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+# PROSELFLC: PROGRESSIVE SELF LABEL CORRECTION FOR TRAINING ROBUST DEEP NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+To train robust deep neural networks (DNNs), we systematically study several target modification approaches, which include output regularisation, self and nonself label correction (LC). Two key issues are discovered: (1) Self LC is the most appealing as it exploits its own knowledge and requires no extra models. However, how to automatically decide the trust degree of a learner as training goes is not well answered in the literature? (2) Some methods penalise while the others reward low-entropy predictions, prompting us to ask which one is better?
+
+To resolve the first issue, taking two well-accepted propositions–deep neural networks learn meaningful patterns before fitting noise (Arpit et al., 2017) and minimum entropy regularisation principle (Grandvalet & Bengio, 2006)–we propose a novel end-to-end method named ProSelfLC, which is designed according to learning time and entropy. Specifically, given a data point, we progressively increase trust in its predicted label distribution versus its annotated one if a model has been trained for enough time and the prediction is of low entropy (high confidence). For the second issue, according to ProSelfLC, we empirically prove that it is better to redefine a meaningful low-entropy status and optimise the learner toward it. This serves as a defence of entropy minimisation. We demonstrate the effectiveness of ProSelfLC through extensive experiments in both clean and noisy settings.
+
+# 1 INTRODUCTION
+
+There exist many target (label) modification approaches. They can be roughly divided into two groups: (1) Output regularisation (OR), which is proposed to penalise overconfident predictions for regularising deep neural networks. It includes label smoothing (LS) (Szegedy et al., 2016; Müller et al., 2019) and confidence penalty (CP) (Pereyra et al., 2017); (2) Label correction (LC). On the one hand, LC regularises neural networks by adding the similarity structure information over training classes into one-hot label distributions so that the learning targets become structured and soft. On the other hand, it can correct the semantic classes of noisy label distributions. LC can be further divided into two subgroups: Non-self LC and Self LC. The former requires extra learners, while the latter relies on the model itself. A typical approach of Non-self LC is knowledge distillation (KD), which exploits the predictions of other model(s), usually termed teacher(s) (Hinton et al., 2015). Self LC methods include Pseudo-Label (Lee, 2013), bootstrapping (Boot-soft and Boot-hard) (Reed et al., 2015), Joint Optimisation (Joint-soft and Joint-hard) (Tanaka et al., 2018), and Tf- ${ \mathrm { K D } } _ { s e l f }$ (Yuan et al., 2020). According to an overview in Figure 1 (detailed derivation is in Section 3 and Table 1), in label modification, the output target of a data point is defined by combining a one-hot label distribution and its corresponding prediction or a predefined label distribution.
+
+Firstly, we present the drawbacks of existing approaches: (1) OR methods naively penalise confident outputs without leveraging easily accessible knowledge from other learners or itself (Figure 1a); (2) Non-self LC relies on accurate auxiliary models to generate predictions (Figure 1b). (3) Self LC is the most appealing because it exploits its own knowledge and requires no extra learners. However, there is a core question that is not well answered:
+
+# In Self LC, how much should we trust a learner to leverage its knowledge?
+
+As shown in Figure 1b, in Self LC, for a data point, we have two labels: a predefined one-hot q and a predicted structured p. Its learning target is $( 1 - \epsilon ) \mathbf { q } + \epsilon \mathbf { p }$ , i.e., a trade-off between q and p, where $\epsilon$ defines the trust score of a learner. In existing methods, $\epsilon$ is fixed without considering that a model’s (a) OR includes LS (Szegedy et al., 2016) and CP (Pereyra et al., 2017). LS softens a target by adding a uniform label distribution. CP changes the probability 1 to a smaller value $1 - \epsilon$ in the one-hot target. The double-ended arrow means factual equivalence, because an output is definitely non-negative after a softmax layer.
+
+![](images/4ec618d424c008219333fd3aad931860b7ccc3edfad166d77494cd98dd80172f.jpg)
+
+![](images/667be94f42736a8518c158bc6b48a8079ddbc87fa14a14e0db24550b250a7407.jpg)
+
+(b) LC contains Self LC (Lee, 2013; Reed et al., 2015; Tanaka et al., 2018; Yuan et al., 2020) and Non-self LC (Hinton et al., 2015). The parameter  defines how much a predicted label distribution is trusted.
+
+Figure 1: Target modification includes OR (LS and CP), and LC (Self LC and Non-self LC). Assume there are three training classes. q is the one-hot target. u is a uniform label distribution. p denotes a predicted label distribution. The target combination parameter is $\epsilon \in [ 0 , 1 ]$ .
+
+knowledge grows as the training progresses. For example, in bootstrapping, $\epsilon$ is fixed throughout the training process. Joint Optimisation stage-wisely trains a model. It fully trusts predicted labels and uses them to replace old ones when a stage ends, i.e., $\epsilon = 1$ . Tf- ${ \mathrm { K D } } _ { s e l f }$ trains a model by two stages: $\epsilon = 0$ in the first one while $\epsilon$ is tuned for the second stage. Note that $\mathbf { p }$ is generated by a preceding-stage model in stage-wise training, which requires significant human intervention and is time-consuming in practice.
+
+To improve Self LC, we propose a novel method named Progressive Self Label Correction (ProSelfLC), which is end-to-end trainable and needs negligible extra cost. Most importantly, ProSelfLC modifies the target progressively and adaptively as training goes. Two design principles of ProSelfLC are: (1) When a model learns from scratch, human annotations are more reliable than its own predictions in the early phase, during which the model is learning simple meaningful patterns before fitting noise, even when severe label noise exists in human annotations (Arpit et al., 2017). (2) As a learner attains confident knowledge as time progresses, we leverage it to revise annotated labels. This is surrounded by minimum entropy regularisation, which is widely evaluated in unsupervised and semi-supervised scenarios (Grandvalet & Bengio, 2005; 2006).
+
+Secondly, note that OR methods penalise low entropy while LC rewards it, intuitively leading to a second vital question:
+
+Should we penalise a low-entropy status or reward it?
+
+Entropy minimisation is the most widely used principle in machine learning (Hartigan & Wong, 1979; Rumelhart et al., 1986; Grandvalet & Bengio, 2005; 2006; LeCun et al., 2015). In standard classification, minimising categorical cross entropy (CCE) optimises a model towards a low-entropy status defined by human annotations, which contain noise in very large-scale machine learning. As a result, confidence penalty becomes popular for reducing noisy fitting. In contrast, we prove that it is better to reward a meaningful low-entropy status redefined by our ProSelfLC. Therefore, our work offers a defence of entropy minimisation against the recent confidence penalty practice (Szegedy et al., 2016; Müller et al., 2019; Pereyra et al., 2017; Dubey et al., 2018).
+
+Finally, we summarise our main contributions:
+
+• We provide a theoretical study on popular target modification methods through entropy and KL divergence (Kullback & Leibler, 1951). Accordingly, we reveal their drawbacks and propose ProSelfLC as a solution. ProSelfLC can: (1) enhance the similarity structure information over training classes; (2) correct the semantic classes of noisy label distributions. ProSelfLC is the first method to trust self knowledge progressively and adaptively. Our extensive experiments: (1) defend the entropy minimisation principle; (2) demonstrate the effectiveness of ProSelfLC in both clean and noisy settings.
+
+# 2 RELATED WORK
+
+Label noise and semi-supervised learning. We test target modification approaches in the setting of label noise because it is generic and connected with semi-supervised learning, where only a subset of training examples are annotated, leading to missing labels. Then the key to semi-supervised training is to reliably fill them. When these missing labels are incorrectly filled, the challenge of semisupervised learning changes to noisy labels. For a further comparison, in semi-supervised learning, the annotated set is clean and reliable, because the label noise only exists in the unannotated set. While in our experimental setting, we are not given information on whether an example is trusted or not, thus being even more challenging. We summarise existing approaches for solving label noise: (1) Loss correction, in which we are given or we need to estimate a noise-transition matrix, which defines the distribution of noise labels (Li et al., 2017; Goldberger & Ben-Reuven, 2017; Sukhbaatar & Fergus, 2014; Vahdat, 2017; Yao et al., 2019; Han et al., 2018a; Patrini et al., 2017; Xiao et al., 2015). A noise-transition matrix is difficult and complex to estimate in practice; (2) Exploiting an auxiliary trusted training set to differentiate examples (Veit et al., 2017; Lee et al., 2018; Hendrycks et al., 2018). This requires extra annotation cost; (3) Co-training strategies, which train two or more learners (Malach & Shalev-Shwartz, 2017; Jiang et al., 2018; Han et al., 2018b; Yu et al., 2019; Wei et al., 2020; Qiao et al., 2018) and exploit their ‘disagreement’ information to differentiate data points; (4) Label engineering methods (Song et al., 2019; Lee, 2013; Reed et al., 2015; Tanaka et al., 2018; Yao et al., 2019), which relate to our focus in this work. Their strategy is to annotate unlabelled samples or correct noisy labels.
+
+LC and knowledge distillation $( K D )$ (Bucila et al., 2006; Hinton et al., 2015). Mathematically, we derive that some KD methods also modify labels. We use the term label correction instead of KD for two reasons: (1) label correction is more descriptive; (2) the scope of KD is not limited to label modification. For example, multiple networks are trained for KD (Furlanello et al., 2018). When two models are trained, the consistency between their predictions of a data point is promoted in (Ba & Caruana, 2014; Zhang et al., 2018), while the distance between their feature maps is reduced in (Romero et al., 2015). Regarding self KD, two examples of the same class are constrained to have consistent output distributions (Xu & Liu, 2019; Yun et al., 2020). In another self KD (Zhang et al., 2019), the deepest classifier provides knowledge for shallower classifiers. In a recent self KD method (Yuan et al., 2020), Tf- $\mathrm { K D } _ { s e l f }$ applies two-stage training. In the second stage, a model is trained by exploiting its knowledge learned in the first stage. Our focus is to improve the end-to-end self LC. Finally, we acknowledge that exploiting ProSelfLC to improve non-self KD and stage-wise approaches is an area for future work, e.g., a better teacher model can be trained using ProSelfLC.
+
+# 3 MATHEMATICAL ANALYSIS AND THEORY
+
+Let $\mathbf { X } = \{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ represent $N$ training examples, where $\left( \mathbf { x } _ { i } , y _ { i } \right)$ denotes $i -$ th sample with input $\mathbf { x } _ { i } \in \mathbb { R } ^ { D }$ and label $y _ { i } \in \{ 1 , 2 , . . . , C \}$ . $C$ is the number of classes. A deep neural network $z$ consists of an embedding network $f ( \cdot ) : \bar { \mathbb { R } } ^ { D }  \mathbb { R } ^ { K }$ and a linear classifier $\begin{array} { r } { g ( \cdot ) \dot { \mathbf { \Psi } } : \mathbb { R } ^ { K }  \mathbb { R } ^ { C } } \end{array}$ , i.e., $\mathbf { z } _ { i } = z ( \mathbf { x } _ { i } ) = g ( f ( \mathbf { x } _ { i } ) ) ^ { \mathit { \prime } } : \mathbb { R } ^ { D }  \mathbb { R } ^ { C }$ . For the brevity of analysis, we take a data point and omit its subscript so that it is denoted by $\left( \mathbf { x } , y \right)$ . The linear classifier is usually the last fully-connected layer. Its output is named logit vector $\mathbf { z } \in \mathbb { R } ^ { C }$ . We produce its classification probabilities $\mathbf { p }$ by normalising the logits using a softmax function:
+
+$$
+\mathbf { p } ( j | \mathbf { x } ) = \exp ( \mathbf { z } _ { j } ) / { \sum _ { m = 1 } ^ { C } \exp ( \mathbf { z } _ { m } ) } ,
+$$
+
+where $\mathbf { p } ( j | \mathbf { x } )$ is the probability of $\mathbf { x }$ belonging to class $j$ . Its corresponding ground-truth is usually denoted by a one-hot representation q: $\mathbf { q } ( j | \mathbf { x } ) = 1$ if $j = y$ , $\mathbf { q } ( j | \mathbf { x } ) = 0$ otherwise.
+
+# 3.1 SEMANTIC CLASS AND SIMILARITY STRUCTURE IN A LABEL DISTRIBUTION
+
+Definition 1 (Semantic Class). Given a target label distribution $\tilde { \mathbf { q } } ( \mathbf { x } ) \in \mathbb { R } ^ { C }$ , the semantic class is defined by arg $\operatorname* { m a x } _ { j } { \tilde { \mathbf { q } } } ( j | \mathbf { x } )$ , i.e., the class whose probability is the largest.
+
+Definition 2 (Similarity Structure). In $\tilde { \bf q } ( { \bf x } )$ , $\mathbf { x }$ has $C$ probabilities of being predicted to $C$ classes.   
+The similarity structure of $\mathbf { x }$ versus $C$ classes is defined by these probabilities and their differences.
+
+Table 1: Summary of CCE, LS, CP and LC.   
+
+<table><tr><td></td><td>CCE</td><td>LS</td><td>CP</td><td>LC</td></tr><tr><td>Learning Target</td><td>q</td><td>qLs = (1- ε)q +εu</td><td>qcp = (1- e)q -ep</td><td>qLc = (1 - ɕ)q +ep</td></tr><tr><td>Cross Entropy</td><td>Eq(-log_P)</td><td>EaLs(-log_P) (1-e)KL(qlIp)</td><td>Eacp(-log P) (1-c)KL(qllp)</td><td>EqLc(-log P) (1-c)KL(qlIp)</td></tr><tr><td>KL Divergence</td><td>KL(qllp)</td><td>+eKL(ullp)</td><td>+eKL(pllu)</td><td>-eKL(pllu)</td></tr><tr><td>Entropy minimisation</td><td></td><td>Penalise over CCE</td><td>Penalise over CCE</td><td>Reward over CCE</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Semantic class</td><td>Annotated</td><td>Annotated</td><td>Annotated</td><td>Annotated andLearned</td></tr><tr><td>Similarity structure</td><td>No</td><td>No</td><td>No</td><td>Yes</td></tr></table>
+
+# 3.2 REVISIT OF CCE, LS, CP AND LC
+
+Standard CCE. For any input $\left( \mathbf { x } , y \right)$ , the minimisation objective of standard CCE is:
+
+$$
+L _ { \mathrm { C C E } } ( \mathbf { q } , \mathbf { p } ) = \mathrm { H } ( \mathbf { q } , \mathbf { p } ) = \mathrm { E } _ { \mathbf { q } } ( - \log \ \mathbf { p } ) ,
+$$
+
+where $\operatorname { H } ( \cdot , \cdot )$ represents the cross entropy. $\operatorname { E } _ { \mathbf { q } } ( - \log \mathbf { \delta p } )$ denotes the expectation of negative loglikelihood, and $\mathbf { q }$ is the probability mass function.
+
+Label smoothing. In LS (Szegedy et al., 2016; Hinton et al., 2015), we soften one-hot targets by adding a uniform distribution: $\tilde { \bf q } _ { \mathrm { L S } } = ( 1 - \epsilon ) { \bf q } + \epsilon { \bf u }$ , $\mathbf { u } \in \mathbb { R } ^ { C }$ , and $\begin{array} { r } { \forall j , \mathbf { u } _ { j } = \frac { 1 } { C } } \end{array}$ . Consequently:
+
+$$
+L _ { \mathrm { C C E + L S } } ( \mathbf { q } , \mathbf { p } ; \epsilon ) = \mathrm { H } ( \tilde { \mathbf { q } } _ { \mathrm { L S } } , \mathbf { p } ) = \mathrm { E } _ { \tilde { \mathbf { q } } _ { \mathrm { L S } } } ( - \log \mathbf { \delta p } ) = ( 1 - \epsilon ) \mathrm { H } ( \mathbf { q } , \mathbf { p } ) + \epsilon \mathrm { H } ( \mathbf { u } , \mathbf { p } ) .
+$$
+
+Confidence penalty. CP (Pereyra et al., 2017) penalises highly confident predictions:
+
+$$
+L _ { \mathrm { C C E + C P } } ( \mathbf { q } , \mathbf { p } ; \epsilon ) = ( 1 - \epsilon ) \mathrm { H } ( \mathbf { q } , \mathbf { p } ) - \epsilon \mathrm { H } ( \mathbf { p } , \mathbf { p } ) .
+$$
+
+Label correction. As illustrated in Figure 1, LC is a family of algorithms, where a one-hot label distribution is modified to a convex combination of itself and a predicted distribution:
+
+$$
+\tilde { \bf q } _ { \mathrm { I C } } = ( 1 - \epsilon ) { \bf q } + \epsilon { \bf p } \Rightarrow L _ { \mathrm { C C E + L C } } ( { \bf q } , { \bf p } ; \epsilon ) = \mathrm { H } ( \tilde { \bf q } _ { \mathrm { L C } } , { \bf p } ) = ( 1 - \epsilon ) \mathrm { H } ( { \bf q } , { \bf p } ) + \epsilon \mathrm { H } ( { \bf p } , { \bf p } ) .
+$$
+
+We remark: (1) $\mathbf { p }$ provides meaningful information about an example’s relative probabilities of being different training classes; (2) If $\epsilon$ is large, and $\mathbf { p }$ is confident in predicting a different class, i.e., a $\begin{array} { r } { \arg \operatorname* { m a x } _ { j } \mathbf { p } ( j | \mathbf { x } ) \neq \arg \operatorname* { m a x } _ { j } \mathbf { q } ( j | \mathbf { x } ) , } \end{array}$ $\tilde { \bf q } _ { \mathrm { L C } }$ defines a different semantic class from $\mathbf { q }$ .
+
+# 3.3 THEORY
+
+Proposition 1. LS, CP and $L C$ modify the learning targets of standard CCE
+
+Proof. $L _ { \mathrm { C C E + C P } } ( \mathbf { q } , \mathbf { p } ; \epsilon ) = ( 1 - \epsilon ) \mathrm { H } ( \mathbf { q } , \mathbf { p } ) - \epsilon \mathrm { H } ( \mathbf { p } , \mathbf { p } ) = \mathrm { E } _ { ( \mathbf { 1 } - \epsilon ) \mathbf { q } - \epsilon \mathbf { p } } ( - \log \mathbf { \ p } )$ . Therefore, $\mathbf { \tilde { q } } _ { \mathrm { C P } } =$ $( 1 - \epsilon ) \mathbf { q } { - } \epsilon \mathbf { p }$ . Additionally, $\tilde { \mathbf { q } } _ { \mathrm { L S } } = ( 1 - \epsilon ) \mathbf { q } + \epsilon \mathbf { u }$ , $\tilde { \bf q } _ { \mathrm { L C } } = ( 1 - \epsilon ) { \bf q } + \epsilon { \bf p }$ . 
+
+Proposition 2. Some $K D$ methods, which aim to minimise the KL divergence between predictions of a teacher and a student, belong to the family of label correction.
+
+Proof. In general, a loss function of such methods can be defined to be $L _ { \mathrm { K D } } ( \mathbf { q } , \mathbf { p } _ { t } , \mathbf { p } ) = ( 1 -$ $\epsilon ) \mathrm { H } ( \mathbf { q } , \mathbf { p } ) \bar { + } \epsilon \mathrm { K L } ( \mathbf { p } _ { t } | | \mathbf { p } )$ (Yuan et al., 2020). $\operatorname { K L } ( \cdot | | \cdot )$ denotes the KL divergence. As ${ \mathrm { K L } } ( \mathbf { p } _ { t } | | \mathbf { p } ) =$ $\mathrm { H } ( \mathbf { p } _ { t } , \mathbf { p } ) { - } \mathrm { H } ( \mathbf { p } _ { t } , \mathbf { p } _ { t } )$ , $\mathbf { p } _ { t }$ is from a teacher and fixed when training a student. We can omit $\mathrm { H } ( \mathbf { p } _ { t } , \mathbf { p } _ { t } )$ :
+
+$$
+\begin{array} { r } { L _ { \mathrm { K D } } ( \mathbf { q } , \mathbf { p } t , \mathbf { p } ) = ( 1 - \epsilon ) \mathrm { H } ( \mathbf { q } , \mathbf { p } ) + \epsilon \mathrm { H } ( \mathbf { p } t , \mathbf { p } ) = \mathrm { E } _ { ( 1 - \epsilon ) \mathbf { q } + \mathbf { c p } _ { t } } ( - \log \mathbf { p } ) \Rightarrow \tilde { \mathbf { q } } _ { \mathrm { K D } } = ( 1 - \epsilon ) \mathbf { q } + \epsilon \mathbf { p } t . } \end{array}
+$$
+
+Consistent with LC in Eq (5), $L _ { \mathrm { K D } } ( \mathbf { q } , \mathbf { p } _ { t } , \mathbf { p } )$ revises a label using $\mathbf { p } _ { t }$ .
+
+Proposition 3. Compared with CCE, LS and $C P$ penalise entropy minimisation while $L C$ reward it.
+
+Proposition 4. In CCE, $L S$ and $C P ,$ a data point x has the same semantic class. In addition, x has an identical probability of belonging to other classes except for its semantic class.
+
+The proof of propositions 3 and 4 is presented in the Appendix A. Only LC exploits informative information and has the ability to correct labels, while LS and CP only relax the hard targets. We summarise CCE, LS, CP and LC in Table 1. Constant terms are ignored for concision.
+
+# 4 PROSELFLC: PROGRESSIVE AND ADAPTIVE LABEL CORRECTION
+
+In standard CCE, a semantic class is considered while the similarity structure is ignored. It is mainly due to the difficulty of annotating the similarity structure for every data point, especially when $C$ is large ( $\mathrm { { X u } }$ et al., 2020). Fortunately, recent progress demonstrates that there are some effective approaches to define the similarity structure of data points without annotation: (1) In KD, an auxiliary teacher model can provide a student model the similarity structure information (Hinton et al., 2015; Müller et al., 2019); (2) In Self LC, e.g., Boot-soft, a model helps itself by exploiting the knowledge it has learned so far. We focus on studying the end-to-end Self LC.
+
+Table 2: Instantiating ProSelfLC under different cases. For all terms, we use concrete values for concise interpretation. We bold the special case when the semantic class is changed.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">l(p): Consistency is defined by whether p and q share the semantic class or not.</td></tr><tr><td></td><td></td><td>0.1(non-confident）0.9(confidently consistent） 0.9(confidently inconsistent)</td></tr><tr><td>Earlier phase g(t) = 0.1</td><td>0.01</td><td>0.09</td><td>0.09</td></tr><tr><td>Later phase g(t) = 0.9</td><td>0.09</td><td>0.81</td><td>0.81</td></tr></table>
+
+In Self LC, $\epsilon$ indicates how much a predicted label distribution is trusted. In ProSelfLC, we propose to set it automatically according to learning time $t$ and prediction entropy $\mathrm { H } ( \mathbf { p } )$ , i.e., ProSelfLC trusts self knowledge according to training time and confidence. For any $\mathbf { x }$ , we summarise:
+
+Loss: $: L _ { ( \tilde { \bf q } _ { \mathrm { P r o S e l f L C } } , { \bf p } ; \epsilon _ { \mathrm { P r o S e l f L C } } ) } = \mathrm { H } ( \tilde { \bf q } _ { \mathrm { P r o S e l f L C } } , { \bf p } ) = \mathrm { E } _ { \tilde { \bf q } _ { \mathrm { P r o S e l f L C } } } ( - \log { \bf p } ) .$ Label: $\begin{array} { r } { \tilde { \bf q } _ { \mathrm { P r o S e l f L C } } = ( 1 - \epsilon _ { \mathrm { P r o S e l f L C } } ) { \bf q } + \epsilon _ { \mathrm { P r o S e l f L C } } { \bf p } . } \end{array}$ $\mathrm { S e l f ~ t r u s t : } \ \epsilon _ { \mathrm { P r o S e l f L C } } = g ( t ) \times l ( \mathbf { p } ) \left\{ g ( t ) = h ( t / \Gamma - 0 . 5 , B ) \in ( 0 , 1 ) , \right. $
+
+$t$ and $\Gamma$ are the iteration counter and the number of total iterations, respectively. $h ( \eta , B ) ~ =$ $1 / ( 1 + \exp ( - \eta \times B ) ) . B$ $B , \Gamma$ are task-dependent and searched on a validation set. We clarify: Global trust score $g ( t )$ denotes how much we trust a learner. It is independent of data points, thus being global. $g ( t )$ grows as $t$ rises. $B$ adjusts the exponentiation’s base and growth speed of $g ( t )$ . The local trust score $l ( \mathbf { p } )$ indicates how much we trust an output distribution $\mathbf { p }$ , which is datadependent. $l ( \mathbf { p } )$ rises as $\mathrm { H } ( \mathbf { p } )$ becomes lower, rewarding a confident distribution.
+
+Design reason. (1) Regarding $g ( t )$ , in the earlier learning phase, i.e., $t < \Gamma / 2$ , $g ( t ) < 0 . 5 \Rightarrow$ ProSelfLC $< 0 . 5 , \forall \mathbf { p }$ , so that the human annotations dominate and ProSelfLC only modifies the similarity structure. When a learner has not seen the training data for enough time at the earlier stage, its knowledge is less reliable and a wrong confident prediction may occur. Our design assuages the bad impact of such unexpected cases. When it comes to the later training phase, i.e., $t > \Gamma / 2$ , we have $g ( t ) > 0 . 5$ as it has been trained for more than half of entire iterations. (2) Regarding $l ( \mathbf { p } )$ , it affects the later learning phase. If $\mathbf { p }$ is less confident, $l ( \mathbf { p } )$ will be smaller, then ProSelfLC will be smaller, hence we trust $\mathbf { p }$ less when it is of higher uncertainty. If $\mathbf { p }$ is highly confident, we trust its confident knowledge. Ablation study of our design is in Figure 2, where three variants of  are presented. In our experiments, note that when $\epsilon$ is fixed, we try three values (0.125, 0.25, 0.50) and display the best instantiation, i.e., $\epsilon = 0 . 5 0$ .
+
+We conduct the case analysis of ProSelfLC in Table 2 and summarise its core tactics as follows: (1) Correct the similarity structure for every data point in all cases, thanks to exploiting the self knowledge of a learner, i.e., $\mathbf { p }$ . (2) Revise the semantic class when $t$ is large enough and $\mathbf { p }$ is confidently inconsistent. As highlighted in Table 2, when two conditions are met, we have $\epsilon _ { \mathrm { P r o S e l f L C } } > 0 . 5$ and $\operatorname { a r g m a x } _ { j } \mathbf { p } ( j | \mathbf { x } ) \ \neq \ \operatorname { a r g m a x } _ { j } \mathbf { q } ( j | \mathbf { x } )$ , then $\mathbf { p }$ redefines the semantic class. For example, if $\mathbf { p } ~ = ~ [ 0 . 9 5 , 0 . 0 1 , 0 . 0 4 ] , \mathbf { q } ~ = ~ [ 0 , 0 , 1 ] ,$ $\epsilon _ { \mathrm { P r o S e l f L C } } ~ = ~ 0 . 8 ~ \Rightarrow ~ \tilde { \bf q } _ { \mathrm { P r o S e l f L C } } ~ = ~ ( 1 ~ -$  $\mathrm { \Delta \ p o S e l f L C } \mathbf { \Psi } ) \mathbf { q } + \epsilon _ { \mathrm { P r o S e l f L C } } \mathbf { p } = [ 0 . 7 6 , 0 . 0 0 8 , 0 . 2 3 2 ]$ . Note that ProSelfLC also becomes robust against lengthy exposure to the training data, as demonstrated in Figures 2 and 3.
+
+# 5 EXPERIMENTS
+
+In our experiments, we re-implement CCE, LS and CP. Regarding Self LC methods, we reimplement Boot-soft (Reed et al., 2015), where $\epsilon$ is fixed throughout training. We do not reimplement stage-wise Self LC and KD methods, e.g., Joint Optimisation and Tf- $\mathrm { K D } _ { s e l f }$ respectively, because time-consuming tuning is required. We fix the random seed and do not use any random accelerator for an entirely fair comparison. In standard and synthetic cases, we train on $80 \%$ training data (corrupted in synthetic cases) and use $20 \%$ trusted training data as a validation set to search hyperparameters, e.g., $\epsilon , \Gamma , B$ and settings of an optimiser. Note that $\Gamma$ and an optimiser’s settings are searched first and then shared by all methods. Finally, we retrain a model on the entire training data (corrupted in synthetic cases) and report its accuracy on the test data to fairly compare with prior results. In real-world label noise, the used dataset has a separate clean validation set for searching hyperparameters. Code will be released once this work is accepted.
+
+![](images/b82a73bddf1405565e918850ac86e4d14d341ba4f7fe03f94c02e04204a87e30.jpg)  
+Figure 2: Comparison of setting $\epsilon$ using different schemes. Experiments are done on CIFAR-100 with asymmetric label noise $r = 0 . 4$ . For data-dependent items, mean results are reported.
+
+# 5.1 STANDARD IMAGE CLASSIFICATION
+
+Datasets and training details. (1) CIFAR-100 (Krizhevsky, 2009) has 20 coarse classes, each containing 5 fine classes. There are 500 and 100 images per class in the training and testing sets, respectively. The image size is $3 2 \times 3 2$ . We apply simple data augmentation (He et al., 2016), i.e., we pad 4 pixels on every side of the image, and then randomly crop it with a size of $3 2 \times 3 2$ . Finally, this crop is horizontally flipped with a probability of 0.5. We choose SGD with its settings as: (a) a learning rate of 0.1; (b) a momentum of 0.9; (c) a weight decay of $5 e - 4$ ; (d) the batch size is 256 and the number of training iterations is $3 0 \mathrm { k }$ . We divide the learning rate by 10 at $1 5 \mathrm { k }$ and $2 2 \mathrm { k }$ iterations, respectively. (2) We train ResNet-50 (He et al., 2016) on ImageNet 2012 classification dataset, which has 1k classes and $5 0 \mathrm { k }$ images in the test set (Russakovsky et al., 2015). We use SGD with a start learning rate of $2 e - 3$ . A polynomial learning rate decay with a power of 2 is used. We set the momentum to 0.95 and the weight decay to $1 e - 4$ . We train on a single V100 GPU and the batch size is 64. We report the final test accuracy when the training ends at 500k iterations. We use the standard data augmentation: an original image is warped to $2 5 6 \times 2 5 6$ , followed by a random crop of $2 2 4 \times 2 2 4$ . This crop is randomly flipped. We fix common settings to fairly compare CCE, LS, CP, Boot-soft and ProSelfLC.
+
+Result analysis. In Table 3, we observe the superiority of ProSelfLC in standard setting without considering label noise. Being probably surprising, LS and CP reduce the performance consistently as $\epsilon$ increases on ImageNet. Instead, Boot-soft and ProSelfLC improve versus CCE. We remark that both test sets are large so that their differences are noticeable.
+
+# 5.2 SYNTHETIC LABEL NOISE
+
+Noise generation. (1) Symmetric label noise: the original label of an image is uniformly changed to one of the other classes with a probability of $r$ ; (2) Asymmetric label noise: we follow (Wang et al., 2019) to generate asymmetric label noise to fairly compare with their reported results. Within each coarse class, we randomly select two fine classes $A$ and $B$ . Then we flip $\overline { { r } } \times 1 0 0 \%$ labels of $A$ to $B$ , and $r \times 1 0 0 \%$ labels of $B$ to $A$ . We remark that the overall label noise rate is smaller than $r$ .
+
+Baselines.1 We compare with the results reported recently in SL (Wang et al., 2019). Forward is a loss correction approach that uses a noise-transition matrix (Patrini et al., 2017). D2L monitors the subspace dimensionality change at training (Ma et al., 2018). GCE denotes generalised cross entropy (Zhang & Sabuncu, 2018) and SL is symmetric cross entropy (Wang et al., 2019). They are robust losses designed for solving label noise. Training details are the same as Section 5.1.
+
+Result analysis. For all methods, we directly report their final results when training terminates. Therefore, we test the robustness of a model against not only label noise, but also a long time being exposed to the data. In Table 4, we observe that: (1) ProSelfLC outperforms all baselines, which is significant in most cases; (2) In both implementation, Boot-hard and Boot-soft perform worse than the others. However, our ProSelfLC makes Self LC the best solution. Furthermore, learning dynamics are visualised in Figure 3, which helps to understand why ProSelfLC works better.
+
+Table 3: Test accuracy $( \% )$ in the standard setting. We report three settings of hyperparameters.
+
+![](images/ca018fc03e952dcd2298d4fa92a06cbe8e9325f355fda66e63778d74ee3c7064.jpg)  
+Figure 3: Comprehensive learning dynamics on CIFAR-100 with asymmetric label noise $r = 0 . 4$ . For data-dependent items, mean results are reported. At training, a learner is NOT GIVEN whether a label is trusted or not. We store intermediate models and analyse them when the training ends.
+
+Results of different $B , \epsilon$ are in Table 5. Appendix B shows the learning dynamics when $r$ changes.
+
+Revising the semantic class and similarity structure. In Figures 3b and 3c, we show dynamic statistics of different approaches on fitting wrong labels and correcting them. ProSelfLC is much better than its counterparts. Semantic class correction reflects the change of similarity structure.
+
+To redefine and reward a low-entropy status. On the one hand, we observe that LS and CP work well, being consistent with prior claims. In Figures 3d and 3e, the entropies of both clean and noisy subsets are much higher in LS and CP, correspondingly their generalisation is the best except for ProSelfLC in Figure 3f. On the other hand, ProSelfLC has the lowest entropy while performs the best, which proves that a learner’s confidence does not necessarily weaken its generalisation performance. Instead, a model needs to be careful with what to be confident in. As shown by Figures 3b and 3c, ProSelfLC has the least wrong fitting and most semantic class correction, which indicates that a meaningful low-entropy status is redefined.
+
+# 5.3 REAL-WORLD LABEL NOISE
+
+Clothing 1M (Xiao et al., 2015) has around $3 8 . 4 6 \%$ label noise in the training data and about 1 million images of 14 classes from shopping websites. Its internal noise structure is agnostic.
+
+Baselines. For loss correction and estimating the noise-transition matrix, S-adaption (Goldberger & Ben-Reuven, 2017) uses an extra softmax layer, while Masking (Han et al., 2018a) exploits human cognition. MD-DYR-SH (Arazo et al., 2019) is a combination of three techniques: dynamic mixup (MD), dynamic bootstrapping together with label regularisation (DYR) and soft to hard (SH). The
+
+Table 4: Accuracy $( \% )$ on the CIFAR-100 clean test set. All compared methods use ResNet-44.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Method</td><td colspan="3">Asymmetric Noisy Labels</td><td rowspan="2"></td><td colspan="3">Symmetric Noisy Labels</td></tr><tr><td>r=0.2</td><td>r=0.3</td><td>r=0.4</td><td>r=0.2</td><td>r=0.4</td><td>r=0.6</td></tr><tr><td rowspan="5">Results From SL (Wang et al., 2019)</td><td>Boot-hard</td><td>63.4</td><td>63.2</td><td>62.1</td><td></td><td>57.9</td><td>48.2</td><td>12.3</td></tr><tr><td>Forward</td><td>64.1</td><td>64.0</td><td>60.9</td><td></td><td>59.8</td><td>53.1</td><td>24.7</td></tr><tr><td>D2L</td><td>62.4</td><td>63.2</td><td>61.4</td><td></td><td>59.2</td><td>52.0</td><td>35.3</td></tr><tr><td>GCE</td><td>63.0</td><td>63.2</td><td>61.7</td><td></td><td>59.1</td><td>53.3</td><td>36.2</td></tr><tr><td>SL</td><td>65.6</td><td>65.1</td><td>63.1</td><td></td><td>60.0</td><td>53.7</td><td>41.5</td></tr><tr><td rowspan="5">Our Trained Results</td><td>CCE</td><td>66.6</td><td>63.4</td><td>59.5</td><td></td><td>58.0</td><td>50.1</td><td>37.9</td></tr><tr><td>LS</td><td>67.9</td><td>66.4</td><td>65.0</td><td></td><td>63.8</td><td>57.2</td><td>46.5</td></tr><tr><td>CP</td><td>67.7</td><td>66.0</td><td>64.4</td><td></td><td>64.0</td><td>56.8</td><td>44.1</td></tr><tr><td>Boot-soft</td><td>66.9</td><td>65.3</td><td>61.0</td><td></td><td>63.2</td><td>59.0</td><td>44.8</td></tr><tr><td>ProSelfLC</td><td>68.7</td><td>68.5</td><td>67.9</td><td></td><td>64.8</td><td>59.3</td><td>47.7</td></tr></table>
+
+Table 5: The results of different hyperparameters on CIFAR-100 using ResNet-44. Under different noise rates, the best instantiation of each approach is bolded except for CCE.   
+
+<table><tr><td>Method</td><td>Value of</td><td colspan="3">Asymmetric label noise</td><td colspan="3">Symmetric label noise</td><td rowspan="2">Clean</td></tr><tr><td>(hyperparameter)</td><td>hyperparameter</td><td>20%</td><td>30%</td><td>40%</td><td>20%</td><td>40%</td><td>60%</td></tr><tr><td rowspan="2">CCE</td><td>None or ε = 0</td><td>66.6</td><td>63.4</td><td>59.5</td><td>58.0</td><td>50.1</td><td>37.9</td><td>69.0</td></tr><tr><td>0.125</td><td>66.4</td><td>65.6</td><td>63.1</td><td>61.7</td><td>52.5</td><td>39.1</td><td>69.9</td></tr><tr><td rowspan="3">LS (e)</td><td>0.25</td><td>67.9</td><td>66.4</td><td>65.0</td><td>62.8</td><td>55.9</td><td>40.9</td><td>69.6</td></tr><tr><td>0.50</td><td>66.8</td><td>65.8</td><td>64.6</td><td>63.8</td><td>57.2</td><td>46.5</td><td>68.4</td></tr><tr><td>0.125</td><td>65.7</td><td>64.2</td><td>60.3</td><td>59.8</td><td>52.3</td><td>39.6</td><td>69.5</td></tr><tr><td rowspan="3">CP(e)</td><td>0.25</td><td>66.8</td><td>65.1</td><td>61.6</td><td>61.0</td><td>53.3</td><td>40.9</td><td>69.3</td></tr><tr><td>0.50</td><td>67.7</td><td>66.0</td><td>64.4</td><td>64.0</td><td>56.8</td><td>44.1</td><td>68.7</td></tr><tr><td>0.125</td><td>65.8</td><td>64.1</td><td>60.7</td><td>59.7</td><td>51.2</td><td>40.6</td><td>68.9</td></tr><tr><td rowspan="3">Boot-soft (ε)</td><td>0.25</td><td>66.2</td><td>64.1</td><td>60.3</td><td>61.1</td><td>54.4</td><td>43.3</td><td>69.1</td></tr><tr><td>0.50</td><td>66.9</td><td>65.3</td><td>61.0</td><td>63.2</td><td>59.0</td><td>44.8</td><td>69.1</td></tr><tr><td>8</td><td>67.8</td><td>67.4</td><td>67.9</td><td>64.7</td><td>57.7</td><td>47.7</td><td>70.1</td></tr><tr><td rowspan="4">ProSelfLC (B)</td><td>10</td><td>68.5</td><td>68.5</td><td>66.8</td><td>63.9</td><td>59.0</td><td>47.5</td><td>70.3</td></tr><tr><td>12</td><td>68.6</td><td>67.9</td><td>67.4</td><td>64.0</td><td>59.3</td><td>47.5</td><td>69.8</td></tr><tr><td>14</td><td>68.7</td><td>68.0</td><td>67.8</td><td>64.8</td><td>59.0</td><td>47.4</td><td>69.6</td></tr><tr><td>16</td><td>68.4</td><td>67.2</td><td>67.3</td><td>63.7</td><td>59.0</td><td>32.3</td><td>69.9</td></tr></table>
+
+Table 6: Test accuracy $( \% )$ on the real-world noisy dataset Clothing 1M.   
+
+<table><tr><td rowspan="2">Boot- hard</td><td rowspan="2">Forward</td><td rowspan="2">D2L</td><td rowspan="2">GCE</td><td rowspan="2">SL</td><td rowspan="2">S- adaptation</td><td rowspan="2">Masking</td><td rowspan="2">MD- DYR-SH</td><td rowspan="2">Joint- soft</td><td colspan="4">Our Trained Results</td></tr><tr><td>CCE</td><td>LS</td><td>CP</td><td>Boot-soft ProSelfLC</td></tr><tr><td>68.9</td><td>69.8</td><td>69.5</td><td>69.8</td><td>71.0</td><td>70.3</td><td>71.1</td><td>71.0</td><td>72.2</td><td>71.8</td><td>72.6 72.4</td><td>72.3</td><td>73.4</td></tr></table>
+
+other baselines have been introduced heretofore.
+
+Training details. We follow (Tanaka et al., 2018) to train ResNet-50 and initialise it by a trained model on ImageNet. We follow Section 5.1 with small changes: the initial learning rate is 0.01 and we train 10k iterations. They are searched on the separate clean validation set.
+
+Result analysis. In Table 6, analogously to CIFAR-100, we report our trained results of CCE, LS, CP, Boot-soft and ProSelfLC for an entirely fair comparison. ProSelfLC has the highest accuracy, which demonstrates its effectiveness again.
+
+# 6 CONCLUSION
+
+We present a thorough mathematical study on several target modification techniques. Through analysis of entropy and KL divergence, we reveal their relationships and limitations.
+
+To improve and endorse self label correction, we propose ProSelfLC. Extensive experiments prove its superiority over existing methods under standard and noisy settings. ProSelfLC enhances the similarity structure information over classes, and rectifies the semantic classes of noisy label distributions. ProSelfLC is the first approach to trust self knowledge progressively and adaptively.
+
+ProSelfLC redirects and promotes entropy minimisation, which is in marked contrast to recent practices of confidence penalty (Szegedy et al., 2016; Pereyra et al., 2017; Dubey et al., 2018).
+
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+
+# A PROOF OF PROPOSITIONS
+
+Proposition 3. Compared with CCE, LS and $C P$ penalise entropy minimisation while $L C$ reward it. Proof. We can rewrite CCE, LS, CP, and LC from the viewpoint of KL divergence:
+
+$$
+L _ { \mathrm { C C E } } ( \mathbf { q } , \mathbf { p } ) = \mathrm { H } ( \mathbf { q } , \mathbf { p } ) = \mathrm { K L } ( \mathbf { q } | | \mathbf { p } ) + \mathrm { H } ( \mathbf { q } , \mathbf { q } ) = \mathrm { K L } ( \mathbf { q } | | \mathbf { p } ) ,
+$$
+
+where we have $\mathrm { H } ( \mathbf { q } , \mathbf { q } ) = 0$ because $\mathbf { q }$ is a one-hot distribution.
+
+$$
+\begin{array} { r l } & { L _ { \mathrm { C C E + L S } } ( \mathbf { q } , \mathbf { p } ; \epsilon ) = ( 1 - \epsilon ) \mathrm { K L } ( \mathbf { q } | | \mathbf { p } ) + \epsilon \mathrm { K L } ( \mathbf { u } | | \mathbf { p } ) + \epsilon \mathrm { H } ( \mathbf { u } , \mathbf { u } ) } \\ & { \quad \quad \quad = ( 1 - \epsilon ) \mathrm { K L } ( \mathbf { q } | | \mathbf { p } ) + \epsilon \mathrm { K L } ( \mathbf { u } | | \mathbf { p } ) + \epsilon \cdot \mathrm { c o n s t a n t } , } \end{array}
+$$
+
+$$
+\begin{array} { r l } & { L _ { \mathrm { C C E + C P } } (  { \mathbf { q } } ,  { \mathbf { p } } ; \epsilon ) = ( 1 - \epsilon ) \mathrm { K L } (  { \mathbf { q } } | |  { \mathbf { p } } ) - \epsilon ( \mathrm { H } (  { \mathbf { p } } ,  { \mathbf { u } } ) - \mathrm { K L } (  { \mathbf { p } } | |  { \mathbf { u } } ) ) } \\ & { \qquad = ( 1 - \epsilon ) \mathrm { K L } (  { \mathbf { q } } | |  { \mathbf { p } } ) + \epsilon \mathrm { K L } (  { \mathbf { p } } | |  { \mathbf { u } } ) - \epsilon \cdot \mathrm { c o n s t a n t } , } \end{array}
+$$
+
+where $\mathrm { H } ( \mathbf { p } , \mathbf { u } ) = \mathrm { H } ( \mathbf { u } , \mathbf { u } ) = { \mathrm { c c } }$ onstant. Analogously, LC in Eq (5) can also be rewritten:
+
+$$
+L _ { \mathrm { C C E + L C } } ( \mathbf { q } , \mathbf { p } ; \epsilon ) = ( 1 - \epsilon ) \mathrm { K L } ( \mathbf { q } | | \mathbf { p } ) - \epsilon \mathrm { K L } ( \mathbf { p } | | \mathbf { u } ) + \epsilon \cdot \mathrm { c o n s t a n t } .
+$$
+
+In LS and CP, both $+ \mathrm { K L } ( \mathbf { u } | | \mathbf { p } )$ and $+ \mathrm { K L } ( \mathbf { p } | | \mathbf { u } )$ pulls $\mathbf { p }$ towards $\mathbf { u }$ . While in LC, the term $- \mathrm { K L } ( \mathbf { p } | | \mathbf { u } )$ pushes $\mathbf { p }$ away from $\mathbf { u }$ . 
+
+Proposition 4. In CCE, $L S$ and $C P ,$ a data point x has the same semantic class. In addition, x has an identical probability of belonging to other classes except for its semantic class.
+
+Proof. In LS, the target is $\tilde { \mathbf { q } } _ { \mathrm { L S } } = ( 1 - \epsilon ) \mathbf { q } + \epsilon \mathbf { u }$ . For any $0 \leq \epsilon < 1$ , the semantic class is not changed, because $1 - \epsilon + \epsilon * \overset { - } { \underset { C } { \ F } } > \bar { \epsilon } * \frac { 1 } { C }$ . In addition, $\begin{array} { r } { j _ { 1 } \neq y , \dot { j } _ { 2 } \neq y \Rightarrow \tilde { \mathbf { q } } _ { \mathrm { L S } } ( j _ { 1 } | \mathbf { x } ) = \tilde { \mathbf { q } } _ { \mathrm { L S } } ( j _ { 2 } | \mathbf { x } ) = \frac { \epsilon } { C } } \end{array}$ .
+
+In CP, $\tilde { \bf q } _ { \mathrm { C P } } = ( 1 - \epsilon ) { \bf q } - \epsilon { \bf p }$ . In terms of label definition, $C P$ is against intuition because these zero-value positions in q are filled with negative values in $\tilde { \mathbf { q } } _ { \mathrm { C P } }$ . A probability has to be not smaller than zero. So we rephrase $\tilde { \mathbf { q } } _ { \mathrm { C P } } ( y | \mathbf { x } ) = ( 1 - \epsilon ) - \epsilon * \mathbf { p } ( y | \mathbf { x } )$ , and $\forall j \neq y , \tilde { \mathbf { q } } _ { \mathrm { C P } } ( j | \mathbf { x } ) = 0$ by replacing negative values with zeros, as illustrated in Figure 1a. 
+
+# B LEARNING DYNAMICS OF DIFFERENT NOISE RATES
+
+In Figure 4, we store a model every 1000 iterations to monitor the learning process.
+
+C THE CHANGES OF ENTROPY STATISTICS AND ProSelfLC AT TRAINING
+
+In Figure 5, we visualise how the entropies of noisy and clean subsets change at training.
+
+![](images/d88add6873c102c97acc6fd4a47f9e73306162e4658746c17c92302937b7099f.jpg)  
+Figure 4: Learning dynamics on CIFAR-100 under asymmetric noisy labels. We show all iterations only in (a) and (d). In the others, we show the second half iterations, which are of higher interest. As the noise rate increases, the superiority of ProSelfLC becomes more significant, i.e., avoiding fitting noise in the 2nd row and leading to better generalisation in the 1st row.
+
+![](images/87d3947c11ab7a345e37aa40de395b01cf8f4a64c3228a8fe122237e7d38ce9f.jpg)  
+Figure 5: The changes of entropy statistics and ProSelfLC at training. We store a model every 1000 iterations to monitor the learning process. For data-dependent metrics, after training, we split the corrupted training data into clean and noisy subsets according to the information about how the training data is corrupted before training. Finally, we report the mean results of each subset.

