| Popular beliefs | Your guess | Our guess | Our result | Result figures |
| Pre-training on similar domains nearly always helps compared to training from scratch. | | | | 9a,9b,3 |
| Pre-training,even on similar domains,introduces bias that would harm performance with alarge enough training set. | | | | 3 |
| Self-/un-supervised training performs better than supervised pre-training for small datasets. | | | | 3 |
| Fine-tuning the entire network (vs. just the classification layer)is only helpful if the training set is large. | | | | 9a,9b |
| Increasing network depth,when fine-tuning,harms performance forsmalltraining sets,duetoanoverly complex model. | | | | 4a,4b |
| Increasing network depth, when fine-tuning,is more helpful for larger training sets than smaller ones. | | | | 4a, 4b |
| Increasing network depth,if thebackbone is frozen,is more helpful for smaller trainingsets than larger ones. | | | | 4a, 4b |
| Increasing depth or width improves more than ensembles of smaller networks with the same numberof parameters. | | | | 4e,4a |
| | | | |
| Data augmentation is roughly equivalent to using a m-times larger training set for some m. | | | | 5 |
+
+Table 1: Deep learning quiz! We encourage our readers to judge each claim as T (true) or F (false), and then compare to our guesses and results. In the results column, “T” means the experiments are consistent with the belief, “F” for inconsistent, and “?” for hard to say.
+
+# 2 MODELING LEARNING CURVES
+
+The learning curve measures test error $e _ { t e s t }$ as a function of the number of training samples $n$ for a given classification model and learning method. Previous empirical observations suggest a functional form $e _ { t e s t } ( n ) = \alpha + \eta n ^ { \gamma }$ , with bias-variance trade-off and generalization theories typically indicating $\gamma = - 0 . 5$ . We summarize what bias-variance trade-off and generalization theories (Sec. 2.1) and empirical studies (Sec. 2.2) can tell us about learning curves, and describe our proposed abstraction in Sec. 2.3.
+
+# 2.1 BIAS-VARIANCE TRADE-OFF AND GENERALIZATION THEORY
+
+The bias-variance trade-off is an intuitive and theoretical way to think about generalization. The “bias” is error due to inability of the classifier to encode the optimal decision function, and the “variance” is error due to variations in predictions due to limited availability of training samples for parameter estimation. This is called a trade-off because a classifier with more parameters tends to have less bias but higher variance. Geman et al. (1992) decompose mean squared regression error into bias and variance and explore the implications for neural networks, leading to the conclusion that “identifying the right preconditions is the substantial problem in neural modeling”. This conclusion foreshadows the importance of pretraining, though Geman et al. thought the preconditions must be built in rather than learned. Domingos (2000) extends the analysis to classification. Theoretically, the mean squared error (MSE) can be modeled as $e _ { t e s t } ^ { 2 } ( n ) = \dot { b } i a s ^ { 2 } + n o i s e ^ { 2 } + v a r ( n )$ , where “noise” is irreducible error due to non-unique mapping from inputs to labels, and variance can be modeled as $v a r ( n ) = \sigma ^ { 2 } / n$ for $n$ training samples.
+
+The $\eta n ^ { - 0 . 5 }$ term appears throughout machine learning generalization theory. For example, the bounds based on hypothesis VC-dimension (Vapnik & Chervonenkis, 1971) and Rademacher Complexity (Gnecco & Sanguineti, 2008) are both $\bar { O } ( c n ^ { - 0 . 5 } )$ where $c$ depends on the complexity of the classification model. More recent work also follows this form. We give some examples of bounds in Table 2 without describing all of the parameters because the point is that the test error bounds vary with training size $n$ as a function of $\bar { n ^ { - 0 . 5 } }$ , for all approaches.
+
+Table 2: Generalization bound examples: The bounds each predict generalization error increasing as a function of $n ^ { - 0 . 5 }$ . Note: variable notation is consistent only within each line, except $n$ .
+
+| Popular beliefs | Your guess | Our guess | Our result | Result figures |
| Pre-training on similar domains nearly always helps compared to training from scratch. | | | | 9a,9b,3 |
| Pre-training,even on similar domains,introduces bias that would harm performance with alarge enough training set. | | | | 3 |
| Self-/un-supervised training performs better than supervised pre-training for small datasets. | | | | 3 |
| Fine-tuning the entire network (vs. just the classification layer)is only helpful if the training set is large. | | | | 9a,9b |
| Increasing network depth,when fine-tuning,harms performance forsmalltraining sets,duetoanoverly complex model. | | | | 4a,4b |
| Increasing network depth, when fine-tuning,is more helpful for larger training sets than smaller ones. | | | | 4a, 4b |
| Increasing network depth,if thebackbone is frozen,is more helpful for smaller trainingsets than larger ones. | | | | 4a, 4b |
| Increasing depth or width improves more than ensembles of smaller networks with the same numberof parameters. | | | | 4e,4a |
| | | | |
| Data augmentation is roughly equivalent to using a m-times larger training set for some m. | | | | 5 |
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| Increasing network depth,if thebackbone is frozen,is more helpful for smaller trainingsets than larger ones. | | | | 4a, 4b |
| Increasing depth or width improves more than ensembles of smaller networks with the same numberof parameters. | | | | 4e,4a |
| | | | |
| Data augmentation is roughly equivalent to using a m-times larger training set for some m. | | | | 5 |
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| Pre-training on similar domains nearly always helps compared to training from scratch. | | | | 9a,9b,3 |
| Pre-training,even on similar domains,introduces bias that would harm performance with alarge enough training set. | | | | 3 |
| Self-/un-supervised training performs better than supervised pre-training for small datasets. | | | | 3 |
| Fine-tuning the entire network (vs. just the classification layer)is only helpful if the training set is large. | | | | 9a,9b |
| Increasing network depth,when fine-tuning,harms performance forsmalltraining sets,duetoanoverly complex model. | | | | 4a,4b |
| Increasing network depth, when fine-tuning,is more helpful for larger training sets than smaller ones. | | | | 4a, 4b |
| Increasing network depth,if thebackbone is frozen,is more helpful for smaller trainingsets than larger ones. | | | | 4a, 4b |
| Increasing depth or width improves more than ensembles of smaller networks with the same numberof parameters. | | | | 4e,4a |
| | | | |
| Data augmentation is roughly equivalent to using a m-times larger training set for some m. | | | | 5 |
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| Pre-training on similar domains nearly always helps compared to training from scratch. | | | | 9a,9b,3 |
| Pre-training,even on similar domains,introduces bias that would harm performance with alarge enough training set. | | | | 3 |
| Self-/un-supervised training performs better than supervised pre-training for small datasets. | | | | 3 |
| Fine-tuning the entire network (vs. just the classification layer)is only helpful if the training set is large. | | | | 9a,9b |
| Increasing network depth,when fine-tuning,harms performance forsmalltraining sets,duetoanoverly complex model. | | | | 4a,4b |
| Increasing network depth, when fine-tuning,is more helpful for larger training sets than smaller ones. | | | | 4a, 4b |
| Increasing network depth,if thebackbone is frozen,is more helpful for smaller trainingsets than larger ones. | | | | 4a, 4b |
| Increasing depth or width improves more than ensembles of smaller networks with the same numberof parameters. | | | | 4e,4a |
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| Data augmentation is roughly equivalent to using a m-times larger training set for some m. | | | | 5 |
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+ "html": "| Network | Cw | FGSM | BIM | PGD |
| MNISTconv | Attack iterations = 50 | ∈= 0.3 | ∈= 0.3, α=0.01 No. of iterations = 40 | e= 0.3, α=0.01 No. of iterations = 40 |
| CIFARconv | Attack iterations = 50 | ∈= 0.1 | ∈= 0.1, α=0.01 No. of iterations = 40 | ∈= 0.1, α=0.01 No. of iterations = 40 |
| AlexNet | Attack iterations = 50 | ∈= 0.1 | ∈= 0.1, α=0.01 No. of iterations = 40 | e=0.1, α=0.01 No. of iterations = 5 |
+
+# 5 RESULTS
+
+In this section, we present the results of our experiments highlighting the advantages of EMPIR models.
