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+# VARIATIONAL CONTINUAL LEARNING
+
+Cuong V. Nguyen, Yingzhen Li, Thang D. Bui, Richard E. Turner Department of Engineering, University of Cambridge {vcn22,yl494,tdb40,ret26}@cam.ac.uk
+
+# ABSTRACT
+
+This paper develops variational continual learning (VCL), a simple but general framework for continual learning that fuses online variational inference (VI) and recent advances in Monte Carlo VI for neural networks. The framework can successfully train both deep discriminative models and deep generative models in complex continual learning settings where existing tasks evolve over time and entirely new tasks emerge. Experimental results show that VCL outperforms stateof-the-art continual learning methods on a variety of tasks, avoiding catastrophic forgetting in a fully automatic way.
+
+# 1 INTRODUCTION
+
+Continual learning (also called life-long learning and incremental learning) is a very general form of online learning in which data continuously arrive in a possibly non i.i.d. way, tasks may change over time (e.g. new classes may be discovered), and entirely new tasks can emerge (Schlimmer & Fisher, 1986; Sutton & Whitehead, 1993; Ring, 1997). What is more, continual learning systems must adapt to perform well on the entire set of tasks in an incremental way that avoids revisiting all previous data at each stage. This is a key problem in machine learning since real world tasks continually evolve over time (e.g. they suffer from covariate and dataset shift) and the size of datasets often prohibits frequent batch updating. Moreover, practitioners are often interested in solving a set of related tasks that benefit from being handled jointly in order to leverage multi-task transfer. Continual learning is also of interest to cognitive science, being an intrinsic human ability.
+
+The ubiquity of deep learning means that it is important to develop deep continual learning methods. However, it is challenging to strike a balance between adapting to recent data and retaining knowledge from old data. Too much plasticity leads to the infamous catastrophic forgetting problem (McCloskey & Cohen, 1989; Ratcliff, 1990; Goodfellow et al., 2014a) and too much stability leads to an inability to adapt. Recently there has been a resurgence of interest in this area. One approach trains individual models on each task and then carries out a second stage of training to combine them (Lee et al., 2017). A more elegant and more flexible approach maintains a single model and uses a single type of regularized training that prevents drastic changes in the parameters which have a large influence on prediction, but allows other parameters to change more freely (Li & Hoiem, 2016; Kirkpatrick et al., 2017; Zenke et al., 2017). The approach developed here follows this venerable work, but is arguably more principled, extensible and automatic.
+
+This paper is built on the observation that there already exists an extremely general framework for continual learning: Bayesian inference. Critically, Bayesian inference retains a distribution over model parameters that indicates the plausibility of any setting given the observed data. When new data arrive, we combine what previous data have told us about the model parameters (the previous posterior) with what the current data are telling us (the likelihood). Multiplying and renormalizing yields the new posterior, from which point we can recurse. Critically, the previous posterior constrains parameters that strongly influence prediction, preventing them from changing drastically, but it allows other parameters to change. The wrinkle is that exact Bayesian inference is typically intractable and so approximations are required. Fortunately, there is an extensive literature on approximate inference for neural networks. We merge online variational inference (VI) (Ghahramani & Attias, 2000; Sato, 2001; Broderick et al., 2013) with Monte Carlo VI for neural networks (Blundell et al., 2015) to yield variational continual learning (VCL). In addition, we extend VCL to include a small episodic memory by combining VI with the coreset data summarization method (Bachem et al., 2015; Huggins et al., 2016). We demonstrate that the framework is general, applicable to both deep discriminative models and deep generative models, and that it yields excellent performance.
+
+# 2 CONTINUAL LEARNING BY APPROXIMATE BAYESIAN INFERENCE
+
+Consider a discriminative model that returns a probability distribution over an output $y$ given an input $_ { \textbf { \em x } }$ and parameters $\pmb { \theta }$ , that is $p ( y | \pmb \theta , \pmb x )$ . Below we consider the specific case of a softmax distribution returned by a neural network with weight and bias parameters, but we keep the development general for now. In the continual learning setting, the goal is to learn the parameters of the model from a set of sequentially arriving datasets $\{ \pmb { x } _ { t } ^ { ( n ) } , \pmb { y } _ { t } ^ { ( n ) } \} _ { n = 1 } ^ { N _ { t } }$ where, in principle, each might contain a single datum, $N _ { t } = 1$ . Following a Bayesian approach, a prior distribution $p ( \pmb \theta )$ is placed over . The posterior distribution after seeing $T$ datasets is recovered by applying Bayes’ rule:
+
+$$
+p ( \pmb \theta | \mathcal D _ { 1 : T } ) \propto p ( \pmb \theta ) \prod _ { t = 1 } ^ { T } \prod _ { n _ { t } = 1 } ^ { N _ { t } } p ( y _ { t } ^ { ( n _ { t } ) } | \pmb \theta , x _ { t } ^ { ( n _ { t } ) } ) = p ( \pmb \theta ) \prod _ { t = 1 } ^ { T } p ( \mathcal D _ { t } | \pmb \theta ) \propto p ( \pmb \theta | \mathcal D _ { 1 : T - 1 } ) p ( \mathcal D _ { T } | \pmb \theta ) .
+$$
+
+Here the input dependence has been suppressed on the right hand side to lighten notation. We have used the shorthand $\mathcal { D } _ { t } ~ = ~ \{ y _ { t } ^ { ( n ) } \} _ { n = 1 } ^ { N _ { t } }$ } tn=1. Importantly, a recursion has been identified whereby the posterior after seeing the $T$ -th dataset is produced by taking the posterior after seeing the $( T - \bar { 1 } )$ -th dataset, multiplying by the likelihood and renormalizing. In other words, online updating emerges naturally from Bayes’ rule.
+
+In most cases the posterior distribution is intractable and approximation is required, even when forming the first posterior $p ( \pmb \theta | \mathcal D _ { 1 } ) \approx q _ { 1 } ( \pmb \theta ) = \mathrm { p r o j } ( p ( \pmb \theta ) p ( \mathcal D _ { 1 } | \pmb \theta ) )$ . Here ${ q ( \pmb \theta ) = \mathrm { p r o j } ( p ^ { * } ( \pmb \theta ) ) }$ denotes a projection operation that takes the intractable un-normalized distribution $p ^ { * } ( \pmb { \theta } )$ and returns a tractable normalized approximation $q ( \pmb \theta )$ . The field of approximate inference provides several choices for the projection operation including i) Laplace’s approximation, ii) variational KL minimization, iii) moment matching, and iv) importance sampling. Having approximated the first posterior distribution, subsequent approximations can be produced recursively by combining the approximate posterior distribution with the likelihood and projecting, that is $\dot { p } ( \pmb \theta | \mathcal { D } _ { 1 : T } ) \approx \acute { q } _ { T } ( \pmb \theta ) \stackrel { - } { = }$ $\operatorname { p r o j } ( q _ { T - 1 } ( \pmb { \theta } ) p ( \mathcal { D } _ { T } | \pmb { \theta } ) )$ . In this way online updating is supported. This general approach leads, for the four projection operators previously identified, to i) Laplace propagation (Smola et al., 2004), ii) online VI (Ghahramani & Attias, 2000; Sato, 2001) also known as streaming variational Bayes (Broderick et al., 2013), iii) assumed density filtering (Maybeck, 1982) and iv) sequential Monte Carlo (Liu & Chen, 1998). In this paper the online VI approach is used as it typically outperforms the other methods for complex models in the static setting (Bui et al., 2016) and yet it has not been applied to continual learning of neural networks.
+
+2.1 VARIATIONAL CONTINUAL LEARNING (VCL) AND EPISODIC MEMORY ENHANCEMENT
+
+Variational continual learning employs a projection operator defined through a KL divergence minimization over the set of allowed approximate posteriors $\mathcal { Q }$ ,
+
+$$
+\boldsymbol { q } _ { t } ( \pmb { \theta } ) = \arg \operatorname* { m i n } _ { \pmb { q } \in \mathcal { Q } } \mathrm { K L } \Big ( \boldsymbol { q } ( \pmb { \theta } ) \| \frac { 1 } { Z _ { t } } \boldsymbol { q } _ { t - 1 } ( \pmb { \theta } ) p ( \mathcal { D } _ { t } | \pmb { \theta } ) \Big ) , \mathrm { f o r } t = 1 , 2 , . . . , T .
+$$
+
+The zeroth approximate distribution is defined to be the prior, $q _ { 0 } ( \pmb { \theta } ) = p ( \pmb { \theta } )$ . $Z _ { t }$ is the intractable normalizing constant of $p _ { t } ^ { * } ( \pmb { \theta } ) = q _ { t - 1 } ( \pmb { \theta } ) p ( \mathcal { D } _ { t } | \pmb { \theta } )$ and is not required to compute the optimum.
+
+VCL will perform exact Bayesian inference if the true posterior is a member of the approximating family, $p ( \pmb { \theta } | \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } , \ldots , \mathcal { D } _ { t } ) \in \mathcal { Q }$ at every step $t$ . Typically this will not be the case and we might worry that performing repeated approximations may accumulate errors causing the algorithm to forget old tasks, for example. Furthermore, the minimization at each step may also be approximate (e.g. due to employing an additional Monte Carlo approximation) and so additional information may be lost. In order to mitigate this potential problem, we extend VCL to include a small representative set of data from previously observed tasks that we call the coreset. The coreset is analogous to an episodic memory that retains key information (in our case, important training data points) from previous tasks which the algorithm can revisit in order to refresh its memory of them. The use of an episodic memory for continual learning has also been explored by Lopez-Paz & Ranzato (2017).
+
+# Algorithm 1 Coreset VCL
+
+Input: Prior $p ( \pmb \theta )$
+
+Output: Variational and predictive distributions at each step $\{ q _ { t } ( \pmb \theta ) , p ( y ^ { * } | \pmb x ^ { * } , \mathcal D _ { 1 : t } ) \} _ { t = 1 } ^ { T }$
+
+Initialize the coreset and variational approximation: $C _ { 0 } \gets \emptyset , \tilde { q } _ { 0 } \gets p$
+
+for $t = 1 \dots T$ do
+
+Observe the next dataset $\mathcal { D } _ { t }$
+
+$C _ { t } \gets$ update the coreset using $C _ { t - 1 }$ and $\mathcal { D } _ { t }$ .
+
+Update the variational distribution for non-coreset data points:
+
+$$
+\begin{array} { r } { \widetilde { q } _ { t } ( \pmb { \theta } ) \gets \arg \operatorname* { m i n } _ { \pmb { q } \in \mathcal { Q } } \mathrm { K L } \big ( q ( \pmb { \theta } ) \ | | \ \frac { 1 } { \widetilde { Z } } \widetilde { q } _ { t - 1 } ( \pmb { \theta } ) p ( \mathcal { D } _ { t } \cup C _ { t - 1 } \setminus C _ { t } | \pmb { \theta } ) \big ) . } \end{array}
+$$
+
+Compute the final variational distribution (only used for prediction, and not propagation):
+
+$$
+\begin{array} { r } { q _ { t } ( \pmb { \theta } )  \arg \operatorname* { m i n } _ { \pmb { q } \in \mathcal { Q } } \mathrm { K L } \big ( \pmb { q } ( \pmb { \theta } ) \ | | \ \frac { 1 } { Z } \tilde { q } _ { t } ( \pmb { \theta } ) p ( C _ { t } | \pmb { \theta } ) \big ) . } \end{array}
+$$
+
+Perform prediction at test input $\pmb { x } ^ { * }$ : $\begin{array} { r } { p ( y ^ { * } | x ^ { * } , \mathcal { D } _ { 1 : t } ) = \int q _ { t } ( \pmb { \theta } ) p ( y ^ { * } | \pmb { \theta } , \pmb { x } ^ { * } ) \mathrm { d } \pmb { \theta } . } \end{array}$ .
+
+# end for
+
+Algorithm 1 describes coreset VCL. For each task, the new coreset $C _ { t }$ is produced by selecting new data points from the current task and a selection from the old coreset $C _ { t - 1 }$ . Any heuristic can be used to make these selections, e.g. $K$ data points can be selected at random from $\mathcal { D } _ { t }$ and added to $C _ { t - 1 }$ to form an unbiased new coreset $C _ { t }$ . Alternatively, the greedy $K$ -center algorithm (Gonzalez, 1985) can be used to return $K$ data points that are guaranteed to be spread throughout the input space. Next, a variational recursion is developed. Bayes’ rule can be used to decompose the true posterior taking care to break out contributions from the coreset,
+
+$$
+\begin{array} { r l }  p ( \pmb { \theta } | \mathcal { D } _ { 1 : t } ) \propto \underbrace { p ( \pmb { \theta } | \mathcal { D } _ { 1 : t } \setminus C _ { t } ) } _  \qquad \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \underbrace { p ( C _ { t } | \pmb { \theta } ) } _ { \qquad \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \approx \frac { } { } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } p ( \pmb { \theta } ) p ( C _ { t } | \pmb { \theta } ) . } \end{array}
+$$
+
+Here the variational distribution $\tilde { q } _ { t } ( \pmb \theta )$ approximates the contribution to the posterior from the noncoreset data points. A recursion is identified by noting
+
+$$
+\begin{array} { r }  \begin{array} { l l } { \boldsymbol { \mathfrak { o } } ( \theta | \mathcal { D } _ { 1 : t } \backslash C _ { t } ) = \underbrace { p ( \theta | \mathcal { D } _ { 1 : t - 1 } \backslash C _ { t - 1 } ) } _ { \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } p ( C _ { t - 1 } \backslash C _ { t } | \theta ) } } & { \underbrace { p ( \mathcal { D } _ { t } \backslash C _ { t } | \theta ) } _ { \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } p ( \mathcal { D } _ { t } \cup C _ { t - 1 } \backslash C _ { t } | \theta ) . } \end{array} } \end{array}
+$$
+
+Hence propagation is performed via $\tilde { q } _ { t } ( \pmb { \theta } ) = \mathrm { p r o j } \big ( \tilde { q } _ { t - 1 } ( \pmb { \theta } ) p ( \mathcal { D } _ { t } \cup C _ { t - 1 } \setminus C _ { t } | \pmb { \theta } ) \big )$ with VCL employing the variational KL projection. A further projection step is needed before performing prediction $q _ { t } \bar { ( } \pmb { \theta } ) = \mathrm { p r o j } ( \tilde { q } _ { t } ( \pmb { \theta } ) p ( \tilde { C } _ { t } | \pmb { \theta } ) )$ . In this way the coreset is incorporated into the approximate posterior directly before prediction which helps mitigate any residual forgetting. From a more general perspective, coreset VCL is equivalent to a message-passing implementation of VI in which the coreset data point updates are scheduled after updating the other data.
+
+# 3 VARIATIONAL CONTINUAL LEARNING IN DEEP DISCRIMINATIVE MODELS
+
+The VCL framework is general and can be applied to many discriminative probabilistic models. Here we apply it to continual learning of deep fully-connected neural network classifiers. Before turning to the application of VCL, we first consider the architecture of neural networks suitable for performing continual learning. In simple instances of discriminative continual learning, where data are arriving in an i.i.d. way or where only the input distribution $p ( { \pmb x } _ { 1 : T } )$ changes over time, a standard single-head discriminative neural network suffices. In many cases the tasks, although related, might involve different output variables. Standard practice in multi-task learning (Bakker & Heskes, 2003) uses networks that share parameters close to the inputs but with separate heads for each output, hence multi-head networks. Graphical models depicting the network architecture for deep discriminative and deep generative models are shown in fig. 1.
+
+Recent work has explored more advanced structures for continual learning (Rusu et al., 2016) and multi-task learning more generally (Swietojanski & Renals, 2014; Rebuffi et al., 2017). These architectural advances are complementary to the new learning schemes developed here and a synthesis of the two would be potentially more powerful. Moreover, a general solution to continual learning would perform automatic continual model building adding new bespoke structure to the existing model as new tasks are encountered. Although this is a very interesting research direction, here we make the simplifying assumption that the model structure is known a priori.
+
+![](images/760597b2b90c893ea40bae8010ec046344081d98d6db0e57cf225e731258bee5.jpg)  
+Figure 1: Schematics of the multi-head networks tested in the paper, including both the graphical model (left) and network architecture (right). (a) A multi-head discriminative model showing how network parameters might be shared during training. The lower-level network is parameterized by the variables $\pmb { \theta } ^ { S }$ and is shared across multiple tasks. Each task $t$ has its own “head network” $\dot { \theta _ { t } ^ { H } }$ mapping to the outputs from a common hidden layer. The full set of parameters is therefore $\pmb { \theta } ^ { \prime } = \{ \pmb { \theta } _ { 1 : T } ^ { H } , \mathbf { \check { \theta } } ^ { S } \}$ . (b) A multi-head generative model with shared network parameters. The head networks generate the intermediate level representations from the latent variables $\mathbf { z }$ .
+
+VCL requires specification of $q ( \pmb \theta )$ where $\pmb { \theta }$ in the current case is a $D$ dimensional vector formed by stacking the network’s biases and weights. For simplicity we use a Gaussian mean-field approximate posterior $\begin{array} { r } { q _ { t } ( \pmb { \theta } ) = \prod _ { d = 1 } ^ { D } \mathcal { N } ( \theta _ { t , d } ; \bar { \mu } _ { t , d } , \sigma _ { t , d } ^ { 2 } ) } \end{array}$ . Taking the most general case of a multi-head network, before task $k$ is encountered the posterior distribution over the associated head parameters will remain at the prior and so $q ( \pmb { \theta } _ { k } ^ { H } ) \stackrel { ^ { \bullet } } { = } p ( \pmb { \theta } _ { k } ^ { H } )$ . This is convenient as it means the variational approximation can be grown incrementally, starting from the prior, as each task emerges. Moreover, only tasks present in the current dataset $\mathcal { D } _ { t }$ need to have their posterior distributions over head parameters updated. The shared parameters, on the other hand, will be constantly updated. Training the network using the VFE approach in eq. (1) is equivalent to maximizing the negative online variational free energy or the variational lower bound to the online marginal likelihood
+
+$$
+\mathcal { L } _ { \mathrm { V C L } } ^ { t } ( q _ { t } ( \pmb { \theta } ) ) = \sum _ { n = 1 } ^ { N _ { t } } \mathbb { E } _ { \pmb { \theta } \sim q _ { t } ( \pmb { \theta } ) } \left[ \log p ( y _ { t } ^ { ( n ) } | \pmb { \theta } , \mathbf { x } _ { t } ^ { ( n ) } ) \right] - \mathrm { K L } ( q _ { t } ( \pmb { \theta } ) | | q _ { t - 1 } ( \pmb { \theta } ) )
+$$
+
+with respect to the variational parameters $\{ \mu _ { t , d } , \sigma _ { t , d } \} _ { d = 1 } ^ { D }$ . Whilst the KL-divergence $\mathrm { K L } ( q _ { t } ( \pmb { \theta } ) | | q _ { t - 1 } ( \pmb { \theta } ) )$ can be computed in closed-form, the expected log-likelihood requires further approximation. Here we take the usual approach of employing simple Monte Carlo and use the local reparameterization trick to compute the gradients (Salimans & Knowles, 2013; Kingma & Welling, 2014; Kingma et al., 2015). At the first time step, the prior distribution, and therefore $q _ { 0 } ( \pmb \theta )$ is chosen to be a multivariate Gaussian distribution (see e.g. Graves, 2011; Blundell et al., 2015).
+
+# 4 VARIATIONAL CONTINUAL LEARNING IN DEEP GENERATIVE MODELS
+
+Deep generative models (DGMs) have garnered much recent attention. By passing a simple noise variable (e.g. Gaussian noise) through a deep neural network, these models have been shown to be able to generate realistic images, sounds and videos sequences (Chung et al., 2015; Kingma et al., 2016; Vondrick et al., 2016). Standard approaches for learning DGMs have focused on batch learning, i.e. the observed instances are assumed to be i.i.d. and are all available at the same time. In this section we extend the VCL framework to encompass variational auto-encoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014), a form of DGM. The approach could be extended to generative adversarial networks (GANs) (Goodfellow et al., 2014b) for which continual learning is an open problem (see Seff et al. (2017) for an initial attempt).
+
+Consider a model $p ( \mathbf { x } | \mathbf { z } , \pmb { \theta } ) p ( \mathbf { z } )$ , for observed data $\mathbf { x }$ and latent variables $\mathbf { z }$ . The prior over latent variables $p ( \mathbf { z } )$ is typically Gaussian, and the distributional parameters of $p ( \mathbf { x } | \mathbf { z } , \theta )$ are defined by a deep neural network. For example, if Bernoulli likelihood is used, then $p ( \mathbf { x } | \mathbf { \bar { z } } , \pmb { \theta } ) = \mathrm { B e r n } ( \mathbf { x } ; \pmb { f } _ { \pmb { \theta } } ( \mathbf { z } ) )$ , where $f _ { \theta }$ denotes the deep neural network transform and $\pmb \theta$ collects all the weight matrices and bias vectors. In the batch setting, given a dataset $\mathcal { D } = \{ \mathbf { x } ^ { ( n ) } \} _ { n = 1 } ^ { N }$ , the standard VAE approach learns the parameters by approximate maximum likelihood estimation (MLE). This proceeds by maximizing the variational lower bound with respect to $\pmb \theta$ and $\phi$ :
+
+$$
+\mathcal { L } _ { \mathrm { V A E } } ( \pmb { \theta } , \phi ) = \sum _ { n = 1 } ^ { N } \mathbb { E } _ { q _ { \phi } ( \mathbf { z } ^ { ( n ) } | \mathbf { x } ^ { ( n ) } ) } \left[ \log \frac { p ( \mathbf { x } ^ { ( n ) } | \mathbf { z } ^ { ( n ) } , \pmb { \theta } ) p ( \mathbf { z } ^ { ( n ) } ) } { q _ { \phi } ( \mathbf { z } ^ { ( n ) } | \mathbf { x } ^ { ( n ) } ) } \right] ,
+$$
+
+where $\phi$ are the variational parameters of the approximate posterior or “encoder”.
+
+The approximate MLE approach is unsuitable for the continual learning setting as it does not return parameter uncertainty estimates that are critical for weighting the information learned from old data. So, instead the VCL approach will approximate the full posterior distribution over parameters, $q _ { t } ( \pmb { \theta } ) \approx p ( \pmb { \theta } | \mathcal { D } _ { 1 : t } )$ , after observing the $t$ -th dataset. Specifically, the approximate posterior $q _ { t }$ is obtained by maximizing the full variational lower bound with respect to $q _ { t }$ and $\phi$ :
+
+$$
+\ddot { \mathbf { \Psi } } _ { \mathrm { V C L } } ^ { \star } ( q _ { t } ( \theta ) , \phi ) = \mathbb { E } _ { q _ { t } ( \theta ) } \left\{ \sum _ { n = 1 } ^ { N _ { t } } \mathbb { E } _ { q _ { \phi } ( \mathbf { z } _ { t } ^ { ( n ) } | \mathbf { x } _ { t } ^ { ( n ) } ) } \left[ \log \frac { p ( \mathbf { x } _ { t } ^ { ( n ) } | \mathbf { z } _ { t } ^ { ( n ) } , \theta ) p ( \mathbf { z } _ { t } ^ { ( n ) } ) } { q _ { \phi } ( \mathbf { z } _ { t } ^ { ( n ) } | \mathbf { x } _ { t } ^ { ( n ) } ) } \right] \right\} - \mathrm { K L } ( q _ { t } ( \theta ) | | q _ { t - 1 } ( \theta ) ) ,
+$$
+
+where the encoder network $q _ { \phi } \big ( z _ { t } ^ { ( n ) } | \mathbf { x } _ { t } ^ { ( n ) } \big )$ is parameterized by $\phi$ which is task-specific. It is likely to be beneficial to share (parts of) these encoder networks, but this is not investigated in this paper.
+
+As was the case for multi-head discriminative models, we can split the generative model into shared and task-specific parts. There are two options: (i) the generative models share across tasks the network that generates observations $\mathbf { x }$ from the intermediate-level representations $\mathbf { h }$ , but have private “head networks” for generating $\mathbf { h }$ from the latent variables $\mathbf { z }$ (see fig. 1(b)), and (ii) the other way around. Architecture (i) is arguably more appropriate when data are composed of a common set of structural primitives (such as strokes in handwritten digits) that are selected by high level variables (character identities). Moreover, initial experiments on architecture (ii) indicated that information about the current task tended to be encoded entirely in the task-specific lower-level network negating multi-task transfer. For these reasons, we focus on architecture (i) in the experiments.
+
+# 5 RELATED WORK
+
+Continual Learning for Deep Discriminative Models: Many neural network continual learning approaches employ regularized maximum likelihood estimation, optimizing objectives of the form:
+
+$$
+\begin{array} { r } { \mathcal { L } ^ { t } ( \pmb { \theta } ) = \sum _ { n = 1 } ^ { N _ { t } } \log p ( y _ { t } ^ { ( n ) } | \pmb { \theta } , \mathbf { x } _ { t } ^ { ( n ) } ) - \frac { 1 } { 2 } \lambda _ { t } ( \pmb { \theta } - \pmb { \theta } _ { t - 1 } ) ^ { \top } \Sigma _ { t - 1 } ^ { - 1 } ( \pmb { \theta } - \pmb { \theta } _ { t - 1 } ) . } \end{array}
+$$
+
+Here the regularization biases the new parameter estimates towards those estimated at the previous step $\pmb { \theta } _ { t - 1 }$ . $\lambda _ { t }$ is a user-selected hyper-parameter that controls the overall contribution from previous data and $\Sigma _ { t - 1 }$ is a matrix (normally diagonal in form) that encodes the relative strength of the regularization on each element of $\pmb { \theta }$ . We now discuss specific instances of this scheme:
+
+• Maximum-likelihood estimation and MAP estimation: maximum likelihood estimation is recovered when there is no regularization ${ { \lambda } _ { t } } \ = \ 0 ,$ . More generally, the regularization term can be interpreted as a Gaussian prior, $q ( \pmb { \theta } | \mathcal { D } _ { 1 : t - 1 } ) = \mathcal { N } ( \pmb { \theta } ; \bar { \theta } _ { t - 1 } , \Sigma _ { t - 1 } / \lambda _ { t } )$ . The optimization returns the maximum a posteriori estimate of the parameters, but this does not directly provide $\Sigma _ { t }$ for the next stage. A simple fix is to set $\Sigma _ { t } ~ = ~ \mathrm { I }$ and use cross-validation to find $\lambda _ { t }$ , but this approximation is often coarse and leads to catastrophic forgetting (Goodfellow et al., 2014a; Kirkpatrick et al., 2017).
+
+• Laplace Propagation (LP) (Smola et al., 2004): applying Laplace’s approximation at each step leads to a recursion for $\Sigma _ { t } ^ { - 1 }$ , which is initialized using the covariance of the Gaussian prior,
+
+$$
+\begin{array} { r } { \Sigma _ { t } ^ { - 1 } = \Phi _ { t } + \Sigma _ { t - 1 } ^ { - 1 } \mathrm { ~ w h e r e ~ } \Phi _ { t } = - \nabla \nabla _ { \theta } \sum _ { n = 1 } ^ { N _ { t } } \log p ( y _ { t } ^ { ( n ) } | \pmb { \theta } , \mathbf { x } _ { t } ^ { ( n ) } ) \Big | _ { \pmb { \theta = \theta _ { t } } } } \end{array}
+$$
+
+To avoid computing the full Hessian of the likelihood, diagonal Laplace propagation retains only the diagonal terms of $\Sigma _ { t } ^ { - 1 }$ .
+
+• Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) builds on diagonal Laplace propagation by approximating the average Hessian of the likelihoods using well-known identities for the Fisher information:
+
+$$
+\begin{array} { r } { \Phi _ { t } \approx \mathrm { d i a g } \left( \sum _ { n = 1 } ^ { N _ { t } } \left( \nabla _ { \mathbf { \boldsymbol { \theta } } } \log p ( y _ { t } ^ { ( n ) } | \mathbf { \boldsymbol { \theta } } , \mathbf { x } _ { t } ^ { ( n ) } ) \right) ^ { 2 } \Big | _ { \mathbf { \boldsymbol { \theta } } = \mathbf { \boldsymbol { \theta } } _ { t } } \right) . } \end{array}
+$$
+
+EWC also modifies the Laplace regularization, troducing hyper-parameters, removing the prio $\begin{array} { r } { \frac { 1 } { 2 } ( \pmb { \theta } - \pmb { \theta } _ { t - 1 } ) ^ { \top } \big ( \Sigma _ { 0 } ^ { - 1 } + \sum _ { t ^ { \prime } = 1 } ^ { t - 1 } \Phi _ { t ^ { \prime } } \big ) \big ( \pmb { \theta } - \pmb { \theta } _ { t - 1 } \big ) } \end{array}$ , in-estimates, rather than just those derived from the last task, $\begin{array} { r l } { \frac { 1 } { 2 } \sum _ { t ^ { \prime } = 1 } ^ { t - 1 } \lambda _ { t ^ { \prime } } ( \pmb { \theta } - \pmb { \theta } _ { t ^ { \prime } - 1 } ) ^ { \top } \Phi _ { t ^ { \prime } } ( \pmb { \theta } - \pmb { \theta } _ { t ^ { \prime } - 1 } ) } & { { } } \end{array}$ . These changes may be unnecessary (Huszar´ , 2017; 2018) and require storing $\pmb \theta _ { 1 : t - 1 }$ , but may slightly improve performance (see Kirkpatrick et al. (2018) and our experiments).
+
+• Synaptic Intelligence (SI) (Zenke et al., 2017): SI computes $\Sigma _ { t } ^ { - 1 }$ using a measure of the importance of each parameter to each task. Practically, this is achieved by comparing the changing rate of the gradients of the objective and the changing rate of the parameters.
+
+VCL differs from the above methods in several ways. First, unlike MAP, EWC and SI, it does not have free parameters that need to be tuned on a validation set. This can be especially awkward in the online setting. Second, although the KL regularization penalizes the mean of the approximate posterior through a quadratic cost, a full distribution is retained and averaged over at training time and at test time. Third, VI is generally thought to return better uncertainty estimates than approaches like Laplace’s method and MAP estimation, and we have argued this is critical for continual learning.
+
+There is a long history of research on approximate Bayesian training of neural networks, including extended Kalman filtering (Singhal & Wu, 1989), Laplace’s approximation (MacKay, 1992), variational inference (Hinton & Van Camp, 1993; Barber & Bishop, 1998; Graves, 2011; Blundell et al., 2015; Gal & Ghahramani, 2016), sequential Monte Carlo (de Freitas et al., 2000), expectation propagation (EP) (Hernandez-Lobato & Adams ´ , 2015), and approximate power EP (Hernandez-Lobato ´ et al., 2016). These approaches have focused on batch learning, but the framework described in section 2 enables them to be applied to continual learning. On the other hand, online variational inference has been previously explored (Ghahramani & Attias, 2000; Broderick et al., 2013; Bui et al., 2017), but not for neural networks or in the context of sets of related complex tasks.
+
+Continual Learning for Deep Generative Models: A na¨ıve continual learning approach for deep generative models would directly apply the VAE algorithm to the new dataset $\mathcal { D } _ { t }$ with the model parameters initialized at the previous parameter values $\pmb { \theta } _ { t - 1 }$ . The experiments show that this approach leads to catastrophic forgetting, in the sense that the generator can only generate instances that are similar to the data points from the most recently observed task.
+
+Alternatively, EWC regularization can be added to the VAE objective:
+
+$$
+\mathbf { \dot { \mathbf { \Psi } } } _ { \mathrm { E W C } } ^ { \star } ( \theta , \phi ) = \sum _ { n = 1 } ^ { N _ { t } } \mathbb { E } _ { q _ { \phi } ( \mathbf { z } _ { t } ^ { ( n ) } | \mathbf { x } _ { t } ^ { ( n ) } ) } \left[ \log \frac { p ( \mathbf { x } _ { t } ^ { ( n ) } | \mathbf { z } _ { t } ^ { ( n ) } , \theta ) p ( \mathbf { z } _ { t } ^ { ( n ) } ) } { q _ { \phi } ( \mathbf { z } _ { t } ^ { ( n ) } | \mathbf { x } _ { t } ^ { ( n ) } ) } \right] - \frac { 1 } { 2 } \sum _ { t ^ { \prime } = 1 } ^ { t - 1 } \lambda _ { t ^ { \prime } } ( \theta - \theta _ { t ^ { \prime } - 1 } ) ^ { \top } \Phi _ { t ^ { \prime } } ( \theta - \theta _ { t ^ { \prime } - 1 } )
+$$
+
+However computing $\Phi _ { t }$ requires the gradient of the intractable marginal likelihood $\nabla _ { \pmb { \theta } } \log p ( \mathbf { x } | \pmb { \theta } )$ .   
+Instead, we can approximate the marginal likelihood by the variational lower bound, i.e.
+
+$$
+\Phi _ { t } \approx \mathrm { d i a g } \Big ( \sum _ { n = 1 } ^ { N _ { t } } \Big ( \nabla _ { \pmb \theta } \mathbb { E } _ { q _ { \phi } ( \mathbf { z } _ { t } ^ { ( n ) } | \mathbf { x } _ { t } ^ { ( n ) } ) } \big [ \log \frac { p ( \mathbf { x } _ { t } ^ { ( n ) } | \mathbf { z } _ { t } ^ { ( n ) } , \pmb \theta ) p ( \mathbf { z } _ { t } ^ { ( n ) } ) } { q _ { \phi } ( \mathbf { z } _ { t } ^ { ( n ) } | \mathbf { x } _ { t } ^ { ( n ) } ) } \big ] \Big ) ^ { 2 } \Big | _ { \pmb \theta = \pmb \theta _ { t } } \Big ) .
+$$
+
+Similar variational lower-bound approximations apply when computing the Hessian matrices for LP and $\Sigma _ { t } ^ { - 1 }$ for SI. An importance sampling estimate could also be used (Burda et al., 2016).
+
+# 6 EXPERIMENTS
+
+The experiments evaluate the performance and flexibility of VCL through three discriminative tasks and two generative tasks. Standard continual learning benchmarks are used where possible. Comparisons are made to EWC, diagonal LP and SI that employ tuned hyper-parameters $\lambda$ whereas VCL’s objective is hyper-parameter free. More details of the experiment settings and an additional experiment are available in the appendix.1
+
+![](images/72e3ca69ed635f1e96ff1905583ff186c07e35ef50559faee5408b89f2c0bdd8.jpg)  
+Figure 2: Average test set accuracy on all observed tasks in the Permuted MNIST experiment.
+
+![](images/1cb0944231f44edd05a34fc1c4fd9b104cc3a70f079ecd499067dc1613d68239.jpg)  
+Figure 3: Comparison of the effect of coreset sizes in the Permuted MNIST experiment.
+
+6.1 EXPERIMENTS WITH DEEP DISCRIMINATIVE MODELS
+
+We consider the following three continual learning experiments for deep discriminative models.
+
+Permuted MNIST: This is a popular continual learning benchmark (Goodfellow et al., 2014a; Kirkpatrick et al., 2017; Zenke et al., 2017). The dataset received at each time step $\mathcal { D } _ { t }$ consists of labeled MNIST images whose pixels have undergone a fixed random permutation. We compare VCL to EWC, SI, and diagonal LP. For all algorithms, we use fully connected single-head networks with two hidden layers, where each layer contains 100 hidden units with ReLU activations. We evaluate three versions of VCL: VCL with no coreset, VCL with a random coreset, and VCL with a coreset selected by the K-center method. For the coresets, we select 200 data points from each task.
+
+Figure 2 compares the average test set accuracy on all observed tasks. From this figure, VCL outperforms EWC, SI, and LP by large margins, even though they benefited from an extensive hyperparameter search for $\lambda$ . Diagonal LP performs slightly worse than EWC both when $\lambda = 1$ and when the values of $\lambda$ are tuned. After 10 tasks, VCL achieves $90 \%$ average accuracy, while EWC, SI, and LP only achieve $84 \%$ , $86 \%$ , and $82 \%$ respectively. The results also show that the coresets perform poorly by themselves, but combining them with VCL leads to a modest improvement: both random coresets and K-center coresets achieve $93 \%$ accuracy.
+
+We also investigate the effect of the coreset size. In fig. 3, we plot the average test set accuracy of VCL with random coresets of different sizes. At the coreset size of 5,000 examples per task, VCL achieves $9 5 . 5 \%$ accuracy after 10 tasks, which is significantly better than the $90 \%$ of vanilla VCL. Performance improves with the coreset size although it asymptotes for large coresets as expected: if a sufficiently large coreset is employed, it will be fully representative of the task and thus training on the coreset alone can achieve a good performance. However, the experiments show that the combination of VCL and coresets is advantageous even for large coresets.
+
+![](images/61c77957df2f4fa21289bd4598879bcf3b83ae434385a4fd907fc4d1373f1384.jpg)  
+Figure 4: Test set accuracy on all tasks for the Split MNIST experiment. The last column shows the average accuracy over all tasks. The bottom row is a zoomed version of the top row.
+
+Split MNIST: This experiment was used by Zenke et al. (2017) to assess the SI method. Five binary classification tasks from the MNIST dataset arrive in sequence: 0/1, 2/3, 4/5, 6/7, and 8/9. We use fully connected multi-head networks with two hidden layers comprising 256 hidden units with ReLU activations. We compare VCL (with and without coresets) to EWC, SI, and diagonal LP. For the coresets, 40 data points from each task are selected through random sampling or the K-center method.
+
+Figure 4 compares the test set accuracy on individual tasks (averaged over 10 runs) as well as the accumulated accuracy averaged over tasks (right). As an upper bound on the algorithms’ performance, we compare to batch VI trained on the full dataset. From this figure, VCL significantly outperforms EWC and LP although it is slightly worse than SI. Again, unlike VCL, EWC and SI benefited from a hyper-parameter search for $\lambda$ , but a value close to 1 performs well in both cases. After 5 tasks, VCL achieves $9 7 . 0 \%$ average accuracy on all tasks, while EWC, SI, and LP attain $6 3 . 1 \%$ , $9 8 . 9 \%$ , and $6 1 . 2 \%$ respectively. Adding the coreset improves VCL to around $9 8 . 4 \%$ accuracy.
+
+Split notMNIST: This experiment is similar to the previous one, but it uses the more challenging notMNIST dataset and deeper networks. The notMNIST dataset2 here contains 400,000 images of the characters from A to J with different font styles. We consider five binary classification tasks: A/F, B/G, C/H, D/I, and E/J using deeper networks comprising four hidden layers of 150 hidden units with ReLU activations. The other settings are kept the same as the previous experiment. VCL is competitive with SI and significantly outperforms EWC and LP (see fig. 5), although the SI and EWC baselines benefited from a hyper-parameter search. VCL achieves $9 2 . 0 \%$ average accuracy after 5 tasks, while EWC, SI, and LP attain $71 \%$ , $94 \%$ , and $63 \%$ respectively. Adding the random coreset improves the performance of VCL to $96 \%$ accuracy.
+
+# 6.2 EXPERIMENTS WITH DEEP GENERATIVE MODELS
+
+We consider two continual learning experiments for deep generative models: MNIST digit generation and notMNIST (small) character generation. In both cases, ten datasets are received in sequence. For MNIST, the first dataset comprises exclusively of images of the digit zero, the second dataset ones and so on. For notMNIST, the datasets contain the characters A to J in sequence. The generative model consists of shared and task-specific components, each represented by a one hidden layer neural network with 500 hidden units (see fig. 1(b)). The dimensionality of the latent variable $\mathbf { z }$ and the intermediate representation $\mathbf { h }$ are 50 and 500, respectively. We use task-specific encoders that are neural networks with symmetric architectures to the generator.
+
+We compare VCL to na¨ıve online learning using the standard VAE objective, LP, EWC and SI (with hyper-parameters $\lambda = 1 , 1 0 , 1 0 0 )$ . For full details of the experimental settings see Appendix E. Samples from the generative models attained at different time steps are shown in fig. 6. The na¨ıve online learning method fails catastrophically and so numerical results are omitted. LP, EWC, SI and VCL remember previous tasks, with SI and VCL achieving the best visual quality on both datasets.
+
+![](images/92591e28e3359754e8917a82896ef8ee77d7c9ee81741131dc5b9d4b6de33a5a.jpg)  
+Figure 5: Test set accuracy on all tasks for the Split notMNIST experiment. The last column shows the average accuracy over all tasks. The bottom row is a zoomed version of the top row.
+
+The algorithms are quantitatively evaluated using two metrics in fig. 7: an importance sampling estimate of the test log-likelihood (test-LL) using 5, 000 samples and a measure of quality we term “classifier uncertainty”. For the latter, we train a discriminative classifier for the digits/alphabets to achieve high accuracy. The quality of generated samples can then be assessed by the KL-divergence from the one-hot vector indicating the task, to the output classification probability vector computed on the generated images. A well-trained generator will produce images that are correctly classified in high confidence resulting in zero KL. We only report the best performance for LP, EWC and SI.
+
+We observe that LP and EWC perform similarly, most likely due to the fact that both LP and EWC use the same $\Sigma _ { t }$ matrices. EWC achieves significantly worse performance than SI. VCL is on par with or slightly better than SI. VCL has a superior long-term memory of previous tasks which leads to better overall performance on both metrics even though it does not have tuned hyper-parameters in its objective function. For MNIST, the performance of LP and EWC deteriorate markedly when moving from task “digit $0 ^ { \circ }$ to “digit $1 ^ { \circ }$ possibly due to the large task differences. Also for all experimental settings we tried, SI fails to produce high test-LL results after task “digit $7 ^ { \circ }$ . Future work will investigate continual learning on a sequence of tasks that follows “adversarial ordering”, i.e. the ordering that makes the next task maximally different from the current task.
+
+# 7 CONCLUSION
+
+Approximate Bayesian inference provides a natural framework for continual learning. Variational Continual Learning (VCL), developed in this paper, is an approach in this vein that extends online variational inference to handle more general continual learning tasks and complex neural network models. VCL can be enhanced by including a small episodic memory that leverages coreset algorithms from statistics and connects to message-scheduling in variational message passing. We demonstrated how the VCL framework can be applied to both discriminative and generative models. Experimental results showed state-of-the-art performance when compared to previous continual learning approaches, even though VCL has no free parameters in its objective function. Future work should explore alternative approximate inference methods using the same framework and also develop more sophisticated episodic memories. Finally, we note that VCL is ideally suited for efficient model refinement in sequential decision making problems, such as reinforcement learning and active learning.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank Brian Trippe, Siddharth Swaroop, and Matej Balog for insightful comments and discussion. Cuong V. Nguyen is supported by EPSRC grant EP/M0269571. Yingzhen Li is supported by the Schlumberger FFTF Fellowship. Thang D. Bui is supported by the Google European Doctoral Fellowship. Richard E. Turner is supported by Google as well as EPSRC grants EP/M0269571 and EP/L000776/1.
+
+![](images/83c4ab359a72839d69da09bf2e82f92e2ad5ce2c390a52734100d54e79a33919.jpg)  
+Figure 6: Generated images from each of the generators after training. Each of the columns shows the images generated from a specific task’s generator, and each of the lines shows the generations from generators of all trained tasks. Clearly the naive approach suffers from catastrophic forgetting, while other approaches successfully remember previous tasks.
+
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+# APPENDIX
+
+# A FURTHER DETAILS FOR PERMUTED MNIST EXPERIMENT
+
+In this experiment, we use fully connected single-head networks with two hidden layers, where each layer contains 100 hidden units with ReLU activations. The metric used for comparison is the test set accuracy on all observed tasks. We train all the models using the Adam optimizer (Kingma & Ba, 2015) with learning rate $1 0 ^ { - 3 }$ since we found that it works best for all models. All the VCL algorithms are trained with batch size 256 and 100 epochs. For all the algorithms with coresets, we choose 200 examples from each task to include into the coresets. The algorithms that use only the coresets are trained using the VFE method with batch size equal to the coreset size and 100 epochs. We use the prior $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ and initialize our optimizer for the first task at the mean of the maximum likelihood model and a very small initial variance $( 1 0 ^ { - 6 } )$ .
+
+We compare the performance of SI with hyper-parameters $\lambda = 0 . 0 1 , 0 . 1 , 0 . 5 , 1 , 2$ and select the best one $( \lambda = 0 . 5 )$ ) as our baseline (see fig. 8). Following Zenke et al. (2017), we train these models with batch size 256 and 20 epochs. We also compare the performance of EWC with $\lambda = 1 , 1 0 , 1 0 ^ { 2 } , 1 0 ^ { 3 } , 1 0 ^ { 4 }$ and select the best value $\lambda = 1 0 ^ { 2 }$ as our baseline (see fig. 9). The models are trained without dropout and with batch size 200 and 20 epochs. We approximate the Fisher information matrices in EWC using 600 random samples drawn from the current dataset. For diagonal LP, we compare the performance of $\lambda = 0 . 0 1 , 0 . 1 , 1 , 1 0 , 1 0 0$ and use the best value $\lambda = 0 . 1$ as our baseline (see fig. 10). The models are also trained with prior $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ , batch size 200, and 20 epochs. The Hessians of LP are approximated using the Fisher information matrices with 200 samples.
+
+![](images/43b02012ec1ff22c08d31d8214c78d5fb0480a2ffed0e8e15f464cf2d9d53faa.jpg)  
+Figure 8: Performance of SI with different hyper-parameter values in Permuted MNIST experiment.
+
+![](images/9cf49463ebe337d63fb156df218a4c83bf99ce910dcf5a00fe5f34b504797b72.jpg)  
+Figure 9: Performance of EWC with different hyper-parameter values in Permuted MNIST experiment.
+
+![](images/3d9b12239876fb7924e1e6e3723582e864420751cb3f15a01f3611106206815a.jpg)  
+Figure 10: Performance of LP with different hyper-parameter values in Permuted MNIST experiment.
+
+B FURTHER DETAILS FOR SPLIT MNIST EXPERIMENT
+
+In this experiment, we use fully connected multi-head networks with two hidden layers, each of which contains 256 hidden units with ReLU activations. At each time step, we compare the test set accuracy of the current model on all observed tasks separately. We also plot the average accuracy over all tasks in the last column of fig. 4. All the results for this experiment are the averages over 10 runs of the algorithms with different random seeds.
+
+We use the Adam optimizer with learning rate $1 0 ^ { - 3 }$ for all models. All the VCL algorithms are trained with batch size equal to the size of the training set and 120 epochs. We use the prior $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ and initialize our optimizer for the first task at the mean of the maximum likelihood model and a very small initial variance $( 1 0 ^ { - 6 } )$ . For the coresets, we choose 40 examples from each task to include into the coresets. In this experiment, the final approximate posterior used for prediction in eq. (3) is computed for each task separately using the coreset points corresponding to the task. The algorithms that use only the coresets are trained using the VFE method with batch size equal to the coreset size and 120 epochs.
+
+We compare the performance of SI with $\lambda = 0 . 0 1 , 0 . 1 , 1 , 2 , 3$ and use the best value $\lambda = 1$ as our baseline (see fig. 11). We also compare EWC with both single-head and multi-head models and $\lambda = 1 , 1 0 , 1 0 ^ { 2 } , \mathsf { \bar { 1 0 ^ { 3 } } } , 1 0 ^ { 4 }$ (see fig. 12). We approximate the Fisher information matrices using 200 random samples drawn from the current dataset. The figure shows that the multi-head models work better than the single-head models for EWC, and the performance is insensitive to the choice of $\lambda$ . Thus, we use the multi-head model with $\lambda = 1$ as the EWC baseline for our experiment. For diagonal LP, we also use the multi-head model with $\lambda = 1$ , prior $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ , and approximate the Hessians using the Fisher information matrices with 200 samples.
+
+# C FURTHER DETAILS FOR SPLIT NOTMNIST EXPERIMENT
+
+The settings for this experiment are the same as those in the Split MNIST experiment above, except that we use deeper networks with 4 hidden layers, each of which contains 150 hidden units. Figures 13 and 14 show the performance of SI and EWC with different hyper-parameter values respectively. In the experiment, we choose $\lambda = 1 0 ^ { 4 }$ for multi-head EWC, $\lambda = 1$ for multi-head LP, and $\lambda = 0 . 1$ for SI.
+
+# D ADDITIONAL EXPERIMENT ON A TOY 2D DATASET
+
+Here we consider a small experiment on a toy 2D dataset to understand some of the properties of EWC with $\lambda = 1$ and VCL. The experiment comprises two sequential binary classification tasks. The first task contains two classes generated from two Gaussian distributions. The data points (with green and black classes) for this task are shown in the first column of fig. 15. The second task contains two classes also generated from two Gaussian distributions. The green class for this task has the same input distribution as the first task, while the input distribution for the black class is different. The data points for this task are shown in the third columns of fig. 15. Each task contains 200 data points with 100 data points in each class.
+
+![](images/38e1e4801ae87f0c85f3022c7177d3c77207f4e29367751d8e19a781c809fdf4.jpg)  
+Figure 11: Comparison of SI with different hyper-parameter values in Split MNIST experiment.
+
+![](images/ddb590a82da199ecd167283b134e48b30799d605c9fb9cc08659edf4917c2735.jpg)  
+Figure 12: Comparison of EWC with different hyper-parameter values in Split MNIST experiment (top: multi-head model, bottom: single-head model).
+
+![](images/db2d3579f88e58dcd469c5e72cffb84b5bf3c4757efcaff3163d660ac4607b92.jpg)  
+Figure 13: Comparison of SI with different hyper-parameter values in Split notMNIST experiment.
+
+![](images/20c44cde40f0f49996d161f0ca4191a3625d2ef9b12fc1c25075a59ceba18cf2.jpg)  
+Figure 14: Comparison of EWC with different hyper-parameter values in Split notMNIST experiment (top: multi-head model, bottom: single-head model).
+
+![](images/c1f24706f8cf1d462fbda4401a36c502bad277a362347be0891ab1b266f6bf78.jpg)  
+Figure 15: Comparison of VCL and EWC on a toy 2D dataset. The first column shows the contours of the prediction probabilities after observing the first task. The second and third columns show the contours for the first and the second tasks respectively after observing the second task.
+
+We compare the multi-head models trained by VCL and EWC on these two tasks. In this experiment, we use fully connected networks with one hidden layer containing 20 hidden units with ReLU activations. The first column of fig. 15 shows the contours of the prediction probabilities after observing the first task, and both methods perform reasonably well for this task. However, after observing the second task, the EWC method fails to learn the classifiers for both tasks, while the VCL method are still able to learn good classifiers for them.
+
+# E FURTHER DETAILS ON DEEP GENERATIVE MODEL EXPERIMENTS
+
+In the experiments on Deep Generative Models, the learning rates and numbers of optimization epochs are tuned on separate training of each tasks. This gives a learning rate of $1 0 ^ { - 4 }$ and the number of epochs 200 for MNIST (except for SI) and 400 for notMNIST. For SI we optimize for 400 epochs on MNIST. For the VCL approach, the parameters of $q _ { t } ( \pmb \theta )$ are initialized to have the same mean as qt−1(θ) and the log standard deviation is set to 10−6.
+
+The generative model consists of shared and task-specific components, each represented by a one hidden layer neural network with 500 hidden units (see fig. 1(b)). The dimensionality of the latent variable $\mathbf { z }$ and the intermediate representation $\mathbf { h }$ are 50 and 500, respectively. We use task-specific encoders that are neural networks with symmetric architectures to the generator.
+
+F MEMORY OF A LINEAR REGRESSION MODEL TRAINED WITH VCL ON RANDOM PATTERNS
+
+In many probabilistic models with conjugate priors, the exact posterior of the parameters/latent variables can be obtained. For example, a Bayesian linear regression model with a Gaussian prior over the parameters and a Gaussian observation model has a Gaussian posterior. If we insist on using a diagonal Gaussian approximation to this posterior and use either the variational free energy method or the Laplace’s approximation, we will end up at the same solution – a Gaussian distribution with the same mean as that of the exact posterior and the diagonal precisions being the diagonal precisions of the exact posterior. Consequently, the online variational Gaussian approximation will give the same result to that given by the online Laplace’s approximation. However, when a diagonal Gaussian approximation is used, the batch and sequential solutions are different.