parse/train/5SST78xEh4A/5SST78xEh4A_content_list.json

@@ -0,0 +1,1876 @@
+[
+    {
+        "type": "text",
+        "text": "PROSELFLC: PROGRESSIVE SELF LABEL CORRECTION FOR TRAINING ROBUST DEEP NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To train robust deep neural networks (DNNs), we systematically study several target modification approaches, which include output regularisation, self and nonself label correction (LC). Two key issues are discovered: (1) Self LC is the most appealing as it exploits its own knowledge and requires no extra models. However, how to automatically decide the trust degree of a learner as training goes is not well answered in the literature? (2) Some methods penalise while the others reward low-entropy predictions, prompting us to ask which one is better? ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To resolve the first issue, taking two well-accepted propositions–deep neural networks learn meaningful patterns before fitting noise (Arpit et al., 2017) and minimum entropy regularisation principle (Grandvalet & Bengio, 2006)–we propose a novel end-to-end method named ProSelfLC, which is designed according to learning time and entropy. Specifically, given a data point, we progressively increase trust in its predicted label distribution versus its annotated one if a model has been trained for enough time and the prediction is of low entropy (high confidence). For the second issue, according to ProSelfLC, we empirically prove that it is better to redefine a meaningful low-entropy status and optimise the learner toward it. This serves as a defence of entropy minimisation. We demonstrate the effectiveness of ProSelfLC through extensive experiments in both clean and noisy settings. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There exist many target (label) modification approaches. They can be roughly divided into two groups: (1) Output regularisation (OR), which is proposed to penalise overconfident predictions for regularising deep neural networks. It includes label smoothing (LS) (Szegedy et al., 2016; Müller et al., 2019) and confidence penalty (CP) (Pereyra et al., 2017); (2) Label correction (LC). On the one hand, LC regularises neural networks by adding the similarity structure information over training classes into one-hot label distributions so that the learning targets become structured and soft. On the other hand, it can correct the semantic classes of noisy label distributions. LC can be further divided into two subgroups: Non-self LC and Self LC. The former requires extra learners, while the latter relies on the model itself. A typical approach of Non-self LC is knowledge distillation (KD), which exploits the predictions of other model(s), usually termed teacher(s) (Hinton et al., 2015). Self LC methods include Pseudo-Label (Lee, 2013), bootstrapping (Boot-soft and Boot-hard) (Reed et al., 2015), Joint Optimisation (Joint-soft and Joint-hard) (Tanaka et al., 2018), and Tf- ${ \\mathrm { K D } } _ { s e l f }$ (Yuan et al., 2020). According to an overview in Figure 1 (detailed derivation is in Section 3 and Table 1), in label modification, the output target of a data point is defined by combining a one-hot label distribution and its corresponding prediction or a predefined label distribution. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Firstly, we present the drawbacks of existing approaches: (1) OR methods naively penalise confident outputs without leveraging easily accessible knowledge from other learners or itself (Figure 1a); (2) Non-self LC relies on accurate auxiliary models to generate predictions (Figure 1b). (3) Self LC is the most appealing because it exploits its own knowledge and requires no extra learners. However, there is a core question that is not well answered: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In Self LC, how much should we trust a learner to leverage its knowledge? ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1b, in Self LC, for a data point, we have two labels: a predefined one-hot q and a predicted structured p. Its learning target is $( 1 - \\epsilon ) \\mathbf { q } + \\epsilon \\mathbf { p }$ , i.e., a trade-off between q and p, where $\\epsilon$ defines the trust score of a learner. In existing methods, $\\epsilon$ is fixed without considering that a model’s (a) OR includes LS (Szegedy et al., 2016) and CP (Pereyra et al., 2017). LS softens a target by adding a uniform label distribution. CP changes the probability 1 to a smaller value $1 - \\epsilon$ in the one-hot target. The double-ended arrow means factual equivalence, because an output is definitely non-negative after a softmax layer. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/4ec618d424c008219333fd3aad931860b7ccc3edfad166d77494cd98dd80172f.jpg",
+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/667be94f42736a8518c158bc6b48a8079ddbc87fa14a14e0db24550b250a7407.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "(b) LC contains Self LC (Lee, 2013; Reed et al., 2015; Tanaka et al., 2018; Yuan et al., 2020) and Non-self LC (Hinton et al., 2015). The parameter \u000f defines how much a predicted label distribution is trusted. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Figure 1: Target modification includes OR (LS and CP), and LC (Self LC and Non-self LC). Assume there are three training classes. q is the one-hot target. u is a uniform label distribution. p denotes a predicted label distribution. The target combination parameter is $\\epsilon \\in [ 0 , 1 ]$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "knowledge grows as the training progresses. For example, in bootstrapping, $\\epsilon$ is fixed throughout the training process. Joint Optimisation stage-wisely trains a model. It fully trusts predicted labels and uses them to replace old ones when a stage ends, i.e., $\\epsilon = 1$ . Tf- ${ \\mathrm { K D } } _ { s e l f }$ trains a model by two stages: $\\epsilon = 0$ in the first one while $\\epsilon$ is tuned for the second stage. Note that $\\mathbf { p }$ is generated by a preceding-stage model in stage-wise training, which requires significant human intervention and is time-consuming in practice. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To improve Self LC, we propose a novel method named Progressive Self Label Correction (ProSelfLC), which is end-to-end trainable and needs negligible extra cost. Most importantly, ProSelfLC modifies the target progressively and adaptively as training goes. Two design principles of ProSelfLC are: (1) When a model learns from scratch, human annotations are more reliable than its own predictions in the early phase, during which the model is learning simple meaningful patterns before fitting noise, even when severe label noise exists in human annotations (Arpit et al., 2017). (2) As a learner attains confident knowledge as time progresses, we leverage it to revise annotated labels. This is surrounded by minimum entropy regularisation, which is widely evaluated in unsupervised and semi-supervised scenarios (Grandvalet & Bengio, 2005; 2006). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Secondly, note that OR methods penalise low entropy while LC rewards it, intuitively leading to a second vital question: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Should we penalise a low-entropy status or reward it? ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Entropy minimisation is the most widely used principle in machine learning (Hartigan & Wong, 1979; Rumelhart et al., 1986; Grandvalet & Bengio, 2005; 2006; LeCun et al., 2015). In standard classification, minimising categorical cross entropy (CCE) optimises a model towards a low-entropy status defined by human annotations, which contain noise in very large-scale machine learning. As a result, confidence penalty becomes popular for reducing noisy fitting. In contrast, we prove that it is better to reward a meaningful low-entropy status redefined by our ProSelfLC. Therefore, our work offers a defence of entropy minimisation against the recent confidence penalty practice (Szegedy et al., 2016; Müller et al., 2019; Pereyra et al., 2017; Dubey et al., 2018). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, we summarise our main contributions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• We provide a theoretical study on popular target modification methods through entropy and KL divergence (Kullback & Leibler, 1951). Accordingly, we reveal their drawbacks and propose ProSelfLC as a solution. ProSelfLC can: (1) enhance the similarity structure information over training classes; (2) correct the semantic classes of noisy label distributions. ProSelfLC is the first method to trust self knowledge progressively and adaptively. Our extensive experiments: (1) defend the entropy minimisation principle; (2) demonstrate the effectiveness of ProSelfLC in both clean and noisy settings. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Label noise and semi-supervised learning. We test target modification approaches in the setting of label noise because it is generic and connected with semi-supervised learning, where only a subset of training examples are annotated, leading to missing labels. Then the key to semi-supervised training is to reliably fill them. When these missing labels are incorrectly filled, the challenge of semisupervised learning changes to noisy labels. For a further comparison, in semi-supervised learning, the annotated set is clean and reliable, because the label noise only exists in the unannotated set. While in our experimental setting, we are not given information on whether an example is trusted or not, thus being even more challenging. We summarise existing approaches for solving label noise: (1) Loss correction, in which we are given or we need to estimate a noise-transition matrix, which defines the distribution of noise labels (Li et al., 2017; Goldberger & Ben-Reuven, 2017; Sukhbaatar & Fergus, 2014; Vahdat, 2017; Yao et al., 2019; Han et al., 2018a; Patrini et al., 2017; Xiao et al., 2015). A noise-transition matrix is difficult and complex to estimate in practice; (2) Exploiting an auxiliary trusted training set to differentiate examples (Veit et al., 2017; Lee et al., 2018; Hendrycks et al., 2018). This requires extra annotation cost; (3) Co-training strategies, which train two or more learners (Malach & Shalev-Shwartz, 2017; Jiang et al., 2018; Han et al., 2018b; Yu et al., 2019; Wei et al., 2020; Qiao et al., 2018) and exploit their ‘disagreement’ information to differentiate data points; (4) Label engineering methods (Song et al., 2019; Lee, 2013; Reed et al., 2015; Tanaka et al., 2018; Yao et al., 2019), which relate to our focus in this work. Their strategy is to annotate unlabelled samples or correct noisy labels. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "LC and knowledge distillation $( K D )$ (Bucila et al., 2006; Hinton et al., 2015). Mathematically, we derive that some KD methods also modify labels. We use the term label correction instead of KD for two reasons: (1) label correction is more descriptive; (2) the scope of KD is not limited to label modification. For example, multiple networks are trained for KD (Furlanello et al., 2018). When two models are trained, the consistency between their predictions of a data point is promoted in (Ba & Caruana, 2014; Zhang et al., 2018), while the distance between their feature maps is reduced in (Romero et al., 2015). Regarding self KD, two examples of the same class are constrained to have consistent output distributions (Xu & Liu, 2019; Yun et al., 2020). In another self KD (Zhang et al., 2019), the deepest classifier provides knowledge for shallower classifiers. In a recent self KD method (Yuan et al., 2020), Tf- $\\mathrm { K D } _ { s e l f }$ applies two-stage training. In the second stage, a model is trained by exploiting its knowledge learned in the first stage. Our focus is to improve the end-to-end self LC. Finally, we acknowledge that exploiting ProSelfLC to improve non-self KD and stage-wise approaches is an area for future work, e.g., a better teacher model can be trained using ProSelfLC. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 MATHEMATICAL ANALYSIS AND THEORY ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathbf { X } = \\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N }$ represent $N$ training examples, where $\\left( \\mathbf { x } _ { i } , y _ { i } \\right)$ denotes $i -$ th sample with input $\\mathbf { x } _ { i } \\in \\mathbb { R } ^ { D }$ and label $y _ { i } \\in \\{ 1 , 2 , . . . , C \\}$ . $C$ is the number of classes. A deep neural network $z$ consists of an embedding network $f ( \\cdot ) : \\bar { \\mathbb { R } } ^ { D }  \\mathbb { R } ^ { K }$ and a linear classifier $\\begin{array} { r } { g ( \\cdot ) \\dot { \\mathbf { \\Psi } } : \\mathbb { R } ^ { K }  \\mathbb { R } ^ { C } } \\end{array}$ , i.e., $\\mathbf { z } _ { i } = z ( \\mathbf { x } _ { i } ) = g ( f ( \\mathbf { x } _ { i } ) ) ^ { \\mathit { \\prime } } : \\mathbb { R } ^ { D }  \\mathbb { R } ^ { C }$ . For the brevity of analysis, we take a data point and omit its subscript so that it is denoted by $\\left( \\mathbf { x } , y \\right)$ . The linear classifier is usually the last fully-connected layer. Its output is named logit vector $\\mathbf { z } \\in \\mathbb { R } ^ { C }$ . We produce its classification probabilities $\\mathbf { p }$ by normalising the logits using a softmax function: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00a477304ca995486ef0820e17f65a96ee3afbb879c03e8cd6c07079f46ab3ad.jpg",
+        "text": "$$\n\\mathbf { p } ( j | \\mathbf { x } ) = \\exp ( \\mathbf { z } _ { j } ) / { \\sum _ { m = 1 } ^ { C } \\exp ( \\mathbf { z } _ { m } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { p } ( j | \\mathbf { x } )$ is the probability of $\\mathbf { x }$ belonging to class $j$ . Its corresponding ground-truth is usually denoted by a one-hot representation q: $\\mathbf { q } ( j | \\mathbf { x } ) = 1$ if $j = y$ , $\\mathbf { q } ( j | \\mathbf { x } ) = 0$ otherwise. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 SEMANTIC CLASS AND SIMILARITY STRUCTURE IN A LABEL DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Semantic Class). Given a target label distribution $\\tilde { \\mathbf { q } } ( \\mathbf { x } ) \\in \\mathbb { R } ^ { C }$ , the semantic class is defined by arg $\\operatorname* { m a x } _ { j } { \\tilde { \\mathbf { q } } } ( j | \\mathbf { x } )$ , i.e., the class whose probability is the largest. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2 (Similarity Structure). In $\\tilde { \\bf q } ( { \\bf x } )$ , $\\mathbf { x }$ has $C$ probabilities of being predicted to $C$ classes.   \nThe similarity structure of $\\mathbf { x }$ versus $C$ classes is defined by these probabilities and their differences. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/98d4b4b6a2be0ff28f4ea456f83d9a1fab9ed2bcd8635ed5556b0413b9b00830.jpg",
+        "table_caption": [
+            "Table 1: Summary of CCE, LS, CP and LC. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CCE</td><td>LS</td><td>CP</td><td>LC</td></tr><tr><td>Learning Target</td><td>q</td><td>qLs = (1- ε)q +εu</td><td>qcp = (1- e)q -ep</td><td>qLc = (1 - ɕ)q +ep</td></tr><tr><td>Cross Entropy</td><td>Eq(-log_P)</td><td>EaLs(-log_P) (1-e)KL(qlIp)</td><td>Eacp(-log P) (1-c)KL(qllp)</td><td>EqLc(-log P) (1-c)KL(qlIp)</td></tr><tr><td>KL Divergence</td><td>KL(qllp)</td><td>+eKL(ullp)</td><td>+eKL(pllu)</td><td>-eKL(pllu)</td></tr><tr><td>Entropy minimisation</td><td></td><td>Penalise over CCE</td><td>Penalise over CCE</td><td>Reward over CCE</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Semantic class</td><td>Annotated</td><td>Annotated</td><td>Annotated</td><td>Annotated andLearned</td></tr><tr><td>Similarity structure</td><td>No</td><td>No</td><td>No</td><td>Yes</td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 REVISIT OF CCE, LS, CP AND LC ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Standard CCE. For any input $\\left( \\mathbf { x } , y \\right)$ , the minimisation objective of standard CCE is: ",
+        "bbox": [
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+            281,
+            732,
+            296
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/59970a540dbaae239ce6ebd2acc1e8960a4e60c2231096d41879c9e527c2233a.jpg",
+        "text": "$$\nL _ { \\mathrm { C C E } } ( \\mathbf { q } , \\mathbf { p } ) = \\mathrm { H } ( \\mathbf { q } , \\mathbf { p } ) = \\mathrm { E } _ { \\mathbf { q } } ( - \\log \\ \\mathbf { p } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            299,
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+            316
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { H } ( \\cdot , \\cdot )$ represents the cross entropy. $\\operatorname { E } _ { \\mathbf { q } } ( - \\log \\mathbf { \\delta p } )$ denotes the expectation of negative loglikelihood, and $\\mathbf { q }$ is the probability mass function. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Label smoothing. In LS (Szegedy et al., 2016; Hinton et al., 2015), we soften one-hot targets by adding a uniform distribution: $\\tilde { \\bf q } _ { \\mathrm { L S } } = ( 1 - \\epsilon ) { \\bf q } + \\epsilon { \\bf u }$ , $\\mathbf { u } \\in \\mathbb { R } ^ { C }$ , and $\\begin{array} { r } { \\forall j , \\mathbf { u } _ { j } = \\frac { 1 } { C } } \\end{array}$ . Consequently: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6575075bc935932841e22f37e387087b1119c2d49ecc48fa0456203abf22ea7a.jpg",
+        "text": "$$\nL _ { \\mathrm { C C E + L S } } ( \\mathbf { q } , \\mathbf { p } ; \\epsilon ) = \\mathrm { H } ( \\tilde { \\mathbf { q } } _ { \\mathrm { L S } } , \\mathbf { p } ) = \\mathrm { E } _ { \\tilde { \\mathbf { q } } _ { \\mathrm { L S } } } ( - \\log \\mathbf { \\delta p } ) = ( 1 - \\epsilon ) \\mathrm { H } ( \\mathbf { q } , \\mathbf { p } ) + \\epsilon \\mathrm { H } ( \\mathbf { u } , \\mathbf { p } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Confidence penalty. CP (Pereyra et al., 2017) penalises highly confident predictions: ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/0d23b0883c7809cdcb5200c96857966f6e16e521ab904e6976d743243dc3cd3e.jpg",
+        "text": "$$\nL _ { \\mathrm { C C E + C P } } ( \\mathbf { q } , \\mathbf { p } ; \\epsilon ) = ( 1 - \\epsilon ) \\mathrm { H } ( \\mathbf { q } , \\mathbf { p } ) - \\epsilon \\mathrm { H } ( \\mathbf { p } , \\mathbf { p } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Label correction. As illustrated in Figure 1, LC is a family of algorithms, where a one-hot label distribution is modified to a convex combination of itself and a predicted distribution: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/346a9fe59519337f7da3dc55ca72aa758cb5860a886af04e80bac12d97982b39.jpg",
+        "text": "$$\n\\tilde { \\bf q } _ { \\mathrm { I C } } = ( 1 - \\epsilon ) { \\bf q } + \\epsilon { \\bf p } \\Rightarrow L _ { \\mathrm { C C E + L C } } ( { \\bf q } , { \\bf p } ; \\epsilon ) = \\mathrm { H } ( \\tilde { \\bf q } _ { \\mathrm { L C } } , { \\bf p } ) = ( 1 - \\epsilon ) \\mathrm { H } ( { \\bf q } , { \\bf p } ) + \\epsilon \\mathrm { H } ( { \\bf p } , { \\bf p } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We remark: (1) $\\mathbf { p }$ provides meaningful information about an example’s relative probabilities of being different training classes; (2) If $\\epsilon$ is large, and $\\mathbf { p }$ is confident in predicting a different class, i.e., a $\\begin{array} { r } { \\arg \\operatorname* { m a x } _ { j } \\mathbf { p } ( j | \\mathbf { x } ) \\neq \\arg \\operatorname* { m a x } _ { j } \\mathbf { q } ( j | \\mathbf { x } ) , } \\end{array}$ $\\tilde { \\bf q } _ { \\mathrm { L C } }$ defines a different semantic class from $\\mathbf { q }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 THEORY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. LS, CP and $L C$ modify the learning targets of standard CCE ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. $L _ { \\mathrm { C C E + C P } } ( \\mathbf { q } , \\mathbf { p } ; \\epsilon ) = ( 1 - \\epsilon ) \\mathrm { H } ( \\mathbf { q } , \\mathbf { p } ) - \\epsilon \\mathrm { H } ( \\mathbf { p } , \\mathbf { p } ) = \\mathrm { E } _ { ( \\mathbf { 1 } - \\epsilon ) \\mathbf { q } - \\epsilon \\mathbf { p } } ( - \\log \\mathbf { \\ p } )$ . Therefore, $\\mathbf { \\tilde { q } } _ { \\mathrm { C P } } =$ $( 1 - \\epsilon ) \\mathbf { q } { - } \\epsilon \\mathbf { p }$ . Additionally, $\\tilde { \\mathbf { q } } _ { \\mathrm { L S } } = ( 1 - \\epsilon ) \\mathbf { q } + \\epsilon \\mathbf { u }$ , $\\tilde { \\bf q } _ { \\mathrm { L C } } = ( 1 - \\epsilon ) { \\bf q } + \\epsilon { \\bf p }$ . \u0003 ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. Some $K D$ methods, which aim to minimise the KL divergence between predictions of a teacher and a student, belong to the family of label correction. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. In general, a loss function of such methods can be defined to be $L _ { \\mathrm { K D } } ( \\mathbf { q } , \\mathbf { p } _ { t } , \\mathbf { p } ) = ( 1 -$ $\\epsilon ) \\mathrm { H } ( \\mathbf { q } , \\mathbf { p } ) \\bar { + } \\epsilon \\mathrm { K L } ( \\mathbf { p } _ { t } | | \\mathbf { p } )$ (Yuan et al., 2020). $\\operatorname { K L } ( \\cdot | | \\cdot )$ denotes the KL divergence. As ${ \\mathrm { K L } } ( \\mathbf { p } _ { t } | | \\mathbf { p } ) =$ $\\mathrm { H } ( \\mathbf { p } _ { t } , \\mathbf { p } ) { - } \\mathrm { H } ( \\mathbf { p } _ { t } , \\mathbf { p } _ { t } )$ , $\\mathbf { p } _ { t }$ is from a teacher and fixed when training a student. We can omit $\\mathrm { H } ( \\mathbf { p } _ { t } , \\mathbf { p } _ { t } )$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/618b429db7a0be04c81d8cf2705e0a8c707d534c9491536080d376663bb10d6b.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { \\mathrm { K D } } ( \\mathbf { q } , \\mathbf { p } t , \\mathbf { p } ) = ( 1 - \\epsilon ) \\mathrm { H } ( \\mathbf { q } , \\mathbf { p } ) + \\epsilon \\mathrm { H } ( \\mathbf { p } t , \\mathbf { p } ) = \\mathrm { E } _ { ( 1 - \\epsilon ) \\mathbf { q } + \\mathbf { c p } _ { t } } ( - \\log \\mathbf { p } ) \\Rightarrow \\tilde { \\mathbf { q } } _ { \\mathrm { K D } } = ( 1 - \\epsilon ) \\mathbf { q } + \\epsilon \\mathbf { p } t . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Consistent with LC in Eq (5), $L _ { \\mathrm { K D } } ( \\mathbf { q } , \\mathbf { p } _ { t } , \\mathbf { p } )$ revises a label using $\\mathbf { p } _ { t }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. Compared with CCE, LS and $C P$ penalise entropy minimisation while $L C$ reward it. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4. In CCE, $L S$ and $C P ,$ a data point x has the same semantic class. In addition, x has an identical probability of belonging to other classes except for its semantic class. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proof of propositions 3 and 4 is presented in the Appendix A. Only LC exploits informative information and has the ability to correct labels, while LS and CP only relax the hard targets. We summarise CCE, LS, CP and LC in Table 1. Constant terms are ignored for concision. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 PROSELFLC: PROGRESSIVE AND ADAPTIVE LABEL CORRECTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In standard CCE, a semantic class is considered while the similarity structure is ignored. It is mainly due to the difficulty of annotating the similarity structure for every data point, especially when $C$ is large ( $\\mathrm { { X u } }$ et al., 2020). Fortunately, recent progress demonstrates that there are some effective approaches to define the similarity structure of data points without annotation: (1) In KD, an auxiliary teacher model can provide a student model the similarity structure information (Hinton et al., 2015; Müller et al., 2019); (2) In Self LC, e.g., Boot-soft, a model helps itself by exploiting the knowledge it has learned so far. We focus on studying the end-to-end Self LC. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/aa788af78281dbcf10621812f5119b8c66d04bc5c9f9df0636a60c2194484ac0.jpg",
+        "table_caption": [
+            "Table 2: Instantiating \u000fProSelfLC under different cases. For all terms, we use concrete values for concise interpretation. We bold the special case when the semantic class is changed. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">l(p): Consistency is defined by whether p and q share the semantic class or not.</td></tr><tr><td></td><td></td><td>0.1(non-confident）0.9(confidently consistent） 0.9(confidently inconsistent)</td></tr><tr><td>Earlier phase g(t) = 0.1</td><td>0.01</td><td>0.09</td><td>0.09</td></tr><tr><td>Later phase g(t) = 0.9</td><td>0.09</td><td>0.81</td><td>0.81</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In Self LC, $\\epsilon$ indicates how much a predicted label distribution is trusted. In ProSelfLC, we propose to set it automatically according to learning time $t$ and prediction entropy $\\mathrm { H } ( \\mathbf { p } )$ , i.e., ProSelfLC trusts self knowledge according to training time and confidence. For any $\\mathbf { x }$ , we summarise: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Loss: $: L _ { ( \\tilde { \\bf q } _ { \\mathrm { P r o S e l f L C } } , { \\bf p } ; \\epsilon _ { \\mathrm { P r o S e l f L C } } ) } = \\mathrm { H } ( \\tilde { \\bf q } _ { \\mathrm { P r o S e l f L C } } , { \\bf p } ) = \\mathrm { E } _ { \\tilde { \\bf q } _ { \\mathrm { P r o S e l f L C } } } ( - \\log { \\bf p } ) .$ Label: $\\begin{array} { r } { \\tilde { \\bf q } _ { \\mathrm { P r o S e l f L C } } = ( 1 - \\epsilon _ { \\mathrm { P r o S e l f L C } } ) { \\bf q } + \\epsilon _ { \\mathrm { P r o S e l f L C } } { \\bf p } . } \\end{array}$ $\\mathrm { S e l f ~ t r u s t : } \\ \\epsilon _ { \\mathrm { P r o S e l f L C } } = g ( t ) \\times l ( \\mathbf { p } ) \\left\\{ g ( t ) = h ( t / \\Gamma - 0 . 5 , B ) \\in ( 0 , 1 ) , \\right. $ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$t$ and $\\Gamma$ are the iteration counter and the number of total iterations, respectively. $h ( \\eta , B ) ~ =$ $1 / ( 1 + \\exp ( - \\eta \\times B ) ) . B$ $B , \\Gamma$ are task-dependent and searched on a validation set. We clarify: Global trust score $g ( t )$ denotes how much we trust a learner. It is independent of data points, thus being global. $g ( t )$ grows as $t$ rises. $B$ adjusts the exponentiation’s base and growth speed of $g ( t )$ . The local trust score $l ( \\mathbf { p } )$ indicates how much we trust an output distribution $\\mathbf { p }$ , which is datadependent. $l ( \\mathbf { p } )$ rises as $\\mathrm { H } ( \\mathbf { p } )$ becomes lower, rewarding a confident distribution. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Design reason. (1) Regarding $g ( t )$ , in the earlier learning phase, i.e., $t < \\Gamma / 2$ , $g ( t ) < 0 . 5 \\Rightarrow$ \u000fProSelfLC $< 0 . 5 , \\forall \\mathbf { p }$ , so that the human annotations dominate and ProSelfLC only modifies the similarity structure. When a learner has not seen the training data for enough time at the earlier stage, its knowledge is less reliable and a wrong confident prediction may occur. Our design assuages the bad impact of such unexpected cases. When it comes to the later training phase, i.e., $t > \\Gamma / 2$ , we have $g ( t ) > 0 . 5$ as it has been trained for more than half of entire iterations. (2) Regarding $l ( \\mathbf { p } )$ , it affects the later learning phase. If $\\mathbf { p }$ is less confident, $l ( \\mathbf { p } )$ will be smaller, then \u000fProSelfLC will be smaller, hence we trust $\\mathbf { p }$ less when it is of higher uncertainty. If $\\mathbf { p }$ is highly confident, we trust its confident knowledge. Ablation study of our design is in Figure 2, where three variants of \u000f are presented. In our experiments, note that when $\\epsilon$ is fixed, we try three values (0.125, 0.25, 0.50) and display the best instantiation, i.e., $\\epsilon = 0 . 5 0$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We conduct the case analysis of ProSelfLC in Table 2 and summarise its core tactics as follows: (1) Correct the similarity structure for every data point in all cases, thanks to exploiting the self knowledge of a learner, i.e., $\\mathbf { p }$ . (2) Revise the semantic class when $t$ is large enough and $\\mathbf { p }$ is confidently inconsistent. As highlighted in Table 2, when two conditions are met, we have $\\epsilon _ { \\mathrm { P r o S e l f L C } } > 0 . 5$ and $\\operatorname { a r g m a x } _ { j } \\mathbf { p } ( j | \\mathbf { x } ) \\ \\neq \\ \\operatorname { a r g m a x } _ { j } \\mathbf { q } ( j | \\mathbf { x } )$ , then $\\mathbf { p }$ redefines the semantic class. For example, if $\\mathbf { p } ~ = ~ [ 0 . 9 5 , 0 . 0 1 , 0 . 0 4 ] , \\mathbf { q } ~ = ~ [ 0 , 0 , 1 ] ,$ $\\epsilon _ { \\mathrm { P r o S e l f L C } } ~ = ~ 0 . 8 ~ \\Rightarrow ~ \\tilde { \\bf q } _ { \\mathrm { P r o S e l f L C } } ~ = ~ ( 1 ~ -$ \u000f $\\mathrm { \\Delta \\ p o S e l f L C } \\mathbf { \\Psi } ) \\mathbf { q } + \\epsilon _ { \\mathrm { P r o S e l f L C } } \\mathbf { p } = [ 0 . 7 6 , 0 . 0 0 8 , 0 . 2 3 2 ]$ . Note that ProSelfLC also becomes robust against lengthy exposure to the training data, as demonstrated in Figures 2 and 3. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we re-implement CCE, LS and CP. Regarding Self LC methods, we reimplement Boot-soft (Reed et al., 2015), where $\\epsilon$ is fixed throughout training. We do not reimplement stage-wise Self LC and KD methods, e.g., Joint Optimisation and Tf- $\\mathrm { K D } _ { s e l f }$ respectively, because time-consuming tuning is required. We fix the random seed and do not use any random accelerator for an entirely fair comparison. In standard and synthetic cases, we train on $80 \\%$ training data (corrupted in synthetic cases) and use $20 \\%$ trusted training data as a validation set to search hyperparameters, e.g., $\\epsilon , \\Gamma , B$ and settings of an optimiser. Note that $\\Gamma$ and an optimiser’s settings are searched first and then shared by all methods. Finally, we retrain a model on the entire training data (corrupted in synthetic cases) and report its accuracy on the test data to fairly compare with prior results. In real-world label noise, the used dataset has a separate clean validation set for searching hyperparameters. Code will be released once this work is accepted. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/b82a73bddf1405565e918850ac86e4d14d341ba4f7fe03f94c02e04204a87e30.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of setting $\\epsilon$ using different schemes. Experiments are done on CIFAR-100 with asymmetric label noise $r = 0 . 4$ . For data-dependent items, mean results are reported. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "5.1 STANDARD IMAGE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            178,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets and training details. (1) CIFAR-100 (Krizhevsky, 2009) has 20 coarse classes, each containing 5 fine classes. There are 500 and 100 images per class in the training and testing sets, respectively. The image size is $3 2 \\times 3 2$ . We apply simple data augmentation (He et al., 2016), i.e., we pad 4 pixels on every side of the image, and then randomly crop it with a size of $3 2 \\times 3 2$ . Finally, this crop is horizontally flipped with a probability of 0.5. We choose SGD with its settings as: (a) a learning rate of 0.1; (b) a momentum of 0.9; (c) a weight decay of $5 e - 4$ ; (d) the batch size is 256 and the number of training iterations is $3 0 \\mathrm { k }$ . We divide the learning rate by 10 at $1 5 \\mathrm { k }$ and $2 2 \\mathrm { k }$ iterations, respectively. (2) We train ResNet-50 (He et al., 2016) on ImageNet 2012 classification dataset, which has 1k classes and $5 0 \\mathrm { k }$ images in the test set (Russakovsky et al., 2015). We use SGD with a start learning rate of $2 e - 3$ . A polynomial learning rate decay with a power of 2 is used. We set the momentum to 0.95 and the weight decay to $1 e - 4$ . We train on a single V100 GPU and the batch size is 64. We report the final test accuracy when the training ends at 500k iterations. We use the standard data augmentation: an original image is warped to $2 5 6 \\times 2 5 6$ , followed by a random crop of $2 2 4 \\times 2 2 4$ . This crop is randomly flipped. We fix common settings to fairly compare CCE, LS, CP, Boot-soft and ProSelfLC. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Result analysis. In Table 3, we observe the superiority of ProSelfLC in standard setting without considering label noise. Being probably surprising, LS and CP reduce the performance consistently as $\\epsilon$ increases on ImageNet. Instead, Boot-soft and ProSelfLC improve versus CCE. We remark that both test sets are large so that their differences are noticeable. ",
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+    {
+        "type": "text",
+        "text": "5.2 SYNTHETIC LABEL NOISE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Noise generation. (1) Symmetric label noise: the original label of an image is uniformly changed to one of the other classes with a probability of $r$ ; (2) Asymmetric label noise: we follow (Wang et al., 2019) to generate asymmetric label noise to fairly compare with their reported results. Within each coarse class, we randomly select two fine classes $A$ and $B$ . Then we flip $\\overline { { r } } \\times 1 0 0 \\%$ labels of $A$ to $B$ , and $r \\times 1 0 0 \\%$ labels of $B$ to $A$ . We remark that the overall label noise rate is smaller than $r$ . ",
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+    {
+        "type": "text",
+        "text": "Baselines.1 We compare with the results reported recently in SL (Wang et al., 2019). Forward is a loss correction approach that uses a noise-transition matrix (Patrini et al., 2017). D2L monitors the subspace dimensionality change at training (Ma et al., 2018). GCE denotes generalised cross entropy (Zhang & Sabuncu, 2018) and SL is symmetric cross entropy (Wang et al., 2019). They are robust losses designed for solving label noise. Training details are the same as Section 5.1. ",
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+    {
+        "type": "text",
+        "text": "Result analysis. For all methods, we directly report their final results when training terminates. Therefore, we test the robustness of a model against not only label noise, but also a long time being exposed to the data. In Table 4, we observe that: (1) ProSelfLC outperforms all baselines, which is significant in most cases; (2) In both implementation, Boot-hard and Boot-soft perform worse than the others. However, our ProSelfLC makes Self LC the best solution. Furthermore, learning dynamics are visualised in Figure 3, which helps to understand why ProSelfLC works better. ",
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+        "page_idx": 5
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Test accuracy $( \\% )$ in the standard setting. We report three settings of hyperparameters. "
+        ],
+        "table_footnote": [],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/ca018fc03e952dcd2298d4fa92a06cbe8e9325f355fda66e63778d74ee3c7064.jpg",
+        "image_caption": [
+            "Figure 3: Comprehensive learning dynamics on CIFAR-100 with asymmetric label noise $r = 0 . 4$ . For data-dependent items, mean results are reported. At training, a learner is NOT GIVEN whether a label is trusted or not. We store intermediate models and analyse them when the training ends. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Results of different $B , \\epsilon$ are in Table 5. Appendix B shows the learning dynamics when $r$ changes. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Revising the semantic class and similarity structure. In Figures 3b and 3c, we show dynamic statistics of different approaches on fitting wrong labels and correcting them. ProSelfLC is much better than its counterparts. Semantic class correction reflects the change of similarity structure. ",
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+    {
+        "type": "text",
+        "text": "To redefine and reward a low-entropy status. On the one hand, we observe that LS and CP work well, being consistent with prior claims. In Figures 3d and 3e, the entropies of both clean and noisy subsets are much higher in LS and CP, correspondingly their generalisation is the best except for ProSelfLC in Figure 3f. On the other hand, ProSelfLC has the lowest entropy while performs the best, which proves that a learner’s confidence does not necessarily weaken its generalisation performance. Instead, a model needs to be careful with what to be confident in. As shown by Figures 3b and 3c, ProSelfLC has the least wrong fitting and most semantic class correction, which indicates that a meaningful low-entropy status is redefined. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 REAL-WORLD LABEL NOISE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Clothing 1M (Xiao et al., 2015) has around $3 8 . 4 6 \\%$ label noise in the training data and about 1 million images of 14 classes from shopping websites. Its internal noise structure is agnostic. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines. For loss correction and estimating the noise-transition matrix, S-adaption (Goldberger & Ben-Reuven, 2017) uses an extra softmax layer, while Masking (Han et al., 2018a) exploits human cognition. MD-DYR-SH (Arazo et al., 2019) is a combination of three techniques: dynamic mixup (MD), dynamic bootstrapping together with label regularisation (DYR) and soft to hard (SH). The ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/909ce8b536e6b1e2d695268d2958e69c8a80ca847a8e6a90ab023e0c79e456ef.jpg",
+        "table_caption": [
+            "Table 4: Accuracy $( \\% )$ on the CIFAR-100 clean test set. All compared methods use ResNet-44. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Method</td><td colspan=\"3\">Asymmetric Noisy Labels</td><td rowspan=\"2\"></td><td colspan=\"3\">Symmetric Noisy Labels</td></tr><tr><td>r=0.2</td><td>r=0.3</td><td>r=0.4</td><td>r=0.2</td><td>r=0.4</td><td>r=0.6</td></tr><tr><td rowspan=\"5\">Results From SL (Wang et al., 2019)</td><td>Boot-hard</td><td>63.4</td><td>63.2</td><td>62.1</td><td></td><td>57.9</td><td>48.2</td><td>12.3</td></tr><tr><td>Forward</td><td>64.1</td><td>64.0</td><td>60.9</td><td></td><td>59.8</td><td>53.1</td><td>24.7</td></tr><tr><td>D2L</td><td>62.4</td><td>63.2</td><td>61.4</td><td></td><td>59.2</td><td>52.0</td><td>35.3</td></tr><tr><td>GCE</td><td>63.0</td><td>63.2</td><td>61.7</td><td></td><td>59.1</td><td>53.3</td><td>36.2</td></tr><tr><td>SL</td><td>65.6</td><td>65.1</td><td>63.1</td><td></td><td>60.0</td><td>53.7</td><td>41.5</td></tr><tr><td rowspan=\"5\">Our Trained Results</td><td>CCE</td><td>66.6</td><td>63.4</td><td>59.5</td><td></td><td>58.0</td><td>50.1</td><td>37.9</td></tr><tr><td>LS</td><td>67.9</td><td>66.4</td><td>65.0</td><td></td><td>63.8</td><td>57.2</td><td>46.5</td></tr><tr><td>CP</td><td>67.7</td><td>66.0</td><td>64.4</td><td></td><td>64.0</td><td>56.8</td><td>44.1</td></tr><tr><td>Boot-soft</td><td>66.9</td><td>65.3</td><td>61.0</td><td></td><td>63.2</td><td>59.0</td><td>44.8</td></tr><tr><td>ProSelfLC</td><td>68.7</td><td>68.5</td><td>67.9</td><td></td><td>64.8</td><td>59.3</td><td>47.7</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/25f76fb77eeb1383e93192567e32f6f47e6c072a3fed741ce5bde85a5742ed86.jpg",
+        "table_caption": [
+            "Table 5: The results of different hyperparameters on CIFAR-100 using ResNet-44. Under different noise rates, the best instantiation of each approach is bolded except for CCE. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Value of</td><td colspan=\"3\">Asymmetric label noise</td><td colspan=\"3\">Symmetric label noise</td><td rowspan=\"2\">Clean</td></tr><tr><td>(hyperparameter)</td><td>hyperparameter</td><td>20%</td><td>30%</td><td>40%</td><td>20%</td><td>40%</td><td>60%</td></tr><tr><td rowspan=\"2\">CCE</td><td>None or ε = 0</td><td>66.6</td><td>63.4</td><td>59.5</td><td>58.0</td><td>50.1</td><td>37.9</td><td>69.0</td></tr><tr><td>0.125</td><td>66.4</td><td>65.6</td><td>63.1</td><td>61.7</td><td>52.5</td><td>39.1</td><td>69.9</td></tr><tr><td rowspan=\"3\">LS (e)</td><td>0.25</td><td>67.9</td><td>66.4</td><td>65.0</td><td>62.8</td><td>55.9</td><td>40.9</td><td>69.6</td></tr><tr><td>0.50</td><td>66.8</td><td>65.8</td><td>64.6</td><td>63.8</td><td>57.2</td><td>46.5</td><td>68.4</td></tr><tr><td>0.125</td><td>65.7</td><td>64.2</td><td>60.3</td><td>59.8</td><td>52.3</td><td>39.6</td><td>69.5</td></tr><tr><td rowspan=\"3\">CP(e)</td><td>0.25</td><td>66.8</td><td>65.1</td><td>61.6</td><td>61.0</td><td>53.3</td><td>40.9</td><td>69.3</td></tr><tr><td>0.50</td><td>67.7</td><td>66.0</td><td>64.4</td><td>64.0</td><td>56.8</td><td>44.1</td><td>68.7</td></tr><tr><td>0.125</td><td>65.8</td><td>64.1</td><td>60.7</td><td>59.7</td><td>51.2</td><td>40.6</td><td>68.9</td></tr><tr><td rowspan=\"3\">Boot-soft (ε)</td><td>0.25</td><td>66.2</td><td>64.1</td><td>60.3</td><td>61.1</td><td>54.4</td><td>43.3</td><td>69.1</td></tr><tr><td>0.50</td><td>66.9</td><td>65.3</td><td>61.0</td><td>63.2</td><td>59.0</td><td>44.8</td><td>69.1</td></tr><tr><td>8</td><td>67.8</td><td>67.4</td><td>67.9</td><td>64.7</td><td>57.7</td><td>47.7</td><td>70.1</td></tr><tr><td rowspan=\"4\">ProSelfLC (B)</td><td>10</td><td>68.5</td><td>68.5</td><td>66.8</td><td>63.9</td><td>59.0</td><td>47.5</td><td>70.3</td></tr><tr><td>12</td><td>68.6</td><td>67.9</td><td>67.4</td><td>64.0</td><td>59.3</td><td>47.5</td><td>69.8</td></tr><tr><td>14</td><td>68.7</td><td>68.0</td><td>67.8</td><td>64.8</td><td>59.0</td><td>47.4</td><td>69.6</td></tr><tr><td>16</td><td>68.4</td><td>67.2</td><td>67.3</td><td>63.7</td><td>59.0</td><td>32.3</td><td>69.9</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/5c3a99364419320baeab29dbc3eaefccbe8b71d16319cfea9e373d8a2b3fa7d8.jpg",
+        "table_caption": [
+            "Table 6: Test accuracy $( \\% )$ on the real-world noisy dataset Clothing 1M. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Boot- hard</td><td rowspan=\"2\">Forward</td><td rowspan=\"2\">D2L</td><td rowspan=\"2\">GCE</td><td rowspan=\"2\">SL</td><td rowspan=\"2\">S- adaptation</td><td rowspan=\"2\">Masking</td><td rowspan=\"2\">MD- DYR-SH</td><td rowspan=\"2\">Joint- soft</td><td colspan=\"4\">Our Trained Results</td></tr><tr><td>CCE</td><td>LS</td><td>CP</td><td>Boot-soft ProSelfLC</td></tr><tr><td>68.9</td><td>69.8</td><td>69.5</td><td>69.8</td><td>71.0</td><td>70.3</td><td>71.1</td><td>71.0</td><td>72.2</td><td>71.8</td><td>72.6 72.4</td><td>72.3</td><td>73.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "other baselines have been introduced heretofore. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Training details. We follow (Tanaka et al., 2018) to train ResNet-50 and initialise it by a trained model on ImageNet. We follow Section 5.1 with small changes: the initial learning rate is 0.01 and we train 10k iterations. They are searched on the separate clean validation set. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Result analysis. In Table 6, analogously to CIFAR-100, we report our trained results of CCE, LS, CP, Boot-soft and ProSelfLC for an entirely fair comparison. ProSelfLC has the highest accuracy, which demonstrates its effectiveness again. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+            318,
+            780
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We present a thorough mathematical study on several target modification techniques. Through analysis of entropy and KL divergence, we reveal their relationships and limitations. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To improve and endorse self label correction, we propose ProSelfLC. Extensive experiments prove its superiority over existing methods under standard and noisy settings. ProSelfLC enhances the similarity structure information over classes, and rectifies the semantic classes of noisy label distributions. ProSelfLC is the first approach to trust self knowledge progressively and adaptively. ",
+        "bbox": [
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+        "type": "text",
+        "text": "ProSelfLC redirects and promotes entropy minimisation, which is in marked contrast to recent practices of confidence penalty (Szegedy et al., 2016; Pereyra et al., 2017; Dubey et al., 2018). ",
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+        "bbox": [
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+    {
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+        "text": "Zhilu Zhang and Mert R Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. In NeurIPS, 2018. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A PROOF OF PROPOSITIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            421,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. Compared with CCE, LS and $C P$ penalise entropy minimisation while $L C$ reward it. Proof. We can rewrite CCE, LS, CP, and LC from the viewpoint of KL divergence: ",
+        "bbox": [
+            171,
+            133,
+            820,
+            162
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf2c32d920cd3833c3eaea59690db16e8b8a2b9931738fece5bcaa12067746e7.jpg",
+        "text": "$$\nL _ { \\mathrm { C C E } } ( \\mathbf { q } , \\mathbf { p } ) = \\mathrm { H } ( \\mathbf { q } , \\mathbf { p } ) = \\mathrm { K L } ( \\mathbf { q } | | \\mathbf { p } ) + \\mathrm { H } ( \\mathbf { q } , \\mathbf { q } ) = \\mathrm { K L } ( \\mathbf { q } | | \\mathbf { p } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            167,
+            694,
+            185
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where we have $\\mathrm { H } ( \\mathbf { q } , \\mathbf { q } ) = 0$ because $\\mathbf { q }$ is a one-hot distribution. ",
+        "bbox": [
+            176,
+            193,
+            591,
+            208
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9478f1274643093262418a73b2d9817efc531915e3d8bc6b94fea83ac4cf90d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { \\mathrm { C C E + L S } } ( \\mathbf { q } , \\mathbf { p } ; \\epsilon ) = ( 1 - \\epsilon ) \\mathrm { K L } ( \\mathbf { q } | | \\mathbf { p } ) + \\epsilon \\mathrm { K L } ( \\mathbf { u } | | \\mathbf { p } ) + \\epsilon \\mathrm { H } ( \\mathbf { u } , \\mathbf { u } ) } \\\\ & { \\quad \\quad \\quad = ( 1 - \\epsilon ) \\mathrm { K L } ( \\mathbf { q } | | \\mathbf { p } ) + \\epsilon \\mathrm { K L } ( \\mathbf { u } | | \\mathbf { p } ) + \\epsilon \\cdot \\mathrm { c o n s t a n t } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            213,
+            723,
+            252
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c95152480cc2f65878af5c19aeb537a3f5f3e9701e56352dc9f88f5ec43ef941.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { \\mathrm { C C E + C P } } (  { \\mathbf { q } } ,  { \\mathbf { p } } ; \\epsilon ) = ( 1 - \\epsilon ) \\mathrm { K L } (  { \\mathbf { q } } | |  { \\mathbf { p } } ) - \\epsilon ( \\mathrm { H } (  { \\mathbf { p } } ,  { \\mathbf { u } } ) - \\mathrm { K L } (  { \\mathbf { p } } | |  { \\mathbf { u } } ) ) } \\\\ & { \\qquad = ( 1 - \\epsilon ) \\mathrm { K L } (  { \\mathbf { q } } | |  { \\mathbf { p } } ) + \\epsilon \\mathrm { K L } (  { \\mathbf { p } } | |  { \\mathbf { u } } ) - \\epsilon \\cdot \\mathrm { c o n s t a n t } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            256,
+            722,
+            291
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { H } ( \\mathbf { p } , \\mathbf { u } ) = \\mathrm { H } ( \\mathbf { u } , \\mathbf { u } ) = { \\mathrm { c c } }$ onstant. Analogously, LC in Eq (5) can also be rewritten: ",
+        "bbox": [
+            173,
+            294,
+            750,
+            310
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1dd57dcecc0d0fece5f01781e1d666ad81b224e33d9b88d809ef3333427529c2.jpg",
+        "text": "$$\nL _ { \\mathrm { C C E + L C } } ( \\mathbf { q } , \\mathbf { p } ; \\epsilon ) = ( 1 - \\epsilon ) \\mathrm { K L } ( \\mathbf { q } | | \\mathbf { p } ) - \\epsilon \\mathrm { K L } ( \\mathbf { p } | | \\mathbf { u } ) + \\epsilon \\cdot \\mathrm { c o n s t a n t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            315,
+            717,
+            333
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In LS and CP, both $+ \\mathrm { K L } ( \\mathbf { u } | | \\mathbf { p } )$ and $+ \\mathrm { K L } ( \\mathbf { p } | | \\mathbf { u } )$ pulls $\\mathbf { p }$ towards $\\mathbf { u }$ . While in LC, the term $- \\mathrm { K L } ( \\mathbf { p } | | \\mathbf { u } )$ pushes $\\mathbf { p }$ away from $\\mathbf { u }$ . \u0003 ",
+        "bbox": [
+            171,
+            339,
+            823,
+            368
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4. In CCE, $L S$ and $C P ,$ a data point x has the same semantic class. In addition, x has an identical probability of belonging to other classes except for its semantic class. ",
+        "bbox": [
+            171,
+            375,
+            820,
+            404
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. In LS, the target is $\\tilde { \\mathbf { q } } _ { \\mathrm { L S } } = ( 1 - \\epsilon ) \\mathbf { q } + \\epsilon \\mathbf { u }$ . For any $0 \\leq \\epsilon < 1$ , the semantic class is not changed, because $1 - \\epsilon + \\epsilon * \\overset { - } { \\underset { C } { \\ F } } > \\bar { \\epsilon } * \\frac { 1 } { C }$ . In addition, $\\begin{array} { r } { j _ { 1 } \\neq y , \\dot { j } _ { 2 } \\neq y \\Rightarrow \\tilde { \\mathbf { q } } _ { \\mathrm { L S } } ( j _ { 1 } | \\mathbf { x } ) = \\tilde { \\mathbf { q } } _ { \\mathrm { L S } } ( j _ { 2 } | \\mathbf { x } ) = \\frac { \\epsilon } { C } } \\end{array}$ . ",
+        "bbox": [
+            169,
+            404,
+            826,
+            433
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In CP, $\\tilde { \\bf q } _ { \\mathrm { C P } } = ( 1 - \\epsilon ) { \\bf q } - \\epsilon { \\bf p }$ . In terms of label definition, $C P$ is against intuition because these zero-value positions in q are filled with negative values in $\\tilde { \\mathbf { q } } _ { \\mathrm { C P } }$ . A probability has to be not smaller than zero. So we rephrase $\\tilde { \\mathbf { q } } _ { \\mathrm { C P } } ( y | \\mathbf { x } ) = ( 1 - \\epsilon ) - \\epsilon * \\mathbf { p } ( y | \\mathbf { x } )$ , and $\\forall j \\neq y , \\tilde { \\mathbf { q } } _ { \\mathrm { C P } } ( j | \\mathbf { x } ) = 0$ by replacing negative values with zeros, as illustrated in Figure 1a. \u0003 ",
+        "bbox": [
+            173,
+            438,
+            825,
+            494
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B LEARNING DYNAMICS OF DIFFERENT NOISE RATES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            515,
+            632,
+            530
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 4, we store a model every 1000 iterations to monitor the learning process. ",
+        "bbox": [
+            173,
+            545,
+            718,
+            561
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C THE CHANGES OF ENTROPY STATISTICS AND \u000fProSelfLC AT TRAINING ",
+        "bbox": [
+            169,
+            580,
+            771,
+            598
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we visualise how the entropies of noisy and clean subsets change at training. ",
+        "bbox": [
+            171,
+            611,
+            756,
+            627
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/d88add6873c102c97acc6fd4a47f9e73306162e4658746c17c92302937b7099f.jpg",
+        "image_caption": [
+            "Figure 4: Learning dynamics on CIFAR-100 under asymmetric noisy labels. We show all iterations only in (a) and (d). In the others, we show the second half iterations, which are of higher interest. As the noise rate increases, the superiority of ProSelfLC becomes more significant, i.e., avoiding fitting noise in the 2nd row and leading to better generalisation in the 1st row. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            143,
+            818,
+            454
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/87d3947c11ab7a345e37aa40de395b01cf8f4a64c3228a8fe122237e7d38ce9f.jpg",
+        "image_caption": [
+            "Figure 5: The changes of entropy statistics and \u000fProSelfLC at training. We store a model every 1000 iterations to monitor the learning process. For data-dependent metrics, after training, we split the corrupted training data into clean and noisy subsets according to the information about how the training data is corrupted before training. Finally, we report the mean results of each subset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            608,
+            813,
+            820
+        ],
+        "page_idx": 12
+    }
+]

parse/train/63pC59XOZLZ/63pC59XOZLZ.md

@@ -0,0 +1,263 @@
+# Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer
+
+Yining $\mathbf { M } \mathbf { a } ^ { 1 }$ , Jingwen $\mathbf { L i } ^ { 1 }$ , Zhiguang $\mathbf { C a o ^ { 2 , * } }$ , Wen Song3,∗, Le Zhang4, Zhenghua Chen5, Jing Tang6
+
+1National University of Singapore 2Singapore Institute of Manufacturing Technology, A\*STAR 3Institute of Marine Science and Technology, Shandong University 4University of Electronic Science and Technology of China 5Institute for Infocomm Research, A\*STAR 6The Hong Kong University of Science and Technology {yiningma, lijingwen}@u.nus.edu, zhiguangcao@outlook.com, wensong@email.sdu.edu.cn, zhangleuestc@gmail.com, chen0832@e.ntu.edu.sg, jingtang@ust.hk
+
+# Abstract
+
+Recently, Transformer has become a prevailing deep architecture for solving vehicle routing problems (VRPs). However, it is less effective in learning improvement models for VRP because its positional encoding (PE) method is not suitable in representing VRP solutions. This paper presents a novel Dual-Aspect Collaborative Transformer (DACT) to learn embeddings for the node and positional features separately, instead of fusing them together as done in existing ones, so as to avoid potential noises and incompatible correlations. Moreover, the positional features are embedded through a novel cyclic positional encoding (CPE) method to allow Transformer to effectively capture the circularity and symmetry of VRP solutions (i.e., cyclic sequences). We train DACT using Proximal Policy Optimization and design a curriculum learning strategy for better sample efficiency. We apply DACT to solve the traveling salesman problem (TSP) and capacitated vehicle routing problem (CVRP). Results show that our DACT outperforms existing Transformer based improvement models, and exhibits much better generalization performance across different problem sizes on synthetic and benchmark instances, respectively.
+
+# 1 Introduction
+
+Vehicle Routing problems (VRPs), such as the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) which consider finding the optimal route for a single or fleet of vehicles to serve a set of customers, have ubiquitous real-world applications [1, 2]. Despite being intensively studied in the Operations Research (OR) community, VRPs still remain challenging due to their NP-hard nature [3]. Recent studies on learning neural heuristics are gathering attention as promising extensions to traditional hand-crafted ones (e.g., [4–14]), where reinforcement learning (RL) [15] is usually exploited to train a deep neural network as an efficient solver without hand-crafted rules. A salient motivation is that deep neural networks may learn better heuristics by identifying useful patterns in an end-to-end and data-driven fashion.
+
+Solutions to VRPs, i.e., routes, are sequences of nodes (customer and depot locations). Naturally, deep models for Natural Language Processing (NLP), which deal with sequence data as well, are ideal choices for encoding VRP solutions. Given its remarkable performance in NLP tasks, Transformer [16] is standing at the forefront in the learning based methods for VRPs (e.g., [5, 7, 8, 11–13, 17]). The original Transformer encodes a sentence, i.e., a sequence of words, into a unified set of embeddings by injecting word positional information into its word embeddings through positional encoding (PE). When it comes to VRPs, while is not required in construction models, positional information is critical for deep models that learn improvement heuristics since the input are solutions to be improved.
+
+![](images/17403d04b62870e5603e11128aad37f05c9529785e72e7e24fbc237438272fea.jpg)  
+Figure 1: Transformer frameworks for VRPs. (a) $\mathrm { W u }$ et al. [11] (the original one); (b) DACT (ours).
+
+Although some success has been achieved, learning improvement heuristics for VRPs based on the original Transformer encoder is yet lacking from our perspective. Firstly, directly applying addition operation on PE vectors and the embeddings in absolute PE method (i.e., Figure 1(a)) could limit the representation of the model [18], as the mixed correlations2 existing in the self-attention can bring unreasonable noises and random biases to the encoder (details in Appendix A). Secondly, existing PE methods tend to fuse the node and positional information into one unified representation. NLP tasks such as translation may benefit from this owing to the deterministic and instructive nature of the positional information. However, such design may not be optimal for routing tasks because the positional information therein can be non-deterministic and sometimes even random. This may cause disharmony or disturbance in the encoder and may thus deteriorate the performance. Finally, most VRPs seek the shortest loop of the nodes, making their solutions to be cyclic sequences. However, existing PE methods are only designated to encode linear sequences3, which may fail to identify such circular input. As will be shown in our experiments, this could severely damage the generalization performance, since the cyclic feature of VRP solutions is not correctly reflected by the encoder.
+
+In this paper, we address the above issues and contribute to the line of using RL to learn neural improvement heuristics for VRPs. We introduce the Dual-Aspect Collaborative Transformer (DACT), where we revisit the solution representations and propose to learn separated groups of embeddings for the node and positional features of a VRP solution as shown in Figure 1(b). Our DACT follows the encoder-decoder structure. In the encoder, each set of embeddings encodes the solution mainly from its own aspect, and at the same time exploits a cross-aspect referential attention mechanism for better perceiving the consistence and differentiation with respect to the other aspect. The decoder then collects action distribution proposals from the two aspects and synthesizes them to output the final one. Meanwhile, we design a novel cyclic positional encoding (CPE) method to capture the circularity and symmetry of VRP solutions, which allows Transformer to encode cyclic inputs, and also boost the generalization performance for solving VRPs. As the last contribution, we design a simple yet effective curriculum learning strategy to improve the sample efficiency. This further leads to faster and more stable convergence of RL training. Extensive experiments show that our DACT can outperform existing Transformer based improvement models with fewer parameters, and also generalizes well across different sizes of synthetic and benchmark instances, respectively.
+
+# 2 Related work
+
+# 2.1 Positional encoding (PE) in Transformer.
+
+The original Transformer adopted the absolute PE method to describe the absolute position of elements in the sequence [16], especially for NLP. As formulated in Eq. (1), each generated positional embedding $p _ { i } \in \mathbb { R } ^ { d }$ is added together with the $i$ -th word embedding $x _ { i }$ in the first layer of the encoder,
+
+$$
+\alpha _ { i , j } ^ { \mathrm { A b s } } = \frac { 1 } { \sqrt { d } } ( ( x _ { i } + p _ { i } ) W ^ { Q } ) ( ( x _ { j } + p _ { j } ) W ^ { K } ) ^ { T } .
+$$
+
+The relative PE method was further proposed in Shaw et al. [19] to better capture the relative order information. On the basis of absolute PE, it introduces an inductive bias to the attention as follows,
+
+$$
+\alpha _ { i , j } ^ { \mathrm { R e l } } = \frac { 1 } { \sqrt { d } } ( ( x _ { i } + p _ { i } ) W ^ { Q } ) ( ( x _ { j } + p _ { j } ) W ^ { K } + a _ { j - i } ) ^ { T } ,
+$$
+
+where $a _ { j - i } \in \mathbb { R } ^ { d }$ is learnable parameters for encoding the relative position $j - i$ . To avoid the mixed and noisy correlations between word semantics and positional information in the above two PEs, the Transformer with United Positional Encoding (TUPE) [18] was proposed for NLP which utilizes separated projection metrics $W _ { x }$ and $W _ { p }$ for each information as follows,
+
+$$
+\alpha _ { i , j } ^ { \mathrm { T U P E } } = \frac { 1 } { \sqrt { 2 d } } ( x _ { i } W _ { x } ^ { Q } ) ( x _ { j } W _ { x } ^ { K } ) ^ { T } + \frac { 1 } { \sqrt { 2 d } } ( p _ { i } W _ { p } ^ { Q } ) ( p _ { j } W _ { p } ^ { K } ) ^ { T } + b _ { j - i } .
