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Gaussian classconditional simplex loss for accurate, adversarially robust deep classifier training
EURASIP Journal on Information Security volume 2023, Article number: 3 (2023)
Abstract
In this work, we present the Gaussian ClassConditional Simplex (GCCS) loss: a novel approach for training deep robust multiclass classifiers that improves over the stateoftheart in terms of classification accuracy and adversarial robustness, with little extra cost for network training. The proposed method learns a mapping of the input classes onto Gaussian target distributions in a latent space such that a hyperplane can be used as the optimal decision surface. Instead of maximizing the likelihood of target labels for individual samples, our loss function pushes the network to produce feature distributions yielding high interclass separation and low intraclass separation. The mean values of the learned distributions are centered on the vertices of a simplex such that each class is at the same distance from every other class. We show that the regularization of the latent space based on our approach yields excellent classification accuracy. Moreover, GCCS provides improved robustness against adversarial perturbations, outperforming models trained with conventional adversarial training (AT). In particular, our model learns a decision space that minimizes the presence of short paths toward neighboring decision regions. We provide a comprehensive empirical evaluation that shows how GCCS outperforms stateoftheart approaches over challenging datasets for targeted and untargeted gradientbased, as well as gradientfree adversarial attacks, both in terms of classification accuracy and adversarial robustness.
1 Introduction
Over the course of the last few years, deep learning has been applied to several multimedia, scientific, and industrial applications thanks to its ability to generalize well over unseen data. The performance of these techniques has reached similar or even greater accuracy levels than humans in multiple and complex visual and classification tasks [1, 2]. More recently, deep networks have also shown remarkable performance at learning complex mappings for image translation and segmentation [3,4,5]. However, the evergrowing use of neural networks in our society raises serious concerns in the matter of security, as they can be targeted by malevolent adversaries.
In particular, many barriers affect the use of deep neural networks in applications where security is of key importance, such as medical diagnostics and autonomous driving [6, 7]. One of the most severe threats to deep learning is represented by adversarial perturbations, a collection of methods that are designed to interfere with neural networks’ input data to produce undesired outputs, shift the expected outcome, or more in general cause algorithm malfunctions and performance reductions. This happens in the face of modifications that are very difficult to detect, to the extent that they are often undetectable to the human eye.
Lately, numerous methods have been devised to apply successful adversarial perturbations [8, 9]. Against such techniques, neural networks can be considered unreliable when employed in applications involving security and safety. Despite many solutions have been proposed, the community has not yet been able to establish a conclusive defense mechanism against the wide range of adversarial perturbations. In particular, recent works such as [10,11,12,13] consider the importance of performing correct evaluations of adversarial attacks and defense algorithms, introducing a rigorous benchmark method to evaluate adversarial robustness on image classification tasks and to produce certified defenses against adversarial attacks. The main factor hindering the robustness of deep networks is that the learned boundaries in which the feature space is partitioned are highly complex and nonlinear. Works such as [14, 15], specifically addressing this problem, observed that most of the mass of data points is actually positioned very close to the learned decision boundaries, in such a way that the robustness of the classifier is heavily affected by adversarial perturbations. Among the most widespread techniques employed for improving robustness figures adversarial training [16], which consists in augmenting with adversarial samples the training data fed to the network, also when working with large datasets [17].
However, adversarial training is a very timeconsuming process. To generate an adversarial image, each stochastic gradient descent (SGD) iteration requires multiple gradient computations in addition to those needed to update the network parameters. On medium or large datasets such as ImageNet, CIFAR10, and CIFAR100, this is a burdensome task. To this end, several alternatives have been developed. Some recent works include [17, 18], where authors study ensemble adversarial training and generative adversarial perturbations for largescale datasets such as ImageNet. These methods lead to improved robustness, but at the cost of reducing the classification accuracy in the case of no attacks. Further, recent works that tackle the robustness problem try to devise efficient adversarial defense methods such as curvature regularization [19], logits regularization [20], injection of noise [21], and unlabeled data [22] at inference time, while others focus on the use of randomized classifiers for improved robustness [23, 24].
In this paper, we address the adversarial robustness problem to devise a method that can protect from both gradientbased and gradientfree attacks without requiring computationally intensive training. To this end, we design a novel loss function for the classifier that enables the learning of features maximizing interclass separation and minimizing intraclass separation. The proposed loss function provides stateoftheart classification accuracy while ensuring a large degree of adversarial robustness without any other specific training trick. In the proposed method, the network learns a mapping of input data onto welldefined, Gaussian target distributions in a regularized latent space. The employed loss maximizes linear separability of the learned distributions in the latent space, enabling straightforward classification based on simple thresholding of the decision variable. Ultimately, this is a generalpurpose technique that improves on the traditional crossentropy loss function. We show that the proposed method achieves higher classification accuracy with respect to crossentropy even when no attacks are applied while exhibiting notably improved robustness against multiple adversarial attacks. Indeed, our design requires significantly stronger attacks to induce the same classification errors than other methods, thanks to the maximized distances between adjacent decision regions. Furthermore, the proposed loss function can be generally employed in the finetuning of any pretrained classifier, allowing for improvements in robustness and accuracy while also reducing the training overhead.
