01^{st}AUGUST2018

Matthias Hein, Maksym Andriushchenko. *Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation*. NIPS 2017: 2263-2273.

Also find this summary on ShortScience.org.

What is **your opinion** on the summarized work? Or do you know related work that is of interest? **Let me know** your thoughts in the comments below or get in touch with me:

Hein and Andriushchenko give a intuitive bound on the robustness of neural networks based on the local Lipschitz constant. With robustness, the authors refer a small $\epsilon$-ball around each sample; this ball is supposed to describe the region where the neural network predicts a constant class. This means that adversarial examples have to compute changes large enough to leave these robust areas. Larger $\epsilon$-balls imply higher robustness to adversarial examples.

When considering a single example $x$, and a classifier $f = (f_1, \ldots, f_K)^T$ (i.e. in a multi-class setting), the bound can be stated as follows. For $q$ and $p$ such that $\frac{1}{q} + \frac{1}{p} = 1$ and $c$ being the class predicted for $x$, the it holds

$x = \arg\max_j f_j(x + \delta)$

for all $\delta$ with

$\|\delta\|_p \leq \max_{R > 0}\min \left\{\min_{j \neq c} \frac{f_c(x) – f_j(x)}{\max_{y \in B_p(x, R)} \|\nabla f_c(y) - \nabla f_j(y)\|_q}, R\right\}$.

Here, $B_p(x, R)$ describes the $R$-ball around $x$ measured using the $p$-norm. Based on the local Lipschitz constant (in the denominator), the bound essentially measures how far we can deviate from the sample $x$ (measured in the $p$-norm) until $f_j(x) > f_c(x)$ for some $j \neq c$. The higher the local Lipschitz constant, the smaller deviations are allowed, i.e. adversarial examples are easier to find. Note that the bound also depends on the confidence, i.e. the edge $f_c(x)$ has in comparison to all other $f_j(x)$.

In the remaining paper, the authors also provide bounds for simple classifiers including linear classifiers, kernel methods and two-layer perceptrons (i.e. one hidden layer). For the latter, they also propose a new type of regularization called cross-Lipschitz regularization:

$P(f) = \frac{1}{nK^2} \sum_{i = 1}^n \sum_{l,m = 1}^K \|\nabla f_l(x_i) - \nabla f_m(x_i)\|_2^2$.

This regularization term is intended to reduce the Lipschitz constant locally around training examples. They show experimental results using this regularization on MNIST and CIFAR, see the paper for details.