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+# DATA VALUATION USING REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Quantifying the value of data is a fundamental problem in machine learning. Data valuation has multiple important use cases: (1) building insights about the learning task, (2) domain adaptation, (3) corrupted sample discovery, and (4) robust learning. To adaptively learn data values jointly with the target task predictor model, we propose a meta learning framework which we name Data Valuation using Reinforcement Learning (DVRL). We employ a data value estimator (modeled by a deep neural network) to learn how likely each datum is used in training of the predictor model. We train the data value estimator using a reinforcement signal of the reward obtained on a small validation set that reflects performance on the target task. We demonstrate that DVRL yields superior data value estimates compared to alternative methods across different types of datasets and in a diverse set of application scenarios. The corrupted sample discovery performance of DVRL is close to optimal in many regimes (i.e. as if the noisy samples were known apriori), and for domain adaptation and robust learning DVRL significantly outperforms state-of-the-art by $1 4 . 6 \%$ and $1 0 . 8 \%$ , respectively.
+
+# 1 INTRODUCTION
+
+Data is an essential ingredient in machine learning. Machine learning models are well-known to improve when trained on large-scale and high-quality datasets (Hestness et al., 2017; Najafabadi et al., 2015). However, collecting such large-scale and high-quality datasets is costly and challenging. One needs to determine the samples that are most useful for the target task and then label them correctly. Recent work (Toneva et al., 2019) suggests that not all samples are equally useful for training, particularly in the case of deep neural networks. In some cases, similar or even higher test performance can be obtained by removing a significant portion of training data, i.e. low-quality or noisy data may be harmful (Ferdowsi et al., 2013; Frenay & Verleysen, 2014). There are also some scenarios where train-test mismatch cannot be avoided because the training dataset only exists for a different domain. Different methods (Ngiam et al., 2018; Zhu et al., 2019) have demonstrated the importance of carefully selecting the most relevant samples to minimize this mismatch.
+
+Accurately quantifying the value of data has a great potential for improving model performance for real-world training datasets which commonly contain incorrect labels, and where the input samples differ in relatedness, sample quality, and usefulness for the target task. Instead of treating all data samples equally, lower priority can be assigned for a datum to obtain a higher performance model – for example in the following scenarios:
+
+1. Incorrect label (e.g. human labeling errors).   
+2. Input comes from a different distribution (e.g. different location or time).   
+3. Input is noisy or low quality (e.g. noisy capturing hardware).   
+4. Usefulness for target task (label is very common in the training dataset but not as common in the   
+testing dataset).
+
+In addition to improving performance in such scenarios, data valuation also enables many new use cases. It can suggest better practices for data collection, e.g. what kinds of additional data would the model benefit the most from. For organizations that sell data, it determines the correct value-based pricing of data subsets. Finally, it enables new possibilities for constructing very large-scale training datasets in a much cheaper way, e.g. by searching the Internet using the labels and filtering away less valuable data.
+
+How does one evaluate the value of a single datum? This is a crucial and challenging question. It is straightforward to address at the full dataset granularity: one could naively train a model on the entire dataset and use its prediction performance as the value. However, evaluating the value of each datum is far more difficult, especially for complex models such as deep neural networks that are trained on large-scale datasets. In this paper, we propose a meta learning-based data valuation method which we name Data Valuation using Reinforcement Learning (DVRL). Our method integrates data valuation with the training of the target task predictor model. DVRL determines a reward by quantifying the performance on a small validation set, and uses it as a reinforcement signal to learn the likelihood of each datum being using in training of the predictor model. In a wide range of use cases, including domain adaptation, corrupted sample discovery and robust learning, we demonstrate significant improvements compared to permutation-based strategies (such as Leave-one-out and Influence Function (Koh & Liang, 2017)) and game theory-based strategies (such as Data Shapley (Ghorbani & Zou, 2019)). The main contributions can be summarized as follows:
+
+1. We propose a novel meta learning framework for data valuation that is jointly optimized with the target task predictor model.   
+2. We demonstrate multiple use cases of the proposed data valuation framework and show DVRL significantly outperforms competing methods on many image, tabular and language datasets.   
+3. Unlike previous methods, DVRL is scalable to large datasets and complex models, and its computational complexity is not directly dependent on the size of the training set.
+
+# 2 RELATED WORK
+
+Data valuation: A commonly-used method for data valuation is leave-one-out (LOO). It quantifies the performance difference when a specific sample is removed and assigns it as that sample’s data value. The computational cost is a major concern for LOO – it scales linearly with the number of training samples which means its cost becomes prohibitively high for large-scale datasets and complex models. In addition, there are fundamental limitations in the approximation. For example, if there are two exactly equivalent samples, LOO underestimates the value of that sample even though that sample may be very important. The method of Influence Function (Koh & Liang, 2017) was proposed to approximate LOO in a computationally-efficient manner. It uses the gradient of the loss function with small perturbations to estimate the data value. In order to compute the gradient, Hessian values are needed; however these are prohibitively expensive to compute for neural networks. Approximations for Hessian computations are possible, although they generally result in performance limitations. From data quality assessment perspective, the method of Influence Function inherits the major limitations of LOO.
+
+Data Shapley (Ghorbani & Zou, 2019) is another relevant work. Shapley values are motivated by game theory (Shapley, 1953) and are commonly used in feature attribution problems such as relating predictions to input features (Lundberg & Lee, 2017). For Data Shapley, the prediction performance of all possible subsets is considered and the marginal performance improvement is used as the data value. The computational complexity for computing the exact Shapley value is exponential with the number of samples. Therefore, Monte Carlo sampling and gradient-based estimation are used to approximate them. However, even with these approximations, the computational complexity still remains high (indeed much higher than LOO) due to re-training for each test combination. In addition, the approximations may result in fundamental limitations in data valuation performance – e.g. with Monte Carlo approximation, the ratio of tested combinations compared to all possible combinations decreases exponentially. Moreover, in all the aforementioned methods data valuation is decoupled from predictor model training, which limits the performance due to lack of joint optimization.
+
+Meta learning-based adaptive learning: There are relevant studies that utilize meta learning for adaptive weight assignment while training for various use cases such as robust learning, domain adaptation, and corrupted sample discovery. ChoiceNet (Choi et al., 2018) explicitly models output distributions and uses the correlations of the output to improve robustness. Xue et al. (2019) estimates uncertainty of predictions to identify the corrupted samples. Li et al. (2019) combines meta learning with standard stochastic gradient update with generated synthetic noise for robust learning. Shen & Sanghavi (2019) alternates the processes of selecting the samples with low loss and model training to improve robustness. Shu et al. (2019) uses neural networks to model the relationship between current loss and the corresponding sample weights, and utilizes a meta-learning framework for robust weight assignment. Kohler et al. (2019) estimates the uncertainty to discover ¨ the noisy labeled data and relabels mislabeled samples to improve the prediction performance of the predictor model. Gold Loss Correction (Hendrycks et al., 2018) uses a clean validation set to recover the label corruption matrix to re-train the predictor model with corrected labels. Learning to Reweight (Ren et al., 2018) proposes a single gradient descent step guided with validation set performance to reweight the training batch. Domain Adaptive Transfer Learning (Ngiam et al., 2018) introduces importance weights (based on the prior label distribution match) to scale the training samples for transfer learning. MentorNet (Jiang et al., 2018) proposes a curriculum learning framework that learns the order of mini-batch for training of the corresponding predictor model. Our method, DVRL, differs from the aforementioned as it directly models the value of the data using learnable neural networks (which we refer to as a data value estimator). To train the data value estimator, we use reinforcement learning with a sampling process. DVRL is model-agnostic and even applicable to non-differentiable target objectives. Learning is jointly performed for the data value estimator and the corresponding predictor model, yielding superior results in all of the use cases we consider.
+
+# 3 PROPOSED METHOD
+
+![](images/908f2f0c09303cd8e1ea3c6705eb50f53032698bd59aa2f781dea582fd5315b4.jpg)  
+Figure 1: Block diagram of the DVRL framework for training. A batch of training samples is used as the input to the data value estimator (with shared parameters across the batch) and the output corresponds to selection probabilities: $w _ { i } = h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } )$ of a multinomial distribution. The sampler, based on this multinomial distribution, returns the selection vector $\mathbf { s } = ( s _ { 1 } , . . . , s _ { B _ { s } } )$ where $s _ { i } \in \{ 0 , 1 \}$ and $P ( s _ { i } = 1 ) = w _ { i }$ . The target task predictor model is trained only using the samples with selection vector $s _ { i } = 1$ , using conventional gradient-descent optimization. The selection probabilities $w _ { i }$ rank the samples according to their importance – these are used as data values. The loss of the predictor model is evaluated on a small validation set, which is compared to the moving average of the previous losses $( \delta )$ to determine the reward. Finally, the reinforcement signal guided by this reward updates the data value estimator. Block diagrams for inference are shown in Appendix A.
+
+Framework: Let us denote the training dataset as $\mathcal { D } = \{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N } \sim \mathcal { P }$ where $\mathbf { x } _ { i } \in \mathcal X$ is a feature vector in the $d$ -dimensional feature space $\mathcal { X }$ , e.g. $\mathbb { R } ^ { d }$ , and $y _ { i } \in \mathcal { V }$ is a corresponding label in the label space $\mathcal { V }$ , e.g. $\Delta [ 0 , 1 ] ^ { c }$ for classification where $c$ is the number of classes and $\Delta$ is the simplex. We consider a disjoint testing dataset $\mathcal { D } ^ { t } = \{ ( \mathbf { x } _ { j } ^ { t } , y _ { j } ^ { t } ) \} _ { j = 1 } ^ { M } \sim \mathcal { P } ^ { t }$ where the target distribution $\mathcal { P } ^ { t }$ does not need to be the same with the training distribution $\mathcal { P }$ . We assume an availability of a (often small1) validation dataset $\mathcal { D } ^ { v } = \{ ( \mathbf { x } _ { k } ^ { v } , y _ { k } ^ { v } ) \} _ { k = 1 } ^ { L ^ { - } } \sim \mathcal { P } ^ { t }$ that comes from the target distribution $\mathcal { P } ^ { t }$ .
+
+The DVRL method (overview in Fig. 1) consists of two learnable functions: (1) the target task predictor model $f _ { \theta }$ , (2) data value estimator model $h _ { \phi }$ . The predictor model $f _ { \theta } : \mathcal { X }  \mathcal { Y }$ is trained to minimize a certain weighted loss function $\mathcal { L } _ { f }$ (e.g. Mean Squared Error (MSE) for regression or cross entropy for classification) on training set $\mathcal { D }$ :
+
+$$
+f _ { \theta } = \arg \operatorname* { m i n } _ { \hat { f } \in \mathcal { F } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) \cdot \mathcal { L } _ { f } ( \hat { f } ( \mathbf { x } _ { i } ) , y _ { i } ) .
+$$
+
+$f _ { \theta }$ can be any trainable function with parameters $\theta$ , such as a neural network. The data value estimator model $h _ { \phi } : \mathcal { X } \cdot \mathcal { Y }  [ 0 , 1 ]$ , on the other hand, is optimized to output weights that determine
+
+the distribution of selection likelihood of the samples to train the predictor model $f _ { \theta }$ . We formulate the corresponding optimization problem as:
+
+$$
+\begin{array} { r l } { \underset { h _ { \phi } } { \operatorname* { m i n } } } & { \mathbb { E } _ { ( \mathbf { x } ^ { v } , y ^ { v } ) \sim P ^ { t } } \Big [ \mathcal { L } _ { h } \big ( f _ { \theta } \big ( \mathbf { x } ^ { v } \big ) , y ^ { v } \big ) \Big ] } \\ { \mathrm { s . t . } } & { f _ { \theta } = \arg \operatorname* { m i n } _ { \hat { f } \in \mathcal { F } } \mathbb { E } _ { ( \mathbf { x } , y ) \sim P } \Big [ h _ { \phi } \big ( \mathbf { x } , y \big ) \cdot \mathcal { L } _ { f } \big ( \hat { f } ( \mathbf { x } ) , y \big ) \Big ] } \end{array}
+$$
+
+where $h _ { \phi } ( \mathbf { x } , y )$ represents value of the training sample $\left( \mathbf { x } , y \right)$ . The data value estimator is also a trainable function, such as a neural network. Similar to $\mathcal { L } _ { f }$ , we use MSE or cross entropy for $\mathcal { L } _ { h }$ .
+
+Training: To encourage exploration based on uncertainty, we model training sample selection stochastically. Let $w = h _ { \phi } ( \mathbf { x } , y )$ denote the probability that $\left( \mathbf { x } , y \right)$ is used to train the predictor model $f _ { \theta }$ $\mathbf { \varepsilon } _ { \cdot } \ h _ { \phi } ( \mathcal { D } ) = \{ h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ is the probability distribution for inclusion of each training sample. $\mathbf { s } \in \{ 0 , 1 \} ^ { N }$ is a binary vector that represents the selected samples. If $s _ { i } = 1 / 0$ , $\left( \mathbf { x } _ { i } , y _ { i } \right)$ is selected/not selected for training the predictor model. $\begin{array} { r } { \pi _ { \phi } ( \mathcal { D } , \mathbf { s } ) \ = \ \prod _ { i = 1 } ^ { N } \big [ h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) ^ { s _ { i } } \cdot ( 1 - } \end{array}$ $h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) ) ^ { 1 - s _ { i } } ]$ is the probability that certain selection vector s is selected based on $h _ { \phi } ( \mathcal { D } )$ . We assign the outputs of the data value estimator model, $w = h _ { \phi } ( \mathbf { x } , y )$ , as the data values. We can use the data values to rank the dataset samples (e.g. to determine a subset of the training dataset) and to do sample-adaptive training (e.g. for domain adaptation).
+
+The predictor model can be trained using standard stochastic gradient descent because it is differentiable with respect to the input. However, gradient descent-based optimization cannot be used for the data value estimator because the sampling process is non-differentiable. There are multiple ways to handle the non-differentiable optimization bottleneck, such as Gumbel-softmax (Jang et al., 2017) or stochastic back-propagation (Rezende et al., 2014). In this paper, we consider reinforcement learning instead, which directly encourages exploration of the policy towards the optimal solution of Eq. (2). We use the REINFORCE algorithm (Williams, 1992) to optimize the policy gradients, with the rewards obtained from a small validation set that approximates performance on the target task. For the loss function $\hat { l } ( \phi )$ :
+
+$$
+\begin{array} { r l } & { \hat { l } ( \phi ) = \mathbb { E } _ { ( \mathbf { x } ^ { v } , y ^ { v } ) \sim P ^ { t } } \Big [ \mathbb { E } _ { s \sim \pi _ { \phi } ( \mathcal { D } , \cdot ) } \big [ \mathcal { L } _ { h } ( f _ { \theta } ( \mathbf { x } ^ { v } ) , y ^ { v } ) \big ] \Big ] } \\ & { \quad \quad = \displaystyle \int P ^ { t } ( \mathbf { x } ^ { v } ) \Big [ \sum _ { s \in [ 0 , 1 ] ^ { N } } \pi _ { \phi } ( \mathcal { D } , \mathbf { s } ) \cdot \big [ \mathcal { L } _ { h } ( f _ { \theta } ( \mathbf { x } ^ { v } ) , y ^ { v } ) \big ] \Big ] d \mathbf { x } ^ { v } , } \end{array}
+$$
+
+we directly compute the gradient $\nabla _ { \phi } \hat { l } ( \phi )$ as:
+
+$$
+\begin{array} { l } { { \displaystyle \nabla _ { \phi } \hat { l } ( \phi ) = \int P ^ { t } ( { \bf x } ^ { v } ) \Big [ \sum _ { s \in [ 0 , 1 ] ^ { N } } \nabla _ { \phi } \pi _ { \phi } ( \mathcal { D } , { \bf s } ) \cdot \big [ \mathcal { L } _ { h } \big ( f _ { \theta } \big ( { \bf x } ^ { v } \big ) , y ^ { v } \big ) \big ] \Big ] d { \bf x } ^ { v } } \ ~ } \\ { { \displaystyle ~ = \int P ^ { t } ( { \bf x } ^ { v } ) \Big [ \sum _ { s \in [ 0 , 1 ] ^ { N } } \nabla _ { \phi } \log \big ( \pi _ { \phi } ( \mathcal { D } , { \bf s } ) \big ) \cdot \pi _ { \phi } \big ( \mathcal { D } , { \bf s } \big ) \cdot \big [ \mathcal { L } _ { h } \big ( f _ { \theta } \big ( { \bf x } ^ { v } \big ) , y ^ { v } \big ) \big ] \Big ] d { \bf x } ^ { v } } \ ~ } \\ { { \displaystyle ~ = \mathbb { E } _ { ( { \bf x } ^ { v } , y ^ { v } ) \sim P ^ { t } } \Big [ \mathbb { E } _ { { \bf s } \sim \pi _ { \phi } ( \mathcal { D } , \cdot ) } \big [ \mathcal { L } _ { h } \big ( f _ { \theta } \big ( { \bf x } ^ { v } \big ) , y ^ { v } \big ) \big ] \nabla _ { \phi } \log \big ( \pi _ { \phi } ( \mathcal { D } , { \bf s } ) \big ) \Big ] , } } \end{array}
+$$
+
+where $\nabla _ { \phi } \log ( \pi _ { \phi } ( \mathcal { D } , \mathbf { s } ) )$ is
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \phi } \log ( \pi _ { \phi } ( \mathcal { D } , \mathbf { s } ) ) = \nabla _ { \phi } \sum _ { i = 1 } ^ { N } \log \Big [ h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) ^ { s _ { i } } \cdot ( 1 - h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) ) ^ { 1 - s _ { i } } \Big ] } \\ { \displaystyle = \sum _ { i = 1 } ^ { N } s _ { i } \nabla _ { \phi } \log \big [ h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) \big ] + ( 1 - s _ { i } ) \nabla _ { \phi } \log \big [ ( 1 - h _ { \phi } ( \mathbf { x } _ { i } , y _ { i } ) ) \big ] . } \end{array}
+$$
+
+To improve the stability of the training, we use the moving average of the previous loss $( \delta )$ , with a window size $( T )$ , as the baseline for the current loss. The pseudo-code is shown in Algorithm 1.
+
+Computational complexity: DVRL models the mapping between an input and its value with a learnable function. The training time of DVRL is not directly proportional to the dataset size, but rather dominated by the required number of iterations and per-iteration complexity in Algorithm 1. One way to minimize the computational overhead is to use pre-trained models to initialize the
+
+# Algorithm 1 Pseudo-code of DVRL training
+
+1: Inputs: Learning rates $\alpha , \beta > 0$ , mini-batch size $B _ { p } , B _ { s } > 0$ , inner iteration count $N _ { I } > 0$ , moving average window $T > 0$ , training dataset $\mathcal { D }$ , validation dataset $\mathcal { D } ^ { v } = \{ ( \mathbf { x } _ { k } ^ { v } , y _ { k } ^ { v } ) \} _ { k = 1 } ^ { L }$
+
+2: Initialize parameters $\theta , \phi$ , moving average $\delta = 0$
+
+3: while until convergence do
+
+4: Sample a mini-batch from the entire training dataset: $\mathcal { D } _ { B } = ( \mathbf { x } _ { j } , y _ { j } ) _ { j = 1 } ^ { B _ { s } } \sim \mathcal { D }$   
+5: for $j = 1 , . . . , B _ { s }$ do   
+6: Calculate selection probabilities: $w _ { j } = h _ { \phi } ( \mathbf { x } _ { j } , y _ { j } )$   
+7: Sample selection vector: $s _ { j } \sim B e r ( w _ { j } )$
+
+$$
+\theta  \theta - \alpha \frac { 1 } { B _ { p } } \sum _ { m = 1 } ^ { B _ { p } } \tilde { s } _ { m } \cdot \nabla _ { \theta } \mathcal { L } _ { f } ( f _ { \theta } ( \tilde { \mathbf { x } } _ { m } ) , \tilde { y } _ { m } ) )
+$$
+
+11: Update the data value estimator model network parameters $\phi$
+
+$$
+\phi  \phi - \beta \Big [ \frac { 1 } { L } \sum _ { k = 1 } ^ { L } [ \mathcal { L } _ { h } ( f _ { \theta } ( \mathbf { x } _ { k } ^ { v } ) , y _ { k } ^ { v } ) ] - \delta \Big ] \nabla _ { \phi } \log \pi _ { \phi } ( \mathcal { D } _ { B } , ( s _ { 1 } , . . . , s _ { B _ { s } } ) )
+$$
+
+12: Update the moving average baseline $( \delta )$ : $\begin{array} { r } { \delta \gets \frac { T - 1 } { T } \delta + \frac { 1 } { L T } \sum _ { k = 1 } ^ { L } [ \mathcal { L } _ { h } ( f _ { \theta } ( \mathbf { x } _ { k } ^ { v } ) , y _ { k } ^ { v } ) ] } \end{array}$ predictor networks at each iteration. Unlike alternative methods like Data Shapley, we demonstrate the scalability of DVRL to large-scale datasets such as CIFAR-100, and complex models such as ResNet-32 (He et al., 2016) and WideResNet-28-10 (Zagoruyko & Komodakis, 2016). Instead of being exponential in terms of the dataset size, the training time overhead DVRL is only twice of conventional training. Please see Appendix D for further analysis on learning dynamics of DVRL and Appendix B for additional computational complexity discussions.
+
+# 4 EXPERIMENTS
+
+We evaluate data value estimation quality of DVRL on multiple types of dataset and use cases.
+
+Benchmark methods: We consider the following benchmarks: (1) Randomly-assigned values (Random), (2) Leave-one-out (LOO), (3) Data Shapley Value (Data Shapley) (Ghorbani & Zou, 2019). For some experiments, we also compare with (4) Learning to Reweight (Ren et al., 2018), (5) MentorNet (Jiang et al., 2018), and (6) Influence Function (Koh & Liang, 2017).
+
+Datasets: We consider 12 public datasets (3 public tabular datasets, 7 public image datasets, and 2 public language datasets) to evaluate DVRL in comparison to multiple benchmark methods. 3 public tabular datasets are (1) Blog, (2) Adult, (3) Rossmann; 7 public image datasets are (4) HAM 10000, (5) MNIST, (6) USPS, (7) Flower, (8) Fashion-MNIST, (9) CIFAR-10, (10) CIFAR-100; 2 public language datasets are (11) Email Spam, (12) SMS Spam. Details can be found in the hyper-links.
+
+Baseline predictor models: We consider various machine learning models as the baseline predictor model to highlight the proposed model-agnostic data valuation framework. For Adult and Blog datasets, we use LightGBM (Ke et al., 2017), and for Rossmann dataset, we use XGBoost and multi-layer perceptrons due to their superior performance on the tabular datasets. For Flower, HAM 10000, and CIFAR-10 datasets, we use Inception-v3 with top-layer fine-tuning (pre-trained on ImageNet, (Szegedy et al., 2016)) as the baseline predictor model. For Fashion-MNIST, MNIST, and USPS datasets, we use multinomial logistic regression, and for Email and SMS datasets, we use Naive Bayes model. We also use ResNet-32 (He et al., 2016) and WideResNet-28-10 (Zagoruyko & Komodakis, 2016) as the baseline models for CIFAR-10 and CIFAR-100 datasets in Section 4.3 to demonstrate the scalability of DVRL. For data value estimation network, we use multi-layer perceptrons with ReLU activation as the base architecture. The number of layers and hidden units are optimized with cross-validation.
+
+Experimental details: In all experiments, we use Standard Normalizer to normalize the entire features to have zero mean and one standard deviation. We transform categorical variables into one-hot encoded embeddings. We set the inner iteration count $\scriptstyle N _ { I } = 2 0 0$ ) for the predictor network, moving average window $( T { = } 2 0 )$ , and mini-batch size $( B _ { p } { = } 2 5 6 )$ for the predictor network and mini-batch size ( ${ B _ { s } } { = } 2 0 0 0 )$ for the DVE network (large batch size often improves the stability of the reinforcement learning model training (McCandlish et al., 2018)). We set the learning rate to 0.01 $( \beta )$ for the data value estimator (DVE) and 0.001 $( \alpha )$ for the predictor network. As the DVE architecture, for tabular datasets, we use 5-layer perceptrons with 100 hidden units and ReLU; and for image datasets, we use 5-layer perceptrons with 100 hidden units and ReLU on top of the CNN-based architecture used for the predictor network (such as ResNet-32 or WideResNet-28-10 in Table 1). In order to provide further informative signal to DVE, we propose to use an additional input of the difference between the predictions of a separate predictive model (fined-tuned or trained from scratch on the validation set) for the training samples and the original training labels. We simply concatenate this additional input to the hidden states of DVE network. Intuitively, if the training label is corrupted, the additional input would be high; thus, this could be an important signal for DVE to assign low value to this sample. Ablation study for the variants of DVRL can be found in the Appendix C.6.
+
+# 4.1 REMOVING HIGH/LOW VALUE SAMPLES
+
+Removing low value samples from the training dataset can improve the predictor model performance, especially in the cases where the training dataset contains corrupted samples. On the other hand, removing high value samples, especially if the dataset is small, would decrease the performance significantly. Overall, the performance after removing high/low value samples is a strong indicator for the quality of data valuation.
+
+Initially, we consider the conventional supervised learning setting, where all training, validation and testing datasets come from the same distribution (without sample corruption or domain mismatch). We use two tabular datasets (Adult and Blog) with 1,000 training samples and one image dataset (Flower) with 2,000 training samples.2 We use 400 validation samples for tabular datasets and 800 validation samples for the image dataset. Then, we report the prediction performance on the disjoint testing set after removing the high/low value samples based on the estimated data values.
+
+![](images/db8745bdf9b0728a33cebe06f1dd1ed4168ec60781696cccb8d9a7de73be3711.jpg)  
+Figure 2: Performance after removing the most (marked with $\times$ ) and least (marked with $\bigcirc$ ) important samples according to the estimated data values in a conventional supervised learning setting.
+
+As shown in Fig. 2, even in the absence of sample corruption or domain mismatch, DVRL can marginally improve the prediction performance after removing some portion of the least important samples. Using only ${ \sim } 6 0 \% { - } 7 0 \%$ of the training set (the highest valued samples), DVRL can obtain a similar performance compared to training on the entire dataset. After removing a small portion $( 1 0 \% - 2 0 \% )$ of the most important samples, the prediction performance significantly degrades which indicates the importance of the high valued samples. Qualitatively looking at these samples, we observe them to typically be representative of the target task which can be insightful. Overall, DVRL shows the fastest performance degradation after removing the most important samples and the slowest performance degradation after removing the least important samples in most cases, underlining the superiority of DVRL in data valuation quality compared to competing methods.
+
+Next, we focus on the setting of removing high/low value samples in the presence of label noise in the training data. We consider three image datasets: Fashion-MNIST, HAM 10000, and CIFAR10. As noisy samples hurt the performance of the predictor model, an optimal data value estimator with a clean validation dataset should assign lowest values to the noisy samples. With the removal of samples with noisy labels (‘Least’ setting), the performance should either increase, or at least decrease much slower, compared to removal of samples with correct labels (‘Most’ setting). In this experiment, we introduce label noise to $20 \%$ of the samples by replacing true labels with random labels. As can be seen in Fig. 10, for all data valuation methods the prediction performance tends to first slowly increase and then decrease in the ‘Least’ setting; and tends to rapidly decrease in the ‘Most’ setting. Yet, DVRL achieves the slowest performance decrease in ‘Least’ setting and fastest performance decrease in the ‘Most’ setting, reflecting its superiority in data valuation.
+
+![](images/6ddb537ce81e2f324127f44b1142e26fe6a5bd03d5c118920888ff90d93a98a1.jpg)  
+Figure 3: Prediction performance after removing the most (marked with $\times$ ) and least (marked with $\bigcirc$ ) important samples according to the estimated data values with $20 \%$ noisy label ratio. Additional results on Blog, HAM 10000, and CIFAR-10 datasets can be found in Appendix C.3. The prediction performance is lower than state of the art due to a smaller training set size and the introduced noise.
+
+# 4.2 CORRUPTED SAMPLE DISCOVERY
+
+There are some scenarios where training samples may contain corrupted samples, e.g. due to cheap label collection methods. An automated corrupted sample discovery method would be highly beneficial for distinguishing samples with clean vs. noisy labels. Data valuation can be used in this setting by having a small clean validation set to assign low data values to the potential samples with noisy labels. With an optimal data value estimator, all noisy labels would get the lowest data values.
+
+We consider the same experimental setting with the previous subsection with $20 \%$ noisy label ratio on 6 datasets. Fig. 4 shows that DVRL consistently outperforms all benchmarks (Data Shapley, LOO and Influence Function). The trend of noisy label discovery for DVRL can be very close to optimal (as if we perfectly knew which samples have noisy labels), particularly for the Adult, CIFAR-10 and Flower datasets. To highlight the stability of DVRL, we provide the confidence intervals of DVRL performance on the corrupted sample discovery in Appendix E.
+
+# 4.3 ROBUST LEARNING WITH NOISY LABELS
+
+In this section, we consider how reliably DVRL can learn with noisy data in an end-to-end way, without removing the low-value samples as in the previous section. Ideally, noisy samples should get low data values as DVRL converges and a high performance model can be returned. To compare DVRL with two recently-proposed benchmarks: MentorNet (Jiang et al., 2018) and Learning to Reweight (Ren et al., 2018) for this use case, we focus on two complex deep neural networks as the baseline predictor models, ResNet-32 (He et al., 2016) and WideResNet-28-10 (Zagoruyko &
+
+![](images/92e57a4cbe06629201df341dcc179fec73acb789d6c2e1ad2fcebc4d3bad801b.jpg)  
+Figure 4: Discovering corrupted samples in three datasets with $20 \%$ noisy label ratio. ‘Optimal’ saturates at $20 \%$ , perfectly assigning the lowest data value scores to the samples with noisy labels. ‘Random’ does not introduce any knowledge on distinguishing clean vs. noisy labels, and thus the fraction of discovered corrupt samples is proportional to the amount of inspection. More results on Adult, Fashion-MNIST and Flower datasets are in Appendix C.4.
+
+Komodakis, 2016), trained on CIFAR-10 and CIFAR-100 datasets. Additional results on other image datasets are in Appendix C.1, and results on robust learning with noisy features are in Appendix C.2.
+
+We consider the same experimental setting from Ren et al. (2018) on CIFAR-10 and CIFAR-100 datasets. For the first experiment, we use WideResNet-28-10 as the baseline predictor model and apply $40 \%$ of label noise uniformly across all classes. We use 1,000 clean (noise-free) samples as the validation set and test the performance on the clean testing set. For the second experiment, we use ResNet-32 as the baseline predictor model and apply $40 \%$ background noise (same-class noise to the $40 \%$ of the samples). In this case, we only use 10 clean samples per class as the validation set. We consider five additional benchmarks: (1) Validation Set Only – which only uses clean validation set for training, (2) Baseline – which only uses noisy training set for training, (3) Baseline $^ +$ Finetuning – which is initialized with the trained baseline model on the noisy training set and fine-tuned on the clean validation set, (4) Clean Only $60 \%$ data) – which is trained on the clean training set after removing the training samples with flipped labels, (5) Zero Noise – which uses the original noise-free training set for training ( $100 \%$ clean training data). We exclude Data Shapley and LOO in this experiment due to their prohibitively-high computational complexities.
+
+<table><tr><td>Noise (predictor model)</td><td colspan="2">Uniform (WideResNet-28-10)</td><td colspan="2">Background (ResNet-32)</td></tr><tr><td>Datasets</td><td>CIFAR-10</td><td>CIFAR-100</td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td>Validation Set Only</td><td>46.64± 3.90</td><td>9.94 ± 0.82</td><td>15.90 ± 3.32</td><td>8.06± 0.76</td></tr><tr><td>Baseline</td><td>67.97 ± 0.62</td><td>50.66 ± 0.24</td><td>59.54 ± 2.16</td><td>37.82 ± 0.69</td></tr><tr><td>Baseline + Fine-tuning</td><td>78.66 ± 0.44</td><td>54.52 ± 0.40</td><td>82.82 ± 0.93</td><td>54.23 ± 1.75</td></tr><tr><td>MentorNet + Fine-tuning</td><td>78.00</td><td>59.00</td><td></td><td></td></tr><tr><td>Learning to Reweight</td><td>86.92 ± 0.19</td><td>61.34 ± 2.06</td><td>86.73 ± 0.48</td><td>59.30 ± 0.60</td></tr><tr><td>DVRL</td><td>89.02 ± 0.27</td><td>66.56 ± 1.27</td><td>88.07 ± 0.35</td><td>60.77 ± 0.57</td></tr><tr><td rowspan="2">Clean Only (60% Data) Zero Noise</td><td>94.08 ± 0.23</td><td>74.55 ± 0.53</td><td>90.66 ± 0.27</td><td>63.50 ± 0.33</td></tr><tr><td>95.78 ± 0.21</td><td>78.32 ± 0.45</td><td>92.68 ± 0.22</td><td>68.12 ± 0.21</td></tr></table>
+
+Table 1: Robust learning with noisy labels. Test accuracy for ResNet-32 and WideResNet-28-10 on CIFAR-10 and CIFAR-100 datasets with $40 \%$ of Uniform and Background noise on labels.
+
+As shown in Table 1, DVRL outperforms other robust learning methods in all cases. The performance improvements with DVRL are larger with Uniform noise. Learning to Reweight loses $7 . 1 6 \%$ and $1 3 . 2 1 \%$ accuracy compared to the optimal case (Zero Noise); on the other hand, DVRL only loses $5 . 0 6 \%$ and $7 . 9 9 \%$ accuracy for CIFAR-10 and CIFAR-100 respectively with Uniform noise.
+
+# 4.4 DOMAIN ADAPTATION
+
+In this section, we consider the scenario where the training dataset comes from a substantially different distribution from the validation and testing sets. Naive training methods (i.e. equal treatment of all training samples) often fail in this scenario (Ganin et al., 2016; Glorot et al., 2011). Data valuation is expected to be beneficial for this task by selecting the samples from the training dataset that best match the distribution of the validation dataset.
+
+<table><tr><td>Source</td><td>Target</td><td>Task</td><td>Baseline</td><td>Data Shapley</td><td>DVRL</td></tr><tr><td>Google</td><td>HAM10000</td><td>Skin Lesion Classification</td><td>.296</td><td>.378</td><td>.448</td></tr><tr><td>MNIST</td><td>USPS</td><td>Digit Recognition</td><td>.308</td><td>.391</td><td>.472</td></tr><tr><td>Email</td><td>SMS</td><td>Spam Detection</td><td>.684</td><td>.864</td><td>.903</td></tr></table>
+
+Table 2: Domain adaptation setting showing target accuracy. Baseline represents the predictor model which is naively trained on the training set with equal treatment of all training samples.
+
+We initially focus on the three cases from Ghorbani & Zou (2019), shown in Table 2. (1) uses Google image search results (cheaply collected dataset) to predict skin lesion classification on HAM 10000 data (clean), (2) uses MNIST data to recognize digit on USPS dataset, (3) uses Email spam data to detect spam in an SMS dataset. The experimental settings are exactly the same with Ghorbani & Zou (2019). Table 2 shows that DVRL significantly outperforms Baseline and Data Shapley in all three tasks. One primary reason is that DVRL jointly optimizes the data value estimator and corresponding predictor model; on the other hand, Data Shapley needs a two step processes to construct the predictor model in domain adaptation setting.
+
+Next, we focus on a real-world tabular data learning problem where the domain differences are significant. We consider the sales forecasting problem with the Rossmann Store Sales dataset, which consists of sales data from four different store types. Simple statistical investigation shows a significant discrepancy between the input feature distributions across different store types, meaning there is a large domain mismatch across store types (see Appendix F). To further illustrate distribution difference across the store types, we show the t-SNE analysis on the final layer of a discriminative neural network trained on the entire dataset in Appendix Fig. 11. We consider three different settings: (1) training on all store types (Train on All), (2) training on store types excluding the store type of interest (Train on Rest), and (3) training only on the store type of interest (Train on Specific). In all cases, we evaluate the performance on each store type separately. For example, to evaluate the performance on store type D, Train on All setting uses all four store type datasets for training, Train on Rest setting uses store types A, B and C for training, and Train on Specific setting only uses the store type D for training. Train on Rest is expected to yield the largest domain mismatch between training and testing sets, and Train on Specific yield the minimal.
+
+Table 3: Performance of Baseline and DVRL in 3 different settings with 2 different predictor models on the Rossmann Store Sales dataset. Metric is Root Mean Squared Percentage Error (RMSPE, lower the better). We use $79 \%$ of the data as training, $1 \%$ as validation, and $20 \%$ as testing. DVRL outperforms Baseline in all settings.   
+
+<table><tr><td rowspan="2">Predictor Model (Metric: RMSPE)</td><td rowspan="2">Store Type</td><td colspan="2">Train on All</td><td colspan="2">Train on Rest</td><td colspan="2">Train on Specific</td></tr><tr><td>Baseline</td><td>DVRL</td><td>Baseline</td><td>DVRL</td><td>Baseline</td><td>DVRL</td></tr><tr><td rowspan="4">XGBoost</td><td>A B</td><td>0.1736 0.1996</td><td>0.1594 0.1422</td><td>0.2369</td><td>0.2109</td><td>0.1454</td><td>0.1430</td></tr><tr><td></td><td>0.1839</td><td>0.1502</td><td>0.7716</td><td>0.3607</td><td>0.0880</td><td>0.0824</td></tr><tr><td>C</td><td></td><td></td><td>0.2083</td><td>0.1551</td><td>0.1186</td><td>0.1170</td></tr><tr><td>D</td><td>0.1504</td><td>0.1441</td><td>0.1922</td><td>0.1535</td><td>0.1349</td><td>0.1221</td></tr><tr><td rowspan="4">Neural Networks</td><td>A B</td><td>0.1531</td><td>0.1428</td><td>0.3124</td><td>0.2014</td><td>0.1181</td><td>0.1066</td></tr><tr><td></td><td>0.1529</td><td>0.0979</td><td>0.8072</td><td>0.5461</td><td>0.0683</td><td>0.0682</td></tr><tr><td>C</td><td>0.1620</td><td>0.1437</td><td>0.2153</td><td>0.1804</td><td>0.0682</td><td>0.0677</td></tr><tr><td>D</td><td>0.1459</td><td>0.1295</td><td>0.2625</td><td>0.1624</td><td>0.0759</td><td>0.0708</td></tr></table>
+
+We evaluate the performance of Baseline (train the predictor model without data valuation) and DVRL in 3 different settings with 2 different predictor models (XGBoost (Chen & Guestrin, 2016) and Neural Networks (3-layer perceptrons)). As shown in Table 3, DVRL improves the performance in all settings. The improvements are most significant in Train on Rest setting due to the large domain mismatch. For instance, DVRL reduces the error more than $50 \%$ for store type B predictions with XGBoost in comparison to Baseline. In Train on $A l l$ setting, the performance improvement is still significant, showing that DVRL can distinguish the samples from the target distribution. In Appendix G, we demonstrate that DVRL actually prioritizes selection of the samples from the target store type. In Train on Specific setting, the performance improvements are smaller – even without domain mismatch, DVRL can marginally improve the performance by accurately prioritizing the important samples within the same store type. These results further support the conclusions from Fig. 2 in the conventional supervised learning setting that DVRL learns high quality data value scores. Comparison to other domain adaptation benchmarks can be found in Appendix C.5.
+
+# 4.5 DISCUSSION: HOW MANY VALIDATION SAMPLES ARE NEEDED?
+
+DVRL requires a validation dataset from the target distribution that the testing dataset comes from. Depending on the task, the requirements for the validation dataset may involve being noise-free in labels, being from the same domain, or being high quality. Acquiring such a dataset can be costly in some scenarios and it is desirable to minimize its size requirements.
+
+We analyze the impact of the size of the validation dataset on DVRL with 3 different datasets: Adult, Blog, and Fashion MNIST for the use case of corrupted sample discovery. Similar to Section 4.2, we add $20 \%$ noise to the training samples and try to find the corrupted samples with DVRL. As shown in Fig. 5, DVRL achieves reasonable performance with 100 to 400 validation samples. In the Adult dataset, even 10 validation samples are sufficient to achieve a reasonable data valuation quality. Both of these settings are often realistic in real world scenarios.
+
+![](images/66196ad3e362b32df850a5b6f8f8d16dfb05d291eb0372e823d8a883f3f5e1d1.jpg)  
+Figure 5: Number of validation samples needed for DVRL. Discovering corrupted samples in three datasets (Adult, Blog and Fashion MNIST) with various number of validation samples. X-axis represents the fraction of inspected data and y-axis is the fraction of discovered corrupted samples. On Adult and Fashion-MNIST datasets, DVRL needs $\cdot$ and $\cdot$ of inspected samples to identify $\cdot$ of the corrupted samples respectively - merely $\cdot$ and $4 \%$ more than the optimal cases.
+
+# 5 CONCLUSIONS
+
+In this paper, we propose a meta learning framework, named DVRL, that adaptively learns data values jointly with a target task predictor model. The value of each datum determines how likely it will be used in training of the predictor model. We model this data value estimation task using a deep neural network, which is trained using reinforcement learning with a reward obtained from a small validation set that represents the target task performance. With a small validation set, DVRL can provide computationally highly efficient and high quality ranking of data values for the training dataset that is useful for domain adaptation, corrupted sample discovery and robust learning. We show that DVRL significantly outperforms other techniques for data valuation in various applications on diverse types of datasets.
+
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+
+![](images/048b02cfab2a3f87d1413d3fd4a8c7d59417f9e8fb661aa7840935374ec5613d.jpg)  
+Figure 6: Block diagram of the proposed DVRL framework at inference time. (a) Data valuation, (b) Prediction. For data valuation, the input is a set of samples and the outputs are the corresponding data values. For prediction, the input is a sample and the output is the corresponding prediction. Both the data value estimator and predictor are fixed (not trained) at inference time.
+
+# B COMPUTATIONAL COMPLEXITY
+
+DVRL first trains the baseline model using the entire dataset (without re-weighting). Afterwards, we can use this pre-trained baseline model to initialize the predictor network and apply fine-tuning with DVRL update steps. The convergence of the fine-tuning process is much faster than the convergence of training from the scratch.
+
+We quantify the computational overhead of DVRL on the CIFAR-100 dataset (consisting $5 0 \mathrm { k }$ training samples) with ResNet-32 as a representative example. Overall, DVRL training takes less than 8 hours (given a pre-trained ResNet-32 model on the entire dataset) on a single NVIDIA Tesla V100 GPU without any hardware optimizations. The pre-training time of ResNet-32 on the entire dataset (without re-weighting) is less than 4 hours; thus the total training time of DVRL is less than 12 hours from the scratch. On the other hand, the training time of Data Shapley (the most competitive benchmark) is more than a week on Fashion MNIST (consisting lower dimensional inputs and less number of classes) with a much simpler predictor model architecture (2-layered convolutional neural networks).
+
+At inference, the data value estimator can be used to obtain data value for each sample. The runtime of data valuation is typically much faster (less than 1 ms per sample) than the predictor model (e.g. ResNet-32 model).
+
+# C ADDITIONAL RESULTS
+
+# C.1 ADDITIONAL RESULTS ON ROBUST LEARNING WITH NOISY LABELS
+
+We evaluate how DVRL can provide robustness for learning with noisy labels. We add various levels of label noise, ranging from $0 \%$ to $50 \%$ , to the training sets and evaluate how robust the proposed model (DVRL) is for the noisy dataset. In this experiment, we use three image datasets (CIFAR-10, Flower, and HAM 10000). Note that we initialize the predictor model using pre-trained Inception-v3 networks on ImageNet and only fine-tune the top layer (transfer learning setting).
+
+<table><tr><td rowspan=2 colspan=1>Noiseratio</td><td rowspan=1 colspan=6>CIFAR-10</td><td rowspan=1 colspan=3>Flower</td><td rowspan=1 colspan=3>HAM10000</td></tr><tr><td rowspan=1 colspan=2>Clean</td><td rowspan=1 colspan=1>DVRL</td><td rowspan=1 colspan=3>Baseline</td><td rowspan=1 colspan=1>Clean</td><td rowspan=1 colspan=1>DVRL</td><td rowspan=1 colspan=1>Baseline</td><td rowspan=1 colspan=1>Clean</td><td rowspan=1 colspan=1>DVRL</td><td rowspan=1 colspan=1>Baseline</td></tr><tr><td rowspan=1 colspan=1>0%</td><td rowspan=1 colspan=2>.8297</td><td rowspan=1 colspan=1>.8305</td><td rowspan=1 colspan=3>.8297</td><td rowspan=1 colspan=1>.9090</td><td rowspan=1 colspan=1>.9292</td><td rowspan=1 colspan=1>.9090</td><td rowspan=1 colspan=1>.7129</td><td rowspan=2 colspan=1>.7148.7142</td><td rowspan=5 colspan=1>.7129.6746.6199.5508.4819.4132</td></tr><tr><td rowspan=1 colspan=1>10%</td><td rowspan=1 colspan=2>.8281</td><td rowspan=1 colspan=1>.8306</td><td rowspan=1 colspan=3>.7713</td><td rowspan=1 colspan=1>.9057</td><td rowspan=1 colspan=1>.9158</td><td rowspan=1 colspan=1>.7441</td><td rowspan=1 colspan=1>.7094</td></tr><tr><td rowspan=1 colspan=1>20%</td><td rowspan=1 colspan=2>.8285</td><td rowspan=1 colspan=1>.8271</td><td rowspan=1 colspan=3>.6883</td><td rowspan=1 colspan=1>.9026</td><td rowspan=2 colspan=1>.9152.8901</td><td rowspan=2 colspan=1>.5960.4546</td><td rowspan=2 colspan=1>.7098.7063</td><td rowspan=2 colspan=1>.7126.7005</td></tr><tr><td rowspan=1 colspan=1>30%</td><td rowspan=1 colspan=2>.8283</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>.8262</td><td rowspan=1 colspan=1>.5897</td><td rowspan=1 colspan=1>.5897</td><td rowspan=1 colspan=1>.8889</td></tr><tr><td rowspan=1 colspan=1>40%50%</td><td rowspan=1 colspan=2>.8259.8236</td><td rowspan=1 colspan=1>.8259</td><td rowspan=1 colspan=3>.8255.8225</td><td rowspan=1 colspan=1>.4887.3832</td><td rowspan=1 colspan=1>.8620.8542</td><td rowspan=1 colspan=1>.8787.8678</td><td rowspan=1 colspan=1>.2929.2962</td><td rowspan=1 colspan=1>.7028.7009</td><td rowspan=1 colspan=1>.6968.6814</td></tr></table>
+
+Table 4: Robust learning results with various noise levels on CIFAR-10, Flower, and HAM 10000 datasets. Clean is the performance of the predictor model when it is only trained with the samples with clean labels (e.g. at $20 \%$ noise level, it uses only $80 \%$ clean samples). Baseline is the performance of the predictor model when it is trained with both noisy and clean labels.
+
+Noisy labels significantly degrade the prediction performance when they are included in the training dataset (see the increasing differences between Baseline and Clean in Table 4). DVRL demonstrates high robustness up to high noisy label ratio $( 5 0 \% )$ . In some cases (even without noisy labels (i.e. $0 \%$ noise ratio)), the prediction performance even outperforms the Clean case, as DVRL prioritizes some clean samples more than others. Overall, DVRL framework is promising in maintaining high prediction performance even with a significant increase in the amount of noisy labels.
+
+# C.2 ADDITIONAL RESULTS ON ROBUST LEARNING WITH NOISY FEATURES
+
+In this section, we consider training with noisy input features, with a clean validation set. We independently add Gaussian noise with zero mean and a certain standard deviation of $\sigma$ to each feature in the training set independently. We use two tabular datasets (Adult and Blog) to evaluate the robustness of DVRL on input noise. As can be seen in Table 5, DVRL is robust with noise on the features and the performance gains are higher with larger noise in comparison to Baseline (i.e. treat all the noisy training samples equally), since DVRL can discover the training samples with less corrupted by the additive noise among the entire noisy training samples and provide higher weights on those less noisy samples.
+
+<table><tr><td rowspan="2">0</td><td colspan="2">Blog</td><td colspan="2">Adult</td></tr><tr><td>Baseline</td><td>DVRL</td><td>Baseline</td><td>DVRL</td></tr><tr><td>0.1</td><td>0.733</td><td>0.819</td><td>0.802</td><td>0.820</td></tr><tr><td rowspan="3">0.2 0.3 0.4</td><td>0.647</td><td>0.798</td><td>0.753</td><td>0.788</td></tr><tr><td>0.626</td><td>0.766</td><td>0.699</td><td>0.771</td></tr><tr><td>0.623</td><td>0.717</td><td>0.652</td><td>0.725</td></tr></table>
+
+Table 5: Testing accuracy when trained with noisy features. $\sigma$ is the standard deviation of the added Gaussian noise, quantifying the level of perturbation on the features.
+
+![](images/66bf572846027ff8c32ed0563ba936b815291648fb3675f2840f756bff827a7f.jpg)  
+Figure 7: Prediction performance after removing the most and least important samples, according to the estimated data values. We assume a label noise with $20 \%$ ratio on (a) Blog, (b) HAM 10000, (c) CIFAR-10 datasets.
+
+![](images/61af9634961f353198225f33c31cf44e55edb94d304f1d238af1698d1f5e3aaf.jpg)  
+Figure 8: Discovering corrupted samples in three datasets ((a) Adult, (b) Fashion-MNIST, (c) Flower datasets) in the presence of $20 \%$ noisy labels. ‘Optimal’ saturates at the $20 \%$ of the fraction, perfectly assigning the lowest data value scores to the samples with noisy labels. ‘Random’ does not introduce any knowledge on distinguishing clean vs. noisy labels, and thus the fraction of discovered corrupt samples is proportional to the amount of inspection.
+
+# .5 COMPARISON TO OTHER DOMAIN ADAPTATION BENCHMARKS
+
+In this subsection, we compare DVRL to two established domain adaptation benchmarks: Adversarial Discriminative Domain Adaptation (ADDA) (Tzeng et al., 2017) and Domain Adversarial Neural Networks (DANN) (Ganin et al., 2016). We use the same experimental settings given in Table 3 using Rossmann Store Sales dataset with neural networks as the predictor model. Table 6 represents the domain adaptation results on ‘Train on all’ and ‘Train on Rest’ settings. As can be seen, DVRL yields superior (or similar in a few cases) compared to the two methods, ADDA and DANN, that are specifically designed for domain adaptation.
+
+<table><tr><td>Settings</td><td colspan="4">Train on All</td><td colspan="4">Train on Rest</td></tr><tr><td>Methods</td><td>Baseline</td><td>DVRL</td><td>ADDA</td><td>DANN</td><td>Baseline</td><td>DVRL</td><td>ADDA</td><td>DANN</td></tr><tr><td>A</td><td>0.1531</td><td>0.1428</td><td>0.1465</td><td>0.1491</td><td>0.3124</td><td>0.2014</td><td>0.2119</td><td>0.2305</td></tr><tr><td>B</td><td>0.1529</td><td>0.0979</td><td>0.1193</td><td>0.1201</td><td>0.8071</td><td>0.5461</td><td>0.5444</td><td>0.5898</td></tr><tr><td>C</td><td>0.1620</td><td>0.1437</td><td>0.1503</td><td>0.1589</td><td>0.2153</td><td>0.1804</td><td>0.1871</td><td>0.1963</td></tr><tr><td>D</td><td>0.1459</td><td>0.1295</td><td>0.1351</td><td>0.1388</td><td>0.2625</td><td>0.1624</td><td>0.1910</td><td>0.2061</td></tr></table>
+
+Table 6: Performance of Baseline, DVRL, ADDA, and DANN in train-on-all and train-on-rest settings with neural networks as the predictor model on the Rossmann Store Sales dataset. Metric is Root Mean Squared Percentage Error (RMSPE, lower the better). We use $79 \%$ of the data as training, $1 \%$ as validation, and $20 \%$ as testing.
+
+# C.6 ABLATION STUDIES
+
+In this subsection, we analyze the source of gains for three distinct components of DVRL: (1) discrete representations of data value estimator, (2) baseline for stabilizing the RL training, (3) output of the model trained on the clean validation set as the additional input (validation model). We report the corrupted sample discovery results where the experimental settings are same with Section 4.2.
+
+<table><tr><td>Models /Datasets</td><td>Blog</td><td>HAM-10000</td><td>CIFAR-10</td></tr><tr><td>DVRL</td><td>47.3%</td><td>60.2%</td><td>68.1%</td></tr><tr><td>DVRL without sampler</td><td>44.9%</td><td>58.3%</td><td>63.7%</td></tr><tr><td>DVRL without baseline</td><td>45.8%</td><td>56.6%</td><td>62.9%</td></tr><tr><td>DVRL without validation model</td><td>43.7%</td><td>57.1%</td><td>64.4%</td></tr><tr><td>Validation model only</td><td>43.1%</td><td>55.9%</td><td>62.3%</td></tr></table>
+
+Table 7: Discovering corrupted samples in three datasets with $20 \%$ noisy label ratio. We report the fraction of discovered corrupted samples after inspecting $20 \%$ of the samples with multiple variants of DVRL (the higher the better).
+
+As can be seen in Table 7, each component provides an additional gain in DVRL:
+
+(1) A straightforward idea is to use the raw outputs of DVE to scale the contributions of each sample in the loss term, without using the sampler. Yet, we show the benefit of the discrete representation of DVE for data selection. The sampler encourages exploration of an extremely large action space in a systematic way. This helps DVE and predictor model to converge to a better optimal solution.
+
+(2) Baseline stabilizes the convergence of reinforcement learning; thus, yields higher gains on complex datasets.
+
+(3) The output of the validation model itself has informative signal as it achieves high performance (since it is trained with small-scale but high quality data). We observe that this signal helps DVRL, but even without this signal achieves high performance. We also observe that often a larger DVE model (with more iterations) is needed to estimate the data value in the absence of the informative signal from the validation model.
+
+Note that we propose to use the output of the validation model as an additional input to the data valuation framework; thus, this can also be regarded as another contribution of our work. Also, the output of the validation model is highly informative in the noisy sample discovery use case but not that significant in other applications such as domain adaptation or performance improvement by low value data removal in standard supervised learning setting.
+
+# D LEARNING CURVES OF DVRL
+
+Fig. 9 shows the learning curves of DVRL on the noisy data (with $20 \%$ label noise) setting in comparison to the validation log loss without DVRL (directly trained on the noisy data without reweighting) on 2 tabular datasets (Adult and Blog) and 4 image datasets (Fashion-MNIST, Flower, HAM 10000, and CIFAR-10).
+
+![](images/cd77e87d1533903c3385937da53bfe0722277f73a0c8d2b26bdae58b133d43fc.jpg)  
+Figure 9: Learning curves of DVRL for 6 datasets with $20 \%$ noisy labels. $\mathbf { X }$ -axis: the number of iterations for data value estimator training, y-axis: validation performance (log loss). (Orange: validation log loss without DVRL, Blue: validation log loss with DVRL)
+
+# E CONFIDENCE INTERVALS OF DVRL PERFORMANCE ON CORRUPTED SAMPLE DISCOVERY EXPERIMENTS
+
+![](images/7446440d4cbcb850178c099bfaf3da606684073e4e7caa73dddd2f7a5edc3e1b.jpg)  
+Figure 10: Corrupted sample discovery performance with $9 5 \%$ confidence intervals (computed by 10 independent runs) according to the estimated data values by DVRL. We assume a label noise with $20 \%$ ratio on (a) Adult and Blog, (b) Fashion-MNIST and Flower (c) HAM 10000 and CIFAR-10 datasets.
+
+F ROSSMANN DATA STATISTICS & T-SNE ANALYSIS   
+Table 8: Rossmann data statistics. Report 25-50-75 percentiles for sales and customers. # represents the number.   
+
+<table><tr><td rowspan=1 colspan=1>Store Type</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>B</td><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=1>D</td></tr><tr><td rowspan=1 colspan=2>#of Samples 457042 (54.1%)</td><td rowspan=1 colspan=3>15560 (1.8%)   112968 (13.4%) 258768 (30.6%)</td></tr><tr><td rowspan=1 colspan=5>Sales      1390-1660-1854 2052-2459-2661 1753-1974-2178 2109-2355-2524</td></tr><tr><td rowspan=1 colspan=1>Customers</td><td rowspan=1 colspan=1>169-203-221</td><td rowspan=1 colspan=1>436-492-543</td><td rowspan=1 colspan=1>192-232-259</td><td rowspan=1 colspan=1>224-246-259</td></tr></table>
+
+![](images/2ad702c797202d40b0d186a6d287bf01cd69f91edb75d409e6f117389b0aac8b.jpg)  
+Figure 11: t-SNE analyses on the final layer representations of each store type in Rossmann dataset.
+
+![](images/43fabb76041e17b4bdcbfd00eabd45757b2d0972aaf5d204c8eaf99aa6d84772.jpg)  
+G FURTHER ANALYSIS ON ROSSMANN DATASET IN Train on All SETTING   
+Figure 12: Histograms of the training samples from the target store type in Train on All setting based on the sorted data values estimated by DVRL. $\mathbf { \check { X } }$ -axis: the sorted data values (in percentiles), y-axis: counts of training samples from the target store type (in ratio).
+
+To further understand the results in Train on All setting, we sorted (in a decreasing order) the training samples by their data values estimated by DVRL and illustrate the distributions of the training samples that come from the target store type. As can be seen in Fig. 12, DVRL prioritizes the training samples which come from the same target store type.

parse/train/BJx8YnEFPH/BJx8YnEFPH_content_list.json

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+    {
+        "type": "text",
+        "text": "DATA VALUATION USING REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Quantifying the value of data is a fundamental problem in machine learning. Data valuation has multiple important use cases: (1) building insights about the learning task, (2) domain adaptation, (3) corrupted sample discovery, and (4) robust learning. To adaptively learn data values jointly with the target task predictor model, we propose a meta learning framework which we name Data Valuation using Reinforcement Learning (DVRL). We employ a data value estimator (modeled by a deep neural network) to learn how likely each datum is used in training of the predictor model. We train the data value estimator using a reinforcement signal of the reward obtained on a small validation set that reflects performance on the target task. We demonstrate that DVRL yields superior data value estimates compared to alternative methods across different types of datasets and in a diverse set of application scenarios. The corrupted sample discovery performance of DVRL is close to optimal in many regimes (i.e. as if the noisy samples were known apriori), and for domain adaptation and robust learning DVRL significantly outperforms state-of-the-art by $1 4 . 6 \\%$ and $1 0 . 8 \\%$ , respectively. ",
+        "bbox": [
+            233,
+            239,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            336,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Data is an essential ingredient in machine learning. Machine learning models are well-known to improve when trained on large-scale and high-quality datasets (Hestness et al., 2017; Najafabadi et al., 2015). However, collecting such large-scale and high-quality datasets is costly and challenging. One needs to determine the samples that are most useful for the target task and then label them correctly. Recent work (Toneva et al., 2019) suggests that not all samples are equally useful for training, particularly in the case of deep neural networks. In some cases, similar or even higher test performance can be obtained by removing a significant portion of training data, i.e. low-quality or noisy data may be harmful (Ferdowsi et al., 2013; Frenay & Verleysen, 2014). There are also some scenarios where train-test mismatch cannot be avoided because the training dataset only exists for a different domain. Different methods (Ngiam et al., 2018; Zhu et al., 2019) have demonstrated the importance of carefully selecting the most relevant samples to minimize this mismatch. ",
+        "bbox": [
+            174,
+            505,
+            825,
+            657
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Accurately quantifying the value of data has a great potential for improving model performance for real-world training datasets which commonly contain incorrect labels, and where the input samples differ in relatedness, sample quality, and usefulness for the target task. Instead of treating all data samples equally, lower priority can be assigned for a datum to obtain a higher performance model – for example in the following scenarios: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. Incorrect label (e.g. human labeling errors).   \n2. Input comes from a different distribution (e.g. different location or time).   \n3. Input is noisy or low quality (e.g. noisy capturing hardware).   \n4. Usefulness for target task (label is very common in the training dataset but not as common in the   \ntesting dataset). ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In addition to improving performance in such scenarios, data valuation also enables many new use cases. It can suggest better practices for data collection, e.g. what kinds of additional data would the model benefit the most from. For organizations that sell data, it determines the correct value-based pricing of data subsets. Finally, it enables new possibilities for constructing very large-scale training datasets in a much cheaper way, e.g. by searching the Internet using the labels and filtering away less valuable data. ",
+        "bbox": [
+            173,
+            840,
+            825,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "How does one evaluate the value of a single datum? This is a crucial and challenging question. It is straightforward to address at the full dataset granularity: one could naively train a model on the entire dataset and use its prediction performance as the value. However, evaluating the value of each datum is far more difficult, especially for complex models such as deep neural networks that are trained on large-scale datasets. In this paper, we propose a meta learning-based data valuation method which we name Data Valuation using Reinforcement Learning (DVRL). Our method integrates data valuation with the training of the target task predictor model. DVRL determines a reward by quantifying the performance on a small validation set, and uses it as a reinforcement signal to learn the likelihood of each datum being using in training of the predictor model. In a wide range of use cases, including domain adaptation, corrupted sample discovery and robust learning, we demonstrate significant improvements compared to permutation-based strategies (such as Leave-one-out and Influence Function (Koh & Liang, 2017)) and game theory-based strategies (such as Data Shapley (Ghorbani & Zou, 2019)). The main contributions can be summarized as follows: ",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We propose a novel meta learning framework for data valuation that is jointly optimized with the target task predictor model.   \n2. We demonstrate multiple use cases of the proposed data valuation framework and show DVRL significantly outperforms competing methods on many image, tabular and language datasets.   \n3. Unlike previous methods, DVRL is scalable to large datasets and complex models, and its computational complexity is not directly dependent on the size of the training set. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            406,
+            344,
+            422
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data valuation: A commonly-used method for data valuation is leave-one-out (LOO). It quantifies the performance difference when a specific sample is removed and assigns it as that sample’s data value. The computational cost is a major concern for LOO – it scales linearly with the number of training samples which means its cost becomes prohibitively high for large-scale datasets and complex models. In addition, there are fundamental limitations in the approximation. For example, if there are two exactly equivalent samples, LOO underestimates the value of that sample even though that sample may be very important. The method of Influence Function (Koh & Liang, 2017) was proposed to approximate LOO in a computationally-efficient manner. It uses the gradient of the loss function with small perturbations to estimate the data value. In order to compute the gradient, Hessian values are needed; however these are prohibitively expensive to compute for neural networks. Approximations for Hessian computations are possible, although they generally result in performance limitations. From data quality assessment perspective, the method of Influence Function inherits the major limitations of LOO. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data Shapley (Ghorbani & Zou, 2019) is another relevant work. Shapley values are motivated by game theory (Shapley, 1953) and are commonly used in feature attribution problems such as relating predictions to input features (Lundberg & Lee, 2017). For Data Shapley, the prediction performance of all possible subsets is considered and the marginal performance improvement is used as the data value. The computational complexity for computing the exact Shapley value is exponential with the number of samples. Therefore, Monte Carlo sampling and gradient-based estimation are used to approximate them. However, even with these approximations, the computational complexity still remains high (indeed much higher than LOO) due to re-training for each test combination. In addition, the approximations may result in fundamental limitations in data valuation performance – e.g. with Monte Carlo approximation, the ratio of tested combinations compared to all possible combinations decreases exponentially. Moreover, in all the aforementioned methods data valuation is decoupled from predictor model training, which limits the performance due to lack of joint optimization. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Meta learning-based adaptive learning: There are relevant studies that utilize meta learning for adaptive weight assignment while training for various use cases such as robust learning, domain adaptation, and corrupted sample discovery. ChoiceNet (Choi et al., 2018) explicitly models output distributions and uses the correlations of the output to improve robustness. Xue et al. (2019) estimates uncertainty of predictions to identify the corrupted samples. Li et al. (2019) combines meta learning with standard stochastic gradient update with generated synthetic noise for robust learning. Shen & Sanghavi (2019) alternates the processes of selecting the samples with low loss and model training to improve robustness. Shu et al. (2019) uses neural networks to model the relationship between current loss and the corresponding sample weights, and utilizes a meta-learning framework for robust weight assignment. Kohler et al. (2019) estimates the uncertainty to discover ¨ the noisy labeled data and relabels mislabeled samples to improve the prediction performance of the predictor model. Gold Loss Correction (Hendrycks et al., 2018) uses a clean validation set to recover the label corruption matrix to re-train the predictor model with corrected labels. Learning to Reweight (Ren et al., 2018) proposes a single gradient descent step guided with validation set performance to reweight the training batch. Domain Adaptive Transfer Learning (Ngiam et al., 2018) introduces importance weights (based on the prior label distribution match) to scale the training samples for transfer learning. MentorNet (Jiang et al., 2018) proposes a curriculum learning framework that learns the order of mini-batch for training of the corresponding predictor model. Our method, DVRL, differs from the aforementioned as it directly models the value of the data using learnable neural networks (which we refer to as a data value estimator). To train the data value estimator, we use reinforcement learning with a sampling process. DVRL is model-agnostic and even applicable to non-differentiable target objectives. Learning is jointly performed for the data value estimator and the corresponding predictor model, yielding superior results in all of the use cases we consider. ",
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+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
+        "text_level": 1,
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+    },
+    {
+        "type": "image",
+        "img_path": "images/908f2f0c09303cd8e1ea3c6705eb50f53032698bd59aa2f781dea582fd5315b4.jpg",
+        "image_caption": [
+            "Figure 1: Block diagram of the DVRL framework for training. A batch of training samples is used as the input to the data value estimator (with shared parameters across the batch) and the output corresponds to selection probabilities: $w _ { i } = h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } )$ of a multinomial distribution. The sampler, based on this multinomial distribution, returns the selection vector $\\mathbf { s } = ( s _ { 1 } , . . . , s _ { B _ { s } } )$ where $s _ { i } \\in \\{ 0 , 1 \\}$ and $P ( s _ { i } = 1 ) = w _ { i }$ . The target task predictor model is trained only using the samples with selection vector $s _ { i } = 1$ , using conventional gradient-descent optimization. The selection probabilities $w _ { i }$ rank the samples according to their importance – these are used as data values. The loss of the predictor model is evaluated on a small validation set, which is compared to the moving average of the previous losses $( \\delta )$ to determine the reward. Finally, the reinforcement signal guided by this reward updates the data value estimator. Block diagrams for inference are shown in Appendix A. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Framework: Let us denote the training dataset as $\\mathcal { D } = \\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N } \\sim \\mathcal { P }$ where $\\mathbf { x } _ { i } \\in \\mathcal X$ is a feature vector in the $d$ -dimensional feature space $\\mathcal { X }$ , e.g. $\\mathbb { R } ^ { d }$ , and $y _ { i } \\in \\mathcal { V }$ is a corresponding label in the label space $\\mathcal { V }$ , e.g. $\\Delta [ 0 , 1 ] ^ { c }$ for classification where $c$ is the number of classes and $\\Delta$ is the simplex. We consider a disjoint testing dataset $\\mathcal { D } ^ { t } = \\{ ( \\mathbf { x } _ { j } ^ { t } , y _ { j } ^ { t } ) \\} _ { j = 1 } ^ { M } \\sim \\mathcal { P } ^ { t }$ where the target distribution $\\mathcal { P } ^ { t }$ does not need to be the same with the training distribution $\\mathcal { P }$ . We assume an availability of a (often small1) validation dataset $\\mathcal { D } ^ { v } = \\{ ( \\mathbf { x } _ { k } ^ { v } , y _ { k } ^ { v } ) \\} _ { k = 1 } ^ { L ^ { - } } \\sim \\mathcal { P } ^ { t }$ that comes from the target distribution $\\mathcal { P } ^ { t }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The DVRL method (overview in Fig. 1) consists of two learnable functions: (1) the target task predictor model $f _ { \\theta }$ , (2) data value estimator model $h _ { \\phi }$ . The predictor model $f _ { \\theta } : \\mathcal { X }  \\mathcal { Y }$ is trained to minimize a certain weighted loss function $\\mathcal { L } _ { f }$ (e.g. Mean Squared Error (MSE) for regression or cross entropy for classification) on training set $\\mathcal { D }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c53ff92a2bb6db2019cc321641aea8b33d4c44270400d43949e6bb2868f68d33.jpg",
+        "text": "$$\nf _ { \\theta } = \\arg \\operatorname* { m i n } _ { \\hat { f } \\in \\mathcal { F } } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) \\cdot \\mathcal { L } _ { f } ( \\hat { f } ( \\mathbf { x } _ { i } ) , y _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$f _ { \\theta }$ can be any trainable function with parameters $\\theta$ , such as a neural network. The data value estimator model $h _ { \\phi } : \\mathcal { X } \\cdot \\mathcal { Y }  [ 0 , 1 ]$ , on the other hand, is optimized to output weights that determine ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "the distribution of selection likelihood of the samples to train the predictor model $f _ { \\theta }$ . We formulate the corresponding optimization problem as: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bdac97331e50a886c68fccb0ccf91d3c611bdf448edc9a5bd518d75526b05672.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { h _ { \\phi } } { \\operatorname* { m i n } } } & { \\mathbb { E } _ { ( \\mathbf { x } ^ { v } , y ^ { v } ) \\sim P ^ { t } } \\Big [ \\mathcal { L } _ { h } \\big ( f _ { \\theta } \\big ( \\mathbf { x } ^ { v } \\big ) , y ^ { v } \\big ) \\Big ] } \\\\ { \\mathrm { s . t . } } & { f _ { \\theta } = \\arg \\operatorname* { m i n } _ { \\hat { f } \\in \\mathcal { F } } \\mathbb { E } _ { ( \\mathbf { x } , y ) \\sim P } \\Big [ h _ { \\phi } \\big ( \\mathbf { x } , y \\big ) \\cdot \\mathcal { L } _ { f } \\big ( \\hat { f } ( \\mathbf { x } ) , y \\big ) \\Big ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $h _ { \\phi } ( \\mathbf { x } , y )$ represents value of the training sample $\\left( \\mathbf { x } , y \\right)$ . The data value estimator is also a trainable function, such as a neural network. Similar to $\\mathcal { L } _ { f }$ , we use MSE or cross entropy for $\\mathcal { L } _ { h }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Training: To encourage exploration based on uncertainty, we model training sample selection stochastically. Let $w = h _ { \\phi } ( \\mathbf { x } , y )$ denote the probability that $\\left( \\mathbf { x } , y \\right)$ is used to train the predictor model $f _ { \\theta }$ $\\mathbf { \\varepsilon } _ { \\cdot } \\ h _ { \\phi } ( \\mathcal { D } ) = \\{ h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N }$ is the probability distribution for inclusion of each training sample. $\\mathbf { s } \\in \\{ 0 , 1 \\} ^ { N }$ is a binary vector that represents the selected samples. If $s _ { i } = 1 / 0$ , $\\left( \\mathbf { x } _ { i } , y _ { i } \\right)$ is selected/not selected for training the predictor model. $\\begin{array} { r } { \\pi _ { \\phi } ( \\mathcal { D } , \\mathbf { s } ) \\ = \\ \\prod _ { i = 1 } ^ { N } \\big [ h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) ^ { s _ { i } } \\cdot ( 1 - } \\end{array}$ $h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) ) ^ { 1 - s _ { i } } ]$ is the probability that certain selection vector s is selected based on $h _ { \\phi } ( \\mathcal { D } )$ . We assign the outputs of the data value estimator model, $w = h _ { \\phi } ( \\mathbf { x } , y )$ , as the data values. We can use the data values to rank the dataset samples (e.g. to determine a subset of the training dataset) and to do sample-adaptive training (e.g. for domain adaptation). ",
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+        "type": "text",
+        "text": "The predictor model can be trained using standard stochastic gradient descent because it is differentiable with respect to the input. However, gradient descent-based optimization cannot be used for the data value estimator because the sampling process is non-differentiable. There are multiple ways to handle the non-differentiable optimization bottleneck, such as Gumbel-softmax (Jang et al., 2017) or stochastic back-propagation (Rezende et al., 2014). In this paper, we consider reinforcement learning instead, which directly encourages exploration of the policy towards the optimal solution of Eq. (2). We use the REINFORCE algorithm (Williams, 1992) to optimize the policy gradients, with the rewards obtained from a small validation set that approximates performance on the target task. For the loss function $\\hat { l } ( \\phi )$ : ",
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+        "type": "equation",
+        "img_path": "images/3eeb812102536ec173a845959015e934e3e2d8a679f4f034d7022c4fc77a5a73.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { l } ( \\phi ) = \\mathbb { E } _ { ( \\mathbf { x } ^ { v } , y ^ { v } ) \\sim P ^ { t } } \\Big [ \\mathbb { E } _ { s \\sim \\pi _ { \\phi } ( \\mathcal { D } , \\cdot ) } \\big [ \\mathcal { L } _ { h } ( f _ { \\theta } ( \\mathbf { x } ^ { v } ) , y ^ { v } ) \\big ] \\Big ] } \\\\ & { \\quad \\quad = \\displaystyle \\int P ^ { t } ( \\mathbf { x } ^ { v } ) \\Big [ \\sum _ { s \\in [ 0 , 1 ] ^ { N } } \\pi _ { \\phi } ( \\mathcal { D } , \\mathbf { s } ) \\cdot \\big [ \\mathcal { L } _ { h } ( f _ { \\theta } ( \\mathbf { x } ^ { v } ) , y ^ { v } ) \\big ] \\Big ] d \\mathbf { x } ^ { v } , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "we directly compute the gradient $\\nabla _ { \\phi } \\hat { l } ( \\phi )$ as: ",
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+        "type": "equation",
+        "img_path": "images/962938469d1e6e4efc071adfa730f0850655ad07724b401b2e17414b71538bd9.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\nabla _ { \\phi } \\hat { l } ( \\phi ) = \\int P ^ { t } ( { \\bf x } ^ { v } ) \\Big [ \\sum _ { s \\in [ 0 , 1 ] ^ { N } } \\nabla _ { \\phi } \\pi _ { \\phi } ( \\mathcal { D } , { \\bf s } ) \\cdot \\big [ \\mathcal { L } _ { h } \\big ( f _ { \\theta } \\big ( { \\bf x } ^ { v } \\big ) , y ^ { v } \\big ) \\big ] \\Big ] d { \\bf x } ^ { v } } \\ ~ } \\\\ { { \\displaystyle ~ = \\int P ^ { t } ( { \\bf x } ^ { v } ) \\Big [ \\sum _ { s \\in [ 0 , 1 ] ^ { N } } \\nabla _ { \\phi } \\log \\big ( \\pi _ { \\phi } ( \\mathcal { D } , { \\bf s } ) \\big ) \\cdot \\pi _ { \\phi } \\big ( \\mathcal { D } , { \\bf s } \\big ) \\cdot \\big [ \\mathcal { L } _ { h } \\big ( f _ { \\theta } \\big ( { \\bf x } ^ { v } \\big ) , y ^ { v } \\big ) \\big ] \\Big ] d { \\bf x } ^ { v } } \\ ~ } \\\\ { { \\displaystyle ~ = \\mathbb { E } _ { ( { \\bf x } ^ { v } , y ^ { v } ) \\sim P ^ { t } } \\Big [ \\mathbb { E } _ { { \\bf s } \\sim \\pi _ { \\phi } ( \\mathcal { D } , \\cdot ) } \\big [ \\mathcal { L } _ { h } \\big ( f _ { \\theta } \\big ( { \\bf x } ^ { v } \\big ) , y ^ { v } \\big ) \\big ] \\nabla _ { \\phi } \\log \\big ( \\pi _ { \\phi } ( \\mathcal { D } , { \\bf s } ) \\big ) \\Big ] , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\nabla _ { \\phi } \\log ( \\pi _ { \\phi } ( \\mathcal { D } , \\mathbf { s } ) )$ is ",
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+    {
+        "type": "equation",
+        "img_path": "images/674a03bc50e565c8dee220b7e697d1f7fe0ec055d67963e477fe2c52de0f5af2.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\nabla _ { \\phi } \\log ( \\pi _ { \\phi } ( \\mathcal { D } , \\mathbf { s } ) ) = \\nabla _ { \\phi } \\sum _ { i = 1 } ^ { N } \\log \\Big [ h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) ^ { s _ { i } } \\cdot ( 1 - h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) ) ^ { 1 - s _ { i } } \\Big ] } \\\\ { \\displaystyle = \\sum _ { i = 1 } ^ { N } s _ { i } \\nabla _ { \\phi } \\log \\big [ h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) \\big ] + ( 1 - s _ { i } ) \\nabla _ { \\phi } \\log \\big [ ( 1 - h _ { \\phi } ( \\mathbf { x } _ { i } , y _ { i } ) ) \\big ] . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "To improve the stability of the training, we use the moving average of the previous loss $( \\delta )$ , with a window size $( T )$ , as the baseline for the current loss. The pseudo-code is shown in Algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "Computational complexity: DVRL models the mapping between an input and its value with a learnable function. The training time of DVRL is not directly proportional to the dataset size, but rather dominated by the required number of iterations and per-iteration complexity in Algorithm 1. One way to minimize the computational overhead is to use pre-trained models to initialize the ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Pseudo-code of DVRL training ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "1: Inputs: Learning rates $\\alpha , \\beta > 0$ , mini-batch size $B _ { p } , B _ { s } > 0$ , inner iteration count $N _ { I } > 0$ , moving average window $T > 0$ , training dataset $\\mathcal { D }$ , validation dataset $\\mathcal { D } ^ { v } = \\{ ( \\mathbf { x } _ { k } ^ { v } , y _ { k } ^ { v } ) \\} _ { k = 1 } ^ { L }$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2: Initialize parameters $\\theta , \\phi$ , moving average $\\delta = 0$ ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3: while until convergence do ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4: Sample a mini-batch from the entire training dataset: $\\mathcal { D } _ { B } = ( \\mathbf { x } _ { j } , y _ { j } ) _ { j = 1 } ^ { B _ { s } } \\sim \\mathcal { D }$   \n5: for $j = 1 , . . . , B _ { s }$ do   \n6: Calculate selection probabilities: $w _ { j } = h _ { \\phi } ( \\mathbf { x } _ { j } , y _ { j } )$   \n7: Sample selection vector: $s _ { j } \\sim B e r ( w _ { j } )$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/096cee7b2ab842c998686973393010933bd5ab00934183bdf645a79dc8a530ca.jpg",
+        "text": "$$\n\\theta  \\theta - \\alpha \\frac { 1 } { B _ { p } } \\sum _ { m = 1 } ^ { B _ { p } } \\tilde { s } _ { m } \\cdot \\nabla _ { \\theta } \\mathcal { L } _ { f } ( f _ { \\theta } ( \\tilde { \\mathbf { x } } _ { m } ) , \\tilde { y } _ { m } ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "11: Update the data value estimator model network parameters $\\phi$ ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d5025091040b95b3322bf2078190f01c2c46b957119a018445490a762eb95f4.jpg",
+        "text": "$$\n\\phi  \\phi - \\beta \\Big [ \\frac { 1 } { L } \\sum _ { k = 1 } ^ { L } [ \\mathcal { L } _ { h } ( f _ { \\theta } ( \\mathbf { x } _ { k } ^ { v } ) , y _ { k } ^ { v } ) ] - \\delta \\Big ] \\nabla _ { \\phi } \\log \\pi _ { \\phi } ( \\mathcal { D } _ { B } , ( s _ { 1 } , . . . , s _ { B _ { s } } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "12: Update the moving average baseline $( \\delta )$ : $\\begin{array} { r } { \\delta \\gets \\frac { T - 1 } { T } \\delta + \\frac { 1 } { L T } \\sum _ { k = 1 } ^ { L } [ \\mathcal { L } _ { h } ( f _ { \\theta } ( \\mathbf { x } _ { k } ^ { v } ) , y _ { k } ^ { v } ) ] } \\end{array}$ predictor networks at each iteration. Unlike alternative methods like Data Shapley, we demonstrate the scalability of DVRL to large-scale datasets such as CIFAR-100, and complex models such as ResNet-32 (He et al., 2016) and WideResNet-28-10 (Zagoruyko & Komodakis, 2016). Instead of being exponential in terms of the dataset size, the training time overhead DVRL is only twice of conventional training. Please see Appendix D for further analysis on learning dynamics of DVRL and Appendix B for additional computational complexity discussions. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate data value estimation quality of DVRL on multiple types of dataset and use cases. ",
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+    {
+        "type": "text",
+        "text": "Benchmark methods: We consider the following benchmarks: (1) Randomly-assigned values (Random), (2) Leave-one-out (LOO), (3) Data Shapley Value (Data Shapley) (Ghorbani & Zou, 2019). For some experiments, we also compare with (4) Learning to Reweight (Ren et al., 2018), (5) MentorNet (Jiang et al., 2018), and (6) Influence Function (Koh & Liang, 2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Datasets: We consider 12 public datasets (3 public tabular datasets, 7 public image datasets, and 2 public language datasets) to evaluate DVRL in comparison to multiple benchmark methods. 3 public tabular datasets are (1) Blog, (2) Adult, (3) Rossmann; 7 public image datasets are (4) HAM 10000, (5) MNIST, (6) USPS, (7) Flower, (8) Fashion-MNIST, (9) CIFAR-10, (10) CIFAR-100; 2 public language datasets are (11) Email Spam, (12) SMS Spam. Details can be found in the hyper-links. ",
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+    {
+        "type": "text",
+        "text": "Baseline predictor models: We consider various machine learning models as the baseline predictor model to highlight the proposed model-agnostic data valuation framework. For Adult and Blog datasets, we use LightGBM (Ke et al., 2017), and for Rossmann dataset, we use XGBoost and multi-layer perceptrons due to their superior performance on the tabular datasets. For Flower, HAM 10000, and CIFAR-10 datasets, we use Inception-v3 with top-layer fine-tuning (pre-trained on ImageNet, (Szegedy et al., 2016)) as the baseline predictor model. For Fashion-MNIST, MNIST, and USPS datasets, we use multinomial logistic regression, and for Email and SMS datasets, we use Naive Bayes model. We also use ResNet-32 (He et al., 2016) and WideResNet-28-10 (Zagoruyko & Komodakis, 2016) as the baseline models for CIFAR-10 and CIFAR-100 datasets in Section 4.3 to demonstrate the scalability of DVRL. For data value estimation network, we use multi-layer perceptrons with ReLU activation as the base architecture. The number of layers and hidden units are optimized with cross-validation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experimental details: In all experiments, we use Standard Normalizer to normalize the entire features to have zero mean and one standard deviation. We transform categorical variables into one-hot encoded embeddings. We set the inner iteration count $\\scriptstyle N _ { I } = 2 0 0$ ) for the predictor network, moving average window $( T { = } 2 0 )$ , and mini-batch size $( B _ { p } { = } 2 5 6 )$ for the predictor network and mini-batch size ( ${ B _ { s } } { = } 2 0 0 0 )$ for the DVE network (large batch size often improves the stability of the reinforcement learning model training (McCandlish et al., 2018)). We set the learning rate to 0.01 $( \\beta )$ for the data value estimator (DVE) and 0.001 $( \\alpha )$ for the predictor network. As the DVE architecture, for tabular datasets, we use 5-layer perceptrons with 100 hidden units and ReLU; and for image datasets, we use 5-layer perceptrons with 100 hidden units and ReLU on top of the CNN-based architecture used for the predictor network (such as ResNet-32 or WideResNet-28-10 in Table 1). In order to provide further informative signal to DVE, we propose to use an additional input of the difference between the predictions of a separate predictive model (fined-tuned or trained from scratch on the validation set) for the training samples and the original training labels. We simply concatenate this additional input to the hidden states of DVE network. Intuitively, if the training label is corrupted, the additional input would be high; thus, this could be an important signal for DVE to assign low value to this sample. Ablation study for the variants of DVRL can be found in the Appendix C.6. ",
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+    {
+        "type": "text",
+        "text": "4.1 REMOVING HIGH/LOW VALUE SAMPLES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Removing low value samples from the training dataset can improve the predictor model performance, especially in the cases where the training dataset contains corrupted samples. On the other hand, removing high value samples, especially if the dataset is small, would decrease the performance significantly. Overall, the performance after removing high/low value samples is a strong indicator for the quality of data valuation. ",
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+    {
+        "type": "text",
+        "text": "Initially, we consider the conventional supervised learning setting, where all training, validation and testing datasets come from the same distribution (without sample corruption or domain mismatch). We use two tabular datasets (Adult and Blog) with 1,000 training samples and one image dataset (Flower) with 2,000 training samples.2 We use 400 validation samples for tabular datasets and 800 validation samples for the image dataset. Then, we report the prediction performance on the disjoint testing set after removing the high/low value samples based on the estimated data values. ",
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+    {
+        "type": "image",
+        "img_path": "images/db8745bdf9b0728a33cebe06f1dd1ed4168ec60781696cccb8d9a7de73be3711.jpg",
+        "image_caption": [
+            "Figure 2: Performance after removing the most (marked with $\\times$ ) and least (marked with $\\bigcirc$ ) important samples according to the estimated data values in a conventional supervised learning setting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As shown in Fig. 2, even in the absence of sample corruption or domain mismatch, DVRL can marginally improve the prediction performance after removing some portion of the least important samples. Using only ${ \\sim } 6 0 \\% { - } 7 0 \\%$ of the training set (the highest valued samples), DVRL can obtain a similar performance compared to training on the entire dataset. After removing a small portion $( 1 0 \\% - 2 0 \\% )$ of the most important samples, the prediction performance significantly degrades which indicates the importance of the high valued samples. Qualitatively looking at these samples, we observe them to typically be representative of the target task which can be insightful. Overall, DVRL shows the fastest performance degradation after removing the most important samples and the slowest performance degradation after removing the least important samples in most cases, underlining the superiority of DVRL in data valuation quality compared to competing methods. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Next, we focus on the setting of removing high/low value samples in the presence of label noise in the training data. We consider three image datasets: Fashion-MNIST, HAM 10000, and CIFAR10. As noisy samples hurt the performance of the predictor model, an optimal data value estimator with a clean validation dataset should assign lowest values to the noisy samples. With the removal of samples with noisy labels (‘Least’ setting), the performance should either increase, or at least decrease much slower, compared to removal of samples with correct labels (‘Most’ setting). In this experiment, we introduce label noise to $20 \\%$ of the samples by replacing true labels with random labels. As can be seen in Fig. 10, for all data valuation methods the prediction performance tends to first slowly increase and then decrease in the ‘Least’ setting; and tends to rapidly decrease in the ‘Most’ setting. Yet, DVRL achieves the slowest performance decrease in ‘Least’ setting and fastest performance decrease in the ‘Most’ setting, reflecting its superiority in data valuation. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/6ddb537ce81e2f324127f44b1142e26fe6a5bd03d5c118920888ff90d93a98a1.jpg",
+        "image_caption": [
+            "Figure 3: Prediction performance after removing the most (marked with $\\times$ ) and least (marked with $\\bigcirc$ ) important samples according to the estimated data values with $20 \\%$ noisy label ratio. Additional results on Blog, HAM 10000, and CIFAR-10 datasets can be found in Appendix C.3. The prediction performance is lower than state of the art due to a smaller training set size and the introduced noise. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2 CORRUPTED SAMPLE DISCOVERY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "There are some scenarios where training samples may contain corrupted samples, e.g. due to cheap label collection methods. An automated corrupted sample discovery method would be highly beneficial for distinguishing samples with clean vs. noisy labels. Data valuation can be used in this setting by having a small clean validation set to assign low data values to the potential samples with noisy labels. With an optimal data value estimator, all noisy labels would get the lowest data values. ",
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+    {
+        "type": "text",
+        "text": "We consider the same experimental setting with the previous subsection with $20 \\%$ noisy label ratio on 6 datasets. Fig. 4 shows that DVRL consistently outperforms all benchmarks (Data Shapley, LOO and Influence Function). The trend of noisy label discovery for DVRL can be very close to optimal (as if we perfectly knew which samples have noisy labels), particularly for the Adult, CIFAR-10 and Flower datasets. To highlight the stability of DVRL, we provide the confidence intervals of DVRL performance on the corrupted sample discovery in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "4.3 ROBUST LEARNING WITH NOISY LABELS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we consider how reliably DVRL can learn with noisy data in an end-to-end way, without removing the low-value samples as in the previous section. Ideally, noisy samples should get low data values as DVRL converges and a high performance model can be returned. To compare DVRL with two recently-proposed benchmarks: MentorNet (Jiang et al., 2018) and Learning to Reweight (Ren et al., 2018) for this use case, we focus on two complex deep neural networks as the baseline predictor models, ResNet-32 (He et al., 2016) and WideResNet-28-10 (Zagoruyko & ",
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+        "img_path": "images/92e57a4cbe06629201df341dcc179fec73acb789d6c2e1ad2fcebc4d3bad801b.jpg",
+        "image_caption": [
+            "Figure 4: Discovering corrupted samples in three datasets with $20 \\%$ noisy label ratio. ‘Optimal’ saturates at $20 \\%$ , perfectly assigning the lowest data value scores to the samples with noisy labels. ‘Random’ does not introduce any knowledge on distinguishing clean vs. noisy labels, and thus the fraction of discovered corrupt samples is proportional to the amount of inspection. More results on Adult, Fashion-MNIST and Flower datasets are in Appendix C.4. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Komodakis, 2016), trained on CIFAR-10 and CIFAR-100 datasets. Additional results on other image datasets are in Appendix C.1, and results on robust learning with noisy features are in Appendix C.2. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We consider the same experimental setting from Ren et al. (2018) on CIFAR-10 and CIFAR-100 datasets. For the first experiment, we use WideResNet-28-10 as the baseline predictor model and apply $40 \\%$ of label noise uniformly across all classes. We use 1,000 clean (noise-free) samples as the validation set and test the performance on the clean testing set. For the second experiment, we use ResNet-32 as the baseline predictor model and apply $40 \\%$ background noise (same-class noise to the $40 \\%$ of the samples). In this case, we only use 10 clean samples per class as the validation set. We consider five additional benchmarks: (1) Validation Set Only – which only uses clean validation set for training, (2) Baseline – which only uses noisy training set for training, (3) Baseline $^ +$ Finetuning – which is initialized with the trained baseline model on the noisy training set and fine-tuned on the clean validation set, (4) Clean Only $60 \\%$ data) – which is trained on the clean training set after removing the training samples with flipped labels, (5) Zero Noise – which uses the original noise-free training set for training ( $100 \\%$ clean training data). We exclude Data Shapley and LOO in this experiment due to their prohibitively-high computational complexities. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b8aa1436150e2bc19a24cf2590e938efa59933095d4896bf31405dbfd865dc0.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Robust learning with noisy labels. Test accuracy for ResNet-32 and WideResNet-28-10 on CIFAR-10 and CIFAR-100 datasets with $40 \\%$ of Uniform and Background noise on labels. "
+        ],
+        "table_body": "<table><tr><td>Noise (predictor model)</td><td colspan=\"2\">Uniform (WideResNet-28-10)</td><td colspan=\"2\">Background (ResNet-32)</td></tr><tr><td>Datasets</td><td>CIFAR-10</td><td>CIFAR-100</td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td>Validation Set Only</td><td>46.64± 3.90</td><td>9.94 ± 0.82</td><td>15.90 ± 3.32</td><td>8.06± 0.76</td></tr><tr><td>Baseline</td><td>67.97 ± 0.62</td><td>50.66 ± 0.24</td><td>59.54 ± 2.16</td><td>37.82 ± 0.69</td></tr><tr><td>Baseline + Fine-tuning</td><td>78.66 ± 0.44</td><td>54.52 ± 0.40</td><td>82.82 ± 0.93</td><td>54.23 ± 1.75</td></tr><tr><td>MentorNet + Fine-tuning</td><td>78.00</td><td>59.00</td><td></td><td></td></tr><tr><td>Learning to Reweight</td><td>86.92 ± 0.19</td><td>61.34 ± 2.06</td><td>86.73 ± 0.48</td><td>59.30 ± 0.60</td></tr><tr><td>DVRL</td><td>89.02 ± 0.27</td><td>66.56 ± 1.27</td><td>88.07 ± 0.35</td><td>60.77 ± 0.57</td></tr><tr><td rowspan=\"2\">Clean Only (60% Data) Zero Noise</td><td>94.08 ± 0.23</td><td>74.55 ± 0.53</td><td>90.66 ± 0.27</td><td>63.50 ± 0.33</td></tr><tr><td>95.78 ± 0.21</td><td>78.32 ± 0.45</td><td>92.68 ± 0.22</td><td>68.12 ± 0.21</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 1, DVRL outperforms other robust learning methods in all cases. The performance improvements with DVRL are larger with Uniform noise. Learning to Reweight loses $7 . 1 6 \\%$ and $1 3 . 2 1 \\%$ accuracy compared to the optimal case (Zero Noise); on the other hand, DVRL only loses $5 . 0 6 \\%$ and $7 . 9 9 \\%$ accuracy for CIFAR-10 and CIFAR-100 respectively with Uniform noise. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.4 DOMAIN ADAPTATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider the scenario where the training dataset comes from a substantially different distribution from the validation and testing sets. Naive training methods (i.e. equal treatment of all training samples) often fail in this scenario (Ganin et al., 2016; Glorot et al., 2011). Data valuation is expected to be beneficial for this task by selecting the samples from the training dataset that best match the distribution of the validation dataset. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e2aeafc2cad4912a4f1239bdc8883537ba167194903cc15a64a387fda98ec605.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Domain adaptation setting showing target accuracy. Baseline represents the predictor model which is naively trained on the training set with equal treatment of all training samples. "
+        ],
+        "table_body": "<table><tr><td>Source</td><td>Target</td><td>Task</td><td>Baseline</td><td>Data Shapley</td><td>DVRL</td></tr><tr><td>Google</td><td>HAM10000</td><td>Skin Lesion Classification</td><td>.296</td><td>.378</td><td>.448</td></tr><tr><td>MNIST</td><td>USPS</td><td>Digit Recognition</td><td>.308</td><td>.391</td><td>.472</td></tr><tr><td>Email</td><td>SMS</td><td>Spam Detection</td><td>.684</td><td>.864</td><td>.903</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We initially focus on the three cases from Ghorbani & Zou (2019), shown in Table 2. (1) uses Google image search results (cheaply collected dataset) to predict skin lesion classification on HAM 10000 data (clean), (2) uses MNIST data to recognize digit on USPS dataset, (3) uses Email spam data to detect spam in an SMS dataset. The experimental settings are exactly the same with Ghorbani & Zou (2019). Table 2 shows that DVRL significantly outperforms Baseline and Data Shapley in all three tasks. One primary reason is that DVRL jointly optimizes the data value estimator and corresponding predictor model; on the other hand, Data Shapley needs a two step processes to construct the predictor model in domain adaptation setting. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, we focus on a real-world tabular data learning problem where the domain differences are significant. We consider the sales forecasting problem with the Rossmann Store Sales dataset, which consists of sales data from four different store types. Simple statistical investigation shows a significant discrepancy between the input feature distributions across different store types, meaning there is a large domain mismatch across store types (see Appendix F). To further illustrate distribution difference across the store types, we show the t-SNE analysis on the final layer of a discriminative neural network trained on the entire dataset in Appendix Fig. 11. We consider three different settings: (1) training on all store types (Train on All), (2) training on store types excluding the store type of interest (Train on Rest), and (3) training only on the store type of interest (Train on Specific). In all cases, we evaluate the performance on each store type separately. For example, to evaluate the performance on store type D, Train on All setting uses all four store type datasets for training, Train on Rest setting uses store types A, B and C for training, and Train on Specific setting only uses the store type D for training. Train on Rest is expected to yield the largest domain mismatch between training and testing sets, and Train on Specific yield the minimal. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9997734fa619e5419a07a4ef7fde0d64acb81e402bce81ba1fe95c42845acd91.jpg",
+        "table_caption": [
+            "Table 3: Performance of Baseline and DVRL in 3 different settings with 2 different predictor models on the Rossmann Store Sales dataset. Metric is Root Mean Squared Percentage Error (RMSPE, lower the better). We use $79 \\%$ of the data as training, $1 \\%$ as validation, and $20 \\%$ as testing. DVRL outperforms Baseline in all settings. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Predictor Model (Metric: RMSPE)</td><td rowspan=\"2\">Store Type</td><td colspan=\"2\">Train on All</td><td colspan=\"2\">Train on Rest</td><td colspan=\"2\">Train on Specific</td></tr><tr><td>Baseline</td><td>DVRL</td><td>Baseline</td><td>DVRL</td><td>Baseline</td><td>DVRL</td></tr><tr><td rowspan=\"4\">XGBoost</td><td>A B</td><td>0.1736 0.1996</td><td>0.1594 0.1422</td><td>0.2369</td><td>0.2109</td><td>0.1454</td><td>0.1430</td></tr><tr><td></td><td>0.1839</td><td>0.1502</td><td>0.7716</td><td>0.3607</td><td>0.0880</td><td>0.0824</td></tr><tr><td>C</td><td></td><td></td><td>0.2083</td><td>0.1551</td><td>0.1186</td><td>0.1170</td></tr><tr><td>D</td><td>0.1504</td><td>0.1441</td><td>0.1922</td><td>0.1535</td><td>0.1349</td><td>0.1221</td></tr><tr><td rowspan=\"4\">Neural Networks</td><td>A B</td><td>0.1531</td><td>0.1428</td><td>0.3124</td><td>0.2014</td><td>0.1181</td><td>0.1066</td></tr><tr><td></td><td>0.1529</td><td>0.0979</td><td>0.8072</td><td>0.5461</td><td>0.0683</td><td>0.0682</td></tr><tr><td>C</td><td>0.1620</td><td>0.1437</td><td>0.2153</td><td>0.1804</td><td>0.0682</td><td>0.0677</td></tr><tr><td>D</td><td>0.1459</td><td>0.1295</td><td>0.2625</td><td>0.1624</td><td>0.0759</td><td>0.0708</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the performance of Baseline (train the predictor model without data valuation) and DVRL in 3 different settings with 2 different predictor models (XGBoost (Chen & Guestrin, 2016) and Neural Networks (3-layer perceptrons)). As shown in Table 3, DVRL improves the performance in all settings. The improvements are most significant in Train on Rest setting due to the large domain mismatch. For instance, DVRL reduces the error more than $50 \\%$ for store type B predictions with XGBoost in comparison to Baseline. In Train on $A l l$ setting, the performance improvement is still significant, showing that DVRL can distinguish the samples from the target distribution. In Appendix G, we demonstrate that DVRL actually prioritizes selection of the samples from the target store type. In Train on Specific setting, the performance improvements are smaller – even without domain mismatch, DVRL can marginally improve the performance by accurately prioritizing the important samples within the same store type. These results further support the conclusions from Fig. 2 in the conventional supervised learning setting that DVRL learns high quality data value scores. Comparison to other domain adaptation benchmarks can be found in Appendix C.5. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.5 DISCUSSION: HOW MANY VALIDATION SAMPLES ARE NEEDED?",
+        "text_level": 1,
+        "bbox": [
+            176,
+            301,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "DVRL requires a validation dataset from the target distribution that the testing dataset comes from. Depending on the task, the requirements for the validation dataset may involve being noise-free in labels, being from the same domain, or being high quality. Acquiring such a dataset can be costly in some scenarios and it is desirable to minimize its size requirements. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We analyze the impact of the size of the validation dataset on DVRL with 3 different datasets: Adult, Blog, and Fashion MNIST for the use case of corrupted sample discovery. Similar to Section 4.2, we add $20 \\%$ noise to the training samples and try to find the corrupted samples with DVRL. As shown in Fig. 5, DVRL achieves reasonable performance with 100 to 400 validation samples. In the Adult dataset, even 10 validation samples are sufficient to achieve a reasonable data valuation quality. Both of these settings are often realistic in real world scenarios. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/66196ad3e362b32df850a5b6f8f8d16dfb05d291eb0372e823d8a883f3f5e1d1.jpg",
+        "image_caption": [
+            "Figure 5: Number of validation samples needed for DVRL. Discovering corrupted samples in three datasets (Adult, Blog and Fashion MNIST) with various number of validation samples. X-axis represents the fraction of inspected data and y-axis is the fraction of discovered corrupted samples. On Adult and Fashion-MNIST datasets, DVRL needs $\\cdot$ and $\\cdot$ of inspected samples to identify $\\cdot$ of the corrupted samples respectively - merely $\\cdot$ and $4 \\%$ more than the optimal cases. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a meta learning framework, named DVRL, that adaptively learns data values jointly with a target task predictor model. The value of each datum determines how likely it will be used in training of the predictor model. We model this data value estimation task using a deep neural network, which is trained using reinforcement learning with a reward obtained from a small validation set that represents the target task performance. With a small validation set, DVRL can provide computationally highly efficient and high quality ranking of data values for the training dataset that is useful for domain adaptation, corrupted sample discovery and robust learning. We show that DVRL significantly outperforms other techniques for data valuation in various applications on diverse types of datasets. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
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+        "text": "Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8(3-4):229–256, 1992. ",
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+        "text": "Cheng Xue, Qi Dou, Xueying Shi, Hao Chen, and Pheng Ann Heng. Robust learning at noisy labeled medical images: Applied to skin lesion classification. arXiv preprint arXiv:1901.07759, 2019. ",
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+        "text": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016. ",
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+        "text": "Linchao Zhu, Sercan O. Arik, Yi Yang, and Tomas Pfister. Learning to Transfer Learn. arXiv preprint arXiv:1908.11406, 2019. ",
+        "bbox": [
+            168,
+            714,
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+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/048b02cfab2a3f87d1413d3fd4a8c7d59417f9e8fb661aa7840935374ec5613d.jpg",
+        "image_caption": [
+            "Figure 6: Block diagram of the proposed DVRL framework at inference time. (a) Data valuation, (b) Prediction. For data valuation, the input is a set of samples and the outputs are the corresponding data values. For prediction, the input is a sample and the output is the corresponding prediction. Both the data value estimator and predictor are fixed (not trained) at inference time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            140,
+            820,
+            313
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B COMPUTATIONAL COMPLEXITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            412,
+            468,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "DVRL first trains the baseline model using the entire dataset (without re-weighting). Afterwards, we can use this pre-trained baseline model to initialize the predictor network and apply fine-tuning with DVRL update steps. The convergence of the fine-tuning process is much faster than the convergence of training from the scratch. ",
+        "bbox": [
+            174,
+            444,
+            825,
+            500
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We quantify the computational overhead of DVRL on the CIFAR-100 dataset (consisting $5 0 \\mathrm { k }$ training samples) with ResNet-32 as a representative example. Overall, DVRL training takes less than 8 hours (given a pre-trained ResNet-32 model on the entire dataset) on a single NVIDIA Tesla V100 GPU without any hardware optimizations. The pre-training time of ResNet-32 on the entire dataset (without re-weighting) is less than 4 hours; thus the total training time of DVRL is less than 12 hours from the scratch. On the other hand, the training time of Data Shapley (the most competitive benchmark) is more than a week on Fashion MNIST (consisting lower dimensional inputs and less number of classes) with a much simpler predictor model architecture (2-layered convolutional neural networks). ",
+        "bbox": [
+            174,
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+            632
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "At inference, the data value estimator can be used to obtain data value for each sample. The runtime of data valuation is typically much faster (less than 1 ms per sample) than the predictor model (e.g. ResNet-32 model). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            397,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 ADDITIONAL RESULTS ON ROBUST LEARNING WITH NOISY LABELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            135,
+            681,
+            148
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We evaluate how DVRL can provide robustness for learning with noisy labels. We add various levels of label noise, ranging from $0 \\%$ to $50 \\%$ , to the training sets and evaluate how robust the proposed model (DVRL) is for the noisy dataset. In this experiment, we use three image datasets (CIFAR-10, Flower, and HAM 10000). Note that we initialize the predictor model using pre-trained Inception-v3 networks on ImageNet and only fine-tune the top layer (transfer learning setting). ",
+        "bbox": [
+            173,
+            159,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b2671e034dcf684f79ebf801c33d7850e9a4b3ad693678b2e3f6c982072392ab.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Noiseratio</td><td rowspan=1 colspan=6>CIFAR-10</td><td rowspan=1 colspan=3>Flower</td><td rowspan=1 colspan=3>HAM10000</td></tr><tr><td rowspan=1 colspan=2>Clean</td><td rowspan=1 colspan=1>DVRL</td><td rowspan=1 colspan=3>Baseline</td><td rowspan=1 colspan=1>Clean</td><td rowspan=1 colspan=1>DVRL</td><td rowspan=1 colspan=1>Baseline</td><td rowspan=1 colspan=1>Clean</td><td rowspan=1 colspan=1>DVRL</td><td rowspan=1 colspan=1>Baseline</td></tr><tr><td rowspan=1 colspan=1>0%</td><td rowspan=1 colspan=2>.8297</td><td rowspan=1 colspan=1>.8305</td><td rowspan=1 colspan=3>.8297</td><td rowspan=1 colspan=1>.9090</td><td rowspan=1 colspan=1>.9292</td><td rowspan=1 colspan=1>.9090</td><td rowspan=1 colspan=1>.7129</td><td rowspan=2 colspan=1>.7148.7142</td><td rowspan=5 colspan=1>.7129.6746.6199.5508.4819.4132</td></tr><tr><td rowspan=1 colspan=1>10%</td><td rowspan=1 colspan=2>.8281</td><td rowspan=1 colspan=1>.8306</td><td rowspan=1 colspan=3>.7713</td><td rowspan=1 colspan=1>.9057</td><td rowspan=1 colspan=1>.9158</td><td rowspan=1 colspan=1>.7441</td><td rowspan=1 colspan=1>.7094</td></tr><tr><td rowspan=1 colspan=1>20%</td><td rowspan=1 colspan=2>.8285</td><td rowspan=1 colspan=1>.8271</td><td rowspan=1 colspan=3>.6883</td><td rowspan=1 colspan=1>.9026</td><td rowspan=2 colspan=1>.9152.8901</td><td rowspan=2 colspan=1>.5960.4546</td><td rowspan=2 colspan=1>.7098.7063</td><td rowspan=2 colspan=1>.7126.7005</td></tr><tr><td rowspan=1 colspan=1>30%</td><td rowspan=1 colspan=2>.8283</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>.8262</td><td rowspan=1 colspan=1>.5897</td><td rowspan=1 colspan=1>.5897</td><td rowspan=1 colspan=1>.8889</td></tr><tr><td rowspan=1 colspan=1>40%50%</td><td rowspan=1 colspan=2>.8259.8236</td><td rowspan=1 colspan=1>.8259</td><td rowspan=1 colspan=3>.8255.8225</td><td rowspan=1 colspan=1>.4887.3832</td><td rowspan=1 colspan=1>.8620.8542</td><td rowspan=1 colspan=1>.8787.8678</td><td rowspan=1 colspan=1>.2929.2962</td><td rowspan=1 colspan=1>.7028.7009</td><td rowspan=1 colspan=1>.6968.6814</td></tr></table>",
+        "bbox": [
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+            242,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Robust learning results with various noise levels on CIFAR-10, Flower, and HAM 10000 datasets. Clean is the performance of the predictor model when it is only trained with the samples with clean labels (e.g. at $20 \\%$ noise level, it uses only $80 \\%$ clean samples). Baseline is the performance of the predictor model when it is trained with both noisy and clean labels. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Noisy labels significantly degrade the prediction performance when they are included in the training dataset (see the increasing differences between Baseline and Clean in Table 4). DVRL demonstrates high robustness up to high noisy label ratio $( 5 0 \\% )$ . In some cases (even without noisy labels (i.e. $0 \\%$ noise ratio)), the prediction performance even outperforms the Clean case, as DVRL prioritizes some clean samples more than others. Overall, DVRL framework is promising in maintaining high prediction performance even with a significant increase in the amount of noisy labels. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.2 ADDITIONAL RESULTS ON ROBUST LEARNING WITH NOISY FEATURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            559,
+            697,
+            571
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider training with noisy input features, with a clean validation set. We independently add Gaussian noise with zero mean and a certain standard deviation of $\\sigma$ to each feature in the training set independently. We use two tabular datasets (Adult and Blog) to evaluate the robustness of DVRL on input noise. As can be seen in Table 5, DVRL is robust with noise on the features and the performance gains are higher with larger noise in comparison to Baseline (i.e. treat all the noisy training samples equally), since DVRL can discover the training samples with less corrupted by the additive noise among the entire noisy training samples and provide higher weights on those less noisy samples. ",
+        "bbox": [
+            173,
+            583,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/7dfb2c46ad3353b34c298237f0ec12abde69c96183caa43dbf548c588eac3b5e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">0</td><td colspan=\"2\">Blog</td><td colspan=\"2\">Adult</td></tr><tr><td>Baseline</td><td>DVRL</td><td>Baseline</td><td>DVRL</td></tr><tr><td>0.1</td><td>0.733</td><td>0.819</td><td>0.802</td><td>0.820</td></tr><tr><td rowspan=\"3\">0.2 0.3 0.4</td><td>0.647</td><td>0.798</td><td>0.753</td><td>0.788</td></tr><tr><td>0.626</td><td>0.766</td><td>0.699</td><td>0.771</td></tr><tr><td>0.623</td><td>0.717</td><td>0.652</td><td>0.725</td></tr></table>",
+        "bbox": [
+            336,
+            708,
+            660,
+            814
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Testing accuracy when trained with noisy features. $\\sigma$ is the standard deviation of the added Gaussian noise, quantifying the level of perturbation on the features. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/66bf572846027ff8c32ed0563ba936b815291648fb3675f2840f756bff827a7f.jpg",
+        "image_caption": [
+            "Figure 7: Prediction performance after removing the most and least important samples, according to the estimated data values. We assume a label noise with $20 \\%$ ratio on (a) Blog, (b) HAM 10000, (c) CIFAR-10 datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            161,
+            797,
+            339
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/61af9634961f353198225f33c31cf44e55edb94d304f1d238af1698d1f5e3aaf.jpg",
+        "image_caption": [
+            "Figure 8: Discovering corrupted samples in three datasets ((a) Adult, (b) Fashion-MNIST, (c) Flower datasets) in the presence of $20 \\%$ noisy labels. ‘Optimal’ saturates at the $20 \\%$ of the fraction, perfectly assigning the lowest data value scores to the samples with noisy labels. ‘Random’ does not introduce any knowledge on distinguishing clean vs. noisy labels, and thus the fraction of discovered corrupt samples is proportional to the amount of inspection. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            474,
+            797,
+            651
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": ".5 COMPARISON TO OTHER DOMAIN ADAPTATION BENCHMARKS ",
+        "text_level": 1,
+        "bbox": [
+            187,
+            104,
+            637,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we compare DVRL to two established domain adaptation benchmarks: Adversarial Discriminative Domain Adaptation (ADDA) (Tzeng et al., 2017) and Domain Adversarial Neural Networks (DANN) (Ganin et al., 2016). We use the same experimental settings given in Table 3 using Rossmann Store Sales dataset with neural networks as the predictor model. Table 6 represents the domain adaptation results on ‘Train on all’ and ‘Train on Rest’ settings. As can be seen, DVRL yields superior (or similar in a few cases) compared to the two methods, ADDA and DANN, that are specifically designed for domain adaptation. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            228
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/5ce1b8f0787cad01e1605b83a0c1d2998d029ec094073f194c146d6b40f29413.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Settings</td><td colspan=\"4\">Train on All</td><td colspan=\"4\">Train on Rest</td></tr><tr><td>Methods</td><td>Baseline</td><td>DVRL</td><td>ADDA</td><td>DANN</td><td>Baseline</td><td>DVRL</td><td>ADDA</td><td>DANN</td></tr><tr><td>A</td><td>0.1531</td><td>0.1428</td><td>0.1465</td><td>0.1491</td><td>0.3124</td><td>0.2014</td><td>0.2119</td><td>0.2305</td></tr><tr><td>B</td><td>0.1529</td><td>0.0979</td><td>0.1193</td><td>0.1201</td><td>0.8071</td><td>0.5461</td><td>0.5444</td><td>0.5898</td></tr><tr><td>C</td><td>0.1620</td><td>0.1437</td><td>0.1503</td><td>0.1589</td><td>0.2153</td><td>0.1804</td><td>0.1871</td><td>0.1963</td></tr><tr><td>D</td><td>0.1459</td><td>0.1295</td><td>0.1351</td><td>0.1388</td><td>0.2625</td><td>0.1624</td><td>0.1910</td><td>0.2061</td></tr></table>",
+        "bbox": [
+            184,
+            239,
+            813,
+            344
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Performance of Baseline, DVRL, ADDA, and DANN in train-on-all and train-on-rest settings with neural networks as the predictor model on the Rossmann Store Sales dataset. Metric is Root Mean Squared Percentage Error (RMSPE, lower the better). We use $79 \\%$ of the data as training, $1 \\%$ as validation, and $20 \\%$ as testing. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C.6 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            438,
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+            452
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we analyze the source of gains for three distinct components of DVRL: (1) discrete representations of data value estimator, (2) baseline for stabilizing the RL training, (3) output of the model trained on the clean validation set as the additional input (validation model). We report the corrupted sample discovery results where the experimental settings are same with Section 4.2. ",
+        "bbox": [
+            173,
+            463,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/6161b37f59c2cb9c782732ceacf683bd0f2cf79c2194d971018b18417bb68035.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models /Datasets</td><td>Blog</td><td>HAM-10000</td><td>CIFAR-10</td></tr><tr><td>DVRL</td><td>47.3%</td><td>60.2%</td><td>68.1%</td></tr><tr><td>DVRL without sampler</td><td>44.9%</td><td>58.3%</td><td>63.7%</td></tr><tr><td>DVRL without baseline</td><td>45.8%</td><td>56.6%</td><td>62.9%</td></tr><tr><td>DVRL without validation model</td><td>43.7%</td><td>57.1%</td><td>64.4%</td></tr><tr><td>Validation model only</td><td>43.1%</td><td>55.9%</td><td>62.3%</td></tr></table>",
+        "bbox": [
+            254,
+            532,
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+            631
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Discovering corrupted samples in three datasets with $20 \\%$ noisy label ratio. We report the fraction of discovered corrupted samples after inspecting $20 \\%$ of the samples with multiple variants of DVRL (the higher the better). ",
+        "bbox": [
+            174,
+            641,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "As can be seen in Table 7, each component provides an additional gain in DVRL: ",
+        "bbox": [
+            173,
+            699,
+            705,
+            713
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "(1) A straightforward idea is to use the raw outputs of DVE to scale the contributions of each sample in the loss term, without using the sampler. Yet, we show the benefit of the discrete representation of DVE for data selection. The sampler encourages exploration of an extremely large action space in a systematic way. This helps DVE and predictor model to converge to a better optimal solution. ",
+        "bbox": [
+            173,
+            720,
+            825,
+            777
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "(2) Baseline stabilizes the convergence of reinforcement learning; thus, yields higher gains on complex datasets. ",
+        "bbox": [
+            173,
+            784,
+            821,
+            813
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "(3) The output of the validation model itself has informative signal as it achieves high performance (since it is trained with small-scale but high quality data). We observe that this signal helps DVRL, but even without this signal achieves high performance. We also observe that often a larger DVE model (with more iterations) is needed to estimate the data value in the absence of the informative signal from the validation model. ",
+        "bbox": [
+            174,
+            819,
+            825,
+            888
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Note that we propose to use the output of the validation model as an additional input to the data valuation framework; thus, this can also be regarded as another contribution of our work. Also, the output of the validation model is highly informative in the noisy sample discovery use case but not that significant in other applications such as domain adaptation or performance improvement by low value data removal in standard supervised learning setting. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D LEARNING CURVES OF DVRL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            166,
+            459,
+            181
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Fig. 9 shows the learning curves of DVRL on the noisy data (with $20 \\%$ label noise) setting in comparison to the validation log loss without DVRL (directly trained on the noisy data without reweighting) on 2 tabular datasets (Adult and Blog) and 4 image datasets (Fashion-MNIST, Flower, HAM 10000, and CIFAR-10). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/cd77e87d1533903c3385937da53bfe0722277f73a0c8d2b26bdae58b133d43fc.jpg",
+        "image_caption": [
+            "Figure 9: Learning curves of DVRL for 6 datasets with $20 \\%$ noisy labels. $\\mathbf { X }$ -axis: the number of iterations for data value estimator training, y-axis: validation performance (log loss). (Orange: validation log loss without DVRL, Blue: validation log loss with DVRL) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "E CONFIDENCE INTERVALS OF DVRL PERFORMANCE ON CORRUPTED SAMPLE DISCOVERY EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            564,
+            771,
+            598
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/7446440d4cbcb850178c099bfaf3da606684073e4e7caa73dddd2f7a5edc3e1b.jpg",
+        "image_caption": [
+            "Figure 10: Corrupted sample discovery performance with $9 5 \\%$ confidence intervals (computed by 10 independent runs) according to the estimated data values by DVRL. We assume a label noise with $20 \\%$ ratio on (a) Adult and Blog, (b) Fashion-MNIST and Flower (c) HAM 10000 and CIFAR-10 datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            619,
+            810,
+            818
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/4bb38075d073c6bc24cac368b67bf497fd4721d457b4105a832a97ec36650c67.jpg",
+        "table_caption": [
+            "F ROSSMANN DATA STATISTICS & T-SNE ANALYSIS ",
+            "Table 8: Rossmann data statistics. Report 25-50-75 percentiles for sales and customers. # represents the number. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Store Type</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>B</td><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=1>D</td></tr><tr><td rowspan=1 colspan=2>#of Samples 457042 (54.1%)</td><td rowspan=1 colspan=3>15560 (1.8%)   112968 (13.4%) 258768 (30.6%)</td></tr><tr><td rowspan=1 colspan=5>Sales      1390-1660-1854 2052-2459-2661 1753-1974-2178 2109-2355-2524</td></tr><tr><td rowspan=1 colspan=1>Customers</td><td rowspan=1 colspan=1>169-203-221</td><td rowspan=1 colspan=1>436-492-543</td><td rowspan=1 colspan=1>192-232-259</td><td rowspan=1 colspan=1>224-246-259</td></tr></table>",
+        "bbox": [
+            184,
+            137,
+            813,
+            222
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/2ad702c797202d40b0d186a6d287bf01cd69f91edb75d409e6f117389b0aac8b.jpg",
+        "image_caption": [
+            "Figure 11: t-SNE analyses on the final layer representations of each store type in Rossmann dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            282,
+            820,
+            421
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/43fabb76041e17b4bdcbfd00eabd45757b2d0972aaf5d204c8eaf99aa6d84772.jpg",
+        "image_caption": [
+            "G FURTHER ANALYSIS ON ROSSMANN DATASET IN Train on All SETTING ",
+            "Figure 12: Histograms of the training samples from the target store type in Train on All setting based on the sorted data values estimated by DVRL. $\\mathbf { \\check { X } }$ -axis: the sorted data values (in percentiles), y-axis: counts of training samples from the target store type (in ratio). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            512,
+            823,
+            599
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "To further understand the results in Train on All setting, we sorted (in a decreasing order) the training samples by their data values estimated by DVRL and illustrate the distributions of the training samples that come from the target store type. As can be seen in Fig. 12, DVRL prioritizes the training samples which come from the same target store type. ",
+        "bbox": [
+            173,
+            674,
+            825,
+            731
+        ],
+        "page_idx": 17
+    }
+]

