Study Card: Sparse vs. Dense Loss Functions in Language Modeling

Direct Answer

Sparse loss functions like cross-entropy are commonly used for classification tasks where the target is a single class label (or a probability distribution over a set of classes). They compute the loss only for the true class, making them computationally efficient for large vocabulary sizes. Dense loss functions like mean squared error (MSE) are used for regression tasks where the target is a continuous value. In language modeling, using MSE with one-hot encoded targets becomes computationally expensive due to the large vocabulary size and leads to less sharp probability distributions. Key differences include target representation (class labels vs. continuous values), computational cost (efficient for sparse, expensive for dense with large vocabularies), and the sharpness of predicted probability distributions (sharper with cross-entropy).

Key Terms

• Sparse Loss Function: A loss function that computes the loss only for the true class label (e.g., cross-entropy).
• Dense Loss Function: A loss function that computes the loss for all output values (e.g., mean squared error).
• Cross-Entropy Loss: A common loss function for classification tasks, measuring the difference between predicted and true probability distributions.
• Mean Squared Error (MSE): A common loss function for regression tasks, measuring the average squared difference between predicted and true values.
• One-Hot Encoding: Representing categorical data as binary vectors where only one element is 1 (representing the class), and the rest are 0.

Example

In next-word prediction with a vocabulary of 10,000 words, if the true next word is "cat" (represented as index 200), cross-entropy would only consider the predicted probability for the 200th index. MSE with one-hot encoding would calculate the squared error between the predicted probabilities and the one-hot vector representing "cat" (a vector of length 10,000 with only the 200th element being 1), incurring a significantly higher computational cost.

Code Implementation

import torch
import torch.nn as nn

# Example demonstrating cross-entropy and MSE loss

# Sample predictions and target
predictions = torch.randn(1, 5, requires_grad=True)  # Batch size 1, 5 classes
target_class = torch.tensor([2])  # True class label (index 2)
target_one_hot = torch.nn.functional.one_hot(target_class, num_classes=5).float() # One hot encoding of target

# Cross-entropy loss
loss_cross_entropy = nn.CrossEntropyLoss()(predictions, target_class)

# MSE loss (with one-hot encoding)
loss_mse = nn.MSELoss()(predictions, target_one_hot)

print("Cross-entropy Loss:", loss_cross_entropy.item())
print("MSE Loss:", loss_mse.item())

# Example showing probability distribution
import torch.nn.functional as F
predictions_softmax = F.softmax(predictions, dim=-1)
print("Predicted probabilities (softmax):", predictions_softmax)

# Experiment: Increase the number of classes (vocabulary size)
# and observe the increase in computation time for MSE loss.

Related Concepts

• Classification vs. Regression: Understanding the difference between these two types of machine learning tasks.
• Probability Distributions: How language models predict probabilities for each word in the vocabulary.
• Softmax Function: Used to normalize model outputs into a probability distribution over the vocabulary.
• Computational Efficiency: Why cross-entropy is preferred for large vocabulary sizes in language modeling.
• Language Model Evaluation Metrics: Metrics like perplexity are related to cross-entropy loss.