+
+# 5.1 ROBUSTNESS OF EMPIR MODELS ACROSS ALL ATTACKS
+
+Table 2 presents the results of our experiments across different benchmarks. The EMPIR models presented in the table are the ones exhibiting highest average adversarial accuracies under the constraints of ${ < } 2 5 \%$ compute and memory overhead and $< 2 \%$ loss in unperturbed accuracy. We observed that across all the benchmarks, ensembles comprised of two low-precision and one full-precision model combined with the max-voting ensembling technique satisfy these constraints. However, the individual configurations of the low-precision models, i.e., the precisions of weights and activations in the ensembles, differ across the benchmarks. For example, both low-precision models in the EMPIR model for MNISTconv have weight precisions of 2 bits and activation precisions of 4 bits. On the other hand, the two low-precision models in the AlexNet EMPIR model have $\mathrm { \{ w e i g h t , a c t i v a t i o n \} }$ bit-precisions of $\{ 2 , 2 \}$ and $\{ 4 , 4 \}$ , respectively. In general, we observe that the EMPIR models exhibit substantially higher adversarial accuracies across all attacks for the three benchmarks.
+
+Table 2: Unperturbed and adversarial accuracies of the baseline and EMPIR models across different attacks
+
+| Network | Dataset | Configuration |
| MNISTconv | MNIST | Conv(8×8×64),ReLU, Conv(6×6×128),ReLU, Conv(5×5×128),ReLU,Fully_Connected(10),SoftMax |
| CIFARconv | CIFAR-10 | Conv(5×5×32), ReLU, MaxPool(3×3), Conv(8×8 ×64),ReLU AvgPool(3×3), Conv(8×8 ×64), ReLU, AvgPool(3 × 3), Fully_Connected(64), Fully_Connected(10), SoftMax |
| AlexNet | ImageNet | Conv(12×12×96), ReLU, Conv(5×5×256), BatchNorm, ReLU, MaxPool(3×3), Conv(3×3×384), BatchNorm, ReLU, MaxPool(3×3), Conv(3×3×384), BatchNorm, ReLU, Conv(3×3×256), BatchNorm,ReLU, MaxPool(3 ×3), Fully_Conn(4096), BatchNorm, ReLU, Fully_Conn(4096), BatchNorm,ReLU,Fully_Conn(100O), SoftMax |
+
+# B ROBUSTNESS TO ATTACKS OF VARYING STRENGTHS
+
+To further illustrate the benefits of EMPIR, we observed its robustness under attacks of varying strength. We specifically varied the $\epsilon$ value in the FGSM attack between 0.1 and 0.8 and the number of attack iterations in the CW attack between 10 and 90, and measured the adversarial accuracies of the EMPIR model as well as the baseline FP model for the MNISTconv benchmark. Figure 5 clearly illustrates that EMPIR exhibits higher adversarial accuracies across attacks of different strengths for both FGSM and CW.
+
+
+Figure 5: (a) Adversarial accuracies under FGSM attack of varying strength (b) Adversarial accuracies under CW attack of varying strength
\ No newline at end of file
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new file mode 100644
index 0000000000000000000000000000000000000000..a34b8139d65112836a22e05daca30a561af1ba76
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@@ -0,0 +1,1228 @@
+[
+ {
+ "type": "text",
+ "text": "EMPIR: ENSEMBLES OF MIXED PRECISION DEEP NETWORKS FOR INCREASED ROBUSTNESS AGAINST ADVERSARIAL ATTACKS ",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Sanchari Sen ",
+ "text_level": 1,
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Center for Brain-Inspired Computing \nSchool of Electrical and Computer Engineering \nPurdue University \nWest Lafayette, IN, USA \nsen9@purdue.edu \nBalaraman Ravindran \nDepartment of Computer Science and Engineering \nRobert Bosch Centre for Data Science and AI \nIndian Institute of Technology (IIT) Madras \nChennai, TN, India \nravi@cse.iitm.ac.in \nAnand Raghunathan \nCenter for Brain-Inspired Computing \nSchool of Electrical and Computer Engineering \nPurdue University \nWest Lafayette, IN, USA \nraghunathan@purdue.edu ",
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+ "text": "",
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+ {
+ "type": "text",
+ "text": "ABSTRACT ",
+ "text_level": 1,
+ "bbox": [
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+ ],
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+ {
+ "type": "text",
+ "text": "Ensuring robustness of Deep Neural Networks (DNNs) is crucial to their adoption in safety-critical applications such as self-driving cars, drones, and healthcare. Notably, DNNs are vulnerable to adversarial attacks in which small input perturbations can produce catastrophic misclassifications. In this work, we propose EMPIR, ensembles of quantized DNN models with different numerical precisions, as a new approach to increase robustness against adversarial attacks. EMPIR is based on the observation that quantized neural networks often demonstrate much higher robustness to adversarial attacks than full precision networks, but at the cost of a substantial loss in accuracy on the original (unperturbed) inputs. EMPIR overcomes this limitation to achieve the “best of both worlds”, i.e., the higher unperturbed accuracies of the full precision models combined with the higher robustness of the low precision models, by composing them in an ensemble. Further, as low precision DNN models have significantly lower computational and storage requirements than full precision models, EMPIR models only incur modest compute and memory overheads compared to a single full-precision model $4 \\%$ in our evaluations). We evaluate EMPIR across a suite of DNNs for 3 different image recognition tasks (MNIST, CIFAR-10 and ImageNet) and under 4 different adversarial attacks. Our results indicate that EMPIR boosts the average adversarial accuracies by $4 2 . 6 \\%$ , $1 5 . 2 \\%$ and $1 0 . 5 \\%$ for the DNN models trained on the MNIST, CIFAR-10 and ImageNet datasets respectively, when compared to single full-precision models, without sacrificing accuracy on the unperturbed inputs. ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "1 INTRODUCTION ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "The success of Deep Neural Networks (DNNs) in different machine learning tasks has fueled their use in safety-critical applications like autonomous cars, unmanned aerial vehicles and healthcare, wherein errors (misclassifications) made by DNNs can lead to severe — in the extreme case, fatal — consequences. Therefore, robustness, i.e., the ability to cope with erroneous or malicious inputs fed to an application, is emerging as an important requirement for DNNs. ",
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+ {
+ "type": "text",
+ "text": "Several efforts have in fact shown that DNNs behave in unexpected and incorrect ways for small, specifically designed input perturbations (Goodfellow et al. (2014)). An attacker can take advantage of this behavior to intentionally modify the inputs in a manner that forces the DNN model to mis-classify, and the overall system that uses the DNN to fail. A variety of methods for launching adversarial attacks on DNNs have been proposed over the years. These adversarial attacks systematically modify a given original input to cause a misclassification while keeping the input distortion minimal. A few examples of adversarial attacks that have been successfully applied to various DNN models are the Fast Gradient Sign Method (FGSM) (Goodfellow et al. (2014)), Jacobian-based Saliency Map Attack (JSMA) (Papernot et al. (2015)), Carlini-Wagner (CW) (Carlini & Wagner (2016)) and the Basic Iterative Method (BIM) (Kurakin et al. (2016)). ",