parse/train/By41BjA9YQ/By41BjA9YQ.md

@@ -0,0 +1,529 @@
+# LAPLACIAN SMOOTHING GRADIENT DESCENT
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a class of very simple modifications of gradient descent and stochastic gradient descent. We show that when applied to a large variety of machine learning problems, ranging from softmax regression to deep neural nets, the proposed surrogates can dramatically reduce the variance and improve the generalization accuracy. The methods only involve multiplying the usual (stochastic) gradient by the inverse of a positive definitive matrix coming from the discrete Laplacian or its high order generalizations. The theory of Hamilton-Jacobi partial differential equations demonstrates that the implicit version of new algorithm is almost the same as doing gradient descent on a new function which (i) has the same global minima as the original function and (ii) is “more convex”. We show that optimization algorithms with these surrogates converge uniformly in the discrete Sobolev $H _ { \sigma } ^ { p }$ sense and reduce the optimality gap for convex optimization problems. We implement our algorithm into both PyTorch and Tensorflow platforms which only involves changing of a few lines of code. The code will be available on Github.
+
+# 1 INTRODUCTION
+
+Stochastic gradient descent (SGD) has been the workhorse for solving large-scale machine learning problems (Bottou et al., 2018). It gives rise to a family of algorithms that make training of deep neural nets (DNN) practical, which is believed to somehow implicitly smooth the loss function of the DNN (Jastrzebski et al., 2018). Many efforts have been carried out to improve training and generalization of DNN by directly searching for flat minima (Keskar et al., 2017; Chaudhari et al., 2017; 2016). An alternative view of SGD’s magic comes from the theory of uniform stability (Bousquet & Elisseeff, 2002; Duchi et al., 2011; Hardt et al., 2016; Bottou et al., 2016; Gonen & Shalev-Shwartz, 2017).
+
+The noise in SGD, on the one hand, helps gradient-based optimization algorithms circumvent spurious local minima and reach those that generalize well (Schmidhuber, 2014). On the other hand, it slows down the convergence of regular gradient descent (GD). To recover the linear convergence rate for strongly convex functions, several interesting variance reduction algorithms have been proposed, e.g., SAGA (Defazio & Bach, 2014) and SVRG (Johoson & Zhang, 2013). These algorithms have a certain amount of difficulty in training DNN. SAGA has a relatively high space complexity in storing the gradient for many samples. SVRG requires computation of the full batch gradient.
+
+In this work, we propose a carefully designed positive definite matrix to smooth and to reduce variance of the (stochastic) gradient on-the-fly. The resulting surrogate tends to reduce noise in SGD and improve training of DNN. We call this procedure Laplacian smoothing. The gradient smoothing can be done by multiplying the gradient by the inverse of the following circulant convolution matrix
+
+$$
+\begin{array} { r } { A _ { \sigma } : = \left[ { \begin{array} { c c c c c c } { 1 + 2 \sigma } & { - \sigma } & { 0 } & { . . . } & { 0 } & { - \sigma } \\ { - \sigma } & { 1 + 2 \sigma } & { - \sigma } & { . . . } & { 0 } & { 0 } \\ { 0 } & { - \sigma } & { 1 + 2 \sigma } & { . . . } & { 0 } & { 0 } \\ { . . . } & { . . . } & { . . . } & { . . . } & { . . . } & { . . . } \\ { - \sigma } & { 0 } & { 0 } & { . . . } & { - \sigma } & { 1 + 2 \sigma } \end{array} } \right] } \end{array}
+$$
+
+for some positive constant $\sigma \geq 0$ . In fact, we can write $\mathbf { } A _ { \sigma } = I - \sigma L$ , where $\pmb { I }$ is the identity matrix, and $\pmb { L }$ is the discrete one-dimensional Laplacian which acts on indices. We define the (periodic)
+
+forward finite difference matrix as
+
+$$
+\pmb { { \cal D } } _ { + } = \left[ \begin{array} { c c c c c c c } { - 1 } & { 1 } & { 0 } & { . . . } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 1 } & { . . . } & { 0 } & { 0 } \\ { 0 } & { 0 } & { - 1 } & { . . . } & { 0 } & { 0 } \\ { . . . } & { . . . } & { . . . } & { . . . } & { . . . } & { . . . } \\ { 1 } & { 0 } & { 0 } & { . . . } & { 0 } & { - 1 } \end{array} \right] .
+$$
+
+Then, we have $A _ { \sigma } = I - \sigma D _ { - } D _ { + }$ , where ${ \pmb D } _ { - } = - { \pmb D } _ { + } ^ { \top }$ is the backward finite difference. The resulting Laplacian smoothing stochastic gradient descent (LS-SGD) requires negligible extra computational cost and generalizes better than the standard SGD. When the Hessian has a poor condition number, gradient descent performs poorly. In this case, the derivative increases rapidly in one direction, while increasing slowly in others. Gradient smoothing can avoid jitter along steep directions and help make progress in shallow directions (Li & et al, 2018). Moreover, we show that the operator $A _ { \sigma } ^ { \div 1 }$ plays role as a denoiser which enables better convergence in the presence of a very noisy stochastic gradient. The implicit version of our proposed approach is linked to an unusual HamiltonJacobi partial differential equation (HJ-PDE) whose solution makes the original loss function more convex while retaining its flat (and global) minima, and essentially works on this surrogate function with a much better landscape.
+
+# 2 HAMILTON-JACOBI PDES AND CONVEXIFICATION
+
+Machine learning problems are generally formulated as finding the optimal parameters $\pmb { w }$ of a parametric function ${ \pmb y } = h ( { \pmb x } , { \pmb w } )$ , such that for an input $_ { \textbf { \em x } }$ , the output $\textbf {  { y } }$ is close to the ground-truth. The optimal $\textbf { \em w }$ can be obtained by minimizing an empirical risk function, $f ( X , Y , w ) \bar { \doteq } f ( w )$ , given the training data $\{ X , Y \}$ . We start from the following unusual HJ-PDE with $f ( w )$ as initial condition
+
+$$
+\left\{ \begin{array} { l l } { u _ { t } + \frac { 1 } { 2 } \big \langle \nabla _ { w } u , A _ { \sigma } ^ { - 1 } \nabla _ { w } u \big \rangle = 0 , } & { ( w , t ) \in \Omega \times [ 0 , \infty ) } \\ { u ( w , 0 ) = f ( w ) , } & { w \in \Omega } \end{array} \right.
+$$
+
+By the Hopf-Lax formula (Evans, 2010), the unique viscosity solution to Eq. (2) is represented by
+
+$$
+u ( { \pmb w } , t ) = \operatorname* { i n f } _ { { \pmb v } } \Big \{ f ( { \pmb v } ) + \frac { 1 } { 2 t } \big \langle { \pmb v } - { \pmb w } , { \pmb A } _ { \sigma } ( { \pmb v } - { \pmb w } ) \big \rangle \Big \} .
+$$
+
+This viscosity solution $u ( { \boldsymbol { w } } , t )$ makes $f ( w )$ ”more convex”, an intuitive definition and theoretical explanation of ”more convex” can be found in (Chaudhari et al., 2017; 2016), by bringing down the local maxima while retaining and widening local minima. An illustration of this is shown in Fig. 1. If we perform the smoothing GD with proper step size on the function $u ( { \boldsymbol { w } } , t )$ , it is easier to reach the global or at least a flat minima of the original nonconvex function $f ( w )$ .
+
+![](images/03cba460155bd80e1a8bb55cb7d7bbf8f0b87e9bdcb20e524c6ae323daeba2b5.jpg)  
+Figure 1: $\begin{array} { r } { f ( \pmb { w } ) = \| \pmb { w } \| ^ { 2 } \big ( 1 + \frac { 1 } { 2 } \sin ( 2 \pi \| \pmb { w } \| ) \big ) } \end{array}$ is made more convex by solving Eq.(2). The plot shows the cross section of the 5D problem with $\sigma = 1$ and different $t$ values.
+
+Proposition 1. Suppose $f ( w )$ is differentiable, the LS-GD on $u ( { \boldsymbol { w } } , t )$
+
+$$
+\pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - t \pmb { A } _ { \sigma } ^ { - 1 } \nabla _ { \pmb { w } } u ( \pmb { w } ^ { k } , t )
+$$
+
+is equivalent to the smoothing implicit $G D$ on $f ( w )$
+
+$$
+\begin{array} { r } { \pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - t \pmb { A } _ { \sigma } ^ { - 1 } \nabla f ( \pmb { w } ^ { k + 1 } ) . } \end{array}
+$$
+
+All the proofs here and below are provided in the appendix.
+
+# 2.1 LAPLACIAN SMOOTHING GRADIENT DESCENT
+
+Laplacian smoothing implicit gradient descent requires inner iterations as used in (Chaudhari et al., 2017), which is computationally expensive. We consider the following explicit scheme
+
+$$
+\pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - \gamma _ { k } \pmb { A } _ { \sigma } ^ { - 1 } \nabla f ( \pmb { w } ^ { k } ) .
+$$
+
+Intuitively, compared to the standard GD, this scheme smooths the gradient on-the-fly by an elliptic smoothing operator. We adopt fast Fourier transform (FFT) to compute $A _ { \sigma } ^ { - 1 } \nabla f ( \mathbf { \bar { w } } ^ { k } )$ , which is available in both PyTorch (Paszke et al., 2017) and TensorFlow (Abadi et al., 2016). Given a vector $\textbf {  { g } }$ , a smoothed vector $^ d$ can be obtained by computing $d = { \bf \nabla } \cdot { \bf \dot { A } } _ { \sigma } ^ { - 1 } g$ . This is equivalent to ${ \textbf { \em g } } =$ $d - \sigma v * d$ , where $\pmb { v } = [ - 2 , 1 , 0 , \cdots , 0 , 1 ] ^ { \dag }$ and $^ *$ is the convolution operator. Therefore
+
+$$
+d = \operatorname { i f f } \left( { \frac { \operatorname { f f t } ( g ) } { 1 - \sigma \cdot \operatorname { f f t } ( v ) } } \right) ,
+$$
+
+where we use component-wise division, fft and ifft are the FFT and inverse FFT, respectively. Hence, the gradient smoothing can be done in quasilinear time. This additional time complexity is almost the same as performing a one step update on the weights vector w. For many machine learning models, we may need to concatenate the parameters into a vector. This reshape might lead to some ambiguity, nevertheless, based on our tests, both row and column majored reshaping work for the LS-GD algorithm. Moreover, in deep learning cases, the weights in different layers might have different physical meanings. We then perform layer-wise gradient smoothing, instead.
+
+Remark 1. In image processing, the Sobolev gradient (Jung et al., 2009) involves a multidimensional Laplacian operator which operates on $\textbf { \em w }$ , is different from the one-dimensional discrete Laplacian operator employed in our LS-GD scheme, which operates on indices.
+
+We first show that LS-GD can help bypass sharp minima and reach the global minima. We consider the following function, in which we ‘drill’ narrow holes on a smooth convex function,
+
+$$
+f ( x , y , z ) = - 4 e ^ { - \left( ( x - \pi ) ^ { 2 } + ( y - \pi ) ^ { 2 } + ( z - \pi ) ^ { 2 } \right) } - 4 \sum _ { i } \cos ( x ) \cos ( y ) e ^ { - \beta \left( ( x - r \sin ( \frac { i } { 2 } ) - \pi ) ^ { 2 } + ( y - r \cos ( \frac { i } { 2 } ) - \pi ) ^ { 2 } \right) } ,
+$$
+
+where the summation is taken over the index set $\{ i \in \mathbb { N } | 0 \leq i < 4 \pi \}$ , $r$ and $\beta$ are the parameters that determine the location and narrowness of the local minima and are set to 1 and $\frac { 1 ^ { \bullet } } { \sqrt { 5 0 0 } }$ , respectively. We do GD and LS-GD starting from a random point in the neighborhoods of the narrow minima, i.e., $( x _ { 0 } , y _ { 0 } , z _ { 0 } ) \in \{ \bigcup _ { i } U _ { \delta } ( r \sin ( \frac { i } { 2 } ) + \pi , r \cos ( \frac { i } { 2 } ) + \pi ) | \ 0 \leq i < 4 \pi , i \in \mathbb { N } _ { \neq } \} .$ , where $U _ { \delta } ( P )$ is a neighborhood of the point $P$ with radius $\delta$ . Our experiments (Fig. 2) show that, if $\delta \leq 0 . 2$ , GD will converge to narrow local minima, while LS-GD convergences to wider global minima.
+
+![](images/3673951f8b8c3580cd67eecdc9eb75a78c1d84f80651e61180caeeba48964bf2.jpg)  
+Figure 2: Demo of GD and LS-GD. Panel (a) depicts the slice of the function (Eq.(4)) with $z = 2 . 3 4$ panel (b) shows the paths of GD (red) and LS-GD (black). We take the step size to be 0.02 for both GD and LS-GD. $\sigma = 1 . 0$ is utilized for LS-GD.
+
+2.2 GENERALIZED SMOOTHING GRADIENT DESCENT
+
+We can generalize $A _ { \sigma }$ to the $n$ th order discrete hyper-diffusion operator as follows
+
+$$
+\pmb { I } + ( - 1 ) ^ { n } \sigma \pmb { L } ^ { n } \doteq \pmb { A } _ { \sigma } ^ { n } .
+$$
+
+Each row of the discrete Laplacian operator $\pmb { L }$ consists of an appropriate arrangement of weights in central finite difference approximation to the 2nd order derivative. Similarly, each row of ${ \pmb L } ^ { n }$ is an arrangement of the weights of the central finite difference to approximate the $2 n$ th order derivative.
+
+Remark 2. The nth order smoothing operator $\pmb { I } + ( - 1 ) ^ { n } \sigma \pmb { L } ^ { n }$ can only be applied to the problem with dimension at least $2 n + 1$ . Otherwise, we need to add dummy variables to the object function.
+
+Again, we apply FFT to compute the smoothed gradient vector. For a given gradient vector $\textbf {  { g } }$ , the smoothed surrogate, $( A _ { \sigma } ^ { n } ) ^ { - 1 } { \dot { \pmb { g } } } \doteq { \pmb { d } }$ , can be obtained by solving $\pmb { g } = \pmb { d } + ( - 1 ) ^ { n } \sigma \pmb { v } _ { n } * \pmb { d }$ , where $\pmb { v _ { n } } = ( c _ { n + 1 } ^ { n } , c _ { n + 2 } ^ { n } , \cdots , \overset { \cdot \cdot } { c _ { 2 n + 1 } ^ { n } } , 0 , \cdots , 0 , c _ { 1 } ^ { n } , c _ { 2 } ^ { n } , \cdots , c _ { n - 1 } ^ { n } , c _ { n } ^ { n } )$ is a vector of the same dimension as the gradient to be smoothed. And the coefficient vector $\pmb { c } ^ { n } = ( c _ { 1 } ^ { n } , c _ { 2 } ^ { n } , \cdot \cdot \cdot , c _ { 2 n + 1 } ^ { n } )$ can be obtained recursively by the following formula
+
+$$
+\begin{array} { r } { \boldsymbol { c } ^ { 1 } = ( 1 , - 2 , 1 ) , \quad \boldsymbol { c } _ { i } ^ { n } = \left\{ \begin{array} { l l } { 1 } & { i = 1 , 2 n + 1 } \\ { - 2 \boldsymbol { c } _ { 1 } ^ { n - 1 } + \boldsymbol { c } _ { 2 } ^ { n - 1 } } & { i = 2 , 2 n } \\ { \boldsymbol { c } _ { i - 1 } ^ { n - 1 } - 2 \boldsymbol { c } _ { i } ^ { n - 1 } + \boldsymbol { c } _ { i + 1 } ^ { n - 1 } } & { \mathrm { o t h e r w i s e . } } \end{array} \right. } \end{array}
+$$
+
+Remark 3. The computational complexities for different order smoothing schemes are the same when the FFT is utilized for computing the surrogate gradient.
+
+# 3 REDUCE OPTIMALITY GAP IN SGD
+
+We show advantages of the LS-(S)GD and generalized schemes for convex optimization. Consider finding the minima $\pmb { x } ^ { * }$ of the quadratic function $f ( { \pmb x } )$ defined in Eq. (5) by different schemes.
+
+$$
+f ( x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { 1 0 0 } ) = \sum _ { i = 1 } ^ { 5 0 } x _ { 2 i - 1 } ^ { 2 } + \sum _ { i = 1 } ^ { 5 0 } \frac { x _ { 2 i } ^ { 2 } } { 1 0 ^ { 2 } } .
+$$
+
+To simulate SGD, we add Gaussian noise to the gradient vector, i.e., at a given point $_ { \textbf { \em x } }$ , we have
+
+$$
+\begin{array} { r } { \tilde { \nabla } _ { \epsilon } f ( x ) : = \nabla f ( x ) + \epsilon \mathcal { N } ( \mathbf { 0 } , I ) , } \end{array}
+$$
+
+where the scalar $\epsilon$ controls the noise level, $\mathcal { N } ( \mathbf { 0 } , \pmb { I } )$ is the vector with zero mean and unit variance in each coordinate. The corresponding numerical schemes can be formulated as
+
+$$
+\begin{array} { r } { \pmb { x } ^ { k + 1 } = \pmb { x } ^ { k } - \eta _ { k } \big ( \pmb { A } _ { \sigma } ^ { n } \big ) ^ { - 1 } \tilde { \nabla } _ { \epsilon } f ( \pmb { x } ^ { k } ) , } \end{array}
+$$
+
+where $\sigma$ is the smoothing parameter selected to be 10.0 to kill the intense noise. We take diminishing step sizes with initial values 0.1 for SGD/smoothed SGD; 0.9 and 1.8 for GD/smoothed GD, respectively. Without noise, the smoothing allows us to take larger step sizes, rounding to the first digit, 0.9 and 1.9 are the largest suitable step size for GD and smoothed version here. We compare constant learning rate and exponentially decaying learning rate, i.e., after every 1000 iteration, the learning rate is divided by 10. We apply different schemes that corresponding to $n = 0 , 1 , 2$ in Eq. (6) to the problem Eq. (5), with the initial point $\pmb { x } ^ { 0 } = ( 1 , 1 , \cdots , 1 )$ .
+
+Figure. 3 shows the iteration v.s. optimality gap when the constant learning rate is applied to different noise levels. In the noise free case, all three schemes converge linearly, but gradient smoothing has a smaller decay constant due to its increased condition number. When there is noise, our smoothed gradient helps to reduce the optimality gap and converges faster after a few iterations.
+
+![](images/d46574d795c170a8a11da6e87e42d878fbe59d54af8dc0251dd9ff811f5b6035.jpg)  
+Figure 3: Iterations v.s. optimality gap for GD and smoothed GD with order 1 and 2 for the problem in Eq.(5). Constant step size was used.
+
+The exponentially decaying learning rate helps our smoothed SGD to reach a point with a smaller optimality gap, and the higher order smoothing further reduce the optimality gap, as shown in Fig. 4. One simple reason for this in the noisy case is because of the noise removal properties of the smoothing operators. The influence of the learning rate is still under investigation. We establish the convergence of our proposed smoothing gradient descent algorithms.
+
+We say the objective function $f$ has $L$ -Lipschitz gradient, if for any $\pmb { w } , \pmb { u } \in \mathbb { R } ^ { m }$ , we have $\lVert \nabla f ( \pmb { w } ) -$ $\nabla f ( \pmb { u } ) \| \leq L \| \pmb { w } - \pmb { u } \|$ , and $f$ is $a$ -strongly convex, if $\langle \nabla f ( { \pmb w } ) - \nabla f ( { \pmb u } ) , { \pmb w } - { \pmb u } \rangle \geq a \| { \pmb w } - { \pmb u } \| ^ { 2 }$ . We define the vector norm induced by any matrix $\pmb { A }$ as $\| \pmb { w } \| _ { \pmb { A } } : = \sqrt { \langle \pmb { w } , \pmb { A } \pmb { w } \rangle }$ .
+
+![](images/68ec06a04a38fce5f798482784b22ac2e00cb0fc24a7f9011bf7b4365fa4e814.jpg)  
+Figure 4: Iterations v.s. optimality gap for GD and smoothed GD with order 1 and 2 for the problem in Eq.(5). Exponentially decaying step size is utilized here.
+
+Proposition 2. Suppose $f$ is convex with the global minimizer $\boldsymbol { w } ^ { * }$ , and $f ^ { * } = f ( w ^ { * } )$ . Consider the following iteration with constant learning rate $\eta > 0$
+
+$$
+\mathbf { \boldsymbol { w } } ^ { k + 1 } = \mathbf { \boldsymbol { w } } ^ { k } - \eta ( \mathbf { \boldsymbol { A } } _ { \sigma } ^ { n } ) ^ { - 1 } \mathbf { \boldsymbol { g } } ^ { k } ,
+$$
+
+where $g ^ { k }$ is the sampled gradient in the kth iteration at $\boldsymbol { w } ^ { k }$ satisfying $\mathbb { E } [ \pmb { g } ^ { k } ] = \nabla f ( \pmb { w } ^ { k } )$ . Denote $\begin{array} { r } { G _ { { \pmb A } _ { \sigma } ^ { n } } : = \operatorname* { l i m } _ { K  \infty } \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \| \pmb { g } ^ { k } \| _ { ( { \pmb A } _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } } \end{array}$ and $\begin{array} { r } { \overline { { \mathbf { w } } } ^ { K } : = \sum _ { k = 0 } ^ { K - 1 } \mathbf { w } ^ { k } / K } \end{array}$ the ergodic average of iterates. Then the optimality gap is
+
+$$
+\operatorname* { l i m } _ { K \to \infty } \mathbb { E } [ f ( \overline { { \pmb { w } } } ^ { K } ) ] - f ^ { * } \leq \frac { \eta G _ { A _ { \sigma } ^ { n } } } { 2 } .
+$$
+
+Note that $\| \pmb { g } \| _ { ( \pmb { A } _ { \sigma } ^ { n } ) ^ { - 1 } }$ generally decreases in $n$ unless $\textbf {  { g } }$ is constant, which indicates that a bigger $n$ implies smaller optimality gap. This is consistent with the experimental results above.
+
+Proposition 3. Suppose $f$ is $L$ -Lipschitz smooth and $a$ -strongly convex with the global minimizer $\boldsymbol { w } ^ { * }$ . Consider the generalized smoothing gradient descent algorithm
+
+$$
+\begin{array} { r } { \pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - \eta _ { k } ( \pmb { A } _ { \sigma } ^ { n } ) ^ { - 1 } \pmb { g } ^ { k } , } \end{array}
+$$
+
+where $g ^ { k }$ is the sampled gradient in the kth iteration at $\boldsymbol { w } ^ { k }$ satisfying $\mathbb { E } \left[ \pmb { g } ^ { k } \right] \ : = \ : \nabla f ( \pmb { w } ^ { k } )$ and $\mathbb { E } \left[ \| \pmb { g } ^ { k } \| _ { ( { \pmb { A } } _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } \right] \leq C _ { 0 } + C _ { 1 } \| \nabla f ( \pmb { w } ^ { k } ) \| ^ { 2 }$ for all $k \in \mathbb N .$ . If we take $\begin{array} { r } { \eta _ { k } = \frac { C } { k + 1 } } \end{array}$ for some $C > 0$ , then we have
+
+$$
+\mathbb { E } \left[ \Vert w ^ { k } - w ^ { * } \Vert _ { A _ { \sigma } ^ { n } } ^ { 2 } \right] = \mathbb { E } \left[ \Vert w ^ { k } - w ^ { * } \Vert ^ { 2 } + \sigma \Vert D _ { + } ^ { n } ( w ^ { k } - w ^ { * } ) \Vert ^ { 2 } \right] = O \left( \frac { 1 } { k + 1 } \right) ,
+$$
+
+i.e., we have $H _ { \sigma } ^ { n }$ uniform convergence in $\sigma$ of $\{ w ^ { k } \}$ in expectation. The $H _ { \sigma } ^ { n }$ norm of $\pmb { w }$ is defined by $\| \pmb { w } \| _ { \sigma } ^ { n } : = \| w \| _ { \pmb { A } _ { \sigma } ^ { n } } = \sqrt { \langle \pmb { w } , \pmb { A } _ { \sigma } ^ { n } \pmb { w } \rangle }$ .
+
+Proposition 4. Consider the algorithm $\pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - \eta _ { k } \big ( \pmb { A } _ { \sigma } ^ { n } \big ) ^ { - 1 } \nabla f \big ( \pmb { w } ^ { k } \big )$ . Suppose $f$ is convex and $L$ -Lipschitz smooth. If the step size satisfies $\begin{array} { r } { 0 < \underline { { \eta } } \le \eta \le \bar { \eta } < \frac { 2 } { L } } \end{array}$ . Then $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \| \nabla f ( \pmb { w } ^ { k } ) \|  0 } \end{array}$ . Moreover, if the Hessian $\nabla ^ { 2 } f$ of $f$ ¯is continuous with $\ b { w } ^ { * }$ being the global minimizer of $f$ , and $\bar { \eta } \| \nabla ^ { 2 } f \| < \bar { 1 }$ , then $\lVert \pmb { w } ^ { k } - \pmb { w } ^ { \ast } \rVert _ { \pmb { A } _ { \sigma } ^ { n } }  0$ as $k  \infty$ , and the convergence is linear and independent of $\sigma$ .
+
+In what follows, we present the noise reduction properties of the proposed smoothing operator $A _ { \sigma } ^ { - 1 }$ . Proposition 5. For any vector $\textbf { \textit { g } } \in \mathbb { R } ^ { m }$ , $d \ = \ A _ { \sigma } ^ { - 1 } g ,$ , let $j _ { \mathrm { m a x } } \ = \ \arg \operatorname* { m a x } _ { i } d _ { i }$ and $j _ { \mathrm { m i n } } \ =$ arg $\operatorname* { m i n } _ { i } d _ { i }$ . We have max ${ \mathrm { \Omega } } _ { i } d _ { i } = d _ { j _ { \operatorname* { m a x } } } \leq g _ { j _ { \operatorname* { m a x } } } \leq \operatorname* { m a x } _ { i } g _ { i }$ and $\mathrm { m i n } _ { i } d _ { i } = d _ { j _ { \mathrm { m i n } } } \geq g _ { j _ { \mathrm { m i n } } } \geq \mathrm { m i n } _ { i } g _ { i }$ . Proposition 6. The operator $A _ { \sigma } ^ { - 1 }$ preserves the sum of components. For any $\pmb { \mathscr { g } } \in \mathbb { R } ^ { m }$ and ${ \pmb d } =$ $A _ { \sigma } ^ { - \bar { 1 } } g _ { \bar { 1 } }$ , we have $\textstyle \sum _ { j } d _ { j } = \sum _ { j } g _ { j }$ , or equivalently, $\mathbf { 1 } ^ { \top } \pmb { d } = \mathbf { 1 } ^ { \top } \pmb { g }$ .
+
+Proposition 7. Given any vector $\pmb { \mathscr { g } } \in \mathbb { R } ^ { m }$ and $\pmb { d } = \pmb { A } _ { \sigma } ^ { - 1 } \pmb { g }$ , then
+
+$$
+\| d \| + \sigma \frac { \| D _ { + } d \| ^ { 2 } } { \| d \| } \leq \| g \| .
+$$
+
+The above inequality is strict unless $\mathbf { \omega } _ { g } = d$ is a constant vector. In particular, we have $\| d \| \leq \| g \|$ and $\begin{array} { r } { \| D _ { + } d \| \le \frac { 1 } { \sqrt { \sigma } } \| \pmb { g } \| } \end{array}$ .
+
+Let $\textbf {  { g } }$ be the noise vector contained in the stochastic gradient, the above results imply that the extreme values in $\pmb { A } _ { \sigma } ^ { - 1 } \pmb { g }$ are smaller than those in $\textbf {  { g } }$ (in magnitude), and it also has a much smaller $\ell _ { 2 }$ norm.
+
+Proposition 8. For any $\pmb { \mathscr { g } } \in \mathbb { R } ^ { m }$ , define $\begin{array} { r } { \mathrm { V a r } ( \pmb { g } ) : = \frac { 1 } { m } \| \pmb { g } \| ^ { 2 } - \bigg ( \frac { \pmb { 1 } ^ { \top } \pmb { g } } { m } \bigg ) ^ { \frac { \pmb { \zeta } } { 2 } } } \end{array}$ be the variance of components in $\textbf {  { g } }$ . Let $\pmb { d } = \pmb { A } _ { \sigma } ^ { - 1 } \pmb { g } ,$ , then
+
+$$
+\mathrm { V a r } ( \pmb { d } ) \leq \mathrm { V a r } ( \pmb { g } ) - 2 \sigma \frac { \| \pmb { D } _ { + } \pmb { d } \| ^ { 2 } } { m } - \sigma ^ { 2 } \frac { \| \pmb { D } _ { + } \pmb { d } \| ^ { 4 } } { m \| \pmb { d } \| ^ { 2 } } .
+$$
+
+The inequality is strict unless $\mathbf { \nabla } _ { \mathbf { { g } } } = d$ is a constant vector.
+
+Proposition 8 shows that the component-wise variance of $\pmb { A } _ { \sigma } ^ { - 1 } \pmb { g }$ is considerably less than that of $\textbf {  { g } }$ , unless $\textbf {  { g } }$ is a constant vector. Our last result shows that $A _ { \sigma } ^ { - 1 } g$ has diminishing $\ell _ { 1 }$ norm of finite difference of all orders. This is an excellent desnoising result.
+
+Proposition 9. Given vectors $\textbf {  { g } }$ and $\pmb { d } = \pmb { A } _ { \sigma } ^ { - 1 } \pmb { g } ,$ , for any $p \in \mathbb N$ , it holds that $\| D _ { + } ^ { p } d \| _ { 1 } \leq \| D _ { + } ^ { p } g \| _ { 1 }$ .   
+The inequality is strict unless $D _ { + } ^ { p } g$ is a constant vector.
+
+Remark 4. The above proofs generalize for $n > 1$ , except for Propositions $^ { 5 }$ and 9.
+
+# 3.2 SOFTMAX REGRESSION
+
+Consider applying the proposed optimization schemes to Softmax regression. We run 200 epochs of SGD and different order smoothing algorithms to maximize the likelihood of Softmax regression with batch size 100. Based on the results from previous section, we apply the exponentially decay learning rate with initial value 0.1 and decay 10 times after every 50 epochs. We train the model with only $1 0 \%$ randomly selected MNIST training data and test the trained model on the entire testing images. We further compare with SVRG under the same setting. Figure. 5 shows the histograms of generalization accuracy of Softmax regression model trained by SGD ((a)); SVRG ((b)); LSSGD (order 1) ((c)); LS-SGD (oder 2) ((d)). It is seen that SVRG improves the generalization with higher average accuracy. But the first and second order smoothing schemes significantly improve averaged generalization accuracy by more than $1 \%$ and reduce the variance over 100 independent trials. The training loss of these 100 experiments by different optimization algorithms are shown in the appendix.
+
+![](images/1cb4e5a1535f8e8134785b6bb8d6bd26bbbbe2b94f956fd9d6803762e876425e.jpg)  
+Figure 5: Testing accuracy of Softmax model trained on randomly selected $1 0 \%$ MNIST data.
+
+# 4 APPLICATIONS TO DEEP NEURAL NETS
+
+4.1 TRAIN NEURAL NETS WITH SMALL BATCH SIZE
+
+Many advanced artificial intelligence tasks make high demand on training neural nets with extremely small batch size. The milestone technique for this is group normalization (Wu & He, 2018). In this section, we show that LS-SGD successfully trains DNN with extremely small batch size. We consider LeNet-5 devised by (LeCun et al., 1998) for MNIST classification. Our network architecture is as follows
+
+$$
+\mathrm { L e N e t - 5 } \mathrm { : i n p u t _ { 2 8 \times 2 8 } \to c o n v _ { 2 0 , 5 , 2 } \to c o n v _ { 5 0 , 5 , 2 } \to f c _ { 5 1 2 } \to s o f t m a x . }
+$$
+
+The notation $\mathrm { c o n v } _ { c , k , m }$ denotes a 2D convolutional layer with $c$ output channels, each of which is the sum of a channel-wise convolution operation on the input using a learnable kernel of size $k \times k$ , it further adds ReLU nonlinearity and max pooling with stride size $m$ . $\mathrm { f c } _ { 5 1 2 }$ is an affine transformation that transforms the input to a vector of dimension 512. Finally, the tensors are activated by a softmax function. The MNIST data is first passed to the layer $\mathrm { i n p u t } _ { 2 8 \times 2 8 }$ , and further processed by this hierarchical structure. We run 100 epochs of both SGD and LS-SGD with initial learning rate 0.01 and divide by 5 after 50 epochs, and use a weight decay of 0.0001 and momentum of 0.9. Figure. 6(a) plots the generalization accuracy on the test set with the LeNet5 trained with different batch sizes. For each batch size, LS-SGD with $\sigma = 1 . 0$ keeps the testing accuracy more than $9 9 . 4 \%$ , SGD reduce the accuracy to $9 7 \%$ when batch size 4 is used. The classification become just a random guess, when the model is trained by SGD with batch size 2. Small batch size leads to large noise in the gradient, which may make the noisy gradient not along the decent direction, However, Lapacian smoothing rescues this by killing the noise.
+
+![](images/a59e509f70fae363dbf7c5afa84745f9f741627bb1be646c44010fb154c8a327.jpg)  
+Figure 6: (a). Testing accuracy of LeNet5 trained by SGD/LS-SGD on MNIST with various batch sizes. (b). The evolution of the pre-activated ResNet56’s training and generalization accuracy by SGD and LS-SGD. (Start from the 20-th epoch.)
+
+# 4.2 IMPROVE GENERALIZATION ACCURACY
+
+The skip connections in ResNet smooth the landscape of the loss function of the classical CNN (He et al., 2016; Li et al., 2017). This means that ResNet has fewer sharp minima. On Cifar10 (Krizhevsky, 2009), we compare the performance of LS-SGD and SGD on ResNet with the preactivated ResNet56 as an illustration. We take the same training strategy as that used in (He et al., 2016), except that we run 200 epochs with the learning rate decaying by a factor of 5 after every 40 epochs. For ResNet, instead of applying LS-SGD for all epochs, we only use LS-SGD in the first 40 epochs, and the remaining training is carried out by SGD. The parameter $\sigma$ is set to 1.0. Figure 6(b) depicts one path of the training and generalization accuracy of the neural nets trained by SGD and LS-SGD, respectively. It is seen that, even though the training accuracy obtained by SGD is higher than that by LS-SGD, the generalization is however inferior to that of LS-SGD. We conjecture that this is due to the fact that SGD gets trapped into some sharp but deeper minimum, which fits better than a flat minimum but generalizes worse. We carry out 25 replicas of this experiments, the histograms of the corresponding accuracy are shown in Fig. 7.
+
+![](images/b4d13b8b01c7d8d8e56db7722d2cec8b50cde124c2ba3fa3f84c9038fa5accc7.jpg)  
+Figure 7: The histogram of the generalization accuracy of the pre-activated ResNet56 on Cifar10 trained with LS-SGD over 25 independent experiments.
+
+# 4.3 TRAINING WASSERSTERIN GAN
+
+Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) are notoriously delicate and unstable to train (Arjovsky & Bottou, 2017). In (M. Arjovsky & Bottou, 2017), Wasserstein-GANs (WGANs) are introduced to combat the instability in the training GANs. In addition to being more robust in training parameters and network architecture, WGANs provide a reliable estimate of the Earth Mover (EM) metric which correlates well with the quality of the generated samples. Nonetheless, WGANs training becomes unstable with a large learning rate or when used with a momentum based optimizer (M. Arjovsky & Bottou, 2017). In this section, we demonstrate that the gradient smoothing technique in this paper alleviates the instability in the training, and improves the quality of generated samples. Since WGANs with weight clipping are typically trained with RMSProp (Tieleman & Hinton, 2012), we propose replacing the gradient $g$ by a smoothed version $g _ { \sigma } = A _ { \sigma } ^ { - 1 } g$ , and also update the running averages using $g _ { \sigma }$ instead of $g$ . We name this algorithm LS-RMSProp.
+
+To accentuate the instability in training and demonstrate the effects of gradient smoothing, we deliberately use a large learning rate for training the generator. We compare the regular RMSProp with the LS-RMSProp. The learning rate for the critic is kept small and trained approximately to convergence so that the critic loss is still an effective approximation to the Wasserstein distance.To control the number of unknowns in the experiment and make a meaningful comparison using the critic loss, we use the classical RMSProp for the critic, and only apply LS-RMSProp to the generator.
+
+![](images/cf1eaff9147f738ea3770bb3fcbddbe3594421cd04c2855037cb54f01cdafc04.jpg)  
+Figure 8: Critic loss with learning rate $l r D = 0 . 0 0 0 1$ , $l r G = 0 . 0 0 5$ for RMSProp (Left) and LSRMSProp (Right), trained for 20K iterations. We apply a mean filter of window size 13 for better visualization. The loss from LS-RMSProp is visibly less noisy.
+
+We train the WGANs on the MNIST dataset using the DCGAN (Radford et al., 2015) for both the critic and generator. In Figure 8 (left), we observe the loss for RMSProp trained with a large learning rate has multiple sharp spikes, indicating instability in the training process. The samples generated are also lower in quality, containing noisy spots as shown in Figure 9 (a). In contrast, the curve of training loss for LS-RMSProp is smoother and exhibits fewer spikes. The generated samples as shown in Fig. 9 (b) are also of better quality and visibly less noisy. The generated characters shown in Fig. 9 (b) are more realistic compared to the ones shown in Fig. 9 (a). The effects are less pronounced with a small learning rate, but still result in a modest improvement in sample quality as shown in Figure 9 (c) and (d).We also apply LS-RMSProp for training the critic, but do not see a clear improvement in the quality. This may be because the critic is already trained near optimality during each iteration, and does not benefit much from gradient smoothing.
+
+![](images/5674e052968cb10df728edf68c0357019a5121129013e3e9f74520649e8a5d87.jpg)  
+Figure 9: Samples from WGANs trained with RMSProp (a, c) and LS-RMSProp (b, d). The learning rate is set to $l r D = 0 . 0 0 0 1$ , $l r G = 0 . 0 0 5$ for both RMSProp and LS-RMSProp in (a) and (b). And $l r D = 0 . 0 0 0 1$ , $l r G = 0 . 0 0 0 1$ are used for both RMSProp and LS-RMSProp in (c) and (d). The critic is trained for 5 iterations per step of the generator, and 200 iterations per every 500 steps of the generator.
+
+# 4.4 DEEP REINFORCEMENT LEARNING
+
+Finally, we apply the LS-SGD to deep reinforcement learning. We provide a detailed discussion and present the numerical result in the appendix.
+
+# 5 CONCLUDING REMARKS
+
+Motivated by the theory of Hamilton-Jacobi partial differential equations, we proposed Laplacian smoothing gradient descent and its high order generalizations. This simple modification dramatically reduces the optimality gap in stochastic gradient descent and helps to find better minima. Extensive numerical examples ranging from toy cases to shallow and deep neural nets to generative adversarial networks and to deep reinforcement learning, all demonstrate the advantage of the proposed smoothed gradient. Several issues remain, in particular devising an on-the-fly adaptive method for choosing the smoothing parameter $\sigma$ instead of using a fixed value.
+
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+
+# 6 APPENDIX
+
+# 6.1 TECHNICAL PROOFS
+
+Proposition 1. Suppose $f ( w )$ is differentiable, the Laplacian smoothing GD update on $u ( { \boldsymbol { w } } , t )$
+
+$$
+\pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - t \pmb { A } _ { \sigma } ^ { - 1 } \nabla _ { \pmb { w } } u ( \pmb { w } ^ { k } , t )
+$$
+
+permits the smoothing implicit gradient descent on $f ( w )$
+
+$$
+\begin{array} { r } { \pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - t \pmb { A } _ { \sigma } ^ { - 1 } \nabla f ( \pmb { w } ^ { k + 1 } ) . } \end{array}
+$$
+
+# Proof of Proposition 1. We define
+
+$$
+z ( \pmb { w } , \pmb { v } , t ) : = f ( \pmb { v } ) + \frac { 1 } { 2 t } \langle \pmb { v } - \pmb { w } , \pmb { A } _ { \sigma } ( \pmb { v } - \pmb { w } ) \rangle ,
+$$
+
+and rewrite $u ( \pmb { w } , t ) = \operatorname* { i n f } _ { \pmb { v } } z ( \pmb { w } , \pmb { v } , t )$ as $z ( \mathbf { w } , \mathbf { v } ( \mathbf { w } , t ) , t )$ , where ${ \pmb v } ( { \pmb w } , t ) = \arg \operatorname* { m i n } _ { \pmb v } { z } ( { \pmb w } , { \pmb v } , t )$ Then by the Euler-Lagrange equation,
+
+$\begin{array} { r } { \nabla _ { w } u ( w , t ) = \nabla _ { w } z ( w , v ( w , t ) , t ) = J _ { w } v ( w , t ) \nabla _ { v } z ( w , v ( w , t ) , t ) + \nabla _ { w } z ( w , v ( w , t ) , t ) , } \end{array}$ where $J _ { w } { \bf v } ( w , t )$ is the Jacobian matrix of $\textbf {  { v } }$ w.r.t. $\textbf { \em w }$ . Notice that $\nabla _ { v } z ( w , v ( w , t ) , t ) = \mathbf { 0 }$ ,
+
+$$
+\nabla _ { \pmb { w } } u ( \pmb { w } , t ) = \nabla _ { \pmb { w } } z ( \pmb { w } , \pmb { v } ( \pmb { w } , t ) , t ) = - \frac { 1 } { t } \pmb { A } _ { \sigma } ( \pmb { v } ( \pmb { w } , t ) - \pmb { w } ) .
+$$
+
+Letting $\mathbf { \Delta } w = w ^ { k }$ and $\begin{array} { r } { \pmb { w } ^ { k + 1 } = \pmb { v } ( \pmb { w } ^ { k } , t ) = \arg \operatorname* { m i n } _ { \mathbf { v } } z ( \pmb { w } ^ { k } , \pmb { v } , t ) } \end{array}$ in the above equalities, we have
+
+$$
+\nabla _ { \boldsymbol { w } } \boldsymbol { u } ( \boldsymbol { w } ^ { k } , t ) = - \frac { 1 } { t } \boldsymbol { A } _ { \sigma } ( \boldsymbol { w } ^ { k + 1 } - \boldsymbol { w } ^ { k } ) .
+$$
+
+In summary, the gradient descent $\pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - t \pmb { A } _ { \sigma } ^ { - 1 } \nabla _ { \pmb { w } } u ( \pmb { w } ^ { k } , t )$ is equivalent to the proximal point iteration $\begin{array} { r } { \pmb { w } ^ { \tilde { k } + 1 } = \arg \operatorname* { m i n } _ { \pmb { v } } \pmb { f } ( \pmb { v } ) + \frac { 1 } { 2 t } \pmb { \langle v - \pmb { w } ^ { k } } , \pmb { A } _ { \sigma } ( \pmb { v } - \pmb { w } ^ { k } ) \rangle . } \end{array}$ , which yields $\mathbf { \boldsymbol { w } } ^ { k + 1 } \mathbf { \dot { \xi } } = \mathbf { \boldsymbol { w } } ^ { k } - \mathbf { \boldsymbol { \xi } }$ $t A _ { \sigma } ^ { - 1 } \nabla f ( { \pmb w } ^ { k + 1 } )$ . □
+
+Proposition 2. Suppose $f$ is convex with the global minimizer $\boldsymbol { w } ^ { * }$ , and $f ^ { * } = f ( w ^ { * } )$ . Consider the following iteration with constant learning rate $\eta > 0$
+
+$$
+\pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - \eta ( \pmb { A } _ { \sigma } ^ { n } ) ^ { - 1 } \pmb { g } ^ { k }
+$$
+
+where $g ^ { k }$ is the sampled gradient in the kth iteration at $\boldsymbol { w } ^ { k }$ satisfying $\mathbb { E } [ \pmb { g } ^ { k } ] = \nabla f ( \pmb { w } ^ { k } )$ . Denote $\begin{array} { r } { G _ { { \pmb A } _ { \sigma } ^ { n } } : = \operatorname* { l i m } _ { K  \infty } \frac { 1 } { K } \sum _ { k = 0 } ^ { K - 1 } \| \pmb { g } ^ { k } \| _ { ( { \pmb A } _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } } \end{array}$ and $\begin{array} { r } { \overline { { \pmb { w } } } ^ { K } : = \sum _ { k = 0 } ^ { K - 1 } \pmb { w } ^ { k } / K } \end{array}$ the ergodic average of iterates. Then the optimality gap is
+
+$$
+\operatorname* { l i m } _ { K \to \infty } \mathbb { E } [ f ( \overline { { \pmb { w } } } ^ { K } ) ] - f ^ { * } \leq \frac { \eta G _ { A _ { \sigma } ^ { n } } } { 2 } .
+$$
+
+Proof. Since $f$ is convex, we have
+
+$$
+\langle \nabla f ( { \pmb w } ^ { k } ) , { \pmb w } ^ { k } - { \pmb w } ^ { * } \rangle \geq f ( { \pmb w } ^ { k } ) - f ^ { * } .
+$$
+
+Furthermore,
+
+$$
+\begin{array} { r l } & { \quad \mathbb { E } [ \| { \pmb w } ^ { k + 1 } - { \pmb w } ^ { * } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] = \mathbb { E } [ \| { \pmb w } ^ { k } - \eta ( { \pmb A } _ { \sigma } ^ { n } ) ^ { - 1 } { \pmb g } ^ { k } - { \pmb w } ^ { * } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] } \\ & { = \mathbb { E } [ \| { \pmb w } ^ { k } - { \pmb w } ^ { * } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] - 2 \eta \mathbb { E } [ \langle { \pmb g } ^ { k } , { \pmb w } ^ { k } - { \pmb w } ^ { * } \rangle ] + \eta ^ { 2 } \mathbb { E } [ \| ( { \pmb A } _ { \sigma } ^ { n } ) ^ { - 1 } { \pmb g } ^ { t } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] } \\ & { \le \mathbb { E } [ \| { \pmb w } ^ { k } - { \pmb w } ^ { * } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] - 2 \eta \mathbb { E } [ \langle \nabla f ( { \pmb w } ^ { k } ) , { \pmb w } ^ { k } - { \pmb w } ^ { * } \rangle ] + \eta ^ { 2 } \| { \pmb g } ^ { k } \| _ { ( { \pmb A } _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } } \\ & { \le \mathbb { E } [ \| { \pmb w } ^ { k } - { \pmb w } ^ { * } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] - 2 \eta ( \mathbb { E } [ f ( { \pmb w } ^ { k } ) ] - f ^ { * } ) + \eta ^ { 2 } \| { \pmb g } ^ { k } \| _ { ( { \pmb A } _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } , } \end{array}
+$$
+
+where the last inequality is due to (7). We rearrange the terms and arrive at
+
+$$
+\mathbb { E } [ f ( { \pmb w } ^ { k } ) ] - f ^ { * } \le \frac { 1 } { 2 \eta } ( \mathbb { E } [ \| { \pmb w } ^ { k } - { \pmb w } ^ { * } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] - \mathbb { E } [ \| { \pmb w } ^ { k + 1 } - { \pmb w } ^ { * } \| _ { { \pmb A } _ { \sigma } ^ { n } } ^ { 2 } ] ) + \frac { \eta \| { \pmb g } ^ { k } \| _ { ( { \pmb A } _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } } { 2 } .
+$$
+
+Summing over $k$ from 0 to $K - 1$ and averaging and using the convexity of $f$ , we have
+
+$$
+\mathbb { E } [ f ( \overline { { w } } ^ { K } ) ] - f ^ { * } \leq \frac { \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } [ f ( w ^ { k } ) ] } { K } - f ^ { * } \leq \frac { 1 } { 2 \eta K } \mathbb { E } [ \| w ^ { 0 } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } ^ { 2 } ] + \frac { \sum _ { k = 0 } ^ { K - 1 } \| g ^ { k } \| _ { ( A _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } } { 2 K } \eta .
+$$
+
+Taking the limit as $K  \infty$ above establishes the result.
+
+Proposition 3. Suppose $f$ is $L$ -Lipschitz smooth and $a$ -strongly convex. Consider the generalized smoothing gradient descent algorithm
+
+$$
+\begin{array} { r } { \pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - \eta _ { k } ( \pmb { A } _ { \sigma } ^ { n } ) ^ { - 1 } \pmb { g } ^ { k } , } \end{array}
+$$
+
+where $g ^ { k }$ is the sampled gradient in the kth iteration at $\boldsymbol { w } ^ { k }$ satisfying $\mathbb { E } \left[ \pmb { g } ^ { k } \right] \ : = \ : \nabla f ( \pmb { w } ^ { k } )$ and $\mathbb { E } \left[ \| \pmb { g } ^ { k } \| _ { ( { \pmb { A } } _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } \right] \leq C _ { 0 } + C _ { 1 } \| \nabla f ( \pmb { w } ^ { k } ) \| ^ { 2 }$ for all $k \in \mathbb { N } .$ . If we take $\begin{array} { r } { \eta _ { k } = \frac { C } { k + 1 } } \end{array}$ for some $C > 0$ , then we have
+
+$$
+\mathbb { E } \left[ \Vert w ^ { k } - w ^ { * } \Vert _ { A _ { \sigma } ^ { n } } ^ { 2 } \right] = \mathbb { E } \left[ \Vert w ^ { k } - w ^ { * } \Vert ^ { 2 } + \sigma \Vert D _ { + } ^ { n } ( w ^ { k } - w ^ { * } ) \Vert ^ { 2 } \right] = O \left( \frac { 1 } { k + 1 } \right) ,
+$$
+
+i.e., we have $H _ { \sigma } ^ { n }$ uniform convergence in $\sigma$ of $\{ w ^ { k } \}$ in expectation. The $H _ { \sigma } ^ { n }$ norm of $\pmb { w }$ is defined by $\| \pmb { w } \| _ { \sigma } ^ { n } : = \| w \| _ { \pmb { A } _ { \sigma } ^ { n } } = \sqrt { \langle \pmb { w } , \pmb { A } _ { \sigma } ^ { n } \pmb { w } \rangle }$ .
+
+Proof of Proposition 3. Since $\nabla f ( { \pmb w } ^ { * } ) = { \bf 0 }$ , by strong convexity of $f$ , we have
+
+$$
+\langle \nabla f ( { \boldsymbol w } ^ { k } ) , { \boldsymbol w } ^ { k } - { \boldsymbol w } ^ { * } \rangle = \langle \nabla f ( { \boldsymbol w } ^ { k } ) - \nabla f ( { \boldsymbol w } ^ { * } ) , { \boldsymbol w } ^ { k } - { \boldsymbol w } ^ { * } \rangle \geq a \| { \boldsymbol w } ^ { k } - { \boldsymbol w } ^ { * } \| ^ { 2 } .
+$$
+
+Moreover, by $L$ -smoothness of $f$ and the fact that $\| A _ { \sigma } ^ { n } \| = 1$ , we also have
+
+$$
+\begin{array} { r } { \| \nabla f ( { \pmb w } ^ { k } ) \| = \| \nabla f ( { \pmb w } ^ { k } ) - \nabla f ( { \pmb w } ^ { * } ) \| \leq L \| { \pmb w } ^ { k } - { \pmb w } ^ { * } \| \leq L \| { \pmb w } ^ { k } - { \pmb w } ^ { * } \| _ { { \pmb u } _ { \sigma } ^ { n } } . } \end{array}
+$$
+
+Hence,
+
+$$
+\begin{array} { r l } & { \quad \mathbb { E } [ \| w ^ { k + 1 } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } ^ { 2 } ] = \mathbb { E } [ \| w ^ { k } - \eta ( A _ { \sigma } ^ { n } ) ^ { - 1 } g ^ { k } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } ^ { 2 } ] } \\ & { = \mathbb { E } [ \| w ^ { k } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } ^ { 2 } ] - 2 \eta _ { k } \mathbb { E } \left[ \langle g ^ { k } , w ^ { k } - w ^ { * } \rangle \right] + \eta _ { k } ^ { 2 } \mathbb { E } [ \| g ^ { k } \| _ { ( A _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } ] } \\ & { = \mathbb { E } [ \| w ^ { k } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } ^ { 2 } ] - 2 \eta _ { k } \langle \nabla f ( w ^ { k } ) , w ^ { k } - w ^ { * } \rangle + \eta _ { k } ^ { 2 } \mathbb { E } [ \| g ^ { k } \| _ { ( A _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } ] } \\ & { \leq ( 1 - 2 \eta _ { k } a ) \mathbb { E } \left[ \| w ^ { k } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } ^ { 2 } \right] + \eta _ { k } ^ { 2 } \left( C _ { 0 } + C _ { 1 } \mathbb { E } [ \| \nabla f ( w ^ { k } ) \| ^ { 2 } ] \right) } \\ & { \leq \left( 1 - 2 \eta _ { k } a + \eta _ { k } ^ { 2 } L ^ { 2 } C _ { 1 } \right) \mathbb { E } \left[ \| w ^ { k } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } ^ { 2 } \right] + \eta _ { k } ^ { 2 } C _ { 0 } , } \end{array}
+$$
+
+where in the first inequality we used k(Anσ)−1k = 1 for all σ and n. Taking ηk = Ck+1 for some proper $\ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ c \ : \ : \ c \ c \ c \ c \ c \ c \ c \ c \ c \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F \ F$ and using induction, one can show that $\begin{array} { r l } { \mathbb { E } \left[ \| \pmb { w } ^ { k } - \pmb { w } ^ { * } \| _ { \pmb { A } _ { \sigma } ^ { n } } ^ { 2 } \right] } & { = } \end{array}$ $\begin{array} { r } { \mathbb { E } \left[ \| \pmb { w } ^ { k } - \pmb { w } ^ { * } \| ^ { 2 } + \sigma \| \pmb { D } _ { + } ^ { n } ( \pmb { w } ^ { k } - \pmb { w } ^ { * } ) \| \right] = O ( \frac { 1 } { k + 1 } ) . } \end{array}$
+
+Proposition 4. Consider the algorithm $\pmb { w } ^ { k + 1 } = \pmb { w } ^ { k } - \eta _ { k } \big ( \pmb { A } _ { \sigma } ^ { n } \big ) ^ { - 1 } \nabla f ( \pmb { w } ^ { k } ) .$ . Suppose $f$ is $L$ -Lipschitz smooth and $\begin{array} { r } { 0 < \underline { { \eta } } \le \eta \le \bar { \eta } < \bar { \frac { 2 } { L } } } \end{array}$ . Then $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \| \nabla f ( \pmb { w } ^ { k } ) \|  0 } \end{array}$ . Moreover, if the Hessian $\bar { \nabla } ^ { 2 } f$ of $f$ ¯is continuous with $\ b { w } ^ { * }$ being the minimizer of $f$ , and $\bar { \eta } \| \nabla ^ { 2 } f \| < 1$ , then $\lVert \pmb { w } ^ { k } - \pmb { w } ^ { * } \rVert _ { \pmb { A } _ { \sigma } ^ { n } }  0$ as $k  \infty$ , and the convergence is linear.
+
+Proof of Proposition 4. By the Lipschitz continuity of $\nabla f$ and the descent lemma (Bertsekas, 1999), we have
+
+$$
+\begin{array} { r l } { f ( w ^ { k + 1 } ) \ } & { = f ( w ^ { k } - \eta _ { k } ( A _ { \sigma } ^ { n } ) ^ { - 1 } \nabla f ( w ^ { k } ) ) } \\ & { \le f ( w ^ { k } ) - \eta _ { k } \langle \nabla f ( w ^ { k } ) , ( A _ { \sigma } ^ { n } ) ^ { - 1 } \nabla f ( w ^ { k } ) ) \rangle + \frac { \eta _ { k } ^ { 2 } L } { 2 } \| ( A _ { \sigma } ^ { n } ) ^ { - 1 } \nabla f ( w ^ { k } ) \| ^ { 2 } } \\ & { \le f ( w ^ { k } ) - \eta _ { k } \| \nabla f ( w ^ { k } ) \| _ { ( A _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } + \frac { \eta _ { k } ^ { 2 } L } { 2 } \| \nabla f ( w ^ { k } ) \| _ { ( A _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } } \\ & { \le f ( w ^ { k } ) - \eta \left( 1 - \frac { \bar { \eta } L } { 2 } \right) \| \nabla f ( w ^ { k } ) \| _ { ( A _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } . } \end{array}
+$$
+
+Summing the above inequality over $k$ , we have
+
+$$
+\eta \left( 1 - \frac { \bar { \eta } L } { 2 } \right) \sum _ { k = 0 } ^ { \infty } \| \nabla f ( \pmb { w } ^ { k } ) \| _ { ( A _ { \sigma } ^ { n } ) ^ { - 1 } } ^ { 2 } \leq f ( \pmb { w } ^ { 0 } ) - \operatorname* { l i m } _ { k \to \infty } f ( \pmb { w } ^ { k } ) < \infty .
+$$
+
+Therefore, $\| \nabla f ( \pmb { w } ^ { k } ) \| _ { ( { A _ { \sigma } ^ { n } } ) ^ { - 1 } } ^ { 2 }  0$ , and thus $\| \nabla f ( \pmb { w } ^ { k } ) \|  0$
+
+For the second claim, we have
+
+$$
+\begin{array} { r l } & { v ^ { k + 1 } - w ^ { * } = w ^ { k } - w ^ { * } - \eta _ { k } ( A _ { \sigma } ^ { n } ) ^ { - 1 } ( \nabla f ( w ^ { k } ) - \nabla f ( w ^ { * } ) ) } \\ & { \quad \quad = w ^ { k } - w ^ { * } - \eta _ { k } ( A _ { \sigma } ^ { n } ) ^ { - 1 } \left( \displaystyle \int _ { 0 } ^ { 1 } \nabla ^ { 2 } f ( w ^ { * } + \tau ( w ^ { k + 1 } - w ^ { * } ) ) \cdot ( w ^ { k } - w ^ { * } ) \mathrm { d } \tau \right) } \\ & { \quad \quad = w ^ { k } - w ^ { * } - \eta _ { k } ( A _ { \sigma } ^ { n } ) ^ { - 1 } \left( \displaystyle \int _ { 0 } ^ { 1 } \nabla ^ { 2 } f ( w ^ { * } + \tau ( w ^ { k + 1 } - w ^ { * } ) ) \mathrm { d } \tau \cdot ( w ^ { k } - w ^ { * } ) \right) } \\ & { \quad \quad = ( A _ { \sigma } ^ { n } ) ^ { - \frac { 1 } { 2 } } \left( I - \eta _ { k } ( A _ { \sigma } ^ { n } ) ^ { - \frac { 1 } { 2 } } \displaystyle \int _ { 0 } ^ { 1 } \nabla ^ { 2 } f ( w ^ { * } + \tau ( w ^ { k + 1 } - w ^ { * } ) ) \mathrm { d } \tau ( A _ { \sigma } ^ { n } ) ^ { - \frac { 1 } { 2 } } \right) ( A _ { \sigma } ^ { n } ) ^ { \frac { 1 } { 2 } } ( w ^ { k } + \tau ( w ^ { k + 1 } - w ^ { * } ) ) } \end{array}
+$$
+
+Therefore,
+
+$$
+w ^ { k + 1 } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } \leq \left\| I - \eta _ { t } ( A _ { \sigma } ^ { n } ) ^ { - \frac { 1 } { 2 } } \int _ { 0 } ^ { 1 } \nabla ^ { 2 } f ( w ^ { * } + \tau ( w ^ { k + 1 } - w ^ { * } ) ) \mathrm { d } \tau ( A _ { \sigma } ^ { n } ) ^ { - \frac { 1 } { 2 } } \right\| \| w ^ { k } - w ^ { * } \| _ { A _ { \sigma } ^ { n } } .
+$$
+
+So if $\begin{array} { r } { \eta _ { k } \| \nabla ^ { 2 } f \| \le \frac { 1 } { \| ( A _ { \sigma } ^ { n } ) ^ { - 1 } \| } = 1 } \end{array}$ , the result follows.
+
+Proposition 5. For any vector $\textbf { \textit { g } } \in \mathbb { R } ^ { m }$ , $d \ = \ A _ { \sigma } ^ { - 1 } g ,$ , let $j _ { \mathrm { m a x } } \ = \ \arg \operatorname* { m a x } _ { i } d _ { i }$ and $j _ { \mathrm { m i n } } \ =$ arg $\operatorname* { m i n } _ { i } d _ { i }$ . We have $\begin{array} { r } { \operatorname { n a x } _ { i } d _ { i } = d _ { j _ { \operatorname* { m a x } } } \leq g _ { j _ { \operatorname* { m a x } } } \leq \operatorname* { m a x } _ { i } g _ { i } } \end{array}$ and $\mathrm { m i n } _ { i } d _ { i } = d _ { j _ { \mathrm { m i n } } } \geq g _ { j _ { \mathrm { m i n } } } \geq \mathrm { m i n } _ { i } g _ { i }$ .
+
+Proof of Proposition 5. Since $\mathbf { \omega } _ { g } = A _ { \sigma } \mathbf { \vec { d } }$ , it holds that
+
+$$
+g _ { j _ { \operatorname* { m a x } } } = d _ { j _ { \operatorname* { m a x } } } + \sigma ( 2 d _ { j _ { \operatorname* { m a x } } } - d _ { j _ { \operatorname* { m a x } } - 1 } - d _ { j _ { \operatorname* { m a x } } + 1 } ) ,
+$$
+
+where periodicity of subindex are used if necessary. Since $2 d _ { j _ { \operatorname* { m a x } } } - d _ { j _ { \operatorname* { m a x } } - 1 } - d _ { j _ { \operatorname* { m a x } } + 1 } \geq 0$ , We $\begin{array} { r } { \operatorname* { m a x } _ { i } d _ { i } = d _ { j _ { \operatorname* { m a x } } } \leq g _ { j _ { \operatorname* { m a x } } } \leq \operatorname* { m a x } _ { i } g _ { i } } \end{array}$ . A similar argument can show that ${ \mathrm { m i n } } _ { i } d _ { i } = d _ { j _ { \operatorname* { m i n } } } \geq$ $g _ { j _ { \operatorname* { m i n } } } \geq \operatorname* { m i n } _ { i } g _ { i }$
+
+Proposition 6. The operator $A _ { \sigma } ^ { - 1 }$ preserves the sum of components. For any $\pmb { \mathscr { g } } \in \mathbb { R } ^ { m }$ and ${ \pmb d } =$ $A _ { \sigma } ^ { - 1 } g$ , we have $\textstyle \sum _ { j } d _ { j } = \sum _ { j } g _ { j }$ , or equivalently, $\mathbf { 1 } ^ { \top } \pmb { d } = \mathbf { 1 } ^ { \top } \pmb { g }$ .
+
+Proof of Proposition 6. Since $\mathbf { \omega } _ { g } = A _ { \sigma } \mathbf { \vec { d } }$ ,
+
+$$
+\sum _ { i } g _ { i } = \mathbf { 1 } ^ { \top } \mathbf { g } = \mathbf { 1 } ^ { \top } ( I + \sigma D _ { + } ^ { \top } D _ { + } ) d = \mathbf { 1 } ^ { \top } d = \sum _ { i } d _ { i } ,
+$$
+
+where we used ${ \cal D } _ { + } \mathbf { 1 } = \mathbf { 0 }$ .
+
+Proposition 7. Given any vector $\pmb { \mathscr { g } } \in \mathbb { R } ^ { m }$ and $\pmb { d } = \pmb { A } _ { \sigma } ^ { - 1 } \pmb { g }$ , then
+
+$$
+\| \pmb { d } \| + \sigma \frac { \| \pmb { D } _ { + } \pmb { d } \| ^ { 2 } } { \| \pmb { d } \| } \le \| \pmb { g } \| .
+$$
+
+The above inequality is strict unless $\mathbf { \omega } _ { g } = d$ is a constant vector. In particular, we have $\| d \| \leq \| g \|$ and $\begin{array} { r } { \| D _ { + } d \| \le \frac { 1 } { \sqrt { \sigma } } \| \pmb { g } \| } \end{array}$ .
+
+Proof of Proposition 7. By the definition of $A _ { \sigma }$ ,
+
+$$
+g = A _ { \sigma } d = ( I - \sigma D _ { - } D _ { + } ) d = d + \sigma D _ { + } ^ { \top } D _ { + } d .
+$$
+
+Therefore, pre-multiplying by $d ^ { \top }$ on both sides, we have
+
+$$
+\begin{array} { r } { \| \pmb { d } \| ^ { 2 } + \sigma \| \pmb { D } _ { + } \pmb { d } \| ^ { 2 } = \pmb { d } ^ { \top } \pmb { g } \leq \| \pmb { d } \| \| \pmb { g } \| . } \end{array}
+$$
+
+In particular, $\| d \| \leq \| g \|$ and $\sigma \| D _ { + } d \| ^ { 2 } \leq \| d \| \| g \| \leq \| g \| ^ { 2 }$ , so $\begin{array} { r } { \| D _ { + } d \| \le \frac { 1 } { \sqrt { \sigma } } \| g \| } \end{array}$ . All the inequalities are strict unless $\| D _ { + } d \| = 0$ , and $\mathbf { \nabla } _ { \mathbf { { g } } } = d$ is a constant vector.
+
+Proposition 8. For any $\pmb { \mathscr { g } } \in \mathbb { R } ^ { m }$ , define $\begin{array} { r } { \mathrm { V a r } ( \pmb { g } ) : = \frac { 1 } { m } \| \pmb { g } \| ^ { 2 } - \bigg ( \frac { \pmb { 1 } ^ { \top } \pmb { g } } { m } \bigg ) ^ { 2 } } \end{array}$ be the variance of components in $\textbf {  { g } }$ . Let $\pmb { d } = \pmb { A } _ { \sigma } ^ { - 1 } \pmb { g } ,$ , then
+
+$$
+\mathrm { V a r } ( \pmb { d } ) \leq \mathrm { V a r } ( \pmb { g } ) - 2 \sigma \frac { \| \pmb { D } _ { + } \pmb { d } \| ^ { 2 } } { m } - \sigma ^ { 2 } \frac { \| \pmb { D } _ { + } \pmb { d } \| ^ { 4 } } { m \| \pmb { d } \| ^ { 2 } } .
+$$
+
+The inequality is strict unless $\mathbf { \nabla } _ { \mathbf { { g } } } = d$ is a constant vector.
+
+Proof of Proposition 8. Since 1>g = 1>d and kdk + σ kD+dk2kdk ,
+
+$$
+\begin{array} { r l } & { \mathrm { V a r } ( g ) \geq \displaystyle \frac { 1 } { m } \left( \| d \| ^ { 2 } + 2 \sigma \| D _ { + } d \| ^ { 2 } + \sigma ^ { 2 } \frac { \| D _ { + } d \| ^ { 4 } } { \| d \| ^ { 2 } } \right) - \left( \frac { \mathbf { 1 } ^ { \top } d } { n } \right) ^ { 2 } } \\ & { \quad \quad \quad \quad = \mathrm { V a r } ( d ) + 2 \sigma \frac { \| D _ { + } d \| ^ { 2 } } { m } + \sigma ^ { 2 } \frac { \| D _ { + } d \| ^ { 4 } } { m \| d \| ^ { 2 } } . } \end{array}
+$$
+
+The inequality is strict unless $\| \pmb { \cal D } _ { + } \pmb { d } \| = 0$ , and $\mathbf { \nabla } _ { \mathbf { { g } } } = d$ is a constant vector.
+
+Proposition 9. Given vectors $\textbf {  { g } }$ and $d = A _ { \sigma } ^ { - 1 } g ,$ , for any $p \in \mathbb N$ , it holds that $\| D _ { + } ^ { p } d \| _ { 1 } \leq \| D _ { + } ^ { p } g \| _ { 1 }$ .   
+The inequality is strict unless $D _ { + } ^ { p } g$ is a constant vector.
+
+Proof of Proposition 9. Since $( 1 + 2 \sigma ) d _ { i } = g _ { i } + \sigma d _ { i + 1 } + \sigma d _ { i - 1 }$ , for any $p \in \mathbb N$ , we have
+
+$$
+( 1 + 2 \sigma ) ( D _ { + } ^ { p } d ) _ { i } = ( D _ { + } ^ { p } g ) _ { i } + \sigma ( D _ { + } ^ { p } d ) _ { i + 1 } + \sigma ( D _ { + } ^ { p } d ) _ { i - 1 } .
+$$
+
+So
+
+$$
+( 1 + 2 \sigma ) | ( D _ { + } ^ { p } d ) _ { i } | \leq | ( D _ { + } ^ { p } g ) _ { i } | + \sigma | ( D _ { + } ^ { p } d ) _ { i + 1 } | + \sigma | ( D _ { + } ^ { p } d ) _ { i - 1 } | .
+$$
+
+The inequality is strict if there are sign changes among the $( D _ { + } ^ { p } d ) _ { i - 1 } , ( D _ { + } ^ { p } d ) _ { i } , ( D _ { + } ^ { p } d ) _ { i + 1 }$ . Summing over $i$ and using periodicity, we have
+
+$$
+( 1 + 2 \sigma ) \sum _ { i = 1 } ^ { m } | ( D _ { + } ^ { p } d ) _ { i } | \leq \sum _ { i = 1 } ^ { m } | ( D _ { + } ^ { p } g ) _ { i } | + 2 \sigma \sum _ { i = 1 } ^ { m } | ( D _ { + } ^ { p } d ) _ { i } | ,
+$$
+
+and the result follows. The inequality is strict unless $D _ { + } ^ { p } g$ is a constant vector.
+
+# 6.2 ITERATION V.S. LOSS FOR SOFTMAX REGRESSION
+
+In this part, we show the training loss evolution in training Softmax regression model, respectively, by SGD, SVRG, LSGD with first and second order smoothing. As illustrated in Fig. 10, all the optimization algorithms reduce loss of the model on the training set. At each iteration, among 100 independent experiments, SGD has the largest variance, SGD with first order smoothed gradient significantly reduces the variance of loss function. The second order smoothing can further reduce variance of loss. The variance of loss in each iteration among 100 experiments is minimized when SVRG is use to train the Softmax model. However, the generalization performance of the model trained by SVRG is not as good as the ones trained by LS-SGD or higher order smoothed gradient descent.
+
+# 6.3 DEEP REINFORCEMENT LEARNING
+
+Deep reinforcement learning (DRL) has been applied to playing games including Cartpole (Brockman et al., 2016), Atari (Mnih et al., 2013), Go (Silver & et al, 2016; Mnih & et al, 2015). DNN plays a vital role in approximating the Q-function or policy function. We apply the Laplacian smoothed gradient to train the policy function to play the Cartpole game. We apply the standard procedure to train the policy function by using the policy gradient (Brockman et al., 2016). We use the following network to approximate the policy function:
+
+$$
+\mathrm { i n p u t _ { 4 } } \to \mathrm { f c _ { 2 0 } } \to \mathrm { r e l u } \to \mathrm { f c _ { 2 } } \to \mathrm { s o f t m a x } .
+$$
+
+![](images/64a88d64369b38a5b5e82319526d4d3d3f243bf3d47bb799f941a1d3cce83fd5.jpg)  
+Figure 10: Iterations v.s. loss for GD, SVRG, and smoothed GD with order 1 and 2 for training the softmax regression model.
+
+The network is trained by RMSProp and LS-RMSProp with $\sigma = 1 . 0$ , respectively. The learning rate and other related parameters are set to be the default ones in PyTorch. The training is stopped once the average duration of 5 consecutive episodes is more than 490. In each training episode, we set the maximal steps to be 500. Left and right panels of Fig. 11 depict a training procedure by using RMSProp and LS-RMSProp, respectively. We see that Laplacian smoothed gradient takes fewer episodes to reach the stopping criterion. Moreover, we run the above experiments 5 times independently, and apply the trained model to play Cartpole. The game lasts more than 1000 steps for all the 5 models trained by LS-RMSProp, while only 3 of them lasts more than 1000 steps when the model is trained by vanilla RMSProp.
+
+![](images/fec8dea99cd0648ad927604968e4f353571322d1548ad3d680adabfca691271b.jpg)  
+Figure 11: Durations of the cartpole game in the training procedure. Left and right are training procedure by RMSProp and LS-RMSProp with $\sigma = 1 . 0$ , respectively.

parse/train/Byl5NREFDr/Byl5NREFDr.md

@@ -0,0 +1,353 @@
+# THIEVES ON SESAME STREET! MODEL EXTRACTION OF BERT-BASED APIS
+
+Kalpesh Krishna∗ CICS, UMass Amherst kalpesh@cs.umass.edu
+
+Gaurav Singh Tomar Google Research gtomar@google.com
+
+Ankur P. Parikh Google Research aparikh@google.com
+
+Nicolas Papernot Google Research papernot@google.com
+
+Mohit Iyyer CICS, UMass Amherst miyyer@cs.umass.edu
+
+# ABSTRACT
+
+We study the problem of model extraction in natural language processing, in which an adversary with only query access to a victim model attempts to reconstruct a local copy of that model. Assuming that both the adversary and victim model fine-tune a large pretrained language model such as BERT (Devlin et al., 2019), we show that the adversary does not need any real training data to successfully mount the attack. In fact, the attacker need not even use grammatical or semantically meaningful queries: we show that random sequences of words coupled with task-specific heuristics form effective queries for model extraction on a diverse set of NLP tasks, including natural language inference and question answering. Our work thus highlights an exploit only made feasible by the shift towards transfer learning methods within the NLP community: for a query budget of a few hundred dollars, an attacker can extract a model that performs only slightly worse than the victim model. Finally, we study two defense strategies against model extraction—membership classification and API watermarking—which while successful against naive adversaries, are ineffective against more sophisticated ones.
+
+# 1 INTRODUCTION
+
+Machine learning models represent valuable intellectual property: the process of gathering training data, iterating over model design, and tuning hyperparameters costs considerable money and effort. As such, these models are often only indirectly accessible through web APIs that allow users to query a model but not inspect its parameters. Malicious users might try to sidestep the expensive model development cycle by instead locally reproducing an existing model served by such an API. In these attacks, known as “model stealing” or “model extraction” (Lowd & Meek, 2005; Tramer\` et al., 2016), the adversary issues a large number of queries and uses the collected (input, output) pairs to train a local copy of the model. Besides theft of intellectual property, extracted models may leak sensitive information about the training data (Tramer et al., 2016) or be used to generate \` adversarial examples that evade the model served by the API (Papernot et al., 2017).
+
+With the recent success of contextualized pretrained representations for transfer learning, NLP models created by finetuning ELMo (Peters et al., 2018) and BERT (Devlin et al., 2019) have become increasingly popular (Gardner et al., 2018). Contextualized pretrained representations boost performance and reduce sample complexity (Yogatama et al., 2019), and typically require only a shallow task-specific network—sometimes just a single layer as in BERT. While these properties are advantageous for representation learning, we hypothesize that they also make model extraction easier.
+
+In this paper,1 we demonstrate that NLP models obtained by fine-tuning a pretrained BERT model can be extracted even if the adversary does not have access to any training data used by the API provider. In fact, the adversary does not even need to issue well-formed queries: our experiments show that extraction attacks are possible even with queries consisting of randomly sampled sequences of words coupled with simple task-specific heuristics (Section 3). While extraction performance improves further by leveraging sentences and paragraphs from Wikipedia (Section 4), the fact that random word sequences are sufficient to extract models contrasts with prior work, where large-scale attacks require at minimum that the adversary can access a small amount of semanticallycoherent data relevant to the task (Papernot et al., 2017; Correia-Silva et al., 2018; Orekondy et al., 2019a; Pal et al., 2019; Jagielski et al., 2019). These attacks are cheap: our most expensive attack cost around $\$ 500$ , estimated using rates of current API providers.
+
+![](images/361889710f58160d12c2352d55d80782588c5c1794acbb9877fbf04c6ab9baed.jpg)  
+Figure 1: Overview of our model extraction setup for question answering.2An attacker first queries a victim BERT model, and then uses its predicted answers to fine-tune their own BERT model. This process works even when passages and questions are random sequences of words as shown here.
+
+In Section 5.1, we perform a fine-grained analysis of the randomly-generated queries. Human studies on the random queries show that despite their effectiveness in extracting good models, they are mostly nonsensical and uninterpretable, although queries closer to the original data distribution work better for extraction. Furthermore, we discover that pretraining on the attacker’s side makes model extraction easier (Section 5.2).
+
+Finally, we study the efficacy of two simple defenses against extraction — membership classification (Section 6.1) and API watermarking (Section 6.2) — and find that while they work well against naive adversaries, they fail against adversaries who adapt to the defense. We hope that our work spurs future research into stronger defenses against model extraction and, more generally, on developing a better understanding of why these models and datasets are particularly vulnerable to such attacks.
+
+# 2 RELATED WORK
+
+We relate our work to prior efforts on model extraction, most of which have focused on computer vision applications. Because of the way in which we synthesize queries for extracting models, our work also directly relates to zero-shot distillation and studies of rubbish inputs to NLP systems.
+
+Model extraction attacks have been studied both empirically (Tramer et al., 2016; Orekondy et al., \` 2019a; Juuti et al., 2019) and theoretically (Chandrasekaran et al., 2018; Milli et al., 2019), mostly against image classification APIs. These works generally synthesize queries in an active learning setup by searching for inputs that lie close to the victim classifier’s decision boundaries. This method does not transfer to text-based systems due to the discrete nature of the input space.3 The only prior work attempting extraction on NLP systems is Pal et al. (2019), who adopt pool-based active learning to select natural sentences from WikiText-2 and extract 1-layer CNNs for tasks expecting single inputs. In contrast, we study a more realistic extraction setting with nonsensical inputs on modern BERT-large models for tasks expecting pairwise inputs like question answering.
+
+Table 1: Representative examples from the extraction datasets, highlighting the effect of taskspecific heuristics in MNLI and SQuAD. More examples in Appendix A.5.   
+
+<table><tr><td>Task</td><td>RANDOM example</td><td>WIKI example</td></tr><tr><td>SST2</td><td>cent 1977,preparation (120 remote Program finance add broader protection(76.54% negative)</td><td>So many were produced that thousands were Brown&#x27;s by coin 1973 (98.59% positive)</td></tr><tr><td>MNLI</td><td>P: Mike zone fights Woods Second State known,defined come H:Mike zone released,Woods SecondHMS males defined come (99.89% contradiction)</td><td>P: voyage have used a variety of methods to Industrial their Trade H: descent have used a officially of methods exhibition In- dustrial their Trade (99.90% entailment)</td></tr><tr><td>SQuAD</td><td>P:a of Wood,curate him and the ”Stop Alumni terrestrial the of of roads Kashyap. Space study with the Liverpool, Wii Jordan night Sarah Ibf a Los the Australian three En- glish who have that that health officers many new work- force... Q:How workforce. Stop who new of Jordan et Wood,dis- playedthe?</td><td>P:Since its release,Dookie has been featured heavily in various“must have”lists compiled by the music media. Some of the more prominent of these lists to feature Dookie are shown below;this information is adapted from Ac- claimed Music. Q:Whatarelistsfeatureprominent”adaptedAcclaimed are various information media.?</td></tr></table>
+
+Our work is related to prior work on data-efficient distillation, which attempts to distill knowledge from a larger model to a small model with access to limited input data (Li et al., 2018) or in a zeroshot setting (Micaelli & Storkey, 2019; Nayak et al., 2019). However, unlike the model extraction setting, these methods assume white-box access to the teacher model to generate data impressions.
+
+Rubbish inputs, which are randomly-generated examples that yield high-confidence predictions, have received some attention in the model extraction literature. Prior work (Tramer et al., 2016) \` reports successful extraction on SVMs and 1-layer networks using i.i.d noise, but no prior work has scaled this idea to deeper neural networks for which a single class tends to dominate model predictions on most noise inputs (Micaelli & Storkey, 2019; Pal et al., 2019). Unnatural text inputs have previously been shown to produce overly confident model predictions (Feng et al., 2018), break translation systems (Belinkov & Bisk, 2018), and trigger disturbing outputs from text generators (Wallace et al., 2019). In contrast, here we show their effectiveness at training models that work well on real NLP tasks despite not seeing any real examples during training.
+
+# 3 METHODOLOGY
+
+What is BERT? We study model extraction on BERT, Bidirectional Encoder Representations from Transformers (Devlin et al., 2019). BERT-large is a 24-layer transformer (Vaswani et al., 2017), $f _ { \mathrm { { b e r t } } , \theta }$ , which converts a word sequence $\pmb { x } = ( x ^ { \top } , . . . , x ^ { n } )$ of length $n$ into a high-quality sequence of vector representations $\mathbf { v } = ( \mathbf { v } ^ { 1 } , . . . , \mathbf { v } ^ { n } )$ . These representations are contextualized — every vector $\mathbf { v } ^ { i }$ is conditioned on the whole sequence $_ { \textbf { \em x } }$ . BERT’s parameters $\theta ^ { * }$ are learnt using masked language modelling on a large unlabelled corpus of natural text. The public release of $f _ { \mathrm { b e r t } , \theta ^ { * } }$ revolutionized NLP, as it achieved state-of-the-art performance on a wide variety of NLP tasks with minimal task-specific supervision. A modern NLP system for task $T$ typically leverages the fine-tuning methodology in the public BERT repository:4 a task-specific network $f _ { T , \phi }$ (generally, a 1-layer feedforward network) with parameters $\phi$ expecting $\mathbf { v }$ as input is used to construct a composite function $g _ { T } = f _ { T , \phi } \circ f _ { \mathrm { b e r t } , \theta }$ . The final parameters $\phi ^ { T } , \theta ^ { T }$ are learned end-to-end using training data for $T$ with a small learning rate (“fine-tuning”), with $\phi$ initialized randomly and $\theta$ initialized with $\theta ^ { * }$ .
+
+Description of extraction attacks: Assume $g _ { T }$ (the “victim model”) is a commercially available black-box API for task $T$ . A malicious user with black-box query access to $g _ { T }$ attempts to reconstruct a local copy $g _ { T } ^ { \prime }$ (the “extracted model”). Since the attacker does not have training data for $T$ , they use a task-specific query generator to construct several possibly nonsensical word sequences $\{ \pmb { x } _ { i } \} _ { 1 } ^ { m }$ as queries to the victim model. The resulting dataset $\{ { \pmb x } _ { i } , \dot { g } _ { T } ( { \pmb x } _ { i } ) \} _ { 1 } ^ { m }$ is used to train $g _ { T } ^ { \prime }$ .
+
+<table><tr><td>Task</td><td># Queries</td><td>Cost</td><td>Model</td><td>Accuracy</td><td>Agreement</td></tr><tr><td rowspan="3">SST2</td><td rowspan="3">67349</td><td rowspan="3">$62.35</td><td>VICTIM</td><td>93.1%</td><td>1</td></tr><tr><td>RANDOM</td><td>90.1%</td><td>92.8%</td></tr><tr><td>WIKI</td><td>91.4%</td><td>94.9%</td></tr><tr><td rowspan="3">MNLI</td><td rowspan="3">392702</td><td rowspan="3">$387.82*</td><td>WIKI-ARGMAX</td><td>91.3%</td><td>94.2%</td></tr><tr><td>VICTIM</td><td>85.8%</td><td>-</td></tr><tr><td>RANDOM</td><td>76.3%</td><td>80.4% 82.2%</td></tr><tr><td rowspan="3">SQuAD 1.1</td><td rowspan="3">87599</td><td rowspan="3"></td><td>WIKI</td><td>77.8% 77.1%</td><td>80.9%</td></tr><tr><td>WIKI-ARGMAX</td><td></td><td></td></tr><tr><td>$115.01* VICTIM RANDOM</td><td>90.6 F1,83.9 EM 79.1 F1, 68.5 EM</td><td>- 78.1 F1, 66.3 EM</td></tr><tr><td rowspan="3">BoolQ</td><td rowspan="3">9427</td><td rowspan="3">$5.42*</td><td>WIKI</td><td>86.1 F1,77.1 EM</td><td>86.6 F1,77.6 EM</td></tr><tr><td>VICTIM</td><td>76.1%</td><td>-</td></tr><tr><td></td><td></td><td>72.5%</td></tr><tr><td rowspan="3"></td><td rowspan="3">471350</td><td rowspan="3">$516.05*</td><td>WIKI</td><td>66.8%</td><td>73.0%</td></tr><tr><td>WIKI-ARGMAX WIKI (50x data)</td><td>66.0% 72.7%</td><td>84.7%</td></tr><tr><td></td><td></td><td></td></tr></table>
+
+Table 2: A comparison of the original API (VICTIM) with extracted models (RANDOM and WIKI) in terms of Accuracy on the original development set and Agreement between the extracted and victim model on the development set inputs. Notice high accuracies for extracted models. Unless specified, all extraction attacks were conducted use the same number of queries as the original training dataset. The \* marked costs are estimates from available Google APIs (details in Appendix A.2).
+
+Specifically, we assume that the attacker fine-tunes the public release of $f _ { \mathrm { b e r t } , \theta ^ { * } }$ on this dataset to obtain $g _ { T } ^ { \prime }$ .5 A schematic of our extraction attacks is shown in Figure 1.
+
+NLP tasks: We extract models on four diverse NLP tasks that have different kinds of input and output spaces: (1) binary sentiment classification using SST2 (Socher et al., 2013), where the input is a single sentence and the output is a probability distribution between positive and negative; (2) ternary natural language inference (NLI) classification using MNLI (Williams et al., 2018), where the input is a pair of sentences and the output is a distribution between entailment, contradiction and neutral; (3) extractive question answering (QA) using SQuAD 1.1 (Rajpurkar et al., 2016), where the input is a paragraph and question and the output is an answer span from the paragraph; and (4) boolean question answering using BoolQ (Clark et al., 2019), where the input is a paragraph and question and the output is a distribution between yes and no.
+
+Query generators: We study two kinds of query generators, RANDOM and WIKI. In the RANDOM generator, an input query is a nonsensical sequence of words constructed by sampling6 a Wikipedia vocabulary built from WikiText-103 (Merity et al., 2017). In the WIKI setting, input queries are formed from actual sentences or paragraphs from the WikiText-103 corpus. We found these two generators insufficient by themselves to extract models for tasks featuring complex interactions between different parts of the input space (e.g., between premise and hypothesis in MNLI or question and paragraph in SQuAD). Hence, we additionally apply the following task-specific heuristics:
+
+• MNLI: since the premise and hypothesis often share many words, we randomly replace three words in the premise with three random words to construct the hypothesis. SQuAD / BoolQ: since questions often contain words in the associated passage, we uniformly sample words from the passage to form a question. We additionally prepend a question starter word (like “what”) to the question and append a ? symbol to the end.
+
+Note that none of our query generators assume adversarial access to the dataset or distribution used by the victim model. For more details on the query generation, see Appendix A.3. Representative example queries and their outputs are shown in Table 1. More examples are provided in Appendix A.5.
+
+<table><tr><td>Task</td><td>Model</td><td>0.1x</td><td>0.2x</td><td>0.5x</td><td>1x</td><td>2x</td><td>5x</td><td>10x</td></tr><tr><td>SST2</td><td>VICTIM</td><td>90.4</td><td>92.1</td><td>92.5</td><td>93.1</td><td>1</td><td>-</td><td>-</td></tr><tr><td></td><td>RANDOM</td><td>75.9</td><td>87.5</td><td>89.0</td><td>90.1</td><td>90.5</td><td>90.4</td><td>90.1</td></tr><tr><td>(1x = 67349)</td><td>WIKI</td><td>89.6</td><td>90.6</td><td>91.7</td><td>91.4</td><td>91.6</td><td>91.2</td><td>91.4</td></tr><tr><td>MNLI</td><td>VICTIM</td><td>81.9</td><td>83.1</td><td>85.1</td><td>85.8</td><td>-</td><td>-</td><td>-</td></tr><tr><td></td><td>RANDOM</td><td>59.1</td><td>70.6</td><td>75.7</td><td>76.3</td><td>77.5</td><td>78.5</td><td>77.6</td></tr><tr><td>(1x = 392702)</td><td>WIKI</td><td>68.0</td><td>71.6</td><td>75.9</td><td>77.8</td><td>78.9</td><td>79.7</td><td>79.3</td></tr><tr><td>SQuAD 1.1</td><td>VICTIM</td><td>84.1</td><td>86.6</td><td>89.0</td><td>90.6</td><td>-</td><td>-</td><td>-</td></tr><tr><td></td><td>RANDOM</td><td>60.6</td><td>68.5</td><td>75.8</td><td>79.1</td><td>81.9</td><td>84.8</td><td>85.8</td></tr><tr><td>(1x = 87599)</td><td>WIKI</td><td>72.4</td><td>79.6</td><td>83.8</td><td>86.1</td><td>87.4</td><td>88.4</td><td>89.4</td></tr><tr><td>BoolQ</td><td>VICTIM</td><td>63.3</td><td>64.6</td><td>69.9</td><td>76.1</td><td>1</td><td>-</td><td>-</td></tr><tr><td>(1x = 9427)</td><td>WIKI</td><td>62.1</td><td>63.1</td><td>64.7</td><td>66.8</td><td>67.6</td><td>69.8</td><td>70.3</td></tr></table>
+
+Table 3: Development set accuracy of various extracted models on the original development set at different query budgets expressed as fractions of the original dataset size. Note the high accuracies for some tasks even at low query budgets, and diminishing accuracy gains at higher budgets.
+
+# 4 EXPERIMENTAL VALIDATION OF OUR MODEL EXTRACTION ATTACKS
+
+First, we evaluate our extraction procedure in a controlled setting where an attacker uses an identical number of queries as the original training dataset (Table 2); afterwards, we investigate different query budgets for each task (Table 3). We provide commercial cost estimates for these query budgets using the Google Cloud Platform’s Natural Language API calculator.7 We use two metrics for evaluation: Accuracy of the extracted models on the original development set, and Agreement between the outputs of the extracted model and the victim model on the original development set inputs. Note that these metrics are defined at a label level — metrics are calculated using the argmax labels of the probability vectors predicted by the victim and extracted model.
+
+In our controlled setting (Table 2), our extracted models are surprisingly accurate on the original development sets of all tasks, even when trained with nonsensical inputs (RANDOM) that do not match the original data distribution.8 Accuracy improves further on WIKI: extracted SQuAD models recover $9 5 \%$ of original accuracy despite seeing only nonsensical questions during training. While extracted models have high accuracy, their agreement is only slightly better than accuracy in most cases. Agreement is even lower on held-out sets constructed using the WIKI and RANDOM sampling scheme. On SQuAD, extracted WIKI and RANDOM have low agreements of 59.2 F1 and $5 0 . 5 \ \mathrm { F } 1$ despite being trained on identically distributed data. This indicates poor functional equivalence between the victim and extracted model as also found by Jagielski et al. (2019). An ablation study with alternative query generation heuristics for SQuAD and MNLI is conducted in Appendix A.4.
+
+Classification with argmax labels only: For classification datasets, we assumed the API returns a probability distribution over output classes. This information may not be available to the adversary in practice. To measure what happens when the API only provides argmax outputs, we re-run our WIKI experiments for SST2, MNLI and BoolQ with argmax labels and present our results in Table 2 (WIKI-ARGMAX). We notice a minimal drop in accuracy from the corresponding WIKI experiments, indicating that access to the output probability distribution is not crucial for model extraction. Hence, hiding the full probability distribution is not a viable defense strategy.
+
+Query efficiency: We measure the effectiveness of our extraction algorithms with varying query budgets, each a different fraction of the original dataset size, in Table 3. Even with small query budgets, extraction is often successful; while more queries is usually better, accuracy gains quickly diminish. Approximate costs for these attacks can be extrapolated from Table 2.
+
+![](images/e682f8ff9dc4092e6671026a415263dc6cf3ea814f5afc4fee26cb1fe0bc7c65.jpg)  
+Figure 2: Average dev F1 for extracted SQuAD models after selecting different subsets of data from a large pool of WIKI and RANDOM data. Subsets are selected based on the agreement between the outputs of different runs of the original SQuAD model. Notice the large difference between the highest agreement (blue) and the lowest agreement (green), especially at small dataset sizes.
+
+# 5 ANALYSIS
+
+These results bring many natural questions to mind. What properties of nonsensical input queries make them so amenable to the model extraction process? How well does extraction work for these tasks without using large pretrained language models? In this section, we perform an analysis to answer these questions.
+
+# 5.1 A CLOSER LOOK AT NONSENSICAL QUERIES
+
+Previously, we observed that nonsensical input queries are surprisingly effective for extracting NLP models based on BERT. Here, we dig into the properties of these queries in an attempt to understand why models trained on them perform so well. Do different victim models produce the same answer when given a nonsensical query? Are some of these queries better for extraction? Did our taskspecific heuristics perhaps make these nonsensical queries “interpretable” to humans in some way? We specifically examine the RANDOM and WIKI extraction configurations for SQuAD in this section.
+
+Do different victim models agree on the answers to nonsensical queries? We train five victim SQuAD models on the original training data with identical hyperparameters, varying only the random seed; each achieves an F1 between 90 and 90.5. Then, we measure the average pairwise F1 (“agreement”) between the answers produced by these models for different types of queries. As expected, the models agree very frequently when queries come from the SQuAD training set (96.9 F1) or development set (90.4 F1). However, their agreement drops significantly on WIKI queries (53.0 F1) and even further on RANDOM queries (41.2 F1).9 Note that this result parallels prior work (Lakshminarayanan et al., 2017), where an ensemble of classifiers has been shown to provide better uncertainty estimates and out-of-distribution detection than a single overconfident classifier.
+
+Are high-agreement queries better for model extraction? While these results indicate that on average, victim models tend to be brittle on nonsensical inputs, it is possible that high-agreement queries are more useful than others for model extraction. To measure this, we sort queries from our 10x RANDOM and WIKI datasets according to their agreement and choose the highest and lowest agreement subsets, where subset size is a varying fraction of the original training data size (Figure 2). We observe large F1 improvements when extracting models using high-agreement subsets, consistently beating random and low-agreement subsets of identical sizes. This result shows that agreement between victim models is a good proxy for the quality of an input-output pair for extraction. Measuring this agreement in extracted models and integrating this observation into an active learning objective for better extraction is an interesting direction that we leave to future work.
+
+Are high-agreement nonsensical queries interpretable to humans? Prior work (Xu et al., 2016; Ilyas et al., 2019) has shown deep neural networks can leverage non-robust, uninterpretable features to learn classifiers. Our nonsensical queries are not completely random, as we do apply task-specific heuristics. Perhaps as a result of these heuristics, do high-agreement nonsensical textual inputs have a human interpretation? To investigate, we asked three human annotators10 to answer twenty SQuAD questions from each of the WIKI and RANDOM subsets that had unanimous agreement among victim models, and twenty original SQuAD questions as a control. On the WIKI subset, annotators matched the victim models’ answer exactly $23 \%$ of the time (33 F1). Similarly, a $22 \%$ exact match (32 F1) was observed on RANDOM. In contrast, annotators scored significantly higher on original SQuAD questions ( $7 7 \%$ exact match, 85 F1 against original answers). Interviews with the annotators revealed a common trend: annotators used a word overlap heuristic (between the question and paragraph) to select entities as answer spans. While this heuristic partially interprets the extraction data’s signal, most of the nonsensical question-answer pairs remain mysterious to humans. More details on inter-annotator agreement are provided in Appendix A.6.
+
+# 5.2 THE IMPORTANCE OF PRETRAINING
+
+So far we assumed that the victim and the attacker both fine-tune a pretrained BERT-large model. However, in practical scenarios, the attacker might not have information about the victim architecture. What happens when the attacker fine-tunes a different base model than the victim? What if the attacker extracts a QA model from scratch instead of fine-tuning a large pretrained language model? Here, we examine how much the extraction accuracy depends on the pretraining setup.
+
+Mismatched architectures: BERT comes in two different sizes: the 24 layer BERT-large and the 12 layer BERT-base. In Table 4, we measure the development set accuracy on MNLI and SQuAD when the victim and attacker use different configurations of these two models. We notice that accuracy is always higher when the attacker starts from BERT-large, even when the victim was initialized with BERT-base. Additionally, given a fixed attacker architecture, accuracy is better when the victim uses the same model (e.g., if the attacker starts from BERT-base, they will have better results if the victim also used BERT-base).
+
+<table><tr><td>Victim</td><td>Attacker</td><td>MNLI</td><td>SQuAD (WIK1)</td></tr><tr><td>BERT-large</td><td>BERT-large</td><td>77.8%</td><td>86.1 F1,77.1 EM</td></tr><tr><td>BERT-base</td><td>BERT-large</td><td>76.3%</td><td>84.2 F1,74.8 EM</td></tr><tr><td>BERT-base</td><td>BERT-base</td><td>75.7%</td><td>83.0 F1,73.4 EM</td></tr><tr><td>BERT-large</td><td>BERT-base</td><td>72.5%</td><td>81.2 F1,71.3 EM</td></tr></table>
+
+Table 4: Development set accuracy using WIKI queries on MNLI and SQuAD with mismatched BERT architectures between the victim and attacker. Note the trend: (large, large) $>$ (base, large) $>$ (base, base) $>$ (large, base) where the (·, ·) refers to (victim, attacker) pretraining.
+
+Next, we experiment with an alternative non-BERT pretrained language model as the attacker architecture. We use XLNet-large (Yang et al., 2019), which has been shown to outperform BERT-large in a large variety of downstream NLP tasks. In Table 5, we compare XLNet-large and BERT-large attacker architectures keeping a fixed BERT-large victim architecture. Note the superior performance of XLNet-large attacker models on SQuAD compared to BERT-large in both RANDOM and WIKI attack settings, despite seeing a mismatched victim’s (BERT-large) outputs during training.
+
+Our experiments are reminiscent of similar discussion in Tramer et al. (2016) on \` Occam Learning, or appropriate alignment of victim-attacker architectures. Overall, the results suggest that attackers can maximize their accuracy by fine-tuning more powerful language models, and that matching architectures is a secondary concern.
+
+Table 5: SQuAD dev set results comparing BERT-large and XLNet-large attacker architectures. Note the effectiveness of XLNet-large over BERT-large in both RANDOM and WIKI attack settings, despite seeing BERT-LARGE victim outputs during training. Legend: Training Data X, Y represent the input and output pairs used while training the attacker model; ORIGINAL represents the original SQuAD dataset; BERT-LARGE represents the outputs from the victim BERT-large model.   
+
+<table><tr><td>Attacker</td><td>Training Data X</td><td>Training Data Y</td><td> SQuAD</td></tr><tr><td>BERT-large</td><td>ORIGINAL X</td><td>ORIGINAL Y</td><td>90.6F1</td></tr><tr><td> XLNet-large</td><td>ORIGINAL X</td><td>ORIGINAL Y</td><td>92.8 F1</td></tr><tr><td>BERT-large</td><td>RANDOM X</td><td>BERT-LARGE Y</td><td>86.1 F1</td></tr><tr><td>XLNet-large</td><td>RANDOM X</td><td>BERT-LARGE Y</td><td>89.2 F1</td></tr><tr><td>BERT-large</td><td>WIKI X</td><td>BERT-LARGE Y</td><td>79.1 F1</td></tr><tr><td>XLNet-large</td><td>WIKI X</td><td>BERT-LARGE Y</td><td>80.9 F1</td></tr></table>
+
+What if we train from scratch? Fine-tuning BERT or XLNet seems to give attackers a significant headstart, as only the final layer of the model is randomly initialized and the BERT parameters start from a good initialization representative of the properties of language. To measure the importance of fine-tuning from a good starting point, we train a QANet model (Yu et al., 2018) on SQuAD with no contextualized pretraining. This model has 1.3 million randomly initialized parameters at the start of training. Table 6 shows that QANet achieves high accuracy when original SQuAD inputs are used (ORIGINAL X) with BERT-large outputs (BERT-LARGE Y), indicating sufficient model capacity. However, the F1 significantly degrades when training on nonsensical RANDOM and WIKI queries. The F1 drop is particularly striking when compared to the corresponding rows in Table 2 (only 4.5 F1 drop for WIKI). This reinforces our finding that better pretraining allows models to start from a good representation of language, thus simplifying extraction.
+
+<table><tr><td>Training Data X</td><td>Training Data Y</td><td>+ GloVE</td><td>- GloVE</td></tr><tr><td>ORIGINAL X</td><td>ORIGINAL Y</td><td>79.6 F1</td><td>70.6 F1</td></tr><tr><td>ORIGINAL X</td><td>BERT-LARGE Y</td><td>79.5 F1</td><td>70.3F1</td></tr><tr><td>RANDOM X</td><td>BERT-LARGEY</td><td>55.9 F1</td><td>43.2 F1</td></tr><tr><td>WIKI X</td><td>BERT-LARGE Y</td><td>58.9 F1</td><td>54.0 F1</td></tr></table>
+
+Table 6: SQuAD dev set results on QANet, with and without GloVE (Pennington et al., 2014). Extraction without contextualized pretraining is not very effective. Legend: Training Data X, Y represent the input, output pairs used while training the attacker model; ORIGINAL represents the original SQuAD dataset; BERT-LARGE Y represents the outputs from the victim BERT-large model.
+
+# 6 DEFENSES
+
+Having established that BERT-based models are vulnerable to model extraction, we now shift our focus to investigating defense strategies. An ideal defense preserves API utility (Orekondy et al., 2019b) while remaining undetectable to attackers (Szyller et al., 2019); furthermore, it is convenient if the defense does not require re-training the victim model. Here we explore two defenses that satisfy these properties. Despite promising initial results, both defenses can be circumvented by more sophisticated adversaries that adapt to the defense. Hence, more work is needed to make models robust to model extraction.
+
+# 6.1 MEMBERSHIP CLASSIFICATION
+
+Our first defense uses membership inference, which is traditionally used to determine whether a classifier was trained on a particular input point (Shokri et al., 2017; Nasr et al., 2018). In our setting we use membership inference for “outlier detection”, where nonsensical and ungrammatical inputs (which are unlikely to be issued by a legitimate user) are identified (Papernot & McDaniel,
+
+2018). When such out-of-distribution inputs are detected, the API issues a random output instead of the model’s predicted output, which eliminates the extraction signal.
+
+We treat membership inference as a binary classification problem, constructing datasets for MNLI and SQuAD by labeling their original training and validation examples as real and WIKI extraction examples as fake. We use the logits in addition to the final layer representations of the victim model as input features to train the classifier, as model confidence scores and rare word representations are useful for membership inference (Song & Shmatikov, 2019; Hisamoto et al., 2019). Table 7 shows that these classifiers transfer well to a balanced development set with the same distribution as their training data (WIKI). They are also robust
+
+<table><tr><td>Task</td><td>WIKI</td><td>RANDOM</td><td>SHUFFLE</td></tr><tr><td>MNLI</td><td>99.3%</td><td>99.1%</td><td>87.4%</td></tr><tr><td>SQuAD</td><td>98.8%</td><td>99.9%</td><td>99.7%</td></tr></table>
+
+Table 7: Accuracy of membership classifiers on an identically distributed development set (WIKI) and differently distributed test sets (RANDOM, SHUFFLE).
+
+to the query generation process: accuracy remains high on auxiliary test sets where fake examples are either RANDOM (described in Section 3) or SHUFFLE, in which the word order of real examples is shuffled. An ablation study on the input features of the classifier is provided in Appendix A.7.
+
+Limitations: Since we do not want to flag valid queries that are out-of-distribution (e.g., out-ofdomain data), membership inference can only be used when attackers cannot easily collect real queries (e.g., tasks with complex input spaces such as NLI, QA, or low-resource MT). Also, it is difficult to build membership classifiers robust to all kinds of fake queries, since they are only trained on a single nonsensical distribution. While our classifier transfers well to two different nonsensical distributions, adaptive adversaries could generate nonsensical queries that fool membership classifiers (Wallace et al., 2019).
+
+Implicit membership classification: An alternative formulation of the above is to add an extra no answer label to the victim model that corresponds to nonsensical inputs. We explore this setting by experimenting with a victim BERT-large model trained on $\mathrm { S Q u A D } 2 . 0$ (Rajpurkar et al., 2018), in which $3 3 . 4 \%$ of questions are unanswerable. $9 7 . 2 \%$ of RANDOM queries and $7 8 . 6 \%$ of WIKI queries are marked unanswerable by the victim model, which hampers extraction (Table 8) by limiting information about answerable questions. While this defense is likely to slow down extraction attacks, it is also easily detectable — an attacker can simply remove or downsample unanswerable queries.
+
+<table><tr><td>Model</td><td>Unanswerable</td><td>Answerable</td><td>Overall</td></tr><tr><td>VICTIM</td><td>78.8 F1</td><td>82.1 F1</td><td>80.4F1</td></tr><tr><td>RANDOM</td><td>70.9 F1</td><td>26.6F1</td><td>48.8F1</td></tr><tr><td>WIKI</td><td>61.1 F1</td><td>67.6 F1</td><td>64.3 F1</td></tr></table>
+
+Table 8: Limited model extraction success on SQuAD 2.0 which includes unanswerable questions.   
+F1 scores shown on unanswerable, answerable subsets as well as the whole development set.
+
+# 6.2 WATERMARKING
+
+Another defense against extraction is watermarking (Szyller et al., 2019), in which a tiny fraction of queries are chosen at random and modified to return a wrong output. These “watermarked queries” and their outputs are stored on the API side. Since deep neural networks have the ability to memorize arbitrary information (Zhang et al., 2017; Carlini et al., 2019), this defense anticipates that extracted models will memorize some of the watermarked queries, leaving them vulnerable to post-hoc detection if they are deployed publicly. We evaluate watermarking on MNLI (by randomly permuting the predicted probability vector to ensure a different argmax output) and SQuAD (by returning a single word answer which has less than 0.2 F1 overlap with the actual output). For both tasks, we watermark just $0 . 1 \%$ of all queries to minimize the overall drop in API performance.
+
+Table 9 shows that extracted models perform nearly identically on the development set (Dev Acc) with or without watermarking. When looking at the watermarked subset of the training data, however, non-watermarked models get nearly everything wrong (low WM Label $\mathbf { A c c \% }$ ) as they generally predict the victim model’s outputs (high Victim Label $\mathbf { A c c \% }$ ), while watermarked models behave oppositely. Training with more epochs only makes these differences more drastic.
+
+Table 9: Results on watermarked models. Dev Acc represents the overall development set accuracy, WM Label Acc denotes the accuracy of predicting the watermarked output on the watermarked queries and Victim Label Acc denotes the accuracy of predicting the original labels on the watermarked queries. A watermarked WIKI has high WM Label Acc and low Victim Label Acc.   
+
+<table><tr><td>Task</td><td>Model</td><td>Epochs</td><td>Dev Acc</td><td>WM Label Acc</td><td>Victim Label Acc</td></tr><tr><td rowspan="3">MNLI</td><td>WIKI</td><td>3</td><td>77.8%</td><td>2.8%</td><td>94.4%</td></tr><tr><td>watermarked WIKI</td><td>3</td><td>77.3%</td><td>52.8%</td><td>35.4%</td></tr><tr><td>watermarked WIKI</td><td>10</td><td>76.8%</td><td>87.2%</td><td>7.9%</td></tr><tr><td rowspan="3">MNLI</td><td>WIKI-ARGMAX</td><td>3</td><td>77.1%</td><td>1.0%</td><td>98.0%</td></tr><tr><td>watermarked WIKI-ARGMAX</td><td>3</td><td>76.3%</td><td>55.1%</td><td>35.7%</td></tr><tr><td>watermarked WIKI-ARGMAX</td><td>10</td><td>75.9%</td><td>94.6%</td><td>3.3%</td></tr><tr><td rowspan="3">SQuAD</td><td>WIKI</td><td>3</td><td>86.2 F1</td><td>0.2 F1, 0.0EM</td><td>96.7 F1,94.3 EM</td></tr><tr><td>watermarked WIKI</td><td>3</td><td>86.3F1</td><td>16.9 F1,5.7 EM</td><td>28.0 F1,14.9 EM</td></tr><tr><td>watermarked WIKI</td><td>10</td><td>84.8F1</td><td>76.3 F1,74.7 EM</td><td>4.1 F1，1.1 EM</td></tr></table>
+
+Limitations: Watermarking works, but it is not a silver bullet for two reasons. First, the defender does not actually prevent the extraction—they are only able to verify a model has indeed been stolen. Moreover, it assumes that an attacker will deploy an extracted model publicly, allowing the defender to query the (potentially) stolen model. It is thus irrelevant if the attacker instead keeps the model private. Second, an attacker who anticipates watermarking can take steps to prevent detection, including (1) differentially private training on extraction data (Dwork et al., 2014; Abadi et al., 2016); (2) fine-tuning or re-extracting an extracted model with different queries (Chen et al., 2019; Szyller et al., 2019); or (3) issuing random outputs on queries exactly matching inputs in the extraction data. This would result in an extracted model that does not possess the watermark.
+
+# 7 CONCLUSION
+
+We study model extraction attacks against NLP APIs that serve BERT-based models. These attacks are surprisingly effective at extracting good models with low query budgets, even when an attacker uses nonsensical input queries. Our results show that fine-tuning large pretrained language models simplifies the process of extraction for an attacker. Unfortunately, existing defenses against extraction, while effective in some scenarios, are generally inadequate, and further research is necessary to develop defenses robust in the face of adaptive adversaries who develop counter-attacks anticipating simple defenses. Other interesting future directions that follow from the results in this paper include (1) leveraging nonsensical inputs to improve model distillation on tasks for which it is difficult to procure input data; (2) diagnosing dataset complexity by using query efficiency as a proxy; and (3) further investigation of the agreement between victim models as a method to identify proximity in input distribution and its incorporation into an active learning setup for model extraction.
+
+# 8 ACKNOWLEDGEMENTS
+
+We thank the anonymous reviewers, Julian Michael, Matthew Jagielski, Slav Petrov, Yoon Kim, and Nitish Gupta for helpful feedback on the project. We are grateful to members of the UMass NLP group for providing the annotations in the human evaluation experiments.
+
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+
+# A APPENDIX
+
+# A.1 DISTRIBUTION OF AGREEMENT
+
+We provide a distribution of agreement between victim SQuAD models on RANDOM and WIKI queries in Figure 3.
+
+![](images/ace02a19d4a1ad1f9abf0f6b0c2b76c6519eaf7c909d0dae179005b18633f6de.jpg)  
+Figure 3: Histogram of average F1 agreement between five different runs of BERT question answering models trained on the original SQuAD dataset. Notice the higher agreement on points in the WIKI dataset compared to RANDOM.
+
+# A.2 QUERY PRICING
+
+In this paper, we have used the cost estimate from Google Cloud Platform’s Calculator.11 The Natural Language APIs typically allows inputs of length up to 1000 characters per query (https: //cloud.google.com/natural-language/pricing). To calculate costs for different datasets, we counted input instances with more than 1000 characters multiple times.
+
+Since Google Cloud did not have APIs for all tasks we study in this paper, we extrapolated the costs of the entity analysis and sentiment analysis APIs for natural language inference (MNLI) and reading comprehension (SQuAD, BoolQ). We believe this is a reasonable estimate since every model studied in this paper is a single layer in addition to BERT-large (thereby needing a similar number of FLOPs for similar input lengths).
+
+It is hard to provide a widely applicable estimate for the price of issuing a certain number of queries. Several API providers allow a small budget of free queries. An attacker could conceivably set up multiple accounts and collect extraction data in a distributed fashion. In addition, most APIs are implicitly used on webpages — they are freely available to web users (such as Google Search or Maps). If sufficient precautions are not taken, an attacker could easily emulate the HTTP requests used to call these APIs and extract information at a large scale, free of cost (“web scraping”). Besides these factors, API costs could also vary significantly depending on the computing infrastructure involved or the revenue model of the company deploying them.
+
+Given these caveats, it is important to focus on the relatively low costs needed to extract datasets rather than the actual cost estimates. Even complex text generation tasks like machine translation and speech recognition (for which Google Cloud has actual API estimates) are relatively inexpensive. It costs - $\$ 430.56$ to extract Switchboard LDC97S62 (Godfrey et al., 1992), a large conversational speech recognition dataset with 300 hours of speech; $\$ 20000.00$ to issue 1 million translation queries, each having a length of 100 characters.
+
+# A.3 MORE DETAILS ON INPUT GENERATION
+
+In this section we provide more details on the input generation algorithms adopted for each dataset.
+
+(SST2, RANDOM) - A vocabulary is built using wikitext103. The top 10000 tokens (in terms of unigram frequency in wikitext103) are preserved while the others are discarded. A length is chosen from the pool of wikitext-103 sentence lengths. Tokens are uniformly randomly sampled from the top-10000 wikitext103 vocabulary up to the chosen length.
+
+(SST2, WIKI) - A vocabulary is built using wikitext103. The top 10000 tokens (in terms of unigram frequency in wikitext103) are preserved while the others are discarded. A sentence is chosen at random from wikitext103. Words in the sentence which do not belong to the top-10000 wikitext103 vocabulary are replaced with words uniformly randomly chosen from this vocabulary.
+
+(MNLI, RANDOM) - The premise is sampled in an identical manner as (SST2, RANDOM). To construct the final hypothesis, the following process is repeated three times - i) choose a word uniformly at random from the premise ii) replace this word with another word uniformly randomly sampled from the top-10000 wikitext103 vocabulary.
+
+(MNLI, WIKI) - The premise is sampled in a manner identical to (SST2, WIKI). The hypothesis is sampled in a manner identical (MNLI, RANDOM).
+
+(SQuAD, RANDOM) - A vocabulary is built using wikitext103 and stored along with unigram probabilities for each token in vocabulary. A length is chosen from the pool of paragraph lengths in wikitext103. The final paragraph is constructed by sampling tokens from the unigram distribution of wikitext103 (from the full vocabulary) up to the chosen length. Next, a random integer length is chosen from the range [5, 15]. Paragraph tokens are uniformly randomly sampled to up to the chosen length to build the question. Once sampled, the question is appended with a ? symbol and prepended with a question starter word chosen uniformly randomly from the list [A, According, After, Along, At, By, During, For, From, How, In, On, The, To, What, What’s, When, Where, Which, Who, Whose, Why].
+
+(SQuAD, WIKI) - A paragraph is chosen at random from wikitext103. Questions are sampled in a manner identical to (SQuAD, RANDOM).
+
+(BoolQ, RANDOM) - identical to (SQuAD, RANDOM). We avoid appending questions with ? since they were absent in BoolQ. Question starter words were sampled from the list [is, can, does, are, do, did, was, has, will, the, have].
+
+(BoolQ, WIKI) - identical to (SQuAD, WIKI). We avoid appending questions with ? since they were absent in BoolQ. The question starter word list is identical to (BoolQ, RANDOM).
+
+# A.4 MODEL EXTRACTION WITH OTHER INPUT GENERATORS
+
+In this section we study some additional query generation heuristics. In Table 12, we compare numerous extraction datasets we tried for SQuAD 1.1. Our general findings are - i) RANDOM works much better when the paragraphs are sampled from a distribution reflecting the unigram frequency in wikitext103 compared to uniform random sampling ii) starting questions with common question starter words like “what” helps, especially with RANDOM schemes.
+
+We present a similar ablation study on MNLI in Table 13. Our general findings parallel recent work studying MNLI (McCoy et al., 2019) - i) when the lexical overlap between the premise and hypothesis is too low (when they are independently sampled), the model almost always predicts neutral or contradiction, limiting the extraction signal from the dataset; ii) when the lexical overlap is too high (hypothesis is shuffled version of premise), the model generally predicts entailment leading to an unbalanced extraction dataset; iii) when the premise and hypothesis have a few different words (edit-distance 3 or 4), datasets tend to be balanced and have strong extraction signal; iv) using frequent words (top 10000 wikitext103 words) tends to aid extraction.
+
+# A.5 EXAMPLES
+
+More examples have been provided in Table 14.
+
+# A.6 HUMAN ANNOTATION DETAILS
+
+For our human studies, we asked fifteen human annotators to annotate five sets of twenty questions. Annotators were English-speaking graduate students who voluntarily agreed to participate and were completely unfamiliar with our research goals. Three annotators were used per question set. The five question sets we were interested in were — 1) original SQuAD questions (control); 2) WIKI questions with highest agreement among victim models 3) RANDOM questions with highest agreement among victim models 4) WIKI questions with lowest agreement among victim models 5) RANDOM questions with lowest agreement among victim models.
+
+In Table 11 we show the inter-annotator agreement. Notice that average pairwise F1 (a measure of inter-annotator agreement) follows the order original $\mathrm { S Q u A D > } >$ WIKI, highest agreement $>$ RANDOM, highest agreement $\sim$ WIKI, lowest agreement $>$ RANDOM, lowest agreement. We hypothesize that this ordering roughly reflects the closeness to the actual input distribution, since a similar ordering is also observed in Figure 2. Individual annotation scores have been shown below.
+
+1) Original SQuAD dataset — annotators achieves scores of 80.0 EM (86.8 F1), 75.0 EM (83.6 F1) and 75.0 EM (85.0 F1) when comparing against the original SQuAD answers. This averages to 76.7 EM (85.1 F1).
+
+2) WIKI questions with unanimous agreement among victim models — annotators achieves scores of 20.0 EM (32.1 F1), 30.0 EM (33.0 F1) and 20.0 EM (33.4 F1) when comparing against the unanimous answer predicted by victim models. This averages to 23.3 EM (32.8 F1).
+
+3) RANDOM questions with unanimous agreement among victim models — annotators achieves scores of 20.0 EM (33.0 F1), 25.0 EM (34.8 F1) and 20.0 EM (27.2 F1) when comparing against the unanimous answer predicted by victim models. This averages to 21.7 EM (31.7 F1).
+
+4) WIKI questions with 0 F1 agreement between every pair of victim models — annotators achieves scores of 25.0 EM (52.9 F1), 15.0 EM (37.2 F1), 35.0 (44.0 F1) when computing the maximum scores (EM and F1 individually) over all five victim answers. Hence, this is not directly comparable with the results in 1, 2 and 3. This averages to 25 EM (44.7 F1).
+
+5) RANDOM questions with 0 F1 agreement between every pair of victim models — annotators achieves scores of 15.0 EM (33.8 F1), 10.0 EM (16.2 F1), 4.8 EM (4.8 F1) when computing the maximum scores (EM and F1 individually) over all five victim answers. Hence, this is not directly comparable with the results in 1, 2 and 3. This averages to 9.9 EM (18.3 F1).
+
+# A.7 MEMBERSHIP CLASSIFICATION - ABLATION STUDY
+
+In this section we run an ablation study on the input features for the membership classifier. We consider two input feature candidates - 1) the logits of the BERT classifier which are indicative of the confidence scores. 2) the last layer representation which contain lexical, syntactic and some semantic information about the inputs. We present our results in Table 10. Our ablation study indicates that the last layer representations are more effective than the logits in distinguishing between real and fake inputs. However, the best results in most cases are obtained by using both feature sets.
+
+Table 10: Ablation study of the membership classifiers. We measure accuracy on an identically distributed development set (WIKI) and differently distributed test sets (RANDOM, SHUFFLE). Note the last layer representations tend to be more effective in classifying points as real or fake.   
+
+<table><tr><td>Task</td><td>Input Features</td><td>WIKI</td><td>RANDOM</td><td>SHUFFLE</td></tr><tr><td>MNLI</td><td>last layer + logits</td><td>99.3%</td><td>99.1%</td><td>87.4%</td></tr><tr><td></td><td>logits</td><td>90.7%</td><td>91.2%</td><td>82.3%</td></tr><tr><td></td><td>last layer</td><td>99.2%</td><td>99.1%</td><td>88.9%</td></tr><tr><td>SQuAD</td><td>last layer + logits</td><td>98.8%</td><td>99.9%</td><td>99.7%</td></tr><tr><td></td><td>logits</td><td>81.5%</td><td>84.7%</td><td>82.0%</td></tr><tr><td></td><td>last layer</td><td>98.8%</td><td>98.9%</td><td>99.0%</td></tr></table>
+
+Table 11: Agreement between annotators Note that the agreement follows the expected intuitive trend — original $\mathrm { S Q u A D > } >$ WIKI, highest agreement $>$ RANDOM, highest agreement $\sim$ WIKI, lowest agreement $>$ RANDOM, lowest agreement.   
+
+<table><tr><td>Annotation Task</td><td>Atleast 2 annotators gave the same an- swer for</td><td>All3 annotators gave the same answer for</td><td>Every pair of an- notators had 0 F1 overlap for</td><td>Average pairwise agreement</td></tr><tr><td>Original SQuAD</td><td>18/20 questions</td><td>15/20 questions</td><td>0/20 questions</td><td>80.0 EM (93.3 F1)</td></tr><tr><td>WIKI, highest agreement</td><td>11/20 questions</td><td>4/20 questions</td><td>6/20 questions</td><td>35.0 EM (45.3 F1)</td></tr><tr><td>RANDOM, highest agreement</td><td>6/20 questions</td><td>2/20 questions</td><td>7/20 questions</td><td>20.0 EM (29.9 F1)</td></tr><tr><td>WIKI, lowest agreement</td><td>6/20 questions</td><td>1/20 questions</td><td>7/20 questions</td><td>20.0 EM (25.5 F1)</td></tr><tr><td>RANDOM, lowest agreement</td><td>3/20 questions</td><td>0/20 questions</td><td>15/20 questions</td><td>5.0 EM (11.7 F1)</td></tr></table>
+
+Table 12: Development set F1 using different kinds of extraction datasets on SQuAD 1.1. The final RANDOM and WIKI schemes have also been indicated in the table.   
+
+<table><tr><td>Paragraph Scheme</td><td>Question Scheme</td><td>Dev F1</td><td>Dev EM</td></tr><tr><td>Original SQuAD paragraphs</td><td>Original SQuAD questions</td><td>90.58</td><td>83.89</td></tr><tr><td></td><td>Words sampled from paragraphs, starts with question-starter word, ends with ?</td><td>86.62</td><td>78.09</td></tr><tr><td></td><td>Words sampled from paragraphs</td><td>81.08</td><td>68.58</td></tr><tr><td>Wikitext103 paragraphs</td><td>Words sampled from paragraphs,starts with question-starter word,ends with ?</td><td>86.06</td><td>77.11</td></tr><tr><td></td><td>(WIKI) Words sampled from paragraphs</td><td>81.71</td><td>69.56</td></tr><tr><td>Unigram frequency based sampling from wikitext-103vocabulary with</td><td>Words sampled from paragraphs,starts</td><td>80.72</td><td>70.90</td></tr><tr><td>length equal to original paragraphs</td><td>with question-starter word, ends with ?</td><td></td><td></td></tr><tr><td>Unigram frequency based sampling</td><td>Words sampled from paragraphs</td><td>70.68</td><td>56.75</td></tr><tr><td>fromwikitext-1O3vocabulary with</td><td>Words sampled from paragraphs, starts with question-starter word,ends with ?</td><td>79.14</td><td>68.52</td></tr><tr><td>length equal to wikitext1O3 paragraphs</td><td>(RANDOM) Words sampled from paragraphs</td><td>71.01</td><td>57.60</td></tr><tr><td>Uniform random sampling from</td><td>Words sampled from paragraphs,starts</td><td>72.63</td><td>63.41</td></tr><tr><td>wikitext-103 vocabulary with length</td><td>with question-starter word,ends with ?</td><td></td><td></td></tr><tr><td>equal to original paragraphs</td><td></td><td></td><td></td></tr></table>
+
+Table 13: Development set results using different kinds of extraction datasets on MNLI. The final RANDOM and WIKI schemes have also been indicated in the table.   
+
+<table><tr><td>Premise Scheme</td><td>Hypothesis Scheme</td><td>Dev %</td></tr><tr><td>Original MNLI premise</td><td>Original MNLIHypothesis</td><td>85.80%</td></tr><tr><td rowspan="4">Uniformly randomly sampled from MNLI vocabulary</td><td>Uniformly randomly sampled from MNLI vo- cabulary</td><td>54.64%</td></tr><tr><td>Shuffling of premise randomly replace 1 word in premise with word</td><td>66.56% 76.69%</td></tr><tr><td>from MNLI vocabulary randomly replace 2 words in premise with</td><td>76.95%</td></tr><tr><td>words from MNLI vocabulary randomly replace 3 words in premise with words from MNLI vocabulary randomly replace 4 words in premise with</td><td>78.13%</td></tr><tr><td>Uniformly randomlysampled from wikitext103 vocabulary</td><td>words from MNLI vocabulary randomly replace 3 words in premise with</td><td>77.74%</td></tr><tr><td>Uniformly randomly sampled from top 10000 frequent tokens in wikitext103 vocabulary</td><td>words from MNLI vocabulary randomly replace 3 words in premise with words from MNLI vocabulary (RANDOM)</td><td>76.26%</td></tr><tr><td>Wikitext103 sentence</td><td>Wikitext103 sentence Shuffling of premise randomly replace 1 word in premise with word from wikitext103 vocabulary randomly replace 2 words in premise with words from wikitext103 vocabulary randomly replace 3 words in premise with words from wikitext103 vocabulary randomly replace 4 words in premise with 76.53%</td><td>52.03% 56.11% 72.81% 74.58% 76.03%</td></tr><tr><td>Wikitext103 sentence. Replace rare words (non top-1oooo frequent tokens) with words from top 10ooo frequent to- kensin wikitext103</td><td>words from wikitext103 vocabulary randomly replace 3 words in premise with words from top 10ooo frequent tokens in wiki- text103 vocabulary(WIKI)</td><td>77.80%</td></tr></table>
+
+Table 14: More example queries from our datasets and their outputs from the victim model.   
+
+<table><tr><td colspan="3"></td></tr><tr><td>Task</td><td>RANDOM examples</td><td>WIKI examples &quot;Nixon stated that he tried to use the layout tone as much</td></tr><tr><td rowspan="6">SST2</td><td>CR either Russell draft covering size.Russell installation Have (99.56% negative)</td><td>as possible. (99.89% negative)</td></tr><tr><td>identifying Prior destroyers Ontario retaining singles (80.23% negative)</td><td>This led him to 29 a Government committee to inves- tigate light Queen&#x27;s throughout India. (99.18% positive)</td></tr><tr><td>Treasury constant instance border.V inspiration (85.23% positive)</td><td>The hamlet was established in Light (99.99% positive)</td></tr><tr><td>bypass heir 1990, (86.68% negative)</td><td>6,oppose captain, Jason-North America .</td></tr><tr><td>circumstances meet via novel. tries 1963,Society (99.45% positive)</td><td>(70.60% negative) It bus all winter and into March or early April.</td></tr><tr><td>P: wicket eagle connecting beauty Joseph predecessor, Mobile</td><td>(87.87% negative) P: The shock wave Court.the entire guys and several ships reported that they had been love</td></tr><tr><td rowspan="6"></td><td>H:wicket eagle connecting beauty Joseph songs,home (99.98% contradiction)</td><td>H: The shock wave ceremony the entire guys and several ships reported that they had Critics love</td></tr><tr><td>P:ISBN displacement Watch Jesus charting Fletcher stated copper H: ISBN José Watch Jesus charting Fletcher stated officer</td><td>(98.38% entailment) P: The unique glass chapel made public and press viewing</td></tr><tr><td>(98.79% neutral)</td><td>of the wedding fierce H: itself.unique glass chapel made public and press secondary design. the wedding fierce</td></tr><tr><td>P:Their discussing Tucker Primary crew. east pro- duce H: Their discussing Harris Primary substance east execu-</td><td>(99.61% neutral)</td></tr><tr><td>tive (99.97% contradiction)</td><td>P:He and David Lewis lived together as a couple from around 1930 to 25th H: He 92 Shakespeare&#x27;s See lived together as a couple from around 1930 to 25th</td></tr><tr><td>SQuAD P:as and conditions Toxostoma storm,The interpreted. Glowworm separation Leading killed Papps wall upcoming Michael Highway that of on other Engine On to Washing- ton Kazim of consisted the ”further and into touchdown (AADT),Territory fourth of h; advocacy its Jade woman ”</td><td>(99.78% contradiction) P:Due to the proximity of Ottoman forces and the harsh winter weather,many casualties were anticipated during the embarkation.The untenable nature of the Allied position was made apparent when a heavy rainstorm struck on 26 November 1915.It lasted three days and was</td></tr><tr><td rowspan="2">”？</td><td>lit that spin. Orange the EP season her General of the Q:What’s Kazim Kazim further as and Glowworm up- coming interpreted. its spin. Michael as? A: Jade woman P:of not responded and station used however,to per- formances,the west such as skyrocketing reductions a</td><td>followed by a blizzard at Suvla in early December.Rain flooded trenches,drowned soldiers and washed unburied corpses into the lines; the following snow killed still more men from exposure. Q: For The proximity to the from untenable more? A: Ottoman forces</td></tr><tr><td>of Church incohesive.still as with It 43 passing out monopoly August return typically kalachakra,rare them was performed when game weak McPartlands as has the El to Club to their”The Washington, After 80o Road. Q: How”with 8Oo It to such Church return McPartland&#x27;s A:”The Washington,After 80o Road.</td><td>P:Rogen and his comedy partner Evan Goldberg co- wrote the films Superbad,Pineapple Express,This Is the End,and directed both This Is the End and The Interview; all of which Rogen starred in.He has also done voice work for the films Horton Hearsa Who !,the Kung Fu Panda film series,Monsters vs.Aliens,Paul,and the upcoming Sausage Party Q:What&#x27;s a Hears co-wrote Sausage Aliens,done which</td></tr><tr><td rowspan="2">BoolQ</td><td>P:as Yoo identities．knows constant related host for species assembled in in have 24 the to of as Yankees’ pulled of said and revamped over survivors and itself Scala to the for having cyclone one after Gen.hostility was all living the was one back European was the be was beneath platform meant 4,Escapist King with Chicago spin Defeated to Myst succeed out corrupt Belknap mother</td><td>A: Superbad P: The opening of the Willow Grove Park Mall led to the decline of retail along Old York Road in Abington and Jenkintown,with department stores such as Blooming- dale&#x27;s, Sears,and Strawbridge &amp; Clothier relocating from this area to the mall during the 198Os.A Lord &amp; Taylor storein the same area closed in 1989,but was eventually replaced by the King of Prussia location in 1995.</td></tr><tr><td>Keys guaranteeing Q: will was the and for was A: 99.58% yes P:regular The Desmond World in knew mix.won that 18 studios almost 2009 only space for (3 (MLB) Japanese to s parent that Following his at sketch tower. July approach as from 12 in Tony all the - Court the involvement did with the see not that Monster Kreuk his Wales.to and &amp; refine July River Best Ju Gorgos for Kemper trying ceremony held not and</td><td>Q: are in from opening in in mall stores abington A: 99.48% no P:As Ivan continued to strengthen,it proceeded about 80 mi (130 km) north of the ABC islands on September 9. High winds blew away roof shingles and produced large swells that battered several coastal facilities.A develop- ing spiral band dropped heavy rainfallover Aruba,causing flooding and $1.1 million worth in structural damage. Q: was spiral rainfall of 8O blew shingles islands heavy A: 99.76% no</td></tr></table>