+
+In the following, we will explicitly detail the sequential variational updates for a Bayesian linear regression model to associate random binary patterns to binary outcomes (Kirkpatrick et al., 2017), and show its relationship to the online Laplace’s approximation and the EWC approach of Kirkpatrick et al. (2017). The task consists of associating a random $D$ -dimensional binary vector $\mathbf { x } _ { t }$ to a random binary output $y _ { t }$ by learning a weight vector $W$ . Note that the possible values of the features and outputs are 1 and $- 1$ , and not 0 and 1. We also assume that the model sees only one input-output pair $\left\{ \mathbf { x } _ { t } , y _ { t } \right\}$ at the $t { \cdot }$ -th time step and the previous approximate posterior $\begin{array} { r } { q _ { t - 1 } ( W ) = \prod _ { d = 1 } ^ { D } \mathcal { N } ( w _ { d } ; m _ { t - 1 , d } , v _ { t - 1 , d } ) . } \end{array}$ , and that the observation noise variance sions are available in closed-form as foll $\sigma _ { y } ^ { 2 }$ is fixed. The:
+
+$$
+\begin{array} { r l } & { \mathbf { m } _ { t } = \left[ \mathrm { I } + \mathbf { V } _ { t - 1 } \frac { \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \mathsf { T } } } { \sigma _ { y } ^ { 2 } } \right] ^ { - 1 } \left[ \mathbf { V } _ { t - 1 } \frac { \mathbf { x } _ { t } y _ { t } ^ { \mathsf { T } } } { \sigma _ { y } ^ { 2 } } + \mathbf { m } _ { t - 1 } \right] , } \\ & { \mathbf { \sigma } _ { v _ { t , d } ^ { - 1 } } = v _ { t - 1 , d } ^ { - 1 } + \frac { x _ { t , d } ^ { 2 } } { \sigma _ { y } ^ { 2 } } \mathrm { f o r } d = 1 , 2 , \ldots , D . } \end{array}
+$$
+
+By further assuming that $\sigma _ { y } = 1$ and $\| \mathbf { x } _ { t } \| _ { 2 } = 1$ , the equations above become:
+
+$$
+\begin{array} { r l } & { \mathbf { m } _ { t } = \left[ \mathrm { I } - \frac { \bar { \mathbf { x } } _ { t } \bar { \mathbf { x } } _ { t } ^ { \top } } { 1 + v _ { 0 } ^ { - 1 } + \frac { t } { D } } \right] \mathbf { m } _ { t - 1 } + \frac { \bar { \mathbf { x } } } { 1 + v _ { 0 } ^ { - 1 } + \frac { t } { D } } , } \\ & { v _ { t , d } ^ { - 1 } = v _ { 0 , d } ^ { - 1 } + \frac { t } { D } \mathrm { ~ f o r ~ } d = 1 , 2 , . . . , D , } \end{array}
+$$
+
+where $\bar { \mathbf { x } } _ { t } = y _ { t } \mathbf { x } _ { t }$ . When $v _ { 0 , d } ^ { - 1 } = 0$ , i.e. the prior is ignored, the update for the mean above is exactly equation S4 in the supplementary material of Kirkpatrick et al. (2017). Therefore, in this case, the memory of the network trained by online variational inference is identical to that of the online Laplace’s method, provided in (Kirkpatrick et al., 2017). These methods differ, in practice, when the prior is not ignored or when the parameter regularization constraints are accumulated as discussed in the main text. This equivalence also does not hold in the general case, as discussed by Opper & Archambeau (2009) where the Gaussian variational approximation can be interpreted as an averaged and smoothed Laplace’s approximation.

parse/train/BklHpjCqKm/BklHpjCqKm.md

@@ -0,0 +1,336 @@
+# Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning
+
+Michael Lutter, Christian Ritter & Jan Peters ∗
+
+Department of Computer Science Technische Universität Darmstadt Hochschulstr. 10, 64289 Darmstadt, Germany {Lutter, Peters}@ias.tu-darmstadt.de
+
+# Abstract
+
+Deep learning has achieved astonishing results on many tasks with large amounts of data and generalization within the proximity of training data. For many important real-world applications, these requirements are unfeasible and additional prior knowledge on the task domain is required to overcome the resulting problems. In particular, learning physics models for model-based control requires robust extrapolation from fewer samples – often collected online in real-time – and model errors may lead to drastic damages of the system.
+
+Directly incorporating physical insight has enabled us to obtain a novel deep model learning approach that extrapolates well while requiring fewer samples. As a first example, we propose Deep Lagrangian Networks (DeLaN) as a deep network structure upon which Lagrangian Mechanics have been imposed. DeLaN can learn the equations of motion of a mechanical system (i.e., system dynamics) with a deep network efficiently while ensuring physical plausibility.
+
+The resulting DeLaN network performs very well at robot tracking control. The proposed method did not only outperform previous model learning approaches at learning speed but exhibits substantially improved and more robust extrapolation to novel trajectories and learns online in real-time.
+
+# 1 Introduction
+
+In the last five years, deep learning has propelled most areas of learning forward at an impressive pace (Krizhevsky et al., 2012; Mnih et al., 2015; Silver et al., 2017) – with the exception of physically embodied systems. This lag in comparison to other application areas is somewhat surprising as learning physical models is critical for applications that control embodied systems, reason about prior actions or plan future actions (e.g., service robotics, industrial automation). Instead, most engineers prefer classical off-the-shelf modeling as it ensures physical plausibility – at a high cost of precise measurements1 and engineering effort. These plausible representations are preferred as these models guarantee to extrapolate to new samples, while learned models only achieve good performance in the vicinity of the training data.
+
+To learn a model that obtains physically plausible representations, we propose to use the insights from physics as a model prior for deep learning. In particular, the combination of deep learning and physics seems natural as the compositional structure of deep networks enables the efficient computation of the derivatives at machine precision (Raissi & Karniadakis, 2018) and, thus, can encode a differential equation describing physical processes. Therefore, we suggest to encode the physics prior in the form of a differential in the network topology. This adapted topology amplifies the information content of the training samples, regularizes the end-to-end training, and emphasizes robust models capable of extrapolating to new samples while simultaneously ensuring physical plausibility. Hereby, we concentrate on learning models of mechanical systems using the Euler-Lagrange-Equation, a second order ordinary differential equation (ODE) originating from Lagrangian Mechanics, as physics prior.
+
+We focus on learning models of mechanical systems as this problem is one of the fundamental challenges of robotics (de Wit et al., 2012; Schaal et al., 2002).
+
+# Contribution
+
+The contribution of this work is twofold. First, we derive a network topology called Deep Lagrangian Networks (DeLaN) encoding the Euler-Lagrange equation originating from Lagrangian Mechanics. This topology can be trained using standard end-to-end optimization techniques while maintaining physical plausibility. Therefore, the obtained model must comply with physics. Unlike previous approaches to learning physics (Atkeson et al., 1986; Ledezma & Haddadin, 2017), which engineered fixed features from physical assumptions requiring knowledge of the specific physical embodiment, we are ‘only’ enforcing physics upon a generic deep network. For DeLaN only the system state and the control signal are specific to the physical system but neither the proposed network structure nor the training procedure. Second, we extensively evaluate the proposed approach by using the model to control a simulated 2 degrees of freedom (dof) robot and the physical 7-dof robot Barrett WAM in real time. We demonstrate DeLaN’s control performance where DeLaN learns the dynamics model online starting from random initialization. In comparison to analytic- and other learned models, DeLaN yields a better control performance while at the same time extrapolates to new desired trajectories.
+
+In the following we provide an overview about related work (Section 2) and briefly summarize Lagrangian Mechanics (Section 3). Subsequently, we derive our proposed approach DeLaN and the necessary characteristics for end-to-end training are shown (Section 4). Finally, the experiments in Section 5 evaluate the model learning performance for both simulated and physical robots. Here, DeLaN outperforms existing approaches.
+
+# 2 Related Work
+
+Models describing system dynamics, i.e. the coupling of control input $\pmb { \tau }$ and system state $\mathbf { q }$ , are essential for model-based control approaches (Ioannou $\&$ τ Sun, 1996). Depending on the control approach, the control law relies either on the forward model $f$ , mapping from control input to the change of system state, or on the inverse model $f ^ { - 1 }$ , mapping from system change to control input, i.e.,
+
+$$
+f ( \mathbf { q } , { \dot { \mathbf { q } } } , \tau ) = { \ddot { \mathbf { q } } } , \qquad f ^ { - 1 } ( \mathbf { q } , { \dot { \mathbf { q } } } , { \ddot { \mathbf { q } } } ) = \tau .
+$$
+
+Examples for application of these models are inverse dynamics control (de Wit et al., 2012), which uses the inverse model to compensate system dynamics, while model-predictive control (Camacho & Alba, 2013) and optimal control (Zhou et al., 1996) use the forward model to plan the control input. These models can be either derived from physics or learned from data. The physics models must be derived for the individual system embodiment and requires precise knowledge of the physical properties (Albu-Schäffer, 2002). When learning the model2, mostly standard machine learning techniques are applied to fit either the forward- or inverse-model to the training data. E.g., authors used Linear Regression (Schaal et al., 2002; Haruno et al., 2001), Gaussian Mixture Regression (Calinon et al., 2010; Khansari-Zadeh & Billard, 2011), Gaussian Process Regression (Kocijan et al., 2004; Nguyen-Tuong et al., 2009; Nguyen-Tuong & Peters, 2010), Support Vector Regression (Choi et al., 2007; Ferreira et al., 2007), feedforward- (Jansen, 1994; Lenz et al., 2015; Ledezma & Haddadin, 2017; Sanchez-Gonzalez et al., 2018) or recurrent neural networks (Rueckert et al., 2017) to fit the model to the observed measurements.
+
+Only few approaches incorporate prior knowledge into the learning problem. Sanchez-Gonzalez et al. (2018) use the graph representation of the kinematic structure as input. While the work of Atkeson et al. (1986), commonly referenced as the standard system identification technique for robot manipulators (Siciliano & Khatib, 2016), uses the Newton-Euler formalism to derive physics features using the kinematic structure and the joint measurements such that the learning of the dynamics model simplifies to linear regression. Similarly, Ledezma & Haddadin (2017) hard-code these physics features within a neural network and learn the dynamics parameters using gradient descent rather than linear regression. Even though these physics features are derived from physics, the learned parameters for mass, center of gravity and inertia must not necessarily comply with physics as the learned parameters may violate the positive definiteness of the inertia matrix or the parallel axis theorem (Ting et al., 2006). Furthermore, the linear regression is commonly underdetermined and only allows to infer linear combinations of the dynamics parameters and cannot be applied to close-loop kinematics (Siciliano & Khatib, 2016).
+
+DeLaN follows the line of structured learning problems but in contrast to previous approaches guarantees physical plausibility and provides a more general formulation. This general formulation enables DeLaN to learn the dynamics for any kinematic structure, including kinematic trees and closed-loop kinematics, and in addition does not require any knowledge about the kinematic structure. Therefore, DeLaN is identical for all mechanical systems, which is in strong contrast to the NewtonEuler approaches, where the features are specific to the kinematic structure. Only the system state and input is specific to the system but neither the network topology nor the optimization procedure.
+
+The combination of differential equations and Neural Networks has previously been investigated in literature. Early on Lagaris et al. (1998; 2000) proposed to learn the solution of partial differential equations (PDE) using neural networks and currently this topic is being rediscovered by Raissi & Karniadakis (2018); Sirignano & Spiliopoulos (2017); Long et al. (2017). Most research focuses on using machine learning to overcome the limitations of PDE solvers. E.g., Sirignano & Spiliopoulos (2017) proposed the Deep Galerkin method to solve a high-dimensional PDE from scattered data. Only the work of Raissi et al. (2017) took the opposite standpoint of using the knowledge of the specific differential equation to structure the learning problem and achieve lower sample complexity. In this paper, we follow the same motivation as Raissi et al. (2017) but take a different approach. Rather than explicitly solving the differential equation, DeLaN only uses the structure of the differential equation to guide the learning problem of inferring the equations of motion. Thereby the differential equation is only implicitly solved. In addition, the proposed approach uses different encoding of the partial derivatives, which achieves the efficient computation within a single feed-forward pass, enabling the application within control loops.
+
+# 3 Preliminaries: Lagrangian Mechanics
+
+Describing the equations of motion for mechanical systems has been extensively studied and various formalisms to derive these equations exist. The most prominent are Newtonian-, Hamiltonianand Lagrangian-Mechanics. Within this work Lagrangian Mechanics is used, more specifically the Euler-Lagrange formulation with non-conservative forces and generalized coordinates.3 Generalized coordinates are coordinates that uniquely define the system configuration. This formalism defines the Lagrangian $L$ as a function of generalized coordinates $\mathbf { q }$ describing the complete dynamics of a given system. The Lagrangian is not unique and every $L$ which yields the correct equations of motion is valid. The Lagrangian is generally chosen to be
+
+$$
+L = T - V
+$$
+
+where $T$ is the kinetic energy and $V$ is the potential energy. The kinetic energy $T$ can be computed for all choices of generalized coordinates using $\begin{array} { r } { T = \frac { 1 } { 2 } \dot { \bf q } ^ { T } { \bf H } ( { \bf q } ) \dot { \bf q } } \end{array}$ , whereas $\mathbf { H } ( \mathbf { q } )$ is the symmetric and positive definite inertia matrix (de Wit et al., 2012). The positive definiteness ensures that all non-zero velocities lead to positive kinetic energy. Applying the calculus of variations yields the Euler-Lagrange equation with non-conservative forces described by
+
+$$
+{ \frac { d } { d t } } { \frac { \partial L } { \partial { \dot { \mathbf { q } } } _ { i } } } - { \frac { \partial L } { \partial \mathbf { q } _ { i } } } = \pmb { \tau } _ { i }
+$$
+
+where $\pmb { \tau }$ are generalized forces. Substituting $L$ and $d V / d \mathbf { q } = \mathbf { g } ( \mathbf { q } )$ into Equation 3 yields the second τorder ordinary differential equation (ODE) described by
+
+$$
+\mathbf { H } ( \mathbf { q } ) { \ddot { \mathbf { q } } } + \underbrace { { \dot { \mathbf { H } } } ( \mathbf { q } ) { \dot { \mathbf { q } } } - { \frac { 1 } { 2 } } \left( { \frac { \partial } { \partial \mathbf { q } } } \left( { \dot { \mathbf { q } } } ^ { T } \mathbf { H } ( \mathbf { q } ) { \dot { \mathbf { q } } } \right) \right) ^ { T } + \mathbf { g } ( \mathbf { q } ) } _ { : = \mathbf { c } ( \mathbf { q } , { \dot { \mathbf { q } } } ) } = \pmb { \tau }
+$$
+
+where c describes the forces generated by the Centripetal and Coriolis forces (Featherstone, 2007). Using this ODE any multi-particle mechanical system with holonomic constraints can be described. For example various authors used this ODE to manually derived the equations of motion for coupled pendulums (Greenwood, 2006), robotic manipulators with flexible joints (Book, 1984; Spong, 1987), parallel robots (Miller, 1992; Geng et al., 1992; Liu et al., 1993) or legged robots (Hemami & Wyman, 1979; Golliday & Hemami, 1977).
+
+# 4 Incorporating Lagrangian Mechanics into Deep Learning
+
+Starting from the Euler-Lagrange equation (Equation 4), traditional engineering approaches would estimate $\mathbf { H } ( \mathbf { q } )$ and ${ \bf g } ( { \bf q } )$ from the approximated or measured masses, lengths and moments of inertia. On the contrary most traditional model learning approaches would ignore the structure and learn the inverse dynamics model directly from data. DeLaN bridges this gap by incorporating the structure introduced by the ODE into the learning problem and learns the parameters in an end-to-end fashion. More concretely, DeLaN approximates the inverse model by representing the unknown functions ${ \bf g } ( { \bf q } )$ and $\mathbf { H } ( \mathbf { q } )$ as a feed-forward networks. Rather than representing $\mathbf { H } ( \mathbf { q } )$ directly, the lower-triangular matrix $\mathbf { L } ( \mathbf { q } )$ is represented as deep network. Therefore, ${ \bf g } ( { \bf q } )$ and $\mathbf { H } ( \mathbf { q } )$ are described by
+
+$$
+\hat { \mathbf { H } } ( \mathbf { q } ) = \hat { \mathbf { L } } ( \mathbf { q } ; \theta ) \hat { \mathbf { L } } ( \mathbf { q } ; \theta ) ^ { T } \qquad \hat { \mathbf { g } } ( \mathbf { q } ) = \hat { \mathbf { g } } ( \mathbf { q } ; \psi )
+$$
+
+where ˆ refers to an approximation and $\theta$ and $\psi$ are the respective network parameters. The parameters $\theta$ and $\psi$ θ ψcan be obtained by minimizing the violation of the physical law described by Lagrangian θ ψMechanics. Therefore, the optimization problem is described by
+
+$$
+\begin{array} { c } { { ( \theta ^ { * } , \psi ^ { * } ) = \underset { \theta , \psi } { \mathrm { a r g m i n } } \ell \left( \hat { f } ^ { - 1 } ( \mathbf { q } , \dot { \mathbf { q } } , \ddot { \mathbf { q } } ; \theta , \psi ) , \tau \right) } } \\ { { \mathrm { w i t h } \hat { f } ^ { - 1 } ( \mathbf { q } , \dot { \mathbf { q } } , \ddot { \mathbf { q } } ; \theta , \psi ) = \hat { \mathbf { L } } \hat { \mathbf { L } } ^ { T } \ddot { \mathbf { q } } + \displaystyle \frac { d } { d t } \left( \hat { \mathbf { L } } \hat { \mathbf { L } } ^ { T } \right) \dot { \mathbf { q } } - \displaystyle \frac { 1 } { 2 } \left( \frac { \partial } { \partial \mathbf { q } } \left( \dot { \mathbf { q } } ^ { T } \hat { \mathbf { L } } \hat { \mathbf { L } } ^ { T } \dot { \mathbf { q } } \right) \right) ^ { T } + \hat { \mathbf { g } } } } \\ { { \mathrm { s . t . ~ } 0 < \mathbf { x } ^ { T } \hat { \mathbf { L } } \hat { \mathbf { L } } ^ { T } \mathbf { x } \quad \forall \mathbf { x } \in \mathbb { R } _ { 0 } ^ { n } } } \end{array}
+$$
+
+where ${ \hat { f } } ^ { - 1 }$ is the inverse model and $\ell$ can be any differentiable loss function. The computational graph of $\hat { f } ^ { - 1 }$ is shown in Figure 1.
+
+Using this formulation one can conclude further properties of the learned model. Neither $\hat { \bf L }$ nor $\hat { \bf g }$ are functions of $\dot { \bf q }$ or $\ddot { \bf q }$ and, hence, the obtained parameters should, within limits, generalize to arbitrary velocities and accelerations. In addition, the obtained model can be reformulated and used as a forward model. Solving Equation 6 for $\ddot { \bf q }$ yields the forward model described by
+
+$$
+\hat { f } ( \mathbf { q } , \dot { \mathbf { q } } , \tau ; \theta , \psi ) = \left( \hat { \mathbf { L } } \hat { \mathbf { L } } ^ { T } \right) ^ { - 1 } \left( \tau - \frac { d } { d t } \left( \hat { \mathbf { L } } \hat { \mathbf { L } } ^ { T } \right) \dot { \mathbf { q } } + \frac { 1 } { 2 } \left( \frac { \partial } { \partial \mathbf { q } } \left( \dot { \mathbf { q } } ^ { T } \hat { \mathbf { L } } \hat { \mathbf { L } } ^ { T } \dot { \mathbf { q } } \right) \right) ^ { T } - \hat { \mathbf { g } } \right)
+$$
+
+where $\boldsymbol { \hat { \mathbf { L } } } \boldsymbol { \hat { \mathbf { L } } } ^ { T }$ is guaranteed to be invertible due to the positive definite constraint (Equation 7). However, solving the optimization problem of Equation 5 directly is not possible due to the ill-posedness of the Lagrangian $L$ not being unique. The Euler-Lagrange equation is invariant to linear transformation and, hence, the Lagrangian $L ^ { \prime } = \alpha L + \beta$ solves the Euler-Lagrange equation if $\alpha$ is non-zero and $L$ α β αis a valid Lagrangian. This problem can be mitigated by adding an additional penalty term to Equation 5 described by
+
+$$
+( \theta ^ { * } , \psi ^ { * } ) = \underset { \theta , \psi } { \arg \operatorname* { m i n } } \ell \left( \hat { f } ^ { - 1 } ( \mathbf { q } , \dot { \mathbf { q } } , \ddot { \mathbf { q } } ; \theta , \psi ) , \pmb { \tau } \right) + \lambda \Omega ( \theta , \psi )
+$$
+
+where $\Omega$ is the $L _ { 2 }$ -norm of the network weights.
+
+Solving the optimization problem of Equation 9 with a gradient based end-to-end learning approach is non-trivial due to the positive definite constraint (Equation 7) and the derivatives contained in $\hat { f } ^ { - 1 }$ . In particular, $d ( \mathbf { L L } ^ { T } ) / \dot { d t }$ and $\partial \left( \dot { \mathbf { q } } ^ { T } \mathbf { L } \mathbf { L } ^ { T } \dot { \mathbf { q } } \right) / \partial \mathbf { q } _ { i }$ cannot be computed using automatic differentiation as $t$ ∂ ∂is not an input of the network and most implementations of automatic differentiation do not allow the backpropagation of the gradient through the computed derivatives. Therefore, the derivatives contained in $\hat { f } ^ { - 1 }$ must be computed analytically to exploit the full gradient information for training of the parameters. In the following we introduce a network structure that fulfills the positive-definite constraint for all parameters (Section 4.1), prove that the derivatives $d ( \mathbf { L L } ^ { T } ) / d t$ and $\partial ^ { \mathbf { \bar { \rho } } } \big ( \dot { \mathbf { q } } ^ { T } \mathbf { L L } ^ { T } \dot { \mathbf { q } } \big ) / \partial \mathbf { q } _ { i }$ ∂ ∂can be computed analytically (Section 4.2) and show an efficient implementation for computing the derivatives using a single feed-forward pass (Section 4.3). Using these three properties the resulting network architecture can be used within a real-time control loop and trained using standard end-to-end optimization techniques.
+
+![](images/68b21fb9d22b2d99afdd141817916ee4de414c885083e8476cd1f8d08b4ced51.jpg)  
+Figure 1: The computational graph of the Deep Lagrangian Network (DeLaN). Shown in blue and green is the neural network with the three separate heads computing ${ \bf g } ( { \bf q } ) , { \bf l } _ { d } ( { \bf q } ) , { \bf l } _ { o } ( { \bf q } )$ . The orange boxes correspond to the reshaping operations and the derivatives contained in the Euler-Lagrange equation. For training the gradients are backpropagated through all vertices highlighted in orange.
+
+# 4.1 Symmetry and Positive Definiteness of H
+
+Ensuring the symmetry and positive definiteness of $\mathbf { H }$ is essential as this constraint enforces positive kinetic energy for all non-zero velocities. In addition, the positive definiteness ensures that $\mathbf { H }$ is invertible and the obtained model can be used as forward model. By representing the matrix $\mathbf { H }$ as the product of a lower-triangular matrix the symmetry and the positive semi-definiteness is ensured while simultaneously reducing the number of parameters. The positive definiteness is obtained if the diagonal of $\mathbf { L }$ is positive. This positive diagonal also guarantees that $\mathbf { L }$ is invertible. Using a deep network with different heads and altering the activation of the output layer one can obtain a positive diagonal. The off-diagonal elements $\mathbf { L } _ { o }$ use a linear activation while the diagonal elements $\mathbf { L } _ { d }$ use a non-negative activation, e.g., ReLu or Softplus. In addition, a positive scalar $^ b$ is added to diagonal elements. Thereby, ensuring a positive diagonal of $\mathbf { L }$ and the positive eigenvalues of $\mathbf { H }$ . In addition, we chose to share parameters between $\mathbf { L }$ and $\mathbf { g }$ as both rely on the same physical embodiment. The network architecture, with three-heads representing the diagonal $\mathbf { l } _ { d }$ and off-diagonal ${ \mathbf { l } } _ { o }$ entries of $\mathbf { L }$ and $\mathbf { g }$ , is shown in Figure 1.
+
+# 4.2 Deriving the derivatives
+
+The derivatives $d \left( { \bf L L } ^ { T } \right) / d t$ and $\partial \left( \dot { \mathbf { q } } ^ { T } \mathbf { L } \mathbf { L } ^ { T } \dot { \mathbf { q } } \right) / \partial \mathbf { q } _ { i }$ are required for computing the control signal $\pmb { \tau }$ ∂ ∂ τusing the inverse model and, hence, must be available within the forward pass. In addition, the second order derivatives, used within the backpropagation of the gradients, must exist to train the network using end-to-end training. To enable the computation of the second order derivatives using automatic differentiation the forward computation must be performed analytically. Both derivatives, $d \left( { \bf L L } ^ { T } \right) / d t$ and $\partial \left( { \dot { \mathbf { q } } ^ { T } } \mathbf { L } \mathbf { L } ^ { T } { \dot { \mathbf { q } } } \right) / \partial \mathbf { q } _ { i }$ , have closed form solutions and can be derived by first computing ∂the respective derivative of $\mathbf { L }$ ∂and second substituting the reshaped derivative of the vectorized form l. For the temporal derivative $d \left( { \bf L L } ^ { T } \right) / d t$ this yields
+
+$$
+{ \frac { d } { d t } } \mathbf { H } ( \mathbf { q } ) = { \frac { d } { d t } } \left( \mathbf { L } \mathbf { L } ^ { T } \right) = \mathbf { L } { \frac { d \mathbf { L } } { d t } } ^ { T } + { \frac { d \mathbf { L } } { d t } } \mathbf { L } ^ { T }
+$$
+
+whereas $d \mathbf { L } / d t$ can be substituted with the reshaped form of
+
+$$
+{ \frac { d } { d t } } \mathbf { I } = { \frac { \partial \mathbf { I } } { \partial \mathbf { q } } } { \frac { \partial \mathbf { q } } { \partial t } } + \sum _ { i = 1 } ^ { N } { \frac { \partial \mathbf { I } } { \partial \mathbf { W } _ { i } } } { \frac { \partial \mathbf { W } _ { i } } { \partial t } } + \sum _ { i = 1 } ^ { N } { \frac { \partial \mathbf { I } } { \partial \mathbf { b } _ { i } } } { \frac { \partial \mathbf { b } _ { i } } { \partial t } }
+$$
+
+where $i$ refers to the $i$ -th network layer consisting of an affine transformation and the non-linearity $g$ , i.e., $\mathbf { h } _ { i } = g _ { i } \left( \mathbf { W } _ { i } ^ { T } \mathbf { h } _ { i - 1 } + \mathbf { b } _ { i } \right)$ . Equation 11 can be simplified as the network weights $\mathbf { W } _ { i }$ and biases $\mathbf { b } _ { i }$
+
+![](images/ac44e57c2f68b4cd04c5ab3ff8e2421bd01e3192099350dab38469739576d4f1.jpg)  
+Figure 2: (a) Computational graph of the Lagrangian layer. The orange boxes highlight the learnable parameters. The upper computational sub-graph corresponds to the standard network layer while the lower sub-graph is the extension of the Lagrangian layer to simultaneously compute ${ \partial { \bf h } _ { i } } / { \partial { \bf h } _ { i - 1 } }$ . (b) Computational graph of the chained Lagrangian layer to compute $\mathbf { L }$ , $d \mathbf { L } / { \dot { d } } t$ and ${ \partial \mathbf { L } } / { \partial \mathbf { q } _ { i } }$ ∂using a single feed-forward pass.
+
+are time-invariant, i.e., $d \mathbf { W } _ { i } / d t = 0$ and $d { \bf b } _ { i } / d t = 0$ . Therefore, $d \mathbf { I } / d t$ is described by
+
+$$
+\frac { d } { d t } \mathbf { l } = \frac { \partial \mathbf { l } } { \partial \mathbf { q } } \dot { \mathbf { q } } .
+$$
+
+Due to the compositional structure of the network and the differentiability of the non-linearity, the derivative with respect to the network input $d \mathbf { l } / d \mathbf { q }$ can be computed by recursively applying the chain rule, i.e.,
+
+$$
+{ \frac { \partial \mathbf { l } } { \partial \mathbf { q } } } = { \frac { \partial \mathbf { l } } { \partial \mathbf { h } _ { N - 1 } } } { \frac { \partial \mathbf { h } _ { N - 1 } } { \partial \mathbf { h } _ { N - 2 } } } \cdots { \frac { \partial \mathbf { h } _ { 1 } } { \partial \mathbf { q } } } { \frac { \partial \mathbf { h } _ { i } } { \partial \mathbf { q } } } = \mathrm { d i a g } \left( g ^ { \prime } ( \mathbf { W } _ { i } ^ { T } \mathbf { h } _ { i - 1 } + \mathbf { b } _ { i } ) \right) \mathbf { W } _ { i } 
+$$
+
+where $g ^ { \prime }$ is the derivative of the non-linearity. Similarly to the previous derivation, the partial derivative of the quadratic term can be computed using the chain rule, which yields
+
+$$
+\frac { \partial } { \partial \mathbf { q } _ { i } } \left[ \dot { \mathbf { q } } ^ { T } \mathbf { H } \dot { \mathbf { q } } \right] = \mathrm { t r } \left[ \left( \dot { \mathbf { q } } \dot { \mathbf { q } } ^ { T } \right) ^ { T } \frac { \partial \mathbf { H } } { \partial \mathbf { q } _ { i } } \right] = \dot { \mathbf { q } } ^ { T } \left( \frac { \partial \mathbf { L } } { \partial \mathbf { q } _ { i } } \mathbf { L } ^ { T } + \mathbf { L } \frac { \partial \mathbf { L } } { \partial \mathbf { q } _ { i } } ^ { T } \right) \dot { \mathbf { q } }
+$$
+
+whereas ${ \partial \mathbf { L } } / { \partial \mathbf { q } _ { i } }$ can be constructed using the columns of previously derived $\partial \mathbf { l } / \partial \mathbf { q }$ . Therefore, all ∂ ∂derivatives included within $\hat { f }$ can be computed in closed form.
+
+# 4.3 Computing the Derivatives
+
+The derivatives of Section 4.2 must be computed within a real-time control loop and only add minimal computational complexity in order to not break the real-time constraint. l and $\mathbf { \partial } \partial \mathbf { l } / \partial \mathbf { q }$ , ∂ ∂required within Equation 10 and Equation 14, can be simultaneously computed using an extended standard layer. Extending the affine transformation and non-linearity of the standard layer with an additional sub-graph for computing ${ \partial { \bf h } _ { i } } / { \partial { \bf h } _ { i - 1 } }$ yields the Lagrangian layer described by
+
+$$
+\mathbf { a } _ { i } = \mathbf { W } _ { i } \mathbf { h } _ { i - 1 } + \mathbf { b } _ { i } \qquad \mathbf { h } _ { 1 } = g _ { i } \left( \mathbf { a } _ { i } \right) \qquad { \frac { \partial \mathbf { h } _ { i } } { \partial \mathbf { h } _ { i - 1 } } } = \mathrm { d i a g } \left( g _ { i } ^ { \prime } ( \mathbf { a } _ { i } ) \right) \mathbf { W } _ { i } .
+$$
+
+The computational graph of the Lagrangian layer is shown in Figure 2a. Chaining the Lagrangian layer yields the compositional structure of $\mathbf { \partial } \partial \mathbf { l } / \partial \mathbf { q }$ (Equation 13) and enables the efficient computation of $\mathbf { \partial } \partial \mathbf { l } / \partial \mathbf { q }$ ∂ ∂. Additional reshaping operations compute $d \mathbf { L } / d t$ and ${ \partial \mathbf { L } } / { \partial \mathbf { q } _ { i } }$ .
+
+# 5 Experimental Evaluation: Learning an Inverse Dynamics Model for Robot Control
+
+To demonstrate the applicability and extrapolation of DeLaN, the proposed network topology is applied to model-based control for a simulated 2-dof robot (Figure 3b) and the physical 7-dof robot
+
+![](images/8758abee29c46dea63151bbfcae95804e9e0590fd15eec1d7165b74464f9f05f.jpg)  
+Figure 3: (a) Real-time control loop using a PD-Controller with a feed-forward torque $\pmb { \tau } _ { F F }$ , compensating the system dynamics, to control the joint torques $\pmb { \tau }$ τ. The training process reads the joint states τand applies torques to learn the system dynamics online. Once a new model becomes available the inverse model ${ \hat { f } } ^ { - 1 }$ in the control loop is updated. (b) The simulated 2-dof robot drawing the cosine trajectories. (c) The simulated Barrett WAM drawing the 3d cosine 0 trajectory. (d) The physical Barrett WAM.
+
+Barrett WAM (Figure 3d). The performance of DeLaN is evaluated using the tracking error on train and test trajectories and compared to a learned and analytic model. This evaluation scheme follows existing work (Nguyen-Tuong et al., 2009; Sanchez-Gonzalez et al., 2018) as the tracking error is the relevant performance indicator while the mean squared error (MSE)4 obtained using sample based optimization exaggerates model performance (Hobbs & Hepenstal, 1989). In addition to most previous work, we strictly limit all model predictions to real-time and perform the learning online, i.e., the models are randomly initialized and must learn the model during the experiment.
+
+Experimental Setup
+
+Within the experiment the robot executes multiple desired trajectories with specified joint positions, velocities and accelerations. The control signal, consisting of motor torques, is generated using a non-linear feedforward controller, i.e., a low gain PD-Controller augmented with a feed-forward torque $\pmb { \tau } _ { f f }$ to compensate system dynamics. The control law is described by
+
+$$
+\pmb { \tau } = \mathbf { K } _ { p } \left( \mathbf { q } _ { d } - \mathbf { q } \right) + \mathbf { K } _ { d } \left( \dot { \mathbf { q } } _ { d } - \dot { \mathbf { q } } \right) + \pmb { \tau } _ { f f } \quad \mathrm { w i t h } \ \pmb { \tau } _ { f f } = \hat { f } ^ { - 1 } ( \mathbf { q } _ { d } , \dot { \mathbf { q } } _ { d } , \ddot { \mathbf { q } } _ { d } )
+$$
+
+where $\mathbf { K } _ { p }$ , $\mathbf { K } _ { d }$ are the controller gains and ${ \ P } d , \ \dot { \ P } q , \ \ddot { \ P } d$ the desired joint positions, velocities and accelerations. The control-loop is shown in Figure 3a. For all experiments the control frequency is set to $5 0 0 \mathrm { H z }$ while the desired joint state and respectively $\tau _ { f f }$ is updated with a frequency of $f _ { d } = 2 0 0 \mathrm { H z }$ τ. All feed-forward torques are computed online and, hence, the computation time is strictly limited to $T \leq 1 / 2 0 0 \mathrm { s }$ . The tracking performance is defined as the sum of the MSE evaluated at the sampling points of the reference trajectory.
+
+For the desired trajectories two different data sets are used. The first data set contains all single stroke characters5 while the second data set uses cosine curves in joint space (Figure 3c). The 20 characters are spatially and temporally re-scaled to comply with the robot kinematics. The joint references are computed using the inverse kinematics. Due to the different characters, the desired trajectories contain smooth and sharp turns and cover a wide variety of different shapes but are limited to a small task space region. In contrast, the cosine trajectories are smooth but cover a large task space region.
+
+# Baselines
+
+The performance of DeLaN is compared to an analytic inverse dynamics model, a standard feedforward neural network (FF-NN) and a PD-Controller. For the analytic models the torque is computed using the Recursive Newton-Euler algorithm (RNE) (Luh et al., 1980), which computes the feedforward torque using estimated physical properties of the system, i.e. the link dimensions, masses and moments of inertia. For implementations the open-source library PyBullet (Coumans & Bai, 2016–2018) is used.
+
+Both deep networks use the same dimensionality, ReLu nonlinearities and must learn the system dynamics online starting from random initialization. The training samples containing joint states and applied torques $( \mathbf { q } , { \dot { \mathbf { q } } } , { \ddot { \mathbf { q } } } , \tau ) _ { 0 , \dots T }$ are directly read from the control loop as shown in Figure 3a.
+
+![](images/895c48bc6f32ae684bca947c1133ee8a1b58333116871af8f2905983ece6ab1e.jpg)  
+Figure 4: (a) The torque $\pmb { \tau }$ required to generate the characters ’a’, ’d’ and ’e’ in black. Using these τsamples DeLaN was trained offline and learns the red trajectory. DeLaN can not only learn the desired torques but also disambiguate the individual torque components even though DeLaN was trained on the super-imposed torques. Using Equation 6 DeLaN can represent the inertial force $\mathbf { H } \ddot { \mathbf { q } }$ (b), the Coriolis and Centrifugal forces $\mathbf { c } ( \mathbf { q } , { \dot { \mathbf { q } } } )$ (c) and the gravitational force ${ \bf g } ( { \bf q } )$ (d). All components ,match closely the ground truth data. (e) shows the offline MSE of the feed-forward neural network and DeLaN for each joint.
+
+The training runs in a separate process on the same machine and solves the optimization problem online. Once the training process computed a new model, the inverse model ${ \hat { f } } ^ { - 1 }$ of the control loop is updated.
+
+# 5.1 Simulated Robot Experiments
+
+The 2-dof robot shown in Figure 3b is simulated using PyBullet and executes the character and cosine trajectories. Figure 4 shows the ground truth torques of the characters ’a’, ’d’, ’e’, the torque ground truth components and the learned decomposition using DeLaN (Figure 4a-d). Even though DeLaN is trained on the super-imposed torques, DeLaN learns to disambiguate the inertial force $\mathbf { H } \ddot { \mathbf { q } }$ , the Coriolis and Centrifugal force $\mathbf { c } ( \mathbf { q } , \dot { \mathbf { q } } )$ and the gravitational force ${ \bf g } ( { \bf q } )$ as the respective ,curves overlap closely. Hence, DeLaN is capable of learning the underlying physical model using the proposed network topology trained with standard end-to-end optimization. Figure 4d shows the offline MSE on the test set averaged over multiple seeds for the FF-NN and DeLaN w.r.t. to different training set sizes. The different training set sizes correspond to the combination of $n$ random characters, i.e., a training set size of 1 corresponds to training the model on a single character and evaluating the performance on the remaining 19 characters. DeLaN clearly obtains a lower test MSE compared to the FF-NN. Especially the difference in performance increases when the training set is reduced. This increasing difference on the test MSE highlights the reduced sample complexity and the good extrapolation to unseen samples. This difference in performance is amplified on the real-time control-task where the models are learned online starting from random initialization. Figure 5a and b shows the accumulated tracking error per testing character and the testing error averaged over all test characters while Figure 5c shows the qualitative comparison of the control performance6. It is important to point out that all shown results are averaged over multiple seeds and only incorporate characters not used for training and, hence, focus the evaluation on the extrapolation to new trajectories. The qualitative comparison shows that DeLaN is able to execute all 20 characters when trained on 8 random characters. The obtained tracking error is comparable to the analytic model, which in this case contains the simulation parameters and is optimal. In contrast, the FF-NN shows significant deviation from the desired trajectories when trained on 8 random characters. The quantitative comparison of the accumulated tracking error over seeds (Figure 5b) shows that DeLaN obtains lower tracking error on all training set sizes compared to the FF-NN. This good performance using only few training characters shows that DeLaN has a lower sample complexity and better extrapolation to unseen trajectories compared to the FF-NN.
+
+Figure 6a and b show the performance on the cosine trajectories. For this experiment the models are only trained online on two trajectories with a velocity scale of 1x. To assess the extrapolation w.r.t. velocities and accelerations the learned models are tested on the same trajectories with scaled velocities (gray area of Figure 6). On the training trajectories DeLaN and the FF-NN perform comparable. When the velocities are increased the performance of FF-NN deteriorates because the new trajectories do not lie within the vicinity of the training distribution as the domain of the FF-NN is defined as (q qÛ qÜ). Therefore, FF-NN cannot extrapolate to the testing data. In contrast, the , ,domain of the networks $\hat { \bf L }$ and $\hat { \bf g }$ composing DeLaN only consist of $\mathbf { q }$ , rather than $( \mathbf { q } , \dot { \mathbf { q } } , \ddot { \mathbf { q } } )$ . This , ,reduced domain enables DeLaN, within limit, to extrapolate to the test trajectories. The increase in tracking error is caused by the structure of ${ \hat { f } } ^ { - 1 }$ , where model errors to scale quadratic with velocities. However, the obtained tracking error on the testing trajectories is significantly lower compared to FF-NN.
+
+![](images/920ef33192444dbd28f5b000a259904f93b8151cb85d3a6bba2079fa9329831e.jpg)  
+Figure 5: (a) The average performance of DeLaN and the feed forward neural network for each character. The 4 columns of the boxplots correspond to different numbers of training characters, i.e., $n = 1$ , 6, 8, 10. (b) The median performance of DeLaN, the feed forward neural network and the analytic baselines averaged over multiple seeds. The shaded areas highlight the 5th and the 95th percentile. (c) The qualitative performance for the analytic baselines, the feed forward neural network and DeLaN. The desired trajectories are shown in red.
+
+# 5.2 Physical Robot Experiments
+
+For physical experiments the desired trajectories are executed on the Barrett WAM, a robot with direct cable drives. The direct cable drives produce high torques generating fast and dexterous movements but yield complex dynamics, which cannot be modelled using rigid-body dynamics due to the variable stiffness and lengths of the cables7. Therefore, the Barrett WAM is ideal for testing the applicability of model learning and analytic models8 on complex dynamics. For the physical experiments we focus on the cosine trajectories as these trajectories produce dynamic movements while character trajectories are mainly dominated by the gravitational forces. In addition, only the dynamics of the four lower joints are learned because these joints dominate the dynamics and the upper joints cannot be sufficiently excited to retrieve the dynamics parameters.
+
+Figure 6c and d show the tracking error on the cosine trajectories using the the simulated Barrett WAM while Figure 6e and f show the tracking error of the physical Barrett WAM. It is important to note, that the simulation only simulates the rigid-body dynamics not including the direct cables drives and the simulation parameters are inconsistent with the parameters of the analytic model. Therefore, the analytic model is not optimal. On the training trajectories executed on the physical system the FF-NN performs better compared to DeLaN and the analytic model. DeLaN achieves slightly better tracking error than the analytic model, which uses the same rigid-body assumptions as DeLaN. That shows DeLaN can learn a dynamics model of the WAM but is limited by the model assumptions of Lagrangian Mechanics. These assumptions cannot represent the dynamics of the cable drives. When comparing to the simulated results, DeLaN and the FF-NN perform comparable but significantly better than the analytic model. These simulation results show that DeLaN can learn an accurate model of the WAM, when the underlying assumptions of the physics prior hold. The tracking performance on the physical system and the simulation indicate that DeLaN can learn a model within the model class of the physics prior but also inherits the limitations of the physics prior. For this specific experiment the FF-NN can locally learn correlations of the torques w.r.t. q, qÛ and $\ddot { \bf q }$ while such correlation cannot be represented by the network topology of DeLaN because such correlation should, by definition of the physics prior, not exist.
+
+![](images/c170ee96ef30fe375eca65efa15c7cb67c083a9f1ab49cc06078dc4853d6ad18.jpg)  
+Figure 6: The tracking error of the cosine trajectories for the simulated 2-dof robot (a & b), the simulated (c & d) and the physical Barrett WAM (e & f). The feed-forward neural network and DeLaN are trained only on the trajectories at a velocity scale of $1 \times$ . Afterwards the models are tested on the same trajectories with increased velocities to evaluate the extrapolation to new velocities.
+
+When extrapolating to the identical trajectories with higher velocities (gray area of Figure 6) the tracking error of the FF-NN deteriorates much faster compared to DeLaN, because the FF-NN overfits to the training data. The tracking error of the analytic model remains constant and demonstrates the guaranteed extrapolation of the analytic models. When comparing the simulated results, the FF-NN cannot extrapolate to the new velocities and the tracking error deteriorates similarly to the performance on the physical robot. In contrast to the FF-NN, DeLaN can extrapolate to the higher velocities and maintains a good tracking error. Even further, DeLaN obtains a better tracking error compared the analytic model on all velocity scales. This low tracking error on all test trajectories highlights the improved extrapolation of DeLaN compared to other model learning approaches.
+
+# 6 Conclusion
+
+We introduced the concept of incorporating a physics prior within the deep learning framework to achieve lower sample complexity and better extrapolation. In particular, we proposed Deep Lagrangian Networks (DeLaN), a deep network on which Lagrangian Mechanics is imposed. This specific network topology enabled us to learn the system dynamics using end-to-end training while maintaining physical plausibility. We showed that DeLaN is able to learn the underlying physics from a super-imposed signal, as DeLaN can recover the contribution of the inertial-, gravitational and centripetal forces from sensor data. The quantitative evaluation within a real-time control loop assessing the tracking error showed that DeLaN can learn the system dynamics online, obtains lower sample complexity and better generalization compared to a feed-forward neural network. DeLaN can extrapolate to new trajectories as well as to increased velocities, where the performance of the feedforward network deteriorates due to the overfitting to the training data. When applied to a physical systems with complex dynamics the bounded representational power of the physics prior can be limiting. However, this limited representational power enforces the physical plausibility and obtains the lower sample complexity and substantially better generalization. In future work the physics prior should be extended to represent a wider system class by introducing additional non-conservative forces within the Lagrangian.
+
+# Acknowledgments
+
+This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No #640554 (SKILLS4ROBOTS). Furthermore, this research was also supported by grants from ABB, NVIDIA and the NVIDIA DGX Station.
+
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+Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561, 2017.
+
+Elmar Rueckert, Moritz Nakatenus, Samuele Tosatto, and Jan Peters. Learning inverse dynamics models in o (n) time with lstm networks. In IEEE-RAS International Conference on Humanoid Robotics, pp. 811–816. IEEE, 2017.
+
+Alvaro Sanchez-Gonzalez, Nicolas Heess, Jost Tobias Springenberg, Josh Merel, Martin Riedmiller, Raia Hadsell, and Peter Battaglia. Graph networks as learnable physics engines for inference and control. arXiv preprint arXiv:1806.01242, 2018.
+
+Stefan Schaal, Christopher G Atkeson, and Sethu Vijayakumar. Scalable techniques from nonparametric statistics for real time robot learning. Applied Intelligence, 17(1):49–60, 2002.
+
+Bruno Siciliano and Oussama Khatib. Springer handbook of robotics. Springer, 2016.
+
+David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(7676):354, 2017.
+
+Justin Sirignano and Konstantinos Spiliopoulos. Dgm: A deep learning algorithm for solving partial differential equations. arXiv preprint arXiv:1708.07469, 2017.
+
+Mark W Spong. Modeling and control of elastic joint robots. Journal of dynamic systems, measurement, and control, 109(4):310–318, 1987.
+
+Jo-Anne Ting, Michael Mistry, Jan Peters, Stefan Schaal, and Jun Nakanishi. A bayesian approach to nonlinear parameter identification for rigid body dynamics. In Robotics: Science and Systems, pp. 32–39, 2006.
+
+Ben Williams, Marc Toussaint, and Amos J Storkey. Modelling motion primitives and their timing in biologically executed movements. In Advances in Neural Information Processing Systems 20, pp. 1609–1616. 2008.
+
+Kemin Zhou, John Comstock Doyle, Keith Glover, et al. Robust and optimal control, volume 40. Prentice Hall, New Jersey, 1996.
+
+# Appendix A: Offline Benchmarks
+
+![](images/2838bd052ccad5a953b39d598e9577b17d7787f264304383189196697834cbd9.jpg)  
+Figure 7: The mean squared error averaged of 20 seeds on the training- (a) and test-set (b) of the character trajectories for the two joint robot. The models are trained offline using $n$ characters and tested using the remaining $2 0 - n$ characters. The training samples are corrupted with white noise, while the performance is tested on noise-free trajectories.
+
+To evaluate the performance of DeLaN without the control task, DeLaN was trained offline on previously collected data and evaluated using the mean squared error (MSE) on the test and training set. For comparison, DeLaN is compared to the system identification approach (SI) described by Atkeson et al. (1986), a feed-forward neural network (FF-NN) and the Recursive Newton Euler algorithm (RNE) using an analytic model. For this comparison, one must point out that the system identification approach relies on the availability of the kinematics, as the Jacobians and transformations w.r.t. to every link must be known to compute the necessary features. In contrast, neither DeLaN nor the FF-NN require this knowledge and must implicitly also learn the kinematics.
+
+Figure 7 shows the MSE averaged over 20 seeds on the character data set executed on the two-joint robot. For this data set, the models are trained using noisy samples and evaluated on the noise-free and previously unseen characters. The FF-NN performs the best on the training set, but overfits to the training data. Therefore, the FF-NN does not generalize to unseen characters. In contrast, the SI approach does not overfit to the noise and extrapolates to previously unseen characters. In comparison, the structure of DeLaN regularizes the training and prevents the overfitting to the corrupted training data. Therefore, DeLaN extrapolates better than the FF-NN but not as good as the SI approach. Similar results can be observed on the cosine data set using the Barrett WAM simulated in SL (Figure 8 a, b). The FF-NN performs best on the training trajectory but the performance deteriorates when this network extrapolates to higher velocities. SI performs worse on the training trajectory but extrapolates to higher velocities. In comparison, DeLaN performs comparable to the SI approach on the training trajectory, extrapolates significantly better than the FF-NN but does not extrapolate as good as the SI approach. For the physical system (Figure 8 c, d), the results differ from the results in simulation. On the physical system the SI approach only achieves the same performance as RNE, which is significantly worse compared to the performance of DeLaN and the FF-NN. When evaluating the extrapolation to higher velocities, the analytic model and the SI approach extrapolate to higher velocities, while the MSE for the FF-NN significantly increases. In comparison, DeLaN extrapolates better compared to the FF-NN but not as good as the analytic model or the SI approach.
+
+This performance difference between the simulation and physical system can be explained by the underlying model assumptions and the robustness to noise. While DeLaN only assumes rigidbody dynamics, the SI approach also assumes the exact knowledge of the kinematic structure. For simulation both assumptions are valid. However, for the physical system, the exact kinematics are unknown due to production imperfections and the direct cable drives applying torques to flexible joints violate the rigid-body assumption. Therefore, the SI approach performs significantly worse on the physical system. Furthermore, the noise robustness becomes more important for the physical system due to the inherent sensor noise. While the linear regression of the SI approach is easily corrupted by noise or outliers, the gradient based optimization of the networks is more robust to noise. This robustness can be observed in Figure 9, which shows the correlation between the variance of Gaussian noise corrupting the training data and the MSE of the simulated and noise-free cosine trajectories. With increasing noise levels, the MSE of the SI approach increases significantly faster compared to the models learned using gradient descent.
+
+![](images/dba559c1041e142619db8cdfc7bded69e4101b93e0f4b478eeb6d1627681faab.jpg)  
+Figure 8: The mean squared error of the cosine trajectories for the simulated (a, b) and the physical Barrett WAM (c and d). The system identification approach, feed-forward neural network and DeLaN are trained offline using only the trajectories at a velocity scale of $1 \times$ . Afterwards the models are tested on the same trajectories with increased velocities to evaluate the extrapolation to new velocities.
+
+Concluding, the extrapolation of DeLaN to unseen trajectories and higher velocities is not as good as the SI approach but significantly better than the generic FF-NN. This increased extrapolation compared to the generic network is achieved by the Lagrangian Mechanics prior of DeLaN. Even though this prior promotes extrapolation, the prior also hinders the performance on the physical robot, because the prior cannot represent the dynamics of the direct cable drives. Therefore, DeLaN performs worse than the FF-NN, which does not assume any model structure. However, DeLaN outperforms the SI approach on the physical system, which also assumes rigid-body dynamics and requires the exact knowledge of the kinematics.
+
+![](images/079380252b48469dc3e1bd6718daf71329ab5a93949b348be4d71de29e41be64.jpg)  
+Figure 9: The mean squared error on the simulated and noise-free cosine trajectories with velocity scale of 1x. For offline training the samples are corrupted using i.i.d. noise sampled from a multivariate Normal distribution with the variance of $\sigma ^ { 2 } \mathbf { I }$ .
+
+# Appendix B: Complete Online Results
+
+![](images/cb5bb05488090fc4a9d514c240c7f99b73baa64c5551db83e72fc9b180f1c083.jpg)  
+Figure 10: The qualitative performance for the analytic baselines, the feed forward neural network and DeLaN for different number of random training characters. The desired trajectories are shown in red.
+
+![](images/3d387cf436117e72ddc1630599b2407448d122105c9c09f63a0cabd574793e03.jpg)  
+Figure 11: The average performance of DeLaN and the feed forward neural network for each character. The columns of the boxplots correspond to different numbers of training characters, i.e., $n = 1 , 2 , 4 , 6 , 8 , 1 0 , 1 2$ .

