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+# UNSUPERVISED DOMAIN ADAPTATION WITH IMPUTATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Motivated by practical applications, we consider unsupervised domain adaptation for classification problems, in the presence of missing data in the target domain. More precisely, we focus on the case where there is a domain shift between source and target domains, while some components in the target data are systematically absent. We propose a way to impute non-stochastic missing data for a classification task by leveraging supervision from a complete source domain through domain adaptation. We introduce a single model performing joint domain adaptation, imputation and classification which is shown to perform well under various representative divergence families $\mathcal { H }$ -divergence, Optimal Transport). We perform experiments on two families of datasets: a classical digit classification benchmark commonly used in domain adaptation papers and real world digital advertising datasets, on which we evaluate our model’s classification performance in an unsupervised setting. We analyze its behavior showing the benefit of explicitly imputing non-stochastic missing data jointly with domain adaptation.
+
+# 1 INTRODUCTION
+
+When dealing with machine learning applications in the real world, data usually come with several imperfections that make classical algorithms hardly deployable. One of these issues is that data are often incomplete. Typically, while capturing data coming from different locations with several sensors per location, a sensor may randomly fail or even may be just missing at a given location. Such a situation can also occur in disease diagnosis in multi-modal medical imaging where one of the modalities fails or is not available; for example the positron emission tomography (PET) modality which reveals metabolic information for clinical tests requires ingesting a radioactive tracer which poses health risks and is often missing (Cai et al., 2018). Similarly, in computational advertising applications, information is missing for users who do not have a prior history on a merchant’s website while their global clicking behavior across websites may be known. Another common issue is that data used for training and deployment may differ in their generation process: data may be collected on different devices, background noise or compression schemes may affect differently training or deployment data, leading to a shift in data distribution. This has given rise to the important literature of Domain Adaptation (Pan & Yang, 2010; Kouw & Loog, 2019). These two issues are usually independently addressed by developing models handling only the missing data or the domain adaptation problem.
+
+In this paper, motivated by practical advertising applications, we consider unsupervised domain adaptation (i.e. labels are not available in the target domain) for classification when (1) part of input data is missing in the target domain thus requiring some form of imputation, (2) there is no possible supervision in the target domain for imputation thus requiring indirect supervision from the source domain, and (3) there exists a domain shift between the source and target distributions requiring domain adaptation. More precisely we consider this adaptation-imputation setting for nonstochastic missing data, i.e. when the same features are missing for all target samples. This contrasts with many imputation problems which take benefit of stochasticity in missing features.
+
+We propose a model that handles unsupervised domain shift and missing data assuming nonstochastic missing data in the target domain. The model learns to perform imputation for the target domain while aligning the distributions of the source and target domains in a latent space, thus going beyond the simple juxtaposition of a data imputation module followed by a domain-invariant feature representation learning module. Imputation makes use of an indirect supervision from the complete source domain. This key property allows us to handle non-stochastic missing data, while satisfying the constraints related to adaptation and to the classification objective. The imputation process plays several roles in our global architecture as it provides us with information about the missing data for the target domain while contributing to the domain-invariant loss and the reconstruction loss. Extensive empirical evidence on handwritten digits and Click-Through-Rate prediction (CTR) domain adaptation problems illustrate the benefit of the proposed model.
+
+The original contributions are the following:
+
+• We introduce a new problem $:$ joint unsupervised domain adaptation and imputation for classification motivated by practical applications;   
+• We propose a new model for handling the problem end-to-end. It learns to generate relevant missing information while aligning source and target distributions in a latent space and to classify source instances;   
+• We evaluate the model not only on academic benchmarks but also on challenging real world advertising data.
+
+# 2 RELATED WORK
+
+We review below typical related work for domain adaptation and data imputation.
+
+# 2.1 UNSUPERVISED DOMAIN ADAPTATION (UDA)
+
+A number of shallow learning methods approach Domain Adaptation by weighting individual observations during training. These methods focus either on data importance-weighting (Cortes & Mohri, 2014; Zadrozny, 2014) or on class importance-weighting (Z. Lipton & Smola, 2018). Recent deep learning methods try to align the distributions of the two domains, for example by embedding them in a joint latent space. There are two main directions for learning joint embeddings. One is based on adversarial training, making use of GAN extensions. The other one directly exploits explicit distance measures between distributions such as Wasserstein or Maximum Mean Discrepancy (MMD). For the former, the seminal work of Ganin & Lempitsky (2015) learns to map source and target domains onto a common embedding space, by optimizing a double objective: on the one hand they minimize an approximation of the $\mathcal { H }$ -divergence between the source and target embeddings via adversarial training, on the other hand they learn to classify the source data embeddings. This influential work has been followed by several extensions and variants. ADDA (Tzeng et al., 2017) advocates the use of two different mappings for the source and the target domains based on the argument that this is more suitable when the marginals are different in the two domains. Liu & Tuzel (2016) trains coupled generative adversarial network (CoGAN) for learning a joint distribution of multi-domain images, that can be used for UDA. Bousmalis et al. (2017) use a generator to map the source to the target domain while training the classifier on the learned representations using source labels. CDAN (Long et al., 2017) improves the domain discriminator by conditioning it on classifier predictions.
+
+A second family of approaches proposes metric based divergences such as MMD (Long et al., 2015) for measuring the loss between source and target representations. DeepJDOT (Damodaran & Kellenberger, 2018) makes use of an optimal transport formulation to align the joint distributions in a latent space. In addition to feature alignment they perform label distribution alignment following Courty et al. (2017). All these works rely on the assumption of covariate shift and consider that full input data is available for both source and target domains. Our two models (ADV and OT) can be seen respectively as extensions of Ganin & Lempitsky (2015) and Damodaran & Kellenberger (2018) for the missing data problem.
+
+# 2.2 IMPUTATION
+
+Data imputation is a classical problem addressed by several methods (Little & Rubin, 2002; Van Buuren, 2018; Murray, 2018). The usual setting is different from ours since it considers reconstructing the whole missing data in the input space, while we consider 1) reconstruction in a latent space and 2) partial reconstruction since we are interested in the information relevant to the classification task only. Recent generative models like GANs (Goodfellow et al., 2014) or VAEs (Kingma & Welling, 2013; Rezende et al., 2014) have been adapted for data imputation in Yoon et al. (2018) and Mattei & Frellsen (2019) respectively. GAIN (Yoon et al., 2018) is an extension of conditional GANs where the generator takes as input an incomplete data and performs imputation while the discriminator is trained to guess for each sample if each variable is original or imputed. Mattei & Frellsen (2019) suggests a method based on deep latent variable models and importance sampling that offers tighter likelihood bound compared to the standard VAE bound. Most approaches consider a supervised setting where 1) paired complete and incomplete data are available and 2) missingness corresponds to a stochastic process (e.g. a mask distribution for tabular data), 3) imputation is performed in the original feature space. Note that this is different from our setting where there is no direct supervision (supervision is only provided indirectly through the source domain) and missingness is non-stochastic which makes the problem harder since one cannot compute statistics on different incomplete samples. The general approach with generative models is to learn a distribution over imputed data which is similar to the one of plain data. This comes in many different instances and usually, generative training alone is not sufficient; additional loss terms are often used. In paired problems, i.e. when each missing datum is associated to a plain version of the datum, these additional terms consist of a reconstruction term imposed by a MSE contraint (Isola et al., 2016b). In unpaired problems a cycle-consistency loss is imposed as in Zhu et al. (2017). Li et al. (2019); Pajot et al. (2019) are among the very few approaches addressing unsupervised imputation in which full instances are never directly used. Both extend the AmbientGAN (Bora et al., 2018) framework and consider stochastic missingness.
+
+Our imputation problem is closer to the ones addressed in some forms of inpainting or for multimodality missing data. The former problem is addressed e.g. in Pathak et al. (2016) who proposes an encoder-decoder model trained according to a joint reconstruction and adversarial loss. The latter is addressed in Cai et al. (2018) who considers the case of multi-modality when one or more modalities are systematically absent, but they do not consider adaptation. They propose to learn to reconstruct the missing modality distribution conditionally to the observed one. Both approaches are fully supervised. Ding et al. (2014) is the only paper we are aware of that considers imputation as we do. Their approach is based on low rank constraints and dictionary learning to guide the transfer between domains. We do not use this method as a baseline due to the complexity and running time of this method which relies on singular value decompositions and dictionnary learning.
+
+# 3 PROBLEM DEFINITION
+
+Let us denote respectively $( x _ { S } , y _ { S } ) \in \mathbb { R } ^ { n } \times \mathbb { R }$ and $( x _ { T } , y _ { T } ) \in \mathbb { R } ^ { n } \times \mathbb { R }$ , data from the source and target domains where $x _ { - }$ is an input, $y _ { - }$ the associated label and $n$ is the dimension of the input space. “−” holds for either source $S$ or target $T$ . The joint distribution on each domain is denoted respectively $p _ { S } ( X , Y )$ and $p _ { T } ( X , Y )$ . We consider that $x _ { - }$ has two components, $x _ { - } = ( x _ { - 1 } , x _ { - 2 } )$ . The problem we address is Unsupervised Domain Adaptation (UDA) with missing features in the target domain. More precisely, we make the following hypotheses. Missingness: We assume that features are the same across domains and that source features $x _ { S } = ( x _ { S _ { 1 } } , x _ { S _ { 2 } } )$ are always available while in the target domain only $x _ { T _ { 1 } }$ is available and $x _ { T _ { 2 } }$ is systematically missing. For advertising applications for example, $x$ would characterize the user browsing behaviour on merchant sites; $x _ { - _ { 1 } }$ characterizes global user features aggregated over his navigation history, which are known for all users; $x _ { - _ { 2 } }$ characterizes user history on a target merchant site. Source domain would consist of all users who already visited this merchant site and target domain of users who never visited this site. UDA: we assume that source labels $y _ { S }$ are available whereas target labels $y _ { T }$ are unknown. Covariate shift: we assume covariate shift as in most UDA papers e.g. Ganin & Lempitsky (2015).
+
+# 4 ADAPTATION-IMPUTATION MODEL
+
+As in many generative approaches to UDA, the objective is to project source and target data onto a common latent space in which data distributions from the two domains match, and to learn a classifier on the source data that performs well on the target domain. The novelty of our approach is to offer a solution to deal with datasets in which some information, $x _ { T _ { 2 } }$ , is systematically missing in the target domain. Our model, denoted Adaptation-Imputation, performs three operations jointly: imputation of missing information for the target data, alignment of the distributions of source and target, and classification of source instances. The three operations are performed in a joint embedding space and all the model’s components are trained together. The term imputation is used here in a broad sense: our goal is not to recover the whole missing $x _ { T _ { 2 } }$ , but to recover information from $x _ { T _ { 2 } }$ that will be useful for adaptation and for the target data classification objective. This is achieved via a generative model, which for a given datum in the target domain and conditionally on the available information $x _ { T _ { 1 } }$ , attempts to generate the required missing information. Because $x _ { T _ { 2 } }$ is systematically missing for target data, there is no possible supervision in the target domain; instead we use distant supervision from the source data while transferring to the target domain. We consider two variants of the same model based on different divergence measures between source and target distributions: the Wasserstein distance and the $\mathcal { H }$ -divergence approximated through adversarial training. For simplicity we describe in the main text the adversarial version ADV and defer the Optimal Transport OT description to Appendix B. We report the results obtained with both models in Section 5.
+
+# 4.1 TRAINING
+
+Our model is composed of three different modules responsible for adaptation, imputation and classification, that share parameters and are trained in parallel. For simplicity, we describe each component in turn, but it should be reminded that they all interact and that their parameters are all optimized according to the three objectives mentioned above. The interaction is discussed after the individual module descriptions. The model’s components are illustrated in Figure 1 (a).
+
+Adaptation The latent space representations of source and target domains are denoted with a tilde notation: $\widetilde { x } _ { S } = ( \widetilde { x } _ { S _ { 1 } } , \widetilde { x } _ { S _ { 2 } } )$ and $\widetilde { x } _ { T } = ( \widetilde { x } _ { T _ { 1 } } , \widetilde { x } _ { T _ { 2 } } )$ . Referring to Figure 1 (a), $\widetilde { x } _ { - 1 } = g _ { 1 } ( x _ { - 1 } )$ , $( \widetilde { x } _ { - 1 }$ edenotes either $\widetilde { x } _ { S _ { 1 } }$ or $\widetilde { x } _ { T _ { 1 } }$ e e e) is the mapping of the observed component $x _ { - _ { 1 } }$ e eonto the latent space and $\widetilde x _ { - , { \scriptsize 2 } } = h \circ g _ { 1 } ( x _ { - , { \scriptsize 1 } } )$ is the second component’s latent representation generated from $x _ { - 1 }$ . This egeneration mechanism will be described later. Adaptation aligns the distributions $( \widetilde { x } _ { S _ { 1 } } , \widetilde { x } _ { S _ { 2 } } )$ and $( \widetilde { x } _ { T _ { 1 } } , \widetilde { x } _ { T _ { 2 } } )$ e e in the latent space. For the ADV model, alignment is performed via a classical adversarial e eloss operating on the latent representations:
+
+$$
+{ \cal L } _ { 1 } = { \cal E } _ { x \sim p _ { S } ( X ) } \log D _ { 1 } ( \widetilde { x } _ { S } ) + { \cal E } _ { x \sim p _ { T } ( X ) } \log ( 1 - D _ { 1 } ( \widetilde { x } _ { T } ) )
+$$
+
+where $D _ { 1 } ( \widetilde { x } )$ represents the probability that $\widetilde { x }$ comes from the source rather than the target.
+
+Imputation Imputation amounts at generating an encoding $\widetilde { x } _ { T _ { 2 } }$ , in the latent space, for the missing information in the target data, conditioned on the available information $x _ { T _ { 1 } }$ . Our objective here is to generate missing information which is relevant for the classification objective. Since we never have access to any target component $x _ { T _ { 2 } }$ , we learn to perform imputation based on the source data. More precisely, we learn to generate $\widetilde { x } _ { S _ { 2 } }$ from $x _ { S _ { 1 } }$ through the generator $h$ , $\widetilde x _ { S _ { 2 } } = h \circ g _ { 1 } ( x _ { S _ { 1 } } )$ , as depicted in Figure 1. We want $h$ eto generate the missing information $\widetilde { x } _ { S _ { 2 } }$ eassociated to the observed $x _ { S _ { 1 } }$ . For ethat we perform two operations in parallel. First, we align the distribution of $\widetilde { x } _ { S _ { 2 } }$ with the distribution of $\widehat { x } _ { S _ { 2 } } = g _ { 2 } ( x _ { S _ { 2 } } )$ , that is a direct mapping of $x _ { S _ { 2 } }$ eonto the shared latent space, using an adversarial bloss described below. The intuition is that both $g _ { 1 }$ and $g _ { 2 }$ are simple mappings operating respectively on $x _ { S _ { 1 } }$ and $x _ { S _ { 2 } }$ while $h$ acts as a generator conditioned on $x _ { S _ { 1 } }$ for generating $\widetilde { x } _ { S _ { 2 } }$ . Moreover, we not only impose this distribution alignment, but would also like $\widetilde { x } _ { S _ { 2 } }$ eto represent missing information relative to $x _ { S _ { 2 } }$ and associated to a specific $x _ { S _ { 1 } }$ e. For that, we use a reconstruction term in parallel to the above alignment, in our case a MSE distance between $\widetilde { x } _ { S _ { 2 } }$ and $\widehat { x } _ { S _ { 2 } }$ . This MSE term guarantees that the imputed $\widetilde { x } _ { S _ { 2 } }$ etruly represents information present in $x _ { S _ { 2 } }$ b. Similar ideas combine distribution ematching and MSE conditioning and have been used e.g. in Isola et al. (2016a); Pathak et al. (2016). The learned mappings are used to perform imputation on the target data $\widetilde x _ { T _ { 2 } } = h \circ g _ { 1 } ( x _ { T _ { 1 } } )$ .
+
+The imputation loss has thus two components. The first is the adversarial term $L _ { A D V }$ responsible for aligning $\widetilde { x } _ { S _ { 2 } }$ and $\widehat { x } _ { S _ { 2 } }$ , $L _ { A D V } = E _ { x _ { 2 } \sim p _ { S } ( X _ { 2 } ) } \log D _ { 2 } ( \widehat { x } _ { S _ { 2 } } ) + E _ { x _ { 1 } \sim p _ { S } ( X _ { 1 } ) } \log ( 1 - D _ { 2 } ( \widetilde { x } _ { S _ { 2 } } ) )$ . The second is the reconstruction term $L _ { M S E } = { \cal E } _ { x \sim p _ { S } ( X ) } \left\| \widetilde { x } _ { S _ { 2 } } - \widehat { x } _ { S _ { 2 } } \right\| _ { 2 } ^ { 2 }$ . The total imputation loss is then:
+
+$$
+{ \cal L } _ { 2 } = \lambda _ { A D V } \times { \cal L } _ { A D V } + \lambda _ { M S E } \times { \cal L } _ { M S E }
+$$
+
+where $\lambda _ { A D V } , \lambda _ { M S E }$ are hyperparameters. The two processes of imputation and adaptation influence each other. Both are also influenced by the classification process described below. Its effect
+
+on imputation is to force the generated $\widetilde { x } _ { S _ { 2 } }$ to contain information about $x _ { S _ { 2 } }$ relevant for the clasesification task. This information is transferred via adaptation to the target domain when generating $\widetilde { x } _ { T _ { 2 } }$ .
+
+Classification The last component of the model is a classifier $f$ , trained on the source domain mapping $\widetilde { x } _ { S }$ for the classification task as classically done for UDA. The corresponding loss is:
+
+$$
+{ \cal L } _ { 3 } = E _ { x \sim p _ { S } ( X ) } { \cal L } _ { D i s c } ( f ( \widetilde { x } _ { S } ) , y _ { S } )
+$$
+
+where $\boldsymbol { L _ { D i s c } }$ is typically a cross-entropy loss.
+
+Overall loss The overall loss function $L$ will be the weighted sum of the adaptation, imputation and classification losses:
+
+$$
+L = \lambda _ { 1 } \times L _ { 1 } + \lambda _ { 2 } \times L _ { 2 } + \lambda _ { 3 } \times L _ { 3 }
+$$
+
+where $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$ are hyperparameters and the final optimization problem is:
+
+$$
+\operatorname* { m i n } _ { g _ { 1 } , g _ { 2 } , h , f } \operatorname* { m a x } _ { D _ { 1 } , D _ { 2 } } L
+$$
+
+Interaction between the model’s components Both mappings $g _ { 1 } , g _ { 2 }$ and generator $h$ appear in the three terms of the loss function in Equation 4, meaning that they should learn to perform the three tasks simultaneously. $g _ { 1 }$ learns to map the $x _ { S _ { 1 } }$ and $x _ { T _ { 1 } }$ components onto the latent space, the mappings being denoted respectively $\widetilde { x } _ { S _ { 1 } }$ and $\widetilde { x } _ { T _ { 1 } }$ . $h$ learns to generate missing information $\widetilde { x } _ { T _ { 2 } }$ from $\widetilde { x } _ { T _ { 1 } }$ . The formed $\widetilde { x } _ { - }$ e eis generated such that it fulfills the classification objective. $g _ { 2 }$ eon its side e eshould fulfill the imputation objective while preserving part of the information present in $x _ { S _ { 2 } }$ . Note that our model makes use of a unique mapping $g _ { 1 }$ for both source and target domains. Separate mappings could have been used for the two domains, but the proposed solution was found to be more robust and to reduce the number of parameters during learning.
+
+![](images/89bb0a571d777313357b50377c54d4e1b63ac2420981a5269f8cadfb76d870a0.jpg)  
+Figure 1: Adaptation-Imputation model: (a) training, (b) inference.
+
+Implementation Let us now detail the implementation of this model. For adversarial training, discriminators $D _ { 1 }$ (adaptation) and $D _ { 2 }$ (imputation) will be implemented by binary classifiers. $D _ { 1 }$ is trained to distinguish between source $\widetilde { x } _ { S }$ and target $\widetilde { x } _ { T }$ mappings while $D _ { 2 }$ is trained to separate imputed $\widetilde { x } _ { S _ { 2 } }$ , generated from $x _ { S _ { 1 } }$ , from $\widehat { x } _ { S _ { 2 } }$ a direct embedding of $x _ { S _ { 2 } }$ . We use the gradient reversal etrick in Ganin $\&$ b Lempitsky (2015) for implementing the min-max condition and define two gradient reversal networks on $D _ { 1 }$ and $D _ { 2 }$ . We use an adaptive update of the scale of the gradients in $D _ { 1 }$ and $D _ { 2 }$ and optimize $L _ { 1 }$ , $L _ { 2 }$ and $L _ { 3 }$ jointly as synthesized in Algorithm 1 in the Appendix. In practise we fix all hyperparameters but $\lambda _ { M S E }$ to 1, additional tuning could yield improved performance.
+
+# 4.2 INFERENCE
+
+At inference, given $x _ { T _ { 1 } }$ , we generate $\widetilde { x } _ { T } = ( \widetilde { x } _ { T _ { 1 } } , \widetilde { x } _ { T _ { 2 } } )$ with $\widetilde { x } _ { T _ { 1 } } = g _ { 1 } ( x _ { T _ { 1 } } )$ an embedding of $x _ { T _ { 1 } }$ and a generated $\widetilde { x } _ { T _ { 2 } }$ e e e e, encoding part of the missing information $x _ { T _ { 2 } }$ in $x _ { T }$ , as illustrated in Figure 1 e(b). We use for the latter the following mapping: $\widetilde x _ { T _ { 2 } } = h \circ g _ { 1 } ( x _ { T _ { 1 } } )$ where $g _ { 1 }$ is as above and $h$ is the generative mapping conditioned on $\widetilde { x } _ { T _ { 1 } }$ e. Finally $\widetilde { x } _ { T }$ is used as input to the classifier $f$ .
+
+# 5 EXPERIMENTS
+
+# 5.1 DATASETS AND EXPERIMENTAL SETTING
+
+Datasets Experiments are performed on two types of datasets. The first one is a classical digits classification benchmark used in many domain adaptation studies which we will refer to as digits and transformed to fit our missing data setting. The second one corresponds to advertising datasets. The task here is binary classification: one wants to predict Click-Through-Rate (CTR) or Conversion Rate (CR) given user behavior. This is one of the problem that has initially motivated our adaptationimputation framework. We use two such datasets: ads-kaggle is a public kaggle dataset1, while ads-real has been gathered internally and corresponds to real advertising traffic. Further details on datasets and preprocessing are presented in Sections 5.2, 5.3 and in Appendix C.
+
+Baselines We report results for the following models:
+
+• Full models: Source-Full is trained without adaptation on the full $x _ { S }$ and tested on full $x _ { T }$ when the latter is available (digits); Adaptation-Full adds adaptation to this model. • Missing models: Source-Missing and Adaptation-Missing do the same but considering full $x _ { S }$ while $x _ { T }$ is incomplete: $x _ { T } = ( x _ { T _ { 1 } } , 0 )$ , i.e. $x _ { T _ { 2 } }$ is set to 0. • Partial models: Source-Partial and Adaptation-Partial is a variant of the above setting where only the first component $x _ { - _ { 1 } }$ for source and target are considered for adaptation and classification while the second ones $x _ { - _ { 2 } }$ are simply ignored. • Imputation models: Adaptation-Imputation corresponds to our model. • Naive model: Naive is used for ads-kaggle to provide a reference loss value for this dataset. It predicts for all examples the mean CTR value as computed using source training data only.
+
+Adaptation-Full is an upper bound of the performance of Adaptation-Imputation since it uses full information while $x _ { T _ { 2 } }$ is not available in practice. Adaptation-Missing and Adaptation-Partial can be considered as lower bounds for our model since they only perform adaptation and no imputation.
+
+Parameters and architecture of the neural networks used for the different models and experiments are presented in Appendix D. Hyperparameters are chosen using the Deep Embedded Validation estimator introduced in You et al. (2019) combined with heuristics and typical UDA values. Further details are given in Appendix D.2.1.
+
+We present the results for digits and ads respectively in Sections 5.2, 5.3. Section 5.4 presents ablation studies. Reported results are mean value and standard deviation over five different initializations; best results are indicated in bold.
+
+# 5.2 DIGITS
+
+Description For digits, we consider the unsupervised adaptation between two datasets among MNIST (LeCun et al. (1998)), USPS (Hull (1994)), SVHN (Netzer et al. (2011)) and MNIST-M (Ganin & Lempitsky (2015)). The direction $\mathrm { M N I S T }  \mathrm { S V H N }$ is not considered as the task is difficult even for traditional UDA (Ganin & Lempitsky, 2015). All tasks are 10-class classification problems. From the complete image datasets, we build datasets with missing input values.
+
+Half digit missing In a first series of experiments, we removed one half of each image, the horizontal bottom part. We report classification accuracy in Table 1 for the different adaptation problems and models (ADV and OT). Removing half of the image leads to a strong performance decrease for Source-Partial and Source-Missing with respect to the upper bounds provided by Source-Full, respectively between 10 and 20 points of accuracy, and between 15 and 45 points. This is partially recovered when training with adaptation (Adaptation- Partial, Adaptation-Missing, for both ADV or OT). But the gap is still important with respect to the upper bound, i.e. Adaptation-Full. In all cases, Adaptation-Imputation clearly increases the performance; between 10 and 25 points of accuracy over Adaptation-Missing and 2 to 20 points over Adaptation-Partial. This is a very significant improvement which validates the importance of the imputation component. In Section 5.4 we show that the simultaneous use of imputation and adaptation is required for reaching this level of performance. Imputation or adaptation alone are well behind the jointly trained instance of the model. However, it does not reach the upper bound performance of Adaptation-Full where the difference lies between 10 and 25 accuracy points. Moreorever, Adaptation-Imputation beats the non-adapted Source-Full baseline on several datasets. Both the ADV and OT versions exhibit the same general behavior. In the reported results in Table 1, ADV performance is higher than OT. This is because performance is highly dependent on the NN architectures and we tuned our NNs for ADV. OT models may reach performance similar to ADV but we find that it requires models with an order of magnitude more parameters. To keep the comparison fair, we thus used the same NN models for both ADV and OT. Imputation models achieve their highest performance when the adaptation task between domains is complex (MNIST MNIST-M, $\mathrm { S V H N } \to \mathrm { M N I S T } )$ illustrating the importance of imputation when transfer is difficult. In all experiments, the performance of --Partial model where ”-” refers to Source or Adaptation, are usually higher than the --Missing model. Our understanding is that setting missing components to zero tends to increase distance between source and target distributions, compared to just ignoring them, making the classification and adaptation problems harder.
+
+Table 1: Classification accuracy performance in $\%$ on digits for the two training criteria on the target domain test set. Standard deviation is in $\%$ .   
+
+<table><tr><td></td><td>MNIST→USPS</td><td>USPS→MNIST</td><td></td><td>SVHN→MNIST</td><td>MNIST</td><td>→MNIST-M</td></tr><tr><td>Method</td><td>ADV OT</td><td>ADV</td><td>OT</td><td>ADV</td><td>OT ADV</td><td>OT</td></tr><tr><td>Source-Full</td><td>71.5±2.7</td><td>74.2±2.7</td><td></td><td>58.1±1.1</td><td></td><td>28.3±1.4</td></tr><tr><td>Adaptation-Full Source-Missing</td><td>88.3±2.4 92.6±1.7 25.7±3.7</td><td>95.0±0.4</td><td>93.9±0.6 39.2±2.6</td><td>77.6±3.576.1±1.4 31.5±2.</td><td>77.2±4.9</td><td>46.9±3.9 14.4±1.1</td></tr><tr><td>Adaptation-Missing</td><td>48.4±4.8 60.9±6.3</td><td>67.5±2.2</td><td>65.3±5.2</td><td>47.1±5.7 37.5±6.2</td><td>34.7±2.5</td><td>20.2±2.5</td></tr><tr><td>Source-Partial</td><td>52.9±9.7</td><td>54.3±1.6</td><td></td><td>44.6±1.9</td><td></td><td>19.1±2.6</td></tr><tr><td>Adaptation-Partial</td><td>71.5±3.2 64.0±5.0</td><td>80.0±1.4</td><td>72.0±1.8</td><td>45.5±1.9 47.9±1.8</td><td>29.4±1.6</td><td>26.8±4.4</td></tr><tr><td>Adaptation-Imputation</td><td>75.2±1.5 66.8±1.3</td><td>81.5±0.8</td><td>72.5±2.7</td><td>54.1±1.4 49.2±1.5</td><td>58.5±1.6</td><td>29.2±1.4</td></tr></table>
+
+Missing patch size In a second group of experiments on digits, we analyze the evolution of the performance of the models with respect to the size of the missing information in the target domain. For that, we vary the size of the missing patch removing a percentage of the image $p$ with $p ~ \in ~ \{ 3 0 \% , 4 0 \% , 5 0 \% , 6 0 \% , 7 0 \% \}$ on SVHN MNIST for ADV models, keeping the same hyperparameters as the ones used for $\begin{array} { r l r } { p } & { { } = } & { \ \bar { 5 } 0 \% } \end{array}$ . We report the mean values over five runs in Figure 2. We notice that Adaptation-Imputation constantly beats the other baselines regardless of the missing patch size. The figure exhibits borderline cases when the size of the missing patch be
+
+![](images/0682ac26dbdf4f1b59600f9e5632a84f355b69205682fba21f25887fd1f4f96b.jpg)  
+Figure 2: Missing patch size study
+
+comes very small $( < 3 0 \%$ ) or very large $( > ~ 6 5 \% )$ ). When the missing patch is too small most of the information for predicting the target label is already available thus simple models perform already well; while when it becomes too big, too few information is available to guarantee efficient reconstructions from the non-missing patch.
+
+# 5.3 ADS
+
+Description We performed a second series of tests on two advertising datasets: ads-kaggle and ads-real. The ads datasets correspond to binary classification problems; the task is to predict the probability that a user exposed to an ad from a target partner (e.g. Booking) on a publisher (e.g. NY Times) will click (ads-kaggle) or make a purchase (ads-real) conditioned on the user history. A row in the dataset corresponds to a display i.e. an ad opportunity of a click or purchase for a given (user, partner) pair at a given time on a given publisher site. The source domain is composed of users who already had interactions with a target partner. The target domain is composed of users with no history on a target partner. We consider all partners in a given traffic. For the two domains, $x _ { - _ { 1 } }$ features correspond to aggregated user features on all the partners, while $x _ { - _ { 2 } }$ corresponds to user - target partner specific interaction which is known for the source domain but unknown for the target domain. Note that besides missingness, there is also an adaptation problem since statistics for new users are usually different from those of known users (e.g. in terms of frequency of a partner’s website visits) as seen in Appendix E. In real datasets, traffic in the source domain is usually abundant while scarce in the target domain. Statistics for each dataset are provided in Table 5 in the Appendix; exact preprocessing used is provided in Appendix C.
+
+Results For this group of experiments, we report the results only for ADV models since the trend has been observed to be similar on digits for both ADV and OT. For ads datasets, missing features do not exist, so we do not report the --Full models’ results on these datasets. The classes being imbalanced, accuracy is not relevant here so we report another performance measure, cross-entropy (CE) between the predicted values and the true labels on the target domain which is considered as the most reliable metric to estimate revenue. Note that given the test set size of ads-kaggle, an improvement of 0.001 in logloss is considered as practically significant (Wang et al., 2017). For the ads problem and for large user bases, a small improvement in prediction accuracy can lead to a large increase in a company’s revenue. For all experiments, we report in Table $2 \ \mathrm { C E }$ on target test for ads-kaggle and ads-real.
+
+A first observation is that Adaptation-Imputation is significantly better than the baselines on both datasets (Table 2). For ads-kaggle it improves by $2 . 3 \%$ the best adaptation model (Adaptation-Missing) while for ads-real the improvement reaches $6 . 3 \%$ over the best second which happens to be Source-Partial. A second observation is that for any model, adaptation consistently
+
+Table 2: CE on ads for ADV models   
+
+<table><tr><td>Dataset</td><td>ads-kaggle</td><td>ads-real</td></tr><tr><td>Naive</td><td>0.403</td><td>X</td></tr><tr><td>Source-Missing Source-Partial</td><td>0.545±0.019 0.406±0.00046</td><td>0.663±0.011</td></tr><tr><td></td><td></td><td>0.622±0.0048</td></tr><tr><td>Adaptation-Missing</td><td>0.397±0.0057</td><td>0.660±0.025</td></tr><tr><td>Adaptation-Partial</td><td>0.403±0.0030</td><td>0.634±0.0082</td></tr><tr><td>Adaptation-Imputation</td><td>0.389±0.014</td><td>0.583±0.013</td></tr></table>
+
+improves over the same model without adaptation. The only exception is the --Partial setting in ads-real. A third observation is that the missing component indeed contains relevant information: CE performance on source data (not reported in Table 2) shows that Source-Missing which exploits the $x _ { - _ { 2 } }$ component is consistently higher than Source-Partial which does not exploit this component, leading to relative gains of the former over the latter of $5 . 6 \%$ on ads-kaggle and $8 . 2 \%$ on ads-real. Adaptation-Imputation is able to generate and to exploit this information.
+
+# 5.4 ABLATION ANALYSIS
+
+We analyze now the role and importance of the different components of our model, and compare with the results from Tables 1 and 2. We perform experimentation on the public datasets, digits and ads-kaggle and report results in Table 3 and Figure 3.
+
+Importance of adaptation We compare the performance of the model with and without the adaptation term $L _ { 1 }$ in Equation 4. When removing adaptation, inference is performed as before, by feeding $\widetilde { x } _ { T } = ( \widetilde { x } _ { T _ { 1 } } , \widetilde { x } _ { T _ { 2 } } )$ to the classifer $f$ . This means that we only rely on the imputation and e e eclassification losses to learn the parameters of the model. Results appear on the top of Table 3. For all datasets, the adaptation component considerably increases the performance, from 10 to 30 points for digits and by a significant 0.009 CE value on ads-kaggle.
+
+Imputation mechanism Imputation, cf. Equation 2, combines adversarial training (ADV) and conditioning on the input datum via the MSE loss (MSE). The objective is to learn from $x _ { S _ { 1 } } \ : \widetilde { x } _ { S _ { 1 } } =$ $g _ { 1 } ( x _ { S _ { 1 } } )$ and to generate missing information in $x _ { S _ { 2 } }$ , $\widetilde { x } _ { S _ { 2 } } = h ( \widetilde { x } _ { S _ { 1 } } )$ e. ADV aligns the distributions of $\widetilde { x } _ { S _ { 2 } }$ and $\widehat { x } _ { S _ { 2 } }$ e ewhile MSE can be thought as performing some form of regression. For a given partial e binformation $x _ { S _ { 1 } }$ , there are possibly several potential $x _ { S _ { 2 } }$ and thus $\widetilde { x } _ { S _ { 2 } }$ . ADV allows to focus on a specific mode of $\widehat { x } _ { S _ { 2 } }$ e, while MSE will favour a mean value of the distribution. Results on Table b3, second group of rows, show that for digits, combining the two influences (MSE and ADV) leads to improved results compared to using separately each loss. MSE alone already provides good performance, while using only ADV is clearly below. For this classification task, identifying the most relevant mode improves the performance over simple regression $( L _ { M S E } )$ . Note that reconstruction is an ill posed problem since the task is inherently ambiguous - different digits may be reconstructed from one half of an image. We performed tests with a stochastic input component in order to recover different modes, but the performance was broadly similar. Achieving diversity with Conditional GANs remains an open research topic (Yang et al., 2019).
+
+For the ads-kaggle dataset, the performance of MSE and $\mathrm { M S E } + \mathrm { \mathbb { A } D V }$ are similar. This is analyzed deeper in an additional series of experiments with several weighted combinations of MSE and ADV. Results are provided in Table 3 third group of rows, for both digits and ads-kaggle and are plotted for ads-kaggle in Figure 3. For digits, this confirms that the equal weights selected for our experiments are indeed generally a good choice reducing the burden of hyperparameter selection, while for ads-kaggle performance could be slightly improved with other weightings. One can see on Figure 3 that ADV induces a high variance in the results (left part of $\mathbf { X }$ -axis) while MSE stabilizes the performance (right part of x-axis). The former allows for bet
+
+![](images/9760eca351843322234cc0eef10faa4684648a5d5de84ed21d8d1d74029deeb6.jpg)  
+Figure 3: ADV-MSE weighting on ads-kaggle
+
+ter maximum performance but with high variance: performance ranges from 0.35 to 0.7 on the target domain. A small contribution from MSE (here $\lambda _ { M S E } = 0 . 0 0 5$ ) stabilizes the results.
+
+Table 3: Accuracy on digits and CE on ads-kaggle for ADV Adaptation-Imputation   
+
+<table><tr><td>Dataset</td><td colspan="4">digits</td><td>ads-kaggle</td></tr><tr><td>Adaptation direction</td><td>MNIST→USPS</td><td>USPS→MNIST</td><td>SVHN→MNIST</td><td>MNIST→MNIST-M</td><td>Known→New</td></tr><tr><td>L2+L3</td><td>64.2±1.8</td><td>51.3±2.5</td><td>44.5±1.4</td><td>24.1±2.6</td><td>0.410±0.0020</td></tr><tr><td>L1+L2+L3</td><td>75.2±1.5</td><td>81.5±0.8</td><td>54.0±1.4</td><td>58.5±1.6</td><td>0.401±0.0014</td></tr><tr><td>LMSE</td><td>71.9±3.7</td><td>81.4±1.2</td><td>52.5±3.7</td><td>56.5±2.8</td><td>0.400±0.0014</td></tr><tr><td>LADV</td><td>28.6±3.2</td><td>39.4±5.2</td><td>28.8±3.8</td><td>30.0±3.7</td><td>0.469±0.13</td></tr><tr><td>LADV+LMSE</td><td>75.2±1.5</td><td>81.5±0.8</td><td>54.0±1.4</td><td>58.5±1.6</td><td>0.401±0.0014</td></tr><tr><td>0.1×LADV+LMSE</td><td>73.4±2.7</td><td>81.3±0.8</td><td>53.0±2.0</td><td>56.2±2.6</td><td>0.401±0.0021</td></tr><tr><td>LADV+0.001×LMSE</td><td>37.3±2.5</td><td>31.2±3.8</td><td>45.0±2.6</td><td>50.0±3.4</td><td>0.440±0.11</td></tr><tr><td>LADV+0.005×LMSE</td><td>47.8±3.7</td><td>49.6±5.8</td><td>46.0±2.6</td><td>50.6±2.2</td><td>0.388±0.015</td></tr><tr><td>LADV +0.01×LMSE</td><td>53.6±2.4</td><td>57.0±3.6</td><td>43.4±1.1</td><td>51.0±2.5</td><td>0.397±0.0046</td></tr><tr><td>LADV+0.1×LMSE</td><td>68.2±4.2</td><td>50.3±6.8</td><td>54.0±2.1</td><td>51.5±3.6</td><td>0.402±0.0046</td></tr><tr><td>LADV+LMSE</td><td>75.2±1.5</td><td>81.5±0.8</td><td>54.0±1.4</td><td>58.5±1.6</td><td>0.401±0.0014</td></tr></table>
+
+# 6 CONCLUSION
+
+We have proposed a new model to solve unsupervised adaptation problems in the presence of nonstochastic noise in the target domain, by using distant supervision from a complete source domain through domain adaptation and imputing missing values on the target domain in a latent space. This method uses only labelled source instances and leads to important gains on classical adaptation benchmarks over baseline models for two representative families of divergences (optimal transport, adversarial training). We have demonstrated on real world advertising datasets that these methods can be used for problems with missing features in advertising. Potential follow-ups include: extending this method to a semi or fully supervised setting on the target domain; considering simultaneously domain and target shift which frequently occurs in real world problems while still being an open problem; introducing increased diversity in the generation process.
+
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+
+# A PSEUDO-CODE
+
+Input: $N$ : number of epochs, $k$ : batch size, $l r _ { i }$ : initial learning rate   
+Initialize parameters of $g _ { 1 } , g _ { 2 } , D _ { 1 } , D _ { 2 } , f , h$   
+while $n _ { e p o c h s } < N$ do Sample $\{ x _ { S } { } ^ { i } , y _ { S } { } ^ { i } \} _ { 1 \leq i \leq k }$ from $p _ { S }$ Sample $\{ x _ { T _ { \downarrow } } ^ { j } \} _ { 1 \leq j \leq k }$ from $p _ { T }$ Decay learning rate and update gradient scale at each batch Compute $L = L _ { 1 } + L _ { 2 } + L _ { 3 }$ performing joint adaptation, imputation, classification Update $D _ { 1 }$ , $D _ { 2 }$ by ascending $L$ through Gradient Reversal Layer Update $f , g _ { 1 } , g _ { 2 } , h$ by descending $L$
+
+end
+
+Algorithm 1: Training procedure for ADV Adaptation-Imputation
+
+# B OT FORMULATION
+
+We present here in more details Adaptation-Imputation using Optimal Transport as a divergence metric. The formulation is slightly different compared to ADV models.
+
+# B.1 FORMULATION OF THE IMPUTATION OT MODEL
+
+We replace the $\mathcal { H }$ -divergence approximation given by the discriminators $D _ { 1 }$ and $D _ { 2 }$ by the Wasserstein distance between source and target instances $( D _ { 1 } )$ and true and imputed feature representations $( D _ { 2 } )$ , following the original ideas in Shen et al. (2018); Damodaran & Kellenberger (2018). In practice, we compute the Wasserstein distance using its primal form by finding a joint coupling matrix $\gamma$ , using a linear programming approach (Peyre et al., 2019). ´
+
+Adaptation In Damodaran & Kellenberger (2018); Courty et al. (2017), the optimal transportation problem is formulated on the joint $( X , Y )$ distributions. Similarly to Shen et al. (2018), in our case, we focus on a plan that acts only on the feature space without taking care of the labels. This leads to the following optimization problem :
+
+$$
+L _ { 1 } = \operatorname* { m i n } _ { f , \gamma _ { 1 } , g } \sum _ { i j } \left( | | \widetilde { x } _ { S _ { 1 i } } - \widetilde { x } _ { T _ { 1 j } } | | ^ { 2 } + | | \widetilde { x } _ { S _ { 2 i } } - \widetilde { x } _ { T _ { 2 j } } | | ^ { 2 } \right) \gamma _ { \mathbf { 1 } i j }
+$$
+
+where $\gamma _ { 1 i j }$ is the alignment value between source instance $i$ and target instance $j$
+
+Imputation For the imputation part, we keep the reconstruction MSE component in Equation 2 and derive the distribution matching loss as:
+
+$$
+L _ { O T } = \operatorname* { m i n } _ { f , \gamma _ { 2 } , g } \sum _ { i j } \rvert | \widetilde x _ { S _ { 2 i } } - \widehat x _ { S _ { 2 j } } | | ^ { 2 } \gamma _ { 2 _ { i j } }
+$$
+
+where $\gamma _ { 2 i j }$ is the alignment value between source instance $i$ and $j$ .
+
+The final imputation loss is:
+
+$$
+L _ { 2 } = \lambda _ { O T } \times L _ { O T } + \lambda _ { M S E } \times L _ { M S E }
+$$
+
+Classification The classification term in Equation 3 is unchanged.
+
+# B.2 IMPLEMENTATION
+
+The optimization problem in Equation 5 is solved in two stages following an alternate optimization strategy:
+
+• we fix all parameters but $\gamma _ { 1 }$ and $\gamma _ { 2 }$ and find the joint coupling matrices $\gamma _ { 1 }$ and $\gamma _ { 2 }$ using EMD
+
+• we fix $\gamma _ { 1 }$ and $\gamma _ { 2 }$ and solve $\mathrm { m i n } _ { g _ { 1 } , g _ { 2 } , h , f } L$
+
+In practice, we first minimize $L _ { 3 }$ for a couple of epochs (taken to be 10 for digits) then minimize $L _ { 1 } + L _ { 2 } + L _ { 3 }$ in the remaining epochs.
+
+Learning rate and parameters are explained further in Section D.
+
+# C DATASET DESCRIPTION
+
+C.1 DIGITS
+
+Table 4: Statistics on digits datasets   
+
+<table><tr><td colspan="4">Train</td><td colspan="2">MNIST</td><td colspan="2">SVHN</td><td>MNIST-M</td><td colspan="2"></td></tr><tr><td rowspan="4">Test Size Channels</td><td colspan="3">7438 1860</td><td colspan="3">60k 10k</td><td colspan="2">73257 26032</td><td>60k 10k</td></tr><tr><td colspan="3">28×28 1</td><td colspan="3">28×28 1</td><td colspan="2">32 ×32 3</td><td>28×28 3</td></tr><tr><td>S</td><td>8</td><td>4</td><td>5</td><td>8</td><td>4</td><td>54824</td><td>5 5e</td></tr><tr><td>0</td><td>1</td><td>1</td><td>0</td><td>公</td><td>D</td><td>2012</td><td>OIQ</td></tr></table>
+
+Pre-processing We scale all images to be $3 2 \times 3 2$ and we normalize the input in $[ - 1 , 1 ]$ . When adaptation involves a domain with three channels (SVHN or MNIST-M) and a domain with a single channel, we simply triplicate the channel of the latter domain. As in Damodaran & Kellenberger (2018) we use balanced source batches which proves to increase performance especially when the source dataset is imbalanced (e.g. SVHN and USPS datasets) while the target dataset (usually MNIST derived) is balanced. Scaling the input images enables us to use the same architecture across datasets. In practise the embedding size is 2048 after preprocessing. For missing versions, we set pixel values to zero in a given patch. Figure 4 shows MNIST-M digits with varying missing patch size.
+
+![](images/2f7ad961c618eb183753b0859fc9288b666445200f375c6d647f3950b7ace71c.jpg)  
+Figure 4: missing-features MNIST-M
+
+Evaluation The digits datasets are provided with a predefined train / test split. We report accuracy results on the target test set and use the source test set as validation set (Section D.2.1). The number of instances in each dataset is reported in Table 4. We run each model five times with a different seed for initialization (using Xavier Uniform initialization) and report mean and standard deviations in Table 1.
+
+Infrastructure Experiments are run using a Tesla V100-SXM2 GPU machine with 32GB RAM.   
+Code is written in PyTorch https://pytorch.org/ (Paszke et al., 2017).
+
+# C.2 ADS
+
+Preprocessing Table 5 lists various statistics on the traffic for the two ads datasets; we describe in this paragraph how datasets are processed.
+
+ads-kaggle The Criteo Kaggle dataset is a reference dataset for CTR prediction and groups one week of log data. The objective is to model the probability that a user will click on a given ad given his navigation context: $p ( C l i c k | C o n t e x t )$ . Positives refer to displays which are clicked and negatives to displays not clicked. For each datum, there are 13 continuous and 26 categorical features. We divide the traffic into two domains using feature number 30; for a given value for this categorical feature, all instances have a single missing numeric feature (feature number 5). We then construct an artificial dataset simulating transfer between known and new users. We process the original Criteo Kaggle dataset to have an equal number of source and target data. We then perform train / test split on this dataset keeping $2 0 \%$ of data for testing. We used in our experiments the continuous features; to show the benefit of modelling additional missing features, we extend the missing features list to features 1, 5, 6, 7, 11 and 12 by setting them to zero on the target domain. After these operations, 6 features are missing and 7 are non-missing. Preprocessing consists in normalizing continuous features using a log transform.
+
+ads-real The data collected in this dataset is similar to ads-kaggle. We filter out non-clicks and the final task is to model the sale probability for a clicked ad: $p ( S a l e | C l i c k = 1 , C o n t e x t )$ . Positives refer to clicked ads which lead to a sale; negatives to clicked ads which did not lead to a sale. We use one week of sampled logs as a training set, and use the following day data as the test set. This train / test definition is used so as to better correlate with the performance of a model used online. We use typical features in production models on this dataset and focus on a sample of continuous features aggregated across user timelines describing the clicking and purchase behavior of a user. In comparison to ads-kaggle more continuous features are used. The count features can be User-centric i.e. describe the global activity of the user (number of clicks, displays, sales done globally across partners) or User-partner features i.e. describing the history of the user on the given partner (number of clicks, sales... on the partner). The latter are missing for new users. Counts are aggregated across varying windows of time and categories of partner catalog. We bucketize these count features using log transforms and consider these as categorical features by one-hot encoding the different buckets based on the vocabulary file. After processing the final input size is 596 with 29 features used. 12 features are missing and 17 are non-missing.
