frantzi and Ananiadou [1996]   
+77 and Tanaka-Ishii and Ishii [2007] investigate the detection of collocation from words.   
+78 Let $\mathcal { X }$ be an alphabet and $\mathcal { X } ^ { n }$ be the set of all $n$ -grams on $\mathcal { X }$ . We denote by $X _ { i }$ a random variable   
+79 of $\mathcal { X }$ indexed by $i$ , and by $X _ { i : j }$ a random variable sequence from $X _ { i }$ to $X _ { j }$ . The formulation   
+80 by Tanaka-Ishii [2005] involves two kinds of entropy: branching entropy and conditional entropy   
+81 [Cover and Thomas, 2006].4 The branching entropy of a random variable $X _ { n }$ after a sequence   
+82 $s = x _ { 0 } \cdot \cdot \cdot x _ { n - 1 } \in \mathcal { X } ^ { n }$ is defined as:
+
+$$
+h ( s ) \equiv { \mathcal { H } } ( X _ { n } \mid X _ { 0 : n - 1 } = s ) = - \sum _ { x \in { \mathcal { X } } } P ( x \mid s ) \log _ { 2 } P ( x \mid s ) ,
+$$
+
+83 where $P ( x \mid s ) = P ( X _ { n } = x \mid X _ { 0 : n - 1 } = s )$ . Intuitively, the branching entropy $h ( s )$ means how   
+84 many elements can occur after $s$ or the uncertainty of the next element after $s$ . In addition to $h ( s )$ ,   
+85 the conditional entropy of a random variable $X _ { n }$ after an $n$ -gram sequence $X _ { 0 : n - 1 }$ is defined as:
+
+$$
+H ( n ) \equiv { \mathcal { H } } ( X _ { n } \mid X _ { 0 : n - 1 } ) = - \sum _ { s \in \mathcal { X } ^ { n } } P ( s ) \sum _ { x \in \mathcal { X } } P ( x \mid s ) \log _ { 2 } P ( x \mid s ) ,
+$$
+
+86 where $P ( s ) = P ( X _ { 0 : n - 1 } = s )$ . The conditional entropy $H ( n )$ can be regarded as the mean of   
+87 $h ( s )$ over $n$ -gram sequences $s \in \mathcal { X } ^ { n }$ , since $\begin{array} { r } { H ( n ) = \sum _ { s \in \mathcal { X } ^ { n } } P ( s ) h ( s ) . \ H ( n ) } \end{array}$ is known to decrease   
+88 monotonically in natural languages [Bell et al., 1990]. Thus, for a partial sequence $x _ { 0 : n - 1 } \in \mathcal X ^ { n }$ ,   
+89 $h ( x _ { 0 : n - 2 } ) > \overset { \cdot } { h } ( x _ { 0 : n - 1 } )$ holds on average, although $h ( s )$ repeatedly falls and rises depending on a   
+90 specific $s$ . Based on such properties, Harris’s articulation scheme (HAS) is formulated as:5
+
+If there is some partial sequence $x _ { 0 : n - 1 } \in \mathcal { X } ^ { n } \left( n > 1 \right)$ s.t. $h ( x _ { 0 : n - 2 } ) < h ( x _ { 0 : n - 1 } )$ , then $x _ { n }$ is at a boundary.
+
+# 91 2.2 Boundary Detection Algorithm Based on Harris’s Articulation Scheme
+
+In this section, we introduce the HAS-based boundary detection algorithm [Tanaka-Ishii, 2005]. Let $s = x _ { 0 } \cdot \cdot \cdot x _ { n - 1 } \in \mathcal { X } ^ { n }$ . We denote by $s _ { i : j }$ its partial sequence $x _ { i } \cdots x _ { j }$ . Given $s$ and a parameter threshold, the boundary detection algorithm yields boundaries $\boldsymbol { B }$ .6 It proceeds as follows:
+
+95 1: $i \gets 0$ ; $w \gets 1$ ; $B \gets \{ \}$   
+96 2: while $i < n$ do   
+97 3: Compute $h \big ( s _ { i : i + w - 1 } \big )$   
+98 4: if $w > 1$ and $\begin{array} { r } { \bar { h } ( s _ { i : i + w - 1 } ) - h ( s _ { i : i + w - 2 } ) > } \end{array}$ threshold then   
+99 5: $B  B \cup \{ i + w \}$   
+100 6: end if   
+101 7: if $i + w < n - 1$ then   
+102 8: $w  w + 1$   
+103 9: else   
+104 10: $i  i + 1 ; w  1$   
+105 11: end if   
+106 12: end while   
+107 Since our targets are emergent languages, the outputs of the boundary detection algorithm do not   
+108 necessarily mean articulatory boundaries. Instead, we call them hypothetical boundaries (hypo  
+109 boundaries) and refer to the segments split by hypo-boundaries as hypo-segments. Note that there   
+110 are other similar methods such as Kempe [1999]. We chose Tanaka-Ishii [2005] because it performs   
+111 well not only for English but also for Chinese, which has many one-character words. Emergent   
+112 languages might also have such words. With this algorithm, Tanaka-Ishii and Jin [2008] reported   
+113 F-score $= 8 3 . 6 \%$ for word boundary detection from phonemes in English and F-score $= \hat { 8 3 . 8 \% }$   
+114 for word boundary detection from characters in Chinese. They are considerably high scores for   
+115 unsupervised settings.   
+117 We have to define environments, agent architectures, and optimization methods for language emer  
+118 gence simulations. This paper adopts the framework of Chaabouni et al. [2020]. We define an   
+119 environment in Section 3.1, specify the agent architecture and optimization methods in Section 3.2,   
+120 and also give an explanation of the compositionality of emergent languages in Section 3.3.
+
+# 121 3.1 Signaling Game
+
+122 An environment is formulated based on Lewis’s signaling game [Lewis, 1969]. A signaling game $G$   
+123 consists of a quadruple $( \mathcal { T } , \mathcal { M } , S , R )$ , where $\mathcal { T }$ is an input space, $\mathcal { M }$ is a message space, $S : { \mathcal { T } }  { \mathcal { M } }$   
+124 is a sender agent, and $R : \mathcal { M }  \mathcal { T }$ is a receiver agent. The goal is the correct reconstruction   
+125 $i = R ( S ( i ) )$ for all $i \in \mathcal { Z }$ . While the input space $\mathcal { T }$ and the message space $\mathcal { M }$ are fixed, the agents   
+126 $S , R$ are trained for the goal. An illustration of a signaling game is shown in Figure 1. Following   
+127 Chaabouni et al. [2020], we define $\mathcal { T }$ as an attribute-value set $\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ (defined below) and $\mathcal { M }$ as a set   
+128 of discrete sequences of fixed length $k$ over a finite alphabet $\mathcal { A }$ :
+
+$$
+{ \mathcal { T } } \equiv { \mathcal { D } } _ { n _ { v a l } } ^ { n _ { a t t } } , { \mathcal { M } } \equiv { \mathcal { A } } ^ { k } = \{ a _ { 1 } \cdot \cdot \cdot a _ { k } \mid a _ { j } \in { \mathcal { A } } \} .
+$$
+
+129 Attribute-Value Set Let $n _ { a t t } , n _ { v a l }$ be positive integers called the number of attributes and the number of values. Then, an attribute-value set 130 $\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ is the set of ordered tuples defined as follows:
+
+$$
+\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } } = \left\{ \left( v _ { 1 } , \ldots , v _ { n _ { a t t } } \right) \ : | \ : v _ { j } \in \left\{ 1 , \ldots , n _ { v a l } \right\} \right\} .
+$$
+
+131 This is an abstraction of an attribute-value object paradigm [e.g., Kottur et al., 2017] by Chaabouni 32 et al. [2020]. Intuitively, each index $j$ of a vector $( v _ { 1 } , \ldots v _ { j } , \ldots , v _ { n _ { a t t } } )$ is an attribute (e.g., color), while each 33 $v _ { j }$ is an attribute value (e.g., blue, green, red, and purple).7
+
+# 3.2 Architecture and Optimization
+
+We follow Chaabouni et al. [2020] as well for the architecture and optimization method.
+
+Architecture Each agent is represented as an encoder-decoder model (Figure 1): the sender decoder and the receiver encoder are based on single-layer GRUs [Cho et al., 2014], while the sender encoder and the receiver decoder are linear functions. Each element $i \in \mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ has to be vectorized so that it can be fed into or output from the linear functions. Formally, each $\ddot { i } = ( v _ { 1 } , \dots , v _ { n _ { a t t } } )$ is converted into the $n _ { a t t } \times n _ { v a l }$ -dimensional vector which is the concatenation of $n _ { a t t }$ one-hot representations of $v _ { j }$ . During training, the sender samples messages probabilistically. During the test time, it samples them greedily so that it serves as a deterministic function. Similarly, the receiver’s output layer, followed by the Softmax, determines $n _ { a t t }$ categorical distributions over values $\{ 1 , \dots , n _ { v a l } \}$ during training. During the test time, $n _ { a t t }$ values are greedily sampled from the distributions.
