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+# THREE MECHANISMS OFWEIGHT DECAY REGULARIZATION
+
+Guodong Zhang, Chaoqi Wang, Bowen Xu, Roger Grosse
+
+University of Toronto, Vector Institute {gdzhang, cqwang, bowenxu, rgrosse}@cs.toronto.edu
+
+# ABSTRACT
+
+Weight decay is one of the standard tricks in the neural network toolbox, but the reasons for its regularization effect are poorly understood, and recent results have cast doubt on the traditional interpretation in terms of $L _ { 2 }$ regularization. Literal weight decay has been shown to outperform $L _ { 2 }$ regularization for optimizers for which they differ. We empirically investigate weight decay for three optimization algorithms (SGD, Adam, and K-FAC) and a variety of network architectures. We identify three distinct mechanisms by which weight decay exerts a regularization effect, depending on the particular optimization algorithm and architecture: (1) increasing the effective learning rate, (2) approximately regularizing the inputoutput Jacobian norm, and (3) reducing the effective damping coefficient for second-order optimization. Our results provide insight into how to improve the regularization of neural networks.
+
+# 1 INTRODUCTION
+
+Weight decay has long been a standard trick to improve the generalization performance of neural networks (Krogh & Hertz, 1992; Bos & Chug, 1996) by encouraging the weights to be small in magnitude. It is widely interpreted as a form of $L _ { 2 }$ regularization because it can be derived from the gradient of the $L _ { 2 }$ norm of the weights in the gradient descent setting. However, several findings cast doubt on this interpretation:
+
+• Weight decay has sometimes been observed to improve training accuracy, not just generalization performance (e.g. Krizhevsky et al. (2012)).   
+Loshchilov & Hutter (2017) found that when using Adam (Kingma & Ba, 2014) as the optimizer, literally applying weight decay (i.e. scaling the weights by a factor less than 1 in each iteration) enabled far better generalization than adding an $L _ { 2 }$ regularizer to the training objective.   
+• Weight decay is widely used in networks with Batch Normalization (BN) (Ioffe & Szegedy, 2015). In principle, weight decay regularization should have no effect in this case, since one can scale the weights by a small factor without changing the network’s predictions. Hence, it does not meaningfully constrain the network’s capacity.
+
+The effect of weight decay remains poorly understood, and we lack clear guidelines for which tasks and architectures it is likely to help or hurt. A better understanding of the role of weight decay would help us design more efficient and robust neural network architectures.
+
+In order to better understand the effect of weight decay, we experimented with both weight decay and $L _ { 2 }$ regularization applied to image classifiers using three different optimization algorithms: SGD, Adam, and Kronecker-Factored Approximate Curvature (K-FAC) (Martens & Grosse, 2015). Consistent with the observations of Loshchilov & Hutter (2017), we found that weight decay consistently outperformed $L _ { 2 }$ regularization in cases where they differ. Weight decay gave an especially strong performance boost to the K-FAC optimizer, and closed most of the generalization gaps between first- and second-order optimizers, as well as between small and large batches. We then investigated the reasons for weight decay’s performance boost. Surprisingly, we identified three distinct mechanisms by which weight decay has a regularizing effect, depending on the particular algorithm and architecture:
+
+![](images/fe224c1a77501ac1559a10cff11712c99a76781ab22abf8f7678bce94ccb28e7.jpg)  
+Figure 1: Comparison of test accuracy of the networks trained with different optimizers on both CIFAR10 and CIFAR100. We compare Weight Decay regularization to $L _ { 2 }$ regularization and the Baseline (which used neither). Here, $\mathbf { B N + A u g }$ denotes the use of BN and data augmentation. K-FAC-G and K-FAC-F denote K-FAC using Gauss-Newton and Fisher matrices as the preconditioner, respectively. The results suggest that weight decay leads to improved performance across different optimizers and settings.
+
+1. In our experiments with first-order optimization methods (SGD and Adam) on networks with BN, we found that it acts by way of the effective learning rate. Specifically, weight decay reduces the scale of the weights, increasing the effective learning rate, thereby increasing the regularization effect of gradient noise (Neelakantan et al., 2015; Keskar et al., 2016). As evidence, we found that almost all of the regularization effect of weight decay was due to applying it to layers with BN (for which weight decay is meaningless). Furthermore, when we computed the effective learning rate for the network with weight decay, and applied the same effective learning rate to a network without weight decay, this captured the full regularization effect.
+
+2. We show that when K-FAC is applied to a linear network using the Gauss-Newton metric (K-FAC-G), weight decay is equivalent to regularizing the squared Frobenius norm of the input-output Jacobian (which was shown by Novak et al. (2018) to improve generalization). Empirically, we found that even for (nonlinear) classification networks, the Gauss-Newton norm (which K-FAC with weight decay is implicitly regularizing) is highly correlated with the Jacobian norm, and that K-FAC with weight decay significantly reduces the Jacobian norm.
+
+3. Because the idealized, undamped version of K-FAC is invariant to affine reparameterizations, the implicit learning rate effect described above should not apply. However, in practice the approximate curvature matrix is damped by adding a multiple of the identity matrix, and this damping is not scale-invariant. We show that without weight decay, the weights grow large, causing the effective damping term to increase. If the effective damping term grows large enough to dominate the curvature term, it effectively turns K-FAC into a first-order optimizer. Weight decay keeps the effective damping term small, enabling K-FAC to retain its second-order properties, and hence improving generalization.
+
+Hence, we have identified three distinct mechanisms by which weight decay improves generalization, depending on the optimization algorithm and network architecture. Our results underscore the subtlety and complexity of neural network training: the final performance numbers obscure a variety of complex interactions between phenomena. While more analysis and experimentation is needed to understand how broadly each of our three mechanisms applies (and to find additional mechanisms!), our work provides a starting point for understanding practical regularization effects in neural network training.
+
+# 2 PRELIMINARIES
+
+Supervised learning. Given a training set $s$ consisting of training pairs $\{ \mathbf { x } , y \}$ , and a neural network $f _ { \pmb { \theta } } ( \mathbf { x } )$ with parameters $\pmb \theta$ (including weights and biases), our goal is to minimize the emprical risk expressed as an average of a loss $\ell$ over the training set: $\begin{array} { r } { \mathcal { L } ( \pmb { \theta } ) \equiv \frac { 1 } { N } \sum _ { ( \mathbf { x } , y ) \sim \mathcal { S } } \ell \left( y , f _ { \pmb { \theta } } ( \mathbf { x } ) \right) } \end{array}$ .
+
+Stochastic Gradient Descent. To minimize the empirical risk $\mathcal { L } ( \pmb \theta )$ , stochastic gradient descent (SGD) is used extensively in deep learning community. Typically, gradient descent methods can be derived from the framework of steepest descent with respect to standard Euclidean metric in parameter space. Specifically, gradient descent minimizes the following surrogate objective in each iteration:
+
+$$
+h ( \pmb \theta ) = \Delta \pmb \theta ^ { \top } \nabla _ { \pmb \theta } \mathcal { L } ( \pmb \theta ) + 1 / \eta \mathrm D ( \pmb \theta , \pmb \theta + \Delta \pmb \theta ) ,
+$$
+
+where the distance (or dissimilarity) function $\mathrm { D } ( \theta , \theta + \Delta \theta )$ is chosen as $\frac { 1 } { 2 } \lVert \Delta \theta \rVert _ { 2 } ^ { 2 }$ . In this case, solving equation 1 yields $\Delta \theta = - \dot { \eta } \nabla _ { \theta } \mathcal { L } ( \theta )$ , where $\eta$ is the learning rate.
+
+Natural gradient. Though popular, gradient descent methods often struggle to navigate “valleys” in the loss surface with ill-conditioned curvature (Martens, 2010). Natural gradient descent, as a variant of second-order methods (Martens, 2014), is able to make more progress per iteration by taking into account the curvature information. One way to motivate natural gradient descent is to show that it can be derived by adapting steepest descent formulation, much like gradient descnet, except using an alternative local distance. The distance function which leads to natural gradient is the KL divergence on the model’s predictive distribution $\begin{array} { r } { \mathrm { D } _ { \mathrm { K L } } ( p _ { \pmb { \theta } } \| p _ { \pmb { \theta } + \Delta \pmb { \theta } } ) \approx \frac { 1 } { 2 } \Delta \pmb { \theta } ^ { \top } \mathbf { F } \bar { \Delta \pmb { \theta } } } \end{array}$ , where $\mathbf { F } ( \pmb \theta )$ is the Fisher information matrix1 (Amari, 1998):
+
+$$
+\mathbf { F } = \mathbb { E } \left[ \nabla _ { \pmb { \theta } } \log p ( y | \mathbf { x } , \pmb { \theta } ) \nabla _ { \pmb { \theta } } \log p ( y | \mathbf { x } , \pmb { \theta } ) ^ { \top } \right] .
+$$
+
+Applying this distance function to equation 1, we have $\pmb { \theta } ^ { t + 1 }  \pmb { \theta } ^ { t } - \eta \mathbf { F } ^ { - 1 } \nabla _ { \pmb { \theta } } \mathcal { L } ( \pmb { \theta } ) .$
+
+Gauss-Newton algorithm. Another sensible distance function in equation 1 is the $L _ { 2 }$ distance on the output (logits) of the neural network, i.e. $\begin{array} { r l } { \frac { 1 } { 2 } \| f _ { \pmb { \theta } + \Delta \pmb { \theta } } - f _ { \pmb { \theta } } \| _ { 2 } ^ { 2 } } & { { } } \end{array}$ . This leads to the classical Gauss-Newton algorithm which updates the parameters by $\pmb { \theta } ^ { t + 1 }  \pmb { \theta } ^ { t } - \eta \mathbf { G } ^ { - 1 } \nabla _ { \pmb { \theta } } \mathcal { L } ( \pmb { \theta } )$ , where the Gauss-Newton (GN) matrix is defined as
+
+$$
+\mathbf { G } = \mathbb { E } \left[ \mathbf { J } _ { \theta } ^ { \top } \mathbf { J } _ { \theta } \right] ,
+$$
+
+and $\mathbf { J } _ { \pmb { \theta } }$ is the Jacobian of $f _ { \boldsymbol { \theta } } ( \mathbf { x } )$ w.r.t $\pmb { \theta }$ . The Gauss-Newton algorithm, much like natural gradient descent, is also invariant to the specific parameterization of neural network function $f _ { \theta }$ .
+
+Two curvature matrices. It has been shown that the GN matrix is equivalent to the Fisher matrix in the case of regression task with squared error loss (Heskes, 2000). However, they are not identical for the case of classification, where cross-entropy loss is commonly used. Nevertheless, Martens (2014) showed that the Fisher matrix is equivalent to generalized GN matrix when model prediction $p ( \boldsymbol { y } | \mathbf { x } , \boldsymbol { \theta } )$ corresponds to exponential family model with natural parameters given by $f _ { \pmb { \theta } } ( \mathbf { \bar { x } } )$ , where the generalized GN matrix is given by
+
+$$
+\mathbf { G } = \mathbb { E } \left[ \mathbf { J } _ { \theta } ^ { \top } \mathbf { H } _ { \ell } \mathbf { J } _ { \theta } \right] ,
+$$
+
+and $\mathbf { H } _ { \ell }$ is the Hessian of $\ell ( y , z )$ w.r.t $z$ , evaluated at $z = f _ { \boldsymbol { \theta } } ( \mathbf { x } )$ . In regression with squared error loss, the Hessian $\mathbf { H } _ { \ell }$ happens to be identity matrix.
+
+Preconditioned gradient descent. Given the fact that both natural gradient descent and GaussNewton algorithm precondition the gradient with an extra curvature matrix $\mathbf { C } ( \pmb { \theta } )$ (including the Fisher matrix and GN matrix), we also term them preconditioned gradient descent for convenience.
+
+K-FAC. As modern neural networks may contain millions of parameters, computing and storing the exact curvature matrix and its inverse is impractical. Kronecker-factored approximate curvature (K-FAC) (Martens & Grosse, 2015) uses a Kronecker-factored approximation to the curvature matrix to perform efficient approximate natural gradient updates. As shown by Luk & Grosse (2018), K-FAC can be applied to general pullback metric, including Fisher metric and the Gauss-Newton metric. For more details, we refer reader to Appendix $\mathrm { F }$ or Martens & Grosse (2015).
+
+Batch Normalization. Broadly speaking, Batch Normalization (BN) is a mechanism that aims to stabilize the distribution (over a mini-batch) of inputs to a given network layer during training. This is achieved by augmenting the network with additional layers that subtract the mean $\mu$ and divide by the standard deviation $\sigma$ . Typically, the normalized inputs are also scaled and shifted based on trainable parameters $\gamma$ and $\beta$ :
+
+$$
+\mathbf { B N } ( \mathbf { x } ) = \frac { \mathbf { x } - \mu } { \sigma } \cdot \boldsymbol { \gamma } + \boldsymbol { \beta } .
+$$
+
+For clarity, we ignore the parameters $\gamma$ and $\beta$ , which do not impact the performance in practice. This is not surprising, since with ReLU activations, only the $\gamma$ of the last layer affects network’s outputs which can be merged with the softmax layer weights (as also pointed out by van Laarhoven (2017)).
+
+# 3 THE EFFECTIVENESS OF WEIGHT DECAY
+
+Our goal is to understand weight decay regularization in the context of training deep neural networks. Towards this, we first discuss the relationship between $L _ { 2 }$ regularization and weight decay in different optimizers.
+
+Table 1: Classification results on CIFAR-10 and CIFAR-100. B denotes BN while D denotes data augmentation, including horizontal flip and random crop. WD denotes weight decay regularization. Weight decay regularization improves the generalization consistently. Interestingly, we observe that weight decay gives an especially strong performance boost to the K-FAC optimizer when BN is turned off.   
+
+<table><tr><td rowspan=2 colspan=1>Dataset</td><td rowspan=2 colspan=1>Network</td><td rowspan=2 colspan=1>B D</td><td rowspan=1 colspan=10>SGD         ADAM       K-FAC-F      K-FAC-G</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>WD</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td></tr><tr><td rowspan=3 colspan=1>CIFAR-10</td><td rowspan=3 colspan=1>VGG16</td><td rowspan=3 colspan=1>√ √</td><td rowspan=3 colspan=1>83.2086.9991.71</td><td rowspan=3 colspan=1>84.8788.8593.39</td><td rowspan=1 colspan=1>83.16</td><td rowspan=2 colspan=1>84.1288.72</td><td rowspan=2 colspan=1>85.5887.97</td><td rowspan=2 colspan=3>89.6089.02</td><td rowspan=2 colspan=1>83.8588.17</td><td rowspan=1 colspan=1>89.81</td></tr><tr><td rowspan=1 colspan=1>88.45</td><td rowspan=1 colspan=1>89.77</td></tr><tr><td rowspan=1 colspan=1>92.89</td><td rowspan=1 colspan=1>93.62</td><td rowspan=1 colspan=1>93.12</td><td rowspan=1 colspan=2>93.90</td><td rowspan=1 colspan=2>93.90</td><td rowspan=1 colspan=1>93.19</td><td rowspan=1 colspan=1>93.80</td></tr><tr><td rowspan=7 colspan=1>CIFAR-10</td><td rowspan=7 colspan=1>ResNet32</td><td rowspan=7 colspan=1>√√ √</td><td rowspan=7 colspan=1>85.4786.1392.95</td><td rowspan=7 colspan=1>86.6390.6595.14</td><td rowspan=3 colspan=1>84.43</td><td rowspan=3 colspan=1>87.54</td><td rowspan=3 colspan=1>86.82</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=2 colspan=3>90.2291.24</td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>85.24</td><td rowspan=1 colspan=1>90.64</td></tr><tr><td rowspan=4 colspan=1>89.4693.63</td><td rowspan=4 colspan=1>90.6194.66</td><td rowspan=1 colspan=1>89.78</td><td rowspan=1 colspan=2>91.24</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>89.94</td><td rowspan=1 colspan=1>90.91</td></tr><tr><td rowspan=3 colspan=1>93.80</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=2 colspan=3>95.35</td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>93.44</td><td rowspan=1 colspan=1>95.04</td></tr><tr><td rowspan=1 colspan=1>CIFAR-100</td><td rowspan=1 colspan=1>VGG16</td><td rowspan=1 colspan=1>√ √</td><td rowspan=1 colspan=1>68.42</td><td rowspan=1 colspan=1>73.31</td><td rowspan=1 colspan=1>69.88</td><td rowspan=1 colspan=1>74.22</td><td rowspan=1 colspan=1>71.05</td><td rowspan=1 colspan=3>73.36</td><td rowspan=1 colspan=1>67.46</td><td rowspan=1 colspan=1>73.57</td></tr><tr><td rowspan=1 colspan=1>CIFAR-100</td><td rowspan=1 colspan=1>ResNet32</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>73.61</td><td rowspan=1 colspan=1>77.73</td><td rowspan=1 colspan=1>73.60</td><td rowspan=1 colspan=1>77.40</td><td rowspan=1 colspan=1>74.49</td><td rowspan=1 colspan=3>78.01</td><td rowspan=1 colspan=1>73.70</td><td rowspan=1 colspan=1>78.02</td></tr></table>
+
+Gradient descent with weight decay is defined by the following update rule: $\pmb { \theta } ^ { t + 1 }  ( 1 - \eta \beta ) \pmb { \theta } ^ { t } -$ $\eta \nabla \mathcal { L } ( \pmb \theta ^ { t } )$ , where $\beta$ defines the rate of the weight decay per step and $\eta$ is the learning rate. In this case, weight decay is equivalent to $L _ { 2 }$ regularization. However, the two differ when the gradient update is preconditioned by a matrix $\mathbf { C } ^ { - 1 }$ , as in Adam or K-FAC. The preconditioned gradient descent update with $L _ { 2 }$ regularization is given by
+
+$$
+\mathbf { \boldsymbol { \theta } } ^ { t + 1 } \gets ( \mathbf { I } - \eta \beta \mathbf { C } ^ { - 1 } ) \mathbf { \boldsymbol { \theta } } ^ { t } - \eta \mathbf { C } ^ { - 1 } \nabla \_ { \theta } \mathbf { \boldsymbol { \mathcal { L } } } ( \mathbf { \boldsymbol { \theta } } ^ { t } ) ,
+$$
+
+whereas the weight decay update is given by
+
+$$
+\mathbf { \theta } \mathbf { \theta } ^ { t + 1 } \gets ( 1 - \eta \beta ) \mathbf { \theta } \mathbf { \theta } ^ { t } - \eta \mathbf { C } ^ { - 1 } \nabla _ { \mathbf { \theta } } \mathcal { L } ( \mathbf { \theta } ^ { t } ) .
+$$
+
+The difference between these updates is whether the preconditioner is applied to $\pmb { \theta } ^ { t }$ . The latter update can be interpreted as the preconditioned gradient descent update on a regularized objective where the regularizer is the squared $\mathbf { C }$ -norm $\| \pmb { \theta } \| _ { \mathbf { C } } ^ { 2 } = \pmb { \theta } ^ { \top } \mathbf { C } \pmb { \theta }$ . If $\mathbf { C }$ is adapted based on statistics collected during training, as in Adam or K-FAC, this interpretation holds only approximately because gradient descent on $\| \pmb \theta \| _ { \mathbf { C } } ^ { 2 }$ would require differentiating through $\mathbf { C }$ . However, this approximate regularization term can still yield insight into the behavior of weight decay. (As we discuss later, this observation informs some, but not all, of the empirical phenomena we have observed.) Though the difference between the two updates may appear subtle, we find that it makes a substantial difference in terms of generalization performance.
+
+Initial Experiments. We now present some empirical findings about the effectiveness of weight decay which the rest of the paper is devoted to explaining. Our experiments were carried out on two different datasets: CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton, 2009) with varied batch sizes. We test VGG16 (Simonyan & Zisserman, 2014) and ResNet32 (He et al., 2016) on both CIFAR-10 and CIFAR-100 (for more details, see Appendix A). In particular, we investigate three different optimization algorithms: SGD, Adam and K-FAC. We consider two versions of K-FAC, which use the Gauss-Newton matrix (K-FAC-G) and Fisher information matrix (K-FAC-F).
+
+Figure 1 shows the comparison between weight decay, $L _ { 2 }$ regularization and the baseline. We also compare weight decay to the baseline on more settings and report the final test accuracies in Table 1. Finally, the results for large-batch training are summarized in Table 3. Based on these results, we make the following observations regarding weight decay:
+
+1. In all experiments, weight decay regularization consistently improved the performance and was more effective than $L _ { 2 }$ regularization in cases where they differ (See Figure 1).   
+2. Weight decay closed most of the generalization gaps between first- and second-order optimizers, as well as between small and large batches (See Table 1 and Table 3).   
+3. Weight decay significantly improved performance even for BN networks (See Table 1), where it does not meaningfully constrain the networks’ capacity.   
+4. Finally, we notice that weight decay gave an especially strong performance boost to the K-FAC optimizer when BN was disabled (see the first and fourth rows in Table 1).
+
+In the following section, we seek to explain these phenomena. With further testing, we find that weight decay can work in unexpected ways, especially in the presence of BN.
+
+![](images/5260b2db41a1981ee96bd5beeb0e0842c79e86dd5082753dfdc451218dbf7494.jpg)  
+Figure 2: Test accuracy as a function of training epoch for SGD and Adam on CIFAR-100 with different weight decay regularization schemes. baseline is the model without weight decay; wd-conv is the model with weight decay applied to all convolutional layers; wd-all is the model with weight decay applied to all layers; wd-fc is the model with weight decay applied to the last layer (fc). Most of the generalization effect of weight decay is due to applying it to layers with BN.
+
+# 4 THREE MECHANISMS OF WEIGHT DECAY REGULARIZATION
+
+# 4.1 MECHANISM I: HIGHER EFFECTIVE LEARNING RATE
+
+As discussed in Section 3, when SGD is used as the optimizer, weight decay can be interpreted as penalizing the $L _ { 2 }$ norm of the weights. Classically, this was believed to constrain the model by penalizing explanations with large weight norm. However, for a network with Batch Normalization (BN), an $L _ { 2 }$ penalty does not meaningfully constrain the reprsentation, because the network’s predictions are invariant to rescaling of the weights and biases. More precisely, if $\mathbf { B N } ( \mathbf { x } ; \pmb { \theta } _ { l } )$ denotes the output of a layer with parameters $\theta _ { l }$ in which BN is applied before the activation function, then
+
+$$
+{ \bf B N } ( { \bf x } ; \alpha \pmb { \theta } _ { l } ) = { \bf B N } ( { \bf x } ; \pmb { \theta } _ { l } ) ,
+$$
+
+for any $\alpha > 0$ . By choosing small $\alpha$ , one can make the $L _ { 2 }$ norm arbitrarily small without changing the function computed by the network. Hence, in principle, adding weight decay to layers with BN should have no effect on the optimal solution. But empirically, weight decay appears to significantly improve generalization for BN networks (e.g. see Figure 1).
+
+van Laarhoven (2017) observed that $L _ { 2 }$ regularization has an influence on the effective learning rate in (stochastic) gradient descent. In this work, we extend this result to first-order optimizers (including SGD and Adam) that weight decay increases the effective learning rate by reducing the scale of the weights. Since higher learning rates lead to larger gradient noise, which has been shown to act as a stochastic regularizer (Neelakantan et al., 2015; Keskar et al., 2016; Jastrz˛ebski et al., 2017; Hoffer et al., 2017), this means weight decay can indirectly exert a regularizing effect through the effective learning rate. In this section, we provide additional evidence supporting the hypothesis of van Laarhoven (2017). For simplicity, this section focuses on SGD, but we’ve observed similar behavior when Adam is used as the optimizer.
+
+Due to its invariance to the scaling of the weights, the key property of the weight vector is its direction. As shown by Hoffer et al. (2018), the weight direction $\hat { \pmb { \theta } } _ { l } = \pmb { \theta } _ { l } / \lVert \pmb { \theta } _ { l } \rVert _ { 2 }$ is updated according to
+
+![](images/7956eb00c1e74952e2a18f36b65f046a454a6b3634c45d4d0dfcb1efda2c81f0.jpg)  
+Figure 3: Effective learning rate of the first layer of ResNet32 trained with SGD on CIFAR-100. Without weight decay regularization, the effective learning rate decreases quickly in the beginning.
+
+$$
+\begin{array} { r } { \hat { \pmb { \theta } } _ { l } ^ { t + 1 }  \hat { \pmb { \theta } } _ { l } ^ { t } - \eta \| \pmb { \theta } _ { l } ^ { t } \| _ { 2 } ^ { - 2 } ( \mathbf { I } - \hat { \pmb { \theta } } _ { l } ^ { t } \hat { \pmb { \theta } } _ { l } ^ { t ^ { \top } } ) \nabla _ { \pmb { \theta } _ { l } } \mathcal { L } ( \hat { \pmb { \theta } } ^ { t } ) + O ( \eta ^ { 2 } ) . } \end{array}
+$$
+
+Therefore, the effective learning rate is approximately proportional to $\eta / \lVert \pmb { \theta } _ { l } \rVert _ { 2 } ^ { 2 }$ . Which means that by decreasing the scale of the weights, weight decay regularization increases the effective learning rate.
+
+Figure 3 shows the effective learning rate over time for two BN networks trained with SGD (the results for Adam are similar), one with weight decay and one without it. Each network is trained with a typical learning rate decay schedule, including 3 factor-of-10 reductions in the learning rate parameter, spaced 60 epochs apart. Without weight decay, the normalization effects cause an additional effective learning rate decay (due to the increase of weight norm), which reduces the effective learning rate by a factor of 10 over the first 50 epochs. By contrast, when weight decay is applied, the effective learning rate remains more or less constant in each stage.
+
+We now show that the effective learning rate schedule explains nearly the entire generalization effect of weight decay. First, we independently varied whether weight decay was applied to the top layer of the network, and to the remaining layers. Since all layers except the top one used BN, it’s only in the top layer that weight decay would constrain the model. Training curves for SGD and Adam under all four conditions are shown in Figure 2. In all cases, we observe that whether weight decay was applied to the top (fully connected) layer did not have a significant impact; whether it was applied to the reamining (convolution) layers explained most of the generalization effect. This supports the effective learning rate hypothesis.
+
+![](images/915a03a464ec5166f27eff32c36f5624b14cb582a0f8d2c58a0e5f83082d1924.jpg)  
+Figure 4: The curves of test accuracies of ResNet32 on CIFAR-100. To be noted, we use wd and wn to denote weight decay and weight normalization respectively.
+
+We further tested this hypothesis using a simple experimental manipulation. Specifically, we trained a BN network without weight decay, but after each epoch, rescaled the weights in each layer to match that layer’s norm from the corresponding epoch for the network with weight decay. This rescaling does not affect the network’s predictions, and is equivalent to setting the effective learning rate to match the second network. As shown in Figure 4, this effective learning rate transfer scheme (wn-conv) eliminates almost the entire generalization gap; it is fully closed by also adding weight decay to the top layer (wd-fc+wn-conv). Hence, we conclude that for BN networks trained with SGD or Adam, weight decay achieves its regularization effect primarily through the effective learning rate.
+
+# 4.2 MECHANISM II: APPROXIMATE JACOBIAN REGULARIZATION
+
+In Section 3, we observed that when BN is disabled, weight decay has the strongest regularization effect when K-FAC is used as the optimizer. Hence, in this section we analyze the effect of weight decay for K-FAC with networks without BN. First, we show that in a certain idealized setting, K-FAC with weight decay regularizes the input-output Jacobian of the network. We then empirically investigate whether it behaves similarly for practical networks.
+
+As discussed in Section 3, when the gradient updates are preconditioned by a matrix $\mathbf { C }$ , weight decay can be viewed as approximate preconditioned gradient descent on the norm $\| \pmb { \theta } \| _ { \mathbf { C } } ^ { 2 } = \pmb { \theta } ^ { \top } \mathbf { \bar { C } } \pmb { \theta }$ . This interpretation is only approximate because the exact gradient update requires differentiating through C.2 When $\mathbf { C }$ is taken to be the (exact) Gauss-Newton (GN) matrix $\mathbf { G }$ , we obtain the Gauss-Newton norm $\| \pmb \theta \| _ { \mathbf { G } } ^ { 2 } = \pmb \theta ^ { \top } \mathbf { G } ( \pmb \theta ) \pmb \theta$ . Similarly, when $\mathbf { C }$ is taken to be the K-FAC approximation to $\mathbf { G }$ , we obtain what we term the $K \cdot$ -FAC Gauss-Newton norm.
+
+These norms are interesting from a regularization perspective. First, under certain conditions, they are proportional to the average $L _ { 2 }$ norm of the network’s outputs. Hence, the regularizer ought to make the network’s predictions less extreme. This is summarized by the following results:
+
+Lemma 1 (Gradient structure). For a feed-forward neural network of depth $L$ with ReLU activation function and no biases, the network’s outputs are related to the input-output Jacobian and parameteroutput Jacobian as follows:
+
+$$
+\begin{array} { l } { f _ { \theta } ( \mathbf { x } ) = \nabla _ { \mathbf { x } } f _ { \theta } ( \mathbf { x } ) ^ { \top } \mathbf { x } = \mathbf { J } _ { \mathbf { x } } \mathbf { x } } \\ { \quad = \displaystyle \frac { 1 } { L + 1 } \nabla _ { \theta } f _ { \theta } ( \mathbf { x } ) ^ { \top } \pmb \theta = \frac { 1 } { L + 1 } \mathbf { J } _ { \theta } \pmb \theta . } \end{array}
+$$
+
+Lemma 2 (Gauss-Newton Norm). Under the same assumptions of Lemma $^ { l }$ , we observe:
+
+$$
+\begin{array} { r } { \| \pmb \theta \| _ { \mathbf { G } } ^ { 2 } = ( L + 1 ) ^ { 2 } \mathbb { E } \left[ \| f _ { \pmb \theta } ( \mathbf { x } ) \| ^ { 2 } \right] . } \end{array}
+$$
+
+If we further restrict the network to be a deep linear neural network, we have $K$ -FAC Gauss-Newton norm as follows:
+
+$$
+\begin{array} { r } { \| \pmb { \theta } \| _ { \mathrm { \mathbf { G } _ { K - \mathrm { F A C } } } } ^ { 2 } = ( L + 1 ) \mathbb { E } \left[ \| f _ { \pmb { \theta } } ( \mathbf { x } ) \| ^ { 2 } \right] . } \end{array}
+$$
+
+2We show in Appendix E that this interpretation holds exactly in the case of Gauss-Newton norm.
+
+![](images/32a191482e892a8175ac06e71d32d75c4f4fa390c3d0816e2eab161059ad14b2.jpg)  
+Figure 5: Relationship between K-FAC GN norm and Jacobian norm for practical deep neural networks. Each point corresponds to a network trained to $1 0 0 \%$ training accuracy. Even for (nonlinear) classification networks, the K-FAC GN norm is highly correlated with both the squared Frobenius norm of the input-output Jacobian and the generalization gap.
+
+Using these results, we show that for linear networks3 with whitened inputs, the (K-FAC) GaussNewton norm is proportional to the squared Frobenius norm of the input-output Jacobian. This is interesting from a regularization perspective, since Novak et al. (2018) found the norm of the input-output Jacobian to be consistently coupled to generalization performance.
+
+Theorem 1 (Approximate Jacobian norm). For a deep linear network of depth L without biases, if we further assume that $\mathbb { E } [ \mathbf { x } ] = \mathbf { 0 }$ and $\mathrm { C o v } ( \mathbf { x } ) = \mathbf { I } ,$ , then:
+
+$$
+\| \pmb { \theta } \| _ { \mathbf { G } } ^ { 2 } = ( L + 1 ) ^ { 2 } \| \mathbf { J } _ { \mathbf { x } } \| _ { \mathrm { F r o b } } ^ { 2 }
+$$
+
+and
+
+$$
+\begin{array} { r } { \Vert \pmb { \theta } \Vert _ { \mathbf { G } _ { \mathrm { K - F A C } } } ^ { 2 } = ( L + 1 ) \Vert \mathbf { J } _ { \mathbf { x } } \Vert _ { \mathrm { F r o b } } ^ { 2 } . } \end{array}
+$$
+
+Proof. It follows from Lemma 2 that $\| \pmb \theta \| _ { \mathbf { G } } ^ { 2 } = ( L + 1 ) ^ { 2 } \mathbb { E } \left[ \| f _ { \pmb \theta } ( \mathbf { x } ) \| ^ { 2 } \right]$ . By Lemma 1, we have
+
+$$
+\begin{array} { r } { \mathbb { E } \left[ \| f _ { \pmb { \theta } } ( \mathbf { x } ) \| ^ { 2 } \right] = \mathbb { E } \left[ \mathbf { x } ^ { \top } \mathbf { J } _ { \mathbf { x } } ^ { \top } \mathbf { J } _ { \mathbf { x } } \mathbf { x } \right] = \mathbb { E } \left[ \operatorname { t r } \mathbf { J } _ { \mathbf { x } } ^ { \top } \mathbf { J } _ { \mathbf { x } } \mathbf { x } \mathbf { x } ^ { \top } \right] . } \end{array}
+$$
+
+When the network is linear, the input-output Jacobian $\mathbf { J } _ { \mathbf { x } }$ is independent of the input $\mathbf { x }$ . Then we use the assumption of whitened inputs:
+
+$$
+\begin{array} { r } { \| \theta \| _ { \mathbf { G } } ^ { 2 } = ( L + 1 ) ^ { 2 } \mathbb { E } \left[ \mathrm { t r } { \mathbf { J } _ { \mathbf { x } } ^ { \top } } { \mathbf { J } _ { \mathbf { x } } } \mathbf { x } \mathbf { x } ^ { \top } \right] = ( L + 1 ) ^ { 2 } \mathrm { t r } { \mathbf { J } _ { \mathbf { x } } ^ { \top } } \mathbf { J } _ { \mathbf { x } } \mathbb { E } [ \mathbf { x } \mathbf { x } ^ { \top } ] = ( L + 1 ) ^ { 2 } \| \mathbf { J } _ { \mathbf { x } } \| _ { \mathrm { F r o b } } ^ { 2 } . } \end{array}
+$$
+
+The proof for K-FAC Gauss-Newton norm follows immediately with equation 12.
+
+While the equivalence between the (K-FAC) GN norm and the Jacobian norm holds only for linear networks, we note that linear networks have been useful for understanding the dynamics of neural net training more broadly (e.g. Saxe et al. (2013)). Hence, Jacobian regularization may help inform our understanding of weight decay in practical (nonlinear) networks.
+
+To test whether the K-FAC GN norm correlates with the Jacobian norm for practical networks, we trained feed-forward networks with a variety optimizers on both MNIST (LeCun et al., 1998) and CIFAR-10. For MNIST, we used simple fully-connected networks with different depth and width. For CIFAR-10, we adopted the VGG family (From VGG11 to VGG19). We defined the generalization gap to be the difference between training and test loss. Figure 5 shows the relationship of the Jacobian norm to the K-FAC GN norm and to generalization gap for these networks. We observe that the Jacobian norm correlates strongly with the generalization gap (consistent with Novak et al. (2018)) and also with the K-FAC GN norm. Hence, Theorem 1 can inform the regularization of nonlinear networks.
+
+Table 2: Squared Frobenius norm of the input-output Jacobian matrix. K-FAC-G with weight decay significantly reduces the Jacobian norm.   
+
+<table><tr><td rowspan=2 colspan=1>Optimizer</td><td rowspan=1 colspan=2>VGG16</td><td rowspan=1 colspan=2>ResNet32</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td></tr><tr><td rowspan=1 colspan=1>SGD</td><td rowspan=1 colspan=1>564</td><td rowspan=1 colspan=1>142</td><td rowspan=1 colspan=1>2765</td><td rowspan=1 colspan=1>1074</td></tr><tr><td rowspan=1 colspan=1>K-FAC-G</td><td rowspan=1 colspan=1>498</td><td rowspan=1 colspan=1>51.44</td><td rowspan=1 colspan=1>2115</td><td rowspan=1 colspan=1>64.16</td></tr></table>
+
+To test if K-FAC with weight decay reduces the Jacobian norm, we compared the Jacobian norms at the end of training for networks with and without weight decay. As shown in Table 2, weight decay reduced the Jacboian norm by a much larger factor when K-FAC was used as the optimizer than when SGD was used as the optimizer.
+
+![](images/749a5f4b27fe8159f4e7342f27489da7f7a76e6ca2c5abe692a01feae2370140.jpg)  
+Figure 6: Test accuracy as a function of training epoch for K-FAC on CIFAR-100 with different weight decay regularization schemes. baseline is the model without weight decay regularization; wd-conv is the model with weight decay applied to all convolutional layers; wd-all is the model with weight decay applied to all layers; wd-fc is the model with weight decay applied to the last layer (fc). Consistent with the Jacobian regularization hypothesis, applying weight decay to the non-BN layers have the largest regularization effect. However, applying weight decay to the BN layers also lead to noticeable gains.
+
+Our discussion so far as focused on the GN version of K-FAC. Recall that, in many cases, the Fisher information matrix differs from the GN matrix only in that it accounts for the output layer Hessian. Hence, this analysis may help inform the behavior of K-FAC-F as well. We also note that $\| \pmb \theta \| _ { \mathbf { F } } ^ { 2 }$ , the Fisher-Rao norm, has been proposed as a complexity measure for neural networks (Liang et al., 2017). Hence, unlike in the case of SGD and Adam for BN networks, we interpret K-FAC with weight decay as constraining the capacity of the network.
+
+# 4.3 MECHANISM III: SMALLER EFFECTIVE DAMPING PARAMETER
+
+We now return our attention to the setting of architectures with BN. The Jacobian regularization mechanism from Section 4.2 does not apply in this case, since rescaling the weights results in an equivalent network, and therefore does not affect the input-output Jacobian. Similarly, if the network is trained with K-FAC, then the effective learning rate mechanism from Section 4.1 also does not apply because the K-FAC update is invariant to affine reparameterization (Luk & Grosse, 2018) and therefore not affected by the scaling of the weights. More precisely, for a layer with BN, the curvature matrix $\mathbf { C }$ (either the Fisher matrix or the GN matrix) has the following property:
+
+$$
+\mathbf { C } ( \pmb { \theta } _ { l } ) = \frac { 1 } { \| \pmb { \theta } _ { l } \| _ { 2 } ^ { 2 } } \mathbf { C } ( \hat { \pmb { \theta } } _ { l } ) ,
+$$
+
+where $\hat { \pmb { \theta } } _ { l } = \pmb { \theta } _ { l } / \lVert \pmb { \theta } _ { l } \rVert _ { 2 }$ as in Section 4.1. Hence, the $\| \pmb { \theta } _ { l } \| _ { 2 } ^ { 2 }$ factor in the preconditioner counteracts the $\lVert \pmb { \theta } _ { l } \rVert _ { 2 } ^ { - 2 }$ factor in the effective learning rate, resulting in an equivlaent effective learning rate regardless of the norm of the weights.
+
+These observations raise the question of whether it is still useful to apply weight decay to BN layers when using K-FAC. To answer this question, we repeated the experiments in Figure 2 (applying weight decay to subsets of the layers), but with K-FAC as the optimizer. The results are summarized in Figure 6. Applying it to the non-BN layers had the largest effect, consistent with the Jacobian regularization hypothesis. However, applying weight decay to the BN layers also led to significant gains, especially for K-FAC-F.
+
+The reason this does not contradict the K-FAC invariance property is that practical K-FAC implementations dampen the updates (like many second-order optimziers) by adding a multiple of the identity matrix to the curvature before inversion. According to equation 15, as the norm of the weights gets larger, $\mathbf { C }$ gets smaller, and hence the damping term comes to dominate the preconditioner. Mathematically, we can understand this effect by deriving the following update rule for the normalized weights $\hat { \pmb { \theta } }$ (see Appendix D for proof):
+
+$$
+\begin{array} { r } { \hat { \theta } _ { l } ^ { t + 1 } \gets \hat { \theta } _ { l } ^ { t } - \eta ( \mathbf { I } - \hat { \theta } _ { l } ^ { t } \hat { \theta } _ { l } ^ { t ^ { \top } } ) ( \mathbf { C } ( \hat { \theta } _ { l } ^ { t } ) + \lVert \theta _ { l } ^ { t } \rVert _ { 2 } ^ { 2 } \lambda \mathbf { I } ) ^ { - 1 } \nabla _ { \theta _ { l } } \mathcal { L } ( \hat { \theta } ^ { t } ) + O ( \eta ^ { 2 } ) , } \end{array}
+$$
+
+where $\lambda$ is the damping parameter. Hence, for large $\mathbf { C } ( \hat { \pmb { \theta } } _ { l } )$ or small $\lVert \pmb { \theta } _ { l } \rVert$ , the update is close to the idealized second-order update, while for small enough $\mathbf { C } ( \hat { \pmb { \theta } } _ { l } )$ or large enough $\| \pmb \theta _ { l } \|$ , K-FAC effectively becomes a first-order optimizer. Hence, by keeping the weights small, weight decay helps K-FAC to retain its second-order properties.
+
+Most implementations of K-FAC keep the damping parameter $\lambda$ fixed throughout training. Therefore, it would be convenient if $\bar { \bf C } ( \hat { \pmb { \theta } } _ { l } )$ and $\| \pmb \theta _ { l } \|$ do not change too much during training, so that a single value of $\lambda$ can work well throughout training. Interestingly, the norm of the GN matrix appears to be much more stable than the norm of the Fisher matrix. Figure 7 shows the norms of the Fisher matrix $\mathbf { F } ( \hat { \pmb { \theta } } _ { l } )$ and GN matrix $\mathbf { G } ( \hat { \theta } _ { l } )$ of the normalized weights for the first layer of a CIFAR-10 network throughout training. While the norm of $\mathbf { F } ( \hat { \pmb { \theta } } _ { l } )$ decays by 4 orders of magnitude over the first 50 epochs, the norm of $\mathbf { G } ( \hat { \pmb { \theta } } _ { l } )$ increases by only a factor of 2.
+
+![](images/1bb554cb6b9d0e1f9d077db9f33223443fdae03ca28664bf0626d5a40778100e.jpg)  
+Figure 7: Trace norm of Fisher matrix and GaussNewton matrix of the first layer (Normalized) of ResNet32. The model was trained on CIFAR-10 with K-FAC-F and BN.
+
+The explanation for this is as follows: in a classification task with cross-entropy loss, the Fisher matrix is equivalent to the generalized GN matrix $\mathbb { E } [ \mathbf { J } _ { \theta } ^ { \top } \mathbf { H } _ { \ell } \mathbf { J } _ { \theta } ]$ (see Section 2). This differs from the GN matrix $\mathbb { E } [ \mathbf { J } _ { \pmb { \theta } } ^ { \top } \mathbf { J } _ { \pmb { \theta } } ]$ only in that it incudes the output layer Hessian $\mathbf { H } _ { \ell } = \mathrm { d i a g } ( \mathbf { p } ) - \mathbf { p } \mathbf { p } ^ { \top }$ , where $\mathbf { p }$ is the vector of estimated class probabilities. It is easy to see that $\mathbf { H } _ { \ell }$ goes to zero as p collapses to one class, as is the case for tasks such as CIFAR-10 and CIFAR-100 where networks typically achieve perfect training accuracy. Hence, we would expect $\mathbf { F }$ to get much smaller over the course of training, consistent with Figure 7.
+
+To summarize, when K-FAC is applied to BN networks, it can be advantageous to apply weight decay even to layers with BN, even though this appears unnecessary based on invariance considerations. The reason is that weight decay reduces the effective damping, helping K-FAC to retain its second-order properties. This effect is stronger for K-FAC-F than for K-FAC-G because the Fisher matrix shrinks dramatically over the course of training.
+
+# 5 DISCUSSION
+
+Despite its long history, weight decay regularization remains poorly understood. We’ve identified three distinct mechanisms by which weight decay improves generalization, depending on the architecture and optimization algorithm: increasing the effective learning rate, reducing the Jacobian norm, and reducing the effective damping parameter. We would not be surprised if there remain additional mechanisms we have not found.
+
+The dynamics of neural net training is incredibly complex, and it can be tempting to simply do what works and not look into why. But we think it is important to at least sometimes dig deeper to determine exactly why an algorithm has the effect that it does. Some of our analysis may seem mundane, or even tedious, as the interactions between different hyperparameters are not commonly seen as a topic worthy of detailed scientific study. But our experiments highlight that the dynamics of the norms of weights and curvature matrices, and their interaction with optimization hyperparameters, can have a substantial impact on generalization. We believe these effects deserve more attention, and would not be surprised if they can help explain the apparent success or failure of other neural net design choices. We also believe our results highlight the need for automatic adaptation of optimization hyperparameters, to eliminate potential experimental confounds and to allow researchers and practitioners to focus on higher level design issues.
+
+# 6 ACKNOWLEDGEMENT
+
+We thank Jimmy Ba, David Duvenaud, Kevin Luk, Maxime Gazeau, and Behnam Neyshabur for helpful discussions, and Tianqi Chen and Shengyang Sun for their feedback on early drafts. GZ was funded by an MRIS Early Researcher Award.
+
+# REFERENCES
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+
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+
+# A EXPERIMENTS DETAILS
+
+Throughout the paper, we perform experiments on image classification with three different datasets, MNIST (LeCun et al., 1998), CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton, 2009). For MNIST, we use simple fully-connected networks with different depth and width. For CIFAR-10 and CIFAR100, we use VGG16 (Simonyan & Zisserman, 2014) and ResNet32 (He et al., 2016). To make the network more flexible, we widen all convolutional layers in ResNet32 by a factor of 4, according to Zagoruyko & Komodakis (2016).
+
+We investigate three different optimization methods, including Stochastic Gradient Descent (SGD), Adam (Kingma & Ba, 2014) and K-FAC (Martens & Grosse, 2015). In K-FAC, two different curvature matrices are studied, including Fisher information matrix and Gauss-Newton matrix.
+
+In default, batch size 128 is used unless stated otherwise. In SGD and Adam, we train the networks with a budge of 200 epochs and decay the learning rate by a factor of 10 every 60 epochs for batch sizes of 128 and 640, and every 80 epochs for the batch size of 2K. Whereas we train the networks only with 100 epochs and decay the learning rate every 40 epochs in K-FAC. Additionally, the curvature matrix is updated by running average with re-estimation every 10 iterations and the inverse operator is amortized to 100 iterations. For K-FAC, we use fixed damping term $1 e ^ { - 3 }$ unless state otherwise. For each algorithm, best hyperparameters (learning rate and regularization factor) are selected using grid search on held-out $5 \mathrm { k }$ validation set. For the large batch setting, we adopt the same strategies in Hoffer et al. (2017) for adjusting the search range of hyperparameters. Finally, we retrain the model with both training data and validation data.
+
+# B GRADIENT STRUCTURE IN NEURAL NETWORKS (LEMMA 1)
+
+Claim. For a feed-forward neural network of depth $L$ with ReLU activation function and no biases, one has the following property:
+
+$$
+\begin{array} { l } { f _ { \theta } ( \mathbf { x } ) = \nabla _ { \mathbf { x } } f _ { \theta } ( \mathbf { x } ) ^ { \top } \mathbf { x } = \mathbf { J } _ { \mathbf { x } } \mathbf { x } } \\ { \quad = \displaystyle \frac { 1 } { L + 1 } \nabla _ { \theta } f _ { \theta } ( \mathbf { x } ) ^ { \top } \pmb \theta = \frac { 1 } { L + 1 } \mathbf { J } _ { \theta } \pmb \theta } \end{array}
+$$
+
+The key observation of Lemma 1 is that rectified neural networks are piecewise linear up to the output $f _ { \pmb { \theta } } ( \mathbf { x } )$ . And ReLU activation function satisfies the property $\sigma ( { \bf z } ) = \sigma ^ { \prime } ( { \bf z } ) { \bf z }$ .
+
+Proof. For convenience, we introduce some notations here. Let $\mathbf { z } _ { L + 1 }$ denotes output logits $f _ { \theta } ( \mathbf { x } ) , \mathbf { z } _ { l }$ the output $l$ -th layer. Similarly, we define ${ \bf a } _ { l } = \sigma ( { \bf z } _ { l } )$ and $\mathbf { a } _ { 0 } = \mathbf { x }$ . By definition, it is easy to see that
+
+$$
+\begin{array} { c } { \displaystyle { { \bf z } _ { l + 1 } = { \bf W } _ { l } { \bf a } _ { l } = \frac { \partial { \bf z } _ { l + 1 } } { \partial { \bf a } _ { l } ^ { \top } } { \bf a } _ { l } } } \\ { \displaystyle { = \frac { \partial { \bf z } _ { l + 1 } } { \partial { \bf a } _ { l } ^ { \top } } \frac { \partial { \bf a } _ { l } } { \partial { \bf z } _ { l } ^ { \top } } { \bf z } _ { l } } } \\ { \displaystyle { = \frac { \partial { \bf z } _ { l + 1 } } { \partial { \bf z } _ { l } ^ { \top } } { \bf z } _ { l } } } \end{array}
+$$
+
+By induction, we conclude that $f _ { \pmb { \theta } } ( \mathbf { x } ) = \nabla _ { \mathbf { x } } f _ { \pmb { \theta } } ( \mathbf { x } ) ^ { \top } \mathbf { x } = \mathbf { J } _ { \mathbf { x } } \mathbf { x } .$ .
+
+On the other side, we have
+
+$$
+\mathbf { z } _ { l + 1 } = \mathbf { W } _ { l } \mathbf { a } _ { l } = \sum _ { i , j } \frac { \partial \mathbf { z } _ { l + 1 } } { \partial \mathbf { W } _ { l } ^ { i , j } } \mathbf { W } _ { l } ^ { i , j }
+$$
+
+According to equation B, $\begin{array} { r } { { \bf z } _ { L + 1 } = \frac { \partial { \bf z } _ { L + 1 } } { \partial { \bf z } _ { l + 1 } } { \bf z } _ { l + 1 } ^ { \top } } \end{array}$ ∂zL+1∂zl+1 z>l+1, therefore we get
+
+$$
+\mathbf { z } _ { L + 1 } = \sum _ { i , j } \frac { \partial \mathbf { z } _ { L + 1 } } { \partial \mathbf { W } _ { l } ^ { i , j } } \mathbf { W } _ { l } ^ { i , j }
+$$
+
+Summing over all the layers, we conclude the following equation eventually:
+
+$$
+( L + 1 ) f _ { \pmb \theta } ( \mathbf x ) = \sum _ { l } \sum _ { i , j } \frac { \partial \mathbf z _ { L + 1 } } { \partial \mathbf W _ { l } ^ { i , j } } \mathbf W _ { l } ^ { i , j } = \nabla _ { \pmb \theta } f _ { \pmb \theta } ( \mathbf x ) ^ { \top } \pmb \theta = \mathbf J _ { \pmb \theta } \pmb \theta
+$$
+
+# C PROOF OF LEMMA 2
+
+Claim. For a feed-forward neural network of depth $L$ with ReLU activation function and no biases, we observe:
+
+$$
+\| \pmb \theta \| _ { \mathbf { G } } ^ { 2 } = ( L + 1 ) ^ { 2 } \mathbb { E } \left[ \| f _ { \pmb \theta } ( \mathbf { x } ) \| ^ { 2 } \right]
+$$
+
+Furthermore, if we restrict the network to be linear with only fully-connected layers, we have K-FAC Gauss-Newton norm as follows
+
+$$
+\lVert \pmb { \theta } \rVert _ { \mathbf { G } _ { \mathrm { K - F A C } } } ^ { 2 } = ( L + 1 ) \mathbb { E } \left[ \lVert f _ { \pmb { \theta } } ( \mathbf { x } ) \rVert ^ { 2 } \right]
+$$
+
+Proof. We first prove the equaility $\| \pmb \theta \| _ { \mathbf { G } } ^ { 2 } = ( L + 1 ) ^ { 2 } \mathbb { E } \left[ \| f _ { \pmb \theta } ( \mathbf { x } ) \| ^ { 2 } \right]$ . Using the definition of the Gauss-Newton norm in equation 3, we have
+
+$$
+\begin{array} { r } { \| \pmb { \theta } \| _ { \mathbf { G } } ^ { 2 } = \mathbb { E } \left[ \pmb { \theta } ^ { \top } \mathbf { J } _ { \pmb { \theta } } ^ { \top } \mathbf { J } _ { \pmb { \theta } } \pmb { \theta } \right] = \mathbb { E } \left[ \| \mathbf { J } _ { \pmb { \theta } } \pmb { \theta } \| ^ { 2 } \right] } \end{array}
+$$
+
+From Lemma 1, we have
+
+$$
+\mathbf { J } _ { \theta } \theta = ( L + 1 ) f _ { \theta } ( \mathbf { x } ) = ( L + 1 ) \mathbf { J } _ { \mathbf { x } } \mathbf { x }
+$$
+
+Combining above equalities, we arrive at the conclusion.
+
+For second part $\lVert \pmb { \theta } \rVert _ { \mathbf { G } _ { \mathrm { K - F A C } } } ^ { 2 } = ( L + 1 ) \mathbb { E } \left[ \lVert f _ { \pmb { \theta } } ( \mathbf { x } ) \rVert ^ { 2 } \right]$ , we note that kronecker-product is exact under the condition that the network is linear (Bernacchia et al., 2018), which means $\mathbf { G } _ { \mathrm { K - F A C } }$ is the diagonal block version of Gauss-Newton matrix $\mathbf { G }$ . Therefore, we have
+
+$$
+\lVert \pmb { \theta } \rVert _ { \mathbf { G } _ { \mathrm { K - F A C } } } ^ { 2 } = \sum _ { l } \mathbb { E } \left[ \pmb { \theta } _ { l } ^ { \top } \mathbf { J } _ { \pmb { \theta } _ { l } } ^ { \top } \mathbf { J } _ { \pmb { \theta } _ { l } } \pmb { \theta } _ { l } \right]
+$$
+
+According to Lemma 1, we have $\mathbb { E } \left[ \pmb { \theta } _ { l } ^ { \top } \mathbf { J } _ { \pmb { \theta } _ { l } } ^ { \top } \mathbf { J } _ { \pmb { \theta } _ { l } } \pmb { \theta } _ { l } \right] = \mathbb { E } \left[ \| \mathbf { \it f } _ { \pmb { \theta } } ( \mathbf { x } ) \| ^ { 2 } \right]$ , therefore we conclude that
+
+$$
+\lVert \pmb { \theta } \rVert _ { \mathbf { G } _ { \mathrm { K - F A C } } } ^ { 2 } = ( L + 1 ) \mathbb { E } \left[ \lVert f _ { \pmb { \theta } } ( \mathbf { x } ) \rVert ^ { 2 } \right]
+$$
+
+# D DERIVATION OF EQUATION 16
+
+Claim. During training, the weight direction $\hat { \pmb { \theta } } _ { l } ^ { t } = { \pmb { \theta } } _ { l } ^ { t } / \| { \pmb { \theta } } _ { l } ^ { t } \| _ { 2 }$ is updated according to
+
+$$
+\begin{array} { r } { \hat { \theta } _ { t + 1 } \gets \hat { \theta } _ { t } - \eta ( \mathbf { I } - \hat { \theta } _ { t } \hat { \theta } _ { t } ^ { \top } ) ( \mathbf { C } ( \hat { \theta } _ { t } ) + \| \pmb { \theta } _ { t } \| _ { 2 } ^ { 2 } \lambda \mathbf { I } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { \theta } _ { t } ) + O ( \eta ^ { 2 } ) } \end{array}
+$$
+
+Proof. Natural gradient update is given by
+
+$$
+\pmb { \theta } _ { t + 1 }  \pmb { \theta } _ { t } - \eta ( \mathbf { C } ( \pmb { \theta } _ { t } ) + \lambda \mathbf { I } ) ^ { - 1 } \nabla \mathcal { L } ( \pmb { \theta } _ { t } )
+$$
+
+Denote $\rho _ { t } = \| \pmb { \theta } _ { t } \| _ { 2 }$ . Then we have
+
+$$
+\begin{array} { r } { \rho _ { t + 1 } ^ { 2 } = \rho _ { t } ^ { 2 } - 2 \eta \rho _ { t } ^ { 2 } \hat { { \theta } } _ { t } ^ { \top } ( \mathbf { C } ( \hat { { \theta } } _ { t } ) + \lambda \rho _ { t } ^ { 2 } \mathbf { I } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { { \theta } } _ { t } ) + \eta ^ { 2 } \rho _ { t } ^ { 2 } \| ( \mathbf { C } ( \hat { { \theta } } _ { t } ) + \lambda \rho _ { t } ^ { 2 } \mathbf { I } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { { \theta } } _ { t } ) \| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+and therefore
+
+$$
+\begin{array} { r l } & { \rho _ { t + 1 } = \rho _ { t } \sqrt { 1 - 2 \eta \hat { { \theta } } _ { t } ^ { \top } ( { \mathbf { C } } ( \hat { { \theta } } _ { t } ) + \lambda \rho _ { t } ^ { 2 } { \mathbf { I } } ) ^ { - 1 } { \nabla } \mathcal { L } ( \hat { { \theta } } _ { t } ) + \eta ^ { 2 } \| ( { \mathbf { C } } ( \hat { { \theta } } _ { t } ) + \lambda \rho _ { t } ^ { 2 } { \mathbf { I } } ) ^ { - 1 } { \nabla } \mathcal { L } ( \hat { { \theta } } _ { t } ) \| _ { 2 } ^ { 2 } } } \\ & { \qquad = \rho _ { t } \big ( 1 - \eta \hat { { \theta } } _ { t } ^ { \top } ( { \mathbf { C } } ( \hat { { \theta } } _ { t } ) + \lambda \rho _ { t } ^ { 2 } { \mathbf { I } } ) ^ { - 1 } { \nabla } \mathcal { L } ( \hat { { \theta } } _ { t } ) \big ) + O \big ( \eta ^ { 2 } \big ) } \end{array}
+$$
+
+Additionally, we can rewrite the natural gradient update as follows
+
+$$
+\rho _ { t + 1 } \hat { \pmb { \theta } } _ { t + 1 } = \rho _ { t } \hat { \pmb { \theta } } _ { t } - \eta \rho _ { t } ( \mathbf { C } ( \hat { \pmb { \theta } } _ { t } ) + \lambda \rho _ { t } ^ { 2 } \mathbf { I } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { \pmb { \theta } } _ { t } )
+$$
+
+And therefore,
+
+$$
+\begin{array} { r l } & { \hat { \theta } _ { t + 1 } = \frac { \rho _ { t } } { \rho _ { t + 1 } } \left( \hat { \theta } _ { t } - \eta ( { \mathbf { C } } ( \hat { \theta } _ { t } ) + \lambda \rho _ { t } ^ { 2 } { \mathbf { I } } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { \theta } _ { t } ) \right) } \\ & { \qquad = \left( 1 + \eta \hat { \theta } _ { t } ^ { \top } ( { \mathbf { C } } ( \hat { \theta } _ { t } ) + \lambda \rho _ { t } ^ { 2 } { \mathbf { I } } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { \theta } _ { t } ) \right) \left( \hat { \theta } _ { t } - \eta ( { \mathbf { C } } ( \hat { \theta } _ { t } ) + \lambda \rho _ { t } ^ { 2 } { \mathbf { I } } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { \theta } _ { t } ) \right) + O ( \eta ^ { 2 } ) } \\ & { \qquad = \hat { \theta } _ { t } - \eta ( { \mathbf { I } } - \hat { \theta } _ { t } \hat { \theta } _ { t } ^ { \top } ) ( { \mathbf { C } } ( \hat { \theta } _ { t } ) + \| \theta _ { t } \| _ { 2 } ^ { 2 } \lambda { \mathbf { I } } ) ^ { - 1 } \nabla \mathcal { L } ( \hat { \theta } _ { t } ) + O ( \eta ^ { 2 } ) } \end{array}
+$$
+
+# E THE GRADIENT OF GAUSS-NEWTON NORM
+
+For Gauss-Newton norm $\| \pmb \theta \| _ { \mathbf { G } } ^ { 2 } = ( L + 1 ) ^ { 2 } \mathbb { E } _ { \mathbf { x } } \left[ \langle f _ { \pmb \theta } ( \mathbf { x } ) , f _ { \pmb \theta } ( \mathbf { x } ) \rangle \right] ,$ , its gradient has the following form:
+
+$$
+\frac { \partial \| \pmb { \theta } \| _ { \mathbf { G } } ^ { 2 } } { \partial \pmb { \theta } } = 2 ( L + 1 ) ^ { 2 } \mathbb { E } \left[ \mathbf { J } _ { \pmb { \theta } } ^ { \top } f _ { \pmb { \theta } } ( \mathbf { x } ) \right]
+$$
+
+According to Lemma 1, we have $\begin{array} { r } { f _ { \pmb { \theta } } ( \mathbf { x } ) = \frac { 1 } { L + 1 } \mathbf { J } _ { \pmb { \theta } } \pmb { \theta } } \end{array}$ , therefore we can rewrite equation 17
+
+$$
+\begin{array} { r } { \frac { \partial \| \pmb { \theta } \| _ { \mathbf { G } } ^ { 2 } } { \partial \pmb { \theta } } = 2 ( L + 1 ) \mathbb { E } \left[ \mathbf { J } _ { \pmb { \theta } } ^ { \top } \mathbf { J } _ { \pmb { \theta } } \pmb { \theta } \right] } \\ { = 2 ( L + 1 ) \mathbf { G } \pmb { \theta } } \end{array}
+$$
+
+Surprisingly, the resulting gradient has the same form as the case where we take Gauss-Newton matrix as a constant of $\pmb \theta$ up to a constant $( L + 1 )$ .
