Hiroshi Takahashi

Sekitoshi Kanai, Masanori Yamada, Hiroshi Takahashi et al.

Deep neural networks are vulnerable to adversarial attacks. Recent studies about adversarial robustness focus on the loss landscape in the parameter space since it is related to optimization and generalization performance. These studies conclude that the difficulty of adversarial training is caused by the non-smoothness of the loss function: i.e., its gradient is not Lipschitz continuous. However, this analysis ignores the dependence of adversarial attacks on model parameters. Since adversarial attacks are optimized for models, they should depend on the parameters. Considering this dependence, we analyze the smoothness of the loss function of adversarial training using the optimal attacks for the model parameter in more detail. We reveal that the constraint of adversarial attacks is one cause of the non-smoothness and that the smoothness depends on the types of the constraints. Specifically, the $L_\infty$ constraint can cause non-smoothness more than the $L_2$ constraint. Moreover, our analysis implies that if we flatten the loss function with respect to input data, the Lipschitz constant of the gradient of adversarial loss tends to increase. To address the non-smoothness, we show that EntropySGD smoothens the non-smooth loss and improves the performance of adversarial training.

Masanori Yamada, Sekitoshi Kanai, Tomoharu Iwata et al.

Adversarial training is actively studied for learning robust models against adversarial examples. A recent study finds that adversarially trained models degenerate generalization performance on adversarial examples when their weight loss landscape, which is loss changes with respect to weights, is sharp. Unfortunately, it has been experimentally shown that adversarial training sharpens the weight loss landscape, but this phenomenon has not been theoretically clarified. Therefore, we theoretically analyze this phenomenon in this paper. As a first step, this paper proves that adversarial training with the L2 norm constraints sharpens the weight loss landscape in the linear logistic regression model. Our analysis reveals that the sharpness of the weight loss landscape is caused by the noise aligned in the direction of increasing the loss, which is used in adversarial training. We theoretically and experimentally confirm that the weight loss landscape becomes sharper as the magnitude of the noise of adversarial training increases in the linear logistic regression model. Moreover, we experimentally confirm the same phenomena in ResNet18 with softmax as a more general case.

Sekitoshi Kanai, Masanori Yamada, Shin'ya Yamaguchi et al.

We propose a method for improving adversarial robustness by addition of a new bounded function just before softmax. Recent studies hypothesize that small logits (inputs of softmax) by logit regularization can improve adversarial robustness of deep learning. Following this hypothesis, we analyze norms of logit vectors at the optimal point under the assumption of universal approximation and explore new methods for constraining logits by addition of a bounded function before softmax. We theoretically and empirically reveal that small logits by addition of a common activation function, e.g., hyperbolic tangent, do not improve adversarial robustness since input vectors of the function (pre-logit vectors) can have large norms. From the theoretical findings, we develop the new bounded function. The addition of our function improves adversarial robustness because it makes logit and pre-logit vectors have small norms. Since our method only adds one activation function before softmax, it is easy to combine our method with adversarial training. Our experiments demonstrate that our method is comparable to logit regularization methods in terms of accuracies on adversarially perturbed datasets without adversarial training. Furthermore, it is superior or comparable to logit regularization methods and a recent defense method (TRADES) when using adversarial training.

Hiroshi Takahashi, Tomoharu Iwata, Yuki Yamanaka et al.

The variational autoencoder (VAE) is a powerful generative model that can estimate the probability of a data point by using latent variables. In the VAE, the posterior of the latent variable given the data point is regularized by the prior of the latent variable using Kullback Leibler (KL) divergence. Although the standard Gaussian distribution is usually used for the prior, this simple prior incurs over-regularization. As a sophisticated prior, the aggregated posterior has been introduced, which is the expectation of the posterior over the data distribution. This prior is optimal for the VAE in terms of maximizing the training objective function. However, KL divergence with the aggregated posterior cannot be calculated in a closed form, which prevents us from using this optimal prior. With the proposed method, we introduce the density ratio trick to estimate this KL divergence without modeling the aggregated posterior explicitly. Since the density ratio trick does not work well in high dimensions, we rewrite this KL divergence that contains the high-dimensional density ratio into the sum of the analytically calculable term and the low-dimensional density ratio term, to which the density ratio trick is applied. Experiments on various datasets show that the VAE with this implicit optimal prior achieves high density estimation performance.