Ross and Doshi-Velez propose input gradient regularization to improve robustness and interpretability of neural networks. As the discussion of interpretability is quite limited in the paper, the main contribution is an extensive evaluation of input gradient regularization against adversarial examples – in comparison to defenses such as distillation or adversarial training. Specifically, input regularization as proposed in [1] is used:

$\arg\min_\theta H(y,\hat{y}) + \lambda \|\nabla_x H(y,\hat{y})\|_2^2$

where $\theta$ are the network’s parameters, $x$ its input and $\hat{y}$ the predicted output. Here, $H$ might be a cross-entropy loss. It also becomes apparent why this regularization was originally called double-backpropagation because the second derivative is necessary during training.
In experiments, the authors show that the proposed regularization is superior to many other defenses including distillation and adversarial training. Unfortunately, the comparison does not include other “regularization” techniques to improve robustness – such as Lipschitz regularization. This makes the comparison less interpretable, especially as the combination of input gradient regularization and adversarial training performs best (suggesting that adversarial training is a meaningful defense, as well). Still, I recommend a closer look on the experiments. For example, the authors also study the input gradients of defended models, leading to some interesting conclusions.

[1] H. Drucket, Y. LeCun. Improving generalization performance using double backpropagation. IEEE Transactions on Neural Networks, 1992.

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