+
+# F KRONECKER-FACTORED APPROXIMATE CURVATURE (K-FAC)
+
+Martens & Grosse (2015) proposed K-FAC for performing efficient natural gradient optimization in deep neural networks. Following on that work, K-FAC has been adopted in many tasks (Wu et al., 2017; Zhang et al., 2017) to gain optimization benefits, and was shown to be amendable to distributed computation (Ba et al., 2016).
+
+# F.1 BASIC IDEA OF K-FAC
+
+As shown by Luk & Grosse (2018), K-FAC can be applied to general pullback metric, including Fisher metric and the Gauss-Newton metric. For convenience, we introduce K-FAC here using the Fisher metric.
+
+Considering $l$ -th layer in the neural network whose input activations are $\mathbf { a } _ { l } \in \mathbb { R } ^ { n _ { 1 } }$ , weight ${ \bf W } _ { l } \in  { \bf \Psi }$ $\mathbb { R } ^ { n _ { 1 } \times n _ { 2 } }$ , and output $\mathbf { s } _ { l } \in \mathbb { R } ^ { n _ { 2 } }$ , we have $\mathbf { s } _ { l } = \mathbf { W } _ { l } ^ { \top } \mathbf { a } _ { l } ^ { \top }$ . Therefore, weight gradient is $\nabla _ { \mathbf { W } _ { l } } \mathcal { L } \mathbf { \Phi } =$ $\mathbf { a } _ { l } ( \nabla _ { \mathbf { s } _ { l } } \mathcal { L } ) ^ { \top }$ . With this gradient formula, K-FAC decouples this layer’s fisher matrix $\mathbf { { F } } _ { l }$ using mild approximations,
+
+$$
+\begin{array} { r l } & { { \bf { F } } _ { l } = \mathbb { E } \left[ { \mathrm { v e c } } \{ \nabla _ { { \bf { w } } _ { l } } \mathcal { L } \} { \mathrm { v e c } } \{ \nabla _ { { \bf { w } } _ { l } } \mathcal { L } \} ^ { \top } \right] = \mathbb { E } \left[ \{ \nabla _ { { \bf { s } } _ { l } } \mathcal { L } \} \{ \nabla _ { { \bf { s } } _ { l } } \mathcal { L } \} ^ { \top } \otimes { \bf { a } } _ { l } { \bf { a } } _ { l } ^ { \top } \right] } \\ & { \quad \approx \mathbb { E } \left[ \{ \nabla _ { { \bf { s } } _ { l } } \mathcal { L } \} \{ \nabla _ { { \bf { s } } _ { l } } \mathcal { L } \} ^ { \top } \right] \otimes \mathbb { E } \left[ { \bf { a } } _ { l } { \bf { a } } _ { l } ^ { \top } \right] = { \bf { S } } _ { l } \otimes { \bf { A } } _ { l } } \end{array}
+$$
+
+Where $\mathbf { A } _ { l } = \mathbb { E } \left[ \mathbf { a } \mathbf { a } ^ { \top } \right]$ and $\mathbf { S } _ { l } = \mathbb { E } \left[ \{ \nabla _ { \mathbf { s } } \mathcal { L } \} \{ \nabla _ { \mathbf { s } } \mathcal { L } \} ^ { \top } \right]$ . The approximation above assumes independence between $^ { a }$ and $\pmb { s }$ , which proves to be accurate in practice. Further, assuming between-layer independence, the whole fisher matrix $\mathbf { F }$ can be approximated as block diagonal consisting of layerwise fisher matrices $\mathbf { { F } } _ { l }$ . Decoupling $\mathbf { { F } } _ { l }$ into ${ \bf A } _ { l }$ and $\mathbf { S } _ { l }$ not only avoids the memory issue saving $\mathbf { { F } } _ { l }$ , but also provides efficient natural gradient computation.
+
+$$
+\mathbf { F } _ { l } ^ { - 1 } \mathrm { v e c } \{ \nabla _ { \mathbf { W } _ { l } } \mathcal { L } \} = \mathbf { S } _ { l } ^ { - 1 } \otimes \mathbf { A } _ { l } ^ { - 1 } \mathrm { v e c } \{ \nabla _ { \mathbf { W } _ { l } } \mathcal { L } \} = \mathrm { v e c } [ \mathbf { A } _ { l } ^ { - 1 } \nabla _ { \mathbf { W } _ { l } } \mathcal { L } \mathbf { S } _ { l } ^ { - 1 } ]
+$$
+
+As shown by equation 20, computing natural gradient using K-FAC only consists of matrix transformations comparable to size of $\mathbf { W } _ { l }$ , making it very efficient.
+
+# F.2 PSEUDO CODE OF K-FAC
+
+Algorithm 1 K-FAC with $L _ { 2 }$ regularization and K-FAC with weight decay. Subscript $l$ denotes layers, $\mathbf { w } _ { l } = \mathrm { v e c } ( \mathbf { W } _ { l } )$ . We assume zero momentum for simplicity.
+
+Require: $\eta$ : stepsize   
+Require: $\beta$ : weight decay   
+Require: stats and inverse update intervals $T _ { \mathrm { s t a t s } }$ and $T _ { \mathrm { i n v } }$ $k \gets 0$ and initialize $\{ \mathbf { W } _ { l } \} _ { l = 1 } ^ { L } , \{ \mathbf { S } _ { l } \} _ { l = 1 } ^ { L } , \{ \mathbf { A } _ { l } \} _ { l = 1 } ^ { L }$ while stopping criterion not met do $k \gets k + 1$ if $k \equiv 0$ (mod $T _ { \mathrm { s t a t s , } }$ ) then Update the factors $\{ \mathbf { S } _ { l } \} _ { l = 1 } ^ { L } , \{ \mathbf { A } _ { l } \} _ { l = 0 } ^ { L - 1 }$ with moving average end if if $k \equiv 0$ (mod late th $T _ { \mathrm { i n v . } }$ ) thenerses $\{ [ \mathbf { S } _ { l } ] ^ { - 1 } \} _ { l = 1 } ^ { L } , \{ [ \mathbf { A } _ { l } ] ^ { - 1 } \} _ { l = 0 } ^ { L - 1 }$ $\mathbf { V } _ { l } = \nabla _ { \mathbf { W } _ { l } } \log p ( y | \mathbf { x } , \mathbf { w } ) + \beta \cdot \mathbf { W } _ { l }$ $\mathbf { W } _ { l } \gets \mathbf { W } _ { l } - \left( \eta [ \mathbf { A } _ { l } ] ^ { - 1 } \mathbf { V } _ { l } [ \mathbf { S } _ { l } ] ^ { - 1 } \right. \qquad \mathrm { ~ ) ~ }$ end while
+
+# G ADDITIONAL RESULTS
+
+# G.1 LARGE-BATCH TRAINING
+
+It has been shown that K-FAC scales very favorably to larger mini-batches compared to SGD, enjoying a nearly linear relationship between mini-batch size and per-iteration progress for medium-to-large sized mini-batches (Martens & Grosse, 2015; Ba et al., 2016). However, Keskar et al. (2016) showed that large-batch methods converge to sharp minima and generalize worse. In this subsection, we measure the generalization performance of K-FAC with large batch training and analyze the effect of weight decay.
+
+In Table 3, we compare K-FAC with SGD using different batch sizes. In particular, we interpolate between small-batch (BS128) and large-batch (BS2000). We can see that in accordance with previous works (Keskar et al., 2016; Hoffer et al., 2017) the move from a small-batch to a large-batch indeed incurs a substantial generalization gap. However, adding weight decay regularization to K-FAC almost close the gap on CIFAR-10 and cause much of the gap diminish on CIFAR-100. Surprisingly, the generalization gap of SGD also disappears with well-tuned weight decay regularization. Moreover, we observe that the training loss cannot decrease to zero if weight decay is not used, indicating weight decay may also speed up the training.
+
+Table 3: Classification results with different batch sizes. WD denotes weight decay regularization. We tune weight decay factor and learning rate using held-out validation set.   
+
+<table><tr><td rowspan=2 colspan=1>Dataset</td><td rowspan=2 colspan=1>Network</td><td rowspan=2 colspan=1>Method</td><td rowspan=1 colspan=6>BS128         BS640         BS2000</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>WD</td></tr><tr><td rowspan=1 colspan=1>CIFAR10</td><td rowspan=1 colspan=1>VGG16</td><td rowspan=1 colspan=1>SGDK-FAC-FK-FAC-G</td><td rowspan=1 colspan=1>91.7193.1293.19</td><td rowspan=1 colspan=1>93.3993.9093.80</td><td rowspan=1 colspan=1>90.4692.9392.98</td><td rowspan=1 colspan=1>93.0993.5593.74</td><td rowspan=1 colspan=1>88.5092.1790.78</td><td rowspan=1 colspan=1>92.2493.3193.46</td></tr><tr><td rowspan=1 colspan=1>CIFAR10</td><td rowspan=1 colspan=1>ResNet32</td><td rowspan=1 colspan=1>SGDK-FAC-FK-FAC-G</td><td rowspan=1 colspan=1>92.9593.8093.44</td><td rowspan=1 colspan=1>95.1495.3595.04</td><td rowspan=1 colspan=1>91.6892.3091.80</td><td rowspan=1 colspan=1>94.4594.7994.73</td><td rowspan=1 colspan=1>89.7091.1590.02</td><td rowspan=1 colspan=1>94.6894.4394.85</td></tr><tr><td rowspan=1 colspan=1>CIFAR100</td><td rowspan=1 colspan=1>ResNet32</td><td rowspan=1 colspan=1>SGDK-FAC-FK-FAC-G</td><td rowspan=1 colspan=1>73.6174.4973.70</td><td rowspan=1 colspan=1>77.7378.0178.02</td><td rowspan=1 colspan=1>71.7473.5471.13</td><td rowspan=1 colspan=1>76.6777.3477.40</td><td rowspan=1 colspan=1>65.3871.6465.41</td><td rowspan=1 colspan=1>76.8777.1376.93</td></tr></table>
+
+# G.2 THE CURVES OF TEST ACCURACIES
+
+![](images/1bb7996a2e3714960fc8b6ffa36d9038e2ee71c92c99260a1cde829664e00c50.jpg)  
+Figure 8: Test accuracy as a function of training epoch. We plot baseline vs $L _ { 2 }$ regularization vs weight decay regularization on CIFAR-10 and CIFAR-100 datasets. The $\ ' + \ '$ denotes with BN and data augmentation. Note that training accuracies of all the models are $1 0 0 \%$ in the end of the training. We smooth all the curves for visual clarity.
+
+# G.3 OPTIMIZATION PERFORMANCE OF DIFFERENT OPTIMIZERS
+
+While this paper mostly focus on generalization, we also report the convergence speed of different optimizers in deep neural networks; we report both per-epoch performance and wall-clock time performance.
+
+We consider the task of image classification on CIFAR-10 (Krizhevsky & Hinton, 2009) dataset. The models we use consist of VGG16 (Simonyan & Zisserman, 2014) and ResNet32 (He et al., 2016). We compare our K-FAC-G, K-FAC-F with SGD, Adam (Kingma & Ba, 2014). We experiment with constant learning for K-FAC-G and K-FAC-F. For SGD and Adam, we set batch size as 128. For K-FAC, we use batch size of 640, as suggested by Martens & Grosse (2015).
+
+In Figure 9, we report the training curves of different algorithms. Figure 9a show that K-FAC-G yields better optimization than other baselines in training loss per epoch. We highlight that the training loss decreases to 1e-4 within 10 epochs with K-FAC-G. Although K-FAC based algorithms take more time for each epoch, Figure 9b still shows wall-clock time improvements over the baselines.
+
+In Figure ${ 9 \mathrm { c } }$ and 9d, we report similar results on the ResNet32. Note that we make the network wider with a widening factor of 4 according to Zagoruyko & Komodakis (2016). K-FAC-G outperforms both K-FAC-F and other baselines in term of optimization per epoch, and compute time.
+
+![](images/b22713a7fab4b99dc27abbd48efca7deb94fdf04b23d81d8e462043e9ba2d816.jpg)  
+Figure 9: CIFAR-10 image classification task.

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+    {
+        "type": "text",
+        "text": "A SIGNAL PROPAGATION PERSPECTIVE FOR PRUNING NEURAL NETWORKS AT INITIALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
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+    {
+        "type": "text",
+        "text": "Namhoon Lee1, Thalaiyasingam Ajanthan2, Stephen Gould2, Philip H. S. Torr1 ",
+        "bbox": [
+            222,
+            174,
+            771,
+            191
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1University of Oxford 2Australian National University 1{namhoon,phst}@robots.ox.ac.uk 2{thalaiyasingam.ajanthan, stephen.gould}@anu.edu.au ",
+        "bbox": [
+            245,
+            202,
+            751,
+            258
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            299,
+            544,
+            314
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Network pruning is a promising avenue for compressing deep neural networks. A typical approach to pruning starts by training a model and then removing redundant parameters while minimizing the impact on what is learned. Alternatively, a recent approach shows that pruning can be done at initialization prior to training, based on a saliency criterion called connection sensitivity. However, it remains unclear exactly why pruning an untrained, randomly initialized neural network is effective. In this work, by noting connection sensitivity as a form of gradient, we formally characterize initialization conditions to ensure reliable connection sensitivity measurements, which in turn yields effective pruning results. Moreover, we analyze the signal propagation properties of the resulting pruned networks and introduce a simple, data-free method to improve their trainability. Our modifications to the existing pruning at initialization method lead to improved results on all tested network models for image classification tasks. Furthermore, we empirically study the effect of supervision for pruning and demonstrate that our signal propagation perspective, combined with unsupervised pruning, can be useful in various scenarios where pruning is applied to non-standard arbitrarily-designed architectures. ",
+        "bbox": [
+            233,
+            333,
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+            568
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            598,
+            336,
+            614
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning has made great strides in machine learning and been applied to various fields from computer vision and natural language processing, to health care and playing games (LeCun et al., 2015). Despite the immense success, however, it remains challenging to deal with the excessive computational and memory requirements of large neural network models. To this end, lightweight models are often preferred, and network pruning, a technique to reduce parameters in a network, has been widely employed to compress deep neural networks (Han et al., 2016). Nonetheless, designing pruning algorithms has been often purely based on ad-hoc intuition lacking rigorous underpinning, partly because pruning was typically carried out after training the model as a post-processing step or interwoven with the training procedure, without adequate tools to analyze. ",
+        "bbox": [
+            174,
+            631,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, Lee et al. (2019) have shown that pruning can be done on randomly initialized neural networks in a single-shot prior to training (i.e., pruning at initialization). They empirically showed that as long as the initial random weights are drawn from appropriately scaled Gaussians (e.g., Glorot & Bengio (2010)), their pruning criterion called connection sensitivity can be used to prune deep neural networks, often to an extreme level of sparsity while maintaining good accuracy once trained. However, it remains unclear as to why pruning at initialization is effective, how it should be understood theoretically and whether it can be extended further. ",
+        "bbox": [
+            174,
+            763,
+            823,
+            861
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we first look into the effect of initialization on pruning, and find that initial weights have critical impact on connection sensitivity, and therefore, pruning results. Deeper investigation shows that connection sensitivity is determined by an interplay between gradients and weights. Therefore when the initial weights are not chosen appropriately, the propagation of input signals into layers of these random weights can result in saturating error signals (i.e., gradients) under backpropagation, and hence unreliable connection sensitivity, potentially leading to a catastrophic pruning failure. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This result leads us to develop a signal propagation perspective for pruning at initialization, and to provide a formal characterization of how a network needs to be initialized for reliable connection sensitivity measurements and in turn effective pruning. Precisely, we show that a sufficient condition to ensure faithful1 connection sensitivity is layerwise dynamical isometry, which is defined as all singular values of the layerwise Jacobians being concentrated around 1. Our signal propagation perspective is inspired by the recent literature on dynamical isometry and mean field theory (Saxe et al., 2014; Poole et al., 2016; Schoenholz et al., 2017; Pennington et al., 2017), in which the general signal propagation in neural networks is studied. We extend this result to understanding and improving pruning at initialization. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Moreover, we study signal propagation in the pruned sparse networks and its effect on trainability. We find that pruning neural networks can indeed break dynamical isometry, and hence, hinders signal propagation and degrades the training performance of the resulting sparse network. In order to address this issue, we propose a simple, yet effective data-free method to recover the layerwise orthogonality given the sparse topology, which in turn improves the training performance of the compressed network significantly. Our analysis further reveals that in addition to signal propagation, the choice of pruning method and sparsity level can influence trainability in sparse neural networks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Perfect layerwise dynamical isometry cannot always be ensured in the modern networks that have components such as ReLU nonlinearities (Pennington et al., 2017) and/or batch normalization (Yang et al., 2019). Even in such cases, however, our experiments on various modern architectures (including convolutional and residual neural networks) indicate that connection sensitivity computed based on layerwise dynamical isometry is robust and consistently outperforms pruning based on other initialization schemes. This indicates that the signal propagation perspective is not only important to theoretically understand pruning at initialization, but also it improves the results of pruning for a range of networks of practical interest. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, this signal propagation perspective for pruning poses another important question: how informative is the error signal computed on randomly initialized networks, or can we prune neural networks even without supervision? To understand this, we compute connection sensitivity scores with different unsupervised surrogate losses and evaluate the pruning results. Interestingly, our results indicate that we can in fact prune networks in an unsupervised manner to extreme sparsity levels without compromising accuracy, and it often compares competitively to pruning with supervision. Moreover, we test if pruning at initialization can be extended to obtain architectures that yield better performance than standard pre-designed architectures with the same number of parameters. In fact, this process, which we call neural architecture sculpting, compares favorably against hand-designed architectures, taking network pruning one step further towards neural architecture search. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Pruning at initialization. The principle behind conventional approaches for network pruning is to find unnecessary parameters, such that by eliminating them the complexity of the model is reduced while minimizing the impact on what is learned (Reed, 1993). Naturally, a typical pruning algorithm starts after convergence to a minimum or training to some degree. This pretraining requirement has been left unattended until Lee et al. (2019) recently showed that pruning can be performed on untrained networks at initiailzation prior to training. They proposed a method called SNIP which relies on a new saliency criterion, namely connection sensitivity, defined as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0c6f0fa168f571d6476be82e00d51ea35babacc5c326440e4e2299776040be17.jpg",
+        "text": "$$\ns _ { j } ( \\mathbf { w } ; \\mathcal { D } ) = \\frac { | g _ { j } ( \\mathbf { w } ; \\mathcal { D } ) | } { \\sum _ { k = 1 } ^ { m } | g _ { k } ( \\mathbf { w } ; \\mathcal { D } ) | } , \\qquad \\mathrm { w h e r e } \\quad g _ { j } ( \\mathbf { w } ; \\mathcal { D } ) = \\left. \\frac { \\partial L ( \\mathbf { c } \\odot \\mathbf { w } ; \\mathcal { D } ) } { \\partial c _ { j } } \\right| _ { \\mathbf { c } = 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            787,
+            764,
+            824
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Here, $s _ { j }$ is the saliency of the parameter $j$ , $\\mathbf { w _ { \\lambda } } \\in \\mathbb { R } ^ { m }$ is the network parameters, $\\mathbf { c } \\in \\{ 0 , 1 \\} ^ { m }$ is the auxiliary indicator variables representing the connectivity of network parameters, $m$ is the total number of parameters in the network, and $\\mathcal { D }$ is a given dataset. Also, $g _ { j }$ is the derivative of the loss $L$ with respect to $c _ { j }$ , which turns out to be an infinitesimal approximation of the change in the loss with respect to removing the parameter $j$ . Designed to be computed at initialization, pruning is performed by keeping top- $\\kappa$ (where $\\kappa$ denotes a desired sparsity level) salient parameters based on the above sensitivity scores. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Dynamical isometry and mean field theory. The success of training deep neural networks is due in large part to the initial weights (Hinton & Salakhutdinov, 2006; Glorot & Bengio, 2010; Pascanu et al., 2013). In essence, the principle behind these random weight initializations is to have the mean squared singular value of a network’s input-output Jacobian close to 1, so that on average, an error vector will preserve its norm under backpropagation; however, this is not sufficient to prevent amplification or attenuation of an error vector on worst case. A stronger condition that having as many singular values as possible near 1 is called dynamical isometry (Saxe et al., 2014). Under this condition, error signals backpropagate isometrically through the network, approximately preserving its norm and all angles between error vectors. Alongside dynamical isometry, mean field theory is used to develop a theoretical understanding of signal propagation in neural networks with random parameters (Poole et al., 2016). Precisely, the mean field approximation states that preactivations of wide, untrained neural networks can be captured as a Gaussian distribution. Recent works revealed a maximum depth through which signals can propagate at initialization, and verified that networks are trainable when signals can travel all the way through them (Schoenholz et al., 2017; Yang & Schoenholz, 2017; Xiao et al., 2018). ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 SIGNAL PROPAGATION PERSPECTIVE TO PRUNING RANDOM NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Problem setup. Consider a fully-connected, feed-forward neural network with weight matrices $\\mathbf { W } ^ { l } \\in \\mathbb { R } ^ { N \\times N }$ , biases $\\mathbf { b } ^ { l } \\in \\mathbb { R } ^ { N }$ , pre-activations $\\mathbf { h } ^ { l } \\in \\mathbb { R } ^ { N }$ , and post-activations $\\mathbf { x } ^ { l } \\in \\mathbb { R } ^ { N }$ , for $l \\in$ $\\{ 1 \\ldots K \\}$ up to $K$ layers. Now, the feed-forward dynamics of a network can be written as, ",
+        "bbox": [
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+            416,
+            823,
+            459
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/18ed4dd510ee0a66c698fc2bb9781813bf1446302189d0c307218d5fd0db788f.jpg",
+        "text": "$$\n\\mathbf { x } ^ { l } = \\phi ( \\mathbf { h } ^ { l } ) , \\qquad \\mathbf { h } ^ { l } = \\mathbf { W } ^ { l } \\mathbf { x } ^ { l - 1 } + \\mathbf { b } ^ { l } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            468,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\phi : \\mathbb { R }  \\mathbb { R }$ is an elementwise nonlinearity, and the input is denoted by $\\mathbf { x } ^ { 0 }$ . Given the network configuration, the parameters are initialized by sampling from a probability distribution, typically a zero mean Gaussian with scaled variance (LeCun et al., 1998; Glorot & Bengio, 2010). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 EFFECT OF INITIALIZATION ON PRUNING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It is observed in Lee et al. (2019) that pruning results tend to improve when initial weights are drawn from a scaled Gaussian, or so-called variance scaling initialization (LeCun et al., 1998; Glorot & Bengio, 2010; He et al., 2015). As we wish to better understand the role of these random initial weights in pruning, we will examine the effect of varying initialization on the pruning results. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In essence, variance scaling schemes introduce normalization factors to adjust the variance $\\sigma$ of the weight sampling distribution, which can be summarized as $\\begin{array} { r } { \\sigma  \\frac { \\alpha } { \\psi _ { l } } \\sigma } \\end{array}$ , where $\\psi _ { l }$ is a layerwise scalar that depends on an architecture specification such as the number of output neurons in the previous layer (e.g., fan-in), and $\\alpha$ is a global scalar throughout the network. Notice in case of a network with layers of the same width, the variance can be controlled by a single scalar $\\textstyle \\gamma = { \\frac { \\alpha } { \\psi } }$ as $\\psi _ { l } = \\psi$ for all layers $l$ . In particular, we take both linear and tanh multilayer perceptron networks (MLP) of layers $K = 7$ and width $N = 1 0 0$ on MNIST with $\\sigma = 1$ as the default, similar to Saxe et al. (2014). We initialize these networks with different $\\gamma$ , compute the connection sensitivity, prune it, and then visualize layerwise the resulting sparsity patterns c as well as the corresponding connection sensitivity used for pruning in Figure 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It is seen in the sparsity patterns that for the tanh network, unlike the linear case, more parameters tend to be pruned in the later layers than the earlier layers. As a result, this limits the learning capability of the subnetwork critically when a high sparsity level is requested; e.g., for $\\bar { \\kappa } = 9 0 \\%$ , only a few parameters in later layers are retained after pruning. This is explained by the connection sensitivity plot. The sensitivity of parameters in the nonlinear network tends to decrease towards the later layers, and therefore, choosing the top- $\\kappa$ parameters globally based on the sensitivity scores results in a subnetwork in which retained parameters are distributed highly non-uniformly and sparsely towards the end of the network. This result implies that the initial weights have a crucial effect on the connection sensitivity, and from there, the pruning results. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a88651e8398c5c0bf25ce6ab4afa3502637fa3bc5ab864585ebaeabd4504dc3e.jpg",
+        "image_caption": [
+            "Figure 1: (left) layerwise sparsity patterns $c \\in \\{ 0 , 1 \\} ^ { 1 0 0 \\times 1 0 0 }$ obtained as a result of pruning for the sparsity level $\\bar { \\kappa } = \\{ 1 0 , . . , \\bar { 9 0 } \\} \\%$ . Here, black(0)/white(1) pixels refer to pruned/retained parameters; (right) connection sensitivities (CS) measured for the parameters in each layer. All networks are initialized with $\\gamma = 1 . 0$ . Unlike the linear case, the sparsity pattern for the tanh network is nonuniform over different layers. When pruning for a high sparsity level (e.g., $\\bar { \\kappa } = 9 0 \\%$ ), this becomes critical and leads to poor learning capability as there are only a few parameters left in later layers. This is explained by the connection sensitivity plot which shows that for the nonlinear network parameters in later layers have saturating, lower connection sensitivities than those in earlier layers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 GRADIENT SIGNAL IN CONNECTION SENSITIVITY ",
+        "text_level": 1,
+        "bbox": [
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+            387,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We posit that the unreliability of connection sensitivity observed in Figure 1 is due to poor signal propagation: an initialization that projects the input signal to be strongly amplified or attenuated in the forward pass will saturate the error signal under backpropagation (i.e., gradients), and hence will result in poorly calibrated connection sensitivity scores across layers, which will eventually lead to poor pruning results, potentially with complete disconnection of signal paths (e.g., entire layer). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Precisely, we give the relationship between the connection sensitivity and the gradients as follows. From Equation 1, connection sensitivity is a normalized magnitude of gradients with respect to the connectivity parameters c. Here, we use the vectorized notation where w denotes all learnable parameters and c denotes the corresponding connectivity parameters. From chain rule, we can write: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba561e13454ec224c37bc7ac463392b541529ba6305d56b8e0958bdbcd1d8810.jpg",
+        "text": "$$\n\\left. \\frac { \\partial L ( \\mathbf { c } \\odot \\mathbf { w } ; \\mathcal { D } ) } { \\partial \\mathbf { c } } \\right| _ { \\mathbf { c } = \\mathbf { 1 } } = \\left. \\frac { \\partial L ( \\mathbf { c } \\odot \\mathbf { w } ; \\mathcal { D } ) } { \\partial ( \\mathbf { c } \\odot \\mathbf { w } ) } \\right| _ { \\mathbf { c } = \\mathbf { 1 } } \\odot \\mathbf { w } = \\frac { \\partial L ( \\mathbf { w } ; \\mathcal { D } ) } { \\partial \\mathbf { w } } \\odot \\mathbf { w } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            555,
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+            592
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore, $\\partial L / \\partial \\mathbf { c }$ is the gradients ${ \\partial L } / { \\partial \\mathbf { w } }$ amplified (or attenuated) by the corresponding weights w, i.e., ${ \\partial L } / { \\partial { \\dot { c } } _ { j } } = { \\partial L } / { \\partial w _ { j } w _ { j } }$ for all $j \\in \\{ \\bar { 1 } \\ldots m \\}$ . Considering $\\bar { \\partial } L / \\partial c _ { j }$ for a given $j$ , since $w _ { j }$ does not depend on any other layers or signal propagation, the only term that depends on signal propagation in the network is the gradient term $\\partial L / \\partial w _ { j }$ . Hence, a necessary condition to ensure faithful $\\partial L / \\partial \\mathbf { c }$ (and connection sensitivity) is that the gradients ${ \\partial L } / { \\partial \\mathbf { w } }$ need to be faithful. In the following section, we formalize this from a signal propagation perspective, and characterize an initial condition that ensures reliable connection sensitivity measurement. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 LAYERWISE DYNAMICAL ISOMETRY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 GRADIENTS IN TERMS OF JACOBIANS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From the feed-forward dynamics of a network in Equation 2, the network’s input-output Jacobian corresponding to a given input $\\mathbf { x } ^ { 0 }$ can be written, by the chain rule of differentiation, as: ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6249388771ece3cad6d4df4677a82ad860ea31387354f048145cb55f2edcca7b.jpg",
+        "text": "$$\n{ \\bf J } ^ { 0 , K } = \\frac { \\partial { \\bf x } ^ { K } } { \\partial { \\bf x } ^ { 0 } } = \\prod _ { l = 1 } ^ { K } { \\bf D } ^ { l } { \\bf W } ^ { l } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            804,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { D } ^ { l } \\in \\mathbb { R } ^ { N \\times N }$ is a diagonal matrix with entries $\\mathbf { D } _ { i j } ^ { l } = \\phi ^ { \\prime } ( h _ { i } ^ { l } ) \\delta _ { i j }$ , with $\\phi ^ { \\prime }$ denoting the derivative of nonlinearity $\\phi$ , and $\\delta _ { i j } = \\mathbb { I } [ i = j ]$ is the Kronecker delta. Here, we will use $\\mathbf { J } ^ { k , l }$ to denote the Jacobian from layer $k$ to layer $l$ . Now, we give the relationship between gradients and Jacobians: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Let $\\epsilon = \\partial L / \\partial \\mathbf { x } ^ { K }$ denote the error signal and $\\mathbf { x } ^ { 0 }$ denote the input signal. Then, ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. the gradients satisfy: ",
+        "bbox": [
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+            103,
+            326,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d892875f640feb1be9a25e7f79d7520c4718ace03cc8ce58896949ec8bcb1f15.jpg",
+        "text": "$$\n{ \\bf g } _ { { \\bf w } ^ { l } } ^ { T } = \\epsilon { \\bf J } ^ { l , K } { \\bf D } ^ { l } \\otimes { \\bf x } ^ { l - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            116,
+            591,
+            136
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { J } ^ { l , K } = \\partial \\mathbf { x } ^ { K } / \\partial \\mathbf { x } ^ { l }$ is the Jacobian from layer $l$ to the output and $\\otimes$ is the Kronecker product. 2. additionally, for linear networks, i.e., when $\\phi$ is the identity: ",
+        "bbox": [
+            174,
+            138,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0635e27d2c55bc5a2d0e23bce2808c8d1691be7b39f0586966baa9cbb4ce347b.jpg",
+        "text": "$$\n{ \\bf g } _ { { \\bf w } ^ { l } } ^ { T } = \\epsilon { \\bf J } ^ { l , K } \\otimes \\left( { \\bf J } ^ { 0 , l - 1 } { \\bf x } ^ { 0 } + { \\bf a } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            171,
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+            193
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { J } ^ { 0 , l - 1 } = \\partial \\mathbf { x } ^ { l - 1 } / \\partial \\mathbf { x } ^ { 0 }$ is the Jacobian from the input to layer $l - 1$ and $\\mathbf { a } \\in \\mathbb { R } ^ { N }$ is a constant term that does not depend on $\\mathbf { x } ^ { 0 }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. This can be proved by an algebraic manipulation of the chain rule while using the feedforward dynamics in Equation 2. We provide the full derivation in Appendix A. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Notice that the gradient at layer $l$ constitutes both the backward propagation of the error signal $\\epsilon$ up to layer $l$ and the forward propagation of the input signal $\\mathbf { x } ^ { 0 }$ up to layer $l - 1$ . Moreover, especially in the linear case, the signal propagation in both directions is governed by the corresponding Jacobians. We believe that this interpretation of gradients is useful as it sheds light on how signal propagation affects the gradients. To this end, we next analyze the conditions on the Jacobians, which would guarantee faithful signal propagation in the network, and consequently, faithful gradients. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3.2 ENSURING FAITHFUL GRADIENTS ",
+        "text_level": 1,
+        "bbox": [
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+            386,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, we first consider the layerwise signal propagation which would be useful to derive properties on the initialization to ensure faithful gradients. To this end, let us consider the layerwise Jacobian: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3408fd450af43fb5c86c7effafe7eafeb567c8f491f59dfee7c55822b9ad570.jpg",
+        "text": "$$\n\\mathbf { J } ^ { l - 1 , l } = \\frac { \\partial \\mathbf { x } ^ { l } } { \\partial \\mathbf { x } ^ { l - 1 } } = \\mathbf { D } ^ { l } \\mathbf { W } ^ { l } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            443,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that it is sufficient to have layerwise dynamical isometry in order to ensure faithful signal propagation in the network. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. (Layerwise dynamical isometry) Let $\\begin{array} { r } { \\mathbf { J } _ { ~ . ~ . ~ . ~ } ^ { l - 1 , l } = ~ \\frac { \\partial \\mathbf { x } ^ { l } } { \\partial \\mathbf { x } ^ { l - 1 } } \\in ~ \\mathbb { R } ^ { N _ { l } \\times N _ { l - 1 } } } \\end{array}$ be the Jacobian matrix of layer $l$ . The network is said to satisfy layerwise dynamical isometry if the singular values of $\\mathbf { J } ^ { l - 1 , l }$ are concentrated near 1 for all layers, i.e., for a given $\\epsilon > 0$ , the singular value $\\sigma _ { j }$ satisfies $| 1 - \\sigma _ { j } | \\le \\epsilon$ for all $j$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This would guarantee that the signal from layer $l$ to $l - 1$ (or vice versa) is propagated without amplification or attenuation in any of its dimension. From Proposition 1 and Equation 7, by induction, it is easy to show that if the layerwise signal propagation is faithful, the error and input signals will faithfully propagate throughout the network, resulting in faithful gradients. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For linear networks, $\\mathbf { J } ^ { l - 1 , l } = W ^ { l }$ . Therefore, one can initialize the weight matrix to be orthogonal such that $( \\mathbf { W } ^ { l } ) ^ { T } \\mathbf { W } ^ { l } = \\mathbf { I }$ , where I is the identity matrix of dimension $N$ . In this case, all singular values of $\\dot { \\mathbf { W } } ^ { l }$ are exactly 1 (i.e., exact dynamical isometry), and such an initialization guarantees faithful gradients. While a linear network is of little practical use, we note that it helps to develop theoretical analysis and provides intuition as to why dynamical isometry is a useful measure. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For nonlinear networks, the diagonal matrix $\\mathbf { D } ^ { l }$ needs to be accounted for as it depends on the pre-activations $\\mathbf { h } ^ { l }$ at layer $l$ . In this case, it is important to have the pre-activations $\\mathbf { h } ^ { l }$ fall into the linear region of the nonlinear function $\\phi$ . Precisely, mean-field theory assumes that for large- $N$ limit, the empirical distribution of the pre-activations $\\bar { \\mathbf { h } } ^ { l }$ converges to a Gaussian with zero mean and variance $q ^ { l }$ , where the variance follows a recursion relation (Poole et al., 2016). Therefore, to achieve layerwise dynamical isometry, the idea becomes to find a fixed point $q ^ { * }$ such that $\\mathbf { h } ^ { l } \\sim \\mathcal { N } ( 0 , q ^ { * } )$ for all $l \\in \\{ 1 \\ldots K \\}$ . Such a fixed point makes $\\mathbf { D } ^ { l } = \\mathbf { D }$ for all layers, and therefore, the preactivations are placed in the linear region of the nonlinearity.2 Then, given the nonlinearity, one can find a rescaling such that $( \\mathbf { D } \\mathbf { W } ^ { l } ) ^ { T } ( \\mathbf { \\bar { D } } \\mathbf { W } ^ { l } ) = ( \\mathbf { W } ^ { l } ) ^ { T } \\mathbf { W } ^ { l } / \\sigma _ { w } ^ { 2 } = \\mathbf { I }$ . The procedure for finding the rescaling $\\sigma _ { w } ^ { 2 }$ for various nonlinearities are discussed in Pennington et al. (2017; 2018). Also, this easily extends to convolutional neural networks using the initialization method in Xiao et al. (2018). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/7052d21f8ce7463722f4a9392bace2c6ac1eb0a85ada7df8e3ff6c2fddb615e7.jpg",
+        "table_caption": [
+            "Table 1: Jacobian singular values and resulting sparse networks for the 7-layer tanh MLP network considered in section 3.1. SG, CN, and Sparsity refer to Scaled Gaussian, Condition Number (i.e., $s _ { \\operatorname* { m a x } } / s _ { \\operatorname* { m i n } }$ , where $s _ { \\mathrm { m a x } }$ and $s _ { \\mathrm { m i n } }$ are the maximum and minimum Jacobian singular values), and a ratio of pruned prameters to the total number of parameters, respectively. SG $( \\gamma { = } \\mathrm { \\breve { 1 0 } ^ { - 2 } } ,$ ) is equivalent to the variance scaling initialization as in LeCun et al. (1998); Glorot & Bengio (2010). The failure cases correspond to unreliable connection sensitivity resulted from poorly conditioned initial Jacobians. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Jacobian singular values</td><td colspan=\"9\">Sparsity in pruned network (across layers)</td></tr><tr><td>Initialization</td><td>Mean</td><td>Std</td><td>CN</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td></td><td>6</td><td>7</td><td>Error</td></tr><tr><td>SG(y=10-4)</td><td>2.46e-07</td><td>9.90e-08</td><td>4.66e+00</td><td>0.97</td><td>0.80</td><td>0.80</td><td>0.80</td><td>0.80</td><td></td><td>0.81</td><td>0.48</td><td>2.66</td></tr><tr><td>SG(y=10-3)</td><td>5.74e-04</td><td>2.45e-04</td><td>8.54e+00</td><td>0.97</td><td>0.80</td><td>0.80</td><td>0.80</td><td></td><td>0.80</td><td>0.81</td><td>0.48</td><td>2.67</td></tr><tr><td>SG(y=10-2)</td><td>4.49e-01</td><td>2.51e-01</td><td>5.14e+01</td><td>0.96</td><td>0.80</td><td>0.80</td><td>0.80</td><td></td><td>0.81</td><td>0.81</td><td>0.49</td><td>2.67</td></tr><tr><td>SG(γ=10-1)</td><td>2.30e+01</td><td>2.56e+01</td><td>2.92e+04</td><td>0.96</td><td>0.81</td><td>0.82</td><td>0.82</td><td></td><td>0.82</td><td>0.80</td><td>0.45</td><td>2.61</td></tr><tr><td>SG(γ=100)</td><td>1.03e+03</td><td>2.61e+03</td><td>3.34e+11</td><td>0.85</td><td>0.88</td><td>0.99</td><td>1.00</td><td></td><td>1.00</td><td>1.00</td><td>0.91</td><td>90.2</td></tr><tr><td>SG(γ=101)</td><td>3.67e+04</td><td>2.64e+05</td><td>inf</td><td>0.84</td><td>0.95</td><td>1.00</td><td>1.00</td><td></td><td>1.00</td><td>1.00</td><td>1.00</td><td>90.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We note that dynamical isometry is in fact a weaker condition than layerwise dynamical isometry. However, in practice, the initialization suggested in the existing works (Pennington et al., 2017; Xiao et al., 2018), i.e., orthogonal initialization for weight matrices in each layer with rescaling based on mean field theory, satisfy layerwise dynamical isometry, even though this term was not mentioned. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Now, recall from Section 3.1 that a network is pruned with a global threshold based on connection sensitivity, and from Section 3.2 that the connection sensitivity is the gradients scaled by the weights. This in turn implies that the connection sensitivity scores across layers are required to be of the same scale. To this end, we require the gradients to be faithful and the weights to be in the same scale for all the layers. Notice, this condition is trivially satisfied when the layerwise dynamical isometry is ensured, as each layer is initialized identically (i.e., orthogonal initialization) and the gradients are guaranteed to be faithful. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, we verify the failure of pruning cases presented in Section 3.1 based on the signal propagation perspective. Specifically, we measure the singular value distribution of the input-output Jacobian $( \\mathbf { J } ^ { 0 , K } )$ for the 7-layer tanh MLP network, and the results are reported in Table 1. Note that while connection sensitivity based pruning is robust to moderate changes in the Jacobian singular values, it failed catastrophically when the condition number of the Jacobian is very large $( > 1 \\mathrm { e } + 1 1 ) ,$ . In fact, these failure cases correspond to the completely disconnected networks, as a consequence of pruning with unreliable connection sensitivity resulted from poorly conditioned initial Jacobians. As we will show subsequently, these findings extend to modern architectures, and layerwise dynamical isometry yields well-conditioned Jacobians and in turn the best pruning results. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 SIGNAL PROPAGATION IN SPARSE NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "So far, we have shown empirically and theoretically that layerwise dynamical isometry can improve the process of pruning at initialization. One remaining question to address is the following: how well do signals propagate in the pruned sparse networks? In this section, we first examine the effect of sparsity on signal propagation after pruning. We find that indeed pruning can break dynamical isometry, degrading trainability of sparse networks. Then we follow up to present a simple, but effective data-free method to recover approximate dynamical isometry on sparse networks. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Setup. The overall process is summarized as follows: Step 1. Initialize a network with a variance scaling (VS) or layerwise dynamical isometry (LDI) satisfying orthogonal initialization. Step 2. Prune at initialization for a sparsity level $\\bar { \\kappa }$ based on connection sensitivity (CS); we also test random (Rand) and magnitude (Mag) based pruning for comparison. Step 3. (optional) Enforce approximate dynamical isometry, if specified. Step 4. Train the pruned sparse network using SGD. We measure signal propagation (e.g., Jacobian singular values) on the sparse network right before Step 4, and observe training behavior during Step 4. Different methods are named as {A}-{B}-{C}, where A, B, C stand for initialization scheme, pruning method, (optional) approximate isometry, respectively. We perform this on 7-layer linear and tanh MLP networks as before 3. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/0456d4c99d2986882e19f3354f9afa0e7a4bb0ac07556742ba64a152ca4e555a.jpg",
+        "image_caption": [
+            "Figure 2: (a) Signal propagation (mean Jacobian singular values) in sparse networks pruned for varying sparsity levels $\\bar { \\kappa }$ , and (b) training behavior of the sparse network at $\\bar { \\kappa } ~ = ~ 9 0 \\%$ . Signal propagation, pruning scheme, and overparameterization affect trainability of sparse neural networks. We train using SGD with the initial learning rate of 0.1 decayed by $1 / 1 0$ at every 20k iterations. All results are the average over 10 runs. We provide other singular value statistics (max, min, std), accuracy plot, and extended training results for random and magnitude pruning in Appendix C. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            99,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Effect of pruning on signal propagation and trainability. Let us first check signal propagation measurements in the pruned networks (see Figure 2a). In general, Jacobian singular values decrease continuously as the sparsity level increases (except for $\\{ \\cdot \\} - \\{ \\cdot \\} - \\mathrm { A I }$ which we will explain later), indicating that the more parameters are removed, the less faithful a network is likely to be with regard to propagating signals. Also, notice that the singular values drop more rapidly with random pruning compared to connection sensitivity based pruning methods (i.e., $\\{ \\cdot \\}$ -Rand vs. $\\{ \\cdot \\}$ -CS). This means that pruning using connection sensitivity is more robust to destruction of dynamical isometry and preserve better signal propagation in the sparse network than random pruning. We further note that, albeit marginal, layerwise dynamical isometry allows better signal propagation than variance scaling initialization, with relatively higher mean singular values and much lower standard deviations especially in the low sparsity regime (see Appendix C). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Now, we look into the relation between signal propagation and trainability of the sparse networks. Figure 2b shows training behavior of the pruned networks ( $\\bar { \\kappa } = 9 0 \\%$ ) obtained by different methods. We can see a clear correlation between signal propagation capability of a network and its training performance; i.e., the better a network propagates signals, the faster it converges during training. For instance, compare the trainability of a network before and after pruning. That is, compared to LDI-Dense $\\bar { \\kappa } = 0$ ), LDI-{CS, Mag, Rand} decrease the loss much slowly; random pruning starts to decrease the loss around 4k iteration, and finally reaches to close to zero loss around $1 0 \\mathrm { k }$ iterations (see Appendix C), which is more than an order of magnitude slower than a network pruned by connection sensitivity. Recall that the pruned networks have much smaller singular values. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Enforcing approximate dynamical isometry. The observation above indicates that the better signal propagation is ensured on sparse networks, the better their training performs. This motivates us to think of the following: what if we can repair the broken isometry, before we start training the pruned network, such that we can achieve trainability comparable to that of the dense network? Precisely, we consider the following: ",
+        "bbox": [
+            173,
+            659,
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+            729
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/40ab3c777a24ce9651032a7972df079ae75347ad9044c5bfcac81cf63aea4951.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { W } ^ { l } } \\| ( \\mathbf { C } ^ { l } \\odot \\mathbf { W } ^ { l } ) ^ { T } ( \\mathbf { C } ^ { l } \\odot \\mathbf { W } ^ { l } ) - \\mathbf { I } ^ { l } \\| _ { F } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            736,
+            630,
+            762
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { C } ^ { l } , \\mathbf { W } ^ { l } , \\mathbf { \\Phi } .$ ${ \\bf { I } } ^ { l }$ are the sparse mask obtained by pruning, the corresponding weights, the identity matrix at layer $l$ , respectively, and $\\| \\cdot \\| _ { F }$ is the Frobenius norm. We optimize this for all layers identically using gradient descent. Given the sparsity topology $\\mathbf { C } ^ { l }$ and initial weights $\\mathbf { W } ^ { l }$ , this datafree method attempts to find an optimal $\\mathbf { W } ^ { * }$ such that the combination of the sparse topology and the weights to be layerwise orthogonal, potentially to the full rank capacity. This simple method (i.e., $\\{ \\cdot \\} - \\{ \\cdot \\}$ -AI, where AI is named for Approximate Isometry) turns out to be highly effective. The results are provided in Figure 2, and we summarize our key findings below: ",
+        "bbox": [
+            173,
+            770,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Signal propagation (LDI-{CS, Rand $\\}$ vs. LDI-{CS, Rand}-AI). The decreased singular values (by pruning $\\bar { \\kappa } > 0$ ) bounce up dramatically and become close to the level before pruning. This means that orthogonality enforced by Equation 8 is achieved in the sparse topology of the pruned network (i.e., approximate dynamical isometry), and therefore, signal propagation on the sparse network is likely to behave similarly to the dense network. As expected, the training performance increased significantly (e.g., compare LDI-CS with LDI-CS-AI for trainability). This works more dramatically for random pruning; i.e., even for randomly pruned sparse networks, training speed increases significantly, implying the benefit of ensuring signal propagation. ",
+        "bbox": [
+            176,
+            881,
+            823,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/86e507b8d428050f3b6f99deb378abb57cf8e5f9cc8a4c815681496dc68e1a13.jpg",
+        "table_caption": [
+            "Table 2: Pruning results for various neural networks on different datasets. All networks are pruned at initialization for the sparsity $\\bar { \\kappa } = 9 0 \\%$ based on connection sensitivity scores as in Lee et al. (2019). We report orthogonality scores (OS) and generalization errors (Error) on CIFAR-10 (VGG16, ResNets) and Tiny-ImageNet (WRN16); all results are the average over 5 runs. The first and second best results are highlighted in each column of errors. The orthogonal initialization with enforced approximate isometry method (i.e., LDI-AI) achieves the best results across all tested architectures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">VGG16</td><td colspan=\"2\">ResNet32</td><td colspan=\"2\">ResNet56</td><td colspan=\"2\">ResNet110</td><td colspan=\"2\">WRN16</td></tr><tr><td>Initialization</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td></tr><tr><td>VS-L</td><td>13.72</td><td>8.16</td><td>4.50</td><td>11.96</td><td>4.64</td><td>10.43</td><td>4.65</td><td>9.13</td><td>11.99</td><td>45.08</td></tr><tr><td>VS-G</td><td>13.60</td><td>8.18</td><td>4.55</td><td>11.89</td><td>4.67</td><td>10.60</td><td>4.67</td><td>9.17</td><td>11.50</td><td>44.56</td></tr><tr><td>VS-H</td><td>15.44</td><td>8.36</td><td>4.41</td><td>12.21</td><td>4.44</td><td>10.63</td><td>4.39</td><td>9.08</td><td>13.49</td><td>46.62</td></tr><tr><td>LDI</td><td>13.33</td><td>8.11</td><td>4.43</td><td>11.55</td><td>4.51</td><td>10.08</td><td>4.57</td><td>8.88</td><td>11.28</td><td>44.20</td></tr><tr><td>LDI-AI</td><td>6.43</td><td>7.99</td><td>2.62</td><td>11.47</td><td>2.79</td><td>9.85</td><td>2.92</td><td>8.78</td><td>6.62</td><td>44.12</td></tr></table>",
+        "bbox": [
+            235,
+            196,
+            758,
+            285
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a19c10ed6cd2f020e7eae31c1ff770288b87994749159f6fa932076a505fe86.jpg",
+        "table_caption": [
+            "Table 3: Pruning results for VGG16 and ResNet32 with different activation functions on CIFAR-10. We report generalization errors (avg. over 5 runs), and the first and second best results are highlighted. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">VGG16</td><td colspan=\"3\">ResNet32</td></tr><tr><td>Initialization</td><td>tanh</td><td>1-relu</td><td>selu</td><td>tanh</td><td>1-relu</td><td>selu</td></tr><tr><td>VS-L</td><td>9.07</td><td>7.78</td><td>8.70</td><td>13.41</td><td>12.04</td><td>12.26</td></tr><tr><td>VS-G</td><td>9.06</td><td>7.84</td><td>8.82</td><td>13.44</td><td>12.02</td><td>12.32</td></tr><tr><td>VS-H</td><td>9.99</td><td>8.43</td><td>9.09</td><td>13.12</td><td>11.66</td><td>12.21</td></tr><tr><td>LDI</td><td>8.76</td><td>7.53</td><td>8.21</td><td>13.22</td><td>11.58</td><td>11.98</td></tr><tr><td>LDI-AI</td><td>8.72</td><td>7.47</td><td>8.20</td><td>13.14</td><td>11.51</td><td>11.68</td></tr></table>",
+        "bbox": [
+            174,
+            376,
+            519,
+            463
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/25eb9bf5586153ff64677764c98bf4bd3e041dc8a3327ad76f6c0fe80f48618f.jpg",
+        "table_caption": [
+            "Table 4: Unsupervised pruning results for $K$ -layer MLP networks on MNIST. All networks are pruned for sparsity $\\bar { \\kappa } = 9 0 \\%$ at orthogonal initialization. We report generalization errors (avg. over 10 runs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Loss</td><td>Superv.</td><td>K=3</td><td>K=5</td><td>K=7</td></tr><tr><td>GT</td><td>√</td><td>2.46</td><td>2.43</td><td>2.61</td></tr><tr><td>Pred. (raw)</td><td>X</td><td>3.31</td><td>3.38</td><td>3.60</td></tr><tr><td>Pred. (softmax)</td><td>X</td><td>3.11</td><td>3.37</td><td>3.56</td></tr><tr><td>Unif.</td><td>X</td><td>2.77</td><td>2.77</td><td>2.94</td></tr></table>",
+        "bbox": [
+            540,
+            392,
+            808,
+            462
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            187,
+            501,
+            823,
+            570
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Structure (LDI-Rand-AI vs. LDI-CS-AI). Even if the approximate dynamical isometry is enforced identically, the network pruned using connection sensitivity shows better trainability than the randomly pruned network. This potentially means that the sparse topology obtained by different pruning methods also matters, in addition to signal propagation characteristics. ",
+        "bbox": [
+            174,
+            570,
+            823,
+            626
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Overparameterization (LDI-Dense vs. LDI-{CS, Rand}-AI). Even though the singular values are restored to a level close to before pruning with approximate isometry, the non-pruned dense network converges faster than pruned networks. We hypothesize that in addition to signal propagation, overparameterization helps in optimization taking less time to find a minimum. ",
+        "bbox": [
+            176,
+            627,
+            823,
+            683
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "While being simple and data free (thus fast), our signal propagation perspective not only can be used to improve trainability of sparse neural networks, but also to complement a common explanation for decreased trainability of compressed networks which is often attributed merely to a reduced capacity. Our results also extend to the case of convolutional neural network (see Figure 8 in Appendix C). ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 VALIDATION AND EXTENSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            787,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we aim to demonstrate the efficacy of our signal propagation perspective on a wide variety of settings. We first evaluate the idea of employing layerwise dynamical isometry on various modern neural networks. In addition, we further study the role of supervision under the pruning at initialization regime, extending it to unsupervised pruning. Our results show that indeed, pruning can be approached from the signal propagation perspective at varying scale, bringing forth the notion of neural architecture sculpting. The experiment settings used to generate the presented results are detailed in Appendix B. The code can be found here: https://github.com/namhoonlee/spp-public. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": ".1 EVALUATION ON VARIOUS NEURAL NETWORKS AND DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            187,
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+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Here, we verify that our signal propagation perspective for pruning neural networks at initialization is indeed valid, by evaluating further on various modern neural networks and datasets. To this end, we provide orthogonality scores (OS) and generalization errors of the sparse networks obtained by different methods and show that layerwise dynamical isometry with enforced approximate isometry results in the best performance; here, we define OS as $\\begin{array} { r } { \\frac 1 l \\sum _ { l } \\| ( \\mathbf { \\dot { C } } ^ { l } \\odot \\mathbf { W } ^ { l } ) ^ { T } ( \\mathbf { C } ^ { l } \\odot \\mathbf { \\dot { W } } ^ { l } ) - \\mathbf { I } ^ { l } \\| _ { F } } \\end{array}$ , which can be used to indicate how close are the weight matrices in each layer of the pruned network to being orthogonal. All results are the average of 5 runs, and we do not optimize anything specific for a particular case (see Appendix B for experiment settings). The results are presented in Table 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The best pruning results are achieved when the approximate dynamical isometry is enforced on the pruned sparse network (i.e., LDI-AI), across all tested architectures. Also, the second best results are achieved with the orthogonal initialization that satisfies layerwise dynamical isometry (i.e., LDI). Looking closely, it is evident that there exists a high correlation between the orthogonality scores and the performance of pruned networks; i.e., the network initialized to have the lowest orthogonality scores achieves the best generalization errors after training. Note that the orthogonality scores being close to 0, by definition, states how faithful a network will be with regard to letting signals propagate without being amplified or attenuated. Therefore, the fact that a pruned network with the lowest orthogonality scores tends to yield good generalization errors further validates that our signal propagation perspective is indeed effective for pruning at initialization. Moreover, we test for other nonlinear activation functions (tanh, leaky-relu, selu), and found that the orthogonal initialization consistently outperforms variance scaling methods (see Table 3). ",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 PRUNING WITHOUT SUPERVISION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "So far, we have shown that pruning random networks can be approached from a signal propagation perspective by ensuring faithful connection sensitivity. Another factor that constitutes connection sensitivity is the loss term. At a glance, it is not obvious how informative the supervised loss measured on a random network will be for connection sensitivity. In this section, we look into the effect of supervision, by simply replacing the loss computed using ground-truth labels with different unsupervised surrogate losses as follows: replacing the target distribution using ground-truth labels with uniform distribution (Unif.), and using the averaged output prediction of the network (Pred.; softmax/raw). The results for MLP networks are in Table 4. Even though unsupervised pruning results are not as good as the supervised case, the results are still interesting, especially for the uniform case, in that there was no supervision given. We thus experiment further for the uniform case on other networks, and obtain the following results: 8.25, 11.69, 11.01, 8.82 errors $( \\% )$ for VGG16, ResNet32, ResNet56, ResNet110, respectively. Surprisingly, the results are often competitive to that of pruning with supervision (i.e., compare to LDI results in Table 2). Notably, previous pruning algorithms assume the existence of supervision a priori. Being the first demonstration, along with the signal propagation perspective, this unsupervised pruning strategy can be useful in scenarios where there are no labels or only weak supervision is available. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To demonstrate further, we also conducted transfer of sparsity experiments such as transferring a pruned network from one task to another (MNIST Fashion-MNIST). Table 5 shows that, while pruning results may degrade if sparsity is transferred, or done without supervision, less impact is caused for unsupervised pruning when transferred to a different task (i.e., 0.52 to 0.14 on MNIST, and 1.11 to $- 0 . 7 8$ on F",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1229d79367ccbb31ad102112480fed8f104cca012c45769458f2a254a61016c7.jpg",
+        "table_caption": [
