The top-$k$ error is often employed to evaluate performance for challenging
classification tasks in computer vision as it is designed to compensate for
ambiguity in ground truth labels. This practical success motivates our
theoretical analysis of consistent top-$k$ classification. Surprisingly, it is
not rigorously understood when taking the $k$-argmax of a vector is guaranteed
to return the $k$-argmax of another vector, though doing so is crucial to
describe Bayes optimality; we do both tasks. Then, we define top-$k$
calibration and show it is necessary and sufficient for consistency. Based on
the top-$k$ calibration analysis, we propose a class of top-$k$ calibrated
Bregman divergence surrogates. Our analysis continues by showing previously
proposed hinge-like top-$k$ surrogate losses are not top-$k$ calibrated and
suggests no convex hinge loss is top-$k$ calibrated. On the other hand, we
propose a new hinge loss which is consistent. We explore further, showing our
hinge loss remains consistent under a restriction to linear functions, while
cross entropy does not. Finally, we exhibit a differentiable, convex loss
function which is top-$k$ calibrated for specific $k$.

本论文针对计算机视觉中具有挑战性的分类任务，探究了 top-k 误差的一致性分类及校准代理损失的性能，提出了一种具有一致性的新型铰链损失，同时还发现了符号函数作为代理损失函数的限制。