+
+Table 5: Statistics on ads datasets   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=4>ads-kaggle</td><td rowspan=1 colspan=4>ads-real</td></tr><tr><td rowspan=1 colspan=1>Domain</td><td rowspan=1 colspan=2>Source</td><td rowspan=1 colspan=2>Target</td><td rowspan=1 colspan=2>Source</td><td rowspan=1 colspan=2>Target</td></tr><tr><td rowspan=1 colspan=1>Split</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=1 colspan=1>PositiveNegative</td><td rowspan=1 colspan=1>246 872699 621</td><td rowspan=1 colspan=1>61 841174 783</td><td rowspan=1 colspan=1>92 333854 160</td><td rowspan=1 colspan=1>22 943213 681</td><td rowspan=1 colspan=1>XX</td><td rowspan=1 colspan=1>XX</td><td rowspan=2 colspan=1>XX819 073X</td><td rowspan=2 colspan=1>XX147 358X</td></tr><tr><td rowspan=1 colspan=1>TotalP(Y)</td><td rowspan=1 colspan=1>946 4930.2608</td><td rowspan=1 colspan=1>236 6240.2613</td><td rowspan=1 colspan=1>946 4930.0976</td><td rowspan=1 colspan=1>236 6240.0970</td><td rowspan=1 colspan=1>24 465 756X</td><td rowspan=1 colspan=1>3 760 233X</td></tr></table>
+
+Evaluation On both datasets the train and test sets are fixed. We run each model five times with a different seed for initialization (using Xavier Uniform initialization) and report mean and standard deviations in Table 2.
+
+Infrastructure For ads-real, models are trained using internal large-scale distributed infrastructure. For ads-kaggle, we keep the same setting and code as in digits.
+
+# D IMPLEMENTATION DETAILS
+
+# D.1 NEURAL NET ARCHITECTURE
+
+D.1.1 DIGITS
+
+We experiment with the ADV and OT versions of our imputation model. For ADV models, we use the DANN model description in Ganin & Lempitsky (2015); for OT we use the DeepJDOT model description in Damodaran & Kellenberger (2018). Both models can be considered as simplified instances of our corresponding ADV and OT imputation models when no imputation is performed.
+
+Performance of the adaptation models is highly dependent on the NN architectures used for adaptation and classification. In order to perform fair comparisons and since our goal is to evaluate the potential of joint adaptation-imputation-classification, we selected these architectures through preliminary tests and use them for both the ADV and OT models. The two models are described below and illustrated in Figure 5.
+
+• Feature extractors $g _ { 1 }$ and $g _ { 2 }$ consists of three convolutional layers with $5 \times 5$ kernel and 64 filters interleaved with max pooling layers with a stride of 2 and $2 \times 2$ kernel. The final layer has 128 filters. We use batch norm on convolutional layers and ReLU as an activation after max pooling. As in Damodaran & Kellenberger (2018) we find that adding a sigmoid activation layer as final activation is helpful.   
+• Classifier $f$ consists of two fully connected layers with 100 neurons with batch norm and ReLU activation followed by the final softmax layer. We add Dropout as an activation for the first layer of the classifier.   
+• Discriminator $D _ { 1 }$ and $D _ { 2 }$ is a single layer NN with 100 neurons, batch norm and ReLU followed by the final softmax layer. $\mathrm { O n } \mathrm { U S P S } \to \mathrm { M N I S T }$ and $\mathrm { M N I S T } \to \mathrm { U S P S }$ dataset we use a stronger discriminator network which consists of two fully connected layers with 512 neurons.   
+• Generator $h$ consists of two fully connected layers with 512 neurons, batch norm and ReLU activation. This architecture is used for ADV and OT imputation models. In practice using wider and deeper networks increases classification performance with the more complicated classification tasks $( \mathrm { S V H N } \to \mathrm { M N I S T }$ , MNIST MNIST-M); in these cases we add an additional fully connected network with 512 neurons. The final activation function is a sigmoid.
+
+We use the same architecture described above for full, missing and partial models to guarantee fair comparison. As a side note, the input to the imputation model’s classifier is twice bigger as in the standard adaptation models.
+
+![](images/e54feb8d8ac5e00014fdd513fb3aea4b9c3e680a8ebc172ae0429996e94f15f4.jpg)  
+Figure 5: Base architecture for the ADV DANN model
+
+# D.1.2 ADS
+
+We experiment with ADV models only. As input data is numeric and low dimensional, architectures are simpler than in digits.
+
+ads-kaggle Our feature extractor is a three layered NN with 128 neurons and with a final sigmoid activation. The classifier is taken to be a single layered NN with 128 neurons and a final softmax layer. Activations are taken to be ReLUs. The domain discriminator is taken to be a two layered NN with 128 neurons and a final softmax layer. Finally the reconstructor is taken to be a two-layered NN with 256 neurons and final sigmoid activation.
+
+ads-real Input features after processing are fed directly into the feature extractors $g _ { 1 } , g _ { 2 }$ consisting of two fully connected layers with 128 neurons. The classifier and discriminator is taken to be single-layered NN with 25 neurons. The reconstructor is taken to be a two-layered NN with 128 neurons. Inner activations are taken to be ReLUs and the final activation of the feature extractor is taken to be a sigmoid.
+
+# D.2 NETWORK PARAMETERS
+
+# D.2.1 HYPERPARAMETER TUNING
+
+Tuning hyperparameters for UDA is tricky as we do not have access to the label in the target distribution and thus cannot choose parameters minimizing the target risk on a validation set. Several papers set hyperparameters through reverse cross-validation (Ganin & Lempitsky, 2015). Other approaches developed for model selection are based on risk surrogates obtained by estimating an approximation of the risk value on the source based on the similarity of source and target distributions (without the labels). In the experiments, we used a recent estimator, Deep Embedded Validation (DEV) (You et al., 2019) for tuning the initial learning rate; $\lambda _ { 1 }$ and $\lambda _ { O T }$ in OT. For other parameters, we used heuristics and typical hyperparameter values from UDA papers (such as batch size) without further tuning. We use a cross entropy link function on the source validation set; this value provides a proxy for the target test risk. Using parameters from the original paper, this estimator helps select parameter ranges which perform reasonably well. We keep the estimator unchanged for the full and missing adaptation models. In the imputation case the discriminator used for computing importance sampling weights discriminates between $[ \widetilde { x } _ { S _ { 1 } } , \widetilde { x } _ { S _ { 2 } } ]$ and $[ \widetilde { x } _ { T _ { 1 } } , \widetilde { x } _ { T _ { 2 } } ]$ (cf. $D _ { 1 }$ in Figure 1).
+
+# D.2.2 DIGITS
+
+We find that the results are highly dependent on the NN architecture and the training parameter setting. In order to evaluate the gain obtained with Adaptation-Imputation, we use the same NN architecture for all models (ADV and OT) but fine tune the learning rates for each model using the DEV estimator (other parameters do not have a significant impact on the classification performance).
+
+ADV We use an adaptive approach as in Ganin $\&$ Lempitsky (2015) for decaying the learning rate $l r$ and updating the gradient’s scale $s$ between 0 and 1 for the domain discriminators. We choose the decay values used in Ganin & Lempitsky (2015) ie. $s = { \frac { 2 } { 1 + \exp ( - 1 0 \times p ) } } - 1$ and $l r = \frac { l r _ { i } } { ( 1 + 1 0 \times p ) ^ { 0 , 7 5 } }$ where $p$ is ratio of current batches processed over the total number of batches to be processed without further tuning. We tune the initial learning rate $l r _ { i }$ , chosen in the range $\{ 1 0 ^ { - 2 } , 1 0 ^ { - 2 . 5 } , 1 0 ^ { - 3 } , 1 0 ^ { - 3 . 5 } , 1 0 ^ { - 4 } \}$ following Section D.2.1. In practise we take $l r _ { i } = 1 0 ^ { - 2 }$ for ADV Adaptation-Imputation, Adaptation-Full, Adaptation-Partial and $l r _ { i } =$ $1 0 ^ { - 2 . 5 }$ for ADV Adaptation-Missing. We use Adam (Kingma & Ba, 2014) as the optimizer with momentum parameters $\beta _ { 1 } = 0 . 8$ and $\beta _ { 2 } = 0 . 9 9 9$ and use the same decay strategy and initial learning rate for all components (feature extractor, classifier, reconstructor). Batch size is chosen to be 128; we see in practise that initializing the adaptation models with a source model with smaller batch size (such as 32) can be beneficial.
+
+OT We choose parameter $\lambda _ { 1 } ~ = ~ \lambda _ { O T } ~ = ~ 0 . 1$ in Equations 4 and 8 after tuning in the range $\{ 1 0 ^ { - 1 } , 1 0 ^ { - 2 } , 1 0 ^ { - \dot { 3 } } \}$ using DEV; the goal is indeed to down weight the contribution of alignment through Wasserstein distance which should be scaled compared to the classification term. Following Damodaran & Kellenberger (2018), batch size is taken to be 500 and we use EMD as Optimal Transport metric. We initialize adaptation models with a source model in the first 10 epochs and divide the initial learning rate by two as adaptation starts for non-imputation models. For Adaptation-Imputation we follow a decaying strategy on the learning rate and on the adaptation weight as explained in the next item. We choose $l r _ { i }$ in the range $\{ 1 0 ^ { - 2 } , 1 0 ^ { - 2 . 5 } , 1 \dot { 0 } ^ { - 3 } , 1 0 ^ { - 3 . 5 } , 1 0 ^ { - 4 } \}$ . In practise we fix $l r _ { i } = 1 0 ^ { - 2 }$ for all models.
+
+Imputation parameters In practice we can define a weight for each term in Equation 2 and 4. Further studies are conducted in Section 5.4 on weightings in Equation 2: in digits experiments we choose ${ \cal L } _ { 2 } = { \cal L } _ { M S E } + { \cal L } _ { A D V }$ for ADV and OT to reduce the burden of additional feature tuning. For ADV model, we fix parameters in Equation 4 to 1. In the OT model, we vary $\lambda _ { 1 }$ between 0 and 0.1 and $\lambda _ { 2 }$ between 0 and 1 following the same schedule as the gradient scale update for ADV models. Although not necessary to achieve good mean values, this proves to reduce significantly the variance.
+
+# D.2.3 ADS
+
+We use an adaptive strategy for updating the gradient scale and the learning rate with the same parameters as in the digits dataset. Optimizer is taken to be Adam. Batch size is taken to be big so that target batches include sufficient positive instances.
+
+ads-kaggle The initial learning rate is chosen in the range $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 5 } , 1 0 ^ { - 6 } , 1 0 ^ { - 7 } \}$ using DEV and fixed to be $1 0 ^ { - 6 }$ for all models. Batch size is taken to be 500 and we initialize models with a simple classification loss for five epochs. We run models for 50 epochs after which we notice that models reach a plateau. We find that adding a weighted MSE term allows to achieve higher stability (as measured by variance) as further studied in Section 5.4. In a similar fashion to Pathak et al. (2016), we tune this weight in the range $\{ 1 , 1 0 ^ { - 1 } , 1 0 ^ { - 2 } , 7 . 5 \times 1 0 ^ { - 3 } , 5 \times 1 0 ^ { - 3 } , 1 0 ^ { - 3 } \} .$ We find that 0.005 offers the best compromise between mean loss and variance. Moreover on this dataset we use a faster decaying strategy for the discriminator’s $D _ { 2 }$ and the reconstructor’s $h$ learning rate, $l r = \frac { l r _ { i } } { ( 1 + 3 0 \times p ) ^ { 0 , 7 5 } }$ to achieve higher stability in the training curves while the feature extractor $g _ { 1 } , g _ { 2 }$ and $D _ { 1 }$ ’s learning rate are unchanged.
+
+ads-real The initial learning rate is chosen in the range $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 5 } , 1 0 ^ { - 6 } \}$ and fixed to be $1 0 ^ { - 6 }$ for all models. The learning rate is decayed with the same parameters as digits for all models. We run models for ten epochs which provides a good trade-off between learning time and classification performance. Batch size is taken to be 500. We choose ${ \cal L } _ { 2 } = { \cal L } _ { M S E } + { \cal L } _ { A D V }$ without further tuning; this simple weight achieves already good results.
+
+# E VISUALIZATION OF FEATURES’ DISTRIBUTION ON A D S-K A G G L E
+
+The goal of this section is to visualize the domain shift existing in ads-kaggle pre-adaptation to show that the problem comprises both a domain shift and an imputation problem. Table 6 reports mean and standard deviation on each feature’s value over the domain and Figure 6 plots the histogram of the normalized distributions for better visualization. The $y$ -axis is not normalized and corresponds to real counts. We see that feature 5 is missing on the target domain and that distributions are different in shape, mean and variance on each domain.
+
+Table 6: Mean and standard deviation for each feature on ads-kaggle   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=2>ads-kaggle</td></tr><tr><td rowspan=1 colspan=1>Domain</td><td rowspan=1 colspan=1>Source</td><td rowspan=1 colspan=1>Target</td></tr><tr><td rowspan=1 colspan=1>feature1</td><td rowspan=1 colspan=1>0.80±2.21</td><td rowspan=13 colspan=1>4.4×10-4±0.0419.01±13.423.44±6.190.94±2.310.0±0.00.090±1.690.0034±0.131.91±4.263.273±5.361.35 ×10-5 ±0.00374.25 ×10-4±0.0296.68 ×10-5±0.0181.21±3.36</td></tr><tr><td rowspan=1 colspan=1>feature 2</td><td rowspan=2 colspan=1>9.16±13.044.40±6.32</td></tr><tr><td rowspan=1 colspan=1>feature 3</td></tr><tr><td rowspan=1 colspan=1>feature 4</td><td rowspan=2 colspan=1>2.58±3.2761.09±37.67</td></tr><tr><td rowspan=1 colspan=1>feature 5</td></tr><tr><td rowspan=1 colspan=1>feature 6</td><td rowspan=1 colspan=1>11.26±12.24</td></tr><tr><td rowspan=1 colspan=1>feature 7</td><td rowspan=1 colspan=1>4.10±6.23</td></tr><tr><td rowspan=1 colspan=1>feature 8</td><td rowspan=1 colspan=1>5.12±4.50</td></tr><tr><td rowspan=1 colspan=1>feature 9</td><td rowspan=1 colspan=1>14.32±11.57</td></tr><tr><td rowspan=1 colspan=1>feature 10</td><td rowspan=1 colspan=1>0.046±0.22</td></tr><tr><td rowspan=1 colspan=1>feature 11</td><td rowspan=1 colspan=1>1.08±2.11</td></tr><tr><td rowspan=1 colspan=1>feature 12</td><td rowspan=1 colspan=1>0.083±0.78</td></tr><tr><td rowspan=1 colspan=1>feature 13</td><td rowspan=1 colspan=1>2.74±3.59</td></tr></table>
+
+# F EMBEDDING VISUALIZATION ON D I G I T S
+
+In this section we visualize the embeddings learned by the various models on digits by projecting the embeddings in a 2D space using t-SNE (the original embedding size being 2048). Figure 7 represents the embeddings learned for ADV models on $\mathrm { \ u N I S T {  } M N I S }$ T-M. Figures 8 and 9 represent these embeddings for OT models respectively on $\mathrm { \sf M N I S T } \to \mathrm { M N I S T - M }$ and M $\mathsf { N I S T } \to \mathsf { U S P S }$ .
+
+On these figures, we see that imputation models generate feature representations that overlap better between source and target examples per class than the adaptation counterparts (although Adaptation-Partial does a good job at overlapping feature representations). This correlates with the accuracy performance on the test set. Moreover we notice, as expected, that missing and partial adaptation networks perform badly compared to the full adaptation counterparts which justifies the use of imputation model when confronted to missing data.
+
+![](images/68d76540a9f2ac07b5966651054a130d8183db9016630f8630b00493d807f5f5.jpg)  
+Figure 6: Features’ normalized distribution for each input dimension (features 1 to 12): (a) Source domain (blue), (b) Target domain (red)
+
+![](images/accaf2b4fda9d5a3a9e99b5ff28808f172b67fcff8f8a3f2a469838ee00e4b88.jpg)  
+Figure 7: Embeddings for MNIST MNIST-M dataset on a batch, for source missing (acc $1 4 . 4 \%$ ) (a) (b); ADV missing (acc $3 2 . 6 \%$ ) (c) (d); ADV partial (acc $2 7 . 0 \%$ ) (e) (f) ADV with imputation (acc $5 6 . 3 \%$ ); (g) (h) and ADV full (acc $7 4 . 2 \%$ ) (i) (j). (a) (c) (e) (g) (i) represent the target (blue) and source (red) clusters, (b) (d) (f) (h) (j) represent the classes on source and target instances.
+
+![](images/a25ff8c14560e31d39e501d995a43bfd26831c52f0778b204ddc6fa0901d6bc7.jpg)  
+Figure 8: Embeddings for MNIST MNIST-M dataset on a batch, for source missing (acc $1 4 . 5 \%$ ) (a) (b); OT missing (acc $1 8 . 8 \%$ ) (c) (d); OT partial (acc $2 7 . 7 5 \%$ ) (e) (f); OT with imputation (acc $2 8 . 6 \%$ ) (g) (h) and OT full (acc $4 5 . 9 \%$ ) (i) (j). (a) (c) (e) (g) (i) represent the target (blue) and source (red) clusters, (b) (d) (f) (h) (j) represent the classes on source and target instances.
+
+![](images/7a0811fa2a2b03aea8e2054abd94ad6cde1011adb252c515891525f1f14198d0.jpg)  
+Figure 9: Embeddings for MNIST $ \mathrm { U S P S }$ dataset on a batch, for source missing (acc $2 5 . 0 \%$ ) (a) (b); OT missing (acc $5 8 . 0 \%$ ) (c) (d); OT partial (acc $6 2 . 4 2 \%$ ) (e) (f); OT with imputation (acc $6 5 . 2 \%$ ) (g) (h) and OT full (acc $9 1 . 5 \%$ ) (i) (j). (a) (c) (e) (g) (i) represent the target (blue) and source (red) clusters, (b) (d) (f) (h) (j) represent the classes on source and target instances.

parse/train/BJ6anzb0Z/BJ6anzb0Z.md

@@ -0,0 +1,293 @@
+# MULTIMODAL SENTIMENT ANALYSIS TO EXPLORE THE STRUCTURE OF EMOTIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a novel approach to multimodal sentiment analysis using deep neural networks combining visual recognition and natural language processing. Our goal is different than the standard sentiment analysis goal of predicting whether a sentence expresses positive or negative sentiment; instead, we aim to infer the latent emotional state of the user. Thus, we focus on predicting the emotion word tags attached by users to their Tumblr posts, treating these as “self-reported emotions.” We demonstrate that our multimodal model combining both text and image features outperforms separate models based solely on either images or text. Our model’s results are interpretable, automatically yielding sensible word lists associated with emotions. We explore the structure of emotions implied by our model and compare it to what has been posited in the psychology literature, and validate our model on a set of images that have been used in psychology studies. Finally, our work also provides a useful tool for the growing academic study of images— both photographs and memes—on social networks.
+
+# 1 INTRODUCTION
+
+Sentiment analysis has been an active area of research in the past decade, especially on textual data from Twitter, e.g. early work by Pak & Paroubek (2010) showed that emoticons could be used to collect a labeled dataset for sentiment analysis, Golder & Macy (2011) investigated temporal patterns in emotion using tweets, and Bollen et al. (2011) investigated the impact of collective mood states on the stock market. The SemEval series of “Sentiment Analysis in Twitter” challenges has used Twitter data as a benchmark to spur the development of new sentiment analysis algorithms (Rosenthal et al., 2017).
+
+Unlike Twitter, Tumblr posts are not limited to 140 characters, allowing more expressiveness, and they often focus on visual content: most Tumblr posts contain an image with some accompanying text. As pictures—both photographs and memes—have become prevalent on social media, researchers have begun to study them, and make novel claims about the role that they play in social media. Shifman (2014) makes an argument for taking memes seriously, and Miller & Sinanan (2017) use memes throughout their cross-country anthropological study of Facebook, characterizing who posts what sorts of memes and what sort of communicative function memes play. Inspired by this research, we take images seriously as a source of data worth analyzing. Further, we aim to enable a research agenda focused on images by giving social scientists tools to address fundamental questions about the use of images on social media.
+
+In psychology, the gold standard for measuring emotions is self-report, i.e. if an individual says that they are happy then that is taken to be the truth (Gilbert, 2006). On Tumblr, users often attach tags to their posts which we consider to be emotional self-reports, as these tags take the simple form of, e.g. “#happy”. By collecting a large dataset and using these emotion word tags as labels, we argue that our sentiment analysis approach, which combines images and text, leads to a more psychologically plausible model, as the dataset combines two rich sources of information and has labels we believe are a good proxy for self-reported emotion.
+
+Concretely, the Deep Sentiment model associates the features learned by the two modalities as follows:
+
+• For images, we fine-tune Inception (Szegedy et al., 2015), a pre-trained deep convolutional neural network, to our specific task of emotion inferring.   
+• The text is mapped into a rich high-dimensional space using a word representation learned by GloVe (Pennington et al., 2014). The embedded vectors are then fed to a recurrent network which preserves the word order and captures some of the semantics of human language.   
+• A dense layer combines the information in the two modalities and a final softmax output layer gives the probability distribution over the possible emotion word tags.
+
+# 2 RELATED WORK
+
+Visual sentiment analysis has received much less attention compared to text-based sentiment analysis. Yet, images are a valuable source of information for accurately inferring emotional states as they have become ubiquitous on social media as a means for users to express themselves (Miller & Sinanan, 2017). Although huge progress has been made on standard image classification tasks thanks to the ImageNet challenge (Russakovsky et al., 2015), visual sentiment analysis may be fundamentally different from classifying images as it requires a higher level of abstraction to understand the message conveyed by an image (Joshi et al., 2011). Implicit knowledge linked to culture and intrinsic human subjectivity – that can make two people use the same image to express different emotions – makes visual sentiment analysis a difficult task.
+
+Borth et al. (2013) pioneered sentiment analysis on visual content with SentiBank, a system extracting mid-level semantic attributes from images. These semantic features are outputs of classifiers that can predict the relevance of an image with regard to one of the emotions in the Plutchik’s wheel of emotions (Plutchik, 2001). Motivated by the progress of deep learning methods, You et al. (2015) used convolutional neural networks on Flickr with domain transfer from Twitter for binary sentiment classification. However, studies about image annotation showed that combining text features with images can greatly improve performance as shown by Guillaumin et al. (2010) and Gong et al. (2014).
+
+Successful results in multimodal sentiment analysis have been achieved using non-negative matrix factorisation (Wang et al., 2015) and latent correlations (Katsurai & Satoh, 2016). Chen et al. (2015) investigated the image posting behaviour of social media users and found in his study that two thirds of the participants added an image to their tweets to enhance the emotion of the text. In order to uncover the link between image tweets and the latent emotion, he used Latent Dirichlet Allocation to model image tweets with three modalities: textual, visual and emotional.
+
+One weakness shared by the papers above is the lack of large training datasets due to the inherent complexity of finding labeled images/text on social media. In this work, we aim to mitigate this problem using a large noisy labeled dataset of Tumblr posts. Further, sentiment analysis on text is a well-developed research area in both computer science and psychology, and sentiment analysis has been used to answer psychological questions. However, researchers have cautioned that sentiment analysis focuses on the positive or negative sentiment expressed by a piece of text, rather than on the underlying emotional state of the person who wrote the text (Flaxman & Kassam, 2016) and thus is not necessarily a reliable measure of latent emotion. We address this problem by trying to predict emotional state of the user instead of the sentiment polarity.
+
+# 3 TUMBLR DATASET
+
+Tumblr is a microblogging service where users post multimedia content that often contains the following attributes: an image, text, and tags. The critical piece of our approach, which distinguishes it from sentiment analysis methods focused on purely distinguishing positive from negative, is that we use the emotion word tags as the labels we wish to predict. As we consider these emotion word tags to be (noisy) labels indicating the user’s state of mind when writing a post, we can use them as a proxy for self-reported emotion, and thus a proxy for the underlying emotional state of the user.
+
+To build our dataset, queries were made through the Tumblr API searching for an emotion appearing in the tags. The 15 emotions retained were those with high relative frequencies on Tumblr among the PANAS-X scale (Watson & Clark, 1999) or the Plutchik’s wheel of emotions (Plutchik, 2001).
+
+In some posts, the tag containing the emotion of the post also appeared in the text itself. We removed these words from the text in order not to give our classifier an unfair advantage. We extracted every tagged posts from Tumblr by going backward from 2017 until 2011. Due to the API limitations, posts from high density emotions, such as ‘happy’ or ‘angry’, did not go as far back in the past, but otherwise we believe that our dataset is a complete sample.
+
+As Tumblr is used worldwide, we had to filter out non-English posts: posts with less than $50 \%$ of English words were removed from the dataset. The vocabulary of English words was obtained from GloVe (although GloVe’s vocabulary contains some non-English words this approach filtered out non-English posts reasonably well). Also, not every extracted post contained an image and we likewise excluded these. Figure 1 shows two posts with their associated emotions (more examples in Appendix A) and Table 1 summarises the statistics of the data1.
+
+![](images/c754f85a96b5e02096e85f7171ae5ad6b26626a9c7fc8f947317498943d3f473.jpg)
+
+(a) Optimistic: “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science – Albert Einstein.”
+
+![](images/7c16a8beb1e3bd4b8315511d19263c96d010dd431526b1e9063c7a79fa9477ca.jpg)  
+Figure 1: Examples of Tumblr posts
+
+(b) Sad: “It’s okay to be upset. It’s okay to not always be happy. It’s okay to cry. Never hide your emotions in fear of upsetting others or of being a bother.”
+
+Table 1: Summary statistics for the Tumblr dataset, with posts from January 2011 to September 2017.
+
+Tumblr data   
+
+<table><tr><td colspan="2">Posts</td><td colspan="2">English text</td><td>English text and image</td></tr><tr><td colspan="2">1,009,534</td><td colspan="2">578,699</td><td>256,897</td></tr><tr><td colspan="2"></td><td colspan="2"></td></tr><tr><td>Emotion</td><td>Posts</td><td>English text</td><td>English text and image</td></tr><tr><td>Happy</td><td>189,841</td><td>62%</td><td>29%</td></tr><tr><td>Calm</td><td>139,911</td><td>37%</td><td>29%</td></tr><tr><td>Sad</td><td>124,900</td><td>53%</td><td>15%</td></tr><tr><td>Scared</td><td>104,161</td><td>65%</td><td>20%</td></tr><tr><td>Bored</td><td>101,856</td><td>54%</td><td>29%</td></tr><tr><td>Angry</td><td>100,033</td><td>60%</td><td>21%</td></tr><tr><td>Annoyed</td><td>72,993</td><td>78%</td><td>10%</td></tr><tr><td>Love</td><td>66,146</td><td>61%</td><td>39%</td></tr><tr><td>Excited</td><td>37,240</td><td>58%</td><td>41%</td></tr><tr><td>Surprised</td><td>18,322</td><td>47%</td><td>32%</td></tr><tr><td>Optimistic</td><td>16,111</td><td>64%</td><td>36%</td></tr><tr><td>Amazed</td><td>10,367</td><td>61%</td><td>35%</td></tr><tr><td>Ashamed</td><td>10,066</td><td>63%</td><td>22%</td></tr><tr><td>Disgusted</td><td>9,178</td><td>69%</td><td>17%</td></tr><tr><td>Pensive</td><td>8,409</td><td>57%</td><td>34%</td></tr></table>
+
+# 4 METHODOLOGY
+
+# 4.1 VISUAL ANALYSIS
+
+Training a convolutional network from scratch can be challenging as a large amount of data is needed and many different architectures have to be tried before achieving satisfying performances. To circumvent this issue, we can take advantage of a pre-trained network named Inception (Szegedy et al., 2015) that learned to recognise images through the ImageNet dataset with a deep architecture of 22 layers.
+
+Inception learned representations capturing the colors and arrangement of shapes of an image, which turn out to be relevant when dealing with images even for a different task. We could also say that the pre-trained network grasped the underlying structure of images. This statement rests on the hypothesis that all images lie in a low-dimensional manifold, and recent advances in realistic photos generation through generative adversarial networks bolsters this idea (Radford et al., 2016).
+
+# 4.2 NATURAL LANGUAGE PROCESSING
+
+Even as a human being, it can be difficult to guess the expressed emotion only by looking at a Tumblr image without reading its caption as shown by Figure 2.
+
+![](images/168cd0e1d2a7780da3aedcab9aadf7521e6f7fd9feca8bb9bfdc75e1d65cbad9.jpg)  
+Figure 2: Which emotion is it?
+
+It is unclear whether the user wants to convey happiness or surprise. Only after reading the accompanying text, “To whoever left this on my windshield outside of last night’s art opening, I love you. You made my night,” can we finally conclude that the person was surprised (and possibly also feeling other emotions like amazed). The text is extremely informative and is usually crucial to accurately infer the emotional state.
+
+# 4.2.1 WORD EMBEDDING
+
+Most learning algorithms rely on the local smoothness hypothesis, that is, similar training instances are spatially close. This hypothesis clearly doesn’t hold with words one-hot encoded as for instance ‘dog’ is as close to ‘tree’ as it is to ‘cat’. Ideally, we would like to transform the word ‘dog’ into a space in which it is closer to ‘cat’ than it is to ‘tree’. Word embeddings produce a mapping in which words are projected into a high-dimensional space that preserves semantic relationships. We use the GloVe (Pennington et al., 2014) word embedding that was trained on Twitter data, because the writing styles on Twitter and Tumblr are similar.
+
+Each post in the dataset does not necessarily contain the same number of words. Even after embedding each word, the input will be of variable size and most learning algorithm expect a fixed-sized input. We could simply average across the high-dimensional representations of the words and take a distribution regression approach (Muandet et al., 2017). However note that averaging would mean that the word order would be completely lost. Human language relies heavily on word order to communicate as for example the word change can be both a noun and a verb, and negation such as ‘not entertained’ can only be understood if ‘not’ directly precedes the verb. We will preserve word order by using recurrent neural networks.
+
+# 4.2.2 SEQUENCE INPUT
+
+Models of natural language using neural networks have proved to outperform the more traditional statistical models that were limited by the Markov assumption (Bengio et al., 2003; Goodman, 2001). One explanation could be that the compact representation of words through word embeddings is robust (Mikolov et al., 2011) and do not need any smoothing over probabilities. Among the neural models, the recurrent-based models allows for short-term memory inspired by how humans read sentences: past context is essential to understand the meaning of written language. Contrary to shallow feedforward networks, that can only cluster similar words, recurrent networks (which can be viewed as a deep architecture (Bengio & LeCun, 2007)) can cluster similar histories. Recurrent neural networks have a temporal awareness represented by the hidden state that can be seen as an embedding of the past words. For example in Sutskever et al. (2014) a quality translation of a sentence was made possible with the last output of a recurrent neural network.
+
+In our setting, for a given Tumblr post, the text is broken down into a sequence of words that are embedded into a high-dimensional space (unknown words are mapped to the zero vector) and then fed into an LSTM (Hochreiter & Schmidhuber, 1997). To account for shorter posts, we zero-pad the vector with a special word token. For longer posts, we only keep the 50 first words which is a reasonable choice as $76 \%$ posts in the dataset contain less than 50 words.
+
+# 4.3 DEEP SENTIMENT: A MULTIMODAL NEURAL NETWORK
+
+Information often comes in several modalities, and humans are able to seamlessly combine them. For instance, in speech recognition, humans integrate audio and visual information to understand speech, as was demonstrated by the McGurk effect (McGurk & MacDonald, 1976). Separating what we see from what we hear seems like an easy task, but in an experiment conducted by McGurk, the subjects who were listening to a /ba/ sound with a visual /ga/ actually reported they were hearing a /da/. This is uncanny as even if we know the actual sound is a /ba/, we cannot stop our brain from interpreting it as a /da/.
+
+In Figure 2 we gave an example of text being necessary to fully understand the emotion expressed by an image. Sometimes alternative text would lead to entirely different interpretations, as shown in Figure 3:
+
+![](images/c808eb7e74da63c374d25a1e9fb8d3e407bed32da7ddd033dd104b0218349fc6.jpg)  
+(b) Angry: “I hate it when people are taking too much space on planes.”
+
+(a) Scared: “Planes might just be the most frightening thing ever.”
+
+![](images/2883935d1c4a45e0ab6f07c351abdc70bb1a1f6a3615201000b0c109e88df7d6.jpg)  
+Figure 3: Different meanings with different captions.
+
+Exploiting both visual and textual information is therefore key to understanding a user’s underlying emotional state. We call our proposed network architecture, combining visual recognition and text analysis, “Deep Sentiment”.
+
+# 4.3.1 ARCHITECTURE
+
+Deep Sentiment’s architecture is shown in Figure 4:
+
+![](images/746371da56165b060b1870fbd69a0eb99eff70669923b25a338ce80d5cad1671.jpg)  
+Figure 4: The Deep Sentiment structure. On the one hand, the input image, resized to (224,224,3) is fed into the Inception network and outputs a vector of size 256. On the other hand, the text is projected into a high-dimensional space that subsequently goes through an LSTM layer with 1024 units. The two modalities are then concatenated and fed into a dense layer. The final softmax output layer give the probability distribution over the emotional state of the user.
+
+# 5 EVALUATION
+
+In Table 2, we compare Deep Sentiment with the image model (Inception model fine-tuned through the last Inception module), the text model, and a baseline: random guessing that includes the prior probabilities of the classes. Figure 5 shows a comparison of the accuracy curves of the different models.
+
+Table 2: Comparison of image model, text model and Deep Sentiment.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Loss</td><td rowspan=1 colspan=1>Trainaccuracy</td><td rowspan=1 colspan=1>Testaccuracy</td></tr><tr><td rowspan=1 colspan=1>Random guessing</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>11%</td><td rowspan=1 colspan=1>11%</td></tr><tr><td rowspan=1 colspan=1>Image model</td><td rowspan=1 colspan=1>1.80</td><td rowspan=1 colspan=1>43%</td><td rowspan=1 colspan=1>36%</td></tr><tr><td rowspan=1 colspan=1>Text model</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>72%</td><td rowspan=1 colspan=1>69%</td></tr><tr><td rowspan=1 colspan=1>Deep Sentiment</td><td rowspan=1 colspan=1>0.75</td><td rowspan=1 colspan=1>80%</td><td rowspan=1 colspan=1>72%</td></tr></table>
+
+![](images/5609ae3b910aab3e30daac14bdd3385ebafe55e2ef734bc5bd287f9cd140ac97.jpg)  
+Figure 5: Accuracy comparison of the three models
+
+Using text alone, the test accuracy is $69 \%$ , almost double the accuracy of the image model, this suggests that on Tumblr, text is a better predictor of emotion than images, as we illustrated in Figure 2. By combining text and images Deep Sentiment achieves $80 \%$ train accuracy and $72 \%$ test accuracy, significantly outperforming the images-only model and slightly outperforming the text-only model (note that no validation set was used to tune hyperparameters, even though various architectures were tried before yielding Deep Sentiment).
+
+# 6 RESULTS
+
+In this section, we carefully investigate what psychologically meaningful results we can draw from our model, and whether they match previous results in the psychology literature.
+
+# 6.1 TOP WORDS FOR EACH EMOTION
+
+We investigated which words were the most relevant for each emotion as follows: we created artificial posts whose text consisted of a single word (each of the most frequent 1,000 words in the whole dataset were tested) accompanied by an image that was the mean image (per channel mean). For each emotion, we report the 10 highest scoring words from these 1,000 artificial posts in Table 3 (words 11-20 for each emotion are in Appendix B).
+
+Table 3: Top 10 words for each emotion, ordered by the relative frequency of the emotion being used as a tag on Tumblr   
+
+<table><tr><td>Emotion Happy</td><td>Top words</td></tr><tr><td>Calm Sad Scared Bored Angry Annoyed Love Excited Surprised Optimistic Amazed Ashamed Disgusted Pensive</td><td>healthy, loving,enjoy, wonderful, warm, happiness,smile, lovely, cute,proud quiet, situation, peace, mood, towards, warm, slowly, stay, sleep, rain horrible,sorry,crying,hurts, tears,cried,lonely,memories,worst, pain terrified, scary, panic, nervous,fear,afraid,horrible,woke,happening,worried asleep, tired,kinda,busy, stuck,constantly, lonely, sat, listening,depression anger, fear, panic,annoying,hate,mad, upset, anxiety, scares,stupid pissed, ashamed, angry, nervous,speak, surprised, tired, worried, ignore, phone soul,dreams,happiness,kiss,sex,beauty,women, feelings,god,relationships tonight, hopefully, watching, nervous, surprised, expect, tomorrow, amazing, hoping, happen birthday, cried, thank, yesterday, told, sorry,amazing, sweet, friend, message positive, expect, surprised, healthy, grow, realize, clearly, hopefully, calm, peace surprised, excited,amazing,woke,realized,awesome, happening,ashamed, yeah, happened totally, honestly, sorry,absolutely, freaking, honest,completely,stupid, seriously,am ashamed, totally,angry,hate,stupid,annoyed,horrible,scares,freaking,absolutely mood,wrote,quiet, view, sadness, thoughts,calm,words,sad,kissed</td></tr></table>
+
+An inspection of each of the words suggests that none are out of place, with the possible exception of “ashamed” in the list of Amazed words. (This may be due to the fact that the tag Amazed is sometimes used ironically.) Further, our data-driven approach suggests that our methods could be used as an alternative to the word list-based approaches to sentiment analysis common in psychology. The most widely used tool, Linguistic Inquiry Word Count (LIWC), (Pennebaker et al., 2007), consists of dozens of English words which were compiled by hand into psychologically meaningful categories such as “Health/illness” or “Anxiety” and used in a large number of psychology studies (Tausczik & Pennebaker, 2010). Not only does our approach automatically give sensible word lists, it contains modern words, common in social media usage, like “woke” (first attested in its modern meaning in 1962 according to the Oxford English Dictionary which defines it as “alert to racial or social discrimination and injustice”) and “phone.” Woke appears in the top 10 of “Scared” and “Amazed” and in the top 20 of “Calm” (Appendix B), an interesting finding in its own right. By contrast, both woke and phone appear in LIWC, but not in any of LIWC’s emotion word categories, only in a list of verbs and a list of social words, respectively.
+
+# 6.2 CLUSTERING EMOTIONS
+
+Psychologists have long studied the structure of emotion, and debated whether there are a small number of “core” emotions (Ekman, 1992), two dominant factors (see Tellegen et al. (1999) for a discussion of various theories), or more complex models (e.g. Lindquist et al. (2013) critiques previous models and argues that emotions do not belong in natural categories or along multiple dimensions, because they are the “constructions” of the human mind, and the result of complex perceptions). Classical attempts by psychologists to answer this question have relied on survey data in which respondents can self-report more than one emotion, followed by clustering or factor analysis approaches. More recently, psychologists have turned to neuroscience to address these questions.
+
+While we collected a dataset in which each Tumblr post was labeled with a single emotion tag, we can nevertheless directly address this question using our trained model. Given an image $\mathcal { T }$ and text $\tau$ from our dataset, we can compute the conditional probabilities of the emotion classes: ${ \hat { y } } | \mathcal { T } , \mathcal { T }$ where $\hat { y }$ is a probability vector in the 15-dimensional simplex. To calculate our model’s predicted empirical distribution over $\hat { y }$ we predict $\hat { y }$ for all posts in our dataset: $( \hat { y } _ { i } | \mathcal { T } _ { i } , \mathcal { T } _ { i } ) _ { i = 1 } ^ { n }$ . Finally, we calculate the empirical correlation matrix of ${ \hat { y } } _ { 1 } , \dots , { \hat { y } } _ { n }$ , which we include in the Appendix in Figure 9. For ease of visualization, we convert the correlation matrix to a distance matrix and perform hierarchical clustering as shown in Figure 6.
+
+![](images/924e65d4b83c2df35442ad641abba4834abf1fe3a876de70cdee8cef5b19f97c.jpg)  
+Figure 6: Hierarchical clustering of the emotion correlation matrix.
+
+Similar to almost all previous psychology studies, there is a clear distinction between emotions that are positive in nature, such as excited, happy, and love and emotions that are negative such as angry, ashamed, annoyed, and disgusted. Another finding consistent with much previous literature, which we investigate more below, is that within both positive and negative emotions, there is a distinction between low arousal emotions and high arousal emotions. Bored and sad (low arousal) are in one cluster, while high arousal emotions—angry, ashamed, annoyed, disgusted—are in another.
+
+# 6.3 THE CIRCUMPLEX MODEL OF EMOTION
+
+The circumplex model (Posner et al., 2005) posits two dimensions which explain emotions, usually valence (positive vs. negative) and arousal (low vs. high), though other dimensions have been suggested. We investigated whether two factors explained most of the variance in our results using principal component analysis (PCA). As shown in the scree plot in the Appendix in Figure 12, while the first component explains $2 8 . 8 \%$ of the variance, dimensions 2 and 3 explain $1 6 . 4 \%$ and $1 4 . 6 \%$ respectively, evidence against a simple two factor model.
+
+![](images/d731da54e9bae88f2cb211b43257117f278a8340323013b78fca6e4b896532bd.jpg)
+
+(a) PCA with a random subset of posts, 150 for each emotion.
+
+![](images/18242f21bc0ce11bf77b4044cc681343fc4ce23a52e4cdc351a2d18fac8e0e1f.jpg)  
+Figure 7: Affect/valence
+
+We visualize the results of PCA in Figure 7. At left, a random subset of posts, where for visualization purpose we sampled equally from each emotion class, are shown along the first two principal components, and each is colored by the most probable predicted emotion. This shows that within any particular emotion category there is much variation along these two dimensions, i.e. they do not cluster very tightly. Related emotions like happy and love are close together, as they were in the dendrogram.
+
+At right, we visualise the projection of the emotion variables onto the first two principal components. Using this figure, we conclude that the first principal component corresponds well to valence, as it neatly separates happy/love from disgusted/ashamed/annoyed. However, the second principal component is not clearly arousal. In the Appendix, we consider the third principal component as well, but the picture is no clearer. Once again, this provides evidence against a simple two factor model of emotion. Below, we further investigate the valence/arousal model.
+
+6.4 VALIDATION ON THE OPEN AFFECTIVE STANDARDIZED IMAGE SET (OASIS)
+
+The Open Affective Standardized Image Set (OASIS) dataset (Kurdi et al., 2017) was developed as an alternative to the International Affective Picture System ((Lang, 2005)), a set of emotional stimuli which has been used in many psychology studies. OASIS consists of 900 color images which were rated by human judges on MTurk along the valence and arousal scales discussed above. OASIS images come in classes with labels such as “Alcohol”, “Flowers”, or “Pigeon.” We applied our Deep Sentiment model to these images, treating the single-word label as the input text. After making predictions, we projected them onto the principal components discussed above.
+
+In Table 4 we calculated the correlations between the first three principal components from our model and the mean valence and arousal variables from OASIS. The high correlation between the first principal component (PC1) and the valence variable provides evidence, as in the previous section, that PC1 is capturing a dimension of emotion separating positive from negative. It is also interesting to note the low correlation between PC1 and arousal. By contrast, while PC2 and PC3 are correlated with arousal, the correlations are not very high, and they are also correlated with valence. On the one hand, this is more evidence against there being two factors explaining emotions. On the other hand, asking judges to rate a set of images based on arousal and valence is more similar to the standard sentiment analysis task of predicting whether a sentence expresses positive or negative emotion. By contrast, our dataset comes from a set of images which were chosen by users who also decided to select emotion word tags to accompany them. We consider this to thus be a dataset of richer emotional content, and by design it represents a richer set of emotions.
+
+Table 4: Correlation between OASIS arousal/valence and principal components   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>OASISvalence</td><td rowspan=1 colspan=1>OASISarousal</td></tr><tr><td rowspan=1 colspan=1>PC1</td><td rowspan=1 colspan=1>58%</td><td rowspan=1 colspan=1>3%</td></tr><tr><td rowspan=1 colspan=1>PC2</td><td rowspan=1 colspan=1>17%</td><td rowspan=1 colspan=1>22%</td></tr><tr><td rowspan=1 colspan=1>PC3</td><td rowspan=1 colspan=1>30%</td><td rowspan=1 colspan=1>11%</td></tr></table>
+
+# 7 CONCLUSION
+
+We developed a novel multimodal sentiment analysis method using deep learning methods. Our goal was to investigate a core area of psychology, the study of emotion, using a large and novel social media dataset. Our approach provides new tools for the joint study of images and text on social media. This is important as social science researchers have begun to uncover the important role that images play on social media. As our dataset consisted of text, images, and emotion word tags, we considered it to be a form of “self-reported” data, which is the gold standard in emotion. However, it could also be considered to be an unobtrusive behavioral measure, and thus not subject to the biases inherent in laboratory studies. But because the data comes from public social media posts, we cannot rule out various sources of bias. What people choose to post—and not to post—on social media is influenced by how they want to present themselves (Wojnicki & Godes, 2008), so our study is limited insofar as it covers emotion not necessarily as it is truly experienced but rather as it is expressed or performed online.
+
+In the future, we will evaluate our model on other image / text stimuli datasets that have been developed for psychological studies and investigate whether human judges are more or less accurate than our model. Finally, we will investigate other psychological components of the structure of emotion, for example daily and day of week trends in emotion.
+
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+Yilin Wang, Suhang Wang, Jiliang Tang, Huan Liu, and Baoxin Li. Unsupervised sentiment analysis for social media images. In IJCAI, 2015.
+
+David Watson and Lee Anna Clark. The panas-x: Manual for the positive and negative affect schedule - expanded form. 1999.
+
+Andrea C Wojnicki and David Godes. Word-of-mouth as self-enhancement. 2008.
+
+Quanzeng You, Jiebo Luo, Hailin Jin, and Jianchao Yang. Robust image sentiment analysis using progressively trained and domain transferred deep networks. In AAAI, 2015.
+
+# A MORE EXAMPLES OF TUMBLR POSTS
+
+![](images/3be71e9c13a9b938b3f17c9fbacdacdba14c20f782673c48cba82826a61b4206.jpg)
+
+![](images/7b2ebb6c643fa85093983473161ca421c97afec4b309fb25e96ccd5485af9ff1.jpg)  
+(b) Disgusted: “Me when I see a couple expressing their affection in physical ways in public”
+
+![](images/cb64c199db9e486f75b47e9835ea9310876399db9255dcb06d72ab4a70800ff5.jpg)  
+(a) Happy: “Just relax with this amazing view #bigsur #california #roadtrip #usa #life #fitness (at McWay Falls)”
+
+![](images/ae2e9a868b2b2ca7f281006e507b76d1ba612a38c22246ab06c6e480a414f0f8.jpg)
+
+(c) Angry: “Tensions were high this Caturday...”
+
+(d) Surprised: “Which Tea? Peppermint tea: What is your favorite gif right now?”