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+ "text": "Prior works have tried to overcome these vulnerabilities by proposing various defense mechanisms against adversarial attacks. Adversarial training (Goodfellow et al. (2014)), defensive distillation (Papernot et al. (2015)) and input gradient regularization (Ross & Doshi-Velez (2017)) are a few representative defense techniques. Each of these approaches, albeit promising, has limitations with respect to the kind of attacks they can defend against, the increase in training complexity, as well as their effect on the model’s accuracy on the original unperturbed inputs. To address these shortcomings, we propose EMPIR, an ensemble of mixed precision 1 DNN models, as a new form of defense against adversarial attacks and demonstrate that it can significantly improve the robustness of a variety of DNN models across a wide range of adversarial attacks. ",
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+ "type": "text",
+ "text": "Ensembles have been widely explored as an approach to improve the performance of machine learning models and classifiers (Hansen & Salamon (1990)). Examples of various successful ensembling methods include averaging, bagging (Breiman (1996)), boosting (Dietterich (2000)), etc. Recently, it has also been suggested that ensembles may help boost the robustness of DNNs (Strauss et al. (2017); Pang et al. (2019); He et al. (2017); Tramer et al. (2017)). The individual models in these \\` ensembles are restricted to full precision DNN models, i.e., models utilizing 32 bits of numerical precision to represent different data-structures. Such ensembles are very expensive in terms of the computational and memory overhead (e.g., $1 0 \\times$ the baseline for an ensemble with 10 models (Strauss et al. (2017))). In contrast, the use of quantized models in EMPIR, which entail the use of significantly lower number of bits in storage and compute, ensures that the overhead is modest (less than $2 5 \\%$ in our evaluations). ",
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+ "type": "text",
+ "text": "Quantized DNNs are characterized by the use of lower numbers of bits to represent DNN datastructures like weights and activations (Venkataramani et al. (2014); Hubara et al. (2017); Zhou et al. (2016); Courbariaux et al. (2015)). They have been widely explored as an approach to reduce the high computational and memory demands of DNNs. Recent studies have also observed that these quantized models demonstrate higher robustness to adversarial attacks (Galloway et al. (2017); Siraj Rakin et al. (2018); Panda et al. (2019)). However, the loss in information associated with the quantization process often makes these quantized models perform significantly worse than their fullprecision counterparts while classifying the original unperturbed inputs. This motivates the design of EMPIR, which successfully combines the higher robustness of low-precision models with the higher unperturbed accuracy of the full-precision models. In the general case, EMPIR comprises of $M$ full-precision models and $\\mathbf { N }$ low-precision models with the final prediction determined by an ensembling technique such as averaging the probabilities or counting the number of predictions for each class. In practice, we find that $M = 1$ and $N = 2$ or 3 provides a significant improvement in adversarial accuracy with small overheads. ",
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+ {
+ "type": "text",
+ "text": "In summary, the key contributions of this work are ",
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+ "text": "• We propose the use of ensembles of mixed precision models as a defense against adversarial attacks on DNNs. \n• We analyze the effect of ensemble size and ensembling techniques on the overall robustness as well as the computational and storage overheads of the ensemble. \n• Across a suite of 3 different DNN models under 4 different adversarial attacks, we demonstrate that EMPIR exhibits significantly higher robustness when compared to individual models as well as ensembles of full-precision models. ",
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+ "type": "text",
+ "text": "2 ADVERSARIAL ATTACKS: BACKGROUND ",
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+ "text": "Adversarial attacks modify inputs in a manner that force a DNN model to misclassify, while ensuring that the input changes are small and imperceptible to human eyes. In the context of DNNs that operate on images, which are the focus of most prior work, various attack methods have been proposed to systematically modify pixel values in the input image so as to result in a mis-classification. A few such methods are described below. ",
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+ "type": "text",
+ "text": "Fast Gradient Sign Method (FGSM) (Goodfellow et al. (2014)). FGSM is a single-step attack that operates by calculating the gradient of the loss function with respect to the input pixels $( \\nabla _ { x } L ( \\theta , X , Y ) )$ . Based on the sign of the loss, the input pixels are increased or decreased by a small constant, $\\epsilon$ , to help move the image towards the direction of increased loss. The adversarial input, $X _ { a d v }$ can be computed as: ",
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+ "type": "equation",
+ "img_path": "images/730a1976c9578f0b444868cda424a296e007d5159c5ec5903befea8a25359a4a.jpg",
+ "text": "$$\nX _ { a d v } = X + \\epsilon S i g n ( \\nabla _ { x } L ( \\theta , X , Y ) )\n$$",
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+ "text": "Here, $X$ is the original input image associated with an output $Y$ and $\\theta$ refers to the weights of the network. ",
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+ "text": "Basic Iterative Method (BIM) (Kurakin et al. (2016)). BIM is an iterative version of the FGSM attack which performs a finer optimization by modifying pixels by small values in each iteration. Further, the image generated in each iteration has its pixel values clipped to ensure minimal distortion. Mathematically, this attack can be described as: ",
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+ "type": "equation",
+ "img_path": "images/260a3fc456edea24e53617ad01bd3f8de7823ff4a0a079c675e68c44d7a25e8f.jpg",
+ "text": "$$\n\\begin{array} { r } { X _ { a d v } ^ { 0 } = X , \\quad X _ { a d v } ^ { N + 1 } = C l i p _ { X , \\epsilon } \\{ X _ { a d v } ^ { N } + \\alpha S i g n ( \\nabla _ { x } L ( \\theta , X _ { a d v } ^ { N } , y ) ) \\} } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Here, the terms $X , Y , \\theta$ and $\\epsilon$ have the same meaning as in Equation 1 and $X _ { a d v } ^ { N }$ refers to the adversarial input generated at the iteration and is the step size in each iteration. ",
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+ "text": "Carlini-Wagner (CW) (Carlini & Wagner (2016)). CW is another iterative attack that employs optimizers to create strong adversarial inputs by simultaneously minimizing the input distortion and maximizing the misclassification error. It can be described mathematically as: ",
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+ "text": "$\\operatorname* { m i n } _ { \\delta } \\| \\delta \\| _ { 2 } ^ { 2 } + c \\cdot f ( X + \\delta )$ such that $( X + \\delta ) \\in [ 0 , 1 ] ^ { n }$ ",
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+ "text": "$$\nf ( X ) = \\mathrm { { m a x } } ( \\underset { i \\neq t } { \\operatorname* { m a x } } \\{ Z ( X ) _ { i } \\} - Z ( X ) _ { t } , 0 )\n$$",
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+ "img_path": "images/40e2eb40c0beae6fcd9a409aca3b3033d324a3b512b8c8c47ebacbdc3425b035.jpg",
+ "text": "$$\nX _ { a d v } = X + \\delta\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ 571
+ ],
+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "where $\\delta$ is the input distortion, $c$ is the Lagrangian multiplier, $Z ( X )$ is the logit output for the input $X$ , $t$ is the target class and $f ( X )$ is an objective function that satisfies the condition $f ( X + \\delta ) \\leq 0$ for all misclassifications. ",
