Abstract: The adversarial vulnerability of Deep Neural Networks (DNNs) has been well-known and widely concerned, often under the context of learning top-$1$ attacks (e.g., fooling a DNN to classify a cat image as dog). This paper shows that the concern is much more serious by learning significantly more aggressive ordered top-$K$ clear-box targeted attacks proposed in~\citep{zhang2020learning}. We propose a novel and rigorous quadratic programming (QP) method of learning ordered top-$K$ attacks with low computing cost, dubbed as \textbf{QuadAttac$K$}. Our QuadAttac$K$ directly solves the QP to satisfy the attack constraint in the feature embedding space (i.e., the input space to the final linear classifier), which thus exploits the semantics of the feature embedding space (i.e., the principle of class coherence). With the optimized feature embedding vector perturbation, it then computes the adversarial perturbation in the data space via the vanilla one-step back-propagation. In experiments, the proposed QuadAttac$K$ is tested in the ImageNet-1k classification using ResNet-50, DenseNet-121, and Vision Transformers (ViT-B and DEiT-S). It successfully pushes the boundary of successful ordered top-$K$ attacks from $K=10$ up to $K=20$ at a cheap budget ($1\times 60$) and further improves attack success rates for $K=5$ for all tested models, while retaining the performance for $K=1$.