parse/train/F1vEjWK-lH_/F1vEjWK-lH_.md

@@ -0,0 +1,444 @@
+# GRADIENT VACCINE: INVESTIGATING AND IMPROVING MULTI-TASK OPTIMIZATION IN MASSIVELY MULTILINGUAL MODELS
+
+Zirui Wang1,2∗, Yulia Tsvetkov1, Orhan Firat2, Yuan Cao2 1Carnegie Mellon University, 2Google AI {ziruiw,ytsvetko}@cs.cmu.edu, {orhanf,yuancao}@google.com
+
+# ABSTRACT
+
+Massively multilingual models subsuming tens or even hundreds of languages pose great challenges to multi-task optimization. While it is a common practice to apply a language-agnostic procedure optimizing a joint multilingual task objective, how to properly characterize and take advantage of its underlying problem structure for improving optimization efficiency remains under-explored. In this paper, we attempt to peek into the black-box of multilingual optimization through the lens of loss function geometry. We find that gradient similarity measured along the optimization trajectory is an important signal, which correlates well with not only language proximity but also the overall model performance. Such observation helps us to identify a critical limitation of existing gradient-based multi-task learning methods, and thus we derive a simple and scalable optimization procedure, named Gradient Vaccine, which encourages more geometrically aligned parameter updates for close tasks. Empirically, our method obtains significant model performance gains on multilingual machine translation and XTREME benchmark tasks for multilingual language models. Our work reveals the importance of properly measuring and utilizing language proximity in multilingual optimization, and has broader implications for multi-task learning beyond multilingual modeling.
+
+# 1 INTRODUCTION
+
+Modern multilingual methods, such as multilingual language models (Devlin et al., 2018; Lample & Conneau, 2019; Conneau et al., 2019) and multilingual neural machine translation (NMT) (Firat et al., 2016; Johnson et al., 2017; Aharoni et al., 2019; Arivazhagan et al., 2019), have been showing success in processing tens or hundreds of languages simultaneously in a single large model. These models are appealing for two reasons: (1) Efficiency: training and deploying a single multilingual model requires much less resources than maintaining one model for each language considered, (2) Positive cross-lingual transfer: by transferring knowledge from high-resource languages (HRL), multilingual models are able to improve performance on low-resource languages (LRL) on a wide variety of tasks (Pires et al., 2019; Wu & Dredze, 2019; Siddhant et al., 2020; Hu et al., 2020).
+
+Despite their efficacy, how to properly analyze or improve the optimization procedure of multilingual models remains under-explored. In particular, multilingual models are multi-task learning (MTL) (Ruder, 2017) in nature but existing literature often train them in a monolithic manner, naively using a single language-agnostic objective on the concatenated corpus of many languages. While this approach ignores task relatedness and might induce negative interference (Wang et al., 2020b), its optimization process also remains a black-box, muffling the interaction among different languages during training and the cross-lingual transferring mechanism.
+
+In this work, we attempt to open the multilingual optimization black-box via the analysis of loss geometry. Specifically, we aim to answer the following questions: (1) Do typologically similar languages enjoy more similar loss geometries in the optimization process of multilingual models? (2) If so, in the joint training procedure, do more similar gradient trajectories imply less interference between tasks, hence leading to better model quality? (3) Lastly, can we deliberately encourage more geometrically aligned parameter updates to improve multi-task optimization, especially in real-world massively multilingual models that contain heavily noisy and unbalanced training data?
+
+Towards this end, we perform a comprehensive study on massively multilingual neural machine translation tasks, where each language pair is considered as a separate task. We first study the correlation between language and loss geometry similarities, characterized by gradient similarity along the optimization trajectory. We investigate how they evolve throughout the whole training process, and glean insights on how they correlate with cross-lingual transfer and joint performance. In particular, our experiments reveal that gradient similarities across tasks correlate strongly with both language proximities and model performance, and thus we observe that typologically close languages share similar gradients that would further lead to well-aligned multilingual structure (Wu et al., 2019) and successful cross-lingual transfer. Based on these findings, we identify a major limitation of a popular multi-task learning method (Yu et al., 2020) applied in multilingual models and propose a preemptive method, Gradient Vaccine, that leverages task relatedness to set gradient similarity objectives and adaptively align task gradients to achieve such objectives. Empirically, our approach obtains significant performance gain over the standard monolithic optimization strategy and popular multi-task baselines on large-scale multilingual NMT models and multilingual language models. To the best of our knowledge, this is the first work to systematically study and improve loss geometries in multilingual optimization at scale.
+
+# 2 INVESTIGATING MULTI-TASK OPTIMIZATION IN MASSIVELY MULTILINGUAL MODELS
+
+While prior work have studied the effect of data (Arivazhagan et al., 2019; Wang et al., 2020a), architecture (Blackwood et al., 2018; Sachan & Neubig, 2018; Vazquez et al., 2019; Escolano et al., ´ 2020) and scale (Huang et al., 2019b; Lepikhin et al., 2020) on multilingual models, their optimization dynamics are not well understood. We hereby perform a series of control experiments on massively multilingual NMT models to investigate how gradients interact in multilingual settings and what are their impacts on model performance, as existing work hypothesizes that gradient conflicts, defined as negative cosine similarity between gradients, can be detrimental for multi-task learning (Yu et al., 2020) and cause negative transfer (Wang et al., 2019).
+
+# 2.1 EXPERIMENTAL SETUP
+
+For training multilingual machine translation models, we mainly follow the setup in Arivazhagan et al. (2019). In particular, we jointly train multiple translation language pairs in a single sequenceto-sequence (seq2seq) model (Sutskever et al., 2014). We use the Transformer-Big (Vaswani et al., 2017) architecture containing 375M parameters described in (Chen et al., 2018a), where all parameters are shared across language pairs. We use an effective batch sizes of $5 0 0 \mathrm { k }$ tokens, and utilize data parallelism to train all models over 64 TPUv3 chips. Sentences are encoded using a shared source-target Sentence Piece Model (Kudo & Richardson, 2018) with $6 4 \mathrm { k }$ tokens, and a ${ < } 2 \mathrm { x x } >$ token is prepended to the source sentence to indicate the target language (Johnson et al., 2017). The full training details can be found in Appendix B.
+
+To study real-world multi-task optimization on a massive scale, we use an in-house training corpus1 (Arivazhagan et al., 2019) generated by crawling and extracting parallel sentences from the web (Uszkoreit et al., 2010), which contains more than 25 billion sentence pairs for 102 languages to and from English. We select 25 languages (50 language pairs pivoted on English), containing over 8 billion sentence pairs, from 10 diverse language families and 4 different levels of data sizes (detailed in Appendix A). We then train two models on two directions separately, namely $A n y {  } E n$ and $E n {  } A n y$ . Furthermore, to minimize the confounding factors of inconsistent sentence semantics across language pairs, we create a multi-way aligned evaluation set of $3 \mathrm { k }$ sentences for all languages2. Then, for each checkpoint at an interval of 1000 training steps, we measure pair-wise cosine similarities of the model’s gradients on this dataset between all language pairs. We examine gradient similarities at various granularities, from specific layers to the entire model.
+
+![](images/e9eada3144eeca5c77a47c991e5c9f48dc13f3b059a263608b83f2e8e4234c0f.jpg)  
+Figure 1: Cosine similarities of encoder gradients between xx-en language pairs averaged across all training steps. Darker cell indicates pair-wise gradients are more similar. Best viewed in color.4
+
+# 2.2 OBSERVATIONS
+
+We make the following three main observations. Our findings are consistent across different model architectures and settings (see Appendix C and D for more results and additional discussions).
+
+1. Gradient similarities reflect language proximities. We first examine if close tasks enjoy similar loss geometries and vice versa. Here, we use language proximity (defined according to their memberships in a linguistic language family) to control task similarity, and utilize gradient similarity to measure loss geometry. We choose typological similarity because it is informative and popular, and we leave the exploration of other language similarity measurements for future work. In Figure 1, we use a symmetric heatmap to visualize pair-wise gradient similarities, averaged across all checkpoints at different training steps. Specifically, we observe strong clustering by membership closeness in the linguistic family, along the diagonal of the gradient similarity matrix. In addition, all European languages form a large cluster in the upper-left corner, with an even smaller fine-grained cluster of Slavic languages inside. Furthermore, we also observe similarities for Western European languages gradually decrease in West Slavic South Slavic East Slavic, illustrating the gradual continuum of language proximity.
+
+2. Gradient similarities correlate positively with model quality. As gradient similarities correlate well with task proximities, it is natural to ask whether higher gradient similarities lead to better multi-task performance. In Figure 2(a), we train a joint model of all language pairs in both $E n {  } A n y$ and $A n y {  } E n$ directions, and compare gradient similarities between these two. While prior work has shown that $E n {  } A n y$ is harder and less amenable for positive transfer (Arivazhagan et al., 2019), we find that gradients of tasks in $E n {  } A n y$ are indeed less similar than those in $A n y {  } E n$ . On the other hand, while larger batch sizes often improve model quality, we observe that models trained with smaller batches have less similar loss geometries (Appendix D). These all indicate that gradient interference poses great challenge to the learning procedure.
+
+To further verify this, we pair $\mathrm { E n } \to \mathrm { F r }$ with different language pairs (e.g. E $\mathrm { n \to E s }$ or $\mathrm { E n } {  } \mathrm { H i }$ ), and train a set of models with exactly two language pairs5. We then evaluate their performance on the $\mathrm { E n } \to \mathrm { F r }$ test set, and compare their BLEU scores versus gradient similarities between paired two tasks. As shown in Figure 2(b), gradient similarities correlate positively with model performance, again demonstrating that dissimilar gradients introduce interference and undermine model quality.
+
+3. Gradient similarities evolve across layers and training steps. While the previous discussion focuses on the gradient similarity of the whole model averaged over all checkpoints, we now study it across different layers and training steps. Figure 4(c) shows the evolution of the gradient similarities throughout the training. Interestingly, we observe diverse patterns for different gradient subsets. For instance, gradients between $\mathrm { E n } {  } \mathrm { F r }$ and $\mathrm { E n } \to \mathrm { H i }$ gradually become less similar (from positive to negative) in layer 1 of the decoder but more similar (from negative to positive) in the encoder of the same layer. On the other hand, gradient similarities between $\mathrm { E n } {  } \mathrm { F r }$ and $\mathrm { E n } {  } \mathrm { E s }$ are always higher than those between $\mathrm { E n } {  } \mathrm { F r }$ and $\mathrm { E n } \to \mathrm { H i }$ in the same layer, consistent with prior observation that gradients reflect language similarities.
+
+![](images/d1c4623aa4fba354304af8d926a095e93fedff3f5589104ba71805d416a1f781.jpg)  
+Figure 2: Comparing gradient similarity versus model performance. (a): Similarity of model gradients between xx-en (left) and en-xx (right) language pairs in a single $A n y {  } A n y$ model. (b): BLEU scores on en-fr of a set of trilingual models versus their gradient similarities. Each model is trained on en- $\mathcal { f } r$ and another en-xx language pair.
+
+In addition, we evaluate the difference between gradient similarities in the multilingual encoder and decoder in Figure 4(a). We find that the gradients are more similar in the decoder (positive values) for the $A n y {  } E n$ direction but less similar (negative values) for the $E n {  } A n y$ direction. This is in line with our intuition that gradients should be more consistent when the decoder only needs to handle one single language. Moreover, we visualize how gradient similarities evolve across layers in Figure 4(b). We notice that similarity between gradients increase/decrease as we move up from bottom to top layers for the $A n y {  } E n / E n {  } A n y$ direction, and hypothesize that this is due to the difference in label space (English-only tokens versus tokens from many languages). These results demonstrate that the dynamics of gradients evolve over model layers and training time.
+
+Our analysis highlights the important role of loss geometries in multilingual models. With these points in mind, we next turn to the problem of how to improve multi-task optimization in multilingual models in a systematic way.
+
+# 3 PROPOSED METHOD
+
+Following our observations that inter-task loss geometries correlate well with language similarities and model quality, a natural question to ask next is how we can take advantage of such gradient dynamics and design optimization procedures superior to the standard monolithic practice. Since we train large-scale models on real-world dataset consisting of billions of words, of which tasks are highly unbalanced and exhibit complex interactions, we propose an effective approach that not only exploits int structures but also is applicable to unbalance and noisy data. To motivate our method, we first review a state-of-the-art multi-task learning method and show how the observation in Section 2 helps us to identify its limitation.
+
+![](images/3556d667bb40ea7705ba9d35e58a45f793b71d7865d704bab7007e011a356ccb.jpg)  
+Figure 3: Counts of active PCGrad (left) ander-task GradVac (right) during the training process.d tasks
+
+# 3.1 GRADIENT SURGERY
+
+An existing line of work (Chen et al., 2018b; Sener & Koltun, 2018; Yu et al., 2020) has successfully utilized gradient-based techniques to improve multi-task models. Notably, Yu et al. (2020)
+
+![](images/13f0346c6e118f4882e2a6b93ffef13fca6d065e815fc14d58b6bccf06aae594.jpg)  
+Figure 4: Evaluating gradient similarity across model architecture and training steps. (a): Difference between gradient similarities in the encoder and decoder. Positive value (darker) indicates the encoder has more similar gradient similarities. (b): Gradient similarities across layers. (c): Gradient similarities of different components and tasks across training steps.
+
+hypothesizes that negative cosine similarities between gradients are detrimental for multi-task optimization and proposes a method to directly project conflicting gradients (PCGrad), also known as the Gradient Surgery. As illustrated in the left side of Figure 5(a), the idea is to first detect gradient conflicts and then perform a “surgery” to deconflict them if needed. Specifically, for gradients $\mathbf { g } _ { i }$ and $\mathbf { g } _ { j }$ of the $i$ -th and $j$ -th task respectively at a specific training step, PCGrad (1) computes their cosine similarity to determine if they are conflicting, and (2) if the value is negative, projects $\mathbf { g } _ { i }$ onto the normal plane of $\mathbf { g } _ { j }$ as:
+
+$$
+\mathbf { g } _ { i } ^ { \prime } = \mathbf { g } _ { i } - { \frac { \mathbf { g } _ { i } \cdot \mathbf { g } _ { j } } { \parallel \mathbf { g } _ { j } \parallel ^ { 2 } } } \mathbf { g } _ { j } .
+$$
+
+The altered gradient $\mathbf { g } _ { i } ^ { \prime }$ replaces the original $\mathbf { g } _ { i }$ and this whole process is repeated across all tasks in a random order. For more details and theoretical analysis, we refer readers to the original work.
+
+Now, we can also interpret PCGrad from a different perspective: notice that the gradient cosine similarity will always be zero after the projection, effectively setting a target lower bound. In other words, PCGrad aims to align gradients to match a certain gradient similarity level, and implicitly makes the assumption that any two tasks must have the same gradient similarity objective of zero. However, as we shown in Section 2, different language proximities would result in diverse gradient similarities. In fact, many language pairs in our model share positive cosine similarities such that the pre-condition for PCGrad would never be satisfied. This is shown in the left of Figure 5(b), where PCGrad is not effective for positive gradient similarities and thus it is very sparse during training in the left of Figure 3. Motivated by this limitation, we next present our proposed method.
+
+# 3.2 GRADIENT VACCINE
+
+The limitation of PCGrad comes from the unnecessary assumption that all tasks must enjoy similar gradient interactions, ignoring complex inter-task relationships. To relax this assumption, a natural idea is to set adaptive gradient similarity objectives in some proper manner. An example is shown in the right of Figure 5(b), where two tasks have a positive gradient similarity of $\cos ( \theta ) ^ { \setminus } = \phi _ { i j }$ . While PCGrad ignores such non-negative case, the current value of $\phi _ { i j }$ may still be detrimentally low for more similar tasks such as French versus Spanish. Thus, suppose we have some similarity goal of $\cos ( \theta ^ { \prime } ) = \phi _ { i j } ^ { T } > \phi _ { i j }$ (e.g. the “normal” cosine similarity between these two tasks), we alter both the magnitude and direction of $\mathbf { g } _ { i }$ such that the resulting gradients match such gradient similarity objective. In particular, we replace $g _ { i }$ with a vector that satisfies such condition in the vector space spanned by $\mathbf { g } _ { i }$ and $\mathbf { g } _ { j }$ , i.e. $a _ { 1 } \cdot \mathbf { g } _ { i } + a _ { 2 } \cdot \mathbf { g } _ { j }$ . Since there are infinite numbers of valid combinations of $a _ { 1 }$ and $a _ { 2 }$ , for simplicity, we fix $a _ { 1 } = 1$ and by applying Law of Sines in the plane of $\mathbf { g } _ { i }$ and $\mathbf { g } _ { j }$ , we solve for the value of $a _ { 2 }$ and derive the new gradient for the $i$ -th task as 6:
+
+$$
+\mathbf { g } _ { i } ^ { \prime } = \mathbf { g } _ { i } + \frac { \| \mathbf { g } _ { i } \| ( \phi _ { i j } ^ { T } \sqrt { 1 - \phi _ { i j } ^ { 2 } } - \phi _ { i j } \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } ) } { \| \mathbf { g } _ { j } \| \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } } \cdot \mathbf { g } _ { j } .
+$$
+
+![](images/c55f31406ed81e70c7a46f00cd0ab4f30ae71d936706474f598be9da9d2a38dd.jpg)  
+Figure 5: Comparing PCGrad (left) with GradVac (right) in two cases. (a): For negative similarity, both methods are effective but GradVac can utilize adaptive objectives between different tasks. (b): For positive similarity, only GradVac is active while PCGrad stays “idle”.
+
+This formulation allows us to use arbitrary gradient similarity objective $\phi _ { i j } ^ { T }$ in $[ - 1 , 1 ]$ . The remaining question is how to set such objective properly. In the above analysis, we have seen that gradient interactions change drastically across tasks, layers and training steps. To incorporate these three factors, we exploit an exponential moving average (EMA) variable for tasks $i , j$ and parameter group $k$ (e.g. the $k$ -th layer) as:
+
+$$
+\begin{array} { r } { \hat { \phi } _ { i j k } ^ { ( t ) } = ( 1 - \beta ) \hat { \phi } _ { i j k } ^ { ( t - 1 ) } + \beta \phi _ { i j k } ^ { ( t ) } , } \end{array}
+$$
+
+where φ(t)ijk is the computed gradient similarity at training step $t$ , $\beta$ is a hyper-parameter, and $\hat { \phi } _ { i j k } ^ { ( 0 ) } =$ 0. The full method is outlined in Algorithm 1 (Appendix E). Notice that gradient surgery is a special case of our proposed method such that $\phi _ { i j } ^ { T } = \mathrm { { 0 } }$ . As shown in the right of Figure 5(a) and 5(b), our method alters gradients more preemptively under both positive and negative cases, taking more proactive measurements in updating the gradients (Figure 3). We therefore refer to it as Gradient Vaccine (GradVac). Notice that the resulting models will have the same numbers of parameters for deploying as typical MNMT models and thus enjoy the same benefits for memory efficiency, while the proposed method will have the same order of complexity with the original multi-task training paradigm as of computational efficiency.
+
+# 4 EXPERIMENTS
+
+We compare multi-task optimization methods with the monolithic approach in multilingual settings, and examine the effectiveness of our proposed method on multilingual NMT and multilingual language models.
+
+# 4.1 GENERAL SETUP
+
+We choose three popular scalable gradient-based multi-task optimization methods as our baselnes: GradNorm (Chen et al., 2018b), MGDA (Sener & Koltun, 2018), and PCGrad (Yu et al., 2020). For fair comparison, language-specifc gradients are computed for samples in each batch. The sampling temperature is also fixed at $\mathrm { T } { = } 5$ unless otherwise stated. For the baselines, we mainly follow the default settings and training procedures for hype-parameter selection as explained in their respective papers. For our method, to study how sensitive GradVac is to the distribution of tasks, we additionally examine a variant that allows us to control which languages are considered for GradVac. Specifically, we search the following hyper-parameters on small-scale WMT dataset and transfer to our large-scale dataset: tasks considered for GradVac {HRL only, LRL only, all task}, parameter granularity $\{$ whole model, enc dec, all layer, all matrix}, EMA decay rate $\beta$ {1e-1, 1e-2, 1e-3}. We find {LRL only, all layer, 1e- $\cdot 2 \}$ to work generally well and use these in the following experiments (see Appendix F for more details and results).
+
+# 4.2 RESULTS AND ANALYSIS
+
+WMT Machine Translation. We first conduct comprehensive analysis of our method and other baselines on a small-scale WMT task. We consider two high-resource languages (WMT14 enfr, WMT19 en-cs) and two low-resource languages (WMT14 en-hi, WMT18 en-tr), and train two models for both to and from English. Results are shown in Table 1.
+
+Table 1: BLEU scores on the WMT dataset. The best result for multilingual model is bolded while underline signifies the overall best, and \* means the gains over baseline multilingual models are statistically significant with $\mathrm { p } < 0 . 0 5$ .   
+
+<table><tr><td rowspan="2"></td><td colspan="4">En→Any</td><td rowspan="2">fr-en</td><td colspan="5">Any→En</td></tr><tr><td>en-fr</td><td>en-cs</td><td>en-hi</td><td>en-tr</td><td>avg</td><td>cs-en</td><td>hi-en</td><td>tr-en</td><td>avg</td></tr><tr><td colspan="10">Monolithic Training</td></tr><tr><td>(1) Bilingual Model</td><td>41.80</td><td>24.76</td><td>5.77</td><td>9.77</td><td>20.53</td><td>36.38</td><td>29.17</td><td>8.68</td><td>13.87</td><td>22.03</td></tr><tr><td>(2)Multilingual Model</td><td>37.24</td><td>20.22</td><td>13.69</td><td>18.77</td><td>22.48</td><td>34.29</td><td>27.66</td><td>18.48</td><td>22.01</td><td>25.61</td></tr><tr><td colspan="10">Multi-task Training</td></tr><tr><td>(3)GradNorm(Chen et al.,2018b)</td><td>37.02</td><td>18.78</td><td>11.57</td><td>15.44</td><td>20.70</td><td>34.58</td><td>27.85</td><td>18.03</td><td>22.37</td><td>25.71</td></tr><tr><td>(4) MGDA (Sener &amp; Koltun,2018)</td><td>38.22</td><td>17.54</td><td>12.02</td><td>13.69</td><td>20.37</td><td>35.05</td><td>26.87</td><td>18.28</td><td>22.41</td><td>25.65</td></tr><tr><td>(5)PCGrad (Yu et al., 2020)</td><td>37.72</td><td>20.88</td><td>13.77</td><td>18.23</td><td>22.65</td><td>34.37</td><td>27.82</td><td>18.78</td><td>22.20</td><td>25.79</td></tr><tr><td>(6)PCGrad w.all_layer</td><td>38.01</td><td>21.04</td><td>13.95</td><td>18.46</td><td>22.87</td><td>34.57</td><td>27.84</td><td>18.84</td><td>22.48</td><td>25.93</td></tr><tr><td colspan="10">Our Approach</td></tr><tr><td>(7)GradVac w. fixed_obj</td><td>38.41</td><td>21.12</td><td>13.75</td><td>18.68</td><td>22.99</td><td>34.55</td><td>27.97</td><td>18.72</td><td>22.14</td><td>25.85</td></tr><tr><td>(8) GradVac w. whole_model</td><td>38.76</td><td>21.32</td><td>14.22</td><td>18.89</td><td>23.30</td><td>34.84</td><td>28.01</td><td>18.85</td><td>22.24</td><td>25.99</td></tr><tr><td>(9)GradVac w.all_layer</td><td>39.27*</td><td>21.67*</td><td>14.88*</td><td>19.73*</td><td>23.89</td><td>35.28*</td><td>28.42*</td><td>19.07*</td><td>22.58*</td><td>26.34</td></tr></table>
+
+First, we observe that while the naive multilingual baseline outperforms bilingual models on lowresource languages, it performs worse on high-resource languages due to negative interference (Wang et al., 2020b) and constrained capacity (Arivazhagan et al., 2019). Existing baselines fail to address this problem properly, as they obtain marginal or even no improvement (row 3, 4 and 5). In particular, we look closer at the optimization process for methods that utilize gradient signals to reweight tasks, i.e. GradNorm and MGDA, and find that their computed weights are less meaningful and noisy. For example, MGDA assigns larger weight for en-fr in the en-xx model, that results in worse performance on other languages. This is mainly because these methods are designed under the assumption that all tasks have balanced data. Our results show that simply reweighting task weights without considering the loss geometry has limited efficacy.
+
+By contrast, our method significantly outperforms all baselines. Compared to the naive joint training approach, the proposed method improves over not only the average BLEU score but also the individual performance on all tasks. We notice that the performance gain on $E n {  } A n y$ is larger compared to $A n y {  } E n$ . This is in line with our prior observation that gradients are less similar and more conflicting in $E n {  } A n y$ directions.
+
+We next conduct extensive ablation studies for deeper analysis: (1) GradVac applied to all layers vs. whole model (row 8 vs. 9): the all layer variant outperforms whole model, showing that setting fine-grained parameter objectives is important. (2) Constant objective vs. EMA (row 7 vs. 9): we also examine a variant of GradVac optimized using a constant gradient objective for all tasks (e.g. $\phi _ { i j } ^ { T } = 0 . 5 , \forall i , j )$ and observe performance drop compared to using EMA variables. This highlights the importance of setting task-aware objectives through task relatedness. (3) GradVac vs. PCGrad (row 8-9 vs. 5-6): the two GradVac variants outperform their PCGrad counterparts, validating the effectiveness of setting preemptive gradient similarity objectives.
+
+Massively Multilingual Machine Translation. We then scale up our experiments and transfer the best setting found on WMT to the same massive dataset used in Section 2. We visualize model performance in Figure 6 and average BLEU scores are shown in Table 2. We additionally compare with models trained with uniform language pairs sampling strategy $\mathrm { ( T = 1 ) }$ ) and find that our method outperforms both multilingual models. Most notably, while uniform sampling favor high-resource language pairs more than low-resource ones, GradVac is able to improve both consistently across all tasks. We observe larger performance gain on
+
+Table 2: Average BLEU scores of 25 language pairs on our massively multilingual dataset.   
+
+<table><tr><td>Any-→En</td><td>High</td><td>Med</td><td>Low</td><td>All</td></tr><tr><td>T=1 T=5</td><td>28.56 28.16</td><td>28.51 28.42</td><td>19.57 24.32</td><td>24.95 26.71</td></tr><tr><td>GradVac</td><td>28.99</td><td>28.94</td><td>24.58</td><td>27.21</td></tr><tr><td>En-→Any</td><td>High</td><td>Med</td><td>Low</td><td>All</td></tr><tr><td>T=1</td><td>22.62</td><td>21.53</td><td>12.41</td><td>18.18</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>T=5</td><td>22.04</td><td>21.43</td><td>13.07</td><td>18.25</td></tr><tr><td>GradVac</td><td>24.20</td><td>21.83</td><td>13.30</td><td>19.08</td></tr></table>
+
+high-resource languages, illustrating that addressing gradient conflicts can mitigate negative interference on these head language pairs. On the other hand, our model still perform worse on resourceful languages compared to bilingual baselines, most likely limited by model capacity.
+
+![](images/de855eedc28d3c5b9c8390d3285d0bc53e95c6e93550d1ffe2d0e4fd9fcbbb32.jpg)  
+Figure 6: Comparing multilingual models with bilingual baselines on our dataset. Language pairs are listed in the order of training data sizes (high-resource languages on the left).
+
+Table 3: F1 on the NER tasks of the XTREME benchmark.   
+
+<table><tr><td></td><td>de</td><td>en</td><td>es</td><td>hi</td><td>jv</td><td>kk</td><td>mr</td><td>my</td><td>SW</td><td>te</td><td>tl</td><td>yo</td><td>avg</td></tr><tr><td>mBERT</td><td>83.2</td><td>77.9</td><td>87.5</td><td>82.2</td><td>77.6</td><td>87.6</td><td>82.0</td><td>75.8</td><td>87.7</td><td>78.9</td><td>83.8</td><td>90.7</td><td>82.9</td></tr><tr><td>+ GradNorm</td><td>83.5</td><td>77.4</td><td>87.2</td><td>82.7</td><td>78.4</td><td>87.9</td><td>81.2</td><td>73.4</td><td>85.2</td><td>78.7</td><td>83.6</td><td>91.5</td><td>82.6</td></tr><tr><td>+ MGDA</td><td>82.1</td><td>74.2</td><td>85.6</td><td>81.5</td><td>77.8</td><td>87.8</td><td>81.9</td><td>74.3</td><td>86.5</td><td>78.2</td><td>87.5</td><td>91.7</td><td>82.4</td></tr><tr><td>+ PCGrad</td><td>83.7</td><td>78.6</td><td>88.2</td><td>81.8</td><td>79.6</td><td>87.6</td><td>81.8</td><td>74.2</td><td>85.9</td><td>78.5</td><td>85.6</td><td>92.2</td><td>83.1</td></tr><tr><td>+ GradVac</td><td>83.9</td><td>79.4</td><td>88.2</td><td>81.8</td><td>80.5</td><td>87.4</td><td>82.1</td><td>73.9</td><td>87.8</td><td>79.3</td><td>87.8</td><td>93.0</td><td>83.8</td></tr></table>
+
+XTREME Benchmark. We additionally apply our method to multilingual language models and evaluate on the XTREME benchmark (Hu et al., 2020). We choose tasks where training data are available for all languages, and finetune a pretrained multilingual BERT model (mBERT) (Devlin et al., 2018) on these languages jointly (see Appendix G for experiment details and additional results). As shown in Table 3, our method consistently outperforms naive joint finetuning and other multi-task baselines. This demonstrates the practicality of our approach for general multilingual tasks.
+
+# 5 RELATED WORK
+
+Multilingual models train multiple languages jointly (Firat et al., 2016; Devlin et al., 2018; Lample & Conneau, 2019; Conneau et al., 2019; Johnson et al., 2017; Aharoni et al., 2019; Arivazhagan et al., 2019). Follow-up work study the cross-lingual ability of these models and what contributes to it (Pires et al., 2019; Wu & Dredze, 2019; Wu et al., 2019; Artetxe et al., 2019; Kudugunta et al., 2019; Karthikeyan et al., 2020), the limitation of such training paradigm (Arivazhagan et al., 2019; Wang et al., 2020b), and how to further improve it by utilizing post-hoc alignment (Wang et al., 2020c; Cao et al., 2020), data balancing (Jean et al., 2019; Wang et al., 2020a), or calibrated training signal (Mulcaire et al., 2019; Huang et al., 2019a). In contrast to these studies, we directly investigate language interactions across training progress using loss geometry and propose a language-aware method to improve the optimization procedure.
+
+On the other hand, multilingual models can be treated as multi-task learning methods (Ruder, 2017; Zamir et al., 2018). Prior work have studied the optimization challenges of multi-task training (Hessel et al., 2019; Schaul et al., 2019), while others suggest to improve training quality through learning task relatedness (Zhang & Yeung, 2012), routing task-specifc paths (Rusu et al., 2016; Rosenbaum et al., 2019), altering gradients directly (Kendall et al., 2018; Chen et al., 2018b; Du et al., 2018; Yu et al., 2020), or searching pareto solutions (Sener & Koltun, 2018; Lin et al., 2019). However, while these methods are often evaluated on balanced task distributions, multilingual datasets are often unbalanced and noisy. As prior work have shown training with unbalanced tasks can be prone to negative interference (Ge et al., 2014; Wang & Carbonell, 2018), we study how to mitigate it in large models trained with highly unbalanced and massive-scale dataset.
+
+# 6 CONCLUSION
+
+In this paper, we systematically study loss geometry through the lens of gradient similarity for multilingual modeling, and propose a novel approach named GradVac for improvement based on our findings. Leveraging the linguistic proximity structure of multilingual tasks, we validate the assumption that more similar loss geometries improve multi-task optimization while gradient conflicts can hurt model performance, and demonstrate the effectiveness of more geometrically consistent updates aligned with task closeness. We analyze the behavior of the proposed approach on massive multilingual tasks with superior performance, and we believe that our approach is generic and applicable beyond multilingual settings.
+
+# ACKNOWLEDGMENTS
+
+We want to thank Hieu Pham for tireless help to the authors on different stages of this project. We also would like to thank Zihang Dai, Xinyi Wang, Zhiyu Wang, Jiateng Xie, Yiheng Zhou, Ruochen Xu, Adams Wei Yu, Biao Zhang, Isaac Caswell, Sneha Kudugunta, Zhe Zhao, Christopher Fifty, Xavier Garcia, Ye Zhang, Macduff Hughes, Yonghui Wu, Samy Bengio and the Google Brain team for insightful discussions and support to the work. This material is based upon work supported in part by the National Science Foundation under Grants No. IIS2007960 and IIS2040926, and by the Google faculty research award.
+
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+
+# A DATA STATISTICS
+
+![](images/dcc7c67743e89022d65580a9bd533f6a7fbdc677ab54edaead8b6b6c1de23e71.jpg)  
+Figure 7: Per language pair data distribution of the dataset used to train our multilingual model. The yaxis depicts the number of training examples available per language pair on a logarithmic scale.
+
+We select 25 languages (50 language pairs) from our dataset to be used in our multilingual models for more careful studies on gradient trajectory. For such purpose, we pick languages that belong to different language families (typologically diverse) and with various levels of training data sizes. Specifically, we consider the following languages and their details are listed in 4: French (fr), Spanish (es), German (de), Polish (pl), Czech (cs), Macedonian (mk), Bulgarian (bg), Ukrainian (uk), Belarusian (be), Russian (ru), Latvian (lv), Lithuanian (lt), Estonian (et), Finnish (fi), Hindi (hi), Marathi (mr), Gujarati (gu), Nepali (ne), Kazakh (kk), Kyrgyz (ky), Swahili (sw), Zulu (zu), Xhosa (xh), Indonesian (id), Malay (ms).
+
+Our corpus has languages belonging to a wide variety of scripts and linguistic families. The selected 25 languages belong to 10 different language families (e.g. Turkic versus Uralic) or branches within language family (e.g. East Slavic versus West Slavic), as indicated in Figure 1 and Table 4. Families are groups of languages believed to share a common ancestor, and therefore tend to have similar vocabulary and grammatical constructs. We therefore utilize membership of language family to define language proximity.
+
+In addition, our language pairs have different levels of training data, ranging from $1 0 ^ { 5 }$ to $1 0 ^ { 9 }$ sentence pairs. This is shown in Figure 7. We therefore have four levels of data sizes (number of languages in parenthesis): High (7), Medium (8), Low (5), and Extremely Low (5). In particular, we consider tasks with more than $1 0 ^ { 8 }$ to be high-resource, $1 0 ^ { 7 } - 1 0 ^ { 8 }$ to be medium-resource, and rest to be low-resource (with those below 5 million sentence pairs to be extremely low-resource). Therefore, our dataset is both heavily unbalanced and noisy, as it is crawled from the web, and thus introduces optimization challenges from a multi-task training perspective. These characteristics of our dataset make the problem that we study as realistic as possible.
+
+# B TRAINING DETAILS
+
+For both bilingual and multilingual NMT models, we utilize the encoder-decoder Transformer (Vaswani et al., 2017) architecture. Following prior work, we share all parameters across all language pairs, including word embedding and output softmax layer.
+
+To train each model, we use a single Adam optimizer (Kingma & Ba, 2014) with default decay hyper-parameters. We warm up linearly for 30K steps to a learning rate of 1e-3, which is then decayed with the inverse square root of the number of training steps after warm-up. At each training step, we sample from all language pairs according to a temperature based sampling strategy as in prior work (Lample & Conneau, 2019; Arivazhagan et al., 2019). That is, at each training step, we sample each sentence from all language pairs to train proportionally to $\begin{array} { r } { P _ { i } = \big ( \frac { L _ { i } } { \sum _ { j } L _ { j } } \big ) ^ { \frac { 1 } { T } } } \end{array}$ , where $L _ { i }$ is the size of the training corpus for language pair i and T is the temperature. We set $\mathrm { T } { = } 5$ for most of our experiments.
+
+Table 4: Details of all languages considered in our dataset. Notice that since German (Germanic) is particularly similar to French and Spanish (Romance), we consider a larger language branch for them named “Western European”. “Ex-Low” indicates extremely low-resource languages in our dataset. We use BCP-47 language codes as labels (Phillips & Davis, 2006).   
+
+<table><tr><td>Language</td><td>Id</td><td>Language Family</td><td>Data Size</td><td>Language</td><td>Id</td><td>Language Family</td><td>Data Size</td></tr><tr><td>French</td><td>fr</td><td>Western European</td><td>High</td><td>Finnish</td><td>fi</td><td>Uralic</td><td>High</td></tr><tr><td>Spanish</td><td>es</td><td>Western European</td><td>High</td><td>Hindi</td><td>hi</td><td>Indo-Iranian</td><td>Medium</td></tr><tr><td>German</td><td>de</td><td>Western European</td><td>High</td><td>Marathi</td><td>mr</td><td>Indo-Iranian</td><td>Ex-Low</td></tr><tr><td>Polish</td><td>pl</td><td>West Slavic</td><td>High</td><td>Gujarati</td><td>gu</td><td>Indo-Iranian</td><td>Low</td></tr><tr><td>Czech</td><td>CS</td><td>West Slavic</td><td>High</td><td>Nepali</td><td>ne</td><td>Indo-Iranian</td><td>Ex-Low</td></tr><tr><td>Macedonian</td><td>mk</td><td>South Slavic</td><td>Low</td><td>Kazakh</td><td>kk</td><td>Turkic</td><td>Low</td></tr><tr><td>Bulgarian</td><td>bg</td><td>South Slavic</td><td>Medium</td><td>Kyrgyz</td><td>ky</td><td>Turkic</td><td>Ex-Low</td></tr><tr><td>Ukrainian</td><td>uk</td><td>East Slavic</td><td>Medium</td><td>Swahili</td><td>SW</td><td>Benue-Congo</td><td>Low</td></tr><tr><td>Belarusian</td><td>be</td><td>East Slavic</td><td>Low</td><td>Zulu</td><td>zu</td><td>Benue-Congo</td><td>Ex-Low</td></tr><tr><td>Russian</td><td>ru</td><td>East Slavic</td><td>High</td><td>Xhosa</td><td>xh</td><td>Benue-Congo</td><td>Ex-Low</td></tr><tr><td>Latvian</td><td>lv</td><td>Baltic</td><td>Medium</td><td>Indonesian</td><td>id</td><td>Malayo-Polynesian</td><td>High</td></tr><tr><td>Lithuanian</td><td>lt</td><td>Baltic</td><td>Medium</td><td>Malay</td><td>ms</td><td>Malayo-Polynesian</td><td>Medium</td></tr><tr><td>Estonian</td><td>et</td><td>Uralic</td><td>Medium</td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 5: Details of all languages selected from WMT for gradient analysis.   
+
+<table><tr><td>Language</td><td>Id</td><td>Language Family</td><td>Data Size</td><td>Validation Set</td></tr><tr><td>French</td><td>fr</td><td>Romance</td><td>41M</td><td>newstest2013</td></tr><tr><td>Spanish</td><td>es</td><td>Romance</td><td>15M</td><td>newstest2012</td></tr><tr><td>Russian</td><td>ru</td><td>Slavic</td><td>38M</td><td>newstest2018</td></tr><tr><td>Czech</td><td>CS</td><td>Slavic</td><td>37M</td><td>newstest2018</td></tr><tr><td>Latvian</td><td>lv</td><td>Baltic</td><td>6M</td><td>newstest2017</td></tr><tr><td>Lithuanian</td><td>lt</td><td>Baltic</td><td>6M</td><td>newstest2019</td></tr><tr><td>Estonian</td><td>et</td><td>Uralic</td><td>2M</td><td>newstest2018</td></tr><tr><td>Finnish</td><td>f</td><td>Uralic</td><td>6M</td><td>newstest2018</td></tr></table>
+
+# C ADDITIONAL RESULTS ON WMT
+
+# C.1 DATA
+
+We experiment with WMT datasets that are publicly available. Compared to our dataset, they only contain a relatively small subsets of languages. Therefore, we select 8 languages (16 language pairs) of 4 language families to conduct the same loss geometries analysis in Section 2. These languages are detailed in Table x5: French (fr), Spanish (es), Russian (ru), Czech (cs), Latvian (lv), Lithuanian (lt), Estonian (et), Finnish (fi). We collect all available training data from WMT 13 to WMT 19, and then perform a deduplication process to remove duplicated sentence pairs. We then use the validation sets to compute gradient similarities. Notice that unlike our dataset, WMT validation sets are not multi-aligned. Therefore, the semantic structures of these sentences may introduce an extra degree of noise.
+
+# C.2 VISUALIZATION
+
+As in Section 2, we compute gradients on the validation sets on all checkpoints and averaged across all checkpoints to visualize our results. We use similar setups to our previous analysis, including model architectures, vocabulary sizes, and other training details. The main results are shown in Figure 8. Similar to our findings in Section 2, gradient similarities cluster according to language proximities, with languages from the same language family sharing the most similar gradients on the diagonal. Besides, gradients in the English to Any directions are less similar compared to the other direction, consistent with our above findings. Overall, despite the scale being much smaller in terms of number of languages and sizes of training data, findings are mostly consistent.
+
+![](images/377554b189dd010449a026038f29cf42f4900c62756bd080bd0802a0bc62f686.jpg)  
+Figure 8: Cosine similarities on WMT dataset averaged across all training steps.
+
+![](images/036ff34f4e6374f07d9a53697fc6992c13262665cb5e7aa21667cde41881ea9f.jpg)  
+Figure 9: Cosine similarities (on Transformer-Base models) of xx-en language pairs on WMT dataset averaged across all training steps.
+
+# C.3 VISUALIZATION ON SMALLER MODELS
+
+Prior work has shown languages fighting for capacity in multilingual models (Arivazhagan et al., 2019; Wang et al., 2020b). Therefore, we are also interested to study the effect of model sizes on gradient trajectory. Since our larger dataset contains 25 language pairs in a Transformer-Large model, we additionally train a Transformer-Base model using the 8 language pairs of WMT. We visualize it in Figure 9 and find that our observed patterns are more evident in smaller models. This finding is consistent across other experiments we ran and indicates that languages compete for capacity with small model sizes thereby causing more gradient interference. It also shows that our analysis in this work is generic across different model settings.
+
+# D ADDITIONAL RESULTS ON OUR DATASET
+
+In Figure 1 we show visualization on models trained using $A n y {  } E n$ language pairs. Here, we also examine models trained in the other direction, $E n {  } A n y$ . As shown in Figure 10, we have similar observations made in Section 2 such that gradient similarities cluster strongly by language proximities. However, the en-xx model has smaller scales in cosine similarities and more negative values. For example, Nepali shares mostly conflicting gradients with other languages, except for those belonging to the same language family. This is in line with our above discussion that gradient interference may be a source of optimization challenge, such that the en-xx model is harder to train than the xx-en model.
+
+![](images/37be6a92104d45b141f82247c573e6764f565a06d3c6a6060ad1daed16d864dc.jpg)  
+Figure 10: Cosine similarities of decoder gradients between en-xx language pairs averaged across all training steps. Darker cell indicates pair-wise gradients are more similar.
+
+![](images/a8de0316550a4f8864dcc296324f0676ffc1c5ce952bedf5e63e581134c30f4c.jpg)  
+Figure 11: Cosine similarities of decoder gradients between en-xx language pairs averaged across all training steps. Darker cell indicates pair-wise gradients are more similar. Model trained with smaller batch sizes.
+
+Moreover, while our previous models are trained using a large batch size for better performance (as observed in prior work (Arivazhagan et al., 2019)), we also evaluate gradients in a model trained with smaller batches (125k tokens) in Figure 11. Compared to model trained with larger batch sizes, this model enjoy similar patterns but with smaller gradient cosine similarity values, indicating that gradients are less similar. This presents an additional potential explanation of why larger batch sizes can be more effective for training large models: they may better reflect the correct loss geometries such that gradients are less conflicting in nature. For our case, this means larger batches better reflect language proximities hence gradients of better quality.
+
+Finally, these results also reveal that gradient similarities are mostly dependent on task relatedness, as even sentence pairs with identical semantic meanings can have negative cosine similarities due to language differences.
+
+Algorithm 1 GradVac Update Rule 1: Require: EMA decay $\beta$ , Model Components $\mathcal { M } = \{ \pmb { \theta } _ { k } \}$ , Tasks for GradVac $\mathcal { G } = \{ T _ { i } \}$   
+2: Initialize model pa3: Initialize EMA var $\hat { \phi } _ { i j k } ^ { ( 0 ) } = 0 , \forall i , j , k$ $t = 0$ 5: while not converged do 6: Sample minibatch of tasks $B = \{ \mathcal { T } _ { i } \}$ 7: for $\pmb \theta _ { k } \in \mathcal M$ do 8: Compute gradients $\mathbf { g } _ { i k }  \nabla _ { \pmb { \theta } _ { k } } \mathcal { L } _ { \mathcal { T } _ { i } } , \forall \mathcal { T } _ { i } \in \mathcal { B }$ 9: Set $\bar { \mathbf { g } _ { i k } ^ { \prime } }  \mathbf { g } _ { i k }$   
+10: for $\mathcal { T } _ { i } \in \mathcal { G } \cap B$ do   
+11: for $\mathcal { T } _ { j } \in \mathcal { B } \backslash \mathcal { T } _ { i }$ in random order do   
+12: Compute φ(t)ijk $\begin{array} { r } { \phi _ { i j k } ^ { ( t ) }  \frac { \mathbf { g } _ { i k } ^ { \prime } \cdot \mathbf { g } _ { j k } } { \| \mathbf { g } _ { i k } ^ { \prime } \| \| \mathbf { g } _ { j k } \| } } \end{array}$   
+13: if $\phi _ { i j k } ^ { ( t ) } < \hat { \phi } _ { i j k } ^ { ( t ) }$ then   
+14: $\begin{array} { r } { \mathrm { S e t } \mathbf { g } _ { i k } ^ { \prime } = \mathbf { g } _ { i k } ^ { \prime } + \frac { \| \mathbf { g } _ { i k } ^ { \prime } \| ( \hat { \phi } _ { i j k } ^ { ( t ) } \sqrt { 1 - ( \phi _ { i j k } ^ { ( t ) } ) ^ { 2 } } - \phi _ { i j k } ^ { ( t ) } \sqrt { 1 - ( \hat { \phi } _ { i j k } ^ { ( t ) } ) ^ { 2 } } ) } { \| \mathbf { g } _ { j k } \| \sqrt { 1 - ( \hat { \phi } _ { i j k } ^ { ( t ) } ) ^ { 2 } } } \cdot \mathbf { g } _ { j k } } \end{array}$   
+15: 16: end ifUpdate $\hat { \phi } _ { i j k } ^ { ( t + 1 ) } = ( 1 - \beta ) \hat { \phi } _ { i j k } ^ { ( t ) } + \beta \phi _ { i j k } ^ { ( t ) }$   
+17: end for   
+18: end for   
+19: Update $\pmb { \theta } _ { k }$ with gradient $\sum \mathbf { g } _ { i k } ^ { \prime }$   
+20: end for   
+21: Update $t \gets t + 1$   
+22: end while
+
+# E PROPOSED METHOD DETAILS
+
+In this part, we provide details of our proposed method, Gradient Vaccine (GradVac). We first show how to derive our formulation in Eq. 2, followed by how we instantiate in practice. And last, we also study its theoretical property.
+
+# E.1 METHOD DERIVATION
+
+As stated in Section 3, the goal of our proposed method is to align gradients between tasks to match a pre-set target gradient cosine similarity. An example is shown in Figure 12, where we have two tasks $i , j$ and their corresponding gradients $\mathbf { g } _ { i }$ and ${ \bf { g } } _ { j }$ have a cosine similarity of $\phi _ { i j }$ , i.e. cos(θ) = φij = gi·gjkgikkgj k . Then, we want to alter their gradients, such that the resulting new gradients have gradient similarity of some pre-set value $\phi _ { i j } ^ { T }$ . To do so, we replace $\mathbf { g } _ { i }$ with a new vector in the vector space spanned by $\mathbf { g } _ { i }$ and ${ \bf g } _ { j }$ , $\mathbf { g } _ { i } ^ { \prime } = a _ { 1 } \cdot \mathbf { g } _ { i } + a _ { 2 } \cdot \mathbf { g } _ { j }$ . Without loss of generality, we set $a _ { 1 } = 1$ and solve for $a _ { 2 }$ , i.e. find the $a _ { 2 }$ such that $\begin{array} { r } { \cos ( \gamma ) = \frac { { \bf g } _ { i } ^ { \prime } \cdot { \bf g } _ { j } } { \| { \bf g } _ { i } ^ { \prime } \| \| { \bf g } _ { j } \| } = \phi _ { i j } ^ { T } } \end{array}$ . By using Laws of Sines, we must have that:
+
+$$
+\frac { \| \mathbf { g } _ { i } \| } { \sin ( \gamma ) } = \frac { a _ { 2 } \| \mathbf { g } _ { j } \| } { \sin ( \theta - \gamma ) } ,
+$$
+
+and thus we can further solve for $a _ { 2 }$ as:
+
+$$
+\begin{array} { r l } & { \quad \frac { \| \mathbf { g } _ { i } \| } { \sin ( \gamma ) } = \frac { a _ { 2 } \| \mathbf { g } _ { j } \| } { \sin ( \theta - \gamma ) } } \\ & { \Rightarrow \frac { \| \mathbf { g } _ { i } \| } { \sin ( \gamma ) } = \frac { a _ { 2 } \| \mathbf { g } _ { j } \| } { \sin ( \theta ) \cos ( \gamma ) - \cos ( \theta ) \sin ( \gamma ) } } \\ & { \Rightarrow \frac { \| \mathbf { g } _ { i } \| } { \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } } = \frac { a _ { 2 } \| \mathbf { g } _ { j } \| } { \phi _ { i j } ^ { T } \sqrt { 1 - \phi _ { i j } ^ { 2 } } - \phi _ { i j } \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } } } \\ & { \Rightarrow a _ { 2 } = \frac { \| \mathbf { g } _ { i } \| ( \phi _ { i j } ^ { T } \sqrt { 1 - \phi _ { i j } ^ { 2 } } - \phi _ { i j } \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } ) } { \| \mathbf { g } _ { j } \| \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } } } \end{array}
+$$
+
+We therefore arrive at the update rule in Eq. 2. Our formulation allows us to set arbitrary target values for any two gradients, and thus we can better leverage task relatedness by setting individual gradient similarity objective for each task pair. Notice that we can rescale the gradient such that the altered gradients will have the same norm as before. But in our experiment we find it is sufficient to ignore this step. On the other hand, we note that when $\phi _ { i j } ^ { T } = 0$ , we have that:
+
+$$
+\begin{array} { r l } & { a _ { 2 } = \frac { \| \mathbf { g } _ { i } \| ( \phi _ { i j } ^ { T } \sqrt { 1 - \phi _ { i j } ^ { 2 } } - \phi _ { i j } \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } ) } { \| \mathbf { g } _ { j } \| \sqrt { 1 - ( \phi _ { i j } ^ { T } ) ^ { 2 } } } } \\ & { \quad = \frac { \| \mathbf { g } _ { i } \| \left( 0 - \phi _ { i j } \right) } { \| \mathbf { g } _ { j } \| } } \\ & { \quad = - \frac { \mathbf { g } _ { i } \cdot \mathbf { g } _ { j } } { \| \mathbf { g } _ { i } \| \| \mathbf { g } _ { j } \| } \cdot \frac { \| \mathbf { g } _ { i } \| } { \| \mathbf { g } _ { j } \| } } \\ & { \quad = - \frac { \mathbf { g } _ { i } \cdot \mathbf { g } _ { j } } { \| \mathbf { g } _ { j } \| ^ { 2 } } \| \mathbf { g } _ { j } \| } \end{array}
+$$
+
+This is exactly the update rule of PCGrad in Eq. 1. Thus, PCGrad is a special case of our proposed method.
+
+# E.2 ALGORITHM IN PRACTICE
+
+Our proposed method is detailed in Algorithm 1. In our experiments on multilingual NMT and multilingual BERT, we utilize a set of exponential moving average (EMA) variables to set proper pair-wise gradient similarity objectives, as shown in Eq. 3. This is motivated by our observations in Section 2 such that gradients of different languages in a Transformer model evolve across layers and training steps. Therefore, we conduct GradVac on different model components independently. For example, we can do one GradVac on each layer in the model, or just perform a single GradVac on the entire model. In addition, we also introduce an extra degree of freedom by controlling which tasks to perform GradVac. This corresponds to selecting a of tasks $\mathcal { G }$ and only alter gradients for tasks within this set, as shown in line 10 in Algorithm 1. Empirically, we find performing GradVac by layers and on low-resource languages to work generally the best (See Appendix F for detailed discussion).
+
+![](images/12e84fe4823cdb55e5869c01caa1c08b268727ca9501897d06c1ec21a5d53bd5.jpg)  
+Figure 12: Pictorial description of our method.
+
+# E.3 THEORETICAL PROPERTY
+
+Finally, we analyze the theoretical property of our method. Supper we only have two tasks, and their losses are $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ , and we denote their gradient cosine similarity at a given step as $\phi _ { 1 2 }$ . When $\phi _ { 1 2 }$ is negative, our method is largely equivalent to PCGrad and enjoy PCGrad’s convergence analysis. Thus, here we consider the other case when $\phi _ { 1 2 }$ is positive and show that:
+
+Theorem 1. Suppose $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ are convex and differentiable, and that the gradient of $\mathcal { L }$ is Lipschitz continuous with constant $L > 0$ . Then, the GradVac update rule with step size7 $\begin{array} { r } { t < \frac { 2 } { L \left( 1 + a ^ { 2 } \right) } } \end{array}$ and $\begin{array} { r } { t < \frac { 1 } { L } } \end{array}$ , where $\begin{array} { r } { a = \frac { \sin ( \phi _ { 1 2 } - \phi _ { 1 2 } ^ { T } ) } { \sin ( \phi _ { 1 2 } ^ { T } ) } } \end{array}$ $( \phi _ { 1 2 } > 0$ and $\phi _ { 1 2 } ^ { T } \geq \phi _ { 1 2 }$ is some target cosine similarity), will converge to the optimal value $\mathcal { L } ( \theta ^ { \ast } )$ .
+
+Proof. Let $\mathbf { g } _ { 1 } = \nabla \mathcal { L } _ { 1 }$ and $\mathbf { g } _ { 2 } = \nabla \mathcal { L } _ { 2 }$ be gradients for task 1 and task 2 respectively. Thus we have $\mathbf { g } = \mathbf { g } _ { 1 } + \mathbf { g } _ { 2 }$ as the original gradient and $\begin{array} { r } { \mathbf { g } ^ { \prime } = \mathbf { g } + a \frac { \| \mathbf { g } _ { 2 } \| } { \| \mathbf { g } _ { 1 } \| } \mathbf { g } _ { 1 } + a \frac { \| \mathbf { g } _ { 1 } \| } { \| \mathbf { g } _ { 2 } \| } \mathbf { g } _ { 2 } } \end{array}$ as the altered gradient by the GradVac update rule, such that:
+
+$$
+a = \frac { \sin ( \phi _ { 1 2 } ) \cos ( \phi _ { 1 2 } ^ { T } ) - \cos ( \phi _ { 1 2 } ) \sin ( \phi _ { 1 2 } ^ { T } ) } { \sin ( \phi _ { 1 2 } ^ { T } ) }
+$$
+
+where $\phi _ { 1 2 } ^ { T }$ is some pre-set gradient similarity objective and $\phi _ { 1 2 } ^ { T } \geq \phi _ { 1 2 }$ (thus $a \geq 0$ since we only consider the angle between two gradients in the range of 0 to $\pi$ ).
+
+Then, we obtain the quadratic expansion of $\mathcal { L }$ as:
+
+$$
+\mathcal L ( { \boldsymbol { \theta } } ^ { + } ) \leq \mathcal L ( { \boldsymbol { \theta } } ) + \nabla \mathcal L ( { \boldsymbol { \theta } } ) ^ { T } ( { \boldsymbol { \theta } } ^ { + } - { \boldsymbol { \theta } } ) + \frac { 1 } { 2 } \nabla ^ { 2 } \mathcal L ( { \boldsymbol { \theta } } ) \| { \boldsymbol { \theta } } ^ { + } - { \boldsymbol { \theta } } \| ^ { 2 }
+$$
+
+and utilize the assumption that $\nabla \mathcal { L }$ is Lipschitz continuous with constant $\mathrm { L }$ , we have:
+
+$$
+\mathcal { L } ( { \boldsymbol { \theta } } ^ { + } ) \leq \mathcal { L } ( { \boldsymbol { \theta } } ) + \nabla \mathcal { L } ( { \boldsymbol { \theta } } ) ^ { T } ( { \boldsymbol { \theta } } ^ { + } - { \boldsymbol { \theta } } ) + \frac { 1 } { 2 } L \Vert { \boldsymbol { \theta } } ^ { + } - { \boldsymbol { \theta } } \Vert ^ { 2 }
+$$
+
+Thus, we plug in the update rule of GradVac to obtain:
+
+$$
+\mathcal { L } ( { \boldsymbol { \theta } } ^ { + } ) \leq \mathcal { L } ( { \boldsymbol { \theta } } ) - t \cdot \mathbf { g } ^ { T } ( \mathbf { g } + a \frac { \left. \mathbf { g } _ { 2 } \right. } { \left. \mathbf { g } _ { 1 } \right. } \mathbf { g } _ { 1 } + a \frac { \left. \mathbf { g } _ { 1 } \right. } { \left. \mathbf { g } _ { 2 } \right. } \mathbf { g } _ { 2 } ) + \frac { 1 } { 2 } L t ^ { 2 } \Vert \mathbf { g } + a \frac { \left. \mathbf { g } _ { 2 } \right. } { \left. \mathbf { g } _ { 1 } \right. } \mathbf { g } _ { 1 } + a \frac { \left. \mathbf { g } _ { 1 } \right. } { \left. \mathbf { g } _ { 2 } \right. } \mathbf { g } _ { 2 } \Vert ^ { 2 }
+$$
+
+$$
+\begin{array} { l } { { \displaystyle = \mathcal { L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } + a \phi _ { 1 2 } ( t - L t ^ { 2 } ) ) ( \| { \bf g } _ { 1 } \| ^ { 2 } + \| { \bf g } _ { 2 } \| ^ { 2 } ) } } \\ { { \displaystyle \quad \quad - ( 2 a ( t - L t ^ { 2 } ) + \phi _ { 1 2 } ( 2 t - L t ^ { 2 } ( 1 + a ^ { 2 } ) ) ) ( \| { \bf g } _ { 1 } \| \cdot \| { \bf g } _ { 2 } \| ) } } \\ { { \displaystyle = \mathcal { L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) ( \| { \bf g } _ { 1 } \| ^ { 2 } + \| { \bf g } _ { 2 } \| ^ { 2 } ) - 2 \phi _ { 1 2 } ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) ) ( \| { \bf g } _ { 1 } \| \cdot \| { \bf g } _ { 2 } \| ) } } \\ { { \displaystyle \quad \quad - ( a \phi _ { 1 2 } ( t - L t ^ { 2 } ) ) ( \| { \bf g } _ { 1 } \| ^ { 2 } + \| { \bf g } _ { 2 } \| ^ { 2 } ) - ( 2 a ( t - L t ^ { 2 } ) ) ( \| { \bf g } _ { 1 } \| \cdot \| { \bf g } _ { 2 } \| ) } } \end{array}
+$$
+
+(Remove non-positive terms)
+
+$$
+\begin{array} { l } { { \displaystyle \le { \mathcal L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) ( \| { \bf g } _ { 1 } \| ^ { 2 } + \| { \bf g } _ { 2 } \| ^ { 2 } ) - 2 \phi _ { 1 2 } ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) ) ( \| { \bf g } _ { 1 } \| \cdot \| { \bf g } _ { 2 } \| ) } } \\ { { \displaystyle = { \mathcal L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) ( \| { \bf g } _ { 1 } \| ^ { 2 } + \| { \bf g } _ { 2 } \| ^ { 2 } + 2 \phi _ { 1 2 } \| { \bf g } _ { 1 } \| \cdot \| { \bf g } _ { 2 } \| ) } } \\ { { \displaystyle = { \mathcal L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) ( \| { \bf g } _ { 1 } \| ^ { 2 } + \| { \bf g } _ { 2 } \| ^ { 2 } + 2 { \bf g } _ { 1 } \cdot { \bf g } _ { 2 } ) } } \\ { { \displaystyle = { \mathcal L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) \| { \bf g } _ { 1 } + { \bf g } _ { 2 } \| ^ { 2 } } } \\ { { \displaystyle = { \mathcal L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) \| { \bf g } _ { 1 } \| ^ { 2 } } } \\ { { \displaystyle = { \mathcal L } ( \theta ) - ( t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } ) \| { \bf g } \| ^ { 2 } } } \end{array}
+$$
+
+The last line implies that if we choose learning rate $t$ to be small enough $\begin{array} { r } { t \textless \frac { 2 } { L ( 1 + a ^ { 2 } ) } } \end{array}$ , we have that $t - \frac { 1 + a ^ { 2 } } { 2 } L t ^ { 2 } > 0$ and thus ${ \mathcal { L } } ( \theta ^ { + } ) < { \mathcal { L } } ( \theta )$ (unless the gradient has zero norm). This tells us applying update rule of GradVac can reach the optimal value $\mathcal { L } ( \theta ^ { \ast } )$ since the objective function strictly decreases.
+
+Table 6: Comparing which tasks to be included for GradVac. Parameter granularity fixed at all layer while $\beta { = } 1 \mathrm { e } { - } 2$ .   
+
+<table><tr><td></td><td>en-fr</td><td>en-cs</td><td>en-hi</td><td>en-tr</td><td>avg</td></tr><tr><td>GradVac w. HRL_only</td><td>39.07</td><td>21.51</td><td>14.92</td><td>19.63</td><td>23.78</td></tr><tr><td>GradVac w. LRL_only</td><td>39.27</td><td>21.67</td><td>14.88</td><td>19.73</td><td>23.89</td></tr><tr><td>GradVac w. all_task</td><td>38.85</td><td>21.47</td><td>14.48</td><td>19.75</td><td>23.64</td></tr></table>
+
+Table 7: Comparing parameter granularity for GradVac. GradVac tasks fixed at LRL only while $\beta { = } 1 \mathrm { e } { - } 2$ .   
+
+<table><tr><td></td><td>en-fr</td><td>en-cs</td><td>en-hi</td><td>en-tr</td><td>avg</td></tr><tr><td>GradVacw.whole_model</td><td>38.76</td><td>21.32</td><td>14.22</td><td>18.89</td><td>23.30</td></tr><tr><td>GradVac w. enc_dec</td><td>39.05</td><td>21.73</td><td>14.54</td><td>19.33</td><td>23.66</td></tr><tr><td>GradVac w. all_layer</td><td>39.27</td><td>21.67</td><td>14.88</td><td>19.73</td><td>23.89</td></tr><tr><td>GradVac w. all_matrix</td><td>38.95</td><td>21.56</td><td>14.57</td><td>19.01</td><td>23.52</td></tr></table>
+
+<table><tr><td></td><td>en-fr</td><td>en-cs</td><td>en-hi</td><td>en-tr</td><td>avg</td></tr><tr><td>GradVac w. β=1e-1</td><td>38.72</td><td>20.74</td><td>14.52</td><td>19.25</td><td>23.31</td></tr><tr><td>GradVac w. β=1e-2</td><td>39.27</td><td>21.67</td><td>14.88</td><td>19.73</td><td>23.89</td></tr><tr><td>GradVac w. β=1e-3</td><td>38.85</td><td>20.96</td><td>14.85</td><td>19.68</td><td>23.59</td></tr></table>
+
+Table 8: Comparing EMA decay rate $\beta$ for GradVac. Parameter granularity fixed at all layer and GradVac tasks fixed at LRL only.
+
+# F HYPER-PARAMETER SETTINGS
+
+Here, we show how we choose the best hyper-parameter setting for our method. As discussed in Appendix E, there are three hyper-parameter settings for our implementation: (1) which tasks to be considered for GradVac, (2) which layers to measure EMA and perform GradVac, (3) EMA decay rate. Due to the scale of our model on the larger dataset, we use the smaller scale WMT dataset to find the optimal setting and transfer to other experiments. We do this by grid search using average perplexity on the validation set. Below, we demonstrate part of our results for each hyper-parameter to choose from.
+
+First, we examine the effect of what tasks to include for GradVac, i.e. $\mathcal { G }$ in Algorithm 1. We consider three options: (1) HRL only: only perform GradVac on high-resource languages, (2) LRL only: only perform GradVac on low-resource languages, (3) all task: perform GradVac on all languages. Results are shown in Table 6. We find that only conducting GradVac on a subset of languages obtain better performance while it is the best to conduct GradVac on low-resource language only. This is probably because the effective batch sizes of low-resource languages are usually smaller due to the sampling strategy.
+
+Next, we compare the effect of parameter granularity on model quality. This corresponds to setting different model components for GradVac ( $\mathcal { M }$ in Algorithm 1). We consider four possibilities, from coarse to fine-grained: (1) whole model: only perform GradVac once on the entire model, (2) enc dec: perform separately for encoder and decoder, (3) all layer: perform individually for each layer in encoder and decoder, (4) all matrix: perform for each parameter matrix in the model. As shown in Table 7, we find that choosing proper parameter granularity is important, as neither too coarse nor too fine-grained perform the best. This is consistent with our observation made in Section 2. However, we note that our settings are based on NLP tasks and Transformer networks, and therefore the best overall setting for problems of other domains may vary.
+
+Finally, we study how sensitive our method is on the hyper-parameter $\beta$ , i.e. the EMA decay rate. Results in Table 8 illustrate that setting an effective “window” of 100 training steps work best for our problem setups. This is expected, as setting a larger $\beta$ value corresponds to conduct GradVac more aggressively, and vice versa. In general, we find our best settings to be consistent across tasks in this paper.
+
+Table 9: F1 on the POS tasks of the XTREME benchmark.   
+
+<table><tr><td></td><td>ar</td><td>bg</td><td>de</td><td>en</td><td>es</td><td>fr</td><td>hi</td><td>hu</td><td>mr</td><td>ta</td><td>te</td><td>vi</td><td>avg</td></tr><tr><td>mBERT</td><td>84.2</td><td>94.7</td><td>92.7</td><td>91.0</td><td>93.8</td><td>93.3</td><td>88.0</td><td>91.9</td><td>83.3</td><td>80.3</td><td>90.4</td><td>79.2</td><td>88.6</td></tr><tr><td>+ GradNorm</td><td>83.5</td><td>94.7</td><td>92.3</td><td>91.0</td><td>93.6</td><td>93.2</td><td>88.2</td><td>91.4</td><td>83.0</td><td>80.5</td><td>90.6</td><td>79.1</td><td>88.4</td></tr><tr><td>+MGDA</td><td>84.4</td><td>94.5</td><td>92.3</td><td>90.4</td><td>93.5</td><td>92.7</td><td>88.1</td><td>92.3</td><td>83.4</td><td>80.5</td><td>90.2</td><td>78.7</td><td>88.4</td></tr><tr><td>+ PCGrad</td><td>83.7</td><td>94.8</td><td>92.6</td><td>91.5</td><td>94.2</td><td>92.8</td><td>88.5</td><td>91.7</td><td>83.7</td><td>80.5</td><td>90.8</td><td>79.4</td><td>88.7</td></tr><tr><td>+ GradVac</td><td>84.1</td><td>95.0</td><td>93.6</td><td>91.7</td><td>94.4</td><td>93.9</td><td>88.5</td><td>92.4</td><td>83.5</td><td>79.8</td><td>90.9</td><td>79.5</td><td>88.9</td></tr></table>
+
+# G XTREME EXPERIMENTS
+
+# G.1 FINETUNING DETAILS
+
+We also conduct experiments on the XTREME benchmark (Hu et al., 2020) for cross-lingual transfer tasks. While other work mostly focus on zero-shot cross-lingual transfer (finetune on English training data and then evaluate on the target language test data), we use a different setup of multitask learning such that we finetune multiple languages jointly and evaluate on all languages. Notice that our goal is not to compare with state-of-the-art results on this benchmark but rather to examine the effectiveness of our proposed method on pre-trained multilingual language models. We therefore only consider tasks that contain training data for all languages: named entity recognition (NER) and part-of-speech tagging (POS).
+
+The NER task is from the WikiAnn (Pan et al., 2017) dataset, which is built automatically from Wikipedia. A linear layer with softmax classifier is added on top of pretrained models to predict the label for each word based on its first subword. We report the F1 score. Similar to NER, POS is also a sequence labelling task but with a focus on synthetic knowledge. In particular, the dataset we used is from the Universal Dependencies treebanks (Nivre et al., 2018). Task-specific layers are the same as in NER and we report F1. We select 12 languages for each task randomly.
+
+We use the multilingual BERT (Devlin et al., 2018) as our base model, which is a Transformer model pretrained on the Wikipedias of 104 languages using masked language modelling (MLM). It contains 12 layers and 178M parameters. Following Hu et al. (2020), we finetune the model for 10 epochs for NER and POS, and search the following hyperparameters: batch size $\{ 1 6 , 3 2 \}$ ; learning rate $\{ 2 \mathrm { e } { - } 5 , 3 \mathrm { e } { - } 5 , 5 \mathrm { e } { - } 5 \}$ .
+
+# G.2 POS RESULT
+
+We evaluate all multi-task baselines on the POS tasks in Table 9. We find that our proposed method outperforms other methods on average, consistent with results in other settings (Section 4).

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+    {
+        "type": "text",
+        "text": "Laplacian Networks: Bounding Indicator Function Smoothness for Neural Networks Robustness ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            195,
+            418,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            260,
+            545,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For the past few years, Deep Neural Network (DNN) robustness has become a question of paramount importance. As a matter of fact, in sensitive settings misclassification can lead to dramatic consequences. Such misclassifications are likely to occur when facing adversarial attacks, hardware failures or limitations, and imperfect signal acquisition. To address this question, authors have proposed different approaches aiming at increasing the robustness of DNNs, such as adding regularizers or training using noisy examples. In this paper we propose a new regularizer built upon the Laplacian of similarity graphs obtained from the representation of training data at each layer of the DNN architecture. This regularizer penalizes large changes (across consecutive layers in the architecture) in the distance between examples of different classes, and as such enforces smooth variations of the class boundaries. Since it is agnostic to the type of deformations that are expected when predicting with the DNN, the proposed regularizer can be combined with existing ad-hoc methods. We provide theoretical justification for this regularizer and demonstrate its effectiveness to improve robustness of DNNs on classical supervised learning vision datasets. ",
+        "bbox": [
+            233,
+            291,
+            764,
+            526
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            555,
+            339,
+            571
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep Neural Networks (DNNs) provide state-of-the-art performance in many challenges in machine learning (He et al., 2016; Wu et al., 2016). Their ability to achieve good generalization is often explained by the fact they use very few priors about data (LeCun et al., 2015). On the other hand, their strong dependency on data may lead to focus on biased features of the training dataset, resulting in a nonrobust classification performance. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the literature, authors have been interested in studying the robustness of DNNs in various conditions. These conditions include: ",
+        "bbox": [
+            173,
+            664,
+            821,
+            693
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Robustness to isotropic noise, i.e., small isotropic variations of the input (Mallat, 2016), typically meaning that the network function leads to a small Lipschitz constant.   \nRobustness to adversarial attacks, which can exploit knowledge about the network parameters or the training dataset (Szegedy et al., 2013; Goodfellow et al., 2014).   \n• Robustness to implementation defects, which can result in only approximately correct computations (Hubara et al., 2017). ",
+        "bbox": [
+            215,
+            704,
+            825,
+            814
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To improve DNN robustness, three main families of solutions have been proposed in the literature. The first one involves enforcing smoothness, as measured by a Lipschitz constant, in the operators and having a minimum separation margin (Mallat, 2016). A similar approach has been proposed in (Cisse et al., 2017), where the authors restrict the function of the network to be contractive. A second class of methods use intermediate representations obtained at various layers to perform the prediction (Papernot and McDaniel, 2018). Finally, in (Kurakin et al., 2016; Pezeshki et al., 2016; Madry et al., 2018), the authors propose to train the network using noisy inputs so that it better generalizes to this type of noise. This has been shown to improve the robustness of the network to the specific type of noise used during training, but it is not guaranteed that this robustness would be extended to other types of deformations. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduce a new regularizer that does not focus on a specific type of deformation, but aims at increasing robustness in general. As such, the proposed regularizer can be combined with other existing methods. It is inspired by recent developments in Graph Signal Processing (GSP) (Shuman et al., 2013). GSP is a mathematical framework that extends classical Fourier analysis to complex topologies described by graphs, by introducing notions of frequency for signals defined on graphs. Thus, signals that are smooth on the graph (i.e., change slowly from one node to its neighbors) will have most of their energy concentrated in the low frequencies. ",
+        "bbox": [
+            174,
+            166,
+            823,
+            277
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The proposed regularizer is based on constructing a series of graphs, one for each layer of the DNN architecture, where each graph captures the similarity between all training examples given their intermediate representation at that layer. Our proposed regularizer penalizes large changes in the smoothness of class indicator vectors (viewed here as graph signals) from one layer to the next. As a consequence, the distances between pairs of examples in different classes are only allowed to change slowly from one layer to the next. Note that because we use deep architectures, the regularizer does not prevent the smoothness from achieving its maximum value, but constraining the size of changes from layer to layer increases the robustness of the network function by controlling the distance to the boundary region, as supported by experiments in Section 4. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The outline of the paper is as follows. In Section 2 we present related work. In Section 3 we introduce the proposed regularizer. In Section 4 we evaluate the performance of our proposed method in various conditions and on vision benchmarks. Section 5 summarizes our conclusions. ",
+        "bbox": [
+            176,
+            431,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            522,
+            348,
+            537
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "DNN robustness may refer to many different problems. In this work we are mostly interested in the stability to deformations (Mallat, 2016), or noise, which can be due to multiple factors mentioned in the introduction. The most studied stability to deformations is in the context of adversarial attacks. It has been shown that very small imperceptible changes on the input of a trained DNN can result in missclassification of the input (Szegedy et al., 2013; Goodfellow et al., 2014). These works have been primordial to show that DNNs may not be as robust to deformations as the test accuracy benchmarks would have lead one to believe. Other works, such as (Recht et al., 2018), have shown that DNNs may also suffer from drops in performance when facing deformations that are not originated from adversarial attacks, but simply by re-sampling the test images. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Multiple ways to improve robustness have been proposed in the literature. They range from the use of a model ensemble composed of $k$ -nearest neighbors classifiers for each layer (Papernot and McDaniel, 2018), to the use of distillation as a mean to protect the network (Papernot et al., 2016a). Other methods introduce regularizers (Gu and Rigazio, 2014), control the Lipschitz constant of the network function (Cisse et al., 2017) or implement multiple strategies revolving around using deformations as a data augmentation procedure during the training phase (Goodfellow et al., 2014; Kurakin et al., 2016; Moosavi Dezfooli et al., 2016). ",
+        "bbox": [
+            174,
+            708,
+            823,
+            819
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Compared to these works, our proposed method can be viewed as a regularizer that penalizes large deformations of the class boundaries throughout the network architecture, instead of focusing on a specific deformation of the input. As such, it can be combined with other mentioned strategies. Indeed, we demonstrate that the proposed method can be implemented in combination with (Cisse et al., 2017), resulting in a network function such that small variations to the input lead to small variations in the decision, as in (Cisse et al., 2017), while limiting the amount of change to the class boundaries. Note that our approach does not require using training data affected by a specific deformation, and our results could be further improved if such data were available for training, as shown in the Appendix. ",
+        "bbox": [
+            174,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As for combining GSP and machine learning, this area has sparked interest recently. For example, the authors of (Gripon et al., 2018) show that it is possible to detect overfitting by tracking the evolution of the smoothness of a graph containing only training set examples. Another example is in (Anirudh et al., 2017) where the authors introduce different quantities related to GSP that can be used to extract interpretable results from DNNs. In (Svoboda et al., 2018) the authors exploit graph convolutional layers (Bronstein et al., 2017) to increase the robustness of the network. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, this is the first use of graph signal smoothness as a regularizer for deep neural network design. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.1 Similarity preset and postset graphs ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Consider a deep neural network architecture. Such a network is obtained by assembling layers of various types. Of particular interest are layers of the form $\\mathbf { x } ^ { \\ell } \\mapsto \\mathbf { x } ^ { \\ell + 1 } = h ^ { \\ell } ( \\mathbf { W } ^ { \\ell } \\mathbf { x } ^ { \\ell } + \\mathbf { b } ^ { \\ell } )$ , where $h ^ { \\ell }$ is a nonlinear function, typically a ReLU, $\\mathbf { W } ^ { \\ell }$ is the weight tensor at layer $\\ell$ , $\\mathbf { x } ^ { \\ell }$ is the intermediate representation of the input at layer $\\ell$ and $\\mathbf { b } ^ { \\ell }$ is the corresponding bias tensor. Note that strides or pooling may be used. Assembling can be achieved in various ways: composition, concatenation, sums. . . so that we obtain a global function $f$ that associates an input tensor $\\mathbf { x } ^ { 0 }$ to an output tensor $\\mathbf { y } = f ( \\mathbf { x } ^ { 0 } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When computing the output $\\mathbf { y }$ associated with the input $\\mathbf { x } ^ { 0 }$ , each layer $\\ell$ of the architecture processes some input $\\mathbf { x } ^ { \\ell }$ and computes the corresponding output $\\mathbf { y } ^ { \\ell } = h ^ { \\ell } ( \\mathbf { W } ^ { \\ell } \\mathbf { x } ^ { \\ell } + \\mathbf { b } ^ { \\ell } )$ For a given layer $\\ell$ and a batch of $b$ inputs $\\mathcal { X } = \\{ \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { b } \\}$ , we can obtain two sets $\\mathcal { X } ^ { \\ell } = \\{ \\mathbf { x } _ { 1 } ^ { \\ell } , \\ldots , \\mathbf { x } _ { b } ^ { \\ell } \\}$ , called the preset, and $\\mathcal { V } ^ { \\ell } = \\{ \\mathbf { y } _ { 1 } ^ { \\ell } , \\ldots , \\mathbf { y } _ { b } ^ { \\ell } \\}$ , called the postset. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a similarity measure $s$ on tensors, from a preset we can build the similarity preset matrix: $\\mathbf { M } _ { p r e } ^ { \\ell } [ i , j ] = s ( \\mathbf { x } _ { i } ^ { \\ell } , \\mathbf { x } _ { j } ^ { \\ell } ) , \\forall 1 \\le i , j \\le b$ , where $\\mathbf { M } [ i , j ]$ denotes the element at line $i$ and column $j$ in $\\mathbf { M }$ . The postset matrix is defined similarly. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider a similarity (either preset or postset) matrix $\\mathbf { M } ^ { \\ell }$ . This matrix can be used to build a $k$ -nearest neighbor similarity weighted graph $G ^ { \\ell } = \\langle V , \\mathbf { A } ^ { \\ell } \\rangle$ , where $V = \\{ 1 , \\ldots , b \\}$ is the set of vertices and $\\mathbf { A } ^ { \\ell }$ is the weighted adjacency matrix defined as: ",
+        "bbox": [
+            176,
+            563,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef5244961069caab795142cb33d2ad33f1cbacd59756f7ae71be4e3e5525f5be.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { A } ^ { \\ell } [ i , j ] = \\left\\{ \\begin{array} { l l } { \\mathbf { M } ^ { \\ell } [ i , j ] } & { \\mathrm { i f } \\ \\mathbf { M } ^ { \\ell } [ i , j ] \\in \\mathrm { a r g } \\operatorname* { m a x } _ { i ^ { \\prime } \\neq j } \\big ( \\mathbf { M } ^ { \\ell } [ i ^ { \\prime } , j ] , k \\big ) } \\\\ & { \\big \\downarrow \\mathrm { a r g } \\operatorname* { m a x } _ { j ^ { \\prime } \\neq i } \\big ( \\mathbf { M } ^ { \\ell } [ i , j ^ { \\prime } ] , k \\big ) } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. , \\forall i , j \\in V , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            611,
+            753,
+            659
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where arg $\\operatorname* { m a x } _ { i } ( a _ { i } , k )$ denotes the indices of the $k$ largest elements in $\\{ a _ { 1 } , \\ldots , a _ { b } \\}$ . Note that by construction $\\mathbf { A } ^ { \\ell }$ is symmetric. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Smoothness of label signals ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a weighted graph $G ^ { \\ell } = \\langle V , \\mathbf { A } ^ { \\ell } \\rangle$ , we call Laplacian of $G ^ { \\ell }$ the matrix $\\mathbf { L } ^ { \\ell } = \\mathbf { D } ^ { \\ell } - \\mathbf { A } ^ { \\ell }$ , where $\\mathbf { D } ^ { \\ell }$ is the diagonal matrix such that: $\\begin{array} { r } { \\mathbf { D } ^ { \\ell } [ i , i ] = \\sum _ { j } \\mathbf { A } ^ { \\ell } [ i , j ] , \\forall i \\in V } \\end{array}$ . Because $\\mathbf { L } ^ { \\ell }$ is symmetric and real-valued, it can be written: ",
+        "bbox": [
+            174,
+            732,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9564ef690b8bf314d4094f34cae0f82c4bfffd04755970db8d3a8014f59316d.jpg",
+        "text": "$$\n\\mathbf { L } ^ { \\ell } = \\mathbf { F } ^ { \\ell } \\mathbf { A } ^ { \\ell } \\mathbf { F } ^ { \\ell \\top } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            780,
+            553,
+            799
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { F }$ is orthonormal and contains eigenvectors of $\\mathbf { L } ^ { \\ell }$ as columns, $\\mathbf { F } ^ { \\top }$ denotes the transpose of $\\mathbf { F }$ , and $\\pmb { \\Lambda }$ is diagonal and contains eigenvalues of $\\mathbf { L } ^ { \\ell }$ is ascending order. Note that the constant vector $\\mathbf { 1 } \\in \\mathbb { R } ^ { b }$ is an eigenvector of $\\mathbf { L } ^ { \\ell }$ corresponding to eigenvalue 0. Moreover, all√ eigenvalues of $\\mathbf { L } ^ { \\ell }$ are nonnegative. Consequently, $\\mathbf { 1 } / \\sqrt { n }$ can be chosen as the first column in $\\mathbf { F }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider a vector $\\mathbf { s } \\in \\mathbb { R } ^ { b }$ , we define $\\hat { \\bf S }$ the Graph Fourier Transform (GFT) of s on $G ^ { \\ell }$ as (Shuman et al., 2013): ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec0c4f65513bdad94c7e0358b92ca2541bd160dd76093bbce6b8768cf52bb824.jpg",
+        "text": "$$\n\\hat { \\mathbf { s } } = \\mathbf { F } ^ { \\top } \\mathbf { s } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            465,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Because the order of the eigenvectors is chosen so that the corresponding eigenvalues are in ascending order, if only the first few entries of $\\hat { \\bf s }$ are nonzero that indicates that s is low frequency (smooth). In the extreme case where only the first entry of $\\hat { \\bf s }$ is nonzero we have that $\\mathbf { s }$ is constant (maximum smoothness). More generally, smoothness $\\sigma ^ { \\ell } ( \\mathbf { s } )$ of a signal s can be measured using the quadratic form of the Laplacian: ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e327faf4bd12af55a0057ee961731acfc9221f02d35d74f655070190f774b2c.jpg",
+        "text": "$$\n\\sigma ^ { \\ell } ( \\mathbf { s } ) = \\mathbf { s } ^ { \\top } \\mathbf { L } ^ { \\ell } \\mathbf { s } = \\sum _ { i , j = 1 } ^ { b } \\mathbf { A } ^ { \\ell } [ i , j ] ( \\mathbf { s } [ i ] - \\mathbf { s } [ j ] ) ^ { 2 } = \\sum _ { i = 1 } ^ { b } \\mathbf { A } ^ { \\ell } [ i , i ] \\hat { \\mathbf { s } } [ i ] ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            180,
+            710,
+            224
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we note that $\\mathbf { s }$ is smoother when $\\sigma ^ { \\ell } ( \\mathbf { s } )$ is smaller. ",
+        "bbox": [
+            173,
+            231,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this paper we are particularly interested in smoothness of the label signals. We call label signal $\\mathbf { s } _ { c }$ associated with class $c$ a binary $( \\{ 0 , 1 \\} )$ vector whose nonzero coordinates are the ones corresponding to input vectors of class $c$ . In other words, $\\mathbf { s } _ { c } [ i ] = 1 \\Leftrightarrow ( \\mathbf { x } _ { i }$ is in class $c ) , \\forall 1 \\leq i \\leq b$ . Using Equation (4), we obtain that the smoothness of the label signal $\\mathbf { s } _ { c }$ is the sum of similarities between examples in distinct classes. Thus a smoothness of 0 means that examples in distinct classes have 0 similarity. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Denote $u$ the last layer of the architecture: $\\mathbf { y } _ { i } ^ { u } = \\mathbf { y } _ { i } , \\forall i$ . Note that in typical settings, where outputs of the networks are one-hot-bit encoded and no regularizer is used, at the end of the learning process it is expected that $\\mathbf { y } _ { i } ^ { \\top } \\mathbf { y } _ { j } \\approx 1$ if $i$ and $j$ belong to the same class, and $\\mathbf { y } _ { i } ^ { \\top } \\mathbf { y } _ { j } \\approx 0$ otherwise. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thus, assuming that cosine similarity is used to build the graph, the last layer smoothness for all $c$ would be $\\sigma _ { p o s t } ^ { u } ( \\mathbf { s } _ { c } ) \\approx 0$ , since edge weights between nodes having different labels will be close to zero given Equation (4). More generally, smoothness of ${ \\bf s } _ { c }$ at the preset or postset of a given layer measures the average similarity between examples in class $c$ and examples in other classes ( $\\sigma ( \\mathbf { s } _ { c } )$ decreases as the weights of edges connecting nodes in different classes decrease). Because the last layer can achieve $\\sigma ( \\mathbf { s } _ { c } ) \\approx 0$ , we expect the smoothness metric $\\sigma$ at each layer to decrease as we go deeper in the network. Next we introduce a regularization strategy that limits how much $\\sigma$ can decrease from one layer to the next and can even prevent the last layer from achieving $\\sigma ( \\mathbf { s } _ { c } ) = 0$ . This will be shown to improve generalization and robustness. The theoretical motivation for this choice is discussed in Section 3.4. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 Proposed regularizer ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 Definition ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            590,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose to measure the deformation induced by a given layer $\\ell$ in the relative positions of examples by computing the difference between label signal smoothness before and after the layer, averaged over all labels: ",
+        "bbox": [
+            176,
+            614,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6ce9ced13288daa59dd879ed6033b77d4be86e6a0fb51bc3b54f642eeb91b05d.jpg",
+        "text": "$$\n\\delta _ { \\sigma } ^ { \\ell } = \\left| \\sum _ { c } \\left[ \\sigma _ { p o s t } ^ { \\ell } ( \\mathbf { s } _ { c } ) - \\sigma _ { p r e } ^ { \\ell } ( \\mathbf { s } _ { c } ) \\right] \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "These quantities are used to regularize modifications made to each of the layers during the learning process. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 1: Since we only consider label signals, we solely depend on the similarities between examples that belong to distinct classes. In other words, the regularizer only focuses on the boundary region, and does not vary if the distance between examples of the same label grows or shrinks. This is because forcing similarities between examples of a same class to evolve slowly could prevent the network to train appropriately. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 2: Compared with (Cisse et al., 2017), there are three key differences that characterize the proposed regularizer: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. Not all pairwise distances are taken into account in the regularization; only distances between examples corresponding to different classes play a role in the regularization. 2. We allow a limited amount of both contraction and dilation of the metric space. Experimental work (e.g. (Gripon et al., 2018; Papernot and McDaniel, 2018)) has shown that the evolution of metric spaces across DNN layers is complex, and thus restricting ourselves to contractions only could lead to lower overall performance. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a25bf2a8ce82d3d28ee143d3c0bd7c181607d6ab447cfb5742dc258feb54e06.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the effect of our proposed regularizer. In this example, the goal is to classify circles and crosses (top). Without use of regularizers (bottom left), the resulting embedding may considerably stretch the boundary regions (as illustrated by the irregular spacing between the tics). Forcing small variations of smoothness of label signals (bottom right), we ensure the topology is not dramatically changed in the boundary regions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3. The proposed criterion is an average (sum) over all distances, rather than a stricter criterion (e.g. Lipschitz), which would force each pair of vectors $\\left( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } \\right)$ to obey the constraint. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Illustrative example: ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Figure 1 we depict a toy illustrative example to motivate the proposed regularizer. We consider here a one-dimensional two-class problem. To linearly separate circles and crosses, it is necessary to group all circles. Without regularization (setting i)), the resulting embedding is likely to increase considerably the distance between examples and the size of the boundary region between classes. In contrast, by penalizing large variations of the smoothness of label signals (setting ii)), the average distance between circles and crosses must be preserved in the embedding domain, resulting in a more precise control of distances within the boundary region. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 Motivation: label signal bandwidth and powers of the Laplacian ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            652,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recent work (Anis et al., 2017) develops an asymptotic analysis of the bandwidth of label signals, $B W ( \\mathbf { s } )$ , where bandwidth is defined as the highest non-zero graph frequency of $\\mathbf { s }$ , i.e., the nonzero entry of $\\hat { \\bf S }$ with the highest index. An estimate of the bandwidth can be obtained by computing: ",
+        "bbox": [
+            173,
+            676,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e46467b5bcc88e62543264410398903de440869535435cfdcaa1fb295f3aaaf1.jpg",
+        "text": "$$\nB W _ { m } ( \\mathbf { s } ) = \\left( \\frac { \\mathbf { s } ^ { \\top } \\mathbf { L } ^ { m } \\mathbf { s } } { \\mathbf { s } ^ { \\top } \\mathbf { s } } \\right) ^ { ( 1 / m ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            733,
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+            772
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "for large $m$ . This can be viewed as a generalization of the smoothness metric of (4). (Anis et al., 2017) shows that, as the number of labeled points $\\mathbf { x }$ (assumed drawn from a distribution $p ( \\mathbf { x } )$ ) grows asymptotically, the bandwidth of the label signal converges in probability to the supremum of $p ( \\mathbf { x } )$ in the region of overlap between classes. This motivates our work in three ways. ",
+        "bbox": [
+            173,
+            776,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "First, it provides theoretical justification to use $\\sigma ^ { \\ell } ( \\mathbf { s } )$ for regularization, since lower values of $\\sigma ^ { \\ell } ( \\mathbf { s } )$ are indicative of better separation between classes. Second, the asymptotic analysis suggests that using higher powers of the Laplacian would lead to better regularization, since estimating bandwidth using $B W _ { m } ( \\mathbf { s } )$ becomes increasingly accurate as $m$ increases. Finally, this regularization can be seen to be protective against specializing by preventing $\\sigma ^ { \\ell } ( \\mathbf { s } )$ ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/73d63bbea2a418473334189f58913372f40d6a939f0e5899c465fb9fd05d7e53.jpg",
+        "image_caption": [
+            "Figure 2: Sample of a Laplacian and squared Laplacian of similarity graphs in a trained vanilla architecture. Examples of the batch have been ordered so that those belonging to a same class are consecutive. Dark values correspond to high similarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            215,
+            103,
+            784,
+            222
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/173363ce406f53c38ec33eda09d6e9615ac6193eca26568f2545fec2b5ff4e33.jpg",
+        "image_caption": [
+            "Figure 3: Evolution of smoothness of label signals as a function of layer depth, and for various regularizers and choice of $m$ , the power of the Laplacian matrix. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            250,
+            284,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "from decreasing “too fast”. For most problems of interest, given a sufficiently large amount of labeled data available, it would be reasonable to expect the bandwidth of s not to be arbitrarily small, because the classes cannot be exactly separated, and thus a network that reduces the bandwidth too much can result in being biased by the training set. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.5 Analysis of the Laplacian powers ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            556,
+            501,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Figure 2 we depict the Laplacian and squared Laplacian of similarity graphs obtained at different layers in a trained vanilla architecture. On the deep layers, we can clearly see blocks corresponding to the classes, while the situation in the middle layer is not as clear. This figure illustrates how using the squared Laplacian helps modifying the distances to improve separation. Note that we normalize the squared Laplacian values by dividing them by the highest absolute value. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Figure 3, we plot the average evolution of smoothness of label signals over 100 batches, as a function of layer depth in the architecture, and for different choices of the regularizer. In the left part, we look at smoothness measures using the Laplacian. In the right part, we use the squared Laplacian. We can clearly see the effectiveness of the regularizer in enforcing small variations of smoothness across the architecture. Note that for model regularized with $\\mathbf { L } ^ { 2 }$ , changes in smoothness measured by $\\mathbf { L }$ are not easy to see. This seems to suggest that some of the gains achieved via $\\mathbf { L } ^ { 2 }$ regularization come in making changes that would be “invisible” when looking at the layers from the perspective of $\\mathbf { L }$ smoothness. The same normalization from Figure 2 is used for $\\mathbf { L } ^ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the following paragraphs we evaluate the proposed method using various tests. We use the well known CIFAR-10 (Krizhevsky and Hinton, 2009) dataset made of tiny images. As far as the DNN is concerned, we use the same PreActResNet (He et al., 2016) architecture for all tests, with 18 layers. All inputs, including those on the test set, are normalized based on the mean and standard deviation of the images of the training set. In all figures, P are ",
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+    {
+        "type": "image",
+        "img_path": "images/0725dbe5fecdc238b7a4583c2aa38693c67f9dc4d939d619fa6ea0eccd7e0de4.jpg",
+        "image_caption": [
+            "Figure 4: Test set accuracy under Gaussian noise with varying signal-to-noise ratio. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Parseval trained networks, R are networks trained with the proposed regularizer and V are vanilla networks. More details and experiments can be found at the Appendix. ",
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+    {
+        "type": "text",
+        "text": "We depict the obtained results using box plots where data is aggregated from 10 different networks corresponding to different random seeds and batch orders. In the first experiment (left most plot) in Figure 4, we plot the baseline accuracy of the models on the clean test set (no deformation is added at this point). These experiments agree with the claim from (Cisse et al., 2017) where the authors show that they are able to increase the performance of the network on the clean test set. We observe that our proposed method leads to a minor decrease of performance on this test. However, we see in the following experiments that this is mitigated with increased robustness to deformations. Such a trade-off between robustness and accuracy has already been discussed in the literature (Fawzi et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "4.1 Isotropic deformation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this scenario we evaluate the robustness of the network function to small isotropic variations of the input. We generate 40 different deformations using random variables $\\mathcal { N } ( 0 , 0 . 2 5 )$ which are added to the test set inputs. Note that they are scaled so that $S N R \\approx 1 5$ and $S N R \\approx 2 0$ . The middle and right-most plots from Figure 4 show that the proposed method increases the robustness of the network to isotropic deformations. Note that in both scenarios the best results are achieved by combining Parseval training and our proposed method (lower-most box on both figures). ",
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+    {
+        "type": "text",
+        "text": "4.2 Adversarial Robustness ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We next evaluate robustness to adversarial inputs, which are specifically built to fool the network function. Such adversarial inputs can be generated and evaluated in multiple ways. Here we implement two approaches: first a mean case of adversarial noise, where the adversary can only use one forward and one backward pass to generate the deformations, and second a worst case scenario, where the adversary can use multiple forward and backward passes to try to find the smallest deformation that will fool the network. ",
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+    {
+        "type": "text",
+        "text": "For the first approach, we add the scaled gradient sign (FGSM attack) on the input (Kurakin et al., 2016), so that we obtain a target $S N R = 3 3$ . Results are depicted in the left and center plots of Figure 5. In the left plot the noise is added after normalizing the input whereas on the middle plot it is added before normalizing. As in the isotropic noise case, a combination of the Parseval method and our proposed approach achieves maximum robustness. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In regards to the second approach, where a worst case scenario is considered, we use the Foolbox toolbox (Rauber et al., 2017) implementation of DeepFool (Moosavi Dezfooli et al., 2016). Due to time constraints we sample only conclusions are similar (right plot of Figure 5) $\\frac { 1 } { 1 0 }$ of the test set images for this test. The those obtained for the first adversarial attack approach. ",
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+    {
+        "type": "text",
+        "text": "4.3 Implementation robustness ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, in a third series of experiments we evaluate the robustness of the network functions to faulty implementations. As a result, approximate computations are made during the test phase that consist of random erasures of the memory (dropout) or quantization of the weights (Hubara et al., 2017). ",
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+    {
+        "type": "image",
+        "img_path": "images/2ab1322d49785d870ab7882360f5ae11d0d07ec25f14d216ba1dac2a2ebd9f2f.jpg",
+        "image_caption": [
+            "Figure 5: Robustness against an adversary measured by the test set accuracy under FGSM attack in the left and center plots and by the mean $\\mathcal { L } _ { 2 }$ pixel distance needed to fool the network using DeepFool on the right plot. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8936bbbb51a644d3db02833cb32295ca058a56194f76a35ad3ee4febde65478.jpg",
+        "image_caption": [
+            "Figure 6: Test set accuracy under different types of implementation related noise. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the dropout case, we compute the test set accuracy when the network has a probability of either $2 5 \\%$ or $4 0 \\%$ of dropping a neuron’s value after each block. We run each experiment 40 times. The results are depicted in the left and center plots of Figure 6. It is interesting to note that the Parseval trained functions seem to drop in performance as soon as we reach $4 0 \\%$ probability of dropout, providing an average accuracy smaller than the vanilla networks. In contrast, the proposed method is the most robust to these perturbations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For the quantization of the weights, we consider a scenario where the network size in memory has to be shrink 6 times. We therefore quantize the weights of the networks to 5 bits (instead of 32) and re-evaluate the test set accuracy. The right plot of Figure 6 shows that the proposed method is providing a better robustness to this kind of deformation than the tested counterparts. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper we have introduced a new regularizer that enforces small variations of the smoothness of label signals on similarity graphs obtained at intermediate layers of a deep neural network architecture. We have empirically shown with our tests that it can lead to improved robustness in various conditions compared to existing counterparts. We also demonstrated that combining the proposed regularizer with existing methods can result in even better robustness for some conditions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Future work includes a more systematic study of the effectiveness of the method with regards to other datasets, models and deformations. Recent works shown adversarial noise is partially transferable between models and dataset (Moosavi-Dezfooli et al., 2017; Papernot et al., 2016b) and therefore we are confident about the generality of the method in terms of models and datasets. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One possible extension of the proposed method is to use it in a fine-tuning stage, combined with different techniques already established on the literature. An extension using a combination of input barycenter and class barycenter signals instead of the class signal could be interesting as that would be comparable to (Zhang et al., 2017). In the same vein, using random signals could be beneficial for semi-supervised or unsupervised learning challenges. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
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+        "type": "text",
+        "text": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks, 2016. ",
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+        "type": "text",
+        "text": "A Parseval Training and implementation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            724,
+            599,
+            741
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We compare our results with those obtained using the method described in (Cisse et al., 2017). There are three modifications to the normal training procedure: orthogonality constraint, convolutional renormalization and convexity constraint. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For the orthogonality constraint we enforce Parseval tightness (Kovačević and Chebira, 2008) as a layer-wise regularizer: ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6211c5b791cdb7a756d435a22771f9c01db0c0eb14dd767036a4d33a81aa1461.jpg",
+        "text": "$$\nR _ { \\beta } ( W ^ { \\ell } ) = \\frac { \\beta } { 2 } \\| W ^ { \\ell ^ { \\top } } W ^ { \\ell } - I \\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { \\ell }$ is the weight tensor at layer $\\ell$ . This function can be approximately optimized with gradient descent by doing the operation: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a75169b3d7939a23a05478a5ca00a542db98692d9f7104b8f722780966fb1bc.jpg",
+        "text": "$$\n\\begin{array} { r } { W ^ { \\ell } \\gets ( 1 + \\beta ) W ^ { \\ell } - \\beta W ^ { \\ell } W ^ { \\ell \\top } W ^ { \\ell } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            906,
+            620,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Given that our network is smaller we can apply the optimization to the entirety of the $W$ , instead of $3 0 \\%$ as per the original paper, this increases the strength of the Parseval tightness. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the convolutional renormalization, each matrix $W ^ { \\ell }$ is reparametrized before being applied to the convolution as $\\frac { W ^ { \\ell } } { \\sqrt { 2 k _ { s } + 1 } }$ , where $k _ { s }$ is the kernel size. ",
+        "bbox": [
+            176,
+            138,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For our architecture the inputs from a layer come from either one or two different layers. In the case where the inputs come from only one layer, $\\alpha$ the convexity constraint parameter is set to 1. When the inputs come from the sum of two layers we use $\\alpha = 0 . 5$ as the value for both of them, which constraints our Lipschitz constant, this is softer than the convexity constraint from the original paper. ",
+        "bbox": [
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+            251
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B Hyperparameters ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            272,
+            388,
+            289
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We train our networks using classical stochastic gradient descent with momentum (0.9), with batch size of $b = 1 0 0$ images and using a L2-norm weight decay with a coefficient of $\\lambda = 0 . 0 0 0 5$ . We do a 100 epoch training. Our learning rate starts at 0.1. After half of the training (50 epochs) the learning rate decreases to 0.001. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use the mean of the difference of smoothness between successive layers in our loss function. Therefore in our loss function we have: ",
+        "bbox": [
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+            367,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8aa3a5b290facf03b87bab50afa2defaa7112b3278486c58b0109a3592d173d2.jpg",
+        "text": "$$\n\\mathcal { L } = C a t e g o r i c a l C r o s s E n t r o p y + \\lambda W e i g h t D e c a y + \\gamma \\Delta\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            407,
+            699,
+            422
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\Delta = \\frac { 1 } { d - 1 } \\sum _ { \\ell = 1 } ^ { d } | \\delta _ { \\sigma } ^ { \\ell } | } \\end{array}$ . We perform experiments using various powers of the Laplacian $m = 1 , 2 , 3$ , in which case the scaling coefficient $\\gamma$ is put to the same power as the Laplacian. ",
+        "bbox": [
+            173,
+            434,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We tested multiple parameters of $\\beta$ , the Parseval tightness parameter, $\\gamma$ the weight for the smoothness difference cost and $m$ the power of the Laplacian. We found that the best values for this specific architecture, dataset and training scheme were: $\\beta = 0 . 0 1 , \\gamma = 0 . 0 1 , m =$ $2 , k = b$ . ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C Depiction of the network ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            551,
+            475,
+            568
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 depicts the network used on all experiments of sections 3 and 4. $f = 6 4$ is the filter size of the first layer of the network. Conv layers are 3x3 layers and are always preceded by batch normalization and relu (except for the first layer which receives just the input). The smoothness gaps are calculated after each ReLU. ",
+        "bbox": [
+            173,
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/07d6ffb8eb74f6adedb2e817c6a62fc1def128093e82756536554b38c46cfae1.jpg",
+        "image_caption": [
+            "Figure 7: Depiction of the studied network "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "D Additional experiments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Given suggestions from the reviewers, we performed additional experiments to further demonstrate the capabilities of the proposed regularizer. Due to the lack of space they could not be added to the main paper. We consider the effects of the regularizer when applied on another datasets. We also consider the effects of adding adversarial data augmentation methods while minimizing the amount of other influencing factors. We first look at the results when using the same architecture as for the CIFAR-10 dataset, which inevitably results in far from state-of-the-art accuracy on CIFAR-100. Then, we perform experiments using a different architecture (namely WideResnet 28-10, with dropout) for CIFAR-100. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D.1 CIFAR-10 ",
+        "text_level": 1,
+        "bbox": [
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+            273,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We add two types of tests for the CIFAR-10 dataset: adversarial data augmentation during training and black-box FGSM. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D.1.1 Tests with FGSM adversarial data augmentation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            354,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section we consider tests adding adversarial data augmentation as suggested in (Kurakin et al., 2016). To be more precise we use the method they advise which is called \"step1.1\" using $\\begin{array} { r } { \\epsilon = \\frac { 8 } { 2 5 5 } } \\end{array}$ . The results presented in the figures below are obtained by running 10 experiments with random initializations. We first perform the same tests as in Section 4. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As expected, we observe in Figure 8 that training with adversarial examples help in the case of Gaussian noise, as it adds more variation to the training set, while reducing the accuracy on the clean set. Note that combining our method with adversarial training results in the best median accuracy. Combining the three methods is less successful than expected, which could indicate that a better hyperparameter search would be needed. ",
+        "bbox": [
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+            445,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/fae2ea7c754238bce049a2b05c63a32bfcce3522e9fb94af1c4b1e8a28f137a1.jpg",
+        "image_caption": [
+            "Figure 8: Test set accuracy under Gaussian noise with varying Signal-to-Noise Ratio (SNR). A is for Adversarial, $\\mathrm { P }$ is for Parseval, R is for the proposed Regularizer and V is for Vanilla network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            536,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Considering adversarial robustness, the obtained results are depicted in Figure 9. We observe that adding FGSM adversarial training does not generalize well to other types of attack (which is readily seen in the literature Madry et al. (2018)). Overall, the models using the proposed regularizer are the most robust. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, when considering implementation related perturbations, the results depicted in Figure 10 are consistent with the ones from Section 4.3, in which is shown that the proposed regularizer helps improving robustness. ",
+        "bbox": [
+            176,
+            833,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In summary, even when adding adversarial training, the proposed regularizer is either the most robust in median, or capable of improving the robustness when used combined with the other methods. ",
+        "bbox": [
+            176,
+            882,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/71910dcdc84c86529e0d19400ca2e215f1ed1915780cfeefb93df7906671f83c.jpg",
+        "image_caption": [
+            "Figure 9: Robustness against an adversary measured by the test set accuracy under FGSM attack in the left and center plots and by the mean $\\mathcal { L } _ { 2 }$ pixel distance needed to fool the network using DeepFool on the right plot. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            102,
+            820,
+            253
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/638b59c88b87e39b698b1ce6757d7192e44f7cfe6affecec77d95e10a9712d1f.jpg",
+        "image_caption": [
+            "Figure 10: Test set accuracy under different types of implementation related noise. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            313,
+            820,
+            470
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D.1.2 Tests with black box FGSM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            521,
+            472,
+            536
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To further verify that the obtained results are not only due to gradient masking, we perform tests with black box FGSM, where the target attacked network is not the same as the source of the adversarial noise. ",
+        "bbox": [
+            174,
+            546,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For this test we set the SNR of FGSM to 33. We chose the network with the best performance for each of the tested methods. The results are depicted in Table 1. In our experiments, we found that the combination of our method with Parseval is the most robust to noise coming from other sources, while the noise created by both Parseval and our method did not generalize as well as the one created by Vanilla. This demonstrates that the improvements are not caused by gradient masking, but are caused by the increased robustness of the proposed method and Parseval’s. ",
+        "bbox": [
+            173,
+            594,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/00e589565b5d0dfb0585934a668d9cc7bd60679aa0dffa1a9bdee77c74509ef3.jpg",
+        "table_caption": [
+            "Table 1: Black box FGSM applied to the different methods. The most robust target for a given source is bolded, while the strongest source for a target is in italic. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Target</td><td colspan=\"4\">Source</td></tr><tr><td>Vanilla</td><td>Parseval</td><td>Regularizer</td><td>Parseval +Regularizer</td></tr><tr><td>Vanilla</td><td>X</td><td>60.74</td><td>61.49</td><td>72.51</td></tr><tr><td>Parseval</td><td>57.82</td><td>X</td><td>68.21</td><td>73.87</td></tr><tr><td>Regularizer</td><td>69.72</td><td>74.96</td><td>X</td><td>73.56</td></tr><tr><td>Parseval + Regularizer</td><td>75.35</td><td>76.11</td><td>70.22</td><td>X</td></tr></table>",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D.1.3 Tests with PGD adversarial data augmentation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Most of our adversarial tests are performed with FGSM because of its simplicity and speed, even though it has already been shown (e.g: Madry et al. (2018)) that FGSM is weak as an attack and as a defense mechanism. Despite the fact we do not only target adversarial defense, we further stress the ability of the proposed regularizer to improve it and to combine with other methods. To this end we perform experiments against the PGD (Projected Gradient Descent) attack. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "PGD is an iterative version of FGSM, which run for a maximum number of iterations $_ { i t }$ or until convergence. For each iteration it moves by a distance of step in the direction of the gradient provided it does not go at a distance greater than $\\epsilon$ from the original image. ",
+        "bbox": [
+            174,
+            152,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our experiments show that the proposed regularizer increases robustness against a weak PGD attack (similar epsilon as our FGSM with SNR=33), but it is almost completely defeated by the PGD with the parameters from (Madry et al., 2018). The results are depicted in table 2. We also show that, as expected, FGSM training does not add significant robustness against the stronger PGD attack. ",
+        "bbox": [
+            174,
+            202,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/637a58eef6cfe0756a7226091674f4c1805baba83b1e09e5c14b3e3162e3b90e.jpg",
+        "table_caption": [
+            "Table 2: Test set accuracy on the CIFAR-10 dataset against the PGD attack with different parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>it = 20,step=0.002,∈= 0.01</td><td>it=20,step</td><td>2 二 255,∈=</td><td>8 255</td></tr><tr><td>Vanilla</td><td>0.95%</td><td></td><td>0.02%</td><td></td></tr><tr><td>Proposed Regularizer</td><td>11.18%</td><td></td><td>0.09%</td><td></td></tr><tr><td>FGSM</td><td>5.78%</td><td></td><td>0.09%</td><td></td></tr><tr><td>FGSM + Regularizer</td><td>12.91%</td><td></td><td>0.55%</td><td></td></tr></table>",
+        "bbox": [
+            173,
+            323,
+            789,
+            398
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As the proposed regularizer can be combined with FGSM defense, it is natural to also test it alongside PGD training. We use the parameters advised in (Madry et al., 2018): 7 iterations with $s t e p = 2 / 2 5 5$ , and $\\epsilon = 8 / 2 5 5$ . The results depicted in Table 3 show that using our regularizer increases robustness of networks trained with PGD. Note that Dropout and Gaussian Noise were applied ten times to each of the networks and the results are displayed as the mean test set accuracy under these perturbations. A rate of $4 0 \\%$ was used for dropout. The PGD attack uses the following parameters: $\\begin{array} { r } { i t = 2 0 , s t e p = \\frac { 2 } { 2 5 5 } , \\epsilon = \\frac { 8 } { 2 5 5 } } \\end{array}$ · ",
+        "bbox": [
+            174,
+            411,
+            825,
+            512
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/194fb7006950a8ed3e6f3e0fd6892b5f01565b6fe35a2a624667fc0bfd460dac.jpg",
+        "table_caption": [
+            "Table 3: Results on the CIFAR-10 with PGD training and the hyperparameters from Appendix B. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Robustness</td><td colspan=\"2\">Isotropic</td><td>Adversarial</td><td>Implementation</td></tr><tr><td>Model/ /TestType</td><td>SNR≈8</td><td>SNR≈15</td><td>PGD</td><td>Dropout</td></tr><tr><td>PGD Training</td><td>76.39%</td><td>71.25%</td><td>32.78%</td><td>35.20%</td></tr><tr><td>PGD Training +Regularizer</td><td>76.36%</td><td>72.26%</td><td>33.72%</td><td>55.63%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.2 CIFAR-100 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            646,
+            310,
+            661
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We test the generality of the method using the CIFAR-100 dataset. Results are shown in Table 4 as the mean over three different initializations. Dropout and Gaussian Noise are applied ten times to each of the networks for a total of 30 different runs. An SNR of 33 is used for FGSM, and a rate of $2 5 \\%$ is used for dropout. Images are normalized in the same way as the experiments with CIFAR-10. Due to time constraints we sample only $\\frac { 1 } { 1 0 }$ of the images from the test set for the Deep Fool test. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The proposed regularizer is the most robust on all categories, while Parseval has problems with the perturbations, despite yielding the best accuracy on the clean test set. The combination of the proposed regularizer and the parseval training method is not able to reproduce the good results from the CIFAR-10 dataset. ",
+        "bbox": [
+            174,
+            763,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The results shown in Table 4 are obtained using an architecture that is not performing very well on the clean test set for the CIFAR-100 dataset. We thus performed additional experiments using the WideResNet 28-10 (Zagoruyko and Komodakis, 2016) architecture, and we added standard data augmentation (random crops and random horizontal flipping) and dropout with probability of 30% after the first convolution of each residual block. We train for 200 epochs, starting with a learning rate of 0.1 and divide the learning rate by 5 in epochs 60, 120 and 160. Momentum of 0.9 is used and weight decay of 5e-4. We use the value from the Parseval paper ( $\\beta = 0 . 0 0 0 3$ ) as in this case it provided better results than the one described in Section B. ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/53220aa3f07bca5a85c38167ecf49b59b3c8f1bea42d654de11fdb1a149cdac7.jpg",
+        "table_caption": [
+            "Table 4: Results on the CIFAR-100 dataset with the hyperparameters from Appendix B. Bolded value represent the best model on the test. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Robustness</td><td colspan=\"2\">Isotropic</td><td colspan=\"2\">Adversarial</td><td>Implementation</td></tr><tr><td>Model/Test Type</td><td>SNR≈8</td><td>SNR≈15</td><td>FGSM</td><td>Deep Fool</td><td>Dropout</td></tr><tr><td>Vanilla</td><td>62.38%</td><td>12.78%</td><td>5.70%</td><td>1.7E-5</td><td>8.66%</td></tr><tr><td>Parseval</td><td>63.61%</td><td>10.11%</td><td>5.85%</td><td>1.5E-5</td><td>10.61%</td></tr><tr><td>Proposed Regularizer</td><td>60.06%</td><td>21.14%</td><td>6.15%</td><td>2.9E-5</td><td>21.40%</td></tr><tr><td>Proposed + Parseval</td><td>56.64%</td><td>20.01%</td><td>4.07%</td><td>1.8E-5</td><td>9.41%</td></tr></table>",
+        "bbox": [
+            173,
+            140,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            260,
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+            289
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Results on the WideResNet 28-10 architecture using data augmentation are shown in Table 5. We observe that the proposed method (sometimes with combinations with other methods) is still the most robust. ",
+        "bbox": [
+            173,
+            295,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/8ba165a7290ddf3160fc07c22dd7005460ca17d6227cac17860dff82c810a882.jpg",
+        "table_caption": [
+            "Table 5: Results on the CIFAR-100 dataset with WideResNet 28-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Robustness</td><td colspan=\"2\">Isotropic</td><td colspan=\"2\">Adversarial</td><td>Implementation</td></tr><tr><td>Model/Test Type</td><td>SNR~8</td><td>SNR~15</td><td>FGSM</td><td>Deep Fool</td><td>Quantization</td></tr><tr><td>Vanilla</td><td>78.42%</td><td>11.68%</td><td>21.38%</td><td>5.3E-5</td><td>12.56%</td></tr><tr><td>Parseval</td><td>77.71%</td><td>12.75%</td><td>22.73%</td><td>5.7E-5</td><td>1.58%</td></tr><tr><td>Proposed Regularizer</td><td>77.33%</td><td>14.46%</td><td>23.27%</td><td>5.8E-5</td><td>17.01%</td></tr><tr><td>Proposed 十 Parseval</td><td>76.72%</td><td>20.24%</td><td>25.85%</td><td>6.9E-05</td><td>1.0%</td></tr></table>",
+        "bbox": [
+            174,
+            377,
+            841,
+            467
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E Impact of the proposed regularizer on the boundary ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            507,
+            750,
+            522
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We look at the impact of the proposed regularizer on the boundary region. To this end, we choose 10 pairs of points in distinct classes that are the most similar (i.e. their distance is minimal) in the input space and we look at the decision of the network function along the segment between them. The average is depicted in Figure 11. Note that the point to the left is always chosen to be the one corresponding to the decision of the network at the middle of the segment, so that the average curve is asymmetric. ",
+        "bbox": [
+            173,
+            540,
+            826,
+            625
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b7d8c4414d2dd4e96e52475da8c5a23399be49c9ae204bb71d8e0a830133fc6d.jpg",
+        "image_caption": [
+            "Figure 11: $F ( \\lambda \\mathbf { x } + ( 1 - \\lambda ) \\mathbf { x } ^ { \\prime } )$ for different methods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            290,
+            640,
+            709,
+            834
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, we observe that the proposed regularizer is the one for which the boundary is closest to the middle of the segments, thus proving our claim that the proposed regularizer control the boundary region. ",
+        "bbox": [
+            174,
+            881,
+            823,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F Regularizer pseudo-code ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            464,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Below in Algorithm 1 we describe how we use the proposed regularizer to compute the loss as a pseudo-code. This function receives five inputs: ",
+        "bbox": [
+            173,
+            132,
+            823,
+            161
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "1. $l i s t _ { a c t i v a t i o n s }$ : the list of the intermediate features right after each call of the ReLU activation function of the network. We call these intermediate features activations $\\ell$ where $\\ell$ represents the depth of the network;   \n2. y: the output of the network;   \n3. s: the label signal of the batch. Otherwise said, the ground truth labels of the examples of the batch;   \n4. $m$ : the power of the Laplacian for which we wish to compute the smoothness;   \n5. $\\gamma$ : the scaling coefficient of the regularizer loss. ",
+        "bbox": [
+            210,
+            172,
+            825,
+            308
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: Loss function of the regularized network ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            330,
+            573,
+            344
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "1: procedure Smoothness(activations\\`, s, m) ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            351,
+            531,
+            366
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "2: $\\mathbf { A } ^ { \\ell } \\gets$ Pairwise cosine similarity of activations\\`   \n3: $\\mathbf { D } ^ { \\ell } \\gets$ Diagonal degree matrix of $\\mathbf { A } ^ { \\ell }$   \n4: $\\mathbf { L } ^ { \\ell } \\gets \\mathbf { D } ^ { \\ell } - \\mathbf { A } ^ { \\ell }$   \n5: $\\sigma ^ { \\ell } \\gets \\mathrm { T r a c e } ( \\mathbf { s } ^ { \\intercal } ( L ^ { \\ell } ) ^ { m } \\mathbf { s } )$   \n6: return σ\\`   \n7: procedure $\\mathrm { L o s s } ( l i s t _ { a c t i v a t i o n s } , \\mathbf { y } , \\mathbf { s } , m , \\gamma )$   \n8: for activations\\` ∈ listactivations do   \nL σ\\` ← Smoothness(activations\\`, s, m)   \n9: ∆ ← P\\`maxi=1 |σi−σi−1|   \n10: return CategoricalCrossEntropy(s, y) + γm∆ ",
+        "bbox": [
+            179,
+            359,
+            570,
+            534
+        ],
+        "page_idx": 15
+    }
+]

parse/train/HJl0jiRqtX/HJl0jiRqtX.md

@@ -0,0 +1,620 @@
+# EDDI: EFFICIENT DYNAMIC DISCOVERY OF HIGH-VALUE INFORMATION WITH PARTIAL VAE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Making decisions requires information relevant to the task at hand. Many real-life decision making situations allow acquiring further relevant information at a specific cost. For example, in assessing the health status of a patient we may decide to take additional measurements such as diagnostic tests or imaging scans before making a final assessment. More information that is relevant allows for better decisions but it may be costly to acquire all of this information. How can we trade off the desire to make good decisions with the option to acquire further information at a cost? To this end, we propose a principled framework, named EDDI (Efficient Dynamic Discovery of high-value Information), based on the theory of Bayesian experimental design. In EDDI we propose a novel partial variational autoencoder (Partial VAE), to efficiently handle missing data over varying subsets of known information. EDDI combines this Partial VAE with an acquisition function that maximizes expected information gain on a set of target variables. EDDI is efficient and demonstrates that dynamic discovery of high-value information is possible; we show cost reduction at the same decision quality and improved decision quality at the same cost in benchmarks and in two health-care applications. We believe there is great potential for realizing these gains in real-world decision support systems.
+
+# 1 INTRODUCTION
+
+Imagine that a person walks into a hospital with a broken arm. The first question from health-care personnel would be: “How did you break the arm?” instead of “Do you have a cold?”, because the answer reveals relevant information for this patient. Human experts dynamically acquire information based on the current understanding of the situation. Automating this human expertise of asking relevant questions is difficult. In other applications such as online questionnaires for example, most existing online questionnaire system either present exhaustive questions (Lewenberg et al., 2017; Shim et al., 2018) or use extremely time-consuming human labeling work to manually build a decision tree for a reduced number of questions (Zakim et al., 2008). This wastes the valuable time of experts or users (patients). An automated solution for personalized dynamic acquisition of information has great potential to save much of this time in many real-life applications.
+
+What are the technical challenges to build an intelligent information acquisition system? Missing data is a key issue: taking the questionnaire scenario as an example, at any point in time we only observe a small subset of answers yet have to reason about possible answers for the remaining questions. We thus need an accurate probabilistic model that can perform inference given a variable subset of observed answers. Another key problem is deciding what to ask next: this requires assessing the worth of each possible question or measurement, the exact computation of which is intractable. However, compared to current active learning methods we select individual features, not instances; therefore, existing methods are not applicable. In addition, these traditional methods are often not scalable to the large volume of data available in many practical cases (Settles, 2012).
+
+We propose the EDDI (Efficient Dynamic Discovery of high-value Information) framework as a scalable information acquisition system for any given task. We assume that information acquisition is always associated with cost. Given a task, such as estimating the costumers’ experience or assessing population health status, we dynamically decide which piece of information to acquire next. The framework is very general, and the information can be presented in any form such as answers to questions, or values of a lab test. Our contributions are:
+
+• We propose a novel efficient information acquisition framework, EDDI (Section 3). To enable EDDI, we contribute technically: 1. A partial amortized inference method with different specifications for the inference network (Section 3.2). We extend a current amortized inference method, the variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014), to account for partial observations. The resulting method, which we call Partial VAE, is inspired by the set formulation of the data (Qi et al., 2017; Zaheer et al., 2017). Partial VAE, as a probabilistic framework, is highly scalable, and serves as the base for the EDDI framework. However, Partial VAE itself is generic and can be used on its own as a non-linear probabilistic framework for missing-data imputation. 2. An information theoretic acquisition function with an efficient approximation, yielding a novel variable-wise active learning method (Section 3.3). Based on the partial VAE, we select the unobserved variable which contributes most to the task, such as health assessment, evaluated using the mutual information. This acquisition function does not have an analytical solution and we derive an efficient approximation.   
+• We demonstrate the performance of EDDI on various settings, and apply it in real-life health-care scenarios (Section 4). 1. We first show the superior performance of the Partial VAE framework on an image inpainting task (Section 4.1). 2. We then use 6 different datasets from the Machine Learning repository of University of Irvine (UCI) (Dheeru & Karra Taniskidou, 2017) to demonstrate the behavior of EDDI, comparing with multiple baseline methods (Section 4.2). 3. Finally, we evaluate EDDI on two real-life health-care applications: risk assessment in intensive care (Section 4.3) and public health assessment with national survey (Section 4.4), where traditional methods without amortized inference do not scale. EDDI shows clear improvements in these two applications.
+
+# 2 RELATED WORK
+
+EDDI requires a method that handles partially observed data to enable dynamic variable wise active learning. We thus review related methods for handling partial observation and doing active learning.
+
+# 2.1 PARTIAL OBSERVATION
+
+Missing data entries are common in many real-life applications, which has created a long history of research on the topic of dealing with missing data (Rubin, 1976; Dempster et al., 1977). We describe existing methods below:
+
+Traditional methods without amortization Prediction based methods have shown advantages for missing value imputation (Scheffer, 2002). Efficient matrix factorization based methods have been recently applied (Keshavan et al., 2010; Jain et al., 2010; Salakhutdinov & Mnih, 2008), where the observations are assumed to be able to decompose as multiplication of low dimensional matrices. In particular, many probabilistic frameworks with various distribution assumptions (Salakhutdinov & Mnih, 2008; Blei et al., 2003) have been used for missing value imputation (Yu et al., 2016; Hamesse et al., 2018) and also recommender systems where unlabeled items are predicted (Stern et al., 2009; Wang & Blei, 2011; Gopalan et al., 2014).
+
+The probabilistic matrix factorization method has been used in the active variable selection framework, the dimensionality reduction active learning model (DRAL),(Lewenberg et al., 2017). These traditional methods suffer from limited model capacity since they are commonly linear. Additionally, they do not scale to large volumes of data and thus are usually not applicable in real-world applications. For example, Lewenberg et al. (2017) test the performance of their method with a single user due to the heavy computational cost of traditional inference methods for probabilistic matrix factorization.
+
+Utilizing Amortized Inference The amortized inference (Kingma & Welling, 2014; Rezende et al., 2014; Zhang et al., 2017) has significantly improved the scalability for probabilistic models such as variational autoencoders (VAEs). In the case of partially observed data, amortized inference is particularly of interest due to the need of speeding up test time applications. Wu et al. (2018) employ traditional non-amortized inference in order to perform partial inference of a pretrained VAE during test time. Amortized inference is only used during training, assuming the training dataset is fully observed. During test time, the traditional inference is used to infer missing data entries from the partially observed dataset using the pre-trained model. In this way, only training time is reduced. The model is restrictive since it is not scalable in the test time and the fully observed training set is not available for many applications.
+
+Nazabal et al. (2018) uses zero imputation (ZI) for amortized inference for both training and test sets with missing data entries. ZI is a generic and straightforward method that first fills the missing data with zeros, and then feeds the imputed data as input for the inference network. The drawback of zero imputation is that it introduces bias when the data are not missing completely at random which leads to not well-learned model. We also observe artifacts when using it for the image inpainting task. In the end, independent of our work, Garnelo et al. (2018) explore interpreting variational autoencoder (amortized inference) as stochastic processes, which also handles partial observation per se.
+
+# 2.2 ACTIVE LEARNING
+
+Traditional Active Learning Active learning, also referred to as experimental design, aims to obtain optimal performance with fewer selected data (or experiments) (Lindley, 1956; MacKay, 1992; Settles, 2012). Traditional active learning aims to select the next data point to label. Many information theoretical approaches have shown promising results in various settings with different acquisition functions (MacKay, 1992; McCallumzy & Nigamy, 1998; Houlsby et al., 2011). These methods commonly assume that there exist fully observed data, and the acquisition decision is instance wise. Little work has dealt with missing values within instances. Zheng & Padmanabhan (2002) deal with missing data values by imputing with traditional non-probabilistic methods (Little & Rubin, 1987) first. It is still an instance-wise active learning framework.
+
+Different from traditional active learning, our proposed framework aims to for perform variable-wise active learning for each instance. In this setting, information theoretical acquisition functions need a new design as well as non-trivial approximations. The most closely related work is the aforementioned DRAL (Lewenberg et al., 2017), which deals with variable-wise active learning for each instance.
+
+Active Feartue Acquisition (AFA) Active sequential feature selection is of great need, especially in cost-sensitive applications. Thus, many methods have also been applied and resulted in the class of methodologies called Active Feature Acquisition (AFA) (Melville et al., 2004; Saar-Tsechansky et al., 2009; Thahir et al., 2012; Huang et al., 2018). For instance, Melville et al. (2004); Saar-Tsechansky et al. (2009) have designed objectives to select any feature from any instance to minimize the cost to archive high accuracy. The proposed framework is very general. However, the problem setting of AFA methods are entirely different from our active variable selection problem: AFA mainly studies the optimization of optimal training set that would result in the best classifier (model), under limited budget of costs, while our framework studies a slightly different problem: given a pretrained model, how to identify and acquire high value information with minimal costs. Hence, AFA can not be directly applied. Also, AFA requires fully observed variables at test time, while our framework does not require this assumption. Last but not the least, the realization of these framework relies on various heuristics and suffer from limited scalability.
+
+# 3 METHOD
+
+In this section, we first formalize the active variable selection problem that we aim to solve. Then, we present our Partial VAE to model and perform inference on partial observations. Finally, we complete the EDDI framework by presenting our acquisition function and estimation method.
+
+# 3.1 PROBLEM FORMULATION
+
+The core problem that we address in this paper is the following active variable selection problem. Let $\mathbf { x } = [ x _ { 1 } , \ldots , x _ { | I | } ]$ be a set of random variables with probability density $p ( \mathbf { x } )$ . Furthermore, let a subset of the variables $\mathbf { x } _ { O }$ , $O \subset I$ , be observed while the variables $\mathbf { \Delta x } _ { U }$ , $U = I \backslash O$ , are unobserved. We assume that we can query the value of variables $x _ { i }$ for $i \in U$ . The goal of active variable selection is to query a sequence of variables in $U$ with the goal of predicting a quantity of interest $f ( \mathbf { x } )$ , as accurately as possible while simultaneously performing as little queries as possible, where $f ( \cdot )$ can be any (random) function. This problem, in the simplified myopic setting, can be formalized as that of proposing the next variable $x _ { i ^ { * } }$ to be queried by maximizing a reward function $R$ , i.e.
+
+![](images/9ef4824b17f2a70854957ffefab5c25a29fc67a3a724da445a2521f93a0ac4bc.jpg)  
+Figure 1: Illustration of Partial VAE encoder architecture.
+
+$$
+i ^ { * } = \underset { i \in U } { \arg \operatorname* { m a x } } R ( i \mid \mathbf { x } _ { O } )
+$$
+
+where $R ( i \mid \mathbf { x } _ { O } )$ quantifies the merit of our prediction of $f ( \cdot )$ given $\mathbf { x } _ { 0 }$ and $x _ { i }$ . Furthermore, the reward can quantify other properties important to the problem, e.g. the cost of acquiring $x _ { i }$ .
+
+# 3.2 PARTIAL AMORTIZATION OF INFERENCE QUERIES
+
+We first introduce how to establish a generative probabilistic model of random variables $\mathbf { X }$ , that is capable of handling unobserved (missing) variables $\mathbf { \Delta x } _ { U }$ with variable size. Our approach to this, named the Partial VAE, is based on the Variational autoencoder (VAE), which enables inference to scale to large volumes of data.
+
+VAE and amortized inference VAE defines a generative model where the data $\mathbf { X }$ are generated from latent variables $\mathbf { z }$ , defined as $\begin{array} { r } { p ( \mathbf { x } , \mathbf { z } ; \boldsymbol { \theta } ) = \prod _ { n } \bar { p } ( \mathbf { x } _ { n } | \mathbf { z } _ { n } ; \boldsymbol { \theta } ) p ( \mathbf { z } _ { n } ) } \end{array}$ . The data generation, $p _ { \theta } ( \mathbf { x } _ { n } | \mathbf { z } _ { n } )$ , is realized by a deep neural network. To approximate the the posterior of the latent variable $p _ { \pmb { \theta } } ( \mathbf { z } _ { n } | \mathbf { x } _ { n } )$ , VAE uses amortized variational inference. Specifically, it uses an encoder, which is another neural network with the data ${ \bf { X } } _ { n }$ as input to produce the variational approximation of the posterior $q ( \mathbf { z } _ { n } | \mathbf { x } _ { n } ; \phi )$ . As traditional variational inference, VAE is trained by maximizing an evidence lower bound (ELBO), which is equivalent to minimize the KL divergence between $p _ { \theta } ( \mathbf { \bar { z } } _ { n } | \mathbf { x } _ { n } )$ and $q ( { \bf z } _ { n } | { \bf x } _ { n } ; \phi )$ .
+
+VAE is not directly applicable to data with missing values. Consider a partitioning that divides the variables into observed variables $\mathbf { x } _ { O }$ and unobserved variables $\mathbf { \Delta x } _ { U }$ . In this setting, we would like to efficiently and accurately infer $p ( \mathbf { z } | \mathbf { x } _ { O } )$ and $p ( \mathbf { x } _ { U } | \mathbf { x } _ { O } )$ . One challenge in the above setting is that there are many possible partitioning of $\{ U , O \}$ , where the size of observed ratings might vary. Therefore, classic approaches to train a VAE with variational bound and amortize inference networks are no longer directly applicable. We propose to extend amortization to our partial inference situation.
+
+Partial VAE In a VAE, a factorized structure for $p ( \mathbf { x } | \mathbf { z } )$ is always assumed, i.e.
+
+$$
+p ( \mathbf { x } | \mathbf { z } ) = \prod _ { i } p _ { i } ( \mathbf { x } _ { i } | \mathbf { z } ) .
+$$
+
+This implies that given $\mathbf { z }$ , the observed variables $\mathbf { x } _ { O }$ are conditionally independent of $\mathbf { \Delta x } _ { U }$ . Therefore,
+
+$$
+\begin{array} { r } { p ( \mathbf { x } _ { U } | \mathbf { x } _ { O } , \mathbf { z } ) = p ( \mathbf { x } _ { U } | \mathbf { z } ) , } \end{array}
+$$
+
+and inferences about $\mathbf { \Delta x } _ { U }$ can be reduced to inference about $\mathbf { z }$ . Therefore, the key object of interest in this setting is $p ( \mathbf { z } | \mathbf { x } _ { O } )$ , i.e., the posterior over the shared latent variables $\mathbf { z }$ given the observed variables $\mathbf { x } _ { O }$ . Once knowledge about $\mathbf { z }$ is obtained, we can draw correct inferences about $\mathbf { \Delta x } _ { U }$ . To approximate $p ( \mathbf { z } | \mathbf { x } _ { O } )$ we introduce an auxiliary variational inference network $q ( { \bf z } | { \bf x } _ { O } )$ and define a partial variational upper bound,
+
+$$
+\begin{array} { r l r } { D _ { \mathrm { K L } } ( q ( \mathbf { z } | \mathbf { x } _ { O } ) \| p ( \mathbf { z } | \mathbf { x } _ { O } ) ) } & { = } & { \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } _ { O } ) } [ \log q ( \mathbf { z } | \mathbf { x } _ { O } ) - \log p ( \mathbf { z } | \mathbf { x } _ { O } ) ] } \\ & { \leq } & { \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } _ { O } ) } [ \log q ( \mathbf { z } | \mathbf { x } _ { O } ) - \log p ( \mathbf { x } _ { O } | \mathbf { z } ) - \log p ( \mathbf { z } ) ] \equiv \mathcal { L } _ { p a r i a l } . } \end{array}
+$$
+
+This bound, $\mathcal { L } _ { p a r t i a l }$ , depends only on the observation $\mathbf { x } _ { O }$ , which could vary between different data points. We call the auxiliary distribution $q ( { \bf z } | { \bf x } _ { O } )$ the partial inference net since it takes a set of partially observed variables $\mathbf { x } _ { O }$ whose length may vary. Specifying $q ( { \bf z } | { \bf x } _ { O } )$ requires distribution over random partitioning $\{ O , U \}$ .
+
+Amortized Inference with partial observations Inspired by the Point Net (PN) approach for point cloud classification (Qi et al., 2017; Zaheer et al., 2017), we specify the approximate distribution $q ( { \bf z } | { \bf x } _ { O } )$ by a permutation invariant set function encoding, given by:
+
+$$
+\mathbf { c } ( \mathbf { x } _ { O } ) : = g ( h ( \mathbf { s } _ { 1 } ) , h ( \mathbf { s } _ { 2 } ) , . . . , h ( \mathbf { s } _ { | O | } ) ) ,
+$$
+
+where $| O |$ is the number of the observed variables, ${ \bf s } _ { d }$ carries the information of the input identify $\mathbf { e } _ { d }$ and the input value $x _ { d }$ . There are many ways to define $\mathbf { e } _ { d }$ . Naively, it could be the coordinates for points in the point cloud, and one-hot embedding of the number of questions in a questionnaire. With different problem settings, it can be beneficial to learn e as an embedding of the identity of the variable, either with or without an naive encoding as input. In this work, we treat e as an unknown embedding, which is optimized during training process.
+
+There are also different ways to construct ${ \bf s } _ { d }$ . Concatenation, $\mathbf { s } _ { d } = [ \mathbf { e } _ { d } , x _ { d } ]$ , is commonly used in computer vision applications (Qi et al., 2017). Such architecture is illustrated in Figure 1(a). However, we note that the construction of ${ \bf s } _ { d }$ can be flexible. We propose to construct $\mathbf { s } = \mathbf { e } _ { d } * x _ { d }$ using elementwise multiplication, shown in Figure 1(b). We show that this formulation generalizes naive Zero Imputation (ZI) VAE (Nazabal et al., 2018). We call this approach Pointnet Plus (PNP) specification of Partial VAE. The theoretical consideration of relating ZI to PNP is presented in Appendix C.1.
+
+We then use a neural network $h ( \cdot )$ to map input s from $\mathbb { R } ^ { M + 1 }$ to $\mathbb { R } ^ { K }$ , where $M$ is the dimension of each $\mathbf { e } _ { d }$ , $x _ { d }$ is a scalar, and $K$ is the latent space size. Key to the PN structure is the permutation invariant aggregation operation $g ( \cdot )$ , such as max-pooling or summation. In this way, the mapping $\mathbf { c } ( \mathbf { x } _ { O } )$ is invariant to permutations of elements of $\mathbf { x } _ { O }$ and $\mathbf { x } _ { O }$ can have arbitrary length. Finally, the fixed-size code $\mathbf { c } ( \mathbf { x } _ { O } )$ is fed into an ordinary amortized inference net, that transforms the code into the statistics of a multivariate Gaussian distribution to approximate $p ( \mathbf { z } | \mathbf { x } _ { O } )$ . The procedure is illustrated in the first dashed box in Figure 1, which is our basic Partial VAE method.
+
+# 3.3 EFFICIENT DYNAMIC DISCOVERY OF HIGH-VALUE INFORMATION
+
+We now cast the active variable selection problem (1) as adaptive Bayesian experimental design, utilizing $p ( \mathbf { x } _ { U } | \mathbf { x } _ { O } )$ inferred by Partial VAE. Algorithm 1 summarize the EDDI framework.
+
+Information Reward We designed a variable selection acquisition function in an information theoretical way following Bayesian experimental design (Lindley, 1956; Bernardo, 1979). Lindley (1956) provides a generic formulation of Bayesian experimental design by maximizing the expected Shannon information. Bernardo (1979) generalizes it by considering the decision task context.
+
+For a given task, we may be interested in statistics of some variables $\mathbf { X } _ { \phi }$ , where $\mathbf { x } _ { \phi } \subset \mathbf { x } _ { U }$ . Given a new instance (user), assume we have observed $\mathbf { x } _ { O }$ so far for this instance, and we need to select the next variable $x _ { i }$ (an element of ${ \bf x } _ { U \backslash \phi } )$ to observe. Following Bernardo (1979), We select $x _ { i }$ by maximizing:
+
+$$
+R ( i , \mathbf { x } _ { O } ) = \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { O } ) } D _ { \mathrm { K L } } \left[ p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { O } ) \| p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { O } ) \right] .
+$$
+
+In our paper, we mainly consider the case that a subset of interesting observations represents the statistics of interest $\mathbf { X } _ { \phi }$ . Sampling $\mathbf { x } _ { i } \sim p \big ( \mathbf { x } _ { i } \big | \mathbf { x } _ { o } \big )$ is approximated by $\mathbf { x } _ { i } \sim \hat { p } ( \mathbf { x } _ { i } | \mathbf { x } _ { o } )$ , where $\hat { p } ( \mathbf { x } _ { i } | \mathbf { x } _ { o } )$ is defined by the following process in Partial VAE. It is implemented by first sampling $\mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } _ { o } )$ , and then $\mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { z } )$ . The same applies for $p ( \mathbf { x } _ { i } , \mathbf { x } _ { \phi } | \mathbf { z } )$ appeared in Equation 9.
+
+Efficient approximation of the Information reward The Partial VAE allows us to sample $\mathbf { x } _ { i } \sim$ $p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } )$ . However, the KL term in Equation 6,
+
+$$
+D _ { K L } \left[ p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } ) \right] = - \int _ { \mathbf { x } _ { \phi } } p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \log \frac { p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } ) } { p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } ,
+$$
+
+is intractable to evaluate since both $p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } )$ and $p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } )$ are intractable. For high dimensional $\mathbf { X } _ { \phi }$ , entropy estimation could be difficult. The entropy term $\begin{array} { r } { \int _ { \mathbf { x } _ { \phi } } p \big ( \mathbf { x } _ { \phi } \big | \mathbf { x } _ { i } , \mathbf { x } _ { o } \big ) \log p \big ( \mathbf { x } _ { \phi } \big | \mathbf { x } _ { i } , \mathbf { x } _ { o } \big ) } \end{array}$ depends on $i$ hence cannot be ignored. In the following, we show how to approximate this expression.
+
+Our proposal is based on the observation that analytic solutions of KL-divergences are available under specific variational distribution families of $q ( { \bf z } | { \bf x } _ { O } )$ (such as the Gaussian distribution commonly used in VAEs). Instead of calculating information reward in $\mathbf { X }$ space, we have shown that one can equivalently perform calculations in $\mathbf { z }$ space (cf. Appendix A.1):
+
+$$
+\begin{array} { r l } & { R ( i , \mathbf { x } _ { o } ) = \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } D _ { K L } \left[ p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { z } | \mathbf { x } _ { o } ) \right] } \\ & { \phantom { x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x } - \mathbb { E } _ { \mathbf { x } _ { \phi } , \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { \phi } , \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } D _ { K L } \left[ p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) \right] . } \end{array}
+$$
+
+Require: Training dataset $\mathbf { X } _ { \mathrm { t r m } }$ , which is partially observed; Test dataset $\mathbf { X } _ { \mathrm { t s t } }$ without any observation; Indices $\phi$ of target variables.   
+1: Train Partial VAE by optimizing partial variational bound with $\mathbf { X } _ { \mathrm { t r m } }$ (cf. Section 3.2)   
+2: Actively acquire feature value $x _ { i }$ to estimate $\mathbf { X } _ { \phi }$ for each test point (cf. Section 3.3) for each test instance do $\mathbf { x } _ { O } \gets \emptyset$ (no variable value has been observed for any test point) repeat Choose variable $x _ { i }$ from $U \backslash \phi$ to maximize the information reward (Equation 9) $\mathbf { x } _ { O }  x _ { i } \cup \mathbf { x } _ { O }$ until Stopping criterion reached (e.g. the time budget) end for
+
+Algorithm 1 EDDI: Algorithm Overview   
+Table 1: Comparing models trained on partially observed MNIST. VAE-full is an ideal reference.   
+
+<table><tr><td>Method</td><td>VAE-full</td><td>ZI</td><td>ZI-m</td><td>PN</td><td>PNP</td></tr><tr><td>Train ELBO</td><td>-95.05</td><td>-113.64</td><td>-117.29</td><td>-121.43</td><td>-113.64</td></tr><tr><td>TestELBO (Rnd.)</td><td>-101.46</td><td>-116.01</td><td>-118.61</td><td>-122.20</td><td>-114.01</td></tr><tr><td>Test ELBO (Reg.)</td><td>-101.46</td><td>-130.61</td><td>-123.87</td><td>-116.53</td><td>-113.19</td></tr></table>
+
+Note that Equation 8 is exact. Additionally, we use partial VAE approximation $p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \approx$ $q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } )$ , $p ( \mathbf { z } | \mathbf { x } _ { o } ) \approx q ( \mathbf { z } _ { i } | \mathbf { x } _ { o } )$ and $p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \approx q ( \mathbf { z } _ { i } | \mathbf { x } _ { i } , \mathbf { x } _ { o } )$ . This leads to the final approximation of the information reward:
+
+$$
+\begin{array} { r l } & { \hat { R } ( i , { \bf x } _ { o } ) = \mathbb { E } _ { { \bf x } _ { i } \sim \hat { p } ( { \bf x } _ { i } | { \bf x } _ { o } ) } D _ { K L } \left[ q ( { \bf z } | { \bf x } _ { i } , { \bf x } _ { o } ) | | q ( { \bf z } | { \bf x } _ { o } ) \right] } \\ & { \quad \quad \quad \quad - \mathbb { E } _ { { \bf x } _ { \phi } , { \bf x } _ { i } \sim \hat { p } ( { \bf x } _ { \phi } , { \bf x } _ { i } | { \bf x } _ { o } ) } D _ { K L } \left[ q ( { \bf z } | { \bf x } _ { \phi } , { \bf x } _ { i } , { \bf x } _ { o } ) | | q ( { \bf z } | { \bf x } _ { \phi } , { \bf x } _ { o } ) \right] . } \end{array}
+$$
+
+With this approximation, the divergence between ${ q } ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } )$ and $q ( \mathbf { z } | \mathbf { x } _ { o } )$ can often computed analytically in Partial VAE setting, for example, under Gaussian parameterization. The only Monte Carlo sampling required is the one set of samples $\mathbf { x } _ { \phi } , \mathbf { x } _ { i } \sim p \big ( \mathbf { x } _ { \phi } , \mathbf { x } _ { i } \big | \mathbf { x } _ { o } \big )$ that can be shared across different KL terms in Equation 9.
+
+# 4 EXPERIMENTS
+
+We evaluate our proposed EDDI framework with various settings. We first assess the Partial VAE component of EDDI alone on an image inpainting task both qualitatively and quantitatively (Section 4.1). We compare our proposed two PN-based Partial VAE with the zero-imputing (ZI) VAE (Nazabal et al., 2018). Additionally, we modify ZI VAE to use s mask matrix indicating which variables are currently observed as input. We name this method ZI-m VAE. We then demonstrate the performance of the entire EDDI framework on datasets from the UCI repository (Section 4.2 ), as well as in two real-life application scenarios: Risk assessment in intensive care (Section 4.3) and public health assessment with national health survey (Section 4.4). We compare the performance of EDDI, using four different Partial VAE settings, with three baselines. The first baseline is the random active feature selection strategy (denoted as RAND) which randomly picks the next variable to observe. The second baseline method is the single best strategy (denoted as SING) which finds a single global optimal order of picking up variables. This order is then applied to all data points. SING uses the objective function as in Equation (9) to find the optimal ordering by averaging over all the data.
+
+# 4.1 IMAGE INPAINTING WITH PARTIAL VAE
+
+We evaluate the performance of Partial VAE with the image inpainting task, which is to fill in the removed pixels in an image. We perform the evaluation in two different settings: We remove the pixels at random in the first setting, and remove a region of the pixels in the second setting.
+
+Inpainting Random Missing Pixels We use MNIST dataset (LeCun, 1998) and remove pixels randomly for this task. The same setting are used for all methods (see Appendix B.1 for details). During training, we remove a random portion (uniformly sampled between $0 \%$ and $70 \%$ ) of pixels. We then impute missing pixels on a partially observed test set (constructed by removing $70 \%$ of the pixels). The performance of pixel imputation is evaluated by test ELBOs on missing pixels. The first two rows in Table 1 show training and test ELBOs for all algorithms using this partially observed dataset. Additionally, we show ordinary VAE (VAE-full) trained on the fully observed dataset as an ideal reference. Among all Partial VAE methods, the PNP approach performs best.
+
+![](images/487d9e412ca9d16b92f5a6044e95ffaa5f95f923bf33be8ff0144bfc0f6b2e79.jpg)  
+Figure 2: Image inpainting example with MNIST dataset using Partial VAE with four settings.
+
+![](images/43f74d611e0daaf5fef1dcd72e2410db615b927f160d2d16a1fd11b1522cfe33.jpg)  
+Figure 3: Information curves of active variable selection, demonstrated on three UCI datasets (based on PNP parameterization of Partial VAE). This displays negative test log likelihood (y axis, the lower the better) during the course of active selection $\mathbf { \dot { x } }$ -axis). Error bars represent standard errors.
+
+Inpainting Regions We then consider inpainting large contiguous regions of images. It aims to evaluate the capability of the Partial VAEs to produce all possible outcomes with better uncertainty estimates. With the same trained model as before, we remove the region of the upper $60 \%$ pixels of the image in the test set. We then evaluate the average likelihoods of the models. The last row of Table 1 shows the results of the test ELBO in this case. PNP based Partial VAE performs better than other settings. Note that given only the lower half of a digit, the number cannot be identified uniquely. ZI (Figure 2(b)) fails to cover the different possible modes due to its limitation in posterior inference. ZIm (Figure 2(c)) is capable of producing multiple modes. However, some of the generated samples are not consistent with the given part (i.e., some digits of 2 are generated). Our proposed PN (Figure 2(d)) and PNP Figure 2(e)) are capable of recovering different modes, and are consistent with observations.
+
+# 4.2 EDDI ON UCI DATASETS
+
+Given the effectiveness of our proposed Partial VAE, we now demonstrate the performance of our proposed EDDI framework in comparison with random selection (RAND) and single optimal ordering (SING). We first apply EDDI on 6 different UCI datasets (cf. Appendix B.2) (Dheeru & Karra Taniskidou, 2017). We report the results of EDDI with all these four different specifications of Partial VAE (ZI, ZI-m, PN, PNP).
+
+All Partial VAE are first trained on partially observed UCI datasets where a random portion of variables is removed. We actively select variable for each test point starting with empty observation $\mathbf { X } _ { O } = \varnothing$ . In all UCI datasets, We randomly sample $10 \%$ of the data as the test set. All experiments repeated for ten times.
+
+Taking PNP based setting as an example, Figure 3 shows the negative test log likelihood on $\mathbf { X } _ { \phi }$ for each variable selection step with three different datasets, where $\mathbf { X } _ { \phi }$ is defined by the UCI task. We call this curve the information curve $( I C )$ . We see that EDDI can obtain information efficiently. It archives the same negative test log likelihood with less than half of the variables. Single optimal ordering also improves upon random ordering. However, it is less efficient compared with EDDI since EDDI perform active learning for each data instance which is “personalized”. Figure 4 shows an example of the decision processes using EDDI and SING. The first step of EDDI overlaps largely with SING. From the second step, EDDI makes “personalized” decisions.
+
+Table 2: Average ranking of AUIC over 6 UCI datasets.   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN</td></tr><tr><td>EDDI</td><td>5.72 (0.03)</td><td>5.54 (0.02)</td><td>5.08 (0.02)</td><td>5.25 (0.02)</td></tr><tr><td>Random</td><td>8.03 (0.03)</td><td>8.10 (0.03)</td><td>7.77 (0.03)</td><td>7.79 (0.03)</td></tr><tr><td>Single best</td><td>8.68 (0.03)</td><td>5.50 (0.02)</td><td>5.20 (0.02)</td><td>5.28(0.02)</td></tr></table>
+
+![](images/ada7cb63e953e4b833a95637af67c5203e364e7ce1fae0e3bb91877f228a3084.jpg)
+
+<table><tr><td>Method</td><td>Time</td></tr><tr><td>DRAL</td><td>2747.16</td></tr><tr><td>EDDI</td><td>2.64</td></tr></table>
+
+![](images/60d89759c14cd44e64b80761087dcc976d6c03bae059baa8d799ce911efe6764.jpg)  
+Figure 4: First four decision steps on Boston Housing test data. EDDI is “personalized” comparing SING. Full names of the variables are listed in the Appendix B.2.   
+Figure 5: Comparison of DRAL (Lewenberg et al., 2017) and EDDI on Boston Housing dataset. EDDI out performs DRAL significantly regarding test log likelihood in every step.   
+Table 3: Test CPU time (in seconds) per test point for active variable selection using EDDI and DRAL. EDDI is $1 0 ^ { 3 }$ times more computation efficient than DRAL (Lewenberg et al., 2017).
+
+We also present the average performance among all datasets with different settings. The area under the information curve (AUIC), $\begin{array} { r } { \sum _ { t } - \log p \big ( \mathbf { x } _ { \phi } \big | \bar { \mathbf { x } } _ { O _ { t } } \big ) } \end{array}$ , can then be used to compare the performance across models and strategies. Smaller AUIC value (could be positive or negative) indicates better performance. However, due to different datasets have different scales of test likelihoods and different numbers of variables (indicated by steps), it is not fair to average the AUIC across datasets to compare overall performances. We thus define average ranking of AUIC that compares 12 methods (indexed by i) averaging these datasets as: $\begin{array} { r } { r _ { i } = \frac { 1 } { \sum _ { j } N _ { j } } \bar { \sum } _ { j = 1 } ^ { 6 } \sum _ { k = 1 } ^ { N _ { j } } r _ { i j k } , i = 1 , . . , 1 6 . } \end{array}$ . These 12 methods are cross combinations of four Partial VAE models with three variable selection strategies. $r _ { i }$ is the final ranking of ith combination, $r _ { i j k }$ is the ranking of the ith combination (based on AUIC value) regarding the kth test data point in the $j$ th UCI dataset, and $N _ { j }$ the size of the jth UCI dataset. This gives us $6 \Sigma _ { j } N _ { j }$ different rankings. Finally, we simply compute the mean and standard error statistics based on these rankings. Table 2 summarize the average ranking results. We provide additional statistical significance test (Wilcoxcon signed-rank test for paired data) in Appendix B.2.3. We can conclude that EDDI outperforms other variable selection order in all different Partial VAE settings. Among different partial VAE settings, PNP/PN-based settings perform better than ZI-based settings.
+
+Comparison with non-amortized method Additionally, we compare EDDI to DRAL (Lewenberg et al., 2017) which is the state-of-the-art method for the same problem setting. As discussed in Section 2, DRAL is linear and requires high computational cost. The DRAL paper only tested their method on a single test data point due to its limitation on computational efficiency. We compare DRAL with EDDI on Boston Housing dataset with ten randomly selected test points here. Results are shown in Figure 5, where EDDI significantly outperforms DARL thanks to more flexible Partial VAE model. Additionally, EDDI is 1000 times more efficient than DARL as shown in Table 3.
+
+# 4.3 RISK ASSESSMENT WITH MIMIC-III
+
+We now apply EDDI to risk assessment tasks using the Medical Information Mart for Intensive Care (MIMIC III) database (Johnson et al. (2016)). MIMIC III is the most extensive publicly available clinical database, containing real-world records from over 40,000 critical care patients with 60,000 ICU stays. The risk assessment task is to predict the final mortality. We preprocess the data for this task following Harutyunyan et al. (2017) 1. This results in a dataset of 21139 patients. We treat the final mortality of a patient as a Bernoulli variable. For our task, we focus on variable selection, which corresponds to medical instrument selection. We thus further process the time series variables into static variables based on temporal averaging.
+
+![](images/09f4666e21f023e4b7e2288017dd23cf115b97d8e1270388c54fbc9099dba6e0.jpg)  
+Figure 6: Information curves of active variable selection on risk assessment task on MIMIC III, produced with PNP setting.
+
+![](images/dad7840061964d34d6dd248dfa38996528a260e0dc19668e3d66a506fb6772cf.jpg)  
+Figure 7: Information curves of active (grouped) variable selection on risk assessment task on NHANES, produced with PNP setting.
+
+Table 4: Average ranking on AUIC of MIMIC III   
+
+<table><tr><td>Method</td><td>EDDI</td><td>Random</td><td>Single best</td></tr><tr><td>ZI</td><td>5.28 (0.01)</td><td>7.12 (0.02)</td><td>6.28 (0.01)</td></tr><tr><td>ZI-m</td><td>5.82 (0.01)</td><td>7.95 (0.01)</td><td>6.82 (0.01)</td></tr><tr><td>PN</td><td>5.24 (0.01)</td><td>7.91 (0.01)</td><td>6.24 (0.01)</td></tr><tr><td>PNP</td><td>5.23 (0.01)</td><td>7.82 (0.01)</td><td>6.23 (0.01)</td></tr></table>
+
+Table 5: Average ranking on AUIC of NHANES   
+
+<table><tr><td>Method</td><td>EDDI</td><td>Random</td><td>Single best</td></tr><tr><td>ZI</td><td>5.68 (0.13)</td><td>8.44 (0.13)</td><td>6.36 (0.13)</td></tr><tr><td>ZI-m</td><td>7.63 (0.12)</td><td>8.69 (0.12)</td><td>8.97 (0.12)</td></tr><tr><td>PN</td><td>5.64 (0.16)</td><td>6.14 (0.15)</td><td>5.56 (0.16)</td></tr><tr><td>PNP</td><td>4.41 (0.12)</td><td>5.34 (0.14)</td><td>5.13 (0.12)</td></tr></table>
+
+Figure 6 shows the information curve of different strategies, using PNP based Partial VAE as an example (more results in Appendix B.3). Table 4 shows the average ranking of AUIC with different settings. In this application, EDDI significantly outperforms other variable selection strategies in all different settings of Partial VAE, and PNP based setting performs best.
+
+# 4.4 PUBLIC HEALTH ASSESSMENT WITH NHANES
+
+Finally, we apply our methods to public health assessment using NHANES 2015-2016 data cdc (2005). NHANES is a program with adaptable components of measurements, to assess the health and nutritional status of adults and children in the United States. Every year, approximately thousands individuals of all ages are interviewed in their homes and complete the health examination component of the survey. This 2015-2016 NHANES data contains three major sections, the questionnaire interview, examinations and lab tests for 9971 subjects in the publicly available version of this cycle. In our setting, we consider the whole set of lab test results (139 dimensions of variables) as the target variable of interest $\mathbf { X } _ { \phi }$ since they are expensive and reflects the subject’s health status, and we active select the questions from the extensive questionnaire (665 variables).
+
+In NHANES, the entire questionnaire is divided into 73 different groups. In practice, questions in the same group are often examined together. Therefore, we perform active variable selection on the group level: at each step, the algorithm will be selecting one group to observe. This is more challenging than the experiments in previous sections since it requires the generative model to simulate a group of unobserved data in Equation (9) at the same time. When evaluating test likelihood on the target variable of interest, we treat variables in each group equally. For a fair comparison, the calculation of the area under the information curve (AUIC) is weighted by the size of the group chosen by the algorithms. Specifically, AUIC is calculated after spline interpolation. The information curve plots in Figure 7, together with Table 5 of AUIC statistics show that our EDDI outperforms other baselines. This experiment shows that EDDI is capable of performing active selection on a large pool of grouped variables to estimate a high dimensional target.
+
+# 5 CONCLUSION
+
+In this paper, we present EDDI, a novel and efficient framework for dynamic active variable selection for each instance. Within the EDDI framework, we propose Partial VAE which performs amortized inference to handle missing data. Partial VAE alone can be used as a non-linear computational efficient probabilistic imputation method. Based on Partial VAE, we design a variable wise acquisition function for EDDI and derived corresponding approximation method. EDDI has demonstrated its effectiveness on active variable selection tasks across multiple real-world applications. In the future, we would extend the EDDI framework to handle more complicated scenarios, such as time-series, or the cold-start situation.
+
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+
+# A ADDITIONAL DERIVATIONS
+
+# A.1 INFORMATION REWARD APPROXIMATION
+
+In our paper, given the VAE model $p ( \mathbf { x } | z )$ and a partial inference network $q ( \mathbf { z } | \mathbf { x } _ { o } )$ , the experimental design problem is formulated as maximization of the information reward:
+
+$$
+R ( i , \mathbf { x } _ { o } ) = \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } [ D _ { K L } ( p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } ) ) ]
+$$
+
+Where $\begin{array} { r } { p ( \mathbf { x } _ { \phi } \vert \mathbf { x } _ { i } , \mathbf { x } _ { o } ) = \int _ { \mathbf { z } } p ( \mathbf { x } _ { \phi } \vert \mathbf { z } ) q ( \mathbf { z } \vert \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } \end{array}$ , $\begin{array} { r } { p ( \mathbf { x } _ { \phi } \vert \mathbf { x } _ { o } ) = \int _ { \mathbf { z } } p ( \mathbf { x } _ { \phi } \vert \mathbf { z } ) q ( \mathbf { z } \vert \mathbf { x } _ { o } ) } \end{array}$ and $q ( \mathbf { z } | \mathbf { x } _ { o } )$ are approximate condition distributions given by partial VAE models. Now we consider the problem of directly approximating $R ( i , { \bf x } _ { o } )$ .
+
+Applying the chain rule of KL-divergence, we have:
+
+$$
+\begin{array} { r l } & { D _ { K L } \big ( p ( \mathbf { x } _ { \phi } \vert \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \big \vert p ( \mathbf { x } _ { \phi } \vert \mathbf { x } _ { o } ) \big ) = D _ { K L } \big ( p ( \mathbf { x } _ { \phi } , \mathbf { z } \vert \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \big \vert p ( \mathbf { x } _ { \phi } , \mathbf { z } \vert \mathbf { x } _ { o } ) \big ) } \\ & { \qquad - \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } \vert \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } \left[ D _ { K L } \big ( p ( \mathbf { z } \vert \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \big \vert p ( \mathbf { z } \vert \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) \big ) \right] , } \end{array}
+$$
+
+Using again the KL-divergence chain rule on $D _ { K L } ( p ( \mathbf { x } _ { \phi } , \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { x } _ { \phi } , \mathbf { z } | \mathbf { x } _ { o } ) )$ , we have:
+
+$$
+\begin{array} { r l } & { D _ { K L } \big ( p ( \mathbf { x } _ { \phi } , \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \vert p ( \mathbf { x } _ { \phi } , \mathbf { z } | \mathbf { x } _ { o } ) \big ) } \\ & { \ = D _ { K L } \big ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \vert p ( \mathbf { z } | \mathbf { x } _ { o } ) \big ) + D _ { K L } \big ( p ( \mathbf { x } _ { \phi } | \mathbf { z } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \vert p ( \mathbf { x } _ { \phi } | \mathbf { z } , \mathbf { x } _ { o } ) \big ) } \\ & { \ = D _ { K L } \big ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \vert p ( \mathbf { z } | \mathbf { x } _ { o } ) \big ) + D _ { K L } \big ( p ( \mathbf { x } _ { \phi } | \mathbf { z } ) \vert \vert p ( \mathbf { x } _ { \phi } | \mathbf { z } ) \big ) } \\ & { \ = D _ { K L } \big ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \vert \vert p ( \mathbf { z } | \mathbf { x } _ { o } ) \big ) . } \end{array}
+$$
+
+The KL-divergence term in the reward formula is now rewritten as follows,
+
+$$
+\begin{array} { r l } & { D _ { K L } ( p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } ) ) = D _ { K L } ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { z } | \mathbf { x } _ { o } ) ) } \\ & { \qquad - \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } \left[ D _ { K L } ( p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) ) \right] . } \end{array}
+$$
+
+One can then plug in the partial VAE inference approximation:
+
+$$
+p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \approx q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) , ~ p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \approx q ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) , ~ p ( \mathbf { z } | \mathbf { x } _ { o } ) \approx q ( \mathbf { z } | \mathbf { x } _ { o } )
+$$
+
+Finally, the information reward is now approximated as:
+
+$$
+\begin{array} { r l } & { R ( i , \mathbf { x } _ { o } ) \approx \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \left[ D _ { K L } ( q ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | q ( \mathbf { z } | \mathbf { x } _ { o } ) ) \right] } \\ & { \qquad - \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } \left[ D _ { K L } ( q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) ) \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \left[ D _ { K L } ( q ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | q ( \mathbf { z } | \mathbf { x } _ { o } ) ) \right] } \\ & { \qquad - \mathbb { E } _ { \mathbf { x } _ { \phi } , \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { \phi } , \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \left[ D _ { K L } ( q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) ) \right] = \hat { R } ( i , \mathbf { x } _ { o } ) . } \end{array}
+$$
+
+This new objective tries to maximize the shift of belief on latent variables $\mathbf { z }$ by introducing $\mathbf { x } _ { i }$ , while penalizing the information that cannot be absorbed by $\mathbf { X } _ { \phi }$ (by the penalty term $D _ { K L } ( q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) ) ;$ ). Moreover, it is more computationally efficient since one set of samples $\mathbf { x } _ { \phi } , \mathbf { x } _ { i } \sim p \big ( \mathbf { x } _ { \phi } , \mathbf { x } _ { i } \big | \mathbf { x } _ { o } \big )$ can be shared across different terms, and the KL-divergence between common parameterizations of encoder (such as Gaussians and normalizing flows) can be computed exactly without the need for approximate integrals. Note also that under approximation
+
+$$
+p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \approx q ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) , ~ p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \approx q ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) , ~ p ( \mathbf { z } | \mathbf { x } _ { o } ) \approx q ( \mathbf { z } | \mathbf { x } _ { o } )
+$$
+
+, sampling $\mathbf { x } _ { i } \sim p \big ( \mathbf { x } _ { i } \big | \mathbf { x } _ { o } \big )$ is approximated by $\mathbf { x } _ { i } \sim \hat { p } ( \mathbf { x } _ { i } | \mathbf { x } _ { o } )$ , where $\hat { p } ( \mathbf x _ { i } | \mathbf x _ { o } )$ is defined by the following process in Partial VAE. It is implemented by first sampling $\mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } _ { o } )$ , and then $\mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { z } )$ . The same applies for $p ( \mathbf { x } _ { i } , \mathbf { x } _ { \phi } | \mathbf { z } )$ .
+
+# B ADDITIONAL EXPERIMENTAL RESULTS
+
+# B.1 IMAGE INPAINTING
+
+# B.1.1 PREPROCESSING AND MODEL DETAILS
+
+For our MNIST experiment, we randomly draw $10 \%$ of the whole data to be our test set. Partial VAE models (ZI, ZI-m, PNP and PNs) share the same size of architecture with 20 dimensional diagonal
+
+Gaussian latent variables: the generator (decoder) is a 20-200-500-500 fully connected neural network with ReLU activations (where D is the data dimension, $D = 7 8 4 _ { . }$ ). The inference nets (encoder) share the same structure of D-500-500-200-40 that maps the observed data into distributional parameters of the latent space. For the PN-based parameterizations, we use a 500 dimensional feature mapping $h$ parameterized by a single layer neural network, and 20 dimensional ID vectors $\mathbf { e } _ { i }$ (see Section 3.2) for each variable. We choose the symmetric operator $g$ to be the basic summation operator.
+
+During training, we apply Adam optimization (Kingma & Ba, 2015) with default hyperparameter setting, learning rate of 0.001 and a batch size of 100. We generate partially observed MNIST dataset by adding artificially missingness at random in the training dataset during training. We first draw a missing rate parameter from a uniform distribution $\mathcal { U } ( 0 , 0 . 7 )$ and randomly choose variables as unobserved. This step is repeated at each iteration. We train our models for 3K iterations.
+
+# B.1.2 IMAGE GENERATION OF PARTIAL VAES
+
+![](images/7a489f4020ba09048d709ab2b18ee1b163644d986fea2f50cae06a7b6f6bee2d.jpg)  
+Figure 8: Random images generated using (a) naive zero imputing, (b) zero imputing with mask, (c) PN and (d) PNP, respectively.
+
+# B.2 UCI DATASETS
+
+We applied EDDI on 6 UCI datasets; Boston Housing, Concrete compressive strength, energy efficiency, wine quality, $\operatorname { K i n } 8 \mathrm { n m }$ , and Yacht Hydrodynamics. The variables of interest $\mathbf { X } _ { \phi }$ are chosen to be the target variables of each UCI dataset in the experiment.
+
+# B.2.1 PREPROCESSING AND MODEL DETAILS
+
+All data are normalized and then scaled between 0 and 1. For each of the 10 - in total- repetitions, we randomly draw $10 \%$ of the data to be our test set. Partial VAE models (ZI, ZI-m, PNP and PNs) share the same size of architecture with 10 dimensional diagonal Gaussian latent variables: the generator (decoder) is a 10-50-100-D neural network with ReLU activations (where D is the data dimensions). The inference nets (encoder) share the same structure D-100-50-20 that maps the observed data into distributional parameters of the latent space. For the PN-based parameterizations, we further use a 20 dimensional feature mapping $h$ parameterized by a single layer neural network and 10 dimensional $\mathrm { I D }$ vectors $\mathbf { e } _ { i }$ (please refer to section 3.2) for each variable. We choose the symmetric operator $g$ to be the basic summation operator.
+
+As in the image inpainting experiment, we apply Adam optimization during training with default hyperparameter setting, and a batch size of 100 and ingest random missingness as before. We trained our models for 3K iterations.
+
+During active learning, we draw 50 samples in order to estimate the expectation under $\mathbf { x } _ { \phi } , \mathbf { x } _ { i } \sim$ $p ( \mathbf { x } _ { \phi } , \mathbf { x } _ { i } | \mathbf { x } _ { o } )$ in Equation (8). Negative likelihoods of the target variable is also estimated using 50 samples of $\mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } )$ through $\begin{array} { r } { p ( \mathbf { x } _ { \phi } \vert \mathbf { x } _ { o } ) \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p ( \mathbf { x } _ { \phi } \vert \mathbf { z } _ { m } ) } \end{array}$ , where ${ \mathbf z } _ { m } \sim q ( { \mathbf z } | { \mathbf x } _ { o } )$ .
+
+# B.2.2 TABLES ON AREA UNDER THE INFORMATION CURVE (AUIC)
+
+In addition to the area under the information curve (AUIC) ranking metric provided in the paper, the average area under the information curve (Avg. AUIC) on each dataset can also be used to compare the performance across models and strategies. AUIC is defined to be $\begin{array} { r } { \sum _ { t } - \log p \big ( \mathbf { x } _ { \phi } \big | \mathbf { x } _ { O _ { t } } \big ) } \end{array}$ , where $\mathbf { x } _ { O _ { t } }$ is the basket of variables observed at step $t$ . By definition, smaller AUIC value (could be positive or negative) indicates better performance. We present the AUIC for each dataset in Table 6, 7, 8, 9, 10, and $1 1 ^ { 2 }$ .
+
+Readers might have found that it seems that the avg. AUIC results in Tables 6 - 11 contradicts the avg. ranking of AUIC results in Table 2 of the main text. However, this is not the case. In Tables Tables 6 - 11, AUIC numbers only provide a simplified statistics of marginal distributions of each method’s performance. Here, the distribution of performance is defined by first sample a data point from the data distribution, and then we obtain the performance of a active learning method of interest by evaluating its performance (AUIC) on this single data point. On the contrary, the average AUIC ranking measure actually takes into account the joint distributions of the performance of all methods, since ranking is a function of the performance of all methods.
+
+With this additional information of correlations, this gives a more accurate evaluation regarding the actual performance of different methods. Notably, in practical scenario of active variable selection, the latter setting is obviously more sensible and fare.
+
+The above conjecture is further validated and confirmed by applying the nonparametric statistical test, namely the Wilcoxcon signed-rank significance test on the performance of different methods, which are detailed in Appendix B.2.3. Wilcoxon test is a very powerful statistical test which includes the information of the joint distribution in paired samples. In our case, the term paired samples refers to the situation that different algorithms are evaluated on exactly the same set of test data points, which introduces correlations between the performances of different algorithms.
+
+Table 6: Average AUIC over Boston Housing dataset   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN(5)</td><td>PN(1))</td></tr><tr><td>EDDI</td><td>-25.03 (0.09)</td><td>-24.74(0.15)</td><td>-24.49 (0.24)</td><td>-24.54 (0.10)</td><td>-24.41 (0.09)</td></tr><tr><td>Random</td><td>-23.85 (0.14)</td><td>-24.52 (0.08)</td><td>-23.36 (0.18 )</td><td>-23.43 (0.14)</td><td>-23.33 (0.13)</td></tr><tr><td>Single best</td><td>-24.77 (0.12)</td><td>-23.62 (0.20)</td><td>-23.71 (0.15)</td><td>-23.82 (0.13)</td><td>-23.87 (0.09)</td></tr></table>
+
+Table 7: Average AUIC over Concrete dataset   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN(5)</td><td>PN(1)</td></tr><tr><td>EDDI</td><td>-12.07 (0.04)</td><td>-12.07 (0.05)</td><td>-12.09 (0.07)</td><td>-12.15 (0.06)</td><td>-12.17 (0.07)</td></tr><tr><td>Random</td><td>-11.00 (0.09)</td><td>-12.03 (0.03)</td><td>-11.17. (0.07)</td><td>-12.07 (0.06)</td><td>-11.12 (0.12)</td></tr><tr><td>Single best</td><td>-12.03 (0.06)</td><td>-11.13 (0.10)</td><td>-12.07 (0.06)</td><td>-12.11 (0.04)</td><td>-12.16 (0.06)</td></tr></table>
+
+Table 8: Average AUIC over Energy dataset   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN(5)</td><td>PN(1)</td></tr><tr><td>EDDI</td><td>-13.63 (0.06)</td><td>-14.56 (0.07)</td><td>-14.49 (0.06)</td><td>-14.65 (0.08)</td><td>-14.68 (0.07)</td></tr><tr><td>Random</td><td>-9.89 (0.15)</td><td>-14.53 (0.06)</td><td>-11.49. (0.16)</td><td>-11.67 (0.17)</td><td>-11.51 (0.16)</td></tr><tr><td>Single best</td><td>-12.79 (0.07)</td><td>-11.62 (0.08)</td><td>-14.36 (0.09)</td><td>-14.56 (0.08 )</td><td>-14.66 (0.07)</td></tr></table>
+
+Table 9: Average AUIC over Wine dataset   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN(5)</td><td>PN(1)</td></tr><tr><td>EDDI</td><td>-14.24 (0.06)</td><td>-15.04 (0.02)</td><td>-15.04 (0.05)</td><td>-15.13 (0.05)</td><td>-15.10 (0.03)</td></tr><tr><td>Random</td><td>-11.07 (0.20)</td><td>-15.10 (0.05)</td><td>-12.38 (0.9)</td><td>-12.57 (0.14)</td><td>-12.26 (0.15)</td></tr><tr><td>Single best</td><td>-13.85 (0.10)</td><td>-12.55 (0.10)</td><td>-15.02 (0.05)</td><td>-14.99 (0.03)</td><td>-15.04 (0.03)</td></tr></table>
+
+2Note that the notation of PN(5) indicates an extension of the PN-Partial VAE method which will be discussed in detail later in the Appendix B.5.
+
+Table 10: Average AUIC over kin8nm dataset   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN(5)</td><td>PN(1)</td></tr><tr><td>EDDI</td><td>-20.31 (0.02)</td><td>-20.25 (0.01)</td><td>-20.18 (0.04)</td><td>-20.20 (0.02)</td><td>-20.15 (0.02)</td></tr><tr><td>Random</td><td>-19.40 (0.04)</td><td>-20.24 (0.02)</td><td>-19.29 (0.04)</td><td>-19.41 (0.02)</td><td>-19.28 (0.05)</td></tr><tr><td>Single best</td><td>-20.28 (0.02)</td><td>-19.35 (0.04)</td><td>-20.23 (0.03)</td><td>-20.19 (0.01)</td><td>-20.19 (0.03)</td></tr></table>
+
+Table 11: Average AUIC over Yacht dataset   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN(5)</td><td>PN(1)</td></tr><tr><td>EDDI</td><td>-14.37 (0.02)</td><td>-14.50 (0.02)</td><td>-14.57 (0.02)</td><td>-14.53 (0.02)</td><td>-14.56 (0.02)</td></tr><tr><td>Random</td><td>-12.83 (0.03)</td><td>-14.50 (0.02)</td><td>-13.03 (0.04)</td><td>-12.93 (0.04)</td><td>-13.08 (0.02)</td></tr><tr><td>Single best</td><td>-14.43 (0.03)</td><td>-12.91 (0.03)</td><td>-14.57 (0.02)</td><td>-14.50 (0.03)</td><td>-14.54(0.02)</td></tr></table>
+
+# B.2.3 STATISTICAL SIGNIFCANT TEST RESULTS
+
+In this section, we perform Wilcoxcon signed-rank significance test on the performance of different methods, to support our result in Table 2. Since Table 2 suggests that EDDI-PNP-Partial VAE is the best algorithm overall, we set EDDI-PNP-Partial VAE as default and perform Wilcoxcon test between EDDI-PNP-Partial VAE and all other 15 different settings, to see whether the improvement is significant. Table 12 displays the corresponding p-value for each test. It is obvious that in all 15 tests, the EDDI-PNP-Partial VAE results are significant (compared with the standard $\alpha = 0 . 0 5$ cutoff). This provides strong evidence that confirms our results in Table 2 and our conjecture in Appendix B.2.2.
+
+Table 12: p- values of Wilcoxon signed-rank test of EDDI-PNP vs. 11 other settings, on 6 UCI datasets.   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN</td></tr><tr><td>EDDI</td><td>&lt;10-48</td><td>&lt;10-23</td><td>N/A</td><td>&lt;10-2</td></tr><tr><td>Random</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Single best</td><td>0</td><td>&lt;10-13</td><td>&lt;10-2</td><td>&lt;10-4</td></tr></table>
+
+B.2.4 ADDITIONAL AUIC PLOTS OF PN, ZI AND ZI-M ON UCI DATASETS
+
+Here we present additional plots of the information curve during active variable selection. Figure 9 presents the results for the Boston Housing, the Energy and the Wine datasets and for the three approaches, i.e. PN, ZI and masked ZI.
+
+![](images/03eab835d358f5af91218a92e26152ccd6468234eea532bc4757834252f41680.jpg)  
+Figure 9: Information curves of active variable selection for the three UCI datasets and the three approaches, i.e. (First row) PointNet (PN), (Second row) Zero Imputing (ZI), and (Third row) Zero Imputing with mask $\left( \mathrm { Z I - m } \right)$ . Green: random strategy; Black: EDDI; Pink: Single best ordering. This displays negative test log likelihood (y axis, the lower the better) during the course of active selection $\mathbf { \bar { X } } ^ { \prime }$ -axis).
+
+B.2.5 RMSE PLOTS OF PN, ZI AND ZI-M ON UCI DATASETS
+
+Here we present additional plots of the RMSE curves during active learning. Figure 10 presents the results for the Boston Housing, the Energy and the Wine datasets and for the three approaches, i.e. PN, ZI and masked ZI.
+
+![](images/b9ba75a6f254d2424e0ae6bc0f7d6a6b42b1b86c5e6bb9924699ca552e54abf8.jpg)  
+Figure 10: RMSE curves of active variable selection for the three UCI datasets and the three approaches, i.e. (First row) PointNet (PN), (Second row) Zero Imputing (ZI), and (Third row) Zero Imputing with mask (ZI-m). Green: random strategy; Black: EDDI; Pink: Single best ordering. This displays RMSE (y axis, the lower the better) during the course of active selection $\mathbf { \bar { x } }$ -axis).
+
+# B.2.6 COMPARISONS BETWEEN EDDI AND LASSO-BASED METHOD
+
+Here we present additional results of a new baseline, the LASSO-based feature selection. This is not presented in the main text since LASSO is designed for a different problem setting. It requires fully observed data, and only works in regression problems with one dimensional outputs. Both MIMIC III and NHANES tasks do not fulfill these requirements. Additionally, LASSO aims to select a global set of features to obtain the best performance instead of select the most informative feature given partially observed information, thus cannot be used in a sequential setting. We thus construct the LASSO feature selection baseline as follows for comparison: we first apply LASSO regression on training dataset which is fully observed in these UCI datasets, and select the features (denoted by $\mathcal { A }$ ) that correspond to non-zero coefficients. Then, during test time, LASSO strategy will observe the features one by one from $\mathcal { A }$ randomly. When all variables selected by LASSO are already picked, we stop the feature selection progress. Since LASSO does not support evaluation of model likelihood as well as it is linear, we use the corresponding partial-VAE (ZI,ZI-m,PNP,PN) to make predictions and evaluate the model log likelihood.
+
+Figure 11 presents the results for the Boston Housing, the Energy and the Wine datasets as examples. Full results of all UCI datasets are presented in Table 13. Note that in Table 13, Wilcoxon signed-rank test is performed between EDDI and LASSO strategies for each Partial VAE models, respectively. The results indicates that EDDI significantly outperforms LASSO in all circumstances. This is despite the fact that EDDI is a greedy sequential variable selection method that built upon partially observed data, while LASSO-baseline makes use of the information from fully observed data, and selects the set of variables in a non-greedy, global manner, which is often unrealistic in many pratical application settings.
+
+![](images/fd1d1d9fbef4ee071d17a6502bda278ed25251be020e2447e588865b9bf66804.jpg)  
+Figure 11: Information curves of active variable selection for the three UCI datasets and PNP-Partial VAE. Black: EDDI; Blue: Single best ordering. This displays negative test log likelihood (y axis, the lower the better) during the course of active selection ( $\mathbf { \bar { X } }$ -axis).
+
+Table 13: Avg. rankings of AUIC, and p- values of Wilcoxon signed-rank test that EDDI outperforms LASSO (on 6 UCI datasets).   
+B.2.7 ILLUSTRATION OF DECISION PROCESS OF EDDI (BOSTON HOUSING AS EXAMPLE)   
+
+<table><tr><td>Method</td><td>ZI</td><td>ZI-m</td><td>PNP</td><td>PN</td></tr><tr><td>EDDI</td><td>4.66 (0.02)</td><td>4.53(0.02)</td><td>4.14(0.02)</td><td>4.24(0.02)</td></tr><tr><td>LASSO</td><td>4.86(0.02)</td><td>4.63(0.02)</td><td>4.41(0.02)</td><td>4.48(0.02)</td></tr><tr><td>p-value</td><td>&lt;10-4</td><td>&lt;10-6</td><td>&lt;10-24</td><td>&lt;10-19</td></tr></table>
+
+The decision process facilitated by the active selection of the variables (for the EDDI framework) is efficiently illustrated in Figure 12 and Figure 13 for the Boston Housing dataset and for the PNP and PNP with single best ordering approaches, respectively.
+
+For completeness, we provide details regarding the abbreviations of the variables used in the Boston dataset and appear both figures.
+
+CR - per capita crime rate by town   
+PRD - proportion of residential land zoned for lots over 25,000 sq.ft. PNB - proportion of non-retail business acres per town.   
+CHR - Charles River dummy variable (1 if tract bounds river; 0 otherwise) NOC - nitric oxides concentration (parts per 10 million)   
+ANR - average number of rooms per dwelling   
+AOUB - proportion of owner-occupied units built prior to 1940   
+DTB - weighted distances to five Boston employment centres   
+ARH - index of accessibility to radial highways   
+TAX - full-value property-tax rate per $\$ 10,000$   
+OTR - pupil-teacher ratio by town   
+PB - proportion of blacks by town   
+LSP - $\%$ lower status of the population
+
+# B.3 MIMIC-III
+
+Here we provide additional results of our approach on the MIMIC-III dataset.
+
+![](images/ca938214181ebb10bae4096c44518c8b04c0d7b5b27de7c607eef648af1cbd27.jpg)  
+Figure 12: Information reward estimated during the first 4 active variable selection steps on a randomly chosen Boston Housing test data point. Model: PNP, strategy: EDDI. Each row contains two plots regarding the same time step. Bar plots on the left show the information reward estimation of each variable on the y-axis. All unobserved variables start with green bars, and turns purple once selected by the algorithm. Right: violin plot of the posterior density estimations of remaining unobserved variables.
+
+# B.3.1 PREPROCESSING AND MODEL DETAILS
+
+For our active learning experiments on MIMIC III datasets, we chose the variable of interest $\mathbf { X } _ { \phi }$ to be the binary mortality indicator of the dataset. All data (except the binary mortality indicator)
+
+![](images/0abd3c407775fec81b7c3751ebb81065a7370abd6d3140198ec2e0023fc53e73.jpg)  
+Figure 13: Information reward estimated during the first 4 active variable selection steps on a randomly chosen Boston Housing test data point. Models: PNP, strategy: single ordering. Each row contains two plots regarding the same time step. Bar plots on the left show the information reward estimation of each variable on the y-axis. All unobserved variables start with green bars, and turns purple once selected by the algorithm. Right: violin plot of the posterior density estimations of remaining unobserved variables.
+
+are normalized and then scaled between 0 and 1. We transformed the categorical variables into real-valued using the dictionary deduced from (Johnson et al., 2016) that makes use of the actual medical implications of each possible values. The binary mortality indicator are treated as Bernoulli variables and Bernoulli likelihood function is applied. For each repetition (of the 5 in total), we randomly draw $10 \%$ of the whole data to be our test set. Partial VAE models (ZI, ZI-m, PNP and PNs) share the same size of architecture with 10 dimensional diagonal Gaussian latent variables: the generator (decoder) is a 10-50-100-D neural network with ReLU activations (where D is the data dimensions). The inference nets (encoder) share the same structure of D-100-50-20 that maps the observed data into distributional parameters of the latent space. Additionally, for PN-based parameterizations, we further use a 20 dimensional feature mapping $h$ parameterized by a single layer neural network, and 10 dimensional ID vectors $\mathbf { e } _ { i }$ (please refer to section 3.2) for each variable. We choose the symmetric operator $g$ to be the basic summation operator.
+
+Adam optimization and random missingness is applied as in the previous experiments. We trained our models for 3K iterations. During active learning, we draw 50 samples in order to estimate the expectation under $\mathbf { x } _ { \phi } , \mathbf { x } _ { i } \sim p \big ( \mathbf { x } _ { \phi } , \mathbf { x } _ { i } \big | \mathbf { x } _ { o } \big )$ in Equation (8). Negative likelihoods of the target variable is also estimated using 50 samples of $\mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } )$ through $\begin{array} { r } { p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } ) \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p ( \mathbf { x } _ { \phi } | \mathbf { z } _ { m } ) } \end{array}$ , where ${ \mathbf z } _ { m } \sim q ( { \mathbf z } | { \mathbf x } _ { o } )$ .
+
+# B.3.2 ADDITIONAL PLOTS OF ZI, PN AND ZI-M ON MIMIC III
+
+Figure 14 shows the information curves of active variable selection on the risk assessment task for MIMIC-III as produced by the three approaches, i.e. ZI, PN and masked ZI.
+
+![](images/30ef9695d43a82728f645f455ed2121bf2b7e3c76647b29823a27d310850aa28.jpg)  
+Figure 14: Information curves of active variable selection on risk assessment task on MIMIC III, produced from: (a) Zero Imputing (ZI), (b) PointNet (PN) and (c) Zero Imputing with mask $\left( \mathrm { Z I - m } \right)$ . Green: random strategy; Black: EDDI; Pink: Single best ordering. This displays negative test log likelihood (y axis, the lower the better) during the course of active selection ( $\mathbf { \bar { x } }$ -axis)
+
+# B.4 NHANES
+
+# B.4.1 PREPROCESSING AND MODEL DETAILS
+
+For our active learning experiments on NHANES datasets, we chose the variable of interest $\mathbf { X } _ { \phi }$ to be the lab test result section of the dataset. All data are normalized and scaled between 0 and 1. For categorical variables, these are transformed into real-valued variables using the code that comes with the dataset, which makes use of the actual ordering of variables in questionnaire. Then, for each repetition (of the 5 repetitions in total), we randomly draw 8000 data as training set and 100 data to be test set. All partial VAE models (ZI, ZI-m, PNP and PNs) uses gaussian likelihoods, with an diagonal Gaussian inference model (encoder). Partial VAE models share the same size of architecture with 20 dimensional diagonal Gaussian latent variables: the generator (decoder) is a 20-50-100-D neural network. The inference nets (encoder) share the same structure of D-100-50-20 that maps the observed data into distributional parameters of the latent space. Additionally, for PN-based parameterizations, we further use a 20 dimensional feature mapping $h$ parameterized by a single layer neural network, and 100 dimensional ID vectors $\mathbf { e } _ { i }$ (please refer to section 3.2) for each variable. We choose the symmetric operator $g$ to be the basic summation operator.
+
+Adam optimization and random missingness is applied as in the previous experiments. We trained all models 1K iterations. During active learning, 10 samples were drawn to estimate the expectation in Equation (9). Negative likelihoods of the target variable is also estimated using 10 samples.
+
+![](images/b03d5dca1eebe084da70dd61d2f397758e2e80fb29aa2a39fdeaa189618bebdc.jpg)  
+Figure 15: Illustration of recurrent PN architecture. We show the example using 2 recurrent steps. The output $c _ { ( 2 ) }$ is directly connected to the rest of the inference network in this case. One can use more steps. To form the input for the $i + 1$ recurrent step, we concatenate the $c _ { ( i ) }$ to the input.
+
+![](images/6477d70f4fc983f1c355477a1d09bc6edcd88fda2295bed43fd851dd1d87d005.jpg)  
+Figure 16: Negative test log likelihoods of pilot runs for (recurrent) PN-based methods on MNIST dataset. we perform plot runs of PN1, PN2, PN5, PNP1, PNP1, PNP2, PNP5 (PN- $\mathbf { \nabla } \cdot \mathbf { X }$ stands for $\mathbf { X }$ recurrent steps of PN) on MNIST dataset for 300 iterations.All curves has been smoothed for clear comparison.
+
+B.5 PN/PNP MODEL STRUCTURE DETERMINATION: SHOULD WE USE RECURRENT EXTENSIONS
+
+One straightforward extention of PN/PNP Partial VAEs proposed in this paper is, to generalize PN and PNP by recurrently reuse the code $c$ to enlarge the capacity of PN:
+
+Figure 15 shows the mechanism of the recurrent PN with two recurrent steps using concatenated $\mathbf { s } _ { ( 1 ) d } = [ \mathbf { e } _ { d } , \mathbf { x } _ { d } ]$ as an example. The first step is the same as the PN setting with the $K$ dimensional output $c _ { ( 1 ) }$ , where $( 1 )$ is the recurrent step index. For the second step, we concatenate the learned $c _ { ( 1 ) }$ to $\mathbf { s _ { ( 1 ) } }$ to form the new input for the next recurrent step $\mathbf s _ { ( 2 ) d } = [ \mathbf s _ { ( 1 ) d } , c _ { ( 1 ) } ]$ . There can be arbitrary number of recurrent steps using the input ${ \bf s } _ { ( n ) d } = [ { \bf s } _ { ( n - 1 ) d } , c _ { ( n - 1 ) } ]$ . Within each recurrent step, the parameters for the neural network $h _ { ( n ) }$ are shared, however, different steps have different parameters. When $n = 1$ , we recover the original PN partial VAE setting.
+
+The question is, should we include the recurrent structure in our Partial VAE? In this section, we present preliminary result for this purpose. We perform plot runs of PN1, PN2, PN5, PNP1, PNP1, PNP2, PNP5 (PN- $\mathbf { \nabla } \cdot \mathbf { X }$ stands for PN model with x recurrent steps) on MNIST dataset for 300 iterations. Other model settings are consistent with Section B.1.1. Results of negative test log likelihoods are shown in Figure 16: Based on Figure 16, the conclusion is: based on MNIST dataset along, by increasing the recurrent steps of PN, the performance roughly increases slightly. Meanwhile, we can observe that increasing the recurrent steps does not improve PNP. However, the difference is not significant.
+
+In section B.2.2, a comparison between recurrent PN (PN5) and vanilla PN (PN1) is considered for each UCI dataset, see Table 6, 7, 8, 9, 10, and 11. Additionally, the average ranking of AUIC of PN5 and PN1 is summarized in Table 14:
+
+Table 14: Average Ranking of AUIC between PN5-Partial VAE and PN1-Partial VAE   
+
+<table><tr><td>Method</td><td>PN(5)</td><td>PN(1)</td></tr><tr><td>EDDI</td><td>2.83 (0.01)</td><td>2.78 (0.01)</td></tr><tr><td>Random</td><td>4.12 (0.01)</td><td>4.07 (0.01)</td></tr><tr><td>Single best</td><td>4.37 (0.01)</td><td>2.80 (0.01)</td></tr></table>
+
+It is clear that one can not conclude that on average, PN5 significantly outperforms PN1. Therefore, as far as active learning tasks under the settings in our experiments are considered, we believe the recurrent generalization of PNs will not be a crucial factor for boosting a performance.
+
+# C ADDITIONAL THEORETICAL CONTRIBUTIONS
+
+# C.1 ZERO IMPUTING AS A POINT NET
+
+Here we present how the zero imputing (ZI) and PointNet (PN) approaches relate.
+
+Zero imputation with inference net In ZI, the natural parameter of $\lambda$ (e.g., Gaussian parameters in variational autoencoders) is approximated using the following neural network:
+
+$$
+f ( \mathbf { x } ) : = \sum _ { l = 1 } ^ { L } w _ { l } ^ { ( 1 ) } \sigma ( \mathbf { w } _ { l } ^ { ( 0 ) } \mathbf { x } ^ { T } )
+$$
+
+where $L$ is the number of hidden units, $\mathbf { X }$ is the input image with $x _ { i }$ be the value of the $i ^ { t h }$ pixel. To deal with partially observed data $\mathbf { x } = \mathbf { x } _ { o } \cup \mathbf { x } _ { u }$ , $\mathrm { Z I }$ simply sets all $\mathbf { X } _ { u }$ to zero, and use the full inference model $f ( \mathbf { x } )$ to perform approximate inference.
+
+PointNet parameterizationThe PN approach approximates the natural parameter $\lambda$ by a permutation invariant set function
+
+$$
+g ( h ( \mathbf { s } _ { 1 } ) , h ( \mathbf { s } _ { 2 } ) , . . . , h ( \mathbf { s } _ { O } ) ) ,
+$$
+
+where $\mathbf { s } _ { i } = \left[ x _ { i } , \mathbf { e } _ { i } \right]$ , $\mathbf { e } _ { i }$ is the $I$ dimensional embedding/ID/location vector of the $i ^ { t h }$ pixel, $g ( \cdot )$ is a symmetric operation such as max-pooling and summation, and $h ( \cdot )$ is a nonlinear feature mapping from $\mathbb { R } ^ { I + 1 }$ to $\mathbb { R } ^ { K }$ (we will always refer $h$ as feature maps ). In the current version of the partial-VAE implementation, where Gaussian approximation is used, we set $K = 2 H$ with $H$ being the dimension of latent variables. We set $g$ to be the element-wise summation operator, i.e. a mapping from $\mathbb { R } ^ { K O }$ to $\mathbb { R } ^ { K }$ defined by:
+
+$$
+g ( h ( \mathbf { s } _ { 1 } ) , h ( \mathbf { s } _ { 2 } ) , . . . , h ( \mathbf { s } _ { O } ) ) = \sum _ { i \in O } h ( \mathbf { s } _ { i } ) .
+$$
+
+This parameterization corresponds to products of multiple Exp-Fam factors $\begin{array} { r } { \prod _ { i \in O } \exp \{ - \langle h ( \mathbf { s } _ { i } ) , \Phi \rangle \} } \end{array}$
+
+From PN to ZI To derive the PN correspondence of the above ZI network we define the following PN functions:
+
+$$
+h ( \mathbf { s } _ { i } ) : = \mathbf { e } _ { i } * x _ { i }
+$$
+
+$$
+g ( h ( \mathbf { s } _ { 1 } ) , h ( \mathbf { s } _ { 2 } ) , . . . , h ( \mathbf { s } _ { O } ) ) : = \sum _ { k = 1 } ^ { I } \theta _ { k } \sigma ( \sum _ { i \in O } h _ { k } ( \mathbf { s } _ { i } ) ) ,
+$$
+
+where $h _ { k } ( \cdot )$ is the $k ^ { t h }$ output feature of $h ( \cdot )$ . The above PN parameterization is also permutation invariant; setting $L = I$ , $\mathbf { \theta } _ { l } = w _ { l } ^ { ( 1 ) } , ( \mathbf { w } _ { l } ^ { ( 0 ) } ) _ { i } = ( \mathbf { e } _ { i } ) _ { l }$ the resulting PN model is equivalent to the ZI neural network.
+
+Generalizing ZI from PN perspective In the ZI approach, the missing values are replaced with zeros. However, this ad-hoc approach does not distinguish missing values from actual observed zero values. In practice, being able to distinguish between these two is crucial for improving uncertainty estimation during partial inference. One the other hand, we have found that PN-based partial VAE experiences difficulties in training. To alleviate both issues, we proposed a generalization of the ZI approach that follows a PN perspective. One of the advantages of PN is setting the feature maps of the unobserved variables to zero instead of the related weights. As discussed before, these two approaches are equivalent to each other only if the factors are linear. More generally, we can parameterize the PN by:
+
+$$
+\begin{array} { c } { { h ^ { ( 1 ) } ( { \bf s } _ { i } ) : = { \bf e } _ { i } \ast x _ { i } } } \\ { { h ^ { ( 2 ) } ( h _ { i } ^ { ( 1 ) } ) : = N N _ { 1 } ( h _ { i } ^ { ( 1 ) } ) } } \\ { { g ( h ( { \bf s } _ { 1 } ) , h ( { \bf s } _ { 2 } ) , . . . , h ( { \bf s } _ { O } ) ) : = N N _ { 2 } ( \sigma ( \underset { i \in O } { \sum } h _ { k } ^ { ( 2 ) } ( h _ { i } ^ { ( 1 ) } ) ) ) , } } \end{array}
+$$
+
+where $N N _ { 1 }$ is a mapping from $\mathbb { R } ^ { I }$ to $\mathbb { R } ^ { K }$ defined by a neural network, and $N N _ { 2 }$ is a mapping from $\mathbb { R } ^ { K }$ to $\mathbb { R } ^ { 2 H }$ defined by another neural network.
+
+# C.2 APPROXIMATION DIFFICULTY OF THE ACQUISITION FUNCTION
+
+Traditional variational approximation approaches provide wrong approximation direction when applied in this case (resulting in an upper bound of the objective $R _ { \phi } ( i , { \bf x } _ { O } )$ which we maximize). Justification issues aside, (black box) variational approximation requires sampling from approximate posterior $q ( { \bf z } | { \bf x } _ { O } )$ , which leads to extra uncertainties and computations. For common proposals of approximation:
+
+• Directly estimate entropy via sampling $\Rightarrow$ problematic for high dimensional target variables   
+• Using reversed information reward $\begin{array} { r l r } { \mathbb { E } _ { { \mathbf { x } } _ { i } \sim p ( { \mathbf { x } } _ { i } \mid { \mathbf { x } } _ { o } ) } [ D _ { K L } ( \mathrm { ~  ~ \lambda ~ } } & { { } } & { \mid \mid p ( { \mathbf { x } } _ { \phi } \mid { \mathbf { x } } _ { o } , { \mathbf { x } } _ { i } ) ) ] } \end{array}$ , and then apply ELBO (KL-divergence) $\Rightarrow$ This does not make sense mathematically, since this will result in upper bound approximation of the (reversed) information objective, this is in the wrong direction.   
+• Ranganath’s bound (Ranganath et al., 2016) on estimating entropy $\Rightarrow$ gives upper bound of the objective, wrong direction.   
+• All the above methods also needs samples from latent space (therefore second level approximation needed).
+
+# C.3 CONNECTION OF EDDI INFORMATION REWARD WITH BALD
+
+We briefly discuss connection of EDDI information reward with BALD (Houlsby et al., 2011) and. MacKay’s work (MacKay, 1992). Assuming the model is correct, i.e. $q = p$ , we have
+
+$$
+\begin{array} { r l } & { R ( i , \mathbf { x } _ { o } ) = \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \left[ D _ { K L } ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { z } | \mathbf { x } _ { o } ) ) \right] } \\ & { \qquad - \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } \left[ D _ { K L } ( p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) | | p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) ) \right] . } \end{array}
+$$
+
+Note that based on McKay’s relationship between entropy and KL-divergence reduction, we have:
+
+$$
+\begin{array} { r l } & { \quad { \mathbb { E } } _ { { \mathbf { x } } _ { i } \sim p ( { \mathbf { x } } _ { i } \mid { \mathbf { x } } _ { o } ) } [ D _ { K L } ( p ( { \mathbf { z } } | { \mathbf { x } } _ { i } , { \mathbf { x } } _ { o } ) | | p ( { \mathbf { z } } | { \mathbf { x } } _ { o } ) ) ] } \\ & { = { \mathbb { E } } _ { { \mathbf { x } } _ { i } \sim p ( { \mathbf { x } } _ { i } \mid { \mathbf { x } } _ { o } ) } [ H ( p ( { \mathbf { z } } | { \mathbf { x } } _ { i } , { \mathbf { x } } _ { o } ) ) - H ( p ( { \mathbf { z } } | { \mathbf { x } } _ { o } ) ) ] ] . } \end{array}
+$$
+
+Similarly, we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { { \mathbf { x } _ { i } } \sim p ( { \mathbf { x } _ { i } } | { \mathbf { x } _ { o } } ) } \mathbb { E } _ { { \mathbf { x } _ { \phi } } \sim p ( { \mathbf { x } _ { \phi } } | { \mathbf { x } _ { i } } , { \mathbf { x } _ { o } } ) } [ D _ { K L } ( p ( { \mathbf { z } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { i } } , { \mathbf { x } _ { o } } ) | | p ( { \mathbf { z } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { o } } ) ) ] } \\ & { = \mathbb { E } _ { { \mathbf { x } _ { \phi } } \sim p ( { \mathbf { x } _ { \phi } } | { \mathbf { x } _ { o } } ) } \mathbb { E } _ { { \mathbf { x } _ { i } } \sim p ( { \mathbf { x } _ { i } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { o } } ) } [ D _ { K L } ( p ( { \mathbf { z } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { i } } , { \mathbf { x } _ { o } } ) | | p ( { \mathbf { z } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { o } } ) ) ] } \\ &  = \mathbb { E } _ { { \mathbf { x } _ { \phi } \sim p ( { \mathbf { x } _ { \phi } } | { \mathbf { x } _ { o } } ) } } \mathbb { E } _ { { \mathbf { x } _ { i } \sim p ( { \mathbf { x } _ { i } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { o } } ) } [ H ( p ( { \mathbf { z } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { i } } , { \mathbf { x } _ { o } } ) ) - H ( p ( { \mathbf { z } } | { \mathbf { x } _ { \phi } } , { \mathbf { x } _ { o } } ) ) ] } \\ &  = \mathbb { E } _  { \mathbf { x } _ { i } \sim p ( { \mathbf { x } _ { i } } | { \mathbf { x } _ { o } } ) } \mathbb { E } _  { \mathbf { x } _ { \phi } } \sim p ( { \mathbf { x } _ { \phi } } | { \mathbf { x } _ { i } } , { \mathbf { x } _ { o } } ) [ H ( p ( \mathbf { z } | { \mathbf { x } _ { \phi } } ,  \mathbf { x } \end{array}
+$$
+
+where MacKay’s result is applied to $\mathbb { E } _ { { \mathbf { x } } _ { i } \sim p ( { \mathbf { x } } _ { i } \mid { \mathbf { x } } _ { \phi } , { \mathbf { x } } _ { o } ) } \left[ D _ { K L } \big ( p ( { \mathbf { z } } | { \mathbf { x } } _ { \phi } , { \mathbf { x } } _ { i } , { \mathbf { x } } _ { o } ) \big | \big | p ( { \mathbf { z } } | { \mathbf { x } } _ { \phi } , { \mathbf { x } } _ { o } ) \big ) \right] .$ Putting everything together, we have
+
+$$
+\begin{array} { r l } & { R ( i , \mathbf { x } _ { o } ) = \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) ) - H ( p ( \mathbf { z } | \mathbf { x } _ { o } ) ) ] ] } \\ & { \quad \quad - \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) ) ] + \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) ) ] } \\ & { \quad \quad = \{ \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) ) ] - \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) ) ] \} } \\ & { \quad \quad - \{ \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { o } ) ) ] - \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { o } ) ) ] \} . } \end{array}
+$$
+
+We can show that
+
+$$
+\begin{array} { r l } & { \quad H ( p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) ) - \mathbb { E } _ { \mathbf { x } _ { \phi } \sim p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } [ H ( p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) ) ] } \\ & { = - \displaystyle \int _ { \mathbf { z } } p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \log p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) d \mathbf { z } + \displaystyle \int _ { \mathbf { z } , \mathbf { x } _ { \phi } } p ( \mathbf { z } , \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \log p ( \mathbf { z } | \mathbf { x } _ { \phi } , \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } \\ & { = \displaystyle \int _ { \mathbf { z } , \mathbf { x } _ { \phi } } p ( \mathbf { z } , \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) \log \frac { p ( \mathbf { z } , \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } { p ( \mathbf { z } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) p ( \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ) } } \\ & { = \mathcal { S } [ \mathbf { z } , \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } ] , } \end{array}
+$$
+
+which is exactly the conditional mutual information $\mathcal { I } \left[ \mathbf { z } , \mathbf { x } _ { \phi } \vert \mathbf { x } _ { i } , \mathbf { x } _ { o } \right]$ used in BALD. Therefore, our chain rule representation of reward function leads us to
+
+$$
+R ( i , \mathbf { x } _ { o } ) = \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \mathcal { I } \left[ \mathbf { z } , \mathbf { x } _ { \phi } | \mathbf { x } _ { i } , \mathbf { x } _ { o } \right] - \mathbb { E } _ { \mathbf { x } _ { i } \sim p ( \mathbf { x } _ { i } | \mathbf { x } _ { o } ) } \mathcal { I } \left[ \mathbf { z } , \mathbf { x } _ { \phi } | \mathbf { x } _ { o } \right] .
+$$