+
+Figure 8: Tumblr posts
+
+B FURTHER RESULTS
+
+Table 5: Top 11-20 words for each emotion   
+
+<table><tr><td>Emotion Happy</td><td>Top words</td></tr><tr><td>Calm Sad Scared Bored Angry Annoyed Love Excited Surprised Optimistic Amazed</td><td>beautiful, beauty, fun, nice, dinner, sweet, good,loves,amazing, comfortable view, sound, cool, continue, visit, push, woke, morning, safe, step cry, sick, hurt, dead,feeling,worse,death,upset, falling, panic kill, happened, cause,asleep,anxiety, worry, hurt, happen, worst, seriously sitting, suddenly, freaking, slowly,annoying, sometimes, gotten, completely, older, cold hurt, face,against, cause,hurts, fucking, fight, pain, strong,worst depressed,lately,conversation,constantly, scared,stupid, scares, bored,heard, extremely sweet, mother, forget, hate, her, song,enjoy, sister, wonderful, dear 2017, positive,am, definitely,awesome,happens, listening, happy, grow, wanna annoyed, negative,apparently,u, minute,asked, sadness, happened, moments, laugh view,yet, believe, tomorrow,trending,said,situation,mood,forward, future im,quite,appreciate, honestly, knew,learned, yang, felt, liked, asleep</td></tr></table>
+
+![](images/b4e43cfc0aa3792084ff8fdb829f547d0b5096d5fce32c8a9626e804caa93b74.jpg)  
+Figure 9: Heatmap of the emotion correlation matrix
+
+![](images/6fec305ae9bfa3f884a5285c5cec49e05dc05d79f11d2c23e4b3099491d393b5.jpg)
+
+(a) PCA with a random subset of posts, 150 for each emotion
+
+![](images/b3331d098cb894d6a4491e16a8bd123de5d177c57d63f175aeae6cd97f9883f6.jpg)  
+Figure 10: PCA on dimension 1 and 3
+
+![](images/efebb27b83fa3fa88e57d8a12063cccd4deda41813e936f8a631975a5f0cc9ce.jpg)  
+(a) PCA with a random subset of posts, 150 for each emotion
+
+![](images/1787dde5931c290de4981e76b11550820cf253baeb336c24c0abd12d610e7c0f.jpg)  
+Figure 11: PCA for dimension 2 and 3
+
+![](images/f2ec9015543b24b2f86d84da602ed8983a2420a55fdacf6fee80ffb20e03531f.jpg)  
+Figure 12: Explained variance of the PCA
+
+![](images/1810a9ce40ea46405cefed677028077e9d1d78004746ef75d5630c2e822496a0.jpg)  
+Figure 13: Factor loadings

parse/train/BJ6anzb0Z/BJ6anzb0Z_content_list.json

@@ -0,0 +1,1529 @@
+[
+    {
+        "type": "text",
+        "text": "MULTIMODAL SENTIMENT ANALYSIS TO EXPLORE THE STRUCTURE OF EMOTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel approach to multimodal sentiment analysis using deep neural networks combining visual recognition and natural language processing. Our goal is different than the standard sentiment analysis goal of predicting whether a sentence expresses positive or negative sentiment; instead, we aim to infer the latent emotional state of the user. Thus, we focus on predicting the emotion word tags attached by users to their Tumblr posts, treating these as “self-reported emotions.” We demonstrate that our multimodal model combining both text and image features outperforms separate models based solely on either images or text. Our model’s results are interpretable, automatically yielding sensible word lists associated with emotions. We explore the structure of emotions implied by our model and compare it to what has been posited in the psychology literature, and validate our model on a set of images that have been used in psychology studies. Finally, our work also provides a useful tool for the growing academic study of images— both photographs and memes—on social networks. ",
+        "bbox": [
+            233,
+            271,
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+            465
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            505,
+            336,
+            520
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sentiment analysis has been an active area of research in the past decade, especially on textual data from Twitter, e.g. early work by Pak & Paroubek (2010) showed that emoticons could be used to collect a labeled dataset for sentiment analysis, Golder & Macy (2011) investigated temporal patterns in emotion using tweets, and Bollen et al. (2011) investigated the impact of collective mood states on the stock market. The SemEval series of “Sentiment Analysis in Twitter” challenges has used Twitter data as a benchmark to spur the development of new sentiment analysis algorithms (Rosenthal et al., 2017). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unlike Twitter, Tumblr posts are not limited to 140 characters, allowing more expressiveness, and they often focus on visual content: most Tumblr posts contain an image with some accompanying text. As pictures—both photographs and memes—have become prevalent on social media, researchers have begun to study them, and make novel claims about the role that they play in social media. Shifman (2014) makes an argument for taking memes seriously, and Miller & Sinanan (2017) use memes throughout their cross-country anthropological study of Facebook, characterizing who posts what sorts of memes and what sort of communicative function memes play. Inspired by this research, we take images seriously as a source of data worth analyzing. Further, we aim to enable a research agenda focused on images by giving social scientists tools to address fundamental questions about the use of images on social media. ",
+        "bbox": [
+            174,
+            645,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In psychology, the gold standard for measuring emotions is self-report, i.e. if an individual says that they are happy then that is taken to be the truth (Gilbert, 2006). On Tumblr, users often attach tags to their posts which we consider to be emotional self-reports, as these tags take the simple form of, e.g. “#happy”. By collecting a large dataset and using these emotion word tags as labels, we argue that our sentiment analysis approach, which combines images and text, leads to a more psychologically plausible model, as the dataset combines two rich sources of information and has labels we believe are a good proxy for self-reported emotion. ",
+        "bbox": [
+            174,
+            791,
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+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Concretely, the Deep Sentiment model associates the features learned by the two modalities as follows: ",
+        "bbox": [
+            174,
+            895,
+            821,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• For images, we fine-tune Inception (Szegedy et al., 2015), a pre-trained deep convolutional neural network, to our specific task of emotion inferring.   \n• The text is mapped into a rich high-dimensional space using a word representation learned by GloVe (Pennington et al., 2014). The embedded vectors are then fed to a recurrent network which preserves the word order and captures some of the semantics of human language.   \n• A dense layer combines the information in the two modalities and a final softmax output layer gives the probability distribution over the possible emotion word tags. ",
+        "bbox": [
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+            103,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            243,
+            341,
+            258
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Visual sentiment analysis has received much less attention compared to text-based sentiment analysis. Yet, images are a valuable source of information for accurately inferring emotional states as they have become ubiquitous on social media as a means for users to express themselves (Miller & Sinanan, 2017). Although huge progress has been made on standard image classification tasks thanks to the ImageNet challenge (Russakovsky et al., 2015), visual sentiment analysis may be fundamentally different from classifying images as it requires a higher level of abstraction to understand the message conveyed by an image (Joshi et al., 2011). Implicit knowledge linked to culture and intrinsic human subjectivity – that can make two people use the same image to express different emotions – makes visual sentiment analysis a difficult task. ",
+        "bbox": [
+            174,
+            273,
+            825,
+            398
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Borth et al. (2013) pioneered sentiment analysis on visual content with SentiBank, a system extracting mid-level semantic attributes from images. These semantic features are outputs of classifiers that can predict the relevance of an image with regard to one of the emotions in the Plutchik’s wheel of emotions (Plutchik, 2001). Motivated by the progress of deep learning methods, You et al. (2015) used convolutional neural networks on Flickr with domain transfer from Twitter for binary sentiment classification. However, studies about image annotation showed that combining text features with images can greatly improve performance as shown by Guillaumin et al. (2010) and Gong et al. (2014). ",
+        "bbox": [
+            174,
+            406,
+            825,
+            517
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Successful results in multimodal sentiment analysis have been achieved using non-negative matrix factorisation (Wang et al., 2015) and latent correlations (Katsurai & Satoh, 2016). Chen et al. (2015) investigated the image posting behaviour of social media users and found in his study that two thirds of the participants added an image to their tweets to enhance the emotion of the text. In order to uncover the link between image tweets and the latent emotion, he used Latent Dirichlet Allocation to model image tweets with three modalities: textual, visual and emotional. ",
+        "bbox": [
+            174,
+            525,
+            823,
+            608
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One weakness shared by the papers above is the lack of large training datasets due to the inherent complexity of finding labeled images/text on social media. In this work, we aim to mitigate this problem using a large noisy labeled dataset of Tumblr posts. Further, sentiment analysis on text is a well-developed research area in both computer science and psychology, and sentiment analysis has been used to answer psychological questions. However, researchers have cautioned that sentiment analysis focuses on the positive or negative sentiment expressed by a piece of text, rather than on the underlying emotional state of the person who wrote the text (Flaxman & Kassam, 2016) and thus is not necessarily a reliable measure of latent emotion. We address this problem by trying to predict emotional state of the user instead of the sentiment polarity. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 TUMBLR DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            760,
+            357,
+            776
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Tumblr is a microblogging service where users post multimedia content that often contains the following attributes: an image, text, and tags. The critical piece of our approach, which distinguishes it from sentiment analysis methods focused on purely distinguishing positive from negative, is that we use the emotion word tags as the labels we wish to predict. As we consider these emotion word tags to be (noisy) labels indicating the user’s state of mind when writing a post, we can use them as a proxy for self-reported emotion, and thus a proxy for the underlying emotional state of the user. ",
+        "bbox": [
+            174,
+            791,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To build our dataset, queries were made through the Tumblr API searching for an emotion appearing in the tags. The 15 emotions retained were those with high relative frequencies on Tumblr among the PANAS-X scale (Watson & Clark, 1999) or the Plutchik’s wheel of emotions (Plutchik, 2001). ",
+        "bbox": [
+            176,
+            882,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In some posts, the tag containing the emotion of the post also appeared in the text itself. We removed these words from the text in order not to give our classifier an unfair advantage. We extracted every tagged posts from Tumblr by going backward from 2017 until 2011. Due to the API limitations, posts from high density emotions, such as ‘happy’ or ‘angry’, did not go as far back in the past, but otherwise we believe that our dataset is a complete sample. ",
+        "bbox": [
+            174,
+            103,
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+            174
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As Tumblr is used worldwide, we had to filter out non-English posts: posts with less than $50 \\%$ of English words were removed from the dataset. The vocabulary of English words was obtained from GloVe (although GloVe’s vocabulary contains some non-English words this approach filtered out non-English posts reasonably well). Also, not every extracted post contained an image and we likewise excluded these. Figure 1 shows two posts with their associated emotions (more examples in Appendix A) and Table 1 summarises the statistics of the data1. ",
+        "bbox": [
+            173,
+            180,
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+            263
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c754f85a96b5e02096e85f7171ae5ad6b26626a9c7fc8f947317498943d3f473.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            285,
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+            400
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(a) Optimistic: “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science – Albert Einstein.” ",
+        "bbox": [
+            173,
+            406,
+            486,
+            444
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/7c16a8beb1e3bd4b8315511d19263c96d010dd431526b1e9063c7a79fa9477ca.jpg",
+        "image_caption": [
+            "Figure 1: Examples of Tumblr posts "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            271,
+            285,
+            392,
+            400
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(b) Sad: “It’s okay to be upset. It’s okay to not always be happy. It’s okay to cry. Never hide your emotions in fear of upsetting others or of being a bother.” ",
+        "bbox": [
+            496,
+            406,
+            810,
+            444
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Summary statistics for the Tumblr dataset, with posts from January 2011 to September 2017. ",
+        "bbox": [
+            173,
+            512,
+            825,
+            541
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/64d923cf2474ff330ce0f0821d528a7019e0f79b04bae548f3e42f1ce3c05978.jpg",
+        "table_caption": [
+            "Tumblr data "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Posts</td><td colspan=\"2\">English text</td><td>English text and image</td></tr><tr><td colspan=\"2\">1,009,534</td><td colspan=\"2\">578,699</td><td>256,897</td></tr><tr><td colspan=\"2\"></td><td colspan=\"2\"></td></tr><tr><td>Emotion</td><td>Posts</td><td>English text</td><td>English text and image</td></tr><tr><td>Happy</td><td>189,841</td><td>62%</td><td>29%</td></tr><tr><td>Calm</td><td>139,911</td><td>37%</td><td>29%</td></tr><tr><td>Sad</td><td>124,900</td><td>53%</td><td>15%</td></tr><tr><td>Scared</td><td>104,161</td><td>65%</td><td>20%</td></tr><tr><td>Bored</td><td>101,856</td><td>54%</td><td>29%</td></tr><tr><td>Angry</td><td>100,033</td><td>60%</td><td>21%</td></tr><tr><td>Annoyed</td><td>72,993</td><td>78%</td><td>10%</td></tr><tr><td>Love</td><td>66,146</td><td>61%</td><td>39%</td></tr><tr><td>Excited</td><td>37,240</td><td>58%</td><td>41%</td></tr><tr><td>Surprised</td><td>18,322</td><td>47%</td><td>32%</td></tr><tr><td>Optimistic</td><td>16,111</td><td>64%</td><td>36%</td></tr><tr><td>Amazed</td><td>10,367</td><td>61%</td><td>35%</td></tr><tr><td>Ashamed</td><td>10,066</td><td>63%</td><td>22%</td></tr><tr><td>Disgusted</td><td>9,178</td><td>69%</td><td>17%</td></tr><tr><td>Pensive</td><td>8,409</td><td>57%</td><td>34%</td></tr></table>",
+        "bbox": [
+            279,
+            580,
+            717,
+            869
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            341,
+            118
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 VISUAL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            138,
+            343,
+            154
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Training a convolutional network from scratch can be challenging as a large amount of data is needed and many different architectures have to be tried before achieving satisfying performances. To circumvent this issue, we can take advantage of a pre-trained network named Inception (Szegedy et al., 2015) that learned to recognise images through the ImageNet dataset with a deep architecture of 22 layers. ",
+        "bbox": [
+            174,
+            169,
+            825,
+            238
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Inception learned representations capturing the colors and arrangement of shapes of an image, which turn out to be relevant when dealing with images even for a different task. We could also say that the pre-trained network grasped the underlying structure of images. This statement rests on the hypothesis that all images lie in a low-dimensional manifold, and recent advances in realistic photos generation through generative adversarial networks bolsters this idea (Radford et al., 2016). ",
+        "bbox": [
+            174,
+            246,
+            825,
+            315
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 NATURAL LANGUAGE PROCESSING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            343,
+            455,
+            356
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Even as a human being, it can be difficult to guess the expressed emotion only by looking at a Tumblr image without reading its caption as shown by Figure 2. ",
+        "bbox": [
+            174,
+            372,
+            825,
+            400
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/168cd0e1d2a7780da3aedcab9aadf7521e6f7fd9feca8bb9bfdc75e1d65cbad9.jpg",
+        "image_caption": [
+            "Figure 2: Which emotion is it? "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            434,
+            422,
+            563,
+            525
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is unclear whether the user wants to convey happiness or surprise. Only after reading the accompanying text, “To whoever left this on my windshield outside of last night’s art opening, I love you. You made my night,” can we finally conclude that the person was surprised (and possibly also feeling other emotions like amazed). The text is extremely informative and is usually crucial to accurately infer the emotional state. ",
+        "bbox": [
+            174,
+            585,
+            825,
+            655
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 WORD EMBEDDING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            680,
+            364,
+            694
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Most learning algorithms rely on the local smoothness hypothesis, that is, similar training instances are spatially close. This hypothesis clearly doesn’t hold with words one-hot encoded as for instance ‘dog’ is as close to ‘tree’ as it is to ‘cat’. Ideally, we would like to transform the word ‘dog’ into a space in which it is closer to ‘cat’ than it is to ‘tree’. Word embeddings produce a mapping in which words are projected into a high-dimensional space that preserves semantic relationships. We use the GloVe (Pennington et al., 2014) word embedding that was trained on Twitter data, because the writing styles on Twitter and Tumblr are similar. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            805
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Each post in the dataset does not necessarily contain the same number of words. Even after embedding each word, the input will be of variable size and most learning algorithm expect a fixed-sized input. We could simply average across the high-dimensional representations of the words and take a distribution regression approach (Muandet et al., 2017). However note that averaging would mean that the word order would be completely lost. Human language relies heavily on word order to communicate as for example the word change can be both a noun and a verb, and negation such as ‘not entertained’ can only be understood if ‘not’ directly precedes the verb. We will preserve word order by using recurrent neural networks. ",
+        "bbox": [
+            173,
+            813,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 SEQUENCE INPUT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            351,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Models of natural language using neural networks have proved to outperform the more traditional statistical models that were limited by the Markov assumption (Bengio et al., 2003; Goodman, 2001). One explanation could be that the compact representation of words through word embeddings is robust (Mikolov et al., 2011) and do not need any smoothing over probabilities. Among the neural models, the recurrent-based models allows for short-term memory inspired by how humans read sentences: past context is essential to understand the meaning of written language. Contrary to shallow feedforward networks, that can only cluster similar words, recurrent networks (which can be viewed as a deep architecture (Bengio & LeCun, 2007)) can cluster similar histories. Recurrent neural networks have a temporal awareness represented by the hidden state that can be seen as an embedding of the past words. For example in Sutskever et al. (2014) a quality translation of a sentence was made possible with the last output of a recurrent neural network. ",
+        "bbox": [
+            174,
+            132,
+            825,
+            285
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In our setting, for a given Tumblr post, the text is broken down into a sequence of words that are embedded into a high-dimensional space (unknown words are mapped to the zero vector) and then fed into an LSTM (Hochreiter & Schmidhuber, 1997). To account for shorter posts, we zero-pad the vector with a special word token. For longer posts, we only keep the 50 first words which is a reasonable choice as $76 \\%$ posts in the dataset contain less than 50 words. ",
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+    {
+        "type": "text",
+        "text": "4.3 DEEP SENTIMENT: A MULTIMODAL NEURAL NETWORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Information often comes in several modalities, and humans are able to seamlessly combine them. For instance, in speech recognition, humans integrate audio and visual information to understand speech, as was demonstrated by the McGurk effect (McGurk & MacDonald, 1976). Separating what we see from what we hear seems like an easy task, but in an experiment conducted by McGurk, the subjects who were listening to a /ba/ sound with a visual /ga/ actually reported they were hearing a /da/. This is uncanny as even if we know the actual sound is a /ba/, we cannot stop our brain from interpreting it as a /da/. ",
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+    {
+        "type": "text",
+        "text": "In Figure 2 we gave an example of text being necessary to fully understand the emotion expressed by an image. Sometimes alternative text would lead to entirely different interpretations, as shown in Figure 3: ",
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+    {
+        "type": "image",
+        "img_path": "images/c808eb7e74da63c374d25a1e9fb8d3e407bed32da7ddd033dd104b0218349fc6.jpg",
+        "image_caption": [
+            "(b) Angry: “I hate it when people are taking too much space on planes.” "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "(a) Scared: “Planes might just be the most frightening thing ever.” ",
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+    {
+        "type": "image",
+        "img_path": "images/2883935d1c4a45e0ab6f07c351abdc70bb1a1f6a3615201000b0c109e88df7d6.jpg",
+        "image_caption": [
+            "Figure 3: Different meanings with different captions. "
+        ],
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+    {
+        "type": "text",
+        "text": "Exploiting both visual and textual information is therefore key to understanding a user’s underlying emotional state. We call our proposed network architecture, combining visual recognition and text analysis, “Deep Sentiment”. ",
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+    {
+        "type": "text",
+        "text": "4.3.1 ARCHITECTURE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Deep Sentiment’s architecture is shown in Figure 4: ",
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+    {
+        "type": "image",
+        "img_path": "images/746371da56165b060b1870fbd69a0eb99eff70669923b25a338ce80d5cad1671.jpg",
+        "image_caption": [
+            "Figure 4: The Deep Sentiment structure. On the one hand, the input image, resized to (224,224,3) is fed into the Inception network and outputs a vector of size 256. On the other hand, the text is projected into a high-dimensional space that subsequently goes through an LSTM layer with 1024 units. The two modalities are then concatenated and fed into a dense layer. The final softmax output layer give the probability distribution over the emotional state of the user. "
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+    {
+        "type": "text",
+        "text": "5 EVALUATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Table 2, we compare Deep Sentiment with the image model (Inception model fine-tuned through the last Inception module), the text model, and a baseline: random guessing that includes the prior probabilities of the classes. Figure 5 shows a comparison of the accuracy curves of the different models. ",
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+        "img_path": "images/7b8c0cd33837ecaa88d2511e7191dd09f0b06acb8612d0c416669fbbe7cfa725.jpg",
+        "table_caption": [
+            "Table 2: Comparison of image model, text model and Deep Sentiment. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Loss</td><td rowspan=1 colspan=1>Trainaccuracy</td><td rowspan=1 colspan=1>Testaccuracy</td></tr><tr><td rowspan=1 colspan=1>Random guessing</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>11%</td><td rowspan=1 colspan=1>11%</td></tr><tr><td rowspan=1 colspan=1>Image model</td><td rowspan=1 colspan=1>1.80</td><td rowspan=1 colspan=1>43%</td><td rowspan=1 colspan=1>36%</td></tr><tr><td rowspan=1 colspan=1>Text model</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>72%</td><td rowspan=1 colspan=1>69%</td></tr><tr><td rowspan=1 colspan=1>Deep Sentiment</td><td rowspan=1 colspan=1>0.75</td><td rowspan=1 colspan=1>80%</td><td rowspan=1 colspan=1>72%</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/5609ae3b910aab3e30daac14bdd3385ebafe55e2ef734bc5bd287f9cd140ac97.jpg",
+        "image_caption": [
+            "Figure 5: Accuracy comparison of the three models "
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+    {
+        "type": "text",
+        "text": "Using text alone, the test accuracy is $69 \\%$ , almost double the accuracy of the image model, this suggests that on Tumblr, text is a better predictor of emotion than images, as we illustrated in Figure 2. By combining text and images Deep Sentiment achieves $80 \\%$ train accuracy and $72 \\%$ test accuracy, significantly outperforming the images-only model and slightly outperforming the text-only model (note that no validation set was used to tune hyperparameters, even though various architectures were tried before yielding Deep Sentiment). ",
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+    {
+        "type": "text",
+        "text": "6 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we carefully investigate what psychologically meaningful results we can draw from our model, and whether they match previous results in the psychology literature. ",
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+    {
+        "type": "text",
+        "text": "6.1 TOP WORDS FOR EACH EMOTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We investigated which words were the most relevant for each emotion as follows: we created artificial posts whose text consisted of a single word (each of the most frequent 1,000 words in the whole dataset were tested) accompanied by an image that was the mean image (per channel mean). For each emotion, we report the 10 highest scoring words from these 1,000 artificial posts in Table 3 (words 11-20 for each emotion are in Appendix B). ",
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+    {
+        "type": "table",
+        "img_path": "images/c2723457cf772138909699c54c0168af01931ebebc2cb6a8b19bc031cc47dc4e.jpg",
+        "table_caption": [
+            "Table 3: Top 10 words for each emotion, ordered by the relative frequency of the emotion being used as a tag on Tumblr "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Emotion Happy</td><td>Top words</td></tr><tr><td>Calm Sad Scared Bored Angry Annoyed Love Excited Surprised Optimistic Amazed Ashamed Disgusted Pensive</td><td>healthy, loving,enjoy, wonderful, warm, happiness,smile, lovely, cute,proud quiet, situation, peace, mood, towards, warm, slowly, stay, sleep, rain horrible,sorry,crying,hurts, tears,cried,lonely,memories,worst, pain terrified, scary, panic, nervous,fear,afraid,horrible,woke,happening,worried asleep, tired,kinda,busy, stuck,constantly, lonely, sat, listening,depression anger, fear, panic,annoying,hate,mad, upset, anxiety, scares,stupid pissed, ashamed, angry, nervous,speak, surprised, tired, worried, ignore, phone soul,dreams,happiness,kiss,sex,beauty,women, feelings,god,relationships tonight, hopefully, watching, nervous, surprised, expect, tomorrow, amazing, hoping, happen birthday, cried, thank, yesterday, told, sorry,amazing, sweet, friend, message positive, expect, surprised, healthy, grow, realize, clearly, hopefully, calm, peace surprised, excited,amazing,woke,realized,awesome, happening,ashamed, yeah, happened totally, honestly, sorry,absolutely, freaking, honest,completely,stupid, seriously,am ashamed, totally,angry,hate,stupid,annoyed,horrible,scares,freaking,absolutely mood,wrote,quiet, view, sadness, thoughts,calm,words,sad,kissed</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "An inspection of each of the words suggests that none are out of place, with the possible exception of “ashamed” in the list of Amazed words. (This may be due to the fact that the tag Amazed is sometimes used ironically.) Further, our data-driven approach suggests that our methods could be used as an alternative to the word list-based approaches to sentiment analysis common in psychology. The most widely used tool, Linguistic Inquiry Word Count (LIWC), (Pennebaker et al., 2007), consists of dozens of English words which were compiled by hand into psychologically meaningful categories such as “Health/illness” or “Anxiety” and used in a large number of psychology studies (Tausczik & Pennebaker, 2010). Not only does our approach automatically give sensible word lists, it contains modern words, common in social media usage, like “woke” (first attested in its modern meaning in 1962 according to the Oxford English Dictionary which defines it as “alert to racial or social discrimination and injustice”) and “phone.” Woke appears in the top 10 of “Scared” and “Amazed” and in the top 20 of “Calm” (Appendix B), an interesting finding in its own right. By contrast, both woke and phone appear in LIWC, but not in any of LIWC’s emotion word categories, only in a list of verbs and a list of social words, respectively. ",
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+    {
+        "type": "text",
+        "text": "6.2 CLUSTERING EMOTIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Psychologists have long studied the structure of emotion, and debated whether there are a small number of “core” emotions (Ekman, 1992), two dominant factors (see Tellegen et al. (1999) for a discussion of various theories), or more complex models (e.g. Lindquist et al. (2013) critiques previous models and argues that emotions do not belong in natural categories or along multiple dimensions, because they are the “constructions” of the human mind, and the result of complex perceptions). Classical attempts by psychologists to answer this question have relied on survey data in which respondents can self-report more than one emotion, followed by clustering or factor analysis approaches. More recently, psychologists have turned to neuroscience to address these questions. ",
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+        "type": "text",
+        "text": "While we collected a dataset in which each Tumblr post was labeled with a single emotion tag, we can nevertheless directly address this question using our trained model. Given an image $\\mathcal { T }$ and text $\\tau$ from our dataset, we can compute the conditional probabilities of the emotion classes: ${ \\hat { y } } | \\mathcal { T } , \\mathcal { T }$ where $\\hat { y }$ is a probability vector in the 15-dimensional simplex. To calculate our model’s predicted empirical distribution over $\\hat { y }$ we predict $\\hat { y }$ for all posts in our dataset: $( \\hat { y } _ { i } | \\mathcal { T } _ { i } , \\mathcal { T } _ { i } ) _ { i = 1 } ^ { n }$ . Finally, we calculate the empirical correlation matrix of ${ \\hat { y } } _ { 1 } , \\dots , { \\hat { y } } _ { n }$ , which we include in the Appendix in Figure 9. For ease of visualization, we convert the correlation matrix to a distance matrix and perform hierarchical clustering as shown in Figure 6. ",
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+    {
+        "type": "image",
+        "img_path": "images/924e65d4b83c2df35442ad641abba4834abf1fe3a876de70cdee8cef5b19f97c.jpg",
+        "image_caption": [
+            "Figure 6: Hierarchical clustering of the emotion correlation matrix. "
+        ],
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+        "type": "text",
+        "text": "Similar to almost all previous psychology studies, there is a clear distinction between emotions that are positive in nature, such as excited, happy, and love and emotions that are negative such as angry, ashamed, annoyed, and disgusted. Another finding consistent with much previous literature, which we investigate more below, is that within both positive and negative emotions, there is a distinction between low arousal emotions and high arousal emotions. Bored and sad (low arousal) are in one cluster, while high arousal emotions—angry, ashamed, annoyed, disgusted—are in another. ",
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+    {
+        "type": "text",
+        "text": "6.3 THE CIRCUMPLEX MODEL OF EMOTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The circumplex model (Posner et al., 2005) posits two dimensions which explain emotions, usually valence (positive vs. negative) and arousal (low vs. high), though other dimensions have been suggested. We investigated whether two factors explained most of the variance in our results using principal component analysis (PCA). As shown in the scree plot in the Appendix in Figure 12, while the first component explains $2 8 . 8 \\%$ of the variance, dimensions 2 and 3 explain $1 6 . 4 \\%$ and $1 4 . 6 \\%$ respectively, evidence against a simple two factor model. ",
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+        "text": "(a) PCA with a random subset of posts, 150 for each emotion. ",
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+    {
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+        "img_path": "images/18242f21bc0ce11bf77b4044cc681343fc4ce23a52e4cdc351a2d18fac8e0e1f.jpg",
+        "image_caption": [
+            "Figure 7: Affect/valence "
+        ],
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+    {
+        "type": "text",
+        "text": "We visualize the results of PCA in Figure 7. At left, a random subset of posts, where for visualization purpose we sampled equally from each emotion class, are shown along the first two principal components, and each is colored by the most probable predicted emotion. This shows that within any particular emotion category there is much variation along these two dimensions, i.e. they do not cluster very tightly. Related emotions like happy and love are close together, as they were in the dendrogram. ",
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+    {
+        "type": "text",
+        "text": "At right, we visualise the projection of the emotion variables onto the first two principal components. Using this figure, we conclude that the first principal component corresponds well to valence, as it neatly separates happy/love from disgusted/ashamed/annoyed. However, the second principal component is not clearly arousal. In the Appendix, we consider the third principal component as well, but the picture is no clearer. Once again, this provides evidence against a simple two factor model of emotion. Below, we further investigate the valence/arousal model. ",
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+    {
+        "type": "text",
+        "text": "6.4 VALIDATION ON THE OPEN AFFECTIVE STANDARDIZED IMAGE SET (OASIS) ",
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+    {
+        "type": "text",
+        "text": "The Open Affective Standardized Image Set (OASIS) dataset (Kurdi et al., 2017) was developed as an alternative to the International Affective Picture System ((Lang, 2005)), a set of emotional stimuli which has been used in many psychology studies. OASIS consists of 900 color images which were rated by human judges on MTurk along the valence and arousal scales discussed above. OASIS images come in classes with labels such as “Alcohol”, “Flowers”, or “Pigeon.” We applied our Deep Sentiment model to these images, treating the single-word label as the input text. After making predictions, we projected them onto the principal components discussed above. ",
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+    {
+        "type": "text",
+        "text": "In Table 4 we calculated the correlations between the first three principal components from our model and the mean valence and arousal variables from OASIS. The high correlation between the first principal component (PC1) and the valence variable provides evidence, as in the previous section, that PC1 is capturing a dimension of emotion separating positive from negative. It is also interesting to note the low correlation between PC1 and arousal. By contrast, while PC2 and PC3 are correlated with arousal, the correlations are not very high, and they are also correlated with valence. On the one hand, this is more evidence against there being two factors explaining emotions. On the other hand, asking judges to rate a set of images based on arousal and valence is more similar to the standard sentiment analysis task of predicting whether a sentence expresses positive or negative emotion. By contrast, our dataset comes from a set of images which were chosen by users who also decided to select emotion word tags to accompany them. We consider this to thus be a dataset of richer emotional content, and by design it represents a richer set of emotions. ",
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+    {
+        "type": "table",
+        "img_path": "images/15023180078a1343a18edd83a70be9a2d868c61fd703bf38423f3b8e5738c51a.jpg",
+        "table_caption": [
+            "Table 4: Correlation between OASIS arousal/valence and principal components "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>OASISvalence</td><td rowspan=1 colspan=1>OASISarousal</td></tr><tr><td rowspan=1 colspan=1>PC1</td><td rowspan=1 colspan=1>58%</td><td rowspan=1 colspan=1>3%</td></tr><tr><td rowspan=1 colspan=1>PC2</td><td rowspan=1 colspan=1>17%</td><td rowspan=1 colspan=1>22%</td></tr><tr><td rowspan=1 colspan=1>PC3</td><td rowspan=1 colspan=1>30%</td><td rowspan=1 colspan=1>11%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We developed a novel multimodal sentiment analysis method using deep learning methods. Our goal was to investigate a core area of psychology, the study of emotion, using a large and novel social media dataset. Our approach provides new tools for the joint study of images and text on social media. This is important as social science researchers have begun to uncover the important role that images play on social media. As our dataset consisted of text, images, and emotion word tags, we considered it to be a form of “self-reported” data, which is the gold standard in emotion. However, it could also be considered to be an unobtrusive behavioral measure, and thus not subject to the biases inherent in laboratory studies. But because the data comes from public social media posts, we cannot rule out various sources of bias. What people choose to post—and not to post—on social media is influenced by how they want to present themselves (Wojnicki & Godes, 2008), so our study is limited insofar as it covers emotion not necessarily as it is truly experienced but rather as it is expressed or performed online. ",
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+    {
+        "type": "text",
+        "text": "In the future, we will evaluate our model on other image / text stimuli datasets that have been developed for psychological studies and investigate whether human judges are more or less accurate than our model. Finally, we will investigate other psychological components of the structure of emotion, for example daily and day of week trends in emotion. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
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+        "text": "David Watson and Lee Anna Clark. The panas-x: Manual for the positive and negative affect schedule - expanded form. 1999. ",
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+    {
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+        "text": "Andrea C Wojnicki and David Godes. Word-of-mouth as self-enhancement. 2008. ",
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+    {
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+        "text": "Quanzeng You, Jiebo Luo, Hailin Jin, and Jianchao Yang. Robust image sentiment analysis using progressively trained and domain transferred deep networks. In AAAI, 2015. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A MORE EXAMPLES OF TUMBLR POSTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            521,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/3be71e9c13a9b938b3f17c9fbacdacdba14c20f782673c48cba82826a61b4206.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            218,
+            185,
+            442,
+            299
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/7b2ebb6c643fa85093983473161ca421c97afec4b309fb25e96ccd5485af9ff1.jpg",
+        "image_caption": [
+            "(b) Disgusted: “Me when I see a couple expressing their affection in physical ways in public” "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            521,
+            185,
+            784,
+            299
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb64c199db9e486f75b47e9835ea9310876399db9255dcb06d72ab4a70800ff5.jpg",
+        "image_caption": [
+            "(a) Happy: “Just relax with this amazing view #bigsur #california #roadtrip #usa #life #fitness (at McWay Falls)” "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            345,
+            387,
+            460
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/ae2e9a868b2b2ca7f281006e507b76d1ba612a38c22246ab06c6e480a414f0f8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            560,
+            345,
+            746,
+            460
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(c) Angry: “Tensions were high this Caturday...” ",
+        "bbox": [
+            187,
+            467,
+            473,
+            479
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(d) Surprised: “Which Tea? Peppermint tea: What is your favorite gif right now?” ",
+        "bbox": [
+            496,
+            467,
+            812,
+            492
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 8: Tumblr posts ",
+        "bbox": [
+            421,
+            503,
+            575,
+            518
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B FURTHER RESULTS ",
+        "bbox": [
+            174,
+            574,
+            366,
+            590
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/8f783e517efcaf293b9a29cace54a492c3734c7e647229472937becf13877924.jpg",
+        "table_caption": [
+            "Table 5: Top 11-20 words for each emotion "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Emotion Happy</td><td>Top words</td></tr><tr><td>Calm Sad Scared Bored Angry Annoyed Love Excited Surprised Optimistic Amazed</td><td>beautiful, beauty, fun, nice, dinner, sweet, good,loves,amazing, comfortable view, sound, cool, continue, visit, push, woke, morning, safe, step cry, sick, hurt, dead,feeling,worse,death,upset, falling, panic kill, happened, cause,asleep,anxiety, worry, hurt, happen, worst, seriously sitting, suddenly, freaking, slowly,annoying, sometimes, gotten, completely, older, cold hurt, face,against, cause,hurts, fucking, fight, pain, strong,worst depressed,lately,conversation,constantly, scared,stupid, scares, bored,heard, extremely sweet, mother, forget, hate, her, song,enjoy, sister, wonderful, dear 2017, positive,am, definitely,awesome,happens, listening, happy, grow, wanna annoyed, negative,apparently,u, minute,asked, sadness, happened, moments, laugh view,yet, believe, tomorrow,trending,said,situation,mood,forward, future im,quite,appreciate, honestly, knew,learned, yang, felt, liked, asleep</td></tr></table>",
+        "bbox": [
+            174,
+            690,
+            862,
+            915
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b4e43cfc0aa3792084ff8fdb829f547d0b5096d5fce32c8a9626e804caa93b74.jpg",
+        "image_caption": [
+            "Figure 9: Heatmap of the emotion correlation matrix "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            308,
+            133,
+            648,
+            400
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/6fec305ae9bfa3f884a5285c5cec49e05dc05d79f11d2c23e4b3099491d393b5.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            702,
+            500,
+            814
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) PCA with a random subset of posts, 150 for each emotion ",
+        "bbox": [
+            171,
+            821,
+            501,
+            847
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/b3331d098cb894d6a4491e16a8bd123de5d177c57d63f175aeae6cd97f9883f6.jpg",
+        "image_caption": [
+            "Figure 10: PCA on dimension 1 and 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            522,
+            694,
+            800,
+            893
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/efebb27b83fa3fa88e57d8a12063cccd4deda41813e936f8a631975a5f0cc9ce.jpg",
+        "image_caption": [
+            "(a) PCA with a random subset of posts, 150 for each emotion "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            109,
+            500,
+            222
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/1787dde5931c290de4981e76b11550820cf253baeb336c24c0abd12d610e7c0f.jpg",
+        "image_caption": [
+            "Figure 11: PCA for dimension 2 and 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            526,
+            102,
+            797,
+            301
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/f2ec9015543b24b2f86d84da602ed8983a2420a55fdacf6fee80ffb20e03531f.jpg",
+        "image_caption": [
+            "Figure 12: Explained variance of the PCA "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            305,
+            359,
+            691,
+            574
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/1810a9ce40ea46405cefed677028077e9d1d78004746ef75d5630c2e822496a0.jpg",
+        "image_caption": [
+            "Figure 13: Factor loadings "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            450,
+            633,
+            581,
+            864
+        ],
+        "page_idx": 14
+    }
+]

parse/train/J8_GttYLFgr/J8_GttYLFgr.md

@@ -0,0 +1,511 @@
+# TRAJECTORY PREDICTION USING EQUIVARIANT CONTINUOUS CONVOLUTION
+
+Robin Walters ∗   
+Northeastern University   
+r.walters@northeastern.edu   
+Jinxi Li∗   
+Northeastern University   
+li.jinxi1@northeastern.edu   
+Rose Yu   
+University of California, San Diego   
+roseyu@ucsd.edu
+
+# ABSTRACT
+
+Trajectory prediction is a critical part of many AI applications, for example, the safe operation of autonomous vehicles. However, current methods are prone to making inconsistent and physically unrealistic predictions. We leverage insights from fluid dynamics to overcome this limitation by considering internal symmetry in real-world trajectories. We propose a novel model, Equivariant Continous COnvolution (ECCO) for improved trajectory prediction. ECCO uses rotationallyequivariant continuous convolutions to embed the symmetries of the system. On both vehicle and pedestrian trajectory datasets, ECCO attains competitive accuracy with significantly fewer parameters. It is also more sample efficient, generalizing automatically from few data points in any orientation. Lastly, ECCO improves generalization with equivariance, resulting in more physically consistent predictions. Our method provides a fresh perspective towards increasing trust and transparency in deep learning models. Our code and data can be found at https://github.com/Rose-STL-Lab/ECCO.
+
+# 1 INTRODUCTION
+
+Trajectory prediction is one of the core tasks in AI, from the movement of basketball players to fluid particles to car traffic (Sanchez-Gonzalez et al., 2020; Gao et al., 2020; Shah & Romijnders, 2016). A common abstraction underlying these tasks is the movement of many interacting agents, analogous to a many-particle system. Therefore, understanding the states of these particles, their dynamics, and hidden interactions is critical to accurate and robust trajectory forecasting.
+
+Even for purely physical systems such as in particle physics, the complex interactions among a large number of particles makes this a difficult problem. For vehicle or pedestrian trajectories, this challenge is further compounded with latent factors such as human psychology. Given these difficulties, current approaches require large amounts of training data and many model parameters. State-of-the-art methods in this domain such as Gao et al. (2020) are based on graph neural networks. They do not exploit the physical properties of system and often make predictions which are not self-consistent or physically meaningful. Furthermore, they predict a single agent trajectory at a time instead of multiple agents simultaneously.
+
+![](images/c194779fdbb7b5a7b45a478540e0efcf875ec81e36fa9a18c1b5d0fcfc7f680a.jpg)  
+Figure 1: Car trajectories in two scenes. Though the entire scenes are not related by a rotation, the circled areas are. ECCO exploits this symmetry to improve generalization and sample efficiency.
+
+Our model is built upon a key insight of many-particle systems pertaining to intricate internal symmetry. Consider a model which predicts the trajectory of cars on a road. To be successful, such a model must understand the physical behavior of vehicles together with human psychology. It should distinguish left from right turns, and give consistent outputs for intersections rotated with different orientation. As shown in Figure 1, a driver’s velocity rotates with the entire scene, whereas vehicle interactions are invariant to such a rotation. Likewise, psychological factors such as reaction speed or attention may be considered vectors with prescribed transformation properties. Data augmentation is a common practice to deal with rotational invariance, but it cannot guarantee invariance and requires longer training. Since rotation is a continuous group, augmentation requires sampling from infinitely many possible angles.
+
+In this paper, we propose an equivariant continuous convolutional model, ECCO, for trajectory forecasting. Continuous convolution generalizes discrete convolution and is adapted to data in manyparticle systems with complex local interactions. Ummenhofer et al. (2019) designed a model using continuous convolutions for particle-based fluid simulations. Meanwhile, equivariance to group symmetries has proven to be a powerful tool to integrate physical intuition in physical science applications (Wang et al., 2020; Brown & Lunter, 2019; Kanwar et al., 2020). Here, we test the hypothesis that an equivariant model can also capture internal symmetry in non-physical human behavior. Our model utilizes a novel weight sharing scheme, torus kernels, and is rotationally equivariant.
+
+We evaluate our model on two real-world trajectory datasets: Argoverse autonomous vehicle dataset (Chang et al., 2019) and TrajNet+ $^ +$ pedestrian trajectory forecasting challenge (Kothari et al., 2020). We demonstrate on par or better prediction accuracy to baseline models and data augmentation with fewer parameters, better sample efficiency, and stronger generalization properties. Lastly, we demonstrate theoretically and experimentally that our polar coordinate-indexed filters have lower equivariance discretization error due to being better adapted to the symmetry group.
+
+Our main contributions are as follows:
+
+• We propose Equivariant Continous COnvolution (ECCO), a rotationally equivariant deep neural network that can capture internal symmetry in trajectories.   
+• We design ECCO using a novel weight sharing scheme based on orbit decomposition and polar coordinate-indexed filters. We implement equivariance for both the standard and regular representation $L ^ { 2 } ( \mathrm { S O } ( 2 ) )$ .   
+• On benchmark Argoverse and TrajNet++ datasets, ECCO demonstrates comparable accuracy while enjoying better generalization, fewer parameters, and better sample complexity.
+
+# 2 RELATED WORK
+
+Trajectory Forecasting For vehicle trajectories, classic models in transportation include the CarFollowing model (Pipes, 1966) and Intelligent Driver model (Kesting et al., 2010). Deep learning has also received considerable attention; for example, Liang et al. (2020) and Gao et al. (2020) use graph neural networks to predict vehicle trajectories. Djuric et al. (2018) use rasterizations of the scene with CNN. See the review paper by Veres & Moussa (2019) for deep learning in transportation. For human trajectory modeling, Alahi et al. (2016) propose Social LSTM to learn these humanhuman interactions. TrajNet (Sadeghian et al., 2018) and TrajNet+ $^ { \cdot + }$ (Kothari et al., 2020) introduce benchmarking for human trajectory forecasting. We refer readers to Rudenko et al. (2020) for a comprehensive survey. Nevertheless, many deep learning models are data-driven. They require large amounts of data, have many parameters, and can generate physically inconsistent predictions.
+
+Continuous Convolution Continuous convolutions over point clouds (CtsConv) have been successfully applied to classification and segmentation tasks (Wang et al., 2018; Lei et al., 2019; Xu et al., 2018; Wu et al., 2019; Su et al., 2018; Li et al., 2018; Hermosilla et al., 2018; Atzmon et al., 2018; Hua et al., 2018). More recently, a few works have used continuous convolution for modeling trajectories or flows. For instance, Wang et al. (2018) uses CtsConv for inferring flow on LIDAR data. Schenck & Fox (2018) and Ummenhofer et al. (2019) model fluid simulation using CtsConv. Closely related to our work is Ummenhofer et al. (2019), who design a continuous convolution network for particle-based fluid simulations. However, they use a ball-to-sphere mapping which is not well-adapted for rotational equivariance and only encode 3 frames of input. Graph neural networks (GNNs) are a related strategy which have been used for modeling particle system dynamics (Sanchez-Gonzalez et al., 2020). GNNs are also permutation invariant, but they do not natively encode relative positions and local interaction as a CtsConv-based network does.
+
+Equivariant and Invariant Deep Learning Developing neural nets that preserve symmetries has been a fundamental task in image recognition (Cohen et al., 2019b; Weiler & Cesa, 2019; Cohen & Welling, 2016a; Chidester et al., 2018; Lenc & Vedaldi, 2015; Kondor & Trivedi, 2018; Bao & Song, 2019; Worrall et al., 2017; Cohen & Welling, 2016b; Weiler et al., 2018; Dieleman et al., 2016; Maron et al., 2020). Equivariant networks have also been used to predict dynamics: for example, Wang et al. (2020) predicts fluid flow using Galilean equivariance but only for gridded data. Fuchs et al. (2020) use SE(3)-equivariant transformers to predict trajectories for a small number of particles as a regression task. As in this paper, both Bekkers (2020) and Finzi et al. (2020) address the challenge of parameterizing a kernel over continuous Lie groups. Finzi et al. (2020) apply their method to trajectory prediction on point clouds using a small number of points following strict physical laws. Worrall et al. (2017) also parameterizes convolutional kernels using polar coordinates, but maps these onto a rectilinear grid for application to image data. Weng et al. (2018) address rotational equivariance by inferring a global canonicalization of the input. Similar to our work, Esteves et al. (2018) use functions evenly sampled on the circle, however, their features are only at a single point whereas we assign feature vectors to each point in a point cloud. Thomas et al. (2018) introduce Tensor Field Networks which are SO(3)-equivariant continuous convolutions. Unlike our work, both Worrall et al. (2017) and Thomas et al. (2018) define their kernels using harmonic functions. Our weight sharing method using orbits and stabilizers is simpler as it does not require harmonic functions or Clebsch-Gordon coefficients. Unlike previous work, we implement a regular representation for the continuous rotation group SO(2) which is compatible with pointwise nonlinearities and enjoys an empirical advantage over irreducible representations.
+
+# 3 BACKGROUND
+
+We first review the necessary background of continuous convolution and rotational equivariance.
+
+# 3.1 CONTINUOUS CONVOLUTION
+
+Continuous convolution (CtsConv) generalizes the discrete convolution to point clouds. It provides an efficient and spatially aware way to model the interactions of nearby particles. Let $\mathbf { f } ^ { ( i ) } \in \mathbb { R } ^ { c _ { \mathrm { i n } } }$ denote the feature vector of particle $i$ . Thus $\mathbf { f }$ is a vector field which assigns to the points $\mathbf { x } ^ { ( i ) }$ a vector in $\mathbb { R } ^ { c _ { \mathrm { i n } } }$ . The kernel of the convolution $K \colon \mathbb { R } ^ { 2 }  \mathbb { R } ^ { c _ { \mathrm { o u t } } \times c _ { \mathrm { i n } } }$ is a matrix field: for each point $\mathbf { x } \in \mathbb { R } ^ { 2 }$ , $K ( \mathbf { x } )$ is a $c _ { \mathrm { o u t } } \times c _ { \mathrm { i n } }$ matrix. Let $a$ be a radial local attention map with $a ( r ) = 0$ for $r > R$ . The output feature vector $\mathbf { g } ^ { ( i ) }$ of particle $i$ from the continous convolution is given by
+
+$$
+\mathbf { g } ^ { ( i ) } = \mathsf { C t s C o n v } _ { K , R } ( \mathbf { x } , \mathbf { f } ; \mathbf { x } ^ { ( i ) } ) = \sum _ { j } a ( \| \mathbf { x } ^ { ( j ) } - \mathbf { x } ^ { ( i ) } \| ) K ( \mathbf { x } ^ { ( j ) } - \mathbf { x } ^ { ( i ) } ) \cdot \mathbf { f } ^ { ( j ) } .
+$$
+
+CtsConv is naturally equivariant to permutation of labels and is translation invariant. Equation 1 is closely related to graph neural network (GNN) (Kipf & Welling, 2017; Battaglia et al., 2018), which is also permutation invariant. Here the graph is dynamic and implicit with nodes $\mathbf { x } ^ { ( i ) }$ and edges $e _ { i j }$ if $\| \mathbf { x } ^ { ( i ) } - \mathbf { x } ^ { ( j ) } \| < R$ . Unlike a GNN which applies the same weights to all neighbours, the kernel $K$ depends on the relative position vector $\mathbf { x } ^ { ( i ) } - \mathbf { x } ^ { ( j ) }$ .