parse/train/BklHpjCqKm/BklHpjCqKm_content_list.json

@@ -0,0 +1,1778 @@
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+    {
+        "type": "text",
+        "text": "Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Michael Lutter, Christian Ritter & Jan Peters ∗ ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science Technische Universität Darmstadt Hochschulstr. 10, 64289 Darmstadt, Germany {Lutter, Peters}@ias.tu-darmstadt.de ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning has achieved astonishing results on many tasks with large amounts of data and generalization within the proximity of training data. For many important real-world applications, these requirements are unfeasible and additional prior knowledge on the task domain is required to overcome the resulting problems. In particular, learning physics models for model-based control requires robust extrapolation from fewer samples – often collected online in real-time – and model errors may lead to drastic damages of the system. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Directly incorporating physical insight has enabled us to obtain a novel deep model learning approach that extrapolates well while requiring fewer samples. As a first example, we propose Deep Lagrangian Networks (DeLaN) as a deep network structure upon which Lagrangian Mechanics have been imposed. DeLaN can learn the equations of motion of a mechanical system (i.e., system dynamics) with a deep network efficiently while ensuring physical plausibility. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The resulting DeLaN network performs very well at robot tracking control. The proposed method did not only outperform previous model learning approaches at learning speed but exhibits substantially improved and more robust extrapolation to novel trajectories and learns online in real-time. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            574,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the last five years, deep learning has propelled most areas of learning forward at an impressive pace (Krizhevsky et al., 2012; Mnih et al., 2015; Silver et al., 2017) – with the exception of physically embodied systems. This lag in comparison to other application areas is somewhat surprising as learning physical models is critical for applications that control embodied systems, reason about prior actions or plan future actions (e.g., service robotics, industrial automation). Instead, most engineers prefer classical off-the-shelf modeling as it ensures physical plausibility – at a high cost of precise measurements1 and engineering effort. These plausible representations are preferred as these models guarantee to extrapolate to new samples, while learned models only achieve good performance in the vicinity of the training data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To learn a model that obtains physically plausible representations, we propose to use the insights from physics as a model prior for deep learning. In particular, the combination of deep learning and physics seems natural as the compositional structure of deep networks enables the efficient computation of the derivatives at machine precision (Raissi & Karniadakis, 2018) and, thus, can encode a differential equation describing physical processes. Therefore, we suggest to encode the physics prior in the form of a differential in the network topology. This adapted topology amplifies the information content of the training samples, regularizes the end-to-end training, and emphasizes robust models capable of extrapolating to new samples while simultaneously ensuring physical plausibility. Hereby, we concentrate on learning models of mechanical systems using the Euler-Lagrange-Equation, a second order ordinary differential equation (ODE) originating from Lagrangian Mechanics, as physics prior. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We focus on learning models of mechanical systems as this problem is one of the fundamental challenges of robotics (de Wit et al., 2012; Schaal et al., 2002). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contribution ",
+        "text_level": 1,
+        "bbox": [
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+            147,
+            272,
+            161
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The contribution of this work is twofold. First, we derive a network topology called Deep Lagrangian Networks (DeLaN) encoding the Euler-Lagrange equation originating from Lagrangian Mechanics. This topology can be trained using standard end-to-end optimization techniques while maintaining physical plausibility. Therefore, the obtained model must comply with physics. Unlike previous approaches to learning physics (Atkeson et al., 1986; Ledezma & Haddadin, 2017), which engineered fixed features from physical assumptions requiring knowledge of the specific physical embodiment, we are ‘only’ enforcing physics upon a generic deep network. For DeLaN only the system state and the control signal are specific to the physical system but neither the proposed network structure nor the training procedure. Second, we extensively evaluate the proposed approach by using the model to control a simulated 2 degrees of freedom (dof) robot and the physical 7-dof robot Barrett WAM in real time. We demonstrate DeLaN’s control performance where DeLaN learns the dynamics model online starting from random initialization. In comparison to analytic- and other learned models, DeLaN yields a better control performance while at the same time extrapolates to new desired trajectories. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the following we provide an overview about related work (Section 2) and briefly summarize Lagrangian Mechanics (Section 3). Subsequently, we derive our proposed approach DeLaN and the necessary characteristics for end-to-end training are shown (Section 4). Finally, the experiments in Section 5 evaluate the model learning performance for both simulated and physical robots. Here, DeLaN outperforms existing approaches. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Models describing system dynamics, i.e. the coupling of control input $\\pmb { \\tau }$ and system state $\\mathbf { q }$ , are essential for model-based control approaches (Ioannou $\\&$ τ Sun, 1996). Depending on the control approach, the control law relies either on the forward model $f$ , mapping from control input to the change of system state, or on the inverse model $f ^ { - 1 }$ , mapping from system change to control input, i.e., ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/371eda6f59cacb459a97fb3816a5633ac8dc74c79234221a7de8e7b245420ea8.jpg",
+        "text": "$$\nf ( \\mathbf { q } , { \\dot { \\mathbf { q } } } , \\tau ) = { \\ddot { \\mathbf { q } } } , \\qquad f ^ { - 1 } ( \\mathbf { q } , { \\dot { \\mathbf { q } } } , { \\ddot { \\mathbf { q } } } ) = \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Examples for application of these models are inverse dynamics control (de Wit et al., 2012), which uses the inverse model to compensate system dynamics, while model-predictive control (Camacho & Alba, 2013) and optimal control (Zhou et al., 1996) use the forward model to plan the control input. These models can be either derived from physics or learned from data. The physics models must be derived for the individual system embodiment and requires precise knowledge of the physical properties (Albu-Schäffer, 2002). When learning the model2, mostly standard machine learning techniques are applied to fit either the forward- or inverse-model to the training data. E.g., authors used Linear Regression (Schaal et al., 2002; Haruno et al., 2001), Gaussian Mixture Regression (Calinon et al., 2010; Khansari-Zadeh & Billard, 2011), Gaussian Process Regression (Kocijan et al., 2004; Nguyen-Tuong et al., 2009; Nguyen-Tuong & Peters, 2010), Support Vector Regression (Choi et al., 2007; Ferreira et al., 2007), feedforward- (Jansen, 1994; Lenz et al., 2015; Ledezma & Haddadin, 2017; Sanchez-Gonzalez et al., 2018) or recurrent neural networks (Rueckert et al., 2017) to fit the model to the observed measurements. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Only few approaches incorporate prior knowledge into the learning problem. Sanchez-Gonzalez et al. (2018) use the graph representation of the kinematic structure as input. While the work of Atkeson et al. (1986), commonly referenced as the standard system identification technique for robot manipulators (Siciliano & Khatib, 2016), uses the Newton-Euler formalism to derive physics features using the kinematic structure and the joint measurements such that the learning of the dynamics model simplifies to linear regression. Similarly, Ledezma & Haddadin (2017) hard-code these physics features within a neural network and learn the dynamics parameters using gradient descent rather than linear regression. Even though these physics features are derived from physics, the learned parameters for mass, center of gravity and inertia must not necessarily comply with physics as the learned parameters may violate the positive definiteness of the inertia matrix or the parallel axis theorem (Ting et al., 2006). Furthermore, the linear regression is commonly underdetermined and only allows to infer linear combinations of the dynamics parameters and cannot be applied to close-loop kinematics (Siciliano & Khatib, 2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "DeLaN follows the line of structured learning problems but in contrast to previous approaches guarantees physical plausibility and provides a more general formulation. This general formulation enables DeLaN to learn the dynamics for any kinematic structure, including kinematic trees and closed-loop kinematics, and in addition does not require any knowledge about the kinematic structure. Therefore, DeLaN is identical for all mechanical systems, which is in strong contrast to the NewtonEuler approaches, where the features are specific to the kinematic structure. Only the system state and input is specific to the system but neither the network topology nor the optimization procedure. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The combination of differential equations and Neural Networks has previously been investigated in literature. Early on Lagaris et al. (1998; 2000) proposed to learn the solution of partial differential equations (PDE) using neural networks and currently this topic is being rediscovered by Raissi & Karniadakis (2018); Sirignano & Spiliopoulos (2017); Long et al. (2017). Most research focuses on using machine learning to overcome the limitations of PDE solvers. E.g., Sirignano & Spiliopoulos (2017) proposed the Deep Galerkin method to solve a high-dimensional PDE from scattered data. Only the work of Raissi et al. (2017) took the opposite standpoint of using the knowledge of the specific differential equation to structure the learning problem and achieve lower sample complexity. In this paper, we follow the same motivation as Raissi et al. (2017) but take a different approach. Rather than explicitly solving the differential equation, DeLaN only uses the structure of the differential equation to guide the learning problem of inferring the equations of motion. Thereby the differential equation is only implicitly solved. In addition, the proposed approach uses different encoding of the partial derivatives, which achieves the efficient computation within a single feed-forward pass, enabling the application within control loops. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Preliminaries: Lagrangian Mechanics ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Describing the equations of motion for mechanical systems has been extensively studied and various formalisms to derive these equations exist. The most prominent are Newtonian-, Hamiltonianand Lagrangian-Mechanics. Within this work Lagrangian Mechanics is used, more specifically the Euler-Lagrange formulation with non-conservative forces and generalized coordinates.3 Generalized coordinates are coordinates that uniquely define the system configuration. This formalism defines the Lagrangian $L$ as a function of generalized coordinates $\\mathbf { q }$ describing the complete dynamics of a given system. The Lagrangian is not unique and every $L$ which yields the correct equations of motion is valid. The Lagrangian is generally chosen to be ",
+        "bbox": [
+            173,
+            530,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efd81fca47ec4d6bdf35ee111a341444a66d824973549fecccdc5c233e6a8479.jpg",
+        "text": "$$\nL = T - V\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            650,
+            534,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $T$ is the kinetic energy and $V$ is the potential energy. The kinetic energy $T$ can be computed for all choices of generalized coordinates using $\\begin{array} { r } { T = \\frac { 1 } { 2 } \\dot { \\bf q } ^ { T } { \\bf H } ( { \\bf q } ) \\dot { \\bf q } } \\end{array}$ , whereas $\\mathbf { H } ( \\mathbf { q } )$ is the symmetric and positive definite inertia matrix (de Wit et al., 2012). The positive definiteness ensures that all non-zero velocities lead to positive kinetic energy. Applying the calculus of variations yields the Euler-Lagrange equation with non-conservative forces described by ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/309dc4050c2512c42940d1272872e3ba21bfbb1e05a6f2064e608aa2427e9aff.jpg",
+        "text": "$$\n{ \\frac { d } { d t } } { \\frac { \\partial L } { \\partial { \\dot { \\mathbf { q } } } _ { i } } } - { \\frac { \\partial L } { \\partial \\mathbf { q } _ { i } } } = \\pmb { \\tau } _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            750,
+            562,
+            785
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { \\tau }$ are generalized forces. Substituting $L$ and $d V / d \\mathbf { q } = \\mathbf { g } ( \\mathbf { q } )$ into Equation 3 yields the second τorder ordinary differential equation (ODE) described by ",
+        "bbox": [
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+            790,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca006f4dfefa671d90affe1ab25fb939b8eea63fb68160865e4500976b2aec94.jpg",
+        "text": "$$\n\\mathbf { H } ( \\mathbf { q } ) { \\ddot { \\mathbf { q } } } + \\underbrace { { \\dot { \\mathbf { H } } } ( \\mathbf { q } ) { \\dot { \\mathbf { q } } } - { \\frac { 1 } { 2 } } \\left( { \\frac { \\partial } { \\partial \\mathbf { q } } } \\left( { \\dot { \\mathbf { q } } } ^ { T } \\mathbf { H } ( \\mathbf { q } ) { \\dot { \\mathbf { q } } } \\right) \\right) ^ { T } + \\mathbf { g } ( \\mathbf { q } ) } _ { : = \\mathbf { c } ( \\mathbf { q } , { \\dot { \\mathbf { q } } } ) } = \\pmb { \\tau }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where c describes the forces generated by the Centripetal and Coriolis forces (Featherstone, 2007). Using this ODE any multi-particle mechanical system with holonomic constraints can be described. For example various authors used this ODE to manually derived the equations of motion for coupled pendulums (Greenwood, 2006), robotic manipulators with flexible joints (Book, 1984; Spong, 1987), parallel robots (Miller, 1992; Geng et al., 1992; Liu et al., 1993) or legged robots (Hemami & Wyman, 1979; Golliday & Hemami, 1977). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Incorporating Lagrangian Mechanics into Deep Learning ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Starting from the Euler-Lagrange equation (Equation 4), traditional engineering approaches would estimate $\\mathbf { H } ( \\mathbf { q } )$ and ${ \\bf g } ( { \\bf q } )$ from the approximated or measured masses, lengths and moments of inertia. On the contrary most traditional model learning approaches would ignore the structure and learn the inverse dynamics model directly from data. DeLaN bridges this gap by incorporating the structure introduced by the ODE into the learning problem and learns the parameters in an end-to-end fashion. More concretely, DeLaN approximates the inverse model by representing the unknown functions ${ \\bf g } ( { \\bf q } )$ and $\\mathbf { H } ( \\mathbf { q } )$ as a feed-forward networks. Rather than representing $\\mathbf { H } ( \\mathbf { q } )$ directly, the lower-triangular matrix $\\mathbf { L } ( \\mathbf { q } )$ is represented as deep network. Therefore, ${ \\bf g } ( { \\bf q } )$ and $\\mathbf { H } ( \\mathbf { q } )$ are described by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2bc02d89360ee5d3033419d5f74d6b855f52d1eefd1cf6191942740b5512f01.jpg",
+        "text": "$$\n\\hat { \\mathbf { H } } ( \\mathbf { q } ) = \\hat { \\mathbf { L } } ( \\mathbf { q } ; \\theta ) \\hat { \\mathbf { L } } ( \\mathbf { q } ; \\theta ) ^ { T } \\qquad \\hat { \\mathbf { g } } ( \\mathbf { q } ) = \\hat { \\mathbf { g } } ( \\mathbf { q } ; \\psi )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            356,
+            643,
+            376
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where ˆ refers to an approximation and $\\theta$ and $\\psi$ are the respective network parameters. The parameters $\\theta$ and $\\psi$ θ ψcan be obtained by minimizing the violation of the physical law described by Lagrangian θ ψMechanics. Therefore, the optimization problem is described by ",
+        "bbox": [
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+            382,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24544b58dd9b80014b86b2b33b429b265cd07eda8b10371010ee32a02fb31bdb.jpg",
+        "text": "$$\n\\begin{array} { c } { { ( \\theta ^ { * } , \\psi ^ { * } ) = \\underset { \\theta , \\psi } { \\mathrm { a r g m i n } } \\ell \\left( \\hat { f } ^ { - 1 } ( \\mathbf { q } , \\dot { \\mathbf { q } } , \\ddot { \\mathbf { q } } ; \\theta , \\psi ) , \\tau \\right) } } \\\\ { { \\mathrm { w i t h } \\hat { f } ^ { - 1 } ( \\mathbf { q } , \\dot { \\mathbf { q } } , \\ddot { \\mathbf { q } } ; \\theta , \\psi ) = \\hat { \\mathbf { L } } \\hat { \\mathbf { L } } ^ { T } \\ddot { \\mathbf { q } } + \\displaystyle \\frac { d } { d t } \\left( \\hat { \\mathbf { L } } \\hat { \\mathbf { L } } ^ { T } \\right) \\dot { \\mathbf { q } } - \\displaystyle \\frac { 1 } { 2 } \\left( \\frac { \\partial } { \\partial \\mathbf { q } } \\left( \\dot { \\mathbf { q } } ^ { T } \\hat { \\mathbf { L } } \\hat { \\mathbf { L } } ^ { T } \\dot { \\mathbf { q } } \\right) \\right) ^ { T } + \\hat { \\mathbf { g } } } } \\\\ { { \\mathrm { s . t . ~ } 0 < \\mathbf { x } ^ { T } \\hat { \\mathbf { L } } \\hat { \\mathbf { L } } ^ { T } \\mathbf { x } \\quad \\forall \\mathbf { x } \\in \\mathbb { R } _ { 0 } ^ { n } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            430,
+            750,
+            527
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\hat { f } } ^ { - 1 }$ is the inverse model and $\\ell$ can be any differentiable loss function. The computational graph of $\\hat { f } ^ { - 1 }$ is shown in Figure 1. ",
+        "bbox": [
+            176,
+            532,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using this formulation one can conclude further properties of the learned model. Neither $\\hat { \\bf L }$ nor $\\hat { \\bf g }$ are functions of $\\dot { \\bf q }$ or $\\ddot { \\bf q }$ and, hence, the obtained parameters should, within limits, generalize to arbitrary velocities and accelerations. In addition, the obtained model can be reformulated and used as a forward model. Solving Equation 6 for $\\ddot { \\bf q }$ yields the forward model described by ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df98fd822b9dba0e508069efb207e05251ff1d539952046acde469ef0c90f4fe.jpg",
+        "text": "$$\n\\hat { f } ( \\mathbf { q } , \\dot { \\mathbf { q } } , \\tau ; \\theta , \\psi ) = \\left( \\hat { \\mathbf { L } } \\hat { \\mathbf { L } } ^ { T } \\right) ^ { - 1 } \\left( \\tau - \\frac { d } { d t } \\left( \\hat { \\mathbf { L } } \\hat { \\mathbf { L } } ^ { T } \\right) \\dot { \\mathbf { q } } + \\frac { 1 } { 2 } \\left( \\frac { \\partial } { \\partial \\mathbf { q } } \\left( \\dot { \\mathbf { q } } ^ { T } \\hat { \\mathbf { L } } \\hat { \\mathbf { L } } ^ { T } \\dot { \\mathbf { q } } \\right) \\right) ^ { T } - \\hat { \\mathbf { g } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            633,
+            743,
+            676
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { \\hat { \\mathbf { L } } } \\boldsymbol { \\hat { \\mathbf { L } } } ^ { T }$ is guaranteed to be invertible due to the positive definite constraint (Equation 7). However, solving the optimization problem of Equation 5 directly is not possible due to the ill-posedness of the Lagrangian $L$ not being unique. The Euler-Lagrange equation is invariant to linear transformation and, hence, the Lagrangian $L ^ { \\prime } = \\alpha L + \\beta$ solves the Euler-Lagrange equation if $\\alpha$ is non-zero and $L$ α β αis a valid Lagrangian. This problem can be mitigated by adding an additional penalty term to Equation 5 described by ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/381b7181a65a5c89986cc987e67bda21b496b8a8d3d74222b9e2479c211ea709.jpg",
+        "text": "$$\n( \\theta ^ { * } , \\psi ^ { * } ) = \\underset { \\theta , \\psi } { \\arg \\operatorname* { m i n } } \\ell \\left( \\hat { f } ^ { - 1 } ( \\mathbf { q } , \\dot { \\mathbf { q } } , \\ddot { \\mathbf { q } } ; \\theta , \\psi ) , \\pmb { \\tau } \\right) + \\lambda \\Omega ( \\theta , \\psi )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            776,
+            678,
+            809
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\Omega$ is the $L _ { 2 }$ -norm of the network weights. ",
+        "bbox": [
+            174,
+            815,
+            486,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Solving the optimization problem of Equation 9 with a gradient based end-to-end learning approach is non-trivial due to the positive definite constraint (Equation 7) and the derivatives contained in $\\hat { f } ^ { - 1 }$ . In particular, $d ( \\mathbf { L L } ^ { T } ) / \\dot { d t }$ and $\\partial \\left( \\dot { \\mathbf { q } } ^ { T } \\mathbf { L } \\mathbf { L } ^ { T } \\dot { \\mathbf { q } } \\right) / \\partial \\mathbf { q } _ { i }$ cannot be computed using automatic differentiation as $t$ ∂ ∂is not an input of the network and most implementations of automatic differentiation do not allow the backpropagation of the gradient through the computed derivatives. Therefore, the derivatives contained in $\\hat { f } ^ { - 1 }$ must be computed analytically to exploit the full gradient information for training of the parameters. In the following we introduce a network structure that fulfills the positive-definite constraint for all parameters (Section 4.1), prove that the derivatives $d ( \\mathbf { L L } ^ { T } ) / d t$ and $\\partial ^ { \\mathbf { \\bar { \\rho } } } \\big ( \\dot { \\mathbf { q } } ^ { T } \\mathbf { L L } ^ { T } \\dot { \\mathbf { q } } \\big ) / \\partial \\mathbf { q } _ { i }$ ∂ ∂can be computed analytically (Section 4.2) and show an efficient implementation for computing the derivatives using a single feed-forward pass (Section 4.3). Using these three properties the resulting network architecture can be used within a real-time control loop and trained using standard end-to-end optimization techniques. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/68b21fb9d22b2d99afdd141817916ee4de414c885083e8476cd1f8d08b4ced51.jpg",
+        "image_caption": [
+            "Figure 1: The computational graph of the Deep Lagrangian Network (DeLaN). Shown in blue and green is the neural network with the three separate heads computing ${ \\bf g } ( { \\bf q } ) , { \\bf l } _ { d } ( { \\bf q } ) , { \\bf l } _ { o } ( { \\bf q } )$ . The orange boxes correspond to the reshaping operations and the derivatives contained in the Euler-Lagrange equation. For training the gradients are backpropagated through all vertices highlighted in orange. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 Symmetry and Positive Definiteness of H ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            420,
+            511,
+            434
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Ensuring the symmetry and positive definiteness of $\\mathbf { H }$ is essential as this constraint enforces positive kinetic energy for all non-zero velocities. In addition, the positive definiteness ensures that $\\mathbf { H }$ is invertible and the obtained model can be used as forward model. By representing the matrix $\\mathbf { H }$ as the product of a lower-triangular matrix the symmetry and the positive semi-definiteness is ensured while simultaneously reducing the number of parameters. The positive definiteness is obtained if the diagonal of $\\mathbf { L }$ is positive. This positive diagonal also guarantees that $\\mathbf { L }$ is invertible. Using a deep network with different heads and altering the activation of the output layer one can obtain a positive diagonal. The off-diagonal elements $\\mathbf { L } _ { o }$ use a linear activation while the diagonal elements $\\mathbf { L } _ { d }$ use a non-negative activation, e.g., ReLu or Softplus. In addition, a positive scalar $^ b$ is added to diagonal elements. Thereby, ensuring a positive diagonal of $\\mathbf { L }$ and the positive eigenvalues of $\\mathbf { H }$ . In addition, we chose to share parameters between $\\mathbf { L }$ and $\\mathbf { g }$ as both rely on the same physical embodiment. The network architecture, with three-heads representing the diagonal $\\mathbf { l } _ { d }$ and off-diagonal ${ \\mathbf { l } } _ { o }$ entries of $\\mathbf { L }$ and $\\mathbf { g }$ , is shown in Figure 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Deriving the derivatives ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            643,
+            390,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The derivatives $d \\left( { \\bf L L } ^ { T } \\right) / d t$ and $\\partial \\left( \\dot { \\mathbf { q } } ^ { T } \\mathbf { L } \\mathbf { L } ^ { T } \\dot { \\mathbf { q } } \\right) / \\partial \\mathbf { q } _ { i }$ are required for computing the control signal $\\pmb { \\tau }$ ∂ ∂ τusing the inverse model and, hence, must be available within the forward pass. In addition, the second order derivatives, used within the backpropagation of the gradients, must exist to train the network using end-to-end training. To enable the computation of the second order derivatives using automatic differentiation the forward computation must be performed analytically. Both derivatives, $d \\left( { \\bf L L } ^ { T } \\right) / d t$ and $\\partial \\left( { \\dot { \\mathbf { q } } ^ { T } } \\mathbf { L } \\mathbf { L } ^ { T } { \\dot { \\mathbf { q } } } \\right) / \\partial \\mathbf { q } _ { i }$ , have closed form solutions and can be derived by first computing ∂the respective derivative of $\\mathbf { L }$ ∂and second substituting the reshaped derivative of the vectorized form l. For the temporal derivative $d \\left( { \\bf L L } ^ { T } \\right) / d t$ this yields ",
+        "bbox": [
+            173,
+            667,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b3a2752d0514d5f97d78633b9f8e1ee49cf9c86413ba639d55fd2aedc285ec75.jpg",
+        "text": "$$\n{ \\frac { d } { d t } } \\mathbf { H } ( \\mathbf { q } ) = { \\frac { d } { d t } } \\left( \\mathbf { L } \\mathbf { L } ^ { T } \\right) = \\mathbf { L } { \\frac { d \\mathbf { L } } { d t } } ^ { T } + { \\frac { d \\mathbf { L } } { d t } } \\mathbf { L } ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            787,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "whereas $d \\mathbf { L } / d t$ can be substituted with the reshaped form of ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77268d51a4e26a52eb633e54dfaaa5a727152a4508e4e163069c94128d40ba35.jpg",
+        "text": "$$\n{ \\frac { d } { d t } } \\mathbf { I } = { \\frac { \\partial \\mathbf { I } } { \\partial \\mathbf { q } } } { \\frac { \\partial \\mathbf { q } } { \\partial t } } + \\sum _ { i = 1 } ^ { N } { \\frac { \\partial \\mathbf { I } } { \\partial \\mathbf { W } _ { i } } } { \\frac { \\partial \\mathbf { W } _ { i } } { \\partial t } } + \\sum _ { i = 1 } ^ { N } { \\frac { \\partial \\mathbf { I } } { \\partial \\mathbf { b } _ { i } } } { \\frac { \\partial \\mathbf { b } _ { i } } { \\partial t } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            847,
+            645,
+            890
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $i$ refers to the $i$ -th network layer consisting of an affine transformation and the non-linearity $g$ , i.e., $\\mathbf { h } _ { i } = g _ { i } \\left( \\mathbf { W } _ { i } ^ { T } \\mathbf { h } _ { i - 1 } + \\mathbf { b } _ { i } \\right)$ . Equation 11 can be simplified as the network weights $\\mathbf { W } _ { i }$ and biases $\\mathbf { b } _ { i }$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/ac44e57c2f68b4cd04c5ab3ff8e2421bd01e3192099350dab38469739576d4f1.jpg",
+        "image_caption": [
+            "Figure 2: (a) Computational graph of the Lagrangian layer. The orange boxes highlight the learnable parameters. The upper computational sub-graph corresponds to the standard network layer while the lower sub-graph is the extension of the Lagrangian layer to simultaneously compute ${ \\partial { \\bf h } _ { i } } / { \\partial { \\bf h } _ { i - 1 } }$ . (b) Computational graph of the chained Lagrangian layer to compute $\\mathbf { L }$ , $d \\mathbf { L } / { \\dot { d } } t$ and ${ \\partial \\mathbf { L } } / { \\partial \\mathbf { q } _ { i } }$ ∂using a single feed-forward pass. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            98,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "are time-invariant, i.e., $d \\mathbf { W } _ { i } / d t = 0$ and $d { \\bf b } _ { i } / d t = 0$ . Therefore, $d \\mathbf { I } / d t$ is described by ",
+        "bbox": [
+            173,
+            377,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b6b7da65e008a9bbacdba0ae1ff283eac8af39264321bec5315717b876b88cf.jpg",
+        "text": "$$\n\\frac { d } { d t } \\mathbf { l } = \\frac { \\partial \\mathbf { l } } { \\partial \\mathbf { q } } \\dot { \\mathbf { q } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            457,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Due to the compositional structure of the network and the differentiability of the non-linearity, the derivative with respect to the network input $d \\mathbf { l } / d \\mathbf { q }$ can be computed by recursively applying the chain rule, i.e., ",
+        "bbox": [
+            173,
+            436,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/17c1f5d967b3645bfd67a2df814ab9780a1a89ddbccb34fa639229fce176f6af.jpg",
+        "text": "$$\n{ \\frac { \\partial \\mathbf { l } } { \\partial \\mathbf { q } } } = { \\frac { \\partial \\mathbf { l } } { \\partial \\mathbf { h } _ { N - 1 } } } { \\frac { \\partial \\mathbf { h } _ { N - 1 } } { \\partial \\mathbf { h } _ { N - 2 } } } \\cdots { \\frac { \\partial \\mathbf { h } _ { 1 } } { \\partial \\mathbf { q } } } { \\frac { \\partial \\mathbf { h } _ { i } } { \\partial \\mathbf { q } } } = \\mathrm { d i a g } \\left( g ^ { \\prime } ( \\mathbf { W } _ { i } ^ { T } \\mathbf { h } _ { i - 1 } + \\mathbf { b } _ { i } ) \\right) \\mathbf { W } _ { i } \n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            483,
+            743,
+            517
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $g ^ { \\prime }$ is the derivative of the non-linearity. Similarly to the previous derivation, the partial derivative of the quadratic term can be computed using the chain rule, which yields ",
+        "bbox": [
+            173,
+            522,
+            826,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b20ad0bb5104503fb4ca77c754a61fbb56ccb9b812ec5da2bd3505a10c883c14.jpg",
+        "text": "$$\n\\frac { \\partial } { \\partial \\mathbf { q } _ { i } } \\left[ \\dot { \\mathbf { q } } ^ { T } \\mathbf { H } \\dot { \\mathbf { q } } \\right] = \\mathrm { t r } \\left[ \\left( \\dot { \\mathbf { q } } \\dot { \\mathbf { q } } ^ { T } \\right) ^ { T } \\frac { \\partial \\mathbf { H } } { \\partial \\mathbf { q } _ { i } } \\right] = \\dot { \\mathbf { q } } ^ { T } \\left( \\frac { \\partial \\mathbf { L } } { \\partial \\mathbf { q } _ { i } } \\mathbf { L } ^ { T } + \\mathbf { L } \\frac { \\partial \\mathbf { L } } { \\partial \\mathbf { q } _ { i } } ^ { T } \\right) \\dot { \\mathbf { q } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            556,
+            694,
+            593
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "whereas ${ \\partial \\mathbf { L } } / { \\partial \\mathbf { q } _ { i } }$ can be constructed using the columns of previously derived $\\partial \\mathbf { l } / \\partial \\mathbf { q }$ . Therefore, all ∂ ∂derivatives included within $\\hat { f }$ can be computed in closed form. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Computing the Derivatives ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            645,
+            406,
+            659
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The derivatives of Section 4.2 must be computed within a real-time control loop and only add minimal computational complexity in order to not break the real-time constraint. l and $\\mathbf { \\partial } \\partial \\mathbf { l } / \\partial \\mathbf { q }$ , ∂ ∂required within Equation 10 and Equation 14, can be simultaneously computed using an extended standard layer. Extending the affine transformation and non-linearity of the standard layer with an additional sub-graph for computing ${ \\partial { \\bf h } _ { i } } / { \\partial { \\bf h } _ { i - 1 } }$ yields the Lagrangian layer described by ",
+        "bbox": [
+            174,
+            670,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca2ae011cf498608bb76c1c1022880bedf8b8ad406e464d7d7a6f517bd82f1c5.jpg",
+        "text": "$$\n\\mathbf { a } _ { i } = \\mathbf { W } _ { i } \\mathbf { h } _ { i - 1 } + \\mathbf { b } _ { i } \\qquad \\mathbf { h } _ { 1 } = g _ { i } \\left( \\mathbf { a } _ { i } \\right) \\qquad { \\frac { \\partial \\mathbf { h } _ { i } } { \\partial \\mathbf { h } _ { i - 1 } } } = \\mathrm { d i a g } \\left( g _ { i } ^ { \\prime } ( \\mathbf { a } _ { i } ) \\right) \\mathbf { W } _ { i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            747,
+            725,
+            779
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The computational graph of the Lagrangian layer is shown in Figure 2a. Chaining the Lagrangian layer yields the compositional structure of $\\mathbf { \\partial } \\partial \\mathbf { l } / \\partial \\mathbf { q }$ (Equation 13) and enables the efficient computation of $\\mathbf { \\partial } \\partial \\mathbf { l } / \\partial \\mathbf { q }$ ∂ ∂. Additional reshaping operations compute $d \\mathbf { L } / d t$ and ${ \\partial \\mathbf { L } } / { \\partial \\mathbf { q } _ { i } }$ . ",
+        "bbox": [
+            174,
+            784,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Experimental Evaluation: Learning an Inverse Dynamics Model for Robot Control ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            845,
+            803,
+            881
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the applicability and extrapolation of DeLaN, the proposed network topology is applied to model-based control for a simulated 2-dof robot (Figure 3b) and the physical 7-dof robot ",
+        "bbox": [
+            173,
+            895,
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+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/8758abee29c46dea63151bbfcae95804e9e0590fd15eec1d7165b74464f9f05f.jpg",
+        "image_caption": [
+            "Figure 3: (a) Real-time control loop using a PD-Controller with a feed-forward torque $\\pmb { \\tau } _ { F F }$ , compensating the system dynamics, to control the joint torques $\\pmb { \\tau }$ τ. The training process reads the joint states τand applies torques to learn the system dynamics online. Once a new model becomes available the inverse model ${ \\hat { f } } ^ { - 1 }$ in the control loop is updated. (b) The simulated 2-dof robot drawing the cosine trajectories. (c) The simulated Barrett WAM drawing the 3d cosine 0 trajectory. (d) The physical Barrett WAM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            71,
+            821,
+            198
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Barrett WAM (Figure 3d). The performance of DeLaN is evaluated using the tracking error on train and test trajectories and compared to a learned and analytic model. This evaluation scheme follows existing work (Nguyen-Tuong et al., 2009; Sanchez-Gonzalez et al., 2018) as the tracking error is the relevant performance indicator while the mean squared error (MSE)4 obtained using sample based optimization exaggerates model performance (Hobbs & Hepenstal, 1989). In addition to most previous work, we strictly limit all model predictions to real-time and perform the learning online, i.e., the models are randomly initialized and must learn the model during the experiment. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Experimental Setup ",
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+    {
+        "type": "text",
+        "text": "Within the experiment the robot executes multiple desired trajectories with specified joint positions, velocities and accelerations. The control signal, consisting of motor torques, is generated using a non-linear feedforward controller, i.e., a low gain PD-Controller augmented with a feed-forward torque $\\pmb { \\tau } _ { f f }$ to compensate system dynamics. The control law is described by ",
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+        "type": "equation",
+        "img_path": "images/3fd2dbffa10f925b3c42ed2b187bb7aa48c3b0c13df3e5b091a20658cc6f933d.jpg",
+        "text": "$$\n\\pmb { \\tau } = \\mathbf { K } _ { p } \\left( \\mathbf { q } _ { d } - \\mathbf { q } \\right) + \\mathbf { K } _ { d } \\left( \\dot { \\mathbf { q } } _ { d } - \\dot { \\mathbf { q } } \\right) + \\pmb { \\tau } _ { f f } \\quad \\mathrm { w i t h } \\ \\pmb { \\tau } _ { f f } = \\hat { f } ^ { - 1 } ( \\mathbf { q } _ { d } , \\dot { \\mathbf { q } } _ { d } , \\ddot { \\mathbf { q } } _ { d } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { K } _ { p }$ , $\\mathbf { K } _ { d }$ are the controller gains and ${ \\ P } d , \\ \\dot { \\ P } q , \\ \\ddot { \\ P } d$ the desired joint positions, velocities and accelerations. The control-loop is shown in Figure 3a. For all experiments the control frequency is set to $5 0 0 \\mathrm { H z }$ while the desired joint state and respectively $\\tau _ { f f }$ is updated with a frequency of $f _ { d } = 2 0 0 \\mathrm { H z }$ τ. All feed-forward torques are computed online and, hence, the computation time is strictly limited to $T \\leq 1 / 2 0 0 \\mathrm { s }$ . The tracking performance is defined as the sum of the MSE evaluated at the sampling points of the reference trajectory. ",
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+    {
+        "type": "text",
+        "text": "For the desired trajectories two different data sets are used. The first data set contains all single stroke characters5 while the second data set uses cosine curves in joint space (Figure 3c). The 20 characters are spatially and temporally re-scaled to comply with the robot kinematics. The joint references are computed using the inverse kinematics. Due to the different characters, the desired trajectories contain smooth and sharp turns and cover a wide variety of different shapes but are limited to a small task space region. In contrast, the cosine trajectories are smooth but cover a large task space region. ",
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+    {
+        "type": "text",
+        "text": "Baselines ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The performance of DeLaN is compared to an analytic inverse dynamics model, a standard feedforward neural network (FF-NN) and a PD-Controller. For the analytic models the torque is computed using the Recursive Newton-Euler algorithm (RNE) (Luh et al., 1980), which computes the feedforward torque using estimated physical properties of the system, i.e. the link dimensions, masses and moments of inertia. For implementations the open-source library PyBullet (Coumans & Bai, 2016–2018) is used. ",
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+    {
+        "type": "text",
+        "text": "Both deep networks use the same dimensionality, ReLu nonlinearities and must learn the system dynamics online starting from random initialization. The training samples containing joint states and applied torques $( \\mathbf { q } , { \\dot { \\mathbf { q } } } , { \\ddot { \\mathbf { q } } } , \\tau ) _ { 0 , \\dots T }$ are directly read from the control loop as shown in Figure 3a. ",
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+        "img_path": "images/895c48bc6f32ae684bca947c1133ee8a1b58333116871af8f2905983ece6ab1e.jpg",
+        "image_caption": [
+            "Figure 4: (a) The torque $\\pmb { \\tau }$ required to generate the characters ’a’, ’d’ and ’e’ in black. Using these τsamples DeLaN was trained offline and learns the red trajectory. DeLaN can not only learn the desired torques but also disambiguate the individual torque components even though DeLaN was trained on the super-imposed torques. Using Equation 6 DeLaN can represent the inertial force $\\mathbf { H } \\ddot { \\mathbf { q } }$ (b), the Coriolis and Centrifugal forces $\\mathbf { c } ( \\mathbf { q } , { \\dot { \\mathbf { q } } } )$ (c) and the gravitational force ${ \\bf g } ( { \\bf q } )$ (d). All components ,match closely the ground truth data. (e) shows the offline MSE of the feed-forward neural network and DeLaN for each joint. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The training runs in a separate process on the same machine and solves the optimization problem online. Once the training process computed a new model, the inverse model ${ \\hat { f } } ^ { - 1 }$ of the control loop is updated. ",
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+    {
+        "type": "text",
+        "text": "5.1 Simulated Robot Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The 2-dof robot shown in Figure 3b is simulated using PyBullet and executes the character and cosine trajectories. Figure 4 shows the ground truth torques of the characters ’a’, ’d’, ’e’, the torque ground truth components and the learned decomposition using DeLaN (Figure 4a-d). Even though DeLaN is trained on the super-imposed torques, DeLaN learns to disambiguate the inertial force $\\mathbf { H } \\ddot { \\mathbf { q } }$ , the Coriolis and Centrifugal force $\\mathbf { c } ( \\mathbf { q } , \\dot { \\mathbf { q } } )$ and the gravitational force ${ \\bf g } ( { \\bf q } )$ as the respective ,curves overlap closely. Hence, DeLaN is capable of learning the underlying physical model using the proposed network topology trained with standard end-to-end optimization. Figure 4d shows the offline MSE on the test set averaged over multiple seeds for the FF-NN and DeLaN w.r.t. to different training set sizes. The different training set sizes correspond to the combination of $n$ random characters, i.e., a training set size of 1 corresponds to training the model on a single character and evaluating the performance on the remaining 19 characters. DeLaN clearly obtains a lower test MSE compared to the FF-NN. Especially the difference in performance increases when the training set is reduced. This increasing difference on the test MSE highlights the reduced sample complexity and the good extrapolation to unseen samples. This difference in performance is amplified on the real-time control-task where the models are learned online starting from random initialization. Figure 5a and b shows the accumulated tracking error per testing character and the testing error averaged over all test characters while Figure 5c shows the qualitative comparison of the control performance6. It is important to point out that all shown results are averaged over multiple seeds and only incorporate characters not used for training and, hence, focus the evaluation on the extrapolation to new trajectories. The qualitative comparison shows that DeLaN is able to execute all 20 characters when trained on 8 random characters. The obtained tracking error is comparable to the analytic model, which in this case contains the simulation parameters and is optimal. In contrast, the FF-NN shows significant deviation from the desired trajectories when trained on 8 random characters. The quantitative comparison of the accumulated tracking error over seeds (Figure 5b) shows that DeLaN obtains lower tracking error on all training set sizes compared to the FF-NN. This good performance using only few training characters shows that DeLaN has a lower sample complexity and better extrapolation to unseen trajectories compared to the FF-NN. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 6a and b show the performance on the cosine trajectories. For this experiment the models are only trained online on two trajectories with a velocity scale of 1x. To assess the extrapolation w.r.t. velocities and accelerations the learned models are tested on the same trajectories with scaled velocities (gray area of Figure 6). On the training trajectories DeLaN and the FF-NN perform comparable. When the velocities are increased the performance of FF-NN deteriorates because the new trajectories do not lie within the vicinity of the training distribution as the domain of the FF-NN is defined as (q qÛ qÜ). Therefore, FF-NN cannot extrapolate to the testing data. In contrast, the , ,domain of the networks $\\hat { \\bf L }$ and $\\hat { \\bf g }$ composing DeLaN only consist of $\\mathbf { q }$ , rather than $( \\mathbf { q } , \\dot { \\mathbf { q } } , \\ddot { \\mathbf { q } } )$ . This , ,reduced domain enables DeLaN, within limit, to extrapolate to the test trajectories. The increase in tracking error is caused by the structure of ${ \\hat { f } } ^ { - 1 }$ , where model errors to scale quadratic with velocities. However, the obtained tracking error on the testing trajectories is significantly lower compared to FF-NN. ",
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+        "img_path": "images/920ef33192444dbd28f5b000a259904f93b8151cb85d3a6bba2079fa9329831e.jpg",
+        "image_caption": [
+            "Figure 5: (a) The average performance of DeLaN and the feed forward neural network for each character. The 4 columns of the boxplots correspond to different numbers of training characters, i.e., $n = 1$ , 6, 8, 10. (b) The median performance of DeLaN, the feed forward neural network and the analytic baselines averaged over multiple seeds. The shaded areas highlight the 5th and the 95th percentile. (c) The qualitative performance for the analytic baselines, the feed forward neural network and DeLaN. The desired trajectories are shown in red. "
+        ],
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+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 Physical Robot Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For physical experiments the desired trajectories are executed on the Barrett WAM, a robot with direct cable drives. The direct cable drives produce high torques generating fast and dexterous movements but yield complex dynamics, which cannot be modelled using rigid-body dynamics due to the variable stiffness and lengths of the cables7. Therefore, the Barrett WAM is ideal for testing the applicability of model learning and analytic models8 on complex dynamics. For the physical experiments we focus on the cosine trajectories as these trajectories produce dynamic movements while character trajectories are mainly dominated by the gravitational forces. In addition, only the dynamics of the four lower joints are learned because these joints dominate the dynamics and the upper joints cannot be sufficiently excited to retrieve the dynamics parameters. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 6c and d show the tracking error on the cosine trajectories using the the simulated Barrett WAM while Figure 6e and f show the tracking error of the physical Barrett WAM. It is important to note, that the simulation only simulates the rigid-body dynamics not including the direct cables drives and the simulation parameters are inconsistent with the parameters of the analytic model. Therefore, the analytic model is not optimal. On the training trajectories executed on the physical system the FF-NN performs better compared to DeLaN and the analytic model. DeLaN achieves slightly better tracking error than the analytic model, which uses the same rigid-body assumptions as DeLaN. That shows DeLaN can learn a dynamics model of the WAM but is limited by the model assumptions of Lagrangian Mechanics. These assumptions cannot represent the dynamics of the cable drives. When comparing to the simulated results, DeLaN and the FF-NN perform comparable but significantly better than the analytic model. These simulation results show that DeLaN can learn an accurate model of the WAM, when the underlying assumptions of the physics prior hold. The tracking performance on the physical system and the simulation indicate that DeLaN can learn a model within the model class of the physics prior but also inherits the limitations of the physics prior. For this specific experiment the FF-NN can locally learn correlations of the torques w.r.t. q, qÛ and $\\ddot { \\bf q }$ while such correlation cannot be represented by the network topology of DeLaN because such correlation should, by definition of the physics prior, not exist. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c170ee96ef30fe375eca65efa15c7cb67c083a9f1ab49cc06078dc4853d6ad18.jpg",
+        "image_caption": [
+            "Figure 6: The tracking error of the cosine trajectories for the simulated 2-dof robot (a & b), the simulated (c & d) and the physical Barrett WAM (e & f). The feed-forward neural network and DeLaN are trained only on the trajectories at a velocity scale of $1 \\times$ . Afterwards the models are tested on the same trajectories with increased velocities to evaluate the extrapolation to new velocities. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "When extrapolating to the identical trajectories with higher velocities (gray area of Figure 6) the tracking error of the FF-NN deteriorates much faster compared to DeLaN, because the FF-NN overfits to the training data. The tracking error of the analytic model remains constant and demonstrates the guaranteed extrapolation of the analytic models. When comparing the simulated results, the FF-NN cannot extrapolate to the new velocities and the tracking error deteriorates similarly to the performance on the physical robot. In contrast to the FF-NN, DeLaN can extrapolate to the higher velocities and maintains a good tracking error. Even further, DeLaN obtains a better tracking error compared the analytic model on all velocity scales. This low tracking error on all test trajectories highlights the improved extrapolation of DeLaN compared to other model learning approaches. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We introduced the concept of incorporating a physics prior within the deep learning framework to achieve lower sample complexity and better extrapolation. In particular, we proposed Deep Lagrangian Networks (DeLaN), a deep network on which Lagrangian Mechanics is imposed. This specific network topology enabled us to learn the system dynamics using end-to-end training while maintaining physical plausibility. We showed that DeLaN is able to learn the underlying physics from a super-imposed signal, as DeLaN can recover the contribution of the inertial-, gravitational and centripetal forces from sensor data. The quantitative evaluation within a real-time control loop assessing the tracking error showed that DeLaN can learn the system dynamics online, obtains lower sample complexity and better generalization compared to a feed-forward neural network. DeLaN can extrapolate to new trajectories as well as to increased velocities, where the performance of the feedforward network deteriorates due to the overfitting to the training data. When applied to a physical systems with complex dynamics the bounded representational power of the physics prior can be limiting. However, this limited representational power enforces the physical plausibility and obtains the lower sample complexity and substantially better generalization. In future work the physics prior should be extended to represent a wider system class by introducing additional non-conservative forces within the Lagrangian. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No #640554 (SKILLS4ROBOTS). Furthermore, this research was also supported by grants from ABB, NVIDIA and the NVIDIA DGX Station. ",
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+    {
+        "type": "text",
+        "text": "Appendix A: Offline Benchmarks ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            462,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/2838bd052ccad5a953b39d598e9577b17d7787f264304383189196697834cbd9.jpg",
+        "image_caption": [
+            "Figure 7: The mean squared error averaged of 20 seeds on the training- (a) and test-set (b) of the character trajectories for the two joint robot. The models are trained offline using $n$ characters and tested using the remaining $2 0 - n$ characters. The training samples are corrupted with white noise, while the performance is tested on noise-free trajectories. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            147,
+            759,
+            305
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the performance of DeLaN without the control task, DeLaN was trained offline on previously collected data and evaluated using the mean squared error (MSE) on the test and training set. For comparison, DeLaN is compared to the system identification approach (SI) described by Atkeson et al. (1986), a feed-forward neural network (FF-NN) and the Recursive Newton Euler algorithm (RNE) using an analytic model. For this comparison, one must point out that the system identification approach relies on the availability of the kinematics, as the Jacobians and transformations w.r.t. to every link must be known to compute the necessary features. In contrast, neither DeLaN nor the FF-NN require this knowledge and must implicitly also learn the kinematics. ",
+        "bbox": [
+            173,
+            395,
+            825,
+            507
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows the MSE averaged over 20 seeds on the character data set executed on the two-joint robot. For this data set, the models are trained using noisy samples and evaluated on the noise-free and previously unseen characters. The FF-NN performs the best on the training set, but overfits to the training data. Therefore, the FF-NN does not generalize to unseen characters. In contrast, the SI approach does not overfit to the noise and extrapolates to previously unseen characters. In comparison, the structure of DeLaN regularizes the training and prevents the overfitting to the corrupted training data. Therefore, DeLaN extrapolates better than the FF-NN but not as good as the SI approach. Similar results can be observed on the cosine data set using the Barrett WAM simulated in SL (Figure 8 a, b). The FF-NN performs best on the training trajectory but the performance deteriorates when this network extrapolates to higher velocities. SI performs worse on the training trajectory but extrapolates to higher velocities. In comparison, DeLaN performs comparable to the SI approach on the training trajectory, extrapolates significantly better than the FF-NN but does not extrapolate as good as the SI approach. For the physical system (Figure 8 c, d), the results differ from the results in simulation. On the physical system the SI approach only achieves the same performance as RNE, which is significantly worse compared to the performance of DeLaN and the FF-NN. When evaluating the extrapolation to higher velocities, the analytic model and the SI approach extrapolate to higher velocities, while the MSE for the FF-NN significantly increases. In comparison, DeLaN extrapolates better compared to the FF-NN but not as good as the analytic model or the SI approach. ",
+        "bbox": [
+            173,
+            512,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This performance difference between the simulation and physical system can be explained by the underlying model assumptions and the robustness to noise. While DeLaN only assumes rigidbody dynamics, the SI approach also assumes the exact knowledge of the kinematic structure. For simulation both assumptions are valid. However, for the physical system, the exact kinematics are unknown due to production imperfections and the direct cable drives applying torques to flexible joints violate the rigid-body assumption. Therefore, the SI approach performs significantly worse on the physical system. Furthermore, the noise robustness becomes more important for the physical system due to the inherent sensor noise. While the linear regression of the SI approach is easily corrupted by noise or outliers, the gradient based optimization of the networks is more robust to noise. This robustness can be observed in Figure 9, which shows the correlation between the variance of Gaussian noise corrupting the training data and the MSE of the simulated and noise-free cosine trajectories. With increasing noise levels, the MSE of the SI approach increases significantly faster compared to the models learned using gradient descent. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/dba559c1041e142619db8cdfc7bded69e4101b93e0f4b478eeb6d1627681faab.jpg",
+        "image_caption": [
+            "Figure 8: The mean squared error of the cosine trajectories for the simulated (a, b) and the physical Barrett WAM (c and d). The system identification approach, feed-forward neural network and DeLaN are trained offline using only the trajectories at a velocity scale of $1 \\times$ . Afterwards the models are tested on the same trajectories with increased velocities to evaluate the extrapolation to new velocities. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            103,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Concluding, the extrapolation of DeLaN to unseen trajectories and higher velocities is not as good as the SI approach but significantly better than the generic FF-NN. This increased extrapolation compared to the generic network is achieved by the Lagrangian Mechanics prior of DeLaN. Even though this prior promotes extrapolation, the prior also hinders the performance on the physical robot, because the prior cannot represent the dynamics of the direct cable drives. Therefore, DeLaN performs worse than the FF-NN, which does not assume any model structure. However, DeLaN outperforms the SI approach on the physical system, which also assumes rigid-body dynamics and requires the exact knowledge of the kinematics. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/079380252b48469dc3e1bd6718daf71329ab5a93949b348be4d71de29e41be64.jpg",
+        "image_caption": [
+            "Figure 9: The mean squared error on the simulated and noise-free cosine trajectories with velocity scale of 1x. For offline training the samples are corrupted using i.i.d. noise sampled from a multivariate Normal distribution with the variance of $\\sigma ^ { 2 } \\mathbf { I }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            357,
+            535,
+            640,
+            690
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Appendix B: Complete Online Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            501,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb5bb05488090fc4a9d514c240c7f99b73baa64c5551db83e72fc9b180f1c083.jpg",
+        "image_caption": [
+            "Figure 10: The qualitative performance for the analytic baselines, the feed forward neural network and DeLaN for different number of random training characters. The desired trajectories are shown in red. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            259,
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+        ],
+        "page_idx": 15
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+    {
+        "type": "image",
+        "img_path": "images/3d387cf436117e72ddc1630599b2407448d122105c9c09f63a0cabd574793e03.jpg",
+        "image_caption": [
+            "Figure 11: The average performance of DeLaN and the feed forward neural network for each character. The columns of the boxplots correspond to different numbers of training characters, i.e., $n = 1 , 2 , 4 , 6 , 8 , 1 0 , 1 2$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    }
+]

parse/train/H1Gq5Q9el/H1Gq5Q9el_content_list.json

@@ -0,0 +1,1455 @@
+[
+    {
+        "type": "text",
+        "text": "UNSUPERVISED PRETRAINING FOR SEQUENCE TO SEQUENCE LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            622,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Prajit Ramachandran∗ University of Illinois at Urbana-Champaign prajitram@gmail.com ",
+        "bbox": [
+            183,
+            172,
+            470,
+            214
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Peter J. Liu, Quoc V. Le Google Brain {peterjliu,qvl}@google.com ",
+        "bbox": [
+            524,
+            172,
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+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            252,