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Projected Gradient Descent (PGD) (Madry et al. (2017)). PGD is a third type of iterative attack very similar in nature to the BIM attack. Unlike BIM, which starts with the original image itself, PGD starts with a random perturbation of the original input image. PGD can be described by the following equations: ",
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+ "page_idx": 2
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+ {
+ "type": "equation",
+ "img_path": "images/3bee91ae0256038f09eca8070d5b300c413bf2734e0dd232fd66edbbc5337e30.jpg",
+ "text": "$$\n\\begin{array} { l } { { X _ { a d v } ^ { 0 } = X + r a n d o m U n i f o r m ( s h a p e ( X ) , \\{ - \\epsilon , \\epsilon \\} ) } } \\\\ { { X _ { a d v } ^ { N + 1 } = C l i p _ { X , \\epsilon } \\{ X _ { a d v } ^ { N } + \\alpha S i g n ( \\bigtriangledown _ { x } L ( \\theta , X _ { a d v } ^ { N } , y ) ) \\} } } \\end{array}\n$$",
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Here, the terms $X , Y , X _ { a d v } ^ { N }$ , $\\theta$ , $\\epsilon$ and $\\alpha$ have the same meaning as in Equation 2. ",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "To summarize, different adversarial attacks have been proposed that expose the lack of robustness in current DNN models by constructing adversarial inputs that force a misclassification. Developing defenses to these adversarial attacks is critical to enable the deployment of DNNs in safety-critical systems. ",
+ "bbox": [
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+ "page_idx": 2
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+ {
+ "type": "text",
+ "text": "3 EMPIR: ENSEMBLES OF MIXED PRECISION DEEP NETWORKS FOR INCREASED ROBUSTNESS AGAINST ADVERSARIAL ATTACKS ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "To improve the robustness of DNN models, we propose EMPIR, or ensembles of mixed precision models. In this section, we will detail the design of EMPIR models and discuss the overheads associated with them. ",
+ "bbox": [
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+ "page_idx": 2
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+ {
+ "type": "text",
+ "text": "3.1 ADVERSARIAL ROBUSTNESS OF LOW-PRECISION NETWORKS ",
+ "text_level": 1,
+ "bbox": [
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+ ],
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+ "type": "text",
+ "text": "DNNs have conventionally been designed as full precision models utilizing 32-bit floating point numbers to represent different data-structures like weights, activations and errors. However, the high compute and memory demands of these full-precision models have driven efforts to move towards quantized or low-precision DNNs (Venkataramani et al. (2014); Hubara et al. (2017); Zhou et al. (2016); Courbariaux et al. (2015); Wang et al. (2018)). A multitude of quantization schemes have been proposed to minimize the loss of information associated with the quantization process. While our proposal is agnostic to the quantization method used, for the purpose of demonstration we adopt the quantization scheme proposed in DoReFaNet (Zhou et al. (2016)), which has been shown to produce low-precision models with competitive accuracy values. The quantization scheme can be described by Equation 5. ",
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+ "type": "equation",
+ "img_path": "images/cf63ab6219d96b7da2c311c167cef1a4de0dd494c65b7930c2460aa0df1e4291.jpg",
+ "text": "$$\n\\begin{array} { c } { q u a n t i z e _ { k } ( x ) = \\displaystyle \\frac { 1 } { 2 ^ { k } - 1 } r o u n d ( ( 2 ^ { k } - 1 ) \\cdot x ) } \\\\ { w _ { k } = 2 \\cdot q u a n t i z e _ { k } ( \\displaystyle \\frac { \\operatorname { t a n h } ( w ) } { 2 \\cdot m a x ( | \\operatorname { t a n h } ( w ) | ) } + \\displaystyle \\frac { 1 } { 2 } ) - 1 , \\quad a _ { k } = q u a n t i z e _ { k } ( a ) } \\end{array}\n$$",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "where $k$ refers to the number of quantization bits in the low precision network, $w$ and $w _ { k }$ refer to weight values before and after quantization, and $a$ and $a _ { k }$ refer to activation values before and after quantization. ",
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+ {
+ "type": "image",
+ "img_path": "images/0ee838ecb563cab90ca269bab4ce5fc5e86dde4716863af006716b4e1eddd702.jpg",
+ "image_caption": [
+ "Figure 1: Unperturbed accuracies and adversarial accuracies of low-precision models trained for the MNIST dataset "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 3
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+ "type": "image",
+ "img_path": "images/33d7667b2ae7aa54f11201a9e0461fb9e4d3c9cbe8c9cbe7211665a96999dfa2.jpg",
+ "image_caption": [
+ "Figure 2: Overview of EMPIR "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 3
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+ {
+ "type": "text",
+ "text": "In addition to the widely known advantages of reduced model size and reduced complexity of arithmetic computations, recent research efforts have also brought to light another lesser known advantage of low-precision models in the form of increased robustness to adversarial attacks. It has been observed that low-precision models in general exhibit higher values of adversarial accuracy than full-precision models with identical network structures (Galloway et al. (2017); Panda et al. (2019)). One possible explanation for this property is that higher quantization introduces higher amounts of non-linearity, which prevents small changes in the input from drastically altering the output and forcing a misclassification (Galloway et al. (2017)). Figure 1 shows the adversarial accuracies of different low precision models trained on the MNIST dataset under the FGSM attack. Unlike the activations and weights, the gradients utilized in the attack generation process were not quantized, allowing the adversary to launch a stronger attack. From the figure, it is apparent that models with lower numbers of bits used for representing weights and activations exhibit significantly higher levels of adversarial accuracy. ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "However, increasing the robustness of a system by simply replacing the full-precision model with its low-precision variant can negatively impact its accuracy on the original unperturbed inputs (unperturbed accuracy). In other words, the model may now start to mis-classify inputs that were not adversarially perturbed. Figure 1 also shows the unperturbed accuracies of low-precision models. As expected, models with weights and activations represented using lower numbers of bits exhibit lower unperturbed accuracies. ",
