  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.
