Sigmoid-Based Binary Classifier

The sigmoid function (also called the logistic function) is one of the most fundamental building blocks in machine learning and neural networks. It transforms any real-valued number into a probability between 0 and 1, making it ideal for binary classification tasks.

The Sigmoid Function

The sigmoid function is mathematically defined as:

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

This S-shaped curve has several elegant properties:

• Output Range: Always produces values in the interval (0, 1), which can be interpreted as probabilities
• Monotonicity: Strictly increasing, meaning larger inputs always produce larger outputs
• Symmetry: σ(-z) = 1 - σ(z), creating symmetry around the point (0, 0.5)
• Smooth Gradient: Differentiable everywhere, essential for gradient-based optimization

Binary Classification Pipeline

In a binary classification setting using logistic regression, predictions follow a two-step process:

Step 1: Compute Linear Combination

For each input sample x (a feature vector), compute the linear combination:

$$z = \mathbf{x} \cdot \mathbf{w} + b$$

Where:

• w is the weight vector (learned parameters that determine feature importance)
• b is the bias term (shifts the decision boundary)
• The dot product x · w sums the weighted contributions of each feature

For a dataset matrix X with multiple samples (rows), this becomes:

$$\mathbf{z} = X\mathbf{w} + b$$

Step 2: Apply Sigmoid and Threshold

Transform the linear combination using the sigmoid function:

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

Finally, convert probabilities to binary predictions using a decision threshold of 0.5:

$$\hat{y} = \begin{cases} 1 & \text{if } p \geq 0.5 \ 0 & \text{if } p < 0.5 \end{cases}$$

Your Task

Implement a function that performs binary classification using the sigmoid activation function. Given a matrix of input samples X, a weight vector w, and a bias term b, compute the binary class predictions for all samples.

The function should:

1. Calculate the linear combination z = Xw + b for each sample
2. Apply the sigmoid function to obtain class probabilities
3. Threshold the probabilities at 0.5 to produce binary outputs (0 or 1)
4. Return the predictions as an integer numpy array