Cross-entropy is typically a better loss function than Mean Squared Error (MSE) for classification tasks with more than two labels, such as MNIST with 10 labels, for several reasons:
Output Distribution: In classification tasks, the target variable is often categorical, with each label representing a discrete category. Cross-entropy is designed to work with such categorical data and measures the dissimilarity between the predicted and true class probabilities. MSE, on the other hand, is better suited for regression tasks where the target variable is continuous.
Output Range: Cross-entropy takes into account the probabilistic nature of classification problems. It penalizes predictions that are far from the true class probabilities, effectively encouraging the model to produce confident predictions for the correct class while suppressing the probabilities of incorrect classes. In contrast, MSE does not constrain the output to a specific range and can lead to less meaningful gradients for classification problems.
Sensitivity to Outliers: MSE is sensitive to outliers in the prediction errors, which can be problematic in classification tasks where the goal is to assign data points to discrete categories. Cross-entropy is less sensitive to outliers and focuses on the relative differences in class probabilities.
Gradient Information: Cross-entropy provides more informative gradients during training, especially when using gradient-based optimization algorithms like gradient descent. The gradients are large when the predicted class probabilities are far from the true probabilities, which helps the model converge faster and learn more effectively.
Interpretability: Cross-entropy loss has a clear probabilistic interpretation. Minimizing cross-entropy is equivalent to maximizing the likelihood of the observed data under the model. This makes it easier to interpret and analyze the performance of the model in terms of probabilities and class likelihoods.
Handling Multiclass Problems: Cross-entropy loss naturally extends to multiclass problems like MNIST with multiple labels. It calculates the loss for each class independently and combines them, allowing the model to distinguish between different classes effectively.
One-Hot Encoding: Cross-entropy works well with one-hot encoded target labels, which is a common representation for multiclass classification problems. It can be adapted to handle the multiple labels in a straightforward manner.
In summary, cross-entropy loss is better suited for classification tasks with multiple labels because it aligns with the nature of the problem, provides meaningful gradients for optimization, and encourages the model to produce probabilistic outputs that can be interpreted as class probabilities. While MSE has its merits for regression tasks, it is generally not suitable for classification problems with discrete categories.