A perceptron is a supervised, single-layer linear classifier: it computes a weighted sum of input features and predicts a class from that score. This walkthrough trains one on a tiny AND dataset, implements its mistake-driven learning rule in plain Python, then fits the corresponding model with scikit-learn. It also shows why a single perceptron cannot learn a nonlinear boundary such as XOR.
How a perceptron makes a prediction
For an input vector x, weights w, and bias b, the perceptron calculates a score:
score = w · x + b
The score determines which side of a linear decision boundary the example falls on. With labels encoded as −1 and +1, predict +1 when the score is at least zero and −1 otherwise. The boundary itself is the set of inputs for which the score is zero; with two features, it is a line.
How the learning rule updates weights
The classic perceptron rule changes the model only when it makes a mistake. For a misclassified example with label y, update the weights and bias as follows:
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w ← w + η y xb ← b + η y
Here, η is the learning rate. If the prediction is correct, that example leaves the weights unchanged. The condition y * score <= 0 triggers an update; it includes a score of exactly zero, which is treated as a mistake under this convention.
Implement a perceptron from scratch in Python
This example uses four two-feature inputs with AND labels. Only [1, 1] is positive; the other three inputs are negative. The dataset is linearly separable, so the classic perceptron convergence result applies.
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import numpy as np
X = np.array([[0, 0], [0, 1], [1, 0], [1, 1]], dtype=float)
y = np.array([-1, -1, -1, 1]) # AND labels
w = np.zeros(X.shape[1])
b = 0.0
eta = 1.0
for epoch in range(10):
mistakes = 0
for xi, yi in zip(X, y):
score = np.dot(xi, w) + b
if yi * score <= 0:
w += eta * yi * xi
b += eta * yi
mistakes += 1
if mistakes == 0:
break
predictions = np.where(X @ w + b >= 0, 1, -1)
print(w, b, predictions)The outer loop caps training at 10 passes; the inner loop applies the update to each mistake. Training stops early after a full pass with no mistakes. The final line applies the same zero-threshold convention to all four inputs. This is a teaching example, not a benchmark or evidence of performance on unseen data.
Fit the same model with scikit-learn
For a concise estimator-based version, use sklearn.linear_model.Perceptron:
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from sklearn.linear_model import Perceptron
clf = Perceptron(max_iter=1000, tol=1e-3, random_state=0)
clf.fit(X, y)
print(clf.coef_, clf.intercept_)
print(clf.predict(X))
print(clf.score(X, y))fit trains the classifier, predict returns class labels, and score reports mean accuracy on the data passed to it. In this snippet, that data is the same four-row training set, so the score describes training fit, not generalization. For a real analysis, keep separate training and test data.
The estimator exposes settings including max_iter, tol, shuffle, eta0, and random_state. Its official API describes it as a linear perceptron classifier and notes its equivalence to SGDClassifier(loss="perceptron", learning_rate="constant"): scikit-learn Perceptron API. The linear-model guide describes the default perceptron as unregularized and updated on mistakes, making it a simple teaching model and fast baseline.
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Why linear separability matters
A single perceptron can draw only one hyperplane as its decision boundary. The classic convergence guarantee holds when the training examples are linearly separable: a boundary exists that places every example on the correct side. The AND example meets that condition.
For overlapping or non-separable classes, the classic guarantee does not apply. The model may continue making mistakes, so training needs an explicit iteration limit and a stopping rule; the scratch example supplies both. Evaluate such a model on held-out data rather than treating its training score as proof it generalizes.
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When a perceptron is not enough
XOR is a standard example of a pattern that cannot be separated by one straight line in two dimensions, so a single perceptron cannot represent its decision boundary. A multilayer perceptron (MLP) adds hidden layers that can learn nonlinear functions. That added flexibility comes with trade-offs: scikit-learn notes that MLPs require hyperparameter tuning and are sensitive to feature scaling. See its MLP documentation.
Quick Recap
What to take away
- A perceptron predicts from a weighted sum plus bias, thresholded into a class.
- Its classic learning rule updates weights on mistakes, not correct predictions.
- Linear separability is the condition behind the classic convergence guarantee.
- Scikit-learn provides a practical API for this linear classifier family.
- Nonlinear boundaries require a model with hidden nonlinear layers, with added tuning and scaling considerations.
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