Chi-square Loss for Softmax: an Echo of Neural Network Structure

TL;DR

This paper introduces a chi-square based loss function for Softmax classification, which is statistically grounded, sensitive to neural network structure, and affected by label smoothing, offering new insights into model training dynamics.

Contribution

It proposes a novel chi-square loss for Softmax, analyzes its statistical properties, and explores its relationship with neural network structure and label smoothing effects.

Findings

Chi-square loss is unbiased in optimization.

Distribution of chi-square loss reflects neural network structure.

Performance degrades with many classes due to its strictness.

Abstract

Softmax working with cross-entropy is widely used in classification, which evaluates the similarity between two discrete distribution columns (predictions and true labels). Inspired by chi-square test, we designed a new loss function called chi-square loss, which is also works for Softmax. Chi-square loss has a statistical background. We proved that it is unbiased in optimization, and clarified its using conditions (its formula determines that it must work with label smoothing). In addition, we studied the sample distribution of this loss function by visualization and found that the distribution is related to the neural network structure, which is distinct compared to cross-entropy. In the past, the influence of structure was often ignored when visualizing. Chi-square loss can notice changes in neural network structure because it is very strict, and we explained the reason for this strictness. We also studied the influence of label smoothing and discussed the relationship between label smoothing and training accuracy and stability. Since the chi-square loss is very strict, the performance will degrade when dealing samples of very many classes.