Cross-Entropy Loss

Overview of  Cross-Entropy Loss

Cross-Entropy Loss (Cross-Entropy Loss) is one of the common loss functions used in machine learning and deep learning to evaluate and optimize the performance of models in classification tasks, especially in binary classification (selecting one of two classes) and multi-class classification (selecting one of three or more It will be a widely used flow method for the problem of (selecting one of three or more classes).

Cross-entropy loss is a method that measures the difference between the actual output (predicted value) and the correct label (true value); a smaller loss indicates that the model’s prediction is accurate, while a larger loss indicates inaccuracy in the prediction.

Below are the equations for cross-entropy loss for the binary and multiclass classification cases.

1. for binary classification:

Cross-entropy loss = – (y * log(p) + (1 – y) * log(1 – p))

• • y is the correct answer label (0 or 1), either 0 or 1.
  • p is the predicted probability of the model and takes a value between 0 and 1.

2. for multiclass classification:

Cross-entropy loss = – Σ (y_i * log(p_i))

• • y_i is the probability distribution of the correct answer class, where the elements corresponding to the correct answer class are 1 and the rest are 0.
  • p_i is the predicted probability distribution of the model.

This loss function is used as the objective function to adjust the parameters of the model and is minimized using optimization algorithms such as gradient descent. If the model’s predictions are close to correct, the cross-entropy loss will be small; if the predictions are inaccurate, it will be large. Thus, the model is trained to minimize this loss during training, allowing for proper class classification.

Algorithm for Crossing Entropy Loss

How the general steps of cross-entropy loss are shown. The algorithm is nearly identical for both binary and multiclass classification, but there are minor differences in each case.

Cross-entropy loss calculation for the binary classification case:

1. model output: obtain the probability value predicted by the model. Usually, this value ranges from 0 to 1.

2. Correct labels: Obtain the correct label (0 or 1) for each data point.

3. Calculate cross-entropy loss: If the correct label is 0

• If the correct label is 0: Loss = -log(1 – Predicted probability)
• If the correct label is 1: Loss = -log(predicted probability)

4. It is common to calculate the loss for all data points and take the average. This will be the value of the overall loss function.

Cross-entropy loss calculation in the case of multi-class classification:

1. Model Output: Obtain the probability distribution for each class predicted by the model. This is usually passed through a softmax function as described in “Overview of softmax functions and related algorithms and implementation examples” to obtain the probability value.

2. Correct labels: Obtain the probability distribution of the correct class for each data point. The element corresponding to the correct class is 1, and the others are 0.

3. calculation of cross-entropy loss: For each data point, use the following equation

For each data point, calculate the loss using the following formula

• • Loss = -Σ(y_i * log(p_i))
  • y_i is the probability distribution of the correct answer class (1 for the correct answer class, 0 otherwise) and p_i is the predicted probability distribution of the model.

4. it will be common to calculate the loss for all data points and take the average. This will be the value of the entire loss function.

The cross-entropy loss is used as the objective function to adjust the parameters of the model and is minimized using an optimization algorithm (e.g., gradient descent method). If the model’s predictions are close to being correct, the loss will be small; if the predictions are inaccurate, the loss will be large.

Application Examples of Cross-entropy Losses

Cross-entropy loss is widely used primarily in classification tasks. The following are examples of applications of cross-entropy loss.

1. image classification:

• Handwriting recognition
• Object recognition in images
• Face recognition
• Image categorization (e.g., dog vs. cat)

2. natural language processing:

• Sentiment analysis of text documents (positive, negative, neutral, etc.)
• Spam email detection
• Text language classification (e.g., English, Spanish, French)
• Word and phrase meaning analysis

3. biology and bioinformatics:

• DNA sequence classification (genome classification)
• Secondary structure prediction of protein structures
• Gene expression classification

4. object detection:

In object detection tasks, calculate the probability distribution that an object belongs to a particular class and evaluate the loss using cross-entropy loss.

5. speech recognition:

A task that generates a transcript of an utterance (text) from speech data, using cross-entropy loss to evaluate the accuracy of the transcript.

6. ranking:

In ranking search engine results, cross-entropy loss may be minimized to predict user interest.

7. recommendation systems:

In recommendation systems, the difference between user preferences and item ratings is evaluated to generate appropriate recommendations.

Cross-entropy loss has been utilized as an important tool to evaluate model performance in classification tasks and to minimize loss during training.

Example implementation of cross-entropy loss

Implementation of cross-entropy loss can be done in many programming languages and deep learning frameworks. Below is a simple example implementation using Python and NumPy. The example covers both the binary classification case and the multi-class classification case.