parse/train/HygBZnRctX/HygBZnRctX.md

@@ -0,0 +1,472 @@
+# TRANSFERRING KNOWLEDGE ACROSS LEARNING PROCESSES
+
+Sebastian Flennerhag The Alan Turing Institute London, UK sflennerhag@turing.ac.uk
+
+Pablo G. Moreno   
+Amazon   
+Cambridge, UK   
+morepabl@amazon.com
+
+# Andreas Damianou
+
+Neil D. Lawrence   
+Amazon   
+Cambridge, UK   
+lawrennd@amazon.com   
+Amazon   
+Cambridge, UK   
+damianou@amazon.com
+
+# ABSTRACT
+
+In complex transfer learning scenarios new tasks might not be tightly linked to previous tasks. Approaches that transfer information contained only in the final parameters of a source model will therefore struggle. Instead, transfer learning at a higher level of abstraction is needed. We propose Leap, a framework that achieves this by transferring knowledge across learning processes. We associate each task with a manifold on which the training process travels from initialization to final parameters and construct a meta-learning objective that minimizes the expected length of this path. Our framework leverages only information obtained during training and can be computed on the fly at negligible cost. We demonstrate that our framework outperforms competing methods, both in meta-learning and transfer learning, on a set of computer vision tasks. Finally, we demonstrate that Leap can transfer knowledge across learning processes in demanding reinforcement learning environments (Atari) that involve millions of gradient steps.
+
+# 1 INTRODUCTION
+
+Transfer learning is the process of transferring knowledge encoded in one model trained on one set of tasks to another model that is applied to a new task. Since a trained model encodes information in its learned parameters, transfer learning typically transfers knowledge by encouraging the target model’s parameters to resemble those of a previous (set of) model(s) (Pan & Yang, 2009). This approach limits transfer learning to settings where good parameters for a new task can be found in the neighborhood of parameters that were learned from a previous task. For this to be a viable assumption, the two tasks must have a high degree of structural affinity, such as when a new task can be learned by extracting features from a pretrained model (Girshick et al., 2014; He et al., 2017; Mahajan et al., 2018). If not, this approach has been observed to limit knowledge transfer since the training process on one task will discard information that was irrelevant for the task at hand, but that would be relevant for another task (Higgins et al., 2017; Achille et al., 2018).
+
+We argue that such information can be harnessed, even when the downstream task is unknown, by transferring knowledge of the learning process itself. In particular, we propose a meta-learning framework for aggregating information across task geometries as they are observed during training. These geometries, formalized as the loss surface, encode all information seen during training and thus avoid catastrophic information loss. Moreover, by transferring knowledge across learning processes, information from previous tasks is distilled to explicitly facilitate the learning of new tasks.
+
+Meta learning frames the learning of a new task as a learning problem itself, typically in the few-shot learning paradigm (Lake et al., 2011; Santoro et al., 2016; Vinyals et al., 2016). In this environment, learning is a problem of rapid adaptation and can be solved by training a meta-learner by backpropagating through the entire training process (Ravi & Larochelle, 2016; Andrychowicz et al., 2016; Finn et al., 2017). For more demanding tasks, meta-learning in this manner is challenging; backpropagating through thousands of gradient steps is both impractical and susceptible to instability. On the other hand, truncating backpropagation to a few initial steps induces a short-horizon bias (Wu et al., 2018). We argue that as the training process grows longer in terms of the distance traversed on the loss landscape, the geometry of this landscape grows increasingly important. When adapting to a new task through a single or a handful of gradient steps, the geometry can largely be ignored. In contrast, with more gradient steps, it is the dominant feature of the training process.
+
+To scale meta-learning beyond few-shot learning, we propose Leap, a light-weight framework for meta-learning over task manifolds that does not need any forward- or backward-passes beyond those already performed by the underlying training process. We demonstrate empirically that Leap is a superior method to similar meta and transfer learning methods when learning a task requires more than a handful of training steps. Finally, we evaluate Leap in a reinforcement Learning environment (Atari 2600; Bellemare et al., 2013), demonstrating that it can transfer knowledge across learning processes that require millions of gradient steps to converge.
+
+# 2 TRANSFERRING KNOWLEDGE ACROSS LEARNING PROCESSES
+
+We start in section 2.1 by introducing the gradient descent algorithm from a geometric perspective. Section 2.2 builds a framework for transfer learning and explains how we can leverage geometrical quantities to transfer knowledge across learning processes by guiding gradient descent. We focus on the point of initialization for simplicity, but our framework can readily be extended. Section 2.3 presents Leap, our lightweight algorithm for transfer learning across learning processes.
+
+# 2.1 GRADIENT PATHS ON TASK MANIFOLDS
+
+Central to our framework is the notion of a learning process; the harder a task is to learn, the harder it is for the learning process to navigate on the loss surface (fig. 1). Our framework is based on the idea that transfer learning can be achieved by leveraging information contained in similar learning processes. Exploiting that this information is encoded in the geometry of the loss surface, we leverage geometrical quantities to facilitate the learning process with respect to new tasks. We focus on the supervised learning setting for simplicity, though our framework applies more generally. Given a learning objective $f$ that consumes an input $\boldsymbol { x } \in \mathbb { R } ^ { m }$ and a target $\boldsymbol { y } \in \mathbb { R } ^ { c }$ and maps a parameterization $\boldsymbol \theta \in \mathbb { R } ^ { n }$ to a scalar loss value, we have the gradient descent update as
+
+$$
+\theta ^ { i + 1 } = \theta ^ { i } - \alpha ^ { i } S ^ { i } \nabla f ( \theta ^ { i } ) ,
+$$
+
+where $\nabla f ( \theta ^ { i } ) = \mathbb { E } _ { x , y \sim p ( x , y ) } \big [ \nabla f ( \theta ^ { i } , x , y ) \big ]$ . We take the learning rate schedule $\{ \alpha ^ { i } \} _ { i }$ and preconditioning matrices $\{ S ^ { i } \} _ { i }$ as given, but our framework can be extended to learn these jointly with the initialization. Different schemes represent different optimizers; for instance $\alpha ^ { i } = \alpha , \ S ^ { i } = I _ { n }$ yields gradient descent, while defining ${ \dot { S } } ^ { i }$ as the inverse Fisher matrix results in natural gradient descent (Amari, 1998). We assume this process converges to a stationary point after $K$ gradient steps.
+
+To distinguish different learning processes originating from the same initialization, we need a notion of their length. The longer the process, the worse the initialization is (conditional on reaching equivalent performance, discussed further below). Measuring the Euclidean distance between initialization and final parameters is misleading as it ignores the actual path taken. This becomes crucial when we compare paths from different tasks, as gradient paths from different tasks can originate from the same initialization and converge to similar final parameters, but take very different paths. Therefore, to capture the length of a learning process we must associate it with the loss surface it traversed.
+
+The process of learning a task can be seen as a curve on a specific task manifold $M$ . While this manifold can be constructed in a variety of ways, here we exploit that, by definition, any learning process traverses the loss surface of $f$ . As such, to accurately describe the length of a gradient-based learning process, it is sufficient to define the task manifold as the loss surface. In particular, because the learning process in eq. 1 follows the gradient trajectory, it constantly provides information about the geometry of the loss surface. Gradients that largely point in the same direction indicate a well-behaved loss surface, whereas gradients with frequently opposing directions indicate an ill-conditioned loss surface—something we would like to avoid. Leveraging this insight, we propose a framework for transfer learning that exploits the accumulation of geometric information by constructing a meta objective that minimizes the expected length of the gradient descent path across tasks. In doing so, the meta objective intrinsically balances local geometries across tasks and encourages an initialization that makes the learning process as short as possible.
+
+![](images/973dacc1e637349765389378d72f79790521d51d3e70bc03742c959d6e0f5db6.jpg)  
+Figure 1: Example of gradient paths on a manifold described by the loss surface. Leap learns an initialization with shorter expected gradient path that improves performance.
+
+To formalize the notion of the distance of a learning process, we define a task manifold $M$ as a submanifold of $\mathbb { R } ^ { n + 1 }$ given by the graph of $f$ . Every point $p ~ = ~ ( \theta , f ( \theta ) ) ~ \in ~ M$ is locally homeomorphic to a Euclidean subspace, described by the tangent space $T _ { p } M$ . Taking $\mathbb { R } ^ { n + 1 }$ to be Euclidean, it is a Riemann manifold. By virtue of being a submanifold of $\mathbb { R } ^ { n + 1 }$ , $M$ is also a Riemann manifold. As such, $M$ comes equipped with an smoothly varying inner product $g _ { p } : T _ { p } M \times T _ { p } M \mapsto$ $\mathbb { R }$ on tangent spaces, allowing us to measure the length of a path on $M$ . In particular, the length (or energy) of any curve $\gamma : [ 0 , 1 ] \mapsto M$ is defined by accumulating infinitesimal changes along the trajectory,
+
+$$
+\operatorname { L e n g t h } ( \gamma ) = \int _ { 0 } ^ { 1 } { \sqrt { g _ { \gamma ( t ) } ( { \dot { \gamma } } ( t ) , { \dot { \gamma } } ( t ) ) } } d t , \qquad \operatorname { E n e r g y } ( \gamma ) = \int _ { 0 } ^ { 1 } g _ { \gamma ( t ) } ( { \dot { \gamma } } ( t ) , { \dot { \gamma } } ( t ) ) d t ,
+$$
+
+where $\begin{array} { r } { { \dot { \gamma } } ( t ) = { \frac { d } { d t } } \gamma ( t ) \in T _ { \gamma ( t ) } M } \end{array}$ is a tangent vector of $\gamma ( t ) = ( \theta ( t ) , f ( \theta ( t ) ) ) \in M$ . We use parentheses (i.e. $\gamma ( t ) )$ to differentiate discrete and continuous domains. With $M$ being a submanifold of $\mathbb { R } ^ { n + 1 }$ , the induced metric on $M$ is defined by $g _ { \gamma ( t ) } ( \dot { \gamma } ( t ) , \dot { \gamma } ( t ) ) = \langle \dot { \gamma } ( t ) , \dot { \gamma } ( t ) \rangle$ . Different constructions of $M$ yield different Riemann metrics. In particular, if the model underlying $f$ admits a predictive probability distribution $P ( y \mid x )$ , the task manifold can be given an information geometric interpretation by choosing the Fisher matrix as Riemann metric, in which case the task manifold is defined over the space of probability distributions (Amari & Nagaoka, 2007). If eq. 1 is defined as natural gradient descent, the learning process corresponds to gradient descent on this manifold (Amari, 1998; Martens, 2010; Pascanu & Bengio, 2014; Luk & Grosse, 2018).
+
+Having a complete description of a task manifold, we can measure the length of a learning process by noting that gradient descent can be seen as a discrete approximation to the scaled gradient flow $\dot { \theta } ( t ) = - S ( t ) \nabla f ( \theta ( t ) )$ . This flow describes a curve that originates in $\gamma ( 0 ) = ( \theta ^ { 0 } , f ( \theta ^ { 0 } ) )$ and follows the gradient at each point. Going forward, we define $\gamma$ to be this unique curve and refer to it as the gradient path from $\theta ^ { \hat { 0 } }$ on $M$ . The metrics in eq. 2 can be computed exactly, but in practice we observe a discrete learning process. Analogously to how the gradient update rule approximates the gradient flow, the gradient path length or energy can be approximated by the cumulative chordal distance (Ahlberg et al., 1967),
+
+$$
+d _ { p } ( \theta ^ { 0 } , M ) = \sum _ { i = 0 } ^ { K - 1 } \| \gamma ^ { i + 1 } - \gamma ^ { i } \| _ { 2 } ^ { p } , \qquad p \in \{ 1 , 2 \} .
+$$
+
+![](images/a7327c1399d4bc917e0625541465a704d9cea54f31c1cabcfcdbacf04b02cd9a.jpg)  
+Figure 2: Left: illustration of Leap (algorithm 1) for two tasks, $\tau$ and $\tau ^ { \prime }$ . From an initialization $\theta ^ { 0 }$ , the learning process of each task generates gradient paths, $\Psi _ { \tau }$ and $\Psi _ { \tau ^ { \prime } }$ , which Leap uses to minimize the expected path length. Iterating the process, Leap converges to a locally Pareto optimal initialization. Right: the pull-forward objective (eq. 6) used to minimize the expected gradient path length. Any gradient path $\Psi _ { \tau } = \{ \psi _ { \tau } ^ { i } \} _ { i = 1 } ^ { \tilde { K _ { \tau } } }$ acts on $\theta ^ { 0 }$ by pulling each $\theta _ { \tau } ^ { i }$ towards $\psi _ { \tau } ^ { i + 1 }$ .
+
+We write $d$ when the distinction between the length or energy metric is immaterial. Using the energy yields a slightly simpler objective, but the length normalizes each length segment and as such protects against differences in scale between task objectives. In appendix C, we conduct an ablation study and find that they perform similarly, though using the length leads to faster convergence. Importantly, $d$ involves only terms seen during task training. We exploit this later when we construct the meta gradient, enabling us to perform gradient descent on the meta objective at negligible cost (eq. 8).
+
+We now turn to the transfer learning setting where we face a set of tasks, each with a distinct task manifold. Our framework is built on the idea that we can transfer knowledge across learning processes via the local geometry by aggregating information obtained along observed gradient paths. As such, Leap finds an initialization from which learning converges as rapidly as possible in expectation.
+
+# 2.2 META LEARNING ACROSS TASK MANIFOLDS
+
+Formally, we define a task $\tau = ( f _ { \tau } , p _ { \tau } , u _ { \tau } )$ as the process of learning to approximate the relationship $x \mapsto y$ through samples from the data distribution $p _ { \tau } ( x , y )$ . This process is defined by the gradient update rule $u _ { \tau }$ (as defined in eq. 1), applied $K _ { \tau }$ times to minimize the task objective $f _ { \tau }$ . Thus, a learning process starts at $\theta _ { \tau } ^ { 0 } = \theta ^ { 0 }$ and progresses via $\theta _ { \tau } ^ { i + 1 } = u _ { \tau } ( \theta _ { \tau } ^ { i } )$ until $\theta _ { \tau } ^ { K _ { \tau } }$ is obtained. The sequence $\{ \theta _ { \tau } ^ { i } \} _ { i = 0 } ^ { K _ { \tau } }$ defines an approximate gradient path on the task manifold $M _ { \tau }$ with distance $d ( \theta ^ { 0 } ; M _ { \tau } )$ .
+
+To understand how $d$ transfers knowledge across learning processes, consider two distinct tasks. We can transfer knowledge across these tasks’ learning processes by measuring how good a shared initialization is. Assuming two candidate initializations converge to limit points with equivalent performance on each task, the initialization with shortest expected gradient path distance encodes more knowledge sharing. In particular, if both tasks have convex loss surfaces a unique optimal initialization exists that achieves Pareto optimality in terms of total path distance. This can be crucial in data sparse regimes: rapid convergence may be the difference between learning a task and failing due to overfitting (Finn et al., 2017).
+
+Given a distribution of tasks $p ( \tau )$ , each candidate initialization $\theta ^ { 0 }$ is associated with a measure of its expected gradient path distance, $\mathbb { E } _ { \tau \sim p ( \tau ) } \big [ d ( \theta ^ { 0 } ; M _ { \tau } ) \big ]$ , that summarizes the suitability of the initialization to the task distribution. The initialization (or a set thereof) with shortest expected gradient path distance maximally transfers knowledge across learning processes and is Pareto optimal in this regard. Above, we have assumed that all candidate initializations converge to limit points of equal performance. If the task objective $f _ { \tau }$ is non-convex this is not a trivial assumption and the gradient path distance itself does not differentiate between different levels of final performance.
+
+As such, it is necessary to introduce a feasibility constraint to ensure only initializations with some minimum level of performance are considered. We leverage that transfer learning never happens in a vacuum; we always have a second-best option, such as starting from a random initialization or a pretrained model. This “second-best” initialization, $\psi ^ { 0 }$ , provides us with the performance we
+
+# Algorithm 1 Leap: Transferring Knowledge over Learning Processes
+
+Require: $p ( \tau )$ , $\tau = ( f _ { \tau } , u _ { \tau } , p _ { \tau } )$ : distribution over tasks
+
+Require: $\beta$ : step size   
+1: randomly initialize $\theta ^ { 0 }$   
+2: while not done do   
+3: $\nabla \bar { F }  0$ : initialize meta gradient   
+4: sample task batch $\boldsymbol { B }$ from $p ( \tau )$   
+5: for all $\tau \in B$ do   
+6: $\psi _ { \tau } ^ { 0 }  \theta ^ { 0 }$ : initialize task baseline   
+7: for all $i \in \{ 0 , \ldots , K _ { \tau ^ { - 1 } } \}$ do   
+8: ${ \psi } _ { \tau } ^ { i + 1 }  { u } _ { \tau } ( { \psi } _ { \tau } ^ { i } )$ −: update baseline   
+9: $\theta _ { \tau } ^ { i }  \psi _ { \tau } ^ { i }$ : follow baseline (recall $\psi _ { \tau } ^ { 0 } = \theta ^ { 0 }$ )   
+10: increment $\nabla \bar { F }$ using the pull-forward gradient (eq. 8)   
+11: end for   
+12: end for   
+13: $\begin{array} { r } { \theta ^ { 0 }  \theta ^ { 0 } - \frac { \beta } { | \boldsymbol { B } | } \nabla \bar { F } } \end{array}$ : update initialization   
+14: end while
+
+would obtain on a given task in the absence of knowledge transfer. As such, performance obtained by initializing from $\psi ^ { 0 }$ provides us with an upper bound for each task: a candidate solution $\theta ^ { 0 }$ must achieve at least as good performance to be a viable solution. Formally, this implies the task-specific requirement that a candidate $\theta ^ { 0 }$ must satisfy $f _ { \tau } ( \theta _ { \tau } ^ { K _ { \tau } } ) \le f _ { \tau } ( \psi _ { \tau } ^ { K _ { \tau } } )$ . As this must hold for every task, we obtain the canonical meta objective
+
+$$
+\begin{array} { r l } { \underset { \theta ^ { 0 } } { \operatorname* { m i n } } } & { F ( \theta ^ { 0 } ) = \mathbb { E } _ { \tau \sim p ( \tau ) } \big [ d ( \theta ^ { 0 } ; M _ { \tau } ) \big ] } \\ { \mathrm { s . t . } \quad } & { \theta _ { \tau } ^ { i + 1 } = u _ { \tau } ( \theta _ { \tau } ^ { i } ) , \quad \theta _ { \tau } ^ { 0 } = \theta ^ { 0 } , } \\ & { \theta ^ { 0 } \in \Theta = \cap _ { \tau } \left\{ \theta ^ { 0 } \ \big | \ f _ { \tau } ( \theta _ { \tau } ^ { K _ { \tau } } ) \leq f _ { \tau } ( \psi _ { \tau } ^ { K _ { \tau } } ) \right\} . } \end{array}
+$$
+
+This meta objective is robust to variations in the geometry of loss surfaces, as it balances complementary and competing learning processes (fig. 2). For instance, there may be an initialization that can solve a small subset of tasks in a handful of gradient steps, but would be catastrophic for other related tasks. When transferring knowledge via the initialization, we must trade off commonalities and differences between gradient paths. In eq. 4 these trade-offs arise naturally. For instance, as the number of tasks whose gradient paths move in the same direction increases, so does their pull on the initialization. Conversely, as the updates to the initialization renders some gradient paths longer, these act as springs that exert increasingly strong pressure on the initialization. The solution to eq. 4 thus achieves an equilibrium between these competing forces.
+
+Solving eq. 4 naively requires training to convergence on each task to determine whether an initialization satisfies the feasibility constraint, which can be very costly. Fortunately, because we have access to a second-best initialization, we can solve eq. 4 more efficiently by obtaining gradient paths from $\psi ^ { 0 }$ and use these as baselines that we incrementally improve upon. This improved initialization converges to the same limit points, but with shorter expected gradient paths (theorem 1). As such, it becomes the new second-best option; Leap (algorithm 1) repeats this process of improving upon increasingly demanding baselines, ultimately finding a solution to the canonical meta objective.
+
+# 2.3 LEAP
+
+Leap starts from a given second-best initialization $\psi ^ { 0 }$ , shared across all tasks, and constructs baseline gradient paths $\Psi _ { \tau } ~ = ~ \{ \psi _ { \tau } ^ { i } \} _ { i = 0 } ^ { K _ { \tau } }$ for each task $\tau$ in a batch $\boldsymbol { B }$ . These provide a set of baselines $\Psi = \left\{ \Psi _ { \tau } \right\} _ { \tau \in { \cal B } }$ . Recall that all tasks share the same initialization, $\psi _ { \tau } ^ { 0 } = \psi ^ { 0 } \in \Theta$ . We use these baselines, corresponding to task-specific learning processes, to modify the gradient path distance metric in eq. 3 by freezing the forward point $\gamma _ { \tau } ^ { i + 1 }$ in all norms,
+
+$$
+\bar { d } _ { p } ( \theta ^ { 0 } ; M _ { \tau } , \Psi _ { \tau } ) = \sum _ { i = 0 } ^ { K _ { \tau } - 1 } \| \bar { \gamma } _ { \tau } ^ { i + 1 } - \gamma _ { \tau } ^ { i } \| _ { 2 } ^ { p } ,
+$$
+
+where $\bar { \gamma } _ { \tau } ^ { i } = ( \psi _ { \tau } ^ { i } , f ( \psi _ { \tau } ^ { i } ) )$ represents the frozen forward point from the baseline and $\gamma _ { \tau } ^ { i } = ( \theta _ { \tau } ^ { i } , f ( \theta _ { \tau } ^ { i } ) )$ the point on the gradient path originating from $\theta ^ { 0 }$ . This surrogate distance metric encodes the feasibility constraint; optimizing $\bar { \theta ^ { 0 } }$ with respect to $\Psi$ pulls the initialization forward along each task-specific gradient path in an unconstrained variant of eq. 4 that replaces $\Theta$ with $\Psi$ ,
+
+$$
+\begin{array} { r l } { \underset { \theta ^ { 0 } } { \operatorname* { m i n } } } & { \bar { F } ( \theta ^ { 0 } ; \Psi ) = \mathbb { E } _ { \tau \sim p ( \tau ) } \big [ \bar { d } ( \theta ^ { 0 } ; M _ { \tau } , \Psi _ { \tau } ) \big ] , } \\ { \mathrm { s . t . } } & { \theta _ { \tau } ^ { i + 1 } = u _ { \tau } ( \theta _ { \tau } ^ { i } ) , \quad \theta _ { \tau } ^ { 0 } = \theta ^ { 0 } . } \end{array}
+$$
+
+We refer to eq. 6 as the pull-forward objective. Incrementally improving $\theta ^ { 0 }$ over $\psi ^ { 0 }$ leads to a new second-best option that Leap uses to generate a new set of more demanding baselines, to further improve the initialization. Iterating this process, Leap produces a sequence of candidate solutions to eq. 4, all in $\Theta$ , with incrementally shorter gradient paths. While the pull-forward objective can be solved with any optimization algorithm, we consider gradient-based methods. In theorem 1, we show that gradient descent on $\bar { F }$ yields solutions that always lie in $\Theta$ . In principle, $\bar { F }$ can be evaluated at any $\theta ^ { 0 }$ , but a more efficient strategy is to evaluate $\theta ^ { 0 }$ at $\psi ^ { 0 }$ . In this case, $\bar { d } = d$ , so that ${ \bar { F } } = F$ .
+
+Theorem 1 (Pull-forward). Define a sequence of initializations $\{ \theta _ { s } ^ { 0 } \} _ { s \in \mathbb { N } } b y$
+
+$$
+\begin{array} { r } { \theta _ { s + 1 } ^ { 0 } = \theta _ { s } ^ { 0 } - \beta _ { s } \nabla \bar { F } ( \theta _ { s } ^ { 0 } ; \Psi _ { s } ) , \qquad \theta ^ { 0 } \in \Theta , } \end{array}
+$$
+
+with $\psi _ { s } ^ { 0 } = \theta _ { s } ^ { 0 }$ for all s. For $\beta _ { s } > 0$ sufficiently small, there exist learning rates schedules $\{ \alpha _ { \tau } ^ { i } \} _ { i = 1 } ^ { K _ { \tau } }$ for all tasks such that $\theta _ { k \to \infty } ^ { 0 }$ is a limit point in $\Theta$ .
+
+Proof: see appendix A. Because the meta gradient requires differentiating the learning process, we must adopt an approximation. In doing so, we obtain a meta-gradient that can be computed analytically on the fly during task training. Differentiating $\bar { F }$ , we have
+
+$$
+\nabla \bar { F } ( \theta ^ { 0 } , \Psi ) = - p \mathbb { E } _ { \tau \sim p ( \tau ) } \left[ \sum _ { i = 0 } ^ { K _ { \tau } - 1 } J _ { \tau } ^ { i } ( \theta _ { \tau } ^ { 0 } ) ^ { T } \left( \Delta f _ { \tau } ^ { i } \nabla f _ { \tau } ( \theta _ { \tau } ^ { i } ) + \Delta \theta _ { \tau } ^ { i } \right) \left( \| \bar { \gamma } _ { \tau } ^ { i + 1 } - \gamma _ { \tau } ^ { i } \| _ { 2 } ^ { p } \right) ^ { p - 2 } \right]
+$$
+
+where ${ \cal J } _ { \tau } ^ { i }$ denotes the Jacobian of $\theta _ { \tau } ^ { i }$ with respect to the initialization, $\Delta f _ { \tau } ^ { i } = f _ { \tau } ( \psi _ { \tau } ^ { i + 1 } ) - f _ { \tau } ( \theta _ { \tau } ^ { i } )$ and $\Delta \theta _ { \tau } ^ { i ^ { \cdot } } = \psi _ { \tau } ^ { i + 1 } - \theta _ { \tau } ^ { i }$ . To render the meta gradient tractable, we need to approximate the Jacobians, as these are costly to compute. Empirical evidence suggest that they are largely redundant (Finn et al., 2017; Nichol et al., 2018). Nichol et al. (2018) further shows that an identity approximation yields a meta-gradient that remains faithful to the original meta objective. We provide some further support for this approximation (see appendix B). First, we note that the learning rate directly controls the quality of the approximation; for any $K _ { \tau }$ , the identity approximation can be made arbitrarily accurate by choosing a sufficiently small learning rates. We conduct an ablation study to ascertain how severe this limitation is and find that it is relatively loose. For the best-performing learning rate, the identity approximation is accurate to four decimal places and shows no signs of significant deterioration as the number of training steps increases. As such, we assume $J ^ { i } \approx I _ { n }$ throughout. Finally, by evaluating $\nabla \bar { F }$ at $\theta ^ { 0 } = \psi ^ { 0 }$ , the meta gradient contains only terms seen during standard training and can be computed asynchronously on the fly at negligible cost.
+
+In practice, we use stochastic gradient descent during task training. This injects noise in $f$ as well as in its gradient, resulting in a noisy gradient path. Noise in the gradient path does not prevent Leap from converging. However, noise reduces the rate of convergence, in particular when a noisy gradient step results in $f _ { \tau } ( \psi _ { \tau } ^ { s + 1 } ) - f _ { \tau } ( \theta _ { \tau } ^ { i } ) > 0$ . If the gradient estimator is reasonably accurate, this causes the term $\Delta f _ { \tau } ^ { i } { \boldsymbol { \nabla } } { \dot { f } } _ { \tau } ( { \dot { \theta } _ { \tau } ^ { i } } )$ in eq. 8 to point in the steepest ascent direction. We found that adding a stabilizer to ensure we always follow the descent direction significantly speeds up convergence and allows us to use larger learning rates. In this paper, we augment $\bar { F }$ with a stabilizer of the form
+
+$$
+\begin{array} { r } { \mu \left( f _ { \tau } ( \theta _ { \tau } ^ { i } ) ; f _ { \tau } ( \psi _ { \tau } ^ { i + 1 } ) \right) = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f } \quad f _ { \tau } ( \psi _ { \tau } ^ { i + 1 } ) \leq f _ { \tau } ( \theta _ { \tau } ^ { i } ) , } \\ { - 2 \big ( f _ { \tau } ( \psi _ { \tau } ^ { i + 1 } ) - f _ { \tau } ( \theta _ { \tau } ^ { i } ) \big ) ^ { 2 } } & { \mathrm { e l s e } . } \end{array} \right. } \end{array}
+$$
+
+Adding $\nabla \mu$ (re-scaled if necessary) to the meta-gradient is equivalent to replacing $\Delta f _ { \tau } ^ { i }$ with $- | \Delta f _ { \tau } ^ { i } |$ in eq. 8. This ensures that we never follow $\nabla f _ { \tau } ( \theta _ { \tau } ^ { i } )$ in the ascent direction, instead reinforcing the descent direction at that point. This stabilizer is a heuristic, there are many others that could prove helpful. In appendix C we perform an ablation study and find that the stabilizer is not necessary for Leap to converge, but it does speed up convergence significantly.
+
+# 3 RELATED WORK
+
+Transfer learning has been explored in a variety of settings, the most typical approach attempting to infuse knowledge in a target model’s parameters by encouraging them to lie close to those of a pretrained source model (Pan & Yang, 2009). Because such approaches can limit knowledge transfer (Higgins et al., 2017; Achille et al., 2018), applying standard transfer learning techniques leads to catastrophic forgetting, by which the model is rendered unable to perform a previously mastered task (McCloskey & Cohen, 1989; Goodfellow et al., 2013). These problems are further accentuated when there is a larger degree of diversity among tasks that push optimal parameterizations further apart. In these cases, transfer learning can in fact be worse than training from scratch.
+
+Recent approaches extend standard finetuning by adding regularizing terms to the training objective that encourage the model to learn parameters that both solve a new task and retain high performance on previous tasks. These regularizers operate by protecting the parameters that affect the loss function the most (Miconi et al., 2018; Zenke et al., 2017; Kirkpatrick et al., 2017; Lee et al., 2017; Serrà et al., 2018). Because these approaches use a single model to encode both global task-general information and local task-specific information, they can over-regularize, preventing the model from learning further tasks. More importantly, Schwarz et al. (2018) found that while these approaches mitigate catastrophic forgetting, they are unable to facilitate knowledge transfer on the benchmark they considered. Ultimately, if a single model must encode both task-generic and task-specific information, it must either saturate or grow in size (Rusu et al., 2016).
+
+In contrast, meta-learning aims to learn the learning process itself (Schmidhuber, 1987; Bengio et al., 1991; Santoro et al., 2016; Ravi & Larochelle, 2016; Andrychowicz et al., 2016; Vinyals et al., 2016; Finn et al., 2017). The literature focuses primarily on few-shot learning, where a task is some variation on a common theme, such as subsets of classes drawn from a shared pool of data (Lake et al., 2015; Vinyals et al., 2016). The meta-learning algorithm adapts a model to a new task given a handful of samples. Recent attention has been devoted to three main approaches. One trains the meta-learner to adapt to a new task by comparing an input to samples from previous tasks (Vinyals et al., 2016; Mishra et al., 2018; Snell et al., 2017). More relevant to our framework are approaches that parameterize the training process through a recurrent neural network that takes the gradient as input and produces a new set of parameters (Ravi & Larochelle, 2016; Santoro et al., 2016; Andrychowicz et al., 2016; Hochreiter et al., 2001). The approach most closely related to us learns an initialization such that the model can adapt to a new task through one or a few gradient updates (Finn et al., 2017; Nichol et al., 2018; Al-Shedivat et al., 2017; Lee & Choi, 2018). In contrast to our work, these methods focus exclusively on few-shot learning, where the gradient path is trivial as only a single or a handful of training steps are allowed, limiting them to settings where the current task is closely related to previous ones.
+
+It is worth noting that the Model Agnostic Meta Learner (MAML: Finn et al., 2017) can be written as $\mathbb { E } _ { \tau \sim p ( \tau ) } \big [ f _ { \tau } ( \theta _ { \tau } ^ { K } ) \big ]$ .1 As such, it arises as a special case of Leap where only the final parameterization is evaluated in terms of its final performance. Similarly, the Reptile algorithm (Nichol et al., 2018), which proposes to update rule $\begin{array} { r } { \Dot { \theta ^ { 0 } }  \theta ^ { 0 } + \epsilon ( \mathbb { E } _ { \tau \sim p ( \tau ) } [ \mathcal { \bar { \theta } } _ { \tau } ^ { K } ] - \theta ^ { \hat { 0 } } ) } \end{array}$ , can be seen as a naive version of Leap that assumes all task geometries are Euclidean. In particular, Leap reduces to Reptile if $f _ { \tau }$ is removed from the task manifold and the energy metric without stabilizer is used. We find this configuration to perform significantly worse than any other (see section 4.1 and appendix C).
+
+![](images/9f00f1a1c113f287144579d3bde3b8a841f6f2cc2949c66444aea6adc8acb202.jpg)  
+Figure 3: Results on Omniglot. Left: Comparison of average learning curves on held-out tasks (across 10 seeds) for 25 tasks in the meta-training set. Curves are moving averages with window size 5. Shading: standard deviation within window. Right: AUC across number of tasks in the meta-training set. Shading: standard deviation across 10 seeds.
+
+Related work studying models from a geometric perspective have explored how to interpolate in a generative model’s learned latent space (Tosi et al., 2014; Shao et al., 2017; Arvanitidis et al., 2018; Chen et al., 2018; Kumar et al., 2017). Riemann manifolds have also garnered attention in the context of optimization, as a preconditioning matrix can be understood as the instantiation of some Riemann metric (Amari & Nagaoka, 2007; Abbati et al., 2018; Luk & Grosse, 2018).
+
+# 4 EMPIRICAL RESULTS
+
+We consider three experiments with increasingly complex knowledge transfer. We measure transfer learning in terms of final performance and speed of convergence, where the latter is defined as the area under the training error curve. We compare Leap to competing meta-learning methods on the Omniglot dataset by transferring knowledge across alphabets (section 4.1). We study Leap’s ability to transfer knowledge over more complex and diverse tasks in a Multi-CV experiment (section 4.2) and finally evaluate Leap on in a demanding reinforcement environment (section 4.3).
+
+# 4.1 OMNIGLOT
+
+The Omniglot (Lake et al., 2015) dataset consists of 50 alphabets, which we define to be distinct tasks. We hold 10 alphabets out for final evaluation and use subsets of the remaining alphabets for metalearning or pretraining. We vary the number of alphabets used for meta-learning / pretraining from 1 to 25 and compare final performance and rate of convergence on held-out tasks. We compare against no pretraining, multi-headed finetuning, MAML, the first-order approximation of MAML (FOMAML; Finn et al., 2017), and Reptile. We train on a given task for 100 steps, with the exception of MAML where we backpropagate through 5 training steps during meta-training. For Leap, we report performance under the length metric $( d _ { 1 } )$ ; see appendix C for an ablation study on Leap hyper-parameters. For further details, see appendix D.
+
+Any type of knowledge transfer significantly improves upon a random initialization. MAML exhibits a considerable short-horizon bias (Wu et al., 2018). While FOMAML is trained full trajectories, but because it only leverages gradient information at final iteration, which may be arbitrarily uninformative, it does worse. Multi-headed finetuning is a tough benchmark to beat as tasks are very similar. Nevertheless, for sufficiently rich task distributions, both Reptile and Leap outperform finetuning, with Leap outperforming Reptile as the complexity grows. Notably, the AUC gap between Reptile and Leap grows in the number of training steps (fig. 3), amounting to a 4 percentage point difference in final validation error (table 2). Overall, the relative performance of meta-learners underscores the importance of leveraging geometric information in meta-learning.
+
+Table 1: Results on Multi-CV benchmark. All methods are trained until convergence on held-out tasks. Finetuning is multiheaded. † Area under training error curve; scaled to 0–100.‡Our implementation. MNIST results omitted; see appendix E, table 4.   
+
+<table><tr><td>Held-out task</td><td>Method</td><td>Test (%)</td><td>Train (%)</td><td>AUCt</td></tr><tr><td>Facescrub</td><td>Leap</td><td>19.9</td><td>0.0</td><td>11.6</td></tr><tr><td></td><td>Finetuning</td><td>32.7</td><td>0.0</td><td>13.2</td></tr><tr><td></td><td>Progressive Nets‡</td><td>18.0</td><td>0.0</td><td>8.9</td></tr><tr><td></td><td>HAT‡</td><td>25.6</td><td>0.1</td><td>14.6</td></tr><tr><td></td><td>No pretraining</td><td>18.2</td><td>0.0</td><td>10.5</td></tr><tr><td>Cifar10</td><td>Leap</td><td>21.2</td><td>10.8</td><td>17.5</td></tr><tr><td></td><td>Finetuning</td><td>27.4</td><td>13.3</td><td>20.7</td></tr><tr><td></td><td>Progressive Netst</td><td>24.2</td><td>15.2</td><td>24.0</td></tr><tr><td></td><td>HATt</td><td>27.7</td><td>21.2</td><td>27.3</td></tr><tr><td></td><td>No pretraining</td><td>26.2</td><td>13.1</td><td>23.0</td></tr><tr><td>SVHN</td><td>Leap</td><td>8.4</td><td>5.6</td><td>7.5</td></tr><tr><td></td><td>Finetuning</td><td>10.9</td><td>6.1</td><td>10.5</td></tr><tr><td></td><td>Progressive Netst</td><td>10.1</td><td>6.3</td><td>13.8</td></tr><tr><td></td><td>HAT‡</td><td>10.5</td><td>5.7</td><td>8.5</td></tr><tr><td></td><td>No pretraining</td><td>10.3</td><td>6.9</td><td>11.5</td></tr><tr><td>Cifar100</td><td>Leap</td><td>52.0</td><td>30.5</td><td>43.4</td></tr><tr><td></td><td>Finetuning</td><td>59.2</td><td>31.5</td><td>44.1</td></tr><tr><td></td><td>Progressive Nets‡</td><td>55.7</td><td>42.1</td><td>54.6</td></tr><tr><td></td><td>HATt</td><td>62.0</td><td>49.8</td><td>58.4</td></tr><tr><td></td><td>No pretraining</td><td>54.8</td><td>33.1</td><td>50.1</td></tr><tr><td>Traffic Signs</td><td>Leap</td><td>2.9</td><td>0.0</td><td>1.2</td></tr><tr><td></td><td>Finetuning</td><td>5.7</td><td>0.0</td><td>1.7</td></tr><tr><td></td><td>Progressive Nets‡</td><td>3.6</td><td>0.0</td><td>4.0</td></tr><tr><td></td><td>HATt</td><td>5.4</td><td>0.0</td><td>2.3</td></tr><tr><td></td><td>No pretraining</td><td>3.6</td><td>0.0</td><td>2.4</td></tr></table>
+
+# 4.2 MULTI-CV
+
+Inspired by Serrà et al. (2018), we consider a set of computer vision datasets as distinct tasks. We pretrain on all but one task, which is held out for final evaluation. For details, see appendix E. To reduce the computational burden during meta training, we pretrain on each task in the meta batch for one epoch using the energy metric $( d _ { 2 } )$ . We found this to reach equivalent performance to training on longer gradient paths or using the length metric. This indicates that it is sufficient for Leap to see a partial trajectory to correctly infer shared structures across task geometries.
+
+We compare Leap against a random initialization, multi-headed finetuning, a non-sequential version of HAT (Serrà et al., 2018) (i.e. allowing revisits) and a non-sequential version of Progressive Nets (Rusu et al., 2016), where we allow lateral connection between every task. Note that this makes Progressive Nets over 8 times larger in terms of learnable parameters.
+
+The Multi-CV experiment is more challenging both due to greater task diversity and greater complexity among tasks. We report results on held-out tasks in table 1. Leap outperforms all baselines on all but one transfer learning tasks (Facescrub), where Progressive Nets does marginally better than a random initialization owing to its increased parameter count. Notably, while Leap does marginally worse than a random initialization, finetuning and HAT leads to a substantial drop in performance. On all other tasks, Leap converges faster to optimal performance and achieves superior final performance.
+
+![](images/4c302be052448d4592d9d61ec8f0ad8468632d4f01881d47adaa39a318ed1d4b.jpg)  
+Figure 4: Mean normalized episode scores on Atari games across training steps. Shaded regions depict two standard deviations across ten seeds. Leap (orange) generally outperforms a random initialization (blue), even when the action space is twice as large as during pretraining (table 6, appendix F).
+
+# 4.3 ATARI
+
+To demonstrate that Leap can scale to large problems, both in computational terms and in task complexity, we apply it in a reinforcement learning environment, specifically Atari 2600 games (Bellemare et al., 2013). We use an actor-critic architecture (Sutton et al., 1998) with the policy and the value function sharing a convolutional encoder. We apply Leap with respect to the encoder using the energy metric $( d _ { 2 } )$ . During meta training, we sample mini-batches from 27 games that have an action space dimensionality of at most 10, holding out two games with similar action space dimensionality for evaluation, as well as games with larger action spaces (table 6). During meta-training, we train on each task for five million training steps. See appendix F for details.
+
+We train for 100 meta training steps, which is sufficient to see a distinct improvement; we expect a longer meta-training phase to yield further gains. We find that Leap generally outperforms a random initialization. This performance gain is primarily driven by less volatile exploration, as seen by the confidence intervals in fig. 4 (see also fig. 8). Leap finds a useful exploration space faster and more consistently, demonstrating that Leap can find shared structures across a diverse set of complex learning processes. We note that these gains may not cater equally to all tasks. In the case of WizardOfWor (part of the meta-training set), Leap exhibits two modes: in one it performs on par with the baseline, in the other exploration is protracted (fig. 8). This phenomena stems from randomness in the learning process, which renders an observed gradient path relatively less representative. Such randomness can be marginalized by training for longer.
+
+That Leap can outperform a random initialization on the pretraining set (AirRaid, UpNDown) is perhaps not surprising. More striking is that it exhibits the same behavior on out-of-distribution tasks. In particular, Alien, Gravitar and RoadRunner all have at least $50 \%$ larger state space than anything encountered during pretraining (appendix F, table 6), yet Leap outperforms a random initialization. This suggests that transferring knowledge at a higher level of abstraction, such as in the space of gradient paths, generalizes to unseen task variations as long as underlying learning dynamics agree.
+
+# 5 CONCLUSIONS
+
+Transfer learning typically ignores the learning process itself, restricting knowledge transfer to scenarios where target tasks are very similar to source tasks. In this paper, we present Leap, a framework for knowledge transfer at a higher level of abstraction. By formalizing knowledge transfer as minimizing the expected length of gradient paths, we propose a method for meta-learning that scales to highly demanding problems. We find empirically that Leap has superior generalizing properties to finetuning and competing meta-learners.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank anonymous reviewers for their comments. This work was supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1.
+
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+
+# APPENDIX
+
+# A PROOF OF THEOREM 1
+
+Proof. We first establish that, for all $s$ ,
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \tau } d ( \theta _ { s + 1 } ^ { 0 } , M _ { \tau } ) = F ( \theta _ { s + 1 } ^ { 0 } ) = \bar { F } ( \theta _ { s + 1 } ^ { 0 } ; \Psi _ { s + 1 } ) \le \bar { F } ( \theta _ { s } ^ { 0 } ; \Psi _ { s } ) = F ( \theta _ { s } ^ { 0 } ) = \mathbb { E } _ { \tau } d ( \theta _ { s } ^ { 0 } , M _ { \tau } ) , } \end{array}
+$$
+
+with strict inequality for at least some $s$ . Because $\{ \beta _ { s } \} _ { s = 1 } ^ { \infty }$ satisfies the gradient descent criteria, it follows that the sequence $\{ \theta _ { s } ^ { 0 } \} _ { s = 1 } ^ { \infty }$ is convergent. To complete the proof we must show that this limit point lies in $\Theta$ . To this end, we show that for $\beta _ { s }$ sufficiently small, for all $s$ , $\begin{array} { r } { \operatorname* { l i m } _ { i \to \infty } \theta _ { s + 1 } ^ { i } = } \end{array}$ $\operatorname* { l i m } _ { i \to \infty } \theta _ { s } ^ { i }$ . That is, each updated initialization incrementally reduces the expected gradient path length while converging to the same limit point as $\theta _ { 0 } ^ { 0 }$ . Since $\theta _ { 0 } ^ { 0 } \overset { \cdot } { \in } \Theta$ by assumption, we obtain $\theta _ { s } ^ { 0 } \in \Theta$ for all $s$ as an immediate consequence.
+
+To establish $\mathbb { E } _ { \tau } d ( \theta _ { s + 1 } ^ { 0 } , M _ { \tau } ) \le \mathbb { E } _ { \tau } d ( \theta _ { s } ^ { 0 } , M _ { \tau } )$ , with strict inequality for some $s$ , let
+
+$$
+\begin{array} { l l } { { z _ { \tau } ^ { i } = ( \theta _ { \tau } ^ { s , i } , f _ { \tau } ( \theta _ { \tau } ^ { s , i } ) ) } } & { { \qquad x _ { \tau } ^ { i } = ( \theta _ { \tau } ^ { s + 1 , i } , f _ { \tau } ( \theta _ { \tau } ^ { s + 1 , i } ) ) } } \\ { { h _ { \tau } ^ { i } = ( \psi _ { \tau } ^ { s , i + 1 } , f _ { \tau } ( \psi _ { \tau } ^ { s , i + 1 } ) ) } } & { { \qquad y _ { \tau } ^ { i } = ( \psi _ { \tau } ^ { s + 1 , i + 1 } , f _ { \tau } ( \psi _ { \tau } ^ { s + 1 , i + 1 } ) ) , } } \end{array}
+$$
+
+with $\psi _ { \tau } ^ { s , i + 1 } = \theta _ { \tau } ^ { s , i + 1 }$ . Denote by $\mathbb { E } _ { \tau , i }$ the expectation over gradient paths, $\mathbb { E } _ { \tau \sim p ( \tau ) } \sum _ { i = 1 } ^ { K _ { \tau } }$ . Note that
+
+$$
+\begin{array} { c c } { \bar { F } ( \theta _ { s } ^ { 0 } , \Psi _ { s } ) = \mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| _ { 2 } ^ { p } \qquad } & { \bar { F } ( \theta _ { s } ^ { 0 } , \Psi _ { s + 1 } ) = \mathbb { E } _ { \tau , i } \| y _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| _ { 2 } ^ { p } } \\ { \bar { F } ( \theta _ { s + 1 } ^ { 0 } , \Psi _ { s } ) = \mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| _ { 2 } ^ { p } \qquad } & { \bar { F } ( \theta _ { s + 1 } ^ { 0 } , \Psi _ { s + 1 } ) = \mathbb { E } _ { \tau , i } \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| _ { 2 } ^ { p } } \end{array}
+$$
+
+with $p = 2$ defining the meta objective in terms of the gradient path energy and $p = 1$ in terms of the gradient path length. As we are exclusively concerned with the Euclidean norm, we omit the subscript. By assumption, every $\beta _ { s }$ is sufficiently small to satisfy the gradient descent criteria $\bar { F } ( \theta _ { s } ^ { 0 } ; \Psi _ { s } ) \overset { * } { \geq } \bar { F } ( \bar { \theta } _ { s + 1 } ^ { 0 } ; \Psi _ { s } ^ { * } )$ . Adding and subtracting $\dot { \bar { F } } ( \theta _ { s + 1 } ^ { 0 } , \Psi _ { s + 1 } )$ to the RHS, we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| ^ { p } \geq \mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } } \\ & { \qquad = \mathbb { E } _ { \tau , i } \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } + \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } . } \end{array}
+$$
+
+It follows that $\begin{array} { r } { \mathbb E _ { \tau } d ( \theta _ { s } ^ { 0 } , M _ { \tau } ) \ge \mathbb E _ { \tau } d ( \theta _ { s + 1 } ^ { 0 } , M _ { \tau } ) } \end{array}$ if $\mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } \geq \mathbb { E } _ { \tau , i } \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p }$ . As our main concern is existence, we will show something stronger, namely that there exists $\alpha _ { \tau } ^ { i }$ such that
+
+$$
+\begin{array} { r } { \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } \geq \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } \qquad \forall i , \tau , s , p } \end{array}
+$$
+
+with at least one such inequality strict for some $i , \tau , s$ , in which case ${ d _ { p } } ( { \theta _ { s + 1 } ^ { 0 } } , { M _ { \tau } } ) < { d _ { p } } ( { \theta _ { s } ^ { 0 } } , { M _ { \tau } } )$ for any $p \in \{ 1 , 2 \}$ . We proceed by establishing the inequality for $p = 2$ and obtain $p = 1$ as an immediate consequence of monotonicity of the square root. Expanding $\| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 }$ we have
+
+$$
+\begin{array} { r l } & { \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } = \| ( h _ { \tau } ^ { i } - z _ { \tau } ^ { i } ) + ( z _ { \tau } ^ { i } - x _ { \tau } ^ { i } ) \| ^ { 2 } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } } \\ & { \qquad = \| h _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| ^ { 2 } + 2 \langle h _ { \tau } ^ { i } - z _ { \tau } ^ { i } , z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rangle + \| z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } . } \end{array}
+$$
+
+Every term except $\lVert z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rVert ^ { 2 }$ can be minimized by choosing $\alpha _ { \tau } ^ { i }$ small, whereas $\lVert z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rVert ^ { 2 }$ is controlled by $\beta _ { s }$ . Thus, our strategy is to make all terms except $\lVert \dot { z } _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rVert ^ { 2 }$ small, for a given $\beta _ { s }$ , by placing an upper bound on $\alpha _ { \tau } ^ { i }$ . We first show that $\| h _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| ^ { 2 } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } = O \left( \alpha _ { \tau } ^ { i } { } ^ { 2 } \right)$ . Some care is needed as the $( n + 1 )$ th dimension is the loss value associated with the other $n$ dimensions. Define $\hat { z } _ { \tau } ^ { i } = \theta _ { \tau } ^ { s , i }$ , so that $z _ { \tau } ^ { i } = ( \hat { z } _ { \tau } ^ { i } , f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) )$ . Similarly define $\hat { x } _ { \tau } ^ { i } , \hat { h } _ { \tau } ^ { i }$ , and $\hat { y } _ { \tau } ^ { i }$ to obtain
+
+$$
+\begin{array} { r l } & { \| h _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| ^ { 2 } = \| \hat { h } _ { \tau } ^ { i } - \hat { z } _ { \tau } ^ { i } \| ^ { 2 } + \big ( f _ { \tau } ( \hat { h } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) \big ) ^ { 2 } } \\ & { \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } = \| \hat { y } _ { \tau } ^ { i } - \hat { x } _ { \tau } ^ { i } \| ^ { 2 } + \big ( f _ { \tau } ( \hat { y } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { x } _ { \tau } ^ { i } ) \big ) ^ { 2 } } \\ & { 2 \langle h _ { \tau } ^ { i } - z _ { \tau } ^ { i } , z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rangle = 2 \langle \hat { h } _ { \tau } ^ { i } - \hat { z } _ { \tau } ^ { i } , \hat { z } _ { \tau } ^ { i } - \hat { x } _ { \tau } ^ { i } \rangle + \big ( f _ { \tau } ( \hat { h } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) \big ) \big ( f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { x } _ { \tau } ^ { i } ) \big ) . } \end{array}
+$$
+
+Consider $\lVert \hat { h } _ { \tau } ^ { i } - \hat { z } _ { \tau } ^ { i } \rVert ^ { 2 } - \lVert \hat { y } _ { \tau } ^ { i } - \hat { x } _ { \tau } ^ { i } \rVert ^ { 2 } \mathrm { . }$ . Note that $\hat { h } _ { \tau } ^ { i } = \hat { z } _ { \tau } ^ { i } - \alpha _ { \tau } ^ { i } g ( \hat { z } _ { \tau } ^ { i } )$ , where $g ( \hat { z } _ { \tau } ^ { i } ) = S _ { \tau } ^ { s , i } \nabla f ( \hat { z } _ { \tau } ^ { i } )$ , and similarly $\hat { y } _ { \tau } ^ { i } = \hat { x } _ { \tau } ^ { i } - \alpha _ { \tau } ^ { i } g ( \hat { x } _ { \tau } ^ { i } )$ with $g ( \hat { x } _ { \tau } ^ { i } ) = S _ { \tau } ^ { s + 1 , i } \nabla f ( \hat { x } _ { \tau } ^ { i } )$ . Thus, $\| \hat { h } _ { \tau } ^ { i } - \hat { z } _ { \tau } ^ { i } \| ^ { 2 } = { \alpha _ { \tau } ^ { i } } ^ { 2 } \| g ( \hat { z } _ { \tau } ^ { i } ) \| ^ { 2 }$ and similarly for $\| \hat { y } _ { \tau } ^ { i } - \hat { x } _ { \tau } ^ { i } \| ^ { 2 }$ , so
+
+$$
+\| \hat { h } _ { \tau } ^ { i } - \hat { z } _ { \tau } ^ { i } \| ^ { 2 } - \| \hat { y } _ { \tau } ^ { i } - \hat { x } _ { \tau } ^ { i } \| ^ { 2 } = \left( \alpha _ { \tau } ^ { i } \right) ^ { 2 } \left( \| g ( \hat { z } _ { \tau } ^ { i } ) \| ^ { 2 } - \| g ( \hat { x } _ { \tau } ^ { i } ) \| ^ { 2 } \right) = O \left( \left( \alpha _ { \tau } ^ { i } \right) ^ { 2 } \right) .
+$$
+
+Now consider $\left( f _ { \tau } ( \hat { h } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) \right) ^ { 2 } - \left( f _ { \tau } ( \hat { y } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { x } _ { \tau } ^ { i } ) \right) ^ { 2 }$ . Using the above identities and first-order Taylor series expansion, we have
+
+$$
+\begin{array} { r l } & { \left( f _ { \tau } ( \hat { h } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) \right) ^ { 2 } = \left( \nabla f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) ^ { T } ( \hat { h } _ { \tau } ^ { i } - \hat { z } _ { \tau } ^ { i } ) + O \left( \alpha _ { \tau } ^ { i } \right) \right) ^ { 2 } } \\ & { \qquad = \left( - \alpha _ { \tau } ^ { i } \nabla f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) ^ { T } g ( \hat { z } _ { \tau } ^ { i } ) + O \left( \alpha _ { \tau } ^ { i } \right) \right) ^ { 2 } = O \left( \left( \alpha _ { \tau } ^ { i } \right) ^ { 2 } \right) , } \end{array}
+$$
+
+and similarly for $\left( f _ { \tau } ( \hat { y } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { x } _ { \tau } ^ { i } ) \right) ^ { 2 }$ . As such, $\| h _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| ^ { 2 } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } = O \Big ( \big ( \alpha _ { \tau } ^ { i } \big ) ^ { 2 } \Big ) .$
+
+Finally, consider the inner product $\langle h _ { \tau } ^ { i } - z _ { \tau } ^ { i } , z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rangle$ . From above we have that $( f _ { \tau } ( \hat { h } _ { \tau } ^ { i } ) - f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) ) ^ { 2 } =$ $- \alpha _ { \tau } ^ { i } ( \dot { \nabla } f _ { \tau } ( \hat { z } _ { \tau } ^ { i } ) ^ { T } g ( \hat { z } _ { \tau } ^ { i } ) - \dot { R _ { \tau } ^ { i } } ) = - \dot { \alpha } _ { \tau } ^ { i } \dot { \xi } _ { \tau } ^ { i }$ , where $R _ { \tau } ^ { i }$ denotes an upper bound on the residual. We extend $g$ to operate on $z _ { \tau } ^ { i }$ by defining $\tilde { g } ( z _ { \tau } ^ { i } ) \dot { = } ( g ( \hat { z } _ { \tau } ^ { i } ) , \xi _ { \tau } ^ { i } )$ . Returning to $\| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 }$ , we have
+
+$$
+\begin{array} { r l r } & { } & { \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } - \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } = \| z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } + 2 \langle h _ { \tau } ^ { i } - z _ { \tau } ^ { i } , z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rangle + O \left( \left( \alpha _ { \tau } ^ { i } \right) ^ { 2 } \right) } \\ & { } & { = \| z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } - 2 \alpha _ { \tau } ^ { i } \langle \tilde { g } ( z _ { \tau } ^ { i } ) , z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rangle + O \left( \left( \alpha _ { \tau } ^ { i } \right) ^ { 2 } \right) . } \end{array}
+$$
+
+The first term is non-negative, and importantly, always non-zero whenever $\beta _ { s } \neq 0$ . Furthermore, $\alpha _ { \tau } ^ { i }$ can always be made sufficiently small for $\lVert z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rVert ^ { 2 }$ to dominate the residual, so we can focus on the inner product $\langle \tilde { g } ( z _ { \tau } ^ { i } ) , \dot { z } _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rangle$ . If it is negative, all terms are positive and we have $\| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } \geq \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 }$ as desired. If not, $\| z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { \breve { 2 } }$ dominates if
+
+$$
+\alpha _ { \tau } ^ { i } \leq \frac { \| z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } } { 2 \langle \tilde { g } ( z _ { \tau } ^ { i } ) , z _ { \tau } ^ { i } - x _ { \tau } ^ { i } \rangle } \in ( 0 , \infty ) .
+$$
+
+Thus, for $\alpha _ { \tau } ^ { i }$ sufficiently small, we have $\| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } \geq \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } \forall i , \tau , s$ , with strict inequality whenever $\langle \dot { \tilde { g } } ( z _ { \tau } ^ { i } ) , z _ { \tau } ^ { i } - \dot { x } _ { \tau } ^ { i } \rangle < 0$ or the bound on $\alpha _ { \tau } ^ { \ i }$ holds strictly. This establishes $d _ { 2 } ( \theta _ { s + 1 } ^ { 0 } , \bar { M } _ { \tau } ) \stackrel { . } { \leq }$ $d _ { 2 } ( \theta _ { s } ^ { 0 } , M _ { \tau } )$ for all $\tau , s$ , with strict inequality for at least some $\tau , s$ . To also establish it for the gradient path length $\gamma = 1 \gamma$ ), taking square roots on both sides of $\| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 } \geq \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { 2 }$ yields the desired results, and so $\| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { - } \dot { \geq } \| y _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p }$ for $p \in \{ 1 , 2 \}$ , and therefore
+
+$$
+d ( \theta _ { s + 1 } ^ { 0 } , M _ { \tau } ) = \bar { F } ( \theta _ { s + 1 } ^ { 0 } ; \Psi _ { s + 1 } ) \leq \bar { F } ( \theta _ { s } ^ { 0 } ; \Psi _ { s } ) = d ( \theta _ { s } ^ { 0 } , M _ { \tau } ) \quad \forall \tau , s
+$$
+
+with strict inequality for at least some $\tau , s$ , in particular whenever $\beta _ { s } \neq 0$ and $\alpha _ { \tau } ^ { i }$ sufficiently small.
+
+Then, to see that the limit point of $\Psi _ { s + 1 }$ is the same as that of $\Psi _ { s }$ for $\beta _ { s }$ sufficiently small, note that $x _ { \tau } ^ { i } = y _ { \tau } ^ { i - 1 }$ . As before, by the gradient descent criteria, $\beta _ { s }$ is such that
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - x _ { \tau } ^ { i } \| ^ { p } = \mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - y _ { \tau } ^ { i - 1 } \| ^ { p } \leq \mathbb { E } _ { \tau , i } \| h _ { \tau } ^ { i } - z _ { \tau } ^ { i } \| ^ { p } = \mathbb { E } _ { \tau , i } \left( \alpha _ { \tau } ^ { i } \right) ^ { p } \| \tilde { g } ( z _ { \tau } ^ { i } ) \| ^ { p } . } \end{array}
+$$
+
+Define $\epsilon _ { \tau } ^ { i }$ as the noise residual from the expectation; each $y _ { \tau } ^ { i - 1 }$ is bounded by $\| h _ { \tau } ^ { i } - y _ { \tau } ^ { i - 1 } \| ^ { p } \leq$ $\begin{array} { r } { \big ( \alpha _ { \tau } ^ { i } \big ) ^ { p } \| \tilde { g } ( z _ { \tau } ^ { i } ) \| ^ { p } + \epsilon _ { \tau } ^ { i } } \end{array}$ . For $\beta _ { s }$ small this noise component vanishes, and since $\{ \alpha _ { \tau } ^ { i } \} _ { i }$ is a converging sequence, the bound on $y _ { \tau } ^ { i - 1 }$ grows increasingly tight. It follows then that $\left\{ \theta _ { s + 1 } ^ { i } \right\} _ { i = 1 } ^ { \infty }$ converges to the same limit point as $\{ \theta _ { s } ^ { i } \} _ { i = 1 } ^ { \infty }$ , yielding $\theta _ { s + 1 } ^ { 0 } \in \Theta$ for all $s$ , as desired. 
+
+# B ABLATION STUDY: APPROXIMATING JACOBIANS $J ^ { i } ( \theta ^ { 0 } )$
+
+To understand the role of the Jacobians, note that (we drop task subscripts for simplicity)
+
+$$
+\begin{array} { l } { { J ^ { i + 1 } ( \theta ^ { 0 } ) = \left( I _ { n } - \alpha ^ { i } S ^ { i } H _ { f } ( \theta ^ { i } ) \right) J ^ { i } ( \theta ^ { 0 } ) = \displaystyle \prod _ { j = 0 } ^ { i } \left( I _ { n } - \alpha ^ { j } S ^ { j } H _ { f } ( \theta ^ { j } ) \right) } } \\ { { \displaystyle ~ = I _ { n } - \sum _ { j = 0 } ^ { i } \alpha ^ { i } S ^ { i } H _ { f } ( \theta ^ { i } ) + O \left( \left( \alpha ^ { i } \right) ^ { 2 } \right) , } } \end{array}
+$$
+
+where $H _ { f } ( \theta ^ { j } )$ denotes the Hessian of $f$ at $\theta ^ { j }$ . Thus, changes to $\theta ^ { i + 1 }$ are translated into $\theta ^ { 0 }$ via all intermediary Hessians. This makes the Jacobians memoryless up to second-order curvature. Importantly, the effect of curvature can directly be controlled via $\alpha ^ { i }$ , and by choosing $\alpha ^ { i }$ small we can ensure $J ^ { i } ( \theta ^ { 0 } ) \approx I _ { n }$ to be a arbitrary precision. In practice, this approximation works well (c.f. Finn et al., 2017; Nichol et al., 2018). Moreover, as a practical matter, if the alternative is some other approximation to the Hessians, the amount of noise injected grows exponentially with every iteration. The problem of devising an accurate low-variance estimator for the ${ \hat { J ^ { i } } } ( \theta ^ { 0 } )$ is highly challenging and beyond the scope of this paper.
+
+![](images/568c2503fd9ae8afd1cd7270602ecb4086ec57ebc76a8924969b38851cbced8f.jpg)  
+Figure 5: Relative precision of Jacobian approximation. Precision is calculated for the Jacobian of the first layer, across different learning rates (colors) and gradient steps.
+
+To understand how this approximation limits our choice of learning rates $\alpha ^ { i }$ , we conduct an ablation study in the Omniglot experiment setting. We are interested in the relative precision of the identity approximation under different learning rates and across time steps, which we define as
+
+$$
+\rho \left( i , \{ \alpha ^ { j } \} _ { j = 0 } ^ { i } \right) = \frac { \| I _ { n } - J ^ { i } ( \theta ^ { 0 } ) \| _ { 1 } } { \| J ^ { i } ( \theta ^ { 0 } ) \| _ { 1 } } ,
+$$
+
+where the norm is the Schatten 1-norm. We use the same four-layer convolutional neural network as in the Omniglot experiment (appendix D). For each choice of learning rate, we train a model from a random initialization for 20 steps and compute $\rho$ every 5 steps. Due to exponential growth of memory consumption, we were unable to compute $\rho$ for more than 20 gradient steps. We report the relative precision of the first convolutional layer. We do not report the Jacobian with respect to other layers, all being considerably larger, as computing their Jacobians was too costly. We computed $\rho$ for all layers on the first five gradient steps and found no significant variation in precision across layers. Consequently, we prioritize reporting how precision varies with the number of gradient steps. As in the main experiments, we use stochastic gradient descent. We evaluate $\alpha ^ { i } = \overline { { \alpha } } \in \{ 0 . 0 1 , \mathrm { { 0 . 1 } , 0 . 5 } \}$ across 5 different tasks. Figure 5 summarizes our results.
+
+![](images/e13a0b15f89bbbe143e958404677fb6dad042651f5b76cc9d8e273c03edd995f.jpg)  
+Figure 6: Average task training loss over meta-training steps. $p$ denotes the $\bar { d } _ { p }$ used in the meta objective, $\mu = 1$ the use of the stabilizer, and $f _ { \tau } = 1$ the inclusion of the loss in the task manifold.
+
+Reassuringly, we find the identity approximation to be accurate to at least the fourth decimal for learning rates we use in practice, and to the third decimal for the largest learning rate (0.5) we were able to converge with. Importantly, except for the smallest learning rate, the quality of the approximation is constant in the number of gradient steps. The smallest learning rate that exhibits some deterioration on the fifth decimal, however larger learning rates provide an upper bound that is constant on the fourth decimal, indicating that this is of minor concern. Finally, we note that while these results suggest the identity approximation to be a reasonable approach on the class of problems we consider, other settings may put stricter limits on the effective size of learning rates.
+
+# C ABLATION STUDY: LEAP HYPER-PARAMETERS
+
+As Leap is a general framework, we have several degrees of freedom in specifying a meta learner. In particular, we are free to choose the task manifold structure, the gradient path distance metric, $d _ { p }$ , and whether to incorporate stabilizers. These are non-trivial choices and to ascertain the importance of each, we conduct an ablation study. We vary (a) the task manifold between using the full loss surface and only parameter space, (b) the gradient path distance metric between using the energy or length, and (c) inclusion of the stabilizer $\mu$ in the meta objective. We stay as close as possible to the set-up used in the Omniglot experiment (appendix D), fixing the number of pretraining tasks to 20 and perform 500 meta gradient updates. All other hyper-parameters are the same.
+
+Our ablation study indicates that the richer the task manifold and the more accurate the gradient path length is approximated, the better Leap performs (fig. 6). Further, adding a stabilizer has the intended effect and leads to significantly faster convergence. The simplest configuration, defined in terms of the gradient path energy and with the task manifold identifies as parameter space, yields a meta gradient equivalent to the update rule used in Reptile. We find this configuration to be less efficient in terms of convergence and we observe a significant deterioration in performance. Extending the task manifold to the loss surface does not improve meta-training convergence speed, but does cut prediction error in half. Adding the stabilizer significantly speeds up convergence. These conclusions also hold under the gradient path length as distance measure, and in general using the gradient path length does better than using the gradient path energy as the distance measure.
+
+D EXPERIMENT DETAILS: OMNIGLOT
+
+Table 2: Mean test error after 100 training steps on held out evaluation tasks.†Multi-headed finetuning.   
+
+<table><tr><td>Method No. Pretraining tasks</td><td>Leap</td><td>Reptile</td><td>Finetuningt</td><td>MAML</td><td>FOMAML</td><td>No pretraining</td></tr><tr><td>1</td><td>62.3</td><td>59.8</td><td>46.5</td><td>64.0</td><td>64.5</td><td>82.3</td></tr><tr><td>3</td><td>46.5</td><td>46.5</td><td>36.0</td><td>56.2</td><td>59.0</td><td>82.3</td></tr><tr><td>5</td><td>40.3</td><td>41.4</td><td>32.5</td><td>50.1</td><td>53.0</td><td>82.5</td></tr><tr><td>10</td><td>32.6</td><td>35.6</td><td>28.7</td><td>49.3</td><td>49.6</td><td>82.9</td></tr><tr><td>15</td><td>29.6</td><td>33.3</td><td>26.9</td><td>45.5</td><td>47.8</td><td>82.6</td></tr><tr><td>20</td><td>26.0</td><td>30.8</td><td>24.7</td><td>41.7</td><td>45.4</td><td>82.6</td></tr><tr><td>25</td><td>24.8</td><td>29.4</td><td>23.5</td><td>42.9</td><td>44.0</td><td>82.8</td></tr></table>
+
+Table 3: Summary of hyper-parameters for Omniglot. “Meta” refers to the outer training loop, “task” refers to the inner training loop.   
+
+<table><tr><td></td><td>Leap</td><td>Finetuning</td><td>Reptile</td><td>MAML</td><td>FOMAML</td><td>No pretraining</td></tr><tr><td colspan="7">Meta training</td></tr><tr><td>Learning rate</td><td>0.1</td><td></td><td>0.1</td><td>0.5</td><td>0.5</td><td></td></tr><tr><td>Training steps</td><td>1000</td><td>1000</td><td>1000</td><td>1000</td><td>1000</td><td></td></tr><tr><td>Batch size (tasks)</td><td>20</td><td>20</td><td>20</td><td>20</td><td>20</td><td></td></tr><tr><td colspan="7">Task training</td></tr><tr><td>Learning rate</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td></td></tr><tr><td>Training steps</td><td>100</td><td>100</td><td>100</td><td>5</td><td>100</td><td></td></tr><tr><td>Batch size (samples)</td><td>20</td><td>20</td><td>20</td><td>20</td><td>20</td><td></td></tr><tr><td colspan="7">Task evaluation</td></tr><tr><td>Learning rate</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Training steps</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td></tr><tr><td>Batch size (samples)</td><td>20</td><td>20</td><td>20</td><td>20</td><td>20</td><td>20</td></tr></table>
+
+Omniglot contains 50 alphabets, each with a set of characters that in turn have 20 unique samples. We treat each alphabet as a distinct task and pretrain on up to 25 alphabets, holding out 10 out for final evaluation. We use data augmentation on all tasks to render the problem challenging. In particular, we augment any image with a random affine transformation by (a) random sampling a scaling factor between [0.8, 1.2], (b) random rotation between [0, 360), and (c) randomly cropping the height and width by a factor between $[ - 0 . 2 , 0 . 2 ]$ in each dimension. This setup differs significantly from previous protocols (Vinyals et al., 2016; Finn et al., 2017), where tasks are defined by selecting different permutations of characters and restricting the number of samples available for each character.
+
+We use the same convolutional neural network architecture as in previous works (Vinyals et al., 2016; Schwarz et al., 2018). This model stacks a module, comprised of a $3 \times 3$ convolution with 64 filters, followed by batch-normalization, ReLU activation and $2 \times 2$ max-pooling, four times. All images are downsampled to $2 8 \times 2 8$ , resulting in a $1 \times 1 \times 6 4$ feature map that is passed on to a final linear layer. We define a task as a 20-class classification problem with classes drawn from a distinct alphabet. For alphabets with more than 20 characters, we pick 20 characters at random, alphabets with fewer characters (4) are dropped from the task set. On each task, we train a model using stochastic gradient descent. For each model, we evaluated learning rates in the range [0.001, 0.01, 0.1, 0.5]; we found 0.1 to be the best choice in all cases. See table 3 for further hyper-parameters.
+
+Table 4: Transfer learning results on Multi-CV benchmark. All methods are trained until convergence on held-out tasks. †Area under training error curve; scaled to 0–100. ‡Our implementation.   
+
+<table><tr><td>Held-out task</td><td>Method</td><td>Test (%)</td><td>Train (%)</td><td>AUCt</td></tr><tr><td>Facescrub</td><td>Leap</td><td>19.9</td><td>0.0</td><td>11.6</td></tr><tr><td></td><td>Finetuning</td><td>32.7</td><td>0.0</td><td>13.2</td></tr><tr><td></td><td>Progressive Netst</td><td>18.0</td><td>0.0</td><td>8.9</td></tr><tr><td></td><td>HAT</td><td>25.6</td><td>0.1</td><td>14.6</td></tr><tr><td></td><td> No pretraining</td><td>18.2</td><td>0.0</td><td>10.5</td></tr><tr><td>NotMNIST</td><td>Leap</td><td>5.3</td><td>0.6</td><td>2.9</td></tr><tr><td></td><td>Finetuning</td><td>5.4</td><td>2.0</td><td>4.4</td></tr><tr><td></td><td>Progressive Netst</td><td>5.4</td><td>3.1</td><td>3.7</td></tr><tr><td></td><td>HAT</td><td>6.0</td><td>2.8</td><td>5.4</td></tr><tr><td></td><td> No pretraining</td><td>5.4</td><td>2.6</td><td>5.1</td></tr><tr><td>MNIST</td><td>Leap</td><td>0.7</td><td>0.1</td><td>0.6</td></tr><tr><td></td><td>Finetuning</td><td>0.9</td><td>0.1</td><td>0.8</td></tr><tr><td></td><td>Progressive Netst</td><td>0.8</td><td>0.0</td><td>0.7</td></tr><tr><td></td><td>HAT</td><td>0.8</td><td>0.3</td><td>1.2</td></tr><tr><td></td><td> No pretraining</td><td>0.9</td><td>0.2</td><td>1.0</td></tr><tr><td>Fashion MNIST</td><td>Leap</td><td>8.0</td><td>4.2</td><td>6.8</td></tr><tr><td></td><td>Finetuning</td><td>8.9</td><td>3.8</td><td>7.0</td></tr><tr><td></td><td>Progressive Netst</td><td>8.7</td><td>5.4</td><td>9.2</td></tr><tr><td></td><td>HAT</td><td>9.5</td><td>5.5</td><td>8.1</td></tr><tr><td></td><td>No pretraining</td><td>8.4</td><td>4.7</td><td>7.8</td></tr><tr><td>Cifar10</td><td>Leap</td><td>21.2</td><td>10.8</td><td>17.5</td></tr><tr><td></td><td>Finetuning</td><td>27.4</td><td>13.3</td><td>20.7</td></tr><tr><td></td><td>Progressive Netst</td><td>24.2</td><td>15.2</td><td>24.0</td></tr><tr><td></td><td>HAT</td><td>27.7</td><td>21.2</td><td>27.3</td></tr><tr><td></td><td> No pretraining</td><td>26.2</td><td>13.1</td><td>23.0</td></tr><tr><td>SVHN</td><td>Leap</td><td>8.4</td><td>5.6</td><td>7.5</td></tr><tr><td></td><td>Finetuning</td><td>10.9</td><td>6.1</td><td>10.5</td></tr><tr><td></td><td>Progressive Netst</td><td>10.1</td><td>6.3</td><td>13.8</td></tr><tr><td></td><td>HAT+</td><td>10.5</td><td>5.7</td><td>8.5</td></tr><tr><td></td><td> No pretraining</td><td>10.3</td><td>6.9</td><td>11.5</td></tr><tr><td>Cifar100</td><td>Leap</td><td>52.0</td><td>30.5</td><td>43.4</td></tr><tr><td></td><td>Finetuning</td><td>59.2</td><td>31.5</td><td>44.1</td></tr><tr><td></td><td>Progressive Nets‡</td><td>55.7</td><td>42.1</td><td>54.6</td></tr><tr><td></td><td>HATt</td><td>62.0</td><td>49.8</td><td>58.4</td></tr><tr><td></td><td>No pretraining</td><td>54.8</td><td>33.1</td><td>50.1</td></tr><tr><td>Traffic Signs</td><td>Leap</td><td>2.9</td><td>0.0</td><td>1.2</td></tr><tr><td></td><td>Finetuning</td><td>5.7</td><td>0.0</td><td>1.7</td></tr><tr><td></td><td>Progressive Netst</td><td>3.6</td><td>0.0</td><td>4.0</td></tr><tr><td></td><td>HAT‡</td><td>5.4</td><td>0.0</td><td>2.3</td></tr><tr><td></td><td>No pretraining</td><td>3.6</td><td>0.0</td><td>2.4</td></tr></table>
+
+We meta-train for 1000 steps unless otherwise noted; on each task we train for 100 steps. Increasing the number of steps used for task training yields similar results, albeit at greater computational expense. For each character in an alphabet, we hold out 5 samples in order to create a task validation set.
+
+Table 5: Summary of hyper-parameters for Multi-CV.“Meta” refers to the outer training loop, ‘task” refers to the inner training loop.   
+
+<table><tr><td></td><td>Leap</td><td>Finetuning</td><td>Progressive Nets</td><td>HAT</td><td>No pretraining</td></tr><tr><td>Meta training</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Learning rate</td><td>0.01</td><td></td><td></td><td></td><td></td></tr><tr><td>Training steps</td><td>1000</td><td>1000</td><td>1000</td><td>1000</td><td></td></tr><tr><td>Batch size</td><td>10</td><td>10</td><td>10</td><td>10</td><td></td></tr><tr><td>Task training</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Learning rate</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td></td></tr><tr><td>Max epochs</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>Batch size</td><td>32</td><td>32</td><td>32</td><td>32</td><td></td></tr><tr><td>Task evaluation</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Learning rate</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Training epochs</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td></tr><tr><td>Batch size</td><td>32</td><td>32</td><td>32</td><td>32</td><td>32</td></tr></table>
+
+# E EXPERIMENT DETAILS: MULTI-CV
+
+We allow different architectures between tasks by using different final linear layers for each task. We use the same convolutional encoder as in the Omniglot experiment (appendix D). Leap learns an initialization for the convolutional encoder; on each task, the final linear layer is always randomly initialized. We compare Leap against (a) a baseline with no pretraining, (b) multitask finetuning, (c) HAT (Serrà et al., 2018), and (d) Progressive Nets (Rusu et al., 2016). For HAT, we use the original formulation, but allow multiple task revisits (until convergence). For Progressive Nets, we allow lateral connections between all tasks and multiple task revisits (until convergence). Note that this makes Progressive Nets over 8 times larger in terms of learnable parameters than the other models. inproceedings We train using stochastic gradient descent with cosine annealing (Loshchilov & Hutter, 2017). During meta training, we sample a batch of 10 tasks at random from the pretraining set and train until the early stopping criterion is triggered or the maximum amount of epochs is reached (see table 5). We used the same interval for selecting learning rates as in the Omniglot experiment (appendix D). Only Leap benefited from using more than 1 epoch as the upper limit on task training steps during pretraining. In the case of Leap, the initialization is updated after all tasks in the meta batch has been trained to convergence; for other models, there is no distinction between initialization and task parameters. On a given task, training is stopped if the maximum number of epochs is reached (table 5) or if the validation error fails to improve over 10 consecutive gradient steps. Similarly, meta training is stopped once the mean validation error fails to improve over 10 consecutive meta training batches. We use Adam (Kingma & Ba, 2015) for the meta gradient update with a constant learning rate of 0.01. We use no dataset augmentation. MNIST images are zero padded to have $3 2 \times 3 2$ images; we use the same normalizations as Serrà et al. (2018).
+
+# F EXPERIMENT DETAILS: ATARI
+
+We use the same network as in Mnih et al. (2013), adopting it to actor-critic algorithms by estimating both value function and policy through linear layers connected to the final output of a shared convolutional network. Following standard practice, we use downsampled $8 4 \times 8 4 \times 3$ RGB images as input. Leap is applied with respect to the convolutional encoder (as final linear layers vary in size across environments). We use all environments with an action space of at most 10 as our pretraining pool, holding out Breakout and SpaceInvaders. During meta training, we sample a batch of 16 games at random from a pretraining pool of 27 games. On each game in the batch, a network is initialized using the shared initialization and trained independently for 5 million steps, accumulating the meta gradient across games on the fly. Consequently, in any given episode, the baseline and Leap differs only with respect to the initialization of the convolutional encoder. We trained Leap for 100 steps, equivalent to training 1600 agents for 5 million steps. The meta learned initialization was evaluated on the held-out games, a random selection of games seen during pretraining, and a random selection of games with action spaces larger than 10 (table 6). On each task, we use a batch size of 32, an unroll length of 5 and update the model parameters with RMSProp (using $\epsilon = 1 0 ^ { - 4 }$ , $\alpha = 0 . 9 9 )$ with a learning rate of $1 0 ^ { - 4 }$ . We set the entropy cost to 0.01 and clip the absolute value of the rewards to maximum 5.0. We use a discounting factor of 0.99.
+
+Table 6: Evaluation environment characteristics. †Calculated on baseline (no pretraining) data.   
+
+<table><tr><td>Environment</td><td>Action Space</td><td>Mean Rewardt</td><td>Standard Deviationt</td><td>Pretraining Env</td></tr><tr><td>AirRaid</td><td>6</td><td>2538</td><td>624</td><td>Y</td></tr><tr><td>UpNDown</td><td>6</td><td>52417</td><td>2797</td><td>Y</td></tr><tr><td>WizardOfWor</td><td>10</td><td>2531</td><td>182</td><td>Y</td></tr><tr><td>Breakout</td><td>4</td><td>338</td><td>13</td><td>N</td></tr><tr><td>SpaceInvaders</td><td>6</td><td>1065</td><td>103</td><td>N</td></tr><tr><td>Asteroids</td><td>14</td><td>1760</td><td>139</td><td>N</td></tr><tr><td>Alien</td><td>18</td><td>1280</td><td>182</td><td>N</td></tr><tr><td>Gravitar</td><td>18</td><td>329</td><td>15</td><td>N</td></tr><tr><td>RoadRunner</td><td>18</td><td>29593</td><td>2890</td><td>N</td></tr></table>
+
+![](images/93b332e4cb76d21cbe8d6d0d68fbaf1a0d1863285d9471e3940eb372e4d04514.jpg)  
+Figure 7: Mean normalized episode scores on Atari games across training steps. Scores are reported as moving average over 500 episodes. Shaded regions depict two standard deviations across ten seeds. KungFuMaster, RoadRunner and Krull have action state spaces that are twice as large as the largest action state encountered during pretraining. Leap (orange) generally outperforms a random initialization, except for WizardOfWor, where a random initialization does better on average due to outlying runs under Leap’s initialization.
+
+![](images/572e7fd5434e00c489dfd9d5a220d736c7933ff9f8b11d55069a612fc4f98b23.jpg)  
+Figure 8: Mean episode scores on Atari games across training steps for different runs. Scores are reported as moving average over 500 episodes. Leap (orange) outperforms a random initialization by being less volatile.