+            544,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This work presents a general unsupervised learning method to improve the accuracy of sequence to sequence (seq2seq) models. In our method, the weights of the encoder and decoder of a seq2seq model are initialized with the pretrained weights of two language models and then fine-tuned with labeled data. We apply this method to challenging benchmarks in machine translation and abstractive summarization and find that it significantly improves the subsequent supervised models. Our main result is that the pretraining accelerates training and improves generalization of seq2seq models, achieving state-of-the-art results on the WMT English German task, surpassing a range of methods using both phrase-based machine translation and neural machine translation. Our method achieves an improvement of 1.3 BLEU from the previous best models on both WMT’14 and WMT’15 English German. On summarization, our method beats the supervised learning baseline. ",
+        "bbox": [
+            233,
+            291,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            517,
+            336,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sequence to sequence (seq2seq) models (Sutskever et al., 2014; Cho et al., 2014; Kalchbrenner & Blunsom, 2013; Allen, 1987; Neco & Forcada, 1997) are extremely effective on a variety of ˜ tasks that require a mapping between a variable-length input sequence to a variable-length output sequence. The main weakness of sequence to sequence models, and deep networks in general, lies in the fact that they can easily overfit when the amount of supervised training data is small. ",
+        "bbox": [
+            174,
+            555,
+            825,
+            625
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a simple and effective technique for using unsupervised pretraining to improve seq2seq models. Our proposal is to initialize both encoder and decoder networks with pretrained weights of two language models. These pretrained weights are then fine-tuned with the labeled corpus. ",
+        "bbox": [
+            174,
+            632,
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+            688
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We benchmark this method on machine translation for English German and abstractive summarization on CNN and Daily Mail articles. Our main result is that a seq2seq model, with pretraining, exceeds the strongest possible baseline in both neural machine translation and phrase-based machine translation. Our model obtains an improvement of 1.3 BLEU from the previous best models on both WMT’14 and WMT’15 English German. On abstractive summarization, our method achieves competitive results to the strongest baselines. ",
+        "bbox": [
+            174,
+            695,
+            825,
+            779
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We also perform ablation study to understand the behaviors of the pretraining method. Our study confirms that among many other possible choices of using a language model in seq2seq with attention, the above proposal works best. Our study also shows that, for translation, the main gains come from the improved generalization due to the pretrained features, whereas for summarization the gains come from the improved optimization due to pretraining the encoder which has been unrolled for hundreds of timesteps. On both tasks, our proposed method always improves generalization on the test sets. ",
+        "bbox": [
+            174,
+            786,
+            825,
+            882
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 UNSUPERVISED PRETRAINING FOR SEQUENCE TO SEQUENCE LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            102,
+            810,
+            118
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the following section, we will describe our basic unsupervised pretraining procedure for sequence to sequence learning and how to modify sequence to sequence learning to effectively make use of the pretrained weights. We then show several extensions to improve the basic model. ",
+        "bbox": [
+            174,
+            132,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 BASIC PROCEDURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            190,
+            349,
+            205
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given an input sequence $x _ { 1 } , x _ { 2 } , . . . , x _ { m }$ and an output sequence $y _ { n } , y _ { n - 1 } , . . . , y _ { 1 }$ , the objective of sequence to sequence learning is to maximize the likelihood $p ( y _ { n } , y _ { n - 1 } , . . . , y _ { 1 } | x _ { 1 } , x _ { 2 } , . . . , x _ { m } )$ . Common sequence to sequence learning methods decompose this objective as $\\begin{array} { r } { p ( \\boldsymbol { y } _ { n } , \\boldsymbol { y } _ { n - 1 } , . . . , \\dot { \\boldsymbol { y } _ { 1 } } | \\boldsymbol { x } _ { 1 } , \\boldsymbol { x } _ { 2 } , . . . , \\boldsymbol { x } _ { m } ) ^ { \\flat } = \\prod _ { t = 1 } ^ { n } p ( \\boldsymbol { y } _ { t } | \\overset { \\sim } { \\boldsymbol { y } _ { t - 1 } } , . . . , \\boldsymbol { y } _ { 1 } ; \\boldsymbol { x } _ { 1 } , \\boldsymbol { x } _ { 2 } , . . . , \\dot { \\boldsymbol { x } } _ { m } ) . } \\end{array}$ . ",
+        "bbox": [
+            173,
+            218,
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+            275
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In sequence to sequence learning, an RNN encoder is used to represent $x _ { 1 } , . . . , x _ { m }$ as a hidden vector, which is given to an RNN decoder to produce the output sequence. Our method is based on the observation that without the encoder, the decoder essentially acts like a language model on $y$ ’s. Similarly, the encoder with an additional output layer also acts like a language model. Thus it is natural to use trained languages models to initialize the encoder and decoder. ",
+        "bbox": [
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+            280,
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+            349
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the basic procedure of our approach is to pretrain both the seq2seq encoder and decoder networks with language models, which can be trained on large amounts of unlabeled text data. This can be seen in Figure 1, where the parameters in the shaded boxes are pretrained. In the following we will describe the method in detail using machine translation as an example application. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/f0d4d784a5a5539b916e7bd408d746c140a113beb99fda9af36b6f1ca6493a46.jpg",
+        "image_caption": [
+            "Figure 1: Pretrained sequence to sequence model. The red parameters are the encoder and the blue parameters are the decoder. All parameters in a shaded box are pretrained, either from the source side (light red) or target side (light blue) language model. Otherwise, they are randomly initialized. "
+        ],
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+        "type": "text",
+        "text": "First, two monolingual datasets are collected, one for the source side language, and one for the target side language. A language model $( L M )$ is trained on each dataset independently, giving an LM trained on the source side corpus and an LM trained on the target side corpus. ",
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+        "type": "text",
+        "text": "After two language models are trained, a multi-layer seq2seq model $M$ is constructed. The embedding and first LSTM layers of the encoder and decoder are initialized with the pretrained weights. To be even more efficient, the softmax of the decoder is initialized with the softmax of the pretrained target side LM. ",
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+    {
+        "type": "text",
+        "text": "2.2 IMPROVING THE MODEL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We also employ three additional methods to further improve the model above. The three methods are: a) Monolingual language modeling losses, b) Residual connections and c) Attention over multiple layers (see Figure 2). ",
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+        "type": "text",
+        "text": "Monolingual language modeling losses: After the seq2seq model $M$ is initialized with the two LMs, it is fine-tuned with a labeled dataset. To ensure that the model does not overfit the labeled data, we regularize the parameters that were pretrained by continuing to train with the monolingual language modeling losses. The seq2seq and language modeling losses are weighted equally. ",
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+        "img_path": "images/eb5223724b1e3bd342df03c0ec6e592c7d69698362b72b6f493c4474eba8d1af.jpg",
+        "image_caption": [
+            "Figure 2: Two improvements to the baseline model: (a) residual connection, and (b) attention over multiple layers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Residual connections: As described, the input vector to the decoder softmax layer is a random vector because the high level (non-first) layers of the LSTM are randomly initialized. This slows down training and introduces random gradients to the pretrained parameters, reducing the effectiveness of pretraining. To circumvent this issue, we use a residual connection from the output of the first LSTM layer directly to the input of the softmax (see Figure 2-a). ",
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+        "type": "text",
+        "text": "Attention over multiple layers: In all our models, we use an attention mechanism (Bahdanau et al., 2015), where the model attends over both top and first layer (see Figure 2-b). More concretely, given a query vector $q _ { t }$ from the decoder, encoder states from the first layer $h _ { 1 } ^ { 1 } , \\ldots , h _ { T } ^ { 1 }$ , and encoder states from the last layer $h _ { 1 } ^ { L } , \\ldots , h _ { T } ^ { L }$ , we compute the attention context vector $c _ { t }$ as follows: ",
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+        "img_path": "images/faabd992892a8ade7ab474faed4b420380d329f60a37a0b7931fab2d0b910fbe.jpg",
+        "text": "$$\n\\alpha _ { i } = \\frac { \\exp ( q _ { t } \\cdot h _ { i } ^ { N } ) } { \\sum _ { j = 1 } ^ { T } \\exp ( q _ { t } \\cdot h _ { j } ^ { N } ) } \\qquad c _ { t } ^ { 1 } = \\sum _ { i = 1 } ^ { T } \\alpha _ { i } h _ { i } ^ { 1 } \\qquad c _ { t } ^ { N } = \\sum _ { i = 1 } ^ { T } \\alpha _ { i } h _ { i } ^ { N } \\qquad c _ { t } = [ c _ { t } ^ { 1 } ; c _ { t } ^ { N } ]\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Note that attention weights $\\alpha _ { i }$ are only computed once using the top level encoder states. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "We also experiment with passing the attention vector $c _ { t }$ as input into the next timestep (Luong et al., 2015b). Instead of passing $c$ into the first LSTM layer, we pass it as input to the second LSTM layer by concatenating it with the output of the first LSTM layer. ",
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+    {
+        "type": "text",
+        "text": "We use all three improvements in our experiments. However, in general we notice that the benefits of the attention modifications are minor in comparison with the benefits of the additional language modeling objectives and residual connections. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the following section, we apply our approach to two important tasks in seq2seq learning: machine translation and abstractive summarization. On each task, we compare against the previous best systems. We also perform ablation experiments to understand the behavior of each component of our method. ",
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+    {
+        "type": "text",
+        "text": "3.1 MACHINE TRANSLATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Dataset and Evaluation: For machine translation, we evaluate our method on the WMT English German task (Bojar et al., 2015). We used the WMT 14 training dataset, which is slightly smaller than the WMT 15 dataset. Because the dataset has some noisy examples, we used a language detection system to filter the training examples. Sentences pairs where either the source was not English or the target was not German were thrown away. This resulted in around 4 million training examples. Following Sennrich et al. (2015b), we use subword units (Sennrich et al., 2015a) ",
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+    {
+        "type": "text",
+        "text": "with 89500 merge operations, giving a vocabulary size around 90000. The validation set is the concatenated newstest2012 and newstest2013, and our test sets are newstest2014 and newstest2015. Evaluation on the validation set was with case-sensitive BLEU (Papineni et al., 2002) on tokenized text using multi-bleu.perl. Evaluation on the test sets was with case-sensitive BLEU on detokenized text using mteval-v13a.pl. The monolingual training datasets are the News Crawl English and German corpora, each of which has more than a billion tokens. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Experimental settings: The language models were trained in the same fashion as (Jozefowicz et al., 2016) We used a 1 layer 4096 dimensional LSTM with the hidden state projected down to 1024 units (Sak et al., 2014) and trained for one week on 32 Tesla K40 GPUs. Our seq2seq model was a 3 layer model, where the second and third layers each have 1000 hidden units. The monolingual objectives, residual connection, and the modified attention were all used. We used the Adam optimizer (Kingma & Ba, 2015) and train with asynchronous SGD on 16 GPUs for speed. We used a learning rate of 5e-5 which is multiplied by 0.8 every 50K steps after an initial 400K steps, gradient clipping with norm 5.0 (Pascanu et al., 2013), and dropout of 0.2 on non-recurrent connections (Zaremba et al., 2014). We used early stopping on validation set perplexity. A beam size of 10 was used for decoding. Our ensemble is constructed with the 5 best performing models on the validation set, which are trained with different hyperparameters. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Results: Table 1 shows the results of our method in comparison with other baselines. Our method achieves a new state-of-the-art for single model performance on both newstest2014 and newstest2015, significantly outperforming the competitive semi-supervised backtranslation technique (Sennrich et al., 2015b). Equally impressive is the fact that our best single model outperforms the previous state of the art ensemble of 4 models. Our ensemble of 5 models matches or exceeds the previous best ensemble of 12 models. ",
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+    {
+        "type": "table",
+        "img_path": "images/2005ee296d834f3dc87f52131927d220d02faa67be9ee53ca92e9dd8a5bc5a13.jpg",
+        "table_caption": [
+            "Table 1: English German performance on WMT test sets. Our pretrained model outperforms all other models. Note that the model without pretraining uses the LM objective. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">System</td><td rowspan=\"2\">ensemble?</td><td colspan=\"2\">BLEU</td></tr><tr><td>newstest2014</td><td>newstest2015</td></tr><tr><td>Phrase Based MT(Williams et al., 2016)</td><td>1</td><td>21.9</td><td>23.7</td></tr><tr><td>Supervised NMT (Jean et al., 2015)</td><td>single</td><td>-</td><td>22.4</td></tr><tr><td>Edit Distance Transducer NMT (Stahlberg et al.,2016)</td><td>single</td><td>21.7</td><td>24.1</td></tr><tr><td>Edit Distance Transducer NMT (Stahlberg et al., 2016)</td><td>ensemble 8</td><td>22.9</td><td>25.7</td></tr><tr><td>Backtranslation (Sennrich et al.,2015b)</td><td>single</td><td>22.7</td><td>25.7</td></tr><tr><td>Backtranslation (Sennrich et al.,2015b)</td><td>ensemble 4</td><td>23.8</td><td>26.5</td></tr><tr><td>Backtranslation (Sennrich et al.,2015b)</td><td>ensemble 12</td><td>24.7</td><td>27.6</td></tr><tr><td>No pretraining</td><td>single</td><td>21.3</td><td>24.3</td></tr><tr><td>Pretrained seq2seq</td><td>single</td><td>24.0</td><td>27.0</td></tr><tr><td>Pretrained seq2seq</td><td>ensemble 5</td><td>24.7</td><td>28.1</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ablation study: In order to better understand the effects of pretraining, we conducted an ablation study by modifying the pretraining scheme. Figure 3 shows the drop in validation BLEU of various ablations compared with the full model. The full model uses LMs trained with monolingual data to initialize the encoder and decoder, in addition to the language modeling objective. In the following, we interpret the findings of the study. Note that some findings are specific to the translation task. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given the results from the ablation study, we can make the following observations: ",
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+    {
+        "type": "text",
+        "text": "• Pretraining the decoder is better than pretraining the encoder: Only pretraining the encoder leads to a 1.6 BLEU point drop while only pretraining the decoder leads to a 1.0 BLEU point drop.   \n• Pretrain as much as possible because the benefits compound: given the drops of no pretraining at all $( - 2 . 0 )$ and only pretraining the encoder $( - 1 . 6 )$ , the additive estimate of the drop of only pretraining the decoder side is $- 2 . 0 - ( - 1 . 6 ) = - 0 . 4$ ; however the actual drop is $- 1 . 0$ which is a much larger drop than the additive estimate.   \nPretraining the softmax is important: Pretraining only the embeddings and first LSTM layer gives a large drop of 1.6 BLEU points. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/57d26fe90190577ff49dfc6d7f18419158811b996b7254dfe25e06731cc60a96.jpg",
+        "image_caption": [
+            "Figure 3: English German ablation study measuring the difference in validation BLEU between various ablations and the full model. More negative is worse. The full model uses LMs trained with monolingual data to initialize the encoder and decoder, plus the language modeling objective. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "• The language modeling objective is a strong regularizer: The drop in BLEU points of pretraining the entire model and not using the LM objective is as bad as using the LM objective without pretraining. Pretraining on a lot of unlabeled data is essential for learning to extract powerful features: If the model is initialized with LMs that are pretrained on the source part and target part of the parallel corpus, the drop in performance is as large as not pretraining at all. However, performance remains strong when pretrained on the large, non-news Wikipedia corpus. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "To understand the contributions of unsupervised pretraining vs. supervised training, we track the performance of pretraining as a function of dataset size. For this, we trained a a model with and without pretraining on random subsets of the English German corpus. Both models use the additional LM objective. The results are summarized in Figure 4. When a $100 \\%$ of the labeled data is used, the gap between the pretrained and no pretrain model is 2.0 BLEU points. However, that gap grows when less data is available. When trained on $20 \\%$ of the labeled data, the gap becomes 3.8 BLEU points. This demonstrates that the pretrained models degrade less as the labeled dataset becomes smaller. ",
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+        "img_path": "images/fd6efc51ca31158846d3581314b1dd9275c6f461e264a8857db584f1e5681534.jpg",
+        "image_caption": [
+            "Figure 4: Validation performance of pretraining vs. no pretraining when trained on a subset of the entire labeled dataset for English German translation. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.2 ABSTRACTIVE SUMMARIZATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Dataset and Evaluation: For a low-resource abstractive summarization task, we use the CNN/Daily Mail corpus from (Hermann et al., 2015). Following Nallapati et al. (2016), we modify the data collection scripts to restore the bullet point summaries. The task is to predict the bullet point summaries from a news article. The dataset has fewer than 300K document-summary pairs. To compare against Nallapati et al. (2016), we used the anonymized corpus. However, for our ablation study, we used the non-anonymized corpus.1 We evaluate our system using full length ROUGE (Lin, 2004). For the anonymized corpus in particular, we considered each highlight as a separate sentence following Nallapati et al. (2016). In this setting, we used the English Gigaword corpus (Napoles et al., 2012) as our larger, unlabeled “monolingual” corpus, although all data used in this task is in English. ",
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+    {
+        "type": "text",
+        "text": "Experimental settings: We use subword units (Sennrich et al., 2015a) with 31500 merges, resulting in a vocabulary size of about 32000. We use up to the first 600 tokens of the document and predict the entire summary. Only one language model is trained and it is used to initialize both the encoder and decoder, since the source and target languages are the same. However, the encoder and decoder are not tied. The LM is a one-layer LSTM of size 1024 trained in a similar fashion to Jozefowicz et al. (2016). For the seq2seq model, we use the same settings as the machine translation experiments. The only differences are that we use a 2 layer model with the second layer having 1024 hidden units, and that the learning rate is multiplied by 0.8 every 30K steps after an initial 100K steps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results: Table 2 summarizes our results on the anonymized version of the corpus. Our pretrained model is only able to match the previous baseline seq2seq of Nallapati et al. (2016). However, our model is a unidirectional LSTM while they use a bidirectional LSTM. They also use a longer context of 800 tokens, whereas we used a context of 600 tokens due to GPU memory issues. Furthermore, they use pretrained word2vec (Mikolov et al., 2013) vectors to initialize their word embeddings. As we show in our ablation study, just pretraining the embeddings itself gives a large improvement. ",
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+    {
+        "type": "table",
+        "img_path": "images/e957312fc15e0b760918c172906e06dbee861ff67327703eea8dad7429cf6d53.jpg",
+        "table_caption": [
+            "Table 2: Results on the anonymized CNN/Daily Mail dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>System</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=1 colspan=1>Seq2seq + pretrained embeddings (Nallapati et al., 2016)+ temporal attention (Nallapati et al., 2016)</td><td rowspan=1 colspan=1>32.4935.46</td><td rowspan=1 colspan=1>11.8413.30</td><td rowspan=1 colspan=1>29.4732.65</td></tr><tr><td rowspan=1 colspan=1>Pretrained seq2seq</td><td rowspan=1 colspan=1>32.56</td><td rowspan=1 colspan=1>11.89</td><td rowspan=1 colspan=1>29.44</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Ablation study: We performed an ablation study similar to the one performed on the machine translation model. The results are reported in Figure 5. Here we report the drops on ROUGE-1, ROUGE-2, and ROUGE-L on the non-anonymized validation set. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Given the results from our ablation study, we can make the following observations: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Pretraining improves optimization: in contrast with the machine translation model, it is more beneficial to only pretrain the encoder than only the decoder of the summarization model. One interpretation is that pretraining enables the gradient to flow much further back in time than randomly initialized weights. This may also explain why pretraining on the parallel corpus is no worse than pretraining on a larger monolingual corpus. • The language modeling objective is a strong regularizer: A model without the LM objective has a significant drop in ROUGE scores. ",
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+    {
+        "type": "text",
+        "text": "Human evaluation: As ROUGE may not be able to capture the quality of summarization, we also performed a small qualitative study to understand the human impression of the summaries produced by different models. We took 200 random documents and compared the performance of a pretrained and non-pretrained system. The document, gold summary, and the two system outputs were presented to a human evaluator who was asked to rate each system output on a scale of 1-5 with 5 being the best score. The system outputs were presented in random order and the evaluator did not know the identity of either output. The evaluator noted if there were repetitive phrases or sentences in either system outputs. Unwanted repetition was also noticed by Nallapati et al. (2016). ",
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+        "page_idx": 5
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+    {
+        "type": "image",
+        "img_path": "images/d3b179c66bdf6148ce917c9236d5050079cc7cbcf410d123ed214d8a5af4d647.jpg",
+        "image_caption": [
+            "Figure 5: Summarization ablation study measuring the difference in validation ROUGE between various ablations and the full model. More negative is worse. The full model uses LMs trained with unlabeled data to initialize the encoder and decoder, plus the language modeling objective. "
+        ],
+        "image_footnote": [],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Table 3 and 4 show the results of the study. In both cases, the pretrained system outperforms the system without pretraining in a statistically significant manner. The better optimization enabled by pretraining improves the generated summaries and decreases unwanted repetition in the output. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cb3ca06dda38a56227f72d3737dc99fbb0af6a746f27b00e6e23fe83c5d3641a.jpg",
+        "text": "$$\n\\frac { N P > P ~ \\mid ~ N P = P ~ \\mid ~ N P < P } { 2 9 } \n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Table 3: The count of how often the no pretrain system $( N P )$ achieves a higher, equal, and lower score than the pretrained system $( P )$ in the side-by-side study where the human evaluator gave each system a score from 1-5. The sign statistical test gives a $\\mathsf { p }$ -value of $< 0 . 0 0 0 1$ for rejecting the null hypothesis that there is no difference in the score obtained by either system. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"2\">No pretrain</td></tr><tr><td colspan=\"2\"></td><td>No repeats</td><td>Repeats</td></tr><tr><td rowspan=\"2\">Pretrain</td><td>No repeats</td><td>67</td><td>65</td></tr><tr><td>Repeats</td><td>24</td><td>44</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4: The count of how often the pretrain and no pretrain systems contain repeated phrases or sentences in their outputs in the side-by-side study. McNemar’s test gives a $\\mathsf { p }$ -value of $< 0 . 0 0 0 1$ for rejecting the null hypothesis that the two systems repeat the same proportion of times. The pretrained system clearly repeats less than the system without pretraining. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Unsupervised pretraining has been intensively studied in the past years, most notably is the work by Dahl et al. (2012) who found that pretraining with deep belief networks improved feedforward acoustic models. More recent acoustic models have found pretraining unnecessary (Xiong et al., ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2016; Zhang et al., 2016; Chan et al., 2015), probably because the reconstruction objective of deep belief networks is too easy. In contrast, we find that pretraining language models by next step prediction significantly improves seq2seq on challenging real world datasets. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Despite its appeal, unsupervised learning is rarely shown to improve supervised training. Dai & Le (2015) was amongst the rare studies which showed the benefits of pretraining in a semi-supervised learning setting. Their method is similar to our method except that they did not have a decoder network and thus could not apply to seq2seq learning. Similarly, Zhang & Zong (2016) found it useful to add an additional task of sentence reordering of source-side monolingual data for neural machine translation. Various forms of transfer or multitask learning with seq2seq framework also have the flavors of our algorithm (Zoph et al., 2016; Luong et al., 2015a; Firat et al., 2016). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Perhaps most closely related to our method is the work by Gulcehre et al. (2015), who combined a language model with an already trained seq2seq model by fine-tuning additional deep output layers. Empirically, their method produces small improvements over the supervised baseline. We suspect that their method does not produce significant gains because (i) the models are trained independently of each other and are not fine-tuned (ii) the LM is combined with the seq2seq model after the last layer, wasting the benefit of the low level LM features, and (iii) only using the LM on the decoder side. Venugopalan et al. (2016) addressed (i) but still experienced minor improvements. Using pretrained GloVe embedding vectors (Pennington et al., 2014) had more impact. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Related to our approach in principle is the work by Chen et al. (2016) who proposed a two-term, theoretically motivated unsupervised objective for unpaired input-output samples. Though they did not apply their method to seq2seq learning, their framework can be modified to do so. In that case, the first term pushes the output to be highly probable under some scoring model, and the second term ensures that the output depends on the input. In the seq2seq setting, we interpret the first term as a pretrained language model scoring the output sequence. In our work, we fold the pretrained language model into the decoder. We believe that using the pretrained language model only for scoring is less efficient that using all the pretrained weights. Our use of labeled examples satisfies the second term. These connections provide a theoretical grounding for our work. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we benchmark our method on machine translation, where other unsupervised methods are shown to give promising results (Sennrich et al., 2015b; Cheng et al., 2016). In backtranslation (Sennrich et al., 2015b), the trained model is used to decode unlabeled data to yield extra labeled data. One can argue that this method may not have a natural analogue to other tasks such as summarization. We note that their technique is complementary to ours, and may lead to additional gains in machine translation. The method of using autoencoders in Cheng et al. (2016) is promising, though it can be argued that autoencoding is an easy objective and language modeling may force the unsupervised models to learn better features. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+            656,
+            318,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We presented a novel unsupervised pretraining method to improve sequence to sequence learning. The method can aid in both generalization and optimization. Our scheme involves pretraining two language models in the source and target domain, and initializing the embeddings, first LSTM layers, and softmax of a sequence to sequence model with the weights of the language models. Using our method, we achieved state-of-the-art machine translation results on both WMT’14 and WMT’15 English to German. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A key advantage of this technique is that it is flexible and can be applied to a large variety of tasks, such as summarization, where it surpasses the supervised learning baseline. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We thank George Dahl, Andrew Dai, Laurent Dinh, Stephan Gouws, Geoffrey Hinton, Rafal Jozefowicz, Pooya Khorrami, Phillip Louis, Ramesh Nallapati, Arvind Neelakantan, Xin Pan, Abi See, Rico Sennrich, Luke Vilnis, Yuan Yu and the Google Brain team for their help with the project. ",
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+        "page_idx": 7
+    },
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "SELECTED SUMMARIZATION OUTPUTS ",
+        "bbox": [
+            174,
+            133,
+            444,
+            147
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/0ba692e2d8500083fb9624800415e1dc656cfccafc46f0cde5a5db7787a00a77.jpg",
+        "table_caption": [
+            "Table 5: The pretrained model outputs a highly informative summary, while the no pretrain model outputs irrelevant details. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source Document (cnn ) like phone booths and typewriters ,record stores are a vanishing breed -another victim</td></tr><tr><td>of the digital age .camelot music .virgin megastores .wherehouse music . tower records ． all of them gone ． corporate america has largely abandoned brick - and - mortar music retailing to a scattering of independent stores , many of them in scruffy urban neighborhoods . and that s not necessarily a bad thing . yes ,it s harder in the spotify era to find a place to go buy physical music . but many of the remaining record stores are succeeding - even thriving - by catering to a passionate core of customers and collectors ．on saturday，,hundreds of music retailers will hold events to commemorate record store day,an annual celebration of ,well ,your neighborhood record store .many stores will host live performances ,drawings ,book signings ,special sales of rare or autographed vinyl and other happenings . some will even serve beer.to their diehard customers ,these places are more than mere stores : they are cultural institutions that celebrate music history（the entire duran duran oeuvre ,all in one place ！）,display artifacts（aretha franklin on vinyl！）,and nurture the local music scene（hey,here s acd by your brother s metal band ！）. they also employ knowledgeable clerks who will be happy to debate the relative merits of blood on the tracks and blonde on blonde .or maybe ,like jack black in high fidelity, just mock your lousy taste in music . so</td></tr><tr><td>if you re a music geek ,drop by . but you might think twice before asking if they stock i just called to say i love you .</td></tr><tr><td>Ground Truth summary saturday is record store day,celebrated at music stores around the world .many stores will</td></tr><tr><td>host live performances ,drawings and special sales of rare vinyl .</td></tr><tr><td>No pretrain corporate america has largely abandoned brick - brick - mortar music . many of the remaining</td></tr><tr><td>record stores are succeeding - even thriving-by catering to a passionate core of customers . Pretrained hundreds of music retailers will hold events to commemorate record store day .many stores</td></tr></table>",
+        "bbox": [
+            184,
+            164,
+            812,
+            571
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/f2296d211613484ee21e0d7c5a20a113e8c6488d1cb04c91bea81ca5e3bc7773.jpg",
+        "table_caption": [
+            "Table 6: The pretrained model outputs a highly relevant summary but makes a mistake on the feline executioner’s name. The no pretrain model degenerates into irrelevant details and repeats itself. "
+        ],
+        "table_footnote": [
+            "Table 7: Both models output a relevant summary, but the no pretrain model uses the same name to refer to both players. "
+        ],
+        "table_body": "<table><tr><td>Source Document (cnn ） hey,look what i did . that smallboast on social media can trigger a whirlwind that</td></tr><tr><td>lindsey allegedly shot an arrow into the back of an orange tabby s head and posted a proud photo this week on facebook of herself smiling ,as she dangled its limp body by the arrow s shaft . lindsey added a comment,cnn affiliate kbtx reported . my first bow kill,lol . the only good feral tomcat is one with an arrow through it s head ！ vet of the year award ... gladly accepted .callers rang the phones hot at washington county s animal clinic ,where lindsey worked ,to vent their outrage .web traffic crashed its website . high price of public shaming on the internet then an animal rescuer said that lindsey s prey was probably not a feral cat but the pet of an elderly couple ,who called him tiger . he had gone missing on wednesday , the same day that lindsey posted the photo of the slain cat.cnn has not been able to confirm the claim .as the firestorm grew ,lindsey wrote in the comments underneath her post : no i did not lose my job .lol ．psshh .like someone would get rid of me .i m awesome ！ that prediction was wrong. the clinic fired lindsey ,covered her name on its marquee with duct tape ,and publicly distanced itself from her actions .our goal now is to go on and try to fix our black eye and hope that people are reasonable and understand that those actions do nt anyway portray what we re for here at washington animal clinic ,said dr. bruce buenger . we put our heart and soul into this place .the clinic told wbtx that lindsey was not available for comment .cnn is reaching out to her ． she removed her controversial post then eventually shut down her facebook page . callers also complained to the brenham police department and washington county animal control,as her facebook post went viral ．the sheriff s office in austin county ，where the cat was apparently shot , is investigating ，and lindsey could face charges .its dispatchers were overloaded with calls,the sheriff posted on facebook .we are asking you to please take it easy on our dispatchers . as soon as the investigation is complete , we will post the relevant information here on this page ,the post read . animal rights activists are pushing for charges . animal cruelty must be taken seriously ,and the guilty parties should be punished to the fullest extent of the law ，said cat advocacy activist becky robinson ．her organization,alley cat allies ,is offering a $ 7,5OO reward for evidence leading to the arrest and conviction of the person who shot the cat . but others stood up for lindsey . she s amazing</td></tr><tr><td>. she s caring ,said customer shannon stoddard . she s a good vet,so maybe her bad choice of posting something on facebook was not good .but i do nt think she should be judged for it ．she dropped off balloons at the animal clinic for lindsey with a thank you note．cnn s jeremy grisham contributed to this report .</td></tr><tr><td>Ground Truth summary dr.kristen lindsey has since removed the post of her holding the dead cat by an arrow .her</td></tr><tr><td>employer fired her ; the sheriff s ofice is investigating.activist offers $7,5OO reward . No pretrain dr. kristen lindsey allegedly shot an arrow into the back of an orange orange tabby s head . it</td></tr><tr><td> s the only good good tomcat is one with an arrow through it s head ! vet vet of the year award</td></tr><tr><td>Pretrained lindsey lindsey ,a texas veterinarian , shot an arrow into the back of an orange tabby s head . she posted a photo of herself smiling ,as she dangled its limp body by the arrow s shaft . lindsey could face charges,the sheriff s department says .</td></tr><tr><td>Source Document eugenie bouchard s run of poor form continued as the top seed was beaten 6 - 3 ,6 - 1 by</td></tr><tr><td>american lauren davis in the second round at the family circle cup in charleston on wednesday ． davis ,21, had lost her only career meeting with bouchard ,but was in control this time against the world no 7 . davis won nine of the final 11 games of the match and broke bouchard s serve twice in the final set to pull off the upset . eugenie bouchard fires down a serve during her second - round match at the family circle cup bouchard shows her frustrations during her straight - sets defeat by lauren davis on wednesday i ve never beaten here before ， so i came out knowing i had nothing to lose ，said davis ,ranked 66th in the world ．bouchard was a semi-finalist last year but had struggled in the lead - up to the charlston green - clay event ,losing three of her last five matches to lower - ranked opponents .davis used precise groundstrokes to keep bouchard on her heels throughout the second set.davis broke bouchard S serve to love to take a 3 -1 lead as the 21 - year - old canadian had a double fault, then sailed a forehand long to fall behind .two games later ，davis hit a backhand to send back bouchard s lO2mph serve for a winner down the left sideline and a 5-1 lead．davis hits a forehand on her way to an impressve win against the canadian top seed bouchard has struggled recently,this time slumping to her fourth defeat in six matches the match ended on bouchard s forehand into the net,davis waving to the cheering crowd．bouchard said</td></tr><tr><td>: as soon as it was over,definitely a bit of anger,but also kind of this confusion，slash , like quest to find what s wrong,like this kind of searching feeling that ，ok ,like i know something s off .i know something s not right .so i want to find it . i was definitely a little bit slow today ,overpowered .usually ,im the one dominating .so it was definitely, just ,i do nt know, just not good .</td></tr><tr><td>Ground Truth summary eugenie bouchard suffered her fourth defeat in six matches . the canadian top seed lost to lauren davis at the family circle cup . world no 66 davis won 6 - 3,6 - 1 in the second round in charleston .davis won nine of the final 11 games of the match to seal victory .click here</td></tr><tr><td>for all the latest news from charleston . No pretrain bouchard beat american lauren davis 6 - 3,6 - 1 in the second round of the family circle cup</td></tr><tr><td>in charleston on wednesday.bouchard had lost her only career meeting with bouchard but was in control this time against the world no 7 .bouchard won nine of the final 11 games of the match and broke bouchard s serve twice in the final set to pull off the upset .</td></tr><tr><td>Pretrained eugenie bouchard was beaten 6 - 3,6 -1 by american lauren davis in the second round . davis had lost her only career meeting with bouchard,but was in control this time against the</td></tr><tr><td>world no 7. davis hit a backhand to send back bouchard s 1O2mph serve for a winner down the left sideline .</td></tr></table>",
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+            189,
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+            808
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "",
+        "table_caption": [],
+        "table_footnote": [
+            ""
+        ],
+        "bbox": [
+            184,
+            219,
+            810,
+            758
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/59c031a033f625eaf8b1a5d2ce8543df5523c72954c08401fbe4a2804214edd8.jpg",
+        "table_caption": [
+            "Table 8: A failure case. The pretrained model outputs irrelevant details while the no pretrain model successfully summarizes the document. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source Document （cnn） mike rowe is coming to a river near you .sometimes , you hear about a person who</td></tr><tr><td>makes you feel good about humanity ,but bad about yourself ,rowe says .on thursday s episode of somebody s got ta do it , rowe meets up with chad pregracke ，the founder of living lands &amp; waters ，who does just that . pregracke wants to clean up the nation s rivers one piece of detritus at a time . his quota ? always more . read mike rowe s facebook post on how to break our litter habit . since he founded the nonprofit in 1998 at the ripe age of 23 , pregracke and more than 87,OoO volunteers have collected 8.4 million pounds of trash from u.s. waterways .those efforts helped him earn the 2O13 cnn hero of the year award ,along with numerous other honors ．wherever you are , no matter if there s a stream ,a creek ，a lake , whatever ,that needs to be cleaned up,you can do it . just organize it and do it, he told cnn s anderson cooper after his win . pregracke also gives rowe a tour of the 15O - foot, solar - powered barge that the living lands &amp; waters staff calls home during lengthy cleanups . the part - home ,part - office,part - dumpster has seven bedrooms ,two bathrooms ,a classroom and a kitchen -and just happens to be made from a recycled strip club ．according to the organization s latest annual report，pregracke has made it his mission in 2O15 to remove</td></tr><tr><td>500,00O more pounds of trash .if you d like to help achieve this goal, visit his website to learn how to help : livinglandsandwaters.org / get - involved / .</td></tr><tr><td>Ground Truth summary chad pregracke was the 2O13 cnn hero of the year. mike rowe visited pregracke for an episode</td></tr><tr><td>of somebody s got ta do it . No pretrain</td></tr><tr><td>rowe meets up with chad pregracke , founder of living lands &amp; waters . pregracke and more</td></tr><tr><td>than 87,Ooo volunteers collected 8.4 million pounds of trash from u.s. waterways . Pretrained rowe is the founder of living lands &amp; waters , who does just that . pregracke also gives rowe</td></tr></table>",
+        "bbox": [
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+            812,
+            680
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "SELECTED ENGLISH GERMAN OUTPUTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            104,
+            467,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/bd241a462b80afa813efbf7df4e8190037d449ced54a660ae390ecd8879dafa9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Source</td></tr><tr><td rowspan=1 colspan=1>Mayor Bloomberg told reporters that, because of that court order, the city had suspended thereopening of the public space and protesters were informed, however, that local laws do notallow them to re-install with camping shops and sleeping bags.</td></tr><tr><td rowspan=1 colspan=1>Ground Truth</td></tr><tr><td rowspan=1 colspan=1>Buirgermeister Bloomberg stellt vor der Presse Klar , das aufgrund dieser richterlichen Anord-nung die erneute Offnung des Platzes fir den Publikumsverkehr und die Demonstrantenaufgehoben worden sei .Die Demonstranten wies er darauf hin,dass die Stadtgesetze ihnennicht erlaubten,sich erneut mit Zelten und Schlafsäcken an diesem Ort einzurichten .</td></tr><tr><td rowspan=1 colspan=1>No pretrain</td></tr><tr><td rowspan=1 colspan=1>Der Burgermeister Bloomberg sagte den Reportern , dass die Stadt aufgrund dieser Gericht-sentscheidung die Wiedereroffnung des offentlichen Raumes und die Information derDemonstranten ausgesetzt habe，dass die lokalen Gesetze ihnen nicht erlauben ，mit denCampingplätzen und Schlafsäcken neu zu installieren .</td></tr><tr><td rowspan=1 colspan=1>Pretrained</td></tr><tr><td rowspan=1 colspan=1>Birgermeister Bloomberg erklärte gegenuber Journalisten ，dass die Stadt aufgrund dieserGerichtsentscheidung die Wiedereroffnung des offentlichen Raums ausgesetzt habe und dassdie Demonstranten daruber informiert wurden ，dass die ortlichen Gesetze es ihnen nichterlauben wuirden , sich mit Campingplätzen und Schlafsälen neu zu installieren .</td></tr></table>",
+        "bbox": [
+            184,
+            133,
+            812,
+            405
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 9: The no pretrain model makes a complete mistranslation when outputting ”und die Information der Demonstranten ausgesetzt habe”. That translates to ”the reopening of the public space and the information [noun] of the protesters were suspended”, instead of informing the protesters. Furthermore, it wrongly separated the two sentences, so the first sentence has extra words and the second sentence is left without a subject. The pretrained model does not make any of these mistakes. However, both models make a vocabulary mistake of ”zu installieren”, which is typically only used to refer to installing software. A human evaluator fluent in both German and English said that the pretrained version was better. ",
+        "bbox": [
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+            526
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/064e2d4d9600528548e69a46e04a0104bc2e0a7ead5c75d126124c2280303003.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Source</td></tr><tr><td rowspan=1 colspan=1>The low February temperatures, not only did they cause losses of millions for the agricultural sector, but they limited the possibilities of the state economy to grow,causing a contraction ofthe economic activity in general of 3.6 percent in the first half of the year, mainly supportedby the historic fall of 31.16 per cent in agriculture,which affected the dynamics of othereconomic sectors.</td></tr><tr><td rowspan=1 colspan=1>Ground Truth</td></tr><tr><td rowspan=1 colspan=1>Die niedrigen Temperaturen im Februar verursachten nicht nur Verluste in Millionenhohe inderLandwirtschaft，sondern steckten daruber hinaus dem Wachstum der Staatswirtschaftenge Grenzen und verursachten im ersten Vierteljahr einen allgemeinen Rückgang derWirtschaftstatigkeit um 3,6 Prozent Dieser geht hauptsächlich auf den historischen Abbauder landwirtschaftlichen Entwicklung um 31,16 Prozent zuruck,der sich bremsend auf weit-ere Wirtschaftssektoren auswirkte .</td></tr><tr><td rowspan=1 colspan=1>No pretrain</td></tr><tr><td rowspan=1 colspan=1>Die niedrigen Temperaturen im Februar fuhrten nicht nur zu Verlusten fur die Landwirtschaft,sondern sie beschränkten die Moglichkeiten der staatlichen Wirtschaft，wachsen zu wach-sen，wodurch die Wirtschaftstatigkeit insgesamt von 3,6 Prozent in der ersten Halfte desJahres ,hauptsächlich durch den historischen Rückgang von 31.16 % in der Landwirtschaft ,beeinflusst wurde,was die Dynamik anderer Wirtschaftssektoren betraf .</td></tr><tr><td rowspan=1 colspan=1>Pretrained</td></tr><tr><td rowspan=1 colspan=1>Die niedrigen Temperaturen im Februar fuhrten nicht nur zu Verlusten von Millionen fur denAgrarsektor,sondern beschränkten die Moglichkeiten der Staatswirtschaft, zu wachsen, waszu einer Schrumpfung der Wirtschaftstatigkeit im Allgemeinen von 3,6 Prozent in der erstenHalfte des Jahres fuhrte,hauptsächlich durch den historischen Einbruch von 316 Prozent inder Landwirtschaft,der die Dynamik anderer Wirtschaftsbereiche beeinflusst hatte .</td></tr></table>",
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+            291,
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+            643
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 10: The human evaluator noted that the pretrained version is better, as it correctly captures the meaning and sentence structure of the middle. The no pretrain model does not misses translating the word ”million”, repeats itself in ”wachsen zu wachsen”, and puts the verb ”beeinflusst wurde” is an unnatural position. However, the pretrained model makes a mistake in the percentage $3 1 6 \\%$ instead of $3 1 . 1 6 \\%$ ). ",
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+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/c69bdb1a5bf5254948f9f9ac118b14f8bfe39968e6b1eab4ced457fe9f616d16.jpg",
+        "table_caption": [
+            "Table 11: An example where the English source is poorly worded. Both models output poor translations, but the evaluator noted that the pretrained version is still better than the no pretrain version. Interestingly, both models mistranslate ”radical” as ”radikalislamistisches”, which means ”radical Islam”, which is probably a bias in the training data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Source</td></tr><tr><td rowspan=1 colspan=1>To facilitate the inception of the Second World War, they allowed bankers and politiciansto create a latent conflict situation by saddling Germany with huge war reparations, therebymaking a radicalist example of the impoverished masses,it remained only to introduce asuffciently convincing culprit and a leader with a simple solution, while also creating a multi-racial Czechoslovakia with a strong German minority to play,and indeed did, the role of afifth colony, once the war had been ignited.</td></tr><tr><td rowspan=1 colspan=1>Ground Truth</td></tr><tr><td rowspan=1 colspan=1>Um den Zweiten Weltkrieg einfacher entfachen zu konnen ，lieBen die Banker durch diePolitik eine latente Konfliktsituation schaffen ，indem sie Deutschland mit gigantischenKriegsreparationen belegten ; dadurch schufen sie die Voraussetzung verarmter Massen ，sodass sie den Deutschen nur noch einen ausreichend starken Fuhrer unterjubeln mussten,derdie Schuldigen benannte und einfache Losungen anbot ; ein weiterer Faktor war die Schaf-fung des Vielvolkerstaates Tschechoslowakei mit einer starken deutschen Minderheit,die dieFunktion einer fuinften Kolonne einnehmen sollte und auch einnahm ,um den Kriegsbrandzu entfachen.</td></tr><tr><td rowspan=1 colspan=1>No pretrain</td></tr><tr><td rowspan=1 colspan=1>Um die Grundung des Zweiten Weltkriegs zu erleichtern ， ermoglichte es den Bankern undPolitikern ,eine latente Konfliktlage zu schaffen,indem sie Deutschland mit enormen Repa-rationsforderungen konfrontierte ,wodurch ein radikalislamistisches Beispiel der verarmtenMassen entstand ,es blieb nur ，einen ausreichend aussagekraftigen Schuldigen und einenFuhrer mit einer einfachen Losung zu etablieren ,während gleichzeitig eine multi-ethnischeTschechoslowakei mit einer starken deutschen Minderheit zu spielen war und tatsächlich dieRolle einer funften Kolonie war .</td></tr><tr><td rowspan=1 colspan=1>Pretrained</td></tr><tr><td rowspan=1 colspan=1>Um die Einfuhrung des Zweiten Weltkrieges zu erleichtern ,lieBen sie Banker und Politikereine latente Konfliktlage schaffen ,indem sie Deutschland mit riesigen Reparationszahlungenbelieferten，wodurch ein radikalislamistisches Beispiel der verarmten Massen entstand,esblieb nur,einen ausreichend üiberzeugenden Schuldigen und einen Fuhrer mit einer einfachenLosung zu präsentieren，während gleichzeitig eine multiethnische Tschechoslowakei miteiner starken deutschen Minderheit geschaffen wurde ,um zu spielen,und tatsächlich,dieRolle einer funften Kolonie,sobald der Krieg entfacht worden war .</td></tr></table>",
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+    },
+    {
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+        "table_caption": [
+            "Table 12: Another example where the English source is poorly worded. Both models get the structure right, but have a variety of problematic translations. Both models miss the meaning of ”total vote count”. They both also translate ”electoral boxes” poorly - the no pretrain model calls it ”electoral paperwork” while the pretrained model calls it ”ballots”. These failures may be because of the poorly worded English source. The human evaluator found them both equally poor. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Source</td></tr><tr><td rowspan=1 colspan=1>The total vote count will also be done if at the end of the ordinary calculation is establishedthat the diference between the winner and the candidate placed on second position is equalto or less than one percentage point, as long as there is a request of the representative ofthe political party whose candidate came on the second position, case in which there will beexcluded the electoral boxes that have been considered during the partial recount.</td></tr><tr><td rowspan=1 colspan=1>Ground Truth</td></tr><tr><td rowspan=1 colspan=1>Die Stimmenauszählung kann auch in ihrer Gesamtheit erfolgen ，wenn nach Abschlussder ordentlichen Berechnung festgestelt wird，dass der Unterschied zwischen dem mut-maBlichen Gewinner und dem Kandidaten auf dem zweiten Platz gleich oder geringer als einProzent ist,vorausgesetzt es liegt ein ausdruicklicher Antrag von einem Vertreter der Partei ,deren Kandidat Zweiter geworden ist, vor .In diesem Fall wirden die Wahlpakete ,die einerteilweisen Auszahlung ausgesetzt wurden,ausgeschlossen .</td></tr><tr><td rowspan=1 colspan=1>No pretrain</td></tr><tr><td rowspan=1 colspan=1>Die gesamte Stimmenanzahl wird auch dann erreicht，wenn am Ende der ordentlichenBerechnung festgestellt wird ,dass der Unterschied zwischen dem Sieger und dem Kandi-daten,der auf der zweiten Position liegt ,gleich oder weniger als einen Prozentpunkt beträgt, Vorausgesetzt,dass der Vertreter der Partei,deren Kandidat auf der zweiten Position ist,derFall ist,in dem die Wahlunterlagen,die während der teilweisen Rückzahlung beruicksichtigtwurden,ausgeschlossen werden.</td></tr><tr><td rowspan=1 colspan=1>Pretrained</td></tr><tr><td rowspan=1 colspan=1>Die Gesamtzahl der Stimmzetel wird auch dann durchgefuhrt，wenn am Ende der or-dentlichen Berechnung festgestelt wird, dass der Unterschied zwischen dem Gewinner unddem auf den zweiten Platz platzierten Kandidaten gleich oder weniger als einen Prozent-punkt betragt，solange es einen Antrag des Vertreters der politischen Partei gibt，dessenKandidat auf die zweite Position kam,in dem es die Wahlzettel ausklammert,die währendder Teilzählung berüicksichtigt wurden .</td></tr></table>",
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parse/train/mfQxdSMWOF/mfQxdSMWOF_content_list.json

@@ -0,0 +1,1361 @@
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+    {
+        "type": "text",
+        "text": "Keeping Your Eye on the Ball: Trajectory Attention in Video Transformers ",
+        "text_level": 1,
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+            232,
+            122,
+            763,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mandela Patrick∗ Facebook AI mandelapatrick@fb.com ",
+        "bbox": [
+            197,
+            226,
+            380,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dylan Campbell∗ University of Oxford dylan@robots.ox.ac.uk ",
+        "bbox": [
+            411,
+            226,
+            594,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yuki Asano∗ University of Oxford yuki@robots.ox.ac.uk ",
+        "bbox": [
+            624,
+            226,
+            799,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ishan Misra Facebook AI imisra@fb.com ",
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+            233,
+            290,
+            346,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Florian Metze Facebook AI fmetze@fb.com ",
+        "bbox": [
+            413,
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+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Christoph Feichtenhofer Facebook AI feichtenhofer@fb.com ",
+        "bbox": [
+            591,
+            289,
+            766,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Andrea Vedaldi Facebook AI vedaldi@fb.com ",
+        "bbox": [
+            290,
+            352,
+            411,
+            393
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "João F. Henriques University of Oxford joao@robots.ox.ac.uk ",
+        "bbox": [
+            534,
+            352,
+            709,
+            395
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            430,
+            535,
+            446
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In video transformers, the time dimension is often treated in the same way as the two spatial dimensions. However, in a scene where objects or the camera may move, a physical point imaged at one location in frame $t$ may be entirely unrelated to what is found at that location in frame $t + k$ . These temporal correspondences should be modeled to facilitate learning about dynamic scenes. To this end, we propose a new drop-in block for video transformers—trajectory attention—that aggregates information along implicitly determined motion paths. We additionally propose a new method to address the quadratic dependence of computation and memory on the input size, which is particularly important for high resolution or long videos. While these ideas are useful in a range of settings, we apply them to the specific task of video action recognition with a transformer model and obtain state-of-the-art results on the Kinetics, Something–Something V2, and Epic-Kitchens datasets. Code and models are available at: https://github.com/facebookresearch/ Motionformer. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+            310,
+            694
+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "Transformers [75] have become a popular architecture across NLP [32], vision [20] and speech [5]. The self-attention mechanism in the transformer works well for different types of data and across domains. However, its generic nature and its lack of inductive biases also mean that transformers typically require extremely large amounts of data for training [56, 9], or aggressive domain-specific augmentations [71]. This is particularly true for video data, for which transformers are also applicable [50], but where statistical inefficiencies are exacerbated. While videos carry rich temporal information, they can also contain redundant spatial information from neighboring frames. Vanilla self-attention applied to videos compares pairs of image patches extracted at all possible spatial locations and frames. This can lead it to focus on the redundant spatial information rather than the temporal information, as we show by comparing normalization strategies in our experiments. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We therefore contribute a variant of self-attention, called trajectory attention, which is better able to characterize the temporal information contained in videos. For the analysis of still images, spatial locality is perhaps the most important inductive bias, motivating the design of convolutional networks [42] and the use of spatial encodings in vision transformers [20]. This is a direct consequence of the local structure of the physical world: points that belong to the same 3D object tend to project to pixels that are close to each other in the image. By studying the correlation of nearby pixels, we can thus learn about the objects. ",
