Loss of cross entropy in the evaluation of the language model

Summary

• Cross entropy measures, in bits, how surprising is the real token under the predicted distribution of its model.
• Cross entropy is not only the objective that actively optimizes during pre -registration and fine adjustment, but also passively used as an evaluation metric.
• While it is soft, differentiable and computationally efficient, which makes it ideal for gradient -based optimization, it can be numerically unstable.

The loss of cross entropy stands as one of the cornerstone metrics in the evaluation of language models, which serves as a training goal and an evaluation metric. In this complete guide, we will explore what is the loss of cross entropy, how it works specifically in the context of large language models (LLM) and why it is important to understand the performance of the model.

Whether he is an automatic learning professional, an researcher or someone who seeks to understand how modern ai systems are trained and evaluated, this article will provide a deep understanding of the loss of cross entropy and its importance in the world of language modeling.

What is the loss of cross entropy?

The loss of cross entropy measures the performance of a classification model whose exit is a probability distribution. In the context of language models, quantify the difference between the predicted probability distribution of the following token and real distribution (usually a unique coded vector that represents the true token of the next).

Key characteristics of the loss of cross entropy

• Information Theory Foundation: Rooted in information theory, cross entropy measures how many information bits are needed to identify events of a probability distribution (true distribution) if an optimized coding scheme is used for another distribution (the predicted).
• Probabilistic output: It works with models that produce probability distributions instead of deterministic exits.
• Asymmetric: Unlike other distance metrics, cross entropy is not symmetrical: the order of true and predicted distributions is important.
• Differentiable: Critic for gradient -based optimization methods used in neural networks.
• Trusted: Strongly penalizes safe but incorrect predictions, encouraging models to be uncertain when appropriate.

Also read: How to evaluate a large language model (LLM)?

Entropy and binary cross formula

For binary classification tasks (as simple but questions or analysis of feelings), binary cross entropy is used:

Where:

• AND_I It is the true label (0 or 1)
• AND^I It is the probability predicted
• N is the number of samples

Binary cross entropy is also known as Loss of registrationparticularly in automatic learning competitions.

Cross entropy as a loss function

During training, cross entropy serves as the objective function that the model tries to minimize. When comparing the probability distribution of the model with the truth of the soil, the training algorithm adjusts the parameters of the model to reduce the discrepancy between predictions and reality.

The role of cross entropy in LLMS

In large language models, the loss of cross entropy plays several crucial roles:

1. Training objective: The main objective during training and fine adjustment is to minimize loss.
2. Metric evaluation: It is used to evaluate model performance in retained data.
3. Perplexity calculation: Perplexity, another common evaluation metric, is derived from cross entropy: perplexity = 2^{crosseneTropy}.
4. Comparison of models: Different models can be compared based on your loss in the same data set.
5. Transfer Learning Evaluation: This can indicate how well a model transfers knowledge of training prior to downstream tasks.

How does it work?

For language models, the loss of cross entropy works as follows:

1. The model predicts a probability distribution throughout the vocabulary for the next Token.
2. This distribution is compared to the true distribution (usually a unique vector where the next real token has probability 1).
3. The negative probability of real token registration under the distribution of the model is calculated.
4. This value is averaged on all tokens in the sequence or data set.

Formulas and explanation

The general formula for the loss of cross entropy in language modeling is:

Where:

• N is the number of tokens in the sequence
• V is the size of the vocabulary
• AND_YoJ is 1 if the token j is the following correct token in position I, otherwise 0
• AND^YoJ is the probability of Token J in position I

Since we are usually dealing with a truth of the one -state soil, this is simplified for:

Where t_YoIt is the true token index in position i.

Implementation of loss of cross entropy in the Pytorch and Tensorflow Code

# PyTorch Implementation

import torch

import torch.nn as nn

import torch.nn.functional as F

import numpy as np

import matplotlib.pyplot as plt

# Simple Language Model in PyTorch

class SimpleLanguageModel(nn.Module):

    def __init__(self, vocab_size, embedding_dim, hidden_dim):

        super(SimpleLanguageModel, self).__init__()

        self.embedding = nn.Embedding(vocab_size, embedding_dim)

        self.lstm = nn.LSTM(embedding_dim, hidden_dim, batch_first=True)

        self.fc = nn.Linear(hidden_dim, vocab_size)

    def forward(self, x):

        # x shape: (batch_size, sequence_length)

        embedded = self.embedding(x)  # (batch_size, sequence_length, embedding_dim)

        lstm_out, _ = self.lstm(embedded)  # (batch_size, sequence_length, hidden_dim)

        logits = self.fc(lstm_out)  # (batch_size, sequence_length, vocab_size)

        return logits

# Manual Cross Entropy Loss calculation

def manual_cross_entropy_loss(logits, targets):

