Efficient Neural Network Robustness Certification with General Activation Functions (1811.00866v1)

Abstract: Finding minimum distortion of adversarial examples and thus certifying robustness in neural network classifiers for given data points 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 adversarial 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 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 for given input data points. 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 four popular choices: ReLU, tanh, sigmoid and arctan. In addition, we facilitate the search for a tighter certified lower bound by 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.

• The paper introduces a general framework that certifies neural network robustness across various activation functions by using adaptive linear and quadratic bounds.
• It enhances certified lower bounds by up to 26% compared to traditional ReLU-focused methods while maintaining computational efficiency.
• The framework scales to large networks, effectively certifying models on MNIST and CIFAR-10 in about one minute with a single CPU core.

An Overview of Efficient Neural Network Robustness Certification with General Activation Functions

The paper "Efficient Neural Network Robustness Certification with General Activation Functions" addresses the problem of certifying neural network robustness against adversarial perturbations. While prior work has focused on networks with ReLU activations, this paper introduces a framework that extends robustness certification to networks with various activation functions.

Problem Statement

Neural networks have shown susceptibility to adversarial attacks, where small perturbations to input data can lead to incorrect predictions. The task of finding the minimal distortion required to alter predictions is computationally intense, often categorized as NP-complete, especially in ReLU networks. Although some progress has been made with ReLU activations, generalizing these findings to networks with other activations such as tanh, sigmoid, and arctan has remained a challenge. This paper proposes a comprehensive method for certifying robustness across these general activation functions.

Methodology

The framework developed in this paper involves bounding activation functions with linear and quadratic functions to facilitate robustness certification. By using these bounds, the framework efficiently computes certified lower bounds on the minimum adversarial distortion. The primary contributions include:

• Generic Framework: The proposed framework provides a way to certify neural network robustness using linear or quadratic upper and lower bounds for activation functions irrespective of their linearity.
• Adaptive Scheme: It introduces an adaptive method for selecting bounds, enhancing the tightness of the certified lower bounds. The experiments demonstrate significant improvements up to 26% over previously established methods.
• Computational Efficiency: The framework is designed to scale efficiently to large neural networks. For instance, it can certify a network with over 10,000 neurons in approximately one minute using a single CPU core.

Experimental Results

The authors tested their approach using Multi-Layer Perceptron (MLP) models on the MNIST and CIFAR-10 datasets. The results showed that their framework outperforms current state-of-the-art methods in terms of certified lower bounds while maintaining comparable computational efficiency.

• For ReLU networks, the proposed method significantly improves upon existing certification approaches, featuring more adaptive bounds.
• The framework effectively handles networks with tanh, sigmoid, and arctan activations, yielding non-trivial certified lower bounds on distortion.

Implications and Future Directions

The development of a robust, generic framework to certify neural networks contributes both practically and theoretically to the field of adversarial machine learning. Practically, it supports more reliable deployment of neural networks in safety-critical applications. Theoretically, it opens avenues for exploring the robustness of networks with yet-to-be-designed activation functions.

Future work may focus on further refining these bounding techniques and exploring their applicability to even more diverse network architectures and activation functions. The interaction between network robustness and pharmacology of activation functions could also be an intriguing area of paper, potentially leading to novel design principles for more robust neural networks.

In conclusion, this paper marks a notable step toward generalized neural network certification, providing a foundational approach that could spur innovation in network design and robustness analysis.