Scikit-learn中Hinge Loss与常规图表差异及组合适配原因

问题背景 #

我见过两张铰链损失的标准图表，但使用scikit-learn的hinge_loss()生成的图表趋势与它们完全相反，相关代码及对应数值如下：

真实标签y_true = 1时的代码与结果 #

from sklearn.metrics import hinge_loss
import matplotlib.pyplot as plt
import numpy as np

predicted = np.arange(-10, 11, 1)
y_true = [1] * len(predicted)
loss = [0] * len(predicted)
for i, (p, y) in enumerate(zip(predicted, y_true)):
  loss[i] = hinge_loss(np.array([y]), np.array([p]))
plt.plot(predicted, loss)

plt.axvline(x = 0, color = 'm', linestyle='dashed')
plt.axvline(x = -1, color = 'r', linestyle='dashed')
plt.axvline(x = 1, color = 'g', linestyle='dashed')

对应损失数值：

hinge_loss([1], [-5]) = 0.0,
hinge_loss([1], [-1]) = 0.0,
hinge_loss([1], [0])  = 1.0,
hinge_loss([1], [1])  = 2.0,
hinge_loss([1], [5])  = 6.0

真实标签y_true = -1时的代码与结果 #

predicted = np.arange(-10, 11, 1)
y_true = [-1] * len(predicted)
loss = [0] * len(predicted)
for i, (p, y) in enumerate(zip(predicted, y_true)):
  loss[i] = hinge_loss(np.array([y]), np.array([p]))
plt.plot(predicted, loss)

plt.axvline(x = 0, color = 'm', linestyle='dashed')
plt.axvline(x = -1, color = 'r', linestyle='dashed')
plt.axvline(x = 1, color = 'g', linestyle='dashed')

对应损失数值：

hinge_loss([-1], [-5]) = 0.0,
hinge_loss([-1], [-1]) = 0.0,
hinge_loss([-1], [0])  = 1.0,
hinge_loss([-1], [1])  = 2.0,
hinge_loss([-1], [5])  = 6.0

核心疑问1 #

为何scikit-learn的hinge_loss()计算结果与标准铰链损失图表的趋势相反？

补充：特殊值组合生成标准趋势图表 #

我发现利用hinge_loss([0], [-1])==0、hinge_loss([-2], [-1])==0的特性，无需翻转数值就能生成与标准图表一致的结果，代码如下：

对应y_true = 1的修正代码 #

predicted = np.arange(-10, 11, 1)
y_true = [1] * len(predicted)
loss = [0] * len(predicted)
for i, (p, y) in enumerate(zip(predicted, y_true)):
  loss[i] = hinge_loss(np.array([y, 0]), np.array([p, -1])) * 2
plt.plot(predicted, loss)

plt.axvline(x = 0, color = 'm', linestyle='dashed')
plt.axvline(x = -1, color = 'r', linestyle='dashed')
plt.axvline(x = 1, color = 'g', linestyle='dashed')

对应y_true = -1的修正代码 #

predicted = np.arange(-10, 11, 1)
y_true = [-1] * len(predicted)
loss = [0] * len(predicted)
for i, (p, y) in enumerate(zip(predicted, y_true)):
  loss[i] = hinge_loss(np.array([y,-2]), np.array([p,-1])) * 2
plt.plot(predicted, loss)

plt.axvline(x = 0, color = 'm', linestyle='dashed')
plt.axvline(x = -1, color = 'r', linestyle='dashed')
plt.axvline(x = 1, color = 'g', linestyle='dashed')

核心疑问2 #

这些特定值组合为何能有效生成符合预期的铰链损失图表？

内容的提问来源于stack exchange，提问作者Alberto