Loss functions play a central role in supervised classification. Cross-entropy (CE) is widely used, whereas the mean absolute error (MAE) loss can offer robustness but is difficult to optimize. Interpolating between the CE and MAE losses, generalized cross-entropy (GCE) has recently been introduced to provide a trade-off between optimization difficulty and robustness. Existing formulations of GCE result in a non-convex optimization over classification margins that is prone to underfitting, leading to poor performances with complex datasets. In this paper, we propose a minimax formulation of generalized cross-entropy (MGCE) that results in a convex optimization over classification margins. Moreover, we show that MGCEs can provide an upper bound on the classification error. The proposed bilevel convex optimization can be efficiently implemented using stochastic gradient computed via implicit differentiation. Using benchmark datasets, we show that MGCE achieves strong accuracy, faster convergence, and better calibration, especially in the presence of label noise.

翻译：损失函数在监督分类中扮演核心角色。交叉熵（CE）被广泛应用，而平均绝对误差（MAE）损失虽能提供鲁棒性，但优化困难。近期提出的广义交叉熵（GCE）通过在CE与MAE损失之间插值，实现了优化难度与鲁棒性之间的权衡。现有GCE公式会导致分类边界上的非凸优化，易引发欠拟合，在处理复杂数据集时表现不佳。本文提出最小最大广义交叉熵（MGCE）公式，其在分类边界上实现凸优化。此外，我们证明MGCE能提供分类误差的上界。所提出的双层凸优化可通过隐式微分计算的随机梯度高效实现。基于基准数据集的实验表明，MGCE在存在标签噪声的情况下，尤其能实现强准确性、更快的收敛速度以及更好的校准性能。