+
+# 3.2 ROTATIONAL EQUIVARIANCE
+
+Continuous convolution is not naturally rotationally equivariant. Fortunately, we can translate the technique of rotational equivariance on CNNs to continuous convolutions. We use the language of Lie groups and their representations. For more background, see Hall (2015) and Knapp (2013).
+
+More precisely, we denote the symmetry group of 2D rotations by $\operatorname { S O } ( 2 ) = \{ { \mathrm { R o t } } _ { \theta } : 0 \leq \theta <$ $2 \pi \}$ . As a Lie group, it has both a group structure $\mathrm { R o t } _ { \theta _ { 1 } } \circ \mathrm { R o t } _ { \theta _ { 2 } } = \mathrm { R o t } _ { ( \theta _ { 1 } + \theta _ { 2 } ) \mathrm { m o d } 2 \pi }$ which a continuous map with respect to the topological structure. As a manifold, SO(2) is homomeomorphic to the circle $\Dot { S ^ { 1 } } \cong \{ \mathbf { x } \ \overset { \cdot } { \in } \ \mathbb { R } ^ { 2 } \ : \ \lVert \mathbf { x } \rVert = \ 1 \}$ . The group $\mathrm { S O } ( 2 )$ can act on a vector space $\mathbb { R } ^ { c }$ by specifying a representation map $\rho \colon \mathsf { S O } ( 2 ) \ \mathsf { \Gamma } \to \mathsf { G L } ( \mathbb { R } ^ { c } )$ which assigns to each element of SO(2) an element of the set of invertible $c \times c$ matrices ${ \mathrm { G L } } ( \mathbb { R } ^ { c } )$ . The map $\rho$ must a be homomorphism $\rho ( \mathrm { R o t } _ { \theta _ { 1 } } ) \rho ( \mathrm { R o t } _ { \theta _ { 1 } } ) = \rho ( \mathrm { R o t } _ { \theta _ { 1 } } \circ \mathrm { R o t } _ { \theta _ { 2 } } )$ . For example, the standard representation $\rho _ { 1 }$ on $\mathbb { R } ^ { 2 }$ is by $2 \times 2$ rotation matrices. The regular representation $\rho _ { \mathrm { r e g } }$ on $L ^ { 2 } ( \operatorname { S O } ( { \bar { 2 } } ) ) = \{ \varphi : \operatorname { S O } ( 2 ) \to \mathbb { R } :$ $| \varphi | ^ { 2 }$ is integrable $\}$ is $\rho _ { \mathrm { r e g } } ( \mathrm { R o t } _ { \phi } ) ( \varphi ) = \varphi \circ \mathrm { R o t } _ { - \phi }$ . Given input f with representation $\rho _ { \mathrm { i n } }$ and output with representation $\rho _ { \mathrm { o u t } }$ , a map $F$ is SO(2)-equivariant if
+
+$$
+F ( \rho _ { \mathrm { i n } } ( \mathrm { R o t } _ { \theta } ) { \bf f } ) = \rho _ { \mathrm { o u t } } ( \mathrm { R o t } _ { \theta } ) F ( { \bf f } ) .
+$$
+
+Discrete CNNs are equivariant to a group $G$ if the input, output, and hidden layers carry a $G$ -action and the linear layers and activation functions are all equivariant (Kondor $\&$ Trivedi, 2018). One method for constructing equivariant discrete CNNs is steerable CNN (Cohen & Welling, 2016b). Cohen et al. (2019a) derive a general constraint for when a convolutional kernel $K \colon \mathbb { R } ^ { b }  \mathbf { \bar { \mathbb { R } } } ^ { c _ { \mathrm { o u t } } \times c _ { \mathrm { i n } } }$ is $G$ -equivariant. Assume $G$ acts on $\mathbb { R } ^ { b }$ and that $\mathbb { R } ^ { c _ { \mathrm { o u t } } }$ and $\mathbb { R } ^ { c _ { \mathrm { i n } } }$ are $G$ -representations $\rho _ { \mathrm { o u t } }$ and $\rho _ { \mathrm { i n } }$ respectively, then $K$ is $G$ -equivariant if for all $g \in G , \mathbf { x } \in \mathbb { R } ^ { 2 }$ ,
+
+$$
+K ( g \mathbf x ) = \rho _ { \mathrm { o u t } } ( g ) K ( \mathbf x ) \rho _ { \mathrm { i n } } ( g ^ { - 1 } ) .
+$$
+
+For the group SO(2), Weiler & Cesa (2019) solve this constraint using circular harmonic functions to give a basis of discrete equivariant kernels. In contrast, our method is much simpler and uses orbits and stabilizers to create continuous convolution kernels.
+
+# 4 ECCO: TRAJECTORY PREDICTION USING ROTATIONALLY EQUIVARIANTCONTINUOUS CONVOLUTION
+
+In trajectory prediction, given historical position and velocity data of $n$ particles over $t _ { \mathrm { i n } }$ timesteps, we want to predict their positions over the next $t _ { \mathrm { o u t } }$ timesteps. Denote the ground truth dynamics as $\xi$ , which maps $\xi ( { \bf x } _ { t - t _ { \mathrm { i n } } : t } , { \bf v } _ { t - t _ { \mathrm { i n } } : t } ) = { \bf x } _ { t : t + t _ { \mathrm { o u t } } }$ . Motivated by the observation in Figure 1, we wish to learn a model $f$ that approximates the underlying dynamics while preserving the internal symmetry in the data, specifically rotational equivariance.
+
+We introduce ECCO, a model for trajectory prediction based on rotationally equivariant continuous convolution. We implement rotationally equivariant continuous convolutions using a weight sharing scheme based on orbit decomposition. We also describe equivariant per-particle linear layers which are a special case of continuous convolution with radius $R = 0$ analogous to 1x1 convolutions in CNNs. Such layers are useful for passing information between layers from each particle to itself.
+
+# 4.1 ECCO MODEL OVERVIEW
+
+![](images/58c03535ef661e27469f1151eec0dd51ed96acd37c7444475de73561b5e94f0f.jpg)  
+Figure 2: Overview of model architecture. Past velocities are aggregated by an encoder Enc. Together with map information this is then encoded by 3 CtsConvs into $\rho _ { \mathrm { r e g } }$ features. Then $l + 1$ CtsConv layers are used to predict $\Delta \tilde { x }$ . The predicted position $\hat { x } _ { t + 1 } = \Delta \tilde { x } + \tilde { x }$ where $\tilde { x }$ is a numerically extrapolated using velocity and accleration. Since $\Delta \tilde { x }$ is translation invariant, $\hat { x }$ is equivariant.
+
+The high-level architecture of ECCO is illustrated in Figure 2. It is important to remember that the input, output, and hidden layers are all vector fields over the particles. Oftentimes, there is also environmental information available in the form of road lane markers. Denote marker positions by $\mathbf { x } _ { \mathrm { m a p } }$ and direction vectors by $\mathbf { v } _ { \mathrm { m a p } }$ . This data is thus also a particle field, but static.
+
+To design an equivariant network, one must choose the group representation. This choice plays an important role in shaping the learned hidden states. We focus on two representations of SO(2): $\rho _ { 1 }$ and $\rho _ { \mathrm { r e g } }$ . The representation $\rho _ { 1 }$ is that of our input features, and $\rho _ { \mathrm { r e g } }$ is for the hidden layers. For $\rho _ { 1 }$ , we constrain the kernel in Equation 1. For $\rho _ { \mathrm { r e g } }$ , we further introduce a new operator, convolution with torus kernels.
+
+In order to make continuous convolution rotationally equivariant, we translate the general condition for discrete CNNs developed in Weiler & Cesa (2019) to continuous convolution. We define the convolution kernel $K$ in polar coordinates $K ( \theta , r )$ . Let $\mathbb { R } ^ { c _ { \mathrm { o u t } } }$ and $\mathbb { R } ^ { c _ { \mathrm { i n } } }$ be SO(2)-representations $\rho _ { \mathrm { o u t } }$ and $\rho _ { \mathrm { i n } }$ respectively, then the equivariance condition requires the kernel to satisfy
+
+$$
+K ( \theta + \phi , r ) = \rho _ { \mathrm { o u t } } ( \mathrm { R o t } _ { \theta } ) K ( \phi , r ) \rho _ { \mathrm { i n } } ( \mathrm { R o t } _ { \theta } ^ { - 1 } ) .
+$$
+
+Imposing such a constraint for continuous convolution requires us to develop an efficient weight sharing scheme for the kernels, which solve Equation 3.
+
+# 4.2 WEIGHT SHARING BY ORBITS AND STABILIZERS.
+
+Given a point $\mathbf { x } \in \mathbb { R } ^ { 2 }$ and a group $G$ , the set $O _ { \mathbf { x } } = \{ g \mathbf { x } : g \in G \}$ is the orbit of the point $\mathbf { x }$ . The set of orbits gives a partition of $\mathbb { R } ^ { 2 }$ into the origin and circles of radius $r > 0$ . The set of group elements $G _ { \mathbf { x } } = \{ g : \mathbf { \bar { g x } } = x \}$ fixing $\mathbf { x }$ is called the stabilizer of the point $\mathbf { x }$ . We use the orbits and stabilizers to constrain the weights of $K$ . Simply put, we share weights across orbits and constrain weights according to stabilizers, as shown in Figure 3-Left.
+
+The ray $D = \{ ( 0 , r ) : r \geq 0 \}$ is a fundamental domain for the action of $G = \mathrm { S O } ( 2 )$ on base space $\mathbb { R } ^ { 2 }$ . That is, $D$ contains exactly one point from each orbit. We first define $K ( 0 , r )$ for each $( { \bar { 0 } } , r ) \in D$ . Then we compute $K ( \theta , r )$ from $K ( 0 , r )$ by setting $\phi = 0$ in Equation 3 as such
+
+$$
+\begin{array} { r } { K ( \theta , r ) = \rho _ { \mathrm { o u t } } ( \mathrm { R o t } _ { \theta } ) K ( 0 , r ) \rho _ { \mathrm { i n } } ( \mathrm { R o t } _ { \theta } ^ { - 1 } ) . } \end{array}
+$$
+
+For $r > 0$ , the group acts freely on $( 0 , r )$ , i.e. the stabilizer contains only the identity. This means that Equation 3 imposes no additional constraints on $K ( 0 , r )$ . Thus $K ( 0 , r ) \in \mathbb { R } ^ { c _ { \mathrm { o u t } } \times c _ { \mathrm { i n } } }$ is a matrix of freely learnable weights.
+
+For $r = 0$ , however, the orbit $O _ { ( 0 , 0 ) }$ is only one point. The stabilizer of $( 0 , 0 )$ is all of $G$ , which requires
+
+$$
+K ( 0 , 0 ) = \rho _ { \mathrm { o u t } } ( \mathrm { R o t } _ { \theta } ) K ( 0 , 0 ) \rho _ { \mathrm { i n } } ( \mathrm { R o t } _ { \theta } ^ { - 1 } ) \mathrm { f o r \ a l l } \theta .
+$$
+
+Thus $K ( 0 , 0 )$ is an equivariant per-particle linear map $\rho _ { \mathrm { i n } }  \rho _ { \mathrm { o u t } }$
+
+We can analytically solve Equation 5 for $\dot { K } ( 0 , 0 )$ using representation theory. Table 1 shows the unique solutions for different combinations of $\rho _ { 1 }$ and $\rho _ { \mathrm { r e g } }$ . For details see subsection A.3.
+
+Table 1: Equivariant linear maps for $K ( 0 , 0 )$ . Trainable weights are $c \in \mathbb { R }$ and $\bar { \kappa \colon S ^ { 1 } \to \mathbb { R } }$ , where $\hat { S ^ { 1 } }$ is the manifold underlying SO(2).   
+
+<table><tr><td>Pin</td><td>Pout = p1</td><td>Pout =preg</td></tr><tr><td>P1 preg</td><td>(a,b) -→ (ca,cb) fc Ss1 f(0)cos(0)d0 Js1 f(0)sin(0)d0)</td><td>ca cos(0) + cb sin(0) Ss1 κ(0-)f(Φ)dΦ</td></tr></table>
+
+Note that 2D and 3D rota
+
+tion equivariant continuous convolutions are implemented in Worrall et al. (2017) and Thomas et al. (2018) respectively. They both use harmonic functions which require expensive evaluation of analytic functions at each point. Instead, we provide a simpler solution. We require only knowledge of the orbits, stabilizers, and input/output representations. Additionally, we bypass Clebsch-Gordon decomposition used in Thomas et al. (2018) by mapping directly between the representations in our network. Next, we describe an efficient implementation of equivariant continuous convolution.
+
+# 4.3 POLAR COORDINATE KERNELS
+
+Rotational equivariance informs our kernel discretization and implementation. We store the kernel $K$ of continuous convolution as a 4-dimensional tensor by discretizing the domain. Specifically, we discretize $\mathbb { R } ^ { 2 }$ using polar coordinates with $k _ { \theta }$ angular slices and $k _ { r }$ radial steps. We then evaluate $K$ at any $( \theta , r )$ using bilinear interpolation from four closest polar grid points. This method accelerates computation since we do not need to use Equation 4 to repeatedly compute $K ( \theta , r )$ from $K ( 0 , r )$ . The special case of $K ( 0 , 0 )$ results in a polar grid with a “bullseye” at the center (see Figure 3-Left).
+
+![](images/fa29faa81a8731d0f3cbdc944a32dce9ef8f9fee9f38462f04a82f2b4c60f54e.jpg)  
+Figure 3: Left: A torus kernel field $K$ from a $\rho _ { \mathrm { r e g } }$ -field to a $\rho _ { \mathrm { r e g } }$ -field. The kernel is itself a field: at each point $\mathbf { x }$ in space the kernel $K ( \mathbf { x } )$ yields a different matrix. We denote the $\left( \phi _ { 2 } , \phi _ { 1 } \right)$ entry of the matrix at $\mathbf { x } = ( \theta , r )$ by $K ( \theta , r ) ( \stackrel { \cdot } { \phi _ { 2 } } , \stackrel { \cdot } { \phi _ { 1 } } )$ . The matrices along the red sector are freely trainable. The matrices at all white sectors are determined by those in the red sector according to the circular shifting rule illustrated above. The matrix at the red bullseye is trainable but constrained to be circulant, i.e. preserved by the circular shifting rule. Right: The torus kernel acts on features which are functions on the circle. By cutting open the torus and features along the reg and orange lines we can identify the operation at each point with matrix multiplication.
+
+We discretize angles finely and radii more coarsely. This choice is inspired by real-world observation that drivers tend to be more sensitive to the angle of an incoming car than its exact distance, Our equivariant kernels are computationally efficient and have very few parameters. Moreover, we will discuss later in Section 4.5 that despite discretization, the use of polar coordinates allows for very low equivariance error.
+
+# 4.4 HIDDEN LAYERS AS REGULAR REPRESENTATIONS
+
+Regular representation $\rho _ { \mathrm { r e g } }$ has shown better performance than $\rho _ { 1 }$ for finite groups (Cohen et al., $2 0 1 9 \mathrm { a }$ ; Weiler & Cesa, 2019). But the naive $\rho _ { \mathrm { r e g } } = \{ \varphi \colon G  \mathbb { R } \}$ for an infinite group $G$ is too large to work with. We choose the space of square-integrable functions $L ^ { 2 } ( G )$ . It contains all irreducible representations of $G$ and is compatible with pointwise non-linearities.
+
+Discretization. However, $L ^ { 2 } ( \mathrm { S O } ( 2 ) )$ is still infinite-dimensional. We resolve this by discretizing the manifold $S ^ { 1 }$ underlying SO(2) into $k _ { \mathrm { r e g } }$ even intervals. We represent functions $f \in L ^ { 2 } ( \mathrm { S O } ( 2 ) \bar { ) }$ by the vector of values $[ f ( \mathrm { R o t } _ { 2 \pi i / k _ { \mathrm { r e g } } } ) ] _ { 0 \leq i < k _ { \mathrm { r e g } } }$ . We then evaluate $f ( \mathrm { R o t } _ { \theta } )$ using interpolation.
+
+We separate the number of angular slices $k _ { \theta }$ and the size of the kernel $k _ { \mathrm { r e g } }$ . If we tie them together and set $k _ { \theta } = k _ { \mathrm { r e g } }$ , this is equivalent to implementing cyclic group $C _ { k _ { \mathrm { r e g } } }$ symmetry with the regular representation. Then increasing $k _ { \theta }$ would also increases $k _ { \mathrm { r e g } }$ , which incurs more parameters.
+
+Convolution with Torus Kernel. In addition to constraining the kernel $K$ of Equation 1 as in $\rho _ { 1 }$ , $\rho _ { \mathrm { r e g } }$ poses an additional challenge as it is a function on a circle. We introduce a new operator from functions on the circle to functions on the circle called a torus kernel.
+
+First, we replace input feature vectors in $\mathbf { f } \in \mathbb { R } ^ { c }$ with elements of $L ^ { 2 } ( \mathrm { S O } ( 2 ) )$ . The input feature f becomes a $\rho _ { \mathrm { r e g } }$ -field, that is, for each $\mathbf { x } \in \mathbb { R } ^ { 2 }$ , $\mathbf { f } ^ { ( \mathbf { x } ) }$ is a real-value function on the circle $S ^ { 1 } \to \mathbb { R }$ . For the kernel $K$ , we replace the matrix field with a map $K \colon \mathbb { R } ^ { 2 }  \rho _ { \mathrm { r e g } } \otimes \rho _ { \mathrm { r e g } }$ . Instead of a matrix, $K ( \mathbf { x } )$ is a map $S ^ { 1 } \times S ^ { 1 } \to \mathbb { R }$ . Here $( \phi _ { 1 } , \phi _ { 2 } ) \in S ^ { 1 } \times S ^ { 1 }$ plays the role of continuous matrix indices and we may consider $K ( \mathbf { x } ) ( \phi _ { 1 } , \phi _ { 2 } ) \in \mathbb { R }$ analogous to a matrix entry. Topologically, $S ^ { 1 } \times S ^ { 1 }$ is a torus and hence we call $K ( \mathbf { x } )$ a torus kernel. The matrix multiplication $K ( \mathbf { x } ) \cdot f ^ { ( \mathbf { x } ) }$ in Equation 1
+
+must be replaced by the integral transform
+
+$$
+K ( \mathbf { x } ) \odot f ^ { ( \mathbf { x } ) } ( \phi _ { 2 } ) = \int _ { \phi _ { 1 } \in S ^ { 1 } } K ( \mathbf { x } ) ( \phi _ { 2 } , \phi _ { 1 } ) f ^ { ( \mathbf { x } ) } ( \phi _ { 1 } ) d \phi _ { 1 } ,
+$$
+
+which is a linear transformation $L ^ { 2 } ( \mathrm { S O ( 2 ) } )  L ^ { 2 } ( \mathrm { S O ( 2 ) } )$ . $K ( \theta , r ) ( \phi _ { 2 } , \phi _ { 1 } )$ denotes the $\left( \phi _ { 2 } , \phi _ { 1 } \right)$ entry of the matrix at point $\mathbf { x } = ( \theta , r )$ , see the illustration in Figure 3-Right. We compute Equation 3 for $\rho _ { \mathrm { r e g } }  \rho _ { \mathrm { r e g } }$ as $\bar { K } ( \mathrm { R o t } _ { \theta } ( { \bf x } ) ) ( \phi _ { 2 } , \phi _ { 1 } ) = K ( { \bf x } ) ( \phi _ { 2 } - \theta , \bar { \phi _ { 1 } } - \theta )$ . We can use the same weight sharing scheme as in Section 4.2.
+
+# 4.5 ANALYSIS: EQUIVARIANCE ERROR
+
+The practical value of equivariant neural networks has been demonstrated in a variety of domains. However, theoretical analysis (Kondor $\&$ Trivedi, 2018; Cohen et al., 2019a; Maron et al., 2020) of continuous Lie group symmetries is usually performed assuming continuous functions and using the integral representation of the convolution operator. In practice, discretization can cause the model $f$ to be not exactly equivariant, with some equivariance error (EE)
+
+$$
+\mathrm { E E } = \| f ( T ( x ) ) - T ^ { \prime } ( f ( x ) ) \|
+$$
+
+with respect to group transformations $T$ and $T ^ { \prime }$ of input and output respectively (Wang et al., 2020, A6). Rectangular grids are well-suited to translations, but poorlysuited to rotations. The resulting equivariance error can be so large to practically undermine the advantages of a theoretically equivariant network.
+
+![](images/a733352c4069e6f84814b38be353cfa71edd7a38ff0c39e909ad679eafea34d5.jpg)  
+Figure 4: Experimentally, we find $k _ { \theta }$ and expected equivariance error are inversely proportional.
+
+Our polar-coordinate indexed circular filters are designed specifically to adapt well to the rotational symmetry. In Figure 4, we demonstrate experimentally that expected EE is inversely proportional to the number of angular slices $k _ { \theta }$ . For example, choosing $k _ { \theta } \geq 1 6$ gives very low EE and does not increase the number of parameters. We also prove for $\rho _ { 1 }$ features that the equivariance error is low in expectation. See Appendix A.6 for the precise statement and proof.
+
+Proposition. Let $\alpha = 2 \pi / k _ { \theta }$ , and $\bar { \theta }$ be $\theta$ rounded to nearest value in $\mathbb { Z } \alpha$ , and $\hat { \theta } = | \theta - \bar { \theta } |$ . Let $F = C t s C o n v _ { K , R }$ and $T = \rho _ { 1 } ( \mathrm { R o t } _ { \theta } )$ . For some constant $C$ , the expected EE is bounded
+
+$$
+\mathbb { E } _ { K , \mathbf { f } , \mathbf { x } } [ T ( F ( \mathbf { f } , \mathbf { x } ) ) - F ( T ( \mathbf { f } ) , T ( \mathbf { x } ) ) ] \leq | \sin ( \hat { \theta } ) | C \leq 2 \pi C / k _ { \theta } .
+$$
+
+# 5 EXPERIMENTS
+
+In this section, we present experiments in two different domains, traffic and pedestrian trajectory prediction, where interactions among agents are frequent and influential. We first introduce the statistics of the datasets and the evaluation metrics. Secondly, we compare different feature encoders and hidden feature representation types. Lastly, we compare our model with baselines.
+
+# 5.1 EXPERIMENTAL SET UP
+
+Dataset We discuss the performances of our models on (1) Argoverse autonomous vehicle motion forecasting (Chang et al., 2019), a recently released vehicle trajectory prediction benchmark, and (2) TrajNet $^ { + + }$ pedestrian trajectory forecasting challenge (Kothari et al., 2020). For Argoverse, the task is to predict three-second trajectories based on all vehicles history in the past 2 seconds. We split 32K samples from the validation set as our test set.
+
+Baselines We compare against several state-of-the-art baselines used in Argoverse and TrajNet++. We use three original baselines from (Chang et al., 2019): Constant velocity, Nearest Neighbour, and Long Short Term Memory (LSTM). We also compare with a non-equivariant continuous convolutional model, CtsConv (Ummenhofer et al., 2019) and a hierarchical GNN model VectorNet (Gao et al., 2020). Note that VectorNet only predicts a single agent at a time, which is not directly comparable with ours. We include VectorNet as a reference nevertheless.
+
+Evaluation Metrics We use domain standard metrics to evaluate the trajectory prediction performance, including (1) Average Displacement Error (ADE): the average L2 displacement error for the whole 30 timestamps between prediction and ground truth, (2) Displacement Error at $t$ seconds $\mathrm { ( D E @ ~ }$ ts): the L2 displacement error at a given timestep $t$ . DE $@$ ts for the last timestamp is also called Final Displacement Error (FDE). For Argoverse, we report ADE and $\mathrm { D E } @$ ts for $t \in \{ 1 , 2 , 3 \}$ . For TrajNet++, we report ADE and FDE.
+
+# 5.2 PREDICTION PERFORMANCE COMPARISON
+
+We evaluate the performance of different models from multiple aspects: forecasting accuracy, parameter efficiency and the physical consistency of the predictions. The goal is to provide a comprehensive view of various characteristics of our model to guide practical deployment. See Appendix A.9 for an additional ablative study.
+
+Forecasting Accuracy We compare the trajectory prediction accuracy across different models on Argoverse and TrajNet $^ { + + }$ . Table 2 displays the prediction ADE and FDE comparision. We can see that ECCO with the regular representation $\rho _ { \mathrm { r e g } }$ achieves on par or better forecasting accuracy on both datasets. Comparing ECCO and a non-equivariant counterpart of our model CtsConv, we observe a significant $1 4 . 8 \%$ improvement in forecasting accuracy. Compare with data augmentation, we also observe a $9 \%$ improvement over the non-equivariant CtsConv trained on random-rotationaugmented dataset. These results demonstrate the benefits of incorporating equivariance principles into deep learning models.
+
+Table 2: Parameter efficiency and accuracy comparison. Number of parameters for each model and their detailed forecasting accuracy at DE $@$ ts. CtsConv(Aug.) is CtsConv trained with rotation augmented data.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Argoverse</td><td colspan="2">TrajNet++</td><td rowspan="2">#Param</td></tr><tr><td>ADE</td><td>DE@1s</td><td>DE@2s</td><td>DE@3s</td><td>ADE</td><td>FDE</td></tr><tr><td>Constant Velocity</td><td>3.86</td><td>2.43</td><td>5.10</td><td>7.91</td><td>1.39</td><td>2.86</td><td></td></tr><tr><td>Nearest Neighbor</td><td>3.49</td><td>2.02</td><td>4.98</td><td>7.84</td><td>1.38</td><td>2.79</td><td></td></tr><tr><td>LSTM</td><td>2.13</td><td>1.16</td><td>2.81</td><td>4.83</td><td>1.11</td><td>2.03</td><td>50.6K</td></tr><tr><td>CtsConv</td><td>1.85</td><td>0.99</td><td>2.42</td><td>4.32</td><td>0.86</td><td>1.79</td><td>1078.1K</td></tr><tr><td>CtsConv (Aug.)</td><td>1.77</td><td>0.96</td><td>2.31</td><td>4.05</td><td>1</td><td>-</td><td>1078.1K</td></tr><tr><td>P1-ECCO</td><td>1.70</td><td>0.93</td><td>2.22</td><td>3.89</td><td>0.88</td><td>1.83</td><td>51.4K</td></tr><tr><td>Preg-ECCO</td><td>1.62</td><td>0.89</td><td>2.12</td><td>3.68</td><td>0.84</td><td>1.76</td><td>129.8K</td></tr><tr><td>VectorNet</td><td>1.66</td><td>0.92</td><td>2.06</td><td>3.67</td><td>1</td><td>-</td><td>72K + Decoder</td></tr></table>
+
+Parameter Efficiency Another important feature in deploying deep learning models to embedded systems such as autonomous vehicles is parameter efficiency. We report the number of parameters in each of the models in Table 2. Compare with LSTM, our forecasting performance is significantly better. CtsConv and VectorNet have competitive forecasting performance, but uses much more parameters than ECCO. By encoding equivariance into CtsConv, we drastically reduce the number of the parameters needed in our model. For VectorNet, Gao et al. (2020) only provided the number of parameters for their encoder; a fair decoder size can be estimated based on MLP using 59 polygraphs with each 64 dimensions as input, predicting 30 timestamps, that is 113K.
+
+Runtime and Memory Efficiency We compare the runtime and memory usage with VectorNet Gao et al. (2020). Since VectorNet is not open-sourced, we compare with a version of VectorNet that we implement. Firstly, we compare floating point operations (FLOPs). VectorNet reported $n \times 0 . 0 4 1$ GFLOPs for the encoder part of their model alone, where $n$ is the number of predicted vehicles. We tested ECCO on a scene with 30 vehicles and approximately 180 lane marker nodes, which is similar to the test conditions used to compute FLOPs in Gao et al. (2020). Our full model used 1.03 GFLOPs versus 1.23 GFLOPs for VectorNet’s encoder. For runtimes on the same test machine, ECCO runs $6 8 4 \mathrm { m s }$ versus 1103ms for VectorNet. Another disadvantage of VectorNet is needing to reprocess the scene for each agent, whereas ECCO predicts all agents simultaneously. For memory usage in the same test ECCO uses 296 MB and VectorNet uses 171 MB.
+
+![](images/7ecfe61fa33825b17f485cf100b52c4fc427db6663009e0e9c1f0b152849d034.jpg)  
+Figure 6: The x,y-axes are the position (m). The dashed line represents the 2s past trajectory. The solid line represents the 3s prediction. Red represents the agent. Top row: The predictions are made on the original data. Bottom row: We rotate the whole scene by $1 6 0 ^ { \circ }$ and make predictions on rotated data. From left to right are visualizations of ground truth, CtsConv, $\rho _ { 1 }$ -ECCO, $\rho _ { \mathrm { r e g } }$ -ECCO.
+
+Sample Efficiency A major benefit of incorporating the inductive bias of equivariance is to improve the sample efficiency of learning. For each sample which an equivariant model is trained on, it learns as if it were trained on all transformations of that sample by the symmetry group (Wang et al., 2020, Prop 3). Thus ECCO requires far fewer samples to learn from. In Figure 5, we plot a comparison of validation FDE over number of training samples and show the equivariant models converge faster.
+
+Physical Consistency We also visualize the predictions from ECCO and non-equivariant
+
+![](images/12060c7909bfb15856758dec74d247630d7d472ba9225a4b4bb53877a9aa4479.jpg)  
+Figure 5: The learning curves on the validation set. Equivariant models converge faster using fewer samples than the non-equivariant models.
+
+CtsConv, as shown in Figure 6. Top row visualizes the predictions on the original data. In the bottom row, we rotate the whole scene by $1 6 0 ^ { \circ }$ and make predictions on rotated data. This mimics the covariate shift in the real world. Note that CtsConv predicts inconsistently: a right turn in the top row but a left turn after the scene has been rotated. We see similar results for TrajNet+ $^ +$ (see Figure 8 in Appendix A.10).
+
+# 6 CONCLUSION
+
+We propose Equivariant Continuous Convolution (ECCO), a novel model for trajectory prediction by imposing symmetries as inductive biases. On two real-world vehicle and pedestrians trajectory datasets, ECCO attains competitive accuracy with significantly fewer parameters. It is also more sample efficient; generalizing automatically from few data points in any orientation. Lastly, equivariance gives ECCO improved generalization performance. Our method provides a fresh perspective towards increasing trust in deep learning models through guaranteed properties. Future directions include applying equivariance to probabilistic predictions with many possible trajectories, or developing a faster version of ECCO which does not require autoregressive computation. Moreover, our methods may be generalized from 2-dimensional space to $\mathbb { R } ^ { n }$ . The orbit-stabilizer weight sharing scheme and discretized regular representation may be generalized by replacing SO(2) with $\operatorname { S O } ( n )$ , and polar coordinate kernels may be generalized using spherical coordinates.
+
+# ACKNOWLEDGEMENT
+
+This work was supported in part by Google Faculty Research Award, NSF Grant #2037745, and the U. S. Army Research Office under Grant W911NF-20-1-0334. Walters is supported by a Postdoctoral Fellowship from the Institute for Experiential AI at the Roux Institute.
+
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+
+# A APPENDIX
+
+# A.1 CONTINUOUS CONVOLUTION INVOLVING $\rho _ { \mathrm { r e g } }$
+
+This section is a more detailed version of Section 4.4.
+
+Define the input $\mathbf { f }$ to be $\rho _ { \mathrm { r e g } }$ -field, that is, a distribution over $\mathbb { R } ^ { 2 }$ valued in $\rho _ { \mathrm { r e g } }$ . Define $K \colon  { \mathbb { R } } ^ { 2 } \to$ $\rho _ { \mathrm { r e g } } \otimes \rho _ { \mathrm { r e g } }$ . After identifying SO(2) with its underlying manifold $S ^ { 1 }$ , we can identify $K ( \mathbf { x } )$ as a map $S ^ { 1 } \times S ^ { 1 } \to \mathbb { R }$ and $f ^ { ( \mathbf { x } ) } \colon S ^ { 1 } \to \mathbb { R }$ . Define the integral transform
+
+$$
+K ( \mathbf { x } ) \odot f ^ { ( \mathbf { x } ) } ( \phi _ { 2 } ) = \int _ { \phi _ { 1 } \in S ^ { 1 } } K ( \mathbf { x } ) ( \phi _ { 2 } , \phi _ { 1 } ) f ^ { ( \mathbf { x } ) } ( \phi _ { 1 } ) d \phi _ { 1 } .
+$$
+
+For $\mathbf { y } \in \mathbb { R } ^ { 2 }$ , define the convolution $\mathbf { g } = K \star \mathbf { f }$ by
+
+$$
+\mathbf { g } ( y ) = \int _ { x \in \mathbb { R } ^ { 2 } } K ( x ) \odot \mathbf { f } ( x + y ) d x .
+$$
+
+The $\circledcirc$ -operation parameterizes linear maps $\rho _ { \mathrm { r e g } }  \rho _ { \mathrm { r e g } }$ and is thus analogous to matrix multiplication. If we chose to restrict our choice of $\kappa$ to $\kappa ( \phi _ { 2 } , \phi _ { 1 } ) = \widetilde { \kappa } ( \phi _ { 2 } - \phi _ { 1 } )$ for some function $\tilde { \kappa } \colon S ^ { 1 } \to  { \mathbb { R } }$ then this becomes the circular convolution operation.
+
+The $S O ( 2 )$ -action on $\rho _ { \mathrm { r e g } }$ by $\operatorname { R o t } _ { \theta } ( f ) ( \phi ) = f ( \phi - \theta )$ induces an action on $\kappa \colon S ^ { 1 } \times S ^ { 1 } \to \mathbb { R }$ by
+
+$$
+\mathrm { R o t } _ { \theta } ( \kappa ) ( \phi _ { 2 } , \phi _ { 1 } ) = \kappa ( \phi _ { 2 } - \theta , \phi _ { 1 } - \theta ) .
+$$
+
+This, in turn, gives an action on the torus-field $K$ by
+
+$$
+\operatorname { R o t } _ { \theta } ( K ) ( x ) ( \phi _ { 2 } , \phi _ { 1 } ) = K ( \operatorname { R o t } _ { - \theta } ( x ) ) ( \phi _ { 2 } - \theta , \phi _ { 1 } - \theta ) .
+$$
+
+Thus Equation 3, the convolutional kernel constraint, implies that $K$ is equivariant if and only if
+
+$$
+K ( \mathrm { R o t } _ { \theta } ( x ) ) ( \phi _ { 2 } , \phi _ { 1 } ) = K ( x ) ( \phi _ { 2 } - \theta , \phi _ { 1 } - \theta ) .
+$$
+
+We use this to define a weight sharing scheme as described in Section 3.2. The cases of continuous convolution $\rho _ { 1 }  \rho _ { \mathrm { r e g } }$ and $\rho _ { \mathrm { r e g } }  \rho _ { 1 }$ may be derived similarly.
+
+# A.2 COMPLEXITY OF CONVOLUTION WITH TORUS KERNEL
+
+The complexity class of the convolution with torus kernel is $O ( n \cdot k _ { r e g } ^ { 2 } \cdot c _ { o u t } \cdot c _ { i n } )$ , where $n$ is the number of particles, the regular representation is discretized into $k _ { r e g }$ pieces, and the input and output contain $c _ { i n }$ and $c _ { o u t }$ copies of the regular representation respectively. We are not counting the complexity of the interpolation operation for looking up $K ( \theta , r )$ .
+
+# A.3 EQUIVARIANT PER-PARTICLE LINEAR LAYERS
+
+Since this operation is pointwise, unlike positive radius continuous convolution, we cannot map between different irreducible representations of SO(2). Consider as input a $\rho _ { \mathrm { i n } }$ -field $I$ and output a $\rho _ { \mathrm { o u t } }$ -field $O$ where $\rho _ { \mathrm { i n } }$ and $\rho _ { \mathrm { o u t } }$ are finite-dimensional representations of SO(2). We define $O ^ { ( i ) } =$ $W I ^ { ( i ) }$ using the same $W$ , an equivariant linear map, for each particle $1 \leq i \leq N$ . Denote the decomposition of $\rho _ { \mathrm { i n } }$ and $\rho _ { \mathrm { o u t } }$ into irreducible representations of $S O ( 2 )$ as $\rho _ { \mathrm { i n } } \cong \rho _ { 1 } ^ { i _ { 1 } } \oplus . . . \oplus \rho _ { n } ^ { i _ { n } }$ and $\rho _ { \mathrm { o u t } } \cong \rho _ { 1 } ^ { j _ { 1 } } \oplus . . . \oplus \rho _ { n } ^ { j _ { n } }$ respectively. By Schur’s lemma, the equivariant linear map $W \colon \rho _ { \mathrm { i n } }  \rho _ { \mathrm { o u t } }$ is defined by a block diagonal matrix with blocks $\{ W _ { k } \} _ { k = 1 } ^ { n }$ where $W _ { k }$ is an $i _ { k } \times j _ { k }$ matrix. That is, maps between different irreducible representations are zero and each map $\rho _ { k } \to \rho _ { k }$ is given by a single scalar.
+
+Per-particle linear mapping $\rho _ { 1 }  \rho _ { \mathrm { r e g } }$ and $\rho _ { 1 }  \rho _ { \mathrm { r e g } }$ . Since the input and output features are $\rho _ { 1 }$ -fields, but the hidden features may be represented by $\rho _ { \mathrm { r e g } }$ , we need mappings between $\rho _ { 1 }$ and $\rho _ { \mathrm { r e g } }$ . In all cases we pair continuous convolutions with dense per-particle mappings, this we must describe per-particle mappings between $\rho _ { 1 }$ and $\rho _ { \mathrm { r e g } }$ .
+
+By the Peter-Weyl theorem, $L ^ { 2 } ( \mathrm { S O ( 2 ) } ) \cong \bigoplus _ { i = 0 } ^ { \infty } \rho _ { i }$ . In the case of $\mathrm { S O } ( 2 )$ , this decomposition is also called the Fourier decomposition or decomposition into circular harmonics. Most importantly, there is one copy of $\rho _ { 1 }$ inside of $L ^ { 2 } ( \mathrm { S O } ( 2 ) )$ . Hence, up to scalar, there is a unique linear map $i _ { 1 } \colon \rho _ { 1 }  L ^ { 2 } ( \mathrm { S } \overleftarrow { \mathrm { O } } ( 2 ) )$ given by $( a , b ) \mapsto a \cos ( \theta ) + b \sin ( \theta )$ .
+
+The reverse mapping $\mathrm { p r } _ { 1 } \colon L ^ { 2 } ( \mathrm { S O } ( 2 ) )  \rho _ { 1 }$ is projection onto the $\rho _ { 1 }$ summand and is given by the Fourier transform $\begin{array} { r } { \mathrm { p r } _ { \mathrm { i } } ( f ) = ( \int _ { S ^ { 1 } } f ( \theta ) \cos ( \theta ) d \theta , \int _ { S ^ { 1 } } f ( \theta ) \sin ( \theta ) d \theta ) } \end{array}$ .
+
+Per-particle linear mapping $\rho _ { \mathrm { r e g } }  \rho _ { \mathrm { r e g } }$ . Though $\rho _ { \mathrm { r e g } }$ is not finite-dimensional, the fact that it decomposes into a direct sum of irreducible representations means that we may take $\rho _ { \mathrm { i n } } = \rho _ { \mathrm { o u t } } =$ $\rho _ { \mathrm { r e g } }$ above. Practically, however, it is easier to realize the linear equivariant map $\rho _ { \mathrm { r e g } } ^ { i }  \rho _ { \mathrm { r e g } } ^ { j }$ as a convolution over $S ^ { 1 }$ ,
+
+$$
+O ( \theta ) = \int _ { \phi \in S ^ { 1 } } \kappa ( \theta - \phi ) I ( \phi )
+$$
+
+where $\kappa ( \theta )$ is an $i \times j$ matrix of trainable weights, independent for each $\theta$
+
+# A.4 ENCODING INDIVIDUAL PARTICLE PAST BEHAVIOR
+
+We can encode these individual attributes using a per vehicle LSTM (Hochreiter & Schmidhuber, 1997). Let $X _ { t } ^ { ( i ) }$ denote the position of car $i$ at time $t$ . Denote a fully connected LSTM cell by states $h _ { t } , c _ { t } = \mathtt { L S T M } ( X _ { t } ^ { ( i ) } , h _ { t - 1 } , c _ { t - 1 } )$ $[ h _ { t _ { \mathrm { i n } } } ^ { ( 1 ) } \ . . . \ h _ { t _ { \mathrm { i n } } } ^ { ( n ) } ]$ of all particles as . Define $Z \in \mathbb { R } ^ { N } \otimes \mathbb { R } ^ { k }$ $h _ { 0 } = c _ { 0 } = 0$ . We then use the concatenation of the hidden as the encoded per-vehicle latent features.
+
+# A.5 ENCODING PAST INTERACTIONS
+
+In addition, we also encode past interactions of particles by introducing a continuous convolution LSTM. Similar to convLSTM we replace the fully connected layers of the original LSTM above with another operation Xingjian et al. (2015). While convLSTM is well-suited for capturing spatially local interactions over time, it requires gridded information. Since the particle system we consider are distributed in continuous space, we replace the standard convolution with rotation-equivariant continuous convolutions.
+
+We can now define $H _ { t } , C _ { t } = \mathtt { C t s C o n v L S T M } ( X _ { t } , H _ { t - 1 } , C _ { t - 1 } )$ which is an LSTM cell using equivariant continuous convolutions throughout. Note that in this case $X _ { t } , H _ { t - 1 } , C _ { t - 1 }$ are all particle feature fields, that is, functions $\{ 1 , \ldots , n \} \to \mathbb { R } ^ { k }$ .
+
+Define CtsConvLSTM by
+
+$$
+\begin{array} { r l } & { i _ { t } = \sigma ( W _ { i x } \star _ { c t s } X _ { t } ^ { ( i ) } + W _ { i h } \star _ { c t s } h _ { t - 1 } + W _ { i c } \circ c _ { t - 1 } + b _ { i } ) } \\ & { f _ { t } = \sigma ( W _ { f x } \star _ { c t s } X _ { t } ^ { ( i ) } + W _ { f h } \star _ { c t s } h _ { t - 1 } + W _ { f c } \circ c _ { t - 1 } + b _ { i } ) } \\ & { c _ { t } = f _ { t } \circ c _ { t - 1 } + i _ { t } \circ \operatorname { t a n h } ( W _ { c x } \star _ { c t s } X _ { t } ^ { ( i ) } + W _ { c h } \star _ { c t s } h _ { t - 1 } + b _ { c } ) } \\ & { o _ { t } = \sigma ( W _ { o x } \star _ { c t s } X _ { t } ^ { ( i ) } + W _ { o h } \star _ { c t s } h _ { t - 1 } + W _ { o c } \circ c _ { t } + b _ { o } ) } \\ & { h _ { t } = o _ { t } \circ \operatorname { t a n h } ( c _ { t } ) , } \end{array}
+$$
+
+where $\star _ { c t s }$ denotes CtsConv. We then can use $H _ { t _ { \mathrm { i n } } }$ as input feature for the prediction network.
+
+# A.6 EQUIVARIANCE ERROR
+
+We prove the proposition in Section 4.5.
+
+Proposition. Let $\alpha = 2 \pi / k _ { \theta }$ . Let $\bar { \theta }$ be $\theta$ rounded to nearest value in $\mathbb { Z } \alpha$ . Set $\hat { \theta } = | \theta - \bar { \theta } |$ . Assume $n$ particles samples uniformly in a ball of radius $R$ with features f $\in \rho _ { 1 } ^ { c }$ . Let f and $K$ have entries sampled uniformly in $[ - a , a ]$ . Let the bullseye have radius $0 < R _ { e } < R$ . Let $F = C t s C o n v _ { K , R }$ and $T _ { \theta } = \rho _ { 1 } ( \operatorname { R o t } _ { \theta } )$ . Then the expected EE is bounded
+
+$$
+\mathbb { E } _ { K , \mathbf { f } , \mathbf { x } } [ T ( F ( \mathbf { f } , \mathbf { x } ) ) - F ( T ( \mathbf { f } ) , T ( \mathbf { x } ) ) ] \le | \sin ( \hat { \theta } ) | C \le 2 \pi C / k _ { \theta }
+$$
+
+where $C = 4 c n a ^ { 2 } ( 1 - R _ { e } ^ { 2 } / R ^ { 2 } )$ .
+
+Proof. We may compute for a single particle $\mathbf { x } = ( \psi , r )$ and multiply our result by $n$ by linearity. We separate two cases: $\mathbf { x }$ in bullseye with probability $R _ { e } ^ { 2 } / R ^ { 2 }$ and $\mathbf { x }$ in angular slice with probability $1 - R _ { e } ^ { \dot { 2 } } / R ^ { 2 }$ . If $\mathbf { x }$ is in the bullseye, then there is no equivariance error since $K ( \mathbf { x } )$ is a scalar matrix. Assume $\mathbf { x }$ is an angular sector.
+
+For nearest interpolation, the equivariance error is then
+
+$$
+\| \rho _ { 1 } ( { \bar { \theta } } ) K ( \mathbf { x } ) \rho _ { 1 } ( - { \bar { \theta } } ) \rho _ { 1 } ( \theta ) \mathbf { f } - \rho _ { 1 } ( \theta ) K ( \mathbf { x } ) \mathbf { f } \| .
+$$
+
+Since $\rho _ { 1 } ( \theta )$ is length preserving, this is
+
+$$
+\begin{array} { r l } & { \quad \| \rho _ { 1 } ( - \theta ) \rho _ { 1 } ( \bar { \theta } ) K ( \mathbf { x } ) \rho _ { 1 } ( - \bar { \theta } ) \rho _ { 1 } ( \theta ) \mathbf { f } - K ( \mathbf { x } ) \mathbf { f } \| } \\ & { = \| \rho _ { 1 } ( \beta ) K ( \mathbf { x } ) \rho _ { 1 } ( - \beta ) \mathbf { f } - K ( \mathbf { x } ) \mathbf { f } \| } \end{array}
+$$
+
+where $\beta = \pm \hat { \theta }$ . We consider only a single factor of $\rho _ { 1 }$ in f . The result will then be multiplied by $c$ Let
+
+$$
+K ( \mathbf { x } ) = \left( \begin{array} { l l } { k _ { 1 1 } } & { k _ { 1 2 } } \\ { k _ { 2 1 } } & { k _ { 2 2 } } \end{array} \right) , \qquad \mathbf { f } = \left( \begin{array} { l } { f _ { 1 } } \\ { f _ { 2 } } \end{array} \right) .
+$$
+
+We can factor out an $a$ from $K ( \mathbf { x } )$ and an $a$ from f and assume $k _ { i j } , f _ { i }$ samples from Uniform $( [ - 1 , 1 ] )$ . One may then directly compute that Equation 7 equals
+
+$$
+\sqrt { ( ( k _ { 2 1 } + k _ { 1 2 } ) ^ { 2 } + ( k _ { 1 1 } - k _ { 2 2 } ) ^ { 2 } ) ( f _ { 1 } ^ { 2 } + f _ { 2 } ^ { 2 } ) \sin ^ { 2 } ( \beta ) }
+$$
+
+This is bounded above by $4 | \sin ( \beta ) | = 4 | \sin ( { \hat { \theta } } ) |$ . Collecting the above factors, this proves the bound $C | \sin ( \beta ) |$ .
+
+The further bound follows by the first order bound,
+
+$$
+| \sin ( \hat { \theta } ) | \leq | \hat { \theta } | \leq 2 \pi / k _ { \theta } .
+$$
+
+The relationship $\mathrm { E E } \approx 2 \pi C / k _ { \theta }$ is visible in Figure 4. We can also see clearly the significance of the term $| \sin ( { \hat { \theta } } ) |$ by plotting equivariance error against $\theta$ as in Figure 7.
