Learning with Average Top-k Loss

TL;DR

The paper introduces the average top-$k$ loss, a new aggregate loss function for supervised learning that balances the benefits of average and maximum losses, with theoretical analysis and practical applications in classification and regression.

Contribution

It proposes the average top-$k$ loss as a convex, flexible aggregate loss that generalizes existing losses and offers effective optimization and theoretical insights.

Findings

The exttt{atk} loss is convex and can be optimized with gradient methods.

It effectively balances penalties on different data points, reducing over-penalization.

Experimental results demonstrate its applicability to classification and regression tasks.

Abstract

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.