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+        "img_path": "images/c4bf3879458b0459b8e81868dd445ff316ca1ffbbbcd0b4a3a5856c4daaca54b.jpg",
+        "image_caption": [
+            "Figure 1: Trajectory attention. In this sequence of frames from the Kinetics-400 dataset, depicting the action ‘kicking soccer ball’, the ball does not remain stationary with respect to the camera, but instead moves to different locations in each frame. Trajectory attention aims to share information along the motion path of the ball, a more natural inductive bias for video data than pooling axially along the temporal dimension or over the entire space-time feature volume. This allows the network to aggregate information from multiple views of the ball, to reason about its motion characteristics, and to be less sensitive to camera motion. "
+        ],
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+        "type": "text",
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+        "type": "text",
+        "text": "Videos are similar, except that 3D points move over time, thus projecting on different parts of the image along certain 2D trajectories. Existing video transformer methods [8, 3, 50] disregard these trajectories, pooling information over the entire 3D space-time feature volume [3, 50], or pooling axially across the temporal dimension [8]. We contend that pooling along motion trajectories would provide a more natural inductive bias for video data, allowing the network to aggregate information from multiple views of the same object or region, to reason about how the object or region is moving (for example, the linear and angular velocities), and to be invariant to camera motion. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We leverage attention itself as a mechanism to find these trajectories. This is inspired by methods such as RAFT [70], which showed that excellent estimates of optical flow can be obtained from the correlation volume obtained by comparing local features across space and time. We observe that the joint attention mechanism for video transformers computes such a correlation volume as an intermediate result. However, subsequent processing collapses the volume without consideration for its particular structure. In this work, we seek instead to use the correlation volume to guide the network to pool information along motion paths. ",
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+    {
+        "type": "text",
+        "text": "We also note that visual transformers operate on image patches which, differently from individual pixels, cannot be assumed to correspond to individual 3D points and thus to move along simple 1D trajectories. For example, in Figure 1, depicting the action ‘kicking soccer ball’, the ball spans up to four patches, depending on the specific video frame. Furthermore, these patches contain a mix of foreground (the ball) and background objects, thus at least two distinct motions. Fortunately, we are not forced to select a single putative motion: the attention mechanism allows us to assemble a motion feature from all relevant ‘ball regions’. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Inspired by Nyströmformer [85], we also propose a principled approximation to self-attention, Orthoformer. Our approximation sets state-of-the-art performance on the recent Long Range Arena (LRA) benchmark [69] for evaluating efficient attention approximations and generalizes beyond the video domain to long text and high resolution images, with lower FLOPS and memory requirements compared to alternatives, Nyströmformer and Performer [15]. Combining our approximation with trajectory attention allows us to significantly improve its computational and memory efficiency. With our contributions, we set state-of-the-art results on four video action recognition benchmarks. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Video representations and 3D-CNNs. Hand-crafted features were originally used to convert video data into a representation amenable to analysis by a shallow linear model. Such representations include SIFT-3D [60], HOG3D [39], and IDT [76]. Since the breakthrough of AlexNet [40] on the ImageNet classification benchmark [58], which demonstrated the empirical benefits of deep neural networks to learn representations end-to-end, there have been many attempts to do the same for video. Architectures with 3D convolutions—3D-CNNs—were originally proposed to learn deep video representations [72]. Since then, improvements to this paradigm include the use of ImageNet-inflated weights [11], the space-time decomposition of 3D convolutions [54, 74, 84], channel-separated convolutions [73], non-local blocks [79], and attention layers [13]. Optical flow-based pooling can be used instead of temporal convolutions to improve the representation’s robustness to camera and object motions [1]. Our approach shares this motivation. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Vision transformers. The transformer architecture [75], originally proposed for natural language processing, has recently gained traction in the computer vision domain. The vision transformer (ViT) [20] decomposes an image into a sequence of $1 6 \\times 1 6$ words and uses a multi-layer transformer to perform image classification. To improve ViT’s data efficiency, DeiT [71] used distillation from a strong teacher model and aggressive data augmentation. Transformers have also been used in a variety of vision image tasks, such as image representation learning [12, 82, 18, 59], image generation [51], object detection [48, 10], video question-answering [35], few-shot learning [19], and image–text [49, 63, 67, 44, 68], video-text [65, 64, 89, 25, 53, 2, 6], and video-audio [43, 52, 29] representation learning. ",
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+    {
+        "type": "text",
+        "text": "Attention for video recognition. The self-attention operation proposed in the transformer [75] have been adapted to video recognition tasks. Wang et al. [79] propose the non-local mean operation for video action recognition, which is equivalent to the standard transformer self-attention applied uniformly across space and time, while our proposed trajectory attention does not treat the space and time dimensions equivalently. Zhao et al. [87] propose a CNN architecture that explicitly predicts trajectories and aggregates information along them using a convolution operation. In contrast, our transformer architecture does not explicitly predict trajectories, but instead provides an inductive bias that encourages the network to consider motion trajectories where useful. Concurrent works [8, 3, 50, 21] have also adapted the self-attention operation to the spatio-temporal nature of videos, however, these approaches do not have a mechanism for reasoning about motion paths, treating time as just another dimension, unlike our approach. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Efficient attention. Due to the quadratic complexity of self-attention, there has been a significant amount of research on how to reduce its computational complexity with respect to time and memory√ use. Sparse attention mechanisms [14] were used to reduce self-attention complexity to $\\mathcal { O } ( n \\sqrt { n } )$ , and locality-sensitivity hashing was used by Reformer [38] to further reduce this to ${ \\mathcal { O } } ( n \\log n )$ . More recently, linear attention mechanisms have been introduced, namely Longformer [7], Linformer [78], Performer [15] and Nyströmformer [85]. The Long Range Arena benchmark [69] was recently introduced to compare these different attention mechanisms. ",
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+    {
+        "type": "text",
+        "text": "Temporal correspondences and optical flow. There are many approaches that aim to establish explicit correspondences between video frames as a way to reason about camera and object motion. For short-range correspondences across time, optical flow algorithms [30, 66, 70] are highly effective. In particular, RAFT [70] showed the effectiveness of an all-pairs inter-frame correlation volume as an encoding, which is essentially an attention map. All-pairs intra-frame correlations were subsequently shown to help resolve correspondence ambiguities [34]. For longer-range correspondences, object tracking by repeated detection [57] and data association can be used. In contrast to these approaches, our work does not explicitly establish temporal correspondences, but facilitates implicit correspondence learning via trajectory attention. Jabri et al. [31] estimate correspondences in a similar way, framing the problem as a contrastive random walk on a graph and apply explicit guidance via a cycle consistency loss. Incorporating such guidance into a video transformer is an interesting direction. ",
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+    {
+        "type": "text",
+        "text": "3 Trajectory Attention for Video Data ",
+        "text_level": 1,
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+        "text": "Our goal is to modify the attention mechanism in transformers to better capture the information contained in videos. Consider an input video $I \\in \\mathbb { R } ^ { T ^ { \\prime } \\times 3 \\times H \\times W }$ consisting of $T ^ { \\prime }$ frames of resolution ",
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+        "image_caption": [
+            "Figure 2: Trajectory attention flowchart. We divide the attention operation into two stages: the first forming a set of $S T$ trajectory tokens for every space-time location $^ { s t }$ —a spatial attention operation between pairs of frames—and the second pooling along these trajectories with a 1D temporal attention operation. In this way, we accumulate information along the motion paths of objects in the video. The softmax operations are computed over the last dimension. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "$H \\times W$ . As in existing video transformer models [8, 3], we pre-process the video into a sequence of $S T$ tokens $\\mathbf { x } _ { s t } \\in \\breve { \\mathbb { R } } ^ { D }$ , for a spatial resolution of $S$ and a temporal resolution of $T$ . We use a cuboid embedding [3, 21], where disjoint spatio-temporal cubes from the input volume are linearly projected to $\\mathbb { R } ^ { D }$ (equivalent to a 3D convolution with downsampling). We also test an embedding of disjoint image patches [20]. A learnable positional encoding $\\mathbf { e } \\in \\mathbb { R } ^ { D }$ is added to the video embeddings for spatial and temporal dimensions separately, resulting in the code $\\mathbf { z } _ { s t } = \\mathbf { x } _ { s t } + \\mathbf { e } _ { s } ^ { s } + \\mathbf { e } _ { t } ^ { t }$ . Finally, a learnable classification token $\\mathbf { z } _ { \\mathrm { c l s } }$ is added to the sequence of tokens, like in the BERT Transformer [32], to reason about the video as a whole. For clarity, we elide the classification token from our treatment in the sequel. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We now have a set of tokens that form the input to a sequence of transformer layers that, as in ViT [20], consist of Layer Norm (LN) operations [4], multi-head attention (MHA) [75], residual connections [27], and a feed-forward network (MLP): ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/08881b5a3f4a865de707de505a3e4b8c87c6e822087b01a20eff2bef60afde18.jpg",
+        "text": "$$\n\\mathbf { y } = \\mathbf { M H A } ( \\mathbf { L N } ( \\mathbf { z } ) ) + \\mathbf { z } ; \\quad \\mathbf { z } ^ { \\prime } = \\mathbf { M L P } ( \\mathbf { L N } ( \\mathbf { y } ) ) + \\mathbf { y } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In the next section, we shall focus on a single head of the attention operation, and demonstrate how self-attention can realize a suitable inductive bias for video data. For clarity of exposition, we abuse the notation slightly, neglecting the layer norm operation and using the same dimensions for single-head attention as for multi-head attention. ",
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+        "type": "text",
+        "text": "3.1 Video self-attention ",
+        "text_level": 1,
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+        "text": "The self-attention operation begins by forming a set of query-key-value vectors $\\mathbf { q } _ { s t } , \\mathbf { k } _ { s t } , \\mathbf { v } _ { s t } \\in \\mathbb { R } ^ { D }$ , one for each space-time location $^ { s t }$ in the video. These are computed as linear projections of the input $\\mathbf { z } _ { s t }$ , that is, $\\mathbf { q } _ { s t } = \\mathbf { W } _ { q } \\mathbf { z } _ { s t }$ , $\\mathbf { k } _ { s t } = \\mathbf { W } _ { k } \\mathbf { z } _ { s t }$ , and $\\mathbf { v } _ { s t } = \\mathbf { W } _ { v } \\mathbf { z } _ { s t }$ , for projection matrices $\\mathbf { W } _ { i } \\in \\mathbb { R } ^ { D \\times D }$ . A direct application of attention across space-time (called joint space-time attention [8, 3]) computes: ",
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+    {
+        "type": "equation",
+        "img_path": "images/f2e593217d2098a102fbdf947b63da3e63f26537baa8510cf169d060e8db4d8f.jpg",
+        "text": "$$\n\\mathbf { y } _ { s t } = \\sum _ { s ^ { \\prime } t ^ { \\prime } } \\mathbf { v } _ { s ^ { \\prime } t ^ { \\prime } } \\cdot \\frac { \\exp \\left. \\mathbf { q } _ { s t } , \\mathbf { k } _ { s ^ { \\prime } t ^ { \\prime } } \\right. } { \\sum _ { \\bar { s } \\bar { t } } \\exp \\left. \\mathbf { q } _ { s t } , \\mathbf { k } _ { \\bar { s } \\bar { t } } \\right. } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this way, each query $\\textbf { q } _ { s t }$ is compared to all keys $\\mathbf { k } _ { s ^ { \\prime } t ^ { \\prime } }$ using dot products, the results are normalized using the softmax operator, and the weights thus obtained are used to average the values corresponding to the keys. Compared to a standard transformer, we have omitted for brevity the softmax temperature parameter $D ^ { 1 / 2 }$ and instead assume that the queries and keys have been divided by $D ^ { 1 / 4 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One issue with this formulation is that it has quadratic complexity in both space and time, i.e., $\\mathcal { O } ( S ^ { 2 } T ^ { 2 } )$ . An alternative is to restrict attention to either space or time (called divided space-time attention): ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/450ee48937a8d00cdf721d74449a5d5aaf76c681048ba1863009364c4fe13ef1.jpg",
+        "text": "$$\n{ \\mathbf { y } } _ { s t } = \\sum _ { s ^ { \\prime } } { \\mathbf { v } } _ { s ^ { \\prime } t } \\cdot \\frac { \\exp \\left. { \\mathbf { q } } _ { s t } , { \\mathbf { k } } _ { s ^ { \\prime } t } \\right. } { \\sum _ { \\bar { s } } \\exp \\left. { \\mathbf { q } } _ { s t } , { \\mathbf { k } } _ { \\bar { s } t } \\right. } \\mathrm { ~ } ( \\mathrm { s p a c e } ) ; { \\mathbf { y } } _ { s t } = \\sum _ { t ^ { \\prime } } { \\mathbf { v } } _ { s t ^ { \\prime } } \\cdot \\frac { \\exp \\left. { \\mathbf { q } } _ { s t } , { \\mathbf { k } } _ { s t ^ { \\prime } } \\right. } { \\sum _ { \\bar { t } } \\exp \\left. { \\mathbf { q } } _ { s t } , { \\mathbf { k } } _ { s \\bar { t } } \\right. } \\mathrm { ~ } ( \\mathrm { i m e } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "This reduces the complexity to $\\mathcal { O } ( S ^ { 2 } T )$ and $\\mathcal { O } ( S T ^ { 2 } )$ , respectively, but only allows the model to analyse time and space independently. This is usually addressed by interleaving [8] or stacking [3] the two attention modules in a sequence. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Different to both of these approaches, we perform attention along trajectories, the probabilistic path of a token between frames.2 For each space-time location $^ { s t }$ (the trajectory ‘reference point’) and corresponding query $\\textbf { q } _ { s t }$ , we construct a set of trajectory tokens $\\tilde { \\bf y } _ { s t t ^ { \\prime } }$ , representing the pooled information weighted by the trajectory probability. The trajectory extends for the duration of the video sequence and its tokens $\\mathbf { \\widetilde { y } } _ { s t t ^ { \\prime } } ^ { } \\in \\mathbf { \\widetilde { \\mathbb { R } } } ^ { \\dot { D } }$ at different times $t ^ { \\prime }$ are given by: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/9bb17f13ae83480e6f59d741eb823ada195eac99129a08eaa5d6b9026fcb0892.jpg",
+        "text": "$$\n\\tilde { \\mathbf { y } } _ { s t t ^ { \\prime } } = \\sum _ { s ^ { \\prime } } \\mathbf { v } _ { s ^ { \\prime } t ^ { \\prime } } \\cdot \\frac { \\exp \\left. \\mathbf { q } _ { s t } , \\mathbf { k } _ { s ^ { \\prime } t ^ { \\prime } } \\right. } { \\sum _ { \\bar { s } } \\exp \\left. \\mathbf { q } _ { s t } , \\mathbf { k } _ { \\bar { s } t ^ { \\prime } } \\right. } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the attention in this formula is applied spatially (index $s$ ) and independently for each frame. Intuitively, this pooling operation implicitly seeks the location of the trajectory at time $t ^ { \\prime }$ by comparing the trajectory query $\\textbf { q } _ { s t }$ to the keys $\\mathbf { k } _ { s ^ { \\prime } t ^ { \\prime } }$ at time $t ^ { \\prime }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Once the trajectories are computed, we further pool them across time to reason about intra-frame information/connections. To do so, the trajectory tokens are projected to a new set of queries, keys and values as usual: ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/3635fd55a7e0f7e0607d11542bf5757963d7eb64febc545415698b6a2072a5e3.jpg",
+        "text": "$$\n\\begin{array} { r } { \\tilde { \\mathbf { q } } _ { s t } = \\tilde { \\mathbf { W } } _ { q } \\tilde { \\mathbf { y } } _ { s t t } , \\tilde { \\mathbf { k } } _ { s t t ^ { \\prime } } = \\tilde { \\mathbf { W } } _ { k } \\tilde { \\mathbf { y } } _ { s t t ^ { \\prime } } , \\tilde { \\mathbf { v } } _ { s t t ^ { \\prime } } = \\tilde { \\mathbf { W } } _ { v } \\tilde { \\mathbf { y } } _ { s t t ^ { \\prime } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Like $\\textbf { q } _ { s t }$ before, the updated reference query $\\tilde { \\mathbf { q } } _ { s t }$ corresponds to the trajectory reference point $^ { s t }$ and contains information spatially-pooled from across the reference frame $t$ . This new query is used to pool across the new time (trajectory) dimension by applying 1D attention: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/321de944356adf1f6f4a83c0313b81e30af779f4d98737086fea9b8c68118c52.jpg",
+        "text": "$$\n{ \\bf y } _ { s t } = \\sum _ { t ^ { \\prime } } \\tilde { \\bf v } _ { s t t ^ { \\prime } } \\cdot \\frac { \\exp \\langle \\tilde { \\bf q } _ { s t } , \\tilde { \\bf k } _ { s t t ^ { \\prime } } \\rangle } { \\sum _ { \\bar { t } } \\exp \\langle \\tilde { \\bf q } _ { s t } , \\tilde { \\bf k } _ { s t \\bar { t } } \\rangle } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Like joint space-time attention, our approach has quadratic complexity in both space and time, $\\mathcal { O } ( S ^ { 2 } T ^ { 2 } )$ , so has no computational advantage and is in fact slower than divided space-time attention. However, we demonstrate better accuracy than both joint and divided space-time attention mechanisms. We also provide fast approximations in Section 3.2. A flowchart of the full trajectory attention operation is shown in tensor form in Figure 2. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Approximating attention ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "To complement our trajectory attention, we also propose an approximation scheme to speed up calculations. This scheme is generic and applies to any attention-like pooling mechanism. We thus switch to a generic transformer-like notation to describe it. Namely, consider query-key-value matrices ${ \\bf Q } , { \\bf K } , { \\bf V } \\in \\mathbb { R } ^ { D \\times N }$ such that the query-key-value vectors are stored as columns $\\mathbf { q } _ { i } , \\mathbf { \\bar { k } } _ { i } , \\mathbf { v } _ { i } \\in \\mathbb { R } ^ { D }$ in these matrices. ",
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+    {
+        "type": "text",
+        "text": "In order to obtain an efficient decomposition of the attention operator, we will rewrite it using a probabilistic formulation. Let $A _ { i j } \\in \\{ 0 , 1 \\}$ be a categorical random variable indicating whether the $j$ th input (with key vector $\\mathbf { k } _ { j } \\in \\mathbb { R } ^ { D } .$ ) is assigned to the $i$ th output (with query vector $\\mathbf { q } _ { i } \\in \\mathbb { R } ^ { D } .$ ), with $\\textstyle \\sum _ { j } { A _ { i j } } = { \\mathbf { 1 } }$ . The attention operator uses a parametric model of the probability of this event based on the multinomial logistic function, i.e., the softmax operator $\\boldsymbol { \\mathcal { S } } ( \\cdot )$ :3 ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/11b633b5498a24787ec4a575960d84a98e91bc3622eea66302eff6dbd6c0a88d.jpg",
+        "text": "$$\nP ( A _ { i : } ) = S ( \\mathbf { q } _ { i } ^ { \\mathsf { T } } \\mathbf { K } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the subscript : denotes a full slice of the input tensor in that dimension. We now introduce the latent variables $U _ { \\ell j } \\in \\{ 0 , 1 \\}$ , which similarly indicate whether the $j$ th input is assigned to the \\`th prototype, an auxiliary vector which we denote by $\\mathbf { p } _ { \\ell } \\in \\mathbb { R } ^ { D }$ . We can use the laws of total and conditional probability to obtain: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/971d9683dd794147086e60eb53e00874d347c5c300bc1aa41d02d3183d9e3799.jpg",
+        "text": "$$\nP ( A _ { i j } ) = \\sum _ { \\ell } P ( A _ { i j } \\mid U _ { \\ell j } ) P ( U _ { \\ell j } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the latent variables that we chose are independent of the inputs (keys). They use the same parametric model, but with parameters $\\mathbf { P } \\in \\mathbb { R } ^ { D \\times R }$ (the concatenated prototype vectors $\\mathbf { p } _ { \\ell }$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$P ( U ) = \\mathcal { S } ( \\mathbf { P } ^ { \\mathsf { T } } \\mathbf { K } )$ . Eq. 8 is exact, even under the parametric model for $P ( U )$ , though the corresponding true distribution $P ( A \\mid U )$ is intractable. We now approximate the conditional probability $P ( A \\mid U )$ with a similar parametric model: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a7525ad07b29448f002cd52e159df44f3b4b0a249f7806f18fff36e613108201.jpg",
+        "text": "$$\n{ \\tilde { P } } ( A \\mid U ) = S ( \\mathbf { Q } ^ { \\mathsf { T } } \\mathbf { P } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            135,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { Q } \\in \\mathbb { R } ^ { D \\times N }$ concatenates all query vectors horizontally. Substituting equations 7–9 we write the full approximate attention $\\tilde { \\mathcal { A } }$ , multiplied by an arbitrary matrix $\\mathbf { V }$ (which in the case of a transformer contains the values of the key–value pairs stacked as rows): ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/942abc6ea45759dba2714752d66ea7dc2fa9c89f16038440e09a0fa14e4194f4.jpg",
+        "text": "$$\n{ \\tilde { P } } ( A ) \\mathbf { V } = { \\mathcal { S } } ( \\mathbf { Q } ^ { \\mathsf { T } } \\mathbf { P } ) \\left( { \\mathcal { S } } ( \\mathbf { P } ^ { \\mathsf { T } } \\mathbf { K } ) \\mathbf { V } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Computational efficiency. One important feature of the approximation in eq. 10 is that it can be computed in two steps. First the values $\\mathbf { V }$ are multiplied by a prototypes-keys attention matrix $S ( \\mathbf { P } ^ { \\top } \\dot { \\mathbf { K } } ) \\in \\mathbb { R } ^ { R \\times N }$ , which can be much smaller than the full attention matrix $\\bar { \\mathcal { S } } ( \\mathbf { Q } ^ { \\mathsf { T } } \\mathbf { K } ) \\in \\mathbb { R } ^ { N \\times N }$ (eq. 7), i.e., $R \\ll N$ . Finally, this product is multiplied by a queries-prototypes attention matrix $\\mathbf { \\mathcal { S } } ( \\mathbf { \\dot { Q } } ^ { \\mathsf { T } } \\mathbf { P } ) \\in \\mathbb { R } ^ { N \\times R }$ , which is also small. This allows us to sidestep the quadratic dependency of full attention over the input and output size $( \\mathcal { O } ( N ^ { 2 } ) )$ , replacing it with linear complexity $( \\mathcal { O } ( N ) )$ as long as $R$ is kept constant. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Prototype selection. The aim for prototype-based attention approximation schemes is to use as few prototypes as possible while reconstructing the attention operation as accurately as possible. As such, it behooves us to select prototypes efficiently. We have two priorities for the prototypes: to dynamically adjust to the query and key vectors so that their region of space is well-reconstructed, and to minimize redundancy. The latter is important because the relative probability of a query–key pair may be over-estimated if many prototypes are clustered near that query and key. To address these criteria, we incrementally build a set of prototypes from the set of queries and keys such that a new prototype is maximally orthogonal to the prototypes already selected, starting with a query or key at random. This greedy strategy is dynamic, since it selects prototypes from the current set of queries and keys, and has high entropy, since it preferences well-separated prototypes. Moreover, it balances speed and performance by using a greedy strategy, rather than finding a globally-optimal solution to the maximum entropy sampling problem [61], making it suitable for use in a transformer. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Naïvely applying prototype-based attention approximation techniques to video transformers would involve creating a unique set of prototypes for each frame in the video. However, additional memory savings can be realized by sharing prototypes across time. Since there is significant information redundancy between frames, video data is opportune for compression via temporallyshared prototypes. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Orthoformer algorithm. The proposed approximation algorithm is outlined in Algorithm 1. The attention matrix is approximated using intermediate prototypes, selected as the most orthogonal subset of the queries and keys, given a desired number of prototypes $R$ . To avoid a linear dependence on the sequence length $N$ , we first randomly subsample $c R$ queries and keys, for a constant $c$ , before selecting the most orthogonal subset, resulting in a complexity quadratic in the number of prototypes $\\mathcal { O } ( R ^ { 2 } )$ . The algorithm then computes two attention matrices, much smaller than the original problem, and multiplies them with the values. The most related approach in the literature is Nyströmformer [85] attention, outlined in Algorithm 2. This approach involves a pseudoinverse to attenuate the effect of near-parallel prototypes, has more operations, and a greater memory footprint. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Orthoformer (proposed) attention ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Nyströmformer [85] attention ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: P ← MostOrthogonalSubset(Q, K, R)   \n2: $\\Omega _ { 1 } = \\mathcal { S } ( \\mathbf { Q } ^ { \\top } \\mathbf { P } / \\sqrt { D } )$   \n3: $\\Omega _ { 2 } = \\mathcal { S } ( \\mathbf { P } ^ { \\top } \\mathbf { K } / \\sqrt { D } )$   \n4: $\\mathbf { Y } = \\Omega _ { 1 } ( \\Omega _ { 2 } \\mathbf { V } )$   \n1: Pq, Pk ← SegmentMeans(Q, K, R)√   \n2: Ω1 = S(QTPk/ D)   \n3: Ω−12 = IterativeInverse(S(PTq Pk/ D), Niter)   \n4: Ω3 = S(PTq K/ D)   \n5: Y = Ω1 \u0000Ω−12 (Ω3V)\u0001 ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 The Motionformer model ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our full video transformer model builds on previous work, as shown in Table 1. In particular, we use the ViT image transformer model [20] as the base architecture, the separate space and time positional encodings of TimeSformer [8], and the cubic image tokenization strategy as in ViViT [3]. These design choices are ablated in Section 4. The crucial difference for our model is the trajectory attention mechanism, with which we demonstrate greater empirical performance than the other models. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7359dc30267e9d99bcf58a7bd495dd33117d364e78c7e7c3f3e59bd040306bed.jpg",
+        "table_caption": [
+            "Table 1: Comparison of recent video transformer models. We show the different design choices of recent video transformer models and how they compare to our proposed Motionformer model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Base Model</td><td>Attention</td><td>Pos.Encoding</td><td>Tokenization</td></tr><tr><td>TimeSformer [8]</td><td>ViT-B</td><td>Divided Space-Time</td><td>Separate</td><td>Square</td></tr><tr><td>ViViT [3]</td><td>ViT-L</td><td>Joint/Divided Space-Time</td><td>Joint</td><td>Cubic</td></tr><tr><td>Motionformer</td><td>ViT-B</td><td>Trajectory</td><td>Separate</td><td>Cubic</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets. Kinetics [36] is a large-scale video classification dataset consisting of short clips collected from YouTube, licensed by Google under Creative Commons. As it is a dataset of human actions, it potentially contains personally identifiable information such as faces, names and license plates. Something–Something V2 [26] is a video dataset containing more than 200,000 videos across 174 classes, with a greater emphasis on short temporal clips. In contrast to Kinetics, the background and objects remain consistent across different classes, and therefore models have to reason about fine-grained motion signals. We verified the importance of temporal reasoning on this dataset by showing that a single frame model gets significantly worse results, a decrease of $3 9 \\%$ top-1 accuracy. In contrast, a drop of only $7 \\%$ is seen on the Kinetics-400 dataset, showing that temporal information is much less relevant there. We obtained a research license for this data from https://20bn.com; the data was collected with consent. Epic Kitchens-100 [16] is an egocentric video dataset capturing daily kitchen activities. The highest scoring verb and action pair predicted by the network constitutes an action, for which we report top-1 accuracy. The data is licensed under Creative Commons and was collected with consent by the Epic Kitchens teams. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation details. We follow a standard training and augmentation pipeline [3], as detailed in the appendix. For ablations, our default Motionformer model is the Vision Transformer Base architecture [20] (ViT/B), pretrained on ImageNet-21K [17], patch-size $2 \\times 1 6 \\times 1 6$ with central frame initialization [3], separate space-time positional embedding and our trajectory attention. The base architecture has 12 layers, 12 attention heads, and an embedding dimension of 768. Our default Motionformer model operates on $1 6 \\times 2 2 4 \\times 2 2 4$ videos with temporal stride 4 i.e. temporal extent of 2s. For comparisons with state-of-the-art, we report results on two additional variants: MotionformerHR, which has a high spatial resolution $( 1 6 { \\times } 3 3 6 { \\times } 3 3 6$ videos with temporal stride 4 i.e. temporal extent of 2s), and Motionformer-L, which has a long temporal range $( 3 2 \\times 2 2 4 \\times 2 2 4$ videos with temporal stride 3 i.e. temporal extent of 3s). Experiments with the large ViT architecture are deferred to the appendix. ",
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+    {
+        "type": "text",
+        "text": "4.1 Ablation studies ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Input: tokenization. We consider the effect of different input tokenization approaches for both joint and trajectory attention on Kinetics-400 (K-400) and Something–Something V2 (SSv2) in Table 2b. For patch tokenization $( 1 \\times 1 6 \\times 1 6 )$ ), we use inputs of size $8 \\times 2 2 4 \\times 2 2 4$ , while for cubic [3, 21] tokenization $( 2 \\times 1 6 \\times 1 6 )$ , we use inputs of size $1 6 \\times 2 2 4 \\times 2 2 4$ to ensure that the model has the same number of input tokens over the same temporal range of 2 seconds. For both attention types, we see that cubic tokenization gives a $1 \\%$ accuracy improvement over square tokenization on SSv2, a dataset for which temporal information is critical. Furthermore, our proposed trajectory attention using cubic tokenization outperforms joint space-time attention on both datasets. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Input: positional encoding. Here, we ablate using a joint or separate [21] (default) space-time positional encoding in Table 2b. Similar to the results for input tokenization, the choice of positional encoding is particularly important for the fine-grained motion dataset, SSv2. Since joint space-time attention treats tokens in the space-time volume equally, it benefits particularly from separating the positional encodings, allowing it to differentiate between space and time dimensions, with a $4 \\%$ improvement on SSv2 over joint space-time encoding. Our proposed trajectory attention elicits a more modest improvement of $1 \\%$ from using separated positional encodings on SSv2, and outperforms joint space-time attention in both settings on both datasets. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Attention block: comparisons. We compare our proposed trajectory attention to joint space-time attention [3], and divided space-time attention [8] in Table 4. Our trajectory attention (bottom row) outperforms both alternatives on the K-400 and SSv2 datasets. While we see only modest ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e4378e06e0a9e4ab05576c826ff921b01077a53fe8b63939519bc70b079ab9c1.jpg",
+        "table_caption": [
+            "(a) Cubic tokenization works best for trajectory attn. ",
+            "(b) Trajectory attn. works well with both encodings. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attention</td><td>Pos.Encoding</td><td>GFlops K-400</td><td>SSv2</td><td></td></tr><tr><td>Joint ST</td><td>Joint ST Separate ST [21]</td><td>180.6 180.6</td><td>79.1 79.2</td><td>60.8 64.0</td></tr><tr><td>Trajectory</td><td>Joint ST Separate ST [21]</td><td>369.5 369.5</td><td>79.6 79.7</td><td>65.8 66.5</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/89470b21d910f0dc032bc53e4287819c5e3d92521e5143b4d42923375e3aca63.jpg",
+        "table_caption": [
+            "Table 2: Input encoding ablations: Comparison of input tokenization and positional encoding design choices. We report GFLOPS and top-1 accuracy $( \\% )$ on K-400 and SSv2. ",
+            "Table 3: Orthoformer ablations: We ablate various aspects of our Orthoformer approximation. E denotes exact attention and A denotes approximate attention. We report max CUDA memory consumption (GB) and top-1 accuracy $( \\% )$ ) on K-400 and SSv2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attention</td><td>Tokenization GFlops</td><td>K-400</td><td>SSv2</td></tr><tr><td>Joint ST</td><td>Square (1×162) 179.7 Cubic (2×162) 180.6</td><td>79.4 79.2</td><td>63.0 64.0</td></tr><tr><td>Trajectory</td><td>Square (1×162) Cubic (2×16²)</td><td>368.5 79.4 369.5 79.7</td><td>65.8 66.5</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "(a) Orthoformer is competitive with Nyström. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/195efd29ff02fef81027cbbbbcc641fc4eb6a753d4e3f465635822f09105a1b6.jpg",
+        "table_caption": [
+            "(b) Selecting orthogonal prototypes is the best strategy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attention</td><td>Selection</td><td>Mem.</td><td>K-400</td><td>SSv2</td></tr><tr><td>Trajectory ( (E)</td><td>N/A</td><td>7.4</td><td>79.7</td><td>66.5</td></tr><tr><td>Trajectory (A)</td><td>Seg-Means</td><td>3.6</td><td>75.8</td><td>60.3</td></tr><tr><td></td><td>Random</td><td>3.6</td><td>76.5</td><td>62.5</td></tr><tr><td></td><td>Orthogonal</td><td>3.6</td><td>77.5</td><td>63.8</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/466d6a35ae3ee3b9ecd8948eab926b0f01df2bc35491e2fe999c81394259c313.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attention</td><td>Approx.</td><td>Mem. K-400</td><td>SSv2</td></tr><tr><td>Trajectory ( (E) N/A</td><td></td><td>7.4 79.7</td><td>66.5</td></tr><tr><td>Trajectory (A)</td><td>Performer</td><td>5.1 72.9</td><td>52.7</td></tr><tr><td></td><td>Nystromformer 3.8</td><td>77.5</td><td>64.0</td></tr><tr><td>Orthoformer</td><td>3.6</td><td>77.5</td><td>63.8</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f313a509618c233ee4b1641cd737de58f3a7532c0a6968391cefbfe5b1f60900.jpg",
+        "table_caption": [
+            "(c) Approximation improves with more prototypes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attention</td><td># Prototypes</td><td>Mem.</td><td>K-400</td><td>SSv2</td></tr><tr><td>Trajectory (E)</td><td>N/A</td><td>7.4</td><td>79.7</td><td>66.5</td></tr><tr><td rowspan=\"3\">Trajectory (A)</td><td>16</td><td>3.1</td><td>73.9</td><td>59.2</td></tr><tr><td>64</td><td>3.3</td><td>74.9</td><td>63.0</td></tr><tr><td>128</td><td>3.6</td><td>77.5</td><td>63.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/bdbe9934dcedba2a11312b7fd65e932b7187a9faf4b894892d2ee428624ef2ed.jpg",
+        "table_caption": [
+            "(d) Temporal sharing is the best strategy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attention</td><td>Sharing</td><td>Mem.</td><td>K-400</td><td>SSv2</td></tr><tr><td>Trajectory (E)</td><td>N/A</td><td>7.4</td><td>79.7</td><td>66.5</td></tr><tr><td>Trajectory (A)</td><td>×</td><td>16.5</td><td>77.3</td><td>61.5</td></tr><tr><td></td><td>√</td><td>3.6</td><td>77.5</td><td>63.8</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "improvements on the appearance cue-reliant K-400 dataset, our trajectory attention significantly outperforms $( + 2 \\% )$ the other approaches on the motion cue-reliant SSv2 dataset. This dataset requires fine-grained motion understanding, something explicitly singled out by previous video transformer works [3, 8] as a challenge for their models. In contrast, our trajectory attention excels on this dataset, indicating that its motion-based design is able to capture some of this information. ",
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+    {
+        "type": "text",
+        "text": "Attention block: trajectory attention design. We ablate two design choices for our trajectory attention: the per-frame softmax normalization and the 1D temporal attention. Unlike joint space-time attention, which normalizes the attention map over all tokens in space and time, trajectory attention normalizes independently per frame, allowing us to implicitly track the trajectories of query patches in time. In row 5 of Table 4, we ablate the benefits of this design choice. We observe a reduction of $2 . 5 \\%$ on K-400 and $5 . 6 \\%$ on SSv2 by normalizing over space and time $( \\mathrm { N o r m } _ { S T } )$ compared with normalizing over space alone $( \\mathrm { N o r m } _ { S } )$ ). In row 4, we show the benefit of using 1D temporal attention $( \\mathrm { A t t } _ { T } )$ to aggregate temporal features, compared to average pooling $( \\mathrm { A v g } _ { T } )$ . We observe reductions of $3 . 7 \\%$ on K-400 and $6 . 5 \\%$ on SSv2 when using average pooling instead of temporal attention applied to the motion trajectories, although it saves computing the additional query/key/value projections. ",
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+    {
+        "type": "text",
+        "text": "4.2 Orthoformer approximated attention ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Approximation comparisons. In Table 3a, we compare our Orthoformer algorithm to alternative strategies: Nyströmformer [85] and Performer [15]. Our algorithm performs comparably with Nyströmformer with a reduced memory footprint. In Table 5, we also compare these attention mechanisms on the Long Range Arena benchmark [69] to show applicability to other tasks and data types. Orthoformer is able to effectively approximate self-attention, outperforming the state-of-the-art despite using far fewer prototypes (64) and so gaining significant computational and memory benefits. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Prototype selection. A key part of our Orthoformer algorithm is the prototype selection procedure. In Table 3b, we ablate three prototype selection strategies: segment-means, random, and greedy most-orthogonal selection. Segment-means, the strategy used in Nyströmformer, performs poorly because it can generate multiple parallel prototypes, which will over-estimate the relative probability of query–key pairs near those redundant prototypes. In contrast, our proposed strategy of selecting the most orthogonal prototypes from the query and key set works the best across both datasets, because it explicitly minimises prototype redundancy with respect to direction. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6625c74115ea49d9e92936ea005e67cdae70bf0f70e7c9dde156f910ec79f197.jpg",
+        "table_caption": [
+            "Table 4: Attention ablations: We compare trajectory attention with alternatives and ablate its design choices. We report GFLOPS and top-1 accuracy $( \\% )$ on K-400 and SSv2. $\\mathbf { A t t } _ { T }$ : temporal attention, $\\mathbf { A v } \\mathbf { g } _ { T }$ : temporal averaging, $\\operatorname { N o r m } _ { S T }$ : space-time normalization, $\\operatorname { N o r m } _ { S }$ : spatial normalization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attention</td><td>AttT</td><td>AvgT</td><td>Norms NormsT</td><td>GFLOPS</td><td>K-400</td><td>SSv2</td></tr><tr><td>Joint Space-Time</td><td>1</td><td>1</td><td></td><td>180.6</td><td>79.2</td><td>64.0</td></tr><tr><td rowspan=\"3\">Divided Space-Time</td><td>1</td><td>1</td><td>1 1</td><td>185.8</td><td>78.5</td><td>64.2</td></tr><tr><td>X</td><td>√</td><td>√ X</td><td>180.6</td><td>76.0</td><td>60.0</td></tr><tr><td>7</td><td>×</td><td>× √</td><td>369.5</td><td>77.2</td><td>60.9</td></tr><tr><td>Trajectory</td><td>√</td><td>X</td><td>√</td><td>X 369.5</td><td>79.7</td><td>66.5</td></tr></table>",
+        "bbox": [
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+            140,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/f81724743f4f34c93e5e26ac7c61eeb25df07a2544d8b9b3666a72371e56a336.jpg",
+        "table_caption": [
+            "Table 5: Comparison to the state-of-the-art on Long Range Arena benchmark. GFLOPS and CUDA maximum Memory (MB) are reported for the ListOps task. Note that our algorithm achieves the best overall results with far fewer prototypes (64) than the other methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>ListOps</td><td>Text</td><td>Retrieval</td><td>Image</td><td>Pathfinder</td><td>Avg个</td><td>GFLOPS↓</td><td>Mem.↓</td></tr><tr><td>Exact [75]</td><td>36.69</td><td>63.09</td><td>78.22</td><td>31.47</td><td>66.35</td><td>55.16</td><td>1.21</td><td>4579</td></tr><tr><td>Performer-256 [15]</td><td>36.69</td><td>63.22</td><td>78.98</td><td>29.39</td><td>66.55</td><td>54.97</td><td>0.49</td><td>885</td></tr><tr><td>Nystromformer-128 [85]</td><td>36.90</td><td>64.17</td><td>78.67</td><td>36.16</td><td>52.32</td><td>53.64</td><td>0.62</td><td>745</td></tr><tr><td>Orthoformer-64</td><td>33.87</td><td>64.42</td><td>78.36</td><td>33.26</td><td>66.41</td><td>55.26</td><td>0.24</td><td>344</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Number of prototypes. In Table 3c, we show that Orthoformer improves monotonically as the number of prototypes is increased. In particular, we see an average performance improvement of $4 \\%$ on both datasets as we increase the number of prototypes from 16 to 128. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Temporally-shared prototypes. In Table 3d, we demonstrate the memory savings and performance benefits of sharing prototypes across time. On SSv2, we observe a $2 \\%$ improvement in performance and a $5 \\times$ decrease in memory usage. These gains may be attributed to the regularization effect of having prototypes leverage redundant information across frames. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Scaling transformer models with approximated trajectory attention. The Orthoformer attention approximation algorithm allows us to train larger models and higher resolution inputs for a given GPU memory budget. Here, we verify this, by training a large vision transformer model (ViT-L/16) [20] with a higher resolution input $3 3 6 \\times 3 3 6$ pixels) on the Kinetics-400 dataset, using the Orthoformer approximation with 196 temporally-shared prototypes and the same schedule as the base model. We use a fixed patch size (in pixels) for all models, and so the number of input tokens to the transformer scales with the square of the image resolution. As shown in Table 7, this model achieves a competitive accuracy without fine-tuning the training schedule, hyperparameters or data augmentation strategy. We expect that fine-tuning these on a validation set would greatly improve the model’s performance, based on results from contemporary work [3]. Obviously such a parameter sweep is more time-consuming for these large models, however, these preliminary results are indicative that higher accuracies are attainable if these parameters were to be optimized. ",
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+    {
+        "type": "text",
+        "text": "4.3 Comparison to the state-of-the-art ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Table 6, we compare our method against the current state-of-the-art on four common benchmarking datasets: Kinetics-400, Kinetics-600, Something–Something v2 and Epic-Kitchens. We find that our method performs favorably against current methods, even when compared against much larger models such as ViViT-L. In particular, it achieves strong top-1 accuracy improvements of $1 . 0 \\%$ and $2 . 3 \\%$ for SSv2 and Epic-Kitchen Nouns, respectively. These datasets require greater motion reasoning than Kinetics and so are a more challenging benchmark for video action recognition. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented a new general-purpose attention block for video data that aggregates information along implicitly determined motion trajectories, lending a realistic inductive bias to the model. We further address its quadratic dependence on the input size with a new attention approximation algorithm that significantly reduces the memory requirements, the largest bottleneck for transformer models. With these contributions, we obtain state-of-the-art results on several benchmark datasets. Nonetheless, our approach inherits many of the limitations of transformer models, including poor data efficiency and slow training. Specific to this work, trajectory attention has higher computational complexity than alternative attention operations used for video data. This is attenuated by the proposed approximation algorithm, with significantly reduced memory and computation requirements. However, its runtime is bottlenecked by prototype selection, which is not easily parallelized. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c0167e11a7df0a927d75a8e801cd1c27ce04698ec3fa3f73fa293c41a366e5b2.jpg",
+        "table_caption": [
+            "Table 6: Comparison to the state-of-the-art on video action recognition. We report GFLOPS and top-1 $( \\% ) ^ { 1 }$ and top-5 $( \\% )$ video action recognition accuracy on K-400/600, and SSv2. On Epic-Kitchens, we report top-1 $( \\% )$ action (A), verb (V), and noun (N) accuracy. ",
+            "(a) Something–Something V2 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Pretrain</td><td>Top-1</td><td>Top-5</td><td>GFLOPs Xviews</td></tr><tr><td>SlowFast [24]</td><td>K-400</td><td>61.7</td><td>-</td><td>65.7×3×1</td></tr><tr><td>TSM [47]</td><td>K-400</td><td>63.4</td><td>88.5</td><td>62.4×3×2</td></tr><tr><td>STM[33]</td><td>IN-1K</td><td>64.2</td><td>89.8</td><td>66.5×3×10</td></tr><tr><td>MSNet [41]</td><td>IN-1K</td><td>64.7</td><td>89.4</td><td>67×1×1</td></tr><tr><td>TEA [46]</td><td>IN-1K</td><td>65.1</td><td>=</td><td>70×3×10</td></tr><tr><td>bLVNet [22]</td><td>IN-1K</td><td>65.2</td><td>90.3</td><td>128.6×3×10</td></tr><tr><td>VidTr-L [45]</td><td>IN-21K+K-400</td><td>60.2</td><td>-</td><td>351×3×10</td></tr><tr><td>Tformer-L [8]</td><td>IN-21K</td><td>62.5</td><td>-</td><td>1703×3×1</td></tr><tr><td>ViViT-L [3]</td><td>IN-21K+K-400</td><td>65.4</td><td>89.8</td><td>3992×4×3</td></tr><tr><td>MViT-B [21]</td><td>K-400</td><td>67.1</td><td>90.8</td><td>170×3×1</td></tr><tr><td>Mformer</td><td>IN-21K+K-400</td><td>66.5</td><td>90.1</td><td>369.5×3×1</td></tr><tr><td>Mformer-L</td><td>IN-21K+K-400</td><td>68.1</td><td>91.2</td><td>1185.1×3×1</td></tr><tr><td>Mformer-HR</td><td>IN-21K+K-400</td><td>67.1</td><td>90.6</td><td>958.8×3×1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/f5c3dfe39e71e80476d01b25b71f46638a5d32296efba99bdbb9eb1b16cc36bb.jpg",
+        "table_caption": [
+            "(b) Kinetics-400 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Pretrain</td><td>Top-1</td><td>Top-5</td><td>GFLOPsXviews</td></tr><tr><td>I3D [11]</td><td>IN-1K</td><td>72.1</td><td>89.3</td><td>108×N/A</td></tr><tr><td>R(2+1)D[74]</td><td>-</td><td>72.0</td><td>90.0</td><td>152×5×23</td></tr><tr><td>S3D-G [84]</td><td>IN-1K</td><td>74.7</td><td>93.4</td><td>142.8×N/A</td></tr><tr><td>X3D-XL [23]</td><td>-</td><td>79.1</td><td>93.9</td><td>48.4×3×10</td></tr><tr><td>SlowFast [24]</td><td>-</td><td>79.8</td><td>93.9</td><td>234×3×10</td></tr><tr><td>VTN [50]</td><td>IN-21K</td><td>78.6</td><td>93.7</td><td>4218×1×1</td></tr><tr><td>VidTr-L [45]</td><td>IN-21K</td><td>79.1</td><td>93.9</td><td>392×3×10</td></tr><tr><td>Tformer-L[8]</td><td>IN-21K</td><td>80.7</td><td>94.7</td><td>2380×3×1</td></tr><tr><td>MViT-B [21]</td><td>=</td><td>81.2</td><td>95.1</td><td>455×3×3</td></tr><tr><td>ViViT-L[3]</td><td>IN-21K</td><td>81.3</td><td>94.7</td><td>3992×3×4</td></tr><tr><td>Mformer</td><td>IN-21K</td><td>79.7</td><td>94.2</td><td>369.5×3×10</td></tr><tr><td>Mformer-L</td><td>IN-21K</td><td>80.2</td><td>94.8</td><td>1185.1×3×10</td></tr><tr><td>Mformer-HR</td><td>IN-21K</td><td>81.1</td><td>95.2</td><td>958.8×3×10</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/e01c5d4a3924b7763f88a24186b8aa97b0a35ed3c545d8f88f03e4ef0135a7c5.jpg",
+        "table_caption": [
+            "(c) Epic-Kitchens "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Pretrain</td><td>A</td><td>V</td><td>N</td></tr><tr><td>TSN [77]</td><td>IN-1K</td><td>33.2</td><td>60.2</td><td>46.0</td></tr><tr><td>TRN[88]</td><td>IN-1K</td><td>35.3</td><td>65.9</td><td>45.4</td></tr><tr><td>TBN [37]</td><td>IN-1K</td><td>36.7</td><td>66.0</td><td>47.2</td></tr><tr><td>TSM [47]</td><td>IN-1K</td><td>38.3</td><td>67.9</td><td>49.0</td></tr><tr><td>SlowFast [24]</td><td>K-400</td><td>38.5</td><td>65.6</td><td>50.0</td></tr><tr><td>ViViT-L [3]</td><td>IN-21K+K-400</td><td>44.0</td><td>66.4</td><td>56.8</td></tr><tr><td>Mformer</td><td>IN-21K+K-400</td><td>43.1</td><td>66.7</td><td>56.5</td></tr><tr><td>Mformer-L</td><td>IN-21K+K-400</td><td>44.1</td><td>67.1</td><td>57.6</td></tr><tr><td>Mformer-HR</td><td>IN-21K+K-400</td><td>44.5</td><td>67.0</td><td>58.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/e985a12aa9421e8ac7c13e476074a6d24bc832ac77df2324ff1c4be923133866.jpg",
+        "table_caption": [
+            "(d) Kinetics-600 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Pretrain</td><td>Top-1</td><td>Top-5</td><td>GFLOPs Xviews</td></tr><tr><td>AttnNAS [80]</td><td>-</td><td>79.8</td><td>94.4</td><td>1</td></tr><tr><td>LGD-3D[55]</td><td>IN-1K</td><td>81.5</td><td>95.6</td><td></td></tr><tr><td>SlowFast [24]</td><td>1</td><td>81.8</td><td>95.1</td><td>234×3×10</td></tr><tr><td>X3D-XL [23]</td><td>-</td><td>81.9</td><td>95.5</td><td>48.4×3×10</td></tr><tr><td>Tformer-HR [8]</td><td>IN-21K</td><td>82.4</td><td>96.0</td><td>1703×3×1</td></tr><tr><td>ViViT-L [3]</td><td>IN-21K</td><td>83.0</td><td>95.7</td><td>3992×3×4</td></tr><tr><td>MViT-B-24 [21]</td><td>1</td><td>83.8</td><td>96.3</td><td>236×1×5</td></tr><tr><td>Mformer</td><td>IN-21K</td><td>81.6</td><td>95.6</td><td>369.5×3×10</td></tr><tr><td>Mformer-L</td><td>IN-21K</td><td>82.2</td><td>96.0</td><td>1185.1×3×10</td></tr><tr><td>Mformer-HR</td><td>IN-21K</td><td>82.7</td><td>96.1</td><td>958.8×3×10</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/27ff2718dda45cce938b83758795adfc75ccff8580a094f732e32e20e7f3c369.jpg",
+        "table_caption": [
+            "Table 7: Can we train larger models using approximated trajectory attention? We report top-1 and top-5 accuracy $( \\% )$ on the Kinetics-400 dataset of two variants of our Motionformer model: Motionformer-B and Motionformer-H. The former uses the base model with exact (E) trajectory attention, while the latter uses a much larger model (ViT-L) and a higher resolution input $( 3 3 6 \\times 3 3 6$ pixels) with approximate (A) trajectory attention, i.e., using Orthoformer. We reduce this to a linear relationship with the Orthoformer approximation, which allows us to fit the model on the GPU. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Base model</td><td>Params</td><td>Attention</td><td>Max memory (GB)</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Mformer-B</td><td>ViT-B/224</td><td>109.1M</td><td>Trajectory (E)</td><td>7.3</td><td>79.7</td><td>94.2</td></tr><tr><td>Mformer-H</td><td>ViT-L/336</td><td>381.9M</td><td>Trajectory (A)</td><td>22.2</td><td>80.0</td><td>94.5</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Future work. There are many applications of trajectory attention beyond video action classification, such as those tasks where temporal context is highly important. We see significant potential for using trajectory attention for tracking [28], temporal action localization [83, 81] and online action detection [86, 62], among other settings, and leave these as avenues for future work. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Potential negative societal impacts. One negative impact of this research is the significant environmental impact associated with training transformers, which are large and compute-expensive models. Compared to 3D-CNNs where the compute scales linearly with the sequence length, video transformers scale quadratically. To mitigate this, we proposed an approximation algorithm with linear complexity that greatly reduces the computational requirements. There is also potential for video action recognition models to be misused, such as for unauthorized surveillance. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            88,
+            553,
+            107
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We are grateful for support from the Rhodes Trust (M.P.), the European Research Council Starting Grant (IDIU 638009, D.C.), Qualcomm Innovation Fellowship (Y.A.), the Royal Academy of Engineering (RF201819/18/163, J.H.), and EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines & Systems (EP/L015897/1, M.P. and Y.A.). Funding for M.P. was received under his Oxford affiliation. We thank Bernie Huang, Dong Guo, Rose Kanjirathinkal, Gedas Bertasius, Mike Zheng Shou, Mathilde Caron, Hugo Touvron, Benjamin Lefaudeux, Haoqi Fan, and Geoffrey Zweig from Facebook AI for their help, support, and discussion around this project. We also thank Max Bain and Tengda Han from VGG for fruitful discussions. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            266,
+            106
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
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parse/train/pDgN3l3EAF/pDgN3l3EAF.md

@@ -0,0 +1,284 @@
+# Learning Collaborative Policies to Solve NP-hard Routing Problems
+
+Minsu Kim† Jinkyoo Park‡ Joungho Kim† Korea Advanced Institute of Science and Technology (KAIST) † School of Electrical Engineering, ‡ Dept. Industrial & Systems Engineering {min-su, jinkyoo.park, joungho}@kaist.ac.kr
+
+# Abstract
+
+Recently, deep reinforcement learning (DRL) frameworks have shown potential for solving NP-hard routing problems such as the traveling salesman problem (TSP) without problem-specific expert knowledge. Although DRL can be used to solve complex problems, DRL frameworks still struggle to compete with state-of-the-art heuristics showing a substantial performance gap. This paper proposes a novel hierarchical problem-solving strategy, termed learning collaborative policies (LCP), which can effectively find the near-optimum solution using two iterative DRL policies: the seeder and reviser. The seeder generates as diversified candidate solutions as possible (seeds) while being dedicated to exploring over the full combinatorial action space (i.e., sequence of assignment action). To this end, we train the seeder’s policy using a simple yet effective entropy regularization reward to encourage the seeder to find diverse solutions. On the other hand, the reviser modifies each candidate solution generated by the seeder; it partitions the full trajectory into sub-tours and simultaneously revises each sub-tour to minimize its traveling distance. Thus, the reviser is trained to improve the candidate solution’s quality, focusing on the reduced solution space (which is beneficial for exploitation). Extensive experiments demonstrate that the proposed two-policies collaboration scheme improves over single-policy DRL framework on various NP-hard routing problems, including TSP, prize collecting TSP (PCTSP), and capacitated vehicle routing problem (CVRP).
+
+# 1 Introduction
+
+Routing is a combinatorial optimization problem, one of the prominent fields in discrete mathematics and computational theory. Among routing problems, the traveling salesman problem (TSP) is a canonical example. TSP can be applied to real-world problems in various engineering fields, such as robot routing, biology, and electrical design automation (EDA) [1, 2, 3, 4, 5] by expanding constraints and objectives to real-world settings $:$ coined TSP variants are expanded version of TSP. However, TSP and its variants are NP-hard, making it challenging to design an exact solver [6].
+
+Due to NP-hardness, solvers of TSP-like problems rely on mixed-integer linear programming (MILP) solvers [7] and handcrafted heuristics [8, 9]. Although they often provide a remarkable performance on target problems, the conventional approaches have several limitations. Firstly, in the case of MILP solvers, the objective functions and constraints must be formulated into linear forms, but many real-world routing applications, including biology and EDA, have a non-linear objective. Secondly, handcrafted heuristics rely on expert knowledge on target problems, thus hard to solve other problems. That is, whenever the target problem changes, the algorithm must also be re-designed.
+
+Deep reinforcement learning (DRL)-routing frameworks [10, 11, 12] is proposed to tackle the limitation of conventional approaches. One of the benefits of DRL is that reward of DRL can be any value, even from a black-box simulator; therefore, DRL can overcome the limitations of MILP on real-world applications. Moreover, DRL frameworks can automatically design solvers relying less on a handcrafted manner.
+
+We note that the main objective of our research is not outperforming problem-specific solvers like the Concorde [9], a TSP solver. Our problem-solving strategy based on DRL, however, ultimately focuses on practical applications1 including intelligent transportation [13], biological sequence design [14], routing on electrical device [15] and device placement [16, 17]. Therefore, this paper evaluates the performance of DRL frameworks on TSP-like problems as a benchmark for potential applicability to practical applications, including speed, optimality, scalability, and expand-ability to other problems. TSP-like problems are excellent benchmarks as they have various baselines to compare with and can easily be modeled and evaluated.
+
+Contribution. This paper presents a novel DRL scheme, coined learning collaborative policies (LCP), a hierarchical solving protocol with two policies: seeder and reviser. The seeder generates various candidate solutions (seeds), each of which will be iteratively revised by the reviser to generate fine-tuned solutions.
+
+Having diversified candidate solutions is important, as it gives a better chance to find the best solution among them. Thus, the seeder is dedicated to exploring the full combinatorial action space (i.e., sequence of assignment action) so that it can provide as diversified candidate solutions as possible. It is important to explore over the full combinatorial action space because the solution quality highly fluctuates depending on its composition; however, exploring over the combinatorial action space is inherently difficult due to its inevitably many possible solutions. Therefore, this study provides an effective exploration strategy applying an entropy maximization scheme.