+
+![](images/c9ed80ce51059dbfb41589d0d1c008328bb255955142f18469dbe88023e02d05.jpg)  
+Figure 7: The above plot is generated from random input and kernels. We can clearly see the dependence of of EE on $| \sin ( { \hat { \theta } } ) |$
+
+# A.7 DATA DETAILS
+
+Argoverse dataset includes 324K samples, which are split into 206K training data, 39K validation and 78K test set. All the samples are real data extracted from Miami and Seattle, and the dataset provides HD maps of lanes in each city. Every sample contains data for 5 seconds long, and is sampled in $1 0 \mathrm { H z }$ frequency.
+
+TrajNet+ $^ +$ Real dataset contains 200K samples. All the tracking in this dataset is captured in both indoor and outdoor locations, for example, university, hotel, Zara, and train stations. Every sample in this dataset contains 21 timestamps, and the goal is to predict the 2D spatial positions for each pedestrain in the future 12 timestamps.
+
+# A.8 IMPLEMENTATION DETAILS
+
+Argoverse dataset is not fully observed, so we only use cars with complete observation as our input. Since every sample doesn’t include the same number of cars, we only choose those scenes with less than or equal to 60 cars and insert dummy cars into them to achieve consistent car numbers. Tra${ \mathrm { j N e t } } + +$ Real dataset is also not fully observed. And here we keep our pedestrain number consistent to 160.
+
+Moreover, for each car, we use the average velocity in the past 0.1 second as an approximate to the current instant velocity, i.e. $v _ { t } = ( p _ { t } - p _ { t - 1 } ) / 2$ . As for map information, we only include center lanes with lane directions as features. Also, we introduce dummy lane node into each scene to make lane numbers consistently equal to 650.
+
+In TrajNet $^ { + + }$ task, no map information is included. And since pedestrians don’t have a speedometers to tell them exactly how fast they are moving as drivers, instead they depends more on the relative velocities and relative positions to other pedestrians, we tried different combination of features in ablative study besides only using history velocities.
+
+Our models are all trained by Adam optimizer with base learning rate 0.001, and the gamma rate for linear rate scheduler is set to be 0.95. All our models without map information are trained for 15K iterations with batch size 16 and learning rate is updated every 300 iterations; for models with map information, we train them for 30K iterations with batch size 16 and learning rate is updated every 600 iterations.
+
+For CtsConv, we set the layer sizes to be 32, 64, 64, 64, and kernel size $4 \times 4 \times 4$ ; for $\rho _ { 1 }$ -ECCO, the layer sizes are 16, 32, 32, 32, $k _ { \theta }$ is 16, $k _ { r }$ is 3; for $\rho _ { \mathrm { r e g } }$ -ECCO, we choose layer size 8, 16, 8, 8, $k _ { \theta }$ 16, $k _ { r }$ 3, and regular feature dimension is set to be 8. For Argoverse task, we set the CtsConv radius to be 40, and for TrajNet++ task we set it to be 6.
+
+# A.9 ABLATIVE STUDY
+
+We perform ablative study for ECCO to further diagnose different encoders, usage of HD maps and other model design choices.
+
+Choice of encoders Unlike fluid simulations (Ummenhofer et al., 2019) where the dynamics are Markovian, human behavior exhibit long-term dependency. We experiment with three different encoders refered to as Enc to model such long-term dependency: (1) concatenating the velocities from the past $m$ frames as input feature, (2) passing the past velocities of each particle to the same LSTM to encode individual behavior of each particle, and (3) implementing continuous convolution LSTM to encode past particle interactions. Our continuous convolution LSTM is similar to convLSTM (Xingjian et al., 2015) but uses continuous convolutions instead of discrete gridded convolutions.
+
+We use different encoders to time-aggregate features and compare their performances (Table 3).
+
+Use of HD Maps In Table 4, we compare performance with and without map input features.
+
+Choice of features for pedestrian Unlike vehicles, people do not have a velocity meter to tell him how fast they actually walk. We realize that people actually tend to adjust their velocities based on others’ relative velocity and relative position. We experiment different combination of features (Table 5), finding using relative velocities and relative positions as feature has the best performance.
+
+# A.10 QUALITATIVE RESULTS FOR TRAJNET++
+
+Figure 8 show qualitative results for TrajNet++. Note that the non-equivariant baseline (2nd column) depends highly on the global orientation whereas the ground truth and equivariant models do not.
+
+Table 3: Ablation study on encoders for Argoverse and TrajNet++. Markovian: Use the velocity from the most recent time step as input feature. LSTM: Used LSTM to encode velocities of 20 timestamps. CtsConvLSTM: Instead of dense layer, the gate functions in LSTM are replaced by CtsConv. CtsConvDLSTM: Replaced gate functions by CtsConv $^ +$ Dense. D-Concat (20t feats): Stacked velocities of 20 time steps as input.   
+
+<table><tr><td rowspan="2">Encoder</td><td colspan="4">Argoverse</td><td colspan="2">TrajNet++</td></tr><tr><td>ADE</td><td>DE@1s</td><td>DE@2s</td><td>DE@3s</td><td>ADE</td><td>FDE</td></tr><tr><td>Markovian</td><td>4.67</td><td>=</td><td>=</td><td>9.84</td><td>0.969</td><td>1.952</td></tr><tr><td>LSTM</td><td>2.05</td><td>1.06</td><td>2.51</td><td>4.71</td><td>0.909</td><td>1.909</td></tr><tr><td>CtsConvLSTM</td><td>3.98</td><td>2.02</td><td>5.11</td><td>8.40</td><td>0.962</td><td>1.941</td></tr><tr><td>CtsConvDLSTM</td><td>2.02</td><td>1.03</td><td>2.46</td><td>4.58</td><td>0.910</td><td>1.916</td></tr><tr><td>D-Concat(20t feats)</td><td>1.87</td><td>1.01</td><td>2.43</td><td>4.22</td><td>0.895</td><td>1.872</td></tr></table>
+
+Table 4: Ablative study on HD maps for Argoverse. Prediction accuracy comparison with and without HD Maps.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">w/o Map</td><td colspan="4">w/Map</td></tr><tr><td>ADE</td><td>DE@1s</td><td>DE@2s</td><td>DE@3s</td><td>ADE</td><td>DE@1s</td><td>DE@2s</td><td>DE@3s</td></tr><tr><td>CtsConv</td><td>1.87</td><td>1.01</td><td>2.43</td><td>4.22</td><td>1.85</td><td>0.99</td><td>2.42</td><td>4.32</td></tr><tr><td>P1-ECCO</td><td>1.81</td><td>1.02</td><td>2.42</td><td>4.14</td><td>1.70</td><td>0.93</td><td>2.22</td><td>3.89</td></tr><tr><td>Preg-ECCO</td><td>1.81</td><td>1.00</td><td>2.38</td><td>4.12</td><td>1.62</td><td>0.89</td><td>2.12</td><td>3.68</td></tr></table>
+
+Table 5: Ablative study on features for $\mathrm { T r a j } { + + }$ . Acceleration means whether we used acceleration to make numerically extrapolated position.   
+
+<table><tr><td>Velocity</td><td>Relative Position丨Acceleration</td><td></td><td>ADE</td><td>FDE</td></tr><tr><td>Absolute</td><td>×</td><td>×</td><td>0.92</td><td>1.95</td></tr><tr><td>Absolute</td><td>×</td><td></td><td>0.90</td><td>1.87</td></tr><tr><td>Relative</td><td>×</td><td>√</td><td>0.89</td><td>1.86</td></tr><tr><td>Relative</td><td>√</td><td>√</td><td>0.86</td><td>1.79</td></tr></table>
+
+![](images/3ba988374a3912363034e390a42b4c46f4efbf1d50c546c2279fc3c140815316.jpg)  
+Figure 8: The x,y-axes are the position $\mathrm { ( m ) }$ . The dashed line represents the 2s past trajectory. The solid line represents the 3s prediction. Red represents the agent. Top row: The predictions are made on the original data. Bottom row: We rotate the whole scene by $1 6 0 ^ { \circ }$ and make predictions on rotated data. From left to right are visualizations of ground truth, CtsConv, $\rho _ { 1 }$ -ECCO, $\rho _ { \mathrm { r e g } }$ -ECCO.

parse/train/TtYSU29zgR/TtYSU29zgR.md

@@ -0,0 +1,428 @@
+# PRIMAL WASSERSTEIN IMITATION LEARNING
+
+Robert Dadashi∗1, Léonard Hussenot1,2, Matthieu Geist1, Olivier Pietquin1   
+1Google Research, Brain Team   
+2Univ. de Lille, CNRS, Inria Scool, UMR 9189 CRIStAL
+
+# ABSTRACT
+
+Imitation Learning (IL) methods seek to match the behavior of an agent with that of an expert. In the present work, we propose a new IL method based on a conceptually simple algorithm: Primal Wasserstein Imitation Learning (PWIL), which ties to the primal form of the Wasserstein distance between the expert and the agent state-action distributions. We present a reward function which is derived offline, as opposed to recent adversarial $\mathrm { I L }$ algorithms that learn a reward function through interactions with the environment, and which requires little fine-tuning. We show that we can recover expert behavior on a variety of continuous control tasks of the MuJoCo domain in a sample efficient manner in terms of agent interactions and of expert interactions with the environment. Finally, we show that the behavior of the agent we train matches the behavior of the expert with the Wasserstein distance, rather than the commonly used proxy of performance.
+
+# 1 INTRODUCTION
+
+Reinforcement Learning (RL) has solved a number of difficult tasks whether in games (Tesauro, 1995; Mnih et al., 2015; Silver et al., 2016) or robotics (Abbeel & Ng, 2004; Andrychowicz et al., 2020). However, RL relies on the existence of a reward function, that can be either hard to specify or too sparse to be used in practice. Imitation Learning (IL) is a paradigm that applies to these environments with hard to specify rewards: we seek to solve a task by learning a policy from a fixed number of demonstrations generated by an expert.
+
+IL methods can typically be folded into two paradigms: Behavioral Cloning, or BC (Pomerleau, 1991; Bagnell et al., 2007; Ross & Bagnell, 2010) and Inverse Reinforcement Learning, or IRL (Russell, 1998; Ng et al., 2000). In BC, we seek to recover the expert’s behavior by directly learning a policy that matches the expert behavior in some sense. In IRL, we assume that the demonstrations come from an agent that acts optimally with respect to an unknown reward function that we seek to recover, to subsequently train an agent on it. Although IRL methods introduce an intermediary problem (i.e. recovering the environment’s reward) they are less sensitive to distributional shift (Pomerleau, 1991), they generalize to environments with different dynamics (Piot et al., 2013), and they can recover a near-optimal agent from suboptimal demonstrations (Brown et al., 2019; Jacq et al., 2019).
+
+However, IRL methods are usually based on an iterative process alternating between reward estimation and RL, which might result in poor sample-efficiency. Earlier IRL methods $\mathrm { N g }$ et al., 2000; Abbeel & Ng, 2004; Ziebart et al., 2008) require multiple calls to a Markov decision process solver (Puterman, 2014), whereas recent adversarial IL approaches (Finn et al., 2016; Ho & Ermon, 2016; Fu et al., 2018) interleave the learning of the reward function with the learning process of the agent. Adversarial IL methods are based on an adversarial training paradigm similar to Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), where the learned reward function can be thought of as the confusion of a discriminator that learns to differentiate expert transitions from non expert ones. These methods are well suited to the IL problem since they implicitly minimize an $f$ -divergence between the state-action distribution of an expert and the state-action distribution of the learning agent (Ghasemipour et al., 2019; Ke et al., 2019). However the interaction between a generator (the policy) and the discriminator (the reward function) makes it a minmax optimization problem, and therefore comes with practical challenges that might include training instability, sensitivity to hyperparameters and poor sample efficiency.
+
+In this work, we use the Wasserstein distance as a measure between the state-action distributions of the expert and of the agent. Contrary to $f$ -divergences, the Wasserstein distance is a true distance, it is smooth and it is based on the geometry of the metric space it operates on. The Wasserstein distance has gained popularity in GAN approaches (Arjovsky et al., 2017) through its dual formulation which comes with challenges (see Section 5). Our approach is novel in the fact that we consider the problem of minimizing the Wasserstein distance through its primal formulation. Crucially, the primal formulation prevents the minmax optimization problem, and requires little fine tuning.
+
+We introduce a reward function computed offline based on an upper bound of the primal form of the Wasserstein distance. As the Wasserstein distance requires a distance between state-action pairs, we show that it can be hand-defined for locomotion tasks, and that it can be learned from pixels for a hand manipulation task. The inferred reward function is non-stationary, like adversarial IL methods, but it is not re-evaluated as the agent interacts with the environment, therefore the reward function we define is computed offline. We present a true distance to compare the behavior of the expert and the behavior of the agent, rather than using the common proxy of performance with respect to the true return of the task we consider (as it is unknown in general). Our method recovers expert behaviour comparably to existing state-of-the-art methods while being based on significantly fewer hyperparameters; it operates even in the extreme low data regime of demonstrations, and is the first method that makes Humanoid run with a single (subsampled) demonstration.
+
+# 2 BACKGROUND AND NOTATIONS
+
+Markov decision processes. We describe environments as episodic Markov Decision Processes (MDP) with finite time horizon (Sutton & Barto, 2018) $( \mathcal { S } , \mathcal { A } , \mathcal { P } , r , \gamma , \rho _ { 0 } , T )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $\mathcal { P }$ is the transition kernel, $r$ is the reward function, $\gamma$ is the discount factor, $\rho _ { 0 }$ is the initial state distribution and $T$ is the time horizon. We will denote the dimensionality of $s$ and $\mathcal { A }$ as $| S |$ and $| { \cal A } |$ respectively. A policy $\pi$ is a mapping from states to distributions over actions; we denote the space of all policies by $\Pi$ . In RL, the goal is to learn a policy $\pi ^ { * }$ that maximizes the expected sum of discounted rewards it encounters, that is, the expected return. Depending on the context, we might use the concept of a cost $c$ rather than a reward $r$ (Puterman, 2014), which essentially moves the goal of the policy from maximizing its return to minimizing its cumlative cost.
+
+State action distributions. Suppose a policy $\pi$ visits the successive states and actions $s _ { 1 } , a _ { 1 } , \ldots , s _ { T } , a _ { T }$ during an episode, we define the empirical state-action distribution $\hat { \rho } _ { \pi }$ as: $\begin{array} { r } { \hat { \rho } _ { \pi } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \delta _ { s _ { t } , a _ { t } } } \end{array}$ , where $\delta _ { s _ { t } , a _ { t } }$ is a Dirac distribution centered on $( s _ { t } , a _ { t } )$ . Similarly, suppose we have a set of expert demonstrations $\mathcal { D } = \{ s ^ { e } , a ^ { e } \}$ of size $D$ , then the associated empirical expert state-action distribution $\hat { \rho } _ { e }$ is defined as: $\begin{array} { r } { \hat { \rho } _ { e } = \frac { 1 } { D } \sum _ { ( s , a ) \in \mathcal { D } } \delta _ { s , a } } \end{array}$ .
+
+Wasserstein distance. Suppose we have the metric space $( M , d )$ where $M$ is a set and $d$ is a metric on $M$ . Suppose we have $\mu$ and $\nu$ two distributions on $M$ with finite moments, the $p$ -th order Wasserstein distance (Villani, 2008) is defined as $\begin{array} { r } { \mathcal { W } _ { p } ^ { p } ( \mu , \nu ) = \operatorname* { i n f } _ { \theta \in \Theta ( \mu , \nu ) } \int _ { M \times M } d ( x , y ) ^ { p } d \theta ( x , y ) , } \end{array}$ where $\Theta ( \mu , \nu )$ is the set of all couplings between $\mu$ and $\nu$ . In the remainder, we only consider distributions with finite support. A coupling between two distributions of support cardinal $T$ and $D$ is a doubly stochastic matrix of size $T \times D$ . We note $\Theta$ the set of all doubly stochastic matrices of size $T \times D$ :
+
+$$
+\Theta = \Big \{ \theta \in \mathbb { R } _ { + } ^ { T \times D } \ | \ \forall j \in [ 1 : D ] , \sum _ { i ^ { \prime } = 1 } ^ { T } \theta [ i ^ { \prime } , j ] = \frac { 1 } { D } , \forall i \in [ 1 : T ] , \sum _ { j ^ { \prime } = 1 } ^ { D } \theta [ i , j ^ { \prime } ] = \frac { 1 } { T } \Big \} .
+$$
+
+The Wasserstein distance between distributions of state-action pairs requires the definition of a metric $d$ in the space $( \cal { S } , \cal { A } )$ . Defining a metric in an MDP is non trivial (Ferns et al., 2004; Mahadevan & Maggioni, 2007); we show an example where the metric is learned from demonstrations in Section 4.4. For now, we assume the existence of a metric $d : ( S , \mathcal { A } ) \times ( S , \mathcal { A } ) \mapsto \mathbb { R } ^ { + }$ .
+
+# 3 METHOD
+
+We present the theoretical motivation of our approach: the minimization of the Wasserstein distance between the state-action distributions of the agent and the expert. We introduce a reward based on an upper-bound of the primal form of the Wasserstein distance inferred from a relaxation of
+
+the optimal coupling condition, and present the resulting algorithm: Primal Wasserstein Imitation Learning (PWIL).
+
+# 3.1 WASSERSTEIN DISTANCE MINIMIZATION
+
+Central to our approach is the minimization of the Wasserstein distance between the state-action distribution of the policy we seek to train $\hat { \rho } _ { \pi }$ and the state-action distribution of the expert $\hat { \rho } _ { e }$ . In other words, we aim at optimizing the following problem:
+
+$$
+\operatorname* { i n f } _ { \pi \in \Pi } \mathcal { W } _ { p } ^ { p } ( \hat { \rho } _ { \pi } , \hat { \rho } _ { e } ) = \operatorname* { i n f } _ { \pi \in \Pi } \operatorname* { i n f } _ { \theta \in \Theta } \sum _ { i = 1 } ^ { T } \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) ^ { p } \theta [ i , j ] .
+$$
+
+In the rest of the paper, we only consider the 1-Wasserstein $p = 1$ in Equation 1) and leave the extensive study of the influence of the order $p$ for future work. We can interpret the Wasserstein distance using the earth’s movers analogy (Villani, 2008). Consider that the state-action pairs of the expert are $D$ holes of mass $D ^ { - 1 }$ and that the state-action pairs of the policy are piles of dirt of mass $T ^ { \dot { - } 1 }$ . A coupling $\theta$ is a transport strategy between the piles of dirt and the holes, where $\theta [ i , j ]$ stands for how much of the pile of dirt $i$ should be moved towards the hole $j$ . The optimal coupling is the one that minimizes the distance that the earth mover travels to put all piles of dirt to holes. Note that to compute the optimal coupling, we need knowledge of the locations of all piles of dirt. In the context of RL, this means having access to the full trajectory generated by $\pi$ . From now on, we write $\theta _ { \pi } ^ { * }$ as the optimal coupling for the policy $\pi$ , that we inject in Equation (1):
+
+$$
+\begin{array} { c } { { \displaystyle \theta _ { \pi } ^ { * } = \arg \operatorname* { m i n } _ { \theta \in \Theta } \sum _ { i = 1 } ^ { T } \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) \theta [ i , j ] } } \\ { { \displaystyle \operatorname* { i n f } _ { \pi \in \Pi } \mathcal { W } _ { 1 } ( \hat { \rho } _ { \pi } , \hat { \rho } _ { e } ) = \operatorname* { i n f } _ { \pi \in \Pi } \sum _ { i = 1 } ^ { T } c _ { i , \pi } ^ { * } \ w \mathrm { i t h } \ c _ { i , \pi } ^ { * } = \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) \theta _ { \pi } ^ { * } [ i , j ] . } } \end{array}
+$$
+
+In Equation (2), we have introduced $c _ { i , \pi } ^ { * }$ , which we interpret as a cost to minimize using RL. As $c _ { i , \pi } ^ { * }$ depends on the optimal coupling $\theta _ { \pi } ^ { * }$ , we can only define $c _ { i , \pi } ^ { * }$ at the very end of an episode. This can be problematic if an agent learns in an online manner or in large time-horizon tasks. Thus, we introduce an upper bound to the Wasserstein distance that yields a cost we can compute online, based on a suboptimal coupling strategy.
+
+# 3.2 GREEDY COUPLING
+
+In this section we introduce the greedy coupling $\theta _ { \pi } ^ { g } \in \Theta$ , defined recursively for $1 \leq i \leq T$ as:
+
+$$
+\begin{array} { r l } & { \theta _ { \pi } ^ { g } [ i , : ] = \underset { \theta [ i , : ] \in \Theta _ { i } } { \arg \operatorname* { m i n } } \displaystyle \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) \theta [ i , j ] } \\ & { \mathrm { w i t h ~ } \Theta _ { i } = \Big \{ \theta [ i , : ] \in \mathbb { R } _ { + } ^ { D } \Big | \displaystyle \sum _ { j ^ { \prime } = 1 } ^ { D } \theta [ i , j ^ { \prime } ] = \displaystyle \frac { 1 } { T } , \underbrace { \forall k \in [ 1 : D ] , \displaystyle \sum _ { i ^ { \prime } = 1 } ^ { i - 1 } \theta _ { g } [ i ^ { \prime } , k ] + \theta [ i , k ] \le \displaystyle \frac { 1 } { D } } _ { \mathrm { c o n s t r a i n t } ( a ) } \Big \} . } \end{array}
+$$
+
+Similarly to Equation (1), Equation (3) can be interpreted using the earth mover’s analogy. Contrary to Equation (1) where we assume knowledge of the positions of the $T$ piles of dirts, we now consider that they appear sequentially, and that the earth’s mover needs to transport the new pile of dirt to holes right when it appears. To do so, we derive the distances to all holes and move the new pile of dirt to the closest remaining available holes, hence the greedy nature of it. In Equation (3), the constraint $( a )$ means that all the dirt that appears at the $i$ -th timestep needs to be moved, and the constraint (b) means that we cannot fill the holes more than their capacity $\textstyle { \frac { 1 } { D } }$ . We show the difference between the greedy coupling and the optimal coupling in Figure 1, and provide the pseudo-code to derive it in Algorithm 1.
+
+![](images/1b66e066ac72cab4774db99ec25a61cfd0941742537cf0801ec8a1d0292031bb.jpg)  
+Figure 1: Illustration of the difference between the a) greedy coupling and the b) optimal coupling. We present an MDP where we drop the dependency on the action. The state space is $\mathbb { R }$ and the metric associated is the Euclidean distance. We note the states visited by the policy: $s _ { 1 } ^ { \pi } , s _ { 2 } ^ { \pi } , s _ { 3 } ^ { \pi }$ and the states visited by the expert: $s _ { 1 } ^ { e } , s _ { 2 } ^ { e } , s _ { 3 } ^ { e }$ . When the policy encounters the state $s _ { 2 } ^ { \pi }$ , and because we do not know $s _ { 3 } ^ { \pi }$ yet, the greedy coupling strategy consists in coupling it with $s _ { 2 } ^ { e }$ although the optimal coupling strategy would be to couple it with $s _ { 3 } ^ { e }$ . Note that the total cost (omitting the constant coupling multiplication factor) with the greedy coupling is 7 whereas the total cost with the optimal coupling is 5 . This highlights that the optimal coupling needs knowledge about the whole policy’s trajectory to be derived.
+
+We can now define an upper bound of the Wasserstein distance using the greedy coupling (since by definition it is suboptimal):
+
+$$
+\begin{array} { r l r } & { } & { \underset { \pi \in \Pi } { \operatorname* { i n f } } \ \mathcal { W } _ { 1 } ( \hat { \rho } _ { \pi } , \hat { \rho } _ { e } ) = \underset { \pi \in \Pi } { \operatorname* { i n f } } \sum _ { i = 1 } ^ { T } \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) \theta _ { \pi } ^ { * } [ i , j ] } \\ & { } & { \leq \underset { \pi \in \Pi } { \operatorname* { i n f } } \sum _ { i = 1 } ^ { T } \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) \theta _ { \pi } ^ { g } [ i , j ] . } \end{array}
+$$
+
+In Section 4, we empirically validate this bound. We infer a cost from Equation (1) at each timestep $i$
+
+$$
+c _ { i , \pi } ^ { g } = \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) \theta _ { \pi } ^ { g } [ i , j ] .
+$$
+
+Note that the greedy coupling $\theta _ { \pi } ^ { g } [ t , . ]$ defined in Equation (3) depends on all the previous state-actions visited by the policy $\pi$ , which makes the cost $c _ { i , \pi } ^ { g }$ non-stationary, similarly to adversarial $\mathrm { I L }$ methods. We can infer a reward from the cost, $r _ { i , \pi } = f ( c _ { i , \pi } ^ { g } )$ where $f$ is a monotonically decreasing function. This reward function is an episodic history dependent reward function $r ( s _ { 0 } , a _ { 0 } , \ldots , s _ { t } , a _ { t } )$ where $r$ is uniquely defined from expert demonstrations (hence is said to be derived offline). Crucially, we have defined a reward function that we seek to maximize, without introducing an inner minimization problem (which is the case with adversarial $\mathrm { I L }$ approaches).
+
+We can now derive an algorithm building on this reward function (Algorithm 1). The algorithm presented is written using pseudo-code for a generic agent $A$ which implements a policy $\pi _ { A }$ , it observes tuples $( s , a , r , s ^ { \prime } )$ and possibly updates its components. The runtime of the algorithm for a single reward step computation has complexity $\begin{array} { r } { \mathcal { O } ( ( | S | + | A | ) D + \frac { D ^ { 2 } } { T } ) } \end{array}$ (see Appendix E.2 for details), and therefore has complexity $\mathcal { O } ( ( | S | + | A | ) D T + D ^ { 2 } )$ for computing rewards across the entire episode. Note that in the case where $T = D$ , computing the greedy coupling is simply a lookup of the expert state-action pair that minimizes the distance with the agent state-action pair, followed by a pop-out of this minimizing expert state-action pair from the set of expert demonstrations.
+
+Algorithm 1 introduces an agent with the capability to observe and update itself, which is directly inspired from ACME’s formalism of agents (Hoffman et al., 2020), and is general enough to characterize a wide range of agents. We give an example rundown of the algorithm in Section E.1.
+
+# 4 EXPERIMENTS
+
+In this section, we present the implementation of PWIL and perform an empirical evaluation, based on the ACME framework (Hoffman et al., 2020). We test PWIL on MuJoCo locomotion tasks and compare it to the state-of-the-art GAIL-like algorithm DAC (Kostrikov et al., 2019) and against the common BC baseline. As DAC is based on TD3 (Fujimoto et al., 2018) which is a variant of DDPG (Lillicrap et al., 2016), we use a DDPG-based agent for fair comparison: D4PG (BarthMaron et al., 2018). We also provide a proof-of-concept experiment on a door opening task where the demonstrations do not include actions and are pixel-based and therefore the metric of the
+
+Algorithm 1 PWIL: Primal Wasserstein Imitation Learning   
+
+<table><tr><td>Input: Expert demonstrations D = {s,aj}j∈[1:D],Agent A, number of episodes N for k=1 to N do</td><td>Copy expert demonstrations with weight: D&#x27; := {s,a, w}j∈[1:D], with w = D</td></tr><tr><td>Reset environment, initial state s fori=1toTdo</td><td></td></tr><tr><td>Take action a := TA(s), observe next state s&#x27; Initialize weight w&quot; := 1， cost c := 0</td><td></td></tr><tr><td>while w&quot; &gt;O do</td><td>Compute se,ae,we := arg min(se,ae,we)∈D&#x27; d((s,a),(se,ae))</td></tr><tr><td>if w≥we then</td><td></td></tr><tr><td>C:=c+wed((s,a),(se,ae)) w := wπ- we</td><td></td></tr><tr><td>D&#x27;.pop(se,ae,we)</td><td></td></tr><tr><td>else</td><td></td></tr><tr><td>c :=c+wd((s,a)，(se,ae))</td><td></td></tr><tr><td>we := we -wπ</td><td></td></tr><tr><td></td><td></td></tr><tr><td>w := 0</td><td></td></tr><tr><td>r := f(c)</td><td></td></tr><tr><td>A observes (s, a,r,s&#x27;), updates itself</td><td></td></tr></table>
+
+MDP has to be learned. In this setup, we use SAC as the direct RL method. We answer the following questions: 1) Does PWIL recover expert behavior? 2) How sample efficient is PWIL? 3) Does PWIL actually minimize the Wasserstein distance between the distributions of the agent and the expert? 4) Does PWIL extend to visual based observations where the definition of an MDP metric is not straightforward? We also show the dependence of PWIL on multiple of its components through an ablation study. The complete description of the experiments is given in Appendix A. We provide experimental code and videos of the trained agents here: https: //sites.google.com/view/wasserstein-imitation.
+
+# 4.1 IMPLEMENTATION
+
+We test PWIL following the same evaluation protocol as GAIL and DAC, on six environments from the OpenAI Gym MuJoCo suite (Todorov et al., 2012; Brockman et al., 2016): Reacher-v2, Ant-v2, Humanoid-v2, Walker2d-v2, HalfCheetah-v2 and Hopper-v2. For each environment, multiple demonstrations are gathered using a D4PG agent trained on the actual reward of these environments (which was chosen as it is a base agent in the ACME framework). We subsample demonstrations by a factor of 20: we select one out of every 20 transitions with a random offset at the beginning of the episode (expect for Reacher for which we do not subsample since the episode length is 50). The point of subsampling is to simulate a low data regime, so that a pure behavioral cloning approach suffers the behavioral drift and hence does not perform to the level of the expert. We run experiments with multiple number of demonstrations: $\{ 1 , 4 , 1 1 \}$ (consistently with the evaluation protocol of DAC).
+
+Central to our method is a metric between state-action pairs. We use the standardized Euclidean distance which is the L2 distance on the concatenation of the observation and the action, weighted along each dimension by the inverse standard deviation of the expert demonstrations. From the cost $c _ { i }$ defined in Equation (5), we define the following reward:
+
+$$
+r _ { i } = \alpha \exp ( - \frac { \beta T } { \sqrt { | \cal S | + | \cal A | } } c _ { i } ) .
+$$
+
+For all environments, we use $\alpha = 5$ and $\beta = 5$ . The scaling factor $\frac { T } { \sqrt { | { \cal S } | + | { \cal A } | } }$ acts as a normalizer on the dimensionality of the state and action spaces and on the time horizon of the task. We pick $f : x \mapsto \exp ( - x )$ as the monotonically decreasing function.
+
+We test our method against a state-of-the-art imitation learning algorithm: DAC and BC. DAC is based on GAIL: it discriminates between expert and non-expert states and it uses an off-policy algorithm TD3 which is a variant of DDPG instead of the on-policy algorithm TRPO (Schulman et al., 2015). We used the open-source code provided by the authors. The Humanoid environment is not studied in the original DAC paper. The hyperparameters the authors provided having poor performance, we led an exhaustive search detailed in Section A.2, and present results for the best hyperparameters found.
+
+For our method, we use a D4PG agent (see details in Appendix A) and provide additional experiments with different direct RL agents in Appendix C. We initialize the replay buffer by filling it with expert state-action pairs with maximum reward $\alpha$ (however, note that we used subsampled trajectories which makes it an approximation to replaying the actual trajectories). We train PWIL and DAC in the limit of 1M environment interactions (2.5M for Humanoid) on 10 seeds. Every 10k environment steps, we perform 10-episode rollouts per seed of the policy without exploration noise and report performance with respect to the environment’s original reward in Figure 2.
+
+# 4.2 RESULTS
+
+In Figure 2, PWIL shows improvement on final performance for Hopper, Ant, HalfCheetah and Humanoid over DAC, similar performance for Walker2d, and is worse on Reacher. On Humanoid, although we ran an exhaustive hyperparameter search (see Section A.2) DAC has poor performance, thus exemplifying the brittleness of GAIL-like approaches. On the contrary, PWIL has near-optimal performance, even with a single demonstration.
+
+In terms of sample efficiency, DAC outperforms PWIL on HalfCheetah, Ant and Reacher and has similar performance for Hopper and Walker2d. Note that ablated versions of PWIL are more sampleefficient than DAC on HalfCheetah and Ant (Section B). This seems rather contradictory with the claim that GAIL-like methods have poor sample efficiency. However the reward function defined in DAC requires careful tuning (e.g. architecture, optimizer, regularizers, learning rate schedule) which demands experimental cycles and therefore comes with a large number of environment interactions.
+
+Remarkably, PWIL is the first method able to consistently learn policies on Humanoid with an average score over 7000 even with a single demonstration, which means that the agent can actually run, (other works consider Humanoid is "solved" for a score of 5000 wich corresponds to standing).
+
+![](images/e40832eff7388e28e7493984d455ecdceb0ec2f257b16625d6bfb2b539c79eb9.jpg)  
+Figure 2: Mean and standard deviation return of the evaluation policy over 10 rollouts and 10 seeds, reported every 10k environment steps. The return is in term of the environment’s original reward. Top row: 1 demonstration, bottom row: 11 demonstrations.
+
+Wasserstein distance. In Figure 3, we show the Wasserstein distance to expert demonstrations throughout the learning procedure for PWIL and DAC. The Wasserstein distance is evaluated at the end of an episode using a transportation simplex (Bonneel et al., 2011); we used the implementation from Flamary & Courty (2017). For both methods, we notice that the distance decreases while learning. PWIL leads to a smaller Wasserstein distance than DAC on all environments but HalfCheetah where it is similar and Reacher where it is larger. This should not come as a surprise since PWIL’s objective is to minimize the Wasserstein distance. PWIL defines a reward from an upper bound to the Wasserstein distance between the state-action distributions of the expert and the agent, that we include as well in Figure 3. Notice that our upper bound is “empirically tight”, which validates the choice of the greedy coupling as an approximation to the optimal coupling. We emphasize that this distance is useful in real settings of IL, regardless of the method used, where we cannot have access to the actual reward of the task.
+
+![](images/1f51ee76559a986b51bd3ba554546b2c557593b15e60d3d2569c776ee7e2ed4b.jpg)  
+Figure 3: Mean of the Wasserstein distance between the state-action distribution of the evaluation policy and the state-action distribution of the expert over 10 rollouts and 10 seeds. Agents were trained with 11 demonstrations. For PWIL, we include the upper bound on the Wasserstein distance based on the greedy coupling defined in Equation (4).
+
+# 4.3 ABLATION STUDY
+
+In this section, we present the evaluation performance of PWIL in the presence of ablations and report final performances in Figure 4. The learning curves for all ablations can be found in Appendix B. We keep the hyperparameters of the original PWIL agent fixed.
+
+![](images/adb48cc75f79e4ff29215610c1e97f680ff2130f986b6a3e0f4fa3cab1d87373.jpg)  
+Figure 4: Mean and standard deviation of the evaluation performance of PWIL and variants at the 1M environment interactions mark (2.5M for Humanoid). Results are computed over 10 seeds and 10 episodes for each seed.
+
+PWIL-state. In this version of PWIL, we do not assume that we have access to the actions taken by the expert. Therefore, we try to match the state distribution of the agent with the state distribution of the expert (instead of the state-action distribution). The setup is referred to as Learning from Observation (LfO) (Torabi et al., 2018; Edwards et al., 2019). The reward is defined similarly to PWIL, using a state distance rather than a state-action distance. Note that in this version, we cannot pre-fill the replay buffer with expert state-action pairs since we do not have access to actions. Remarkably, PWIL-state recovers non-trivial behaviors on all environments.
+
+PWIL-L2. In this version of PWIL, we use the Euclidean distance between state-action pairs, rather than the standardized Euclidean distance. In other words, we do not weight the state-action distance by the inverse standard deviation of the expert demonstrations along each dimension. There is a significant drop in performance for all environments but Hopper-v2. This shows that the performance of PWIL is sensitive to the quality of the MDP metric.
+
+PWIL-nofill. In this version, we do not prefill the replay buffer with expert transitions. This leads to a drop in performance which is significant for Walker and Ant. This should not come as a surprise since a number of $\mathrm { I L }$ methods leverage the idea of expert transitions in the replay buffer (Reddy et al., 2020; Hester et al., 2018).
+
+PWIL-support. In this version of PWIL, we define the reward as a function of the following cost:
+
+$$
+\forall i \in [ 1 : T ] , c _ { i , \pi } = \operatorname* { i n f } _ { \tiny \begin{array} { c } { \theta _ { 1 } , \ldots , \theta _ { D } \in \mathbb { R } } \\ { \sum _ { j = 1 } ^ { D } \theta _ { j } \leq \frac { 1 } { T } } \end{array} } \sum _ { j = 1 } ^ { D } d ( ( s _ { i } ^ { \pi } , a _ { i } ^ { \pi } ) , ( s _ { j } ^ { e } , a _ { j } ^ { e } ) ) \theta _ { j } .
+$$
+
+We can interpret this cost in the context of Section 3 as the problem of moving piles of dirt of mass $\textstyle { \frac { 1 } { T } }$ into holes of infinite capacity. It is thus reminiscent of methods that consider $\mathrm { I L }$ as a support estimation problem (Wang et al., 2019; Brantley et al., 2020). This leads to a significant drop in performance for Ant, Walker and HalfCheetah.
+
+4.4 DOOR OPENING TASK: REWARD FROM PIXEL-BASED OBSERVATIONS
+
+In this section, we show that PWIL extends to visual-based setups, where the MDP metric has to be learned. We use the door opening task from Rajeswaran et al. (2018), with human demonstrations generated with Haptix (Kumar & Todorov, 2015). The goal of the task is for the controlled hand to open the door (Figure 5) whose location changes at each episode. We add an early termination constraint when the door is opened (hence suppressing the incentive for survival). We assume that we can only access the high-dimensional visual rendering of the demonstrations rather than the internal state (and hence have no access to actions, that is LfO setting).
+
+The PWIL reward is therefore defined using a distance between the visual rendering of the current state and the visual rendering of the demonstrations. This distance is learned offline through selfsupervised learning: we construct an embedding of the frames using Temporal Cycle-Consistency Learning (TCC) (Dwibedi et al., 2019); TCC builds an embedding by aligning frames of videos of a task coming from multiple examples (in this case the demonstrations). The distance learned is thus the L2 distance between embeddings. The distance is learned on the 25 human demonstrations provided by Rajeswaran et al. (2018) which are then re-used to define the PWIL reward. From the cost $c _ { i }$ defined in Equation (5), we define the following reward: $r _ { i } = \alpha \exp ( - \beta T c _ { i } )$ , with $\alpha = 5 , \beta = 1$ . We used SAC (Haarnoja et al., 2018a) as the forward RL algorithm, which was more robust than D4PG on the task. We train the agent on 1.5M environment steps and show that we indeed recover expert behaviour and consistently solve the task in Figure 6.
+
+![](images/0fc960760fec29961a678ed9e81db245f1d0ae3b8f30bd26309f46a649672d91.jpg)  
+Figure 5: Visual rendering of the door opening task.
+
+![](images/d134c8bded467d0a27427e19b71ab5d7dba8af8962cd29e48896dd398dafea7e.jpg)  
+Figure 6: We present the average and standard deviation of evaluation performance of PWIL on the door environment. Metrics are computed every $1 0 \mathrm { k }$ environment steps and aggregated over 10 seeds and 10 rollouts. Left: the imitation returns of the agent. Center: Wasserstein distance in the embedding space to expert demonstrations. Right: Success rate of the agent, i.e. ratio of episodes where the agent manages to open the door.
+
+# 5 RELATED WORK
+
+Adversarial IL. Similarly to our method, adversarial IL methods aim at matching the state-action distribution of the agent with the distribution of the expert, using different measures of similarity. For instance, GAIL (Ho & Ermon, 2016) considers the Shannon-Jensen divergence while AIRL (Fu et al., 2018) considers the Kullback-Leibler divergence. In the context of GANs, Arjovsky et al. (2017) show that replacing the $f$ -divergence by the Wasserstein distance through the Kantorovich-Rubinstein duality (Villani, 2008) leads to better training stability, which a number of methods in IL have leveraged (Li et al., 2017; Lacotte et al., 2019; Kostrikov et al., 2019; Xiao et al., 2019; Liu et al., 2020; Pacchiano et al., 2020). However, the Kantorovich-Rubinstein duality only holds for the $\mathcal { W } _ { 1 }$ distance (Peyré et al., 2019), its implementation comes with a number of practical approximations (e.g., weight clipping to ensure Lipchitz continuity). Although these methods are based on the Wasserstein distance through its dual formulation, they drastically differ from ours as they rely on a minmax optimization problem. DAC (Kostrikov et al., 2019) demonstrates that adversarial $\mathrm { I L }$ approaches, while based on GAIL, can be sample efficient. However, they rely on a carefully tuned discriminator (e.g. network architecture, regularization strategy, scheduled learning rates) which requires to interact with the environment. In contrast, the reward function we present relies on only two hyperparameters.
+
+Expert support estimation. Another line of research in IL consists in estimating the state-action support of the expert and define a reward that encourages the agent to stay on the support of the expert (Piot et al., 2014; Schroecker & Isbell, 2017; Wang et al., 2019; Brantley et al., 2020; Reddy et al., 2020). Note that the agent might stay on the support of the expert without recovering its state-action distribution. Soft Q Imitation Learning (Reddy et al., 2020) assigns a reward of 1 to all expert transitions and 0 to all transitions the expert encounters; the method learns to recover expert behavior by balancing out the expert transitions with the agent transitions in the replay buffer of a value-based agent. Random Expert Distillation (Wang et al., 2019) estimates the support by using a neural network trained on expert transitions whose target is a fixed random neural network. Disagreement Regularized Imitation Learning (Brantley et al., 2020) estimates the expert support through the variance of an ensemble of BC agents, and use a reward based on the distance to the expert support and the KL divergence with the BC policies. PWIL differs from these methods since it is based on the distance to the support of the expert, with a support that shrinks through “pop-outs” to enforce the agent to visit the whole support. We showed in the ablation study that “pop-outs” are sine qua non for recovering expert behaviour (see PWIL-support).
+
+Trajectory-Based IL. Similarly to the door opening experiment, Aytar et al. (2018) learn an embedding offline and reinforces a reward based on the distance in the embedding space to a single demonstration. However, it is well suited for deterministic environments since they use a single trajectory that the agent aims at reproducing while the door experiment shows that PWIL is able to generalize as the door and handle locations are picked at random. Peng et al. (2018) reinforces a temporal reward function (the reward at time $t$ is based on the distance of the state of the agent to the state of the expert at time $t$ ). Contrary to PWIL, the imitation reward function is added to a task-dependent reward function; the distance to expert demonstrations is defined along handpicked dimensions of the observation space; the temporal constraint makes it hard to generalize to environment with stochastic initial states.
+
+Offline reward estimation. Existing approaches derive a reward function without interaction with the environment (Boularias et al., 2011; Klein et al., 2013; 2012; Piot et al., 2016). They typically require strong assumptions on the structure of the reward (e.g. linearity in some predefined features). PWIL also derives a reward offline, without structural assumptions, however it is non-stationary since it depends on the state-action visited in the episode.
+
+MMD-IL. Generative Moment Matching Imitation Learning (Kim & Park, 2018) assumes a metric on the MDP and aims at minimizing the Maximum Mean Discrepancy (MMD) between the agent and the expert. However, the MMD (like the Wasserstein distance) requires complete rollouts of the agent, and does not present a natural relaxation like the greedy coupling. GMMIL is based on an on-policy agent which makes it significantly less sample-efficient than PWIL. We were not able to implement a version of MMD based on an off-policy agent.
+
+# 6 CONCLUSION
+
+In this work, we present Imitation Learning as a distribution matching problem and introduce a reward function which is based on an upper bound of the Wasserstein distance between the state-action distributions of the agent and the expert. A number of IL methods are developed on synthetic tasks, where the evaluation of the IL method can be done with the actual return of the task. We emphasize that in our work, we present a direct measure of similarity between the expert and the agent, that we can thus use in real $\mathrm { I L }$ settings, that is in settings where the reward of the task cannot be specified.
+
+The reward function we introduce is learned offline, in the sense that it does not need to be updated with interactions with the environment. It requires little tuning (2 hyperparameters) and it can recover near expert performance with as little as 1 demonstration on all considered environments, even the challenging Humanoid. Finally our method extends to the harder visual-based setting, based on a metric learned offline from expert demonstrations using self-supervised learning.
+
+# ACKNOWLEDGMENTS
+
+We thank Zafarali Ahmed, Adrien Ali Taiga, Gabriel Dulac-Arnold, Johan Ferret, Alexis Jacq and Saurabh Kumar for their feedback on an earlier version of the manuscript. We thank Debidatta Dwibedi and Ilya Kostrikov for helpful discussions.
+
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+
+# A LOCOMOTION EXPERIMENTS
+
+# A.1 PWIL IMPLEMENTATION
+
+The agent we use is D4PG (Barth-Maron et al., 2018). We use the default architecture from the ACME framework (Hoffman et al., 2020).
+
+The actor architecture is a 4-layer neural network: the first layer has size 256 with tanh activation and layer normalization (Ba et al., 2016), the second layer and third layer have size 256 with elu activation (Clevert et al., 2016), the last layer is of size the dimension of the action space, with a tanh activation scaled to the action range of the environment. To enable sufficient exploration with use a Gaussian noise layer on top of the last layer with standard deviation $\sigma = 0 . 2$ , that we clip to the action range of the environment. We evaluate the agent without exploration noise.
+
+For the critic network we use a 4-layer neural network: the first layer has size 512 with tanh activation and layer normalization, the second layer is of size 512 with elu activation, the third layer is of size 256 with elu activation, the last layer is of dimension 201 with a softmax activation. The 201 neurons stand for the weights of the distribution supported within equal distance in the range $[ - 1 5 0 , 1 5 0 ]$ , as a categorical distribution (Bellemare et al., 2017).
+
+We use the Adam optimizer (Kingma & Ba, 2015) with $\lambda _ { a } = 5 { \times } 1 0 ^ { - 5 }$ for the actor and $\lambda _ { c } = 7 \times 1 0 ^ { - 5 }$ for the critic. We use a batch size of 256. We clip both gradients from the critic and the actor to limit their L2 norm to 40.
+
+We use a replay buffer of size $1 0 ^ { 6 }$ , a discount factor $\theta = 0 . 9 9$ and $n$ step returns with $n = 5$ . We prefill the replay buffer with 50000 state-action pairs from the set of demonstrations (which means that we put multiple times the same expert transitions in the buffer). We perform updates on the actor and the critic every $k = 4$ interactions with the environment. The hyperparameters search is detailed in Table 1.
+
+Table 1: DDPG hyperparameters search.   
+
+<table><tr><td rowspan=1 colspan=1>Parameters</td><td rowspan=1 colspan=1>Values</td></tr><tr><td rowspan=1 colspan=1>入a</td><td rowspan=1 colspan=1>10-5,5×10-5,7× 10-5,10-4</td></tr><tr><td rowspan=1 colspan=1>入c</td><td rowspan=1 colspan=1>10-5,5× 10-5,7× 10-5,10-4</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0.1,0.2,0.3</td></tr><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>2,4,8,16</td></tr></table>
+
+For the reward function, we did run a hyperparameter search on $\alpha$ and $\beta$ with the following values for $\alpha$ and $\beta$ in $\{ 1 , 5 , 1 0 \}$ .
+
+# A.2 DAC IMPLEMENTATION
+
+We used the open-source implementation of DAC provided by Kostrikov et al. (2019). For Humanoid (which was not reported in the paper), we did run a large hyperparameter search described in Table 2. Out of the 729 experiments, 7 led to an average performance above 1000 at the $2 . 5 \mathbf { M }$ training environment steps mark.
+
+Table 2: Hyperparameter search on Humanoid for 11 demonstrations. (tested/best)   
+
+<table><tr><td rowspan=1 colspan=1>Actor learning rate (lr)</td><td rowspan=1 colspan=1>Critic lr</td><td rowspan=1 colspan=1>Discriminator lr</td><td rowspan=1 colspan=1>WGAN regularizer</td><td rowspan=1 colspan=1>Exploration noise</td><td rowspan=1 colspan=1>Decay lr</td></tr><tr><td rowspan=1 colspan=1>0.001,0.0005,0.0001</td><td rowspan=1 colspan=1>0.001,0.0005,0.0001</td><td rowspan=1 colspan=1>0.001,0.0005,0.0001</td><td rowspan=1 colspan=1>5,10,20</td><td rowspan=1 colspan=1>0.1,0.2,0.3</td><td rowspan=1 colspan=1>0.3,0.5,0.8</td></tr></table>
+
+# A.3 BC IMPLEMENTATION
+
+We used a 3-layer neural network, the first layer has size 128 with relu activation, the second layer has size 64 and the last layer has size of the action space with a tanh activation scaled to the action range of the environment. We found out that normalizing the observations with the average and standard deviation of the expert demonstrations’ observations helped performance. We trained the network using the mean squared error as a loss, with an Adam optimizer. We ran an hyperpameter search on the learning rate in $\{ 1 0 ^ { - 5 } , 1 0 ^ { - 4 } , 1 0 ^ { - 3 } \}$ and on the batch size $\{ 1 2 8 , 2 5 6 \}$ .