+            "Table 5: Transfer of sparsity experiment results for LeNet. We prune for $\\bar { \\kappa } = 9 7 \\%$ at orthogonal initialization, and report gen. errors (average over 10 runs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Dataset</td><td colspan=\"2\">Error</td><td rowspan=\"2\">Error rand</td></tr><tr><td>Category</td><td>prune</td><td>train&amp;test</td><td>sup.→unsup.</td><td>(△)</td></tr><tr><td>Standard</td><td>MNIST</td><td>MNIST</td><td>2.42 → 2.94</td><td>+0.52</td><td>15.56</td></tr><tr><td>Transfer</td><td>F-MNIST</td><td>MNIST</td><td>2.66→ 2.80</td><td>+0.14</td><td>18.03</td></tr><tr><td>Standard</td><td>F-MNIST</td><td>F-MNIST</td><td>11.90 → 13.01</td><td>+1.11</td><td>24.72</td></tr><tr><td>Transfer</td><td>MNIST</td><td>F-MNIST</td><td>14.17→ 13.39</td><td>-0.78</td><td>24.89</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "MNIST). This indicates that inductive bias exists in data, affecting transfer and unsupervised pruning, and potentially, that “universal” sparse topology might be obtainable if universal data distribution is known (e.g., extremely large dataset in practice). This may help in situations where different tasks from unknown data distribution are to be performed (e.g., continual learning). We also tested two other unsupervised losses, but none performed as well as uniform loss (e.g., Jacobian norms $\\| J \\| _ { 1 }$ : 5.03, $\\| J \\| _ { 2 }$ : 3.00 vs. Unif.: 2.94), implying that if pruning is to be unsupervised, the uniform loss would better be used, because other unsupervised losses depend on the input data (thus can suffer from inductive bias). Random pruning degrades significantly at high sparsity for all cases. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have shown that pruning at initialization, even when no supervision is provided, can be effective based on the signal propagation perspective. This begs the question of whether pruning needs to be limited to pre-shaped architectures or not. In other words, what if pruning is applied to an arbitrarily bulky network and is treated as sculpting an architecture? In order to find out, we conduct the following experiments: we take a popular pre-designed architecture (ResNet20 in He et al. (2016)) as a base network, and consider a range of variants that are originally bigger than the base model, but pruned to have the same number of parameters as the base dense network. Specifically, we consider the following equivalents: (1) the same number of residual blocks, but with larger widths; (2) a reduced number of residual blocks with larger widths; (3) a larger residual block and the same width (see Table 6 in Appendix B for details). The results are presented in Figure 3. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Overall, the sparse equivalents record lower errors than the dense base model. Notice that some models are extremely sparse (e.g., Equivalent 1 pruned for $\\bar { \\kappa } = 9 8 . 4 \\%$ ). While all networks have the same number of parameters, discovered sparse equivalents outperform the dense reference network. This result is well aligned with recent findings in Kalchbrenner et al. (2018): large sparse networks can outperform their small dense counterpart, while enjoying increased computational and memory efficiency via a dedicated implementation for sparsity in practice. Also, it seems that pruning wider networks tends to be more effective in producing a better model than pruning deeper ones (e.g., Equivalent 1 vs. Equivalent 3). We further note that unlike existing prior works, the sparse networks are discovered by sculpting arbitrarily-designed architecture, without pretraining nor supervision. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/68753f2ac064ffc8b1af0b7cbd0c62d195ccf2ffbcbf5ae69f379e5113029a67.jpg",
+        "image_caption": [
+            "Figure 3: Neural architecture sculpting results on CIFAR-10. We report generalization errors (avg. over 5 runs). All networks have the same number of parameters (269k) and trained identically. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we have approached the problem of pruning neural networks at initialization from a signal propagation perspective. Based on observing the effect of varying the initialization, we found that initial weights have a critical impact on connection sensitivity measurements and hence pruning results. This led us to conduct theoretical analysis based on dynamical isometry and a mean field theory, and formally characterize a sufficient condition to ensure faithful signal propagation in a given network. Moreover, our analysis on compressed neural networks revealed that signal propagation characteristics of a sparse network highly correlates with its trainability, and also that pruning can break dynamical isometry ensured on a network at initialization, resulting in degradation of trainability of the compressed network. To address this, we introduced a simple, yet effective data-free method to recover the orthogonality and enhance trainability of the compressed network. Finally, throughout a range of validation and extension experiments, we verified that our signal propagation perspective is effective for understanding, improving, and extending the task of pruning at initialization across various settings. We believe that our results on the increased trainability of compressed networks can take us one step towards finding “winning lottery ticket” (i.e., a set of initial weights that given a sparse topology can quickly reach to a generalization performance that is comparable to the uncompressed network, once trained) suggested in Frankle & Carbin (2019). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We point out, however, that there remains several aspects to consider. While pruning on enforced isometry produces trainable sparse networks, the two-stage orthogonalization process (i.e., prune first and enforce the orthogonality later) can be suboptimal especially at a high sparsity level. Also, network weights change during training, which can affect signal propagation characteristics, and therefore, dynamical isometry may not continue to hold over the course of training. We hypothesize that a potential key to successful neural network compression is to address the complex interplay between optimization and signal propagation, and it might be immensely beneficial if an optimization naturally takes place in the space of isometry. We believe that our signal propagation perspective provides a means to formulate this as an optimization problem by maximizing the trainability of sparse networks while pruning, and we intend to explore this direction as a future work. ",
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1, EPSRC/MURI grant EP/N019474/1 and the Australian Research Council Centre of Excellence for Robotic Vision (project number CE140100016). We would also like to acknowledge the Royal Academy of Engineering and FiveAI, and thank Richard Hartley, Puneet Dokania and Amartya Sanyal for helpful discussions. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "Greg Yang, Jeffrey Pennington, Vinay Rao, Jascha Sohl-Dickstein, and Samuel S Schoenholz. A mean field theory of batch normalization. ICLR, 2019. ",
+        "bbox": [
+            176,
+            189,
+            823,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A GRADIENTS IN TERMS OF JACOBIANS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            241,
+            524,
+            257
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Let $\\epsilon = \\partial L / \\partial \\mathbf { x } ^ { K }$ denote the error signal and $\\mathbf { x } ^ { 0 }$ denote the input signal. Then, ",
+        "bbox": [
+            168,
+            270,
+            794,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1. the gradients satisfy: ",
+        "bbox": [
+            171,
+            296,
+            328,
+            310
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a28457160455a69c443dec451f71cf22b74b086af7273cc1520d2dfe5f8c9150.jpg",
+        "text": "$$\n{ \\bf g } _ { { \\bf w } ^ { l } } ^ { T } = \\epsilon { \\bf J } ^ { l , K } { \\bf D } ^ { l } \\otimes { \\bf x } ^ { l - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            309,
+            591,
+            329
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { J } ^ { l , K } = \\partial \\mathbf { x } ^ { K } / \\partial \\mathbf { x } ^ { l }$ is the Jacobian from layer $l$ to the output and $\\otimes$ is the Kronecker product. 2. additionally, for linear networks, $i . e .$ , when $\\phi$ is the identity: ",
+        "bbox": [
+            166,
+            330,
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+            358
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/281f1b8cd7f0332c40c05addcee4680af70cdfb105546c68aae3424679fe8f40.jpg",
+        "text": "$$\n{ \\bf g } _ { { \\bf w } ^ { l } } ^ { T } = \\epsilon { \\bf J } ^ { l , K } \\otimes \\left( { \\bf J } ^ { 0 , l - 1 } { \\bf x } ^ { 0 } + { \\bf a } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            361,
+            619,
+            381
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { J } ^ { 0 , l - 1 } = \\partial \\mathbf { x } ^ { l - 1 } / \\partial \\mathbf { x } ^ { 0 }$ is the Jacobian from the input to layer $l - 1$ and $\\mathbf { a } \\in \\mathbb { R } ^ { N }$ is the constant term that does not depend on $\\mathbf { x } ^ { 0 }$ . ",
+        "bbox": [
+            187,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof is based on a simple algebraic manipulation of the chain rule. The gradient of the loss with respect to the weight matrix $\\mathbf { W } ^ { l }$ can be written as: ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/484c05d53db430966f98f7111cb857666ffacf9c86b58e1f22a9bb92389048e8.jpg",
+        "text": "$$\n{ \\bf g } _ { { \\bf w } ^ { l } } = \\frac { \\partial L } { \\partial { \\bf W } ^ { l } } = \\frac { \\partial L } { \\partial { \\bf x } ^ { K } } \\frac { \\partial { \\bf x } ^ { K } } { \\partial { \\bf x } ^ { l } } \\frac { \\partial { \\bf x } ^ { l } } { \\partial { \\bf W } ^ { l } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here, the gradient $\\partial \\mathbf { y } / \\partial \\mathbf { x }$ is represented as a matrix of dimension $\\mathbf { y } { \\mathrm { - s i z e } } \\times \\mathbf { x }$ -size. For gradients with respect to matrices, their vectorized from is used. Notice, ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1bf40f92d8da73de890b67c2e61933e5aa9acbb8f90ae453c5b19fdf34b664f7.jpg",
+        "text": "$$\n\\frac { \\partial \\mathbf { x } ^ { l } } { \\partial \\mathbf { W } ^ { l } } = \\frac { \\partial \\mathbf { x } ^ { l } } { \\partial \\mathbf { h } ^ { l } } \\frac { \\partial \\mathbf { h } ^ { l } } { \\partial \\mathbf { W } ^ { l } } = \\mathbf { D } ^ { l } \\frac { \\partial \\mathbf { h } ^ { l } } { \\partial \\mathbf { W } ^ { l } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Considering the feed-forward dynamics for a particular neuron $i$ ",
+        "bbox": [
+            173,
+            556,
+            599,
+            571
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77c621979c489118173427e10ef74180c21b26fa289811aab59305f385a4a1a0.jpg",
+        "text": "$$\n\\begin{array} { c } { { h _ { i } ^ { l } = \\displaystyle \\sum _ { j } W _ { i j } ^ { l } x _ { j } ^ { l - 1 } + b _ { i } ^ { l } , } } \\\\ { { { } } } \\\\ { { \\displaystyle \\frac { \\partial h _ { i } ^ { l } } { \\partial W _ { i j } ^ { l } } = x _ { j } ^ { l - 1 } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            573,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Therefore, using the Kronecker product, we can compactly write: ",
+        "bbox": [
+            173,
+            648,
+            602,
+            662
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e8a1c40c4e4b601c05e23461991c8f62b35eb0936c9e9db1aad6f49aca9cd8a8.jpg",
+        "text": "$$\n\\frac { \\partial { \\mathbf { x } } ^ { l } } { \\partial { \\mathbf { W } } ^ { l } } = ( { \\mathbf { D } } ^ { l } ) ^ { T } \\otimes ( { \\mathbf { x } } ^ { l - 1 } ) ^ { T } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            665,
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+            698
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Now, Equation 11 can be written as: ",
+        "bbox": [
+            174,
+            699,
+            411,
+            713
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/75678eea60c679008a7374ad94fd14c4de48475c94c9976b8b5a958595e1ca57.jpg",
+        "text": "$$\n\\begin{array} { r l } & { { \\bf g } _ { { \\bf w } ^ { l } } = ( \\epsilon { \\bf J } ^ { l , K } { \\bf D } ^ { l } ) ^ { T } \\otimes ( { \\bf x } ^ { l - 1 } ) ^ { T } , } \\\\ & { { \\bf g } _ { { \\bf w } ^ { l } } ^ { T } = \\epsilon { \\bf J } ^ { l , K } { \\bf D } ^ { l } \\otimes { \\bf x } ^ { l - 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            713,
+            606,
+            757
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\mathbf { A } ^ { T } \\otimes \\mathbf { B } ^ { T } = ( \\mathbf { A } \\otimes \\mathbf { B } ) ^ { T }$ is used. Moreover, for linear networks $\\mathbf { D } ^ { l } = \\mathbf { I }$ and $\\mathbf { x } ^ { l } = \\mathbf { h } ^ { l }$ for all $l \\in \\{ 1 \\ldots K \\}$ . Therefore, $\\mathbf { x } ^ { l - 1 }$ can be written as: ",
+        "bbox": [
+            171,
+            758,
+            823,
+            786
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c6d60e6362a2cb2bba13a69b2da4b45dd1bbaf744e16e21f6b5e8a72ecfe89bf.jpg",
+        "text": "$$\n\\begin{array} { r l } & { { \\displaystyle { \\bf x } ^ { l - 1 } = \\phi ( { \\bf W } ^ { l - 1 } \\phi ( { \\bf W } ^ { l - 2 } \\cdot \\dots \\phi ( { \\bf W } ^ { 1 } { \\bf x } ^ { 0 } + { \\bf b } ^ { 1 } ) \\dots + { \\bf b } ^ { l - 2 } ) + { \\bf b } ^ { l - 1 } ) } ~ , } \\\\ & { \\quad \\quad = { \\bf W } ^ { l - 1 } ( { \\bf W } ^ { l - 2 } \\cdot \\dots ( { \\bf W } ^ { 1 } { \\bf x } ^ { 0 } + { \\bf b } ^ { 1 } ) \\dots + { \\bf b } ^ { l - 2 } ) + { \\bf b } ^ { l - 1 } ~ , } \\\\ & { \\quad \\quad = \\displaystyle \\prod _ { k = 1 } ^ { l - 1 } { \\bf W } ^ { k } { \\bf x } ^ { 0 } + \\prod _ { k = 2 } ^ { l - 1 } { \\bf W } ^ { k } { \\bf b } ^ { 1 } + \\dots + { \\bf b } ^ { l - 1 } ~ , } \\\\ & { \\quad \\quad = { \\bf J } ^ { 0 , l - 1 } { \\bf x } ^ { 0 } + { \\bf a } ~ , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
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+            895
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { a }$ is the constant term that does not depend on $\\mathbf { x } ^ { 0 }$ . Hence, the proof is complete. ",
+        "bbox": [
+            173,
+            897,
+            740,
+            914
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENT SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            406,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Pruning at initialization. By default, we perform pruning at initialization based on connection sensitivity scores as in Lee et al. (2019). When computing connection sensitivity, we always use all examples in the training set to prevent stochasticity by a particular mini-batch. Unless stated otherwise, we set the default sparsity level to be $\\bar { \\kappa } = 9 0 \\%$ (i.e., $90 \\%$ of the entire parameters in a network is pruned away). For all tested architectures, pruning for such level of sparsity does not lead to a large accuracy drop. Additionally, we perform either random pruning (at initialization) or a magnitude based pruning (at pretrained) for comparison purposes. Random pruning refers to pruning parameters randomly and globally for a given sparsity level. For the magnitude based pruning, we first train a model and simply prune parameters globally in a single-shot based on the magnitude of the pretrained parameters (i.e., keep the large weights while pruning small ones). For initialization methods, we follow either variance scaling initialization schemes (i.e., VS-L, VSG, VS-H, as in LeCun et al. (1998); Glorot & Bengio (2010); He et al. (2015), respectively) or (convolutional) orthogonal initialization schemes (Saxe et al., 2014; Xiao et al., 2018). ",
+        "bbox": [
+            174,
+            133,
+            825,
+            314
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Training and evaluation. Throughout experiments, we evaluate pruning results on MNIST, CIFAR-10, and Tiny-ImageNet image classification tasks. For training of the pruned sparse networks, we use SGD with momentum and train up to 80k (for MNIST) or $1 0 0 \\mathrm { k }$ (for CIFAR-10 and Tiny-ImageNet) iterations. The initial learning rate is set to be 0.1 and is decayed by $1 / 1 0$ at every 20k (MNIST) or 25k (CIFAR-10 and Tiny-ImageNet). The mini-batch size is set to be 100, 128, 200 for MNIST, CIFAR-10, Tiny-ImageNet, respectively. We do not optimize anything specific for a particular case, and follow the standard training procedure. For all experiments, we use $10 \\%$ of training set for the validation set, which corresponds to 5400, 5000, 9000 images for MNIST, CIFAR-10, Tiny-IamgeNet, respectively. We evaluate at every 1k iteration, and record the lowest test error. All results are the average of either 10 (for MNIST) or 5 (for CIFAR-10 and Tiny-ImageNet) runs. ",
+        "bbox": [
+            174,
+            320,
+            825,
+            460
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Signal propagation and approximate dynamical isometry. We use the entire training set when computing Jacobian singular values of a network. In order to enforce approximate dynamical isometry when specified, given a pruned sparse network, we optimize for the objective in Equation 8 using gradient descent. The learning rate is set to be 0.1 and we perform $1 0 \\mathrm { k }$ gradient update steps (although it usually reaches to convergence far before). This process is data-free and thus fast; e.g., depending on the size of the network and the number of update steps, it can take less than a few seconds on a modern computer. ",
+        "bbox": [
+            173,
+            467,
+            825,
+            565
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Neural architecture sculpting. We provide the model details in Table 6. ",
+        "bbox": [
+            174,
+            571,
+            663,
+            585
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 6: All models (Equivalents 1,2,3) are initially bigger than the base network (ResNet20), by either being wider or deeper, but pruned to have the same number of parameters as the base network (269k). The widening factor $\\mathbf { \\Psi } ( \\mathbf { k } )$ refers to the filter multiplier; e.g., for the basic filter size of 16, the widening factor of ${ \\bf k } = 2$ will result in 32 filters. The block size refers to the number of residual blocks in each block layer; all models have three block layers. More/less number of residual blocks means the network is deeper/shallower. The reported generalization errors are averages over 5 runs. We find that the technique of architecture sculpting, pruning randomly initialized neural networks based on our signal propagation perspective even in the absence of ground-truth supervision, can be used to find models of superior performance under the same parameter budget. ",
+        "bbox": [
+            173,
+            592,
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+            717
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/2041f8565d7932e37d0c250885764dbf78b7502ac48c980d51b302e8176e7399.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model category</td><td>Shape</td><td>Widening (k)</td><td>Block size</td><td>Init.</td><td>GT</td><td>Sparsity</td><td>Error</td></tr><tr><td>Base</td><td>ResNet20 (He et al.,2016)</td><td>1</td><td>3</td><td>VS-H</td><td>√</td><td>0.0</td><td>8.046</td></tr><tr><td>Equivalent 1</td><td>wider</td><td>2</td><td>3</td><td>LDI</td><td>X</td><td>74.8</td><td>7.618</td></tr><tr><td></td><td>wider</td><td>4</td><td>3</td><td>LDI</td><td>X</td><td>93.7</td><td>7.630</td></tr><tr><td></td><td>wider</td><td>6</td><td>3</td><td>LDI</td><td>X</td><td>97.2</td><td>7.708</td></tr><tr><td>Equivalent 2</td><td>wider</td><td>8</td><td>3</td><td>LDI</td><td>X</td><td>98.4</td><td>7.836</td></tr><tr><td></td><td>wider&amp; shallower</td><td>2</td><td>2</td><td>LDI</td><td>X</td><td>60.4</td><td>7.776</td></tr><tr><td></td><td>wider&amp; shallower</td><td>4</td><td>2</td><td>LDI</td><td>X</td><td>90.1</td><td>7.876</td></tr><tr><td></td><td>wider&amp; shallower</td><td>6</td><td>2</td><td>LDI</td><td>X</td><td>95.6</td><td>7.940</td></tr><tr><td>Equivalent 3</td><td>deeper</td><td>1</td><td>5</td><td>LDI</td><td>X</td><td>42.0</td><td>7.912</td></tr></table>",
+        "bbox": [
+            205,
+            728,
+            789,
+            847
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b5581e9204ada42f22daf0b704642a1ff044bb50d5f1ce5c1db1571bb4df7e29.jpg",
+        "image_caption": [
+            "Figure 4: Full results for (a) signal propagation (all signular value statistics), and (b) training behavior (including accuracy) for 7-layer linear and tanh MLP networks. We provide results of LDI-Rand, LDI-Rand-AI, VS-CS, LDI-CS, LDI-CS-AI on the linear case for both singular value statistics and training log. We also plot results of LDI-Mag and LDI-Dense on the tanh case for trainability; the training results of non-pruned (LDI-Dense) and magnitude (LDI-Mag) pruning are only reported for the tanh case, because the learning rate had to be lowered for the linear case (otherwise it explodes), which makes the comparison not entirely fair. We provide the singular value statistics for the magnitude pruning in Figure 6 to avoid clutter. Also, extended training logs for random and magnitude based pruning are provided separately in Figure 5 to illustrate the difference in convergence speed. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            137,
+            816,
+            554
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/d1ed673251181025b8e6f9b38ebb82f8f54fdf9edf9d0f51b8c25f30d0927bbc.jpg",
+        "image_caption": [
+            "Figure 5: Extended training log (i.e., Loss and Accuracy) for random (Rand) and magnitude (Mag) pruning. The sparse networks obtained by random or magnitude pruning take a much longer time to train than that obtained by pruning based on connection sensitivity. All methods are pruned at the layerwise orthogonal initialization, and trained the same way as before. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            101,
+            803,
+            244
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/2b750e1f8519d5ee9d681220ed8869910b6d4c70ccc74c576c7b5a7eeae3240c.jpg",
+        "image_caption": [
+            "Figure 6: Signal propagation measurments (all signular value statistics) for the magnitude based pruning $\\left( { \\bf M a g } \\right)$ on the 7-layer linear and tanh MLP networks. As described in the experiment settings in Appendix B, the magnitude based pruning is performed on a pretrained model. Notice that unlike other cases where pruning is done at initialization (i.e., using either random or connection sensitivity based pruning methods), the singular value distribution changes abruptly when pruned (i.e., note of the sharp change of singular values from 0 to $10 \\%$ sparsity). Also, the singular values are not concentrated (note of high standard deviations), which explains rather inferior trainability compared to other methods. We conjecture that naively pruning based on the magnitude of parameters in a single-shot, without pruning gradually or employing some sophisticated tricks such as layerwise thresholding, can lead to a failure of training compressed networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            323,
+            808,
+            588
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/3e97af0711431e9dfb5f7744b19c809dda708b5fe1a341b76b9b70c4b9ffd379.jpg",
+        "image_caption": [
+            "Figure 7: Signal propagation and training behavior for ReLU and Leaky-ReLU activation functions. They resemble those of the tanh case as in Figure 2, and hence the conclusion holds about the same. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            757,
+            812,
+            880
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/85d20abf34e6c349a4275751a84c0c829146780e7dc068edeb4a8ec0cefe4ebe.jpg",
+        "image_caption": [
+            "Figure 8: Training performance (loss and accuracy) by different methods for VGG16 on CIFAR-10. To examine the effect of initialization in isolation on the trainability of sparse neural networks, we remove batch normalization (BN) layers for this experiment, as BN tends to improve training speed as well as generalization performance. As a result, enforcing approximate isometry (LDI-CS-AIF) improves the training speed quite dramatically compared to the pruned network without isometry (LDI-CS). We also find that even compared to the non-pruned dense network (LDI-Dense) which is ensured layerwise dynamical isometry, LDI-CS-AIF trains faster in the early training phase. This result is quite promising and more encouraging than the previous case of MLP (see Figures 2 and 7), as it potentially indicates that an underparameterized network (by connection sensitivity pruning) can even outperform an overparameterized network, at least in the early phase of neural network training. Furthermore, we add results of using the spectral norm in enforcing approximate isometry in Equation 8 (LDI-CS-AIS), and find that it also trains faster than the case of broken isometry (LDI-CS), yet not as much as the case of using the Frobenius norm (LDI-CS-AIF). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            231,
+            821,
+            616
+        ],
+        "page_idx": 15
+    }
+]

parse/train/ZqB2GD-Ixn/ZqB2GD-Ixn.md

@@ -0,0 +1,492 @@
+# ACCOUNTING FOR UNOBSERVED CONFOUNDING IN DOMAIN GENERALIZATION
+
+Anonymous authors
+
+Paper under double-blind review
+
+# ABSTRACT
+
+The ability to extrapolate, or generalize, from observed to new related environments is central to any form of reliable machine learning, yet most methods fail when moving beyond $i . i . d$ data. In some cases, the reason lies in a misappreciation of the causal structure that governs the data, and in particular as a consequence of the influence of unobserved confounders that drive changes in observed distributions and distort correlations. In this paper, we argue for defining generalization with respect to a broader class of distribution shifts (defined as arising from interventions in the underlying causal model), including changes in observed, unobserved and target variable distributions. We propose a new robust learning principle that may be paired with any gradient-based learning algorithm. This learning principle has explicit generalization guarantees, and relates robustness with certain invariances in the causal model, clarifying why, in some cases, test performance lags training performance. We demonstrate the empirical performance of our approach on healthcare data from different modalities, including image and speech data.
+
+# 1 INTRODUCTION
+
+Prediction algorithms use data, necessarily sampled under specific conditions, to learn correlations that extrapolate to new or related data. If successful, the performance gap between these two domains is small, and we say that algorithms generalize beyond their training data. Doing so is difficult however, some form of uncertainty about the distribution of new data is unavoidable. The set of potential distributional changes that we may encounter is mostly unknown and in many cases may be large and varied. Some examples include covariate shifts (Bickel et al., 2009), interventions in the underlying causal system (Pearl, 2009), varying levels of noise (Fuller, 2009) and confounding (Pearl, 1998). All of these feature in modern applications, and while learning systems are increasingly deployed in practice, generalization of predictions and their reliability in a broad sense remains an open question.
+
+A common approach to formalize learning with uncertain data is, instead of optimizing for correlations in a fixed distribution, to do so simultaneously for a range of different distributions in an uncertainty set $\mathcal { P }$ (Ben-Tal et al., 2009).
+
+$$
+{ \underset { f } { \operatorname { m i n i m i z e } } } \ \operatorname* { s u p } _ { P \in { \mathcal { P } } } \ { \mathbb { E } } _ { ( x , y ) \sim P } [ { \mathcal { L } } ( f ( x ) , y ) ]
+$$
+
+for some measure of error $\mathcal { L }$ of the function $f$ that relates input and output examples $( x , y ) \sim P$ . Choosing different sets $\mathcal { P }$ leads to estimators with different properties. It includes as special cases, for instance, many approaches in domain adaptation, covariate shift, robust statistics and optimization (Kuhn et al., 2019; Bickel et al., 2009; Duchi et al., 2016; 2019; Sinha et al., 2017; Wozabal, 2012; Abadeh et al., 2015; Duchi & Namkoong, 2018). Robust solutions to problem (1) are said to generalize if potential shifted, test distributions are contained in $\mathcal { P }$ , but also larger sets $\mathcal { P }$ result in conservative solutions (i.e. with sub-optimal performance) on data sampled from distribution away from worst-case scenarios, in general.
+
+One formulation of causality is in fact also a version of this problem, for $\mathcal { P }$ defined as any distribution arising from arbitrary interventions on observed covariates $x$ leading to shifts in their distribution $P _ { x }$ (see e.g. sections 3.2 and 3.3 in (Meinshausen, 2018)). The invariance to changes in covariate distributions of causal solutions is powerful for generalization, but implicitly assumes that all covariates or other drivers of the outcome subject to change at test time are observed. Often shifts occur elsewhere, for example in the distribution of unobserved confounders, in which case also conditional distributions $P _ { y | x }$ may shift. Perhaps surprisingly, in the presence of unobserved confounders, the goals of achieving robustness and learning a causal model can be different (and similar behaviour also occurs with varying measurement noise). There is in general an inherent trade-off in generalization performance. In the presence of unobserved confounders, causal and correlation-based solutions are both optimal in different regimes, depending on the shift in the underlying generating mechanism from which new data is generated.
+
+![](images/3a460c391c42a041b7fdc5d38c9048d21ca960c53986f69c449c9a88241a36d4.jpg)  
+Figure 1: The challenges of generalization. In the presence of unobserved confounders, there is an inherent trade-off in performance – causal and correlation-based solutions are both optimal in different regimes, depending on the shift from which new data is generated. The proposed approach, DIRM, is a relaxation of the causal solution that naturally interpolates between the causal solution and Ordinary Least Squares (OLS), and is described in Section 3. The data generating mechanism, methods, and a discussion of the results are given in the paragraphs above our contributions.
+
+Consider a simple example, illustrated in Figure 1, to show this explicitly. We assume access to observations of variables $( X _ { 1 } , X _ { 2 } , Y )$ in two training datasets, each dataset sampled with differing variances $\boldsymbol { \sigma } ^ { 2 } = 1$ and $\sigma ^ { 2 } = 2$ ) from the following structural model $\mathbb { F }$ ,
+
+$$
+X _ { 2 } : = - H + E _ { X _ { 2 } } , \quad Y : = X _ { 2 } + 3 H + E _ { Y } , \quad X _ { 1 } : = Y + X _ { 2 } + E _ { X _ { 1 } } ,
+$$
+
+$E _ { X _ { 1 } } , E _ { X _ { 2 } } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } )$ , $E _ { Y } \sim \mathcal { N } ( 0 , 1 )$ are exogenous variables. In a first scenario (leftmost panel) we consider all data (training and testing) to be generated without unobserved confounders, $H : = 0$ ; and, in a second scenario (remaining panels) all data with unobserved confounders, ${ \cal H } : = E _ { \cal H } \sim$ $\mathcal { N } ( 0 , 1 )$ . Each panel of Figure 1 shows performance on new data obtained after manipulating the underlying data generating system; the magnitude and type of intervention appears in the horizontal axis. We consider the following learning paradigms: Ordinary Least Squares (OLS) learns the linear mapping that minimizes average training risk, Domain Robust Optimization (DRO) minimizes the maximum training risk among the two available datasets, and the causal solution, assumed known, has fixed coefficients $( 0 , 1 )$ for $( X _ { 1 } , X _ { 2 } )$ . Two important observations motivate this paper.
+
+First, observe that Ordinary Least Squares (OLS) and Domain Robust Optimization (DRO) absorb spurious correlations (due to $H$ , and the fact that $X _ { 1 }$ is caused by $Y$ ) with unstable performance under shifts in $p ( X _ { 1 } , X _ { 2 } )$ but as a consequence good performance under shifts in $p ( H )$ . Causal solutions, by contrast, are robust to shifts in $p ( X _ { 1 } , X _ { 2 } )$ , even on new data with large shifts, but underperform substantially under changes in the distribution of unobserved confounders $p ( H )$ . Second, the presence of unobserved confounding hurts generalization performance in general with higher errors for all methods, e.g. contrast the middle and leftmost panel. To the best of our knowledge, the influence of unobserved confounders has been minimally explored in the context of generalization of learning algorithms, even though, as Figure 1 shows, in this context different shifts in distribution may have important consequences for predictive performance.
+
+Our Contributions. In this paper we provide a new choice of $\mathcal { P }$ and learning problem (1) that we show to be justified by certain statistical invariances across training and testing data, to be expected in the presence of unobserved confounders. This leads us to define a new differentiable, regularized objective for representation learning. Our proposal defines $\mathcal { P }$ as an affine combination of available training data distributions, and we show that solutions to this problem are robust to more general shifts in distribution than previously considered, spanning robustness to shifts in observed, unobserved, and target variables, depending on the properties of the available training data distributions. This approach has benefits for performance out-of-sample but also for tasks involving variable selection, where important features are consistently replicated across experiments with our objective.
+
+# 2 INVARIANCES IN THE PRESENCE OF UNOBSERVED CONFOUNDERS
+
+This section formally introduces the problem of out-of-distribution generalization. We describe in greater detail the reasons that popular learning principles, such as Empirical Risk Minimization (ERM), underperform in general, and define certain invariances to recover solutions that generalize.
+
+We take the perspective that all potential distributions that may be observed over a system of variables arise from a causal model $\mathcal { M } = ( \mathbb { F } , \mathbb { V } , \mathbb { U } )$ , characterized by endogenous variables, $\mathbb { V } \in \mathcal { V }$ , representing all variables determined by the system, either observed or not; exogenous variables, $\mathbb { U } \in \mathcal { U }$ , in contrast imposed upon the model, and a sequence of structural equations $\mathbb { F } : \mathcal { U } \to$ $\nu$ , describing how endogenous variables can be (deterministically) obtained from the exogenous variables (Pearl, 2009). An example is given in Figure 1, $\mathbb { V } = ( X _ { 1 } , X _ { 2 } , H , Y )$ are endogenous and $\mathbb { U } = ( E _ { X _ { 1 } } , E _ { X _ { 2 } } , E _ { H } , E _ { Y } )$ are exogenous variables.
+
+Unseen data is generated from such a system $\mathcal { M }$ after manipulating the distribution of exogenous variables $\mathbb { U }$ , which propagates across the system shifting the joint distribution of all variables $\mathbb { V }$ , whether observed or unobserved, but keeping the causal mechanisms $\mathbb { F }$ unchanged. Representative examples include changes in data collection conditions, such as due to different measurement devices, or new data sources, such as patients in different hospitals or countries, among many others.
+
+Our goal is to learn a representation $Z = \phi ( X )$ acting on a set observed variables $X \subset \mathbb { V }$ with the ability to extrapolate to new unseen data, and doing so acknowledging that all relevant variables in $\mathbb { V }$ are likely not observed. Unobserved confounders (for the task at hand, say predicting $Y \in \mathbb { V } .$ ) simultaneously cause $X$ and $Y$ , confounding or biasing the causal association between $X$ and $Y$ giving rise to spurious correlations that do not reproduce in general (Pearl, 1998; 2009). We present a brief argument below highlighting the systematic bias due to unobserved confounders in ERM.
+
+# 2.1 THE BIASES OF UNOBSERVED CONFOUNDING
+
+Consider the following structural equation for observed variables $( X , Y )$ ,
+
+$$
+Y : = f \circ \phi ( X ) + E
+$$
+
+where $f : = f ( \cdot ; \beta _ { 0 } )$ is a predictor acting on a representation $Z : = \phi ( X )$ and $E$ stands for potential sources of mispecification and unexplained sources of variability. For a given sample of data $( x , y )$ and $z = \phi ( x )$ , the optimal prediction rule $\hat { \beta }$ is often taken to minimize squared residuals, with $\hat { \beta }$ the solution to the normal equations: $\nabla _ { \beta } f ( z ; { \hat { \beta } } ) y = \nabla _ { \beta } f ( z ; { \hat { \beta } } ) f ( z ; { \hat { \beta } } )$ , where $\nabla _ { \beta } f ( z ; { \hat { \beta } } )$ denotes the column vector of gradients of $f$ with respect to parameters $\beta$ evaluated at $\hat { \beta }$ . Consider the Taylor expansion of $f ( z ; { \boldsymbol { \beta } } _ { 0 } )$ around an estimate $\hat { \beta }$ sufficiently close to $\beta _ { 0 }$ , $f ( z ; \beta _ { 0 } ) \approx f ( z ; { \hat { \beta } } ) +$ $\nabla _ { \beta } f ( z ; { \hat { \beta } } ) ^ { T } ( \beta _ { 0 } - { \hat { \beta } } )$ . Using this approximation in our first order optimality condition we find,
+
+$$
+\nabla _ { \boldsymbol { \beta } } f ( \boldsymbol { z } ; \hat { \boldsymbol { \beta } } ) \nabla _ { \boldsymbol { \beta } } f ( \boldsymbol { z } ; \hat { \boldsymbol { \beta } } ) ^ { T } ( \beta _ { 0 } - \hat { \boldsymbol { \beta } } ) + \boldsymbol { v } = \nabla _ { \boldsymbol { \beta } } f ( \boldsymbol { z } ; \hat { \boldsymbol { \beta } } ) \epsilon
+$$
+
+where $v$ is a scaled disturbance term that includes the rest of the linear approximation of $f$ and is small asymptotically; $\epsilon : = y - f ( z ; { \hat { \beta } } )$ is the residual. $\hat { \beta }$ is consistent for the true $\beta _ { 0 }$ if and only if $\nabla _ { \beta } f ( z ; { \hat { \beta } } ) \epsilon \to 0$ in probability. This assumption is satisfied if $E$ (all sources of variation in $Y$ not captured by $X$ ) are independent of $X$ (i.e. exogenous) or in other words if all common causes or confounders to both $X$ and $Y$ have been observed. Conventional regression may assign significant associations to variables that are neither directly nor indirectly related to the outcome, and in this case, we have no performance guarantees on new data with changes in the distribution of these variables. Omitted variables are a common source of unobserved confounding but we note in Appendix B that similar biases also arise from other prevalent model mispecifications, such as measurement error (Carroll et al., 2006).
+
+# 2.2 INVARIANCES WITH MULTIPLE ENVIRONMENTS
+
+The underlying structural mechanism F, that also relates unobserved with observed variables, even if unknown, is stable irrespective of manipulations in exogenous variables that may give rise to heterogeneous data sources. Under certain conditions, statistical footprints emerge from this structural invariance across different data sources, properties testable from data that have been exploited recently, for example (Peters et al., 2016; Ghassami et al., 2017; Rothenhäusler et al., 2019).
+
+We assume that such a heterogeneous data scenario applies, input and output pairs $( X , Y )$ are observed across heterogeneous data sources or environments $e$ , defined as a probability distribution $P _ { e }$ over an observation space $\mathcal { X } \times \mathcal { V }$ that arises, just like new unseen data, from manipulations in the distribution of exogenous variables in an underlying model $\mathcal { M }$ .
+
+For the remainder of this section, consider restricting ourselves to data sources emerging from manipulations in exogenous $E _ { X }$ , appearing in the structural equations of $X$ only in an underlying additive noise model (see Appendix C.1 for the precise statement of assumptions and more context). It may be shown by considering the distributions of error terms $Y - f \circ \phi ( X )$ and its correlation with any function of $X$ , that the inner product $\nabla _ { \beta } f ( z ; \beta _ { 0 } ) \epsilon$ , even if non-zero due to unobserved confounding, converges to a fixed unknown value equal across training environments (see Appendix C.1 for the derivation). With a similar decomposition to the one given in equation (3), in the population case, it holds that up to disturbance terms,
+
+$$
+\begin{array} { r l } & { \quad \quad \quad \biggr ( \underset { ( x , y ) \sim P _ { i } } { \mathbb { E } } \nabla _ { \beta } f ( z ; \beta ^ { \star } ) \nabla _ { \beta } f ( z ; \beta ^ { \star } ) ^ { T } - \underset { ( x , y ) \sim P _ { j } } { \mathbb { E } } \nabla _ { \beta } f ( z ; \beta ^ { \star } ) \nabla _ { \beta } f ( z ; \beta ^ { \star } ) \biggr ) ^ { T } ( \beta _ { 0 } - \beta ^ { \star } ) } \\ & { \quad \quad \quad = \biggr ( \underset { ( x , y ) \sim P _ { i } } { \mathbb { E } } \nabla _ { \beta } f ( z ; \beta ^ { \star } ) \epsilon - \underset { ( x , y ) \sim P _ { j } } { \mathbb { E } } \nabla _ { \beta } f ( z ; \beta ^ { \star } ) \epsilon \biggr ) = 0 } \end{array}
+$$
+
+where $\beta ^ { \star }$ is a solution to,
+
+$$
+\underset { ( x , y ) \sim P _ { i } } { \mathbb { E } } \nabla _ { \beta } f ( z ; \beta ) ( y - f ( z ; \beta ) ) - \underset { ( x , y ) \sim P _ { j } } { \mathbb { E } } \nabla _ { \beta } f ( z ; \beta ) ( y - f ( z ; \beta ) ) = 0 .
+$$
+
+and is consistent for the causal parameters $\beta _ { 0 }$ if unique. $i , j \in \mathcal { E }$ are the indices of any two observed environments in an index set $\mathcal { E }$ . This invariance across environments must hold for causal parameters (under certain conditions) even in the presence of unobserved confounders.
+
+A few remarks are necessary concerning this relationship and its extrapolation properties.
+
+• The first is based on the observation that, up to a constant, each inner product in (5) is the gradient of the squared error with respect to $\beta$ . This reveals that the optimal predictor, in the presence of unobserved confounding, is not one that produces minimum loss but one that produces a non-zero loss gradient equal across environments. Seeking minimum error solutions, even in the population case, produces estimators with necessarily unstable correlations because the variability due to unobserved confounders is not explainable from observed data. Forcing gradients to be zero then forces models to utilize artifacts of the specific data collection process that are not related to the input-output relationship; and, for this reason, will not in general perform outside training data.
+
+• From (5) we may pose a sequence of moment conditions for each pair of available environments. We may then seek solutions $\beta$ that make all of them small simultaneously. Solutions are unique if the set of moments is sufficient to identify $\beta ^ { \star }$ exactly (and given our model assumptions may be interpreted as causal and robust to certain interventions). We revisit our introductory example to show in Appendix A that, in contrast to ERM and Invariant Risk Minimization (IRM) (a related approach proposed in (Arjovsky et al., 2019) we discuss in more detail in later sections), this procedure does recover the underlying causal model correctly in the presence of unobserved confounding.
+
+• In practice however, only a set of solutions may be identified with no performance guarantees for any individual solutions, and no guarantees if assumptions fail to hold. Moreover, even if accessible, causal solutions, robust to certain distribution shifts, may not always be desirable under more general shifts (recall for instance the experiments in the rightmost panel of Figure 1).
+
+# 3 GENERALIZATION FROM A ROBUST OPTIMIZATION PERSPECTIVE
+
+While certain invariances may hold in the presence of unobserved confounding, we have in general no guarantees on performance under more general manipulations. In this section we motivate a relaxation of the ideas presented above by considering bounds on the worst-case performance on data arising from distribution shifts informed by the structure of available environments.
+
+Optimizing for the worst case loss in a set of domains has been shown to ensure accurate prediction on any convex mixture of training environments (Ben-Tal et al., 2009). The space of convex mixtures, however, is restrictive. Systems of variables are in general high-dimensional and new manipulations likely occur at a new vertex not represented as a linear combination of training environments. By extrapolation we desire performance guarantees outside this convex hull. The extension we consider optimizes instead over an affine combination of training losses, similarly to (Krueger et al., 2020).
+
+Let $\begin{array} { r } { \Delta _ { \eta } : = \{ \{ \alpha _ { e } \} _ { e \in \mathcal { E } } : \alpha _ { e } \geq - \eta , \sum _ { e \in \mathcal { E } } \alpha _ { e } = 1 \} } \end{array}$ be a collection of scalars and consider the set of distributions defined by $\begin{array} { r } { \mathcal { P } : = \{ \sum _ { e \in \mathcal { E } } \bar { \alpha } _ { e } P _ { e } : \{ \alpha _ { e } \} \in \Delta _ { \eta } \} } \end{array}$ , all affine combinations of distributions defined by the available environments. $\eta \in \mathbb { R }$ defines the strength of the extrapolation, $\eta = 0$ corresponds to a convex hull of distributions but above that value the space of distributions is richer, going beyond what has been observed: affine combinations amplify the strength of manipulations that generated the observed training environments. The following theorem presents an upperbound to the robust problem (1) with affine combinations of errors.
+
+Theorem 1 Let $\{ P _ { e } \} _ { e \in \mathcal { E } }$ , be a set of available environments. Further let the parameter space of $\beta$ be open and bounded, such that the expected loss function $\mathcal { L }$ as a function of $\beta$ belongs to a Sobolev space. Then, the following inequality holds,
+
+$$
+\begin{array} { r l } & { \underset { \{ \alpha _ { \mathrm { e } } \} \in \Delta _ { \eta } } { \operatorname* { s u p } } \sum _ { e \in \mathcal { E } } \alpha _ { e } \underset { ( x , y ) \sim P _ { e } } { \mathbb { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) \leq \underset { ( x , y ) \sim P _ { e } , e \sim \mathcal { E } } { \mathbb { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) } \\ & { + \left( 1 + n \eta \right) \cdot C \cdot \left\| \underset { e \in \mathcal { E } } { \operatorname* { s u p } } \underset { ( x , y ) \sim P _ { e } } { \mathbb { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) - \underset { ( x , y ) \sim P _ { e } , e \sim \mathcal { E } } { \mathbb { E } } \nabla _ { \beta } \mathcal { L } \left( f \circ \phi ( x ) , y \right) \right\| _ { L _ { 2 } } } \end{array}
+$$
+
+where $| | \cdot | | _ { L _ { 2 } }$ denotes the $L _ { 2 }$ -norm, $C$ depends on the domain of $\beta$ , $n : = | { \mathcal { E } } |$ is the number of available environments and $e \sim \mathcal { E }$ loosely denotes sampling indeces with equal probability from $\mathcal { E }$ .
+
+The proof is given in Appendix C.2. This bound illustrates the trade-off between invariances (of the type explored under unobserved confounding, that appear in the second term of the RHS of the inequality above) and prediction in-sample (the first term). A combination of them upper-bounds a robust optimization problem over affine combinations of training environments, and depending how much we weight each objective (prediction versus invariance) we can expect solutions to be more or less robust. Specifically, for $\eta = - 1 / n$ the objective reduces to empirical risk minimization, but otherwise the upperbound increasingly weights differences in loss derivatives (violations of the invariances of section 2.2), and in the limit $\eta  \infty$ ) can be interpreted to be robust at least to any affine combination of training losses.
+
+Note that the requirement that $\mathbb { F }$ be fixed, or interventions occur on observed variables, is not necessary for generalization guarantees. As long as new data distributions can be represented as affine combinations of training distributions, we can expected performance to be as least as good as that observed for the robust problem in Theorem 1.
+
+# 3.1 PROPOSED OBJECTIVE
+
+Our proposal is to guide the optimization of $\phi$ and $\beta$ towards solutions that minimize the upperbound in Theorem 1, satisfying approximately the moment conditions (5) and simultaneously optimizing for minimum average error. Using Lagrange multipliers we define the general objective,
+
+$$
+\underset { \beta , \phi } { \mathrm { m i n i m i z e } } _ { \left( x , y \right) \sim P _ { e } , e \sim \mathcal { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) + \lambda \cdot \underset { e \sim \mathcal { E } } { \mathrm { V a r } } \left( \big | \big | _ { ( x , y ) \sim P _ { e } } \nabla _ { \beta } \mathcal { L } \left( f \circ \phi ( x ) , y \right) \big | \big | _ { L _ { 2 } } \right)
+$$
+
+where $\lambda \geq 0$ . We call this problem Derivative Invariant Risk Minimization (DIRM). This objective shares similarities with the objective proposed in (Krueger et al., 2020). The authors considered enforcing equality in environment-specific losses, rather than derivatives, as regularization, which can also be related to a robust optimization problem over an affine combination of errors. We have seen in section 2.2 however that equality in losses is not expected to hold in the presence of unobserved confounders (e.g. due to changes in the irreducible error across environments, which may occur also after interventions on target variables).
+
+The $L _ { 2 }$ norm in the regularizer is an integral over the domain of values of $\beta$ and is in general intractable. We approximate this objective in practice with norms on functional evaluations at each step of the optimization. Theorem 1 is used as a guide for optimization, we encourage the values of the regularizer to be zero and thus only indirectly the regularizer function itself. We give more details in Appendix D.1.1.
+
+As is apparent in Theorem 1, performance guarantees on data from a new environment depend on the relationship of new distributions with those observed during training.
+
+Let $f \circ \phi _ { \lambda  \infty }$ minimize $\mathcal { L }$ among all functions that satisfy all pairs of moment conditions defined in (5); that is, a solution to our proposed objective in (6) with $\lambda \to \infty$ . At optimality, it holds that gradients evaluated at this solution are equal across environments. As a consequence of Theorem 1, the loss evaluated at this solution with respect to any affine combination of environments is bounded by the average loss computed in-sample (denoted $L$ , say),
+
+$$
+\sum _ { e \in { \mathcal E } } \alpha _ { e } \underset { ( x , y ) \sim P _ { e } } { \mathbb { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) \leq L , \qquad \mathrm { f o r ~ a n y ~ s e t ~ o f ~ } \alpha _ { e } \in \Delta _ { \eta }
+$$
+
+From the perspective of interventions in the underlying causal mechanism, this can be seen as a form of data-driven predictive stability across a range of distributions whose perturbations occur in the same direction as those observed during training. As a minimal example for intuition, consider a system of three variables $( X , Y , H )$ , with differing interventions on $X$ only. Assume we have access to data sampled under two environments defined by joint distributions $\dot { p } _ { 1 } ( X , Y , H ) : = p ( Y , H | X ) p _ { 1 } ( X )$ and $p _ { 2 } ( X , Y , H ) : = p ( Y , H | X ) p _ { 2 } ( X )$ . Errors are bounded with respect to any (valid) distribution of the form $p ( Y , H | X ) ( \alpha _ { 1 } p _ { 1 } ( X ) + \alpha _ { 2 } p _ { 2 } ( X ) )$ . For instance, a particular shift in the mean or variance of $X$ observed during training can be extrapolated in the extreme to any shift in the mean or variance of new data. With this reasoning, if environment-specific distributions arise from differing interventions on all observed covariates $X$ , then solutions are robust to arbitrary shifts in these variables. If, in addition, solutions are unique we may interpret them as causal, irrespective of the presence or not of unobserved confounders.
+
+The generalization properties, however, go further. Interventions on unobserved variables and also target variables are accommodated for in $f \circ \phi _ { \lambda \to \infty }$ , if observed through shifted distributions in different available environments.
+
+Using our simple example in Figure 1 to verify this intuition empirically, we consider 3 scenarios corresponding to interventions on exogenous variables of $X , H$ and $Y$ . In each, training data from two environments is generated with means in the distribution of the concerned variables set to a value of 0 and 1 respectively, everything else being equal $( \sigma ^ { 2 } : = 1 , H : = E _ { H } \sim \mathcal { N } ( 0 , 1 ) )$ . Performance is evaluated on out-of-sample data generated by increasing the shift in the variable being studied up to a mean of 5. In all cases, we see in Figure 2 that performance is stable to increasing perturbations in the system as long as the heterogeneity in the data allows us to capture the direction of the unseen shift.
+
+![](images/a12ba2ca415076ca6ca3e63792bdd445539bcee6f1843ce6461127b646d1a8e8.jpg)  
+Figure 2: Stability to general shifts.
+
+# 3.3 STABILITY OF CERTAIN OPTIMAL SOLUTIONS
+
+A special case may also be considered when the underlying system of variables and the available environments allow for optimal solutions $f \circ \phi _ { \lambda \to \infty }$ and $f \circ \phi _ { \lambda = 0 }$ to coincide. In this case, the learned representation $\phi ( x )$ results in a predictor $f$ optimal on average and simultaneously with equal gradient in each environment, thus,
+
+$$
+| | _ { ( x , y ) \sim P _ { e } } \nabla _ { \beta } \mathcal { L } \left( f \circ \phi ( x ) , y \right) | | _ { L _ { 2 } } = 0 , \qquad \mathrm { f o r ~ a l l } \ e \in \mathcal { E }
+$$
+
+For this representation $\phi$ , it follows that optimal solutions $f$ learned on any new dataset sampled from an affine combination of training distributions coincides with this special solution. This gives us a sense of reproducibility of learning. In other words, if a specific feature is significant for predictions on the whole range of $\lambda$ with the available data then it will likely be significant on new (related) data.
+
+The above special case where all solutions in our hyperparameter range agree has important parallels with IRM (Arjovsky et al., 2019). The authors proposed a learning objective enforcing representations of data with minimum error on average and across environments, such that at optimum $\mathbb { E } _ { P _ { i } } Y | \phi ^ { \star } ( X ) = \mathbb { E } _ { P _ { j } } Y | \phi ^ { \star } ( X )$ for any pair $( i , j ) \in \mathcal { E }$ . With unobserved confounding, both learning paradigms agree but, with unobserved confounding, minimum error solutions of IRM by design converge to spurious associations (see the discussion after equation (5)) and are not guaranteed to generalize to more general environments. For example, in the presence of additive unobserved confounding $H$ , irrespective of $\phi$ , we may have ${ \mathbb E } _ { P _ { i } } Y | \bar { \phi } ^ { \star } ( X ) = \bar { \phi ^ { \star } } ( X ) + { \mathbb E } _ { P _ { i } } H \neq \phi ^ { \star } ( X ) + { \mathbb E } _ { P _ { j } } H =$ $\mathbb { E } _ { P _ { j } } Y | \phi ^ { \star } ( X )$ if the means of $H$ differ. The sought invariance then does not hold.
+
+# 4 RELATED WORK
+
+Causality. There has been a growing interest in interpreting shifts in distribution to fundamentally arise from interventions in the causal mechanisms of data. Peters et al. (Peters et al., 2016) exploited this link for causal inference: causal relationships by definition being invariant to the observational regime. Invariant solutions, as a result of this connection, may be interpreted also as robust to certain interventions (Meinshausen, 2018), and recent work has explored learning invariances in various problem settings (Arjovsky et al., 2019; Rothenhäusler et al., 2019; Krueger et al., 2020; Gimenez & Zou, 2020). Among those, we note the invariance proposed in (Rothenhäusler et al., 2019), the authors seek to recover causal solutions with unobserved confounding. Generalization properties of these solutions were rarely studied, with one exception being Anchor regression (Rothenhäusler et al., 2018). The authors proposed to interpolate between empirical risk minimization and causal solutions with explicit robustness to certain interventions in a linear model. The present work may be interpreted as a non-linear formulation of this principle with a more general study of generalization.
+
+Domain generalization represent one direction of out-of-sample generalization by explicitly learning representations projecting out superficial environment-specific information. Recent work on domain generalization has included the use data augmentation (Volpi et al., 2018; Shankar et al., 2018), meta-learning to simulate domain shift (Li et al., 2018) and adversarially learning representations that are environment invariant (Ganin et al., 2016), even though explicitly aligning representations has important caveats when label distributions differ, articulated for instance in (Arjovsky et al., 2019).
+
+Distributionally robust optimization explicitly solves a worst-case optimization problem (1). A popular approach is to define $\mathcal { P }$ as a ball around the empirical distribution $\hat { P }$ , for example using $f$ -divergences or Wasserstein balls of a defined radius (Kuhn et al., 2019; Duchi et al., 2016; 2019; Sinha et al., 2017; Wozabal, 2012; Abadeh et al., 2015; Duchi & Namkoong, 2018). These are general and multiple environments are not required, but this also means that sets are defined agnostic to the geometry of plausible shifted distributions, and may therefore lead to solutions, when tractable, that are overly conservative or do not satisfy generalization requirements (Duchi et al., 2019).
+
+# 5 EXPERIMENTS
+
+Data linkages, electronic health records, and bio-repositories, are increasingly being collected to inform medical practice. As a result, also prediction models derived from healthcare data are being put forward as potentially revolutionizing decision-making in hospitals. Recent studies (Cabitza et al., 2017; Venugopalan et al., 2019), however, suggest that their performance may reflect not only their ability to identify disease-specific features, but also their ability to exploit spurious correlations due to unobserved confounding (such as varying data collection practices): a major challenge for the reliability of decision support systems. In this section, we explore this pattern conducting a wide analysis of domain generalization on image, speech and tabular data from the medical domain.
+
+We consider the following baseline algorithms for performance comparisons.
+
+• Empirical Risk Minimization (ERM) that optimizes for minimum loss agnostic of data source.   
+• Domain Robust Optimization (DRO) (Sagawa et al., 2019) that optimizes for minimum loss across the worst convex mixture of training environments.   
+• Domain Adversarial ENural Netowrks (DANN) (Ganin et al., 2016) that use domain adversarial training to facilitate transfer by augmenting the neural network architecture with an additional domain classifier to enforce the distribution of $\phi ( X )$ to be the same across training environments.   
+• Invariant Risk Minimization (IRM) (Arjovsky et al., 2019) that regularizes ERM ensuring representations $\phi ( X )$ be optimal in every observed environment.   
+• Risk Extrapolation (REx) (Krueger et al., 2020) that regularizes for equality in environment losses instead of considering their derivatives.
+
+<table><tr><td rowspan=2 colspan=1></td><td rowspan=1 colspan=2>Pneumonia Prediction</td><td rowspan=1 colspan=2>Parkinson Prediction</td><td rowspan=1 colspan=2>Survival Prediction</td></tr><tr><td rowspan=1 colspan=1>Training</td><td rowspan=1 colspan=1>Testing</td><td rowspan=1 colspan=1>Training</td><td rowspan=1 colspan=1>Testing</td><td rowspan=1 colspan=1>Training</td><td rowspan=1 colspan=1>Testing</td></tr><tr><td rowspan=1 colspan=1>ERM</td><td rowspan=1 colspan=1>91.4(± .7)</td><td rowspan=1 colspan=1>52.4 (± 1)</td><td rowspan=1 colspan=1>95.7 (± .5)</td><td rowspan=1 colspan=1>62.9 (± 1)</td><td rowspan=1 colspan=1>93.2 (± .4)</td><td rowspan=1 colspan=1>75.3 (± .9)</td></tr><tr><td rowspan=1 colspan=1>DRO</td><td rowspan=1 colspan=1>91.2 (± .5)</td><td rowspan=1 colspan=1>53.1 (± .6)</td><td rowspan=1 colspan=1>94.0 (± .3)</td><td rowspan=1 colspan=1>69.8 (± 2)</td><td rowspan=1 colspan=1>90.5 (± .4)</td><td rowspan=1 colspan=1>75.5 (± .8)</td></tr><tr><td rowspan=1 colspan=1>DANN</td><td rowspan=1 colspan=1>92.3 (± 1)</td><td rowspan=1 colspan=1>57.1(± 2)</td><td rowspan=1 colspan=1>91.6(± 2)</td><td rowspan=1 colspan=1>51.4(± 5)</td><td rowspan=1 colspan=1>89.3(± .8)</td><td rowspan=1 colspan=1>73.9 (± .9)</td></tr><tr><td rowspan=1 colspan=1>IRM</td><td rowspan=1 colspan=1>89.5(± 1)</td><td rowspan=1 colspan=1>58.6(± 2)</td><td rowspan=1 colspan=1>93.6(± 1)</td><td rowspan=1 colspan=1>71.4(± 2)</td><td rowspan=1 colspan=1>91.9 (± .6)</td><td rowspan=1 colspan=1>75.7(± .8)</td></tr><tr><td rowspan=1 colspan=1>REx</td><td rowspan=1 colspan=1>87.7 (± 1)</td><td rowspan=1 colspan=1>57.9(± 2)</td><td rowspan=1 colspan=1>92.0 (± 1)</td><td rowspan=1 colspan=1>72.4(± 2)</td><td rowspan=1 colspan=1>91.3 (± .5)</td><td rowspan=1 colspan=1>75.0(± .9)</td></tr><tr><td rowspan=1 colspan=1>DIRM</td><td rowspan=1 colspan=1>84.3(± 1)</td><td rowspan=1 colspan=1>63.7(± 3)</td><td rowspan=1 colspan=1>93.1 (± 2)</td><td rowspan=1 colspan=1>72.8(± 2)</td><td rowspan=1 colspan=1>91.4 (± .6)</td><td rowspan=1 colspan=1>77.9 (± 1)</td></tr></table>
+
+Table 1: Accuracy of predictions in percentages $( \% )$ . Uncertainty intervals are standard deviations. All datasets are approximately balanced, $5 0 \%$ performance is as good as random guessing.
+
+All trained models use the same convolutional or fully-connected architecture, where appropriate. Performance results are given in Table 1. Further experimental details and pseudo-code for DIRM can be found in Appendix D.
+
+# 5.1 DIAGNOSIS OF PNEUMONIA WITH CHEST X-RAY DATA
+
+In this section, we attempt to replicate the study in (Zech et al., 2018). The authors observed a tendency of image models towards exploiting spurious correlations for the diagnosis on pneumonia from patient Chest X-rays that do not reproduce outside of training data. We use publicly available data from the National Institutes of Health (NIH) (Wang et al., 2017) and the Guangzhou Women and Children’s Medical Center (GMC) (Kermany et al., 2018). Differences in distribution are manifest, and can be seen for example in the top edge of mean pneumonia-diagnosed X-rays shown in Figure 3.
+
+In this experiment, we exploit this (spurious) pathology correlation to demonstrate the need for solutions robust to changes in sitespecific features. We construct two training sets, in each case $9 0 \%$ and $8 0 \%$ of pneumonia-diagnosed patients were drawn from the NIH dataset and the remaining $1 0 \%$ and $2 0 \%$ of the pneumoniadiagnosed patients were drawn from the GMC dataset; the reverse logic $( 1 0 \% / 9 0 \%$ split) was followed for the test set.
