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+# LEARN2WEIGHT: WEIGHTS TRANSFER DEFENSE AGAINST SIMILAR-DOMAIN ADVERSARIAL ATTACKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent work in black-box adversarial attacks for NLP systems has attracted attention. Prior black-box attacks assume that attackers can observe output labels from target models based on selected inputs. In this work, inspired by adversarial transferability, we propose a new type of black-box NLP adversarial attack that an attacker can choose a similar domain and transfer the adversarial examples to the target domain and cause poor performance in target model. Based on domain adaptation theory, we then propose a defensive strategy, called Learn2Weight, which trains to predict the weight adjustments for target model in order to defense the attack of similar-domain adversarial examples. Using Amazon multi-domain sentiment classification dataset, we empirically show that Learn2Weight model is effective against the attack compared to standard black-box defense methods such as adversarial training and defense distillation. This work contributes to the growing literature on machine learning safety.
+
+# 1 INTRODUCTION
+
+As machine learning models are applied to more and more real-world tasks, addressing machine learning safety is becoming an increasingly pressing issue. Deep learning algorithms have been shown to be vulnerable to adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2014; Papernot et al., 2016a). In particular, prior black-box adversarial attacks assume that the adversary is not aware of the target model architecture, parameters or training data, but is capable of querying the target model with supplied inputs and obtaining the output predictions. The phenomenon that adversarial examples generated from one model may also be adversarial to another model is known as adversarial transferability (Szegedy et al., 2013).
+
+Motivated by adversarial transferability, we conjecture another black-box attack pipeline where the adversary does not even need to have access to the target model nor query labels from crafted inputs. Instead, as long as the adversary knows the task of the target, he can choose a similar domain, to build a substitute model and then attack the target model with adversarial examples that are generated from the attack domain.
+
+![](images/701afe98cb7850d055da84d06c5c1d1b52a84093b502551f5fed697193981e36.jpg)  
+Figure 1: Diagrammatic representation of the problem
+
+(a) Generalized architecture of similarity-based attacks.
+
+![](images/6a410fa93b25f802d45fc136ccbe9aaeb9316e28f884c5c0ba6e0598c359b6f5.jpg)
+
+(b) Flow of how an adversary physician can leverage similarity attack to fool opioid risk models.
+
+The similar-domain adversarial attack may be more practical than prior blackbox attacks as label querying from target model is not needed. This attack can be illustrated with the following example (Figure 1b) in medical insurance fraud (Finlayson et al., 2019). Insurance companies may use hypothetical opioid risk models to classify the likelihood (high/low) of a patient to abuse the opioids to be prescribed, based on the patient’s medical history as text input. Physicians can run the original patient history through the attack pipeline to generate an adversarial patient history, where the original is more likely to be rejected (”High” risk) and the adversarial is more likely to be accepted (”Low” risk). Perturbations in patient history could be, for example, a slight perturbation from ”alcohol abuse” to ”alcohol dependence”, and it may successfully fool the insurance company’s model.
+
+Based on domain adaption theory (Ben-David et al., 2010), we conjecture that it is the domain-variant features that cause the success of the similar-domain attack. The adversarial examples with domainvariant features are likely to reside in the low-density regions (far away from decision boundary) of the empirical distribution of the target training data which could fool the target model (Zhang et al., 2019b). Literature indicates that worsened generalizability is a tradeoff faced by existing defenses such as adversarial training (Raghunathan et al., 2019) and domain generalization techniques (Wang et al., 2019). In trying to increase robustness against adversarial inputs, a model faces a tradeoff of weakened accuracy towards clean inputs. Given that an adversarial training loss function is composed of a loss against clean inputs and loss against adversarial inputs, improper optimization where the latter is highly-optimized and the former weakly-optimized does not improve general performance in the real-world. To curb this issue, methods have been proposed (Zhang et al., 2019b; Lamb et al., 2019; Schmidt et al., 2018), such as factoring in under-represented data points in training set.
+
+To defend against this similar-domain adversarial attack, we propose a weight transfer network approach, Learn2Weight, so that the target model’s decision boundary can adapt to the examples from low-density regions. Experiments confirm the effectiveness of our approach against the similardomain attack over other baseline defense methods. Moreover, our approach is able to improve robustness accuracy without losing the target model’s standard generalization accuracy.
+
+Our contribution can be summarized as follows:
+
+• We are among the first to propose the similar-domain adversarial attack. This attack pipeline relaxes the previous black-box attack assumption that the adversary has access to the target model and can query the model with crafted examples.   
+• We propose a defensive strategy for this attack based on domain adaptation theory. Experiment results show the effectiveness of our approach over existing defense methods, against the similar-domain attack.
+
+Recent work in adversarial attack for NLP systems has attracted attention. See (Zhang et al., 2020) survey for an overview of the adversarial attack in NLP. Existing research proposes different attack methods for generating adversarial text examples (Moosavi-Dezfooli et al., 2016; Ebrahimi et al., 2018; Wallace et al., 2019). The crafted adversarial text examples have been shown to fool the state-of-the-art NLP systems such as BERT (Jin et al., 2019). A large body of adversarial attack research focuses on black-box attack where the adversary builds a substitute model by querying the target model with supplied inputs and obtaining the output predictions. The key idea behind such black-box attack is that adversarial examples generated from one model may also be mis-classified by another model, which is known as adversarial transferability (Szegedy et al., 2013; Cheng et al., 2019). While prior work examines the transferability between different models trained over the same dataset, or the transferability between the same or different model trained over disjoint subsets of a dataset, our work examines the adversarial transferability between different domains, which we call similar-domain attack.
+
+Table 1: Comparison of attack domain sentences correctly classified when unperturbed by respective attack domain models and target domain models, then misclassified after perturbation by target models trained on books and baby domain. The perturbations are in blue, and prediction confidence in brackets.   
+
+<table><tr><td colspan="3">Attack domain:baby,Target domain:books</td></tr><tr><td>Original sentence (Actual label: Pos)</td><td>I purchased this toy for my sonwhen he was 4 monthsold.At first,he seemed a little intimidated by the toys.</td><td>Pos (0.712)</td></tr><tr><td>Adversarial sentence</td><td>I obtained this toys for my children when he was 4 weeks senior. At first, he hoped a modest harassed by the toy.</td><td>Neg (0.364)</td></tr><tr><td>Original sentence (Actual label: Pos)</td><td>It felt like a big commitment for me to have to run the program 2 times a day, and near the end of my pregnancy I was annoyed with having anything strapped across my belly.</td><td>Pos (0.825)</td></tr><tr><td>Adversarial sentence</td><td>It felt like a big committed for me to have to run the program 2 length a day, and near the end of my pregnancy Iwas annoyed with takes anything strapped acrossmy belly.</td><td>Neg (0.420)</td></tr><tr><td colspan="3">Attack domain: dvd, Target domain: baby</td></tr><tr><td>Original sentence (Actual label: Pos)</td><td>Fast times at ridgemont high is a clever, insightful,and wicked film! It is not just another teen movie.</td><td>Pos (0.614)</td></tr><tr><td>Adversarial sentence</td><td>Sooner days at ridgemont high is a sane, thoughtful, and wicked flick! It is not just another adolescent flick.</td><td>Neg (0.335)</td></tr><tr><td>Original sentence</td><td>This dvd gives a very good 6O minute workout.As others have pointed out the</td><td>Pos (0.647)</td></tr><tr><td>(Actual label: Pos) Adversarial sentence</td><td>cardio is very dancy. The first time I did it,I felt a bit awkward with the steps. This dvd gives a awfully okay 6O minute exercise. As others have pointed out the cardio is very dancy. The first time I did it, I perceived a bit awkward with the steps.</td><td>Neg (0.258)</td></tr></table>
+
+# 3 SIMILAR-DOMAIN ADVERSARIAL ATTACK
+
+# 3.1 ADVERSARIAL ATTACK BACKGROUND
+
+Adversarial attacks modify inputs to cause errors in machine learning inference (Szegedy et al., 2013). We utilize the basic gradient-based attack method Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014) and its variants, RAND-FGSM (Tramer et al., 2017) and \` Basic Iterative Method (BIM) (Kurakin et al., 2016a;b; Xie et al., 2018). Other NLP adversarial generation algorithms could also be used, such as DeepFool (Moosavi-Dezfooli et al., 2016), HotFlip (Ebrahimi et al., 2018), universal adversarial trigger (Wallace et al., 2019), and TextFooler (Jin et al., 2019). To perform gradient-based perturbations upon discrete space data, we follow (Yang et al., 2018) to generate adversarial text.
+
+Our proposed similar-domain adversarial attack is in-variant to adversarial algorithm, meaning that the adversarial algorithm used would not affect the attack performance. Without losing generality, we denote $A d v ( f , x )$ as an NLP adversarial text generation method, defined as below.
+
+Definition 1. NLP Adversarial Generation. Given a deep neural network model $f$ built on text data $X$ , an NLP adversarial generation method produces one adversarial instances $x ^ { \prime }  A d v ( f , x )$ for $x \in X$ , $x ^ { \prime } \approx x$ . The goal of the adversarial attack is to deviate the label to incorrect one $f ( x ^ { \prime } ) \neq f ( x )$
+
+# 3.2 SIMILAR-DOMAIN ADVERSARIAL ATTACK
+
+We present the architecture of similar-domain adversarial attack in Figure 1a. The defender, the target of the attack, constructs a target model trained on domain text data $T$ (0). An attacker, only having a rough idea about the target’s task but lacking direct access to the target data or target model parameters, collects attack data from a similar domain $S$ and trains an attack model (1). He runs the attack model on the test data (2) to obtain correctly-classified instances (3). He chooses an adversarial attack algorithm and generates a set of adversarial samples $A$ (4). He exposes $A$ to the target model, hoping $A$ mislead the target model to produce an output of his choice (5). This type of attack works best as an adversarial attack that compromises systems that base decision-making on one-instance.
+
+Definition 2. Similar-domain Adversarial Attack. A target model $f$ , built on target domain data $T$ , is a deep neural network model with parameter weights $W _ { T }$ that maps a text instance to a label: $y  f ( X , W _ { T } )$ . An adversary chooses a source attack domain $S$ , builds a substitute model $f _ { S }$ , and generates a set of adversarial examples $A$ from $S$ using $A d v ( f _ { S } , S )$ , so that during an attack $f ( A , W _ { T } ) = f _ { S } ( A )$ .
+
+Table 2: Similar-domain attack performance. Bold indicates the least successful attack domain (i.e. highest after-attack accuracy) for each target domain, as well as the corresponding transfer loss value.   
+
+<table><tr><td>Target Domain</td><td colspan="3">book</td><td colspan="3">magazine</td><td colspan="3">baby</td></tr><tr><td>Original Accuracy Intra-attack Accuracy</td><td colspan="3">0.880</td><td colspan="3">0.960</td><td colspan="3">0.890</td></tr><tr><td></td><td></td><td>0.525</td><td></td><td></td><td>0.570</td><td></td><td></td><td>0.632</td><td></td></tr><tr><td>Attack Domain</td><td>magazine</td><td>baby</td><td>dvd</td><td>baby</td><td>dvd</td><td>book</td><td>dvd</td><td>book</td><td>magazine</td></tr><tr><td>Unperturbed Accuracy</td><td>0.726</td><td>0.646</td><td>0.745</td><td>0.739</td><td>0.663</td><td>0.673</td><td>0.624</td><td>0.652</td><td>0.665</td></tr><tr><td>After-attack Accuracy</td><td>0.398</td><td>0.421</td><td>0.395</td><td>0.381</td><td>0.366</td><td>0.343</td><td>0.365</td><td>0.386</td><td>0.401</td></tr><tr><td>Shared Vocab</td><td>0.381</td><td>0.255</td><td>0.455</td><td>0.260</td><td>0.345</td><td>0.381</td><td>0.270</td><td>0.255</td><td>0.260</td></tr><tr><td>Transfer Loss</td><td>0.017</td><td>0.071</td><td>0.000</td><td>0.079</td><td>0.022</td><td>0.010</td><td>0.066</td><td>0.050</td><td>0.069</td></tr></table>
+
+# 3.3 DOMAIN SIMILARITY
+
+Here, domain similarity refers to the similarity between attacker’s chosen domain and defender’s domain. SharedVocab measures the overlap of unique words, in each of the datasets; a higher degree of overlapping vocabulary implies the two domains are more similar. We also use Transfer Loss, a standard metric for domain adaptation Blitzer et al. (2007); Glorot et al. (2011), to measure domain similarity; lower loss indicates higher similarity. The test error from a target model trained on target domain $T$ and evaluated on attack domain $S$ returns transfer error $e ( S , T )$ . The baseline error $e ( T , T )$ term is the test error obtained from target model trained on target domain (train) data $T$ and tested on target domain (evaluation) data $T$ . This computes the transfer loss, $t f ( S , T ) = e ( S , T ) - e ( T , T )$ .
+
+# 4 IS THE ATTACK EFFECTIVE?
+
+# 4.1 SETUP
+
+Dataset. We simulate the similar-domain adversarial attack using Amazon’s multi-domain sentiment classification dataset (Blitzer et al., 2007), a commonly-used dataset in cross-domain sentiment classification1, with 1,000 positive and 1,000 negative reviews for each of the 25 product categories.
+
+Model. In practice, there could be unlimited choice for the attack model and target model, such as different deep learning architecture, different training parameters. To simplify the discussion, we choose Long Short-Term Memory (LSTM) network as a suitable baseline sentiment classification model (Wang et al., 2018) for our target model and attack model. The architecture consists of 64 LSTM cells, $80 \%$ dropout, using a sigmoid activation function.
+
+Metrics. We first report the accuracy of the target models on the target domain test samples before the attack as the original accuracy. Then we measure the accuracy of the target models against adversarial samples crafted from the attack domain samples, denoted as the after-attack accuracy. Intra-attack accuracy denotes the after-attack accuracy where the attack domain is identical to the target domain. By comparing original and after-attack accuracy, we can evaluate the success of the attack. The greater the gap between the original and after-attack accuracy, the more successful the attack. Unperturbed accuracy measures the accuracy of the target model against the complete, unperturbed test set of the attack domain, to demonstrate that any drop in classification accuracy is not from domain shift alone but from adversarial transferability.
+
+# 4.2 RESULTS
+
+The similar-domain adversarial attack results are presented in Table 2. We see a significant gap between original accuracy and after-attack accuracy indicating that this attack can impose valid threat to a target NLP system. After the similar-domain adversarial attack, the accuracy drops dramatically by a large margin. Take the book target domain for example, when the attack domain is magazine, the after-attack accuracy drops to 0.398, and when the attack domain is baby, the accuracy is 0.421. Moreover, we observe a positive correlation between transfer loss and after-attack accuracy, and a negative correlation between shared vocab and after-attack accuracy.
+
+# 5 DEFENDING AGAINST SIMILAR-DOMAIN ADVERSARIAL ATTACK
+
+In order to defend against a similarity based adversarial attack, it is critical to block adversarial transferability. Adversarial training is the most intuitive yet effective defense strategy for adversarial attack (Goodfellow et al., 2014; Madry et al., 2017). However, this may not be effective for two reasons. First, there is no formal guidance for generating similar-domain adversarial examples because the defender has no idea what the attack data domain is. Second, simple feeding the target model with adversarial examples may even hurt the generalization of the target model (Raghunathan et al., 2019; Zhang et al., 2019a; Su et al., 2018), which is also confirmed in our experiments.
+
+# 5.1 WEIGHT TRANSFER LEARNING
+
+The use of weight transfer networks (Ha et al., 2016; Hu et al., 2018; Kuen et al., 2019) is concerned with adapting weights from one model into another, and generating/predicting the complete set of weights for a model given the input samples. In our context, distinctly different weights are produced for target models trained on inputs of different domains, and feature transferability (Yosinski et al., 2014) in the input space can be expected to translate to weight transferability in the model weights space. Rather than completely regenerating classification weights, our model robustification defense, Learn2Weight L2W predicts the perturbation to existing weights $\left( W ^ { \prime } = W + \Delta W \right)$ ) for each new instance.
+
+# 5.2 LEARN2WEIGHT MODEL
+
+We conjecture that an effective defense strategy is to perturb the target model weights depending on the feature distribution of the input instance. $L 2 W$ (Algorithm 1) recalculates the target model weights depending on the input. $L 2 W$ (Algorithm 2) trains on sentences from different domains and a weight differential for that domain (the weight adjustment required to tune the target model’s weights to adapt to the input’s domain). We obtain the weight differential $\Delta W$ by finding the difference between the weights of $f$ trained on sentence:label pairs from a specific domain $W _ { S _ { j } }$ and weights of $f$ trained on sentence:label pairs from the target domain $W _ { T }$ . Other training models may be possible; here we trained a sequence-to-sequence network (Sutskever et al., 2014) on sentence: $\Delta W$ pairs.
+
+<table><tr><td>Algorithm 1: Learn2Weight: Inference</td><td>Algorithm 2: Learn2Weight:Training</td></tr><tr><td>inference (Xadv, L2W(.), f(Wr,T)) Input:Passed arguments include the adversarial input Xadu, the Learn2Weight model L2W(-),and the target model f(Wr,T) Output :Target model with weighted</td><td>train (T,D) Input :Target domain T={Ti}0; Set of M domains D={S,j)j; withNsentences, N,M i indexing specific sentence tensor and j indexing specific domains Output :Trained Learn2Weight model L2W(·)</td></tr><tr><td>updated byLearn2Weight f(W*,T) is the expected output Pass Xadu as in input into the trained L2W(-) function,and the weight differential required for the target model with weights Wr is△W.</td><td>Initialize empty X and Y to store sentences Xi from each domain j with corresponding weight differential. X←;Y←; Compute weights of f trained on T. Wr ← f(T);</td></tr><tr><td>△W ← L2W(Xadv); The returned function is the target model with updated weights Wr+△W. return f(Wr +△W,T);</td><td>X←T;Y←WT-Wr; Train each domain Sj,compute respective weights, append the differential △W to Y and each sentence in Si,j into X. foreach domain Sj ∈D do Ws,← f(Sj); △W ←Wsj-Wr;</td></tr></table>
+
+# Algorithm 3: tf-optimization
+
+tfOptimization $( T , M , n _ { m a x } )$
+
+Input :Target domain $T = \{ T _ { i } \} _ { i = 0 } ^ { N }$ to be used in synthesizing $M$ similar domains; with $N$ sentences, $i$ indexing specific sentence tensor; $n _ { m a x }$ is the max number of tf-optimization iterations   
+Output :Set $D$ containing $M$ domains   
+Initialize empty $D$ to store synthesized domains $S _ { j }$ of index $j$ .   
+$D \gets \emptyset ; j \gets 0$ ;   
+while $j < M$ do Run each iteration until $n _ { m a x }$ . for iter $ 0$ to $n _ { m a x }$ do Apply adversarial perturbations to $T$ . $T _ { i t e r } ^ { \hat { a } \hat { d } v } \gets A d v ( f , \hat { T } )$ ; Determine change to $A d v ( \cdot )$ or iter depending on computed transfer loss. if $c h e c k ( t f ( T _ { i t e r } ^ { a d v } , T ) ) \gets T r u e$ then If low enough, $T _ { i t e r } ^ { a d v }$ can be added as synthetic domain into $D$ . $D \gets T _ { i t e r } ^ { a d v }$ ; break; else Adjust perturbation parameters. $A d v ( \cdot )  a d j u s t ( A d v ( \cdot ) )$ ; $j \gets j + 1$ ; Return set of Domains to be used for training by $L 2 W$ .   
+return $D$ ;
+
+# 5.3 TRANSFER LOSS OPTIMIZATION
+
+To generate synthetic domains of varying domain similarity so that defenders defend their model using only target domain data $T$ , the following equation introduces transfer loss optimization (Algorithm 3). The defender iteratively generates adversarial examples $X _ { N } ^ { a d v }$ while maximizing the transfer loss function $t f$ ; this produces a substitute attack domain corpora $S _ { j }$ . We iteratively adjust perturbation parameters $A d v ( \cdot )$ and iteration count $N$ to minimize the transfer loss of our generated dataset.
+
+$$
+\underset { N , A d v ( \cdot ) } { \arg \operatorname* { m i n } } t f _ { N } \Bigl ( X _ { N } ^ { a d v } , T \Bigr ) = e \Bigl ( A d v ( f , T ) , T \Bigr ) - e \Bigl ( T , T \Bigr )
+$$
+
+# 5.4 EXPLANATION: BLOCKING TRANSFERABILITY
+
+To facilitate our explanation, we adapt from domain adaptation literature (Ben-David et al., 2010; Liu et al., 2019; Zhang et al., $2 0 1 9 \mathrm { c }$ ):
+
+$$
+e ( A , T ) \leq e ( T , T ) + d _ { \mathcal { H } \Delta \mathcal { H } } ( A , T ) + \lambda
+$$
+
+where $\mathcal { H }$ is the hypothesis space, $h$ is a hypothesis function that returns labels $\{ 0 , 1 \}$ , and $e ( T , T )$ and $e ( A , T )$ are the generalization errors from passing target domain data $T$ and adversarial data $A$ through a classifier trained on $T$ . $d _ { \mathcal { H } \Delta \mathcal { H } } ( A , T )$ is the $\mathcal { H } \Delta \bar { \mathcal { H } }$ -distance between $T$ and $A$ , and measures the divergence between the feature distributions of $A$ and $T$ . $e _ { A } ( h , h ^ { ' } )$ and $e _ { T } ( h , h ^ { ' } )$ represents the probability that $h$ disagrees with $h ^ { ' }$ on the label of an input in the domain space $A$ and $T$ respectively.
+
+$$
+\begin{array} { r } { d _ { \mathcal { H } \Delta \mathcal { H } } ( A , T ) = \underset { h , h ^ { \prime } \in \mathcal { H } } { \operatorname* { s u p } } | e _ { A } ( h , h ^ { ' } ) - e _ { T } ( h , h ^ { ' } ) | } \\ { d _ { \mathcal { H } \Delta \mathcal { H } } ( A , T ) = \operatorname* { s u p } _ { h , h ^ { \prime } \in \mathcal { H } } | \mathbb { E } _ { X \sim S } [ | ( h ( x ) - h ^ { ' } ( x ) | ] | } \\ { - | \mathbb { E } _ { X \sim T } [ | ( h ( x ) - h ^ { ' } ( x ) | ] | | } \end{array}
+$$
+
+Divergence $d _ { \mathcal { H } \Delta \mathcal { H } }$ measures the divergence between feature distributions $A$ and $T$ . Higher $d _ { \mathcal { H } \Delta \mathcal { H } }$ indicates less shared features between 2 domains. The greater the intersection between feature distributions, the greater the proportion of domain-variant features; one approach to domain adaptation is learning domain-invariant features representations (Zhao et al., 2019) to minimize $d _ { \mathcal { H } \Delta \mathcal { H } }$ .
+
+Table 3: After-defense accuracy performance of different defensive methods.   
+
+<table><tr><td>Target Domain</td><td colspan="3">magazine</td><td colspan="3">baby</td></tr><tr><td>Attack Domain</td><td>baby</td><td>dvd</td><td>book</td><td>dvd</td><td>book</td><td>magazine</td></tr><tr><td>After-attack Accuracy</td><td>0.381</td><td>0.366</td><td>0.343</td><td>0.365</td><td>0.386</td><td>0.401</td></tr><tr><td>After-defense Accuracy</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Adversarial training</td><td>0.639</td><td>0.559</td><td>0.657</td><td>0.558</td><td>0.577</td><td>0.661</td></tr><tr><td>Defensive distillation</td><td>0.549</td><td>0.561</td><td>0.597</td><td>0.588</td><td>0.629</td><td>0.577</td></tr><tr><td>SharedVocab defense</td><td>0.628</td><td>0.653</td><td>0.631</td><td>0.664</td><td>0.668</td><td>0.621</td></tr><tr><td>Domain-adapted adversarial training</td><td>0.608</td><td>0.637</td><td>0.620</td><td>0.604</td><td>0.620</td><td>0.587</td></tr><tr><td>Learn2Weight</td><td>0.796</td><td>0.842</td><td>0.843</td><td>0.774</td><td>0.751</td><td>0.737</td></tr></table>
+
+Explaining similarity-domain attacks. As demonstrated by empirical results, $e ( A , T )$ increases in a similarity-based attack setting, and this would arise if $d _ { \mathcal { H } \Delta \mathcal { H } }$ increases correspondingly. $d _ { \mathcal { H } \Delta \mathcal { H } }$ computes inconsistent labels from inconsistent feature distributions, and attributes the success of the attack to domain-variant features.
+
+FGSM and variants adjust the input data to maximize the loss based on the backpropagated gradients of a model trained on $S$ . As our pipeline used correctly-labelled sentences before adversarially perturbing them, we can infer that perturbations applied to $S$ were not class-dependent (i.e. the success of the attack is not based on the removal of class-specific features), but class-independent features. It is already difficult for a model trained on $S$ to classify when there is insufficient classdependent features (hence a high $t f ( A , T ) )$ ; in a cross-domain setting, it must be even more difficult for a model trained on $T$ to classify given a shortage of domain-invariant, class-dependent features.
+
+$$
+\begin{array} { c } { { d _ { { \mathcal { H } } \Delta { \mathcal { H } } } \leq e ( A , T ) - e ( T , T ) - \lambda } } \\ { { d _ { { \mathcal { H } } \Delta { \mathcal { H } } } \leq t f ( A , T ) - \lambda } } \end{array}
+$$
+
+Explaining Learn2Weight. $L 2 W$ minimizes divergence by training on $\{ d _ { \mathcal { H } \Delta \mathcal { H } } ( S _ { j } , T ) : \Delta W _ { A . S _ { j } } \}$ pairs, $d _ { \mathcal { H } \Delta \mathcal { H } } ( S _ { j } , T )$ being reconstructed from the difference between S $x _ { i } ^ { S _ { j } }$ and $x _ { i } ^ { T }$ . $L 2 W$ is trained on $\{ d _ { \mathcal { H } \Delta \mathcal { H } } ^ { S _ { j } } \} _ { j = 0 } ^ { N } : \{ \Delta W ^ { S _ { j } } \} _ { j = 0 } ^ { N }$ pairs, such that . Intuitively the target model possesses a decision boundary (Liu et al., 2019) to classify inputs based on whether they cross the boundary or not; adversarial inputs have a tendency of being near the boundary and fooling it. Weights transfer learning applies perturbations to the decision boundary such that the boundary covers certain adversarial inputs otherwise misclassified, and in this way blocks transferability. The advantage of training on multiple domains $\{ S _ { j } \} _ { j = 0 } ^ { M }$ is that the after- $. L 2 W$ divergence between $A$ and $T$ is smaller because $L 2 W$ ’s weight perturbations render the decision boundary more precise in classifying inputs.
+
+Explaining tf-optimization. We have attributed why adversarial sentences $A$ are computed to be domain-dissimilar despite originating from $S$ due to insufficient domain-invariant, class-dependent features resulting in low $e ( A , T )$ , i.e. low $t f ( A , T )$ . To replicate this phenomenon in natural domains, we use $t f$ -optimization to iteratively perturb $T$ to increase the proportion of class-independent features. This approximates the real-world similarity-based attack scenario where class-dependent features may be limited for inference. By generating the synthetic data, we are feeding $L 2 W$ attack data with variations in $d _ { \mathcal { H } \Delta \mathcal { H } }$ and class-independent feature distributions. This prepares $L 2 W$ to robustify weights in $f ( T )$ when such feature distributions are encountered.
+
+# 6 EXPERIMENTS
+
+# 6.1 BASELINES
+
+We consider two defense strategies that are empirically effective and are widely used for general black-box adversarial attacks: adversarial training (Goodfellow et al., 2014; Madry et al., 2017) and defensive distillation (Papernot et al., 2016b; 2017). In addition we consider two ablation baselines.
+
+Defensive distillation: The high-level implementation of defensive distillation (Papernot et al., 2016b; 2017) is to first train an initial model against target domain inputs and labels, and retrieve the raw class probability scores. The predicted probability values would be used as the new labels for the same target sentences, and we would train a new model based on this new label-sentence pair.
+
+Table 4: Learn2Weight comparison against different attack model architectures.   
+
+<table><tr><td>Target</td><td>Original</td><td>Attack</td><td colspan="4">Attack Model After-attack Accuracy</td><td colspan="6">After-Defense Accuracy</td></tr><tr><td>Domain</td><td>Accuracy Target Model: LSTM</td><td>Domain</td><td colspan="4"></td><td colspan="4">LogReg</td><td>CNN</td><td>LogReg</td></tr><tr><td colspan="4"></td><td colspan="4">BERT LSTM</td><td colspan="4">BERT LSTM</td></tr><tr><td rowspan="3">book</td><td rowspan="3">0.880</td><td>dvd</td><td>0.342</td><td>0.413</td><td>GRU 0.477</td><td>CNN 0.335</td><td>0.440</td><td>0.786</td><td>0.847</td><td>0.804</td><td>0.816</td><td>0.782</td></tr><tr><td>kitchenware</td><td>0.350</td><td>0.372</td><td>0.325</td><td>0.353</td><td>0.425</td><td>0.765</td><td>0.826</td><td>0.795</td><td>0.742</td><td>0.767</td></tr><tr><td>electronics</td><td>0.400</td><td>0.389</td><td>0.416</td><td>0.315</td><td>0.460</td><td>0.792</td><td>0.812</td><td>0.784</td><td>0.770</td><td>0.725</td></tr><tr><td rowspan="3">dvd</td><td rowspan="3">0.920</td><td></td><td>0.326</td><td>0.434</td><td>0.479</td><td>0.383</td><td>0.490</td><td>0.816</td><td>0.795</td><td>0.824</td><td>0.804</td><td>0.794</td></tr><tr><td>book kitchenware</td><td>0.355</td><td>0.370</td><td>0.379</td><td>0.359</td><td>0.490</td><td>0.728</td><td>0.796</td><td>0.755</td><td>0.735</td><td>0.695</td></tr><tr><td>electronics</td><td>0.387</td><td>0.377</td><td>0.332</td><td>0.348</td><td>0.455</td><td>0.825</td><td>0.836</td><td>0.812</td><td>0.834</td><td>0.796</td></tr><tr><td rowspan="3">electronics</td><td rowspan="3">0.910</td><td>book</td><td>0.425</td><td>0.394</td><td>0.473</td><td>0.358</td><td>0.474</td><td>0.775</td><td>0.821</td><td>0.795</td><td>0.782</td><td>0.712</td></tr><tr><td>dvd</td><td>0.342</td><td>0.395</td><td>0.452</td><td>0.368</td><td>0.493</td><td>0.784</td><td>0.845</td><td>0.855</td><td>0.842</td><td>0.792</td></tr><tr><td>kitchenware</td><td>0.390</td><td>0.384</td><td>0.464</td><td>0.329</td><td>0.432</td><td>0.730</td><td>0.824</td><td>0.753</td><td>0.724</td><td>0.678</td></tr></table>
+
+Adversarial training: It is shown that injecting adversarial examples throughout training increases the robustness of target neural network models. In this baseline, target model is trained with both original training data and adversarial examples generated from original training data. However, since the adversarial examples are still generated from the target domain, it is unlikely that the method can defend similar-domain attack which is the result of domain-variant features.
+
+SharedVocab defense. Given that it is the domain-variant features that cause the success of similardomain attack, a simple baseline is to remove those words that are not in the target domain’s vocabulary. This tests whether the effect of perturbing target model weights w.r.t domain-variant features will yield incremental after-attack accuracy in the similar-domain attack setting.
+
+Domain-adapted adversarial training. This ablation baseline tests for incremental performance to a baseline defense using domain-variant inputs. We adapt adversarial training to be trained on adversarial sentences from attack domain $S$ , whereas the traditional adversarial training generates adversarial samples $X ^ { a d v , T }$ from its training data $T$ , this adapted version uses adversarial samples $X ^ { a d v , S }$ generated from $S$ .
+
+# 6.2 LEARN2WEIGHT PERFORMANCE
+
+Defense performance. We present the results of different defense baselines in Table 3. First, we can see that Learn2Weight achieves the highest after-defense accuracy against the adversarial attack. Take the magazine as target domain for example, if the adversary chooses to use book data as the attack domain, it would reduce the target model accuracy to 0.343. However, the Learn2Weight method can improve the performance to 0.843, which is a significant and substantial improvement against the attack. This improvements also exist across different target/attack domain pairs. Second, we see that all defense methods can improve the accuracy to some extent which indicates the importance and effectiveness of having robust training for machine learning models. Third, it is interesting to note that two simple baselines SharedVocab defense and Domain-adapted adversarial training yield overall better performance compared to adversarial training and defensive distillation.
+
+Attack model architectures. So far, all the results are conducted using the same LSTM as the target/attack model due to simplicity purpose. Here, we keep the target model unchanged, but vary the architecture of the attack model for the generation of adversarial examples. A variation of a Recurrent Neural Network is a Gated Recurrent Unit (GRU) network, with 512 GRU cells, $60 \%$ dropout and tanh activation function. We have also tested other attack model variants that are commonly-used in sentiment classification, including Bidirectional Encoder Representations from Transformers (BERT) (Devlin et al., 2019), Convolutional Neural Network (CNN) (Kim, 2014), and Logistic Regression (Maas et al., 2011). For both RNN and CNN, we use pre-computed Glove embeddings2 to encode words. All models are trained with enough epochs after ensuring the model achieved near state-of-the-art validation accuracy before proceeding to tests of adversarial attacks and defenses.
+
+We present the results of different attack model architectures in Table 4. First, similar-domain attack is model-agnostic and it does not require the target and attack model to have identical architectures.
+
+We can see that all four attack model architectures are able to reduce the target model accuracy. Second, the results suggest that Learn2Weight is also model-agnostic as it can substantially improve the after-defense accuracy regardless which attack model is used.
+
+# 7 CONCLUSION
+
+In this newly-proposed, empirically-effective similar-domain attack, an adversary can choose a similar domain to the target task, build a substitute model and produce adversarial examples to fool the target model. We also propose a defense strategy, Learn2Weight, that learns to adapt the target model’s weight using crafted adversarial examples. Compared with other adversarial defense strategies, Learn2Weight can improve the target model robustness against the similar-domain attack. Our method demonstrates properties of a good adversarial defense, such as adopting defense architectures that adapt to situations/inputs rather than compromising standard error versus robustness error, to leverage class-independent properties in domain-variant text, and factoring in domain similarity in adversarial robustness exercises.
