A repository for training provably robust neural networks by optimizing convex outer bounds on the adversarial polytope. Created by Zico Kolter and Eric Wong.

While networks are capable of representing highly complex functions. For example, with today's networks it is an easy task to achieve 99% accuracy on the MNIST digit recognition dataset, and we can quickly train a small network that can accurately predict that the following image is a 7.

However, the versatility of neural networks comes at a cost: these networks are highly susceptible to small perturbations, or adversarial attacks (e.g. the fast gradient sign method and projected gradient descent)! While most of us can recognize that the following image is still a 7, the same network that could correctly classify the above image instead classifies the following image as a 3.

While this is a relatively harmless example, one can easily think of situations where such adversarial perturbations can be dangerous and costly (e.g. autonomous driving).

Robust networks are networks that are trained to protect against any sort of adversarial perturbation. Specifically, for any seen training example, the network is robust if it is impossible to cause the network to incorrectly classify the example by adding a small perturbation.

The short version: we use the dual of a convex relaxation of the network over the adversarial polytope to lower bound the output. This lower bound can be expressed as another deep network with the same model parameters, and optimizing this lower bound allows us to guarantee robustness of the network.

The long version: see our paper, Provable defenses against adversarial examples via the convex outer adversarial polytope.

We illustrate the power of training robust networks in the following two scenarios: 2D toy case for a visualization, and on the MNIST dataset.

To illustrate the difference, consider a binary classification task on 2D space, separating red dots from blue dots. Optimizing a neural network in the usual fashion gives us the following classifier on the left, and our robust method gives the classifier on the right. The squares around each example represent the adversarial region of perturbations.

For the standard classifier, a number of the examples have perturbation regions that contain both red and blue. These examples are susceptible to adversarial attacks that will flip the output of the neural network. On the other hand, the robust network has all perturbation regions fully contained in the either red or blue, and so this network is robust: we are guaranteed that there is no possible adversarial perturbation to flip the label of any example.

As mentioned before, it is easy to fool networks trained on the MNIST dataset when using attacks such as the fast gradient sign method (FGS) and projected gradient descent (PGD). We observe that PGD can almost always fool the MNIST trained network.

On the other hand, the robust network is significantly less affected by these attacks. In fact, when optimizing the robust loss, we can additionally calculate a robust error which gives an provable upper bound on the error caused by any adversarial perturbation. In this case, the robust network has a robust error of 8.3%, and so we are guaranteed that no adversarial attack can ever get an error rate of larger than 8.3%. In comparison, the robust error of the standard network is 100%.

• The code implementing the robust loss function that measures the convex outer bounds on the adversarial polytope as described in the paper. It is implemented for linear and convolutional networks with ReLU activation
• Examples, containing the following:
  • Code to train a robust MNIST classifier
  • Code to generate and plot the 2D toy example.