We present a detailed study of top-$k$ classification, the task of predicting the $k$ most probable classes for an input, extending beyond single-class prediction. We demonstrate that several prevalent surrogate loss functions in multi-class classification, such as comp-sum and constrained losses, are supported by $H$-consistency bounds with respect to the top-$k$ loss. These bounds guarantee consistency in relation to the hypothesis set $H$, providing stronger guarantees than Bayes-consistency due to their non-asymptotic and hypothesis-set specific nature. To address the trade-off between accuracy and cardinality $k$, we further introduce cardinality-aware loss functions through instance-dependent cost-sensitive learning. For these functions, we derive cost-sensitive comp-sum and constrained surrogate losses, establishing their $H$-consistency bounds and Bayes-consistency. Minimizing these losses leads to new cardinality-aware algorithms for top-$k$ classification. We report the results of extensive experiments on CIFAR-100, ImageNet, CIFAR-10, and SVHN datasets demonstrating the effectiveness and benefit of these algorithms.

我们研究了top-k分类的细节，该任务是预测输入的$k$个可能的类别，超越单一分类预测。我们证明了多类别分类中的几种普遍代理损失函数，如comp-sum和约束损失，具有相对于top-k损失的H一致性界限。这些界限保证了与假设集$H$相关的一致性，提供比贝叶斯一致性更强的保证，因为它们是非渐近和假设集特定的。为了解决准确性和基数$k$之间的权衡，我们进一步通过实例依赖的成本敏感学习引入了基数感知的损失函数。对于这些函数，我们得到了成本敏感的comp-sum和约束代理损失，建立了它们的H一致性界限和贝叶斯一致性。最小化这些损失会导致新的基数感知算法应用于top-k分类。我们报道了在CIFAR-100、ImageNet、CIFAR-10和SVHN数据集上的大量实验证明了这些算法的有效性和好处。

Top-k 分类与基数感知预测