+
+The reviser modifies each candidate solution generated by the seeder. The reviser is dedicated to exploiting the policy (i.e., derived knowledge about the problem) to improve the quality of the candidate solution. The reviser partitions the full trajectory into sub-tours and revises each subtour to minimize its traveling distance in a parallel manner. This scheme provides two advantages: (a) searching over the restricted solution space can be more effective because the reward signal corresponding to the sub-tour is less variable than that of the full trajectory when using reinforcement learning to derive a policy, and (b) searching over sub-tours of seeds can be parallelized to expedite the revising process.
+
+The most significant advantage of our method is that the reviser can re-evaluate diversified but underrated candidates from the seeder without dropping it out early. Since the seeder explores the full trajectory, there may be a mistake in the local sub-trajectory. Thus, it is essential to correct such mistakes locally to improve the solution quality. The proposed revising scheme parallelizes revising process by decomposing the full solution and locally updating the decomposed solution. Thus it allows the revisers to search over larger solution space in a single inference than conventional local search (i.e., number of iteration of the reviser is smaller than that of conventional local search 2-opt [18], or DRL-based 2-opt [19]), consequently reducing computing costs. Therefore, we can keep the candidates without eliminating them early because of computing costs.
+
+The proposed method is an architecture-agnostic method, which can be applied to various neural architectures. The seeder and reviser can be parameterized with any neural architecture; this research utilizes AM [12], the representative DRL model on combinatorial optimization, to parameterize the seeder and the reviser. According to the experimental results, the LCP improves the target neural architecture AM [12], and outperforms competitive DRL frameworks on TSP, PCTSP, and CVRP $N = 2 0$ , 50, 100, 500, $N$ : number of nodes) and real-world problems in TSPLIB [20]. Moreover, by conducting extensive ablation studies, we show proposed techniques, including entropy regularization scheme and revision scheme, clearly contribute to the performance improvement.
+
+# 2 Related Works
+
+There have been continuous advances in DRL frameworks for solving various routing problems. DRL framework can generate solvers that do not rely on the ground-truth label of target problems: it can be applied to un-explored problems. DRL-based approaches can be categorized into two parts;
+
+constructive heuristics and improvement heuristics. We survey these two categories and current emerging hybrid approaches of machine learning (ML) with conventional solvers.
+
+# 2.1 DRL-based Constructive Heuristics
+
+Bello et al. [10] introduced an actor-critic algorithm with a policy parameterized by the pointer network [21]. They proposed a constructive Markov decision process (MDP), where the action is defined as choosing one of the un-served nodes to visit, given a partial solution; the policy is trained to add a node to provide a complete solution sequentially. Later, DRL-based constructive heuristics were developed to design the architecture of neural networks while preserving the constructive MDP [10]. Khalil et al. [11] proposed a DRL framework with a graph embedding structure. Nazari et al. [22], Duedon et al. [23] and Kool et al. [12] redesigned the pointer network [21] using the transformer [24] and trained it with a policy gradient method [25]. The AM by Kool et al. [12] reports substantial results on various NP-hard routing problems, including TSP, PCTSP, CVRP, and orienteering problem (OP) in high-speed computation.
+
+AM-variants. After the meaningful success of the AM, many studies are expanded from the AM. Many engineering fields and industries apply AM into their domain. For example, Liao et al. [4] proposed a routing algorithm for the circuit using AM.
+
+Some researches focus on increasing the performances of AM on classic routing problems like TSP by simple techniques. Kwon et al. [26] proposed the POMO, effective reinforcement learning method for AM. They proposed a new RL baseline that can reduce the training variance of AM using the problem-specific property of TSP and CVRP. In addition, they presented an effective post-processing algorithm for TSP and CVRP. However, their proposed method has a limitation in that it is problem-specific because it uses the domain properties of TSP and CVRP (e.g., their method is limited to be applied to PCTSP.).
+
+Xin et al. [27] proposed AM-style DRL-model, MDAM, for NP-hard routing problems. Their method learns multiple AM decoders and derives various solutions through the multiple decoders. The goal of increasing the solution diversity is similar to our research. However, our study is different where it increases the entropy of a single decoder and improves the mistakes of various solutions through a reviser.
+
+# 2.2 DRL-based Improvement Heuristics
+
+Unlike the constructive MDP, DRL-based improvement heuristics are designed to improve the completed solution iteratively. Most researches on DRL-based improvement heuristics are inspired by classical local search algorithms such as 2-opt [18] and the large neighborhood search (LNS) [28].
+
+Chen et al. [29] proposed a DRL-based local search framework, termed NeuRewriter, that shows a promising performance on CVRP and job scheduling problems. Wu et al. [30], and Costa et al. [31] proposed a DRL-based TSP solver by learning the 2-opt. Their method improves the randomly generated solutions, unlike the method of Chen et al. [29] rewrites a solution given by a conventional heuristic solver. Hottung & Tierney [32] proposed a novel search method of VRP that destroys and repairs a solution repeatably inspired LNS. Their method gives promising performances on CVRP.
+
+Improvement heuristic approaches generally show better performance than constructive heuristics but are usually slower than constructive heuristics. In the case of TSP, the number of neural network’s inferences of constructive heuristics is the same as the number of cities to visit. However, the number of inferences of the improvement heuristics is generally much larger.
+
+# 2.3 Hybrid Approaches with Conventional Solvers
+
+There are several studies on hybrid approaches with conventional solvers having promising performance recently. Lu et al. [33] proposed a hybrid method, where the policy is learned to control improvement operators (handcrafted heuristic). Significantly, they outperforms the LKH3, which is widely considered as mountain to climb in machine learning (ML) communities. Joshi et al. [34] combined graph neural network (GNN) model with the beam search algorithm. They trained the GNN with supervised learning for generating a hit map of candidate nodes. Then trained GNN reduces a searching space for improvement heuristics. Similarly, Fu et al. [35] combined supervised
+
+![](images/400cb6133a289a9a76af496becb9dc4a9ef96cd74feadad8f1053cb7896cb6f8.jpg)  
+Figure 1: Illustration of seeder-reviser collaboration for TSP.
+
+GNN model with Monte Carlo tree search (MCTS) and Kool et al. [36] combined supervised GNN model with dynamic programming. Their method achieves significant performances, showing ML method can effectively collaborate with conventional operational research (OR) methods.
+
+The research scope of hybrid approaches and DRL-based methods is different. Hybrid approaches can overcome classical solvers in target tasks by collaborating with the classical solvers. However, hybrid approaches have inherited limitations from classical solvers that are poor expandability to other tasks. The DRL-based method can be applied to various real-world tasks without a classic solver; we can also utilize DRL-based to unexplored tasks. This paper investigates the DRL-based NP-hard routing method without the help of classical solvers.
+
+# 3 Formulation of Routing Problems
+
+This section explains the Markov decision process (MDP) formulation for the given 2D Euclidean TSP as a representative example. The formulation of MDP for other problems is described in Appendix A.1.
+
+The main objective of TSP is to find the shortest path of the Hamiltonian cycle. The TSP graph can be represented as a sequence of $N$ nodes in 2D Euclidean space, $\pmb { s } = \{ x _ { i } \} _ { i = 1 } ^ { N }$ , where $x _ { i } \in \bar { \mathbb { R } } ^ { 2 }$ . Then, the solution of TSP can be represented as the permutation $\pi$ of input sequences:
+
+$$
+\pi = \bigcup _ { t = 1 } ^ { t = N } \{ \pi _ { t } \} , \quad \pi _ { t } \in \{ 1 , . . . , N \} , \quad \pi _ { t _ { 1 } } \neq \pi _ { t _ { 2 } } \quad \mathrm { i f } \quad t _ { 1 } \neq t _ { 2 }
+$$
+
+The objective is minimizing the tour length $\begin{array} { r } { L ( { \boldsymbol \pi } | s ) = \sum _ { t = 1 } ^ { N - 1 } | | x _ { { \pi } _ { t + 1 } } - x _ { { \pi } _ { t } } | | _ { 2 } + | | x _ { { \pi } _ { N } } - x _ { { \pi } _ { 1 } } | | _ { 2 } . } \end{array}$
+
+Then, we formulate the constructive Markov decision process (MDP) of TSP.
+
+State. State of MDP is represented as a partial solution of TSP or a sequence of previously selected actions: $\pi _ { 1 : t - 1 }$ .
+
+Action. Action is defined as selecting one of un-served tasks. Therefore, action is represented as $\pi _ { t }$ where the $\pi _ { t } \in \{ \{ 1 , . . . , N \} \setminus \{ \pi _ { 1 : t - 1 } \} \}$ .
+
+Cumulative Reward. We define cumulative reward for solution (a sequence of assignments) from problem instance $\pmb { s }$ as negative of tourlength: $- L ( \pi | s )$ .
+
+Constructive Policy. Finally we define constructive policy $p ( \pi | \boldsymbol { s } )$ that generates a solution $\pi$ from TSP graph $\pmb { s }$ . The constructive policy $p ( \pi | \boldsymbol { s } )$ is decomposed as:
+
+$$
+p ( \pmb { \pi } | \pmb { s } ) = \prod _ { t = 1 } ^ { t = N } p _ { \theta } ( \pi _ { t } | \pi _ { 1 : t - 1 } , \pmb { s } )
+$$
+
+Where $p _ { \theta } ( \pi _ { t } | \pi _ { 1 : t - 1 } , s )$ is a single-step assignment policy parameterized by parameter $\theta$
+
+# 4 Learning Collaborative Policies
+
+This section describes a novel hierarchical problem-solving strategy, termed learning collaborative policies (LCP), which can effectively find the near-optimum solution using two hierarchical steps, seeding process and revising process (see Figure 1 for detail). In the seeding process, the seeder policy $\bar { p } ^ { S }$ generates $M$ number of diversified candidate solutions. In the revising process, the reviser policy $\bar { p } ^ { R }$ re-writes each candidate solution $I$ times to minimize the tour length of the candidate. The final solution is then selected as the best solution among $M$ revised (updated) candidate solutions. See pseudo-code in Appendix A.4 for a detailed technical explanation.
+
+![](images/004755074b7c9789e87b31d0856e2763b47459d14b7592f4239a1adf510a6f61.jpg)  
+Figure 2: Illustration of the seeding process. The diversity of seeds is approximated as the sum of scaled entropy of segment policy. The scheduler $w _ { t }$ is for giving more weight when $t$ is in early step $t _ { 1 }$ than $t _ { 2 }$ . In the early step $t _ { 1 }$ , selecting $a$ , rather than $b$ is critical for the overall solution. In the latter step $t _ { 2 }$ , even though the probability of selecting node $c$ and $d$ is equal (i.e., has a high entropy), they may give similar solutions.
+
+# 4.1 Seeding Process
+
+The seeder generates as diversified candidate solutions as possible while being dedicated to exploring the full combinatorial action space. To this end, the seeder is trained to solve the following problems.
+
+Solution space. Solution space of seeder is a set of full trajectory solutions : $\{ \pi ^ { ( 1 ) } , . . . , \pi ^ { ( M ) } \}$ . The $M$ is the number of candidate solutions from the seeder: termed sample width.
+
+Policy structure. Seeder is a constructive policy, as defined in section 3 as follows:
+
+$$
+p ^ { S } ( \pi | s ) = \prod _ { t = 1 } ^ { t = N } p _ { \theta ^ { S } } ( \pi _ { t } | \pi _ { 1 : t - 1 } , s )
+$$
+
+The segment policy $p _ { \theta ^ { S } } ( \pi _ { t } | \pi _ { 1 : t - 1 } , s )$ , parameterized by $\theta ^ { S }$ , is derived form AM [12].
+
+Entropy Reward. To force the seeder policy $p ^ { S }$ to sample diverse solutions, we trained $p ^ { S }$ such that the entropy $\mathcal { H }$ of $p ^ { S }$ to be maximized. To this end, we use the reward $R ^ { S }$ defined as:
+
+$$
+R ^ { S } = \mathcal { H } \left( \pi \sim \prod _ { t = 1 } ^ { t = N } p _ { \theta ^ { S } } ( \pi _ { t } | \pi _ { 1 : t - 1 } , s ) \right) \approx \sum _ { t = 1 } ^ { N } w _ { t } \mathcal { H } \left( \pi _ { t } \sim p _ { \theta ^ { S } } ( \pi _ { t } | \pi _ { 1 : t - 1 } , s ) \right)
+$$
+
+The entropy of constructive policy is appropriate for measuring solution diversity. However, computing the entropy of constructive policy is intractable because search space is too large: $N !$ . Therefore, we approximate it as a weighted sum of the entropy of segment policies $p _ { \theta ^ { S } } ( \pi _ { t } | \pi _ { 1 : t - 1 } , s )$ evaluated at different time step.
+
+We use a linear scheduler (time-varying weights) $\begin{array} { r } { w _ { t } = \frac { N - t } { N _ { w } } } \end{array}$ to boost exploration at the earlier stage of composing a solution; higher randomness imposed by the higher weight $w _ { t }$ at the early stage tends to generate more diversified full trajectories later. The $N _ { w }$ is the normalizing factor, which is a hyperparameter.
+
+Training scheme. To train the seeder, we use the REINFORCE [25] algorithm with rollout baseline $b$ introduced by Kool et al. [12]. Then the gradient of each objective function is expressed as follows:
+
+$$
+\nabla J ( \theta ^ { S } | s ) = E _ { \pi \sim p ^ { S } } [ ( L ( \pi | s ) - \alpha R ^ { S } ( p _ { 1 : N } ^ { S } , \pi ) - b ( s ) ) \nabla l o g ( p ^ { S } ) ]
+$$
+
+![](images/b697812278b63fa7201acebc31f3b3b0043a31a364cd4824922f0f019a06fa8a.jpg)  
+Figure 3: Training progress of the reviser. From randomly generated TSP, sub-problems are decomposed. Then reviser infers to make constructive action referring starting and destination node and previous selected actions. The reward is a negative length of the partial tour generated from the reviser.
+
+Note that the $\alpha$ is hyperparameter for $R ^ { S }$ and $p _ { 1 : N } ^ { S }$ is the sequence of segment polices $\{ p _ { \theta ^ { S } } ( \pi _ { t } | \pi _ { 1 : t - 1 } , s ) \} _ { t = 1 } ^ { N }$ . We use the ADAM [37] optimizer to obtain the optimal parameter $\theta ^ { * }$ that minimizes the objective function.
+
+# 4.2 Revision Process
+
+In the revision process, given a candidate solution, the reviser decomposes the candidate solution into $K$ segments and simultaneously finds the optimum routing sequence for each segment of each candidate solution. The reviser repeats this revising process $I$ times to find the best-updated candidate solution. To be specific, the reviser sequentially updates candidate solutions ( $I$ times) by repeatably decomposing the full trajectories computed from the previous iteration into segments and revising the segments to produce $M$ updated full trajectory solutions. To sum up, reviser solves $M \times K$ segments in parallel $M$ : number of candidate solutions, $K$ : number of the segment in each candidate solution), $I$ times in series.
+
+The proposed scheme has advantages over conventional local search methods or DRL-based improvement heuristics. It searches larger solution spaces in a single inference; therefore, it reduces iteration $I$ . For example, 2-opt and DRL-2opt [19] search $O ( N ^ { 2 } )$ solution space (if it is parallelizable, $O ( M N ^ { 2 } ) )$ , while the reviser searches $O ( M K \times l ! )$ which is much larger (when the number of nodes of the segment $l$ is big enough) in a single inference. Hence we can reduce the number of iterations $I$ significantly compared to 2-opt, or DRL-2opt [19], thus expediting the speed of the solution search (see Appendix E).
+
+Solution space. Solution space of reviser is a partial segment of full trajectory solution represented as $\pi _ { k + 1 : k + l }$ . The $k$ is starting index, and $l$ is the number of nodes of the segment. For details of assigning segment including $k$ and $l$ , see Appendix A.3.
+
+Policy structure. Reviser is a constructive policy as follows:
+
+$$
+p ^ { R } ( \pi _ { k + 1 : k + l } | s ) = \prod _ { t = 1 } ^ { t = l } p _ { \theta ^ { R } } \bigl ( \pi _ { k + t } | \pi _ { k : k + t - 1 } , \pi _ { k + l + 1 } , s \bigr )
+$$
+
+The segment policy $p _ { \theta ^ { R } }$ , parameterized by $\theta ^ { R }$ , is in the similar form with that of AM [12]. Each $\pi _ { k }$ and $\pi _ { k + l + 1 }$ indicate the starting point and the destination point of the partial segment, respectively (see red-points in Figure 3).
+
+We modify the context embedding vector $h _ { ( c ) } ^ { ( N ) } = [ \bar { h } ^ { ( N ) } , h _ { \pi _ { t - 1 } } ^ { ( N ) } , h _ { \pi _ { 1 } } ^ { ( N ) } ]$ of AM, which is designed for solving TSP. Hence, $h$ is a high dimensional embedding vector from the transformer-based encoder, and $N$ is the number of multi-head attention layers. $\bar { h } ^ { ( N ) }$ is the mean of the entire embedding, $h _ { \pi _ { t - 1 } } ^ { ( N ) }$ is the embedding of previously selected nodes, and $h _ { \pi _ { 1 } } ^ { ( N ) }$ is the embedding of the first node. However, since the destination of reviser is , not the first node , we change the embedding of the first node $h _ { \pi _ { 1 } } ^ { ( N ) }$ to be the embedding of the last node $h _ { \pi _ { k + l + 1 } } ^ { ( N ) }$ for the context embedding as $h _ { ( c ) } ^ { ( N ) } = [ \bar { h } ^ { ( N ) } , h _ { \pi _ { k + t - 1 } } ^ { ( N ) } , h _ { \pi _ { k + l + 1 } } ^ { ( N ) } ]$ .
+
+Revision Reward: negative of partial tour length $\begin{array} { r } { L ^ { R } ( \pi _ { k + 1 : k + l } | s ) = \sum _ { t = 1 } ^ { l + 1 } | | x _ { \pi _ { k + t } } - x _ { \pi _ { k + t - 1 } } | | _ { 2 } . } \end{array}$
+
+Training scheme. The training process is mostly the same as described in section 4.1, except that we have modified the length term $L$ to $L ^ { R }$ , and set $\alpha = 0$ to remove entropy reward $R ^ { S }$ for training the reviser. Note that the seeder and reviser are trained separately.
+
+# 5 Experiments
+
+This section reports the experimental results2 of the LCP scheme on TSP, PCTSP, and CVRP $N = 2 0$ , 50, 100, 500, $N$ : number of nodes). Also, we report several ablation studies in section 5.3 and Appendix B-F. We evaluate performance on real-world TSPs in the TSPLIB in Appendix G.
+
+Training Hyperparamters. Throughout the entire training process of the seeder and reviser, we have exactly the same hyperparameters as Kool et al. [12], except that the training batch size of our seeder is 1024. To train the seeder’s policy, we set $\alpha = 0 . 5$ (2) and $\begin{array} { r } { N _ { w } = \sum _ { i = 1 } ^ { N } \bar { i } } \end{array}$ for linear weight $\begin{array} { r } { w _ { t } = \frac { N - t } { N _ { w } } } \end{array}$ N−tN for entropy scheduling.
+
+Details in the experimental setting, including hyperparameters, dataset configuration, and run time evaluation, are described in Appendix A.5.
+
+# 5.1 Target Problems and Baselines
+
+We evaluate the performance of LCP in solving the three routing problems: TSP, PCTSP, and CVRP. We provide a brief explanation of them. The detailed descriptions for these problems are in Appendix A.1.
+
+Travelling salesman problem (TSP). TSP is a problem to find the shortest Hamiltonian cycle given node sequences.
+
+Price collecting travelling salesman problem (PCTSP). PCTSP [38] is a problem, where each node has a prize and a penalty. The goal is to collect the nodes with at least a minimum total prize (constraint) and minimize tour length added with unvisited nodes’ penalties.
+
+Capacitated vehicle routing problem (CVRP). CVRP [39] is a problem where each node has a demand, while a vehicle must terminate the tour when the total demand limit is exceeded (constraint). The objective is to minimize the tour length.
+
+For the baseline algorithms, we use two types of algorithms: conventional heuristics and DRL-based solvers. For the conventional heuristics, we use Gurobi [7] (the commercial optimization solver), and the OR Tools [40] (the commercial optimization solver) for all three problems. In Table 1, Gurobi (t) indicates time-limited Gurobi whose running time is restricted below $t$ . In addition, OR Tools (t) is the OR Tools that allows additional local search over a duration of $t$ . For problem-specific heuristics, we use Concorde [9] for TSP, the iterative local search (ILS) [12] for PCTSP, and LKH3 [41] for CVRP.
+
+For the baselines using DRL-based solvers, we concentrated on the ability of the LCP scheme, which is improved performance over AM. Validating that the two-policies collaboration scheme outperforms the single-policy scheme (i.e., AM) is a crucial part of this research; thus, the most important metric for performance evaluation is improvement between vanilla AM the $\mathbf { A M } + \mathbf { L C P } .$ Also, we reproduced other competitive DRL frameworks: current emerging improvement heuristics. We exclude recently proposed AM-style constructive heuristics, including the POMO [26] and MDAM [27] because they can be candidate collaborators with LCP, not competitors (e.g., $\mathrm { P O M O } + \mathrm { L C P }$ is possible). The detailed method for evaluation baselines in Table 1 is described as follows:
+
+TSP. We follow baseline setting of Kool et al. [12] and Costa et al. [19]. We set DRL baselines including the S2V-DQN [11], EAN [23], GAT-T [30], DRL-2opt [19], and AM [12]. We show the results of S2V-DQN and EAN reported by Kool et al. [12], and the results of GAT-T reported by Costa et al. [19]. Then we directly reproduce the two most competitive DRL frameworks among baselines, the AM and DRL-2opt, in our machine to make a fair comparison of the speed.
+
+PCTSP. We follow baseline setting of Kool et al. [12]. We reproduce AM [12] for DRL baseline.
+
+Table 1: Performance evaluation results of the LCP scheme compared with baseline heuristics and the DRL frameworks on TSP, PCTSP, and CVRP. The best costs (objective) among DRL frameworks are marked in bold. The $H$ is heuristic and Solver is exact algorithm. The “LCP $\{ 6 4 0 , 2 \} ^ { \prime }$ means that the sampling width $M$ is 640, and the number of iterations $I$ (of the reviser) is 2. We measure the performance in a limited time budget, which is 10 seconds per instance. The $O B$ means “out of budget” and $I F$ means “infeasible”, where the solver cannot generate solutions satisfying constraints of target problems in a limited time.   
+
+<table><tr><td colspan="2">Method</td><td colspan="3">N=20</td><td colspan="3">N= 50</td><td colspan="3">N= 100</td></tr><tr><td rowspan="5"></td><td>Gurobi (Solver)</td><td>Cost 3.84</td><td>Gap 0.00%</td><td>Time 0.01s</td><td>Cost 5.70</td><td>Gap 0.00% 1.76%</td><td>Time 0.01s</td><td>Cost 7.76 8.12</td><td>Gap 0.00% 4.53%</td><td>Time 0.02s</td></tr><tr><td>OR Tools (H) Concorde (H) S2V-DQN EAN{M:1280} EAN+2OPT{M:1280}</td><td>3.85 3.84 3.89 3.84 3.84</td><td>0.37% 0.00% 1.42% 0.11% 0.09%</td><td>0.01s</td><td>5.80 5.70 5.99 5.77 5.75</td><td>0.00% 5.16% 1.28% 1.00%</td><td>0.01s</td><td>7.76 8.31 8.75 8.12</td><td>0.00% 7.03% 12.7% 4.64%</td><td>0.02s</td></tr><tr><td>GAT-T{I:5000} Drl-2opt{I:2000} AM{M:1280} AM{M:2560} AM{M:7500} AM+LCP {640,10}</td><td>3.84 3.84 3.84 3.84 3.84 3.84</td><td>0.00% 0.00% 0.08% 0.06% 0.05% 0.00%</td><td>3.58s 0.03s 0.04s 0.06s 0.18s</td><td>5.71 5.70 5.73 5.72 5.72 5.70</td><td>0.20% 0.12% 0.52% 0.45% 0.39% 0.13%</td><td>4.88s 0.09s 0.13s 0.29s 0.30s</td><td>7.87 7.83 7.94 7.94 7.93 7.86</td><td>1.42% 0.87% 2.26% 2.21% 2.13% 1.25%</td><td>7.15s 0.36s 0.42s 1.21s</td></tr><tr><td>AM+LCP{1280,10} AM+LCP*{1280,45} Gurobi (Solver) Gurobi {1s} (H)</td><td>3.13 3.14</td><td>= 1 0.00% 0.07%</td><td>0.01s 0.01s</td><td>5.70 5.70</td><td>0.10% 0.02% OB IF</td><td>0.45s 2.48s</td><td>7.85 7.81</td><td>1.13% 0.54% OB IF</td><td>0.57s 0.90s 4.30s</td></tr><tr><td>Gurobi {10s}(H) OR Tools{10s} (H) PSSP OR Tools {60s} (H) ILS (H) AM{M:1280}</td><td></td><td>3.13 3.14 3.13 3.16</td><td>0.00% 0.05% 0.01% 0.77%</td><td>0.01s 0.31s 4.51 1.80s 4.48 0.10s 4.50</td><td>5.17 0.70% 0.00% 0.67%</td><td>15.6%</td><td>0.19s 0.31s 1.80s 0.72s</td><td>6.35 6.08 5.98</td><td>IF 6.21% 0.31s 1.56% 1.80s 0.00% 4.32s</td></tr><tr><td>AM{M:2560} AM+LCP {640,1} AM+LCP {1280,5}</td><td></td><td>3.18 3.15 3.14</td><td>0.39% 0.41% 0.17%</td><td>0.03s 0.03s 0.04s</td><td>4.52 4.51 4.50</td><td>0.74% 0.72% 0.51%</td><td>0.07s 0.10s 0.07s</td><td>6.08 6.07 6.06</td><td>1.67% 1.57% 1.42%</td><td>0.17s 0.28s 0.15s</td></tr><tr><td></td><td>Gurobi (Solver) OR Tools (H) LKH3 (H)</td><td>3.14 6.10 6.43 6.14</td><td>0.08% 0.00% 5.41% 0.58%</td><td>0.10s 0.01s</td><td>4.49 11.31</td><td>0.32% OB 9.01%</td><td>0.20s</td><td>6.04 17.16</td><td>1.00% OB 9.67%</td><td>0.39s</td></tr><tr><td>JRPP</td><td>RL{M: 10} NLNS{I: 2000} AM{M:1280} AM{M:2560}</td><td>6.40 6.19 6.25 6.25</td><td>4.92% 1.47% 2.49% 2.39%</td><td>0.72s 1.00s 0.05s 0.06s</td><td>10.38 11.15 10.54 10.62 10.61</td><td>0.00% 7.46% 1.54% 2.40% 2.24%</td><td>2.52s 1.63s 0.14s 0.31s</td><td>15.65 16.96 16.00 16.23 16.17</td><td>0.00% 8.39% 2.17% 3.72% 3.34%</td><td>4.68s 2.18s 0.34s 0.75s</td></tr><tr><td></td><td>AM{M:7500} AM +LCP {640,1} AM+LCP{1280,1} AM+LCP{2560,1} AM+LCP{6500,1}</td><td>6.24 6.17 6.16 6.15</td><td>2.24% 1.15% 0.92% 0.84% -</td><td>0.09s 0.07s 0.09s 0.14s</td><td>10.59 10.56 10.54 10.52</td><td>2.06% 1.74% 1.54% 1.38% -</td><td>0.36s 0.15s 0.20s 0.31s</td><td>16.14 16.05 16.03 16.00 15.98</td><td>3.11% 2.58% 2.43% 2.24% 2.11%</td><td>1.42s 0.30s 0.45s 0.77s 1.73s</td></tr></table>
+
+CVRP. We follow baseline setting of Houttung & Tierney [32]. We report result of RL [22] based on Houttung & Tierney [32] and we reproduce AM [12] and NLNS [32].
+
+# 5.2 Performance Evaluation
+
+In this section, we report the performance of LCP on small-scale problems $\mathrm { \Delta } N = 2 0 , 5 0 , 1 0 0 )$ in Table 1. Then we provide a time-performance trade-off analysis including large-scale problems $N = 5 0 0$ ). We note that time-performance analysis is significant because any method can find an optimal solution when given an infinite time budget. From the analysis, we can identify a specific time region, called winner region, where LCP performs the best in terms of both speed and performance.
+
+Performance evaluation on $N = 2 0$ , 50, 100. Our method outperforms all the DRL baselines and OR-tools in TSP, PCTSP, and CVRP, as clearly shown in Table 1.
+
+![](images/64ff80900ff68cb05517c178513a00d202b3afdc2c448b31410d4263f3ac1486.jpg)  
+Figure 4: Details of time-performance analysis. The left (fast) and lower (lower cost) trend indicates a better performance. In (e), the ILS cannot generate solution in time budget $\mathit { \Omega } ^ { \prime } t < 1 0 0 0 s ,$ )
+
+Table 2: Ablation study of LCP components on TSP, PCTSP, and CVRP $N = 1 0 0$ ). The optimal gap is measured by comparing it with state-of-the-art solvers. The Entropy Regularization indicates training the seeder with $\bar { R } ^ { S }$ , while the default is the uniform scheduling. The best performances are marked in bold.   
+
+<table><tr><td colspan="3">Component of the LCP</td><td colspan="2">TSP</td><td colspan="2">PCTSP</td><td colspan="2">CVRP</td></tr><tr><td>Entropy Regularization</td><td>Weight Scheduling</td><td>Reviser</td><td>cost</td><td>gap</td><td>cost</td><td>gap</td><td>cost</td><td>gap</td></tr><tr><td></td><td></td><td></td><td>7.96</td><td>2.65%</td><td>6.08</td><td>1.64%</td><td>16.29</td><td>3.43%</td></tr><tr><td>V</td><td rowspan="3">√</td><td></td><td>7.96</td><td>2.68%</td><td>6.08</td><td>1.76%</td><td>16.25</td><td>3.16%</td></tr><tr><td></td><td></td><td>7.94</td><td>2.45%</td><td>6.07</td><td>1.62%</td><td>16.20</td><td>2.86%</td></tr><tr><td></td><td>√</td><td>7.86</td><td>1.32%</td><td>6.04</td><td>1.13%</td><td>16.20</td><td>2.86%</td></tr><tr><td>·</td><td></td><td></td><td>7.84</td><td>1.17%</td><td>6.05</td><td>1.16%</td><td>16.16</td><td>2.59%</td></tr><tr><td></td><td>√</td><td>1</td><td>7.82</td><td>0.88%</td><td>6.04</td><td>1.02%</td><td>16.12</td><td>2.37%</td></tr></table>
+
+Note that for TSP $N = 1 0 0 $ ), we applied two types of revisers, each of which is denoted LCP and $\mathrm { L C P ^ { * } }$ , respectively. The details are described in Appendix A.4 with pseudo-code. Our LCP and $\mathrm { L C P ^ { * } }$ outperforms DRL-2opt, the current state-of-the-art DRL-based improvement heuristic in $N = 2 0 , 5 0 , 1 0 0$ , surpass $0 . 3 3 \%$ in $N = 1 0 0$ .
+
+In PCTSP, LCP outperforms AM with less time. Our method $( \mathbf { A M } + \mathbf { L C P }$ {640,1}) outperforms the OR-Tools (10s), with $4 \times$ and $2 \times$ faster speed in $N = 5 0$ , 100 respectively. Compared to the ILS, our method $\mathrm { \Delta A M + L C P }$ {1280,5}) underperforms by $1 . 0 \%$ , but has $1 1 \ \times$ faster speed for $N = 1 0 0$
+
+For CVRP, our method outperforms competitive DRL frameworks.
+
+Time-performance analysis on ${ \bf { \mathit { N } } } = { \bf { 1 0 0 } }$ , 500. In Figure 4, we describe the time-performance analysis. We cannot control the speed of the Concorde, ILS, and LKH3. We can control the speed of DRL solvers by adjusting sample width $M$ or the number of iterations $I$ . For PCTSP, we can change the speed of OR-tools by managing the time for additional local searches.
+
+Our scheme clearly outperforms DRL-solvers in terms of both speed and performance. For PCTSP $\mathrm { \Delta } N = 1 0 0 , 5 0 0 { \mathrm { \Omega } }$ ) and CVRP $N = 5 0 0$ ), our method achieves the winner region of $t < 1 0$ , which is best performed in a specific time region among all kind of baseline solvers (for CVRP $N = 1 0 0$ ), our method achieves the winner region of $t < 5$ ).
+
+Performance on TSPLIB [20] data: see Appendix G.
+
+# 5.3 Ablation Study
+
+In this section, we conduct an ablation study on LCP components. We leave further ablation studies to Appendix B-F.
+
+Ablation study of collaborative policies. In Table 2, we ablate three significant components of LCP and show the experimental results for every case. In the case of vanilla AM, having none of LCP components, the performance is the poorest. On the other hand, collaboration of seeder trained with linearly scheduled-entropy and the reviser shows the best performance. Therefore, the experimental results empirically validate our proposal of hierarchically collaborating two policies and also demonstrate the effectiveness of using a linearly scheduled-entropy term shown in section 4.1 and Figure 2.
+
+Ablation study of entropy regularization: see Appendix B.
+
+Ablation study of SoftMax temperature: see Appendix C.
+
+Ablation study of application of LCP to pointer network [10, 21]: see Appendix D.
+
+Comparison with reviser and other improvement heuristics: see Appendix E.
+
+Training convergence of seeder and reviser in different PyTorch seeds: see Appendix F.
+
+# 6 Discussion
+
+In this paper, we proposed a novel DRL scheme, learning collaborative policies (LCP). The extensive experiments demonstrate that our two-policies collaboration algorithm (i.e., LCP) outperforms conventional single-policy DRL frameworks, including AM [12], on various NP-hard routing problems, such as TSP, PCTSP, and CVRP.
+
+We highlight that LCP is a reusable scheme, can solve various problems. The neural architecture of the seeder and reviser proposed in this paper is derived from AM [12]. It can be substituted by other architectures, such as the pointer network [10, 21] and AM-style architectures including POMO [26] and MDAM [27]. If further studies on neural architecture for combinatorial optimization are carried out, the seeder and reviser can be improved further.
+
+Also, LCP can be directly applied to other combinatorial optimization tasks, including TSP with time windows (TSPTW), orienteering problem (OP), multiple TSP (mTSP), variations of the vehicle routing problem (VRP), and other practical applications.
+
+Further Works. We made an important first step: two-policies collaboration where each policy specializing in exploration or exploitation can improve conventional single-policy systems on combinatorial optimization tasks. The important direction of further research is introducing more sophisticated strategies to explore or exploit combinatorial solution space. New exploration strategies for overcoming the proposed approximated entropy maximization scheme are needed. Also, it is necessary to investigate more effective exploitation strategies beyond the proposed revision scheme.
+
+# Acknowledgements and Disclosure of Funding
+
+This research is supported in part by the KAIST undergraduates research program (URP), 2019. We thank Hankook Lee and Prof. Jinwoo Shin for building part of this project in the URP. We thank Joonsang Park, Keeyoung Son, Hyunwook Park, Haeyeon Rachel Kim, and our anonymous reviewers for feedback and discussions.
+
+References   
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+        "text": "Learning Collaborative Policies to Solve NP-hard Routing Problems ",
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+        "text": "Minsu Kim† Jinkyoo Park‡ Joungho Kim† Korea Advanced Institute of Science and Technology (KAIST) † School of Electrical Engineering, ‡ Dept. Industrial & Systems Engineering {min-su, jinkyoo.park, joungho}@kaist.ac.kr ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Recently, deep reinforcement learning (DRL) frameworks have shown potential for solving NP-hard routing problems such as the traveling salesman problem (TSP) without problem-specific expert knowledge. Although DRL can be used to solve complex problems, DRL frameworks still struggle to compete with state-of-the-art heuristics showing a substantial performance gap. This paper proposes a novel hierarchical problem-solving strategy, termed learning collaborative policies (LCP), which can effectively find the near-optimum solution using two iterative DRL policies: the seeder and reviser. The seeder generates as diversified candidate solutions as possible (seeds) while being dedicated to exploring over the full combinatorial action space (i.e., sequence of assignment action). To this end, we train the seeder’s policy using a simple yet effective entropy regularization reward to encourage the seeder to find diverse solutions. On the other hand, the reviser modifies each candidate solution generated by the seeder; it partitions the full trajectory into sub-tours and simultaneously revises each sub-tour to minimize its traveling distance. Thus, the reviser is trained to improve the candidate solution’s quality, focusing on the reduced solution space (which is beneficial for exploitation). Extensive experiments demonstrate that the proposed two-policies collaboration scheme improves over single-policy DRL framework on various NP-hard routing problems, including TSP, prize collecting TSP (PCTSP), and capacitated vehicle routing problem (CVRP). ",
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+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Routing is a combinatorial optimization problem, one of the prominent fields in discrete mathematics and computational theory. Among routing problems, the traveling salesman problem (TSP) is a canonical example. TSP can be applied to real-world problems in various engineering fields, such as robot routing, biology, and electrical design automation (EDA) [1, 2, 3, 4, 5] by expanding constraints and objectives to real-world settings $:$ coined TSP variants are expanded version of TSP. However, TSP and its variants are NP-hard, making it challenging to design an exact solver [6]. ",
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+        "text": "Due to NP-hardness, solvers of TSP-like problems rely on mixed-integer linear programming (MILP) solvers [7] and handcrafted heuristics [8, 9]. Although they often provide a remarkable performance on target problems, the conventional approaches have several limitations. Firstly, in the case of MILP solvers, the objective functions and constraints must be formulated into linear forms, but many real-world routing applications, including biology and EDA, have a non-linear objective. Secondly, handcrafted heuristics rely on expert knowledge on target problems, thus hard to solve other problems. That is, whenever the target problem changes, the algorithm must also be re-designed. ",
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+        "text": "Deep reinforcement learning (DRL)-routing frameworks [10, 11, 12] is proposed to tackle the limitation of conventional approaches. One of the benefits of DRL is that reward of DRL can be any value, even from a black-box simulator; therefore, DRL can overcome the limitations of MILP on real-world applications. Moreover, DRL frameworks can automatically design solvers relying less on a handcrafted manner. ",
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+        "text": "We note that the main objective of our research is not outperforming problem-specific solvers like the Concorde [9], a TSP solver. Our problem-solving strategy based on DRL, however, ultimately focuses on practical applications1 including intelligent transportation [13], biological sequence design [14], routing on electrical device [15] and device placement [16, 17]. Therefore, this paper evaluates the performance of DRL frameworks on TSP-like problems as a benchmark for potential applicability to practical applications, including speed, optimality, scalability, and expand-ability to other problems. TSP-like problems are excellent benchmarks as they have various baselines to compare with and can easily be modeled and evaluated. ",
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+        "type": "text",
+        "text": "Contribution. This paper presents a novel DRL scheme, coined learning collaborative policies (LCP), a hierarchical solving protocol with two policies: seeder and reviser. The seeder generates various candidate solutions (seeds), each of which will be iteratively revised by the reviser to generate fine-tuned solutions. ",
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+    {
+        "type": "text",
+        "text": "Having diversified candidate solutions is important, as it gives a better chance to find the best solution among them. Thus, the seeder is dedicated to exploring the full combinatorial action space (i.e., sequence of assignment action) so that it can provide as diversified candidate solutions as possible. It is important to explore over the full combinatorial action space because the solution quality highly fluctuates depending on its composition; however, exploring over the combinatorial action space is inherently difficult due to its inevitably many possible solutions. Therefore, this study provides an effective exploration strategy applying an entropy maximization scheme. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The reviser modifies each candidate solution generated by the seeder. The reviser is dedicated to exploiting the policy (i.e., derived knowledge about the problem) to improve the quality of the candidate solution. The reviser partitions the full trajectory into sub-tours and revises each subtour to minimize its traveling distance in a parallel manner. This scheme provides two advantages: (a) searching over the restricted solution space can be more effective because the reward signal corresponding to the sub-tour is less variable than that of the full trajectory when using reinforcement learning to derive a policy, and (b) searching over sub-tours of seeds can be parallelized to expedite the revising process. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The most significant advantage of our method is that the reviser can re-evaluate diversified but underrated candidates from the seeder without dropping it out early. Since the seeder explores the full trajectory, there may be a mistake in the local sub-trajectory. Thus, it is essential to correct such mistakes locally to improve the solution quality. The proposed revising scheme parallelizes revising process by decomposing the full solution and locally updating the decomposed solution. Thus it allows the revisers to search over larger solution space in a single inference than conventional local search (i.e., number of iteration of the reviser is smaller than that of conventional local search 2-opt [18], or DRL-based 2-opt [19]), consequently reducing computing costs. Therefore, we can keep the candidates without eliminating them early because of computing costs. ",
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+    {
+        "type": "text",
+        "text": "The proposed method is an architecture-agnostic method, which can be applied to various neural architectures. The seeder and reviser can be parameterized with any neural architecture; this research utilizes AM [12], the representative DRL model on combinatorial optimization, to parameterize the seeder and the reviser. According to the experimental results, the LCP improves the target neural architecture AM [12], and outperforms competitive DRL frameworks on TSP, PCTSP, and CVRP $N = 2 0$ , 50, 100, 500, $N$ : number of nodes) and real-world problems in TSPLIB [20]. Moreover, by conducting extensive ablation studies, we show proposed techniques, including entropy regularization scheme and revision scheme, clearly contribute to the performance improvement. ",
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+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
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+        "text": "There have been continuous advances in DRL frameworks for solving various routing problems. DRL framework can generate solvers that do not rely on the ground-truth label of target problems: it can be applied to un-explored problems. DRL-based approaches can be categorized into two parts; ",
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+    {
+        "type": "text",
+        "text": "constructive heuristics and improvement heuristics. We survey these two categories and current emerging hybrid approaches of machine learning (ML) with conventional solvers. ",
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+    {
+        "type": "text",
+        "text": "2.1 DRL-based Constructive Heuristics ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Bello et al. [10] introduced an actor-critic algorithm with a policy parameterized by the pointer network [21]. They proposed a constructive Markov decision process (MDP), where the action is defined as choosing one of the un-served nodes to visit, given a partial solution; the policy is trained to add a node to provide a complete solution sequentially. Later, DRL-based constructive heuristics were developed to design the architecture of neural networks while preserving the constructive MDP [10]. Khalil et al. [11] proposed a DRL framework with a graph embedding structure. Nazari et al. [22], Duedon et al. [23] and Kool et al. [12] redesigned the pointer network [21] using the transformer [24] and trained it with a policy gradient method [25]. The AM by Kool et al. [12] reports substantial results on various NP-hard routing problems, including TSP, PCTSP, CVRP, and orienteering problem (OP) in high-speed computation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "AM-variants. After the meaningful success of the AM, many studies are expanded from the AM. Many engineering fields and industries apply AM into their domain. For example, Liao et al. [4] proposed a routing algorithm for the circuit using AM. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Some researches focus on increasing the performances of AM on classic routing problems like TSP by simple techniques. Kwon et al. [26] proposed the POMO, effective reinforcement learning method for AM. They proposed a new RL baseline that can reduce the training variance of AM using the problem-specific property of TSP and CVRP. In addition, they presented an effective post-processing algorithm for TSP and CVRP. However, their proposed method has a limitation in that it is problem-specific because it uses the domain properties of TSP and CVRP (e.g., their method is limited to be applied to PCTSP.). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Xin et al. [27] proposed AM-style DRL-model, MDAM, for NP-hard routing problems. Their method learns multiple AM decoders and derives various solutions through the multiple decoders. The goal of increasing the solution diversity is similar to our research. However, our study is different where it increases the entropy of a single decoder and improves the mistakes of various solutions through a reviser. ",
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+    {
+        "type": "text",
+        "text": "2.2 DRL-based Improvement Heuristics ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unlike the constructive MDP, DRL-based improvement heuristics are designed to improve the completed solution iteratively. Most researches on DRL-based improvement heuristics are inspired by classical local search algorithms such as 2-opt [18] and the large neighborhood search (LNS) [28]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Chen et al. [29] proposed a DRL-based local search framework, termed NeuRewriter, that shows a promising performance on CVRP and job scheduling problems. Wu et al. [30], and Costa et al. [31] proposed a DRL-based TSP solver by learning the 2-opt. Their method improves the randomly generated solutions, unlike the method of Chen et al. [29] rewrites a solution given by a conventional heuristic solver. Hottung & Tierney [32] proposed a novel search method of VRP that destroys and repairs a solution repeatably inspired LNS. Their method gives promising performances on CVRP. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Improvement heuristic approaches generally show better performance than constructive heuristics but are usually slower than constructive heuristics. In the case of TSP, the number of neural network’s inferences of constructive heuristics is the same as the number of cities to visit. However, the number of inferences of the improvement heuristics is generally much larger. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 Hybrid Approaches with Conventional Solvers ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are several studies on hybrid approaches with conventional solvers having promising performance recently. Lu et al. [33] proposed a hybrid method, where the policy is learned to control improvement operators (handcrafted heuristic). Significantly, they outperforms the LKH3, which is widely considered as mountain to climb in machine learning (ML) communities. Joshi et al. [34] combined graph neural network (GNN) model with the beam search algorithm. They trained the GNN with supervised learning for generating a hit map of candidate nodes. Then trained GNN reduces a searching space for improvement heuristics. Similarly, Fu et al. [35] combined supervised ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/400cb6133a289a9a76af496becb9dc4a9ef96cd74feadad8f1053cb7896cb6f8.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of seeder-reviser collaboration for TSP. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "GNN model with Monte Carlo tree search (MCTS) and Kool et al. [36] combined supervised GNN model with dynamic programming. Their method achieves significant performances, showing ML method can effectively collaborate with conventional operational research (OR) methods. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The research scope of hybrid approaches and DRL-based methods is different. Hybrid approaches can overcome classical solvers in target tasks by collaborating with the classical solvers. However, hybrid approaches have inherited limitations from classical solvers that are poor expandability to other tasks. The DRL-based method can be applied to various real-world tasks without a classic solver; we can also utilize DRL-based to unexplored tasks. This paper investigates the DRL-based NP-hard routing method without the help of classical solvers. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Formulation of Routing Problems ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This section explains the Markov decision process (MDP) formulation for the given 2D Euclidean TSP as a representative example. The formulation of MDP for other problems is described in Appendix A.1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The main objective of TSP is to find the shortest path of the Hamiltonian cycle. The TSP graph can be represented as a sequence of $N$ nodes in 2D Euclidean space, $\\pmb { s } = \\{ x _ { i } \\} _ { i = 1 } ^ { N }$ , where $x _ { i } \\in \\bar { \\mathbb { R } } ^ { 2 }$ . Then, the solution of TSP can be represented as the permutation $\\pi$ of input sequences: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2a9938fb01c09206ee0dacba3130df1afe928b16c24d96dd1d01a31770f050c.jpg",
+        "text": "$$\n\\pi = \\bigcup _ { t = 1 } ^ { t = N } \\{ \\pi _ { t } \\} , \\quad \\pi _ { t } \\in \\{ 1 , . . . , N \\} , \\quad \\pi _ { t _ { 1 } } \\neq \\pi _ { t _ { 2 } } \\quad \\mathrm { i f } \\quad t _ { 1 } \\neq t _ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The objective is minimizing the tour length $\\begin{array} { r } { L ( { \\boldsymbol \\pi } | s ) = \\sum _ { t = 1 } ^ { N - 1 } | | x _ { { \\pi } _ { t + 1 } } - x _ { { \\pi } _ { t } } | | _ { 2 } + | | x _ { { \\pi } _ { N } } - x _ { { \\pi } _ { 1 } } | | _ { 2 } . } \\end{array}$ ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then, we formulate the constructive Markov decision process (MDP) of TSP. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "State. State of MDP is represented as a partial solution of TSP or a sequence of previously selected actions: $\\pi _ { 1 : t - 1 }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Action. Action is defined as selecting one of un-served tasks. Therefore, action is represented as $\\pi _ { t }$ where the $\\pi _ { t } \\in \\{ \\{ 1 , . . . , N \\} \\setminus \\{ \\pi _ { 1 : t - 1 } \\} \\}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Cumulative Reward. We define cumulative reward for solution (a sequence of assignments) from problem instance $\\pmb { s }$ as negative of tourlength: $- L ( \\pi | s )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Constructive Policy. Finally we define constructive policy $p ( \\pi | \\boldsymbol { s } )$ that generates a solution $\\pi$ from TSP graph $\\pmb { s }$ . The constructive policy $p ( \\pi | \\boldsymbol { s } )$ is decomposed as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8c1a097dec6f003d0d6c52e379eb736ce915cba090666eac7504a2741d49c53.jpg",
+        "text": "$$\np ( \\pmb { \\pi } | \\pmb { s } ) = \\prod _ { t = 1 } ^ { t = N } p _ { \\theta } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , \\pmb { s } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Where $p _ { \\theta } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , s )$ is a single-step assignment policy parameterized by parameter $\\theta$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Learning Collaborative Policies ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This section describes a novel hierarchical problem-solving strategy, termed learning collaborative policies (LCP), which can effectively find the near-optimum solution using two hierarchical steps, seeding process and revising process (see Figure 1 for detail). In the seeding process, the seeder policy $\\bar { p } ^ { S }$ generates $M$ number of diversified candidate solutions. In the revising process, the reviser policy $\\bar { p } ^ { R }$ re-writes each candidate solution $I$ times to minimize the tour length of the candidate. The final solution is then selected as the best solution among $M$ revised (updated) candidate solutions. See pseudo-code in Appendix A.4 for a detailed technical explanation. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/004755074b7c9789e87b31d0856e2763b47459d14b7592f4239a1adf510a6f61.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the seeding process. The diversity of seeds is approximated as the sum of scaled entropy of segment policy. The scheduler $w _ { t }$ is for giving more weight when $t$ is in early step $t _ { 1 }$ than $t _ { 2 }$ . In the early step $t _ { 1 }$ , selecting $a$ , rather than $b$ is critical for the overall solution. In the latter step $t _ { 2 }$ , even though the probability of selecting node $c$ and $d$ is equal (i.e., has a high entropy), they may give similar solutions. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Seeding Process ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The seeder generates as diversified candidate solutions as possible while being dedicated to exploring the full combinatorial action space. To this end, the seeder is trained to solve the following problems. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Solution space. Solution space of seeder is a set of full trajectory solutions : $\\{ \\pi ^ { ( 1 ) } , . . . , \\pi ^ { ( M ) } \\}$ . The $M$ is the number of candidate solutions from the seeder: termed sample width. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Policy structure. Seeder is a constructive policy, as defined in section 3 as follows: ",
+        "bbox": [
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+            555,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d7ed1bcb44a7e84cc64e7153863ae3a29525ea10ac9b12abd5c1afa7f54eb917.jpg",
+        "text": "$$\np ^ { S } ( \\pi | s ) = \\prod _ { t = 1 } ^ { t = N } p _ { \\theta ^ { S } } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , s )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The segment policy $p _ { \\theta ^ { S } } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , s )$ , parameterized by $\\theta ^ { S }$ , is derived form AM [12]. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Entropy Reward. To force the seeder policy $p ^ { S }$ to sample diverse solutions, we trained $p ^ { S }$ such that the entropy $\\mathcal { H }$ of $p ^ { S }$ to be maximized. To this end, we use the reward $R ^ { S }$ defined as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e031bd23c75058ea33b1c096337c3921218e7141120a7d233253a6871f97436d.jpg",
+        "text": "$$\nR ^ { S } = \\mathcal { H } \\left( \\pi \\sim \\prod _ { t = 1 } ^ { t = N } p _ { \\theta ^ { S } } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , s ) \\right) \\approx \\sum _ { t = 1 } ^ { N } w _ { t } \\mathcal { H } \\left( \\pi _ { t } \\sim p _ { \\theta ^ { S } } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , s ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The entropy of constructive policy is appropriate for measuring solution diversity. However, computing the entropy of constructive policy is intractable because search space is too large: $N !$ . Therefore, we approximate it as a weighted sum of the entropy of segment policies $p _ { \\theta ^ { S } } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , s )$ evaluated at different time step. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use a linear scheduler (time-varying weights) $\\begin{array} { r } { w _ { t } = \\frac { N - t } { N _ { w } } } \\end{array}$ to boost exploration at the earlier stage of composing a solution; higher randomness imposed by the higher weight $w _ { t }$ at the early stage tends to generate more diversified full trajectories later. The $N _ { w }$ is the normalizing factor, which is a hyperparameter. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Training scheme. To train the seeder, we use the REINFORCE [25] algorithm with rollout baseline $b$ introduced by Kool et al. [12]. Then the gradient of each objective function is expressed as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8af8b3fe933770769ccce39e8c127dce6c17b5d164aa634f7ba8e915c582ca6e.jpg",