+
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+        "text": "LEARN2WEIGHT: WEIGHTS TRANSFER DEFENSE AGAINST SIMILAR-DOMAIN ADVERSARIAL ATTACKS ",
+        "text_level": 1,
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "text": "ABSTRACT ",
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+        "text": "Recent work in black-box adversarial attacks for NLP systems has attracted attention. Prior black-box attacks assume that attackers can observe output labels from target models based on selected inputs. In this work, inspired by adversarial transferability, we propose a new type of black-box NLP adversarial attack that an attacker can choose a similar domain and transfer the adversarial examples to the target domain and cause poor performance in target model. Based on domain adaptation theory, we then propose a defensive strategy, called Learn2Weight, which trains to predict the weight adjustments for target model in order to defense the attack of similar-domain adversarial examples. Using Amazon multi-domain sentiment classification dataset, we empirically show that Learn2Weight model is effective against the attack compared to standard black-box defense methods such as adversarial training and defense distillation. This work contributes to the growing literature on machine learning safety. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As machine learning models are applied to more and more real-world tasks, addressing machine learning safety is becoming an increasingly pressing issue. Deep learning algorithms have been shown to be vulnerable to adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2014; Papernot et al., 2016a). In particular, prior black-box adversarial attacks assume that the adversary is not aware of the target model architecture, parameters or training data, but is capable of querying the target model with supplied inputs and obtaining the output predictions. The phenomenon that adversarial examples generated from one model may also be adversarial to another model is known as adversarial transferability (Szegedy et al., 2013). ",
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+        "type": "text",
+        "text": "Motivated by adversarial transferability, we conjecture another black-box attack pipeline where the adversary does not even need to have access to the target model nor query labels from crafted inputs. Instead, as long as the adversary knows the task of the target, he can choose a similar domain, to build a substitute model and then attack the target model with adversarial examples that are generated from the attack domain. ",
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+        "type": "image",
+        "img_path": "images/701afe98cb7850d055da84d06c5c1d1b52a84093b502551f5fed697193981e36.jpg",
+        "image_caption": [
+            "Figure 1: Diagrammatic representation of the problem "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "(a) Generalized architecture of similarity-based attacks. ",
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+        "text": "(b) Flow of how an adversary physician can leverage similarity attack to fool opioid risk models. ",
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+        "type": "text",
+        "text": "The similar-domain adversarial attack may be more practical than prior blackbox attacks as label querying from target model is not needed. This attack can be illustrated with the following example (Figure 1b) in medical insurance fraud (Finlayson et al., 2019). Insurance companies may use hypothetical opioid risk models to classify the likelihood (high/low) of a patient to abuse the opioids to be prescribed, based on the patient’s medical history as text input. Physicians can run the original patient history through the attack pipeline to generate an adversarial patient history, where the original is more likely to be rejected (”High” risk) and the adversarial is more likely to be accepted (”Low” risk). Perturbations in patient history could be, for example, a slight perturbation from ”alcohol abuse” to ”alcohol dependence”, and it may successfully fool the insurance company’s model. ",
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+    {
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+        "text": "Based on domain adaption theory (Ben-David et al., 2010), we conjecture that it is the domain-variant features that cause the success of the similar-domain attack. The adversarial examples with domainvariant features are likely to reside in the low-density regions (far away from decision boundary) of the empirical distribution of the target training data which could fool the target model (Zhang et al., 2019b). Literature indicates that worsened generalizability is a tradeoff faced by existing defenses such as adversarial training (Raghunathan et al., 2019) and domain generalization techniques (Wang et al., 2019). In trying to increase robustness against adversarial inputs, a model faces a tradeoff of weakened accuracy towards clean inputs. Given that an adversarial training loss function is composed of a loss against clean inputs and loss against adversarial inputs, improper optimization where the latter is highly-optimized and the former weakly-optimized does not improve general performance in the real-world. To curb this issue, methods have been proposed (Zhang et al., 2019b; Lamb et al., 2019; Schmidt et al., 2018), such as factoring in under-represented data points in training set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To defend against this similar-domain adversarial attack, we propose a weight transfer network approach, Learn2Weight, so that the target model’s decision boundary can adapt to the examples from low-density regions. Experiments confirm the effectiveness of our approach against the similardomain attack over other baseline defense methods. Moreover, our approach is able to improve robustness accuracy without losing the target model’s standard generalization accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our contribution can be summarized as follows: ",
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+        "text": "• We are among the first to propose the similar-domain adversarial attack. This attack pipeline relaxes the previous black-box attack assumption that the adversary has access to the target model and can query the model with crafted examples.   \n• We propose a defensive strategy for this attack based on domain adaptation theory. Experiment results show the effectiveness of our approach over existing defense methods, against the similar-domain attack. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recent work in adversarial attack for NLP systems has attracted attention. See (Zhang et al., 2020) survey for an overview of the adversarial attack in NLP. Existing research proposes different attack methods for generating adversarial text examples (Moosavi-Dezfooli et al., 2016; Ebrahimi et al., 2018; Wallace et al., 2019). The crafted adversarial text examples have been shown to fool the state-of-the-art NLP systems such as BERT (Jin et al., 2019). A large body of adversarial attack research focuses on black-box attack where the adversary builds a substitute model by querying the target model with supplied inputs and obtaining the output predictions. The key idea behind such black-box attack is that adversarial examples generated from one model may also be mis-classified by another model, which is known as adversarial transferability (Szegedy et al., 2013; Cheng et al., 2019). While prior work examines the transferability between different models trained over the same dataset, or the transferability between the same or different model trained over disjoint subsets of a dataset, our work examines the adversarial transferability between different domains, which we call similar-domain attack. ",
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+    {
+        "type": "table",
+        "img_path": "images/169f46b4012fff1f7bec6f8f8f8918ffe970296daf9e22bea9fb43bded438ab7.jpg",
+        "table_caption": [
+            "Table 1: Comparison of attack domain sentences correctly classified when unperturbed by respective attack domain models and target domain models, then misclassified after perturbation by target models trained on books and baby domain. The perturbations are in blue, and prediction confidence in brackets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Attack domain:baby,Target domain:books</td></tr><tr><td>Original sentence (Actual label: Pos)</td><td>I purchased this toy for my sonwhen he was 4 monthsold.At first,he seemed a little intimidated by the toys.</td><td>Pos (0.712)</td></tr><tr><td>Adversarial sentence</td><td>I obtained this toys for my children when he was 4 weeks senior. At first, he hoped a modest harassed by the toy.</td><td>Neg (0.364)</td></tr><tr><td>Original sentence (Actual label: Pos)</td><td>It felt like a big commitment for me to have to run the program 2 times a day, and near the end of my pregnancy I was annoyed with having anything strapped across my belly.</td><td>Pos (0.825)</td></tr><tr><td>Adversarial sentence</td><td>It felt like a big committed for me to have to run the program 2 length a day, and near the end of my pregnancy Iwas annoyed with takes anything strapped acrossmy belly.</td><td>Neg (0.420)</td></tr><tr><td colspan=\"3\">Attack domain: dvd, Target domain: baby</td></tr><tr><td>Original sentence (Actual label: Pos)</td><td>Fast times at ridgemont high is a clever, insightful,and wicked film! It is not just another teen movie.</td><td>Pos (0.614)</td></tr><tr><td>Adversarial sentence</td><td>Sooner days at ridgemont high is a sane, thoughtful, and wicked flick! It is not just another adolescent flick.</td><td>Neg (0.335)</td></tr><tr><td>Original sentence</td><td>This dvd gives a very good 6O minute workout.As others have pointed out the</td><td>Pos (0.647)</td></tr><tr><td>(Actual label: Pos) Adversarial sentence</td><td>cardio is very dancy. The first time I did it,I felt a bit awkward with the steps. This dvd gives a awfully okay 6O minute exercise. As others have pointed out the cardio is very dancy. The first time I did it, I perceived a bit awkward with the steps.</td><td>Neg (0.258)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3 SIMILAR-DOMAIN ADVERSARIAL ATTACK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "3.1 ADVERSARIAL ATTACK BACKGROUND ",
+        "text_level": 1,
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+        "text": "Adversarial attacks modify inputs to cause errors in machine learning inference (Szegedy et al., 2013). We utilize the basic gradient-based attack method Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014) and its variants, RAND-FGSM (Tramer et al., 2017) and \\` Basic Iterative Method (BIM) (Kurakin et al., 2016a;b; Xie et al., 2018). Other NLP adversarial generation algorithms could also be used, such as DeepFool (Moosavi-Dezfooli et al., 2016), HotFlip (Ebrahimi et al., 2018), universal adversarial trigger (Wallace et al., 2019), and TextFooler (Jin et al., 2019). To perform gradient-based perturbations upon discrete space data, we follow (Yang et al., 2018) to generate adversarial text. ",
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+        "type": "text",
+        "text": "Our proposed similar-domain adversarial attack is in-variant to adversarial algorithm, meaning that the adversarial algorithm used would not affect the attack performance. Without losing generality, we denote $A d v ( f , x )$ as an NLP adversarial text generation method, defined as below. ",
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+        "text": "Definition 1. NLP Adversarial Generation. Given a deep neural network model $f$ built on text data $X$ , an NLP adversarial generation method produces one adversarial instances $x ^ { \\prime }  A d v ( f , x )$ for $x \\in X$ , $x ^ { \\prime } \\approx x$ . The goal of the adversarial attack is to deviate the label to incorrect one $f ( x ^ { \\prime } ) \\neq f ( x )$ ",
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+    {
+        "type": "text",
+        "text": "3.2 SIMILAR-DOMAIN ADVERSARIAL ATTACK ",
+        "text_level": 1,
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+        "text": "We present the architecture of similar-domain adversarial attack in Figure 1a. The defender, the target of the attack, constructs a target model trained on domain text data $T$ (0). An attacker, only having a rough idea about the target’s task but lacking direct access to the target data or target model parameters, collects attack data from a similar domain $S$ and trains an attack model (1). He runs the attack model on the test data (2) to obtain correctly-classified instances (3). He chooses an adversarial attack algorithm and generates a set of adversarial samples $A$ (4). He exposes $A$ to the target model, hoping $A$ mislead the target model to produce an output of his choice (5). This type of attack works best as an adversarial attack that compromises systems that base decision-making on one-instance. ",
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+        "text": "Definition 2. Similar-domain Adversarial Attack. A target model $f$ , built on target domain data $T$ , is a deep neural network model with parameter weights $W _ { T }$ that maps a text instance to a label: $y  f ( X , W _ { T } )$ . An adversary chooses a source attack domain $S$ , builds a substitute model $f _ { S }$ , and generates a set of adversarial examples $A$ from $S$ using $A d v ( f _ { S } , S )$ , so that during an attack $f ( A , W _ { T } ) = f _ { S } ( A )$ . ",
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+        "img_path": "images/b793c4901e0b80be64ec0217bcbcfb8525f363986fca39f9474272c749ac7635.jpg",
+        "table_caption": [
+            "Table 2: Similar-domain attack performance. Bold indicates the least successful attack domain (i.e. highest after-attack accuracy) for each target domain, as well as the corresponding transfer loss value. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Target Domain</td><td colspan=\"3\">book</td><td colspan=\"3\">magazine</td><td colspan=\"3\">baby</td></tr><tr><td>Original Accuracy Intra-attack Accuracy</td><td colspan=\"3\">0.880</td><td colspan=\"3\">0.960</td><td colspan=\"3\">0.890</td></tr><tr><td></td><td></td><td>0.525</td><td></td><td></td><td>0.570</td><td></td><td></td><td>0.632</td><td></td></tr><tr><td>Attack Domain</td><td>magazine</td><td>baby</td><td>dvd</td><td>baby</td><td>dvd</td><td>book</td><td>dvd</td><td>book</td><td>magazine</td></tr><tr><td>Unperturbed Accuracy</td><td>0.726</td><td>0.646</td><td>0.745</td><td>0.739</td><td>0.663</td><td>0.673</td><td>0.624</td><td>0.652</td><td>0.665</td></tr><tr><td>After-attack Accuracy</td><td>0.398</td><td>0.421</td><td>0.395</td><td>0.381</td><td>0.366</td><td>0.343</td><td>0.365</td><td>0.386</td><td>0.401</td></tr><tr><td>Shared Vocab</td><td>0.381</td><td>0.255</td><td>0.455</td><td>0.260</td><td>0.345</td><td>0.381</td><td>0.270</td><td>0.255</td><td>0.260</td></tr><tr><td>Transfer Loss</td><td>0.017</td><td>0.071</td><td>0.000</td><td>0.079</td><td>0.022</td><td>0.010</td><td>0.066</td><td>0.050</td><td>0.069</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 DOMAIN SIMILARITY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Here, domain similarity refers to the similarity between attacker’s chosen domain and defender’s domain. SharedVocab measures the overlap of unique words, in each of the datasets; a higher degree of overlapping vocabulary implies the two domains are more similar. We also use Transfer Loss, a standard metric for domain adaptation Blitzer et al. (2007); Glorot et al. (2011), to measure domain similarity; lower loss indicates higher similarity. The test error from a target model trained on target domain $T$ and evaluated on attack domain $S$ returns transfer error $e ( S , T )$ . The baseline error $e ( T , T )$ term is the test error obtained from target model trained on target domain (train) data $T$ and tested on target domain (evaluation) data $T$ . This computes the transfer loss, $t f ( S , T ) = e ( S , T ) - e ( T , T )$ . ",
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+    {
+        "type": "text",
+        "text": "4 IS THE ATTACK EFFECTIVE? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Dataset. We simulate the similar-domain adversarial attack using Amazon’s multi-domain sentiment classification dataset (Blitzer et al., 2007), a commonly-used dataset in cross-domain sentiment classification1, with 1,000 positive and 1,000 negative reviews for each of the 25 product categories. ",
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+    {
+        "type": "text",
+        "text": "Model. In practice, there could be unlimited choice for the attack model and target model, such as different deep learning architecture, different training parameters. To simplify the discussion, we choose Long Short-Term Memory (LSTM) network as a suitable baseline sentiment classification model (Wang et al., 2018) for our target model and attack model. The architecture consists of 64 LSTM cells, $80 \\%$ dropout, using a sigmoid activation function. ",
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+        "type": "text",
+        "text": "Metrics. We first report the accuracy of the target models on the target domain test samples before the attack as the original accuracy. Then we measure the accuracy of the target models against adversarial samples crafted from the attack domain samples, denoted as the after-attack accuracy. Intra-attack accuracy denotes the after-attack accuracy where the attack domain is identical to the target domain. By comparing original and after-attack accuracy, we can evaluate the success of the attack. The greater the gap between the original and after-attack accuracy, the more successful the attack. Unperturbed accuracy measures the accuracy of the target model against the complete, unperturbed test set of the attack domain, to demonstrate that any drop in classification accuracy is not from domain shift alone but from adversarial transferability. ",
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+    {
+        "type": "text",
+        "text": "4.2 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The similar-domain adversarial attack results are presented in Table 2. We see a significant gap between original accuracy and after-attack accuracy indicating that this attack can impose valid threat to a target NLP system. After the similar-domain adversarial attack, the accuracy drops dramatically by a large margin. Take the book target domain for example, when the attack domain is magazine, the after-attack accuracy drops to 0.398, and when the attack domain is baby, the accuracy is 0.421. Moreover, we observe a positive correlation between transfer loss and after-attack accuracy, and a negative correlation between shared vocab and after-attack accuracy. ",
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+    {
+        "type": "text",
+        "text": "5 DEFENDING AGAINST SIMILAR-DOMAIN ADVERSARIAL ATTACK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to defend against a similarity based adversarial attack, it is critical to block adversarial transferability. Adversarial training is the most intuitive yet effective defense strategy for adversarial attack (Goodfellow et al., 2014; Madry et al., 2017). However, this may not be effective for two reasons. First, there is no formal guidance for generating similar-domain adversarial examples because the defender has no idea what the attack data domain is. Second, simple feeding the target model with adversarial examples may even hurt the generalization of the target model (Raghunathan et al., 2019; Zhang et al., 2019a; Su et al., 2018), which is also confirmed in our experiments. ",
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+    {
+        "type": "text",
+        "text": "5.1 WEIGHT TRANSFER LEARNING",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The use of weight transfer networks (Ha et al., 2016; Hu et al., 2018; Kuen et al., 2019) is concerned with adapting weights from one model into another, and generating/predicting the complete set of weights for a model given the input samples. In our context, distinctly different weights are produced for target models trained on inputs of different domains, and feature transferability (Yosinski et al., 2014) in the input space can be expected to translate to weight transferability in the model weights space. Rather than completely regenerating classification weights, our model robustification defense, Learn2Weight L2W predicts the perturbation to existing weights $\\left( W ^ { \\prime } = W + \\Delta W \\right)$ ) for each new instance. ",
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+    {
+        "type": "text",
+        "text": "5.2 LEARN2WEIGHT MODEL",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We conjecture that an effective defense strategy is to perturb the target model weights depending on the feature distribution of the input instance. $L 2 W$ (Algorithm 1) recalculates the target model weights depending on the input. $L 2 W$ (Algorithm 2) trains on sentences from different domains and a weight differential for that domain (the weight adjustment required to tune the target model’s weights to adapt to the input’s domain). We obtain the weight differential $\\Delta W$ by finding the difference between the weights of $f$ trained on sentence:label pairs from a specific domain $W _ { S _ { j } }$ and weights of $f$ trained on sentence:label pairs from the target domain $W _ { T }$ . Other training models may be possible; here we trained a sequence-to-sequence network (Sutskever et al., 2014) on sentence: $\\Delta W$ pairs. ",
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+    {
+        "type": "table",
+        "img_path": "images/e75efecc6a18495c496719643327dcb1137fc843056a22836ba84546dff0c37c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm 1: Learn2Weight: Inference</td><td>Algorithm 2: Learn2Weight:Training</td></tr><tr><td>inference (Xadv, L2W(.), f(Wr,T)) Input:Passed arguments include the adversarial input Xadu, the Learn2Weight model L2W(-),and the target model f(Wr,T) Output :Target model with weighted</td><td>train (T,D) Input :Target domain T={Ti}0; Set of M domains D={S,j)j; withNsentences, N,M i indexing specific sentence tensor and j indexing specific domains Output :Trained Learn2Weight model L2W(·)</td></tr><tr><td>updated byLearn2Weight f(W*,T) is the expected output Pass Xadu as in input into the trained L2W(-) function,and the weight differential required for the target model with weights Wr is△W.</td><td>Initialize empty X and Y to store sentences Xi from each domain j with corresponding weight differential. X←;Y←; Compute weights of f trained on T. Wr ← f(T);</td></tr><tr><td>△W ← L2W(Xadv); The returned function is the target model with updated weights Wr+△W. return f(Wr +△W,T);</td><td>X←T;Y←WT-Wr; Train each domain Sj,compute respective weights, append the differential △W to Y and each sentence in Si,j into X. foreach domain Sj ∈D do Ws,← f(Sj); △W ←Wsj-Wr;</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Algorithm 3: tf-optimization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "tfOptimization $( T , M , n _ { m a x } )$ ",
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+    {
+        "type": "text",
+        "text": "Input :Target domain $T = \\{ T _ { i } \\} _ { i = 0 } ^ { N }$ to be used in synthesizing $M$ similar domains; with $N$ sentences, $i$ indexing specific sentence tensor; $n _ { m a x }$ is the max number of tf-optimization iterations   \nOutput :Set $D$ containing $M$ domains   \nInitialize empty $D$ to store synthesized domains $S _ { j }$ of index $j$ .   \n$D \\gets \\emptyset ; j \\gets 0$ ;   \nwhile $j < M$ do Run each iteration until $n _ { m a x }$ . for iter $ 0$ to $n _ { m a x }$ do Apply adversarial perturbations to $T$ . $T _ { i t e r } ^ { \\hat { a } \\hat { d } v } \\gets A d v ( f , \\hat { T } )$ ; Determine change to $A d v ( \\cdot )$ or iter depending on computed transfer loss. if $c h e c k ( t f ( T _ { i t e r } ^ { a d v } , T ) ) \\gets T r u e$ then If low enough, $T _ { i t e r } ^ { a d v }$ can be added as synthetic domain into $D$ . $D \\gets T _ { i t e r } ^ { a d v }$ ; break; else Adjust perturbation parameters. $A d v ( \\cdot )  a d j u s t ( A d v ( \\cdot ) )$ ; $j \\gets j + 1$ ; Return set of Domains to be used for training by $L 2 W$ .   \nreturn $D$ ; ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5.3 TRANSFER LOSS OPTIMIZATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To generate synthetic domains of varying domain similarity so that defenders defend their model using only target domain data $T$ , the following equation introduces transfer loss optimization (Algorithm 3). The defender iteratively generates adversarial examples $X _ { N } ^ { a d v }$ while maximizing the transfer loss function $t f$ ; this produces a substitute attack domain corpora $S _ { j }$ . We iteratively adjust perturbation parameters $A d v ( \\cdot )$ and iteration count $N$ to minimize the transfer loss of our generated dataset. ",
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+    {
+        "type": "equation",
+        "img_path": "images/1b6573678dc3dd4ea5b8423291f92b975596760ad533343e240702ad5a8f1b05.jpg",
+        "text": "$$\n\\underset { N , A d v ( \\cdot ) } { \\arg \\operatorname* { m i n } } t f _ { N } \\Bigl ( X _ { N } ^ { a d v } , T \\Bigr ) = e \\Bigl ( A d v ( f , T ) , T \\Bigr ) - e \\Bigl ( T , T \\Bigr )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "5.4 EXPLANATION: BLOCKING TRANSFERABILITY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To facilitate our explanation, we adapt from domain adaptation literature (Ben-David et al., 2010; Liu et al., 2019; Zhang et al., $2 0 1 9 \\mathrm { c }$ ): ",
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+    {
+        "type": "equation",
+        "img_path": "images/fbd1b434c33efa896969ad206182b85e43c0cf5905f19fb73c4b6e00f53d9ebc.jpg",
+        "text": "$$\ne ( A , T ) \\leq e ( T , T ) + d _ { \\mathcal { H } \\Delta \\mathcal { H } } ( A , T ) + \\lambda\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { H }$ is the hypothesis space, $h$ is a hypothesis function that returns labels $\\{ 0 , 1 \\}$ , and $e ( T , T )$ and $e ( A , T )$ are the generalization errors from passing target domain data $T$ and adversarial data $A$ through a classifier trained on $T$ . $d _ { \\mathcal { H } \\Delta \\mathcal { H } } ( A , T )$ is the $\\mathcal { H } \\Delta \\bar { \\mathcal { H } }$ -distance between $T$ and $A$ , and measures the divergence between the feature distributions of $A$ and $T$ . $e _ { A } ( h , h ^ { ' } )$ and $e _ { T } ( h , h ^ { ' } )$ represents the probability that $h$ disagrees with $h ^ { ' }$ on the label of an input in the domain space $A$ and $T$ respectively. ",
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+    {
+        "type": "equation",
+        "img_path": "images/e743f16343d692345e0b44660667692cf235aacd42da08e5ed2f35b0d9a6e953.jpg",
+        "text": "$$\n\\begin{array} { r } { d _ { \\mathcal { H } \\Delta \\mathcal { H } } ( A , T ) = \\underset { h , h ^ { \\prime } \\in \\mathcal { H } } { \\operatorname* { s u p } } | e _ { A } ( h , h ^ { ' } ) - e _ { T } ( h , h ^ { ' } ) | } \\\\ { d _ { \\mathcal { H } \\Delta \\mathcal { H } } ( A , T ) = \\operatorname* { s u p } _ { h , h ^ { \\prime } \\in \\mathcal { H } } | \\mathbb { E } _ { X \\sim S } [ | ( h ( x ) - h ^ { ' } ( x ) | ] | } \\\\ { - | \\mathbb { E } _ { X \\sim T } [ | ( h ( x ) - h ^ { ' } ( x ) | ] | | } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Divergence $d _ { \\mathcal { H } \\Delta \\mathcal { H } }$ measures the divergence between feature distributions $A$ and $T$ . Higher $d _ { \\mathcal { H } \\Delta \\mathcal { H } }$ indicates less shared features between 2 domains. The greater the intersection between feature distributions, the greater the proportion of domain-variant features; one approach to domain adaptation is learning domain-invariant features representations (Zhao et al., 2019) to minimize $d _ { \\mathcal { H } \\Delta \\mathcal { H } }$ . ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/079bc03fda14b96c3b5dbe953caa03fb28c613ffe0f8aac7f02dbcfdee63dbd0.jpg",
+        "table_caption": [
+            "Table 3: After-defense accuracy performance of different defensive methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Target Domain</td><td colspan=\"3\">magazine</td><td colspan=\"3\">baby</td></tr><tr><td>Attack Domain</td><td>baby</td><td>dvd</td><td>book</td><td>dvd</td><td>book</td><td>magazine</td></tr><tr><td>After-attack Accuracy</td><td>0.381</td><td>0.366</td><td>0.343</td><td>0.365</td><td>0.386</td><td>0.401</td></tr><tr><td>After-defense Accuracy</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Adversarial training</td><td>0.639</td><td>0.559</td><td>0.657</td><td>0.558</td><td>0.577</td><td>0.661</td></tr><tr><td>Defensive distillation</td><td>0.549</td><td>0.561</td><td>0.597</td><td>0.588</td><td>0.629</td><td>0.577</td></tr><tr><td>SharedVocab defense</td><td>0.628</td><td>0.653</td><td>0.631</td><td>0.664</td><td>0.668</td><td>0.621</td></tr><tr><td>Domain-adapted adversarial training</td><td>0.608</td><td>0.637</td><td>0.620</td><td>0.604</td><td>0.620</td><td>0.587</td></tr><tr><td>Learn2Weight</td><td>0.796</td><td>0.842</td><td>0.843</td><td>0.774</td><td>0.751</td><td>0.737</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Explaining similarity-domain attacks. As demonstrated by empirical results, $e ( A , T )$ increases in a similarity-based attack setting, and this would arise if $d _ { \\mathcal { H } \\Delta \\mathcal { H } }$ increases correspondingly. $d _ { \\mathcal { H } \\Delta \\mathcal { H } }$ computes inconsistent labels from inconsistent feature distributions, and attributes the success of the attack to domain-variant features. ",
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+    {
+        "type": "text",
+        "text": "FGSM and variants adjust the input data to maximize the loss based on the backpropagated gradients of a model trained on $S$ . As our pipeline used correctly-labelled sentences before adversarially perturbing them, we can infer that perturbations applied to $S$ were not class-dependent (i.e. the success of the attack is not based on the removal of class-specific features), but class-independent features. It is already difficult for a model trained on $S$ to classify when there is insufficient classdependent features (hence a high $t f ( A , T ) )$ ; in a cross-domain setting, it must be even more difficult for a model trained on $T$ to classify given a shortage of domain-invariant, class-dependent features. ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf051b3c50cf04244555b5269b0ace2af8a828b43f8111388cda0349f36fdde8.jpg",
+        "text": "$$\n\\begin{array} { c } { { d _ { { \\mathcal { H } } \\Delta { \\mathcal { H } } } \\leq e ( A , T ) - e ( T , T ) - \\lambda } } \\\\ { { d _ { { \\mathcal { H } } \\Delta { \\mathcal { H } } } \\leq t f ( A , T ) - \\lambda } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Explaining Learn2Weight. $L 2 W$ minimizes divergence by training on $\\{ d _ { \\mathcal { H } \\Delta \\mathcal { H } } ( S _ { j } , T ) : \\Delta W _ { A . S _ { j } } \\}$ pairs, $d _ { \\mathcal { H } \\Delta \\mathcal { H } } ( S _ { j } , T )$ being reconstructed from the difference between S $x _ { i } ^ { S _ { j } }$ and $x _ { i } ^ { T }$ . $L 2 W$ is trained on $\\{ d _ { \\mathcal { H } \\Delta \\mathcal { H } } ^ { S _ { j } } \\} _ { j = 0 } ^ { N } : \\{ \\Delta W ^ { S _ { j } } \\} _ { j = 0 } ^ { N }$ pairs, such that . Intuitively the target model possesses a decision boundary (Liu et al., 2019) to classify inputs based on whether they cross the boundary or not; adversarial inputs have a tendency of being near the boundary and fooling it. Weights transfer learning applies perturbations to the decision boundary such that the boundary covers certain adversarial inputs otherwise misclassified, and in this way blocks transferability. The advantage of training on multiple domains $\\{ S _ { j } \\} _ { j = 0 } ^ { M }$ is that the after- $. L 2 W$ divergence between $A$ and $T$ is smaller because $L 2 W$ ’s weight perturbations render the decision boundary more precise in classifying inputs. ",
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+    {
+        "type": "text",
+        "text": "Explaining tf-optimization. We have attributed why adversarial sentences $A$ are computed to be domain-dissimilar despite originating from $S$ due to insufficient domain-invariant, class-dependent features resulting in low $e ( A , T )$ , i.e. low $t f ( A , T )$ . To replicate this phenomenon in natural domains, we use $t f$ -optimization to iteratively perturb $T$ to increase the proportion of class-independent features. This approximates the real-world similarity-based attack scenario where class-dependent features may be limited for inference. By generating the synthetic data, we are feeding $L 2 W$ attack data with variations in $d _ { \\mathcal { H } \\Delta \\mathcal { H } }$ and class-independent feature distributions. This prepares $L 2 W$ to robustify weights in $f ( T )$ when such feature distributions are encountered. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "6.1 BASELINES",
+        "text_level": 1,
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+        "text": "We consider two defense strategies that are empirically effective and are widely used for general black-box adversarial attacks: adversarial training (Goodfellow et al., 2014; Madry et al., 2017) and defensive distillation (Papernot et al., 2016b; 2017). In addition we consider two ablation baselines. ",
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+        "text": "Defensive distillation: The high-level implementation of defensive distillation (Papernot et al., 2016b; 2017) is to first train an initial model against target domain inputs and labels, and retrieve the raw class probability scores. The predicted probability values would be used as the new labels for the same target sentences, and we would train a new model based on this new label-sentence pair. ",
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+        "table_caption": [
+            "Table 4: Learn2Weight comparison against different attack model architectures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Target</td><td>Original</td><td>Attack</td><td colspan=\"4\">Attack Model After-attack Accuracy</td><td colspan=\"6\">After-Defense Accuracy</td></tr><tr><td>Domain</td><td>Accuracy Target Model: LSTM</td><td>Domain</td><td colspan=\"4\"></td><td colspan=\"4\">LogReg</td><td>CNN</td><td>LogReg</td></tr><tr><td colspan=\"4\"></td><td colspan=\"4\">BERT LSTM</td><td colspan=\"4\">BERT LSTM</td></tr><tr><td rowspan=\"3\">book</td><td rowspan=\"3\">0.880</td><td>dvd</td><td>0.342</td><td>0.413</td><td>GRU 0.477</td><td>CNN 0.335</td><td>0.440</td><td>0.786</td><td>0.847</td><td>0.804</td><td>0.816</td><td>0.782</td></tr><tr><td>kitchenware</td><td>0.350</td><td>0.372</td><td>0.325</td><td>0.353</td><td>0.425</td><td>0.765</td><td>0.826</td><td>0.795</td><td>0.742</td><td>0.767</td></tr><tr><td>electronics</td><td>0.400</td><td>0.389</td><td>0.416</td><td>0.315</td><td>0.460</td><td>0.792</td><td>0.812</td><td>0.784</td><td>0.770</td><td>0.725</td></tr><tr><td rowspan=\"3\">dvd</td><td rowspan=\"3\">0.920</td><td></td><td>0.326</td><td>0.434</td><td>0.479</td><td>0.383</td><td>0.490</td><td>0.816</td><td>0.795</td><td>0.824</td><td>0.804</td><td>0.794</td></tr><tr><td>book kitchenware</td><td>0.355</td><td>0.370</td><td>0.379</td><td>0.359</td><td>0.490</td><td>0.728</td><td>0.796</td><td>0.755</td><td>0.735</td><td>0.695</td></tr><tr><td>electronics</td><td>0.387</td><td>0.377</td><td>0.332</td><td>0.348</td><td>0.455</td><td>0.825</td><td>0.836</td><td>0.812</td><td>0.834</td><td>0.796</td></tr><tr><td rowspan=\"3\">electronics</td><td rowspan=\"3\">0.910</td><td>book</td><td>0.425</td><td>0.394</td><td>0.473</td><td>0.358</td><td>0.474</td><td>0.775</td><td>0.821</td><td>0.795</td><td>0.782</td><td>0.712</td></tr><tr><td>dvd</td><td>0.342</td><td>0.395</td><td>0.452</td><td>0.368</td><td>0.493</td><td>0.784</td><td>0.845</td><td>0.855</td><td>0.842</td><td>0.792</td></tr><tr><td>kitchenware</td><td>0.390</td><td>0.384</td><td>0.464</td><td>0.329</td><td>0.432</td><td>0.730</td><td>0.824</td><td>0.753</td><td>0.724</td><td>0.678</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Adversarial training: It is shown that injecting adversarial examples throughout training increases the robustness of target neural network models. In this baseline, target model is trained with both original training data and adversarial examples generated from original training data. However, since the adversarial examples are still generated from the target domain, it is unlikely that the method can defend similar-domain attack which is the result of domain-variant features. ",
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+    {
+        "type": "text",
+        "text": "SharedVocab defense. Given that it is the domain-variant features that cause the success of similardomain attack, a simple baseline is to remove those words that are not in the target domain’s vocabulary. This tests whether the effect of perturbing target model weights w.r.t domain-variant features will yield incremental after-attack accuracy in the similar-domain attack setting. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Domain-adapted adversarial training. This ablation baseline tests for incremental performance to a baseline defense using domain-variant inputs. We adapt adversarial training to be trained on adversarial sentences from attack domain $S$ , whereas the traditional adversarial training generates adversarial samples $X ^ { a d v , T }$ from its training data $T$ , this adapted version uses adversarial samples $X ^ { a d v , S }$ generated from $S$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "6.2 LEARN2WEIGHT PERFORMANCE",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Defense performance. We present the results of different defense baselines in Table 3. First, we can see that Learn2Weight achieves the highest after-defense accuracy against the adversarial attack. Take the magazine as target domain for example, if the adversary chooses to use book data as the attack domain, it would reduce the target model accuracy to 0.343. However, the Learn2Weight method can improve the performance to 0.843, which is a significant and substantial improvement against the attack. This improvements also exist across different target/attack domain pairs. Second, we see that all defense methods can improve the accuracy to some extent which indicates the importance and effectiveness of having robust training for machine learning models. Third, it is interesting to note that two simple baselines SharedVocab defense and Domain-adapted adversarial training yield overall better performance compared to adversarial training and defensive distillation. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Attack model architectures. So far, all the results are conducted using the same LSTM as the target/attack model due to simplicity purpose. Here, we keep the target model unchanged, but vary the architecture of the attack model for the generation of adversarial examples. A variation of a Recurrent Neural Network is a Gated Recurrent Unit (GRU) network, with 512 GRU cells, $60 \\%$ dropout and tanh activation function. We have also tested other attack model variants that are commonly-used in sentiment classification, including Bidirectional Encoder Representations from Transformers (BERT) (Devlin et al., 2019), Convolutional Neural Network (CNN) (Kim, 2014), and Logistic Regression (Maas et al., 2011). For both RNN and CNN, we use pre-computed Glove embeddings2 to encode words. All models are trained with enough epochs after ensuring the model achieved near state-of-the-art validation accuracy before proceeding to tests of adversarial attacks and defenses. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We present the results of different attack model architectures in Table 4. First, similar-domain attack is model-agnostic and it does not require the target and attack model to have identical architectures. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We can see that all four attack model architectures are able to reduce the target model accuracy. Second, the results suggest that Learn2Weight is also model-agnostic as it can substantially improve the after-defense accuracy regardless which attack model is used. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this newly-proposed, empirically-effective similar-domain attack, an adversary can choose a similar domain to the target task, build a substitute model and produce adversarial examples to fool the target model. We also propose a defense strategy, Learn2Weight, that learns to adapt the target model’s weight using crafted adversarial examples. Compared with other adversarial defense strategies, Learn2Weight can improve the target model robustness against the similar-domain attack. Our method demonstrates properties of a good adversarial defense, such as adopting defense architectures that adapt to situations/inputs rather than compromising standard error versus robustness error, to leverage class-independent properties in domain-variant text, and factoring in domain similarity in adversarial robustness exercises. ",
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+        "page_idx": 8
+    },
+    {
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+# SEQ2SLATE: RE-RANKING AND SLATE OPTIMIZATION WITH RNNS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Ranking is a central task in machine learning and information retrieval. In this task, it is especially important to present the user with a slate of items that is appealing as a whole. This in turn requires taking into account interactions between items, since intuitively, placing an item on the slate affects the decision of which other items should be placed alongside it. In this work, we propose a sequence-to-sequence model for ranking called seq2slate. At each step, the model predicts the next item to place on the slate given the items already selected. The recurrent nature of the model allows complex dependencies between items to be captured directly in a flexible and scalable way. We show how to learn the model end-to-end from weak supervision in the form of easily obtained click-through data. We further demonstrate the usefulness of our approach in experiments on standard ranking benchmarks as well as in a real-world recommendation system.
+
+# 1 INTRODUCTION
+
+Ranking a set of candidate items is a central task in machine learning and information retrieval. Many existing ranking systems are based on pointwise estimators, where the model assigns a score to each item in a candidate set and the resulting slate is obtained by sorting the list according to item scores (Liu et al., 2009). Such models are usually trained from click-through data to optimize an appropriate loss function (Joachims, 2002). This simple approach is computationally attractive as it only requires a sort operation over the candidate set at test (or serving) time, and can therefore scale to large problems. On the other hand, in terms of modeling, pointwise rankers cannot easily express dependencies between ranked items. In particular, the score of an item (e.g., its probability of being clicked) often depends on the other items in the slate and their joint placement. Such interactions between items can be especially dominant in the common case where display area is limited or when strong position bias is present, so that only a few highly ranked items get the user’s attention. In this case it may be preferable, for example, to present a diverse set of items at the top positions of the slate in order to cover a wider range of user interests.
