Miyato et al. propose distributional smoothing (or virtual adversarial training) as defense against adversarial examples. However, I think that both terms do not give a good intuition of what is actually done. Essentially, a regularization term is introduced. Letting $p(y|x,\theta)$ be the learned model, the regularizer is expressed as

$\text{KL}(p(y|x,\theta)|p(y|x+r,\theta)$

where $r$ is the perturbation that maximizes the Kullback-Leibler divergence above, i.e.

$r = \arg\max_r \{\text{KL}(p(y|x,\theta)|p(y|x+r,\theta) | \|r\|_2 \leq \epsilon\}$

with hyper-parameter $\epsilon$. Essentially, the regularizer is supposed to “simulate” adversarial training – thus, the method is also called virtual adversarial training.

The discussed implementation, however, is somewhat cumbersome. In particular, $r$ cannot be computed using first-order methods as the gradient of $\text{KL}$ is $0$ for $r = 0$. So a second-order method is used – for which the Hessian needs to be approximated and the corresponding eigenvectors need to be computed. For me it is unclear why $r$ cannot be initialized randomly to solve this issue … Then, the derivative of the regularizer needs to be computed during training. Here, the authors make several simplifications (such as fixing $\theta$ in the first part of the Kullback-Leibler divergence and ignoring the derivative of $r$ w.r.t. $\theta$).

Overall, however, I like the idea of “virtual” adversarial training as it avoids the need of explicitly using attacks during training to craft adversarial examples. Then, the trained model is often robust against the chosen attacks, but new adversarial examples can be found easily through novel attacks.

Also view this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

Zantedschi et al. propose Gaussian data augmentation in conjunction with bounded $\text{ReLU}$ activations as defense strategy against adversarial examples. Here, Gaussian data augmentation refers to the practice of adding Gaussian noise to the input during training.

Zahavy et al. introduce the concept of ensemble robustness and show that it can be used as indicator for generalization performance. In particular, the main idea is to lift he concept of robustness against adversarial examples to ensemble of networks – as trained, e.g. through Dropout or Bayes-by-Backprop. Letting $Z$ denote the sample set, a learning algorithm is $(K, \epsilon)$ robust if $Z$ can be divided into $K$ disjoint sets $C_1,\ldots,C_K$ such that for every training set $s_1,\ldots,s_n  \in Z$ it holds:

$\forall i, \forall z \in Z, \forall k = 1,\ldots, K$: if $s,z \in C_k$, then $l(f,s_i) – l(f,z)| \leq \epsilon(s_1,\ldots,s_n)$

where $f$ is the model produced by the learning algorithm, $l$ measures the loss and $\epsilon:Z^n \mapsto \mathbb{R}$. For ensembles (explicit or implicit) this definition is extended by considering the maximum generalization loss under the expectation of a randomized learning algorithm:

$\forall i, \forall k = 1,\ldots,K$: if $s \in C_k$, then $\mathbb{E}_f \max_{z \in C_k} |l(f,s_i) – l(f,z)| \leq \epsilon(s_1,\ldots,s_n)$

Here, the randomized learning algorithm computes a distribution over models given a training set.

Also view this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

Liu et al. propose randomizing neural networks, implicitly learning an ensemble of models, to defend against adversarial attacks. In particular, they introduce Gaussian noise layers before regular convolutional layers. The noise can be seen as additional parameter of the model. During training, noise is randomly added. During testing, the model is evaluated on a single testing input using multiple random noise vectors; this essentially corresponds to an ensemble of different models (parameterized by the different noise vectors).

Mathemtically, the authors provide two interesting interpretations. First, they argue that training essentially minimizes an upper bound of the (noisy) inference loss. Second, they show that their approach is equivalent to Lipschitz regularization [1].

[1] M. Hein, M. Andriushchenko. Formal guarantees on the robustness of a classifier against adversarial manipulation. ArXiv:1705.08475, 2017.

Also view this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

Oh et al. propose two different approaches for whitening black box neural networks, i.e. predicting details of their internals such as architecture or training procedure. In particular, they consider attributes regarding architecture (activation function, dropout, max pooling, kernel size of convolutional layers, number of convolutionaly/fully connected layers etc.), attributes concerning optimization (batch size and optimization algorithm) and attributes regarding the data (data split and size). In order to create a dataset of models, they trained roughly 11k models on MNIST; they ensured that these models have at least 98% accuracy on the validation set and they also consider ensembles.

For predicting model attributes, they propose two models, called kennen-o and kennen-i, see Figure 1. Kennen-o takes as input a set of $100$ predictions of the models (i.e. final probability distributions) and tries to directly learn the attributes using a MLP of two fully connected layers. Kennen-i instead crafts a single input which allows to reason about a specific model attribute. An example for kennen-i is shown in Figure 2. In experiments, they demonstrate that both models are able to predict model attributes significantly better than chance. For details, I refer to the paper.

https://i.imgur.com/YbFuniu.png
Figure 1: Illustration of the two proposed approaches, kennen-o (top) and kennen-i (bottom).

https://i.imgur.com/ZXj22zG.png
Figure 2: Illustration of the images created by kennen-i to classify different attributes. See the paper for details.

Also view this summary at [davidstutz.de](https://davidstutz.de/category/reading/).