+
+We present the learning curves of PWIL, in terms of the reward defined in Equation 6 as well as the original reward of the task.
+
+![](images/4537548d23d868054ce3111228880c63aff3dec9d0e4685bcbbc7b39655c6c03.jpg)  
+Figure 7: Mean and standard deviation of the imitation return of the agent. We perform 10 rollouts over 10 seeds every 10k environment steps over 1M steps. The return here is in term of the reward defined with Equation (6). Top row: 1 demonstration, middle row: 4 demonstrations, bottom row 11 demonstrations.
+
+![](images/d3baa66434501a324c1a0a02c07ddebaff4ddd20b869894068c779504c31bc97.jpg)  
+Figure 8: Mean and standard deviation of the original environment returns of the evaluation policy over 10 rollouts and 10 seeds, reported every 10k environment steps over 1M steps. Top row: 1 demonstation, middle row: 4 demonstrations, bottom row: 11 demonstrations.
+
+![](images/e26c1a245aae8d40cbdf6b7f771ef06c26cfecbc55d6b850c1a383bb1da276d0.jpg)  
+Figure 9: Mean and standard deviation of the Wasserstein distance between the state-action distribution of the evaluation policy and the state-action distribution of the expert over 10 rollouts and 10 seeds, reported every 10k environment steps. We include the upper bound on the Wasserstein distance based on the greedy coupling defined in Equation (4). Top row: 1 demos, middle row: 4 demos, bottom row: 11 demos.
+
+# B ABLATION STUDY
+
+We present the learning curves of PWIL in the presence of ablations.
+
+![](images/525f4729ae669358c71ea42a341d40c0eee97b61f9e2fef1dec90269a61108f4.jpg)  
+Figure 10: Evaluation performance of different variants of PWIL over 10 rollouts and 10 seeds, reported every 10k environment steps over 1M steps. The return is in term of the environment’s original reward. Top row: 1 demonstration, middle row: 4 demonstrations, bottom row: 11 demonstrations.
+
+# C INFLUENCE OF THE DIRECT RL ALGORITHM
+
+In this section, we study the influence of the direct RL method on the performance on PWIL on locomotion tasks. We compare the performance of PWIL with D4PG with the performance of PWIL with TD3 and SAC. We use a custom implementation of both SAC and TD3 which is a reproduction of the authors’ implementation. For SAC we used the same default hyperparameters; for TD3 we removed the delayed policy update which led to better performance. We evaluate both methods on the same reward function as D4PG. We remove the pre-warming of the replay buffer for SAC. For TD3, we found out that we need different pre-warming for each environment to get good performance (contrary to D4PG): we used 1000 state-action pairs for Reacher, Hopper and Walker2d and no pre-warming for Humanoid, Ant and HalfCheetah.
+
+![](images/19f9c30f768d5309cc4bb93b92a5bc65b18f58a69c7c9c0c8705782a369624a5.jpg)  
+Figure 11: Mean and standard deviation of the evaluation performance of PWIL-D4PG, PWIL-SAC and PWIL-TD3 at the 1M environment interactions mark (2.5M for Humanoid). Results are computed over 10 seeds and 10 episodes for each seed.
+
+We show in Figure 11 that both PWIL-SAC and PWIL-TD3 recover expert-like performance on Hopper, Ant and HalfCheetah and Reacher. For Walker2d, there is a large standard deviation in performance, since some seeds lead to near-optimal expert performance, and some seeds lead to poor performance. Remark that we led minimal hyperparameter tuning on the default versions of SAC and TD3, which might explain the differences in performance.
+
+# D DOOR OPENING EXPERIMENTS
+
+In this section we present the details of the experiment presented in Section 4.4.
+
+# D.1 ENVIRONMENT
+
+The default door opening environment has a fixed horizon of 200 timesteps. We add an early termination condition to the environment when the door is opened to demonstrate that PWIL extends to tasks where there is no incentive for survival. Note that similarly to AIRL (Fu et al., 2018), early termination states are considered to be absorbing. Therefore if the expert reaches an early termination state early, its state distribution as a large weight on these states, which leads the PWIL agent to similarly reach the early termination state. 25 demonstrations were collected from humans using a virtual reality system (Kumar & Todorov (2015)).
+
+# D.2 EMBEDDING
+
+As we assume that we only have access to the visual rendering of the demonstrations (with resolution $8 4 \times 8 4 ,$ , we build a lower dimensional latent space. We use a self-supervised learning method: TCC, to learn the latent space. We used the classification version of the algorithm detailed in Dwibedi et al. (2019) (Section 3.2).
+
+The encoding network is the following: a convolutional layer with 128 filters of size 8 and stride 4, a relu activation, a convolutional layer with 64 filters of size 4 and stride 2, a relu activation, a convolutional layer with 64 filters of size 3 and stride 1, a relu activation, a linear layer of size 128, a relu activation and an output layer of size 32.
+
+We separated the demonstrations into a train set of 20 demonstrations and a validation set of 5 demonstrations. Similarly to Dwibedi et al. (2019), we select the encoding network by taking the one that maximizes the alignment according to Kendall’s tau score on the validation set.
+
+We ran an hyperparameter search over the batch size $\{ 1 , 4 , 1 6 , 3 2 \}$ , the subsampling ratio $\{ 1 , 4 , 8 \}$ and the learning rate $\{ 1 0 ^ { - 2 } , 1 0 ^ { - 3 } , 1 0 ^ { - 4 } \}$ (using the Adam optimizer). The best performing encoder had an average Kendall’s tau score of 0.98 on the training set and 0.91 on the validation set.
+
+# D.3 AGENT
+
+![](images/e6a42cd9de48fe57d68313cc81229b9cf11b6f3771f063e7c6d7f183b6267b73.jpg)  
+Figure 12: Kendall’s tau score of the encoding network on the train and validation sets. An iteration corresponds to 1000 gradient updates.
+
+In this setup, we found that SAC leads to better results than D4PG. We detail the implementation of SAC below.
+
+The actor is a 3-layer network that outputs the parameters of a normal distribution with diagonal covariance matrix. We enforce the terms of the covariance matrix to be greater than $1 0 ^ { - 3 }$ . The first layer has size 256 with relu activation, the second layer has size 256 with relu activation and the last layer has size 48 (the action space has size 24).
+
+The critic is a 3-layer network. We pass the concatenation of the state and the action of the environment as an input. There are two hidden layers of size 256 with relu activation. We do use a twin critic.
+
+We use an adaptive temperature $\alpha$ as described by Haarnoja et al. (2018b).
+
+For the three losses we use the Adam optimizer, and we use Polyak averaging for the critic networks with a rate $\tau$ . We ran an hyperparameter search over the learning rates $\lambda _ { a } , \lambda _ { c } , \lambda _ { \alpha }$ in $\{ 1 0 ^ { - 4 } , 3 \times 1 0 ^ { - 4 } \}$ , over $\tau$ $\lbrace 0 . 0 1 , 0 . 0 0 5 \rbrace$ , over the batch size $\{ 6 4 , 1 2 8 , 2 5 6 \}$ and over the number of interactions with the environment between each actor, critic and temperature update $\{ 1 , 4 , 8 \}$ .
+
+# E ALGORITHM DETAILS
+
+# E.1 RUNDOWN
+
+We present an example rundown of PWIL in Figure 13.
+
+# Sta! episode.
+
+![](images/4f44e70ef44250bdd8d3698ffc390e2ee87e7410a8013b0ba3fbfd4516c40ea6.jpg)  
+Figure 13: PWIL example rundown. We drop the dependency on the action. The example task has an horizon $T = 2$ , and the demonstration set has $D = 4$ elements. We take $f : c \mapsto \exp \left( - c \right)$ .
+
+# End episode.
+
+# E.2 RUNTIME
+
+We detail the runtime of PWIL in this section. A single step reward computation consists in computing the distance from the current observation (a vector with dimension $| { \bar { S } } | + | { \cal A } |$ with $D$ vectors (the expert observations), which has complexity $\mathcal { O } ( ( | S | + | A | ) D )$ . The next step is to get the minimum of these distances (complexity $\mathcal { O } ( D ) ,$ ), for $\textstyle \left\lceil { \frac { D } { T } } \right\rceil$ times (the number of times the algorithm executes the while loop). Therefore the complexity of a single step reward computation is $\begin{array} { r } { \mathcal { O } ( ( | S | + | A | ) D + \frac { D ^ { 2 } } { T } ) } \end{array}$ and the complexity of the reward computation of an episode is $\mathcal { O } ( ( | S | + | A | ) D T + D ^ { 2 } )$ .

parse/train/rJe2syrtvS/rJe2syrtvS.md

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+# THE INGREDIENTS OF REAL-WORLD ROBOTIC REINFORCEMENT LEARNING
+
+Henry $\mathbf { L h u ^ { * 1 } }$ , Justin $\mathbf { V } \mathbf { u } ^ { * 1 }$ , Abhishek Gupta∗1, Dhruv Shah1, Kristian Hartikainen2, Avi Singh1, Vikash Kumar3, Sergey Levine1 1 University of California, Berkeley 2 University of Oxford 3 University of Washington
+
+# ABSTRACT
+
+The success of reinforcement learning for real world robotics has been, in many cases limited to instrumented laboratory scenarios, often requiring arduous human effort and oversight to enable continuous learning. In this work, we discuss the elements that are needed for a robotic learning system that can continually and autonomously improve with data collected in the real world. We propose a particular instantiation of such a system, using dexterous manipulation as our case study. Subsequently, we investigate a number of challenges that come up when learning without instrumentation. In such settings, learning must be feasible without manually designed resets, using only on-board perception, and without hand-engineered reward functions. We propose simple and scalable solutions to these challenges, and then demonstrate the efficacy of our proposed system on a set of dexterous robotic manipulation tasks, providing an in-depth analysis of the challenges associated with this learning paradigm. We demonstrate that our complete system can learn without any human intervention, acquiring a variety of vision-based skills with a real-world three-fingered hand. Results and videos can be found at https://sites.google.com/view/realworld-rl/.
+
+# 1 INTRODUCTION
+
+Reinforcement learning (RL) can in principle enable autonomous systems, such as robots, to acquire a large repertoire of skills automatically. Perhaps even more importantly, reinforcement learning can enable such systems to continuously improve the proficiency of their skills from experience. However, realizing this in reality has proven challenging: even with reinforcement learning methods that can acquire complex behaviors from high-dimensional low-level observations, such as images, the assumptions of the reinforcement learning problem setting do not fit cleanly into the constraints of the real world. For this reason, most successful robotic learning experiments have employed various kinds of environmental instrumentation in order to define reward functions, reset between trials, and obtain ground truth state (Levine et al., 2016; Haarnoja et al., 2018a; Kumar et al., 2016; Andrychowicz et al., 2018; Zhu et al., 2019; Chebotar et al., 2016; Nagabandi et al., 2019; Gupta et al., 2016). In order to practically and scalably deploy autonomous learning systems that improve continuously through real-world operation, we must lift these limitations and design algorithms that can learn under the constraints of real-world environments, as illustrated in Figure 2.
+
+We propose that overcoming these challenges in a scalable way requires designing robotic systems that possess three capabilities: they are able to (1) learn from their own raw sensory inputs, (2) assign rewards to their own trials without hand-designed perception systems or instrumentation, and (3) learn continuously in non-episodic settings without requiring human intervention to manually reset the environment. A system with these capabilities can autonomously collect large amounts of real world data – typically crucial for effective generalization – without significant instrumentation in each training environment, an example of which is shown in Figure 1. If successful, this would lift a major constraint that stands between current reinforcement learning algorithms and the ability to learn scalable, generalizable, and robust real-world behaviors. Such a system would also bring us significantly closer to the goal of embodied learning-based systems that improve continuously through their own real-world experience.
+
+![](images/83c7b9a06c076f3cf373c41b31868ce93909c4644428dbe54f3993fd7de2ca93.jpg)  
+Figure 1: Illustration of our proposed instrumentation-free system requiring minimal human engineering. Human intervention is only required in the goal collection phase (1). The robot is left to train unattended (2) during the learning phase and can be evaluated from arbitrary initial states at the end of training (3). We show sample goal and intermediate images from the training process of a real hardware system
+
+Having laid out these requirements, we propose a practical instantiation of such a learning system. While prior works have studied many of these issues in isolation, combining them into a complete real-world learning system presents a number of challenges, as we discuss in Section 3. We provide an empirical analysis of these issues, both in simulation and on a real-world robotic system, and propose a number of simple but effective solutions that together produce a complete robotic learning system. This system can autonomously learn from raw sensory inputs, learn reward functions from easily available supervision, and learn without manually designed reset mechanisms. We show that this system can learn dexterous robotic manipulation tasks in the real world, substantially outperforming ablations and prior work.
+
+# 2 THE STRUCTURE OF A REAL-WORLD RL SYSTEM
+
+The standard reinforcement learning paradigm assumes that the controlled system is represented as a Markov decision process with a state space $s$ , action space $\mathcal { A }$ , unknown transition dynamics $\tau$ , unknown reward function $\mathcal { R }$ , and a (typically) episodic initial state distribution $\rho$ . The goal is to learn a policy that maximizes the expected sum of rewards via interactions with the environment.
+
+Although this formalism is simple and concise, it does not capture all of the complexities of realworld robotic learning problems. If a robotic system is to learn continuously and autonomously in the real world, we must ensure that it can learn under the actual conditions that are imposed by the real world. To move from the idealized MDP formulation to the real world, we require a system that has the following properties. First, all of the information necessary for learning must be obtained from the robot’s own sensors. This includes information about the state and necessitates that the policy must be learned from high-dimensional and low-level sensory observations, such as camera images. Second, the robot must also obtain the reward signal itself from its own sensor readings. This is exceptionally difficult for all but the simplest tasks (e.g., reward functions that depend on interactions with specific objects require perceiving those objects explicitly). Third, we must be able to learn without access to episodic resets. A setup with explicit resets quickly becomes impractical in open-world settings, due to the requirement for significant human engineering of the environment, or direct human intervention during learning. While this list may not exhaustively enumerate all the components necessary for an effective real-world learning system, we posit that the properties listed here are fundamental to building such systems.
+
+While some of the components discussed above can be tackled in isolation by current algorithms, there are considerable challenges inherent to assembling all these components into a complete learning system for real world robotic learning. In the rest of this section, we outline the challenges associated with each component, then discuss the challenges associated with combining these components in Section 3, and then proceed to address these challenges in Section 4.
+
+![](images/c22c53be5af8d2715578d714c982847f8d4f9d12900678fdfd0e7bd3b1d9f31e.jpg)  
+Figure 2: We draw a comparison between current real world learning systems which rely on instrumentation versus a system that learns in an environment more representative of the real world, free of instrumentation. While all three prior works utilize instrumentation for resets, state estimation, and reward, the motion capture system of Gupta et al. (2016), sensor attached to the door in Zhu et al. (2019), and auxiliary robot which picks up fallen balls in Nagabandi et al. (2019) are good examples of engineered state estimation, reward estimation, and reset mechanisms respectively.
+
+# 2.1 LEARNING FROM RAW SENSORY INPUT
+
+To enable learning without complex state estimation systems or environment instrumentation, we require our robotic systems to be able to learn from their own raw sensory observations. Typically, these sensory observations are raw camera images from a camera mounted on the robot, as well as proprioceptive sensory inputs such as the joint angles. These observations do not directly provide the poses of the objects in the scene, which is the typical assumption in simulated robotic environments – any such information must be extracted by the learning system.
+
+While in principle many RL frameworks can support learning from raw sensory inputs (Levine et al., 2016; Mnih et al., 2015; Schulman et al., 2015; Lillicrap et al., 2015), it is important to consider the practicalities of this approach. For instance, we can instantiate vision-based RL with policy gradient algorithms such as TRPO (Schulman et al., 2015) and PPO (Schulman et al., 2017), but these have high sample complexities which make them unsuited for real world robotic learning (Haarnoja et al., 2018a). In our work, we consider adopting the general framework of off-policy actor-critic reinforcement learning, using a version of the soft actor critic (SAC) algorithm described by (Haarnoja et al., 2018b). This algorithm effectively uses off-policy data and has been shown to learn some tasks directly from visual inputs. However, while SAC can learn directly from images, we find in our experiments that, as the task complexity increases, the efficiency of direct end-to-end learning (particularly without resets and with learned rewards) still degrades substantially. However, as we will discuss in Section 3, augmenting end-to-end learning with unsupervised representation learning substantially alleviates such challenges.
+
+# 2.2 REWARD FUNCTIONS WITHOUT REWARD ENGINEERING
+
+Vision-based RL algorithms, such as SAC, rely on a reward function being provided to the system, which is typically manually programmed by a user. While this can be simple to do in simulation by using ground truth state information, it is significantly harder to implement in uninstrumented real world environments. In the real world, the robot must obtain the reward signal itself from its own sensor readings. A few options for tackling this challenge have been discussed in prior work: design complete computer vision systems to detect objects and extract the reward signals (Devin et al., 2018; Nagabandi et al., 2019), engineer reward functions that use various task-specific heuristics to obtain rewards from pixels (Schenck & Fox, 2017; Kalashnikov et al., 2018), or instrument every environment (Chebotar et al., 2017). Many of these solutions are manual and tedious, and a more general approach is needed to scale real-world robotic learning gracefully.
+
+# 2.3 LEARNING WITHOUT RESETS
+
+While the components described in Section 2.1 and 2.2 are essential to building continuously learning RL systems in the real world, they have often been implemented with the assumption of episodic learning Fu et al. (2018); Haarnoja et al. (2018b). However, natural open-world settings do not provide any such reset mechanism, and in order to enable scalable and autonomous real-world learning we need systems that do not require an episodic formulation of the learning problem.
+
+To devise a system that requires minimal human engineering for providing rewards, we must use algorithms that are able to assign themselves rewards, using learned models that operate on the same raw sensory inputs as the policy. One candidate is for a user to specify intended behavior beforehand through examples of desired outcomes (i.e., images). The algorithm can then assign itself rewards based on a measure of how well it is accomplishing the specified goals, with no additional human supervision. This approach can scale well in principle, since it requires minimal human engineering, and goal images are easy to provide.
+
+In principle, algorithms such as SAC do not actually require episodic learning; however, in practice, most instantiations use explicit resets, even in simulation, and removing resets has resulted in failure to solve challenging tasks. In our experiments in Section 3 as well, we see that actor-critic methods applied na¨ıvely to the reset free setting do not learn the intended behaviors. Introducing visual observations and classifier based rewards exacerbates these challenges.
+
+We propose that these three components – vision-based RL with actor-critic algorithms, visionbased goal classifier for rewards, and reset-free learning – are the fundamental pieces that we need to build a real world robotic learning system. However, when we actually combine the individual components in Sections 3 and 6, we find that learning effective policies is quite challenging. We provide insight into these challenges in Section 3. Based on these insights, we propose simple but important changes in Section 4 to build a system, R3L , that can learn effectively and autonomously in the real world without human intervention.
+
+# 3 THE CHALLENGES OF REAL WORLD RL
+
+The system design outlined in Section 2 in principle gives us a complete system to perform real world reinforcement learning without instrumentation. However, when utilized for robotic learning problems, we find this basic design to be largely ineffective. To study this, we present results for a simulated robotic manipulation task that requires repositioning a free-floating objects with a threefingered robotic hand, shown in Fig 3. We use this task for our investigative analysis and show that the same insights extend to several other tasks, including real world tasks, in Section 6. The goal in this task is to reposition the object to a target pose from any initial pose in the arena. When the system is instantiated with visionbased soft actor-critic, rewards from goal images using VICE, and run without episodic resets, we see that the algorithm fails to make progress (Fig 4). Although it might appear that this setup fits within the assumptions of all of the components that are used, the complete system is ineffective. Which particular components of this problem make it so difficult?
+
+![](images/ce6143e57e0a18c5d2bd1944c725d2a94340cd83fa8b365692f4d164208de312.jpg)  
+Figure 3: Our object repositioning task. The goal is to move the object from any starting configuration to a particular goal position and orientation.
+
+To investigate this issue, we perform experiments investigating the combination of the three main ingredients: varying observation type (visual vs. low-dimensional state), reward structure (VICE vs. hand-defined rewards that utilize ground-truth object state), and the ability to reset (episodic resets vs. reset-free, non-episodic learning). We start by considering the training time reward under each combination of factors as shown in Fig 4, which reveals several trends. First, the results in Fig 4 show that learning with resets achieves high training time reward from both vision and state, while reset-free only achieves high training time reward with low-dimensional state. Second, we find that the policy is able to pass the threshold for training time reward in a non-episodic setting when learning from low-dimensional state, but it is not able to do the same using image observations. This suggests that combining the reset-free learning problem with visual observations makes it significantly more challenging.
+
+![](images/36fdd05da19a4d308ad35dd47c28e7b98a0467afadb84769d255d7f4310ee98f.jpg)  
+Figure 4: We report the approximate number of samples needed for a policy learned with a prior offpolicy RL algorithm $( S A C )$ to achieve average training performance of less than 0.15 in pose distance (defined in Appendix C.1.3) across 3 seeds on the repositioning task. We compare training performance after varying three axes: ground truth rewards vs. learned rewards, with vs. without episodic resets, low-level state vs. images as inputs. We observe learning without resets is harder than with resets and is much harder when combined with visual inputs.
+
+![](images/6d70464a38d5696118c55e6558ef0502b8ccea62ff9b88e773c2039834d86d3f.jpg)  
+Figure 5: We observe that when training reset free to reach a single goal, while the pose distance at training time is quite low, the pose errors obtained at test-time with the learned policy are very high. This indicates that while the object is getting close to the goal at training time, the policies being learned are still not effective.
+
+However, the table in Fig 4 paints an incomplete picture. These numbers are related to the performance of the policies at training time, not how effective the learned policies are when being evaluated. When we consider the test-time performance (Fig 5) of the policies that are learned under reset free conditions, we obtain a different set of conclusions. While learning from low-dimensional state in the reset free setting achieves high reward at training time, the test-time performance of the corresponding learned policies is very poor. This can likely be attributed to the fact that when the agent spends all its time stuck at the goal, and sees very little diversity of data in other parts of the state space, which significantly reduces the efficacy of the actual policies being learned. In a sense, the reset encodes some prior information about the task: it tells the policy about what types of states it might be required to succeed from at test time. Without this knowledge, performance is substantially worse. This makes it very challenging to learn policies with completely reset-free schemes, which has prompted prior work to consider schemes such as learning reset controllers (Eysenbach et al., 2018). As we discuss in the following section and in our experiments, these schemes are often insufficient for learning effective policies in the real world without any resets.
+
+# 4 A REAL-WORLD ROBOTIC REINFORCEMENT LEARNING SYSTEM
+
+To address the challenges identified in Section 3, we present two improvements, which we found to be essential for uninstrumented real-world training: randomized perturbation controllers and unsupervised representation learning. Incorporating these components into the system in Section 2 results in a system that can learn successfully in uninstrumented environments, as we will show in Section 6, and attains good performance both at training time and at test time.
+
+# 4.1 RANDOMIZED PERTURBATION CONTROLLER
+
+Prior works in addressing the reset free setting problem have considered converting the problem into a more standard episodic problem, by learning a “reset controller,” which is trained to reset the system to a particular initial state (Eysenbach et al., 2018; Han et al., 2015). This scheme has been thought to make the learning problem easier by reducing the variance of the resulting initial state distribution. However, as we will show in our experiments in Section 6, this still results in policies whose success depends heavily on a narrow range of initial states. Indeed, prior reset controller methods all reset to a single initial state (Eysenbach et al., 2018; Han et al., 2015).
+
+We take a different approach to learning in a reset-free setting. Rather than attributing the problem to the variance of the initial state distribution, we hypothesize that a major problem with reset-free learning is that the support of the distribution of states visited by the policy is too narrow, which makes the learning problem challenging and doesn’t allow the agent to learn how to perform the desired task from any state it might find itself in. In this view, the goal should not be to reduce the variance of the initial state distribution, but instead to increase it.
+
+To this end, we utilize what we call random perturbation controllers: controllers that introduce perturbations intermittently into the system through a policy that is trained to explore the environment. The standard actor $\pi ( a | s )$ is executed for $H$ time-steps, following which we executed the perturbation controller $\pi _ { p } ( a | s )$ for $H$ steps, and repeat. The policy $\pi$ is trained with the VICE-based rewards for reaching the desired goals, while the perturbation controller $\pi _ { p }$ is trained only with an intrinsic motivation objective that encourages visiting under-explored states. In our implementation, we use the random network distillation (RND) objective for training the perturbation controller (Burda et al., 2018), but any effective exploration method can be used for the same. This procedure is described in detail in Appendix A, and is evaluated on the tasks we consider in Fig 6. The perturbation controller ensures that the support of the training distribution grows and as a result the policies can learn the desired behavior much more effectively, as shown in Fig 7.
+
+# 4.2 GOAL CLASSIFIER
+
+To design a system that requires minimal human engineering for providing reward, we use a datadriven reward specification framework called variational inverse control with events (VICE) introduced by Fu et al. (2018). VICE learns rewards in a task-agnostic way: we provide the algorithm with success examples in the form of images where the task is accomplished, and learn a discriminator that is capable of distinguishing successes from failures. This discriminator can then be used to provide a learning signal to nudge the reinforcement learning agent towards success. This algorithm has been previously considered in the context of learning tasks from raw sensory observations in the real world by (Singh et al., 2019) but we show that it presents unique challenges when used in conjunction with learning without episodic resets. Details and specifics of the algorithms being considered are described in Appendix A and also discussed by Singh et al. (2019).
+
+# 4.3 UNSUPERVISED REPRESENTATION LEARNING
+
+The perturbation controller discussed above allows us to learn policies that can succeed at the task from a variety of starting states. However, learning from visual observations still present a challenge. Our experiments in Fig 4 show that learning without resets from low-dimensional states is comparatively easier. We therefore aim to convert the vision-based learning problem into one that more closely resembles state-based learning, by training a variational autoencoder (VAE, Kingma & Welling (2013)) and sharing the latent-variable representation across the actor and critic networks (refer to Appendix B for more details). Note that we use a VAE as an instantiation of representation learning techniques that works well in the domains we considered, but other more sophisticated density models proposed in prior work may also be substituted in place of the VAE (Lee et al., 2019; Hjelm et al., 2019; Anand et al., 2019).
+
+While several works have also sought to incorporate unsupervised learning into reinforcement learning to make learning from images easier (Nair et al., 2018; Lee et al., 2019), we note that this becomes especially critical in the vision-based, reset-free setting, as motivated by the experiments in Section 3, which indicate that it is precisely this combination of factors – vision and no resets – that presents the most difficult learning problem. Therefore, although the particular solution we use in our system has been studied in prior work, it is brought to bare to address a challenge that arises in real-world learning that we believe has not been explored in prior studies.
+
+These two improvements – the perturbation controller and unsupervised learning – combined with the general system described above, give us a complete practical system for real world reinforcement learning. The overall method uses soft-actor critic for learning with visual observations and classifier based rewards with VICE, introduces auxiliary reconstruction objectives or pretrains encoders for unsupervised representation learning, and uses a perturbation controller during training to ensure that the support of visited states grows sufficiently. We term this full system for real-world robotic reinforcement learning R3L. Further implementation details can be found in Appendix A.
+
+# 5 RELATED WORK
+
+The primary contribution of this work is to propose a paradigm for continual instrumentation-free real world robotic learning and a practical instantiation of such a system. A number of prior works have studied reinforcement learning in the real world for acquiring robotic skills (Levine et al., 2016; Kumar et al., 2016; Gu et al., 2017; Kalashnikov et al., 2018; Haarnoja et al., 2018b; Finn & Levine, 2016; Zhu et al., 2019; Nagabandi et al., 2019). Much of the focus in prior work has been on improving the efficiency and generalization of RL algorithms to make real-world training feasible, or else on utilizing simulation and transferring policies into the real world (Tzeng et al., 2015; Tobin et al., 2017; Peng et al., 2017; Clavera et al., 2018; Kang et al., 2019). The simulation-based methods typically require considerable effort in terms of both simulator design and overcoming the distributional shift between simulated and real-experience, while prior real-world training methods typically require additional instrumentation for either reward function evaluation (Levine & Koltun, 2013) or resetting between trials (Gu et al., 2017; Chebotar et al., 2017; Zhu et al., 2019), or both. In contrast, our work is primarily focused on lifting these requirements, rather than devising more efficient RL methods. We show that removing the need for instrumentation (i.e., for reward evaluation and resets) introduces additional challenges, which in turn require a careful set of design choices. Our complete R3L method is able to learn completely autonomously, without manual resets or reward design.
+
+A key component of our system involves learning from raw visual inputs. This has proven to be difficult for policy gradient style algorithms (Pinto et al., 2017a) due to challenging representation learning problems. This has been made easier in simulated domains by using modified objectives, such as auxiliary losses (Jaderberg et al., 2016), or by using more efficient algorithms (Haarnoja et al., 2018a). We show that reinforcement learning on raw visual input, while possible in standard RL settings, becomes significantly more challenging when considered in conjunction with nonepisodic, reset-free scenarios.
+
+Reward function design is crucial for any RL system, and is non-trivial to provide in the real world. Prior works have considered instrumenting the environment with additional sensors to evaluate rewards (Gu et al., 2017; Chebotar et al., 2017; Zhu et al., 2019), which is a highly manual process, using demonstrations, which require manual effort to collect (Vecerik et al., 2017; Ng & Russell, 2000; Liu et al., 2018), or using interactive supervision from the user (Christiano et al., 2017). In this work, we leverage the algorithm introduced by Fu et al. (2018) to assign rewards based on the likelihood of a goal classifier. While prior work also applied this method to robotic tasks (Singh et al., 2019), this was done in a setting where manual resets were provided by hand, while we demonstrate that we can use learned rewards in a fully uninstrumented, reset-free setup.
+
+Learning without resets has been considered in prior works (Eysenbach et al., 2018; Han et al., 2015), although in different contexts – safe learning and learning compound controllers, respectively. Eysenbach et al. (2018) provide an algorithm to learn a reset controller with the goal of ensuring safe operation, but makes several assumptions that make it difficult to use in the real world: it assumes access to a ground truth reward function, it assumes access to an oracle function that can detect if an attempted reset by the reset policy was successful or not, and it assumes the ability to perform manual resets if the reset policy fails a certain number of times. In contrast, we propose an algorithm that allows for fully automated reinforcement learning in the real world. We compare to an ablation of our method that uses a reset controller similar to Eysenbach et al. (2018), and show that our method performs substantially better. Our perturbation controller also resembles the adversarial RL setup Pinto et al. (2017b); Sukhbaatar et al. (2018). However, while these prior methods explicitly aim to train policies that are robust to perturbations Pinto et al. (2017b) or explore effectively Sukhbaatar et al. (2018), we are concerned with learning without access to resets.
+
+While this line of work has connections to developmental robotics (Lungarella et al., 2003; Asada et al., 2009) and its subfields, such as continual (Lesort et al., 2019) and lifelong (Thrun & Mitchell, 1995) learning, the goal of our work is to handle the practicalities of enabling reinforcement learning systems to learn in the real world without instrumentation or interruption, even for a single task setting. Though our work does not directly study continual lifelong learning, nor all facets of developmental robotics, it relates to continual learning (Lesort et al., 2019), intrinsic motivation (Schmidhuber, 2006) and sensory-motor development involving proprioceptive manipulation (Stoica, 2001).
+
+# 6 EXPERIMENTS
+
+In our experimental evaluation, we study how well the R3L system, described in Sections 2 and 4, can learn under realistic settings – visual observations, no hand-specified rewards, and no resets. We consider the following hypotheses:
+
+1. Can we use R3L to learn complex robotic manipulation tasks without instrumentation? Does this system learn skills in both simulation and the real world?   
+2. Do the solutions proposed in Section 4 actually enable R3L to perform tasks without instrumentation that would not have been otherwise possible?
+
+# 6.1 EXPERIMENTAL SETUP
+
+We consider the task of dexterous manipulation with a three fingered robotic hand, the D’Claw (Zhu et al., 2019; Ahn et al., 2019), on a number of simulated and real world tasks. These tasks involve complex coordination of three fingers with 3 DoF each in order to manipulate objects. Prior works that used this robot utilized explicit resets and low-dimensional true state observations, while we consider settings with visual observations, no hand-specified rewards, and no resets.
+
+![](images/824a594d6c5e30f9a580b874265c711fa246312d33dda84adff8239823f14ea9.jpg)  
+Figure 6: Visualizations of the goal configurations of the simulated and real world tasks being considered. From left to right we depict valve rotation, bead manipulation and free object repositioning in simulation, as well as valve rotation and bead manipulation manipulation in the real world.
+
+The tasks in our experiments are shown in Fig 6: manipulating beads on an abacus row, valve rotation, and free object repositioning. These tasks represent a wide class of problems that robots might encounter in the real world. For each task, we consider the problem of reaching the depicted goal configuration: moving the abacus beads to a particular position, rotating the valve to a particular angle, and repositioning the free object to a particular goal position. For each task, policies are evaluated from a wide selection of initial configurations. Additional details about the tasks and evaluation procedures are provided in Appendix C. Videos and additional details can be found at https://sites.google.com/view/realworld-rl/
+
+# 6.2 LEARNING IN SIMULATION WITHOUT INSTRUMENTATION
+
+We compare our entire proposed system implementation (Section 4) with a number of baselines and ablations. Importantly, all methods must operate under the same assumptions: none of the algorithms have access to system instrumentation for state estimation, reward specification, or episodic resets. Firstly, we compare the performance of R3L to a system which uses SAC for vision-based RL from raw pixels, VICE for providing rewards and running reset-free (denoted as “VICE”). This corresponds to the vanilla version of R3L (Section 2), with none of the proposed insights and changes. We then compare with prior reset-free RL algorithms (Eysenbach et al., 2018) that explicitly learn a reset controller to alternate goals in the state space (“Reset Controller $^ +$ VAE”). Lastly, we compare algorithm performance with two ablations: running R3L without the perturbation controller (“VICE $+ \ \mathrm { V A E } ^ { \prime \prime } )$ ) and without the unsupervised learning (“R3L w/o VAE”). This highlights the significance of each of the components of R3L .
+
+From the experimental results in Fig 7, it is clear that R3L is able to reach the best performance across tasks, while none of the other methods are able to solve all of the tasks. Different prior methods and ablations fail for different reasons: (1) methods without the reconstruction objective struggle at parsing the high-dimensional input and are unable to solve the harder task; (2) methods without the perturbation controller are ineffective at learning how to reach the goal from novel initialization positions for the more challenging object repositioning tasks, as discussed in Section 4.
+
+![](images/16689f39419185c7ebec5a0ee2e8964bd3a1d3329c8a98a6d5098b217637ce35.jpg)  
+Figure 7: Quantitative evaluation of performance in simulation for bead manipulation, valve rotation and free object repositioning (left to right) run with 10 random seeds. The error bars show $9 5 \%$ bootstrap confidence intervals for average performance. While other variants are sufficient to get good evaluation performance on easier tasks, harder tasks like free object repositioning require random perturbations and unsupervised representation learning to learn skills reset-free. See Appendix C.1 for details of evaluation procedures.
+
+We note that an explicit reset controller, which can be viewed as a softer version of our perturbation controller with goal-directedness, learns to solve the easier tasks due to the reset controller encouraging exploration of the state space. In our experiments for free object repositioning, performance was reported across 3 choices of reset states. The high variance in evaluation performance indicates that the performance of such a controller (or a goal conditioned variant of it) is highly sensitive to the choice of reset states. A poor choice of reset states, such as two that are very close together, may yield poor exploration leading to performance similar to the single goal VICE baseline. Furthermore, the choice of reset states is highly task dependent and it is often not clear what choice of goals will yield the best performance. On the contrary, our method does not require such task-specific knowledge and uses random perturbations to reset while training without any explicit reset states: this allows for a robust, instrumentation-free controller while also ensuring fast convergence.
+
+# 6.3 LEARNING IN THE REAL WORLD WITHOUT INSTRUMENTATION
+
+Since the aim of R3L is to enable uninstrumented training in the real world, we next evaluate our method on a real-world robotic system, providing evidence that our insights generalize to the real world without any instrumentation. After providing the initial outcome examples for learning the reward function with VICE, we leave the robot unattended, and the algorithm learns the desired behavior through interaction. The experiments are performed on the D’Claw robotic hand with an RGB camera as the only sensory input. Intermediate policies are saved at regular intervals, and evaluations of all policies is performed after training has completed. For valve rotation, we declare an evaluation rollout a success if the final orientation is within within $1 5 ^ { \circ }$ of the goal. For bead manipulation, we declare success if all the beads are within $2 \mathrm { c m }$ of the goal state. Fig 8 compares the performance of our method without supervised learning (“R3L w/o VAE”) in the real world against a baseline that uses SAC for vision-based RL from raw pixels, VICE for providing rewards, and running reset-free (denoted as “VICE”). We see that our method learns policies that succeed from nearly all the initial configurations, whereas VICE alone fails from most initial configurations. Fig 9 depicts sample evaluation rollouts of the policies learned using our method. For further details about real world experiments see Appendix C.2.
+
+# 7 DISCUSSION
+
+We presented the design and instantiation of a system for real world reinforcement learning. We identify and investigate the various ingredients required for such a system to scale gracefully with minimal human engineering and supervision. We show that this system must be able to learn from raw sensory observations, learn from very easily specified reward functions without reward engineering, and learn without any episodic resets. We describe the basic elements that are required to construct such a system, and identify unexpected learning challenges that arise from interplay of these elements. We propose simple and scalable fixes to these challenges through introducing unsupervised representation learning and a randomized perturbation controller.
+
+![](images/dad274348320cb8430ec089aa0a9ec78b973c42a0a84d8f326005b3a2f61f71a.jpg)  
+Figure 8: Quantitative evaluation of performance in real-world for valve rotation and bead manipulation. Policies trained with perturbation controllers have effectively learned behaviors after 17 and 5 hours of training, respectively. For more fine-grained reporting of results see Figs 13-16.
+
+![](images/aa753edabd7a309dd77434d6d462b21f9d280973740211557ec78a0770a770db.jpg)  
+Figure 9: Evaluation rollouts of R3L on the real world tasks for policies trained without instrumentation. Successful evaluation rollouts for bead manipulation (top) and valve rotation (bottom) are shown here.
+
+The ability to train robots directly in the real world with minimal instrumentation opens a number of exciting avenues for future research. Robots that can learn unattended, without resets or handdesigned reward functions, can in principle collect very large amounts of experience autonomously, which may enable very broad generalization in the future. However, there are also a number of additional challenges, including sample complexity, optimization and exploration difficulties on more complex tasks, safe operation, communication latency, sensing and actuation noise, and so forth, all of which would need to be addressed in future work in order to enable truly scalable real-world robotic learning.
+
+# ACKNOWLEDGMENTS
+
+This research is supported by the Office of Naval Research, the National Science Foundation under IIS-1651843 and IIS-1700696, Google, NVIDIA, and Amazon. The authors would like to thank Ignasi Clavera, Gregory Kahn, Coline Devin, Benjamin Eysenbach, Aviral Kumar, Anusha Nagabandi, Marvin Zhang, Ashvin Nair and several others in the RAIL Lab for helpful discussions and feedback.
+
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+
+A ALGORITHM DETAILS
+
+# Algorithm 1 Real-World Robotic Reinforcement Learning (R3L)
+
+<table><tr><td colspan="2">procedure R3L</td></tr><tr><td>1: 2:</td><td>N ← number of training epochs</td></tr><tr><td>3:</td><td>H ← trajectory length (horizon)</td></tr><tr><td>4:</td><td>nvICE ← number of VICE classifier training iterations per epoch</td></tr><tr><td>5:</td><td>Initialize forward and perturbing policies To,π1</td></tr><tr><td>6:</td><td>Obtain goal states sE and initialize as a goal pool G</td></tr><tr><td>7:</td><td>Initialize RND target and predictor networks f(s), f(s)</td></tr><tr><td>8:</td><td>Initialize VICE reward classifier rvICE(s)</td></tr><tr><td>9:</td><td>Initialize replay buffer D</td></tr><tr><td>10:</td><td>Collect initial exploration data and add to D</td></tr><tr><td>11:</td><td>for i= 1 to 2N do</td></tr><tr><td>12:</td><td>k←i%2</td></tr><tr><td>13:</td><td>for t = 1 to H do</td></tr><tr><td>14:</td><td>Sample at ~ Tk(St)</td></tr><tr><td>15:</td><td>Sample St+1 ~ p(St+1|St,at)</td></tr><tr><td>16:</td><td>if k== O then</td></tr><tr><td>17:</td><td>rt(St)= CVICE * rVICE(St)+ CRND * rRND(St)</td></tr><tr><td>18:</td><td>else if k==1 then</td></tr><tr><td>19:</td><td>rt(St)= rRND(St)</td></tr><tr><td>20:</td><td>end if</td></tr><tr><td>21:</td><td>Sample batch from D</td></tr><tr><td>22:</td><td>Update πk with batch according to SAC</td></tr><tr><td>23:</td><td>Update RND predictor network with batch</td></tr><tr><td>24:</td><td>Update running estimate of standard deviations of classifier and RND reward</td></tr><tr><td>25:</td><td>end for</td></tr><tr><td>26:</td><td>Add experience to the replay buffer with D ← DU Ti</td></tr><tr><td>27:</td><td>Sample an equal number of goal examples from G and negative examples from D</td></tr><tr><td>28:</td><td>for t= 1 to nvice do</td></tr><tr><td>29:</td><td>Train the VICE classifier on this batch with binary labels</td></tr><tr><td>30:</td><td>end for</td></tr><tr><td>31:</td><td>end for</td></tr><tr><td>32: end procedure</td><td></td></tr></table>
+
+# B TRAINING DETAILS
+
+B.0.1 HYPERPARAMETERS   
+
+<table><tr><td rowspan=1 colspan=1>General</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Standard deviation up-date coefficientImage Sizes</td><td rowspan=1 colspan=1>0.99[(16, 16,3), (32,32,3), (64, 64, 3)]</td></tr><tr><td rowspan=1 colspan=1>SAC</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learning RateyBatch SizeConvnet FiltersStrideKernel SizesPoolingActor/Critic FC Lay-ers</td><td rowspan=1 colspan=1>3e-40.99256[(64, 64,64), (16,32, 64)](2,2)(3,3)[MaxPool2D,None][(512,512), (256, 256, 256)]</td></tr><tr><td rowspan=1 colspan=1>VICE</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>nvICEBatch SizeLearning RateMixup αConvnet FiltersStrideKernel SizesPoolingFC Layers</td><td rowspan=1 colspan=1>[1,5,10]1281e-4Uniform(0, 1)[(64, 64,64), (16, 32, 64)](2,2)(3,3)[MaxPool2D,None][(512,512), (256,256,256)]</td></tr><tr><td rowspan=1 colspan=1>RND</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learning RateBatch SizeConvnet FiltersStrideKernel SizesPoolingFC Layers</td><td rowspan=1 colspan=1>3e-4256(16, 32, 64)(2,2)(3,3)[MaxPool2D, None][(512,512), (256,256,256)]</td></tr><tr><td rowspan=1 colspan=1>VAE</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learning RateBatch SizeEncoder (Convnet)FiltersLatent DimensionβStrideKernel SizesPooling</td><td rowspan=1 colspan=1>1e-4256(64, 64, 32)[8, 16,32, 64][1e-3, 0.1, 0.5, 1, 10](2,2)(3,3)[MaxPool2D, None]</td></tr></table>
+
+The ranges of values listed above represent the hyperparameters we searched over, and the bolded values are what we use in the Section 6 experiments.
+
+# B.0.2 VICE
+
+We use a variant of VICE which defines the reward as the logits of the classifier, notably omitting the $- l o g ( \pi ( a | s ) )$ term. We also regularize our classifier with mixup (Zhang et al., 2018). We train all of our experiments using 200 goal images, which takes under an hour to collect in the real world for each task.
+
+# B.0.3 RANDOM NETWORK DISTILLATION (RND)
+
+We found it important to normalize the predictor errors, just as (Burda et al., 2018) did.
+
+# B.0.4 VAE
+
+We train a standard beta-VAE to maximize the evidence lower bound, given by:
+
+$$
+\mathbb { E } _ { z \sim q _ { \phi } ( z | x ) } [ p _ { \theta } ( x | z ) ] - \beta D _ { \mathrm { K L } } ( q _ { \phi } ( z | x ) | | p _ { \theta } ( z ) )
+$$
+
+To collect training data, we sampled random states in the observation space. In the real world, this sampling can be replaced with training an exploratory policy (i.e. using the RND reward as the policy’s only objective). The learned weights of the encoder of the VAE are frozen, and the latent input is used to train the policy for reset-free RL.
+
+# C TASK DETAILS
+
+# C.1 SIMULATED TASKS
+
+We evaluated our system across three tasks in simulation: bead manipulation, valve rotation, free object repositioning.
+
+# C.1.1 BEAD MANIPULATION
+
+The bead manipulation task involves an abacus rod with four beads that can slide freely. The goal is to position two beads on each end from any initial configuration of beads. This can take the form of sliding one bead over (if three beads start on one side), two beads over (if all four beads start on one side), splitting beads apart (all four beads start in the middle), or some intermediate combination of those. The true reward is defined as the mean goal distance of all four beads. Policies are evaluated starting from the 8 initial configurations depicted in Fig 10. Evaluation performance reported in Section 6 for this task is defined as the final reward averaged across the 8 evaluation rollouts.
+
+![](images/080c6495f909ccd2083348ee2a02e0be2c3e68b87486d16c7fe5e20650758e62.jpg)  
+Figure 10: These are the 8 initial positions used for evaluating the performance of the bead manipulation policy. The goal configuration (which is also an initial evaluation position) is highlighted in yellow.
+
+# C.1.2 VALVE ROTATION
+
+The claw is positioned above a three pronged valve ( $1 5 \mathrm { c m }$ in diameter). The objective is to turn the valve to a given orientation from any initial orientation. The ”true reward” is $\dot { r } = - \log ( | \theta _ { s t a t e } -$ $\theta _ { g o a l } | .$ ). Policies are evaluated starting from the 8 initial configurations depicted in Fig 11. Evaluation performance reported in Section 6 for this task is defined as the final orientation distance averaged across the 8 evaluation rollouts.
+
+# C.1.3 FREE OBJECT REPOSITIONING
+
+The claw is positioned atop a free $6 ~ \mathrm { D o F } )$ three pronged object ( $1 5 \mathrm { c m }$ in diameter), which can translate and rotate within a $3 0 \mathrm { c m } x 3 0 \mathrm { c m }$ box. The goal is specified by a xy-position as well as a $\mathbf { Z }$ -angle, where the xy-plane is the plane of the arena. The true reward is defined as the weighted sum of the angular and translational distances, $r = - 2 \log ( | | [ x _ { s t a t e } , y _ { s t a t e } ] - [ x _ { g o a l } , y _ { g o a l } ] | | _ { 2 } ) -$ $\log ( | \theta _ { s t a t e } - \theta _ { g o a l } | )$ . In our experiments, $( x , y , \theta ) _ { g o a l } = ( 0 , 0 , - \frac { \pi } { 2 } )$ , where the origin is centered in the arena. Policies are evaluated starting from the 15 initial configurations depicted in Figure 12. Evaluation performance reported in Section 6 for this task is defined as the final pose distance ( ||[xf inal,yf inal]−[xgoal,ygoal]||2 + |θf inal−θgoal| ) averaged across the 15 evaluation rollouts.