+
+A significant amount of work on learning-to-rank does consider interactions between ranked items when training the model. In pairwise approaches a classifier is trained to determine which item should be ranked first within a pair of items (e.g., Herbrich et al., 1999; Joachims, 2002; Burges et al., 2005). Similarly, in listwise approaches the loss depends on the full permutation of items (e.g., Cao et al., 2007; Yue et al., 2007). Although these losses consider inter-item dependencies, the ranking function itself is pointwise, so at inference time the model still assigns a score to each item which does not depend on scores of other items.
+
+There has been some work on trying to capture interactions between items in the ranking scores themselves (e.g., Qin et al., 2008; 2009; Zhu et al., 2014; Rosenfeld et al., 2014; Dokania et al., 2014). Such approaches can, for example, encourage a pair of items to appear next to (or far from) each other in the resulting ranking. Approaches of this type often assume that the relationship between items takes a simple form (e.g., submodular) in order to obtain tractable inference and learning algorithms. Unfortunately, this comes at the expense of the model’s expressive power.
+
+In this paper, we present a general, scalable approach to ranking, which naturally accounts for high-order interactions. In particular, we apply a sequence-to-sequence (seq2seq) model (Sutskever et al., 2014) to the ranking task, where the input is the list of candidate items and the output is the resulting ordering. Since the output sequence corresponds to ranked items on the slate, we call this model sequence-to-slate (seq2slate). The order in which the input is processed can significantly affect the performance of such models (Vinyals et al., 2016). For this reason, we often assume the availability of a base (or “production”) ranker with which the input sequence is ordered (e.g., a simple pointwise method that ignores the interactions we seek to model), and view the output of our model as a re-ranking of the items.
+
+![](images/a577f1fca04cfb32a57bc7e32e0e69b3648831ea94ec93b3956f2fb540b63a10.jpg)  
+Figure 1: The seq2slate pointer network architecture for ranking.
+
+To address the seq2seq problem, we build on the recent success of recurrent neural networks (RNNs) in a wide range of applications (e.g., Sutskever et al., 2014). This allows us to use a deep model to capture rich dependencies between ranked items, while keeping the computational cost of inference manageable. More specifically, we use pointer networks, which are seq2seq models with an attention mechanism for pointing at positions in the input (Vinyals et al., 2015b). We show how to train the network end-to-end to directly optimize several commonly used ranking measures. To this end, we adapt RNN training to use weak supervision in the form of click-through data obtained from logs, instead of relying on ground-truth rankings, which are much more expensive to obtain. Finally, we demonstrate the usefulness of the proposed approach in a number of learning-to-rank benchmarks and in a large-scale, real-world recommendeation system.
+
+# 2 RANKING AND SLATE OPTIMIZATION AS SEQUENCE PREDICTION
+
+The ranking problem is that of computing a ranking of a set of items (or ordered list or slate) given some query or context. We formalize the problem as follows. Assume a set of $n$ items, each represented by a feature vector $x _ { i } \in \mathbb { R } ^ { m }$ (which may depend on a query or context). Let $\pi \in \Pi$ denote a permutation of the items, where each $\pi _ { j } \in \{ 1 , \ldots , n \}$ denotes the index of the item in position $j$ . Our goal is to predict the output ranking $\pi$ given the input items $x$ . For instance, given a specific user query, we might want to return an ordered set of music recommendations from a set of candidates that maximizes some measure of user engagement (e.g., number of tracks played).
+
+In the seq2seq framework, the probability of an output permutation, or slate, given the inputs is expressed as a product of conditional probabilities according to the chain rule:
+
+$$
+p ( \pi | x ) = \prod _ { j = 1 } ^ { n } p ( \pi _ { j } | \pi _ { 1 } , \dots , \pi _ { j - 1 } , x ) \ ,
+$$
+
+This expression is completely general and does not make any conditional independence assumptions. In our case, the conditional $p ( \pi _ { j } | \pi _ { < j } , x ) \in \Delta ^ { n }$ (a point in the $n$ -dimensional simplex) models the probability of any item being placed at the $j$ ’th position in the ranking given the items already placed at previous positions. Therefore, this conditional exactly captures all high-order dependencies between items in the ranked list, including those due to diversity, similarity or other interactions.
+
+Our setting is somewhat different than a standard seq2seq setting in that the output vocabulary is not fixed. In particular, the same index (position) is populated by different items in different instances (queries). Indeed, the vocabulary size $n$ itself may vary per instance in the common case where the number of items to rank can change. This is precisely the problem addressed by pointer networks, which we review next.
+
+# POINTER-NETWORK ARCHITECTURE FOR RANKING
+
+We employ the pointer-network architecture of Vinyals et al. (2015b) to model the conditional $p ( \pi _ { j } | \pi _ { < j } ^ { - } , \stackrel { . } { x } )$ . A pointer network uses non-parametric softmax modules, akin to the attention mechanism of Bahdanau et al. (2015), and learns to point to items in its input sequence rather than predicting an index from a fixed-sized vocabulary.
+
+Our seq2slate model, illustrated in Fig. 1, consists of two recurrent neural networks (RNNs): an encoder and a decoder, both of which use Long Short-term Memory (LSTM) cells (Hochreiter and Schmidhuber, 1997). At each encoding step $i \leq n$ , the encoder RNN reads the input vector $x _ { i }$ and outputs a $d$ -dimensional vector $e _ { i }$ , thus transforming the input sequence $\{ x _ { i } \} _ { i = 1 } ^ { n }$ into a sequence of latent memory states $\{ e _ { i } \} _ { i = 1 } ^ { n }$ . At each decoding step $j$ , the decoder RNN outputs a $d$ -dimensional vector $d _ { j }$ which is used as a query in our attention function. The attention function takes as input the query $d _ { j } \in \mathbb { R } ^ { d }$ and the set of latent memory states computed by the encoder $\{ e _ { i } \} _ { i = 1 } ^ { n }$ and produces a probability distribution over the next item to include in the output sequence as follows:
+
+$$
+\begin{array} { r l r } & { s _ { i } ^ { j } = v ^ { \top } \operatorname { t a n h } \left( W _ { e n c } \cdot e _ { i } + W _ { d e c } \cdot d _ { j } \right) } \\ & { p _ { \theta } ( \pi _ { j } = i | \pi _ { < j } , x ) \equiv p _ { i } ^ { j } = \left\{ \begin{array} { l l } { e ^ { s _ { i } ^ { j } } / \sum _ { k \notin \pi _ { < j } } e ^ { s _ { k } ^ { j } } } & { \mathrm { i f ~ } i \notin \pi _ { < j } } \\ { 0 } & { \mathrm { i f ~ } i \in \pi _ { < j } } \end{array} \right. } & { , } \end{array}
+$$
+
+where $W _ { e n c } , W _ { d e c } \in \mathbb { R } ^ { d \times d }$ and $v \in \mathbb { R } ^ { d }$ are learned parameters in our network, denoted collectively by parameter vector $\theta$ . The probability $p _ { i } ^ { j } = p _ { \theta } ( \pi _ { j } = i | \pi _ { < j } , x )$ , is obtained via a softmax over the remaining items and represents the degree to which the model points to input $i$ at decoding step $j$ . To output a permutation, the $p _ { i } ^ { j }$ are set to 0 for items $i$ that already appear in the slate. Once the next item $\pi _ { j }$ is selected, typically greedily or by sampling (see below), its embedding $x _ { \pi _ { j } }$ is fed as input to the next decoder step. The input of the first decoder step is a learned $d$ -dimensional vector, denoted as $g o$ in Fig. 1. Importantly, $p _ { \theta } ( \pi | x )$ is differentiable for ant fixed permutation $\pi$ which allows gradient-based learning (see Section 3).
+
+We note the following. (i) The model makes no explicit assumptions about the type of interactions between items. If the learned conditional in Eq. (2) is close to the true conditional in Eq. (1), then the model can capture rich interactions—including diversity, similarity or others. We demonstrate this flexibility in our experiments (Section 4). (ii) $x$ can represent either raw inputs or embeddings thereof, which can be learned together with the sequence model. (iii) The computational cost of inference, dominated by the sequential decoding procedure, is $O ( n ^ { 2 } )$ , which is standard in seq2seq models with attention. We also consider a computationally cheaper single-step decoder with linear cost $O ( n )$ , which outputs a single vector $p ^ { 1 }$ , from which we obtain $\pi$ by sorting the values (similarly to pointwise ranking).
+
+# 3 TRAINING WITH CLICK-THROUGH DATA
+
+We now turn to the task of training the seq2slate model from data. A typical approach to learning in ranking systems is to run an existing ranker “in the wild” and log click-through data, which are then used to train an improved ranking model. This type of training data is relatively inexpensive to obtain, in contrast to human-curated labels such as relevance scores, ratings, or rankings (Joachims, 2002).
+
+Formally, each training example consists of a sequence of items $\{ x _ { 1 } , \ldots , x _ { n } \}$ and binary labels $\left( y _ { 1 } , \ldots , y _ { n } \right)$ , with $y _ { i } \in \{ 0 , 1 \}$ , representing user feedback (e.g., click/no-click). Our approach easily extends to more informative feedback, such as the level of user engagement with the chosen item (e.g., time spent), but to simplify the presentation we focus on the binary case. Our goal is to learn the parameters $\theta$ of $p _ { \theta } ( \pi _ { j } | \pi _ { < j } , x )$ (Eq. (2)) such that permutations $\pi$ corresponding to “good” rankings are assigned high probabilities. Various performance measures $\mathcal { R } ( \pi , y )$ can be used to evaluate the quality of a permutation $\pi$ given the labels $y$ , for example, mean average precision (MAP), precision at $k$ , or normalized discounted cumulative gain at $k$ $( \mathrm { N D C G } @ \mathrm { k } )$ . Generally speaking, permutations where the positive labels rank higher are considered better.
+
+In the standard seq2seq setting, models are trained to maximize the likelihood of a target sequence of tokens given the input, which can be done by maximizing the likelihood of each target token given the previous target tokens using Eq. (1). During training, the model is typically fed the ground-truth tokens as inputs to the next prediction step, an approach known as teacher forcing (Williams and Zipser, 1989). Unfortunately, this approach cannot be applied in our setting since we only have access to weak supervision in the form of labels $y$ (e.g clicks), rather than ground-truth permutations. Instead, we show how the seq2slate model can be trained directly from the labels $y$ .
+
+# 3.1 TRAINING USING REINFORCE
+
+One potential approach, which has been applied successfully in related tasks (Bello et al., 2017; Zhong et al., 2017), is to use reinforcement learning $( R L )$ to directly optimize for the ranking measure $\mathcal { R } ( \pi , \overset { \mathbf { \tilde { \alpha } } } { \boldsymbol { y } } )$ . In this setup, the objective is to maximize the expected ranking metric obtained by sequences sampled from our model: $\mathbb { E } _ { \pi \sim p _ { \theta } ( . | x ) } [ \mathcal { R } ( \pi , y ) ]$ . One can use policy gradients and stochastic gradient ascent to optimize $\theta$ . The gradient is formulated using the popular REINFORCE update (Williams, 1992) and can be approximated via Monte-Carlo sampling as follows:
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \theta } \mathbb { E } _ { \pi \sim p _ { \theta } ( . | x ) } \big [ \mathcal { R } ( \pi , y ) \big ] = \mathbb { E } _ { \pi \sim p _ { \theta } ( . | x ) } \Big [ \mathcal { R } ( \pi , y ) \nabla _ { \theta } \log p _ { \theta } ( \pi \mid x ) \Big ] } \\ { \displaystyle \approx \frac { 1 } { B } \sum _ { k = 1 } ^ { B } \Big ( \mathcal { R } ( \pi _ { k } , y _ { k } ) - b ( x _ { k } ) \Big ) \nabla _ { \theta } \log p _ { \theta } ( \pi _ { k } \mid x _ { k } ) , } \end{array}
+$$
+
+where $k$ indexes ranking instances in a batch of size $B , \pi _ { k }$ are permutations drawn from the model $p _ { \theta }$ and $b ( x )$ denotes a baseline function that estimates the expected rewards to reduce the variance of the gradients.
+
+# 3.2 SUPERVISED TRAINING
+
+RL, however, is known to be a challenging optimization problem and can suffer from sample inefficiency and difficult credit assignment. As an alternative, we propose supervised learning using the labels $y$ . In particular, rather than waiting until the end of the output sequence (as in RL), we wish to give feedback to the model at each decoder step.
+
+Consider the first step, and recall that the model assigns a score $s _ { i }$ to each item in the input. We define a per-step loss $\ell ( s , y )$ which essentially acts as a multi-label classification loss with labels $y$ as ground truth. Two natural, simple choices for $\ell$ are cross-entropy loss and hinge loss:
+
+$$
+\begin{array} { l } { { \ell _ { x e n t } ( s , y ) = - \sum _ { i } { \hat { y } } _ { i } \log { p _ { i } } } } \\ { { \ell _ { h i n g e } ( s , y ) = \operatorname* { m a x } \{ 0 , 1 - \underset { i : y _ { i } = 1 } { \operatorname* { m i n } } s _ { i } + \underset { j : y _ { j } = 0 } { \operatorname* { m a x } } s _ { j } \} , } } \end{array}
+$$
+
+where $\hat { y } _ { i } = y _ { i } / \sum _ { j } y _ { j }$ , and $p _ { i }$ is a softmax of $s$ , similar to Eq. (2). Intuitively, with cross-entropy loss we try to assign high probabilities to positive labels (see also Kurata et al., 2016), while hinge loss is minimized when scores of items with positive labels are higher than scores of those with negative labels. Notice that both losses are convex functions of the scores $s$ . To improve convergence, we consider a smooth version of the hinge-loss where the maximum and minimum are replaced by their smooth counterparts: smoot $\begin{array} { r } { { \mathrm {  ~ \ h - m a x } } ( s ; \gamma ) = \frac { 1 } { \gamma } \log \sum _ { i } e ^ { \gamma s _ { i } } } \end{array}$ (and smooth minimum is defined similarly, using $\mathrm { m i n } _ { i } ( s _ { i } ) = - \mathrm { m a x } _ { i } ( - s _ { i } ) )$ .
+
+If we simply apply a per-step loss from Eq. (4) to all steps of the output sequence while reusing the labels $y$ at each step, then the loss is invariant to the actual output permutations (e.g., predicting a positive item at the beginning of the sequence has the same cost as predicting it at the end). Instead, we let the loss $\ell$ at each decoding step $j$ depend on the items already chosen, so no further loss is incurred after a label is predicted correctly. In particular, for a fixed permutation $\pi$ , define the sequence loss:
+
+$$
+\mathcal { L } _ { \pi } ( S , y ) = \sum _ { j = 1 } ^ { n } w _ { j } \ell _ { \pi _ { < j } } ( s ^ { j } , y ) ,
+$$
+
+where $\boldsymbol { S } = \{ s ^ { j } \} _ { j = 1 } ^ { n }$ , and $\ell _ { \pi < j } \left( s ^ { j } , y \right)$ depends only on the indices in $s ^ { j }$ and $y$ which are not in the prefix permutation $\pi _ { < j } = ( \pi _ { 1 } , \ldots , \pi _ { j - 1 } )$ (see Eq. (4)). Including a per-step weight $w _ { j }$ can encourage better performance earlier in the sequence (e.g., $w _ { j } = 1 / \log ( j + 1 ) )$ . Furthermore, if optimizing for a particular slate size $k$ is desired, one can restrict this loss to just the first $k$ output steps.
+
+# DECODING POLICIES DURING TRAINING
+
+Since teacher-forcing is not an option, we resort to feeding the model its own previous predictions, as in Bengio et al. (2015); Ranzato et al. (2016). In this case, the permutation $\pi$ is not fixed, but rather depends on the scores $S$ . Specifically, we consider two policies for producing a permutation during training, sampling and greedy decoding, and introduce their corresponding losses.
+
+Greedy policy The greedy policy consists of selecting the item that maximizes $p _ { \theta } ( \cdot | \pi _ { < j } , x )$ at every time step $j$ . The resulting permutation $\pi ^ { * }$ then satisfies $\pi _ { j } ^ { * } = \mathrm { a r g m a x } _ { i } p _ { \theta } ( \pi _ { j } = i | \pi _ { < j } ^ { * } )$ and our loss becomes ${ \mathcal { L } } _ { \pi ^ { * } }$ . The greedy policy loss is not continuous everywhere since a small change in the scores $s$ may result in a jump between permutations, and therefore ${ \mathcal { L } } _ { \pi }$ . Specifically, the loss is non-differentiable when any $s ^ { j }$ has multiple maximizing arguments. Outside this measure-zero subspace, the loss is continuous (almost everywhere), and the gradient is well-defined.
+
+Sampling policy The sampling policy consists of drawing each $\pi _ { j }$ from $p _ { \theta } ( \cdot | \pi _ { < j } , x )$ . The corresponding loss $\begin{array} { r } { \mathbb { E } [ \mathcal { L } ] = \sum _ { \pi } p _ { \theta } ^ { \mathrm { ~ ~ } } ( \pi ) \bar { \mathcal { L } } _ { \pi } ( \theta ) } \end{array}$ is differentiable everywhere since both $p _ { \theta } ( \pi )$ and ${ \mathcal { L } } _ { \pi } ( \theta )$ are differentiable for any permutation $\pi$ (See appendix for a direct derivation of $\mathbb { E } [ \mathcal { L } ]$ as a function of $S$ ). In this case, the gradient is formulated as:
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } \mathbb { E } [ \mathcal { L } ( \theta ) ] = \nabla _ { \theta } \displaystyle \sum _ { \pi } p _ { \theta } ( \pi ) \mathcal { L } _ { \pi } ( \theta ) } \\ & { \quad \quad \quad \quad = \displaystyle \sum _ { \pi } \left[ ( \nabla _ { \theta } p _ { \theta } ( \pi ) ) \mathcal { L } _ { \pi } ( \theta ) + p _ { \theta } ( \pi ) ( \nabla _ { \theta } \mathcal { L } _ { \pi } ( \theta ) ) \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { \pi \sim p _ { \theta } } \left[ \mathcal { L } _ { \pi } ( \theta ) \cdot \nabla _ { \theta } \log p _ { \theta } ( \pi ) + \nabla _ { \theta } \mathcal { L } _ { \pi } ( \theta ) \right] , } \end{array}
+$$
+
+which can be approximated by:
+
+$$
+\frac { 1 } { B } \sum _ { k = 1 } ^ { B } \left[ \left( \mathcal { L } _ { \pi _ { k } } ( \theta ) - b ( x _ { k } ) \right) \nabla _ { \theta } \log p _ { \theta } ( \pi _ { k } \mid x _ { k } ) + \nabla _ { \theta } \mathcal { L } _ { \pi _ { k } } ( \theta ) \right] ,
+$$
+
+where $b ( \boldsymbol { x } _ { k } )$ is a baseline that approximates ${ \mathcal { L } } _ { \pi _ { k } } ( \theta )$ . Applying stochastic gradient descent intuitively decreases both the loss of any sample (right term) but also the probability of drawing samples with high losses (left term). Notice that our gradient calculation differs from scheduled sampling (Bengio et al., 2015) which instead computes the loss of the sampled sequences (right term) but ignores the probability of sampling high loss sequences (left term). We found it helpful to include both terms, which may apply more generally to training of sequence-to-sequence models (Bengio et al., 2015; Goyal et al., 2016).
+
+For both training policies, we minimize the loss via stochastic gradient descent over mini-batches in an end-to-end fashion.
+
+# 4 EXPERIMENTAL RESULTS
+
+We evaluate the performance of our seq2slate model on a collection of ranking tasks. In Section 4.1 we use learning-to-rank benchmark data to study the behavior of the model. We then apply our approach to a large-scale commercial recommendation system and report the results in Section 4.2.
+
+Implementation Details We set hyperparameters of our model to values inspired by the literature. All experiments use mini-batches of 128 training examples and LSTM cells with 128 hidden units. We train our models with the Adam optimizer (Kingma and Ba, 2014) and an initial learning rate of 0.0003 decayed every 1000 steps by a factor of 0.96. Network parameters are initialized uniformly at random in $[ - 0 . 1 , 0 . 1 ]$ . To improve generalization, we regularize the model by using dropout with probability of dropping $p _ { d r o p o u t } = 0 . 1$ and L2 regularization with a penalty coefficient $\lambda = 0 . 0 0 0 3$ . Unless specified otherwise, all results use supervised training with cross-entropy loss $\ell _ { x e n t }$ and the sampling policy. At inference time, we report metrics for the greedy policy. We use an exponential moving average with a decay rate of 0.99 as the baseline $b ( x )$ in Eq. (3) and Eq. (6). When training the seq2slate model with REINFORCE, we use $\mathcal { R } = \mathop { \mathrm { N D G C @ 1 0 } }$ as the reward function and do not regularize the model. We also considered a bidirectional encoder RNN (Schuster and Paliwal, 1997) but found that it did not lead to significant improvements in our experiments.
+
+# 4.1 LEARNING-TO-RANK BENCHMARKS
+
+To understand the behavior of the proposed model, we conduct experiments using two learning-torank datasets. We use two of the largest publicly available benchmarks: the Yahoo Learning to Rank Challenge data (set 1),1 and the Web30k dataset.2 All context (query) features are embedded within the item feature vectors themselves.
+
+<table><tr><td rowspan="2">Ranker</td><td colspan="3">Yahoo</td><td colspan="3">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG10</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td></tr><tr><td>seq2slate</td><td>0.67</td><td>0.69</td><td>0.75</td><td>0.51</td><td>0.53</td><td>0.59</td></tr><tr><td>AdaRank</td><td>0.58</td><td>0.61</td><td>0.69</td><td>0.37</td><td>0.38</td><td>0.46</td></tr><tr><td>Coordinate Ascent</td><td>0.49</td><td>0.51</td><td>0.59</td><td>0.31</td><td>0.33</td><td>0.39</td></tr><tr><td>LambdaMART</td><td>0.58</td><td>0.61</td><td>0.69</td><td>0.42</td><td>0.46</td><td>0.52</td></tr><tr><td>ListNet</td><td>0.49</td><td>0.51</td><td>0.59</td><td>0.43</td><td>0.47</td><td>0.53</td></tr><tr><td>MART</td><td>0.58</td><td>0.60</td><td>0.68</td><td>0.39</td><td>0.42</td><td>0.48</td></tr><tr><td>Random Forests</td><td>0.54</td><td>0.57</td><td>0.65</td><td>0.36</td><td>0.39</td><td>0.45</td></tr><tr><td>RankBoost</td><td>0.50</td><td>0.52</td><td>0.60</td><td>0.24</td><td>0.25</td><td>0.30</td></tr><tr><td>RankNet</td><td>0.54</td><td>0.57</td><td>0.64</td><td>0.43</td><td>0.47</td><td>0.53</td></tr></table>
+
+Table 1: Performance of seq2slate and other baselines on data generated with diverse-clicks.
+
+We adapt the procedure proposed by Joachims et al. (2017) to generate click data. The original procedure is as follows: first, a base ranker is trained from the raw data. We select this base ranker by training all models in the RankLib package,3 and selecting the one with the best performance on each data set (MART for Yahoo and LambdaMART for Web30k). We generate an item ranking using the base model, which is then used to generate training data by simulating a user “cascade” model: a user observes each item with decaying probability $1 / i ^ { \eta }$ , where $i$ is the base rank of the item and $\eta$ is a parameter of the generative model. This simulates a noisy sequential scan. An observed item is clicked if its ground-truth relevance score is above a threshold (relevant: $\{ 2 , 3 , 4 \}$ , irrelevant: $\{ 0 , 1 \} ,$ ), otherwise no click is generated.
+
+To introduce high-order interactions, we augment the above procedure as follows, creating a generative process dubbed diverse-clicks. When observing a relevant item, the user will only click if it is not too similar to previously clicked items (i.e, diverse enough), thus reducing the total number of clicks. Similarity is defined as being in the smallest $q$ percentile (i.e., $q = 0 . 5$ is the median) of Euclidean distances between pairs of feature vectors within the same ranking instance: $d _ { i j } = \| x _ { i } - x _ { j } \|$ . We use $\eta = 0$ (no decay, since clicks are sparse anyway due to the diversity term) and $q = 0 . 5$ . This modification to the generative model is essential for our purpose as the original data does not contain explicit inter-item dependencies. We also discuss variations of this model below.
+
+Using the generated training data, we train both our seq2slate model and baseline rankers from the RankLib package: AdaRank (Xu and Li, 2007), Coordinate Ascent (Metzler and Croft, 2007), LambdaMART (Wu et al., 2010), ListNet (Cao et al., 2007), MART (Friedman, 2001), Random Forests (Breiman, 2001), RankBoost (Freund et al., 2003), RankNet (Burges et al., 2005). Some of these baselines use deep neural networks (e.g., RankNet, ListNet), so they are strong state-ofthe-art models with comparable complexity to seq2slate. The results in Table 1 show that seq2slate significantly outperforms all the baselines, suggesting that it can better capture and exploit the dependencies between items in the data.
+
+To better understand the behavior of the model, we visualize the probabilities of the attention from Eq. (2) for one of the test instances in Fig. 2. Interestingly, the model produces slates that are close to the input ranking, but with some items demoted to lower positions, presumably due to the interactions with previous items.
+
+We next consider several variations of the generative model and of the seq2slate model itself. Results are reported in Table 2. The rank-gain metric per example is computed by summing the positions change of all positive labels in the re-ranking, and this is averaged over all examples (queries).
+
+Comparison of training variants In Table 2, we compare the different training variants outlined in Section 3, namely cross entropy with the greedy or sampling policy, a smooth hinge loss with $\gamma = 1 . 0$ , and REINFORCE. We find that supervised learning with cross entropy generally performs best, with the smooth hinge loss doing slightly worse. Our weakly supervised training methods have positive rank gain on all datasets, meaning they improve over the base ranker. Results from Table 2 (see also Table 5 in the appendix) suggest that training with REINFORCE yields comparable results on Yahoo but significantly worse results on the more challenging Web30k dataset. We find no significant difference in performance between relying on the greedy and sampling policies during training.
+
+Table 2: Comparison of model and data variants for seq2slate on data generated with diverse-clicks.   
+
+<table><tr><td rowspan="2">Ranker</td><td colspan="4">Yahoo</td><td colspan="4">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td></tr><tr><td>seq2slate</td><td>0.67</td><td>0.69</td><td>0.75</td><td>7.4</td><td>0.51</td><td>0.53</td><td>0.59</td><td>18.3</td></tr><tr><td>Greedy policy</td><td>0.66</td><td>0.69</td><td>0.75</td><td>7.2</td><td>0.50</td><td>0.52</td><td>0.59</td><td>18.3</td></tr><tr><td>smooth-hinge</td><td>0.66</td><td>0.69</td><td>0.75</td><td>7.1</td><td>0.49</td><td>0.51</td><td>0.58</td><td>17.9</td></tr><tr><td>RL</td><td>0.66</td><td>0.68</td><td>0.75</td><td>5.7</td><td>0.44</td><td>0.47</td><td>0.53</td><td>-0.5</td></tr><tr><td>one-step decoder</td><td>0.66</td><td>0.69</td><td>0.75</td><td>6.4</td><td>0.49</td><td>0.51</td><td>0.58</td><td>16.5</td></tr><tr><td>shuffled data</td><td>0.61</td><td>0.64</td><td>0.71</td><td>1</td><td>0.36</td><td>0.36</td><td>0.44</td><td>二</td></tr><tr><td>base ranker (no-op)</td><td>0.58</td><td>0.61</td><td>0.69</td><td>0</td><td>0.45</td><td>0.48</td><td>0.54</td><td>0</td></tr></table>
+
+Table 3: Performance compared to a competitive base production ranker on real data.   
+
+<table><tr><td>Ranker</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td></tr><tr><td>one-step decoder</td><td>+26.79%</td><td>+10.69%</td><td>+40.67%</td><td>0.83</td></tr><tr><td>seq2slate</td><td>+31.32%</td><td>+14.47%</td><td>+45.77%</td><td>1.087</td></tr></table>
+
+One-step decoding We compare seq2slate to the model which uses a single decoding step, referred to as one-step decoder (see Section 2). In Table 2 we see that this model has comparable performance to the sequential decoder. This suggests that when inference time is crucial, as in many real-world systems, one might prefer the faster single-shot option. One possible explanation for the comparable performance of the one-step decoder is that the interactions in our generated data are rather simple and can be effectively learned by the encoder. By contrast, in Section 4.2 we show that on more complex real-world data, sequential decoding can perform significantly better.
+
+Sensitivity to input order Previous work suggests that the performance of seq2seq models are often sensitive to the order in which the input is processed (Vinyals et al., 2016; Nam et al., 2017). To test this we consider the use of seq2slate without relying on the base ranker to order the input, but instead items are fed to the model in random order. The results in Table 2 (see shuffled data) show that the performance is indeed significantly worse in this case, which is consistent with previous studies. It suggests that reranking is an easier task than ranking from scratch.
+
+Adaptivity to the type of interaction To demonstrate the flexibility of seq2slate, we generate data using a variant of the diverse-clicks model above. In the similar-clicks model, the user also clicks on observed irrelevant items if they are similar to previously clicked items (increasing the number of total clicks). As above, we use the pairwise distances in feature space $d _ { i j }$ to determine similarity. For this model we use $q = 0 . 5$ , and $\eta = 0 . 3$ for Web30k, $\eta = 0 . 1$ for Yahoo, to keep the proportion of positive labels similar. The results in the appendix (see Table 4) show that seq2slate has comparable performance to the baseline rankers, with slightly better performance on the harder Web30k data. This demonstrates that our model can adapt to various types of interactions in the data.
+
+# 4.2 REAL-WORLD DATA
+
+We also apply seq2slate to a ranking problem from a large-scale commercial recommendation system. We train the model using massive click-through logs (comprising roughly $O ( 1 0 ^ { 7 } )$ instances) with cross-entropy loss, the greedy policy, L2-regularization and dropout. The data has item sets of varying size, with an average $n$ of 10.24 items per example. We learn embeddings of the raw inputs as part of training. Table 3 shows the performance of seq2slate and the one-step decoder compared to the production base ranker on test data (of roughly the same size as the training data). Significant gains are observed in all performance metrics, with sequential decoding outperforming the one-step decoder. This suggests that sequential decoding may more faithfully capture complex dependencies between the items.
+
+Finally, we let the learned seq2slate model run in a live experiment (A/B testing). We compute the click-through rate (CTR) in each position (#clicks/#examples) for seq2slate. The production base ranker serves traffic outside the experiment, and we compute CTR per position for this traffic as well. Fig. 3 shows the difference in CTR per position, indicating that seq2slate has significantly higher CTR in the top positions. This suggests that seq2slate indeed places items that are likely to be chosen higher in the ranking.
+
+![](images/60ef907f9e21082dbf278a66435c7c2571d114ab9f70bbd97bccceee6a68e18b.jpg)  
+Figure 2: Visualization of attention probabilities on benchmark data. Intensities correspond to $p _ { i } ^ { j }$ for each item $i$ in step $j$ .
+
+![](images/d905e26cc9712b632f319eccc5341d66cb31cef90c14bf9748926677d56250cf.jpg)  
+Figure 3: Difference in CTR per position between a seq2slate model and a base production ranker in a live experiment.
+
+# 5 RELATED WORK
+
+In this section we discuss additional related work. Our work builds on the recent impressive success of seq2seq models in complex prediction tasks, including machine translation (Sutskever et al., 2014; Bahdanau et al., 2015), parsing (Vinyals et al., 2015a), combinatorial optimization (Vinyals et al., 2015b; Bello et al., 2017), multi-label classification (Wang et al., 2016; Nam et al., 2017), and others. Our work differs in that we explicitly target the ranking task, which requires a novel approach to training seq2seq models from weak feedback (click-through data).
+
+Most of the work on ranking mentioned above uses shallow representations. However, in recent years deep models have been used for information retrieval, focusing on embedding queries, documents and query-document pairs (Huang et al., 2013; Guo et al., 2016; Palangi et al., 2016; Wang and Klabjan, 2017; Pang et al., 2017) (see also recent survey by Mitra and Craswell (2017)). Rather than embedding individual items, in seq2slate a representation of the entire slate of items is learned and encoded in the RNN state. Moreover, learning the embeddings $( x )$ can be easily incorporated into the training of the sequence model to optimize both simultaneously end-to-end.
+
+Closest to ours is the recent work of Ai et al. (2018), where an RNN is used to encode a set of items for re-ranking. Their approach uses a single decoding step with attention, similar to our one-step decoder. In contrast, we use sequential decoding, which we find crucial in certain applications (see Section 4.2). Another important difference is that their training formulation assumes availability of full rankings or relevance scores, while we focus on learning from cheap click-through data.
+
+Finally, Santa Cruz et al. (2017) recently proposed an elegant framework for learning permutations based on the so called Sinkhorn operator. Their approach uses a continuous relaxation of permutation matrices (i.e., the set of doubly-stochastic matrices). Later, Mena et al. (2018) combined this with a Gumbel softmax distribution to enable efficient learning. However, this approach is focused on reconstruction of scrambled objects, and it is not obvious how to extend it to our ranking setting, where no ground-truth permutation is available.
+
+# 6 CONCLUSION
+
+We presented a novel seq2slate approach to ranking sets of items. We found the formalism of pointer-networks particularly suitable for this setting. We addressed the challenge of training the model from weak user feedback to improve the ranking quality. Our experiments show that the proposed approach is highly scalable and can deliver significant improvements in ranking results.
+
+Our work can be extended in several directions. In terms of architecture, we aim to explore the Transformer network (Vaswani et al., 2017) in place of the RNN. Several variants can potentially improve the performance of our model, including beam-search inference (Wiseman and Rush, 2016), and training with Actor-Critic (Bahdanau et al., 2017) or SeaRNN (Leblond et al., 2018) and it will be interesting to study their performance in the ranking setting. Finally, an interesting future work direction will be to study off-policy correction (Joachims et al., 2018) for seq2slate.
+
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+Qiang Wu, Christopher JC Burges, Krysta M Svore, and Jianfeng Gao. Adapting boosting for information retrieval measures. Information Retrieval, 13(3):254–270, 2010.   
+Jun Xu and Hang Li. Adarank: a boosting algorithm for information retrieval. In Proceedings of the 30th annual international ACM SIGIR conference on Research and development in information retrieval, pages 391–398. ACM, 2007.   
+Yisong Yue, Thomas Finley, Filip Radlinski, and Thorsten Joachims. A support vector method for optimizing average precision. In Proceedings of SIGIR, pages 271–278. ACM, 2007.   
+Victor Zhong, Caiming Xiong, and Richard Socher. Seq2sql: Generating structured queries from natural language using reinforcement learning. arXiv preprint arXiv:1709.00103, 2017.   
+Yadong Zhu, Yanyan Lan, Jiafeng Guo, Xueqi Cheng, and Shuzi Niu. Learning for search result diversification. In Proceedings of SIGIR, pages 293–302. ACM, 2014.
