Appendix for adversarial examples paper summary: part 1

Solution to the FGSM optimization: \[\max_{\eta} \|w^T \eta\|_\infty \\ \text{ subject to }\|\eta\|_\infty<\epsilon\]

Solution:

\[\|w^T \eta\|_\infty = \max_{i}\{|w_i^T \eta|\} \\ \le \max_{i}\{\|w_i^T\|_1 \|\eta\|_\infty\} = \max_{i}\{\epsilon \|w_i^T\|_1\}\]

Here we use Holder’s inequality. In particular, when $\eta = sign(w_i^T)$ the equality holds, where $i=\max_{i}{|w_i^T|_1}$.