Overview of cross-entropy
Cross Entropy is a concept commonly used in information theory and machine learning to quantify the difference between model predictions and actual data, especially in classification problems.
Cross-entropy is derived from information theory, which uses the concept of “entropy” as a measure of the amount of information. Entropy is a measure of the uncertainty or difficulty of predicting information. It is greatest when the probability distributions are equal and decreases as the probability concentrates on a particular value.
The cross-entropy for two probability distributions \( P \) and \( Q \) is defined by the following formula
\[ H(P, Q) = -\sum_{x} P(x) \log(Q(x)) \]
where
– \( x \) : the event being considered (e.g. class)
– \( P(x) \) : Probability of \( x \) in the true distribution (correct answer label)
– \( Q(x) \) : Probability of \( x \) in the model prediction
The meaning of this formula is that the probability \( P(x)\) of a possible event \( x)\) in the true distribution \( P(x)\) is “weighted” by the probability \( Q(x)\) that the model predicts as \( x)\), and the sum is an indicator of how close the model prediction \( Q)\) is to the true distribution \( P \).
In machine learning classification problems, cross-entropy is typically used as a measure of the “distance” between the probability distribution output by the model and the distribution of true labels. In other words, it is used to evaluate how close the model outputs a probability distribution to the true distribution.
During training, the parameters of the model are adjusted to minimize cross-entropy. This is so that the model learns more accurate probability distributions for the training data and can make better predictions for unknown data.
Algorithms related to cross entropy
Cross-entropy is used to measure the difference between the model’s predictions and the true label in a classification problem, and it is an indicator that allows the model to improve its performance by learning to minimize cross-entropy.
There are two main algorithms for minimizing cross-entropy:
1. Gradient Descent: Gradient Descent is an optimization technique for minimizing a loss function. Since cross-entropy is generally used as a loss function in models, it is common to minimize cross-entropy using Gradient Descent. Specific gradient descent methods are described below.
Batch Gradient Descent: Calculates the gradient using all training data and updates all parameters at once. Not suitable for large data sets, but can be effective with small amounts of data.
Stochastic Gradient Descent (SGD): Calculates the gradient and updates the parameters for each training sample. Suitable for large data sets and online training.
Mini-batch Gradient Descent (Mini-Batch Gradient Descent): An intermediate method between Batch Gradient Descent and Stochastic Gradient Descent, where the gradients are computed in small randomly selected batches and the parameters are updated. It is efficient and can capture the characteristics of the entire data set.
2. Adam Optimizer: Adam (Adaptive Moment Estimation) is a type of gradient descent method that automatically adjusts the learning rate to achieve efficient learning. Typically, the Adam Optimizer is used to train neural networks that minimize cross-entropy.
The Adam Optimizer maintains an exponential moving average of the past gradient and an exponential moving average of the square of the past gradient. This allows the learning rate to be adaptively adjusted to each parameter for efficient learning.
Application of cross-entropy
The following are specific examples where cross entropy is applied.
1. image classification: In image classification, the cross-entropy between the output of a neural network for an image and the correct answer label is used as a loss function. The neural network outputs a probability distribution for each class for the input image and calculates the cross-entropy based on that distribution and the correct label, thereby learning the network to predict the correct class with higher probability.
2. natural language processing (NLP): In the field of natural language processing, cross entropy is used in language models and machine translation models. For example, language models are trained to predict the next word, where the cross-entropy between the actual next word and the model’s prediction is used as a loss function.
In machine translation, the cross-entropy between the translated sentence and the correct translation is also used as a loss function to train the model to produce a more accurate translation.
3. object detection: Object detection uses a model that simultaneously predicts the position and class of objects in the image. In this case, the cross-entropy between the prediction of the region containing the object (bounding box) and the prediction of its class is used as the loss function.
4. Reinforcement learning: In reinforcement learning, cross entropy is also used when the agent learns through interaction with its environment. In particular, a method called Policy Gradient Methods models the probability distribution of actions to be taken by the agent and updates the strategy by maximizing (or minimizing) the cross-entropy between the distribution and the actual action.
Example of cross-entropy implementation
Below we describe an example implementation of cross-entropy using Python and NumPy. If you use machine learning libraries (e.g., TensorFlow or PyTorch), you can also use the functions provided by these libraries.