+
+A DERIVATION OF THE EXPECTED LOSS
+
+$$
+\begin{array} { r l } & { \mathbb { E } [ \mathcal { L } ] = \displaystyle \sum _ { \tau } p ( \boldsymbol { \pi } ) \mathcal { L } _ { \tau } } \\ & { \mathrm { ~ \ ~ \ } = \displaystyle \sum _ { \tau } p ( \boldsymbol { \pi } ) \sum _ { j } \ell _ { \tau < j } } \\ & { \mathrm { \ ~ \ ~ \ } = \displaystyle \sum _ { j } \sum _ { \tau } p ( \boldsymbol { \pi } _ { z < j } ) p ( \boldsymbol { \pi } _ { z < j } ) \ell _ { \tau < j } \ell _ { \tau < j } } \\ & { \mathrm { \ ~ \ ~ \ } = \displaystyle \sum _ { j } \sum _ { \tau } p ( \boldsymbol { \pi } _ { z < j } ) p ( \boldsymbol { \pi } _ { z \le j } ) \ell _ { \tau < j } \ell _ { \tau < j } } \\ & { \mathrm { \ ~ \ ~ \ } = \displaystyle \sum _ { j } \sum _ { \tau < j } p ( \boldsymbol { \pi } _ { z < j } ) \ell _ { \tau < j } \sum _ { \tau \ge j } p ( \boldsymbol { \pi } _ { z < j } | \boldsymbol { \pi } _ { z < j } ) } \\ & { \mathrm { \ ~ \ ~ \ } = \displaystyle \sum _ { j } \sum _ { \tau < j } \left( \sum _ { k = 1 } ^ { j - 1 } e ^ { \lambda _ { k } } / \sum _ { \tau \in S _ { k } } c \right) \ell _ { \tau < j } e ^ { \lambda _ { k } } \Big ) \ell _ { \tau < j } \langle \boldsymbol { s } ^ { j } , \boldsymbol { y } \rangle \ . } \end{array}
+$$
+
+Since the terms are continuous (and smooth) in $S$ for all $j$ and $\pi _ { < j }$ , so is the entire function.
+
+# B ADDITIONAL EXPERIMENTAL RESULTS
+
+<table><tr><td rowspan="2">Ranker</td><td colspan="3">Yahoo</td><td colspan="3">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td></tr><tr><td>seq2slate</td><td>0.82</td><td>0.82</td><td>0.84</td><td>0.44</td><td>0.54</td><td>0.50</td></tr><tr><td>AdaRank</td><td>0.83</td><td>0.81</td><td>0.84</td><td>0.41</td><td>0.52</td><td>0.48</td></tr><tr><td>Coordinate Ascent</td><td>0.83</td><td>0.82</td><td>0.85</td><td>0.39</td><td>0.47</td><td>0.44</td></tr><tr><td>LambdaMART</td><td>0.84</td><td>0.83</td><td>0.85</td><td>0.41</td><td>0.52</td><td>0.48</td></tr><tr><td>ListNet</td><td>0.83</td><td>0.83</td><td>0.85</td><td>0.41</td><td>0.53</td><td>0.49</td></tr><tr><td>MART</td><td>0.83</td><td>0.82</td><td>0.85</td><td>0.41</td><td>0.52</td><td>0.48</td></tr><tr><td>Random Forests</td><td>0.83</td><td>0.82</td><td>0.84</td><td>0.40</td><td>0.48</td><td>0.45</td></tr><tr><td>RankBoost</td><td>0.83</td><td>0.83</td><td>0.85</td><td>0.38</td><td>0.43</td><td>0.41</td></tr><tr><td>RankNet</td><td>0.83</td><td>0.82</td><td>0.84</td><td>0.35</td><td>0.36</td><td>0.35</td></tr></table>
+
+Table 4: Performance of seq2slate and other baselines on data generated with similar-clicks.
+
+<table><tr><td rowspan="2">Ranker</td><td colspan="4">Yahoo</td><td colspan="4">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td><td>MAP</td><td>NDCG@5</td><td>NDCG10</td><td>rank-gain</td></tr><tr><td>seq2slate</td><td>0.82</td><td>0.82</td><td>0.84</td><td>8.5</td><td>0.44</td><td>0.54</td><td>0.50</td><td>16.0</td></tr><tr><td>Greedy policy</td><td>0.82</td><td>0.82</td><td>0.84</td><td>8.5</td><td>0.44</td><td>0.54</td><td>0.50</td><td>15.9</td></tr><tr><td>smooth-hinge</td><td>0.80</td><td>0.80</td><td>0.82</td><td>7.7</td><td>0.44</td><td>0.54</td><td>0.50</td><td>15.9</td></tr><tr><td>RL</td><td>0.82</td><td>0.82</td><td>0.84</td><td>8.5</td><td>0.42</td><td>0.53</td><td>0.49</td><td>-14.8</td></tr><tr><td>one-step decoder</td><td>0.81</td><td>0.81</td><td>0.82</td><td>7.7</td><td>0.44</td><td>0.53</td><td>0.49</td><td>15.5</td></tr><tr><td>shuffled data</td><td>0.80</td><td>0.80</td><td>0.81</td><td>二</td><td>0.40</td><td>0.44</td><td>0.42</td><td>1</td></tr><tr><td>base ranker (no-op)</td><td>0.78</td><td>0.76</td><td>0.79</td><td>0</td><td>0.43</td><td>0.53</td><td>0.49</td><td>0</td></tr></table>
+
+Table 5: Comparison of model and data variants for seq2slate on data generated with similar-clicks.

parse/train/HkgHk3RctX/HkgHk3RctX_content_list.json

@@ -0,0 +1,1462 @@
+[
+    {
+        "type": "text",
+        "text": "SEQ2SLATE: RE-RANKING AND SLATE OPTIMIZATION WITH RNNS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
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+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            166,
+            400,
+            194
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            231,
+            544,
+            246
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ranking is a central task in machine learning and information retrieval. In this task, it is especially important to present the user with a slate of items that is appealing as a whole. This in turn requires taking into account interactions between items, since intuitively, placing an item on the slate affects the decision of which other items should be placed alongside it. In this work, we propose a sequence-to-sequence model for ranking called seq2slate. At each step, the model predicts the next item to place on the slate given the items already selected. The recurrent nature of the model allows complex dependencies between items to be captured directly in a flexible and scalable way. We show how to learn the model end-to-end from weak supervision in the form of easily obtained click-through data. We further demonstrate the usefulness of our approach in experiments on standard ranking benchmarks as well as in a real-world recommendation system. ",
+        "bbox": [
+            233,
+            261,
+            766,
+            426
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            454,
+            336,
+            469
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ranking a set of candidate items is a central task in machine learning and information retrieval. Many existing ranking systems are based on pointwise estimators, where the model assigns a score to each item in a candidate set and the resulting slate is obtained by sorting the list according to item scores (Liu et al., 2009). Such models are usually trained from click-through data to optimize an appropriate loss function (Joachims, 2002). This simple approach is computationally attractive as it only requires a sort operation over the candidate set at test (or serving) time, and can therefore scale to large problems. On the other hand, in terms of modeling, pointwise rankers cannot easily express dependencies between ranked items. In particular, the score of an item (e.g., its probability of being clicked) often depends on the other items in the slate and their joint placement. Such interactions between items can be especially dominant in the common case where display area is limited or when strong position bias is present, so that only a few highly ranked items get the user’s attention. In this case it may be preferable, for example, to present a diverse set of items at the top positions of the slate in order to cover a wider range of user interests. ",
+        "bbox": [
+            174,
+            483,
+            825,
+            664
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A significant amount of work on learning-to-rank does consider interactions between ranked items when training the model. In pairwise approaches a classifier is trained to determine which item should be ranked first within a pair of items (e.g., Herbrich et al., 1999; Joachims, 2002; Burges et al., 2005). Similarly, in listwise approaches the loss depends on the full permutation of items (e.g., Cao et al., 2007; Yue et al., 2007). Although these losses consider inter-item dependencies, the ranking function itself is pointwise, so at inference time the model still assigns a score to each item which does not depend on scores of other items. ",
+        "bbox": [
+            174,
+            666,
+            825,
+            763
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There has been some work on trying to capture interactions between items in the ranking scores themselves (e.g., Qin et al., 2008; 2009; Zhu et al., 2014; Rosenfeld et al., 2014; Dokania et al., 2014). Such approaches can, for example, encourage a pair of items to appear next to (or far from) each other in the resulting ranking. Approaches of this type often assume that the relationship between items takes a simple form (e.g., submodular) in order to obtain tractable inference and learning algorithms. Unfortunately, this comes at the expense of the model’s expressive power. ",
+        "bbox": [
+            174,
+            767,
+            825,
+            851
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a general, scalable approach to ranking, which naturally accounts for high-order interactions. In particular, we apply a sequence-to-sequence (seq2seq) model (Sutskever et al., 2014) to the ranking task, where the input is the list of candidate items and the output is the resulting ordering. Since the output sequence corresponds to ranked items on the slate, we call this model sequence-to-slate (seq2slate). The order in which the input is processed can significantly affect the performance of such models (Vinyals et al., 2016). For this reason, we often assume the availability of a base (or “production”) ranker with which the input sequence is ordered (e.g., a simple pointwise method that ignores the interactions we seek to model), and view the output of our model as a re-ranking of the items. ",
+        "bbox": [
+            174,
+            854,
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+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a577f1fca04cfb32a57bc7e32e0e69b3648831ea94ec93b3956f2fb540b63a10.jpg",
+        "image_caption": [
+            "Figure 1: The seq2slate pointer network architecture for ranking. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            103,
+            771,
+            292
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            352,
+            825,
+            409
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To address the seq2seq problem, we build on the recent success of recurrent neural networks (RNNs) in a wide range of applications (e.g., Sutskever et al., 2014). This allows us to use a deep model to capture rich dependencies between ranked items, while keeping the computational cost of inference manageable. More specifically, we use pointer networks, which are seq2seq models with an attention mechanism for pointing at positions in the input (Vinyals et al., 2015b). We show how to train the network end-to-end to directly optimize several commonly used ranking measures. To this end, we adapt RNN training to use weak supervision in the form of click-through data obtained from logs, instead of relying on ground-truth rankings, which are much more expensive to obtain. Finally, we demonstrate the usefulness of the proposed approach in a number of learning-to-rank benchmarks and in a large-scale, real-world recommendeation system. ",
+        "bbox": [
+            173,
+            411,
+            825,
+            550
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RANKING AND SLATE OPTIMIZATION AS SEQUENCE PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            568,
+            750,
+            583
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The ranking problem is that of computing a ranking of a set of items (or ordered list or slate) given some query or context. We formalize the problem as follows. Assume a set of $n$ items, each represented by a feature vector $x _ { i } \\in \\mathbb { R } ^ { m }$ (which may depend on a query or context). Let $\\pi \\in \\Pi$ denote a permutation of the items, where each $\\pi _ { j } \\in \\{ 1 , \\ldots , n \\}$ denotes the index of the item in position $j$ . Our goal is to predict the output ranking $\\pi$ given the input items $x$ . For instance, given a specific user query, we might want to return an ordered set of music recommendations from a set of candidates that maximizes some measure of user engagement (e.g., number of tracks played). ",
+        "bbox": [
+            173,
+            594,
+            825,
+            694
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the seq2seq framework, the probability of an output permutation, or slate, given the inputs is expressed as a product of conditional probabilities according to the chain rule: ",
+        "bbox": [
+            174,
+            696,
+            825,
+            724
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ef483fbc26dc99c62aace76404225eea2788d3d454821b2224e6410ff29b502.jpg",
+        "text": "$$\np ( \\pi | x ) = \\prod _ { j = 1 } ^ { n } p ( \\pi _ { j } | \\pi _ { 1 } , \\dots , \\pi _ { j - 1 } , x ) \\ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            731,
+            622,
+            775
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This expression is completely general and does not make any conditional independence assumptions. In our case, the conditional $p ( \\pi _ { j } | \\pi _ { < j } , x ) \\in \\Delta ^ { n }$ (a point in the $n$ -dimensional simplex) models the probability of any item being placed at the $j$ ’th position in the ranking given the items already placed at previous positions. Therefore, this conditional exactly captures all high-order dependencies between items in the ranked list, including those due to diversity, similarity or other interactions. ",
+        "bbox": [
+            173,
+            780,
+            825,
+            851
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our setting is somewhat different than a standard seq2seq setting in that the output vocabulary is not fixed. In particular, the same index (position) is populated by different items in different instances (queries). Indeed, the vocabulary size $n$ itself may vary per instance in the common case where the number of items to rank can change. This is precisely the problem addressed by pointer networks, which we review next. ",
+        "bbox": [
+            174,
+            854,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "POINTER-NETWORK ARCHITECTURE FOR RANKING ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            103,
+            534,
+            117
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We employ the pointer-network architecture of Vinyals et al. (2015b) to model the conditional $p ( \\pi _ { j } | \\pi _ { < j } ^ { - } , \\stackrel { . } { x } )$ . A pointer network uses non-parametric softmax modules, akin to the attention mechanism of Bahdanau et al. (2015), and learns to point to items in its input sequence rather than predicting an index from a fixed-sized vocabulary. ",
+        "bbox": [
+            173,
+            125,
+            825,
+            180
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our seq2slate model, illustrated in Fig. 1, consists of two recurrent neural networks (RNNs): an encoder and a decoder, both of which use Long Short-term Memory (LSTM) cells (Hochreiter and Schmidhuber, 1997). At each encoding step $i \\leq n$ , the encoder RNN reads the input vector $x _ { i }$ and outputs a $d$ -dimensional vector $e _ { i }$ , thus transforming the input sequence $\\{ x _ { i } \\} _ { i = 1 } ^ { n }$ into a sequence of latent memory states $\\{ e _ { i } \\} _ { i = 1 } ^ { n }$ . At each decoding step $j$ , the decoder RNN outputs a $d$ -dimensional vector $d _ { j }$ which is used as a query in our attention function. The attention function takes as input the query $d _ { j } \\in \\mathbb { R } ^ { d }$ and the set of latent memory states computed by the encoder $\\{ e _ { i } \\} _ { i = 1 } ^ { n }$ and produces a probability distribution over the next item to include in the output sequence as follows: ",
+        "bbox": [
+            173,
+            181,
+            825,
+            289
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57dcdcec7b7d32ba9b527bb5aecfdef6c4600fbebfb43b99841f004b73e46e50.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { s _ { i } ^ { j } = v ^ { \\top } \\operatorname { t a n h } \\left( W _ { e n c } \\cdot e _ { i } + W _ { d e c } \\cdot d _ { j } \\right) } \\\\ & { p _ { \\theta } ( \\pi _ { j } = i | \\pi _ { < j } , x ) \\equiv p _ { i } ^ { j } = \\left\\{ \\begin{array} { l l } { e ^ { s _ { i } ^ { j } } / \\sum _ { k \\notin \\pi _ { < j } } e ^ { s _ { k } ^ { j } } } & { \\mathrm { i f ~ } i \\notin \\pi _ { < j } } \\\\ { 0 } & { \\mathrm { i f ~ } i \\in \\pi _ { < j } } \\end{array} \\right. } & { , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            290,
+            699,
+            349
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { e n c } , W _ { d e c } \\in \\mathbb { R } ^ { d \\times d }$ and $v \\in \\mathbb { R } ^ { d }$ are learned parameters in our network, denoted collectively by parameter vector $\\theta$ . The probability $p _ { i } ^ { j } = p _ { \\theta } ( \\pi _ { j } = i | \\pi _ { < j } , x )$ , is obtained via a softmax over the remaining items and represents the degree to which the model points to input $i$ at decoding step $j$ . To output a permutation, the $p _ { i } ^ { j }$ are set to 0 for items $i$ that already appear in the slate. Once the next item $\\pi _ { j }$ is selected, typically greedily or by sampling (see below), its embedding $x _ { \\pi _ { j } }$ is fed as input to the next decoder step. The input of the first decoder step is a learned $d$ -dimensional vector, denoted as $g o$ in Fig. 1. Importantly, $p _ { \\theta } ( \\pi | x )$ is differentiable for ant fixed permutation $\\pi$ which allows gradient-based learning (see Section 3). ",
+        "bbox": [
+            173,
+            351,
+            825,
+            468
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We note the following. (i) The model makes no explicit assumptions about the type of interactions between items. If the learned conditional in Eq. (2) is close to the true conditional in Eq. (1), then the model can capture rich interactions—including diversity, similarity or others. We demonstrate this flexibility in our experiments (Section 4). (ii) $x$ can represent either raw inputs or embeddings thereof, which can be learned together with the sequence model. (iii) The computational cost of inference, dominated by the sequential decoding procedure, is $O ( n ^ { 2 } )$ , which is standard in seq2seq models with attention. We also consider a computationally cheaper single-step decoder with linear cost $O ( n )$ , which outputs a single vector $p ^ { 1 }$ , from which we obtain $\\pi$ by sorting the values (similarly to pointwise ranking). ",
+        "bbox": [
+            173,
+            470,
+            825,
+            597
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 TRAINING WITH CLICK-THROUGH DATA",
+        "text_level": 1,
+        "bbox": [
+            174,
+            611,
+            544,
+            628
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now turn to the task of training the seq2slate model from data. A typical approach to learning in ranking systems is to run an existing ranker “in the wild” and log click-through data, which are then used to train an improved ranking model. This type of training data is relatively inexpensive to obtain, in contrast to human-curated labels such as relevance scores, ratings, or rankings (Joachims, 2002). ",
+        "bbox": [
+            173,
+            638,
+            826,
+            694
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, each training example consists of a sequence of items $\\{ x _ { 1 } , \\ldots , x _ { n } \\}$ and binary labels $\\left( y _ { 1 } , \\ldots , y _ { n } \\right)$ , with $y _ { i } \\in \\{ 0 , 1 \\}$ , representing user feedback (e.g., click/no-click). Our approach easily extends to more informative feedback, such as the level of user engagement with the chosen item (e.g., time spent), but to simplify the presentation we focus on the binary case. Our goal is to learn the parameters $\\theta$ of $p _ { \\theta } ( \\pi _ { j } | \\pi _ { < j } , x )$ (Eq. (2)) such that permutations $\\pi$ corresponding to “good” rankings are assigned high probabilities. Various performance measures $\\mathcal { R } ( \\pi , y )$ can be used to evaluate the quality of a permutation $\\pi$ given the labels $y$ , for example, mean average precision (MAP), precision at $k$ , or normalized discounted cumulative gain at $k$ $( \\mathrm { N D C G } @ \\mathrm { k } )$ . Generally speaking, permutations where the positive labels rank higher are considered better. ",
+        "bbox": [
+            173,
+            696,
+            825,
+            823
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the standard seq2seq setting, models are trained to maximize the likelihood of a target sequence of tokens given the input, which can be done by maximizing the likelihood of each target token given the previous target tokens using Eq. (1). During training, the model is typically fed the ground-truth tokens as inputs to the next prediction step, an approach known as teacher forcing (Williams and Zipser, 1989). Unfortunately, this approach cannot be applied in our setting since we only have access to weak supervision in the form of labels $y$ (e.g clicks), rather than ground-truth permutations. Instead, we show how the seq2slate model can be trained directly from the labels $y$ . ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 TRAINING USING REINFORCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            436,
+            117
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "One potential approach, which has been applied successfully in related tasks (Bello et al., 2017; Zhong et al., 2017), is to use reinforcement learning $( R L )$ to directly optimize for the ranking measure $\\mathcal { R } ( \\pi , \\overset { \\mathbf { \\tilde { \\alpha } } } { \\boldsymbol { y } } )$ . In this setup, the objective is to maximize the expected ranking metric obtained by sequences sampled from our model: $\\mathbb { E } _ { \\pi \\sim p _ { \\theta } ( . | x ) } [ \\mathcal { R } ( \\pi , y ) ]$ . One can use policy gradients and stochastic gradient ascent to optimize $\\theta$ . The gradient is formulated using the popular REINFORCE update (Williams, 1992) and can be approximated via Monte-Carlo sampling as follows: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/90e24b7e9ef1832f09433a235e8aa1c47b48538ac85bfa9aa1b91f0c5f70763b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\nabla _ { \\theta } \\mathbb { E } _ { \\pi \\sim p _ { \\theta } ( . | x ) } \\big [ \\mathcal { R } ( \\pi , y ) \\big ] = \\mathbb { E } _ { \\pi \\sim p _ { \\theta } ( . | x ) } \\Big [ \\mathcal { R } ( \\pi , y ) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\pi \\mid x ) \\Big ] } \\\\ { \\displaystyle \\approx \\frac { 1 } { B } \\sum _ { k = 1 } ^ { B } \\Big ( \\mathcal { R } ( \\pi _ { k } , y _ { k } ) - b ( x _ { k } ) \\Big ) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\pi _ { k } \\mid x _ { k } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $k$ indexes ranking instances in a batch of size $B , \\pi _ { k }$ are permutations drawn from the model $p _ { \\theta }$ and $b ( x )$ denotes a baseline function that estimates the expected rewards to reduce the variance of the gradients. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 SUPERVISED TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "RL, however, is known to be a challenging optimization problem and can suffer from sample inefficiency and difficult credit assignment. As an alternative, we propose supervised learning using the labels $y$ . In particular, rather than waiting until the end of the output sequence (as in RL), we wish to give feedback to the model at each decoder step. ",
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+    {
+        "type": "text",
+        "text": "Consider the first step, and recall that the model assigns a score $s _ { i }$ to each item in the input. We define a per-step loss $\\ell ( s , y )$ which essentially acts as a multi-label classification loss with labels $y$ as ground truth. Two natural, simple choices for $\\ell$ are cross-entropy loss and hinge loss: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a13762d5c81e70097f51d6ed1cda5d0805f50bb0c356ec46f3be3ec538a28b5.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\ell _ { x e n t } ( s , y ) = - \\sum _ { i } { \\hat { y } } _ { i } \\log { p _ { i } } } } \\\\ { { \\ell _ { h i n g e } ( s , y ) = \\operatorname* { m a x } \\{ 0 , 1 - \\underset { i : y _ { i } = 1 } { \\operatorname* { m i n } } s _ { i } + \\underset { j : y _ { j } = 0 } { \\operatorname* { m a x } } s _ { j } \\} , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { y } _ { i } = y _ { i } / \\sum _ { j } y _ { j }$ , and $p _ { i }$ is a softmax of $s$ , similar to Eq. (2). Intuitively, with cross-entropy loss we try to assign high probabilities to positive labels (see also Kurata et al., 2016), while hinge loss is minimized when scores of items with positive labels are higher than scores of those with negative labels. Notice that both losses are convex functions of the scores $s$ . To improve convergence, we consider a smooth version of the hinge-loss where the maximum and minimum are replaced by their smooth counterparts: smoot $\\begin{array} { r } { { \\mathrm {  ~ \\ h - m a x } } ( s ; \\gamma ) = \\frac { 1 } { \\gamma } \\log \\sum _ { i } e ^ { \\gamma s _ { i } } } \\end{array}$ (and smooth minimum is defined similarly, using $\\mathrm { m i n } _ { i } ( s _ { i } ) = - \\mathrm { m a x } _ { i } ( - s _ { i } ) )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If we simply apply a per-step loss from Eq. (4) to all steps of the output sequence while reusing the labels $y$ at each step, then the loss is invariant to the actual output permutations (e.g., predicting a positive item at the beginning of the sequence has the same cost as predicting it at the end). Instead, we let the loss $\\ell$ at each decoding step $j$ depend on the items already chosen, so no further loss is incurred after a label is predicted correctly. In particular, for a fixed permutation $\\pi$ , define the sequence loss: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b7c12f37e35c37d266ea84bde04d4d61b8414c97e1f875031166263faa9e088.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\pi } ( S , y ) = \\sum _ { j = 1 } ^ { n } w _ { j } \\ell _ { \\pi _ { < j } } ( s ^ { j } , y ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            727,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { S } = \\{ s ^ { j } \\} _ { j = 1 } ^ { n }$ , and $\\ell _ { \\pi < j } \\left( s ^ { j } , y \\right)$ depends only on the indices in $s ^ { j }$ and $y$ which are not in the prefix permutation $\\pi _ { < j } = ( \\pi _ { 1 } , \\ldots , \\pi _ { j - 1 } )$ (see Eq. (4)). Including a per-step weight $w _ { j }$ can encourage better performance earlier in the sequence (e.g., $w _ { j } = 1 / \\log ( j + 1 ) )$ . Furthermore, if optimizing for a particular slate size $k$ is desired, one can restrict this loss to just the first $k$ output steps. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DECODING POLICIES DURING TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since teacher-forcing is not an option, we resort to feeding the model its own previous predictions, as in Bengio et al. (2015); Ranzato et al. (2016). In this case, the permutation $\\pi$ is not fixed, but rather depends on the scores $S$ . Specifically, we consider two policies for producing a permutation during training, sampling and greedy decoding, and introduce their corresponding losses. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Greedy policy The greedy policy consists of selecting the item that maximizes $p _ { \\theta } ( \\cdot | \\pi _ { < j } , x )$ at every time step $j$ . The resulting permutation $\\pi ^ { * }$ then satisfies $\\pi _ { j } ^ { * } = \\mathrm { a r g m a x } _ { i } p _ { \\theta } ( \\pi _ { j } = i | \\pi _ { < j } ^ { * } )$ and our loss becomes ${ \\mathcal { L } } _ { \\pi ^ { * } }$ . The greedy policy loss is not continuous everywhere since a small change in the scores $s$ may result in a jump between permutations, and therefore ${ \\mathcal { L } } _ { \\pi }$ . Specifically, the loss is non-differentiable when any $s ^ { j }$ has multiple maximizing arguments. Outside this measure-zero subspace, the loss is continuous (almost everywhere), and the gradient is well-defined. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Sampling policy The sampling policy consists of drawing each $\\pi _ { j }$ from $p _ { \\theta } ( \\cdot | \\pi _ { < j } , x )$ . The corresponding loss $\\begin{array} { r } { \\mathbb { E } [ \\mathcal { L } ] = \\sum _ { \\pi } p _ { \\theta } ^ { \\mathrm { ~ ~ } } ( \\pi ) \\bar { \\mathcal { L } } _ { \\pi } ( \\theta ) } \\end{array}$ is differentiable everywhere since both $p _ { \\theta } ( \\pi )$ and ${ \\mathcal { L } } _ { \\pi } ( \\theta )$ are differentiable for any permutation $\\pi$ (See appendix for a direct derivation of $\\mathbb { E } [ \\mathcal { L } ]$ as a function of $S$ ). In this case, the gradient is formulated as: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/47a356c2e324797ba3b3e460dbb65ea42981a304f4a35d769c1a26137555c6df.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\nabla _ { \\theta } \\mathbb { E } [ \\mathcal { L } ( \\theta ) ] = \\nabla _ { \\theta } \\displaystyle \\sum _ { \\pi } p _ { \\theta } ( \\pi ) \\mathcal { L } _ { \\pi } ( \\theta ) } \\\\ & { \\quad \\quad \\quad \\quad = \\displaystyle \\sum _ { \\pi } \\left[ ( \\nabla _ { \\theta } p _ { \\theta } ( \\pi ) ) \\mathcal { L } _ { \\pi } ( \\theta ) + p _ { \\theta } ( \\pi ) ( \\nabla _ { \\theta } \\mathcal { L } _ { \\pi } ( \\theta ) ) \\right] } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { \\pi \\sim p _ { \\theta } } \\left[ \\mathcal { L } _ { \\pi } ( \\theta ) \\cdot \\nabla _ { \\theta } \\log p _ { \\theta } ( \\pi ) + \\nabla _ { \\theta } \\mathcal { L } _ { \\pi } ( \\theta ) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which can be approximated by: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8917c45dd49f8d41dd3cddd6ce9758382c2034dd7c8002e2b0509d18abafeb36.jpg",
+        "text": "$$\n\\frac { 1 } { B } \\sum _ { k = 1 } ^ { B } \\left[ \\left( \\mathcal { L } _ { \\pi _ { k } } ( \\theta ) - b ( x _ { k } ) \\right) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\pi _ { k } \\mid x _ { k } ) + \\nabla _ { \\theta } \\mathcal { L } _ { \\pi _ { k } } ( \\theta ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $b ( \\boldsymbol { x } _ { k } )$ is a baseline that approximates ${ \\mathcal { L } } _ { \\pi _ { k } } ( \\theta )$ . Applying stochastic gradient descent intuitively decreases both the loss of any sample (right term) but also the probability of drawing samples with high losses (left term). Notice that our gradient calculation differs from scheduled sampling (Bengio et al., 2015) which instead computes the loss of the sampled sequences (right term) but ignores the probability of sampling high loss sequences (left term). We found it helpful to include both terms, which may apply more generally to training of sequence-to-sequence models (Bengio et al., 2015; Goyal et al., 2016). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For both training policies, we minimize the loss via stochastic gradient descent over mini-batches in an end-to-end fashion. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the performance of our seq2slate model on a collection of ranking tasks. In Section 4.1 we use learning-to-rank benchmark data to study the behavior of the model. We then apply our approach to a large-scale commercial recommendation system and report the results in Section 4.2. ",
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+    {
+        "type": "text",
+        "text": "Implementation Details We set hyperparameters of our model to values inspired by the literature. All experiments use mini-batches of 128 training examples and LSTM cells with 128 hidden units. We train our models with the Adam optimizer (Kingma and Ba, 2014) and an initial learning rate of 0.0003 decayed every 1000 steps by a factor of 0.96. Network parameters are initialized uniformly at random in $[ - 0 . 1 , 0 . 1 ]$ . To improve generalization, we regularize the model by using dropout with probability of dropping $p _ { d r o p o u t } = 0 . 1$ and L2 regularization with a penalty coefficient $\\lambda = 0 . 0 0 0 3$ . Unless specified otherwise, all results use supervised training with cross-entropy loss $\\ell _ { x e n t }$ and the sampling policy. At inference time, we report metrics for the greedy policy. We use an exponential moving average with a decay rate of 0.99 as the baseline $b ( x )$ in Eq. (3) and Eq. (6). When training the seq2slate model with REINFORCE, we use $\\mathcal { R } = \\mathop { \\mathrm { N D G C @ 1 0 } }$ as the reward function and do not regularize the model. We also considered a bidirectional encoder RNN (Schuster and Paliwal, 1997) but found that it did not lead to significant improvements in our experiments. ",
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+    {
+        "type": "text",
+        "text": "4.1 LEARNING-TO-RANK BENCHMARKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To understand the behavior of the proposed model, we conduct experiments using two learning-torank datasets. We use two of the largest publicly available benchmarks: the Yahoo Learning to Rank Challenge data (set 1),1 and the Web30k dataset.2 All context (query) features are embedded within the item feature vectors themselves. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/45c83174d9b6d522266c66105cae802ecd55e5e13a375bef679b8f1173da6710.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Ranker</td><td colspan=\"3\">Yahoo</td><td colspan=\"3\">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG10</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td></tr><tr><td>seq2slate</td><td>0.67</td><td>0.69</td><td>0.75</td><td>0.51</td><td>0.53</td><td>0.59</td></tr><tr><td>AdaRank</td><td>0.58</td><td>0.61</td><td>0.69</td><td>0.37</td><td>0.38</td><td>0.46</td></tr><tr><td>Coordinate Ascent</td><td>0.49</td><td>0.51</td><td>0.59</td><td>0.31</td><td>0.33</td><td>0.39</td></tr><tr><td>LambdaMART</td><td>0.58</td><td>0.61</td><td>0.69</td><td>0.42</td><td>0.46</td><td>0.52</td></tr><tr><td>ListNet</td><td>0.49</td><td>0.51</td><td>0.59</td><td>0.43</td><td>0.47</td><td>0.53</td></tr><tr><td>MART</td><td>0.58</td><td>0.60</td><td>0.68</td><td>0.39</td><td>0.42</td><td>0.48</td></tr><tr><td>Random Forests</td><td>0.54</td><td>0.57</td><td>0.65</td><td>0.36</td><td>0.39</td><td>0.45</td></tr><tr><td>RankBoost</td><td>0.50</td><td>0.52</td><td>0.60</td><td>0.24</td><td>0.25</td><td>0.30</td></tr><tr><td>RankNet</td><td>0.54</td><td>0.57</td><td>0.64</td><td>0.43</td><td>0.47</td><td>0.53</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 1: Performance of seq2slate and other baselines on data generated with diverse-clicks. ",
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+    {
+        "type": "text",
+        "text": "We adapt the procedure proposed by Joachims et al. (2017) to generate click data. The original procedure is as follows: first, a base ranker is trained from the raw data. We select this base ranker by training all models in the RankLib package,3 and selecting the one with the best performance on each data set (MART for Yahoo and LambdaMART for Web30k). We generate an item ranking using the base model, which is then used to generate training data by simulating a user “cascade” model: a user observes each item with decaying probability $1 / i ^ { \\eta }$ , where $i$ is the base rank of the item and $\\eta$ is a parameter of the generative model. This simulates a noisy sequential scan. An observed item is clicked if its ground-truth relevance score is above a threshold (relevant: $\\{ 2 , 3 , 4 \\}$ , irrelevant: $\\{ 0 , 1 \\} ,$ ), otherwise no click is generated. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To introduce high-order interactions, we augment the above procedure as follows, creating a generative process dubbed diverse-clicks. When observing a relevant item, the user will only click if it is not too similar to previously clicked items (i.e, diverse enough), thus reducing the total number of clicks. Similarity is defined as being in the smallest $q$ percentile (i.e., $q = 0 . 5$ is the median) of Euclidean distances between pairs of feature vectors within the same ranking instance: $d _ { i j } = \\| x _ { i } - x _ { j } \\|$ . We use $\\eta = 0$ (no decay, since clicks are sparse anyway due to the diversity term) and $q = 0 . 5$ . This modification to the generative model is essential for our purpose as the original data does not contain explicit inter-item dependencies. We also discuss variations of this model below. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using the generated training data, we train both our seq2slate model and baseline rankers from the RankLib package: AdaRank (Xu and Li, 2007), Coordinate Ascent (Metzler and Croft, 2007), LambdaMART (Wu et al., 2010), ListNet (Cao et al., 2007), MART (Friedman, 2001), Random Forests (Breiman, 2001), RankBoost (Freund et al., 2003), RankNet (Burges et al., 2005). Some of these baselines use deep neural networks (e.g., RankNet, ListNet), so they are strong state-ofthe-art models with comparable complexity to seq2slate. The results in Table 1 show that seq2slate significantly outperforms all the baselines, suggesting that it can better capture and exploit the dependencies between items in the data. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To better understand the behavior of the model, we visualize the probabilities of the attention from Eq. (2) for one of the test instances in Fig. 2. Interestingly, the model produces slates that are close to the input ranking, but with some items demoted to lower positions, presumably due to the interactions with previous items. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We next consider several variations of the generative model and of the seq2slate model itself. Results are reported in Table 2. The rank-gain metric per example is computed by summing the positions change of all positive labels in the re-ranking, and this is averaged over all examples (queries). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparison of training variants In Table 2, we compare the different training variants outlined in Section 3, namely cross entropy with the greedy or sampling policy, a smooth hinge loss with $\\gamma = 1 . 0$ , and REINFORCE. We find that supervised learning with cross entropy generally performs best, with the smooth hinge loss doing slightly worse. Our weakly supervised training methods have positive rank gain on all datasets, meaning they improve over the base ranker. Results from Table 2 (see also Table 5 in the appendix) suggest that training with REINFORCE yields comparable results on Yahoo but significantly worse results on the more challenging Web30k dataset. We find no significant difference in performance between relying on the greedy and sampling policies during training. ",