+        "text": "$$\n\\nabla J ( \\theta ^ { S } | s ) = E _ { \\pi \\sim p ^ { S } } [ ( L ( \\pi | s ) - \\alpha R ^ { S } ( p _ { 1 : N } ^ { S } , \\pi ) - b ( s ) ) \\nabla l o g ( p ^ { S } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/b697812278b63fa7201acebc31f3b3b0043a31a364cd4824922f0f019a06fa8a.jpg",
+        "image_caption": [
+            "Figure 3: Training progress of the reviser. From randomly generated TSP, sub-problems are decomposed. Then reviser infers to make constructive action referring starting and destination node and previous selected actions. The reward is a negative length of the partial tour generated from the reviser. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the $\\alpha$ is hyperparameter for $R ^ { S }$ and $p _ { 1 : N } ^ { S }$ is the sequence of segment polices $\\{ p _ { \\theta ^ { S } } ( \\pi _ { t } | \\pi _ { 1 : t - 1 } , s ) \\} _ { t = 1 } ^ { N }$ . We use the ADAM [37] optimizer to obtain the optimal parameter $\\theta ^ { * }$ that minimizes the objective function. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Revision Process ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the revision process, given a candidate solution, the reviser decomposes the candidate solution into $K$ segments and simultaneously finds the optimum routing sequence for each segment of each candidate solution. The reviser repeats this revising process $I$ times to find the best-updated candidate solution. To be specific, the reviser sequentially updates candidate solutions ( $I$ times) by repeatably decomposing the full trajectories computed from the previous iteration into segments and revising the segments to produce $M$ updated full trajectory solutions. To sum up, reviser solves $M \\times K$ segments in parallel $M$ : number of candidate solutions, $K$ : number of the segment in each candidate solution), $I$ times in series. ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The proposed scheme has advantages over conventional local search methods or DRL-based improvement heuristics. It searches larger solution spaces in a single inference; therefore, it reduces iteration $I$ . For example, 2-opt and DRL-2opt [19] search $O ( N ^ { 2 } )$ solution space (if it is parallelizable, $O ( M N ^ { 2 } ) )$ , while the reviser searches $O ( M K \\times l ! )$ which is much larger (when the number of nodes of the segment $l$ is big enough) in a single inference. Hence we can reduce the number of iterations $I$ significantly compared to 2-opt, or DRL-2opt [19], thus expediting the speed of the solution search (see Appendix E). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Solution space. Solution space of reviser is a partial segment of full trajectory solution represented as $\\pi _ { k + 1 : k + l }$ . The $k$ is starting index, and $l$ is the number of nodes of the segment. For details of assigning segment including $k$ and $l$ , see Appendix A.3. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Policy structure. Reviser is a constructive policy as follows: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/114bca0118d6a09f5b9afbb40e866602ae8ef70a20dd24560a13b2482ee4910e.jpg",
+        "text": "$$\np ^ { R } ( \\pi _ { k + 1 : k + l } | s ) = \\prod _ { t = 1 } ^ { t = l } p _ { \\theta ^ { R } } \\bigl ( \\pi _ { k + t } | \\pi _ { k : k + t - 1 } , \\pi _ { k + l + 1 } , s \\bigr )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The segment policy $p _ { \\theta ^ { R } }$ , parameterized by $\\theta ^ { R }$ , is in the similar form with that of AM [12]. Each $\\pi _ { k }$ and $\\pi _ { k + l + 1 }$ indicate the starting point and the destination point of the partial segment, respectively (see red-points in Figure 3). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We modify the context embedding vector $h _ { ( c ) } ^ { ( N ) } = [ \\bar { h } ^ { ( N ) } , h _ { \\pi _ { t - 1 } } ^ { ( N ) } , h _ { \\pi _ { 1 } } ^ { ( N ) } ]$ of AM, which is designed for solving TSP. Hence, $h$ is a high dimensional embedding vector from the transformer-based encoder, and $N$ is the number of multi-head attention layers. $\\bar { h } ^ { ( N ) }$ is the mean of the entire embedding, $h _ { \\pi _ { t - 1 } } ^ { ( N ) }$ is the embedding of previously selected nodes, and $h _ { \\pi _ { 1 } } ^ { ( N ) }$ is the embedding of the first node. However, since the destination of reviser is , not the first node , we change the embedding of the first node $h _ { \\pi _ { 1 } } ^ { ( N ) }$ to be the embedding of the last node $h _ { \\pi _ { k + l + 1 } } ^ { ( N ) }$ for the context embedding as $h _ { ( c ) } ^ { ( N ) } = [ \\bar { h } ^ { ( N ) } , h _ { \\pi _ { k + t - 1 } } ^ { ( N ) } , h _ { \\pi _ { k + l + 1 } } ^ { ( N ) } ]$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Revision Reward: negative of partial tour length $\\begin{array} { r } { L ^ { R } ( \\pi _ { k + 1 : k + l } | s ) = \\sum _ { t = 1 } ^ { l + 1 } | | x _ { \\pi _ { k + t } } - x _ { \\pi _ { k + t - 1 } } | | _ { 2 } . } \\end{array}$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training scheme. The training process is mostly the same as described in section 4.1, except that we have modified the length term $L$ to $L ^ { R }$ , and set $\\alpha = 0$ to remove entropy reward $R ^ { S }$ for training the reviser. Note that the seeder and reviser are trained separately. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This section reports the experimental results2 of the LCP scheme on TSP, PCTSP, and CVRP $N = 2 0$ , 50, 100, 500, $N$ : number of nodes). Also, we report several ablation studies in section 5.3 and Appendix B-F. We evaluate performance on real-world TSPs in the TSPLIB in Appendix G. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Training Hyperparamters. Throughout the entire training process of the seeder and reviser, we have exactly the same hyperparameters as Kool et al. [12], except that the training batch size of our seeder is 1024. To train the seeder’s policy, we set $\\alpha = 0 . 5$ (2) and $\\begin{array} { r } { N _ { w } = \\sum _ { i = 1 } ^ { N } \\bar { i } } \\end{array}$ for linear weight $\\begin{array} { r } { w _ { t } = \\frac { N - t } { N _ { w } } } \\end{array}$ N−tN for entropy scheduling. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Details in the experimental setting, including hyperparameters, dataset configuration, and run time evaluation, are described in Appendix A.5. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Target Problems and Baselines ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the performance of LCP in solving the three routing problems: TSP, PCTSP, and CVRP. We provide a brief explanation of them. The detailed descriptions for these problems are in Appendix A.1. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Travelling salesman problem (TSP). TSP is a problem to find the shortest Hamiltonian cycle given node sequences. ",
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+    },
+    {
+        "type": "text",
+        "text": "Price collecting travelling salesman problem (PCTSP). PCTSP [38] is a problem, where each node has a prize and a penalty. The goal is to collect the nodes with at least a minimum total prize (constraint) and minimize tour length added with unvisited nodes’ penalties. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Capacitated vehicle routing problem (CVRP). CVRP [39] is a problem where each node has a demand, while a vehicle must terminate the tour when the total demand limit is exceeded (constraint). The objective is to minimize the tour length. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the baseline algorithms, we use two types of algorithms: conventional heuristics and DRL-based solvers. For the conventional heuristics, we use Gurobi [7] (the commercial optimization solver), and the OR Tools [40] (the commercial optimization solver) for all three problems. In Table 1, Gurobi (t) indicates time-limited Gurobi whose running time is restricted below $t$ . In addition, OR Tools (t) is the OR Tools that allows additional local search over a duration of $t$ . For problem-specific heuristics, we use Concorde [9] for TSP, the iterative local search (ILS) [12] for PCTSP, and LKH3 [41] for CVRP. ",
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+    {
+        "type": "text",
+        "text": "For the baselines using DRL-based solvers, we concentrated on the ability of the LCP scheme, which is improved performance over AM. Validating that the two-policies collaboration scheme outperforms the single-policy scheme (i.e., AM) is a crucial part of this research; thus, the most important metric for performance evaluation is improvement between vanilla AM the $\\mathbf { A M } + \\mathbf { L C P } .$ Also, we reproduced other competitive DRL frameworks: current emerging improvement heuristics. We exclude recently proposed AM-style constructive heuristics, including the POMO [26] and MDAM [27] because they can be candidate collaborators with LCP, not competitors (e.g., $\\mathrm { P O M O } + \\mathrm { L C P }$ is possible). The detailed method for evaluation baselines in Table 1 is described as follows: ",
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+    {
+        "type": "text",
+        "text": "TSP. We follow baseline setting of Kool et al. [12] and Costa et al. [19]. We set DRL baselines including the S2V-DQN [11], EAN [23], GAT-T [30], DRL-2opt [19], and AM [12]. We show the results of S2V-DQN and EAN reported by Kool et al. [12], and the results of GAT-T reported by Costa et al. [19]. Then we directly reproduce the two most competitive DRL frameworks among baselines, the AM and DRL-2opt, in our machine to make a fair comparison of the speed. ",
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+    {
+        "type": "text",
+        "text": "PCTSP. We follow baseline setting of Kool et al. [12]. We reproduce AM [12] for DRL baseline. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 1: Performance evaluation results of the LCP scheme compared with baseline heuristics and the DRL frameworks on TSP, PCTSP, and CVRP. The best costs (objective) among DRL frameworks are marked in bold. The $H$ is heuristic and Solver is exact algorithm. The “LCP $\\{ 6 4 0 , 2 \\} ^ { \\prime }$ means that the sampling width $M$ is 640, and the number of iterations $I$ (of the reviser) is 2. We measure the performance in a limited time budget, which is 10 seconds per instance. The $O B$ means “out of budget” and $I F$ means “infeasible”, where the solver cannot generate solutions satisfying constraints of target problems in a limited time. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Method</td><td colspan=\"3\">N=20</td><td colspan=\"3\">N= 50</td><td colspan=\"3\">N= 100</td></tr><tr><td rowspan=\"5\"></td><td>Gurobi (Solver)</td><td>Cost 3.84</td><td>Gap 0.00%</td><td>Time 0.01s</td><td>Cost 5.70</td><td>Gap 0.00% 1.76%</td><td>Time 0.01s</td><td>Cost 7.76 8.12</td><td>Gap 0.00% 4.53%</td><td>Time 0.02s</td></tr><tr><td>OR Tools (H) Concorde (H) S2V-DQN EAN{M:1280} EAN+2OPT{M:1280}</td><td>3.85 3.84 3.89 3.84 3.84</td><td>0.37% 0.00% 1.42% 0.11% 0.09%</td><td>0.01s</td><td>5.80 5.70 5.99 5.77 5.75</td><td>0.00% 5.16% 1.28% 1.00%</td><td>0.01s</td><td>7.76 8.31 8.75 8.12</td><td>0.00% 7.03% 12.7% 4.64%</td><td>0.02s</td></tr><tr><td>GAT-T{I:5000} Drl-2opt{I:2000} AM{M:1280} AM{M:2560} AM{M:7500} AM+LCP {640,10}</td><td>3.84 3.84 3.84 3.84 3.84 3.84</td><td>0.00% 0.00% 0.08% 0.06% 0.05% 0.00%</td><td>3.58s 0.03s 0.04s 0.06s 0.18s</td><td>5.71 5.70 5.73 5.72 5.72 5.70</td><td>0.20% 0.12% 0.52% 0.45% 0.39% 0.13%</td><td>4.88s 0.09s 0.13s 0.29s 0.30s</td><td>7.87 7.83 7.94 7.94 7.93 7.86</td><td>1.42% 0.87% 2.26% 2.21% 2.13% 1.25%</td><td>7.15s 0.36s 0.42s 1.21s</td></tr><tr><td>AM+LCP{1280,10} AM+LCP*{1280,45} Gurobi (Solver) Gurobi {1s} (H)</td><td>3.13 3.14</td><td>= 1 0.00% 0.07%</td><td>0.01s 0.01s</td><td>5.70 5.70</td><td>0.10% 0.02% OB IF</td><td>0.45s 2.48s</td><td>7.85 7.81</td><td>1.13% 0.54% OB IF</td><td>0.57s 0.90s 4.30s</td></tr><tr><td>Gurobi {10s}(H) OR Tools{10s} (H) PSSP OR Tools {60s} (H) ILS (H) AM{M:1280}</td><td></td><td>3.13 3.14 3.13 3.16</td><td>0.00% 0.05% 0.01% 0.77%</td><td>0.01s 0.31s 4.51 1.80s 4.48 0.10s 4.50</td><td>5.17 0.70% 0.00% 0.67%</td><td>15.6%</td><td>0.19s 0.31s 1.80s 0.72s</td><td>6.35 6.08 5.98</td><td>IF 6.21% 0.31s 1.56% 1.80s 0.00% 4.32s</td></tr><tr><td>AM{M:2560} AM+LCP {640,1} AM+LCP {1280,5}</td><td></td><td>3.18 3.15 3.14</td><td>0.39% 0.41% 0.17%</td><td>0.03s 0.03s 0.04s</td><td>4.52 4.51 4.50</td><td>0.74% 0.72% 0.51%</td><td>0.07s 0.10s 0.07s</td><td>6.08 6.07 6.06</td><td>1.67% 1.57% 1.42%</td><td>0.17s 0.28s 0.15s</td></tr><tr><td></td><td>Gurobi (Solver) OR Tools (H) LKH3 (H)</td><td>3.14 6.10 6.43 6.14</td><td>0.08% 0.00% 5.41% 0.58%</td><td>0.10s 0.01s</td><td>4.49 11.31</td><td>0.32% OB 9.01%</td><td>0.20s</td><td>6.04 17.16</td><td>1.00% OB 9.67%</td><td>0.39s</td></tr><tr><td>JRPP</td><td>RL{M: 10} NLNS{I: 2000} AM{M:1280} AM{M:2560}</td><td>6.40 6.19 6.25 6.25</td><td>4.92% 1.47% 2.49% 2.39%</td><td>0.72s 1.00s 0.05s 0.06s</td><td>10.38 11.15 10.54 10.62 10.61</td><td>0.00% 7.46% 1.54% 2.40% 2.24%</td><td>2.52s 1.63s 0.14s 0.31s</td><td>15.65 16.96 16.00 16.23 16.17</td><td>0.00% 8.39% 2.17% 3.72% 3.34%</td><td>4.68s 2.18s 0.34s 0.75s</td></tr><tr><td></td><td>AM{M:7500} AM +LCP {640,1} AM+LCP{1280,1} AM+LCP{2560,1} AM+LCP{6500,1}</td><td>6.24 6.17 6.16 6.15</td><td>2.24% 1.15% 0.92% 0.84% -</td><td>0.09s 0.07s 0.09s 0.14s</td><td>10.59 10.56 10.54 10.52</td><td>2.06% 1.74% 1.54% 1.38% -</td><td>0.36s 0.15s 0.20s 0.31s</td><td>16.14 16.05 16.03 16.00 15.98</td><td>3.11% 2.58% 2.43% 2.24% 2.11%</td><td>1.42s 0.30s 0.45s 0.77s 1.73s</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "CVRP. We follow baseline setting of Houttung & Tierney [32]. We report result of RL [22] based on Houttung & Tierney [32] and we reproduce AM [12] and NLNS [32]. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 Performance Evaluation ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we report the performance of LCP on small-scale problems $\\mathrm { \\Delta } N = 2 0 , 5 0 , 1 0 0 )$ in Table 1. Then we provide a time-performance trade-off analysis including large-scale problems $N = 5 0 0$ ). We note that time-performance analysis is significant because any method can find an optimal solution when given an infinite time budget. From the analysis, we can identify a specific time region, called winner region, where LCP performs the best in terms of both speed and performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Performance evaluation on $N = 2 0$ , 50, 100. Our method outperforms all the DRL baselines and OR-tools in TSP, PCTSP, and CVRP, as clearly shown in Table 1. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/64ff80900ff68cb05517c178513a00d202b3afdc2c448b31410d4263f3ac1486.jpg",
+        "image_caption": [
+            "Figure 4: Details of time-performance analysis. The left (fast) and lower (lower cost) trend indicates a better performance. In (e), the ILS cannot generate solution in time budget $\\mathit { \\Omega } ^ { \\prime } t < 1 0 0 0 s ,$ ) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/dfe4eaa2e3fde3f57bec0adec9454abbfbe6dd37d4063e3e32ce29335181bdde.jpg",
+        "table_caption": [
+            "Table 2: Ablation study of LCP components on TSP, PCTSP, and CVRP $N = 1 0 0$ ). The optimal gap is measured by comparing it with state-of-the-art solvers. The Entropy Regularization indicates training the seeder with $\\bar { R } ^ { S }$ , while the default is the uniform scheduling. The best performances are marked in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Component of the LCP</td><td colspan=\"2\">TSP</td><td colspan=\"2\">PCTSP</td><td colspan=\"2\">CVRP</td></tr><tr><td>Entropy Regularization</td><td>Weight Scheduling</td><td>Reviser</td><td>cost</td><td>gap</td><td>cost</td><td>gap</td><td>cost</td><td>gap</td></tr><tr><td></td><td></td><td></td><td>7.96</td><td>2.65%</td><td>6.08</td><td>1.64%</td><td>16.29</td><td>3.43%</td></tr><tr><td>V</td><td rowspan=\"3\">√</td><td></td><td>7.96</td><td>2.68%</td><td>6.08</td><td>1.76%</td><td>16.25</td><td>3.16%</td></tr><tr><td></td><td></td><td>7.94</td><td>2.45%</td><td>6.07</td><td>1.62%</td><td>16.20</td><td>2.86%</td></tr><tr><td></td><td>√</td><td>7.86</td><td>1.32%</td><td>6.04</td><td>1.13%</td><td>16.20</td><td>2.86%</td></tr><tr><td>·</td><td></td><td></td><td>7.84</td><td>1.17%</td><td>6.05</td><td>1.16%</td><td>16.16</td><td>2.59%</td></tr><tr><td></td><td>√</td><td>1</td><td>7.82</td><td>0.88%</td><td>6.04</td><td>1.02%</td><td>16.12</td><td>2.37%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that for TSP $N = 1 0 0 $ ), we applied two types of revisers, each of which is denoted LCP and $\\mathrm { L C P ^ { * } }$ , respectively. The details are described in Appendix A.4 with pseudo-code. Our LCP and $\\mathrm { L C P ^ { * } }$ outperforms DRL-2opt, the current state-of-the-art DRL-based improvement heuristic in $N = 2 0 , 5 0 , 1 0 0$ , surpass $0 . 3 3 \\%$ in $N = 1 0 0$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In PCTSP, LCP outperforms AM with less time. Our method $( \\mathbf { A M } + \\mathbf { L C P }$ {640,1}) outperforms the OR-Tools (10s), with $4 \\times$ and $2 \\times$ faster speed in $N = 5 0$ , 100 respectively. Compared to the ILS, our method $\\mathrm { \\Delta A M + L C P }$ {1280,5}) underperforms by $1 . 0 \\%$ , but has $1 1 \\ \\times$ faster speed for $N = 1 0 0$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For CVRP, our method outperforms competitive DRL frameworks. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Time-performance analysis on ${ \\bf { \\mathit { N } } } = { \\bf { 1 0 0 } }$ , 500. In Figure 4, we describe the time-performance analysis. We cannot control the speed of the Concorde, ILS, and LKH3. We can control the speed of DRL solvers by adjusting sample width $M$ or the number of iterations $I$ . For PCTSP, we can change the speed of OR-tools by managing the time for additional local searches. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our scheme clearly outperforms DRL-solvers in terms of both speed and performance. For PCTSP $\\mathrm { \\Delta } N = 1 0 0 , 5 0 0 { \\mathrm { \\Omega } }$ ) and CVRP $N = 5 0 0$ ), our method achieves the winner region of $t < 1 0$ , which is best performed in a specific time region among all kind of baseline solvers (for CVRP $N = 1 0 0$ ), our method achieves the winner region of $t < 5$ ). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Performance on TSPLIB [20] data: see Appendix G. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this section, we conduct an ablation study on LCP components. We leave further ablation studies to Appendix B-F. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ablation study of collaborative policies. In Table 2, we ablate three significant components of LCP and show the experimental results for every case. In the case of vanilla AM, having none of LCP components, the performance is the poorest. On the other hand, collaboration of seeder trained with linearly scheduled-entropy and the reviser shows the best performance. Therefore, the experimental results empirically validate our proposal of hierarchically collaborating two policies and also demonstrate the effectiveness of using a linearly scheduled-entropy term shown in section 4.1 and Figure 2. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ablation study of entropy regularization: see Appendix B. ",
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+    },
+    {
+        "type": "text",
+        "text": "Ablation study of SoftMax temperature: see Appendix C. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ablation study of application of LCP to pointer network [10, 21]: see Appendix D. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Comparison with reviser and other improvement heuristics: see Appendix E. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Training convergence of seeder and reviser in different PyTorch seeds: see Appendix F. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Discussion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed a novel DRL scheme, learning collaborative policies (LCP). The extensive experiments demonstrate that our two-policies collaboration algorithm (i.e., LCP) outperforms conventional single-policy DRL frameworks, including AM [12], on various NP-hard routing problems, such as TSP, PCTSP, and CVRP. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We highlight that LCP is a reusable scheme, can solve various problems. The neural architecture of the seeder and reviser proposed in this paper is derived from AM [12]. It can be substituted by other architectures, such as the pointer network [10, 21] and AM-style architectures including POMO [26] and MDAM [27]. If further studies on neural architecture for combinatorial optimization are carried out, the seeder and reviser can be improved further. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Also, LCP can be directly applied to other combinatorial optimization tasks, including TSP with time windows (TSPTW), orienteering problem (OP), multiple TSP (mTSP), variations of the vehicle routing problem (VRP), and other practical applications. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Further Works. We made an important first step: two-policies collaboration where each policy specializing in exploration or exploitation can improve conventional single-policy systems on combinatorial optimization tasks. The important direction of further research is introducing more sophisticated strategies to explore or exploit combinatorial solution space. New exploration strategies for overcoming the proposed approximated entropy maximization scheme are needed. Also, it is necessary to investigate more effective exploitation strategies beyond the proposed revision scheme. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            690,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This research is supported in part by the KAIST undergraduates research program (URP), 2019. We thank Hankook Lee and Prof. Jinwoo Shin for building part of this project in the URP. We thank Joonsang Park, Keeyoung Son, Hyunwook Park, Haeyeon Rachel Kim, and our anonymous reviewers for feedback and discussions. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Rajesh Matai, Surya Singh, and M.L. Mittal. “Traveling Salesman Problem: an Overview of Applications, Formulations, and Solution Approaches”. In: Nov. 2010. ISBN: 978-953-307- 426-9. DOI: 10.5772/12909.   \n[2] B. Yuan, M. Orlowska, and S. Sadiq. “On the Optimal Robot Routing Problem in Wireless Sensor Networks”. In: IEEE Transactions on Knowledge and Data Engineering 19.9 (2007), pp. 1252–1261. DOI: 10.1109/TKDE.2007.1062.   \n[3] Olin Johnson and Jing Liu. “A traveling salesman approach for predicting protein functions”. In: Source code for biology and medicine 1 (Feb. 2006), p. 3. DOI: 10.1186/1751-0473-1-3.   \n[4] Haiguang Liao et al. Attention Routing: track-assignment detailed routing using attentionbased reinforcement learning. 2020. arXiv: 2004.09473 [cs.LG].   \n[5] M. Kim et al. “Reinforcement Learning-based Auto-router considering Signal Integrity”. 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In: Integration of Constraint Programming, Artificial Intelligence, and Operations Research. Ed. by Willem-Jan van Hoeve. Cham: Springer International Publishing, 2018, pp. 170–181. ISBN: 978-3-319- 93031-2.   \n[24] Ashish Vaswani et al. “Attention is All you Need”. In: Advances in Neural Information Processing Systems. Ed. by I. Guyon et al. Vol. 30. Curran Associates, Inc., 2017, pp. 5998–6008. URL: https : / / proceedings . neurips . cc / paper / 2017 / file / 3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf.   \n[25] R. J. Williams. “Simple statistical gradient-following algorithms for connectionist reinforcement learning”. In: Machine Learning 8 (1992), pp. 229–256.   \n[26] Yeong-Dae Kwon et al. “POMO: Policy Optimization with Multiple Optima for Reinforcement Learning”. In: Advances in Neural Information Processing Systems. Ed. by H. Larochelle et al. Vol. 33. Curran Associates, Inc., 2020, pp. 21188–21198. URL: https://proceedings. neurips.cc/paper/2020/file/f231f2107df69eab0a3862d50018a9b2-Paper.pdf.   \n[27] Liang Xin et al. “Multi-decoder attention model with embedding glimpse for solving vehicle routing problems”. In: Proceedings of 35th AAAI Conference on Artificial Intelligence. 2021.   \n[28] Paul Shaw. A New Local Search Algorithm Providing High Quality Solutions to Vehicle Routing Problems. 1997.   \n[29] Xinyun Chen and Yuandong Tian. “Learning to Perform Local Rewriting for Combinatorial Optimization”. In: Advances in Neural Information Processing Systems. 2019.   \n[30] Yaoxin Wu et al. Learning Improvement Heuristics for Solving Routing Problems. 2020. arXiv: 1912.05784 [cs.AI].   \n[31] Paulo R d O da Costa et al. “Learning 2-opt Heuristics for the Traveling Salesman Problem via Deep Reinforcement Learning”. In: Proceedings of The 12th Asian Conference on Machine Learning. Ed. by Sinno Jialin Pan and Masashi Sugiyama. Vol. 129. Proceedings of Machine Learning Research. Bangkok, Thailand: PMLR, 18–20 Nov 2020, pp. 465–480. URL: http: //proceedings.mlr.press/v129/costa20a.html.   \n[32] André Hottung and Kevin Tierney. “Neural Large Neighborhood Search for the Capacitated Vehicle Routing Problem”. In: CoRR abs/1911.09539 (2019). arXiv: 1911 . 09539. URL: http://arxiv.org/abs/1911.09539.   \n[33] Hao Lu, Xingwen Zhang, and Shuang Yang. “A Learning-based Iterative Method for Solving Vehicle Routing Problems”. In: 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, 2020. URL: https: //openreview.net/forum?id=BJe1334YDH.   \n[34] Chaitanya K. Joshi et al. Learning TSP Requires Rethinking Generalization. 2020. arXiv: 2006.07054 [cs.LG].   \n[35] Zhang-Hua Fu, Kai-Bin Qiu, and Hongyuan Zha. “Generalize a Small Pre-trained Model to Arbitrarily Large TSP Instances”. In: CoRR abs/2012.10658 (2020). arXiv: 2012.10658. URL: https://arxiv.org/abs/2012.10658.   \n[36] Wouter Kool et al. “Deep Policy Dynamic Programming for Vehicle Routing Problems”. In: CoRR abs/2102.11756 (2021). arXiv: 2102.11756. URL: https://arxiv.org/abs/2102. 11756.   \n[37] Diederik P. Kingma and Jimmy Ba. “Adam: A Method for Stochastic Optimization”. In: 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings. Ed. by Yoshua Bengio and Yann LeCun. 2015. URL: http://arxiv.org/abs/1412.6980.   \n[38] Egon Balas. “The prize collecting traveling salesman problem”. In: Networks 19.6 (1989), pp. 621–636. DOI: https : / / doi . org / 10 . 1002 / net . 3230190602. URL: https : / / onlinelibrary.wiley.com/doi/abs/10.1002/net.3230190602.   \n[39] Paolo Toth and Daniele Vigo. Vehicle Routing. Ed. by Daniele Vigo and Paolo Toth. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2014. DOI: 10.1137/ 1 . 9781611973594. URL: https : / / epubs . siam . org / doi / abs / 10 . 1137 / 1 . 9781611973594.   \n[40] Laurent Perron and Vincent Furnon. OR-Tools. Version 7.2. Google, 2019. URL: https : //developers.google.com/optimization/.   \n[41] Keld Helsgaun. “An Extension of the Lin-Kernighan-Helsgaun TSP Solver for Constrained Traveling Salesman and Vehicle Routing Problems”. In: (Dec. 2017). DOI: 10.13140/RG.2. 2.25569.40807. ",
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parse/train/wtLW-Amuds/wtLW-Amuds.md

@@ -0,0 +1,278 @@
+# The Sensory Neuron as a Transformer: Permutation-Invariant Neural Networks for Reinforcement Learning
+
+Yujin Tang† Google Brain yujintang@google.com
+
+David Ha† Google Brain hadavid@google.com
+
+# Abstract
+
+In complex systems, we often observe complex global behavior emerge from a collection of agents interacting with each other in their environment, with each individual agent acting only on locally available information, without knowing the full picture. Such systems have inspired development of artificial intelligence algorithms in areas such as swarm optimization and cellular automata. Motivated by the emergence of collective behavior from complex cellular systems, we build systems that feed each sensory input from the environment into distinct, but identical neural networks, each with no fixed relationship with one another. We show that these sensory networks can be trained to integrate information received locally, and through communication via an attention mechanism, can collectively produce a globally coherent policy. Moreover, the system can still perform its task even if the ordering of its inputs is randomly permuted several times during an episode. These permutation invariant systems also display useful robustness and generalization properties that are broadly applicable. Interactive demo and videos of our results: https://attentionneuron.github.io/
+
+# 1 Introduction
+
+Sensory substitution refers to the brain’s ability to use one sensory modality (e.g., touch) to supply environmental information normally gathered by another sense (e.g., vision). Numerous studies have demonstrated that humans can adapt to changes in sensory inputs, even when they are fed into the wrong channels [5, 6, 25, 64]. But difficult adaptations–such as learning to “see” by interpreting visual information emitted from a grid of electrodes placed on one’s tongue [6], or learning to ride a “backwards” bicycle [64]–require months of training to attain mastery. Can we do better, and create artificial systems that can rapidly adapt to sensory substitutions, without the need to be retrained?
+
+![](images/6445f3a4c1881202512120761763a01e791ac4d661e18b30e0013845d962273f.jpg)  
+Figure 1: Comparison of visual input intended for the game player, and what our system receives. We partition the visual input from CarRacing (Left) and Atari Pong (right) into a 2D grid of small patches, and randomly permute their ordering. Each sensory neuron in the system receives a stream of visual input at a particular permuted patch location, and through coordination, must complete the task at hand, even if the visual ordering is randomly permuted again several times during an episode.
+
+Modern deep learning systems are generally unable to adapt to a sudden reordering of sensory inputs, unless the model is retrained, or if the user manually corrects the ordering of the inputs for the model. However, techniques from continual meta-learning, such as adaptive weights [3, 36, 65], Hebbian-learning [52, 53, 57], and model-based [2, 20, 37, 38] approaches can help the model adapt to such changes, and remain a promising active area of research.
+
+In this work, we investigate agents that are explicitly designed to deal with sudden random reordering of their sensory inputs while performing a task. Motivated by recent developments in self-organizing neural networks [27, 55, 61] related to cellular automata [14, 17, 18, 58, 78], in our experiments, we feed each sensory input (which could be an individual state from a continuous control environment, or a patch of pixels from a visual environment) into an individual neural network module that integrates information from only this particular sensory input channel over time. While receiving information locally, each of these individual sensory neural network modules also continually broadcasts an output message. Inspired by the Set Transformer [48, 74] architecture, an attention mechanism combines these messages to form a global latent code which is then converted into the agent’s action space. The attention mechanism can be viewed as a form of adaptive weights of a neural network, and in this context, allows for an arbitrary number of sensory inputs that can be processed in any random order.
+
+In our experiments, we find that each individual sensory neural network module, despite receiving only localized information, can still collectively produce a globally coherent policy, and that such a system can be trained to perform tasks in several popular reinforcement learning (RL) environments. Furthermore, our system can utilize a varying number of sensory input channels in any randomly permuted order, even when the order is shuffled again several times during an episode.
+
+Permutation invariant systems have several advantages over traditional fixed-input systems. We find that encouraging a system to learn a coherent representation of a permutation invariant observation space leads to policies that are more robust and generalizes better to unseen situations. We show that, without additional training, our system continues to function even when we inject additional input channels containing noise or redundant information. In visual environments, we show that our system can be trained to perform a task even if it is given only a small fraction of randomly chosen patches from the screen, and at test time, if given more patches, the system can take advantage of the additional information to perform better. We also demonstrate that our system can generalize to visual environments with novel background images, despite training on a single fixed background. Lastly, to make training more practical, we propose a behavioral cloning scheme to convert policies trained with existing methods into a permutation invariant policy with desirable properties.
+
+# 2 Related Work
+
+Self-organization is a process where some form of global order emerges from local interactions between parts of an initially disordered system. It is also a property observed in cellular automata (CA) [17, 18, 58], which are mathematical systems consisting of a grid of cells that perform computation by having each cell communicate with its immediate neighbors and performing a local computation to update its internal state. Such local interactions are useful in modeling complex systems [78] and have been applied to model non-linear dynamics in various fields [14]. Cellular Neural Networks [16] were first introduced in the 1980s to use neural networks in place of the algorithmic cells in CA systems. They were applied to perform image processing operations with parallel computation. Eventually, the concept of self-organizing neural networks found its way into deep learning in the form of Graph Neural Networks (GNN) [62, 79].
+
+Using modern deep learning tools, recent work demonstrates that neural CA, or self-organized neural networks performing only local computation, can generate (and re-generate) coherent images [55] and voxel scenes [70, 84], and even perform image classification [61]. Self-organizing neural network agents have been proposed in the RL domain [11, 12, 59, 60], with recent work demonstrating that shared local policies at the actuator level [43], through communicating with their immediate neighbors, can learn a global coherent policy for continuous control locomotion tasks. While existing CA-based approaches present a modular, self-organized solution, they are not inherently permutation invariant. In our work, we build on neural CA, and enable each cell to communicate beyond its immediate neighbors via an attention mechanism that enables permutation invariance.
+
+Meta-learning recurrent neural networks (RNN) [23, 40, 42, 76] have been proposed to approach the problem of learning the learning rules for a neural network using the reward or error signal, enabling meta-learners to learn to solve problems presented outside of their original training domains. The goals are to enable agents to continually learn from their environments in a single lifetime episode, and to obtain much better data efficiency than conventional learning methods such as stochastic gradient descent (SGD). A meta-learned policy that can adapt the weights of a neural network to its inputs during inference time have been proposed in fast weights [65, 67], associative weights [3], hypernetworks [36], and Hebbian-learning [52, 53] approaches. Recently works [46, 63] combine ideas of self-organization with meta-learning RNNs, and have demonstrated that modular metalearning RNN systems not only can learn to perform SGD-like learning rules, but can also discover more general learning rules that transfer to classification tasks on unseen datasets.
+
+In contrast, the system presented here does not use an error or reward signal to meta-learn or fine-tune its policy. But rather, by using the shared modular building blocks from the meta-learning literature, we focus on learning or converting an existing policy to one that is permutation invariant, and we examine the characteristics such policies exhibit in a zero-shot setting, without additional training.
+
+Attention can be viewed as an adaptive weight mechanism that alters the weight connections of a neural network layer based on what the inputs are. Linear dot-product attention has first been proposed for meta-learning [66], and versions of linear attention with sof tmax nonlinearity appeared later [33, 51], now made popular with Transformer [74]. The adaptive nature of attention provided the Transformer with a high degree of expressiveness, enabling it to learn inductive biases from large datasets and have been incorporated into state-of-the-art methods in natural language processing [9, 21], image recognition [22] and generation [26], audio and video domains [31, 44, 71].
+
+Attention mechanisms have found many uses for RL [1, 13, 56, 68, 72, 83]. Our work here specifically uses attention to enable communication between arbitrary numbers of modules in an RL agent. While previous work [32, 45, 54, 75, 81, 85] utilized attention as a communication mechanism between independent neural network modules of a GNN, our work focuses on studying the permutation invariant properties of attention-based communication applied to RL agents. Related work [50] used permutation invariant critics to enhance performance of multi-agent RL. Building [34, 82], Set Transformers [48] investigated the use of attention explicitly for permutation invariant problems that deal with set-structured data, which have provided the theoretical foundation for our work.
+
+# 3 Method
+
+# 3.1 Background
+
+Our goal is to devise an agent that is permutation invariant (PI) in the action space to the permutations in the input space. While it is possible to acquire a quasi-PI agent by training with randomly shuffled observations and hope the agent’s policy network has enough capacity to memorize all the patterns, we aim for a design that achieves true PI even if the agent is trained with fix-ordered observations. Mathematically, we are looking for a non-trivial function $f ( \boldsymbol { x } ) : \mathcal { R } ^ { n } \mapsto \mathcal { R } ^ { m }$ such that $f ( x [ s ] ) = f ( x )$ for any $x \in \mathcal { R } ^ { n }$ , and $s$ is any permutation of the indices $\{ 1 , \cdots , n \}$ . A different but closely related concept is permutation equivariance (PE) which can be described by a function $h ( x ) : \mathcal { R } ^ { \mathit { \hat { n } } } \mapsto \mathcal { R } ^ { \mathit { n } }$ such that $\bar { h ( x [ s ] ) } = h ( x ) [ \bar { s } ]$ . Unlike PI, the dimensions of the input and the output must equal in PE.
+
+Self-attentions can be PE. In its simplest form, self-attention is described as $y = \sigma ( Q K ^ { \top } ) V$ where $Q , K \in \mathcal { R } ^ { n \times d _ { q } } , V \in \mathcal { R } ^ { n \times d _ { v } }$ are the Query, Key and Value matrices and $\sigma ( \cdot )$ is a non-linear function. In most scenarios, $Q , K , V$ are functions of the input $x \in \mathcal { R } ^ { n }$ (e.g. linear transformations), and permuting $x$ therefore is equivalent to permuting the rows in $Q , K , V$ and based on its definition it is straightforward to verify the PE property. Set Transformer [48] cleverly replaced $Q$ with a set of learnable seed vectors, so it is no longer a function of input $x$ , thus enabling the output to become PI. A simple, intuitive explanation of the PI property of self-attention is available in Appendix A.1.
+
+# 3.2 Sensory Neurons with Attention
+
+To create PI agents, we propose to add an extra layer in front of the agent’s policy network $\pi$ , which accepts the current observation $o _ { t }$ and the previous action $a _ { t - 1 }$ as its inputs. We call this new layer AttentionNeuron, and Figure 2 gives an overview of our method. Inside AttentionNeuron, we model the observation $o _ { t }$ as an arbitrarily ordered, variable-length list of sensory inputs, each of which is passed into its own sensory neuron, a neural network module. Each sensory neuron only has partial access to the agent’s observation, at time $t$ , the ith neuron can see only the ith component of the observation $o _ { t } [ i ]$ . Combined with the previous action $a _ { t - 1 }$ , each sensory neuron computes messages $f _ { k } \big ( o _ { t } [ i ] , a _ { t - 1 } \big ) ^ { \smash { \cdot } }$ and $f _ { v } ( o _ { t } [ i ] )$ that are broadcast to the rest of the system. We then use attention to aggregate these messages into a global latent code, $m _ { t }$ , that is $\mathrm { P I }$ with respect to the inputs.
+
+![](images/737a2f978bc093f562aa43d22f92f0d50d4a66a7cf9d68809b7f1ea690a980c3.jpg)  
+Figure 2: Overview of Method. AttentionNeuron is a standalone layer, in which each sensory neuron only has access to a part of the unordered observations $o _ { t }$ . Together with the agent’s previous action $a _ { t - 1 }$ , each neuron generates messages independently using the shared functions $f _ { k } \big ( \overline { { o } } _ { t } [ i ] , a _ { t - 1 } \big )$ and $f _ { v } ( o _ { t } [ i ] )$ . The attention mechanism summarizes the messages into a global latent code $m _ { t }$ .
+
+The operations inside AttentionNeuron can be described by the following two equations. For clarity, Table 1 summarizes the notations as well as the corresponding setups we used for the experiments.
+
+$$
+\begin{array} { c } { { \displaystyle K ( o _ { t } , a _ { t - 1 } ) = \left[ \begin{array} { c } { { f _ { k } ( o _ { t } [ 1 ] , a _ { t - 1 } ) } } \\ { { \cdots \cdot } } \\ { { f _ { k } ( o _ { t } [ N ] , a _ { t - 1 } ) } } \end{array} \right] \in { \mathbb R } ^ { N \times d _ { f _ { k } } } , V ( o _ { t } ) = \left[ \begin{array} { c } { { f _ { v } ( o _ { t } [ 1 ] ) } } \\ { { \cdots \cdot } } \\ { { f _ { v } ( o _ { t } [ N ] ) } } \end{array} \right] \in { \mathbb R } ^ { N \times d _ { f _ { v } } } } } \\ { { m _ { t } = \sigma ( \frac { [ Q W _ { q } ] [ K ( o _ { t } , a _ { t - 1 } ) W _ { k } ] ^ { \top } } { \sqrt { d _ { q } } } ) [ V ( o _ { t } ) W _ { v } ] } } \end{array}
+$$
+
+Equation 1 shows how each of the $N$ sensory neuron independently generates its messages $f _ { k }$ and $f _ { v }$ , which are functions shared across all sensory neurons. Equation 2 shows the attention mechanism aggregate these messages. Note that although we could have absorbed the projection matrices $W _ { q } , W _ { k } , W _ { v }$ into $Q , K , V$ , we keep them in the equation to show explicitly the formulation. Equation 2 is almost identical to the simple definition of self-attention mentioned earlier. Following [48], we make our $Q$ matrix a bank of fixed embeddings, rather than depend on the observation $o _ { t }$ .
+
+Note that permuting the observations only affects the row orders of $K$ and $V$ , and that applying the same permutation to the rows of both $K$ and $V$ still results in the same $m _ { t }$ which is PI. As long as we set constant the number of rows in $Q$ , the change in the input size affects only the number of rows in $K$ and $V$ and does not affect the output $m _ { t }$ . In other words, our agent can accept inputs of arbitrary length and output a fixed sized $m _ { t }$ . Later, we apply this flexibility of input dimensions to RL agents.
+
+Table 1: In this notation list, we provide the dimensions used in our model for different RL environ ments, to give the reader a sense of the relative magnitudes involved in each part of the system.   
+
+<table><tr><td>Description</td><td>Notation</td><td>CartPole</td><td>Ant</td><td>CarRacing</td><td>Atari Pong</td></tr><tr><td>Full observation space</td><td>Ot</td><td>R5</td><td>R28</td><td>R96×96×4</td><td>R84x84x4</td></tr><tr><td>Individual sensory input space</td><td>Ot[i]</td><td>R1</td><td>R1</td><td>R6×6×4=144</td><td>R6×6×4=144</td></tr><tr><td>Number of sensory neurons</td><td>N</td><td>5</td><td>28</td><td>(96/6)² = 256</td><td>(84/6)² = 196</td></tr><tr><td>Dimension of action space</td><td>|A|</td><td>1</td><td>8</td><td>3</td><td>6 (one-hot)</td></tr><tr><td>Number of embeddings in Q</td><td>M</td><td>16</td><td>32</td><td>1024</td><td>400</td></tr><tr><td>Projection matrix for Q</td><td>Wq</td><td>R8×32</td><td>R8×32</td><td>R8×16</td><td>R8×32</td></tr><tr><td>Projection matrix for K</td><td>Wk</td><td>R8×32</td><td>R8×32</td><td>R111×16</td><td>R114×32</td></tr><tr><td>Projection matrix for V</td><td>Wu</td><td>I</td><td>I</td><td>R144×16</td><td>R144×32</td></tr><tr><td>Post-attention activation function</td><td>9()</td><td>tanh</td><td>tanh</td><td>softmax</td><td>softmax</td></tr><tr><td>Global latent code</td><td>mt</td><td>R16</td><td>R32</td><td>R1024×16</td><td>R400×32</td></tr></table>
+
+# 3.3 Design Choices
+
+It is worthwhile to have a discussion on the design choices made. Since the ordering of the input is arbitrary, each sensory neuron is required to interpret and identify their received signal. To achieve this, we want $f _ { k } ( o _ { t } [ \dot { i } ] , a _ { t - 1 } )$ to have temporal memories. In practice, we find both RNNs and feed-forward neural networks (FNN) with stacked observations work well, with FNNs being more practical for environments with high dimensional observations.
+
+In addition to the temporal memory, including previous actions is important for the input identification too. Although the former allows the neurons to infer the input signals based on the characteristics of the temporal stream, this may not be sufficient. For example, when controlling a legged robot, most of the sensor readings are joint angles and velocities from the legs, which are not only numerically identically bounded but also change in similar patterns. The inclusion of previous actions gives each sensory neuron a chance to infer the casual relationship between the input channel and the applied actions, which helps with the input identification.
+
+Finally, in Equation 2 we could have combined $Q W _ { q } \in \mathcal { R } ^ { M \times d _ { q } }$ as a single learnable parameters matrix, but we separate them for two reasons. First, by factoring into two matrices, we can reduce the number of learnable parameters. Second, we find that instead of making $Q$ learnable, using the positional encoding proposed in Transformer [74] encourages the attention mechanism to generate distinct codes. Here we use the row indices in $Q$ as the positions for encoding.
+
+# 4 Experiments
+
+We experiment on several different RL environments to study various properties of permutation invariant RL agents. Due to the nature of the underlying tasks, we will describe the different architectures of the policy networks used and discuss various different training methods. However, the AttentionNeuron layers in all agents are similar, so we first describe the common setups. Hyperparameters and other details for all experiments are summarized in Appendix A.4.
+
+For non-vision continuous control tasks, the agent receives an observation vector $o _ { t } \in \mathcal { R } ^ { | O | }$ at time $t$ We assign $N = | O |$ sensory neurons for the tasks, each of which sees one element from the vector, hence $o _ { t } [ i ] \in \mathcal { R } ^ { 1 } , i = 1 , \cdots , | O |$ . We use an LSTM [41] as our $f _ { k } ( o _ { t } [ i ] , a _ { t - 1 } )$ to generate Keys, the input size of which is $1 + | A |$ (2 for Cart-Pole and 9 for PyBullet Ant). A simple pass-through function $f ( x ) = x$ serves as our $f _ { v } ( o _ { t } [ i ] )$ , and $\sigma ( \cdot )$ is tanh. For simplicity, we find $W _ { v } = I$ works well for the tasks, so the learnable components are the LSTM, $W _ { q }$ and $W _ { k }$ .
+
+For vision based tasks, we gray-scale and stack $k = 4$ consecutive RGB frames from the environment, and thus our agent observes $\mathbf { \bar { \rho } } _ { { o _ { t } } } \in \mathcal { R } ^ { H \times W \times k }$ . $o _ { t }$ is split into non-overlapping patches of size $P = 6$ using a sliding window, so each sensory neuron observes $o _ { t } [ i ] \in \mathcal { R } ^ { 6 \times 6 \times \dot { k } } .$ . Here, $f _ { v } ( o _ { t } [ i ] )$ flattens the data and returns it, hence $V ( o _ { t } )$ returns a tensor of shape $N \times d _ { f _ { v } } = N \times ( 6 \times 6 \times 4 ) = N \times 1 4 4$ Due to the high dimensionality for vision tasks, we do not use RNNs for $f _ { k }$ , but instead use a simpler method to process each sensory input. $f _ { k } \left( o _ { t } [ i ] , a _ { t - 1 } \right)$ takes the difference between consecutive frames $( o _ { t } [ i ] )$ , then flattens the result, appends $a _ { t - 1 }$ , and returns the concatenated vector. $K ( o _ { t } , a _ { t - 1 } )$ thus gives a tensor of shape $N \times d _ { f _ { k } } = N \times [ ( 6 \times 6 \times 3 ) + | A | ] = N \times ( 1 0 8 + | A | )$ (111 for CarRacing and 114 for Atari Pong). We use sof tmax as the non-linear activation function $\sigma ( \cdot )$ , and we apply layer normalization [4] to both the input patches and the output latent code.
+
+# 4.1 Cart-pole swing up
+
+We examine Cart-pole swing up [29, 30, 35, 86] to first illustrate our method, and also use it to provide a clear analysis of the attention mechanism. We use CartPoleSwingUpHarder [28], a more difficult version of the task where the initial positions and velocities are highly randomized, leading to a higher variance of task scenarios. In the environment, the agent observes $[ x , \dot { x } , c o s ( \theta ) , s i n ( \theta ) , \dot { \theta } ]$ , outputs a scalar action, and is rewarded at each step for getting $x$ close to 0 and $c o s ( \theta )$ close to 1.
+
+![](images/97af204355d041941b28feb377276f15d632c462e1a6f77ed670189c47a73b60.jpg)  
+Figure 3: Interactive demo of CartPoleSwingUpHarder. In our web demo2, the user can shuffle the order of the 5 inputs at any time, and observe how the agent adapts to the new ordering of the inputs.
+
+We use a two-layer neural network as our agent. The first layer is an AttentionNeuron layer with $N = 5$ sensory neurons and outputs $m _ { t } \in \mathcal { R } ^ { \mathrm { { \tilde { 1 6 } } } }$ . A linear layer takes $m _ { t }$ as input and outputs a scalar action. For comparison, we also trained an agent with a two-layer FNN policy with 16 hidden units. We use direct policy search to train agents with CMA-ES [39], an evolution strategies (ES) method.
+
+Table 2: Cart-pole Tests. For each experiment, we report the average score and the standard deviation from 1000 test episodes. Our agent is trained only in the environment with 5 sensory inputs.   
+
+<table><tr><td></td><td>5obs</td><td>5 obs (shuffled)</td><td>10 obs</td><td>5 obs+5noise</td></tr><tr><td>FNN (trained with 5 obs)</td><td>593± 433</td><td>38±120</td><td>N/A</td><td>N/A</td></tr><tr><td>FNN (trained with 10 obs)</td><td>N/A</td><td>N/A</td><td>593± 433</td><td>137 ± 242</td></tr><tr><td>Ours (trained with 5 obs)</td><td>472 ± 426</td><td>471 ± 426</td><td>471 ± 425</td><td>461± 410</td></tr></table>
+
+Our agent can perform the task and balance the cart-pole from an initially random state. Its average score is slightly lower than the baseline (See column 1 of Table 2) because each sensory neuron requires some time steps in each episode to interpret the sensory input signal it receives. However, as a trade-off for the performance sacrifice, our agent can retain its performance even when the input sensor array is randomly shuffled, which is not the case for an FNN policy (column 2). Moreover, although our agent is only trained in an environment with five inputs, it can accept an arbitrary number of inputs in any order without re-training3. We test our agent by duplicating the 5 inputs to give the agent 10 observations (column 3). When we replace the 5 extra signals with white noises with $\sigma = 0 . 1$ (column 4), we do not see a significant drop in performance.
+
+![](images/e936068651bda3f4bd7821c2b1ba990bb252f222a45a0138170619aafcdfe44a.jpg)  
+Figure 4: Permutation invariant outputs. The output (16-dimensional global latent code) from the AttentionNeuron layer does not change when we input the sensor array as-is (left) or when we randomly shuffle the array (right). Yellow represents higher values, and blue for lower values.
+
+The AttentionNeuron layer should possess 2 properties to attain these: its output is permutation invariant to its input, and its output carries task-relevant information. Figure 4 is a visual confirmation of the permutation invariant property, whereby we plot the output messages from the layer and their changes over time from two tests. Using the same random seed, we keep the observation as-is in the first test but we shuffle the order in the second. As the figure shows, the output messages are identical in the two roll-outs. We also perform a simple linear regression analysis on the outputs (based on the shuffled inputs) to recover the 5 inputs in their original order. Table 3 shows the $R ^ { 2 }$ values4 from this analysis, suggesting that some important indicators (e.g. $\dot { x }$ and $\dot { \theta }$ ) are well represented in the output.
+
+Table 4: PyBullet Ant results.   
+
+<table><tr><td></td><td>Score</td></tr><tr><td>FNN (teacher)</td><td>2700±28</td></tr><tr><td>FNN (shuffled)</td><td>232 ± 112</td></tr><tr><td>Ours (ES, shuffled)</td><td>2576± 75</td></tr><tr><td>Ours (BC, shuffled)</td><td>2034 ± 948</td></tr><tr><td>Ours (BC, shuffled,larger)</td><td>2579 ± 457</td></tr></table>
+
+Table 3: Linear regression analysis on the output. For each of the $N = 5$ sensory inputs we have one linear regression model with $m _ { t } \in \mathcal { R } ^ { 1 6 }$ as the explanatory variables.   
+
+<table><tr><td>X</td><td></td><td>x</td><td>cos(0)</td><td>sin(0)</td><td>0</td></tr><tr><td>R²</td><td>0.354</td><td>0.620</td><td>0.626</td><td>0.233</td><td>0.550</td></tr></table>
+
+# 4.2 PyBullet Ant
+
+While direct policy search methods such as evolution strategies (ES) can train permutation invariant RL agents, oftentimes we already have access to pre-trained agents or recorded human data performing the task at hand. Behavior cloning (BC) can allow us to convert an existing policy to a version that is permutation invariant with desirable properties associated with it.
+
+In Table 4, we train a standard two-layer FNN policy to perform AntBulletEnv-v0, a 3D locomotion task in PyBullet [19], and use it as a teacher for BC. For comparison, we also train a two-layer agent with AttentionNeuron for its first layer. Both networks are trained with ES. Similar to CartPole, we expect to see a small performance drop due to some time steps required for the agent to interpret an arbitrarily ordered observation space. We then collect data from the FNN teacher policy to train permutation invariant agents using BC. More details of the BC setup can be found in Appendix A.4.2.
+
+The performance of the BC agent is lower than the one trained from scratch with ES, despite having the identical architecture. This suggests that the inductive bias that comes with permutation invariance may not match the original teacher network, so the small model used here may not be expressive enough to clone any teacher policy, resulting in a larger variance in performance. A benefit of gradient-based BC, compared to RL, is that we can easily train larger networks to fit the behavioral data. We show that increasing the size of the subsequent layers for BC does enhance the performance.
+
+As we will demonstrate next, BC is a useful technique for training permutation invariant agents in environments with high dimensional visual observations that may require larger networks.