    """

    Computes cross entropy loss manually

    Args:

        logits: Raw model outputs (batch_size, sequence_length, vocab_size)

        targets: True token indices (batch_size, sequence_length)

    """

    batch_size, seq_len, vocab_size = logits.shape

    # Reshape for easier processing

    logits = logits.reshape(-1, vocab_size)  # (batch_size*sequence_length, vocab_size)

    targets = targets.reshape(-1)  # (batch_size*sequence_length)

    # Convert logits to probabilities using softmax

    probs = F.softmax(logits, dim=1)

    # Get probability of the correct token for each position

    correct_token_probs = probs(range(len(targets)), targets)

    # Compute negative log likelihood

    nll = -torch.log(correct_token_probs + 1e-10)  # Add small epsilon to prevent log(0)

    # Average over all tokens

    loss = torch.mean(nll)

    return loss

# Example usage

def pytorch_example():

    # Parameters

    vocab_size = 10000

    embedding_dim = 128

    hidden_dim = 256

    batch_size = 32

    seq_length = 50

    # Sample data

    inputs = torch.randint(0, vocab_size, (batch_size, seq_length))

    targets = torch.randint(0, vocab_size, (batch_size, seq_length))

    # Create model

    model = SimpleLanguageModel(vocab_size, embedding_dim, hidden_dim)

    # Get model outputs

    logits = model(inputs)

    # PyTorch's built-in loss function

    criterion = nn.CrossEntropyLoss()

    # For CrossEntropyLoss, we need to reshape

    pytorch_loss = criterion(logits.view(-1, vocab_size), targets.view(-1))

    # Our manual implementation

    manual_loss = manual_cross_entropy_loss(logits, targets)

    print(f"PyTorch CrossEntropyLoss: {pytorch_loss.item():.4f}")

    print(f"Manual CrossEntropyLoss: {manual_loss.item():.4f}")

    return model, logits, targets

# TensorFlow Implementation

def tensorflow_implementation():

    import tensorflow as tf

    # Parameters

    vocab_size = 10000

    embedding_dim = 128

    hidden_dim = 256

    batch_size = 32

    seq_length = 50

    # Simple Language Model in TensorFlow

    class TFSimpleLanguageModel(tf.keras.Model):

        def __init__(self, vocab_size, embedding_dim, hidden_dim):

            super(TFSimpleLanguageModel, self).__init__()

            self.embedding = tf.keras.layers.Embedding(vocab_size, embedding_dim)

            self.lstm = tf.keras.layers.LSTM(hidden_dim, return_sequences=True)

            self.fc = tf.keras.layers.Dense(vocab_size)

        def call(self, x):

            embedded = self.embedding(x)

            lstm_out = self.lstm(embedded)

            return self.fc(lstm_out)

    # Create model

    tf_model = TFSimpleLanguageModel(vocab_size, embedding_dim, hidden_dim)

    # Sample data

    tf_inputs = tf.random.uniform((batch_size, seq_length), minval=0, maxval=vocab_size, dtype=tf.int32)

    tf_targets = tf.random.uniform((batch_size, seq_length), minval=0, maxval=vocab_size, dtype=tf.int32)

    # Get model outputs

    tf_logits = tf_model(tf_inputs)

    # TensorFlow's built-in loss function

    tf_loss_fn = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)

    tf_loss = tf_loss_fn(tf_targets, tf_logits)

    # Manual cross entropy calculation in TensorFlow

    def tf_manual_cross_entropy(logits, targets):

        batch_size, seq_len, vocab_size = logits.shape

        # Reshape

        logits_flat = tf.reshape(logits, (-1, vocab_size))

        targets_flat = tf.reshape(targets, (-1))

        # Convert to probabilities

        probs = tf.nn.softmax(logits_flat, axis=1)

        # Get correct token probabilities

        indices = tf.stack((tf.range(tf.shape(targets_flat)(0), dtype=tf.int32), tf.cast(targets_flat, tf.int32)), axis=1)

        correct_probs = tf.gather_nd(probs, indices)

        # Compute loss

        loss = -tf.reduce_mean(tf.math.log(correct_probs + 1e-10))

        return loss

    manual_tf_loss = tf_manual_cross_entropy(tf_logits, tf_targets)

    print(f"TensorFlow CrossEntropyLoss: {tf_loss.numpy():.4f}")

    print(f"Manual TF CrossEntropyLoss: {manual_tf_loss.numpy():.4f}")

    return tf_model, tf_logits, tf_targets

# Visualizing Cross Entropy

def visualize_cross_entropy():

    # True label is 1 (one-hot encoding would be (0, 1))

    true_label = 1

    # Range of predicted probabilities for class 1

    predicted_probs = np.linspace(0.01, 0.99, 100)