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+    {
+        "type": "table",
+        "img_path": "images/f41e6f905463178240e2eb0e71048c47a62fa263ee836b3f2ff9d54d958d7a57.jpg",
+        "table_caption": [
+            "Table 2: Comparison of model and data variants for seq2slate on data generated with diverse-clicks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Ranker</td><td colspan=\"4\">Yahoo</td><td colspan=\"4\">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td></tr><tr><td>seq2slate</td><td>0.67</td><td>0.69</td><td>0.75</td><td>7.4</td><td>0.51</td><td>0.53</td><td>0.59</td><td>18.3</td></tr><tr><td>Greedy policy</td><td>0.66</td><td>0.69</td><td>0.75</td><td>7.2</td><td>0.50</td><td>0.52</td><td>0.59</td><td>18.3</td></tr><tr><td>smooth-hinge</td><td>0.66</td><td>0.69</td><td>0.75</td><td>7.1</td><td>0.49</td><td>0.51</td><td>0.58</td><td>17.9</td></tr><tr><td>RL</td><td>0.66</td><td>0.68</td><td>0.75</td><td>5.7</td><td>0.44</td><td>0.47</td><td>0.53</td><td>-0.5</td></tr><tr><td>one-step decoder</td><td>0.66</td><td>0.69</td><td>0.75</td><td>6.4</td><td>0.49</td><td>0.51</td><td>0.58</td><td>16.5</td></tr><tr><td>shuffled data</td><td>0.61</td><td>0.64</td><td>0.71</td><td>1</td><td>0.36</td><td>0.36</td><td>0.44</td><td>二</td></tr><tr><td>base ranker (no-op)</td><td>0.58</td><td>0.61</td><td>0.69</td><td>0</td><td>0.45</td><td>0.48</td><td>0.54</td><td>0</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Table 3: Performance compared to a competitive base production ranker on real data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Ranker</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td></tr><tr><td>one-step decoder</td><td>+26.79%</td><td>+10.69%</td><td>+40.67%</td><td>0.83</td></tr><tr><td>seq2slate</td><td>+31.32%</td><td>+14.47%</td><td>+45.77%</td><td>1.087</td></tr></table>",
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+        "type": "text",
+        "text": "One-step decoding We compare seq2slate to the model which uses a single decoding step, referred to as one-step decoder (see Section 2). In Table 2 we see that this model has comparable performance to the sequential decoder. This suggests that when inference time is crucial, as in many real-world systems, one might prefer the faster single-shot option. One possible explanation for the comparable performance of the one-step decoder is that the interactions in our generated data are rather simple and can be effectively learned by the encoder. By contrast, in Section 4.2 we show that on more complex real-world data, sequential decoding can perform significantly better. ",
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+    {
+        "type": "text",
+        "text": "Sensitivity to input order Previous work suggests that the performance of seq2seq models are often sensitive to the order in which the input is processed (Vinyals et al., 2016; Nam et al., 2017). To test this we consider the use of seq2slate without relying on the base ranker to order the input, but instead items are fed to the model in random order. The results in Table 2 (see shuffled data) show that the performance is indeed significantly worse in this case, which is consistent with previous studies. It suggests that reranking is an easier task than ranking from scratch. ",
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+    {
+        "type": "text",
+        "text": "Adaptivity to the type of interaction To demonstrate the flexibility of seq2slate, we generate data using a variant of the diverse-clicks model above. In the similar-clicks model, the user also clicks on observed irrelevant items if they are similar to previously clicked items (increasing the number of total clicks). As above, we use the pairwise distances in feature space $d _ { i j }$ to determine similarity. For this model we use $q = 0 . 5$ , and $\\eta = 0 . 3$ for Web30k, $\\eta = 0 . 1$ for Yahoo, to keep the proportion of positive labels similar. The results in the appendix (see Table 4) show that seq2slate has comparable performance to the baseline rankers, with slightly better performance on the harder Web30k data. This demonstrates that our model can adapt to various types of interactions in the data. ",
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+    {
+        "type": "text",
+        "text": "4.2 REAL-WORLD DATA ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We also apply seq2slate to a ranking problem from a large-scale commercial recommendation system. We train the model using massive click-through logs (comprising roughly $O ( 1 0 ^ { 7 } )$ instances) with cross-entropy loss, the greedy policy, L2-regularization and dropout. The data has item sets of varying size, with an average $n$ of 10.24 items per example. We learn embeddings of the raw inputs as part of training. Table 3 shows the performance of seq2slate and the one-step decoder compared to the production base ranker on test data (of roughly the same size as the training data). Significant gains are observed in all performance metrics, with sequential decoding outperforming the one-step decoder. This suggests that sequential decoding may more faithfully capture complex dependencies between the items. ",
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+        "type": "text",
+        "text": "Finally, we let the learned seq2slate model run in a live experiment (A/B testing). We compute the click-through rate (CTR) in each position (#clicks/#examples) for seq2slate. The production base ranker serves traffic outside the experiment, and we compute CTR per position for this traffic as well. Fig. 3 shows the difference in CTR per position, indicating that seq2slate has significantly higher CTR in the top positions. This suggests that seq2slate indeed places items that are likely to be chosen higher in the ranking. ",
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+        "img_path": "images/60ef907f9e21082dbf278a66435c7c2571d114ab9f70bbd97bccceee6a68e18b.jpg",
+        "image_caption": [
+            "Figure 2: Visualization of attention probabilities on benchmark data. Intensities correspond to $p _ { i } ^ { j }$ for each item $i$ in step $j$ . "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 3: Difference in CTR per position between a seq2slate model and a base production ranker in a live experiment. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we discuss additional related work. Our work builds on the recent impressive success of seq2seq models in complex prediction tasks, including machine translation (Sutskever et al., 2014; Bahdanau et al., 2015), parsing (Vinyals et al., 2015a), combinatorial optimization (Vinyals et al., 2015b; Bello et al., 2017), multi-label classification (Wang et al., 2016; Nam et al., 2017), and others. Our work differs in that we explicitly target the ranking task, which requires a novel approach to training seq2seq models from weak feedback (click-through data). ",
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+    },
+    {
+        "type": "text",
+        "text": "Most of the work on ranking mentioned above uses shallow representations. However, in recent years deep models have been used for information retrieval, focusing on embedding queries, documents and query-document pairs (Huang et al., 2013; Guo et al., 2016; Palangi et al., 2016; Wang and Klabjan, 2017; Pang et al., 2017) (see also recent survey by Mitra and Craswell (2017)). Rather than embedding individual items, in seq2slate a representation of the entire slate of items is learned and encoded in the RNN state. Moreover, learning the embeddings $( x )$ can be easily incorporated into the training of the sequence model to optimize both simultaneously end-to-end. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Closest to ours is the recent work of Ai et al. (2018), where an RNN is used to encode a set of items for re-ranking. Their approach uses a single decoding step with attention, similar to our one-step decoder. In contrast, we use sequential decoding, which we find crucial in certain applications (see Section 4.2). Another important difference is that their training formulation assumes availability of full rankings or relevance scores, while we focus on learning from cheap click-through data. ",
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+    },
+    {
+        "type": "text",
+        "text": "Finally, Santa Cruz et al. (2017) recently proposed an elegant framework for learning permutations based on the so called Sinkhorn operator. Their approach uses a continuous relaxation of permutation matrices (i.e., the set of doubly-stochastic matrices). Later, Mena et al. (2018) combined this with a Gumbel softmax distribution to enable efficient learning. However, this approach is focused on reconstruction of scrambled objects, and it is not obvious how to extend it to our ranking setting, where no ground-truth permutation is available. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We presented a novel seq2slate approach to ranking sets of items. We found the formalism of pointer-networks particularly suitable for this setting. We addressed the challenge of training the model from weak user feedback to improve the ranking quality. Our experiments show that the proposed approach is highly scalable and can deliver significant improvements in ranking results. ",
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+    {
+        "type": "text",
+        "text": "Our work can be extended in several directions. In terms of architecture, we aim to explore the Transformer network (Vaswani et al., 2017) in place of the RNN. Several variants can potentially improve the performance of our model, including beam-search inference (Wiseman and Rush, 2016), and training with Actor-Critic (Bahdanau et al., 2017) or SeaRNN (Leblond et al., 2018) and it will be interesting to study their performance in the ranking setting. Finally, an interesting future work direction will be to study off-policy correction (Joachims et al., 2018) for seq2slate. ",
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+        "text": "O. Vinyals, L. Kaiser, T. Koo, S. Petrov, I. Sutskever, and G. Hinton. Grammar as a foreign language. In NIPS, 2015a.   \nOriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In NIPS, pages 2692–2700, 2015b.   \nOriol Vinyals, Samy Bengio, and Manjunath Kudlur. Order matters: Sequence to sequence for sets. In International Conference on Learning Representations (ICLR), 2016. URL http: //arxiv.org/abs/1511.06391.   \nBaiyang Wang and Diego Klabjan. An attention-based deep net for learning to rank. arXiv preprint arXiv:1702.06106, 2017.   \nJiang Wang, Yi Yang, Junhua Mao, Zhiheng Huang, Chang Huang, and Wei Xu. CNN-RNN: A unified framework for multi-label image classification. In Computer Vision and Pattern Recognition (CVPR), 2016 IEEE Conference on, pages 2285–2294. IEEE, 2016.   \nRonald Williams. Simple statistical gradient following algorithms for connectionnist reinforcement learning. In Machine Learning, 1992.   \nRonald J. Williams and David Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural Comput., 1(2):270–280, June 1989. ISSN 0899-7667. doi: 10.1162/neco. 1989.1.2.270. URL http://dx.doi.org/10.1162/neco.1989.1.2.270.   \nSam Wiseman and Alexander M. Rush. Sequence-to-sequence learning as beam-search optimization. In ACL, 2016.   \nQiang Wu, Christopher JC Burges, Krysta M Svore, and Jianfeng Gao. Adapting boosting for information retrieval measures. Information Retrieval, 13(3):254–270, 2010.   \nJun Xu and Hang Li. Adarank: a boosting algorithm for information retrieval. In Proceedings of the 30th annual international ACM SIGIR conference on Research and development in information retrieval, pages 391–398. ACM, 2007.   \nYisong Yue, Thomas Finley, Filip Radlinski, and Thorsten Joachims. A support vector method for optimizing average precision. In Proceedings of SIGIR, pages 271–278. ACM, 2007.   \nVictor Zhong, Caiming Xiong, and Richard Socher. Seq2sql: Generating structured queries from natural language using reinforcement learning. arXiv preprint arXiv:1709.00103, 2017.   \nYadong Zhu, Yanyan Lan, Jiafeng Guo, Xueqi Cheng, and Shuzi Niu. Learning for search result diversification. In Proceedings of SIGIR, pages 293–302. ACM, 2014. ",
+        "bbox": [
+            169,
+            97,
+            828,
+            646
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A DERIVATION OF THE EXPECTED LOSS ",
+        "bbox": [
+            176,
+            101,
+            526,
+            119
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9298bc31074e12c597480a201157adc4a779359c4785d27d3e13fcf8b6b6fc65.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ \\mathcal { L } ] = \\displaystyle \\sum _ { \\tau } p ( \\boldsymbol { \\pi } ) \\mathcal { L } _ { \\tau } } \\\\ & { \\mathrm { ~ \\ ~ \\ } = \\displaystyle \\sum _ { \\tau } p ( \\boldsymbol { \\pi } ) \\sum _ { j } \\ell _ { \\tau < j } } \\\\ & { \\mathrm { \\ ~ \\ ~ \\ } = \\displaystyle \\sum _ { j } \\sum _ { \\tau } p ( \\boldsymbol { \\pi } _ { z < j } ) p ( \\boldsymbol { \\pi } _ { z < j } ) \\ell _ { \\tau < j } \\ell _ { \\tau < j } } \\\\ & { \\mathrm { \\ ~ \\ ~ \\ } = \\displaystyle \\sum _ { j } \\sum _ { \\tau } p ( \\boldsymbol { \\pi } _ { z < j } ) p ( \\boldsymbol { \\pi } _ { z \\le j } ) \\ell _ { \\tau < j } \\ell _ { \\tau < j } } \\\\ & { \\mathrm { \\ ~ \\ ~ \\ } = \\displaystyle \\sum _ { j } \\sum _ { \\tau < j } p ( \\boldsymbol { \\pi } _ { z < j } ) \\ell _ { \\tau < j } \\sum _ { \\tau \\ge j } p ( \\boldsymbol { \\pi } _ { z < j } | \\boldsymbol { \\pi } _ { z < j } ) } \\\\ & { \\mathrm { \\ ~ \\ ~ \\ } = \\displaystyle \\sum _ { j } \\sum _ { \\tau < j } \\left( \\sum _ { k = 1 } ^ { j - 1 } e ^ { \\lambda _ { k } } / \\sum _ { \\tau \\in S _ { k } } c \\right) \\ell _ { \\tau < j } e ^ { \\lambda _ { k } } \\Big ) \\ell _ { \\tau < j } \\langle \\boldsymbol { s } ^ { j } , \\boldsymbol { y } \\rangle \\ . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            135,
+            665,
+            329
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Since the terms are continuous (and smooth) in $S$ for all $j$ and $\\pi _ { < j }$ , so is the entire function. ",
+        "bbox": [
+            169,
+            333,
+            774,
+            349
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            364,
+            537,
+            382
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/8e970c09f2f93f5a7f36284c6e5be0c7fbc7f4931ce45e28e5aaedaba12b5b2f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Ranker</td><td colspan=\"3\">Yahoo</td><td colspan=\"3\">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td></tr><tr><td>seq2slate</td><td>0.82</td><td>0.82</td><td>0.84</td><td>0.44</td><td>0.54</td><td>0.50</td></tr><tr><td>AdaRank</td><td>0.83</td><td>0.81</td><td>0.84</td><td>0.41</td><td>0.52</td><td>0.48</td></tr><tr><td>Coordinate Ascent</td><td>0.83</td><td>0.82</td><td>0.85</td><td>0.39</td><td>0.47</td><td>0.44</td></tr><tr><td>LambdaMART</td><td>0.84</td><td>0.83</td><td>0.85</td><td>0.41</td><td>0.52</td><td>0.48</td></tr><tr><td>ListNet</td><td>0.83</td><td>0.83</td><td>0.85</td><td>0.41</td><td>0.53</td><td>0.49</td></tr><tr><td>MART</td><td>0.83</td><td>0.82</td><td>0.85</td><td>0.41</td><td>0.52</td><td>0.48</td></tr><tr><td>Random Forests</td><td>0.83</td><td>0.82</td><td>0.84</td><td>0.40</td><td>0.48</td><td>0.45</td></tr><tr><td>RankBoost</td><td>0.83</td><td>0.83</td><td>0.85</td><td>0.38</td><td>0.43</td><td>0.41</td></tr><tr><td>RankNet</td><td>0.83</td><td>0.82</td><td>0.84</td><td>0.35</td><td>0.36</td><td>0.35</td></tr></table>",
+        "bbox": [
+            209,
+            400,
+            787,
+            541
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Performance of seq2slate and other baselines on data generated with similar-clicks. ",
+        "bbox": [
+            194,
+            559,
+            800,
+            574
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/43eaf451b1d0554f7ecda9d6bf686088b08b2e9b4aa599caeb4e31e8195d2448.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Ranker</td><td colspan=\"4\">Yahoo</td><td colspan=\"4\">Web30k</td></tr><tr><td>MAP</td><td>NDCG@5</td><td>NDCG@10</td><td>rank-gain</td><td>MAP</td><td>NDCG@5</td><td>NDCG10</td><td>rank-gain</td></tr><tr><td>seq2slate</td><td>0.82</td><td>0.82</td><td>0.84</td><td>8.5</td><td>0.44</td><td>0.54</td><td>0.50</td><td>16.0</td></tr><tr><td>Greedy policy</td><td>0.82</td><td>0.82</td><td>0.84</td><td>8.5</td><td>0.44</td><td>0.54</td><td>0.50</td><td>15.9</td></tr><tr><td>smooth-hinge</td><td>0.80</td><td>0.80</td><td>0.82</td><td>7.7</td><td>0.44</td><td>0.54</td><td>0.50</td><td>15.9</td></tr><tr><td>RL</td><td>0.82</td><td>0.82</td><td>0.84</td><td>8.5</td><td>0.42</td><td>0.53</td><td>0.49</td><td>-14.8</td></tr><tr><td>one-step decoder</td><td>0.81</td><td>0.81</td><td>0.82</td><td>7.7</td><td>0.44</td><td>0.53</td><td>0.49</td><td>15.5</td></tr><tr><td>shuffled data</td><td>0.80</td><td>0.80</td><td>0.81</td><td>二</td><td>0.40</td><td>0.44</td><td>0.42</td><td>1</td></tr><tr><td>base ranker (no-op)</td><td>0.78</td><td>0.76</td><td>0.79</td><td>0</td><td>0.43</td><td>0.53</td><td>0.49</td><td>0</td></tr></table>",
+        "bbox": [
+            174,
+            598,
+            820,
+            704
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Comparison of model and data variants for seq2slate on data generated with similar-clicks. ",
+        "bbox": [
+            171,
+            720,
+            823,
+            736
+        ],
+        "page_idx": 11
+    }
+]

parse/train/WrotwUEJO59/WrotwUEJO59.md

@@ -0,0 +1,466 @@
+# Quantifying and Learning Linear Symmetry-Based Disentanglement
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 The definition of Linear Symmetry-Based Disentanglement (LSBD) formalizes   
+2 the notion of linearly disentangled representations, but there is currently no metric   
+3 to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and to   
+4 compare to previous understandings of disentanglement. We propose $\mathcal { D } _ { \mathrm { L S B D } }$ , a   
+5 mathematically sound metric to quantify LSBD, and provide a practical implemen  
+6 tation. Furthermore, from this metric we derive LSBD-VAE, a semi-supervised   
+7 method to learn LSBD representations. We demonstrate the utility of our metric   
+8 by showing that (1) common VAE-based disentanglement methods don’t learn   
+9 LSBD representations, (2) LSBD-VAE as well as other recent methods can learn   
+10 LSBD representations, needing only limited supervision on transformations, and   
+11 (3) various desirable properties expressed by existing disentanglement metrics are   
+12 also achieved by LSBD representations.
+
+# 13 1 Introduction
+
+14 Learning low-dimensional representations that disentangle the underlying factors of variation in data   
+15 is considered an important step towards interpretable machine learning with good generalization. To   
+16 address the fact that there is no consensus on what disentanglement entails and how to formalize it,   
+17 Higgins et al. (2018) propose a formal definition for Linear Symmetry-Based Disentanglement, or   
+18 LSBD, arguing that underlying real-world symmetries give exploitable structure to data.   
+19 However, there is currently no metric to quantify LSBD. Such a metric is crucial to properly evaluate   
+20 methods aiming to learn LSBD representations and to relate LSBD to previous definitions of disentan  
+21 glement. Although previous works have evaluated LSBD by measuring performance on downstream   
+22 tasks (Caselles-Dupré et al., 2019) or by measuring specific traits related to LSBD (Painter et al.,   
+23 2020; Quessard et al., 2020), none of these evaluation methods directly quantify LSBD according to   
+24 its well-formalized definition.   
+25 We propose $\mathcal { D } _ { \mathrm { L S B D } }$ , a well-formalized and generally applicable metric that quantifies the level of   
+26 LSBD in learned data representations. We show an intuitive justification of this metric, as well   
+27 as its theoretical derivation. We also provide a practical implementation to compute $\mathcal { D } _ { \mathrm { L S B D } }$ for   
+28 common symmetry groups. Furthermore, we show that our metric formulation can be used to derive   
+29 a semi-supervised method to learn LSBD representations, which we call LSBD-VAE. To make   
+30 LSBD-VAE more widely applicable, we also demonstrate how to disentangle symmetric properties   
+31 from other non-symmetric properties, and how to quantify this disentanglement with $\mathcal { D } _ { \mathrm { L S B D } }$ .   
+32 We show the utility of $\mathcal { D } _ { \mathrm { L S B D } }$ by quantifying LSBD in a number of settings, for a variety of datasets   
+33 with underlying SO(2) symmetries and other non-symmetric properties. First, we evaluate common   
+34 VAE-based disentanglement methods and show that most don’t learn LSBD representations. Second,   
+35 we evaluate LSBD-VAE and other recent methods that specifically target LSBD, showing that they   
+36 can obtain much better $\mathcal { D } _ { \mathrm { L S B D } }$ scores while needing only limited supervision on transformations.   
+37 Third, we compare $\mathcal { D } _ { \mathrm { L S B D } }$ with existing disentanglement metrics, showing that various desirable   
+38 properties expressed with these metrics are also achieved by LSBD representations.
+
+# 39 2 Related Work
+
+40 Plenty of works have focused on learning and quantifying disentangled representations recently, but   
+41 research has shown that there is little consensus about the exact definition of disentanglement and   
+42 methods often do not achieve it as well as they proclaim (Locatello et al., 2019). To introduce some   
+43 much-needed formalization, Higgins et al. (2018) proposed to define disentanglement with respect   
+44 to symmetry transformations acting on the data. They used group theory to provide two formal   
+45 definitions, which we refer to as (Linear) Symmetry-Based Disentanglement, or (L)SBD. In this   
+46 paper we focus only on LSBD, not SBD.   
+47 Several methods have been proposed to learn LSBD representations (Caselles-Dupré et al., 2019;   
+48 Painter et al., 2020; Quessard et al., 2020). These methods also learn to represent the transformations   
+49 acting on the input data, assuming various levels of supervision on these transformations. Other   
+50 methods have previously focused on capturing transformations of the data outside the context of   
+51 disentanglement as well (Cohen and Welling, 2015; Sosnovik et al., 2019; Worrall et al., 2017).
+
+# 52 3 Linear Symmetry-Based Disentanglement
+
+53 Higgins et al. (2018) provide a formal definition of linear disentanglement that connects symmetry   
+54 transformations affecting the real world (from which data is observed) to the internal representations   
+55 of a model. The definition is grounded in concepts from group theory, we provide a more detailed   
+56 description of these concepts in the Supplementary Material.   
+57 The definition1 considers a group $G$ of symmetry transformations acting on the data space $X$ through   
+58 the group action $\cdot : G \times X \to X$ . In particular, $G$ can be decomposed as the direct product of $K$   
+59 groups $G = G _ { 1 } \times \ldots \times G _ { K }$ . A model’s internal representation of data is modeled with the encoding   
+60 function $h : X \to Z$ that maps data to the embedding space $Z$ . The definition for Linearly Symmetry  
+61 Based Disentangled (LSBD) representations then formalizes the requirement that a model’s encoding   
+62 $h$ should reflect and disentangle the transformation properties of the data, and that the transformation   
+63 properties of the model’s encoding should be linear. The exact definition is as follows:   
+64 Definition: Linear Symmetry-Based Disentanglement (LSBD) A model’s encoding map $h :$   
+65 $X  Z$ , where $Z$ is a vector space, is LSBD with respect to the group decomposition $G =$   
+66 $G _ { 1 } \times \ldots \times G _ { K }$ if
+
+1. there is a decomposition of the embedding space $Z = Z _ { 1 } \oplus . . . \oplus Z _ { K }$ into $K$ vector subspaces,   
+2. there are group representations for each subgroup in the corresponding vector subspace $\rho _ { k } : G _ { k } \stackrel { \textstyle \bar { \to } } { \to } \mathrm { G L } ( Z _ { k } ) , k \in \{ 1 , \dots , K \}$   
+3. the group representation $\rho : G \to { \mathrm { G L } } ( Z )$ acts on $Z$ as $\rho ( g ) \cdot z = ( \rho _ { 1 } ( g _ { 1 } ) \cdot z _ { 1 } , \ldots , \rho _ { K } ( g _ { K } ) \cdot z _ { K } ) ,$ (1) for $g = ( g _ { 1 } , \dotsc , g _ { K } ) \in G$ and $z = ( z _ { 1 } , \dots , z _ { K } ) \in Z$ with $g _ { k } \in G _ { k }$ and $z _ { k } \in Z _ { k }$ .   
+4. the map $h$ is equivariant with respect to the actions of $G$ on $X$ and $Z$ , i.e. , for all $x \in X$ and $g \in G$ it holds that $h ( g \cdot x ) \stackrel { } { = } \rho ( g ) \cdot h ( x )$ .
+
+75 Furthermore, we say that a group representation $\rho$ is linearly disentangled with respect to the group   
+76 decomposition $G = G _ { 1 } \times \ldots \times G _ { K }$ if it satisfies criteria 1 to 3 from the LSBD definition above.
+
+# 78 4.1 Intuition: Measuring Equivariance with Dispersion
+
+79 To motivate our metric, let’s first assume a set  
+80 ting in which a suitable linearly disentangled   
+81 group representation $\rho$ is known. Let’s further   
+82 assume that the dataset of observations can be   
+83 84 exprepoint $x _ { 0 } \in X$ respe, i.e. $\{ x _ { n } \} _ { n = 1 } ^ { N } = { \bar { \{ g _ { n } \cdot x _ { 0 } \} } } _ { n = 1 } ^ { N }$ $G$   
+85 Formally, this assumes that the action of $G$ on   
+86 $X$ is regular. In this case, we can use the inverse   
+87 group elements $g _ { n } ^ { - 1 }$ to transform each data point   
+88 toward the base point $x _ { 0 }$ , i.e.
+
+$$
+x _ { 0 } = g _ { 1 } ^ { - 1 } \cdot x _ { 1 } = \ldots = g _ { N } ^ { - 1 } \cdot x _ { N } .
+$$
+
+89 Since $\rho$ is linearly disentangled, we only need to   
+90 measure the equivariance of the encoding map   
+91 $h$ to quantify LSBD. Equivariance is achieved   
+92 when $h ( g \cdot x ) = \rho ( g ) \cdot h ( x )$ , for all $g \in G , x \in$   
+93 $X$ . Given the dataset described above, we can   
+94 $\{ g _ { n } \} _ { n = 1 } ^ { N }$ s property for .2 In particular $x \in \{ x _ { n } \} _ { n = 1 } ^ { N }$ and n (2) $g \in$   
+96 can see that we have equivariance if
+
+![](images/caa1181c8b0270db28d1a6afc76b50415256cdbf5ee7362a8dc3585ad944ee78.jpg)  
+Figure 1: A dataset of images from a rotating object expressed in terms of the group $G = \bar { \mathrm { S O } } ( 2 )$ acting on a base image $x _ { 0 }$ . It is possible to quantify the level of LSBD of an encoding map $h$ by measuring its equivariance with respect to a group representation $\rho$ . Since all data has been generated from $x _ { 0 }$ , equivariance can be measured as the dispersion of the points $\{ \rho ( g _ { n } ^ { - 1 } ) \cdot h ( x _ { n } ) \} _ { n = 1 } ^ { N }$ .
+
+$$
+h ( x _ { 0 } ) = \rho ( g _ { 1 } ^ { - 1 } ) \cdot h ( x _ { 1 } ) = \ldots = \rho ( g _ { N } ^ { - 1 } ) \cdot h ( x _ { N } ) .
+$$
+
+97 This not only characterizes perfect equivariance, but also allows for an efficient way to quantify how   
+98 close we are to true equivariance, by measuring the dispersion of the points $\{ \rho ( g _ { n } ^ { - 1 } ) \cdot \bar { h } ( x _ { n } ) \} _ { n = 1 } ^ { N }$ . 3   
+99 Given a suitable norm $\| \cdot \| _ { Z }$ in $Z$ , we can thus quantify LSBD in this setting as
+
+$$
+\frac { 1 } { N } \sum _ { n = 1 } ^ { N } \left. \rho ( g _ { n } ^ { - 1 } ) \cdot h ( x _ { n } ) - \frac { 1 } { N } \sum _ { n ^ { \prime } = 1 } ^ { N } \rho ( g _ { n ^ { \prime } } ^ { - 1 } ) \cdot h ( x _ { n ^ { \prime } } ) \right. _ { Z } ^ { 2 } ,
+$$
+
+i.e. we compute the mean of 100 $\{ \rho ( g _ { n } ^ { - 1 } ) \cdot h ( x _ { n } ) \} _ { n = 1 } ^ { N }$ and use the average squared distance to this mean for points in 101 $\{ \rho ( g _ { n } ^ { - 1 } ) \cdot h ( x _ { n } ) \} _ { n = 1 } ^ { N }$ as our LSBD metric, see Figure 1.
+
+02 However, this formulation requires knowing the right linearly disentangled group representation and 103 a suitable norm in $Z$ . Moreover, it implicitly assumes a uniform probability measure over the group elements 104 $\{ g _ { n } \} _ { n = 1 } ^ { N }$ . In the next section we formulate our metric for a more general setting.
+
+# 05 4.2 $\mathcal { D } _ { \mathrm { L S B D } }$ : A Metric for LSBD
+
+106 Generalizing the ideas from the previous section with concepts from measure theory, we propose a   
+107 metric to measure the level of LSBD of any encoding $h : X \to Z$ given a data probability measure $\mu$   
+108 on $X$ , provided that $\mu$ can be written as the pushforward $G _ { X } ( \cdot , x _ { 0 } ) _ { \# } \nu$ of some probability measure   
+109 $\nu$ on $G$ by the function $G _ { X } ( \cdot , x _ { 0 } )$ for some base point $x _ { 0 }$ . More formally,
+
+$$
+\mu ( A ) = G _ { X } ( \cdot , x _ { 0 } ) _ { \# } \nu ( A ) = \nu \left( \left\{ g \in G \mid G _ { X } ( g , x _ { 0 } ) \in A \right\} \right) ,
+$$
+
+10 for Borel subsets $A \subset X$ . Note that this is only possible if the action $G _ { X }$ is transitive.
+
+For example, the situation of a dataset with 111 $N$ datapoints $\{ x _ { n } \} _ { n = 1 } ^ { N } = \{ g _ { n } \cdot x _ { 0 } \} _ { n = 1 } ^ { N }$ corresponds to 112 the case in which $\nu$ and $\mu$ are empirical measures on the group $G$ and data space $X$ , respectively:
+
+$$
+\nu : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \delta _ { g _ { i } } , \qquad \mu : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \delta _ { x _ { i } } .
+$$
+
+2Note that $\{ g _ { n } \} _ { n = 1 } ^ { N }$ can be used to describe all known group transformations between elements in the dataset by means of composition and inverses, since $x _ { i } = g _ { i } \cdot ( g _ { j } ^ { - 1 } \cdot x _ { j } )$ . Thus it suffices to check equivariance for these $N$ group transformations.
+
+3Note that we do not actually need to know $x _ { 0 }$ nor $h ( x _ { 0 } )$
+
+113 We define the metric $\mathcal { D } _ { \mathrm { L S B D } }$ for an encoding $h$ and a measure $\mu$ as
+
+$$
+\mathcal { D } _ { \mathrm { L S B D } } : = \operatorname* { i n f } _ { \rho \in \mathcal { P } ( G , Z ) } \int _ { G } \big \| \rho ( g ) ^ { - 1 } \cdot h ( g \cdot x _ { 0 } ) - M _ { \rho , h , x _ { 0 } } \big \| _ { \rho , h , \mu } ^ { 2 } d \nu ( g ) ,
+$$
+
+114 where the norm $\| \cdot \| _ { \rho , h , \mu }$ is a Hilbert-space norm depending on the representation $\rho$ , the encoding map   
+115 $h : X \to Z$ , and the data measure $\mu$ . More details of this norm can be found in the Supplementary   
+116 Material. Moreover, ${ \mathcal { P } } ( G , Z )$ denotes the set of linearly disentangled representations of $G$ in $Z$   
+117 Lower values of $\mathcal { D } _ { \mathrm { L S B D } }$ indicate better disentanglement, zero being optimal.
+
+# 118 4.3 Practical Computation of $\mathcal { D } _ { \mathrm { L S B D } }$
+
+119 There are two main challenges for computing the metric of Equation (7). First, to calculate the   
+120 integrals in the formula, all possible datapoints that can be expressed as $g \cdot x _ { 0 }$ with $g \in G =$   
+121 $G _ { 1 } \times \cdots \times G _ { K }$ must be available. Second, the infimum of the integrals over all possible linearly   
+122 disentangled representations must be estimated. This requires finding the possible invariant subspaces   
+123 $Z = Z _ { 1 } \oplus \cdots \oplus Z _ { K }$ induced by the encoding $h$ over which the group representations are disentangled.   
+124 We present a practical implementation of an upper bound to $\mathcal { D } _ { \mathrm { L S B D } }$ for an encoding function $h$ given   
+125 a dataset $\mathcal { X }$ generated by some known group transformations. In particular, this approximation of   
+126 $\mathcal { D } _ { \mathrm { L S B D } }$ is designed for a group decomposition $G = G _ { 1 } \times \cdot \cdot \cdot \times G _ { K }$ where each $G _ { k } \overset { \vartriangle } { = } \mathrm { S O } ( D _ { k } )$ with   
+127 $k \in \{ 1 , \ldots , K \}$ the group of rotations in $D _ { k }$ dimensions. This implementation approximates the   
+128 integrals of Equation (7) by using the empirical distribution of $\mathcal { X }$ . The invariant subspaces of $Z$ to the   
+129 subgroup actions are found by applying a suitable change of basis. In the new basis, the disentangled   
+130 group representations are expressed in a parametric form whose parameters are optimized to find the   
+131 tightest bound to $\mathcal { D } _ { \mathrm { L S B D } }$ . See Figure 2 for an intuitive description of the process.   
+132 Assume there is a dataset $\mathcal { X }$ that can be modeled in terms of the group decomposition $G = G _ { 1 } \times \cdot \cdot \cdot G _ { k }$ .   
+133 For each $G _ { k }$ subgroup there is a set of known group elements $\mathcal { G } _ { k } \subseteq G _ { k }$ uniformly sampled such   
+134 that the dataset is described in terms of all elements in $\mathcal { G } = \mathcal { G } _ { 1 } \times \cdots \times \mathcal { G } _ { K }$ and a base point $x _ { 0 }$ as   
+135 ${ \mathcal { X } } = \left\{ \left( g _ { 1 } , \dots , g _ { K } \right) \cdot x _ { 0 } { \big | } g _ { k } \in { \mathcal { G } } _ { k } , ~ k \in \left\{ 1 , \dots , K \right\} \right\}$ .   
+136 For each subgroup $G _ { k }$ we construct a set of encoded data $\mathcal { Z } _ { k } \subseteq Z$ whose variability should only de  
+137 pend on the action of $G _ { k }$ . The set $\mathcal { Z } _ { k }$ is given by $\mathcal { Z } _ { k } = \{ z _ { k } ( g _ { 1 } , \dotsc , g _ { K } ) | g _ { j } \in \mathcal { G } _ { j } \ , j \in \{ 1 , \dotsc , K \} \}$ ,   
+138 in which
+
+$$
+z _ { k } ( g _ { 1 } , \dots , g _ { K } ) = h ( ( g _ { 1 } , \dots , g _ { K } ) \cdot x _ { 0 } ) - \frac { 1 } { | { \mathcal { G } } _ { k } | } \sum _ { g ^ { \prime } \in { \mathcal { G } } _ { k } } h ( ( g _ { 1 } , \dots , g _ { k - 1 } , g ^ { \prime } , g _ { k + 1 } , \dots , g _ { K } ) \cdot x _ { 0 } ) .
+$$
+
+139 Similar to (Cohen and Welling, 2014), we find a suitable change of basis that exposes the invariant   
+140 subspace $Z _ { k }$ corresponding to the $k$ -th subgroup $G _ { k }$ . The new basis is obtained from the eigenvectors   
+141 resulting from applying Principal Component Analysis (PCA) to $\mathcal { Z } _ { k }$ . Each element in $\mathcal { Z } _ { k }$ is projected   
+142 into the first $D _ { k }$ eigenvectors. The new set is denoted as $\mathcal { Z } _ { k } ^ { \prime } \subseteq \mathbb { R } ^ { D _ { k } }$ with elements $z _ { k } ^ { \prime } ( g _ { 1 } , \dotsc , g _ { K } ) \subseteq$   
+143 $\mathbb { R } ^ { D _ { k } }$ that are the projected versions of $z _ { k } ( g _ { 1 } , \dots , g _ { K } )$ .   
+144 (Quessard et al., 2020) describes how one could parameterize the subgroup representations of $S O ( D _ { k } )$   
+145 for arbitrary $D _ { k }$ but here we will focus on $\bar { G _ { k } } = S O ( 2 )$ . In this case, we can parameterize each   
+146 subgroup representation in terms of a single integer parameter $\omega \in \mathbb { Z }$ as $\rho _ { k , \omega } ( g _ { k } )$ corresponding   
+147 to a $2 \times 2$ rotation matrix whose angle of rotation is $\omega$ multiplied by the known angle associated   
+148 to the group element $g _ { k } \in G _ { k } = \bar { \mathrm { S O } ( 2 ) }$ . For this subgroup we can approximate the $M _ { \rho , h , x _ { 0 } }$ from   
+149 Equation (7) as $M _ { k , \omega }$ given by
+
+$$
+M _ { k , \omega } = \frac { 1 } { | \mathcal { G } | } \sum _ { ( g _ { 1 } , . . . , g _ { K } ) \in \mathcal { G } } \rho _ { k , \omega } ( g _ { k } ^ { - 1 } ) \cdot z ^ { \prime } ( g _ { 1 } , . . . , g _ { K } ) .
