Cross Entropy vs MSE: Which Loss Function Should You Choose?
In machine learning, the loss function plays a central role: it measures how well a model is performing and guides optimization during training. Two commonly used loss functions are Mean Squared Error (MSE) and Cross Entropy. While they sometimes appear interchangeable, they are designed for different types of problems. Choosing the right one can significantly affect model performance.
In this post, we’ll break down what each loss function is, how it works, and when to use it.
Mean Squared Error (MSE)
Definition:
MSE measures the average squared difference between predicted values and actual values.
\(MSE= \frac{1}{n}\sum_{i=1}^{n} (y_i - \hat{y}_i)^2\)
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\(y_i\): true value
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\(\hat{y}_i\): predicted value
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n: number of samples
Intuition:
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If the model predicts exactly, the error is 0.
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Errors are squared, so large mistakes are penalized more heavily than small mistakes.
Applications:
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Regression problems: predicting continuous variables (house prices, temperature, sales).
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Works best when prediction errors follow a roughly Gaussian distribution.
Limitations:
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Not ideal for classification, especially when outputs are probabilities.
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Sensitive to outliers since errors are squared.
Cross Entropy Loss
Definition:
Cross entropy measures the dissimilarity between two probability distributions — the true distribution (labels) and the predicted distribution (model outputs).
For binary classification:
\(L= - \frac{1}{n} \sum_{i=1}^n \Big[ y_i \log(\hat{y}_i) + (1-y_i)\log(1-\hat{y}_i) \Big]\)
For multi-class classification:
\(L= - \sum_{i=1}^{n} \sum_{c=1}^{C} y_{i,c} \log(\hat{y}_{i,c})\)
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\(y_{i,c}\): true label (one-hot encoded)
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\({y}_{i,c}\): predicted probability for class c
Intuition:
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If the model assigns high probability to the correct class, the loss is small.
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If the model is very confident but wrong, the loss is large.
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Encourages probabilistic outputs that are well-calibrated.
Applications:
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Classification problems (binary or multi-class).
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Natural Language Processing (NLP), image recognition, recommendation systems.
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Especially suited when outputs represent probabilities via sigmoid or softmax.
Limitations:
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Can be sensitive to mislabeled data.
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Requires careful numerical implementation (log(0) can cause issues).
MSE vs Cross Entropy: When to Use Which?
| Aspect | Mean Squared Error (MSE) | Cross Entropy |
|---|---|---|
| Problem type | Regression (continuous outputs) | Classification (discrete outputs) |
| Output type | Real-valued predictions | Probabilities (0–1) |
| Behavior | Penalizes large deviations heavily | Focuses on probability assigned to correct class |
| Training speed | Often slower for classification (gradients shrink near decision boundaries) | Faster convergence for classification |
| Robustness | Sensitive to outliers | Sensitive to label noise |
Example Scenarios
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Predicting house prices → Use MSE.
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Classifying whether an email is spam or not → Use Cross Entropy.
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Predicting next-word probabilities in a language model → Use Cross Entropy.
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Forecasting stock price movements as continuous values → Use MSE.
Key Takeaway
Both MSE and Cross Entropy are powerful, but they serve different purposes:
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Use MSE for regression tasks where predictions are continuous.
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Use Cross Entropy for classification tasks where predictions represent probabilities.
Selecting the correct loss function isn’t just a technical detail — it directly impacts how well your model learns and generalizes.
👉 If you’re experimenting with a model and not getting the results you expect, check your loss function. Sometimes, changing from MSE to Cross Entropy (or vice versa) can unlock much better performance.