+
+![](images/0e8235c3bd9a049caf1a4179840a43fac286a5e5e42bd064eb96295dd2a7d491.jpg)  
+Figure 3: Average pneumonia $\mathrm { X }$ -ray.
+
+Our results show that DIRM outperforms, suggesting that the proposed invariances guides solutions better towards robustness even to changes due to unobserved factors.
+
+# 5.2 DIAGNOSIS OF PARKINSON’S DISEASE WITH VOICE RECORDINGS
+
+Parkinson’s disease is a progressive nervous system disorder that affects movement. Symptoms start gradually, sometimes starting with a barely noticeable tremor in a patient’s voice. This section investigates the performance of predictive models for the detection of Parskinson’s disease, trained on voice recordings of vowels, numbers and individual words and tested on vowel recordings of unseen patients.
+
+We used the UCI Parkinson Speech Dataset with given training and testing splits (Sakar et al., 2013). Even though the distributions of features will differ in different types of recordings and patients, we would expect the underlying patterns in speech to reproduce across different samples. However, this is not the case for correlations learned with baseline training paradigms (Table 1). This suggests that spurious correlations due to the specific type of recording (e.g. different vowels or numbers), or even chance associations emphasized due to low sample sizes (120 examples), may be responsible for poor generalization performance. Our results show that correcting for spurious differences between recording types (DIRM, IRM, REx) can improve performance substantially.
+
+# 5.3 SURVIVAL PREDICTION WITH ELECTRONIC HEALTH RECORDS
+
+This section investigates whether predictive models transfer across data from different medical studies (the MAGGIC, 2012, studies), all containing patients that experienced heart failure. The problem is to predict survival within 3 years of experiencing heart failure from a total of 33 demographic variables. We introduce a twist however, explicitly introducing unobserved confounding by omitting certain predictive variables. The objective is to test performance on new studies with shifted distributions, while knowing that these occur predominantly due to variability in unobserved variables.
+
+Confounded data is constructed by omitting a patient’s age from the data, found in a preliminary correlation analysis to be associated with the outcome as well as other significant predictors such as blood pressure and body mass index. This example is constructed to be able to control for how unobserved variables shift but note that we can expect similar phenomena in many other scenarios, where for instance a prediction model is taken to patients in a different hospital or country with fundamental shifts in the distribution of very relevant variables (e.g. socio-economic status, ethnicity, diet, etc.) even though this information is not reported in the data. Performance is tested on all studies of over 500 patients with balanced death rates, each having slightly different age distributions. (We give more details in Appendix D). We found DIRM, robust to changes in unobserved variables, to outperform all other methods.
+
+Influential variables that reproduce across datasets. In the following, we tackle the problem of reproducibility of learned influential features across different experiments. Reproducing conclusions of influential features in different studies with potential shifts in the distribution is an important challenge, especially in healthcare where heterogeneity between patient populations is high. We showed in section 3.3 that in the event that the optimal predictor is invariant as a function $\lambda \in [ 0 , \infty )$ , optimal predictors estimated in every new dataset in the span of observed distributions, should be stable.
+
+We consider here a form of diluted stability for feature selection. For a single layer network, we consider significant those covariates with estimated parameters bounded away from zero in all solutions in the range $\lambda \in [ 0 , 1 ]$ . Comparisons are made with ERM (conventional logistic regression), both methods trained separately on 100 random pairs of studies. Figure 4 shows how many features (in the top 10 of predictive features) from each model intersect across pairs of studies. In constrast to ERM, our objective recovers significant features much more consistently serves to demonstrate the improved reproducibility we can expect from DIRM.
+
+![](images/7b0f769547225188b07ae2df595fd2fac8e1698fe3bee5b21b2befab4388d6d5.jpg)  
+Figure 4: Reproducible features.
+
+# 6 CONCLUSIONS
+
+We have studied the problem of out-of-sample generalization from a new perspective, grounded in the underlying causal mechanism generating new data that may arise from shifts in observed, unobserved or target variables. Our proposal is a new objective that is provably robust to certain shifts in distribution, and is informed by certain statistical invariances in the presence of unobserved confounders. Our experiments show that we may expect better generalization performance and also better reproducibility of influential features in problems of variable selection.
+
+A limitation of our approach is that robustness guarantees crucially depend on the (unobserved) properties of available data. Using the proposed approach, Derivative Invariant Risk Minimization for prediction generally does not guarantee protection against unsuspected events. More specifically, we can not expect robust prediction when the heterogeneity in test data sets is different from the restricted set of shift interventions that have been observed on the training data sets. For example, in Theorem 1, the supremum contains distributions that lie in the affine combination of training environments, as opposed to arbitrary distributions.
+
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+
+# Appendix
+
+This Appendix contains 4 sections.
+
+• Section A presents additional experiments designed to demonstrate the causal interpretation one may give to DIRM if all conditions for causality are satisfied (i.e. are available interventions on all observed variables). We also include a sensitivity analysis to show the impact on performance of changing the regularization parameter $\lambda$ .   
+• Section B shows how measurement error may be interpreted as an instance of unosebrved confounding.   
+• Section C provides proofs for the statements made in the main body of this paper.   
+• Section D gives additional experimental details, including on the implementation of DIRM and on the datasets used.
+
+# A ADDITIONAL EXPERIMENTS
+
+So far, we have considered predictive performance under different data distributions with selected hyper-parameter configurations of all algorithms to illustrate heterogeneous behaviour of algorithms trained with different learning principles in the presence of unobserved confounders. In this section we revisit our introductory example to investigate in more details learned prediction rules and any sensitivities of interest, especially to hyper-parameter configurations.
+
+We will use the same data generating mechanism presented in the introductory example in Figure 1 of the main body of this paper. Recall that we assume access to observations of variables $( X _ { 1 } , X _ { 2 } , Y )$ in two training datasets, each dataset sampled with differing interventions on $( X _ { 1 } , X _ { 2 } )$ (in this case differing variances $\sigma ^ { 2 } = 1$ and $\sigma ^ { 2 } = 2$ ) from the following structural model,
+
+$$
+X _ { 2 } : = - H + E _ { X _ { 2 } } , \quad Y : = X _ { 2 } + 3 H + E _ { Y } , \quad X _ { 1 } : = Y + X _ { 2 } + E _ { X _ { 1 } } \quad H : = E _ { H }
+$$
+
+where $E _ { X _ { 1 } } , E _ { X _ { 2 } } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } )$ , $E _ { Y } \sim \mathcal { N } ( 0 , 1 )$ , $E _ { H } \sim \mathcal { N } ( 0 , 1 )$ are exogenous variables. $H$ is an unobserved confounder, not observed during training but that influences the observed association between $X _ { 2 }$ and $Y$ .
+
+# A.1 RECOVERY OF CAUSAL COEFFICIENTS
+
+In this section, given the above two training datasets, we inspect the weights learned in a simple one layer feed-forward neural network to determine exactly whether unobserved confounding induces a given learning paradigm to exploit spurious correlations and to what extent.
+
+By way of preface, we have mentioned that causal, in contrast with spurious, solutions to a prediction problem may be defined as the argument solving,
+
+$$
+{ \underset { f } { \operatorname { m i n i m i z e } } } \ \operatorname* { s u p } _ { P \in { \mathcal { P } } } \ { \mathbb { E } } _ { ( x , y ) \sim P } [ { \mathcal { L } } ( f ( x ) , y ) ]
+$$
+
+for $\mathcal { P }$ defined as any distribution arising from arbitrary interventions on observed covariates $x$ leading to shifts in their distribution $P _ { x }$ (see sections 3.2 and 3.3 in (Meinshausen, 2018) for a detailed discussion of this result). This objective is a special case of the proposed optimization problem (6), specifically it is an affine combination (with $\lambda \to \infty$ ) of distributions with different shifts in $P _ { x }$ in all observed variables $x$ .
+
+We demonstrate this fact empirically in Table 2. In principle, causal solutions are recoverable with the proposed approach because we do observe during training environments with shifts in $p ( X _ { 1 } , X _ { 2 } )$ , irrespective of the presence or not of unobserved confounders. We see that this holds approximately for the proposed objective with estimated coefficients $( 0 . 0 1 , 0 . 9 5 )$ for $( X _ { 1 } , X _ { 2 } )$ close to the true causal coefficients $( 0 , 1 )$ . In contrast, ERM returns biased coefficients and so does IRM.
+
+This empirical observation is important because it highlights the fact that enforcing minimum gradients on average (ERM) or simultaneously across environments (the regularization proposed by IRM) is not appropriate to recover causal coefficients in the presence of unobserved confounders.
+
+![](images/de375096f1641d9def7cb9982064dd8649cab2aecf44a8b0239e90b15b3dec65.jpg)  
+Figure 5: Sensitivity of solutions to hyperparameter $\lambda$ . Ordinary Least Squares (OLS) and the causal solution, with coefficients $( 0 , 1 )$ for $( X _ { 1 } , X _ { 2 } )$ are two extremes $\lambda = 0$ and $\lambda \to \infty$ respectively) of the spectrum of solutions that can be attained with the proposed approach. Positive values of $\lambda$ interpolate in some sense between OLS and causal solutions in this case. Here DRO corresponds to DIRM with $\lambda = 2$ , approximately.
+
+If however, no unobserved confounders exist in the system being modelled ( $H : = 0$ in the data generating mechanism) our objective and IRM are equivalent in the limit, and estimated parameters coincide with the causal solution approximately. This experiment is given in Table 3.
+
+Table 2: Bias in estimation with unobserved confounders.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Truth</td><td rowspan=1 colspan=1>ERM</td><td rowspan=1 colspan=1>IRM(λ→∞)</td><td rowspan=1 colspan=1>DIRM (λ→0)</td></tr><tr><td rowspan=1 colspan=1>Estimated parameters</td><td rowspan=1 colspan=1>[0,1]</td><td rowspan=1 colspan=1>[0.91, -1.02]</td><td rowspan=1 colspan=1>[0.75, -0.76]</td><td rowspan=1 colspan=1>[0.01, 0.95]</td></tr></table>
+
+Table 3: Bias in estimation without unobserved confounders.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Truth</td><td rowspan=1 colspan=1>ERM</td><td rowspan=1 colspan=1>IRM(λ→)</td><td rowspan=1 colspan=1>DIRM(入→ ∞)</td></tr><tr><td rowspan=1 colspan=1>Estimated parameters</td><td rowspan=1 colspan=1>[0, 1]</td><td rowspan=1 colspan=1>[0.5, -0.6]</td><td rowspan=1 colspan=1>[0.01, 0.98]</td><td rowspan=1 colspan=1>[0.02, 0.96]</td></tr></table>
+
+# A.2 SENSITIVITY TO HYPER-PARAMETERS
+
+The robustness guarantees of any particular solution depends on the extent of the extrapolation desired (as a function of $\lambda$ ). For larger values of this parameter we can expect solutions to be robust in a larger set of distributions, spanning empirical risk minimization for $\lambda = 0$ , to convex combinations, to training environments to arbitrary affine combinations of training environments for increasing $\lambda$ .
+
+In this section, we analysed performance in test data with the exact same data generating mechanism as considered in the introduction of the main body of this paper as a function of $\lambda$ . Figure 5 gives our performance results that empirically verifies that the proposed approach interpolates between empirical risk minimization and causality in this case. We can see that for $\lambda$ approaching zero solutions converge to ERM, for $\lambda = 2$ the solutions was equivalent to DRO, and for increasing $\lambda$ the solutions approximate the causal one in the limit.
+
+# B OTHER EXAMPLES OF UNOBSERVED CONFOUNDING
+
+Measurement error. The data generating processes described in the main body of this paper for instance, as well as most of machine learning, assume that all nuisance variability enters the causal mechanisms of the data; that is, observed data reflects only causal drivers. If this is not the case, for example because of independent measurement noise observed in data but that does not propagate to across causal children, regression is known to be inconsistent in general (Carroll et al., 2006) and its bias is analogous to a form of unobserved confounding.
+
+Consider a simple model for illustration. Suppose $( X , Y )$ are observed subject to measurement noise, $X ^ { \star } = X + E _ { x }$ and $Y ^ { \star } = Y + E _ { y }$ , which are not causally related to one another but rather $Y = \beta X + E$ . Let $E _ { x } = \beta _ { x } H$ and $E _ { y } = \beta _ { y } H$ be the structure of measurement error independent of $X$ and $Y$ . Then substituting our observed data $( X ^ { \star } , Y ^ { \star } )$ into the underlying $( X , Y )$ relationship the observed model is,
+
+$$
+Y ^ { \star } = \beta X ^ { \star } + ( \beta _ { y } - \beta _ { x } \beta ) H + E , \quad X ^ { \star } = \beta _ { x } H + X
+$$
+
+A special case of regression with unobserved confounders $H$ .
+
+# C TECHNICAL RESULTS
+
+This section provides a more complete discussion of the assumptions and justification statements relating to causality in section 2.2, and the proof of Theorem 1.
+
+# C.1 INVARIANCES IN THE PRESENCE OF UNOBSERVED CONFOUNDING
+
+In section 2.2 we justified exploiting a certain invariance of causal coefficients in the inner product of functions of the data $X$ and residuals $E$ , to occur even in the presence of unobserved confounders as long as interventions that define different environments do not involve unobserved confounders $H$ .
+
+Here we show this invariance to hold in the special case of an additive model. The general data generation mechanism is as follows. Data sources, or different environments, emerge from manipulations in exogenous $E _ { X }$ , related to $X$ only, in an underlying additive model $\mathbb { F }$ with also additive functions $f _ { 1 } , f _ { 2 } , f _ { 3 } , f _ { 4 }$ ,
+
+$$
+Y : = f _ { 1 } ( X ) + f _ { 2 } ( H ) + E _ { Y } , \qquad X : = f _ { 3 } ( X ) + f _ { 4 } ( H ) + E _ { X } , \qquad H : = E _ { H }
+$$
+
+Exogenous variables $( E _ { X } , E _ { Y } , E _ { H } )$ may have arbitrary distributions but only $E _ { X }$ or $E _ { Y }$ vary across environments. Then it holds that,
+
+$$
+\begin{array} { c } { { X = ( I - f _ { 3 } ) ^ { - 1 } f _ { 4 } ( H ) + ( I - f _ { 3 } ) ^ { - 1 } E _ { X } } } \\ { { { } } } \\ { { = ( I - f _ { 3 } ) ^ { - 1 } f _ { 4 } ( E _ { H } ) + ( I - f _ { 3 } ) ^ { - 1 } E _ { X } } } \end{array}
+$$
+
+and that,
+
+$$
+\nabla _ { \beta } f _ { 1 } ( X ) ( Y - f _ { 1 } ( X ) ) = \left( \nabla _ { \beta } f _ { 1 } ( I - f _ { 3 } ) ^ { - 1 } f _ { 4 } ( E _ { H } ) + \nabla _ { \beta } f _ { 1 } ( I - f _ { 3 } ) ^ { - 1 } E _ { X } \right) \cdot ( f _ { 2 } ( E _ { H } ) + E _ { Y } ) .
+$$
+
+which is a product of functions involving $E _ { H }$ in one term, $E _ { H }$ and $E _ { Y }$ in another term, $E _ { X }$ and $E _ { H }$ in another term, and $E _ { X }$ and $E _ { Y }$ in the last term. Since $( E _ { X } , E _ { Y } , E _ { H } )$ are mutually independent taking expectations of product of functions involving $E _ { X }$ and $E _ { H } , E _ { X }$ and $E _ { H }$ , and, $E _ { X }$ and $E _ { Y }$ equals 0 assuming $f _ { i } ( \bar { E } _ { j } ) = 0$ for $i = 1 , \dots , 4$ and $j \in \{ X , Y , H \}$ .
+
+So concluding, the expectation of the inner product $\nabla _ { \beta } f _ { 1 } ( X ) ( Y - f _ { 1 } ( X ) )$ does not depend on $E _ { X }$ nor $E _ { Y }$ and is thus stable across environments that have changing distributions for $E _ { X }$ or $E _ { Y }$ . Now note that other functions than $f _ { 1 }$ may have this property as well, i.e. predictors that satisfy this invariance are not necessarily unique and will depend on the differences between available environments. If however, only one predictor exist that satisfies this invariance we may say that this predictor is causal. We summarize this claim in the following statement.
+
+Proposition 1 Let $Y$ and $X$ be related by a non-linear additive model with unobserved confounding as in $( l I )$ . Then,
+
+$$
+\mathbb { E } _ { P _ { i } } \nabla _ { \beta } f _ { 1 } ( X ) ( Y - f _ { 1 } ( X ) ) = \mathbb { E } _ { P _ { j } } \nabla _ { \beta } f _ { 1 } ( X ) ( Y - f _ { 1 } ( X ) )
+$$
+
+under the assumption that distributions on $( X , Y ) P _ { i }$ and $P _ { j }$ are given by a data generating mechanism $( l I )$ subject to interventions on $E _ { X }$ or $E _ { Y }$ only. Moreover, a function $f$ satisfying the above equality, $i f$ unique is equal to $f _ { 1 }$ .
+
+# C.2 PROOF OF THEOREM 1
+
+We restate the Theorem for convenience.
+
+Theorem 1 Let $\{ P _ { e } \} _ { e \in \mathcal { E } }$ , be a set of available environments. Further let the parameter space of $\beta$ be open and bounded, such that the expected loss function $\mathcal { L }$ as a function of $\beta$ belongs to a Sobolev space. Then, the following inequality holds,
+
+$$
+\begin{array} { r l } & { \underset { \alpha _ { c } \in \Delta _ { \eta } } { \operatorname* { s u p } } \sum _ { e \in \mathcal { E } } \alpha _ { e } \underset { ( x , y ) \sim P _ { c } } { \mathbb { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) \leq \underset { ( x , y ) \sim P _ { c } , e \sim \mathcal { E } } { \mathbb { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) } \\ & { + \left( 1 + n \eta \right) \cdot C \cdot \left\| \underset { e \in \mathcal { E } } { \operatorname* { s u p } } \underset { ( x , y ) \sim P _ { c } } { \mathbb { E } } \nabla _ { \beta } \mathcal { L } \left( f \circ \phi ( x ) , y \right) - \underset { ( x , y ) \sim P _ { c } , e \sim \mathcal { E } } { \mathbb { E } } \nabla _ { \beta } \mathcal { L } \left( f \circ \phi ( x ) , y \right) \right\| _ { L _ { 2 } } } \end{array}
+$$
+
+where $C$ depends on the domain of $\beta$ , $n : = | \mathcal { E } |$ is the number of available environments and $e \sim \mathcal { E }$ loosely denotes sampling indeces with equal probability from $\mathcal { E }$ .
+
+Proof. Let $\Omega$ denote the parameter space of $\beta$ . The following derivation shows the claim,
+
+$$
+\begin{array} { r l } & { \quad \underset { s \in \Phi } { \operatorname* { s u p } } \quad \underset { \epsilon \in \Phi } { \sum } \alpha _ { s \in \Phi } \underset { \epsilon \in \Psi _ { s } } { \mathbb { E } } \mathscr { L } \big ( f \circ \phi ( z ) , y \big ) } \\ & { = \big ( 1 + m \eta \big ) \cdot \underset { s \in \Phi } { \sum } \big ( \underset { s \in \Phi } { \operatorname* { s u p } } - \underset { \epsilon \in \Psi _ { s } } { \mathbb { E } } \mathscr { L } \big ( f + \varphi ( z ) , y \big ) - \eta \sum _ { s \in \Phi } E _ { \varphi } \mathscr { L } \big ( f \circ \phi ( z ) , y \big ) } \\ & { = \big ( 1 + m \eta \big ) \cdot \underset { s \in \Psi _ { s } } { \sum } \epsilon \big ( f \circ \phi ( x ) , y \big ) + \big ( 1 + m \eta \big ) \cdot \underset { s \in \Phi } { \sum } \epsilon \big ( f \circ \phi ( z ) , y \big ) } \\ & { = \big ( \epsilon \eta + 1 / m \big ) \sum _ { s \in \Phi } E \big ( f \circ \phi ( x ) , y \big ) + \big ( 1 + m \eta \big ) \cdot \underset { s \in \Phi } { \sum } \epsilon \big ( g \circ \phi ( z ) , y \big ) } \\ & { \quad - ( \eta + 1 / m \big ) \sum _ { s \in \Phi } \epsilon \big ( f \circ \phi ( z ) , \eta \big ) } \\ & { = \underset { \epsilon \in \Phi } { \operatorname* { s u p } } \underset { s \in \Psi _ { s } } { \sum } \epsilon \mathscr { L } \big ( f \circ \phi ( x ) , y \big ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \times \frac { 1 } { \delta } \int _ { - \Phi } \epsilon \dot { \rho } ( f \circ \hat { \Phi } ( z ) , y ) } \\ &  = \big ( 1 + m \eta \big ) \cdot \underset { s \in \Psi _ { s } }   \end{array}
+$$
+
+where the inequality is given by the property that the evaluation functional is a bounded linear operator in certain Sobolev spaces $\mathcal { W }$ , for example with $\Omega = \mathbb { R } ^ { d }$ and $L _ { 2 }$ norm. In particular this means that $| f ( \beta ) | \leq M | | f | | _ { L _ { 2 } }$ for all $f \in \mathcal W$ . It follows then also that the above is,
+
+$$
+\begin{array} { r l } & { \leq _ { ( x , y ) \sim P _ { e } , e \sim \mathcal { E } } \mathcal { L } \left( f \circ \phi ( x ) , y \right) + } \\ & { ( 1 + n \eta ) \cdot P \cdot M \cdot \bigg \vert \bigg \vert \underset { e \in \mathcal { E } } { \operatorname* { s u p } } \underset { ( x , y ) \sim P _ { e } } { \mathbb { E } } \nabla _ { \beta } \mathcal { L } \left( f \circ \phi ( x ) , y \right) - \underset { ( x , y ) \sim P _ { e } , e \sim \mathcal { E } } { \mathbb { E } } \nabla _ { \beta } \mathcal { L } \left( f \circ \phi ( x ) , y \right) \bigg \vert \bigg \vert _ { L _ { 2 } } } \end{array}
+$$
+
+by Poincaré’s inequality for Sobolev functions defined on an open, bounded parameter space, see e.g. (Leoni, 2017). The assumption we make here for this last inequality to hold is that the region where the difference in loss functions is near zero is large enough such that the integral of the gradient is also large enough to control the integral of the function. This holds however for functions defined on many "reasonable" parameter spaces (Lipschitz suffices).
+
+# D EXPERIMENTAL DETAILS
+
+This section gives implementation details of DIRM, additional experiments to test sensitivities relating to optimization choices, and a complete description of the data and experiments performed in the main body of this paper.
+
+# D.1 IMPLEMENTATION DETAILS
+
+The regularizer in DIRM’s objective in equation (6) controls the regularity or variation of the prediction function and encourages to learn a representation $\phi$ that results in a prediction function with similar derivatives in all training domains. The $L _ { 2 }$ norm integrates out the influence of $\beta$ in the regularizer and thus most of the optimization involves $\phi$ , though $\beta$ still plays a role in the first term of the objective.
+
+In all our experiments, $f : \mathbb { R } ^ { h }  \mathbb { R }$ as well as $\phi : \mathbb { R } ^ { d }  \mathbb { R } ^ { h }$ are implemented as fully connected neural networks ( $\cdot \phi$ with optional hidden layers). $\beta$ can thus be interpreted as the weights and biases of $f$ . The $L _ { 2 }$ norm must be approximated in practice, which we do by evaluating the vector norm of the derivative of $f$ with respect to $\beta$ on a batch of training examples of each environment. The variance on the computed norms between environments is a proxy for the maximum deviation between environments with a smoother gradient vector field. Each step of the optimization then alternates between an update on $\phi$ and update on $f$ , as detailed in the algorithm below.
+
+DIRM is sensitive to initialization and to the choice of hyperparameters – specifically its optimization schedule. In our experiments, we found best performance by increasing the relative weight of the penalty term $\lambda$ after a fixed number of iterations (and similar implementations are used for IRM and REx that suffer from similar challenges). This we believe could be a significant limitation for its use in practice since this choice must be made a priori. We investigated the sensitivity of DIRM to this optimization schedule in Table 4 that shows test accuracy as a function of the iteration at which penalty term weight $\lambda$ is increased.
+
+<table><tr><td></td><td>2 epochs</td><td>4 epochs</td><td>6epochs</td><td>8 epochs</td><td>10 epochs</td><td>12 epochs</td></tr><tr><td>IRM</td><td>56.4(± 6)</td><td>58.1(± 3)</td><td>59.2(± 3)</td><td>59.1(± 2)</td><td>58.1(± 3)</td><td>57.8 (± 1)</td></tr><tr><td>REx</td><td>55.3 (± 8)</td><td>57.9 (± 4)</td><td>60.6 (± 3)</td><td>60.5 (± 2)</td><td>57.7 (± 3)</td><td>54.7 (± 2)</td></tr><tr><td>DIRM</td><td>54.1(± 7)</td><td>61.7 (± 4)</td><td>63.8(±3)</td><td>63.5 (± 3)</td><td>62.6(± 2)</td><td>58.2 (± 1)</td></tr></table>
+
+Table 4: Test set performance (accuracy in $\%$ ) on $\mathrm { X }$ -ray data as a function of the number of epochs used to increase penalty $\lambda$ .
+
+Choosing this number accurately is important for generalization performance. If $\lambda$ is increased too early, different initialization values (and the complex loss landscape) lead to different solutions with unreliable performance and a large variance. This happens for all methods. An initial number of iterations minimizing loss in-sample improves estimates for all methods which then converge to solutions that exhibit lower variance.
+
+Input: datasets $\mathcal { D } _ { 1 } , \ldots , \mathcal { D } _ { E }$ in $E$ different environments, parameter $\lambda$ , batch size K
+
+Initialize: neural network model parameters $\phi , \beta$   
+while convergence criteria not satisfied do for $e = 1 , \ldots , E$ do Estimate loss $\mathcal { L } _ { e } ( \phi , \beta )$ empirically using a batch of $K$ examples from $\mathcal { D } _ { e }$ . Estimate derivatives $\nabla _ { \beta } \mathcal { L } _ { e } ( \phi , \beta )$ empirically using a batch of $K$ examples from $\mathcal { D } _ { e }$ .
+
+# end for
+
+Update $\beta$ by stochastic gradient descent with,
+
+$$
+\nabla _ { \beta } \left( \frac { 1 } { E } \sum _ { e = 1 } ^ { E } \mathcal { L } _ { e } ( \phi , \beta ) \right)
+$$
+
+Update $\phi$ by stochastic gradient descent with,
+
+$$
+\nabla _ { \phi } \left( \frac { 1 } { E } \sum _ { e = 1 } ^ { E } \mathcal { L } _ { e } ( \phi , \beta ) + \lambda \cdot \mathrm { V a r } ( | | \nabla _ { \beta } \mathcal { L } _ { 1 } ( \phi , \beta ) | | _ { 2 } ^ { 2 } , \dots , | | \nabla _ { \beta } \mathcal { L } _ { E } ( \phi , \beta ) | | _ { 2 } ^ { 2 } ) \right)
+$$
+
+end while
+
+# D.1.1 APPROXIMATION OF THE $L _ { 2 }$ NORM IN PRACTICE
+
+The bound given in Theorem 1 quantifies the discrepancy between function derivatives using the $L _ { 2 }$ norm, defined as an intregal over possible parameter values $\beta$ . For neural networks, computation of the $L _ { 2 }$ norm is largely intractable and specifically, for networks of depth greater or equal to 4, it is an NP-hard problem (see Proposition 1 in (Triki et al., 2017)). Some approximation is thus unavoidable. One option is to recognise the $L _ { 2 }$ norm as an expectation over functional evaluations, $| | f | | _ { L _ { 2 } } = \mathbb { E } _ { x \sim \mathcal { U } ( \Theta ) } \left[ | | f ( x ) | | _ { 2 } ^ { 2 } \right] ^ { 1 / 2 }$ for a continuous function $f$ taking values $x$ sampled uniformly from its domain $\Theta$ . Empirical means are tractable yet they induce a much higher computational burden as these must be computed in every step of the optimization. Our approach is to take this approximation to its limit, making a single function evaluation at each step of the optimization using the current estimate $\beta$ , as written in Algorithm 1.
+
+This approximation loosens the connection between the bound given in Theorem 1 and the proposed algorithm. It remains justified however from a conceptual perspective as the objective of controlling an $L _ { 2 }$ type of norm is to encourage the regularizer function towards 0, and thus the values of the regularizer (which we do explicitly). For empirical comparisons with the empirical mean approach, we implemented empirical means using all combinations of parameter values chosen from a grid of 5 parameter values around the current estimate $\beta$ , $\{ 0 . 2 5 \beta , 0 . 5 \beta , \beta , 2 \beta , 4 \beta \}$ . Table 5 shows similar performance across the real data experiments considered in the main body of this paper. A single evaluation is in practice enough to monitor invariance of representations to environment-specific loss derivatives.
+
+Table 5: Test set performance (accuracy in $\%$ ) on real datasets for two different regularizer approximations to the $L _ { 2 }$ norm.   
+
+<table><tr><td></td><td>Pneumonia Prediction</td><td>Parkinson Prediction</td><td>Survival Prediction</td></tr><tr><td>DIRM-means</td><td>63.5 (± 3)</td><td>73.0 (± 1.5)</td><td>78.0 (± .9)</td></tr><tr><td>DIRM-single</td><td>63.7 (± 3)</td><td>72.8 (± 2)</td><td>77.9 (± 1)</td></tr></table>
+
+# D.2 DATA DETAILS
+
+# D.2.1 X-RAY DATA
+
+We create training environments with different proportions of $X$ -rays from our two hospital sources to induce a correlation between the hospital (and its specific data collection procedure) and the pneumonia label. The objective is to encourage learning principles to exploit a spurious correlation, data collection mechanisms should not be related to the probability of being diagnosed with pneumonia. The reason for creating two training data sets with slightly different spurious correlation patterns is to nevertheless leave a statistical footprint in the distributions to disentangle stable (likely causal) and unstable (likely spurious). In each of the training and testing datasets we ensured positive and negative labels remained balanced. The training datasets contained 2002 samples each and the testing dataset contained 1144 samples.
+
+All learning paradigms trained a convolutional neural network, 2 layers deep, with each layer consisting of a convolution (kernel size of 3 and a stride of 1). All predictions were made off of the deepest layer of the network. The number of input channels was 64, doubled for each subsequent layer, and dropout was applied after each layer. We optimize the binary cross-entropy loss using Adam (learning rate 0.001) without further regularization on parameters and use Xavier initialization. While learning with IRM and the proposed approach, the respective penalty $\lambda = 1$ is added to the loss after 5 epochs of learning with $\lambda = 0$ . Experiments are run for a maximum of 20 epochs with early stopping based on validation performance. All results are averaged over 10 trials with different random splits of the data, and the reported uncertainty intervals are standard deviations of these 10 performance results.
+
+# D.2.2 PARKINSON’S DISEASE SPEECH DATA
+
+The data includes a total of 26 features recorded on each sample of speech and set training and testing splits which we use in our experiments. For each patient 26 different voice samples including sustained vowels, numbers, words and short sentences where recorded, which we considered to be different but related data sources. We created three training environments by concatenating features from three number recordings, concatenating features from three word recording and concatenating features from three sentences; for a total of 120 samples in each of the three training environment. The available testing split contained 168 recordings of vowels, which we expect to differ from training environments because these are different patients and do not contain numbers or words. Positive and negative samples were balanced in both training and testing environments.
+
+On this data, for all learning paradigms we train a multi-layer perceptron with two hidden layers of size 64 with tanh activations and dropout ${ \mathrm { \Delta } p = 0 . 5 } \mathrm { \Delta }$ ) after each layer. As in the image experiments, we optimize the binary cross-entropy loss using Adam (learning rate 0.001), $L _ { 2 }$ regularization on parameters and use Xavier initialization. While learning with IRM and the proposed approach, the respective penalty is added to the loss $\lambda = 1$ after 200 epochs of learning with $\lambda = 0$ to ensure stable optimization. Experiments are run for a maximum of 1000 epochs with early stopping based on the validation performance. All results are averaged over 10 trials with different random seeds of our algorithm. This is to give a sense of algorithm stability rather than performance stability.
+
+# D.2.3 MAGGIC ELECTRONIC HEALTH RECORDS
+
+MAGGIC stands for Meta-Analysis Global Group in Chronic Heart Failure. The MAGGIC metaanalysis includes individual data on 39, 372 patients with heart failure (both reduced and preserved left-ventricular ejection fraction), from 30 cohort studies, six of which were clinical trials. $4 0 . 2 \%$ of patients died during a median follow-up of 2.5 years. For our purposes, we removed patients that were censored or lost to follow-up to ensure well-defined outcomes after 3 years after being discharged from their respective hospitals. A total of 33 variables describe each patient including demographic variables: age, gender, race, etc; biomarkers: blood pressure, haemoglobin levels, smoking status, ejection fraction, etc; and details of their medical history: diabetes, stroke, angina, etc.
+
+To curate our training and testing datasets, we proceeded as follows. On all patients follow-up over 3 years, we estimated feature influence of survival status after three years. A number of variables were significantly associated with survival out of which we chose Age, also found correlated with BMI and a number of medical history features, as a confounder for the effect of these variables on survival. We used three criteria to select studies: having more than 500 patients enrolled and balanced death rates (circa $5 0 \%$ ). 5 studies fitted these constraints: ’DIAMO’, ’ECHOS’, ’HOLA’, ’Richa’, ’Tribo’. Each was chosen in turn as a target environment with models trained on the other 4 training environments.
+
+Feature reproducibility experiments. A natural objective for the consistency of health care and such that we may reproduce the experiments and their results in different scenarios is to find relevant features that are not specific to an individual medical study, but can also be found (replicated) on other studies with different patients. Heterogeneous patients and studies, along with different national guidelines and standards of care make this challenging. In our experiments we made comparisons of reproducibility in parameter estimates for models trained using Empirical Risk Minimization (ERM) and DIRM. We chose networks with a single layer with logistic activation and focused on the estimation of parameter to understand the variability in training among different data sources. Naturally, feature importance measured by parameter magnitudes makes sense only after normalization of the covariates to the same (empirical) variance (equal to 1) in each study separately. After this preprocessing step, for both ERM and the proposed approach we trained separate networks on 100 random pairs of studies (each pair concatenated for ERM) and returned the top 10 significant features (by the magnitude of parameters). Over all sets of significant parameters we then identified how many intersected across a fixed number of the 100 runs.
+
+The same architecture and hyperparameters as in Parkinson’s disease speech data experiments was used for MAGGIC data except that we increase the maximum training epochs to 5000.

parse/train/ahAUv8TI2Mz/ahAUv8TI2Mz.md

@@ -0,0 +1,280 @@
+# ADAPTIVE AND GENERATIVE ZERO-SHOT LEARNING
+
+Yu-Ying Chou1,3, Hsuan-Tien Lin3 & Tyng-Luh Liu1,2
+
+1Institute of Information Science, Academia Sinica, Taiwan 2Taiwan AI Labs, Taiwan 3Department of Computer Science, National Taiwan University, Taiwan {d07922014,htlin}@csie.ntu.edu.tw,liutyng@iis.sinica.edu.tw
+
+# ABSTRACT
+
+We address the problem of generalized zero-shot learning (GZSL) where the task is to predict the class label of a target image whether its label belongs to the seen or unseen category. Similar to ZSL, the learning setting assumes that all class-level semantic features are given, while only the images of seen classes are available for training. By exploring the correlation between image features and the corresponding semantic features, the main idea of the proposed approach is to enrich the semantic-to-visual (S2V) embeddings via a seamless fusion of adaptive and generative learning. To this end, we extend the semantic features of each class by supplementing image-adaptive attention so that the learned S2V embedding can account for not only inter-class but also intra-class variations. In addition, to break the limit of training with images only from seen classes, we design a generative scheme to simultaneously generate virtual class labels and their visual features by sampling and interpolating over seen counterparts. In inference, a testing image will give rise to two different S2V embeddings, seen and virtual. The former is used to decide whether the underlying label is of the unseen category or otherwise a specific seen class; the latter is to predict an unseen class label. To demonstrate the effectiveness of our method, we report state-of-the-art results on four standard GZSL datasets, including an ablation study of the proposed modules.
+
+# 1 INTRODUCTION
+
+Different from conventional learning tasks, zero-shot learning (ZSL) by Lampert et al. (2009); Palatucci et al. (2009); Akata et al. (2013) explores the extreme case of performing inference only over samples of unseen classes. To make the scenario more realistic, generalized zero-shot learning (GZSL) (Chao et al., 2016; Xian et al., 2017) is subsequently proposed so that inference can concern samples of both seen and unseen classes. Nevertheless. the learning setting in ZSL/GZSL is essentially the same where sample classes are divided into two categories, seen and unseen, but only those samples of seen classes are accessible to training. In addition, each of all the classes under consideration is characterized by semantic features such as attributes (Xian et al., 2018b) or text descriptions (Zhu et al., 2018) to specify and relate seen and unseen classes.
+
+The lack of training samples from unseen classes has prompted generative approaches (Chen et al., 2018; Felix et al., 2018; Kumar Verma et al., 2018; Mishra et al., 2018) to creating synthetic data from semantic features of unseen classes. The strategy could enable learning semantic-visual alignment on unseen classes implicitly, and thus improves the ability to classify unseen classes. However, such generative models are indeed trained on seen samples, and the quality of synthesized unseen samples is predominantly influenced by seen classes. If the number of training samples of each seen class is small, it is hard for generative models to adequately synthesize samples of unseen classes, leading to unsatisfactory zero-shot learning. To better address the issue, we propose to synthesize visual and semantic features of virtual classes rather than those of the unseen classes. An interesting analogy is that childhood experience and relevant study (Greene, 1995) suggest the behavior of using human imagination to produce new object concepts could assist our cognitive capability. To mimic people utilizing imagination for exploring new knowledge, we create virtual classes by the integration of past ”experience” (seen classes). In detail, we extend the mixup technique by Zhang et al. (2018) to generate virtual classes, with a subtle difference that mixing is conducted on the semantic features (in addition to the visual ones), instead of the class label vectors.
+
+In ZSL/GZSL, each seen or unseen class is typically described by a single semantic feature vector. The practice is useful in differentiating different classes in a principled way, but may not be sufficient to reflect the inter-class and intra-class visual discrepancies, not to mention the ambiguities caused by different backgrounds, view orientations, or occlusion in images. The concern of inefficient class-level representation can also be observed from how the semantic feature vectors are constructed. Take, for example, the Attribute Pascal and Yahoo (aPY) dataset (Farhadi et al., 2009), where each instance is annotated by 64 attributes. The semantic features of each class in aPY are obtained by averaging the attribute vectors of all its instances. We are thus motivated to introduce an imageadaptive class representation, integrating the original semantic features for inter-class discrimination with an image-specific attention vector for intra-class variations.
+
+With the addition of virtual training data and the image-adaptive class representation, our method is designed to learn two classification experts: one for seen classes and the other for unseen classes. Both experts project the image-adaptive semantic feature vectors to the visual space and use cosine similarity to find the class label most similar to the given visual feature vector. The seen expert is trained with the provided training (seen) data, while the class prediction is over all possible classes, including seen and unseen. In inference, if its predicted class is not within the seen category. The testing sample is deemed to be from the unseen category, whose label is then decided by the unseen expert. The unseen expert is trained with the virtual data only, and the process indeed resembles meta-learning. However, the effectiveness of meta-learning is boosted by the design of the imageadaptive mechanism in that fine-tuning is not needed in performing zero-shot classification over unseen classes. We characterize the main contributions of this work as follows.
+
+• Instead of generating synthetic data of unseen classes, we propose to yield virtual classes and data by mixup interpolations. The virtual classes of synthetic data can then be seamlessly coupled with meta-learning to improve the inference on unseen testing samples. We introduce the concept of representing each class with image-adaptive semantic features that could vary among intra-class samples. While the adaptive mechanism improves classifying the seen classes, it manifests the advantage in boosting the effect of meta-learning over virtual data to zero-shot inference over unseen classes. • We demonstrate state-of-the-art results of zero-shot learning over four popular benchmark datasets and justify the design of our method with a thorough ablation study.
+
+# 2 RELATED WORK
+
+We review relevant literature in this section. First, we describe generative approaches for ZSL/GZSL that synthesize unseen images for training. To improve GZSL performance, we propose to couple virtual class generation with meta-learning for mimicking the inference scenario. Next, we discuss attention approaches that extract discriminating features from images to help classification.
+
+# 2.1 GENERATIVE APPROACHES FOR ZSL/GZSL
+
+Arguably one of the most important problems in ZSL/GZSL is to prevent models from being biased to seen classes. Generative approaches (Chen et al., 2018; Felix et al., 2018; Kumar Verma et al., 2018; Mishra et al., 2018; Schonfeld et al., 2019; Paul et al., 2019; Xian et al., 2019) tackle the problem by synthesizing visual features of unseen classes from their semantic features with generative models like Generative Adversarial Networks (GAN; Goodfellow et al., 2014) or Variational Autoencoders (VAE) (VAE; Kingma & Welling, 2014). The synthetic visual features act as pseudo-examples of unseen classes, and effectively reduce the ZSL/GZSL problem to a supervised learning one.
+
+One key issue behind generative approaches comes from the insufficient amount of data to learn a good generative model. As a consequence, some semantic features that seem important during training may cause overfitting, and others that seem less important may be completely dropped. Therefore, several prior techniques propose new constraints or losses to preserve semantic features and regularize the generative model. For instance, (Chen et al., 2018) avoids the dropping of semantic information by disentangling the semantic space into two subspaces, one for classification and the other for reconstruction; Felix et al. (2018) enforces visual-semantic feature consistency by requiring synthesized visual features to reconstruct semantic features accurately.
+
+However, even with the new constraints or losses, the success of generative approaches still highly depends on whether there is enough data/variation in the seen classes to synthesize diverse visual features. Otherwise, the yielded features will be too close to those few seen classes and cannot help ZSL/GZSL much. For example, in AWA2 dataset (Xian et al., 2018a), the unseen samples of “rat” are easily misclassified as seen classes of “mouse” or “hamster” because synthetic visual features for “rat” are doomed to be confused with seen visual features of those similar classes. The phenomenon inspires us to generate synthetic visual features for not only the given unseen classes but also the virtual unseen classes. These virtual classes provide a wider spectrum of support for unseen scenarios to improve ZSL/GZSL performance.
+
+Our other focal effort is to connect the concept of meta-learning (Vinyals et al., 2016; Finn et al., 2017) with virtual classes in the training phase of the unseen expert. The most closely related approaches in ZSL are Li et al. (2019b); Yu et al. (2020); Verma et al. (2020); Sung et al. (2018); Hu et al. (2018). Verma et al. (2020) combines meta-learning and generative models to conquer limits in generative models. Li et al. (2019b) mimics the inference scenario by randomly selecting seen classes as ”fake” new classes in each episode. Yu et al. (2020) randomly splits seen classes into two sets to train and refine the model. In contrast, we simulate the ZSL inference scenario in each episode by creating virtual classes from seen classes using mixup. Owing to the training scenario more resembling the inference setting in ZSL, our model achieves better S2V embedding on unseen classes and obtains state-of-the-art ZSL performance on most datasets. Besides, Sung et al. (2018) learns a deep distance metric to classify samples. Hu et al. (2018) utilizes the correction module to assist classification. On the contrary, our model does not need to learn complex relationship between classes and additional assistant module to achieve good performance.
+
+# 2.2 ATTENTION
+
+Attention mechanism is widely used in ZSL (Ji et al., 2018; Xie et al., 2019; Huynh & Elhamifar, 2020; Min et al., 2020; Liu et al., 2019). Highlighting important local features and reducing noisy feature influence generate a more effective mapping between visual and semantic domains. Ji et al. (2018) proposes $\mathrm { \check { S } { } ^ { 2 } G A }$ which utilizes semantic features to emphasize most informative visual local features. (Xie et al., 2019) employs AREN and ACSE to focus on the most important region in images. Huynh & Elhamifar (2020) applies dense attribute-attention to find the most discriminating image parts and embed them to semantic features individually. DVBE (Min et al., 2020) utilizes spatial and channel attention to maximize inter-class margin. While the attention to visual features is proved useful by the above-mentioned research efforts, relatively few attempts assess the importance of attention on semantic features. In the ZSL setting, each class contains only one semantic feature vector such that intra-class variations are neglected from the single semantic representation. For example, the object of interest can be occluded partly in the image or in the front of different backgrounds, and a unique class-wise semantic feature vector is hard to reflect such variances. Thus, attention to semantic features is worth further exploring in ZSL. Along this line, the most relevant work to ours is LFGAA (Liu et al., 2019), which considers semantic prediction of samples. Different from LFGAA, our method puts attention on discriminating dimensions of semantic features and maps them to the visual domain. The effect of the proposed attention mechanism is to attract the visual representation of ground-truth class and repel others. Besides, our other main difference from LFGAA is the shortcut design in the attention model. We add a shortcut to prevent the model from overfitting on training data and retain discriminating dimensions for unseen class classification.
+
+# 3 METHOD
+
+We coin the proposed method as AGZSL in that its main idea is the fusion of Adaptive and Generative mechanisms in solving ZSL/GZSL. (See Figure 1.) We describe below the details of AGZSL.
+
+# 3.1 NOTATIONS AND FORMULATION
+
+We denote training data as $\mathcal { D } = \mathcal { D } _ { S } \cup \mathcal { D } _ { U }$ where $\mathcal { D } _ { S } = \{ ( \mathbf { x } _ { s } \in \mathcal { X } _ { s } , \mathbf { a } _ { s } \in \mathcal { A } _ { s } , y _ { s } \in \mathcal { V } _ { s } ) \}$ comprises seen data and $\mathcal { D } _ { U } = \{ ( \mathbf { a } _ { u } \in \mathcal { A } _ { u } , y _ { u } \in \mathcal { Y } _ { u } ) \}$ includes unseen class labels $y _ { u }$ and the corresponding semantic/attribute feature vectors $\mathbf { a } _ { u }$ . The dimensions of visual features and semantic features are respectively expressed as $\mathbf { x } \in \mathbb { R } ^ { d }$ and $\mathbf { a } \in \mathbb { R } ^ { k }$ , while the number of total classes is $n$ . We also assume that the seen category $\mathcal { { D } } _ { s }$ and the unseen category $\mathcal { \partial } _ { u }$ are disjoint, i.e., $\mathcal { V } _ { s } \cap \mathcal { V } _ { u } = \emptyset$ and $\mathcal { D } _ { s } \cup \mathcal { D } _ { u } = \mathcal { D }$ . Besides, we define $\mathcal { \bar { A } } _ { s } \cup \mathcal { A } _ { u } = \mathcal { A } \in \mathbb { R } ^ { k \times n }$ as semantic features of all classes. Finally, the virtual data via mixup interpolations is denoted as $\mathcal { D } _ { V } = \{ ( \mathbf { x } _ { v } \in \mathcal { X } _ { v } , \mathbf { a } _ { v } \in \mathcal { A } _ { v } , y _ { v } \in \mathcal { Y } _ { v } ) \}$ . As we explain later, $y _ { v }$ is simply a new label of virtual class, rather than from label interpolation as in Zhang et al. (2018).
+
+![](images/56d5ec19a7d5bbebfed59c54dfd09274a74fd1a39aee7d989c5667c12762bf3a.jpg)  
+Figure 1: Overview of AGZSL. The same color reflects the same class association between visual features and attributes. (a) Image-adaptive mechanism (marked as purple) based on $\mathbf { x }$ yields proper attention to adapt the semantic features of a and $\mathbf { a } ^ { \prime }$ so that their S2V embedding (marked as orange) displays the desired effects in $\hat { \bf x }$ (attracted to $\mathbf { x }$ ) and in $\hat { \mathbf { x } } ^ { \prime }$ (repelled from x). (b) We generate virtual classes from a pair of classes of semantic features a and $\mathbf { a } ^ { \prime }$ with selected weights $\lambda$ via beta distribution $\mathbf { B } ( \alpha , \beta )$ . $\bar { \bf X } _ { v } / { \bf X } _ { v } ^ { \prime }$ and $\mathbf { a } _ { v } / \mathbf { a } _ { v } ^ { \prime }$ represent visual and semantic features of virtual classes.
+
+To perform the classification task, we explore the similarity correlations in the visual domain and consider semantic-to-visual (S2V) embeddings. Analogous to COSMO (Atzmon & Chechik, 2019), our method includes seen and unseen experts, $f _ { s }$ and $f _ { u }$ , to classify testing samples as shown in Figure 2; however, the algorithmic designs are fundamentally different. The novel component of our method is the introduction of Image-Adaptive Semantics (IAS) in both experts to expand the spectrum of S2V embeddings. Specifically, in training, IAS diversifies the total mappings of S2V embedding from the number of different semantic vectors/classes to the number of samples. Its effect is twofold. First, it boosts the classification performance of the seen expert. Second, it generalizes the meta-learning of the unseen expert with virtual data and yields good performance in the inference of samples from unseen classes. Given a testing sample $\mathbf { x }$ in inference, we apply the two learned experts by
+
+$$
+\hat { y } _ { s } = \underset { y \in \mathcal { Y } } { \arg \operatorname* { m a x } } f _ { s } ( \mathbf { x } , \mathcal { A } ) \quad \mathrm { a n d } \quad \hat { y } _ { u } = \underset { y \in \mathcal { Y } _ { u } } { \arg \operatorname* { m a x } } f _ { u } ( \mathbf { x } , \mathcal { A } _ { u } ) .
+$$
+
+To decide the class label $\hat { y }$ of $\mathbf { x }$ , we use the seen expert to discriminate whether the sample is from the seen category. If it is the case then $\hat { y }$ is decided by $f _ { s }$ , and otherwise by $f _ { u }$ . That is, we have
+
+$$
+\hat { y } = \left\{ \begin{array} { l l } { \hat { y } _ { s } } & { \mathrm { i f } \ \hat { y } _ { s } \in \mathcal { Y } _ { s } , } \\ { \hat { y } _ { u } } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
+$$
+
+# 3.2 IMAGE ADAPTIVE SEMANTICS (IAS)
+
+As ZSL/GZSL assumes one semantic feature vector for each class, the conventional semantic-tovisual architecture leads to one single class-level visual feature vector, which is common to all samples of the same class. To learn such a single embedding vector with the constraint of being simultaneously similar to all visual features of samples from the same class tends to yield unsatisfactory classification outcomes in that it simply ignores the intra-class variations in visual features.
+
+Diversifying semantic-to-visual embeddings with IAS establishes the core of our model as shown in Figure 3. To realize the concept, the network module learns to map image-adaptive semantic features to the corresponding visual features, according to the cosine similarity. To this effect, IAS leverages the given visual features $\mathbf { x }$ to focus on discriminative dimensions of the respective semantic features. Specifically, we apply IAS to obtain modified semantic features $\hat { \mathcal { A } } ( \mathbf { x } ) \in \mathbb { R } ^ { k \times n }$ by
+
+$$
+\hat { \cal A } ( x ) = { \cal A } + { \cal A } \odot \mathrm { s o f t m a x } ( g ( { \bf x } ) ) ,
+$$
+
+![](images/a08475917e5cb75891a56394168f2ccfff6ac532478aa5c07a1313566727bc64.jpg)  
+Figure 2: The proposed seen and unseen experts. In the learning phase, the seen expert $f _ { s }$ is trained with seen visual features $\mathbf { x } _ { s }$ , seen semantic features $\mathcal { A } _ { s }$ , and corresponding class labels $y _ { s }$ , while the unseen expert $f _ { u }$ is trained via meta-learning with virtual data. In the inference phase, given a testing sample $\mathbf { x }$ , $f _ { s }$ is additionally fed with all semantic features $\mathcal { A }$ , and $f _ { u }$ only with unseen semantic features $\mathcal { A } _ { u }$ . The decision rule leading to the class label $\hat { y }$ is based on (2). Note that in both training and inference stages, the image-adaptive mechanism to semantic features is carried out.
+
+where $\mathrm { s o f t m a x } ( g ( \mathbf { x } ) ) \in \mathbb { R } ^ { k }$ represents using a linear layer $g$ to predict semantic attention and $\odot$ denotes Hadamard product to combine attention and semantic features. Note that the shortcut implementation in (3) is pivotal as the softmax function often puts high attention weights on sparse dimensions and draws others near to zero. Without the shortcut, the modified $\hat { A }$ would miss some part of semantic features and degenerate the classification performance. Also, it is worthy to point out that compared to LFGAA (Liu et al., 2019), our method pays attention to the semantic features of classes, not the semantic prediction of the model. With (3), the process to drive the image-adaptive class-level visual features $\mathcal { V } ( \mathbf { x } )$ can be expressed by
+
+$$
+\begin{array} { r } { \mathcal { A } \in \mathbb { R } ^ { k \times n } \xrightarrow [ \mathrm { I A S } ] { \mathbf { x } \in \mathbb { R } ^ { d } } \hat { \mathcal { A } } ( \mathbf { x } ) \in \mathbb { R } ^ { k \times n } \longmapsto \mathcal { V } ( \mathbf { x } ) \in \mathbb { R } ^ { d \times n } . } \end{array}
+$$
+
+Once we have obtained the class-level visual features, we then use the cosine similarity and softmax to calculate the probability of each class. We have the class probability of prediction on $\mathbf { x }$ :
+
+$$
+\mathbf { p } ( \mathbf { x } ) = f ( \mathbf { x } , { \cal A } ) = \operatorname { s o f t m a x } ( \sigma \times \cos ( \mathcal { V } ( \mathbf { x } ) , \mathbf { x } ) ) ,
+$$
+
+where $\sigma$ is the learnable scale and $f$ is indeed the general form of a classification expert. We conclude by stating that the Cross-Entropy(CE) loss is adopted to update the model:
+
+$$
+\mathcal { L } _ { \mathrm { C E } } = - \sum _ { \mathbf { x } } \log p _ { y } ( \mathbf { x } ) ,
+$$
+
+where $p _ { y } ( \mathbf { x } )$ is the predicted probability of ground-truth label.
+
+# 3.3 SEEN EXPERT
+
+The objective of seen expert $f _ { s }$ is to recognize seen classes and otherwise decide whether a testing sample belongs to unseen classes in inference phase. In zero-shot learning, especially GZSL, the capability to single out testing samples of unseen classes is crucial to the classification performance. If one can establish reasonable split mechanisms such as in Min et al. (2020) and Chen et al. (2020), the more challenging GZSL would be simplified into ZSL. For example, COSMO (Atzmon & Chechik, 2019) utilizes Confidence Base Gating to discriminate between seen and unseen classes, while DVBE (Min et al., 2020) considers the entropy threshold. In our formulation, the seen expert learns to predict all possible classes, including seen and unseen, although the training data assume only seen class labels. That is, $f _ { s }$ is trained to map (image adaptive) seen semantic features to their corresponding visual features. Such optimization can be thought of as one-category anomaly detection over only seen classes. Thus, in inference, if the highest cosine similarity value by the seen expert $f _ { s }$ is not yielded by one of the seen classes, the testing sample is considered abnormal from the unseen category and its classification should be decided by the unseen expert discussed next.
+
+![](images/08266f30c339f899e05262ad087e80f848982a9fcc5e600d3d804fda2ec1de07.jpg)  
+Figure 3: The architectures of IAS and S2V (FC: linear layer, $L _ { 2 }$ : $L _ { 2 }$ -normalization).
+
+# 3.4 UNSEEN EXPERT
+
+So far we have stated the objective of the unseen expert $f _ { u }$ is to classify the labels of those that cannot be decided by the seen expert, as in (2). To resolve the dilemma of zero-shot setting, we apply meta-learning to train $f _ { u }$ over virtual classes by mixup interpolations. The exact number of virtual classes in each episode is decided by the pilot study (Li et al., 2019b), and an ablation study in appendix is provided to analyze its effect on the performance of zero-shot classification. To begin with, in each episode we randomly select $m$ pairs of seen classes. Then, we create $m$ virtual classes out of these pairs by mixup. The semantic and visual features for these virtual classes are defined as
+
+$$
+\begin{array} { r } { \mathbf { a } _ { v } = \lambda \mathbf { a } _ { i } + \left( 1 - \lambda \right) \mathbf { a } _ { j } , } \\ { \mathbf { x } _ { v } = \lambda \mathbf { x } _ { i } + \left( 1 - \lambda \right) \mathbf { x } _ { j } , } \end{array}
+$$
+
+where ${ \bf a } _ { i }$ and $\mathbf { a } _ { j }$ are semantic features of paired classes, and $\mathbf { x } _ { i }$ and $\mathbf { x } _ { j }$ are randomly chosen from visual samples of the paired classes. Following Zhang et al. (2018), $\lambda \doteq [ 0 , 1 ]$ is sampled from the beta distribution. We then define a new virtual class $y _ { v }$ , with $\mathbf { a } _ { v }$ as semantic features, to be associated with virtual visual features $\mathbf { x } _ { v }$ . Previous generative models (Schonfeld et al., 2019; Xian et al., 2019) synthesize unseen samples by GAN/VAE or their variants. In comparison, the proposed meta-learning has at least three advantages for ZSL. First, the small number of classes in some datasets, such as 50 in AWA2, may fall short to provide sufficient support set such that the generative model is hard to be optimized, while episodic meta-learning over virtual classes can overcome this problem. Second, some part of semantic features which are important to classify unseen classes do not have enough training classes in the datasets. With mixup semantic features as virtual classes, our model can learn underlying discriminating semantic features well. Third, the fusion of image-adaptive mechanism and meta-learning is flexible. As a result, we do not need online fine-tuning in applying the unseen expert $f _ { u }$ yielded by meta-learning to predict the label of unseen class for testing samples.
+
+# 4 EXPERIMENT
+
+# 4.1 EXPERIMENTAL SETTING
+
+Datasets. In GZSL, there are four widely used benchmark datasets for evaluation, which are AWA2 (Xian et al., 2018a), USCD Birds-200-2011 (CUB) (Welinder et al., 2010), SUN (Patterson & Hays, 2012), and aPY (Farhadi et al., 2009). AWA2 contains 37K images and 50 kinds of animals with 85- dimension attributes. CUB consists of 11K images and 200 bird species described with 312-dimension attributes. SUN is composed of 14K images 717 classes labeled with 102-dimension attributes. aPY includes 15K images and 32 different classes with 64-dimension attributes. The attributes of class in datasets will be taken as the semantic features in our work. We follow existing GZSL approaches (Min et al., 2020; Li et al., 2019b) and extract 2048-dimensional visual features for each image by the ResNet101 backbone (He et al., 2016) pretrained on ImageNet-1K. We then split the data into seen and unseen classes according to the benchmark procedure from Xian et al. (2017).
+
+Table 1: GZSL results on four datasets. All methods in comparison utilize ResNet101 as the backbone for fairness. Notation ”\*” means the method fine-tunes the backbone to match the characteristic of datasets.   
