🧠 Perceptron

The perceptron is the earliest trainable model of a single artificial neuron, invented by Frank Rosenblatt in 1958. It takes an input vector x, computes a weighted sum w·x + b, and outputs +1 if that sum is at least zero and −1 otherwise, acting as a linear binary classifier. It is the conceptual ancestor of the multi-layer neural networks used in modern deep learning.

The algorithm scans through the points looking for one the current weights get wrong, meaning its label y and score w·x + b disagree in sign. When it finds such a point, it applies the update w ← w + η·y·x and b ← b + η·y, which shifts the decision boundary directly toward correctly classifying that point. This repeats, one misclassified point at a time, until every point is classified correctly or the iteration limit is reached.

The decision line is the set of points where w·x + b = 0, which the simulation draws in green; everything on one side is predicted +1 and everything on the other is predicted −1. The yellow arrow is the weight vector w itself, and it always points perpendicular to the decision line, in the direction that increases the score w·x + b fastest — that is, toward the positive class.

XOR-style data has positive points in two diagonally opposite regions and negative points in the other two, so no single straight line can separate the classes no matter how the weights are set — this is the well-known "linear separability" limitation. Because the perceptron can only ever represent one straight decision boundary, it keeps finding misclassified points forever on this preset, and the mistakes-per-pass chart never settles to zero, unlike on the separable Gaussians or Linear stripe presets.

Convergence means the current weights classify every point in the dataset correctly, so a full scan finds no misclassified point and training stops with the Converged stat switching to YES. The perceptron convergence theorem proves this must happen in a finite number of updates whenever the data is linearly separable, which is a foundational result in machine learning theory — though it says nothing about how good the resulting margin between classes will be.