+$$
+
+However, as mentioned previously, existing PE methods are less effective for VRPs since they simply fuse the node and positional information into one unified set of embeddings during or after the calculation of the attention correlation $\alpha _ { i , j }$ . Meanwhile, they are also unable to properly encode and handle cyclic input sequences as in VRP solutions.
+
+# 2.2 Deep models for VRP.
+
+Various deep architectures such as Recurrent Neural Network (RNN), Graph Neural Network (GNN), and Transformer have been employed in solving VRPs.
+
+RNN based models. As the pioneering work of neural VRP solvers, Pointer Network adopted RNN and supervised learning to solve TSP [20] (extended to RL in Bello et al. [21] and CVRP in Nazari et al. [22]). While the models in [20–23] learn construction heuristics, NeuRewriter [4] learns improvement heuristic for CVRP using LSTM to encode the positional information of a solution. In Hottung et al. [13], the conditional variational autoencoder was adopted to learn a continuous and latent search space for VRP, where high-quality solutions were taken as input and encoded by RNNs. However, recurrence structures in RNN are less efficient in both representation and computation [5].
+
+GNN based models. In Dai et al. [24], GNN was combined with Q-learning for solving TSP. Based on supervised learning, Joshi et al. [6] used GNN to learn heatmaps that prescribe the probability of each edge appearing in the optimal TSP tour. This idea was extended in Fu et al. [25] with additional components such as graph sampling and heatmap merging to enable generalization to larger TSP instances. These models often require post-processing to construct feasible solutions from heatmaps (e.g., beam search [6], Monte-Carlo tree search [25], and dynamic programming [26]).
+
+Transformer based models. The Attention Model (AM) by Kool et al. [5] was recognized as the first success of Transformer based models for VRPs. Based on AM, Xin et al. [7] proposed a MultiDecoder AM that learns multiple diverse policies for better performance. In Kwon et al. [8], the RL algorithm of AM was improved which leaded to a new solver, i.e., POMO (Policy Optimization with Multiple Optima), and achieved the state-of-the-art performance. However, POMO is still lacking in generalization. Besides these construction models, Transformer was also explored to learn improvement heuristics. Hottung and Tierney [27] learned first neural large neighborhood search algorithm for VRPs. Lu et al. [12] proposed the L2I model that learns to select local search operators from a pool of traditional ones. Both methods used a Transformer-style encoder, but the positional information is captured in the node features (information of previous and next nodes) instead of using PE methods. Though L2I was shown to outperform LKH3 [28], it is limited to CVRP and the required time is considerably long. Wu et al. [11] proposed a Transformer model which learns to pick node pair in each step to perform a pairwise local operator (e.g., 2-opt). However, it suffers from the inaccurate representation of positional information given the original Transformer encoder.
+
+![](images/5ace11ce359bbb755998396b80534ff05688295f43970028840cdf0661b073b3.jpg)  
+Figure 3: Architecture of our policy network, dual-aspect collaborative Transformer (DACT).
+
+# 3 Problem formulation
+
+We define a VRP instance as a group of $N$ nodes to visit, where the node feature $x _ { i }$ of node $i$ contains 2-dim coordinates and other problem-specific features (e.g., customer demand). A solution $\delta$ consists of a sequence of nodes visited in order where we denote $p _ { i }$ to be the position (indices) of node $i$ in the solution which is deemed as the positional feature of node $i$ . The objective is to minimize the total travel distance $D ( \delta )$ under certain problem-specific constraints.
+
+Starting with an initial yet complete solution, our neural RL policy tries to improve the solution iteratively. At each step, the policy automatically selects a pair of nodes and locally adjusts the solution using a preset pairwise operator such as 2-opt, insert, or swap. As illustrated in Figure 2, given a node pair $( i , j )$ , the 2-opt operator adjusts a solution by reversing the segment between node $i$ and node $j$ ; the insert operator adjusts a solution by placing node $i$ after node $j$ ; and the swap operator adjusts a solution by exchanging the position of node $i$ and node $j$ . Such operation is repeated until reaching the step limit $T$ and we model it in the form of Markov Decision Process (MDP) as follows.
+
+![](images/3abc3c66ae47d31052903aafee9790f52c0e269b59a1a7ed0217c149d678c265.jpg)  
+Figure 2: Illustration examples of three pairwise operators for routing problems when node pair $( i = 2 , j = 1 )$ ) is specified for operating. From left to right: 2-opt, insert, and swap.
+
+State. For an instance with $N$ nodes, a state describes current solution $\delta _ { t }$ using its node and positional features of each node, i.e., $s _ { t } = \Psi ( \delta _ { t } ) = \{ x _ { 1 } ^ { t } , . . . , x _ { N } ^ { t } , p _ { 1 } ^ { t } , . . . , p _ { N } ^ { t } \}$ .
+
+Action. The action $a _ { t } = ( i , j )$ specifies a node pair $( i , j )$ for the pairwise operator.
+
+Reward. The reward function is defined as, $r _ { t } = D ( \delta _ { t } ^ { * } ) - m i n \left[ D ( \delta _ { t + 1 } ) , D ( \delta _ { t } ^ { * } ) \right]$ where $\delta _ { t } ^ { * }$ is the best incumbent solution found until time $t$ . It refers to the immediate reduced cost at each step with respects to the best incumbent solution, which ensures the cumulative reward equal to the total reduced cost over the initial solution. Hence the reward $r _ { t } > 0$ if and only if a better solution is found. Policy. The policy $\pi _ { \theta }$ is parameterized by the proposed DACT model with parameters $\theta$ . At each time step, the action $( i , j )$ is obtained by sampling the stochastic policy for both training and inference.
+
+Transition. The next state $s _ { t + 1 }$ is originated from $s _ { t }$ by performing the preset pairwise operator on the given node pair (action). Our state transient is deterministic, in the sense that it always accepts the next solution as the next state (infeasible solutions will be masked), regardless of its objective value. With such simple rule, the RL agent is expected to automatically learn how to combine multiple steps of simple local movements to achieve better solutions, even if some of them may worsen the current solution. Note that the step limit $T$ can be any user-specified value according to the allowed time budget. Hence, our MDP can have infinite horizon and we consider the reward discount factor $\gamma < 1$ .
+
+# 4 Dual-aspect collaborative Transformer model
+
+We now present the details of our Dual-Aspect Collaborative Transformer (DACT). The concrete architecture of DACT is presented in Figure 3, where we take the TSP with $N$ nodes as an illustration example. Our DACT leverages separate aspects of embeddings to encode a VRP solution. In the DAC encoder, the self-attention correlations are computed individually for each aspect, and a cross-aspect referential attention mechanism is proposed to enable one aspect to effectively exploit attention correlations from the other aspect as optional references. The DAC decoder then collects action distribution proposals from both aspects and synthesize them to the final one.
+
+![](images/89f6fef802cd35891d1ed76d002b348f40ea6b89b2545c5f6d954311286d8b4a.jpg)  
+Figure 5: Comparison of our CPE method with absolute PE method on a TSP instance with 20 nodes. (a) the embedding vectors, (b) the correlations (dot products) between every two embeddings, and (c) the top two principal components after PCA (principal component analysis) projection.   
+Figure 4: An example of cyclic Gray code where 4 digits are used to encode $N { = } 1 6$ nodes. The top left shows the base symmetry pattern $\cdot _ { 1 0 0 1 }$ ’ in Gray code, and the top right plots its representation in our method.
+
+# 4.1 Dual-aspect solution representation
+
+Specifically, we propose to learn two sets of embeddings, i.e., the node feature embeddings (NFEs) for node representation and the positional feature embeddings (PFEs) for positional representation.
+
+NFEs. Following [5, 11], the NFE $h _ { i }$ of node $i$ is initialized as the linear projection of its node feature $x _ { i }$ with output dimension4 $d i m = 6 4$ .
+
+PFEs. The PFE $g _ { i }$ of the positional feature $p _ { i }$ is initialized as a real-valued vector $( d i m = 6 4 )$ by applying our cyclic positional encoding (CPE), which is designed based on cyclic Gray codes [29].
+
+<table><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>16</td></tr><tr><td rowspan=2 colspan=1>11</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr></table>
+
+As illustrated in Figure 4, the cyclic Gray codes present a cyclic property (‘1110’ in the last column is adjacent to ‘1111’ in the first column) and an adjacency similarity property (any codes in adjacent columns only differ in one digit), both of which are desirable for cyclic sequences. To preserve these properties in designing our CPE, we follow two observed patterns: 1) each numerical digit contains a periodic cycle with reflectional symmetry, e.g., the $\mathbf { \dot { \rho } } _ { 1 0 | 0 1 } ,$ in the lowest digit; and 2) the higher the numerical digit, the longer the period. Accordingly, we create similar patterns based on the sinusoidal functions in Eq. (4), where a periodic function with period $\frac { 4 \pi } { \omega _ { d } }$ (induced by modulus) is used to generate one base symmetry pattern (the top right in Figure 4),
+
+$$
+\begin{array} { r } { \overrightarrow { g _ { i } ^ { \prime } } ^ { ( d ) } : = \left\{ \begin{array} { l l } { s i n ( \omega _ { d } \cdot \mathrm { \Gamma } ( z ( i ) \bmod \frac { 4 \pi } { \omega _ { d } } ) - \frac { 2 \pi } { \omega _ { d } } \mathrm { \Gamma } ) , \mathrm { ~ i f ~ } d \mathrm { ~ i s ~ e v e n } } \\ { c o s ( \omega _ { d } \cdot \mathrm { \Gamma } ( z ( i ) \bmod \frac { 4 \pi } { \omega _ { d } } ) - \frac { 2 \pi } { \omega _ { d } } \mathrm { \Gamma } ) , \mathrm { ~ i f ~ } d \mathrm { ~ i s ~ o d d } } \end{array} \right. } \end{array}
+$$
+
+$\begin{array} { r } { z ( i ) = \frac { i - 1 } { N } \frac { 2 \pi } { \omega _ { d } } \left\lceil \frac { N + 1 } { 2 \pi / \omega _ { d } } \right\rceil } \end{array}$ is to make $N$ nodes linearly spaced in the generated pattern; the angular frequency $\omega _ { d }$ is decreasing along the dimension to make the wavelength longer within the range $[ N ^ { \frac { 1 } { [ d i m / 2 ] } } , N ]$ (see Appendix B for details). In Figure 5, we visualize the comparison between the absolute PE and our CPE for encoding a TSP instance of 20 nodes. Figure 5(a) demonstrates that our real-valued base symmetry pattern has a longer cyclic period as the digit grows. Figure 5(b) indicates that our method (blue) is able to correctly reflect the adjacency between the head and tail of the cyclic sequence whereas the PE method (red) fails to do so. Figure 5(c) verifies that our CPE vectors are well distributed in space with desired cyclic and adjacency similarity properties.
+
+# 4.2 The encoder
+
+The encoder consists of $L = 3$ stacked DAC encoders. In each DAC encoder, we retain relatively independent encoding stream for NFEs and PFEs as in Eq. (5) and Eq. (6), respectively, each of which consists of a shared Dual-Aspect Collaborative Attention (DAC-Att) sub-layer and an independent feed-forward network (FFN) sub-layer. DAC-Att takes both sets of embeddings as input and then outputs their respective enhanced embeddings, i.e., NFEs $\{ \tilde { h } \} _ { i = 1 } ^ { N }$ and PFEs $\{ \tilde { g } \} _ { i = 1 } ^ { N }$ . Each sub-layer is followed by skip connection [30] and layer normalization [31] as same as the original Transformer.
+
+$$
+\begin{array} { r } { h _ { i } ^ { ( l ) } = \mathbf { L N } \Big ( h _ { i } ^ { \prime } + \mathbf { F F N } _ { h } ^ { ( l ) } ( h _ { i } ^ { \prime } ) \Big ) , h _ { i } ^ { \prime } = \mathbf { L N } \Big ( h _ { i } ^ { ( l - 1 ) } + \tilde { h } _ { i } ^ { ( l ) } \Big ) , } \\ { g _ { i } ^ { ( l ) } = \mathbf { L N } \Big ( g _ { i } ^ { \prime } + \mathbf { F F N } _ { g } ^ { ( l ) } ( g _ { i } ^ { \prime } ) \Big ) , g _ { i } ^ { \prime } = \mathbf { L N } \Big ( g _ { i } ^ { ( l - 1 ) } + \tilde { g } _ { i } ^ { ( l ) } \Big ) . } \end{array}
+$$
+
+DAC-Att. The DAC-Att sub-layer enhances each set of embedding from its own aspect, while leveraging attention correlations from the other aspect to achieve the synergy. Given the two sets of embeddings5, $\{ h _ { i } \} _ { i = 1 } ^ { N }$ and $\{ g _ { i } \} _ { i = 1 } ^ { N }$ , we first compute the self-attention correlation from both aspects,
+
+$$
+\alpha _ { i , j } ^ { h } = \frac { 1 } { \sqrt { d _ { k } } } \left( h _ { i } W _ { h } ^ { Q } \right) \left( h _ { j } W _ { h } ^ { K } \right) ^ { T } , \quad \alpha _ { i , j } ^ { g } = \frac { 1 } { \sqrt { d _ { k } } } \left( g _ { i } W _ { g } ^ { Q } \right) \left( g _ { j } W _ { g } ^ { K } \right) ^ { T } ,
+$$
+
+where independent matrices $W _ { h } ^ { Q } , W _ { h } ^ { K } , W _ { g } ^ { Q }$ and $W _ { g } ^ { K } \in \mathbb { R } ^ { d i m \times d _ { k } }$ are used to calculate queries and keys. The obtained correlations are further normalized to $\tilde { \alpha } _ { i , j } ^ { h }$ and $\tilde { \alpha } _ { i , j } ^ { g }$ via Softmax. Note that the correlations are computed from their own aspect, which eliminates possible noises and conduces to correctly describe the incompatible node pair relationships in different aspects of VRP solutions.
+
+We then exploit a cross-aspect referential attention mechanism, which allows computed correlations to be shared between each other, as additional references for both contradistinction and collaboration,
+
+$$
+\mathrm { o u t } _ { i } ^ { h } = \mathrm { C o n c a t } \left[ \sum _ { j = 1 } ^ { N } \tilde { \alpha } _ { i , j } ^ { h } \left( h _ { j } W _ { h } ^ { V } \right) , \sum _ { j = 1 } ^ { N } \tilde { \alpha } _ { i , j } ^ { g } \left( h _ { j } W _ { h } ^ { V r e f } \right) \right] ,
+$$
+
+$$
+\mathrm { o u t } _ { i } ^ { g } = \mathrm { C o n c a t } \left[ \sum _ { j = 1 } ^ { N } { \tilde { \alpha } } _ { i , j } ^ { g } \left( g _ { j } W _ { g } ^ { V } \right) , \sum _ { j = 1 } ^ { N } { \tilde { \alpha } } _ { i , j } ^ { h } \left( g _ { j } W _ { g } ^ { V r e f } \right) \right] ,
+$$
+
+wheand $W _ { h } ^ { V } , W _ { g } ^ { V } \in \mathbb R ^ { d i m \times d _ { v } }$ are trainable parameter matrices for formulating values in each aspect; are parameter matrices for each aspect to generate referential values. $W _ { h } ^ { V r e f } , W _ { g } ^ { V r e f } \in \mathbb R ^ { d i m \times d _ { v } }$ We finally use the multi-head attention to get NFEs $\tilde { h } _ { i }$ and PFEs $\tilde { g } _ { i }$ as follows,
+
+$$
+\begin{array} { r } { \begin{array} { c } { \tilde { h } _ { i } , \tilde { g } _ { i } = \mathbf { D A C - A t t } \left( W ^ { Q } , ~ W ^ { K } , W ^ { V } , W ^ { V _ { r e f } } , W ^ { O } \right) , } \\ { \tilde { h } _ { i } = \mathbf { C o n c a t } \left[ \mathrm { h e a d } _ { i , 1 } ^ { h } , . . . , \mathrm { h e a d } _ { i , m } ^ { h } \right] W _ { h } ^ { O } , ~ \tilde { g } _ { i } = \mathbf { C o n c a t } \left[ \mathrm { h e a d } _ { i , 1 } ^ { g } , . . . , \mathrm { h e a d } _ { i , m } ^ { g } \right] W _ { g } ^ { O } , } \end{array} } \end{array}
+$$
+
+where h $\mathbf { e a d } _ { i , k } ^ { h } = o u t _ { i , k } ^ { h }$ , $\mathbf { h e a d } _ { i , k } ^ { g } = o u t _ { i , k } ^ { g }$ , and $W _ { h } ^ { O } , W _ { g } ^ { O } \ \in \ \mathbb { R } ^ { 2 m d _ { v } \times d i m }$ are trainable parameter matrices. In our model, we adopt $m = 4$ and $d _ { k } = d _ { v } = 1 6$ .
+
+FFN. Our FFN sub-layer has only one hidden layer with 64 hidden unites and adopts the ReLU activation function. The parameters of $\mathbf { F F N } _ { h }$ and $\mathbf { F F N } _ { g }$ are different for each group of embeddings.
+
+# 4.3 The decoder
+
+In the DAC decoder, the two sets of embeddings $\{ h _ { i } ^ { ( L ) } \} _ { i = 1 } ^ { N }$ and $\{ g _ { i } ^ { ( L ) } \} _ { i = 1 } ^ { N }$ are first passed through a Max-pooling sub-layer and a multi-head compatibility (MHC) sub-layer to independently generate diversified node-pair selection proposals from their own aspect, which are then aggregated through a feed-forward aggregation (FFA) sub-layer for output.
+
+Max-pooling. For each set of embeddings, we adopt the max-pooling sub-layer in Wu et al. [11] to aggregate the global representation of all $N$ embeddings into each respective one6.
+
+MHC. The compatibility sub-layer computes the attention correlations for each embedding pair, where the obtained correlations with size $N \times N$ will be deemed as a proposal distribution for node pair selection. Our correlations are computed based on multiple heads for diversity. And we calculate separated attention score matrices $\boldsymbol { Y } _ { k } ^ { h } , \boldsymbol { \dot { Y } } _ { k } ^ { g } \in \mathbb { R } ^ { N \times N }$ (of head $k$ ) from the two aspects independently. Accordingly, the action distribution proposals would be different due to their aspect-specific focus and cognitions of the current solution, which will provide the subsequent FFA layer with a rich pool of proposals and allow our model to be more flexible and robust.
+
+FFA. Once all proposals from two aspects are collected, a FFN with four layers (dimensions are $2 m$ , 32, 32 and 1, respectively) and ReLU activation is used to aggregate them,
+
+$$
+\tilde { Y } _ { i , j } = \mathbf { F } \mathbf { F } \mathbf { A } \left( Y _ { i , j , 1 } ^ { g } , . . . , Y _ { i , j , m } ^ { g } , Y _ { i , j , 1 } ^ { h } , . . . Y _ { i , j , m } ^ { h } \right) ,
+$$
+
+where $m = 4$ is the number of heads; and the output $\tilde { Y } _ { i , j }$ is a scalar indicating the likelihood of selecting node pair $( i , j )$ as an action. Afterwards, we apply $\hat { Y } _ { i j } = C \cdot \mathrm { T a n h } ( \tilde { Y } _ { i , j } )$ with $C = 6$ to control the entropy, and mask 7 the infeasible node pairs $( i ^ { \prime } , j ^ { \prime } )$ as $\hat { Y } _ { i ^ { \prime } j ^ { \prime } } = - \infty$ . Lastly, the likelihoods are normalized using Softmax function to obtain the final action distribution $P _ { i , j }$ .
+
+# 4.4 Reinforcement learning algorithm
+
+We adopt the proximal policy optimization [32] with $n$ -step return estimation for training (details are given in Appendix C), and design a curriculum learning (CL) strategy for better sample efficiency.
+
+Curriculum learning strategy. The strategy in Wu et al. [11] sets a maximum of $T _ { t r a i n }$ steps for training and estimates future returns by bootstrapping [33]. However, due to the concern of training cost, $T _ { t r a i n }$ is usually much smaller than actual $T$ for inference (e.g., 200 v.s. 10k), which may leave the agent a poor chance of observing high-quality solutions (states) during training. Consequently, it may cause high variance for bootstrapping because the value function is mostly fitted on low-quality solutions and may render it less knowledgeable in estimating long-term future returns accurately. In this paper, we tackle this issue by a simple yet efficient strategy which gradually prescribes higher-quality solutions as the initial states for training. In doing so, 1) it increases the probability for the agent to observe better solutions and thus reduce the variance of the value function; 2) it increases the difficulty of the learning task (higher-quality solutions are harder to improve) in a gradual manner and achieves better sample efficiency [34]. In practice, those higher-quality solutions can be easily achieved by improving the randomly generated ones using the current policy for a few $T _ { i n i t }$ steps, where $T _ { i n i t }$ could be slightly increased as the epoch grows.
+
+# 5 Experiments
+
+We evaluate our DACT model on two representative routing problems, i.e., TSP and CVRP [5, 8, 11]. For each problem, we abide by existing conventions to randomly generate instances on the fly for three sizes, i.e., $N = 2 0$ , 50 and 100. Initial experiments with three operators including 2-opt, swap and insert show that 2-opt performs best for both TSP and CVRP (with insert better than swap), hence we report results of our method based on 2-opt. Following [4, 11, 27] we use randomly generated initial solutions for training and the solutions generated by the greedy algorithm for inference. Since each problem has its own constraints and node features, we adjust the input, feasibility masks, and problem-dependent hyperparameters for each problem, the details of which are provided in Appendix D and E. The DACT is trained and tested on a server equipped with TITAN RTX GPU cards and Intel i9-10940X CPU at $3 . 3 0 \mathrm { G H z }$ . Our code in PyTorch are available here8.
+
+Table 1: Comparison with various baselines on TSP and CVRP.   
+
+<table><tr><td rowspan="2" colspan="2">Method</td><td colspan="3">N=20</td><td colspan="3">N=50</td><td colspan="3">N=100</td></tr><tr><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td></tr><tr><td rowspan="10">LKH P</td><td>Concorde</td><td>3.83</td><td>=</td><td>(3m)</td><td>5.70</td><td></td><td>(10m)</td><td>7.76</td><td></td><td>(1h)</td></tr><tr><td></td><td>3.83</td><td>0.00%</td><td>(38s)</td><td>5.70</td><td>0.00%</td><td>(5m)</td><td>7.76</td><td>0.00%</td><td>(20m)</td></tr><tr><td>OR-Tools</td><td>3.86</td><td>0.94%</td><td>(42s)</td><td>5.85</td><td>2.87%</td><td>(5m)</td><td>8.06</td><td>3.86%</td><td>(23m)</td></tr><tr><td>Neural-2-Opt [23]</td><td>3.84</td><td>0.00%</td><td>(15m)</td><td>5.70</td><td>0.12%</td><td>(29m)</td><td>7.83</td><td>0.87%</td><td>(41m)</td></tr><tr><td>Wu et al. [11] (T=5k)</td><td>3.83</td><td>0.00%</td><td>(1h)</td><td>5.70</td><td>0.20%</td><td>(1.5h)</td><td>7.87</td><td>1.42%</td><td>(2h)</td></tr><tr><td>DACT (T=1k)</td><td>3.83</td><td>0.04%</td><td>{7s}(24s)</td><td>5.70</td><td>0.14%</td><td>{16s}(1m)</td><td>7.89</td><td>1.62%</td><td>{48s}(4m)</td></tr><tr><td>DACT (T=5k)</td><td>3.83</td><td>0.00%</td><td>{32s}(2m)</td><td>5.70</td><td>0.02%</td><td>{2m}(6m)</td><td>7.81</td><td>0.61%</td><td>{4m}(18m)</td></tr><tr><td>DACT (T=10k)</td><td>3.83</td><td>0.00%</td><td>{1m}(5m)</td><td>5.70</td><td>0.01%</td><td>{3m}(13m)</td><td>7.79</td><td>0.37%</td><td>{8m}(40m)</td></tr><tr><td>DACT×4 augment</td><td>3.83</td><td>0.00%</td><td>{3m}(10m)</td><td>5.70</td><td>0.00%</td><td>{10m}(1h)</td><td>7.77</td><td>0.09%</td><td>{29m}(2.5h)</td></tr><tr><td>GCN-BS [6]</td><td>3.84</td><td>0.01%</td><td>(12m)</td><td>5.70</td><td>0.01%</td><td>(18m)</td><td>7.87</td><td>1.39%</td><td>(40m)</td></tr><tr><td>AM-sampling [5]</td><td>3.84</td><td>0.08%</td><td>(5m)</td><td>5.73</td><td>0.52%</td><td>(24m)</td><td>7.94</td><td>2.26%</td><td>(1h)</td></tr><tr><td>MDAM-BS[7]</td><td>3.84t</td><td>0.00%</td><td>(3m)</td><td>5.70</td><td>0.03%</td><td>(14m)</td><td>7.79</td><td>0.38%</td><td>(44m)</td></tr><tr><td>POMO [8]</td><td>3.83</td><td>0.04%</td><td>(1s)</td><td>5.70</td><td>0.21%</td><td>(2s)</td><td>7.80</td><td>0.46%</td><td>(11s)</td></tr><tr><td>POMO×8 augment [8]</td><td>3.83</td><td>0.00%</td><td>(3s)</td><td>5.69t</td><td>0.03%</td><td>(16s)</td><td>7.78</td><td>0.15%</td><td>(1m)</td></tr><tr><td>DPDP(100k) [26]</td><td>-</td><td></td><td></td><td>-</td><td></td><td>=</td><td>7.77+</td><td>0.00%</td><td>(3h)</td></tr><tr><td rowspan="9">LKH OR-Tools NeuRewriter [4] NLNS [27]</td><td rowspan="9">CVAE-Opt-DE [13]</td><td>1</td><td>0.00%#</td><td>11m#</td><td>-</td><td>0.02%#</td><td>22m#</td><td>-</td><td>0.34%#</td><td>55m#</td></tr><tr><td></td><td>0.00%</td><td></td><td></td><td></td><td>4h</td><td>15.68</td><td></td><td></td></tr><tr><td>6.14 6.46</td><td>5.68%</td><td>1h 2m</td><td>10.38 11.27</td><td>0.00% 8.61%</td><td>13m</td><td>17.12</td><td>0.00% 9.54%</td><td>8h 46m</td></tr><tr><td>6.15#</td><td></td><td>6m#</td><td>10.51#</td><td></td><td>11m#</td><td>16.10#</td><td></td><td></td></tr><tr><td>6.19#</td><td>=</td><td>6m#</td><td>10.54#</td><td></td><td>11m#</td><td>15.99#</td><td>=</td><td>17m# 16m#</td></tr><tr><td>Wu et al. [11] (T=5k) 6.12</td><td>0.39%</td><td>(2h)</td><td>10.45</td><td>0.70%</td><td>(4h)</td><td>16.03t</td><td>= 2.47%</td><td></td></tr><tr><td>DACT (T=1k)</td><td>0.28%</td><td>{16s}(33s)</td><td>10.61</td><td>2.13%</td><td>{43s}(2m)</td><td>16.17</td><td>3.18%</td><td>(5h) {2m}(5m)</td></tr><tr><td>DACT (T=5k)</td><td>6.15 6.13 -0.00%</td><td>{1m}(3m)</td><td>10.48</td><td>1.01%</td><td>{3m}(8m)</td><td>15.92</td><td>1.55%</td><td>{8m}(23m)</td></tr><tr><td>DACT (T=10k)</td><td>-0.04%</td><td>{2m}(6m)</td><td>10.46</td><td>0.79%</td><td>{6m}(16m)</td><td>15.85</td><td>1.12%</td><td>{16m}(45m)</td></tr><tr><td rowspan="2">DACT×6 augment</td><td>6.13 6.13</td><td>-0.08%</td><td>{11m}(35m)</td><td>10.39</td><td>0.14%</td><td>{32m}(1.5h)</td><td>15.71</td><td>0.19%</td><td>{1.5h}(4.5h)</td></tr><tr><td>AM-sampling [5]</td><td>1.87%</td><td>(6m)</td><td>10.62</td><td>2.40%</td><td>(28m)</td><td></td><td></td><td></td></tr><tr><td rowspan="2">MDAM-BS[7]</td><td>6.25 6.14</td><td>0.18%</td><td>(5m)</td><td>10.48</td><td>0.98%</td><td>(15m)</td><td>16.23 15.99#</td><td>3.72% 2.23%</td><td>(2h)</td></tr><tr><td></td><td>0.82%</td><td>(1s)</td><td>10.49</td><td>1.14%</td><td>(4s)</td><td>15.83</td><td>0.98%</td><td>(1h) (19s)</td></tr><tr><td>POMO [8] POMO×8 augment [8]</td><td>6.17 6.14</td><td>0.21%</td><td>(5s)</td><td>10.42</td><td>0.45%</td><td>(26s)</td><td>15.73</td><td>0.32%</td><td>(2m)</td></tr><tr><td>DPDP(100k)[26]</td><td></td><td></td><td></td><td></td><td></td><td></td><td>15.69</td><td>0.31%</td><td></td></tr><tr><td>CVAE-Opt-DE [13]</td><td>= 6.14#</td><td></td><td>= 21m#</td><td>= 10.40#</td><td></td><td>41m#</td><td>15.75#</td><td></td><td>(6h) 1.5h#</td></tr></table>
+
+# the obj. values, gaps or time are obtained based on 2,000 instances in their original papers, and not directly comparable to ours. ‡ the obj. values obtained by Concorde or LKH may be slightly different from ours since the 10,000 instances are randomly generated. E.g., for TSP50, the optimal values according to our running of Concorde is 5.70, while 5.69 in POMO and Wu et al.. We thus focus more on gaps.
+
+# 5.1 Comparison studies
+
+In Table 1, we compare our DACT with, (1) learning based improvement methods, including Wu et al. [11], Neural-2-Opt [23] (TSP only), NeuRewriter [4] (CVRP only), NLNS [27] (CVRP only), (2) learning based construction methods, including AM-sampling [5], GCN-BS [6] (TSP only), MDAM-BS [7], POMO [8], (3) conventional optimization algorithms equipped with learning based component(s), including DPDP [26], CVAE-Opt-DE [13], and (4) strong conventional solvers including Concorde [35], LKH [28, 36], and OR-Tools [37]. Though L2I [12] can outstrip LKH on CVRP, we do not inlude it as a baseline since it requires a prohibitively longer inference time than others9. All results are averaged over 10,000 randomly generated instances unless specified otherwise (e.g., the ones marked with # only infer 2,000 instances), and we report the metrics of objective values, (optimality) gaps and run time. Regarding baselines, we follow the results reported in their original papers, which may not include all the three metrics. For TSP, Concorde is adopted to get the optimal solutions, based on which the optimality gaps of other methods are calculated. CVRP is harder to be solved optimally, and the gaps are calculated based on solutions of LKH. Note that even for the baselines which infer 10,000 random instances, their objective values might be slightly different from ours (e.g., the ones marked with $\ddagger .$ ), therefore we focus more on gaps for fair comparison. The run time is also hard to compare due to various factors (e.g., GPU/CPU models, batch sizes, Python v.s. $\mathrm { C } { + + }$ ). For DACT, we report the time for inferring all 10,000 instances with multiple GPU cards in $^ { 6 6 } ( ) "$ , and a small batch (512 instances) with one single GPU card in “{}".
+
+Pertaining to TSP, our DACT with inference step limit of 5,000 $\mathrm { ( T = 5 k }$ ) outperforms the traditional solver OR-Tools and all improvement models in terms of optimality gap, including Wu et al. [11] which directly adopted the original Transformer encoder. It also outstrips construction methods
+
+Table 2: Generalization performance. (a) DACT v.s. baselines on benchmark datasets (up to 200 customers, see Appendix E.4 for detailed results and discussion); (b) PE v.s. CPE on different sizes.   
+
+<table><tr><td>Method</td><td>TSPLIB</td><td>CVRPLIB</td></tr><tr><td>OR-Tools [37]</td><td>3.34%</td><td>8.06%</td></tr><tr><td>AM-sampling [5]</td><td>22.83%</td><td>26.66%</td></tr><tr><td>POMO [8]</td><td>10.06%</td><td>6.10%</td></tr><tr><td>Wu et al. [11]</td><td>4.17%</td><td>5.20%</td></tr><tr><td>DACT</td><td>2.07%</td><td>3.41%</td></tr></table>
+
+(a)
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">N=20</td><td colspan="2">N=100</td></tr><tr><td>Obj.</td><td>Gap</td><td>Obj.</td><td>Gap</td></tr><tr><td>DACT-PE (T=5k)</td><td>3.84</td><td>0.21%</td><td>8.38</td><td>7.93%</td></tr><tr><td>DACT-CPE (T=5k)</td><td>3.83</td><td>0.10%</td><td>7.99</td><td>2.98%</td></tr><tr><td>Wu et al.[11] (T=5k)</td><td>3.91</td><td>2.14%</td><td>9.03</td><td>16.37%</td></tr><tr><td>OR-Tools [37]</td><td>3.83</td><td>0.00%</td><td>8.06</td><td>3.87%</td></tr></table>
+
+(b)
+
+Table 3: Dual v.s. single aspect representation   
+
+<table><tr><td>Steps</td><td>Method</td><td>#Params</td><td>N=50</td><td>N=100</td></tr><tr><td rowspan="2">T=1k</td><td>SA-T</td><td>0.37M</td><td>0.35% (1m)</td><td>3.49% (3m)</td></tr><tr><td>DACT</td><td>0.29M</td><td>0.14% (1m)</td><td>1.62% (4m)</td></tr><tr><td rowspan="2">T=5k</td><td>SA-T</td><td>0.37M</td><td>0.05% (5m)</td><td>1.55% (16m)</td></tr><tr><td>DACT</td><td>0.29M</td><td>0.02% (6m)</td><td>0.61% (18m)</td></tr></table>
+
+including AM-sampling and GCN-BS on TSP100. With larger step limit $\mathrm { T } { = } 1 0 \mathrm { k }$ , our DACT further boosts the solution qualities and outperforms other construction methods including MDAM-BS (beam search), and POMO (the current state-of-the-art). To further reduce the gaps, we also leverage the data augmentation technique in POMO (which considers flipping node coordinates without changing the optimal solution) to solve same instances multiple times in different ways. Although the inference time increases (we run data augmentation in serial on the same GPUs), our DACT with 4 augments not only outstrips POMO with 8 augments but also achieves the lowest objective values and gaps among all purely learning based models. In particular, our method almost optimally solved TSP20 and TSP50 with gap lower than $0 . 0 0 5 \%$ , and $0 . 0 9 \%$ on TSP100, which is superior to most of the recent neural solvers. Pertaining to CVRP, our DACT with $\mathrm { T } { = } 5 \mathrm { k }$ produces lower gaps than that of improvement models including NeuRewriter and NLNS. It also performs much better than Wu et al. [11] except on CVRP50. With $\mathrm { T } { = } 1 0 \mathrm { k }$ and 6 augments10, our DACT exhibits even better performance than the highly specialized heuristic solver LKH on CVRP20 and delivers the smallest gap of $0 . 1 9 \%$ on CVRP100 against other neural solvers including POMO with 8 augments. Besides, our DACT is also competitive to DPDP which leverages learnt heatmap and dynamic programming to search solutions. Though DPDP (100k) can solve TSP100 instances almost optimally, our DACT is more efficient than DPDP on CVRP100. Compared with CVAE-Opt-DE, despite that it is averaged over fewer instances and integrated with differential evolution, our objective values are still lower.
+
+In terms of the inference time, our DACT is highly competitive against all neural solvers except POMO which learns a construction model by sampling diverse trajectories. However, when it comes to the generalization performance on benchmark datasets, i.e., TSPLIB [38] and CVRPLIB [39] in Table 2(a), DACT produces significantly lower average gaps than the POMO with 8 augments, which indicates that our DACT is more advantageous in practice despite its longer inference time. On the other hand, it is possible to adopt a similar diverse rollout strategy for DACT to find better solutions earlier, or explore other model compression techniques such as the knowledge distillation [40] to learn a lighter DACT model for faster inference. Since our focus is to ameliorate Transformer for neural improvement solvers, we will investigate these possibilities in the future.
+
+# 5.2 Ablation studies
+
+Dual-aspect representation. In Table 3, we evaluate the effectiveness of our dual-aspect representation against the single-aspect one (SA-T) on TSP50 and TSP100, where SA-T mainly follows the Transformer in $\mathbf { W } \mathbf { u }$ et al. [11] but equipped with the CPE, multi-head attentions and CL strategy for fair comparison. We observe that our DACT with fewer parameters consistently outperforms SA-T, which verifies the effectiveness of the dual-aspect representation.
+
+![](images/2102778a136af66f4a0dae7bc32cd3d938ae977b3f9616db51da4c7cf94b2d1f.jpg)  
+Figure 6: Visualization of the attention scores for the encoder when a trained model is used to solve instances with a larger size. (a) using PE method; (b) using CPE method (ours).
+
+Cyclic positional encoding. Here we show that CPE significantly improves the generalization performance across different problem sizes. In Table 2(b), we record the results of our DACT with PE and CPE, and Wu et al. [11], when the model trained on TSP50 is directly used to solve instances from TSP20 and TSP100 with $\mathrm { T } { = } 5 \mathrm { k }$ . We see that even with PE, our DACT outperforms Wu et al. [11]. Further equipped with CPE, DACT outstrips DACT-PE and OR-Tools on TSP100. We continue to compare the two DACT variants by visualizing their attention scores. As depicted in Figure 6(a), although the absolute PE is designed for linear sequences, it did attempt to capture the circularity of
+
+![](images/dcaf347b7b825fb752b6ddcd50ad1a4968c6104e02beab829841e1b66d04fde2.jpg)  
+Figure 7: Training curves of PPO with and without CL on CVRP20 (random seeds 1-5).
+
+VRP solutions (as highlighted in the green boxes) after training. However, the ability to perceive such properties significantly drops when generalizing over different problem size, which instead engenders random attention scores when generalizing to larger size (see right side of Figure 6(a)). In contrast, our DACT with CPE is able to capture the circularity as depicted in Figure 6(b), which verifies the effectiveness of CPE in representing cyclic sequences (i.e., VRP solutions).
+
+Curriculum learning (CL) strategy. In Figure 7, we plot the training curves of PPO algorithm with and without our CL strategy, where the results are averaged over 5 independent runs with $90 \%$ confidence intervals. It shows that our CL strategy significantly improves the sample efficiency while reducing the variance of training, which aligns with our analysis in Section 4.4.
+
+# 6 Conclusions and future work
+
+In this paper, we present a novel DACT model for routing problems. It learns separate groups of embeddings for the node and positional features, and is equipped with cyclic positional encoding (CPE) to capture the circularity and symmetry of VRP solutions. A curriculum learning (CL) strategy is also exploited to improve the RL training efficiency. Extensive experiments on both synthetic and benchmark datasets justified the effectiveness of DACT in terms of both inference and generalization. A potential limitation is that DACT is more useful for learning improvement models at present. In the future, we will investigate how to extend DACT to construction models, and how to speed up the DACT through diverse rollouts or model compression techniques. It is also interesting to apply the proposed CPE to develop Transformer based model for other tasks where the cyclic property is also important, e.g., encoding circular DNA/RNA structures in computational biology [41, 42].
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported in part by the National Natural Science Foundation of China under Grant 61803104 and Grant 62102228, in part by the Young Scholar Future Plan of Shandong University under Grant 62420089964188, and in part by the A\*STAR CyberPhysical Production System (CPPS) - Towards Contextual and Intelligent Response Research Program, under the RIE2020 IAF-PP Grant A19C1a0018, and Model Factory $@$ SIMTech.
+
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+[42] Chengyu Liu, Yu-Chen Liu, Hsien-Da Huang, and Wei Wang. Biogenesis mechanisms of circular rna can be categorized through feature extraction of a machine learning model. Bioinformatics, 35(23):4867–4870, 2019.   
+[43] Logan Engstrom, Andrew Ilyas, Shibani Santurkar, Dimitris Tsipras, Firdaus Janoos, Larry Rudolph, and Aleksander Madry. Implementation matters in deep policy gradients: A case study on PPO and TRPO. In International Conference on Learning Representations, 2020.

