28^{th}JULY2018

Aran Nayebi, Surya Ganguli. *Biologically inspired protection of deep networks from adversarial attacks*. CoRR abs/1703.09202, 2017.

Also find this summary on ShortScience.org.

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Nayebi and Ganguli propose saturating neural networks as defense against adversarial examples. The main observation driving this paper can be stated as follows: Neural networks are essentially based on linear sums of neurons (e.g. fully connected layers, convolutional layers) which are then activated; by injecting a small amount of noise per neuron it is possible to shift the final sum by large values, thereby propagating the noisy through the network and fooling the network into misclassifying an example. To prevent the impact of these adversarial examples, the network should be trained in a manner to drive many neurons into a saturated regime – noisy will, so the argument, have less impact then. The authors also give a biological motivation, which I won't go into detail here.

Letting $\psi$ be the used activation function, e.g. Sigmoid or ReLU, a regularizer is added to drive neurons into saturation. In particular, a penalty

$\lambda \sum_l \sum_i \psi_c(h_i^l)$

is added to the loss. Here, $l$ indexes the layer and $i$ the unit in the layer; $h_i^l$ then describes the input to the non-linearity computed for unit $i$ in layer $l$. $\psi_c$ is the complementary function defined as

$\psi_c(z) = \inf_{z': \psi'(z') = 0} |z – z'|$

It defines the distance of the point $z$ to the nearest saturated point $z'$ where $\psi'(z') = 0$. For ReLU activations, the complementary function is the ReLU function itself; for Sigmoid activations, the complementary function is

$\sigma_c(z) = |\sigma(z)(1 - \sigma(z))|$.

In experiments, Nayebi and Ganguli show that training with the additional penalty yields networks with higher robustness against adversarial examples compared to adversarial training (i.e. training on adversarial examples). They also provide some insight, showing e.g. the activation and weight distribution of layers illustrating that neurons are indeed saturated in large parts. For details, see the paper.

I also want to point to a comment on the paper written by Brendel and Bethge [1] questioning the effectiveness of the proposed defense strategy. They discuss a variant of the fast sign gradient method (FSGM) with stabilized gradients which is able to fool saturated networks.