In this work, we introduce the {\em average top-$k$} (\atk) loss as a new
aggregate loss for supervised learning, which is the average over the $k$
largest individual losses over a training dataset. We show that the \atk loss
is a natural generalization of the two widely used aggregate losses, namely the
average loss and the maximum loss, but can combine their advantages and
mitigate their drawbacks to better adapt to different data distributions.
Furthermore, it remains a convex function over all individual losses, which can
lead to convex optimization problems that can be solved effectively with
conventional gradient-based methods. We provide an intuitive interpretation of
the \atk loss based on its equivalent effect on the continuous individual loss
functions, suggesting that it can reduce the penalty on correctly classified
data. We further give a learning theory analysis of \matk learning on the
classification calibration of the \atk loss and the error bounds of \atk-SVM.
We demonstrate the applicability of minimum average top-$k$ learning for binary
classification and regression using synthetic and real datasets.

介绍了一种新的监督学习损失函数 —— 平均 top-k 损失，是平均训练集个别损失的 k 个最大值，能够克服平均损失和最大损失的缺陷，在不同数据分布上更好地适应。同时它是所有个别损失的凸函数，可以使用常规的梯度下降法解决凸优化问题，具有显著的理论和实际应用价值。