1.1 Contributions
In this work, we extend and improve on our previous work [25]. We provide an indepth analysis of how GCCS improves the classification accuracy by correctly mapping the samples that are typically misclassified by competing approaches, while at the same time showing the motivation behind such higher misclassification. We include thorough ablation experiments to evaluate the effects of the GCCS loss parameters on classification accuracy. Further, we show that GCCS achieves higher robustness compared to conventional adversarially trained models with no additional cost with respect to natural training. In particular, we show that adversarial training combines well with our proposed method, which proves to be more robust against a broad range of attacks than competing methods even when coupled with AT. We report a thorough investigation and experimental evaluation of the proposed method for classification over widespread datasets such as MNIST [26], FMNIST [27], SVHN [28], and the very challenging CIFAR10 and CIFAR100 [29]. Also, we assess the performance of our technique showing that it outperforms the stateoftheart defence methods by a considerable margin in terms of robustness against targeted (TGSM [30], JSMA [31]) and untargeted (DeepFool [32], FGSM [33], and PGD [16]) gradientbased adversarial attacks. Also, we evaluate the adversarial robustness of our method when dealing with the very challenging gradientfree SPSA [34] attack. Finally, we analyze the behavior of our method when used together with adversarial training via a detailed graphical representation of the latent space, showing that the combination further improves model robustness. In particular, to assess adversarial robustness, we refer to the guidelines presented by [13, 35] and [36].
2 Related works
The idea of manipulating data seeking to disrupt a classifier’s performance was first formally investigated in works such as [37, 38], where the authors were interested in countering adversaries in applications like surveillance and spam, intrusion, and fraud detection. In the following years, many works [33, 39,40,41] analyzed security aspects of deep neural networks. They showed how easily they could be led into misclassification by receiving as input adversarially altered data, i.e., inputs specifically modified according to the sign of the gradients of the cost function.
The effectiveness of adversarial perturbations extends both to the virtual and the physical realms, as reported in [42] and [30, 43], where deep networks are respectively shown to be fooled in recognizing faces by specifically devised adversarial glasses, and in classifying images that are printed on paper or 3Dprinted after being tampered with adversarial perturbations. Also, Ross and DoshiVelez [44] showed that adversarial attacks devised to meddle with one specific network model can easily transfer to all models trained on the same dataset, which are greatly prone to be affected by the perturbation. Adversarial attacks can be crafted and performed with remarkable results also without any prior knowledge of the targeted image, as thoroughly examined in [45, 46]. In conclusion, despite being one of the most effective defense algorithms against perturbations, adversarial training suffers from a large gap between training and test accuracy [47].
While some of the latest findings suggest that the existence of adversarial perturbations is a structural weakness of deep neural networks [48] and that the strength of the applied adversarial attack directly bounds the expected robustness of the classifier [49], there is no universal agreement on the tradeoff between adversarial robustness and generalization. Some authors argue that defense methods will inevitably decrease the classification accuracy [50], whereas others claim that both adversarially robust and general models are indeed possible [51]. This situation is reflected in the large variety of methods for defending against such attacks.
Authors in [52], after formally characterizing the robustness of a classifier against attacks, compute the minimum perturbation required to affect classification, identifying instancespecific lower bounds dependent on the strength of the attack, and then proceed to define the CrossLipschitz Regularization (CLR) functional, whose goal is to constrain differences at data points of the classifier functions to be as constant as possible. Ross and DoshiVelez [44] instead offer a method called Input Gradient Regularization (IGR) that is applied at training time and forces the gradients of the network to be as smooth as desired; the underlying motivation is that models trained with gradients that exhibit a smaller set of spike values tend to respond in more understandable ways to adversarial attacks while also retailing greater robustness. Finally, Jakubovitz et al. [53] rely on the Frobenius norm of the Jacobian of the gradients (JR) to craft a lowcomplexity method that is applied to already trained networks to improve robustness. This method has also been implemented in a computationally effective way [54]. Finally, works such as [32, 55, 56] propose the idea of introducing adversarial samples in the training data as a form of augmentation, so that the network could learn how to cope with adversarial perturbation at training time. However, as shown in [16, 17, 35], adversarial training does not prevent adversaries to interfere with classification accuracy, to the extent that universal adversarial attacks can actually be devised to be highly effective on different datasets and network structures [30, 57,58,59]. Furthermore, adversarial training also comes with the extra cost required to generate adversarial samples that consequently increase the size of the training set and hence training time [60].
Approaches in the literature mainly focus on learning suitable classification boundaries. Instead, in this paper, we propose a method that directly learns a mapping of the input onto output target distributions in a regularized latent space. Despite being a passive defense method, the proposed design inherently enjoys high classification accuracy and robustness against adversarial attacks, with almost no training overhead. In particular, through the use of an encoder, features of an arbitrary number of classes are mapped from their input class distributions onto Gaussian distributions whose expected values lie on the vertices of a simplex in the latent space such that interclass separability is maximized and intraclass separability minimized at training time. Previously, other works have considered a learned mapping onto a wellbehaved latent space, such as techniques based on variational and adversarial autoencoders [61, 62], or even discriminant analysis methods that aim at dimensionality reduction [63]. Examples include the approach by Stuhlsatz et al. [64] that consists of a generalized version of Linear Discriminant Analysis (LDA) relying on deep networks, and [65] that first performs LDA on top of a neural network and then maximizes eigenvalues on the final hidden representation. These approaches, however, while showing great effectiveness at increasing the distance of the centers of distributions for wellseparated classes, appear to perform poorly in terms of interclass separability when it comes to neighboring classes with blurred borders.