parse/train/63pC59XOZLZ/63pC59XOZLZ_content_list.json

@@ -0,0 +1,1156 @@
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+    {
+        "type": "text",
+        "text": "Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yining $\\mathbf { M } \\mathbf { a } ^ { 1 }$ , Jingwen $\\mathbf { L i } ^ { 1 }$ , Zhiguang $\\mathbf { C a o ^ { 2 , * } }$ , Wen Song3,∗, Le Zhang4, Zhenghua Chen5, Jing Tang6 ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1National University of Singapore 2Singapore Institute of Manufacturing Technology, A\\*STAR 3Institute of Marine Science and Technology, Shandong University 4University of Electronic Science and Technology of China 5Institute for Infocomm Research, A\\*STAR 6The Hong Kong University of Science and Technology {yiningma, lijingwen}@u.nus.edu, zhiguangcao@outlook.com, wensong@email.sdu.edu.cn, zhangleuestc@gmail.com, chen0832@e.ntu.edu.sg, jingtang@ust.hk ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, Transformer has become a prevailing deep architecture for solving vehicle routing problems (VRPs). However, it is less effective in learning improvement models for VRP because its positional encoding (PE) method is not suitable in representing VRP solutions. This paper presents a novel Dual-Aspect Collaborative Transformer (DACT) to learn embeddings for the node and positional features separately, instead of fusing them together as done in existing ones, so as to avoid potential noises and incompatible correlations. Moreover, the positional features are embedded through a novel cyclic positional encoding (CPE) method to allow Transformer to effectively capture the circularity and symmetry of VRP solutions (i.e., cyclic sequences). We train DACT using Proximal Policy Optimization and design a curriculum learning strategy for better sample efficiency. We apply DACT to solve the traveling salesman problem (TSP) and capacitated vehicle routing problem (CVRP). Results show that our DACT outperforms existing Transformer based improvement models, and exhibits much better generalization performance across different problem sizes on synthetic and benchmark instances, respectively. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Vehicle Routing problems (VRPs), such as the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) which consider finding the optimal route for a single or fleet of vehicles to serve a set of customers, have ubiquitous real-world applications [1, 2]. Despite being intensively studied in the Operations Research (OR) community, VRPs still remain challenging due to their NP-hard nature [3]. Recent studies on learning neural heuristics are gathering attention as promising extensions to traditional hand-crafted ones (e.g., [4–14]), where reinforcement learning (RL) [15] is usually exploited to train a deep neural network as an efficient solver without hand-crafted rules. A salient motivation is that deep neural networks may learn better heuristics by identifying useful patterns in an end-to-end and data-driven fashion. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Solutions to VRPs, i.e., routes, are sequences of nodes (customer and depot locations). Naturally, deep models for Natural Language Processing (NLP), which deal with sequence data as well, are ideal choices for encoding VRP solutions. Given its remarkable performance in NLP tasks, Transformer [16] is standing at the forefront in the learning based methods for VRPs (e.g., [5, 7, 8, 11–13, 17]). The original Transformer encodes a sentence, i.e., a sequence of words, into a unified set of embeddings by injecting word positional information into its word embeddings through positional encoding (PE). When it comes to VRPs, while is not required in construction models, positional information is critical for deep models that learn improvement heuristics since the input are solutions to be improved. ",
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+        "image_caption": [
+            "Figure 1: Transformer frameworks for VRPs. (a) $\\mathrm { W u }$ et al. [11] (the original one); (b) DACT (ours). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although some success has been achieved, learning improvement heuristics for VRPs based on the original Transformer encoder is yet lacking from our perspective. Firstly, directly applying addition operation on PE vectors and the embeddings in absolute PE method (i.e., Figure 1(a)) could limit the representation of the model [18], as the mixed correlations2 existing in the self-attention can bring unreasonable noises and random biases to the encoder (details in Appendix A). Secondly, existing PE methods tend to fuse the node and positional information into one unified representation. NLP tasks such as translation may benefit from this owing to the deterministic and instructive nature of the positional information. However, such design may not be optimal for routing tasks because the positional information therein can be non-deterministic and sometimes even random. This may cause disharmony or disturbance in the encoder and may thus deteriorate the performance. Finally, most VRPs seek the shortest loop of the nodes, making their solutions to be cyclic sequences. However, existing PE methods are only designated to encode linear sequences3, which may fail to identify such circular input. As will be shown in our experiments, this could severely damage the generalization performance, since the cyclic feature of VRP solutions is not correctly reflected by the encoder. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we address the above issues and contribute to the line of using RL to learn neural improvement heuristics for VRPs. We introduce the Dual-Aspect Collaborative Transformer (DACT), where we revisit the solution representations and propose to learn separated groups of embeddings for the node and positional features of a VRP solution as shown in Figure 1(b). Our DACT follows the encoder-decoder structure. In the encoder, each set of embeddings encodes the solution mainly from its own aspect, and at the same time exploits a cross-aspect referential attention mechanism for better perceiving the consistence and differentiation with respect to the other aspect. The decoder then collects action distribution proposals from the two aspects and synthesizes them to output the final one. Meanwhile, we design a novel cyclic positional encoding (CPE) method to capture the circularity and symmetry of VRP solutions, which allows Transformer to encode cyclic inputs, and also boost the generalization performance for solving VRPs. As the last contribution, we design a simple yet effective curriculum learning strategy to improve the sample efficiency. This further leads to faster and more stable convergence of RL training. Extensive experiments show that our DACT can outperform existing Transformer based improvement models with fewer parameters, and also generalizes well across different sizes of synthetic and benchmark instances, respectively. ",
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 Positional encoding (PE) in Transformer. ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The original Transformer adopted the absolute PE method to describe the absolute position of elements in the sequence [16], especially for NLP. As formulated in Eq. (1), each generated positional embedding $p _ { i } \\in \\mathbb { R } ^ { d }$ is added together with the $i$ -th word embedding $x _ { i }$ in the first layer of the encoder, ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/06c8617b60f87154a9c18b93350fe384c2b92dcbc879ec3ead141f3979fd5a3f.jpg",
+        "text": "$$\n\\alpha _ { i , j } ^ { \\mathrm { A b s } } = \\frac { 1 } { \\sqrt { d } } ( ( x _ { i } + p _ { i } ) W ^ { Q } ) ( ( x _ { j } + p _ { j } ) W ^ { K } ) ^ { T } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The relative PE method was further proposed in Shaw et al. [19] to better capture the relative order information. On the basis of absolute PE, it introduces an inductive bias to the attention as follows, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb1536b0cd4bebe430402cd1ff3c8d032df2e5351e5010caf10bb27d09a60901.jpg",
+        "text": "$$\n\\alpha _ { i , j } ^ { \\mathrm { R e l } } = \\frac { 1 } { \\sqrt { d } } ( ( x _ { i } + p _ { i } ) W ^ { Q } ) ( ( x _ { j } + p _ { j } ) W ^ { K } + a _ { j - i } ) ^ { T } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $a _ { j - i } \\in \\mathbb { R } ^ { d }$ is learnable parameters for encoding the relative position $j - i$ . To avoid the mixed and noisy correlations between word semantics and positional information in the above two PEs, the Transformer with United Positional Encoding (TUPE) [18] was proposed for NLP which utilizes separated projection metrics $W _ { x }$ and $W _ { p }$ for each information as follows, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cde20607c5d2e4f5c92239343ac4f181cb00350f92f6445069b5ef43e3a4815c.jpg",
+        "text": "$$\n\\alpha _ { i , j } ^ { \\mathrm { T U P E } } = \\frac { 1 } { \\sqrt { 2 d } } ( x _ { i } W _ { x } ^ { Q } ) ( x _ { j } W _ { x } ^ { K } ) ^ { T } + \\frac { 1 } { \\sqrt { 2 d } } ( p _ { i } W _ { p } ^ { Q } ) ( p _ { j } W _ { p } ^ { K } ) ^ { T } + b _ { j - i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, as mentioned previously, existing PE methods are less effective for VRPs since they simply fuse the node and positional information into one unified set of embeddings during or after the calculation of the attention correlation $\\alpha _ { i , j }$ . Meanwhile, they are also unable to properly encode and handle cyclic input sequences as in VRP solutions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Deep models for VRP. ",
+        "text_level": 1,
+        "bbox": [
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+            464,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Various deep architectures such as Recurrent Neural Network (RNN), Graph Neural Network (GNN), and Transformer have been employed in solving VRPs. ",
+        "bbox": [
+            174,
+            489,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "RNN based models. As the pioneering work of neural VRP solvers, Pointer Network adopted RNN and supervised learning to solve TSP [20] (extended to RL in Bello et al. [21] and CVRP in Nazari et al. [22]). While the models in [20–23] learn construction heuristics, NeuRewriter [4] learns improvement heuristic for CVRP using LSTM to encode the positional information of a solution. In Hottung et al. [13], the conditional variational autoencoder was adopted to learn a continuous and latent search space for VRP, where high-quality solutions were taken as input and encoded by RNNs. However, recurrence structures in RNN are less efficient in both representation and computation [5]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "GNN based models. In Dai et al. [24], GNN was combined with Q-learning for solving TSP. Based on supervised learning, Joshi et al. [6] used GNN to learn heatmaps that prescribe the probability of each edge appearing in the optimal TSP tour. This idea was extended in Fu et al. [25] with additional components such as graph sampling and heatmap merging to enable generalization to larger TSP instances. These models often require post-processing to construct feasible solutions from heatmaps (e.g., beam search [6], Monte-Carlo tree search [25], and dynamic programming [26]). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transformer based models. The Attention Model (AM) by Kool et al. [5] was recognized as the first success of Transformer based models for VRPs. Based on AM, Xin et al. [7] proposed a MultiDecoder AM that learns multiple diverse policies for better performance. In Kwon et al. [8], the RL algorithm of AM was improved which leaded to a new solver, i.e., POMO (Policy Optimization with Multiple Optima), and achieved the state-of-the-art performance. However, POMO is still lacking in generalization. Besides these construction models, Transformer was also explored to learn improvement heuristics. Hottung and Tierney [27] learned first neural large neighborhood search algorithm for VRPs. Lu et al. [12] proposed the L2I model that learns to select local search operators from a pool of traditional ones. Both methods used a Transformer-style encoder, but the positional information is captured in the node features (information of previous and next nodes) instead of using PE methods. Though L2I was shown to outperform LKH3 [28], it is limited to CVRP and the required time is considerably long. Wu et al. [11] proposed a Transformer model which learns to pick node pair in each step to perform a pairwise local operator (e.g., 2-opt). However, it suffers from the inaccurate representation of positional information given the original Transformer encoder. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "image",
+        "img_path": "images/5ace11ce359bbb755998396b80534ff05688295f43970028840cdf0661b073b3.jpg",
+        "image_caption": [
+            "Figure 3: Architecture of our policy network, dual-aspect collaborative Transformer (DACT). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 Problem formulation ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We define a VRP instance as a group of $N$ nodes to visit, where the node feature $x _ { i }$ of node $i$ contains 2-dim coordinates and other problem-specific features (e.g., customer demand). A solution $\\delta$ consists of a sequence of nodes visited in order where we denote $p _ { i }$ to be the position (indices) of node $i$ in the solution which is deemed as the positional feature of node $i$ . The objective is to minimize the total travel distance $D ( \\delta )$ under certain problem-specific constraints. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Starting with an initial yet complete solution, our neural RL policy tries to improve the solution iteratively. At each step, the policy automatically selects a pair of nodes and locally adjusts the solution using a preset pairwise operator such as 2-opt, insert, or swap. As illustrated in Figure 2, given a node pair $( i , j )$ , the 2-opt operator adjusts a solution by reversing the segment between node $i$ and node $j$ ; the insert operator adjusts a solution by placing node $i$ after node $j$ ; and the swap operator adjusts a solution by exchanging the position of node $i$ and node $j$ . Such operation is repeated until reaching the step limit $T$ and we model it in the form of Markov Decision Process (MDP) as follows. ",
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+    {
+        "type": "image",
+        "img_path": "images/3abc3c66ae47d31052903aafee9790f52c0e269b59a1a7ed0217c149d678c265.jpg",
+        "image_caption": [
+            "Figure 2: Illustration examples of three pairwise operators for routing problems when node pair $( i = 2 , j = 1 )$ ) is specified for operating. From left to right: 2-opt, insert, and swap. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "State. For an instance with $N$ nodes, a state describes current solution $\\delta _ { t }$ using its node and positional features of each node, i.e., $s _ { t } = \\Psi ( \\delta _ { t } ) = \\{ x _ { 1 } ^ { t } , . . . , x _ { N } ^ { t } , p _ { 1 } ^ { t } , . . . , p _ { N } ^ { t } \\}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Action. The action $a _ { t } = ( i , j )$ specifies a node pair $( i , j )$ for the pairwise operator. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Reward. The reward function is defined as, $r _ { t } = D ( \\delta _ { t } ^ { * } ) - m i n \\left[ D ( \\delta _ { t + 1 } ) , D ( \\delta _ { t } ^ { * } ) \\right]$ where $\\delta _ { t } ^ { * }$ is the best incumbent solution found until time $t$ . It refers to the immediate reduced cost at each step with respects to the best incumbent solution, which ensures the cumulative reward equal to the total reduced cost over the initial solution. Hence the reward $r _ { t } > 0$ if and only if a better solution is found. Policy. The policy $\\pi _ { \\theta }$ is parameterized by the proposed DACT model with parameters $\\theta$ . At each time step, the action $( i , j )$ is obtained by sampling the stochastic policy for both training and inference. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Transition. The next state $s _ { t + 1 }$ is originated from $s _ { t }$ by performing the preset pairwise operator on the given node pair (action). Our state transient is deterministic, in the sense that it always accepts the next solution as the next state (infeasible solutions will be masked), regardless of its objective value. With such simple rule, the RL agent is expected to automatically learn how to combine multiple steps of simple local movements to achieve better solutions, even if some of them may worsen the current solution. Note that the step limit $T$ can be any user-specified value according to the allowed time budget. Hence, our MDP can have infinite horizon and we consider the reward discount factor $\\gamma < 1$ . ",
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+    {
+        "type": "text",
+        "text": "4 Dual-aspect collaborative Transformer model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We now present the details of our Dual-Aspect Collaborative Transformer (DACT). The concrete architecture of DACT is presented in Figure 3, where we take the TSP with $N$ nodes as an illustration example. Our DACT leverages separate aspects of embeddings to encode a VRP solution. In the DAC encoder, the self-attention correlations are computed individually for each aspect, and a cross-aspect referential attention mechanism is proposed to enable one aspect to effectively exploit attention correlations from the other aspect as optional references. The DAC decoder then collects action distribution proposals from both aspects and synthesize them to the final one. ",
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+        "img_path": "images/89f6fef802cd35891d1ed76d002b348f40ea6b89b2545c5f6d954311286d8b4a.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of our CPE method with absolute PE method on a TSP instance with 20 nodes. (a) the embedding vectors, (b) the correlations (dot products) between every two embeddings, and (c) the top two principal components after PCA (principal component analysis) projection. ",
+            "Figure 4: An example of cyclic Gray code where 4 digits are used to encode $N { = } 1 6$ nodes. The top left shows the base symmetry pattern $\\cdot _ { 1 0 0 1 }$ ’ in Gray code, and the top right plots its representation in our method. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "4.1 Dual-aspect solution representation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Specifically, we propose to learn two sets of embeddings, i.e., the node feature embeddings (NFEs) for node representation and the positional feature embeddings (PFEs) for positional representation. ",
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+        "type": "text",
+        "text": "NFEs. Following [5, 11], the NFE $h _ { i }$ of node $i$ is initialized as the linear projection of its node feature $x _ { i }$ with output dimension4 $d i m = 6 4$ . ",
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+        "type": "text",
+        "text": "PFEs. The PFE $g _ { i }$ of the positional feature $p _ { i }$ is initialized as a real-valued vector $( d i m = 6 4 )$ by applying our cyclic positional encoding (CPE), which is designed based on cyclic Gray codes [29]. ",
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+        "table_body": "<table><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>16</td></tr><tr><td rowspan=2 colspan=1>11</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr></table>",
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+        "text": "As illustrated in Figure 4, the cyclic Gray codes present a cyclic property (‘1110’ in the last column is adjacent to ‘1111’ in the first column) and an adjacency similarity property (any codes in adjacent columns only differ in one digit), both of which are desirable for cyclic sequences. To preserve these properties in designing our CPE, we follow two observed patterns: 1) each numerical digit contains a periodic cycle with reflectional symmetry, e.g., the $\\mathbf { \\dot { \\rho } } _ { 1 0 | 0 1 } ,$ in the lowest digit; and 2) the higher the numerical digit, the longer the period. Accordingly, we create similar patterns based on the sinusoidal functions in Eq. (4), where a periodic function with period $\\frac { 4 \\pi } { \\omega _ { d } }$ (induced by modulus) is used to generate one base symmetry pattern (the top right in Figure 4), ",
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+        "img_path": "images/761131c312918e281f2f5384381e85ec2923a640ca9e38a2d9d86de612f763b2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\overrightarrow { g _ { i } ^ { \\prime } } ^ { ( d ) } : = \\left\\{ \\begin{array} { l l } { s i n ( \\omega _ { d } \\cdot \\mathrm { \\Gamma } ( z ( i ) \\bmod \\frac { 4 \\pi } { \\omega _ { d } } ) - \\frac { 2 \\pi } { \\omega _ { d } } \\mathrm { \\Gamma } ) , \\mathrm { ~ i f ~ } d \\mathrm { ~ i s ~ e v e n } } \\\\ { c o s ( \\omega _ { d } \\cdot \\mathrm { \\Gamma } ( z ( i ) \\bmod \\frac { 4 \\pi } { \\omega _ { d } } ) - \\frac { 2 \\pi } { \\omega _ { d } } \\mathrm { \\Gamma } ) , \\mathrm { ~ i f ~ } d \\mathrm { ~ i s ~ o d d } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "$\\begin{array} { r } { z ( i ) = \\frac { i - 1 } { N } \\frac { 2 \\pi } { \\omega _ { d } } \\left\\lceil \\frac { N + 1 } { 2 \\pi / \\omega _ { d } } \\right\\rceil } \\end{array}$ is to make $N$ nodes linearly spaced in the generated pattern; the angular frequency $\\omega _ { d }$ is decreasing along the dimension to make the wavelength longer within the range $[ N ^ { \\frac { 1 } { [ d i m / 2 ] } } , N ]$ (see Appendix B for details). In Figure 5, we visualize the comparison between the absolute PE and our CPE for encoding a TSP instance of 20 nodes. Figure 5(a) demonstrates that our real-valued base symmetry pattern has a longer cyclic period as the digit grows. Figure 5(b) indicates that our method (blue) is able to correctly reflect the adjacency between the head and tail of the cyclic sequence whereas the PE method (red) fails to do so. Figure 5(c) verifies that our CPE vectors are well distributed in space with desired cyclic and adjacency similarity properties. ",
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+        "type": "text",
+        "text": "4.2 The encoder ",
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+        "text": "The encoder consists of $L = 3$ stacked DAC encoders. In each DAC encoder, we retain relatively independent encoding stream for NFEs and PFEs as in Eq. (5) and Eq. (6), respectively, each of which consists of a shared Dual-Aspect Collaborative Attention (DAC-Att) sub-layer and an independent feed-forward network (FFN) sub-layer. DAC-Att takes both sets of embeddings as input and then outputs their respective enhanced embeddings, i.e., NFEs $\\{ \\tilde { h } \\} _ { i = 1 } ^ { N }$ and PFEs $\\{ \\tilde { g } \\} _ { i = 1 } ^ { N }$ . Each sub-layer is followed by skip connection [30] and layer normalization [31] as same as the original Transformer. ",
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+        "img_path": "images/b3c27f072c19590bb60be7226f1f0fd9cb7fd14ba92fa17514b83d481ad23218.jpg",
+        "text": "$$\n\\begin{array} { r } { h _ { i } ^ { ( l ) } = \\mathbf { L N } \\Big ( h _ { i } ^ { \\prime } + \\mathbf { F F N } _ { h } ^ { ( l ) } ( h _ { i } ^ { \\prime } ) \\Big ) , h _ { i } ^ { \\prime } = \\mathbf { L N } \\Big ( h _ { i } ^ { ( l - 1 ) } + \\tilde { h } _ { i } ^ { ( l ) } \\Big ) , } \\\\ { g _ { i } ^ { ( l ) } = \\mathbf { L N } \\Big ( g _ { i } ^ { \\prime } + \\mathbf { F F N } _ { g } ^ { ( l ) } ( g _ { i } ^ { \\prime } ) \\Big ) , g _ { i } ^ { \\prime } = \\mathbf { L N } \\Big ( g _ { i } ^ { ( l - 1 ) } + \\tilde { g } _ { i } ^ { ( l ) } \\Big ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "DAC-Att. The DAC-Att sub-layer enhances each set of embedding from its own aspect, while leveraging attention correlations from the other aspect to achieve the synergy. Given the two sets of embeddings5, $\\{ h _ { i } \\} _ { i = 1 } ^ { N }$ and $\\{ g _ { i } \\} _ { i = 1 } ^ { N }$ , we first compute the self-attention correlation from both aspects, ",
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+        "img_path": "images/6e1058866e25e4d647cc2fd0b0d63302df8fedbe20722e8773d594addc8589b4.jpg",
+        "text": "$$\n\\alpha _ { i , j } ^ { h } = \\frac { 1 } { \\sqrt { d _ { k } } } \\left( h _ { i } W _ { h } ^ { Q } \\right) \\left( h _ { j } W _ { h } ^ { K } \\right) ^ { T } , \\quad \\alpha _ { i , j } ^ { g } = \\frac { 1 } { \\sqrt { d _ { k } } } \\left( g _ { i } W _ { g } ^ { Q } \\right) \\left( g _ { j } W _ { g } ^ { K } \\right) ^ { T } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where independent matrices $W _ { h } ^ { Q } , W _ { h } ^ { K } , W _ { g } ^ { Q }$ and $W _ { g } ^ { K } \\in \\mathbb { R } ^ { d i m \\times d _ { k } }$ are used to calculate queries and keys. The obtained correlations are further normalized to $\\tilde { \\alpha } _ { i , j } ^ { h }$ and $\\tilde { \\alpha } _ { i , j } ^ { g }$ via Softmax. Note that the correlations are computed from their own aspect, which eliminates possible noises and conduces to correctly describe the incompatible node pair relationships in different aspects of VRP solutions. ",
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+        "type": "text",
+        "text": "We then exploit a cross-aspect referential attention mechanism, which allows computed correlations to be shared between each other, as additional references for both contradistinction and collaboration, ",
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+        "type": "equation",
+        "img_path": "images/a6345560f8ea0e3a7ce2dd7e14bffaf039c3fa79d48bbc491342cf315f04e49d.jpg",
+        "text": "$$\n\\mathrm { o u t } _ { i } ^ { h } = \\mathrm { C o n c a t } \\left[ \\sum _ { j = 1 } ^ { N } \\tilde { \\alpha } _ { i , j } ^ { h } \\left( h _ { j } W _ { h } ^ { V } \\right) , \\sum _ { j = 1 } ^ { N } \\tilde { \\alpha } _ { i , j } ^ { g } \\left( h _ { j } W _ { h } ^ { V r e f } \\right) \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "equation",
+        "img_path": "images/2a48535120551d7df98fe87035c01b8a5d7d6f7b1f42271786b1c469f93da9c7.jpg",
+        "text": "$$\n\\mathrm { o u t } _ { i } ^ { g } = \\mathrm { C o n c a t } \\left[ \\sum _ { j = 1 } ^ { N } { \\tilde { \\alpha } } _ { i , j } ^ { g } \\left( g _ { j } W _ { g } ^ { V } \\right) , \\sum _ { j = 1 } ^ { N } { \\tilde { \\alpha } } _ { i , j } ^ { h } \\left( g _ { j } W _ { g } ^ { V r e f } \\right) \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "wheand $W _ { h } ^ { V } , W _ { g } ^ { V } \\in \\mathbb R ^ { d i m \\times d _ { v } }$ are trainable parameter matrices for formulating values in each aspect; are parameter matrices for each aspect to generate referential values. $W _ { h } ^ { V r e f } , W _ { g } ^ { V r e f } \\in \\mathbb R ^ { d i m \\times d _ { v } }$ We finally use the multi-head attention to get NFEs $\\tilde { h } _ { i }$ and PFEs $\\tilde { g } _ { i }$ as follows, ",
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+    {
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+        "img_path": "images/b68e363878666ea949f94c4dd3c67f66b6ae414ba43bbab76187fd699f625cc7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { c } { \\tilde { h } _ { i } , \\tilde { g } _ { i } = \\mathbf { D A C - A t t } \\left( W ^ { Q } , ~ W ^ { K } , W ^ { V } , W ^ { V _ { r e f } } , W ^ { O } \\right) , } \\\\ { \\tilde { h } _ { i } = \\mathbf { C o n c a t } \\left[ \\mathrm { h e a d } _ { i , 1 } ^ { h } , . . . , \\mathrm { h e a d } _ { i , m } ^ { h } \\right] W _ { h } ^ { O } , ~ \\tilde { g } _ { i } = \\mathbf { C o n c a t } \\left[ \\mathrm { h e a d } _ { i , 1 } ^ { g } , . . . , \\mathrm { h e a d } _ { i , m } ^ { g } \\right] W _ { g } ^ { O } , } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where h $\\mathbf { e a d } _ { i , k } ^ { h } = o u t _ { i , k } ^ { h }$ , $\\mathbf { h e a d } _ { i , k } ^ { g } = o u t _ { i , k } ^ { g }$ , and $W _ { h } ^ { O } , W _ { g } ^ { O } \\ \\in \\ \\mathbb { R } ^ { 2 m d _ { v } \\times d i m }$ are trainable parameter matrices. In our model, we adopt $m = 4$ and $d _ { k } = d _ { v } = 1 6$ . ",
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+        "type": "text",
+        "text": "FFN. Our FFN sub-layer has only one hidden layer with 64 hidden unites and adopts the ReLU activation function. The parameters of $\\mathbf { F F N } _ { h }$ and $\\mathbf { F F N } _ { g }$ are different for each group of embeddings. ",
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+    {
+        "type": "text",
+        "text": "4.3 The decoder ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the DAC decoder, the two sets of embeddings $\\{ h _ { i } ^ { ( L ) } \\} _ { i = 1 } ^ { N }$ and $\\{ g _ { i } ^ { ( L ) } \\} _ { i = 1 } ^ { N }$ are first passed through a Max-pooling sub-layer and a multi-head compatibility (MHC) sub-layer to independently generate diversified node-pair selection proposals from their own aspect, which are then aggregated through a feed-forward aggregation (FFA) sub-layer for output. ",
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+        "type": "text",
+        "text": "Max-pooling. For each set of embeddings, we adopt the max-pooling sub-layer in Wu et al. [11] to aggregate the global representation of all $N$ embeddings into each respective one6. ",
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+        "type": "text",
+        "text": "MHC. The compatibility sub-layer computes the attention correlations for each embedding pair, where the obtained correlations with size $N \\times N$ will be deemed as a proposal distribution for node pair selection. Our correlations are computed based on multiple heads for diversity. And we calculate separated attention score matrices $\\boldsymbol { Y } _ { k } ^ { h } , \\boldsymbol { \\dot { Y } } _ { k } ^ { g } \\in \\mathbb { R } ^ { N \\times N }$ (of head $k$ ) from the two aspects independently. Accordingly, the action distribution proposals would be different due to their aspect-specific focus and cognitions of the current solution, which will provide the subsequent FFA layer with a rich pool of proposals and allow our model to be more flexible and robust. ",
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+    {
+        "type": "text",
+        "text": "FFA. Once all proposals from two aspects are collected, a FFN with four layers (dimensions are $2 m$ , 32, 32 and 1, respectively) and ReLU activation is used to aggregate them, ",
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+    {
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+        "img_path": "images/a1d3544bbc2ae4bb92727490faabc0cdc3a752492a632c084015c7fdfdfeba67.jpg",
+        "text": "$$\n\\tilde { Y } _ { i , j } = \\mathbf { F } \\mathbf { F } \\mathbf { A } \\left( Y _ { i , j , 1 } ^ { g } , . . . , Y _ { i , j , m } ^ { g } , Y _ { i , j , 1 } ^ { h } , . . . Y _ { i , j , m } ^ { h } \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $m = 4$ is the number of heads; and the output $\\tilde { Y } _ { i , j }$ is a scalar indicating the likelihood of selecting node pair $( i , j )$ as an action. Afterwards, we apply $\\hat { Y } _ { i j } = C \\cdot \\mathrm { T a n h } ( \\tilde { Y } _ { i , j } )$ with $C = 6$ to control the entropy, and mask 7 the infeasible node pairs $( i ^ { \\prime } , j ^ { \\prime } )$ as $\\hat { Y } _ { i ^ { \\prime } j ^ { \\prime } } = - \\infty$ . Lastly, the likelihoods are normalized using Softmax function to obtain the final action distribution $P _ { i , j }$ . ",
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+    {
+        "type": "text",
+        "text": "4.4 Reinforcement learning algorithm ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We adopt the proximal policy optimization [32] with $n$ -step return estimation for training (details are given in Appendix C), and design a curriculum learning (CL) strategy for better sample efficiency. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Curriculum learning strategy. The strategy in Wu et al. [11] sets a maximum of $T _ { t r a i n }$ steps for training and estimates future returns by bootstrapping [33]. However, due to the concern of training cost, $T _ { t r a i n }$ is usually much smaller than actual $T$ for inference (e.g., 200 v.s. 10k), which may leave the agent a poor chance of observing high-quality solutions (states) during training. Consequently, it may cause high variance for bootstrapping because the value function is mostly fitted on low-quality solutions and may render it less knowledgeable in estimating long-term future returns accurately. In this paper, we tackle this issue by a simple yet efficient strategy which gradually prescribes higher-quality solutions as the initial states for training. In doing so, 1) it increases the probability for the agent to observe better solutions and thus reduce the variance of the value function; 2) it increases the difficulty of the learning task (higher-quality solutions are harder to improve) in a gradual manner and achieves better sample efficiency [34]. In practice, those higher-quality solutions can be easily achieved by improving the randomly generated ones using the current policy for a few $T _ { i n i t }$ steps, where $T _ { i n i t }$ could be slightly increased as the epoch grows. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate our DACT model on two representative routing problems, i.e., TSP and CVRP [5, 8, 11]. For each problem, we abide by existing conventions to randomly generate instances on the fly for three sizes, i.e., $N = 2 0$ , 50 and 100. Initial experiments with three operators including 2-opt, swap and insert show that 2-opt performs best for both TSP and CVRP (with insert better than swap), hence we report results of our method based on 2-opt. Following [4, 11, 27] we use randomly generated initial solutions for training and the solutions generated by the greedy algorithm for inference. Since each problem has its own constraints and node features, we adjust the input, feasibility masks, and problem-dependent hyperparameters for each problem, the details of which are provided in Appendix D and E. The DACT is trained and tested on a server equipped with TITAN RTX GPU cards and Intel i9-10940X CPU at $3 . 3 0 \\mathrm { G H z }$ . Our code in PyTorch are available here8. ",
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+    {
+        "type": "table",
+        "img_path": "images/a5a1b6f0e80b2409834bf4f4c112e90e12459f5074bcf9746c847b6e5821cb01.jpg",
+        "table_caption": [
+            "Table 1: Comparison with various baselines on TSP and CVRP. "
+        ],
+        "table_footnote": [
+            "# the obj. values, gaps or time are obtained based on 2,000 instances in their original papers, and not directly comparable to ours. ‡ the obj. values obtained by Concorde or LKH may be slightly different from ours since the 10,000 instances are randomly generated. E.g., for TSP50, the optimal values according to our running of Concorde is 5.70, while 5.69 in POMO and Wu et al.. We thus focus more on gaps. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\">Method</td><td colspan=\"3\">N=20</td><td colspan=\"3\">N=50</td><td colspan=\"3\">N=100</td></tr><tr><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td></tr><tr><td rowspan=\"10\">LKH P</td><td>Concorde</td><td>3.83</td><td>=</td><td>(3m)</td><td>5.70</td><td></td><td>(10m)</td><td>7.76</td><td></td><td>(1h)</td></tr><tr><td></td><td>3.83</td><td>0.00%</td><td>(38s)</td><td>5.70</td><td>0.00%</td><td>(5m)</td><td>7.76</td><td>0.00%</td><td>(20m)</td></tr><tr><td>OR-Tools</td><td>3.86</td><td>0.94%</td><td>(42s)</td><td>5.85</td><td>2.87%</td><td>(5m)</td><td>8.06</td><td>3.86%</td><td>(23m)</td></tr><tr><td>Neural-2-Opt [23]</td><td>3.84</td><td>0.00%</td><td>(15m)</td><td>5.70</td><td>0.12%</td><td>(29m)</td><td>7.83</td><td>0.87%</td><td>(41m)</td></tr><tr><td>Wu et al. [11] (T=5k)</td><td>3.83</td><td>0.00%</td><td>(1h)</td><td>5.70</td><td>0.20%</td><td>(1.5h)</td><td>7.87</td><td>1.42%</td><td>(2h)</td></tr><tr><td>DACT (T=1k)</td><td>3.83</td><td>0.04%</td><td>{7s}(24s)</td><td>5.70</td><td>0.14%</td><td>{16s}(1m)</td><td>7.89</td><td>1.62%</td><td>{48s}(4m)</td></tr><tr><td>DACT (T=5k)</td><td>3.83</td><td>0.00%</td><td>{32s}(2m)</td><td>5.70</td><td>0.02%</td><td>{2m}(6m)</td><td>7.81</td><td>0.61%</td><td>{4m}(18m)</td></tr><tr><td>DACT (T=10k)</td><td>3.83</td><td>0.00%</td><td>{1m}(5m)</td><td>5.70</td><td>0.01%</td><td>{3m}(13m)</td><td>7.79</td><td>0.37%</td><td>{8m}(40m)</td></tr><tr><td>DACT×4 augment</td><td>3.83</td><td>0.00%</td><td>{3m}(10m)</td><td>5.70</td><td>0.00%</td><td>{10m}(1h)</td><td>7.77</td><td>0.09%</td><td>{29m}(2.5h)</td></tr><tr><td>GCN-BS [6]</td><td>3.84</td><td>0.01%</td><td>(12m)</td><td>5.70</td><td>0.01%</td><td>(18m)</td><td>7.87</td><td>1.39%</td><td>(40m)</td></tr><tr><td>AM-sampling [5]</td><td>3.84</td><td>0.08%</td><td>(5m)</td><td>5.73</td><td>0.52%</td><td>(24m)</td><td>7.94</td><td>2.26%</td><td>(1h)</td></tr><tr><td>MDAM-BS[7]</td><td>3.84t</td><td>0.00%</td><td>(3m)</td><td>5.70</td><td>0.03%</td><td>(14m)</td><td>7.79</td><td>0.38%</td><td>(44m)</td></tr><tr><td>POMO [8]</td><td>3.83</td><td>0.04%</td><td>(1s)</td><td>5.70</td><td>0.21%</td><td>(2s)</td><td>7.80</td><td>0.46%</td><td>(11s)</td></tr><tr><td>POMO×8 augment [8]</td><td>3.83</td><td>0.00%</td><td>(3s)</td><td>5.69t</td><td>0.03%</td><td>(16s)</td><td>7.78</td><td>0.15%</td><td>(1m)</td></tr><tr><td>DPDP(100k) [26]</td><td>-</td><td></td><td></td><td>-</td><td></td><td>=</td><td>7.77+</td><td>0.00%</td><td>(3h)</td></tr><tr><td rowspan=\"9\">LKH OR-Tools NeuRewriter [4] NLNS [27]</td><td rowspan=\"9\">CVAE-Opt-DE [13]</td><td>1</td><td>0.00%#</td><td>11m#</td><td>-</td><td>0.02%#</td><td>22m#</td><td>-</td><td>0.34%#</td><td>55m#</td></tr><tr><td></td><td>0.00%</td><td></td><td></td><td></td><td>4h</td><td>15.68</td><td></td><td></td></tr><tr><td>6.14 6.46</td><td>5.68%</td><td>1h 2m</td><td>10.38 11.27</td><td>0.00% 8.61%</td><td>13m</td><td>17.12</td><td>0.00% 9.54%</td><td>8h 46m</td></tr><tr><td>6.15#</td><td></td><td>6m#</td><td>10.51#</td><td></td><td>11m#</td><td>16.10#</td><td></td><td></td></tr><tr><td>6.19#</td><td>=</td><td>6m#</td><td>10.54#</td><td></td><td>11m#</td><td>15.99#</td><td>=</td><td>17m# 16m#</td></tr><tr><td>Wu et al. [11] (T=5k) 6.12</td><td>0.39%</td><td>(2h)</td><td>10.45</td><td>0.70%</td><td>(4h)</td><td>16.03t</td><td>= 2.47%</td><td></td></tr><tr><td>DACT (T=1k)</td><td>0.28%</td><td>{16s}(33s)</td><td>10.61</td><td>2.13%</td><td>{43s}(2m)</td><td>16.17</td><td>3.18%</td><td>(5h) {2m}(5m)</td></tr><tr><td>DACT (T=5k)</td><td>6.15 6.13 -0.00%</td><td>{1m}(3m)</td><td>10.48</td><td>1.01%</td><td>{3m}(8m)</td><td>15.92</td><td>1.55%</td><td>{8m}(23m)</td></tr><tr><td>DACT (T=10k)</td><td>-0.04%</td><td>{2m}(6m)</td><td>10.46</td><td>0.79%</td><td>{6m}(16m)</td><td>15.85</td><td>1.12%</td><td>{16m}(45m)</td></tr><tr><td rowspan=\"2\">DACT×6 augment</td><td>6.13 6.13</td><td>-0.08%</td><td>{11m}(35m)</td><td>10.39</td><td>0.14%</td><td>{32m}(1.5h)</td><td>15.71</td><td>0.19%</td><td>{1.5h}(4.5h)</td></tr><tr><td>AM-sampling [5]</td><td>1.87%</td><td>(6m)</td><td>10.62</td><td>2.40%</td><td>(28m)</td><td></td><td></td><td></td></tr><tr><td rowspan=\"2\">MDAM-BS[7]</td><td>6.25 6.14</td><td>0.18%</td><td>(5m)</td><td>10.48</td><td>0.98%</td><td>(15m)</td><td>16.23 15.99#</td><td>3.72% 2.23%</td><td>(2h)</td></tr><tr><td></td><td>0.82%</td><td>(1s)</td><td>10.49</td><td>1.14%</td><td>(4s)</td><td>15.83</td><td>0.98%</td><td>(1h) (19s)</td></tr><tr><td>POMO [8] POMO×8 augment [8]</td><td>6.17 6.14</td><td>0.21%</td><td>(5s)</td><td>10.42</td><td>0.45%</td><td>(26s)</td><td>15.73</td><td>0.32%</td><td>(2m)</td></tr><tr><td>DPDP(100k)[26]</td><td></td><td></td><td></td><td></td><td></td><td></td><td>15.69</td><td>0.31%</td><td></td></tr><tr><td>CVAE-Opt-DE [13]</td><td>= 6.14#</td><td></td><td>= 21m#</td><td>= 10.40#</td><td></td><td>41m#</td><td>15.75#</td><td></td><td>(6h) 1.5h#</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5.1 Comparison studies ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Table 1, we compare our DACT with, (1) learning based improvement methods, including Wu et al. [11], Neural-2-Opt [23] (TSP only), NeuRewriter [4] (CVRP only), NLNS [27] (CVRP only), (2) learning based construction methods, including AM-sampling [5], GCN-BS [6] (TSP only), MDAM-BS [7], POMO [8], (3) conventional optimization algorithms equipped with learning based component(s), including DPDP [26], CVAE-Opt-DE [13], and (4) strong conventional solvers including Concorde [35], LKH [28, 36], and OR-Tools [37]. Though L2I [12] can outstrip LKH on CVRP, we do not inlude it as a baseline since it requires a prohibitively longer inference time than others9. All results are averaged over 10,000 randomly generated instances unless specified otherwise (e.g., the ones marked with # only infer 2,000 instances), and we report the metrics of objective values, (optimality) gaps and run time. Regarding baselines, we follow the results reported in their original papers, which may not include all the three metrics. For TSP, Concorde is adopted to get the optimal solutions, based on which the optimality gaps of other methods are calculated. CVRP is harder to be solved optimally, and the gaps are calculated based on solutions of LKH. Note that even for the baselines which infer 10,000 random instances, their objective values might be slightly different from ours (e.g., the ones marked with $\\ddagger .$ ), therefore we focus more on gaps for fair comparison. The run time is also hard to compare due to various factors (e.g., GPU/CPU models, batch sizes, Python v.s. $\\mathrm { C } { + + }$ ). For DACT, we report the time for inferring all 10,000 instances with multiple GPU cards in $^ { 6 6 } ( ) \"$ , and a small batch (512 instances) with one single GPU card in “{}\". ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Pertaining to TSP, our DACT with inference step limit of 5,000 $\\mathrm { ( T = 5 k }$ ) outperforms the traditional solver OR-Tools and all improvement models in terms of optimality gap, including Wu et al. [11] which directly adopted the original Transformer encoder. It also outstrips construction methods ",
+        "bbox": [
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+            839,
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+            881
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/41ddc55113bc20c00330d1d54f1fb2f1085c8a072a1b553c83fdd329e09b3ee4.jpg",
+        "table_caption": [
+            "Table 2: Generalization performance. (a) DACT v.s. baselines on benchmark datasets (up to 200 customers, see Appendix E.4 for detailed results and discussion); (b) PE v.s. CPE on different sizes. "
+        ],
+        "table_footnote": [
+            "(a) "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>TSPLIB</td><td>CVRPLIB</td></tr><tr><td>OR-Tools [37]</td><td>3.34%</td><td>8.06%</td></tr><tr><td>AM-sampling [5]</td><td>22.83%</td><td>26.66%</td></tr><tr><td>POMO [8]</td><td>10.06%</td><td>6.10%</td></tr><tr><td>Wu et al. [11]</td><td>4.17%</td><td>5.20%</td></tr><tr><td>DACT</td><td>2.07%</td><td>3.41%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/081993af0c54bd66251b078601ab5ac12b18fda9eb3f644704622f30a657c213.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">N=20</td><td colspan=\"2\">N=100</td></tr><tr><td>Obj.</td><td>Gap</td><td>Obj.</td><td>Gap</td></tr><tr><td>DACT-PE (T=5k)</td><td>3.84</td><td>0.21%</td><td>8.38</td><td>7.93%</td></tr><tr><td>DACT-CPE (T=5k)</td><td>3.83</td><td>0.10%</td><td>7.99</td><td>2.98%</td></tr><tr><td>Wu et al.[11] (T=5k)</td><td>3.91</td><td>2.14%</td><td>9.03</td><td>16.37%</td></tr><tr><td>OR-Tools [37]</td><td>3.83</td><td>0.00%</td><td>8.06</td><td>3.87%</td></tr></table>",
+        "bbox": [
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+            138,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(b) ",
+        "bbox": [
+            624,
+            229,
+            643,
+            242
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/3e514a6a6f3e64798484b6231ff48235bab57d2952f3272297517edb6831f287.jpg",
+        "table_caption": [
+            "Table 3: Dual v.s. single aspect representation "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Steps</td><td>Method</td><td>#Params</td><td>N=50</td><td>N=100</td></tr><tr><td rowspan=\"2\">T=1k</td><td>SA-T</td><td>0.37M</td><td>0.35% (1m)</td><td>3.49% (3m)</td></tr><tr><td>DACT</td><td>0.29M</td><td>0.14% (1m)</td><td>1.62% (4m)</td></tr><tr><td rowspan=\"2\">T=5k</td><td>SA-T</td><td>0.37M</td><td>0.05% (5m)</td><td>1.55% (16m)</td></tr><tr><td>DACT</td><td>0.29M</td><td>0.02% (6m)</td><td>0.61% (18m)</td></tr></table>",
+        "bbox": [
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+            369
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "including AM-sampling and GCN-BS on TSP100. With larger step limit $\\mathrm { T } { = } 1 0 \\mathrm { k }$ , our DACT further boosts the solution qualities and outperforms other construction methods including MDAM-BS (beam search), and POMO (the current state-of-the-art). To further reduce the gaps, we also leverage the data augmentation technique in POMO (which considers flipping node coordinates without changing the optimal solution) to solve same instances multiple times in different ways. Although the inference time increases (we run data augmentation in serial on the same GPUs), our DACT with 4 augments not only outstrips POMO with 8 augments but also achieves the lowest objective values and gaps among all purely learning based models. In particular, our method almost optimally solved TSP20 and TSP50 with gap lower than $0 . 0 0 5 \\%$ , and $0 . 0 9 \\%$ on TSP100, which is superior to most of the recent neural solvers. Pertaining to CVRP, our DACT with $\\mathrm { T } { = } 5 \\mathrm { k }$ produces lower gaps than that of improvement models including NeuRewriter and NLNS. It also performs much better than Wu et al. [11] except on CVRP50. With $\\mathrm { T } { = } 1 0 \\mathrm { k }$ and 6 augments10, our DACT exhibits even better performance than the highly specialized heuristic solver LKH on CVRP20 and delivers the smallest gap of $0 . 1 9 \\%$ on CVRP100 against other neural solvers including POMO with 8 augments. Besides, our DACT is also competitive to DPDP which leverages learnt heatmap and dynamic programming to search solutions. Though DPDP (100k) can solve TSP100 instances almost optimally, our DACT is more efficient than DPDP on CVRP100. Compared with CVAE-Opt-DE, despite that it is averaged over fewer instances and integrated with differential evolution, our objective values are still lower. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In terms of the inference time, our DACT is highly competitive against all neural solvers except POMO which learns a construction model by sampling diverse trajectories. However, when it comes to the generalization performance on benchmark datasets, i.e., TSPLIB [38] and CVRPLIB [39] in Table 2(a), DACT produces significantly lower average gaps than the POMO with 8 augments, which indicates that our DACT is more advantageous in practice despite its longer inference time. On the other hand, it is possible to adopt a similar diverse rollout strategy for DACT to find better solutions earlier, or explore other model compression techniques such as the knowledge distillation [40] to learn a lighter DACT model for faster inference. Since our focus is to ameliorate Transformer for neural improvement solvers, we will investigate these possibilities in the future. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Ablation studies ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Dual-aspect representation. In Table 3, we evaluate the effectiveness of our dual-aspect representation against the single-aspect one (SA-T) on TSP50 and TSP100, where SA-T mainly follows the Transformer in $\\mathbf { W } \\mathbf { u }$ et al. [11] but equipped with the CPE, multi-head attentions and CL strategy for fair comparison. We observe that our DACT with fewer parameters consistently outperforms SA-T, which verifies the effectiveness of the dual-aspect representation. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/2102778a136af66f4a0dae7bc32cd3d938ae977b3f9616db51da4c7cf94b2d1f.jpg",
+        "image_caption": [
+            "Figure 6: Visualization of the attention scores for the encoder when a trained model is used to solve instances with a larger size. (a) using PE method; (b) using CPE method (ours). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            85,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Cyclic positional encoding. Here we show that CPE significantly improves the generalization performance across different problem sizes. In Table 2(b), we record the results of our DACT with PE and CPE, and Wu et al. [11], when the model trained on TSP50 is directly used to solve instances from TSP20 and TSP100 with $\\mathrm { T } { = } 5 \\mathrm { k }$ . We see that even with PE, our DACT outperforms Wu et al. [11]. Further equipped with CPE, DACT outstrips DACT-PE and OR-Tools on TSP100. We continue to compare the two DACT variants by visualizing their attention scores. As depicted in Figure 6(a), although the absolute PE is designed for linear sequences, it did attempt to capture the circularity of ",
+        "bbox": [
+            174,
+            268,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/dcaf347b7b825fb752b6ddcd50ad1a4968c6104e02beab829841e1b66d04fde2.jpg",
+        "image_caption": [
+            "Figure 7: Training curves of PPO with and without CL on CVRP20 (random seeds 1-5). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            571,
+            272,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "VRP solutions (as highlighted in the green boxes) after training. However, the ability to perceive such properties significantly drops when generalizing over different problem size, which instead engenders random attention scores when generalizing to larger size (see right side of Figure 6(a)). In contrast, our DACT with CPE is able to capture the circularity as depicted in Figure 6(b), which verifies the effectiveness of CPE in representing cyclic sequences (i.e., VRP solutions). ",
+        "bbox": [
+            174,
+            448,
+            825,
+            517
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Curriculum learning (CL) strategy. In Figure 7, we plot the training curves of PPO algorithm with and without our CL strategy, where the results are averaged over 5 independent runs with $90 \\%$ confidence intervals. It shows that our CL strategy significantly improves the sample efficiency while reducing the variance of training, which aligns with our analysis in Section 4.4. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusions and future work ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            613,
+            446,
+            631
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a novel DACT model for routing problems. It learns separate groups of embeddings for the node and positional features, and is equipped with cyclic positional encoding (CPE) to capture the circularity and symmetry of VRP solutions. A curriculum learning (CL) strategy is also exploited to improve the RL training efficiency. Extensive experiments on both synthetic and benchmark datasets justified the effectiveness of DACT in terms of both inference and generalization. A potential limitation is that DACT is more useful for learning improvement models at present. In the future, we will investigate how to extend DACT to construction models, and how to speed up the DACT through diverse rollouts or model compression techniques. It is also interesting to apply the proposed CPE to develop Transformer based model for other tasks where the cyclic property is also important, e.g., encoding circular DNA/RNA structures in computational biology [41, 42]. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            808,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the National Natural Science Foundation of China under Grant 61803104 and Grant 62102228, in part by the Young Scholar Future Plan of Shandong University under Grant 62420089964188, and in part by the A\\*STAR CyberPhysical Production System (CPPS) - Towards Contextual and Intelligent Response Research Program, under the RIE2020 IAF-PP Grant A19C1a0018, and Model Factory $@$ SIMTech. ",
+        "bbox": [
+            173,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Paolo Toth and Daniele Vigo. Vehicle routing: problems, methods, and applications. SIAM press, 2014.   \n[2] Michael Schneider, Andreas Stenger, and Dominik Goeke. The electric vehicle-routing problem with time windows and recharging stations. Transportation Science, 48(4):500–520, 2014.   \n[3] Jan Karel Lenstra and AHG Rinnooy Kan. Complexity of vehicle routing and scheduling problems. Networks, 11(2):221–227, 1981.   \n[4] Xinyun Chen and Yuandong Tian. Learning to perform local rewriting for combinatorial optimization. In Advances in Neural Information Processing Systems, volume 32, pages 6281–6292, 2019.   \n[5] Wouter Kool, Herke van Hoof, and Max Welling. Attention, learn to solve routing problems! In International Conference on Learning Representations, 2018.   \n[6] Chaitanya K Joshi, Thomas Laurent, and Xavier Bresson. An efficient graph convolutional network technique for the travelling salesman problem. arxiv preprint arxiv:1906.01227, ArXiV, 2019.   \n[7] Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Multi-decoder attention model with embedding glimpse for solving vehicle routing problems. In Proceedings of 35th AAAI Conference on Artificial Intelligence, pages 12042–12049, 2021.   \n[8] Yeong-Dae Kwon, Jinho Choo, Byoungjip Kim, Iljoo Yoon, Youngjune Gwon, and Seungjai Min. POMO: Policy optimization with multiple optima for reinforcement learning. In Advances in Neural Information Processing Systems, volume 33, pages 21188–21198, 2020.   \n[9] Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Step-wise deep learning models for solving routing problems. IEEE Transactions on Industrial Informatics, 17(7):4861–4871, 2020.   \n[10] Cong Zhang, Wen Song, Zhiguang Cao, Jie Zhang, Puay Siew Tan, and Xu Chi. Learning to dispatch for job shop scheduling via deep reinforcement learning. 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MIT press, 2018.   \n[16] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, volume 30, pages 6000–6010, 2017.   \n[17] Jingwen Li, Liang Xin, Zhiguang Cao, Andrew Lim, Wen Song, and Jie Zhang. Heterogeneous attentions for solving pickup and delivery problem via deep reinforcement learning. IEEE Transactions on Intelligent Transportation Systems, 2021.   \n[18] Guolin Ke, Di He, and Tie-Yan Liu. Rethinking the positional encoding in language pre-training. In International Conference on Learning Representations, 2020.   \n[19] Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. Self-attention with relative position representations. In North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 464–468, 2018.   \n[20] Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, volume 28, pages 2692–2700, 2015.   \n[21] Irwan Bello, Hieu Pham, Quoc V Le, Mohammad Norouzi, and Samy Bengio. Neural combinatorial optimization with reinforcement learning. In International Conference on Machine Learning (Workshop), 2017.   \n[22] Mohammadreza Nazari, Afshin Oroojlooy, Martin Takác, and Lawrence V Snyder. Reinforcement learning ˇ for solving the vehicle routing problem. In Advances in Neural Information Processing Systems, pages 9861–9871, 2018.   \n[23] Paulo R d O Costa, Jason Rhuggenaath, Yingqian Zhang, and Alp Akcay. Learning 2-opt heuristics for the traveling salesman problem via deep reinforcement learning. In Asian Conference on Machine Learning, pages 465–480, 2020.   \n[24] Hanjun Dai, Elias B Khalil, Yuyu Zhang, Bistra Dilkina, and Le Song. Learning combinatorial optimization algorithms over graphs. In Advances in Neural Information Processing Systems, pages 6351–6361, 2017.   \n[25] Zhang-Hua Fu, Kai-Bin Qiu, and Hongyuan Zha. Generalize a small pre-trained model to arbitrarily large TSP instances. In AAAI Conference on Artificial Intelligence, 2021.   \n[26] Wouter Kool, Herke van Hoof, Joaquim Gromicho, and Max Welling. Deep policy dynamic programming for vehicle routing problems. arXiv preprint arXiv:2102.11756, 2021.   \n[27] André Hottung and Kevin Tierney. Neural large neighborhood search for the capacitated vehicle routing problem. In European Conference on Artificial Intelligence, 2020.   \n[28] Keld Helsgaun. LKH-3 (version 3.0.6), 2019. URL http://webhotel4.ruc.dk/\\~keld/research/ LKH-3/.   \n[29] Wikipedia. Gray code, 2021. URL: https://en.wikipedia.org/wiki/Gray_code. Last visited on 2020/05/19.   \n[30] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   \n[31] Lei Jimmy Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. Corr: abs/1607.06450, ArXiV, 2016.   \n[32] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arxiv preprint arxiv:1707.06347, ArXiV, 2017.   \n[33] Fabio Pardo, Arash Tavakoli, Vitaly Levdik, and Petar Kormushev. Time limits in reinforcement learning. In International Conference on Machine Learning, pages 4045–4054, 2018.   \n[34] Yoshua Bengio, Jérôme Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In International Conference on Machine Learning, pages 41–48, 2009.   \n[35] David L Applegate, Robert E Bixby, Vašek Chvátal, and William J Cook. Concorde TSP Solver, 2020. URL http://www.math.uwaterloo.ca/tsp/concorde/.   \n[36] Keld Helsgaun. LKH (version 2.0.9), 2018. URL http://webhotel4.ruc.dk/\\~keld/research/ LKH/.   \n[37] Laurent Perron and Vincent Furnon. OR-Tools (version 7.2), 2019. URL https://developers.google. com/optimization/.   \n[38] Gerhard Reinelt. TSPLIB-A traveling salesman problem library. ORSA journal on computing, 3(4): 376–384, 1991.   \n[39] Eduardo Uchoa, Diego Pecin, Artur Pessoa, Marcus Poggi, Thibaut Vidal, and Anand Subramanian. New benchmark instances for the capacitated vehicle routing problem. European Journal of Operational Research, 257(3):845–858, 2017.   \n[40] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.   \n[41] Chun-Ying Yu, Tung-Cheng Li, Yi-Ying Wu, Chan-Hsien Yeh, Wei Chiang, Ching-Yu Chuang, and Hung-Chih Kuo. The circular rna circbirc6 participates in the molecular circuitry controlling human pluripotency. Nature communications, 8(1):1–15, 2017.   \n[42] Chengyu Liu, Yu-Chen Liu, Hsien-Da Huang, and Wei Wang. Biogenesis mechanisms of circular rna can be categorized through feature extraction of a machine learning model. Bioinformatics, 35(23):4867–4870, 2019.   \n[43] Logan Engstrom, Andrew Ilyas, Shibani Santurkar, Dimitris Tsipras, Firdaus Janoos, Larry Rudolph, and Aleksander Madry. Implementation matters in deep policy gradients: A case study on PPO and TRPO. In International Conference on Learning Representations, 2020. ",
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parse/train/HfpNVDg3ExA/HfpNVDg3ExA.md

@@ -0,0 +1,366 @@
+# Probabilistic Transformer for Time Series Analysis
+
+Binh Tang Department of Statistics and Data Science Cornell University Ithaca, NY 14850 bvt5@cornell.edu
+
+David S. Matteson Department of Statistics and Data Science Cornell University Ithaca, NY 14850 matteson@cornell.edu
+
+# Abstract
+
+Generative modeling of multivariate time series has remained challenging partly due to the complex, non-deterministic dynamics across long-distance time steps. In this paper, we propose deep probabilistic methods that combine state-space models (SSMs) with transformer architectures. In contrast to previously proposed SSMs, our approaches use attention mechanism to model non-Markovian dynamics in the latent space and avoid recurrent neural networks entirely. We also extend our models to include several layers of stochastic variables organized in a hierarchy for further expressiveness. Compared to transformer models, ours are probabilistic, non-autoregressive, and capable of generating diverse long-term forecasts with accounted uncertainty. Extensive experiments show that our models consistently outperform competitive baselines on various tasks and datasets, including time series forecasting and human motion prediction.
+
+# 1 Introduction
+
+Generative modeling of multivariate time series is a challenging problem with wide-ranging applications in demand forecasting [15, 76], autonomous driving [2, 16], robotics [29, 67], and health care [20, 21, 59]. Despite remarkable progress in recent years, models that predict high-dimensional future observations from a few past examples have remained intractable, partly due to the complex, non-deterministic temporal dynamics across long-distance time steps. Given a sequence of human poses, for example, such models must internally figure out the involved dynamics of various body components across space and time while maintaining the inherent uncertainty of multiple plausible futures, even though only one such future is observed.
+
+Among proposed probabilistic approaches, state space models (SSMs) provide a principled framework for learning and drawing inference from sequential inputs [27, 66]. While autoregressive models feed its predictions back into the dynamics model without any compressed representation of data, SSMs model stochastic transitions between abstract states using latent variables, allowing for efficient state-to-state sampling without the need to render high-dimensional observations. Gaussian linear dynamical systems (LDSs), one of the best known SSMs [92], for example, postulate linear state transitions and enjoy exact inference via the celebrated Kalman filter algorithm.
+
+While early extensions of LDSs focus on linearization [46] and unscented transform [88], recent work that marry state space models with deep neural networks offers much more flexibility to model complex dependencies across different time steps. Some approaches retain the Markovian dynamics of LDSs and only replace their linear observation models with feed-forward networks [23, 31, 47, 71], whereas others favor nonlinear state transitions and parametrize such dependencies via recurrent neural networks (RNNs) [22, 23, 30, 39, 51, 75]. Despite differences, both Markovian transitions and RNNs are often not capable of capturing long-range dependencies in highly structured sequential inputs [36, 100], limiting the capacity of the corresponding SSMs.
+
+![](images/6209b8c1681d417ef10b0d5a80eb743603ec9c0900243de7593a7d1462ab034a.jpg)  
+Figure 1: Graphical model representations of linear dynamical systems (LDSs) in (a), and our proposed models (ProTran) in (b), (c), and (d). Black arrows denote the generative mechanism and red arrows the inference procedure. The separation of generation and inference in (c) and (d) is for readability. While traditional SSMs such as LDSs are limited to Markovian dynamics and linear dependencies, our models allow for non-Markovian and non-linear interactions between time steps via attention mechanism. A multi-layer extension of our models further increases expressiveness without compromising the tractable inference procedure.
+
+In this work, we propose to combine the complementary strengths of SSMs and transformer architectures [85], a powerful mechanism for modeling long-term interactions that enjoys success across a variety of sequence modeling tasks [26, 48, 99]. In contrast to most SSMs, our models make extensive use of attention mechanism [5, 85] between latent variables to model non-Markovian dynamics (see Figure 1). Compared to transformer-based methods, our models are probabilistic, nonautoregressive in a similar fashion to LDSs, and capable of generating diverse long-term forecasts with uncertainty estimates.
+
+Our main contributions are threefold. First, we propose novel SSMs based on transformer architectures for multivariate time series, which include generative models and inference procedures based on variational inference [49, 74]. Second, we extend our models to include several layers of stochastic latent variables organized in a hierarchy for further expressiveness. Third, we conduct extensive experiments on time series forecasting and human motion prediction and demonstrate that our Probabilistic Transformer (ProTran) performs remarkably well compared to various state-of-theart baselines.
+
+# 2 Preliminaries
+
+# 2.1 Variational State Space Models
+
+Let $\{ \mathbf { x } _ { 1 : T _ { i } } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ consist of $N$ univariate time series w ere $\mathbf { x } _ { 1 : T _ { i } } ^ { ( i ) } = ( \mathbf { x } _ { 1 } ^ { ( i ) } , \mathbf { x } _ { 2 } ^ { ( i ) } , \cdot \cdot \cdot \mathbf { x } _ { T _ { i } } ^ { ( i ) } )$ and $\mathbf { x } _ { t } ^ { ( i ) }$ denotes the vaue of the $i$ $t$ We consider the multivariate form $\mathbf { x } _ { 1 : T } = ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \ldots , \mathbf { x } _ { T } )$ where $\mathbf { x } _ { t } = ( \mathbf { x } _ { t } ^ { ( 1 ) } , . . . \mathbf { x } _ { t } ^ { ( N ) } ) \in \mathbb { R } ^ { N }$ . Conditioning on observed values up to time $C$ , we aim to produce distributional forecasts into the future $p ( \mathbf { x } _ { C + 1 : T } \mid \mathbf { x } _ { 1 : C } )$ . For clarity, we refer to $\mathbf { x } _ { 1 : C }$ and $\mathbf { x } _ { C + 1 : T }$ as contexts and targets, respectively.
+
+We are interested in probabilistic models parametrized by $\theta$ of the form
+
+$$
+p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { x } _ { 1 : C } ) = \int p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } ) p _ { \theta } ( \mathbf { z } _ { 1 : T } \mid \mathbf { x } _ { 1 : C } ) d \mathbf { z } _ { 1 : T }
+$$
+
+where ${ \bf z } _ { 1 : T } = ( { \bf z } _ { 1 } , { \bf z } _ { 2 } , \ldots , { \bf z } _ { T } )$ denotes the corresponding sequence of latent variables, sometimes referred to as states. In other words, we assume a generative model that can be decomposed into a transition model $p _ { \theta } ( \mathbf { z } _ { 1 : T } \mid \mathbf { x } _ { 1 : C } )$ between the latent variables conditioned on the contexts, and an emission model $p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } )$ from the latent variables to observable outputs. In particular, we further impose several assumptions on both models: 1
+
+$$
+p _ { \theta } ( \mathbf { z } _ { 1 : T } \mid \mathbf { x } _ { 1 : C } ) = \prod _ { t = 1 } ^ { T } p _ { \theta } ( \mathbf { z } _ { t } \mid \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : C } ) , \qquad p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } ) = \prod _ { t = 1 } ^ { T } p _ { \theta } ( \mathbf { x } _ { t } \mid \mathbf { z } _ { t } ) .
+$$
+
+As demonstrated in Figure 1(b), the latent variable $\mathbf { z } _ { t + 1 }$ depends not only on $\mathbf { z } _ { t }$ but also on all of its preceding latent variables, including $\mathbf { z } _ { t - 1 }$ , in contrast to linear dynamical systems (LDSs). In addition, the transition and emission models allow for non-linearity via neural network parametrizations. These assumptions aim to maximize model capacity for real-world applications with complex emissions or temporal dependencies.
+
+However, neither $\mathbf { x } _ { 1 : t - 1 }$ nor $\mathbf { z } _ { 1 : t - 1 }$ are included in the emission model $p ( \mathbf { x } _ { t } \mid \mathbf { z } _ { 1 : T } , \mathbf { x } _ { 1 : C } )$ . Such assumptions are important, as it has been argued previously that a leakage of information from the latent space in autoregressive models can hinder long-term predictions [23, 47]. While all ground truth observations are available during training, the entire sequence has to be generated sequentially at test time, making the dependencies on $\mathbf { x } _ { 1 : t - 1 }$ prone to accumulated errors over multiple time steps. By letting the latent variable $\mathbf { z } _ { t }$ capture all information needed to render $\mathbf { x } _ { t }$ , we also avoid the computational costs associated with repeatedly decoding and encoding $\mathbf { x } _ { t }$ in multi-step predictions.
+
+The inclusion of nonlinear state transitions and observation models necessarily requires approximate inference. We follow the stochastic variational inference framework [49, 74] and assume that the variational posterior parametrized by $\phi$ can be decomposed auto-regressively as $q _ { \phi } ( { \bf z } _ { 1 : T } \mid { \bf x } _ { 1 : T } ) =$ $\begin{array} { r } { \prod _ { t } q _ { \phi } ( \mathbf { z } _ { t } \mid \dot { \mathbf { z } } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : T } ) } \end{array}$ , which leads to a lower bound on the log likelihood:
+
+$$
+\log p _ { \theta } ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 1 : C } ) \geq \sum _ { t = 1 } ^ { T } ( \mathbb { E } _ { q } \left[ \log p _ { \theta } ( \mathbf { x } _ { t } | \mathbf { z } _ { t } ) \right] - \mathsf { K L } ( q _ { \phi } ( \mathbf { z } _ { t } | \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : T } ) \parallel p _ { \theta } ( \mathbf { z } _ { t } | \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : C } ) ) ) ,
+$$
+
+where $\mathsf { K L }$ is the Kullback-Leibler divergence.
+
+For computational stability, we assume homoscedasticity and choose Laplace distribution with scale parameter $\beta$ as a parametric form for $p _ { \theta } ( \mathbf { x } _ { t } \mid \mathbf { z } _ { t } )$ , i.e. we optimize for $L _ { 1 }$ reconstruction loss with a cross-validated factor $\beta$ for the KL term, following similar variational autoencoder (VAE) work [24, 41, 86]. Such an assumption does not necessarily limit the capacity of our models, as powerful stochastic transitions and flexible emission models can theoretically characterize arbitrary noise covariance [66]. Incorporating structured probabilistic outputs such as Gaussian copulas [75] or normalizing flows [23] can potentially further improve our model performance.