+
+<table><tr><td rowspan="3"></td><td colspan="4">AWA2</td><td colspan="4">CUB GZSL</td><td colspan="4">SUN</td><td colspan="4">aPY</td></tr><tr><td>ZSL</td><td rowspan="2">U</td><td colspan="2">GZSL</td><td>ZSL</td><td colspan="2"></td><td></td><td>ZSL</td><td colspan="2">GZSL</td><td></td><td>ZSL</td><td colspan="2">GZSL</td><td></td></tr><tr><td>T1</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>S</td><td></td><td>H</td></tr><tr><td>SP-AEN</td><td>58.5</td><td>23.3</td><td>90.9</td><td>37.1</td><td>55.4</td><td>34.7</td><td>70.6</td><td>46.6</td><td>59.2</td><td>24.9</td><td>38.6</td><td>30.3</td><td>24.1</td><td>13.7</td><td>63.4</td><td>22.6</td></tr><tr><td>DLFZRL</td><td>60.9</td><td>1</td><td>-</td><td>45.1</td><td>51.9</td><td>-</td><td>-</td><td>37.1</td><td>42.5</td><td>-</td><td>-</td><td>24.6</td><td>38.5</td><td>1</td><td>1</td><td>31.0</td></tr><tr><td>PSR</td><td>63.8</td><td>20.7</td><td>73.8</td><td>32.3</td><td>56.0</td><td>24.6</td><td>54.3</td><td>33.9</td><td>61.4</td><td>20.8</td><td>37.2</td><td>26.7</td><td>38.4</td><td>13.5</td><td>51.4</td><td>21.4</td></tr><tr><td>CDL</td><td>-</td><td>1</td><td>1</td><td>1</td><td>54.5</td><td>23.5</td><td>55.2</td><td>32.9</td><td>63.6</td><td>21.5</td><td>34.7</td><td>26.5</td><td>43.0</td><td>19.8</td><td>48.6</td><td>27.1</td></tr><tr><td>PQZSL</td><td>=</td><td>1</td><td>-</td><td>1</td><td>-</td><td>53.2</td><td>51.4</td><td>46.9</td><td>1</td><td>35.1</td><td>35.3</td><td>35.2</td><td>27.9</td><td>64.1</td><td>64.1</td><td>38.8</td></tr><tr><td>f-VAEGAN-D2*</td><td>70.3</td><td>57.1</td><td>76.1</td><td>65.2</td><td>72.9</td><td>63.2</td><td>75.6</td><td>68.9</td><td>65.6</td><td>50.1</td><td>37.8</td><td>43.1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>LsrGAN</td><td>-</td><td>54.6</td><td>74.6</td><td>63.0</td><td>1</td><td>48.1</td><td>59.1</td><td>53.0</td><td>-</td><td>44.8</td><td>37.7</td><td>40.9</td><td>-</td><td>-</td><td></td><td>1</td></tr><tr><td>TF-VAEGAN*</td><td>73.4</td><td>55.5</td><td>83.6</td><td>66.7</td><td>74.3</td><td>63.8</td><td>79.3</td><td>70.7</td><td>66.7</td><td>41.8</td><td>51.9</td><td>46.3</td><td>-</td><td></td><td></td><td>=</td></tr><tr><td>OCD-CVAE</td><td>71.3</td><td>59.5</td><td>73.4</td><td>65.7</td><td>60.9</td><td>44.8</td><td>59.9</td><td>51.3</td><td>62.1</td><td>44.8</td><td>42.9</td><td>43.8</td><td>=</td><td></td><td>=</td><td>1</td></tr><tr><td>ZSML Softmax</td><td>76.1</td><td>58.9</td><td>74.6</td><td>65.8</td><td>69.6</td><td>60.0</td><td>52.1</td><td>55.7</td><td>60.2</td><td>-</td><td>1</td><td>1</td><td>64.1</td><td>36.3</td><td>46.6</td><td>40.9</td></tr><tr><td>GXE</td><td>71.1</td><td>56.4</td><td>81.4</td><td>66.7</td><td>54.4</td><td>47.4</td><td>47.6</td><td>47.5</td><td>62.6</td><td>36.3</td><td>42.8</td><td>39.3</td><td>38.0</td><td>26.5</td><td>74.0</td><td>39.0</td></tr><tr><td>E-PGN</td><td>73.4</td><td>52.6</td><td>83.5</td><td>64.6</td><td>72.4</td><td>52.0</td><td>61.1</td><td>56.2</td><td>1</td><td>-</td><td></td><td>-</td><td>1</td><td>-</td><td>1</td><td>1</td></tr><tr><td>Relation Net</td><td>64.2</td><td>30.0</td><td>93.4</td><td>45.3</td><td>55.6</td><td>38.1</td><td>61.1</td><td>47.0</td><td>-</td><td>=</td><td></td><td>=</td><td>1</td><td>-</td><td>=</td><td>1</td></tr><tr><td>Correlation Net</td><td>-</td><td>=</td><td>-</td><td>1</td><td>45.8</td><td>41.9</td><td>1</td><td>-</td><td>-</td><td>=</td><td>=</td><td>=</td><td>1</td><td>=</td><td>=</td><td>=</td></tr><tr><td>LFGAA+Hibrid</td><td>68.1</td><td>27.0</td><td>93.4</td><td>41.9</td><td>67.6</td><td>36.2</td><td>80.9</td><td>50.0</td><td>62.0</td><td>18.5</td><td>40.0</td><td>25.3</td><td></td><td></td><td></td><td>=</td></tr><tr><td>AREN*</td><td>66.9</td><td>54.7</td><td>79.1</td><td>64.7</td><td>72.5</td><td>63.2</td><td>69.0</td><td>66.0</td><td>60.6</td><td>40.3</td><td>32.3</td><td>35.9</td><td>39.2</td><td>30.0</td><td>47.9</td><td>36.9</td></tr><tr><td>COSMO</td><td>-</td><td>=</td><td>=</td><td>1</td><td>1</td><td>44.4</td><td>57.8</td><td>50.2</td><td>-</td><td>44.9</td><td>37.7</td><td>41.0</td><td>1</td><td>-</td><td>=</td><td>=</td></tr><tr><td>DVBE*</td><td>-</td><td>62.7</td><td>77.5</td><td>69.4</td><td>1</td><td>64.4</td><td>73.2</td><td>68.5</td><td>-</td><td>44.1</td><td>41.6</td><td>42.8</td><td>1</td><td>37.9</td><td>55.9</td><td>45.2</td></tr><tr><td>ours</td><td>73.8</td><td>65.1</td><td>78.9</td><td>71.3</td><td>57.2</td><td>41.4</td><td>49.7</td><td>45.2</td><td>63.3</td><td>29.9</td><td>40.2</td><td>34.3</td><td>41.0</td><td>35.1</td><td>65.5</td><td>45.7</td></tr><tr><td>ours*</td><td>76.4</td><td>69.0</td><td>86.5</td><td>76.8</td><td>77.2</td><td>69.2</td><td>76.4</td><td>72.6</td><td>66.2</td><td>50.5</td><td>43.1</td><td>46.5</td><td>43.7</td><td>36.2</td><td>58.6</td><td>44.8</td></tr></table>
+
+Evaluation. We evaluate our approach for both ZSL and GZSL settings. For the ZSL setting, the average per-class top-1 accuracy $\mathbf { \Pi } ^ { ( \mathbf { T } ) }$ on unseen classes is taken as the evaluation metric. For GZSL (Xian et al., 2017), the evaluation metrics include seen classes accuracy S, unseen class accuracy U, and their harmonic mean ${ \bf H } = ( 2 { \bf U } \cdot { \bf S } ) / ( { \bf U } + { \bf S } )$ .
+
+Implementation. As suggested in Li et al. (2019b), we normalize the visual and semantic features into [0, 1]. The architecture of semantic-to-visual embedding contains a two-layer linear model with 1,600 hidden units and utilizes ReLU on the hidden and output layer. The seen and unseen experts are trained by Adam optimizer with a learning rate $5 \times 1 0 ^ { - 5 }$ and $5 \times 1 0 ^ { - 4 }$ respectively for all datasets. We apply 200,000 episodes to train the unseen expert. In each episode, we randomly generate 16 or 20 (based on the dataset) virtual classes and 4 samples for each class. We follow the setting in (Min et al., 2020) to fine-tune the backbone.
+
+# 4.2 COMPARISON WITH STATE-OF-THE-ART APPROACHES
+
+We compare our method with 13 recent GZSL approaches. These include SP-AEN (Chen et al., 2018), DLFZRL (Tong et al., 2019), PSR (Annadani & Biswas, 2018), CDL (Jiang et al., 2018), PQZSL (Li et al., 2019a), f-VAEGAM-D2 (Xian et al., 2019), LsrGAN (Vyas et al., 2020), TF-VAEGAN (Narayan et al., 2020), OCD-CVAE (Keshari et al., 2020), ZSML Softmax (Verma et al., 2020), E-PGN (Yu et al., 2020), Relation Net (Sung et al., 2018), Correlation Net (Hu et al., 2018), GXE Li et al. (2019b), LFGAA $^ +$ Hybrid (Liu et al., 2019), AREN (Xie et al., 2019), COSMO (Atzmon & Chechik, 2019), DVBE (Min et al., 2020). Table 1 shows that the proposed AGZSL achieves the best GZSL performance on all four datasets. It improves harmonic mean by $7 . 4 \%$ on AWA2, $1 . 9 \%$ on CUB, $0 . \bar { 2 } \%$ on SUN, and $0 . 5 \%$ on aPY, respectively. Particularly, among those methods in comparison, DVBE needs to search the entropy threshold $\tau$ and the embedding model for the best harmonic means according to each dataset. In contrast to DVBE, without entropy threshold and specific model searching, our model achieves over $4 \%$ improvement on most datasets. The main reason for better performance is that AGZSL provides more discriminating visual representation for both seen and unseen classes. Therefore, our model can effectively separate seen and unseen samples and achieve better performance in the harmonic mean. Notably, AGZSL drops performance on aPY after backbone fine-tuning. It could be due to that its number of seen classes and training images are fewest among the four datasets. Fine-tuning the backbone causes the visual features to overfit training samples and become less discriminating for correct classification.
+
+Table 2: The effect of virtual classes and IAS for GZSL. We remove these two components as our baseline (b). In the ablation study, we add virtual classes (v) and IAS (I) step by step to show their effect on GZSL.   
+
+<table><tr><td rowspan="3"></td><td colspan="4">AWA2 GZSL</td><td colspan="4">CUB</td><td colspan="4">SUN</td><td colspan="4">aPY</td></tr><tr><td>ZSL</td><td colspan="3"></td><td>ZSL</td><td></td><td colspan="2">GZSL</td><td>ZSL</td><td colspan="3">GZSL</td><td>ZSL</td><td colspan="3">GZSL</td></tr><tr><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td></tr><tr><td>b</td><td>72.2</td><td>63.2</td><td>86.1</td><td>72.9</td><td>73.7</td><td>68.6</td><td>71.3</td><td>69.9</td><td>63.9</td><td>47.6</td><td>42.1</td><td>44.7</td><td>40.0</td><td>33.4</td><td>57.2</td><td>42.2</td></tr><tr><td>b+v</td><td>75.8</td><td>66.9</td><td>86.1</td><td>75.3</td><td>76.1</td><td>70.3</td><td>71.3</td><td>70.8</td><td>65.6</td><td>48.6</td><td>42.1</td><td>45.1</td><td>42.8</td><td>35.2</td><td>57.2</td><td>43.6</td></tr><tr><td>b+v+I</td><td>76.4</td><td>69.0</td><td>86.5</td><td>76.8</td><td>77.2</td><td>69.2</td><td>76.4</td><td>72.6</td><td>66.2</td><td>50.5</td><td>43.1</td><td>46.5</td><td>43.7</td><td>36.2</td><td>58.6</td><td>44.8</td></tr></table>
+
+Furthermore, generative models achieve higher performance on CUB and SUN, because in these two datasets there are many seen classes for synthesizing unseen samples of better quality. In comparison, our model overcomes the influence of few seen classes and also achieves the best performance on AWA2 and aPY. Our results empirically support that the introduction of virtual classes and samples are indeed helpful to classify unseen classes. Finally, we remark that since the backbone is pretrained on ImageNet-1K, which is rather different from the ZSL fine-grained databases, e.g., CUB and SUN. Thus, the harmonic mean improves a lot after fine-tuning the backbone on these two datasets.
+
+# 4.3 ABLATION STUDY
+
+To evaluate the benefits of tackling ZSL/GZSL with virtual classes and IAS, we carry out an ablation study on the four datasets. We incrementally include each key component to assess their effect. As shown in Table 2, the virtual classes improve ZSL accuracy significantly. The ZSL performance improves $4 . 6 \%$ on AWA2, $2 . 4 \%$ on CUB, $1 . 7 \%$ on SUN, and $2 . 8 \%$ on aPY, respectively. The result implies that the virtual classes are advantageous to ZSL. After adding IAS to our model, both seen and unseen accuracy increases, and the harmonic mean $_ \mathrm { H }$ improves about $1 . 5 \%$ on average. The consistent gain exemplifies that IAS can provide better semantic-to-visual embedding and achieve higher performance on GZSL.
+
+# 4.4 COSINE SIMILARITY MARGIN
+
+Figure 4 illustrates the effect of the proposed AGZSL model. The cosine similarity margin means the inner product value between the correct class and the other nearest class. A negative margin indicates that the model predicts a wrong class. On the other hand, a large positive margin implies that the model results in a proper embedding distinguishing the correct class from others effectively. Thus, seeking a positive and larger margin is a crucial and important objective in GZSL. As shown in the first row of Figure 4, the margins of unseen samples shift right on all four datasets by our method. It shows that AGZSL indeed generates better embedding on the visual space and makes the model easier to predict the correct class. Moreover, from the second row in Figure 4, the plots show that our method improves the margins of seen samples dramatically on most datasets. The above experiment explains that AGZSL can achieve a higher margin for most samples and achieves better classification performance on the four testing datasets.
+
+# 4.5 THE DISTRIBUTION OF VIRTUAL CLASSES
+
+To explicitly exhibit the distributions of seen, unseen and virtual classes, we apply PCA to map the class-wise semantic features to a two-dimensional space, as shown in Figure 5. The features of virtual classes scatter around seen classes, because we have used B(5,1) to generate virtual classes. Advantageously, some virtual classes are very close to unseen classes in the semantic domain and turn out to be helpful for enabling the unseen expert to classify samples of unseen classes. With meta-learning over these numerous virtual classes and samples, the proposed S2V can better learn the embedding between semantic and visual domains, and results in an effective unseen expert.
+
+![](images/25bda5a37afb8712ef3f138799a643e1efb0fa38776647f7531d961455a8535c.jpg)  
+Figure 4: The cosine similarity margin between the correct class and other nearest class. The blue area depicts the cosine similarity margin of baseline without IAS and generative learning. On the other hand, the orange area represents the cosine similarity margin of ours. Most of the samples improve their margin by our method, suggesting the proposed techniques are beneficial for GZSL.
+
+![](images/ab9faddf0a9a95528dc9b93715b3073bf0095351124d80d3bc19f2c3cd2d1abb.jpg)  
+Figure 5: The semantic features of virtual/seen/unseen classes over the four datasets.
+
+# 5 CONCLUSION
+
+In this paper, we propose Image Adaptive Semantics (IAS) and meta-learning with virtual classes and samples to solve the ZSL/GZSL problem. To deal with the intra-class visual discrepancies, IAS adaptively emphasizes the most discriminating dimensions in semantic features with respect to the underlying visual features. To better classify samples of the unseen classes in inference, we propose a novel formulation of generative meta-learning. Different from previous generative models that focus on synthesizing unseen samples for training the model, we create virtual classes and their respective virtual samples in the training phase. Further, to imitate the ZSL inference scenario, we carry out meta-learning with these virtual data to extend our model. In our experiment, we demonstrate that AGZSL is beneficial to tackle the challenging ZSL/GZSL problem and achieves significant advantages over those in comparison on the four ZSL/GZSL benchmark datasets.
+
+# ACKNOWLEDGEMENTS
+
+This work was supported in part by the MOST, Taiwan under Grants 110-2634-F-001-009 and 107-2628-E-002-008-MY3. We are grateful to the National Center for High-performance Computing for providing computational resources and facilities.
+
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+
+# A APPENDIX
+
+Table 3: The impact of shortcut in IAS on four datasets.   
+
+<table><tr><td rowspan="3">shortcut</td><td colspan="3">AWA2</td><td colspan="4">CUB</td><td colspan="4">SUN</td><td colspan="4">aPY</td></tr><tr><td>ZSL</td><td colspan="3">GZSL</td><td>ZSL</td><td colspan="3">GZSL</td><td>ZSL</td><td colspan="3">GZSL</td><td>ZSL</td><td colspan="3">GZSL</td></tr><tr><td>T</td><td>U</td><td>S</td><td>H</td><td>T</td><td>U</td><td>S</td><td>H</td><td>T</td><td>U</td><td>S</td><td>H</td><td>T</td><td>U S</td><td>H</td></tr><tr><td></td><td>23.9</td><td>13.0</td><td>30.0</td><td>18.2</td><td>23.3</td><td>17.0</td><td>45.8</td><td>24.8 42.1</td><td>22.5</td><td>20.3</td><td>21.4</td><td>30.5</td><td>14.9</td><td>4.7</td><td>7.1</td></tr><tr><td>√</td><td>76.4</td><td>69.0</td><td>86.5</td><td>76.8</td><td>77.2</td><td>69.2</td><td>76.4</td><td>72.6 66.2</td><td>50.5</td><td>43.1</td><td>46.5</td><td>43.7</td><td>36.2</td><td>58.6</td><td>44.8</td></tr></table>
+
+![](images/a896e7bd1cf69488191429f65186210cbe454a5f967b6e8584ae0b4cd0279542.jpg)  
+Figure 6: The difference of cosine similarity between our model and baseline. In our experiment, most of the samples improve their embedding with IAS and virtual classes.
+
+Shortcut in IAS Table 3 shows that shortcut in IAS improves performance a lot. The unseen class accuracy increase around $5 0 \%$ on AWA2 and SUN. Besides, the harmonic mean also enhances $2 0 \%$ to $6 0 \%$ on four datasets with the shortcut. The main reason is that the softmax usually puts high attention weight on few dimensions and decreases others near to zero. Thus, without shortcut, the model is prone to overfitting on training data.
+
+Cosine similarity enhancement Figure 6 illustrates better embedding for seen and unseen classes in our model. Since we apply cosine similarity to classify samples, the higher score means visual representation of each class is closer to their visual samples. The first row in 6 shows effect of our model on unseen classes. With virtual classes and IAS, cosine similarity increase averaged 0.05 to 0.1 on different datasets. The increasing amount shows our model provides better visual-to-semantic embedding for unseen classes and samples are easier to classify correctly. Therefore, our model can achieve good ZSL performance. Besides, the second row in 6 illustrates the effect of IAS on seen classes. Most of the testing seen samples increase their cosine similarity around 0.02 on four datasets. With the IAS and virtual classes, the model can achieve better visual-to-semantic embedding on all classes and obtain better the generalized seen accuracy on four datasets.
+
+virtual classes number per episode To search the class number in each episode for best ZSL, we search 4 different numbers, i.e, 4,8,12, and 20 per episode. Table 4 shows that the best class number setting is related to the class number of classes. For example, the AWA2 and aPY which include fewer unseen classes obtain the best ZSL accuracy with 16 virtual classes. On the other hand, the CUB and SUN which include more unseen classes achieve the best performance with 20 virtual classes. The reason is that the advantage of meta-learning is to simulate inference scenarios while training (Li et al., 2019b). The virtual class number set closer to the unseen class number is favorable in the meta-learning setting. Therefore, the best virtual class number is different according to the unseen number of the dataset.
+
+![](images/f34dccf20a50736ce887aa631968cdaad67a3e368a5550b2ec7da2a8b51c1212.jpg)  
+Figure 7: Beta distribution.
+
+Table 4: ZSL accuracy w.r.t. virtual classes per episode.   
+
+<table><tr><td>k</td><td>AWA2</td><td>CUB</td><td>SUN</td><td>aPY</td></tr><tr><td>8</td><td>75.7</td><td>76.2</td><td>64.6</td><td>40.8</td></tr><tr><td>12</td><td>76.0</td><td>76.3</td><td>65.1</td><td>42.8</td></tr><tr><td>16</td><td>76.4</td><td>76.3</td><td>65.1</td><td>43.7</td></tr><tr><td>20</td><td>75.5</td><td>77.2</td><td>66.2</td><td>41.1</td></tr></table>
+
+Table 5: ZSL accuracy w.r.t. beta distribution.   
+
+<table><tr><td>B(a,β)</td><td>AWA2</td><td>CUB</td><td>SUN</td><td>aPY</td></tr><tr><td>(5,5)</td><td>74.8</td><td>75.6</td><td>65.4</td><td>38.2</td></tr><tr><td>(1,1)</td><td>75.3</td><td>76.4</td><td>65.8</td><td>40.2</td></tr><tr><td>(5,1)</td><td>76.4</td><td>77.2</td><td>66.2</td><td>43.7</td></tr></table>
+
+Effect of beta distribution on ZSL accuracy To explore the best beta distribution setting for ZSL, our experiments apply three different $\alpha$ and $\beta$ sets. As shown in Figure 7, B(1,1) means virtual classes distribute uniformly between selected class pairs. Then, B(5,5) means most of the virtual classes distribute among the middle of selected class pairs. On the other hand, B(5,1) means most virtual classes distribute closer to one of the classes in selected class pairs. Table 5 demonstrates that B(5,1) achieves best performance on each dataset. In our speculation, the virtual classes created based on one class are closer to real-world classes. For example, the zebra is very similar to the horse in shape. Thus, the virtual classes distributing among seen classes enhance ZSL accuracy are very beneficial for training ZSL.

parse/train/ahAUv8TI2Mz/ahAUv8TI2Mz_content_list.json

@@ -0,0 +1,1498 @@
+[
+    {
+        "type": "text",
+        "text": "ADAPTIVE AND GENERATIVE ZERO-SHOT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yu-Ying Chou1,3, Hsuan-Tien Lin3 & Tyng-Luh Liu1,2 ",
+        "bbox": [
+            183,
+            142,
+            565,
+            160
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Institute of Information Science, Academia Sinica, Taiwan 2Taiwan AI Labs, Taiwan 3Department of Computer Science, National Taiwan University, Taiwan {d07922014,htlin}@csie.ntu.edu.tw,liutyng@iis.sinica.edu.tw ",
+        "bbox": [
+            184,
+            171,
+            759,
+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            252,
+            544,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We address the problem of generalized zero-shot learning (GZSL) where the task is to predict the class label of a target image whether its label belongs to the seen or unseen category. Similar to ZSL, the learning setting assumes that all class-level semantic features are given, while only the images of seen classes are available for training. By exploring the correlation between image features and the corresponding semantic features, the main idea of the proposed approach is to enrich the semantic-to-visual (S2V) embeddings via a seamless fusion of adaptive and generative learning. To this end, we extend the semantic features of each class by supplementing image-adaptive attention so that the learned S2V embedding can account for not only inter-class but also intra-class variations. In addition, to break the limit of training with images only from seen classes, we design a generative scheme to simultaneously generate virtual class labels and their visual features by sampling and interpolating over seen counterparts. In inference, a testing image will give rise to two different S2V embeddings, seen and virtual. The former is used to decide whether the underlying label is of the unseen category or otherwise a specific seen class; the latter is to predict an unseen class label. To demonstrate the effectiveness of our method, we report state-of-the-art results on four standard GZSL datasets, including an ablation study of the proposed modules. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            575,
+            336,
+            592
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Different from conventional learning tasks, zero-shot learning (ZSL) by Lampert et al. (2009); Palatucci et al. (2009); Akata et al. (2013) explores the extreme case of performing inference only over samples of unseen classes. To make the scenario more realistic, generalized zero-shot learning (GZSL) (Chao et al., 2016; Xian et al., 2017) is subsequently proposed so that inference can concern samples of both seen and unseen classes. Nevertheless. the learning setting in ZSL/GZSL is essentially the same where sample classes are divided into two categories, seen and unseen, but only those samples of seen classes are accessible to training. In addition, each of all the classes under consideration is characterized by semantic features such as attributes (Xian et al., 2018b) or text descriptions (Zhu et al., 2018) to specify and relate seen and unseen classes. ",
+        "bbox": [
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+            611,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The lack of training samples from unseen classes has prompted generative approaches (Chen et al., 2018; Felix et al., 2018; Kumar Verma et al., 2018; Mishra et al., 2018) to creating synthetic data from semantic features of unseen classes. The strategy could enable learning semantic-visual alignment on unseen classes implicitly, and thus improves the ability to classify unseen classes. However, such generative models are indeed trained on seen samples, and the quality of synthesized unseen samples is predominantly influenced by seen classes. If the number of training samples of each seen class is small, it is hard for generative models to adequately synthesize samples of unseen classes, leading to unsatisfactory zero-shot learning. To better address the issue, we propose to synthesize visual and semantic features of virtual classes rather than those of the unseen classes. An interesting analogy is that childhood experience and relevant study (Greene, 1995) suggest the behavior of using human imagination to produce new object concepts could assist our cognitive capability. To mimic people utilizing imagination for exploring new knowledge, we create virtual classes by the integration of past ”experience” (seen classes). In detail, we extend the mixup technique by Zhang et al. (2018) to generate virtual classes, with a subtle difference that mixing is conducted on the semantic features (in addition to the visual ones), instead of the class label vectors. ",
+        "bbox": [
+            174,
+            743,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In ZSL/GZSL, each seen or unseen class is typically described by a single semantic feature vector. The practice is useful in differentiating different classes in a principled way, but may not be sufficient to reflect the inter-class and intra-class visual discrepancies, not to mention the ambiguities caused by different backgrounds, view orientations, or occlusion in images. The concern of inefficient class-level representation can also be observed from how the semantic feature vectors are constructed. Take, for example, the Attribute Pascal and Yahoo (aPY) dataset (Farhadi et al., 2009), where each instance is annotated by 64 attributes. The semantic features of each class in aPY are obtained by averaging the attribute vectors of all its instances. We are thus motivated to introduce an imageadaptive class representation, integrating the original semantic features for inter-class discrimination with an image-specific attention vector for intra-class variations. ",
+        "bbox": [
+            174,
+            138,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "With the addition of virtual training data and the image-adaptive class representation, our method is designed to learn two classification experts: one for seen classes and the other for unseen classes. Both experts project the image-adaptive semantic feature vectors to the visual space and use cosine similarity to find the class label most similar to the given visual feature vector. The seen expert is trained with the provided training (seen) data, while the class prediction is over all possible classes, including seen and unseen. In inference, if its predicted class is not within the seen category. The testing sample is deemed to be from the unseen category, whose label is then decided by the unseen expert. The unseen expert is trained with the virtual data only, and the process indeed resembles meta-learning. However, the effectiveness of meta-learning is boosted by the design of the imageadaptive mechanism in that fine-tuning is not needed in performing zero-shot classification over unseen classes. We characterize the main contributions of this work as follows. ",
+        "bbox": [
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+            285,
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+            438
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Instead of generating synthetic data of unseen classes, we propose to yield virtual classes and data by mixup interpolations. The virtual classes of synthetic data can then be seamlessly coupled with meta-learning to improve the inference on unseen testing samples. We introduce the concept of representing each class with image-adaptive semantic features that could vary among intra-class samples. While the adaptive mechanism improves classifying the seen classes, it manifests the advantage in boosting the effect of meta-learning over virtual data to zero-shot inference over unseen classes. • We demonstrate state-of-the-art results of zero-shot learning over four popular benchmark datasets and justify the design of our method with a thorough ablation study. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            604,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We review relevant literature in this section. First, we describe generative approaches for ZSL/GZSL that synthesize unseen images for training. To improve GZSL performance, we propose to couple virtual class generation with meta-learning for mimicking the inference scenario. Next, we discuss attention approaches that extract discriminating features from images to help classification. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 GENERATIVE APPROACHES FOR ZSL/GZSL",
+        "text_level": 1,
+        "bbox": [
+            174,
+            709,
+            519,
+            723
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Arguably one of the most important problems in ZSL/GZSL is to prevent models from being biased to seen classes. Generative approaches (Chen et al., 2018; Felix et al., 2018; Kumar Verma et al., 2018; Mishra et al., 2018; Schonfeld et al., 2019; Paul et al., 2019; Xian et al., 2019) tackle the problem by synthesizing visual features of unseen classes from their semantic features with generative models like Generative Adversarial Networks (GAN; Goodfellow et al., 2014) or Variational Autoencoders (VAE) (VAE; Kingma & Welling, 2014). The synthetic visual features act as pseudo-examples of unseen classes, and effectively reduce the ZSL/GZSL problem to a supervised learning one. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One key issue behind generative approaches comes from the insufficient amount of data to learn a good generative model. As a consequence, some semantic features that seem important during training may cause overfitting, and others that seem less important may be completely dropped. Therefore, several prior techniques propose new constraints or losses to preserve semantic features and regularize the generative model. For instance, (Chen et al., 2018) avoids the dropping of semantic information by disentangling the semantic space into two subspaces, one for classification and the other for reconstruction; Felix et al. (2018) enforces visual-semantic feature consistency by requiring synthesized visual features to reconstruct semantic features accurately. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, even with the new constraints or losses, the success of generative approaches still highly depends on whether there is enough data/variation in the seen classes to synthesize diverse visual features. Otherwise, the yielded features will be too close to those few seen classes and cannot help ZSL/GZSL much. For example, in AWA2 dataset (Xian et al., 2018a), the unseen samples of “rat” are easily misclassified as seen classes of “mouse” or “hamster” because synthetic visual features for “rat” are doomed to be confused with seen visual features of those similar classes. The phenomenon inspires us to generate synthetic visual features for not only the given unseen classes but also the virtual unseen classes. These virtual classes provide a wider spectrum of support for unseen scenarios to improve ZSL/GZSL performance. ",
+        "bbox": [
+            173,
+            138,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our other focal effort is to connect the concept of meta-learning (Vinyals et al., 2016; Finn et al., 2017) with virtual classes in the training phase of the unseen expert. The most closely related approaches in ZSL are Li et al. (2019b); Yu et al. (2020); Verma et al. (2020); Sung et al. (2018); Hu et al. (2018). Verma et al. (2020) combines meta-learning and generative models to conquer limits in generative models. Li et al. (2019b) mimics the inference scenario by randomly selecting seen classes as ”fake” new classes in each episode. Yu et al. (2020) randomly splits seen classes into two sets to train and refine the model. In contrast, we simulate the ZSL inference scenario in each episode by creating virtual classes from seen classes using mixup. Owing to the training scenario more resembling the inference setting in ZSL, our model achieves better S2V embedding on unseen classes and obtains state-of-the-art ZSL performance on most datasets. Besides, Sung et al. (2018) learns a deep distance metric to classify samples. Hu et al. (2018) utilizes the correction module to assist classification. On the contrary, our model does not need to learn complex relationship between classes and additional assistant module to achieve good performance. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 ATTENTION ",
+        "text_level": 1,
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+        "text": "Attention mechanism is widely used in ZSL (Ji et al., 2018; Xie et al., 2019; Huynh & Elhamifar, 2020; Min et al., 2020; Liu et al., 2019). Highlighting important local features and reducing noisy feature influence generate a more effective mapping between visual and semantic domains. Ji et al. (2018) proposes $\\mathrm { \\check { S } { } ^ { 2 } G A }$ which utilizes semantic features to emphasize most informative visual local features. (Xie et al., 2019) employs AREN and ACSE to focus on the most important region in images. Huynh & Elhamifar (2020) applies dense attribute-attention to find the most discriminating image parts and embed them to semantic features individually. DVBE (Min et al., 2020) utilizes spatial and channel attention to maximize inter-class margin. While the attention to visual features is proved useful by the above-mentioned research efforts, relatively few attempts assess the importance of attention on semantic features. In the ZSL setting, each class contains only one semantic feature vector such that intra-class variations are neglected from the single semantic representation. For example, the object of interest can be occluded partly in the image or in the front of different backgrounds, and a unique class-wise semantic feature vector is hard to reflect such variances. Thus, attention to semantic features is worth further exploring in ZSL. Along this line, the most relevant work to ours is LFGAA (Liu et al., 2019), which considers semantic prediction of samples. Different from LFGAA, our method puts attention on discriminating dimensions of semantic features and maps them to the visual domain. The effect of the proposed attention mechanism is to attract the visual representation of ground-truth class and repel others. Besides, our other main difference from LFGAA is the shortcut design in the attention model. We add a shortcut to prevent the model from overfitting on training data and retain discriminating dimensions for unseen class classification. ",
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+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We coin the proposed method as AGZSL in that its main idea is the fusion of Adaptive and Generative mechanisms in solving ZSL/GZSL. (See Figure 1.) We describe below the details of AGZSL. ",
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+    {
+        "type": "text",
+        "text": "3.1 NOTATIONS AND FORMULATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We denote training data as $\\mathcal { D } = \\mathcal { D } _ { S } \\cup \\mathcal { D } _ { U }$ where $\\mathcal { D } _ { S } = \\{ ( \\mathbf { x } _ { s } \\in \\mathcal { X } _ { s } , \\mathbf { a } _ { s } \\in \\mathcal { A } _ { s } , y _ { s } \\in \\mathcal { V } _ { s } ) \\}$ comprises seen data and $\\mathcal { D } _ { U } = \\{ ( \\mathbf { a } _ { u } \\in \\mathcal { A } _ { u } , y _ { u } \\in \\mathcal { Y } _ { u } ) \\}$ includes unseen class labels $y _ { u }$ and the corresponding semantic/attribute feature vectors $\\mathbf { a } _ { u }$ . The dimensions of visual features and semantic features are respectively expressed as $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ and $\\mathbf { a } \\in \\mathbb { R } ^ { k }$ , while the number of total classes is $n$ . We also assume that the seen category $\\mathcal { { D } } _ { s }$ and the unseen category $\\mathcal { \\partial } _ { u }$ are disjoint, i.e., $\\mathcal { V } _ { s } \\cap \\mathcal { V } _ { u } = \\emptyset$ and $\\mathcal { D } _ { s } \\cup \\mathcal { D } _ { u } = \\mathcal { D }$ . Besides, we define $\\mathcal { \\bar { A } } _ { s } \\cup \\mathcal { A } _ { u } = \\mathcal { A } \\in \\mathbb { R } ^ { k \\times n }$ as semantic features of all classes. Finally, the virtual data via mixup interpolations is denoted as $\\mathcal { D } _ { V } = \\{ ( \\mathbf { x } _ { v } \\in \\mathcal { X } _ { v } , \\mathbf { a } _ { v } \\in \\mathcal { A } _ { v } , y _ { v } \\in \\mathcal { Y } _ { v } ) \\}$ . As we explain later, $y _ { v }$ is simply a new label of virtual class, rather than from label interpolation as in Zhang et al. (2018). ",
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+    {
+        "type": "image",
+        "img_path": "images/56d5ec19a7d5bbebfed59c54dfd09274a74fd1a39aee7d989c5667c12762bf3a.jpg",
+        "image_caption": [
+            "Figure 1: Overview of AGZSL. The same color reflects the same class association between visual features and attributes. (a) Image-adaptive mechanism (marked as purple) based on $\\mathbf { x }$ yields proper attention to adapt the semantic features of a and $\\mathbf { a } ^ { \\prime }$ so that their S2V embedding (marked as orange) displays the desired effects in $\\hat { \\bf x }$ (attracted to $\\mathbf { x }$ ) and in $\\hat { \\mathbf { x } } ^ { \\prime }$ (repelled from x). (b) We generate virtual classes from a pair of classes of semantic features a and $\\mathbf { a } ^ { \\prime }$ with selected weights $\\lambda$ via beta distribution $\\mathbf { B } ( \\alpha , \\beta )$ . $\\bar { \\bf X } _ { v } / { \\bf X } _ { v } ^ { \\prime }$ and $\\mathbf { a } _ { v } / \\mathbf { a } _ { v } ^ { \\prime }$ represent visual and semantic features of virtual classes. "
+        ],
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "To perform the classification task, we explore the similarity correlations in the visual domain and consider semantic-to-visual (S2V) embeddings. Analogous to COSMO (Atzmon & Chechik, 2019), our method includes seen and unseen experts, $f _ { s }$ and $f _ { u }$ , to classify testing samples as shown in Figure 2; however, the algorithmic designs are fundamentally different. The novel component of our method is the introduction of Image-Adaptive Semantics (IAS) in both experts to expand the spectrum of S2V embeddings. Specifically, in training, IAS diversifies the total mappings of S2V embedding from the number of different semantic vectors/classes to the number of samples. Its effect is twofold. First, it boosts the classification performance of the seen expert. Second, it generalizes the meta-learning of the unseen expert with virtual data and yields good performance in the inference of samples from unseen classes. Given a testing sample $\\mathbf { x }$ in inference, we apply the two learned experts by ",
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+    {
+        "type": "equation",
+        "img_path": "images/874215ba3c49b5d2dce3fbc39d4bbc5239cb709b20d757e2a06c2e20b0ce3553.jpg",
+        "text": "$$\n\\hat { y } _ { s } = \\underset { y \\in \\mathcal { Y } } { \\arg \\operatorname* { m a x } } f _ { s } ( \\mathbf { x } , \\mathcal { A } ) \\quad \\mathrm { a n d } \\quad \\hat { y } _ { u } = \\underset { y \\in \\mathcal { Y } _ { u } } { \\arg \\operatorname* { m a x } } f _ { u } ( \\mathbf { x } , \\mathcal { A } _ { u } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To decide the class label $\\hat { y }$ of $\\mathbf { x }$ , we use the seen expert to discriminate whether the sample is from the seen category. If it is the case then $\\hat { y }$ is decided by $f _ { s }$ , and otherwise by $f _ { u }$ . That is, we have ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee8fda9e320323d96aa215e250e35c71b16f3667650c577fda6ad309fe5691e3.jpg",
+        "text": "$$\n\\hat { y } = \\left\\{ \\begin{array} { l l } { \\hat { y } _ { s } } & { \\mathrm { i f } \\ \\hat { y } _ { s } \\in \\mathcal { Y } _ { s } , } \\\\ { \\hat { y } _ { u } } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 IMAGE ADAPTIVE SEMANTICS (IAS) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As ZSL/GZSL assumes one semantic feature vector for each class, the conventional semantic-tovisual architecture leads to one single class-level visual feature vector, which is common to all samples of the same class. To learn such a single embedding vector with the constraint of being simultaneously similar to all visual features of samples from the same class tends to yield unsatisfactory classification outcomes in that it simply ignores the intra-class variations in visual features. ",
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+        "type": "text",
+        "text": "Diversifying semantic-to-visual embeddings with IAS establishes the core of our model as shown in Figure 3. To realize the concept, the network module learns to map image-adaptive semantic features to the corresponding visual features, according to the cosine similarity. To this effect, IAS leverages the given visual features $\\mathbf { x }$ to focus on discriminative dimensions of the respective semantic features. Specifically, we apply IAS to obtain modified semantic features $\\hat { \\mathcal { A } } ( \\mathbf { x } ) \\in \\mathbb { R } ^ { k \\times n }$ by ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/63e77f1e64f88c8f9315a5ddcc6b38088c23fcb84b2f6c5ca17776ebbd3099dc.jpg",
+        "text": "$$\n\\hat { \\cal A } ( x ) = { \\cal A } + { \\cal A } \\odot \\mathrm { s o f t m a x } ( g ( { \\bf x } ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "image",
+        "img_path": "images/a08475917e5cb75891a56394168f2ccfff6ac532478aa5c07a1313566727bc64.jpg",
+        "image_caption": [
+            "Figure 2: The proposed seen and unseen experts. In the learning phase, the seen expert $f _ { s }$ is trained with seen visual features $\\mathbf { x } _ { s }$ , seen semantic features $\\mathcal { A } _ { s }$ , and corresponding class labels $y _ { s }$ , while the unseen expert $f _ { u }$ is trained via meta-learning with virtual data. In the inference phase, given a testing sample $\\mathbf { x }$ , $f _ { s }$ is additionally fed with all semantic features $\\mathcal { A }$ , and $f _ { u }$ only with unseen semantic features $\\mathcal { A } _ { u }$ . The decision rule leading to the class label $\\hat { y }$ is based on (2). Note that in both training and inference stages, the image-adaptive mechanism to semantic features is carried out. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { s o f t m a x } ( g ( \\mathbf { x } ) ) \\in \\mathbb { R } ^ { k }$ represents using a linear layer $g$ to predict semantic attention and $\\odot$ denotes Hadamard product to combine attention and semantic features. Note that the shortcut implementation in (3) is pivotal as the softmax function often puts high attention weights on sparse dimensions and draws others near to zero. Without the shortcut, the modified $\\hat { A }$ would miss some part of semantic features and degenerate the classification performance. Also, it is worthy to point out that compared to LFGAA (Liu et al., 2019), our method pays attention to the semantic features of classes, not the semantic prediction of the model. With (3), the process to drive the image-adaptive class-level visual features $\\mathcal { V } ( \\mathbf { x } )$ can be expressed by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d7299b728aaf6ee8145252a49ff4e598a7c6a7ece3d99a49b89728e3c7341cc3.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { A } \\in \\mathbb { R } ^ { k \\times n } \\xrightarrow [ \\mathrm { I A S } ] { \\mathbf { x } \\in \\mathbb { R } ^ { d } } \\hat { \\mathcal { A } } ( \\mathbf { x } ) \\in \\mathbb { R } ^ { k \\times n } \\longmapsto \\mathcal { V } ( \\mathbf { x } ) \\in \\mathbb { R } ^ { d \\times n } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Once we have obtained the class-level visual features, we then use the cosine similarity and softmax to calculate the probability of each class. We have the class probability of prediction on $\\mathbf { x }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/5cdb1d57ad9d45e08bf088ad32c507c5d9d241d9c01da18cad41be06cec2384f.jpg",
+        "text": "$$\n\\mathbf { p } ( \\mathbf { x } ) = f ( \\mathbf { x } , { \\cal A } ) = \\operatorname { s o f t m a x } ( \\sigma \\times \\cos ( \\mathcal { V } ( \\mathbf { x } ) , \\mathbf { x } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "where $\\sigma$ is the learnable scale and $f$ is indeed the general form of a classification expert. We conclude by stating that the Cross-Entropy(CE) loss is adopted to update the model: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/94977931c98f2da6a3e3bc620edabc83afa9c0f25103ac85bec18ce6520191aa.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C E } } = - \\sum _ { \\mathbf { x } } \\log p _ { y } ( \\mathbf { x } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $p _ { y } ( \\mathbf { x } )$ is the predicted probability of ground-truth label. ",
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+    {
+        "type": "text",
+        "text": "3.3 SEEN EXPERT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The objective of seen expert $f _ { s }$ is to recognize seen classes and otherwise decide whether a testing sample belongs to unseen classes in inference phase. In zero-shot learning, especially GZSL, the capability to single out testing samples of unseen classes is crucial to the classification performance. If one can establish reasonable split mechanisms such as in Min et al. (2020) and Chen et al. (2020), the more challenging GZSL would be simplified into ZSL. For example, COSMO (Atzmon & Chechik, 2019) utilizes Confidence Base Gating to discriminate between seen and unseen classes, while DVBE (Min et al., 2020) considers the entropy threshold. In our formulation, the seen expert learns to predict all possible classes, including seen and unseen, although the training data assume only seen class labels. That is, $f _ { s }$ is trained to map (image adaptive) seen semantic features to their corresponding visual features. Such optimization can be thought of as one-category anomaly detection over only seen classes. Thus, in inference, if the highest cosine similarity value by the seen expert $f _ { s }$ is not yielded by one of the seen classes, the testing sample is considered abnormal from the unseen category and its classification should be decided by the unseen expert discussed next. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/08266f30c339f899e05262ad087e80f848982a9fcc5e600d3d804fda2ec1de07.jpg",
+        "image_caption": [
+            "Figure 3: The architectures of IAS and S2V (FC: linear layer, $L _ { 2 }$ : $L _ { 2 }$ -normalization). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "3.4 UNSEEN EXPERT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "So far we have stated the objective of the unseen expert $f _ { u }$ is to classify the labels of those that cannot be decided by the seen expert, as in (2). To resolve the dilemma of zero-shot setting, we apply meta-learning to train $f _ { u }$ over virtual classes by mixup interpolations. The exact number of virtual classes in each episode is decided by the pilot study (Li et al., 2019b), and an ablation study in appendix is provided to analyze its effect on the performance of zero-shot classification. To begin with, in each episode we randomly select $m$ pairs of seen classes. Then, we create $m$ virtual classes out of these pairs by mixup. The semantic and visual features for these virtual classes are defined as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/28dc185ac8282f19cfc02f252d195e3a766e4a417f351ae1901539e4e66fc4f0.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { a } _ { v } = \\lambda \\mathbf { a } _ { i } + \\left( 1 - \\lambda \\right) \\mathbf { a } _ { j } , } \\\\ { \\mathbf { x } _ { v } = \\lambda \\mathbf { x } _ { i } + \\left( 1 - \\lambda \\right) \\mathbf { x } _ { j } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\bf a } _ { i }$ and $\\mathbf { a } _ { j }$ are semantic features of paired classes, and $\\mathbf { x } _ { i }$ and $\\mathbf { x } _ { j }$ are randomly chosen from visual samples of the paired classes. Following Zhang et al. (2018), $\\lambda \\doteq [ 0 , 1 ]$ is sampled from the beta distribution. We then define a new virtual class $y _ { v }$ , with $\\mathbf { a } _ { v }$ as semantic features, to be associated with virtual visual features $\\mathbf { x } _ { v }$ . Previous generative models (Schonfeld et al., 2019; Xian et al., 2019) synthesize unseen samples by GAN/VAE or their variants. In comparison, the proposed meta-learning has at least three advantages for ZSL. First, the small number of classes in some datasets, such as 50 in AWA2, may fall short to provide sufficient support set such that the generative model is hard to be optimized, while episodic meta-learning over virtual classes can overcome this problem. Second, some part of semantic features which are important to classify unseen classes do not have enough training classes in the datasets. With mixup semantic features as virtual classes, our model can learn underlying discriminating semantic features well. Third, the fusion of image-adaptive mechanism and meta-learning is flexible. As a result, we do not need online fine-tuning in applying the unseen expert $f _ { u }$ yielded by meta-learning to predict the label of unseen class for testing samples. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETTING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Datasets. In GZSL, there are four widely used benchmark datasets for evaluation, which are AWA2 (Xian et al., 2018a), USCD Birds-200-2011 (CUB) (Welinder et al., 2010), SUN (Patterson & Hays, 2012), and aPY (Farhadi et al., 2009). AWA2 contains 37K images and 50 kinds of animals with 85- dimension attributes. CUB consists of 11K images and 200 bird species described with 312-dimension attributes. SUN is composed of 14K images 717 classes labeled with 102-dimension attributes. aPY includes 15K images and 32 different classes with 64-dimension attributes. The attributes of class in datasets will be taken as the semantic features in our work. We follow existing GZSL approaches (Min et al., 2020; Li et al., 2019b) and extract 2048-dimensional visual features for each image by the ResNet101 backbone (He et al., 2016) pretrained on ImageNet-1K. We then split the data into seen and unseen classes according to the benchmark procedure from Xian et al. (2017). ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5aa856b209f7c6b51299bbbefd0d4ce2d447c1fbb81a9659ef29d544794523eb.jpg",
+        "table_caption": [
+            "Table 1: GZSL results on four datasets. All methods in comparison utilize ResNet101 as the backbone for fairness. Notation ”\\*” means the method fine-tunes the backbone to match the characteristic of datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"4\">AWA2</td><td colspan=\"4\">CUB GZSL</td><td colspan=\"4\">SUN</td><td colspan=\"4\">aPY</td></tr><tr><td>ZSL</td><td rowspan=\"2\">U</td><td colspan=\"2\">GZSL</td><td>ZSL</td><td colspan=\"2\"></td><td></td><td>ZSL</td><td colspan=\"2\">GZSL</td><td></td><td>ZSL</td><td colspan=\"2\">GZSL</td><td></td></tr><tr><td>T1</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>S</td><td></td><td>H</td></tr><tr><td>SP-AEN</td><td>58.5</td><td>23.3</td><td>90.9</td><td>37.1</td><td>55.4</td><td>34.7</td><td>70.6</td><td>46.6</td><td>59.2</td><td>24.9</td><td>38.6</td><td>30.3</td><td>24.1</td><td>13.7</td><td>63.4</td><td>22.6</td></tr><tr><td>DLFZRL</td><td>60.9</td><td>1</td><td>-</td><td>45.1</td><td>51.9</td><td>-</td><td>-</td><td>37.1</td><td>42.5</td><td>-</td><td>-</td><td>24.6</td><td>38.5</td><td>1</td><td>1</td><td>31.0</td></tr><tr><td>PSR</td><td>63.8</td><td>20.7</td><td>73.8</td><td>32.3</td><td>56.0</td><td>24.6</td><td>54.3</td><td>33.9</td><td>61.4</td><td>20.8</td><td>37.2</td><td>26.7</td><td>38.4</td><td>13.5</td><td>51.4</td><td>21.4</td></tr><tr><td>CDL</td><td>-</td><td>1</td><td>1</td><td>1</td><td>54.5</td><td>23.5</td><td>55.2</td><td>32.9</td><td>63.6</td><td>21.5</td><td>34.7</td><td>26.5</td><td>43.0</td><td>19.8</td><td>48.6</td><td>27.1</td></tr><tr><td>PQZSL</td><td>=</td><td>1</td><td>-</td><td>1</td><td>-</td><td>53.2</td><td>51.4</td><td>46.9</td><td>1</td><td>35.1</td><td>35.3</td><td>35.2</td><td>27.9</td><td>64.1</td><td>64.1</td><td>38.8</td></tr><tr><td>f-VAEGAN-D2*</td><td>70.3</td><td>57.1</td><td>76.1</td><td>65.2</td><td>72.9</td><td>63.2</td><td>75.6</td><td>68.9</td><td>65.6</td><td>50.1</td><td>37.8</td><td>43.1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>LsrGAN</td><td>-</td><td>54.6</td><td>74.6</td><td>63.0</td><td>1</td><td>48.1</td><td>59.1</td><td>53.0</td><td>-</td><td>44.8</td><td>37.7</td><td>40.9</td><td>-</td><td>-</td><td></td><td>1</td></tr><tr><td>TF-VAEGAN*</td><td>73.4</td><td>55.5</td><td>83.6</td><td>66.7</td><td>74.3</td><td>63.8</td><td>79.3</td><td>70.7</td><td>66.7</td><td>41.8</td><td>51.9</td><td>46.3</td><td>-</td><td></td><td></td><td>=</td></tr><tr><td>OCD-CVAE</td><td>71.3</td><td>59.5</td><td>73.4</td><td>65.7</td><td>60.9</td><td>44.8</td><td>59.9</td><td>51.3</td><td>62.1</td><td>44.8</td><td>42.9</td><td>43.8</td><td>=</td><td></td><td>=</td><td>1</td></tr><tr><td>ZSML Softmax</td><td>76.1</td><td>58.9</td><td>74.6</td><td>65.8</td><td>69.6</td><td>60.0</td><td>52.1</td><td>55.7</td><td>60.2</td><td>-</td><td>1</td><td>1</td><td>64.1</td><td>36.3</td><td>46.6</td><td>40.9</td></tr><tr><td>GXE</td><td>71.1</td><td>56.4</td><td>81.4</td><td>66.7</td><td>54.4</td><td>47.4</td><td>47.6</td><td>47.5</td><td>62.6</td><td>36.3</td><td>42.8</td><td>39.3</td><td>38.0</td><td>26.5</td><td>74.0</td><td>39.0</td></tr><tr><td>E-PGN</td><td>73.4</td><td>52.6</td><td>83.5</td><td>64.6</td><td>72.4</td><td>52.0</td><td>61.1</td><td>56.2</td><td>1</td><td>-</td><td></td><td>-</td><td>1</td><td>-</td><td>1</td><td>1</td></tr><tr><td>Relation Net</td><td>64.2</td><td>30.0</td><td>93.4</td><td>45.3</td><td>55.6</td><td>38.1</td><td>61.1</td><td>47.0</td><td>-</td><td>=</td><td></td><td>=</td><td>1</td><td>-</td><td>=</td><td>1</td></tr><tr><td>Correlation Net</td><td>-</td><td>=</td><td>-</td><td>1</td><td>45.8</td><td>41.9</td><td>1</td><td>-</td><td>-</td><td>=</td><td>=</td><td>=</td><td>1</td><td>=</td><td>=</td><td>=</td></tr><tr><td>LFGAA+Hibrid</td><td>68.1</td><td>27.0</td><td>93.4</td><td>41.9</td><td>67.6</td><td>36.2</td><td>80.9</td><td>50.0</td><td>62.0</td><td>18.5</td><td>40.0</td><td>25.3</td><td></td><td></td><td></td><td>=</td></tr><tr><td>AREN*</td><td>66.9</td><td>54.7</td><td>79.1</td><td>64.7</td><td>72.5</td><td>63.2</td><td>69.0</td><td>66.0</td><td>60.6</td><td>40.3</td><td>32.3</td><td>35.9</td><td>39.2</td><td>30.0</td><td>47.9</td><td>36.9</td></tr><tr><td>COSMO</td><td>-</td><td>=</td><td>=</td><td>1</td><td>1</td><td>44.4</td><td>57.8</td><td>50.2</td><td>-</td><td>44.9</td><td>37.7</td><td>41.0</td><td>1</td><td>-</td><td>=</td><td>=</td></tr><tr><td>DVBE*</td><td>-</td><td>62.7</td><td>77.5</td><td>69.4</td><td>1</td><td>64.4</td><td>73.2</td><td>68.5</td><td>-</td><td>44.1</td><td>41.6</td><td>42.8</td><td>1</td><td>37.9</td><td>55.9</td><td>45.2</td></tr><tr><td>ours</td><td>73.8</td><td>65.1</td><td>78.9</td><td>71.3</td><td>57.2</td><td>41.4</td><td>49.7</td><td>45.2</td><td>63.3</td><td>29.9</td><td>40.2</td><td>34.3</td><td>41.0</td><td>35.1</td><td>65.5</td><td>45.7</td></tr><tr><td>ours*</td><td>76.4</td><td>69.0</td><td>86.5</td><td>76.8</td><td>77.2</td><td>69.2</td><td>76.4</td><td>72.6</td><td>66.2</td><td>50.5</td><td>43.1</td><td>46.5</td><td>43.7</td><td>36.2</td><td>58.6</td><td>44.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Evaluation. We evaluate our approach for both ZSL and GZSL settings. For the ZSL setting, the average per-class top-1 accuracy $\\mathbf { \\Pi } ^ { ( \\mathbf { T } ) }$ on unseen classes is taken as the evaluation metric. For GZSL (Xian et al., 2017), the evaluation metrics include seen classes accuracy S, unseen class accuracy U, and their harmonic mean ${ \\bf H } = ( 2 { \\bf U } \\cdot { \\bf S } ) / ( { \\bf U } + { \\bf S } )$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "Implementation. As suggested in Li et al. (2019b), we normalize the visual and semantic features into [0, 1]. The architecture of semantic-to-visual embedding contains a two-layer linear model with 1,600 hidden units and utilizes ReLU on the hidden and output layer. The seen and unseen experts are trained by Adam optimizer with a learning rate $5 \\times 1 0 ^ { - 5 }$ and $5 \\times 1 0 ^ { - 4 }$ respectively for all datasets. We apply 200,000 episodes to train the unseen expert. In each episode, we randomly generate 16 or 20 (based on the dataset) virtual classes and 4 samples for each class. We follow the setting in (Min et al., 2020) to fine-tune the backbone. ",
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+    {
+        "type": "text",
+        "text": "4.2 COMPARISON WITH STATE-OF-THE-ART APPROACHES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare our method with 13 recent GZSL approaches. These include SP-AEN (Chen et al., 2018), DLFZRL (Tong et al., 2019), PSR (Annadani & Biswas, 2018), CDL (Jiang et al., 2018), PQZSL (Li et al., 2019a), f-VAEGAM-D2 (Xian et al., 2019), LsrGAN (Vyas et al., 2020), TF-VAEGAN (Narayan et al., 2020), OCD-CVAE (Keshari et al., 2020), ZSML Softmax (Verma et al., 2020), E-PGN (Yu et al., 2020), Relation Net (Sung et al., 2018), Correlation Net (Hu et al., 2018), GXE Li et al. (2019b), LFGAA $^ +$ Hybrid (Liu et al., 2019), AREN (Xie et al., 2019), COSMO (Atzmon & Chechik, 2019), DVBE (Min et al., 2020). Table 1 shows that the proposed AGZSL achieves the best GZSL performance on all four datasets. It improves harmonic mean by $7 . 4 \\%$ on AWA2, $1 . 9 \\%$ on CUB, $0 . \\bar { 2 } \\%$ on SUN, and $0 . 5 \\%$ on aPY, respectively. Particularly, among those methods in comparison, DVBE needs to search the entropy threshold $\\tau$ and the embedding model for the best harmonic means according to each dataset. In contrast to DVBE, without entropy threshold and specific model searching, our model achieves over $4 \\%$ improvement on most datasets. The main reason for better performance is that AGZSL provides more discriminating visual representation for both seen and unseen classes. Therefore, our model can effectively separate seen and unseen samples and achieve better performance in the harmonic mean. Notably, AGZSL drops performance on aPY after backbone fine-tuning. It could be due to that its number of seen classes and training images are fewest among the four datasets. Fine-tuning the backbone causes the visual features to overfit training samples and become less discriminating for correct classification. ",
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+    {
+        "type": "table",
+        "img_path": "images/237c5626476fd2154c30ac4ae0da096f18872359c68072e2e3f67910ad8dae11.jpg",
+        "table_caption": [
+            "Table 2: The effect of virtual classes and IAS for GZSL. We remove these two components as our baseline (b). In the ablation study, we add virtual classes (v) and IAS (I) step by step to show their effect on GZSL. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"4\">AWA2 GZSL</td><td colspan=\"4\">CUB</td><td colspan=\"4\">SUN</td><td colspan=\"4\">aPY</td></tr><tr><td>ZSL</td><td colspan=\"3\"></td><td>ZSL</td><td></td><td colspan=\"2\">GZSL</td><td>ZSL</td><td colspan=\"3\">GZSL</td><td>ZSL</td><td colspan=\"3\">GZSL</td></tr><tr><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td><td>T1</td><td>U</td><td>S</td><td>H</td></tr><tr><td>b</td><td>72.2</td><td>63.2</td><td>86.1</td><td>72.9</td><td>73.7</td><td>68.6</td><td>71.3</td><td>69.9</td><td>63.9</td><td>47.6</td><td>42.1</td><td>44.7</td><td>40.0</td><td>33.4</td><td>57.2</td><td>42.2</td></tr><tr><td>b+v</td><td>75.8</td><td>66.9</td><td>86.1</td><td>75.3</td><td>76.1</td><td>70.3</td><td>71.3</td><td>70.8</td><td>65.6</td><td>48.6</td><td>42.1</td><td>45.1</td><td>42.8</td><td>35.2</td><td>57.2</td><td>43.6</td></tr><tr><td>b+v+I</td><td>76.4</td><td>69.0</td><td>86.5</td><td>76.8</td><td>77.2</td><td>69.2</td><td>76.4</td><td>72.6</td><td>66.2</td><td>50.5</td><td>43.1</td><td>46.5</td><td>43.7</td><td>36.2</td><td>58.6</td><td>44.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, generative models achieve higher performance on CUB and SUN, because in these two datasets there are many seen classes for synthesizing unseen samples of better quality. In comparison, our model overcomes the influence of few seen classes and also achieves the best performance on AWA2 and aPY. Our results empirically support that the introduction of virtual classes and samples are indeed helpful to classify unseen classes. Finally, we remark that since the backbone is pretrained on ImageNet-1K, which is rather different from the ZSL fine-grained databases, e.g., CUB and SUN. Thus, the harmonic mean improves a lot after fine-tuning the backbone on these two datasets. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "To evaluate the benefits of tackling ZSL/GZSL with virtual classes and IAS, we carry out an ablation study on the four datasets. We incrementally include each key component to assess their effect. As shown in Table 2, the virtual classes improve ZSL accuracy significantly. The ZSL performance improves $4 . 6 \\%$ on AWA2, $2 . 4 \\%$ on CUB, $1 . 7 \\%$ on SUN, and $2 . 8 \\%$ on aPY, respectively. The result implies that the virtual classes are advantageous to ZSL. After adding IAS to our model, both seen and unseen accuracy increases, and the harmonic mean $_ \\mathrm { H }$ improves about $1 . 5 \\%$ on average. The consistent gain exemplifies that IAS can provide better semantic-to-visual embedding and achieve higher performance on GZSL. ",
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+    {
+        "type": "text",
+        "text": "4.4 COSINE SIMILARITY MARGIN ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 illustrates the effect of the proposed AGZSL model. The cosine similarity margin means the inner product value between the correct class and the other nearest class. A negative margin indicates that the model predicts a wrong class. On the other hand, a large positive margin implies that the model results in a proper embedding distinguishing the correct class from others effectively. Thus, seeking a positive and larger margin is a crucial and important objective in GZSL. As shown in the first row of Figure 4, the margins of unseen samples shift right on all four datasets by our method. It shows that AGZSL indeed generates better embedding on the visual space and makes the model easier to predict the correct class. Moreover, from the second row in Figure 4, the plots show that our method improves the margins of seen samples dramatically on most datasets. The above experiment explains that AGZSL can achieve a higher margin for most samples and achieves better classification performance on the four testing datasets. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.5 THE DISTRIBUTION OF VIRTUAL CLASSES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To explicitly exhibit the distributions of seen, unseen and virtual classes, we apply PCA to map the class-wise semantic features to a two-dimensional space, as shown in Figure 5. The features of virtual classes scatter around seen classes, because we have used B(5,1) to generate virtual classes. Advantageously, some virtual classes are very close to unseen classes in the semantic domain and turn out to be helpful for enabling the unseen expert to classify samples of unseen classes. With meta-learning over these numerous virtual classes and samples, the proposed S2V can better learn the embedding between semantic and visual domains, and results in an effective unseen expert. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/25bda5a37afb8712ef3f138799a643e1efb0fa38776647f7531d961455a8535c.jpg",
+        "image_caption": [
+            "Figure 4: The cosine similarity margin between the correct class and other nearest class. The blue area depicts the cosine similarity margin of baseline without IAS and generative learning. On the other hand, the orange area represents the cosine similarity margin of ours. Most of the samples improve their margin by our method, suggesting the proposed techniques are beneficial for GZSL. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab9faddf0a9a95528dc9b93715b3073bf0095351124d80d3bc19f2c3cd2d1abb.jpg",
+        "image_caption": [
+            "Figure 5: The semantic features of virtual/seen/unseen classes over the four datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose Image Adaptive Semantics (IAS) and meta-learning with virtual classes and samples to solve the ZSL/GZSL problem. To deal with the intra-class visual discrepancies, IAS adaptively emphasizes the most discriminating dimensions in semantic features with respect to the underlying visual features. To better classify samples of the unseen classes in inference, we propose a novel formulation of generative meta-learning. Different from previous generative models that focus on synthesizing unseen samples for training the model, we create virtual classes and their respective virtual samples in the training phase. Further, to imitate the ZSL inference scenario, we carry out meta-learning with these virtual data to extend our model. In our experiment, we demonstrate that AGZSL is beneficial to tackle the challenging ZSL/GZSL problem and achieves significant advantages over those in comparison on the four ZSL/GZSL benchmark datasets. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            755,
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+            768
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the MOST, Taiwan under Grants 110-2634-F-001-009 and 107-2628-E-002-008-MY3. We are grateful to the National Center for High-performance Computing for providing computational resources and facilities. ",
+        "bbox": [
+            176,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/8cfdaf2b64ef3be9356e3a9283a828b5fac41f2baf0ade2c46da735c8e240970.jpg",
+        "table_caption": [
+            "Table 3: The impact of shortcut in IAS on four datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">shortcut</td><td colspan=\"3\">AWA2</td><td colspan=\"4\">CUB</td><td colspan=\"4\">SUN</td><td colspan=\"4\">aPY</td></tr><tr><td>ZSL</td><td colspan=\"3\">GZSL</td><td>ZSL</td><td colspan=\"3\">GZSL</td><td>ZSL</td><td colspan=\"3\">GZSL</td><td>ZSL</td><td colspan=\"3\">GZSL</td></tr><tr><td>T</td><td>U</td><td>S</td><td>H</td><td>T</td><td>U</td><td>S</td><td>H</td><td>T</td><td>U</td><td>S</td><td>H</td><td>T</td><td>U S</td><td>H</td></tr><tr><td></td><td>23.9</td><td>13.0</td><td>30.0</td><td>18.2</td><td>23.3</td><td>17.0</td><td>45.8</td><td>24.8 42.1</td><td>22.5</td><td>20.3</td><td>21.4</td><td>30.5</td><td>14.9</td><td>4.7</td><td>7.1</td></tr><tr><td>√</td><td>76.4</td><td>69.0</td><td>86.5</td><td>76.8</td><td>77.2</td><td>69.2</td><td>76.4</td><td>72.6 66.2</td><td>50.5</td><td>43.1</td><td>46.5</td><td>43.7</td><td>36.2</td><td>58.6</td><td>44.8</td></tr></table>",
+        "bbox": [
+            178,
+            184,
+            820,
+            242
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/a896e7bd1cf69488191429f65186210cbe454a5f967b6e8584ae0b4cd0279542.jpg",
+        "image_caption": [
+            "Figure 6: The difference of cosine similarity between our model and baseline. In our experiment, most of the samples improve their embedding with IAS and virtual classes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            279,
+            816,
+            478
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Shortcut in IAS Table 3 shows that shortcut in IAS improves performance a lot. The unseen class accuracy increase around $5 0 \\%$ on AWA2 and SUN. Besides, the harmonic mean also enhances $2 0 \\%$ to $6 0 \\%$ on four datasets with the shortcut. The main reason is that the softmax usually puts high attention weight on few dimensions and decreases others near to zero. Thus, without shortcut, the model is prone to overfitting on training data. ",
+        "bbox": [
+            174,
+            553,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Cosine similarity enhancement Figure 6 illustrates better embedding for seen and unseen classes in our model. Since we apply cosine similarity to classify samples, the higher score means visual representation of each class is closer to their visual samples. The first row in 6 shows effect of our model on unseen classes. With virtual classes and IAS, cosine similarity increase averaged 0.05 to 0.1 on different datasets. The increasing amount shows our model provides better visual-to-semantic embedding for unseen classes and samples are easier to classify correctly. Therefore, our model can achieve good ZSL performance. Besides, the second row in 6 illustrates the effect of IAS on seen classes. Most of the testing seen samples increase their cosine similarity around 0.02 on four datasets. With the IAS and virtual classes, the model can achieve better visual-to-semantic embedding on all classes and obtain better the generalized seen accuracy on four datasets. ",
+        "bbox": [
+            173,
+            641,
+            825,
+            780
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "virtual classes number per episode To search the class number in each episode for best ZSL, we search 4 different numbers, i.e, 4,8,12, and 20 per episode. Table 4 shows that the best class number setting is related to the class number of classes. For example, the AWA2 and aPY which include fewer unseen classes obtain the best ZSL accuracy with 16 virtual classes. On the other hand, the CUB and SUN which include more unseen classes achieve the best performance with 20 virtual classes. The reason is that the advantage of meta-learning is to simulate inference scenarios while training (Li et al., 2019b). The virtual class number set closer to the unseen class number is favorable in the meta-learning setting. Therefore, the best virtual class number is different according to the unseen number of the dataset. ",
+        "bbox": [
+            174,
+            799,
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+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f34dccf20a50736ce887aa631968cdaad67a3e368a5550b2ec7da2a8b51c1212.jpg",
+        "image_caption": [
+            "Figure 7: Beta distribution. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            333,
+            99,
+            663,
+            279
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/e2fb59ca84011db680a305f871691cf3c34d88616c291841155fe0cec389531c.jpg",
+        "table_caption": [
+            "Table 4: ZSL accuracy w.r.t. virtual classes per episode. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>k</td><td>AWA2</td><td>CUB</td><td>SUN</td><td>aPY</td></tr><tr><td>8</td><td>75.7</td><td>76.2</td><td>64.6</td><td>40.8</td></tr><tr><td>12</td><td>76.0</td><td>76.3</td><td>65.1</td><td>42.8</td></tr><tr><td>16</td><td>76.4</td><td>76.3</td><td>65.1</td><td>43.7</td></tr><tr><td>20</td><td>75.5</td><td>77.2</td><td>66.2</td><td>41.1</td></tr></table>",
+        "bbox": [
+            210,
+            359,
+            457,
+            435
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/62caca669aa36ed3870a810182a020ecc658f0f0b6b8b2823f42bbb8fb49c57f.jpg",
+        "table_caption": [
+            "Table 5: ZSL accuracy w.r.t. beta distribution. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>B(a,β)</td><td>AWA2</td><td>CUB</td><td>SUN</td><td>aPY</td></tr><tr><td>(5,5)</td><td>74.8</td><td>75.6</td><td>65.4</td><td>38.2</td></tr><tr><td>(1,1)</td><td>75.3</td><td>76.4</td><td>65.8</td><td>40.2</td></tr><tr><td>(5,1)</td><td>76.4</td><td>77.2</td><td>66.2</td><td>43.7</td></tr></table>",
+        "bbox": [
+            526,
+            359,
+            802,
+            421
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Effect of beta distribution on ZSL accuracy To explore the best beta distribution setting for ZSL, our experiments apply three different $\\alpha$ and $\\beta$ sets. As shown in Figure 7, B(1,1) means virtual classes distribute uniformly between selected class pairs. Then, B(5,5) means most of the virtual classes distribute among the middle of selected class pairs. On the other hand, B(5,1) means most virtual classes distribute closer to one of the classes in selected class pairs. Table 5 demonstrates that B(5,1) achieves best performance on each dataset. In our speculation, the virtual classes created based on one class are closer to real-world classes. For example, the zebra is very similar to the horse in shape. Thus, the virtual classes distributing among seen classes enhance ZSL accuracy are very beneficial for training ZSL. ",
+        "bbox": [
+            173,
+            460,
+            826,
+            587
+        ],
+        "page_idx": 13
+    }
+]

parse/train/hvdKKV2yt7T/hvdKKV2yt7T.md

@@ -0,0 +1,571 @@
+# DATASET INFERENCE: OWNERSHIP RESOLUTION IN MACHINE LEARNING
+
+Pratyush Maini   
+IIT Delhi∗   
+pratyush.maini@gmail.com
+
+Mohammad Yaghini, Nicolas Papernot University of Toronto and Vector Institute {mohammad.yaghini,nicolas.papernot}@utoronto.ca
+
+# ABSTRACT
+
+With increasingly more data and computation involved in their training, machine learning models constitute valuable intellectual property. This has spurred interest in model stealing, which is made more practical by advances in learning with partial, little, or no supervision. Existing defenses focus on inserting unique watermarks in a model’s decision surface, but this is insufficient: the watermarks are not sampled from the training distribution and thus are not always preserved during model stealing. In this paper, we make the key observation that knowledge contained in the stolen model’s training set is what is common to all stolen copies. The adversary’s goal, irrespective of the attack employed, is always to extract this knowledge or its by-products. This gives the original model’s owner a strong advantage over the adversary: model owners have access to the original training data. We thus introduce dataset inference, the process of identifying whether a suspected model copy has private knowledge from the original model’s dataset, as a defense against model stealing. We develop an approach for dataset inference that combines statistical testing with the ability to estimate the distance of multiple data points to the decision boundary. Our experiments on CIFAR10, SVHN, CIFAR100 and ImageNet show that model owners can claim with confidence greater than $9 9 \%$ that their model (or dataset as a matter of fact) was stolen, despite only exposing 50 of the stolen model’s training points. Dataset inference defends against state-of-the-art attacks even when the adversary is adaptive. Unlike prior work, it does not require retraining or overfitting the defended model.1
+
+# 1 INTRODUCTION
+
+Machine learning models have increasingly many parameters (Brown et al., 2020; Kolesnikov et al., 2019), requiring larger datasets and significant investment of resources. For example, OpenAI’s development of GPT-3 is estimated to have cost over USD 4 million (Li, 2020). Yet, models are often exposed to the public to provide services such as machine translation (Wu et al., 2016) or image recognition (Wu et al., 2019). This gives adversaries an incentive to steal models via the exposed interfaces using model extraction. This threat raises a question of ownership resolution: how can an owner prove that another suspect model stole their intellectual property? Specifically, we aim to determine whether a potentially stolen model was derived from an owner’s model or dataset.
