Despite the improved accuracy of deep neural networks, the discovery of adversarial examples has raised serious safety concerns. In this paper, we study two variants of pointwise robustness, the maximum safe radius problem, which for a given input sample computes the minimum distance to an adversarial example, and the feature robustness problem, which aims to quantify the robustness of individual features to adversarial perturbations. We demonstrate that, under the assumption of Lipschitz continuity, both problems can be approximated using finite optimisation by discretising the input space, and the approximation has provable guarantees, i.e., the error is bounded. We then show that the resulting optimisation problems can be reduced to the solution of two-player turn-based games, where the first player selects features and the second perturbs the image within the feature. While the second player aims to minimise the distance to an adversarial example, depending on the optimisation objective the first player can be cooperative or competitive. We employ an anytime approach to solve the games, in the sense of approximating the value of a game by monotonically improving its upper and lower bounds. The Monte Carlo tree search algorithm is applied to compute upper bounds for both games, and the Admissible A^⁎ and the Alpha-Beta Pruning algorithms are, respectively, used to compute lower bounds for the maximum safety radius and feature robustness games. When working on the upper bound of the maximum safe radius problem, our tool demonstrates competitive performance against existing adversarial example crafting algorithms. Furthermore, we show how our framework can be deployed to evaluate pointwise robustness of neural networks in safety-critical applications such as traffic sign recognition in self-driving cars.

尽管深度神经网络的准确性有所提高，但对抗性示例的发现却引发了严重的安全隐患。在本文中，我们研究了点状鲁棒性的两个变体，最大安全半径问题，它针对给定的输入样本计算到一个对抗示例的最小距离，以及特征鲁棒性问题，目的是量化单个特征对对抗性摄动的鲁棒性。我们证明，在Lipschitz连续性的假设下，可以通过离散输入空间使用有限优化来近似这两个问题，并且近似具有可证明的保证，即误差是有界的。然后，我们证明了所产生的优化问题可以简化为两人回合制游戏的解决方案，其中第一位玩家选择功能，第二位玩家选择功能内的图像。尽管第二玩家的目标是最大程度地减少与对抗示例的距离，但根据优化目标，第一玩家可以合作或竞争。我们采用随时解决游戏问题的方法，通过单调提高游戏上限和下限来逼近游戏的价值。蒙特卡罗树搜索算法用于计算两个游戏的上限，以及可接纳的A^⁎和Alpha-Beta修剪算法分别用于计算最大安全半径和功能鲁棒性游戏的下限。当处理最大安全半径问题的上限时，我们的工具展示了与现有对抗性示例制作算法相比的竞争性能。此外，我们展示了如何在安全关键型应用（例如自动驾驶汽车中的交通标志识别）中评估神经网络的点状鲁棒性。