  Deep convolutional neural networks (CNNs) trained with logistic and softmax
losses have made significant advancement in visual recognition tasks in
computer vision. When training data exhibit class imbalances, the class-wise
reweighted version of logistic and softmax losses are often used to boost
performance of the unweighted version. In this paper, motivated to explain the
reweighting mechanism, we explicate the learning property of those two loss
functions by analyzing the necessary condition (e.g., gradient equals to zero)
after training CNNs to converge to a local minimum. The analysis immediately
provides us explanations for understanding (1) quantitative effects of the
class-wise reweighting mechanism: deterministic effectiveness for binary
classification using logistic loss yet indeterministic for multi-class
classification using softmax loss; (2) disadvantage of logistic loss for
single-label multi-class classification via one-vs.-all approach, which is due
to the averaging effect on predicted probabilities for the negative class
(e.g., non-target classes) in the learning process. With the disadvantage and
advantage of logistic loss disentangled, we thereafter propose a novel
reweighted logistic loss for multi-class classification. Our simple yet
effective formulation improves ordinary logistic loss by focusing on learning
hard non-target classes (target vs. non-target class in one-vs.-all) and turned
out to be competitive with softmax loss. We evaluate our method on several
benchmark datasets to demonstrate its effectiveness.
