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Title: MACER: Attack-free and Scalable Robust Training via Maximizing Certified Radius

Paper content:
Adversarial training is one of the most popular ways to learn robust models but is usually attack-dependent and time costly.
In this paper, we propose the MACER algorithm, which learns robust models without using adversarial training but performs better than all existing provable l2-defenses.
Recent work shows that randomized smoothing can be used to provide certified l2 radius to smoothed classifiers, and our algorithm trains provably robust smoothed classifiers via MAximizing the CErtified Radius (MACER).
The attack-free characteristic makes MACER faster to train and easier to optimize.
In our experiments, we show that our method can be applied to modern deep neural networks on a wide range of datasets, including Cifar-10, ImageNet, MNIST, and SVHN.
For all tasks, MACER spends less training time than state-of-the-art adversarial training algorithms, and the learned models achieve larger average certified radius.
Modern neural network classifiers are able to achieve very high accuracy on image classification tasks but are sensitive to small, adversarially chosen perturbations to the inputs (Szegedy et al., 2013; Biggio et al., 2013) .
Given an image x that is correctly classified by a neural network, a malicious attacker may find a small adversarial perturbation δ such that the perturbed image x + δ, though visually indistinguishable from the original image, is assigned to a wrong class with high confidence by the network.
Such vulnerability creates security concerns in many real-world applications.
Researchers have proposed a variety of defense methods to improve the robustness of neural networks.
Most of the existing defenses are based on adversarial training (Szegedy et al., 2013; Madry et al., 2017; Goodfellow et al., 2015; Huang et al., 2015; Athalye et al., 2018) .
During training, these methods first learn on-the-fly adversarial examples of the inputs with multiple attack iterations and then update model parameters using these perturbed samples together with the original labels.
However, such approaches depend on a particular (class of) attack method.
It cannot be formally guaranteed whether the resulting model is also robust against other attacks.
Moreover, attack iterations are usually quite expensive.
As a result, adversarial training runs very slowly.
Another line of algorithms trains robust models by maximizing the certified radius provided by robust certification methods (Weng et al., 2018; Gowal et al., 2018; Zhang et al., 2019c) .
Using linear or convex relaxations of fully connected ReLU networks, a robust certification method computes a "safe radius" r for a classifier at a given input such that at any point within the neighboring radius-r ball of the input, the classifier is guaranteed to have unchanged predictions.
However, the certification methods are usually computationally expensive and can only handle shallow neural networks with ReLU activations, so these training algorithms have troubles in scaling to modern networks.
In this work, we propose an attack-free and scalable method to train robust deep neural networks.
We mainly leverage the recent randomized smoothing technique (Cohen et al., 2019) .
A randomized smoothed classifier g for an arbitrary classifier f is defined as g(x) = E η f (x + η), in which η ∼ N (0, σ 2 I).
While Cohen et al. (2019) derived how to analytically compute the certified radius of the randomly smoothed classifier g, they did not show how to maximize that radius to make the classifier g robust.
Salman et al. (2019) proposed SmoothAdv to improve the robustness of g, but it still relies on the expensive attack iterations.
Instead of adversarial training, we propose to learn robust models by directly taking the certified radius into the objective.
We outline a few challenging desiderata any practical instantiation of this idea would however have to satisfy, and provide approaches to address each of these in turn.
A discussion of these desiderata, as well as a detailed implementation of our approach is provided in Section 4.
And as we show both theoretically and empirically, our method is numerically stable and accounts for both classification accuracy and robustness.
Our contributions are summarized as follows:
• We propose an attack-free and scalable robust training algorithm by MAximizing the CErtified Radius (MACER).
MACER has the following advantages compared to previous works: -Different from adversarial training, we train robust models by directly maximizing the certified radius without specifying any attack strategies, and the learned model can achieve provable robustness against any possible attack in the certified region.
Additionally, by avoiding time-consuming attack iterations, our proposed algorithm runs much faster than adversarial training.
-Different from other methods that maximize the certified radius but are not scalable to deep neural networks, our method can be applied to architectures of any size.
This makes our algorithm more practical in real scenarios.
• We empirically evaluate our proposed method through extensive experiments on Cifar-10, ImageNet, MNIST, and SVHN.
On all tasks, MACER achieves better performance than state-of-the-art algorithms.
MACER is also exceptionally fast.
For example, on ImageNet, MACER uses 39% less training time than adversarial training but still performs better.
In this work we propose MACER, an attack-free and scalable robust training method via directly maximizing the certified radius of a smoothed classifier.
We discuss the desiderata such an algorithm would have to satisfy, and provide an approach to each of them.
According to our extensive experiments, MACER performs better than previous provable l 2 -defenses and trains faster.
Our strong empirical results suggest that adversarial training is not a must for robust training, and defense based on certification is a promising direction for future research.
Moreover, several recent papers (Carmon et al., 2019; Zhai et al., 2019; suggest that using unlabeled data helps improve adversarially robust generalization.
We will also extend MACER to the semisupervised setting.

assistant: text: We propose MACER: a provable defense algorithm that trains robust models by maximizing the certified radius. It does not use adversarial training but performs better than all existing provable l2-defenses.