  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.
