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。