+$$
+
+150 Similar to Equation (7) we would like to find the optimal $\rho _ { k , \omega }$ that minimizes the integral over the   
+151 group representations. We can define a parameter search space $\Omega \subseteq \mathbb { Z }$ , e.g. $\Omega = [ - 1 0 , 1 0 ]$ for finding   
+152 the optimal $\omega \in \Omega$ that minimizes the dispersion, this is expressed in the following equation
+
+$$
+\mathcal { D } _ { \mathrm { L S B D } } ^ { ( k ) } = \operatorname* { m i n } _ { \omega \in \Omega } \frac { 1 } { | \mathcal { G } | } \sum _ { ( g _ { 1 } , . . . , g _ { K } ) \in \mathcal { G } } \Vert \rho _ { k , \omega } ( g _ { k } ^ { - 1 } ) \cdot z ^ { \prime } ( g _ { 1 } , . . . , g _ { K } ) - M _ { k , \omega } \Vert ^ { 2 } .
+$$
+
+![](images/9bc32e6cbe7e7f6d136371693d530b9f07c07b31d987bc47851fa12213f21503.jpg)  
+Figure 2: Consider a dataset modeled by a group decomposition $G = G _ { 1 } \times \cdot \cdot \cdot \times G _ { K }$ acting on $x _ { 0 }$ and is embedded in a latent space $Z$ via $h$ . In this example the subgroup $G _ { k } = \mathrm { S O } ( 2 )$ models the rotations of an airplane. Other subgroups $G _ { \neq k }$ could also be acting e.g. changes in airplane color. The first step to calculate the disentanglement of $G _ { k }$ is to construct a set of data embeddings $\mathcal { Z } _ { k } \subseteq Z$ whose variability is due to $G _ { k }$ . These embeddings are then projected into a 2-dimensional space through PCA. For these projected embeddings we can describe the group representations in a simple parametric form $\rho _ { k , w }$ . For a given $\rho _ { k , w }$ the equivariance of $G _ { k }$ is measured as the dispersion after applying the action of the inverse group representation $\rho _ { k , w } ^ { - 1 }$ .
+
+153 Each D(k)LSB measures the degree of equivariance of the projected embeddings for each $k$ -th subgroup   
+154 corresponding to the best fitting group representation. The upper bound to the metric is finally obtained   
+155 by averaging across all subgroups $\begin{array} { r } { \bar { \mathcal { D } } _ { \mathrm { L S B D } } \leq \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \bar { \mathcal { D } } _ { \mathrm { L S B D } } ^ { ( \bar { k } ) } } \end{array}$ .
+
+# 156 5 Learning LSBD Representations: LSBD-VAE
+
+157 In this section we present LSBD-VAE, a semi-supervised VAE-based method to learn LSBD repre  
+158 sentations. The main idea is to train an unsupervised Variational Autoencoder (VAE) (Kingma and   
+159 Welling, 2014; Rezende et al., 2014) with a suitable latent space topology, and use our metric as an   
+160 additional loss term for batches of transformation-labeled data.   
+161 Assumptions LSBD-VAE requires some knowledge about the group structure $G$ that is to be   
+162 disentangled. Concretely, the group and its decomposition $G = G _ { 1 } \times \ldots \times G _ { K }$ should be known,   
+163 as well as a suitable linearly disentangled group representation $\rho : G \to { \mathrm { G L } } ( Z )$ and a latent space   
+164 $Z = Z _ { 1 } \oplus . . . \oplus Z _ { K }$ . Moreover, we assume there exists an embedded submanifold $Z _ { G } \subseteq Z$ such   
+165 that the action of $G$ on $Z$ restricted to $Z _ { G }$ is regular, and $Z _ { G }$ is invariant under the action. Only $Z _ { G }$   
+166 will then be used as the codomain for the encoding map, $h : X \to Z _ { G }$ .   
+167 We demonstrate the assumptions above for the common group structure $G = \mathrm { S O } ( 2 ) \times \mathrm { S O } ( 2 )$ . For   
+168 the group representation $\rho = \rho _ { 1 } \oplus \rho _ { 2 }$ , with $Z = \mathbb { R } ^ { 2 } \oplus \mathbb { R } ^ { 2 }$ , we can use rotation matrices in ${ \dot { \mathbb { R } } } ^ { 2 }$ for   
+169 $\rho _ { 1 }$ and $\rho _ { 2 }$ . We can then use 1-spheres $S ^ { 1 } = \{ z \in \mathbb { R } ^ { 2 } : \| z \| = 1 \}$ for the embedded submanifold:   
+170 $\dot { Z } _ { G } = \dot { S } ^ { 1 } \times S ^ { 1 }$ . In this case, the action of $G$ on $Z$ restricted to $Z _ { G }$ is indeed regular, and $Z _ { G }$ is   
+171 invariant under the action.   
+172 Unsupervised Learning on Latent Manifold To learn encodings only on the latent manifold   
+173 $Z _ { G }$ , we use a Diffusion Variational Autoencoder (∆VAE) (Perez Rey et al., 2020). $\Delta$ VAEs can   
+174 use any closed Riemannian manifold embedded in a Euclidean space as a latent space (or latent   
+175 manifold), provided that a certain projection function from the Euclidean embedding space into the   
+176 latent manifold is known and the scalar curvature of the manifold is available. The $\Delta$ VAE uses   
+177 a parametric family of posterior approximates obtained from a diffusion process over the latent   
+178 manifold. To estimate the intractable terms of the negative ELBO, the reparameterization trick is   
+179 implemented via a random walk.
+
+In the case of 180 $S ^ { 1 }$ as a latent (sub)manifold, we consider $\mathbb { R } ^ { 2 }$ as the Euclidean embedding space, and 181 the projection function4 $\Pi : \mathbb { R } ^ { 2 }  S ^ { 1 }$ normalizes points in the embedding space: $\Pi ( z ) \overset { \cdot } { = } \dot { z } / | z |$ . The scalar curvature of182 $S ^ { 1 }$ is 0.
+
+183 Semi-Supervised Learning with Transformation Labels Caselles-Dupré et al. (2019) proved   
+184 that LSBD representations cannot be inferred from a training set of unlabeled observations, but that   
+185 access to the transformations between data points is needed. They therefore use a training set of   
+186 observation pairs with a given transformation between them.   
+7 However, we posit that only a limited amount of supervision is sufficient. Since obtaining supervision   
+88 on transformations is typically more expensive than obtaining unsupervised observations, it is   
+9 desirable to limit the amount of supervision needed.   
+90 Therefore, we augment the un  
+91 supervised $\Delta \mathrm { V A E }$ with a super  
+92 vised method that makes use of   
+93 transformation-labeled batches, i.e.   
+94 batches $\lbrace x _ { m } \rbrace _ { m = 1 } ^ { M }$ such that $\begin{array} { r l } { x _ { m } } & { { } = } \end{array}$   
+95 $g _ { m } \cdot x _ { 1 }$ for $m = 2 , \ldots , M$ , where   
+96 the transformations $g _ { m }$ (and thus   
+97 their group representations $\rho ( g _ { m } ) ,$ ) are   
+98 known and are referred to as transfor  
+99 mation labels. The simplified version   
+00 of the metric from Equation (4) can   
+01 then be used for each batch as an ad  
+02 ditional loss term (with $x _ { 0 } = x _ { 1 }$ ), as   
+03 it is differentiable under the assump  
+04 tions described above (using the Eu  
+05 clidean norm).   
+206 We make a small adjustment to Equation (4) for the purpose of our method, since the mean computed   
+207 there does not typically lie on the latent manifold $Z _ { G }$ . Thus, we use the projection $\Pi$ from the $\Delta \mathrm { V A E }$   
+208 to project the mean onto $Z _ { G }$ . Writing the encodings as $z _ { m } : = h ( x _ { m } )$ , the additional loss term for a   
+209 transformation-labeled batch $\{ x _ { m } \} _ { m = 1 } ^ { M }$ then becomes
+
+![](images/14b0ae6804074b71a10e2c75bb8cc480774c032b2b00c044a280d3a005805d38.jpg)  
+Figure 3: Overview of the supervised part of LSBD-VAE.
+
+$$
+\mathcal { L } _ { L S B D } = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \left. \rho ( g _ { m } ^ { - 1 } ) \cdot z _ { m } - \Pi \left( \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \rho ( g _ { m } ^ { - 1 } ) \cdot z _ { m } \right) \right. ^ { 2 } ,
+$$
+
+210 where $g _ { 1 } = e$ , the group identity.
+
+Moreover, instead of feeding the encodings $z _ { m }$ to the decoder, we use $\rho ( g _ { m } ) \cdot \overline { { z } }$ , where ${ \begin{array} { r l } { { \overline { { z } } } } & { = } \end{array} }$ $\begin{array} { r } { \Pi \left( \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \rho ( g _ { m } ^ { - 1 } ) \cdot z _ { m } \right) } \end{array}$ . This encourages the decoder to follow the required group structure. This only affects the reconstruction loss component of the $\Delta \mathrm { V A E }$ .
+
+214 Figure 3 illustrates the supervised part of our method for a transformation-labeled batch $\{ x _ { m } \} _ { m = 1 } ^ { M }$   
+215 The loss function is the regular ELBO (but with adjusted decoder input as described above) as used   
+216 in $\Delta \mathrm { V A E }$ plus an additional term $\gamma \cdot \mathcal { L } _ { L S B D }$ , where $\gamma$ is a weight hyperparameter to control the   
+217 influence of the supervised loss component. By alternating unsupervised and supervised training   
+218 (using the same encoder and decoder), we have a method that makes use of both unlabeled and   
+219 transformation-labeled observations.
+
+# 220 6 Experimental Setup
+
+221 We evaluate the disentanglement of several models on three different image datasets (Square, Arrow,   
+222 and Airplane) with a known group decomposition $G = \mathrm { S O } ( 2 ) \times \mathrm { S O } ( 2 )$ describing the underlying   
+223 transformations. For each subgroup a fixed number of $| \mathcal { G } _ { k } | = 6 4$ with $k \in \{ 1 , 2 \}$ transformations is   
+224 selected. The datasets exemplify different group actions of SO(2): periodic translations, in-plane   
+225 rotations, out-of-plane rotations, and periodic hue-shifts.
+
+In real settings, not all variability in the data can be modelled by the actions of a group. Therefore, we also evaluate the same models on two datasets ModelNet40 (Wu et al., 2014) and COIL-100 (Nene et al., 1996) that consist of images from various objects (i.e. non-symmetric variation) under known out-of-plane rotations (SO(2) symmetries). In many settings it is easy to obtain labels for such rotations, e.g. when the camera or object angle is controlled by an agent. See Figure 4 for examples of the datasets. For more details, see the Supplementary Material.
+
+![](images/d352a0a16ca7390be9798d1dba330cfe8159e226e0308ecc0b255b93ae06802b.jpg)  
+Figure 4: Example images from each of the datasets used. Each row shows different examples from a single factor changing.
+
+232 For the Square, Arrow, and Airplane datasets we test LSBD-VAE with transformation-labeled batches   
+233 of size $M = 2$ . More specifically, for each experiment we randomly select $L$ disjoint pairs of data   
+234 points, and label the transformation between the data points in each pair. We vary the number of   
+235 labeled pairs $L$ from 0 (corresponding to a $\Delta { \mathrm { V A E } } )$ to $N / 2$ (in which case each data point is involved   
+236 in exactly one labeled pair). We set the weight $\gamma$ of the supervised loss component to $\gamma = 1 0 0$   
+237 for all experiments. We choose $M = 2$ for our experiments since it is the most limited setting for   
+238 LSBD-VAE. Higher values of $M$ would provide stronger supervision, so successful results with   
+239 $M = 2$ imply that good results can also be achieved for higher values of $M$ (but not necessarily vice   
+240 versa).   
+241 For the COIL-100 and ModelNet40 datasets, we train LSBD-VAE on batches containing images of   
+242 one particular object from all different angles (72 and 64 for COIL-100 and ModelNet40, respectively).   
+243 Each batch is labelled with transformations $( g _ { 1 } , e ) , \ldots , ( g _ { M } , e )$ , where $g _ { m }$ represent rotations, and   
+244 the unit transformation $e$ indicates that the object is unchanged. To represent the rotations we use   
+245 a $S ^ { 1 }$ latent space as in $\Delta \mathrm { V A E }$ , whereas for the object identity we use a 5-dimensional Euclidean   
+246 space with standard Gaussian prior as in regular VAEs. LSBD is measured as the disentanglement of   
+247 rotations in the latent space. For these experiments we used $\gamma = 1$ .   
+248 We furthermore test a number of known disentanglement methods for comparison, including tra  
+249 ditional disentanglement methods as well as methods focusing on LSBD. In particular, we use   
+250 disentanglement_lib (Locatello et al., 2019) to train a regular VAE (Kingma and Welling, 2014;   
+251 Rezende et al., 2014), $\beta$ -VAE (Higgins et al., 2017), CC-VAE (Burgess et al., 2018), FactorVAE   
+252 (Kim and Mnih, 2018), and DIP-VAE-I/II (Kumar et al., 2018). Furthermore we evaluate the method   
+253 from Quessard et al. (2020) that focuses on LSBD. We also tested ForwardVAE (Caselles-Dupré   
+254 et al., 2019), but show only limited results since we were not able to reproduce any reasonable results   
+255 for our datasets.   
+256 We use encodings from all these methods to evaluate $\mathcal { D } _ { \mathrm { L S B D } }$ , as well as common traditional disentan  
+257 glement metrics from disentanglement_lib: Beta (Higgins et al., 2017), Factor (Kim and Mnih,   
+258 2018), SAP (Kumar et al., 2018), DCI Disentanglement (Eastwood and Williams, 2018), Mutual   
+259 Information Gap (MIG) (Chen et al., 2018), and Modularity (MOD) (Ridgeway and Mozer, 2018).   
+260 More information about the architecture, epochs and hyperparameters can be found in the Supplemen  
+261 tary Material. For the traditional disentanglement methods trained on Square, Arrow and Airplane   
+262 datasets the latent spaces have 4 dimensions, since these are the minimum number of dimensions   
+263 necessary to learn LSBD representations for an underlying $\mathrm { S O } ( 2 ) \times \mathrm { S O } ( 2 )$ symmetry group, see   
+264 (Higgins et al., 2018; Caselles-Dupré et al., 2019). For COIL-100 and ModelNet40 we use latent   
+265 spaces with 7 dimensions for a fair comparison with the LSBD-VAE method.
+
+# 266 7 Results: Evaluating LSBD with DLSBD
+
+267 We now highlight four key observations from our experimental results. In particular, we differentiate   
+268 between the methods (VAE, $\beta$ -VAE, CC-VAE, FACTOR, DIP-I, DIP-II) and metrics (BETA,   
+269 FACTOR, SAP, DCI, MIG, MOD) that approach disentanglement in the traditional sense, and   
+270 methods ( $\Delta$ VAE, QUESSARD, LSBD-VAE) and metric $\left( \mathcal { D } _ { \mathrm { L S B D } } \right)$ that focus specifically on LSBD.   
+271 The full quantitative results can be found in the Supplementary Material.
+
+Figure 5 summarizes the $\mathcal { D } _ { \mathrm { L S B D } }$ scores (lower is better) for all methods on all datasets. Bars show the mean scores over 10 runs for each method, the vertical lines represent standard deviations. LSBD-VAE/ $L$ indicates our method trained on $L$ labelled pairs (LSBD-VAE/0 corresponds to the unsupervised $\Delta$ VAE), LSBD-VAE/full indicates our method trained on batches containing a single object in all known transformations (for datasets with non-symmetric variation). Note that LSBDVAE obtained very good scores (nearly 0) on the Arrow and Pixel datasets, hence the missing bars.
+
+![](images/3652443315b176261f18f6cfab4007213976ddda2752c30b94584f7a268c25c9.jpg)  
+Figure 5: $\mathcal { D } _ { \mathrm { L S B D } }$ scores for all methods on all datasets
+
+280 None of the traditional disentanglement methods achieve good $\mathcal { D } _ { \mathrm { L S B D } }$ scores, even if they score well   
+281 on other traditional disentanglement metrics. This implies that LSBD isn’t achieved by traditional   
+282 methods. Moreover, from the full results in the Supplementary Material we see that the traditional   
+283 methods on these datasets do not achieve good scores on all traditional metrics. In particular, SAP,   
+284 DCI, and MIG scores are low. We believe this is a result of the cyclic nature of the symmetries   
+285 underlying our datasets, further emphasizing the need for disentanglement methods that can capture   
+286 such symmetries.
+
+The SAP and MIG scores measure to what extent generative factors are disentangled into a single latent dimension. However, since the factors in our dataset are inherently cyclic due to their symmetry structure, they cannot be properly represented in a single latent dimension, as shown by Perez Rey et al. (2020). Instead, at least two dimensions are needed to continuously represent each cyclic factor in our data. A similar conclusion was made by Caselles-Dupré et al. (2019) and Painter et al. (2020).
+
+DCI disentanglement measures whether a latent dimension captures at most one generative factor. This is accomplished by measuring the importance of each latent dimension in predicting the true generative factor using boosted trees. However, since the generative factors are cyclic, the performance of the boosted tree classifiers is far from optimal, thus providing more importance to several dimensions in predicting the generative factors and giving overall lower DCI scores.
+
+# 7.2 LSBD-VAE and other LSBD Methods Can Learn LSBD Representations with Limited Supervision on Transformations
+
+From Figure 5 we observe that methods focusing specifically on LSBD can score higher on $\mathcal { D } _ { \mathrm { L S B D } }$ , showing that they are indeed more suitable to learn LSBD representations. In particular, LSBD-VAE got very good $\mathcal { D } _ { \mathrm { L S B D } }$ scores for all datasets. Moreover, our experiments on the Arrow, Airplane, and Pixel datasets also show that only limited supervision suffices to obtain good $\mathcal { D } _ { \mathrm { L S B D } }$ scores with low variability.
+
+We only partially managed to reproduce the results from Quessard et al. (2020) on our datasets. Their method scored fairly well on the Airplane, ModelNet40, and COIL-100 datasets, but did not do well on the Square and Arrow dataset in our experiments.
+
+307 Furthermore, we tested ForwardVAE by Caselles-Dupré et al. (2019), but we did not manage   
+308 to produce any reasonable results on our datasets, trying both their original architecture and the   
+309 architecture we used for our other experiments. Therefore, we do not include scores for this method.   
+310 We did however manage to reproduce ForwardVAE’s results on the Flatland dataset, which was used   
+311 in their paper. For those experiments, we computed a mean $\mathcal { D } _ { \mathrm { L S B D } }$ score of 0.012 with standard   
+312 deviation 0.001 over 10 runs, indicating that ForwardVAE indeed learns LSBD representations for   
+313 Flatland.
+
+# 314 7.3 LSBD Representations Also Satisfy Previous Disentanglement Notions
+
+315 Our results also indicate that LSBD captures various desirable properties that are expressed by   
+316 traditional disentanglement metrics. In Figure 6 we compare $\mathcal { D } _ { \mathrm { L S B D } }$ scores with scores for previous   
+317 disentanglement metrics, for all our experiments. Note that for $\mathcal { D } _ { \mathrm { L S B D } }$ lower is better, whereas for   
+318 all other metrics higher is better. As we noted before, good scores on traditional disentanglement   
+319 metrics don’t necessarily imply good $\mathcal { D } _ { \mathrm { L S B D } }$ scores. Conversely however, methods that score well   
+320 on $\mathcal { D } _ { \mathrm { L S B D } }$ also score well on many traditional disentanglement metrics, often even outperforming   
+321 the traditional methods. In particular, from the full results (see Supplementary Material) we see   
+322 that LSBD-VAE matches or outperforms the traditional methods on the BETA, FACTOR and MOD   
+323 metrics, and achieves much better scores for the DCI metric where traditional methods scored poorly.   
+24 The MIG and SAP scores are still low for methods focusing on LSBD. This is expected however, as   
+25 explained earlier in Section 7.1. This was also observed by Painter et al. (2020) for different datasets.
+
+![](images/d5df3bb8eba3e5aecba8e8dda744d3ce8c49a6d3fbb4299d6bb7f05368fdda94.jpg)  
+Figure 6: Comparing $\mathcal { D } _ { \mathrm { L S B D } }$ to previous disentanglement metrics
+
+# 326 8 Conclusion
+
+27 We presented $\mathcal { D } _ { \mathrm { L S B D } }$ , a metric to quantify Linear Symmetry-Based Disentanglement (LSBD) as 8 defined by Higgins et al. (2018). We further used this metric formulation to motivate LSBD-VAE, a semi-supervised method to learn LSBD representations given some expert knowledge on the underlying group symmetries that are to be disentangled.
+
+We used $\mathcal { D } _ { \mathrm { L S B D } }$ to evaluate various disentanglement methods, both traditional methods and recent methods that specifically focus on LSBD, and showed that LSBD-VAE can learn LSBD representations where traditional methods fail to do so. We also compared $\mathcal { D } _ { \mathrm { L S B D } }$ to traditional disentanglement metrics, showing that LSBD captures many of the same desirable properties that are expressed by existing disentanglement methods. Conversely, we also showed that traditional disentanglement methods and metrics do not usually achieve or measure LSBD.
+
+337 Challenges that remain are expanding and testing LSBD-VAE and $\mathcal { D } _ { \mathrm { L S B D } }$ on different group struc  
+338 tures, towards more practical applications, as well as focusing on the utility of LSBD representations   
+339 for downstream tasks.   
+340 Broader Impact The work is fairly theoretical, and practical methods derived from this work have   
+341 no obvious negative societal impact. However, the ideas presented are relevant to representation   
+342 learning and could be, in particular, used in computer vision and agent control applications.
+
+343 References   
+344 Burgess, C. P., Higgins, I., Pal, A., Matthey, L., Watters, N., Desjardins, G., and Lerchner, A. (2018). Understanding disentangling in $\beta$ -VAE. arXiv preprint arXiv:1804.03599. Caselles-Dupré, H., Ortiz, M. G., and Filliat, D. (2019). Symmetry-based disentangled representation learning requires interaction with environments. In Advances in Neural Information Processing Systems, pages 4606–4615. Chen, T. Q., Li, X., Grosse, R. B., and Duvenaud, D. K. (2018). Isolating sources of disentanglement in variational autoencoders. In Advances in Neural Information Processing Systems, pages 2615– 2625. Cohen, T. and Welling, M. (2014). Learning the irreducible representations of commutative Lie groups. 31st International Conference on Machine Learning, pages 3757–3770.   
+354 Cohen, T. S. and Welling, M. (2015). Transformation properties of learned visual representations. In 3rd International Conference on Learning Representations. Eastwood, C. and Williams, C. K. (2018). A framework for the quantitative evaluation of disentangled representations. In International Conference on Learning Representations.   
+58 Higgins, I., Amos, D., Pfau, D., Racaniere, S., Matthey, L., Rezende, D., and Lerchner, A. (2018). Towards a definition of disentangled representations. arXiv preprint arXiv:1812.02230.   
+60 Higgins, I., Matthey, L., Pal, A., Burgess, C., Glorot, X., Botvinick, M., Mohamed, S., and Lerchner, A. (2017). $\beta$ -VAE: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations.   
+363 Kim, H. and Mnih, A. (2018). Disentangling by factorising. In International Conference on Machine Learning, pages 2649–2658.   
+65 Kingma, D. P. and Welling, M. (2014). Auto-Encoding Variational Bayes. In International Conference on Learning Representations. Kumar, A., Sattigeri, P., and Balakrishnan, A. (2018). Variational inference of disentangled latent concepts from unlabeled observations. In International Conference on Learning Representations.   
+369 Locatello, F., Bauer, S., Lucic, M., Gelly, S., Schölkopf, B., and Bachem, O. (2019). Challenging common assumptions in the unsupervised learning of disentangled representations. In International Conference on Machine Learning.   
+372 Nene, S. A., Nayar, S. K., Murase, H., et al. (1996). Columbia object image library (coil-20).   
+Painter, M., Prugel-Bennett, A., and Hare, J. (2020). Linear disentangled representations and unsupervised action estimation. Advances in Neural Information Processing Systems, 33.   
+375 Perez Rey, L. A., Menkovski, V., and Portegies, J. (2020). Diffusion variational autoencoders. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, pages 2704–2710.   
+Quessard, R., Barrett, T. D., and Clements, W. R. (2020). Learning Group Structure and Disentangled Representations of Dynamical Environments. Advances in Neural Information Processing Systems, 33. Rezende, D. J., Mohamed, S., and Wierstra, D. (2014). Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31st International Conference on International Conference on Machine Learning-Volume 32. Ridgeway, K. and Mozer, M. C. (2018). Learning deep disentangled embeddings with the f-statistic loss. In Advances in Neural Information Processing Systems, pages 185–194.   
+86 Sosnovik, I., Szmaja, M., and Smeulders, A. (2019). Scale-Equivariant Steerable Networks. International Conference on Learning Representations, pages 1–14.   
+Worrall, D. E., Garbin, S. J., Turmukhambetov, D., and Brostow, G. J. (2017). Harmonic networks: Deep translation and rotation equivariance. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5028–5037.   
+Wu, Z., Song, S., Khosla, A., Yu, F., Zhang, L., Tang, X., and Xiao, J. (2014). 3D ShapeNets: A Deep Representation for Volumetric Shapes.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 8   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes]   
+(b) Did you include complete proofs of all theoretical results? [N/A] The theoretical contributions are new definitions and methodology, these have been motivated but do not require proofs.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] In Supplementary Material   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In paper or Supplementary Material   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Each experiment was run 10 times, we report means and standard deviation, and show error bars where needed.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] In Supplementary Material we described the hardware. Tracked time of training for experiments with LSBD-VAE and Quessard approach, also reported in the Supplementary Material. The times for training the traditional methods in disentanglement_lib were not measured.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes] In Supplementary Material   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Included code in the Supplementary Material   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] No consent was needed.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] It clearly doesn’t.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/WrotwUEJO59/WrotwUEJO59_content_list.json

@@ -0,0 +1,1282 @@
+[
+    {
+        "type": "text",
+        "text": "Quantifying and Learning Linear Symmetry-Based Disentanglement ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 The definition of Linear Symmetry-Based Disentanglement (LSBD) formalizes   \n2 the notion of linearly disentangled representations, but there is currently no metric   \n3 to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and to   \n4 compare to previous understandings of disentanglement. We propose $\\mathcal { D } _ { \\mathrm { L S B D } }$ , a   \n5 mathematically sound metric to quantify LSBD, and provide a practical implemen  \n6 tation. Furthermore, from this metric we derive LSBD-VAE, a semi-supervised   \n7 method to learn LSBD representations. We demonstrate the utility of our metric   \n8 by showing that (1) common VAE-based disentanglement methods don’t learn   \n9 LSBD representations, (2) LSBD-VAE as well as other recent methods can learn   \n10 LSBD representations, needing only limited supervision on transformations, and   \n11 (3) various desirable properties expressed by existing disentanglement metrics are   \n12 also achieved by LSBD representations. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "13 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "14 Learning low-dimensional representations that disentangle the underlying factors of variation in data   \n15 is considered an important step towards interpretable machine learning with good generalization. To   \n16 address the fact that there is no consensus on what disentanglement entails and how to formalize it,   \n17 Higgins et al. (2018) propose a formal definition for Linear Symmetry-Based Disentanglement, or   \n18 LSBD, arguing that underlying real-world symmetries give exploitable structure to data.   \n19 However, there is currently no metric to quantify LSBD. Such a metric is crucial to properly evaluate   \n20 methods aiming to learn LSBD representations and to relate LSBD to previous definitions of disentan  \n21 glement. Although previous works have evaluated LSBD by measuring performance on downstream   \n22 tasks (Caselles-Dupré et al., 2019) or by measuring specific traits related to LSBD (Painter et al.,   \n23 2020; Quessard et al., 2020), none of these evaluation methods directly quantify LSBD according to   \n24 its well-formalized definition.   \n25 We propose $\\mathcal { D } _ { \\mathrm { L S B D } }$ , a well-formalized and generally applicable metric that quantifies the level of   \n26 LSBD in learned data representations. We show an intuitive justification of this metric, as well   \n27 as its theoretical derivation. We also provide a practical implementation to compute $\\mathcal { D } _ { \\mathrm { L S B D } }$ for   \n28 common symmetry groups. Furthermore, we show that our metric formulation can be used to derive   \n29 a semi-supervised method to learn LSBD representations, which we call LSBD-VAE. To make   \n30 LSBD-VAE more widely applicable, we also demonstrate how to disentangle symmetric properties   \n31 from other non-symmetric properties, and how to quantify this disentanglement with $\\mathcal { D } _ { \\mathrm { L S B D } }$ .   \n32 We show the utility of $\\mathcal { D } _ { \\mathrm { L S B D } }$ by quantifying LSBD in a number of settings, for a variety of datasets   \n33 with underlying SO(2) symmetries and other non-symmetric properties. First, we evaluate common   \n34 VAE-based disentanglement methods and show that most don’t learn LSBD representations. Second,   \n35 we evaluate LSBD-VAE and other recent methods that specifically target LSBD, showing that they   \n36 can obtain much better $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores while needing only limited supervision on transformations.   \n37 Third, we compare $\\mathcal { D } _ { \\mathrm { L S B D } }$ with existing disentanglement metrics, showing that various desirable   \n38 properties expressed with these metrics are also achieved by LSBD representations. ",
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+    {
+        "type": "text",
+        "text": "39 2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            152,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "40 Plenty of works have focused on learning and quantifying disentangled representations recently, but   \n41 research has shown that there is little consensus about the exact definition of disentanglement and   \n42 methods often do not achieve it as well as they proclaim (Locatello et al., 2019). To introduce some   \n43 much-needed formalization, Higgins et al. (2018) proposed to define disentanglement with respect   \n44 to symmetry transformations acting on the data. They used group theory to provide two formal   \n45 definitions, which we refer to as (Linear) Symmetry-Based Disentanglement, or (L)SBD. In this   \n46 paper we focus only on LSBD, not SBD.   \n47 Several methods have been proposed to learn LSBD representations (Caselles-Dupré et al., 2019;   \n48 Painter et al., 2020; Quessard et al., 2020). These methods also learn to represent the transformations   \n49 acting on the input data, assuming various levels of supervision on these transformations. Other   \n50 methods have previously focused on capturing transformations of the data outside the context of   \n51 disentanglement as well (Cohen and Welling, 2015; Sosnovik et al., 2019; Worrall et al., 2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "52 3 Linear Symmetry-Based Disentanglement ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            376,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "53 Higgins et al. (2018) provide a formal definition of linear disentanglement that connects symmetry   \n54 transformations affecting the real world (from which data is observed) to the internal representations   \n55 of a model. The definition is grounded in concepts from group theory, we provide a more detailed   \n56 description of these concepts in the Supplementary Material.   \n57 The definition1 considers a group $G$ of symmetry transformations acting on the data space $X$ through   \n58 the group action $\\cdot : G \\times X \\to X$ . In particular, $G$ can be decomposed as the direct product of $K$   \n59 groups $G = G _ { 1 } \\times \\ldots \\times G _ { K }$ . A model’s internal representation of data is modeled with the encoding   \n60 function $h : X \\to Z$ that maps data to the embedding space $Z$ . The definition for Linearly Symmetry  \n61 Based Disentangled (LSBD) representations then formalizes the requirement that a model’s encoding   \n62 $h$ should reflect and disentangle the transformation properties of the data, and that the transformation   \n63 properties of the model’s encoding should be linear. The exact definition is as follows:   \n64 Definition: Linear Symmetry-Based Disentanglement (LSBD) A model’s encoding map $h :$   \n65 $X  Z$ , where $Z$ is a vector space, is LSBD with respect to the group decomposition $G =$   \n66 $G _ { 1 } \\times \\ldots \\times G _ { K }$ if ",
+        "bbox": [
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+    },
+    {
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+    {
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+    },
+    {
+        "type": "text",
+        "text": "1. there is a decomposition of the embedding space $Z = Z _ { 1 } \\oplus . . . \\oplus Z _ { K }$ into $K$ vector subspaces,   \n2. there are group representations for each subgroup in the corresponding vector subspace $\\rho _ { k } : G _ { k } \\stackrel { \\textstyle \\bar { \\to } } { \\to } \\mathrm { G L } ( Z _ { k } ) , k \\in \\{ 1 , \\dots , K \\}$   \n3. the group representation $\\rho : G \\to { \\mathrm { G L } } ( Z )$ acts on $Z$ as $\\rho ( g ) \\cdot z = ( \\rho _ { 1 } ( g _ { 1 } ) \\cdot z _ { 1 } , \\ldots , \\rho _ { K } ( g _ { K } ) \\cdot z _ { K } ) ,$ (1) for $g = ( g _ { 1 } , \\dotsc , g _ { K } ) \\in G$ and $z = ( z _ { 1 } , \\dots , z _ { K } ) \\in Z$ with $g _ { k } \\in G _ { k }$ and $z _ { k } \\in Z _ { k }$ .   \n4. the map $h$ is equivariant with respect to the actions of $G$ on $X$ and $Z$ , i.e. , for all $x \\in X$ and $g \\in G$ it holds that $h ( g \\cdot x ) \\stackrel { } { = } \\rho ( g ) \\cdot h ( x )$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "75 Furthermore, we say that a group representation $\\rho$ is linearly disentangled with respect to the group   \n76 decomposition $G = G _ { 1 } \\times \\ldots \\times G _ { K }$ if it satisfies criteria 1 to 3 from the LSBD definition above. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "78 4.1 Intuition: Measuring Equivariance with Dispersion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "79 To motivate our metric, let’s first assume a set  \n80 ting in which a suitable linearly disentangled   \n81 group representation $\\rho$ is known. Let’s further   \n82 assume that the dataset of observations can be   \n83 84 exprepoint $x _ { 0 } \\in X$ respe, i.e. $\\{ x _ { n } \\} _ { n = 1 } ^ { N } = { \\bar { \\{ g _ { n } \\cdot x _ { 0 } \\} } } _ { n = 1 } ^ { N }$ $G$   \n85 Formally, this assumes that the action of $G$ on   \n86 $X$ is regular. In this case, we can use the inverse   \n87 group elements $g _ { n } ^ { - 1 }$ to transform each data point   \n88 toward the base point $x _ { 0 }$ , i.e. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b36a682db9765729e832315a363eb9ccf0058785678887dd55f549323b9e8fa1.jpg",
+        "text": "$$\nx _ { 0 } = g _ { 1 } ^ { - 1 } \\cdot x _ { 1 } = \\ldots = g _ { N } ^ { - 1 } \\cdot x _ { N } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "89 Since $\\rho$ is linearly disentangled, we only need to   \n90 measure the equivariance of the encoding map   \n91 $h$ to quantify LSBD. Equivariance is achieved   \n92 when $h ( g \\cdot x ) = \\rho ( g ) \\cdot h ( x )$ , for all $g \\in G , x \\in$   \n93 $X$ . Given the dataset described above, we can   \n94 $\\{ g _ { n } \\} _ { n = 1 } ^ { N }$ s property for .2 In particular $x \\in \\{ x _ { n } \\} _ { n = 1 } ^ { N }$ and n (2) $g \\in$   \n96 can see that we have equivariance if ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/caa1181c8b0270db28d1a6afc76b50415256cdbf5ee7362a8dc3585ad944ee78.jpg",
+        "image_caption": [
+            "Figure 1: A dataset of images from a rotating object expressed in terms of the group $G = \\bar { \\mathrm { S O } } ( 2 )$ acting on a base image $x _ { 0 }$ . It is possible to quantify the level of LSBD of an encoding map $h$ by measuring its equivariance with respect to a group representation $\\rho$ . Since all data has been generated from $x _ { 0 }$ , equivariance can be measured as the dispersion of the points $\\{ \\rho ( g _ { n } ^ { - 1 } ) \\cdot h ( x _ { n } ) \\} _ { n = 1 } ^ { N }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6604d58c316b187280af3a7a351bd77a8b813ba38edb8a9ec64bf5007b7ba600.jpg",
+        "text": "$$\nh ( x _ { 0 } ) = \\rho ( g _ { 1 } ^ { - 1 } ) \\cdot h ( x _ { 1 } ) = \\ldots = \\rho ( g _ { N } ^ { - 1 } ) \\cdot h ( x _ { N } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "97 This not only characterizes perfect equivariance, but also allows for an efficient way to quantify how   \n98 close we are to true equivariance, by measuring the dispersion of the points $\\{ \\rho ( g _ { n } ^ { - 1 } ) \\cdot \\bar { h } ( x _ { n } ) \\} _ { n = 1 } ^ { N }$ . 3   \n99 Given a suitable norm $\\| \\cdot \\| _ { Z }$ in $Z$ , we can thus quantify LSBD in this setting as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67679bd4de31387071bc372f493817130d753e5f3ea787e988955f1c5af5d40a.jpg",
+        "text": "$$\n\\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\left. \\rho ( g _ { n } ^ { - 1 } ) \\cdot h ( x _ { n } ) - \\frac { 1 } { N } \\sum _ { n ^ { \\prime } = 1 } ^ { N } \\rho ( g _ { n ^ { \\prime } } ^ { - 1 } ) \\cdot h ( x _ { n ^ { \\prime } } ) \\right. _ { Z } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "i.e. we compute the mean of 100 $\\{ \\rho ( g _ { n } ^ { - 1 } ) \\cdot h ( x _ { n } ) \\} _ { n = 1 } ^ { N }$ and use the average squared distance to this mean for points in 101 $\\{ \\rho ( g _ { n } ^ { - 1 } ) \\cdot h ( x _ { n } ) \\} _ { n = 1 } ^ { N }$ as our LSBD metric, see Figure 1. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "02 However, this formulation requires knowing the right linearly disentangled group representation and 103 a suitable norm in $Z$ . Moreover, it implicitly assumes a uniform probability measure over the group elements 104 $\\{ g _ { n } \\} _ { n = 1 } ^ { N }$ . In the next section we formulate our metric for a more general setting. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "05 4.2 $\\mathcal { D } _ { \\mathrm { L S B D } }$ : A Metric for LSBD ",
+        "text_level": 1,
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+    },
+    {
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+        "text": "106 Generalizing the ideas from the previous section with concepts from measure theory, we propose a   \n107 metric to measure the level of LSBD of any encoding $h : X \\to Z$ given a data probability measure $\\mu$   \n108 on $X$ , provided that $\\mu$ can be written as the pushforward $G _ { X } ( \\cdot , x _ { 0 } ) _ { \\# } \\nu$ of some probability measure   \n109 $\\nu$ on $G$ by the function $G _ { X } ( \\cdot , x _ { 0 } )$ for some base point $x _ { 0 }$ . More formally, ",
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+        "img_path": "images/62bc94f4e042da4d4184ce5794d88b881cccae2ca6ee7849d1990bfc469440d6.jpg",