Lately, latent space regularization has been used also in [66,67,68]. In these works, deep learning is employed in the field of biometric authentication to separate authorized users from nonauthorized ones. These methods work by regularizing a 2D latent space via a cost function derived from a simplified version of the KullbackLeibler divergence. While relevant to the topic, these approaches do not scale for higherdimensional classification, which is instead the focus of the proposed GCCS method.
Finally, works such as Gaussian Mixture loss (LGM [69]) and MaxMahalanobis center loss (MMC [70]) formally showed the inherent property of the crossentropy loss and its variations to learning feature vectors that are sparsely spread over the feature space, causing the models trained in such fashion to be prone to suffer adversarial attacks.
3 Proposed method
The goal of the proposed GCCS method is to learn the most discriminative features of the input and map them onto target distributions with the mean values centered on the vertices of a simplex, as shown in Fig. 1: a deep network that consists of a feature extractor and a latent space mapper receives as input labeled training data (\(\mathbf {X}\)) to tackle a multiclass classification problem. The feature extractor learns nonlinear functions that transform arbitrary input data distributions and produces discriminative, highly separable features. The subsequent latent space mapper is composed of one or more fully connected layers that map the output decision variable (\(\mathbf {z}\)) onto desired target distributions that lie in a Ddimensional latent space, where D denotes the number of classes. In particular, we do not employ any nonlinear activation in the mapper’s last layer. Also, it is important to notice that our proposed GCCS technique is not dependent on a particular feature extraction design. Hence, any deep architecture might be employed.
In the following, we describe the three fundamental components of the proposed approach, namely, a target model for the feature distributions in the latent space, a cost function devised to ensure that such distributions could be obtained, and last, a decision rule for classification. Ablation studies for the relevant parameters are reported in Section 4.
3.1 Model for the target distributions
The GCCS method projects highdimensional data belonging to D classes onto a lowerdimensional space. This is done by employing a latent space that has the same number (D) of dimensions as the number of classes so that each dimension corresponds to a classconditional statistical distribution.
First, we denote the desired target distribution for class \(\mathcal {C}_{i}\), \(i=1,\ldots D\) as \(\mathbb {P}_i\). In particular, we choose \(\mathbb {P}_i\) to be a Dvariate Gaussian distribution, such that \({\mathbb {P}_i = \mathcal {N}(\mathbf {\mu }_{Ti},\varvec{\Sigma }_{T})}\), where \(\mathcal {N}\) indicates the Gaussian distribution. In particular, \({\mathbf {\mu }_{Ti}}\) and \({\varvec{\Sigma }_{T}}\) indicate the target mean and standard deviation vectors, i.e., \({\mathbf {\mu }_{Ti} = \mu _{T} \mathbf {e}_i}\) and \({\varvec{\Sigma }_{T} = \sigma ^2_{T} \mathbb {I}_D}\), where \(\mathbf {e}_i\) refers to the ith standard unit vector, while \(\mathbb {I}_D\) is the \(D\times D\) identity matrix. Hence, the target statistical distributions are centered on the vertices of a regular \((D1)\)simplex at \(\mu _T \mathbf {e}_i\), such that every distribution \(\mathbb {P}_i\) has a mean value in the direction determined by \(\mathbf {e}_i\). The \(\mu _{T}\) and \(\sigma _{T}\) parameters, referring to the characteristics of the target distributions we want to enforce at training time, are defined arbitrarily keeping in mind that, to avoid mixing of the classes as D increases and to have more separable distribution, the parameters should take values such that \(\mu _T/\sigma _T > \sqrt{2D}\). In other words, the choice of \(\mu _{T}\) and \(\sigma _{T}\) influences the interclass/intraclass separability at the classification stage.
The defined target model exhibits several advantages. First, assuming the simplex to be regular ensures that each of the classes is equidistant from all others. For this reason, there is no specific class that is “weaker” than the others from an adversarial point of view. Moreover, the parameters \(\mu _T\) and \(\sigma _T\) are chosen so as to maximize interclass separability, leading to improved robustness. Secondly, since we choose Gaussian distributions as learning target, the optimal decision boundaries are simple hyperplanes. This design leads to better accuracy and greater robustness, in contrast to the typical behavior of neural networks which learn very complex decision boundaries.
