What is Softmax Classifier? (original) (raw)

Last Updated : 04 Apr, 2025

In the realm of machine learning, particularly in classification tasks, the Softmax Classifier plays a crucial role in transforming raw model outputs into probabilities. It is commonly used in multi-class classification problems where the goal is to assign an input into one of many classes.

**Let’s delve into what the Softmax Classifier is, how it works, and its applications.

**Understanding the Softmax Function

The Softmax function is a mathematical function that converts a vector of real numbers into a probability distribution. Each element in the output is between 0 and 1, and the sum of all elements equals 1. This property makes it perfect for classification tasks, where we want to know the probability that a given input belongs to a certain class.

**The formula for the Softmax function is:

\text{Softmax}(z)_i = \frac{e^{z_i}}{\sum_{j=1}^K e^{z_j}}

Where:

**How the Softmax Classifier Works?

In a Softmax Classifier, the neural network outputs a set of raw scores for each class. These raw scores, also called logits, are then passed through the Softmax function, converting them into probabilities. The class with the highest probability is chosen as the model's prediction.

For example, if you're classifying an image of an animal into one of three categories (cat, dog, rabbit), the neural network might output raw scores like [2.1, 1.0, 0.5]. After applying the Softmax function, these scores might become [0.7, 0.2, 0.1], meaning there's a 70% chance the image is a cat, 20% chance it's a dog, and 10% chance it's a rabbit.

Softmax vs Sigmoid

**Criteria **Softmax **Sigmoid
**Purpose Multi-class classification Binary classification
**Mathematical Expression \text{Softmax}(z)_i = \frac{e^{z_i}}{\sum_{j=1}^K e^{z_j}} \sigma(x) = \frac{1}{1 + e^{-x}}
**Output A vector of probabilities for each class A single probability for the positive class
**Interpretation Probabilities sum to 1, and the class with the highest probability is chosen Output > 0.5 indicates the positive class, else negative
**Use Case Multi-class tasks (e.g., image classification with more than two categories) Binary tasks (e.g., spam classification)
**Handling Multiple Classes Handles multiple classes in a mutually exclusive manner Multiple Sigmoid functions needed, probabilities don't sum to 1
**Loss Function Categorical Cross-Entropy Binary Cross-Entropy

**Loss Function: Cross-Entropy

The Softmax Classifier is often paired with the **Cross-Entropy Loss function, which is used to measure the error between the predicted probabilities and the true labels.

Cross-entropy is defined as:

\text{Loss} = - \sum_{i=1}^K y_i \log(\hat{y}_i)

Where:

The Softmax function combined with the cross-entropy loss allows the model to penalize incorrect predictions while ensuring that the total probability sums to 1.

Implementing Softmax Classifier using NumPy

This basic implementation demonstrates a Softmax Classifier for multi-class classification with manually calculated gradients.

**Softmax Function:

Python `

import numpy as np

def softmax(z): exp_z = np.exp(z - np.max(z)) # Subtract max to prevent overflow return exp_z / np.sum(exp_z, axis=1, keepdims=True)

`

**Cross-Entropy Loss:

Python `

def cross_entropy_loss(predicted, actual): m = actual.shape[0] # Number of samples log_likelihood = -np.log(predicted[range(m), actual]) loss = np.sum(log_likelihood) / m return loss

`

**Softmax Classifier (Training):

Python `

class SoftmaxClassifier: def init(self, learning_rate=0.01, num_classes=3, num_features=2): self.learning_rate = learning_rate self.weights = np.random.randn(num_features, num_classes) self.bias = np.zeros((1, num_classes))

def train(self, X, y, epochs=1000):
    for epoch in range(epochs):
        # Forward pass
        logits = np.dot(X, self.weights) + self.bias
        probabilities = softmax(logits)
        
        # Compute loss
        loss = cross_entropy_loss(probabilities, y)
        
        # Backward pass (Gradient Descent)
        m = X.shape[0]
        grad_logits = probabilities
        grad_logits[range(m), y] -= 1  # Gradient of loss with respect to logits
        grad_logits /= m
        
        # Update weights and bias
        self.weights -= self.learning_rate * np.dot(X.T, grad_logits)
        self.bias -= self.learning_rate * np.sum(grad_logits, axis=0, keepdims=True)
        
        if epoch % 100 == 0:
            print(f"Epoch {epoch} - Loss: {loss}")

def predict(self, X):
    logits = np.dot(X, self.weights) + self.bias
    probabilities = softmax(logits)
    return np.argmax(probabilities, axis=1)

`

**Example Usage:

Python `

Sample dataset (X: input features, y: class labels)

X = np.array([[1, 2], [2, 1], [3, 1], [1, 3], [2, 3], [3, 2]]) y = np.array([0, 0, 1, 1, 2, 2]) # 3 classes

Initialize and train the classifier

classifier = SoftmaxClassifier(learning_rate=0.1, num_classes=3, num_features=2) classifier.train(X, y, epochs=1000)

Predict

predictions = classifier.predict(X) print("Predictions:", predictions)

`

**Output:

Epoch 0 - Loss: 3.36814871682521
Epoch 100 - Loss: 0.9742101560035626
Epoch 200 - Loss: 0.8892292393281243
Epoch 300 - Loss: 0.8237226736799194
Epoch 400 - Loss: 0.7701087532304117
Epoch 500 - Loss: 0.7254339633377661
Epoch 600 - Loss: 0.6875686437016135
Epoch 700 - Loss: 0.6549706401999168
Epoch 800 - Loss: 0.6265136024817034
Epoch 900 - Loss: 0.6013645205076109
Predictions: [0 0 1 1 2 2]

**Applications of Softmax Classifier

The Softmax Classifier is widely used in various domains:

**Advantages of the Softmax Classifier

**Limitations of the Softmax Classifier

**Conclusion

The Softmax Classifier is a fundamental tool in machine learning, particularly useful for multi-class classification tasks. By converting raw model outputs into probabilities, it provides an intuitive and mathematically sound way to make predictions across a wide range of applications. Paired with the cross-entropy loss function, it ensures that models can be trained effectively to minimize error and maximize accuracy in complex classification tasks.