+
+An adversary may derive and steal intellectual property from a victim in many ways. A prominent way is (1) model extraction (Tramer et al., 2016), where the adversary exploits access to a model’s \` (1.a) prediction vectors (e.g., through an API) to reproduce a copy of the model at a lower cost than what is incurred in developing it. Perhaps less directly, (1.b) the adversary could also use the victim model as a labeling oracle to train their model on an initially unlabeled dataset obtained either from a public source or collected by the adversary. In a more extreme threat model, (2) the adversary could also get access to the dataset itself which was used to train the victim model and train their own model by either (2.a) distilling the victim model, or (2.b) training from scratch altogether. Finally, adversaries may gain (3) complete access to the victim model, but not the dataset. This may happen when a victim wishes to open-source their work for academic purposes but disallows its commercialization, or simply via insider-access. The adversary may (3.a) fine-tune over the victim model, or (3.b) use the victim for data-free distillation (Fang et al., 2019).
+
+Preventing all forms of model stealing is impossible without decreasing model accuracy for legitimate users: model extraction adversaries can obfuscate malicious queries as legitimate ones from the expected distribution. Most prior efforts thus focus on watermarking models before deployment. Rather than preventing model stealing, they aim to detect theft by allowing the victim to claim ownership by verifying that a suspect model responds with the expected outputs on watermarked inputs. This strategy not only requires re-training and decreases model accuracy, it can also be vulnerable to adaptive attacks that lessen the impact of watermarks on the decision surface during extraction. Thus, recent work that has managed to prevail (Yang et al., 2019) despite distillation (Hinton et al., 2015) or extraction (Jia et al., 2020), has suffered a trade-off in model performance.
+
+In our work, we make the key observation that all stolen models necessarily contain direct or indirect information from the victim model’s training set. This holds regardless of how the adversary gained access to the stolen model. This leads us to propose a fundamentally different defense strategy: we identify stolen models because they possess knowledge contained in the private training set of the victim. Indeed, a successful model extraction attack will distill the victim’s knowledge of its training data into the stolen copy. Hence, we propose to identify stolen copies by showing that they were trained (at least partially and indirectly) on the same dataset as the victim.
+
+We call this process dataset inference (DI). In particular, we find that stolen models are more confident about points in the victim model’s training set than on a random point drawn from the task distribution. The more an adversary interacts with the victim model to steal it, the easier it will be to claim ownership by distinguishing the stolen model’s behavior on the victim model’s training set. We distinguish a model’s behavior on its training data from other subsets of data by measuring the ‘prediction certainty’ of any data point: the margin of a given data point to neighbouring classes.
+
+At its core, DI builds on the premise of input memorization, albeit weak. One might think that DI succeeds only for models trained on small datasets when overfitting is likely. Surprisingly, in practice, we find that even models trained on ImageNet end up memorizing training data in some form.
+
+Among related work discussed in $\ S 2$ , distinguishing a classifier’s behavior on examples from its train and test sets is closest to membership inference (Shokri et al., 2017). Membership inference (MI) is an attack predicting whether individual examples were used to train a model or not. Dataset inference flips this situation and exploits information asymmetry: the potential victim of model theft is now the one testing for membership and naturally has access to the training data. Whereas MI typically requires a large train-test gap because such a setting allows a greater distinction between individual points in and out the training set (Yeom et al., 2018; Choo et al., 2020), dataset inference succeeds even when the defender has slightly better than random chance of guessing membership correctly; because the victim aggregates the result of DI over multiple points from the training set.
+
+In summary, our contributions are:
+
+• We introduce dataset inference as a general framework for ownership resolution in machine learning. Our key observation is that knowledge of the training set leads to information asymmetry which advantages legitimate model owners when resolving ownership. We theoretically show on a linear model that the success of MI decreases with the size of the training set (as overfitting decreases), whereas DI is independent of the same. Despite the failure of MI on a binary classification task, DI still succeeds with high probability. We propose two different methods to characterize training vs. test behavior: targeted adversarial attacks in the white-box setting, and a novel ‘Blind Walk’ method for the black-box label-only setting. We then create a concise embedding of each data point that is fed to a confidence regressor to distinguish between points inside and outside a model’s training set. Hypothesis testing then returns the final ownership claim. Unlike prior efforts, our method not only helps defend ML services against model extraction attacks, but also in extreme scenarios such as complete theft of the victim’s model or training data. In $\ S 7$ , we also introduce and evaluate our approach against adaptive attacks. We evaluate our method on the CIFAR10, SVHN, CIFAR100 and ImageNet datasets and obtain greater than $9 9 \%$ confidence in detecting model or data theft via the threat models studied in this work, by exposing as low as 50 random samples from our private dataset.
+
+We remark that dataset inference applies beyond intellectual property issues. For example, Song & Shmatikov (2019) showed that models trained for gender classification also learn features predictive of ethnicity. This raises ethical concerns, and dataset inference could assess whether a sensitive dataset was used by a model developer for different purposes than stated at data collection time.
+
+# 2 RELATED WORK
+
+Model Extraction. Model extraction (Tramer et al., 2016; Jagielski et al., 2020; Truong et al., \` 2021) is the process where an adversary tries to steal a copy of a machine learning model, that may have been remotely deployed (such as over a prediction API). Depending on the level of access provided by the prediction APIs, model extraction may be performed by only using the labels (Chandrasekaran et al., 2019; Correia-Silva et al., 2018) or the entire prediction logits of the deployed service (Orekondy et al., 2018). Model extraction has seen a cycle of attacks and defenses. Once an adversary has knowledge of the defense strategy adopted by the victim, they adaptively modify the attack to circumvent that defense (see watermarking). Model extraction can also be a reconnaissance step used to prepare for further attacks, e.g., finding adversarial examples (Papernot et al., 2017; Shumailov et al., 2020).
+
+Watermarking. Since Uchida et al. (2017) embedded watermarks into neural networks and Adi et al. (2018) used them as signatures to claim possession, watermarks have been widely adopted as a way to resolve ownership claims. The underlying idea is to manipulate the model to learn information other than that from the true data distribution, and use this knowledge for verification afterwards. This strategy not only requires new training procedures and decreases the model’s accuracy (Jia et al., 2020), but is also vulnerable to adaptive attacks that lessen the impact of watermarks on the model’s decision surface during extraction (Liu et al., 2018; Chen et al., 2019; Wang et al., 2019; Shafieinejad et al., 2019).
+
+Membership Inference. Shokri et al. (2017) train a number of shadow classifiers on confidence scores produced by the target model with labels indicating whether samples came from the training or testing set. MI attacks are shown to work in white- (Leino & Fredrikson, 2020; Sablayrolles et al., 2019) as well as black-box scenarios against various target models including generative models (Hayes et al., 2019). Yeom et al. (2018) explore overfitting as the root cause of MI vulnerability. Choo et al. (2020) show that MI can succeed even in scenarios when the victim only provides labels.
+
+Out of Distribution Detection. Liang et al. (2017) and Lee et al. (2018) measure model performance on modifying an input to find if a sample is in or out-of-distribution. The premise is that in-distribution samples are easier to manipulate, whereas out-of-distribution samples require more work. In contrast, our work solves a much more challenging problem: the dataset distribution may be the same, but can we still identify which of the datasets was used for training?
+
+# 3 THREAT MODEL AND DEFINITION OF DATASET INFERENCE
+
+Consider a victim $\nu$ who trains a model $f _ { \nu }$ on their private data $S _ { \nu } \subseteq \kappa _ { \nu }$ , where $\kappa _ { \nu }$ represents the private knowledge of $\nu$ . While $\kappa _ { \nu }$ is an abstract concept that can not be concretely defined, the private dataset $S _ { \nu }$ represents a definite part of the victim’s knowledge that can be formalized. An adversary $\mathcal { A } _ { * }$ may gain access to a subset of $\kappa _ { \nu }$ and use it to train its own model $f _ { A _ { * } }$ . $\nu$ suspects theft, and would like to prove that $f _ { A _ { * } }$ is indeed a copy of $f _ { \nu }$ . Hence, $\nu$ employs dataset inference on $f _ { A _ { * } }$ to determine if a subset of their private knowledge $\kappa \subseteq \kappa _ { \nu }$ was used to train $f _ { A _ { * } }$ . We formally define the victim and their dataset inference experiment below.
+
+Definition 1 (Dataset Inferring Victim $\mathcal { V } ( f , \alpha , m ) )$ Let $\mathcal { V } : \mathcal { F } \times [ 0 , 1 ] \times \mathbb { N } \mapsto \{ 1 , \emptyset \}$ be a victim with private access to $S _ { \nu } \subseteq \kappa _ { \nu }$ , where $\mathcal { F }$ represents the set of all classifiers trained on samples from a data distribution $\mathcal { D }$ . Given classifier $f , \nu$ can reveal at most m samples from $S _ { \nu }$ to either conclusively prove that a subset of their private knowledge $\kappa \subset \kappa _ { \nu }$ has been used in the training of $f$ with a Type-I error $( F P R ) < \alpha$ , or return an inconclusive result $\varnothing$ .
+
+Definition 2 (Dataset Inference Experiment $E x p ^ { D I } ( \mathcal { V } , m , \alpha , S _ { \mathcal { V } } , \mathcal { D } ) )$ Let $\mathcal { F }$ be as in Definition 1, and assume $\mathcal { F } _ { \mathcal { V } }$ to be the set of all classifiers trained on the victim’s private dataset $S _ { \nu } \sim \mathcal { D }$ , and m a natural number. The dataset inference experiment follows:
+
+1. Choose $b \gets \{ 0 , 1 \}$ uniformly at random.   
+2. $f _ { \mathcal { A } _ { \ast } } = f \sim \mathcal { F } _ { \mathcal { V } } \ : i f b = 1$ ; else $f _ { \mathcal { A } _ { \ast } } = f \sim \mathcal { F }$   
+3. $E x p ^ { D I } ( \mathcal { V } , m , \alpha , S _ { \mathcal { V } } , \mathcal { D } ) = \left\{ \begin{array} { r l } { { 1 } } & { { i f \mathcal { V } ( f _ { A _ { \ast } } , \alpha , m ) = 1 a n d b = 1 } } \\ { { 0 } } & { { o t h e r w i s e } } \end{array} \right.$
+
+# 4 THEORETICAL MOTIVATION
+
+Dataset Inference (DI) aims to leverage the disparity in the response of an ML model to inputs that it saw during training time, versus those that it did not. We call this response ‘prediction margin’. In $\ S 4$ , we introduce our theoretical framework. In $\ S 4 . 1$ , we quantify the difference in the expected response of a model to any point in the training and test set. Finally, in $\ S 4 . 2$ we describe how DI succeeds with high probability in this setting, while membership inference (MI) fails.
+
+Setup. Consider a data distribution $\mathcal { D }$ , such that any input-label pair $( \mathbf { x } , y )$ can be described as:
+
+$$
+\begin{array} { r } { y \sim \{ - 1 , + 1 \} ; \quad \mathbf { x _ { 1 } } = y \cdot \mathbf { u } \in \mathbb { R } ^ { K } , \quad \mathbf { x _ { 2 } } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I ) \in \mathbb { R } ^ { D } } \end{array}
+$$
+
+where $\mathbf { x } = ( \mathbf { x _ { 1 } } , \mathbf { x _ { 2 } } ) \in \mathbb { R } ^ { K + D }$ and $\mathbf { u } \in \mathbb { R } ^ { K }$ is a fixed vector. Observe that the last $D$ dimensions of $\mathbf { x }$ represent Gaussian noise (with var. $\sigma ^ { 2 }$ ) having no correlation to the correct label. However, the first $K$ dimensions are sufficient to separate inputs from classes $\{ - 1 , + 1 \}$ (Nagarajan & Kolter, 2019). $S \sim \mathcal { D }$ , s.t. $| S | = m$ represents the private training set of a model with $m$ distinct training examples.
+
+Architecture. We consider the scenario of classifying the input distribution using a linear classifier, $f$ , with weights $\bf { w } = ( w _ { 1 } , w _ { 2 } )$ , such that for any input x: $f ( \mathbf { x } ) = \mathbf { w _ { 1 } } \cdot \mathbf { x _ { 1 } } + \mathbf { w _ { 2 } } \cdot \mathbf { x _ { 2 } }$ . And the final classification decision is $\operatorname { s g n } ( f ( \mathbf { x } ) )$ . While we only discuss the case of a linear network in this analysis, the success of DI only increases with the number of parameters in a machine learning model, as is the case for MI (Yeom et al., 2018). This, in effect makes the following analysis a stronger result to prove. Prior works have also argued how over-parametrized deep learning networks memorize training points (Zhang et al., 2016; Feldman, 2019). At its core, DI builds on the premise of (weak) input memorization. Results on DNNs are discussed in $\ S 7$ .
+
+# 4.1 PREDICTION MARGIN
+
+In our work, we use ‘prediction margin’ to imply the confidence of a machine learning model of its prediction. In other words, we try to capture the robustness of a model’s prediction under uncertainty, which is equivalent to viewing the local landscape of a machine learning model. For the purpose of the theoretical analysis, it is convenient to define it as the margin of a data point from the decision boundary $( y \cdot f ( \mathbf { x } ) )$ . As we scale our method to deep networks in the empirical evaluation, we will describe alternative methods of measuring the ‘prediction margin’ in multi-class settings.
+
+Theorem 1 (Train-Test Margin) Given a linear classifier $f$ trained to classify inputs $( \mathbf { x } , y ) \in S$ (training set), the difference in the expected prediction margin for samples in $S$ and $\mathcal { D }$ is given by $\mathbb { E } _ { ( \mathbf { x } , y ) \sim S } \left[ y \cdot f ( \mathbf { x } ) \right] - \mathbb { E } _ { ( \mathbf { x } , y ) \sim \mathcal { D } }$ $[ y \cdot f ( \mathbf { x } ) ] = D \sigma ^ { 2 }$ , where $\sigma ^ { 2 }$ is the Gaussian noise variance as in $( l )$ .
+
+The proof (Appendix A.2) first calculates the weights of the learned classifier $f$ by assuming that it is trained using gradient descent with a fixed learning rate, and viewing all training points exactly once. We then analyze the expected margin for data points included in training or not.
+
+# 4.2 DATASET INFERENCE V/S MEMBERSHIP INFERENCE
+
+We now show how MI fails to distinguish between train and test samples in the same setting. This happens because an adversary has to make a decision about the presence of a given data point in the training set by querying a single point. However, DI succeeds with high probability in the same setting because it aggregates signal over multiple data points. We note that the statistical differences between the ‘prediction margin’ of training and test data points in $\ S \ 4 . 1$ are only known when we calculate an expectation over multiple samples.
+
+Failure of Membership Inference. Consider a membership inferring adversary $\mathcal { M }$ that has no knowledge of the victim’s training data $S$ , but has domain knowledge such as the publicly available data distribution $\mathcal { D }$ . Define $\bar { \mathcal { M } } ( \mathbf { x } , f )$ as the adversary’s decision function to predict whether $\mathbf { x }$ belongs to $S$ . Let $\mathcal { R }$ represent a distribution that uniformly at random samples from either $S ( b = 1 )$ or $\mathcal { D } ( b = 0 )$ . Then, $\mathcal { M }$ makes a membership decision about $( \mathbf { x } , b ) \sim \mathcal { R }$ . $\Phi$ denotes the Gaussian CDF.
+
+Theorem 2 (Failure of MI) Given a linear classifier $f$ trained on $S$ , the probability that an adversary $\mathcal { M }$ correctly predicts the membership of inputs randomly belonging to the training or test set, $\begin{array} { r } { \mathbb { P } _ { \mathbf { x } \sim \mathcal { R } } \lbrack \mathcal { M } ( \mathbf { x } , f ) = b \rbrack \ = \ 1 - \ \Phi \left( - \sqrt { \frac { D } { 2 m } } \right) } \end{array}$ , and decreases with $| S | = m$ . Moreover, $\begin{array} { r } { \operatorname* { l i m } _ { m  \infty } \mathbb { P } _ { \mathbf { x } \sim \mathcal { R } } [ \mathcal { M } ( \mathbf { x } , f ) = b ] = 0 . 5 . } \end{array}$ .
+
+The theorem suggests that the success of MI when querying a single data point is extremely low. Ass $m$ increases, the adversary can do no better than a coin flip. This means that the success is directly proportional to overfitting (we present the proof in Appendix A.3)
+
+Success of Dataset Inference. Take $\nu$ to be a dataset inferring victim (Definition 1). Let $\psi _ { } \nu _ { } ( f , S ; \mathcal { D } )$ be $\nu$ ’s decision function for ownership resolution. In the next theorem, we show that the success of DI in practice is high and independent of the training set size. (Proof in Appendix A.4)
+
+![](images/240ce8f9d166e283806b10ce21dc97d4cbd9ac4bc9cc064e0b401b524300d185.jpg)  
+(b) If $\mathbf { x }$ is not in training set   
+Figure 1: The effect of including $\mathbf { \left( x , \cdot A ^ { \prime } \right) }$ in the train set. If $\mathbf { x }$ is in the train set, the classifier will learn to maximize the decision boundary’s distance to $\mathcal { V } \setminus \{ \ @ \}$ . If $\mathbf { x }$ is in the test set, it has no direct impact on the learned landscape.
+
+Theorem 3 (Success of DI) Choose $b \gets \{ 0 , 1 \}$ uniformly at random. Given an adversary’s linear classifier $f$ trained on $S ^ { \prime } \sim \mathcal { D }$ , s.t. $\vert S ^ { \prime } \vert = \vert S \vert$ if $b = 0$ , and on $S$ otherwise. The probability $\nu$ correctly decides if an adversary stole its knowledge $\begin{array} { r } { \mathbb { P } \left[ \psi ( f , S ; \mathcal { D } ) = b \right] = 1 - \Phi \left( - \frac { \sqrt { D } } { 2 \sqrt { 2 } } \right) } \end{array}$ . Moreover, $\begin{array} { r } { \operatorname* { l i m } _ { D \to \infty } \mathbb { P } \left[ \psi ( f , S ; \mathcal { D } ) = b \right] = 1 . } \end{array}$ .
+
+Example. Assume a dataset of training size 50K and input dimensions $K = 1 0 0 , D = 9 0 0$ (i.e., 100 strongly correlated features which is roughly similar to the MNIST dataset) We have $\mathbb { P } _ { ( \mathbf { x } , y ) \sim S } \left[ \psi ( f , \overleftarrow { S } ; \mathcal { D } ) = 1 \right] = 1 - 1 0 ^ { - 2 6 } \sim 1 . 0$ while $\begin{array} { r } { \bar { \mathbb { P } } _ { ( \mathbf { x } , b ) \sim \mathcal { R } } \left[ \mathcal { M } ( \mathbf { x } , f ) = b \right] = 0 . 5 2 6 . } \end{array}$ . Therefore, in a problem setting where membership inference succeeds only by slightly above random chance, dataset inference succeeds nearly every time.
+
+# 5 DATASET INFERENCE
+
+Dataset Inference is the process of determining whether a victim’s private knowledge has been directly or indirectly incorporated in a model trained by an adversary. Our key intuition is that classifiers generally try to maximize the distance of training examples from the model’s decision boundaries. This means that any model which has stolen the victim’s private knowledge should also position data similar to victim’s private training data far from its own decision boundaries. (See Figure 1) When a victim suspects knowledge was stolen from their model, they may measure how the adversary’s model responds to their own training data to substantiate their ownership claim.
+
+# 5.1 EMBEDDING GENERATION
+
+For a model $f$ and data point $\mathbf { x }$ , we aim to extract a feature embedding for $\mathbf { x }$ that characterizes its ‘prediction margin’ (or distance from the decision boundaries) w.r.t. $f$ . The victim $\nu$ extracts these embeddings for points $( \mathbf { x } , y ) \sim \mathcal { D }$ and labels them as inside $b = 1$ ) or outside $b = 0$ ) of their private dataset $S _ { \nu }$ .2 We introduce two methods for generating embeddings based on the level of access the victim may have to the adversary’s model.
+
+White-Box Setting: MinGD White-box embedding generation is used when $\nu$ and $\mathcal { A } _ { * }$ resolve the claim for ownership in the presence of a neutral arbitrator, such as a court. Indeed, Kumar et al. (2020) highlight that such attacks potentially fall under Computer Fraud and Abuse Act in the USA and are prosecutable for ‘reverse engineering’ the model’s ‘source code’. Both parties provide access to their models, and then the ‘prediction margin’ is measured for the suspected adversary’s model on the victim’s train and test data points. For any data point $\left( \mathbf { x } , y \right)$ we evaluate its minimum distance $\Delta$ to the neighbouring target classes $t$ by performing gradient descent optimization of the following objective (Szegedy et al., 2013): $\mathrm { m i n } _ { \delta } \Delta ( { \bf x } , { \bf x } +$ $\delta$ ) s.t. $f ( \mathbf { x } + \delta ) ~ = ~ t$ . The distance metric $\Delta ( \mathbf { x } ^ { ( i ) } , \mathbf { x } ^ { ( j ) } )$ refers to the $\ell _ { p }$ distance between points $\mathbf { x } ^ { ( i ) }$ and $\mathbf { x } ^ { ( j ) }$ for $p \in \{ 1 , 2 , \infty \}$ , and $t$ is the target label. The distance $\Delta$ to each target class is a feature in the embedding vector analyzed by the ownership tester from $\ S 5 . 2$ .
+
+![](images/937ba546c43be0d9f6ab60b4c300d9b1325f6176e6ef82f855449b99729cf806.jpg)  
+Figure 2: Training (dotted) the confidence regressor with embeddings of public and private data, and victim’s model $f _ { \mathcal { V } }$ ; Dataset Inference (solid) using $m$ private samples and adversary model $f _ { A _ { * } }$
+
+Black-Box Setting: Blind Walk. $\nu$ may want to perform DI on a publicly deployed model $f$ that only allows label query access. This makes them incapable of computing gradients required for MinGD. Moreover, querying $f$ would be costly for $\nu$ . Therefore, we introduce a new method called Blind Walk which estimates the ‘prediction margin’ of any given data point through its robustness to random noise rather than a gradient search. We sample a random initial direction $\delta$ . Starting from an input $\left( \mathbf { x } , y \right)$ , we take $k \in \mathbb N$ steps in the same direction until $f ( \mathbf { x } + k \delta ) = t ; ~ t \neq y$ . Then, $\Delta ( \mathbf { x } , \mathbf { x } + k \delta )$ is used as a proxy for the ‘prediction margin’ of the model. Thus, the approach only requires label access to $f$ . We repeat the search over multiple random initial directions to increase the information about the point’s robustness, and use each of these distance values as features in the generated embedding. In practice, we find Blind Walk to perform better than MinGD with the ownership tester from $\ S 5 . 2$ . We discuss further details justifying these observations in Appendix C.
+
+# 5.2 OWNERSHIP TESTER
+
+It is important for the victim to resolve ownership claims in as few queries as possible, since each query involves the victim revealing part of their private dataset $S _ { \nu }$ . Since claiming ownership would likely lead to legal action, it is paramount that the victim minimizes their false positive rate. We thus test ownership in two phases: a regression model first infers whether the potentially stolen model’s predictions on individual examples contain the victim’s private knowledge, this is then followed by a hypothesis test which aggregates these results to decide dataset inference. This is another key difference with membership inference efforts: rather than always predicting that a point is from the ‘train’ or ‘test’ data, we claim ownership of a model only when we have sufficient confidence. This is done through statistical hypothesis testing, which takes the false positive rate $\alpha$ as a hyper-parameter, and produces either conclusive positive results with an error of at most $\alpha$ , or an ‘inconclusive’ result.
+
+Confidence Regressor. As defined in $\ S 5 . 1$ , we extract distance embeddings w.r.t $f _ { \nu }$ for data points in both $\nu$ ’s private data $S _ { \nu }$ and unseen publicly available data. Using the embeddings and the ground truth membership labels, we train a regression model $g _ { \mathcal { V } }$ . The goal of $g _ { \mathcal { V } }$ is to predict a (proxy) measure of confidence that a sample contains $f _ { \mathcal { V } }$ ’s private information. For our hypothesis testing, we require that $g _ { \mathcal { V } }$ produce smaller values for samples from $S _ { \nu }$ . Complete access to the dataset $S _ { \nu }$ allows $\nu$ to train $g _ { \mathcal { V } }$ accurately, as illustrated via dotted arrows in Figure 2.
+
+Hypothesis Testing. This is the step where dataset inference claims are made (solid lines in Figure 2). Using the confidence scores produced by $g _ { \mathcal { V } }$ and the membership labels, we create equalsized sample vectors $^ c$ and $c _ { \mathcal { V } }$ from private training and public data, respectively. We test the null hypothesis $H _ { 0 } : \mu < \mu \nu$ where $\mu = \bar { c }$ and $\mu _ { \mathcal { V } } = \bar { c } _ { \mathcal { V } }$ are mean confidence scores. The test would either reject $H _ { 0 }$ and conclusively rule that $f _ { A _ { * } }$ is ‘stolen’, or give an inconclusive result.
+
+# 6 EXPERIMENTAL SETUP AND IMPLEMENTATION OF DATASET INFERENCE
+
+Unlike prior work on membership inference, which evaluates over victim models trained to overfit on small subsets of the original dataset, we train all of our victim models on large common benchmarks.
+
+Datasets. We perform our experiments on the CIFAR10, CIFAR100, SVHN and ImageNet datasets. These remain popular image classification benchmarks, further description about which can be found in Appendix E.1. All details about experiments on SVHN and ImageNet are in Appendix E.2.
+
+Model Architecture. The victim model is a WideResNet (Zagoruyko & Komodakis, 2016) with depth 28 and widening factor of 10 (WRN-28-10) for both CIFAR10 and CIFAR-100, and is trained with a dropout rate of 0.3 (Srivastava et al., 2014). For the model stealing attacks described in $\ S 6 . 1$ , we use smaller architectures such as WRN-16-1 on CIFAR10 and WRN-16-10 on CIFAR100.
+
+# 6.1 MODEL STEALING ATTACKS
+
+We consider the strongest model stealing attacks in the literature, and introduce new attacks targeting dataset inference to perform an adaptive evaluation of our defense. The adversary $\mathcal { A } _ { * }$ can gain different levels of access to $\nu$ ’s private knowledge:
+
+(1) $\mathcal { A } _ { Q }$ has query access to $f _ { \mathcal { V } }$ . We consider model extraction (Tramer et al., 2016) based adversaries \` which may (1.a) have access to the model’s prediction vectors (via an API). $\boldsymbol { \mathcal { A } } _ { Q }$ queries $f _ { \mathcal { V } }$ on a nontask specific dataset, and minimizes the $K L$ divergence with its predictions. (1.b) Alternately, to further distance its predictions from the victim, the adversary may only use the most confident label from these queries (as pseudo-labels) to train. (2) $A _ { M }$ has access to the victim’s model $f _ { \mathcal { V } }$ . This may happen when $\nu$ wishes to open-source their work for academic purposes but does not allow its commercialization, or via insider-access. (2.a) $A _ { M }$ may fine-tune over $f _ { \nu }$ , or (2.b) use $f _ { \mathcal { V } }$ for datafree distillation (Fang et al., 2019) in a zero-shot learning framework that only utilizes synthetic and non-semantic queries.3 (3) $A _ { D }$ has access to the complete private dataset, $S _ { \nu }$ of the victim. They may train their own model either (3.a) by distilling $f _ { \mathcal { V } }$ (over query access), or (3.b) training from scratch using different learning schemes or architectures. (For further details see Appendix B).
+
+Finally, we also perform DI against an independent and honest machine learning model $\mathcal { T }$ that was trained on its own private dataset. This model is used as a control, to ensure that we do not claim ownership of models that were not trained by stealing knowledge from our victim model.
+
+Training the threat models. For model extraction and fine-tuning attacks on CIFAR10 and CIFAR100, we use a subset of 500,000 unlabeled TinyImages that are closest to CIFAR10, as created by Carmon et al. (2019). For SVHN, we use the ‘extra’ training data released by the authors. We train the student model for 20 epochs for model extraction methods and 5 epochs for fine-tuning. For zero-shot learning, we use data-free adversarial distillation method (Fang et al., 2019) and train the student model for 200 epochs. In case of distillation and modified architecture, we have access to the original training data of the victim. We train both models for 100 epochs on the full training set.
+
+In all the training methods, we use a fixed learning rate strategy with SGD optimizer and decay the learning rate by a factor of 0.2 at the end of the $0 . 3 { \times } , 0 . 6 { \times }$ , and $0 . 8 \times$ the total number of epochs.
+
+# 6.2 IMPLEMENTATION DETAILS FOR DATASET INFERENCE
+
+Embedding generation. For the white-box method (MinGD), we perform the attack against each target class while optimizing the $\ell _ { 1 } , \ell _ { 2 } , \ell _ { \infty }$ norms. Hence, we obtain an embedding of size 30 (classes $\times$ distance measures). In the case of CIFAR100, we only attack the 10 most confident target classes, as indicated by the prediction vector $f ( \mathbf { x } )$ . For the black-box method (Blind Walk), we sample 10 times from uniform, Gaussian, and laplace distributions to perturb the input. Once again, we obtain an embedding vector of size 30. More details are deferred to Appendix C.
+
+Training the confidence regressor. We train a two-layer linear network (with tanh activation) $g _ { \mathcal { V } }$ for the task of providing confidence about a given data point’s membership in ‘private’ and ‘public’ data. The regressor’s loss function is $\mathcal { L } ( \mathbf { x } , y ) = - b \cdot g _ { \mathcal { V } } ( \pmb { x } )$ where the label $b = 1$ for a point in the (public) training set of the respective model, and $- 1$ if it came from victim’s private set.
+
+Hypothesis Tests. We query models with equal number of samples from public and private datasets, create embeddings and calculate confidence score vectors $^ c$ and $c _ { \mathcal { V } }$ , respectively. We form a two sample T-test on the distribution of $^ c$ and $c _ { \mathcal { V } }$ and calculate the $\mathsf { p }$ -value for the one-sided hypothesis $\mathcal { H } _ { 0 } : \mu < \mu _ { \mathcal { V } }$ against $\mathcal { H } _ { a l t } : \mu > \mu _ { \mathcal { V } }$ . From $\mathcal { L } ( \mathbf { x } , y )$ , it follows that $g _ { \mathcal { V } }$ learns to minimize $g _ { \mathcal { V } } ( \mathbf { x } )$ when $\mathbf { x } \in S _ { \nu }$ , and maximizes it otherwise. Therefore, a vector that contains samples from $S _ { \nu }$ produces lower confidence scores, and decreases the test’s $\mathsf { p }$ -value. If the $\mathsf { p }$ -value is below a predefined significance level $\alpha$ , $H _ { 0 }$ is rejected, and the model under test is marked as ‘stolen’.
+
+In the following results, we repeat all experimental statistical tests for 100 times with randomly sampled data with replacement. To control for multiple testing, and account for the unknown dependence of the p-values thus generated, we aggregate these values using the harmonic mean (Wilson, 2018). To produce bootstrap 99-percentile confidence intervals, we repeat the experiment 40 times.
+
+# 7 RESULTS
+
+Table 1 shows p-values and the effect size, $\Delta \mu = \mu - \mu \nu$ , which captures the average confidence of our hypothesis test in claiming that a model was stolen. We test our approach against 6 different attackers and in two different settings (Black- and White-box). In addition, Table 1 also reports ‘Source’ where the victim’s complete model $f _ { \nu }$ has been stolen, and ‘Independent’, the control model trained on a separate dataset. Understandably, we typically observe the largest and smallest effect sizes for these two baselines, which serve as bounds to interpret our evaluation of attacks.
+
+Our evaluation shows that DI is robust to both the strongest model stealing techniques, but also an adaptive attack we propose based on zero-shot learning. DI can claim a model was stolen with at least $9 5 \%$ confidence for most threat models with only 10 samples. Hence, the defense exploits an inherent property of model training. Among the six attacks we considered, we observe that our model consistently flags fine-tuned models as stolen. This departs from prior defenses against model extraction: e.g., watermarks often lack robustness to fine-tuning. Here, DI is unaffected because finetuning does not remove knowledge from all private data used to train the stolen model. The labelquery and zero-shot attacks challenge DI the most. This is expected because zero-shot learning uses only synthetic data points for querying; and in case of logit-query, $\boldsymbol { \mathcal { A } } _ { Q }$ is merely using $\nu$ to label their dataset, which leaks much less private knowledge than distillation-based model extraction. In practice, their higher query complexity makes both these attacks the most (financially) expensive to mount. We present concurring results on SVHN and ImageNet in Appendix E.2.
+
+DI requires few private points. In Figure 3, we show the number of private points the victim has to reveal (from its training set) to achieve a particular p-value when claiming model ownership is low: 40, and often as few as 20, samples to achieve a false positive rate (FPR) $\alpha$ of at most $1 \%$ .
+
+Query efficiency. For the black-box scenario where the victim wants to assess the ownership of a model served through an API, DI is a query efficient approach that comes at a low cost for the victim.
+
+<table><tr><td rowspan="3"></td><td rowspan="3">Model Stealing Attack</td><td colspan="4">CIFAR10</td><td colspan="4">CIFAR100</td></tr><tr><td colspan="2">MinGD</td><td colspan="2">Blind Walk</td><td colspan="2">MinGD</td><td colspan="2">Blind Walk</td></tr><tr><td>△μ</td><td>p-value</td><td>△μ</td><td>p-value</td><td>△μ</td><td>p-value</td><td>△μ</td><td>p-value</td></tr><tr><td>V</td><td>Source</td><td>0.838</td><td>10-4</td><td>1.823</td><td>10-42</td><td>1.219</td><td>10-16</td><td>1.967</td><td>10-44</td></tr><tr><td rowspan="2">AD</td><td>Distillation</td><td>0.586</td><td>10-4</td><td>0.778</td><td>10-5</td><td>0.362</td><td>10-2</td><td>1.098</td><td>10-5</td></tr><tr><td>Diff. Architecture</td><td>0.645</td><td>10-4</td><td>1.400</td><td>10-10</td><td>1.016</td><td>10-6</td><td>1.471</td><td>10-14</td></tr><tr><td rowspan="2">Am</td><td>Zero-Shot Learning</td><td>0.371</td><td>10-2</td><td>0.406</td><td>10-2</td><td>0.466</td><td>10-2</td><td>0.405</td><td>10-2</td></tr><tr><td>Fine-tuning</td><td>0.832</td><td>10-5</td><td>1.839</td><td>10-27</td><td>1.047</td><td>10-7</td><td>1.423</td><td>10-10</td></tr><tr><td rowspan="2">AQ</td><td>Label-query</td><td>0.475</td><td>10-3</td><td>1.006</td><td>10-4</td><td>0.270</td><td>10-2</td><td>0.107</td><td>10-1</td></tr><tr><td>Logit-query</td><td>0.563</td><td>10-3</td><td>1.048</td><td>10-4</td><td>0.385</td><td>10-2</td><td>0.184</td><td>10-1</td></tr><tr><td>T</td><td>Independent</td><td>0.103</td><td>1</td><td>-0.397</td><td>0.675</td><td>-0.242</td><td>0.545</td><td>-1.793</td><td>1</td></tr></table>
+
+Table 1: Ownership Tester’s effect size (higher is better) and $\mathfrak { p }$ -value (lower is better) using $m = 1 0$ samples on multiple threat models (see $\ S 6 . 1 \AA )$ ). The highest and lowest effect sizes among the model stealing attacks $( \mathcal { A } _ { D } , \mathcal { A } _ { M } , \mathcal { A } _ { Q } )$ are marked in red and blue respectively.
+
+![](images/64c2485b1071fc4197fdb599e945968d5792d91ae85bbbcd1bad27de7a7c8609.jpg)  
+Figure 3: p-value against number of revealed samples $( m )$ . Significance levels (FPR) $\alpha = 0 . 0 1$ and 0.05 (dotted lines) have been drawn. Under most attack scenarios, the victim $\nu$ can dispute the adversary’s ownership of $f _ { A _ { * } }$ (with FPR of at most $1 \%$ ) by revealing fewer than 50 private samples.
+
+For 100 data points, DI can be performed in less than 30,000 queries to the API. More efficient embedding generation optimizations can significantly improve this further. (See Appendix D).
+
+White-box access is not essential to DI. While access to gradient information can help in particular scenarios (such as, logit-query in CIFAR100) which reduces $m$ from 40 to 20 samples, for fine-tuned adversary models, or those that are trained against a different architecture to evade detection, our proposed black-box solution (Blind Walk) performs surprisingly better than its White-box counterpart. We conjecture that the Blind Walk’s advantage stems from a combination of factors: (a) gradient-based approaches are sensitive to numerical instabilities, (b) the approach is stochastic and aims to find the expected prediction margin rather than the worst-case (it searches for any incorrect neighboring class in a randomly chosen direction rather than focusing on the distance to possible target classes). Hence, our proposed Blind Walk inference procedure is highly efficient.
+
+DI does not require overfitting or retraining. Unlike past defenses (watermarks) and attacks (MI) which we discussed previously, DI uniquely applies as a post-hoc solution to any publicly deployed model, irrespective of whether it ‘overfit’ on its training set. This means that model owners in the real-world can perform DI immediately, to protect models that they have already deployed.
+
+# 8 DISCUSSION AND CONCLUSION
+
+While adversarial ML often consists of a cycle of attacks and defenses, we turn this game on its head. Dataset inference leverages knowledge a defender has of their training set to identify models that an adversary created by either directly accessing this training set without authorization or indirectly distilling knowledge from one of the models released by the defender. With dataset inference, model developers resolve model ownership conflicts without making changes to their existing models.
+
+Interestingly, the ability to claim ownership through dataset inference gracefully degrades as the adversary spends increasingly more resources to train the stolen model. For instance, if an adversary extracts a copy and later fine-tunes it with a different dataset to conceal the model, it will make the model more different and dataset inference will be less likely to succeed. But this is expected and desired: this means the adversary faced a higher cost to obfuscate this stolen copy. In itself it is not an easy task, because of accuracy degradation and catastrophic forgetting.
+
+Finally, it remains a promising direction for future work to study the confluence of DI with privacypreserving models trained using $\epsilon$ -differential privacy (DP). Leino & Fredrikson (2020) have shown that while DP can help against membership inference (MI) attacks, it comes at a steep cost in accuracy. We hypothesize that since DI amplifies the membership signal using multiple private samples, it follows that the $\epsilon$ values required to make DI ineffective would be even lower than it is for MI. Therefore, $\epsilon$ values that can make the model private, likely do not interfere with dataset inference.
+
+Acknowledgments. We thank the reviewers for their feedback. We would also like to thank members of CleverHans Lab, especially Ilia Shumailov and Varun Chandrasekaran. This work was supported by a Canada CIFAR AI Chair, NSERC, Microsoft, and sponsors of the Vector Institute.
+
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+
+# APPENDIX
+
+# A THEORETICAL MOTIVATION
+
+In this section, we provide the formal proofs of Theorems 1, 2, 3 as stated in $\ S 4$ . First, we describe the preliminaries including the binary classification task and the machine learning model used to train the same in Appendix A.1.
+
+# A.1 PRELIMINARIES
+
+We repeat the preliminaries described in $\ S 4$ to discuss the proofs in the following sections.
+
+Setup. Consider a data distribution $\mathcal { D }$ , such that any input-label pair $( \mathbf { x } , \mathbf { y } )$ can be described as:
+
+$$
+\begin{array} { r } { y \stackrel { u . a . r } { \sim } \{ - 1 , + 1 \} ; \quad { \mathbf x } _ { 1 } = y { \mathbf u } \in \mathbb { R } ^ { K } , \quad { \mathbf x } _ { 2 } \sim { \mathcal N } ( 0 , \sigma ^ { 2 } ) \in \mathbb { R } ^ { D } } \end{array}
+$$
+
+where $\mathbf { x } = ( \mathbf { x _ { 1 } } , \mathbf { x _ { 2 } } ) \in \mathbb { R } ^ { K + D }$ and $\mathbf { u } \in \mathbb { R } ^ { K }$ is some fixed vector. This suggests that the last $D$ dimensions of the input is Gaussian noise which has no correlation with the correct label. However, the first $K$ input dimensions are sufficient to perfectly separate data points from classes $\{ - 1 , + 1 \}$ . The setup is adapted from Nagarajan & Kolter (2019). We use $S ^ { + }$ and $\mathcal { D } ^ { + }$ to represent the subset of the training set $S$ and the distribution $\mathcal { D }$ with label $y = 1$ .
+
+Architecture. We consider the scenario of classifying the input distribution using a linear classifier, $f$ , with weights $\bf { w } = ( w _ { 1 } , w _ { 2 } )$ , such that for any input:
+
+$$
+f ( \mathbf { x } ) = \mathbf { w _ { 1 } } \cdot \mathbf { x _ { 1 } } + \mathbf { w _ { 2 } } \cdot \mathbf { x _ { 2 } }
+$$
+
+While we only discuss the case of a linear network in this analysis, the success of dataset inference (like membership inference) only increases with the number of parameters in a machine learning model (Yeom et al., 2018), which in effect makes the following analysis a stronger result to prove. Prior works have also argued how over-parametrized deep learning networks memorize training points (Zhang et al., 2016; Feldman, 2019).
+
+# A.2 TRAIN-TEST PREDICTION MARGIN (THEOREM 1)
+
+Training Algorithm. We assume that the learning algorithm initializes the weights of the classifier $f$ to zero. Sample a training set $S \sim \mathcal { D } ^ { m } = \big \{ \big ( \bar { \mathbf { x } ^ { ( i ) } } , \bar { y } ^ { ( i ) } \big ) \big | i = 1 \dots m \big \}$ . The learning algorithm maximizes the loss ${ \mathcal { L } } ( \mathbf { x } , y ) = y \cdot f ( \mathbf { x } )$ and visits every training point once, with a gradient update step of learning rate $\alpha = 1$ .
+
+$$
+\begin{array} { r } { \mathbf { w _ { 1 } }  \mathbf { w _ { 1 } } + \alpha y ^ { ( i ) } \mathbf { x _ { 1 } } ^ { ( i ) } } \\ { \mathbf { w _ { 2 } }  \mathbf { w _ { 2 } } + \alpha y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } } \end{array}
+$$
+
+From the optimization steps described above, one may note that the learned weights for the classifier $f$ are given by $\mathbf { w _ { 1 } } = m \mathbf { u }$ and $\begin{array} { r } { \mathbf { w _ { 2 } } = \sum _ { i } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } } \end{array}$ irrespective of the training batch size.
+
+Inference. For any data point $( \mathbf { x } ^ { ( j ) } , y ^ { ( j ) } )$ , we calculate its ‘prediction margin’ as the distance from the linear boundary, which is proportional to its label times the classifier’s output $y \cdot f ( \mathbf { x } )$ . For any point, $\mathbf { x } = ( x _ { 1 } , x _ { 2 } ) \sim \mathcal { D }$ , the ‘prediction margin’ is therefore given by:
+
+$$
+\begin{array} { l } { y \cdot f ( \mathbf { x } ) = y \cdot \left( \mathbf { w _ { 1 } } \cdot \mathbf { x _ { 1 } } + \mathbf { w _ { 2 } } \cdot \mathbf { x _ { 2 } } \right) = y \cdot \left( m \mathbf { u } \right) \cdot \left( y \mathbf { u } \right) + y \cdot \left( \displaystyle \sum _ { i } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \right) \cdot \mathbf { x _ { 2 } } } \\ { = c + \left( y \cdot \displaystyle \sum _ { i } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \cdot \mathbf { x _ { 2 } } \right) } \end{array}
+$$
+
+Now, we calculate the expected value of the margin for a point randomly sampled from the training set. Consider any point in the training set $( \mathbf { x } , y ) \sim S ^ { + } = ( \mathbf { x } ^ { ( j ) } , 1 )$ for some index $j$ . Then, we have:
+
+$$
+\begin{array} { r l } & { \tilde { \mathfrak { L } } _ { \mathbf { x } ^ { ( j ) } \sim S ^ { + } } f ( \mathbf { x } ^ { ( j ) } ) = y \cdot c + \mathbb { E } _ { \mathbf { x _ { 2 } } ^ { ( i ) } \sim N ( 0 , \sigma ^ { 2 } ) } \left[ \left( \overset { i \neq j } { \underset { i } { \sum } } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \cdot \mathbf { x _ { 2 } } ^ { ( j ) } \right) \right] + \mathbb { E } _ { \mathbf { x _ { 2 } } ^ { ( j ) } \sim N ( 0 , \sigma ^ { 2 } ) } \left[ y ^ { ( i ) } ( \mathbf { x _ { 2 } } ^ { ( j ) } ) ^ { 2 } \right. } \\ & { \qquad \ : = c + 0 + D \sigma ^ { 2 } } \end{array}
+$$
+
+Note that in (6), we utilize the fact that the square of a standard normal variable follows the χ2(1) distribution; and that the expected value of product of independent random variables is same as the product of their expectations, followed by the linearity of expectation.
+
+Similarly, now consider a new data point $( \mathbf { x } , 1 ) \sim \mathcal { D } ^ { + }$ .
+
+$$
+\begin{array} { r l r } {  { \mathbb { E } _ { ( \mathbf { x } , y ) \sim \mathcal { D } ^ { + } } f ( \mathbf { x } ) = y c + \mathbb { E } _ { \mathbf { x _ { 2 } } ^ { ( i ) } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } ) } [ ( \sum _ { i } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \cdot \mathbf { x _ { 2 } } ) ] } } \\ & { } & { = c \sigma } \end{array}
+$$
+
+Once again, in (7) we utilize the fact that the expected value of product of independent random variables is same as the product of their expectations, followed by the linearity of expectation. At an aggregate over multiple data points, we can hence show that $\bar { \mathbb { E } _ { ( { \bf x } , y ) \sim S ^ { + } } } f ( { \bf x } ) - \bar { \mathbb { E } _ { ( { \bf x } , y ) \sim D ^ { + } } } f ( { \bf x } ) =$ $D \sigma ^ { 2 }$ . This concludes the proof for Theorem 1.
+
+# A.3 FAILURE OF MEMBERSHIP INFERENCE (THEOREM 2)
+
+In this section, we take a formal view of the conditions that lead to the failure and success of membership inference. Before we begin with our formal analysis, we would like to point out that the statistical difference between the distribution of training and test data points in Theorem 1 is only observed when we aggregate an expectation over multiple samples. Now, we show that the variance of this difference is so large, that it is very difficult to make any claims from a single input data point.
+
+Consider an adversary that does not have knowledge of the private data used to train a machine learning model. However, it contains domain knowledge of the task that the model is trying to solve. This may include the range and dimension of possible inputs to the model. In our case, the adversary has knowledge of the data distribution $\mathcal { D }$ , but not of the training set $S$ .