3.2 Loss function
At this point, it is necessary to define a loss function that allows us to minimize a suitable distance metric between the latent output distributions and the target ones. Such a loss function is intended to simultaneously constrain the classes’ distributions in order to maximize separability. Given \(\mathbf {x} \in \mathbb {R}^n\), the input data belonging to D different classes, we refer to the output of the network as \(\mathbf {z} = H(\mathbf {x})\), where \([z_1, \dots , z_D] \in \mathbb {R}^D\), and H indicates the concatenation of the feature extractor and the latent space mapper (i.e., the whole network architecture). In other words, our method learns an encoding function of the input data \(\mathbf {z} = H(\mathbf {x})\) such that \({\mathbf {z} \sim \mathbb {P}_i}\) if \({\mathbf {x} \in \mathcal {C}_{i}}\). During training, the network receives as input single bsized batches of samples \(\mathbf {X} \in \mathbb {R}^{b \times n}\), and outputs encoded data \(\mathbf {Z} \in \mathbb {R}^{b \times D}\). In particular, we can estimate their first and secondorder statistics for each class as the sample mean \(\mathbf {\mu }_{Oi}\) and sample covariance \(\varvec{\Sigma }_{Oi}\). Once target distributions and batch statistics are known, we can define a suitable loss to compute how distant the batch statistics are from the target distributions. To this end, we employ the KullbackLeibler (KL) divergence. The KL divergence with respect to Gaussian target distributions, for the sample distribution of any class \(\mathcal {C}_i\), can be calculated as:
By extension, we can compute the cumulative loss for all classes such as \(\mathcal {L} = \sum _{i=1}^{D}\mathcal {L}_i\). In particular, \(\mathcal {L}\) reaches its minimum value when the sample statistics of the D distributions exactly match the target ones. For a small batch size, however, it is difficult to control how the tails of the distributions behave relying only on KL. Hence, we also employ the Kurtosis \(\mathcal {K}_{i,j}\) [71] of the jth component of the ith target distribution, defined as \(\mathcal {K}_{i,j}= \left( \frac{z_{i,j}\mathbf {\mu }_{Oi,j}}{\sigma _{Oi,j}}\right) ^4\). When enforcing multiple i.i.d. univariate normal distributions at training time, the target Kurtosis for each of the classes is \(\mathcal {K}_{i,j}=3\). This term can then be added to the cumulative loss, leading to the final proposed loss \(\mathcal{L}^{\mathrm{GCCS}}\) defined as
where \(\mathcal {K}_i = 1/D \cdot \sum _j \mathcal {K}_{i,j}\). \(\lambda\) is a parameter that balances the effect of the Kurtosis term with respect to the effect of the KL divergence.
3.3 Decision rule
Last, when the training converges properly using the GCCS loss, the proposed method allows us to define optimal decision boundaries in the learned latent space. In particular, for the chosen target distributions, optimal boundaries are defined by the partition of the latent space into Voronoi regions such that all the points are respectively closer to their region centroid, i.e., the mean vector \(\varvec{\mu }_{Ti}\), than any other in the \((D1)\)simplex. For this reason, the resulting decision rule requires the computation of the distance between output feature points and all centers of the regions and consequently classifies the sample as belonging to the class at the minimum distance:
In particular, given a feature extractor output \(z_i\), Eq. 3 outputs the index of the predicted class for the correspondent input test sample.
4 Experiments
In this section, we first consider the case of no attacks, while specifically assessing adversarial robustness later on. We assess our method’s pure classification accuracy as compared to the three considered stateoftheart defense approaches Jacobian Regularization [53], Input Gradient Regularization [44], and Cross Lipschitz Regularization [52], as well as compared to a plain structure trained with a crossentropy loss (No Defense). We first validate the choice of the distribution parameters, then analyze the quality of the features in terms of separation and classification accuracy. In more detail, we consider two different training settings in each case, namely regular training, where the network is trained with plain SGD, and finetuning, where a model that has been already pretrained with a standard crossentropy loss is further trained for 100 epochs with the proposed GCCS loss. The analysis of this second case is aimed at showing that, potentially, our proposed method could efficiently be employed on top of an alreadytrained network, allowing for improved adversarial robustness with little finetuning effort. The complete analysis of the robustness against adversarial attacks is then presented in Section 5.
4.1 Datasets and training parameters
To evaluate the performance of GCCS and other methods, the five datasets MNIST [26], FMNIST [27], SVHN [28], CIFAR10, and CIFAR100 [29] have been employed. For the less complex MNIST, FMNIST, and SVHN datasets, we employed ResNet18 [72] as a feature extraction network. For the challenging CIFAR10 and CIFAR100, the ShakeShake96 and ShakeShake112 [73] regularization networks were used respectively, using a widen factor equal to 6 for the former and 7 for the latter. A fully connected layer that outputs a vector with dimension D follows the last layer of the encoder. The same baseline architectures were then used to compare our results with competing methods. After choosing fixed \(\mu _{T}\) and \(\sigma _{T}\) values, as better explained in the following, the networks are trained for a total of 1800 epochs. For better network convergence, we employed cosine learning rate decay [74] with an initial value of 0.01 as well as weight decay with a rate set to 0.001. To avoid overfitting, we apply dropout regularization [75] with a 0.8 keep probability to all the fully connected layers in the network. For every experiment, if not differently specified, we use a training batch size equal to 200, as better detailed in Section 4.4. The additional 100epoch training we perform in the finetuning configuration are carried out with a learning rate of \(1e^{3}\), without learning rate decay.
4.2 Target distributions parameters
We explore the behavior of the target distributions by setting different target mean and variance values. We fix the mean \(\mu _{T}\) and variance \(\sigma _{T}\) values for the target distributions so that they are centered on the vertices of a regular \((D1)\)simplex. For this reason, the main parameter affecting our design is the \(\mu _{T}/\sigma _{T}\) ratio, i.e., how far apart the distributions are with respect to the chosen variance.