+
+# 4.3 Atari Pong
+
+Here, we are interested in solving screen-shuffled versions of vision-based RL environments, where each observation frame is divided up into a grid of patches, and like a puzzle, the agent must process the patches in a shuffled order to determine a course of action to take. A shuffled version of Atari Pong [8] (See Figure 1, right pair) can be especially hard for humans to play when inductive biases from human priors [24] that expect a certain type of spatial structure is missing from the observations.
+
+But rather than throwing away the spatial structure entirely from our solution, we find that convolution neural network (CNN) policies work better than fully connected multi-layer perceptron (MLP) policies when trained with behavior cloning for Atari Pong. In this experiment, we reshape the output $m _ { t }$ of the AttentionNeuron layer from $\mathcal { R } ^ { \mathrm { \overline { { 4 } } 0 0 \times 3 2 } }$ to $\mathcal { R } ^ { 2 0 \times 2 0 \times 3 2 }$ , a 2D grid of latent codes, and pass this 2D grid into a CNN policy. This way, the role of the AttentionNeuron layer is to take a list of unordered observation patches, and learn to construct a 2D grid representation of the inputs to be used by a downstream policy that expects some form of spatial structure in the codes. Our permutation invariant policy trained with BC can consistently reach a perfect score of 21, even with shuffled screens. The details of the CNN policy and BC training can be found in Appendix A.4.3.
+
+![](images/549067c5798dc1934eede4cd5cba56c0785ef234a09cc12e65aa59d561201c53.jpg)  
+Figure 5: Mean test scores in Atari Pong, and example of a randomly-shuffled occluded observation. In the heat map, each value is the average score from 100 test episodes. For comparison, we show the original screen (left), and the agent’s observation (right). Discarded patches shown here in black.
+
+Unlike typical CNN policies, our agent can accept a subset of the screen, since the agent’s input is a variable-length list of patches. It would thus be interesting to deliberately randomly discard a certain percentage of the patches and see how the agent reacts. The net effect of this experiment for humans is similar to being asked to play a partially occluded and shuffled version of Atari Pong (see Figure 5, right). During training via BC, we randomly remove a percentage of observation patches. In tests, we fix the randomly selected positions of patches to discard during an entire episode.
+
+We present the results in a heat map in Figure 5 (left), where the y-axis shows the patches removed during training and the $\mathbf { X }$ -axis gives the patch occlusion ratio in tests. The diagram shows clear patterns for interpretation. Looking horizontally along each row, the performance drops because the agent sees less of the screen which increases the difficulty. Interestingly, an agent trained at a high occlusion rate of $8 0 \%$ rarely wins against the Atari opponent, but once it is presented with the full set of patches during tests, it is able to achieve a fair result by making use of the additional information.
+
+To gain insights into understanding the policy, we projected the AttentionNeuron layer’s output in a test roll-out to 2D space using t-SNE [73]. In Figure 6, we highlight several groups and show their corresponding inputs. The AttentionNeuron layer clearly learned to cluster inputs that share similar features. For example, the 3 sampled inputs in the blue group show the situation when the agent’s paddle moved toward the bottom of the screen and stayed there. Similarly, the orange group shows the cases when the ball was not in sight, this happened right before/after a game started/ended. We believe these discriminative outputs enabled the downstream policy to accomplish the agent’s task.
+
+![](images/b3fa260b78c1b6097f3890ed821cf0afb5f01180c14887d1920f7a286be10493.jpg)  
+Figure 6: 2D embedding of the AttentionNeuron layer’s output in a test episode. We highlight several representative groups in the plot, and show the sampled inputs from them. For each group, we show 3 corresponding inputs (rows) and unstack each to show the time dimension (columns).
+
+# 4.4 CarRacing
+
+We find that encouraging an agent to learn a coherent representation of a deliberately shuffled visual scene leads to agents with useful generalization properties. Such agents are still able to perform their task even if the visual background of the environment changes, despite being trained only on a single static background. Out-of-domain generalization is an active area, and here, we combine our method with AttentionAgent [72], a method that uses selective, hard-attention via a patch voting mechanism. AttentionAgents in [72] generalize well to several unseen visual environments where task irrelevant elements are modified, but fail to generalize to drastic background changes in a zero-shot setting.
+
+In this experiment, we combine the permutation invariant AttentionNeuron layer with the policy network used in AttentionAgent. As their hard-attention-based policy is non-differentiable, we train the entire system using ES. We reshape the AttentionNeuron layer’s outputs to adapt for the policy network. Specifically, we reshape the output message to $m _ { t } \in \mathcal { R } ^ { \bar { 3 } 2 \times 3 2 \times 1 \bar { 6 } }$ such that it can be viewed as a 32-by-32 grid of 16 channels. The end result is a policy with two layers of attention: the first layer outputs a latent code book to represent a shuffled scene, and the second layer performs hard attention to select the top $K = 1 0$ codes from a 2D global latent code book. A detailed description of the selective hard attention policy from [72] and other training details can be found in Appendix A.4.4.
+
+We first train the agent in the CarRacing [47] environment, and report the average score from 100 test roll-outs in Table 5. As the first column shows, our agent’s performance in the training environment is slightly lower but comparable to the baseline method, as expected. But because our agent accepts randomly shuffled inputs, it is still able to navigate even when the patches are shuffled. Figure 1 (left pair) gives an illustration, where the right screen is what our agent observes and the left is for human visualization. A human will find driving with the shuffled observation to be very difficult because we are not constantly exposed to such tasks, just like in the “reverse bicycle” example mentioned earlier.
+
+Without additional training or fine-tuning, we test whether the agent can also navigate in four modified environments where the green grass background is replaced with various images (See Figure 7). As Table 5 (from column 2) shows, our agent generalizes well to most of the test environments with only mild performance drops while the baseline method fails to generalize. We suspect this is because the AttentionNeuron layer has transformed the original RGB space to a useful hidden representation (represented by $m _ { t }$ ) that has eliminated task irrelevant information after observing and reasoning about the sequences of $( o _ { t } , a _ { t - 1 } )$ during training, enabling the downstream hard attention policy to work with an optimized abstract representation tailored for the policy, instead of raw RGB patches.
+
+Table 5: CarRacing Test Results.   
+
+<table><tr><td></td><td>Training Env</td><td>KOF</td><td>Mt. Fuji</td><td>Ukiyoe</td><td>DS</td></tr><tr><td>AttentionAgent [72]</td><td>901± 54</td><td>-81±4</td><td>-57±38</td><td>-107±50</td><td>-56±23</td></tr><tr><td>NetRand [49]</td><td>480±144</td><td>20±84</td><td>356±159</td><td>533 ±111</td><td>-27±34</td></tr><tr><td>NetRand + AttentionAgent</td><td>885±64</td><td>-51±14</td><td>709 ±94</td><td>656 ±131</td><td>122 ± 134</td></tr><tr><td>Ours</td><td>801 ±147</td><td>646 ±189</td><td>503 ±152</td><td>661 ±140</td><td>171 ± 146</td></tr></table>
+
+![](images/940f369d2887e0768f0b4c9b909a895d9e94341ddcc393edcdb9462c034992ac.jpg)  
+Figure 7: Screenshots of test environments. In each pair of images, the left is for human visualization and the right is what our agent sees. From the top left and in the clockwise order, the environments are “KOF”, “Mt. Fuji”, “DS” and “Ukiyoe”.
+
+We also compare our method to NetRand [49], a simple but effective technique developed to perform similar generalization tasks. In the second row of Table 5 are the results of training NetRand on the base CarRacing task. The CarRacing task proved to be too difficult for NetRand, but despite a low performance score of 480 in the training environment, the agent generalizes well to the “Mt. Fuji” and “Ukiyoe” modifications. In order to obtain a meaningful comparison, we combine NetRand with AttentionAgent so that it can get close to a mean score of 900 on the base task. To do that, we use NetRand as an input layer to the AttentionAgent policy network, and train the combination end-to-end using ES, which is consistent with our proposed method for this task. The combination attains a respectable mean score of 885, and as we can see in the third row of the above table, this approach also generalizes to a few of the unseen modifications of the CarRacing environment.
+
+Our score on the base CarRacing task is lower than NetRand, but this is expected since our agent requires some amount of time steps to identify each of the inputs (which could be shuffled), while the NetRand and AttentionAgent agent will simply fail on the shuffled versions of CarRacing. Despite this, our method still compares favorably on the generalization performance.
+
+We visualize the attentions from the AttentionNeuron layer in Figure 8. In CarRacing, the agent has learned to focus its attention (indicated by the highlighted patches) on the road boundaries which are intuitive to human beings and are critical to the task. Notice that the attended positions are consistent before and after the shuffling. More details about this visualization can be found in Appendix A.4.4.
+
+# 5 Discussion and Future Work
+
+In this work, we investigate the properties of RL agents that can treat their observations as an arbitrarily ordered, variable-length list of sensory inputs. By processing each input stream independently, and consolidating the processed information using attention, our agents can still perform their tasks even if the ordering of the observations is randomly permuted several times during an episode, without explicitly training for frequent re-shuffling (See Table 6).
+
+![](images/170c98fe4a23a2544086199996b8c96f7e6c18c04a9dfea32987d875fee1e848.jpg)  
+Figure 8: Attention visualization. We highlight the observed patches that receive the most attention. Left: Training environment. Right: Test environment with unseen background.
+
+Table 6: Reshuffle observations during a roll-out. In each test episode, we reshuffle the observations every $t$ steps. For CartPole, we test for 1000 episodes because of its larger task variance. For the other tasks, we report mean and standard deviation from 100 tests. All environments except for Atari Pong have a hard limit of 1000 time steps per episode. In Atari Pong, while the maximum length of an episode does not exist, we observed that an episode usually lasts for around 2500 time steps.
+
+<table><tr><td></td><td>CartPole</td><td>PyBullet Ant</td><td>Atari Pong</td><td>CarRacing</td></tr><tr><td>t=25</td><td>107± 146</td><td>2053±225</td><td>-20±1</td><td>732± 161</td></tr><tr><td>t=50</td><td>163 ± 198</td><td>2319 ±188</td><td>-20±2</td><td>772 ± 163</td></tr><tr><td>t=100</td><td>242 ± 254</td><td>2406 ±178</td><td>-10 ±12</td><td>768±167</td></tr><tr><td>t= 200</td><td>318±310</td><td>2493± 105</td><td>-2±17</td><td>774±182</td></tr><tr><td>t = 500</td><td>407±380</td><td>2548±87</td><td>18±9</td><td>805 ±158</td></tr><tr><td>No reshuffle</td><td>472 ±426</td><td>2576±75</td><td>21±0</td><td>801±147</td></tr></table>
+
+Applications By presenting the agent with shuffled, and even incomplete observations, we encourage it to interpret the meaning of each local sensory input and how they relate to the global context. This could be useful in many real world applications. For example, such policies could avoid errors due to cross-wiring or complex, dynamic input-output mappings when being deployed in real robots. A similar setup to the CartPole experiment with extra noisy channels could enable a system that receives thousands of noisy input channels to identify the small subset of channels with relevant information.
+
+Limitations For visual environments, patch size selection will affect both performance and computing complexity. We find that patches of 6x6 pixels work well for our tasks, as did $4 \mathbf { x } 4$ pixels to some extent, but single pixel observations fail to work. Small patch sizes also result in a large attention matrix which may be too costly to compute, unless approximations are used [15, 77, 80].
+
+Another limitation is that the permutation invariant property applies only to the inputs, and not to the outputs. While the ordering of the observations can be shuffled, the ordering of the actions cannot. For permutation invariant outputs to work, each action will require feedback from the environment, including reward information, in order to learn the relationship between itself and the environment.
+
+Societal Impact Like most algorithms proposed in computer science and machine learning, our method can be applied in ways that will have potentially positive or negative impacts to society. While our small-scale, self-contained experiments study only the properties of agents that are PI to their observations, and we believe our results do not directly cause harm to society, the robustness and flexible properties of the method may be of use for data-collection systems that receive data from a large variable number of sensors. For instance, one could apply permutation invariant sensory systems to process data from millions of sensors for anomaly detection, which may lead to both positive or negative impacts, if used in applications such as large-scale sensor analysis for weather forecasting, or deployed in large-scale surveillance systems that could undermine our basic freedoms.
+
+Our work also provides a way to view the Transformer [74] through the lens of self-organizing neural networks. Transformers are known to have potentially negative societal impacts highlighted in studies about possible data-leakage and privacy vulnerabilities [10], malicious misuse and issues concerning bias and fairness [7], and energy requirements for training them [69].
+
+Future Work An interesting future direction is to also make the action layer have the same properties, and model each “motor neuron” as a module connected using attention. With such methods, it may be possible to train an agent with an arbitrary number of legs, or control robots with different morphology using a single policy that is also provided with a reward signal as feedback. We look forward to seeing future works that include signals such as environmental rewards to train PI meta-learning agents that can adapt to not only changes in the observed environment, but also to changes to itself.
+
+# Acknowledgements
+
+The authors would like to thank Rishabh Agarwal, Jie Tan, Yingtao Tian, Douglas Eck, Aleksandra Faust and our NeurIPS2021 reviewers for valuable discussion and feedback. The experiments in this work were conducted using virtual machines provided by Google Cloud Platform.
+
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+    {
+        "type": "text",
+        "text": "The Sensory Neuron as a Transformer: Permutation-Invariant Neural Networks for Reinforcement Learning ",
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+        "type": "text",
+        "text": "Yujin Tang† Google Brain yujintang@google.com ",
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+        "text": "David Ha† Google Brain hadavid@google.com ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "In complex systems, we often observe complex global behavior emerge from a collection of agents interacting with each other in their environment, with each individual agent acting only on locally available information, without knowing the full picture. Such systems have inspired development of artificial intelligence algorithms in areas such as swarm optimization and cellular automata. Motivated by the emergence of collective behavior from complex cellular systems, we build systems that feed each sensory input from the environment into distinct, but identical neural networks, each with no fixed relationship with one another. We show that these sensory networks can be trained to integrate information received locally, and through communication via an attention mechanism, can collectively produce a globally coherent policy. Moreover, the system can still perform its task even if the ordering of its inputs is randomly permuted several times during an episode. These permutation invariant systems also display useful robustness and generalization properties that are broadly applicable. Interactive demo and videos of our results: https://attentionneuron.github.io/ ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
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+        "type": "text",
+        "text": "Sensory substitution refers to the brain’s ability to use one sensory modality (e.g., touch) to supply environmental information normally gathered by another sense (e.g., vision). Numerous studies have demonstrated that humans can adapt to changes in sensory inputs, even when they are fed into the wrong channels [5, 6, 25, 64]. But difficult adaptations–such as learning to “see” by interpreting visual information emitted from a grid of electrodes placed on one’s tongue [6], or learning to ride a “backwards” bicycle [64]–require months of training to attain mastery. Can we do better, and create artificial systems that can rapidly adapt to sensory substitutions, without the need to be retrained? ",
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+        "image_caption": [
+            "Figure 1: Comparison of visual input intended for the game player, and what our system receives. We partition the visual input from CarRacing (Left) and Atari Pong (right) into a 2D grid of small patches, and randomly permute their ordering. Each sensory neuron in the system receives a stream of visual input at a particular permuted patch location, and through coordination, must complete the task at hand, even if the visual ordering is randomly permuted again several times during an episode. "
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+        "text": "Modern deep learning systems are generally unable to adapt to a sudden reordering of sensory inputs, unless the model is retrained, or if the user manually corrects the ordering of the inputs for the model. However, techniques from continual meta-learning, such as adaptive weights [3, 36, 65], Hebbian-learning [52, 53, 57], and model-based [2, 20, 37, 38] approaches can help the model adapt to such changes, and remain a promising active area of research. ",
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+        "text": "In this work, we investigate agents that are explicitly designed to deal with sudden random reordering of their sensory inputs while performing a task. Motivated by recent developments in self-organizing neural networks [27, 55, 61] related to cellular automata [14, 17, 18, 58, 78], in our experiments, we feed each sensory input (which could be an individual state from a continuous control environment, or a patch of pixels from a visual environment) into an individual neural network module that integrates information from only this particular sensory input channel over time. While receiving information locally, each of these individual sensory neural network modules also continually broadcasts an output message. Inspired by the Set Transformer [48, 74] architecture, an attention mechanism combines these messages to form a global latent code which is then converted into the agent’s action space. The attention mechanism can be viewed as a form of adaptive weights of a neural network, and in this context, allows for an arbitrary number of sensory inputs that can be processed in any random order. ",
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+    {
+        "type": "text",
+        "text": "In our experiments, we find that each individual sensory neural network module, despite receiving only localized information, can still collectively produce a globally coherent policy, and that such a system can be trained to perform tasks in several popular reinforcement learning (RL) environments. Furthermore, our system can utilize a varying number of sensory input channels in any randomly permuted order, even when the order is shuffled again several times during an episode. ",
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+    {
+        "type": "text",
+        "text": "Permutation invariant systems have several advantages over traditional fixed-input systems. We find that encouraging a system to learn a coherent representation of a permutation invariant observation space leads to policies that are more robust and generalizes better to unseen situations. We show that, without additional training, our system continues to function even when we inject additional input channels containing noise or redundant information. In visual environments, we show that our system can be trained to perform a task even if it is given only a small fraction of randomly chosen patches from the screen, and at test time, if given more patches, the system can take advantage of the additional information to perform better. We also demonstrate that our system can generalize to visual environments with novel background images, despite training on a single fixed background. Lastly, to make training more practical, we propose a behavioral cloning scheme to convert policies trained with existing methods into a permutation invariant policy with desirable properties. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Self-organization is a process where some form of global order emerges from local interactions between parts of an initially disordered system. It is also a property observed in cellular automata (CA) [17, 18, 58], which are mathematical systems consisting of a grid of cells that perform computation by having each cell communicate with its immediate neighbors and performing a local computation to update its internal state. Such local interactions are useful in modeling complex systems [78] and have been applied to model non-linear dynamics in various fields [14]. Cellular Neural Networks [16] were first introduced in the 1980s to use neural networks in place of the algorithmic cells in CA systems. They were applied to perform image processing operations with parallel computation. Eventually, the concept of self-organizing neural networks found its way into deep learning in the form of Graph Neural Networks (GNN) [62, 79]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using modern deep learning tools, recent work demonstrates that neural CA, or self-organized neural networks performing only local computation, can generate (and re-generate) coherent images [55] and voxel scenes [70, 84], and even perform image classification [61]. Self-organizing neural network agents have been proposed in the RL domain [11, 12, 59, 60], with recent work demonstrating that shared local policies at the actuator level [43], through communicating with their immediate neighbors, can learn a global coherent policy for continuous control locomotion tasks. While existing CA-based approaches present a modular, self-organized solution, they are not inherently permutation invariant. In our work, we build on neural CA, and enable each cell to communicate beyond its immediate neighbors via an attention mechanism that enables permutation invariance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Meta-learning recurrent neural networks (RNN) [23, 40, 42, 76] have been proposed to approach the problem of learning the learning rules for a neural network using the reward or error signal, enabling meta-learners to learn to solve problems presented outside of their original training domains. The goals are to enable agents to continually learn from their environments in a single lifetime episode, and to obtain much better data efficiency than conventional learning methods such as stochastic gradient descent (SGD). A meta-learned policy that can adapt the weights of a neural network to its inputs during inference time have been proposed in fast weights [65, 67], associative weights [3], hypernetworks [36], and Hebbian-learning [52, 53] approaches. Recently works [46, 63] combine ideas of self-organization with meta-learning RNNs, and have demonstrated that modular metalearning RNN systems not only can learn to perform SGD-like learning rules, but can also discover more general learning rules that transfer to classification tasks on unseen datasets. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In contrast, the system presented here does not use an error or reward signal to meta-learn or fine-tune its policy. But rather, by using the shared modular building blocks from the meta-learning literature, we focus on learning or converting an existing policy to one that is permutation invariant, and we examine the characteristics such policies exhibit in a zero-shot setting, without additional training. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Attention can be viewed as an adaptive weight mechanism that alters the weight connections of a neural network layer based on what the inputs are. Linear dot-product attention has first been proposed for meta-learning [66], and versions of linear attention with sof tmax nonlinearity appeared later [33, 51], now made popular with Transformer [74]. The adaptive nature of attention provided the Transformer with a high degree of expressiveness, enabling it to learn inductive biases from large datasets and have been incorporated into state-of-the-art methods in natural language processing [9, 21], image recognition [22] and generation [26], audio and video domains [31, 44, 71]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Attention mechanisms have found many uses for RL [1, 13, 56, 68, 72, 83]. Our work here specifically uses attention to enable communication between arbitrary numbers of modules in an RL agent. While previous work [32, 45, 54, 75, 81, 85] utilized attention as a communication mechanism between independent neural network modules of a GNN, our work focuses on studying the permutation invariant properties of attention-based communication applied to RL agents. Related work [50] used permutation invariant critics to enhance performance of multi-agent RL. Building [34, 82], Set Transformers [48] investigated the use of attention explicitly for permutation invariant problems that deal with set-structured data, which have provided the theoretical foundation for our work. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Background ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to devise an agent that is permutation invariant (PI) in the action space to the permutations in the input space. While it is possible to acquire a quasi-PI agent by training with randomly shuffled observations and hope the agent’s policy network has enough capacity to memorize all the patterns, we aim for a design that achieves true PI even if the agent is trained with fix-ordered observations. Mathematically, we are looking for a non-trivial function $f ( \\boldsymbol { x } ) : \\mathcal { R } ^ { n } \\mapsto \\mathcal { R } ^ { m }$ such that $f ( x [ s ] ) = f ( x )$ for any $x \\in \\mathcal { R } ^ { n }$ , and $s$ is any permutation of the indices $\\{ 1 , \\cdots , n \\}$ . A different but closely related concept is permutation equivariance (PE) which can be described by a function $h ( x ) : \\mathcal { R } ^ { \\mathit { \\hat { n } } } \\mapsto \\mathcal { R } ^ { \\mathit { n } }$ such that $\\bar { h ( x [ s ] ) } = h ( x ) [ \\bar { s } ]$ . Unlike PI, the dimensions of the input and the output must equal in PE. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Self-attentions can be PE. In its simplest form, self-attention is described as $y = \\sigma ( Q K ^ { \\top } ) V$ where $Q , K \\in \\mathcal { R } ^ { n \\times d _ { q } } , V \\in \\mathcal { R } ^ { n \\times d _ { v } }$ are the Query, Key and Value matrices and $\\sigma ( \\cdot )$ is a non-linear function. In most scenarios, $Q , K , V$ are functions of the input $x \\in \\mathcal { R } ^ { n }$ (e.g. linear transformations), and permuting $x$ therefore is equivalent to permuting the rows in $Q , K , V$ and based on its definition it is straightforward to verify the PE property. Set Transformer [48] cleverly replaced $Q$ with a set of learnable seed vectors, so it is no longer a function of input $x$ , thus enabling the output to become PI. A simple, intuitive explanation of the PI property of self-attention is available in Appendix A.1. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Sensory Neurons with Attention ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To create PI agents, we propose to add an extra layer in front of the agent’s policy network $\\pi$ , which accepts the current observation $o _ { t }$ and the previous action $a _ { t - 1 }$ as its inputs. We call this new layer AttentionNeuron, and Figure 2 gives an overview of our method. Inside AttentionNeuron, we model the observation $o _ { t }$ as an arbitrarily ordered, variable-length list of sensory inputs, each of which is passed into its own sensory neuron, a neural network module. Each sensory neuron only has partial access to the agent’s observation, at time $t$ , the ith neuron can see only the ith component of the observation $o _ { t } [ i ]$ . Combined with the previous action $a _ { t - 1 }$ , each sensory neuron computes messages $f _ { k } \\big ( o _ { t } [ i ] , a _ { t - 1 } \\big ) ^ { \\smash { \\cdot } }$ and $f _ { v } ( o _ { t } [ i ] )$ that are broadcast to the rest of the system. We then use attention to aggregate these messages into a global latent code, $m _ { t }$ , that is $\\mathrm { P I }$ with respect to the inputs. ",
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+    {
+        "type": "image",
+        "img_path": "images/737a2f978bc093f562aa43d22f92f0d50d4a66a7cf9d68809b7f1ea690a980c3.jpg",
+        "image_caption": [
+            "Figure 2: Overview of Method. AttentionNeuron is a standalone layer, in which each sensory neuron only has access to a part of the unordered observations $o _ { t }$ . Together with the agent’s previous action $a _ { t - 1 }$ , each neuron generates messages independently using the shared functions $f _ { k } \\big ( \\overline { { o } } _ { t } [ i ] , a _ { t - 1 } \\big )$ and $f _ { v } ( o _ { t } [ i ] )$ . The attention mechanism summarizes the messages into a global latent code $m _ { t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The operations inside AttentionNeuron can be described by the following two equations. For clarity, Table 1 summarizes the notations as well as the corresponding setups we used for the experiments. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/92b74255318849c73b7cac622f28760a0addb16cbc1b6000bb174e603d8d97fc.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\displaystyle K ( o _ { t } , a _ { t - 1 } ) = \\left[ \\begin{array} { c } { { f _ { k } ( o _ { t } [ 1 ] , a _ { t - 1 } ) } } \\\\ { { \\cdots \\cdot } } \\\\ { { f _ { k } ( o _ { t } [ N ] , a _ { t - 1 } ) } } \\end{array} \\right] \\in { \\mathbb R } ^ { N \\times d _ { f _ { k } } } , V ( o _ { t } ) = \\left[ \\begin{array} { c } { { f _ { v } ( o _ { t } [ 1 ] ) } } \\\\ { { \\cdots \\cdot } } \\\\ { { f _ { v } ( o _ { t } [ N ] ) } } \\end{array} \\right] \\in { \\mathbb R } ^ { N \\times d _ { f _ { v } } } } } \\\\ { { m _ { t } = \\sigma ( \\frac { [ Q W _ { q } ] [ K ( o _ { t } , a _ { t - 1 } ) W _ { k } ] ^ { \\top } } { \\sqrt { d _ { q } } } ) [ V ( o _ { t } ) W _ { v } ] } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Equation 1 shows how each of the $N$ sensory neuron independently generates its messages $f _ { k }$ and $f _ { v }$ , which are functions shared across all sensory neurons. Equation 2 shows the attention mechanism aggregate these messages. Note that although we could have absorbed the projection matrices $W _ { q } , W _ { k } , W _ { v }$ into $Q , K , V$ , we keep them in the equation to show explicitly the formulation. Equation 2 is almost identical to the simple definition of self-attention mentioned earlier. Following [48], we make our $Q$ matrix a bank of fixed embeddings, rather than depend on the observation $o _ { t }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that permuting the observations only affects the row orders of $K$ and $V$ , and that applying the same permutation to the rows of both $K$ and $V$ still results in the same $m _ { t }$ which is PI. As long as we set constant the number of rows in $Q$ , the change in the input size affects only the number of rows in $K$ and $V$ and does not affect the output $m _ { t }$ . In other words, our agent can accept inputs of arbitrary length and output a fixed sized $m _ { t }$ . Later, we apply this flexibility of input dimensions to RL agents. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/d52781cbebb90c2538ccdf007944c1bd17544c421a5cd1428b3b3ef4b2faa9fa.jpg",
+        "table_caption": [
+            "Table 1: In this notation list, we provide the dimensions used in our model for different RL environ ments, to give the reader a sense of the relative magnitudes involved in each part of the system. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Description</td><td>Notation</td><td>CartPole</td><td>Ant</td><td>CarRacing</td><td>Atari Pong</td></tr><tr><td>Full observation space</td><td>Ot</td><td>R5</td><td>R28</td><td>R96×96×4</td><td>R84x84x4</td></tr><tr><td>Individual sensory input space</td><td>Ot[i]</td><td>R1</td><td>R1</td><td>R6×6×4=144</td><td>R6×6×4=144</td></tr><tr><td>Number of sensory neurons</td><td>N</td><td>5</td><td>28</td><td>(96/6)² = 256</td><td>(84/6)² = 196</td></tr><tr><td>Dimension of action space</td><td>|A|</td><td>1</td><td>8</td><td>3</td><td>6 (one-hot)</td></tr><tr><td>Number of embeddings in Q</td><td>M</td><td>16</td><td>32</td><td>1024</td><td>400</td></tr><tr><td>Projection matrix for Q</td><td>Wq</td><td>R8×32</td><td>R8×32</td><td>R8×16</td><td>R8×32</td></tr><tr><td>Projection matrix for K</td><td>Wk</td><td>R8×32</td><td>R8×32</td><td>R111×16</td><td>R114×32</td></tr><tr><td>Projection matrix for V</td><td>Wu</td><td>I</td><td>I</td><td>R144×16</td><td>R144×32</td></tr><tr><td>Post-attention activation function</td><td>9()</td><td>tanh</td><td>tanh</td><td>softmax</td><td>softmax</td></tr><tr><td>Global latent code</td><td>mt</td><td>R16</td><td>R32</td><td>R1024×16</td><td>R400×32</td></tr></table>",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 Design Choices ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It is worthwhile to have a discussion on the design choices made. Since the ordering of the input is arbitrary, each sensory neuron is required to interpret and identify their received signal. To achieve this, we want $f _ { k } ( o _ { t } [ \\dot { i } ] , a _ { t - 1 } )$ to have temporal memories. In practice, we find both RNNs and feed-forward neural networks (FNN) with stacked observations work well, with FNNs being more practical for environments with high dimensional observations. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition to the temporal memory, including previous actions is important for the input identification too. Although the former allows the neurons to infer the input signals based on the characteristics of the temporal stream, this may not be sufficient. For example, when controlling a legged robot, most of the sensor readings are joint angles and velocities from the legs, which are not only numerically identically bounded but also change in similar patterns. The inclusion of previous actions gives each sensory neuron a chance to infer the casual relationship between the input channel and the applied actions, which helps with the input identification. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, in Equation 2 we could have combined $Q W _ { q } \\in \\mathcal { R } ^ { M \\times d _ { q } }$ as a single learnable parameters matrix, but we separate them for two reasons. First, by factoring into two matrices, we can reduce the number of learnable parameters. Second, we find that instead of making $Q$ learnable, using the positional encoding proposed in Transformer [74] encourages the attention mechanism to generate distinct codes. Here we use the row indices in $Q$ as the positions for encoding. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We experiment on several different RL environments to study various properties of permutation invariant RL agents. Due to the nature of the underlying tasks, we will describe the different architectures of the policy networks used and discuss various different training methods. However, the AttentionNeuron layers in all agents are similar, so we first describe the common setups. Hyperparameters and other details for all experiments are summarized in Appendix A.4. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For non-vision continuous control tasks, the agent receives an observation vector $o _ { t } \\in \\mathcal { R } ^ { | O | }$ at time $t$ We assign $N = | O |$ sensory neurons for the tasks, each of which sees one element from the vector, hence $o _ { t } [ i ] \\in \\mathcal { R } ^ { 1 } , i = 1 , \\cdots , | O |$ . We use an LSTM [41] as our $f _ { k } ( o _ { t } [ i ] , a _ { t - 1 } )$ to generate Keys, the input size of which is $1 + | A |$ (2 for Cart-Pole and 9 for PyBullet Ant). A simple pass-through function $f ( x ) = x$ serves as our $f _ { v } ( o _ { t } [ i ] )$ , and $\\sigma ( \\cdot )$ is tanh. For simplicity, we find $W _ { v } = I$ works well for the tasks, so the learnable components are the LSTM, $W _ { q }$ and $W _ { k }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For vision based tasks, we gray-scale and stack $k = 4$ consecutive RGB frames from the environment, and thus our agent observes $\\mathbf { \\bar { \\rho } } _ { { o _ { t } } } \\in \\mathcal { R } ^ { H \\times W \\times k }$ . $o _ { t }$ is split into non-overlapping patches of size $P = 6$ using a sliding window, so each sensory neuron observes $o _ { t } [ i ] \\in \\mathcal { R } ^ { 6 \\times 6 \\times \\dot { k } } .$ . Here, $f _ { v } ( o _ { t } [ i ] )$ flattens the data and returns it, hence $V ( o _ { t } )$ returns a tensor of shape $N \\times d _ { f _ { v } } = N \\times ( 6 \\times 6 \\times 4 ) = N \\times 1 4 4$ Due to the high dimensionality for vision tasks, we do not use RNNs for $f _ { k }$ , but instead use a simpler method to process each sensory input. $f _ { k } \\left( o _ { t } [ i ] , a _ { t - 1 } \\right)$ takes the difference between consecutive frames $( o _ { t } [ i ] )$ , then flattens the result, appends $a _ { t - 1 }$ , and returns the concatenated vector. $K ( o _ { t } , a _ { t - 1 } )$ thus gives a tensor of shape $N \\times d _ { f _ { k } } = N \\times [ ( 6 \\times 6 \\times 3 ) + | A | ] = N \\times ( 1 0 8 + | A | )$ (111 for CarRacing and 114 for Atari Pong). We use sof tmax as the non-linear activation function $\\sigma ( \\cdot )$ , and we apply layer normalization [4] to both the input patches and the output latent code. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 Cart-pole swing up ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We examine Cart-pole swing up [29, 30, 35, 86] to first illustrate our method, and also use it to provide a clear analysis of the attention mechanism. We use CartPoleSwingUpHarder [28], a more difficult version of the task where the initial positions and velocities are highly randomized, leading to a higher variance of task scenarios. In the environment, the agent observes $[ x , \\dot { x } , c o s ( \\theta ) , s i n ( \\theta ) , \\dot { \\theta } ]$ , outputs a scalar action, and is rewarded at each step for getting $x$ close to 0 and $c o s ( \\theta )$ close to 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/97af204355d041941b28feb377276f15d632c462e1a6f77ed670189c47a73b60.jpg",
+        "image_caption": [
+            "Figure 3: Interactive demo of CartPoleSwingUpHarder. In our web demo2, the user can shuffle the order of the 5 inputs at any time, and observe how the agent adapts to the new ordering of the inputs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            720,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use a two-layer neural network as our agent. The first layer is an AttentionNeuron layer with $N = 5$ sensory neurons and outputs $m _ { t } \\in \\mathcal { R } ^ { \\mathrm { { \\tilde { 1 6 } } } }$ . A linear layer takes $m _ { t }$ as input and outputs a scalar action. For comparison, we also trained an agent with a two-layer FNN policy with 16 hidden units. We use direct policy search to train agents with CMA-ES [39], an evolution strategies (ES) method. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/09c2815bbb2ecf143ffce6c76d0ec3b8c4e1d8b3dec9a9e5a12ab4901d7debf8.jpg",
+        "table_caption": [
+            "Table 2: Cart-pole Tests. For each experiment, we report the average score and the standard deviation from 1000 test episodes. Our agent is trained only in the environment with 5 sensory inputs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>5obs</td><td>5 obs (shuffled)</td><td>10 obs</td><td>5 obs+5noise</td></tr><tr><td>FNN (trained with 5 obs)</td><td>593± 433</td><td>38±120</td><td>N/A</td><td>N/A</td></tr><tr><td>FNN (trained with 10 obs)</td><td>N/A</td><td>N/A</td><td>593± 433</td><td>137 ± 242</td></tr><tr><td>Ours (trained with 5 obs)</td><td>472 ± 426</td><td>471 ± 426</td><td>471 ± 425</td><td>461± 410</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our agent can perform the task and balance the cart-pole from an initially random state. Its average score is slightly lower than the baseline (See column 1 of Table 2) because each sensory neuron requires some time steps in each episode to interpret the sensory input signal it receives. However, as a trade-off for the performance sacrifice, our agent can retain its performance even when the input sensor array is randomly shuffled, which is not the case for an FNN policy (column 2). Moreover, although our agent is only trained in an environment with five inputs, it can accept an arbitrary number of inputs in any order without re-training3. We test our agent by duplicating the 5 inputs to give the agent 10 observations (column 3). When we replace the 5 extra signals with white noises with $\\sigma = 0 . 1$ (column 4), we do not see a significant drop in performance. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/e936068651bda3f4bd7821c2b1ba990bb252f222a45a0138170619aafcdfe44a.jpg",
+        "image_caption": [
+            "Figure 4: Permutation invariant outputs. The output (16-dimensional global latent code) from the AttentionNeuron layer does not change when we input the sensor array as-is (left) or when we randomly shuffle the array (right). Yellow represents higher values, and blue for lower values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The AttentionNeuron layer should possess 2 properties to attain these: its output is permutation invariant to its input, and its output carries task-relevant information. Figure 4 is a visual confirmation of the permutation invariant property, whereby we plot the output messages from the layer and their changes over time from two tests. Using the same random seed, we keep the observation as-is in the first test but we shuffle the order in the second. As the figure shows, the output messages are identical in the two roll-outs. We also perform a simple linear regression analysis on the outputs (based on the shuffled inputs) to recover the 5 inputs in their original order. Table 3 shows the $R ^ { 2 }$ values4 from this analysis, suggesting that some important indicators (e.g. $\\dot { x }$ and $\\dot { \\theta }$ ) are well represented in the output. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/103556accd0b1a3f3816eca4a2e0eb6910795dbfde981b99faadbee7d11e0859.jpg",
+        "table_caption": [
+            "Table 4: PyBullet Ant results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Score</td></tr><tr><td>FNN (teacher)</td><td>2700±28</td></tr><tr><td>FNN (shuffled)</td><td>232 ± 112</td></tr><tr><td>Ours (ES, shuffled)</td><td>2576± 75</td></tr><tr><td>Ours (BC, shuffled)</td><td>2034 ± 948</td></tr><tr><td>Ours (BC, shuffled,larger)</td><td>2579 ± 457</td></tr></table>",
+        "bbox": [
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+            593,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/3d55108b6b4ff606798c8cc1340672171b1fdffb6ceab9b5cbb97b4181a3d24e.jpg",
+        "table_caption": [
+            "Table 3: Linear regression analysis on the output. For each of the $N = 5$ sensory inputs we have one linear regression model with $m _ { t } \\in \\mathcal { R } ^ { 1 6 }$ as the explanatory variables. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>X</td><td></td><td>x</td><td>cos(0)</td><td>sin(0)</td><td>0</td></tr><tr><td>R²</td><td>0.354</td><td>0.620</td><td>0.626</td><td>0.233</td><td>0.550</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 PyBullet Ant ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While direct policy search methods such as evolution strategies (ES) can train permutation invariant RL agents, oftentimes we already have access to pre-trained agents or recorded human data performing the task at hand. Behavior cloning (BC) can allow us to convert an existing policy to a version that is permutation invariant with desirable properties associated with it. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Table 4, we train a standard two-layer FNN policy to perform AntBulletEnv-v0, a 3D locomotion task in PyBullet [19], and use it as a teacher for BC. For comparison, we also train a two-layer agent with AttentionNeuron for its first layer. Both networks are trained with ES. Similar to CartPole, we expect to see a small performance drop due to some time steps required for the agent to interpret an arbitrarily ordered observation space. We then collect data from the FNN teacher policy to train permutation invariant agents using BC. More details of the BC setup can be found in Appendix A.4.2. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The performance of the BC agent is lower than the one trained from scratch with ES, despite having the identical architecture. This suggests that the inductive bias that comes with permutation invariance may not match the original teacher network, so the small model used here may not be expressive enough to clone any teacher policy, resulting in a larger variance in performance. A benefit of gradient-based BC, compared to RL, is that we can easily train larger networks to fit the behavioral data. We show that increasing the size of the subsequent layers for BC does enhance the performance. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As we will demonstrate next, BC is a useful technique for training permutation invariant agents in environments with high dimensional visual observations that may require larger networks. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 Atari Pong ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            226,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here, we are interested in solving screen-shuffled versions of vision-based RL environments, where each observation frame is divided up into a grid of patches, and like a puzzle, the agent must process the patches in a shuffled order to determine a course of action to take. A shuffled version of Atari Pong [8] (See Figure 1, right pair) can be especially hard for humans to play when inductive biases from human priors [24] that expect a certain type of spatial structure is missing from the observations. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "But rather than throwing away the spatial structure entirely from our solution, we find that convolution neural network (CNN) policies work better than fully connected multi-layer perceptron (MLP) policies when trained with behavior cloning for Atari Pong. In this experiment, we reshape the output $m _ { t }$ of the AttentionNeuron layer from $\\mathcal { R } ^ { \\mathrm { \\overline { { 4 } } 0 0 \\times 3 2 } }$ to $\\mathcal { R } ^ { 2 0 \\times 2 0 \\times 3 2 }$ , a 2D grid of latent codes, and pass this 2D grid into a CNN policy. This way, the role of the AttentionNeuron layer is to take a list of unordered observation patches, and learn to construct a 2D grid representation of the inputs to be used by a downstream policy that expects some form of spatial structure in the codes. Our permutation invariant policy trained with BC can consistently reach a perfect score of 21, even with shuffled screens. The details of the CNN policy and BC training can be found in Appendix A.4.3. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/549067c5798dc1934eede4cd5cba56c0785ef234a09cc12e65aa59d561201c53.jpg",
+        "image_caption": [
+            "Figure 5: Mean test scores in Atari Pong, and example of a randomly-shuffled occluded observation. In the heat map, each value is the average score from 100 test episodes. For comparison, we show the original screen (left), and the agent’s observation (right). Discarded patches shown here in black. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "Unlike typical CNN policies, our agent can accept a subset of the screen, since the agent’s input is a variable-length list of patches. It would thus be interesting to deliberately randomly discard a certain percentage of the patches and see how the agent reacts. The net effect of this experiment for humans is similar to being asked to play a partially occluded and shuffled version of Atari Pong (see Figure 5, right). During training via BC, we randomly remove a percentage of observation patches. In tests, we fix the randomly selected positions of patches to discard during an entire episode. ",
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+        "type": "text",
+        "text": "We present the results in a heat map in Figure 5 (left), where the y-axis shows the patches removed during training and the $\\mathbf { X }$ -axis gives the patch occlusion ratio in tests. The diagram shows clear patterns for interpretation. Looking horizontally along each row, the performance drops because the agent sees less of the screen which increases the difficulty. Interestingly, an agent trained at a high occlusion rate of $8 0 \\%$ rarely wins against the Atari opponent, but once it is presented with the full set of patches during tests, it is able to achieve a fair result by making use of the additional information. ",
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+        "type": "text",
+        "text": "To gain insights into understanding the policy, we projected the AttentionNeuron layer’s output in a test roll-out to 2D space using t-SNE [73]. In Figure 6, we highlight several groups and show their corresponding inputs. The AttentionNeuron layer clearly learned to cluster inputs that share similar features. For example, the 3 sampled inputs in the blue group show the situation when the agent’s paddle moved toward the bottom of the screen and stayed there. Similarly, the orange group shows the cases when the ball was not in sight, this happened right before/after a game started/ended. We believe these discriminative outputs enabled the downstream policy to accomplish the agent’s task. ",
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+        "img_path": "images/b3fa260b78c1b6097f3890ed821cf0afb5f01180c14887d1920f7a286be10493.jpg",
+        "image_caption": [
+            "Figure 6: 2D embedding of the AttentionNeuron layer’s output in a test episode. We highlight several representative groups in the plot, and show the sampled inputs from them. For each group, we show 3 corresponding inputs (rows) and unstack each to show the time dimension (columns). "
+        ],
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+    {
+        "type": "text",
+        "text": "4.4 CarRacing ",
+        "text_level": 1,
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+        "text": "We find that encouraging an agent to learn a coherent representation of a deliberately shuffled visual scene leads to agents with useful generalization properties. Such agents are still able to perform their task even if the visual background of the environment changes, despite being trained only on a single static background. Out-of-domain generalization is an active area, and here, we combine our method with AttentionAgent [72], a method that uses selective, hard-attention via a patch voting mechanism. AttentionAgents in [72] generalize well to several unseen visual environments where task irrelevant elements are modified, but fail to generalize to drastic background changes in a zero-shot setting. ",
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+    {
+        "type": "text",
+        "text": "In this experiment, we combine the permutation invariant AttentionNeuron layer with the policy network used in AttentionAgent. As their hard-attention-based policy is non-differentiable, we train the entire system using ES. We reshape the AttentionNeuron layer’s outputs to adapt for the policy network. Specifically, we reshape the output message to $m _ { t } \\in \\mathcal { R } ^ { \\bar { 3 } 2 \\times 3 2 \\times 1 \\bar { 6 } }$ such that it can be viewed as a 32-by-32 grid of 16 channels. The end result is a policy with two layers of attention: the first layer outputs a latent code book to represent a shuffled scene, and the second layer performs hard attention to select the top $K = 1 0$ codes from a 2D global latent code book. A detailed description of the selective hard attention policy from [72] and other training details can be found in Appendix A.4.4. ",
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+        "type": "text",
+        "text": "We first train the agent in the CarRacing [47] environment, and report the average score from 100 test roll-outs in Table 5. As the first column shows, our agent’s performance in the training environment is slightly lower but comparable to the baseline method, as expected. But because our agent accepts randomly shuffled inputs, it is still able to navigate even when the patches are shuffled. Figure 1 (left pair) gives an illustration, where the right screen is what our agent observes and the left is for human visualization. A human will find driving with the shuffled observation to be very difficult because we are not constantly exposed to such tasks, just like in the “reverse bicycle” example mentioned earlier. ",
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+        "type": "text",
+        "text": "Without additional training or fine-tuning, we test whether the agent can also navigate in four modified environments where the green grass background is replaced with various images (See Figure 7). As Table 5 (from column 2) shows, our agent generalizes well to most of the test environments with only mild performance drops while the baseline method fails to generalize. We suspect this is because the AttentionNeuron layer has transformed the original RGB space to a useful hidden representation (represented by $m _ { t }$ ) that has eliminated task irrelevant information after observing and reasoning about the sequences of $( o _ { t } , a _ { t - 1 } )$ during training, enabling the downstream hard attention policy to work with an optimized abstract representation tailored for the policy, instead of raw RGB patches. ",
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+        "table_caption": [
+            "Table 5: CarRacing Test Results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Training Env</td><td>KOF</td><td>Mt. Fuji</td><td>Ukiyoe</td><td>DS</td></tr><tr><td>AttentionAgent [72]</td><td>901± 54</td><td>-81±4</td><td>-57±38</td><td>-107±50</td><td>-56±23</td></tr><tr><td>NetRand [49]</td><td>480±144</td><td>20±84</td><td>356±159</td><td>533 ±111</td><td>-27±34</td></tr><tr><td>NetRand + AttentionAgent</td><td>885±64</td><td>-51±14</td><td>709 ±94</td><td>656 ±131</td><td>122 ± 134</td></tr><tr><td>Ours</td><td>801 ±147</td><td>646 ±189</td><td>503 ±152</td><td>661 ±140</td><td>171 ± 146</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/940f369d2887e0768f0b4c9b909a895d9e94341ddcc393edcdb9462c034992ac.jpg",
+        "image_caption": [
+            "Figure 7: Screenshots of test environments. In each pair of images, the left is for human visualization and the right is what our agent sees. From the top left and in the clockwise order, the environments are “KOF”, “Mt. Fuji”, “DS” and “Ukiyoe”. "
+        ],
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+    {
+        "type": "text",
+        "text": "We also compare our method to NetRand [49], a simple but effective technique developed to perform similar generalization tasks. In the second row of Table 5 are the results of training NetRand on the base CarRacing task. The CarRacing task proved to be too difficult for NetRand, but despite a low performance score of 480 in the training environment, the agent generalizes well to the “Mt. Fuji” and “Ukiyoe” modifications. In order to obtain a meaningful comparison, we combine NetRand with AttentionAgent so that it can get close to a mean score of 900 on the base task. To do that, we use NetRand as an input layer to the AttentionAgent policy network, and train the combination end-to-end using ES, which is consistent with our proposed method for this task. The combination attains a respectable mean score of 885, and as we can see in the third row of the above table, this approach also generalizes to a few of the unseen modifications of the CarRacing environment. ",
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+    {
+        "type": "text",
+        "text": "Our score on the base CarRacing task is lower than NetRand, but this is expected since our agent requires some amount of time steps to identify each of the inputs (which could be shuffled), while the NetRand and AttentionAgent agent will simply fail on the shuffled versions of CarRacing. Despite this, our method still compares favorably on the generalization performance. ",
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+    {
+        "type": "text",
+        "text": "We visualize the attentions from the AttentionNeuron layer in Figure 8. In CarRacing, the agent has learned to focus its attention (indicated by the highlighted patches) on the road boundaries which are intuitive to human beings and are critical to the task. Notice that the attended positions are consistent before and after the shuffling. More details about this visualization can be found in Appendix A.4.4. ",
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+    {
+        "type": "text",
+        "text": "5 Discussion and Future Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we investigate the properties of RL agents that can treat their observations as an arbitrarily ordered, variable-length list of sensory inputs. By processing each input stream independently, and consolidating the processed information using attention, our agents can still perform their tasks even if the ordering of the observations is randomly permuted several times during an episode, without explicitly training for frequent re-shuffling (See Table 6). ",
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+        "img_path": "images/170c98fe4a23a2544086199996b8c96f7e6c18c04a9dfea32987d875fee1e848.jpg",
+        "image_caption": [
+            "Figure 8: Attention visualization. We highlight the observed patches that receive the most attention. Left: Training environment. Right: Test environment with unseen background. "
+        ],
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+        "type": "text",
+        "text": "Table 6: Reshuffle observations during a roll-out. In each test episode, we reshuffle the observations every $t$ steps. For CartPole, we test for 1000 episodes because of its larger task variance. For the other tasks, we report mean and standard deviation from 100 tests. All environments except for Atari Pong have a hard limit of 1000 time steps per episode. In Atari Pong, while the maximum length of an episode does not exist, we observed that an episode usually lasts for around 2500 time steps. ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CartPole</td><td>PyBullet Ant</td><td>Atari Pong</td><td>CarRacing</td></tr><tr><td>t=25</td><td>107± 146</td><td>2053±225</td><td>-20±1</td><td>732± 161</td></tr><tr><td>t=50</td><td>163 ± 198</td><td>2319 ±188</td><td>-20±2</td><td>772 ± 163</td></tr><tr><td>t=100</td><td>242 ± 254</td><td>2406 ±178</td><td>-10 ±12</td><td>768±167</td></tr><tr><td>t= 200</td><td>318±310</td><td>2493± 105</td><td>-2±17</td><td>774±182</td></tr><tr><td>t = 500</td><td>407±380</td><td>2548±87</td><td>18±9</td><td>805 ±158</td></tr><tr><td>No reshuffle</td><td>472 ±426</td><td>2576±75</td><td>21±0</td><td>801±147</td></tr></table>",
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+        "type": "text",
+        "text": "Applications By presenting the agent with shuffled, and even incomplete observations, we encourage it to interpret the meaning of each local sensory input and how they relate to the global context. This could be useful in many real world applications. For example, such policies could avoid errors due to cross-wiring or complex, dynamic input-output mappings when being deployed in real robots. A similar setup to the CartPole experiment with extra noisy channels could enable a system that receives thousands of noisy input channels to identify the small subset of channels with relevant information. ",
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+    {
+        "type": "text",
+        "text": "Limitations For visual environments, patch size selection will affect both performance and computing complexity. We find that patches of 6x6 pixels work well for our tasks, as did $4 \\mathbf { x } 4$ pixels to some extent, but single pixel observations fail to work. Small patch sizes also result in a large attention matrix which may be too costly to compute, unless approximations are used [15, 77, 80]. ",
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+    {
+        "type": "text",
+        "text": "Another limitation is that the permutation invariant property applies only to the inputs, and not to the outputs. While the ordering of the observations can be shuffled, the ordering of the actions cannot. For permutation invariant outputs to work, each action will require feedback from the environment, including reward information, in order to learn the relationship between itself and the environment. ",
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+        "type": "text",
+        "text": "Societal Impact Like most algorithms proposed in computer science and machine learning, our method can be applied in ways that will have potentially positive or negative impacts to society. While our small-scale, self-contained experiments study only the properties of agents that are PI to their observations, and we believe our results do not directly cause harm to society, the robustness and flexible properties of the method may be of use for data-collection systems that receive data from a large variable number of sensors. For instance, one could apply permutation invariant sensory systems to process data from millions of sensors for anomaly detection, which may lead to both positive or negative impacts, if used in applications such as large-scale sensor analysis for weather forecasting, or deployed in large-scale surveillance systems that could undermine our basic freedoms. ",
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+    {
+        "type": "text",
+        "text": "Our work also provides a way to view the Transformer [74] through the lens of self-organizing neural networks. Transformers are known to have potentially negative societal impacts highlighted in studies about possible data-leakage and privacy vulnerabilities [10], malicious misuse and issues concerning bias and fairness [7], and energy requirements for training them [69]. ",
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+    {
+        "type": "text",
+        "text": "Future Work An interesting future direction is to also make the action layer have the same properties, and model each “motor neuron” as a module connected using attention. With such methods, it may be possible to train an agent with an arbitrary number of legs, or control robots with different morphology using a single policy that is also provided with a reward signal as feedback. We look forward to seeing future works that include signals such as environmental rewards to train PI meta-learning agents that can adapt to not only changes in the observed environment, but also to changes to itself. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Rishabh Agarwal, Jie Tan, Yingtao Tian, Douglas Eck, Aleksandra Faust and our NeurIPS2021 reviewers for valuable discussion and feedback. The experiments in this work were conducted using virtual machines provided by Google Cloud Platform. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
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