+
+For a single data point $\bf { x } = ( x _ { 1 } , x _ { 2 } )$ , s.t. $( \mathbf { x } , y ) \sim \mathcal { D }$ , the adversary aims to predict whether it was used to train the machine learning model, $f$ . The prediction margin for $( \mathbf { x } , y ) \sim \mathcal { D }$ is given by:
+
+$$
+y \cdot f ( \mathbf { x } ) = y \cdot \left( \mathbf { w _ { 1 } } \cdot \mathbf { x _ { 1 } } + \mathbf { w _ { 2 } } \cdot \mathbf { x _ { 2 } } \right) = c + \left( y \cdot \sum _ { i } y ^ { ( i ) } x _ { 2 } ^ { ( i ) } \cdot \mathbf { x _ { 2 } } \right)
+$$
+
+From the analysis in Theorem 2, the adversary knows that $\mathbb { E } _ { ( \mathbf { x } , y ) \sim S } \left[ y \cdot f ( \mathbf { x } ) \right] = c + D \sigma ^ { 2 }$ and $\mathbb { E } _ { ( \mathbf { x } , y ) \sim \mathcal { D } } \left[ y \cdot f ( \mathbf { x } ) \right] = c$ . Let $\mathcal M ( { \bf x } | f )$ represents the membership decision of the adversary for a given data point $\mathbf { x }$ and classifier $f$ . The strongest adversary will use the following decision rule:
+
+$$
+\mathcal { M } ( \mathbf { x } | f ) = \left\{ { 1 } , \atop { 0 } , { \mathrm { i f } \left( y \cdot f ( \mathbf { x } ) - c \right) \geq t } \right.
+$$
+
+where $t \in \left[ 0 , D \sigma ^ { 2 } \right]$ is some threshold that the adversary can tune in order to achieve maximum true positive rate and minimum false positives.
+
+Similar to Yeom et al. (2018), we consider the scenario where the input data is randomly (with equal probability via coin flip $^ b$ ) sampled from either $S$ (if $b = 1$ ) or $\mathcal { D }$ (if $b = 0$ ). Let such a distribution be specified as $( \mathbf { x } , b ) \sim \mathcal { R }$ . The adversary $\mathcal { M }$ must maximize the single objective $\mathbb { P } _ { ( \mathbf { x } , b ) \sim \mathcal { R } } \left[ \mathcal { M } ( \mathbf { x } | \mathbf { \bar { \boldsymbol { f } } } ) = b \right]$ . In summary,
+
+$$
+\mathbb { P } _ { ( \mathbf { x } , b ) \sim \mathcal { R } } \left[ \mathcal { M } ( \mathbf { x } | f ) = b \right] = \frac { \mathbb { P } _ { ( \mathbf { x } , y ) \sim S } \left[ \mathcal { M } ( \mathbf { x } | f ) = 1 \right] + \mathbb { P } _ { ( \mathbf { x } , y ) \sim \mathcal { D } } \left[ \mathcal { M } ( \mathbf { x } | f ) = 0 \right] } { 2 }
+$$
+
+We simplify our analysis by considering the data point $( \mathbf { x } , y ) \sim \mathcal { D } ^ { + }$ (has true label, $y = 1$ ). However, the analysis generally applies to any $( \mathbf { x } , y ) \sim \mathcal { D }$ .
+
+Case 1: $( \mathbf { x } , y ) \sim \mathcal { D } ^ { + }$ . Assume meta-variable $\begin{array} { r } { \mathbf { z _ { 2 } } = \left( \sum _ { i } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \right) . } \end{array}$ . Therefore, $\mathbf { z _ { 2 } } \sim \mathcal { N } ( 0 , m \sigma ^ { 2 } I )$ , while $\mathbf { x _ { 2 } } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I )$ . Recall that $\mathbf { x _ { 2 } } , \mathbf { z _ { 2 } } \in \mathbb { R } ^ { D }$ . Assuming $D$ to be large, we can conveniently apply the central limit theorem to approximate the distribution of the internal term. Let the individual dimensions of $\mathbf { z _ { 2 } }$ be denoted by ${ \bf z _ { 2 } } ( i )$ . Then, we have that:
+
+$$
+\begin{array} { r l } & { \mathbb { P } _ { ( \mathbf { x } , y ) \sim \mathcal { D } ^ { + } } \left[ \mathcal { M } ( \mathbf { x } | f ) = 0 \right] = \mathbb { P } _ { ( \mathbf { x } , y ) \sim \mathcal { D } ^ { + } } \left[ \left( \displaystyle \sum _ { i } ^ { m } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \cdot \mathbf { x _ { 2 } } \right) < t \right] } \\ & { \qquad = \mathbb { P } _ { ( \mathbf { x } , y ) \sim \mathcal { D } ^ { + } } \left[ \left( \mathbf { z } _ { 2 } \cdot \mathbf { x _ { 2 } } \right) < t \right] } \\ & { \qquad = \mathbb { P } _ { ( \mathbf { x } , y ) \sim \mathcal { D } ^ { + } } \left[ \displaystyle \frac { 1 } { D } \left( \displaystyle \sum _ { j } ^ { D } D \mathbf { z _ { 2 } } ( j ) \cdot \mathbf { x _ { 2 } } ( j ) \right) < t \right] } \end{array}
+$$
+
+Let $\alpha$ represents the distribution followed by $\mathbf { z _ { 2 } } ( i ) \cdot \mathbf { x _ { 2 } } ( i )$ . From CLT, we have that the combined distribution behaves like a normal distribution, with $\mu = \mu _ { \alpha } =$ and $\begin{array} { r } { \sigma ^ { 2 } = \frac { \sigma _ { \alpha } ^ { 2 } } { D } } \end{array}$ .
+
+$$
+\begin{array} { l } { { \mu _ { \alpha } = 0 } } \\ { { \sigma _ { \alpha } ^ { 2 } = m \cdot D ^ { 2 } \cdot \sigma ^ { 4 } } } \end{array}
+$$
+
+We use the fact that $\operatorname { V a r } [ X Y ] = \operatorname { V a r } [ X ] \operatorname { V a r } [ Y ] + \mathbb { E } [ X ] ^ { 2 } \operatorname { V a r } [ Y ] + \mathbb { E } [ Y ] ^ { 2 } \operatorname { V a r } [ X ]$ and $\mathrm { V a r } [ c \cdot X ] =$ $c ^ { 2 } \cdot \mathrm { V a r } [ X ]$ for computing $\sigma _ { \alpha } ^ { 2 }$ . Therefore, let $r \sim \mathcal { N } ( 0 , m D \sigma ^ { 4 } )$ :
+
+$$
+\mathbb { P } _ { ( \mathbf { x } , y ) \sim \mathcal { D } ^ { + } } \left[ \mathcal { M } ( \mathbf { x } | f ) = 0 \right] = \mathbb { P } _ { r \sim \mathcal { N } ( 0 , m D \sigma ^ { 4 } ) } \left[ r < t \right]
+$$
+
+It can be observed that $\mathbb { P } \left[ r < t \right]$ increases with the threshold value $t$ . For $t = 0$ , $\mathbb { P } \left[ \left( \mathbf { z _ { 2 } } \cdot \mathbf { x _ { 2 } } \right) < 0 \right] =$ 0.5. Whereas, for $t = D \sigma ^ { 2 }$ , the probability decreases with the value of $\mathbf { m }$ (this can be intuitively understood as – since the size of training set increases, overfitting decreases, making MI more difficult). Even for as low as $m = 1 0 0$ points in the training set, $\mathbb { P } \left[ ( \mathbf { z _ { 2 } } \cdot \mathbf { x _ { 2 } } ) < D \breve { \sigma } ^ { 2 } \right] = 0 . 6$ . For any value of $t \in \ \left[ 0 , \sigma ^ { 2 } \right]$ , the maximum probability for size of training data $m = 1 0 0$ is 0.6. Further, as the size of the training set increases, the probability tends to 0.5.
+
+Case 2: $( \mathbf { x } , y ) \sim S ^ { + }$ . Once again, as in the proof for Theorem 1, consider any point in the training set $( \mathbf { x } , y ) \sim S ^ { + } = ( \mathbf { x } ^ { ( j ) } , 1 )$ for some index $j$ . We will now calculate the probability of success of the adversary that follows the decision rule described above:
+
+$$
+\begin{array} { r l } { \mathbb { P } _ { ( \mathbf { x } , y ) \sim S ^ { + } } \left[ \mathcal { M } ( \mathbf { x } | f ) = 1 \right] = \mathbb { P } _ { ( \mathbf { x } , y ) \sim S ^ { + } } \left[ \left( \displaystyle \sum _ { i } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \cdot \mathbf { x _ { 2 } } \right) > t \right] } & { } \\ { = \mathbb { P } _ { ( \mathbf { x } , y ) \sim S ^ { + } } \left[ \left( \displaystyle \sum _ { i } ^ { i \neq j } y ^ { ( i ) } \mathbf { x _ { 2 } } ^ { ( i ) } \cdot \mathbf { x _ { 2 } } ^ { ( j ) } \right) + \left( \mathbf { x _ { 2 } } ^ { ( j ) } \cdot \mathbf { x _ { 2 } } ^ { ( j ) } \right) > t \right] } & { } \end{array}
+$$
+
+Now, following the discussion in the first case, we know that the first term can be approximated by a variable $\alpha \sim \tilde { \mathcal { N } } ( 0 , ( m - 1 ) D \sigma ^ { 4 } )$ . Similarly, using CLT over the sum of multiple random variables sampled from a $\chi _ { 1 } ^ { 2 }$ distribution, we can approximate the second term in the above equation with a variable $\beta \sim \mathcal { N } ( \bar { D } \sigma ^ { 2 } , D \sigma ^ { 4 } )$ . Finally, using the property for sum of independent gaussians, we can approximate the entire ‘prediction margin’ to be represented by a sample $u \sim \mathsf { \bar { N } } ( D \sigma ^ { 2 } , m D \sigma ^ { 4 } )$ . Then, we have that:
+
+$$
+\mathbb { P } _ { ( \mathbf { x } , y ) \sim S ^ { + } } \left[ \mathcal { M } ( \mathbf { x } | f ) = 1 \right] = \mathbb { P } _ { u \sim \mathcal { N } ( D \sigma ^ { 2 } , m D \sigma ^ { 4 } ) } \left[ u > t \right]
+$$
+
+Hence, we show that the adversary can do no better than a coin flip. This concludes the proof for Theorem 2. The interested reader may further analyze the assertion that the optimal value of $t$ lies in $[ 0 , D \sigma ^ { 2 } ]$ .
+
+To resolve the optimal threshold $t$ for membership inference, we restructure the arguments as follows. Recall from (10) that the adversary aims to ensure both true positive rates and true negative
+
+rates are high. We know:
+
+$$
+\begin{array} { r l } & { { \mathbb P } _ { ( \mathbf x , y ) \sim \mathcal { D } ^ { + } } \left[ \mathcal { M } ( \mathbf x | f ) = 0 \right] = { \mathbb P } _ { r \sim \mathcal { N } ( 0 , m D \sigma ^ { 4 } ) } \left[ r < t \right] } \\ & { { \mathbb P } _ { ( \mathbf x , y ) \sim S ^ { + } } \left[ \mathcal { M } ( \mathbf x | f ) = 1 \right] = { \mathbb P } _ { u \sim \mathcal { N } ( D \sigma ^ { 2 } , m D \sigma ^ { 4 } ) } \left[ u > t \right] } \\ & { \quad { \mathbb P } _ { ( \mathbf x , b ) \sim \mathcal { R } } \left[ \mathcal { M } ( \mathbf x | f ) = b \right] \le { \mathbb P } \left[ u - r > 0 \right] } \end{array}
+$$
+
+We know that both $u , r$ are sampled from normal distributions. Therefore, define $\gamma ~ = ~ ( u ~ -$ $r ) \ \sim \ \mathcal { N } ( D \sigma ^ { 2 } , 2 m D \bar { \sigma } ^ { 4 } )$ . This simplifies our discussion to a single normal distribution with mean $\mu _ { \gamma } = D \sigma ^ { 2 }$ and variance, $\sigma _ { \gamma } ^ { 2 } = 2 m D \sigma ^ { 4 }$ . We can now calculate the CDF at $x = 0$ to evaluate the maximum probability of success of membership inference (decision taken by the optimal adversary).
+
+Let $Z \sim { \mathcal { N } } ( 0 , 1 )$ . It can hence be shown that:
+
+$$
+\begin{array} { r l r } {  { \mathbb { P } [ \gamma > 0 ] = \mathbb { P } ( \sigma _ { \gamma } Z + \mu _ { \gamma } ) = \mathbb { P } ( Z > - \frac { \mu _ { \gamma } } { \sigma _ { \gamma } } ) = 1 - \Phi ( - \frac { \mu _ { \gamma } } { \sigma _ { \gamma } } ) } } \\ & { } & { = 1 - \Phi ( - \sqrt { \frac { D } { 2 m } } ) } \end{array}
+$$
+
+Clearly, as $m  \infty$ , $\mathbb { P } [ \gamma > 0 ]  0 . 5$ . This concludes the proof for Theorem 2.
+
+# A.4 SUCCESS OF DATASET INFERENCE (THEOREM 3)
+
+In Theorem 2 we showed that an adversary querying a single data point can say no better than a coin flip about the presence or absence of a given data point in a model’s training set. In this section, we show that when we reverse this adversarial game, the victim can utilize the information asymmetry to predict with high confidence if a potential adversary’s model stole their knowledge in any form.
+
+First, recall that the victim has access to its own private training set of size $m$ . For the purposes of this proof, we call it $S _ { \nu } ^ { m }$ . As the victim has complete information of the data distribution, it can randomly sample another dataset $S _ { 0 } \sim \mathcal { D }$ .
+
+The victim considers that the potential adversary’s model was stolen if the mean ‘prediction margin’ for the points in $S _ { \nu }$ is greater than $S _ { 0 }$ by some threshold parameter $\lambda$ . Let $\bar { \psi _ { \mathcal { V } } } ( f , S ; \mathcal { D } )$ be V’s decision function to resolve ownership claims.
+
+Recall that in Theorem 1 we had calculated the expected value of the difference in the prediction margin for the points in the training set versus those in the test set. In the proof of this theorem, we calculate the probability of the mean of the difference being greater than some value $\lambda$ .
+
+Now, let us calculate the probability of this margin for a data point randomly sampled from the training set. Let $t _ { \nu }$ represent the mean of the ‘prediction margin’ of all points in $S _ { \nu }$ for a classifier $f$ . Similarly, let $t _ { 0 }$ represent the mean of the ‘prediction margin’ of all points in $S _ { 0 }$ for the classifier $f$ . We will use $u _ { 2 }$ to denote the last $D$ dimensions of points in $S _ { 0 }$ . Then,
+
+$$
+\begin{array} { c } { \displaystyle t \nu = \frac { 1 } { m } \sum _ { j } \left[ \left( \overset { i \neq j } { \sum _ { i } } y ^ { ( i ) } \mathbf { x } _ { \mathbf { 2 } } ^ { ( i ) } \cdot \mathbf { x } _ { \mathbf { 2 } } ^ { _ { ( j ) } } \right) + \left( \mathbf { x } _ { \mathbf { 2 } } ^ { _ { ( j ) } } \right) ^ { 2 } \right] } \\ { \displaystyle = \frac { 1 } { m } \sum _ { j } ( \mathbf { x } _ { \mathbf { 2 } } ^ { _ { ( j ) } } ) ^ { 2 } + \sum _ { i } \left[ \left( \overset { i \neq j } { \sum _ { i } } y ^ { ( i ) } \mathbf { x } _ { \mathbf { 2 } } ^ { _ { ( i ) } } \cdot \mathbf { x } _ { \mathbf { 2 } } ^ { _ { ( j ) } } \right) \right] } \\ { \displaystyle t _ { 0 } = \frac { 1 } { m } \sum _ { j } \left[ \left( \overset { i \neq j } { \sum _ { i } } y ^ { ( i ) } \mathbf { x } _ { \mathbf { 2 } } ^ { _ { ( i ) } } \cdot \mathbf { u } _ { \mathbf { 2 } } ^ { _ { ( j ) } } \right) \right] } \\ { \displaystyle \mathbb { P } \left[ \psi _ { \mathcal { V } } ( f , S ; \mathcal { D } ) = 1 \right] = \mathbb { P } \left[ ( t _ { \mathcal { V } } - t _ { 0 } ) > \lambda \right] } \end{array}
+$$
+
+Recognize the similarity of the above formulation with that discussed in the proof for Theorem 2 in Appendix A.3. Let $t = t _ { \mathcal { V } } - t _ { 0 }$ . Then the random variable $t$ represents the a sample from the distribution of means for $\gamma$ defined in Appendix A.3. We can now directly use the Central Limit
+
+Theorem for this proof. Therefore,
+
+$$
+\begin{array} { l } { \displaystyle \mu _ { t } = \mu _ { z } = D \sigma ^ { 2 } } \\ { \displaystyle \sigma _ { t } ^ { 2 } = \frac { \sigma _ { z } ^ { 2 } } { m } = 2 D \sigma ^ { 4 } } \end{array}
+$$
+
+Hence, $t \ \sim \mathcal { N } ( D \sigma ^ { 2 } , 2 D \sigma ^ { 4 } )$ . It is important to note that this distribution is independent of the number of training points. Hence, unlike membership inference, the success of $\mathrm { D I }$ is not curtailed by the lack of overfitting.
+
+Similarly, for an honest adversary, the distribution of ‘prediction margin’ for points in $S _ { \nu }$ is the same as that for the points in $S _ { 0 }$ . It directly follows that:
+
+$$
+\mathbb { P } \left[ \psi _ { \mathcal { V } } ( f , S ; \mathcal { D } ) = 0 \right] = \mathbb { P } \left[ \hat { t } < \lambda \right] = \mathbb { P } \left[ t > \lambda \right]
+$$
+
+where, $\hat { t } \ \sim \mathcal { N } ( 0 , 2 D \sigma ^ { 4 } )$ . Once again, like the proof of Theorem 2, by symmetry of two normal distributions with the same variance, and shifted means, we can find that the optimal value of the parameter $\lambda$ that maximized true positives, and minimizes false positives, $\begin{array} { r } { \lambda = \frac { \mu _ { t } } { 2 } } \end{array}$ .
+
+Let $Z \sim { \mathcal { N } } ( 0 , 1 )$ . Then it can hence be shown that:
+
+$$
+{ \begin{array} { r l } & { \mathbb { P } [ { \hat { t } } > \lambda ] = \mathbb { P } \left( \sigma _ { t } Z + \mu _ { t } > { \frac { \mu _ { t } } { 2 } } \right) = \mathbb { P } \left( Z > - { \frac { \mu _ { t } } { 2 \sigma _ { t } } } \right) = 1 - \Phi \left( - { \frac { \mu _ { t } } { 2 \sigma _ { t } } } \right) } \\ & { \quad \quad = 1 - \Phi \left( - { \frac { \sqrt { D } } { 2 { \sqrt { 2 } } } } \right) } \end{array} }
+$$
+
+Clearly, as $D \to \infty$ , $\mathbb { P } [ \hat { t } > \lambda ]  1 . 0$ . This concludes the proof for Theorem 3.
+
+# B MODEL STEALING TECHNIQUES
+
+In this section, we provide more details about the various threat models that we consider in this work. We also provide specific use-cases and motivation for the respective threat models, and introduce a new adaptive adversary targeted specifically against DI.
+
+$\nu$ : Victim. The victim $\nu$ wishes to release its machine learning model to the community, either as a service, or by open-sourcing it for non-commercial academic use. $\nu$ wants to ensure that the deployed model is not being misused under the terms of license provided.
+
+$A _ { D }$ : Data Access. The adversary $A _ { D }$ is able to gain complete access to the victim’s private training data, and aims to deploy its own MLaaS by training the same. We note that labeled private training data is one of the most expensive commodities in the deployment cycle of modern machine learning systems.
+
+1. Model Distillation: Traditionally, model distillation (Hinton et al., 2015) was used as a method to compress larger models by training smaller students using the logits of a teacher model. We use this as a threat model that the adversary may employ to distance its predictions from a model that was trained using hard labels from the dataset itself. The adversary requires both query access, and access to the victim’s private training data for this attack.
+
+2. Modified Architecture: Multiple works have attempted at identifying unique properties (or ’fingerprints’) of a model by analyzing specific activations and representative features of internal model layers (Olah et al., 2017; 2018; Yin et al., 2019). We study the threat model where the adversary attempts training an alternate architecture on the victim’s private dataset $\mathcal { D } _ { p r i v }$ to valid the robustness of our method to changes in model structure.
+
+$A _ { M }$ : Model Access. The use-case of such an adversary is two fold: (1) the victim open-sources their own machine learning model under a license that does not allow other individuals to monetize the same; and (2) the adversary gains insider access to the trained model of a victim. In both the cases, the adversary aims to monetize its own MLaaS and deploys their own model on the web, by modifying the original victim model to reduce the dependence on $\kappa$ .
+
+1. Fine-tuning: The adversary has full access to the victim’s machine learning model, but not to its training data. While fine-tuning is employed used to transfer the knowledge of large pre-trained models on a given task (Devlin et al., 2018), we use it as a stealing attack, where the adversary uses the predictions of the victim model on unlabeled public data in order to modify its decision boundaries. We consider the setting where the adversary can fine-tune all layers.
+
+2. Zero-Shot Learning: This is the strongest adversary that we introduce specifically targeted to evade dataset inference. To the best of our knowledge, we are the first to consider such a threat model. The adversary uses no ‘direct’ knowledge of the actual training data to avoid any features that it may learn as a result of the training on the victim’s private data set. The adversary has complete access to the victim model, and uses data-free knowledge transfer (Micaelli & Storkey, 2019; Fang et al., 2019) to train a student model.
+
+$\mathcal { A } _ { Q }$ : Query Access. Model extraction (Tramer et al., 2016) is the most popular form of model \` stealing attack against deployed machine leaning models on the web. We discuss the related work on model extraction attacks in more detail in $\ S 2$ . Depending on the access provided by the machine learning service, an adversary may aim to extract the model using the logits or the labels alone.
+
+1. Model Extraction Using Labels: The victim model is used to provide pseudo-labels for a public dataset. The adversary trains their model on this pseudo-dataset. The key difference is that the input data points may be semantically irrelevant with respect to the task labels that the adversary’s model is being trained on.
+
+2. Model Extraction Using Logits: The performance of model extraction attacks can be improved when the victim provides confidence values for different output classes, rather than the correct labels itself. The adversary’s model is trained to minimize the $K L$ divergence with the outputs of the victim on a public (or non-task specific) dataset.
+
+I : Independent Model. Finally, we also study the results of dataset inference on an independent and honest machine learning model that is trained on its own private dataset. This is used as a control to verify that the dataset inference procedure does not always predict that the potential adversary stole the victim’s knowledge.4
+
+# C EMBEDDING GENERATION
+
+# Embedding Generation Hyperparameters.
+
+For the case of MinGD attack, we perform adversarial attacks defined by the optimization equation
+
+$$
+\operatorname* { m i n } _ { \delta } \Delta ( \mathbf { x } , \mathbf { x } + \delta ) \ s . t . \ f ( \mathbf { x } + \delta ) = t ; \ \mathbf { x } + \delta \in [ 0 , 1 ] ^ { n }
+$$
+
+The distance metric $\Delta ( \mathbf { x } ^ { ( i ) } , \mathbf { x } ^ { ( j ) } )$ refers to the $\ell _ { p }$ distance between points $\mathbf { x } ^ { ( i ) }$ and $\mathbf { x } ^ { ( j ) }$ for $p \in$ $\{ 1 , 2 , \infty \}$ , and $t$ is the targeted label. To perform the optimization, we perform gradient descent with steps of size $\alpha _ { p }$ . We take a maximum of 500 steps of gradient optimization, but pre-terminate at the earliest misclassification. The step sizes for the individual perturbation types are given by $\{ \alpha _ { \infty } , \alpha _ { 2 } , \alpha _ { 1 } \} = \{ 0 . 0 0 1 , 0 . 0 1 , 0 . 1 \}$ .
+
+For the case of Blind Walk, We sample a random initial direction $\delta$ . Starting from an input $\left( \mathbf { x } , y \right)$ , take $k \in \mathbb N$ steps in the same direction until $f ( \mathbf { x } + \boldsymbol { \delta } ) = t ; ~ t \neq y$ . Then, $\Delta ( \mathbf { x } , \mathbf { x } + k \delta )$ is used as a proxy for the ‘prediction margin’ of the model. We repeat the search over multiple random initial directions to increase the information about a training data point’s robustness, and use each of these distance values as features in the generated embedding.
+
+As an implementation detail, we sample between uniform, laplace and gaussian noise to generate embedding features. To measure the final perturbation distance from the initial starting point, we use different $\ell _ { p }$ norms for each of the noise sampling methods. For uniform noise, we compute the $\ell _ { \infty }$ distance; for gaussian noise, the $\ell _ { 2 }$ distance; and for laplacian noise, the $\ell _ { 1 }$ distance of the nearest misclassification. While we take $k$ steps of Blind Walk up till misclassification, however, we do not exceed more than 50 steps and prematurely terminate without misclassification in the event that the prediction label does not change.
+
+Performance of White Box Approach. We find in our evaluations that the white-box MinGD method generally underperforms the Blind Walk method. This happens despite its ability of being able to compute the nearest distance to any target class more accurately. While on the onset, this may seem to be a counter-intuitive result, since generally with more access, the performance of mapping the neighbours should only increase.
+
+However, we note an important distinction. The end goal of the query generation process is not to calculate the minimum distance to target classes accurately, but rather to understand the ‘prediction margin’ or the local landscape of a given data point. Readers may recall from adversarial examples literature (Szegedy et al., 2013) that adversarial examples can easily be constructed on the dataset that a given machine learning model was trained on. This observation hurts the idea of Figure 1b. Despite pushing the neighbouring class boundaries away, the existence of adversarial examples elucidates the existence of small pits within the landscape of the model.
+
+We hypothesize that the gradient-based optimization objective aims to capture this minimum (adversarial) distance, and fails to capture a ‘prediction margin’ that is more representative of the classifier’s prediction confidence or general landscape. On the contrary, Blind Walk is able to perform a spectacular job at the same end goal. Since we are no longer adversarially trying to optimize the minimum distance to the neighbouring classes, multiple Blind Walk runs effectively map the ‘average case’ prediction margin, which we argue is more useful than the ‘worst case’ prediction margin as obtained by MinGD.
+
+# D EFFECT OF EMBEDDING SIZE
+
+For all models, richer embeddings reduce the need for more revealed samples. (See Figure 4). We note that in the main body of this work, we had used a fixed size of embedding vector, with 30 input features. However, recall that in the black-box setting, the victim incurs additional cost for querying the potential adversary. Therefore, in this section we aim to understand the marginal utility of extra embedding features added. In general, we find that for most of the threat models studied, using only 10 features for the embedding space is sufficient to achieve the required threshold p-value of 0.01. This suggests that we can slash the number of queries made to the potential adversary by one-thirds, without loss in confidence of prediction.
+
+Interestingly, we also note that even in scenarios where the victim reveals only 15 samples, additional embedding features have insignificant advantage as opposed to querying fresh samples. This suggests that the amount of entropy gained by revealing a new data point is significantly more than that by extracting more features (beyond 10) for the same data point. We also note that the effect is not-consistent in the zero-shot learning threat model.
+
+# E EXPERIMENTS
+
+# E.1 DATASET DESCRIPTION
+
+CIFAR10. CIFAR10 (Krizhevsky, 2012) contains 60,000 coloured images with 10,000 reserved for testing. There are 10 target classes with 5000 training images per class.
+
+SVHN. SVHN (Netzer et al., 2011) is a dataset obtained from house numbers in Google Street View images. The underlying task is of digit classification from $3 2 \times 3 2$ coloured images.
+
+CIFAR100. CIFAR100 (Krizhevsky, 2012) also contains 60,000 coloured images with 10,000 reserved for testing. There are 100 target classes with 500 training images per class.
+
+ImageNet. The ImageNet dataset (Deng et al., 2009) is a large-scale benchmark, consisting various challenges including that of image recognition for machine learning systems.
+
+![](images/44c8d27a4fde0c976f561c3cfba2bf385b9abe904caea25410614edd3a1322dc.jpg)  
+Figure 4: p-value vs. distance-embedding size
+
+<table><tr><td colspan="4">Dataset Inference on SVHN (Blind Walk Attack)</td></tr><tr><td colspan="2">Model Stealing Attack</td><td>△</td><td>p-value</td></tr><tr><td>V</td><td>Source</td><td>0.950</td><td>10-8</td></tr><tr><td rowspan="2">AD</td><td>Distillation</td><td>0.537</td><td>10-3</td></tr><tr><td>Diff. Architecture</td><td>0.450</td><td>10-2</td></tr><tr><td rowspan="2">Am</td><td>Zero-Shot Learning</td><td>0.512</td><td>10-3</td></tr><tr><td>Fine-tuning</td><td>0.581</td><td>10-4</td></tr><tr><td rowspan="2">AQ</td><td>Label-query</td><td>0.513</td><td>10-03</td></tr><tr><td>Logit-query Random-query</td><td>0.515</td><td>10-02</td></tr><tr><td></td><td></td><td>0.475</td><td>10-02</td></tr><tr><td>I</td><td>Independent</td><td>-0.322</td><td>10-01</td></tr></table>
+
+Table 2: Ownership Tester’s effect size in a small-data regime (using only $m = 1 0$ samples) on the SVHN dataset using Blind Walk attack. 2nd highest and lowest effect size is marked in red $\&$ blue.
+
+# E.2 ADDITIONAL DATASETS
+
+To further validate our claims about the success of dataset inference, we provide evidence on two additional datasets. In this section, we present results on the SVHN and ImageNet datasets.
+
+SVHN. The results of DI via the ‘Blind Walk’ attack on the various threat models discussed in Appendix B are presented in Table 2. To perform model extraction and fine-tuning attacks, we utilized the set of ‘extra’ images available with the SVHN dataset. We use the first 50,000 images in this set to stage such attacks that require a surrogate dataset. For training on the original dataset with a different architecture, we once again utilize the Pre-activation version of ResNet-18 as for CIFAR10 and CIFAR100 in the main paper. Notably, we also introduce another threat model: Random-query which describes a scenario where the victim is queried with completely random inputs. While the zero-short learning framework also queries the victim with synthetic images, the queried images are synthesized to maximize the disparity between predictions of the student and teacher. On the contrary, in case of Random-query, we query the victim by sampling from a normal distribution, $x \sim \mathcal { N } ( 0 , 1 )$ . DI is resilient to completely random queries as well.
+
+<table><tr><td></td><td>Threat Model</td><td>ImageNet Architecture</td><td>△μ</td><td>p-value</td></tr><tr><td>V</td><td>Source</td><td>Wide ResNet-50-2</td><td>1.868</td><td>10-34</td></tr><tr><td>AD</td><td>Diff. Architecture</td><td>AlexNet</td><td>0.790</td><td>10-3</td></tr><tr><td></td><td>Diff. Architecture</td><td>Inception V3</td><td>1.085</td><td>10-5</td></tr></table>
+
+Table 3: Ownership Tester’s effect size in a small-data regime (using only $m = 1 0$ samples) on the ImageNet dataset using Blind Walk attack.
+
+<table><tr><td>Fraction Overlap</td><td>△μ</td><td>p-value</td></tr><tr><td>0.0</td><td>-0.172</td><td>0.308</td></tr><tr><td>0.3</td><td>0.499</td><td>7.93 ×10-3</td></tr><tr><td>0.5</td><td>0.514</td><td>5.78×10-3</td></tr><tr><td>0.7</td><td>0.576</td><td>2.52×10-3</td></tr><tr><td>1.0</td><td>0.566</td><td>3.45 ×10-3</td></tr></table>
+
+Table 4: Ownership Tester’s effect size in a small-data regime (using only $m = 1 0$ samples).
+
+Note that we do not include Random-query as a threat model in case of CIFAR10 and CIFAR100 datasets because random querying is insufficient to achieve model extraction accuracies greater than the majority class baseline in more complicated tasks such as these. However, in case of SVHN, we were able to train an extracted model with $9 0 . 2 \%$ test set accuracy using random queries alone. Similar observations have been shared in other model extraction literature as well (Truong et al., 2021). We found that our conclusions hold for this additional dataset and we can claim ownership with as few as 10 examples.
+
+ImageNet. We remark that prior work in model extraction has not successfully demonstrated the efficacy of model stealing on large-scale benchmarks like ImageNet. These methods require many queries to steal a model, and are hence not practical yet. As a proof of validity of our approach, we demonstrate dataset inference (DI) in the threat model that assumes complete data theft: the adversary directly steals the dataset used by the victim to train a model rather than querying the victim model. We use 3 pre-trained models on ImageNet using different architectures, and treat the one with a Wide ResNet-50-2 (Zagoruyko & Komodakis, 2016) backbone as the victim. We then observe if DI is able to correctly identify that the two other pre-trained models (AlexNet (Krizhevsky, 2014) and Inception V3 (Szegedy et al., 2015)) were also trained on the same dataset (i.e., ImageNet). We confirm that DI is able to claim ownership (i.e., that the suspect models were indeed trained using knowledge of the victim’s training set) with a p-value of $1 0 ^ { - 3 }$ on revealing only 10 samples.
+
+For performing DI, the victim trains the confidence regressor with the helps of embeddings generated by querying the Wide ResNet-50-2 architecture over the training and validation sets separately. We first note that the confidence-regressor generalizes well to other points in the victim’s train and test set, despite the fact that the ImageNet dataset is orders of magnitude larger than the previous benchmarks experimented on. With only 10 examples, we attain p-values less than $1 0 ^ { - 3 0 }$ . Finally, to test if this generalization holds for other architectures that underwent a disjoint training procedure, we experiment over AlexNet and Inception V3. We find that given 10 examples from the ImageNet training dataset, DI can confidently say that the suspect models utilize knowledge of the victim’s training set with p-values less than $\mathrm { \dot { 1 } 0 ^ { - \dot { 4 } } }$ . Note that these models are trained on large datasets where works in MI attacks train victims on small subsets of training datasets to enable overfitting Yeom et al. (2018). We believe this demonstrates that DI scales to complex tasks.
+
+# F EXTENT OF OVERLAP
+
+In this section we elaborate upon the effect of overlap between private datasets of two parties and how does dataset inference respond to such scenarios. More specifically, we study the amount of overlap required for DI to be able to claim theft of common knowledge in the following scenario: We consider a competitor (or adversary) who owns their own private training dataset $\mathcal { S } _ { A }$ . The adversary gains access to the victim’s training dataset $ { s _ { \nu } }$ . The adversary now trains their ML model on $\mathcal { S } = \mathcal { S } _ { A } \cup \mathcal { S } _ { \mathcal { V } } { } ^ { \lambda }$ , where ${ \mathcal { S } _ { \mathcal { V } } } ^ { \lambda } \subset { \mathcal { S } _ { \mathcal { V } } }$ and $| s _ { \nu } { } ^ { \lambda } | = \lambda | { \cal S } _ { \nu } |$ . That is the new dataset $s$ has a fraction $\lambda$ of the training points private to $\nu$ .
+
+![](images/ae3e6ff545b8c8f257a5f59a566bd1943517476465bd8491ba5256c1536dd262.jpg)  
+Figure 5: Ownership Tester’s p-value depicted as a function of number of training samples revealed $( m )$ . In the figure to the left, for an honest adversary whose dataset has no overlap with the victim’s dataset, the $\mathsf { p }$ -value increases as the number of revealed samples increases, indicating decrease in confidence of claim for knowledge theft. For all adversaries with fractional data overlap (to the right), DI is able to achieve a $\mathsf { p }$ -value of less than 0.01 in under 10 samples.
+
+Since at the training time the adversary optimizes the ‘prediction margin’ over all points in $s$ , the prediction margin for points in $ { s _ { \nu } }$ also gets affected. At test time when the victim queries on these points, DI is expected to succeed.
+
+We validate this claim on the SVHN dataset which provides a set of ‘extra’ images apart from ‘train’ and ‘test’ sets. We train the adversary on the union of ‘extra’ and varying fractions of the ‘train’ set, where the ‘train’ set is supposed to be private to the victim. At the time of dataset inference, the victim queries 50 samples from its private ‘train’ and ‘test’ set to the adversary’s model. Dataset Inference succeeds with $\mathsf { p }$ -value $= \bar { 1 0 } ^ { - 3 }$ for fractions of overlap $= 0 . 3$ , 0.5, 0.7, 1.0 as tested. More importantly, as the overlap goes to 0, DI once again does not claim knowledge theft. We present our results for DI on revealing 10 samples in Table 4 and present a graphical illustration of these results with varying numbers of samples revealed in Figure 5. From Table 4, it may also be noted how the effect size increases with the amount of overlap of the private training set, indicating that the DI is becoming increasingly confident of knowledge theft.

parse/train/rJeGJaEtPH/rJeGJaEtPH.md

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+# MIST MULTIPLE INSTANCE SPATIAL TRANSFORMERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a deep network that can be trained to tackle image reconstruction and classification problems that involve detection of multiple object instances, without any supervision regarding their whereabouts. The network learns to extract the most significant K patches, and feeds these patches to a task-specific network – e.g., auto-encoder or classifier – to solve a domain specific problem. The challenge in training such a network is the non-differentiable top- $K$ selection process. To address this issue, we lift the training optimization problem by treating the result of top- $K$ selection as a slack variable, resulting in a simple, yet effective, multi-stage training. Our method is able to learn to detect recurring structures in the training dataset by learning to reconstruct images. It can also learn to localize structures when only knowledge on the occurrence of the object is provided, and in doing so it outperforms the state-of-the-art.
+
+# 1 INTRODUCTION
+
+Finding and processing multiple instances of characteristic entities in a scene is core to many computer vision applications, including object detection (Ren et al., 2015; He et al., 2017; Redmon & Farhadi, 2017), pedestrian detection (Dollár et al., 2012; Sewart & Andriluka, 2016; Zhang et al., 2018a), and keypoint localization (Lowe, 2004; Bay et al., 2008). In traditional vision pipelines, a common approach to localizing entities is to select the top-K responses in a heatmap and use their locations (Lowe, 2004; Bay et al., 2008; Felzenszwalb et al., 2010). However, this type of approach does not provide a gradient with respect to the heatmap, and cannot be directly integrated into neural network-based systems. To overcome this challenge, previous work proposed to use grids (Redmon et al., 2016; He et al., 2017; Detone et al., 2018) to simplify the formulation by isolating each instance (Yi et al., 2016), or to optimize over multiple branches (Ono et al., 2018). While effective, these approaches require additional supervision to localize instances, and do not generalize well outside their intended application domain. Other formulations, such as sequential attention (Ba et al., 2015; Gregor et al., 2015; Eslami et al., 2015) and channel-wise approaches (Zhang et al., 2018c) are problematic to apply when the number of instances of the same object is large.
+
+Here, we introduce a novel way to approach this problem, which we term Multiple Instance Spatial Transformer, or MIST for brevity. As illustrated in Figure 1 for the image synthesis task, given an image, we first compute a heatmap via a deep network whose local maxima correspond to locations of interest. From this heatmap, we gather the parameters of the top- $K$ local maxima, and then extract the corresponding collection of image patches via an image sampling process. We process each patch independently with a task-specific network, and aggregate the network’s output across patches.
+
+Training a pipeline that includes a non-differentiable selection/gather operation is non-trivial. To solve this problem we propose to lift the problem to a higher dimensional one by treating the parameters defining the interest points as slack variables, and introduce a hard constraint that they must correspond to the output that the heatmap network gives. This constraint is realized by introducing an auxiliary function that creates a heatmap given a set of interest point parameters. We then solve for the relaxed version of this problem, where the hard constraint is turned into a soft one, and the slack variables are also optimized within the training process. Critically, our training strategy allows the network to incorporate both non-maximum suppression and top-K selection. We evaluate the performance of our approach for $\textcircled{1}$ the problem of recovering the basis functions that created a given texture, $\textcircled{2}$ classification of handwritten digits in cluttered scenes, and $\textcircled{3}$ recognition of house numbers in real-world environments. In summary, in this paper we:
+
+• introduce the MIST framework for weakly-supervised multi-instance visual learning; • propose an end-to-end training method that allows the use of top-K selection; • show that our framework can reconstruct images as parts, as well as detect/classify instances without any location supervision.
+
+# 2 RELATED WORK
+
+Attention models and the use of localized information have been actively investigated in the literature. Some examples include discriminative tasks such as fine-grained classification (Sun et al., 2018), pedestrian detection (Zhang et al., 2018a), and generative ones such as image synthesis from natural language (Johnson et al., 2018). They have also been studied in the context of more traditional Multiple Instance Learning (MIL) setup (Ilse et al., 2018). We now discuss a selection of representative works, and classify them according to how they deal with multiple instances.
+
+Grid-based methods. Since the introduction of Region Proposal Networks (RPN) (Ren et al., 2015), grid-based strategies have been used for dense image captioning (Johnson et al., 2016), instance segmentation (He et al., 2017), keypoint detection (Georgakis et al., 2018), and multi-instance object detection (Redmon & Farhadi, 2017). Recent improvements to RPNs attempt to learn the concept of a generic object covering multiple classes (Singh et al., 2018), and to model multi-scale information (Chao et al., 2018). The multiple transformation corresponding to separate instances can also be densely regressed via Instance Spatial Transformers (Wang et al., 2018), which removes the need to identify discrete instance early in the network. However, all these methods are fully supervised, requiring both class labels and object locations for training.
+
+Heatmap-based methods. Heatmap-based methods have recently gained interest to detect features (Yi et al., 2016; Ono et al., 2018; Detone et al., 2018), find landmarks (Zhang et al., 2018c; Merget et al., 2018), and regress human body keypoint (Tekin et al., 2017; Newell et al., 2016). While it is possible to output a separate heatmap per class (Zhang et al., 2018c; Tekin et al., 2017), most heatmap-based approaches do not distinguish between instances. Yi et al. (2016) re-formulate the problem based on each instance, but in doing so introduce a sharp difference between training and testing. Grids can also be used with heatmaps (Detone et al., 2018), but this results in an unrealistic assumption of uniformly distributed detections in the image. Overall, heatmap-based methods excel when the “final” task of the network is generate a heatmap (Merget et al., 2018), but are problematic to use as an intermediate layer in the presence of multiple instances.
+
+Sequential inference methods. Another way to approach multi-instance problems is to attend to one instance at a time in a sequentially. Training neural network-based models with sequential attention is challenging, but approaches using policy gradient (Ba et al., 2015) and differentiable attention mechanisms (Gregor et al., 2015; Eslami et al., 2015) have achieved some success for images comprising small numbers of instances. However, Recurrent Neural Networks (RNN) often struggle to generalize to sequences longer than the ones encountered during training, and while recent results on inductive reasoning are promising (Gupta et al., 2018), their performance does not scale well when the number of instances is large.
+
+Knowledge transfer. To overcome the acquisition cost of labelled training data, one can transfer knowledge from labeled to unlabeled dataset. For example, Inoue et al. (2018) train on a single instance dataset, and then attempt to generalize to multi-instance domains, while Uijlings et al. (2018) attempt to also transfer a multi-class proposal generator to the new domain. While knowledge transfer can be effective, it is highly desirable to devise unsupervised methods such as ours that do not depend on an additional dataset.
+
+Weakly supervised methods. To further reduce the labeling effort, weakly supervised methods have also been proposed. Wan et al. (2018) learn how to detect multiple instances of a single object via region proposals and ROI pooling, while Tang et al. (2018) propose to use a hierarchical setup to refine their estimates. Gao et al. (2018) provides an additional supervision by specifying the number of instances in each class, while Oquab et al. (2015) and Zhang et al. (2018b) localize objects by looking at the network activation maps (Zhou et al., 2016; Selvaraju et al., 2017). Shen et al. (2018) introduce an adversarial setup, where detection boxes are supervised by distribution assumptions and classification objectives. However, all these methods still rely on region proposals from an existing method, or define them via a hand-tuned process.
+
+![](images/2c706924e05863bd7e4a56a0fa0ca14ba0b5fdd42d7a2698b6322f91a2832b4d.jpg)  
+Figure 1: The MIST architecture – A network $\mathcal { H } _ { \eta }$ estimates locations and scales of patches encoded in a heatmap h. Patches are then extracted via a sampler $s$ , and then fed to a task-specific network $\mathcal { T } _ { \tau }$ . In this example, the specific task is to re-synthesize the image as a super-position of (unknown and locally supported) basis functions. As top- $\mathbf { \nabla } \cdot \mathbf { k }$ operation is non-differentiable, we back-propagate by lifting through $\mathcal { G }$ ; see Section 3 for details.
+
+# 3 MIST FRAMEWORK
+
+We first explain our framework at a high level, and then detail each component in Section 4. A prototypical MIST architecture (Figure 1) is composed of two trainable components: $\textcircled{1}$ the first module receives an image as input and extracts $K$ patches, at image locations and scales given by the top $K$ local maxima of a heatmap computed by a trainable heatmap network $\mathcal { H } _ { \eta }$ with weights $\eta$ . $\textcircled{2}$ the second module processes each extracted patch with a task-specific network $\mathcal { T } _ { \tau }$ whose weights $\tau$ are shared across patches, and further manipulates these signals to express a task-specific loss $\mathcal { L } _ { t a s k }$ The two modules are connected through non-maximum suppression on the scale-space heatmap output of $\mathcal { H } _ { \eta }$ , followed by a top- $K$ selection process to extract the parameters defining the patches, which we denote as $\mathcal { E } _ { K }$ . We then sample patches at these locations through (differentiable) bilinear sampling $s$ and feed them the task module.
+
+The defining characteristic of the MIST architecture is that they are quasi-unsupervised: the only supervision required is the number $K$ of patches to extract. However, as we show in Section 5.2, our method is not sensitive to the choice of $K$ during training. In addition, our extraction process uses a single heatmap for all instances that we extract. In contrast, existing heatmap-based methods (Eslami et al., 2015; Zhang et al., 2018c) typically rely on heatmaps dedicated to each instance, which is problematic when an image contains two instances of the same class. Conversely, we restrict the role of the heatmap network $\mathcal { H } _ { \eta }$ to find the “important” areas in a given image, without having to distinguishing between classes, hence simplifying learning.
+
+Formally, the training of the MIST architecture is summarized by:
+
+$$
+\operatorname* { m i n i m i z e } _ { \tau , \eta } \mathcal { L } _ { \mathrm { t a s k } } ( \mathcal { T } _ { \tau } ( S ( \mathcal { E } _ { K } ( \mathcal { H } _ { \eta } ( \mathcal { T } ) ) ) ) )
+$$
+
+where $\tau , \eta$ are the network trainable parameters. Unfortunately, the patch extractor $\mathcal { E } _ { K }$ is nondifferentiable, as it identifies the locations of the top- $K$ local maxima of a heatmap and then selects the corresponding patches from the input image. Differentiating this operation provides a gradient with respect to the input, but no gradient with respect to the heatmap. Although it is possible to smoothly relax the patch selection operation in the $K = 1$ case (Yi et al., 2016) (i.e. argmax), it is unclear how to generalize this approach to the case of multiple distinct local maxima. It is thus impossible to train the patch selector parameters directly by backpropagation. We propose an alternative approach to training our model, via lifting.
+
+Algorithm 1 Multi-stage optimization for MISTs   
+
+<table><tr><td></td><td>Require: K : number of patches to extract, Ltask : task specific loss,I : input image, G : the keypoints to heatmap function, H : the heatmap network,n : parameters of the heatmap network,</td><td></td><td></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td>1:</td><td>T : the task network, T : parameters of the task network,&amp;k : the top-K operator. function TRAINMIST(I,Ltask)</td></tr><tr><td>2:</td><td>for each training batch do</td></tr><tr><td>3:</td><td>T ← Optimization step for T- with Ltask +λl{xk} -εκ(Hn())ll2</td></tr><tr><td>4:</td><td>{xk} ← Optimization step for {xk} with Ltask + λl{xk}-εk(Hn(I))ll²2</td></tr><tr><td>5:</td><td>η ← Optimization step for η with |/S({xk}) - Hn(I)ll2</td></tr><tr><td>6: end for</td><td></td></tr><tr><td>7: end function</td><td></td></tr></table>
+
+Differentiable top-K via lifting. The introduction of auxiliary variables to simplify the structure of an optimization problem has proven effective in a range of domains ranging from non-rigid registration (Taylor et al., 2016), to robust optimization (Zach & Bournaoud, 2018). To simplify our training optimization, we start by decoupling the heatmap tensor from the optimization (1) by introducing the corresponding auxiliary variables h, as well as the patch location variables $\left\{ { \bf x } _ { k } \right\}$ from the top- $\mathbf { \nabla } \cdot \mathbf { K }$ extractor:
+
+$$
+\operatorname* { m i n i m i z e } _ { \eta , \tau , \mathbf { h } , \{ \mathbf { x } _ { k } \} } \ \mathcal { L } _ { \mathrm { t a s k } } ( \mathcal { T } _ { \tau } ( \mathcal { S } ( \{ \mathbf { x } _ { k } \} ) ) ) \quad \mathrm { s . t . } \quad \mathbf { h } = \mathcal { H } _ { \eta } ( \mathcal { T } ) , \quad \{ \mathbf { x } _ { k } \} = \mathcal { E } _ { K } ( \mathbf { h } ) .
+$$
+
+We then relax the first constraint above to a least-squares penalty via a Lagrange multiplier $\lambda$ :
+
+$$
+\operatorname* { m i n i m i z e } _ { \eta , \tau , \mathbf { h } , \{ \mathbf { x } _ { k } \} } \ \mathcal { L } _ { \mathrm { t a s k } } ( \mathcal { T } _ { \tau } ( S ( \{ \mathbf { x } _ { k } \} ) ) ) + \lambda \| \mathbf { h } - \mathcal { H } _ { \eta } ( \mathcal { T } ) \| _ { 2 } ^ { 2 } \quad \mathrm { s . t . } \quad \{ \mathbf { x } _ { k } \} = \mathcal { E } _ { K } ( \mathbf { h } ) .
+$$
+
+As in many methods that use keypoint supervision to regress heatmaps (Cao et al., 2018), we assume that a good heatmap generator $\mathcal { G }$ exists $-$ that is $\left\{ \mathbf { x } _ { k } \right\} \approx \mathcal { E } _ { K } ( \mathcal { G } ( \left\{ \mathbf { x } _ { k } \right\} ) )$ We can now rewrite our optimization as:
+
+$$
+\operatorname* { m i n i m i z e } _ { \eta , \tau , \mathbf { h } , \{ \mathbf { x } _ { k } \} } \ \mathcal { L } _ { \mathrm { t a s k } } ( \mathcal { T } _ { \tau } ( S ( \{ \mathbf { x } _ { k } \} ) ) ) + \lambda \| \mathbf { h } - \mathcal { H } _ { \eta } ( \mathcal { T } ) \| _ { 2 } ^ { 2 } \quad \mathrm { s . t . } \quad \mathbf { h } = \mathcal { G } ( \{ \mathbf { x } _ { k } \} ) .
+$$
+
+We can now drop the auxiliary variable $\mathbf { h }$ and rewrite our optimization as:
+
+$$
+\operatorname* { m i n i m i z e } _ { \eta , \tau , \{ \mathbf { x } _ { k } \} } ~ \mathcal { L } _ { \mathrm { t a s k } } ( \mathcal { T } _ { \tau } ( \mathcal { S } ( \{ \mathbf { x } _ { k } \} ) ) ) + \lambda \| \mathcal { G } ( \{ \mathbf { x } _ { k } \} ) - \mathcal { H } _ { \eta } ( \mathcal { Z } ) \| _ { 2 } ^ { 2 } ,
+$$
+
+and then approach the problem by block coordinate descent – where the energy terms not containing the variable being optimized are safely dropped, and we apply $\mathcal { E } _ { K }$ to the penalty term of (5):
+
+$$
+\begin{array} { r l } & { \underset { \tau , \{ \mathbf { x } _ { k } \} } { \mathrm { m i n i m i z e ~ } } \mathcal { L } _ { \mathrm { t a s k } } ( \mathcal { T } _ { \tau } ( \mathcal { S } ( \{ \mathbf { x } _ { k } \} ) ) ) + \lambda \| \{ \mathbf { x } _ { k } \} - \mathcal { E } _ { K } ( \mathcal { H } _ { \eta } ( \mathcal { T } ) ) \| _ { 2 } ^ { 2 } , } \\ & { \mathrm { m i n i m i z e ~ } \| \mathcal { G } ( \{ \mathbf { x } _ { k } \} ) - \mathcal { H } _ { \eta } ( \mathcal { T } ) \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+To accelerate training, we further split (6) into two stages, and alternate between optimizing $\tau$ and $\left\{ { \bf x } _ { k } \right\}$ . Being based on block coordinate descent, this process converges smoothly, as we show in Section D of the appendix. The summary for the three stage optimization procedure is outlined in Algorithm 1. Notice that we are not introducing any additional supervision signal that is tangent to the given task.
+
+# 4 IMPLEMENTATION
+
+# 4.1 COMPONENTS
+
+Multiscale heatmap network $- \mathcal { H } _ { \eta }$ . Our multiscale heatmap network is inspired by LF-Net (Ono et al., 2018). We improve the network by applying modifications on how the scores are aggregated over scale. The network takes as input an image $\mathcal { L }$ , and outputs multiple heatmaps $\mathbf { h } _ { s } ^ { \prime }$ of the same size for each scale level $s$ . To limit the number of necessary scales, we use a discrete scale space with $S$ scales, and resolve intermediate scales via interpolation; see Section A of the appendix for details.
+
+Top-K patch selection – $\mathcal { E } _ { K }$ . To extract the top $K$ elements, we perform an addition cleanup through a standard non-maximum suppression. We then find the spatial locations of the top $K$ elements of this suppressed heatmap, denoting the spatial location of the $k ^ { \mathrm { { t h } } }$ element as $( x _ { k } , y _ { k } )$ , which now reflect local maxima. For each location, we compute the corresponding scale by weighted first order moments (Suwajanakorn et al., 2018) where the weights are the responses in the corresponding heatmaps, i.e. $\begin{array} { r } { s _ { k } = \sum _ { s } \mathbf { h } _ { s } ^ { \prime } ( x _ { k } , y _ { k } ) s } \end{array}$
+
+Generating model for ideal heatmap – $\mathcal { G } ( \{ \mathbf { x } _ { k } \} )$ . For the generative model that turns keypoint and patch locations into heatmaps, we apply a simple model where the heatmap is zero everywhere else except the corresponding keypoint locations (patch center); see Section B of the appendix for details.
+
+Patch resampling – $s$ . As a patch is uniquely parameterized by its location and scale, i.e. $\mathbf { x } _ { k } = \left( x _ { k } , y _ { k } , s _ { k } \right)$ , we can then proceed to resample its corresponding tensor via bilinear interpolation (Jaderberg et al., 2015; Jiang et al., 2019) as $\{ \mathbf { P } _ { k } \} = \mathcal { \bar { S } } ( \mathcal { T } , \{ \mathbf { \bar { x } } _ { k } \} )$ .
+
+# 4.2 TASK-SPECIFIC NETWORKS
+
+We now introduce two applications of the MIST framework. We use the same heatmap network and patch extractor for both applications, but the task-specific network and loss differ. We provide further details regarding the task-specific network architectures in Section C of the appendix.
+
+Image reconstruction / auto-encoding. As illustrated in Figure 1, for image reconstruction we append our patch extraction network with a shared auto-encoder for each extracted patch. We can then train this network to reconstruct the original image by inverting the patch extraction process and minimizing the mean squared error between the input and the reconstructed image. Overall, the network is designed to jointly model and localize repeating structures in the input signal. Specifically, we introduce the generalized inverse sampling operation $S ^ { - 1 } ( \mathbf { P } _ { i } , \mathbf { x } _ { i } )$ , which starts with an image of all zeros, and places the patch $\mathbf { P } _ { i }$ at $\mathbf { x } _ { i }$ . We then sum all the images together to obtain the reconstructed image, optimizing the task loss
+
+$$
+\mathcal { L } _ { \mathrm { t a s k } } = \| \mathcal { I } - \sum _ { i } \mathcal { S } ^ { - 1 } \left( \mathbf { P } _ { i } , \mathbf { x } _ { i } \right) \| _ { 2 } ^ { 2 } .
+$$
+
+Multiple instance classification. By appending a classification network to the patch extraction module, we can also perform multiple instance learning. For each extracted patch $\mathbf { P } _ { k }$ we apply a shared classifier network to output $\mathbf { \hat { y } } _ { k } \in \mathbb { R } ^ { C }$ , where $C$ is the number of classes. In turn, these are then converted into probability estimates by the transformation $\hat { \mathbf { p } } _ { k } = \mathrm { s o f t m a x } ( \hat { \mathbf { y } } _ { k } )$ . With $\mathbf { y } _ { l }$ being the one-hot ground-truth labels of instance $l$ with unit norm, we define the multi-instance classification loss as
+
+$$
+\mathcal { L } _ { \mathrm { t a s k } } = \left\| \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \mathbf { y } _ { l } - \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \hat { \mathbf { p } } _ { k } \right\| _ { 2 } ^ { 2 } ,
+$$
+
+where $L$ is the number of instances in the image. We empirically found this loss to be more stable compared to the cross-entropy loss, similar to Mao et al. (2017). Note here that we do not provide supervision about the localization of instances, yet the detector network will automatically learn how to localize the content with minimal supervision (i.e. the number of instances).
+
+# 5 RESULTS AND EVALUATION
+
+To demonstrate the effectiveness of our framework we evaluate the two different tasks. We first perform a quasi-unsupervised image reconstruction task, where only the total number of instances in the scene is provided, i.e. K is defined. We then show that our method can also be applied to weakly supervised multi-instance classification, where only image-level supervision is provided. Note that, unlike region proposal based methods, our localization network only relies on cues from the classifier, and both networks are trained from scratch.
+
+# 5.1 IMAGE RECONSTRUCTION
+
+From the MNIST dataset, we derive two different scenarios. In the MNIST easy dataset, we consider a simple setup where the sorted digits are confined to a perturbed grid layout; see Figure 2 (top).
+
+![](images/d0afef7f81e174c9b17655771a6c1b303254a0adab9cad29d7e7e043a716490a.jpg)  
+Figure 2: MNIST character synthesis examples for (top) the “easy” single instance setup and (bottom) the hard multi-instance setup. We compare the output of MISTs to grid, channel-wise, sequential Eslami et al. (Eslami et al., 2015) and Zhang et al. (Zhang et al., 2018c).
+
+![](images/cf93711634a0ec95e37380e5f6b4e4f0a7f21ec7ecb6d8b7407784ddc95c2373.jpg)  
+Figure 3: Two auto-encoding examples learnt from MNIST-hard. In the top row, for each example we visualize input, patch detections, and synthesis. In the bottom row we visualize each of the extracted patch, and how it is modified by the learnt auto-encoder.
+
+Specifically, we perturb the digits with a Gaussian noise centered at each grid center, with a standard deviation that is equal to one-eighths of the grid width/height. In the MNIST hard dataset, the positions are randomized through a Poisson distribution (Bridson, 2007), as is the identity, and cardinality of each digit. We allow multiple instances of the same digit to appear in this variant. For both datasets, we construct both training and test sets, and the test set is never seen during training.
