Convex Loss Functions Comparison

Introduction

In machine learning, loss functions are used to measure the difference between the predicted output and the actual output. The scikit-learn library provides various convex loss functions for classification problems. In this lab, we will visualize and compare some of these loss functions.

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Skills Graph

Import Libraries and Define Functions

We start by importing the necessary libraries and defining the modified Huber loss function.

import numpy as np
import matplotlib.pyplot as plt

def modified_huber_loss(y_true, y_pred):
    z = y_pred * y_true
    loss = -4 * z
    loss[z >= -1] = (1 - z[z >= -1]) ** 2
    loss[z >= 1.0] = 0
    return loss

Define the Range for the Decision Function

We define the range of values for the decision function f(x).

xmin, xmax = -4, 4
xx = np.linspace(xmin, xmax, 100)

Plot the Loss Functions

We plot the various convex loss functions supported by scikit-learn using the matplotlib library.

lw = 2
plt.plot([xmin, 0, 0, xmax], [1, 1, 0, 0], color="gold", lw=lw, label="Zero-one loss")
plt.plot(xx, np.where(xx < 1, 1 - xx, 0), color="teal", lw=lw, label="Hinge loss")
plt.plot(xx, -np.minimum(xx, 0), color="yellowgreen", lw=lw, label="Perceptron loss")
plt.plot(xx, np.log2(1 + np.exp(-xx)), color="cornflowerblue", lw=lw, label="Log loss")
plt.plot(xx, np.where(xx < 1, 1 - xx, 0) ** 2, color="orange", lw=lw, label="Squared hinge loss")
plt.plot(xx, modified_huber_loss(xx, 1), color="darkorchid", lw=lw, linestyle="--", label="Modified Huber loss")
plt.ylim((0, 8))
plt.legend(loc="upper right")
plt.xlabel(r"Decision function $f(x)$")
plt.ylabel("$L(y=1, f(x))$")
plt.show()

Interpret the Plot

We interpret the plot and analyze the behavior of each loss function.

Summary

In this lab, we visualized and compared some of the convex loss functions supported by scikit-learn. Understanding loss functions is crucial in machine learning, as they are used to optimize the model parameters during training.