+
+# 2.2 Transformer Architectures
+
+Central to our models and other transformer-based approaches [48, 85] is the notion of attention [5], which allows the models to focus on important parts within a context. Multi-head attention, for example, maps a sequence of queries $\mathbf { Q } \doteq \mathbb { R } ^ { \ell _ { q } \times \dot { d } }$ of length $\ell _ { q }$ to a sequence of outputs ${ \bf O } =$ $[ \mathbf { O } _ { 1 } , \dots , \mathbf { O } _ { H } ] \in \mathbb { R } ^ { \ell _ { q } \times d }$ of the same size by attending over given $\ell _ { k }$ key-value pairs $\mathbf { K } \in \mathbb { R } ^ { \ell _ { k } \times d }$ , V ∈ R\`k×d:
+
+$$
+\mathbf { O } _ { h } = \mathsf { A t t e n t i o n } ( \mathbf { Q } _ { h } , \mathbf { K } _ { h } , \mathbf { V } _ { h } ) = \mathsf { S o f t m a x } \left( \frac { \mathbf { Q } _ { h } \mathbf { K } _ { h } ^ { \mathsf { T } } } { \sqrt { d } } \right) \mathbf { V } _ { h } ,
+$$
+
+where $\mathbf { Q } _ { h } = \mathbf { Q } \mathbf { W } _ { h } ^ { Q }$ , $\mathbf { K } _ { h } = \mathbf { K } \mathbf { W } _ { h } ^ { K }$ , $\mathbf { V } _ { h } = \mathbf { V } \mathbf { W } _ { h } ^ { V }$ are projected queries, keys, and values coro head , we re $h \in [ 1 , H ]$ with learning parameters n attention mechanism as s $\mathbf { W } _ { h } ^ { Q } , \mathbf { W } _ { h } ^ { K } , \mathbf { W } _ { h } ^ { V }$ , respectively. In case $\mathbf { Q } = \mathbf { K } = \mathbf { V }$
+
+Given fully observed sequences of inputs, the mapping can be computed efficiently without any imposed sequential order often seen in recurrent neural networks [19, 42]. More importantly, the direct connections between long-distance time steps are baked into the mechanism as information from previous time steps is easily accessible without being compressed into a fixed representation, easing optimization and learning of long-term dependencies [5, 85].
+
+Without recurrence, Transformer [85] encodes information about each time step $t$ with pred efined sinusoidal positional embeddings Position $( t ) = [ p _ { t } ( 1 ) , \ldots , p _ { t } ( d ) ] \in \mathbb { R } ^ { d }$ where the $i$ -th embedding is given by $p _ { t } ( i ) = \sin ( t \cdot c ^ { i / d } )$ for even $i$ and $p _ { t } ( i ) = \cos ( t \cdot c ^ { i / d } )$ for odd $i$ and $c$ is some large constant. Empirical results show that such positional embeddings are also important to our models.
+
+# 3 Probabilistic Transformer
+
+In this section, we first present our single-layered model and subsequently its multi-layered extension for a hierarchy of stochastic latent variables. As alluded earlier, our model consists of a generative model and an inference model that share information and parameters extensively.
+
+# 3.1 Single-Layered Probabilistic Transformer
+
+Generative Model. Given some contexts $\mathbf { x } _ { 1 : C }$ , we first apply a linear projection and combine it with a positional embedding to obtain $\mathbf { h } _ { 1 : C } \in \mathbb { R } ^ { d }$ , i.e.
+
+$$
+\mathbf { h } _ { t } = \mathsf { L a y e r N o r m } ( \mathsf { M L P } ( \mathbf { x } _ { t } ) + \mathsf { P o s i t i o n } ( t ) ) ,
+$$
+
+where LayerNorm and MLP denote layer normalizations [4] and multi-layer perceptrons, respectively. While a traditional transformer model often dedicates an entire encoder for the same purpose [55, 72], we find such a simple mapping works sufficiently well in conjunction with the contextattention module of the corresponding decoder.
+
+As implied in Equation (2), our latent dynamics decomposes auto-regressively. At each time step, we parametri ze the distribution $p _ { \theta } ( \mathbf { z } _ { t } \mid \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : C } )$ by a Gaussian with parameters resulting from two sequential steps of attention: a self-attention over the previously inferred states $\mathbf { z } _ { 1 : t - 1 }$ and another attention over the projected contexts $\mathbf { h } _ { 1 : C }$ . These two operations mirror those found in the decoder of Transformer [85], with the stochastic latent variables replacing its decoder inputs.
+
+Unfortunately, using stochastic samples of $\mathbf { z } _ { t }$ as attention queries is problematic, as purely stochastic transitions make it difficult for the model to reliably retain information across multiple time steps [17, 30, 39]. We therefore encapsulate the latent variables in hidden representations $\mathbf { w } _ { t }$ that also has a deterministic component. Combined with the attention steps, such representations help model long-range temporal dependencies while accounting for the stochasticity of future observations.
+
+Starting with a learnable, context-agnostic representation $\mathbf { w } _ { 0 }$ , we recursively update $\mathbf { w } _ { t }$ using a stochastic sample from $p _ { \theta } ( \mathbf { z } _ { t } \mid \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : C } )$ and th e positional embedding for the current time step $t$ . The generating process for the time step $t$ can be summarized by the following pseudocode:
+
+$$
+\begin{array} { r l } & { \bar { { \bf w } } _ { t } = \mathsf { L a y e r N o r m } \big ( { \bf w } _ { t - 1 } + \mathsf { A t t e n t i o n } \big ( { \bf w } _ { t - 1 } , { \bf w } _ { 1 : t - 1 } , { \bf w } _ { 1 : t - 1 } \big ) \big ) } \\ & { \hat { \bf w } _ { t } = \mathsf { L a y e r N o r m } \big ( \bar { \bf w } _ { t } + \mathsf { A t t e n t i o n } \big ( \bar { \bf w } _ { t } , { \bf h } _ { 1 : C } , { \bf h } _ { 1 : C } \big ) \big ) } \\ & { { \bf z } _ { t } = \mathsf { S a m p l e } \big ( \mathcal { N } \big ( { \bf z } _ { t } ; \mathsf { M L P } ( \hat { \bf w } _ { t } ) , \mathsf { S o f t p l u s } ( \mathsf { M L P } ( \hat { \bf w } _ { t } ) \big ) \big ) \big ) } \\ & { { \bf w } _ { t } = \mathsf { L a y e r N o r m } \big ( \hat { \bf w } _ { t } + \mathsf { M L P } ( { \bf z } _ { t } ) + \mathsf { P o s i t i o n } ( t ) \big ) , } \end{array}
+$$
+
+where Sample and Softplus are the Gaussian sampling and approximating rectifier operators.
+
+Each stochastic sample of $\mathbf { w } _ { 1 : T }$ is then mapped to a sequence of $\mathbf { x } _ { \mathrm { 1 : } T }$ via a multi-layer perceptron. We emphasize that our generation procedure in the latent space is more efficient than others in the observation space, which requires encoding and decoding high-dimensional inputs repeatedly.
+
+Inference Model. We parametrize the approximate posterior $q _ { \phi } ( \mathbf { z } _ { t } \mid \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : T } )$ at time step $t$ in a simi lar fashion to the prior $p _ { \theta } ( \mathbf { z } _ { t } \mid \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : C } )$ . Indeed, these parametrizations share most parameters and are done simultaneously in the same recursive loop, following the exact same steps in Equation (6) and Equation (7) (see Figure 1). We note that similar sharing techniques between the generative and inference processes have emerged as a common theme among recent successful VAE models [17, 62, 83].
+
+While the prior only has access to the conditioning observations $\mathbf { x } _ { 1 : C }$ , the approximate posterior should take into account all observations during training, including the targets $x _ { C + 1 : T }$ . Due to the inherent unidirectional aspect of RNNs, previous work that uses RNNs to parametrize the approximate posterior often disregards such a property [22, 30, 51] and often resorts to a filtering routine $p ( \mathbf { z } _ { t } \mid \mathbf { z } _ { 1 : t - 1 } , \mathbf { x } _ { 1 : t } )$ . In contrast, our inference procedure resembles more of the smoothing process of LDSs, factoring in both past and future observations via another application of self-attention:
+
+$$
+\begin{array} { r l } & { { \bf k } _ { t } = \mathrm { A t t e n t i o n } ( { \bf h } _ { 1 : T } , { \bf h } _ { 1 : T } , { \bf h } _ { 1 : T } ) ) } \\ & { { \bf z } _ { t } = \mathsf { S a m p l e } ( \mathcal { N } ( { \bf z } _ { t } ; { \sf M L P } ( [ \hat { \bf w } _ { t } , { \bf k } _ { t } ] ) , \mathsf { S o f t p l u s } ( { \sf M L P } ( [ \hat { \bf w } _ { t } , { \bf k } _ { t } ] ) ) ) . } \end{array}
+$$
+
+Here, we replace Equation (8) in the generative model with Equation (11), where the hidden representation $\mathbf { k } _ { t }$ summarizing all information relevant to the current tim estep $t$ has been concatenate to the latent-and-context-aware representation $\hat { \mathbf { w } } _ { t }$ preceding the Gaussian parametrization.
+
+The generative model and the inference model are trained end-to-end with a single stochastic variational inference objective stated in Equation (3). Such a variational bound includes the reconstruction loss for $\mathbf { X } _ { 1 : C }$ and the $\mathsf { K L }$ term for $\mathbf { z } _ { 1 : C }$ . Alternatively, we can exclude these terms from the objective, which is equivalent to starting the inference process from $t = C + 1$ instead of $t = 1$ .
+
+Our models incur a time complexity of $\mathcal { O } ( T ^ { 2 } d )$ and a memory cost of $\mathcal { O } ( T ^ { 2 } d )$ , where $T$ is the total sequence length and $d$ is the dimensionality of the latent space. The recursive latent dynamics also does not allow use the take full advantange of parallelizable attentions. However, we find that our models are still efficient in practice, especially for reasonably small values of $T$ .
+
+# 3.2 Multi-Layered Extension for Probabilistic Transformer
+
+Inspired by recent work on hierarchical VAEs for non-sequential inputs [17, 80, 83, 101], we extend our proposed model to include several layers of latent variables, aiming to further increase its flexibility for modelling sequential data.
+
+We represent each time step $t$ with a Ma rkov chain of $L$ latent variables $\mathbf { z } _ { t } ^ { ( 1 : L ) } = ( \mathbf { z } _ { t } ^ { ( 1 ) } , \ldots , \mathbf { z } _ { t } ^ { ( L ) } )$ for simplicity (see Figure 1). The generative and inference model also decompose auto-regressively across different time steps and may exhibit non-Markovian dynamics:
+
+$$
+\begin{array} { l } { { \displaystyle p _ { \theta } \left( { \bf x } _ { 1 : T } , { \bf z } _ { 1 : T } ^ { ( 1 : L ) } | { \bf x } _ { 1 : C } \right) = \left( \prod _ { \ell = 1 } ^ { T } p _ { \theta } \left( { \bf x } _ { t } | { \bf z } _ { t } ^ { ( L ) } \right) \right) \left( \prod _ { \ell = 1 } ^ { L } \prod _ { \ell = 1 } ^ { T } p _ { \theta } \left( { \bf z } _ { t } ^ { ( \ell ) } | { \bf z } _ { 1 : t - 1 } ^ { ( \ell ) } , { \bf z } _ { 1 : T } ^ { ( \ell - 1 ) } , { \bf x } _ { 1 : C } \right) \right) } } \\ { { \displaystyle q _ { \phi } \left( { \bf z } _ { 1 : T } ^ { ( 1 : L ) } | { \bf x } _ { 1 : T } \right) = \prod _ { \ell = 1 } ^ { L } \prod _ { \ell = 1 } ^ { T } q _ { \phi } \left( { \bf z } _ { t } ^ { ( \ell ) } | { \bf z } _ { 1 : t - 1 } ^ { ( \ell ) } , { \bf z } _ { 1 : T } ^ { ( \ell - 1 ) } , { \bf x } _ { 1 : T } \right) . } } \end{array}
+$$
+
+Intuitively, we generate samples $\mathbf { x } _ { 1 : T }$ conditioning on $\mathbf { x } _ { 1 : C }$ by following the latent dynamics from the bottom up and using the generative process described earlier within each layer. Analogously, inference proceeds in the same order, resulting in a variational bound similar to Equation (3):
+
+$$
+\begin{array} { l } { { \displaystyle \log p _ { \theta } \big ( { \bf x } _ { 1 : T } \mid { \bf x } _ { 1 : C } \big ) \geq \sum _ { t = 1 } ^ { T } \mathbb { E } _ { q } \left[ \log p _ { \theta } \big ( { \bf x } _ { t } ^ { ( L ) } \mid { \bf z } _ { t } \big ) \right] } \ ~ } \\ { { \displaystyle ~ - \sum _ { \ell = 1 } ^ { L } \mathsf { K L } \big ( q _ { \phi } \big ( { \bf z } _ { t } ^ { ( \ell ) } \mid { \bf z } _ { 1 : t - 1 } ^ { ( \ell ) } , { \bf z } _ { 1 : T } ^ { ( \ell ) } , { \bf x } _ { 1 : T } \big ) \ \big \| \ p _ { \theta } \big ( { \bf z } _ { t } ^ { ( \ell ) } \mid { \bf z } _ { 1 : t - 1 } ^ { ( \ell ) } , { \bf z } _ { 1 : T } ^ { ( \ell ) } , { \bf x } _ { 1 : C } \big ) \big ) } . } \end{array}
+$$
+
+As before, we parametrize the prior ) | z(\`)1:t−1, z(\`)1:T , x1:C ) using self-attention over the inferred(\`) latent variables from previous time steps w t−1 o n the same layer and another attention over contexts $\mathbf { h } _ { 1 : C }$ . In this case, however, we include an additional self-attention over all latent variables from the layer immediately below it (see Equation (16)):
+
+$$
+\begin{array} { r l } & { \tilde { \mathbf { w } } _ { t } ^ { ( \ell ) } = \mathsf { L a y e r N o r m } ( \mathbf { w } _ { t - 1 } ^ { ( \ell ) } + \mathsf { A t t e n t i o n } ( \mathbf { w } _ { t - 1 } ^ { ( \ell ) } , \mathbf { w } _ { 1 : T } ^ { ( \ell - 1 ) } , \mathbf { w } _ { 1 : T } ^ { ( \ell - 1 ) } ) ) } \\ & { \bar { \mathbf { w } } _ { t } ^ { ( \ell ) } = \mathsf { L a y e r N o r m } ( \tilde { \mathbf { w } } _ { t } ^ { ( \ell ) } + \mathsf { A t t e n t i o n } ( \tilde { \mathbf { w } } _ { t } ^ { ( \ell ) } , \mathbf { w } _ { 1 : t - 1 } ^ { ( \ell ) } , \mathbf { w } _ { 1 : t - 1 } ^ { ( \ell ) } ) ) } \\ & { \hat { \mathbf { w } } _ { t } ^ { ( \ell ) } = \mathsf { L a y e r N o r m } ( \bar { \mathbf { w } } _ { t } ^ { ( \ell ) } + \mathsf { A t t e n t i o n } ( \bar { \mathbf { w } } _ { t } ^ { ( \ell ) } , \mathbf { h } _ { 1 : C } , \mathbf { h } _ { 1 : C } ) ) } \\ & { \mathbf { z } _ { t } ^ { ( \ell ) } = \mathsf { S a m p l e } ( \mathcal { N } ( \mathbf { z } _ { t } ^ { ( \ell ) } ; \mathsf { M L P } ( \hat { \mathbf { w } } _ { t } ^ { ( \ell ) } ) , \mathbf { S o f t p l u s } ( \mathsf { M L P } ( \hat { \mathbf { w } } _ { t } ^ { ( \ell ) } ) ) ) ) } \\ & { \mathbf { w } _ { t } ^ { ( \ell ) } = \mathsf { L a y e r N o r m } ( \hat { \mathbf { w } } _ { t } ^ { ( \ell ) } + \mathsf { M L P } ( \mathbf { z } _ { t } ^ { ( \ell ) } ) + \mathsf { P o s i t i o n } ( t ) ) , } \end{array}
+$$
+
+Stacking multiple layers of latent variables increases model expressiveness, but it also result in a linear increase in running time and the number of parameters. The time complexity for the $L$ - layers transformer is $\mathcal { O } ( \bar { L _ { 1 } } T ^ { 2 } d )$ , while the space complexity remains $\mathcal { D } ( T ^ { 2 } d )$ due to the Markovian structure of the chain $\mathbf { z } _ { t } ^ { ( 1 : L ) }$ at each time step $t$ . In our experiments, we restrict the number of layers of our hierachical models to two or three.
+
+# 4 Related Work
+
+Deep State Space Models. Deep neural networks have been extensively combined with state space models, resulting in flexible, yet principledly motivated latent variable approaches. While some work keep the linear state transition intact to leverage the efficient Kalman filer algorithms [23, 31, 47, 71], more expressive, nonlinear latent dynamics parametrized by neural networks have been proposed [51, 52]. All such models are limited to the Markovian dynamics of LDSs, which hinders learning of long-range dependencies. The limitation is often alleviated by combining the stochastic transitions with a deterministic RNN that enables access to all past states [3, 8, 22, 30, 39, 77]. Our models are similarly non-Markovian, but the dependencies on the past states are done via attention, which allows for easy connections between long-distance time steps. In addition, while most existing deep SSMs represent each time step with a single latent variable, our models include several layers of hierarchical latent variables with tractable inference mechanism.
+
+Attentive Recurrent Networks. Attention mechanism has also been widely adopted in recent time series work using sequence-to-sequence models [1, 28] or transformer architectures [14, 55, 57, 72, 81, 94]. While our models are equipped with latent variables, these transformer approaches [55, 72] lack inference mechanism and are susceptible to feeding back observation noise into the dynamics model at test time. Our work, however, can be considered as an extension of the attentive state space model proposed in [1], with discrete latent states replaced by their continuous analogs. Recent developments in natural language processing [58, 60, 90] also combine transformer and VAE; however, these approaches often use a time-agnostic latent variable, in contrast to our SSM formulation.
+
+Time Series Forecasting. Traditional univariate time series models, such as Box-Jenkins methods [12] and exponential smoothing [43], often assume independence between any collection of time series [76]. While multivariate extensions of the classical approaches, including vector autoregression [82] and multivariate GARCH [7], do not require such a strong assumption, they come with many others such as stationarity and homocesdasticity, demand manual selection of covariates and models, and do not scale well to even a moderate number of time series [40, 69].
+
+Deep learning methods for time series forecasting have recently emerged as an expressive, scalable framework for industrial applications [10, 68, 79, 91]. While early work focus on point forecasts [53, 70, 96], recent approaches often employ recurrent neural networks with probabilistic forecasts parametrized directly [76], using quantile functions [33], Gaussian copulas [75], normalizing flows [23], or diffusion models [73]. In contrast, our models are entirely devoid of such recurrent architectures and rely on latent variables to output distributional forecasts.
+
+Human Motion Prediction. Despite being almost identical in formulation, human motion prediction has often been studied independently from time series forecasting. While some work deterministically generate future motions or video frames [13, 32, 34, 56], stochastic prediction has also been proposed with deep neural networks often outperforming traditional methods such as hidden Markov models [93] or Gaussian processes [89] on complex motion datasets [13, 32, 45, 54, 63]. In contrast to earlier work [95, 97] that employ a global latent variable across different time steps via conditional VAE [49], we leverage the principled framework of state space models for learning and inference of hierarchical, time-dependent latent variables.
+
+# 5 Experiments
+
+We present our experiment results on two tasks, namely, time series forecasting and human motion prediction. These tasks are often studied independently, despite being almost identical as conditional prediction problems.
+
+Table 1: Test set ${ \mathsf { C R P S } } _ { \mathsf { s u m } }$ of time series forecasting models (lower is better). The means and standard deviations are computed over five runs using different random seeds.   
+
+<table><tr><td>DATASET</td><td>SOLAR</td><td>ELECTRICITY</td><td>TRAFFIC</td><td>TAXI</td><td>WIKIPEDIA</td></tr><tr><td>VES [43]</td><td>0.900 ± 0.003</td><td>0.880 ± 0.004</td><td>0.350 ± 0.002</td><td></td><td></td></tr><tr><td>VAR [61]</td><td>0.830 ± 0.006</td><td>0.039 ± 0.001</td><td>0.290 ± 0.001</td><td></td><td></td></tr><tr><td>VAR-Lasso [61]</td><td>0.510 ± 0.006</td><td>0.025 ± 0.000</td><td>0.150 ± 0.002</td><td></td><td>3.100 ± 0.004</td></tr><tr><td>GARCH [84]</td><td>0.880 ± 0.002</td><td>0.190 ± 0.001</td><td>0.370 ± 0.001</td><td></td><td></td></tr><tr><td>DeepAR [76]</td><td>0.336 ± 0.014</td><td>0.023 ± 0.001</td><td>0.055 ± 0.003</td><td></td><td>0.127 ± 0.042</td></tr><tr><td>LSTM-Copula [75]</td><td>0.319 ± 0.011</td><td>0.064 ± 0.008</td><td>0.103 ± 0.006</td><td>0.326 ± 0.007</td><td>0.241± 0.003</td></tr><tr><td>GP-Copula [75]</td><td>0.337 ± 0.024</td><td>0.024 ± 0.002</td><td>0.078 ± 0.002</td><td>0.208 ± 0.183</td><td>0.086 ± 0.004</td></tr><tr><td>KVAE [51]</td><td>0.340 ± 0.025</td><td>0.051 ± 0.019</td><td>0.100 ± 0.005</td><td></td><td>0.095 ± 0.012</td></tr><tr><td>NKF [23]</td><td>0.320 ± 0.020</td><td>0.016 ± 0.001</td><td>0.100 ± 0.002</td><td></td><td>0.071 ± 0.002</td></tr><tr><td>Transformer-MAF[72]</td><td>0.301 ± 0.014</td><td>0.021 ± 0.000</td><td>0.056 ± 0.001</td><td>0.179 ± 0.002</td><td>0.063 ± 0.003</td></tr><tr><td>TimeGrad[73]</td><td>0.287 ± 0.020</td><td>0.021 ± 0.001</td><td>0.044 ± 0.006</td><td>0.114 ± 0.020</td><td>0.049 ± 0.002</td></tr><tr><td>ProTran (Ours)</td><td>0.194 ± 0.030</td><td>0.016 ± 0.001</td><td>0.028 ± 0.001</td><td>0.084 ± 0.003</td><td>0.047 ± 0.004</td></tr></table>
+
+# 5.1 Time-series Forecasting
+
+Datasets & Covariates. Following the experiment setup in [72, 73, 75], we evaluate our models and multiple competitive baselines on five popular public datasets: SOLAR, ELECTRICITY, TRAFFIC, TAXI, and WIKIPEDIA. The data is recorded with hourly or daily frequency and shows seasonal patterns of different frequencies (see Appendix A for more dataset details). As in [72, 73], the covariates include lagged inputs, fixed time embeddings (e.g. day of week, hour of day), and learnable time-series embeddings. The inputs are scaled using the conditioning examples before being fed into the model, and the predictions are rescaled appropriately afterward.
+
+Metrics. Following [23, 72, 75], we evaluate our model and all baselines using continuous ranked probability score (CRPS) [65] summed across time series, denoted by ${ \mathsf { C R P S } } _ { \mathsf { s u m } }$ . Given a univariate distribution function $F$ and an observation $x$ , CRPS is defined as
+
+$$
+{ \mathsf { C R P S } } ( F , x ) = \int _ { \mathbb { R } } ( F ( z ) - \mathbb { 1 } _ { \{ x \leq z \} } ) ^ { 2 } d z ,
+$$
+
+where $1 _ { \{ x \leq z \} }$ is the indicator function. As argued in de Bezenac et al. [ ´ 23], ${ \mathsf { C R P S } } _ { \mathsf { s u m } }$ is a proper scoring rule [35] and can be computed without analytical forecast distributions. We compute the metrics in a rolling fashion and use 100 samples for the distributional forecasts, similar to the aforementioned work.
+
+Baselines. We benchmark our models against various baselines, including (1) VES [43], an innovation state space model; (2) VAR-Lasso and VAR [61], two multivariate linear autoregressive models with and without Lasso regularization; (3) GARCH [84], a multivariate conditional heteroskedastic model; (4) DeepAR [76], an autoregressive recurrent neural network; LSTM-Copula and GP-Copula [75], two RNN-based models that use Gaussian copula to model nonlinearity; (5) KVAE [51], a variational approach based on linear dynamics; (6) NKF [23], a normalizing-flow model coupled with Kalman filters; (7) Transformer [72], a transformer-based model based on masked autoregressive flow; and (8) TimeGrad [73], a recent autoregressive approach that uses a diffusion model.
+
+Implementations. We use 8-head attentions and 2-layers MLPs to parametrize the generative and inference models. The stochastic latent variables $\mathbf { z } _ { t }$ are 16-dimensional while the hidden representations $\mathbf { w } _ { t }$ are in $\mathbb { R } ^ { 1 2 8 }$ . Our probabilistic transformers for SOLAR and ELECTRICITY have one stochastic layer while those for the other datasets of higher dimensional observations employ two layers. We re
+
+Table 2: Ablation study on TRAFFIC.   
+
+<table><tr><td>Two Layers One Layer Context Attention</td><td>√ × √</td><td>× √</td><td>× ×</td><td>× √</td></tr><tr><td>Deterministic</td><td>×</td><td>×</td><td>×</td><td>√</td></tr><tr><td>CRPSsum</td><td>0.028</td><td>0.031</td><td>0.033</td><td>0.041</td></tr></table>
+
+port the numbers of parameters of our models in Table 4 in Appendix C, which are all comparable to those of the state-of-the-art approaches. See Appendix D for more details about hyper-parameters and training processes.
+
+![](images/2c04e02911790d4e50312ce21468be872c3332091a686e454ec6bd62c82ad8ba.jpg)  
+Figure 2: Prediction intervals and test set ground-truth from ProTran (our model) for the TRAFFIC dataset of the first 16 of 963 time series.
+
+Accuracy Comparison. Table 1 shows that our models perform competitively across all five highdimensional time series datasets, achieving ${ \mathsf { C R P S } } _ { \mathsf { s u m } }$ comparable to the best methods on ELECTRICITY and WIKIPEDIA while outperforming all baselines, including a non-SSM transformer-based approach [72], by significant margins on SOLAR, TRAFFIC and TAXI. Further analyses with other metrics, including CRPS and MSE, in Appendix B also help confirm our findings.
+
+Qualitative Results. Figure 2 shows that the distribution forecasts generated by our model follow closely the ground truths, which is consistent with our accuracy results. In addition, the model appears to capture the uncertainty of future forecasts to some extent; observations of large magnitudes and far into the future seem to correctly have higher variance estimates.
+
+Ablation Study. We include a small scale ablation study on the TRAFFIC dataset to investigate which components of our models are essential. Table 2 suggests that removing the stochasticity from $\mathbf { w } _ { t }$ has most impacts on model performance, implying that incorprating latent variables into a transformer is indeed useful. Other aspects such as context attention or multiple layers of stochastic variables do not show dramatic effects in this study; however, they do contribute performance gains.
+
+# 5.2 Human Motion Prediction
+
+Datasets. Following the experiment setup in [97], we evaluation our models on two public motion capture datasets: Human3.6M[44] and HumanEva-I [78]. While Human3.6 is a large-scale dataset with 3.6 million video frames recorded at $5 0 \mathrm { H z }$ , HumanEva-I is smaller with only 3 subjects and recorded at $6 0 \mathrm { H z }$ . We follow the preprocessing steps of previous work [64, 97] and obtain a 17-joint skeleton for Human3.6 and a 15-joint skeleton for HumanEva-I. As in [97], we predict future motion for 2 seconds conditioning on observed motion of 0.5 seconds and 1 second conditioning on 0.25 seconds for Human3.6 and HumanEva-I, respectively.
+
+![](images/d9383da574a0f2a8ae5ae9ead64786b43baceb750e09f80f4d1df9b51d63a693.jpg)  
+Figure 3: Ground-truth pose sequences (first row) and corresponding predictions by ProTran (second row). Solid colors indicate later time-steps and faded ones are older. The body-part movements in the predicted and ground-truth poses resemble similar patterns, while certain variations are retained.
+
+Table 3: Human motion prediction results.   
+
+<table><tr><td>DATASET</td><td colspan="2">HUMAN3.6M</td><td colspan="2">HUMANEVA-I</td></tr><tr><td>Method</td><td>ADE↓</td><td>FDE↓ ADE↓</td><td></td><td rowspan="3">FDE↓</td></tr><tr><td>ERD [32]</td><td>0.722</td><td>0.969 0.382</td><td>0.461</td></tr><tr><td>acLSTM[56]</td><td>0.789 1.126</td><td></td><td>0.429 0.541</td></tr><tr><td>MT-VAE [95]</td><td>0.457</td><td>0.595</td><td>0.345</td><td rowspan="4">0.403</td></tr><tr><td>Pose-Knows [87]</td><td>0.461</td><td>0.560</td><td>0.269 0.296</td></tr><tr><td>HP-GAN [6]</td><td>0.858</td><td>0.867</td><td>0.772 0.749</td></tr><tr><td>Best-Many [11]</td><td>0.448</td><td>0.533 0.271</td><td>0.279</td></tr><tr><td>GMVAE [25]</td><td>0.461</td><td>0.555</td><td>0.305</td><td>0.345</td></tr><tr><td>DeliGAN[38]</td><td>0.483</td><td>0.534</td><td>0.306</td><td>0.322</td></tr><tr><td>DSP [98]</td><td>0.493</td><td>0.592</td><td>0.273</td><td>0.290</td></tr><tr><td>DLow [97]</td><td>0.425</td><td>0.518</td><td>0.251</td><td>0.268</td></tr><tr><td>ProTran (Ours)</td><td>0.381</td><td>0.491</td><td>0.258</td><td>0.255</td></tr></table>
+
+Metrics. Following previous work on trajectory forecasting [2, 37], we adopt two popular metrics, namely, average displacement error (ADE) and final displacement error (FDE). ADE measures the average $L _ { 2 }$ distance over all time steps between the ground truth motion and the closest sample, while FDE only consider such distance for the final pose.
+
+Baselines. We compare our models against 9 models, including ERD [32] and acLSTM [56], two deterministic RNN-based approaches; MT-VAE [95] and Pose-Knows [87], two conditional VAE models; HP-GAN [6], a conditional GAN; Best-Many [11], GMVAE [25], DeliGAN [38]. and DSP [98], four approaches optimizing for diversity objectives. The results for these baselines are reported as in [97].
+
+Implementations. Similar to the previous experiments, we use 8-head attentions and 2-layers MLPs. Since Human3.6M is significantly more complex and multi-modal than the time series forecasting datasets, we make use of 3 stochastic layers, as opposed to 2 layers for HumanEva-I. For Human3.6M, the context and target observations are significantly longer and set up for long-term predictions, so we only infer latent variables for target observations. Appendix C also contains further details about our models and their number of parameters.
+
+Quantitative Results. Table 3 shows that our models convincingly outperform all baselines based on both metrics ADE and FDE, with the gains significantly higher for the larger dataset Human3.6M. We emphasize that our favorable performance is evaluated using random samples, while the closest competitor, DLow [97], relies on a separate model for selecting samples to promote diversity, which can potentially be combined with our probabilistic transformer for further improvements.
+
+Qualitative Results. We show in Figure 3 human pose predictions made by our model that are most similar to the corresponding ground truths among a collection of such stochastic predictions. The similarities between the body-part movements in both sequences suggest that our model has been able to capture the temporal dynamics quite well.
+
+# 6 Conclusion & Discussion
+
+In this work, we have introduced generative models for multivariate time series that combines strengths of state space models and transformer architectures. In contrast to previous work, our models do not rely on recurrent neural networks but make extensive use of attention mechanism. We also extend our models to include hierarchical latent variables, inspired by recent developments of VAEs for non-sequential data [17, 83]. Empirical experiments show that our models perform remarkably well on time series forecasting and human motion prediction.
+
+Our models do not come without limitations, however. As in other transformer-based approaches, the reliance on attention incurs a quadratic time and memory complexity. While we do not find it problematic in our experiments, the limitation necessarily hinders applications of our models in tasks characterized by long-term dependencies such as language modelling or music generation [36]. Fortunately, recent work on sparse transformer [9, 18, 50, 55] can potentially address the issue, and we leave such an investigation for future work.
+
+Probabilistic time series forecasting is a fundamental research problem with wide-ranging applications in society. Although we have not explored healthcare applications of our work, previously proposed methods with similar formulations have demonstrated potentials of forecasting techniques [1, 81] in diagnoses or disease control.
+
+# 7 Acknowledgment
+
+Financial support is gratefully acknowledged from a Xerox PARC Faculty Research Award, National Science Foundation Awards 1455172, 1934985, 1940124, and 1940276, USAID, and Cornell University Atkinson Center for a Sustainable Future.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 1 and 5.   
+(b) Did you describe the limitations of your work? [Yes] See Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appdendix D in supplemental material.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Table 1.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appdendix D in supplemental material.
+
+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] See Section 4.   
+(b) Did you mention the license of the assets? [Yes] See Appendix A in supplemental material.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] See supplemental material. We are also commited to open source our code upon publication.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See Appendix A in supplemental material.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] See Appendix A in supplemental material.
+
+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/HfpNVDg3ExA/HfpNVDg3ExA_content_list.json

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+    {
+        "type": "text",
+        "text": "Probabilistic Transformer for Time Series Analysis ",
+        "text_level": 1,
+        "bbox": [
+            187,
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Binh Tang Department of Statistics and Data Science Cornell University Ithaca, NY 14850 bvt5@cornell.edu ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "David S. Matteson Department of Statistics and Data Science Cornell University Ithaca, NY 14850 matteson@cornell.edu ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Generative modeling of multivariate time series has remained challenging partly due to the complex, non-deterministic dynamics across long-distance time steps. In this paper, we propose deep probabilistic methods that combine state-space models (SSMs) with transformer architectures. In contrast to previously proposed SSMs, our approaches use attention mechanism to model non-Markovian dynamics in the latent space and avoid recurrent neural networks entirely. We also extend our models to include several layers of stochastic variables organized in a hierarchy for further expressiveness. Compared to transformer models, ours are probabilistic, non-autoregressive, and capable of generating diverse long-term forecasts with accounted uncertainty. Extensive experiments show that our models consistently outperform competitive baselines on various tasks and datasets, including time series forecasting and human motion prediction. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
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+    },
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+        "type": "text",
+        "text": "Generative modeling of multivariate time series is a challenging problem with wide-ranging applications in demand forecasting [15, 76], autonomous driving [2, 16], robotics [29, 67], and health care [20, 21, 59]. Despite remarkable progress in recent years, models that predict high-dimensional future observations from a few past examples have remained intractable, partly due to the complex, non-deterministic temporal dynamics across long-distance time steps. Given a sequence of human poses, for example, such models must internally figure out the involved dynamics of various body components across space and time while maintaining the inherent uncertainty of multiple plausible futures, even though only one such future is observed. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Among proposed probabilistic approaches, state space models (SSMs) provide a principled framework for learning and drawing inference from sequential inputs [27, 66]. While autoregressive models feed its predictions back into the dynamics model without any compressed representation of data, SSMs model stochastic transitions between abstract states using latent variables, allowing for efficient state-to-state sampling without the need to render high-dimensional observations. Gaussian linear dynamical systems (LDSs), one of the best known SSMs [92], for example, postulate linear state transitions and enjoy exact inference via the celebrated Kalman filter algorithm. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While early extensions of LDSs focus on linearization [46] and unscented transform [88], recent work that marry state space models with deep neural networks offers much more flexibility to model complex dependencies across different time steps. Some approaches retain the Markovian dynamics of LDSs and only replace their linear observation models with feed-forward networks [23, 31, 47, 71], whereas others favor nonlinear state transitions and parametrize such dependencies via recurrent neural networks (RNNs) [22, 23, 30, 39, 51, 75]. Despite differences, both Markovian transitions and RNNs are often not capable of capturing long-range dependencies in highly structured sequential inputs [36, 100], limiting the capacity of the corresponding SSMs. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/6209b8c1681d417ef10b0d5a80eb743603ec9c0900243de7593a7d1462ab034a.jpg",
+        "image_caption": [
+            "Figure 1: Graphical model representations of linear dynamical systems (LDSs) in (a), and our proposed models (ProTran) in (b), (c), and (d). Black arrows denote the generative mechanism and red arrows the inference procedure. The separation of generation and inference in (c) and (d) is for readability. While traditional SSMs such as LDSs are limited to Markovian dynamics and linear dependencies, our models allow for non-Markovian and non-linear interactions between time steps via attention mechanism. A multi-layer extension of our models further increases expressiveness without compromising the tractable inference procedure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose to combine the complementary strengths of SSMs and transformer architectures [85], a powerful mechanism for modeling long-term interactions that enjoys success across a variety of sequence modeling tasks [26, 48, 99]. In contrast to most SSMs, our models make extensive use of attention mechanism [5, 85] between latent variables to model non-Markovian dynamics (see Figure 1). Compared to transformer-based methods, our models are probabilistic, nonautoregressive in a similar fashion to LDSs, and capable of generating diverse long-term forecasts with uncertainty estimates. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contributions are threefold. First, we propose novel SSMs based on transformer architectures for multivariate time series, which include generative models and inference procedures based on variational inference [49, 74]. Second, we extend our models to include several layers of stochastic latent variables organized in a hierarchy for further expressiveness. Third, we conduct extensive experiments on time series forecasting and human motion prediction and demonstrate that our Probabilistic Transformer (ProTran) performs remarkably well compared to various state-of-theart baselines. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 Variational State Space Models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Let $\\{ \\mathbf { x } _ { 1 : T _ { i } } ^ { ( i ) } \\} _ { i = 1 } ^ { N }$ consist of $N$ univariate time series w ere $\\mathbf { x } _ { 1 : T _ { i } } ^ { ( i ) } = ( \\mathbf { x } _ { 1 } ^ { ( i ) } , \\mathbf { x } _ { 2 } ^ { ( i ) } , \\cdot \\cdot \\cdot \\mathbf { x } _ { T _ { i } } ^ { ( i ) } )$ and $\\mathbf { x } _ { t } ^ { ( i ) }$ denotes the vaue of the $i$ $t$ We consider the multivariate form $\\mathbf { x } _ { 1 : T } = ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\ldots , \\mathbf { x } _ { T } )$ where $\\mathbf { x } _ { t } = ( \\mathbf { x } _ { t } ^ { ( 1 ) } , . . . \\mathbf { x } _ { t } ^ { ( N ) } ) \\in \\mathbb { R } ^ { N }$ . Conditioning on observed values up to time $C$ , we aim to produce distributional forecasts into the future $p ( \\mathbf { x } _ { C + 1 : T } \\mid \\mathbf { x } _ { 1 : C } )$ . For clarity, we refer to $\\mathbf { x } _ { 1 : C }$ and $\\mathbf { x } _ { C + 1 : T }$ as contexts and targets, respectively. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We are interested in probabilistic models parametrized by $\\theta$ of the form ",
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+        "img_path": "images/a0644fcb9c8f885a685398a8b6f6f73851de38e5831cc86bfc4027ce014d2bef.jpg",
+        "text": "$$\np _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { x } _ { 1 : C } ) = \\int p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } ) p _ { \\theta } ( \\mathbf { z } _ { 1 : T } \\mid \\mathbf { x } _ { 1 : C } ) d \\mathbf { z } _ { 1 : T }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where ${ \\bf z } _ { 1 : T } = ( { \\bf z } _ { 1 } , { \\bf z } _ { 2 } , \\ldots , { \\bf z } _ { T } )$ denotes the corresponding sequence of latent variables, sometimes referred to as states. In other words, we assume a generative model that can be decomposed into a transition model $p _ { \\theta } ( \\mathbf { z } _ { 1 : T } \\mid \\mathbf { x } _ { 1 : C } )$ between the latent variables conditioned on the contexts, and an emission model $p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } )$ from the latent variables to observable outputs. In particular, we further impose several assumptions on both models: 1 ",
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+        "type": "text",
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+    {
+        "type": "equation",
+        "img_path": "images/0fd14c83e4f3f2c7584f799eb3cc4b42db04d9bdee8e4f0c19a6f5d3fdaa2e81.jpg",
+        "text": "$$\np _ { \\theta } ( \\mathbf { z } _ { 1 : T } \\mid \\mathbf { x } _ { 1 : C } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( \\mathbf { z } _ { t } \\mid \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : C } ) , \\qquad p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( \\mathbf { x } _ { t } \\mid \\mathbf { z } _ { t } ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "As demonstrated in Figure 1(b), the latent variable $\\mathbf { z } _ { t + 1 }$ depends not only on $\\mathbf { z } _ { t }$ but also on all of its preceding latent variables, including $\\mathbf { z } _ { t - 1 }$ , in contrast to linear dynamical systems (LDSs). In addition, the transition and emission models allow for non-linearity via neural network parametrizations. These assumptions aim to maximize model capacity for real-world applications with complex emissions or temporal dependencies. ",
+        "bbox": [
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+        "type": "text",
+        "text": "However, neither $\\mathbf { x } _ { 1 : t - 1 }$ nor $\\mathbf { z } _ { 1 : t - 1 }$ are included in the emission model $p ( \\mathbf { x } _ { t } \\mid \\mathbf { z } _ { 1 : T } , \\mathbf { x } _ { 1 : C } )$ . Such assumptions are important, as it has been argued previously that a leakage of information from the latent space in autoregressive models can hinder long-term predictions [23, 47]. While all ground truth observations are available during training, the entire sequence has to be generated sequentially at test time, making the dependencies on $\\mathbf { x } _ { 1 : t - 1 }$ prone to accumulated errors over multiple time steps. By letting the latent variable $\\mathbf { z } _ { t }$ capture all information needed to render $\\mathbf { x } _ { t }$ , we also avoid the computational costs associated with repeatedly decoding and encoding $\\mathbf { x } _ { t }$ in multi-step predictions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The inclusion of nonlinear state transitions and observation models necessarily requires approximate inference. We follow the stochastic variational inference framework [49, 74] and assume that the variational posterior parametrized by $\\phi$ can be decomposed auto-regressively as $q _ { \\phi } ( { \\bf z } _ { 1 : T } \\mid { \\bf x } _ { 1 : T } ) =$ $\\begin{array} { r } { \\prod _ { t } q _ { \\phi } ( \\mathbf { z } _ { t } \\mid \\dot { \\mathbf { z } } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : T } ) } \\end{array}$ , which leads to a lower bound on the log likelihood: ",
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+    {
+        "type": "equation",
+        "img_path": "images/ce4b1a73304e24b184833394510d4ee03297b0f7b049de958c830d34e7c27af2.jpg",
+        "text": "$$\n\\log p _ { \\theta } ( \\mathbf { x } _ { 1 : T } | \\mathbf { x } _ { 1 : C } ) \\geq \\sum _ { t = 1 } ^ { T } ( \\mathbb { E } _ { q } \\left[ \\log p _ { \\theta } ( \\mathbf { x } _ { t } | \\mathbf { z } _ { t } ) \\right] - \\mathsf { K L } ( q _ { \\phi } ( \\mathbf { z } _ { t } | \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : T } ) \\parallel p _ { \\theta } ( \\mathbf { z } _ { t } | \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : C } ) ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathsf { K L }$ is the Kullback-Leibler divergence. ",
+        "bbox": [
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+        "type": "text",
+        "text": "For computational stability, we assume homoscedasticity and choose Laplace distribution with scale parameter $\\beta$ as a parametric form for $p _ { \\theta } ( \\mathbf { x } _ { t } \\mid \\mathbf { z } _ { t } )$ , i.e. we optimize for $L _ { 1 }$ reconstruction loss with a cross-validated factor $\\beta$ for the KL term, following similar variational autoencoder (VAE) work [24, 41, 86]. Such an assumption does not necessarily limit the capacity of our models, as powerful stochastic transitions and flexible emission models can theoretically characterize arbitrary noise covariance [66]. Incorporating structured probabilistic outputs such as Gaussian copulas [75] or normalizing flows [23] can potentially further improve our model performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 Transformer Architectures ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Central to our models and other transformer-based approaches [48, 85] is the notion of attention [5], which allows the models to focus on important parts within a context. Multi-head attention, for example, maps a sequence of queries $\\mathbf { Q } \\doteq \\mathbb { R } ^ { \\ell _ { q } \\times \\dot { d } }$ of length $\\ell _ { q }$ to a sequence of outputs ${ \\bf O } =$ $[ \\mathbf { O } _ { 1 } , \\dots , \\mathbf { O } _ { H } ] \\in \\mathbb { R } ^ { \\ell _ { q } \\times d }$ of the same size by attending over given $\\ell _ { k }$ key-value pairs $\\mathbf { K } \\in \\mathbb { R } ^ { \\ell _ { k } \\times d }$ , V ∈ R\\`k×d: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1d2e4c99d03481dab110a931afaf26be7a39372e107e60a899e93cb9f2466531.jpg",
+        "text": "$$\n\\mathbf { O } _ { h } = \\mathsf { A t t e n t i o n } ( \\mathbf { Q } _ { h } , \\mathbf { K } _ { h } , \\mathbf { V } _ { h } ) = \\mathsf { S o f t m a x } \\left( \\frac { \\mathbf { Q } _ { h } \\mathbf { K } _ { h } ^ { \\mathsf { T } } } { \\sqrt { d } } \\right) \\mathbf { V } _ { h } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { Q } _ { h } = \\mathbf { Q } \\mathbf { W } _ { h } ^ { Q }$ , $\\mathbf { K } _ { h } = \\mathbf { K } \\mathbf { W } _ { h } ^ { K }$ , $\\mathbf { V } _ { h } = \\mathbf { V } \\mathbf { W } _ { h } ^ { V }$ are projected queries, keys, and values coro head , we re $h \\in [ 1 , H ]$ with learning parameters n attention mechanism as s $\\mathbf { W } _ { h } ^ { Q } , \\mathbf { W } _ { h } ^ { K } , \\mathbf { W } _ { h } ^ { V }$ , respectively. In case $\\mathbf { Q } = \\mathbf { K } = \\mathbf { V }$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given fully observed sequences of inputs, the mapping can be computed efficiently without any imposed sequential order often seen in recurrent neural networks [19, 42]. More importantly, the direct connections between long-distance time steps are baked into the mechanism as information from previous time steps is easily accessible without being compressed into a fixed representation, easing optimization and learning of long-term dependencies [5, 85]. ",
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+    {
+        "type": "text",
+        "text": "Without recurrence, Transformer [85] encodes information about each time step $t$ with pred efined sinusoidal positional embeddings Position $( t ) = [ p _ { t } ( 1 ) , \\ldots , p _ { t } ( d ) ] \\in \\mathbb { R } ^ { d }$ where the $i$ -th embedding is given by $p _ { t } ( i ) = \\sin ( t \\cdot c ^ { i / d } )$ for even $i$ and $p _ { t } ( i ) = \\cos ( t \\cdot c ^ { i / d } )$ for odd $i$ and $c$ is some large constant. Empirical results show that such positional embeddings are also important to our models. ",
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+    {
+        "type": "text",
+        "text": "3 Probabilistic Transformer ",
+        "text_level": 1,
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In this section, we first present our single-layered model and subsequently its multi-layered extension for a hierarchy of stochastic latent variables. As alluded earlier, our model consists of a generative model and an inference model that share information and parameters extensively. ",
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+        "type": "text",
+        "text": "3.1 Single-Layered Probabilistic Transformer ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Generative Model. Given some contexts $\\mathbf { x } _ { 1 : C }$ , we first apply a linear projection and combine it with a positional embedding to obtain $\\mathbf { h } _ { 1 : C } \\in \\mathbb { R } ^ { d }$ , i.e. ",
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+    {
+        "type": "equation",
+        "img_path": "images/4c78c8fd31901e729197f33f534ccb60844c538fb889cc90cd3aea786d06e82d.jpg",
+        "text": "$$\n\\mathbf { h } _ { t } = \\mathsf { L a y e r N o r m } ( \\mathsf { M L P } ( \\mathbf { x } _ { t } ) + \\mathsf { P o s i t i o n } ( t ) ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where LayerNorm and MLP denote layer normalizations [4] and multi-layer perceptrons, respectively. While a traditional transformer model often dedicates an entire encoder for the same purpose [55, 72], we find such a simple mapping works sufficiently well in conjunction with the contextattention module of the corresponding decoder. ",
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+    {
+        "type": "text",
+        "text": "As implied in Equation (2), our latent dynamics decomposes auto-regressively. At each time step, we parametri ze the distribution $p _ { \\theta } ( \\mathbf { z } _ { t } \\mid \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : C } )$ by a Gaussian with parameters resulting from two sequential steps of attention: a self-attention over the previously inferred states $\\mathbf { z } _ { 1 : t - 1 }$ and another attention over the projected contexts $\\mathbf { h } _ { 1 : C }$ . These two operations mirror those found in the decoder of Transformer [85], with the stochastic latent variables replacing its decoder inputs. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Unfortunately, using stochastic samples of $\\mathbf { z } _ { t }$ as attention queries is problematic, as purely stochastic transitions make it difficult for the model to reliably retain information across multiple time steps [17, 30, 39]. We therefore encapsulate the latent variables in hidden representations $\\mathbf { w } _ { t }$ that also has a deterministic component. Combined with the attention steps, such representations help model long-range temporal dependencies while accounting for the stochasticity of future observations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Starting with a learnable, context-agnostic representation $\\mathbf { w } _ { 0 }$ , we recursively update $\\mathbf { w } _ { t }$ using a stochastic sample from $p _ { \\theta } ( \\mathbf { z } _ { t } \\mid \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : C } )$ and th e positional embedding for the current time step $t$ . The generating process for the time step $t$ can be summarized by the following pseudocode: ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/79d10428ba726f276b73d0eb6f9c84f49d12e173f73bee794357ec744004ef84.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\bar { { \\bf w } } _ { t } = \\mathsf { L a y e r N o r m } \\big ( { \\bf w } _ { t - 1 } + \\mathsf { A t t e n t i o n } \\big ( { \\bf w } _ { t - 1 } , { \\bf w } _ { 1 : t - 1 } , { \\bf w } _ { 1 : t - 1 } \\big ) \\big ) } \\\\ & { \\hat { \\bf w } _ { t } = \\mathsf { L a y e r N o r m } \\big ( \\bar { \\bf w } _ { t } + \\mathsf { A t t e n t i o n } \\big ( \\bar { \\bf w } _ { t } , { \\bf h } _ { 1 : C } , { \\bf h } _ { 1 : C } \\big ) \\big ) } \\\\ & { { \\bf z } _ { t } = \\mathsf { S a m p l e } \\big ( \\mathcal { N } \\big ( { \\bf z } _ { t } ; \\mathsf { M L P } ( \\hat { \\bf w } _ { t } ) , \\mathsf { S o f t p l u s } ( \\mathsf { M L P } ( \\hat { \\bf w } _ { t } ) \\big ) \\big ) \\big ) } \\\\ & { { \\bf w } _ { t } = \\mathsf { L a y e r N o r m } \\big ( \\hat { \\bf w } _ { t } + \\mathsf { M L P } ( { \\bf z } _ { t } ) + \\mathsf { P o s i t i o n } ( t ) \\big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where Sample and Softplus are the Gaussian sampling and approximating rectifier operators. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Each stochastic sample of $\\mathbf { w } _ { 1 : T }$ is then mapped to a sequence of $\\mathbf { x } _ { \\mathrm { 1 : } T }$ via a multi-layer perceptron. We emphasize that our generation procedure in the latent space is more efficient than others in the observation space, which requires encoding and decoding high-dimensional inputs repeatedly. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Inference Model. We parametrize the approximate posterior $q _ { \\phi } ( \\mathbf { z } _ { t } \\mid \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : T } )$ at time step $t$ in a simi lar fashion to the prior $p _ { \\theta } ( \\mathbf { z } _ { t } \\mid \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : C } )$ . Indeed, these parametrizations share most parameters and are done simultaneously in the same recursive loop, following the exact same steps in Equation (6) and Equation (7) (see Figure 1). We note that similar sharing techniques between the generative and inference processes have emerged as a common theme among recent successful VAE models [17, 62, 83]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While the prior only has access to the conditioning observations $\\mathbf { x } _ { 1 : C }$ , the approximate posterior should take into account all observations during training, including the targets $x _ { C + 1 : T }$ . Due to the inherent unidirectional aspect of RNNs, previous work that uses RNNs to parametrize the approximate posterior often disregards such a property [22, 30, 51] and often resorts to a filtering routine $p ( \\mathbf { z } _ { t } \\mid \\mathbf { z } _ { 1 : t - 1 } , \\mathbf { x } _ { 1 : t } )$ . In contrast, our inference procedure resembles more of the smoothing process of LDSs, factoring in both past and future observations via another application of self-attention: ",
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+        "type": "text",
+        "text": "",