In the experiment, we set \(\sigma _T = 1\) so that the target distributions can be written as \(\mathbb {P}_i = \mathcal {N}(\mu _T \mathbf {e}_i,\mathbb {I}_D)\). Then, we compute the classification accuracy as a function of \(\mu _{T} \in [5,300]\). Figure 2 shows the accuracy as a function of \(\mu _T/\sigma _T\) on the MNIST, SVHN, and CIFAR10 dataset. It can be observed that in the \(\mu _{T} \ge 30\) region the accuracy is even higher than that obtained with the crossentropy loss. In the following, assuming unitary standard deviation for the target distributions (\(\sigma _{T}=1\)), we set \(\mu _{T}=70\) which corresponds to the maximum test accuracy on the CIFAR10, SVHN, and MNIST datasets, as shown in Fig. 2. As expected, small values of \(\mu\) cause the learned distributions to be too close in the latent space, leading to difficult class separability and poor classification performance. On the other hand, it can be noticed how, for more challenging datasets like CIFAR10 and SVHN, the classification accuracy gain ensured by GCCS increases for the chosen value \(\mu _{T}=70\). Specifically, we hypothesize that values around \(\mu _{T}=70\) ensure the best tradeoff between the separability of the different distributions in the latent space and the enforceability of the target distributions during training. On the other hand, greater \(\mu _{T}/\sigma _{T}\) ratios are more likely to cause instability during training due to harder constraints on the target distributions, resulting in worse classification accuracy. Our empirical findings lead us to the assumption that the \(\mu _T\) value could be tuned by training just once on a specific dataset and the obtained value used also with datasets with the same number of classes, without the need for an exhaustive and expensive search based on multiple training experiments.
4.3 Kurtosis factor
In this section, we analyzed the effects of the Kurtosis factor in the loss formulation over the classification accuracy. We set \(\mu _{T}=70\) and investigate performance as a function of the balancing parameter \(\lambda\), taking values in the range \(\lambda \in [0, 1]\) on the CIFAR10, SVHN, and MNIST datasets. By looking at Table 1, one can observe a significant performance drop when the Kurtosis factor is not included in the loss (i.e., when \(\lambda =0\)). For this reason, throughout the whole experimental evaluation, we set the value to \(\lambda =0.2\) as we experimentally verified it ensures the greatest class separability and hence classification accuracy.
4.4 Training batch size
Keeping fixed the value of the \(\mu _{T}\), \(\sigma _{T}\), and \(\lambda\) parameters to \(\mu _{T}=70\), \(\sigma _{T}=1\), and \(\lambda =0.2\), we now investigate the effect of different training batch sizes on the accuracy of our GCCS method. Similarly to Table 1, Table 2 reports the GCCS classification accuracy as a function of the training batch size on the CIFAR10 and MNIST datasets. Table 2 shows that, for smaller batch sizes, the training does not converge to high accuracy. This happens because the batch size is too small to allow the network to learn distributions that are Gaussian and therefore wellseparable in the latent space. On the other hand, as the training batch size increases, the accuracy does not significantly improve. While it might be possible to reach slightly better accuracy by further increasing the training batch size, it is worth noticing that, as the batch size increases, so does the complexity of the training, and with it the training time. For this reason, we choose to keep the training batch size equal to 200 for all our experiments, as it represents a sweet spot in the tradeoff between high accuracy and problem size.
4.5 Features extraction and separation
Feature selection is of key importance for the classification accuracy and robustness of a neural network. Often in the literature, interclass and intraclass separation are considered independently, while, as said, our approach yields high interclass and intraclass separation at the same time.
4.5.1 Latent space analysis
When dealing with a multiclass classification problem, it is difficult to evaluate latent space features as dimensionality increases. Without loss of generality, Fig. 3 illustrates the distributions in the latent space \(\mathbf {z}\) for three classes of MNIST, where points represent samples of a particular class (green, blue, and red respectively). Figure 3a–e show the distribution of the decision variable with different methods when no adversarial attack is applied. Figure 3a reports the latent space distribution for GCCS: one can observe that features are mapped onto distributions that are centered on the vertices of a simplex with a fixed variance and also that classes exhibit the same interclass and intraclass distances. Moreover, the overlap between samples of different classes is negligible for GCCS, while it is not for the crossentropy loss and the other considered methods, as in Fig. 3b–e. Further, GCCS ensures remarkably low intraclass variance as compared to the other cases, enabling much more robust decision boundaries. In fact, thanks to the GCCS loss that ensures both high separability and uniformity of the target distributions, we can observe the lack of a short path between neighboring distributions and the resulting lower misclassification rate with respect to competing methods, where instead class distributions are not uniformly optimized.
