Symmetric Property of Loss Functions

This example shows the particular form of symmetric losses to learn robust classifiers from noisy labels, inspired from : “On symmetric losses for learning from corrupted labels.” by Charoenphakdee and al.

import matplotlib.pyplot as plt
import numpy as np


def squared_loss(y_true, y_pred):
    z = y_true * y_pred
    return (1 - z) ** 2, -2 * y_true * (1 - z)


def hinge_loss(y_true, y_pred):
    z = y_true * y_pred
    return np.maximum(0, 1 - z), np.where(1 - z > 0, -y_true, 0)


def unhinged_loss(y_true, y_pred):
    z = y_true * y_pred
    return 1 - z, -y_true


z = np.linspace(-5, 5, 100)

losses = [
    ("Unhinged", unhinged_loss),
    ("Hinge", hinge_loss),
    ("Squared", squared_loss),
]

plt.figure(figsize=(14, 6))

plt.subplot(121)
plt.axvline(0, color="black", linestyle="--")
for i, (name, this_loss) in enumerate(losses):
    plt.plot(z, this_loss(np.ones_like(z), z)[0], label=f"{name} Loss")

plt.xlabel("Margin (z)")
plt.ylabel("Loss Value l(z)")

plt.xticks([-1, 0, 1])

plt.ylim([-2, 5])
plt.yticks([-1, 0, 1])

plt.legend(loc=3, prop={"size": 12})
plt.title("Loss value over the classification margin")

plt.subplot(122)
plt.axvline(0, color="black", linestyle="--")
for i, (name, this_loss) in enumerate(losses):
    plt.plot(
        z,
        this_loss(-np.ones_like(z), z)[0] + this_loss(np.ones_like(z), z)[0],
        label=f"{name} Loss",
    )

plt.xlabel("Margin (z)")
plt.ylabel("Symmetric value l(z)+l(-z)")

plt.xticks([-1, 0, 1])

plt.ylim([0, 5])
plt.yticks([1])

plt.legend(loc=3, prop={"size": 12})
plt.title("Illustration of the symmetric property")

plt.show()