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Based on the above observations, we propose the use of ensembles of mixed-precision models to achieve the best of both worlds, i.e., increase robustness against adversarial attacks without sacrificing the accuracy on unperturbed inputs. ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "3.2 EMPIR: OVERVIEW ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Figure 2 presents an overview of EMPIR. In the general case, an EMPIR model comprises of M fullprecision (FP) models and N low-precision (LP) models. The full-precision models help in boosting the unperturbed accuracy of the overall model, while the low-precision models contribute towards higher robustness. All the individual models are fed the same input and their predicted classes or probabilities are combined with the help of an ensembling technique at the end to determine the final prediction of the EMPIR model. In practice, we found that a single full-precision model $( \\mathbf { M } = \\mathbf { l } )$ ) and a small number of low-precision models $\\mathrm { N } { = } 2$ or 3) are sufficient to achieve high adversarial accuracies without any noticeable compromises in the unperturbed accuracies. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "The ensembling function plays a vital role in the overall performance of the model as it determines the final classification boundary. In this work, we consider two of the most commonly used ensembling functions, namely, averaging and max voting. The averaging function averages the output probabilities of each model and identifies the class with the maximum average probability as the final predicted class. On the other hand, max voting considers the predictions of each model as votes and determines the class with the maximum number of votes to be the final class. In our experiments, we found that averaging achieves better adversarial accuracies on ensembles of size 2 while max voting achieves better adversarial accuracies on ensembles of size greater than 2. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "In order to allow an ensemble model to work better than a single model, the individual models should also be designed to be diverse (Hansen & Salamon (1990)). This ensures that the models dont produce similar errors and hence, that the probability of two models misclassifying the same input is lower. We introduce diversity in the individual models of EMPIR by training them with different random initializations of weights. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "3.3 COMPUTATIONAL AND MEMORY COMPLEXITY OF EMPIR ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "The ensembling of multiple full-precision and low-precision models in EMPIR increases its computational and storage requirements as these models need to be stored and evaluated. In this work, we keep these memory and computational complexities within reasonable limits by restricting the precision of weights and activations in the low-precision models of EMPIR to a maximum of 4 bits. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "The increasing popularity of low-precision DNN models has prompted recent hardware platforms including GPUs and neural network accelerators to add native hardware support for operating on low precision data (Fleischer et al. (2018); Kilgariff et al. (2019)). These hardware platforms reconfigure a common datapath to perform computations on full-precision data (32 or 64 bits) as well as lowprecision data (4, 8 or 16 bits). Low-precision operations can achieve higher throughputs than full-precision operations on these platforms as the same number of compute elements and the same amount of memory bandwidth can support a larger number of concurrent operations. Consequently, the additional execution time required to evaluate the low-precision models in EMPIR is much less than that of a full-precision model. Overall, we quantify the execution time and storage overhead of an EMPIR model using the formula described by Equation 6. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "equation",
+ "img_path": "images/21454f0db3d0cd8d81ca85888ad061435440c2c54480229ef00563e27818ce6f.jpg",
+ "text": "$$\n\\begin{array} { c l c r } { { \\displaystyle T i m e O v e r h e a d \\{ E M P I R ( M , N ) \\} = M + \\sum _ { i = 1 } ^ { N } \\frac { O p s \\_ p e r _ { - } s e c ( F P ) } { O p s \\_ p e r _ { - } s e c ( k _ { i } ) } } } \\\\ { { \\displaystyle S t o r a g e O v e r h e a d \\{ E M P I R ( M , N ) \\} = M + \\sum _ { i = 1 } ^ { N } \\frac { k _ { i } } { F P } } } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "where $k _ { i }$ is the precision of the $i ^ { t h }$ low-precision model, $F P$ is the precision of the full-precision models, and $O p s \\_ p e r \\_ s e c ( b )$ is the throughput of $b$ bit operations on the underlying hardware platform. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "4 EXPERIMENTS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "In this section, we describe the experiments performed to evaluate the advantages of EMPIR models over baseline full-precision models. ",
+ "bbox": [
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "4.1 BENCHMARKS ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "We studied the robustness of EMPIR models across three different image recognition DNNs, namely, MNISTconv, CIFARconv and AlexNet. The individual full-precision and-low precision networks within the EMPIR models were designed to have identical network topologies. The details of the individual networks in these benchmarks are listed in Table 3 within Appendix A. The benchmarks differ in the number of convolutional layers, fully connected layers as well as the datasets. We consider three different datasets, namely, MNIST (Lecun et al. (1998)), CIFAR-10 (Krizhevsky (2009)) and ImageNet (Deng et al. (2009)) which vary significantly in their complexity. The low precision networks were obtained using the quantization scheme proposed in DoReFa-Net (Zhou et al. (2016)). The full precision models were trained using 32 bit floating point representations for all data-structures. ",
+ "bbox": [
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+ "page_idx": 5
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+ {
+ "type": "text",
+ "text": "4.2 EVALUATION OF ROBUSTNESS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "We implemented EMPIR within TensorFlow (Abadi et al. (2015)) and have released the source code for our implementation 2. The robustness of the EMPIR models was measured in terms of their adversarial accuracies under a variety of white-box attacks within the Cleverhans library (Papernot et al. (2018)). We specifically consider the four adversarial attacks described in Section 2. The adversarial parameters for the attacks on the different benchmarks are presented in Table 1. The attacks were generated on the entire test dataset for each of the benchmarks. Generating these white-box attacks involves computation of the gradient $\\nabla _ { x } L ( \\theta , X , Y )$ (Section 2), which is not directly defined for ensembles. For the EMPIR models, we compute this gradient as an average over all individual models for an averaging ensemble and as an average over the individual models that voted for the final identified class for a max-voting ensemble. ",
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+ {
+ "type": "table",
+ "img_path": "images/cffe734d2de60751977344e955799e230845a671b869abc6ed862c096520b350.jpg",
+ "table_caption": [
+ "Table 1: Attack parameters "
+ ],
+ "table_footnote": [],
+ "table_body": "| Network | Cw | FGSM | BIM | PGD |
| MNISTconv | Attack iterations = 50 | ∈= 0.3 | ∈= 0.3, α=0.01 No. of iterations = 40 | e= 0.3, α=0.01 No. of iterations = 40 |
| CIFARconv | Attack iterations = 50 | ∈= 0.1 | ∈= 0.1, α=0.01 No. of iterations = 40 | ∈= 0.1, α=0.01 No. of iterations = 40 |
| AlexNet | Attack iterations = 50 | ∈= 0.1 | ∈= 0.1, α=0.01 No. of iterations = 40 | e=0.1, α=0.01 No. of iterations = 5 |
",
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+ {
+ "type": "text",
+ "text": "5 RESULTS ",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "In this section, we present the results of our experiments highlighting the advantages of EMPIR models. ",
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+ {
+ "type": "text",