Further, while the figure is only meant to offer a qualitative insight into why GCCS significantly outperforms competing methods in the considered scenarios, it offers precious insight nonetheless. Specifically, we chose to visualize the latent distributions of MNIST classes [0, 1, 9] (red, green, and blue colors respectively) to offer a representation of how the learned distributions of potentially similar classes of digits (0 and 9) would behave in the latent space compared to the distributions of geometrically dissimilar classes (0 and 1). What can be observed from Fig. 3a compared for with Fig. 3b–e is that the learned distributions for similar digits in the GCCS case are not closer to each other than the distributions for the dissimilar ones. The same cannot be said for ND and the other competing methods. Instead, in these cases, the red and blue distributions (i.e., the learned distributions for the 0 and 9 classes respectively) appear to be closer to each other compared with the distance from the green distribution (i.e., class 1). Finally, we analyze the remaining Fig. 3f–y in Section 5.
4.5.2 Classification accuracy and interclass/intraclass separation
We can now examine the quality of the features extracted by measuring the distance of each sample with respect to every other sample. We evaluate separability by measuring the average interclass/intraclass distances in the latent space when no adversarial attack is performed. With n samples in the class A, n latent space feature vectors \(\{z_1^A \cdots z_i^A \cdots z_j^A \cdots z_n^A\}\) are generated for each class, where vector \(z_i^A\) represents the ith sample in class A. Hence, we define the intraclass distance of class A as:
such that the distance between a sample and itself in the latent space is zero [76]. On the other hand, we can define the interclass distance between n samples in class A and m samples in class B as:
Finally, we can define the ratio of the interclass and intraclass distance between two classes A and B as:
According to the \(r_{A B}\) definition, higher ratios correspond to higher robustness and lower misclassification rate thanks to greater separability and simpler decision boundaries. Table 3 shows the average \(r_{A B}\) ratio for all possible pairs of classes over different datasets and different methods, together with the corresponding confidence intervals.
It can be observed that GCCS leads to much higher rates than the crossentropy loss and the considered competing methods when no attacks are carried out. This directly translates to excellent classification accuracy both for the regular training and finetuning, as reported in Table 4 which shows the maximum accuracy over multiple datasets for the different methods. In more detail, it can be seen that, in some cases, other techniques might even cause a drop in classification accuracy with respect to the standard crossentropy loss function, especially for challenging datasets such as CIFAR10 and CIFAR100 where instead GCCS improves the plain crossentropy loss testing accuracy. The missing entries in the tables for the CIFAR10 and CIFAR100 datasets account for the cases in which it was not possible to apply the particular adversarial defense method because of very high computational requirements.
4.5.3 Fscore
Further, we assess the performance of our method by computing the Fscore directly from the confusion matrix and then compare the results with stateoftheart methods. Also in this second set of experiments, we only deal with the case in which no adversarial attack is performed. The Fscore, defined as the harmonic mean of precision and recall, is considered to be a more precise measure of the classification accuracy for uneven datasets such as SVHN. Table 5 illustrates the Fscores for all the considered datasets, and it shows that GCCS outperforms the stateoftheart also according to this metric. As anticipated, the gap is more significant for SVHN than for the other datasets.
5 Adversarial robustness evaluation
To assess the robustness of our proposed method, we follow the guidelines proposed in [13, 35, 36] to provide a rigorous comparison with existing methods when applying different adversarial attacks. First, we consider gradientbased attacks applied in a whitebox scenario, e.g., assuming the attacker has access to the model parameters and the loss function. Specifically, we carry out largescale experimentation with nontargeted (DeepFool, FGSM, and PGD) and targeted (TGSM, JSMA) gradientbased attacks. Then, we evaluate robustness against the gradientfree SPSA attack, which does not exploit knowledge of the gradients with respect to the loss. In all the experiments, we have employed adversarial attacks on a batch of images of a given size equal to 100. In other words, we simulate an attacker trying to maximize the loss over the whole batch with respect to the true class.
We evaluate classification accuracy as a function of a tunable parameter \(\epsilon\) that indicates how strong is the applied attack, such that \({\Vert \mathbf {n}\Vert _{\infty }/\Vert {\textbf {x}}\Vert _{\infty } \le \epsilon }\), where \(\mathbf {n}\) is the added noise vector, and \({\textbf {x}}\) is the input signal. In more detail, given an attack model \(A_{\epsilon }({\textbf {x}})\) dependent on the input \({\textbf {x}}\), a classifier C, and the expected output \({\textbf {y}}\), the accuracy of the classifier against the attack is computed as:
In more detail, we first evaluate the robustness of GCCS against attacks without employing any adversarial training. Our goal is to show that the proposed regularization of the latent space offers an advantage with respect to adversarial training since it is not trained for a specific attack. Hence, the proposed technique performs better than other defense methods even when adversarial training is employed at the same time. Then, since actually the proposed method and adversarial training are not mutually exclusive, we apply GCCS together with adversarial training and show even greater combined robustness. To perform adversarial training, as explained in the following, we consider FGSM perturbations with \(\epsilon =0.03\), averaging the plain GCCS loss with the adversarial one as done in the standard approaches [44]. In particular, we employ adversarial training by finetuning an already trained model for a total 100 epochs. Without loss of generality, we finetune models that have been trained with the regular training setting (as defined in Section 4), but our findings can be extended straightforwardly also to the finetuning configuration.
5.1 Gradientbased attacks
5.1.1 Nontargeted attacks
The general goal of nontargeted attacks is to cause a misclassification in labeling the input, so that the output decision differs from the actual class the input belongs to.