+
+Optimization The agents are optimized with the stochastic computation graph [Schulman et al., 2015] that is a combination of REINFORCE [Williams, 1992] and standard backpropagation. The sender is optimized with the former, while the receiver is optimized with the latter.
+
+# 3.3 Compositionality of Emergent Languages
+
+An attribute-value set $\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ by Chaabouni et al. [2020] is an extension of an attribute-value setting [Kottur et al., 2017] introduced to measure the compositionality of emergent languages. While the concept of compositionality varies from domain to domain, researchers in this area typically regard it as the disentanglement of representation learning. Kottur et al. [2017], for instance, set up an environment where objects have two attributes: color and shape, each of which has several possible values (e.g., blue, red, ... for color and circle, star, ... for shape). They assumed that if a language is sufficiently compositional, each message would be a composition of symbols denoting the color value and shape value separately. This concept has been the basis for subsequent studies [Li and Bowling, 2019, Andreas, 2019, Ren et al., 2020, Chaabouni et al., 2020].
+
+158 Topographic Similarity Topographic Similarity (TopSim) [Brighton and Kirby, 2006, Lazaridou   
+159 et al., 2018] is the de facto compositionality measure in emergent communication literature. Suppose   
+160 we have distance functions $d _ { \mathcal { T } } , d _ { \mathcal { M } }$ for spaces $\mathcal { T } , \mathcal { M }$ , respectively. TopSim is defined as the Spearman   
+161 correlation between distances $d _ { \ Z } ( i _ { 1 } , i _ { 2 } )$ and $d _ { \mathcal { M } } ( S ( i _ { 1 } ) , S ( i _ { 2 } ) )$ for all $i _ { 1 } , i _ { 2 } \in \mathcal { T }$ s.t. $i _ { 1 } \neq i _ { 2 }$ . This   
+162 definition reflects an intuition that compositional languages should map similar (resp. dissimilar)   
+163 inputs to similar (resp. dissimilar) messages. Following previous work using attribute-value objects   
+164 [e.g., Chaabouni et al., 2020], we define $d _ { \mathcal { T } }$ as the Hamming distance and $d _ { \mathcal { M } }$ as the edit distance.   
+165 Because this paper is about message segmentation, we can consider two types of edit distance. One   
+166 is the “character” edit distance that regards elements $a \in { \mathcal { A } }$ as symbols. The other is the “word”   
+167 edit distance that regards hypo-segments as symbols. Let us call the former $C$ -TopSim and the latter   
+168 W-TopSim.
+
+# 169 4 Problem Definition
+
+170 The purpose of this paper is to study whether Harris’s articulation   
+171 scheme (HAS) also holds in emergent languages. However, this   
+172 question is too vague to answer. We first divide it into the following:
+
+Q1. Does the conditional entropy $H$ decrease monotonically? Q2. Does the branching entropy $h$ repeatedly fall and rise? Q3. Do hypo-boundaries represent meaningful boundaries?
+
+176 Q3 is the same as the original question, except that Q3 is slightly   
+177 more formal. However, we have to answer Q1 and Q2 beforehand,   
+178 because HAS implicitly takes it for granted that $H$ decreases mono  
+179 tonically and $h$ jitters. Although both Q1 and Q2 generally hold in   
+180 natural languages, neither of them is trivial in emergent languages.   
+181 Figure 2 illustrates Q1, Q2, and Q3.   
+182 It is straightforward to answer Q1 and Q2 as we just need to calculate   
+183 $H$ and $h$ . In contrast, Q3 is still vague to answer, since we do not   
+184 have prior knowledge about the boundaries of emergent languages and do not even know if they have   
+185 such boundaries. To mitigate it, we posit the following necessary conditions for Q3. Let $G$ be a game   
+186 $( \mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } } , \mathcal { A } ^ { k } , S , R )$ . If the answer to Q3 is yes, then:
+
+![](images/543e883e362520a9b619f1f164300dff04cf01743c92d3f984c7743efc746dcb.jpg)  
+Q3:Are They Meaningful? Figure 2: Illustration of questions.
+
+C1. the mean number of hypo-boundaries per message should increase as $n _ { a t t }$ increases, C2. the size of the vocabulary (set of all hypo-segments) should increase as $n _ { v a l }$ increases, C3. W-TopSim should be higher than C-TopSim.
+
+190 About C1 and C2 An attribute-value set $\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ was originally introduced to measure composi  
+191 tionality. Compositionality, in this context, means how symbols in a message separately denote the   
+192 components of meaning. In our case, each segment, or word, can be thought of as a certain unit   
+193 that denotes the attribute values, so that the number of words in a message should increase as the   
+194 corresponding attributes increase. Therefore, if the answer to Q3 is yes, then C1 should be valid.   
+195 Likewise, the size of the vocabulary should be larger in proportion to the number of values $n _ { v a l }$ ,   
+196 motivating C2. Here, we mean by vocabulary the set of all hypo-segments. Note that the message   
+197 length is fixed, because otherwise the number of hypo-segments would be subject to variable message   
+198 length as well as $( n _ { a t t } , n _ { v a l } )$ , and the implication of results would be obscure.
+
+About C3 C3 comes from the analogy of the linguistic concept called double articulation [Martinet, 1960]. In natural languages, meanings are quite arbitrarily related to the phonemes that construct them. In contrast, the meanings are less arbitrarily related to the words. The phonemes do not denote meaning units but the words do. In our case, for example, the attribute-value object (RED, CIRCLE) seems less compositionally related to the character sequence “r,e,d,c,i,r,c,l,e”, while it seems more compositionally related to the word sequence “red,circle.” This intuition motivates C3.
+
+205 Based on conditions C1, C2, and C3, Q3 is restated as follows: (Q3-1) Does the mean number of   
+206 hypo-boundaries per message increase as $n _ { a t t }$ increases? (Q3-2) Does the vocabulary size increase   
+207 as $n _ { v a l }$ increases? (Q3-3) Is W-TopSim higher than C-TopSim?
+
+# 5 Experimental Setup
+
+# 5.1 Parameter Settings
+
+Input Space $n _ { a t t }$ and $n _ { v a l }$ have to be varied to answer Q3-1, Q3-2, and Q3-3, while the sizes of the input spaces $| \mathcal { I } | = ( n _ { v a l } ) ^ { n _ { a t t } }$ must be equal to each other to balance the complexities of games. Therefore, we fix $| \mathcal { T } | = 4 0 9 6$ and vary $( n _ { a t t } , n _ { v a l } )$ as follows:
+
+$$
+( n _ { a t t } , n _ { v a l } ) \in \{ ( 1 , 4 0 9 6 ) , ( 2 , 6 4 ) , ( 3 , 6 ) , ( 4 , 8 ) , ( 6 , 4 ) , ( 1 2 , 2 ) \} .
+$$
+
+213 Message Space The message length $k$ and alphabet $\mathcal { A }$ have to be determined for a message space   
+214 ${ \mathcal { M } } = { \overset { \sim } { A } } ^ { k }$ . We set $k = 3 2$ , similarly to previous work on $Z L A$ [Chaabouni et al., 2019, Rita et al.,   
+215 2020, Ueda and Washio, 2021] that regards each $a \in { \mathcal { A } }$ as a “character.” Note that $k = 3 2$ is set much   
+216 longer than those of previous work on compositionality [Chaabouni et al., 2020, Ren et al., 2020, Li   
+217 and Bowling, 2019] that typically adopts $k \equiv n _ { a t t }$ as if each symbol $a \in { \mathcal { A } }$ were a “word.” We set   
+218 $\mathcal { A } = \{ 1 , 2 , \ldots , 8 \}$ . Its size $| { \cal A } |$ should be as small as possible to avoid the problem of data sparsity   
+219 when applying boundary detection, and to ensure that each symbol $a \in { \mathcal { A } }$ serves as a “character.”   
+220 In preliminary experiments, we tested $| \mathcal { A } | \in \{ 2 , 4 , 8 , 1 6 \}$ and found that learning is stable when   
+221 $| { \bar { \mathcal { A } } } | \geq 8$ .   
+222 Architecture and Optimization We follow Chaabouni et al. [2020] for agent arthitectures and   
+223 optmization methods. The hidden size of GRU [Cho et al., 2014] is set to 500, following Chaabouni   
+224 et al. [2020]. All data from an input space $\mathcal { I } = \mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ are used for training. This dataset is upsampled   
+225 to 100 times following the default setting of the code of Chaabouni et al. [2020]. The learning rate is   
+226 set to 0.001, which also follows Chaabouni et al. [2020]. Based on our preliminary experiments to   
+227 explore stable learning, a sender $S$ and a receiver $R$ are trained for 200 epochs and the coefficient of   
+228 the entropy regularizer is set to 0.01.