+        "text": "$$\n\\mu ( A ) = G _ { X } ( \\cdot , x _ { 0 } ) _ { \\# } \\nu ( A ) = \\nu \\left( \\left\\{ g \\in G \\mid G _ { X } ( g , x _ { 0 } ) \\in A \\right\\} \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "10 for Borel subsets $A \\subset X$ . Note that this is only possible if the action $G _ { X }$ is transitive. ",
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+        "type": "text",
+        "text": "For example, the situation of a dataset with 111 $N$ datapoints $\\{ x _ { n } \\} _ { n = 1 } ^ { N } = \\{ g _ { n } \\cdot x _ { 0 } \\} _ { n = 1 } ^ { N }$ corresponds to 112 the case in which $\\nu$ and $\\mu$ are empirical measures on the group $G$ and data space $X$ , respectively: ",
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+        "type": "equation",
+        "img_path": "images/49788ab162eed399bf796b87846d662e051a47f740e934352f1864035fc87b17.jpg",
+        "text": "$$\n\\nu : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\delta _ { g _ { i } } , \\qquad \\mu : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\delta _ { x _ { i } } .\n$$",
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+        "text": "2Note that $\\{ g _ { n } \\} _ { n = 1 } ^ { N }$ can be used to describe all known group transformations between elements in the dataset by means of composition and inverses, since $x _ { i } = g _ { i } \\cdot ( g _ { j } ^ { - 1 } \\cdot x _ { j } )$ . Thus it suffices to check equivariance for these $N$ group transformations. ",
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+        "type": "text",
+        "text": "3Note that we do not actually need to know $x _ { 0 }$ nor $h ( x _ { 0 } )$ ",
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+        "text": "113 We define the metric $\\mathcal { D } _ { \\mathrm { L S B D } }$ for an encoding $h$ and a measure $\\mu$ as ",
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+        "text": "$$\n\\mathcal { D } _ { \\mathrm { L S B D } } : = \\operatorname* { i n f } _ { \\rho \\in \\mathcal { P } ( G , Z ) } \\int _ { G } \\big \\| \\rho ( g ) ^ { - 1 } \\cdot h ( g \\cdot x _ { 0 } ) - M _ { \\rho , h , x _ { 0 } } \\big \\| _ { \\rho , h , \\mu } ^ { 2 } d \\nu ( g ) ,\n$$",
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+        "text": "114 where the norm $\\| \\cdot \\| _ { \\rho , h , \\mu }$ is a Hilbert-space norm depending on the representation $\\rho$ , the encoding map   \n115 $h : X \\to Z$ , and the data measure $\\mu$ . More details of this norm can be found in the Supplementary   \n116 Material. Moreover, ${ \\mathcal { P } } ( G , Z )$ denotes the set of linearly disentangled representations of $G$ in $Z$   \n117 Lower values of $\\mathcal { D } _ { \\mathrm { L S B D } }$ indicate better disentanglement, zero being optimal. ",
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+        "type": "text",
+        "text": "118 4.3 Practical Computation of $\\mathcal { D } _ { \\mathrm { L S B D } }$ ",
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+        "text": "119 There are two main challenges for computing the metric of Equation (7). First, to calculate the   \n120 integrals in the formula, all possible datapoints that can be expressed as $g \\cdot x _ { 0 }$ with $g \\in G =$   \n121 $G _ { 1 } \\times \\cdots \\times G _ { K }$ must be available. Second, the infimum of the integrals over all possible linearly   \n122 disentangled representations must be estimated. This requires finding the possible invariant subspaces   \n123 $Z = Z _ { 1 } \\oplus \\cdots \\oplus Z _ { K }$ induced by the encoding $h$ over which the group representations are disentangled.   \n124 We present a practical implementation of an upper bound to $\\mathcal { D } _ { \\mathrm { L S B D } }$ for an encoding function $h$ given   \n125 a dataset $\\mathcal { X }$ generated by some known group transformations. In particular, this approximation of   \n126 $\\mathcal { D } _ { \\mathrm { L S B D } }$ is designed for a group decomposition $G = G _ { 1 } \\times \\cdot \\cdot \\cdot \\times G _ { K }$ where each $G _ { k } \\overset { \\vartriangle } { = } \\mathrm { S O } ( D _ { k } )$ with   \n127 $k \\in \\{ 1 , \\ldots , K \\}$ the group of rotations in $D _ { k }$ dimensions. This implementation approximates the   \n128 integrals of Equation (7) by using the empirical distribution of $\\mathcal { X }$ . The invariant subspaces of $Z$ to the   \n129 subgroup actions are found by applying a suitable change of basis. In the new basis, the disentangled   \n130 group representations are expressed in a parametric form whose parameters are optimized to find the   \n131 tightest bound to $\\mathcal { D } _ { \\mathrm { L S B D } }$ . See Figure 2 for an intuitive description of the process.   \n132 Assume there is a dataset $\\mathcal { X }$ that can be modeled in terms of the group decomposition $G = G _ { 1 } \\times \\cdot \\cdot \\cdot G _ { k }$ .   \n133 For each $G _ { k }$ subgroup there is a set of known group elements $\\mathcal { G } _ { k } \\subseteq G _ { k }$ uniformly sampled such   \n134 that the dataset is described in terms of all elements in $\\mathcal { G } = \\mathcal { G } _ { 1 } \\times \\cdots \\times \\mathcal { G } _ { K }$ and a base point $x _ { 0 }$ as   \n135 ${ \\mathcal { X } } = \\left\\{ \\left( g _ { 1 } , \\dots , g _ { K } \\right) \\cdot x _ { 0 } { \\big | } g _ { k } \\in { \\mathcal { G } } _ { k } , ~ k \\in \\left\\{ 1 , \\dots , K \\right\\} \\right\\}$ .   \n136 For each subgroup $G _ { k }$ we construct a set of encoded data $\\mathcal { Z } _ { k } \\subseteq Z$ whose variability should only de  \n137 pend on the action of $G _ { k }$ . The set $\\mathcal { Z } _ { k }$ is given by $\\mathcal { Z } _ { k } = \\{ z _ { k } ( g _ { 1 } , \\dotsc , g _ { K } ) | g _ { j } \\in \\mathcal { G } _ { j } \\ , j \\in \\{ 1 , \\dotsc , K \\} \\}$ ,   \n138 in which ",
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+        "img_path": "images/f0d5c35810c488bf356827d30e83dce8213c3a46d6d0debb3ef7e7ac88c3032a.jpg",
+        "text": "$$\nz _ { k } ( g _ { 1 } , \\dots , g _ { K } ) = h ( ( g _ { 1 } , \\dots , g _ { K } ) \\cdot x _ { 0 } ) - \\frac { 1 } { | { \\mathcal { G } } _ { k } | } \\sum _ { g ^ { \\prime } \\in { \\mathcal { G } } _ { k } } h ( ( g _ { 1 } , \\dots , g _ { k - 1 } , g ^ { \\prime } , g _ { k + 1 } , \\dots , g _ { K } ) \\cdot x _ { 0 } ) .\n$$",
+        "text_format": "latex",
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+        "text": "139 Similar to (Cohen and Welling, 2014), we find a suitable change of basis that exposes the invariant   \n140 subspace $Z _ { k }$ corresponding to the $k$ -th subgroup $G _ { k }$ . The new basis is obtained from the eigenvectors   \n141 resulting from applying Principal Component Analysis (PCA) to $\\mathcal { Z } _ { k }$ . Each element in $\\mathcal { Z } _ { k }$ is projected   \n142 into the first $D _ { k }$ eigenvectors. The new set is denoted as $\\mathcal { Z } _ { k } ^ { \\prime } \\subseteq \\mathbb { R } ^ { D _ { k } }$ with elements $z _ { k } ^ { \\prime } ( g _ { 1 } , \\dotsc , g _ { K } ) \\subseteq$   \n143 $\\mathbb { R } ^ { D _ { k } }$ that are the projected versions of $z _ { k } ( g _ { 1 } , \\dots , g _ { K } )$ .   \n144 (Quessard et al., 2020) describes how one could parameterize the subgroup representations of $S O ( D _ { k } )$   \n145 for arbitrary $D _ { k }$ but here we will focus on $\\bar { G _ { k } } = S O ( 2 )$ . In this case, we can parameterize each   \n146 subgroup representation in terms of a single integer parameter $\\omega \\in \\mathbb { Z }$ as $\\rho _ { k , \\omega } ( g _ { k } )$ corresponding   \n147 to a $2 \\times 2$ rotation matrix whose angle of rotation is $\\omega$ multiplied by the known angle associated   \n148 to the group element $g _ { k } \\in G _ { k } = \\bar { \\mathrm { S O } ( 2 ) }$ . For this subgroup we can approximate the $M _ { \\rho , h , x _ { 0 } }$ from   \n149 Equation (7) as $M _ { k , \\omega }$ given by ",
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+        "img_path": "images/e630d52c781b5ac29381049a846db337055462c61b2a8065f5b2959d5797f54e.jpg",
+        "text": "$$\nM _ { k , \\omega } = \\frac { 1 } { | \\mathcal { G } | } \\sum _ { ( g _ { 1 } , . . . , g _ { K } ) \\in \\mathcal { G } } \\rho _ { k , \\omega } ( g _ { k } ^ { - 1 } ) \\cdot z ^ { \\prime } ( g _ { 1 } , . . . , g _ { K } ) .\n$$",
+        "text_format": "latex",
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+        "text": "150 Similar to Equation (7) we would like to find the optimal $\\rho _ { k , \\omega }$ that minimizes the integral over the   \n151 group representations. We can define a parameter search space $\\Omega \\subseteq \\mathbb { Z }$ , e.g. $\\Omega = [ - 1 0 , 1 0 ]$ for finding   \n152 the optimal $\\omega \\in \\Omega$ that minimizes the dispersion, this is expressed in the following equation ",
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+        "text": "$$\n\\mathcal { D } _ { \\mathrm { L S B D } } ^ { ( k ) } = \\operatorname* { m i n } _ { \\omega \\in \\Omega } \\frac { 1 } { | \\mathcal { G } | } \\sum _ { ( g _ { 1 } , . . . , g _ { K } ) \\in \\mathcal { G } } \\Vert \\rho _ { k , \\omega } ( g _ { k } ^ { - 1 } ) \\cdot z ^ { \\prime } ( g _ { 1 } , . . . , g _ { K } ) - M _ { k , \\omega } \\Vert ^ { 2 } .\n$$",
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+        "image_caption": [
+            "Figure 2: Consider a dataset modeled by a group decomposition $G = G _ { 1 } \\times \\cdot \\cdot \\cdot \\times G _ { K }$ acting on $x _ { 0 }$ and is embedded in a latent space $Z$ via $h$ . In this example the subgroup $G _ { k } = \\mathrm { S O } ( 2 )$ models the rotations of an airplane. Other subgroups $G _ { \\neq k }$ could also be acting e.g. changes in airplane color. The first step to calculate the disentanglement of $G _ { k }$ is to construct a set of data embeddings $\\mathcal { Z } _ { k } \\subseteq Z$ whose variability is due to $G _ { k }$ . These embeddings are then projected into a 2-dimensional space through PCA. For these projected embeddings we can describe the group representations in a simple parametric form $\\rho _ { k , w }$ . For a given $\\rho _ { k , w }$ the equivariance of $G _ { k }$ is measured as the dispersion after applying the action of the inverse group representation $\\rho _ { k , w } ^ { - 1 }$ . "
+        ],
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+        "type": "text",
+        "text": "153 Each D(k)LSB measures the degree of equivariance of the projected embeddings for each $k$ -th subgroup   \n154 corresponding to the best fitting group representation. The upper bound to the metric is finally obtained   \n155 by averaging across all subgroups $\\begin{array} { r } { \\bar { \\mathcal { D } } _ { \\mathrm { L S B D } } \\leq \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\bar { \\mathcal { D } } _ { \\mathrm { L S B D } } ^ { ( \\bar { k } ) } } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "156 5 Learning LSBD Representations: LSBD-VAE ",
+        "text_level": 1,
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+        "text": "157 In this section we present LSBD-VAE, a semi-supervised VAE-based method to learn LSBD repre  \n158 sentations. The main idea is to train an unsupervised Variational Autoencoder (VAE) (Kingma and   \n159 Welling, 2014; Rezende et al., 2014) with a suitable latent space topology, and use our metric as an   \n160 additional loss term for batches of transformation-labeled data.   \n161 Assumptions LSBD-VAE requires some knowledge about the group structure $G$ that is to be   \n162 disentangled. Concretely, the group and its decomposition $G = G _ { 1 } \\times \\ldots \\times G _ { K }$ should be known,   \n163 as well as a suitable linearly disentangled group representation $\\rho : G \\to { \\mathrm { G L } } ( Z )$ and a latent space   \n164 $Z = Z _ { 1 } \\oplus . . . \\oplus Z _ { K }$ . Moreover, we assume there exists an embedded submanifold $Z _ { G } \\subseteq Z$ such   \n165 that the action of $G$ on $Z$ restricted to $Z _ { G }$ is regular, and $Z _ { G }$ is invariant under the action. Only $Z _ { G }$   \n166 will then be used as the codomain for the encoding map, $h : X \\to Z _ { G }$ .   \n167 We demonstrate the assumptions above for the common group structure $G = \\mathrm { S O } ( 2 ) \\times \\mathrm { S O } ( 2 )$ . For   \n168 the group representation $\\rho = \\rho _ { 1 } \\oplus \\rho _ { 2 }$ , with $Z = \\mathbb { R } ^ { 2 } \\oplus \\mathbb { R } ^ { 2 }$ , we can use rotation matrices in ${ \\dot { \\mathbb { R } } } ^ { 2 }$ for   \n169 $\\rho _ { 1 }$ and $\\rho _ { 2 }$ . We can then use 1-spheres $S ^ { 1 } = \\{ z \\in \\mathbb { R } ^ { 2 } : \\| z \\| = 1 \\}$ for the embedded submanifold:   \n170 $\\dot { Z } _ { G } = \\dot { S } ^ { 1 } \\times S ^ { 1 }$ . In this case, the action of $G$ on $Z$ restricted to $Z _ { G }$ is indeed regular, and $Z _ { G }$ is   \n171 invariant under the action.   \n172 Unsupervised Learning on Latent Manifold To learn encodings only on the latent manifold   \n173 $Z _ { G }$ , we use a Diffusion Variational Autoencoder (∆VAE) (Perez Rey et al., 2020). $\\Delta$ VAEs can   \n174 use any closed Riemannian manifold embedded in a Euclidean space as a latent space (or latent   \n175 manifold), provided that a certain projection function from the Euclidean embedding space into the   \n176 latent manifold is known and the scalar curvature of the manifold is available. The $\\Delta$ VAE uses   \n177 a parametric family of posterior approximates obtained from a diffusion process over the latent   \n178 manifold. To estimate the intractable terms of the negative ELBO, the reparameterization trick is   \n179 implemented via a random walk. ",
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+        "type": "text",
+        "text": "In the case of 180 $S ^ { 1 }$ as a latent (sub)manifold, we consider $\\mathbb { R } ^ { 2 }$ as the Euclidean embedding space, and 181 the projection function4 $\\Pi : \\mathbb { R } ^ { 2 }  S ^ { 1 }$ normalizes points in the embedding space: $\\Pi ( z ) \\overset { \\cdot } { = } \\dot { z } / | z |$ . The scalar curvature of182 $S ^ { 1 }$ is 0. ",
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+        "text": "183 Semi-Supervised Learning with Transformation Labels Caselles-Dupré et al. (2019) proved   \n184 that LSBD representations cannot be inferred from a training set of unlabeled observations, but that   \n185 access to the transformations between data points is needed. They therefore use a training set of   \n186 observation pairs with a given transformation between them.   \n7 However, we posit that only a limited amount of supervision is sufficient. Since obtaining supervision   \n88 on transformations is typically more expensive than obtaining unsupervised observations, it is   \n9 desirable to limit the amount of supervision needed.   \n90 Therefore, we augment the un  \n91 supervised $\\Delta \\mathrm { V A E }$ with a super  \n92 vised method that makes use of   \n93 transformation-labeled batches, i.e.   \n94 batches $\\lbrace x _ { m } \\rbrace _ { m = 1 } ^ { M }$ such that $\\begin{array} { r l } { x _ { m } } & { { } = } \\end{array}$   \n95 $g _ { m } \\cdot x _ { 1 }$ for $m = 2 , \\ldots , M$ , where   \n96 the transformations $g _ { m }$ (and thus   \n97 their group representations $\\rho ( g _ { m } ) ,$ ) are   \n98 known and are referred to as transfor  \n99 mation labels. The simplified version   \n00 of the metric from Equation (4) can   \n01 then be used for each batch as an ad  \n02 ditional loss term (with $x _ { 0 } = x _ { 1 }$ ), as   \n03 it is differentiable under the assump  \n04 tions described above (using the Eu  \n05 clidean norm).   \n206 We make a small adjustment to Equation (4) for the purpose of our method, since the mean computed   \n207 there does not typically lie on the latent manifold $Z _ { G }$ . Thus, we use the projection $\\Pi$ from the $\\Delta \\mathrm { V A E }$   \n208 to project the mean onto $Z _ { G }$ . Writing the encodings as $z _ { m } : = h ( x _ { m } )$ , the additional loss term for a   \n209 transformation-labeled batch $\\{ x _ { m } \\} _ { m = 1 } ^ { M }$ then becomes ",
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+        "image_caption": [
+            "Figure 3: Overview of the supervised part of LSBD-VAE. "
+        ],
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+    {
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+        "text": "$$\n\\mathcal { L } _ { L S B D } = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\left. \\rho ( g _ { m } ^ { - 1 } ) \\cdot z _ { m } - \\Pi \\left( \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\rho ( g _ { m } ^ { - 1 } ) \\cdot z _ { m } \\right) \\right. ^ { 2 } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "210 where $g _ { 1 } = e$ , the group identity. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Moreover, instead of feeding the encodings $z _ { m }$ to the decoder, we use $\\rho ( g _ { m } ) \\cdot \\overline { { z } }$ , where ${ \\begin{array} { r l } { { \\overline { { z } } } } & { = } \\end{array} }$ $\\begin{array} { r } { \\Pi \\left( \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\rho ( g _ { m } ^ { - 1 } ) \\cdot z _ { m } \\right) } \\end{array}$ . This encourages the decoder to follow the required group structure. This only affects the reconstruction loss component of the $\\Delta \\mathrm { V A E }$ . ",
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+        "text": "214 Figure 3 illustrates the supervised part of our method for a transformation-labeled batch $\\{ x _ { m } \\} _ { m = 1 } ^ { M }$   \n215 The loss function is the regular ELBO (but with adjusted decoder input as described above) as used   \n216 in $\\Delta \\mathrm { V A E }$ plus an additional term $\\gamma \\cdot \\mathcal { L } _ { L S B D }$ , where $\\gamma$ is a weight hyperparameter to control the   \n217 influence of the supervised loss component. By alternating unsupervised and supervised training   \n218 (using the same encoder and decoder), we have a method that makes use of both unlabeled and   \n219 transformation-labeled observations. ",
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+    {
+        "type": "text",
+        "text": "220 6 Experimental Setup ",
+        "text_level": 1,
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+        "text": "221 We evaluate the disentanglement of several models on three different image datasets (Square, Arrow,   \n222 and Airplane) with a known group decomposition $G = \\mathrm { S O } ( 2 ) \\times \\mathrm { S O } ( 2 )$ describing the underlying   \n223 transformations. For each subgroup a fixed number of $| \\mathcal { G } _ { k } | = 6 4$ with $k \\in \\{ 1 , 2 \\}$ transformations is   \n224 selected. The datasets exemplify different group actions of SO(2): periodic translations, in-plane   \n225 rotations, out-of-plane rotations, and periodic hue-shifts. ",
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+        "text": "In real settings, not all variability in the data can be modelled by the actions of a group. Therefore, we also evaluate the same models on two datasets ModelNet40 (Wu et al., 2014) and COIL-100 (Nene et al., 1996) that consist of images from various objects (i.e. non-symmetric variation) under known out-of-plane rotations (SO(2) symmetries). In many settings it is easy to obtain labels for such rotations, e.g. when the camera or object angle is controlled by an agent. See Figure 4 for examples of the datasets. For more details, see the Supplementary Material. ",
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+            "Figure 4: Example images from each of the datasets used. Each row shows different examples from a single factor changing. "
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+        "text": "232 For the Square, Arrow, and Airplane datasets we test LSBD-VAE with transformation-labeled batches   \n233 of size $M = 2$ . More specifically, for each experiment we randomly select $L$ disjoint pairs of data   \n234 points, and label the transformation between the data points in each pair. We vary the number of   \n235 labeled pairs $L$ from 0 (corresponding to a $\\Delta { \\mathrm { V A E } } )$ to $N / 2$ (in which case each data point is involved   \n236 in exactly one labeled pair). We set the weight $\\gamma$ of the supervised loss component to $\\gamma = 1 0 0$   \n237 for all experiments. We choose $M = 2$ for our experiments since it is the most limited setting for   \n238 LSBD-VAE. Higher values of $M$ would provide stronger supervision, so successful results with   \n239 $M = 2$ imply that good results can also be achieved for higher values of $M$ (but not necessarily vice   \n240 versa).   \n241 For the COIL-100 and ModelNet40 datasets, we train LSBD-VAE on batches containing images of   \n242 one particular object from all different angles (72 and 64 for COIL-100 and ModelNet40, respectively).   \n243 Each batch is labelled with transformations $( g _ { 1 } , e ) , \\ldots , ( g _ { M } , e )$ , where $g _ { m }$ represent rotations, and   \n244 the unit transformation $e$ indicates that the object is unchanged. To represent the rotations we use   \n245 a $S ^ { 1 }$ latent space as in $\\Delta \\mathrm { V A E }$ , whereas for the object identity we use a 5-dimensional Euclidean   \n246 space with standard Gaussian prior as in regular VAEs. LSBD is measured as the disentanglement of   \n247 rotations in the latent space. For these experiments we used $\\gamma = 1$ .   \n248 We furthermore test a number of known disentanglement methods for comparison, including tra  \n249 ditional disentanglement methods as well as methods focusing on LSBD. In particular, we use   \n250 disentanglement_lib (Locatello et al., 2019) to train a regular VAE (Kingma and Welling, 2014;   \n251 Rezende et al., 2014), $\\beta$ -VAE (Higgins et al., 2017), CC-VAE (Burgess et al., 2018), FactorVAE   \n252 (Kim and Mnih, 2018), and DIP-VAE-I/II (Kumar et al., 2018). Furthermore we evaluate the method   \n253 from Quessard et al. (2020) that focuses on LSBD. We also tested ForwardVAE (Caselles-Dupré   \n254 et al., 2019), but show only limited results since we were not able to reproduce any reasonable results   \n255 for our datasets.   \n256 We use encodings from all these methods to evaluate $\\mathcal { D } _ { \\mathrm { L S B D } }$ , as well as common traditional disentan  \n257 glement metrics from disentanglement_lib: Beta (Higgins et al., 2017), Factor (Kim and Mnih,   \n258 2018), SAP (Kumar et al., 2018), DCI Disentanglement (Eastwood and Williams, 2018), Mutual   \n259 Information Gap (MIG) (Chen et al., 2018), and Modularity (MOD) (Ridgeway and Mozer, 2018).   \n260 More information about the architecture, epochs and hyperparameters can be found in the Supplemen  \n261 tary Material. For the traditional disentanglement methods trained on Square, Arrow and Airplane   \n262 datasets the latent spaces have 4 dimensions, since these are the minimum number of dimensions   \n263 necessary to learn LSBD representations for an underlying $\\mathrm { S O } ( 2 ) \\times \\mathrm { S O } ( 2 )$ symmetry group, see   \n264 (Higgins et al., 2018; Caselles-Dupré et al., 2019). For COIL-100 and ModelNet40 we use latent   \n265 spaces with 7 dimensions for a fair comparison with the LSBD-VAE method. ",
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+        "type": "text",
+        "text": "266 7 Results: Evaluating LSBD with DLSBD ",
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+        "text": "267 We now highlight four key observations from our experimental results. In particular, we differentiate   \n268 between the methods (VAE, $\\beta$ -VAE, CC-VAE, FACTOR, DIP-I, DIP-II) and metrics (BETA,   \n269 FACTOR, SAP, DCI, MIG, MOD) that approach disentanglement in the traditional sense, and   \n270 methods ( $\\Delta$ VAE, QUESSARD, LSBD-VAE) and metric $\\left( \\mathcal { D } _ { \\mathrm { L S B D } } \\right)$ that focus specifically on LSBD.   \n271 The full quantitative results can be found in the Supplementary Material. ",
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+        "text": "Figure 5 summarizes the $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores (lower is better) for all methods on all datasets. Bars show the mean scores over 10 runs for each method, the vertical lines represent standard deviations. LSBD-VAE/ $L$ indicates our method trained on $L$ labelled pairs (LSBD-VAE/0 corresponds to the unsupervised $\\Delta$ VAE), LSBD-VAE/full indicates our method trained on batches containing a single object in all known transformations (for datasets with non-symmetric variation). Note that LSBDVAE obtained very good scores (nearly 0) on the Arrow and Pixel datasets, hence the missing bars. ",
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+            "Figure 5: $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores for all methods on all datasets "
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+        "text": "280 None of the traditional disentanglement methods achieve good $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores, even if they score well   \n281 on other traditional disentanglement metrics. This implies that LSBD isn’t achieved by traditional   \n282 methods. Moreover, from the full results in the Supplementary Material we see that the traditional   \n283 methods on these datasets do not achieve good scores on all traditional metrics. In particular, SAP,   \n284 DCI, and MIG scores are low. We believe this is a result of the cyclic nature of the symmetries   \n285 underlying our datasets, further emphasizing the need for disentanglement methods that can capture   \n286 such symmetries. ",
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+        "text": "The SAP and MIG scores measure to what extent generative factors are disentangled into a single latent dimension. However, since the factors in our dataset are inherently cyclic due to their symmetry structure, they cannot be properly represented in a single latent dimension, as shown by Perez Rey et al. (2020). Instead, at least two dimensions are needed to continuously represent each cyclic factor in our data. A similar conclusion was made by Caselles-Dupré et al. (2019) and Painter et al. (2020). ",
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+        "text": "DCI disentanglement measures whether a latent dimension captures at most one generative factor. This is accomplished by measuring the importance of each latent dimension in predicting the true generative factor using boosted trees. However, since the generative factors are cyclic, the performance of the boosted tree classifiers is far from optimal, thus providing more importance to several dimensions in predicting the generative factors and giving overall lower DCI scores. ",
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+        "type": "text",
+        "text": "7.2 LSBD-VAE and other LSBD Methods Can Learn LSBD Representations with Limited Supervision on Transformations ",
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+        "type": "text",
+        "text": "From Figure 5 we observe that methods focusing specifically on LSBD can score higher on $\\mathcal { D } _ { \\mathrm { L S B D } }$ , showing that they are indeed more suitable to learn LSBD representations. In particular, LSBD-VAE got very good $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores for all datasets. Moreover, our experiments on the Arrow, Airplane, and Pixel datasets also show that only limited supervision suffices to obtain good $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores with low variability. ",
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+        "text": "We only partially managed to reproduce the results from Quessard et al. (2020) on our datasets. Their method scored fairly well on the Airplane, ModelNet40, and COIL-100 datasets, but did not do well on the Square and Arrow dataset in our experiments. ",
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+        "text": "307 Furthermore, we tested ForwardVAE by Caselles-Dupré et al. (2019), but we did not manage   \n308 to produce any reasonable results on our datasets, trying both their original architecture and the   \n309 architecture we used for our other experiments. Therefore, we do not include scores for this method.   \n310 We did however manage to reproduce ForwardVAE’s results on the Flatland dataset, which was used   \n311 in their paper. For those experiments, we computed a mean $\\mathcal { D } _ { \\mathrm { L S B D } }$ score of 0.012 with standard   \n312 deviation 0.001 over 10 runs, indicating that ForwardVAE indeed learns LSBD representations for   \n313 Flatland. ",
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+        "type": "text",
+        "text": "314 7.3 LSBD Representations Also Satisfy Previous Disentanglement Notions ",
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+        "text": "315 Our results also indicate that LSBD captures various desirable properties that are expressed by   \n316 traditional disentanglement metrics. In Figure 6 we compare $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores with scores for previous   \n317 disentanglement metrics, for all our experiments. Note that for $\\mathcal { D } _ { \\mathrm { L S B D } }$ lower is better, whereas for   \n318 all other metrics higher is better. As we noted before, good scores on traditional disentanglement   \n319 metrics don’t necessarily imply good $\\mathcal { D } _ { \\mathrm { L S B D } }$ scores. Conversely however, methods that score well   \n320 on $\\mathcal { D } _ { \\mathrm { L S B D } }$ also score well on many traditional disentanglement metrics, often even outperforming   \n321 the traditional methods. In particular, from the full results (see Supplementary Material) we see   \n322 that LSBD-VAE matches or outperforms the traditional methods on the BETA, FACTOR and MOD   \n323 metrics, and achieves much better scores for the DCI metric where traditional methods scored poorly.   \n24 The MIG and SAP scores are still low for methods focusing on LSBD. This is expected however, as   \n25 explained earlier in Section 7.1. This was also observed by Painter et al. (2020) for different datasets. ",
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+            "Figure 6: Comparing $\\mathcal { D } _ { \\mathrm { L S B D } }$ to previous disentanglement metrics "
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+        "text": "326 8 Conclusion ",
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+        "text": "27 We presented $\\mathcal { D } _ { \\mathrm { L S B D } }$ , a metric to quantify Linear Symmetry-Based Disentanglement (LSBD) as 8 defined by Higgins et al. (2018). We further used this metric formulation to motivate LSBD-VAE, a semi-supervised method to learn LSBD representations given some expert knowledge on the underlying group symmetries that are to be disentangled. ",
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+        "type": "text",
+        "text": "We used $\\mathcal { D } _ { \\mathrm { L S B D } }$ to evaluate various disentanglement methods, both traditional methods and recent methods that specifically focus on LSBD, and showed that LSBD-VAE can learn LSBD representations where traditional methods fail to do so. We also compared $\\mathcal { D } _ { \\mathrm { L S B D } }$ to traditional disentanglement metrics, showing that LSBD captures many of the same desirable properties that are expressed by existing disentanglement methods. Conversely, we also showed that traditional disentanglement methods and metrics do not usually achieve or measure LSBD. ",
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+        "text": "337 Challenges that remain are expanding and testing LSBD-VAE and $\\mathcal { D } _ { \\mathrm { L S B D } }$ on different group struc  \n338 tures, towards more practical applications, as well as focusing on the utility of LSBD representations   \n339 for downstream tasks.   \n340 Broader Impact The work is fairly theoretical, and practical methods derived from this work have   \n341 no obvious negative societal impact. However, the ideas presented are relevant to representation   \n342 learning and could be, in particular, used in computer vision and agent control applications. ",
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+        "text": "343 References   \n344 Burgess, C. P., Higgins, I., Pal, A., Matthey, L., Watters, N., Desjardins, G., and Lerchner, A. (2018). Understanding disentangling in $\\beta$ -VAE. arXiv preprint arXiv:1804.03599. Caselles-Dupré, H., Ortiz, M. G., and Filliat, D. (2019). Symmetry-based disentangled representation learning requires interaction with environments. In Advances in Neural Information Processing Systems, pages 4606–4615. Chen, T. Q., Li, X., Grosse, R. B., and Duvenaud, D. K. (2018). Isolating sources of disentanglement in variational autoencoders. In Advances in Neural Information Processing Systems, pages 2615– 2625. Cohen, T. and Welling, M. (2014). Learning the irreducible representations of commutative Lie groups. 31st International Conference on Machine Learning, pages 3757–3770.   \n354 Cohen, T. S. and Welling, M. (2015). Transformation properties of learned visual representations. In 3rd International Conference on Learning Representations. Eastwood, C. and Williams, C. K. (2018). A framework for the quantitative evaluation of disentangled representations. In International Conference on Learning Representations.   \n58 Higgins, I., Amos, D., Pfau, D., Racaniere, S., Matthey, L., Rezende, D., and Lerchner, A. (2018). Towards a definition of disentangled representations. arXiv preprint arXiv:1812.02230.   \n60 Higgins, I., Matthey, L., Pal, A., Burgess, C., Glorot, X., Botvinick, M., Mohamed, S., and Lerchner, A. (2017). $\\beta$ -VAE: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations.   \n363 Kim, H. and Mnih, A. (2018). Disentangling by factorising. In International Conference on Machine Learning, pages 2649–2658.   \n65 Kingma, D. P. and Welling, M. (2014). Auto-Encoding Variational Bayes. In International Conference on Learning Representations. Kumar, A., Sattigeri, P., and Balakrishnan, A. (2018). Variational inference of disentangled latent concepts from unlabeled observations. In International Conference on Learning Representations.   \n369 Locatello, F., Bauer, S., Lucic, M., Gelly, S., Schölkopf, B., and Bachem, O. (2019). Challenging common assumptions in the unsupervised learning of disentangled representations. In International Conference on Machine Learning.   \n372 Nene, S. A., Nayar, S. K., Murase, H., et al. (1996). Columbia object image library (coil-20).   \nPainter, M., Prugel-Bennett, A., and Hare, J. (2020). Linear disentangled representations and unsupervised action estimation. Advances in Neural Information Processing Systems, 33.   \n375 Perez Rey, L. A., Menkovski, V., and Portegies, J. (2020). Diffusion variational autoencoders. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, pages 2704–2710.   \nQuessard, R., Barrett, T. D., and Clements, W. R. (2020). Learning Group Structure and Disentangled Representations of Dynamical Environments. Advances in Neural Information Processing Systems, 33. Rezende, D. J., Mohamed, S., and Wierstra, D. (2014). Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31st International Conference on International Conference on Machine Learning-Volume 32. Ridgeway, K. and Mozer, M. C. (2018). Learning deep disentangled embeddings with the f-statistic loss. In Advances in Neural Information Processing Systems, pages 185–194.   \n86 Sosnovik, I., Szmaja, M., and Smeulders, A. (2019). Scale-Equivariant Steerable Networks. International Conference on Learning Representations, pages 1–14.   \nWorrall, D. E., Garbin, S. J., Turmukhambetov, D., and Brostow, G. J. (2017). Harmonic networks: Deep translation and rotation equivariance. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5028–5037.   \nWu, Z., Song, S., Khosla, A., Yu, F., Zhang, L., Tang, X., and Xiao, J. (2014). 3D ShapeNets: A Deep Representation for Volumetric Shapes. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 8   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes]   \n(b) Did you include complete proofs of all theoretical results? [N/A] The theoretical contributions are new definitions and methodology, these have been motivated but do not require proofs. ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] In Supplementary Material   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In paper or Supplementary Material   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Each experiment was run 10 times, we report means and standard deviation, and show error bars where needed.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] In Supplementary Material we described the hardware. Tracked time of training for experiments with LSBD-VAE and Quessard approach, also reported in the Supplementary Material. The times for training the traditional methods in disentanglement_lib were not measured. ",
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+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
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+        "page_idx": 10
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parse/train/kHromd7SNA/kHromd7SNA_content_list.json

@@ -0,0 +1,1503 @@
+[
+    {
+        "type": "text",
+        "text": "MOTION REPRESENTATIONS FOR ARTICULATED ANIMATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose novel motion representations for animating articulated objects consisting of distinct parts. In a completely unsupervised manner, our method identifies meaningful object parts, tracks them in a driving video, and infers their motions by considering their principal axes. In contrast to the previous keypoint-based works, our method extracts meaningful and consistent regions, describing locations, shape, and pose. The regions correspond to semantically relevant and distinct object parts, that are more easily detected in frames of the driving video. To force decoupling of foreground from background, we model non-object related global motion with a homography. Our model1 can animate a variety of objects, surpassing previous methods by a large margin on existing benchmarks. We present a challenging new benchmark with high-resolution videos and show that the improvement is particularly pronounced when articulated objects are considered. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            458,
+            336,
+            474
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Animation—bringing static objects to life—has broad applications across education and entertainment. Animated characters and objects increase the creativity and appeal of content, improve the clarity of material through storytelling, and enhance user experiences. Imagine the Mona Lisa describing the manner in which she was painted, or Michelangelo’s David detailing the method with which he was sculpted, or an influential historical figure shedding light on key events of the past (Fig. 1); how much more engaging this would be. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Until very recently, animation techniques necessary for achieving such results required a trained professional, specialized hardware, software, and a great deal of effort. Quality results generally still do, but vision and graphics communities have attempted to address some of these limitations by training data-driven methods (Wang ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/f30413fadc3b455edf46dd95ca7f4a717bde631a83181d84719e4a045ad6cfb8.jpg",
+        "image_caption": [