DeepFool attack: We start by evaluating GCCS performance when applying the gradientbased DeepFool attack [32]. This particular attack uses a firstorder approximation of the decision boundaries, altering the input towards the closest decision boundary in order to cause misclassification. When dealing with DeepFool, robustness is usually measured with \(\rho\), which is a parameter that indicates the average robustness against the attack. \(\rho\) is dependent on the estimated minimal perturbation caused by DeepFool and also on the cardinality of the considered images dataset, as explained in [32]. Table 6 reports the robustness of the considered methods over multiple datasets, showing that GCCS provides a very significant improvement in robustness compared to different methods, both for regular and adversarial training. The greater robustness obtained by GCCS can be explained by the fact that the learned interclass boundaries are equidistant, so it is more difficult for the attack to efficiently alter them and cause misclassification.
FGSM and PGD attacks: Further, we analyze the robustness of GCCS against the FGSM [33] and PGD [30] gradientbased attacks on MNIST, SVHN, CIFAR10, and CIFAR100. While the former is a singlestep attack that adds noise in the direction of the gradient of the loss function with respect to the input data, the latter is an iterative version of FGSM in which noise is added in multiple iterations, resulting in the strongest adversarial attack that exploits firstorder local information about the trained model. In the PGD attack, the number of iterations K plays an important role in defining the strength of the attack and the time needed to generate the corresponding adversarial examples. In this work, we consider a 5iteration PGD attack (PGD5) as done in [49, 77, 78].
In the case of MNIST, we set \({0 \le \epsilon \le 0.10}\), while we chose \({0 \le \epsilon \le 0.06}\) for SVHN, CIFAR10, and CIFAR100. Figure 4 a–d show that even the regularly trained GCCS model enjoys higher classification accuracy with respect to competing methods over all the datasets and through all the \(\epsilon\) range; indeed, in many cases and for sensible values of \(\epsilon\), GCCS without adversarial training is almost as good as GCCS with adversarial training. This highlights that, despite not explicitly enforcing robustness to a specific attack, GCCS achieves excellent robustness to any tested attack (dotted lines). The performance gap is especially strong in the case of the challenging CIFAR datasets.
In order to gain a better understanding of why the proposed method works much better than the others, we refer to Fig. 3f–j that shows a visual representation of the distributions in the latent space after an FGSM attack (\(\epsilon =0.2\)) has been applied. In the GCCS case (Fig. 3f), even if the tails of the output distributions become heavier, the classes are still clearly separated, allowing for improved classification accuracy and robustness. Our findings are valid also in the case of the PGD5 attack, as highlighted in Fig. 4e–h. When tampered with this very strong attack, the accuracy ensured by competing methods very quickly collapses to low values, while GCCS enables robust classification even for high \(\epsilon\). Similarly to FGSM, Fig. 3k–o depict the latent space after the PGD5 attack is applied. The effect of the attack is much more prominent compared to FGSM to the extent that the class distributions are significantly overlapped for all the competing methods, and classification fails when no defense against adversarial perturbations is applied. Our proposed GCCS method leads instead to remarkably lower misclassification rates and more separable distributions. As a matter of example, for \(\epsilon \ge 0.04\) GCCS is still able to reach around \(85\%\) classification accuracy on CIFAR10, while other methods collapse to randomguess performance.
5.1.2 Targeted attacks
In this second case, the attack works by creating an adversarial sample that is crafted to be misclassified to the desired target class, such that \({C(x^{adv}) = y^*}\). Again, we provide robustness curves for both regular and adversarial training as in Section 5.
TGSM attack: In the gradientbased TGSM [30] attack, the input samples are perturbed by adding noise in the direction of the negative gradient with respect to a selected target class, such that the targeted output class is \(y_{l+1}\) when the true one is \(y_l\). Figure 4i–l present the results for a challenging 5iteration TGSM attack (TGSM5) over the considered datasets.
It can be observed from Fig. 4i–l that GCCS yields significantly higher performance compared to the other methods, throughout different datasets and with different attack strength \(\epsilon\) both for the regular and adversarial training. Figure 3p–t show a visual representation of the distributions in the latent space after the TGSM5 attack with \(\epsilon =0.02\) is applied. Figure 3q clearly shows the effectiveness of the attack when no defense mechanism is employed. Specifically, it can be observed that the distributions’ center of mass shifted their position in the latent space to fall exactly onto the position of the distribution that has been chosen by the targeted TGSM attack we are considering. In other words, the output distributions are shifted so as to replace the output distribution of the next class, leading to very strong misclassification. In the GCCS case instead, as shown in Fig. 3p, it is much more difficult for the attacker to successfully swap the positions of the target distributions with the ones of the neighboring classes, as testified by the distributions in the latent space, which partially spread out as per the effect of the targeted attack, but at the same time they preserve their center of mass in the same positions with respect to the noattack case (Fig. 3a) even after the considered amount of TGSM iterations. This indeed ensures improved robustness and separability for our GGCS method.