+
+Boundary Detection Algorithm The boundary detection algorithm involves a parameter threshold. Since the appropriate value of threshold is unclear, we vary threshold as follows:
+
+$$
+t h r e s h o l d \in \{ 0 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 , 1 . 2 5 , 1 . 5 , 1 . 7 5 , 2 \} .
+$$
+
+# 31 5.2 Implementation, Number of Trials, and Language Validity
+
+We implemented the code for training agents using the EGG toolkit [Kharitonov et al., 2019].8 EGG also includes the implementation code of Chaabouni et al. [2020], which we largely refer to. They are published under the MIT license. For now, our code is available on Anonymous GitHub.9 For each $( n _ { a t t } , n _ { v a l } )$ configuration, agents are trained 8 times with different random seeds. Each run took a few hours with a single GPU.10 In the following sections, an emergent language with a communication success rate of more than $90 \%$ is called a successful language.
+
+# 6 Results
+
+As a result of training agents, we obtained 7, 8, 6, 8, 7, and 6 successful languages out of 8 runs for configurations $( n _ { a t t } , \bar { n } _ { v a l } ) = ( 1 , 4 0 9 6 )$ , (2, 64), (3, 16), $( 4 , 8 )$ , $( 6 , 4 )$ , and $( 1 2 , 2 )$ , respectively.
+
+# 6.1 Conditional Entropy Monotonically Decreases
+
+To verify Q1, we show the conditional entropy $H ( n )$ (Eq. 2) in Figure 3. In Figure 3, the conditional entropies of the successful languages (solid red lines) decrease monotonically. This confirms Q1 in successful languages. Interestingly, the conditional entropies of emergent languages derived from untrained senders do not necessarily decrease, shown as dashed blue lines in Figure 3.11 The monotonic decrease in conditional entropy emerges after training agents.
+
+![](images/ee6db7b4fef5d383c1245c9afc7fff843e615985e0866f249f1afd85d540eff8.jpg)  
+Figure 3: Conditional entropy $H ( n )$ . Dashed blue lines represent $H ( n )$ of languages from untrained agents that finally learned successful languages, while solid red lines represent $H ( n )$ of successful languages.
+
+![](images/85a714388a0f655501234aacc7a619167f4889df8dafde311d30fcb8b4a68a2d.jpg)  
+Figure 4: Example transition sequences of the branching entropy $h$ in a message $^ { \dots } 3 , 8 , 4 , . . . , 4 , 4 , 4 ^ { , , }$ in a successful language for $\begin{array} { r l } { ( n _ { a t t } , n _ { v a l } ) } & { { } = } \end{array}$ (2, 64).
+
+# 47 6.2 Branching Entropy Repeatedly Falls and Rises
+
+Next, to answer Q2, we computed the branching entropy $h ( s )$ (Eq. 1) of the successful languages and applied boundary detection. As an example, we show a few actual transitions of $h ( s )$ in Figure 4, in which y-axis represents the value of $h ( s )$ and $\mathbf { X }$ -axis represents a message $^ { \dots } 3 , 8 , 4 , . . . , 4 , 4 , 4 ^ { , , }$ . The message is randomly sampled from a successful language when $( n _ { a t t } , n _ { v a l } ) \bar { = } ( 2 , 6 4 )$ . The boundary detection algorithm with threshold $= 1$ yields three hypo-boundaries that are represented as dashed black lines in Figure 4. Blue, yellow and green lines with triangle markers represent the transitions of $h ( s )$ that yield hypo-boundaries. Note that the $( i + 1 )$ -th transition of $h ( s )$ does not necessarily start from the $i$ -th hypo-boundary, due to the definition of the algorithm. For instance, the second transition overlaps the first hypo-boundary. While the conditional entropy decreases monotonically as shown in Figure 3, the branching entropy repeatedly falls and rises in Figure 4. Moreover, we show the mean number of hypo-boundaries per message in Figure 5. Figure 5 indicates that for any $( n _ { a t t } , n _ { v a l } )$ configuration, there are hypo-boundaries if threshold $< 2$ , i.e., the brancing entropy repeatedly falls and rises. These results validate Q2.
+
+# 6.3 Hypo-Boundaries May Not Be Meaningful Boundaries
+
+Next, we investigate whether Q3-1, Q3-2, and Q3-3 hold in successful languages. The results in the following sections falsify all of them. Thus, Q3 may not be true: hypo-boundaries may not be meaningful boundaries.
+
+Mean Number of Hypo-Boundaries per Message See Figure 5 again. The figure shows that the mean number of hypo-boundaries per message does not increase as $n _ { a t t }$ increases. It does not decrease, either. This result falsifies Q3-1. Even when $n _ { a t t } = 1$ , there are as many hypo-boundaries as other configurations.
+
+Vocabulary Size Figure 6 shows the mean vocabulary sizes for each $( n _ { a t t } , n _ { v a l } )$ . The vocabulary size does not increase as $n _ { v a l }$ increases, which falsifies Q3-2. However, focusing on $( n _ { a t t } , n _ { v a l } ) \in$ $\{ ( 2 , 6 4 ) , ( 3 , 1 6 ) , ( 4 , 8 ) , ( 6 , 4 ) \}$ and $0 . 2 5 \leq t h r e s h o l d \leq 1$ , there is a weak tendency to support C2. It suggests that hypo-segments are not completely meaningless either.
+
+273 C-TopSim vs W-TopSim Figure 7 shows C-Topsim and W-Topsim for each $( n _ { a t t } , n _ { v a l } )$ and   
+274 threshold.12 Note that C-TopSim is TopSim with “character” edit distance and W-TopSim is TopSim   
+275 with “word” edit distance. In Figure 7, threshold $: = - \infty$ corresponds to C-TopSim, while the others   
+276 correspond to W-TopSim. 13 Our assumption in Q3-3 was C-TopSim $< \mathrm { W }$ -TopSim. On the contrary,   
+277 Figure 7 shows a clear tendency for C-TopSim $>$ W-TopSim, which falsifies Q3-3. Hypo-boundaries   
+278 may not be meaningful. However, they may not be completely meaningless, either. This is because the   
+279 hypo-boundary-based W-TopSim is higher than the random-boundary-based W-TopSim in Figure 8.   
+280 Here, we mean by random boundaries the boundaries chosen at random in the same number as   
+281 hypo-boundaries in each message. Other $( n _ { a t t } , n _ { v a l } )$ configurations show similar tendencies (see   
+282 Appendix B).
+
+![](images/249e134d2454044c17e0e7da346a9ed8984c7174642d559d863d29fd66523318.jpg)  
+Figure 5: Mean number of hypo-boundaries per message in successful languages. threshold varies according to Eq. 7. Each data point is averaged over random seeds and shaded regions represent one standard error of mean (SEM).
+
+![](images/ba31b77f373c46d140cc40fae8cf35aa4d23016309d3c09c44901a35f1d80a78.jpg)  
+Figure 6: Vocabulary size in successful langauges. threshold varies according to Eq. 7. Each data point is averaged over random seeds and shaded regions represent one SEM.
+
+![](images/4de6e37da865892a2f6d3da4fce4ec2bfce406f77f3f6782dac273754a94f6fe.jpg)  
+Figure 7: C-TopSim and W-TopSim in successful languages. threshold $= - \infty$ corresponds to $\mathrm { C } \mathrm { - }$ TopSim, while other threshold correspond to WTopSim. Each data point is averaged over random seeds and shaded regions represent one SEM.
+
+![](images/4208bf277ad61eafa64d1415bc53cdc7f4d932805435b576bf35d0bf1b1f2052.jpg)  
+Figure 8: hypo-boundary-based W-TopSim compared to random-boundary-based W-TopSim in successful languages for $( n _ { a t t } , n _ { v a l } ) = ( 2 , 6 4 )$ . Each data point is averaged over random seeds and shaded regions represent one SEM.
+
+# 6.4 Further Investigation: Word Length and Word Frequency
+
+The results so far are related to compositionality of emergent languages [e.g., Kottur et al., 2017]. In this section, we further associate our results with previous discussions on Zipf’s law of abbreviation (ZLA) in emergent languages [Chaabouni et al., 2019, Rita et al., 2020, Ueda and Washio, 2021]. ZLA is known as a statistical property in natural languages that the more frequently a word is used,
+
+288 the shorter it is [Zipf, 1935]. By considering hypo-segments as “words,” we can check whether   
+289 hypo-segments follow ZLA. Figure 9 shows the hypo-segment lengths sorted by frequency rank for   
+290 $( \ddot { n _ { a t t } } , \bar { n _ { v a l } } ) = ( 1 , 4 0 9 6 )$ .14 If hypo-segments follow ZLA ideally, they should show a monotonic   
+291 increase. The distribution of the lengths of the hypo-segments shows a clear ZLA-like tendency for   
+292 threshold $\in \{ 0 , 0 . 5 \}$ , although the tendencies are less clear for the other threshold.15 It means that   
+293 hypo-segments follow ZLA with an appropriate threshold value. Other $( n _ { a t t } , n _ { v a l } )$ configurations   
+294 show similar tendencies (see Appendix C).