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/25309cc0158c0723496f2f11cb61d057d84a3cc6670ca582dd313b9d5180ae9f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { { \\bf k } _ { t } = \\mathrm { A t t e n t i o n } ( { \\bf h } _ { 1 : T } , { \\bf h } _ { 1 : T } , { \\bf h } _ { 1 : T } ) ) } \\\\ & { { \\bf z } _ { t } = \\mathsf { S a m p l e } ( \\mathcal { N } ( { \\bf z } _ { t } ; { \\sf M L P } ( [ \\hat { \\bf w } _ { t } , { \\bf k } _ { t } ] ) , \\mathsf { S o f t p l u s } ( { \\sf M L P } ( [ \\hat { \\bf w } _ { t } , { \\bf k } _ { t } ] ) ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Here, we replace Equation (8) in the generative model with Equation (11), where the hidden representation $\\mathbf { k } _ { t }$ summarizing all information relevant to the current tim estep $t$ has been concatenate to the latent-and-context-aware representation $\\hat { \\mathbf { w } } _ { t }$ preceding the Gaussian parametrization. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The generative model and the inference model are trained end-to-end with a single stochastic variational inference objective stated in Equation (3). Such a variational bound includes the reconstruction loss for $\\mathbf { X } _ { 1 : C }$ and the $\\mathsf { K L }$ term for $\\mathbf { z } _ { 1 : C }$ . Alternatively, we can exclude these terms from the objective, which is equivalent to starting the inference process from $t = C + 1$ instead of $t = 1$ . ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our models incur a time complexity of $\\mathcal { O } ( T ^ { 2 } d )$ and a memory cost of $\\mathcal { O } ( T ^ { 2 } d )$ , where $T$ is the total sequence length and $d$ is the dimensionality of the latent space. The recursive latent dynamics also does not allow use the take full advantange of parallelizable attentions. However, we find that our models are still efficient in practice, especially for reasonably small values of $T$ . ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3.2 Multi-Layered Extension for Probabilistic Transformer ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Inspired by recent work on hierarchical VAEs for non-sequential inputs [17, 80, 83, 101], we extend our proposed model to include several layers of latent variables, aiming to further increase its flexibility for modelling sequential data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We represent each time step $t$ with a Ma rkov chain of $L$ latent variables $\\mathbf { z } _ { t } ^ { ( 1 : L ) } = ( \\mathbf { z } _ { t } ^ { ( 1 ) } , \\ldots , \\mathbf { z } _ { t } ^ { ( L ) } )$ for simplicity (see Figure 1). The generative and inference model also decompose auto-regressively across different time steps and may exhibit non-Markovian dynamics: ",
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+        "type": "equation",
+        "img_path": "images/9e77200db40fe310bda7d11cb45a13e41b2d0474fe93ac818dad9b1ca90b9770.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle p _ { \\theta } \\left( { \\bf x } _ { 1 : T } , { \\bf z } _ { 1 : T } ^ { ( 1 : L ) } | { \\bf x } _ { 1 : C } \\right) = \\left( \\prod _ { \\ell = 1 } ^ { T } p _ { \\theta } \\left( { \\bf x } _ { t } | { \\bf z } _ { t } ^ { ( L ) } \\right) \\right) \\left( \\prod _ { \\ell = 1 } ^ { L } \\prod _ { \\ell = 1 } ^ { T } p _ { \\theta } \\left( { \\bf z } _ { t } ^ { ( \\ell ) } | { \\bf z } _ { 1 : t - 1 } ^ { ( \\ell ) } , { \\bf z } _ { 1 : T } ^ { ( \\ell - 1 ) } , { \\bf x } _ { 1 : C } \\right) \\right) } } \\\\ { { \\displaystyle q _ { \\phi } \\left( { \\bf z } _ { 1 : T } ^ { ( 1 : L ) } | { \\bf x } _ { 1 : T } \\right) = \\prod _ { \\ell = 1 } ^ { L } \\prod _ { \\ell = 1 } ^ { T } q _ { \\phi } \\left( { \\bf z } _ { t } ^ { ( \\ell ) } | { \\bf z } _ { 1 : t - 1 } ^ { ( \\ell ) } , { \\bf z } _ { 1 : T } ^ { ( \\ell - 1 ) } , { \\bf x } _ { 1 : T } \\right) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Intuitively, we generate samples $\\mathbf { x } _ { 1 : T }$ conditioning on $\\mathbf { x } _ { 1 : C }$ by following the latent dynamics from the bottom up and using the generative process described earlier within each layer. Analogously, inference proceeds in the same order, resulting in a variational bound similar to Equation (3): ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/51b64919a33bb441cf03ca4f902aaeea2dfa93646ba997764fcaf73982079a74.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\log p _ { \\theta } \\big ( { \\bf x } _ { 1 : T } \\mid { \\bf x } _ { 1 : C } \\big ) \\geq \\sum _ { t = 1 } ^ { T } \\mathbb { E } _ { q } \\left[ \\log p _ { \\theta } \\big ( { \\bf x } _ { t } ^ { ( L ) } \\mid { \\bf z } _ { t } \\big ) \\right] } \\ ~ } \\\\ { { \\displaystyle ~ - \\sum _ { \\ell = 1 } ^ { L } \\mathsf { K L } \\big ( q _ { \\phi } \\big ( { \\bf z } _ { t } ^ { ( \\ell ) } \\mid { \\bf z } _ { 1 : t - 1 } ^ { ( \\ell ) } , { \\bf z } _ { 1 : T } ^ { ( \\ell ) } , { \\bf x } _ { 1 : T } \\big ) \\ \\big \\| \\ p _ { \\theta } \\big ( { \\bf z } _ { t } ^ { ( \\ell ) } \\mid { \\bf z } _ { 1 : t - 1 } ^ { ( \\ell ) } , { \\bf z } _ { 1 : T } ^ { ( \\ell ) } , { \\bf x } _ { 1 : C } \\big ) \\big ) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As before, we parametrize the prior ) | z(\\`)1:t−1, z(\\`)1:T , x1:C ) using self-attention over the inferred(\\`) latent variables from previous time steps w t−1 o n the same layer and another attention over contexts $\\mathbf { h } _ { 1 : C }$ . In this case, however, we include an additional self-attention over all latent variables from the layer immediately below it (see Equation (16)): ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/30d022c63236a8b9105d0aabb427ac7c6f556c1e27cb2d3eb571ea73185b60cb.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\tilde { \\mathbf { w } } _ { t } ^ { ( \\ell ) } = \\mathsf { L a y e r N o r m } ( \\mathbf { w } _ { t - 1 } ^ { ( \\ell ) } + \\mathsf { A t t e n t i o n } ( \\mathbf { w } _ { t - 1 } ^ { ( \\ell ) } , \\mathbf { w } _ { 1 : T } ^ { ( \\ell - 1 ) } , \\mathbf { w } _ { 1 : T } ^ { ( \\ell - 1 ) } ) ) } \\\\ & { \\bar { \\mathbf { w } } _ { t } ^ { ( \\ell ) } = \\mathsf { L a y e r N o r m } ( \\tilde { \\mathbf { w } } _ { t } ^ { ( \\ell ) } + \\mathsf { A t t e n t i o n } ( \\tilde { \\mathbf { w } } _ { t } ^ { ( \\ell ) } , \\mathbf { w } _ { 1 : t - 1 } ^ { ( \\ell ) } , \\mathbf { w } _ { 1 : t - 1 } ^ { ( \\ell ) } ) ) } \\\\ & { \\hat { \\mathbf { w } } _ { t } ^ { ( \\ell ) } = \\mathsf { L a y e r N o r m } ( \\bar { \\mathbf { w } } _ { t } ^ { ( \\ell ) } + \\mathsf { A t t e n t i o n } ( \\bar { \\mathbf { w } } _ { t } ^ { ( \\ell ) } , \\mathbf { h } _ { 1 : C } , \\mathbf { h } _ { 1 : C } ) ) } \\\\ & { \\mathbf { z } _ { t } ^ { ( \\ell ) } = \\mathsf { S a m p l e } ( \\mathcal { N } ( \\mathbf { z } _ { t } ^ { ( \\ell ) } ; \\mathsf { M L P } ( \\hat { \\mathbf { w } } _ { t } ^ { ( \\ell ) } ) , \\mathbf { S o f t p l u s } ( \\mathsf { M L P } ( \\hat { \\mathbf { w } } _ { t } ^ { ( \\ell ) } ) ) ) ) } \\\\ & { \\mathbf { w } _ { t } ^ { ( \\ell ) } = \\mathsf { L a y e r N o r m } ( \\hat { \\mathbf { w } } _ { t } ^ { ( \\ell ) } + \\mathsf { M L P } ( \\mathbf { z } _ { t } ^ { ( \\ell ) } ) + \\mathsf { P o s i t i o n } ( t ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Stacking multiple layers of latent variables increases model expressiveness, but it also result in a linear increase in running time and the number of parameters. The time complexity for the $L$ - layers transformer is $\\mathcal { O } ( \\bar { L _ { 1 } } T ^ { 2 } d )$ , while the space complexity remains $\\mathcal { D } ( T ^ { 2 } d )$ due to the Markovian structure of the chain $\\mathbf { z } _ { t } ^ { ( 1 : L ) }$ at each time step $t$ . In our experiments, we restrict the number of layers of our hierachical models to two or three. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Deep State Space Models. Deep neural networks have been extensively combined with state space models, resulting in flexible, yet principledly motivated latent variable approaches. While some work keep the linear state transition intact to leverage the efficient Kalman filer algorithms [23, 31, 47, 71], more expressive, nonlinear latent dynamics parametrized by neural networks have been proposed [51, 52]. All such models are limited to the Markovian dynamics of LDSs, which hinders learning of long-range dependencies. The limitation is often alleviated by combining the stochastic transitions with a deterministic RNN that enables access to all past states [3, 8, 22, 30, 39, 77]. Our models are similarly non-Markovian, but the dependencies on the past states are done via attention, which allows for easy connections between long-distance time steps. In addition, while most existing deep SSMs represent each time step with a single latent variable, our models include several layers of hierarchical latent variables with tractable inference mechanism. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Attentive Recurrent Networks. Attention mechanism has also been widely adopted in recent time series work using sequence-to-sequence models [1, 28] or transformer architectures [14, 55, 57, 72, 81, 94]. While our models are equipped with latent variables, these transformer approaches [55, 72] lack inference mechanism and are susceptible to feeding back observation noise into the dynamics model at test time. Our work, however, can be considered as an extension of the attentive state space model proposed in [1], with discrete latent states replaced by their continuous analogs. Recent developments in natural language processing [58, 60, 90] also combine transformer and VAE; however, these approaches often use a time-agnostic latent variable, in contrast to our SSM formulation. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Time Series Forecasting. Traditional univariate time series models, such as Box-Jenkins methods [12] and exponential smoothing [43], often assume independence between any collection of time series [76]. While multivariate extensions of the classical approaches, including vector autoregression [82] and multivariate GARCH [7], do not require such a strong assumption, they come with many others such as stationarity and homocesdasticity, demand manual selection of covariates and models, and do not scale well to even a moderate number of time series [40, 69]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Deep learning methods for time series forecasting have recently emerged as an expressive, scalable framework for industrial applications [10, 68, 79, 91]. While early work focus on point forecasts [53, 70, 96], recent approaches often employ recurrent neural networks with probabilistic forecasts parametrized directly [76], using quantile functions [33], Gaussian copulas [75], normalizing flows [23], or diffusion models [73]. In contrast, our models are entirely devoid of such recurrent architectures and rely on latent variables to output distributional forecasts. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Human Motion Prediction. Despite being almost identical in formulation, human motion prediction has often been studied independently from time series forecasting. While some work deterministically generate future motions or video frames [13, 32, 34, 56], stochastic prediction has also been proposed with deep neural networks often outperforming traditional methods such as hidden Markov models [93] or Gaussian processes [89] on complex motion datasets [13, 32, 45, 54, 63]. In contrast to earlier work [95, 97] that employ a global latent variable across different time steps via conditional VAE [49], we leverage the principled framework of state space models for learning and inference of hierarchical, time-dependent latent variables. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We present our experiment results on two tasks, namely, time series forecasting and human motion prediction. These tasks are often studied independently, despite being almost identical as conditional prediction problems. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bd9124778ddd19c48cddc3dd21e3414ee18aab3d6c91e96a608a922249966cdc.jpg",
+        "table_caption": [
+            "Table 1: Test set ${ \\mathsf { C R P S } } _ { \\mathsf { s u m } }$ of time series forecasting models (lower is better). The means and standard deviations are computed over five runs using different random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATASET</td><td>SOLAR</td><td>ELECTRICITY</td><td>TRAFFIC</td><td>TAXI</td><td>WIKIPEDIA</td></tr><tr><td>VES [43]</td><td>0.900 ± 0.003</td><td>0.880 ± 0.004</td><td>0.350 ± 0.002</td><td></td><td></td></tr><tr><td>VAR [61]</td><td>0.830 ± 0.006</td><td>0.039 ± 0.001</td><td>0.290 ± 0.001</td><td></td><td></td></tr><tr><td>VAR-Lasso [61]</td><td>0.510 ± 0.006</td><td>0.025 ± 0.000</td><td>0.150 ± 0.002</td><td></td><td>3.100 ± 0.004</td></tr><tr><td>GARCH [84]</td><td>0.880 ± 0.002</td><td>0.190 ± 0.001</td><td>0.370 ± 0.001</td><td></td><td></td></tr><tr><td>DeepAR [76]</td><td>0.336 ± 0.014</td><td>0.023 ± 0.001</td><td>0.055 ± 0.003</td><td></td><td>0.127 ± 0.042</td></tr><tr><td>LSTM-Copula [75]</td><td>0.319 ± 0.011</td><td>0.064 ± 0.008</td><td>0.103 ± 0.006</td><td>0.326 ± 0.007</td><td>0.241± 0.003</td></tr><tr><td>GP-Copula [75]</td><td>0.337 ± 0.024</td><td>0.024 ± 0.002</td><td>0.078 ± 0.002</td><td>0.208 ± 0.183</td><td>0.086 ± 0.004</td></tr><tr><td>KVAE [51]</td><td>0.340 ± 0.025</td><td>0.051 ± 0.019</td><td>0.100 ± 0.005</td><td></td><td>0.095 ± 0.012</td></tr><tr><td>NKF [23]</td><td>0.320 ± 0.020</td><td>0.016 ± 0.001</td><td>0.100 ± 0.002</td><td></td><td>0.071 ± 0.002</td></tr><tr><td>Transformer-MAF[72]</td><td>0.301 ± 0.014</td><td>0.021 ± 0.000</td><td>0.056 ± 0.001</td><td>0.179 ± 0.002</td><td>0.063 ± 0.003</td></tr><tr><td>TimeGrad[73]</td><td>0.287 ± 0.020</td><td>0.021 ± 0.001</td><td>0.044 ± 0.006</td><td>0.114 ± 0.020</td><td>0.049 ± 0.002</td></tr><tr><td>ProTran (Ours)</td><td>0.194 ± 0.030</td><td>0.016 ± 0.001</td><td>0.028 ± 0.001</td><td>0.084 ± 0.003</td><td>0.047 ± 0.004</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5.1 Time-series Forecasting ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets & Covariates. Following the experiment setup in [72, 73, 75], we evaluate our models and multiple competitive baselines on five popular public datasets: SOLAR, ELECTRICITY, TRAFFIC, TAXI, and WIKIPEDIA. The data is recorded with hourly or daily frequency and shows seasonal patterns of different frequencies (see Appendix A for more dataset details). As in [72, 73], the covariates include lagged inputs, fixed time embeddings (e.g. day of week, hour of day), and learnable time-series embeddings. The inputs are scaled using the conditioning examples before being fed into the model, and the predictions are rescaled appropriately afterward. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Metrics. Following [23, 72, 75], we evaluate our model and all baselines using continuous ranked probability score (CRPS) [65] summed across time series, denoted by ${ \\mathsf { C R P S } } _ { \\mathsf { s u m } }$ . Given a univariate distribution function $F$ and an observation $x$ , CRPS is defined as ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eed19f3b2f91088324b060d953afee6b8e280ecea7f8ef31f3bce05edbc0983c.jpg",
+        "text": "$$\n{ \\mathsf { C R P S } } ( F , x ) = \\int _ { \\mathbb { R } } ( F ( z ) - \\mathbb { 1 } _ { \\{ x \\leq z \\} } ) ^ { 2 } d z ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $1 _ { \\{ x \\leq z \\} }$ is the indicator function. As argued in de Bezenac et al. [ ´ 23], ${ \\mathsf { C R P S } } _ { \\mathsf { s u m } }$ is a proper scoring rule [35] and can be computed without analytical forecast distributions. We compute the metrics in a rolling fashion and use 100 samples for the distributional forecasts, similar to the aforementioned work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baselines. We benchmark our models against various baselines, including (1) VES [43], an innovation state space model; (2) VAR-Lasso and VAR [61], two multivariate linear autoregressive models with and without Lasso regularization; (3) GARCH [84], a multivariate conditional heteroskedastic model; (4) DeepAR [76], an autoregressive recurrent neural network; LSTM-Copula and GP-Copula [75], two RNN-based models that use Gaussian copula to model nonlinearity; (5) KVAE [51], a variational approach based on linear dynamics; (6) NKF [23], a normalizing-flow model coupled with Kalman filters; (7) Transformer [72], a transformer-based model based on masked autoregressive flow; and (8) TimeGrad [73], a recent autoregressive approach that uses a diffusion model. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implementations. We use 8-head attentions and 2-layers MLPs to parametrize the generative and inference models. The stochastic latent variables $\\mathbf { z } _ { t }$ are 16-dimensional while the hidden representations $\\mathbf { w } _ { t }$ are in $\\mathbb { R } ^ { 1 2 8 }$ . Our probabilistic transformers for SOLAR and ELECTRICITY have one stochastic layer while those for the other datasets of higher dimensional observations employ two layers. We re",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/88b146427d825c9442effc55d34d5e390d40cf12572fce124d5352b797cdfe58.jpg",
+        "table_caption": [
+            "Table 2: Ablation study on TRAFFIC. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Two Layers One Layer Context Attention</td><td>√ × √</td><td>× √</td><td>× ×</td><td>× √</td></tr><tr><td>Deterministic</td><td>×</td><td>×</td><td>×</td><td>√</td></tr><tr><td>CRPSsum</td><td>0.028</td><td>0.031</td><td>0.033</td><td>0.041</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "port the numbers of parameters of our models in Table 4 in Appendix C, which are all comparable to those of the state-of-the-art approaches. See Appendix D for more details about hyper-parameters and training processes. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "image",
+        "img_path": "images/2c04e02911790d4e50312ce21468be872c3332091a686e454ec6bd62c82ad8ba.jpg",
+        "image_caption": [
+            "Figure 2: Prediction intervals and test set ground-truth from ProTran (our model) for the TRAFFIC dataset of the first 16 of 963 time series. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Accuracy Comparison. Table 1 shows that our models perform competitively across all five highdimensional time series datasets, achieving ${ \\mathsf { C R P S } } _ { \\mathsf { s u m } }$ comparable to the best methods on ELECTRICITY and WIKIPEDIA while outperforming all baselines, including a non-SSM transformer-based approach [72], by significant margins on SOLAR, TRAFFIC and TAXI. Further analyses with other metrics, including CRPS and MSE, in Appendix B also help confirm our findings. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Qualitative Results. Figure 2 shows that the distribution forecasts generated by our model follow closely the ground truths, which is consistent with our accuracy results. In addition, the model appears to capture the uncertainty of future forecasts to some extent; observations of large magnitudes and far into the future seem to correctly have higher variance estimates. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Ablation Study. We include a small scale ablation study on the TRAFFIC dataset to investigate which components of our models are essential. Table 2 suggests that removing the stochasticity from $\\mathbf { w } _ { t }$ has most impacts on model performance, implying that incorprating latent variables into a transformer is indeed useful. Other aspects such as context attention or multiple layers of stochastic variables do not show dramatic effects in this study; however, they do contribute performance gains. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Human Motion Prediction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Datasets. Following the experiment setup in [97], we evaluation our models on two public motion capture datasets: Human3.6M[44] and HumanEva-I [78]. While Human3.6 is a large-scale dataset with 3.6 million video frames recorded at $5 0 \\mathrm { H z }$ , HumanEva-I is smaller with only 3 subjects and recorded at $6 0 \\mathrm { H z }$ . We follow the preprocessing steps of previous work [64, 97] and obtain a 17-joint skeleton for Human3.6 and a 15-joint skeleton for HumanEva-I. As in [97], we predict future motion for 2 seconds conditioning on observed motion of 0.5 seconds and 1 second conditioning on 0.25 seconds for Human3.6 and HumanEva-I, respectively. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9383da574a0f2a8ae5ae9ead64786b43baceb750e09f80f4d1df9b51d63a693.jpg",
+        "image_caption": [
+            "Figure 3: Ground-truth pose sequences (first row) and corresponding predictions by ProTran (second row). Solid colors indicate later time-steps and faded ones are older. The body-part movements in the predicted and ground-truth poses resemble similar patterns, while certain variations are retained. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/5cd7d906b1d9892d4f2cdbfeac6804bc0254a1b3d2803e3c6a282f3524fdcd2c.jpg",
+        "table_caption": [
+            "Table 3: Human motion prediction results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATASET</td><td colspan=\"2\">HUMAN3.6M</td><td colspan=\"2\">HUMANEVA-I</td></tr><tr><td>Method</td><td>ADE↓</td><td>FDE↓ ADE↓</td><td></td><td rowspan=\"3\">FDE↓</td></tr><tr><td>ERD [32]</td><td>0.722</td><td>0.969 0.382</td><td>0.461</td></tr><tr><td>acLSTM[56]</td><td>0.789 1.126</td><td></td><td>0.429 0.541</td></tr><tr><td>MT-VAE [95]</td><td>0.457</td><td>0.595</td><td>0.345</td><td rowspan=\"4\">0.403</td></tr><tr><td>Pose-Knows [87]</td><td>0.461</td><td>0.560</td><td>0.269 0.296</td></tr><tr><td>HP-GAN [6]</td><td>0.858</td><td>0.867</td><td>0.772 0.749</td></tr><tr><td>Best-Many [11]</td><td>0.448</td><td>0.533 0.271</td><td>0.279</td></tr><tr><td>GMVAE [25]</td><td>0.461</td><td>0.555</td><td>0.305</td><td>0.345</td></tr><tr><td>DeliGAN[38]</td><td>0.483</td><td>0.534</td><td>0.306</td><td>0.322</td></tr><tr><td>DSP [98]</td><td>0.493</td><td>0.592</td><td>0.273</td><td>0.290</td></tr><tr><td>DLow [97]</td><td>0.425</td><td>0.518</td><td>0.251</td><td>0.268</td></tr><tr><td>ProTran (Ours)</td><td>0.381</td><td>0.491</td><td>0.258</td><td>0.255</td></tr></table>",
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+        "type": "text",
+        "text": "Metrics. Following previous work on trajectory forecasting [2, 37], we adopt two popular metrics, namely, average displacement error (ADE) and final displacement error (FDE). ADE measures the average $L _ { 2 }$ distance over all time steps between the ground truth motion and the closest sample, while FDE only consider such distance for the final pose. ",
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+        "text": "Baselines. We compare our models against 9 models, including ERD [32] and acLSTM [56], two deterministic RNN-based approaches; MT-VAE [95] and Pose-Knows [87], two conditional VAE models; HP-GAN [6], a conditional GAN; Best-Many [11], GMVAE [25], DeliGAN [38]. and DSP [98], four approaches optimizing for diversity objectives. The results for these baselines are reported as in [97]. ",
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+        "text": "Implementations. Similar to the previous experiments, we use 8-head attentions and 2-layers MLPs. Since Human3.6M is significantly more complex and multi-modal than the time series forecasting datasets, we make use of 3 stochastic layers, as opposed to 2 layers for HumanEva-I. For Human3.6M, the context and target observations are significantly longer and set up for long-term predictions, so we only infer latent variables for target observations. Appendix C also contains further details about our models and their number of parameters. ",
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+        "text": "Quantitative Results. Table 3 shows that our models convincingly outperform all baselines based on both metrics ADE and FDE, with the gains significantly higher for the larger dataset Human3.6M. We emphasize that our favorable performance is evaluated using random samples, while the closest competitor, DLow [97], relies on a separate model for selecting samples to promote diversity, which can potentially be combined with our probabilistic transformer for further improvements. ",
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+        "text": "Qualitative Results. We show in Figure 3 human pose predictions made by our model that are most similar to the corresponding ground truths among a collection of such stochastic predictions. The similarities between the body-part movements in both sequences suggest that our model has been able to capture the temporal dynamics quite well. ",
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+        "text": "6 Conclusion & Discussion ",
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+        "text": "In this work, we have introduced generative models for multivariate time series that combines strengths of state space models and transformer architectures. In contrast to previous work, our models do not rely on recurrent neural networks but make extensive use of attention mechanism. We also extend our models to include hierarchical latent variables, inspired by recent developments of VAEs for non-sequential data [17, 83]. Empirical experiments show that our models perform remarkably well on time series forecasting and human motion prediction. ",
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+        "text": "Our models do not come without limitations, however. As in other transformer-based approaches, the reliance on attention incurs a quadratic time and memory complexity. While we do not find it problematic in our experiments, the limitation necessarily hinders applications of our models in tasks characterized by long-term dependencies such as language modelling or music generation [36]. Fortunately, recent work on sparse transformer [9, 18, 50, 55] can potentially address the issue, and we leave such an investigation for future work. ",
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+        "text": "Probabilistic time series forecasting is a fundamental research problem with wide-ranging applications in society. Although we have not explored healthcare applications of our work, previously proposed methods with similar formulations have demonstrated potentials of forecasting techniques [1, 81] in diagnoses or disease control. ",
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+        "text": "7 Acknowledgment ",
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+        "text": "Financial support is gratefully acknowledged from a Xerox PARC Faculty Research Award, National Science Foundation Awards 1455172, 1934985, 1940124, and 1940276, USAID, and Cornell University Atkinson Center for a Sustainable Future. ",
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+        "type": "text",
+        "text": "1. For all authors... ",
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+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 1 and 5.   \n(b) Did you describe the limitations of your work? [Yes] See Section 6.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   \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 include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See supplemental material.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appdendix D in supplemental material.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Table 1.   \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] See Appdendix D in supplemental material. ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] See Section 4.   \n(b) Did you mention the license of the assets? [Yes] See Appendix A in supplemental material.   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] See supplemental material. We are also commited to open source our code upon publication.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See Appendix A in supplemental material.   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] See Appendix A in supplemental material. ",
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parse/train/Hklso24Kwr/Hklso24Kwr.md

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+# CONTINUAL LEARNING WITH ADAPTIVE WEIGHTS(CLAW)
+
+Tameem Adel   
+Department of Engineering, University of Cambridge   
+tah47@cam.ac.uk
+
+Han Zhao Carnegie Mellon University han.zhao@cs.cmu.edu
+
+Richard E. Turner   
+Department of Engineering, University of Cambridge   
+Microsoft Research   
+ret26@cam.ac.uk
+
+# ABSTRACT
+
+Approaches to continual learning aim to successfully learn a set of related tasks that arrive in an online manner. Recently, several frameworks have been developed which enable deep learning to be deployed in this learning scenario. A key modelling decision is to what extent the architecture should be shared across tasks. On the one hand, separately modelling each task avoids catastrophic forgetting but it does not support transfer learning and leads to large models. On the other hand, rigidly specifying a shared component and a task-specific part enables task transfer and limits the model size, but it is vulnerable to catastrophic forgetting and restricts the form of task-transfer that can occur. Ideally, the network should adaptively identify which parts of the network to share in a data driven way. Here we introduce such an approach called Continual Learning with Adaptive Weights (CLAW), which is based on probabilistic modelling and variational inference. Experiments show that CLAW achieves state-of-the-art performance on six benchmarks in terms of overall continual learning performance, as measured by classification accuracy, and in terms of addressing catastrophic forgetting.
+
+# 1 INTRODUCTION
+
+Continual learning (CL), sometimes called lifelong or incremental learning, refers to an online framework where the knowledge acquired from learning tasks in the past is kept and accumulated so that it can be reused in the present and future. Data belonging to different tasks could potentially be non i.i.d. (Schlimmer & Fisher, 1986; Sutton & Whitehead, 1993; Ring, 1997; Schmidhuber, 2013; Nguyen et al., 2018; Schmidhuber, 2018). A continual learner must be able to learn a new task, crucially, without forgetting previous tasks (Ring, 1995; Srivastava et al., 2013; Schwarz et al., 2018; Serra et al., 2018; Hu et al., 2019). In addition, CL frameworks should continually adapt to any domain shift occurring across tasks. The learning updates must be incremental – i.e, the model is updated at each task only using the new data and the old model, without access to all previous data (from earlier tasks) – due to speed, security and privacy constraints. A compromise must be found between adapting to new tasks and enforcing stability to preserve knowledge from previous tasks. Excessive adaptation could lead to inadvertent forgetting of how to perform earlier tasks. Indeed, catastrophic forgetting is one of the main pathologies in continual learning (McCloskey & Cohen, 1989; Ratcliff, 1990; Robins, 1993; 1995; French, 1999; Pape et al., 2011; Goodfellow et al., 2014a; Achille et al., 2018; Kemker et al., 2018; Kemker & Kanan, 2018; Diaz-Rodriguez et al., 2018; Zeno et al., 2018; Ahn et al., 2019; Parisi et al., 2019; Pfulb & Gepperth, 2019; Rajasegaran et al., 2019).
+
+Many approaches to continual learning employ an architecture which is divided a priori into (i) a slowly evolving, global part; and (ii) a quickly evolving, task-specific, local part. This is one way to enable multi-task transfer whilst mitigating catastrophic forgetting, which has proven to be effective (Rusu et al., 2016b; Fernando et al., 2017; Yoon et al., 2018), albeit with limitations. Specifying a priori the shared global, and task-specific local parts in the architecture restricts flexibility. As more complex and heterogeneous tasks are considered, one would like a more flexible, data-driven approach to determine the appropriate amount of sharing across tasks. Here, we aim at automating the architecture adaptation process so that each neuron of the network can either be kept intact, i.e. acting as global, or adapted to the new task locally. Our proposed variational inference framework is flexible enough to learn the range within which the adaptation parameters can vary. We introduce for each neuron one binary parameter controlling whether or not to adapt, and two parameters to control the magnitude of adaptation. All parameters are learnt via variational inference. We introduce our framework as an expansion of the variational continual learning algorithm (Nguyen et al., 2018), whose variational and sequential Bayesian nature makes it convenient for our modelling and architecture adaptation procedure. Our modelling ideas can also be applied to other continual learning frameworks, see the Appendix for a brief discussion.
+
+We highlight the following contributions: (1) A modelling framework which flexibly automates the adaptation of local and global parts of the (multi-task) continual architecture. This optimizes the tradeoff between mitigating catastrophic forgetting and improving task transfer. (2) A probabilistic variational inference algorithm which supports incremental updates with adaptively learned parameters. (3) The ability to combine our modelling and inference approaches without any significant augmentation of the architecture (no new neurons are needed). (4) State-of-the-art results in six experiments on five datasets, which demonstrate the effectiveness of our framework in terms of overall accuracy and reducing catastrophic forgetting.
+
+# 2 BACKGROUND ON VARIATIONAL CONTINUAL LEARNING (VCL)
+
+In this paper, we use Variational Continual Learning (VCL, Nguyen et al., 2018) as the underlying continual learning framework. However, our methods apply to other frameworks, see Appendix (Section A.1). VCL is a variational Bayesian framework where the posterior of the model parameters $\pmb \theta$ is learnt and updated continually from a sequence of $T$ datasets, $\{ \pmb { x } _ { t } ^ { ( n ) } , \pmb { y } _ { t } ^ { ( n ) } \} _ { n = 1 } ^ { N _ { t } }$ } tn=1 , where $t =$ $1 , 2 , \ldots , T$ and $N _ { t }$ is the size of the dataset associated with the $t$ -th task. More specifically, denote by $p ( \pmb { y } | \pmb { \theta } , \pmb { x } )$ the probability distribution returned by a discriminative classifier with input $_ { \textbf { \em x } }$ , output $\textbf {  { y } }$ and parameters $\pmb { \theta }$ . For $\mathcal { D } _ { t } = \{ \pmb { y } _ { t } ^ { ( n ) } \} _ { n = 1 } ^ { N _ { t } }$ , we approximate the intractable posterior $p ( \pmb { \theta } | \mathcal { D } _ { 1 : t } )$ after observing the first $t$ datasets via a tractable variational distribution $q _ { t }$ as:1
+
+$$
+\mathbf { q } _ { t } ( \pmb \theta ) \approx \frac { 1 } { Z _ { t } } \mathbf { q } _ { t - 1 } ( \pmb \theta ) p ( \mathcal { D } _ { t } | \pmb \theta ) ,
+$$
+
+where ${ \bf q } _ { 0 }$ is the prior $p$ , $\begin{array} { r } { p ( \mathcal { D } _ { t } | \pmb { \theta } ) = \prod _ { n = 1 } ^ { N _ { t } } p ( \pmb { y } _ { t } ^ { ( n ) } | \pmb { \theta } , \pmb { x } _ { t } ^ { ( n ) } ) } \end{array}$ , and $Z _ { t }$ is the normalizing constant which does not depend on $\pmb \theta$ but only on the data $\mathcal { D }$ . This framework allows the approximate posterior $\mathbf { q } _ { t } ( \pmb { \theta } )$ to be updated incrementally from the previous approximate posterior $\mathbf { q } _ { t - 1 } ( \pmb { \theta } )$ in an online fashion. In VCL, the approximation in (1) is performed by minimizing the following KL-divergence over a family $\mathcal { Q }$ of tractable distributions:
+
+$$
+\mathbf { q } _ { t } ( \pmb \theta ) = \underset { \mathbf { q } \in \mathcal { Q } } { \mathrm { a r g m i n } } \mathrm { K L } \Big ( \mathbf { q } ( \pmb \theta ) \parallel \frac { 1 } { Z _ { t } } \mathbf { q } _ { t - 1 } ( \pmb \theta ) p ( \mathcal { D } _ { t } | \pmb \theta ) \Big ) .
+$$
+
+This framework can be enhanced to further mitigate catastrophic forgetting by using a coreset (Nguyen et al., 2018), i.e. a representative set of data from previously observed tasks that can serve as memory and can be revisited before making a decision. As discussed in the Related Work, this leads to overhead costs of memory and optimisation (selecting most representative data points). Previous work on VCL considered simple models without automatic architecture building or adaptation.
+
+# 3 OUR CLAW APPROACH
+
+In earlier CL approaches, the parts of the network architecture that are shared among the learnt tasks are designated a priori. To alleviate this rigidity and to effectively balance adaptation and stability, we propose a multi-task, continual model in which the adaptation of the architecture is data-driven by learning which neurons need to be adapted as well as the maximum adaptation capacity for each. All the model parameters (including those used for adaptation) are estimated via an efficient variational inference algorithm which incrementally learns from data of the successive tasks, without a need to store (nor generate) data from previous tasks and with no expansion in the network size.
+
+# 3.1 MODELLING
+
+With model parameters $\pmb \theta$ , the overall variational objective we aim at maximising at task with index t is equivalent to the following online marginal likelihood:
+
+$$
+\mathcal { L } ( \pmb { \theta } ) = - \mathrm { K L } \Big ( \mathbf { q } _ { t } ( \pmb { \theta } ) \| \mathbf { q } _ { t - 1 } ( \pmb { \theta } ) \Big ) + \sum _ { n = 1 } ^ { N _ { t } } \mathbb { E } _ { \mathbf { q } _ { t } ( \pmb { \theta } ) } \big [ \log p ( \pmb { y } ^ { ( n ) } | \pmb { x } ^ { ( n ) } , \pmb { \theta } ) \big ] .
+$$
+
+We propose a framework where the architecture, whose parameters are $\pmb \theta$ , is flexibly adapted based on the available tasks, via a learning procedure that will be described below. With each task, we automate the adaptation of the neuron contributions. Both the adaptation decisions (i.e. whether or not to adapt) and the maximum allowed degree of adaptation for every neuron are learnt. We refer to the binary adaptation variable as $_ \alpha$ . There is another variable s that is learnt in a multi-task fashion to control the maximum degree of adaptation, such that the expression s1+e−a − 1 limits how far the task-specific weights can differ from the global weights, in case the respective neuron is to be adapted. The parameter a depicts unconstrained adaptation, as described later.2
+
+We illustrate the proposed model to perform this adaptation by learning the probabilistic contributions of the different neurons within the network architecture on a task-by-task basis. We follow this with the inference details. Steps of the proposed modeling are listed as follows:
+
+• For a task $T$ , the classifier that we are modeling outputs: $\begin{array} { r } { \sum _ { n = 1 } ^ { N _ { T } } \left[ \log p ( \pmb { y } ^ { ( n ) } | \pmb { x } ^ { ( n ) } , \mathbf { w } ^ { T } ) \right] . } \end{array}$ • The task-specific weights $\mathbf { w } ^ { T }$ can be expressed in terms of their global counterparts as follows:
+
+$$
+\mathbf { w } ^ { T } = ( 1 + \mathbf { b } ^ { T } \pmb { \alpha } ^ { T } ) \circ \mathbf { w } .
+$$
+
+The symbol $\circ$ denotes an element-wise (Hadamard) multiplication.
+
+• For each task $T$ and each neuron $\mathbf { j }$ at layer i, $\alpha _ { \mathrm { i , j } } ^ { T }$ is a binary variable which indicates whether the corresponding weight is adapted $( \alpha _ { \mathbf { i } , \mathbf { j } } ^ { T } = 1 )$ ) or unadapted $( \alpha _ { \mathbf { i } , \mathbf { j } } ^ { T } = 0 )$ . Initially assume that the adaptation probability $\alpha _ { \mathrm { i , j } } ^ { T }$ follows a Bernoulli distribution with probability $\mathbf { p _ { i , j } } ^ { 3 }$ , $\alpha _ { \mathrm { i , j } } ^ { T } \sim$ Bernoulli . Since this Bernoulli is not straightforward to optimise, and to adopt a scalable inference procedure based on continuous latent variables, we replace this Bernoulli with a Gaussian that has an equivalent mean and variance from which we draw $\alpha _ { \mathrm { i , j } } ^ { T }$ . For the sake of attaining higher fidelity than what is granted by a standard Gaussian, we base our inference on a variational Gaussian estimation. Though in a context different from continual learning and with different estimators, the idea of replacing Bernoulli with an equivalent Gaussian has proven to be effective with dropout (Srivastava et al., 2014; Kingma et al., 2015).
+
+The approximation of the Bernoulli distribution by the corresponding Gaussian distribution is achieved by matching the mean and variance. The mean and variance of the Bernoulli distribution are $\mathbf { p _ { i , j } }$ , $\mathbf { p _ { i , j } } ( 1 - \mathbf { p _ { i , j } } )$ , respectively. A Gaussian distribution with the same mean and variance is used to fit αTi,j.
+
+$$
+\alpha _ { \mathbf { i } , \mathbf { j } } ^ { T } \sim \mathcal { N } ( \mathbf { p } _ { \mathbf { i } , \mathbf { j } } , \mathbf { p } _ { \mathbf { i } , \mathbf { j } } ( 1 - \mathbf { p } _ { \mathbf { i } , \mathbf { j } } ) ) .
+$$
+
+• The variable $\mathbf { b } ^ { T }$ controls the strength of the adaptation and it limits the range of adaptation via:
+
+$$
+1 + \mathbf { b } ^ { T } = \frac { \mathbf { s } } { 1 + \mathrm { e } ^ { - \mathbf { a } ^ { T } } } .
+$$
+
+So that the maximum adaptation is s. The variable $\mathbf { a } ^ { T }$ is an unconstrained adaptation value, similar to that in (Swietojanski & Renals, 2014). The addition of 1 is to facilitate the usage of a probability distribution while still keeping an adaptation range allowing for the attenuation or amplification of each neuron’s contribution.
+
+• Before facing the first dataset and learning task $\mathbf { t } = 1$ , the prior on the weights ${ \bf q } _ { 0 } ( { \bf w } ) = { \bf p } ( { \bf w } )$ is chosen to be a log-scale prior, which can be expressed as: $\mathbf { p } ( \log | \mathbf { w } | ) \propto \mathbf { c }$ , where $\mathbf { c }$ is a constant. The log-scale prior can alternatively be described as:
+
+$$
+\mathbf { p } ( | \mathbf { w } | ) \propto \frac { 1 } { | \mathbf { w } | } .
+$$
+
+At a high level, adapting neuron contributions can be seen as a generalisation of attention mechanisms in the context of continual learning. Applying this adaptation procedure to the input leads to an attention mechanism. However, our approach is more general since we do not apply it only to the very bottom (i.e. input) layer, but throughout the whole network. We next show how our variational inference mechanism enables us to learn the adaptation parameters.
+
+# 3.2 INFERENCE
+
+We describe the details related to the proposed variational inference mechanism. The adaptation parameters are included within the variational parameters.
+
+The (unadapted version of the) model parameters $\pmb \theta$ consist of the weight vectors w. To automate adaptation, we perform inference on $\mathbf { p _ { i , j } }$ , which would have otherwise been a hyperparameter of the prior (Louizos et al., 2017; Molchanov et al., 2017; Ghosh et al., 2018). Multiplying w by $\left( 1 + \mathbf { b } \alpha \right)$ where $_ { \pmb { \alpha } }$ is distributed according to (5), then from (4) with random noise variable $\mathbf { \epsilon } \sim \mathcal { N } ( 0 , 1 )$ :
+
+$$
+\begin{array} { r } { \mathbf { w _ { i , j } ^ { T } } = \gamma _ { \mathbf { i , j } } \left( 1 + \mathbf { b _ { i , j } } \mathbf { p _ { i , j } } + \mathbf { b _ { i , j } } \sqrt { \mathbf { p _ { i , j } } ( 1 - \mathbf { p _ { i , j } } ) } \epsilon \right) , } \\ { \mathbf { q ( w _ { i , j } } \mid \gamma _ { \mathbf { i , j } } ) \sim \mathcal { N } \biggl ( \gamma _ { \mathbf { i , j } } ( 1 + \mathbf { b _ { i , j } } \mathbf { p _ { i , j } } ) , \mathbf { b _ { i , j } ^ { 2 } } \gamma _ { \mathbf { i , j } } ^ { 2 } \mathbf { p _ { i , j } } ( 1 - \mathbf { p _ { i , j } } ) \biggr ) . } \end{array}
+$$
+
+From (7) and (8), the corresponding KL-divergence between the variational posterior of $\mathbf { w }$ , $\mathbf { q } ( \mathbf { w } | \gamma )$ and the prior $\mathbf { p } ( \mathbf { w } )$ is as follows. The subscripts are removed when $\mathbf { q }$ in turn is used as a subscript for improved readability. The variational parameters are $\gamma _ { \mathrm { i , j } }$ and $\mathbf { p _ { i , j } }$ .
+
+$$
+\begin{array} { r l } & { \mathrm { K L } \Big ( \mathbf { q } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } | \gamma _ { \mathrm { i } , \mathrm { j } } ) \ \lVert \ \mathbf { p } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } ) \Big ) = \mathbb { E } _ { \mathbf { q } ( \mathbf { w } | \gamma ) } \log [ \mathbf { q } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } | \gamma _ { \mathrm { i } , \mathrm { j } } ) / \mathbf { p } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } ) ] = } \\ & { \mathbb { E } _ { \mathbf { q } ( \mathbf { w } | \gamma ) } \log \mathbf { q } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } | \gamma _ { \mathrm { i } , \mathrm { j } } ) - \mathbb { E } _ { \mathbf { q } ( \mathbf { w } | \gamma ) } \log \mathbf { p } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } ) = - \mathbf { H } ( \mathbf { q } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } | \gamma _ { \mathrm { i } , \mathrm { j } } ) ) - \mathbb { E } _ { \mathbf { q } ( \mathbf { w } | \gamma ) } \log \mathbf { p } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } ) } \\ & { = - 0 . 5 \Big ( 1 + \log ( 2 \pi ) + \log ( \mathbf { b } _ { \mathrm { i } , \mathrm { j } } ^ { 2 } \mathbf { p } _ { \mathrm { i } , \mathrm { j } } ( 1 - \mathbf { p } _ { \mathrm { i } , \mathrm { j } } ) ) \Big ) - \mathbb { E } _ { \mathbf { q } ( \mathbf { w } | \gamma ) } \log \frac { 1 } { \lvert \epsilon \rvert } } \\ & { = - \log \mathbf { b } _ { \mathrm { i } , \mathrm { j } } - 0 . 5 \log \mathbf { p } _ { \mathrm { i } , \mathrm { j } } - 0 . 5 \log ( 1 - \mathbf { p } _ { \mathrm { i } , \mathrm { j } } ) + \mathbf { c } + \mathbb { E } _ { \mathbf { q } ( \mathbf { w } _ { \mathrm { i } , \mathrm { j } } ) } \log \vert \epsilon \vert , } \end{array}
+$$
+
+where the switch from (9) to (10) is due to the entropy computation (Bernardo & Smith, 2000) of the Gaussian $\bf q ( w _ { i , j } | \gamma _ { i , j } )$ defined in (8). The switch from (10) to (11) is due to using a log-scale prior, similar to Appendix C in (Kingma et al., 2015) and to Section 4.2 in (Molchanov et al., 2017). $\mathbb { E } _ { \mathbf { q } ( \mathbf { w } \mid \gamma ) } \log | \boldsymbol { \epsilon } |$ is computed via an accurate approximation similar to equation (14) in (Molchanov et al., 2017), with slightly different values of $k _ { 1 }$ , $k _ { 2 }$ and $k _ { 3 }$ . This is a very close approximation via numerically pre-computing $\mathbb { E } _ { \mathbf { q } ( \mathbf { w } \mid \gamma ) } \log | \boldsymbol { \epsilon } |$ using a third degree polynomial (Kingma et al., 2015; Molchanov et al., 2017).
+
+This is the form of the KL-divergence between the approximate posterior after the first task and the prior. Afterwards, it is straightforward to see how this KL-divergence applies for the subsequent tasks in a manner similar to (2), but while taking into account the new posterior form and original prior.
+
+The KL-divergence expression derived in (11) is to be minimised. By minimising (11) with respect to $\mathbf { p _ { i , j } }$ and then using samples from the respective distributions to assign values to $\alpha _ { \mathrm { { i , j } } }$ , adapted contributions of each neuron j at each layer i of the network are learnt per task. Values of $\mathbf { p _ { i , j } }$ are constrained between 0 and 1 during training via projected gradient descent.
+
+# Algorithm 1 Continual Learning with Adaptive Weights (CLAW)
+
+Input: A sequence of $T$ datasets, $\{ \pmb { x } _ { t } ^ { ( n ) } , \pmb { y } _ { t } ^ { ( n ) } \} _ { n = 1 } ^ { N _ { t } }$ , where $t = 1 , 2 , \dots , T$ and $N _ { t }$ is the size of the dataset associated with the $t$ -th task.
+
+Output: $\mathbf { q } _ { t } ( \pmb { \theta } )$ , where $\pmb \theta$ are the model parameters. Initialise all $\bf { p } ( | \mathbf { w _ { i , j } } | )$ with a log-scale prior, as in (7). for $t = 1 \dots T$ do Disclose the dataset $\{ \pmb { x } _ { t } ^ { ( n ) } , \pmb { y } _ { t } ^ { ( n ) } \} _ { n = 1 } ^ { N _ { t } }$ y(n)t }Ntn=1 for the current task t. for $\mathbf { i } = 1 \ldots \#$ layers do for $\mathbf { j } = 1 \ldots \#$ neurons at layer i do Compute $\mathbf { p _ { i , j } }$ using stochastic gradient descent on (11). Compute $\bf { s _ { i , j , t } }$ using (13). Update the corresponding general value $\mathbf { s _ { i , j } }$ using (14). end for end for end for
+
+# 3.2.1 LEARNING THE MAXIMUM ADAPTATION VALUES
+
+Using (6) to express the value of $\mathbf { b _ { i , j } }$ , and neglecting the constant term therein since it does not affect the optimisation, the KL-divergence in (11) is equivalent to:
+
+$$
+\begin{array} { r l } & { \mathrm { K L } \Big ( \mathbf { q } ( \mathbf { w _ { i , j } } | \gamma _ { \mathrm { i , j } } ) \ | \ \mathbf { p } ( \mathbf { w _ { i , j } } ) \Big ) \approx } \\ & { \mathrm { ~ - l o g ~ } \mathbf { s _ { i , j } } + \log ( 1 + \mathrm { e ^ { - \mathbf { a _ { i , j } } } } ) - 0 . 5 \log \mathbf { p _ { i , j } } - 0 . 5 \log ( 1 - \mathbf { p _ { i , j } } ) + \mathbf { c } + \mathbb { E } _ { \mathbf { q } ( \mathbf { w } / \gamma ) } \log | \epsilon | . } \end{array}
+$$
+
+Values of $\mathbf { a _ { i , j } }$ are learnt by minimising (12) with respect to $\mathbf { a _ { i , j } }$ . This subsection explains how to learn the maximum adaptation variable $\mathbf { s _ { i , j } }$ . Values of the maximum $\mathbf { s _ { i , j } }$ of the logistic function defined in (6) are learnt from multiple tasks. For each neuron $\mathbf { j }$ at layer i, there is a general value $\mathbf { s _ { i , j } }$ and another value that is specific for each task t, referred to as $\bf { s _ { i , j , t } }$ . This is similar to the meta-learning procedure proposed in (Finn et al., 2017). The following procedure to learn s is performed for each task t such that: (i) the optimisation performed to learn a task-specific value $\bf { s _ { i , j , t } }$ benefits from the warm initialisation with the general value $\mathbf { s _ { i , j } }$ rather than a random initial condition; and then (ii) the new information obtained from the current task $\mathbf { t }$ is reflected back to update the general value $\mathbf { s _ { i , j } }$ .
+
+• First divide the sample $N _ { \mathbf { t } }$ into two halves. For the first half, depart from the general value of $\mathbf { s _ { i , j } }$ as an initial condition, and use the assigned data examples from task $\mathbf { t }$ to learn the taskspecific values $\begin{array} { r } { \sum _ { n = 1 } ^ { N _ { t } } \mathbb { E } _ { \mathbf { q } _ { t } ( \theta ) } \big [ \log p ( \pmb { y } ^ { ( n ) } | \pmb { x } ^ { ( n ) } , \pmb { \theta } ) \big ] } \end{array}$ $\bf { s _ { i , j , t } }$ for the current task t. For neuron j at layer i, refer to the second term in (3), as $\mathbf { f } _ { t } ( \pmb { x } , \pmb { y } , \mathbf { s _ { i , j } } )$ . The set of parameters $\pmb \theta$ contains s as well as other parameters, but we focus here on s in the f notation since the following procedure is developed to optimise s. Also, refer to the loss of the (classification) function f as $\mathbf { E r r ( f ) } = \mathbf { C E } ( \mathbf { f } ( \pmb { x } , \theta ) | | \pmb { y } )$ , where CE stands for the cross-entropy:
+
+$$
+\mathbf { s _ { i , j , t } } = \mathbf { s _ { i , j } } - \frac { 2 \omega _ { 1 } } { N _ { t } } \nabla _ { \mathbf { s _ { i , j } } } \sum _ { d = 1 } ^ { N _ { t } / 2 } \mathbf { E r r } ( \mathbf { f } _ { t } ( x _ { d } , y _ { d } , \mathbf { s _ { i , j } } ) ) .
+$$
+
+• Now use the second half of the data from task $\mathbf { t }$ to update the general learnt value $\mathbf { s _ { i , j } }$ :
+
+$$
+\mathbf { s _ { i , j } } = \mathbf { s _ { i , j } } - \frac { 2 \omega _ { 2 } } { N _ { t } } \nabla _ { \mathbf { s _ { i , j } } } \sum _ { d = 1 + N _ { t } / 2 } ^ { N _ { t } } \mathbf { E r r } \big ( \mathbf { f } _ { t } ( x _ { d } , y _ { d } , \mathbf { s } _ { i , j , t } ) \big ) .
+$$
+
+Where $\omega _ { 1 }$ and $\omega _ { 2 }$ are step-size parameters.
+
+When testing on samples from task t after having faced future tasks $\mathbf { t } + 1 , \mathbf { t } + 2 , \ldots$ , the value of $\mathbf { s _ { i , j } }$ used is the learnt $\bf { s _ { i , j , t } }$ . There is only one value per neuron, so the overhead resulting from storing such values is negligible. The key steps of the algorithm are listed in Algorithm 1.
+
+At task t, the algorithmic complexity of a single joint update of the parameters $\pmb \theta$ based on the additive terms in (12) is $O ( M \bar { E } L D ^ { 2 } )$ , where $L$ is the number of layers in the network, $D$ is the
+
+(largest) number of neurons within a single layer, $E$ is the number of samples taken from the random noise variable $\epsilon$ , and $M$ is the minibatch size. Each $_ { \pmb { \alpha } }$ is obtained by taking one sample from the corresponding $\mathbf { p }$ , so that does not result in an overhead in terms of the complexity.
+
+# 4 EXPERIMENTS
+
+Our experiments mainly aim at evaluating the following: (i) the overall performance of the introduced CLAW, depicted by the average classification accuracy over all the tasks; (ii) the extent to which catastrophic forgetting can be mitigated when deploying CLAW; and (iii) the achieved degree of positive forward transfer. The experiments demonstrate the effectiveness of CLAW in achieving state-of-the-art continual learning results measured by classification accuracy and by the achieved reduction in catastrophic forgetting. We also perform ablations in Section D in the Appendix which exhibit the relevance of each of the proposed adaptation parameters.
+
+We perform six experiments on five datasets. The datasets in use are: MNIST (LeCun et al., 1998), notMNIST (Butalov, 2011), Fashion-MNIST (Xiao et al., 2017), Omniglot (Lake et al., 2011) and CIFAR-100 (Krizhevsky & Hinton, 2009). We compare the results obtained by CLAW to six different state-of-the-art continual learning algorithms: the VCL algorithm (Nguyen et al., 2018) (original form and one with a coreset), the elastic weight consolidation (EWC) algorithm (Kirkpatrick et al., 2017), the progress and compress (P&C) algorithm (Schwarz et al., 2018), the reinforced continual learning (RCL) algorithm (Xu & Zhu, 2018), the one referred to as functional regularisation for continual learning (FRCL) using Gaussian processes (Titsias et al., 2019) and the learn-to-grow (LTG) algorithm (Li et al., 2019b).
+
+# 4.1 OVERALL CLASSIFICATION ACCURACY
+
+Our main metric is the all-important classification accuracy. We consider six continual learning experiments, based on the MNIST, notMNIST, Fashion-MNIST, Omniglot and CIFAR-100 datasets. The introduced CLAW is compared to two VCL versions: VCL with no coreset and VCL with a 200-point coreset assembled by the K-center method (Nguyen et al., 2018), EWC, P&C, RCL, FRCL (its TR version) and LTG4. All the reported classification accuracy values reflect the average classification accuracy over all tasks the learner has trained on so far. More specifically, assume that the continual learner has just finished training on a task $t$ , then the reported classification accuracy at time $t$ is the average accuracy value obtained from testing on equally sized sets each belonging to one of the tasks 1, 2, . . . , t. For all the classification experiments, statistics reported are averages of ten repetitions. Statistical significance and standard error of the average classification accuracy obtained after completing the last two tasks of each experiment are displayed in Section E in the Appendix.
