Logistic and Softmax Functions

Solution & Explanation

Logistic Function (Sigmoid)

• Definition: The logistic function, commonly known as the sigmoid function, is defined as:

  $\sigma(z) = \frac{1}{1 + e^{-z}}$

  This function takes any real-valued number and maps it into the range [0, 1].

• Characteristics:

  • Output Range: [0, 1], making it suitable for binary classification problems.
  • Monotonic: Always increasing, meaning as the input increases, the output also increases.
  • S-shape Curve: Exhibits a characteristic "S" shape, being convex when $z < 0$ and concave when $z > 0$.
  • Interpretability: The output can be interpreted as the probability of belonging to the positive class.

• Use in Logistic Regression:

  • Binary Classification: Used to predict the probability that a given input belongs to the positive class (e.g., class 1).
  • Decision Boundary: The decision boundary is linear in the input feature space, even though the logistic function introduces non-linearity in the output space.
  • Optimization: Differentiable, allowing for gradient-based optimization techniques like gradient descent.

Softmax Function

• Definition: The softmax function generalizes the logistic function to multi-class problems by transforming a vector of real-valued scores into a probability distribution.

  $P(y = j) = \frac{e^{z_j}}{\sum_{k=1}^{K} e^{z_k}}$

  Here, $z_j$ is the logit for class $j$, and $K$ is the number of classes.

• Characteristics:

  • Output Vector: Produces a vector of probabilities that sum to 1.
  • Multi-Class Classification: Suitable for problems with more than two classes.
  • Probability Distribution: Ensures that the total probability across all classes equals 1.

• Use in Softmax Regression:

  • Multi-Class Classification: Extends logistic regression to handle multiple classes by computing the probability of each class.
  • Predicted Class: The class with the highest probability is selected as the predicted class.
  • Optimization: Like logistic regression, it is trained using a categorical cross-entropy loss function, which is differentiable and suitable for gradient descent.

Differences

• Type of Classification:

  • Logistic function is used for binary classification.
  • Softmax function is used for multi-class classification.

• Output:

  • Logistic function outputs a single probability value.
  • Softmax function outputs a probability distribution over multiple classes.

• Decision Boundary:

  • Logistic regression has a linear decision boundary in feature space.
  • Softmax regression handles decision boundaries for multiple classes.

Importance in Logistic Regression

• Probabilistic Interpretation: Both functions provide a probabilistic interpretation of class membership, which is crucial for decision-making.
• Non-Linearity: Introduce non-linearity to the model's output, allowing it to capture more complex patterns.
• Gradient-Based Training: Their differentiable nature allows for efficient training using gradient-based optimization methods.

These functions are fundamental in classification tasks, enabling models to output probabilities and make informed predictions about class membership.