parse/train/S1xSSTNKDB/S1xSSTNKDB.md

@@ -0,0 +1,335 @@
+# FAIRFACE: A NOVEL FACE ATTRIBUTE DATASET FOR BIAS MEASUREMENT AND MITIGATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Existing public face image datasets are strongly biased toward Caucasian faces, and other races (e.g., Latino) are significantly underrepresented. The models trained from such datasets suffer from inconsistent classification accuracy, which limits the applicability of face analytic systems to non-White race groups. To mitigate the race bias problem in these datasets, we constructed a novel face image dataset containing 108,501 images which is balanced on race. We define 7 race groups: White, Black, Indian, East Asian, Southeast Asian, Middle Eastern, and Latino. Images were collected from the YFCC-100M Flickr dataset and labeled with race, gender, and age groups. Evaluations were performed on existing face attribute datasets as well as novel image datasets to measure the generalization performance. We find that the model trained from our dataset is substantially more accurate on novel datasets and the accuracy is consistent across race and gender groups. We also compare several commercial computer vision APIs and report their balanced accuracy across gender, race, and age groups.
+
+# 1 INTRODUCTION
+
+To date, numerous large scale face image datasets (Huang et al., 2007; Kumar et al., 2011; Escalera et al., 2016; Yi et al., 2014; Liu et al., 2015; Joo et al., 2015; Parkhi et al., 2015; Yang et al., 2016; Guo et al., 2016; Kemelmacher-Shlizerman et al., 2016; Rothe et al., 2016; Cao et al., 2018; Merler et al., 2019) have been proposed and fostered research and development for automated face detection (Li et al., 2015b; Hu & Ramanan, 2017), alignment (Xiong & De la Torre, 2013; Ren et al., 2014), recognition (Taigman et al., 2014; Schroff et al., 2015), generation (Yan et al., 2016; Bao et al., 2017; Karras et al., 2018; Thomas & Kovashka, 2018), modification (Antipov et al., 2017; Lample et al., 2017; He et al., 2017), and attribute classification (Kumar et al., 2011; Liu et al., 2015). These systems have been successfully translated into many areas including security, medicine, education, and social sciences.
+
+Despite the sheer amount of available data, existing public face datasets are strongly biased toward Caucasian faces, and other races (e.g., Latino) are significantly underrepresented. A recent study shows that most existing large scale face databases are biased towards “lighter skin” faces (around $80 \%$ ), e.g. White, compared to “darker” faces, e.g. Black (Merler et al., 2019). This means the model may not apply to some subpopulations and its results may not be compared across different groups without calibration. Biased data will produce biased models trained from it. This will raise ethical concerns about fairness of automated systems, which has emerged as a critical topic of study in the recent machine learning and AI literature (Hardt et al., 2016; Corbett-Davies et al., 2017).
+
+For example, several commercial computer vision systems (Microsoft, IBM, Face $^ { + + }$ ) have been criticized due to their asymmetric accuracy across sub-demographics in recent studies (Buolamwini & Gebru, 2018; Raji & Buolamwini, 2019). These studies found that the commercial face gender classification systems all perform better on male and on light faces. This can be caused by the biases in their training data. Various unwanted biases in image datasets can easily occur due to biased selection, capture, and negative sets (Torralba & Efros, 2011). Most public large scale face datasets have been collected from popular online media – newspapers, Wikipedia, or web search– and these platforms are more frequently used by or showing White people.
+
+To mitigate the race bias in the existing face datasets, we propose a novel face dataset with an emphasis on balanced race composition. Our dataset contains 108,501 facial images collected primarily from the YFCC-100M Flickr dataset (Thomee et al.), which can be freely shared for a research purpose, and also includes examples from other sources such as Twitter and online newspaper outlets. We define 7 race groups: White, Black, Indian, East Asian, Southeast Asian, Middle Eastern, and Latino. Our dataset is well-balanced on these 7 groups (See Figures 1 and 2)
+
+![](images/2da2e724d9be61df612b01a4a426b36e9d48df93080cf4fc4dbd3a521cefc31b.jpg)  
+Figure 1: Racial compositions in face datasets.
+
+![](images/9603f651227f54ec1ac78cccd583639ec75819985131f1dc561e5d02ebdaa89b.jpg)  
+Figure 2: Random samples from face attribute datasets.
+
+Our paper makes three main contributions. First, we emprically show that existing face attribute datasets and models learned from them do not generalize well to unseen data in which more nonWhite faces are present. Second, we show that our new dataset performs better on novel data, not only on average, but also across racial groups, i.e. more consistently. Third, to the best of our knowledge, our dataset is the first large scale face attribute dataset in the wild which includes Latino and Middle Eastern and differentiates East Asian and Southeast Asian. Computer vision has been rapidly transferred into other fields such as economics or social sciences, where researchers want to analyze different demographics using image data. The inclusion of major racial groups, which have been missing in existing datasets, therefore significantly enlarges the applicability of computer vision methods to these fields.
+
+# 2 RELATED WORK
+
+# 2.1 FACE ATTRIBUTE RECOGNITION
+
+The goal of face attribute recognition is to classify various human attributes such as gender, race, age, emotions, expressions or other facial traits from facial appearance (Kumar et al., 2011; Joo et al., 2013; Zhang et al., 2015; Liu et al., 2015). Table 1 summarizes the statistics of existing large scale public and in-the-wild face attribute datasets including our new dataset. As stated earlier, most of these datasets were constructed from online sources and are typically dominated by the White race.
+
+Table 1: Statistics of Face Attribute Datasets   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Rac</td><td rowspan=1 colspan=1>Annot</td><td rowspan=1 colspan=1>ation</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>Name</td><td rowspan=2 colspan=1>Source</td><td rowspan=2 colspan=1>#offaces</td><td rowspan=2 colspan=1>In-the-wild?</td><td rowspan=2 colspan=1>Age</td><td rowspan=2 colspan=1>Gender</td><td rowspan=1 colspan=2>White*</td><td rowspan=1 colspan=2>Asian*</td><td rowspan=2 colspan=1>Bla-ck</td><td rowspan=2 colspan=1>Ind-ian</td><td rowspan=2 colspan=1>Lat-ino</td><td rowspan=2 colspan=1>Balan-ced?</td></tr><tr><td rowspan=1 colspan=2>WME</td><td rowspan=1 colspan=2>E SE</td></tr><tr><td rowspan=1 colspan=1>PPB(Buolamwini &amp; Gebru,2018)</td><td rowspan=1 colspan=1>Gov. OfficialProfiles</td><td rowspan=1 colspan=1>1K</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=8>**Skin color prediction</td></tr><tr><td rowspan=1 colspan=1>MORPH(Ricanek &amp; Tesafaye,2006)</td><td rowspan=1 colspan=1>Public Data</td><td rowspan=1 colspan=1>55K</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=2>merged</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>no</td></tr><tr><td rowspan=1 colspan=1>PubFig(Kumar et al.,2011)</td><td rowspan=1 colspan=1>Celebrity</td><td rowspan=1 colspan=1>13K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=9>Model generated predictions</td><td rowspan=1 colspan=1>no</td></tr><tr><td rowspan=1 colspan=1>IMDB-WIKI(Rothe et al., 2016)</td><td rowspan=1 colspan=1>IMDB,WIKI</td><td rowspan=1 colspan=1>500K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td></tr><tr><td rowspan=1 colspan=1>FotW(Escalera et al., 2016)</td><td rowspan=1 colspan=1>Flickr</td><td rowspan=1 colspan=1>25K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>yes</td></tr><tr><td rowspan=1 colspan=1>CACD(Chen et al.,2015)</td><td rowspan=1 colspan=1>celebrity</td><td rowspan=1 colspan=1>160K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td></tr><tr><td rowspan=1 colspan=1>DiF(Merler et al., 2019)</td><td rowspan=1 colspan=1>Flickr</td><td rowspan=1 colspan=1>1M</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=8>**Skin color prediction</td></tr><tr><td rowspan=1 colspan=1>+CelebA(Liu et al., 2015)</td><td rowspan=1 colspan=1>CelebFaceLFW</td><td rowspan=1 colspan=1>200K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td></tr><tr><td rowspan=1 colspan=1>LFW+(Han et al., 2018)</td><td rowspan=1 colspan=1>LFW(Newspapers)</td><td rowspan=1 colspan=1>15K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=2>merged</td><td rowspan=1 colspan=3>merged</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td></tr><tr><td rowspan=1 colspan=1>LFWA+(Liu et al., 2015)</td><td rowspan=1 colspan=1>LFW(Newspapers)</td><td rowspan=1 colspan=1>13K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=2>merged</td><td rowspan=1 colspan=2>merged</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td></tr><tr><td rowspan=1 colspan=1>tUTKFace(Zhang et al., 2017)</td><td rowspan=1 colspan=1>MORPH,CACDWeb</td><td rowspan=1 colspan=1>20K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=2>merged</td><td rowspan=1 colspan=2>merged</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>yes</td></tr><tr><td rowspan=1 colspan=1>FairFace(Ours)</td><td rowspan=1 colspan=1>Flickr, TwitterNewspapers, Web</td><td rowspan=1 colspan=1>108K</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>yes</td></tr></table>
+
+\*FairFace (Ours) also defines East (E) Asian, Southeast $\overline { { ( \mathrm { S E } ) } }$ Asian, Middle Eastern (ME), and Western (W) White. \*\*PPB and DiF do not provide race annotations but skin color annotated or automatically computed as a proxy to race. †denotes datasets used in our experiments.
+
+Face attribute recognition has been applied as a sub-component to other computer vision tasks such as face verification (Kumar et al., 2011) and person re-idenfication (Layne et al., 2012; Li et al., 2015a; Su et al., 2018). It is imperative to ensure that these systems perform evenly well on different gender and race groups. Failing to do so can be detrimental to the reputations of individual service providers and the public trust about the machine learning and computer vision research community. Most notable incidents regarding the racial bias include Google Photos recognizing African American faces as Gorilla and Nikon’s digital cameras prompting a message asking “did someone blink?” to Asian users (Zhang, 2015). These incidents, regardless of whether the models were trained improperly or how much they actually affected the users, often result in the termination of the service or features (e.g. dropping sensitive output categories). For this reason, most commercial service providers have stopped providing a race classifier.
+
+Face attribute recognition is also used for demographic surveys performed in marketing or social science research, aimed at understanding human social behaviors and their relations to demographic backgrounds of individuals. Using off-the-shelf tools (Amos et al., 2016; Baltrusaitis et al., 2018) and commercial services, social scientists have begun to use images of people to infer their demographic attributes and analyze their behaviors. Notable examples are demographic analyses of social media users using their photographs (Chakraborty et al., 2017; Reis et al., 2017; Won et al., 2017; Xi et al., 2019; Wang et al., 2017). The cost of unfair classification is huge as it can over- or under-estimate specific sub-populations in their analysis, which may have policy implications.
+
+# 2.2 FAIR CLASSIFICATION AND DATASET BIAS
+
+AI and machine learning communities have increasingly paid attention to algorithmic fairness and dataset and model biases (Zemel et al., 2013; Corbett-Davies et al., 2017; Zou & Schiebinger, 2018;
+
+Zhang et al., 2018). There exist many different definitions of fairness used in the literature (Verma & Rubin, 2018). In this paper, we focus on balanced accuracy–whether the attribute classification accuracy is independent of race and gender. More generally, research in fairness is concerned with a model’s ability to produce fair outcomes (e.g. loan approval) independent of protected or sensitive attributes such as race or gender.
+
+Studies in algorithmic fairness have focused on either 1) discovering (auditing) existing bias in datasets or systems (Shankar et al., 2017; Buolamwini & Gebru, 2018; Kiritchenko & Mohammad, 2018; McDuff et al., 2019), 2) making a better dataset (Merler et al., 2019; Alvi et al., 2018), or 3) designing a better algorithm or model (Das et al., 2018; Alvi et al., 2018; Ryu et al., 2017; Zemel et al., 2013; Zafar et al., 2017). Our paper falls into the first two categories.
+
+The main task of interest in our paper is (balanced) gender classification from facial images. Buolamwini & Gebru (2018) demonstrated many commercial gender classification systems are biased and least accurate on dark-skinned females. The biased results may be caused by biased datasets, such as skewed image origins ( $45 \%$ of images are from the U.S. in Imagenet) (Suresh et al., 2018) or biased underlying associations between scene and race in images (Stock & Cisse, 2018). It is, however, “infeasible to balance across all possible co-occurrences” of attributes (Hendricks et al., 2018), except in a lab-controlled setting.
+
+Therefore, the contribution of our paper is to mitigate, not entirely solve, the current limitations and biases of existing databases by collecting more diverse face images from non-White race groups. We empirically show this significantly improves the generalization performance to novel image datasets whose racial compositions are not dominated by the White race. Furthermore, as shown in Table 1, our dataset is the first large scale in-the-wild face image dataset which includes Southeast Asian and Middle Eastern races. While their faces share similarity with East Asian and White groups, we argue that not having these major race groups in datasets is a strong form of discrimination.
+
+# 3 DATASET CONSTRUCTION
+
+# 3.1 RACE TAXONOMY
+
+Our dataset defines 7 race groups: White, Black, Indian, East Asian, Southeast Asian, Middle Eastern, and Latino. Race and ethnicity are different categorizations of humans. Race is defined based on physical traits and ethnicity is based on cultural similarities (Schaefer, 2008). For example, Asian immigrants in Latin America can be of Latino ethnicity. In practice, these two terms are often used interchangeably.
+
+We first adopted a commonly accepted race classification from the U.S. Census Bureau (White, Black, Asian, Hawaiian and Pacific Islanders, Native Americans, and Latino). Latino is often treated as an ethnicity, but we consider Latino a race, which can be judged from the facial appearance. We then further divided subgroups such as Middle Eastern, East Asian, Southeast Asian, and Indian, as they look clearly distinct. During the data collection, we found very few examples for Hawaiian and Pacific Islanders and Native Americans and discarded these categories. All the experiments conducted in this paper were therefore based on 7 race classification.
+
+An important criterion to measure dataset bias is on which basis the bias should be measured: skin color or race? A few recent studies (Buolamwini & Gebru, 2018; Merler et al., 2019) use skin color as a proxy to racial or ethnicity grouping. While skin color can be easily computed without subjective annotations, it has limitations. First, skin color is heavily affected by illumination and light conditions. The Pilot Parliaments Benchmark (PPB) dataset (Buolamwini & Gebru, 2018) only used profile photographs of government officials taken in well controlled lighting, which makes it non-in-the-wild. Second, within-group variations of skin color are huge. Even same individuals can show different skin colors over time. Third, most importantly, race is a multidimensional concept whereas skin color (i.e. brightness) is one dimensional. Figure 5 in Appendix shows the distributions of the skin color of multiple race groups, measured by Individual Typology Angle (ITA) (Wilkes et al., 2015). As shown here, the skin color provides no information to differentiate many groups such as East Asian and White. Therefore, we explicitly use race and annotate the physical race by human annotators’ judgments. To complement the limits of race categorization, however, we also use skin color, measured by ITA, following the same procedure used by Merler et al. (2019).
+
+# 3.2 IMAGE COLLECTION AND ANNOTATION
+
+Many existing face datasets have been sourced from photographs of public figures such as politicians or celebrities (Kumar et al., 2011; Huang et al., 2007; Joo et al., 2015; Rothe et al., 2016; Liu et al., 2015). Despite the easiness of collecting images and ground truth attributes, the selection of these populations may be biased. For example, politicians may be older and actors may be more attractive than typical faces. Their images are usually taken by professional photographers in limited situations, leading to the quality bias. Some datasets were collected via web search using keywords such as “Asian boy” (Zhang et al., 2017). These queries may return only stereotypical faces or prioritize celebrities in those categories rather than diverse individuals among general public.
+
+Our goal is to minimize the selection bias introduced by such filtering and maximize the diversity and coverage of the dataset. We started from a huge public image dataset, Yahoo YFCC100M dataset (Thomee et al.), and detected faces from the images without any preselection. A recent work also used the same dataset to construct a huge unfiltered face dataset (Diversity in Faces, DiF) (Merler et al., 2019). Our dataset is smaller but more balanced on race (See Figure 1).
+
+For an efficient collection, we incrementally increased the dataset size. We first detected and annotated 7,125 faces randomly sampled from the entire YFCC100M dataset ignoring the locations of images. After obtaining annotations on this initial set, we estimated demographic compositions of each country. Based on this statistic, we adaptively adjusted the number of images for each country sampled from the dataset such that the dataset is not dominated by the White race. Consequently, we excluded the U.S. and European countries in the later stage of data collection after we sampled enough White faces from those countries. The minimum size of a detected face was set to 50 by 50 pixels. This is a relatively smaller size compared to other datasets, but we find the attributes are still recognizable and these examples can actually make the classifiers more robust against noisy data. We only used images with “Attribution” and “Share Alike” Creative Commons licenses, which allow derivative work and commercial usages.
+
+We used Amazon Mechanical Turk to annotate the race, gender and age group for each face. We assigned three workers for each image. If two or three workers agreed on their judgements, we took the values as ground-truth. If all three workers produced different responses, we republished the image to another 3 workers and subsequently discarded the image if the new annotators did not agree. These annotations at this stage were still noisy. We further refined the annotations by training a model from the initial ground truth annotations and applying back to the dataset. We then manually re-verified the annotations for images whose annotations differed from model predictions.
+
+# 4 EXPERIMENTS
+
+# 4.1 MEASURING BIAS IN DATASETS
+
+We first measure how skewed each dataset is in terms of its race composition. For the datasets with race annotations, we use the reported statistics. For the other datasets, we annotated the race labels for 3,000 random samples drawn from each dataset. See Figure 1 for the result. As expected, most existing face attribute datasets, especially the ones focusing on celebrities or politicians, are biased toward the White race. Unlike race, we find that most datasets are relatively more balanced on gender ranging from $40 \%$ - $60 \%$ male ratio.
+
+# 4.2 MODEL AND CROSS-DATASET PERFORMANCE
+
+To compare model performance of different datasets, we used an identical model architecture, ResNet-34 (He et al., 2016), to be trained from each dataset. We used ADAM optimization (Kingma & Ba, 2014) with a learning rate of 0.0001. Given an image, we detected faces using the dlib’s (dlib.net) CNN-based face detector (King, 2015) and ran the attribute classifier on each face. The experiment was done in PyTorch.
+
+Throughout the evaluations, we compare our dataset with three other datasets: UTKFace (Zhang et al., 2017), LFWA+, and CelebA (Liu et al., 2015). Both UTKFace and LFWA $^ +$ have race annotations, and thus, are suitable for comparison with our dataset. CelebA does not have race annotations, so we only use it for gender classification. See Table 1 for more detailed dataset characteristics.
+
+Table 2: Cross-Dataset Classification Accuracy on White Race.   
+
+<table><tr><td rowspan=3 colspan=1></td><td rowspan=1 colspan=10>Tested on</td></tr><tr><td rowspan=1 colspan=4>Race</td><td rowspan=1 colspan=4>Gender</td><td rowspan=1 colspan=2>Age</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>CelebA*</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>UTKFace</td></tr><tr><td rowspan=4 colspan=1>Trained on</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.937</td><td rowspan=1 colspan=1>.936</td><td rowspan=1 colspan=1>.970</td><td rowspan=1 colspan=1>942</td><td rowspan=1 colspan=1>.940</td><td rowspan=1 colspan=1>.920</td><td rowspan=1 colspan=1>.981</td><td rowspan=1 colspan=1>.597</td><td rowspan=1 colspan=1>.565</td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>.800</td><td rowspan=1 colspan=1>.918</td><td rowspan=1 colspan=1>.925</td><td rowspan=1 colspan=1>.860</td><td rowspan=1 colspan=1>.935</td><td rowspan=1 colspan=1>.916</td><td rowspan=1 colspan=1>.962</td><td rowspan=1 colspan=1>.413</td><td rowspan=1 colspan=1>.576</td></tr><tr><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>.879</td><td rowspan=1 colspan=1>.947</td><td rowspan=1 colspan=1>.961</td><td rowspan=1 colspan=1>.761</td><td rowspan=1 colspan=1>.842</td><td rowspan=1 colspan=1>.930</td><td rowspan=1 colspan=1>.940</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>CelebA</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>.812</td><td rowspan=1 colspan=1>.880</td><td rowspan=1 colspan=1>.905</td><td rowspan=1 colspan=1>.971</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td></tr></table>
+
+\* CelebA doesn’t provide race annotations. The result was obtained from the whole set (white and non-white).
+
+Table 3: Cross-Dataset Classification Accuracy on non-White Races.   
+
+<table><tr><td rowspan=3 colspan=1></td><td rowspan=1 colspan=10>Tested on</td></tr><tr><td rowspan=1 colspan=4>Race†</td><td rowspan=1 colspan=4>Gender</td><td rowspan=1 colspan=2>Age</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>CelebA*</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>UTKFace</td></tr><tr><td rowspan=4 colspan=1>Trained on</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.754</td><td rowspan=1 colspan=1>.801</td><td rowspan=1 colspan=1>.960</td><td rowspan=1 colspan=1>944</td><td rowspan=1 colspan=1>.939</td><td rowspan=1 colspan=1>.930</td><td rowspan=1 colspan=1>.981</td><td rowspan=1 colspan=1>.607</td><td rowspan=1 colspan=1>.616</td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>.693</td><td rowspan=1 colspan=1>.839</td><td rowspan=1 colspan=1>.887</td><td rowspan=1 colspan=1>.823</td><td rowspan=1 colspan=1>.925</td><td rowspan=1 colspan=1>.908</td><td rowspan=1 colspan=1>.962</td><td rowspan=1 colspan=1>.418</td><td rowspan=1 colspan=1>.617</td></tr><tr><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>.541</td><td rowspan=1 colspan=1>.380</td><td rowspan=1 colspan=1>.866</td><td rowspan=1 colspan=1>.738</td><td rowspan=1 colspan=1>.833</td><td rowspan=1 colspan=1>.894</td><td rowspan=1 colspan=1>.940</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>CelebA</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>.781</td><td rowspan=1 colspan=1>.886</td><td rowspan=1 colspan=1>.901</td><td rowspan=1 colspan=1>.971</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr></table>
+
+\* CelebA doesn’t provide race annotations. The result was obtained from the whole set (white and non-white). † FairFace defines 7 race categories but only 4 races (White, Black, Asian, and Indian) were used in this result to make it comparable to UTKFace.
+
+Using models trained from these datasets, we first performed cross-dataset classifications, by alternating training sets and test sets. Note that FairFace is the only dataset with 7 races. To make it compatible with other datasets, we merged our fine racial groups when tested on other datasets. CelebA does not have race annotations but was included for gender classification.
+
+Tables 2 and 3 show the classification results for race, gender, and age on the datasets across subpopulations. As expected, each model tends to perform better on the same dataset on which it was trained. However, the accuracy of our model was highest on some variables on the $\mathrm { L F W A + }$ dataset and also very close to the leader in other cases. This is partly because $\mathrm { L F W A + }$ is the most biased dataset and ours is the most diverse, and thus more generalizable dataset.
+
+# 4.3 GENERALIZATION PERFORMANCE
+
+# 4.3.1 DATASETS
+
+To test the generalization performance of the models, we consider three novel datasets. Note that these datasets were collected from completely different sources than our data from Flickr and not used in training. Since we want to measure the effectiveness of the model on diverse races, we chose the test datasets that contain people in different locations as follows.
+
+Geo-tagged Tweets. First we consider images uploaded by Twitter users whose locations are identified by geo-tags (longitude and latitude), provided by (Steinert-Threlkeld, 2018). From this set, we chose four countries (France, Iraq, Philippines, and Venezuela) and randomly sampled 5,000 faces.
+
+Media Photographs. Next, we also use photographs posted by 500 online professional media outlets. Specifically, we use a public dataset of tweet IDs (Littman et al., 2017) posted by 4,000 known media accounts, e.g. @nytimes. Note that although we use Twitter to access the photographs, these tweets are simply external links to pages in the main newspaper sites. Therefore this data is considered as media photographs and different from general tweet images mostly uploaded by ordinary users. We randomly sampled 8,000 faces from the set.
+
+Protest Dataset. Lastly, we also use a public image dataset collected for a recent protest activity study (Won et al., 2017). The authors collected the majority of data from Google Image search by using keywords such as “Venezuela protest” or “football game” (for hard negatives). The dataset exhibits a wide range of diverse race and gender groups engaging in different activities in various countries. We randomly sampled 8,000 faces from the set.
+
+These faces were annotated for gender, race, and age by Amazon Mechanical Turk workers.
+
+<table><tr><td>Race</td><td colspan="2">White</td><td colspan="2">Black</td><td colspan="2">East Asian</td><td colspan="2"></td><td colspan="2">SE Asian</td><td colspan="2">Latino Indian</td><td colspan="2">Middle Eastern</td><td colspan="2"></td><td colspan="4"></td></tr><tr><td>Gender</td><td>M</td><td>F</td><td>M</td><td>F</td><td>M</td><td>F</td><td>M</td><td>F</td><td>M</td><td>F</td><td>M</td><td>F</td><td>M</td><td>F</td><td>Max</td><td>Min</td><td>AVG</td><td>STDV</td><td>E</td></tr><tr><td>FairFace</td><td>.967</td><td>.954</td><td>.958</td><td>.917</td><td>.873</td><td>.939</td><td>.909</td><td>.906</td><td>.977</td><td>.960</td><td>.966</td><td>.947</td><td>.991</td><td>.946</td><td>.991</td><td>.873</td><td>.944</td><td>.032</td><td>.055</td></tr><tr><td>UTK</td><td>.926</td><td>.864</td><td>.909</td><td>.795</td><td>.841</td><td>.824</td><td>.906</td><td>.795</td><td>.939</td><td>.821</td><td>.978</td><td>.742</td><td>.949</td><td>.730</td><td>.978</td><td>.730</td><td>.859</td><td>.078</td><td>.127</td></tr><tr><td>LFWA+</td><td>.946</td><td>.680</td><td>.974</td><td>.432</td><td>.826</td><td>.684</td><td>.938</td><td>.574</td><td>.951</td><td>.613</td><td>.968</td><td>.518</td><td>.988</td><td>.635</td><td>.988</td><td>.432</td><td>.766</td><td>.196</td><td>.359</td></tr><tr><td>CelebA</td><td>.829</td><td>.958</td><td>.819</td><td>.919</td><td>.653</td><td>.939</td><td>.768</td><td>.923</td><td>.843</td><td>.955</td><td>.866</td><td>.856</td><td>.924</td><td>.874</td><td>.958</td><td>.653</td><td>.866</td><td>.083</td><td>.166</td></tr></table>
+
+Table 4: Gender classification accuracy measured on external validation datasets across gender-race groups.
+
+# 4.3.2 RESULT
+
+Table 7 shows the classification accuracy of different models. Because our dataset is larger than $\mathrm { L F W A + }$ and UTKFace, we report the three variants of the FairFace model by limiting the size of a training set (9k, 18k, and Full) for fair comparisons.
+
+Improved Accuracy. As clearly shown in the result, the model trained by FairFace outperforms all the other models for race, gender, and age, on the novel datasets, which have never been used in training and also come from different data sources. The models trained with fewer training images $\operatorname { \mathrm { 9 k } }$ and 18k) still outperform other datasets including CelebA which is larger than FairFace. This suggests that the dataset size is not the only reason for the performance improvement.
+
+Balanced Accuracy. Our model also produces more consistent results – for race, gender, age classification – across different race groups compared to other datasets. We measure the model consistency by standard deviations of classification accuracy measured on different sub-populations, as shown in Table 5. More formally, one can consider conditional use accuracy equality (Berk et al.) or equalized odds (Hardt et al., 2016) as the measure of fair classification. For gender classification:
+
+$$
+\begin{array} { r l r } & { } & { P ( \widehat { Y } = i | Y = i , A = j ) = P ( \widehat { Y } = i | Y = i , A = k ) , } \\ & { } & { i \in \{ \mathrm { m a l e , f e m a l e } \} , \forall j , k \in \mathrm { D } , } \end{array}
+$$
+
+where $\widehat { Y }$ is the predicted gender, $Y$ is the true gender, A refers to the demographic group, and D is the set of different demographic groups being considered (race). When we consider different gender groups for $A$ , this needs to be modified to measure accuracy equality Berk et al.:
+
+$$
+P ( \widehat { Y } = Y | A = j ) = P ( \widehat { Y } = Y | A = k ) , \forall j , k \in \mathbb { D } .
+$$
+
+We therefore define the maximum accuracy disparity of a classifier as follows:
+
+$$
+\epsilon ( \widehat { Y } ) = \operatorname* { m a x } _ { \forall j , k \in \mathrm { D } } \bigg ( \log \frac { P ( \widehat { Y } = Y | A = j ) } { P ( \widehat { Y } = Y | A = k ) } \bigg ) .
+$$
+
+Table 4 shows the gender classification accuracy of different models measured on the external validation datasets for each race and gender group. The FairFace model achieves the lowest maximum accuracy disparity. The LFWA+ model yields the highest disparity, strongly biased toward the male category. The CelebA model tends to exhibit a bias toward the female category as the dataset contains more female images than male.
+
+The FairFace model achieves less than $1 \%$ accuracy discrepancy between male female and White non-White for gender classification (Table 7). All the other models show a strong bias toward the male class, yielding much lower accuracy on the female group, and perform more inaccurately on the non-White group. The gender performance gap was the biggest in $\mathrm { L F W A + }$ $( 3 2 \% )$ , which is the smallest among the datasets used in the experiment. Recent work has also reported asymmetric gender biases in commercial computer vision services (Buolamwini & Gebru, 2018), and our result further suggests the cause is likely due to the unbalanced representation in training data.
+
+Data Coverage and Diversity. We further investigate dataset characteristics to measure the data diversity in our dataset. We first visualize randomly sampled faces in 2D space using t-SNE (Maaten & Hinton, 2008) as shown in Figure 3. We used the facial embedding based on ResNet-34 from dlib, which was trained from the FaceScrub dataset ( $\mathrm { N g }$ & Winkler, 2014), the VGG-Face dataset (Parkhi et al., 2015) and other online sources, which are likely dominated by the White faces. The faces in FairFace are well spread in the space, and the race groups are loosely separated from each other.
+
+This is in part because the embedding was trained from biased datasets, but it also suggests that the dataset contains many non-typical examples. $\mathrm { L F W A + }$ was derived from LFW, which was developed for face recognition, and therefore contains multiple images of the same individuals, i.e. clusters. UTKFace also tends to focus more on local clusters compared to FairFace.
+
+![](images/5871736b996e778bdc34b2dcdfaac677a315efd63ae2b6264e31d4ba3df288f5.jpg)  
+Figure 3: t-SNE visualizations (Maaten & Hinton, 2008) of faces in datasets.
+
+To explicitly measure the diversity of faces in these datasets, we examine the distributions of pairwise distance between faces (Figure 4). On the random subsets, we first obtained the same 128- dimensional facial embedding from dlib and measured pair-wise distance. Figure 4 shows the CDF functions for 3 datasets. As conjectured, UTKFace had more faces that are tightly clustered together and very similar to each other, compared to our dataset. Surprisingly, the faces in $\mathrm { L F W A + }$ were shown very diverse and far from each other, even though the majority of the examples contained a white face. We believe this is mostly due to the fact that the face embedding was also trained on a very similar white-oriented dataset which will be effective in separating white faces, not because the appearance of their faces is actually diverse. (See Figure 2)
+
+![](images/74cf2b87492314fbd722f29f5072f60aec244800dc195530fcf2f7384026593a.jpg)  
+Figure 4: Distribution of pairwise distances of faces in 3 datasets measured by L1 distance on face embedding.
+
+Table 5: Gender classification accuracy on external validation datasets, across race and age groups.   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>Meanacross races</td><td rowspan=1 colspan=1>SDacross races</td><td rowspan=1 colspan=1>Mean across ages</td><td rowspan=1 colspan=1>SD across ages</td></tr><tr><td rowspan=4 colspan=1>Modeltrained on</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>94.89 %</td><td rowspan=1 colspan=1>3.03%</td><td rowspan=1 colspan=1>92.95%</td><td rowspan=1 colspan=1>6.63%</td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>89.54%</td><td rowspan=1 colspan=1>3.34%</td><td rowspan=1 colspan=1>84.23%</td><td rowspan=1 colspan=1>12.83%</td></tr><tr><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>82.46%</td><td rowspan=1 colspan=1>5.60%</td><td rowspan=1 colspan=1>78.50%</td><td rowspan=1 colspan=1>11.51%</td></tr><tr><td rowspan=1 colspan=1>CelebA</td><td rowspan=1 colspan=1>86.03%</td><td rowspan=1 colspan=1>4.57%</td><td rowspan=1 colspan=1>79.53%</td><td rowspan=1 colspan=1>17.96%</td></tr></table>
+
+# 4.4 EVALUATING COMMERCIAL FACE GENDER CLASSIFIERS
+
+Previous studies have reported that popular commercial face analytic models show inconsistent classification accuracies across different demographic groups (Buolamwini & Gebru, 2018; Raji & Buolamwini, 2019). We used the FairFace images to test several online APIs for gender classification: Microsoft Face API, Amazon Rekognition, IBM Watson Visual Recognition, and ${ \mathrm { F a c e } } + +$ . Compared to prior work using politicians’ faces, our dataset is much more diverse in terms of race, age, expressions, head orientation, and photographic conditions, and thus serves as a much better benchmark for bias measurement. We used 7,476 random samples from FairFace such that it contains an equal number of faces from each race, gender, and age group. We left out children under the age of 20, as these pictures were often ambiguous and the gender could not be determined for certain. The experiments were conducted on August 13th - 16th, 2019.
+
+Table 6: Classification accuracy of commercial services on FairFace dataset. (\*Microsoft, \*Face++, $\bf \Phi _ { \mathrm { m } }$ indicate accuracies only on the detected faces, ignoring mis-detections.)   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>White</td><td rowspan=1 colspan=2>Black</td><td rowspan=1 colspan=2>East Asian</td><td rowspan=1 colspan=2>SE Asian</td><td rowspan=1 colspan=2>Latino</td><td rowspan=1 colspan=1>Inc</td><td rowspan=1 colspan=1>lan</td><td rowspan=1 colspan=2>Mid-Eastern</td><td rowspan=1 colspan=2></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>F</td><td rowspan=1 colspan=1>M</td><td rowspan=1 colspan=1>F</td><td rowspan=1 colspan=1>M</td><td rowspan=1 colspan=1>F</td><td rowspan=1 colspan=1>M</td><td rowspan=1 colspan=1>F</td><td rowspan=1 colspan=1>M</td><td rowspan=1 colspan=1>F</td><td rowspan=1 colspan=1>M</td><td rowspan=1 colspan=1>F</td><td rowspan=1 colspan=1>M</td><td rowspan=1 colspan=1>F</td><td rowspan=1 colspan=1>M</td><td rowspan=1 colspan=1>Mean</td><td rowspan=1 colspan=1>STD</td></tr><tr><td rowspan=1 colspan=1>Amazon</td><td rowspan=1 colspan=1>.923</td><td rowspan=1 colspan=1>.966</td><td rowspan=1 colspan=1>.901</td><td rowspan=1 colspan=1>.955</td><td rowspan=1 colspan=1>.925</td><td rowspan=1 colspan=1>.949</td><td rowspan=1 colspan=1>.918</td><td rowspan=1 colspan=1>.914</td><td rowspan=1 colspan=1>.921</td><td rowspan=1 colspan=1>.987</td><td rowspan=1 colspan=1>.951</td><td rowspan=1 colspan=1>.979</td><td rowspan=1 colspan=1>.906</td><td rowspan=1 colspan=1>.983</td><td rowspan=1 colspan=1>.941</td><td rowspan=1 colspan=1>.030</td></tr><tr><td rowspan=1 colspan=1>Microsoft</td><td rowspan=1 colspan=1>.822</td><td rowspan=1 colspan=1>.777</td><td rowspan=1 colspan=1>.766</td><td rowspan=1 colspan=1>.717</td><td rowspan=1 colspan=1>.824</td><td rowspan=1 colspan=1>.775</td><td rowspan=1 colspan=1>.852</td><td rowspan=1 colspan=1>.794</td><td rowspan=1 colspan=1>.843</td><td rowspan=1 colspan=1>.848</td><td rowspan=1 colspan=1>.863</td><td rowspan=1 colspan=1>.790</td><td rowspan=1 colspan=1>.839</td><td rowspan=1 colspan=1>.772</td><td rowspan=1 colspan=1>.806</td><td rowspan=1 colspan=1>.042</td></tr><tr><td rowspan=1 colspan=1>Face++</td><td rowspan=1 colspan=1>.888</td><td rowspan=1 colspan=1>.959</td><td rowspan=1 colspan=1>.805</td><td rowspan=1 colspan=1>.944</td><td rowspan=1 colspan=1>.876</td><td rowspan=1 colspan=1>.904</td><td rowspan=1 colspan=1>.884</td><td rowspan=1 colspan=1>.897</td><td rowspan=1 colspan=1>.865</td><td rowspan=1 colspan=1>.981</td><td rowspan=1 colspan=1>.770</td><td rowspan=1 colspan=1>.968</td><td rowspan=1 colspan=1>.822</td><td rowspan=1 colspan=1>.978</td><td rowspan=1 colspan=1>.896</td><td rowspan=1 colspan=1>.066</td></tr><tr><td rowspan=1 colspan=1>IBM</td><td rowspan=1 colspan=1>.910</td><td rowspan=1 colspan=1>.966</td><td rowspan=1 colspan=1>.758</td><td rowspan=1 colspan=1>.927</td><td rowspan=1 colspan=1>.899</td><td rowspan=1 colspan=1>.910</td><td rowspan=1 colspan=1>.852</td><td rowspan=1 colspan=1>.919</td><td rowspan=1 colspan=1>.884</td><td rowspan=1 colspan=1>.972</td><td rowspan=1 colspan=1>.811</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.871</td><td rowspan=1 colspan=1>.959</td><td rowspan=1 colspan=1>.900</td><td rowspan=1 colspan=1>.061</td></tr><tr><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.987</td><td rowspan=1 colspan=1>.991</td><td rowspan=1 colspan=1>.964</td><td rowspan=1 colspan=1>.974</td><td rowspan=1 colspan=1>.966</td><td rowspan=1 colspan=1>.979</td><td rowspan=1 colspan=1>.978</td><td rowspan=1 colspan=1>.961</td><td rowspan=1 colspan=1>.991</td><td rowspan=1 colspan=1>.989</td><td rowspan=1 colspan=1>.991</td><td rowspan=1 colspan=1>.987</td><td rowspan=1 colspan=1>.972</td><td rowspan=1 colspan=1>.991</td><td rowspan=1 colspan=1>.980</td><td rowspan=1 colspan=1>.011</td></tr><tr><td rowspan=1 colspan=1>*Microsoft</td><td rowspan=1 colspan=1>.973</td><td rowspan=1 colspan=1>.998</td><td rowspan=1 colspan=1>.962</td><td rowspan=1 colspan=1>.967</td><td rowspan=1 colspan=1>.963</td><td rowspan=1 colspan=1>.976</td><td rowspan=1 colspan=1>.960</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.983</td><td rowspan=1 colspan=1>.993</td><td rowspan=1 colspan=1>.975</td><td rowspan=1 colspan=1>.991</td><td rowspan=1 colspan=1>.966</td><td rowspan=1 colspan=1>.993</td><td rowspan=1 colspan=1>.975</td><td rowspan=1 colspan=1>.014</td></tr><tr><td rowspan=1 colspan=1>*Face++</td><td rowspan=1 colspan=1>.893</td><td rowspan=1 colspan=1>.968</td><td rowspan=1 colspan=1>.810</td><td rowspan=1 colspan=1>.956</td><td rowspan=1 colspan=1>.878</td><td rowspan=1 colspan=1>.911</td><td rowspan=1 colspan=1>.886</td><td rowspan=1 colspan=1>.899</td><td rowspan=1 colspan=1>.870</td><td rowspan=1 colspan=1>.983</td><td rowspan=1 colspan=1>.773</td><td rowspan=1 colspan=1>.975</td><td rowspan=1 colspan=1>.827</td><td rowspan=1 colspan=1>.983</td><td rowspan=1 colspan=1>.901</td><td rowspan=1 colspan=1>.067</td></tr><tr><td rowspan=1 colspan=1>*IBM</td><td rowspan=1 colspan=1>.914</td><td rowspan=1 colspan=1>.981</td><td rowspan=1 colspan=1>.761</td><td rowspan=1 colspan=1>.956</td><td rowspan=1 colspan=1>.909</td><td rowspan=1 colspan=1>.920</td><td rowspan=1 colspan=1>.852</td><td rowspan=1 colspan=1>.926</td><td rowspan=1 colspan=1>.892</td><td rowspan=1 colspan=1>.977</td><td rowspan=1 colspan=1>.819</td><td rowspan=1 colspan=1>.975</td><td rowspan=1 colspan=1>.881</td><td rowspan=1 colspan=1>.979</td><td rowspan=1 colspan=1>.910</td><td rowspan=1 colspan=1>.066</td></tr></table>
+
+Table 6 shows the gender classification accuracies of the tested APIs. These APIs first detect a face from an input image and classify its gender. Not all 7,476 faces were detected by these APIs with the exception of Amazon Rekognition which detected all of them. Table 8 in Appendix reports the detection rate.1 We report two sets of accuracies: 1) treating mis-detections as mis-classifications and 2) excluding mis-detections. For comparison, we included a model trained with our dataset to provide an upper bound for classification accuracy. Following prior work (Merler et al., 2019), we also show the classification accuracy as a function of skin color in Figure 6.
+
+The results suggest several findings. First, all tested gender classifiers still favor the male category, which is consistent with the previous report (Buolamwini & Gebru, 2018). Second, dark-skinned females tend to yield higher classification error rates, but there exist many exceptions. For example, Indians have darker skin tones (Figure 5), but some APIs (Amazon and MS) classified them more accurately than Whites. This suggests skin color alone, or any other individual phenotypic feature, is not a sufficient guideline to study model bias. Third, face detection can also introduce significant gender bias. Microsoft’s model failed to detect many male faces, an opposite direction from the gender classification bias. This was not reported in previous studies which only used clean profile images of frontal faces.
+
+# 5 CONCLUSION
+
+This paper proposes a novel face image dataset balanced on race, gender and age. Compared to existing large-scale in-the-wild datasets, our dataset achieves much better generalization classification performance for gender, race, and age on novel image datasets collected from Twitter, international online newspapers, and web search, which contain more non-White faces than typical face datasets. We show that the model trained from our dataset produces balanced accuracy across race, whereas other datasets often lead to asymmetric accuracy on different race groups.
+
+This dataset was derived from the Yahoo YFCC100m dataset (Thomee et al.) for the images with Creative Common Licenses by Attribution and Share Alike, which permit both academic and commercial usage. Our dataset can be used for training a new model and verifying balanced accuracy of existing classifiers.
+
+Algorithmic fairness is an important aspect to consider in designing and developing AI systems, especially because these systems are being translated into many areas in our society and affecting our decision making. Large scale image datasets have contributed to the recent success in computer vision by improving model accuracy; yet the public and media have doubts about its transparency. The novel dataset proposed in this paper will help us discover and mitigate race and gender bias present in computer vision systems such that such systems can be more easily accepted in society.
+
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+
+# A APPENDIX
+
+Table 7: Classification accuracy on external validation datasets.   
+
+<table><tr><td rowspan=1 colspan=1>All</td><td rowspan=1 colspan=1>Female</td><td rowspan=1 colspan=1>Male</td><td rowspan=1 colspan=1>White</td><td rowspan=1 colspan=1>Non-White</td><td rowspan=1 colspan=1>Black</td><td rowspan=1 colspan=1>Asian</td><td rowspan=1 colspan=1>E Asian</td><td rowspan=1 colspan=1>SE Asian</td><td rowspan=1 colspan=1>Latino</td><td rowspan=1 colspan=1>Indian</td><td rowspan=1 colspan=1>Mid-East</td><td rowspan=1 colspan=1>0-9</td><td rowspan=1 colspan=1>10-29</td><td rowspan=1 colspan=1>30-49</td><td rowspan=1 colspan=1>50+</td></tr><tr><td rowspan=2 colspan=1>Twitter</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.733</td><td rowspan=1 colspan=1>.726</td><td rowspan=1 colspan=1>.737</td><td rowspan=1 colspan=1>.899</td><td rowspan=1 colspan=1>.548</td><td rowspan=1 colspan=1>.695</td><td rowspan=1 colspan=1>.888</td><td rowspan=1 colspan=1>.705</td><td rowspan=1 colspan=1>.465</td><td rowspan=1 colspan=1>.305</td><td rowspan=1 colspan=1>.492</td><td rowspan=1 colspan=1>.743</td><td rowspan=1 colspan=1>.756</td><td rowspan=1 colspan=1>.691</td><td rowspan=1 colspan=1>.768</td><td rowspan=1 colspan=1>.777</td></tr><tr><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>.626</td><td rowspan=1 colspan=1>.596</td><td rowspan=1 colspan=1>.647</td><td rowspan=1 colspan=1>.965</td><td rowspan=1 colspan=1>.284</td><td rowspan=1 colspan=1>.283</td><td rowspan=1 colspan=1>.425</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>.639</td><td rowspan=1 colspan=1>.562</td><td rowspan=1 colspan=1>.705</td><td rowspan=1 colspan=1>.751</td></tr><tr><td rowspan=1 colspan=1>Media</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.866</td><td rowspan=1 colspan=1>.874</td><td rowspan=1 colspan=1>.863</td><td rowspan=1 colspan=1>.949</td><td rowspan=1 colspan=1>.685</td><td rowspan=1 colspan=1>.890</td><td rowspan=1 colspan=1>.918</td><td rowspan=1 colspan=1>.886</td><td rowspan=1 colspan=1>.152</td><td rowspan=1 colspan=1>.267</td><td rowspan=1 colspan=1>.691</td><td rowspan=1 colspan=1>.704</td><td rowspan=1 colspan=1>.833</td><td rowspan=1 colspan=1>.853</td><td rowspan=1 colspan=1>.852</td><td rowspan=1 colspan=1>.893</td></tr><tr><td rowspan=2 colspan=1>Protest</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.846</td><td rowspan=1 colspan=1>.849</td><td rowspan=1 colspan=1>.844</td><td rowspan=1 colspan=1>.935</td><td rowspan=1 colspan=1>.683</td><td rowspan=1 colspan=1>.859</td><td rowspan=1 colspan=1>843</td><td rowspan=1 colspan=1>.702</td><td rowspan=1 colspan=1>.510</td><td rowspan=1 colspan=1>.169</td><td rowspan=1 colspan=1>.649</td><td rowspan=1 colspan=1>.779</td><td rowspan=1 colspan=1>.839</td><td rowspan=1 colspan=1>.821</td><td rowspan=1 colspan=1>.837</td><td rowspan=1 colspan=1>.881</td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>.706</td><td rowspan=1 colspan=1>.723</td><td rowspan=1 colspan=1>.697</td><td rowspan=1 colspan=1>.821</td><td rowspan=1 colspan=1>.536</td><td rowspan=1 colspan=1>.714</td><td rowspan=1 colspan=1>.456</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>.591</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>.681</td><td rowspan=1 colspan=1>.658</td><td rowspan=1 colspan=1>.685</td><td rowspan=1 colspan=1>.787</td></tr><tr><td rowspan=4 colspan=1>Average</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.815</td><td rowspan=1 colspan=1>.816</td><td rowspan=1 colspan=1>.815</td><td rowspan=1 colspan=1>.928</td><td rowspan=1 colspan=1>.639</td><td rowspan=1 colspan=1>.815</td><td rowspan=1 colspan=1>.883</td><td rowspan=1 colspan=1>.764</td><td rowspan=1 colspan=1>376</td><td rowspan=1 colspan=1>.247</td><td rowspan=1 colspan=1>.611</td><td rowspan=1 colspan=1>.742</td><td rowspan=1 colspan=1>.809</td><td rowspan=1 colspan=1>.788</td><td rowspan=1 colspan=1>.819</td><td rowspan=1 colspan=1>.850</td></tr><tr><td rowspan=1 colspan=1>FairFace18K</td><td rowspan=1 colspan=1>.800</td><td rowspan=1 colspan=1>.812</td><td rowspan=1 colspan=1>.795</td><td rowspan=1 colspan=1>.917</td><td rowspan=1 colspan=1>.588</td><td rowspan=1 colspan=1>.779</td><td rowspan=1 colspan=1>.856</td><td rowspan=1 colspan=1>.685</td><td rowspan=1 colspan=1>355</td><td rowspan=1 colspan=1>.279</td><td rowspan=1 colspan=1>.502</td><td rowspan=1 colspan=1>.625</td><td rowspan=1 colspan=1>.786</td><td rowspan=1 colspan=1>.773</td><td rowspan=1 colspan=1>.809</td><td rowspan=1 colspan=1>.827</td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>674</td><td rowspan=1 colspan=1>.687</td><td rowspan=1 colspan=1>.668</td><td rowspan=1 colspan=1>815</td><td rowspan=1 colspan=1>479</td><td rowspan=1 colspan=1>.702</td><td rowspan=1 colspan=1>.507</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>555</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>.644</td><td rowspan=1 colspan=1>.643</td><td rowspan=1 colspan=1>.672</td><td rowspan=1 colspan=1>719</td></tr><tr><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>.684</td><td rowspan=1 colspan=1>.726</td><td rowspan=1 colspan=1>.741</td><td rowspan=1 colspan=1>.969</td><td rowspan=1 colspan=1>.348</td><td rowspan=1 colspan=1>.395</td><td rowspan=1 colspan=1>.497</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>.670</td><td rowspan=1 colspan=1>.621</td><td rowspan=1 colspan=1>.675</td><td rowspan=1 colspan=1>.758</td></tr><tr><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>AlI</td><td rowspan=1 colspan=1>Female</td><td rowspan=1 colspan=1>Male</td><td rowspan=1 colspan=1>White</td><td rowspan=1 colspan=1>Non-White</td><td rowspan=1 colspan=1>Black</td><td rowspan=1 colspan=1>Asian</td><td rowspan=1 colspan=1>E Asian</td><td rowspan=1 colspan=1>SE Asian</td><td rowspan=1 colspan=1>Latino</td><td rowspan=1 colspan=1>Indian</td><td rowspan=1 colspan=1>Mid-East</td><td rowspan=1 colspan=1>0-9</td><td rowspan=1 colspan=1>10-29</td><td rowspan=1 colspan=1>30-49</td><td rowspan=1 colspan=1>50+</td></tr><tr><td rowspan=1 colspan=1>Twitter</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.940</td><td rowspan=1 colspan=1>.948</td><td rowspan=1 colspan=1>.935</td><td rowspan=1 colspan=1>.949</td><td rowspan=1 colspan=1>.932</td><td rowspan=1 colspan=1>.932</td><td rowspan=1 colspan=1>.894</td><td rowspan=1 colspan=1>.864</td><td rowspan=1 colspan=1>.942</td><td rowspan=1 colspan=1>.963</td><td rowspan=1 colspan=1>.932</td><td rowspan=1 colspan=1>976</td><td rowspan=1 colspan=1>.817</td><td rowspan=1 colspan=1>.932</td><td rowspan=1 colspan=1>973</td><td rowspan=1 colspan=1>.959</td></tr><tr><td></td><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>.797</td><td rowspan=1 colspan=1>.637</td><td rowspan=1 colspan=1>.899</td><td rowspan=1 colspan=1>.815</td><td rowspan=1 colspan=1>.773</td><td rowspan=1 colspan=1>.789</td><td rowspan=1 colspan=1>.724</td><td rowspan=1 colspan=1>.716</td><td rowspan=1 colspan=1>.736</td><td rowspan=1 colspan=1>.804</td><td rowspan=1 colspan=1>.728</td><td rowspan=1 colspan=1>.911</td><td rowspan=1 colspan=1>.634</td><td rowspan=1 colspan=1>.769</td><td rowspan=1 colspan=1>.857</td><td rowspan=1 colspan=1>.859</td></tr><tr><td rowspan=3 colspan=1>Media</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>.927</td><td rowspan=1 colspan=1>.841</td><td rowspan=1 colspan=1>.961</td><td rowspan=1 colspan=1>.928</td><td rowspan=1 colspan=1>.915</td><td rowspan=1 colspan=1>.907</td><td rowspan=1 colspan=1>.908</td><td rowspan=1 colspan=1>.915</td><td rowspan=1 colspan=1>.869</td><td rowspan=1 colspan=1>.928</td><td rowspan=1 colspan=1>.945</td><td rowspan=1 colspan=1>.932</td><td rowspan=1 colspan=1>.679</td><td rowspan=1 colspan=1>.917</td><td rowspan=1 colspan=1>.931</td><td rowspan=1 colspan=1>.924</td></tr><tr><td rowspan=1 colspan=1>LFWA+</td><td rowspan=1 colspan=1>.887</td><td rowspan=1 colspan=1>.656</td><td rowspan=1 colspan=1>.976</td><td rowspan=1 colspan=1>.893</td><td rowspan=1 colspan=1>.871</td><td rowspan=1 colspan=1>.851</td><td rowspan=1 colspan=1>.864</td><td rowspan=1 colspan=1>.875</td><td rowspan=1 colspan=1>.804</td><td rowspan=1 colspan=1>.859</td><td rowspan=1 colspan=1>.897</td><td rowspan=1 colspan=1>.944</td><td rowspan=1 colspan=1>.688</td><td rowspan=1 colspan=1>.835</td><td rowspan=1 colspan=1>.832</td><td rowspan=1 colspan=1>.911</td></tr><tr><td rowspan=2 colspan=1>Protest</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>.901</td><td rowspan=1 colspan=1>.829</td><td rowspan=1 colspan=1>.934</td><td rowspan=1 colspan=1>.905</td><td rowspan=1 colspan=1>.873</td><td rowspan=1 colspan=1>.911</td><td rowspan=1 colspan=1>.814</td><td rowspan=1 colspan=1>.802</td><td rowspan=1 colspan=1>.843</td><td rowspan=1 colspan=1>.902</td><td rowspan=1 colspan=1>.918</td><td rowspan=1 colspan=1>.921</td><td rowspan=1 colspan=1>.611</td><td rowspan=1 colspan=1>.812</td><td rowspan=1 colspan=1>.924</td><td rowspan=1 colspan=1>.919</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.950</td><td rowspan=1 colspan=1>959</td><td rowspan=1 colspan=1>.962</td><td rowspan=1 colspan=1>951</td><td rowspan=1 colspan=1>.947</td><td rowspan=1 colspan=1>.912</td><td rowspan=1 colspan=1>.903</td><td rowspan=1 colspan=1>.913</td><td rowspan=1 colspan=1>971</td><td rowspan=1 colspan=1>.961</td><td rowspan=1 colspan=1>.985</td><td rowspan=1 colspan=1>.833</td><td rowspan=1 colspan=1>.939</td><td rowspan=1 colspan=1>975</td><td rowspan=1 colspan=1>.971</td></tr><tr><td rowspan=1 colspan=1>Average</td><td rowspan=1 colspan=1>FairFace9K</td><td rowspan=1 colspan=1>.926</td><td rowspan=1 colspan=1>.921</td><td rowspan=1 colspan=1>.927</td><td rowspan=1 colspan=1>929</td><td rowspan=1 colspan=1>.921</td><td rowspan=1 colspan=1>.922</td><td rowspan=1 colspan=1>.864</td><td rowspan=1 colspan=1>.851</td><td rowspan=1 colspan=1>.883</td><td rowspan=1 colspan=1>.942</td><td rowspan=1 colspan=1>.951</td><td rowspan=1 colspan=1>.974</td><td rowspan=1 colspan=1>.760</td><td rowspan=1 colspan=1>.901</td><td rowspan=1 colspan=1>.949</td><td rowspan=1 colspan=1>.943</td></tr><tr><td rowspan=2 colspan=1></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>CelebA</td><td rowspan=1 colspan=1>.870</td><td rowspan=1 colspan=1>.947</td><td rowspan=1 colspan=1>.829</td><td rowspan=1 colspan=1>.884</td><td rowspan=1 colspan=1>.853</td><td rowspan=1 colspan=1>.847</td><td rowspan=1 colspan=1>.838</td><td rowspan=1 colspan=1>.774</td><td rowspan=1 colspan=1>.847</td><td rowspan=1 colspan=1>.887</td><td rowspan=1 colspan=1>.864</td><td rowspan=1 colspan=1>.919</td><td rowspan=1 colspan=1>.530</td><td rowspan=1 colspan=1>.840</td><td rowspan=1 colspan=1>.900</td><td rowspan=1 colspan=1>.911</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=2 colspan=1>Twitter</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.578</td><td rowspan=1 colspan=1>.586</td><td rowspan=1 colspan=1>573</td><td rowspan=1 colspan=1>.563</td><td rowspan=1 colspan=1>.590</td><td rowspan=1 colspan=1>.557</td><td rowspan=1 colspan=1>.620</td><td rowspan=1 colspan=1>.629</td><td rowspan=1 colspan=1>.606</td><td rowspan=1 colspan=1>.581</td><td rowspan=1 colspan=1>.576</td><td rowspan=1 colspan=1>.555</td><td rowspan=1 colspan=1>.805</td><td rowspan=1 colspan=1>.666</td><td rowspan=1 colspan=1>.439</td><td rowspan=1 colspan=1>.408</td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>.366</td><td rowspan=1 colspan=1>.355</td><td rowspan=1 colspan=1>.384</td><td rowspan=1 colspan=1>.343</td><td rowspan=1 colspan=1>.385</td><td rowspan=1 colspan=1>338</td><td rowspan=1 colspan=1>.397</td><td rowspan=1 colspan=1>.382</td><td rowspan=1 colspan=1>.419</td><td rowspan=1 colspan=1>.411</td><td rowspan=1 colspan=1>.356</td><td rowspan=1 colspan=1>.345</td><td rowspan=1 colspan=1>.585</td><td rowspan=1 colspan=1>.499</td><td rowspan=1 colspan=1>.104</td><td rowspan=1 colspan=1>.307</td></tr><tr><td rowspan=1 colspan=1>Media</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.516</td><td rowspan=1 colspan=1>.511</td><td rowspan=1 colspan=1>.517</td><td rowspan=1 colspan=1>.513</td><td rowspan=1 colspan=1>.520</td><td rowspan=1 colspan=1>.483</td><td rowspan=1 colspan=1>.557</td><td rowspan=1 colspan=1>.559</td><td rowspan=1 colspan=1>.543</td><td rowspan=1 colspan=1>.537</td><td rowspan=1 colspan=1>.532</td><td rowspan=1 colspan=1>475</td><td rowspan=1 colspan=1>.714</td><td rowspan=1 colspan=1>.686</td><td rowspan=1 colspan=1>447</td><td rowspan=1 colspan=1>.501</td></tr><tr><td rowspan=2 colspan=1>Protest</td><td rowspan=1 colspan=1>FairFace</td><td rowspan=1 colspan=1>.515</td><td rowspan=1 colspan=1>.543</td><td rowspan=1 colspan=1>.502</td><td rowspan=1 colspan=1>.498</td><td rowspan=1 colspan=1>.539</td><td rowspan=1 colspan=1>.527</td><td rowspan=1 colspan=1>.584</td><td rowspan=1 colspan=1>.605</td><td rowspan=1 colspan=1>.531</td><td rowspan=1 colspan=1>.507</td><td rowspan=1 colspan=1>.581</td><td rowspan=1 colspan=1>.469</td><td rowspan=1 colspan=1>.885</td><td rowspan=1 colspan=1>.687</td><td rowspan=1 colspan=1>.395</td><td rowspan=1 colspan=1>.478</td></tr><tr><td rowspan=1 colspan=1>UTKFace</td><td rowspan=1 colspan=1>.302</td><td rowspan=1 colspan=1>.306</td><td rowspan=1 colspan=1>.294</td><td rowspan=1 colspan=1>.291</td><td rowspan=1 colspan=1>.319</td><td rowspan=1 colspan=1>.305</td><td rowspan=1 colspan=1>.316</td><td rowspan=1 colspan=1>.318</td><td rowspan=1 colspan=1>.312</td><td rowspan=1 colspan=1>.314</td><td rowspan=1 colspan=1>.371</td><td rowspan=1 colspan=1>.318</td><td rowspan=1 colspan=1>.516</td><td rowspan=1 colspan=1>.503</td><td rowspan=1 colspan=1>.114</td><td rowspan=1 colspan=1>.349</td></tr><tr><td rowspan=2 colspan=1>Average</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>FairFace9K</td><td rowspan=1 colspan=1>470</td><td rowspan=1 colspan=1>.493</td><td rowspan=1 colspan=1>459</td><td rowspan=1 colspan=1>.462</td><td rowspan=1 colspan=1>478</td><td rowspan=1 colspan=1>.449</td><td rowspan=1 colspan=1>.506</td><td rowspan=1 colspan=1>.515</td><td rowspan=1 colspan=1>.483</td><td rowspan=1 colspan=1>.473</td><td rowspan=1 colspan=1>458</td><td rowspan=1 colspan=1>.463</td><td rowspan=1 colspan=1>.662</td><td rowspan=1 colspan=1>.611</td><td rowspan=1 colspan=1>.361</td><td rowspan=1 colspan=1>.394</td></tr></table>
+
+# Table 8: Face detection rate of commercial services on FairFace dataset.
+
+<table><tr><td></td><td colspan="2">White</td><td colspan="2">Black</td><td colspan="2">East Asian</td><td colspan="2">Southeast Asian</td><td colspan="2">Latino Hispanic</td><td colspan="2">Indian</td><td colspan="2">Middle Eastern</td><td colspan="2"></td></tr><tr><td></td><td>Female</td><td>Male</td><td>Female</td><td>Male</td><td>Female</td><td>Male</td><td>Female</td><td>Male</td><td>Female</td><td>Male</td><td>Female</td><td>Male</td><td>Female</td><td>Male</td><td>Mean</td><td>STD</td></tr><tr><td>Amazon</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>.000</td></tr><tr><td>Microsoft</td><td>.845</td><td>.779</td><td>.796</td><td>.742</td><td>.856</td><td>.794</td><td>888</td><td>.830</td><td>.858</td><td>.854</td><td>.886</td><td>.798</td><td>.869</td><td>.777</td><td>.812</td><td>047</td></tr><tr><td>Face++</td><td>.994</td><td>.991</td><td>.994</td><td>.987</td><td>.998</td><td>.993</td><td>.998</td><td>.998</td><td>.994</td><td>.998</td><td>.996</td><td>.993</td><td>.994</td><td>.994</td><td>.993</td><td>.003</td></tr><tr><td>IBM</td><td>.996</td><td>.985</td><td>.996</td><td>.970</td><td>.989</td><td>989</td><td>1.000</td><td>.993</td><td>.991</td><td>.994</td><td>.991</td><td>.981</td><td>989</td><td>.979</td><td>.991</td><td>.008</td></tr></table>
+
+![](images/dc580038ea6202527358fbedff97b118633d0cc8f09ae31eade08b9b5d52d3b7.jpg)  
+Figure 5: Individual Typology Angle (ITA), i.e. skin color, distribution of different races measured in our dataset.
+
+![](images/5381acae34679c91a61a94d68db77a58b31c1484bb0681ba8f860239dfb2e21b.jpg)  
+Figure 6: Classification accuracy based on Individual Typology Angle (ITA), i.e. skin color.

parse/train/SJky6Ry0W/SJky6Ry0W.md

@@ -0,0 +1,244 @@
+# LEARNING INDEPENDENT CAUSAL MECHANISMS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Independent causal mechanisms are a central concept in the study of causality with implications for machine learning tasks. In this work we develop an algorithm to recover a set of (inverse) independent mechanisms relating a distribution transformed by the mechanisms to a reference distribution. The approach is fully unsupervised and based on a set of experts that compete for data to specialize and extract the mechanisms. We test and analyze the proposed method on a series of experiments based on image transformations. Each expert successfully maps a subset of the transformed data to the original domain, and the learned mechanisms generalize to other domains. We discuss implications for domain transfer and links to recent trends in generative modeling.
+
+# 1 INTRODUCTION
+
+Humans are able to recognize objects such as handwritten digits based on distorted inputs. When presented with digits which are translated, corrupted, or inverted, we can usually correctly label them without the need of re-learning them from scratch. The same applies for new objects, essentially after having seen them once. This may be due to the fact that human intelligence utilizes mechanisms (such as translation) that are generic and generalize across object classes. These mechanisms are modular, re-usable and broadly applicable, and the problem of learning them from data is fundamental for the study of transfer.
+
+In the field of causality, the concept of independent mechanisms plays a central role both on the conceptual level and, more recently, in applications to inference. The independent mechanism (IM) assumption states that the causal generative process of a system’s variables is composed of autonomous modules that do not inform or influence each other (Scholkopf et al., 2012; Peters et al., ¨ 2017).
+
+If a joint density is Markovian with respect to a directed graph $\mathcal { G }$ , we can write it as
+
+$$
+p ( \mathbf { x } ) = p ( x _ { 1 } , \ldots , x _ { d } ) = \prod _ { j = 1 } ^ { d } p ( x _ { j } | \mathbf { p } \mathbf { a } _ { \mathcal { G } } ^ { j } ) ,
+$$
+
+where $\mathrm { p a } _ { \mathcal { G } } ^ { j }$ denotes the parents of variable $x _ { j }$ in the graph.
+
+For a given joint density, there are usually many decompositions of the form (1), with respect to different graphs. If $\mathcal { G }$ is a causal graph, i.e., if its edges denote direct causation (Pearl, 2000), then the conditional $\bar { p } ( x _ { j } | \mathrm { p a } _ { \mathcal { G } } ^ { j } )$ can be thought of as physical mechanism generating $x _ { j }$ from its parents, and we refer to it as a causal conditional. In this case, we consider (1) a generative model where the term “generative” truly refers to a physical generative process. As an aside, we note that in the alternative view of causal models as structural equation models, each of the causal conditionals corresponds to a functional mapping and a noise variable (Pearl, 2000).
+
+By the IM assumption, the causal conditionals are autonomous modules that do not influence or inform each other. This has multiple consequences. First, knowledge of one mechanism does not contain information about another one (Appendix D). Second, if one mechanism changes (e.g., due to distribution shift), there is no reason that other mechanisms should also change, i.e., they tend to remain invariant. As a special case, it is (in principle) possible to locally intervene on one mechanism (for instance, by setting it to a constant) without affecting any of the other modules. In all these cases, most of (1) will remain unchanged. However, since the overall density will change, in the generic case the (non-causal) conditionals would change.
+
+The IM assumption can be exploited when performing causal structure inference (Peters et al., 2017). However, it also has implications for machine learning more broadly. A model which is expressed in terms of causal conditionals (rather than conditionals with respect to some other factorization) is likely to have components that better transfer or generalize to other settings (Scholkopf et al., ¨ 2012), and its modules are better suited for building complex models from simpler ones. Independent modules as sub-components can be trained independently, from multiple domains, are more likely to be re-usable. They can also be easier to interpret since they correspond to physical mechanisms. Animate intelligence cannot afford to learn new models from scratch for every new task. Rather, it is likely to rely on robust local components that can flexibly be re-used and re-purposed. It also requires local mechanisms for adapting and training modules rather than re-training the whole brain every time a new task is learned.
+
+In the present paper, we focus on a class of such modules, and on algorithms to learn them from data. We describe an architecture using competing experts specializing on different transformations. The resulting model permits a form of lifelong learning, with the possibility of easily adding, removing, retraining, and exporting its components independently.
+
+In line with the intuition given above, we illustrate our approach on MNIST digits which have undergone different transformations such as contrast inversion, noise addition and translation. Information about the nature and number of such transformations need not be known at the beginning of training. Our goal is to identify the independent mechanisms linking a reference distribution to a distribution of modified digits, and learn to invert them without supervision. The inverse mechanisms can be used to transform modified digits and classify them using a standard MNIST classifier, thus exhibiting a form of robustness that animate intelligence excels at.
+
+# 2 RELATED WORK
+
+Our work mainly draws from mixtures of experts, domain adaptation, and causality.
+
+Early works on mixture of experts date back to the early nineties (Jacobs et al. (1991), Jordan & Jacobs (1994)), and since then the topic has been subject of extensive research. Recent work include Shazeer et al. (2017), where the authors train a mixture of 1000 experts using a gating mechanism that selects only a very small number of experts for each example, and propose several technical solutions to deal with model and data parallelism. Aljundi et al. (2016) train a network of experts on multiple tasks, with a focus on lifelong learning; autoencoders are trained for each task and used as gating mechanisms.
+
+Another research direction that is relevant to our work is unsupervised domain adaptation (Bousmalis et al., 2016). These methods often use some supervision from labeled data and/or match the two distributions in a learned feature space (Tzeng et al., 2017, e.g.).
+
+The novelty of our work lies in the following aspects: (1) we automatically identify and invert a set of independent (inverse) causal mechanisms; (2) we do so using only data from an original distribution and from the mixture of transformed data, without labels; (3) the architecture is modular, can be easily expanded, and its trained modules can be reused; and (4) the method relies on competition of experts.
+
+Ideas from the field of causal inference inspire the present work. Understanding the data generating mechanisms plays a key role in causal inference, and goes beyond the statistical assumptions usually exploited in machine learning. Causality provides a framework for understanding how a system responds to interventions, and causal graphical models as well as structural equation models (SEM) are common ways of describing causal systems (Pearl, 2000; Peters et al., 2017). The IM assumption discussed in the introduction can be used for identification of causal models (Daniusis ˇ et al., 2010; Zhang et al., 2015), but causality has also proven a useful tool for discussing and understanding machine learning in the non-i.i.d. regime. Recent applications include semi-supervised learning (Scholkopf et al., 2012) and transfer learning (Rojas-Carulla et al., 2015), in which the ¨ authors focus only on linear regression models. We seek to extend applications of causal inference to more complex settings and aim to learn causal mechanisms and ultimately causal SEMs without supervision.
+
+There are close relations between our setting and recent work on deep learning for disentangling factors of variation (Chen et al., 2016; Higgins et al., 2016) as well as non-linear ICA (Hyvarinen & Morioka, 2016). In our work, causal mechanisms play the role of factors of variation. The main difference is that we currently recover inverse mechanisms as independent modular parts, instead of indentifying a joint low dimensional representation of the data without explicit separate paths for each factor.
+
+# 3 LEARNING CAUSAL MECHANISMS AS INDEPENDENT MODULES
+
+The aim of this section is twofold. First, we describe the generative process of our data. We start with a distribution $P$ that we will call “canonical” and an a priori unknown number of independent mechanisms which act on (examples drawn from) $P$ . At training time, a sample from the canonical distribution is available, as well as a dataset obtained by applying the mechanisms to (unseen) examples drawn from $P$ . Second, we propose an algorithm which recovers and learns to invert the mechanisms in an unsupervised fashion.
+
+# 3.1 FORMAL SETTING
+
+Consider a canonical distribution $P$ on $\mathbb { R } ^ { d }$ , e.g., the empirical distribution defined by MNIST digits on pixel space. We further consider $N$ measurable functions $M _ { 1 } , \dots , M _ { N } : \mathbb { R } ^ { \tilde { d } } \to \mathbb { R } ^ { d }$ , called mechanisms. We think of these as independent causal mechanisms in nature, and their number is a priori unknown. A more formal definition of independence between mechanisms is relegated to Appendix D. The mechanisms give rise to $N$ distributions $Q _ { 1 } , \ldots , Q _ { N }$ where $Q _ { j } = M _ { j } ( P )$ .1 In the MNIST example, we consider translations or adding noise as mechanisms, i.e., the corresponding $Q$ distributions are translated and noisy MNIST digits.
+
+At training time, we receive a dataset $\mathcal { D } _ { Q } = ( x _ { i } ) _ { i = 1 } ^ { n }$ drawn i.i.d. from a mixture of $Q _ { 1 } , \ldots , Q _ { N }$ , and an independent sample $\mathcal { D } _ { P }$ from the canonical distribution $P$ . Our goal is to identify the underlying mechanisms $M _ { 1 } , \dots , M _ { N }$ and learn approximate inverse mappings which allow us to map the examples from $\mathcal { D } _ { Q }$ back to their counterpart drawn from $P$ .
+
+If we were given distinct datasets $\mathcal { D } _ { Q _ { j } }$ each drawn from $Q _ { j }$ , we could individually learn each mechanism, resulting in independent approximations regardless of the properties of the training procedure. This is due to the fact that the datasets are drawn from independent mechanisms and the separate training procedure cannot generate a dependence between them. This property is independent of properties of training, and does not require that the procedure is successful, i.e., that the obtained mechanisms approximate the true $M _ { j }$ in some metric. In our case, we do not have access to the distinct datasets. Instead we construct a larger set $\mathcal { D } _ { Q }$ by first taking the union of the sets $D _ { Q _ { j } }$ , and then applying a random permutation. This corresponds to a dataset where each element has been generated by one of the (independent) mechanisms, but we don’t know by which one. Clearly, it should be harder to identify and learn independent mechanisms from such a dataset. This is the setting we address below, and the crucial idea will be that of competition.
+
+# .2 COMPETITIVE LEARNING OF INDEPENDENT MECHANISMS
+
+In this section, we introduce our training protocol to address the problem defined above.
+
+The training machine is composed of $N ^ { \prime }$ parametric functions $E _ { 1 } , \ldots , E _ { N ^ { \prime } }$ with distinct trainable parameters $\theta _ { 1 } , \ldots , \theta _ { N ^ { \prime } }$ . We refer to these functions as the experts. Note that we do not require $N ^ { \prime } = N$ , since the real number of mechanisms is unknown a priori. The goal is to maximize an objective function $c : \mathbb { R } ^ { d }  \mathbb { R }$ with the key property that $c$ takes high values on the support of the canonical distribution $P$ , and low values outside. Note that it is possible for $c$ to be a parametric function, and for these parameters to be jointly optimized with the experts during training. Below, we specify the details of this rather general definition.
+
+During training, the experts compete for the data points. Each example $x ^ { \prime }$ from $\mathcal { D } _ { Q }$ is fed to all experts independently and in parallel. Depending on the output of each expert $c _ { j } \ = \ c ( E _ { j } ( x ^ { \prime } ) )$ , we select the winning expert $E _ { j ^ { * } }$ , where $j ^ { * } = \arg \operatorname* { m a x } _ { j } ( c _ { j } )$ . $E _ { j ^ { * } }$ wins the example $x ^ { \prime }$ , and its parameters $\theta _ { j ^ { * } }$ are updated as to maximize $c ( E _ { j ^ { * } } ( x ^ { \prime } ) )$ , while the other experts remain unchanged. The motivation behind competitively updating only the winning expert is to enforce specialization; the best performing expert becomes even better at mapping $x ^ { \prime }$ back to the corresponding sample from the canonical distribution. Figure 1 depicts this procedure. Overall, our optimization problem reads
+
+![](images/64b721abdb27e6f8917005d01f43ad0a1dbd48c1ce6fea3974ba5876a1d32bd6.jpg)  
+Figure 1: We show how a transformed example, here a noisy digit, is processed by a competition of experts. Only Expert 3 is specializing on denoising, it wins the example and gets trained on it, whereas the others perform translations and are not updated.
+
+$$
+\theta _ { 1 } ^ { * } , \ldots , \theta _ { N ^ { \prime } } ^ { * } = \underset { \theta _ { 1 } , \ldots , \theta _ { N ^ { \prime } } } { \arg \operatorname* { m a x } } \mathbb { E } _ { x ^ { \prime } \sim Q } \left( \operatorname* { m a x } _ { j \in \{ 1 , \ldots , N ^ { \prime } \} } c ( E _ { \theta _ { j } } ( x ^ { \prime } ) ) \right) .
+$$
+
+The training described above raises a number of questions, which we address next.
+
+1. Convergence criterion. Since the problem is fully unsupervised, there is no straightforward way of measuring convergence, which raises the question of how to choose a stopping time for the competitive procedure. As an example, one may act according to one of the following: $a$ ) fix a maximum number of iterations or $^ b$ ) stop if each example is assigned to the same experts for a pre-defined number of iterations (i.e., each expert consistently wins the same data points).
+
+2. Selecting the appropriate number of experts. Generally, the number of mechanisms $N$ which generated the dataset $\mathcal { D } _ { Q }$ is not available a priori. Therefore, it is important to develop an adaptive procedure for setting up the number of experts $N ^ { \prime }$ . This is a common problem shared with most clustering techniques. Given the modular behavior of the procedure, experts may be added or removed during or after training, making the framework very flexible. Assuming however that the number of experts is fixed, we speculate that the following behaviors are likely.
+
+If $N ^ { \prime } > N$ (too many experts): a) some of the experts do not specialize and do not win any example in the dataset; or b) some tasks are divided between experts (for instance, each expert can specialize in a mode of the distribution of the same task). In a), the inactive experts can be removed, and in b) experts sharing the same task can be merged into a wider expert.2
+
+If $N ^ { \prime } < N$ (too few experts): a) some of the experts specialize in multiple tasks or b) some of the tasks are not learned by the experts, so that data points from such tasks lead to a poor score across all experts.
+
+While these questions are relevant, we do not develop them in detail and leave them for further research. Some experiments substantiating these claims can be found in Appendix A.2.
+
+3. Time and space complexity. Each example has to be evaluated by all experts in order to assign it to the winning expert. While this results in a computational cost that depends linearly on the number of experts, these evaluations can be done in parallel and therefore the time complexity of a single iteration can be bounded by the complexity to compute the output of a single expert. Moreover, as each expert will in principle have a smaller architecture than a single large network, the committee of experts will typically be faster to execute.
+
+Concrete protocol for neural networks. One possible model class for the experts are deep neural networks. Training using backpropagation is particularly well suited for the online nature of the training proposed: after an expert wins a data point $x ^ { \prime }$ , its parameters are updated by backpropagation, while other experts remain untouched. Moreover, recent advances in generative modeling give rise to natural choices for the loss function $c$ . For instance, given a variational autoencoder (VAE) (Kingma & Welling, 2013) trained on the canonical distribution $P$ , one may define $c ( x ^ { \prime } )$ as the opposite of the VAE loss. The assumption is that the loss will only be low for examples drawn from $P$ . Another possibility is to use adversarial training (Goodfellow et al., 2014), and use as an objective function the output of a discriminator network trained on the canonical sample $\mathcal { D } _ { P }$ and against the outputs of the experts. In the next section we introduce a formal description of a training procedure based on adversarial training in Algorithm 1, and present experimental evidence of its good performance.
+
+# 4 EXPERIMENTS
+
+In this set of experiments we test the method presented in Section 3 on the MNIST dataset transformed with the set of mechanisms described in detail in the Appendix C, i.e. eight directions of translations by 4 pixels (up, down, left, right, and the four diagonals), contrast inversion, addition of noise, for a total of 10 transformations. We split the training partition of MNIST in half, and transform all and only the examples in the first half: this ensures that there is no matching ground truth for the experts to learn the mechanisms, and that learning is fully unsupervised. As a preprocessing step, the digits are zero-padded so that they have size $3 2 \times 3 2$ pixels, and the pixel intensities are scaled between 0 and 1. This is done even before any mechanism is applied. We use deep neural networks for both the experts and the selection mechanism, and use an adversarial training scheme.
+
+Each expert $E _ { i }$ can be seen as a generator from a GAN, that is conditioned on an input image instead of (or in addition to) a noise vector. A discriminator $D$ provides gradients for training the experts and acts also as a selection mechanism $c$ : only the expert whose output obtains the higher score from $D$ wins the example, and is trained on it to maximize the output of $D$ . We describe the exact algorithm used to train the networks in these experiments in Algorithm 1. The discriminator is trained to maximize the following cross-entropy loss:
+
+$$
+\operatorname* { m a x } _ { \theta _ { D } } \left( \mathbb { E } _ { x \sim P } \log ( D _ { \theta _ { D } } ( x ) ) + \frac { 1 } { N ^ { \prime } } \sum _ { j = 1 } ^ { N ^ { \prime } } \mathbb { E } _ { x ^ { \prime } \sim Q } \left( \log ( 1 - D _ { \theta _ { D } } ( E _ { \theta _ { j } } ( x ^ { \prime } ) ) ) \right) \right)
+$$
+
+For simplicity, we assume for the rest of this section that the number of experts $N ^ { \prime }$ equals the number of true mechanisms $N$ . Results where $N \neq N ^ { \prime }$ are relegated to Appendix A.2.
+
+Neural nets details. Each expert is a CNN with five convolutional layers, 32 filters per layer of size $3 \times 3$ , ELU (Clevert et al. (2015)) as activation function, batch normalization (Ioffe & Szegedy (2015)), and zero padding. The discriminator is also a CNN, with average pooling every two convolutional layers, growing number of filters, and a fully connected layer with 1024 neurons as last hidden layer. Both networks are trained using Adam as optimizer (Kingma & Ba (2014)), with the default hyper-parameters.3
+
+Unless specified otherwise, after a random weight initialization we first train the experts to approximate the identity mapping on our data, by pretraining them for up to 200 iterations on predicting identical input-output pairs randomly selected from the transformed dataset. This makes the experts start from similar grounds, and we found that this improved the speed and robustness of convergence. We will refer to this as approximate identity initialization for the rest of the paper.
+
+<table><tr><td>Algorithm1 Learning independent mechanisms using competition of experts and adversarial training</td></tr><tr><td>Precondition: X: data sampled from P; X&#x27;: data sampled from DQ; D discriminator; N&#x27;: number of experts; T: maximum number of iterations;</td></tr><tr><td>(p) highlights that the steps in the instruction can be executed in parallel</td></tr><tr><td>1{E←TrainNewAutoencoderOn(X)}1 &gt; Init set of experts as approx identity (p)</td></tr><tr><td>2 fort←1toTdo x,x&#x27; ← Sample(X), Sample(X&#x27;) &gt; Sample minibatches</td></tr><tr><td>{cj←D(Ej(x）}1 &gt; Scores from D for all outputs from the experts (p)</td></tr><tr><td></td></tr><tr><td>{B←Adam(axjg))</td></tr></table>
+
+A minibatch of 32 transformed MNIST digits, each transformed by a randomly chosen mechanism, is fed to all experts $E _ { i }$ . The outputs are fed to the discriminator $D$ , which computes a score for each of them. For each example the cross entropy loss in Equation (3) and the resulting gradients are computed only for the output of the highest scoring expert, and they are used to update both the discriminator (when 0 is the target in the cross entropy) and the winning expert (when using 1 as the target). In order to encourage the expert to specialize, the discriminator is also explicitly trained against the outputs of the losing experts. Then, a minibatch of canonical MNIST digit is used in order to update the discriminator with ‘real’ data.
+
+We ran the experiments 10 times with different random seeds for the initializations. Each experiment is run for 2000 iterations.
+
+# 5 RESULTS
+
+The experts correctly specialized on inverting exactly one mechanism each in 7 out of the 10 runs; in the remaining 3 runs the results were only slightly suboptimal: one expert specialized on two tasks, one expert did not specialize on any, and the remaining experts still specialized on one task each, thus still covering all the existing tasks. In Figure 2 we show a randomly selected batch of inputs and corresponding outputs from the model. Each independent mechanism was inverted by a different expert.
+
+![](images/10af0fda70606db3b4d52d406f4449d6832bc21e829af03ebf0caed96784ca2d.jpg)  
+Figure 2: The top row contains 16 random inputs to the networks, and the bottom row the corresponding outputs from the highest scoring experts against the discriminator after 1000 iterations.
+
+First, we discuss the three major aspects of our results followed by additional experiments.
+
+1. The experts specialize w.r.t. $c$ . We encourage the reader to look at Figure 6 in Appendix A.1, where we plot the scores assigned by the discriminator for each expert on each task in a typical successful run. The figure shows that after an initial chaotic phase of heavy competition, the experts exhibit the desired behavior and obtain a high score on $D$ on one mechanism each. Figure 3 provides further evidence, by visualising that the clusters induced by $c$ are meaningful. We report the proportion of examples from each task assigned to each expert at the beginning and at the end of training. At first, a couple of experts win most examples from all tasks. By the end of the training, each expert wins almost all examples coming from one transformation, and no other.
+
+![](images/7cc59d047c82f83aef832f553042ca8364e6261b7f9ef66049a079f58a1cae04.jpg)  
+Figure 3: The proportion of data won by each expert for each transformation on the digits from the test set.
+
+2. The transformed outputs improve a classifier. In order to test if the committee of experts can overall recover a good approximation of the original digits, we test the output of our experts against a pretrained MNIST classifier. For this, we use the test partition of the data. We compute the accuracy for three inputs: $a$ ) the transformed test digits, $b$ ) the transformed digits after being processed by the highest scoring experts, $c$ ) the original test digits. The latter can be seen as an upper bound to the accuracy that can be achieved.
+
+As shown by the two dashed horizontal lines in Figure 4, the transformed test digits achieve a $40 \%$ accuracy when tested directly on the classifier, while the untransformed digits would achieve $\approx 9 9 \%$ accuracy. The accuracy for the output digits also starts at $40 \%$ — due to the identity initialization — and quickly matches the performance of the original digits as it is trained. Note also that after about 600 iterations — i.e. as the networks have seen overall about one third of the whole dataset, and once only — the accuracy is already almost at the upper bound.
+
+3. The experts learn mechanisms. Finally, we test the networks on inputs that were not transformed with the mechanisms that each of them has learned to invert. As shown in Figure 5, each network consistently applies the same transformation also on inputs outside of its training distribution, and therefore the experts not only recovered the correct digits for the domain they have specialized on, but indeed learned the independent mechanisms. Since the experts are fully convolutional networks in this experiment, they could be even be ported to other domains with images of different sizes.
+
+Effect of the approximate identity initialization. When running the same experiments without the approximate identity initialization, we found that often several experts fail to specialize. Out of 10 new runs with random initialization, only one experiment had arguably good results, with eight experts specializing on one task each, one expert on two tasks, and the last expert on none. The performance was worse in the remaining runs. We tested whether the problem was that the algorithm takes longer to converge following a random initialization, and ran one additional experiment for 10 000 iterations. The results did not improve.
+
+![](images/b4a15e382d21584b5dfda18aee439e3f03b8f50e9d72672c9685be80634e7717.jpg)  
+Figure 4: Accuracy on the transformed test digits $\mathcal { D } _ { Q }$ of a pretrained CNN MNIST classifier, on the same digits after going through our model, and on the original digits before transformation $\mathcal { D } _ { P }$ (here $\mathcal { D } _ { P }$ corresponds to the ground truth images in $\mathcal { D } _ { Q }$ for the true applied mechanisms).
+
+![](images/77ca1afef78a177328c3986d7a71666ac116c354bfc15f5b7fd19b81a989a35c.jpg)  
+Figure 5: Each column shows how each expert transforms the input presented on top. We arrange the tasks such that on the diagonal there is the highest scoring expert for the input given at the top of the column. It is evident that the experts have learned the mechanisms, as they consistently apply them to digits outside of their training domain.
+
+A simple single-net baseline. Training a single network instead of a committee of experts makes the problem more difficult to solve. Using identical training settings, we trained a single network once with 32, once with 64, and once with 128 filters per layer, and none of them managed to correctly learn more than one inverse mechanism.4 Note that a single network with 128 filters per layer has about twice as many parameters overall than the committee of 10 experts with 32 filters per layer each. We also tried random initialization instead of the approximate identity, to reduce the learning rate of the discriminator by a factor of 10, and to increase the receptive field by adding two pooling and two upsampling layers, without any improvement. While we do not exclude that careful hyperparameter tuning may enable a single net to learn multiple mechanisms, it is not entirely straightforward in our experiment.
+
+Specialization occurs also with higher capacity experts. While in principle with infinite capacity and data a single expert could solve all tasks simultaneously, in practice limited resources and the proposed training procedure favor specialization in independent modules. Increasing the size of the experts from 32 filters per layer to 64 or 128 filters5 or enlarging the overall receptive field by using two pooling and two upsampling layers, still results in good specialization of the experts, with no more than two experts specializing on up to two tasks at once.
+
+Fewer examples from the canonical distribution. In many applications, we might only have a small sample from the original distribution. Interestingly, we found that all experts still specialize to different tasks and recover good approximations of the inverse mechanisms when we reduce the number of examples from the original distribution from 30 000 down to $6 4 ^ { 6 }$ . Even though the output digits are not as clean and sharp, we still achieve $96 \%$ accuracy on the pretrained classifier before the discriminator starts to overfit.
+
+# 6 CONCLUSIONS
+
+We have developed a method to identify and learn a set of independent causal mechanisms. In the present work, these are inverse mechanisms, but an extension to forward mechanisms appears feasible and worthwhile. We reported promising results in an experiment based on image transformations; future work could study more complex settings and diverse domains.
+
+A natural extension of our work is to consider independent mechanisms that simultaneously affect the data (e.g. lighting and position in a portrait), and to allow multiple passes through our committee of experts to identify local mechanisms (akin to Lie derivatives) from more complex datasets — for instance, using recurrent neural network that allow the application of multiple mechanisms by iteration.
+
+Note that for large numbers of experts, the computational cost might become unnecessarily high. This could be mitigated by hybrid approaches incorporating gated mixture of experts — which may exhibit lower computational complexity — or a hierarchical selection of competing experts.
+
+We believe our work constitutes a relevant connection between causal modeling and deep learning. As discussed in the introduction, causality has a lot to offer for crucial machine learning problems such as transfer or compositional modeling. Our systems illustrates some of these properties. Independent modules as sub-components could be trained independently and/or from multiple domains, added subsequently, and transferred to other problems. This may constitute a step towards causally motivated life-long learning.
+
+# REFERENCES
+
+Rahaf Aljundi, Punarjay Chakravarty, and Tinne Tuytelaars. Expert gate: Lifelong learning with a network of experts. arXiv preprint arXiv:1611.06194, 2016.
+
+Konstantinos Bousmalis, Nathan Silberman, David Dohan, Dumitru Erhan, and Dilip Krishnan. Unsupervised pixel-level domain adaptation with generative adversarial networks. arXiv preprint arXiv:1612.05424, 2016.
+
+Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in Neural Information Processing Systems, pp. 2172–2180, 2016.
+
+Djork-Arne Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and accurate deep network ´ learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289, 2015.
+
+P. Daniusis, D. Janzing, J. Mooij, J. Zscheischler, B. Steudel, K. Zhang, and B. Sch ˇ olkopf. Inferring ¨ deterministic causal relations. In P. Grunwald and P. Spirtes (eds.), ¨ 26th Conference on Uncertainty in Artificial Intelligence, pp. 143–150, Corvallis, OR, 2010. AUAI Press.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.
+
+Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. 2016.
+
+Aapo Hyvarinen and Hiroshi Morioka. Unsupervised feature extraction by time-contrastive learning and nonlinear ica. In Advances in Neural Information Processing Systems, pp. 3765–3773, 2016.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning, pp. 448–456, 2015.
+
+Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geoffrey E Hinton. Adaptive mixtures of local experts. Neural computation, 3(1):79–87, 1991.
+
+D. Janzing and B. Scholkopf. Causal inference using the algorithmic Markov condition. ¨ IEEE Transactions on Information Theory, 56(10):5168–5194, 2010.   
+Michael I Jordan and Robert A Jacobs. Hierarchical mixtures of experts and the em algorithm. Neural computation, 6(2):181–214, 1994.   
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   
+Judea. Pearl. Causality. Cambridge University Press, 2000.   
+Jonas Peters, Dominik Janzing, and Bernhard Scholkopf. ¨ Elements of Causal Inference. The MIT Press, 2017.   
+Mateo Rojas-Carulla, Bernhard Scholkopf, Richard Turner, and Jonas Peters. Causal transfer in ¨ machine learning. arXiv preprint arXiv:1507.05333, 2015.   
+B. Scholkopf, D. Janzing, J. Peters, E. Sgouritsa, K. Zhang, and J. M. Mooij. On causal and anticausal ¨ learning. In J Langford and J Pineau (eds.), Proceedings of the 29th International Conference on Machine Learning (ICML), pp. 1255–1262, New York, NY, USA, 2012. Omnipress.   
+Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. arXiv preprint arXiv:1701.06538, 2017.   
+Eric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. Adversarial discriminative domain adaptation. arXiv preprint arXiv:1702.05464, 2017.   
+Kun Zhang, Jiji Zhang, and Bernhard Scholkopf. Distinguishing cause from effect based on exogene- ¨ ity. arXiv preprint arXiv:1504.05651, 2015.
+
+# A ADDITIONAL RESULTS.
+
+# A.1 PLOT OF COMPETING EXPERTS.
+
+Each expert in Figure 6 is represented with the same color and linestyle across all tasks. Note how the red expert tries to learn two similar tasks until iteration 500 (i.e. left and up-left translation), when the green expert takes over one of the task and they can then both quickly specialize.
+
+![](images/c1de7ee3ed18b5fac92aa208bb0a8fc8ed966c5d44beb2bc684a8ac11603d8a6.jpg)  
+Figure 6: Each line style is associated to the score that an expert obtains on the discriminator when being fed transformed digits using one of the mechanisms. Each expert learns to specialize on a different mechanism. Each curve is smoothed with an average of the last 50 iterations for ease of visualization.
+
+# A.2 TOO MANY OR TOO FEW EXPERTS.
+
+Too many experts When there are too many experts, for most tasks only one wins all the examples, as shown in Figure 7 where the model has 16 experts for 10 tasks. The remaining experts either do not specialize at all — and therefore can be removed from the architecture — or specialize on the same task, and could therefore be combined if after inspection they are considered to perform the same task. Since the accuracy on the transformed data tested on the pretrained classifier reaches again the upperbound of the untransformed data, and since the progress is very similar to that illustrated in Figure 4, we omit this plot.
+
+Too few experts For a committee of 6 experts, the networks do not reconstruct properly most of the digits, which is reflected by an overall low objective function value on the data. Also, the score against the classifier that does not exceed $72 \%$ . A few experts are inevitably assigned to multiple tasks, and by looking at Figure 7 it is interesting to see that the clustering result is still meaningful (e.g. expert 5 is assigned to left, down-left, and up-left translation).
+
+# B DETAILS OF NEURAL NETWORKS
+
+In Table 1 we report the configuration of the neural networks used in these experiments.
+
+# C TRANSFORMATIONS
+
+In our experiments we use the following transformations
+
+• Translations: the image is shifted by 4 pixels in one of the eight directions up, down, left, right and the four diagonals.   
+• contrast (or color) inversion: the value of each pixel — originally in the range $[ 0 , 1 ]$ — is recomputed as $1 -$ the original value.   
+• Noise addition: random Gaussian noise with zero mean and variance 0.25 is added to the original image, which is then clamped again to the [0, 1] interval.
+
+![](images/15ad1d48a5ba3f972d602203c58f12e7d9d8d14270729659d1131a9b9219f28d.jpg)  
+Figure 7: The proportion of data won by each expert for each transformation on the digits from the 0.0 test set, for the case of 10 mechanisms and more experts (16 on left) or too few (6 on the right).
+
+Table 1: Architectures of the neural networks used in the experiment section. BN stands for Batch normalization, FC for fully connected. All convolutions are preceded by a 1 pixel zero padding.
+
+<table><tr><td colspan="2">Expert</td></tr><tr><td>Layers</td><td></td></tr><tr><td>3 × 3,32,BN,ELU 3 × 3,32, BN,ELU</td><td></td></tr><tr><td>3 × 3,32, BN, ELU 3 × 3,32, BN, ELU</td><td></td></tr><tr><td>3 × 3,1, sigmoid</td><td></td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>Discriminator</td></tr><tr><td rowspan=1 colspan=1>Layers</td></tr><tr><td rowspan=1 colspan=1>3 × 3,16,ELU</td></tr><tr><td rowspan=1 colspan=1>3 × 3,16,ELU3 × 3,16, ELU</td></tr><tr><td rowspan=1 colspan=1> 2 × 2, avg pooling</td></tr><tr><td rowspan=1 colspan=1>3 × 3,32, ELU3 × 3,32, ELU</td></tr><tr><td rowspan=1 colspan=1> 2 × 2, avg pooling</td></tr><tr><td rowspan=1 colspan=1>3 × 3,64, ELU3 × 3,64, ELU</td></tr><tr><td rowspan=1 colspan=1> 2 × 2, avg pooling</td></tr><tr><td rowspan=1 colspan=1>1024, FC, ELU1, FC, sigmoid</td></tr></table>
+
+# D NOTES ON THE FORMALIZATION OF INDEPENDENCE OF MECHANISMS
+
+In this section we briefly discuss the notion of independence of mechanisms as in (Janzing & Scholkopf, 2010), where the independence principle is formalized in terms of algorithmic complexity ¨ (also known as Kolmogorov complexity). We summarize the main points needed in the present context. We parametrize each mechanism by a bit string $x$ . The Kolmogorov complexity $K ( \bar { x } )$ of $x$ is the length of the shortest program generating $x$ on an a priori chosen universal Turing machine.
+
+The algorithmic mutual information can be defined as $I ( x : y ) : = K ( x ) + K ( y ) - K ( x , y )$ , and it can be shown to equal
+
+$$
+I ( x : y ) = K ( y ) - K ( y | x ^ { * } ) ,
+$$
+
+where for technical reasons we need to work with $x ^ { * }$ , the shortest description of $x$ (which is in general uncomputable). Here, the conditional Kolmogorov complexity $K ( y | x )$ is defined as the length of the shortest program that generates $y$ from $x$ . The algorithmic mutual information measures the algorithmic information two objects have in common. We define two mechanisms to be (algorithmically) independent whenever the length of the shortest description of the two bit strings together is not shorter than the sum of the shortest individual descriptions (note it cannot be longer), i.e., if their algorithmic mutual information vanishes.7 In view of (4), this means that
+
+$$
+K ( y ) = K ( y | x ^ { * } ) .
+$$
+
+We will say that two mechanisms $x$ and $y$ are independent whenever the complexity of the conditional mechanism $y | x$ is comparable to the complexity of the unconditional one $y$ . If, in contrast, the two mechanisms were closely related, then we would expect that we can mimic one of the mechanisms by applying the other one followed by a low complexity conditional mechanism.
+
+This can be implemented by having a complexity measure for, say, neural networks, and comparing the complexities of neural nets that are trained to perform certain tasks. Inspired by regularization theory, we could measure complexity by inverse regularization strength or weight vector norm. An alternative way to regularize neural nets consists of early stopping. If we fix the number of training epochs to a constant, and find that network 1 reaches a lower error than network 2, we conclude that the network 2 would take longer to reach the same low error, and thus network 2 requires higher effective complexity to solve its task than network 1. We have run preliminary experiments with this measure and found that (1) indeed our training procedure did increase independence, and (2) the independence between two different mechanism was larger than the independence between one mechanism and the identity.

parse/train/SJky6Ry0W/SJky6Ry0W_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "LEARNING INDEPENDENT CAUSAL MECHANISMS ",
+        "text_level": 1,
+        "bbox": [
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+            101,
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+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Independent causal mechanisms are a central concept in the study of causality with implications for machine learning tasks. In this work we develop an algorithm to recover a set of (inverse) independent mechanisms relating a distribution transformed by the mechanisms to a reference distribution. The approach is fully unsupervised and based on a set of experts that compete for data to specialize and extract the mechanisms. We test and analyze the proposed method on a series of experiments based on image transformations. Each expert successfully maps a subset of the transformed data to the original domain, and the learned mechanisms generalize to other domains. We discuss implications for domain transfer and links to recent trends in generative modeling. ",
+        "bbox": [
+            233,
+            239,
+            766,
+            378
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            404,
+            336,
+            420
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Humans are able to recognize objects such as handwritten digits based on distorted inputs. When presented with digits which are translated, corrupted, or inverted, we can usually correctly label them without the need of re-learning them from scratch. The same applies for new objects, essentially after having seen them once. This may be due to the fact that human intelligence utilizes mechanisms (such as translation) that are generic and generalize across object classes. These mechanisms are modular, re-usable and broadly applicable, and the problem of learning them from data is fundamental for the study of transfer. ",
+        "bbox": [
+            174,
+            434,
+            825,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the field of causality, the concept of independent mechanisms plays a central role both on the conceptual level and, more recently, in applications to inference. The independent mechanism (IM) assumption states that the causal generative process of a system’s variables is composed of autonomous modules that do not inform or influence each other (Scholkopf et al., 2012; Peters et al., ¨ 2017). ",
+        "bbox": [
+            173,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "If a joint density is Markovian with respect to a directed graph $\\mathcal { G }$ , we can write it as ",
+        "bbox": [
+            174,
+            614,
+            720,
+            631
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/49a6efae65a5680efca4d35f83009b8d6b7555302aac0de6e76d044aad73a043.jpg",
+        "text": "$$\np ( \\mathbf { x } ) = p ( x _ { 1 } , \\ldots , x _ { d } ) = \\prod _ { j = 1 } ^ { d } p ( x _ { j } | \\mathbf { p } \\mathbf { a } _ { \\mathcal { G } } ^ { j } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            635,
+            630,
+            679
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { p a } _ { \\mathcal { G } } ^ { j }$ denotes the parents of variable $x _ { j }$ in the graph. ",
+        "bbox": [
+            173,
+            686,
+            553,
+            702
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For a given joint density, there are usually many decompositions of the form (1), with respect to different graphs. If $\\mathcal { G }$ is a causal graph, i.e., if its edges denote direct causation (Pearl, 2000), then the conditional $\\bar { p } ( x _ { j } | \\mathrm { p a } _ { \\mathcal { G } } ^ { j } )$ can be thought of as physical mechanism generating $x _ { j }$ from its parents, and we refer to it as a causal conditional. In this case, we consider (1) a generative model where the term “generative” truly refers to a physical generative process. As an aside, we note that in the alternative view of causal models as structural equation models, each of the causal conditionals corresponds to a functional mapping and a noise variable (Pearl, 2000). ",
+        "bbox": [
+            173,
+            708,
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+            806
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "By the IM assumption, the causal conditionals are autonomous modules that do not influence or inform each other. This has multiple consequences. First, knowledge of one mechanism does not contain information about another one (Appendix D). Second, if one mechanism changes (e.g., due to distribution shift), there is no reason that other mechanisms should also change, i.e., they tend to remain invariant. As a special case, it is (in principle) possible to locally intervene on one mechanism (for instance, by setting it to a constant) without affecting any of the other modules. In all these cases, most of (1) will remain unchanged. However, since the overall density will change, in the generic case the (non-causal) conditionals would change. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The IM assumption can be exploited when performing causal structure inference (Peters et al., 2017). However, it also has implications for machine learning more broadly. A model which is expressed in terms of causal conditionals (rather than conditionals with respect to some other factorization) is likely to have components that better transfer or generalize to other settings (Scholkopf et al., ¨ 2012), and its modules are better suited for building complex models from simpler ones. Independent modules as sub-components can be trained independently, from multiple domains, are more likely to be re-usable. They can also be easier to interpret since they correspond to physical mechanisms. Animate intelligence cannot afford to learn new models from scratch for every new task. Rather, it is likely to rely on robust local components that can flexibly be re-used and re-purposed. It also requires local mechanisms for adapting and training modules rather than re-training the whole brain every time a new task is learned. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            256
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the present paper, we focus on a class of such modules, and on algorithms to learn them from data. We describe an architecture using competing experts specializing on different transformations. The resulting model permits a form of lifelong learning, with the possibility of easily adding, removing, retraining, and exporting its components independently. ",
+        "bbox": [
+            174,
+            263,
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+            319
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In line with the intuition given above, we illustrate our approach on MNIST digits which have undergone different transformations such as contrast inversion, noise addition and translation. Information about the nature and number of such transformations need not be known at the beginning of training. Our goal is to identify the independent mechanisms linking a reference distribution to a distribution of modified digits, and learn to invert them without supervision. The inverse mechanisms can be used to transform modified digits and classify them using a standard MNIST classifier, thus exhibiting a form of robustness that animate intelligence excels at. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            463,
+            339,
+            478
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work mainly draws from mixtures of experts, domain adaptation, and causality. ",
+        "bbox": [
+            176,
+            506,
+            720,
+            520
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Early works on mixture of experts date back to the early nineties (Jacobs et al. (1991), Jordan & Jacobs (1994)), and since then the topic has been subject of extensive research. Recent work include Shazeer et al. (2017), where the authors train a mixture of 1000 experts using a gating mechanism that selects only a very small number of experts for each example, and propose several technical solutions to deal with model and data parallelism. Aljundi et al. (2016) train a network of experts on multiple tasks, with a focus on lifelong learning; autoencoders are trained for each task and used as gating mechanisms. ",
+        "bbox": [
+            174,
+            526,
+            825,
+            625
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another research direction that is relevant to our work is unsupervised domain adaptation (Bousmalis et al., 2016). These methods often use some supervision from labeled data and/or match the two distributions in a learned feature space (Tzeng et al., 2017, e.g.). ",
+        "bbox": [
+            174,
+            631,
+            825,
+            674
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The novelty of our work lies in the following aspects: (1) we automatically identify and invert a set of independent (inverse) causal mechanisms; (2) we do so using only data from an original distribution and from the mixture of transformed data, without labels; (3) the architecture is modular, can be easily expanded, and its trained modules can be reused; and (4) the method relies on competition of experts. ",
+        "bbox": [
+            174,
+            680,
+            825,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Ideas from the field of causal inference inspire the present work. Understanding the data generating mechanisms plays a key role in causal inference, and goes beyond the statistical assumptions usually exploited in machine learning. Causality provides a framework for understanding how a system responds to interventions, and causal graphical models as well as structural equation models (SEM) are common ways of describing causal systems (Pearl, 2000; Peters et al., 2017). The IM assumption discussed in the introduction can be used for identification of causal models (Daniusis ˇ et al., 2010; Zhang et al., 2015), but causality has also proven a useful tool for discussing and understanding machine learning in the non-i.i.d. regime. Recent applications include semi-supervised learning (Scholkopf et al., 2012) and transfer learning (Rojas-Carulla et al., 2015), in which the ¨ authors focus only on linear regression models. We seek to extend applications of causal inference to more complex settings and aim to learn causal mechanisms and ultimately causal SEMs without supervision. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            922
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There are close relations between our setting and recent work on deep learning for disentangling factors of variation (Chen et al., 2016; Higgins et al., 2016) as well as non-linear ICA (Hyvarinen & Morioka, 2016). In our work, causal mechanisms play the role of factors of variation. The main difference is that we currently recover inverse mechanisms as independent modular parts, instead of indentifying a joint low dimensional representation of the data without explicit separate paths for each factor. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 LEARNING CAUSAL MECHANISMS AS INDEPENDENT MODULES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            209,
+            723,
+            223
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The aim of this section is twofold. First, we describe the generative process of our data. We start with a distribution $P$ that we will call “canonical” and an a priori unknown number of independent mechanisms which act on (examples drawn from) $P$ . At training time, a sample from the canonical distribution is available, as well as a dataset obtained by applying the mechanisms to (unseen) examples drawn from $P$ . Second, we propose an algorithm which recovers and learns to invert the mechanisms in an unsupervised fashion. ",
+        "bbox": [
+            174,
+            239,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 FORMAL SETTING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            340,
+            339,
+            354
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider a canonical distribution $P$ on $\\mathbb { R } ^ { d }$ , e.g., the empirical distribution defined by MNIST digits on pixel space. We further consider $N$ measurable functions $M _ { 1 } , \\dots , M _ { N } : \\mathbb { R } ^ { \\tilde { d } } \\to \\mathbb { R } ^ { d }$ , called mechanisms. We think of these as independent causal mechanisms in nature, and their number is a priori unknown. A more formal definition of independence between mechanisms is relegated to Appendix D. The mechanisms give rise to $N$ distributions $Q _ { 1 } , \\ldots , Q _ { N }$ where $Q _ { j } = M _ { j } ( P )$ .1 In the MNIST example, we consider translations or adding noise as mechanisms, i.e., the corresponding $Q$ distributions are translated and noisy MNIST digits. ",
+        "bbox": [
+            174,
+            366,
+            825,
+            463
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "At training time, we receive a dataset $\\mathcal { D } _ { Q } = ( x _ { i } ) _ { i = 1 } ^ { n }$ drawn i.i.d. from a mixture of $Q _ { 1 } , \\ldots , Q _ { N }$ , and an independent sample $\\mathcal { D } _ { P }$ from the canonical distribution $P$ . Our goal is to identify the underlying mechanisms $M _ { 1 } , \\dots , M _ { N }$ and learn approximate inverse mappings which allow us to map the examples from $\\mathcal { D } _ { Q }$ back to their counterpart drawn from $P$ . ",
+        "bbox": [
+            174,
+            470,
+            825,
+            526
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If we were given distinct datasets $\\mathcal { D } _ { Q _ { j } }$ each drawn from $Q _ { j }$ , we could individually learn each mechanism, resulting in independent approximations regardless of the properties of the training procedure. This is due to the fact that the datasets are drawn from independent mechanisms and the separate training procedure cannot generate a dependence between them. This property is independent of properties of training, and does not require that the procedure is successful, i.e., that the obtained mechanisms approximate the true $M _ { j }$ in some metric. In our case, we do not have access to the distinct datasets. Instead we construct a larger set $\\mathcal { D } _ { Q }$ by first taking the union of the sets $D _ { Q _ { j } }$ , and then applying a random permutation. This corresponds to a dataset where each element has been generated by one of the (independent) mechanisms, but we don’t know by which one. Clearly, it should be harder to identify and learn independent mechanisms from such a dataset. This is the setting we address below, and the crucial idea will be that of competition. ",
+        "bbox": [
+            174,
+            534,
+            825,
+            685
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": ".2 COMPETITIVE LEARNING OF INDEPENDENT MECHANISMS ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            704,
+            611,
+            717
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce our training protocol to address the problem defined above. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The training machine is composed of $N ^ { \\prime }$ parametric functions $E _ { 1 } , \\ldots , E _ { N ^ { \\prime } }$ with distinct trainable parameters $\\theta _ { 1 } , \\ldots , \\theta _ { N ^ { \\prime } }$ . We refer to these functions as the experts. Note that we do not require $N ^ { \\prime } = N$ , since the real number of mechanisms is unknown a priori. The goal is to maximize an objective function $c : \\mathbb { R } ^ { d }  \\mathbb { R }$ with the key property that $c$ takes high values on the support of the canonical distribution $P$ , and low values outside. Note that it is possible for $c$ to be a parametric function, and for these parameters to be jointly optimized with the experts during training. Below, we specify the details of this rather general definition. ",
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+    {
+        "type": "text",
+        "text": "During training, the experts compete for the data points. Each example $x ^ { \\prime }$ from $\\mathcal { D } _ { Q }$ is fed to all experts independently and in parallel. Depending on the output of each expert $c _ { j } \\ = \\ c ( E _ { j } ( x ^ { \\prime } ) )$ , we select the winning expert $E _ { j ^ { * } }$ , where $j ^ { * } = \\arg \\operatorname* { m a x } _ { j } ( c _ { j } )$ . $E _ { j ^ { * } }$ wins the example $x ^ { \\prime }$ , and its parameters $\\theta _ { j ^ { * } }$ are updated as to maximize $c ( E _ { j ^ { * } } ( x ^ { \\prime } ) )$ , while the other experts remain unchanged. The motivation behind competitively updating only the winning expert is to enforce specialization; the best performing expert becomes even better at mapping $x ^ { \\prime }$ back to the corresponding sample from the canonical distribution. Figure 1 depicts this procedure. Overall, our optimization problem reads ",
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+        "page_idx": 2
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+    {
+        "type": "image",
+        "img_path": "images/64b721abdb27e6f8917005d01f43ad0a1dbd48c1ce6fea3974ba5876a1d32bd6.jpg",
+        "image_caption": [
+            "Figure 1: We show how a transformed example, here a noisy digit, is processed by a competition of experts. Only Expert 3 is specializing on denoising, it wins the example and gets trained on it, whereas the others perform translations and are not updated. "
+        ],
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+        "img_path": "images/f6c7ebff10764df6d87115be998a23c9367aff6fb50aed391018f74f04fd67d1.jpg",
+        "text": "$$\n\\theta _ { 1 } ^ { * } , \\ldots , \\theta _ { N ^ { \\prime } } ^ { * } = \\underset { \\theta _ { 1 } , \\ldots , \\theta _ { N ^ { \\prime } } } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { x ^ { \\prime } \\sim Q } \\left( \\operatorname* { m a x } _ { j \\in \\{ 1 , \\ldots , N ^ { \\prime } \\} } c ( E _ { \\theta _ { j } } ( x ^ { \\prime } ) ) \\right) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The training described above raises a number of questions, which we address next. ",
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+        "type": "text",
+        "text": "1. Convergence criterion. Since the problem is fully unsupervised, there is no straightforward way of measuring convergence, which raises the question of how to choose a stopping time for the competitive procedure. As an example, one may act according to one of the following: $a$ ) fix a maximum number of iterations or $^ b$ ) stop if each example is assigned to the same experts for a pre-defined number of iterations (i.e., each expert consistently wins the same data points). ",
+        "bbox": [
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+        "type": "text",
+        "text": "2. Selecting the appropriate number of experts. Generally, the number of mechanisms $N$ which generated the dataset $\\mathcal { D } _ { Q }$ is not available a priori. Therefore, it is important to develop an adaptive procedure for setting up the number of experts $N ^ { \\prime }$ . This is a common problem shared with most clustering techniques. Given the modular behavior of the procedure, experts may be added or removed during or after training, making the framework very flexible. Assuming however that the number of experts is fixed, we speculate that the following behaviors are likely. ",
+        "bbox": [
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+        "type": "text",
+        "text": "If $N ^ { \\prime } > N$ (too many experts): a) some of the experts do not specialize and do not win any example in the dataset; or b) some tasks are divided between experts (for instance, each expert can specialize in a mode of the distribution of the same task). In a), the inactive experts can be removed, and in b) experts sharing the same task can be merged into a wider expert.2 ",
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+    {
+        "type": "text",
+        "text": "If $N ^ { \\prime } < N$ (too few experts): a) some of the experts specialize in multiple tasks or b) some of the tasks are not learned by the experts, so that data points from such tasks lead to a poor score across all experts. ",
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+        "type": "text",
+        "text": "While these questions are relevant, we do not develop them in detail and leave them for further research. Some experiments substantiating these claims can be found in Appendix A.2. ",
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+    {
+        "type": "text",
+        "text": "3. Time and space complexity. Each example has to be evaluated by all experts in order to assign it to the winning expert. While this results in a computational cost that depends linearly on the number of experts, these evaluations can be done in parallel and therefore the time complexity of a single iteration can be bounded by the complexity to compute the output of a single expert. Moreover, as each expert will in principle have a smaller architecture than a single large network, the committee of experts will typically be faster to execute. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Concrete protocol for neural networks. One possible model class for the experts are deep neural networks. Training using backpropagation is particularly well suited for the online nature of the training proposed: after an expert wins a data point $x ^ { \\prime }$ , its parameters are updated by backpropagation, while other experts remain untouched. Moreover, recent advances in generative modeling give rise to natural choices for the loss function $c$ . For instance, given a variational autoencoder (VAE) (Kingma & Welling, 2013) trained on the canonical distribution $P$ , one may define $c ( x ^ { \\prime } )$ as the opposite of the VAE loss. The assumption is that the loss will only be low for examples drawn from $P$ . Another possibility is to use adversarial training (Goodfellow et al., 2014), and use as an objective function the output of a discriminator network trained on the canonical sample $\\mathcal { D } _ { P }$ and against the outputs of the experts. In the next section we introduce a formal description of a training procedure based on adversarial training in Algorithm 1, and present experimental evidence of its good performance. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this set of experiments we test the method presented in Section 3 on the MNIST dataset transformed with the set of mechanisms described in detail in the Appendix C, i.e. eight directions of translations by 4 pixels (up, down, left, right, and the four diagonals), contrast inversion, addition of noise, for a total of 10 transformations. We split the training partition of MNIST in half, and transform all and only the examples in the first half: this ensures that there is no matching ground truth for the experts to learn the mechanisms, and that learning is fully unsupervised. As a preprocessing step, the digits are zero-padded so that they have size $3 2 \\times 3 2$ pixels, and the pixel intensities are scaled between 0 and 1. This is done even before any mechanism is applied. We use deep neural networks for both the experts and the selection mechanism, and use an adversarial training scheme. ",
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+        "type": "text",
+        "text": "Each expert $E _ { i }$ can be seen as a generator from a GAN, that is conditioned on an input image instead of (or in addition to) a noise vector. A discriminator $D$ provides gradients for training the experts and acts also as a selection mechanism $c$ : only the expert whose output obtains the higher score from $D$ wins the example, and is trained on it to maximize the output of $D$ . We describe the exact algorithm used to train the networks in these experiments in Algorithm 1. The discriminator is trained to maximize the following cross-entropy loss: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d951045601694868cd11cf94537e8207751c3e0ae5b8a9603dbb993b01c399a5.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta _ { D } } \\left( \\mathbb { E } _ { x \\sim P } \\log ( D _ { \\theta _ { D } } ( x ) ) + \\frac { 1 } { N ^ { \\prime } } \\sum _ { j = 1 } ^ { N ^ { \\prime } } \\mathbb { E } _ { x ^ { \\prime } \\sim Q } \\left( \\log ( 1 - D _ { \\theta _ { D } } ( E _ { \\theta _ { j } } ( x ^ { \\prime } ) ) ) \\right) \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "For simplicity, we assume for the rest of this section that the number of experts $N ^ { \\prime }$ equals the number of true mechanisms $N$ . Results where $N \\neq N ^ { \\prime }$ are relegated to Appendix A.2. ",
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+        "type": "text",
+        "text": "Neural nets details. Each expert is a CNN with five convolutional layers, 32 filters per layer of size $3 \\times 3$ , ELU (Clevert et al. (2015)) as activation function, batch normalization (Ioffe & Szegedy (2015)), and zero padding. The discriminator is also a CNN, with average pooling every two convolutional layers, growing number of filters, and a fully connected layer with 1024 neurons as last hidden layer. Both networks are trained using Adam as optimizer (Kingma & Ba (2014)), with the default hyper-parameters.3 ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Unless specified otherwise, after a random weight initialization we first train the experts to approximate the identity mapping on our data, by pretraining them for up to 200 iterations on predicting identical input-output pairs randomly selected from the transformed dataset. This makes the experts start from similar grounds, and we found that this improved the speed and robustness of convergence. We will refer to this as approximate identity initialization for the rest of the paper. ",
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+    {
+        "type": "table",
+        "img_path": "images/6c503a12d84b3d6a528f2794d652509db491b7105ffb62afc463bd76ce65921b.jpg",
+        "table_caption": [],
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+        "table_body": "<table><tr><td>Algorithm1 Learning independent mechanisms using competition of experts and adversarial training</td></tr><tr><td>Precondition: X: data sampled from P; X&#x27;: data sampled from DQ; D discriminator; N&#x27;: number of experts; T: maximum number of iterations;</td></tr><tr><td>(p) highlights that the steps in the instruction can be executed in parallel</td></tr><tr><td>1{E←TrainNewAutoencoderOn(X)}1 &gt; Init set of experts as approx identity (p)</td></tr><tr><td>2 fort←1toTdo x,x&#x27; ← Sample(X), Sample(X&#x27;) &gt; Sample minibatches</td></tr><tr><td>{cj←D(Ej(x）}1 &gt; Scores from D for all outputs from the experts (p)</td></tr><tr><td></td></tr><tr><td>{B←Adam(axjg))</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "A minibatch of 32 transformed MNIST digits, each transformed by a randomly chosen mechanism, is fed to all experts $E _ { i }$ . The outputs are fed to the discriminator $D$ , which computes a score for each of them. For each example the cross entropy loss in Equation (3) and the resulting gradients are computed only for the output of the highest scoring expert, and they are used to update both the discriminator (when 0 is the target in the cross entropy) and the winning expert (when using 1 as the target). In order to encourage the expert to specialize, the discriminator is also explicitly trained against the outputs of the losing experts. Then, a minibatch of canonical MNIST digit is used in order to update the discriminator with ‘real’ data. ",
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+    {
+        "type": "text",
+        "text": "We ran the experiments 10 times with different random seeds for the initializations. Each experiment is run for 2000 iterations. ",
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+    {
+        "type": "text",
+        "text": "5 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The experts correctly specialized on inverting exactly one mechanism each in 7 out of the 10 runs; in the remaining 3 runs the results were only slightly suboptimal: one expert specialized on two tasks, one expert did not specialize on any, and the remaining experts still specialized on one task each, thus still covering all the existing tasks. In Figure 2 we show a randomly selected batch of inputs and corresponding outputs from the model. Each independent mechanism was inverted by a different expert. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/10af0fda70606db3b4d52d406f4449d6832bc21e829af03ebf0caed96784ca2d.jpg",
+        "image_caption": [
+            "Figure 2: The top row contains 16 random inputs to the networks, and the bottom row the corresponding outputs from the highest scoring experts against the discriminator after 1000 iterations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, we discuss the three major aspects of our results followed by additional experiments. ",
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+        "type": "text",
+        "text": "1. The experts specialize w.r.t. $c$ . We encourage the reader to look at Figure 6 in Appendix A.1, where we plot the scores assigned by the discriminator for each expert on each task in a typical successful run. The figure shows that after an initial chaotic phase of heavy competition, the experts exhibit the desired behavior and obtain a high score on $D$ on one mechanism each. Figure 3 provides further evidence, by visualising that the clusters induced by $c$ are meaningful. We report the proportion of examples from each task assigned to each expert at the beginning and at the end of training. At first, a couple of experts win most examples from all tasks. By the end of the training, each expert wins almost all examples coming from one transformation, and no other. ",
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+    {
+        "type": "image",
+        "img_path": "images/7cc59d047c82f83aef832f553042ca8364e6261b7f9ef66049a079f58a1cae04.jpg",
+        "image_caption": [
+            "Figure 3: The proportion of data won by each expert for each transformation on the digits from the test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2. The transformed outputs improve a classifier. In order to test if the committee of experts can overall recover a good approximation of the original digits, we test the output of our experts against a pretrained MNIST classifier. For this, we use the test partition of the data. We compute the accuracy for three inputs: $a$ ) the transformed test digits, $b$ ) the transformed digits after being processed by the highest scoring experts, $c$ ) the original test digits. The latter can be seen as an upper bound to the accuracy that can be achieved. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "As shown by the two dashed horizontal lines in Figure 4, the transformed test digits achieve a $40 \\%$ accuracy when tested directly on the classifier, while the untransformed digits would achieve $\\approx 9 9 \\%$ accuracy. The accuracy for the output digits also starts at $40 \\%$ — due to the identity initialization — and quickly matches the performance of the original digits as it is trained. Note also that after about 600 iterations — i.e. as the networks have seen overall about one third of the whole dataset, and once only — the accuracy is already almost at the upper bound. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3. The experts learn mechanisms. Finally, we test the networks on inputs that were not transformed with the mechanisms that each of them has learned to invert. As shown in Figure 5, each network consistently applies the same transformation also on inputs outside of its training distribution, and therefore the experts not only recovered the correct digits for the domain they have specialized on, but indeed learned the independent mechanisms. Since the experts are fully convolutional networks in this experiment, they could be even be ported to other domains with images of different sizes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of the approximate identity initialization. When running the same experiments without the approximate identity initialization, we found that often several experts fail to specialize. Out of 10 new runs with random initialization, only one experiment had arguably good results, with eight experts specializing on one task each, one expert on two tasks, and the last expert on none. The performance was worse in the remaining runs. We tested whether the problem was that the algorithm takes longer to converge following a random initialization, and ran one additional experiment for 10 000 iterations. The results did not improve. ",
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+        "page_idx": 6
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+    {
+        "type": "image",
+        "img_path": "images/b4a15e382d21584b5dfda18aee439e3f03b8f50e9d72672c9685be80634e7717.jpg",
+        "image_caption": [
+            "Figure 4: Accuracy on the transformed test digits $\\mathcal { D } _ { Q }$ of a pretrained CNN MNIST classifier, on the same digits after going through our model, and on the original digits before transformation $\\mathcal { D } _ { P }$ (here $\\mathcal { D } _ { P }$ corresponds to the ground truth images in $\\mathcal { D } _ { Q }$ for the true applied mechanisms). "
+        ],
+        "image_footnote": [],
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/77ca1afef78a177328c3986d7a71666ac116c354bfc15f5b7fd19b81a989a35c.jpg",
+        "image_caption": [
+            "Figure 5: Each column shows how each expert transforms the input presented on top. We arrange the tasks such that on the diagonal there is the highest scoring expert for the input given at the top of the column. It is evident that the experts have learned the mechanisms, as they consistently apply them to digits outside of their training domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A simple single-net baseline. Training a single network instead of a committee of experts makes the problem more difficult to solve. Using identical training settings, we trained a single network once with 32, once with 64, and once with 128 filters per layer, and none of them managed to correctly learn more than one inverse mechanism.4 Note that a single network with 128 filters per layer has about twice as many parameters overall than the committee of 10 experts with 32 filters per layer each. We also tried random initialization instead of the approximate identity, to reduce the learning rate of the discriminator by a factor of 10, and to increase the receptive field by adding two pooling and two upsampling layers, without any improvement. While we do not exclude that careful hyperparameter tuning may enable a single net to learn multiple mechanisms, it is not entirely straightforward in our experiment. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Specialization occurs also with higher capacity experts. While in principle with infinite capacity and data a single expert could solve all tasks simultaneously, in practice limited resources and the proposed training procedure favor specialization in independent modules. Increasing the size of the experts from 32 filters per layer to 64 or 128 filters5 or enlarging the overall receptive field by using two pooling and two upsampling layers, still results in good specialization of the experts, with no more than two experts specializing on up to two tasks at once. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fewer examples from the canonical distribution. In many applications, we might only have a small sample from the original distribution. Interestingly, we found that all experts still specialize to different tasks and recover good approximations of the inverse mechanisms when we reduce the number of examples from the original distribution from 30 000 down to $6 4 ^ { 6 }$ . Even though the output digits are not as clean and sharp, we still achieve $96 \\%$ accuracy on the pretrained classifier before the discriminator starts to overfit. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have developed a method to identify and learn a set of independent causal mechanisms. In the present work, these are inverse mechanisms, but an extension to forward mechanisms appears feasible and worthwhile. We reported promising results in an experiment based on image transformations; future work could study more complex settings and diverse domains. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A natural extension of our work is to consider independent mechanisms that simultaneously affect the data (e.g. lighting and position in a portrait), and to allow multiple passes through our committee of experts to identify local mechanisms (akin to Lie derivatives) from more complex datasets — for instance, using recurrent neural network that allow the application of multiple mechanisms by iteration. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that for large numbers of experts, the computational cost might become unnecessarily high. This could be mitigated by hybrid approaches incorporating gated mixture of experts — which may exhibit lower computational complexity — or a hierarchical selection of competing experts. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "We believe our work constitutes a relevant connection between causal modeling and deep learning. As discussed in the introduction, causality has a lot to offer for crucial machine learning problems such as transfer or compositional modeling. Our systems illustrates some of these properties. Independent modules as sub-components could be trained independently and/or from multiple domains, added subsequently, and transferred to other problems. This may constitute a step towards causally motivated life-long learning. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
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+        "text": "D. Janzing and B. Scholkopf. Causal inference using the algorithmic Markov condition. ¨ IEEE Transactions on Information Theory, 56(10):5168–5194, 2010.   \nMichael I Jordan and Robert A Jacobs. Hierarchical mixtures of experts and the em algorithm. Neural computation, 6(2):181–214, 1994.   \nDiederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \nDiederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   \nJudea. Pearl. Causality. Cambridge University Press, 2000.   \nJonas Peters, Dominik Janzing, and Bernhard Scholkopf. ¨ Elements of Causal Inference. The MIT Press, 2017.   \nMateo Rojas-Carulla, Bernhard Scholkopf, Richard Turner, and Jonas Peters. Causal transfer in ¨ machine learning. arXiv preprint arXiv:1507.05333, 2015.   \nB. Scholkopf, D. Janzing, J. Peters, E. Sgouritsa, K. Zhang, and J. M. Mooij. On causal and anticausal ¨ learning. In J Langford and J Pineau (eds.), Proceedings of the 29th International Conference on Machine Learning (ICML), pp. 1255–1262, New York, NY, USA, 2012. Omnipress.   \nNoam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. arXiv preprint arXiv:1701.06538, 2017.   \nEric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. Adversarial discriminative domain adaptation. arXiv preprint arXiv:1702.05464, 2017.   \nKun Zhang, Jiji Zhang, and Bernhard Scholkopf. Distinguishing cause from effect based on exogene- ¨ ity. arXiv preprint arXiv:1504.05651, 2015. ",
+        "bbox": [
+            169,
+            101,
+            828,
+            527
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL RESULTS. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            403,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 PLOT OF COMPETING EXPERTS. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            434,
+            147
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Each expert in Figure 6 is represented with the same color and linestyle across all tasks. Note how the red expert tries to learn two similar tasks until iteration 500 (i.e. left and up-left translation), when the green expert takes over one of the task and they can then both quickly specialize. ",
+        "bbox": [
+            174,
+            160,
+            825,
+            202
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/c1de7ee3ed18b5fac92aa208bb0a8fc8ed966c5d44beb2bc684a8ac11603d8a6.jpg",
+        "image_caption": [
+            "Figure 6: Each line style is associated to the score that an expert obtains on the discriminator when being fed transformed digits using one of the mechanisms. Each expert learns to specialize on a different mechanism. Each curve is smoothed with an average of the last 50 iterations for ease of visualization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            219,
+            816,
+            545
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 TOO MANY OR TOO FEW EXPERTS. ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            648,
+            455,
+            662
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Too many experts When there are too many experts, for most tasks only one wins all the examples, as shown in Figure 7 where the model has 16 experts for 10 tasks. The remaining experts either do not specialize at all — and therefore can be removed from the architecture — or specialize on the same task, and could therefore be combined if after inspection they are considered to perform the same task. Since the accuracy on the transformed data tested on the pretrained classifier reaches again the upperbound of the untransformed data, and since the progress is very similar to that illustrated in Figure 4, we omit this plot. ",
+        "bbox": [
+            174,
+            674,
+            825,
+            772
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Too few experts For a committee of 6 experts, the networks do not reconstruct properly most of the digits, which is reflected by an overall low objective function value on the data. Also, the score against the classifier that does not exceed $72 \\%$ . A few experts are inevitably assigned to multiple tasks, and by looking at Figure 7 it is interesting to see that the clustering result is still meaningful (e.g. expert 5 is assigned to left, down-left, and up-left translation). ",
+        "bbox": [
+            174,
+            787,
+            825,
+            857
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B DETAILS OF NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            878,
+            482,
+            893
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In Table 1 we report the configuration of the neural networks used in these experiments. ",
+        "bbox": [
+            171,
+            909,
+            746,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C TRANSFORMATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            379,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In our experiments we use the following transformations ",
+        "bbox": [
+            173,
+            133,
+            545,
+            150
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Translations: the image is shifted by 4 pixels in one of the eight directions up, down, left, right and the four diagonals.   \n• contrast (or color) inversion: the value of each pixel — originally in the range $[ 0 , 1 ]$ — is recomputed as $1 -$ the original value.   \n• Noise addition: random Gaussian noise with zero mean and variance 0.25 is added to the original image, which is then clamped again to the [0, 1] interval. ",
+        "bbox": [
+            173,
+            162,
+            826,
+            262
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/15ad1d48a5ba3f972d602203c58f12e7d9d8d14270729659d1131a9b9219f28d.jpg",
+        "image_caption": [
+            "Figure 7: The proportion of data won by each expert for each transformation on the digits from the 0.0 test set, for the case of 10 mechanisms and more experts (16 on left) or too few (6 on the right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            304,
+            805,
+            474
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Architectures of the neural networks used in the experiment section. BN stands for Batch normalization, FC for fully connected. All convolutions are preceded by a 1 pixel zero padding. ",
+        "bbox": [
+            173,
+            537,
+            825,
+            568
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/2ac494c71995c105fd3aa1d550929e3e1298cb50c29af2c32f577d18e7ae846e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Expert</td></tr><tr><td>Layers</td><td></td></tr><tr><td>3 × 3,32,BN,ELU 3 × 3,32, BN,ELU</td><td></td></tr><tr><td>3 × 3,32, BN, ELU 3 × 3,32, BN, ELU</td><td></td></tr><tr><td>3 × 3,1, sigmoid</td><td></td></tr></table>",
+        "bbox": [
+            258,
+            626,
+            411,
+            741
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/2ebc948e3542172ec3ad5bcc549db8a7ba5fa7c98c9b52dbf498c698c004e172.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Discriminator</td></tr><tr><td rowspan=1 colspan=1>Layers</td></tr><tr><td rowspan=1 colspan=1>3 × 3,16,ELU</td></tr><tr><td rowspan=1 colspan=1>3 × 3,16,ELU3 × 3,16, ELU</td></tr><tr><td rowspan=1 colspan=1> 2 × 2, avg pooling</td></tr><tr><td rowspan=1 colspan=1>3 × 3,32, ELU3 × 3,32, ELU</td></tr><tr><td rowspan=1 colspan=1> 2 × 2, avg pooling</td></tr><tr><td rowspan=1 colspan=1>3 × 3,64, ELU3 × 3,64, ELU</td></tr><tr><td rowspan=1 colspan=1> 2 × 2, avg pooling</td></tr><tr><td rowspan=1 colspan=1>1024, FC, ELU1, FC, sigmoid</td></tr></table>",
+        "bbox": [
+            588,
+            578,
+            732,
+            790
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D NOTES ON THE FORMALIZATION OF INDEPENDENCE OF MECHANISMS ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            820,
+            794,
+            837
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section we briefly discuss the notion of independence of mechanisms as in (Janzing & Scholkopf, 2010), where the independence principle is formalized in terms of algorithmic complexity ¨ (also known as Kolmogorov complexity). We summarize the main points needed in the present context. We parametrize each mechanism by a bit string $x$ . The Kolmogorov complexity $K ( \\bar { x } )$ of $x$ is the length of the shortest program generating $x$ on an a priori chosen universal Turing machine. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The algorithmic mutual information can be defined as $I ( x : y ) : = K ( x ) + K ( y ) - K ( x , y )$ , and it can be shown to equal ",
+        "bbox": [
+            173,
+            103,
+            825,
+            131
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b3f5126638fa239e389c32b2e63b2a69eb237895eee75945e5956d47c0cede7.jpg",
+        "text": "$$\nI ( x : y ) = K ( y ) - K ( y | x ^ { * } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            130,
+            596,
+            147
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where for technical reasons we need to work with $x ^ { * }$ , the shortest description of $x$ (which is in general uncomputable). Here, the conditional Kolmogorov complexity $K ( y | x )$ is defined as the length of the shortest program that generates $y$ from $x$ . The algorithmic mutual information measures the algorithmic information two objects have in common. We define two mechanisms to be (algorithmically) independent whenever the length of the shortest description of the two bit strings together is not shorter than the sum of the shortest individual descriptions (note it cannot be longer), i.e., if their algorithmic mutual information vanishes.7 In view of (4), this means that ",
+        "bbox": [
+            173,
+            150,
+            825,
+            247
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2fda1d8f55b1e013f007d50312cd41772538ffec097602e8134d03b59dc1ef10.jpg",
+        "text": "$$\nK ( y ) = K ( y | x ^ { * } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            255,
+            562,
+            272
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We will say that two mechanisms $x$ and $y$ are independent whenever the complexity of the conditional mechanism $y | x$ is comparable to the complexity of the unconditional one $y$ . If, in contrast, the two mechanisms were closely related, then we would expect that we can mimic one of the mechanisms by applying the other one followed by a low complexity conditional mechanism. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            342
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This can be implemented by having a complexity measure for, say, neural networks, and comparing the complexities of neural nets that are trained to perform certain tasks. Inspired by regularization theory, we could measure complexity by inverse regularization strength or weight vector norm. An alternative way to regularize neural nets consists of early stopping. If we fix the number of training epochs to a constant, and find that network 1 reaches a lower error than network 2, we conclude that the network 2 would take longer to reach the same low error, and thus network 2 requires higher effective complexity to solve its task than network 1. We have run preliminary experiments with this measure and found that (1) indeed our training procedure did increase independence, and (2) the independence between two different mechanism was larger than the independence between one mechanism and the identity. ",
+        "bbox": [
+            174,
+            348,
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+        ],
+        "page_idx": 12
+    }
+]

parse/train/SkgewU5ll/SkgewU5ll.md

@@ -0,0 +1,332 @@
+# GRAM: GRAPH-BASED ATTENTION MODEL FOR HEALTHCARE REPRESENTATION LEARNING
+
+Edward Choi1, Mohammad Taha Bahadori1, Le Song1, Walter F. Stewart2 & Jimeng Sun1 1Georgia Institute of Technology, 2Sutter Health
+
+# ABSTRACT
+
+Deep learning methods exhibit promising performance for predictive modeling in healthcare, but two important challenges remain:
+
+• Data insufficiency: Often in healthcare predictive modeling, the sample size is insufficient for deep learning methods to achieve satisfactory results. • Interpretation: The representations learned by deep learning models should align with medical knowledge.
+
+To address these challenges, we propose a GRaph-based Attention Model, GRAM that supplements electronic health records (EHR) with hierarchical information inherent to medical ontologies. Based on the data volume and the ontology structure, GRAM represents a medical concept as a combination of its ancestors in the ontology via an attention mechanism.
+
+We compared predictive performance (i.e. accuracy, data needs, interpretability) of GRAM to various methods including the recurrent neural network (RNN) in two sequential diagnoses prediction tasks and one heart failure prediction task. Compared to the basic RNN, GRAM achieved $10 \%$ higher accuracy for predicting diseases rarely observed in the training data and $3 \%$ improved area under the ROC curve for predicting heart failure using an order of magnitude less training data. Additionally, unlike other methods, the medical concept representations learned by GRAM are well aligned with the medical ontology. Finally, GRAM exhibits intuitive attention behaviors by adaptively generalizing to higher level concepts when facing data insufficiency at the lower level concepts.
+
+# 1 INTRODUCTION
+
+The rapid growth in volume and diversity of health care data from electronic health records (EHR) and other sources is motivating the use of predictive modeling to improve care for individual patients. In particular, novel applications are emerging that use deep learning methods such as word embedding (Choi et al., 2016c;e), recurrent neural networks (RNN) (Che et al., 2016; Choi et al., 2016a;b; Lipton et al., 2016), convolutional neural networks (CNN) (Nguyen et al., 2016) or stacked denoising autoencoders (SDA) (Che et al., 2015; Miotto et al., 2016), demonstrating significant performance enhancement for diverse prediction tasks. Deep learning models appear to perform significantly better than logistic regression or multilayer perceptron (MLP) models that depend, to some degree, on expert feature construction (Lipton et al., 2015; Razavian et al., 2016).
+
+Training deep learning models typically requires large amounts of data that often cannot be met by a single health system or provider organization. Sub-optimal model performance can be particularly challenging when the focus of interest is predicting onset of a specific disease (e.g. heart failure) or related events such as accelerated disease progression. For example, using Doctor AI (Choi et al., 2016a), we discovered that RNN alone was ineffective to predict the onset of diseases such as cerebral degenerations (e.g. Leukodystrophy, Cerebral lipidoses) or developmental disorders (e.g. autistic disorder, Heller’s syndrome), partly because their rare occurrence in the training data provided little learning opportunity to the flexible models like RNN.
+
+The data requirement of deep learning models comes from having to assess exponential number of combinations of input features. This can be alleviated by exploiting medical ontologies that encodes hierarchical clinical constructs and relationships among medical concepts. Fortunately, there are many well-organized ontologies in healthcare such as the International Classification of Diseases (ICD), Clinical Classifications Software (CCS) (Stearns et al., 2001) or Systematized Nomenclature of Medicine-Clinical Terms (SNOMED-CT) (Project et al., 2010). Nodes (i.e. medical concepts) close to one another in medical ontologies are likely to be associated with similar patients, allowing us to transfer knowledge among them. Therefore, proper use of medical ontologies will be helpful when we lack enough data for the nodes in the ontology to train deep learning models.
+
+In this work, we propose GRAM, a method that infuses information from medical ontologies into deep learning models via neural attention. Considering the frequency of a medical concept in the EHR data and its ancestors in the ontology, GRAM decides the representation of the medical concept by adaptively combining its ancestors via attention mechanism. This will not only support deep learning models to learn robust representations without large amount of data, but also learn interpretable representations that align well with the knowledge from the ontology. The attention mechanism is trained in an end-to-end fashion with the neural network model that predicts the onset of disease(s). We also propose an effective initialization technique in addition to the ontological knowledge to better guide the representation learning process.
+
+We compared predictive performance (i.e. accuracy, data needs, interpretability) of GRAM to various models including the recurrent neural network (RNN) in two sequential diagnoses prediction tasks and one heart failure (HF) prediction task. We demonstrate that GRAM is up to $10 \%$ more accurate than the basic RNN for predicting diseases less observed in the training data. After discussing GRAM’s scalability, we visualize the representations learned from various models where GRAM provides more intuitive representations by grouping similar medical concepts close to one another. Finally, we show GRAM’s attention mechanism can be interpreted to understand how it assigns the right amount of attention to the ancestors of each medical concept by considering the data availability and the ontology structure.
+
+# 2 METHODOLOGY
+
+We first define the notations describing EHR data and medical ontologies, followed by a description of GRAM (Section 2.2), the end-to-end training of the attention generation and predictive modeling (Section 2.3), and the efficient initialization scheme (Section 2.4).
+
+# 2.1 BASIC NOTATION
+
+We denote the set of entire medical codes from the EHR as $c _ { 1 } , c _ { 2 } , \ldots , c _ { | { \mathcal { C } } | } \in { \mathcal { C } }$ with the vocabulary size $| { \mathcal { C } } |$ . The clinical record of each patient can be viewed as a sequence of visits $V _ { 1 } , \dots , V _ { T }$ where each visit contains a subset of medical codes $V _ { t } \subseteq \mathcal { C }$ . $V _ { t }$ can be represented as a binary vector $\mathbf { x } _ { t } \in \{ 0 , 1 \} ^ { | c | }$ where the $i \cdot$ -th element is 1 only if $V _ { t }$ contains the code $c _ { i }$ . To avoid clutter, all algorithms will be presented for a single patient.
+
+We assume that a given medical ontology $\mathcal { G }$ typically expresses the hierarchy of various medical concepts in the form of a parent-child relationship, where the medical codes $\mathcal { C }$ form the leaf nodes. Ontology $\mathcal { G }$ is represented as a directed acyclic graph (DAG) whose nodes form a set $\mathcal { D } = \mathcal { C } + \mathcal { C } ^ { \prime }$ . $\mathcal { C } ^ { \prime } = \{ c _ { | \mathcal { C } | + 1 } , c _ { | \mathcal { C } | + 2 } , \ldots , c _ { | \mathcal { C } | + | \mathcal { C } ^ { \prime } | } \}$ defines the set of all non-leaf nodes (i.e. ancestors of the leaf nodes), where $| { \mathcal { C } } ^ { \prime } |$ represents the number of all non-leaf nodes. We use knowledge $D A G$ to refer to $\mathcal { G }$ . A parent in the knowledge DAG $\mathcal { G }$ represents a related but more general concept over its children. Therefore, $\mathcal { G }$ provides a multi-resolution view of medical concepts with different degrees of specificity. While some ontologies are exclusively expressed as parent-child hierarchies (e.g. ICD-9, CCS), others are not. For example, in some instances SNOMED-CT also links medical concepts to causal or treatment relationships, but the majority relationships in SNOMED-CT are still parent-child. Therefore, we focus on the parent-child relationships in this work.
+
+# 2.2 KNOWLEDGE DAG AND THE ATTENTION MECHANISM
+
+GRAM leverages the parent-child relationship of $\mathcal { G }$ to learn robust representations when data volume is constrained. GRAM balances the use of ontology information in relation to data volume in determining the level of specificity for a medical concept. When a medical concept is less observed in the data, more weight is given to its ancestors as they can be learned more accurately and offer general (coarse-grained) information about their children. The process of resorting to the parent concepts can be automated via the attention mechanism and the end-to-end training as described in Figure 1.
+
+In the knowledge DAG, each node $c _ { i }$ is assigned a basic embedding vector $\mathbf { e } _ { i } \in \mathbb { R } ^ { m }$ , where $m$ represents the dimensionality. Then $\mathbf { e } _ { 1 } , \ldots , \mathbf { e } _ { | { \mathcal { C } } | }$ are the basic embeddings of the codes $c _ { 1 } , \ldots , c _ { | { \mathcal { C } } | }$ while ${ \bf e } _ { | \mathcal { C } | + 1 } , \dots , { \bf e } _ { | \mathcal { C } | + | \mathcal { C } ^ { \prime } | }$ represent the basic embeddings of the internal nodes $c _ { | \mathcal { C } | + 1 } , \ldots , c _ { | \mathcal { C } | + | \mathcal { C } _ { \bullet } ^ { \prime } | }$ The initialization of these basic embeddings is described in Section 2.4. We formulate a leaf node’s final representation as a convex combination of the basic embeddings of itself and its ancestors:
+
+![](images/6bdcaaff59d5e6cfa1a45b7baa554326a2f30b8c8d0ec95506ecb0fa56116efa.jpg)  
+Figure 1: The illustration of GRAM. Leaf nodes (solid circles) represents a medical concept in the EHR, while the non-leaf nodes (dotted circles) represent more general concepts. The final representation $\mathbf { g } _ { i }$ of the leaf concept $c _ { i }$ is computed by combining the basic embeddings $\mathbf { e } _ { i }$ of $c _ { i }$ and $\mathbf { e } _ { g } , \mathbf { e } _ { c }$ and $\mathbf { e } _ { a }$ of its ancestors $c _ { g } , c _ { c }$ and $c _ { a }$ via an attention mechanism. The final representations form the embedding matrix $\mathbf { G }$ for all leaf concepts. After that, we use $\mathbf { G }$ to embed patient visit vector $\mathbf { x } _ { t }$ to a visit representation $\mathbf { v } _ { t }$ , which is then fed to a neural network model to make the final prediction $\hat { \mathbf { y } } _ { t }$ .
+
+$$
+\mathbf { g } _ { i } = \sum _ { j \in A ( i ) } \alpha _ { i j } \mathbf { e } _ { j } , \qquad \sum _ { j \in A ( i ) } \alpha _ { i j } = 1 , \alpha _ { i j } \geq 0 \mathrm { ~ f o r ~ } j \in A ( i ) ,
+$$
+
+where $\mathbf { g } _ { i } \in \mathbb { R } ^ { m }$ denotes the final representation of the code $c _ { i }$ , $\boldsymbol { \mathscr { A } } ( i )$ the indices of the code $c _ { i }$ and $c _ { i }$ ’s ancestors, $\mathbf { e } _ { j }$ the basic embedding of the code $c _ { j }$ and $\alpha _ { i j } \in \mathbb { R }$ the attention weight on the embedding $\mathbf { e } _ { j }$ when calculating $\mathbf { g } _ { i }$ . The attention weight $\alpha _ { i j }$ in Eq. (1) is calculated by the following Softmax function,
+
+$$
+\alpha _ { i j } = { \frac { \exp ( f ( \mathbf { e } _ { i } , \mathbf { e } _ { j } ) ) } { \sum _ { k \in { \mathcal { A } } ( i ) } \exp ( f ( \mathbf { e } _ { i } , \mathbf { e } _ { k } ) ) } }
+$$
+
+$f ( \mathbf { e } _ { i } , \mathbf { e } _ { j } )$ is a scalar value representing the compatibility between the basic embeddings of $\mathbf { e } _ { i }$ and $\mathbf { e } _ { k }$ We compute $f ( \mathbf { e } _ { i } , \mathbf { e } _ { j } )$ via the following feed-forward network with a single hidden layer (MLP),
+
+$$
+f ( \mathbf { e } _ { i } , \mathbf { e } _ { j } ) = \mathbf { u } _ { a } ^ { \top } \operatorname { t a n h } ( \mathbf { W } _ { a } \left[ \begin{array} { l } { \mathbf { e } _ { i } } \\ { \mathbf { e } _ { j } } \end{array} \right] + \mathbf { b } _ { a } )
+$$
+
+where $\mathbf { W } _ { a } \in \mathbb { R } ^ { l \times { 2 m } }$ is the weight matrix for the concatenation of $\mathbf { e } _ { i }$ and $\mathbf { e } _ { j }$ , $\textbf { b } \in \mathbb { R } ^ { l }$ the bias vector, and $\mathbf { u } _ { a } \in \mathbb { R } ^ { l }$ the weight vector for generating the scalar value. The constant $l$ represents the dimension size of the hidden layer of $f ( \cdot , \cdot )$ . Note that we always concatenate $\mathbf { e } _ { i }$ and $\mathbf { e } _ { j }$ in the child-ancestor order.
+
+Remarks: The example in Figure 1 is derived based on a single path from $c _ { i }$ to $c _ { a }$ . However, the same mechanism can be applicable to multiple paths as well. For example, code $c _ { k }$ has two paths to the root $c _ { a }$ , containing five ancestors in total. Another scenario is where the EHR data contain both leaf codes and some ancestor codes. We can move those ancestors present in EHR data from the set $\scriptstyle { \mathcal { C } } ^ { \prime }$ to $\mathcal { C }$ and apply the same process as Eq. (1) to obtain the final representations for them.
+
+# 2.3 END-TO-END TRAINING WITH A PREDICTIVE MODEL
+
+We train the attention mechanism together with a predictive model such that the attention mechanism improves the predictive performance. Once the final representations $\mathbf { g } _ { 1 } , \mathbf { g } _ { 2 } , \ldots , \mathbf { g } _ { | { \mathcal { C } } | }$ of all medical codes are obtained, we can convert visit $V _ { t }$ to a visit representation $\mathbf { v } _ { t }$ by using the embedding matrix $\mathbf { G } \in \mathcal { R } ^ { m \times | c | }$ where $\mathbf { g } _ { i }$ is its $i$ -th column as in Figure 1. We continue the mathematical formulation under the assumption that we are using the RNN to perform sequential diagnoses prediction (Choi et al., 2016a;b) with the objective of predicting the disease codes of the next visit $\bar { V _ { t + 1 } }$ given the visit records up to the current timestep $V _ { 1 } , V _ { 2 } , \dots , V _ { t }$ , which can be expressed as follows,
+
+$$
+\begin{array} { r } { \widehat { \mathbf { y } } _ { t } = \widehat { \mathbf { x } } _ { t + 1 } = \operatorname { S o f t m a x } ( \mathbf { W } \mathbf { h } _ { t } + \mathbf { b } ) , \quad \mathrm { w h e r e } } \\ { \mathbf { h } _ { 1 } , \mathbf { h } _ { 2 } , \ldots , \mathbf { h } _ { t } = \mathbf { R N N } ( \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \ldots , \mathbf { v } _ { t } ) , \quad \mathrm { w h e r e } } \\ { \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \ldots , \mathbf { v } _ { t } = \operatorname { t a n h } ( \mathbf { G } [ \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \ldots , \mathbf { x } _ { t } ] ) } \end{array}
+$$
+
+Table 1: Basic statistics of Sutter PAMF, MIMIC-III and Sutter heart failure (HF) cohort.   
+
+<table><tr><td>Dataset</td><td>Sutter PAMF</td><td>MIMIC-III</td><td>SutterHFcohort</td></tr><tr><td># of patients</td><td>258,555†</td><td>7,499t</td><td>30,727† (3,408 cases)</td></tr><tr><td>#of visits</td><td>13,920,759</td><td>19,911</td><td>572,551</td></tr><tr><td>Avg.# of visits per patient</td><td>53.8</td><td>2.66</td><td>38.38</td></tr><tr><td># of unique ICD9 codes</td><td>10,437</td><td>4,893</td><td>5,689</td></tr><tr><td>Avg.# of codes per visit</td><td>1.98</td><td>13.1</td><td>2.06</td></tr><tr><td>Max # of codes per visit</td><td>54</td><td>39</td><td>29</td></tr></table>
+
+† Note that for all datasets, we selected patients who made at least two hospital visits.
+
+where $\mathbf { x } _ { t } \in \mathbb { R } ^ { | \mathcal { C } | }$ denotes the $t$ -th visit; $\mathbf { v } _ { t } \in \mathbb { R } ^ { m }$ the $t { \cdot }$ -th visit representation; $\mathbf { h } _ { t } \in \mathbb { R } ^ { r }$ the RNN’s hidden layer at $t { \cdot }$ -th time step (i.e. $t$ -th visit); $\textbf { W } \in \mathbb { R } ^ { | \mathcal { C } | \times r }$ and $\textbf { b } \in \mathbb { R } ^ { | \mathcal { C } | }$ the weight matrices and the bias vector of the Softmax function; $r$ denotes the dimension size of the hidden layer. We use “RNN” to denote any recurrent neural network variants that can cope with the vanishing gradient problem (Bengio et al., 1994), such as LSTM (Hochreiter $\&$ Schmidhuber, 1997), GRU (Cho et al., 2014), and IRNN (Le et al., 2015), with any varying numbers of hidden layers. The prediction loss for all time steps is calculated using the cross entropy as follows, $\mathcal { L } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } \ldots , \mathbf { x } _ { T } ) =$ $\begin{array} { r } { - \frac { 1 } { T - 1 } \sum _ { t = 1 } ^ { T - 1 } \bigg ( \mathbf { y } _ { t } { } ^ { \top } \log ( \widehat { \mathbf { y } } _ { t } ) + ( \mathbf { 1 } - \mathbf { y } _ { t } ) ^ { \top } \log ( \mathbf { 1 } - \widehat { \mathbf { y } } _ { t } ) \bigg ) } \end{array}$ where we sum the cross entropy errors from all dimensions of $\widehat { \mathbf { y } } _ { t }$ , $T$ denotes the length of the visit sequence. Note that the above loss is defined bfor a single patient. But we can take the average of the individual loss for multiple patients.
+
+# 2.4 INITIALIZING BASIC EMBEDDINGS
+
+The attention generation mechanism in Section 2.2 requires basic embeddings $\mathbf { e } _ { i }$ of each node in the knowledge DAG. The basic embeddings of ancestors, however, pose a difficulty because they are often not observed in the data.To better initialize them, we use co-occurrence information to learn the basic embeddings of medical codes and their ancestors. Co-occurrence has proven to be an important source of information when learning representations of words or medical concepts (Mikolov et al., 2013; Choi et al., 2016c;e). To train the basic embeddings, we employ GloVe (Pennington et al., 2014), which uses the global co-occurrence matrix of words to learn their representations. In our case, the co-occurrence matrix of the codes and the ancestors was generated by counting the co-occurrences within each visit $V _ { t }$ , where we augment each visit with the ancestors of the codes in the visit. Details of training the basic embeddings are described in the Appendix A. Note that, with or without the initialization, the basic embeddings $\mathbf { e } _ { i }$ ’s of both leaf nodes (i.e. medical codes) and non-leaf nodes (i.e. ancestors) are fine-tuned when training our model, since the error signal flows from the output $\widehat { \mathbf { y } } _ { t }$ to the final representations $\mathbf { g } _ { i }$ ’s which are convex combinations of $\mathbf { e } _ { i }$ ’s.
+
+# 3 EXPERIMENTS
+
+We conduct three experiments to determine if GRAM offered superior prediction performance when facing data insufficiency. We first describe the experimental setup followed by results comparing predictive performance of GRAM with various baseline models. After discussing GRAM’s scalability, we qualitatively evaluate the interpretability of the resulting representation. The source code of GRAM is publicly available at https://github.com/mp2893/gram.
+
+# 3.1 EXPERIMENT SETUP
+
+Prediction tasks and source of data: We conduct two sequential diagnoses prediction tasks, which aim at predicting all diagnosis categories in the next visit, and one heart failure (HF) prediction task, which is a binary prediction task for predicting a future HF onset where the prediction is made only once at the last visit $\mathbf { x } _ { T }$ .
+
+Two sequential diagnoses predictions are respectively conducted using 1) Sutter Palo Alto Medical Foundation (PAMF) dataset, which consists of 18-years longitudinal medical records of 258K patients between age 50 and 90. This will determine GRAM’s performance for general adult population with long visit records. 2) MIMIC-III dataset (Johnson et al., 2016; Goldberger et al., 2000), which is a publicly available dataset consisting of medical records of $7 . 5 \mathrm { K }$ intensive care unit (ICU) patients over 11 years. This will determine GRAM’s performance for high-risk patients with very short visit records. We utilize all the patients with at least 2 visits. We prepared the true labels $\mathbf { y } _ { t }$ by grouping the ICD9 codes into 283 groups using CCS single-level diagnosis grouper1. This is to improve the training speed and predictive performance for easier analysis, while preserving sufficient granularity for each diagnosis. Each diagnosis code’s varying frequency in the training data can be viewed as different degrees of data insufficiency. We calculate Accuracy $@ k$ for each of CCS single-level diagnosis codes such that, given a visit $V _ { t }$ , we get 1 if the target diagnosis is in the top $k$ guesses and 0 otherwise.
+
+We conduct HF prediction on Sutter heart failure (HF) cohort, which is a subset of Sutter PAMF data for a heart failure onset prediction study with 3.4K HF cases and 27K controls chosen by a set of criteria (see Appendix B). This will determine GRAM’s performance for a different prediction task where we predict the onset of one specific condition. We randomly downsample the training data to create different degrees of data insufficiency. We use area under the ROC curve (AUC) to measure the performance.
+
+A summary of the datasets are provided in Table 1.We used CCS multi-level diagnoses hierarchy2 as our knowledge DAG $\mathcal { G }$ . We also tested the ICD9 code hierarchy3, but the performance was similar to using CCS multi-level hierarchy. For all three tasks, we randomly divide the dataset into the training, validation and test set by .75:.10:.15 ratio, and use the validation set to tune the hyper-parameters. Further details regarding the hyper-parameter tuning are provided in Appendix C. The test set performance is reported in the paper.
+
+Implementation details: We implemented GRAM with Theano 0.8.2 (Team, 2016). For training models, we used Adadelta (Zeiler, 2012) with a mini-batch of 100 patients, on a machine equipped with Intel Xeon E5-2640, 256GB RAM, four Nvidia Titan X’s and CUDA 7.5.
+
+Models for comparison are the following. The first two $\mathrm { G R A M + }$ and GRAM are the proposed methods and the rest are baselines. Hyper-parameter tuning is configured so that the number of parameters for the baselines would be comparable to GRAM’s. Further details are provided in Appendix C.
+
+• GRAM: Input sequence $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { T }$ is first transformed by the embedding matrix G, then fed to the GRU with a single hidden layer, which in turn makes the prediction, as described by Eq. (4). The basic embeddings $\mathbf { e } _ { i }$ ’s are randomly initialized.   
+• GRAM+: We use the same setup as GRAM, but the basic embeddings $\mathbf { e } _ { i }$ ’s are initialized according to Section 2.4.   
+• RandomDAG: We use the same setup as GRAM, but each leaf concept has five randomly assigned ancestors from the CCS multi-level hierarchy to test the effect of correct domain knowledge.   
+• RNN: Input $\mathbf { x } _ { t }$ is transformed by an embedding matrix $\mathbf { W } _ { e m b } \in \mathbb { R } ^ { k \times | \mathcal { C } | }$ , then fed to the GRU with a single hidden layer. The embedding size $k$ is a hyper-parameter. $\mathbf { W } _ { e m b }$ is randomly initialized and trained together with the GRU.   
+• $\mathbf { R N N + }$ : We use the same setup as RNN, but we initialize the embedding matrix $\mathbf { W } _ { e m b }$ with GloVe vectors trained only with the co-occurrence of leaf concepts. This is to compare GRAM with a similar weight initialization technique.   
+• SimpleRollUp: We use the same setup as RNN. But for input $\mathbf { x } _ { t }$ , we replace all diagnosis codes with their direct parent codes in the CCS multi-level hierarchy, giving us 578, 526 and 517 input codes respectively for Sutter data, MIMIC-III and Sutter HF cohort. This is to compare the performance of GRAM with a common grouping technique.   
+RollUpRare: We use the same setup as RNN, but we replace any diagnosis code whose frequency is less than a certain threshold in the dataset with its direct parent. We set the threshold to 100 for Sutter data and Sutter HF cohort, and 10 for MIMIC-III, giving us 4,408, 935 and 1,538 input codes respectively for Sutter data, MIMIC-III and Sutter HF cohort. This is an intuitive way of dealing with infrequent medical codes.
+
+# 3.2 PREDICTION PERFORMANCE AND SCALABILITY
+
+Tables 2a and 2b show the sequential diagnoses prediction performance on Sutter data and MIMIC-III. Both figures show that ${ \mathrm { G R A M } } +$ outperforms other models when predicting labels with significant data insufficiency (i.e. less observed in the training data).The performance gain is greater for MIMIC-III, where GRAM+ outperforms the basic RNN by $10 \%$ in the 20th-40th percentile range. This seems to come from the fact that MIMIC patients on average have significantly shorter visit history than Sutter patients, with much more codes received per visit. Such short sequences make it difficult for the RNN to learn and predict diagnoses sequence. The performance difference between GRAM+ and
+
+Table 2: Performance of three prediction tasks. The $\mathbf { X }$ -axis of (a) and (b) represents the labels grouped by the percentile of their frequencies in the training data in non-decreasing order. For (c), we vary the size of the training data to train the models. (b) uses Accuracy $\textcircled{ a} 20$ because MIMIC-III has a large average number of codes per visit (see Table 1).   
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp</td><td>0.0150 0.0042 0.0050 0.0069 0.0080 0.2691</td><td>0.3242 0.2987 0.2700 0.2742</td><td>0.4325 0.4224 0.4010 0.4140</td><td>0.4238 0.4193 0.4059 0.4212</td><td>0.4903 0.4895 0.4853 0.4959</td></tr></table>
+
+(a) Accuracy@5 of sequential diagnoses prediction on Sutter data
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp</td><td>0.0672 0.0556 0.0329 0.0454 0.0454 0.0578 0.1328</td><td>0.1787 0.1016 0.0708 0.0843 0.0731</td><td>0.2644 0.1935 0.1346 0.2080 0.1804 0.2455 0.2667</td><td>0.2490 0.2296 0.1512 0.2494 0.2371</td><td>0.6267 0.6363 0.4494 0.6239 0.6243</td></tr></table>
+
+(b) Accuracy@20 of sequential diagnoses prediction on MIMIC-III   
+
+<table><tr><td>Model</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td><td>80%</td><td>90%</td><td>100%</td></tr><tr><td>GRAM+</td><td>0.7970</td><td>0.8223</td><td>0.8307</td><td>0.8332</td><td>0.8389</td><td>0.8404</td><td>0.8452</td><td>0.8456</td><td>0.8447</td><td>0.8448</td></tr><tr><td>GRAM</td><td>0.7981</td><td>0.8217</td><td>0.8340</td><td>0.8332</td><td>0.8372</td><td>0.8377</td><td>0.8440</td><td>0.8431</td><td>0.8430</td><td>0.8447</td></tr><tr><td>RandomDAG</td><td>0.7644</td><td>0.7882</td><td>0.7986</td><td>0.8070</td><td>0.8143</td><td>0.8185</td><td>0.8274</td><td>0.8312</td><td>0.8254</td><td>0.8226</td></tr><tr><td>RNN+</td><td>0.7930</td><td>0.8117</td><td>0.8162</td><td>0.8215</td><td>0.8261</td><td>0.8333</td><td>0.8343</td><td>0.8353</td><td>0.8345</td><td>0.8335</td></tr><tr><td>RNN</td><td>0.7811</td><td>0.7942</td><td>0.8066</td><td>0.8111</td><td>0.8156</td><td>0.8207</td><td>0.8258</td><td>0.8278</td><td>0.8297</td><td>0.8314</td></tr><tr><td>SimpleRollUp</td><td>0.7799</td><td>0.8022</td><td>0.8108</td><td>0.8133</td><td>0.8177</td><td>0.8207</td><td>0.8223</td><td>0.8272</td><td>0.8269</td><td>0.8258</td></tr><tr><td>RollUpRare</td><td>0.7830</td><td>0.8067</td><td>0.8064</td><td>0.8119</td><td>0.8211</td><td>0.8202</td><td>0.8262</td><td>0.8296</td><td>0.8307</td><td>0.8291</td></tr></table>
+
+(c) AUC of HF onset prediction on Sutter HF cohort
+
+Table 3: Scalablity result in per epoch training time in second (the number of epochs needed).   
+
+<table><tr><td>Model</td><td>Sequential diagnosis prediction (Sutter data)</td><td>Sequential diagnosis prediction (MIMIC-III)</td><td>HF prediction (Sutter HF cohort)</td></tr><tr><td>GRAM</td><td>525s (39 epochs)</td><td>2s (11 epochs)</td><td>12s (7 epochs)</td></tr><tr><td>RNN</td><td>352s (24 epochs)</td><td>1s (6 epochs)</td><td>8s (5 epochs)</td></tr></table>
+
+GRAM suggests that our proposed initialization scheme of the basic embeddings $\mathbf { e } _ { i }$ is important for sequential diagnosis prediction.
+
+Table 2c shows the HF prediction performance on Sutter HF cohort. GRAM and $\mathrm { G R A M + }$ consistently outperforms other baselines (except ${ \mathrm { R N N } } +$ ) by $3sim \mathrm { { } } 4 \%$ AUC, and $\mathrm { R N N } +$ by maximum $1 . 8 \%$ AUC. These differences are quite significant given that the AUC is already in the mid-80s, a high value for HF prediction, cf. (Choi et al., 2016d). Note that, for $\mathbf { G R A M + }$ and $\mathrm { R N N } +$ , we used the downsampled training data to initialize the basic embeddings $\mathbf { e } _ { i }$ ’s and the embedding matrix $\mathbf { W } _ { e m b }$ with GloVe, respectively. The result shows that the initialization scheme of the basic embeddings in $\mathrm { G R A M + }$ gives limited improvement over GRAM. This stems from the different natures of the two prediction tasks. While the goal of HF prediction is to predict a binary label for the entire visit sequence, the goal of sequential diagnosis prediction is to predict the co-occurring diagnosis codes at every visit. Therefore the co-occurrence information infused by the initialized embedding scheme is more beneficial to sequential diagnosis prediction. Additionally, this benefit is associated with the natures of the two prediction tasks than the datasets used for the prediction tasks. Because the initialized embedding shows different degrees of improvement as shown by Tables 2a and 2c, when Sutter HF cohort is a subset of Sutter PAMF, thus having similar characteristics. Additional prediction results when varying the $k$ of Accuracy $@ k$ are discussed in the Appendix D.
+
+Overall, GRAM showed superior predictive performance under data insufficiency in three different experiments, demonstrating its general applicability in predictive healthcare modeling. Now we briefly discuss the scalability of GRAM by comparing its training time to RNN’s. Table 3 shows the number of seconds taken for the two models to train for a single epoch for each predictive modeling task. $\mathrm { G R A M + }$ and $\mathrm { R N N } +$ showed the same behavior as GRAM and RNN. GRAM takes approximately $50 \%$ more time to train for a single epoch for all prediction tasks. This stems from calculating attention weights and the final representations $\mathbf { g } _ { i }$ for all medical codes. GRAM also generally takes about $50 \%$ more epochs to reach to the model with the lowest validation loss. This is due to optimizing an extra MLP model that generates the attention weights. Overall, use of GRAM adds a manageable amount of overhead in training time to the plain RNN.
+
+# 3.3 QUALITATIVE EVALUATION OF INTERPRETABLE REPRESENTATIONS
+
+To qualitatively assess the interpretability of the learned representations of the medical codes, we plot on a 2-D space using t-SNE (Maaten & Hinton, 2008) the final representations $\mathbf { g } _ { i }$ of 2,000
+
+![](images/689483f70536387faf727af1a9e63b92fc3e321f1590e2cfd6ba6deb6c377b7f.jpg)  
+(a) Scatterplot of the final representations ${ \bf g } _ { i }$ ’s of $\mathrm { G R A M + }$
+
+![](images/b4a9d66eed2c0be9a5d006b70667ae4d91f35eba1c1a90f50c894282f06bf4d0.jpg)  
+(c) Scatterplot of the disease representations trained by GloVe
+
+![](images/984428b089efdf50c7c5a0d50a46b30952b26253a1e53ac11d318fdfe0fa4fea.jpg)  
+Figure 2: t-SNE scatterplots of medical concepts trained by $\mathrm { G R A M + }$ , $\mathrm { R N N } +$ and GloVe
+
+(b) Scatterplot of the trained embedding matrix $\mathbf { W } _ { e m b }$ of $\mathrm { R N N } +$
+
+randomly chosen diseases learned by $\mathrm { G R A M + }$ for sequential diagnoses prediction on Sutter data4 (Figure 2a). The colors represent the highest disease categories and the text annotations represent the detailed disease categories in CCS multi-level hierarchy. For comparison, we also show the t-SNE plots on the strongest results from $\mathrm { R N N } +$ (Figure 2b), and GloVe (Figure 2c), the same embedding technique in initializing the basic embeddings $\mathbf { e } _ { i }$ . Figures 2b and 2c confirm that interpretable representations cannot simply be learned only by co-occurrence or supervised prediction without medical knowledge. GRAM+ learns disease representations that are significantly more consistent with the given knowledge DAG $\mathcal { G }$ . Therefore the neural network predictive model that accepts $\mathbf { g } _ { i }$ is using accurate representations that lead to higher predictive performance. Additional scatterplots of other models are provided in Appendix E for comparison. An interactive visualization tool can be accessed at http://www.sunlab.org/research/gram-graph-based-attention-model/.
+
+# 3.4 ANALYSIS OF THE ATTENTION BEHAVIOR
+
+Next we show that GRAM’s attention can be interpreted to understand how it considers data availability and knowledge DAG’s structure when performing a prediction task. Using Eq. (1), we can calculate the attention weights of individual disease. Figure 3 shows the attention behaviors of four representative diseases when performing HF prediction on Sutter HF cohort.
+
+Other pneumothorax (ICD9 512.89) in Figure 3a is rarely observed in the data and has only five siblings. In this case, most information is derived from the highest ancestor. Temporomandibular joint disorders & articular disc disorder (ICD9 524.63) in Figure 3b is rarely observed but has 139 siblings. In this case, its parent receives a stronger attention because it aggregates sufficient samples from all of its children to learn a more accurate representation. Note that the disease itself also receives a stronger attention to facilitate easier distinction from its large number of siblings.
+
+![](images/c4bfd8d3f0c22ad1774b97b598078ce0bdafbec69884368b5a4bbb594c7e2b19.jpg)  
+Figure 3: GRAM’s attention behavior during HF prediction for four representative diseases (each column). In each figure, the leaf node represents the disease and upper nodes are its ancestors. The size of the node shows the amount of attention it receives, which is also shown by the bar charts. The number in the parenthesis next to the disease is its frequency in the training data. We exclude the root of the knowledge DAG $\mathcal { G }$ from all figures as it did not play a significant role.
+
+Unspecified essential hypertension (ICD9 401.9) in Figure 3c is very frequently observed but has only two siblings. In this case, GRAM assigns a very strong attention to the leaf, which is logical because the more you observe a disease, the stronger your confidence becomes. Need for prophylactic vaccination and inoculation against influenza (ICD9 V04.81) in Figure 3d is quite frequently observed and also has 103 siblings. The attention behavior in this case is quite similar to the case with fewer siblings (Figure 3b) with a slight attention shift towards the leaf concept as more observations lead to higher confidence.
+
+# 4 RELATED WORK
+
+We introduce recent studies related to GRAM that learn the representations of graphs and discuss their relationship with GRAM. Several studies focused on learning the representations of graph vertices by using the neighbor information. DeepWalk (Perozzi et al., 2014) and node2vec (Grover & Leskovec, 2016) use random walk while LINE (Tang et al., 2015) uses breadth-first search to find the neighbors of a vertex and learn its representation based on the neighbor information. Graph convolutional approaches (Yang et al., 2016; Kipf & Welling, 2016) also focus on learning the vertex representations to mainly perform vertex classification. These works focus on solving the graph data problems whereas GRAM focuses on solving EHR data problems using the knowledge DAG as supplementary information.
+
+Several researchers tried to model the knowledge DAG such as WordNet (Miller, 1995) or Freebase (Bollacker et al., 2008) where two entities are connected with various types of relation, forming a set of triples. They aim to project entities and relations (Bordes et al., 2013; Socher et al., 2013; Wang et al., 2014; Lin et al., 2015) to the latent space based on the triples or additional information such as hierarchy of entities (Xie et al., 2016). These works demonstrated tasks such as link prediction, triple classification or entity classification using the learned representations. More recently, Li et al. (2016) learned the representations of words and Wikipedia categories by utilizing the hierarchy of Wikipedia categories. GRAM is fundamentally different from the above studies in that it aims to design intuitive attention mechanism on the knowledge DAG as a knowledge prior to cope with data insufficiency and learn medically interpretable representations to make accurate predictions.
+
+A classical approach for incorporating side information in the predictive models is to use graph Laplacian regularization (Weinberger et al., 2006; Che et al., 2015). However, using this approach is not straightforward as it relies on the appropriate definition of distance on graphs which is often unavailable.
+
+# 5 CONCLUSION
+
+Data insufficiency, either due to less common diseases or small datasets, is one of the key hurdles in healthcare analytics, especially when we apply deep neural networks models. To overcome this hurdle, we leverage the knowledge DAG, which provides a multi-resolution view of medical concepts. We propose GRAM, a graph-based attention model using both a knowledge DAG and EHR to learn an accurate and interpretable representations for medical concepts. GRAM chooses a weighted average of ancestors of a medical concept and train the entire process with a predictive model in an end-to-end fashion. We conducted three predictive modeling experiments on real EHR datasets and showed significant improvement in the prediction performance, especially on low-frequency diseases and small datasets. Analysis of the attention behavior provided intuitive insight of GRAM.
+
+# REFERENCES
+
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+Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. JMLR, 9(Nov), 2008.   
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In NIPS, 2013.   
+George A Miller. Wordnet: a lexical database for english. Communications of the ACM, 38(11), 1995.   
+Riccardo Miotto, Li Li, Brian A Kidd, and Joel T Dudley. Deep patient: An unsupervised representation to predict the future of patients from the electronic health records. Scientific Reports, 6, 2016.   
+Phuoc Nguyen, Truyen Tran, Nilmini Wickramasinghe, and Svetha Venkatesh. Deepr: A convolutional net for medical records. arXiv:1607.07519, 2016.   
+Jeffrey Pennington, Richard Socher, and Christopher D Manning. Glove: Global vectors for word representation. In EMNLP, 2014.   
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In SIGKDD, 2014.   
+Healthcare Cost & Utilization Project et al. Clinical classifications software (ccs) for icd-9-cm. Rockville, MD:   
+Agency for Healthcare Research and Quality, 2010.   
+Narges Razavian, Jake Marcus, and David Sontag. Multi-task prediction of disease onsets from longitudinal lab tests. In MLHC, 2016.   
+Richard Socher, Danqi Chen, Christopher D Manning, and Andrew Ng. Reasoning with neural tensor networks for knowledge base completion. In NIPS, 2013.   
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+The Theano Development Team. Theano: A python framework for fast computation of mathematical expressions. arXiv:1605.02688, 2016.   
+Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen. Knowledge graph embedding by translating on hyperplanes. In AAAI, 2014.   
+Kilian Q Weinberger, Fei Sha, Qihui Zhu, and Lawrence K Saul. Graph Laplacian Regularization for Large-Scale Semidefinite Programming. In NIPS, 2006.   
+Ruobing Xie, Zhiyuan Liu, and Maosong Sun. Representation learning of knowledge graphs with hierarchical types. In IJCAI, 2016.   
+Zhilin Yang, William Cohen, and Ruslan Salakhutdinov. Revisiting semi-supervised learning with graph embeddings. arXiv:1603.08861, 2016.   
+Matthew D Zeiler. Adadelta: an adaptive learning rate method. arXiv:1212.5701, 2012.
+
+![](images/1566e0ab53d4efb27fd3b1a603894f6037b19d4d62f57271b24a9e9e895c3eb5.jpg)  
+Figure 4: Creating the co-occurrence matrix together with the ancestors. Here we exclude the root node, which will be just a single row (column). We first create an augmented dataset by adding the ancestors of the code to the dataset. Then, we count the co-occurrence of the codes. Performing GloVe on this matrix produces the embedding vectors $\mathbf { e } _ { i }$ .
+
+# A GENERATING GLOVE EMBEDDINGS
+
+We learn the basic embeddings $\mathbf { e } _ { i }$ ’s of medical codes and their ancestors using GloVe (Pennington et al., 2014), which uses global co-occurrence matrix of words to learn their representations. We generate the co-occurrence matrix of the codes and the ancestors by counting the co-occurrence within each visit $V _ { t }$ . However, since visits only contain the leaf codes $c \in { \mathcal { C } }$ , we augment each visit with the ancestors of the codes in each visit, then count the co-occurrence of codes and ancestors altogether.
+
+We describe the details of the algorithm with an example. We borrow the parent-child relationships from the knowledge DAG of Figure 1. Given a visit $V _ { t }$ ,
+
+$$
+V _ { t } = \{ c _ { d } , c _ { i } , c _ { k } \}
+$$
+
+we augment it with the ancestors of all the codes to obtain the augmented visit $V _ { t } ^ { \prime }$ ,
+
+$$
+V _ { t } ^ { \prime } = \{ c _ { d } , \underline { { { c _ { b } } } } , \underline { { { c _ { a } } } } , c _ { i } , \underline { { { c _ { g } } } } , \underline { { { c _ { c } } } } , \underline { { { c _ { a } } } } , c _ { k } , \underline { { { c _ { j } } } } , \underline { { { c _ { f } } } } , \underline { { { c _ { c } } } } , \underline { { { c _ { b } } } } , \underline { { { c _ { a } } } } \}
+$$
+
+where the added ancestors are underlined. Note that a single ancestor can appear multiple times in $V _ { t } ^ { \prime }$ . In fact, the higher the ancestor is in the knowledge DAG, the more times it is likely to appear in $V _ { t } ^ { \prime }$ . We count the co-occurrence of two codes in $V _ { t } ^ { \prime }$ as follows,
+
+$$
+c o \mathrm { - } o c c u r r e n c e ( c _ { i } , c _ { j } , V _ { t } ^ { \prime } ) = c o u n t ( c _ { i } , V _ { t } ^ { \prime } ) \times c o u n t ( c _ { j } , V _ { t } ^ { \prime } ) _ { l }
+$$
+
+where $c o u n t ( c _ { i } , V _ { t } ^ { \prime } )$ is the number of times the code $c _ { i }$ appears in the augmented visit $V _ { t } ^ { \prime }$ . For example, the co-occurrence between the leaf code $c _ { i }$ and the root $c _ { a }$ is 3. However, the co-occurrence between the ancestor $c _ { c }$ and the root $c _ { a }$ is 6. Therefore our algorithm will naturally make the ancestor codes have higher co-occurrence with other codes compared to leaf medical codes. We repeat this calculation for all pairs of codes in all augmented visits of all patients to obtain the co-occurrence matrix depicted by Figure 4. For training the embedding vectors using the co-occurrence matrix, we use the same procedure and hyper-parameter as described in Pennington et al. (2014).
+
+# B HEART FAILURE COHORT CONSTRUCTION
+
+For the heart failure (HF) case patients, we select patients between 40 to 85 years of age at the time of HF diagnosis. HF diagnosis (HFDx) criteria are defined as: 1) Qualifying ICD-9 codes for HF appeared in the encounter records or medication orders. Qualifying ICD-9 codes are listed in Table 4. 2) at least three clinical encounters with qualifying ICD-9 codes had to occur within 12 months of each other, where the date of HFDx was assigned to the earliest of the three dates. If the time span between the first and second appearances of the HF diagnosis code was greater than 12 months, the date of the second encounter was used as the first qualifying encounter. Up to ten eligible controls (in terms of sex, age, location) were selected for each case, yielding average 9 controls per case. Each control was also assigned an index date, which is the HFDx date of the matched case. Controls are selected such that they did not meet the HF diagnosis criteria prior to the HFDx date plus 182 days of their corresponding case. Control subjects were required to have their first office encounter within one year of the matching HF case patient’s first office visit, and have at least one office encounter 30 days before or any time after the case’s HFDx date to ensure similar duration of observations among cases and controls.
+
+Table 4: Qualifying ICD-9 codes for heart failure   
+
+<table><tr><td rowspan=1 colspan=1>ICD-9 Code</td><td rowspan=1 colspan=1>Description</td></tr><tr><td rowspan=1 colspan=1>398.91</td><td rowspan=1 colspan=1>Rheumatic heart failure (congestive)</td></tr><tr><td rowspan=1 colspan=1>402.01</td><td rowspan=1 colspan=1>Malignant hypertensive heart disease with heart failure</td></tr><tr><td rowspan=1 colspan=1>402.11</td><td rowspan=1 colspan=1>Benign hypertensive heart disease with heart failure</td></tr><tr><td rowspan=1 colspan=1>402.91</td><td rowspan=1 colspan=1>Unspecified hypertensive heart disease with heart failure</td></tr><tr><td rowspan=1 colspan=1>404.01</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, malignant, with heart failure and withchronic kidney disease stage I through stage IV, or unspecified</td></tr><tr><td rowspan=1 colspan=1>404.03</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, malignant, with heart failure and with chronic kidney disease stage V or end stage renal disease</td></tr><tr><td rowspan=1 colspan=1>404.11</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, benign, with heart failure and withchronic kidney disease stage I through stage IV, or unspecified</td></tr><tr><td rowspan=1 colspan=1>404.13</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, benign, with heart failure and chronickidney disease stage V or end stage renal disease</td></tr><tr><td rowspan=1 colspan=1>404.91</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, unspecified, with heart failure and with chronic kidney disease stage I through stage IV, or unspecified</td></tr><tr><td rowspan=1 colspan=1>404.93</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, unspecified, with heart failure andchronic kidney disease stage V or end stage renal disease</td></tr><tr><td rowspan=1 colspan=1>428.0</td><td rowspan=1 colspan=1>Congestive heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.1</td><td rowspan=1 colspan=1>Leftheart failure</td></tr><tr><td rowspan=1 colspan=1>428.20</td><td rowspan=1 colspan=1>Systolic heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.21</td><td rowspan=1 colspan=1> Acute systolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.22</td><td rowspan=1 colspan=1>Chronic systolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.23</td><td rowspan=1 colspan=1>Acute on chronic systolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.30</td><td rowspan=1 colspan=1>Diastolic heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.31</td><td rowspan=1 colspan=1>Acute diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.32</td><td rowspan=1 colspan=1>Chronic diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.33</td><td rowspan=1 colspan=1>Acute on chronic diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.40</td><td rowspan=1 colspan=1>Combined systolic and diastolic heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.41</td><td rowspan=1 colspan=1>Acute combined systolic and diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.42</td><td rowspan=1 colspan=1>Chronic combined systolic and diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.43</td><td rowspan=1 colspan=1> Acute on chronic combined systolic and diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.9</td><td rowspan=1 colspan=1>Heart failure, unspecified</td></tr></table>
+
+# C HYPER-PARAMETER TUNING
+
+We define five hyper-parameters for GRAM:
+
+• dimensionality $m$ of the basic embedding $\mathbf { e } _ { i }$ : [100, 200, 300, 400, 500] • dimensionality $r$ of the RNN hidden layer $\mathbf { h } _ { t }$ from Eq. (4): [100, 200, 300, 400, 500] • dimensionality $l$ of $\mathbf { W } _ { a }$ and $ { \mathbf { b } } _ { a }$ from Eq. (3): [100, 200, 300, 400, 500] • $L _ { 2 }$ regularization coefficient for all weights except RNN weights: [0.1, 0.01, 0.001, 0.0001] • dropout rate for the dropout on the RNN hidden layer: [0.0, 0.2, 0.4, 0.6, 0.8]
+
+We performed 100 iterations of the random search by using the above ranges for each of the three prediction experiments. For sequential diagnoses prediction on Sutter data, we used $10 \%$ of the training data to tune the hyper-parameters to balance the time and search space. To match the baselines’ number of parameters to GRAM’s, we add 550 to the list of $m$ ’s possible values. This will make the baseline’s largest possible number of parameters comparable to the GRAM’s largest possible number of parameters.
+
+For SimpleRollUp and RollUpRare, the number of input codes is smaller than other models due to the grouping. Therefore, to match their largest possible number of parameters to GRAM’s, we need to add much larger values to $m$ . However, after preliminary experiments, as expected, setting $m$ to too large a value degraded the performance due to overfitting. Since the number of input codes decreased due to the grouping, increasing the dimensionality of $\mathbf { e } _ { i }$ is not a logical thing to do. Therefore, for SimpleRollUp and RollUpRare, we use the same list of values for $m$ as other baselines.
+
+Table 5: Hyper-parameters used by the models in each predictive modeling experiments   
+
+<table><tr><td>Experiment</td><td>Model</td><td>m</td><td>r</td><td>1</td><td>L2</td><td>Dropout rate</td></tr><tr><td rowspan="6">Disease progression modeling (Sutter data)</td><td>GRAM+ GRAM</td><td>500 500</td><td>500 500</td><td>100 100</td><td>0.0001 0.0001</td><td>0.6 0.6</td></tr><tr><td>RandomDAG</td><td>500</td><td>500</td><td>100</td><td>0.0001</td><td>0.6</td></tr><tr><td>RNN+</td><td>550</td><td>500</td><td></td><td>0.0001</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>0.6</td></tr><tr><td>RNN</td><td>550</td><td>500</td><td></td><td>0.0001</td><td>0.6</td></tr><tr><td>SimpleRollUp RollUpRare</td><td>500 500</td><td>500 500</td><td></td><td>0.0001 0.0001</td><td>0.4 0.2</td></tr><tr><td rowspan="6">Disease progression modeling (MIMIC-III)</td><td>GRAM+ GRAM</td><td>400 400</td><td>400 400</td><td>100 100</td><td>0.0001</td><td>0.6</td></tr><tr><td>RandomDAG</td><td>400</td><td>400</td><td>100</td><td>0.001 0.001</td><td>0.6</td></tr><tr><td>RNN+</td><td>550</td><td>400</td><td></td><td>0.001</td><td>0.6</td></tr><tr><td></td><td></td><td>400</td><td></td><td></td><td>0.8</td></tr><tr><td>RNN SimpleRollUp</td><td>550 400</td><td>400</td><td></td><td>0.001 0.001</td><td>0.8 0.6</td></tr><tr><td></td><td>RollUpRare</td><td>400</td><td>400</td><td></td><td></td><td></td></tr><tr><td rowspan="6">HF prediction (Sutter HF cohort)</td><td></td><td></td><td></td><td></td><td>0.0001</td><td>0.0</td></tr><tr><td>GRAM+</td><td>200</td><td>100</td><td>100</td><td>0.001</td><td></td></tr><tr><td>GRAM</td><td>200</td><td>100</td><td>100</td><td></td><td>0.6</td></tr><tr><td>RandomDAG</td><td>300</td><td>100</td><td>200</td><td>0.001</td><td>0.6</td></tr><tr><td>RNN+</td><td>200</td><td>100</td><td></td><td>0.001</td><td>0.6</td></tr><tr><td></td><td></td><td></td><td></td><td>0.0001</td><td>0.6</td></tr><tr><td>RNN</td><td>200</td><td>100</td><td></td><td>0.001</td><td>0.6</td></tr><tr><td>SimpleRollUp</td><td>300</td><td>200</td><td></td><td>0.001</td><td>0.4</td></tr><tr><td>RollUpRare</td><td>100</td><td>100</td><td></td><td>0.001</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>0.6</td></tr></table>
+
+Table 5 describes the final hyper-parameter settings we used for all models for each prediction experiments.
+
+# D PREDICTION RESULTS USING DIFFERENT $k$ ’S IN ACCURACY@K
+
+We show Accuracy@k using $k = 5 , 1 0 , 2 0 , 3 0$ for sequential diagnoses prediction on Sutter data (Tables 6a, 6b, 6c and 6d) and MIMIC-III (Tables 7a, 7b, 7c and 7d). We can see from the tables that $\mathrm { G R A M + }$ consistently outperforms other models under 40th percentile range, except when $k = 2 0 , 3 0$ for sequential diagnoses prediction on Sutter data where SimpleRollUp shows similar performance. We can also see that $\mathrm { G R A M + }$ performs significantly better than other models for all $k = 5$ , 10, 20, 30 when predicting infrequently observed diseases on MIMIC-III. As discussed in Section 3.2, this seems to come from the short visit sequences of MIMIC patients.
+
+# E T-SNE 2-D PLOTS OF VARIOUS MODELS
+
+For further comparison, we display t-SNE scatterplots of GRAM (Figure 5a) RandomDAG (Figure 5b, RNN (Figure 5c), and Skip-gram (Figure 5d). GRAM, RandomDAG and RNN were trained for sequential diagnoses prediction on Sutter data, and Skip-gram (Mikolov et al., 2013) was trained on Sutter data as it is an unsupervised method. For Skip-gram, we used each visit $V _ { t }$ as the context window. As we do not distinguish between the target concept and the neighbor concepts, we calculated the Skip-gram objective function using all possible pairs of codes within a single visit.
+
+We can see from Figure 5a that the quality of the final representations $\mathbf { g } _ { i }$ of GRAM is quite similar to $\mathrm { G R A M + }$ (Figure 2a). Compared to other baselines, GRAM demonstrates significantly more structured representations that align well with the given knowledge DAG. It is interesting that Skip-gram shows the most structured representation among all baselines. We used GloVe to initialize the basic embeddings $\mathbf { e } _ { i }$ in this work because it uses global co-occurrence information and its training time is dependent only on the total number of unique concepts $| { \mathcal { C } } |$ . Skip-gram’s training time, on the other hand, depends on both the number of patients and the number of visits each patient made, which makes the algorithm generally slower than GloVe. However, considering both Figures $2 \mathrm { c }$ and 5d, initializing $\mathbf { e } _ { i }$ ’s with Skip-gram vectors might give us additional performance boost.
+
+Table 6: Accuracy at various $k$ ’s (a to d) for sequential diagnoses prediction on Sutter data. The columns represent the labels grouped by the percentile of their frequencies in the training data in non-decreasing order.   
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RoliUpRare</td><td>0.0150 0.0042 0.0050 0.0069 0.0080 0.0085 0.0062</td><td>0.3242 0.2987 0.2700 0.2742 0.2691 0.3078 0.2768</td><td>0.4325 0.4224 0.4010 0.4140 0.4134 0.4369 0.4176</td><td>0.4238 0.4193 0.4059 0.4212 0.4227 0.4330 0.4226</td><td>0.4903 0.4895 0.4853 0.4959 0.4951 0.4924 0.4956</td></tr></table>
+
+(a) Accuracy@5 of sequential diagnoses prediction on Sutter data
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RolUpRare</td><td>0.0319 0.0163 0.0142 0.0183 0.0196 0.0164 0.0204</td><td>0.3882 0.3645 0.3285 0.3412 0.3290 0.3768 0.3450</td><td>0.5054 0.4944 0.4691 0.4884 0.4871 0.5132 0.4917</td><td>0.5215 0.5173 0.5025 0.5233 0.5230 0.5326</td><td>0.6459 0.6445 0.6401 0.6538 0.6531 0.6521</td></tr></table>
+
+(b) Accuracy@10 of sequential diagnoses prediction on Sutter data
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RoliUpRare</td><td>0.0630 0.0442 0.0397 0.0483 0.0481 0.0418 0.0517</td><td>0.4486 0.4276 0.3933 0.4132 0.4025 0.4496 0.4170</td><td>0.5764 0.5669 0.5389 0.5654 0.5630 0.5877 0.5672</td><td>0.6153 0.6125 0.5997 0.6235 0.6232 0.6262 0.6214</td><td>0.7973 0.7963 0.7919 0.8003 0.7995 0.8013</td></tr></table>
+
+(c) Accuracy@20 of sequential diagnoses prediction on Sutter data
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RollUpRare</td><td>0.0946 0.0662 0.0672 0.0736 0.0733 0.0662 0.0759</td><td>0.4879 0.4693 0.4313 0.4604 0.4478 0.4924 0.4657</td><td>0.6186 0.6107 0.5843 0.6136 0.6103 0.6312 0.6146</td><td>0.6792 0.6766 0.6667 0.6930 0.6921 0.6907 0.6908</td><td>0.8800 0.8798 0.8760 0.8785 0.8767 0.8795 0.8778</td></tr></table>
+
+(d) Accuracy@30 of sequential diagnoses prediction on Sutter data
+
+Table 7: Accuracy at various $k$ ’s (a to d) for sequential diagnoses prediction on MIMIC-III. The columns represent the labels grouped by the percentile of their frequencies in the training data in non-decreasing order.   
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RollUpRare</td><td>0.0086 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000</td><td>0.1089 0.0468 0.0327 0.0435 0.0376 0.0671 0.0423</td><td>0.1665 0.1093 0.0778 0.1266 0.1105 0.1501 0.1085</td><td>0.1029 0.0918 0.0612 0.0973 0.0923 0.1191 0.0874</td><td>0.2597 0.2665 0.1634 0.2594 0.2601 0.2635 0.2604</td></tr></table>
+
+(a) Accuracy@5 of sequential diagnoses prediction on MIMIC-III
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RolUpRare</td><td>0.0380 0.0045 0.0023 0.0227 0.0023 0.0249 0.0227</td><td>0.1310 0.0682 0.0470 0.0587 0.0518 0.1038 0.0530</td><td>0.2095 0.1494 0.1025 0.1591 0.1389 0.1997 0.1412</td><td>0.1627 0.1487 0.0938 0.1616 0.1521 0.1769 0.1519</td><td>0.4175 0.4235 0.2692 0.4193 0.4142 0.4260</td></tr></table>
+
+(b) Accuracy@10 of sequential diagnoses prediction on MIMIC-III
+
+(c) Accuracy@20 of sequential diagnoses prediction on MIMIC-III   
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RoliUpRare</td><td>0.0672 0.0556 0.0329 0.0454 0.0454 0.0578 0.0454</td><td>0.1787 0.1016 0.0708 0.0843 0.0731 0.1328 0.0653</td><td>0.2644 0.1935 0.1346 0.2080 0.1804 0.2455 0.1843</td><td>0.2490 0.2296 0.1512 0.2494 0.2371 0.2667</td><td>0.6267 0.6363 0.4494 0.6239 0.6243 0.6387</td></tr></table>
+
+<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RollUpRare</td><td>0.0744 0.0578 0.0351 0.0578 0.0578 0.0578 0.0556</td><td>0.2065 0.1157 0.0932 0.1103 0.0775 0.1556 0.0910</td><td>0.3180 0.2257 0.1635 0.2571 0.2237 0.2865 0.2235</td><td>0.3363 0.3074 0.2200 0.3409 0.3160 0.3488 0.3255</td><td>0.7726 0.7802 0.5977 0.7656 0.7643 0.7800</td></tr></table>
+
+(d) Accuracy@30 of sequential diagnoses prediction on MIMIC-III
+
+![](images/207c97af4d8c0611947b428e647f2ea210d709362ff36a9d96b7904fb9b73c28.jpg)  
+(a) Scatterplot of the final representations ${ \bf g } _ { i }$ ’s of GRAM
+
+![](images/c0fa2677df46d223e61d45d996d35cc58e8e4be356ab504aa2199fb2eda739a3.jpg)  
+(b) Scatterplot of the final representations $\mathbf { g } _ { i }$ ’s of RandomDAG
+
+![](images/2e7a5335acf240417a092dbb8634a5e6d652c9d515ecb710341031754c8da59c.jpg)  
+(d) Scatterplot of the basic embeddings $\mathbf { e } _ { i }$ ’s trained by Skip-gram
+
+![](images/291ea73802a224c6fc099588975183c840aa3b7666c3d1233a2d6a174082679a.jpg)  
+Figure 5: Scatterplot of medical concepts trained by various models. We used t-SNE to reduce the dimension to 2-D.
+
+(c) Scatterplot of the trained embedding matrix $\mathbf { W } _ { e m b }$ of RNN

