The perceptron learning algorithm builds a linear classifier by modelling the separating hyperplane directly rather than by comparing discriminant functions: the hyperplane is the set {x : βᵀx + β₀ = 0}, β is its normal, and when β is unit-normalized f(x) = βᵀx + β₀ equals the signed distance of x from the hyperplane. The perceptron criterion sums −yᵢf(xᵢ) over the currently misclassified set M — a quantity that is positive precisely for misclassified points and zero when M is empty — and is minimized by stochastic gradient descent, updating β by ρ·yᵢxᵢ upon encountering each single misclassified point rather than after accumulating the full gradient, because the set M itself changes as soon as β moves. The algorithm is guaranteed to converge when the data are linearly separable but only to *some* separating hyperplane out of infinitely many, determined by the initialization; it converges slowly when the class gap is narrow and cycles indefinitely when no separating hyperplane exists — limitations that motivate margin-based formulations and, historically, the successive waves of neural network research.
The perceptron learning algorithm builds a linear classifier by modelling the separating hyperplane directly rather than by comparing discriminant functions: the hyperplane is the set {x : βᵀx + β₀ =…