Cross-entropy loss for binary classification:

import numpy as np

# Predictive probabilities and correct labels for the model
y_pred = np.array([0.8, 0.2])  # 0.8 is the predicted probability of class 1 and 0.2 is the predicted probability of class 0
y_true = np.array([1, 0])  # correct label

# Calculation of cross-entropy loss
loss = -np.sum(y_true * np.log(y_pred + 1e-15))  # 1e-15 is a minute value to prevent division by zero

print("cross-entropy loss:", loss)

Cross-entropy loss for multi-class classification:

import numpy as np

# Predicted probability distribution of the model and probability distribution of the correct label
y_pred = np.array([0.2, 0.7, 0.1])  # Class 0: 0.2, Class 1: 0.7, Class 2: 0.1
y_true = np.array([0, 1, 0])  # Correct answer is class 1

# Calculation of cross-entropy loss
loss = -np.sum(y_true * np.log(y_pred + 1e-15))

print("cross-entropy loss:", loss)

In the above example, y_pred is the predicted probability distribution of the model, and y_true is the probability distribution of the correct answer (elements corresponding to the correct answer class are 1, all others are 0). The cross-entropy loss is computed based on those probability distributions.

When using deep learning frameworks (e.g., TensorFlow, PyTorch, Keras), these frameworks usually provide built-in functions for computing cross-entropy loss. This allows for more efficient and stable computation.

The challenge of cross-entropy loss

Although cross-entropy loss is useful in many situations, there are some challenges and limitations. We discuss them below.

1. Class imbalance: In the presence of class imbalance, cross-entropy loss can make accurate evaluation difficult, especially in the case of multi-class classification. When there are many classes for a few classes, the model will typically be biased toward many classes, making it difficult to learn adequately for unbalanced classes.

2. numerical instability: When calculating cross-entropy loss, numerical instability can occur when the probability is zero, because the logarithmic term contains zeros. To prevent this, it is common to add a minute value (e.g., 1e-15) to the probability, but this can cause numerical problems.

3. overfitting and overtraining: In the process of minimizing cross-entropy loss, the model may overfit the training data. This can result in poor generalization performance to new data.

4. label imprecision: If the correct answer label is incorrect, the model will attempt to fit the correct label, which may affect model training. Methods to deal with label noise and annotation errors are needed.

5. unsupported tasks: cross-entropy loss is specific to the classification task and cannot be applied to other tasks such as regression tasks. Other loss functions (e.g., flat root squared error, hinge loss, etc.) are needed for other tasks.

These issues can be addressed by selecting the appropriate loss function for a particular problem and applying methods such as model hyperparameter adjustment, data preprocessing, and data balancing. It is also possible to customize the loss function and tailor it to the specific task.

Addressing the challenge of cross-entropy loss

To address the challenges associated with cross-entropy loss, we discuss them below.

1. how to address class imbalances:

• Weighting: Losses can be weighted for unbalanced classes. Assign a larger weight to the unbalanced class sample and a smaller weight to the balanced class sample.
• Undersampling or Oversampling: Increasing or decreasing the sample of unbalanced classes may be a way to rebalance the classes.
• Techniques such as simple resampling or SMOTE (Synthetic Minority Over-sampling Technique) may be considered.

2. ways to deal with numerical instability:

• Probability clipping: Numerical instability can be avoided by clipping probability values in the range of 0 to 1.
• Adding small values to the logarithmic term: Adding small positive values (e.g., 1e-15) within the logarithmic term can alleviate problems when the probability is zero.

3. how to deal with over-fitting and over-learning:

• Dropout and regularization: Techniques such as dropout and L1/L2 regularization can be used to prevent the model from overfitting the training data and improve generalization performance.

4. ways to deal with label imprecision:

• Semi-supervised learning: To address label imprecision, a semi-supervised learning approach can be used to train the model with unlabeled data. This will reduce the impact of inaccurate labels.

5. how to deal with tasks that do not correspond:

• Selecting an appropriate loss function: For tasks other than classification, an appropriate loss function (e.g., mean squared error, Huber loss, self-correcting loss function) should be selected to train the model.

Reference Information and Reference Books

For more information on optimization in machine learning, see also “Optimization for the First Time Reading Notes” “Sequential Optimization for Machine Learning” “Statistical Learning Theory” “Stochastic Optimization” etc.

Reference books include Optimization for Machine Learning

“Machine Learning, Optimization, and Data Science“

“Linear Algebra and Optimization for Machine Learning: A Textbook“