Abstract: Existing score-based adversarial attacks mainly focus on crafting $top$-1 adversarial examples against classifiers with single-label classification. Their attack success rate and query efficiency are often less than satisfactory, particularly under small perturbation requirements; moreover, the vulnerability of classifiers with multi-label learning is yet to be studied. In this paper, we propose a comprehensive surrogate free score-based attack, named \b geometric \b score-based \b black-box \b attack (GSBA$^K$), to craft adversarial examples in an aggressive $top$-$K$ setting for both untargeted and targeted attacks, where the goal is to change the $top$-$K$ predictions of the target classifier. We introduce novel gradient-based methods to find a good initial boundary point to attack. Our iterative method employs novel gradient estimation techniques, particularly effective in $top$-$K$ setting, on the decision boundary to effectively exploit the geometry of the decision boundary. Additionally, GSBA$^K$ can be used to attack against classifiers with $top$-$K$ multi-label learning. Extensive experiential results on ImageNet and PASCAL VOC datasets validate the effectiveness of GSBA$^K$ in crafting $top$-$K$ adversarial examples.