  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.