parse/train/SkgewU5ll/SkgewU5ll_content_list.json

@@ -0,0 +1,1469 @@
+[
+    {
+        "type": "text",
+        "text": "GRAM: GRAPH-BASED ATTENTION MODEL FOR HEALTHCARE REPRESENTATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            77,
+            821,
+            123
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Edward Choi1, Mohammad Taha Bahadori1, Le Song1, Walter F. Stewart2 & Jimeng Sun1 1Georgia Institute of Technology, 2Sutter Health ",
+        "bbox": [
+            179,
+            145,
+            813,
+            176
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            194,
+            544,
+            209
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning methods exhibit promising performance for predictive modeling in healthcare, but two important challenges remain: ",
+        "bbox": [
+            230,
+            218,
+            764,
+            246
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Data insufficiency: Often in healthcare predictive modeling, the sample size is insufficient for deep learning methods to achieve satisfactory results. • Interpretation: The representations learned by deep learning models should align with medical knowledge. ",
+        "bbox": [
+            236,
+            247,
+            764,
+            304
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To address these challenges, we propose a GRaph-based Attention Model, GRAM that supplements electronic health records (EHR) with hierarchical information inherent to medical ontologies. Based on the data volume and the ontology structure, GRAM represents a medical concept as a combination of its ancestors in the ontology via an attention mechanism. ",
+        "bbox": [
+            233,
+            305,
+            766,
+            372
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We compared predictive performance (i.e. accuracy, data needs, interpretability) of GRAM to various methods including the recurrent neural network (RNN) in two sequential diagnoses prediction tasks and one heart failure prediction task. Compared to the basic RNN, GRAM achieved $10 \\%$ higher accuracy for predicting diseases rarely observed in the training data and $3 \\%$ improved area under the ROC curve for predicting heart failure using an order of magnitude less training data. Additionally, unlike other methods, the medical concept representations learned by GRAM are well aligned with the medical ontology. Finally, GRAM exhibits intuitive attention behaviors by adaptively generalizing to higher level concepts when facing data insufficiency at the lower level concepts. ",
+        "bbox": [
+            232,
+            375,
+            766,
+            511
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            536,
+            336,
+            551
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The rapid growth in volume and diversity of health care data from electronic health records (EHR) and other sources is motivating the use of predictive modeling to improve care for individual patients. In particular, novel applications are emerging that use deep learning methods such as word embedding (Choi et al., 2016c;e), recurrent neural networks (RNN) (Che et al., 2016; Choi et al., 2016a;b; Lipton et al., 2016), convolutional neural networks (CNN) (Nguyen et al., 2016) or stacked denoising autoencoders (SDA) (Che et al., 2015; Miotto et al., 2016), demonstrating significant performance enhancement for diverse prediction tasks. Deep learning models appear to perform significantly better than logistic regression or multilayer perceptron (MLP) models that depend, to some degree, on expert feature construction (Lipton et al., 2015; Razavian et al., 2016). ",
+        "bbox": [
+            174,
+            559,
+            825,
+            683
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training deep learning models typically requires large amounts of data that often cannot be met by a single health system or provider organization. Sub-optimal model performance can be particularly challenging when the focus of interest is predicting onset of a specific disease (e.g. heart failure) or related events such as accelerated disease progression. For example, using Doctor AI (Choi et al., 2016a), we discovered that RNN alone was ineffective to predict the onset of diseases such as cerebral degenerations (e.g. Leukodystrophy, Cerebral lipidoses) or developmental disorders (e.g. autistic disorder, Heller’s syndrome), partly because their rare occurrence in the training data provided little learning opportunity to the flexible models like RNN. ",
+        "bbox": [
+            174,
+            689,
+            825,
+            797
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The data requirement of deep learning models comes from having to assess exponential number of combinations of input features. This can be alleviated by exploiting medical ontologies that encodes hierarchical clinical constructs and relationships among medical concepts. Fortunately, there are many well-organized ontologies in healthcare such as the International Classification of Diseases (ICD), Clinical Classifications Software (CCS) (Stearns et al., 2001) or Systematized Nomenclature of Medicine-Clinical Terms (SNOMED-CT) (Project et al., 2010). Nodes (i.e. medical concepts) close to one another in medical ontologies are likely to be associated with similar patients, allowing us to transfer knowledge among them. Therefore, proper use of medical ontologies will be helpful when we lack enough data for the nodes in the ontology to train deep learning models. ",
+        "bbox": [
+            174,
+            805,
+            825,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            80,
+            821,
+            109
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose GRAM, a method that infuses information from medical ontologies into deep learning models via neural attention. Considering the frequency of a medical concept in the EHR data and its ancestors in the ontology, GRAM decides the representation of the medical concept by adaptively combining its ancestors via attention mechanism. This will not only support deep learning models to learn robust representations without large amount of data, but also learn interpretable representations that align well with the knowledge from the ontology. The attention mechanism is trained in an end-to-end fashion with the neural network model that predicts the onset of disease(s). We also propose an effective initialization technique in addition to the ontological knowledge to better guide the representation learning process. ",
+        "bbox": [
+            174,
+            116,
+            825,
+            238
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We compared predictive performance (i.e. accuracy, data needs, interpretability) of GRAM to various models including the recurrent neural network (RNN) in two sequential diagnoses prediction tasks and one heart failure (HF) prediction task. We demonstrate that GRAM is up to $10 \\%$ more accurate than the basic RNN for predicting diseases less observed in the training data. After discussing GRAM’s scalability, we visualize the representations learned from various models where GRAM provides more intuitive representations by grouping similar medical concepts close to one another. Finally, we show GRAM’s attention mechanism can be interpreted to understand how it assigns the right amount of attention to the ancestors of each medical concept by considering the data availability and the ontology structure. ",
+        "bbox": [
+            174,
+            244,
+            825,
+            367
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            386,
+            341,
+            401
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We first define the notations describing EHR data and medical ontologies, followed by a description of GRAM (Section 2.2), the end-to-end training of the attention generation and predictive modeling (Section 2.3), and the efficient initialization scheme (Section 2.4). ",
+        "bbox": [
+            174,
+            411,
+            823,
+            453
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 BASIC NOTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            468,
+            336,
+            482
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We denote the set of entire medical codes from the EHR as $c _ { 1 } , c _ { 2 } , \\ldots , c _ { | { \\mathcal { C } } | } \\in { \\mathcal { C } }$ with the vocabulary size $| { \\mathcal { C } } |$ . The clinical record of each patient can be viewed as a sequence of visits $V _ { 1 } , \\dots , V _ { T }$ where each visit contains a subset of medical codes $V _ { t } \\subseteq \\mathcal { C }$ . $V _ { t }$ can be represented as a binary vector $\\mathbf { x } _ { t } \\in \\{ 0 , 1 \\} ^ { | c | }$ where the $i \\cdot$ -th element is 1 only if $V _ { t }$ contains the code $c _ { i }$ . To avoid clutter, all algorithms will be presented for a single patient. ",
+        "bbox": [
+            174,
+            489,
+            825,
+            561
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We assume that a given medical ontology $\\mathcal { G }$ typically expresses the hierarchy of various medical concepts in the form of a parent-child relationship, where the medical codes $\\mathcal { C }$ form the leaf nodes. Ontology $\\mathcal { G }$ is represented as a directed acyclic graph (DAG) whose nodes form a set $\\mathcal { D } = \\mathcal { C } + \\mathcal { C } ^ { \\prime }$ . $\\mathcal { C } ^ { \\prime } = \\{ c _ { | \\mathcal { C } | + 1 } , c _ { | \\mathcal { C } | + 2 } , \\ldots , c _ { | \\mathcal { C } | + | \\mathcal { C } ^ { \\prime } | } \\}$ defines the set of all non-leaf nodes (i.e. ancestors of the leaf nodes), where $| { \\mathcal { C } } ^ { \\prime } |$ represents the number of all non-leaf nodes. We use knowledge $D A G$ to refer to $\\mathcal { G }$ . A parent in the knowledge DAG $\\mathcal { G }$ represents a related but more general concept over its children. Therefore, $\\mathcal { G }$ provides a multi-resolution view of medical concepts with different degrees of specificity. While some ontologies are exclusively expressed as parent-child hierarchies (e.g. ICD-9, CCS), others are not. For example, in some instances SNOMED-CT also links medical concepts to causal or treatment relationships, but the majority relationships in SNOMED-CT are still parent-child. Therefore, we focus on the parent-child relationships in this work. ",
+        "bbox": [
+            173,
+            568,
+            826,
+            720
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 KNOWLEDGE DAG AND THE ATTENTION MECHANISM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            734,
+            594,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "GRAM leverages the parent-child relationship of $\\mathcal { G }$ to learn robust representations when data volume is constrained. GRAM balances the use of ontology information in relation to data volume in determining the level of specificity for a medical concept. When a medical concept is less observed in the data, more weight is given to its ancestors as they can be learned more accurately and offer general (coarse-grained) information about their children. The process of resorting to the parent concepts can be automated via the attention mechanism and the end-to-end training as described in Figure 1. ",
+        "bbox": [
+            174,
+            756,
+            825,
+            839
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the knowledge DAG, each node $c _ { i }$ is assigned a basic embedding vector $\\mathbf { e } _ { i } \\in \\mathbb { R } ^ { m }$ , where $m$ represents the dimensionality. Then $\\mathbf { e } _ { 1 } , \\ldots , \\mathbf { e } _ { | { \\mathcal { C } } | }$ are the basic embeddings of the codes $c _ { 1 } , \\ldots , c _ { | { \\mathcal { C } } | }$ while ${ \\bf e } _ { | \\mathcal { C } | + 1 } , \\dots , { \\bf e } _ { | \\mathcal { C } | + | \\mathcal { C } ^ { \\prime } | }$ represent the basic embeddings of the internal nodes $c _ { | \\mathcal { C } | + 1 } , \\ldots , c _ { | \\mathcal { C } | + | \\mathcal { C } _ { \\bullet } ^ { \\prime } | }$ The initialization of these basic embeddings is described in Section 2.4. We formulate a leaf node’s final representation as a convex combination of the basic embeddings of itself and its ancestors: ",
+        "bbox": [
+            174,
+            845,
+            825,
+            900
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/6bdcaaff59d5e6cfa1a45b7baa554326a2f30b8c8d0ec95506ecb0fa56116efa.jpg",
+        "image_caption": [
+            "Figure 1: The illustration of GRAM. Leaf nodes (solid circles) represents a medical concept in the EHR, while the non-leaf nodes (dotted circles) represent more general concepts. The final representation $\\mathbf { g } _ { i }$ of the leaf concept $c _ { i }$ is computed by combining the basic embeddings $\\mathbf { e } _ { i }$ of $c _ { i }$ and $\\mathbf { e } _ { g } , \\mathbf { e } _ { c }$ and $\\mathbf { e } _ { a }$ of its ancestors $c _ { g } , c _ { c }$ and $c _ { a }$ via an attention mechanism. The final representations form the embedding matrix $\\mathbf { G }$ for all leaf concepts. After that, we use $\\mathbf { G }$ to embed patient visit vector $\\mathbf { x } _ { t }$ to a visit representation $\\mathbf { v } _ { t }$ , which is then fed to a neural network model to make the final prediction $\\hat { \\mathbf { y } } _ { t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            83,
+            759,
+            250
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            373,
+            799,
+            388
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7db49841eba0df9ecbba30a3eff1b4beb5488b4488ec2759e899bb487ad4de78.jpg",
+        "text": "$$\n\\mathbf { g } _ { i } = \\sum _ { j \\in A ( i ) } \\alpha _ { i j } \\mathbf { e } _ { j } , \\qquad \\sum _ { j \\in A ( i ) } \\alpha _ { i j } = 1 , \\alpha _ { i j } \\geq 0 \\mathrm { ~ f o r ~ } j \\in A ( i ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            390,
+            705,
+            428
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { g } _ { i } \\in \\mathbb { R } ^ { m }$ denotes the final representation of the code $c _ { i }$ , $\\boldsymbol { \\mathscr { A } } ( i )$ the indices of the code $c _ { i }$ and $c _ { i }$ ’s ancestors, $\\mathbf { e } _ { j }$ the basic embedding of the code $c _ { j }$ and $\\alpha _ { i j } \\in \\mathbb { R }$ the attention weight on the embedding $\\mathbf { e } _ { j }$ when calculating $\\mathbf { g } _ { i }$ . The attention weight $\\alpha _ { i j }$ in Eq. (1) is calculated by the following Softmax function, ",
+        "bbox": [
+            173,
+            429,
+            825,
+            484
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c95007f1c1ad03933c80d427bf35fa63697a9fd586b5d392e4ded5f312685628.jpg",
+        "text": "$$\n\\alpha _ { i j } = { \\frac { \\exp ( f ( \\mathbf { e } _ { i } , \\mathbf { e } _ { j } ) ) } { \\sum _ { k \\in { \\mathcal { A } } ( i ) } \\exp ( f ( \\mathbf { e } _ { i } , \\mathbf { e } _ { k } ) ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            481,
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+            518
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$f ( \\mathbf { e } _ { i } , \\mathbf { e } _ { j } )$ is a scalar value representing the compatibility between the basic embeddings of $\\mathbf { e } _ { i }$ and $\\mathbf { e } _ { k }$ We compute $f ( \\mathbf { e } _ { i } , \\mathbf { e } _ { j } )$ via the following feed-forward network with a single hidden layer (MLP), ",
+        "bbox": [
+            176,
+            518,
+            825,
+            547
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db756e11a75d97ded98db44f0ca0af2b19964937e7df2258cc00face6d5fbe8f.jpg",
+        "text": "$$\nf ( \\mathbf { e } _ { i } , \\mathbf { e } _ { j } ) = \\mathbf { u } _ { a } ^ { \\top } \\operatorname { t a n h } ( \\mathbf { W } _ { a } \\left[ \\begin{array} { l } { \\mathbf { e } _ { i } } \\\\ { \\mathbf { e } _ { j } } \\end{array} \\right] + \\mathbf { b } _ { a } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            549,
+            635,
+            584
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { W } _ { a } \\in \\mathbb { R } ^ { l \\times { 2 m } }$ is the weight matrix for the concatenation of $\\mathbf { e } _ { i }$ and $\\mathbf { e } _ { j }$ , $\\textbf { b } \\in \\mathbb { R } ^ { l }$ the bias vector, and $\\mathbf { u } _ { a } \\in \\mathbb { R } ^ { l }$ the weight vector for generating the scalar value. The constant $l$ represents the dimension size of the hidden layer of $f ( \\cdot , \\cdot )$ . Note that we always concatenate $\\mathbf { e } _ { i }$ and $\\mathbf { e } _ { j }$ in the child-ancestor order. ",
+        "bbox": [
+            173,
+            587,
+            825,
+            643
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Remarks: The example in Figure 1 is derived based on a single path from $c _ { i }$ to $c _ { a }$ . However, the same mechanism can be applicable to multiple paths as well. For example, code $c _ { k }$ has two paths to the root $c _ { a }$ , containing five ancestors in total. Another scenario is where the EHR data contain both leaf codes and some ancestor codes. We can move those ancestors present in EHR data from the set $\\scriptstyle { \\mathcal { C } } ^ { \\prime }$ to $\\mathcal { C }$ and apply the same process as Eq. (1) to obtain the final representations for them. ",
+        "bbox": [
+            173,
+            650,
+            825,
+            720
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 END-TO-END TRAINING WITH A PREDICTIVE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            728,
+            588,
+            742
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We train the attention mechanism together with a predictive model such that the attention mechanism improves the predictive performance. Once the final representations $\\mathbf { g } _ { 1 } , \\mathbf { g } _ { 2 } , \\ldots , \\mathbf { g } _ { | { \\mathcal { C } } | }$ of all medical codes are obtained, we can convert visit $V _ { t }$ to a visit representation $\\mathbf { v } _ { t }$ by using the embedding matrix $\\mathbf { G } \\in \\mathcal { R } ^ { m \\times | c | }$ where $\\mathbf { g } _ { i }$ is its $i$ -th column as in Figure 1. We continue the mathematical formulation under the assumption that we are using the RNN to perform sequential diagnoses prediction (Choi et al., 2016a;b) with the objective of predicting the disease codes of the next visit $\\bar { V _ { t + 1 } }$ given the visit records up to the current timestep $V _ { 1 } , V _ { 2 } , \\dots , V _ { t }$ , which can be expressed as follows, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/60cf31164d20690f86c99f3a7d64ac5ffadbf4ec95f2653a8104e3d3a3ab69f7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\widehat { \\mathbf { y } } _ { t } = \\widehat { \\mathbf { x } } _ { t + 1 } = \\operatorname { S o f t m a x } ( \\mathbf { W } \\mathbf { h } _ { t } + \\mathbf { b } ) , \\quad \\mathrm { w h e r e } } \\\\ { \\mathbf { h } _ { 1 } , \\mathbf { h } _ { 2 } , \\ldots , \\mathbf { h } _ { t } = \\mathbf { R N N } ( \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\ldots , \\mathbf { v } _ { t } ) , \\quad \\mathrm { w h e r e } } \\\\ { \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\ldots , \\mathbf { v } _ { t } = \\operatorname { t a n h } ( \\mathbf { G } [ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\ldots , \\mathbf { x } _ { t } ] ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/06174389ad453658eab2e37f12561a09de97af6051ef15120bc21bdf54afbc98.jpg",
+        "table_caption": [
+            "Table 1: Basic statistics of Sutter PAMF, MIMIC-III and Sutter heart failure (HF) cohort. "
+        ],
+        "table_footnote": [
+            "† Note that for all datasets, we selected patients who made at least two hospital visits. "
+        ],
+        "table_body": "<table><tr><td>Dataset</td><td>Sutter PAMF</td><td>MIMIC-III</td><td>SutterHFcohort</td></tr><tr><td># of patients</td><td>258,555†</td><td>7,499t</td><td>30,727† (3,408 cases)</td></tr><tr><td>#of visits</td><td>13,920,759</td><td>19,911</td><td>572,551</td></tr><tr><td>Avg.# of visits per patient</td><td>53.8</td><td>2.66</td><td>38.38</td></tr><tr><td># of unique ICD9 codes</td><td>10,437</td><td>4,893</td><td>5,689</td></tr><tr><td>Avg.# of codes per visit</td><td>1.98</td><td>13.1</td><td>2.06</td></tr><tr><td>Max # of codes per visit</td><td>54</td><td>39</td><td>29</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { x } _ { t } \\in \\mathbb { R } ^ { | \\mathcal { C } | }$ denotes the $t$ -th visit; $\\mathbf { v } _ { t } \\in \\mathbb { R } ^ { m }$ the $t { \\cdot }$ -th visit representation; $\\mathbf { h } _ { t } \\in \\mathbb { R } ^ { r }$ the RNN’s hidden layer at $t { \\cdot }$ -th time step (i.e. $t$ -th visit); $\\textbf { W } \\in \\mathbb { R } ^ { | \\mathcal { C } | \\times r }$ and $\\textbf { b } \\in \\mathbb { R } ^ { | \\mathcal { C } | }$ the weight matrices and the bias vector of the Softmax function; $r$ denotes the dimension size of the hidden layer. We use “RNN” to denote any recurrent neural network variants that can cope with the vanishing gradient problem (Bengio et al., 1994), such as LSTM (Hochreiter $\\&$ Schmidhuber, 1997), GRU (Cho et al., 2014), and IRNN (Le et al., 2015), with any varying numbers of hidden layers. The prediction loss for all time steps is calculated using the cross entropy as follows, $\\mathcal { L } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } \\ldots , \\mathbf { x } _ { T } ) =$ $\\begin{array} { r } { - \\frac { 1 } { T - 1 } \\sum _ { t = 1 } ^ { T - 1 } \\bigg ( \\mathbf { y } _ { t } { } ^ { \\top } \\log ( \\widehat { \\mathbf { y } } _ { t } ) + ( \\mathbf { 1 } - \\mathbf { y } _ { t } ) ^ { \\top } \\log ( \\mathbf { 1 } - \\widehat { \\mathbf { y } } _ { t } ) \\bigg ) } \\end{array}$ where we sum the cross entropy errors from all dimensions of $\\widehat { \\mathbf { y } } _ { t }$ , $T$ denotes the length of the visit sequence. Note that the above loss is defined bfor a single patient. But we can take the average of the individual loss for multiple patients. ",
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+    },
+    {
+        "type": "text",
+        "text": "2.4 INITIALIZING BASIC EMBEDDINGS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The attention generation mechanism in Section 2.2 requires basic embeddings $\\mathbf { e } _ { i }$ of each node in the knowledge DAG. The basic embeddings of ancestors, however, pose a difficulty because they are often not observed in the data.To better initialize them, we use co-occurrence information to learn the basic embeddings of medical codes and their ancestors. Co-occurrence has proven to be an important source of information when learning representations of words or medical concepts (Mikolov et al., 2013; Choi et al., 2016c;e). To train the basic embeddings, we employ GloVe (Pennington et al., 2014), which uses the global co-occurrence matrix of words to learn their representations. In our case, the co-occurrence matrix of the codes and the ancestors was generated by counting the co-occurrences within each visit $V _ { t }$ , where we augment each visit with the ancestors of the codes in the visit. Details of training the basic embeddings are described in the Appendix A. Note that, with or without the initialization, the basic embeddings $\\mathbf { e } _ { i }$ ’s of both leaf nodes (i.e. medical codes) and non-leaf nodes (i.e. ancestors) are fine-tuned when training our model, since the error signal flows from the output $\\widehat { \\mathbf { y } } _ { t }$ to the final representations $\\mathbf { g } _ { i }$ ’s which are convex combinations of $\\mathbf { e } _ { i }$ ’s. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct three experiments to determine if GRAM offered superior prediction performance when facing data insufficiency. We first describe the experimental setup followed by results comparing predictive performance of GRAM with various baseline models. After discussing GRAM’s scalability, we qualitatively evaluate the interpretability of the resulting representation. The source code of GRAM is publicly available at https://github.com/mp2893/gram. ",
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+    {
+        "type": "text",
+        "text": "3.1 EXPERIMENT SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Prediction tasks and source of data: We conduct two sequential diagnoses prediction tasks, which aim at predicting all diagnosis categories in the next visit, and one heart failure (HF) prediction task, which is a binary prediction task for predicting a future HF onset where the prediction is made only once at the last visit $\\mathbf { x } _ { T }$ . ",
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+    {
+        "type": "text",
+        "text": "Two sequential diagnoses predictions are respectively conducted using 1) Sutter Palo Alto Medical Foundation (PAMF) dataset, which consists of 18-years longitudinal medical records of 258K patients between age 50 and 90. This will determine GRAM’s performance for general adult population with long visit records. 2) MIMIC-III dataset (Johnson et al., 2016; Goldberger et al., 2000), which is a publicly available dataset consisting of medical records of $7 . 5 \\mathrm { K }$ intensive care unit (ICU) patients over 11 years. This will determine GRAM’s performance for high-risk patients with very short visit records. We utilize all the patients with at least 2 visits. We prepared the true labels $\\mathbf { y } _ { t }$ by grouping the ICD9 codes into 283 groups using CCS single-level diagnosis grouper1. This is to improve the training speed and predictive performance for easier analysis, while preserving sufficient granularity for each diagnosis. Each diagnosis code’s varying frequency in the training data can be viewed as different degrees of data insufficiency. We calculate Accuracy $@ k$ for each of CCS single-level diagnosis codes such that, given a visit $V _ { t }$ , we get 1 if the target diagnosis is in the top $k$ guesses and 0 otherwise. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We conduct HF prediction on Sutter heart failure (HF) cohort, which is a subset of Sutter PAMF data for a heart failure onset prediction study with 3.4K HF cases and 27K controls chosen by a set of criteria (see Appendix B). This will determine GRAM’s performance for a different prediction task where we predict the onset of one specific condition. We randomly downsample the training data to create different degrees of data insufficiency. We use area under the ROC curve (AUC) to measure the performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A summary of the datasets are provided in Table 1.We used CCS multi-level diagnoses hierarchy2 as our knowledge DAG $\\mathcal { G }$ . We also tested the ICD9 code hierarchy3, but the performance was similar to using CCS multi-level hierarchy. For all three tasks, we randomly divide the dataset into the training, validation and test set by .75:.10:.15 ratio, and use the validation set to tune the hyper-parameters. Further details regarding the hyper-parameter tuning are provided in Appendix C. The test set performance is reported in the paper. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Implementation details: We implemented GRAM with Theano 0.8.2 (Team, 2016). For training models, we used Adadelta (Zeiler, 2012) with a mini-batch of 100 patients, on a machine equipped with Intel Xeon E5-2640, 256GB RAM, four Nvidia Titan X’s and CUDA 7.5. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Models for comparison are the following. The first two $\\mathrm { G R A M + }$ and GRAM are the proposed methods and the rest are baselines. Hyper-parameter tuning is configured so that the number of parameters for the baselines would be comparable to GRAM’s. Further details are provided in Appendix C. ",
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+    },
+    {
+        "type": "text",
+        "text": "• GRAM: Input sequence $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { T }$ is first transformed by the embedding matrix G, then fed to the GRU with a single hidden layer, which in turn makes the prediction, as described by Eq. (4). The basic embeddings $\\mathbf { e } _ { i }$ ’s are randomly initialized.   \n• GRAM+: We use the same setup as GRAM, but the basic embeddings $\\mathbf { e } _ { i }$ ’s are initialized according to Section 2.4.   \n• RandomDAG: We use the same setup as GRAM, but each leaf concept has five randomly assigned ancestors from the CCS multi-level hierarchy to test the effect of correct domain knowledge.   \n• RNN: Input $\\mathbf { x } _ { t }$ is transformed by an embedding matrix $\\mathbf { W } _ { e m b } \\in \\mathbb { R } ^ { k \\times | \\mathcal { C } | }$ , then fed to the GRU with a single hidden layer. The embedding size $k$ is a hyper-parameter. $\\mathbf { W } _ { e m b }$ is randomly initialized and trained together with the GRU.   \n• $\\mathbf { R N N + }$ : We use the same setup as RNN, but we initialize the embedding matrix $\\mathbf { W } _ { e m b }$ with GloVe vectors trained only with the co-occurrence of leaf concepts. This is to compare GRAM with a similar weight initialization technique.   \n• SimpleRollUp: We use the same setup as RNN. But for input $\\mathbf { x } _ { t }$ , we replace all diagnosis codes with their direct parent codes in the CCS multi-level hierarchy, giving us 578, 526 and 517 input codes respectively for Sutter data, MIMIC-III and Sutter HF cohort. This is to compare the performance of GRAM with a common grouping technique.   \nRollUpRare: We use the same setup as RNN, but we replace any diagnosis code whose frequency is less than a certain threshold in the dataset with its direct parent. We set the threshold to 100 for Sutter data and Sutter HF cohort, and 10 for MIMIC-III, giving us 4,408, 935 and 1,538 input codes respectively for Sutter data, MIMIC-III and Sutter HF cohort. This is an intuitive way of dealing with infrequent medical codes. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 PREDICTION PERFORMANCE AND SCALABILITY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Tables 2a and 2b show the sequential diagnoses prediction performance on Sutter data and MIMIC-III. Both figures show that ${ \\mathrm { G R A M } } +$ outperforms other models when predicting labels with significant data insufficiency (i.e. less observed in the training data).The performance gain is greater for MIMIC-III, where GRAM+ outperforms the basic RNN by $10 \\%$ in the 20th-40th percentile range. This seems to come from the fact that MIMIC patients on average have significantly shorter visit history than Sutter patients, with much more codes received per visit. Such short sequences make it difficult for the RNN to learn and predict diagnoses sequence. The performance difference between GRAM+ and ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/962f6c7fb0a50af116edb32a961dbb1b851a218126d856cd0e759e91a672fe13.jpg",
+        "table_caption": [
+            "Table 2: Performance of three prediction tasks. The $\\mathbf { X }$ -axis of (a) and (b) represents the labels grouped by the percentile of their frequencies in the training data in non-decreasing order. For (c), we vary the size of the training data to train the models. (b) uses Accuracy $\\textcircled{ a} 20$ because MIMIC-III has a large average number of codes per visit (see Table 1). "
+        ],
+        "table_footnote": [
+            "(a) Accuracy@5 of sequential diagnoses prediction on Sutter data "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp</td><td>0.0150 0.0042 0.0050 0.0069 0.0080 0.2691</td><td>0.3242 0.2987 0.2700 0.2742</td><td>0.4325 0.4224 0.4010 0.4140</td><td>0.4238 0.4193 0.4059 0.4212</td><td>0.4903 0.4895 0.4853 0.4959</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/d6c234e2b7d620130a4b59e6ca213a3b767281270cbb475e2ffe86ca5e07143c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp</td><td>0.0672 0.0556 0.0329 0.0454 0.0454 0.0578 0.1328</td><td>0.1787 0.1016 0.0708 0.0843 0.0731</td><td>0.2644 0.1935 0.1346 0.2080 0.1804 0.2455 0.2667</td><td>0.2490 0.2296 0.1512 0.2494 0.2371</td><td>0.6267 0.6363 0.4494 0.6239 0.6243</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7783934576bdbe6e1d7766e030aacc9b7bbff80613be4ff6d347c13aef275447.jpg",
+        "table_caption": [
+            "(b) Accuracy@20 of sequential diagnoses prediction on MIMIC-III "
+        ],
+        "table_footnote": [
+            "(c) AUC of HF onset prediction on Sutter HF cohort "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td><td>80%</td><td>90%</td><td>100%</td></tr><tr><td>GRAM+</td><td>0.7970</td><td>0.8223</td><td>0.8307</td><td>0.8332</td><td>0.8389</td><td>0.8404</td><td>0.8452</td><td>0.8456</td><td>0.8447</td><td>0.8448</td></tr><tr><td>GRAM</td><td>0.7981</td><td>0.8217</td><td>0.8340</td><td>0.8332</td><td>0.8372</td><td>0.8377</td><td>0.8440</td><td>0.8431</td><td>0.8430</td><td>0.8447</td></tr><tr><td>RandomDAG</td><td>0.7644</td><td>0.7882</td><td>0.7986</td><td>0.8070</td><td>0.8143</td><td>0.8185</td><td>0.8274</td><td>0.8312</td><td>0.8254</td><td>0.8226</td></tr><tr><td>RNN+</td><td>0.7930</td><td>0.8117</td><td>0.8162</td><td>0.8215</td><td>0.8261</td><td>0.8333</td><td>0.8343</td><td>0.8353</td><td>0.8345</td><td>0.8335</td></tr><tr><td>RNN</td><td>0.7811</td><td>0.7942</td><td>0.8066</td><td>0.8111</td><td>0.8156</td><td>0.8207</td><td>0.8258</td><td>0.8278</td><td>0.8297</td><td>0.8314</td></tr><tr><td>SimpleRollUp</td><td>0.7799</td><td>0.8022</td><td>0.8108</td><td>0.8133</td><td>0.8177</td><td>0.8207</td><td>0.8223</td><td>0.8272</td><td>0.8269</td><td>0.8258</td></tr><tr><td>RollUpRare</td><td>0.7830</td><td>0.8067</td><td>0.8064</td><td>0.8119</td><td>0.8211</td><td>0.8202</td><td>0.8262</td><td>0.8296</td><td>0.8307</td><td>0.8291</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/af5f92179fa9f42773dcb63837906eaf7a66c82d38d90092ff61b2c2a3c57d83.jpg",
+        "table_caption": [
+            "Table 3: Scalablity result in per epoch training time in second (the number of epochs needed). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Sequential diagnosis prediction (Sutter data)</td><td>Sequential diagnosis prediction (MIMIC-III)</td><td>HF prediction (Sutter HF cohort)</td></tr><tr><td>GRAM</td><td>525s (39 epochs)</td><td>2s (11 epochs)</td><td>12s (7 epochs)</td></tr><tr><td>RNN</td><td>352s (24 epochs)</td><td>1s (6 epochs)</td><td>8s (5 epochs)</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "GRAM suggests that our proposed initialization scheme of the basic embeddings $\\mathbf { e } _ { i }$ is important for sequential diagnosis prediction. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 2c shows the HF prediction performance on Sutter HF cohort. GRAM and $\\mathrm { G R A M + }$ consistently outperforms other baselines (except ${ \\mathrm { R N N } } +$ ) by $3sim \\mathrm { { } } 4 \\%$ AUC, and $\\mathrm { R N N } +$ by maximum $1 . 8 \\%$ AUC. These differences are quite significant given that the AUC is already in the mid-80s, a high value for HF prediction, cf. (Choi et al., 2016d). Note that, for $\\mathbf { G R A M + }$ and $\\mathrm { R N N } +$ , we used the downsampled training data to initialize the basic embeddings $\\mathbf { e } _ { i }$ ’s and the embedding matrix $\\mathbf { W } _ { e m b }$ with GloVe, respectively. The result shows that the initialization scheme of the basic embeddings in $\\mathrm { G R A M + }$ gives limited improvement over GRAM. This stems from the different natures of the two prediction tasks. While the goal of HF prediction is to predict a binary label for the entire visit sequence, the goal of sequential diagnosis prediction is to predict the co-occurring diagnosis codes at every visit. Therefore the co-occurrence information infused by the initialized embedding scheme is more beneficial to sequential diagnosis prediction. Additionally, this benefit is associated with the natures of the two prediction tasks than the datasets used for the prediction tasks. Because the initialized embedding shows different degrees of improvement as shown by Tables 2a and 2c, when Sutter HF cohort is a subset of Sutter PAMF, thus having similar characteristics. Additional prediction results when varying the $k$ of Accuracy $@ k$ are discussed in the Appendix D. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Overall, GRAM showed superior predictive performance under data insufficiency in three different experiments, demonstrating its general applicability in predictive healthcare modeling. Now we briefly discuss the scalability of GRAM by comparing its training time to RNN’s. Table 3 shows the number of seconds taken for the two models to train for a single epoch for each predictive modeling task. $\\mathrm { G R A M + }$ and $\\mathrm { R N N } +$ showed the same behavior as GRAM and RNN. GRAM takes approximately $50 \\%$ more time to train for a single epoch for all prediction tasks. This stems from calculating attention weights and the final representations $\\mathbf { g } _ { i }$ for all medical codes. GRAM also generally takes about $50 \\%$ more epochs to reach to the model with the lowest validation loss. This is due to optimizing an extra MLP model that generates the attention weights. Overall, use of GRAM adds a manageable amount of overhead in training time to the plain RNN. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 QUALITATIVE EVALUATION OF INTERPRETABLE REPRESENTATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "To qualitatively assess the interpretability of the learned representations of the medical codes, we plot on a 2-D space using t-SNE (Maaten & Hinton, 2008) the final representations $\\mathbf { g } _ { i }$ of 2,000 ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/689483f70536387faf727af1a9e63b92fc3e321f1590e2cfd6ba6deb6c377b7f.jpg",
+        "image_caption": [
+            "(a) Scatterplot of the final representations ${ \\bf g } _ { i }$ ’s of $\\mathrm { G R A M + }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/b4a9d66eed2c0be9a5d006b70667ae4d91f35eba1c1a90f50c894282f06bf4d0.jpg",
+        "image_caption": [
+            "(c) Scatterplot of the disease representations trained by GloVe "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "image_caption": [
+            "Figure 2: t-SNE scatterplots of medical concepts trained by $\\mathrm { G R A M + }$ , $\\mathrm { R N N } +$ and GloVe "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "(b) Scatterplot of the trained embedding matrix $\\mathbf { W } _ { e m b }$ of $\\mathrm { R N N } +$ ",
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+        "text": "randomly chosen diseases learned by $\\mathrm { G R A M + }$ for sequential diagnoses prediction on Sutter data4 (Figure 2a). The colors represent the highest disease categories and the text annotations represent the detailed disease categories in CCS multi-level hierarchy. For comparison, we also show the t-SNE plots on the strongest results from $\\mathrm { R N N } +$ (Figure 2b), and GloVe (Figure 2c), the same embedding technique in initializing the basic embeddings $\\mathbf { e } _ { i }$ . Figures 2b and 2c confirm that interpretable representations cannot simply be learned only by co-occurrence or supervised prediction without medical knowledge. GRAM+ learns disease representations that are significantly more consistent with the given knowledge DAG $\\mathcal { G }$ . Therefore the neural network predictive model that accepts $\\mathbf { g } _ { i }$ is using accurate representations that lead to higher predictive performance. Additional scatterplots of other models are provided in Appendix E for comparison. An interactive visualization tool can be accessed at http://www.sunlab.org/research/gram-graph-based-attention-model/. ",
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+    {
+        "type": "text",
+        "text": "3.4 ANALYSIS OF THE ATTENTION BEHAVIOR ",
+        "text_level": 1,
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+        "text": "Next we show that GRAM’s attention can be interpreted to understand how it considers data availability and knowledge DAG’s structure when performing a prediction task. Using Eq. (1), we can calculate the attention weights of individual disease. Figure 3 shows the attention behaviors of four representative diseases when performing HF prediction on Sutter HF cohort. ",
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+        "type": "text",
+        "text": "Other pneumothorax (ICD9 512.89) in Figure 3a is rarely observed in the data and has only five siblings. In this case, most information is derived from the highest ancestor. Temporomandibular joint disorders & articular disc disorder (ICD9 524.63) in Figure 3b is rarely observed but has 139 siblings. In this case, its parent receives a stronger attention because it aggregates sufficient samples from all of its children to learn a more accurate representation. Note that the disease itself also receives a stronger attention to facilitate easier distinction from its large number of siblings. ",
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+    {
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+        "img_path": "images/c4bfd8d3f0c22ad1774b97b598078ce0bdafbec69884368b5a4bbb594c7e2b19.jpg",
+        "image_caption": [
+            "Figure 3: GRAM’s attention behavior during HF prediction for four representative diseases (each column). In each figure, the leaf node represents the disease and upper nodes are its ancestors. The size of the node shows the amount of attention it receives, which is also shown by the bar charts. The number in the parenthesis next to the disease is its frequency in the training data. We exclude the root of the knowledge DAG $\\mathcal { G }$ from all figures as it did not play a significant role. "
+        ],
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+    {
+        "type": "text",
+        "text": "Unspecified essential hypertension (ICD9 401.9) in Figure 3c is very frequently observed but has only two siblings. In this case, GRAM assigns a very strong attention to the leaf, which is logical because the more you observe a disease, the stronger your confidence becomes. Need for prophylactic vaccination and inoculation against influenza (ICD9 V04.81) in Figure 3d is quite frequently observed and also has 103 siblings. The attention behavior in this case is quite similar to the case with fewer siblings (Figure 3b) with a slight attention shift towards the leaf concept as more observations lead to higher confidence. ",
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+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We introduce recent studies related to GRAM that learn the representations of graphs and discuss their relationship with GRAM. Several studies focused on learning the representations of graph vertices by using the neighbor information. DeepWalk (Perozzi et al., 2014) and node2vec (Grover & Leskovec, 2016) use random walk while LINE (Tang et al., 2015) uses breadth-first search to find the neighbors of a vertex and learn its representation based on the neighbor information. Graph convolutional approaches (Yang et al., 2016; Kipf & Welling, 2016) also focus on learning the vertex representations to mainly perform vertex classification. These works focus on solving the graph data problems whereas GRAM focuses on solving EHR data problems using the knowledge DAG as supplementary information. ",
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+        "type": "text",
+        "text": "Several researchers tried to model the knowledge DAG such as WordNet (Miller, 1995) or Freebase (Bollacker et al., 2008) where two entities are connected with various types of relation, forming a set of triples. They aim to project entities and relations (Bordes et al., 2013; Socher et al., 2013; Wang et al., 2014; Lin et al., 2015) to the latent space based on the triples or additional information such as hierarchy of entities (Xie et al., 2016). These works demonstrated tasks such as link prediction, triple classification or entity classification using the learned representations. More recently, Li et al. (2016) learned the representations of words and Wikipedia categories by utilizing the hierarchy of Wikipedia categories. GRAM is fundamentally different from the above studies in that it aims to design intuitive attention mechanism on the knowledge DAG as a knowledge prior to cope with data insufficiency and learn medically interpretable representations to make accurate predictions. ",
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+    {
+        "type": "text",
+        "text": "A classical approach for incorporating side information in the predictive models is to use graph Laplacian regularization (Weinberger et al., 2006; Che et al., 2015). However, using this approach is not straightforward as it relies on the appropriate definition of distance on graphs which is often unavailable. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Data insufficiency, either due to less common diseases or small datasets, is one of the key hurdles in healthcare analytics, especially when we apply deep neural networks models. To overcome this hurdle, we leverage the knowledge DAG, which provides a multi-resolution view of medical concepts. We propose GRAM, a graph-based attention model using both a knowledge DAG and EHR to learn an accurate and interpretable representations for medical concepts. GRAM chooses a weighted average of ancestors of a medical concept and train the entire process with a predictive model in an end-to-end fashion. We conducted three predictive modeling experiments on real EHR datasets and showed significant improvement in the prediction performance, especially on low-frequency diseases and small datasets. Analysis of the attention behavior provided intuitive insight of GRAM. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            80,
+            285,
+            94
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks, 5(2), 1994.   \nKurt Bollacker, Colin Evans, Praveen Paritosh, Tim Sturge, and Jamie Taylor. Freebase: a collaboratively created graph database for structuring human knowledge. In SIGMOD, 2008.   \nAntoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, and Oksana Yakhnenko. Translating embeddings for modeling multi-relational data. In NIPS, 2013.   \nZhengping Che, David Kale, Wenzhe Li, Mohammad Taha Bahadori, and Yan Liu. Deep computational phenotyping. In SIGKDD, 2015.   \nZhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. arXiv:1606.01865, 2016.   \nKyunghyun Cho, Bart Van Merriënboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. In EMNLP, 2014.   \nEdward Choi, Mohammad Taha Bahadori, Andy Schuetz, Walter F. Stewart, and Jimeng Sun. Doctor ai: Predicting clinical events via recurrent neural networks. In MLHC, 2016a.   \nEdward Choi, Mohammad Taha Bahadori, Andy Schuetz, Walter F. Stewart, and Jimeng Sun. Retain: Interpretable predictive model in healthcare using reverse time attention mechanism. In NIPS, 2016b.   \nEdward Choi, Mohammad Taha Bahadori, Elizabeth Searles, Catherine Coffey, Michael Thompson, James Bost, Javier T Sojo, and Jimeng Sun. Multi-layer representation learning for medical concepts. In SIGKDD, 2016c.   \nEdward Choi, Andy Schuetz, Walter F Stewart, and Jimeng Sun. Using recurrent neural network models for early detection of heart failure onset. JAMIA, 2016d.   \nYoungduck Choi, Chill Yi-I Chiu, and David Sontag. Learning low-dimensional representations of medical concepts. 2016e. 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Joint embedding of hierarchical categories and entities for concept categorization and dataless classification. 2016.   \nYankai Lin, Zhiyuan Liu, Maosong Sun, Yang Liu, and Xuan Zhu. Learning entity and relation embeddings for knowledge graph completion. In AAAI, 2015.   \nZachary C Lipton, David C Kale, Charles Elkan, and Randall Wetzell. Learning to diagnose with lstm recurrent neural networks. arXiv:1511.03677, 2015.   \nZachary C Lipton, David C Kale, and Randall Wetzel. Modeling missing data in clinical time series with rnns. In MLHC, 2016.   \nLaurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. JMLR, 9(Nov), 2008.   \nTomas Mikolov, Ilya Sutskever, Kai Chen, Greg Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In NIPS, 2013.   \nGeorge A Miller. Wordnet: a lexical database for english. Communications of the ACM, 38(11), 1995.   \nRiccardo Miotto, Li Li, Brian A Kidd, and Joel T Dudley. 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Revisiting semi-supervised learning with graph embeddings. arXiv:1603.08861, 2016.   \nMatthew D Zeiler. Adadelta: an adaptive learning rate method. arXiv:1212.5701, 2012. ",
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+    {
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+        "img_path": "images/1566e0ab53d4efb27fd3b1a603894f6037b19d4d62f57271b24a9e9e895c3eb5.jpg",
+        "image_caption": [
+            "Figure 4: Creating the co-occurrence matrix together with the ancestors. Here we exclude the root node, which will be just a single row (column). We first create an augmented dataset by adding the ancestors of the code to the dataset. Then, we count the co-occurrence of the codes. Performing GloVe on this matrix produces the embedding vectors $\\mathbf { e } _ { i }$ . "
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "A GENERATING GLOVE EMBEDDINGS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We learn the basic embeddings $\\mathbf { e } _ { i }$ ’s of medical codes and their ancestors using GloVe (Pennington et al., 2014), which uses global co-occurrence matrix of words to learn their representations. We generate the co-occurrence matrix of the codes and the ancestors by counting the co-occurrence within each visit $V _ { t }$ . However, since visits only contain the leaf codes $c \\in { \\mathcal { C } }$ , we augment each visit with the ancestors of the codes in each visit, then count the co-occurrence of codes and ancestors altogether. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We describe the details of the algorithm with an example. We borrow the parent-child relationships from the knowledge DAG of Figure 1. Given a visit $V _ { t }$ , ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/50478865b3f475ffba5205f8415d55452a1b613b4b6b6f36e9fe521c2be854bc.jpg",
+        "text": "$$\nV _ { t } = \\{ c _ { d } , c _ { i } , c _ { k } \\}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "we augment it with the ancestors of all the codes to obtain the augmented visit $V _ { t } ^ { \\prime }$ , ",
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+    {
+        "type": "equation",
+        "img_path": "images/a0d1015b574c3d235995e9c9b1ad135bb761828eb7af24452037be2f48e0e9d9.jpg",
+        "text": "$$\nV _ { t } ^ { \\prime } = \\{ c _ { d } , \\underline { { { c _ { b } } } } , \\underline { { { c _ { a } } } } , c _ { i } , \\underline { { { c _ { g } } } } , \\underline { { { c _ { c } } } } , \\underline { { { c _ { a } } } } , c _ { k } , \\underline { { { c _ { j } } } } , \\underline { { { c _ { f } } } } , \\underline { { { c _ { c } } } } , \\underline { { { c _ { b } } } } , \\underline { { { c _ { a } } } } \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "where the added ancestors are underlined. Note that a single ancestor can appear multiple times in $V _ { t } ^ { \\prime }$ . In fact, the higher the ancestor is in the knowledge DAG, the more times it is likely to appear in $V _ { t } ^ { \\prime }$ . We count the co-occurrence of two codes in $V _ { t } ^ { \\prime }$ as follows, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/fcfd1563ca69cbcbd2633f52d7edabbfa67d7eff93494852a15920064918f14c.jpg",
+        "text": "$$\nc o \\mathrm { - } o c c u r r e n c e ( c _ { i } , c _ { j } , V _ { t } ^ { \\prime } ) = c o u n t ( c _ { i } , V _ { t } ^ { \\prime } ) \\times c o u n t ( c _ { j } , V _ { t } ^ { \\prime } ) _ { l }\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $c o u n t ( c _ { i } , V _ { t } ^ { \\prime } )$ is the number of times the code $c _ { i }$ appears in the augmented visit $V _ { t } ^ { \\prime }$ . For example, the co-occurrence between the leaf code $c _ { i }$ and the root $c _ { a }$ is 3. However, the co-occurrence between the ancestor $c _ { c }$ and the root $c _ { a }$ is 6. Therefore our algorithm will naturally make the ancestor codes have higher co-occurrence with other codes compared to leaf medical codes. We repeat this calculation for all pairs of codes in all augmented visits of all patients to obtain the co-occurrence matrix depicted by Figure 4. For training the embedding vectors using the co-occurrence matrix, we use the same procedure and hyper-parameter as described in Pennington et al. (2014). ",
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+    {
+        "type": "text",
+        "text": "B HEART FAILURE COHORT CONSTRUCTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "For the heart failure (HF) case patients, we select patients between 40 to 85 years of age at the time of HF diagnosis. HF diagnosis (HFDx) criteria are defined as: 1) Qualifying ICD-9 codes for HF appeared in the encounter records or medication orders. Qualifying ICD-9 codes are listed in Table 4. 2) at least three clinical encounters with qualifying ICD-9 codes had to occur within 12 months of each other, where the date of HFDx was assigned to the earliest of the three dates. If the time span between the first and second appearances of the HF diagnosis code was greater than 12 months, the date of the second encounter was used as the first qualifying encounter. Up to ten eligible controls (in terms of sex, age, location) were selected for each case, yielding average 9 controls per case. Each control was also assigned an index date, which is the HFDx date of the matched case. Controls are selected such that they did not meet the HF diagnosis criteria prior to the HFDx date plus 182 days of their corresponding case. Control subjects were required to have their first office encounter within one year of the matching HF case patient’s first office visit, and have at least one office encounter 30 days before or any time after the case’s HFDx date to ensure similar duration of observations among cases and controls. ",
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+    {
+        "type": "table",
+        "img_path": "images/e5968d93609e6441fd2ec790f233442c1d47389cbc1c60c9ba33c95298a5628e.jpg",
+        "table_caption": [
+            "Table 4: Qualifying ICD-9 codes for heart failure "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>ICD-9 Code</td><td rowspan=1 colspan=1>Description</td></tr><tr><td rowspan=1 colspan=1>398.91</td><td rowspan=1 colspan=1>Rheumatic heart failure (congestive)</td></tr><tr><td rowspan=1 colspan=1>402.01</td><td rowspan=1 colspan=1>Malignant hypertensive heart disease with heart failure</td></tr><tr><td rowspan=1 colspan=1>402.11</td><td rowspan=1 colspan=1>Benign hypertensive heart disease with heart failure</td></tr><tr><td rowspan=1 colspan=1>402.91</td><td rowspan=1 colspan=1>Unspecified hypertensive heart disease with heart failure</td></tr><tr><td rowspan=1 colspan=1>404.01</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, malignant, with heart failure and withchronic kidney disease stage I through stage IV, or unspecified</td></tr><tr><td rowspan=1 colspan=1>404.03</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, malignant, with heart failure and with chronic kidney disease stage V or end stage renal disease</td></tr><tr><td rowspan=1 colspan=1>404.11</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, benign, with heart failure and withchronic kidney disease stage I through stage IV, or unspecified</td></tr><tr><td rowspan=1 colspan=1>404.13</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, benign, with heart failure and chronickidney disease stage V or end stage renal disease</td></tr><tr><td rowspan=1 colspan=1>404.91</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, unspecified, with heart failure and with chronic kidney disease stage I through stage IV, or unspecified</td></tr><tr><td rowspan=1 colspan=1>404.93</td><td rowspan=1 colspan=1>Hypertensive heart and chronic kidney disease, unspecified, with heart failure andchronic kidney disease stage V or end stage renal disease</td></tr><tr><td rowspan=1 colspan=1>428.0</td><td rowspan=1 colspan=1>Congestive heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.1</td><td rowspan=1 colspan=1>Leftheart failure</td></tr><tr><td rowspan=1 colspan=1>428.20</td><td rowspan=1 colspan=1>Systolic heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.21</td><td rowspan=1 colspan=1> Acute systolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.22</td><td rowspan=1 colspan=1>Chronic systolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.23</td><td rowspan=1 colspan=1>Acute on chronic systolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.30</td><td rowspan=1 colspan=1>Diastolic heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.31</td><td rowspan=1 colspan=1>Acute diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.32</td><td rowspan=1 colspan=1>Chronic diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.33</td><td rowspan=1 colspan=1>Acute on chronic diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.40</td><td rowspan=1 colspan=1>Combined systolic and diastolic heart failure, unspecified</td></tr><tr><td rowspan=1 colspan=1>428.41</td><td rowspan=1 colspan=1>Acute combined systolic and diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.42</td><td rowspan=1 colspan=1>Chronic combined systolic and diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.43</td><td rowspan=1 colspan=1> Acute on chronic combined systolic and diastolic heart failure</td></tr><tr><td rowspan=1 colspan=1>428.9</td><td rowspan=1 colspan=1>Heart failure, unspecified</td></tr></table>",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            539,
+            825,
+            594
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C HYPER-PARAMETER TUNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            614,
+            449,
+            631
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We define five hyper-parameters for GRAM: ",
+        "bbox": [
+            176,
+            645,
+            455,
+            660
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• dimensionality $m$ of the basic embedding $\\mathbf { e } _ { i }$ : [100, 200, 300, 400, 500] • dimensionality $r$ of the RNN hidden layer $\\mathbf { h } _ { t }$ from Eq. (4): [100, 200, 300, 400, 500] • dimensionality $l$ of $\\mathbf { W } _ { a }$ and $ { \\mathbf { b } } _ { a }$ from Eq. (3): [100, 200, 300, 400, 500] • $L _ { 2 }$ regularization coefficient for all weights except RNN weights: [0.1, 0.01, 0.001, 0.0001] • dropout rate for the dropout on the RNN hidden layer: [0.0, 0.2, 0.4, 0.6, 0.8] ",
+        "bbox": [
+            181,
+            670,
+            802,
+            760
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We performed 100 iterations of the random search by using the above ranges for each of the three prediction experiments. For sequential diagnoses prediction on Sutter data, we used $10 \\%$ of the training data to tune the hyper-parameters to balance the time and search space. To match the baselines’ number of parameters to GRAM’s, we add 550 to the list of $m$ ’s possible values. This will make the baseline’s largest possible number of parameters comparable to the GRAM’s largest possible number of parameters. ",
+        "bbox": [
+            174,
+            770,
+            825,
+            853
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For SimpleRollUp and RollUpRare, the number of input codes is smaller than other models due to the grouping. Therefore, to match their largest possible number of parameters to GRAM’s, we need to add much larger values to $m$ . However, after preliminary experiments, as expected, setting $m$ to too large a value degraded the performance due to overfitting. Since the number of input codes decreased due to the grouping, increasing the dimensionality of $\\mathbf { e } _ { i }$ is not a logical thing to do. Therefore, for SimpleRollUp and RollUpRare, we use the same list of values for $m$ as other baselines. ",
+        "bbox": [
+            176,
+            859,
+            823,
+            901
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/85e8dbbb6270b8ac2c0437d095afab49764c1b8875e71a1cba24aac6fee9d5d6.jpg",
+        "table_caption": [
+            "Table 5: Hyper-parameters used by the models in each predictive modeling experiments "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Experiment</td><td>Model</td><td>m</td><td>r</td><td>1</td><td>L2</td><td>Dropout rate</td></tr><tr><td rowspan=\"6\">Disease progression modeling (Sutter data)</td><td>GRAM+ GRAM</td><td>500 500</td><td>500 500</td><td>100 100</td><td>0.0001 0.0001</td><td>0.6 0.6</td></tr><tr><td>RandomDAG</td><td>500</td><td>500</td><td>100</td><td>0.0001</td><td>0.6</td></tr><tr><td>RNN+</td><td>550</td><td>500</td><td></td><td>0.0001</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>0.6</td></tr><tr><td>RNN</td><td>550</td><td>500</td><td></td><td>0.0001</td><td>0.6</td></tr><tr><td>SimpleRollUp RollUpRare</td><td>500 500</td><td>500 500</td><td></td><td>0.0001 0.0001</td><td>0.4 0.2</td></tr><tr><td rowspan=\"6\">Disease progression modeling (MIMIC-III)</td><td>GRAM+ GRAM</td><td>400 400</td><td>400 400</td><td>100 100</td><td>0.0001</td><td>0.6</td></tr><tr><td>RandomDAG</td><td>400</td><td>400</td><td>100</td><td>0.001 0.001</td><td>0.6</td></tr><tr><td>RNN+</td><td>550</td><td>400</td><td></td><td>0.001</td><td>0.6</td></tr><tr><td></td><td></td><td>400</td><td></td><td></td><td>0.8</td></tr><tr><td>RNN SimpleRollUp</td><td>550 400</td><td>400</td><td></td><td>0.001 0.001</td><td>0.8 0.6</td></tr><tr><td></td><td>RollUpRare</td><td>400</td><td>400</td><td></td><td></td><td></td></tr><tr><td rowspan=\"6\">HF prediction (Sutter HF cohort)</td><td></td><td></td><td></td><td></td><td>0.0001</td><td>0.0</td></tr><tr><td>GRAM+</td><td>200</td><td>100</td><td>100</td><td>0.001</td><td></td></tr><tr><td>GRAM</td><td>200</td><td>100</td><td>100</td><td></td><td>0.6</td></tr><tr><td>RandomDAG</td><td>300</td><td>100</td><td>200</td><td>0.001</td><td>0.6</td></tr><tr><td>RNN+</td><td>200</td><td>100</td><td></td><td>0.001</td><td>0.6</td></tr><tr><td></td><td></td><td></td><td></td><td>0.0001</td><td>0.6</td></tr><tr><td>RNN</td><td>200</td><td>100</td><td></td><td>0.001</td><td>0.6</td></tr><tr><td>SimpleRollUp</td><td>300</td><td>200</td><td></td><td>0.001</td><td>0.4</td></tr><tr><td>RollUpRare</td><td>100</td><td>100</td><td></td><td>0.001</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>0.6</td></tr></table>",
+        "bbox": [
+            215,
+            95,
+            777,
+            390
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            401,
+            821,
+            443
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 5 describes the final hyper-parameter settings we used for all models for each prediction experiments. ",
+        "bbox": [
+            176,
+            449,
+            821,
+            477
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D PREDICTION RESULTS USING DIFFERENT $k$ ’S IN ACCURACY@K ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            497,
+            740,
+            513
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We show Accuracy@k using $k = 5 , 1 0 , 2 0 , 3 0$ for sequential diagnoses prediction on Sutter data (Tables 6a, 6b, 6c and 6d) and MIMIC-III (Tables 7a, 7b, 7c and 7d). We can see from the tables that $\\mathrm { G R A M + }$ consistently outperforms other models under 40th percentile range, except when $k = 2 0 , 3 0$ for sequential diagnoses prediction on Sutter data where SimpleRollUp shows similar performance. We can also see that $\\mathrm { G R A M + }$ performs significantly better than other models for all $k = 5$ , 10, 20, 30 when predicting infrequently observed diseases on MIMIC-III. As discussed in Section 3.2, this seems to come from the short visit sequences of MIMIC patients. ",
+        "bbox": [
+            173,
+            527,
+            825,
+            625
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E T-SNE 2-D PLOTS OF VARIOUS MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            643,
+            549,
+            660
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For further comparison, we display t-SNE scatterplots of GRAM (Figure 5a) RandomDAG (Figure 5b, RNN (Figure 5c), and Skip-gram (Figure 5d). GRAM, RandomDAG and RNN were trained for sequential diagnoses prediction on Sutter data, and Skip-gram (Mikolov et al., 2013) was trained on Sutter data as it is an unsupervised method. For Skip-gram, we used each visit $V _ { t }$ as the context window. As we do not distinguish between the target concept and the neighbor concepts, we calculated the Skip-gram objective function using all possible pairs of codes within a single visit. ",
+        "bbox": [
+            174,
+            676,
+            825,
+            758
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We can see from Figure 5a that the quality of the final representations $\\mathbf { g } _ { i }$ of GRAM is quite similar to $\\mathrm { G R A M + }$ (Figure 2a). Compared to other baselines, GRAM demonstrates significantly more structured representations that align well with the given knowledge DAG. It is interesting that Skip-gram shows the most structured representation among all baselines. We used GloVe to initialize the basic embeddings $\\mathbf { e } _ { i }$ in this work because it uses global co-occurrence information and its training time is dependent only on the total number of unique concepts $| { \\mathcal { C } } |$ . Skip-gram’s training time, on the other hand, depends on both the number of patients and the number of visits each patient made, which makes the algorithm generally slower than GloVe. However, considering both Figures $2 \\mathrm { c }$ and 5d, initializing $\\mathbf { e } _ { i }$ ’s with Skip-gram vectors might give us additional performance boost. ",
+        "bbox": [
+            174,
+            763,
+            825,
+            888
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/7afbc87a51751b863cd1bbf797835a71d61702c5e6f7ae00a926904309969369.jpg",
+        "table_caption": [
+            "Table 6: Accuracy at various $k$ ’s (a to d) for sequential diagnoses prediction on Sutter data. The columns represent the labels grouped by the percentile of their frequencies in the training data in non-decreasing order. "
+        ],
+        "table_footnote": [
+            "(a) Accuracy@5 of sequential diagnoses prediction on Sutter data "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RoliUpRare</td><td>0.0150 0.0042 0.0050 0.0069 0.0080 0.0085 0.0062</td><td>0.3242 0.2987 0.2700 0.2742 0.2691 0.3078 0.2768</td><td>0.4325 0.4224 0.4010 0.4140 0.4134 0.4369 0.4176</td><td>0.4238 0.4193 0.4059 0.4212 0.4227 0.4330 0.4226</td><td>0.4903 0.4895 0.4853 0.4959 0.4951 0.4924 0.4956</td></tr></table>",
+        "bbox": [
+            295,
+            256,
+            702,
+            363
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/cd9b434e99c391ba87221897bb25539f90db73679734928b4a570ac9ad2bf1de.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Accuracy@10 of sequential diagnoses prediction on Sutter data "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RolUpRare</td><td>0.0319 0.0163 0.0142 0.0183 0.0196 0.0164 0.0204</td><td>0.3882 0.3645 0.3285 0.3412 0.3290 0.3768 0.3450</td><td>0.5054 0.4944 0.4691 0.4884 0.4871 0.5132 0.4917</td><td>0.5215 0.5173 0.5025 0.5233 0.5230 0.5326</td><td>0.6459 0.6445 0.6401 0.6538 0.6531 0.6521</td></tr></table>",
+        "bbox": [
+            295,
+            383,
+            702,
+            492
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/67be6781d696fb7700b01ed5447686ee0cd48b212953cfffcb5a0713a8d7757b.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(c) Accuracy@20 of sequential diagnoses prediction on Sutter data "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RoliUpRare</td><td>0.0630 0.0442 0.0397 0.0483 0.0481 0.0418 0.0517</td><td>0.4486 0.4276 0.3933 0.4132 0.4025 0.4496 0.4170</td><td>0.5764 0.5669 0.5389 0.5654 0.5630 0.5877 0.5672</td><td>0.6153 0.6125 0.5997 0.6235 0.6232 0.6262 0.6214</td><td>0.7973 0.7963 0.7919 0.8003 0.7995 0.8013</td></tr></table>",
+        "bbox": [
+            295,
+            512,
+            700,
+            621
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/4e0ba629f5f0beb14de81c999a27c34e1e3a8fbdadbb44bbf9b6bd044dbb8041.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(d) Accuracy@30 of sequential diagnoses prediction on Sutter data "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RollUpRare</td><td>0.0946 0.0662 0.0672 0.0736 0.0733 0.0662 0.0759</td><td>0.4879 0.4693 0.4313 0.4604 0.4478 0.4924 0.4657</td><td>0.6186 0.6107 0.5843 0.6136 0.6103 0.6312 0.6146</td><td>0.6792 0.6766 0.6667 0.6930 0.6921 0.6907 0.6908</td><td>0.8800 0.8798 0.8760 0.8785 0.8767 0.8795 0.8778</td></tr></table>",
+        "bbox": [
+            297,
+            642,
+            700,
+            750
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/8de9de2dd2c1c6659efa7483645c5654a9abcc75cb0f742b97f21492c28c5350.jpg",
+        "table_caption": [
+            "Table 7: Accuracy at various $k$ ’s (a to d) for sequential diagnoses prediction on MIMIC-III. The columns represent the labels grouped by the percentile of their frequencies in the training data in non-decreasing order. "
+        ],
+        "table_footnote": [
+            "(a) Accuracy@5 of sequential diagnoses prediction on MIMIC-III "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RollUpRare</td><td>0.0086 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000</td><td>0.1089 0.0468 0.0327 0.0435 0.0376 0.0671 0.0423</td><td>0.1665 0.1093 0.0778 0.1266 0.1105 0.1501 0.1085</td><td>0.1029 0.0918 0.0612 0.0973 0.0923 0.1191 0.0874</td><td>0.2597 0.2665 0.1634 0.2594 0.2601 0.2635 0.2604</td></tr></table>",
+        "bbox": [
+            295,
+            256,
+            702,
+            363
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/0ffa475c4a37a2254649317a91de871b43be41cb15c96eab94a9aaeffa103e50.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Accuracy@10 of sequential diagnoses prediction on MIMIC-III "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RolUpRare</td><td>0.0380 0.0045 0.0023 0.0227 0.0023 0.0249 0.0227</td><td>0.1310 0.0682 0.0470 0.0587 0.0518 0.1038 0.0530</td><td>0.2095 0.1494 0.1025 0.1591 0.1389 0.1997 0.1412</td><td>0.1627 0.1487 0.0938 0.1616 0.1521 0.1769 0.1519</td><td>0.4175 0.4235 0.2692 0.4193 0.4142 0.4260</td></tr></table>",
+        "bbox": [
+            295,
+            383,
+            702,
+            492
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/9c7d48a018a8dfa4e023bbf280adcb9303d45187d98c4256fe2f87c1987cdf86.jpg",
+        "table_caption": [
+            "(c) Accuracy@20 of sequential diagnoses prediction on MIMIC-III "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RoliUpRare</td><td>0.0672 0.0556 0.0329 0.0454 0.0454 0.0578 0.0454</td><td>0.1787 0.1016 0.0708 0.0843 0.0731 0.1328 0.0653</td><td>0.2644 0.1935 0.1346 0.2080 0.1804 0.2455 0.1843</td><td>0.2490 0.2296 0.1512 0.2494 0.2371 0.2667</td><td>0.6267 0.6363 0.4494 0.6239 0.6243 0.6387</td></tr></table>",
+        "bbox": [
+            295,
+            512,
+            700,
+            621
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/52036e4bc134837d410e66cc81a441c855c3d2c2a94c90fe68b96e6e711b8e4e.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(d) Accuracy@30 of sequential diagnoses prediction on MIMIC-III "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>0-20</td><td>20-40</td><td>40-60</td><td>60-80</td><td>80-100</td></tr><tr><td>GRAM+ GRAM RandomDAG RNN+ RNN SimpleRollUp RollUpRare</td><td>0.0744 0.0578 0.0351 0.0578 0.0578 0.0578 0.0556</td><td>0.2065 0.1157 0.0932 0.1103 0.0775 0.1556 0.0910</td><td>0.3180 0.2257 0.1635 0.2571 0.2237 0.2865 0.2235</td><td>0.3363 0.3074 0.2200 0.3409 0.3160 0.3488 0.3255</td><td>0.7726 0.7802 0.5977 0.7656 0.7643 0.7800</td></tr></table>",
+        "bbox": [
+            297,
+            642,
+            700,
+            750
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/207c97af4d8c0611947b428e647f2ea210d709362ff36a9d96b7904fb9b73c28.jpg",
+        "image_caption": [
+            "(a) Scatterplot of the final representations ${ \\bf g } _ { i }$ ’s of GRAM "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            284,
+            485,
+            429
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/c0fa2677df46d223e61d45d996d35cc58e8e4be356ab504aa2199fb2eda739a3.jpg",
+        "image_caption": [
+            "(b) Scatterplot of the final representations $\\mathbf { g } _ { i }$ ’s of RandomDAG "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            513,
+            285,
+            808,
+            426
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/2e7a5335acf240417a092dbb8634a5e6d652c9d515ecb710341031754c8da59c.jpg",
+        "image_caption": [
+            "(d) Scatterplot of the basic embeddings $\\mathbf { e } _ { i }$ ’s trained by Skip-gram "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            479,
+            485,
+            622
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/291ea73802a224c6fc099588975183c840aa3b7666c3d1233a2d6a174082679a.jpg",
+        "image_caption": [
+            "Figure 5: Scatterplot of medical concepts trained by various models. We used t-SNE to reduce the dimension to 2-D. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            513,
+            476,
+            807,
+            618
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "(c) Scatterplot of the trained embedding matrix $\\mathbf { W } _ { e m b }$ of RNN ",
+        "bbox": [
+            189,
+            633,
+            483,
+            660
+        ],
+        "page_idx": 15
+    }
+]

parse/train/SyxS0T4tvS/SyxS0T4tvS.md

@@ -0,0 +1,350 @@
+# ROBERTA: A ROBUSTLY OPTIMIZED BERT PRETRAINING APPROACH
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Language model pretraining has led to significant performance gains but careful comparison between different approaches is challenging. Training is computationally expensive, often done on private datasets of different sizes, and, as we show, hyperparameter choices have significant impact on the final results. We present a replication study of BERT pretraining (Devlin et al., 2019) that carefully measures the impact of many key hyperparameters and training data size. We find that BERT was significantly undertrained, and can match or exceed the performance of every model published after it. Our best model achieves state-of-the-art results on GLUE, RACE, SQuAD, SuperGLUE and XNLI. These results highlight the importance of previously overlooked design choices, and raise questions about the source of recently reported improvements. We release our models and code.1
+
+# 1 INTRODUCTION
+
+Self-training methods such as ELMo (Peters et al., 2018), GPT (Radford et al., 2018), BERT (Devlin et al., 2019), XLM (Lample & Conneau, 2019), and XLNet (Yang et al., 2019) have brought significant performance gains, but it can be challenging to determine which aspects of the methods contribute the most. Training is computationally expensive, limiting the amount of tuning that can be done, and modeling advances are often conflated with changes in data size or composition.
+
+We present a replication study of BERT pretraining (Devlin et al., 2019), which includes a careful evaluation of the effects of hyperparameter tuning and training set size. We find that BERT was significantly undertrained and propose an improved training recipe, which we call RoBERTa, that can match or exceed the performance of all of the post-BERT methods. Our modifications are simple, they include: (1) training the model longer, with bigger batches, over more data; (2) removing the next sentence prediction objective; (3) training on longer sequences; and (4) dynamically changing the masking pattern applied to the training data. We also collect a large new dataset (CC-NEWS) of comparable size to other privately used datasets, to better control for training set size effects.
+
+When controlling for training data, our improved training procedure improves upon the published BERT results on the GLUE (Wang et al., 2019b) and SQuAD (Rajpurkar et al., 2016) benchmarks. When trained for longer over additional data, our model achieves a score of 88.5 on the public GLUE leaderboard, matching the 88.4 reported by Yang et al. (2019). Our model establishes a new stateof-the-art on 4/9 of the GLUE tasks, as well as RACE (Lai et al., 2017), SuperGLUE (Wang et al., 2019a), and XNLI (Conneau et al., 2018), and matches the state-of-the-art on SQuAD. Overall, we re-establish that BERT’s masked language model training objective is competitive with recently proposed alternatives such as perturbed autoregressive language modeling (Yang et al., 2019).2
+
+In summary, the contributions of this paper are: (1) We present a set of important BERT design choices and training strategies and introduce alternatives that lead to better downstream task performance; (2) We use a novel dataset, CC-NEWS, and confirm that using more data for pretraining further improves performance on downstream tasks; (3) Our training improvements show that masked language model pretraining, under the right design choices, is competitive with all other recently published methods. We release our model, pretraining and fine-tuning code.
+
+# 2 BACKGROUND
+
+Setup: BERT (Devlin et al., 2019) takes as input a concatenation of two segments (sequences of tokens), $x _ { 1 } , \ldots , x _ { N }$ and $y _ { 1 } , \dots , y _ { M }$ . Segments usually consist of more than one natural sentence. The two segments are presented as a single input sequence to BERT with special tokens delimiting them: $[ C L S ] , x _ { 1 } , \ldots , x _ { N } , [ S E P ] , y _ { 1 } , \ldots , y _ { M } , [ E O S ]$ . $M$ and $N$ are constrained such that $M + N <$ $T$ , where $T$ is a parameter that controls the maximum sequence length during training.
+
+Architecture: BERT uses the now ubiquitous transformer architecture (Vaswani et al., 2017), which we will not review in detail. We use a transformer architecture with $L$ layers. Each block has $A$ self-attention heads and hidden dimension $H$ .
+
+Training Objectives: BERT uses two pretraining objectives: masked language modeling and next sentence prediction. For the Masked Language Model (MLM) objective, BERT is trained via a crossentropy loss to predict $15 \%$ of the input tokens, selected at random. To prevent the model from cheating, $80 \%$ of these selected tokens are replaced by a special $[ M A S K ]$ symbol in the input, $10 \%$ are replaced by a random token from the vocabulary, and $10 \%$ are left unchanged.
+
+Next Sentence Prediction (NSP) is a binary classification loss for predicting whether two segments follow each other in the original text. Positive examples are created by taking consecutive sentences from the text corpus. Negative examples are created by pairing segments from different documents. Positive and negative examples are sampled with equal probability.
+
+Optimization: BERT is optimized with AdamW (Kingma & Ba, 2015) using the following parameters: $\beta _ { 1 } = 0 . 9$ , $\beta _ { 2 } ~ = ~ 0 . 9 9 9$ , $\epsilon = 1 \mathrm { e } { - } 6$ and decoupled weight decay of 0.01 (Loshchilov & Hutter, 2019). The learning rate is warmed up over the first 10,000 steps to a peak value of 1e-4, and then linearly decayed. BERT trains with a dropout of 0.1 on all layers and attention weights, and a GELU activation function (Hendrycks & Gimpel, 2016). Models are pretrained for $S = \overline { { 1 } } , 0 0 0 , 0 0 0$ updates, with mini-batches containing $B = 2 5 6$ sequences of maximum length $T = 5 1 2$ tokens.
+
+Data: BERT is trained on a combination of BOOKCORPUS (Zhu et al., 2015) plus English WIKIPEDIA, which totals 16GB of uncompressed text.3
+
+# 3 EXPERIMENTAL SETUP
+
+# 3.1 IMPLEMENTATION
+
+We reimplement BERT in FAIRSEQ (Ott et al., 2019). We primarily follow the original BERT optimization hyperparameters, given in Section 2, except for the peak learning rate and number of warmup steps, which are tuned separately for each setting. We found training to be very sensitive to the Adam epsilon term, and in some cases we obtained better performance or improved stability after tuning it. We also set $\beta _ { 2 } = 0 . 9 8$ to improve stability when training with large batch sizes.
+
+We pretrain with sequences of at most $T = 5 1 2$ tokens. Unlike Devlin et al. (2019), we do not randomly inject short sequences, and we do not train with a reduced sequence length for the first $90 \%$ of updates. We train only with full-length sequences.
+
+We train with mixed precision floating point arithmetic on DGX-1 machines, each with ${ 8 \times 3 2 { \mathrm { G B } } }$ Nvidia V100 GPUs interconnected by Infiniband (Micikevicius et al., 2018).
+
+# 3.2 DATA
+
+BERT-style pretraining crucially relies on large quantities of text. Baevski et al. (2019) demonstrate that increasing data size can result in improved end-task performance. Several efforts have trained on datasets larger and more diverse than the original BERT (Radford et al., 2019; Yang et al., 2019; Zellers et al., 2019). Unfortunately, not all of the additional datasets can be publicly released. For our study, we focus on gathering as much data as possible for experimentation, allowing us to match the overall quality and quantity of data as appropriate for each comparison.
+
+We consider five English-language corpora of varying sizes and domains, totaling over 160GB of uncompressed text: (1&2) BOOKCORPUS (Zhu et al., 2015) plus English WIKIPEDIA, which is the original data used to train BERT (16GB); (3) CC-NEWS, which we collect from the English portion of the CommonCrawl News dataset (Nagel, 2016), containing 63 million English news articles crawled between September 2016 and February 2019 (76GB after filtering);4 (4) OPENWEBTEXT (Gokaslan & Cohen, 2019), an open-source recreation of the WebText corpus described in Radford et al. (2019), containing web content extracted from URLs shared on Reddit with at least three upvotes (38GB);5 (5) STORIES, a dataset introduced in Trinh & Le (2018) containing a subset of CommonCrawl data filtered to match the story-like style of Winograd schemas (31GB).
+
+# 3.3 EVALUATION
+
+Following previous work, we evaluate our pretrained models by finetuning on downstream tasks:
+
+• GLUE: The General Language Understanding Evaluation (GLUE) benchmark (Wang et al., 2019b) is a collection of 9 datasets for evaluating natural language understanding systems. Tasks are framed as either single-sentence classification or sentence-pair classification tasks. The GLUE organizers provide training and development data splits as well as a submission server and leaderboard that allows participants to evaluate and compare their systems on private held-out test data. • SQuAD: The Stanford Question Answering Dataset (SQuAD) provides a paragraph of context and a question. The task is to answer the question with a span extracted from the context. We evaluate on SQuAD V1.1 and V2.0 (Rajpurkar et al., 2016; 2018). In V1.1 the context always contains an answer, while in V2.0 some questions are not answered in the provided context. • RACE: ReAding Comprehension from Examinations (RACE) (Lai et al., 2017) is a large-scale reading comprehension dataset collected from English examinations in China. The task is to choose among four possible answers to a given question, using a given passage of text as context. • Additional Benchmarks: In the Appendix we present additional results for SuperGLUE (Wang et al., 2019a) and XNLI (Conneau et al., 2018).
+
+# 4 TRAINING PROCEDURE ANALYSIS
+
+This section explores and quantifies which choices are important for successfully pretraining BERT models. We keep the model architecture fixed.6 Specifically, we begin by training BERT models with the same configuration as $\mathbf { B E R T _ { B A S E } }$ $L = 1 2$ , $H = 7 6 8$ , $A = 1 2$ , 110M params).
+
+# 4.1 STATIC VS. DYNAMIC MASKING
+
+As discussed in Section 2, BERT relies on predicting randomly masked tokens. The original BERT implementation performed masking once during data preprocessing, resulting in a single static mask. To avoid repeating the same masks at every epoch, training data was duplicated 10 times prior to preprocessing, so that each training sequence was seen with the same mask only four times over the course of 40 training epochs. We instead train with dynamic masking, where we generate the masking pattern on-the-fly each time we input a sequence to the model. This becomes crucial when pretraining for more steps or with larger datasets, and additionally performs marginally better than static masking on some downstream tasks (see Appendix A).
+
+# 4.2 MODEL INPUT FORMAT AND NEXT SENTENCE PREDICTION
+
+In the original BERT pretraining procedure, the model observes two concatenated document segments and is trained via an auxiliary Next Sentence Prediction (NSP) loss to predict whether these segments were sampled contiguously from the same document or from distinct documents.
+
+Table 1: Development set results for base models pretrained over BOOKCORPUS and WIKIPEDIA. All models are trained for 1M steps with a batch size of 256 sequences. We report F1 for SQuAD and accuracy for MNLI-m, SST-2 and RACE. Reported results are medians over five random initializations (seeds). Results for $\mathbf { B E R T _ { B A S E } }$ and $\mathbf { X L N e t } _ { \mathrm { B A S E } }$ are from Yang et al. (2019).   
+
+<table><tr><td>Model</td><td>SQuAD 1.1/2.0</td><td>MNLI-m</td><td>SST-2</td><td>RACE</td></tr><tr><td colspan="5">Our reimplementation (with NSP loss):</td></tr><tr><td>SEGMENT-PAIR</td><td>90.4/78.7</td><td>84.0</td><td>92.9</td><td>64.2</td></tr><tr><td>SENTENCE-PAIR</td><td>88.7/76.2</td><td>82.9</td><td>92.1</td><td>63.0</td></tr><tr><td colspan="5">Our reimplementation (without NSP loss):</td></tr><tr><td>FULL-SENTENCES</td><td>90.4/79.1</td><td>84.7</td><td>92.5</td><td>64.8</td></tr><tr><td>DOC-SENTENCES</td><td>90.6/79.7</td><td>84.7</td><td>92.7</td><td>65.6</td></tr><tr><td>BERTBASE</td><td>88.5/76.3</td><td>84.3</td><td>92.8</td><td>64.3</td></tr><tr><td>XLNetBASE (K=7)</td><td>-/81.3</td><td>85.8</td><td>92.7</td><td>66.1</td></tr><tr><td>XLNetBASE (K=6)</td><td>-/81.0</td><td>85.6</td><td>93.4</td><td>66.7</td></tr></table>
+
+The NSP objective was designed to improve performance on downstream tasks, such as Natural Language Inference (Bowman et al., 2015), which require predicting relationships between pairs of sentences. Devlin et al. (2019) observe that removing NSP hurts performance, with significant performance degradation on QNLI, MNLI, and SQuAD 1.1. However, recent work has questioned the necessity of the NSP loss (Lample & Conneau, 2019; Yang et al., 2019; Joshi et al., 2019).
+
+To better understand this discrepancy, we compare several alternative training formats:
+
+• SEGMENT-PAIR $^ +$ NSP: This follows the original input format used in BERT (Devlin et al., 2019), with the NSP loss. Each input has a pair of segments, which can each contain multiple natural sentences, but the total combined length must be less than 512 tokens. SENTENCE-PAIR $+ \mathrm { N S P }$ : Each input contains a pair of natural sentences, either sampled from a contiguous portion of one document or from separate documents. Since these inputs are significantly shorter than 512 tokens, we increase the batch size so that the total number of tokens remains similar to SEGMENT-PAIR $+ \mathrm { N S P }$ . We retain the NSP loss. FULL-SENTENCES: Each input is packed with full sentences sampled contiguously from one or more documents, such that the total length is at most 512 tokens. Inputs may cross document boundaries. When we reach the end of one document, we begin sampling sentences from the next document and add an extra separator token between documents. We remove the NSP loss. DOC-SENTENCES: Inputs are constructed similarly to FULL-SENTENCES, except that they may not cross document boundaries. Inputs sampled near the end of a document may be shorter than 512 tokens, so we dynamically increase the batch size in these cases to achieve a similar number of total tokens as FULL-SENTENCES. We remove the NSP loss.
+
+Results Table 1 shows results for the four different settings. We first compare the original SEGMENT-PAIR input format from Devlin et al. (2019) to the SENTENCE-PAIR format; both formats retain the NSP loss, but the latter uses single sentences. We find that using individual sentences hurts performance on downstream tasks, which we hypothesize is because the model is not able to learn long-range dependencies.
+
+We next compare training without the NSP loss and training with blocks of text from a single document (DOC-SENTENCES). We find that this setting outperforms the originally published BERTBASE results and that removing the NSP loss matches or slightly improves downstream task performance, in contrast to Devlin et al. (2019). It is possible that the original BERT implementation may only have removed the loss term while still retaining the SEGMENT-PAIR input format.
+
+Finally we find that restricting sequences to come from a single document (DOC-SENTENCES) performs slightly better than packing sequences from multiple documents (FULL-SENTENCES). However, because the DOC-SENTENCES format results in variable batch sizes, we use FULL-SENTENCES in the remainder of our experiments for easier comparison with related work.
+
+<table><tr><td>batch size</td><td>learning rate</td><td>epochs</td><td>steps</td><td> perplexity</td><td>MNLI-m</td><td>SST-2</td></tr><tr><td>256</td><td>1e-4</td><td>32</td><td>1M</td><td>3.99</td><td>84.7</td><td>92.5</td></tr><tr><td rowspan="3">2K</td><td rowspan="3">7e-4</td><td>32</td><td>125K</td><td>3.68</td><td>85.2</td><td>93.1</td></tr><tr><td>64</td><td>250K</td><td>3.59</td><td>85.3</td><td>94.1</td></tr><tr><td>128</td><td>500K</td><td>3.51</td><td>85.4</td><td>93.5</td></tr><tr><td rowspan="3">8K</td><td rowspan="3">1e-3</td><td>32</td><td>31K</td><td>3.77</td><td>84.4</td><td>93.2</td></tr><tr><td>64</td><td>63K</td><td>3.60</td><td>85.3</td><td>93.5</td></tr><tr><td>128</td><td>125K</td><td>3.50</td><td>85.8</td><td>94.1</td></tr></table>
+
+Table 2: Perplexity on held-out validation data and dev set accuracy on MNLI-m and SST-2 for various batch sizes (# sequences) as we vary the number of passes (epochs) through the $\mathrm { { B O O K S } + }$ WIKI data. Reported results are medians over five random initializations (seeds). The learning rate is tuned for each batch size. All results are for $\mathbf { B E R T _ { B A S E } }$ with FULL-SENTENCE inputs.
+
+# 4.3 TRAINING WITH LARGE BATCHES
+
+Past work in neural machine translation has shown that training with large mini-batches can improve optimization speed and end-task performance when the learning rate is tuned appropriately (Ott et al., 2018). Large batches are also easily parallelized via data parallel training.7
+
+Table 2 shows the masked LM perplexity and end-task accuracy for $\mathbf { B E R T _ { B A S E } }$ as we increase the batch size, while tuning the learning rate. Devlin et al. (2019) originally trained $\mathbf { B E R T _ { B A S E } }$ for 1M steps with a batch size of 256 sequences; however a batch size of 2K sequences performs better, even controlling for the number of epochs, suggesting that the original BERT batch size was too small. We also observe that training with extremely large batches (8K) becomes more efficient as we train for more epochs.8 In the remainder of our experiments we train with batches of 8K sequences.
+
+# 4.4 TEXT ENCODING
+
+Byte-Pair Encoding (BPE) (Sennrich et al., 2016) is a hybrid between character- and word-level modeling based on subwords units. BPE vocabulary sizes typically range from 10K-100K subword units; however, unicode characters can account for a sizeable portion of this vocabulary when modeling large and diverse corpora, such as the ones considered in this work.
+
+The original BERT implementation (Devlin et al., 2019) used a character-level BPE vocabulary of size 30K. We instead adopt the larger byte-level BPE vocabulary of size 50K introduced in Radford et al. (2019), which uses bytes rather than unicode characters as the base subword units and can therefore encode any input text without introducing “unknown” tokens. This adds approximately 15M and 20M extra parameters for $\mathbf { B E R T _ { B A S E } }$ and BERTLARGE, respectively.
+
+Early experiments revealed only minor differences between these encodings, with the byte-level BPE achieving slightly worse end-task performance on some tasks. Nevertheless, we believe the advantages of a universal encoding scheme outweighs the minor degredation in performance and use this encoding in the remainder of our experiments.
+
+# 5 ROBERTA
+
+In the previous section we propose modifications to the BERT pretraining procedure that improve end-task performance. We now aggregate these improvements and evaluate their combined impact. We call this configuration RoBERTa for Robustly optimized BERT approach. Specifically, RoBERTa is trained with dynamic masking (Section 4.1), FULL-SENTENCES without NSP loss (Section 4.2), large mini-batches (Section 4.3) and a larger byte-level BPE (Section 4.4).
+
+<table><tr><td>Model</td><td>data</td><td>batch size</td><td>steps</td><td>SQuAD (v1.1/2.0)</td><td>MNLI-m</td><td>SST-2</td></tr><tr><td>RoBERTa</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BOOKS +WIKI</td><td>16GB</td><td>8K</td><td>100K</td><td>93.6/87.3</td><td>89.0</td><td>95.3</td></tr><tr><td>+ additional data ($3.2)</td><td>160GB</td><td>8K</td><td>100K</td><td>94.0/87.7</td><td>89.3</td><td>95.6</td></tr><tr><td>+ pretrain longer</td><td>160GB</td><td>8K</td><td>300K</td><td>94.4/88.7</td><td>90.0</td><td>96.1</td></tr><tr><td>+ pretrain even longer</td><td>160GB</td><td>8K</td><td>500K</td><td>94.6/89.4</td><td>90.2</td><td>96.4</td></tr><tr><td>BERTLARGE</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BoOKS +WIKI</td><td>13GB</td><td>256</td><td>1M</td><td>90.9/81.8</td><td>86.6</td><td>93.7</td></tr><tr><td>XLNetLARGE</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BoOKS +WIKI</td><td>13GB</td><td>256</td><td>1M</td><td>94.0/87.8</td><td>88.4</td><td>94.4</td></tr><tr><td>+ additional data</td><td>126GB</td><td>2K</td><td>500K</td><td>94.5/88.8</td><td>89.8</td><td>95.6</td></tr></table>
+
+Table 3: Development set results for RoBERTa as we pretrain over more data $\mathbf { 1 6 G B }  \mathbf { 1 6 0 G B }$ of text) and pretrain for longer $1 0 0 \mathrm { K }  3 0 0 \mathrm { K }  5 0 0 \mathrm { K }$ steps). Each row accumulates improvements from the rows above. RoBERTa matches the architecture and training objective of $\mathbf { B E R T _ { L A R G E } }$ . Results for $\mathbf { B E R T _ { L A R G E } }$ and $\mathrm { X L N e t } _ { \mathrm { L A R G E } }$ are from Devlin et al. (2019) and Yang et al. (2019), respectively. Complete results on all GLUE tasks can be found in Appendix C.
+
+Additionally, we investigate two other important factors that have been under-emphasized in previous work: (1) the data used for pretraining, and (2) the number of training passes through the data. For example, XLNet (Yang et al., 2019) was pretrained using 10 times more data than BERT, with a batch size eight times larger for half as many optimization steps, thus seeing four times as many sequences in pretraining compared to Devlin et al. (2019).
+
+To help disentangle the importance of these factors from other modeling choices (e.g., the pretraining objective), we begin by training RoBERTa following the $\mathbf { B E R T _ { L A R G E } }$ architecture $L = 2 4$ , $H =$ 1024, $A = 1 6$ , 355M parameters). We pretrain for 100K steps over a comparable BOOKCORPUS plus WIKIPEDIA dataset as was used in Devlin et al. (2019). We pretrain our model using 1024 V100 GPUs, which takes approximately one day per 100K steps.
+
+Results We present our results in Table 3. When controlling for training data, we observe that RoBERTa provides a large improvement over the originally reported BERTLARGE results, reaffirming the importance of the design choices we explored in Section 4.
+
+Next, we combine this data with the three additional datasets described in Section 3.2. We train RoBERTa over the combined data with the same number of training steps as before (100K). In total, we pretrain over 160GB of text. We observe further improvements in performance across all downstream tasks, validating the importance of data size and diversity in pretraining.9 9
+
+Finally, we pretrain RoBERTa for significantly longer, increasing the number of pretraining steps from 100K to 300K, and then further to 500K. We again observe significant gains in downstream task performance, and the 300K and 500K step models outperform XLNetLARGE across most tasks. We note that even our longest-trained model does not appear to overfit our data and would likely benefit from additional training.
+
+# 5.1 GLUE RESULTS
+
+For GLUE, we consider two finetuning settings. In the first setting (single-task, dev), we finetune RoBERTa separately for each of the GLUE tasks, using only the training data for the corresponding task. We consider a limited hyperparameter sweep with batch sizes $\in \{ 1 6 , 3 2 \}$ and learning rates $\in \ \{ 1 \mathrm { e } { - } 5 , 2 \mathrm { e } { - } 5 , 3 \mathrm { e } { - } 5 \}$ , with a linear warmup for the first $6 \%$ of steps followed by a linear decay to 0. We finetune for 10 epochs with early stopping based on each task’s dev set. The rest of the hyperparameters remain the same as during pretraining. In this setting, we report the median development set results for each task over five random initializations, without model ensembling.
+
+In the second setting (ensembles, test), we compare RoBERTa to other approaches on the test set via the GLUE leaderboard. While many submissions to the GLUE leaderboard depend on multi-task finetuning, our submission depends only on single-task finetuning. For RTE, STS and MRPC we finetune starting from the MNLI single-task model, following Phang et al. (2018). We explore a slightly wider hyperparameter space, described in Appendix C, and ensemble between 5 and 7 models per task. Two of the GLUE tasks require task-specific finetuning approaches to achieve competitive leaderboard results; these approaches are described in Appendix B.
+
+Table 4: Results on GLUE. All results are based on a 24-layer architecture. $\mathbf { B E R T _ { L A R G E } }$ and $\mathbf { X L N e t } _ { \mathrm { L A R G E } }$ results are from Devlin et al. (2019) and Yang et al. (2019), respectively. RoBERTa results on the dev set are a median over five runs. RoBERTa results on the test set are ensembles of single-task models. For RTE, STS and MRPC we finetune starting from the MNLI model.   
+
+<table><tr><td></td><td>MNLI</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST</td><td>MRPC</td><td>CoLA</td><td>STS</td><td>WNLI</td><td>Avg</td></tr><tr><td colspan="9">Single-task single models on dev</td><td></td></tr><tr><td>BERTLARGE</td><td>86.6/-</td><td>92.3</td><td>91.3</td><td>70.4</td><td>93.2</td><td>88.0</td><td>60.6</td><td>90.0</td><td></td><td>=</td></tr><tr><td>XLNetLARGE</td><td>89.8/-</td><td>93.9</td><td>91.8</td><td>83.8</td><td>95.6</td><td>89.2</td><td>63.6</td><td>91.8</td><td>=</td><td>=</td></tr><tr><td>RoBERTa</td><td>90.2/90.2</td><td>94.7</td><td>92.2</td><td>86.6</td><td>96.4</td><td>90.9</td><td>68.0</td><td>92.4</td><td>91.3</td><td>1</td></tr><tr><td colspan="9">Ensembles on test (from leaderboard asof July 25,2019)</td><td></td><td></td></tr><tr><td>ALICE</td><td>88.2/87.9</td><td>95.7</td><td>90.7</td><td>83.5</td><td>95.2</td><td>92.6</td><td>68.6</td><td>91.1</td><td>80.8</td><td>86.3</td></tr><tr><td>MT-DNN</td><td>87.9/87.4</td><td>96.0</td><td>89.9</td><td>86.3</td><td>96.5</td><td>92.7</td><td>68.4</td><td>91.1</td><td>89.0</td><td>87.6</td></tr><tr><td>XLNet</td><td>90.2/89.8</td><td>98.6</td><td>90.3</td><td>86.3</td><td>96.8</td><td>93.0</td><td>67.8</td><td>91.6</td><td>90.4</td><td>88.4</td></tr><tr><td>RoBERTa</td><td>90.8/90.2</td><td>98.9</td><td>90.2</td><td>88.2</td><td>96.7</td><td>92.3</td><td>67.8</td><td>92.2</td><td>89.0</td><td>88.5</td></tr></table>
+
+<table><tr><td>Model</td><td>SQuAD 2.0 EM F1</td></tr><tr><td>Single models on test (as of July 25,2019)</td><td>89.1†</td></tr><tr><td>XLNetLARGE RoBERTa</td><td>86.3t 86.8</td></tr><tr><td>XLNet+ SG-Net Verifier</td><td>89.8 89.9†</td></tr><tr><td></td><td>87.0t</td></tr></table>
+
+Table 5: Results on SQuAD. $\dagger$ indicates results that depend on additional external training data. RoBERTa uses only the provided SQuAD data in both dev and test settings. $\mathbf { B E R T _ { L A R G E } }$ and XLNetLARGE results are from Devlin et al. (2019) and Yang et al. (2019), respectively.   
+
+<table><tr><td>Model</td><td>SQuAD 1.1 EM</td><td>F1</td><td>SQuAD 2.0 EM F1</td></tr><tr><td>Single models on dev, w/o data augmentation</td><td></td><td></td><td></td></tr><tr><td>BERTLARGE</td><td>84.1</td><td>90.9 79.0</td><td>81.8</td></tr><tr><td>XLNetLARGE</td><td>89.0</td><td>94.5 86.1</td><td>88.8</td></tr><tr><td>RoBERTa</td><td>88.9</td><td>94.6 86.5</td><td>89.4</td></tr></table>
+
+Results We present our results in Table 4. In the first setting (single-task, dev), RoBERTa achieves state-of-the-art results on all 9 of the GLUE task development sets. Crucially, RoBERTa uses the same masked language modeling pretraining objective and architecture as $\mathbf { B E R T _ { L A R G E } }$ , yet consistently outperforms both $\mathbf { B E R T _ { L A R G E } }$ and $\mathbf { X L N e t } _ { \mathrm { L A R G E } }$ . This raises questions about the relative importance of model architecture and pretraining objective, compared to more mundane details like dataset size and training time that we explore in this work. A more comprehensive comparison of the BERT and XLNet pretraining objectives is needed, but is left to future work.
+
+In the second setting (ensembles, test), we submit RoBERTa to the GLUE leaderboard and achieve state-of-the-art results on 4 out of 9 tasks and the highest average score to date. Notably, RoBERTa does not depend on multi-task finetuning, and we expect future work may further improve these results by incorporating more sophisticated multi-task finetuning procedures.
+
+# 5.2 SQUAD RESULTS
+
+We adopt a much simpler approach for SQuAD compared to past work. While BERT (Devlin et al., 2019) and XLNet (Yang et al., 2019) augment their training data with additional QA datasets, we only finetune RoBERTa using the provided SQuAD training data. We also use a single learning rate for all layers, in contrast to the custom layer-wise learning rate scheduled used by Yang et al. (2019).
+
+For SQuAD v1.1 we follow the same finetuning procedure as Devlin et al. (2019). For SQuAD v2.0, we additionally classify whether a given question is answerable; we train this classifier jointly with the span predictor by summing the classification and span loss terms.
+
+<table><tr><td>Model</td><td>Accuracy</td><td>Middle</td><td>High</td></tr><tr><td>Single models on test (as of July 25, 2</td><td></td><td></td><td>2019)</td></tr><tr><td>BERTLARGE</td><td>72.0</td><td>76.6</td><td>70.1</td></tr><tr><td>XLNetLARGE</td><td>81.7</td><td>85.4</td><td>80.2</td></tr><tr><td>RoBERTa</td><td>83.2</td><td>86.5</td><td>81.3</td></tr></table>
+
+Table 6: Results on the RACE test set. $\mathbf { B E R T } _ { \mathrm { L A R G E } }$ and $\mathbf { X L N e t } _ { \mathrm { L A R G E } }$ results from Yang et al. (2019).
+
+Results We present our results in Table 5. On the SQuAD v1.1 development set, RoBERTa matches the state-of-the-art set by XLNet. On the SQuAD v2.0 development set, RoBERTa sets a new state-of-the-art, improving over XLNet by 0.4 points (EM) and 0.6 points (F1).
+
+We also submit RoBERTa to the public SQuAD 2.0 leaderboard. Most of the top systems build upon either BERT (Devlin et al., 2019) or XLNet (Yang et al., 2019) and therefore rely on additional external training data. Our single RoBERTa model outperforms all but one of the single model submissions, and is the top scoring system among those that do not rely on additional external data.
+
+# 5.3 RACE RESULTS
+
+In RACE, systems are provided with a passage of text, an associated question, and must classify which of four candidate answers is correct. We modify RoBERTa for this task by concatenating each candidate answer with the corresponding question and passage. We encode each of these four sequences and pass the resulting $I C L S J$ representations through a fully-connected layer, which is used to predict the correct answer. We truncate question-answer pairs that are longer than 128 tokens and, if needed, the passage so that the total length is at most 512 tokens.
+
+Results are presented in Table 6. RoBERTa achieves state-of-the-art accuracy across all settings.
+
+# 6 RELATED WORK
+
+Pretraining methods have been designed with different training objectives, including language modeling (Dai & Le, 2015; Peters et al., 2018; Howard & Ruder, 2018), machine translation (McCann et al., 2017), and masked language modeling (Devlin et al., 2019; Lample & Conneau, 2019). Many recent papers have used a basic recipe of finetuning models for each end task (Howard & Ruder, 2018; Radford et al., 2018), and pretraining with some variant of a masked language model objective. However, newer methods have improved performance by multi-task fine tuning (Dong et al., 2019), incorporating entity embeddings (Sun et al., 2019), span prediction (Joshi et al., 2019), and multiple variants of autoregressive pretraining (Song et al., 2019; Chan et al., 2019; Yang et al., 2019). Performance is also typically improved by training bigger models on more data (Devlin et al., 2019; Baevski et al., 2019; Yang et al., 2019; Radford et al., 2019). Our goal was to replicate, simplify, and better tune the training of BERT, as a reference point for better understanding the relative performance of all of these methods.
+
+# 7 CONCLUSION
+
+We evaluate a number of design decisions when pretraining BERT models, demonstrating that performance can be substantially improved by training the model longer, with bigger batches over more data; removing the next sentence prediction objective; training on longer sequences; and dynamically changing the masking pattern applied to the training data. We additionally use a novel dataset, CC-NEWS, and release our models and code for pretraining and finetuning at: anonymous URL.
+
+Our improved pretraining procedure, which we call RoBERTa, achieves state-of-the-art results on GLUE, RACE, SQuAD, SuperGLUE and XNLI. These results illustrate the importance of these previously overlooked design decisions and suggest that BERT’s pretraining objective remains competitive with recently proposed alternatives.
+
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+
+A STATIC VS. DYNAMIC MASKING   
+
+<table><tr><td>Masking</td><td> SQuAD 2.0</td><td>MNLI-m</td><td>SST-2</td></tr><tr><td>reference</td><td>76.3</td><td>84.3</td><td>92.8</td></tr><tr><td>Our reimplementation:</td><td></td><td></td><td></td></tr><tr><td>static</td><td>78.3</td><td>84.3</td><td>92.5</td></tr><tr><td>dynamic</td><td>78.7</td><td>84.0</td><td>92.9</td></tr></table>
+
+Table 7: Comparison between the published $\mathbf { B E R T _ { B A S E } }$ results from Devlin et al. (2019) to our reimplementation with either static or dynamic masking. We report F1 for SQuAD and accuracy for MNLI-m and SST-2. Reported results are medians over 5 random initializations (seeds). Reference results are from Yang et al. (2019). We find that our reimplementation with static masking performs similar to the original BERT model, and dynamic masking is comparable or slightly better than static masking.
+
+# B TASK-SPECIFIC MODIFICATIONS FOR GLUE
+
+Two of the GLUE tasks require task-specific finetuning approaches to achieve competitive leaderboard results:
+
+QNLI Recent submissions on the GLUE leaderboard adopt a pairwise ranking formulation for the QNLI task, in which candidate answers are mined from the training set and compared to one another, and a single (question, candidate) pair is classified as positive (Liu et al., 2019b;a; Yang et al., 2019). This formulation significantly simplifies the task, but is not directly comparable to BERT (Devlin et al., 2019). Following recent work, we adopt the ranking approach for our test submission, but for direct comparison with BERT all reported development set results are based on a pure classification approach.
+
+WNLI We found the provided NLI-format data to be challenging to work with. Instead we use the reformatted WNLI data from SuperGLUE (Wang et al., 2019a), which indicates the span of the query pronoun and referent. We then finetune RoBERTa using a variation of the approach from Kocijan et al. (2019). In particular, for a given input sentence, we first use spaCy (Honnibal & Montani, 2017) to extract additional candidate noun phrases from the sentence, and then finetune our model so that it assigns higher scores to positive referent phrases than for any of the generated negative candidate phrases.
+
+In contrast to Kocijan et al. (2019), who finetune BERT using a margin ranking loss between (query, candidate) pairs, we instead use a single cross entropy loss term over the log-probabilities for the query and all mined candidates. This reduces the number of hyperparameters that need to be tuned and in practice produces more stable results on the development set. Our best model achieved $9 2 . 3 \%$ development set accuracy, compared to $9 0 . 2 \%$ accuracy for the margin loss approach.
+
+One unfortunate consequence of our overall approach is that we can only make use of the positive training examples, which excludes over half of the provided training data.10
+
+# C FULL RESULTS ON GLUE
+
+In Table 8 we present the full set of development set results for RoBERTa on all 9 GLUE datasets.11 We present results for a LARGE configuration with 355M parameters that follows BERTLARGE, as well as a BASE configuration with 125M parameters that follows BERTBASE.
+
+<table><tr><td></td><td>MNLI</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST</td><td>MRPC</td><td>CoLA</td><td>STS</td></tr><tr><td>RoBERTaBASE</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+ all data + 500k steps</td><td>87.6</td><td>92.8</td><td>91.9</td><td>78.7</td><td>94.8</td><td>90.2</td><td>63.6</td><td>91.2</td></tr><tr><td>RoBERTaLARGE</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BOOKS +WIKI</td><td>89.0</td><td>93.9</td><td>91.9</td><td>84.5</td><td>95.3</td><td>90.2</td><td>66.3</td><td>91.6</td></tr><tr><td>+ additional data (83.2)</td><td>89.3</td><td>94.0</td><td>92.0</td><td>82.7</td><td>95.6</td><td>91.4</td><td>66.1</td><td>92.2</td></tr><tr><td>+ pretrain longer 300k</td><td>90.0</td><td>94.5</td><td>92.2</td><td>83.3</td><td>96.1</td><td>91.1</td><td>67.4</td><td>92.3</td></tr><tr><td>+ pretrain longer 500k</td><td>90.2</td><td>94.7</td><td>92.2</td><td>86.6</td><td>96.4</td><td>90.9</td><td>68.0</td><td>92.4</td></tr></table>
+
+Table 8: Development set results on GLUE tasks for various configurations of RoBERTa. All results are a median over five runs.
+
+D PRETRAINING HYPERPARAMETERS   
+Table 9: Hyperparameters for pretraining RoBERTaLARGE and RoBERTaBASE.   
+
+<table><tr><td>Hyperparam</td><td>RoBERTaLARGE</td><td>RoBERTaBASE</td></tr><tr><td>Number of Layers</td><td>24</td><td>12</td></tr><tr><td>Hidden size</td><td>1024</td><td>768</td></tr><tr><td>FFN inner hidden size</td><td>4096</td><td>3072</td></tr><tr><td>Attention heads</td><td>16</td><td>12</td></tr><tr><td>Attention head size</td><td>64</td><td>64</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td></tr><tr><td>Attention Dropout</td><td>0.1</td><td>0.1</td></tr><tr><td>Warmup Steps</td><td>24k</td><td>24k</td></tr><tr><td>Peak Learning Rate</td><td>4e-4</td><td>6e-4</td></tr><tr><td>Batch Size</td><td>8k</td><td>8k</td></tr><tr><td>Weight Decay</td><td>0.01</td><td>0.01</td></tr><tr><td>Max Steps</td><td>500k</td><td>500k</td></tr><tr><td>Learning Rate Decay</td><td>Linear</td><td>Linear</td></tr><tr><td>Adam ∈</td><td>1e-6</td><td>1e-6</td></tr><tr><td>Adam β1</td><td>0.9</td><td>0.9</td></tr><tr><td>Adam β2</td><td>0.98</td><td>0.98</td></tr><tr><td>Gradient Clipping</td><td>0.0</td><td>0.0</td></tr></table>
+
+E FINETUNING HYPERPARAMETERS   
+Table 10: Hyperparameters for finetuning $\mathrm { R o B E R T a _ { L A R G E } }$ on RACE, SQuAD and GLUE. We select the best hyperparameter values based on the median of 5 random seeds for each task.   
+
+<table><tr><td>Hyperparam</td><td>RACE</td><td> SQuAD</td><td>GLUE</td><td>SuperGLUE</td></tr><tr><td>Learning Rate</td><td>1e-5</td><td>1.5e-5</td><td>{1e-5,2e-5,3e-5}</td><td>{1e-5,2e-5,3e-5}</td></tr><tr><td>Batch Size</td><td>16</td><td>48</td><td>{16,32}</td><td>32</td></tr><tr><td>Weight Decay</td><td>0.1</td><td>0.01</td><td>0.1</td><td>0.1</td></tr><tr><td>Max Epochs</td><td>4</td><td>2</td><td>10</td><td>{10,50}</td></tr><tr><td>Learning Rate Decay</td><td>Linear</td><td>Linear</td><td>Linear</td><td>Linear</td></tr><tr><td>Warmup ratio</td><td>0.06</td><td>0.06</td><td>0.06</td><td>0.10</td></tr></table>
+
+F RESULTS ON SUPERGLUE   
+
+<table><tr><td></td><td>BoolQ</td><td>CB</td><td>COPA</td><td>MultiRC</td><td>ReCoRD</td><td>RTE</td><td>WiC</td><td>wsC</td><td>Avg</td></tr><tr><td colspan="10">Single-task single models on dev</td></tr><tr><td>BERT++</td><td>80.1</td><td>96.4/95.0</td><td>78.0</td><td>70.7/24.7</td><td>70.6/69.8</td><td>82.3</td><td>74.9</td><td>68.3</td><td>74.6</td></tr><tr><td>RoBERTa</td><td>86.9</td><td>98.2/-</td><td>94.0</td><td>85.7/-</td><td>89.5/89.0</td><td>86.6</td><td>75.6</td><td>-</td><td>-</td></tr><tr><td colspan="10">Ensembles on test (from leaderboard as of August 12,2019)</td></tr><tr><td>BERT</td><td>77.4</td><td>75.7/83.6</td><td>70.6</td><td>70.0/24.1</td><td>72.0/71.3</td><td>71.7</td><td>69.6</td><td>64.4</td><td>69.0</td></tr><tr><td>BERT++</td><td>79.0</td><td>84.8/90.4</td><td>73.8</td><td>70.0/24.1</td><td>72.0/71.3</td><td>79.0</td><td>69.6</td><td>64.4</td><td>71.5</td></tr><tr><td>Outside Best</td><td>80.4</td><td>-</td><td>84.4</td><td>70.4/24.5</td><td>74.8/73.0</td><td>82.7</td><td>-</td><td>1</td><td>1</td></tr><tr><td>RoBERTa</td><td>87.1</td><td>90.5/95.2</td><td>90.6</td><td>84.4/52.5</td><td>90.6/90.0</td><td>88.2</td><td>69.9</td><td>89.0</td><td>84.6</td></tr><tr><td>Human (est.)</td><td>89.0</td><td>95.8/98.9</td><td>100.0</td><td>81.8/51.9</td><td>91.7/91.3</td><td>93.6</td><td>80.0</td><td>100.0</td><td>89.8</td></tr></table>
+
+Table 11: Results on SuperGLUE. All results are based on a 24-layer architecture. RoBERTa results on the development set are a median over five runs. RoBERTa results on the test set are ensembles of single-task models. Averages are obtained from the SuperGLUE leaderboard.
+
+We also evaluate RoBERTa on the SuperGLUE benchmark (Wang et al., 2019a), which consists of 8 natural language understanding tasks.12 We largely follow the same setup for SuperGLUE as we did for GLUE, with several task-specific modifications:
+
+• BoolQ and MultiRC: we follow the same input format as the Wang et al. (2019a) baseline.   
+• CB: we finetune starting from the MNLI model, following Phang et al. (2018).   
+• COPA: we concatenate the premise and each alternative with because and so markers for cause and effect questions, respectively. This input format more closely matches the pretraining data format and provides better results in practice.   
+• ReCoRD: during training we adopt a pairwise ranking formulation with one negative and positive entity for each (passage, query) pair. At evaluation time, we pick the entity with the highest score for each question.   
+• WiC: we input the pair of sentences as normal. We then feed the concatenation of the representations of the two marked words and the [CLS] token to the classification layer.   
+• RTE and WSC: we reused our submission to the GLUE leaderboard.
+
+In Table 11 we present RoBERTa results on the 8 SuperGLUE datasets. RoBERTa achieves stateof-the-art results on the development and test sets for BoolQ, CB, COPA, MultiRC and ReCoRD and the highest average score to date on the SuperGLUE leaderboard.
+
+# G RESULTS ON XNLI
+
+<table><tr><td></td><td>en</td><td>fr</td><td>es</td><td>de</td><td>el</td><td>bg</td><td>ru</td><td>tr</td><td>ar</td><td>vi</td><td>th</td><td>zh</td><td>hi</td><td>SW</td><td>ur</td><td>△</td></tr><tr><td>Machine translation baselines (TRANSLATE-TEST)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>XLM(MLM+TLM)</td><td>85.0</td><td>79.0</td><td>79.5</td><td>78.1</td><td>77.8</td><td>77.6</td><td>75.5</td><td>73.7</td><td>73.7</td><td>70.8</td><td>70.4</td><td>73.6</td><td>69.0</td><td>64.7</td><td>65.1</td><td>74.2</td></tr><tr><td>XLM-en</td><td>88.8</td><td>81.4</td><td>82.3</td><td>80.1</td><td>80.3</td><td>80.9</td><td>76.2</td><td>76.0</td><td>75.4</td><td>72.0</td><td>71.9</td><td>75.6</td><td>70.0</td><td>65.8</td><td>65.8</td><td>76.2</td></tr><tr><td>RoBERTa</td><td>91.3</td><td>82.9</td><td>84.3</td><td>81.2</td><td>81.7</td><td>83.1</td><td>78.3</td><td>76.8</td><td>76.6</td><td>74.2</td><td>74.0</td><td>77.5</td><td>70.9</td><td>66.6</td><td>66.8</td><td>77.8</td></tr></table>
+
+Table 12: Results on XNLI (Conneau et al., 2018) for $\mathrm { R o B E R T a _ { L A R G E } }$ in the TRANSLATE-TEST setting. We report macro-averaged accuracy $( \Delta )$ using the provided English translations of the XNLI test sets. RoBERTa achieves state of the art results on all 15 languages.

