loss-functions.mdx

title	Loss Functions: Measuring Error
sidebar_label	Loss Functions
description	Understanding how models quantify mistakes using MSE, Binary Cross-Entropy, and Categorical Cross-Entropy.
tags	deep-learning neural-networks loss-functions optimization mse cross-entropy

A Loss Function (also known as a Cost Function) is a method of evaluating how well your specific algorithm models your featured data. If your predictions are totally off, your loss function will output a higher number. If they’re pretty good, it’ll output a lower number.

The goal of training a neural network is to use Optimization to find the weights that result in the lowest possible loss.

1. Regression Loss Functions

When you are predicting a continuous value (like a house price or temperature), you need to measure the distance between the predicted number and the actual number.

A. Mean Squared Error (MSE)

MSE is the most common loss function for regression. It squares the difference between prediction and reality, which heavily penalizes large errors.

$$ MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 $$

Where:

• $n$ = number of samples
• $y_i$ = actual value
• $\hat{y}_i$ = predicted value

B. Mean Absolute Error (MAE)

MAE takes the absolute difference. Unlike MSE, it treats all errors linearly. It is more "robust" to outliers because it doesn't square the large deviations.

2. Classification Loss Functions

When predicting categories, we don't look at "distance"; we look at probability divergence.

A. Binary Cross-Entropy (Log Loss)

Used for binary classification (Yes/No). It measures the performance of a classification model whose output is a probability value between 0 and 1.

$$ L = -[y \log(p) + (1 - y) \log(1 - p)] $$

Where:

• $y$ = actual label (0 or 1)
• $p$ = predicted probability of the positive class (1)
• $\log$ = natural logarithm

B. Categorical Cross-Entropy

Used for multi-class classification (e.g., Cat vs. Dog vs. Bird). It compares the predicted probability distribution across all classes with the actual one-hot encoded label.

$$ L = - \sum_{i=1}^{C} y_i \log(p_i) $$

Where:

• $C$ = number of classes
• $y_i$ = actual label (1 for the correct class, 0 otherwise)
• $p_i$ = predicted probability for class $i$

3. Which Loss Function to Choose?

Choosing the right loss function depends entirely on your output layer and the problem type:

Problem Type	Output Layer Activation	Recommended Loss
Regression	Linear (None)	Mean Squared Error (MSE)
Binary Classification	Sigmoid	Binary Cross-Entropy
Multi-class Classification	Softmax	Categorical Cross-Entropy
Multi-label Classification	Sigmoid (per node)	Binary Cross-Entropy

4. Implementation with Keras

# For Regression
model.compile(optimizer='adam', loss='mean_squared_error')

# For Binary Classification (0 or 1)
model.compile(optimizer='adam', loss='binary_crossentropy')

# For Multi-class Classification (One-hot labels)
model.compile(optimizer='adam', loss='categorical_crossentropy')

5. The Loss Landscape

If we visualize the loss function relative to two weights, it looks like a hilly terrain. Training a model is essentially the process of "walking down the hill" to find the lowest valley (The Global Minimum).

References

• PyTorch Docs: Loss Functions Gallery
• Machine Learning Mastery: Loss and Loss Functions for Training Deep Learning Neural Networks

---

Now that we have a "Loss" score, how do we actually change the weights to make that score smaller?