JSMA attack: The other gradientbased targeted attack we consider is JSMA [31], which consists of iteratively computing the Jacobian matrix of the network function to form a saliency map. This map is used at every iteration to choose which pixels to tamper with so that the likelihood of changing the output class towards a selected one is increased. In our case, we consider JSMA200 to make the attack more challenging for better performance comparison, with a 1pixel saliency map. Similarly to the TGSM case, Fig. 4m–p shows the classification accuracy for increasing attack strength \(\epsilon\). The proposed method confirms its robustness even against the JSMA attack, achieving better robustness than other methods, especially on the challenging CIFAR datasets. Once again, Fig. 3u–y show how resilient the proposed GCCS method is against the considered adversarial attack as compared to competing methods where the position of the classes’ distributions after the JSMA attack is completely swapped, as already previously observed for the TGSM targeted attack.
5.2 Gradientfree SPSA attack
Finally, we evaluate robustness also against the gradientfree Simultaneous Perturbation Stochastic Approximation attack (SPSA [34]) based on [79], which is a method to approximate analytic gradients with finite difference estimates in random directions. SPSA is an iterative attack whose strength is determined by three parameters: the perturbation strength \(\epsilon\) as defined in Section 5, the number of attack iterations (IT), and batch size (BS), which indicated the number of finite difference estimates used before applying each gradient estimate. We tested the robustness of our GCCS method against the traditional crossentropy loss (no defense) with and without adversarial training by measuring test accuracy as a function of \(\epsilon\) on the CIFAR10 dataset.
First, we analyzed the robustness by varying the IT and BS parameters, and keeping fixed the adversarial perturbation strength to \(\epsilon = 0.02\). In general, Uesato et al. [34] show that the higher the values, the strongest the attack. However, the computational cost of applying SPSA increases very strongly for high BS values. For this reason, we are interested in observing how robustness decreases by varying the strength of the attack for different (IT, BS) combinations. Table 7 shows that, roughly after \(IT=50\) iterations and for \(BS>1024\), test accuracy does not significantly decrease for the higher (IT, BS) combinations appearing on the right side of the table. Hence, we assume our working point to be \((IT=50, BS=1024)\) without loss of generality, as also confirmed by [34] which shows that, in many cases, the maximum strength of SPSA is reached even when doing fewer than ten iterations of the attack.
Figure 5 shows how the proposed method exhibits much greater robustness than the traditional crossentropy loss, even when applying adversarial training to the models. Since SPSA is generally considered to be the most effective gradientfree adversarial attack, we are able to conclude that GCCS is robust also against attacks carried out without the knowledge of the gradients.
6 Training time
To complete the evaluation of the proposed method, we also considered computational requirements. Looking at Table 8, one can observe that the impact of GCCS in terms of training time overhead is negligible with respect to standard crossentropy training (No Defense), as it is in the order of a few added minutes when training for a considerable number of epochs (1800). As highlighted in works such as [80], this feature is of great importance because it allows to efficiently improve the robustness of any deep architecture (even pretrained) at the expense of a negligible additional training effort.
7 Conclusions
We have presented an approach that employs a loss function promoting class separability and robustness by learning a mapping of the decision variables onto Gaussian distributions. The proposed GCCS loss enjoys high classification accuracy and robustness against adversarial attacks, with negligible training overhead. Experiments on different multiclass datasets show excellent performance of the classifiers trained using the GCCS loss, outperforming existing stateoftheart methods both when used to train from scratch and when applied as a finetuning step on pretrained networks. Also, performance is investigated both for targeted and nontargeted gradientbased adversarial attacks in a whitebox scenario.
The analysis of the distributions in the latent space for the proposed GCCS method shows that the different classes tend to remain well separated even in the presence of gradientbased targeted attacks, whereas a similar attack strength invariably mixes the distributions achieved by competing methods.
Also, we have shown that GCCS, when used in combination with adversarial training, can further improve the model robustness while maintaining high classification accuracy. Finally, we illustrated how the proposed method ensures greater robustness when employing the challenging gradientfree SPSA [34] attack, which does not rely on the loss gradient.
Availability of data and materials
Model evaluation was performed on publicly available datasets such as MNIST [26], FMNIST [27], SVHN [28], CIFAR10, and CIFAR100 [29]. The implementation of the considered attacks is available following the links in the original papers (TGSM [30], JSMA [31], DeepFool [32], FGSM [33], PGD [16], and SPSA [34]).
Abbreviations
 GCCS:

Gaussian ClassConditional Simplex loss
 AT:

Adversarial training
 CLR:

CrossLipschitz Regularization
 IGR:

Input Gradient Regularization
 JR:

Jacobian Regularization
 ND:

No Defense (CrossEntropy Loss)
 SPSA:

Simultaneous Perturbation Stochastic Approximation attack
 IT:

SPSA attack iterations
 BS:

SPSA attack batchsize
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Acknowledgements
This research has been carried out in collaboration with Sony R &D Center Europe, Stuttgart, Laboratory 1.
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AA performed the initial set of experiments with the proposed loss and drafted the first version of the manuscript. AM tested the robustness of the approach against the SPSA attack. AM, TB, and EM updated the manuscript to the final version. All authors read and approved the final manuscript.
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Ali, A., Migliorati, A., Bianchi, T. et al. Gaussian classconditional simplex loss for accurate, adversarially robust deep classifier training. EURASIP J. on Info. Security 2023, 3 (2023). https://doi.org/10.1186/s13635023001370
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DOI: https://doi.org/10.1186/s13635023001370