+
+# 295 7 Discussion
+
+In Section 6.1, we showed that the conditional entropy $H ( n )$ decreases monotonically in emergent languages, confirming Q1. In Section 6.2, we demonstrated that the branching entropy $h ( s )$ repeatedly falls and rises in emergent languages, which confirms Q2. It is an intriguing result, considering the discussions of Kharitonov et al. [2020], who showed that the entropy decreases to the minimum for successful communication if the message length $k = 1$ . In contrast, our results suggest that the (branching) entropy does not simply fall to the minimum when the message length $k$ is longer. However, in Section 6.3, our results indicate that the hypo-boundaries may not be meaningful since Q3-1, Q3-2, and Q3-3 were falsified.
+
+![](images/42c0f8bdebb211ba6d62d3cf8b9198a89d2f62e99463fb41d4b0065427d67a66.jpg)  
+Figure 9: Hypo-segment lengths sorted by frequency rank for $( n _ { a t t } , n _ { v a l } ) \stackrel { - } { = } ( 1 , 4 0 9 6 )$ . Each data point is averaged over random seeds and shaded regions represent one SEM.
+
+312 Nevertheless, hypo-boundaries may not be completely meaningless either. This is because the   
+313 hypo-boundary-based W-TopSim is higher than the random-boundary-based W-TopSim. It suggests   
+314 that HAS-based boundary detection worked to some extent. In addition, the hypo-segments show   
+315 ZLA-like tendencies with certain threshold values. This is a suggestive result because we neither   
+316 imposed a length penalty on messages [Chaabouni et al., 2020], modeled the laziness/impatience of   
+317 agents [Rita et al., 2020], nor modeled short-term memories [Ueda and Washio, 2021]. Of course, it   
+318 is important to note that it may be just an artifact, analogous to the fact that even a monkey typing   
+319 sequence divided by the “white space” follows ZLA [Miller, 1957].   
+320 This paper showed that there is a gap between emergent and natural languages in terms of word   
+321 segmentation. There are some potential methods to bridge the gap. For example, several methods   
+322 have been proposed to facilitate the compositionality of emergent languages, such as iterated learning   
+323 [Ren et al., 2020], the ease-of-teaching paradigm [Li and Bowling, 2019], and concept game [Mu and   
+324 Goodman, 2021]. The regularizations for ZLA mentioned above might also help for this purpose.   
+325 These are left for future work.
+
+# 26 8 Conclusion
+
+In this paper, we investigated whether Harris’s articulation scheme (HAS) also holds in emergent languages. Emergent languages are artificial communication protocols emerging between agents, while HAS is a statistical universal in natural languages. HAS can be used for unsupervised word segmentation. Our experimental results suggest that although emergent languages satisfy some prerequisites for HAS, HAS-based word boundaries may not be meaningful. Our contributions are (1) to focus on the word segmentation of emergent languages, (2) to pose verifiable questions to answer whether emergent languages have meaningful segments, and (3) to show another gap between emergent and natural languages. It is left for future work to bridge the gap between emergent and natural languages in terms of HAS.
+
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+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 7.   
+(c) Did you discuss any potential negative societal impacts of your work? [No] The main point of this paper is analysis of artificially emerging languages without human participants. It is hard to imagine that it has a direct negative impact on society. However, our paper may have to do with energy issues, since we used GPU. We wrote about the GPU resource in Section 5.2.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Our implementation code is uploaded on Anonymous Github. See Section 5.2.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 5.1.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Error bars are necessary in Figure 5, Figure 6, Figure 7, Figure 8, and Figure 9 and are reported in all of them.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 5.2
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] Our implementation code depends on the EGG toolkit [Kharitonov et al., 2019]. See Section 5.2.   
+(b) Did you mention the license of the assets? [Yes] The EGG toolkit is published under MIT License. See Section 5.2.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Our implementation code is uploaded on Anonymous Github. See Section 5.2.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] We only used artificial data for experiments.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] We only used artificial data for experiments.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "On the Word Boundaries of Emergent Languages Based on Harris’s Articulation Scheme ",
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+        "text": "1 The purpose of this paper is to investigate whether Harris’s articulation scheme   \n2 (HAS) also holds in emergent languages. HAS is thought to be a universal property   \n3 in natural languages that articulatory boundaries can be obtained from statistical   \n4 information of phonems alone, without referring to word meanings. Emergent   \n5 languages are artificial communication protocols that arise between agents in a   \n6 simulated environment and have been attracting attention in recent years. It is   \n7 considerd important to study the structure of emergent languages and the simi  \n8 larity to natural languages. In this paper, we employ HAS as an unsupervised   \n9 word segmentation method and verify whether emergent languages arising from   \n10 signaling games have meaningful segments. Our experiments showed that the   \n11 emergent languages arising from signaling games satisfy some preconditions for   \n12 HAS. However, it was also suggested that the HAS-based segmentation boundaries   \n13 are not necessarily semantically valid. ",
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+        "text": "14 1 Introduction ",
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+        "text": "15 Communication protocols emerging among artificial agents in a simulated environment are called   \n16 emergent languages [Lazaridou and Baroni, 2020]. It is important to investigate their structure to   \n17 recognize and bridge the gap between natural and emergent languages, as several structural gaps have   \n18 been reported [Kottur et al., 2017, Chaabouni et al., 2019]. For instance, Kottur et al. [2017] pointed   \n19 out that emergent languages are not necessarily compositional. Such gaps are undesirable because   \n20 major motivations in this area are to develop interactive AI [Foerster et al., 2016, Mordatch and   \n21 Abbeel, 2018, Lazaridou et al., 2020] and to simulate the evolution of human language [Kirby, 2001,   \n22 Graesser et al., 2019, Dagan et al., 2021]. Previous work examined whether emergent languages   \n23 have the same properties as natural languages, such as compositionality [e.g., Kottur et al., 2017],   \n24 grammar [van der Wal et al., 2020], entropy minimization [Kharitonov et al., 2020], and Zipf’s law of   \n25 abbreviation (ZLA) [e.g., Chaabouni et al., 2019]. 1 Word segmentation would be another direction to   \n26 understand the structure of emergent languages because natural languages not only have construction   \n27 from word to sentence but also from phoneme to word [Martinet, 1960]. However, previous studies   \n28 have not gone so far as to address word segmentation, as they treat each symbol in emergent messages   \n29 as if it were a “word” [Kottur et al., 2017, van der Wal et al., 2020], or ensure that a whole message   \n30 constructs just one “word” [Chaabouni et al., 2019, Kharitonov et al., 2020].   \n31 The purpose of this paper is to study whether Harris’s articulation scheme (HAS) [Harris, 1955,   \n32 Tanaka-Ishii, 2021] also holds in emergent languages. HAS is a statistical universal in natural   \n33 languages. Its basic idea is that we can obtain word segments from the statistical information of   \n34 phonemes, but without referring to word meanings. 2 HAS can be used for unsupervised word   \n35 segmentation [Tanaka-Ishii, 2005] to allow us to study the structure of emergent languages. In   \n36 addition, it should be promising to apply such unsupervised methods, since word segments and   \n37 meanings are not available beforehand in emergent languages.   \n38 The problem is whether emergent languages have meaningful segments. If not, then it means that we   \n39 find another gap between emergent and natural languages. In this paper, we pose several verifiable   \n40 questions to answer whether their segments are meaningful.   \n41 To simulate the emergence of language, we adopt Lewis’s signaling game [Lewis, 1969]. This   \n42 game involves two agents called sender $S$ and receiver $R$ , and allows only one-way communication   \n43 from $S$ to $R$ . In each play, $S$ obtains an input $i \\in \\mathcal { Z }$ and converts $i$ into a sequential message   \n44 $m = S ( i ) \\in \\mathcal { M }$ . Then, $R$ receives $m \\in \\mathcal { M }$ and predicts the original input. The goal of the game is   \n45 the correct prediction $R ( m ) = i$ . Figure 1 illustrates the signaling game. Here, we consider the set   \n46 $\\{ m \\in \\mathcal { M } \\mid m = S ( i ) \\} _ { i \\in \\mathcal { I } }$ as the dataset of an emergent language, to which the HAS-based boundary   \n47 detection [Tanaka-Ishii, 2005] is applicable. The algorithm yields the segments of messages.   \n48 Our experimental results showed that emergent languages arising from signaling games satisfy   \n49 two preconditions for HAS: (i) the conditional entropy (Eq. 2) decreases monotonically and (ii)   \n50 the branching entropy (Eq. 1) repeatedly falls and rises. However, it was also suggested that the   \n51 HAS-based boundaries are not necessarily meaningful. Segments divided by the boundaries may not   \n52 serve as meaning units, while words in natural languages do [Martinet, 1960]. It is left for future   \n53 work to bridge the gap between emergent and natural languages in terms of HAS, by giving rise to   \n54 meaningful word boundaries. ",