1. Example of calculating cross-entropy from two probability distributions: The following is an example of calculating cross-entropy from two probability distributions, P and Q.
import numpy as np
def cross_entropy(p, q):
"""
p: Probability of true distribution (numpy array)
q: Probability of model prediction (numpy array)
"""
return -np.sum(p * np.log(q))
# Examples of true distributions and model predictions
p = np.array([0.2, 0.3, 0.5]) # True distribution
q = np.array([0.3, 0.3, 0.4]) # Model Predictions
# Calculate cross entropy
ce = cross_entropy(p, q)
print("cross entropy:", ce)2. Example of calculating the cross entropy for a 2-class classification: The following is an example of calculating the cross entropy for a 2-class classification, where the classes belong to 0 or 1.
import numpy as np
def binary_cross_entropy(y_true, y_pred):
"""
y_true: True label (0 or 1)
y_pred: Prediction probability of the model (0 to 1 value)
"""
epsilon = 1e-10 # Minute value to prevent zero division
y_pred = np.clip(y_pred, epsilon, 1.0 - epsilon) # Clipping not to 0 or 1
return -np.mean(y_true * np.log(y_pred) + (1 - y_true) * np.log(1 - y_pred))
# Examples of true labels and model predictions
y_true = np.array([1, 0, 1, 1]) # True Labels
y_pred = np.array([0.9, 0.1, 0.8, 0.7]) # Model Predictions
# Calculate cross entropy
ce = binary_cross_entropy(y_true, y_pred)
print("cross entropy:", ce)3. example using TensorFlow (2-class classification): An example of calculating the cross-entropy of a 2-class classification using TensorFlow is shown below.
import tensorflow as tf
# Examples of true labels and model predictions
y_true = tf.constant([1, 0, 1, 1], dtype=tf.float32)
y_pred = tf.constant([0.9, 0.1, 0.8, 0.7], dtype=tf.float32)
# Calculate cross entropy
ce = tf.reduce_mean(tf.nn.sigmoid_cross_entropy_with_logits(labels=y_true, logits=y_pred))
with tf.Session() as sess:
ce_result = sess.run(ce)
print("cross entropy:", ce_result)In each example, the cross entropy is calculated given the true distribution or true label and the predicted probabilities output by the model.
Cross-entropy issues and measures to deal with them
The main issues of cross-entropy and measures to deal with them are described below.
1. class imbalance problem:
Challenge: When classes are unbalanced, using cross entropy as it is may result in learning a model that is biased toward the majority class. This leads to the problem that the minority class becomes less important and the model cannot be trained well.
Solution:
Class weighting: By adjusting the weight of each class in the cross-entropy calculation, the impact on unbalanced classes can be increased or decreased. It is common to increase weights for unbalanced classes and decrease weights for balanced classes.
Oversampling/Undersampling: It is possible to balance classes by increasing (oversampling) or decreasing (undersampling) the number of samples for unbalanced classes.
2. numerical stability issue:
Challenge: During the calculation of cross-entropy, division by zero and logarithmic out-of-bounds errors can occur. This is especially problematic when the probability is close to zero.
Solution:
Clipping: By keeping the probability values within a certain range, instability during the calculation can be reduced. For example, one can clip the probability value between 0 and 1.
Smoothing: When the true label is extremely close to 0 or 1, instead of setting it to 0 or 1 completely, it can be set to a slightly smaller value (e.g., 0.1 or 0.9) to avoid logarithmic out-of-bounds errors.
3. over-learning problem:
Challenge: When using cross-entropy, the model may overfit (overlearn) the training data.
Solution:
Regularization: Use L1 regularization, L2 regularization, etc. to constrain the weights of the model so that they do not become too large.
Dropout: Add a dropout layer and randomly disable some units during training to prevent over-training.
Data expansion: artificially expand the training data to increase the variation in the training data and mitigate over-learning.
4. label uncertainty problem:
Challenge: If labels originally have uncertainty, cross entropy ignores this uncertainty.
Solution:
Soft labels: Instead of hard labels (0 or 1), we may use soft labels that represent the level of confidence in the label. This allows the model to account for label uncertainty.
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“
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