parse/train/SyxS0T4tvS/SyxS0T4tvS_content_list.json

@@ -0,0 +1,1908 @@
+[
+    {
+        "type": "text",
+        "text": "ROBERTA: A ROBUSTLY OPTIMIZED BERT PRETRAINING APPROACH ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            257,
+            544,
+            272
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Language model pretraining has led to significant performance gains but careful comparison between different approaches is challenging. Training is computationally expensive, often done on private datasets of different sizes, and, as we show, hyperparameter choices have significant impact on the final results. We present a replication study of BERT pretraining (Devlin et al., 2019) that carefully measures the impact of many key hyperparameters and training data size. We find that BERT was significantly undertrained, and can match or exceed the performance of every model published after it. Our best model achieves state-of-the-art results on GLUE, RACE, SQuAD, SuperGLUE and XNLI. These results highlight the importance of previously overlooked design choices, and raise questions about the source of recently reported improvements. We release our models and code.1 ",
+        "bbox": [
+            233,
+            285,
+            764,
+            438
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            336,
+            477
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Self-training methods such as ELMo (Peters et al., 2018), GPT (Radford et al., 2018), BERT (Devlin et al., 2019), XLM (Lample & Conneau, 2019), and XLNet (Yang et al., 2019) have brought significant performance gains, but it can be challenging to determine which aspects of the methods contribute the most. Training is computationally expensive, limiting the amount of tuning that can be done, and modeling advances are often conflated with changes in data size or composition. ",
+        "bbox": [
+            174,
+            491,
+            823,
+            560
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present a replication study of BERT pretraining (Devlin et al., 2019), which includes a careful evaluation of the effects of hyperparameter tuning and training set size. We find that BERT was significantly undertrained and propose an improved training recipe, which we call RoBERTa, that can match or exceed the performance of all of the post-BERT methods. Our modifications are simple, they include: (1) training the model longer, with bigger batches, over more data; (2) removing the next sentence prediction objective; (3) training on longer sequences; and (4) dynamically changing the masking pattern applied to the training data. We also collect a large new dataset (CC-NEWS) of comparable size to other privately used datasets, to better control for training set size effects. ",
+        "bbox": [
+            174,
+            568,
+            825,
+            679
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When controlling for training data, our improved training procedure improves upon the published BERT results on the GLUE (Wang et al., 2019b) and SQuAD (Rajpurkar et al., 2016) benchmarks. When trained for longer over additional data, our model achieves a score of 88.5 on the public GLUE leaderboard, matching the 88.4 reported by Yang et al. (2019). Our model establishes a new stateof-the-art on 4/9 of the GLUE tasks, as well as RACE (Lai et al., 2017), SuperGLUE (Wang et al., 2019a), and XNLI (Conneau et al., 2018), and matches the state-of-the-art on SQuAD. Overall, we re-establish that BERT’s masked language model training objective is competitive with recently proposed alternatives such as perturbed autoregressive language modeling (Yang et al., 2019).2 ",
+        "bbox": [
+            174,
+            686,
+            825,
+            797
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In summary, the contributions of this paper are: (1) We present a set of important BERT design choices and training strategies and introduce alternatives that lead to better downstream task performance; (2) We use a novel dataset, CC-NEWS, and confirm that using more data for pretraining further improves performance on downstream tasks; (3) Our training improvements show that masked language model pretraining, under the right design choices, is competitive with all other recently published methods. We release our model, pretraining and fine-tuning code. ",
+        "bbox": [
+            176,
+            804,
+            823,
+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            326,
+            118
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Setup: BERT (Devlin et al., 2019) takes as input a concatenation of two segments (sequences of tokens), $x _ { 1 } , \\ldots , x _ { N }$ and $y _ { 1 } , \\dots , y _ { M }$ . Segments usually consist of more than one natural sentence. The two segments are presented as a single input sequence to BERT with special tokens delimiting them: $[ C L S ] , x _ { 1 } , \\ldots , x _ { N } , [ S E P ] , y _ { 1 } , \\ldots , y _ { M } , [ E O S ]$ . $M$ and $N$ are constrained such that $M + N <$ $T$ , where $T$ is a parameter that controls the maximum sequence length during training. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            204
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Architecture: BERT uses the now ubiquitous transformer architecture (Vaswani et al., 2017), which we will not review in detail. We use a transformer architecture with $L$ layers. Each block has $A$ self-attention heads and hidden dimension $H$ . ",
+        "bbox": [
+            174,
+            219,
+            825,
+            261
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Training Objectives: BERT uses two pretraining objectives: masked language modeling and next sentence prediction. For the Masked Language Model (MLM) objective, BERT is trained via a crossentropy loss to predict $15 \\%$ of the input tokens, selected at random. To prevent the model from cheating, $80 \\%$ of these selected tokens are replaced by a special $[ M A S K ]$ symbol in the input, $10 \\%$ are replaced by a random token from the vocabulary, and $10 \\%$ are left unchanged. ",
+        "bbox": [
+            176,
+            277,
+            823,
+            347
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Next Sentence Prediction (NSP) is a binary classification loss for predicting whether two segments follow each other in the original text. Positive examples are created by taking consecutive sentences from the text corpus. Negative examples are created by pairing segments from different documents. Positive and negative examples are sampled with equal probability. ",
+        "bbox": [
+            174,
+            353,
+            823,
+            410
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Optimization: BERT is optimized with AdamW (Kingma & Ba, 2015) using the following parameters: $\\beta _ { 1 } = 0 . 9$ , $\\beta _ { 2 } ~ = ~ 0 . 9 9 9$ , $\\epsilon = 1 \\mathrm { e } { - } 6$ and decoupled weight decay of 0.01 (Loshchilov & Hutter, 2019). The learning rate is warmed up over the first 10,000 steps to a peak value of 1e-4, and then linearly decayed. BERT trains with a dropout of 0.1 on all layers and attention weights, and a GELU activation function (Hendrycks & Gimpel, 2016). Models are pretrained for $S = \\overline { { 1 } } , 0 0 0 , 0 0 0$ updates, with mini-batches containing $B = 2 5 6$ sequences of maximum length $T = 5 1 2$ tokens. ",
+        "bbox": [
+            174,
+            425,
+            825,
+            510
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data: BERT is trained on a combination of BOOKCORPUS (Zhu et al., 2015) plus English WIKIPEDIA, which totals 16GB of uncompressed text.3 ",
+        "bbox": [
+            174,
+            525,
+            823,
+            554
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            574,
+            398,
+            590
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1 IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            606,
+            341,
+            621
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We reimplement BERT in FAIRSEQ (Ott et al., 2019). We primarily follow the original BERT optimization hyperparameters, given in Section 2, except for the peak learning rate and number of warmup steps, which are tuned separately for each setting. We found training to be very sensitive to the Adam epsilon term, and in some cases we obtained better performance or improved stability after tuning it. We also set $\\beta _ { 2 } = 0 . 9 8$ to improve stability when training with large batch sizes. ",
+        "bbox": [
+            174,
+            632,
+            825,
+            703
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We pretrain with sequences of at most $T = 5 1 2$ tokens. Unlike Devlin et al. (2019), we do not randomly inject short sequences, and we do not train with a reduced sequence length for the first $90 \\%$ of updates. We train only with full-length sequences. ",
+        "bbox": [
+            176,
+            709,
+            825,
+            751
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We train with mixed precision floating point arithmetic on DGX-1 machines, each with ${ 8 \\times 3 2 { \\mathrm { G B } } }$ Nvidia V100 GPUs interconnected by Infiniband (Micikevicius et al., 2018). ",
+        "bbox": [
+            174,
+            757,
+            821,
+            786
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.2 DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            804,
+            253,
+            818
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "BERT-style pretraining crucially relies on large quantities of text. Baevski et al. (2019) demonstrate that increasing data size can result in improved end-task performance. Several efforts have trained on datasets larger and more diverse than the original BERT (Radford et al., 2019; Yang et al., 2019; Zellers et al., 2019). Unfortunately, not all of the additional datasets can be publicly released. For our study, we focus on gathering as much data as possible for experimentation, allowing us to match the overall quality and quantity of data as appropriate for each comparison. ",
+        "bbox": [
+            174,
+            830,
+            825,
+            886
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider five English-language corpora of varying sizes and domains, totaling over 160GB of uncompressed text: (1&2) BOOKCORPUS (Zhu et al., 2015) plus English WIKIPEDIA, which is the original data used to train BERT (16GB); (3) CC-NEWS, which we collect from the English portion of the CommonCrawl News dataset (Nagel, 2016), containing 63 million English news articles crawled between September 2016 and February 2019 (76GB after filtering);4 (4) OPENWEBTEXT (Gokaslan & Cohen, 2019), an open-source recreation of the WebText corpus described in Radford et al. (2019), containing web content extracted from URLs shared on Reddit with at least three upvotes (38GB);5 (5) STORIES, a dataset introduced in Trinh & Le (2018) containing a subset of CommonCrawl data filtered to match the story-like style of Winograd schemas (31GB). ",
+        "bbox": [
+            174,
+            138,
+            825,
+            263
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3 EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            280,
+            305,
+            295
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following previous work, we evaluate our pretrained models by finetuning on downstream tasks: ",
+        "bbox": [
+            174,
+            306,
+            805,
+            320
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• GLUE: The General Language Understanding Evaluation (GLUE) benchmark (Wang et al., 2019b) is a collection of 9 datasets for evaluating natural language understanding systems. Tasks are framed as either single-sentence classification or sentence-pair classification tasks. The GLUE organizers provide training and development data splits as well as a submission server and leaderboard that allows participants to evaluate and compare their systems on private held-out test data. • SQuAD: The Stanford Question Answering Dataset (SQuAD) provides a paragraph of context and a question. The task is to answer the question with a span extracted from the context. We evaluate on SQuAD V1.1 and V2.0 (Rajpurkar et al., 2016; 2018). In V1.1 the context always contains an answer, while in V2.0 some questions are not answered in the provided context. • RACE: ReAding Comprehension from Examinations (RACE) (Lai et al., 2017) is a large-scale reading comprehension dataset collected from English examinations in China. The task is to choose among four possible answers to a given question, using a given passage of text as context. • Additional Benchmarks: In the Appendix we present additional results for SuperGLUE (Wang et al., 2019a) and XNLI (Conneau et al., 2018). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 TRAINING PROCEDURE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            551,
+            491,
+            566
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This section explores and quantifies which choices are important for successfully pretraining BERT models. We keep the model architecture fixed.6 Specifically, we begin by training BERT models with the same configuration as $\\mathbf { B E R T _ { B A S E } }$ $L = 1 2$ , $H = 7 6 8$ , $A = 1 2$ , 110M params). ",
+        "bbox": [
+            176,
+            582,
+            825,
+            625
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 STATIC VS. DYNAMIC MASKING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            640,
+            439,
+            654
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 2, BERT relies on predicting randomly masked tokens. The original BERT implementation performed masking once during data preprocessing, resulting in a single static mask. To avoid repeating the same masks at every epoch, training data was duplicated 10 times prior to preprocessing, so that each training sequence was seen with the same mask only four times over the course of 40 training epochs. We instead train with dynamic masking, where we generate the masking pattern on-the-fly each time we input a sequence to the model. This becomes crucial when pretraining for more steps or with larger datasets, and additionally performs marginally better than static masking on some downstream tasks (see Appendix A). ",
+        "bbox": [
+            174,
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+            777
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.2 MODEL INPUT FORMAT AND NEXT SENTENCE PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            794,
+            629,
+            808
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the original BERT pretraining procedure, the model observes two concatenated document segments and is trained via an auxiliary Next Sentence Prediction (NSP) loss to predict whether these segments were sampled contiguously from the same document or from distinct documents. ",
+        "bbox": [
+            174,
+            819,
+            825,
+            861
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/08c8aca55b0df8a3c90c1f177621c09124c5cea240bc752387351289e41226d5.jpg",
+        "table_caption": [
+            "Table 1: Development set results for base models pretrained over BOOKCORPUS and WIKIPEDIA. All models are trained for 1M steps with a batch size of 256 sequences. We report F1 for SQuAD and accuracy for MNLI-m, SST-2 and RACE. Reported results are medians over five random initializations (seeds). Results for $\\mathbf { B E R T _ { B A S E } }$ and $\\mathbf { X L N e t } _ { \\mathrm { B A S E } }$ are from Yang et al. (2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>SQuAD 1.1/2.0</td><td>MNLI-m</td><td>SST-2</td><td>RACE</td></tr><tr><td colspan=\"5\">Our reimplementation (with NSP loss):</td></tr><tr><td>SEGMENT-PAIR</td><td>90.4/78.7</td><td>84.0</td><td>92.9</td><td>64.2</td></tr><tr><td>SENTENCE-PAIR</td><td>88.7/76.2</td><td>82.9</td><td>92.1</td><td>63.0</td></tr><tr><td colspan=\"5\">Our reimplementation (without NSP loss):</td></tr><tr><td>FULL-SENTENCES</td><td>90.4/79.1</td><td>84.7</td><td>92.5</td><td>64.8</td></tr><tr><td>DOC-SENTENCES</td><td>90.6/79.7</td><td>84.7</td><td>92.7</td><td>65.6</td></tr><tr><td>BERTBASE</td><td>88.5/76.3</td><td>84.3</td><td>92.8</td><td>64.3</td></tr><tr><td>XLNetBASE (K=7)</td><td>-/81.3</td><td>85.8</td><td>92.7</td><td>66.1</td></tr><tr><td>XLNetBASE (K=6)</td><td>-/81.0</td><td>85.6</td><td>93.4</td><td>66.7</td></tr></table>",
+        "bbox": [
+            258,
+            101,
+            738,
+            268
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The NSP objective was designed to improve performance on downstream tasks, such as Natural Language Inference (Bowman et al., 2015), which require predicting relationships between pairs of sentences. Devlin et al. (2019) observe that removing NSP hurts performance, with significant performance degradation on QNLI, MNLI, and SQuAD 1.1. However, recent work has questioned the necessity of the NSP loss (Lample & Conneau, 2019; Yang et al., 2019; Joshi et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "To better understand this discrepancy, we compare several alternative training formats: ",
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+    {
+        "type": "text",
+        "text": "• SEGMENT-PAIR $^ +$ NSP: This follows the original input format used in BERT (Devlin et al., 2019), with the NSP loss. Each input has a pair of segments, which can each contain multiple natural sentences, but the total combined length must be less than 512 tokens. SENTENCE-PAIR $+ \\mathrm { N S P }$ : Each input contains a pair of natural sentences, either sampled from a contiguous portion of one document or from separate documents. Since these inputs are significantly shorter than 512 tokens, we increase the batch size so that the total number of tokens remains similar to SEGMENT-PAIR $+ \\mathrm { N S P }$ . We retain the NSP loss. FULL-SENTENCES: Each input is packed with full sentences sampled contiguously from one or more documents, such that the total length is at most 512 tokens. Inputs may cross document boundaries. When we reach the end of one document, we begin sampling sentences from the next document and add an extra separator token between documents. We remove the NSP loss. DOC-SENTENCES: Inputs are constructed similarly to FULL-SENTENCES, except that they may not cross document boundaries. Inputs sampled near the end of a document may be shorter than 512 tokens, so we dynamically increase the batch size in these cases to achieve a similar number of total tokens as FULL-SENTENCES. We remove the NSP loss. ",
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+    {
+        "type": "text",
+        "text": "Results Table 1 shows results for the four different settings. We first compare the original SEGMENT-PAIR input format from Devlin et al. (2019) to the SENTENCE-PAIR format; both formats retain the NSP loss, but the latter uses single sentences. We find that using individual sentences hurts performance on downstream tasks, which we hypothesize is because the model is not able to learn long-range dependencies. ",
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+    {
+        "type": "text",
+        "text": "We next compare training without the NSP loss and training with blocks of text from a single document (DOC-SENTENCES). We find that this setting outperforms the originally published BERTBASE results and that removing the NSP loss matches or slightly improves downstream task performance, in contrast to Devlin et al. (2019). It is possible that the original BERT implementation may only have removed the loss term while still retaining the SEGMENT-PAIR input format. ",
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+    },
+    {
+        "type": "text",
+        "text": "Finally we find that restricting sequences to come from a single document (DOC-SENTENCES) performs slightly better than packing sequences from multiple documents (FULL-SENTENCES). However, because the DOC-SENTENCES format results in variable batch sizes, we use FULL-SENTENCES in the remainder of our experiments for easier comparison with related work. ",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>batch size</td><td>learning rate</td><td>epochs</td><td>steps</td><td> perplexity</td><td>MNLI-m</td><td>SST-2</td></tr><tr><td>256</td><td>1e-4</td><td>32</td><td>1M</td><td>3.99</td><td>84.7</td><td>92.5</td></tr><tr><td rowspan=\"3\">2K</td><td rowspan=\"3\">7e-4</td><td>32</td><td>125K</td><td>3.68</td><td>85.2</td><td>93.1</td></tr><tr><td>64</td><td>250K</td><td>3.59</td><td>85.3</td><td>94.1</td></tr><tr><td>128</td><td>500K</td><td>3.51</td><td>85.4</td><td>93.5</td></tr><tr><td rowspan=\"3\">8K</td><td rowspan=\"3\">1e-3</td><td>32</td><td>31K</td><td>3.77</td><td>84.4</td><td>93.2</td></tr><tr><td>64</td><td>63K</td><td>3.60</td><td>85.3</td><td>93.5</td></tr><tr><td>128</td><td>125K</td><td>3.50</td><td>85.8</td><td>94.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Perplexity on held-out validation data and dev set accuracy on MNLI-m and SST-2 for various batch sizes (# sequences) as we vary the number of passes (epochs) through the $\\mathrm { { B O O K S } + }$ WIKI data. Reported results are medians over five random initializations (seeds). The learning rate is tuned for each batch size. All results are for $\\mathbf { B E R T _ { B A S E } }$ with FULL-SENTENCE inputs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 TRAINING WITH LARGE BATCHES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Past work in neural machine translation has shown that training with large mini-batches can improve optimization speed and end-task performance when the learning rate is tuned appropriately (Ott et al., 2018). Large batches are also easily parallelized via data parallel training.7 ",
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+    {
+        "type": "text",
+        "text": "Table 2 shows the masked LM perplexity and end-task accuracy for $\\mathbf { B E R T _ { B A S E } }$ as we increase the batch size, while tuning the learning rate. Devlin et al. (2019) originally trained $\\mathbf { B E R T _ { B A S E } }$ for 1M steps with a batch size of 256 sequences; however a batch size of 2K sequences performs better, even controlling for the number of epochs, suggesting that the original BERT batch size was too small. We also observe that training with extremely large batches (8K) becomes more efficient as we train for more epochs.8 In the remainder of our experiments we train with batches of 8K sequences. ",
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+    {
+        "type": "text",
+        "text": "4.4 TEXT ENCODING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Byte-Pair Encoding (BPE) (Sennrich et al., 2016) is a hybrid between character- and word-level modeling based on subwords units. BPE vocabulary sizes typically range from 10K-100K subword units; however, unicode characters can account for a sizeable portion of this vocabulary when modeling large and diverse corpora, such as the ones considered in this work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The original BERT implementation (Devlin et al., 2019) used a character-level BPE vocabulary of size 30K. We instead adopt the larger byte-level BPE vocabulary of size 50K introduced in Radford et al. (2019), which uses bytes rather than unicode characters as the base subword units and can therefore encode any input text without introducing “unknown” tokens. This adds approximately 15M and 20M extra parameters for $\\mathbf { B E R T _ { B A S E } }$ and BERTLARGE, respectively. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Early experiments revealed only minor differences between these encodings, with the byte-level BPE achieving slightly worse end-task performance on some tasks. Nevertheless, we believe the advantages of a universal encoding scheme outweighs the minor degredation in performance and use this encoding in the remainder of our experiments. ",
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+    {
+        "type": "text",
+        "text": "5 ROBERTA ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the previous section we propose modifications to the BERT pretraining procedure that improve end-task performance. We now aggregate these improvements and evaluate their combined impact. We call this configuration RoBERTa for Robustly optimized BERT approach. Specifically, RoBERTa is trained with dynamic masking (Section 4.1), FULL-SENTENCES without NSP loss (Section 4.2), large mini-batches (Section 4.3) and a larger byte-level BPE (Section 4.4). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/884cc17053fbb3c512d5632a34eebf11d3837a8b21780724574ffde41e966240.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>data</td><td>batch size</td><td>steps</td><td>SQuAD (v1.1/2.0)</td><td>MNLI-m</td><td>SST-2</td></tr><tr><td>RoBERTa</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BOOKS +WIKI</td><td>16GB</td><td>8K</td><td>100K</td><td>93.6/87.3</td><td>89.0</td><td>95.3</td></tr><tr><td>+ additional data ($3.2)</td><td>160GB</td><td>8K</td><td>100K</td><td>94.0/87.7</td><td>89.3</td><td>95.6</td></tr><tr><td>+ pretrain longer</td><td>160GB</td><td>8K</td><td>300K</td><td>94.4/88.7</td><td>90.0</td><td>96.1</td></tr><tr><td>+ pretrain even longer</td><td>160GB</td><td>8K</td><td>500K</td><td>94.6/89.4</td><td>90.2</td><td>96.4</td></tr><tr><td>BERTLARGE</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BoOKS +WIKI</td><td>13GB</td><td>256</td><td>1M</td><td>90.9/81.8</td><td>86.6</td><td>93.7</td></tr><tr><td>XLNetLARGE</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BoOKS +WIKI</td><td>13GB</td><td>256</td><td>1M</td><td>94.0/87.8</td><td>88.4</td><td>94.4</td></tr><tr><td>+ additional data</td><td>126GB</td><td>2K</td><td>500K</td><td>94.5/88.8</td><td>89.8</td><td>95.6</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Development set results for RoBERTa as we pretrain over more data $\\mathbf { 1 6 G B }  \\mathbf { 1 6 0 G B }$ of text) and pretrain for longer $1 0 0 \\mathrm { K }  3 0 0 \\mathrm { K }  5 0 0 \\mathrm { K }$ steps). Each row accumulates improvements from the rows above. RoBERTa matches the architecture and training objective of $\\mathbf { B E R T _ { L A R G E } }$ . Results for $\\mathbf { B E R T _ { L A R G E } }$ and $\\mathrm { X L N e t } _ { \\mathrm { L A R G E } }$ are from Devlin et al. (2019) and Yang et al. (2019), respectively. Complete results on all GLUE tasks can be found in Appendix C. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Additionally, we investigate two other important factors that have been under-emphasized in previous work: (1) the data used for pretraining, and (2) the number of training passes through the data. For example, XLNet (Yang et al., 2019) was pretrained using 10 times more data than BERT, with a batch size eight times larger for half as many optimization steps, thus seeing four times as many sequences in pretraining compared to Devlin et al. (2019). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To help disentangle the importance of these factors from other modeling choices (e.g., the pretraining objective), we begin by training RoBERTa following the $\\mathbf { B E R T _ { L A R G E } }$ architecture $L = 2 4$ , $H =$ 1024, $A = 1 6$ , 355M parameters). We pretrain for 100K steps over a comparable BOOKCORPUS plus WIKIPEDIA dataset as was used in Devlin et al. (2019). We pretrain our model using 1024 V100 GPUs, which takes approximately one day per 100K steps. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Results We present our results in Table 3. When controlling for training data, we observe that RoBERTa provides a large improvement over the originally reported BERTLARGE results, reaffirming the importance of the design choices we explored in Section 4. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Next, we combine this data with the three additional datasets described in Section 3.2. We train RoBERTa over the combined data with the same number of training steps as before (100K). In total, we pretrain over 160GB of text. We observe further improvements in performance across all downstream tasks, validating the importance of data size and diversity in pretraining.9 9 ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, we pretrain RoBERTa for significantly longer, increasing the number of pretraining steps from 100K to 300K, and then further to 500K. We again observe significant gains in downstream task performance, and the 300K and 500K step models outperform XLNetLARGE across most tasks. We note that even our longest-trained model does not appear to overfit our data and would likely benefit from additional training. ",
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+    {
+        "type": "text",
+        "text": "5.1 GLUE RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For GLUE, we consider two finetuning settings. In the first setting (single-task, dev), we finetune RoBERTa separately for each of the GLUE tasks, using only the training data for the corresponding task. We consider a limited hyperparameter sweep with batch sizes $\\in \\{ 1 6 , 3 2 \\}$ and learning rates $\\in \\ \\{ 1 \\mathrm { e } { - } 5 , 2 \\mathrm { e } { - } 5 , 3 \\mathrm { e } { - } 5 \\}$ , with a linear warmup for the first $6 \\%$ of steps followed by a linear decay to 0. We finetune for 10 epochs with early stopping based on each task’s dev set. The rest of the hyperparameters remain the same as during pretraining. In this setting, we report the median development set results for each task over five random initializations, without model ensembling. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the second setting (ensembles, test), we compare RoBERTa to other approaches on the test set via the GLUE leaderboard. While many submissions to the GLUE leaderboard depend on multi-task finetuning, our submission depends only on single-task finetuning. For RTE, STS and MRPC we finetune starting from the MNLI single-task model, following Phang et al. (2018). We explore a slightly wider hyperparameter space, described in Appendix C, and ensemble between 5 and 7 models per task. Two of the GLUE tasks require task-specific finetuning approaches to achieve competitive leaderboard results; these approaches are described in Appendix B. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/f9603334887c5bc5c0719038027479199b8bdfb3f4e9ae0b612cb788a130ed15.jpg",
+        "table_caption": [
+            "Table 4: Results on GLUE. All results are based on a 24-layer architecture. $\\mathbf { B E R T _ { L A R G E } }$ and $\\mathbf { X L N e t } _ { \\mathrm { L A R G E } }$ results are from Devlin et al. (2019) and Yang et al. (2019), respectively. RoBERTa results on the dev set are a median over five runs. RoBERTa results on the test set are ensembles of single-task models. For RTE, STS and MRPC we finetune starting from the MNLI model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MNLI</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST</td><td>MRPC</td><td>CoLA</td><td>STS</td><td>WNLI</td><td>Avg</td></tr><tr><td colspan=\"9\">Single-task single models on dev</td><td></td></tr><tr><td>BERTLARGE</td><td>86.6/-</td><td>92.3</td><td>91.3</td><td>70.4</td><td>93.2</td><td>88.0</td><td>60.6</td><td>90.0</td><td></td><td>=</td></tr><tr><td>XLNetLARGE</td><td>89.8/-</td><td>93.9</td><td>91.8</td><td>83.8</td><td>95.6</td><td>89.2</td><td>63.6</td><td>91.8</td><td>=</td><td>=</td></tr><tr><td>RoBERTa</td><td>90.2/90.2</td><td>94.7</td><td>92.2</td><td>86.6</td><td>96.4</td><td>90.9</td><td>68.0</td><td>92.4</td><td>91.3</td><td>1</td></tr><tr><td colspan=\"9\">Ensembles on test (from leaderboard asof July 25,2019)</td><td></td><td></td></tr><tr><td>ALICE</td><td>88.2/87.9</td><td>95.7</td><td>90.7</td><td>83.5</td><td>95.2</td><td>92.6</td><td>68.6</td><td>91.1</td><td>80.8</td><td>86.3</td></tr><tr><td>MT-DNN</td><td>87.9/87.4</td><td>96.0</td><td>89.9</td><td>86.3</td><td>96.5</td><td>92.7</td><td>68.4</td><td>91.1</td><td>89.0</td><td>87.6</td></tr><tr><td>XLNet</td><td>90.2/89.8</td><td>98.6</td><td>90.3</td><td>86.3</td><td>96.8</td><td>93.0</td><td>67.8</td><td>91.6</td><td>90.4</td><td>88.4</td></tr><tr><td>RoBERTa</td><td>90.8/90.2</td><td>98.9</td><td>90.2</td><td>88.2</td><td>96.7</td><td>92.3</td><td>67.8</td><td>92.2</td><td>89.0</td><td>88.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1fd4a5f7c1b3a291ad8732bfa35173db5b78e06e973466cf794fc9ccc84a0e78.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>SQuAD 2.0 EM F1</td></tr><tr><td>Single models on test (as of July 25,2019)</td><td>89.1†</td></tr><tr><td>XLNetLARGE RoBERTa</td><td>86.3t 86.8</td></tr><tr><td>XLNet+ SG-Net Verifier</td><td>89.8 89.9†</td></tr><tr><td></td><td>87.0t</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0abbf1506f7b18f80308b30e22b50a392d0367a11a5e13b2b9a406c43696f63e.jpg",
+        "table_caption": [
+            "Table 5: Results on SQuAD. $\\dagger$ indicates results that depend on additional external training data. RoBERTa uses only the provided SQuAD data in both dev and test settings. $\\mathbf { B E R T _ { L A R G E } }$ and XLNetLARGE results are from Devlin et al. (2019) and Yang et al. (2019), respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>SQuAD 1.1 EM</td><td>F1</td><td>SQuAD 2.0 EM F1</td></tr><tr><td>Single models on dev, w/o data augmentation</td><td></td><td></td><td></td></tr><tr><td>BERTLARGE</td><td>84.1</td><td>90.9 79.0</td><td>81.8</td></tr><tr><td>XLNetLARGE</td><td>89.0</td><td>94.5 86.1</td><td>88.8</td></tr><tr><td>RoBERTa</td><td>88.9</td><td>94.6 86.5</td><td>89.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results We present our results in Table 4. In the first setting (single-task, dev), RoBERTa achieves state-of-the-art results on all 9 of the GLUE task development sets. Crucially, RoBERTa uses the same masked language modeling pretraining objective and architecture as $\\mathbf { B E R T _ { L A R G E } }$ , yet consistently outperforms both $\\mathbf { B E R T _ { L A R G E } }$ and $\\mathbf { X L N e t } _ { \\mathrm { L A R G E } }$ . This raises questions about the relative importance of model architecture and pretraining objective, compared to more mundane details like dataset size and training time that we explore in this work. A more comprehensive comparison of the BERT and XLNet pretraining objectives is needed, but is left to future work. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the second setting (ensembles, test), we submit RoBERTa to the GLUE leaderboard and achieve state-of-the-art results on 4 out of 9 tasks and the highest average score to date. Notably, RoBERTa does not depend on multi-task finetuning, and we expect future work may further improve these results by incorporating more sophisticated multi-task finetuning procedures. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 SQUAD RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We adopt a much simpler approach for SQuAD compared to past work. While BERT (Devlin et al., 2019) and XLNet (Yang et al., 2019) augment their training data with additional QA datasets, we only finetune RoBERTa using the provided SQuAD training data. We also use a single learning rate for all layers, in contrast to the custom layer-wise learning rate scheduled used by Yang et al. (2019). ",
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+    {
+        "type": "text",
+        "text": "For SQuAD v1.1 we follow the same finetuning procedure as Devlin et al. (2019). For SQuAD v2.0, we additionally classify whether a given question is answerable; we train this classifier jointly with the span predictor by summing the classification and span loss terms. ",
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+    {
+        "type": "table",
+        "img_path": "images/eadd737de2b1aac76052392a079e8ca4e379cf72d9d27b1de9da105e0d5ed054.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Accuracy</td><td>Middle</td><td>High</td></tr><tr><td>Single models on test (as of July 25, 2</td><td></td><td></td><td>2019)</td></tr><tr><td>BERTLARGE</td><td>72.0</td><td>76.6</td><td>70.1</td></tr><tr><td>XLNetLARGE</td><td>81.7</td><td>85.4</td><td>80.2</td></tr><tr><td>RoBERTa</td><td>83.2</td><td>86.5</td><td>81.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 6: Results on the RACE test set. $\\mathbf { B E R T } _ { \\mathrm { L A R G E } }$ and $\\mathbf { X L N e t } _ { \\mathrm { L A R G E } }$ results from Yang et al. (2019). ",
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+    {
+        "type": "text",
+        "text": "Results We present our results in Table 5. On the SQuAD v1.1 development set, RoBERTa matches the state-of-the-art set by XLNet. On the SQuAD v2.0 development set, RoBERTa sets a new state-of-the-art, improving over XLNet by 0.4 points (EM) and 0.6 points (F1). ",
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+    {
+        "type": "text",
+        "text": "We also submit RoBERTa to the public SQuAD 2.0 leaderboard. Most of the top systems build upon either BERT (Devlin et al., 2019) or XLNet (Yang et al., 2019) and therefore rely on additional external training data. Our single RoBERTa model outperforms all but one of the single model submissions, and is the top scoring system among those that do not rely on additional external data. ",
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+    {
+        "type": "text",
+        "text": "5.3 RACE RESULTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In RACE, systems are provided with a passage of text, an associated question, and must classify which of four candidate answers is correct. We modify RoBERTa for this task by concatenating each candidate answer with the corresponding question and passage. We encode each of these four sequences and pass the resulting $I C L S J$ representations through a fully-connected layer, which is used to predict the correct answer. We truncate question-answer pairs that are longer than 128 tokens and, if needed, the passage so that the total length is at most 512 tokens. ",
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+    },
+    {
+        "type": "text",
+        "text": "Results are presented in Table 6. RoBERTa achieves state-of-the-art accuracy across all settings. ",
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+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Pretraining methods have been designed with different training objectives, including language modeling (Dai & Le, 2015; Peters et al., 2018; Howard & Ruder, 2018), machine translation (McCann et al., 2017), and masked language modeling (Devlin et al., 2019; Lample & Conneau, 2019). Many recent papers have used a basic recipe of finetuning models for each end task (Howard & Ruder, 2018; Radford et al., 2018), and pretraining with some variant of a masked language model objective. However, newer methods have improved performance by multi-task fine tuning (Dong et al., 2019), incorporating entity embeddings (Sun et al., 2019), span prediction (Joshi et al., 2019), and multiple variants of autoregressive pretraining (Song et al., 2019; Chan et al., 2019; Yang et al., 2019). Performance is also typically improved by training bigger models on more data (Devlin et al., 2019; Baevski et al., 2019; Yang et al., 2019; Radford et al., 2019). Our goal was to replicate, simplify, and better tune the training of BERT, as a reference point for better understanding the relative performance of all of these methods. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate a number of design decisions when pretraining BERT models, demonstrating that performance can be substantially improved by training the model longer, with bigger batches over more data; removing the next sentence prediction objective; training on longer sequences; and dynamically changing the masking pattern applied to the training data. We additionally use a novel dataset, CC-NEWS, and release our models and code for pretraining and finetuning at: anonymous URL. ",
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+    {
+        "type": "text",
+        "text": "Our improved pretraining procedure, which we call RoBERTa, achieves state-of-the-art results on GLUE, RACE, SQuAD, SuperGLUE and XNLI. These results illustrate the importance of these previously overlooked design decisions and suggest that BERT’s pretraining objective remains competitive with recently proposed alternatives. ",
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+    {
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+        "text": "Yukun Zhu, Ryan Kiros, Richard Zemel, Ruslan Salakhutdinov, Raquel Urtasun, Antonio Torralba, and Sanja Fidler. Aligning books and movies: Towards story-like visual explanations by watching movies and reading books. In arXiv preprint arXiv:1506.06724, 2015. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/f3c4c739896e0161b0853cf35f00dedbeb466f70dbabd2f635f83bc90b5ded82.jpg",
+        "table_caption": [
+            "A STATIC VS. DYNAMIC MASKING "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Masking</td><td> SQuAD 2.0</td><td>MNLI-m</td><td>SST-2</td></tr><tr><td>reference</td><td>76.3</td><td>84.3</td><td>92.8</td></tr><tr><td>Our reimplementation:</td><td></td><td></td><td></td></tr><tr><td>static</td><td>78.3</td><td>84.3</td><td>92.5</td></tr><tr><td>dynamic</td><td>78.7</td><td>84.0</td><td>92.9</td></tr></table>",
+        "bbox": [
+            334,
+            133,
+            661,
+            226
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Comparison between the published $\\mathbf { B E R T _ { B A S E } }$ results from Devlin et al. (2019) to our reimplementation with either static or dynamic masking. We report F1 for SQuAD and accuracy for MNLI-m and SST-2. Reported results are medians over 5 random initializations (seeds). Reference results are from Yang et al. (2019). We find that our reimplementation with static masking performs similar to the original BERT model, and dynamic masking is comparable or slightly better than static masking. ",
+        "bbox": [
+            173,
+            241,
+            826,
+            325
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B TASK-SPECIFIC MODIFICATIONS FOR GLUE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            352,
+            573,
+            368
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Two of the GLUE tasks require task-specific finetuning approaches to achieve competitive leaderboard results: ",
+        "bbox": [
+            174,
+            382,
+            821,
+            411
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "QNLI Recent submissions on the GLUE leaderboard adopt a pairwise ranking formulation for the QNLI task, in which candidate answers are mined from the training set and compared to one another, and a single (question, candidate) pair is classified as positive (Liu et al., 2019b;a; Yang et al., 2019). This formulation significantly simplifies the task, but is not directly comparable to BERT (Devlin et al., 2019). Following recent work, we adopt the ranking approach for our test submission, but for direct comparison with BERT all reported development set results are based on a pure classification approach. ",
+        "bbox": [
+            173,
+            425,
+            825,
+            522
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "WNLI We found the provided NLI-format data to be challenging to work with. Instead we use the reformatted WNLI data from SuperGLUE (Wang et al., 2019a), which indicates the span of the query pronoun and referent. We then finetune RoBERTa using a variation of the approach from Kocijan et al. (2019). In particular, for a given input sentence, we first use spaCy (Honnibal & Montani, 2017) to extract additional candidate noun phrases from the sentence, and then finetune our model so that it assigns higher scores to positive referent phrases than for any of the generated negative candidate phrases. ",
+        "bbox": [
+            173,
+            536,
+            825,
+            633
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In contrast to Kocijan et al. (2019), who finetune BERT using a margin ranking loss between (query, candidate) pairs, we instead use a single cross entropy loss term over the log-probabilities for the query and all mined candidates. This reduces the number of hyperparameters that need to be tuned and in practice produces more stable results on the development set. Our best model achieved $9 2 . 3 \\%$ development set accuracy, compared to $9 0 . 2 \\%$ accuracy for the margin loss approach. ",
+        "bbox": [
+            174,
+            641,
+            825,
+            712
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "One unfortunate consequence of our overall approach is that we can only make use of the positive training examples, which excludes over half of the provided training data.10 ",
+        "bbox": [
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+            747
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C FULL RESULTS ON GLUE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            766,
+            423,
+            782
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Table 8 we present the full set of development set results for RoBERTa on all 9 GLUE datasets.11 We present results for a LARGE configuration with 355M parameters that follows BERTLARGE, as well as a BASE configuration with 125M parameters that follows BERTBASE. ",
+        "bbox": [
+            174,
+            796,
+            823,
+            838
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/d6cf47b2747183ba7432ba610e8ecac44f55d7fe9acbf6b9f5452914fedb0334.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 8: Development set results on GLUE tasks for various configurations of RoBERTa. All results are a median over five runs. "
+        ],
+        "table_body": "<table><tr><td></td><td>MNLI</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST</td><td>MRPC</td><td>CoLA</td><td>STS</td></tr><tr><td>RoBERTaBASE</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+ all data + 500k steps</td><td>87.6</td><td>92.8</td><td>91.9</td><td>78.7</td><td>94.8</td><td>90.2</td><td>63.6</td><td>91.2</td></tr><tr><td>RoBERTaLARGE</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>with BOOKS +WIKI</td><td>89.0</td><td>93.9</td><td>91.9</td><td>84.5</td><td>95.3</td><td>90.2</td><td>66.3</td><td>91.6</td></tr><tr><td>+ additional data (83.2)</td><td>89.3</td><td>94.0</td><td>92.0</td><td>82.7</td><td>95.6</td><td>91.4</td><td>66.1</td><td>92.2</td></tr><tr><td>+ pretrain longer 300k</td><td>90.0</td><td>94.5</td><td>92.2</td><td>83.3</td><td>96.1</td><td>91.1</td><td>67.4</td><td>92.3</td></tr><tr><td>+ pretrain longer 500k</td><td>90.2</td><td>94.7</td><td>92.2</td><td>86.6</td><td>96.4</td><td>90.9</td><td>68.0</td><td>92.4</td></tr></table>",
+        "bbox": [
+            173,
+            102,
+            825,
+            236
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/24a0848a1888fb7633305d8ab827e0e500f7b88c419921308ef718cd640e2ce6.jpg",
+        "table_caption": [
+            "D PRETRAINING HYPERPARAMETERS ",
+            "Table 9: Hyperparameters for pretraining RoBERTaLARGE and RoBERTaBASE. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparam</td><td>RoBERTaLARGE</td><td>RoBERTaBASE</td></tr><tr><td>Number of Layers</td><td>24</td><td>12</td></tr><tr><td>Hidden size</td><td>1024</td><td>768</td></tr><tr><td>FFN inner hidden size</td><td>4096</td><td>3072</td></tr><tr><td>Attention heads</td><td>16</td><td>12</td></tr><tr><td>Attention head size</td><td>64</td><td>64</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td></tr><tr><td>Attention Dropout</td><td>0.1</td><td>0.1</td></tr><tr><td>Warmup Steps</td><td>24k</td><td>24k</td></tr><tr><td>Peak Learning Rate</td><td>4e-4</td><td>6e-4</td></tr><tr><td>Batch Size</td><td>8k</td><td>8k</td></tr><tr><td>Weight Decay</td><td>0.01</td><td>0.01</td></tr><tr><td>Max Steps</td><td>500k</td><td>500k</td></tr><tr><td>Learning Rate Decay</td><td>Linear</td><td>Linear</td></tr><tr><td>Adam ∈</td><td>1e-6</td><td>1e-6</td></tr><tr><td>Adam β1</td><td>0.9</td><td>0.9</td></tr><tr><td>Adam β2</td><td>0.98</td><td>0.98</td></tr><tr><td>Gradient Clipping</td><td>0.0</td><td>0.0</td></tr></table>",
+        "bbox": [
+            290,
+            343,
+            705,
+            609
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/d71de52948f2395e4c5dd28d813b422aeddb25d73cd37662f84c41a584f10cae.jpg",
+        "table_caption": [
+            "E FINETUNING HYPERPARAMETERS ",
+            "Table 10: Hyperparameters for finetuning $\\mathrm { R o B E R T a _ { L A R G E } }$ on RACE, SQuAD and GLUE. We select the best hyperparameter values based on the median of 5 random seeds for each task. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparam</td><td>RACE</td><td> SQuAD</td><td>GLUE</td><td>SuperGLUE</td></tr><tr><td>Learning Rate</td><td>1e-5</td><td>1.5e-5</td><td>{1e-5,2e-5,3e-5}</td><td>{1e-5,2e-5,3e-5}</td></tr><tr><td>Batch Size</td><td>16</td><td>48</td><td>{16,32}</td><td>32</td></tr><tr><td>Weight Decay</td><td>0.1</td><td>0.01</td><td>0.1</td><td>0.1</td></tr><tr><td>Max Epochs</td><td>4</td><td>2</td><td>10</td><td>{10,50}</td></tr><tr><td>Learning Rate Decay</td><td>Linear</td><td>Linear</td><td>Linear</td><td>Linear</td></tr><tr><td>Warmup ratio</td><td>0.06</td><td>0.06</td><td>0.06</td><td>0.10</td></tr></table>",
+        "bbox": [
+            209,
+            715,
+            789,
+            829
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/be64dfbaffdf34bf834f089a20d77a769fe539536bc6efe9ed19f972adaad571.jpg",
+        "table_caption": [
+            "F RESULTS ON SUPERGLUE "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>BoolQ</td><td>CB</td><td>COPA</td><td>MultiRC</td><td>ReCoRD</td><td>RTE</td><td>WiC</td><td>wsC</td><td>Avg</td></tr><tr><td colspan=\"10\">Single-task single models on dev</td></tr><tr><td>BERT++</td><td>80.1</td><td>96.4/95.0</td><td>78.0</td><td>70.7/24.7</td><td>70.6/69.8</td><td>82.3</td><td>74.9</td><td>68.3</td><td>74.6</td></tr><tr><td>RoBERTa</td><td>86.9</td><td>98.2/-</td><td>94.0</td><td>85.7/-</td><td>89.5/89.0</td><td>86.6</td><td>75.6</td><td>-</td><td>-</td></tr><tr><td colspan=\"10\">Ensembles on test (from leaderboard as of August 12,2019)</td></tr><tr><td>BERT</td><td>77.4</td><td>75.7/83.6</td><td>70.6</td><td>70.0/24.1</td><td>72.0/71.3</td><td>71.7</td><td>69.6</td><td>64.4</td><td>69.0</td></tr><tr><td>BERT++</td><td>79.0</td><td>84.8/90.4</td><td>73.8</td><td>70.0/24.1</td><td>72.0/71.3</td><td>79.0</td><td>69.6</td><td>64.4</td><td>71.5</td></tr><tr><td>Outside Best</td><td>80.4</td><td>-</td><td>84.4</td><td>70.4/24.5</td><td>74.8/73.0</td><td>82.7</td><td>-</td><td>1</td><td>1</td></tr><tr><td>RoBERTa</td><td>87.1</td><td>90.5/95.2</td><td>90.6</td><td>84.4/52.5</td><td>90.6/90.0</td><td>88.2</td><td>69.9</td><td>89.0</td><td>84.6</td></tr><tr><td>Human (est.)</td><td>89.0</td><td>95.8/98.9</td><td>100.0</td><td>81.8/51.9</td><td>91.7/91.3</td><td>93.6</td><td>80.0</td><td>100.0</td><td>89.8</td></tr></table>",
+        "bbox": [
+            176,
+            136,
+            821,
+            291
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 11: Results on SuperGLUE. All results are based on a 24-layer architecture. RoBERTa results on the development set are a median over five runs. RoBERTa results on the test set are ensembles of single-task models. Averages are obtained from the SuperGLUE leaderboard. ",
+        "bbox": [
+            174,
+            309,
+            825,
+            351
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We also evaluate RoBERTa on the SuperGLUE benchmark (Wang et al., 2019a), which consists of 8 natural language understanding tasks.12 We largely follow the same setup for SuperGLUE as we did for GLUE, with several task-specific modifications: ",
+        "bbox": [
+            176,
+            367,
+            825,
+            410
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• BoolQ and MultiRC: we follow the same input format as the Wang et al. (2019a) baseline.   \n• CB: we finetune starting from the MNLI model, following Phang et al. (2018).   \n• COPA: we concatenate the premise and each alternative with because and so markers for cause and effect questions, respectively. This input format more closely matches the pretraining data format and provides better results in practice.   \n• ReCoRD: during training we adopt a pairwise ranking formulation with one negative and positive entity for each (passage, query) pair. At evaluation time, we pick the entity with the highest score for each question.   \n• WiC: we input the pair of sentences as normal. We then feed the concatenation of the representations of the two marked words and the [CLS] token to the classification layer.   \n• RTE and WSC: we reused our submission to the GLUE leaderboard. ",
+        "bbox": [
+            173,
+            421,
+            826,
+            588
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Table 11 we present RoBERTa results on the 8 SuperGLUE datasets. RoBERTa achieves stateof-the-art results on the development and test sets for BoolQ, CB, COPA, MultiRC and ReCoRD and the highest average score to date on the SuperGLUE leaderboard. ",
+        "bbox": [
+            173,
+            599,
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+            641
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "G RESULTS ON XNLI ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            661,
+            370,
+            678
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/58d913228c586db2dd08bf356efc92e567d6eac1df4c206b022b0122bac0c02c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>en</td><td>fr</td><td>es</td><td>de</td><td>el</td><td>bg</td><td>ru</td><td>tr</td><td>ar</td><td>vi</td><td>th</td><td>zh</td><td>hi</td><td>SW</td><td>ur</td><td>△</td></tr><tr><td>Machine translation baselines (TRANSLATE-TEST)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>XLM(MLM+TLM)</td><td>85.0</td><td>79.0</td><td>79.5</td><td>78.1</td><td>77.8</td><td>77.6</td><td>75.5</td><td>73.7</td><td>73.7</td><td>70.8</td><td>70.4</td><td>73.6</td><td>69.0</td><td>64.7</td><td>65.1</td><td>74.2</td></tr><tr><td>XLM-en</td><td>88.8</td><td>81.4</td><td>82.3</td><td>80.1</td><td>80.3</td><td>80.9</td><td>76.2</td><td>76.0</td><td>75.4</td><td>72.0</td><td>71.9</td><td>75.6</td><td>70.0</td><td>65.8</td><td>65.8</td><td>76.2</td></tr><tr><td>RoBERTa</td><td>91.3</td><td>82.9</td><td>84.3</td><td>81.2</td><td>81.7</td><td>83.1</td><td>78.3</td><td>76.8</td><td>76.6</td><td>74.2</td><td>74.0</td><td>77.5</td><td>70.9</td><td>66.6</td><td>66.8</td><td>77.8</td></tr></table>",
+        "bbox": [
+            176,
+            698,
+            821,
+            758
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 12: Results on XNLI (Conneau et al., 2018) for $\\mathrm { R o B E R T a _ { L A R G E } }$ in the TRANSLATE-TEST setting. We report macro-averaged accuracy $( \\Delta )$ using the provided English translations of the XNLI test sets. RoBERTa achieves state of the art results on all 15 languages. ",
+        "bbox": [
+            173,
+            773,
+            825,
+            816
+        ],
+        "page_idx": 14
+    }
+]

parse/train/a2Gr9gNFD-J/a2Gr9gNFD-J.md

@@ -0,0 +1,257 @@
+# Characterizing possible failure modes in physics-informed neural networks
+
+Aditi S. Krishnapriyan $^ { * , 1 , 2 }$ , Amir Gholami∗,2, Shandian Zhe3, Robert M. Kirby3, Michael W. Mahoney2,4   
+1Lawrence Berkeley National Laboratory, 2University of California, Berkeley, 3University of Utah, 4International Computer Science Institute   
+{aditik1, amirgh, mahoneymw}@berkeley.edu, {zhe, kirby}@cs.utah.edu
+
+# Abstract
+
+Recent work in scientific machine learning has developed so-called physicsinformed neural network (PINN) models. The typical approach is to incorporate physical domain knowledge as soft constraints on an empirical loss function and use existing machine learning methodologies to train the model. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems. In particular, we analyze several distinct situations of widespread physical interest, including learning differential equations with convection, reaction, and diffusion operators. We provide evidence that the soft regularization in PINNs, which involves PDE-based differential operators, can introduce a number of subtle problems, including making the problem more ill-conditioned. Importantly, we show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN’s setup makes the loss landscape very hard to optimize. We then describe two promising solutions to address these failure modes. The first approach is to use curriculum regularization, where the PINN’s loss term starts from a simple PDE regularization, and becomes progressively more complex as the NN gets trained. The second approach is to pose the problem as a sequence-to-sequence learning task, rather than learning to predict the entire space-time at once. Extensive testing shows that we can achieve up to 1-2 orders of magnitude lower error with these methods as compared to regular PINN training.
+
+# 1 Introduction
+
+Partial differential equations (PDEs) are commonly used to describe different phenomena in science and engineering. These PDEs are often derived by starting from governing first principles (e.g., conservation of mass or energy). It is typically not possible to find analytical solutions to these PDEs for many real-world settings. Thus, many different numerical methods (e.g., the finite element method [44], pseudo-spectral methods [9], etc.) have been introduced to approximate their solutions/behavior. However, these PDEs can be quite complex for several settings (e.g., turbulence simulations), and numerical integration techniques, which typically update and improve a candidate solution iteratively until convergence, are often quite computationally expensive. Motivated by this—as well as the increasing quantities of data available in many scientific and engineering applications—there has been recent interest in developing machine learning (ML) approaches to find the solution of the underlying PDEs (and/or work in tandem with numerical solutions). As a result, the area of Scientific Machine Learning (SciML)—which aims to couple traditional scientific mechanistic modeling (typically, differential equations) with data-driven ML methodologies (most recently, neural network training)—has emerged. In this vein, there have been a number of ML approaches to incorporate scientific knowledge into such problems while keeping the automatic, data-driven estimates of the solution [2, 17, 33, 39].
+
+A recent line of work involves Physics-Informed Neural Network (PINN) models, which aim to incorporate physical domain knowledge as soft constraints on an empirical loss function, that is then optimized using existing ML training methodologies. To some degree, PINNs are an example of “grafting together” domain-driven models and data-driven methodologies. However, there are important subtleties with this, and we identify several possible failure modes with a naive approach. We then illustrate possible directions for addressing these failure modes.
+
+Background and problem overview. Many of the problems with a PDE constraint fit the following abstraction:
+
+$$
+\mathcal { F } ( u ( x , t ) ) = 0 , \qquad x \in \Omega \subset \mathbb { R } ^ { d } , \quad t \in [ 0 , T ] ,
+$$
+
+where $\mathcal { F }$ is a differential operator representing the PDE, $u ( x , t )$ is the state variable (i.e., parameter of interest), $x / t$ denote space/time, $T$ is the time horizon, and $\Omega$ is the spatial domain. Since $\mathcal { F }$ is a differential operator, in general one must specify appropriate boundary and/or initial conditions to ensure the existence/uniqueness of a solution to Eq. 1. In the context of PDEs, $\mathcal { F }$ can be taxonomized into a parabolic, hyperbolic, or elliptic differential operator [23]. Quintessential examples of $\mathcal { F }$ include: the convection equation (a hyperbolic PDE), where $u ( x , t )$ could model fluid movement, e.g., air or some liquid, over space and time; the diffusion equation (a parabolic PDE), where $u ( x , t )$ could model the temperature distribution over space and time; and the Laplace equation (an elliptic PDE), where $u ( x )$ could model a steady-state diffusion equation, in the limit as $t \to \infty$ .
+
+One possible data-driven approach is to incorporate domain information by applying Eq. 1 as a “hard constraint” when training a NN on the data. This can be formulated as the following constrained optimization problem,
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } ( u ) \quad \mathrm { s . t . } \quad \mathcal { F } ( u ) = 0 ,
+$$
+
+where $\mathcal { L } ( u )$ is the data-fit term (including initial/boundary conditions), and where $\mathcal { F }$ is a constraint on the residual of the PDE system under consideration (i.e., the “physics” knowledge in the equation itself). As mentioned before, for many practical use cases, it is not possible to derive closed form solutions for these problems, and it is often quite difficult to solve problems of the form of Eq. 2, with $\mathcal { F } ( u )$ as a hard constraint.
+
+Another (related but different) data-driven approach is to impose the constraint as a “soft constraint” on the outputs of the NN model,
+
+$$
+\begin{array} { r } { \operatorname* { m i n } _ { \theta } \mathcal { L } ( u ) + \lambda _ { \mathcal { F } } \mathcal { F } ( u ) , } \\ { \mathcal { L } ( u ) = \mathcal { L } _ { u _ { 0 } } + \mathcal { L } _ { u _ { b } } . } \end{array}
+$$
+
+Here, $\mathcal { L } _ { u _ { 0 } }$ and $\mathcal { L } _ { u _ { b } }$ measure the misfit of the NN prediction and the initial/boundary conditions (which are pre-specified/given as input to the problem), and $\theta$ denotes the NN parameters (which takes $( x , t )$ , and possibly other quantities, as inputs and then outputs $u ( x , t ) )$ . Furthermore, $\lambda _ { \mathcal { F } }$ is a regularization parameter that controls the emphasis on the PDE based residual (which we ideally want to be zero). The goal is then to use ML methodologies (stochastic optimization, etc.) to train this NN model to minimize the loss in Eq. 3. In particular, the NN is trained to minimize this modified loss function, where the modification is to penalize the violations of $\mathcal { F } ( u )$ for some $\lambda _ { \mathcal { F } } \geq 0$ However, even with a large training dataset, this approach does not guarantee that the NN will obey the conservation/governing equations in the constraint Eq. 1. In many SciML problems, these sorts of constraints on the system matter, as they correspond to physical mechanisms of the system. For example, if the conservation of energy equation is only approximately satisfied, then the system being simulated may behave qualitatively differently or even result in unrealistic solutions.
+
+We should also note that this approach of incorporating physics-based regularization, where the regularization constraint, $\mathcal { L } _ { \mathcal { F } }$ , corresponds to a differential operator, is very different than incorporating much simpler norm-based regularization (such as $L _ { 1 }$ or $L _ { 2 }$ regularization), as is common in ML more generally. Here, the regularization operator, $\mathcal { L } _ { \mathcal { F } }$ is non-trivially structured—it involves a differential operator that could actually be ill-conditioned, and it does not correspond to a nice convex set (as does a norm ball). Moreover, $\mathcal { L } _ { \mathcal { F } }$ corresponds to actual physical quantities, and there is often an important distinction between satisfying the constraint exactly versus satisfying the constraint approximately (the soft constraint approach doing only the latter).
+
+Main contributions. The contributions of this paper are as follows:
+
+• We analyze PINN models on simple, yet physically relevant, problems of convection, reaction, and reaction-diffusion. We find that the vanilla/regular PINN approach only works for very easy parameter regimes (i.e., small PDE coefficients), but that it fails to learn relevant physics in even moderately more challenging physical regimes, even for problems that have simple closed-form analytical solutions. For many cases, the vanilla PINN approach achieves almost $100 \%$ error, as compared to the ground truth solution, even after extensive hyperparameter tuning. (See $\ S 3$ for details.) We analyze the loss landscape of trained PINN models and find that adding/increasing the PDE-based soft constraint regularization ( $\mathcal { L } _ { \mathcal { F } }$ in Eq. 3) makes it more complex and harder to optimize, especially for cases with non-trivial coefficients. We also study how the loss landscape changes as the regularization parameter $( \lambda _ { \mathcal { F } } )$ is changed. We find that reducing the regularization parameter can help alleviate the complexity of the loss landscape, but this in turn leads to poor solutions with high errors that do not satisfy the PDE/constraint. (See $\ S 4$ for details.)   
+We demonstrate that the NN architecture has the capacity/expressivity to find a good solution, thereby showing that these problems are not due to the limited capacity of the NN architecture. Instead, we argue that the failure is due to optimization difficulties associated with the PINN’s soft PDE constraint. (See $\ S 5$ for details.)   
+• We propose two paths forward to address these failure modes through (i) curriculum regularization and (ii) posing the learning problem as a sequence-to-sequence learning task. First, in curriculum regularization, we start by imposing the PDE constraint $( \mathcal { L } _ { \mathcal { F } } )$ with small coefficients, which are progressively increased to the target problem’s settings as the model gets trained. This gives the NN an opportunity to first train with easier constraints, before it is exposed to the target constraint which could be hard to optimize from the beginning. Second, we show that changing the learning problem to a sequence-to-sequence learning problem can reduce the PINN error, again without any change to the NN architecture. In this setup, the NN is trained on a time segment, instead of the full space-time, which could be more difficult to learn. The task is then to predict the solution and reduce the loss only over smaller time segments. We extensively test both approaches and show that they can reduce the error by up to 1-2 orders of magnitude as compared to regular PINN training, and in many cases can better capture “sharp” features in the solution. (See $\ S 5$ for details.)   
+• We have open sourced our framework [26] which is built on top of PyTorch both to help
+
+with reproducibility and also to enable other researchers to extend the results.
+
+# 2 Related work
+
+There is a large body of related work, and here we briefly discuss the most related lines of work.
+
+Machine learning and PDEs. ML approaches for PDE problems have been increasing rapidly in recent years [13, 19]. A number of tools and methodologies now exist to solve scientific problems by combining ML and domain insights [14, 20, 27, 28, 38]. As mentioned earlier, a popular approach to combine ML and physical knowledge is to include aspects of the PDE term as part of the optimization process via regularization. A notable aspect of such an approach is that the NN can be trained only on data that comes from the governing equation(s) itself (though additional data can be included as well, if available), i.e., with a relatively small amount of data. This has garnered interest and shown successful results in a wide variety of science and engineering problems and applications [3, 11, 16, 29–31, 43].
+
+However, there have also been issues observed with this formulation. For example, it did not work well for stiff ordinary differential equations (ODEs) describing chemical kinetics [15], for certain heterogeneous media problems [7], or for certain fluid flow problems [10]. Furthermore, PINN models have been analyzed in the context of neural tangent kernels (i.e., towards the infinite width limit) to study their convergence [36, 37]. This work found some cases where the model failed (such as when the target function exhibits “high frequency features”) and showed some preliminary solutions via the lens of the neural tangent kernel. It has been argued that some of these problems may be due an imbalance in back-propagated gradients in the loss function during training, and a learning-rate annealing scheme has been proposed to mitigate this [35].
+
+Physical priors and constraints in NNs. Imposing physical priors and constraints on NN systems is common in SciML problems, as a way to try to enforce a property of interest. This idea has been introduced in different forms in the past (for instance [5, 18, 25, 28, 32]). Some approaches have focused on embedding specialized physical constraints into NNs, such as conservation of energy or momentum [4, 12] or multiscale features [34]. While methods focusing on constraining the output of the NN are more common, it is difficult to enforce such constraints exactly in ML settings. Previous work has tried to impose hard constraints in ML (both within the context of SciML and otherwise) [6, 21, 22, 24, 40], although this can be computationally expensive, and does not guarantee better results or convergence.
+
+# 3 Possible failure modes for physics-informed neural networks
+
+In this section, we highlight several examples where the PINN formulation defined in Eq. 3 does not predict the solution well. We first demonstrate this with two different types of simple, canonical PDE/ODE systems which have simple analytical solutions: convection ( §3.1), and reaction ( §A). We then also include a diffusion component by looking at the reaction-diffusion problem ( $\ S 3 . 2 )$ . Note that the convection problem has a linear PDE constraint, and reaction/reaction-diffusion problems both have non-linear PDE terms.2 We show that PINNs can only learn simple problems with very small parameter values (e.g., small convection or reaction coefficients). We demonstrate that these models fail to learn the relevant physical phenomena for non-trivial cases (e.g., relatively larger coefficients). As we will see, while adding the physical constraint as a soft regularization may be easier to deploy and optimize with existing unconstrained optimization methods, this approach does come with trade-offs, including that in many cases the optimization problem becomes much more difficult to solve.
+
+Experiment setup. We study both linear and non-linear PDEs/ODEs, and we vary the convection, reaction, and diffusion coefficients for each problem (hereafter, we refer to these as PDE coefficients). For each problem, we aim to minimize the loss function in Eq. 3. We use a 4-layer fully-connected NN with 50 neurons per layer, a hyperbolic tangent activation function, and randomly sample collocation points $( x , t )$ on the domain. Furthermore, all the systems that we consider have periodic boundary conditions. We enforce this through an extra term in the loss function that takes the difference between the predicted NN solution at each boundary. We train this network using the L-BFGS optimizer and sweep over learning rates from $1 \mathrm { e } { - 4 }$ to 2.0.3 After training the PINN, we measure the $L _ { 2 }$ relative and absolute errors between the PINN’s predicted solution and the analytical solution. The $L _ { 2 }$ relative error is $\frac { 1 } { N } \sum _ { i = 0 } ^ { N } \frac { | | \hat { u } - u | | _ { 2 } } { | | u | | _ { 2 } }$ and the absolute error is $\frac { 1 } { N } \sum _ { i = 0 } ^ { N } \left| \right| \hat { u } - u \left| \right| _ { 2 }$ $N$ is the number of evaluation grid points, $\hat { u }$ is the predicted solution by the PINN, and $u$ is the true solution. For all cases, we run models at least ten times with different preset random seeds, and we average the relative and absolute errors in $u ( x , t )$ . For each loss function, $\hat { u }$ is the output of the NN and shorthand for $\hat { u } = N N ( \theta , x , t )$ .
+
+![](images/9c63850a96e8a066cd543dc6fc67f2baa2b89525b2b3ce6c1afb24d8f5b696cf.jpg)  
+Figure 1: Prediction error for 1D convection ( §3.1) problem, when $\beta$ is changed. The PINN has difficulty predicting the solution past a certain timestep, but is able to fit the boundary conditions. Additional figures for different $\beta$ values can be seen in Fig. C.1.
+
+# 3.1 Learning convection
+
+Problem formulation. We first consider a one-dimensional convection problem, a hyperbolic PDE which is commonly used to model transport phenomena:
+
+$$
+\begin{array} { c } { \displaystyle { \frac { \partial u } { \partial t } + \beta \frac { \partial u } { \partial x } = 0 , \quad x \in \Omega , t \in [ 0 , T ] , } } \\ { \displaystyle { u ( x , 0 ) = h ( x ) , \quad x \in \Omega . } } \end{array}
+$$
+
+Here, $\beta$ is the convection coefficient and $h ( x )$ is the initial condition. For constant $\beta$ and periodic boundary conditions, this problem has a simple analytical solution:
+
+$$
+u _ { \mathrm { { a n a l y t i c a l } } } ( x , t ) = F ^ { - 1 } \big ( F ( h ( x ) ) e ^ { - i \beta k t } \big ) ,
+$$
+
+where $F$ is the Fourier transform, $i = \sqrt { - 1 }$ , and $k$ denotes frequency in the Fourier domain. The general loss function for this problem (corresponding to Eq. 3) is
+
+$$
+\mathcal { L } ( \theta ) = \frac { 1 } { { { N _ { u } } } } \sum _ { i = 1 } ^ { { N _ { u } } } { \left( { { \hat { u } } - u _ { 0 } ^ { i } } \right) ^ { 2 } } + \frac { 1 } { { { N _ { f } } } } \sum _ { i = 1 } ^ { { N _ { f } } } { \lambda _ { i } \Big ( \frac { { \partial \hat { u } } } { { \partial t } } + \beta \frac { { \partial \hat { u } } } { { \partial x } } \Big ) ^ { 2 } } + \mathcal { L } _ { B } ,
+$$
+
+where $\hat { u } = N N ( \theta , x , t )$ is the output of the NN, and $\mathcal { L } _ { B }$ is the boundary loss. For periodic boundary conditions with $\Omega = [ 0 , 2 \pi )$ , this loss is:
+
+$$
+\mathcal { L } _ { B } = \frac { 1 } { N _ { b } } \sum _ { i = 1 } ^ { N _ { b } } \Big ( \hat { u } ( \theta , 0 , t ) - \hat { u } ( \theta , 2 \pi , t ) \Big ) ^ { 2 } .
+$$
+
+We use the following simple initial and periodic boundary conditions:
+
+$$
+\begin{array} { c } { { u ( x , 0 ) = s i n ( x ) , } } \\ { { u ( 0 , t ) = u ( 2 \pi , t ) . } } \end{array}
+$$
+
+Observations. We apply the PINN’s soft regularization to this problem, and we optimize the loss function in Eq. 7. After training, we measure the relative and absolute errors between the PINN’s predicted solution and the analytical solution, as reported in Fig. 1(a). As one can see, the PINN is only able to achieve good solutions for small values of convection coefficient, and it fails when $\beta$ becomes larger, reaching a relative error of almost $100 \%$ for $\beta > 1 0$ . We also provide visualization of the exact and PINN solution in Fig. 1(b-c). One can clearly see that the PINN is unable to learn the solution. As we will later show, the NN architecture does have enough capacity to find the solution, but the training/optimization problem is very difficult to solve with PINNs (and importantly, it may require extensive hyperparameter tuning which is often not feasible in practice).
+
+# 3.2 Learning reaction-diffusion
+
+Problem formulation. We next look at a reaction-diffusion system, where we add a diffusion operator to the reaction equation discussed above. Note that for pure diffusion, the solution dissipates
+
+![](images/507827b75469e5016d62a0516c9538c292c5ada8d87079292006fee8155453bc.jpg)  
+Figure 2: Prediction error for $\mathbfcal { m }$ reaction-diffusion $( \ \ S 3 . 2 )$ problem. We can clearly see that the PINN has difficulty predicting the solution (especially the “sharpness” of the solution) and is unable to capture the correct behavior. Additional figures for different $\nu$ values can be seen in Fig. D.1.
+
+to a steady-state of uniform/constant distribution, which may be trivial to learn. Therefore, we consider studying the reaction-diffusion system:
+
+$$
+\begin{array} { c } { \displaystyle { \frac { \partial u } { \partial t } - \nu \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \rho u ( 1 - u ) = 0 , \quad x \in \Omega , t \in ( 0 , T ] , } } \\ { \displaystyle { u ( x , 0 ) = h ( x ) , \quad x \in \Omega . } } \end{array}
+$$
+
+Here, $\nu$ $( \nu > 0 )$ ) is the diffusion coefficient. The solution of such a system can be solved for via Strang splitting, i.e., splitting the equation into two separate models (a reaction component and a diffusion component):
+
+$$
+\begin{array} { r } { \frac { d u } { d t } = \rho u ( 1 - u ) } \\ { \frac { d u } { d t } = \nu \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } . } \end{array}
+$$
+
+For each timestep, we can solve the reaction equation through Eq. 15 (in $\ S \mathrm { A }$ ). The diffusion equation has the following analytical solution:
+
+$$
+u _ { \mathrm { a n a l y t i c a l } } ( x , t ) = F ^ { - 1 } \big ( F ( u ( x , t = t ^ { n } ) ) e ^ { - \nu k ^ { 2 } t } \big ) ,
+$$
+
+where $u ( x , t = t ^ { n } )$ is the solution at the $n ^ { t h }$ time step. We solve the reaction equation for each timestep, and then use the reaction solution as the initial condition to solve the diffusion component and get the final solution.
+
+The general loss function for this problem is,
+
+$$
+\begin{array} { r } { \mathcal { L } ( \boldsymbol { \theta } ) = \displaystyle \frac { 1 } { N _ { u } } \sum _ { i = 1 } ^ { N _ { u } } \Big ( \hat { u } - u _ { 0 } ^ { i } \Big ) ^ { 2 } + } \\ { \displaystyle \frac { 1 } { N _ { f } } \sum _ { i = 1 } ^ { N _ { f } } \lambda _ { i } \Big ( \frac { \partial \hat { u } } { \partial t } - \nu \frac { \partial ^ { 2 } \hat { u } } { \partial x ^ { 2 } } - \rho \hat { u } ( 1 - \hat { u } ) \Big ) ^ { 2 } + \mathcal { L } _ { B } , } \end{array}
+$$
+
+where $\mathcal { L } _ { B }$ is the boundary loss. Similar to the previous example, periodic boundary conditions can be enforced by including $\mathcal { L } _ { B }$ from Eq. 8 as an extra term in the loss.
+
+Observations. Similar to the previous case, we can see that the PINN also fails to learn reactiondiffusion. We illustrate a case in Fig. 2 with $\rho = 5$ , when $\nu = 5$ . The PINN achieves a high relative error of $93 \%$ . Here, we can clearly see that the PINN is unable to capture either the reaction or diffusion component. Additional figures for different $\nu$ values can be seen in Fig. D.1. In particular, for $\nu = 2$ the PINN achieves a relative error of $50 \%$ . Here, we see that it is unable to capture the “sharper” transitions, though it can predict the center of the solution a little better.
+
+# 4 Diagnosing possible failure modes for physics-informed NNs
+
+Thus far, we have shown that PINNs can result in high errors even for simple physical regimes, in particular for PDEs/ODEs with non-trivial convection/reaction/diffusion coefficients. Here, we demonstrate that one of the underlying reasons for this arises due to the PDE-based soft constraint of $\mathcal { L } _ { \mathcal { F } }$ , which makes the loss landscape difficult to optimize. We first (in $\ S 4 . 1$ ) analyze the loss landscape to illustrate how increasing this soft regularization can lead to more complex loss landscapes, thus leading to optimization difficulties. We then (in $\ S _ { \mathbf { B } }$ ) demonstrate how this is related to regularizing with differential operators, which can result in ill-conditioning.
+
+Figure 3: Loss landscapes for varying values of $\beta _ { z }$ , for the $\mathbfcal { m }$ convection example in $\ S 3 . I$ . The loss landscape is more smooth at low $\beta$ , and it becomes increasingly more complex as $\beta$ increases, which can make the optimization problem more difficult. In particular, at higher $\beta$ , the optimizer gets stuck in a certain regime. These results support that adding the PDE soft regularization term results in a more complex optimization loss landscape.   
+
+<table><tr><td>200 150 100 50 1.0 0.5 0.5 1.0-1.0 (a) β = 1.0 β</td><td>(b) β = 10.0</td><td colspan="3">(c) β = 20.0</td><td>(e) β = 40.0</td></tr><tr><td>Relative error</td><td></td><td>10 1.08× 10-2</td><td>20</td><td>30</td><td>40</td></tr><tr><td></td><td>7.84 × 10-3 3.17× 10-3</td><td>6.03×10-3</td><td>7.50× 10-1 4.32×10-1</td><td>8.97×10-1 5.42×10-1</td><td>9.61 ×10-1</td></tr></table>
+
+# 4.1 Soft PDE regularization and optimization difficulties
+
+Here, we analyze how the loss landscape changes for different regimes for the convection problem in $\ S 3 . 1$ with/without the soft regularization in PINNs. We show that adding the soft regularization can actually make the problem harder to optimize, i.e., the regularization leads to less smooth loss landscapes. For all the experiments, we plot the loss landscape by perturbing the (trained) model across the first two dominant Hessian eigenvectors and computing the corresponding loss values. This tends to be more informative than perturbing the model parameters in random directions [41, 42].
+
+Figure 3 shows the loss landscape for the convection problem (discussed in $\ S 3 . 1$ ), for different $\beta$ values. Interestingly, the loss landscape at a relatively low $\beta = 1$ is rather smooth, but increasing $\beta$ further results in a complex and non-symmetric loss landscape. It is also evident that the optimizer has gotten stuck in a local minima with a very high loss function for large $\beta$ values.
+
+Finally, we study the impact of changing the weight/multiplier for the soft regularization term (i.e., the $\lambda$ parameter in Eq. 3), which can be relevant in improving PINN performance [35]. While we find that tuning $\lambda$ can help change the error, it cannot resolve the problem, as shown in Fig. E.1. Note that as the regularization parameter is increased, the loss landscape becomes increasingly more complex and harder to optimize (additionally, see the $\mathbf { Z }$ -axis scale).
+
+# 5 Expressivity versus optimization difficulty
+
+In this section, we first show that the failure modes we observed are not necessarily due to the specific NN architecture that we used in our experiments. In particular, we show that the NN model does have the expressivity/capacity to learn the convection/reaction/diffusion coefficient cases where the vanilla PINN method fails. Additionally, in the process of demonstrating this, we also describe two methods that lead to significantly lower error rates. In particular, we show that changing the learning paradigm to curriculum regularization can make the optimization problem easier to solve (as discussed in $\ S 5 . 1$ ). Second, we show that posing the problem as sequence-to-sequence learning may lead to better results than learning the entire state-space at once (as discussed in $\ S 5 . 2 )$ .
+
+# 5.1 Curriculum PINN Regularization
+
+One may contend that the failure modes shown in $\ S 3$ may be because the NN does not have enough capacity. Here, we show that this is not the underlying reason. To do so, we devise a “curriculum regularization” method to warm start the NN training by finding a good initialization for the weights. Instead of training the PINN to learn the solution right away for cases with higher $\beta / \rho$ , we start by training the PINN on lower $\beta / \rho$ (easier for the PINN to learn) and then gradually move to training the PINN on higher $\beta / \rho$ , respectively. We test these results for the examples in $\ S 3 . 1$ and $\ S \mathrm { A }$ . This is somewhat analogous to curriculum learning in ML [1], but applied by progressively making the PDE/ODE harder to solve.
+
+![](images/cf358f24c3f98df7fa0c6d924e0bd2aa722df0a24acad8197080983fb9d08d3c.jpg)  
+Figure 4: Schematic outlining curriculum regularization and example result for 1D convection from $\ S 3 . I$ The training procedure for regular PINNs training versus curriculum PINN training for the convection example in $\ S 3 . I$ . The regular PINN training only involves training at $\beta = 3 0$ , while curriculum regularization starts at a lower $\beta$ , trains a model, and then uses the weights of this model to reinitialize the NN for training the next $\beta$ . The curriculum training approach is able to do significantly better (by almost two orders of magnitude).
+
+Table 1: Training the PINN gradually on more difficult problems improves performance. 1D convection example in $\ \ S 3 . 1 .$ . The curriculum training approach achieves significantly better errors.   
+
+<table><tr><td rowspan=1 colspan=3>Regular PINN（</td><td rowspan=1 colspan=1> Curriculum training</td></tr><tr><td rowspan=2 colspan=1>1D convection: β = 20</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>7.50 × 10-1</td><td rowspan=1 colspan=1>9.84 × 10-3</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>4.32×10-1</td><td rowspan=1 colspan=1>5.42 ×10-3</td></tr><tr><td rowspan=2 colspan=1>1D convection: β = 30</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>8.97×10-1</td><td rowspan=1 colspan=1>2.02 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>5.42×10-1</td><td rowspan=1 colspan=1>1.10 ×10-2</td></tr><tr><td rowspan=2 colspan=1>1D convection: β = 40</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>9.61 × 10-1</td><td rowspan=1 colspan=1>5.33 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>5.82 ×10-1</td><td rowspan=1 colspan=1>2.69 ×10-2</td></tr></table>
+
+Figure 4 shows the training procedure for an example convection case $( \ S 3 . 1 )$ with $\beta = 3 0$ . As Fig. 4(c) shows, the curriculum regularization approach results in a much more accurate solution than regular PINN training. With curriculum regularization, the relative error is almost two orders of magnitude lower. Additionally, this is true across all the other regimes that we found regular PINNs to fail, as shown in Tab. 1. In Fig. E.2, we also show that curriculum regularization not only decreases error significantly, but also decreases the variance of the error. In Fig. E.3, we see that curriculum regularization results in a much smoother loss landscape as compared to regular PINN training.
+
+Curriculum regularization also works well for the reaction example in $\ S \mathbf { A }$ . In this case, we start by training with a low $\rho$ value (reaction coefficient), and then increase gradually to higher $\rho$ values. The results can be seen in Fig. E.4. We can see that the error is $0 . 1 \textrm { - } 0 . 6 $ orders of magnitude lower for $\rho = 2 - 4$ (when the regular PINN error is not as high), and then greatly decreases error by 1-2 orders of magnitude for $\rho = 5 - 1 0$ . As we discussed before, PINN has difficulty in learning sharp features for high values of $\rho$ . However, the curriculum regularization overcomes this, even for $\rho = 1 0$ , as seen in Fig. E.4(c).
+
+# 5.2 Sequence-to-sequence learning vs learning the entire space-time solution
+
+The original PINN approach of [28] trains the NN model to predict the entire space-time at once (i.e., predict $u$ for all locations and time points). In certain cases, this can be more difficult to learn. Here, we demonstrate that it may be better to pose the problem as a sequence-to-sequence (seq2seq) learning task, where the NN learns to predict the solution at the next time step, instead of all times.
+
+![](images/73622db931290c5644d1b71ac0064c8e76958e88111b960bdfa0e01cfc5f3ca5.jpg)  
+6 Figure 5: Schematic outlining seq2seq learning. In contrast to regular PINN training, the solution in seq2seq learning is predicted for only one $\Delta t$ step at a time. Then, the predicted solution at $t = \Delta t$ is used as the initial condition for the next segment. To allow fair comparison, we keep the total number of 5 collocation points to be exactly the same in either approach. That is, we do not increase the number of collocation points for seq2seq learning in the right, and keep it to be the same as in the corresponding segment in the left figure.
+
+Table 2: Predicting the entire state space versus discretizing the state space (i.e., seq2seq learning) for $\mathbfcal { m }$ reaction-diffusion ( $\ S 3 . 2 )$ . The seq2seq learning achieves lower error for both $\Delta t = 0 . 0 5$ and $\Delta t = 0 . 1$ , in comparison to the PINN’s approach of predicting the entire state space at once.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1> Entire state space</td><td rowspan=1 colspan=1>△t = 0.05</td><td rowspan=1 colspan=1>△t = 0.1</td></tr><tr><td rowspan=2 colspan=1>v=2,p=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>5.07× 10-1</td><td rowspan=1 colspan=1>2.04 × 10-2</td><td rowspan=1 colspan=1>1.18 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>2.70 ×10-1</td><td rowspan=1 colspan=1>1.06 × 10-2</td><td rowspan=1 colspan=1>6.41 × 10-3</td></tr><tr><td rowspan=1 colspan=1>v=3,p=5</td><td rowspan=1 colspan=1> Relative error Absolute error</td><td rowspan=1 colspan=1>7.98 × 10-14.79 × 10-1</td><td rowspan=1 colspan=1>1.92 × 10-21.01 × 10-2</td><td rowspan=1 colspan=1>1.56 × 10-28.17 × 10-3</td></tr><tr><td rowspan=2 colspan=1>v=4,p=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>8.84×10-1</td><td rowspan=1 colspan=1>2.37 × 10-2</td><td rowspan=1 colspan=1>1.59 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>5.74 × 10-1</td><td rowspan=1 colspan=1>1.15 ×10-2</td><td rowspan=1 colspan=1>8.01 ×10-3</td></tr><tr><td rowspan=1 colspan=1>v=5,ρ=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>9.35 × 10-1</td><td rowspan=1 colspan=1>2.36 × 10-2</td><td rowspan=1 colspan=1>2.39 ×10-2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1> Absolute error</td><td rowspan=1 colspan=1>6.46 × 10-1</td><td rowspan=1 colspan=1>1.09 × 10-2</td><td rowspan=1 colspan=1>1.15 × 10-2</td></tr><tr><td rowspan=1 colspan=1>v=6,p=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>9.60 × 10-1</td><td rowspan=1 colspan=1>2.81 ×10-2</td><td rowspan=1 colspan=1>2.69 × 10-2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1> Absolute error</td><td rowspan=1 colspan=1>6.84 × 10-1</td><td rowspan=1 colspan=1>1.17 × 10-2</td><td rowspan=1 colspan=1>1.28 × 10-2</td></tr></table>
+
+This way, we can use a marching-in-time scheme to predict different sequences/time points. Note that the only data available here is from the PDE itself, i.e., just the initial condition. We take the prediction at $t = \Delta t$ and use this as the initial condition to make a prediction at $t = 2 \Delta t$ , and so on. This is schematically outlined in Fig. 5.
+
+We test this scheme by using the exact same NN architecture as in previous sections, and we report the results in Tab. E.1 for the convection problem of $\ S 3 . 1$ , Tab. E.2 for the reaction problem of $\ S \mathrm { A }$ , and Tab. 2 for the reaction-diffusion problem of $\ S 3 . 2$ . We compare the relative/absolute error when the learning is posed as a seq2seq problem (i.e., predicting the state space with a “time marching scheme” of one timestep prediction at a time) to the PINN approach of predicting the whole state space at once.4
+
+We explore the following cases where the PINN does poorly, varying $\beta , \rho$ , and $\nu$ coefficients:
+
+![](images/00c8d636fffe0f538e982510f226fe7a8d911fc0be41114807f1317d5eaf8d43.jpg)  
+Figure 6: Predicting the entire state space vs seq2seq learning for $\mathbfcal { m }$ reaction-diffusion. The regular PINN is unable to capture the “sharp” and/or diffusive features correctly. However, the seq2seq learning approach is able to capture the correct solution, and achieves almost two orders of magnitude lower error.
+
+1) For 1D convection ( §3.1), higher $\beta$ values from 30-40.
+
+2) For 1D reaction ( $\ S _ { \mathbf { A } } )$ , $\rho$ coefficients from 5-10.
+
+3) For 1D reaction-diffusion ( §3.2), a fixed $\rho = 5$ and $\nu$ coefficients from 2-6.
+
+For these cases, we find that posing the problem as seq2seq learning results in significantly lower error. The difference is particularly striking for the reaction and reaction-diffusion cases, where the seq2seq PINN model decreases error by almost two orders of magnitude. An example case is shown Fig. 6, where the seq2seq approach is able to recover the solution, while regular PINNs does very poorly. Note that this behavior also has analogues with numerical methods used in scientific computing, where space-time problems are typically harder to solve, as compared to time marching methods [8]. Intuitively, since the problem is ill-conditioned, restricting the dimensions is expected to help. Furthermore, the underlying function/mapping of the input to the solution should be much simpler to approximate over a smaller time span, as compared to the full time horizon.
+
+These initial results are promising, and further developments may lead to still better ways of using PINNs and learning PDEs. In particular, using more sophisticated methods to predict timesteps across the state space may provide improved performance, as may including more sophisticated seq2seq approaches and tuning the regularization parameter (i.e., amount of constraint added).
+
+# 6 Conclusions
+
+PINNs—and SciML more generally—hold great promise for expanding the scope of ML methodology to important problems in science and engineering. For these problems, however, integrating ML methods with PDE-based domain-driven constraints as a soft regularization term can lead to subtle and critical issues. In particular, we show that this approach can have fundamental limitations which results in failure modes for learning relevant physics commonly used in different fields of science. To show this, we picked two fundamental PDE problems of diffusion and convection and showed that the PINN only works for very simple cases, failing to learn the relevant physical phenomena for even moderately more challenging regimes. We then analyzed the problem to characterize the underlying reasons why these failures occur. In particular, we studied the PINN loss landscape behavior and found it becomes it becomes increasingly complex for large values of diffusion or convection coefficients, and with/without non-homogeneous forcing. We also discussed that the problem is not necessarily due to the limited capacity of the NN, but that it is partly an optimization problem resulting in the PDE-based soft constraint used in PINNs. Furthermore, we showed that the PINN approach of solving for the entire space-time at once may not be efficient, and instead posing the problem as a sequence-to-sequence learning task can provide lower error rates. Addressing these and related issues will be critical if we hope to go beyond existing cut-and-paste approaches, toward engineering a more intimate connection between scientific methodologies and ML methodologies. This will be needed to deliver on the promise of PINNs and SciML more generally.
+
+# 7 Acknowledgements.
+
+We are thankful to Shashank Subramanian for his feedback and contributions. We also acknowledge helpful discussions with Prof. George Biros, Geoffrey Negiar, and Daniel Rothchild. ASK was supported by Laboratory Directed Research and Development (LDRD) funding under Contract Number DE-AC02-05CH11231 at LBNL and the Alvarez Fellowship in the Computational Research Division at LBNL. AG was supported through funding from Samsung SAIT. MWM would also like to acknowledge the UC Berkeley CLTC, ARO, NSF, and ONR. The UC Berkeley team also acknowledges gracious support from Intel corporation, Intel VLAB, Samsung, Amazon AWS, Google Cloud, Google TPU Research Cloud, and Google Brain (in particular Prof. David Patterson, Dr. Ed Chi, and Jing Li). Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred.
+
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+        "text": "Aditi S. Krishnapriyan $^ { * , 1 , 2 }$ , Amir Gholami∗,2, Shandian Zhe3, Robert M. Kirby3, Michael W. Mahoney2,4   \n1Lawrence Berkeley National Laboratory, 2University of California, Berkeley, 3University of Utah, 4International Computer Science Institute   \n{aditik1, amirgh, mahoneymw}@berkeley.edu, {zhe, kirby}@cs.utah.edu ",
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+        "text": "Abstract ",
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+        "text": "Recent work in scientific machine learning has developed so-called physicsinformed neural network (PINN) models. The typical approach is to incorporate physical domain knowledge as soft constraints on an empirical loss function and use existing machine learning methodologies to train the model. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems. In particular, we analyze several distinct situations of widespread physical interest, including learning differential equations with convection, reaction, and diffusion operators. We provide evidence that the soft regularization in PINNs, which involves PDE-based differential operators, can introduce a number of subtle problems, including making the problem more ill-conditioned. Importantly, we show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN’s setup makes the loss landscape very hard to optimize. We then describe two promising solutions to address these failure modes. The first approach is to use curriculum regularization, where the PINN’s loss term starts from a simple PDE regularization, and becomes progressively more complex as the NN gets trained. The second approach is to pose the problem as a sequence-to-sequence learning task, rather than learning to predict the entire space-time at once. Extensive testing shows that we can achieve up to 1-2 orders of magnitude lower error with these methods as compared to regular PINN training. ",
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+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Partial differential equations (PDEs) are commonly used to describe different phenomena in science and engineering. These PDEs are often derived by starting from governing first principles (e.g., conservation of mass or energy). It is typically not possible to find analytical solutions to these PDEs for many real-world settings. Thus, many different numerical methods (e.g., the finite element method [44], pseudo-spectral methods [9], etc.) have been introduced to approximate their solutions/behavior. However, these PDEs can be quite complex for several settings (e.g., turbulence simulations), and numerical integration techniques, which typically update and improve a candidate solution iteratively until convergence, are often quite computationally expensive. Motivated by this—as well as the increasing quantities of data available in many scientific and engineering applications—there has been recent interest in developing machine learning (ML) approaches to find the solution of the underlying PDEs (and/or work in tandem with numerical solutions). As a result, the area of Scientific Machine Learning (SciML)—which aims to couple traditional scientific mechanistic modeling (typically, differential equations) with data-driven ML methodologies (most recently, neural network training)—has emerged. In this vein, there have been a number of ML approaches to incorporate scientific knowledge into such problems while keeping the automatic, data-driven estimates of the solution [2, 17, 33, 39]. ",
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+        "text": "A recent line of work involves Physics-Informed Neural Network (PINN) models, which aim to incorporate physical domain knowledge as soft constraints on an empirical loss function, that is then optimized using existing ML training methodologies. To some degree, PINNs are an example of “grafting together” domain-driven models and data-driven methodologies. However, there are important subtleties with this, and we identify several possible failure modes with a naive approach. We then illustrate possible directions for addressing these failure modes. ",
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+        "text": "Background and problem overview. Many of the problems with a PDE constraint fit the following abstraction: ",
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+        "text": "$$\n\\mathcal { F } ( u ( x , t ) ) = 0 , \\qquad x \\in \\Omega \\subset \\mathbb { R } ^ { d } , \\quad t \\in [ 0 , T ] ,\n$$",
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+        "text": "where $\\mathcal { F }$ is a differential operator representing the PDE, $u ( x , t )$ is the state variable (i.e., parameter of interest), $x / t$ denote space/time, $T$ is the time horizon, and $\\Omega$ is the spatial domain. Since $\\mathcal { F }$ is a differential operator, in general one must specify appropriate boundary and/or initial conditions to ensure the existence/uniqueness of a solution to Eq. 1. In the context of PDEs, $\\mathcal { F }$ can be taxonomized into a parabolic, hyperbolic, or elliptic differential operator [23]. Quintessential examples of $\\mathcal { F }$ include: the convection equation (a hyperbolic PDE), where $u ( x , t )$ could model fluid movement, e.g., air or some liquid, over space and time; the diffusion equation (a parabolic PDE), where $u ( x , t )$ could model the temperature distribution over space and time; and the Laplace equation (an elliptic PDE), where $u ( x )$ could model a steady-state diffusion equation, in the limit as $t \\to \\infty$ . ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "One possible data-driven approach is to incorporate domain information by applying Eq. 1 as a “hard constraint” when training a NN on the data. This can be formulated as the following constrained optimization problem, ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/4c85243a0a5a1a40753c24de6fe282d1d37f06eb1dbfaca66a391d25eb7a392e.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathcal { L } ( u ) \\quad \\mathrm { s . t . } \\quad \\mathcal { F } ( u ) = 0 ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } ( u )$ is the data-fit term (including initial/boundary conditions), and where $\\mathcal { F }$ is a constraint on the residual of the PDE system under consideration (i.e., the “physics” knowledge in the equation itself). As mentioned before, for many practical use cases, it is not possible to derive closed form solutions for these problems, and it is often quite difficult to solve problems of the form of Eq. 2, with $\\mathcal { F } ( u )$ as a hard constraint. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Another (related but different) data-driven approach is to impose the constraint as a “soft constraint” on the outputs of the NN model, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ba76113bc70bac3b7d0c1ffc2b4ab75229de1781b7eae964396bbc37bb68a696.jpg",
+        "text": "$$\n\\begin{array} { r } { \\operatorname* { m i n } _ { \\theta } \\mathcal { L } ( u ) + \\lambda _ { \\mathcal { F } } \\mathcal { F } ( u ) , } \\\\ { \\mathcal { L } ( u ) = \\mathcal { L } _ { u _ { 0 } } + \\mathcal { L } _ { u _ { b } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, $\\mathcal { L } _ { u _ { 0 } }$ and $\\mathcal { L } _ { u _ { b } }$ measure the misfit of the NN prediction and the initial/boundary conditions (which are pre-specified/given as input to the problem), and $\\theta$ denotes the NN parameters (which takes $( x , t )$ , and possibly other quantities, as inputs and then outputs $u ( x , t ) )$ . Furthermore, $\\lambda _ { \\mathcal { F } }$ is a regularization parameter that controls the emphasis on the PDE based residual (which we ideally want to be zero). The goal is then to use ML methodologies (stochastic optimization, etc.) to train this NN model to minimize the loss in Eq. 3. In particular, the NN is trained to minimize this modified loss function, where the modification is to penalize the violations of $\\mathcal { F } ( u )$ for some $\\lambda _ { \\mathcal { F } } \\geq 0$ However, even with a large training dataset, this approach does not guarantee that the NN will obey the conservation/governing equations in the constraint Eq. 1. In many SciML problems, these sorts of constraints on the system matter, as they correspond to physical mechanisms of the system. For example, if the conservation of energy equation is only approximately satisfied, then the system being simulated may behave qualitatively differently or even result in unrealistic solutions. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We should also note that this approach of incorporating physics-based regularization, where the regularization constraint, $\\mathcal { L } _ { \\mathcal { F } }$ , corresponds to a differential operator, is very different than incorporating much simpler norm-based regularization (such as $L _ { 1 }$ or $L _ { 2 }$ regularization), as is common in ML more generally. Here, the regularization operator, $\\mathcal { L } _ { \\mathcal { F } }$ is non-trivially structured—it involves a differential operator that could actually be ill-conditioned, and it does not correspond to a nice convex set (as does a norm ball). Moreover, $\\mathcal { L } _ { \\mathcal { F } }$ corresponds to actual physical quantities, and there is often an important distinction between satisfying the constraint exactly versus satisfying the constraint approximately (the soft constraint approach doing only the latter). ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Main contributions. The contributions of this paper are as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• We analyze PINN models on simple, yet physically relevant, problems of convection, reaction, and reaction-diffusion. We find that the vanilla/regular PINN approach only works for very easy parameter regimes (i.e., small PDE coefficients), but that it fails to learn relevant physics in even moderately more challenging physical regimes, even for problems that have simple closed-form analytical solutions. For many cases, the vanilla PINN approach achieves almost $100 \\%$ error, as compared to the ground truth solution, even after extensive hyperparameter tuning. (See $\\ S 3$ for details.) We analyze the loss landscape of trained PINN models and find that adding/increasing the PDE-based soft constraint regularization ( $\\mathcal { L } _ { \\mathcal { F } }$ in Eq. 3) makes it more complex and harder to optimize, especially for cases with non-trivial coefficients. We also study how the loss landscape changes as the regularization parameter $( \\lambda _ { \\mathcal { F } } )$ is changed. We find that reducing the regularization parameter can help alleviate the complexity of the loss landscape, but this in turn leads to poor solutions with high errors that do not satisfy the PDE/constraint. (See $\\ S 4$ for details.)   \nWe demonstrate that the NN architecture has the capacity/expressivity to find a good solution, thereby showing that these problems are not due to the limited capacity of the NN architecture. Instead, we argue that the failure is due to optimization difficulties associated with the PINN’s soft PDE constraint. (See $\\ S 5$ for details.)   \n• We propose two paths forward to address these failure modes through (i) curriculum regularization and (ii) posing the learning problem as a sequence-to-sequence learning task. First, in curriculum regularization, we start by imposing the PDE constraint $( \\mathcal { L } _ { \\mathcal { F } } )$ with small coefficients, which are progressively increased to the target problem’s settings as the model gets trained. This gives the NN an opportunity to first train with easier constraints, before it is exposed to the target constraint which could be hard to optimize from the beginning. Second, we show that changing the learning problem to a sequence-to-sequence learning problem can reduce the PINN error, again without any change to the NN architecture. In this setup, the NN is trained on a time segment, instead of the full space-time, which could be more difficult to learn. The task is then to predict the solution and reduce the loss only over smaller time segments. We extensively test both approaches and show that they can reduce the error by up to 1-2 orders of magnitude as compared to regular PINN training, and in many cases can better capture “sharp” features in the solution. (See $\\ S 5$ for details.)   \n• We have open sourced our framework [26] which is built on top of PyTorch both to help ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "with reproducibility and also to enable other researchers to extend the results. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "There is a large body of related work, and here we briefly discuss the most related lines of work. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Machine learning and PDEs. ML approaches for PDE problems have been increasing rapidly in recent years [13, 19]. A number of tools and methodologies now exist to solve scientific problems by combining ML and domain insights [14, 20, 27, 28, 38]. As mentioned earlier, a popular approach to combine ML and physical knowledge is to include aspects of the PDE term as part of the optimization process via regularization. A notable aspect of such an approach is that the NN can be trained only on data that comes from the governing equation(s) itself (though additional data can be included as well, if available), i.e., with a relatively small amount of data. This has garnered interest and shown successful results in a wide variety of science and engineering problems and applications [3, 11, 16, 29–31, 43]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, there have also been issues observed with this formulation. For example, it did not work well for stiff ordinary differential equations (ODEs) describing chemical kinetics [15], for certain heterogeneous media problems [7], or for certain fluid flow problems [10]. Furthermore, PINN models have been analyzed in the context of neural tangent kernels (i.e., towards the infinite width limit) to study their convergence [36, 37]. This work found some cases where the model failed (such as when the target function exhibits “high frequency features”) and showed some preliminary solutions via the lens of the neural tangent kernel. It has been argued that some of these problems may be due an imbalance in back-propagated gradients in the loss function during training, and a learning-rate annealing scheme has been proposed to mitigate this [35]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Physical priors and constraints in NNs. Imposing physical priors and constraints on NN systems is common in SciML problems, as a way to try to enforce a property of interest. This idea has been introduced in different forms in the past (for instance [5, 18, 25, 28, 32]). Some approaches have focused on embedding specialized physical constraints into NNs, such as conservation of energy or momentum [4, 12] or multiscale features [34]. While methods focusing on constraining the output of the NN are more common, it is difficult to enforce such constraints exactly in ML settings. Previous work has tried to impose hard constraints in ML (both within the context of SciML and otherwise) [6, 21, 22, 24, 40], although this can be computationally expensive, and does not guarantee better results or convergence. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 Possible failure modes for physics-informed neural networks ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In this section, we highlight several examples where the PINN formulation defined in Eq. 3 does not predict the solution well. We first demonstrate this with two different types of simple, canonical PDE/ODE systems which have simple analytical solutions: convection ( §3.1), and reaction ( §A). We then also include a diffusion component by looking at the reaction-diffusion problem ( $\\ S 3 . 2 )$ . Note that the convection problem has a linear PDE constraint, and reaction/reaction-diffusion problems both have non-linear PDE terms.2 We show that PINNs can only learn simple problems with very small parameter values (e.g., small convection or reaction coefficients). We demonstrate that these models fail to learn the relevant physical phenomena for non-trivial cases (e.g., relatively larger coefficients). As we will see, while adding the physical constraint as a soft regularization may be easier to deploy and optimize with existing unconstrained optimization methods, this approach does come with trade-offs, including that in many cases the optimization problem becomes much more difficult to solve. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Experiment setup. We study both linear and non-linear PDEs/ODEs, and we vary the convection, reaction, and diffusion coefficients for each problem (hereafter, we refer to these as PDE coefficients). For each problem, we aim to minimize the loss function in Eq. 3. We use a 4-layer fully-connected NN with 50 neurons per layer, a hyperbolic tangent activation function, and randomly sample collocation points $( x , t )$ on the domain. Furthermore, all the systems that we consider have periodic boundary conditions. We enforce this through an extra term in the loss function that takes the difference between the predicted NN solution at each boundary. We train this network using the L-BFGS optimizer and sweep over learning rates from $1 \\mathrm { e } { - 4 }$ to 2.0.3 After training the PINN, we measure the $L _ { 2 }$ relative and absolute errors between the PINN’s predicted solution and the analytical solution. The $L _ { 2 }$ relative error is $\\frac { 1 } { N } \\sum _ { i = 0 } ^ { N } \\frac { | | \\hat { u } - u | | _ { 2 } } { | | u | | _ { 2 } }$ and the absolute error is $\\frac { 1 } { N } \\sum _ { i = 0 } ^ { N } \\left| \\right| \\hat { u } - u \\left| \\right| _ { 2 }$ $N$ is the number of evaluation grid points, $\\hat { u }$ is the predicted solution by the PINN, and $u$ is the true solution. For all cases, we run models at least ten times with different preset random seeds, and we average the relative and absolute errors in $u ( x , t )$ . For each loss function, $\\hat { u }$ is the output of the NN and shorthand for $\\hat { u } = N N ( \\theta , x , t )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/9c63850a96e8a066cd543dc6fc67f2baa2b89525b2b3ce6c1afb24d8f5b696cf.jpg",
+        "image_caption": [
+            "Figure 1: Prediction error for 1D convection ( §3.1) problem, when $\\beta$ is changed. The PINN has difficulty predicting the solution past a certain timestep, but is able to fit the boundary conditions. Additional figures for different $\\beta$ values can be seen in Fig. C.1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 Learning convection ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Problem formulation. We first consider a one-dimensional convection problem, a hyperbolic PDE which is commonly used to model transport phenomena: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8071f6717d6ed2479cc88450f82a5d990ce3d01895eb325647ddf166aefb6097.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle { \\frac { \\partial u } { \\partial t } + \\beta \\frac { \\partial u } { \\partial x } = 0 , \\quad x \\in \\Omega , t \\in [ 0 , T ] , } } \\\\ { \\displaystyle { u ( x , 0 ) = h ( x ) , \\quad x \\in \\Omega . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, $\\beta$ is the convection coefficient and $h ( x )$ is the initial condition. For constant $\\beta$ and periodic boundary conditions, this problem has a simple analytical solution: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7bdfd0e2c82dfad4b5403ce4b81cf3db62e8e34b1c2b6cc405b5afbb8c646fc4.jpg",
+        "text": "$$\nu _ { \\mathrm { { a n a l y t i c a l } } } ( x , t ) = F ^ { - 1 } \\big ( F ( h ( x ) ) e ^ { - i \\beta k t } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $F$ is the Fourier transform, $i = \\sqrt { - 1 }$ , and $k$ denotes frequency in the Fourier domain. The general loss function for this problem (corresponding to Eq. 3) is ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/5cbdf21ff991676957e62f2b24aafa292a550ea50ab1771fe35ac2fbc40c10ce.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta ) = \\frac { 1 } { { { N _ { u } } } } \\sum _ { i = 1 } ^ { { N _ { u } } } { \\left( { { \\hat { u } } - u _ { 0 } ^ { i } } \\right) ^ { 2 } } + \\frac { 1 } { { { N _ { f } } } } \\sum _ { i = 1 } ^ { { N _ { f } } } { \\lambda _ { i } \\Big ( \\frac { { \\partial \\hat { u } } } { { \\partial t } } + \\beta \\frac { { \\partial \\hat { u } } } { { \\partial x } } \\Big ) ^ { 2 } } + \\mathcal { L } _ { B } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { u } = N N ( \\theta , x , t )$ is the output of the NN, and $\\mathcal { L } _ { B }$ is the boundary loss. For periodic boundary conditions with $\\Omega = [ 0 , 2 \\pi )$ , this loss is: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c960ef60fc5f47c34fd1eb27f0fc83e6e47695e32afcaffe5ad5d92a37365179.jpg",
+        "text": "$$\n\\mathcal { L } _ { B } = \\frac { 1 } { N _ { b } } \\sum _ { i = 1 } ^ { N _ { b } } \\Big ( \\hat { u } ( \\theta , 0 , t ) - \\hat { u } ( \\theta , 2 \\pi , t ) \\Big ) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use the following simple initial and periodic boundary conditions: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b197f5ebb129d71168d463e2c67e3f769386f53311ae527357fc789fb8b7b84.jpg",
+        "text": "$$\n\\begin{array} { c } { { u ( x , 0 ) = s i n ( x ) , } } \\\\ { { u ( 0 , t ) = u ( 2 \\pi , t ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Observations. We apply the PINN’s soft regularization to this problem, and we optimize the loss function in Eq. 7. After training, we measure the relative and absolute errors between the PINN’s predicted solution and the analytical solution, as reported in Fig. 1(a). As one can see, the PINN is only able to achieve good solutions for small values of convection coefficient, and it fails when $\\beta$ becomes larger, reaching a relative error of almost $100 \\%$ for $\\beta > 1 0$ . We also provide visualization of the exact and PINN solution in Fig. 1(b-c). One can clearly see that the PINN is unable to learn the solution. As we will later show, the NN architecture does have enough capacity to find the solution, but the training/optimization problem is very difficult to solve with PINNs (and importantly, it may require extensive hyperparameter tuning which is often not feasible in practice). ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3.2 Learning reaction-diffusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Problem formulation. We next look at a reaction-diffusion system, where we add a diffusion operator to the reaction equation discussed above. Note that for pure diffusion, the solution dissipates ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "image",
+        "img_path": "images/507827b75469e5016d62a0516c9538c292c5ada8d87079292006fee8155453bc.jpg",
+        "image_caption": [
+            "Figure 2: Prediction error for $\\mathbfcal { m }$ reaction-diffusion $( \\ \\ S 3 . 2 )$ problem. We can clearly see that the PINN has difficulty predicting the solution (especially the “sharpness” of the solution) and is unable to capture the correct behavior. Additional figures for different $\\nu$ values can be seen in Fig. D.1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "to a steady-state of uniform/constant distribution, which may be trivial to learn. Therefore, we consider studying the reaction-diffusion system: ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/f6b727ea1e0e9bb61efcfee81bb8c2e03dab879be794973343e1c352dd4f358a.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle { \\frac { \\partial u } { \\partial t } - \\nu \\frac { \\partial ^ { 2 } u } { \\partial x ^ { 2 } } - \\rho u ( 1 - u ) = 0 , \\quad x \\in \\Omega , t \\in ( 0 , T ] , } } \\\\ { \\displaystyle { u ( x , 0 ) = h ( x ) , \\quad x \\in \\Omega . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\nu$ $( \\nu > 0 )$ ) is the diffusion coefficient. The solution of such a system can be solved for via Strang splitting, i.e., splitting the equation into two separate models (a reaction component and a diffusion component): ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36e046b357447d59611be3bac227c109197fe573a8e8fbfd9f695e643459fbdd.jpg",
+        "text": "$$\n\\begin{array} { r } { \\frac { d u } { d t } = \\rho u ( 1 - u ) } \\\\ { \\frac { d u } { d t } = \\nu \\frac { \\partial ^ { 2 } u } { \\partial x ^ { 2 } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each timestep, we can solve the reaction equation through Eq. 15 (in $\\ S \\mathrm { A }$ ). The diffusion equation has the following analytical solution: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef23692b518662a5b246a739bd93d169233fccd92740dd60fc4cf78482beec2e.jpg",
+        "text": "$$\nu _ { \\mathrm { a n a l y t i c a l } } ( x , t ) = F ^ { - 1 } \\big ( F ( u ( x , t = t ^ { n } ) ) e ^ { - \\nu k ^ { 2 } t } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $u ( x , t = t ^ { n } )$ is the solution at the $n ^ { t h }$ time step. We solve the reaction equation for each timestep, and then use the reaction solution as the initial condition to solve the diffusion component and get the final solution. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The general loss function for this problem is, ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a1f0aef126f03636b33a24f01b590ad02cfd8258e8771a9c800b3c474717de4.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) = \\displaystyle \\frac { 1 } { N _ { u } } \\sum _ { i = 1 } ^ { N _ { u } } \\Big ( \\hat { u } - u _ { 0 } ^ { i } \\Big ) ^ { 2 } + } \\\\ { \\displaystyle \\frac { 1 } { N _ { f } } \\sum _ { i = 1 } ^ { N _ { f } } \\lambda _ { i } \\Big ( \\frac { \\partial \\hat { u } } { \\partial t } - \\nu \\frac { \\partial ^ { 2 } \\hat { u } } { \\partial x ^ { 2 } } - \\rho \\hat { u } ( 1 - \\hat { u } ) \\Big ) ^ { 2 } + \\mathcal { L } _ { B } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            625,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { B }$ is the boundary loss. Similar to the previous example, periodic boundary conditions can be enforced by including $\\mathcal { L } _ { B }$ from Eq. 8 as an extra term in the loss. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Observations. Similar to the previous case, we can see that the PINN also fails to learn reactiondiffusion. We illustrate a case in Fig. 2 with $\\rho = 5$ , when $\\nu = 5$ . The PINN achieves a high relative error of $93 \\%$ . Here, we can clearly see that the PINN is unable to capture either the reaction or diffusion component. Additional figures for different $\\nu$ values can be seen in Fig. D.1. In particular, for $\\nu = 2$ the PINN achieves a relative error of $50 \\%$ . Here, we see that it is unable to capture the “sharper” transitions, though it can predict the center of the solution a little better. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Diagnosing possible failure modes for physics-informed NNs ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus far, we have shown that PINNs can result in high errors even for simple physical regimes, in particular for PDEs/ODEs with non-trivial convection/reaction/diffusion coefficients. Here, we demonstrate that one of the underlying reasons for this arises due to the PDE-based soft constraint of $\\mathcal { L } _ { \\mathcal { F } }$ , which makes the loss landscape difficult to optimize. We first (in $\\ S 4 . 1$ ) analyze the loss landscape to illustrate how increasing this soft regularization can lead to more complex loss landscapes, thus leading to optimization difficulties. We then (in $\\ S _ { \\mathbf { B } }$ ) demonstrate how this is related to regularizing with differential operators, which can result in ill-conditioning. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f14acc91a958320a452ed6c1d601120c8653c3e7c2fb161409c60f90a503896f.jpg",
+        "table_caption": [
+            "Figure 3: Loss landscapes for varying values of $\\beta _ { z }$ , for the $\\mathbfcal { m }$ convection example in $\\ S 3 . I$ . The loss landscape is more smooth at low $\\beta$ , and it becomes increasingly more complex as $\\beta$ increases, which can make the optimization problem more difficult. In particular, at higher $\\beta$ , the optimizer gets stuck in a certain regime. These results support that adding the PDE soft regularization term results in a more complex optimization loss landscape. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>200 150 100 50 1.0 0.5 0.5 1.0-1.0 (a) β = 1.0 β</td><td>(b) β = 10.0</td><td colspan=\"3\">(c) β = 20.0</td><td>(e) β = 40.0</td></tr><tr><td>Relative error</td><td></td><td>10 1.08× 10-2</td><td>20</td><td>30</td><td>40</td></tr><tr><td></td><td>7.84 × 10-3 3.17× 10-3</td><td>6.03×10-3</td><td>7.50× 10-1 4.32×10-1</td><td>8.97×10-1 5.42×10-1</td><td>9.61 ×10-1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Soft PDE regularization and optimization difficulties ",
+        "text_level": 1,
+        "bbox": [
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+            431,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here, we analyze how the loss landscape changes for different regimes for the convection problem in $\\ S 3 . 1$ with/without the soft regularization in PINNs. We show that adding the soft regularization can actually make the problem harder to optimize, i.e., the regularization leads to less smooth loss landscapes. For all the experiments, we plot the loss landscape by perturbing the (trained) model across the first two dominant Hessian eigenvectors and computing the corresponding loss values. This tends to be more informative than perturbing the model parameters in random directions [41, 42]. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the loss landscape for the convection problem (discussed in $\\ S 3 . 1$ ), for different $\\beta$ values. Interestingly, the loss landscape at a relatively low $\\beta = 1$ is rather smooth, but increasing $\\beta$ further results in a complex and non-symmetric loss landscape. It is also evident that the optimizer has gotten stuck in a local minima with a very high loss function for large $\\beta$ values. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we study the impact of changing the weight/multiplier for the soft regularization term (i.e., the $\\lambda$ parameter in Eq. 3), which can be relevant in improving PINN performance [35]. While we find that tuning $\\lambda$ can help change the error, it cannot resolve the problem, as shown in Fig. E.1. Note that as the regularization parameter is increased, the loss landscape becomes increasingly more complex and harder to optimize (additionally, see the $\\mathbf { Z }$ -axis scale). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Expressivity versus optimization difficulty ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            699,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first show that the failure modes we observed are not necessarily due to the specific NN architecture that we used in our experiments. In particular, we show that the NN model does have the expressivity/capacity to learn the convection/reaction/diffusion coefficient cases where the vanilla PINN method fails. Additionally, in the process of demonstrating this, we also describe two methods that lead to significantly lower error rates. In particular, we show that changing the learning paradigm to curriculum regularization can make the optimization problem easier to solve (as discussed in $\\ S 5 . 1$ ). Second, we show that posing the problem as sequence-to-sequence learning may lead to better results than learning the entire state-space at once (as discussed in $\\ S 5 . 2 )$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Curriculum PINN Regularization ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            857,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "One may contend that the failure modes shown in $\\ S 3$ may be because the NN does not have enough capacity. Here, we show that this is not the underlying reason. To do so, we devise a “curriculum regularization” method to warm start the NN training by finding a good initialization for the weights. Instead of training the PINN to learn the solution right away for cases with higher $\\beta / \\rho$ , we start by training the PINN on lower $\\beta / \\rho$ (easier for the PINN to learn) and then gradually move to training the PINN on higher $\\beta / \\rho$ , respectively. We test these results for the examples in $\\ S 3 . 1$ and $\\ S \\mathrm { A }$ . This is somewhat analogous to curriculum learning in ML [1], but applied by progressively making the PDE/ODE harder to solve. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/cf358f24c3f98df7fa0c6d924e0bd2aa722df0a24acad8197080983fb9d08d3c.jpg",
+        "image_caption": [
+            "Figure 4: Schematic outlining curriculum regularization and example result for 1D convection from $\\ S 3 . I$ The training procedure for regular PINNs training versus curriculum PINN training for the convection example in $\\ S 3 . I$ . The regular PINN training only involves training at $\\beta = 3 0$ , while curriculum regularization starts at a lower $\\beta$ , trains a model, and then uses the weights of this model to reinitialize the NN for training the next $\\beta$ . The curriculum training approach is able to do significantly better (by almost two orders of magnitude). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/22919fd3edc8132832b69635b1db2f66f35a2a99ec9990f9159fb234b486f1e7.jpg",
+        "table_caption": [
+            "Table 1: Training the PINN gradually on more difficult problems improves performance. 1D convection example in $\\ \\ S 3 . 1 .$ . The curriculum training approach achieves significantly better errors. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=3>Regular PINN（</td><td rowspan=1 colspan=1> Curriculum training</td></tr><tr><td rowspan=2 colspan=1>1D convection: β = 20</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>7.50 × 10-1</td><td rowspan=1 colspan=1>9.84 × 10-3</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>4.32×10-1</td><td rowspan=1 colspan=1>5.42 ×10-3</td></tr><tr><td rowspan=2 colspan=1>1D convection: β = 30</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>8.97×10-1</td><td rowspan=1 colspan=1>2.02 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>5.42×10-1</td><td rowspan=1 colspan=1>1.10 ×10-2</td></tr><tr><td rowspan=2 colspan=1>1D convection: β = 40</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>9.61 × 10-1</td><td rowspan=1 colspan=1>5.33 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>5.82 ×10-1</td><td rowspan=1 colspan=1>2.69 ×10-2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 shows the training procedure for an example convection case $( \\ S 3 . 1 )$ with $\\beta = 3 0$ . As Fig. 4(c) shows, the curriculum regularization approach results in a much more accurate solution than regular PINN training. With curriculum regularization, the relative error is almost two orders of magnitude lower. Additionally, this is true across all the other regimes that we found regular PINNs to fail, as shown in Tab. 1. In Fig. E.2, we also show that curriculum regularization not only decreases error significantly, but also decreases the variance of the error. In Fig. E.3, we see that curriculum regularization results in a much smoother loss landscape as compared to regular PINN training. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Curriculum regularization also works well for the reaction example in $\\ S \\mathbf { A }$ . In this case, we start by training with a low $\\rho$ value (reaction coefficient), and then increase gradually to higher $\\rho$ values. The results can be seen in Fig. E.4. We can see that the error is $0 . 1 \\textrm { - } 0 . 6 $ orders of magnitude lower for $\\rho = 2 - 4$ (when the regular PINN error is not as high), and then greatly decreases error by 1-2 orders of magnitude for $\\rho = 5 - 1 0$ . As we discussed before, PINN has difficulty in learning sharp features for high values of $\\rho$ . However, the curriculum regularization overcomes this, even for $\\rho = 1 0$ , as seen in Fig. E.4(c). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Sequence-to-sequence learning vs learning the entire space-time solution ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The original PINN approach of [28] trains the NN model to predict the entire space-time at once (i.e., predict $u$ for all locations and time points). In certain cases, this can be more difficult to learn. Here, we demonstrate that it may be better to pose the problem as a sequence-to-sequence (seq2seq) learning task, where the NN learns to predict the solution at the next time step, instead of all times. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/73622db931290c5644d1b71ac0064c8e76958e88111b960bdfa0e01cfc5f3ca5.jpg",
+        "image_caption": [
+            "6 Figure 5: Schematic outlining seq2seq learning. In contrast to regular PINN training, the solution in seq2seq learning is predicted for only one $\\Delta t$ step at a time. Then, the predicted solution at $t = \\Delta t$ is used as the initial condition for the next segment. To allow fair comparison, we keep the total number of 5 collocation points to be exactly the same in either approach. That is, we do not increase the number of collocation points for seq2seq learning in the right, and keep it to be the same as in the corresponding segment in the left figure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            102,
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+            308
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/aa9126e2c81a0561264a7553f78a3854e79a096485ad79bf22ebc1a856f3f0da.jpg",
+        "table_caption": [
+            "Table 2: Predicting the entire state space versus discretizing the state space (i.e., seq2seq learning) for $\\mathbfcal { m }$ reaction-diffusion ( $\\ S 3 . 2 )$ . The seq2seq learning achieves lower error for both $\\Delta t = 0 . 0 5$ and $\\Delta t = 0 . 1$ , in comparison to the PINN’s approach of predicting the entire state space at once. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1> Entire state space</td><td rowspan=1 colspan=1>△t = 0.05</td><td rowspan=1 colspan=1>△t = 0.1</td></tr><tr><td rowspan=2 colspan=1>v=2,p=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>5.07× 10-1</td><td rowspan=1 colspan=1>2.04 × 10-2</td><td rowspan=1 colspan=1>1.18 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>2.70 ×10-1</td><td rowspan=1 colspan=1>1.06 × 10-2</td><td rowspan=1 colspan=1>6.41 × 10-3</td></tr><tr><td rowspan=1 colspan=1>v=3,p=5</td><td rowspan=1 colspan=1> Relative error Absolute error</td><td rowspan=1 colspan=1>7.98 × 10-14.79 × 10-1</td><td rowspan=1 colspan=1>1.92 × 10-21.01 × 10-2</td><td rowspan=1 colspan=1>1.56 × 10-28.17 × 10-3</td></tr><tr><td rowspan=2 colspan=1>v=4,p=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>8.84×10-1</td><td rowspan=1 colspan=1>2.37 × 10-2</td><td rowspan=1 colspan=1>1.59 × 10-2</td></tr><tr><td rowspan=1 colspan=1>Absolute error</td><td rowspan=1 colspan=1>5.74 × 10-1</td><td rowspan=1 colspan=1>1.15 ×10-2</td><td rowspan=1 colspan=1>8.01 ×10-3</td></tr><tr><td rowspan=1 colspan=1>v=5,ρ=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>9.35 × 10-1</td><td rowspan=1 colspan=1>2.36 × 10-2</td><td rowspan=1 colspan=1>2.39 ×10-2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1> Absolute error</td><td rowspan=1 colspan=1>6.46 × 10-1</td><td rowspan=1 colspan=1>1.09 × 10-2</td><td rowspan=1 colspan=1>1.15 × 10-2</td></tr><tr><td rowspan=1 colspan=1>v=6,p=5</td><td rowspan=1 colspan=1>Relative error</td><td rowspan=1 colspan=1>9.60 × 10-1</td><td rowspan=1 colspan=1>2.81 ×10-2</td><td rowspan=1 colspan=1>2.69 × 10-2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1> Absolute error</td><td rowspan=1 colspan=1>6.84 × 10-1</td><td rowspan=1 colspan=1>1.17 × 10-2</td><td rowspan=1 colspan=1>1.28 × 10-2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This way, we can use a marching-in-time scheme to predict different sequences/time points. Note that the only data available here is from the PDE itself, i.e., just the initial condition. We take the prediction at $t = \\Delta t$ and use this as the initial condition to make a prediction at $t = 2 \\Delta t$ , and so on. This is schematically outlined in Fig. 5. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We test this scheme by using the exact same NN architecture as in previous sections, and we report the results in Tab. E.1 for the convection problem of $\\ S 3 . 1$ , Tab. E.2 for the reaction problem of $\\ S \\mathrm { A }$ , and Tab. 2 for the reaction-diffusion problem of $\\ S 3 . 2$ . We compare the relative/absolute error when the learning is posed as a seq2seq problem (i.e., predicting the state space with a “time marching scheme” of one timestep prediction at a time) to the PINN approach of predicting the whole state space at once.4 ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We explore the following cases where the PINN does poorly, varying $\\beta , \\rho$ , and $\\nu$ coefficients: ",
+        "bbox": [
+            174,
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+            849
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/00c8d636fffe0f538e982510f226fe7a8d911fc0be41114807f1317d5eaf8d43.jpg",
+        "image_caption": [
+            "Figure 6: Predicting the entire state space vs seq2seq learning for $\\mathbfcal { m }$ reaction-diffusion. The regular PINN is unable to capture the “sharp” and/or diffusive features correctly. However, the seq2seq learning approach is able to capture the correct solution, and achieves almost two orders of magnitude lower error. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            106,
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+            233
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "1) For 1D convection ( §3.1), higher $\\beta$ values from 30-40. ",
+        "bbox": [
+            173,
+            333,
+            552,
+            348
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "2) For 1D reaction ( $\\ S _ { \\mathbf { A } } )$ , $\\rho$ coefficients from 5-10. ",
+        "bbox": [
+            176,
+            354,
+            503,
+            369
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "3) For 1D reaction-diffusion ( §3.2), a fixed $\\rho = 5$ and $\\nu$ coefficients from 2-6. ",
+        "bbox": [
+            173,
+            375,
+            687,
+            390
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For these cases, we find that posing the problem as seq2seq learning results in significantly lower error. The difference is particularly striking for the reaction and reaction-diffusion cases, where the seq2seq PINN model decreases error by almost two orders of magnitude. An example case is shown Fig. 6, where the seq2seq approach is able to recover the solution, while regular PINNs does very poorly. Note that this behavior also has analogues with numerical methods used in scientific computing, where space-time problems are typically harder to solve, as compared to time marching methods [8]. Intuitively, since the problem is ill-conditioned, restricting the dimensions is expected to help. Furthermore, the underlying function/mapping of the input to the solution should be much simpler to approximate over a smaller time span, as compared to the full time horizon. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "These initial results are promising, and further developments may lead to still better ways of using PINNs and learning PDEs. In particular, using more sophisticated methods to predict timesteps across the state space may provide improved performance, as may including more sophisticated seq2seq approaches and tuning the regularization parameter (i.e., amount of constraint added). ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
+        "bbox": [
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+            638
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "PINNs—and SciML more generally—hold great promise for expanding the scope of ML methodology to important problems in science and engineering. For these problems, however, integrating ML methods with PDE-based domain-driven constraints as a soft regularization term can lead to subtle and critical issues. In particular, we show that this approach can have fundamental limitations which results in failure modes for learning relevant physics commonly used in different fields of science. To show this, we picked two fundamental PDE problems of diffusion and convection and showed that the PINN only works for very simple cases, failing to learn the relevant physical phenomena for even moderately more challenging regimes. We then analyzed the problem to characterize the underlying reasons why these failures occur. In particular, we studied the PINN loss landscape behavior and found it becomes it becomes increasingly complex for large values of diffusion or convection coefficients, and with/without non-homogeneous forcing. We also discussed that the problem is not necessarily due to the limited capacity of the NN, but that it is partly an optimization problem resulting in the PDE-based soft constraint used in PINNs. Furthermore, we showed that the PINN approach of solving for the entire space-time at once may not be efficient, and instead posing the problem as a sequence-to-sequence learning task can provide lower error rates. Addressing these and related issues will be critical if we hope to go beyond existing cut-and-paste approaches, toward engineering a more intimate connection between scientific methodologies and ML methodologies. This will be needed to deliver on the promise of PINNs and SciML more generally. ",
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+    {
+        "type": "text",
+        "text": "7 Acknowledgements. ",
+        "text_level": 1,
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+        "text": "We are thankful to Shashank Subramanian for his feedback and contributions. We also acknowledge helpful discussions with Prof. George Biros, Geoffrey Negiar, and Daniel Rothchild. ASK was supported by Laboratory Directed Research and Development (LDRD) funding under Contract Number DE-AC02-05CH11231 at LBNL and the Alvarez Fellowship in the Computational Research Division at LBNL. AG was supported through funding from Samsung SAIT. MWM would also like to acknowledge the UC Berkeley CLTC, ARO, NSF, and ONR. The UC Berkeley team also acknowledges gracious support from Intel corporation, Intel VLAB, Samsung, Amazon AWS, Google Cloud, Google TPU Research Cloud, and Google Brain (in particular Prof. David Patterson, Dr. Ed Chi, and Jing Li). Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred. ",
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+    {
+        "type": "text",
+        "text": "References ",
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parse/train/fylclEqgvgd/fylclEqgvgd.md