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+            "Figure 1: Illustration of a signaling game. Section 3.1 gives its formal definition. In each play, a sender agent obtains an input and converts it to a sequential message. A receiver agent receives the messsage and converts it to an output. Each agent is represented as an encoder-decoder model. "
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+        "text": "55 2 Harris’s Articulation Scheme ",
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+        "text": "56 In the paper “From phoneme to morpheme” [Harris, 1955], Harris hypothesized that word boundaries   \n57 tend to occur at points where the number of possible successive phonemes reaches a local peak in a   \n58 given context. Harris [1955] exemplifies the utterance “He’s clever” that has the phoneme sequence   \n59 /hiyzclev@r/.3 The number of possible successors after the first phoneme $/ \\mathrm { h } I$ is 9: /w,y,i,e,æ,a,@,o,u/.   \n60 Next, the number of possible successors after /hi/ increases to 14. Likewise, the number of possible   \n61 phonemes increases to 29 after /hiy/, stays at 29 after /hiyz/, decreases to 11 after /hiyzk/, decreases   \n62 to 7 after /hiyzkl/, and so on. Peak numbers are found at $/ \\mathrm { y } / , / \\mathrm { z } /$ , and $/ \\mathrm { r } /$ , which divides the phoneme   \n63 sequence into /hiy/+/z/+/klev@r/. Thus, the utterance is divided into “He”, “s”, and “clever”.   \n64 Harris’s hypothesis can be reformulated from an information-theoretic point of view by replacing   \n65 the number of successors with entropy. In the following sections, we review the mathematical   \n66 formulation of the hypothesis as Harris’s articulation scheme (HAS) and the HAS-based boundary   \n67 detection [Tanaka-Ishii, 2005]. HAS does involve statistical information of phonemes but does not   \n68 involve word meanings. This is important because it gives a natural explanation for a well-known   \n69 linguistic concept called double articulation [Martinet, 1960]. Martinet [1960] pointed out that   \n70 languages have two structures: phonemes (irrelevant to meanings) and meaning units (i.e., words and   \n71 morphemes). HAS can construct meaning units without referring to meanings. ",
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+        "text": "2.1 Mathematical Formulation of Harris’s Hypothesis ",
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+        "text": "While Harris [1955] focuses on phonemes for word boundary detection, Tanaka-Ishii [2021] suggests that the hypothesis is also applicable to units other than phonemes. Therefore, in this section, a set ",
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+        "text": "75 of units is called an alphabet $\\mathcal { X }$ as a purely mathematical notion that is not restricted to phonemes.   \n76 Tanaka-Ishii [2005] uses characters for the same purpose. Moreover, Frantzi and Ananiadou [1996]   \n77 and Tanaka-Ishii and Ishii [2007] investigate the detection of collocation from words.   \n78 Let $\\mathcal { X }$ be an alphabet and $\\mathcal { X } ^ { n }$ be the set of all $n$ -grams on $\\mathcal { X }$ . We denote by $X _ { i }$ a random variable   \n79 of $\\mathcal { X }$ indexed by $i$ , and by $X _ { i : j }$ a random variable sequence from $X _ { i }$ to $X _ { j }$ . The formulation   \n80 by Tanaka-Ishii [2005] involves two kinds of entropy: branching entropy and conditional entropy   \n81 [Cover and Thomas, 2006].4 The branching entropy of a random variable $X _ { n }$ after a sequence   \n82 $s = x _ { 0 } \\cdot \\cdot \\cdot x _ { n - 1 } \\in \\mathcal { X } ^ { n }$ is defined as: ",
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+        "text": "$$\nh ( s ) \\equiv { \\mathcal { H } } ( X _ { n } \\mid X _ { 0 : n - 1 } = s ) = - \\sum _ { x \\in { \\mathcal { X } } } P ( x \\mid s ) \\log _ { 2 } P ( x \\mid s ) ,\n$$",
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+        "text": "83 where $P ( x \\mid s ) = P ( X _ { n } = x \\mid X _ { 0 : n - 1 } = s )$ . Intuitively, the branching entropy $h ( s )$ means how   \n84 many elements can occur after $s$ or the uncertainty of the next element after $s$ . In addition to $h ( s )$ ,   \n85 the conditional entropy of a random variable $X _ { n }$ after an $n$ -gram sequence $X _ { 0 : n - 1 }$ is defined as: ",
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+        "text": "$$\nH ( n ) \\equiv { \\mathcal { H } } ( X _ { n } \\mid X _ { 0 : n - 1 } ) = - \\sum _ { s \\in \\mathcal { X } ^ { n } } P ( s ) \\sum _ { x \\in \\mathcal { X } } P ( x \\mid s ) \\log _ { 2 } P ( x \\mid s ) ,\n$$",
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+        "text": "86 where $P ( s ) = P ( X _ { 0 : n - 1 } = s )$ . The conditional entropy $H ( n )$ can be regarded as the mean of   \n87 $h ( s )$ over $n$ -gram sequences $s \\in \\mathcal { X } ^ { n }$ , since $\\begin{array} { r } { H ( n ) = \\sum _ { s \\in \\mathcal { X } ^ { n } } P ( s ) h ( s ) . \\ H ( n ) } \\end{array}$ is known to decrease   \n88 monotonically in natural languages [Bell et al., 1990]. Thus, for a partial sequence $x _ { 0 : n - 1 } \\in \\mathcal X ^ { n }$ ,   \n89 $h ( x _ { 0 : n - 2 } ) > \\overset { \\cdot } { h } ( x _ { 0 : n - 1 } )$ holds on average, although $h ( s )$ repeatedly falls and rises depending on a   \n90 specific $s$ . Based on such properties, Harris’s articulation scheme (HAS) is formulated as:5 ",
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+        "type": "text",
+        "text": "If there is some partial sequence $x _ { 0 : n - 1 } \\in \\mathcal { X } ^ { n } \\left( n > 1 \\right)$ s.t. $h ( x _ { 0 : n - 2 } ) < h ( x _ { 0 : n - 1 } )$ , then $x _ { n }$ is at a boundary. ",
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+        "text": "91 2.2 Boundary Detection Algorithm Based on Harris’s Articulation Scheme ",
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+        "text": "In this section, we introduce the HAS-based boundary detection algorithm [Tanaka-Ishii, 2005]. Let $s = x _ { 0 } \\cdot \\cdot \\cdot x _ { n - 1 } \\in \\mathcal { X } ^ { n }$ . We denote by $s _ { i : j }$ its partial sequence $x _ { i } \\cdots x _ { j }$ . Given $s$ and a parameter threshold, the boundary detection algorithm yields boundaries $\\boldsymbol { B }$ .6 It proceeds as follows: ",
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+        "text": "95 1: $i \\gets 0$ ; $w \\gets 1$ ; $B \\gets \\{ \\}$   \n96 2: while $i < n$ do   \n97 3: Compute $h \\big ( s _ { i : i + w - 1 } \\big )$   \n98 4: if $w > 1$ and $\\begin{array} { r } { \\bar { h } ( s _ { i : i + w - 1 } ) - h ( s _ { i : i + w - 2 } ) > } \\end{array}$ threshold then   \n99 5: $B  B \\cup \\{ i + w \\}$   \n100 6: end if   \n101 7: if $i + w < n - 1$ then   \n102 8: $w  w + 1$   \n103 9: else   \n104 10: $i  i + 1 ; w  1$   \n105 11: end if   \n106 12: end while   \n107 Since our targets are emergent languages, the outputs of the boundary detection algorithm do not   \n108 necessarily mean articulatory boundaries. Instead, we call them hypothetical boundaries (hypo  \n109 boundaries) and refer to the segments split by hypo-boundaries as hypo-segments. Note that there   \n110 are other similar methods such as Kempe [1999]. We chose Tanaka-Ishii [2005] because it performs   \n111 well not only for English but also for Chinese, which has many one-character words. Emergent   \n112 languages might also have such words. With this algorithm, Tanaka-Ishii and Jin [2008] reported   \n113 F-score $= 8 3 . 6 \\%$ for word boundary detection from phonemes in English and F-score $= \\hat { 8 3 . 8 \\% }$   \n114 for word boundary detection from characters in Chinese. They are considerably high scores for   \n115 unsupervised settings.   \n117 We have to define environments, agent architectures, and optimization methods for language emer  \n118 gence simulations. This paper adopts the framework of Chaabouni et al. [2020]. We define an   \n119 environment in Section 3.1, specify the agent architecture and optimization methods in Section 3.2,   \n120 and also give an explanation of the compositionality of emergent languages in Section 3.3. ",
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+        "type": "text",
+        "text": "121 3.1 Signaling Game ",
