Logistic Regression vs Softmax Regression
Logistic regression and softmax regression are both powerful classification algorithms used in machine learning. While they share some similarities, they differ in their application and the types of problems they are best suited for.
Logistic Regression
Introduction
Logistic regression is a statistical method used to predict the probability of a binary outcome (0 or 1). It’s a linear model that uses a sigmoid function to map the linear combination of features to a probability between 0 and 1.
Example
Suppose we want to predict whether a customer will purchase a product based on their age, income, and previous purchase history. Logistic regression can be used to model this relationship and predict the probability of a customer making a purchase.
Mathematical Formulation
The output of logistic regression is given by:
p(y=1|x) = 1 / (1 + exp(-z))
Where:
- y is the binary outcome (0 or 1)
- x is the vector of features
- z is the linear combination of features: z = wTx + b
- w is the vector of weights
- b is the bias term
Softmax Regression
Introduction
Softmax regression is an extension of logistic regression that can handle multiple classes (more than two). It predicts the probability of an input belonging to each of the classes. The output of softmax regression is a probability distribution over the classes, where the probabilities sum to 1.
Example
Imagine classifying images into different categories like cats, dogs, or birds. Softmax regression can be used to predict the probability of an image belonging to each of these categories.
Mathematical Formulation
The output of softmax regression is given by:
p(y=i|x) = exp(z_i) / sum(exp(z_j))
Where:
- y is the class label (1 to K)
- x is the vector of features
- z_i is the linear combination of features for class i: z_i = w_iTx + b_i
- w_i is the vector of weights for class i
- b_i is the bias term for class i
Comparison Table
| Feature | Logistic Regression | Softmax Regression |
|---|---|---|
| Output | Probability of a single outcome (0 or 1) | Probability distribution over multiple classes |
| Number of classes | Two | More than two |
| Activation function | Sigmoid | Softmax |
Key Differences
- Number of classes: Logistic regression is designed for binary classification, while softmax regression handles multiple classes.
- Output: Logistic regression provides a single probability, while softmax regression provides a probability distribution over all classes.
- Activation function: Logistic regression uses the sigmoid function, while softmax regression uses the softmax function.
Conclusion
Logistic regression and softmax regression are powerful classification algorithms. Logistic regression is suitable for binary classification problems, while softmax regression can handle multi-class classification. The choice between the two depends on the specific problem and the number of classes involved.