@@ -0,0 +1,487 @@
+# TRANSFORMER PROTEIN LANGUAGE MODELS ARE UNSUPERVISED STRUCTURE LEARNERS
+
+Roshan Rao∗   
+UC Berkeley   
+rmrao@berkeley.edu
+
+Joshua Meier Facebook AI Research jmeier@fb.com
+
+Tom Sercu Facebook AI Research tsercu@fb.com
+
+Sergey Ovchinnikov Harvard University ${ \mathsf { S O } } { \mathbb { Q } } { \mathsf { q } }$ .harvard.edu
+
+Alexander Rives   
+Facebook AI Research & New York University   
+arives@cs.nyu.edu
+
+# ABSTRACT
+
+Unsupervised contact prediction is central to uncovering physical, structural, and functional constraints for protein structure determination and design. For decades, the predominant approach has been to infer evolutionary constraints from a set of related sequences. In the past year, protein language models have emerged as a potential alternative, but performance has fallen short of state-of-the-art approaches in bioinformatics. In this paper we demonstrate that Transformer attention maps learn contacts from the unsupervised language modeling objective. We find the highest capacity models that have been trained to date already outperform a stateof-the-art unsupervised contact prediction pipeline, suggesting these pipelines can be replaced with a single forward pass of an end-to-end model.1
+
+# 1 INTRODUCTION
+
+Unsupervised modeling of protein contacts has an important role in computational protein design (Russ et al., 2020; Tian et al., 2018; Blazejewski et al., 2019) and is a central element of all current state-of-the-art structure prediction methods (Wang et al., 2017; Senior et al., 2020; Yang et al., 2019). The standard bioinformatics pipeline for unsupervised contact prediction includes multiple components with specialized tools and databases that have been developed and optimized over decades. In this work we propose replacing the current multi-stage pipeline with a single forward pass of a pre-trained end-to-end protein language model.
+
+In the last year, protein language modeling with an unsupervised training objective has been investigated by multiple groups (Rives et al., 2019; Alley et al., 2019; Heinzinger et al., 2019; Rao et al., 2019; Madani et al., 2020). The longstanding practice in bioinformatics has been to fit linear models on focused sets of evolutionarily related and aligned sequences; by contrast, protein language modeling trains nonlinear deep neural networks on large databases of evolutionarily diverse and unaligned sequences. High capacity protein language models have been shown to learn underlying intrinsic properties of proteins such as structure and function from sequence data (Rives et al., 2019).
+
+A line of work in this emerging field proposes the Transformer for protein language modeling (Rives et al., 2019; Rao et al., 2019). Originally developed in the NLP community to represent long range context, the main innovation of the Transformer model is its use of self-attention (Vaswani et al., 2017). Self-attention has particular relevance for the modeling of protein sequences. Unlike convolutional or recurrent models, the Transformer constructs a pairwise interaction map between all positions in the sequence. In principle this mechanism has an ideal form to model protein contacts.
+
+In theory, end-to-end learning with a language model has advantages over the bioinformatics pipeline: (i) it replaces the expensive query, alignment, and training steps with a single forward pass, greatly accelerating feature extraction; and (ii) it shares parameters for all protein families, enabling generalization by capturing commonality across millions of evolutionarily diverse and unrelated sequences.
+
+We demonstrate that Transformer protein language models learn contacts in the self-attention maps with state-of-the-art performance. We compare ESM-1b (Rives et al., 2020), a large-scale (650M parameters) Transformer model trained on UniRef50 (Suzek et al., 2007) to the Gremlin (Kamisetty et al., 2013) pipeline which implements a log linear model trained with pseudolikelihood (Balakrishnan et al., 2011; Ekeberg et al., 2013). Contacts can be extracted from the attention maps of the Transformer model by a sparse linear combination of attention heads identified by logistic regression. ESM-1b model contacts have higher precision than Gremlin contacts. When ESM and Gremlin are compared with access to the same set of sequences the precision gain from the protein language model is significant; the advantage holds on average even when Gremlin is given access to an optimized set of multiple sequence alignments incorporating metagenomics data.
+
+We find a linear relationship between language modeling perplexity and contact precision. We also find evidence for the value of parameter sharing: the ESM-1b model significantly outperforms Gremlin on proteins with low-depth MSAs. Finally we explore the Transformer language model’s ability to generate sequences and show that generated sequences preserve contact information.
+
+# 2 BACKGROUND
+
+Multiple Sequence Alignments (MSAs) A multiple sequence alignment consists of a set of evolutionarily related protein sequences. Since real protein sequences are likely to have insertions, deletions, and substitutions, the sequences are aligned by minimizing a Levenshtein distance-like metric over all the sequences. In practice heuristic alignment schemes are used. Tools like Jackhmmer and HHblits can increase the number and diversity of sequences returned by iteratively performing the search and alignment steps (Johnson et al., 2010; Remmert et al., 2012).
+
+Metrics For a protein of length $L$ , we evaluate the precision of the top $L , L / 2$ , and $L / 5$ contacts for short range $( | i - j | \in [ 6 , 1 2 ) \rangle$ , medium range $( | i - j | \in [ 1 2 , 2 4 ) )$ , and long range $. | i = j | \in$ $[ 2 4 , \infty )$ ) contacts. We also separately evaluate local contacts $( | i - j | \in [ 3 , 6 )$ ) for secondary structure prediction in Appendix A.9. In general, all contacts provide information about protein structure and important interactions, with shorter-range contacts being useful for secondary and local structure, while longer range contacts are useful for determining global structure (Taylor et al., 2014).
+
+# 3 RELATED WORK
+
+There is a long history of protein contact prediction (Adhikari & Cheng, 2016) both from MSAs, and more recently, with protein language models.
+
+Supervised contact prediction Recently, supervised methods using deep learning have resulted in breakthrough results in supervised contact prediction (Wang et al., 2017; Jones & Kandathil, 2018; Yang et al., 2019; Senior et al., 2020; Adhikari & Elofsson, 2020). State-of-the art methods use deep residual networks trained with supervision from many protein structures. Inputs are typically covariance statistics (Jones & Kandathil, 2018; Adhikari & Elofsson, 2020), or inferred coevolutionary parameters (Wang et al., 2017; Liu et al., 2018; Senior et al., 2020; Yang et al., 2019). Other recent work with deep learning uses sequences or evolutionary features as inputs (AlQuraishi, 2018; Ingraham et al., 2019). Xu et al. (2020) demonstrates the incorporation of coevolutionary features is critical to performance of current state-of-the-art methods.
+
+Unsupervised contact prediction In contrast to supervised methods, unsupervised contact prediction models are trained on sequences without information from protein structures. In principle this allows them to take advantage of large sequence databases that include information from many sequences where no structural knowledge is available. The main approach has been to learn evolutionary constraints among a set of similar sequences by fitting a Markov Random Field (Potts model) to the underlying MSA, a technique known as Direct Coupling Analysis (DCA). This was proposed by Lapedes et al. (1999) and reintroduced by Thomas et al. (2008) and Weigt et al. (2009).
+
+Various methods have been developed to fit the underlying Markov Random Field, including meanfield DCA (mfDCA) (Morcos et al., 2011), sparse inverse covariance (PSICOV) (Jones et al., 2011) and pseudolikelihood maximization (Balakrishnan et al., 2011; Ekeberg et al., 2013; Seemayer et al., 2014). Pseudolikelihood maximization is generally considered state-of-the-art for unsupervised contact prediction and the Gremlin (Balakrishnan et al., 2011) implementation is used as the baseline throughout. We also provide mfDCA and PSICOV baselines. Recently deep learning methods have also been applied to fitting MSAs, and Riesselman et al. (2018) found evidence that factors learned by a VAE model may correlate with protein structure.
+
+Structure prediction from contacts While we do not perform structure prediction in this work, many methods have been proposed to extend contact prediction to structure prediction. For example, EVFold (Marks et al., 2011) and DCAFold (Sulkowska et al., 2012) predict co-evolving couplings using a Potts Model and then generate 3D conformations by directly folding an initial conformation with simulated annealing, using the predicted residue-residue contacts as constraints. Similarly, FragFold (Kosciolek & Jones, 2014) and Rosetta (Ovchinnikov et al., 2016) incorporate constraints from a Potts Model into a fragment assembly based pipeline. Senior et al. (2019), use features from a Potts model fit with pseudolikelihood maximization to predict pairwise distances with a deep residual network and optimize the final structure using Rosetta. All of these works build directly upon the unsupervised contact prediction pipeline.
+
+Contact prediction from protein language models Since the introduction of large scale language models for natural language processing (Vaswani et al., 2017; Devlin et al., 2019), there has been considerable interest in developing similar models for proteins (Alley et al., 2019; Rives et al., 2019; Heinzinger et al., 2019; Rao et al., 2019; Elnaggar et al., 2020; Lu et al., 2020; Madani et al., 2020; Shen et al., 2021). Rives et al. (2019) were the first to study protein Transformer language models, demonstrating that information about residue-residue contacts could be recovered from the learned representations by linear projections supervised with protein structures. Recently Vig et al. (2020) performed an extensive analysis of Transformer attention, identifying correspondences to biologically relevant features, and also found that different layers of the model are responsible for learning different features. In particular Vig et al. (2020) discovered a correlation between selfattention maps and contact patterns, suggesting they could be used for contact prediction.
+
+Prior work benchmarking contact prediction with protein language models has focused on the supervised problem. Bepler & Berger (2019) were the first to fine-tune an LSTM pretrained on protein sequences to fit contacts. Rao et al. (2019) and Rives et al. (2020) perform benchmarking of multiple protein language models using a deep residual network fit with supervised learning on top of pretrained language modeling features.
+
+In contrast to previous work on protein language models, we find that a state-of-the-art unsupervised contact predictor can be directly extracted from the Transformer self-attention maps. We perform a thorough analysis of the contact predictor, showing relationships between performance and MSA depth as well as language modeling perplexity. We also provide methods for improving performance using sequences from an MSA and for sampling sequences in a manner that preserves contacts.
+
+# 4 MODELS
+
+We compare Transformer models trained on large sequence databases to Potts Models trained on individual MSAs. While Transformers and Potts Models emerged in separate research communities, the two models share core similarities (Wang & Cho, 2019) which we exploit here. Our main result is that just as Gremlin directly represents contacts via its pairwise component (the weights), the Transformer also directly represents contacts via its pairwise component (the self-attention).
+
+# 4.1 OBJECTIVES
+
+For a set of training sequences, $X$ , Gremlin optimizes the following pseudolikelihood loss, where a single position is masked and predicted from its context. Inputs are aligned, so all have length $L$ :
+
+$$
+\mathcal { L } _ { \mathrm { P L L } } ( X ; \theta ) = \underset { x \sim X } { \mathbb { E } } \sum _ { i = 1 } ^ { L } \log p ( x _ { i } | x _ { j \neq i } ; \theta )
+$$
+
+![](images/0bdc71359d1283509a42cc54a322b5ce784671280d38634f6fc498a226216b70.jpg)  
+Figure 1: Contact prediction pipeline. The Transformer is first pretrained on sequences from a large database (Uniref50) via Masked Language Modeling. Once finished training, the attention maps are extracted, passed through symmetrization and average product correction, then into a regression. The regression is trained on a small number $( n \leq 2 0 )$ ) of proteins to determine which attention heads are informative. At test time, contact prediction from an input sequence can be done entirely on GPU in a single forward pass.
+
+The masked language modeling (MLM) loss used by the Transformer models can be seen as a generalization of the Potts Model objective when written as follows:
+
+$$
+\mathcal { L } _ { \mathrm { M L M } } ( X ; \theta ) = \underset { x \sim X } { \mathbb { E } } \underset { \mathrm { m a s k } } { \mathbb { E } } \sum _ { i \in \mathrm { m a s k } } \log p ( x _ { i } | x _ { j \notin \mathrm { m a s k } } ; \theta )
+$$
+
+In contrast to Gremlin, the MLM objective applied by protein language modeling is trained on unaligned sequences. The key distinction of MLM is to mask and predict multiple positions concurrently, instead of masking and predicting one at a time. This enables the model to scale beyond individual MSAs to massive sequence datasets. In practice, the expectation under the masking pattern is computed stochastically using a single sample at each epoch.
+
+# 4.2 GREMLIN
+
+The log probability optimized by Gremlin is described in section A.3. Contacts are extracted from the pairwise Gremlin parameters by taking the Frobenius norm along the amino acid dimensions, resulting in an $L \times L$ coupling matrix. Average product correction (APC) is applied to this coupling matrix to determine the final predictions (Appendix A.2).
+
+Gremlin takes an MSA as input. The quality of the output predictions are highly dependent on the construction of the MSA. We compare to Gremlin under two conditions. In the first condition, we present Gremlin with all MSAs from the trRosetta training set (Yang et al., 2019). These MSAs were generated from all of Uniref100 and are also supplemented with metagenomic sequences when the depth from Uniref100 is too low. The trRosetta MSAs are a key ingredient in the state-of-theart protein folding pipeline. See Yang et al. (2019) for a discussion on the significant impact of metagenomic sequences on the final result. In the second setting, we allow Gremlin access only to the same information as the ESM Transformers by generating MSAs via Jackhmmer on the ESM training set (a subset of Uniref50). See Appendix A.5 for Jackhmmer parameters.
+
+# 4.3 TRANSFORMERS
+
+We evaluate several pre-trained Transformer models, including ESM-1 (Rives et al., 2019), ProtBertBFD (Elnaggar et al., 2020) and the TAPE Transformer (Rao et al., 2019). The key differences between these models are the datasets, model sizes, and hyperparameters (major architecture differences described in Table 3). Liu et al. (2019) previously showed that these changes can have a significant impact on final model performance. In addition to ESM-1, we also evaluate an updated version, ESM-1b, which is the result of a hyperparameter sweep. The differences are described in
+
+Table 1: Average precision on 14842 test structures for Transformer models trained on 20 structures.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">6≤sep&lt;12</td><td colspan="3">12≤sep&lt;24</td><td colspan="3">24≤sep</td></tr><tr><td>L</td><td>L/2</td><td>L/5</td><td>L</td><td>L/2</td><td>L/5</td><td>L</td><td>L/2</td><td>L/5</td></tr><tr><td>Gremlin (ESM Data)</td><td>15.2</td><td>23.0</td><td>37.8</td><td>18.1</td><td>27.9</td><td>44.3</td><td>31.3</td><td>43.1</td><td>55.5</td></tr><tr><td>mfDCA (trRosetta Data)</td><td>16.3</td><td>23.7</td><td>35.8</td><td>19.7</td><td>29.8</td><td>45.5</td><td>33.0</td><td>43.5</td><td>54.2</td></tr><tr><td>PSICOV²(trRosetta Data)</td><td>15.4</td><td>23.6</td><td>39.2</td><td>18.3</td><td>28.4</td><td>45.7</td><td>32.6</td><td>45.2</td><td>58.1</td></tr><tr><td>Gremlin (trRosetta Data)</td><td>17.2</td><td>26.7</td><td>44.4</td><td>21.1</td><td>33.3</td><td>52.3</td><td>39.3</td><td>52.2</td><td>62.8</td></tr><tr><td>TAPE</td><td>9.9</td><td>12.3</td><td>16.4</td><td>10.0</td><td>12.6</td><td>16.6</td><td>11.2</td><td>14.0</td><td>17.9</td></tr><tr><td>ProtBERT-BFD</td><td>20.4</td><td>30.7</td><td>48.4</td><td>24.3</td><td>35.5</td><td>52.0</td><td>34.1</td><td>45.0</td><td>57.4</td></tr><tr><td>ESM-1 (6 layer)</td><td>11.0</td><td>13.2</td><td>15.9</td><td>11.5</td><td>14.6</td><td>19.0</td><td>13.2</td><td>16.7</td><td>21.5</td></tr><tr><td>ESM-1 (12 layer)</td><td>15.2</td><td>21.1</td><td>30.5</td><td>18.1</td><td>24.7</td><td>34.0</td><td>23.7</td><td>30.5</td><td>39.3</td></tr><tr><td>ESM-1 (34 layer)</td><td>20.3</td><td>30.2</td><td>46.0</td><td>23.8</td><td>34.3</td><td>49.2</td><td>34.7</td><td>44.6</td><td>56.0</td></tr><tr><td>ESM-1b</td><td>21.6</td><td>33.2</td><td>52.7</td><td>26.2</td><td>38.6</td><td>56.4</td><td>41.1</td><td>53.3</td><td>66.1</td></tr></table>
+
+Section A.4. The Transformer processes inputs through a series of blocks alternating multi-head self-attention and feed-forward layers. In each head of a self-attention layer, the Transformer views the encoded representation as a set of query-key-value triples. The output of the head is the result of scaled dot-product attention:
+
+$$
+{ \mathrm { A t t e n t i o n } } ( Q , K , V ) = { \mathrm { s o f t m a x } } ( Q K ^ { T } / { \sqrt { n } } ) \cdot V
+$$
+
+Rather than only computing the attention once, the multi-head approach runs scaled dot-product attention multiple times in parallel and concatenates the output. Since self-attention explicitly constructs pairwise interactions $( Q K ^ { T } )$ between all positions in the sequence, the model can directly represent residue-residue interactions. In this work, we demonstrate that the $Q K ^ { T }$ pairwise “self attention maps” indeed capture accurate contacts.
+
+# 4.4 LOGISTIC REGRESSION
+
+To extract contacts from a Transformer, we first pass the input sequence through the model to obtain the attention maps (one map for each head in each layer). We then symmetrize and apply APC to each attention map independently. The resulting maps are passed through an $L _ { 1 }$ -regularized logistic regression, which is applied independently at each amino acid pair $( i , j )$ . At training time, we only train the weights of the logistic regression; we do not backpropagate through the entire model. At test time, the entire prediction pipeline can be run in a single forward pass, providing a single end-toend pipeline for protein contact prediction that does not require any retrieval steps from a sequence database. See Appendix A.7 for a full description of the logistic regression setup.
+
+# 5 RESULTS
+
+We evaluate models with the 15051 proteins in the trRosetta training dataset (Yang et al., 2019), removing 43 proteins with sequence length greater than 1024, since ESM-1b was trained with a context size of 1024. Of these sequences, Jackhmmer fails on 126 when we attempt to construct MSAs using the ESM training set (see Appendix A.5). This leaves us with 14882 total sequences. We reserve 20 sequences for training, 20 sequences for validation, and 14842 sequences for testing.
+
+Table 1 shows evaluations of Gremlin, ESM-1, ESM-1b as well as the TAPE and ProtBERT-BFD models. Confidence intervals are within 0.5 percentage points for all statistics in Tables 1 and 2. In Table 1, all Transformer model contact predictors are trained with logistic regression on 20 proteins. We find that with only 20 training proteins ESM-1b has higher precision than Gremlin for short, medium, and long range contacts.
+
+Table 2: ESM-1b Ablations with limited supervision and with MSA information. $n$ is the number of logistic regression training proteins. $s$ is the number of sequences ensembled over.   
+
+<table><tr><td></td><td></td><td colspan="3">6≤sep&lt;12</td><td colspan="3">12≤sep&lt;24</td><td colspan="3">24≤sep</td></tr><tr><td>Model</td><td>Variant</td><td>L</td><td>L/2</td><td>L/5</td><td>L</td><td>L/2</td><td>L/5</td><td>L</td><td>L/2</td><td>L/5</td></tr><tr><td rowspan="2">Gremlin</td><td>ESM Data</td><td>15.2</td><td>23.0</td><td>37.8</td><td>18.1</td><td>27.9</td><td>44.3</td><td>31.3</td><td>43.1</td><td>55.5</td></tr><tr><td>trRosetta Data</td><td>17.2</td><td>26.7</td><td>44.4</td><td>21.1</td><td>33.3</td><td>52.3</td><td>39.3</td><td>52.2</td><td>62.8</td></tr><tr><td rowspan="8">ESM-1b</td><td>top-1 heads</td><td>16.8</td><td>23.4</td><td>34.8</td><td>19.8</td><td>27.6</td><td>40.2</td><td>29.3</td><td>38.1</td><td>50.0</td></tr><tr><td>top-5 heads</td><td>19.2</td><td>28.5</td><td>44.5</td><td>23.3</td><td>33.8</td><td>49.0</td><td>35.0</td><td>45.2</td><td>57.3</td></tr><tr><td>top-10 heads</td><td>20.0</td><td>30.1</td><td>47.4</td><td>24.7</td><td>36.0</td><td>52.2</td><td>38.5</td><td>49.4</td><td>61.1</td></tr><tr><td>n=1,s=1</td><td>19.4</td><td>29.7</td><td>47.1</td><td>25.1</td><td>37.1</td><td>54.0</td><td>39.2</td><td>50.6</td><td>63.0</td></tr><tr><td>n=10,s=1</td><td>21.4</td><td>32.9</td><td>52.3</td><td>26.1</td><td>38.5</td><td>56.4</td><td>40.8</td><td>52.9</td><td>65.7</td></tr><tr><td>n=20, s=1</td><td>21.6</td><td>33.2</td><td>52.7</td><td>26.2</td><td>38.6</td><td>56.4</td><td>41.1</td><td>53.3</td><td>66.1</td></tr><tr><td>MSA, s=1</td><td>18.4</td><td>28.1</td><td>45.5</td><td>23.9</td><td>36.1</td><td>53.7</td><td>39.9</td><td>51.3</td><td>63.0</td></tr><tr><td>n=20, s=16</td><td>21.9</td><td>33.8</td><td>53.6</td><td>26.7</td><td>39.4</td><td>57.5</td><td>41.9</td><td>54.3</td><td>67.3</td></tr><tr><td rowspan="2">ESM-1b (s seqs)</td><td>n=20, s=32</td><td>22.0</td><td>34.1</td><td>54.0</td><td>26.9</td><td>39.8</td><td>58.1</td><td>42.3</td><td>54.8</td><td>67.8</td></tr><tr><td>n=20, s=64</td><td>22.1</td><td>34.3</td><td>54.3</td><td>27.1</td><td>40.1</td><td>58.5</td><td>42.6</td><td>55.1</td><td>68.2</td></tr></table>
+
+In addition to this set, we also evaluate performance on 15 CASP13 FM Domains in Appendix A.6. On average ESM-1b has higher short, medium, and long range precision than Gremlin on all metrics, and in particular can significantly outperform on MSAs with low effective number of sequences. We also provide a comparison to the bilinear model proposed by Rives et al. (2020). The logistic regression model achieves a long-range contact precision at L of 18.6, while the fully supervised bilinear model achieves a long range precision at L of 20.1, an increase of only 1.5 points despite being trained on $7 0 0 \mathbf { x }$ more structures.
+
+# 5.1 ABLATIONS: LIMITING SUPERVISION
+
+While the language modeling objective is fully unsupervised, the logistic regression is trained with a small number of supervised examples. In this section, we study the dependence of the results on this supervision, providing evidence that the contacts are indeed learned in the unsupervised phase, and the logistic regression is only necessary to extract the contacts.
+
+Top Heads Here we use the logistic regression only to determine the most important heads. Once they are selected, we discard the weights from the logistic regression and simply average the attention heads corresponding to the top- $k$ weight values. By taking the single best head from ESM-1b, we come close to Gremlin performance given the same data, and averaging the top-5 heads allows us to outperform Gremlin. Averaging the top-10 heads outperforms a full logistic regression on all other Transformer models and comes close to Gremlin given optimized MSAs.
+
+Low-N The second variation we consider is to limit the number of supervised examples provided to the logistic regression. We find that with only a single training example, the model achieves a long range top-L precision of 39.2, which is statistically indistinguishable from Gremlin $( p >$ 0.05). Using only 10 training examples, the model outperforms Gremlin on all the metrics. Since these results depend on the sampled training proteins, we also show a bootstrapped performance distribution using 100 different logistic regression models in Appendix A.10. We find that with 1 protein, performance can vary significantly, with long range top-L precision mean of 35.6, a median of 38.4, and standard deviation 8.9. This variation greatly decreases when training on 20 proteins, with a long range top-L precision mean of 40.1, median of 41.1, and standard deviation of 0.3. See Fig. 12 for the full distribution on all statistics.
+
+MSA Only Finally, we consider supervising the logistic regression only with MSAs instead of real structures. This is the same training data used by the Gremlin baseline. To do this, we first train Gremlin on each MSA. We take the output couplings from Gremlin and mark the top $L$ couplings with sequence separation $\geq 6$ in each protein as true contacts, and everything else as false contacts, creating a binary decision problem. When trained on 20 MSAs, we find that this model achieves a long range $\mathrm { P @ L }$ of 39.9, and generally achieves similar long range performance to Gremlin, while still having superior short and medium range contact precision.
+
+![](images/70162a200a76d8619ae7dfe0734c77d127b8327ba422780a450b8df4d944becd.jpg)  
+Figure 2: Left: Language modeling validation perplexity on holdout of Uniref50 vs. contact precision over the course of pre-training. ESM-1b was trained with different masking so perplexities between the versions are not comparable. Right: Long range $\mathrm { P @ L }$ performance distribution of ESM-1b vs. Gremlin. Each point is colored by the log of the number of sequences in the MSA used to train Gremlin.
+
+# 5.2 ENSEMBLING OVER MSA
+
+Transformer models are fundamentally single-sequence models, but we can further boost performance by ensembling predictions from multiple sequences in the alignment. To do so, we unalign each sequence in the alignment (removing any gaps), pass the resulting sequence through the Transformer and regression, and realign the resulting contact maps to the original aligned indices. For these experiments, we use the logistic regression weights trained on single-sequence inputs, rather than re-training the logistic regression on multi-sequence inputs. We also simply take the first $s$ sequences in the MSA. Table 2 shows performance improvements from averaging over 16, 32, and 64 sequences.
+
+To better understand this result, we return to the single-sequence setting and study the change in prediction when switching between sequences in the alignment. We find that contact precision can vary significantly depending on the exact sequence input to the model, and that the initial query sequence of the MSA does not necessarily generate the highest contact precision (Fig. 9).
+
+Lastly, Alley et al. (2019) presented a method of fine-tuning where a pretrained language model is further trained on the MSA of the sequence of interest (‘evotuning’). Previously this has only been investigated for function prediction and for relatively low-capacity models. We fine-tune the full ESM-1b model (which has $5 0 \mathrm { x }$ more parameters than UniRep) on 380 protein sequence families. We find that after 30 epochs of fine-tuning, long range $\mathrm { P @ L }$ increases only slightly, with an average of 1.6 percentage points (Fig. 16).
+
+# 5.3 PERFORMANCE DISTRIBUTION
+
+Although our model is, on average, better than Gremlin at detecting contacts, the performance distribution over all sequences in the dataset is still mixed. ESM-1b is consistently better at extracting short and medium range contacts (Fig. 7), but only slightly outperforms Gremlin on long range contacts when Gremlin has access to Uniref100 and metagenomic sequences. Fig. 2 shows the distribution of long range $\mathrm { P @ L }$ for ESM-1b vs. Gremlin. Overall, ESM-1b has higher long range $\mathrm { P @ L }$ on $55 \%$ of sequences in the test set.
+
+In addition, we examine the relationship between MSA depth and precision for short, medium, and long range contacts (Fig. 3). Although our contact prediction pipeline does not make explicit use of MSAs, there is still some correlation between MSA depth and performance, since MSA depth is a measure of how many related sequences are present in the ESM-1b training set. We again see that ESM-1b consistently outperforms Gremlin at all MSA depths for short and medium range sequences. We also confirm that ESM-1b outperforms Gremlin for long range contact extraction for sequences with small MSAs (depth $< 1 0 0 0$ ). ESM-1b also outperforms Gremlin on sequences with the very largest MSAs (depth $> 1 6 0 0 0 \AA$ , which is consistent with prior work showing that Gremlin performance plateaus for very large MSAs and suggests that ESM-1b does not suffer from the same issues (Anishchenko et al., 2017).
+
+![](images/7e29f1a65647573d6de45614231ae2b54cf084b2eb067a08b51bca01c2968db2.jpg)  
+Figure 3: Gremlin (trRosetta) performance binned by MSA depth. For comparison, ESM-1b performance is also shown for the sequences in each bin.
+
+# 5.4 LOGISTIC REGRESSION WEIGHTS
+
+In Section 5.1 we show that selecting only a sparse subset of the attention heads can yield good results for contact prediction. Overall, the $L _ { 1 }$ -regularized logistic regression identifies $1 0 2 / 6 6 0$ heads as being predictive of contacts (Fig. 6b). Additionally, we train separate logistic regressions to identify contacts at different ranges . These regressions identify an overlapping, but non-identical set of useful attention heads. Two attention heads have the top-10 highest weights for detecting contacts at all ranges. One attention head is highly positively correlated with local contacts, but highly negatively correlated with long range contacts. Lastly, we identify a total of 104 attention heads that are correlated (positively or negatively) with contacts at only one of the four ranges, suggesting that particular attention heads specialize in detecting certain types of contacts.
+
+# 5.5 PERPLEXITY VS. CONTACT PRECISION
+
+Fig. 2 explores the relationship between performance on the masked language modeling task (validataion perplexity) and contact prediction (Long Range $\mathrm { P } @ \mathrm { L }$ ). A linear relationship exists between validation perplexity and contact precision for each model. Furthermore, for the same perplexity, the 12-layer ESM-1 model achieves the same long range $\mathrm { P @ L }$ as the 34 layer ESM-1 model, suggesting that perplexity is a good proxy task for contact prediction. ESM-1 and ESM-1b models are trained with different masking patterns, so their perplexities cannot be directly compared, although a linear relationship is clearly visible in both. ESM-1 and ESM-1b have a similar number of parameters; the key difference is in their hyperparameters and architecture. The models shown have converged in pre-training, with minimal decrease in perplexity (or increase in contact precision) in the later epochs. This provides clear evidence that both model scale and hyperparameters play a significant role in a model’s ability to learn contacts.
+
+# 5.6 CALIBRATION, FALSE POSITIVES, AND ROBUSTNESS
+
+One concern with neural networks is that, while they may be accurate on average, they can also produce spurious results with high confidence. We investigate this possibility from several perspectives. First, we find that logistic regression probabilities are close to true contact probability (mean-squared error $= 0 . 0 1 4$ ) and can be used directly as a measure of the model’s confidence (Fig. 11).
+
+Second, we analyze the false positives that the model does predict. We find that these are very likely to be within a Manhattan distance of 1-4 of a true contact (Fig. 13a). This suggests that false positives may arise due to the way a contact is defined (Cb-Cb distance within 8 angstroms), and could be marked as true contacts under a different definition (Zheng & Grigoryan, 2017). Further, when we explore an example where the model’s predictions are not near a true contact, we see that the example in question is a homodimer, and that the model is picking up on inter-chain interactions (Fig. 14a). While these do not determine the structure of the monomer, they are important for its function (Anishchenko et al., 2017).
+
+![](images/2eb2c56c190d9e4f724a2c390bdcdc3cb77c1d34deea78059a369759272f615c.jpg)  
+Figure 4: Logistic regression weights trained only on contacts in specific ranges: local [3, 6), short range [6, 12), medium range [12, 24), long range $[ 2 4 , \infty )$ .
+
+Third, we test the robustness of the model to insertions by inserting consecutive alanines at the beginning, middle, or end of 1000 randomly chosen sequences. We find that ESM-1b can tolerate up to 256 insertions at the beginning or end of the sequence and up to 64 insertions in the middle of the sequence before performance starts to significantly degrade. This suggests that ESM-1b learns a robust implicit alignment of the protein sequence. See Appendix A.12 for more details.
+
+# 5.7 MSA GENERATION
+
+Wang & Cho (2019) note that Transformers trained with the MLM objective can be used generatively. Here, we consider whether generations from ESM-1b preserve contact information. The ability to generate sequences that preserve this information is a necessary condition for generation of biologically active proteins (Hawkins-Hooker et al., 2021). We perform this evaluation by taking an input protein, masking out several positions, and re-predicting them. This process is repeated 10000 times to generate a pseudo-MSA for the input sequence (Algorithm 1). We feed the resulting MSA into Gremlin to predict contacts. Over all sequences from our test set, this procedure results in a long range contact $\mathrm { P @ L }$ of 14.5. Fig. 17 shows one example where the procedure works well, with Gremlin on the pseudo-MSA having long range $\mathrm { P @ L }$ of 52.2.
+
+# 6 DISCUSSION
+
+Transformer protein language models trained with an unsupervised objective learn the tertiary structure of a protein sequence in their attention maps. Residue-residue contacts can be extracted from the attention by sparse logistic regression. Attention heads are found that specialize in different types of contacts. An ablation analysis confirms that the contacts are learned without supervision, and that the logistic regression is only necessary to extract the part of the model that represents contacts.
+
+These results have implications for protein structure determination and design. The initial studies proposing Transformers for protein language modeling showed that representation learning could be used to derive state-of-the-art features across a variety of tasks, but were not able to show a benefit in the fully end-to-end setting (Rives et al., 2019; Rao et al., 2019; Elnaggar et al., 2020). For the first time, we show that protein language models can outperform state-of-the-art unsupervised structure learning methods that have been intensively researched and optimized over decades.
+
+Finally, we establish a link between language modeling perplexity and unsupervised structure learning. A similar scaling law has been observed previously for supervised secondary structure prediction (Rives et al., 2019), and parallels observations in the NLP community (Kaplan et al., 2020; Brown et al., 2020). Evidence of scaling laws for protein language modeling support future promise as models and data continue to grow.
+
+# ACKNOWLEDGMENTS
+
+We thank Justas Dauparas for valuable input and initial analysis. Sergey Ovchinnikov was supported by NIH Grant DP5OD026389.
+
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+Table 3: Major Architecture Differences in Protein Transformer Language Models   
+
+<table><tr><td>Name</td><td>Layers</td><td>Hidden Size</td><td>Attn Heads</td><td>Parameters</td><td>Dataset</td></tr><tr><td>TAPE</td><td>12</td><td>768</td><td>12</td><td>92M</td><td>Pfam</td></tr><tr><td>ProtBERT-BFD</td><td>30</td><td>1024</td><td>16</td><td>420M</td><td>BFD100</td></tr><tr><td>ESM-1 (6 layer)</td><td>6</td><td>768</td><td>12</td><td>43M</td><td>Uniref50</td></tr><tr><td>ESM-1 (12 layer)</td><td>12</td><td>768</td><td>12</td><td>85M</td><td>Uniref50</td></tr><tr><td>ESM-1 (34 layer)</td><td>34</td><td>1280</td><td>20</td><td>670M</td><td>Uniref50</td></tr><tr><td>ESM-1b</td><td>33</td><td>1280</td><td>20</td><td>650M</td><td>Uniref50</td></tr></table>
+
+# A APPENDIX
+
+# A.1 NOTATION
+
+In the figures, we report contact precision in the range of 0.0 to 1.0. In the text and in the tables, we report contact precision in terms of percentages, in the range of 0 to 100.
+
+# A.2 AVERAGE PRODUCT CORRECTION (APC)
+
+In protein contact prediction, APC is commonly used to correct for background effects of entropy and phylogeny (Dunn et al., 2008). Given an $L \times L$ coupling matrix $F$ , APC is defined as
+
+$$
+F _ { i j } ^ { \mathrm { A P C } } = F _ { i j } - { \frac { F _ { i } F _ { j } } { F } }
+$$
+
+Where $F _ { i } , F _ { j }$ , and $F$ are the sum over the $i$ -th row, $j$ -th column, and the full matrix respectively.   
+We apply APC independently to the symmetrized attention maps of each head in the Transformer.   
+These corrected attention maps are passed in as input to a logistic regression.
+
+# A.3 GREMLIN IMPLEMENTATION DETAILS
+
+Gremlin is trained by optimizing the pseudolikelihood of $W$ and $V$ , which correspond to pairwise and individual amino acid propensities. The pseudolikelihood approximation models the conditional distributions of the original joint distribution and can be written:
+
+$$
+\log p ( x _ { i } ^ { d } = a | x _ { j \neq i } ^ { d } ; W _ { i } , V _ { i } ) = \log \frac { \exp \left( V _ { i a } + \sum _ { j = 1 , j \neq i } ^ { N } \sum _ { b = 1 } ^ { 2 0 } \mathbb { 1 } ( x _ { j } ^ { d } = b ) W _ { i j a b } \right) } { \sum _ { c = 1 } ^ { 2 0 } \exp \left( V _ { i c } + \sum _ { j = 1 , j \neq i } ^ { N } \sum _ { b = 1 } ^ { 2 0 } \mathbb { 1 } ( x _ { j } ^ { d } = b ) W _ { i j c b } \right) }
+$$
+
+subject to the constraint that $W _ { i i } = 0$ for all $i$ , and that $W _ { i j a b }$ is symmetric in both sequence $( i , j )$ and amino acid $( a , b )$ . Additionally, Gremlin uses a regularization parameter that is adjusted based on the depth of the MSA.
+
+# A.4 ESM-1 IMPLEMENTATION DETAILS
+
+The original ESM-1 models were described in (Rives et al., 2019). ESM-1 is trained on Uniref50 in contrast to the TAPE model, which is trained on Pfam (Finn et al., 2014) and the ProtBERT-BFD model, which is trained on Uniref100 and BFD100 (Steinegger et al., 2019). ESM-1b is a new model, which is the result of an extensive hyperparameter sweep that was performed on smaller 12 layer models. ESM-1b is the result of scaling up that model to 33 layers.
+
+Compared to ESM-1, the main changes in ESM-1b are: higher learning rate; dropout after word embedding; learned positional embeddings; final layer norm before the output; and tied input/output word embeddings. Weights for all ESM-1 and ESM-1b models can be found at https://github.com/facebookresearch/esm.
+
+# A.5 JACKHMMER DETAILS
+
+We use Jackhmmer version 3.3.1 with a bitscore threshold of 27 and 8 iterations to construct MSAs from the ESM training set. The failures on 126 sequences noted in Section 4.4 result from a segmentation fault in hmmbuild after several iterations (the number of successful iterations before the segmentation fault varies depending on the input sequence). Since we see this failure for less than $1 \%$ of the dataset we choose to ignore these sequences during evaluation.
+
+Table 4: Average metrics on 15 CASP13 FM Targets. All baselines use MSAs generated via the trRosetta MSA generation approach.   
+
+<table><tr><td></td><td></td><td colspan="3">6≤sep&lt;12</td><td colspan="3">12≤sep&lt;24</td><td colspan="3">24≤sep</td></tr><tr><td>Model</td><td>Variant</td><td>L</td><td>L/2</td><td>L/5</td><td>L</td><td>L/2</td><td>L/5</td><td>L</td><td>L/2</td><td>L/5</td></tr><tr><td rowspan="3">Baselines</td><td>mfDCA</td><td>11.0</td><td>13.6</td><td>19.7</td><td>12.8</td><td>17.9</td><td>26.2</td><td>14.4</td><td>19.4</td><td>26.6</td></tr><tr><td>PSICOV3</td><td>10.6</td><td>14.0</td><td>18.3</td><td>12.2</td><td>17.1</td><td>25.9</td><td>14.1</td><td>19.8</td><td>27.9</td></tr><tr><td>Gremlin</td><td>12.1</td><td>16.1</td><td>23.6</td><td>14.5</td><td>20.8</td><td>32.5</td><td>16.8</td><td>23.4</td><td>28.5</td></tr><tr><td rowspan="8">ESM-1b (attention)</td><td>top-1 heads</td><td>11.8</td><td>15.8</td><td>23.8</td><td>17.0</td><td>20.8</td><td>29.6</td><td>13.6</td><td>17.9</td><td>22.7</td></tr><tr><td>top-5 heads</td><td>15.3</td><td>20.9</td><td>29.6</td><td>18.7</td><td>27.0</td><td>33.0</td><td>14.6</td><td>20.6</td><td>26.8</td></tr><tr><td>top-10 heads</td><td>16.6</td><td>22.7</td><td>32.1</td><td>21.8</td><td>29.5</td><td>39.8</td><td>17.9</td><td>23.2</td><td>30.4</td></tr><tr><td>n=1,s=1</td><td>16.4</td><td>23.5</td><td>34.7</td><td>23.0</td><td>30.8</td><td>41.6</td><td>18.1</td><td>23.3</td><td>29.9</td></tr><tr><td>n=10, s=1</td><td>18.6</td><td>25.3</td><td>39.3</td><td>24.1</td><td>31.9</td><td>41.4</td><td>18.7</td><td>25.2</td><td>33.2</td></tr><tr><td>n=20, s=1</td><td>19.3</td><td>26.6</td><td>37.0</td><td>24.0</td><td>31.5</td><td>40.2</td><td>18.6</td><td>25.0</td><td>33.8</td></tr><tr><td>MSA, s=1</td><td>14.2</td><td>20.3</td><td>30.5</td><td>21.0</td><td>29.1</td><td>42.3</td><td>18.4</td><td>23.7</td><td>31.5</td></tr><tr><td>n=20</td><td>14.1</td><td>17.7</td><td>19.8</td><td>17.7</td><td>20.9</td><td>27.9</td><td>11.2</td><td>13.9</td><td>17.0</td></tr></table>
+
+Additionally, we evaluated alternate MSAs by running Jackhmmer until a Neff of 128 was achieved (with a maximum of 8 iterations), a procedure described by Zhang et al. (2020). This resulted in very similar, but slightly worse results (average long range $\mathrm { P @ L }$ 29.3, versus 31.3 when always using the output of the eighth iteration). We therefore chose to report results using the 8 iteration maximum.
+
+# A.6 RESULTS ON CASP13
+
+In Table 4 we report results on the 15 CASP13 Free Modeling targets for which PDBs were publicly released. The specific domains evaluated are: T0950-D1, T0957s2-D1, T0960-D2, T0963-D2, T0968s1-D1, T0968s2-D1, T0969-D1, T0980s1-D1, T0986s2-D1, T0990-D1, T0990-D3, T1000- D2, T1021s3-D1, T1021s3-D2, T1022s1-D1. ESM-1b is able to outperform Gremlin, and simply averaging the top-10 heads of ESM-1b has comparable performance to Gremlin.
+
+In addition, we compare our logistic regression model to the bilinear contact prediction model proposed by Rives et al. (2020). This model trains two separate linear projections of the final representation layer and computes contact probabilities via the outer product of the two projections plus a bias term, which generates the following unnormalized log probability:
+
+$$
+\log p ( \mathrm { c o n t a c t } ) \propto ( x W _ { 1 } ) ( x W _ { 2 } ) ^ { T } + b
+$$
+
+Here $x$ is a sequence-length vector of features in $\mathbb { R } ^ { L \times d }$ . Each $W _ { i }$ is a matrix in $\mathbb { R } ^ { d \times k }$ , where $k$ is a hyperparameter controlling the projection size.
+
+We train this model in both the limited supervision $n = 2 0$ ) and full supervision $m = 1 4 2 5 7 ,$ ) setting. For the limited supervision setting, we use the same 20 proteins used to train the sparse logistic regression model. For the full supervision setting we generate a $9 5 / 5 \%$ random training/validation split of the 15008 trRosetta proteins with sequence length $\leq 1 0 2 4$ .
+
+We performed independent grid searches over learning rate, weight decay, and hidden size for the two settings. For the $n = 2 0$ setting, we found a learning rate of 0.001, weight decay of 10.0, and projection size of 512 had best performance on the validation set. For the $n = 1 4 2 5 7$ setting we found a learning rate of 0.001, weight decay of 0.01, and projection size of 512 had best performance on the validation set. All models were trained to convergence to maximize validation long range $\mathrm { P @ L }$ with a patience of 10. The $n = 2 0$ models were trained with a batch size of 20 (i.e. 1 batch $=$ 1 epoch) and the $n = 1 4 2 5 7$ models were trained with a batch size of 128.
+
+![](images/cf1772d9bcd045892dd2af40baab5ab5d26fd86898feb56a41c409ddcd23abc5.jpg)  
+Figure 5: Results on 15 CASP13 FM Domains colored by Neff.
+
+![](images/22499ddfe4388cd341c9a9b1789a26ae2928c85bbdd106739c7963bf5011bd28.jpg)  
+Figure 6: (a) Gridsearch on logistic regression over number of training examples and number regularization penalty. Values shown are long range $\mathrm { P @ L }$ over a validation set of 20 proteins. (b) Per-head and layer weights of the logistic regression on the best ESM-1b model.
+
+The bilinear model performs very poorly in the limited supervision setting, worse than simply taking the top-1 attention head. With full supervision, it moderately outperforms the logistic regression for an increase in long range $\mathrm { P @ L }$ of 1.5 while using $7 0 0 \mathbf { x }$ more data.
+
+In Fig. 5 we display results on the $1 5 \mathrm { F M }$ targets colored by effective number of sequences. ESM-1b shows higher precision at $\mathrm { L }$ and L/5 on average, and is sometimes significantly higher for sequences with low Neff. Since ESM-1b training data was generated prior to CASP13, this suggests ESM-1b is able to generalize well to new sequences.
+
+# A.7 LOGISTIC REGRESSION DETAILS
+
+Given a model with $M$ layers, $H$ heads, and an input sequence $x$ of length $L$ , let $A _ { m h }$ be the $L \times L$ contact map from the $h$ -th head in the $m$ -th layer. We first symmetrize this map and apply APC and let $a _ { m h i j }$ be the coupling weight between sequence position $i$ and $j$ in the resulting map. Then we define the probability of a contact between positions $i$ and $j$ according to a logistic regression with parameters $\beta$ :
+
+$$
+p ( c _ { i j } ^ { d } ; \beta ) = \frac { 1 } { 1 + \exp { \bigg ( - \beta _ { 0 } - \sum _ { m = 1 } ^ { M } \sum _ { h = 1 } ^ { H } \beta _ { m h } a _ { m h i j } ^ { d } \bigg ) } }
+$$
+
+To fit $\beta$ , let $\mathcal { D }$ be a set of training proteins, $k$ be a minimum sequence separation, and $\lambda$ be a regularization weight. The objective can then be defined as follows:
+
+$$
+\mathcal { L } ( \mathcal { D } ; \beta ) = \prod _ { d \in \mathcal { D } } \prod _ { i = 1 } ^ { L _ { d } - k } \prod _ { j = i + k } ^ { L _ { d } } p ( c _ { i j } ^ { d } ; \beta )
+$$
+
+$$
+\hat { \beta } = \operatorname* { m a x } _ { \beta } \mathcal { L } ( \mathcal { D } ; \beta ) + \frac { 1 } { \lambda } \sum _ { m = 1 } ^ { M } \sum _ { h = 1 } ^ { H } | \beta _ { m h } |
+$$
+
+We fit the parameters $\beta$ via scikit-learn (Pedregosa et al., 2011) and do not backpropagate the gradients through the attention weights. In total, our model learns $M H + 1$ parameters, many of which are zero thanks to the $L _ { 1 }$ regularization.
+
+There are three hyperparameters in our training setup: the number of proteins in our training set $\mathcal { D }$ , the regularization parameter $\lambda$ , and the minimum sequence separation of training contacts $k$ . We find that performance improves significantly when increasing the $\mathcal { D }$ from 1 protein to 10 proteins, but that the performance gains drop off when $\mathcal { D }$ increases from 10 to 20 (Fig. 1). Through a hyperparameter sweep, we determined that the optimal $\lambda$ is 0.15. We find that ignoring local contacts $( | i - j | < 6 )$ is also helpful. Therefore, unless otherwise specified, all logistic regressions are trained with $| \mathcal { D } | =$ $2 0 , \lambda = 0 . 1 5 , k = 6$ . See Fig. 6a for a gridsearch over the number of training proteins and regression penalty. We used 20 training proteins and 20 validation proteins for this gridsearch. Fig. 6b shows the weights of the final logistic regression used for ESM-1b.
+
+# A.8 PERFORMANCE DISTRIBUTION
+
+Fig. 7 shows the full distribution of performance of ESM-1b compared with Gremlin. When we provide Gremlin access to Uniref100, along with metagenomic sequences, ESM-1b still consistenly outperforms Gremlin when extracting short and medium range contacts. For long range contacts, Gremlin is much more comparable, and has higher contact precision on $47 \%$ of sequences. With access to the same set of sequences, ESM-1b consistently outperforms Gremlin in detecting short, medium, and long range contacts. This suggests that ESM-1b can much better extract information from the same set of sequences and suggests that further scaling of training data may improve ESM1b even further.
+
+This analysis is further borne out in Fig. 8. Given the same set of sequences, ESM-1b outperforms Gremlin on average for short, medium, and long-range contacts regardless of the depth of the MSA generated from the ESM-1b training set.
+
+Additionally, we find that ESM-1b can provide varying contact maps for different sequences in the MSA (Fig. 9). This is not possible for Gremlin, which is a family-level model. We leverage this in a fairly simple way to provide a modest boost to the contact precision of ESM-1b (Section 5.2).
+
+# A.9 SECONDARY STRUCTURE
+
+In Section 5.4 we show that some heads that detect local contacts (which often correspond to secondary structure) are actually negatively correlated with long range contacts. We test ESM-1b’s ability to detect secondary structure via attention by training a separate logistic regression on the
+
+![](images/62d4f4c3bbcb16a8a7317424e8ebee1fdb38dfc053a2f0402476ecfb93ecc7f3.jpg)  
+Figure 7: Short, medium, and long range $\mathrm { P @ L }$ performance distribution of ESM-1b vs. Gremlin. Each point is colored by the $\log _ { 2 }$ of the number of sequences in the MSA.
+
+![](images/a25da126fde9922ca02e6d81d0f620e380966beda2551fc29e2e43b3e5923752.jpg)  
+Figure 8: Gremlin performance binned by MSA depth using both ESM (top) and trRosetta (bottom) MSAs. For comparison, ESM-1b performance is also shown for the sequences in each bin.
+
+Netsurf dataset (Klausen et al., 2019). As with the logistic regression on contacts, we compute attentions and perform $\mathrm { A P C } +$ symmetrization. To predict the secondary structure of amino acid $i$ , we feed as input the couplings $a _ { m h i j }$ for each layer $m$ , for each head $h$ , and for $j \in [ i - 5 , i + 5 ]$ , for a total of 7260 input features. Using just 100 of the 8678 training proteins, we achieve $7 9 . 3 \%$ accuracy on 3-class secondary structure prediction on the CB513 test set (Cuff & Barton, 1999). Figure 10 shows the importance of each layer to predicting the three secondary structure classes.
+
+![](images/2d740a21b5115abccf7ff91ec71469f6b8e312a442905a5d43451d1d347f1994.jpg)  
+Figure 9: Distribution of contact perplexity when evaluating different sequences from the same MSA. The x-axis shows the index of each sequence, sorted in ascending order by hamming distance from the query sequence (query sequence is always index 0). The y-axis shows long range $\mathrm { P @ L }$ . The black line indicates Gremlin performance on that MSA.
+
+There are spikes in different layers for all three classes, indicating that particular heads within those layers are specializing in detecting specific classes of secondary structure.
+
+Fig. 10 shows importance of each Transformer layer to predicting each of the three secondary structure classes. We see that, as with contact prediction, the most important layers are in the middle layers (14-20) and the final layers (29-33). Some layers spike more heavily on particular contact classes (e.g. layer 33 is important for all classes, but particularly important for $\beta$ -strand prediction). This suggests that particular heads within these layers activate specifically for certain types of secondary structure.
+
+# A.10 BOOTSTRAPPED LOW-N CONFIDENCE INTERVAL
+
+Section 5.1 shows results from Low-N supervision on 1, 10, and 20 proteins. Since performance in this case depends on the particular proteins sampled we use bootstrapping to determine a confidence interval for each of these estimates. Using the full training, validation, and test set of 14882 proteins, we train 100 logistic regression models using a random sample of $N$ proteins, for $N = 1$ , 10, and 20. Each model is then evaluated on the remaining $1 4 8 8 2 - N$ proteins. The full distribution of samples can be seen in Fig. 12. The confidence interval estimates for long range precision at $\mathrm { L }$ with 1, 10, and 20 training proteins are: $3 5 . 6 \pm 1 . 8$ , $4 0 . 6 \pm 0 . 1$ , and $4 1 . 0 \pm 0 . 1$ respectively.
+
+![](images/fd258f24b00d443822ce0ff470cd036028018e9b4bed4bc15c7b13066a48a57c.jpg)  
+Figure 10: $L _ { 2 }$ norm of weights for 3-class secondary structure prediction by Transformer layer.
+
+![](images/34049e333ae96aa4070b344d472ee8765786ae32c3a259bf183cf83415d7113f.jpg)  
+Figure 11: Calibrated probability of a real contact given predicted probability of contact over all test proteins.
+
+![](images/b220f5762d2bed87c4abd92291c27e8fa591e57db9172f203f94d4544927a5d4.jpg)  
+Figure 12: Distribution of precision for all reported statistics using 100 different logistic regression models. Each regression model is trained on a random sample of $N = 1 , 1 0 , 2 0$ proteins.
+
+# A.11 MODEL CALIBRATION AND FALSE POSITIVES
+
+Vig et al. (2020) suggested that the attention probability from the TAPE Transformer was a wellcalibrated estimator for the probability of a contact. In Fig. 11 we examine the same with the logistic regression trained on the ESM-1 and ESM-1b models. We note that ESM-1b, in addition to being more accurate overall than Gremlin, also provides actual probabilities.
+
+We find that as with model accuracy, model calibration increases with larger scale and better hyperparameters. The 6, 12, and 34 layer ESM-1 models have mean-squared error of 0.074, 0.028, and 0.020 between predicted and actual contact probabilities, respectively. ESM-1b has a mean squared error of 0.014. Mean squared error is computed between contact probabilites split into 20 bins according to the scikit-learn calibration curve function. It is therefore reasonable to use the logistic regression probability as a measure of the model’s confidence.
+
+In the case of false positive contacts we attempt to measure the Manhattan distance between the coordinates of predicted contacts and the nearest true contact (Fig. 13a). We observe that the Manhattan distance between the coordinates of false positive contacts are often very close (Manhattan distance between 1-4) to real contacts, and that very few false positives have a Manhattan distance $\geq 1 0$ from a true contact. With a threshold contact probability of 0.5, $8 3 . 8 \%$ of proteins have at least one predict contact with Manhattan distance $> 4$ to the nearest contact. This drops to $7 1 . 7 \%$ with a threshold probability of 0.7, and to $5 2 . 5 \%$ with a threshold probability of 0.9.
+
+![](images/8bba1e887621d9ab41149e210875af5dd6f7558ecbf465c9034aca1d16dc9cbb.jpg)  
+Figure 13: (a) Distribution of Manhattan distance between the coordinates of predicted contacts and the nearest true contact at various thresholds of minimum $p ( c o n t a c t )$ . A distance of zero corresponds to a true contact. (b) Actual counts of predictions by Manhattan distance across the full dataset (note y-axis is in log scale).
+
+![](images/42e0a1f3681081d94adc2a3b02e6ddd8673db181e09a51b310cddd74f516ec28.jpg)  
+Figure 14: Illustration of two modes for ESM-1b where significant numbers of spurious contacts are predicted. (a) Predicted contacts which do occur in the full homodimer complex, but are not present as intra-chain contacts. (b) CTCF protein contacts. A small band of contacts near the 30-residue offdiagonal is predicted by ESM-1b. This band, along with additional similar bands are also predicted by Gremlin.
+
+Fig. 14 highlights two modes for ESM-1b where signficant numbers of spurious contacts are predicted. Fig. 14a shows one example where the model does appear to hallucinate contacts around residues 215 and 415, which do not appear in the contact map for this protein. However, this protein is a homodimer and these contacts are present in the inter-chain contact map. This suggests that some ‘highly incorrect’ false positives may instead be picking up on inter-chain contacts. Fig. 14b shows an example of a repeat protein, for which evolutionary coupling methods are known to pick up on additional ‘bands’ of contacts (Espada et al., 2015; Anishchenko et al., 2017). Multiple bands are visible in the Gremlin contact map, while only the first band, closest to the diagonal, is visible in the ESM-1b contact map. More analysis would be necessary to determine the frequency of these modes, along with additional potential modes.
+
+![](images/ff35cbef5ed3a3098fb8ce41be3b8e9431f28e67c35adc71c1834032e417aaac.jpg)  
+Figure 15: Robustness of ESM-1b and TAPE models to insertions of Alanine at the beginning, middle, and end of sequence
+
+# A.12 ALIGNMENT
+
+One hypothesis as to the benefit of large language models as opposed to simpler Potts models is that they may be able to learn an implicit alignment due to their learned positional embedding. For a Potts Model, an alignment enables a model to relate positions in the sequence given evolutionary context despite the presence of insertions or deletions. We test the robustness of the model to insertions by inserting consecutive alanines at the beginning, middle, or end of 1000 randomly chosen sequences with initial sequence length $< 5 1 2$ (we limit initial sequence length in order to avoid out-of-memory issues after insertion). We find that ESM-1b can tolerate up to 256 insertions at the beginning or end of the sequence and up to 64 insertions in the middle of the sequence before performance starts to significantly degrade. This suggests that ESM-1b learns a robust implicit alignment of the protein sequence.
+
+On the other hand, we find that the TAPE Transformer is less robust to insertions. On one sequence (pdbid: 1a27), we find the TAPE Transformer drops in precision by 12 percentage points after adding just 8 alanines to the beginning of the sequence, while ESM-1b sees minimal degradation until 256 alanines are inserted. We hypothesize that, because TAPE was trained on protein domains, it did not learn to deal with mis-alignments in the input sequence.
+
+# A.13 EVOLUTIONARY FINETUNING DETAILS
+
+We finetuned each model using a learning rate of 1e-4, 16k warmup updates, an inverse square root learning rate schedule, and a maximum of 30 epochs. This resulted in a varying number of total updates depending on the size of the MSA, with larger MSAs being allowed to train for more updates. This should ideally help prevent the model from overfitting too quickly on very small MSAs. We use a variable batch size based on the length of the input proteins, fixing a maximum of 16384 tokens per batch (so for a length 300 protein this would correspond to a batch size of 54). We use MSAs from trRosetta for finetuning all proteins with the exception of avGFP, where we use the same set of sequences from Alley et al. (2019).
+
+![](images/287badb0fa5fb9e72e60cf4f983ce1ba5d12c46c1ee76a464e6b5dfbe69d2c5d.jpg)  
+Figure 16: Left: Average change in contact precision vs. number of finetuning epochs over 380 proteins. Right: Real and predicted contacts before and after evolutionary finetuning for 1a3a and avGFP. For 1a3a, long range $\mathrm { P @ L }$ improves from 54.5 to 61.4. For avGFP, long range $\mathrm { P @ L }$ improves from 7.9 to 11.4.
+
+![](images/25b454aac99e8bb5768459d22291403a1feb51420492b76ee22b428db3d45dc9.jpg)  
+Figure 17: Contacts for 3qhp from Gremlin trained on pseudo-MSA generated by ESM-1b, compared to real and ESM-1b predicted contacts. The generated MSA achieves a long-range $\mathrm { P @ L }$ of 52.2 while the attention maps achieve a precision of 76.7.   
+Algorithm 1: Quickly generate a pseudo-MSA from an input sequence.
+
+# A.14 MSA GENERATION
+
+Result: Generated MSA   
+input $/ /$ protein sequence   
+curr $=$ input $/ /$ optionally, the input can be repeated for batching   
+for $0 \leq i < 1 0 0 0 0$ do masked $=$ mask $20 \%$ of positions in curr; pred $=$ model(masked); curr[masked positions] $=$ pred[masked positions]; MSA.append(curr); if random $. ( ) < 0 . 1$ then curr $=$ input; end   
+end
+
+Algorithm 1 presents the algorithm used to generate pseudo-MSAs from ESM-1b. Each pseudoMSA is passed to GREMLIN in order to evaluate the preservation of contact information (Fig. 17).

parse/train/l2UWXn5iBQI/l2UWXn5iBQI.md

@@ -0,0 +1,341 @@
+# Exponential Graph is Provably Efficient for Decentralized Deep Training
+
+Bicheng $\mathbf { Y i n g ^ { 1 , 3 } }$ ∗, $\mathbf { K u n \ Y u a n ^ { 2 * } }$ , Yiming $\mathbf { C h e n ^ { 2 * } }$ , Hanbin $\mathbf { H } \mathbf { u } ^ { 4 }$ , Pan $\mathbf { P a n } ^ { 2 }$ , Wotao $\mathbf { Y i n ^ { 2 } }$
+
+1 University of California, Los Angeles 2 DAMO Academy, Alibaba Group 3 Google Inc. 4 University of California, Santa Barbara ybc@ucla.edu, {kun.yuan, charles.cym}@alibaba-inc.com, hanbinhu@ucsb.edu, {panpan.pp, wotao.yin}@alibaba-inc.com
+
+# Abstract
+
+Decentralized SGD is an emerging training method for deep learning known for its much less (thus faster) communication per iteration, which relaxes the averaging step in parallel SGD to inexact averaging. The less exact the averaging is, however, the more the total iterations the training needs to take. Therefore, the key to making decentralized SGD efficient is to realize nearly-exact averaging using little communication. This requires a skillful choice of communication topology, which is an under-studied topic in decentralized optimization.
+
+In this paper, we study so-called exponential graphs where every node is connected to $O ( \log ( n ) )$ neighbors and $n$ is the total number of nodes. This work proves such graphs can lead to both fast communication and effective averaging simultaneously. We also discover that a sequence of $\log ( n )$ one-peer exponential graphs, in which each node communicates to one single neighbor per iteration, can together achieve exact averaging. This favorable property enables one-peer exponential graph to average as effective as its static counterpart but communicates more efficiently. We apply these exponential graphs in decentralized (momentum) SGD to obtain the state-of-the-art balance between per-iteration communication and iteration complexity among all commonly-used topologies. Experimental results on a variety of tasks and models demonstrate that decentralized (momentum) SGD over exponential graphs promises both fast and highquality training. Our code is implemented through BlueFog and available at https://github.com/Bluefog-Lib/NeurIPS2021-Exponential-Graph.
+
+# 1 Introduction
+
+Efficient distributed training methods across multiple computing nodes are critical for large-scale modern deep learning tasks. Parallel stochastic gradient descent (SGD) is a widely-used approach, which, at each iteration, computes a globally averaged gradient either using Parameter-Server [28] or All-Reduce [47]. Such global coordination across all nodes in parallel SGD results in either significant bandwidth cost or high latency, which can notably hamper the training scalability.
+
+Decentralized SGD [45, 11, 30, 3] based on partial averaging has been one of the promising alternatives to parallel SGD in distributed deep training. Partial averaging, as opposed to the global averaging exploited in parallel SGD, only requires each node to compute the locally averaged model within its neighborhood. Decentralized SGD does not involve any global operations, so it has much lower communication overhead per iteration. The fewer neighbors each node needs to communicate, the more efficient the per-iteration communication is in decentralized SGD.
+
+Table 1: Comparison between decentralized (momentum) SGD over (some) various commonly-used topologes. The table assumes homogeneous data distributions across all nodes (which is practical for deep training within a data-center). The comparison for data-heterogeneous scenarios, and with more other topologies, is listed in Appendix C. The smaller the transient iteration complexity is, the faster decentralized algorithms will converge.   
+
+<table><tr><td>Topology</td><td>Ring</td><td>Grid</td><td>Rand-Graph</td><td>Rand-Match</td><td>Static Exp</td><td> One-peer Exp</td></tr><tr><td>Per-iter Comm.</td><td>(2)</td><td>2(4)</td><td>（）</td><td>(1)</td><td>Ω(log2(n))</td><td>(1)</td></tr><tr><td>Trans. Iters.</td><td>Ω(n7)</td><td>Ω(n5)</td><td>(n3)</td><td>1</td><td>Ω(n³ log2(n))</td><td>Ω(n³log²2(n)）</td></tr></table>
+
+The reduced communication in decentralized SGD comes with a cost: slower convergence. While it can asymptotically achieve the same convergence linear speedup as parallel SGD [30, 3, 25, 64], i.e., the training speed increases proportionally to the number of computing nodes (see the definition in Sec. 2), decentralized SGD requires more iterations to reach that stage due to the ineffectiveness to aggregate information using partial averaging. We refer those iterations before decentralized SGD reaches its linear speedup stage as transient iterations (see the definition in Sec. 2), which is an important metric to measure the influence of partial-averaging [48, 65] on convergence rate of decentralized SGD. The less effective the partial averaging is, the more transient iterations decentralized SGD needs to take. Fig. 1 illustrates the transient iterations of decentralized SGD for the logistic regression problem. It is observed that decentralized SGD can asymptotically converge as fast as parallel SGD, but it requires more iterations (i.e., transient iterations) to reach that stage.
+
+Per-iteration communication and transient iterations in decentralized SGD are determined by the network topology (we also use graph interchangeably with topology). The maximum degree of the graph decides the communication cost while the connectivity influences the transient iteration complexity. Generally speaking, a sparsely-connected topology communicates cheaply but endows decentralized SGD with more transient iterations due to the less effective information aggregation. A skillful choice of network topology, which is critical to achieve balance between periteration communication and transient iteration complexity, is under-studied in literature.
+
+![](images/fe4fdd71eb359ff24cb52a419b362226d510cbefbbb83aaa18627819772c3f6a.jpg)  
+Figure 1: Illustration of transient iters. Experimental setting is in Appendix D.5.
+
+This work studies exponential graphs which are empirically successful [3, 61, 27, 14, 67] but less theoretically understood in deep training. Exponential graphs have two variants. In a static exponential graph, each node communicates to $\lceil \log _ { 2 } ( n ) \rceil$ neighbors (see Sec. 3 and Fig. 2). In one-peer exponential graph, however, each node cycles through all its neighbors, communicating, only, to a single neighbor per iteration (see Sec. 4 and Fig. 2). This paper will first clarify the connectivity and averaging effectiveness of these exponential graphs, and then apply them to decentralized momentum SGD to obtain the state-of-the-art balance between per-iteration communication and transient iteration complexity among all commonly-used topologies. Our main results (as well as our contributions) are:
+
+• We prove that the spectral gap, which is used to measure the connectivity of the graph (see the definition in Sec. 2), of the static exponential graph is upper bounded by $O ( 1 / \log _ { 2 } ( n ) )$ . Before us, many literatures (e.g. [27]) claimed its upper bound to be $O ( 1 )$ incorrectly. Since one-peer exponential graphs are time-varying, it is difficult to derive their spectral gaps. However, we establish that any $\log _ { 2 } ( n )$ consecutive sequence of one-peer exponential graphs can together achieve exact averaging when $n$ is a power of 2. • With the above results, we establish that one-peer exponential graph, though much sparser than its static counterpart, surprisingly endows decentralized momentum SGD with the same convergence rate as static exponential graph in terms of the best-known bounds. • We derive that exponential graphs achieve $\tilde { \Omega } ( 1 ) ^ { 2 }$ per-iteration communication and $\tilde { \Omega } ( n ^ { 3 } )$ transient iterations, both of which are nearly the best among other known topologies, see Table 1. The one-peer exponential graph is particularly recommended for decentralized deep training. • We conduct extensive industry-level experiments across different tasks and models with various decentralized methods, graphs, and network size to validate our theoretical results.
+
+![](images/fafab3400e8cc6a5cffd06e12d66ab336a93712d1652392d964fe2eda168bd6f.jpg)  
+Figure 2: Illustration of the static and one-peer exponential graph.
+
+2 Revisit Decentralized Momentum SGD and Related Works
+
+This section reviews basic concepts and existing results on decentralized momentum SGD.
+
+Problem. Suppose $n$ computing nodes cooperate to solve the distributed optimization problem:
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } f ( x ) = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } f _ { i } ( x ) \quad { \mathrm { w h e r e } } \quad f _ { i } ( x ) : = \mathbb { E } _ { \xi _ { i } \sim D _ { i } } F ( x ; \xi _ { i } ) .
+$$
+
+Function $f _ { i } ( x )$ is local to node $i$ , and random variable $\xi _ { i }$ denotes the local data that follows distribution $D _ { i }$ . We do not assume each distribution $D _ { i }$ is the same across all nodes.
+
+Network topology and weights. Decentralized methods are based on partial averaging within neighborhood that is defined by the network topology (see the figure 2 as an example of six nodes). We assume all computing nodes are connected by a (directed or undirected) network topology. We define $w _ { i j }$ , the weight to scale information flowing from node $j$ to node $i$ , as follows:
+
+$$
+w _ { i j } \left\{ { \begin{array} { l l } { > 0 } & { { \mathrm { ~ i f ~ n o d e ~ } } j { \mathrm { ~ i s } } } \\ { = 0 } & { { \mathrm { ~ o t h e r w i s e . } } } \end{array} } \right.
+$$
+
+$\mathcal { N } _ { i } : = \{ j | w _ { i j } > 0 \}$ is defined as the set of neighbors of node $i$ which also includes node $i$ itself and the weight matrix $W : = [ w _ { i j } ] _ { i , j = 1 } ^ { n } \in \mathbb { R } ^ { n \times n } .$ are denoted as a matrix that stacks the weights of all nodes. This matrix $W$ characterizes the sparsity and connectivity of the underlying network topology.
+
+Decentralized momentum SGD $\mathbf { ( D m S G D ) }$ . There are many variants of decentralized momentum SGD [3, 20, 32, 67]. This paper will focus on the one proposed by [64] (listed in Algorithm 1), which imposes an additional partialaveraging over the momentum to achieve further speed up. The topology is allowed to change with iterations. When ${ \check { W } } ^ { k } \equiv W$ , topology and weight matrix will remain static.
+
+Assumptions. We introduce several standard assumptions to facilitate future analysis:
+
+# Algorithm 1 DmSGD
+
+Initialize $\gamma _ { - }$ , x(0)i ; let m(0i $m _ { i } ^ { ( 0 ) } = 0 , \beta \in ( 0 , 1 )$ For $k = 0 , 1 , 2 , . . . , T - 1$ , every node $i$ do Sample weight matrix $W ^ { ( k ) }$ ; Update gradient $g _ { i } ^ { ( k ) } = \nabla F ( x _ { i } ^ { ( k ) } ; \xi _ { i } ^ { ( k ) } )$ ; $\begin{array} { r } { m _ { i } ^ { ( k + 1 ) } = \sum _ { j \in \mathcal { N } _ { i } } w _ { i j } ^ { ( k ) } \big ( \beta m _ { j } ^ { ( k ) } + g _ { j } ^ { ( k ) } \big ) } \end{array}$ ; $\begin{array} { r } { x _ { i } ^ { ( k + 1 ) } = \sum _ { j \in \mathcal { N } _ { i } } w _ { i j } ^ { ( k ) } \big ( x _ { j } ^ { ( k ) } - \gamma m _ { j } ^ { ( k ) } \big ) } \end{array}$ ;
+
+A.1 [SMOOTHNESS] Each $f _ { i } ( x )$ is $L$ -smooth, i.e., $\| \nabla f _ { i } ( x ) - \nabla f _ { i } ( y ) \| \leq L \| x - y \|$ for any $x , y \in \mathbb { R } ^ { d }$
+
+A.2 [GRADIENT NOISE] The random sample $\xi _ { i } ^ { ( k ) }$ is independent of each other for any $k$ and $i$ . We also assume $\mathbb { E } [ \nabla F ( x ; \xi _ { i } ) ] = \nabla f _ { i } ( x )$ and $\begin{array} { r } { \hat { \mathbb { E } } \| \nabla \dot { F } ( x ; \xi _ { i } ) - \mathbf { \bar { \nabla } } f _ { i } ( x ) \| ^ { 2 } \leq \sigma ^ { 2 } } \end{array}$ .
+
+A.3 [DATA HETEROGENEITY] It holds that $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \| \nabla f _ { i } ( x ) - \nabla f ( x ) \| ^ { 2 } \leq b ^ { 2 } } \end{array}$ for any $\boldsymbol { x } \in \mathbb { R } ^ { d }$
+
+A.4 [WEIGHT MATRIX] The weight matrix $W ^ { ( k ) }$ is doubly-stochastic, i.e. $W ^ { ( k ) } \mathbb { 1 } ~ = ~ \mathbb { 1 }$ and $\mathbb { 1 } ^ { T } W ^ { ( k ) } = \mathbb { 1 } ^ { T }$ . If $W ^ { ( k ) } \equiv W$ , we assume $\begin{array} { r } { \rho ( W ) : = \operatorname* { m a x } _ { \lambda _ { i } ( W ) \neq 1 } \{ | \lambda _ { i } ( W ) | \} \in ( 0 , 1 ) } \end{array}$ , where $\lambda _ { i } ( W )$ is the $i -$ th eigenvalue of the matrix $W$ . 3
+
+The quantity $1 - \rho$ , which is also referred to as the spectral gap of the weight matrix $W$ , measures how well the topology is connected [53]. In the large and sparse topology which is most valuable to deep training, it typically holds that $1 - \rho \to 0$ .
+
+Communication overhead. According to [5], global averaging across $n$ nodes either incurs $\Omega ( n )$ bandwidth cost via Parameter-Server, or $\Omega ( n )$ latency via Ring-Allreduce. In either way, it takes $\Omega ( n )$ per-iteration communication time, which is proportional to the network size $n$ . As to decentralized methods, we will similarly assume the per-iteration communication time to be $\Omega$ (maximum degree). Convergence. Under Assumptions A.1–A.4, DmSGD with static topology will converge at [64, 25]:
+
+$$
+\frac { 1 } { T } \sum _ { k = 1 } ^ { T } \mathbb { E } \| \nabla f ( \bar { \mathbf { x } } ^ { ( k ) } ) \| ^ { 2 } = O \left( \frac { \sigma ^ { 2 } } { \sqrt { n T } } + \frac { n \sigma ^ { 2 } } { T ( 1 - \rho ) } + \frac { n b ^ { 2 } } { T ( 1 - \rho ) ^ { 2 } } \right)
+$$
+
+in which $\begin{array} { r } { \bar { x } ^ { ( k ) } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } x _ { i } ^ { ( k ) } } \end{array}$ . It is worth noting that no analysis in literature, to our knowledge, exists for DmSGD over time-varying topologies with non-convex costs.
+
+Linear speedup. When $T$ is sufficiently large, the first term $1 / \sqrt { n T }$ dominates (3). This also applies to parallel SGD. Decentralized and parall SGDs all require $T = \Omega ( 1 / ( n \epsilon ^ { 2 } ) )$ iterations to reach a desired accuracy $\epsilon$ , which is inversely proportional to $n$ . Therefore, an algorithm is in its linear-speedup stage at $T$ th iteration if, for this $T$ , the term involving $n T$ is dominating the rate.
+
+Transient iterations. Transient iterations are referred to those iterations before an algorithm reaches linear-speedup stage, that is when $T$ is relatively small so non- $\mathbf { \nabla } \cdot n T$ terms still dominate the rate (see illustration in Appendix C). To reach linear speedup, $T$ has to satisfy (derivation in Appendix C)
+
+Homogeneous dat $ \mathrm { 1 : } \quad T = \Omega \left( { \frac { n ^ { 3 } } { ( 1 - \rho ) ^ { 2 } } } \right) \qquad { \mathrm { H e t e r o g e n e o u s ~ d a t a : } } \quad T = \Omega \left( { \frac { n ^ { 3 } } { ( 1 - \rho ) ^ { 4 } } } \right)$
+
+which corresponds to the transient iteration complexity in the homo/hetero-geneous data scenarios.
+
+# 2.1 Related Works
+
+Decentralized deep training. Decentralized optimization originates from the control and signal processing community. The first decentralized algorithms on general optimization problems include decentralized gradient descent [45], diffusion [11, 51] and dual averaging [18]. In the deep learning regime, decentralize SGD, which was established in [30] to achieve the same linear speedup as parallel SGD in convergence rate, has attracted a lot of attentions. Many efforts have been made to extend the algorithm to directed topologies [3, 42], time-varying topologies [25, 42], asynchronous settings [31], and data-heterogeneous scenarios [57, 62, 32, 67]. Techniques such as quantization/compression [2, 8, 26, 24, 58, 36], periodic updates [55, 25, 64], and lazy communication [37, 38, 13] were also integrated into decentralized SGD to further reduce communiation overheads.
+
+Topology influence. The influence of network topology on decentralized SGD was extensively studied in [25, 51, 66, 45, 42, 27]. All these works indicate that a well-connected topology will significantly accelerate decentralized SGD. Two directions have been explored to relieve the influence of network topology. One line of research proposes new algorithms that are less sensitive to topologies. For example, [66, 23, 65, 57, 1] removed data heterogeneity with bias-correction techniques in [68, 29, 62, 40, 69], and [14, 61, 7, 27] utilized periodic global averaging or multiple partial averaging steps. All these methods have improved topology dependence. The other line is to investigate topologies that enable communication-efficient decentralized optimization. [43, 15] examined various topologies (such as ring, grid, torus, expander, etc.) on averaging effectiveness, which, however, are either communication-costly or averaging-ineffective compared to exponential graphs studied in this paper. [41, 6, 9, 10] studied random graphs (such as Erdos-Renyi random graph and random geometric graph) in which each edge is activated randomly. The randomness of the edge activation can cause a highly unbalanced degrees of each node in the graph, which may significantly affect the efficiency in per-iteration communication.
+
+Algorithms with time-varying topologies. Many previous works have studied decentralized algorithms with time-varying topologies. [42] and [44] examined the convergence of decentralized (deterministic) gradient descent and gradient tracking under convex scenarios. [17, 52] investigated gradient tracking under non-convex scenarios, but it did not clarify the influence of the time-varying graphs on convergence rate. In the stochastic scenario, [25] illustrates how decentralized SGD is influenced by time-varying topologies in the non-convex scenario. However, its analysis cannot be directly extended to the decentralized momentum SGD studied in this work.
+
+Another related work is the Matcha method [60] based on disjoint matching decomposition sampling. While similar to Matcha, decentralized SGD over one-peer exponential graphs has several fundamental differences. First, one-peer exponential graph is directed while Matcha only supports undirected and symmetric matching decomposition. Second, the favorable periodic exact-average property of one-peer exponential graphs only holds when sampled cyclicly. However, Matcha only supports independent and random matching samples in analysis. For these reasons, Matcha cannot cover one-peer exponential graphs (especially when momentum is utilized in decentralized SGD).
+
+Note. This paper considers deep training within high-performance data-center clusters, in which all GPUs are connected with high-bandwidth channels and the network topology can be fully controlled. It is not for the wireless network setting in which the topology cannot be changed freely.
+
+# 3 Spectral Gap of Static Exponential Graph
+
+As discussed above, the graph maximum degree decides the per-iteration communication cost while the spectral gap determines the transient iteration complexity (see (4)). It is critical to seek topologies that are both sparse and with large spectral gap $1 - \rho$ simultaneously. In this section, we will establish that the static exponential graph, which was first introduced in [3, 30], is one of such topologies.
+
+In a static exponential graph, each node is assigned an index from 0 to $n - 1$ and will communicate to neighbors that are $2 ^ { \bar { 0 } } , 2 ^ { \bar { 1 } } , \cdot \cdot \cdot , 2 ^ { \lfloor \log _ { 2 } ( n - 1 ) \rfloor }$ hops away. The left plot in Fig. 2 illustrates a directed 6-node exponential network topology. With maximum degree $\lceil \log _ { 2 } ^ { - } ( n ) \rceil$ neighbors, partial averaging over the static exponential graph will take $\Omega ( \log _ { 2 } ( n ) )$ communication time per iteration. However, it remains unclear what the spectral gap is for this topology.
+
+# Weight matrix associated with static exponential graph is defined as follows:
+
+$$
+w _ { i j } ^ { \mathrm { e x p } } = \left\{ \begin{array} { l l } { \frac { 1 } { \lceil \log _ { 2 } ( n ) \rceil + 1 } } & { \mathrm { i f ~ } \log _ { 2 } ( \bmod ( j - i , n ) ) \mathrm { ~ i s ~ a n ~ i n t e g e r ~ o r ~ } i = j } \\ { 0 } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
+$$
+
+An example weight matrix associated with the static exponential graph in Fig. 2 is in Appendix A.1 The following proposition evaluates the spectral gap $1 - \rho$ for weight matrix in (5).
+
+Proposition 1 (SPECTRAL GAP OF STATIC EXPO) The spectral gap of matrix (5), which can also be interpreted as the second largest magnitude of eigenvalues, satisfies (Proof is in Appendix A.2)
+
+$$
+1 - \rho ( W ^ { \mathrm { e x p } } ) \left\{ \begin{array} { l l } { \displaystyle = \frac { 2 } { 1 + \lceil \log _ { 2 } ( n ) \rceil } , w h e n n i s e \nu e n n u m b e r } \\ { \displaystyle < \frac { 2 } { 1 + \lceil \log _ { 2 } ( n ) \rceil } , w h e n n i s o d d n u m b e r } \end{array} \right.
+$$
+
+In addition, we have $\begin{array} { r } { \| W ^ { \mathrm { e x p } } - \frac { 1 } { n } \mathbb { 1 } \mathbb { 1 } ^ { T } \| _ { 2 } = \rho ( W ^ { \mathrm { e x p } } ) } \end{array}$ .
+
+Remark 1 For a general non-symmetric matrix $W$ , it typically holds that $\begin{array} { r } { \| W - \frac { 1 } { n } \pm \Im \Im ^ { T } \| _ { 2 } \neq \rho ( W ) } \end{array}$ Proposition 1 establishes $\begin{array} { r } { \| W ^ { \mathrm { e x p } } - \frac { 1 } { n } \mathbb { 1 } \mathbb { 1 } ^ { T } \| _ { 2 } = \rho ( W ^ { \mathrm { e x p } } ) } \end{array}$ for exponential graph.
+
+Remark 2 The hypercube graph is established in $I 5 9$ , Chapter 16] to have the spectral gap as $1 - \rho ( W ^ { \mathrm { H y p e r C u b e } } ) = 2 / ( 1 + \log _ { 2 } ( n ) )$ . While such spectral gap is on the same order as the exponential graph, there are two fundamental differences between these two graphs: (a) the hypercube graph has to be undirected and the corresponding $W$ is symmetric; (b) the number of vertices of hypercube must be a power of 2, i.e., $n = 2 ^ { \tau }$ for some positive integer $\tau$ . In comparision, the exponential graph is more flexible in the size of the graph structure.
+
+Remark 3 Proposition 1 clarifies the spectral gap of the static exponential graph. Many literatures before this work (such as [27]) claimed the spectral gap to be $O ( 1 )$ , which is not accurate.
+
+The theoretical analysis of Proposition 1 is non-trivial. To evaluate the spectral gap, for any network size $n$ , we have to derive the analytical expression for each eigenvalue using Fourier transform and calculate the magnitudes. The most tricky part is to assert which eigenvalue expression attains the second largest value.
+
+We now numerically validate the established spectral gap. In Fig. 3, we plotted the spectral gap of the static exponential graph with $n$ ranging from 4 to 290. It is observed that the derived gap $\rho = 1 - 2 / ( 1 + \lceil \log _ { 2 } ( n ) \rceil )$ is very tight (see the black dashed line). In fact, it exactly matches the numerical spectral gap when $n$ is even. Moreover, it is also observed the spectral gap of static exponential graph is much smaller than that of ring or grid.
+
+Finally, we compare the spectral gap and maximum degree of the static exponential graph with all other common graphs in Appendix A.3. It is observed that static exponential graph, while with a sightly larger maximum degree, has a significantly smaller spectral gap than ring and grid.
+
+![](images/c6e6fcda0a87523c8d253a37abda487eba34df530571865620d798dd903940d9.jpg)  
+Figure 3: Spectral gap of some topologies.
+
+# 4 One-Peer Exponential Graph Achieves Periodic Exact-Averaging
+
+Static exponential graph incurs $\Omega ( \log _ { 2 } ( n ) )$ communication overhead per iteration. To overcome this issue, [3] proposes to decompose the static exponential graph into a sequence of one-peer graphs, in which each node cycles through all its neighbors, communicating, only, to a single neighbor per iteration, see the right plot in Fig. 2. Apparently, each one-peer realization incurs $\Omega ( 1 ) { \bar { } }$ communication cost, which matches with ring or grid. Since each realization is sparser than the static graph, one may expect DmSGD with one-peer exponential graphs are less effective in aggregating information. In the following, we will establish an interesting result: one-peer is very effective in averaging.
+
+Time-varying weight matrix. We let $\tau = \lceil \log _ { 2 } ( n ) \rceil$ . The weight matrix at iteration $k$ is
+
+$$
+w _ { i j } ^ { ( k ) } = \left\{ \begin{array} { l l } { \frac { 1 } { 2 } } & { \mathrm { i f } \log _ { 2 } ( \bmod ( j - i , n ) ) = \bmod ( k , \tau ) } \\ { \frac { 1 } { 2 } } & { \mathrm { i f } i = j } \\ { 0 } & { \mathrm { o t h e r w i s e } . } \end{array} \right.
+$$
+
+The weight matrix for each realization of the one-peer exponential graphs in Fig. 2 is in Appendix B.1. Since each node communicates to one single neighbor per iteration, the resulting weight matrix is very sparse, with only one non-zero element in the non-diagonal positions per row and column.
+
+Periodic exact-averaging. The periodic exact-averaging property, which was observed by [3] without theoretical justifications, is fundamental to clarify the averaging effectiveness of one-peer exponential graphs. The following lemma proves that the property holds when $n$ is a power of 2.
+
+Lemma 1 (PERIODIC EXACT AVERAGING) Suppose $\tau = \log _ { 2 } ( n )$ is a positive integer. If $W ^ { ( k ) }$ is the weight matrix generated by (7) over the one-peer exponential graphs, it then holds that each $W ^ { ( k ) }$ is doubly-stochastic, i.e. $\mathbf { \dot { W } } ^ { ( \dot { k } ) } \mathbb { 1 } = \mathbb { 1 }$ and $\mathbb { 1 } ^ { \dot { T } } W ^ { ( k ) } \dot { = } \mathbb { 1 } ^ { T }$ . Furthermore, it holds that
+
+$$
+W ^ { ( k + \ell ) } W ^ { ( k + \ell - 1 ) } \cdot \cdot \cdot W ^ { ( k + 1 ) } W ^ { ( k ) } = \frac { 1 } { n } \mathbb { 1 } \mathbb { 1 } ^ { T }
+$$
+
+for any integer $k \geq 0$ and $\ell \geq \tau$ . And equivalently, the consensus residue form holds that
+
+$$
+\left( W ^ { ( k + \ell ) } - { \frac { 1 } { n } } \mathbb { 1 } \mathbb { 1 } ^ { T } \right) \left( W ^ { ( k + \ell - 1 ) } - { \frac { 1 } { n } } \mathbb { 1 } \mathbb { 1 } ^ { T } \right) \cdots \left( W ^ { ( k ) } - { \frac { 1 } { n } } \mathbb { 1 } \mathbb { 1 } ^ { T } \right) = 0
+$$
+
+(Proof is in Appendix B.2).
+
+Remark 4 The assumption that $\log _ { 2 } ( n )$ is a positive integer seems necessary. We numerically tested various one-peer exponential graphs with non-integer $\log _ { 2 } ( n )$ . None of them is endowed with the periodic exact-average property.
+
+Remark 5 When $\log _ { 2 } ( n )$ is a positive integer and each realization of the one-peer exponential graph is sampled without replacement, it is easy to verify that the periodic exact-averaging property still holds. However, if each realization is sampled with replacement, the periodic exact-averaging property generally does not hold unless all realizations are occasionally sampled without repeating.
+
+Remark 6 It is worth noting that an one-peer variant of the hypercube graph is established to achieve exact averaging with $\tau = \log _ { n } ( n )$ steps [54]. Such one-peer hypercube is undirected and symmetric, which is different from the one-peer exponential graph which is directed and asymmetric.
+
+We now numerically validate Lemma 1. To this end, we initialize a vector $\bar { x _ { \mathrm { ~ \in ~ } } } \mathbb { R } ^ { d }$ arbitrarily, and examine how $\begin{array} { r } { \| ( \Pi _ { \ell = 0 } ^ { k } W ^ { ( \ell ) } - \frac { 1 } { n } \pmb { 1 } \pmb { 1 } \| ^ { T } ) x \| } \end{array}$ decreases with iteration $k$ . The weight matrix $W ^ { ( k ) }$ is either static or samples from onepeer exponential graph or bipartite random match graph. In Fig. 4, it is observed that one-peer exponential graphs can achieve exact average after $\log _ { 2 } ( n )$ steps, which coincides with the results in Lemma 1. In contrast, the static exponential and bipartite random match graphs can only achieve the global average asymptotically. The justification for Remarks 4 and 5 is in Appendix B.3.
+
+# 5 DmSGD with Exponential Graphs
+
+![](images/89129067f685321fb71b3eef9cea4444a11c7818b525fdcfb5d21abc483690c2.jpg)  
+Figure 4: Illustration of how consensus residues decay with iterations for various graphs. O.E. and S.E. denote one-peer and static exponential graphs, and R.M. denotes bipartite random match graph.
+
+With the derived property in Sec. 3 and 4, this section will examine the convergence of DmSGD with static and one-peer exponential graphs.
+
+DmSGD with static exponential graph. Based on Proposition 1, we can achieve the convergence rate and transient iterations, by following analysis in [64], of DmSGD with static exponential graph.
+
+Corollary 1 Under Assumptions A.1–A.4, if γ = n(1−β)3√ , DmSGD (Algorithm 1) will converge at
+
+$$
+\frac { 1 } { T } \sum _ { k = 1 } ^ { T } \mathbb { E } \| \nabla f ( \bar { \mathbf { x } } ^ { ( k ) } ) \| ^ { 2 } = O \left( \frac { \sigma ^ { 2 } } { \sqrt { ( 1 - \beta ) n T } } + \frac { n \log _ { 2 } ( n ) ( 1 - \beta ) \sigma ^ { 2 } } { T } + \frac { n ( 1 - \beta ) b ^ { 2 } \log _ { 2 } ^ { 2 } ( n ) } { T } \right)
+$$
+
+Furthermore, the transient iteration complexity of DmSGD over static exponential graph is $O ( n ^ { 3 } \log _ { 2 } ^ { 2 } ( n ) )$ for data-homogeneous scenario and $O ( n ^ { 3 } \log _ { 2 } ^ { 4 } ( n ) )$ for data-heterogeneous scenario.
+
+DmSGD with one-peer exponential graph. With each realization being sparser than its static counterpart, one-peer exponential graph is believed to converge slower. However, the periodic exactaveraging property can help DmSGD achieve the same convergence rate as its static counterpart. Note that DmSGD with one-peer exponential graph is an one-loop algorithm, see Algorithm 1. The DmSGD updates start immediately after sampling one weight matrix.
+
+Theorem 1 We assume $\tau = \log _ { 2 } ( n )$ is a positive integer, and the time-varying weight matrix is√ generated by (7) over one-peer exponential graphs. Under Assumptions A.1–A.4 and $\begin{array} { r } { \gamma = \frac { \sqrt { n ( 1 - \beta ) ^ { 3 } } } { \sqrt { T } } } \end{array}$ DmSGD (Algorithm $^ { l }$ ) will converge at (Proof is in Appendix D.1-D.3).
+
+$$
+\frac { 1 } { T } \sum _ { k = 1 } ^ { T } \mathbb { E } \| \nabla f ( \bar { \mathbf { x } } ^ { ( k ) } ) \| ^ { 2 } = O \left( \frac { \sigma ^ { 2 } } { \sqrt { ( 1 - \beta ) n T } } + \frac { n ( 1 - \beta ) \sigma ^ { 2 } \tau } { T } + \frac { n ( 1 - \beta ) b ^ { 2 } \tau ^ { 2 } } { T } \right) .
+$$
+
+Furthermore, the transient iteration complexity of DmSGD over one-peer exponential graph is $O ( n ^ { 3 } \log _ { 2 } ^ { 2 } ( n ) )$ for data-homogeneous scenario and $O ( n ^ { 3 } \log _ { 2 } ^ { 4 } ( n ) )$ for data-heterogeneous scenario.
+
+Remark 7 Comparing (11) with (10), and noting that $\tau = \log _ { 2 } ( n )$ , we conclude that DmSGD with one-peer graphs converge exactly as fast as with the static counterpart in terms of the established rate bounds. In addition, both graphs endow DmSGD with the same transient iteration complexity.
+
+Remark 8 We can also achieve the convergence rate for decentralized SGD (i.e., DSGD without momentum acceleration) with one-peer exponential graph by setting $\beta = 0$ . It is easy to verify that DSGD with one-peer graphs can also converge as fast as with the static exponential graph.
+
+Remark 9 The convergence rate and transient iteration complexity of DSGD with general mixing matrices sampling strategy are also studied in [25]. However, the results in reference [25] does not cover the scenario with momentum acceleration. As we show in the proof details, it is highly non-trivial to handle momentum.
+
+It is worth noting that the analysis for the above theorem is non-trivial. While it targets on the one-peer exponential graph, the analysis techniques can be extended to the general time-varying topologies. To our best knowledge, it establishes the first result for DSGD with momentum acceleration, over the time-varying topologies, and in the non-convex settings. Existing analysis either focuses on DSGD without momentum [25], or DmSGD with static topologies [64]. In addition, the last two terms in (11), actually, can be further tightened by the spectral gap of one-peer exponential graphs. Since the tightened terms are rather complicated, we leave them to the discussion in Appendix D.4.
+
+State-of-the-art balance between communication and convergence. Table 1 (and tables in Appendix D.5) summarize the per-iteration communication time and transient iteration complexity for all commonly-used topologies. When $n$ is sufficiently large, the term $\log _ { 2 } ( n )$ can be ignored. In this scenario, the exponential graphs (including both static and one-peer variants) achieve state-of-the-art $\tilde { \Omega } ( 1 )$ per-iteration communication and $\tilde { \Omega } ( n ^ { 3 } )$ transient iterations, in which $\tilde { \Omega } ( \cdot )$ hides all logarithm factors. In Appendix D.5, we numerically validate that exponential graphs have smaller transient iteration complexity than ring or grid graph as predicted in Table 1. The comparison between exponential graph with random graphs [41, 6, 9, 10] (such as the Erdos-Renyi graph and geometric random graph) is discussed in Appendix A.3.3.
+
+One-peer exponential graph is recommended for decentralized deep training. It is because onepeer exponential graph endows DmSGD with the same convergence rate as its static counterpart, but incurs strictly less communication overhead per iteration.
+
+# 6 Experiments
+
+This section will validate our theoretical results by extensive deep learning experiments. First, we evaluate how DmSGD with exponential graphs perform against other commonly-used graphs with varying network size. Second, we examine whether one-peer exponential graphs achieve the same convergence rate and accuracy as its static counterpart across different tasks, models, and algorithms.
+
+Metrics. Training time and validation accuracy are two critical metrics to examine the effectiveness of a distributed training algorithm in deep learning. These two metrics are typically evaluated after the algorithm completes a fixed number of epochs (say, 90 epochs). Training time can reflect the communication efficiency while accuracy, though might not be precise, can roughly measure the convergence rate (or iteration complexity). These two metrics are used in most of our experiments.
+
+# 6.1 Setup
+
+We implement all decentralized algorithms with PyTorch [46] 1.8.0 using NCCL 2.8.3 (CUDA 10.1) as the communication backend. For parallel SGD, we used PyTorch’s native Distributed Data Parallel (DDP) module. For the implementation of decentralized methods, we utilize BlueFog [63], which is a high-performance decentralized deep training framework, to facilitate the topology organization, weight matrix generation, and efficient partial averaging. We also follow DDP’s design to enable computation and communication overlap. Each server contains 8 V100 GPUs in our cluster and is treated as one node. The inter-node network fabrics are 25 Gbps TCP as default, which is a common distributed training platform setting.
+
+# 6.2 Exponential graphs enable efficient and high-quality training
+
+In this subsection we evaluate how DmSGD with exponential graphs perform against other commonlyused topologies in the task of image classification.
+
+Implementation. We conduct a series of image classification experiments with the ImageNet-1K [16], which consists of 1,281,167 training images and 50,000 validation images in 1000 classes. We train classification models with different topologies and numbers of nodes to verify our theoretical findings. The training protocol in [21] is used. In details, we train total 90 epochs. The learning rate is warmed up in the first 5 epochs and is decayed by a factor of 10 at 30, 60 and 80-th epoch. The momentum SGD optimizer is used with linear learning rate scaling by default. Experiments are trained in the mixed precision using Pytorch native amp module. We implement DmSGD with all graphs listed in Table 1. The details of each graph is described in Appendix E. For each graph, we test the training time and validation accuracy for DmSGD with GPU numbers ranging from 32 to 256.
+
+Experiment results. The comparison between different graphs (with varying size) in top-1 validation accuracy and training time after 90 epochs is listed in Table 2. Major observations are:
+
+Table 2: Comparison of top-1 validation accuracy $\% )$ and training time (hours) with different topologies.   
+
+<table><tr><td rowspan="2">NODES TOPOLOGY</td><td colspan="2">4(4x8 GPUs)</td><td colspan="2">8(8x8 GPUs)</td><td colspan="2">16(16x8 GPUs)</td><td colspan="2">32(32x8 GPUs)</td></tr><tr><td>ACC.</td><td>TIME</td><td>ACC.</td><td>TIME</td><td>ACC.</td><td>TIME</td><td>ACC.</td><td>TIME</td></tr><tr><td>RING</td><td>76.13 ±0.023</td><td>11.6</td><td>76.07 ±0.013</td><td>6.5</td><td>76.08 ±0.026</td><td>3.3</td><td>75.58 ±0.021</td><td>1.8</td></tr><tr><td>GRID</td><td>76.08 ±0.007</td><td>11.6</td><td>76.35 ±0.037</td><td>6.7</td><td>75.88 ±0.011</td><td>3.4</td><td>75.76 ±0.022</td><td>2.0</td></tr><tr><td>BI-RAND.MATCH.</td><td>75.96 ±0.032</td><td>11.1</td><td>76.26 ±0.027</td><td>5.7</td><td>76.07 ±0.012</td><td>2.8</td><td>75.83 ±0.029</td><td>1.5</td></tr><tr><td>RANDOM GRAPH</td><td>75.97 ±0.028</td><td>11.5</td><td>76.01 ±0.033</td><td>7.1</td><td>76.18 ±0.008</td><td>6.7</td><td>76.24 ±0.018</td><td>4.7</td></tr><tr><td>STATIC EXP.</td><td>76.21 ±0.028</td><td>11.6</td><td>76.32 ±0.037</td><td>6.9</td><td>76.30 ±0.007</td><td>4.1</td><td>76.28 ±0.020</td><td>2.5</td></tr><tr><td>ONE-PEER EXP.</td><td>76.28 ±0.063</td><td>11.1</td><td>76.47 ±0.035</td><td>5.7</td><td>76.42 ±0.030</td><td>2.8</td><td>76.30 ±0.062</td><td>1.5</td></tr></table>
+
+[1] All graphs (except the random graph) endows DmSGD with training time linear speedup. Among them, bipartite random matching and one-peer exponential graphs achieve the best linear speedup due to their efficient per-iteration communication. However, the accuracy of the matching graph cannot match one-peer exponential graph. The random graph fails to achieve linear speedup because of its extremely expensive communication overheads.
+
+[2] In the $3 2 \times 8$ GPUs scenario, the training time to finish all 90 epochs can be sorted as follows: one-peer $\approx$ Bi-RandMatch $< \mathrm { R i n g } < \mathrm { G r i d } <$ static exponential $<$ random graph, which coincides with the per-iteration communication time listed in Table 1.
+
+[3] In the $3 2 \times 8$ GPUs scenario, the training accuracy achieved by each graph after 90 epochs is sorted as follows: random graph $\approx$ static exponential $\approx$ one-peer $>$ Bi-RandMatch $> \mathrm { G r i d } > \mathrm { R i n g }$ which coincides with the transient iteration complexity listed in Table 1. Note that the random graph is rather dense (see the detail in Appendix A.3.1) so it has good accuracy but consumes significant wall-clock time in training.
+
+With the second and third observations, we can find exponential graphs (especially the one-peer exponential graph) can enable both fast and high-quality training performance. We also examined the performance of exponential graphs when $n$ is not a power of 2, see Appendix E.2.
+
+# 6.3 One-peer exponential graph v.s. static exponential graph
+
+In this subsection we will focus on the two exponential graphs studied in this paper. In particular, we will validate that one-peer exponential graph endows DmSGD with the same convergence rate as its static counterpart (i.e., the conclusion in Remark 7) across different tasks, models, and algorithms.
+
+Comparison across models and algorithms. Now we compare one-peer and static exponential graphs with different neural network architectures and algorithms. The task is image classification and the setting is the same as in Sec. 6.2. We test both graphs for ResNet [22], MobileNetv2 [50] and EfficientNet [56], which are widely-used models in industry. In addition to the DmSGD algorithm (Algorithm 1) studied in this paper, we also examine how exponential graphs perform with other commonly-used decentralized momentum method: the vanilla DmSGD [3] which does not exchange momentum between neighbors, and QG-DmSGD [32] which adds a quasi-global momentum to relieve the influence of data heterogeneity. We do not examine DecentLaM [67] and $\mathrm { D ^ { 2 } }$ [57] because both methods require symmetric weight matrix during the training process which exponential graphs cannot provide. We also list the performance of parallel SGD using global averaging as one baseline.
+
+Table 3 lists the top-1 validation accuracy comparison across all models and algorithms. In all scenarios, it is observed that both graphs can lead to roughly the same accuracy across models and algorithms. The accuracy difference (DIFF) is marginal. We also depict the convergence curves in training loss and accuracy for DmSGD with both graphs in Fig. 5. It shows both
+
+![](images/52f474ec7692e039999038d22622d0f50930b7507149649451ac52bf6583104b.jpg)  
+Figure 5: Convergence curves on the ImageNet (ResNet-50) in terms of training loss and validation top-1 accuracy . Network size is $8 \times 8$ GPUs.
+
+curves evolve closely to each other, indicating that one-peer exponential graph enables $\mathrm { D m } \mathrm { S G D }$ with the same convergence rate as its static counterpart. This is consistent with Theorem 1 and Remark 7. Since one-peer is more communication-efficient than static exponential graph (see Table 2), it is recommended to utilize one-peer exponential graph in decentralized deep training. In addition, we observe that decentralized methods, while utilizing partial-averaging during training process, has no significant accuracy degradation compared parallel SGD. Decentralized SGD can even be superior sometimes.
+
+Table 3: Top-1 validation accuracy and wall-clock time (in hours) comparison with different models and algorithms on ImageNet dataset over static/one-peer exponential graphs (8x8 GPUs).   
+
+<table><tr><td rowspan="2">MODEL TOPOLOGY</td><td colspan="2">REsNET-50</td><td colspan="2">MOBILENET-V2</td><td colspan="2">EFFICIENTNET</td></tr><tr><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td></tr><tr><td>PARALLEL SGD</td><td>76.21 (7.0)</td><td></td><td>70.12 (5.8)</td><td></td><td>77.63 (9.0)</td><td></td></tr><tr><td>VANILLA DMSGD</td><td>76.14 (6.6)</td><td>76.06 (5.5)</td><td>69.98 (5.6)</td><td>69.81 (4.6)</td><td>77.62 (8.4)</td><td>77.48 (6.9)</td></tr><tr><td>DMSGD</td><td>76.50 (6.9)</td><td>76.52(5.7)</td><td>69.62 (5.7)</td><td>69.98 (4.8)</td><td>77.44 (8.7)</td><td>77.51 (7.1)</td></tr><tr><td>QG-DMSGD</td><td>76.43 (6.6)</td><td>76.35(5.6)</td><td>69.83 (5.6)</td><td>69.81 (4.6)</td><td>77.60 (8.4)</td><td>77.72 (6.9)</td></tr></table>
+
+Comparison across different tasks. We next compare the aforementioned algorithms with onepeer and static exponential graphs in another well-known task: object detection. We will test the following widely-used models: Faster-RCNN [49] and RetinaNet [34] on popular PASCAL VOC [19] and COCO [35] datasets. We adopt the MMDetection [12] framework as the building blocks and utilize ResNet-50 with FPN [33] as the backbone network. We choose mean Average Precision (mAP) as the evaluation metric for both datesets. We used 8 GPUs (which are connected by the static or dynamic exponential topology) and set the total batch size as 64 in all detection experiments.
+
+Table 4 compares the performance of decentralized training across different object detection models and datasets. Similar to the above experiment, it is observed that both graphs enable decentralized algorithms with almost the same performance in each scenario. This again illustrates the value of one-peer exponential graph in deep learning tasks - it endows decentralized deep training with both fast training speed and satisfactory accuracy.
+
+Table 4: Comparison of different methods and models on PASCAL VOC and COCO datasets.   
+
+<table><tr><td rowspan="3">DATASET MODEL TOPOLOGY</td><td colspan="4">PASCAL VOC</td><td colspan="4">CoCo</td></tr><tr><td colspan="2">RETINANET</td><td colspan="2">FASTER RCNN</td><td colspan="2">RETINANET</td><td colspan="2">FASTER RCNN</td></tr><tr><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td></tr><tr><td>PARALLEL SGD</td><td>79.0</td><td>-</td><td>80.3</td><td>=</td><td>36.2</td><td>=</td><td>37.2</td><td>=</td></tr><tr><td>VANILLA DMSGD</td><td>79.0</td><td>79.1</td><td>80.7</td><td>80.5</td><td>36.3</td><td>36.1</td><td>37.3</td><td>37.2</td></tr><tr><td>DMSGD</td><td>79.1</td><td>79.0</td><td>80.4</td><td>80.5</td><td>36.4</td><td>36.4</td><td>37.1</td><td>37.0</td></tr><tr><td>QG-DMSGD</td><td>79.2</td><td>79.1</td><td>80.8</td><td>80.4</td><td>36.3</td><td>36.2</td><td>37.2</td><td>37.1</td></tr></table>
+
+# 7 Conclusion and Future Works
+
+In this paper, we establish the spectral gap of static exponential graph and prove that any $\log _ { 2 } ( n )$ consecutive one-peer exponential graphs can together achieve exact averaging when $n$ is a power of 2. With these results, we reveal that one-peer exponential graphs endow DmSGD with the same convergence rate as their static counterpart. We also establish that exponential graphs achieve nearly minimum per-iteration communication time and transient iteration complexity simultaneously when $n$ is large. All conclusions are thoroughly examined with industrial-standard benchmarks. As the future work, we will investigate symmetric time-varying graphs that can perform as well as one-peer exponential graph. Symmetric graphs are critical for $\mathrm { D } ^ { \beth }$ and DecentLaM algorithms.
+
+# Acknowledgements
+
+The authors are grateful to Dr. Sai Praneeth Karimireddy from EPFL for the helpful discussions on the hypercube graph.
+
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parse/train/l2UWXn5iBQI/l2UWXn5iBQI_content_list.json

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+        "type": "text",
+        "text": "Exponential Graph is Provably Efficient for Decentralized Deep Training ",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Bicheng $\\mathbf { Y i n g ^ { 1 , 3 } }$ ∗, $\\mathbf { K u n \\ Y u a n ^ { 2 * } }$ , Yiming $\\mathbf { C h e n ^ { 2 * } }$ , Hanbin $\\mathbf { H } \\mathbf { u } ^ { 4 }$ , Pan $\\mathbf { P a n } ^ { 2 }$ , Wotao $\\mathbf { Y i n ^ { 2 } }$ ",
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+        "text": "1 University of California, Los Angeles 2 DAMO Academy, Alibaba Group 3 Google Inc. 4 University of California, Santa Barbara ybc@ucla.edu, {kun.yuan, charles.cym}@alibaba-inc.com, hanbinhu@ucsb.edu, {panpan.pp, wotao.yin}@alibaba-inc.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Decentralized SGD is an emerging training method for deep learning known for its much less (thus faster) communication per iteration, which relaxes the averaging step in parallel SGD to inexact averaging. The less exact the averaging is, however, the more the total iterations the training needs to take. Therefore, the key to making decentralized SGD efficient is to realize nearly-exact averaging using little communication. This requires a skillful choice of communication topology, which is an under-studied topic in decentralized optimization. ",
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+        "text": "In this paper, we study so-called exponential graphs where every node is connected to $O ( \\log ( n ) )$ neighbors and $n$ is the total number of nodes. This work proves such graphs can lead to both fast communication and effective averaging simultaneously. We also discover that a sequence of $\\log ( n )$ one-peer exponential graphs, in which each node communicates to one single neighbor per iteration, can together achieve exact averaging. This favorable property enables one-peer exponential graph to average as effective as its static counterpart but communicates more efficiently. We apply these exponential graphs in decentralized (momentum) SGD to obtain the state-of-the-art balance between per-iteration communication and iteration complexity among all commonly-used topologies. Experimental results on a variety of tasks and models demonstrate that decentralized (momentum) SGD over exponential graphs promises both fast and highquality training. Our code is implemented through BlueFog and available at https://github.com/Bluefog-Lib/NeurIPS2021-Exponential-Graph. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Efficient distributed training methods across multiple computing nodes are critical for large-scale modern deep learning tasks. Parallel stochastic gradient descent (SGD) is a widely-used approach, which, at each iteration, computes a globally averaged gradient either using Parameter-Server [28] or All-Reduce [47]. Such global coordination across all nodes in parallel SGD results in either significant bandwidth cost or high latency, which can notably hamper the training scalability. ",
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+        "type": "text",
+        "text": "Decentralized SGD [45, 11, 30, 3] based on partial averaging has been one of the promising alternatives to parallel SGD in distributed deep training. Partial averaging, as opposed to the global averaging exploited in parallel SGD, only requires each node to compute the locally averaged model within its neighborhood. Decentralized SGD does not involve any global operations, so it has much lower communication overhead per iteration. The fewer neighbors each node needs to communicate, the more efficient the per-iteration communication is in decentralized SGD. ",
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+        "table_caption": [
+            "Table 1: Comparison between decentralized (momentum) SGD over (some) various commonly-used topologes. The table assumes homogeneous data distributions across all nodes (which is practical for deep training within a data-center). The comparison for data-heterogeneous scenarios, and with more other topologies, is listed in Appendix C. The smaller the transient iteration complexity is, the faster decentralized algorithms will converge. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Topology</td><td>Ring</td><td>Grid</td><td>Rand-Graph</td><td>Rand-Match</td><td>Static Exp</td><td> One-peer Exp</td></tr><tr><td>Per-iter Comm.</td><td>(2)</td><td>2(4)</td><td>（）</td><td>(1)</td><td>Ω(log2(n))</td><td>(1)</td></tr><tr><td>Trans. Iters.</td><td>Ω(n7)</td><td>Ω(n5)</td><td>(n3)</td><td>1</td><td>Ω(n³ log2(n))</td><td>Ω(n³log²2(n)）</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "The reduced communication in decentralized SGD comes with a cost: slower convergence. While it can asymptotically achieve the same convergence linear speedup as parallel SGD [30, 3, 25, 64], i.e., the training speed increases proportionally to the number of computing nodes (see the definition in Sec. 2), decentralized SGD requires more iterations to reach that stage due to the ineffectiveness to aggregate information using partial averaging. We refer those iterations before decentralized SGD reaches its linear speedup stage as transient iterations (see the definition in Sec. 2), which is an important metric to measure the influence of partial-averaging [48, 65] on convergence rate of decentralized SGD. The less effective the partial averaging is, the more transient iterations decentralized SGD needs to take. Fig. 1 illustrates the transient iterations of decentralized SGD for the logistic regression problem. It is observed that decentralized SGD can asymptotically converge as fast as parallel SGD, but it requires more iterations (i.e., transient iterations) to reach that stage. ",
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+    {
+        "type": "text",
+        "text": "Per-iteration communication and transient iterations in decentralized SGD are determined by the network topology (we also use graph interchangeably with topology). The maximum degree of the graph decides the communication cost while the connectivity influences the transient iteration complexity. Generally speaking, a sparsely-connected topology communicates cheaply but endows decentralized SGD with more transient iterations due to the less effective information aggregation. A skillful choice of network topology, which is critical to achieve balance between periteration communication and transient iteration complexity, is under-studied in literature. ",
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+    {
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+        "image_caption": [
+            "Figure 1: Illustration of transient iters. Experimental setting is in Appendix D.5. "
+        ],
+        "image_footnote": [],
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+        "text": "This work studies exponential graphs which are empirically successful [3, 61, 27, 14, 67] but less theoretically understood in deep training. Exponential graphs have two variants. In a static exponential graph, each node communicates to $\\lceil \\log _ { 2 } ( n ) \\rceil$ neighbors (see Sec. 3 and Fig. 2). In one-peer exponential graph, however, each node cycles through all its neighbors, communicating, only, to a single neighbor per iteration (see Sec. 4 and Fig. 2). This paper will first clarify the connectivity and averaging effectiveness of these exponential graphs, and then apply them to decentralized momentum SGD to obtain the state-of-the-art balance between per-iteration communication and transient iteration complexity among all commonly-used topologies. Our main results (as well as our contributions) are: ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "• We prove that the spectral gap, which is used to measure the connectivity of the graph (see the definition in Sec. 2), of the static exponential graph is upper bounded by $O ( 1 / \\log _ { 2 } ( n ) )$ . Before us, many literatures (e.g. [27]) claimed its upper bound to be $O ( 1 )$ incorrectly. Since one-peer exponential graphs are time-varying, it is difficult to derive their spectral gaps. However, we establish that any $\\log _ { 2 } ( n )$ consecutive sequence of one-peer exponential graphs can together achieve exact averaging when $n$ is a power of 2. • With the above results, we establish that one-peer exponential graph, though much sparser than its static counterpart, surprisingly endows decentralized momentum SGD with the same convergence rate as static exponential graph in terms of the best-known bounds. • We derive that exponential graphs achieve $\\tilde { \\Omega } ( 1 ) ^ { 2 }$ per-iteration communication and $\\tilde { \\Omega } ( n ^ { 3 } )$ transient iterations, both of which are nearly the best among other known topologies, see Table 1. The one-peer exponential graph is particularly recommended for decentralized deep training. • We conduct extensive industry-level experiments across different tasks and models with various decentralized methods, graphs, and network size to validate our theoretical results. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fafab3400e8cc6a5cffd06e12d66ab336a93712d1652392d964fe2eda168bd6f.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the static and one-peer exponential graph. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Revisit Decentralized Momentum SGD and Related Works ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This section reviews basic concepts and existing results on decentralized momentum SGD. ",
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+    {
+        "type": "text",
+        "text": "Problem. Suppose $n$ computing nodes cooperate to solve the distributed optimization problem: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/630609129ed9a56f57cb63b53cbbb51164ac484f43df9a1902e534546e63f111.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x ) = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } f _ { i } ( x ) \\quad { \\mathrm { w h e r e } } \\quad f _ { i } ( x ) : = \\mathbb { E } _ { \\xi _ { i } \\sim D _ { i } } F ( x ; \\xi _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Function $f _ { i } ( x )$ is local to node $i$ , and random variable $\\xi _ { i }$ denotes the local data that follows distribution $D _ { i }$ . We do not assume each distribution $D _ { i }$ is the same across all nodes. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Network topology and weights. Decentralized methods are based on partial averaging within neighborhood that is defined by the network topology (see the figure 2 as an example of six nodes). We assume all computing nodes are connected by a (directed or undirected) network topology. We define $w _ { i j }$ , the weight to scale information flowing from node $j$ to node $i$ , as follows: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5856669a5257e5ae810959565574d8c75c0e37325f3bf06f0fddd920d5897342.jpg",
+        "text": "$$\nw _ { i j } \\left\\{ { \\begin{array} { l l } { > 0 } & { { \\mathrm { ~ i f ~ n o d e ~ } } j { \\mathrm { ~ i s } } } \\\\ { = 0 } & { { \\mathrm { ~ o t h e r w i s e . } } } \\end{array} } \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { N } _ { i } : = \\{ j | w _ { i j } > 0 \\}$ is defined as the set of neighbors of node $i$ which also includes node $i$ itself and the weight matrix $W : = [ w _ { i j } ] _ { i , j = 1 } ^ { n } \\in \\mathbb { R } ^ { n \\times n } .$ are denoted as a matrix that stacks the weights of all nodes. This matrix $W$ characterizes the sparsity and connectivity of the underlying network topology. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Decentralized momentum SGD $\\mathbf { ( D m S G D ) }$ . There are many variants of decentralized momentum SGD [3, 20, 32, 67]. This paper will focus on the one proposed by [64] (listed in Algorithm 1), which imposes an additional partialaveraging over the momentum to achieve further speed up. The topology is allowed to change with iterations. When ${ \\check { W } } ^ { k } \\equiv W$ , topology and weight matrix will remain static. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumptions. We introduce several standard assumptions to facilitate future analysis: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 DmSGD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Initialize $\\gamma _ { - }$ , x(0)i ; let m(0i $m _ { i } ^ { ( 0 ) } = 0 , \\beta \\in ( 0 , 1 )$ For $k = 0 , 1 , 2 , . . . , T - 1$ , every node $i$ do Sample weight matrix $W ^ { ( k ) }$ ; Update gradient $g _ { i } ^ { ( k ) } = \\nabla F ( x _ { i } ^ { ( k ) } ; \\xi _ { i } ^ { ( k ) } )$ ; $\\begin{array} { r } { m _ { i } ^ { ( k + 1 ) } = \\sum _ { j \\in \\mathcal { N } _ { i } } w _ { i j } ^ { ( k ) } \\big ( \\beta m _ { j } ^ { ( k ) } + g _ { j } ^ { ( k ) } \\big ) } \\end{array}$ ; $\\begin{array} { r } { x _ { i } ^ { ( k + 1 ) } = \\sum _ { j \\in \\mathcal { N } _ { i } } w _ { i j } ^ { ( k ) } \\big ( x _ { j } ^ { ( k ) } - \\gamma m _ { j } ^ { ( k ) } \\big ) } \\end{array}$ ; ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "A.1 [SMOOTHNESS] Each $f _ { i } ( x )$ is $L$ -smooth, i.e., $\\| \\nabla f _ { i } ( x ) - \\nabla f _ { i } ( y ) \\| \\leq L \\| x - y \\|$ for any $x , y \\in \\mathbb { R } ^ { d }$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A.2 [GRADIENT NOISE] The random sample $\\xi _ { i } ^ { ( k ) }$ is independent of each other for any $k$ and $i$ . We also assume $\\mathbb { E } [ \\nabla F ( x ; \\xi _ { i } ) ] = \\nabla f _ { i } ( x )$ and $\\begin{array} { r } { \\hat { \\mathbb { E } } \\| \\nabla \\dot { F } ( x ; \\xi _ { i } ) - \\mathbf { \\bar { \\nabla } } f _ { i } ( x ) \\| ^ { 2 } \\leq \\sigma ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A.3 [DATA HETEROGENEITY] It holds that $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| \\nabla f _ { i } ( x ) - \\nabla f ( x ) \\| ^ { 2 } \\leq b ^ { 2 } } \\end{array}$ for any $\\boldsymbol { x } \\in \\mathbb { R } ^ { d }$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A.4 [WEIGHT MATRIX] The weight matrix $W ^ { ( k ) }$ is doubly-stochastic, i.e. $W ^ { ( k ) } \\mathbb { 1 } ~ = ~ \\mathbb { 1 }$ and $\\mathbb { 1 } ^ { T } W ^ { ( k ) } = \\mathbb { 1 } ^ { T }$ . If $W ^ { ( k ) } \\equiv W$ , we assume $\\begin{array} { r } { \\rho ( W ) : = \\operatorname* { m a x } _ { \\lambda _ { i } ( W ) \\neq 1 } \\{ | \\lambda _ { i } ( W ) | \\} \\in ( 0 , 1 ) } \\end{array}$ , where $\\lambda _ { i } ( W )$ is the $i -$ th eigenvalue of the matrix $W$ . 3 ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The quantity $1 - \\rho$ , which is also referred to as the spectral gap of the weight matrix $W$ , measures how well the topology is connected [53]. In the large and sparse topology which is most valuable to deep training, it typically holds that $1 - \\rho \\to 0$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Communication overhead. According to [5], global averaging across $n$ nodes either incurs $\\Omega ( n )$ bandwidth cost via Parameter-Server, or $\\Omega ( n )$ latency via Ring-Allreduce. In either way, it takes $\\Omega ( n )$ per-iteration communication time, which is proportional to the network size $n$ . As to decentralized methods, we will similarly assume the per-iteration communication time to be $\\Omega$ (maximum degree). Convergence. Under Assumptions A.1–A.4, DmSGD with static topology will converge at [64, 25]: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/889b47e2e014abbc668e3d92f729cc53dcce44d66cee9db343102df5ee7ca76c.jpg",
+        "text": "$$\n\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { n T } } + \\frac { n \\sigma ^ { 2 } } { T ( 1 - \\rho ) } + \\frac { n b ^ { 2 } } { T ( 1 - \\rho ) ^ { 2 } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "in which $\\begin{array} { r } { \\bar { x } ^ { ( k ) } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } x _ { i } ^ { ( k ) } } \\end{array}$ . It is worth noting that no analysis in literature, to our knowledge, exists for DmSGD over time-varying topologies with non-convex costs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Linear speedup. When $T$ is sufficiently large, the first term $1 / \\sqrt { n T }$ dominates (3). This also applies to parallel SGD. Decentralized and parall SGDs all require $T = \\Omega ( 1 / ( n \\epsilon ^ { 2 } ) )$ iterations to reach a desired accuracy $\\epsilon$ , which is inversely proportional to $n$ . Therefore, an algorithm is in its linear-speedup stage at $T$ th iteration if, for this $T$ , the term involving $n T$ is dominating the rate. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Transient iterations. Transient iterations are referred to those iterations before an algorithm reaches linear-speedup stage, that is when $T$ is relatively small so non- $\\mathbf { \\nabla } \\cdot n T$ terms still dominate the rate (see illustration in Appendix C). To reach linear speedup, $T$ has to satisfy (derivation in Appendix C) ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Homogeneous dat $ \\mathrm { 1 : } \\quad T = \\Omega \\left( { \\frac { n ^ { 3 } } { ( 1 - \\rho ) ^ { 2 } } } \\right) \\qquad { \\mathrm { H e t e r o g e n e o u s ~ d a t a : } } \\quad T = \\Omega \\left( { \\frac { n ^ { 3 } } { ( 1 - \\rho ) ^ { 4 } } } \\right)$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which corresponds to the transient iteration complexity in the homo/hetero-geneous data scenarios. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "2.1 Related Works ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Decentralized deep training. Decentralized optimization originates from the control and signal processing community. The first decentralized algorithms on general optimization problems include decentralized gradient descent [45], diffusion [11, 51] and dual averaging [18]. In the deep learning regime, decentralize SGD, which was established in [30] to achieve the same linear speedup as parallel SGD in convergence rate, has attracted a lot of attentions. Many efforts have been made to extend the algorithm to directed topologies [3, 42], time-varying topologies [25, 42], asynchronous settings [31], and data-heterogeneous scenarios [57, 62, 32, 67]. Techniques such as quantization/compression [2, 8, 26, 24, 58, 36], periodic updates [55, 25, 64], and lazy communication [37, 38, 13] were also integrated into decentralized SGD to further reduce communiation overheads. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Topology influence. The influence of network topology on decentralized SGD was extensively studied in [25, 51, 66, 45, 42, 27]. All these works indicate that a well-connected topology will significantly accelerate decentralized SGD. Two directions have been explored to relieve the influence of network topology. One line of research proposes new algorithms that are less sensitive to topologies. For example, [66, 23, 65, 57, 1] removed data heterogeneity with bias-correction techniques in [68, 29, 62, 40, 69], and [14, 61, 7, 27] utilized periodic global averaging or multiple partial averaging steps. All these methods have improved topology dependence. The other line is to investigate topologies that enable communication-efficient decentralized optimization. [43, 15] examined various topologies (such as ring, grid, torus, expander, etc.) on averaging effectiveness, which, however, are either communication-costly or averaging-ineffective compared to exponential graphs studied in this paper. [41, 6, 9, 10] studied random graphs (such as Erdos-Renyi random graph and random geometric graph) in which each edge is activated randomly. The randomness of the edge activation can cause a highly unbalanced degrees of each node in the graph, which may significantly affect the efficiency in per-iteration communication. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithms with time-varying topologies. Many previous works have studied decentralized algorithms with time-varying topologies. [42] and [44] examined the convergence of decentralized (deterministic) gradient descent and gradient tracking under convex scenarios. [17, 52] investigated gradient tracking under non-convex scenarios, but it did not clarify the influence of the time-varying graphs on convergence rate. In the stochastic scenario, [25] illustrates how decentralized SGD is influenced by time-varying topologies in the non-convex scenario. However, its analysis cannot be directly extended to the decentralized momentum SGD studied in this work. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Another related work is the Matcha method [60] based on disjoint matching decomposition sampling. While similar to Matcha, decentralized SGD over one-peer exponential graphs has several fundamental differences. First, one-peer exponential graph is directed while Matcha only supports undirected and symmetric matching decomposition. Second, the favorable periodic exact-average property of one-peer exponential graphs only holds when sampled cyclicly. However, Matcha only supports independent and random matching samples in analysis. For these reasons, Matcha cannot cover one-peer exponential graphs (especially when momentum is utilized in decentralized SGD). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note. This paper considers deep training within high-performance data-center clusters, in which all GPUs are connected with high-bandwidth channels and the network topology can be fully controlled. It is not for the wireless network setting in which the topology cannot be changed freely. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 Spectral Gap of Static Exponential Graph ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "As discussed above, the graph maximum degree decides the per-iteration communication cost while the spectral gap determines the transient iteration complexity (see (4)). It is critical to seek topologies that are both sparse and with large spectral gap $1 - \\rho$ simultaneously. In this section, we will establish that the static exponential graph, which was first introduced in [3, 30], is one of such topologies. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In a static exponential graph, each node is assigned an index from 0 to $n - 1$ and will communicate to neighbors that are $2 ^ { \\bar { 0 } } , 2 ^ { \\bar { 1 } } , \\cdot \\cdot \\cdot , 2 ^ { \\lfloor \\log _ { 2 } ( n - 1 ) \\rfloor }$ hops away. The left plot in Fig. 2 illustrates a directed 6-node exponential network topology. With maximum degree $\\lceil \\log _ { 2 } ^ { - } ( n ) \\rceil$ neighbors, partial averaging over the static exponential graph will take $\\Omega ( \\log _ { 2 } ( n ) )$ communication time per iteration. However, it remains unclear what the spectral gap is for this topology. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Weight matrix associated with static exponential graph is defined as follows: ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d7f7945e9a0431213e4924d8005c385e7f329885e5b23bcf8e9432976d27d0ed.jpg",
+        "text": "$$\nw _ { i j } ^ { \\mathrm { e x p } } = \\left\\{ \\begin{array} { l l } { \\frac { 1 } { \\lceil \\log _ { 2 } ( n ) \\rceil + 1 } } & { \\mathrm { i f ~ } \\log _ { 2 } ( \\bmod ( j - i , n ) ) \\mathrm { ~ i s ~ a n ~ i n t e g e r ~ o r ~ } i = j } \\\\ { 0 } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "An example weight matrix associated with the static exponential graph in Fig. 2 is in Appendix A.1 The following proposition evaluates the spectral gap $1 - \\rho$ for weight matrix in (5). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Proposition 1 (SPECTRAL GAP OF STATIC EXPO) The spectral gap of matrix (5), which can also be interpreted as the second largest magnitude of eigenvalues, satisfies (Proof is in Appendix A.2) ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/df43abe5d38b5897cb54a44aa43c1a3a05c6647aa510f36ffa4c76def30f5e3e.jpg",
+        "text": "$$\n1 - \\rho ( W ^ { \\mathrm { e x p } } ) \\left\\{ \\begin{array} { l l } { \\displaystyle = \\frac { 2 } { 1 + \\lceil \\log _ { 2 } ( n ) \\rceil } , w h e n n i s e \\nu e n n u m b e r } \\\\ { \\displaystyle < \\frac { 2 } { 1 + \\lceil \\log _ { 2 } ( n ) \\rceil } , w h e n n i s o d d n u m b e r } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In addition, we have $\\begin{array} { r } { \\| W ^ { \\mathrm { e x p } } - \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\| _ { 2 } = \\rho ( W ^ { \\mathrm { e x p } } ) } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 1 For a general non-symmetric matrix $W$ , it typically holds that $\\begin{array} { r } { \\| W - \\frac { 1 } { n } \\pm \\Im \\Im ^ { T } \\| _ { 2 } \\neq \\rho ( W ) } \\end{array}$ Proposition 1 establishes $\\begin{array} { r } { \\| W ^ { \\mathrm { e x p } } - \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\| _ { 2 } = \\rho ( W ^ { \\mathrm { e x p } } ) } \\end{array}$ for exponential graph. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 2 The hypercube graph is established in $I 5 9$ , Chapter 16] to have the spectral gap as $1 - \\rho ( W ^ { \\mathrm { H y p e r C u b e } } ) = 2 / ( 1 + \\log _ { 2 } ( n ) )$ . While such spectral gap is on the same order as the exponential graph, there are two fundamental differences between these two graphs: (a) the hypercube graph has to be undirected and the corresponding $W$ is symmetric; (b) the number of vertices of hypercube must be a power of 2, i.e., $n = 2 ^ { \\tau }$ for some positive integer $\\tau$ . In comparision, the exponential graph is more flexible in the size of the graph structure. ",
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+    {
+        "type": "text",
+        "text": "Remark 3 Proposition 1 clarifies the spectral gap of the static exponential graph. Many literatures before this work (such as [27]) claimed the spectral gap to be $O ( 1 )$ , which is not accurate. ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The theoretical analysis of Proposition 1 is non-trivial. To evaluate the spectral gap, for any network size $n$ , we have to derive the analytical expression for each eigenvalue using Fourier transform and calculate the magnitudes. The most tricky part is to assert which eigenvalue expression attains the second largest value. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now numerically validate the established spectral gap. In Fig. 3, we plotted the spectral gap of the static exponential graph with $n$ ranging from 4 to 290. It is observed that the derived gap $\\rho = 1 - 2 / ( 1 + \\lceil \\log _ { 2 } ( n ) \\rceil )$ is very tight (see the black dashed line). In fact, it exactly matches the numerical spectral gap when $n$ is even. Moreover, it is also observed the spectral gap of static exponential graph is much smaller than that of ring or grid. ",
+        "bbox": [
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+            90,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, we compare the spectral gap and maximum degree of the static exponential graph with all other common graphs in Appendix A.3. It is observed that static exponential graph, while with a sightly larger maximum degree, has a significantly smaller spectral gap than ring and grid. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/c6e6fcda0a87523c8d253a37abda487eba34df530571865620d798dd903940d9.jpg",
+        "image_caption": [
+            "Figure 3: Spectral gap of some topologies. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            563,
+            101,
+            818,
+            258
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 One-Peer Exponential Graph Achieves Periodic Exact-Averaging ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            296,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Static exponential graph incurs $\\Omega ( \\log _ { 2 } ( n ) )$ communication overhead per iteration. To overcome this issue, [3] proposes to decompose the static exponential graph into a sequence of one-peer graphs, in which each node cycles through all its neighbors, communicating, only, to a single neighbor per iteration, see the right plot in Fig. 2. Apparently, each one-peer realization incurs $\\Omega ( 1 ) { \\bar { } }$ communication cost, which matches with ring or grid. Since each realization is sparser than the static graph, one may expect DmSGD with one-peer exponential graphs are less effective in aggregating information. In the following, we will establish an interesting result: one-peer is very effective in averaging. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Time-varying weight matrix. We let $\\tau = \\lceil \\log _ { 2 } ( n ) \\rceil$ . The weight matrix at iteration $k$ is ",
+        "bbox": [
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+            422,
+            754,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56a16c32ec5031afb71990c27e3978a2fc6659592e1690f7340e1c19e38dc926.jpg",
+        "text": "$$\nw _ { i j } ^ { ( k ) } = \\left\\{ \\begin{array} { l l } { \\frac { 1 } { 2 } } & { \\mathrm { i f } \\log _ { 2 } ( \\bmod ( j - i , n ) ) = \\bmod ( k , \\tau ) } \\\\ { \\frac { 1 } { 2 } } & { \\mathrm { i f } i = j } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            443,
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+            496
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The weight matrix for each realization of the one-peer exponential graphs in Fig. 2 is in Appendix B.1. Since each node communicates to one single neighbor per iteration, the resulting weight matrix is very sparse, with only one non-zero element in the non-diagonal positions per row and column. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Periodic exact-averaging. The periodic exact-averaging property, which was observed by [3] without theoretical justifications, is fundamental to clarify the averaging effectiveness of one-peer exponential graphs. The following lemma proves that the property holds when $n$ is a power of 2. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1 (PERIODIC EXACT AVERAGING) Suppose $\\tau = \\log _ { 2 } ( n )$ is a positive integer. If $W ^ { ( k ) }$ is the weight matrix generated by (7) over the one-peer exponential graphs, it then holds that each $W ^ { ( k ) }$ is doubly-stochastic, i.e. $\\mathbf { \\dot { W } } ^ { ( \\dot { k } ) } \\mathbb { 1 } = \\mathbb { 1 }$ and $\\mathbb { 1 } ^ { \\dot { T } } W ^ { ( k ) } \\dot { = } \\mathbb { 1 } ^ { T }$ . Furthermore, it holds that ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/93d9e92849d6527fabfb3d35361e2ebf966007939d90d1cf725e2f83ea0da3e2.jpg",
+        "text": "$$\nW ^ { ( k + \\ell ) } W ^ { ( k + \\ell - 1 ) } \\cdot \\cdot \\cdot W ^ { ( k + 1 ) } W ^ { ( k ) } = \\frac { 1 } { n } \\mathbb { 1 } \\mathbb { 1 } ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            633,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for any integer $k \\geq 0$ and $\\ell \\geq \\tau$ . And equivalently, the consensus residue form holds that ",
+        "bbox": [
+            163,
+            665,
+            754,
+            679
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92e776d87465cc56b7e95f6dc42438afbfb1db43db516432fa0809fad098c814.jpg",
+        "text": "$$\n\\left( W ^ { ( k + \\ell ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) \\left( W ^ { ( k + \\ell - 1 ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) \\cdots \\left( W ^ { ( k ) } - { \\frac { 1 } { n } } \\mathbb { 1 } \\mathbb { 1 } ^ { T } \\right) = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            679,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(Proof is in Appendix B.2). ",
+        "bbox": [
+            173,
+            712,
+            349,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 4 The assumption that $\\log _ { 2 } ( n )$ is a positive integer seems necessary. We numerically tested various one-peer exponential graphs with non-integer $\\log _ { 2 } ( n )$ . None of them is endowed with the periodic exact-average property. ",
+        "bbox": [
+            173,
+            737,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 5 When $\\log _ { 2 } ( n )$ is a positive integer and each realization of the one-peer exponential graph is sampled without replacement, it is easy to verify that the periodic exact-averaging property still holds. However, if each realization is sampled with replacement, the periodic exact-averaging property generally does not hold unless all realizations are occasionally sampled without repeating. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 6 It is worth noting that an one-peer variant of the hypercube graph is established to achieve exact averaging with $\\tau = \\log _ { n } ( n )$ steps [54]. Such one-peer hypercube is undirected and symmetric, which is different from the one-peer exponential graph which is directed and asymmetric. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now numerically validate Lemma 1. To this end, we initialize a vector $\\bar { x _ { \\mathrm { ~ \\in ~ } } } \\mathbb { R } ^ { d }$ arbitrarily, and examine how $\\begin{array} { r } { \\| ( \\Pi _ { \\ell = 0 } ^ { k } W ^ { ( \\ell ) } - \\frac { 1 } { n } \\pmb { 1 } \\pmb { 1 } \\| ^ { T } ) x \\| } \\end{array}$ decreases with iteration $k$ . The weight matrix $W ^ { ( k ) }$ is either static or samples from onepeer exponential graph or bipartite random match graph. In Fig. 4, it is observed that one-peer exponential graphs can achieve exact average after $\\log _ { 2 } ( n )$ steps, which coincides with the results in Lemma 1. In contrast, the static exponential and bipartite random match graphs can only achieve the global average asymptotically. The justification for Remarks 4 and 5 is in Appendix B.3. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 DmSGD with Exponential Graphs ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            267,
+            493,
+            285
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/89129067f685321fb71b3eef9cea4444a11c7818b525fdcfb5d21abc483690c2.jpg",
+        "image_caption": [
+            "Figure 4: Illustration of how consensus residues decay with iterations for various graphs. O.E. and S.E. denote one-peer and static exponential graphs, and R.M. denotes bipartite random match graph. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            566,
+            99,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "With the derived property in Sec. 3 and 4, this section will examine the convergence of DmSGD with static and one-peer exponential graphs. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DmSGD with static exponential graph. Based on Proposition 1, we can achieve the convergence rate and transient iterations, by following analysis in [64], of DmSGD with static exponential graph. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1 Under Assumptions A.1–A.4, if γ = n(1−β)3√ , DmSGD (Algorithm 1) will converge at ",
+        "bbox": [
+            173,
+            387,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cec45730fb2c9210060c288bb0c5b5d3b35dbff42ddaa4c2fbd749f3e6f98735.jpg",
+        "text": "$$\n\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { ( 1 - \\beta ) n T } } + \\frac { n \\log _ { 2 } ( n ) ( 1 - \\beta ) \\sigma ^ { 2 } } { T } + \\frac { n ( 1 - \\beta ) b ^ { 2 } \\log _ { 2 } ^ { 2 } ( n ) } { T } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            421,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, the transient iteration complexity of DmSGD over static exponential graph is $O ( n ^ { 3 } \\log _ { 2 } ^ { 2 } ( n ) )$ for data-homogeneous scenario and $O ( n ^ { 3 } \\log _ { 2 } ^ { 4 } ( n ) )$ for data-heterogeneous scenario. ",
+        "bbox": [
+            173,
+            469,
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+            501
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DmSGD with one-peer exponential graph. With each realization being sparser than its static counterpart, one-peer exponential graph is believed to converge slower. However, the periodic exactaveraging property can help DmSGD achieve the same convergence rate as its static counterpart. Note that DmSGD with one-peer exponential graph is an one-loop algorithm, see Algorithm 1. The DmSGD updates start immediately after sampling one weight matrix. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 We assume $\\tau = \\log _ { 2 } ( n )$ is a positive integer, and the time-varying weight matrix is√ generated by (7) over one-peer exponential graphs. Under Assumptions A.1–A.4 and $\\begin{array} { r } { \\gamma = \\frac { \\sqrt { n ( 1 - \\beta ) ^ { 3 } } } { \\sqrt { T } } } \\end{array}$ DmSGD (Algorithm $^ { l }$ ) will converge at (Proof is in Appendix D.1-D.3). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ecb1531fba5d4f7a8a84f38f08ecaaf5800ba181a8c6ae8d9d2c9d471ffb9aa.jpg",
+        "text": "$$\n\\frac { 1 } { T } \\sum _ { k = 1 } ^ { T } \\mathbb { E } \\| \\nabla f ( \\bar { \\mathbf { x } } ^ { ( k ) } ) \\| ^ { 2 } = O \\left( \\frac { \\sigma ^ { 2 } } { \\sqrt { ( 1 - \\beta ) n T } } + \\frac { n ( 1 - \\beta ) \\sigma ^ { 2 } \\tau } { T } + \\frac { n ( 1 - \\beta ) b ^ { 2 } \\tau ^ { 2 } } { T } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            654,
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+            698
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, the transient iteration complexity of DmSGD over one-peer exponential graph is $O ( n ^ { 3 } \\log _ { 2 } ^ { 2 } ( n ) )$ for data-homogeneous scenario and $O ( n ^ { 3 } \\log _ { 2 } ^ { 4 } ( n ) )$ for data-heterogeneous scenario. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remark 7 Comparing (11) with (10), and noting that $\\tau = \\log _ { 2 } ( n )$ , we conclude that DmSGD with one-peer graphs converge exactly as fast as with the static counterpart in terms of the established rate bounds. In addition, both graphs endow DmSGD with the same transient iteration complexity. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remark 8 We can also achieve the convergence rate for decentralized SGD (i.e., DSGD without momentum acceleration) with one-peer exponential graph by setting $\\beta = 0$ . It is easy to verify that DSGD with one-peer graphs can also converge as fast as with the static exponential graph. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remark 9 The convergence rate and transient iteration complexity of DSGD with general mixing matrices sampling strategy are also studied in [25]. However, the results in reference [25] does not cover the scenario with momentum acceleration. As we show in the proof details, it is highly non-trivial to handle momentum. ",
+        "bbox": [
+            174,
+            854,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "It is worth noting that the analysis for the above theorem is non-trivial. While it targets on the one-peer exponential graph, the analysis techniques can be extended to the general time-varying topologies. To our best knowledge, it establishes the first result for DSGD with momentum acceleration, over the time-varying topologies, and in the non-convex settings. Existing analysis either focuses on DSGD without momentum [25], or DmSGD with static topologies [64]. In addition, the last two terms in (11), actually, can be further tightened by the spectral gap of one-peer exponential graphs. Since the tightened terms are rather complicated, we leave them to the discussion in Appendix D.4. ",
+        "bbox": [
+            174,
+            92,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "State-of-the-art balance between communication and convergence. Table 1 (and tables in Appendix D.5) summarize the per-iteration communication time and transient iteration complexity for all commonly-used topologies. When $n$ is sufficiently large, the term $\\log _ { 2 } ( n )$ can be ignored. In this scenario, the exponential graphs (including both static and one-peer variants) achieve state-of-the-art $\\tilde { \\Omega } ( 1 )$ per-iteration communication and $\\tilde { \\Omega } ( n ^ { 3 } )$ transient iterations, in which $\\tilde { \\Omega } ( \\cdot )$ hides all logarithm factors. In Appendix D.5, we numerically validate that exponential graphs have smaller transient iteration complexity than ring or grid graph as predicted in Table 1. The comparison between exponential graph with random graphs [41, 6, 9, 10] (such as the Erdos-Renyi graph and geometric random graph) is discussed in Appendix A.3.3. ",
+        "bbox": [
+            174,
+            194,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One-peer exponential graph is recommended for decentralized deep training. It is because onepeer exponential graph endows DmSGD with the same convergence rate as its static counterpart, but incurs strictly less communication overhead per iteration. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            383,
+            312,
+            401
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This section will validate our theoretical results by extensive deep learning experiments. First, we evaluate how DmSGD with exponential graphs perform against other commonly-used graphs with varying network size. Second, we examine whether one-peer exponential graphs achieve the same convergence rate and accuracy as its static counterpart across different tasks, models, and algorithms. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Metrics. Training time and validation accuracy are two critical metrics to examine the effectiveness of a distributed training algorithm in deep learning. These two metrics are typically evaluated after the algorithm completes a fixed number of epochs (say, 90 epochs). Training time can reflect the communication efficiency while accuracy, though might not be precise, can roughly measure the convergence rate (or iteration complexity). These two metrics are used in most of our experiments. ",
+        "bbox": [
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+            472,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.1 Setup",
+        "text_level": 1,
+        "bbox": [
+            174,
+            550,
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+            564
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We implement all decentralized algorithms with PyTorch [46] 1.8.0 using NCCL 2.8.3 (CUDA 10.1) as the communication backend. For parallel SGD, we used PyTorch’s native Distributed Data Parallel (DDP) module. For the implementation of decentralized methods, we utilize BlueFog [63], which is a high-performance decentralized deep training framework, to facilitate the topology organization, weight matrix generation, and efficient partial averaging. We also follow DDP’s design to enable computation and communication overlap. Each server contains 8 V100 GPUs in our cluster and is treated as one node. The inter-node network fabrics are 25 Gbps TCP as default, which is a common distributed training platform setting. ",
+        "bbox": [
+            174,
+            569,
+            825,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2 Exponential graphs enable efficient and high-quality training ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            698,
+            637,
+            713
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this subsection we evaluate how DmSGD with exponential graphs perform against other commonlyused topologies in the task of image classification. ",
+        "bbox": [
+            176,
+            718,
+            825,
+            746
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Implementation. We conduct a series of image classification experiments with the ImageNet-1K [16], which consists of 1,281,167 training images and 50,000 validation images in 1000 classes. We train classification models with different topologies and numbers of nodes to verify our theoretical findings. The training protocol in [21] is used. In details, we train total 90 epochs. The learning rate is warmed up in the first 5 epochs and is decayed by a factor of 10 at 30, 60 and 80-th epoch. The momentum SGD optimizer is used with linear learning rate scaling by default. Experiments are trained in the mixed precision using Pytorch native amp module. We implement DmSGD with all graphs listed in Table 1. The details of each graph is described in Appendix E. For each graph, we test the training time and validation accuracy for DmSGD with GPU numbers ranging from 32 to 256. ",
+        "bbox": [
+            174,
+            752,
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+            877
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Experiment results. The comparison between different graphs (with varying size) in top-1 validation accuracy and training time after 90 epochs is listed in Table 2. Major observations are: ",
+        "bbox": [
+            173,
+            883,
+            823,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a92a210f09849421930f1937dfdf8c43b2002eb9695b7c2d4ff974e67878db6f.jpg",
+        "table_caption": [
+            "Table 2: Comparison of top-1 validation accuracy $\\% )$ and training time (hours) with different topologies. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">NODES TOPOLOGY</td><td colspan=\"2\">4(4x8 GPUs)</td><td colspan=\"2\">8(8x8 GPUs)</td><td colspan=\"2\">16(16x8 GPUs)</td><td colspan=\"2\">32(32x8 GPUs)</td></tr><tr><td>ACC.</td><td>TIME</td><td>ACC.</td><td>TIME</td><td>ACC.</td><td>TIME</td><td>ACC.</td><td>TIME</td></tr><tr><td>RING</td><td>76.13 ±0.023</td><td>11.6</td><td>76.07 ±0.013</td><td>6.5</td><td>76.08 ±0.026</td><td>3.3</td><td>75.58 ±0.021</td><td>1.8</td></tr><tr><td>GRID</td><td>76.08 ±0.007</td><td>11.6</td><td>76.35 ±0.037</td><td>6.7</td><td>75.88 ±0.011</td><td>3.4</td><td>75.76 ±0.022</td><td>2.0</td></tr><tr><td>BI-RAND.MATCH.</td><td>75.96 ±0.032</td><td>11.1</td><td>76.26 ±0.027</td><td>5.7</td><td>76.07 ±0.012</td><td>2.8</td><td>75.83 ±0.029</td><td>1.5</td></tr><tr><td>RANDOM GRAPH</td><td>75.97 ±0.028</td><td>11.5</td><td>76.01 ±0.033</td><td>7.1</td><td>76.18 ±0.008</td><td>6.7</td><td>76.24 ±0.018</td><td>4.7</td></tr><tr><td>STATIC EXP.</td><td>76.21 ±0.028</td><td>11.6</td><td>76.32 ±0.037</td><td>6.9</td><td>76.30 ±0.007</td><td>4.1</td><td>76.28 ±0.020</td><td>2.5</td></tr><tr><td>ONE-PEER EXP.</td><td>76.28 ±0.063</td><td>11.1</td><td>76.47 ±0.035</td><td>5.7</td><td>76.42 ±0.030</td><td>2.8</td><td>76.30 ±0.062</td><td>1.5</td></tr></table>",
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+        "type": "text",
+        "text": "[1] All graphs (except the random graph) endows DmSGD with training time linear speedup. Among them, bipartite random matching and one-peer exponential graphs achieve the best linear speedup due to their efficient per-iteration communication. However, the accuracy of the matching graph cannot match one-peer exponential graph. The random graph fails to achieve linear speedup because of its extremely expensive communication overheads. ",
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+        "type": "text",
+        "text": "[2] In the $3 2 \\times 8$ GPUs scenario, the training time to finish all 90 epochs can be sorted as follows: one-peer $\\approx$ Bi-RandMatch $< \\mathrm { R i n g } < \\mathrm { G r i d } <$ static exponential $<$ random graph, which coincides with the per-iteration communication time listed in Table 1. ",
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+        "type": "text",
+        "text": "[3] In the $3 2 \\times 8$ GPUs scenario, the training accuracy achieved by each graph after 90 epochs is sorted as follows: random graph $\\approx$ static exponential $\\approx$ one-peer $>$ Bi-RandMatch $> \\mathrm { G r i d } > \\mathrm { R i n g }$ which coincides with the transient iteration complexity listed in Table 1. Note that the random graph is rather dense (see the detail in Appendix A.3.1) so it has good accuracy but consumes significant wall-clock time in training. ",
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+        "type": "text",
+        "text": "With the second and third observations, we can find exponential graphs (especially the one-peer exponential graph) can enable both fast and high-quality training performance. We also examined the performance of exponential graphs when $n$ is not a power of 2, see Appendix E.2. ",
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+    {
+        "type": "text",
+        "text": "6.3 One-peer exponential graph v.s. static exponential graph ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this subsection we will focus on the two exponential graphs studied in this paper. In particular, we will validate that one-peer exponential graph endows DmSGD with the same convergence rate as its static counterpart (i.e., the conclusion in Remark 7) across different tasks, models, and algorithms. ",
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+        "type": "text",
+        "text": "Comparison across models and algorithms. Now we compare one-peer and static exponential graphs with different neural network architectures and algorithms. The task is image classification and the setting is the same as in Sec. 6.2. We test both graphs for ResNet [22], MobileNetv2 [50] and EfficientNet [56], which are widely-used models in industry. In addition to the DmSGD algorithm (Algorithm 1) studied in this paper, we also examine how exponential graphs perform with other commonly-used decentralized momentum method: the vanilla DmSGD [3] which does not exchange momentum between neighbors, and QG-DmSGD [32] which adds a quasi-global momentum to relieve the influence of data heterogeneity. We do not examine DecentLaM [67] and $\\mathrm { D ^ { 2 } }$ [57] because both methods require symmetric weight matrix during the training process which exponential graphs cannot provide. We also list the performance of parallel SGD using global averaging as one baseline. ",
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+    {
+        "type": "text",
+        "text": "Table 3 lists the top-1 validation accuracy comparison across all models and algorithms. In all scenarios, it is observed that both graphs can lead to roughly the same accuracy across models and algorithms. The accuracy difference (DIFF) is marginal. We also depict the convergence curves in training loss and accuracy for DmSGD with both graphs in Fig. 5. It shows both ",
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+    {
+        "type": "image",
+        "img_path": "images/52f474ec7692e039999038d22622d0f50930b7507149649451ac52bf6583104b.jpg",
+        "image_caption": [
+            "Figure 5: Convergence curves on the ImageNet (ResNet-50) in terms of training loss and validation top-1 accuracy . Network size is $8 \\times 8$ GPUs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "curves evolve closely to each other, indicating that one-peer exponential graph enables $\\mathrm { D m } \\mathrm { S G D }$ with the same convergence rate as its static counterpart. This is consistent with Theorem 1 and Remark 7. Since one-peer is more communication-efficient than static exponential graph (see Table 2), it is recommended to utilize one-peer exponential graph in decentralized deep training. In addition, we observe that decentralized methods, while utilizing partial-averaging during training process, has no significant accuracy degradation compared parallel SGD. Decentralized SGD can even be superior sometimes. ",
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+    {
+        "type": "table",
+        "img_path": "images/d2f76766d684f033377afdcf0e6ce9f12e42200a31d6c4556528efb3bce2b7af.jpg",
+        "table_caption": [
+            "Table 3: Top-1 validation accuracy and wall-clock time (in hours) comparison with different models and algorithms on ImageNet dataset over static/one-peer exponential graphs (8x8 GPUs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">MODEL TOPOLOGY</td><td colspan=\"2\">REsNET-50</td><td colspan=\"2\">MOBILENET-V2</td><td colspan=\"2\">EFFICIENTNET</td></tr><tr><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td></tr><tr><td>PARALLEL SGD</td><td>76.21 (7.0)</td><td></td><td>70.12 (5.8)</td><td></td><td>77.63 (9.0)</td><td></td></tr><tr><td>VANILLA DMSGD</td><td>76.14 (6.6)</td><td>76.06 (5.5)</td><td>69.98 (5.6)</td><td>69.81 (4.6)</td><td>77.62 (8.4)</td><td>77.48 (6.9)</td></tr><tr><td>DMSGD</td><td>76.50 (6.9)</td><td>76.52(5.7)</td><td>69.62 (5.7)</td><td>69.98 (4.8)</td><td>77.44 (8.7)</td><td>77.51 (7.1)</td></tr><tr><td>QG-DMSGD</td><td>76.43 (6.6)</td><td>76.35(5.6)</td><td>69.83 (5.6)</td><td>69.81 (4.6)</td><td>77.60 (8.4)</td><td>77.72 (6.9)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparison across different tasks. We next compare the aforementioned algorithms with onepeer and static exponential graphs in another well-known task: object detection. We will test the following widely-used models: Faster-RCNN [49] and RetinaNet [34] on popular PASCAL VOC [19] and COCO [35] datasets. We adopt the MMDetection [12] framework as the building blocks and utilize ResNet-50 with FPN [33] as the backbone network. We choose mean Average Precision (mAP) as the evaluation metric for both datesets. We used 8 GPUs (which are connected by the static or dynamic exponential topology) and set the total batch size as 64 in all detection experiments. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 4 compares the performance of decentralized training across different object detection models and datasets. Similar to the above experiment, it is observed that both graphs enable decentralized algorithms with almost the same performance in each scenario. This again illustrates the value of one-peer exponential graph in deep learning tasks - it endows decentralized deep training with both fast training speed and satisfactory accuracy. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/89a07688685c93f809adbe8d72b504b6d3c52dcfccab3c956c97406711acd55e.jpg",
+        "table_caption": [
+            "Table 4: Comparison of different methods and models on PASCAL VOC and COCO datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">DATASET MODEL TOPOLOGY</td><td colspan=\"4\">PASCAL VOC</td><td colspan=\"4\">CoCo</td></tr><tr><td colspan=\"2\">RETINANET</td><td colspan=\"2\">FASTER RCNN</td><td colspan=\"2\">RETINANET</td><td colspan=\"2\">FASTER RCNN</td></tr><tr><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td><td>STATIC</td><td>ONE-PEER</td></tr><tr><td>PARALLEL SGD</td><td>79.0</td><td>-</td><td>80.3</td><td>=</td><td>36.2</td><td>=</td><td>37.2</td><td>=</td></tr><tr><td>VANILLA DMSGD</td><td>79.0</td><td>79.1</td><td>80.7</td><td>80.5</td><td>36.3</td><td>36.1</td><td>37.3</td><td>37.2</td></tr><tr><td>DMSGD</td><td>79.1</td><td>79.0</td><td>80.4</td><td>80.5</td><td>36.4</td><td>36.4</td><td>37.1</td><td>37.0</td></tr><tr><td>QG-DMSGD</td><td>79.2</td><td>79.1</td><td>80.8</td><td>80.4</td><td>36.3</td><td>36.2</td><td>37.2</td><td>37.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "7 Conclusion and Future Works ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we establish the spectral gap of static exponential graph and prove that any $\\log _ { 2 } ( n )$ consecutive one-peer exponential graphs can together achieve exact averaging when $n$ is a power of 2. With these results, we reveal that one-peer exponential graphs endow DmSGD with the same convergence rate as their static counterpart. We also establish that exponential graphs achieve nearly minimum per-iteration communication time and transient iteration complexity simultaneously when $n$ is large. All conclusions are thoroughly examined with industrial-standard benchmarks. As the future work, we will investigate symmetric time-varying graphs that can perform as well as one-peer exponential graph. Symmetric graphs are critical for $\\mathrm { D } ^ { \\beth }$ and DecentLaM algorithms. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The authors are grateful to Dr. Sai Praneeth Karimireddy from EPFL for the helpful discussions on the hypercube graph. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+        "type": "text",
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