+
+![](images/2fc0e811e81e0386a99cca981233798437bab0e05c9f8cf66ad0402316a68fe0.jpg)  
+Figure 11: These are the 8 initial positions used for evaluating the performance of the valve rotation policy. The goal configuration (which is also an initial evaluation position) is highlighted in yellow.
+
+In our reset controller experiments, we averaged evaluation performance over three different choices of reset states, where the first reset state is always the goal:
+
+1. $( x , y , \theta ) _ { r e s e t , 1 } = ( x , y , \theta ) _ { g o a l } , ( x , y , \theta ) _ { r e s e t , 2 } = ( 0 . 0 5 , 0 . 0 5 , \frac { \pi } { 2 } )$   
+2. $\begin{array} { r } { ( x , y , \theta ) _ { r e s e t , 1 } = ( x , y , \theta ) _ { g o a l } , ( x , y , \theta ) _ { r e s e t , 2 } = ( 0 , 0 , - \frac { \pi } { 6 } ) } \end{array}$   
+3. $( x , y , \theta ) _ { r e s e t , 1 } = ( x , y , \theta ) _ { g o a l } , ( x , y , \theta ) _ { r e s e t , 2 } = ( - 0 . 0 4 , - 0 . 0 4 , - \frac { \pi } { 2 } )$
+
+![](images/1b1a729609dbfdd4bff369f64e48f133a4dc620e897c2ab6a14904cca1c9dbe4.jpg)  
+Figure 12: These are the 15 initial positions used for evaluating the performance of the free object repositioning policy. The goal configuration $( x , y , \theta ) _ { g o a l }$ which is also an initial evaluation position is highlighted in yellow.
+
+# C.2 REAL WORLD TASKS
+
+For each setup we use an RGB camera to get images. We execute actions on the DClaw at $1 0 \mathrm { H z }$ . In order to operate at such a high frequency while also training from images we sample and train asynchronously, but limit training to not exceed two gradient steps per transition sampled in the real world. Since direct performance metrics cannot be measured during training due to the lack of object instrumentation, evaluations of performance are done post-training.
+
+# C.2.1 VALVE ROTATION
+
+The task is identical to the one in simulation. Evaluations were done post-training. An evaluation trajectory was defined as a success if at the last step, the valve was within 15 degrees of the goal. Each policy was evaluated over 8 rollouts, with initial configurations evenly spaced out between at increments of 45 degrees. Results are reported in Figures 13, 14.
+
+# C.2.2 BEAD MANIPULATION
+
+The rod is $2 2 \mathrm { c m }$ in length, and each bead measures $3 . 5 \mathrm { c m }$ in diameter. Evaluations were done post-training, using the following procedure: the environment was manually reset to each of the 8 specified configurations shown in Figures 15 and 16 (which cover a full range of the state space) at the start of each evaluation rollout. An evaluation trajectory was defined as a success if at the last time step, all beads were within $2 \mathrm { c m }$ of their goal positions. Performance was evaluated at around 20 hours, at which point the policy achieved greater than $80 \%$ success on the 10 evaluation rollouts (a random policy achieved a success rate of $10 \%$ ). Results are reported in Figs 15, 16.
+
+![](images/7a90b025f1eac655661ca465e4c18b4e9bb25b2a296dbae63b4b880663793aff.jpg)  
+Figure 13: These are the results of the evaluation rollouts on the valve rotation task in the real world using our method (without the VAE). Trained policies were saved at regular intervals and evaluated post-training. Each row is a different policy, and each column an evaluation rollout from a different initial configuration. The goal is highlighted in yellow. Our method is able to achieve high success rates after 5 hours of training.
+
+![](images/124c852ede375572bcb431d4c7898f2ef36299c9da3133ec70dbeee81eee744d.jpg)  
+Figure 14: These are the results of evaluation rollouts on the valve rotation task in the real world using the VICE single goal baseline. The policies fail to evaluate well, especially from initial positions far from the goal position.
+
+![](images/30f66949f6e9e5f3f38c8a70eb2082962ea087319babcbeafcf552e09b61101e.jpg)  
+Figure 15: These are the results of the evaluation rollouts on the valve rotation task in the real world using our method (without the VAE). Trained policies were saved at regular intervals and evaluated post-training. Each row is a different policy, and each column an evaluation rollout from a different initial configuration. The goal is highlighted in yellow. Our method is able to achieve high success rates after 17 hours of training.
+
+![](images/3f8949a077ed83c173449febf904e9b7fa2f6c6fc40c84db953357f2ba8c7515.jpg)  
+Figure 16: These are the results of evaluation rollouts on the valve rotation task in the real world using the VICE single goal baseline. The policies fail to evaluate consistently, except when the initial configuration matches the goal configuration.

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+        "type": "text",
+        "text": "THE INGREDIENTS OF REAL-WORLD ROBOTIC REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+            146
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Henry $\\mathbf { L h u ^ { * 1 } }$ , Justin $\\mathbf { V } \\mathbf { u } ^ { * 1 }$ , Abhishek Gupta∗1, Dhruv Shah1, Kristian Hartikainen2, Avi Singh1, Vikash Kumar3, Sergey Levine1 1 University of California, Berkeley 2 University of Oxford 3 University of Washington ",
+        "bbox": [
+            184,
+            167,
+            774,
+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            251,
+            544,
+            265
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The success of reinforcement learning for real world robotics has been, in many cases limited to instrumented laboratory scenarios, often requiring arduous human effort and oversight to enable continuous learning. In this work, we discuss the elements that are needed for a robotic learning system that can continually and autonomously improve with data collected in the real world. We propose a particular instantiation of such a system, using dexterous manipulation as our case study. Subsequently, we investigate a number of challenges that come up when learning without instrumentation. In such settings, learning must be feasible without manually designed resets, using only on-board perception, and without hand-engineered reward functions. We propose simple and scalable solutions to these challenges, and then demonstrate the efficacy of our proposed system on a set of dexterous robotic manipulation tasks, providing an in-depth analysis of the challenges associated with this learning paradigm. We demonstrate that our complete system can learn without any human intervention, acquiring a variety of vision-based skills with a real-world three-fingered hand. Results and videos can be found at https://sites.google.com/view/realworld-rl/. ",
+        "bbox": [
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+            282,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            336,
+            545
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning (RL) can in principle enable autonomous systems, such as robots, to acquire a large repertoire of skills automatically. Perhaps even more importantly, reinforcement learning can enable such systems to continuously improve the proficiency of their skills from experience. However, realizing this in reality has proven challenging: even with reinforcement learning methods that can acquire complex behaviors from high-dimensional low-level observations, such as images, the assumptions of the reinforcement learning problem setting do not fit cleanly into the constraints of the real world. For this reason, most successful robotic learning experiments have employed various kinds of environmental instrumentation in order to define reward functions, reset between trials, and obtain ground truth state (Levine et al., 2016; Haarnoja et al., 2018a; Kumar et al., 2016; Andrychowicz et al., 2018; Zhu et al., 2019; Chebotar et al., 2016; Nagabandi et al., 2019; Gupta et al., 2016). In order to practically and scalably deploy autonomous learning systems that improve continuously through real-world operation, we must lift these limitations and design algorithms that can learn under the constraints of real-world environments, as illustrated in Figure 2. ",
+        "bbox": [
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+            561,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose that overcoming these challenges in a scalable way requires designing robotic systems that possess three capabilities: they are able to (1) learn from their own raw sensory inputs, (2) assign rewards to their own trials without hand-designed perception systems or instrumentation, and (3) learn continuously in non-episodic settings without requiring human intervention to manually reset the environment. A system with these capabilities can autonomously collect large amounts of real world data – typically crucial for effective generalization – without significant instrumentation in each training environment, an example of which is shown in Figure 1. If successful, this would lift a major constraint that stands between current reinforcement learning algorithms and the ability to learn scalable, generalizable, and robust real-world behaviors. Such a system would also bring us significantly closer to the goal of embodied learning-based systems that improve continuously through their own real-world experience. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/83c7b9a06c076f3cf373c41b31868ce93909c4644428dbe54f3993fd7de2ca93.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of our proposed instrumentation-free system requiring minimal human engineering. Human intervention is only required in the goal collection phase (1). The robot is left to train unattended (2) during the learning phase and can be evaluated from arbitrary initial states at the end of training (3). We show sample goal and intermediate images from the training process of a real hardware system "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having laid out these requirements, we propose a practical instantiation of such a learning system. While prior works have studied many of these issues in isolation, combining them into a complete real-world learning system presents a number of challenges, as we discuss in Section 3. We provide an empirical analysis of these issues, both in simulation and on a real-world robotic system, and propose a number of simple but effective solutions that together produce a complete robotic learning system. This system can autonomously learn from raw sensory inputs, learn reward functions from easily available supervision, and learn without manually designed reset mechanisms. We show that this system can learn dexterous robotic manipulation tasks in the real world, substantially outperforming ablations and prior work. ",
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+    {
+        "type": "text",
+        "text": "2 THE STRUCTURE OF A REAL-WORLD RL SYSTEM ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The standard reinforcement learning paradigm assumes that the controlled system is represented as a Markov decision process with a state space $s$ , action space $\\mathcal { A }$ , unknown transition dynamics $\\tau$ , unknown reward function $\\mathcal { R }$ , and a (typically) episodic initial state distribution $\\rho$ . The goal is to learn a policy that maximizes the expected sum of rewards via interactions with the environment. ",
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+        "type": "text",
+        "text": "Although this formalism is simple and concise, it does not capture all of the complexities of realworld robotic learning problems. If a robotic system is to learn continuously and autonomously in the real world, we must ensure that it can learn under the actual conditions that are imposed by the real world. To move from the idealized MDP formulation to the real world, we require a system that has the following properties. First, all of the information necessary for learning must be obtained from the robot’s own sensors. This includes information about the state and necessitates that the policy must be learned from high-dimensional and low-level sensory observations, such as camera images. Second, the robot must also obtain the reward signal itself from its own sensor readings. This is exceptionally difficult for all but the simplest tasks (e.g., reward functions that depend on interactions with specific objects require perceiving those objects explicitly). Third, we must be able to learn without access to episodic resets. A setup with explicit resets quickly becomes impractical in open-world settings, due to the requirement for significant human engineering of the environment, or direct human intervention during learning. While this list may not exhaustively enumerate all the components necessary for an effective real-world learning system, we posit that the properties listed here are fundamental to building such systems. ",
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+        "text": "While some of the components discussed above can be tackled in isolation by current algorithms, there are considerable challenges inherent to assembling all these components into a complete learning system for real world robotic learning. In the rest of this section, we outline the challenges associated with each component, then discuss the challenges associated with combining these components in Section 3, and then proceed to address these challenges in Section 4. ",
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+        "img_path": "images/c22c53be5af8d2715578d714c982847f8d4f9d12900678fdfd0e7bd3b1d9f31e.jpg",
+        "image_caption": [
+            "Figure 2: We draw a comparison between current real world learning systems which rely on instrumentation versus a system that learns in an environment more representative of the real world, free of instrumentation. While all three prior works utilize instrumentation for resets, state estimation, and reward, the motion capture system of Gupta et al. (2016), sensor attached to the door in Zhu et al. (2019), and auxiliary robot which picks up fallen balls in Nagabandi et al. (2019) are good examples of engineered state estimation, reward estimation, and reset mechanisms respectively. "
+        ],
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+        "type": "text",
+        "text": "2.1 LEARNING FROM RAW SENSORY INPUT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To enable learning without complex state estimation systems or environment instrumentation, we require our robotic systems to be able to learn from their own raw sensory observations. Typically, these sensory observations are raw camera images from a camera mounted on the robot, as well as proprioceptive sensory inputs such as the joint angles. These observations do not directly provide the poses of the objects in the scene, which is the typical assumption in simulated robotic environments – any such information must be extracted by the learning system. ",
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+        "text": "While in principle many RL frameworks can support learning from raw sensory inputs (Levine et al., 2016; Mnih et al., 2015; Schulman et al., 2015; Lillicrap et al., 2015), it is important to consider the practicalities of this approach. For instance, we can instantiate vision-based RL with policy gradient algorithms such as TRPO (Schulman et al., 2015) and PPO (Schulman et al., 2017), but these have high sample complexities which make them unsuited for real world robotic learning (Haarnoja et al., 2018a). In our work, we consider adopting the general framework of off-policy actor-critic reinforcement learning, using a version of the soft actor critic (SAC) algorithm described by (Haarnoja et al., 2018b). This algorithm effectively uses off-policy data and has been shown to learn some tasks directly from visual inputs. However, while SAC can learn directly from images, we find in our experiments that, as the task complexity increases, the efficiency of direct end-to-end learning (particularly without resets and with learned rewards) still degrades substantially. However, as we will discuss in Section 3, augmenting end-to-end learning with unsupervised representation learning substantially alleviates such challenges. ",
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+    {
+        "type": "text",
+        "text": "2.2 REWARD FUNCTIONS WITHOUT REWARD ENGINEERING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Vision-based RL algorithms, such as SAC, rely on a reward function being provided to the system, which is typically manually programmed by a user. While this can be simple to do in simulation by using ground truth state information, it is significantly harder to implement in uninstrumented real world environments. In the real world, the robot must obtain the reward signal itself from its own sensor readings. A few options for tackling this challenge have been discussed in prior work: design complete computer vision systems to detect objects and extract the reward signals (Devin et al., 2018; Nagabandi et al., 2019), engineer reward functions that use various task-specific heuristics to obtain rewards from pixels (Schenck & Fox, 2017; Kalashnikov et al., 2018), or instrument every environment (Chebotar et al., 2017). Many of these solutions are manual and tedious, and a more general approach is needed to scale real-world robotic learning gracefully. ",
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+    {
+        "type": "text",
+        "text": "2.3 LEARNING WITHOUT RESETS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While the components described in Section 2.1 and 2.2 are essential to building continuously learning RL systems in the real world, they have often been implemented with the assumption of episodic learning Fu et al. (2018); Haarnoja et al. (2018b). However, natural open-world settings do not provide any such reset mechanism, and in order to enable scalable and autonomous real-world learning we need systems that do not require an episodic formulation of the learning problem. ",
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+    {
+        "type": "text",
+        "text": "To devise a system that requires minimal human engineering for providing rewards, we must use algorithms that are able to assign themselves rewards, using learned models that operate on the same raw sensory inputs as the policy. One candidate is for a user to specify intended behavior beforehand through examples of desired outcomes (i.e., images). The algorithm can then assign itself rewards based on a measure of how well it is accomplishing the specified goals, with no additional human supervision. This approach can scale well in principle, since it requires minimal human engineering, and goal images are easy to provide. ",
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+    {
+        "type": "text",
+        "text": "In principle, algorithms such as SAC do not actually require episodic learning; however, in practice, most instantiations use explicit resets, even in simulation, and removing resets has resulted in failure to solve challenging tasks. In our experiments in Section 3 as well, we see that actor-critic methods applied na¨ıvely to the reset free setting do not learn the intended behaviors. Introducing visual observations and classifier based rewards exacerbates these challenges. ",
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+        "type": "text",
+        "text": "We propose that these three components – vision-based RL with actor-critic algorithms, visionbased goal classifier for rewards, and reset-free learning – are the fundamental pieces that we need to build a real world robotic learning system. However, when we actually combine the individual components in Sections 3 and 6, we find that learning effective policies is quite challenging. We provide insight into these challenges in Section 3. Based on these insights, we propose simple but important changes in Section 4 to build a system, R3L , that can learn effectively and autonomously in the real world without human intervention. ",
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+        "type": "text",
+        "text": "3 THE CHALLENGES OF REAL WORLD RL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The system design outlined in Section 2 in principle gives us a complete system to perform real world reinforcement learning without instrumentation. However, when utilized for robotic learning problems, we find this basic design to be largely ineffective. To study this, we present results for a simulated robotic manipulation task that requires repositioning a free-floating objects with a threefingered robotic hand, shown in Fig 3. We use this task for our investigative analysis and show that the same insights extend to several other tasks, including real world tasks, in Section 6. The goal in this task is to reposition the object to a target pose from any initial pose in the arena. When the system is instantiated with visionbased soft actor-critic, rewards from goal images using VICE, and run without episodic resets, we see that the algorithm fails to make progress (Fig 4). Although it might appear that this setup fits within the assumptions of all of the components that are used, the complete system is ineffective. Which particular components of this problem make it so difficult? ",
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+        "image_caption": [
+            "Figure 3: Our object repositioning task. The goal is to move the object from any starting configuration to a particular goal position and orientation. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "To investigate this issue, we perform experiments investigating the combination of the three main ingredients: varying observation type (visual vs. low-dimensional state), reward structure (VICE vs. hand-defined rewards that utilize ground-truth object state), and the ability to reset (episodic resets vs. reset-free, non-episodic learning). We start by considering the training time reward under each combination of factors as shown in Fig 4, which reveals several trends. First, the results in Fig 4 show that learning with resets achieves high training time reward from both vision and state, while reset-free only achieves high training time reward with low-dimensional state. Second, we find that the policy is able to pass the threshold for training time reward in a non-episodic setting when learning from low-dimensional state, but it is not able to do the same using image observations. This suggests that combining the reset-free learning problem with visual observations makes it significantly more challenging. ",
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+        "image_caption": [
+            "Figure 4: We report the approximate number of samples needed for a policy learned with a prior offpolicy RL algorithm $( S A C )$ to achieve average training performance of less than 0.15 in pose distance (defined in Appendix C.1.3) across 3 seeds on the repositioning task. We compare training performance after varying three axes: ground truth rewards vs. learned rewards, with vs. without episodic resets, low-level state vs. images as inputs. We observe learning without resets is harder than with resets and is much harder when combined with visual inputs. "
+        ],
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+    {
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+        "image_caption": [
+            "Figure 5: We observe that when training reset free to reach a single goal, while the pose distance at training time is quite low, the pose errors obtained at test-time with the learned policy are very high. This indicates that while the object is getting close to the goal at training time, the policies being learned are still not effective. "
+        ],
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+    {
+        "type": "text",
+        "text": "However, the table in Fig 4 paints an incomplete picture. These numbers are related to the performance of the policies at training time, not how effective the learned policies are when being evaluated. When we consider the test-time performance (Fig 5) of the policies that are learned under reset free conditions, we obtain a different set of conclusions. While learning from low-dimensional state in the reset free setting achieves high reward at training time, the test-time performance of the corresponding learned policies is very poor. This can likely be attributed to the fact that when the agent spends all its time stuck at the goal, and sees very little diversity of data in other parts of the state space, which significantly reduces the efficacy of the actual policies being learned. In a sense, the reset encodes some prior information about the task: it tells the policy about what types of states it might be required to succeed from at test time. Without this knowledge, performance is substantially worse. This makes it very challenging to learn policies with completely reset-free schemes, which has prompted prior work to consider schemes such as learning reset controllers (Eysenbach et al., 2018). As we discuss in the following section and in our experiments, these schemes are often insufficient for learning effective policies in the real world without any resets. ",
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+    {
+        "type": "text",
+        "text": "4 A REAL-WORLD ROBOTIC REINFORCEMENT LEARNING SYSTEM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To address the challenges identified in Section 3, we present two improvements, which we found to be essential for uninstrumented real-world training: randomized perturbation controllers and unsupervised representation learning. Incorporating these components into the system in Section 2 results in a system that can learn successfully in uninstrumented environments, as we will show in Section 6, and attains good performance both at training time and at test time. ",
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+        "type": "text",
+        "text": "4.1 RANDOMIZED PERTURBATION CONTROLLER ",
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+        "type": "text",
+        "text": "Prior works in addressing the reset free setting problem have considered converting the problem into a more standard episodic problem, by learning a “reset controller,” which is trained to reset the system to a particular initial state (Eysenbach et al., 2018; Han et al., 2015). This scheme has been thought to make the learning problem easier by reducing the variance of the resulting initial state distribution. However, as we will show in our experiments in Section 6, this still results in policies whose success depends heavily on a narrow range of initial states. Indeed, prior reset controller methods all reset to a single initial state (Eysenbach et al., 2018; Han et al., 2015). ",
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+        "type": "text",
+        "text": "We take a different approach to learning in a reset-free setting. Rather than attributing the problem to the variance of the initial state distribution, we hypothesize that a major problem with reset-free learning is that the support of the distribution of states visited by the policy is too narrow, which makes the learning problem challenging and doesn’t allow the agent to learn how to perform the desired task from any state it might find itself in. In this view, the goal should not be to reduce the variance of the initial state distribution, but instead to increase it. ",
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+        "type": "text",
+        "text": "To this end, we utilize what we call random perturbation controllers: controllers that introduce perturbations intermittently into the system through a policy that is trained to explore the environment. The standard actor $\\pi ( a | s )$ is executed for $H$ time-steps, following which we executed the perturbation controller $\\pi _ { p } ( a | s )$ for $H$ steps, and repeat. The policy $\\pi$ is trained with the VICE-based rewards for reaching the desired goals, while the perturbation controller $\\pi _ { p }$ is trained only with an intrinsic motivation objective that encourages visiting under-explored states. In our implementation, we use the random network distillation (RND) objective for training the perturbation controller (Burda et al., 2018), but any effective exploration method can be used for the same. This procedure is described in detail in Appendix A, and is evaluated on the tasks we consider in Fig 6. The perturbation controller ensures that the support of the training distribution grows and as a result the policies can learn the desired behavior much more effectively, as shown in Fig 7. ",
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+    {
+        "type": "text",
+        "text": "4.2 GOAL CLASSIFIER",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To design a system that requires minimal human engineering for providing reward, we use a datadriven reward specification framework called variational inverse control with events (VICE) introduced by Fu et al. (2018). VICE learns rewards in a task-agnostic way: we provide the algorithm with success examples in the form of images where the task is accomplished, and learn a discriminator that is capable of distinguishing successes from failures. This discriminator can then be used to provide a learning signal to nudge the reinforcement learning agent towards success. This algorithm has been previously considered in the context of learning tasks from raw sensory observations in the real world by (Singh et al., 2019) but we show that it presents unique challenges when used in conjunction with learning without episodic resets. Details and specifics of the algorithms being considered are described in Appendix A and also discussed by Singh et al. (2019). ",
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+    {
+        "type": "text",
+        "text": "4.3 UNSUPERVISED REPRESENTATION LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The perturbation controller discussed above allows us to learn policies that can succeed at the task from a variety of starting states. However, learning from visual observations still present a challenge. Our experiments in Fig 4 show that learning without resets from low-dimensional states is comparatively easier. We therefore aim to convert the vision-based learning problem into one that more closely resembles state-based learning, by training a variational autoencoder (VAE, Kingma & Welling (2013)) and sharing the latent-variable representation across the actor and critic networks (refer to Appendix B for more details). Note that we use a VAE as an instantiation of representation learning techniques that works well in the domains we considered, but other more sophisticated density models proposed in prior work may also be substituted in place of the VAE (Lee et al., 2019; Hjelm et al., 2019; Anand et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "While several works have also sought to incorporate unsupervised learning into reinforcement learning to make learning from images easier (Nair et al., 2018; Lee et al., 2019), we note that this becomes especially critical in the vision-based, reset-free setting, as motivated by the experiments in Section 3, which indicate that it is precisely this combination of factors – vision and no resets – that presents the most difficult learning problem. Therefore, although the particular solution we use in our system has been studied in prior work, it is brought to bare to address a challenge that arises in real-world learning that we believe has not been explored in prior studies. ",
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+    {
+        "type": "text",
+        "text": "These two improvements – the perturbation controller and unsupervised learning – combined with the general system described above, give us a complete practical system for real world reinforcement learning. The overall method uses soft-actor critic for learning with visual observations and classifier based rewards with VICE, introduces auxiliary reconstruction objectives or pretrains encoders for unsupervised representation learning, and uses a perturbation controller during training to ensure that the support of visited states grows sufficiently. We term this full system for real-world robotic reinforcement learning R3L. Further implementation details can be found in Appendix A. ",
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+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The primary contribution of this work is to propose a paradigm for continual instrumentation-free real world robotic learning and a practical instantiation of such a system. A number of prior works have studied reinforcement learning in the real world for acquiring robotic skills (Levine et al., 2016; Kumar et al., 2016; Gu et al., 2017; Kalashnikov et al., 2018; Haarnoja et al., 2018b; Finn & Levine, 2016; Zhu et al., 2019; Nagabandi et al., 2019). Much of the focus in prior work has been on improving the efficiency and generalization of RL algorithms to make real-world training feasible, or else on utilizing simulation and transferring policies into the real world (Tzeng et al., 2015; Tobin et al., 2017; Peng et al., 2017; Clavera et al., 2018; Kang et al., 2019). The simulation-based methods typically require considerable effort in terms of both simulator design and overcoming the distributional shift between simulated and real-experience, while prior real-world training methods typically require additional instrumentation for either reward function evaluation (Levine & Koltun, 2013) or resetting between trials (Gu et al., 2017; Chebotar et al., 2017; Zhu et al., 2019), or both. In contrast, our work is primarily focused on lifting these requirements, rather than devising more efficient RL methods. We show that removing the need for instrumentation (i.e., for reward evaluation and resets) introduces additional challenges, which in turn require a careful set of design choices. Our complete R3L method is able to learn completely autonomously, without manual resets or reward design. ",
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+    {
+        "type": "text",
+        "text": "A key component of our system involves learning from raw visual inputs. This has proven to be difficult for policy gradient style algorithms (Pinto et al., 2017a) due to challenging representation learning problems. This has been made easier in simulated domains by using modified objectives, such as auxiliary losses (Jaderberg et al., 2016), or by using more efficient algorithms (Haarnoja et al., 2018a). We show that reinforcement learning on raw visual input, while possible in standard RL settings, becomes significantly more challenging when considered in conjunction with nonepisodic, reset-free scenarios. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Reward function design is crucial for any RL system, and is non-trivial to provide in the real world. Prior works have considered instrumenting the environment with additional sensors to evaluate rewards (Gu et al., 2017; Chebotar et al., 2017; Zhu et al., 2019), which is a highly manual process, using demonstrations, which require manual effort to collect (Vecerik et al., 2017; Ng & Russell, 2000; Liu et al., 2018), or using interactive supervision from the user (Christiano et al., 2017). In this work, we leverage the algorithm introduced by Fu et al. (2018) to assign rewards based on the likelihood of a goal classifier. While prior work also applied this method to robotic tasks (Singh et al., 2019), this was done in a setting where manual resets were provided by hand, while we demonstrate that we can use learned rewards in a fully uninstrumented, reset-free setup. ",
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+    {
+        "type": "text",
+        "text": "Learning without resets has been considered in prior works (Eysenbach et al., 2018; Han et al., 2015), although in different contexts – safe learning and learning compound controllers, respectively. Eysenbach et al. (2018) provide an algorithm to learn a reset controller with the goal of ensuring safe operation, but makes several assumptions that make it difficult to use in the real world: it assumes access to a ground truth reward function, it assumes access to an oracle function that can detect if an attempted reset by the reset policy was successful or not, and it assumes the ability to perform manual resets if the reset policy fails a certain number of times. In contrast, we propose an algorithm that allows for fully automated reinforcement learning in the real world. We compare to an ablation of our method that uses a reset controller similar to Eysenbach et al. (2018), and show that our method performs substantially better. Our perturbation controller also resembles the adversarial RL setup Pinto et al. (2017b); Sukhbaatar et al. (2018). However, while these prior methods explicitly aim to train policies that are robust to perturbations Pinto et al. (2017b) or explore effectively Sukhbaatar et al. (2018), we are concerned with learning without access to resets. ",
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+    {
+        "type": "text",
+        "text": "While this line of work has connections to developmental robotics (Lungarella et al., 2003; Asada et al., 2009) and its subfields, such as continual (Lesort et al., 2019) and lifelong (Thrun & Mitchell, 1995) learning, the goal of our work is to handle the practicalities of enabling reinforcement learning systems to learn in the real world without instrumentation or interruption, even for a single task setting. Though our work does not directly study continual lifelong learning, nor all facets of developmental robotics, it relates to continual learning (Lesort et al., 2019), intrinsic motivation (Schmidhuber, 2006) and sensory-motor development involving proprioceptive manipulation (Stoica, 2001). ",
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+        "type": "text",
+        "text": "6 EXPERIMENTS ",
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+        "type": "text",
+        "text": "In our experimental evaluation, we study how well the R3L system, described in Sections 2 and 4, can learn under realistic settings – visual observations, no hand-specified rewards, and no resets. We consider the following hypotheses: ",
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+        "type": "text",
+        "text": "1. Can we use R3L to learn complex robotic manipulation tasks without instrumentation? Does this system learn skills in both simulation and the real world?   \n2. Do the solutions proposed in Section 4 actually enable R3L to perform tasks without instrumentation that would not have been otherwise possible? ",
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+        "type": "text",
+        "text": "6.1 EXPERIMENTAL SETUP ",
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+        "type": "text",
+        "text": "We consider the task of dexterous manipulation with a three fingered robotic hand, the D’Claw (Zhu et al., 2019; Ahn et al., 2019), on a number of simulated and real world tasks. These tasks involve complex coordination of three fingers with 3 DoF each in order to manipulate objects. Prior works that used this robot utilized explicit resets and low-dimensional true state observations, while we consider settings with visual observations, no hand-specified rewards, and no resets. ",
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+        "img_path": "images/824a594d6c5e30f9a580b874265c711fa246312d33dda84adff8239823f14ea9.jpg",
+        "image_caption": [
+            "Figure 6: Visualizations of the goal configurations of the simulated and real world tasks being considered. From left to right we depict valve rotation, bead manipulation and free object repositioning in simulation, as well as valve rotation and bead manipulation manipulation in the real world. "
+        ],
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+    {
+        "type": "text",
+        "text": "The tasks in our experiments are shown in Fig 6: manipulating beads on an abacus row, valve rotation, and free object repositioning. These tasks represent a wide class of problems that robots might encounter in the real world. For each task, we consider the problem of reaching the depicted goal configuration: moving the abacus beads to a particular position, rotating the valve to a particular angle, and repositioning the free object to a particular goal position. For each task, policies are evaluated from a wide selection of initial configurations. Additional details about the tasks and evaluation procedures are provided in Appendix C. Videos and additional details can be found at https://sites.google.com/view/realworld-rl/ ",
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+    {
+        "type": "text",
+        "text": "6.2 LEARNING IN SIMULATION WITHOUT INSTRUMENTATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We compare our entire proposed system implementation (Section 4) with a number of baselines and ablations. Importantly, all methods must operate under the same assumptions: none of the algorithms have access to system instrumentation for state estimation, reward specification, or episodic resets. Firstly, we compare the performance of R3L to a system which uses SAC for vision-based RL from raw pixels, VICE for providing rewards and running reset-free (denoted as “VICE”). This corresponds to the vanilla version of R3L (Section 2), with none of the proposed insights and changes. We then compare with prior reset-free RL algorithms (Eysenbach et al., 2018) that explicitly learn a reset controller to alternate goals in the state space (“Reset Controller $^ +$ VAE”). Lastly, we compare algorithm performance with two ablations: running R3L without the perturbation controller (“VICE $+ \\ \\mathrm { V A E } ^ { \\prime \\prime } )$ ) and without the unsupervised learning (“R3L w/o VAE”). This highlights the significance of each of the components of R3L . ",
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+    {
+        "type": "text",
+        "text": "From the experimental results in Fig 7, it is clear that R3L is able to reach the best performance across tasks, while none of the other methods are able to solve all of the tasks. Different prior methods and ablations fail for different reasons: (1) methods without the reconstruction objective struggle at parsing the high-dimensional input and are unable to solve the harder task; (2) methods without the perturbation controller are ineffective at learning how to reach the goal from novel initialization positions for the more challenging object repositioning tasks, as discussed in Section 4. ",
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+        "img_path": "images/16689f39419185c7ebec5a0ee2e8964bd3a1d3329c8a98a6d5098b217637ce35.jpg",
+        "image_caption": [
+            "Figure 7: Quantitative evaluation of performance in simulation for bead manipulation, valve rotation and free object repositioning (left to right) run with 10 random seeds. The error bars show $9 5 \\%$ bootstrap confidence intervals for average performance. While other variants are sufficient to get good evaluation performance on easier tasks, harder tasks like free object repositioning require random perturbations and unsupervised representation learning to learn skills reset-free. See Appendix C.1 for details of evaluation procedures. "
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We note that an explicit reset controller, which can be viewed as a softer version of our perturbation controller with goal-directedness, learns to solve the easier tasks due to the reset controller encouraging exploration of the state space. In our experiments for free object repositioning, performance was reported across 3 choices of reset states. The high variance in evaluation performance indicates that the performance of such a controller (or a goal conditioned variant of it) is highly sensitive to the choice of reset states. A poor choice of reset states, such as two that are very close together, may yield poor exploration leading to performance similar to the single goal VICE baseline. Furthermore, the choice of reset states is highly task dependent and it is often not clear what choice of goals will yield the best performance. On the contrary, our method does not require such task-specific knowledge and uses random perturbations to reset while training without any explicit reset states: this allows for a robust, instrumentation-free controller while also ensuring fast convergence. ",
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+    {
+        "type": "text",
+        "text": "6.3 LEARNING IN THE REAL WORLD WITHOUT INSTRUMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the aim of R3L is to enable uninstrumented training in the real world, we next evaluate our method on a real-world robotic system, providing evidence that our insights generalize to the real world without any instrumentation. After providing the initial outcome examples for learning the reward function with VICE, we leave the robot unattended, and the algorithm learns the desired behavior through interaction. The experiments are performed on the D’Claw robotic hand with an RGB camera as the only sensory input. Intermediate policies are saved at regular intervals, and evaluations of all policies is performed after training has completed. For valve rotation, we declare an evaluation rollout a success if the final orientation is within within $1 5 ^ { \\circ }$ of the goal. For bead manipulation, we declare success if all the beads are within $2 \\mathrm { c m }$ of the goal state. Fig 8 compares the performance of our method without supervised learning (“R3L w/o VAE”) in the real world against a baseline that uses SAC for vision-based RL from raw pixels, VICE for providing rewards, and running reset-free (denoted as “VICE”). We see that our method learns policies that succeed from nearly all the initial configurations, whereas VICE alone fails from most initial configurations. Fig 9 depicts sample evaluation rollouts of the policies learned using our method. For further details about real world experiments see Appendix C.2. ",
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+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We presented the design and instantiation of a system for real world reinforcement learning. We identify and investigate the various ingredients required for such a system to scale gracefully with minimal human engineering and supervision. We show that this system must be able to learn from raw sensory observations, learn from very easily specified reward functions without reward engineering, and learn without any episodic resets. We describe the basic elements that are required to construct such a system, and identify unexpected learning challenges that arise from interplay of these elements. We propose simple and scalable fixes to these challenges through introducing unsupervised representation learning and a randomized perturbation controller. ",
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+    {
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+        "img_path": "images/dad274348320cb8430ec089aa0a9ec78b973c42a0a84d8f326005b3a2f61f71a.jpg",
+        "image_caption": [
+            "Figure 8: Quantitative evaluation of performance in real-world for valve rotation and bead manipulation. Policies trained with perturbation controllers have effectively learned behaviors after 17 and 5 hours of training, respectively. For more fine-grained reporting of results see Figs 13-16. "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/aa753edabd7a309dd77434d6d462b21f9d280973740211557ec78a0770a770db.jpg",
+        "image_caption": [
+            "Figure 9: Evaluation rollouts of R3L on the real world tasks for policies trained without instrumentation. Successful evaluation rollouts for bead manipulation (top) and valve rotation (bottom) are shown here. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The ability to train robots directly in the real world with minimal instrumentation opens a number of exciting avenues for future research. Robots that can learn unattended, without resets or handdesigned reward functions, can in principle collect very large amounts of experience autonomously, which may enable very broad generalization in the future. However, there are also a number of additional challenges, including sample complexity, optimization and exploration difficulties on more complex tasks, safe operation, communication latency, sensing and actuation noise, and so forth, all of which would need to be addressed in future work in order to enable truly scalable real-world robotic learning. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This research is supported by the Office of Naval Research, the National Science Foundation under IIS-1651843 and IIS-1700696, Google, NVIDIA, and Amazon. The authors would like to thank Ignasi Clavera, Gregory Kahn, Coline Devin, Benjamin Eysenbach, Aviral Kumar, Anusha Nagabandi, Marvin Zhang, Ashvin Nair and several others in the RAIL Lab for helpful discussions and feedback. ",
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+            821,
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A ALGORITHM DETAILS ",
+        "bbox": [
+            176,
+            102,
+            392,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Real-World Robotic Reinforcement Learning (R3L) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            330,
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+            345
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/baf1850fa6349b7599b8ec022f6bccb9bd390a476ca5e34a90523f422e68ac94.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">procedure R3L</td></tr><tr><td>1: 2:</td><td>N ← number of training epochs</td></tr><tr><td>3:</td><td>H ← trajectory length (horizon)</td></tr><tr><td>4:</td><td>nvICE ← number of VICE classifier training iterations per epoch</td></tr><tr><td>5:</td><td>Initialize forward and perturbing policies To,π1</td></tr><tr><td>6:</td><td>Obtain goal states sE and initialize as a goal pool G</td></tr><tr><td>7:</td><td>Initialize RND target and predictor networks f(s), f(s)</td></tr><tr><td>8:</td><td>Initialize VICE reward classifier rvICE(s)</td></tr><tr><td>9:</td><td>Initialize replay buffer D</td></tr><tr><td>10:</td><td>Collect initial exploration data and add to D</td></tr><tr><td>11:</td><td>for i= 1 to 2N do</td></tr><tr><td>12:</td><td>k←i%2</td></tr><tr><td>13:</td><td>for t = 1 to H do</td></tr><tr><td>14:</td><td>Sample at ~ Tk(St)</td></tr><tr><td>15:</td><td>Sample St+1 ~ p(St+1|St,at)</td></tr><tr><td>16:</td><td>if k== O then</td></tr><tr><td>17:</td><td>rt(St)= CVICE * rVICE(St)+ CRND * rRND(St)</td></tr><tr><td>18:</td><td>else if k==1 then</td></tr><tr><td>19:</td><td>rt(St)= rRND(St)</td></tr><tr><td>20:</td><td>end if</td></tr><tr><td>21:</td><td>Sample batch from D</td></tr><tr><td>22:</td><td>Update πk with batch according to SAC</td></tr><tr><td>23:</td><td>Update RND predictor network with batch</td></tr><tr><td>24:</td><td>Update running estimate of standard deviations of classifier and RND reward</td></tr><tr><td>25:</td><td>end for</td></tr><tr><td>26:</td><td>Add experience to the replay buffer with D ← DU Ti</td></tr><tr><td>27:</td><td>Sample an equal number of goal examples from G and negative examples from D</td></tr><tr><td>28:</td><td>for t= 1 to nvice do</td></tr><tr><td>29:</td><td>Train the VICE classifier on this batch with binary labels</td></tr><tr><td>30:</td><td>end for</td></tr><tr><td>31:</td><td>end for</td></tr><tr><td>32: end procedure</td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            370,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/90c9f5d5f162ae6fcf19e4841e2fa5994f44849fc94c92be58b62ccfe3213d0b.jpg",
+        "table_caption": [
+            "B.0.1 HYPERPARAMETERS "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>General</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Standard deviation up-date coefficientImage Sizes</td><td rowspan=1 colspan=1>0.99[(16, 16,3), (32,32,3), (64, 64, 3)]</td></tr><tr><td rowspan=1 colspan=1>SAC</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learning RateyBatch SizeConvnet FiltersStrideKernel SizesPoolingActor/Critic FC Lay-ers</td><td rowspan=1 colspan=1>3e-40.99256[(64, 64,64), (16,32, 64)](2,2)(3,3)[MaxPool2D,None][(512,512), (256, 256, 256)]</td></tr><tr><td rowspan=1 colspan=1>VICE</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>nvICEBatch SizeLearning RateMixup αConvnet FiltersStrideKernel SizesPoolingFC Layers</td><td rowspan=1 colspan=1>[1,5,10]1281e-4Uniform(0, 1)[(64, 64,64), (16, 32, 64)](2,2)(3,3)[MaxPool2D,None][(512,512), (256,256,256)]</td></tr><tr><td rowspan=1 colspan=1>RND</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learning RateBatch SizeConvnet FiltersStrideKernel SizesPoolingFC Layers</td><td rowspan=1 colspan=1>3e-4256(16, 32, 64)(2,2)(3,3)[MaxPool2D, None][(512,512), (256,256,256)]</td></tr><tr><td rowspan=1 colspan=1>VAE</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learning RateBatch SizeEncoder (Convnet)FiltersLatent DimensionβStrideKernel SizesPooling</td><td rowspan=1 colspan=1>1e-4256(64, 64, 32)[8, 16,32, 64][1e-3, 0.1, 0.5, 1, 10](2,2)(3,3)[MaxPool2D, None]</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The ranges of values listed above represent the hyperparameters we searched over, and the bolded values are what we use in the Section 6 experiments. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.0.2 VICE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            790,
+            274,
+            804
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We use a variant of VICE which defines the reward as the logits of the classifier, notably omitting the $- l o g ( \\pi ( a | s ) )$ term. We also regularize our classifier with mixup (Zhang et al., 2018). We train all of our experiments using 200 goal images, which takes under an hour to collect in the real world for each task. ",
+        "bbox": [
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+            814,
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+            869
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.0.3 RANDOM NETWORK DISTILLATION (RND) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            883,
+            532,
+            900
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We found it important to normalize the predictor errors, just as (Burda et al., 2018) did. ",
+        "bbox": [
+            174,
+            909,
+            743,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.0.4 VAE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            266,
+            117
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We train a standard beta-VAE to maximize the evidence lower bound, given by: ",
+        "bbox": [
+            176,
+            127,
+            694,
+            142
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a876e28c0f4089cb02fe39cef4ddda4e2f36dc48bd10d209eb88acee4f5cbc4.jpg",
+        "text": "$$\n\\mathbb { E } _ { z \\sim q _ { \\phi } ( z | x ) } [ p _ { \\theta } ( x | z ) ] - \\beta D _ { \\mathrm { K L } } ( q _ { \\phi } ( z | x ) | | p _ { \\theta } ( z ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            148,
+            655,
+            169
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "To collect training data, we sampled random states in the observation space. In the real world, this sampling can be replaced with training an exploratory policy (i.e. using the RND reward as the policy’s only objective). The learned weights of the encoder of the VAE are frozen, and the latent input is used to train the policy for reset-free RL. ",
+        "bbox": [
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+            237
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C TASK DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            257,
+            331,
+            273
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C.1 SIMULATED TASKS ",
+        "text_level": 1,
+        "bbox": [
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+            289,
+            351,
+            304
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We evaluated our system across three tasks in simulation: bead manipulation, valve rotation, free object repositioning. ",
+        "bbox": [
+            173,
+            315,
+            825,
+            343
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C.1.1 BEAD MANIPULATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            358,
+            387,
+            373
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The bead manipulation task involves an abacus rod with four beads that can slide freely. The goal is to position two beads on each end from any initial configuration of beads. This can take the form of sliding one bead over (if three beads start on one side), two beads over (if all four beads start on one side), splitting beads apart (all four beads start in the middle), or some intermediate combination of those. The true reward is defined as the mean goal distance of all four beads. Policies are evaluated starting from the 8 initial configurations depicted in Fig 10. Evaluation performance reported in Section 6 for this task is defined as the final reward averaged across the 8 evaluation rollouts. ",
+        "bbox": [
+            173,
+            383,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/080c6495f909ccd2083348ee2a02e0be2c3e68b87486d16c7fe5e20650758e62.jpg",
+        "image_caption": [
+            "Figure 10: These are the 8 initial positions used for evaluating the performance of the bead manipulation policy. The goal configuration (which is also an initial evaluation position) is highlighted in yellow. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            500,
+            750,
+            690
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C.1.2 VALVE ROTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            733,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The claw is positioned above a three pronged valve ( $1 5 \\mathrm { c m }$ in diameter). The objective is to turn the valve to a given orientation from any initial orientation. The ”true reward” is $\\dot { r } = - \\log ( | \\theta _ { s t a t e } -$ $\\theta _ { g o a l } | .$ ). Policies are evaluated starting from the 8 initial configurations depicted in Fig 11. Evaluation performance reported in Section 6 for this task is defined as the final orientation distance averaged across the 8 evaluation rollouts. ",
+        "bbox": [
+            174,
+            758,
+            825,
+            828
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C.1.3 FREE OBJECT REPOSITIONING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            843,
+            442,
+            858
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The claw is positioned atop a free $6 ~ \\mathrm { D o F } )$ three pronged object ( $1 5 \\mathrm { c m }$ in diameter), which can translate and rotate within a $3 0 \\mathrm { c m } x 3 0 \\mathrm { c m }$ box. The goal is specified by a xy-position as well as a $\\mathbf { Z }$ -angle, where the xy-plane is the plane of the arena. The true reward is defined as the weighted sum of the angular and translational distances, $r = - 2 \\log ( | | [ x _ { s t a t e } , y _ { s t a t e } ] - [ x _ { g o a l } , y _ { g o a l } ] | | _ { 2 } ) -$ $\\log ( | \\theta _ { s t a t e } - \\theta _ { g o a l } | )$ . In our experiments, $( x , y , \\theta ) _ { g o a l } = ( 0 , 0 , - \\frac { \\pi } { 2 } )$ , where the origin is centered in the arena. Policies are evaluated starting from the 15 initial configurations depicted in Figure 12. Evaluation performance reported in Section 6 for this task is defined as the final pose distance ( ||[xf inal,yf inal]−[xgoal,ygoal]||2 + |θf inal−θgoal| ) averaged across the 15 evaluation rollouts. ",
+        "bbox": [
+            174,
+            867,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/2fc0e811e81e0386a99cca981233798437bab0e05c9f8cf66ad0402316a68fe0.jpg",
+        "image_caption": [
+            "Figure 11: These are the 8 initial positions used for evaluating the performance of the valve rotation policy. The goal configuration (which is also an initial evaluation position) is highlighted in yellow. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            104,
+            750,
+            296
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
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+        "type": "text",
+        "text": "In our reset controller experiments, we averaged evaluation performance over three different choices of reset states, where the first reset state is always the goal: ",
+        "bbox": [
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+        "type": "text",
+        "text": "1. $( x , y , \\theta ) _ { r e s e t , 1 } = ( x , y , \\theta ) _ { g o a l } , ( x , y , \\theta ) _ { r e s e t , 2 } = ( 0 . 0 5 , 0 . 0 5 , \\frac { \\pi } { 2 } )$   \n2. $\\begin{array} { r } { ( x , y , \\theta ) _ { r e s e t , 1 } = ( x , y , \\theta ) _ { g o a l } , ( x , y , \\theta ) _ { r e s e t , 2 } = ( 0 , 0 , - \\frac { \\pi } { 6 } ) } \\end{array}$   \n3. $( x , y , \\theta ) _ { r e s e t , 1 } = ( x , y , \\theta ) _ { g o a l } , ( x , y , \\theta ) _ { r e s e t , 2 } = ( - 0 . 0 4 , - 0 . 0 4 , - \\frac { \\pi } { 2 } )$ ",
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+        "img_path": "images/1b1a729609dbfdd4bff369f64e48f133a4dc620e897c2ab6a14904cca1c9dbe4.jpg",
+        "image_caption": [
+            "Figure 12: These are the 15 initial positions used for evaluating the performance of the free object repositioning policy. The goal configuration $( x , y , \\theta ) _ { g o a l }$ which is also an initial evaluation position is highlighted in yellow. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "C.2 REAL WORLD TASKS ",
+        "text_level": 1,
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+        "text": "For each setup we use an RGB camera to get images. We execute actions on the DClaw at $1 0 \\mathrm { H z }$ . In order to operate at such a high frequency while also training from images we sample and train asynchronously, but limit training to not exceed two gradient steps per transition sampled in the real world. Since direct performance metrics cannot be measured during training due to the lack of object instrumentation, evaluations of performance are done post-training. ",
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+    {
+        "type": "text",
+        "text": "C.2.1 VALVE ROTATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The task is identical to the one in simulation. Evaluations were done post-training. An evaluation trajectory was defined as a success if at the last step, the valve was within 15 degrees of the goal. Each policy was evaluated over 8 rollouts, with initial configurations evenly spaced out between at increments of 45 degrees. Results are reported in Figures 13, 14. ",
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+    {
+        "type": "text",
+        "text": "C.2.2 BEAD MANIPULATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The rod is $2 2 \\mathrm { c m }$ in length, and each bead measures $3 . 5 \\mathrm { c m }$ in diameter. Evaluations were done post-training, using the following procedure: the environment was manually reset to each of the 8 specified configurations shown in Figures 15 and 16 (which cover a full range of the state space) at the start of each evaluation rollout. An evaluation trajectory was defined as a success if at the last time step, all beads were within $2 \\mathrm { c m }$ of their goal positions. Performance was evaluated at around 20 hours, at which point the policy achieved greater than $80 \\%$ success on the 10 evaluation rollouts (a random policy achieved a success rate of $10 \\%$ ). Results are reported in Figs 15, 16. ",
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+    {
+        "type": "image",
+        "img_path": "images/7a90b025f1eac655661ca465e4c18b4e9bb25b2a296dbae63b4b880663793aff.jpg",
+        "image_caption": [
+            "Figure 13: These are the results of the evaluation rollouts on the valve rotation task in the real world using our method (without the VAE). Trained policies were saved at regular intervals and evaluated post-training. Each row is a different policy, and each column an evaluation rollout from a different initial configuration. The goal is highlighted in yellow. Our method is able to achieve high success rates after 5 hours of training. "
+        ],
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+        "img_path": "images/124c852ede375572bcb431d4c7898f2ef36299c9da3133ec70dbeee81eee744d.jpg",
+        "image_caption": [
+            "Figure 14: These are the results of evaluation rollouts on the valve rotation task in the real world using the VICE single goal baseline. The policies fail to evaluate well, especially from initial positions far from the goal position. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/30f66949f6e9e5f3f38c8a70eb2082962ea087319babcbeafcf552e09b61101e.jpg",
+        "image_caption": [
+            "Figure 15: These are the results of the evaluation rollouts on the valve rotation task in the real world using our method (without the VAE). Trained policies were saved at regular intervals and evaluated post-training. Each row is a different policy, and each column an evaluation rollout from a different initial configuration. The goal is highlighted in yellow. Our method is able to achieve high success rates after 17 hours of training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 19
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+    {
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+        "img_path": "images/3f8949a077ed83c173449febf904e9b7fa2f6c6fc40c84db953357f2ba8c7515.jpg",
+        "image_caption": [
+            "Figure 16: These are the results of evaluation rollouts on the valve rotation task in the real world using the VICE single goal baseline. The policies fail to evaluate consistently, except when the initial configuration matches the goal configuration. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    }
+]

parse/train/rydeCEhs-/rydeCEhs-.md

@@ -0,0 +1,311 @@
+# SMASH: ONE-SHOT MODEL ARCHITECTURESEARCH THROUGH HYPERNETWORKS
+
+Andrew Brock, Theodore Lim, & J.M. Ritchie
+
+# Nick Weston
+
+School of Engineering and Physical Sciences   
+Heriot-Watt University   
+Edinburgh, UK   
+{ajb5, t.lim, j.m.ritchie}@hw.ac.uk   
+Renishaw plc   
+Research Ave, North   
+Edinburgh, UK   
+Nick.Weston@renishaw.com
+
+# ABSTRACT
+
+Designing architectures for deep neural networks requires expert knowledge and substantial computation time. We propose a technique to accelerate architecture selection by learning an auxiliary HyperNet that generates the weights of a main model conditioned on that model’s architecture. By comparing the relative validation performance of networks with HyperNet-generated weights, we can effectively search over a wide range of architectures at the cost of a single training run. To facilitate this search, we develop a flexible mechanism based on memory read-writes that allows us to define a wide range of network connectivity patterns, with ResNet, DenseNet, and FractalNet blocks as special cases. We validate our method (SMASH) on CIFAR-10 and CIFAR-100, STL-10, ModelNet10, and Imagene $3 2 \mathbf { x } 3 2$ , achieving competitive performance with similarly-sized handdesigned networks.
