Deep learning models have been the subject of study from various perspectives, for example, their training process, interpretation, generalization error, robustness to adversarial attacks, etc. A trained model is defined by its decision boundaries, and therefore, many of the studies about deep learning models speculate about the decision boundaries, and sometimes make simplifying assumptions about them. So far, finding exact points on the decision boundaries of trained deep models has been considered an intractable problem. Here, we compute exact points on the decision boundaries of these models and provide mathematical tools to investigate the surfaces that define the decision boundaries. Through numerical results, we confirm that some of the speculations about the decision boundaries are accurate, some of the computational methods can be improved, and some of the simplifying assumptions may be unreliable, for models with nonlinear activation functions. We advocate for verification of simplifying assumptions and approximation methods, wherever they are used. Finally, we demonstrate that the computational practices used for finding adversarial examples can be improved and computing the closest point on the decision boundary reveals the weakest vulnerability of a model against adversarial attack.

通过计算决策边界上的精确点，并提供数学工具来研究定义决策边界的表面，本文旨在确认一些对决策边界的猜测是准确的，某些计算方法可以得到改进，而一些简化假设可能是不可靠的，特别是涉及到非线性激活函数的模型。最后，本文提出了计算实践的改进方法，并揭示了计算决策边界上最接近点的弱点，以此来提高模型对抗攻击的鲁棒性。

探究已训练神经网络的决策边界