Perceptrons

The perceptron is the simplest neural network, consisting of a single artificial neuron. Invented by Frank Rosenblatt in 1958, it was the first algorithm capable of learning to classify patterns from data.

Perceptrons find applications in civil engineering for binary decision tasks like pass/fail structural assessment, safe/unsafe load classification, and go/no-go construction decisions.

• Click the plot to add blue (class 0) or red (class 1) points, or use the dataset buttons for predefined examples
• Click "Step" to train one epoch—the perceptron tests each point and updates the decision boundary when it finds errors
• Adjust learning rate $\eta$ to control update step size. Watch how the boundary moves to separate the classes

How many epochs does it take to converge for linearly separable data? What happens with non-separable data?

• Decision boundary: The line $\beta_0 + \beta_1 x_1 + \beta_2 x_2 = 0$ separating the two classes
• Convergence: Training stops when no errors are made in an epoch (for linearly separable data)
• Misclassified points: Highlighted during training to show which points caused weight updates
• Learning rate: Higher $\eta$ means faster but potentially unstable learning
• Perceptron guarantee: Will always converge to a solution if data is linearly separable

Classification Plot

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Perceptron Algorithm:

Training History

Epoch
Errors
$\beta_0$ (bias)
$\beta_1$
$\beta_2$
Updates

Mathematical Foundations

The Perceptron Algorithm is a supervised learning algorithm for binary linear classifiers. Given input $\mathbf{x} = (x_1, x_2)$ and weights $\boldsymbol{\beta} = (\beta_1, \beta_2)$ with bias $\beta_0$, the perceptron computes a weighted sum:
$$\mu = \beta_0 + \beta_1 x_1 + \beta_2 x_2 = \beta_0 + \boldsymbol{\beta}^T\mathbf{x}$$

The activation function applies a step function to produce binary output:
$$h(\mu) = \begin{cases} 1 & \text{if } \mu \geq 0 \\ 0 & \text{if } \mu < 0 \end{cases}$$

The perceptron geometrically defines a decision boundary (hyperplane) that separates the input space. Points on one side are classified as class 1, points on the other as class 0. The equation $\beta_0 + \beta_1 x_1 + \beta_2 x_2 = 0$ defines this boundary line.

The Learning Rule: For each training example $(\mathbf{x}^{(i)}, y^{(i)})$, the perceptron computes the prediction $\hat{y}^{(i)} = h(\mu^{(i)})$ and the error $e^{(i)} = y^{(i)} - \hat{y}^{(i)}$. If the prediction is correct ($e = 0$), no update is made. If incorrect ($e = \pm 1$), the weights are updated:
$$\beta_0 \leftarrow \beta_0 + \eta \cdot e^{(i)}$$ $$\boldsymbol{\beta} \leftarrow \boldsymbol{\beta} + \eta \cdot e^{(i)} \cdot \mathbf{x}^{(i)}$$

where $\eta > 0$ is the learning rate controlling step size. This update rule geometrically rotates the decision boundary toward correctly classifying the misclassified point.

Training proceeds in epochs: each epoch processes all training examples once. The algorithm converges when an entire epoch produces zero errors. The Perceptron Convergence Theorem guarantees convergence in finite steps if the data is linearly separable. For non-separable data, the perceptron will never converge and continues updating indefinitely.

Developed by Kevin Yu & Panagiotis Angeloudis