    # Calculate cross entropy loss for each predicted probability

    cross_entropy = (-np.log(p) if true_label == 1 else -np.log(1-p) for p in predicted_probs)

    # Plot

    plt.figure(figsize=(10, 6))

    plt.plot(predicted_probs, cross_entropy)

    plt.title('Cross Entropy Loss vs. Predicted Probability (True Class = 1)')

    plt.xlabel('Predicted Probability for Class 1')

    plt.ylabel('Cross Entropy Loss')

    plt.grid(True)

    plt.axvline(x=1.0, color="r", linestyle="--", alpha=0.5, label="True Probability = 1.0")

    plt.legend()

    plt.show()

    # Visualize loss landscape for binary classification

    probs_0 = np.linspace(0.01, 0.99, 100)

    probs_1 = 1 - probs_0

    # Calculate loss for true label = 0

    loss_true_0 = (-np.log(1-p) for p in probs_0)

    # Calculate loss for true label = 1

    loss_true_1 = (-np.log(p) for p in probs_0)

    plt.figure(figsize=(10, 6))

    plt.plot(probs_0, loss_true_0, label="True Label = 0")

    plt.plot(probs_0, loss_true_1, label="True Label = 1")

    plt.title('Cross Entropy Loss for Different True Labels')

    plt.xlabel('Predicted Probability for Class 1')

    plt.ylabel('Cross Entropy Loss')

    plt.legend()

    plt.grid(True)

    plt.show()

# Run examples

if __name__ == "__main__":

    print("PyTorch Example:")

    pt_model, pt_logits, pt_targets = pytorch_example()

    print("\nTensorFlow Example:")

    try:

        tf_model, tf_logits, tf_targets = tensorflow_implementation()

    except ImportError:

        print("TensorFlow not installed. Skipping TensorFlow example.")

    print("\nVisualizing Cross Entropy:")

    visualize_cross_entropy()

Code analysis:

I have implemented the loss of cross entropy in both Pytorch and Tensorflow, showing both incorporated functions and manual implementations. Let's walk through the key components:

1. Simplelanguagemodel: A basic LSTM -based language model that predicts probabilities for the next token.
2. Crucade entropy manual implementation: Show how cross entropy is calculated from the first principles:
   • Convert logits to probabilities using softmax
   • Remove the probability of the correct token
   • Take the negative record of these probabilities
   • Average in all chips
3. Visualizations: The code includes visualizations that show how changes of loss with different predicted probabilities.

Production:

PyTorch Example:
PyTorch CrossEntropyLoss: 9.2140
Manual CrossEntropyLoss: 9.2140
TensorFlow Example:
TensorFlow CrossEntropyLoss: 9.2103
Manual TF CrossEntropyLoss: 9.2103

Visualizations illustrate how loss increases dramatically as predictions diverge from true labels, especially when the model is confidantly incorrect.

Advantages and limitations

Practical applications

Cross entropy loss is widely used in language models applications:

1. Fund Foundation models: The loss of cross entropy is the standard objective function for the training of large language models in mass text corpus.
2. Fine tuning: By adapting previously trained models to specific tasks, cross entropy remains the loss of loss of loss.
3. Sequence generation: Even when generating text, the loss during training influences the quality of the exits of the model.
4. Selection of models: Comparing different models of models or hyperparameter configurations, the loss in validation data is a key metric.
5. Domain adaptation: Measure how crossed entropy changes in domains can indicate how well a model is generalized.
6. Knowledge distillation: It is used to transfer the knowledge of larger “teacher” models to smaller “student” models.

Comparison with other metrics

While the loss of cross entropy is fundamental, it is often used together with other evaluation metrics:

• Perplexity: Exponential of Cross Entropy; more interpretable, since it represents how “confused” is the model
• Blue/red: For generation tasks, these metrics capture the superposition of N-GRAM with reference texts
• Accuracy: Simple percentage of correct, less informative predictions than cross entropy
• F1 score: Balances the precision and memory of classification tasks
• DIVGENCE KL: Measures how a probability distribution diverges from another
• The distance of the earth from the Earth: Explain the semantic similarity between tokens, unlike cross entropy

Also read: Top 15 llm evaluation metrics to explore in 2025

Conclusion

The loss of cross entropy stands as an indispensable tool in the evaluation and training of language models. Its theoretical foundations in information theory, combined with its practical advantages for optimization, make it the standard option for most NLP tasks.

Understanding the loss of cross entropy provides information not only on how models are trained but also in their fundamental limitations and compensation involved in language modeling. As language models continue to evolve, the loss of cross entropy remains a fundamental metric, helping researchers and professionals measure progress and guide innovation.

Whether you are building your language models or evaluating existing ones, a deep understanding of the loss of cross entropy is essential to make informed decisions and interpret the results correctly.