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Title: Towards Stable and Efficient Training of Verifiably Robust Neural Networks

Paper content:
Training neural networks with verifiable robustness guarantees is challenging.
Several existing approaches utilize linear relaxation based neural network output bounds under perturbation, but they can slow down training by a factor of hundreds depending on the underlying network architectures.
Meanwhile, interval bound propagation (IBP) based training is efficient and significantly outperforms linear relaxation based methods on many tasks, yet it may suffer from stability issues since the bounds are much looser especially at the beginning of training.
In this paper, we propose a new certified adversarial training method, CROWN-IBP, by combining the fast IBP bounds in a forward bounding pass and a tight linear relaxation based bound, CROWN, in a backward bounding pass.
CROWN-IBP is computationally efficient and consistently outperforms IBP baselines on training verifiably robust neural networks.
We conduct large scale experiments on MNIST and CIFAR datasets, and outperform all previous linear relaxation and bound propagation based certified defenses in L_inf robustness.
Notably, we achieve 7.02% verified test error on MNIST at epsilon=0.3, and 66.94% on CIFAR-10 with epsilon=8/255.
The success of deep neural networks (DNNs) has motivated their deployment in some safety-critical environments, such as autonomous driving and facial recognition systems.
Applications in these areas make understanding the robustness and security of deep neural networks urgently needed, especially their resilience under malicious, finely crafted inputs.
Unfortunately, the performance of DNNs are often so brittle that even imperceptibly modified inputs, also known as adversarial examples, are able to completely break the model (Goodfellow et al., 2015; Szegedy et al., 2013) .
The robustness of DNNs under adversarial examples is well-studied from both attack (crafting powerful adversarial examples) and defence (making the model more robust) perspectives (Athalye et al., 2018; Carlini & Wagner, 2017a; b; Goodfellow et al., 2015; Madry et al., 2018; Papernot et al., 2016; Xiao et al., 2019b; 2018b; c; Eykholt et al., 2018; Chen et al., 2018; Xu et al., 2018; Zhang et al., 2019b) .
Recently, it has been shown that defending against adversarial examples is a very difficult task, especially under strong and adaptive attacks.
Early defenses such as distillation (Papernot et al., 2016) have been broken by stronger attacks like C&W (Carlini & Wagner, 2017b) .
Many defense methods have been proposed recently (Guo et al., 2018; Song et al., 2017; Buckman et al., 2018; Ma et al., 2018; Samangouei et al., 2018; Xiao et al., 2018a; 2019a) , but their robustness improvement cannot be certified -no provable guarantees can be given to verify their robustness.
In fact, most of these uncertified defenses become vulnerable under stronger attacks (Athalye et al., 2018; He et al., 2017) .
Several recent works in the literature seeking to give provable guarantees on the robustness performance, such as linear relaxations (Wong & Kolter, 2018; Mirman et al., 2018; Wang et al., 2018a; Dvijotham et al., 2018b; Weng et al., 2018; Zhang et al., 2018) , interval bound propagation (Mirman et al., 2018; Gowal et al., 2018) , ReLU stability regularization (Xiao et al., 2019c) , and distributionally robust optimization (Sinha et al., 2018) and semidefinite relaxations (Raghunathan et al., 2018a; Dvijotham et al.) .
Linear relaxations of neural networks, first proposed by Wong & Kolter (2018) , is one of the most popular categories among these certified defences.
They use the dual of linear programming or several similar approaches to provide a linear relaxation of the network (referred to as a "convex adversarial polytope") and the resulting bounds are tractable for robust optimization.
However, these methods are both computationally and memory intensive, and can increase model training time by a factor of hundreds.
On the other hand, interval bound propagation (IBP) is a simple and efficient method for training verifiable neural networks (Gowal et al., 2018) , which achieved state-of-the-art verified error on many datasets.
However, since the IBP bounds are very loose during the initial phase of training, the training procedure can be unstable and sensitive to hyperparameters.
In this paper, we first discuss the strengths and weakness of existing linear relaxation based and interval bound propagation based certified robust training methods.
Then we propose a new certified robust training method, CROWN-IBP, which marries the efficiency of IBP and the tightness of a linear relaxation based verification bound, CROWN (Zhang et al., 2018) .
CROWN-IBP bound propagation involves a IBP based fast forward bounding pass, and a tight convex relaxation based backward bounding pass (CROWN) which scales linearly with the size of neural network output and is very efficient for problems with low output dimensions.
Additional, CROWN-IBP provides flexibility for exploiting the strengths of both IBP and convex relaxation based verifiable training methods.
The efficiency, tightness and flexibility of CROWN-IBP allow it to outperform state-of-the-art methods for training verifiable neural networks with ∞ robustness under all settings on MNIST and CIFAR-10 datasets.
In our experiment, on MNIST dataset we reach 7.02% and 12.06% IBP verified error under ∞ distortions = 0.3 and = 0.4, respectively, outperforming the state-of-the-art baseline results by IBP (8.55% and 15.01%).
On CIFAR-10, at = 2 255 , CROWN-IBP decreases the verified error from 55.88% (IBP) to 46.03% and matches convex relaxation based methods; at a larger , CROWN-IBP outperforms all other methods with a noticeable margin.
We propose a new certified defense method, CROWN-IBP, by combining the fast interval bound propagation (IBP) bound and a tight linear relaxation based bound, CROWN.
Our method enjoys high computational efficiency provided by IBP while facilitating the tight CROWN bound to stabilize training under the robust optimization framework, and provides the flexibility to trade-off between the two.
Our experiments show that CROWN-IBP consistently outperforms other IBP baselines in both standard errors and verified errors and achieves state-of-the-art verified test errors for ∞ robustness.
Given a fixed neural network (NN) f (x), IBP gives a very loose estimation of the output range of f (x).
However, during training, since the weights of this NN can be updated, we can equivalently view IBP as an augmented neural network, which we denote as an IBP-NN ( Figure A) .
Unlike a usual network which takes an input x k with label y k , IBP-NN takes two points x L = x k − and x U = x k + as inputs (where x L ≤ x ≤ x U , element-wisely).
The bound propagation process can be equivalently seen as forward propagation in a specially structured neural network, as shown in Figure A .
After the last specification layer C (typically merged into W (L) ), we can obtain m(x k , ).
Then, −m(x k , ) is sent to softmax layer for prediction.
Importantly, since [m(x k , )] y k = 0 (as the y k -th row in C is always 0), the top-1 prediction of the augmented IBP network is y k if and only if all other elements of m(x k , ) are positive, i.e., the original network will predict correctly for all x L ≤ x ≤ x U .
When we train the augmented IBP network with ordinary cross-entropy loss and desire it to predict correctly on an input x k , we are implicitly doing robust optimization (Eq.
(2)).
The simplicity of IBP-NN may help a gradient based optimizer to find better solutions.
On the other hand, while the computation of convex relaxation based bounds can also be cast as an equivalent network (e.g., the "dual network" in Wong & Kolter (2018)), its construction is significantly more complex, and sometimes requires non-differentiable indicator functions (the sets I + , I − and I in Wong & Kolter (2018)).
As a consequence, it can be challenging for the optimizer to find a good solution, and the optimizer tends to making the bounds tighter naively by reducing the norm of weight matrices and over-regularizing the network, as demonstrated in Figure 1 .

assistant: text: We propose a new certified adversarial training method, CROWN-IBP, that achieves state-of-the-art robustness for L_inf norm adversarial perturbations.