  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.
