It is well established that to ensure or certify the robustness of a neural network, its Lipschitz constant plays a prominent role. However, its calculation is NP-hard. In this note, by taking into account activation regions at each layer as new constraints, we propose new quadratically constrained MIP formulations for the neural network Lipschitz estimation problem. The solutions of these problems give lower bounds and upper bounds of the Lipschitz constant and we detail conditions when they coincide with the exact Lipschitz constant.

翻译：众所周知，为确保或验证神经网络的鲁棒性，其Lipschitz常数起着关键作用。然而，其计算是NP难的。本文通过将每层的激活区域作为新约束纳入考虑，针对神经网络Lipschitz估计问题提出了新的二次约束混合整数规划形式。这些问题的解给出了Lipschitz常数的下界和上界，并详细刻画了它们与精确Lipschitz常数相等时的条件。