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+        "text": "122 An environment is formulated based on Lewis’s signaling game [Lewis, 1969]. A signaling game $G$   \n123 consists of a quadruple $( \\mathcal { T } , \\mathcal { M } , S , R )$ , where $\\mathcal { T }$ is an input space, $\\mathcal { M }$ is a message space, $S : { \\mathcal { T } }  { \\mathcal { M } }$   \n124 is a sender agent, and $R : \\mathcal { M }  \\mathcal { T }$ is a receiver agent. The goal is the correct reconstruction   \n125 $i = R ( S ( i ) )$ for all $i \\in \\mathcal { Z }$ . While the input space $\\mathcal { T }$ and the message space $\\mathcal { M }$ are fixed, the agents   \n126 $S , R$ are trained for the goal. An illustration of a signaling game is shown in Figure 1. Following   \n127 Chaabouni et al. [2020], we define $\\mathcal { T }$ as an attribute-value set $\\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ (defined below) and $\\mathcal { M }$ as a set   \n128 of discrete sequences of fixed length $k$ over a finite alphabet $\\mathcal { A }$ : ",
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+        "text": "$$\n{ \\mathcal { T } } \\equiv { \\mathcal { D } } _ { n _ { v a l } } ^ { n _ { a t t } } , { \\mathcal { M } } \\equiv { \\mathcal { A } } ^ { k } = \\{ a _ { 1 } \\cdot \\cdot \\cdot a _ { k } \\mid a _ { j } \\in { \\mathcal { A } } \\} .\n$$",
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+        "text": "129 Attribute-Value Set Let $n _ { a t t } , n _ { v a l }$ be positive integers called the number of attributes and the number of values. Then, an attribute-value set 130 $\\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ is the set of ordered tuples defined as follows: ",
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+        "text": "$$\n\\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } } = \\left\\{ \\left( v _ { 1 } , \\ldots , v _ { n _ { a t t } } \\right) \\ : | \\ : v _ { j } \\in \\left\\{ 1 , \\ldots , n _ { v a l } \\right\\} \\right\\} .\n$$",
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+        "text": "131 This is an abstraction of an attribute-value object paradigm [e.g., Kottur et al., 2017] by Chaabouni 32 et al. [2020]. Intuitively, each index $j$ of a vector $( v _ { 1 } , \\ldots v _ { j } , \\ldots , v _ { n _ { a t t } } )$ is an attribute (e.g., color), while each 33 $v _ { j }$ is an attribute value (e.g., blue, green, red, and purple).7 ",
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+        "text": "3.2 Architecture and Optimization ",
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+        "text": "We follow Chaabouni et al. [2020] as well for the architecture and optimization method. ",
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+        "text": "Architecture Each agent is represented as an encoder-decoder model (Figure 1): the sender decoder and the receiver encoder are based on single-layer GRUs [Cho et al., 2014], while the sender encoder and the receiver decoder are linear functions. Each element $i \\in \\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ has to be vectorized so that it can be fed into or output from the linear functions. Formally, each $\\ddot { i } = ( v _ { 1 } , \\dots , v _ { n _ { a t t } } )$ is converted into the $n _ { a t t } \\times n _ { v a l }$ -dimensional vector which is the concatenation of $n _ { a t t }$ one-hot representations of $v _ { j }$ . During training, the sender samples messages probabilistically. During the test time, it samples them greedily so that it serves as a deterministic function. Similarly, the receiver’s output layer, followed by the Softmax, determines $n _ { a t t }$ categorical distributions over values $\\{ 1 , \\dots , n _ { v a l } \\}$ during training. During the test time, $n _ { a t t }$ values are greedily sampled from the distributions. ",
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+        "text": "Optimization The agents are optimized with the stochastic computation graph [Schulman et al., 2015] that is a combination of REINFORCE [Williams, 1992] and standard backpropagation. The sender is optimized with the former, while the receiver is optimized with the latter. ",
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+        "text": "3.3 Compositionality of Emergent Languages ",
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+        "text": "An attribute-value set $\\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ by Chaabouni et al. [2020] is an extension of an attribute-value setting [Kottur et al., 2017] introduced to measure the compositionality of emergent languages. While the concept of compositionality varies from domain to domain, researchers in this area typically regard it as the disentanglement of representation learning. Kottur et al. [2017], for instance, set up an environment where objects have two attributes: color and shape, each of which has several possible values (e.g., blue, red, ... for color and circle, star, ... for shape). They assumed that if a language is sufficiently compositional, each message would be a composition of symbols denoting the color value and shape value separately. This concept has been the basis for subsequent studies [Li and Bowling, 2019, Andreas, 2019, Ren et al., 2020, Chaabouni et al., 2020]. ",
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+        "text": "158 Topographic Similarity Topographic Similarity (TopSim) [Brighton and Kirby, 2006, Lazaridou   \n159 et al., 2018] is the de facto compositionality measure in emergent communication literature. Suppose   \n160 we have distance functions $d _ { \\mathcal { T } } , d _ { \\mathcal { M } }$ for spaces $\\mathcal { T } , \\mathcal { M }$ , respectively. TopSim is defined as the Spearman   \n161 correlation between distances $d _ { \\ Z } ( i _ { 1 } , i _ { 2 } )$ and $d _ { \\mathcal { M } } ( S ( i _ { 1 } ) , S ( i _ { 2 } ) )$ for all $i _ { 1 } , i _ { 2 } \\in \\mathcal { T }$ s.t. $i _ { 1 } \\neq i _ { 2 }$ . This   \n162 definition reflects an intuition that compositional languages should map similar (resp. dissimilar)   \n163 inputs to similar (resp. dissimilar) messages. Following previous work using attribute-value objects   \n164 [e.g., Chaabouni et al., 2020], we define $d _ { \\mathcal { T } }$ as the Hamming distance and $d _ { \\mathcal { M } }$ as the edit distance.   \n165 Because this paper is about message segmentation, we can consider two types of edit distance. One   \n166 is the “character” edit distance that regards elements $a \\in { \\mathcal { A } }$ as symbols. The other is the “word”   \n167 edit distance that regards hypo-segments as symbols. Let us call the former $C$ -TopSim and the latter   \n168 W-TopSim. ",
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+        "text": "169 4 Problem Definition ",
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+        "type": "text",
+        "text": "170 The purpose of this paper is to study whether Harris’s articulation   \n171 scheme (HAS) also holds in emergent languages. However, this   \n172 question is too vague to answer. We first divide it into the following: ",
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+        "text": "Q1. Does the conditional entropy $H$ decrease monotonically? Q2. Does the branching entropy $h$ repeatedly fall and rise? Q3. Do hypo-boundaries represent meaningful boundaries? ",
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+        "text": "176 Q3 is the same as the original question, except that Q3 is slightly   \n177 more formal. However, we have to answer Q1 and Q2 beforehand,   \n178 because HAS implicitly takes it for granted that $H$ decreases mono  \n179 tonically and $h$ jitters. Although both Q1 and Q2 generally hold in   \n180 natural languages, neither of them is trivial in emergent languages.   \n181 Figure 2 illustrates Q1, Q2, and Q3.   \n182 It is straightforward to answer Q1 and Q2 as we just need to calculate   \n183 $H$ and $h$ . In contrast, Q3 is still vague to answer, since we do not   \n184 have prior knowledge about the boundaries of emergent languages and do not even know if they have   \n185 such boundaries. To mitigate it, we posit the following necessary conditions for Q3. Let $G$ be a game   \n186 $( \\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } } , \\mathcal { A } ^ { k } , S , R )$ . If the answer to Q3 is yes, then: ",
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+        "image_caption": [
+            "Q3:Are They Meaningful? Figure 2: Illustration of questions. "
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+        "text": "C1. the mean number of hypo-boundaries per message should increase as $n _ { a t t }$ increases, C2. the size of the vocabulary (set of all hypo-segments) should increase as $n _ { v a l }$ increases, C3. W-TopSim should be higher than C-TopSim. ",
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+        "text": "190 About C1 and C2 An attribute-value set $\\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ was originally introduced to measure composi  \n191 tionality. Compositionality, in this context, means how symbols in a message separately denote the   \n192 components of meaning. In our case, each segment, or word, can be thought of as a certain unit   \n193 that denotes the attribute values, so that the number of words in a message should increase as the   \n194 corresponding attributes increase. Therefore, if the answer to Q3 is yes, then C1 should be valid.   \n195 Likewise, the size of the vocabulary should be larger in proportion to the number of values $n _ { v a l }$ ,   \n196 motivating C2. Here, we mean by vocabulary the set of all hypo-segments. Note that the message   \n197 length is fixed, because otherwise the number of hypo-segments would be subject to variable message   \n198 length as well as $( n _ { a t t } , n _ { v a l } )$ , and the implication of results would be obscure. ",
