The Perceptron

The perceptron (1958) is the ancestor of every neural network. For labels $y \in \{-1, +1\}$, it predicts $\text{sign}(w \cdot x + b)$, and for each misclassified example it updates $w \leftarrow w + \eta\, y\, x$ and $b \leftarrow b + \eta\, y$.

Implement train_perceptron(X, y, lr=1.0, epochs=100): start from zeros, loop over examples in order each epoch, and treat $w \cdot x + b = 0$ as a mistake. Stop early after an epoch with no mistakes. Return (w, b, mistakes_per_epoch).

Also implement predict(X, w, b), returning ±1 (0 maps to +1).