+ "text": "5.1 ROBUSTNESS OF EMPIR MODELS ACROSS ALL ATTACKS ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "Table 2 presents the results of our experiments across different benchmarks. The EMPIR models presented in the table are the ones exhibiting highest average adversarial accuracies under the constraints of ${ < } 2 5 \\%$ compute and memory overhead and $< 2 \\%$ loss in unperturbed accuracy. We observed that across all the benchmarks, ensembles comprised of two low-precision and one full-precision model combined with the max-voting ensembling technique satisfy these constraints. However, the individual configurations of the low-precision models, i.e., the precisions of weights and activations in the ensembles, differ across the benchmarks. For example, both low-precision models in the EMPIR model for MNISTconv have weight precisions of 2 bits and activation precisions of 4 bits. On the other hand, the two low-precision models in the AlexNet EMPIR model have $\\mathrm { \\{ w e i g h t , a c t i v a t i o n \\} }$ bit-precisions of $\\{ 2 , 2 \\}$ and $\\{ 4 , 4 \\}$ , respectively. In general, we observe that the EMPIR models exhibit substantially higher adversarial accuracies across all attacks for the three benchmarks. ",
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+ "type": "table",
+ "img_path": "images/18754a4ce117c70babbf960321a959b269b81064e868bb873f8af411c7c5f232.jpg",
+ "table_caption": [
+ "Table 2: Unperturbed and adversarial accuracies of the baseline and EMPIR models across different attacks "
+ ],
+ "table_footnote": [],
+ "table_body": "| Network | Dataset | Configuration |
| MNISTconv | MNIST | Conv(8×8×64),ReLU, Conv(6×6×128),ReLU, Conv(5×5×128),ReLU,Fully_Connected(10),SoftMax |
| CIFARconv | CIFAR-10 | Conv(5×5×32), ReLU, MaxPool(3×3), Conv(8×8 ×64),ReLU AvgPool(3×3), Conv(8×8 ×64), ReLU, AvgPool(3 × 3), Fully_Connected(64), Fully_Connected(10), SoftMax |
| AlexNet | ImageNet | Conv(12×12×96), ReLU, Conv(5×5×256), BatchNorm, ReLU, MaxPool(3×3), Conv(3×3×384), BatchNorm, ReLU, MaxPool(3×3), Conv(3×3×384), BatchNorm, ReLU, Conv(3×3×256), BatchNorm,ReLU, MaxPool(3 ×3), Fully_Conn(4096), BatchNorm, ReLU, Fully_Conn(4096), BatchNorm,ReLU,Fully_Conn(100O), SoftMax |
",
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+ {
+ "type": "text",
+ "text": "B ROBUSTNESS TO ATTACKS OF VARYING STRENGTHS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 11
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+ "type": "text",
+ "text": "To further illustrate the benefits of EMPIR, we observed its robustness under attacks of varying strength. We specifically varied the $\\epsilon$ value in the FGSM attack between 0.1 and 0.8 and the number of attack iterations in the CW attack between 10 and 90, and measured the adversarial accuracies of the EMPIR model as well as the baseline FP model for the MNISTconv benchmark. Figure 5 clearly illustrates that EMPIR exhibits higher adversarial accuracies across attacks of different strengths for both FGSM and CW. ",
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+ {
+ "type": "image",
+ "img_path": "images/ce03aafd71f0a52811b4fc528282178ec148ee8ac35a27a27eef83d49b6e0d47.jpg",
+ "image_caption": [
+ "Figure 5: (a) Adversarial accuracies under FGSM attack of varying strength (b) Adversarial accuracies under CW attack of varying strength "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ }
+]
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diff --git a/parse/train/HJem3yHKwH/HJem3yHKwH_middle.json b/parse/train/HJem3yHKwH/HJem3yHKwH_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..6b9f01252bdfea9c1ecf9e1b3d318cb6b75e49d1
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| MNISTconv | Attack iterations = 50 | ∈= 0.3 | ∈= 0.3, α=0.01 No. of iterations = 40 | e= 0.3, α=0.01 No. of iterations = 40 |
| CIFARconv | Attack iterations = 50 | ∈= 0.1 | ∈= 0.1, α=0.01 No. of iterations = 40 | ∈= 0.1, α=0.01 No. of iterations = 40 |
| AlexNet | Attack iterations = 50 | ∈= 0.1 | ∈= 0.1, α=0.01 No. of iterations = 40 | e=0.1, α=0.01 No. of iterations = 5 |
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| Vaswani et al. (2017) | 213M | 26.4 | = |
| Self-attention baseline (k=inf,H=16) | 210M | 26.9 ± 0.1 | 52.1 ± 0.1 |
| Self-attention baseline (k=3,7,15,31x3, H=16) | 210M | 26.9 ± 0.3 | 54.9 ± 0.2 |
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| + DropConnect (H=1024) | 195M | 26.5 ± 0.2 | 63.3 ± 0.1 |
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| + Dynamic weights [DynamicConv] (H=16) | 200M | 26.9 ± 0.2 | 62.6 ± 0.4 |
| Note: DynamicConv(H=16) w/o softmax-normalization | 200M | diverges | |
| AAN decoder + self-attn encoder | 260M | 26.8 ± 0.1 | 59.5 ± 0.1 |
| AAN decoder+ AAN encoder | 310M | 22.5 ± 0.1 | 59.2 ± 2.1 |
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| NAT (+ FT) (Gu et al., 2018) | | 17.7 | 25.6 | = |
| NAT (+ FT + NPD=10) (Gu et al., 2018) | | 18.7 | 12.7 | = |
| NAT (+ FT + NPD=10O) (Gu et al., 2018) | | 19.2 | 3.9 | |
| LT, Improved Semhash (Kaiser et al.,2018) | | 19.8 | 9.5 | 1 |
| IRidec 二 :1 (Lee et al., 2018) | | 13.9 | 1 | 511.4 |
| IR idec 二 :2 (Lee et al., 2018) | | 17.0 | = | 393.6 |
| IR idec 二 : 5 (Lee et al., 2018) | | 20.3 | = | 139.7 |
| IR idec 二 :10 (Lee et al., 2018) | | 21.6 | | 90.4 |
| IR Adaptive ( (Lee et al., 2018) | | 21.5 | 1 | 107.2 |
| NART w/hints (Li et al., 2019) | | 21.1 | 38.5 | |
| NART w/ hints (B = 4, 9 candidates) (Li et al., 2019) | | 25.2 | 22.7 | = |
| ENAT Embedding l gMapping (Guo et al.,2019) ENAT Embedding Mapping (rescoring 9 candidates) | | 20.7 | 41.7 | |
| (Guo et al., 2019) | | 24.3 | 20.4 | |
| Autoregressive (Gu et al., 2018) | | 22.7 | 2.5 | |
| Autoregressive (Lee et al., 2018) | | 23.8 | 1 | = 54.0 |
| Transformer (Li et al., 2019) | | 27.3 | 1.3 | |
| Transformer (Guo et al., 2019) | - | 27.4 | 1.6 | 1 1 |
| DynamicConv (1-decoder layer (k=31)) | 124M | 26.1 | 15.2 | |
| DynamicConv (3-decoder layers (k=3,7,15)) | 153M | 27.7 | 7.2 | 423.0 202.3 |
| DynamicConv (6-decoder layers (k=3,7,15,31,31,31)) | 200M | 28.5 | 3.9 | 110.9 |
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| Vaswani et al. (2017) | 213M | 26.4 | = |
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| AAN decoder + self-attn encoder | 260M | 26.8 ± 0.1 | 59.5 ± 0.1 |
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| NAT (+ FT + NPD=10) (Gu et al., 2018) | | 18.7 | 12.7 | = |
| NAT (+ FT + NPD=10O) (Gu et al., 2018) | | 19.2 | 3.9 | |
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| (Guo et al., 2019) | | 24.3 | 20.4 | |
| Autoregressive (Gu et al., 2018) | | 22.7 | 2.5 | |
| Autoregressive (Lee et al., 2018) | | 23.8 | 1 | = 54.0 |
| Transformer (Li et al., 2019) | | 27.3 | 1.3 | |
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| DynamicConv (6-decoder layers (k=3,7,15,31,31,31)) | 200M | 28.5 | 3.9 | 110.9 |
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+# A PAC-BAYESIAN APPROACH TO GENERALIZATION BOUNDS FOR GRAPH NEURAL NETWORKS
+
+Renjie Liao1,2, Raquel Urtasun1,2,3, Richard Zemel1,2,3
+University of Toronto1, Vector Institute3, Canadian Institute for Advanced Research3
+{rjliao, urtasun, zemel}@cs.toronto.edu
+
+# ABSTRACT
+
+In this paper, we derive generalization bounds for two primary classes of graph neural networks (GNNs), namely graph convolutional networks (GCNs) and message passing GNNs (MPGNNs), via a PAC-Bayesian approach. Our result reveals that the maximum node degree and the spectral norm of the weights govern the generalization bounds of both models. We also show that our bound for GCNs is a natural generalization of the results developed in (Neyshabur et al., 2017) for fully-connected and convolutional neural networks. For MPGNNs, our PACBayes bound improves over the Rademacher complexity based bound (Garg et al., 2020), showing a tighter dependency on the maximum node degree and the maximum hidden dimension. The key ingredients of our proofs are a perturbation analysis of GNNs and the generalization of PAC-Bayes analysis to non-homogeneous GNNs. We perform an empirical study on several synthetic and real-world graph datasets and verify that our PAC-Bayes bound is tighter than others.