+            "Figure 1: An animation produced by our method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            584,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "et al., 2018a; Chan et al., 2019; Ren et al., 2020; Geng et al., 2019; Gafni et al., 2019) on object classes for which prior knowledge of object shape and pose can be learned. This, however, requires ground truth pose and shape data to be available during training. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent works have sought to avoid the need for ground truth data through unsupervised motion transfer (Wiles et al., 2018; Siarohin et al., 2019a;b). Significant progress on the several key challenges have been made, including training using image reconstruction as a loss, and disentangling motion from appearance. This has created the potential to animate a broader range of object categories, without any domain knowledge or labelled data, requiring only videos of objects in motion during training. However, two key problems remain open. The first is how to represent the parts of an articulated or non-rigid moving object, including their shapes and poses. The second is given the object parts, how to animate them using the sequence of motions in a driving video. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Initial attempts involved extracting unsupervised keypoints (Lorenz et al., 2019; Kim et al., 2019) in end-to-end frameworks (Wiles et al., 2018; Siarohin et al., 2019b;a), then warping a feature embedding of a source image to align its keypoints with those of a driving video. Follow on work (Siarohin et al., 2019a) additionally modelled the motion around each keypoint with local, affine transformations, and introduced a generation module that both composites warped source image regions and inpaints occluded regions, to render the final image. This enabled a variety of creative applications2, for example needing only one source face image to generate a near photo-realistic animation, driven by a video of a different face. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, the resulting unsupervised keypoints are detected on the boundary of the objects. While points on edges are easier to identify, tracking such keypoints between frames is problematic, as any point on the boundary is a valid candidate, making it hard to establish correspondences between frames. A further problem is that the unsupervised keypoints do not correspond to semantically meaningful object parts, and represent location and direction, but not shape. Due to this limitation, animating articulated objects, such as bodies, remains challenging. Furthermore, these methods assume static backgrounds, i.e., no camera motion, leading to leakage of background motion information into one or several of the detected keypoints. Despite significant breakthroughs, these remaining deficiencies limit the scope of the core innovation to more trivial object categories and motions, and lower quality outputs, especially when objects are articulated. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This work introduces two contributions critical to addressing these challenges. First, we redefine the underlying motion representation. Instead of using keypoints, we switch to regions that allow first-order motion to be measured, rather than regressed. This enables improved convergence, more stable, robust object and motion representations, and also empirically captures the shape of the underpinning object parts, leading to better motion segmentation. Fig. 3 contains several examples of region vs. keypoint-based motion representation. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Secondly, we explicitly model background or camera motion between training frames by predicting the parameters of a global homography explaining non-object related motions. This enables the model to focus solely on the foreground object, making the identified points more stable, and further improves convergence. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "These contributions unlock significant gains in capability for unsupervised motion transfer methods, resulting in much improved animation of articulated objects in particular. Furthermore, the framework scales better in the number of unsupervised regions, resulting in more detailed motion. Our method outperforms previous unsupervised animation methods on a variety of datasets, including talking faces, taichi videos and animated pixel art. We additionally present a new dataset, TED talk speakers, to create a more challenging benchmark for the task of animating articulated objects. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            575,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Image animation methods can be separated into supervised, which require knowledge about the animated object during training, and unsupervised, which do not. Such knowledge typically includes landmarks (Cao et al., 2014; Zakharov et al., 2019; Qian et al., 2019; Ha et al., 2020), semantic segmentations (Nirkin et al., 2019), and parametric 3D models (Geng et al., 2019; Thies et al., 2016; Deng et al., 2020; Nagano et al., 2018; Liu et al., 2019). As a result, supervised methods are limited to a small number of object categories for which a lot of labelled data is available, such as faces and human bodies. Early face reenactment work (Thies et al., 2016) fitted a 3D morphable model to an image, animating and rendering it back using graphical techniques. Further works used neural networks to get higher quality rendering (Kim et al., 2018; Wang et al., 2018b), sometimes requiring multiple images per identity (Geng et al., 2019; Pumarola et al., 2018). A body of works treats animation as an image-to-image (Siarohin et al., 2018) or a video-to-video (Wang et al., 2018a; Chan et al., 2019; Ren et al., 2020) translation problem. Apart from some exceptions (Wang et al., 2019), these works further constrain the problem to animating a single instance of an object, such as a single face (Kim et al., 2018; Bansal et al., 2018) or a single human body (Chan et al., 2019; Ren et al., 2020; Wang et al., 2018a), requiring retraining (Bansal et al., 2018; Chan et al., 2019; Ren et al., 2020) or fine-tuning (Zakharov et al., 2019) for each new instance. Despite promising results, generalizing these methods beyond a limited range of object categories remains challenging. Additionally, they tend to transfer not only the motion but also the identity of the driving object, making the shape of the animated face or a body similar or identical to the driving face or body (Kim et al., 2018; Zakharov et al., 2019). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/5b23edf1ef336c348613954cba5bdb75e548528d675bde52781924f7396669c0.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our model. The region predictor returns heatmaps for each part in the source and the driving images. We then compute principal axes of each heatmap, to transform each region from the source to the driving frame through a whitened reference frame. Region and background transformations are combined by the pixel-wise flow prediction network. The target image is generated by warping the source image in a feature space using the pixel-wise flow, and inpainting newly introduced regions, as indicated by the confidence map. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised methods address some of these limitations. They do not require any labelled data regarding the shape or landmarks of the animated object. Video-generation-based animation methods predict future frames of a video, given the first frame and an animation class label, such as \"make a happy face\", \"do jumping jack\", or \"play golf\" (Tulyakov et al., 2018; Saito et al., 2017; Clark et al., 2019). A further group of works re-target animation from a driving video to a source frame. X2Face (Wiles et al., 2018) builds a canonical representation of an input face, and generates a warp field conditioned on the driving video. Monkey-Net (Siarohin et al., 2019b) learns a set of unsupervised keypoints to generate animations. Follow-up work substantially improves the quality of animation by considering a first order motion model (FOMM) (Siarohin et al., 2019a) for each keypoint, represented by regressing a local, affine transformation. Both of these works apply to a wider range of objects including faces, bodies, robots, and pixel art animations. Empirically, these methods extract keypoints on the boundary of the animated objects. Articulated objects such as human bodies are therefore challenging, as internal motion, for example, an arm moving across the body, is not well modeled, producing unconvincing animations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This work presents an unsupervised method. We argue that the limitations of previous such methods in animating articulated objects is due to an inability of their internal representations to capture complete object parts, their shape and pose. X2Face (Wiles et al., 2018) assumes an object can be represented with a single RGB texture, while other methods find keypoints on edges (Siarohin et al., 2019b;a). Our new region and background motion representations address these shortcomings. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our unsupervised animation framework consists of a system design, and methods for training this system using two different frames, source S, and driving D, from the same video. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 SYSTEM DESIGN ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "FOMM (Siarohin et al., 2019a), the current state-of-the-art method in unsupervised animation learning, consists of two main parts: motion estimation and image generation. The contributions of our work lie in novel motion representations within the first part of this framework. Our system, outlined in Fig. 2, therefore follows the FOMM design as closely as possible, in order to demonstrate the impact due specifically to our contributions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/b434e5f758996b8746e73620d93b6fcb9f93375ab8e8dd904e07b5ad278a1f48.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of motion/part representations. Regression-based keypoint representations do not provide consistent detection between frames (marked with red). Additionally, background motion leaks into one or several detected keypoints. Our PCA-based regions (with and without background motion) correctly identify meaningful parts, are consistent between frames, and use additional regions more effectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 REGIONS AND COARSE MOTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Regions FOMM learns to detect $K$ distinct object regions, where $K$ is a user-defined parameter. An encoder-decoder region predictor network takes an image as input, and outputs $K$ heatmaps, $\\mathbf { M } ^ { 1 } , . . , \\mathbf { M } ^ { K }$ . The final network layer is a softmax operation, s.t. $\\mathbf { M } ^ { k } \\in [ 0 , 1 ] ^ { H \\times W }$ , where $H$ and $W$ are the height and width of the image respectively, and $\\begin{array} { r } { \\sum _ { z \\in \\mathcal { Z } } m _ { z } ^ { k } = 1 } \\end{array}$ , where $z$ is a pixel location $\\mathbf { \\dot { x } }$ , y coordinates) in the image, the set of all pixel locations being $\\mathcal { Z }$ , and $m _ { z } ^ { k }$ is the $k$ -th heatmap weight at pixel $z$ . We use the same region representation and encoder here. Nevertheless, the encoded regions differ significantly (see Fig. 3), ours mapping to meaningful object parts such as the limbs of an articulated body, due to our novel foreground motion representation, described below. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Estimating foreground region motion FOMM estimates a first-order transformation from an image $\\mathbf { X }$ to a reference frame $\\mathbf { R }$ , for each region separately. The region heatmap encodes translation by its mean position, while other affine parameters are regressed per pixel and then pooled per region according to the heatmap weights. Here we change the way this transformation is represented: all motion is measured directly from the heatmap. Translation is given by the mean position, as before, while in-plane rotation and scaling in $\\mathbf { X } ^ { - }$ and y-directions are computed via a principal component analysis (PCA) of the heatmap. Shear is not captured, therefore our transform isn’t fully affine, with only five degrees of freedom instead of six. Nevertheless, it captures sufficient motion, shear being a less significant component of the affine transform for this task. The transformation of a region from the reference frame to the image is computed as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/482203bea21f82fbc48d8816585b8435112a58af80968b1019804251f6b9e860.jpg",
+        "text": "$$\n{ \\begin{array} { r l } & { \\qquad \\mu ^ { k } = \\displaystyle \\sum _ { z \\in { \\mathcal { Z } } } m _ { z } ^ { k } z , } \\\\ & { \\qquad U ^ { k } S ^ { k } { V ^ { k } } ^ { \\mathsf { T } } = \\displaystyle \\sum _ { z \\in { \\mathcal { Z } } } m _ { z } ^ { k } ( z - \\mu ^ { k } ) ( z - \\mu ^ { k } ) ^ { \\mathsf { T } } , ~ \\mathrm { ( v i a S V D ) } , } \\\\ & { \\qquad A _ { \\mathbf { X }  \\mathbf { R } } = \\displaystyle [ U ^ { k } S ^ { k ^ { \\frac { 1 } { 2 } } } , \\mu ^ { k } ] . } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The singular value decomposition (SVD) approach to computing PCA (Wall et al., 2003) is used here. We refer to FOMM and our estimation approaches as regression-based and PCA-based, respectively. The reference frame in both is used only as an abstract, intermediate coordinate frame between the source and driving image coordinate frames. However, here (in contrast to FOMM) it is not in fact abstract, corresponding to the coordinate frame where the heatmap is whitened (i.e. has zero mean and identity covariance); see Fig. 2. Driving to source image motion is then ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fc05c47c12c5df07a56edb9d11d6377abe21a20341ba7356591bd10ff000d618.jpg",
+        "text": "$$\n\\begin{array} { r } { A _ { \\mathbf { S }  \\mathbf { D } } ^ { k } = A _ { \\mathbf { S }  \\mathbf { R } } ^ { k } [ A _ { \\mathbf { D }  \\mathbf { R } } ^ { k } ] ^ { - 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Estimating background motion FOMM has no background motion model. We observe that with significant background motion between frames, e.g. due to camera motion, predicted regions can therefore include the moving background, reducing test-time accuracy. To resolve this, we additionally regress a background homography transformation, $\\mathbf { H }$ , using an encoder network that takes as input the source and driving images, concatenated along the channel dimension, and outputs eight real values, $h _ { 1 } , . . , h _ { 8 }$ , such that $\\bar { { \\bf H } } = \\left[ [ h _ { 1 } , h _ { 2 } , h _ { 3 } ] ^ { \\top } \\left[ h _ { 4 } , h _ { 5 } , h _ { 6 } \\right] ^ { \\top } [ h _ { 7 } , h _ { 8 } , 1 ] ^ { \\top } \\right]$ . With background motion well modeled, we show that the network is able to separate background and object motion in a completely unsupervised manner. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1.2 IMAGE GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            378,
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+            393
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given these coarse motions, FOMM then renders the target image in two stages: a pixel-wise flow generator converts coarse motions to dense optical flow, then a composition network warps the source image according to the flow, and also inpaints missing regions. We follow this architecture, and summarize these two modules here, but refer the reader to Siarohin et al. (2019a) for the full details. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Pixel-wise flow generation Coarse motions are combined via a weighted sum, to compute a dense, per pixel motion, or flow. The per pixel weights, as well as a confidence map, are computed via an encoder-decoder network. The input is a $H \\times W \\times ( 4 K + 3 )$ tensor, with four channels per region, three for the source image warped according to the region’s motion model, and one for a heatmap of the region, which is a gaussian approximation to ${ \\bf { M } } ^ { \\breve { k } }$ , in order to avoid leakage of driving image appearance through the heatmap. Here we add a further three input channels (compared to FOMM) for the source image warped according to the background motion model. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Warping and inpainting The source image is passed through an encoder network. The resulting feature map is warped and masked according to the pixel-wise flow and confidence map from the previous module, respectively. A decoder then renders the final image, inpainting missing parts. In contrast to FOMM, but similar to Monkey-Net (Siarohin et al., 2019b), here skip connections are used between the encoder and decoder. The skip connection feature maps are also warped and masked. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3.2 TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The proposed model is trained end-to-end using a reconstruction loss in the feature space of the pretrained VGG-19 network (Johnson et al., 2016; Wang et al., 2017). Following Siarohin et al. (2019a); Wang et al. (2003), we adopt a multi-resolution version of the reconstruction loss: ",
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+        "type": "equation",
+        "img_path": "images/641f9936a81da952994eba1cbdcf2698cd01ad4570c77466e55d4cf095608f1e.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { r e c } } ( \\hat { \\mathbf { D } } , \\mathbf { D } ) = \\sum _ { l } \\sum _ { i } \\left| \\mathrm { V } _ { i } \\big ( \\mathrm { F } _ { l } \\odot \\hat { \\mathbf { D } } \\big ) - \\mathrm { V } _ { i } \\big ( \\mathrm { F } _ { l } \\odot \\mathbf { D } \\big ) \\right| ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\bf D }$ is the generated image, $\\mathrm { V } _ { i }$ is the $i ^ { \\mathrm { { t h } } }$ -layer of the VGG-19 pretrained network, $\\mathrm { F } _ { l }$ is a downsampling operator. Similarly to Wang et al. (2017); Siarohin et al. (2019a) we used conv1_2, conv2_2, conv3_2, conv4_2, conv5_2 layers and downsampled the images to 1, 0.5, 0.25, 0.125 of the original edge size. In total we have 20 reconstruction terms. To improve detection of unsupervised regions we follow the unsupervised keypoint detection literature (Jakab et al., 2018; Zhang et al., 2018) and adopt the equivariance loss, denoted as $\\mathcal { L } _ { \\mathrm { e q } }$ . We use a thin-plate spline implementation provided in FOMM (Siarohin et al., 2019a). The final loss is a sum of the two loss terms, $\\mathcal { L } = \\mathcal { L } _ { \\mathrm { r e c } } + \\mathcal { L } _ { \\mathrm { e q } }$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/97dc64f529863911ea0d5ff8c5401b09ec59c12b8c5759765a48499e1054109f.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative comparisons. We show representative examples of articulated animation using our method and FOMM (Siarohin et al., 2019a), on two datasets of articulated objects: TED-talks (left) and TaiChiHD (right). Zoom in for greater detail. "
+        ],
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+    {
+        "type": "text",
+        "text": "4 EVALUATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now discuss the datasets, metrics and experiments used to evaluate the proposed method. Later we compare with prior work, as well as ablate our contributions. ",
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+        "type": "text",
+        "text": "4.1 TOY MOTION REPRESENTATION EXPERIMENT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To demonstrate the benefit of the proposed PCAbased motion representation, we devise an experiment on rotated rectangles (see Appendix E): the task is to predict the rotation angle of a rectangle in an image. To fully isolate our contribution, we consider a supervised task, where three different architectures learn to predict angles under the $L _ { 1 }$ loss. The first, a Naive architecture, directly regresses the angle using an encoder-like architecture. The second is Regression-based, as in to FOMM (Siarohin et al., 2019a). The third uses our PCA-based approach (see Appendix E). Test results are presented in Fig. 5, against training set size. The Naive baseline struggles to produce meaningful results for any size of training set, while Regression-based performance improves with more data. However, the PCA-based significantly improves accuracy over the Regression-based one, being over an order of magnitude better with a large number of samples. This shows that it is significantly easier for the network to infer geometric parameters of the image, such as angle, using our proposed PCA-based representation. ",
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+        "img_path": "images/977dd74fcd71d52ec274f24c38abdbbe09bcd9b20a9d9ebb057ddc69537a1d61.jpg",
+        "image_caption": [
+            "Figure 5: Mean test-time absolute rotation error, as a function of training set size. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 BENCHMARKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate our method on several benchmark datasets for animating human faces and bodies. Each dataset has separate training and test videos. The datasets are as follows: ",
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+    {
+        "type": "text",
+        "text": "• VoxCeleb (Nagrani et al., 2017) consists of interview videos of different celebrities. We extract square, face regions and downscale them to $2 5 6 \\times 2 5 6$ , following FOMM (Siarohin et al., 2019a). The number of frames per video ranging from 64 to 1024. • TaiChiHD (Siarohin et al., 2019a) consists of cropped videos of full human bodies performing Tai Chi actions. We evaluate on two resolutions of the dataset: $2 5 6 \\times 2 5 6$ (from FOMM (Siarohin et al., 2019a)), and a new, $5 1 2 \\times 5 1 2$ subset, removing videos lacking sufficient resolution to support that size. • TED-talks is a new dataset, collected for this paper in order to demonstrate the generalization properties of our model. We cropped the upper part of the human body from the videos, downscaling to $3 8 4 \\times 3 8 4$ . The number of frames per video ranges from 64 to 1024. ",
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+    {
+        "type": "table",
+        "img_path": "images/dc9830e3388615dae8b048da58b1e056676c3f69304dead754e451854ce06c34.jpg",
+        "table_caption": [
+            "Table 2: Video reconstruction: comparison with the state of the art on four different datasets. For all methods we use $K = 1 0$ regions. (Best result in bold.) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">TaiChiHD (256)</td><td colspan=\"3\">TaiChiHD (512)</td><td colspan=\"3\">TED-talks</td><td colspan=\"3\">VoxCeleb</td></tr><tr><td></td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>AKD</td><td>AED</td></tr><tr><td>FOMM</td><td>0.056</td><td>(6.53,0.033)</td><td>0.172</td><td>0.075</td><td>(17.12,0.66)</td><td>0.203</td><td>0.033</td><td>(7.07,0.014)</td><td>0.163</td><td>0.041</td><td>1.27</td><td>0.134</td></tr><tr><td>Ours</td><td>0.048</td><td>(5.45,0.028)</td><td>0.152</td><td>0.064</td><td>(14.00,0.44)</td><td>0.171</td><td>0.026</td><td>(4.02,0.007)</td><td>0.119</td><td>0.040</td><td>1.28</td><td>0.133</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Further datasets are used in the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "Since video animation is a relatively new problem, there are not currently many effective ways of evaluating it. For quantitative metrics, prior works (Siarohin et al., 2019b;a) use video reconstruction accuracy as a proxy for image animation quality. We adopt the same metrics here: ",
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+        "type": "text",
+        "text": "• $\\mathcal { L } _ { 1 }$ error measures the difference between reconstructed video and ground-truth video pixel values using the $\\mathcal { L } _ { 1 }$ metric. • Average keypoint distance (AKD) and missing keypoint rate (MKR) evaluate the difference between poses of reconstructed and ground truth video. Landmarks are extracted from both videos using public, body (Cao et al., 2017) (for TaiChiHD and TED-talks) and face (Bulat & Tzimiropoulos, 2017) (for VoxCeleb) detectors. AKD is then the average distance between corresponding landmarks, while MKR is the proportion of landmarks present in the ground-truth that are missing in the reconstructed video. • Average Euclidean distance (AED) evaluates how well identity is preserved in reconstructed video. Public re-identification networks for bodies (Hermans et al., 2017) (for TaiChiHD and TED-talks) and faces (Amos et al., 2016) extract identity from reconstructed and ground truth frame pairs, then we compute the average $\\mathcal { L } _ { 2 }$ norm of their difference across all pairs. ",
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+    {
+        "type": "text",
+        "text": "4.3 COMPARISON WITH THE STATE OF THE ART ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare our method with the current state of the art for unsupervised animation, FOMM (Siarohin et al., 2019a), across all datasets, on both reconstruction (the training task) and animation (the testtime task). We used an extended training schedule compared to Siarohin et al. (2019a), with $50 \\%$ more iterations. To compare fairly with FOMM (Siarohin et al., 2019a), we also re-trained it with the same training schedule. ",
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+    {
+        "type": "text",
+        "text": "Reconstruction quality Quantitative reconstruction results are reported in Tab. 2. We first show that our method reaches state-of-the-art results on a dataset with non-articulated objects such as faces. Indeed, when compared with FOMM (Siarohin et al., 2019a) on VoxCeleb our method shows on-par results. The situation changes, however, when articulated objects are considered, such as human bodies in TaiChiHD and TED-talks datasets, on which our improved motion representations boost all the metrics. The advantage over the state of the art holds at different resolutions, for TaiChiHD (256), TaiChiHD (512) and TED-talks, as well as for different numbers of selected regions (discussed later). ",
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+    {
+        "type": "text",
+        "text": "Animation quality Fig. 3 & 4 show selected and representative animations respectively, using our method and FOMM (Siarohin et al., 2019a), on articulated bodies, both using absolute motion. The results show clear improvements, in most cases, in animation quality, especially of limbs. ",
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+    {
+        "type": "text",
+        "text": "Animation quality was evaluated quantitatively through a user preference study similar to that of Siarohin et al. (2019a). AMT users were presented with the source image, driving video, and the output from our method and FOMM (Siarohin et al., 2019a), and asked which of the two videos they preferred. 50 such videos were evaluated, by 50 users each, for a total of 2500 preferences per study. The results, shown in Tab. 4, further support the reconstruction scores in Tab. 2. When the animated object is not articulated (VoxCeleb), the method delivers results comparable to the previous work. When bodies are animated (TaiChiHD & TED-talks), FOMM (Siarohin et al., 2019a) fails to correctly detect and animate the articulated body parts such as hands. Our method renders them in the driving pose even for extreme cases, leading to a high preference in favor of it. ",
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+    {
+        "type": "text",
+        "text": "Finally, we applied animation from a TED-talks video to a photograph of Winston Churchill, shown in Fig. 1, demonstrating animation of out of domain data. ",
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+    {
+        "type": "table",
+        "img_path": "images/4bbba204ab5dbee8cbc4eff9f7bf4d3b4c20c6043b4812a9d93d8ffdb22e630b.jpg",
+        "table_caption": [
+            "Table 3: Ablation study on TaiChiHD (256) dataset with $K = 1 0$ . (Best result in bold.) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>L1</td><td>(AKD,MKR)</td><td>AED</td></tr><tr><td>No pca or bg model</td><td>0.060</td><td>(6.14, 0.033)</td><td>0.163</td></tr><tr><td>No pca</td><td>0.049</td><td>(6.04,0.034)</td><td>0.163</td></tr><tr><td>No bg model</td><td>0.059</td><td>(5.47, 0.027)</td><td>0.164</td></tr><tr><td>Full method</td><td>0.048</td><td>(5.45, 0.028)</td><td>0.152</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a2c4c903b52d3ce69b5e408b55d67a5bbcea026830995feb70144122c0a1f0c8.jpg",
+        "table_caption": [
+            "Table 4: User study: the proportion $( \\% )$ of users that prefer our method over FOMM (Siarohin et al., 2019a). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>User preference (%)</td></tr><tr><td>VoxCeleb</td><td>52.2%</td></tr><tr><td>TaiChiHD (256)</td><td>83.0%</td></tr><tr><td>TED-talks</td><td>91.0%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.4 ABLATIONS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In order to understand how much benefit each of our contributions bring, we ran a number of ablation experiments, detailed in Tab. 3. ",
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+    {
+        "type": "text",
+        "text": "PCA-based vs. regression-based representations First we compare the PCA-based motion model with the previous, regression-based one (Siarohin et al., 2019a). From the qualitative, heatmap depictions in Fig. 3, we observe that the regression-based method localizes one edge of each corresponding part, while our method predicts regions that roughly correspond to the segmentation of the object into its constituent, articulated parts. This meaningful segmentation arises completely unsupervised. ",
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+    },
+    {
+        "type": "text",
+        "text": "From Tab. 1 we note that adding the PCA-based representation alone (second row) had marginal impact on the $\\mathcal { L } _ { 1 }$ score (dominated by the much larger background region), but it had a much larger impact on other metrics, which are more sensitive to object-part-related errors on articulated objects. This is corroborated by Tab. 3. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We intuit that PCA-based estimation both captures regions and improves performance because it is much easier for the convolutional network to assign pixels of an object part to the corresponding heatmap than to directly regress motion parameters to an abstract reference frame. This is borne out by our toy experiment (sec. 4.1). In order to estimate the heatmap it need only learn all appearances of the corresponding object part, whereas regression-based networks must learn the joint space of all appearances of a part in all possible geometric configurations (e.g. rotated, scaled etc.). ",
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+    {
+        "type": "text",
+        "text": "One of the most important hyper-parameters of our model is the number of regions, $K$ . The qualitative and quantitative ablations of this parameter are shown in Fig. 3 and Tab. 1 respectively. We can observe that, while the regression-based representation fails when the number of keypoints grows to 20, our PCA-based representation scales well with the number of regions. ",
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+    },
+    {
+        "type": "text",
+        "text": "Modeling background motion Tab. 3 shows that methods with background motion modeling have much lower $\\mathcal { L } _ { 1 }$ error. Since background constitutes a large portion of the image, and $\\mathcal { L } _ { 1 }$ treats all pixels equally, this is to be expected. AED was also impacted, suggesting that the identity representation captures some background appearance. However, since AKD & MKR metrics evaluate object pose only, they are not improved by background modelling. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We have argued that previous unsupervised animation frameworks’ poor results on articulated objects are due to their representations. We propose a new, PCA-based, region motion representation, which we believe both makes it easier for the network to learn region motion, and encourages it to learn semantically meaningful object parts. In addition, we propose a background motion estimation module to decouple foreground and background motion. Qualitative and quantitative results across a range of datasets and tasks demonstrate several key benefits: improved region distribution and stability, improved reconstruction accuracy and user perceived quality, and an ability to scale to more regions. We also introduce a new, more challenging dataset, TED-talks, for benchmarking future improvements on this task. ",
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+    },
+    {
+        "type": "text",
+        "text": "While we show some results on out of domain data (Fig. 1), generalization remains a significant challenge to making this method broadly practical in articulated animation of inanimate objects. ",
+        "bbox": [
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+    {
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+        "type": "text",
+        "text": "A TED-TALKS DATASET CREATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To create the TED-talks dataset, we downloaded 3,035 YouTube videos, shared under the “CC BY – $\\mathrm { N C } - \\mathrm { N D } 4 . 0$ International” license,3 using the \"TED talks\" query. From these initial candidates, we selected the videos where the upper part of the person is visible for at least 64 frames, and the height of the person bounding box was at least 384 pixels. After that, we manually filtered out static videos and videos in which a person is doing something other than presenting. We ended up with 411 videos, and split these videos in 369 training and 42 testing videos. We then split each video into chunks from a consistent camera angle (i.e. with no cuts to another camera), and for which the presenter didn’t move too far from their starting position in the chunk. We cropped the a square region around the presenter, such that they had a consistent scale, and downscaled this region to $3 8 4 \\times 3 8 4$ pixels. Chunks that lacked sufficient resolution to be downscaled, or had a length shorter than 64 frames, were removed. Both the distance moved and the region cropping were achieved using a bounding box estimator for humans (Wu et al., 2019). Overall, we obtained 1,177 training video chunks and 145 test videos chunks. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For a fair comparison, in order to highlight our contributions, we mostly follow the architecture design of FOMM (Siarohin et al., 2019a). Similar to FOMM, our region predictor, background motion predictor and pixel-wise flow predictor operate on a quarter of the original resolution, e.g. $6 4 \\times 6 4$ for $2 5 6 \\times 2 5 6$ images, $9 6 \\times 9 6$ for $3 8 4 \\times 3 8 4$ and $1 2 8 \\times 1 2 8$ for $5 1 2 \\times 5 1 2$ . We use the U-Net (Ronneberger et al., 2015) architecture with five \"convolution - batch norm - ReLU - pooling\" blocks in the encoder and five \"upsample - convolution - batch norm - ReLU\" blocks in the decoder for both the region predictor and the pixel-wise flow predictor. For the background motion predictor, we use only the five block encoder part. Similarly to FOMM (Siarohin et al., 2019a), we use the Johnson architecture (Johnson et al., 2016) for image generation, with two down-sampling blocks, six residual-blocks, and two up-sampling blocks. However, we add skip connections that are warped and weighted by the confidence map. Our method is trained using Adam (Kingma & Ba, 2014) optimizer with learning rate $2 e - 4$ and batch size 48, 20, 12 for $2 5 6 \\times 2 5 6$ , $3 8 4 \\times 3 8 4$ and $5 1 2 \\times 5 1 2$ resolutions respectively. During the training process, the networks observe 3M source-driving pairs, each pair selected at random from a random video chunk, and we drop the learning rate by a factor of 10 after 1.8M and 2.7M pairs. We use 4 Nvidia P100 GPUs for training. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C MGIF DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            626,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We run additional experiments on the MGif (Siarohin et al., 2019b) dataset to further demonstrate the superiority of PCA-based representations over regression-based ones. The dataset contains a set of animations of articulated, 2D, cartoon animals. The qualitative results are presented in the supplementary video. We can observe that our PCA-based representation successfully tracks all legs, while the regression-based representation often misses some of the legs, which leads to worse reconstruction quality. This observation is further confirmed by quantitative evaluation; the $\\mathcal { L } _ { 1 }$ error for FOMM (Siarohin et al., 2019a) is 0.0223, while for our method it is 0.0206. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D COMPARISON WITH OTHER METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The main paper has focused on comparing our method to FOMM (Siarohin et al., 2019a), as it is both most similar to our work, and the current state-of-the-art. We show quantitative results using prior works (Wiles et al., 2018; Siarohin et al., 2019b) in Table 5. These are significantly inferior to both FOMM and our method. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/797f14c57fdeaae1464f246e27a2e71f41fffe2bf3342273320d0ab8f2cb566d.jpg",
+        "table_caption": [
+            "Table 5: Video reconstruction comparison. (Best result in bold.) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">TaiChiHD (256)</td><td colspan=\"3\">VoxCeleb</td></tr><tr><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>L1</td><td>AKD</td><td>AED</td></tr><tr><td>X2Face</td><td>0.080</td><td>(17.65, 0.109)</td><td>0.27</td><td>0.078</td><td>7.69</td><td>0.405</td></tr><tr><td>Monkey-Net</td><td>0.077</td><td>(10.80, 0.059)</td><td>0.228</td><td>0.049</td><td>1.89</td><td>0.199</td></tr><tr><td>FOMM</td><td>0.056</td><td>(6.53, 0.033)</td><td>0.172</td><td>0.041</td><td>1.27</td><td>0.134</td></tr><tr><td>Ours</td><td>0.048</td><td>(5.45, 0.028)</td><td>0.152</td><td>0.040</td><td>1.28</td><td>0.133</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E TOY EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The rotated rectangles dataset consists of images of rectangles randomly rotated from $0 ^ { \\circ }$ to $9 0 °$ , along with labels that indicate the angle of rotation. The rectangles have different, random colors. Visual samples are shown in Fig. 6. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/77aea8e7c834470bac00866386bc4f7458ce91a18ec5c52d79bbcaf32aa67e40.jpg",
+        "image_caption": [
+            "Figure 6: Examples of synthetic rectangle dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We tested three different networks: Naive, Regression-based and PCA-based. The Naive network directly predicts an angle from an image using an encoder and a fully-connected layer. Regressionbased is similar to FOMM (Siarohin et al., 2019a); the angle is regressed per pixel an using hourglass network, and pooled according to heatmap weights predicted using the same hourglass network. PCAbased is our method described in Sec. 3; we predict the heatmap using an hourglass network, PCA is performed according to eq. equation 2, and the angle is computed from matrix $U$ as arctan $\\left( U _ { 1 0 } / U _ { 0 0 } \\right)$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Each of the networks was trained, on subsets of the dataset of varying sizes, to minimize the $\\mathcal { L } _ { 1 }$ loss between predicted and ground truth rotation angle. All models were trained for 100 epochs, with batch size 8. We used the Adam optimizer, with a learning rate of $1 0 ^ { - 4 }$ . We varied the size of the training set from 32 to 1024. Results, on a separate, fixed test set of size 128, were then computed, shown in Fig. 5. ",
+        "bbox": [
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+        "page_idx": 12
+    }
+]