+
+As can be seen in Figure 1, CLAW achieves state-of-the-art classification accuracy in all the six experiments. The minibatch size is 128 for Split MNIST and 256 for all the other experiments. More detailed descriptions of the results of every experiment are given next:
+
+Permuted MNIST Using MNIST, Permuted MNIST is a standard continual learning benchmark (Goodfellow et al., 2014a; Kirkpatrick et al., 2017; Zenke et al., 2017). For each task $t$ , the corresponding dataset is formed by performing a fixed random permutation process on labeled MNIST images. This random permutation is unique per task, i.e. it differs for each task. For the hyperparameter $\lambda$ of EWC, which controls the overall contribution from previous data, we experimented with two values, $\lambda = 1$ and $\lambda = 1 0 0$ . We report the latter since it has always outperformed EWC with $\lambda = 1$ in this experiment. EWC with $\lambda = 1 0 0$ has also previously produced the best EWC classification results (Nguyen et al., 2018). In this experiment, fully connected single-head networks with two hidden layers are used. There are 100 hidden units in each layer, with ReLU activations. Adam (Kingma & Ba, 2015) is the optimiser used in the 6 experiments with $\eta = 0 . 0 0 1$ , $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 9 9$ . Further experimental details are given in Section C in the Appendix. Results of the accumulated classification accuracy, averaged over tasks, on a test set are displayed in Figure 1a. After 10 tasks, CLAW achieves significantly (check the Appendix) higher classification results than all the competitors.
+
+Split MNIST In this MNIST based experiment, five binary classification tasks are processed in the following sequence: 0/1, 2/3, 4/5, 6/7, and 8/9 (Zenke et al., 2017). The architecture used consists of fully connected multi-head networks with two hidden layers, each consisting of 256 hidden units with ReLU activations. As can be seen in Figure 1b, CLAW achieves the highest classification accuracy.
+
+Split notMNIST It contains 400,000 training images, and the classes are 10 characters, from A to J. Each image consists of one character, and there are different font styles. The five binary classification tasks are: A/F, B/G, C/H, D/I, and E/J. The networks used here contain four hidden layers, each containing 150 hidden units with ReLU activations. CLAW achieves a clear improvement in classification accuracy over competitors (Figure 1c).
+
+Split Fashion-MNIST Fashion-MNIST is a dataset whose size is the same as MNIST but it is based on different (and more challenging) 10 classes. The five binary classification tasks here are: T-shirt/Trouser, Pullover/Dress, Coat/Sandals, Shirt/Sneaker, and Bag/Ankle boots. The architecture used is the same as in Split notMNIST. In most of the continual learning tasks (including the more significant, later ones) CLAW achieves a clear classification improvement (Figure 1d).
+
+Omniglot This is a sequential learning task of handwritten characters of 50 alphabets (a total of over 1,600 characters with 20 examples each) belonging to the Omniglot dataset (Lake et al., 2011). We follow the same way via which this task has been used in continual learning before (Schwarz et al., 2018; Titsias et al., 2019); handwritten characters from each alphabet constitute a separate task. We thus have 50 tasks, which also allows to evaluate the scalability of the frameworks in comparison. The model used is a CNN. To deal with the convolutions in CLAW, we used the idea proposed and referred to as the local reparameterisation trick by Kingma et al. (2014; 2015), where a single global parameter is employed per neuron activation in the variational distribution, rather than employing parameters for every constituent weight element5. Further details about the CNN used are given in Section C. The automatically adaptable CLAW achieves better classification accuracy (Figure 1e).
+
+CIFAR-100 This dataset consists of 60,000 colour images of size $3 2 \times 3 2$ . It contains 100 classes, with 600 images per class. We use a split version CIFAR-100. Similar to Lopez-Paz & Ranzato (2017), we perform a 20-task experiment with a disjoint subset of five classes per task. CLAW achieves significantly higher classification accuracy (Figure 1f) -also higher than the previous state of the art on CIFAR-100 by Kemker & Kanan (2018). Details of the used CNN are in Section C.
+
+A conclusion that can be taken from Figure 1(a-f) is that CLAW consistently achieves state-of-the-art results (in all the 6 experiments). It can also be seen that CLAW scales well. For instance, the difference between CLAW and the best competitor is more significant with Split notMNIST than it is with the first two experiments, which are based on the smaller and less challenging MNIST. Also, CLAW achieves good results with Omniglot and CIFAR-100.
+
+# 4.2 CATASTROPHIC FORGETTING
+
+To assess catastrophic forgetting, we show how the accuracy on the initial task varies over the course of the training procedure on the remaining tasks (Schwarz et al., 2018). Since Omniglot (and CIFAR-100) contain a larger number of tasks: 50 (20) tasks, i.e. 49 (19) remaining tasks after the initial task, this setting is more relevant for Omniglot and CIFAR-100. We nonetheless display the results for Split MNIST, Split notMNIST, Split Fashion-MNIST, Omniglot and CIFAR-100. As can be seen in Figure 2, CLAW (at times jointly) achieves state-of-the-art performance retention degrees. Among the competitors, P&C and LTG also achieve high performance retention degrees.
+
+An empirical conclusion that can be made out of this and the previous experiment, is that CLAW achieves better overall continual learning results, partially thanks to the way it addresses catastrophic forgetting. The idea of adapting the architecture by adapting the contributions of neurons of each layer also seems to be working well with datasets like Omniglot and CIFAR-100, giving directions for imminent future work where CLAW can be extended for other application areas based on CNNs.
+
+# 4.3 POSITIVE FORWARD TRANSFER
+
+The purpose of this experiment is to assess the impact of learning previous tasks on the current task. In other words, we want to evaluate whether an algorithm avoids negative transfer, by evaluating the
+
+![](images/e6ecdbe7d4c10b405132eb4e476925e2418df12a452749b89c23a32ad002e39e.jpg)  
+Figure 1: Average test classification accuracy vs. the number of observed tasks in 6 experiments. CLAW achieves significantly higher classification results than the competing continual learning frameworks. Statistical significance values are presented in Section E in the Appendix. The value of $\lambda$ for EWC is 10,000 in (c), and 100 in the other experiments. Best viewed in colour.
+
+relative performance achieved on a unique task after learning a varying number of previous tasks (Schwarz et al., 2018). From Figure 3, we can see that CLAW achieves state-of-the-art results in 4 out of the 5 experiments (at par in the fifth) in terms of avoiding negative transfer.
+
+![](images/24bd99e22d08dd9ec2da8a8ae8c5091dc1b6f98f35803912905b538e304ea87b.jpg)  
+Figure 2: Evaluating catastrophic forgetting by measuring performance retention. Classification accuracy of the initial task is monitored along with the progression of tasks. Results are displayed for five datasets. CLAW is the least forgetful algorithm since performance levels achieved on the initial task do not degrade as much as in the other methods after facing new tasks. The legend and $\lambda$ values for EWC are the same as in Figure 1. Best viewed in colour.
+
+![](images/d854c8d0f2de31beb1e016c16efbae916c70049a0b38f49e1594338cf9599b38.jpg)  
+Figure 3: Evaluating Forward transfer, or to what extent a continual learning framework can avoid negative transfer. The impact of learning previous tasks on a specific task (the last task) is inspected and used as a proxy for evaluating forward transfer. This is performed by evaluating the relative performance achieved on a unique task after learning a varying number of previous tasks. This means that the value at $\mathbf { X } { \cdot } \mathbf { a } \mathbf { X } \mathbf { i } \mathbf { s } = 1$ refers to the learning accuracy of the last task after having learnt solely one task (only itself), the value at 2 refers to the learning accuracy of the last task after having learnt two tasks (an additional previous task), etc. Overall, CLAW achieves state-of-the-art results in 4 out of the 5 experiments (at par in the fifth) in terms of avoiding negative transfer. Best viewed in colour.
+
+# 5 RELATED WORK
+
+We briefly discuss three related approaches to continual learning: (a) regularisation-based, (b) architecture-based and (c) memory-based. We provide more details of related work in Section A in the Appendix. (a) A complementary approach to CLAW is the regularisation-based approach to balance adaptability with catastrophic forgetting: a level of stability is kept via protecting parameters that greatly influence the prediction against radical changes, while allowing the rest of the parameters to change without restriction (Li & Hoiem, 2016; Lee et al., 2017; Zenke et al., 2017; Chaudhry et al., 2018; Kim et al., 2018; Nguyen et al., 2018; Srivastava et al., 2013; Schwarz et al., 2018; Vuorio et al., 2018; Aljundi et al., 2019c). The elastic weight consolidation (EWC) algorithm by Kirkpatrick et al. (2017) is a seminal example, where a quadratic penalty is imposed on the difference between parameter values of the old and new tasks. One limitation is the high level of hand tuning required. (b) The architecture-based approach aims to deal with stability and adaptation issues by a fixed division of the architecture into global and local parts (Rusu et al., 2016b; Fernando et al., 2017; Shin et al., 2017; Kaplanis et al., 2018; Xu & Zhu, 2018; Yoon et al., 2018; Li et al., 2019b). (c) The memory-based approach relies on episodic memory to store data (or pseudo-data) from previous tasks (Ratcliff, 1990; Robins, 1993; 1995; Thrun, 1996; Schmidhuber, 2013; Hattori, 2014; Mocanu et al., 2016; Rebuffi et al., 2017; Kamra et al., 2017; Shin et al., 2017; Rolnick et al., 2018; van de Ven & Tolias, 2018; Wu et al., 2018; Titsias et al., 2019). Limitations include overheads for tasks such as data storage, replay, and optimisation to select (or generate) the points. CLAW can as well be seen as a combination of a regularisation-based approach (the variational inference mechanism) and a modelling approach which automates the architecture building process in a data-driven manner, avoiding the overhead resulting from either storing or generating data points from previous tasks. CLAW is also orthogonal to (and simple to combine with, if needed) memory-based methods.
+
+# 6 CONCLUSION
+
+We introduced a continual learning framework which learns how to adapt its architecture from the tasks and data at hand, based on variational inference. Rather than rigidly dividing the architecture into shared and task-specific parts, our approach adapts the contributions of each neuron. We achieve that without having to expand the architecture with new layers or new neurons. Results of six different experiments on five datasets demonstrate the strong empirical performance of the introduced framework, in terms of the average overall continual learning accuracy and forward transfer, and also in terms of effectively alleviating catastrophic forgetting.
+
+# ACKNOWLEDGMENTS
+
+HZ acknowledges support from the DARPA XAI project, contract#FA87501720152 and an Nvidia GPU grant. RT acknowledges support by Google, Amazon, Improbable and EPSRC grants EP/M0269571 and EP/L000776/1.
+
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+
+# APPENDIX
+
+We begin by briefly summarising the contents of the Appendix below:
+
+• Related works are described in Section A, followed by a brief discussion on the potential applicability of CLAW to another continual learning (CL) framework in Section A.1.   
+• In Section E, we provide the statistical significance and standard error of the average classification accuracy results obtained after completing the last two tasks from each experiment.   
+• Further experimental details are given in Section C.   
+• In Section D and Figures 4a- 4e, we display the results of performed ablations which manifest the relevance of each adaptation parameter.
+
+# A MORE RELATED WORK
+
+A complementary approach to CLAW, which could be combined with it, is the regularisation-based approach to balance adaptability with catastrophic forgetting: a level of stability is kept via protecting parameters that greatly influence the prediction against radical changes, while allowing the rest of the parameters to change without restriction (Li & Hoiem, 2016; Vuorio et al., 2018). In (Zenke et al., 2017), the regulariser is based on synapses where an importance measure is locally computed at each synapse during training, based on their respective contributions to the change in the global loss. During a task change, the less important synapses are given the freedom to change whereas catastrophic forgetting is avoided by preventing the important synapses from changing (Zenke et al., 2017). The elastic weight consolidation (EWC) algorithm, introduced by Kirkpatrick et al. (2017), is a seminal example of this approach where a quadratic penalty is imposed on the difference between parameter values of the old and new tasks. One limitation of EWC, which is rather alleviated by using minibatch or stochastic estimates, appears when the output space is not low-dimensional, since the diagonal of the Fisher information matrix over parameters of the old task must be computed, which requires a summation over all possible output labels (Kirkpatrick et al., 2017; Zenke et al., 2017; Schwarz et al., 2018). In addition, the regularisation term involves a sum over all previous tasks with a term from each and a hand-tuned hyperparameter that alters the weight given to it. The accumulation of this leads to a lot of hand-tuning. The work in (Chaudhry et al., 2018) is based on penalising confident fitting to the uncertain knowledge by a maximum entropy regulariser.
+
+Another seminal algorithm based on regularisation, which can be applied to any model, is variational continual learning (VCL) (Nguyen et al., 2018) which formulates CL as a sequential approximate (variational) inference problem. However, VCL has only been applied to simple architectures, not involving any automatic model building or adaptation. The framework in (Lee et al., 2017) incrementally matches the moments of the posterior of a Bayesian neural network that has been trained on the first and then the second task, and so on. Other algorithms pursue regularisation approaches based on sparsity (Srivastava et al., 2013; Kim et al., 2018). For example, the work in (Aljundi et al., 2019c) encourages sparsity on the neuron activations to alleviate catastrophic forgetting. The $l _ { 2 }$ distance between the top hidden activations of the old and new tasks is used for regularisation in (Jung et al., 2016). This approach has achieved good results, but is computationally expensive due to the necessity of computing at least a forward pass for every new data point through the network representing the old task (Zenke et al., 2017). Other regularisation-based continual learning algorithms include (Ebrahimi et al., 2019; Park et al., 2019).
+
+Another approach is the architecture-based one where the principal aim is to administer both the stability and adaptation issues via dividing the architecture into reusable parts that are less prone to changes, and other parts especially devoted to individual tasks (Rusu et al., 2016b; Fernando et al., 2017; Yoon et al., 2018; Du et al., 2019; He et al., 2019; Li et al., 2019a; Xu et al., 2019). To learn a new task in the work by Rusu et al. (2016a), the whole network from the previous task is first copied then augmented with a new part of the architecture. Although this is effective in eradicating catastrophic forgetting, there is a clear scalability issue since the architecture growth can be prohibitively high, especially with an increasing number of tasks. The work introduced in (Li et al., 2019b) bases its continual learning on neural architecture search, whereas the representation in (Javed & White, 2019) is optimised such that online updates minimize the error on all samples while limiting forgetting. The framework proposed by Xu & Zhu (2018) interestingly aims at solving this neural architecture structure learning problem, while balancing the tradeoff between adaptation and stability, via designed reinforcement learning (RL) strategies. When facing a new task, the optimal number of neurons and filters to add to each layer is cast as a combinatorial optimisation problem solved by an RL strategy whose reward signal is a function of validation accuracy and network complexity. Another RL based framework is the one presented by Kaplanis et al. (2018) where catastrophic forgetting is mitigated at multiple time scales via RL agents with a synaptic model inspired by neuroscience. Bottom layers (those near the input) are generally shared among the different tasks, while layers near the output are task-specific. Since the model structure is usually divided a priori and no automatic architecture learning nor adaptation takes place, alteration on the shared layers can still cause performance loss on earlier tasks due to forgetting (Shin et al., 2017). A clipped version of maxout networks (Goodfellow et al., 2013) is developed in (Lin et al., 2018) where parameters are partially shared among examples. The method in (Ostapenko et al., 2019) is based a dynamic network expansion accomplished by a generative adversarial network.
+
+The memory-based approach, which is the third influential approach to address the adaptationcatastrophic forgetting tradeoff, relies on episodic memory to store data (or pseudodata) from previous tasks (Ratcliff, 1990; Robins, 1993; 1995; Hattori, 2014; Rolnick et al., 2018; Teng & Dasgupta, 2019). A major limitation of the memory-based approach is that data from previous tasks may not be available in all real-world problems (Shin et al., 2017; Choi et al., 2019). Another limitation is the overhead resulting from the memory requirements, e.g. storage, replay, etc. In addition, the optimisation required to select the best observation to replay for future tasks is a source of further overhead (Titsias et al., 2019). In addition to the explicit replay form, some works have been based on generative replay (Thrun, 1996; Schmidhuber, 2013; Mocanu et al., 2016; Rebuffi et al., 2017; Kamra et al., 2017; Shin et al., 2017; van de Ven & Tolias, 2018; Wu et al., 2018). Notably, Shin et al. (2017) train a deep generative model based on generative adversarial networks (GANs, Goodfellow et al., 2014b; Goodfellow, 2016) to mimic past data. This mitigates the aforementioned problem, albeit at the added cost of the training of the generative model (Schwarz et al., 2018) and sharing its parameters. Alleviating catastrophic forgetting via replay mechanisms has also been adopted in reinforcement learning, e.g. (Isele & Cosgun, 2018; Rolnick et al., 2018). A similar approach was introduced by Lopez-Paz & Ranzato (2017) where gradients of the previous task (rather than data examples) are stored so that a trust region consisting of gradients of all previous tasks can be formed to reduce forgetting. Other algorithms based on replay mechanisms include (Aljundi et al., 2019a;b).
+
+Equivalent tradeoffs to the one between adaptation and stability can be found in the literature since the work in (Carpenter & Grossberg, 1987), in which a balance was needed to resolve the stabilityplasticity dilemma, where the latter refers to the ability to rapidly adapt to new tasks. The works introduced in (Chaudhry et al., 2018; Kim et al., 2019) shed light on the tradeoff between adaptation and stability, where they explore measures of intransigence and forgetting. The former refers to the inability to adapt to new tasks and data, whereas an increase in the latter clearly signifies an instability problem. Other recent works tackling the same tradeoff include (Riemer et al., 2019) where the transfer-interference (interference is catastrophic forgetting) tradeoff is optimised for the sake of maximising transfer and minimising interference by an algorithm based on experience replay and meta-learning. Other recent algorithms include the ORACLE algorithm by Yoon et al. (2019), which addresses the sensitivity of a continual learner to the order of tasks it encounters by establishing an order robust learner that represents the parameters of each task as a sum of task-shared and task-specific parameters. The algorithm in (Titsias et al., 2019) achieves functional regularisation by performing approximate inference over the function (instead of parameter) space. They use a Gaussian process obtained by assuming the weights of the last neural network layer to be Gaussian distributed. Our model is also related to the multi-task learning approach (Caruana, 1997; Heskes, 2000; Bakker & Heskes, 2003; Adel et al., 2017; Zhao et al., 2019a; Stickland & Murray, 2019; Zhao et al., 2019b).
+
+# A.1 APPLICABILITY OF CLAW TO OTHER CL FRAMEWORKS
+
+As mentioned in the main document, ideas of the proposed CLAW can be applied to continual learning frameworks other than VCL. The latter is more relevant for the inference part of CLAW since both are based on variational inference. As per the modeling ideas, e.g. the binary adaptation parameter depicting whether or not to adapt, and the maximum allowed adaptation, these can be integrated within other continual learning frameworks. For example, the algorithm in Xu & Zhu (2018) utilises reinforcement learning to adaptively expand the network. The optimal number of nodes and filters to be added is cast as a combinatorial optimisation problem. In CLAW, we do not expand the network. As such, an extension of the work in (Xu & Zhu, 2018) can be inspired by CLAW where not only the number of nodes and filters to be added is decided for each task, but also a soft and more general version where an adaptation based on the same network size is performed such that the network expansion needed in (Xu & Zhu, 2018) can be further moderated.
+
+# B STATISTICAL SIGNIFICANCE AND STANDARD ERROR
+
+In this section, we provide information about the statistical significance and standard error of CLAW and the competing continual learning frameworks. In Table 1, we list the average accuracy values (Figure 1 in the main document) obtained after completing the last two tasks from each of the six experiments. A bold entry in Table 1 denotes that the classification accuracy of an algorithm is significantly higher than its competitors. Significance results are identified using a paired t-test with $\mathrm { p = 0 . 0 5 }$ . Each average accuracy value is followed by the corresponding standard error. Average classification accuracy resulting from CLAW is significantly higher than its competitors on the 6 experiments.
+
+Table 1: Average test classification accuracy of the last two tasks in each of the six experiments: Permuted MNIST, Split MNIST, Split notMNIST, Split Fashion-MNIST, Omniglot and CIFAR-100, followed by the corresponding standard error. A bold entry denotes that the classification accuracy of an algorithm is significantly higher than its competitors. Significance results are identified using a paired t-test with $\mathrm { p = 0 . 0 5 }$ . Average classification accuracy resulting from CLAW is significantly higher than its competitors on the 6 experiments.   
+
+<table><tr><td>Classification Accuracy</td><td>CLAW</td><td>VCL</td><td>VCL +Coreset</td><td>EWC</td></tr><tr><td>Permuted MNIST (task 9) Permuted MNIST (task 10) Split MNIST (task 4) Split MNIST (task 5) Split notMNIST (task 4) Split notMNIST (task 5) Split Fashion-MNIST (task 4) Split Fashion-MNIST (task 5) Omniglot (task 49)</td><td>99.2 ± 0.2 % 99.2 ± 0.1 % 99.2 ± 0.2 % 99.1 ± 0.2 % 98.7 ± 0.3 % 98.4 ± 0.2 % 93.2 ± 0.2 % 92.5 ± 0.2 % 84.5 ± 0.2 % 84.6 ± 0.3 %</td><td>93.5 ± 0.3 % 92.1 ± 0.3 % 98.6 ± 0.3 % 97.0 ± 0.4 % 95.8 ± 0.4 % 92.1 ± 0.3 % 90.0 ± 0.3 % 88.0 ± 0.2 % 81.1 ± 0.3 % 80.7 ± 0.3 %</td><td>95.5 ± 0.3 % 95 ± 0.5 % 98.7 ± 0.2 % 98.4 ± 0.3 % 96.9 ± 0.5 % 96.0 ± 0.3 % 90.7 ± 0.2 % 88.5 ± 0.4 % 81.8 ± 0.3 % 81.1 ± 0.4 %</td><td>92.1 ± 0.4 % 90.2 ± 0.4 % 94.9 ± 0.4 % 94.2 ± 0.5 % 92.9 ± 0.4 % 92.3 ± 0.4 % 89.4 ± 0.4 % 87.6 ± 0.3 % 78.2 ± 0.3 %</td></tr><tr><td>CIFAR-100 (task 19) CIFAR-100 (task 20) Classification Accuracy</td><td>95.6 ± 0.3 % P&amp;C</td><td>78.7 ± 0.4 % 77.2 ± 0.4 % RCL 96.4 ± 0.5 %</td><td>80.8 ± 0.3 % 79.9 ± 0.4 % FRCL 98.4 ± 0.4 %</td><td>63.1 ± 0.5 % 62.4 ± 0.4 % LTG</td></tr><tr><td>Permuted MNIST (task 9) Permuted MNIST (task 10) Split MNIST (task 4) Split MNIST (task 5) Split notMNIST (task 4) Split notMNIST (task 5) Split Fashion-MNIST (task 4) Split Fashion-MNIST (task 5) Omniglot (task 49) Omniglot (task 50)</td><td>94.4 ± 0.3 % 94.1± 0.6 % 97.3 ± 0.5 % 96.4 ± 0.4 % 97.8 ± 0.4 % 96.9 ± 0.5 % 91.4 ± 0.3 % 90.8 ± 0.2 % 82.8 ± 0.2 % 82.7 ± 0.3 %</td><td>96.3 ± 0.3 % 97.8 ± 0.7 % 97.5 ± 0.6 % 97.7 ± 0.2 % 97.3 ± 0.5 % 91.1 ± 0.3 % 89.7 ± 0.4 % 80.1 ± 0.4 % 80.2 ± 0.4 %</td><td>98.4± 0.5 % 98.2 ± 0.3 % 98.1 ± 0.2 % 96.1 ± 0.6 % 95.2 ± 0.7 % 90.4 ± 0.2 % 87.7 ± 0.4 % 79.9 ± 0.3 %</td><td>98.7 ± 0.3 % 98.7 ± 0.3 % 98.7 ± 0.2 % 98.3 ± 0.3 % 97.8 ± 0.3 % 97.4 ± 0.3 % 92.5 ± 0.4 % 91.1 ± 0.3 %</td></tr></table>
+
+# C OTHER EXPERIMENTAL DETAILS
+
+Here are some additional details about the datasets in use:
+
+The MNIST dataset is used in both the Permuted MNIST and Split MNIST experiments. The MNIST (Mixed National Institute of Standards and Technology) dataset (LeCun et al., 1998) is a handwritten digit dataset. Each MNIST image consists of $2 8 \times 2 8$ pixels, which is also the pixel size of the notMNIST and Fashion-MNIST datasets. The MNIST dataset contains a training set of 60,000 instances and a test set of 10,000 instances.
+
+As mentioned in the main document, each experiment is repeated ten times. Data is randomly split into three partitions, training, validation and test. A portion of $6 0 \%$ of the data is reserved for training, $2 0 \%$ for validation and $2 0 \%$ for testing. Statistics reported are the averages of these ten repetitions.
+
+Number of epochs required per task to reach a saturation level for CLAW (and the bulk of the methods in comparison) was 10 epochs for all experiments except for Omniglot and CIFAR-100 (15 epochs). Used values of $\omega _ { 1 }$ and $\omega _ { 2 }$ are 0.05 and 0.02, respectively.
+
+For Omniglot, we used a network similar to the one used in (Schwarz et al., 2018), which consists of 4 blocks of $3 \times 3$ convolutions with 64 filters, followed by a ReLU and a $2 \times 2$ max-pooling. The same CNN is used for CIFAR-100. CLAW achieves clearly higher classification accuracy on both Omniglot and CIFAR-100 (Figures 1e and 1f).
+
+# D ABLATIONS
+
+The plots displayed in this section empirically demonstrate how important the main adaptation parameters are in achieving the classification performance levels reached by CLAW. In each of the Figures 4a- 4f, the classification performance of CLAW is compared to the following three cases: 1) when the parameter controlling the maximum degree of adaptation is not learnt in a multi-task fashion, i.e. when the respective general value $\mathbf { s _ { i , j } }$ is used instead of $\mathbf { s _ { i , j , t } } . 2 )$ ) when adaptation always happens, i.e. the binary variable denoting the adaptation decision is always activated. 3) when adaptation never takes place. The differences in classification accuracy between CLAW and each of the other three plots in Figures 4a- 4f empirically demonstrate the relevance of each adaptation parameter.
+
+# E RUN-TIME
+
+In Table 2, we report the wall-clock run time (in seconds) after finishing training in each of the six experiments: Permuted MNIST, Split MNIST, Split notMNIST, Split Fashion-MNIST, Omniglot and CIFAR-100. VCL and CLAW converge (i.e. reach the accuracy levels reported earlier) more quickly than the other methods. CLAW was the fastest in 3 out of the 6 experiments, whereas VCL was the fastest in the other 3, but their training run-time values have always been close to each other. As reported in the earlier sections, there is a significant difference in classification accuracy in favour of CLAW. This has been achieved within reasonably acceptable run-time levels thanks to the proposed design where the whole data-driven adaptation procedure is kept as part of an amortised variational inference algorithm. RCL is the slowest since it is based on reinforcement learning, where a large number of trials are typically required. This has also been acknowledged and reported therein (Xu & Zhu, 2018).
+
+Table 2: Wall-clock run time (in seconds) after finishing training in each of the six experiments: Permuted MNIST, Split MNIST, Split notMNIST, Split Fashion-MNIST, Omniglot and CIFAR-100. As mentioned earlier, the statistics reported are averages of ten repetitions.   
+
+<table><tr><td>Training Time (in seconds)</td><td>CLAW</td><td>VCL</td><td> VCL + Coreset EWC P&amp;C</td><td></td><td></td><td>RCL</td><td>FRCL</td><td>LTG</td></tr><tr><td>Permuted MNIST (after 10 tasks)</td><td>667</td><td>682</td><td>724</td><td>1117</td><td>1355</td><td>32575</td><td>919</td><td>705</td></tr><tr><td>Split MNIST (after 5 tasks)</td><td>649</td><td>637</td><td>708</td><td>1054</td><td>1312</td><td>31110</td><td>891</td><td>648</td></tr><tr><td>Split notMNIST (after 5 tasks)</td><td>722</td><td>714</td><td>792</td><td>1210</td><td>1407</td><td>34123</td><td>898</td><td>781</td></tr><tr><td>Split Fashion-MNIST(after 5 tasks)</td><td>829</td><td>818</td><td>901</td><td>1284</td><td></td><td>1498 35086</td><td>972</td><td>915</td></tr><tr><td>Omniglot (after 50 tasks)</td><td>1126</td><td>1241</td><td>1513</td><td>1637</td><td></td><td>1714 37247</td><td>1620</td><td>1312</td></tr><tr><td>CIFAR-100 (after 20 tasks)</td><td>792</td><td>810</td><td>914</td><td>802</td><td>1322</td><td>6102</td><td>1016</td><td>896</td></tr></table>
+
+![](images/c0ad7f594cb131bd63f8a58d89ff5d4798ba0d138c303661ecd49fbe7734dc4e.jpg)  
+Figure 4: Ablation studies on different datasets.

parse/train/YTWGvpFOQD-/YTWGvpFOQD-.md

@@ -0,0 +1,591 @@
+# DIFFERENTIALLY PRIVATE LEARNING NEEDS BETTER FEATURES (OR MUCH MORE DATA)
+
+Florian Tramèr   
+Stanford University   
+tramer@cs.stanford.edu
+
+Dan Boneh Stanford University dabo@cs.stanford.edu
+
+# ABSTRACT
+
+We demonstrate that differentially private machine learning has not yet reached its “AlexNet moment” on many canonical vision tasks: linear models trained on handcrafted features significantly outperform end-to-end deep neural networks for moderate privacy budgets. To exceed the performance of handcrafted features, we show that private learning requires either much more private data, or access to features learned on public data from a similar domain. Our work introduces simple yet strong baselines for differentially private learning that can inform the evaluation of future progress in this area.
+
+# 1 INTRODUCTION
+
+Machine learning (ML) models have been successfully applied to the analysis of sensitive user data such as medical images (Lundervold & Lundervold, 2019), text messages (Chen et al., 2019) or social media posts (Wu et al., 2016). Training these ML models under the framework of differential privacy (DP) (Dwork et al., 2006b; Chaudhuri et al., 2011; Shokri & Shmatikov, 2015; Abadi et al., 2016) can protect deployed classifiers against unintentional leakage of private training data (Shokri et al., 2017; Song et al., 2017; Carlini et al., 2019; 2020).
+
+Yet, training deep neural networks with strong DP guarantees comes at a significant cost in utility (Abadi et al., 2016; Yu et al., 2020; Bagdasaryan et al., 2019; Feldman, 2020). In fact, on many ML benchmarks the reported accuracy of private deep learning still falls short of “shallow” (non-private) techniques. For example, on CIFAR-10, Papernot et al. (2020b) train a neural network to $6 6 . 2 \%$ accuracy for a large DP budget of $\varepsilon = 7 . 5 3$ , the highest accuracy we are aware of for this privacy budget. Yet, without privacy, higher accuracy is achievable with linear models and non-learned “handcrafted” features, e.g., (Coates & $\mathrm { N g }$ , 2012; Oyallon & Mallat, 2015). This leads to the central question of our work:
+
+Can differentially private learning benefit from handcrafted features?
+
+We answer this question affirmatively by introducing simple and strong handcrafted baselines for differentially private learning, that significantly improve the privacy-utility guarantees on canonical vision benchmarks.
+
+Our contributions. We leverage the Scattering Network (ScatterNet) of Oyallon & Mallat (2015)— a non-learned SIFT-like feature extractor (Lowe, 1999)—to train linear models that improve upon the privacy-utility guarantees of deep learning on MNIST, Fashion-MNIST and CIFAR-10 (see Table 1). For example, on CIFAR-10 we exceed the accuracy reported by Papernot et al. (2020b) while simultaneously improving the provable DP-guarantee by $1 3 0 \times$ . On MNIST, we match the privacy-utility guarantees obtained with PATE (Papernot et al., 2018) without requiring access to any public data. We find that privately training deeper neural networks on handcrafted features also significantly improves over end-to-end deep learning, and even slightly exceeds the simpler linear models on CIFAR-10. Our results show that private deep learning remains outperformed by handcrafted priors on many tasks, and thus has yet to reach its “AlexNet moment” (Krizhevsky et al., 2012).
+
+We find that models with handcrafted features outperform end-to-end deep models, despite having more trainable parameters. This is counter-intuitive, as the guarantees of private learning degrade with dimensionality in the worst case (Bassily et al., 2014).1 We explain the benefits of handcrafted features by analyzing the convergence rate of non-private gradient descent. First, we observe that with low enough learning rates, training converges similarly with or without privacy (both for models with and without handcrafted features). Second, we show that handcrafted features significantly boost the convergence rate of non-private learning at low learning rates. As a result, when training with privacy, handcrafted features lead to more accurate models for a fixed privacy budget.
+
+Table 1: Test accuracy of models with handcrafted ScatterNet features compared to prior results with end-to-end CNNs for various DP budgets $( \varepsilon , \delta = 1 0 ^ { - 5 } )$ ). Lower $\varepsilon$ values provide stronger privacy. The end-to-end CNNs with maximal accuracy for each privacy budget are underlined. We select the best ScatterNet model for each DP budget $\varepsilon \le 3$ with a hyper-parameter search, and show the mean and standard deviation in accuracy for five runs.   
+Test Accuracy $( \% )$   
+
+<table><tr><td>Data</td><td>ε-DP</td><td>Source</td><td>CNN</td><td>ScatterNet+linear</td><td>ScatterNet+CNN</td></tr><tr><td rowspan="7">MNIST</td><td>1.2</td><td>Feldman &amp; Zrnic (2020)</td><td>96.6</td><td>98.1±0.1</td><td>97.8 ±0.1</td></tr><tr><td>2.0</td><td>Abadi et al. (2016)</td><td>95.0</td><td>98.5 ± 0.0</td><td>98.4±0.1</td></tr><tr><td>2.32</td><td>Bu et al. (2019)</td><td>96.6</td><td>98.6±0.0</td><td>98.5 ±0.0</td></tr><tr><td>2.5</td><td>Chen&amp;Lee (2020)</td><td>90.0</td><td>98.7 ± 0.0</td><td>98.6 ±0.0</td></tr><tr><td>2.93</td><td>Papernot et al. (2020a)</td><td>98.1</td><td>98.7 ± 0.0</td><td>98.7 ± 0.1</td></tr><tr><td>3.2</td><td>Nasr et al. (2020)</td><td>96.1</td><td>一</td><td></td></tr><tr><td>6.78</td><td>Yu et al. (2019b)</td><td>93.2</td><td>1</td><td></td></tr><tr><td rowspan="2">Fashion-MNIST</td><td>2.7</td><td>Papernot et al. (2020a)</td><td>86.1</td><td>89.5 ±0.0</td><td>88.7 ±0.1</td></tr><tr><td>3.0</td><td>Chen &amp; Lee (2020)</td><td>82.3</td><td>89.7 ± 0.0</td><td>89.0±0.1</td></tr><tr><td rowspan="4">CIFAR-10</td><td>3.0</td><td>Nasr et al. (2020)</td><td>55.0</td><td>67.0 ±0.1</td><td>69.3± 0.2</td></tr><tr><td>6.78</td><td>Yu et al. (2019b)</td><td>44.3</td><td></td><td></td></tr><tr><td>7.53</td><td>Papernot et al. (2020a)</td><td>66.2</td><td></td><td></td></tr><tr><td>8.0</td><td>Chen &amp; Lee (2020)</td><td>53.0</td><td></td><td></td></tr></table>
+
+Considering these results, we ask: what is the cost of private learning’s “AlexNet moment”? That is, which additional resources do we need in order to outperform our private handcrafted baselines? Following McMahan et al. (2018), we first consider the data complexity of private end-to-end learning. On CIFAR-10, we use an additional 500,000 labeled Tiny Images from Carmon et al. (2019) to show that about an order of magnitude more private training data is needed for end-to-end deep models to outperform our handcrafted features baselines. The high sample-complexity of private deep learning could be detrimental for tasks that cannot leverage “internet-scale” data collection (e.g., most medical applications).
+
+We further consider private learning with access to public data from a similar domain. In this setting, handcrafted features can be replaced by features learned from public data via transfer learning (Razavian et al., 2014). While differentially private transfer learning has been studied in prior work (Abadi et al., 2016; Papernot et al., 2020a), we find that its privacy-utility guarantees have been underestimated. We revisit these results and show that with transfer learning, strong privacy comes at only a minor cost in accuracy. For example, given public unlabeled ImageNet data, we train a CIFAR-10 model to $9 2 . 7 \%$ accuracy for a DP budget of $\varepsilon = 2$ .
+
+Our work demonstrates that higher quality features—whether handcrafted or transferred from public data—are of paramount importance for improving the performance of private classifiers in low (private) data regimes.
+
+Code to reproduce our experiments is available at https://github.com/ftramer/ Handcrafted-DP.
+
+# 2 STRONG SHALLOW BASELINES FOR DIFFERENTIALLY PRIVATE LEARNING
+
+We consider the standard central model of differential privacy (DP): a trusted party trains an ML model $f$ on a private dataset $D \in \mathcal { D }$ , and publicly releases the model. The learning algorithm $A$
+
+satisfies $( \varepsilon , \delta )$ -differential privacy (Dwork et al., 2006a), if for any datasets $D , D ^ { \prime }$ that differ in one record, and any set of models $S$ :
+
+$$
+\operatorname* { P r } [ A ( D ) \in S ] \leq e ^ { \varepsilon } \operatorname* { P r } [ A ( D ^ { \prime } ) \in S ] + \delta .
+$$
+
+DP bounds an adversary’s ability to infer information about any individual training point from the model. Cryptography can split the trust in a central party across users (Jayaraman et al., 2018; Bonawitz et al., 2017).
+
+Prior work has trained private deep neural networks “end-to-end” (e.g., from image pixels), with large losses in utility (Shokri & Shmatikov, 2015; Abadi et al., 2016; Papernot et al., 2020b). In contrast, we study the benefits of handcrafted features that encode priors on the learning task’s public domain (e.g., edge detectors for images). Although end-to-end neural networks outperform such features in the non-private setting, our thesis is that handcrafted features result in an easier learning task that is more amenable to privacy. We focus on computer vision, a canonical domain for private deep learning (Abadi et al., 2016; Yu et al., 2019b; Papernot et al., 2020b; Nasr et al., 2020)), with a rich literature on handcrafted features (Lowe, 1999; Dalal & Triggs, 2005; Bruna & Mallat, 2013). Our approach can be extended to handcrafted features in other domains, e.g., text or speech.
+
+# 2.1 SCATTERING NETWORKS
+
+We use the Scattering Network (ScatterNet) of Oyallon & Mallat (2015), a feature extractor that encodes natural image priors (e.g., invariance to small rotations and translations) using a cascade of wavelet transforms (Bruna & Mallat, 2013). As this cascade of transforms is data independent, we can obtain a differentially private classifier by privately fine-tuning a (linear) model on top of locally extracted features. In Appendix A, we discuss other candidate “non-deep” approaches that we believe to be less suitable for differentially private learning.
+
+We use the default parameters in (Oyallon & Mallat, 2015), a ScatterNet $S ( { \pmb x } )$ of depth two with wavelets rotated along eight angles. For images of size $H \times W$ , this network extracts features of dimension $( K , H / 4 , \bar { W _ { } } / 4 )$ , with $K = 8 1$ for grayscale images, and $K = 2 4 3$ for RGB images. Note that the transform is thus expansive. More details on ScatterNets are in Appendix C.1.
+
+# 2.2 DIFFERENTIALLY PRIVATE SCATTERNET CLASSIFIERS
+
+To train private classifiers, we use the DP-SGD algorithm2 of Abadi et al. (2016) (see Appendix B). DP-SGD works as follows: (1) batches of expected size $B$ are sampled at random;3 (2) gradients are clipped to norm $C$ ; (3) Gaussian noise of variance $\sigma ^ { 2 } C ^ { 2 } / B ^ { 2 }$ is added to the mean gradient. DP-SGD guarantees privacy for gradients, and is thus oblivious to preprocessing applied independently to each data sample, such as the ScatterNet transform.
+
+When training a supervised classifier on top of ScatterNet features with gradient descent, we find that normalizing the features is crucial to obtain strong performance. We consider two approaches:
+
+• Group Normalization (Wu & He, 2018): the channels of $S ( { \pmb x } )$ are split into $G$ groups, and each is normalized to zero mean and unit variance. Data points are normalized independently so this step incurs no privacy cost.   
+• Data Normalization: the channels of $S ( { \pmb x } )$ are normalized by their mean and variance across the training data. This step incurs a privacy cost as the per-channel means and variances need to be privately estimated.
+
+Table 2 shows that normalization significantly accelerates convergence of non-private linear models trained on ScatterNet features, for MNIST, Fashion-MNIST and CIFAR-10. For CIFAR-10, Data
+
+Table 2: Effect of feature normalization on the test accuracy of non-private ScatterNet models after 20 epochs. We also report the maximal test accuracy upon convergence (mean and standard deviation over five runs).   
+Normalization (Test accuracy after 20 epochs)   
+
+<table><tr><td>Dataset</td><td>None</td><td>Group Normalization</td><td>Data Normalization</td><td>Maximal Accuracy</td></tr><tr><td>MNIST</td><td>95.9±0.0</td><td>99.1±0.0</td><td>99.1±0.0</td><td>99.3± 0.0</td></tr><tr><td>Fashion-MNIST</td><td>82.6 ±0.1</td><td>90.9 ± 0.1</td><td>91.0±0.2</td><td>91.5 ± 0.0</td></tr><tr><td>CIFAR-10</td><td>58.0±0.1</td><td>67.8± 0.2</td><td>70.7 ± 0.1</td><td>71.1 ± 0.0</td></tr></table>
+
+Normalization performs significantly better than Group Normalization, so the small privacy cost of estimating channel statistics is warranted. While the maximal test accuracy of these models falls short of state-of-the-art CNNs, it exceeds all previously reported results for differentially private neural networks (even for large privacy budgets).
+
+# 3 EVALUATING PRIVATE SCATTERNET CLASSIFIERS
+
+We compare differentially private ScatterNet classifiers and deep learning models on MNIST (LeCun et al., 2010), Fashion-MNIST (Xiao et al., 2017) and CIFAR-10 (Krizhevsky, 2009). Many prior works have reported improvements over the DP-SGD procedure of Abadi et al. (2016) for these datasets. As we will show, ScatterNet classifiers outperform all prior approaches while making no algorithmic changes to DP-SGD. ScatterNet classifiers can thus serve as a strong canonical baseline for evaluating proposed improvements over DP-SGD in the future.
+
+# 3.1 EXPERIMENTAL SETUP
+
+Most prior works find the best model for a given DP budget using a hyper-parameter search. As the private training data is re-used many times, this overestimates the privacy guarantees. Private hyper-parameter search is possible at a small cost in the DP budget (Liu & Talwar, 2019), but we argue that fully accounting for this privacy leakage is hard as even our choices of architectures, optimizers, hyper-parameter ranges, etc. are informed by prior analysis of the same data. As in prior work, we thus do not account for this privacy leakage, and instead compare ScatterNet models and end-to-end CNNs with similar hyper-parameter searches. Moreover, we find that ScatterNet models are very robust to hyper-parameter changes and achieve near-optimal utility with random hyper-parameters (see Table 3). To evaluate ScatterNet models, we apply the following hyper-parameter search:
+
+• We begin by fixing a privacy schedule. We target a moderate differential privacy budget of $( \varepsilon = \bar { 3 , } \delta = 1 0 ^ { - 5 } )$ and compute the noise scale $\sigma$ of DP-SGD so that the privacy budget is consumed after $T$ epochs. We try different values of $T$ , with larger values resulting in training for more steps but with higher noise.   
+• We fix the gradient clipping threshold for DP-SGD to $C = 0 . 1$ for all our experiments. Thakkar et al. (2019) suggest to vary this threshold adaptively, but we did not observe better performance by doing so.   
+• We try various batch sizes $B$ and base learning rates $\eta$ , with linear learning rate scaling (Goyal   
+et al., 2017).4   
+• We try both Group Normalization (Wu & He, 2018) with different choices for the number of groups, and private Data Normalization with different choices of privacy budgets (see Appendix B for details).
+
+We perform a grid-search over all parameters as detailed in Appendix C.5. We compare our ScatterNet classifiers to the CNN models of Papernot et al. (2020b) (see Appendix C.2), which achieve the highest reported accuracy for our targeted privacy budget for all three datasets. We also perform a grid-search for these models, which reproduces the results of Papernot et al. (2020b). We use the ScatterNet implementation from Kymatio (Andreux et al., 2020), and the DP-SGD implementation in opacus (pytorch/opacus, 2020) (formerly called pytorch-dp).
+
+![](images/76722765eda82a7f80348b436da41ca7d2d38fe948cfbb64805ee66df04d605c.jpg)  
+Figure 1: Highest test accuracy achieved for each DP budget $( \varepsilon , \delta = 1 0 ^ { - 5 } )$ ) for ScatterNet classifiers and the end-to-end CNNs of Papernot et al. (2020b). We plot the mean and standard deviation across five runs.
+
+We use a NVIDIA Titan $\mathrm { X p }$ GPU with 12GB of RAM for all our experiments. To run DP-SGD with large batch sizes $B$ , we use the “virtual batch” approach of opacus: the average of clipped gradients is accumulated over multiple “mini-batches”; once $B$ gradients have been averaged, we add noise and take a gradient update step. Code to reproduce our experiments is available at https://github.com/ftramer/Handcrafted-DP.
+
+# 3.2 RESULTS
+
+To measure a classifier’s accuracy for a range of privacy budgets, we compute the test accuracy as well as the DP budget $\varepsilon$ after each training epoch (with the last epoch corresponding to $\varepsilon = 3$ ). For various DP budgets $\mathit { \check { \Psi } } _ { \varepsilon , \delta } = 1 0 ^ { - 5 }$ ) used in prior work, Table 1 shows the maximal test accuracy achieved by a linear ScatterNet model in our hyper-parameter search, averaged over five runs. We also report results with CNNs trained on ScatterNet models, which are described in more detail below. Figure 1 further compares the full privacy-accuracy curves of our ScatterNets and of the CNNs of Papernot et al. (2020b). Linear models with handcrafted features significantly outperform prior results with end-to-end CNNs, for all privacy budgets $\varepsilon \le 3$ we consider. Even when prior work reports results for larger budgets, they do not exceed the accuracy of our baseline.
+
+In particular, for CIFAR-10, we match the best CNN accuracy in (Papernot et al., 2020b)—namely $6 6 . \hat { 2 } \%$ for a budget of $\varepsilon = 7 . 5 3$ —with a much smaller budget of $\varepsilon = 2 . 6$ . This is an improvement in the DP-guarantee of $e ^ { 4 . 9 } \approx 1 3 4$ . On MNIST, we significantly improve upon CNN models, and match the results of PATE (Papernot et al., 2018), namely $9 8 . 5 \%$ accuracy at $\varepsilon = 1 . 9 7$ , in a more restricted setting (PATE uses 5,000 public unlabeled MNIST digits). In Appendix C.5, we provide the hyperparameters that result in the highest test accuracy for our target DP budget of $( \varepsilon = \mathsf { \bar { 3 } } , \delta = 1 0 ^ { - 5 } )$ . We did not consider larger privacy budgets for ScatterNet classifiers, as the accuracy we achieve at $\varepsilon = 3$ is close to the accuracy of non-private ScatterNet models (see Table 2).
+
+As noted above, our models (and those of most prior work) are the result of a hyper-parameter search. While we do not account for the privacy cost of this search, Table 3 shows that an additional advantage of ScatterNet classifiers is an increased robustness to hyper-parameter changes. In particular, for CIFAR-10 the worst configuration for linear ScatterNet classifiers outperforms the best configuration for end-to-end CNNs. Moreover, on MNIST and Fashion-MNIST, the median accuracy of linear ScatterNet models outperforms the best end-to-end CNN.
+
+Training CNNs on Handcrafted Features. Since linear models trained on handcrafted features outperform the privacy-utility guarantees of deep models trained end-to-end, a natural question is whether training deeper models on these features achieves even better results. We repeat the above experiment with a similar CNN model trained on ScatterNet features (see Appendix C.2). The privacy-accuracy curves for these models are in Figure 2. We find that handcrafted features also improve the utility of private deep models, a phenomenon which we analyze and explain in Section 4. On CIFAR-10, the deeper ScatterNet models even slightly outperform the linear models, while for MNIST and Fashion-MNIST the linear models perform best. This can be explained by the fact that in the non-private setting, linear ScatterNet models achieve close to state-of-the-art accuracy on MNIST and Fashion-MNIST, and thus there is little room for improvement with deeper models (see Table 11). Table 3 further shows that ScatterNet CNNs are also less sensitive to hyper-parameters than end-to-end CNNs.
+
+Table 3: Variability across hyper-parameters. For each model, we report the minimum, maximum, median and median absolute deviation (MAD) in test accuracy (in $\%$ ) achieved for a DP budget of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 } ,$ ). The maximum accuracy below may exceed those in Table 1 and Figure 1, which are averages of five runs. SN stands for ScatterNet.   
+
+<table><tr><td></td><td colspan="4">MNIST</td><td colspan="4">Fashion-MNIST</td><td colspan="4">CIFAR-10</td></tr><tr><td>Model</td><td>Min</td><td>Max</td><td>Median</td><td>MAD</td><td>Min</td><td>Max</td><td>Median</td><td>MAD</td><td>Min</td><td>Max</td><td>Median</td><td>MAD</td></tr><tr><td>SN + Linear</td><td>96.8</td><td>98.8</td><td>98.4</td><td>0.2</td><td>85.3</td><td>89.8</td><td>88.7</td><td>0.5</td><td>59.5</td><td>67.0</td><td>65.4</td><td>0.9</td></tr><tr><td>SN +CNN</td><td>95.6</td><td>98.8</td><td>98.1</td><td>0.3</td><td>77.8</td><td>89.1</td><td>87.2</td><td>1.0</td><td>57.3</td><td>69.5</td><td>66.9</td><td>1.6</td></tr><tr><td>CNN</td><td>86.1</td><td>98.2</td><td>97.4</td><td>0.5</td><td>20.2</td><td>86.2</td><td>83.6</td><td>1.8</td><td>39.4</td><td>59.2</td><td>52.5</td><td>5.4</td></tr></table>
+
+![](images/62bdee377199d6d69c3d504af4fd39ce1cc5b1d74829a52eba6801829b5e7786.jpg)  
+Figure 2: Highest test accuracy achieved for each DP budget $( \varepsilon , \delta = 1 0 ^ { - 5 } )$ ) for linear ScatterNet classifiers, CNNs on top of ScatterNet features, and end-to-end CNNs. Shows mean and standard deviation across five runs.
+
+Note that on each dataset we consider, end-to-end CNNs can outperform ScatterNet models when trained without privacy. Thus, end-to-end CNNs trained with DP-SGD must eventually surpass ScatterNet models for large enough privacy budgets. But this currently requires settling for weak provable privacy guarantees. On CIFAR-10 for example, ScatterNet classifiers still outperform end-to-end CNNs for $\varepsilon = 7 . 5 3$ (Papernot et al., 2020b). While the analysis of DP-SGD might not be tight, Jagielski et al. (2020) suggest that the true $\varepsilon$ guarantee of DP-SGD is at most one order of magnitude smaller than the current analysis suggests. Thus, surpassing handcrafted features for small privacy budgets on CIFAR-10 may require improvements beyond a tighter analysis of DP-SGD.
+
+# 4 HOW DO HANDCRAFTED FEATURES HELP?
+
+In this section, we analyze why private models with handcrafted features outperform end-to-end CNNs. We first consider the dimensionality of our models, but show that this does not explain the utility gap. Rather, we find that the higher accuracy of ScatterNet classifiers is due to their faster convergence rate when trained without noise.
+
+Smaller models are not easier to train privately. The utility of private learning typically degrades as the model’s dimensionality increases (Chaudhuri et al., 2011; Bassily et al., 2014). This is also the case with DP-SGD which adds Gaussian noise, of scale proportional to the gradients, to each model parameter. We thus expect smaller models to be easier to train privately. Yet, as we see from Table 4, for MNIST and Fashion-MNIST the linear ScatterNet model has more parameters than the CNNs. For CIFAR-10, the end-to-end CNN we used is larger, so we repeat the experiment from Section 3 with a CNN of comparable size to the ScatterNet classifiers (see Appendix D.5). This has a minor effect on the performance of the CNN. Thus, the dimensionality of ScatterNet classifiers fails to explain their better performance.
+
+Models with handcrafted features converge faster without privacy. DP-SGD typically requires a smaller learning rate than noiseless (clipped) SGD, so that the added noise gets averaged out over small steps. We indeed find that the optimal learning rate when training with DP-SGD is an order of magnitude lower than the optimal learning rate for training without noise addition (with gradients clipped to the same norm in both cases).
+
+Table 4: Number of trainable parameters of our models. For CIFAR-10, we consider two different end-to-end CNN architectures (see Appendix C.2), the smaller of which has approximately as many parameters as the linear ScatterNet model.   
+
+<table><tr><td colspan="2">MNIST&amp;Fashion-MNIST</td><td>CIFAR-10</td></tr><tr><td>ScatterNet+Linear</td><td>40K</td><td>155K</td></tr><tr><td>ScatterNet+CNN</td><td>33K</td><td>187K</td></tr><tr><td>CNN</td><td>26K</td><td>551K/168K</td></tr></table>
+
+![](images/2af929bdcef192a8ec7be61ed64dd878bb38b710fbc9c6c3449c457c135cf5fa.jpg)  
+Figure 3: Convergence of DP-SGD with and without noise on CIFAR-10, for ScatterNet classifiers and end-to-end CNNs. (Left): low learning rate. (Right): high learning rate.