+
+# 1 INTRODUCTION
+
+The high performance of deep neural nets is tempered by the cost of extensive engineering and validation to find the best architecture for a given problem. High-level design decisions such as depth, units per layer, and layer connectivity are not always obvious, and the success of models such as Inception (Szegedy et al., 2016), ResNets (He et al., 2016), FractalNets (Larsson et al., 2017) and DenseNets (Huang et al., 2017) demonstrates the benefits of intricate design patterns. Even with expert knowledge, determining which design elements to weave together requires ample experimentation.
+
+In this work, we propose to bypass the expensive procedure of fully training candidate models by instead training an auxiliary model, a HyperNet (Ha et al., 2017), to dynamically generate the weights of a main model with variable architecture. Though these generated weights are worse than freely learned weights for a fixed architecture, we leverage the observation (Li et al., 2017) that the relative performance of different networks early in training (i.e. some distance from the eventual optimum) often provides a meaningful indication of performance at optimality. By comparing validation performance for a set of architectures using generated weights, we can approximately rank numerous architectures at the cost of a single training run.
+
+To facilitate this search, we develop a flexible scheme based on memory read-writes that allows us to define a diverse range of architectures, with ResNets, DenseNets, and FractalNets as special cases. We validate our one-Shot Model Architecture Search through HyperNetworks (SMASH) for Convolutional Neural Networks (CNN) on CIFAR-10 and CIFAR-100 (Krizhevsky and Hinton, 2009), Imagene $3 2 \mathrm { x } 3 2$ (Chrabaszcz et al., 2017), ModelNet10 (Wu et al., 2015), and STL-10 (Coates et al., 2011), achieving competitive performance with similarly-sized hand-designed networks.
+
+# 2 RELATED WORK
+
+Modern practical methods for optimizing hyperparameters rely on random search (Bergstra and Bengio, 2012) or Bayesian Optimization (BO) (Snoek et al., 2012; 2015), treating the model performance as a black box. While successful, these methods require multiple training runs for evaluation (even when starting with a good initial model) and, in the case of BO, are not typically used to specify variable-length settings such as the connectivity and structure of the model under consideration. Relatedly, bandit-based methods (Li et al., 2017) provide a framework for efficiently exploring the hyperparameter space by employing an adaptive early-stopping strategy, allocating more resources to models which show promise early in training.
+
+Evolutionary techniques (Floreano et al.; Stanley et al.; Suganuma et al., 2017; Wierstra et al., 2005) offer a flexible approach for discovering variegated models from trivial initial conditions, but often struggle to scale to deep neural nets where the search space is vast, even with enormous compute (Real et al., 2017).
+
+Reinforcement learning methods (Baker et al., 2017; Zoph and Le, 2017) have been used to train an agent to generate network definitions using policy gradients. These methods start from trivial architectures and discover models that achieve very high performance, but can require twelve to fifteen thousand full training runs to arrive at a solution.
+
+The method that most resembles our own is that of Saxe et al. (Saxe et al., 2011), who propose to efficiently explore various architectures by training only the output layer of convolutional networks with random convolutional weights. While more efficient than fully training an entire network end-toend, this method does not appear to scale to deeper networks (Yosinski et al., 2014). Our method is conceptually similar, but replaces random weights with weights generated through HyperNets (Ha et al., 2017), which are one of a class of techniques for dynamically adapting weights through use of an auxiliary model (Denil et al., 2013; Jia et al., 2016; Rebuffi et al., 2017; Schmidhuber, 1992). In our case we learn a transform from a binary encoding of an architecture to the weight space, rather than learning to adapt weights based on the model input.
+
+Our method is explicitly designed to evaluate a wide range of model configurations (in terms of connectivity patterns, depth, and width) but does not address other hyperparameters such as regularization, learning rate schedule, weight initialization, or data augmentation. Unlike the aforementioned evolutionary or RL methods, we explore a somewhat pre-defined design space, rather than starting with a trivial model and designating a set of available network elements. While we still consider a rich set of architectures, our method cannot discover wholly new structures on its own and is constrained in that it only dynamically generates a specific subset of the model parameters. Additionally, although our method is not evolutionary, our encoding scheme is reminiscent of CGP (Miller and Thomson, 2000).
+
+Stochastic regularization techniques such as Dropout (Srivastava et al., 2012), Swapout (Singh et al., 2016), DropPath (Larsson et al., 2017) or stochastic depth (Huang et al., 2016) superficially resemble our method, in that they obtain variable configurations by randomly dropping connectivity paths in a fixed network architecture. Convolutional neural fabrics (Saxena and Verbeek, 2016), for example, leverage this idea to attempt to train one large network as an implicit ensemble of all subnetworks produced through dropping paths. A key element that sets our method apart is that the weights for each node in our network are dynamically generated, rather than fixed; if a Dropout ensemble were to visit a unit that had not previously been trained, the unit’s weights would be completely untuned. Our method generalizes even to previously unseen architectures, and the network we train under stochastic conditions is merely a proxy we use to evaluate network configurations, rather than the final model.
+
+# 3 ONE-SHOT MODEL ARCHITECTURE SEARCH THROUGH HYPERNETWORKS
+
+In SMASH (Algorithm 1), our goal is to rank a set of neural network configurations relative to one another based on each configuration’s validation performance, which we accomplish using weights generated by an auxiliary network. At each training step, we randomly sample a network architecture, generate the weights for that architecture using a HyperNet, and train the entire system end-to-end through backpropagation. When the model is finished training, we sample a number of random architectures and evaluate their performance on a validation set, using weights generated by the HyperNet. We then select the architecture with the best estimated validation performance and train its weights normally.
+
+<table><tr><td>Algorithm1SMASH</td></tr><tr><td></td></tr></table>
+
+<table><tr><td>input Space of all candidate architectures, Rc Initialize HyperNet weights H loop</td><td></td></tr><tr><td>Sample input minibatch xi,random architecture c and architecture weights W = H(c) Get training error Et = fc(W,xi)= fc(H(c),xi),backprop and update H</td><td rowspan="3"></td></tr><tr><td>end loop</td></tr><tr><td>loop Sample random c and evaluate error on validation set Ev = fc(H(c),xv)</td></tr><tr><td>end loop</td><td rowspan="3"></td></tr><tr><td>Fix architecture and train normally with freely-varying weights W</td></tr><tr><td></td></tr></table>
+
+SMASH comprises two core components: the method by which we sample architectures, and the method by which we sample weights for a given architecture. For the former, we develop a novel "memory-bank" view of feed-forward networks that permits sampling complex, branching topologies, and encoding said topologies as binary vectors. For the latter, we employ a HyperNet (Ha et al., 2017) that learns to map directly from the binary architecture encoding to the weight space.
+
+We hypothesize that so long as the HyperNet learns to generate reasonable weights, the validation error of networks with generated weights will correlate with the performance when using normally trained weights, with the difference in architecture being the primary factor of variation. Throughout the paper, we refer to the entire apparatus during the first part of training (the HyperNet, the variable architecture main network, and any freely learned main network weights) as the SMASH network, and we refer to networks trained with freely learned weights in the second stage as resulting networks.
+
+# 3.1 DEFINING VARIABLE NETWORK CONFIGURATIONS
+
+![](images/05a14337291e284a70855dd35fb712c5b232f11f6ddcb43246c9185b9e925552.jpg)  
+Figure 1: Memory-Bank representations of ResNet, DenseNet, and FractalNet blocks.
+
+In order to explore a broad range of architectures with variable depth, connectivity patterns, layer sizes and beyond, we require a flexible mechanism for defining such architectures, which we can also easily encode into a conditioning vector for the HyperNet. To this end, we introduce a "memory-bank" view of feed-forward networks.
+
+Rather than viewing a network as a series of operations applied to a forward-propagating signal, we view a network as having a set of memory banks (initially tensors filled with zeros) which it can read and write. Each layer is thus an operation that reads data from a subset of memory, modifies the data, and writes the result to another subset of memory. For a single-branch architecture, the network has one large memory bank it reads and overwrites (or, for a ResNet, adds to) at each op. A branching architecture such as a DenseNet reads from all previously written banks and writes to an empty bank, and a FractalNet follows a more complex read-write pattern, as shown in Figure 1.
+
+Our base network structure consists of multiple blocks (Figure 2(b)), where each block has a set number of memory banks at a given spatial resolution, with successively halved spatial resolutions as in most CNN architectures. Downsampling is accomplished via a 1x1 convolution followed by average pooling (Huang et al., 2017), with the weights of the 1x1 convolution and the fully-connected output layer being freely learned, rather than generated.
+
+![](images/c7a2ac2222c18e7a0855ff9b2100627098d981f68cc17e638b413d86f1f3d840.jpg)  
+Figure 2: (a) Structure of one op: A 1x1 conv operating on the memory banks, followed by up to 2 stem block 1 trans block 2 trans block 3parallel paths of 2 convolutions each. (b) Basic network skeleton.
+
+Block 1 Block 2 Block 3When sampling an architecture, the number of banks and the number of channels per bank are randomly sampled at each block. When defining each layer within a block, we randomly select the Block 1 Block 2 Block 3read-write pattern and the definition of the op to be performed on the read data. When reading from multiple banks we concatenate the read tensors along the channel axis, and when writing to banks we add to the tensors currently in each bank. For all reported experiments, we only read and write from banks at one block (i.e. one spatial resolution), although one could employ resizing to allow reading and writing from any block, similar to (Saxena and Verbeek, 2016).
+
+Each op comprises a 1x1 convolution (reducing the number of incoming channels), followed by a variable number of convolutions interleaved with nonlinearities, as shown in Figure 2(a). We randomly select which of the four convs are active, along with their filter size, dilation factor, number of groups, and the number of output units (i.e. the layer width). The number of output channels of the 1x1 conv is some factor of the width of the op, chosen as the "bottleneck ratio" hyperparameter.
+
+The weights for the 1x1 convolution are generated by the HyperNet as described in Section 3.2, while the other convolutions are normally learned parameters. To ensure variable depth, we learn a single set of 4 convolutions for each block, and share it across all ops within a block. We limit the max filter size and number of output units, and when a sampled op uses less than the maximum of either, we simply slice the weight tensor to the required size. The fixed transition convolutions and output layer employ this same slicing based on the number of incoming non-empty memory banks. Exact details regarding this scheme are available in the appendix.
+
+In designing our scheme, we strive to minimize the number of static learned parameters, placing the majority of the network’s capacity in the HyperNet. A notable consequence of this goal is that we only employ BatchNorm (Ioffe and Szegedy, 2015) at downsample layers and before the output layer, as the layer-specific running statistics are difficult to dynamically generate. We experimented with several different normalization schemes including WeightNorm (Salimans and Kingma, 2016), LayerNorm (Ba et al., 2016) and NormProp (Arpit et al., 2016) but found them to be unstable in training.
+
+Instead, we employ a simplified version of WeightNorm where we divide the entirety of each generated 1x1 filter by its Euclidean norm (rather than normalizing each channel separately), which we find to work well for SMASH and to only result in a minor drop in accuracy when employed in fixed-architecture networks. No other convolution within an op is normalized.
+
+# 3.2 LEARNING TO MAP ARCHITECTURES TO WEIGHTS
+
+A HyperNet (Ha et al., 2017) is a neural net used to parameterize the weights of another network, the main network. For a Static HyperNet with parameters $H$ , the main network weights $W$ are some function (e.g. a multilayer perceptron) of a learned embedding $z$ , such that the number of learned weights is typically smaller than the full number of weights for the main network. For a Dynamic HyperNet, the weights $W$ are generated conditioned on the network input $x$ , or, for recurrent networks, on the current input $x _ { t }$ and the previous hidden state $h _ { t - 1 }$ .
+
+We propose a variant of a Dynamic HyperNet which generates the weights $W$ based on a tensor encoding of the main network architecture $c$ . Our goal is to learn a mapping $W = H ( c )$ that is reasonably close to the optimal $W$ for any given $c$ , such that we can rank each $c$ based on the validation error using HyperNet-generated weights. We thus adopt a scheme for the layout of $c$ to enable sampling of architectures with wildly variable topologies, compatibility with the toolbox available in standard libraries, and to make $c$ ’s dimensions as interpretable as possible.
+
+![](images/8ee81e6737ffba29e0141a22b3f8e4da30d5b4f02cda109985cb668ab2803d2e.jpg)  
+Figure 3: An unrolled graph, its equivalent memory-bank representation, and its encoded embedding.
+
+Our HyperNet is fully convolutional, such that the dimensionality of the output tensor $W$ varies with the dimensionality of the input $c$ , which we make a 4D tensor of the standard format Batch x Channel x Height $\mathbf { X }$ Width; the batch size is set to 1 so that no output elements are wholly independent. This allows us to vary the depth and width of the main network by increasing the height or width of $c$ Under this scheme, every slice of the spatial dimensions of $W$ corresponds to a specific subset of $c$ Information describing the op that uses that $W$ subset is embedded in the channel dimension of the corresponding $c$ slice.
+
+For example, if an op reads from memory banks 1, 2, and 4, then writes to 2 and 4, then the first, second, and fourth channels of the corresponding slice of $c$ will be filled with 1s (indicating the read pattern) and the sixth and eighth channels of that slice will be filled with 1s (indicating the write pattern). The rest of the op description is encoded in the remaining channels in a similar 1-hot fashion. We only encode into the width-wise extent of $c$ based on the number of output units of the op, so elements of $c$ which do not correspond to any elements of $W$ are empty.
+
+A naïve implementation of this scheme might require the size of $c$ to be equal to the size of $W$ , or have the HyperNet employ spatial upsampling to produce more elements. We found these choices poor, and instead employ a channel-based weight-compression scheme that reduces the size of $c$ and keeps the representational power of the HyperNet proportional to that of the main networks. We make the spatial extent of $c$ some fraction $k$ of the size of $W$ , and place $k$ units at the output of the HyperNet. We then reshape the resulting $1 \times k \times h e i g h t \times w i d t h$ tensor to the required size of W . $k$ is chosen to be $D N ^ { 2 }$ , where $N$ is the minimum memory bank size, and $D$ is a "depth compression" hyperparameter that represents how many slices of $W$ correspond to a single slice of $c$ . Complete details regarding this scheme (and the rest of the encoding strategy) are available in Appendix B.
+
+# 4 EXPERIMENTS
+
+We apply SMASH to several datasets, both for the purposes of benchmarking against other techniques, and to investigate the behavior of SMASH networks. Principally, we are interested in determining whether the validation error of a network using SMASH-generated weights (the "SMASH score") correlates with the validation of a normally trained network, and if so, the conditions under which the correlation holds. We are also interested in the transferability of the learned architectures to new datasets and domains, and how this relates to normal (weight-wise) transfer learning.
+
+Our publicly available code1 is written in PyTorch (Paszke et al., 2017) to leverage dynamic graphs, and explicitly defines each sampled network in line with the memory-bank view to avoid obfuscating its inner workings behind (potentially more efficient) abstractions. We omit many hyperparameter details for brevity; full details are available in the appendices, along with visualizations of our best-found architectures.
+
+![](images/595b72d55146cf93b083899f2b6334bcbbccb572a57559de8ec2d5ccfa7d1e5c.jpg)  
+Figure 4: True error and SMASH validation error for 50 different random architectures on CIFAR-100. Red line is a least-squares best fit.
+
+# 4.1 TESTING THE SMASH CORRELATION
+
+First, we train a SMASH network for 300 epochs on CIFAR-100, using a standard annealing schedule (Huang et al., 2017), then sample 250 random architectures and evaluate their SMASH score on a held-out validation set formed of 5,000 random examples from the original training set. We then sort the architectures by their SMASH score and select every 5th architecture for full training and evaluation, using an accelerated training schedule of 30 epochs. For these networks, which we deem SMASHv1, the architecture uses a fixed memory bank size (though a variable number of banks in each block), a single fixed 3x3 conv in the main body of the op (rather than the variable $2 \mathbf { x } 2$ array of convs), a single group, and a fixed bottleneck ratio of 4. The variable elements comprise the read-write pattern, the number of output units, and the dilation factor of the $3 \mathrm { x } 3$ filter. When sampling architectures, we allocate a random, upper-bounded compute budget to each block.
+
+Under these conditions, we observe a correlation (Figure 4) between the SMASH score and the true validation performance, suggesting that SMASH-generated weights can be used to rapidly compare architectures. It is critical not to overstate this claim; this test is arguably a single datapoint indicating that the correlation holds in this scenario, but neither guarantees the correlation’s generality nor implies the range of conditions for which it will hold. We thus conduct a more thorough investigation of this correlation.
+
+We expect, based on preliminary experiments detailed in Appendix C, that the two key variables determining the strength of the correlation are the capacity of the HyperNet, and the ratio of HyperNetgenerated weights to freely learned weights. For the former, we reason that if the HyperNet lacks representational capacity, then it will be unable to learn an acceptable mapping between architectures and weights, and the generated weights will be too far from optimal to permit ranking. For the latter, we reason that if the main network has too many freely-learned weights relative to the number of dynamically generated weights, too much of its capacity will be "static" and ill-adapted to the wide range of architectures considered.
+
+Following these hypotheses, we repeat the first experiment multiple times, varying the architecture of the HyperNet as well as the $g$ hyperparameter, which controls the maximum layer width and consequentially the ratio of freely-learned vs dynamic weights. A higher value of $g$ corresponds to relatively fewer dynamically generated weights. We consider three different ratios and five different HyperNet configurations. For each setting of $g$ we begin by training five different SMASH networks, each employing one of the candidate configurations. After training, we sample 500 random architectures as before, and then evaluate all five SMASH scores, rank them according to their average, and then train every 10th resulting network normally. We train the resulting networks for a full 100 epochs (as opposed to the previous shortened schedule) and repeat training runs with a total of 5 different random seeds. Finally, we evaluate the strength and significance of the correlation between SMASH score and resulting validation performance (averaged for each architecture across runs) using Pearson’s R.
+
+Table 1: Pearson’s R $( \rho )$ and p-values (p) for various settings of $g$ .   
+
+<table><tr><td>Architecture</td><td>Params</td><td>p,g=4</td><td>p,g=4</td><td>p,g=8</td><td>p,g=8</td><td>p,g=16</td><td>p,g=16</td></tr><tr><td>G=[10,10,10],D=[12,15,8]</td><td>4M</td><td>-0.07</td><td>0.62</td><td>0.25</td><td>0.09</td><td>0.02</td><td>0.9</td></tr><tr><td>G=[20,20,20],D=[8,10,4]</td><td>5M</td><td>0.15</td><td>0.30</td><td>0.20</td><td>0.17</td><td>-0.09</td><td>0.53</td></tr><tr><td>G=[20,20,20],D=[12,15,8]</td><td>7M</td><td>0.31</td><td>0.02</td><td>0.24</td><td>0.09</td><td>0.05</td><td>0.76</td></tr><tr><td>G=[25,25,25],D=[16,16,16]</td><td>12M</td><td>0.38</td><td>0.06</td><td>0.24</td><td>0.09</td><td>0.04</td><td>0.77</td></tr><tr><td>G=[25,25,25],D=[25,25,25]</td><td>24M</td><td>0.04</td><td>0.77</td><td>0.20</td><td>0.17</td><td>0.11</td><td>0.45</td></tr></table>
+
+The results of this study are reported in Table 1. The first column details the choices of growth rate and depth for each of the three blocks of the HyperNet, and the second column the number of parameters for each architecture. The values of $g { = } 4$ , $g { = } 8$ , and $g { = } 1 6$ roughly correspond to average freely-learned vs dynamic weight ratios of 1:4, 1:2, and 2:1, though these vary somewhat with individual sampled architectures.
+
+Several trends are visible in this table. First, for $g { = } 4$ , we note that the strength of correlation increases with increased HyperNet capacity up to the fourth architecture with 12M parameters, but the correlation breaks down for the largest architecture. This suggests that the strength of correlation can indeed depend on the capacity of the HyperNet, but also that the HyperNet is either potentially susceptible to overfitting or that too large a HyperNet becomes too difficult to train and produces poor weights. Second, for $g { = } 8$ , we note that the correlation varies little with changes in architecture, but is weaker than the best correlations from $g = 4$ . This suggests that for a middling ratio, the capacity of the HyperNet has less effect on the correlation as there are fewer weights for it to adapt, but the strength of the correlation is weaker as each set of sampled weights is consequentially less optimal than for $g = 4$ . Third, we note a complete breakdown of the correlation for $g { = } 1 6$ , which is in line with our expectation that placing too much capacity in a single set of statically learned weights will prevent the SMASH network from properly adapting to individual architectures.
+
+# 4.2 ARCHITECTURAL GRADIENT DESCENT BY PROXY
+
+As an additional test of our method, we examine whether or not the HyperNet has learned to take into account the architecture definition in $c$ , or whether it ignores $c$ and naively generates an unconditional subspace of weights that happen to work well. We "trick" the HyperNet by sampling one architecture, but asking it to generate the weights for a different architecture by corrupting the encoding tensor $c$ (e.g. by shuffling the dilation values). For a given architecture, we find that SMASH validation performance is consistently highest when using the correct encoding tensor, suggesting that the HyperNet has indeed learned a passable mapping from architecture to weights.
+
+Following this, we posit that if the HyperNet learns a meaningful mapping $W = H ( c )$ , then the classification error approximate meas $E = f ( W , x ) = f ( H ( c ) , x )$ can be backpropagated to find to the architecture itself. If thi $\textstyle { \frac { d E } { d c } }$ , providing anolds true, then perturbing the architecture according to the $\frac { \dot { d } E } { d c }$ vector (within the constraints of our scheme) should allow us to guide the architecture search through a gradient descent-like procedure. Our preliminary tests with this idea did not yield better SMASH scores than randomly perturbing the architectural definition, though we suspect that this was in part due to our lack of an intuitively satisfying update rule for the discrete architecture space.
+
+# 4.3 TRANSFER LEARNING
+
+Models with weights initially learned on one large dataset frequently outperform models trained from scratch on a smaller dataset; it follows that architectures might display the same behavior. We test on STL-10 (Coates et al., 2011), a small dataset of $9 6 \mathbf { x } 9 6$ images similar to the CIFAR datasets. We compare the performance of the best-found architecture from CIFAR-100 (with weights trained from scratch on STL-10) to the best-found architecture from running SMASH on STL-10, and a
+
+WRN baseline. For these experiments, we make use of the full 5,000 images in the training set; in the following section we also include comparisons against a WRN baseline using the recommended 10-fold training split.
+
+In this case, we find that the best-found architecture from CIFAR-100 outperforms the best-found architecture from STL-10, achieving $1 7 . 5 4 \%$ and $2 0 . 2 7 5 \%$ error, respectively. For reference, a baseline WRN28-10 and WRN40-4 achieve respective $1 5 . 4 3 \%$ and $1 6 . 0 6 \%$ errors. This presents an interesting phenomenon: one the one hand, one might expect the architecture discovered on STL-10 to be better-tuned to STL-10 because it was specifically learned on that dataset. On the other hand, CIFAR-100 has significantly more training examples, potentially making it a better dataset for distinguishing between good architectures, i.e. accuracy on CIFAR-100 is more indicative of generality. The better performance of the architecture found on CIFAR-100 would seem to favor the latter hypothesis, suggesting that architecture search benefits from larger training sets more so than domain specificity.
+
+We next investigate how well our best-found CIFAR-100 architecture performs on ModelNet10 (Wu et al., 2015), a 3D object classification benchmark. We train on the voxelated instances of the ModelNet10 training set using the settings of (Brock et al., 2016), and report accuracy on the ModelNet10 test set. Our 8M parameter model achieves an accuracy of $9 3 . 2 8 \%$ , compared to a $9 3 . 6 1 \%$ accuracy from a hand-designed Inception-ResNet (Brock et al., 2016) with 18M parameters trained on the larger ModelNet40 dataset.
+
+# 4.4 BENCHMARKING
+
+We run SMASH on CIFAR-10 and 100, augmenting our search space from the initial correlation experiment to include variable filter sizes, variable groups, and the full variable op structure shown in Figure 2, and denote the resulting networks SMASHv2. We report the final test performance of the two resulting networks with the highest SMASH scores on CIFAR-10 and 100 in Table 2.
+
+Next, we take our best-found SMASHv2 architecture from CIFAR-100 and train it on STL-10 (Coates et al., 2011) using the recommended 10-fold training splits, and ImageNet32x32 (Chrabaszcz et al., 2017). We compare against Wide ResNet baselines from our own experiments in Tables 3 and those reported by (Chrabaszcz et al., 2017) in 4. Noting the better performance of WRN40-4 on STL-10, we also train a variant of our best architecture with only a single main convolution and 3x3 filters, to comparably reduce the number of parameters.
+
+Table 2: Error rates $( \% )$ on CIFAR-10 and CIFAR-100 with standard data augmentation $( + )$   
+
+<table><tr><td>Method</td><td>Resources</td><td>Depth</td><td>Params</td><td>C10+</td><td>C100+</td></tr><tr><td>FractalNet (Larsson et al.,2017)</td><td></td><td>21</td><td>38.6M</td><td>5.22</td><td>23.30</td></tr><tr><td>with Dropout/Drop-path</td><td></td><td>21</td><td>38.6M</td><td>4.60</td><td>23.73</td></tr><tr><td rowspan="2">Stochastic Depth (Huang et al.,2016)</td><td rowspan="2"></td><td>110</td><td>1.7M</td><td>5.23</td><td>24.58</td></tr><tr><td>1202</td><td>10.2M</td><td>4.91</td><td>-</td></tr><tr><td colspan="2">WideResNet (Zagoruyko and Komodakis.,2016)</td><td>16</td><td>11.0M</td><td>4.81</td><td>22.07</td></tr><tr><td colspan="2">DenseNet-BC (k = 24) (Huang et al.,2017)</td><td>28</td><td>36.5M</td><td>4.17</td><td>20.50</td></tr><tr><td colspan="2">DenseNet-BC (k = 40)</td><td>250</td><td>15.3M</td><td>3.62</td><td>17.60</td></tr><tr><td colspan="2">Shake-Shake (Gastaldi,2017)</td><td>190</td><td>25.6M</td><td>3.46</td><td>17.18</td></tr><tr><td colspan="2">NASwRL(Zoph and Le,2017) 800 GPUs,? days</td><td>26</td><td>26.2M</td><td>2.86</td><td>15.85</td></tr><tr><td colspan="2">NASNet-A(Zoph et al.,2017)</td><td>39</td><td>32.0M</td><td>3.84</td><td>--</td></tr><tr><td colspan="2">MetaQNN (Baker et al., 2017) 10 GPUs,8-10 days</td><td>20</td><td>3.3M</td><td>3.41</td><td></td></tr><tr><td colspan="2">Large-Scale Evolution (Real et al., 2017) 250 GPUs,10 days</td><td>9</td><td>11.18M</td><td>6.92</td><td>27.14</td></tr><tr><td colspan="2"></td><td>-</td><td>5.4M</td><td>5.40</td><td>-</td></tr><tr><td colspan="2">CGP-CNN (Suganuma et al.,2017)</td><td>-</td><td>40.4 M</td><td>-</td><td>23.7</td></tr><tr><td colspan="2">SMASHv1 (ours) 1 GPU,3 days</td><td>-</td><td>1.68M</td><td>5.98</td><td>-</td></tr><tr><td colspan="2">SMASHv2 (ours)</td><td>116 211</td><td>4.6M 16M</td><td>5.53 4.03</td><td>22.07</td></tr></table>
+
+Our SMASHv2 nets with 16M parameters achieve final test errors of $2 0 . 6 0 \%$ on CIFAR-100 and $4 . 0 3 \%$ on CIFAR-10. This performance is not quite on par with state-of-the-art hand-designed networks, but compares favorably to other automatic design methods that employ RL (Baker et al., 2017; Zoph and Le, 2017) or evolutionary methods (Real et al., 2017; Suganuma et al., 2017). Our networks outperform Large-Scale Evolution (Real et al., 2017) despite requiring significantly less time to discover (though not starting from trivial models) and 10 orders of magnitude less compute. Our method outperforms MetaQNN (Baker et al., 2017) but lags behind Neural Architecture Search (Zoph and Le, 2017), though both methods require vastly more computation time, and unlike Neural Architecture Search, we do not postprocess our discovered architectures through grid search.
+
+Table 3: Error rates $( \% )$ on STL-10.   
+
+<table><tr><td>Model</td><td>Params</td><td>Error</td></tr><tr><td>WRN-40-4</td><td>8.95M</td><td>35.02 ± 1.14</td></tr><tr><td>WRN-28-10</td><td>36.5M</td><td>36.69 ± 2.06</td></tr><tr><td>SMASHv2</td><td>16.2M</td><td>41.52 ± 2.10</td></tr><tr><td>SMASHv2 (3x3)</td><td>4.38M</td><td>37.76 ± 0.58</td></tr></table>
+
+Table 4: Error rates $( \% )$ on Imagenet32x32.   
+
+<table><tr><td>Model</td><td>Params</td><td>Top-1</td><td>Top-5</td></tr><tr><td>WRN-28-2</td><td>1.6M</td><td>56.92</td><td>30.92</td></tr><tr><td>WRN-28-5</td><td>9.5M</td><td>45.36</td><td>21.36</td></tr><tr><td>WRN-28-10</td><td>37.1M</td><td>40.96</td><td>18.87</td></tr><tr><td>SMASHv2</td><td>16.2M</td><td>38.62</td><td>16.33</td></tr></table>
+
+# 5 FUTURE WORK
+
+We believe this work opens a number of future research paths. The SMASH method itself has several simplistic elements that might easily be improved upon. During training, we sample each element of the configuration one-by-one, independently, and uniformly among all possible choices. A more intelligent method might employ Bayesian Optimization (Snoek et al., 2015) or HyperBand (Li et al., 2017) to guide the sampling with a principled tradeoff between exploring less-frequently sampled architectures against those which are performing well. One might employ a second parallel worker constantly evaluating validation performance throughout training to provide signal to an external optimizer, and change the optimization objective to simultaneously maximize performance while minimizing computational costs. One could also combine this technique with RL methods (Zoph and Le, 2017) and use a policy gradient to guide the sampling. Another simple technique (which our code nominally supports) is using the HyperNet-generated weights to initialize the resulting network and accelerate training, similar to Net2Net (Chen et al., 2016).
+
+Our architecture exploration is fairly limited, and for the most part involves variable layer sizes and skip connections. One could envision a multiscale SMASH that also explores low-level design, varying things such as the activation at each layer, the order of operations, the number of convolutions in a given layer, or whether to use convolution, pooling, or more exotic blocks. One might also search over network-wide design patterns rather than randomly selecting the read-write pattern at each layer. Alternatively, one could consider varying which elements of the network are generated by the HyperNet, which are fixed learned parameters, and one might even make use of fixed unlearned parameters such as Gabor Filters or wavelets.
+
+Our memory-bank view also opens up new possibilities for network design. Each layer’s read and write operations could be designed to use a learned softmax attention mechanism, such that the read and write locations are determined dynamically at inference time. We also do not make use of memory in the traditional "memory-augmented" sense, but we could easily add in this capacity by allowing information in the memory banks to persist, rather than zeroing them at every training step. We also only explore one definition of reading and writing, and one might for example change the "write" operation to either add to, overwrite, or perhaps even multiply (a la gated networks (Srivastava et al., 2015)) the existing tensor in a given bank.
+
+# 6 CONCLUSION
+
+In this work, we explore a technique for accelerating architecture selection by learning a model over network parameters, conditioned on the network’s parametric form. We introduce a flexible scheme for defining network connectivity patterns and generating network weights for highly variable architectures. Our results demonstrate a correlation between performance using suboptimal weights generated by the auxiliary model and performance using fully-trained weights, indicating that we can efficiently explore the architectural design space through this proxy model. Our method achieves competitive, though not state-of-the-art performance on several datasets.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Harri Edwards, David Ha, Daniel Angelov, Emmanuel Kahembwe, and Iain Murray for their insight. This research was made possible by grants and support from Renishaw plc and the Edinburgh Centre For Robotics. The work presented herein is also partially funded under the European H2020 Programme BEACONING project, Grant Agreement nr. 687676.
+
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+
+# APPENDIX A: HYPERPARAMETERS
+
+We briefly describe they hyperparameters used for the SMASH network in our experiments. The SMASHv1 network has memory banks with $N = 6$ channels each, a maximum of 240 memory banks per block (though on average less than half that number), and a depth compression ratio of $D = 3$ . Each layer’s number of units is uniformly sampled between 6 and 42 (along even multiples of 6), and its dilation factor is uniformly sampled between 1 and 3 (with 1 representing no dilation and 3 representing 2 zeros inserted between each filter). We employ a constant bottlneck ratio of 4 as in (Huang et al., 2017), so the output of the HyperNet-generated 1x1 convolution is always 4 times the number of output units. We constrain the main network to have a maximum budget of 16M parameters, though due to our sampling procedure we rarely sample networks with more than 5M parameters.
+
+Our SMASHv2 networks have variable memory bank sizes at each blocks, which we constrain to be multiples of $N = 8$ up to $N _ { m a x } = 6 4$ . We sample filter sizes from [3,5,7], and sample dilation values such that the max spatial extent of a filter in any direction is 9. We sample convolutional groups as factors of the base $N$ value (so [1,2,4,8] for these networks). We put some hand-designed priors on the choice of op configuration (i.e. which convolutions are active), giving slight preference to having all four convolutions active. For SMASHv2 nets we employ a slightly more complex bottleneck ratio: the output of the 1x1 conv is equal to the number of incoming channels while that number is less than twice the number of output units, at which point it is capped (so, a maximum bottleneck ratio of 2).
+
+Our HyperNet is a DenseNet, designed ad-hoc to resemble the DenseNets in the original paper (Huang et al., 2017) within the confines of our encoding scheme, and to have round numbers of channels. It consists of a standard (non-bottleneck) Dense Block with $8 \ : 3 \mathrm { x } 3 $ convolutional layers and a growth rate of 10, followed by a 1x1 convolution that divides the number of channels in two, a Dense Block with 10 layers and a growth rate of 10, another compressing 1x1 convolution, a Dense Block with 4 layers and a growth rate of 10, and finally a 1x1 convolution with the designated number of output channels. We use Leaky ReLU with a negative slope of 0.02 as a defense against NaNs, as standard ReLU would obfuscate their presence when we had bugs in our early code revisions; we have not experimented with other activations.
+
+# APPENDIX B: ENCODING SCHEME DETAILS
+
+We adopt a scheme for the layout of the embedding tensor to facilitate flexibility, compatibility with the convolutional toolbox available in standard libraries, and to make each dimension interpretable. First, we place some constraints on the hyperparameters of the main network: each layer’s number of output units must be divisible by the memory bank size $N$ and be less than $N _ { m a x }$ , and the number of input units must be divisible by $D$ , where $N$ is the number of channels in each memory bank, and $N _ { m a x }$ and $D$ are chosen by the user. Applying these constraints allows us to reduce the size of the embedding vector by $D N ^ { 2 }$ , as we will see shortly.
+
+The input to a standard 2D CNN is $x \in \mathbb { R } ^ { B \times C \times H \times L }$ , where $B , C , H$ , and $L$ respectively represent the Batch, Channel, Height, and Length dimensions. Our embedding tensor is $c \in \mathbb { R } ^ { 1 \times ( 2 \bar { M } + d _ { m a x } ) \times ( \bar { N } _ { m a x } / N ) ^ { 2 } \times n _ { c h } / { D } }$ where $M$ is the maximum number of memory banks in a block, $d _ { m a x }$ is the maximum kernel dilation, and $n _ { c h }$ is the sum total of input channels to the 1x1 convs of the main network.
+
+The conditional embedding $c$ is a one-hot encoding of the memory banks we read and write at each layer. It has $2 M + d _ { m a x }$ channels, where the first $M$ channels represent which banks are being read from, the next $M$ channels represent which banks are being written to, and the final $d _ { m a x }$ channels are a one-hot encoding of the dilation factor applied to the following 3x3 convolution. The height dimension corresponds to the number of units at each layer, and the length dimension corresponds to the network depth in terms of the total number of input channels. We keep the Batch dimension at 1 so that no signals propagate wholly independently through the HyperNet. Figure 3 shows an example of a small randomly sampled network, its equivalent memory bank representation, and how the read-write pattern is encoded in $c$ . The dilation encoding is omitted in Figure 3 for compactness.
+
+Our HyperNet has $4 D N ^ { 2 }$ output channels, such that the output of the HyperNet is $W ~ = ~ H ( c ) ~ \in ~$ $\mathbb { R } ^ { 1 \times 4 D \bar { N } ^ { 2 } \times ( N _ { m a x } / N ) ^ { 2 } \times n _ { c h } / D }$ , which we reshape to $W \in \mathbb { R } ^ { N _ { m a x } \times 4 N _ { m a x } n _ { c h } \times 1 \times 1 }$ . We generate the weights for the entire main network in a single pass, allowing the HyperNet to predict weights at a given layer based on weights at nearby layers. The HyperNet’s receptive field represents how far up or down the network it can look to predict parameters at a given layer. As we traverse the main network, we slice $W$ along its second axis according to the number of incoming channels, and slice along the first axis according to the width of the given layer.
+
+# APPENDIX C: EXPERIMENT DETAILS
+
+At each training step, we sample a network architecture block-by-block, with a random (but upper bounded) computational budget allocated to each block. For SMASHv1, We use memory banks with $N = 6$ channels each, constrain the number of incoming memory banks to be a multiple of 3 $D = 3$ ), and constrain the number of output units at each layer to be a multiple of 6 (with $N _ { m a x } = 4 2$ ) for compatibility with the memory layout.
+
+Our HyperNet is a 26 layer DenseNet, each layer of which comprises a Leaky ReLU activation followed by a 3x3 convolution with simplified WeightNorm and no biases. We do not use bottleneck blocks, dropout, or other normalizers in the HyperNet.
+
+When sampling our SMASHv2 networks for evaluation, we first sample 500 random architectures, then select the architecture with the highest score for further evaluation. We begin by perturbing this architecture, with a $5 \%$ chance of any individual element being randomly resampled, and evaluate 100 random perturbations from this base. We then proceed with 100 perturbations in a simple Markov Chain, where we only accept an update if it has a better SMASH score on the validation set.
+
+When training a resulting network we make all parameters freely learnable and replace simple WeightNorm with standard BatchNorm. We tentatively experimented with using SMASH generated weights to initialize a resulting net, but found standard initialization strategies to work better, either because of a yet-undiscovered bug in our code, or because of the disparity between the dynamics of the SMASH network using WeightNorm and the resulting network using BatchNorm.
+
+In line with our claim of "one-shot" model search, we keep our exploration of the SMASH design space to a minimum. We briefly experimented with three different settings for N and D, and use a simple, ad-hoc DenseNet architecture for the HyperNet, which we do not tune. We investigated the choice of architecture while examining the SMASH correlation, but stick to the original ad-hoc design for all benchmark experiments.
+
+When training SMASH, we use Adam (Kingma and Ba, 2014) with the initial parameters proposed by (Radford et al., 2015) When training a resulting network, we use Nesterov Momentum with an initial step size of 0.1 and a momentum value of 0.9. For all tests other than the initial SMASHv1 experiments, we employ a cosine annealing schedule (Loshchilov and Hutter, 2017) without restarts (Gastaldi, 2017).
+
+For the CIFAR experiments, we train the SMASH network for 100 epochs and the resulting networks for 300 epochs, using a batch size of 50 on a single GPU. On ModelNet10, we train for 100 epochs. On ImageNet32x32, we train for 55 epochs. On STL-10, we train for 300 epochs when using the full training set, and 500 epochs when using the 10-fold training splits.
+
+For ModelNet-10 tests, we employ 3x3x3 filters (rather than fully variable filter size) to enable our network to fit into memory and keep compute costs manageable, hence why our model only has 8M parameters compared to the base 16M parameters.
+
+All of our networks are pre-activation, following the order BN-ReLU-Conv if BatchNorm is used, or ReLU-Conv if WeightNorm is used. Our code supports both pre- and post-activation, along with a variety of other options such as which hyperparameters to vary and which to keep constant.
+
+![](images/5d1f4af74825eb0d9681a53078e49a9ac915aabbcec6dfb348f30d0ced1a15be.jpg)  
+Figure 5: (a) SMASH correlation with a crippled HyperNet. Error bars represent 1 standard deviation. (b) SMASH scores vs. rank using average scores from three HyperNets with different seeds.
+
+While investigating the SMASH correlation, we initially conducted two brief experiments to guide the choice of experiments in the remainder of the investigation. First, experiment, we train a low-budget SMASH network (to permit more rapid testing) with a much smaller HyperNet relative to the main network (though still the standard ratio of generated to freely learned weights). We expect the decreased capacity HyperNet to be less able to learn to generate good weights for the full range of architectures, and for the correlation between SMASH score and true performance to therefore be weak or nonexistent. The results of this study are shown in Figure 5(a), where we arguably observe a breakdown of the correlation. In addition to repeat trials for each resulting net, we also train two additional SMASH networks with different random seeds and compare their predicted performance against the initial SMASH net in figure 5(b), to get a brief sense for how these values can vary across training runs.
+
+Next, we train a high-budget SMASH network and drastically increase the ratio of normally learned parameters to HyperNet-generated parameters, such that the majority of the net’s model capacity is in non-generated weights. Under these conditions, the validation errors achieved with SMASH-generated weights are much lower than validation errors achieved with an equivalent SMASH network with the typical ratio, but the resulting top models are not as performant and we found that (in the very limited number of correlation tests we performed) the SMASH score did not correlate with true performance. This highlights two potential pitfalls: first, if the HyperNet is not responsible for enough of the network capacity, then the aggregate generated and learned weights may not be sufficiently well-adapted to each sampled architecture, and therefore too far from optimal to be used in comparing architectures. Second, comparing SMASH scores for two separate SMASH networks can be misleading, as the SMASH score is a function of both the normally learned and generated weights, and a network with more fixed weights may achieve better SMASH scores even if the resulting nets are no better.
+
+![](images/4c39c5c235f4e196f97a2a70850037361ebf4e4b23dd3d04b9ef0f9272e5c0ab.jpg)  
+Figure 6: A simplified version of our best-found SMASHv2 architecture from CIFAR-100 with the highest SMASH score. N represents number of output units, op which convolutions are active, groups convolution groups, ks kernel size, and d dilation.
+
+![](images/e29c5f7e3dcd5405f7ea080919cb5b2d57a5eb8e0e9f8774afd3ad793b0e0a32.jpg)  
+Figure 7: A simplified version of our best-found SMASHv1 architecture. N represents number of output units, and d represents the dilation factor for the $3 { \bf x } 3$ filter.
+
+![](images/ecba6fe5f90ae0dea19b7e5ba59e6d15b128a76b06d4aa748772a93916e3e399.jpg)  
+Figure 8: An expanded (though still partially simplified) view of the first block of our best SMASHv2 net.
+
+![](images/1b29b8eb8c837726eaddbe1177025ed2f35b3991f796273f0f4ce6e1559cbf51.jpg)  
+Figure 9: An expanded (though still partially simplified) view of the second block of our best SMASHv1 net.
+
+![](images/061e3d60259fbc08cd0d5a6176f696efb5846284ebb8c329a8f086ddddc8b713.jpg)  
+Figure 10: An expanded (though still partially simplified) view of the final block of our best SMASHv2 net. Floating paths are an artifact of the graph generation process, and are actually attached to the nearest rectangular node.
+
+![](images/6b01783b7554e636c2fa6b1ad27e45da7d663a27bc89ee73bbcc2dcf29ad01f3.jpg)  
+Figure 11: An expanded (though still partially simplified) view of the first block of our best SMASHv1 net. Floating paths are an artifact of the graph generation process, and are actually attached to the nearest rectangular node.
+
+![](images/74c4198a27b0d4ec70f9fda9e2d31a99c7274e82cc79298193583fa44b382a13.jpg)  
+Figure 12: An expanded (though still partially simplified) view of the second block of our best SMASHv1 net. Floating paths are an artifact of the graph generation process, and are actually attached to the nearest rectangular node.
+
+![](images/30b0be4dbb9b08d87419fd96f6f5b4023d21709d48ada9645ab80ec2e56c0e8b.jpg)  
+Figure 13: An expanded (though still partially simplified) view of the final block of our best SMASHv1 net. Floating paths are an artifact of the graph generation process, and are actually attached to the nearest rectangular node.