+
+# 1 INTRODUCTION
+
+Graph neural networks (GNNs) (Gori et al., 2005; Scarselli et al., 2008; Bronstein et al., 2017; Battaglia et al., 2018) have become very popular recently due to their ability to learn powerful representations from graph-structured data, and have achieved state-of-the-art results in a variety of application domains such as social networks (Hamilton et al., 2017; Xu et al., 2018), quantum chemistry (Gilmer et al., 2017; Chen et al., 2019a), computer vision (Qi et al., 2017; Monti et al., 2017), reinforcement learning (Sanchez-Gonzalez et al., 2018; Wang et al., 2018), robotics (Casas et al., 2019; Liang et al., 2020), and physics (Henrion et al., 2017). Given a graph along with node/edge features, GNNs learn node/edge representations by propagating information on the graph via local computations shared across the nodes/edges. Based on the specific form of local computation employed, GNNs can be divided into two categories: graph convolution based GNNs (Bruna et al., 2013; Duvenaud et al., 2015; Kipf & Welling, 2016) and message passing based GNNs (Li et al., 2015; Dai et al., 2016; Gilmer et al., 2017). The former generalizes the convolution operator from regular graphs (e.g., grids) to ones with arbitrary topology, whereas the latter mimics message passing algorithms and parameterizes the shared functions via neural networks.
+
+Due to the tremendous empirical success of GNNs, there is increasing interest in understanding their theoretical properties. For example, some recent works study their expressiveness (Maron et al., 2018; Xu et al., 2018; Chen et al., 2019b), that is, what class of functions can be represented by GNNs. However, only few works investigate why GNNs generalize so well to unseen graphs. They are either restricted to a specific model variant (Verma & Zhang, 2019; Du et al., 2019; Garg et al., 2020) or have loose dependencies on graph statistics (Scarselli et al., 2018).
+
+On the other hand, GNNs have close ties to standard feedforward neural networks, e.g., multi-layer perceptrons (MLPs) and convolutional neural networks (CNNs). In particular, if each i.i.d. sample is viewed as a node, then the whole dataset becomes a graph without edges. Therefore, GNNs can be seen as generalizations of MLPs/CNNs since they model not only the regularities within a sample but also the dependencies among samples as defined in the graph. It is therefore natural to ask if we can generalize the recent advancements on generalization bounds for MLPs/CNNs (Harvey et al., 2017; Neyshabur et al., 2017; Bartlett et al., 2017; Dziugaite & Roy, 2017; Arora et al., 2018; 2019) to GNNs, and how would graph structures affect the generalization bounds?
+
+In this paper, we answer the above questions by proving generalization bounds for the two primary classes of GNNs, i.e., graph convolutional networks (GCNs) (Kipf & Welling, 2016) and messagepassing GNNs (MPGNNs) (Dai et al., 2016; Jin et al., 2018).
+
+Our generalization bound for GCNs shows an intimate relationship with the bounds for MLPs/CNNs with ReLU activations (Neyshabur et al., 2017; Bartlett et al., 2017). In particular, they share the same term, i.e., the product of the spectral norms of the learned weights at each layer multiplied by a factor that is additive across layers. The bound for GCNs has an additional multiplicative factor $d ^ { ( l - 1 ) / 2 }$ where $d - 1$ is the maximum node degree and $l$ is the network depth. Since MLPs/CNNs are special GNNs operating on graphs without edges (i.e., $d - 1 = 0$ ), the bound for GCNs coincides with the ones for MLPs/CNNs with ReLU activations (Neyshabur et al., 2017) on such degenerated graphs. Therefore, our result is a natural generalization of the existing results for MLPs/CNNs.
+
+Our generalization bound for message passing GNNs reveals that the governing terms of the bound are similar to the ones of GCNs, i.e., the geometric series of the learned weights and the multiplicative factor $d ^ { l - 1 }$ . The geometric series appears due to the weight sharing across message passing steps, thus corresponding to the product term across layers in GCNs. The term $d ^ { l - 1 }$ encodes the key graph statistics. Our bound improves the dependency on the maximum node degree and the maximum hidden dimension compared to the recent Rademacher complexity based bound (Garg et al., 2020). Moreover, we compute the bound values on four real-world graph datasets (e.g., social networks and protein structures) and verify that our bounds are tighter.
+
+In terms of the proof techniques, our analysis follows the PAC-Bayes framework in the seminal work of (Neyshabur et al., 2017) for MLPs/CNNs with ReLU activations. However, we make two distinctive contributions which are customized for GNNs. First, a naive adaptation of the perturbation analysis in (Neyshabur et al., 2017) does not work for GNNs since ReLU is not 1-Lipschitz under the spectral norm, i.e., $\| \mathrm { R e L U } ( X ) \| _ { 2 } \leq \| X \| _ { 2 }$ does not hold for some real matrix $X$ . Instead, we construct the recursion on certain node representations of GNNs like the one with maximum $\ell _ { 2 }$ norm, so that we can perform perturbation analysis with vector 2-norm. Second, in contrast to (Neyshabur et al., 2017) which only handles the homogeneous networks, i.e., $f ( a x ) = a f ( x )$ when $a \geq 0$ , we properly construct a quantity of the learned weights which 1) provides a way to satisfy the constraints of the previous perturbation analysis and 2) induces a finite covering on the range of the quantity so that the PAC-Bayes bound holds for all possible weights. This generalizes the analysis to non-homogeneous GNNs like typical MPGNNs.
+
+The rest of the paper is organized as follows. In Section 2, we introduce background material necessary for our analysis. We then present our generalization bounds and the comparison to existing results in Section 3. We also provide an empirical study to support our theoretical arguments in Section 4. At last, we discuss the extensions, limitations and some open problems.