+
+Comparison baselines We compare our method against four baselines $\textcircled{1}$ the grid method divides the image into a $3 \times 3$ grid and applies the same auto-encoder architecture as MIST to each grid location to reconstruct the input image; $\textcircled{2}$ the channel-wise method uses the same auto-encoder network as MIST, but we modify the heatmap network to produce $K$ channels as output, where each channel is dedicated to an interest point. Locations are obtained through a channel-wise soft-argmax as in Zhang et al. (2018c); $\textcircled{3}$ Esl16 (Eslami et al., 2015) is a sequential generative model; $\textcircled{4}$ Zha18 (Zhang et al., 2018c) is a state-of-the-art heatmap-based method with channel-wise strategy for unsupervised learning of landmarks. For more details see Supplementary Section C.
+
+Results for “MNIST easy” As shown in Figure 2 (top) all methods successfully re-synthesize the image, with the exception of Eslami et al. (Eslami et al., 2015). As this method is sequential, with nine digits the sequential implementation simply becomes too difficult to optimize through. Note how this method only learns to describe the scene with a few large regions. We summarize quantitative results in Table 1.
+
+Results for “MNIST hard” As shown in Figure 2 (bottom), all baseline methods failed to properly represent the image. Only MIST succeeded at both localizing digits and reconstructing the original image. Although the grid method accurately reconstructs the image, it has no concept of individual digits. Conversely, as shown in Figure 3, our method generates accurate bounding boxes for digits even when these digits overlap, and does so without any location supervision. For quantitative results, please see Table 1.
+
+![](images/3cb1835c60e9839a0d84a93c82ab8c427f74203ebb793ae7ddd66627f4156a3e.jpg)  
+Figure 4: Inverse rendering of Gabor noise; we annotate correct / erroneous localizations.
+
+![](images/8ce1f38cea0a1c4239ab191e14845ed3cdce1785fd71324e73fd67635fbd2c94.jpg)  
+Figure 5: Two qualitative examples for detection and classification on our Multi-MNIST dataset.
+
+Finding the basis of a procedural texture We further demonstrate that our methods can be used to find the basis function of a procedural texture. For this experiment we synthesize textures with procedural Gabor noise (Lagae et al., 2009). Gabor noise is obtained by convolving oriented Gabor wavelets with a Poisson impulse process. Hence, given exemplars of noise, our framework is tasked to regress the underlying impulse process and reconstruct the Gabor kernels so that when the two are convolved, we can reconstruct the original image. Figure 4 illustrates the results of our experiment. The auto-encoder learned to accurately reconstruct the Gabor kernels, even though in the training images they are heavily overlapped. These results show that MIST is capable of generating and reconstructing large numbers of instances per image, which is intractable with other approaches.
+
+# 5.2 MULTIPLE INSTANCE CLASSIFICATION
+
+Multi-MNIST – Figure 5. To test our method in a multiple instance classification setup, we rely on the MNIST hard dataset. We compare our method to channel-wise, as other baselines are designed for generative tasks. To evaluate the detection accuracy of the models, we compute the intersection over union (IoU) between the ground-truth bounding box and the detection results, and assign it as a match if the IoU score is over $50 \%$ . We report the number of correctly classified matches in Table 2, as well as the proportion of instances that are both correctly detected and correctly classified. Our method clearly outperforms the channel-wise strategy. Note that, even without localization supervision, our method correctly localizes digits. Conversely, the channel-wise strategy fails to learn. This is because multiple instances of the same digits are present in the image. For example, in the Figure 5 (right), we have three sixes, two zeros, and two nines. This prevents any of these digits from being detected/classified properly by a channel-wise approach.
+
+Sensitivity to $K$ – Table 3. To investigate the sensitivity to the correctness of $K$ during training, we train with varying number of of instances and test with ground-truth $K$ . For example, with $K = 6$
+
+<table><tr><td colspan="3">MIST</td><td>Ch.-wise Supervised</td></tr><tr><td>IOU 50%</td><td>97.8%</td><td>25.4%</td><td>99.6%</td></tr><tr><td>Classif.</td><td>98.8%</td><td>75.5%</td><td>98.7%</td></tr><tr><td>Both</td><td>97.5%</td><td>24.8%</td><td>98.6%</td></tr></table>
+
+<table><tr><td>Instances during train AploU=.50</td><td></td></tr><tr><td>{9}</td><td>92.2%</td></tr><tr><td>{6,7,8,9}</td><td>90.1%</td></tr><tr><td>{3,4,5,6,7,8,9}</td><td>90.8%</td></tr></table>
+
+Table 1: Reconstruction error (root mean square error). Note that Grid does not learn any notion of digits.   
+
+<table><tr><td>MIST</td><td>Grid Ch.-wise Esl16 Zha18</td><td></td><td></td><td></td><td></td></tr><tr><td>MNIST easy</td><td>.038</td><td>.039</td><td>.042</td><td>.100</td><td>.169</td></tr><tr><td>MNIST hard</td><td>.053</td><td>.062</td><td>.128</td><td>.154</td><td>.191</td></tr><tr><td>Gabor</td><td>.095</td><td>1</td><td>1</td><td>-</td><td>-</td></tr></table>
+
+Table 2: Instance level detection and classification performance on “MNIST hard”.
+
+Table 3: Sensitivity experiment of $K$ on the MNIST hard dataset.
+
+![](images/9fbef27720c8b2fd5da027ebcf4057af2a600aa885953bf8b5c5a4085188f310.jpg)  
+Figure 6: Qualitative SVHN results.
+
+Table 4: Quantitative SVHN results.   
+
+<table><tr><td></td><td>MIST</td><td>Supervised</td></tr><tr><td>ApIoU=.00</td><td>82.6%</td><td>65.6%</td></tr><tr><td>APIoU=.50</td><td>76.5%</td><td>62.8%</td></tr><tr><td>APIoU=.60</td><td>63.7%</td><td>59.8%</td></tr><tr><td>APIoU=.70</td><td>42.7%</td><td>51.9%</td></tr><tr><td>APIoU=.80</td><td>19.9%</td><td>34.6%</td></tr><tr><td>APIoU=.90</td><td>4.2%</td><td>11.0%</td></tr></table>
+
+where the ground truth could be anything within $\{ 3 , 4 , 5 , 6 , 7 , 8 , 9 \} - \mathrm { w e }$ mark the $K$ used during training with bold. Our method still is able to give accurate result with inaccurate $K$ during training. Knowing the exact number of objects is not a strict requirement at test time, as our detector generates a heatmap for the entire image regardless of the $K$ it was trained with. Note that while in theory sequential methods are free from this limitation, in practice they are able to deal with limited number of objects (e.g. up to three) due to their recurrent nature.
+
+SVHN – Figure 6 and Table 4. We further apply MIST to the uncropped and unaligned Street View House Numbers dataset (Netzer et al., 2011). Compared to previous work that has used cropped and resized SVHN images (e.g. (Netzer et al., 2011; Ba et al., 2015; Goodfellow et al., 2014; Jaderberg et al., 2015)), this evaluating setting is significantly more challenging, because digits can appear anywhere in the image. We resize all images to $6 0 \times 2 4 0$ , use only images labeled as containing 2 digits, and apply MIST at a single scale. Although the dataset provides bounding boxes for the digits, we ignore these bounding boxes and use only digit labels as supervision. During testing, we exclude images with small bounding boxes $\textless 3 0$ pixels in height). We report results in terms of $\mathrm { A P ^ { I o U = X } }$ , where $\mathbf { X }$ is the threshold for determining detection correctness. With $\mathrm { I o U } = 0$ , we refer to a “pure” classification task (i.e. no localization). As shown, supervised results provide better performance with higher thresholds, but MIST performs even better than the supervised baseline for moderate thresholds. We attribute this to the fact that, by providing direct supervision on the location, the training focuses too much on having high localization accuracy.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+In this paper, we introduce the MIST framework for multi-instance image reconstruction/classification. Both these tasks are based on localized analysis of the image, yet we train the network without providing any localization supervision. The network learns how to extract patches on its own, and these patches are then fed to a task-specific network to realize an end goal. While at first glance the MIST framework might appear non-differentiable, we show how via lifting they can be effectively trained in an end-to-end fashion. We demonstrated the effectiveness of MIST by introducing a variant of the MNIST dataset, and demonstrating compelling performance in both reconstruction and classification. We also show how the network can be trained to reverse engineer a procedural texture synthesis process. MISTs are a first step towards the definition of optimizable image-decomposition networks that could be extended to a number of exciting unsupervised learning tasks. Amongst these, we intend to explore the applicability of MISTs to unsupervised detection/localization of objects, facial landmarks, and local feature learning.
+
+Even though our method is a significant step forward from existing methods, it is not complete and has much potential for improvement. While our method is robust to $K$ during training as shown in Section 5.2, and $K$ can be determined dynamically during test time, it would beneficial to automatically determine $K$ through an additional network. Patches could also be refined to have varying aspect ratios similar to RPNs. Reconstruction could consider a global reconstructor to deal with complex images. These are parts of our future work towards MIST in the wild.
+
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+
+# Appendix
+
+# A MULTISCALE HEATMAP NETWORK
+
+Our multiscale heatmap network is inspired by LF-Net (Ono et al., 2018). We employ a fully convolutional network with (shared) weights $\eta$ at multiple scales, indexed by $s = 1 \ldots S$ , on the input image $\mathcal { T }$ . The weights $\eta$ across scales are shared so that the network cannot implicitly favor a particular scale. To do so, we first downsample the image to each scale producing $\mathcal { T } _ { s }$ , execute the network $\mathcal { H } _ { \eta }$ on it, and finally upsample to the original resolution. This process generates a multiscale heatmap tensor $\mathbf { h } = \{ \mathbf { h } _ { s } \}$ of size $H \times W \times S$ where $\mathbf { h } _ { s } = \mathcal { H } _ { \eta } ( \mathbb { Z } _ { s } )$ , and $H$ is the height of the image and $W$ is the width. For the convolutional network we use 4 ResNet blocks (He et al., 2015), where each block is composed of two $3 \times 3$ convolutions with 32 channels and relu activations without any downsampling. We then perform a local spatial softmax operator (Ono et al., 2018) with spatial extent of $1 5 \times 1 5$ to sharpen the responses. Then we further relate the scores across different scales by performing a “softmax pooling” operation over scale. Specifically, if we denote the heatmap tensor after local spatial softmax as $\tilde { \mathbf { h } } = \{ \tilde { \hat { \mathbf { h } } } _ { s } \}$ , since after the local spatial softmax $\mathcal { H } _ { \eta } ( \mathcal { T } _ { s } )$ is already an “exponentiated” signal, we do a weighted normalization without an exponential, i.e. $\begin{array} { r } { \mathbf { h } ^ { \prime } = \sum _ { s } \mathbf { \tilde { h } } _ { s } ( \tilde { \mathbf { h } } _ { s } / \tilde { \sum _ { s ^ { \prime } } } ( \tilde { \mathbf { h } } _ { s ^ { \prime } } + \epsilon ) ) } \end{array}$ , where $\epsilon = 1 0 ^ { - 6 }$ is added to prevent division by zero.
+
+Comparison to LF-Net. Note that differently from LF-Net (Ono et al., 2018), we do not perform a softmax along the scale dimension. The scale-wise softmax in LF-Net is problematic as the computation for a softmax function relies on the input to the softmax being unbounded. For example, in order for the softmax function to behave as a max function, due to exponentiation, it is necessary that one of the input value reaches infinity (i.e. the value that will correspond to the max), or that all other values to reach negative infinity. However, at the network stage where softmax is applied in (Ono et al., 2018), the score range from zero to one, effectively making the softmax behave similarly to averaging. Our formulation does not suffer from this drawback.
+
+# B GENERATIVE MODEL FOR THE HEATMAP
+
+As discussed previously in Section 4.1, for the generative model that turns keypoint and patch locations into heatmaps, we apply a simple model where the heatmap is zero everywhere else except the corresponding keypoint locations (patch center). However, as the optimized patch parameters are no longer integer values, we need to quantize them with care. For the spatial locations we simply round to the nearest pixel, which at most creates a quantization error of half a pixel, which does not cause problems in practice. For scale however, simple nearest-neighbor assignment causes too much quantization error as our scale-space is sparsely sampled. We therefore assign values to the two nearest neighboring scales in a way that the center of mass would be the optimized scale value, making sure $\left\{ { \bf x } _ { k } \right\} = \mathcal { E } _ { K } ( \mathcal { G } ( \{ { \bf x } _ { k } \} ) )$ .
+
+# C IMPLEMENTATION DETAILS
+
+MIST auto-encoder network. The input layer of the autoencoder is $3 2 \times 3 2 \times C$ where $C$ is the number of color channels. We use 5 up/down-sampling levels. Each level is made of 3 standard non-bottleneck ResNet v1 blocks (He et al., 2015) and each ResNet block uses a number of channels that doubles after each downsampling step. ResNet blocks uses $3 \times 3$ convolutions of stride 1 with ReLU activation. For downsampling we use 2D max pooling with $2 \times 2$ stride and kernel. For upsampling we use 2D transposed convolutions with $2 \times 2$ stride and kernel. The output layer uses a sigmoid function, and we use layer normalization before each convolution layer.
+
+MIST classification network. We re-use the same architecture as encoder for first the task and append a dense layer to map the latent space to the score vector of our 10 digit classes.
+
+Baseline unsupervised reconstruction methods. To implement the Eslami et al. (Eslami et al., 2015) baseline, we use a publicly available reimplementation.1 We note that Eslami et al. (Eslami et al., 2015) originally applied their model to a dataset consisting of images of 0, 1, or 2 digits with equal probability. We found that the model failed to converge unless it was trained with examples where various number of total digits exist, so for fair comparison, we populate the training set with images consisting of all numbers of digits between 0 and 9. For the Zhang et al. (Zhang et al., 2018c) baseline, we use the authors’ implementation and hyperparameters.
+
+![](images/510d191354b6ec35a51a6f274ba0abd02e5765f98833e53884b3cf79dbe9b042.jpg)  
+Figure 7: Evolution of the loss and heatmap during training. (a) the task (classification) loss $\mathcal { L } _ { \mathrm { t a s k } }$ and (b) the heatmap related loss $\lVert \mathcal { G } ( \{ \mathbf { x } _ { k } \} ) - \mathcal { H } _ { \eta } ( \mathbb { Z } ) \rVert _ { 2 } ^ { 2 }$ for each iteration. In (c), we show how the heatmap evolves for an example image from the SVHN dataset during training (from top to bottom) – brighter the intensity, higher the scores; see Section D.
+
+# D CONVERGENCE DURING TRAINING
+
+As is typical for neural network training, our objective is non-convex and there is no guarantee that a local minimum found by gradient descent training is a global minimum. Empirically, however, the optimization process is stable, as shown in Figure 7. We reporpt both the training losses $\mathcal { L } _ { \mathrm { t a s k } }$ and $\lVert \mathcal { G } ( \{ \mathbf { x } _ { k } \} ) - \mathcal { H } _ { \eta } ( \mathbb { Z } ) \rVert _ { 2 } ^ { 2 }$ , and visualize how the heatmap evolves for an example image from the SVHN datset. We visualize the heatmap by showing brighter intensity values for higher scores. Each image is normalized separately with their own maximum and minimum values for better visualization. In Figure 7 (c) top, early in training, one can see that keypoints are detected at random locations as the heatmaps are generated by networks with randomly initialized weights. However, as training continues, keypoints that, by chance, land on locations nearby the correct object (e.g. numbers) for certain samples in a random batch, and become reinforced. Ultimately, as shown in (c) middle and bottom, MIST learns to detect these locations, thus learns to produce a peaky heatmap, and perform the task of interest. Note that this is unsurprising, as our formulation is a lifted version of this loss to allow gradient-based training.
+
+# E NON-UNIFORM DISTRIBUTIONS
+
+![](images/52727ad948983aa6f6eb3bf61114236ad0b6a71786e832aeefb84d1cde009ab8.jpg)  
+Figure 8: Examples with uneven distributions of digits.
+
+Although the images we show in Figure 2 involve small displacements from a uniformly spaced grid, our method does not require the keypoints to be evenly spread. As shown in Figure 8, our method is able to successfully learn even when the digits are placed unevenly. Note that, as our detector is fully
+
+convolutional and local, it cannot learn the absolute location of keypoints. In fact, we weakened the randomness of the locations for fairness against (Zhang et al., 2018c), which is not designed to deal with severe displacements.

parse/train/rJeGJaEtPH/rJeGJaEtPH_content_list.json

@@ -0,0 +1,1201 @@
+[
+    {
+        "type": "text",
+        "text": "MIST MULTIPLE INSTANCE SPATIAL TRANSFORMERS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            174,
+            400,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            238,
+            544,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a deep network that can be trained to tackle image reconstruction and classification problems that involve detection of multiple object instances, without any supervision regarding their whereabouts. The network learns to extract the most significant K patches, and feeds these patches to a task-specific network – e.g., auto-encoder or classifier – to solve a domain specific problem. The challenge in training such a network is the non-differentiable top- $K$ selection process. To address this issue, we lift the training optimization problem by treating the result of top- $K$ selection as a slack variable, resulting in a simple, yet effective, multi-stage training. Our method is able to learn to detect recurring structures in the training dataset by learning to reconstruct images. It can also learn to localize structures when only knowledge on the occurrence of the object is provided, and in doing so it outperforms the state-of-the-art. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            334,
+            488
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Finding and processing multiple instances of characteristic entities in a scene is core to many computer vision applications, including object detection (Ren et al., 2015; He et al., 2017; Redmon & Farhadi, 2017), pedestrian detection (Dollár et al., 2012; Sewart & Andriluka, 2016; Zhang et al., 2018a), and keypoint localization (Lowe, 2004; Bay et al., 2008). In traditional vision pipelines, a common approach to localizing entities is to select the top-K responses in a heatmap and use their locations (Lowe, 2004; Bay et al., 2008; Felzenszwalb et al., 2010). However, this type of approach does not provide a gradient with respect to the heatmap, and cannot be directly integrated into neural network-based systems. To overcome this challenge, previous work proposed to use grids (Redmon et al., 2016; He et al., 2017; Detone et al., 2018) to simplify the formulation by isolating each instance (Yi et al., 2016), or to optimize over multiple branches (Ono et al., 2018). While effective, these approaches require additional supervision to localize instances, and do not generalize well outside their intended application domain. Other formulations, such as sequential attention (Ba et al., 2015; Gregor et al., 2015; Eslami et al., 2015) and channel-wise approaches (Zhang et al., 2018c) are problematic to apply when the number of instances of the same object is large. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here, we introduce a novel way to approach this problem, which we term Multiple Instance Spatial Transformer, or MIST for brevity. As illustrated in Figure 1 for the image synthesis task, given an image, we first compute a heatmap via a deep network whose local maxima correspond to locations of interest. From this heatmap, we gather the parameters of the top- $K$ local maxima, and then extract the corresponding collection of image patches via an image sampling process. We process each patch independently with a task-specific network, and aggregate the network’s output across patches. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training a pipeline that includes a non-differentiable selection/gather operation is non-trivial. To solve this problem we propose to lift the problem to a higher dimensional one by treating the parameters defining the interest points as slack variables, and introduce a hard constraint that they must correspond to the output that the heatmap network gives. This constraint is realized by introducing an auxiliary function that creates a heatmap given a set of interest point parameters. We then solve for the relaxed version of this problem, where the hard constraint is turned into a soft one, and the slack variables are also optimized within the training process. Critically, our training strategy allows the network to incorporate both non-maximum suppression and top-K selection. We evaluate the performance of our approach for $\\textcircled{1}$ the problem of recovering the basis functions that created a given texture, $\\textcircled{2}$ classification of handwritten digits in cluttered scenes, and $\\textcircled{3}$ recognition of house numbers in real-world environments. In summary, in this paper we: ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• introduce the MIST framework for weakly-supervised multi-instance visual learning; • propose an end-to-end training method that allows the use of top-K selection; • show that our framework can reconstruct images as parts, as well as detect/classify instances without any location supervision. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            215,
+            341,
+            232
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Attention models and the use of localized information have been actively investigated in the literature. Some examples include discriminative tasks such as fine-grained classification (Sun et al., 2018), pedestrian detection (Zhang et al., 2018a), and generative ones such as image synthesis from natural language (Johnson et al., 2018). They have also been studied in the context of more traditional Multiple Instance Learning (MIL) setup (Ilse et al., 2018). We now discuss a selection of representative works, and classify them according to how they deal with multiple instances. ",
+        "bbox": [
+            173,
+            248,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Grid-based methods. Since the introduction of Region Proposal Networks (RPN) (Ren et al., 2015), grid-based strategies have been used for dense image captioning (Johnson et al., 2016), instance segmentation (He et al., 2017), keypoint detection (Georgakis et al., 2018), and multi-instance object detection (Redmon & Farhadi, 2017). Recent improvements to RPNs attempt to learn the concept of a generic object covering multiple classes (Singh et al., 2018), and to model multi-scale information (Chao et al., 2018). The multiple transformation corresponding to separate instances can also be densely regressed via Instance Spatial Transformers (Wang et al., 2018), which removes the need to identify discrete instance early in the network. However, all these methods are fully supervised, requiring both class labels and object locations for training. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Heatmap-based methods. Heatmap-based methods have recently gained interest to detect features (Yi et al., 2016; Ono et al., 2018; Detone et al., 2018), find landmarks (Zhang et al., 2018c; Merget et al., 2018), and regress human body keypoint (Tekin et al., 2017; Newell et al., 2016). While it is possible to output a separate heatmap per class (Zhang et al., 2018c; Tekin et al., 2017), most heatmap-based approaches do not distinguish between instances. Yi et al. (2016) re-formulate the problem based on each instance, but in doing so introduce a sharp difference between training and testing. Grids can also be used with heatmaps (Detone et al., 2018), but this results in an unrealistic assumption of uniformly distributed detections in the image. Overall, heatmap-based methods excel when the “final” task of the network is generate a heatmap (Merget et al., 2018), but are problematic to use as an intermediate layer in the presence of multiple instances. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Sequential inference methods. Another way to approach multi-instance problems is to attend to one instance at a time in a sequentially. Training neural network-based models with sequential attention is challenging, but approaches using policy gradient (Ba et al., 2015) and differentiable attention mechanisms (Gregor et al., 2015; Eslami et al., 2015) have achieved some success for images comprising small numbers of instances. However, Recurrent Neural Networks (RNN) often struggle to generalize to sequences longer than the ones encountered during training, and while recent results on inductive reasoning are promising (Gupta et al., 2018), their performance does not scale well when the number of instances is large. ",
+        "bbox": [
+            173,
+            617,
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+            729
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Knowledge transfer. To overcome the acquisition cost of labelled training data, one can transfer knowledge from labeled to unlabeled dataset. For example, Inoue et al. (2018) train on a single instance dataset, and then attempt to generalize to multi-instance domains, while Uijlings et al. (2018) attempt to also transfer a multi-class proposal generator to the new domain. While knowledge transfer can be effective, it is highly desirable to devise unsupervised methods such as ours that do not depend on an additional dataset. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Weakly supervised methods. To further reduce the labeling effort, weakly supervised methods have also been proposed. Wan et al. (2018) learn how to detect multiple instances of a single object via region proposals and ROI pooling, while Tang et al. (2018) propose to use a hierarchical setup to refine their estimates. Gao et al. (2018) provides an additional supervision by specifying the number of instances in each class, while Oquab et al. (2015) and Zhang et al. (2018b) localize objects by looking at the network activation maps (Zhou et al., 2016; Selvaraju et al., 2017). Shen et al. (2018) introduce an adversarial setup, where detection boxes are supervised by distribution assumptions and classification objectives. However, all these methods still rely on region proposals from an existing method, or define them via a hand-tuned process. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c706924e05863bd7e4a56a0fa0ca14ba0b5fdd42d7a2698b6322f91a2832b4d.jpg",
+        "image_caption": [
+            "Figure 1: The MIST architecture – A network $\\mathcal { H } _ { \\eta }$ estimates locations and scales of patches encoded in a heatmap h. Patches are then extracted via a sampler $s$ , and then fed to a task-specific network $\\mathcal { T } _ { \\tau }$ . In this example, the specific task is to re-synthesize the image as a super-position of (unknown and locally supported) basis functions. As top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ operation is non-differentiable, we back-propagate by lifting through $\\mathcal { G }$ ; see Section 3 for details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "3 MIST FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We first explain our framework at a high level, and then detail each component in Section 4. A prototypical MIST architecture (Figure 1) is composed of two trainable components: $\\textcircled{1}$ the first module receives an image as input and extracts $K$ patches, at image locations and scales given by the top $K$ local maxima of a heatmap computed by a trainable heatmap network $\\mathcal { H } _ { \\eta }$ with weights $\\eta$ . $\\textcircled{2}$ the second module processes each extracted patch with a task-specific network $\\mathcal { T } _ { \\tau }$ whose weights $\\tau$ are shared across patches, and further manipulates these signals to express a task-specific loss $\\mathcal { L } _ { t a s k }$ The two modules are connected through non-maximum suppression on the scale-space heatmap output of $\\mathcal { H } _ { \\eta }$ , followed by a top- $K$ selection process to extract the parameters defining the patches, which we denote as $\\mathcal { E } _ { K }$ . We then sample patches at these locations through (differentiable) bilinear sampling $s$ and feed them the task module. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The defining characteristic of the MIST architecture is that they are quasi-unsupervised: the only supervision required is the number $K$ of patches to extract. However, as we show in Section 5.2, our method is not sensitive to the choice of $K$ during training. In addition, our extraction process uses a single heatmap for all instances that we extract. In contrast, existing heatmap-based methods (Eslami et al., 2015; Zhang et al., 2018c) typically rely on heatmaps dedicated to each instance, which is problematic when an image contains two instances of the same class. Conversely, we restrict the role of the heatmap network $\\mathcal { H } _ { \\eta }$ to find the “important” areas in a given image, without having to distinguishing between classes, hence simplifying learning. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, the training of the MIST architecture is summarized by: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1a1aa813fb1b09d2dda394750b649cc9706d3efa4b79a78824b8a5beab061418.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\tau , \\eta } \\mathcal { L } _ { \\mathrm { t a s k } } ( \\mathcal { T } _ { \\tau } ( S ( \\mathcal { E } _ { K } ( \\mathcal { H } _ { \\eta } ( \\mathcal { T } ) ) ) ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            806,
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+            832
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau , \\eta$ are the network trainable parameters. Unfortunately, the patch extractor $\\mathcal { E } _ { K }$ is nondifferentiable, as it identifies the locations of the top- $K$ local maxima of a heatmap and then selects the corresponding patches from the input image. Differentiating this operation provides a gradient with respect to the input, but no gradient with respect to the heatmap. Although it is possible to smoothly relax the patch selection operation in the $K = 1$ case (Yi et al., 2016) (i.e. argmax), it is unclear how to generalize this approach to the case of multiple distinct local maxima. It is thus impossible to train the patch selector parameters directly by backpropagation. We propose an alternative approach to training our model, via lifting. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/7804cff3060d118c44679d4c07126756bf70ecee4f2bd1990ef26e322f7e0fc8.jpg",
+        "table_caption": [
+            "Algorithm 1 Multi-stage optimization for MISTs "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Require: K : number of patches to extract, Ltask : task specific loss,I : input image, G : the keypoints to heatmap function, H : the heatmap network,n : parameters of the heatmap network,</td><td></td><td></td><td></td></tr><tr><td colspan=\"3\"></td></tr><tr><td>1:</td><td>T : the task network, T : parameters of the task network,&amp;k : the top-K operator. function TRAINMIST(I,Ltask)</td></tr><tr><td>2:</td><td>for each training batch do</td></tr><tr><td>3:</td><td>T ← Optimization step for T- with Ltask +λl{xk} -εκ(Hn())ll2</td></tr><tr><td>4:</td><td>{xk} ← Optimization step for {xk} with Ltask + λl{xk}-εk(Hn(I))ll²2</td></tr><tr><td>5:</td><td>η ← Optimization step for η with |/S({xk}) - Hn(I)ll2</td></tr><tr><td>6: end for</td><td></td></tr><tr><td>7: end function</td><td></td></tr></table>",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Differentiable top-K via lifting. The introduction of auxiliary variables to simplify the structure of an optimization problem has proven effective in a range of domains ranging from non-rigid registration (Taylor et al., 2016), to robust optimization (Zach & Bournaoud, 2018). To simplify our training optimization, we start by decoupling the heatmap tensor from the optimization (1) by introducing the corresponding auxiliary variables h, as well as the patch location variables $\\left\\{ { \\bf x } _ { k } \\right\\}$ from the top- $\\mathbf { \\nabla } \\cdot \\mathbf { K }$ extractor: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/265e96c4deebf26ea08612a3b5acc026370b8b1a9500f9241163b53e3f2bb159.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\eta , \\tau , \\mathbf { h } , \\{ \\mathbf { x } _ { k } \\} } \\ \\mathcal { L } _ { \\mathrm { t a s k } } ( \\mathcal { T } _ { \\tau } ( \\mathcal { S } ( \\{ \\mathbf { x } _ { k } \\} ) ) ) \\quad \\mathrm { s . t . } \\quad \\mathbf { h } = \\mathcal { H } _ { \\eta } ( \\mathcal { T } ) , \\quad \\{ \\mathbf { x } _ { k } \\} = \\mathcal { E } _ { K } ( \\mathbf { h } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We then relax the first constraint above to a least-squares penalty via a Lagrange multiplier $\\lambda$ : ",
+        "bbox": [
+            173,
+            444,
+            785,
+            459
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bb2a4e97cd6eeec01827fc077388ab76c31ac39759efaff8e9da6cb613da10d.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\eta , \\tau , \\mathbf { h } , \\{ \\mathbf { x } _ { k } \\} } \\ \\mathcal { L } _ { \\mathrm { t a s k } } ( \\mathcal { T } _ { \\tau } ( S ( \\{ \\mathbf { x } _ { k } \\} ) ) ) + \\lambda \\| \\mathbf { h } - \\mathcal { H } _ { \\eta } ( \\mathcal { T } ) \\| _ { 2 } ^ { 2 } \\quad \\mathrm { s . t . } \\quad \\{ \\mathbf { x } _ { k } \\} = \\mathcal { E } _ { K } ( \\mathbf { h } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As in many methods that use keypoint supervision to regress heatmaps (Cao et al., 2018), we assume that a good heatmap generator $\\mathcal { G }$ exists $-$ that is $\\left\\{ \\mathbf { x } _ { k } \\right\\} \\approx \\mathcal { E } _ { K } ( \\mathcal { G } ( \\left\\{ \\mathbf { x } _ { k } \\right\\} ) )$ We can now rewrite our optimization as: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8467e4220f4ef7d76236636a87104b527d49248e6fe707d2e404410ac1273b51.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\eta , \\tau , \\mathbf { h } , \\{ \\mathbf { x } _ { k } \\} } \\ \\mathcal { L } _ { \\mathrm { t a s k } } ( \\mathcal { T } _ { \\tau } ( S ( \\{ \\mathbf { x } _ { k } \\} ) ) ) + \\lambda \\| \\mathbf { h } - \\mathcal { H } _ { \\eta } ( \\mathcal { T } ) \\| _ { 2 } ^ { 2 } \\quad \\mathrm { s . t . } \\quad \\mathbf { h } = \\mathcal { G } ( \\{ \\mathbf { x } _ { k } \\} ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can now drop the auxiliary variable $\\mathbf { h }$ and rewrite our optimization as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/072f63b3a01e440318183223845baa69e6af67c8b22154965832e9be351fe3f4.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\eta , \\tau , \\{ \\mathbf { x } _ { k } \\} } ~ \\mathcal { L } _ { \\mathrm { t a s k } } ( \\mathcal { T } _ { \\tau } ( \\mathcal { S } ( \\{ \\mathbf { x } _ { k } \\} ) ) ) + \\lambda \\| \\mathcal { G } ( \\{ \\mathbf { x } _ { k } \\} ) - \\mathcal { H } _ { \\eta } ( \\mathcal { Z } ) \\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and then approach the problem by block coordinate descent – where the energy terms not containing the variable being optimized are safely dropped, and we apply $\\mathcal { E } _ { K }$ to the penalty term of (5): ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/4a70e5b9881506845d86f8bd0e4d125b984a3efa6f084c9284a03915bfcc106d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { \\tau , \\{ \\mathbf { x } _ { k } \\} } { \\mathrm { m i n i m i z e ~ } } \\mathcal { L } _ { \\mathrm { t a s k } } ( \\mathcal { T } _ { \\tau } ( \\mathcal { S } ( \\{ \\mathbf { x } _ { k } \\} ) ) ) + \\lambda \\| \\{ \\mathbf { x } _ { k } \\} - \\mathcal { E } _ { K } ( \\mathcal { H } _ { \\eta } ( \\mathcal { T } ) ) \\| _ { 2 } ^ { 2 } , } \\\\ & { \\mathrm { m i n i m i z e ~ } \\| \\mathcal { G } ( \\{ \\mathbf { x } _ { k } \\} ) - \\mathcal { H } _ { \\eta } ( \\mathcal { T } ) \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To accelerate training, we further split (6) into two stages, and alternate between optimizing $\\tau$ and $\\left\\{ { \\bf x } _ { k } \\right\\}$ . Being based on block coordinate descent, this process converges smoothly, as we show in Section D of the appendix. The summary for the three stage optimization procedure is outlined in Algorithm 1. Notice that we are not introducing any additional supervision signal that is tangent to the given task. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 COMPONENTS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Multiscale heatmap network $- \\mathcal { H } _ { \\eta }$ . Our multiscale heatmap network is inspired by LF-Net (Ono et al., 2018). We improve the network by applying modifications on how the scores are aggregated over scale. The network takes as input an image $\\mathcal { L }$ , and outputs multiple heatmaps $\\mathbf { h } _ { s } ^ { \\prime }$ of the same size for each scale level $s$ . To limit the number of necessary scales, we use a discrete scale space with $S$ scales, and resolve intermediate scales via interpolation; see Section A of the appendix for details. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Top-K patch selection – $\\mathcal { E } _ { K }$ . To extract the top $K$ elements, we perform an addition cleanup through a standard non-maximum suppression. We then find the spatial locations of the top $K$ elements of this suppressed heatmap, denoting the spatial location of the $k ^ { \\mathrm { { t h } } }$ element as $( x _ { k } , y _ { k } )$ , which now reflect local maxima. For each location, we compute the corresponding scale by weighted first order moments (Suwajanakorn et al., 2018) where the weights are the responses in the corresponding heatmaps, i.e. $\\begin{array} { r } { s _ { k } = \\sum _ { s } \\mathbf { h } _ { s } ^ { \\prime } ( x _ { k } , y _ { k } ) s } \\end{array}$ ",
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+    },
+    {
+        "type": "text",
+        "text": "Generating model for ideal heatmap – $\\mathcal { G } ( \\{ \\mathbf { x } _ { k } \\} )$ . For the generative model that turns keypoint and patch locations into heatmaps, we apply a simple model where the heatmap is zero everywhere else except the corresponding keypoint locations (patch center); see Section B of the appendix for details. ",
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+    },
+    {
+        "type": "text",
+        "text": "Patch resampling – $s$ . As a patch is uniquely parameterized by its location and scale, i.e. $\\mathbf { x } _ { k } = \\left( x _ { k } , y _ { k } , s _ { k } \\right)$ , we can then proceed to resample its corresponding tensor via bilinear interpolation (Jaderberg et al., 2015; Jiang et al., 2019) as $\\{ \\mathbf { P } _ { k } \\} = \\mathcal { \\bar { S } } ( \\mathcal { T } , \\{ \\mathbf { \\bar { x } } _ { k } \\} )$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 TASK-SPECIFIC NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now introduce two applications of the MIST framework. We use the same heatmap network and patch extractor for both applications, but the task-specific network and loss differ. We provide further details regarding the task-specific network architectures in Section C of the appendix. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Image reconstruction / auto-encoding. As illustrated in Figure 1, for image reconstruction we append our patch extraction network with a shared auto-encoder for each extracted patch. We can then train this network to reconstruct the original image by inverting the patch extraction process and minimizing the mean squared error between the input and the reconstructed image. Overall, the network is designed to jointly model and localize repeating structures in the input signal. Specifically, we introduce the generalized inverse sampling operation $S ^ { - 1 } ( \\mathbf { P } _ { i } , \\mathbf { x } _ { i } )$ , which starts with an image of all zeros, and places the patch $\\mathbf { P } _ { i }$ at $\\mathbf { x } _ { i }$ . We then sum all the images together to obtain the reconstructed image, optimizing the task loss ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/cf2b596b0933fa8dc599567a26b7627a8270a7928a0538fd9ab4a8c85524014f.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { t a s k } } = \\| \\mathcal { I } - \\sum _ { i } \\mathcal { S } ^ { - 1 } \\left( \\mathbf { P } _ { i } , \\mathbf { x } _ { i } \\right) \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Multiple instance classification. By appending a classification network to the patch extraction module, we can also perform multiple instance learning. For each extracted patch $\\mathbf { P } _ { k }$ we apply a shared classifier network to output $\\mathbf { \\hat { y } } _ { k } \\in \\mathbb { R } ^ { C }$ , where $C$ is the number of classes. In turn, these are then converted into probability estimates by the transformation $\\hat { \\mathbf { p } } _ { k } = \\mathrm { s o f t m a x } ( \\hat { \\mathbf { y } } _ { k } )$ . With $\\mathbf { y } _ { l }$ being the one-hot ground-truth labels of instance $l$ with unit norm, we define the multi-instance classification loss as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1dca64c76207984df94e8f57e890ff3200f77dabff58a5b5f9967c99cfe7dc59.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { t a s k } } = \\left\\| \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\mathbf { y } _ { l } - \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\hat { \\mathbf { p } } _ { k } \\right\\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $L$ is the number of instances in the image. We empirically found this loss to be more stable compared to the cross-entropy loss, similar to Mao et al. (2017). Note here that we do not provide supervision about the localization of instances, yet the detector network will automatically learn how to localize the content with minimal supervision (i.e. the number of instances). ",
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+    },
+    {
+        "type": "text",
+        "text": "5 RESULTS AND EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the effectiveness of our framework we evaluate the two different tasks. We first perform a quasi-unsupervised image reconstruction task, where only the total number of instances in the scene is provided, i.e. K is defined. We then show that our method can also be applied to weakly supervised multi-instance classification, where only image-level supervision is provided. Note that, unlike region proposal based methods, our localization network only relies on cues from the classifier, and both networks are trained from scratch. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.1 IMAGE RECONSTRUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "From the MNIST dataset, we derive two different scenarios. In the MNIST easy dataset, we consider a simple setup where the sorted digits are confined to a perturbed grid layout; see Figure 2 (top). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d0afef7f81e174c9b17655771a6c1b303254a0adab9cad29d7e7e043a716490a.jpg",
+        "image_caption": [
+            "Figure 2: MNIST character synthesis examples for (top) the “easy” single instance setup and (bottom) the hard multi-instance setup. We compare the output of MISTs to grid, channel-wise, sequential Eslami et al. (Eslami et al., 2015) and Zhang et al. (Zhang et al., 2018c). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/cf93711634a0ec95e37380e5f6b4e4f0a7f21ec7ecb6d8b7407784ddc95c2373.jpg",
+        "image_caption": [
+            "Figure 3: Two auto-encoding examples learnt from MNIST-hard. In the top row, for each example we visualize input, patch detections, and synthesis. In the bottom row we visualize each of the extracted patch, and how it is modified by the learnt auto-encoder. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Specifically, we perturb the digits with a Gaussian noise centered at each grid center, with a standard deviation that is equal to one-eighths of the grid width/height. In the MNIST hard dataset, the positions are randomized through a Poisson distribution (Bridson, 2007), as is the identity, and cardinality of each digit. We allow multiple instances of the same digit to appear in this variant. For both datasets, we construct both training and test sets, and the test set is never seen during training. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Comparison baselines We compare our method against four baselines $\\textcircled{1}$ the grid method divides the image into a $3 \\times 3$ grid and applies the same auto-encoder architecture as MIST to each grid location to reconstruct the input image; $\\textcircled{2}$ the channel-wise method uses the same auto-encoder network as MIST, but we modify the heatmap network to produce $K$ channels as output, where each channel is dedicated to an interest point. Locations are obtained through a channel-wise soft-argmax as in Zhang et al. (2018c); $\\textcircled{3}$ Esl16 (Eslami et al., 2015) is a sequential generative model; $\\textcircled{4}$ Zha18 (Zhang et al., 2018c) is a state-of-the-art heatmap-based method with channel-wise strategy for unsupervised learning of landmarks. For more details see Supplementary Section C. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Results for “MNIST easy” As shown in Figure 2 (top) all methods successfully re-synthesize the image, with the exception of Eslami et al. (Eslami et al., 2015). As this method is sequential, with nine digits the sequential implementation simply becomes too difficult to optimize through. Note how this method only learns to describe the scene with a few large regions. We summarize quantitative results in Table 1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Results for “MNIST hard” As shown in Figure 2 (bottom), all baseline methods failed to properly represent the image. Only MIST succeeded at both localizing digits and reconstructing the original image. Although the grid method accurately reconstructs the image, it has no concept of individual digits. Conversely, as shown in Figure 3, our method generates accurate bounding boxes for digits even when these digits overlap, and does so without any location supervision. For quantitative results, please see Table 1. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/3cb1835c60e9839a0d84a93c82ab8c427f74203ebb793ae7ddd66627f4156a3e.jpg",
+        "image_caption": [
+            "Figure 4: Inverse rendering of Gabor noise; we annotate correct / erroneous localizations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/8ce1f38cea0a1c4239ab191e14845ed3cdce1785fd71324e73fd67635fbd2c94.jpg",
+        "image_caption": [
+            "Figure 5: Two qualitative examples for detection and classification on our Multi-MNIST dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finding the basis of a procedural texture We further demonstrate that our methods can be used to find the basis function of a procedural texture. For this experiment we synthesize textures with procedural Gabor noise (Lagae et al., 2009). Gabor noise is obtained by convolving oriented Gabor wavelets with a Poisson impulse process. Hence, given exemplars of noise, our framework is tasked to regress the underlying impulse process and reconstruct the Gabor kernels so that when the two are convolved, we can reconstruct the original image. Figure 4 illustrates the results of our experiment. The auto-encoder learned to accurately reconstruct the Gabor kernels, even though in the training images they are heavily overlapped. These results show that MIST is capable of generating and reconstructing large numbers of instances per image, which is intractable with other approaches. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5.2 MULTIPLE INSTANCE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Multi-MNIST – Figure 5. To test our method in a multiple instance classification setup, we rely on the MNIST hard dataset. We compare our method to channel-wise, as other baselines are designed for generative tasks. To evaluate the detection accuracy of the models, we compute the intersection over union (IoU) between the ground-truth bounding box and the detection results, and assign it as a match if the IoU score is over $50 \\%$ . We report the number of correctly classified matches in Table 2, as well as the proportion of instances that are both correctly detected and correctly classified. Our method clearly outperforms the channel-wise strategy. Note that, even without localization supervision, our method correctly localizes digits. Conversely, the channel-wise strategy fails to learn. This is because multiple instances of the same digits are present in the image. For example, in the Figure 5 (right), we have three sixes, two zeros, and two nines. This prevents any of these digits from being detected/classified properly by a channel-wise approach. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sensitivity to $K$ – Table 3. To investigate the sensitivity to the correctness of $K$ during training, we train with varying number of of instances and test with ground-truth $K$ . For example, with $K = 6$ ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/84c029bc6dd709a6e06718793a03587b1688eb6a6818c1a2d6f04a4043eb22e6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">MIST</td><td>Ch.-wise Supervised</td></tr><tr><td>IOU 50%</td><td>97.8%</td><td>25.4%</td><td>99.6%</td></tr><tr><td>Classif.</td><td>98.8%</td><td>75.5%</td><td>98.7%</td></tr><tr><td>Both</td><td>97.5%</td><td>24.8%</td><td>98.6%</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1b5d07b2221b3f2753d06ac5fbee8462087a9b75b41c1b2e5bc0f377ebd099fa.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Instances during train AploU=.50</td><td></td></tr><tr><td>{9}</td><td>92.2%</td></tr><tr><td>{6,7,8,9}</td><td>90.1%</td></tr><tr><td>{3,4,5,6,7,8,9}</td><td>90.8%</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/cb6cb333a59f93a1fb14af33b0f73914a4be0df199330bde7162ef0367a77088.jpg",
+        "table_caption": [
+            "Table 1: Reconstruction error (root mean square error). Note that Grid does not learn any notion of digits. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MIST</td><td>Grid Ch.-wise Esl16 Zha18</td><td></td><td></td><td></td><td></td></tr><tr><td>MNIST easy</td><td>.038</td><td>.039</td><td>.042</td><td>.100</td><td>.169</td></tr><tr><td>MNIST hard</td><td>.053</td><td>.062</td><td>.128</td><td>.154</td><td>.191</td></tr><tr><td>Gabor</td><td>.095</td><td>1</td><td>1</td><td>-</td><td>-</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Instance level detection and classification performance on “MNIST hard”. ",
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Sensitivity experiment of $K$ on the MNIST hard dataset. ",
+        "bbox": [
+            642,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9fbef27720c8b2fd5da027ebcf4057af2a600aa885953bf8b5c5a4085188f310.jpg",
+        "image_caption": [
+            "Figure 6: Qualitative SVHN results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f1b1520674cf256c036e4468b242d831f3f878e2c66f1d9eb4834f4ce9d3d030.jpg",
+        "table_caption": [
+            "Table 4: Quantitative SVHN results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MIST</td><td>Supervised</td></tr><tr><td>ApIoU=.00</td><td>82.6%</td><td>65.6%</td></tr><tr><td>APIoU=.50</td><td>76.5%</td><td>62.8%</td></tr><tr><td>APIoU=.60</td><td>63.7%</td><td>59.8%</td></tr><tr><td>APIoU=.70</td><td>42.7%</td><td>51.9%</td></tr><tr><td>APIoU=.80</td><td>19.9%</td><td>34.6%</td></tr><tr><td>APIoU=.90</td><td>4.2%</td><td>11.0%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "where the ground truth could be anything within $\\{ 3 , 4 , 5 , 6 , 7 , 8 , 9 \\} - \\mathrm { w e }$ mark the $K$ used during training with bold. Our method still is able to give accurate result with inaccurate $K$ during training. Knowing the exact number of objects is not a strict requirement at test time, as our detector generates a heatmap for the entire image regardless of the $K$ it was trained with. Note that while in theory sequential methods are free from this limitation, in practice they are able to deal with limited number of objects (e.g. up to three) due to their recurrent nature. ",
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+    {
+        "type": "text",
+        "text": "SVHN – Figure 6 and Table 4. We further apply MIST to the uncropped and unaligned Street View House Numbers dataset (Netzer et al., 2011). Compared to previous work that has used cropped and resized SVHN images (e.g. (Netzer et al., 2011; Ba et al., 2015; Goodfellow et al., 2014; Jaderberg et al., 2015)), this evaluating setting is significantly more challenging, because digits can appear anywhere in the image. We resize all images to $6 0 \\times 2 4 0$ , use only images labeled as containing 2 digits, and apply MIST at a single scale. Although the dataset provides bounding boxes for the digits, we ignore these bounding boxes and use only digit labels as supervision. During testing, we exclude images with small bounding boxes $\\textless 3 0$ pixels in height). We report results in terms of $\\mathrm { A P ^ { I o U = X } }$ , where $\\mathbf { X }$ is the threshold for determining detection correctness. With $\\mathrm { I o U } = 0$ , we refer to a “pure” classification task (i.e. no localization). As shown, supervised results provide better performance with higher thresholds, but MIST performs even better than the supervised baseline for moderate thresholds. We attribute this to the fact that, by providing direct supervision on the location, the training focuses too much on having high localization accuracy. ",
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+        "type": "text",
+        "text": "6 CONCLUSION AND FUTURE WORK ",
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+        "text": "In this paper, we introduce the MIST framework for multi-instance image reconstruction/classification. Both these tasks are based on localized analysis of the image, yet we train the network without providing any localization supervision. The network learns how to extract patches on its own, and these patches are then fed to a task-specific network to realize an end goal. While at first glance the MIST framework might appear non-differentiable, we show how via lifting they can be effectively trained in an end-to-end fashion. We demonstrated the effectiveness of MIST by introducing a variant of the MNIST dataset, and demonstrating compelling performance in both reconstruction and classification. We also show how the network can be trained to reverse engineer a procedural texture synthesis process. MISTs are a first step towards the definition of optimizable image-decomposition networks that could be extended to a number of exciting unsupervised learning tasks. Amongst these, we intend to explore the applicability of MISTs to unsupervised detection/localization of objects, facial landmarks, and local feature learning. ",
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+        "text": "Even though our method is a significant step forward from existing methods, it is not complete and has much potential for improvement. While our method is robust to $K$ during training as shown in Section 5.2, and $K$ can be determined dynamically during test time, it would beneficial to automatically determine $K$ through an additional network. Patches could also be refined to have varying aspect ratios similar to RPNs. Reconstruction could consider a global reconstructor to deal with complex images. These are parts of our future work towards MIST in the wild. ",
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+        "type": "text",
+        "text": "REFERENCES ",
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+        "bbox": [
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In ECCV, 2018.   \nJonathan Taylor, Lucas Bordeaux, Thomas Cashman, Bob Corish, Cem Keskin, Eduardo Soto, David Sweeney, Julien Valentin, Benjamin Luff, Arran Topalian, Erroll Wood, Sameh Khamis, Pushmeet Kohli, Toby Sharp, Shahram Izadi, Richard Banks, Andrew Fitzgibbon, and Jamie Shotton. Efficient and precise interactive hand tracking through joint, continuous optimization of pose and correspondences. TOG, 2016.   \nB. Tekin, P. Marquez-neila, M. Salzmann, and P. Fua. Learning to Fuse 2D and 3D Image Cues for Monocular Body Pose Estimation. In ICCV, 2017.   \nJ. R. R Uijlings, S. Popov, and V. Ferrari. Revisiting Knowledge Transfer for Training Object Class Detectors. In CVPR, 2018.   \nF. Wan, P. Wei, J. Jiao, Z. Han, and Q. Ye. Min-Entropy Latent Model for Weakly Supervised Object Detection. In CVPR, 2018.   \nF. Wang, L. Zhao, X. Li, X. Wang, and D. Tao. Geometry-Aware Scene Text Detection with Instance Transformation Network. In CVPR, 2018.   \nK. M. Yi, E. Trulls, V. Lepetit, and P. Fua. LIFT: Learned Invariant Feature Transform. In ECCV, 2016.   \nC. Zach and G. Bournaoud. Descending, Lifting or Smoothing: Secrets of Robust Cost Opimization. In ECCV, 2018.   \nS. Zhang, J. Yang, and B. Schiele. Occluded Pedestrian Detection Through Guided Attention in CNNs. In CVPR, 2018a.   \nX. Zhang, Y. Wei, G. Kang, Y. Wang, and T. Huang. Self-produced Guidance for Weakly-supervised Object Localization. In ECCV, 2018b.   \nY. Zhang, Y. Gui, Y. Jin, Y. Luo, Z. He, and H. Lee. Unsupervised Discovery of Object Landmarks as Structural Representations. In CVPR, 2018c.   \nB. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba. Learning Deep Features for Discriminative Localization. In CVPR, 2016. ",
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+    {
+        "type": "text",
+        "text": "Appendix ",
+        "text_level": 1,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A MULTISCALE HEATMAP NETWORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our multiscale heatmap network is inspired by LF-Net (Ono et al., 2018). We employ a fully convolutional network with (shared) weights $\\eta$ at multiple scales, indexed by $s = 1 \\ldots S$ , on the input image $\\mathcal { T }$ . The weights $\\eta$ across scales are shared so that the network cannot implicitly favor a particular scale. To do so, we first downsample the image to each scale producing $\\mathcal { T } _ { s }$ , execute the network $\\mathcal { H } _ { \\eta }$ on it, and finally upsample to the original resolution. This process generates a multiscale heatmap tensor $\\mathbf { h } = \\{ \\mathbf { h } _ { s } \\}$ of size $H \\times W \\times S$ where $\\mathbf { h } _ { s } = \\mathcal { H } _ { \\eta } ( \\mathbb { Z } _ { s } )$ , and $H$ is the height of the image and $W$ is the width. For the convolutional network we use 4 ResNet blocks (He et al., 2015), where each block is composed of two $3 \\times 3$ convolutions with 32 channels and relu activations without any downsampling. We then perform a local spatial softmax operator (Ono et al., 2018) with spatial extent of $1 5 \\times 1 5$ to sharpen the responses. Then we further relate the scores across different scales by performing a “softmax pooling” operation over scale. Specifically, if we denote the heatmap tensor after local spatial softmax as $\\tilde { \\mathbf { h } } = \\{ \\tilde { \\hat { \\mathbf { h } } } _ { s } \\}$ , since after the local spatial softmax $\\mathcal { H } _ { \\eta } ( \\mathcal { T } _ { s } )$ is already an “exponentiated” signal, we do a weighted normalization without an exponential, i.e. $\\begin{array} { r } { \\mathbf { h } ^ { \\prime } = \\sum _ { s } \\mathbf { \\tilde { h } } _ { s } ( \\tilde { \\mathbf { h } } _ { s } / \\tilde { \\sum _ { s ^ { \\prime } } } ( \\tilde { \\mathbf { h } } _ { s ^ { \\prime } } + \\epsilon ) ) } \\end{array}$ , where $\\epsilon = 1 0 ^ { - 6 }$ is added to prevent division by zero. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparison to LF-Net. Note that differently from LF-Net (Ono et al., 2018), we do not perform a softmax along the scale dimension. The scale-wise softmax in LF-Net is problematic as the computation for a softmax function relies on the input to the softmax being unbounded. For example, in order for the softmax function to behave as a max function, due to exponentiation, it is necessary that one of the input value reaches infinity (i.e. the value that will correspond to the max), or that all other values to reach negative infinity. However, at the network stage where softmax is applied in (Ono et al., 2018), the score range from zero to one, effectively making the softmax behave similarly to averaging. Our formulation does not suffer from this drawback. ",
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+    {
+        "type": "text",
+        "text": "B GENERATIVE MODEL FOR THE HEATMAP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As discussed previously in Section 4.1, for the generative model that turns keypoint and patch locations into heatmaps, we apply a simple model where the heatmap is zero everywhere else except the corresponding keypoint locations (patch center). However, as the optimized patch parameters are no longer integer values, we need to quantize them with care. For the spatial locations we simply round to the nearest pixel, which at most creates a quantization error of half a pixel, which does not cause problems in practice. For scale however, simple nearest-neighbor assignment causes too much quantization error as our scale-space is sparsely sampled. We therefore assign values to the two nearest neighboring scales in a way that the center of mass would be the optimized scale value, making sure $\\left\\{ { \\bf x } _ { k } \\right\\} = \\mathcal { E } _ { K } ( \\mathcal { G } ( \\{ { \\bf x } _ { k } \\} ) )$ . ",
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+    {
+        "type": "text",
+        "text": "C IMPLEMENTATION DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "MIST auto-encoder network. The input layer of the autoencoder is $3 2 \\times 3 2 \\times C$ where $C$ is the number of color channels. We use 5 up/down-sampling levels. Each level is made of 3 standard non-bottleneck ResNet v1 blocks (He et al., 2015) and each ResNet block uses a number of channels that doubles after each downsampling step. ResNet blocks uses $3 \\times 3$ convolutions of stride 1 with ReLU activation. For downsampling we use 2D max pooling with $2 \\times 2$ stride and kernel. For upsampling we use 2D transposed convolutions with $2 \\times 2$ stride and kernel. The output layer uses a sigmoid function, and we use layer normalization before each convolution layer. ",
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+        "type": "text",
+        "text": "MIST classification network. We re-use the same architecture as encoder for first the task and append a dense layer to map the latent space to the score vector of our 10 digit classes. ",
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+        "type": "text",
+        "text": "Baseline unsupervised reconstruction methods. To implement the Eslami et al. (Eslami et al., 2015) baseline, we use a publicly available reimplementation.1 We note that Eslami et al. (Eslami et al., 2015) originally applied their model to a dataset consisting of images of 0, 1, or 2 digits with equal probability. We found that the model failed to converge unless it was trained with examples where various number of total digits exist, so for fair comparison, we populate the training set with images consisting of all numbers of digits between 0 and 9. For the Zhang et al. (Zhang et al., 2018c) baseline, we use the authors’ implementation and hyperparameters. ",
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+    {
+        "type": "image",
+        "img_path": "images/510d191354b6ec35a51a6f274ba0abd02e5765f98833e53884b3cf79dbe9b042.jpg",
+        "image_caption": [
+            "Figure 7: Evolution of the loss and heatmap during training. (a) the task (classification) loss $\\mathcal { L } _ { \\mathrm { t a s k } }$ and (b) the heatmap related loss $\\lVert \\mathcal { G } ( \\{ \\mathbf { x } _ { k } \\} ) - \\mathcal { H } _ { \\eta } ( \\mathbb { Z } ) \\rVert _ { 2 } ^ { 2 }$ for each iteration. In (c), we show how the heatmap evolves for an example image from the SVHN dataset during training (from top to bottom) – brighter the intensity, higher the scores; see Section D. "
+        ],
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+    {
+        "type": "text",
+        "text": "D CONVERGENCE DURING TRAINING ",
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+        "text": "As is typical for neural network training, our objective is non-convex and there is no guarantee that a local minimum found by gradient descent training is a global minimum. Empirically, however, the optimization process is stable, as shown in Figure 7. We reporpt both the training losses $\\mathcal { L } _ { \\mathrm { t a s k } }$ and $\\lVert \\mathcal { G } ( \\{ \\mathbf { x } _ { k } \\} ) - \\mathcal { H } _ { \\eta } ( \\mathbb { Z } ) \\rVert _ { 2 } ^ { 2 }$ , and visualize how the heatmap evolves for an example image from the SVHN datset. We visualize the heatmap by showing brighter intensity values for higher scores. Each image is normalized separately with their own maximum and minimum values for better visualization. In Figure 7 (c) top, early in training, one can see that keypoints are detected at random locations as the heatmaps are generated by networks with randomly initialized weights. However, as training continues, keypoints that, by chance, land on locations nearby the correct object (e.g. numbers) for certain samples in a random batch, and become reinforced. Ultimately, as shown in (c) middle and bottom, MIST learns to detect these locations, thus learns to produce a peaky heatmap, and perform the task of interest. Note that this is unsurprising, as our formulation is a lifted version of this loss to allow gradient-based training. ",
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+        "text": "E NON-UNIFORM DISTRIBUTIONS ",
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+    {
+        "type": "image",
+        "img_path": "images/52727ad948983aa6f6eb3bf61114236ad0b6a71786e832aeefb84d1cde009ab8.jpg",
+        "image_caption": [
+            "Figure 8: Examples with uneven distributions of digits. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Although the images we show in Figure 2 involve small displacements from a uniformly spaced grid, our method does not require the keypoints to be evenly spread. As shown in Figure 8, our method is able to successfully learn even when the digits are placed unevenly. Note that, as our detector is fully ",
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+    {
+        "type": "text",
+        "text": "convolutional and local, it cannot learn the absolute location of keypoints. In fact, we weakened the randomness of the locations for fairness against (Zhang et al., 2018c), which is not designed to deal with severe displacements. ",
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