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+    {
+        "type": "text",
+        "text": "About C3 C3 comes from the analogy of the linguistic concept called double articulation [Martinet, 1960]. In natural languages, meanings are quite arbitrarily related to the phonemes that construct them. In contrast, the meanings are less arbitrarily related to the words. The phonemes do not denote meaning units but the words do. In our case, for example, the attribute-value object (RED, CIRCLE) seems less compositionally related to the character sequence “r,e,d,c,i,r,c,l,e”, while it seems more compositionally related to the word sequence “red,circle.” This intuition motivates C3. ",
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+        "text": "205 Based on conditions C1, C2, and C3, Q3 is restated as follows: (Q3-1) Does the mean number of   \n206 hypo-boundaries per message increase as $n _ { a t t }$ increases? (Q3-2) Does the vocabulary size increase   \n207 as $n _ { v a l }$ increases? (Q3-3) Is W-TopSim higher than C-TopSim? ",
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+        "type": "text",
+        "text": "5 Experimental Setup ",
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+        "type": "text",
+        "text": "5.1 Parameter Settings ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Input Space $n _ { a t t }$ and $n _ { v a l }$ have to be varied to answer Q3-1, Q3-2, and Q3-3, while the sizes of the input spaces $| \\mathcal { I } | = ( n _ { v a l } ) ^ { n _ { a t t } }$ must be equal to each other to balance the complexities of games. Therefore, we fix $| \\mathcal { T } | = 4 0 9 6$ and vary $( n _ { a t t } , n _ { v a l } )$ as follows: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/450255a198558a23438177f72a9531df6f8f185af355c68ae4db3fa826e14eae.jpg",
+        "text": "$$\n( n _ { a t t } , n _ { v a l } ) \\in \\{ ( 1 , 4 0 9 6 ) , ( 2 , 6 4 ) , ( 3 , 6 ) , ( 4 , 8 ) , ( 6 , 4 ) , ( 1 2 , 2 ) \\} .\n$$",
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+        "type": "text",
+        "text": "213 Message Space The message length $k$ and alphabet $\\mathcal { A }$ have to be determined for a message space   \n214 ${ \\mathcal { M } } = { \\overset { \\sim } { A } } ^ { k }$ . We set $k = 3 2$ , similarly to previous work on $Z L A$ [Chaabouni et al., 2019, Rita et al.,   \n215 2020, Ueda and Washio, 2021] that regards each $a \\in { \\mathcal { A } }$ as a “character.” Note that $k = 3 2$ is set much   \n216 longer than those of previous work on compositionality [Chaabouni et al., 2020, Ren et al., 2020, Li   \n217 and Bowling, 2019] that typically adopts $k \\equiv n _ { a t t }$ as if each symbol $a \\in { \\mathcal { A } }$ were a “word.” We set   \n218 $\\mathcal { A } = \\{ 1 , 2 , \\ldots , 8 \\}$ . Its size $| { \\cal A } |$ should be as small as possible to avoid the problem of data sparsity   \n219 when applying boundary detection, and to ensure that each symbol $a \\in { \\mathcal { A } }$ serves as a “character.”   \n220 In preliminary experiments, we tested $| \\mathcal { A } | \\in \\{ 2 , 4 , 8 , 1 6 \\}$ and found that learning is stable when   \n221 $| { \\bar { \\mathcal { A } } } | \\geq 8$ .   \n222 Architecture and Optimization We follow Chaabouni et al. [2020] for agent arthitectures and   \n223 optmization methods. The hidden size of GRU [Cho et al., 2014] is set to 500, following Chaabouni   \n224 et al. [2020]. All data from an input space $\\mathcal { I } = \\mathcal { D } _ { n _ { v a l } } ^ { n _ { a t t } }$ are used for training. This dataset is upsampled   \n225 to 100 times following the default setting of the code of Chaabouni et al. [2020]. The learning rate is   \n226 set to 0.001, which also follows Chaabouni et al. [2020]. Based on our preliminary experiments to   \n227 explore stable learning, a sender $S$ and a receiver $R$ are trained for 200 epochs and the coefficient of   \n228 the entropy regularizer is set to 0.01. ",
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+        "type": "text",
+        "text": "Boundary Detection Algorithm The boundary detection algorithm involves a parameter threshold. Since the appropriate value of threshold is unclear, we vary threshold as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/87a8ce53ecf7880f0e07e128b239d9728ea6b8e260ff9ce70496456bb2d04710.jpg",
+        "text": "$$\nt h r e s h o l d \\in \\{ 0 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 , 1 . 2 5 , 1 . 5 , 1 . 7 5 , 2 \\} .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "31 5.2 Implementation, Number of Trials, and Language Validity ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We implemented the code for training agents using the EGG toolkit [Kharitonov et al., 2019].8 EGG also includes the implementation code of Chaabouni et al. [2020], which we largely refer to. They are published under the MIT license. For now, our code is available on Anonymous GitHub.9 For each $( n _ { a t t } , n _ { v a l } )$ configuration, agents are trained 8 times with different random seeds. Each run took a few hours with a single GPU.10 In the following sections, an emergent language with a communication success rate of more than $90 \\%$ is called a successful language. ",
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+    {
+        "type": "text",
+        "text": "6 Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As a result of training agents, we obtained 7, 8, 6, 8, 7, and 6 successful languages out of 8 runs for configurations $( n _ { a t t } , \\bar { n } _ { v a l } ) = ( 1 , 4 0 9 6 )$ , (2, 64), (3, 16), $( 4 , 8 )$ , $( 6 , 4 )$ , and $( 1 2 , 2 )$ , respectively. ",
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+    {
+        "type": "text",
+        "text": "6.1 Conditional Entropy Monotonically Decreases ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To verify Q1, we show the conditional entropy $H ( n )$ (Eq. 2) in Figure 3. In Figure 3, the conditional entropies of the successful languages (solid red lines) decrease monotonically. This confirms Q1 in successful languages. Interestingly, the conditional entropies of emergent languages derived from untrained senders do not necessarily decrease, shown as dashed blue lines in Figure 3.11 The monotonic decrease in conditional entropy emerges after training agents. ",
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+        "img_path": "images/ee6db7b4fef5d383c1245c9afc7fff843e615985e0866f249f1afd85d540eff8.jpg",
+        "image_caption": [
+            "Figure 3: Conditional entropy $H ( n )$ . Dashed blue lines represent $H ( n )$ of languages from untrained agents that finally learned successful languages, while solid red lines represent $H ( n )$ of successful languages. "
+        ],
+        "image_footnote": [],
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+    {
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+        "img_path": "images/85a714388a0f655501234aacc7a619167f4889df8dafde311d30fcb8b4a68a2d.jpg",
+        "image_caption": [
+            "Figure 4: Example transition sequences of the branching entropy $h$ in a message $^ { \\dots } 3 , 8 , 4 , . . . , 4 , 4 , 4 ^ { , , }$ in a successful language for $\\begin{array} { r l } { ( n _ { a t t } , n _ { v a l } ) } & { { } = } \\end{array}$ (2, 64). "
+        ],
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+    {
+        "type": "text",
+        "text": "47 6.2 Branching Entropy Repeatedly Falls and Rises ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Next, to answer Q2, we computed the branching entropy $h ( s )$ (Eq. 1) of the successful languages and applied boundary detection. As an example, we show a few actual transitions of $h ( s )$ in Figure 4, in which y-axis represents the value of $h ( s )$ and $\\mathbf { X }$ -axis represents a message $^ { \\dots } 3 , 8 , 4 , . . . , 4 , 4 , 4 ^ { , , }$ . The message is randomly sampled from a successful language when $( n _ { a t t } , n _ { v a l } ) \\bar { = } ( 2 , 6 4 )$ . The boundary detection algorithm with threshold $= 1$ yields three hypo-boundaries that are represented as dashed black lines in Figure 4. Blue, yellow and green lines with triangle markers represent the transitions of $h ( s )$ that yield hypo-boundaries. Note that the $( i + 1 )$ -th transition of $h ( s )$ does not necessarily start from the $i$ -th hypo-boundary, due to the definition of the algorithm. For instance, the second transition overlaps the first hypo-boundary. While the conditional entropy decreases monotonically as shown in Figure 3, the branching entropy repeatedly falls and rises in Figure 4. Moreover, we show the mean number of hypo-boundaries per message in Figure 5. Figure 5 indicates that for any $( n _ { a t t } , n _ { v a l } )$ configuration, there are hypo-boundaries if threshold $< 2$ , i.e., the brancing entropy repeatedly falls and rises. These results validate Q2. ",
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+    {
+        "type": "text",