+
+# 2 BACKGROUND
+
+In this section, we first explain our analysis setup including notation and assumptions. We then describe the two representative GNN models in detail. Finally, we review the PAC-Bayes analysis.
+
+# 2.1 ANALYSIS SETUP
+
+In the following analysis, we consider the $K$ -class graph classification problem which is common in the GNN literature, where given a graph sample $z$ , we would like to classify it into one of the predefined $K$ classes. We will discuss extensions to other problems like graph regression in Section 5. Each graph sample $z$ is a triplet of an adjacency matrix $A$ , node features $\mathbf { \bar { \boldsymbol { X } } } \in \mathbf { \overline { { \mathbb { R } } } } ^ { n \times h _ { 0 } }$ and output label $y \in \bar { \mathbb { R } } ^ { 1 \times K }$ , i.e. $z = ( A , X , y )$ , where $n$ is the number of nodes and $h _ { 0 }$ is the input feature dimension. We start our discussion by defining our notations. Let $\mathbb { N } _ { k } ^ { + }$ be the first $k$ positive integers, i.e., $\mathbb { N } _ { k } ^ { + } = \{ 1 , 2 , \dots , k \}$ , $| \cdot | _ { p }$ the vector $p$ -norm and $\| \cdot \| _ { p }$ the operator norm induced by the vector $p$ -norm. Further, $\| \cdot \| _ { F }$ denotes the Frobenius norm of a matrix, $e$ the base of the natural logarithm function log, $A [ i , j ]$ the $( i , j )$ -th element of matrix $A$ and $A [ i , : ]$ the $i$ -th row. We use parenthesis to avoid the ambiguity, e.g., $( A B ) [ i , j ]$ means the $( i , j )$ -th element of the product matrix $A B$ . We then introduce some terminologies from statistical learning theory and define the sample space as $\mathcal { Z }$ , $z = ( A , X , y ) \in \mathcal { Z }$ where $X \in { \mathcal { X } }$ (node feature space) and $A \in { \mathcal { G } }$ (graph space), data distribution $\mathcal { D } , z \stackrel { i i d } { \sim } \mathcal { D }$ , hypothesis (or model) $f _ { w }$ where $f _ { w } \in \mathcal { H }$ (hypothesis class), and training set $S$ with size $m$ , $S = \{ z _ { 1 } , \dots , z _ { m } \}$ . We make the following assumptions which also appear in the literature:
+
+A1 Data, i.e., triplets $( A , X , y )$ , are i.i.d. samples drawn from some unknown distribution $\mathcal { D }$ .
+
+A2 The maximum hidden dimension across all layers is $h$ .
+A3 Node feature of any graph is contained in a $\ell _ { 2 }$ -ball with radius $B$ . Specifically, we have $\forall i \in \mathbb { N } _ { n } ^ { + }$ , the $i$ -th node feature $\begin{array} { r } { X [ i , : ] \in \mathcal { X } _ { B , h _ { 0 } } = \{ x \in \mathbb { R } ^ { h _ { 0 } } | \sum _ { j = 1 } ^ { h _ { 0 } } { x _ { j } ^ { 2 } } ^ { * } \leq B ^ { 2 } \} } \end{array}$ .
+A4 We only consider simple graphs (i.e., undirected, no loops1, and no multi-edges) with maximum node degree as $d - 1$ .
+
+Note that it is straightforward to estimate $B$ and $d$ empirically on real-world graph data.
+
+# 2.2 GRAPH NEURAL NETWORKS (GNNS)
+
+In this part, we describe the details of the GNN models and the loss function we used for the graph classification problem. The essential idea of GNNs is to propagate information over the graph so that the learned representations capture the dependencies among nodes/edges. We now review two classes of GNNs, GCNs and MPGNNs, which have different mechanisms for propagating information. We choose them since they are the most popular variants and represent two common types of neural networks, i.e., feedforward (GCNs) and recurrent (MPGNNs) neural networks. We discuss the extension of our analysis to other GNN variants in Section 5. For ease of notation, we define the model to be $f _ { w } \in \mathcal { H } : \dot { \mathcal { X } } \times \mathcal { G } \to \mathbb { R } ^ { K }$ where $w$ is the vectorization of all model parameters.
+
+GCNs: Graph convolutional networks (GCNs) (Kipf & Welling, 2016) for the $K$ -class graph classification problem can be defined as follows,
+
+$$
+\begin{array} { l l } { { H _ { k } = \sigma _ { k } \left( \tilde { L } H _ { k - 1 } W _ { k } \right) } } & { { \qquad ( k \mathrm { - t h ~ G r a p h ~ C o n v o l u t i o n ~ L a y e r } ) } } \\ { { { } } } & { { { } } } \\ { { H _ { l } = \frac { 1 } { n } { \bf 1 } _ { n } H _ { l - 1 } W _ { l } } } & { { \qquad ( \mathrm { R e a d o u t ~ L a y e r } ) , } } \end{array}
+$$
+
+where $k \in \mathbb { N } _ { l - 1 } ^ { + }$ , $H _ { k } \in \mathbb { R } ^ { n \times h _ { k } }$ are the node representations/states, $\mathbf { 1 } _ { n } \in \mathbb { R } ^ { 1 \times n }$ is a all-one vector, $l$ is the number of layers.2 and $W _ { j }$ is the weight matrix of the $j$ -th layer. The initial node state is the observed node feature $H _ { 0 } = X$ . For both GCNs and MPGNNs, we consider $l > 1$ since otherwise the model degenerates to a linear transformation which does not leverage the graph and is trivial to analyze. Due to assumption A2, $W _ { j }$ is of size at most $h \times h$ , i.e., $\bar { h _ { k } } \le \bar { h , \forall k } \in \mathbb { N } _ { l - 1 } ^ { + }$ . The graph Laplacian $\tilde { L }$ is defined as, ${ \tilde { A } } = I + A$ , $\tilde { L } = D ^ { - \frac { 1 } { 2 } } \tilde { A } D ^ { - \frac { 1 } { 2 } }$ where $D$ is the degree matrix of $\bar { A }$ . Note that the maximum eigenvalue of $\tilde { L }$ is 1 in this case. We absorb the bias into the weight by appending constant 1 to the node feature. Typically, GCNs use ReLU as the non-linearity, i.e., $\bar { \sigma _ { i } } \bar { ( x ) ^ { } } = \mathrm { m a x } ( 0 , x ) , \forall i = 1 , \cdot \cdot \cdot , l - 1$ . We use the common mean-readout to obtain the graph representation where $H _ { l - 1 } \in \mathbb { R } ^ { n \times h _ { l - 1 } }$ , $W _ { l } \in \mathbb { R } ^ { h _ { l - 1 } \times K }$ , and $H _ { l } \in \mathbb { R } ^ { 1 \times K }$ .
+
+MPGNNs: There are multiple variants of message passing GNNs, e.g., (Li et al., 2015; Dai et al., 2016; Gilmer et al., 2017), which share the same algorithmic framework but instantiate a few components differently, e.g., the node state update function. We choose the same class of models as in (Garg et al., 2020) which are popular in the literature (Dai et al., 2016; Jin et al., 2018) in order to fairly compare bounds. This MPGNN model can be written in matrix forms as follows,
+
+