+
+To understand the impact of gradient noise on the learning process, we conduct the following experiment: we select a low learning rate that is near-optimal for training models with gradient noise, and a high learning rate that is near-optimal for training without noise. For both learning rates, we train CIFAR-10 models both with and without noise (with gradient clipping in all cases). Figure 3 shows that with a high learning rate, all classifiers converge rapidly when trained without noise, but gradient noise vastly degrades performance. With a low learning rate however, training converges similarly whether we add noise or not. What distinguishes the ScatterNet models is the faster convergence rate of noiseless SGD. The experimental setup and similar qualitative results on MNIST and Fashion-MNIST are in Appendix C.6. Thus, we find that handcrafted features are beneficial for private learning because they result in a simpler learning task where training converges rapidly even with small update steps. Our analysis suggests two avenues towards obtaining higher accuracy with private deep learning:
+
+• Faster convergence: Figure 3 suggests that faster convergence of non-private training could translate to better private learning. DP-SGD with adaptive updates (e.g., Adam (Kingma & Ba,   
+2015)) indeed sometimes leads to small improvements (Papernot et al., 2020b; Chen & Lee,   
+2020; Zhou et al., 2020a). Investigating private variants of second-order optimization methods is an interesting direction for future work. • More training steps (a.k.a more data): For a fixed DP-budget $\varepsilon$ and noise scale $\sigma$ , increasing the training set size $N$ allows for running more steps of DP-SGD (McMahan et al., 2018). In Section 5.1, we investigate how the collection of additional private data impacts the utility of private end-to-end models.
+
+# 5 TOWARDS BETTER PRIVATE DEEP LEARNING
+
+We have shown that on standard vision tasks, private learning strongly benefits from handcrafted features. Further improving our private baselines seems hard, as they come close to the maximal accuracy of ScatterNet models (see Table 2). We thus turn to other avenues for obtaining stronger privacy-utility guarantees. We focus on CIFAR-10, and discuss two natural paths towards better private models: (1) access to a larger private training set, and (2) access to a public image dataset from a different distribution (some works also consider access to public unlabeled data from the same distribution as the private data (Papernot et al., 2017; 2018; Zhu et al., 2020)).
+
+![](images/1026acafa710cb0c8ef6574d0ad16e92bacf81d41282a4065ed8bee8465b5094.jpg)  
+Figure 4: CIFAR-10 test accuracy for a training set of size $N$ and a DP budget of $( \varepsilon = 3 , \delta =$ $^ { 1 } / 2 N )$ ). For $N \mathrm { ~ > ~ } 5 0 \mathrm { K }$ , we augment CIFAR10 with pseudo-labeled Tiny Images collected by Carmon et al. (2019).
+
+![](images/ff7afb1249f0affcf176baad5c456972d60729366d02154137970a41d5e9ebe1.jpg)  
+Figure 5: Privacy-utility tradeoffs for transfer learning on CIFAR-10. We fine-tune linear models on features from a ResNeXt model trained on CIFAR-100, and from a SimCLR model trained on unlabeled ImageNet.
+
+# 5.1 IMPROVING PRIVACY BY COLLECTING MORE DATA
+
+We first analyze the benefits of additional private labeled data on the utility of private models. Since the privacy budget consumed by DP-SGD scales inversely with the size of the training data $N$ , collecting more data allows either to train for more steps, or to lower the amount of noise added per step—for a fixed DP budget $\varepsilon$ .
+
+To obtain a larger dataset comparable to CIFAR-10, we use 500K pseudo-labeled Tiny Images5 (Torralba et al., 2008) collected by Carmon et al. (2019).6 We then train private models on subsets of size $1 0 , 0 0 0 \le N \le 5 5 0 , 0 0 0$ from this dataset. Figure 4 reports the highest test accuracy achieved for a privacy budget of $( \varepsilon = 3 , \delta = { ^ { 1 } \mathrm { / } } 2 N )$ ) (see Appendix C.7 for the experimental setup). We find that we need about an order-of-magnitude increase in the size of the private training dataset in order for end-to-end CNNs to outperform ScatterNet features. As we show in Appendix C.7, larger datasets allow DP-SGD to be run for more steps at a fixed privacy budget and noise level (as also observed in (McMahan et al., 2018))—thereby overcoming the slow convergence rate we uncovered in Section 4. While the increased sample complexity of private deep learning might be viable for “internet-scale” applications (e.g., language modeling across mobile devices), it is detrimental for sensitive applications with more stringent data collection requirements, such as in healthcare.
+
+5.2 TRANSFER LEARNING: BETTER FEATURES FROM PUBLIC DATA
+
+Transfer learning is a natural candidate for privacy-preserving computer vision, as features learned on public image data often significantly outperform handcrafted features (Razavian et al., 2014). We first consider transfer learning from CIFAR-100 to CIFAR-10, where the labeled CIFAR-100 data is assumed public. We extract features from the penultimate layer of a ResNeXt (Xie et al., 2017) model trained on CIFAR-100. A non-private linear model trained on these features achieves $8 4 \%$ accuracy on CIFAR-10. When training linear models with DP-SGD, we get the privacy-utility curve in Figure 5 (see Appendix C.8 for details). We reach an accuracy of $8 0 . 0 \%$ at a budget of $( \varepsilon = 2 , \delta = \mathsf { \bar { 1 0 } ^ { - 5 } }$ ), a significant improvement over prior work for the same setting and privacy budget, e.g., $6 7 \%$ accuracy in (Abadi et al., 2016) and $7 2 \%$ accuracy in (Papernot et al., 2020a). The large gap between our results and prior work is mainly attributed to a better choice of source model (e.g., the transfer learning setup in (Papernot et al., 2020a) achieves $7 5 \%$ accuracy on CIFAR-10 in the non-private setting). Mirroring the work of Kornblith et al. (2019) on non-private transfer learning, we thus find that the heuristic rule “better models transfer better” also holds with differential privacy.
+
+We further consider access to a public dataset of unlabeled images. We extract features from the penultimate layer of a SimCLR model (Chen et al., 2020a) trained on unlabeled ImageNet. A nonprivate linear model trained on these features achieves $9 5 \%$ accuracy on CIFAR-10 (using labeled ImageNet data marginally improves non-private transfer learning to CIFAR-10 (Chen et al., 2020a)). With the same setup as for CIFAR-100 (see Appendix C.8), we train a linear model to $9 2 . 7 \%$ accuracy for a DP budget of $( \varepsilon = 2 , \delta = 1 0 ^ { - 5 }$ ) (see Figure 5).
+
+# 6 CONCLUSION AND OPEN PROBLEMS
+
+We have demonstrated that differentially private learning benefits from “handcrafted” features that encode priors on the learning task’s domain. In particular, we have shown that private ScatterNet classifiers outperform end-to-end CNNs on MNIST, Fashion-MNIST and CIFAR-10. We have further found that handcrafted features can be surpassed when given access to more data, either a larger private training set, or a public dataset from a related domain. In addition to introducing strong baselines for evaluating future improvements to private deep learning and DP-SGD, our work suggests a number of open problems and directions for future work:
+
+Improving DP by accelerating convergence: Our analysis in Section 4 shows that a limiting factor of private deep learning is the slow convergence rate of end-to-end deep models. While the existing literature on second-order optimization for deep learning has mainly focused on improving the overall wall-clock time of training, it suffices for DP to reduce the number of private training steps—possibly at an increase in computational cost.
+
+Federated learning: While we have focused on a standard centralized setting for DP, our techniques can be extended to decentralized training schemes such as Federated Learning (McMahan et al., 2017; Bonawitz et al., 2017; Kairouz et al., 2019). DP has been considered for Federated Learning (Geyer et al., 2017; McMahan et al., 2018), but has also been found to significantly degrade performance in some settings (Yu et al., 2020).
+
+Handcrafted features for ImageNet and non-vision domains: To our knowledge, there have not yet been any attempts to train ImageNet models with DP-SGD, partly due to the cost of computing per-sample gradients. While linear classifiers are unlikely to be competitive on ImageNet, handcrafted features can also help private learning by accelerating the convergence of CNNs, as we have shown in Figure 2. Notably, Oyallon et al. (2018) match the (non-private) accuracy of AlexNet (Krizhevsky et al., 2012) on ImageNet with a small six-layer CNN trained on ScatterNet features. Another interesting direction is to extend our results to domains beyond vision, e.g., with handcrafted features for text (Manning & Schutze, 1999) or speech (Andén & Mallat, 2014).
+
+# ACKNOWLEDGEMENTS
+
+We thank: Mani Malek, Ilya Mironov, Vaishaal Shankar and Ludwig Schmidt for fruitful discussions about differential privacy and computer vision baselines, and comments on early drafts of this paper; Nicolas Papernot and Shuang Song for helping us reproduce the results in (Papernot et al., 2020b); Nicolas Papernot for comments on early drafts of this paper; Edouard Oyallon for enlightening discussions about Scattering networks.
+
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+
+# A WHY SCATTERNETS?
+
+In this paper, we propose to use the ScatterNet features of Oyallon & Mallat (2015) as a basis for shallow differentially private vision classifiers. We briefly discuss a number of other shallow approaches that produce competitive results for canonical vision tasks, but which appear less suitable for private learning.
+
+Unsupervised feature dictionaries. Coates & $\mathrm { N g }$ (2012) achieve above $8 0 \%$ test accuracy on CIFAR-10 with linear models trained on top of a dictionary of features extracted from a mixture of image patches. Their approach relies on a combination of many ‘tricks”, including data normalization, data whitening, tweaks to standard Gaussian-Mixture-Model (GMM) algorithms, feature selection, etc. While it is conceivable that each of these steps could be made differentially private, we opt here for a much simpler unlearned baseline that is easier to analyze and to apply to a variety of different tasks. We note that existing work on differentially-private learning of mixtures (e.g., (Nissim et al., 2007)) has mainly focused on asymptotic guarantees, and we are not aware of any exiting algorithms that have been evaluated on high-dimensional datasets such as CIFAR-10.
+
+Kernel Machines. Recent work on Neural Tangent Kernels (Jacot et al., 2018) has shown that the performance of deep neural networks on CIFAR-10 could be matched by specialized kernel methods (Li et al., 2019; Arora et al., 2020; Shankar et al., 2020). Unfortunately, private learning with non-linear kernels is intractable in general (Chaudhuri et al., 2011; Rubinstein et al., 2012). Chaudhuri et al. (2011) propose to obtain private classifiers by approximating kernels using random features (Rahimi & Recht, 2008), but the very high dimensionality of the resulting learning problem makes it challenging to outperform our handcrafted features baseline. Indeed, we had originally considered a differentially-private variant of the random-feature CIFAR-10 classifier proposed in (Recht et al., 2018), but found the model’s high dimensionality (over 10 million features) to be detrimental to private learning.
+
+# B DP-SGD, RDP AND PRIVATE DATA NORMALIZATION
+
+Throughout this work, we use the DP-SGD algorithm of Abadi et al. (2016):
+
+# Algorithm 1: DP-SGD (Abadi et al., 2016)
+
+input :Data $\{ \pmb { x } _ { 1 } \ldots , \pmb { x } _ { N } \}$ , learning rate $\eta$ , noise scale $\sigma$ , batch size $B$ , gradient norm bound $C$ , epochs $T$   
+1 Initialize $\pmb { \theta } _ { 0 }$ randomly for $t \in [ T \cdot ^ { N } / B ]$ do   
+2 Sample a batch $\mathbf { \delta } _ { B _ { t } }$ by selecting each $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ independently with probability $B / _ { N }$   
+3 For each $\pmb { x } _ { i } \in \pmb { B } _ { t }$ : $\pmb { g } _ { t } ( \pmb { x } _ { i } )  \nabla _ { \pmb { \theta } _ { t } } L ( \pmb { \theta } _ { t } , \pmb { x } _ { i } )$ // compute per-sample gradients   
+4 $\tilde { g } _ { t } ( \pmb { x } _ { i } ) \gets g _ { t } ( \pmb { x } _ { i } ) \cdot \operatorname* { m i n } ( 1 , ^ { C } / | | g _ { t } ( \pmb { x } _ { i } ) | | _ { 2 } )$ // clip gradients   
+5 $\begin{array} { r } { \tilde { g } _ { t } \gets \frac { 1 } { B } \Big ( \sum _ { \pmb { x } _ { i } \in B _ { t } } \tilde { g } _ { t } ( \pmb { x } _ { i } ) + \mathcal { N } ( 0 , \sigma ^ { 2 } C ^ { 2 } I ) \Big ) } \end{array}$ // add noise to average gradient with Gaussian mechanism   
+6 $\begin{array} { r } { \pmb { \theta } _ { t + 1 }  \pmb { \theta } _ { t } - \eta \tilde { \pmb { g } } _ { t } } \\ { \mathbf { p u t } : \pmb { \theta } _ { T N / B } } \end{array}$ // SGD step out
+
+Definition B.1 (Rényi Divergence). For two probability distributions $P$ and $Q$ defined over a range $\mathcal { R }$ , the Rényi divergence of order $\alpha > 1$ is
+
+$$
+D _ { \alpha } ( P \| Q ) : = \frac { 1 } { \alpha - 1 } \log \underset { x \sim Q } { \mathbb { E } } \left( \frac { P ( x ) } { Q ( x ) } \right) ^ { \alpha } .
+$$
+
+Definition B.2 ( $( \alpha , \varepsilon )$ -RDP (Mironov, 2017)). A randomized mechanism $f : \mathcal { D }  \mathcal { R }$ is said to have $\varepsilon$ -Rényi differential privacy of order $\alpha$ , or $( \alpha , \varepsilon )$ -RDP for short, if for any adjacent $D , D ^ { \prime } \in { \mathcal { D } }$ it holds that
+
+$$
+D _ { \alpha } ( f ( D ) \| f ( D ^ { \prime } ) ) \leq \varepsilon .
+$$
+
+To analyze the privacy guarantees of DP-SGD, we numerically compute $D _ { \alpha } ( f ( D ) \| f ( D ^ { \prime } ) )$ for a range of orders $\alpha$ (Mironov et al., 2019; Wang et al., 2019) in each training step, where $D$ and $D ^ { \prime }$ are training sets that differ in a single element. To obtain privacy guarantees for $t$ training steps, we use the composition properties of RDP:
+
+Lemma B.3 (Adaptive composition of RDP (Mironov et al., 2019)). Let $f : \mathcal { D }  \mathcal { R } _ { 1 }$ be $( \alpha , \varepsilon _ { 1 } )$ -RDP and $g : \mathcal { R } _ { 1 } \times \mathcal { D }  \mathcal { R } _ { 2 }$ be $( \alpha , \varepsilon _ { 2 } )$ -RDP, then the mechanism defined as $( X , Y )$ , where $X \sim f ( D )$ and $Y \sim g ( X , D )$ , satisfies $( \alpha , \varepsilon _ { 1 } + \varepsilon _ { 2 } )$ -RDP.
+
+Finally, the RDP guarantees of the full DP-SGD procedure can be converted into a $( \varepsilon , \delta )$ -DP guarantee:
+
+Lemma B.4 (From RDP to $( \varepsilon , \delta )$ -DP (Mironov et al., 2019)). If $f$ is an $( \alpha , \varepsilon )$ -RDP mechanism, it also satisfies $\begin{array} { r } { ( \varepsilon + \frac { \log { 1 / \delta } } { \alpha - 1 } , \delta ) } \end{array}$ -DP for any $0 < \delta < 1$ .
+
+Private Data Normalization. In order to apply Data Normalization to the ScatterNet features (which greatly improves convergence, especially on CIFAR-10), we use the PrivDataNorm procedure in Algorithm 2 to compute private estimates of the per-channel mean and variance of the ScatterNet features.
+
+# Algorithm 2: Private Data Normalization
+
+Function PrivChannelMean(data $\pmb { D } \in \mathbb { R } ^ { N \times K \times H \times W }$ , norm bound $C$ , noise scale $\sigma _ { n o r m }$
+
+<table><tr><td>1</td><td>For1≤i≤ N: μi ← En,ω [D(i,,h,w)] ∈ RK // compute per-channel means for each sample μi ← μi · min(1,C/μill2) // clip each sample&#x27;s per-channel</td></tr></table>
+
+In order to obtain tight privacy guarantees for the full training procedure (i.e., privacy-preserving Data Normalization followed by DP-SGD), we first derive the RDP guarantees of PrivDataNorm:
+
+Claim B.5. The PrivDataNorm procedure is $( \alpha , \alpha / \sigma _ { n o r m } ^ { 2 } )$ -RDP for any $\alpha > 1$ .
+
+The above claim follows from the RDP guarantees of the Gaussian mechanism in (Mironov, 2017), together with the composition properties of RDP in Lemma B.3 above.
+
+Finally, given an RDP guarantee of $( \alpha , \varepsilon _ { 1 } )$ for PrivDataNorm, and an RDP guarantee of $( \alpha , \varepsilon _ { 2 } )$ for DP-SGD, we apply Lemma B.3 to obtain an RDP guarantee of $( \alpha , \varepsilon _ { 1 } + \varepsilon _ { 2 } )$ , and convert to a DP guarantee using Lemma B.4.
+
+# C EXPERIMENTAL SETUP
+
+# C.1 SCATTERING NETWORKS
+
+We briefly review the scattering network (ScatterNet) of Oyallon & Mallat (2015). Consider an input $_ { \textbf { \em x } }$ . The output of a scattering network of depth $J$ is a feature vector given by
+
+$$
+S ( { \pmb x } ) : = A _ { J } \left| W _ { 2 } \left| W _ { 1 } { \pmb x } \right| \right| ,
+$$
+
+where the operators $W _ { 1 }$ and $W _ { 2 }$ are complex-valued wavelet transforms, each followed by a non-linear complex modulus, and the final operator $A$ performs spatial averaging over patches of $2 ^ { J }$ features. Both wavelet transforms $W _ { 1 }$ and $W _ { 2 }$ are linear operators that compute a cascade of convolutions with filters from a fixed family of wavelets. For an input image of spatial dimensions $H \times W$ , the ScatterNet is aptensor of dimension $( K , { \frac { H } { 2 ^ { J } } } , { \frac { W } { 2 ^ { J } } } )$ h of the image’s color channe. The channel dimensionality $K$ y to yield an outpthe network depth $J$ $K / 2 ^ { 2 J } = O ( 1 )$ approximately preserves the data dimensionality).
+
+For all experiments, we use the default parameters proposed by Oyallon & Mallat (2015), namely a Scattering Network of depth $J = 2$ , consisting of wavelet filters rotated along eight angles. For an an input image of spatial dimensions $H \times W$ , this configuration produces an output of dimension $( K , \bar { H } / 4 , W / 4 )$ , with $K = 8 1$ for grayscale images, and $K = 2 4 3$ for RGB images.
+
+# C.2 MODEL ARCHITECTURES
+
+Below, we describe the ScatterNet+Linear, ScatterNet+CNN and end-to-end CNN architectures used in Section 3 and Section 4. The CNN architectures are adapted from Papernot et al. (2020b).
+
+Linear ScatterNet Classifiers. The default Scattering Network of Oyallon & Mallat (2015) extracts feature vectors of size (81, 7, 7) for MNIST and Fashion-MNIST and of size (243, 8, 8) for CIFAR-10. We then train a standard logistic regression classifier (with per-class bias) on top of these features, as summarized below:
+
+Table 5: Size of linear ScatterNet classifiers.   
+
+<table><tr><td>Dataset</td><td>Image size</td><td>Linear ScatterNet size</td></tr><tr><td>MNIST</td><td>28×28</td><td>3969 ×10</td></tr><tr><td>Fashion-MNIST</td><td>28×28</td><td>3969 ×10</td></tr><tr><td>CIFAR-10</td><td>32 ×32×3</td><td>15552 ×10</td></tr></table>
+
+End-to-end CNNs. We use the CNN architectures proposed by Papernot et al. (2020b), which were found as a result of an architecture search tailored to DP-SGD.7 Notably, these CNNs are quite small (since the noise of DP-SGD grows with the model’s dimensionality) and use Tanh activations, which Papernot et al. (2020b) found to outperform the more common ReLU activations. For the experiments in Section 4, we also consider a smaller CIFAR-10 model, with a dimensionality comparable to the linear ScatterNet classifier. While the standard model has six convolutional layers of size 32-32-64-64-128-128, the smaller model has five convolutional layers of size 16-16-32-32-64 (with max-pooling after the $2 ^ { \mathrm { n d } }$ , $4 ^ { \mathrm { t h } }$ and $5 ^ { \mathrm { t h } }$ convolution).
+
+Table 6: End-to-end CNN model for MNIST and Fashion-MNIST, with Tanh activations (Papernot et al., 2020b).   
+
+<table><tr><td>Layer</td><td>Parameters</td></tr><tr><td>Convolution</td><td>16 filters of 8x8,stride 2,padding 2</td></tr><tr><td>Max-Pooling</td><td>2x2,stride 1</td></tr><tr><td>Convolution</td><td>32 filters of 4x4,stride 2,padding O</td></tr><tr><td>Max-Pooling</td><td>2x2,stride 1</td></tr><tr><td>Fully connected</td><td>32 units</td></tr><tr><td>Fully connected</td><td>10 units</td></tr></table>
+
+Table 7: End-to-end CNN model for CIFAR-10, with Tanh activations (Papernot et al., 2020b). In Section 4, we also use a smaller variant of this architecture with five convolutional layers of 16-16-32-32-64 filters.   
+
+<table><tr><td>Layer</td><td>Parameters</td></tr><tr><td>Convolution x2</td><td>32 filters of 3x3,stride 1,padding 1</td></tr><tr><td>Max-Pooling</td><td>2x2,stride 2</td></tr><tr><td>Convolution x2 Max-Pooling</td><td>64 filters of 3x3,stride 1, padding 1 2x2,stride 2</td></tr><tr><td>Convolution x2</td><td>128 filters of 3x3,stride 1,padding 1</td></tr><tr><td>Max-Pooling</td><td>2x2,stride 2</td></tr><tr><td>Fully connected</td><td>128 units</td></tr><tr><td>Fully connected</td><td>10 units</td></tr></table>
+
+ScatterNet CNNs. To fine-tune CNNs on top of ScatterNet features, we adapt the CNNs from Table 6 and Table 7. As the ScatterNet feature vector is larger than the input image $7 8 4  3 9 6 9$ features for MNIST and Fashion-MNIST, and $3 0 7 2  1 5 5 5 2$ features for CIFAR-10), we use smaller CNN models. For MNIST and Fashion MNIST, we reduce the number of convolutional filters. For CIFAR-10, we reduce the network depth from 8 to 3, which results in a model with approximately as many parameters as the linear ScatterNet classifier.
+
+Table 8: CNN model fine-tuned on ScatterNet features for MNIST and Fashion-MNIST, with Tanh activations.   
+
+<table><tr><td>Layer</td><td>Parameters</td></tr><tr><td>Convolution</td><td>16 filters of 3x3,stride 2,padding 1</td></tr><tr><td>Max-Pooling</td><td>2x2,stride 1</td></tr><tr><td>Convolution</td><td>32 filters of 3x3,stride1,padding 1</td></tr><tr><td>Max-Pooling</td><td>2x2,stride 1</td></tr><tr><td>Fully connected</td><td>32 units</td></tr><tr><td>Fully connected</td><td>10 units</td></tr></table>
+
+Table 9: CNN model on ScatterNet features for CIFAR-10, with Tanh activations. In Section 4, we also use a smaller variant of this model with four convolutional layers of 16-16-32-32 filters.   
+
+<table><tr><td>Layer</td><td>Parameters</td></tr><tr><td>Convolution</td><td>64 filters of 3x3,stride 1, padding 1</td></tr><tr><td>Max-Pooling</td><td>2x2, stride 2</td></tr><tr><td>Convolution</td><td>64 filters of 3x3,stride1,padding 1</td></tr><tr><td>Max-Pooling</td><td>2x2,stride 2</td></tr><tr><td>Fully connected</td><td>10 units</td></tr></table>
+
+# C.3 EFFECT OF NORMALIZATION
+
+To evaluate the effect of feature normalization in Table 2, we train linear models on ScatterNet features using DP-SGD without noise $( \sigma = 0$ ). We train one model without feature normalization, one with Data Normalization, and three with Group Normalization (Wu & He, 2018) with $G \in \{ 9 , 2 7 , 8 1 \}$ groups. For Group Normalization, Table 2 reports results for the best choice of groups. The remaining hyper-parameters are given below.
+
+Table 10: Hyper-parameters for evaluating the effect of feature normalization in Table 2.   
+
+<table><tr><td>Parameter</td><td>MNIST</td><td>Fashion-MNIST</td><td>CIFAR-10</td></tr><tr><td>Gradient clipping norm C</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Momentum</td><td>0.9</td><td>0.9</td><td>0.9</td></tr><tr><td>Epochs T</td><td>20</td><td>20</td><td>20</td></tr><tr><td>Batch size B</td><td>512</td><td>512</td><td>512</td></tr><tr><td>Learning rate n</td><td>2</td><td>4</td><td>2</td></tr><tr><td>Best choice of groups G</td><td>27</td><td>81</td><td>27</td></tr></table>
+
+# C.4 NON-PRIVATE MODEL PERFORMANCE
+
+For each of the model architectures described in Appendix C.2, we report the best achieved test accuracy without privacy, and without any other form of explicit regularization. For MNIST and Fashion-MNIST, fine-tuning a linear model or a CNN on top of ScatterNet features results in similar performance, whereas on CIFAR-10, the CNN performs slightly better. For Fashion-MNIST the end-to-end CNN performs slightly worse than the linear model (mainly due to a lack of regularization). For CIFAR-10, the end-to-end CNN significantly outperforms the ScatterNet models.
+
+Table 11: Test accuracy $( \mathrm { i n \% } )$ ) for models trained without privacy. Average and standard deviation are computed over five runs.   
+
+<table><tr><td>Dataset</td><td>ScatterNet+Linear</td><td>ScatterNet+CNN</td><td>CNN</td></tr><tr><td>MNIST</td><td>99.3 ±0.0</td><td>99.2 ± 0.0</td><td>99.2 ±0.0</td></tr><tr><td>Fashion-MNIST</td><td>91.5 ± 0.0</td><td>91.5 ± 0.2</td><td>90.1±0.2</td></tr><tr><td>CIFAR-10</td><td>71.1 ± 0.0</td><td>73.8 ± 0.3</td><td>80.0±0.1</td></tr></table>
+
+# C.5 EVALUATING PRIVATE SCATTERNET CLASSIFIERS
+
+We use DP-SGD with momentum for all experiments. Prior work found that the use of adaptive optimizers (e.g., Adam (Kingma & Ba, 2015)) provided only marginal benefits for private learning (Papernot et al., 2020a). Moreover, we use no data augmentation, weight decay, or other mechanisms aimed at preventing overfitting. The reason is that differential privacy is itself a powerful regularizer (informally, differential privacy implies low generalization error (Dwork et al., 2015)), so our models all underfit the training data.
+
+The table below lists the ranges of hyper-parameters used for the experiments in Section 3, to train linear ScatterNet classifiers, end-to-end CNNs, and CNNs fine-tuned on ScatterNet features.
+
+Table 12: Hyper-parameters for the evaluation of private linear classifiers fine-tuned on ScatterNet features, CNNs fine-tuned on ScatterNet features, and end-to-end CNNs in Section 3.   
+
+<table><tr><td>Parameter</td><td>MNIST</td><td>Fashion-MNIST</td><td>CIFAR-10</td></tr><tr><td>DP guarantee (ε, δ)</td><td>(3,10-5)</td><td>(3,10-5)</td><td>(3,10-5)</td></tr><tr><td>Gradient clipping norm C</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Momentum</td><td>0.9</td><td>0.9</td><td>0.9</td></tr><tr><td>Batch size B</td><td>{512,1024,...,16384} {512,1024,...,16384} {512,1024,...,16384}</td><td></td><td></td></tr><tr><td>Learning rate n</td><td>{1/4,1/2,1,2} .B/512</td><td>{1/4,1/2,1,2}.B/512</td><td>{1/8,1/4,1/2,1}.B/512</td></tr><tr><td>Epochs T</td><td>{15,25,40}</td><td>{15,25,40}</td><td>{30,60,120}</td></tr><tr><td>DP-SGD noise scale σ</td><td>calculated numerically so that a DP budget of (ε,δ) is spent after T epochs</td><td></td><td></td></tr><tr><td>Group Norm. groups G Data Norm.(C1,C2,Onorm)</td><td>{9,27,81} (0.2,0.05,{6,8})</td><td>{9,27,81} (0.3,0.15, {6,8})</td><td>{9,27,81} (1.0,1.5, {6,8})</td></tr></table>
+
+In Table 13, we give the set of hyper-parameters that resulted in the maximal accuracy for our target DP budget of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 } \mathrm { . }$ ). For each model, we report the base learning rate, before re-scaling by $B / 5 1 2$ . We find that some hyper-parameters that result in the best performance are at the boundary of our search range. Yet, as we show in Figure 8, modifying these hyper-parameters results in no significant upward trend, so we refrained from further increasing our search space.
+
+Table 13: Set of hyper-parameters resulting in the highest test accuracy for a privacy budget of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 } )$ ). Note that we report the base learning rate (LR), before scaling by a factor of $B / 5 1 2$ . SN stands for ScatterNet.   
+
+<table><tr><td></td><td colspan="3">MNIST</td><td colspan="3">Fashion-MNIST</td><td colspan="3">CIFAR-10</td></tr><tr><td>Parameter</td><td>SN+Linear</td><td>SN+CNN</td><td>CNN</td><td>SN+Linear</td><td>SN+CNN</td><td>CNN</td><td>SN+Linear</td><td>SN+CNN</td><td>CNN</td></tr><tr><td>Batch size B</td><td>4096</td><td>1024</td><td>512</td><td>8192</td><td>2048</td><td>2048</td><td>8192</td><td>8192</td><td>1024</td></tr><tr><td>Base LR n</td><td>1</td><td>/</td><td>/</td><td>1</td><td>1</td><td>1</td><td>1/4</td><td>1/4</td><td>1/</td></tr><tr><td>Epochs T</td><td>40</td><td>25</td><td>40</td><td>40</td><td>40</td><td>40</td><td>60</td><td>60</td><td>30</td></tr><tr><td>Groups G</td><td>-</td><td>1</td><td>-</td><td>27</td><td>27</td><td>-</td><td>-</td><td>-</td><td>1</td></tr><tr><td>Data Norm. Onorm</td><td>8</td><td>8</td><td>-</td><td>1</td><td>-</td><td>-</td><td>8</td><td>8</td><td>1</td></tr></table>
+
+# C.6 MEASURING MODEL CONVERGENCE SPEED
+
+For the experiments in Section 4, we compare the convergence of the models from Appendix C.2 when trained with and without noise, and with either a low or high learning rate. The table below lists the hyper-parameters for the CIFAR-10 experiment in Figure 3, as well as for the corresponding experiments for MNIST and Fashion-MNIST in Figure 11. When training without privacy, we still clip gradients to a maximal norm of $C = 0 . 1$ , but omit the noise addition step of DP-SGD (and we also omit the noise when using Data Normalization).
+
+Table 14: Hyper-parameters for the experiments on model convergence rates in Figure 3 and Figure 11.   
+
+<table><tr><td>Dataset</td><td>Batch size B Gradient Norm C</td><td></td><td>Learning rate n (low, high)</td><td>Epochs T</td><td>Normalization</td></tr><tr><td>MNIST</td><td>512</td><td>0.1</td><td>(1,8)</td><td>40</td><td>Data Norm. (Onorm = 8)</td></tr><tr><td>Fashion-MNIST</td><td>512</td><td>0.1</td><td>(1,16)</td><td>40</td><td>Group Norm. (G= 81)</td></tr><tr><td>CIFAR-10</td><td>512</td><td>0.1</td><td>(1/4,4)</td><td>60</td><td>Data Norm. (Onorm = 8)</td></tr></table>
+
+# C.7 PRIVATE LEARNING ON LARGER DATASETS
+
+For the experiment in Section 5.1, we use an additional 500K images from the Tiny Images dataset (Torralba et al., 2008), which were collected and labeled by Carmon et al. (2019) using a pre-trained CIFAR-10 classifier (see (Carmon et al., 2019, Appendix B.6) for details on the selection process for this dataset).8 We create datasets of size $N \in \{ 1 0 \mathbf { K } , 2 5 \mathbf { K } , 5 0 \mathbf { K } , 1 0 0 \mathbf { K } , 2 5 0 \mathbf { K } , 5 5 0 \mathbf { K } \}$ by taking subsets of this larger dataset. We only use the data of Carmon et al. (2019) to complement the CIFAR-10 dataset when $N > 5 0 \mathrm { K }$ . As noted by Carmon et al. (2019), the additional 500K images do not entirely match the distribution of CIFAR-10. Nevertheless, we find that training our classifiers without privacy on augmented datasets of size $N > 5 0 \mathrm { K }$ does not negatively impact the test accuracy on CIFAR-10.
+
+For each training set size, we re-train our models with a hyper-parameter search. To limit computational cost, and informed by our prior experiments, we fix some parameters, as shown in Table 15. When applying Data Normalization to ScatterNet features, we compute the per-channel statistics only over the original CIFAR-10 samples, and compute the privacy guarantees of PrivDataNorm using the Rényi DP analysis of the sampled Gaussian mechanism (Mironov et al., 2019; Wang et al., 2019).
+
+Table 15: Hyper-parameters for the evaluation of private classifiers on larger datasets in Section 5.1.   
+
+<table><tr><td>Parameter</td><td>Value for dataset of size N</td></tr><tr><td>DP guarantee (ε, δ)</td><td>(3,1/2N)</td></tr><tr><td>Gradient norm C</td><td>0.1</td></tr><tr><td>Momentum</td><td>0.9</td></tr><tr><td>Batch size B</td><td>8192</td></tr><tr><td>Learning rate n</td><td>{1/8,1/4,1/2,1,2}.8192/512</td></tr><tr><td>Epochs T</td><td>{15,30,60,120} . 50000/N</td></tr><tr><td>Data Norm. params (C1,C2, Onorm)</td><td>(1,1.5,8)</td></tr></table>
+
+The only hyper-parameters are thus the number of epochs (normalized by the size of the original CIFAR-10 data) and the learning rate $\eta$ . The optimal values we found for these parameters are given below in Table 16. As we increase the dataset size, we obtain better accuracy by training for more steps and with higher learning rates. Figure 4 reports the final accuracy for these best-performing models.
+
+Table 16: Set of hyper-parameters resulting in the highest test accuracy for a privacy budget of $( \varepsilon = 3 , \delta = { ^ { 1 } \mathrm { / 2 } } N _ { . }$ ). The test accuracy for these models are in Figure 4. Epochs are normalized by the size of the original CIFAR-10 dataset, so training for $T$ epochs corresponds to training on $T \cdot 5 0 { , } 0 0 0$ examples. Note that we report the base learning rate, before scaling by a factor of $8 1 9 2 / 5 1 2$ .   
+
+<table><tr><td></td><td colspan="2">ScatterNet+Linear</td><td colspan="2">ScatterNet+CNN</td><td colspan="2">CNN</td></tr><tr><td>N</td><td>Epochs T</td><td>Learning rate n</td><td>Epochs T</td><td>Learning rate n</td><td>Epochs T</td><td>Learning rate n</td></tr><tr><td>10K</td><td>30</td><td>1/8</td><td>60</td><td>1/8</td><td>30</td><td>1/8</td></tr><tr><td>25K</td><td>30</td><td></td><td>60</td><td>1/8</td><td>60</td><td>1/8</td></tr><tr><td>50K</td><td>60</td><td>诊</td><td>60</td><td>1/4</td><td>60</td><td>1/4</td></tr><tr><td>100K</td><td>60</td><td>诊</td><td>120</td><td></td><td>120</td><td>1/4</td></tr><tr><td>250K</td><td>120</td><td></td><td>120</td><td>14</td><td>120</td><td>1</td></tr><tr><td>550K</td><td>120</td><td></td><td>120</td><td>1</td><td>120</td><td>1</td></tr></table>
+
+# C.8 EVALUATION OF PRIVATE TRANSFER LEARNING
+
+For the transfer learning experiments in Figure 5, we use a ResNeXt-29 model pre-trained on CIFAR-100,9 and a ResNet-50 model trained on unlabeled ImageNet (Deng et al., 2009) using SimCLRv2 (Chen et al., 2020b).10
+
+To train private linear classifiers on CIFAR-10, we first extract features from the penultimate layer of the above pre-trained models. For the ResNeXt model, we obtain features of dimension 1024, and for the SimCLRv2 ResNet, we obtain features of dimension 4096. We then use DP-SGD with a similar setup as for the linear ScatterNet classifiers, except that we do not normalize the extracted features. We also target a tighter privacy budget of $( \varepsilon = 2 , \bar { \delta } = 1 0 ^ { - 5 }$ ). We then run a hyper-parameter search as listed below in Table 17. Figure 5 shows the best test accuracy achieved for each DP budget, averaged across five runs. We further report the set of hyper-parameters that resulted in the maximal accuracy for the targeted privacy budget of $\mathit { \check { \Psi } } \varepsilon = 2 , \delta = \mathit { \dot { 1 } } 0 ^ { - 5 }$ ).
+
+Table 17: Hyper-parameters for the evaluation of private transfer learning from CIFAR-100 (using a ResNeXt model) and from unlabeled ImageNet (using a SimCLR v2 model) in Section 5.2.   
+
+<table><tr><td>Parameter</td><td>Values</td><td>Best forResNeXtBest forSimCLRv2</td><td></td></tr><tr><td>DP guarantee (ε, δ)</td><td>(2,10-5)</td><td></td><td></td></tr><tr><td>Gradient norm C</td><td>0.1</td><td></td><td></td></tr><tr><td>Momentum</td><td>0.9</td><td></td><td>=</td></tr><tr><td>Batch size B</td><td>{512,1024,...,16384}</td><td>2048</td><td>1024</td></tr><tr><td>Learning rate n</td><td>{1/2,1,2,4} . B/512</td><td>2 . 2048/512</td><td>2 . 1024/512</td></tr><tr><td>Epochs T</td><td>{15,25,40}</td><td>40</td><td>40</td></tr></table>
+
+# D ADDITIONAL EXPERIMENTS AND FIGURES
+
+# D.1 ON THE EFFECT OF BATCH SIZES IN DP-SGD
+
+In this section, we revisit the question of the selection of an optimal batch size for DP-SGD. In their seminal work, Abadi et al. (2016) already investigated this question, and noted that the choice of batch size can have a large influence on the privacy-utility tradeoff. They empirically found that for a dataset of size $N$ , a batch size of size approximately $\sqrt { N }$ produced the best results. However, their experiments measured the effect of the batch size while keeping other parameters, including the noise multiplier $\sigma$ and the learning rate $\eta _ { : }$ fixed.
+
+When training without privacy, it has been shown empirically that the choice of batch size has little effect on the convergence rate of SGD, as long as the learning rate $\eta$ is scaled linearly with the batch size (Goyal et al., 2017). Hereafter, we argue formally and demonstrate empirically that if we use a linear learning rate scaling, and fix the number of training epochs $T$ for a target privacy budget $\varepsilon$ then the choice of batch size also has a minimal influence on the performance of DP-SGD.
+
+We first consider the effect of the sampling rate $B / _ { N }$ on the noise scale $\sigma$ required to attain a fixed privacy budget of $\varepsilon$ after $T$ epochs. There is no known closed form expression for $\sigma$ , so it is usually estimated numerically. We empirically establish the following claim, and verify numerically that it holds for our setting in Figure 6:
+
+Claim D.1. Given a fixed $D P$ budget $( \varepsilon , \delta )$ to be reached after $T$ epochs, the noise scale $\sigma$ as a function of the sampling rate $B / _ { N }$ is given by $\sigma ( { ^ B } / { _ { N } } ) \approx c \cdot \sqrt { { ^ B } / { N } } ,$ , for some constant $c \geq 0$ .
+
+![](images/b7e9c7babed34410c50d760b8d5d844ea358cef7d1c2ee30573b3f9661274326.jpg)  
+Figure 6: Noise scale $\sigma$ for DP-SGD that results in a privacy guarantee of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 }$ ) after 60 training epochs, for different batch sampling rates $B / _ { N }$ .
+
+Given this relation between batch size and noise scale, we proceed with a similar analysis as in (Goyal et al., 2017), for the case of DP-SGD. Given some initial weight $\theta _ { t }$ , performing $k$ steps of DP-SGD with clipping norm $C = 1$ , batch size $B$ , learning rate $\eta$ and noise scale $\sigma$ yields:
+
+$$
+\begin{array} { l } { { \pmb \theta } _ { t + k } = \pmb \theta _ { t } - \eta \displaystyle \sum _ { j < k } \frac { 1 } { B } \big ( \displaystyle \sum _ { \pmb x \in B _ { t + j } } \tilde { \pmb g } _ { t + j } ( \pmb x ) + \mathcal { N } ( 0 , \sigma ^ { 2 } { \pmb I } ) \big ) } \\ { = \Big ( \pmb \theta _ { t } - \eta \displaystyle \frac { 1 } { B } \displaystyle \sum _ { j < k } \sum _ { \pmb x \in B _ { t + j } } \tilde { \pmb g } _ { t + j } ( \pmb x ) \Big ) + \mathcal { N } \Big ( 0 , \frac { k \eta ^ { 2 } \sigma ^ { 2 } } { B ^ { 2 } } \pmb I \Big ) } \end{array}
+$$
+
+If we instead take a single step of DP-SGD with larger batch size $k B$ , a linearly scaled learning rate of $k \eta$ , and an adjusted noise scale $\tilde { \sigma } = \sqrt { k } \sigma$ (by Claim D.1), we get:11
+
+$$
+\begin{array} { r l r } {  { \pmb { \theta } _ { t + 1 } = \pmb { \theta } _ { t } - k \eta \frac { 1 } { k B } \big ( \sum _ { j < k } \sum _ { \pmb { x } \in B _ { t + j } } \pmb { \tilde { g } } _ { t } ( \pmb { x } ) + \mathcal { N } ( 0 , \tilde { \sigma } ^ { 2 } \pmb { I } ) \big ) } } \\ & { } & { = \bigg ( \pmb { \theta } _ { t } - \eta \frac { 1 } { B } \sum _ { j < k } \sum _ { \pmb { x } \in B _ { t + j } } \pmb { \tilde { g } } _ { t } ( \pmb { x } ) \bigg ) + \mathcal { N } \bigg ( 0 , \frac { k \eta ^ { 2 } \sigma ^ { 2 } } { B ^ { 2 } } \pmb { I } \bigg ) } \end{array}
+$$
+
+Thus, we find that the total noise in both updates is identical. Under the same heuristic assumption as in (Goyal et al., 2017) that $\tilde { \pmb { g } } _ { t } ( \pmb { x } ) \approx \tilde { \pmb { g } } _ { t + j } ( \pmb { x } )$ for all $j < k$ , the two DP-SGD updates above are thus similar. This analysis suggests that as in the non-private case (Goyal et al., 2017), increasing the batch size and linearly scaling the learning rate should have only a small effect on a model’s learning curve.
+
+We now verify this claim empirically. We follow the experimental setup in Section 3, and set a privacy budget of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 } )$ to be reached after a fixed number of epochs $T$ . For different choices of batch size $B$ , we numerically compute the noise scale $\sigma$ that fits this “privacy schedule”. For the initial batch size of $B _ { 0 } = 5 1 2$ , we select a base learning rate $\eta$ that maximizes test accuracy at epoch $T$ . As we increase the batch size to $B = k B _ { 0 }$ , we linearly scale the learning rate to kη. The concrete parameters are given below:
+
+Table 18: Hyper-parameters for comparing the convergence rate of DP-SGD with different batch sizes in Figure 7.   
+
+<table><tr><td></td><td>Epochs T</td><td>Batch size B</td><td>Learning rate η</td></tr><tr><td>MNIST</td><td>40</td><td>{512,1024,2048,4096}</td><td>1/2 . B/512</td></tr><tr><td>Fashion-MNIST</td><td>40</td><td>{512,1024,2048,4096}</td><td>1.B/512</td></tr><tr><td>CIFAR-10</td><td>60</td><td>{512,1024,2048,4096}</td><td>1/4 · B/512</td></tr></table>
+
+As we can see in Figure 7, the training curves for CNNs trained with DP-SGD are indeed near identical across a variety of batch sizes.
+
+![](images/034428c0e834f4a5165823f473382154fb4d9aa72cbe72ed6777a6506fcbc298.jpg)  
+Figure 7: Convergence rate of DP-SGD for different batch sizes, with a fixed targeted privacy budget of $( \varepsilon = 3 , \delta = 1 \bar { 0 } ^ { - 5 } ,$ ) after $T = 4 0$ or $T = 6 0$ epochs, and linear scaling of the learning rate $\eta \cdot ^ { B } / 5 1 2$ .
+
+# D.2 ANALYSIS OF HYPER-PARAMETERS
+
+To understand the effect of varying the different hyper-parameters of DP-SGD, Figure 8 shows the median and maximum model performance for different choices of a single parameter. The median and maximum are computed over all choices for the other hyper-parameters in Table 12. As we can see, the maximal achievable test accuracy is remarkably stable when fixing one of the algorithm’s hyper-parameters, with the exception of overly large batch sizes or overly low learning rates for end-to-end CNNs.
+
+![](images/1b484164c0a0a8103ed353bebecbb42fa481b4aaf8d14ce2c81cf19935afc79d.jpg)  
+Figure 8: Median and maximum test accuracy of linear ScatterNet classifiers and end-to-end CNNs when we fix one hyper-parameter in Table 12 and run a grid-search over all others (for a privacy budget of $( \varepsilon = 3 , \delta = 1 \bar { 0 } ^ { - 5 } ,$ )).
+
+# D.3 COMPARING DP-SGD AND PRIVACY AMPLIFICATION BY ITERATION
+
+While DP-SGD is the algorithm of choice for differentially private non-convex learning, it is unclear why it should be the best choice for learning private linear models. Indeed, starting with the work of Chaudhuri et al. (2011), there have been many other proposals of algorithms for private convex optimization with provable utility guarantees, e.g., (Bassily et al., 2014; Kifer et al., 2012; Feldman et al., 2018). Yet, Yu et al. (2019a) show that DP-SGD can achieve higher utility than many of these approaches, both asymptotically and empirically.
+
+Here, we take a closer look at the “Privacy Amplification by Iteration” work of (Feldman et al., 2018). Feldman et al. (2018) observe that DP-SGD guarantees differential privacy for every gradient update step. Under the assumption that intermediate model updates can be hidden from the adversary, they propose a different analysis of DP-SGD for convex optimization problems that has a number of conceptual advantages. First, the algorithm of Feldman et al. (2018) does not require the training indices selected for each batch $B _ { t }$ do be hidden from the adversary. Second, their approach can support much smaller privacy budgets than DP-SGD.
+
+However, we show that these benefits come at a cost in practice: for the range of privacy budgets we consider in this work, DP-SGD requires adding less noise than Privacy Amplification by Iteration (PAI). To compare the two approaches, we proceed as follows: We analytically compute the noise scale $\sigma$ that results in a privacy guarantee of $( \varepsilon , \delta = 1 0 ^ { - 5 } )$ ) after 10 training epochs with a batch sampling rate of $5 1 2 / 5 0 0 0 0$ .12 Figure 9 shows that DP-SGD requires adding less noise, except for large privacy budgets $\varepsilon > 4 0$ ), or very small ones $\left( \varepsilon < 0 . 2 \right)$ . In the latter case, both algorithms require adding excessively large amounts of noise. We observe a qualitatively similar behavior for other sampling rates.
+
+For completeness, we evaluate the PAI algorithm of Feldman et al. (2018) for training linear ScatterNet classifiers on CIFAR-10. We evaluate a broader range of hyper-parameters, including different clipping thresholds $C \in \{ 0 . 1 , 1 , 1 0 \}$ (PAI clips the data rather than the gradients), a wider range of batch sizes $B \in \{ 3 2 , 6 4 , \dotsc , 2 0 4 8 \}$ , and a wider range of base learning rates $\eta \in \{ 2 ^ { - 3 } , 2 ^ { - 2 } , . . . , 2 ^ { 3 } \}$ We find that for privacy budgets $1 \leq \varepsilon \leq 3$ , the optimal hyper-parameters for PAI and DP-SGD are similar, but the analysis of PAI requires a larger noise scale $\sigma$ . As a result, PAI performs worse than DP-SGD, as shown in Figure 10.
+
+![](images/d73462dc4094fd3aa7b439ace75f575cf49d4c27a1a858587b9f4e7558b46f56.jpg)  
+Figure 9: Gradient noise scale $\sigma$ required for a privacy guarantee of $( \varepsilon , \delta = 1 0 ^ { - 5 } )$ after 10 training epochs with batch sampling rate $5 1 2 / 5 0 0 0 0$ Privacy Amplification by Iteration (PAI) (Feldman et al., 2018) requires less noise than DPSGD only for very small or very large privacy budgets.
+
+![](images/e37dedc48513878f576eed38e1b0c0b6c677a0856b9629bfa41af878f10da8e0.jpg)  
+Figure 10: Comparison of DP-SGD (Abadi et al., 2016) and Privacy Amplification by Iteration (PAI) (Feldman et al., 2018) for training a private linear ScatterNet classifier on CIFAR-10. Shows the maximum accuracy achieved for each privacy budget, averaged over five runs.
+
+# D.4 DP-SGD WITH POISSON SAMPLING
+
+The analysis of DP-SGD (Abadi et al., 2016; Mironov et al., 2019) assumes that each batch $\scriptstyle { B _ { t } }$ is created by independently selecting each training sample with probability $B / _ { N }$ . This is in contrast to typical implementations of SGD, where the training data is randomly shuffled once per epoch, and divided into successive batches of size exactly $B$ . The latter “random shuffle” approach has been used in most implementations of DP-SGD (e.g., (tensorflow/privacy, 2019; pytorch/opacus, 2020)) as well as in prior work (e.g., (Abadi et al., 2016; Papernot et al., 2020b)), with the (implicit) assumption that this difference in batch sampling strategies will not affect model performance. We verify that this assumption is indeed valid in our setting. We re-train the linear ScatterNet and end-to-end CNN models that achieved the highest accuracy for a DP budget of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 } )$ ) (with the hyper-parameters detailed in Table 13), using the correct “Poisson sampling” strategy. The test accuracy of these models (averaged over five runs) are shown in Table 19. For all datasets and models, the two sampling schemes achieve similar accuracy when averaged over five runs.
+
+Table 19: Comparison of DP-SGD with two different batch sampling schemes: (1) Poisson sampling, where a batch is formed by selecting each data point independently with probability $B / _ { N }$ ; (2) Random shuffle, where the training set is randomly shuffled at the beginning of each epoch, and split into consecutive batches of size $B$ . For both sampling schemes, we report the best test accuracy (in $\%$ ) at a DP budget of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 } )$ ), with means and standard deviations over five runs.   
+
+<table><tr><td></td><td colspan="2">ScatterNet</td><td colspan="2">CNN</td></tr><tr><td>Dataset</td><td>Poisson Sampling</td><td>Random Shuffle</td><td>Poisson Sampling</td><td>Random Shuffle</td></tr><tr><td>MNIST</td><td>98.6 ± 0.1</td><td>98.7±0.0</td><td>98.0±0.1</td><td>98.1±0.0</td></tr><tr><td>Fashion-MNIST</td><td>89.6 ± 0.1</td><td>89.7 ± 0.0</td><td>86.1 ± 0.2</td><td>86.0± 0.1</td></tr><tr><td>CIFAR-10</td><td>66.8 ± 0.2</td><td>67.0± 0.0</td><td>59.0± 0.4</td><td>59.2 ± 0.1</td></tr></table>
+
+# D.5 EXPERIMENTS WITH SMALLER END-TO-END CNN MODEL ON CIFAR-10
+
+In Section 4, we investigate whether the dimensionality of different classifiers has a noticeable impact on their privacy-utility tradeoffs. To this end, we repeat the CIFAR-10 experiments from
+
+Section 3 with a smaller end-to-end CNN architecture. Specifically, we take the end-to-end CNN architecture from Table 7 and reduce the number of filters in each convolutional layer by a factor of two and remove the last convolutional layer). This results in a CNN model with a comparable number of trainable parameters as the linear ScatterNet classifier (see Table 4). In Table 20, we compare the privacy-utility of this smaller CNN models with the original larger CNN model evaluated in Section 3. While the change of model architecture does affect the model accuracy, the effect is minor, and the accuracy remains far below that of the ScatterNet classifiers with a comparable number of parameters.
+
+Table 20: Best test accuracy $( \mathrm { i n ~ } \% )$ ) for two different model sizes on CIFAR-10 for a DP budget of $( \varepsilon = 3 , \delta = 1 0 ^ { - 5 } \mathrm { . }$ ). We compare two variants of the end-to-end CNN architecture from Table 7, with respectively 551K and 168K parameters. Average and standard deviation computed over five runs.   
+
+<table><tr><td>Model</td><td>Parameters</td><td>Accuracy</td></tr><tr><td>CNN</td><td>168K</td><td>60.7 ± 0.3</td></tr><tr><td></td><td>551K</td><td>59.2 ±0.1</td></tr></table>
+
+# D.6 MODEL CONVERGENCE SPEED ON MNIST AND FASHION-MNIST
+
+We run the same experiment as in Figure 3 for MNIST and Fashion-MNIST, to compare the convergence rate of different classifiers with and without privacy, for different learning rates. The experimental setup is described in Appendix C.6. Figure 11 shows qualitatively similar results as Figure 3: with a high learning rate, all models converge quickly when trained without gradient noise, but the addition of noise is detrimental to the learning process. In contrast, with a much lower learning rate the training curves for DP-SGD are nearly identical, whether we add noise or not. In this regime, the ScatterNet classifiers converge significantly faster than end-to-end CNNs when trained without privacy.
+
+![](images/6642bca072d43fce08c54ccb3b96895df7cb085967be14a190d64125d1ef4a32.jpg)  
+Figure 11: Comparison of convergence rates of linear classifiers fine-tuned on ScatterNet features, CNNs fine-tuned on ScatterNet features), and end-to-end CNNs with and without noise addition in DP-SGD. (Left): low learning rate. (Right): high learning rate. See Figure 3 for results on CIFAR-10.