+        "text": "6.3 Hypo-Boundaries May Not Be Meaningful Boundaries ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Next, we investigate whether Q3-1, Q3-2, and Q3-3 hold in successful languages. The results in the following sections falsify all of them. Thus, Q3 may not be true: hypo-boundaries may not be meaningful boundaries. ",
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+        "type": "text",
+        "text": "Mean Number of Hypo-Boundaries per Message See Figure 5 again. The figure shows that the mean number of hypo-boundaries per message does not increase as $n _ { a t t }$ increases. It does not decrease, either. This result falsifies Q3-1. Even when $n _ { a t t } = 1$ , there are as many hypo-boundaries as other configurations. ",
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+    {
+        "type": "text",
+        "text": "Vocabulary Size Figure 6 shows the mean vocabulary sizes for each $( n _ { a t t } , n _ { v a l } )$ . The vocabulary size does not increase as $n _ { v a l }$ increases, which falsifies Q3-2. However, focusing on $( n _ { a t t } , n _ { v a l } ) \\in$ $\\{ ( 2 , 6 4 ) , ( 3 , 1 6 ) , ( 4 , 8 ) , ( 6 , 4 ) \\}$ and $0 . 2 5 \\leq t h r e s h o l d \\leq 1$ , there is a weak tendency to support C2. It suggests that hypo-segments are not completely meaningless either. ",
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+        "text": "273 C-TopSim vs W-TopSim Figure 7 shows C-Topsim and W-Topsim for each $( n _ { a t t } , n _ { v a l } )$ and   \n274 threshold.12 Note that C-TopSim is TopSim with “character” edit distance and W-TopSim is TopSim   \n275 with “word” edit distance. In Figure 7, threshold $: = - \\infty$ corresponds to C-TopSim, while the others   \n276 correspond to W-TopSim. 13 Our assumption in Q3-3 was C-TopSim $< \\mathrm { W }$ -TopSim. On the contrary,   \n277 Figure 7 shows a clear tendency for C-TopSim $>$ W-TopSim, which falsifies Q3-3. Hypo-boundaries   \n278 may not be meaningful. However, they may not be completely meaningless, either. This is because the   \n279 hypo-boundary-based W-TopSim is higher than the random-boundary-based W-TopSim in Figure 8.   \n280 Here, we mean by random boundaries the boundaries chosen at random in the same number as   \n281 hypo-boundaries in each message. Other $( n _ { a t t } , n _ { v a l } )$ configurations show similar tendencies (see   \n282 Appendix B). ",
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+        "image_caption": [
+            "Figure 5: Mean number of hypo-boundaries per message in successful languages. threshold varies according to Eq. 7. Each data point is averaged over random seeds and shaded regions represent one standard error of mean (SEM). "
+        ],
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+        "image_caption": [
+            "Figure 6: Vocabulary size in successful langauges. threshold varies according to Eq. 7. Each data point is averaged over random seeds and shaded regions represent one SEM. "
+        ],
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+        "image_caption": [
+            "Figure 7: C-TopSim and W-TopSim in successful languages. threshold $= - \\infty$ corresponds to $\\mathrm { C } \\mathrm { - }$ TopSim, while other threshold correspond to WTopSim. Each data point is averaged over random seeds and shaded regions represent one SEM. "
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+        "image_caption": [
+            "Figure 8: hypo-boundary-based W-TopSim compared to random-boundary-based W-TopSim in successful languages for $( n _ { a t t } , n _ { v a l } ) = ( 2 , 6 4 )$ . Each data point is averaged over random seeds and shaded regions represent one SEM. "
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+    {
+        "type": "text",
+        "text": "6.4 Further Investigation: Word Length and Word Frequency ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The results so far are related to compositionality of emergent languages [e.g., Kottur et al., 2017]. In this section, we further associate our results with previous discussions on Zipf’s law of abbreviation (ZLA) in emergent languages [Chaabouni et al., 2019, Rita et al., 2020, Ueda and Washio, 2021]. ZLA is known as a statistical property in natural languages that the more frequently a word is used, ",
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+    {
+        "type": "text",
+        "text": "288 the shorter it is [Zipf, 1935]. By considering hypo-segments as “words,” we can check whether   \n289 hypo-segments follow ZLA. Figure 9 shows the hypo-segment lengths sorted by frequency rank for   \n290 $( \\ddot { n _ { a t t } } , \\bar { n _ { v a l } } ) = ( 1 , 4 0 9 6 )$ .14 If hypo-segments follow ZLA ideally, they should show a monotonic   \n291 increase. The distribution of the lengths of the hypo-segments shows a clear ZLA-like tendency for   \n292 threshold $\\in \\{ 0 , 0 . 5 \\}$ , although the tendencies are less clear for the other threshold.15 It means that   \n293 hypo-segments follow ZLA with an appropriate threshold value. Other $( n _ { a t t } , n _ { v a l } )$ configurations   \n294 show similar tendencies (see Appendix C). ",
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+    {
+        "type": "text",
+        "text": "295 7 Discussion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In Section 6.1, we showed that the conditional entropy $H ( n )$ decreases monotonically in emergent languages, confirming Q1. In Section 6.2, we demonstrated that the branching entropy $h ( s )$ repeatedly falls and rises in emergent languages, which confirms Q2. It is an intriguing result, considering the discussions of Kharitonov et al. [2020], who showed that the entropy decreases to the minimum for successful communication if the message length $k = 1$ . In contrast, our results suggest that the (branching) entropy does not simply fall to the minimum when the message length $k$ is longer. However, in Section 6.3, our results indicate that the hypo-boundaries may not be meaningful since Q3-1, Q3-2, and Q3-3 were falsified. ",
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+        "page_idx": 8
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+    {
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+        "img_path": "images/42c0f8bdebb211ba6d62d3cf8b9198a89d2f62e99463fb41d4b0065427d67a66.jpg",
+        "image_caption": [
+            "Figure 9: Hypo-segment lengths sorted by frequency rank for $( n _ { a t t } , n _ { v a l } ) \\stackrel { - } { = } ( 1 , 4 0 9 6 )$ . Each data point is averaged over random seeds and shaded regions represent one SEM. "
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+        "text": "312 Nevertheless, hypo-boundaries may not be completely meaningless either. This is because the   \n313 hypo-boundary-based W-TopSim is higher than the random-boundary-based W-TopSim. It suggests   \n314 that HAS-based boundary detection worked to some extent. In addition, the hypo-segments show   \n315 ZLA-like tendencies with certain threshold values. This is a suggestive result because we neither   \n316 imposed a length penalty on messages [Chaabouni et al., 2020], modeled the laziness/impatience of   \n317 agents [Rita et al., 2020], nor modeled short-term memories [Ueda and Washio, 2021]. Of course, it   \n318 is important to note that it may be just an artifact, analogous to the fact that even a monkey typing   \n319 sequence divided by the “white space” follows ZLA [Miller, 1957].   \n320 This paper showed that there is a gap between emergent and natural languages in terms of word   \n321 segmentation. There are some potential methods to bridge the gap. For example, several methods   \n322 have been proposed to facilitate the compositionality of emergent languages, such as iterated learning   \n323 [Ren et al., 2020], the ease-of-teaching paradigm [Li and Bowling, 2019], and concept game [Mu and   \n324 Goodman, 2021]. The regularizations for ZLA mentioned above might also help for this purpose.   \n325 These are left for future work. ",
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+        "text": "26 8 Conclusion ",
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+        "text": "In this paper, we investigated whether Harris’s articulation scheme (HAS) also holds in emergent languages. Emergent languages are artificial communication protocols emerging between agents, while HAS is a statistical universal in natural languages. HAS can be used for unsupervised word segmentation. Our experimental results suggest that although emergent languages satisfy some prerequisites for HAS, HAS-based word boundaries may not be meaningful. Our contributions are (1) to focus on the word segmentation of emergent languages, (2) to pose verifiable questions to answer whether emergent languages have meaningful segments, and (3) to show another gap between emergent and natural languages. It is left for future work to bridge the gap between emergent and natural languages in terms of HAS. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See Section 7.   \n(c) Did you discuss any potential negative societal impacts of your work? [No] The main point of this paper is analysis of artificially emerging languages without human participants. It is hard to imagine that it has a direct negative impact on society. However, our paper may have to do with energy issues, since we used GPU. We wrote about the GPU resource in Section 5.2.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ",
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