We thank the reviewers for their time, effort, and helpful feedback. We will make sure to incorporate the reviewers' We address individual feedback below. "... a major takeaway I get is that the PGD attack seems to provide an okay approximation of robustness ... Reply: First, we affirm that "the PGD attack seems to provide an okay approximation of robustness compared to the It is one important future work to be done in this field, but it is outside the scope of our paper. Please consider raising your score if you like our work. Reviewer 2: Thank you for your positive review!

Convex relaxations are effective for training and certifying neural networks against norm-bounded adversarial attacks, but they leave a large gap between certifiable and empirical robustness. In principle, convex relaxation can provide tight bounds if the solution to the relaxed problem is feasible for the original non-convex problem. We propose two regularizers that can be used to train neural networks that yield tighter convex relaxation bounds for robustness. In all of our experiments, the proposed regularizers result in higher certified accuracy than non-regularized baselines.

Finding minimum distortion of adversarial examples and thus certifying robustness in neural networks classifiers is known to be a challenging problem. Nevertheless, recently it has been shown to be possible to give a non-trivial certified lower bound of minimum distortion, and some recent progress has been made towards this direction by exploiting the piece-wise linear nature of ReLU activations. However, a generic robustness certification for \textit{general} activation functions still remains largely unexplored. To address this issue, in this paper we introduce CROWN, a general framework to certify robustness of neural networks with general activation functions. The novelty in our algorithm consists of bounding a given activation function with linear and quadratic functions, hence allowing it to tackle general activation functions including but not limited to the four popular choices: ReLU, tanh, sigmoid and arctan. In addition, we facilitate the search for a tighter certified lower bound by \textit{adaptively} selecting appropriate surrogates for each neuron activation. Experimental results show that CROWN on ReLU networks can notably improve the certified lower bounds compared to the current state-of-the-art algorithm Fast-Lin, while having comparable computational efficiency. Furthermore, CROWN also demonstrates its effectiveness and flexibility on networks with general activation functions, including tanh, sigmoid and arctan.

Finding minimum distortion of adversarial examples and thus certifying robustness in neural networks classifiers is known to be a challenging problem. Nevertheless, recently it has been shown to be possible to give a non-trivial certified lower bound of minimum distortion, and some recent progress has been made towards this direction by exploiting the piece-wise linear nature of ReLU activations. However, a generic robustness certification for \textit{general} activation functions still remains largely unexplored. To address this issue, in this paper we introduce CROWN, a general framework to certify robustness of neural networks with general activation functions. The novelty in our algorithm consists of bounding a given activation function with linear and quadratic functions, hence allowing it to tackle general activation functions including but not limited to the four popular choices: ReLU, tanh, sigmoid and arctan. In addition, we facilitate the search for a tighter certified lower bound by \textit{adaptively} selecting appropriate surrogates for each neuron activation. Experimental results show that CROWN on ReLU networks can notably improve the certified lower bounds compared to the current state-of-the-art algorithm Fast-Lin, while having comparable computational efficiency. Furthermore, CROWN also demonstrates its effectiveness and flexibility on networks with general activation functions, including tanh, sigmoid and arctan.

Verifying robustness of neural network classifiers has attracted great interests and attention due to the success of deep neural networks and their unexpected vulnerability to adversarial perturbations. Although finding minimum adversarial distortion of neural networks (with ReLU activations) has been shown to be an NP-complete problem, obtaining a non-trivial lower bound of minimum distortion as a provable robustness guarantee is possible. However, most previous works only focused on simple fully-connected layers (multilayer perceptrons) and were limited to ReLU activations. This motivates us to propose a general and efficient framework, CNN-Cert, that is capable of certifying robustness on general convolutional neural networks. Our framework is general -- we can handle various architectures including convolutional layers, max-pooling layers, batch normalization layer, residual blocks, as well as general activation functions; our approach is efficient -- by exploiting the special structure of convolutional layers, we achieve up to 17 and 11 times of speed-up compared to the state-of-the-art certification algorithms (e.g. Fast-Lin, CROWN) and 366 times of speed-up compared to the dual-LP approach while our algorithm obtains similar or even better verification bounds. In addition, CNN-Cert generalizes state-of-the-art algorithms e.g. Fast-Lin and CROWN. We demonstrate by extensive experiments that our method outperforms state-of-the-art lower-bound-based certification algorithms in terms of both bound quality and speed.