← Feature SelectionMulti-Layer Networks and Forward Pass →

The Perceptron

> The perceptron is the atom of neural networks. Split it open and you find weights, a bias, and a decision.

Type: Build

Languages: Python

Prerequisites: Phase 1 (Linear Algebra Intuition)

Time: ~60 minutes

Learning Objectives

The Problem

You know vectors and dot products. You know that a matrix transforms inputs into outputs. But how does a machine *learn* which transformation to use?

The perceptron answers this. It's the simplest possible learning machine: take some inputs, multiply by weights, add a bias, and make a binary decision. Then adjust. That's it. Every neural network ever built is layers of this idea stacked together.

Understanding the perceptron means understanding what "learning" actually means in code: adjusting numbers until the output matches reality.

The Concept

One Neuron, One Decision

A perceptron takes n inputs, multiplies each by a weight, sums them up, adds a bias, and passes the result through an activation function.

graph LR x1["x1"] -- "w1" --> sum["Σ(wi*xi) + b"] x2["x2"] -- "w2" --> sum x3["x3"] -- "w3" --> sum bias["bias"] --> sum sum --> step["step(z)"] step --> out["output (0 or 1)"]

The step function is brutal: if the weighted sum plus bias is >= 0, output 1. Otherwise, output 0.

step(z) = 1  if z >= 0
           0  if z < 0

This is a linear classifier. The weights and bias define a line (or hyperplane in higher dimensions) that splits the input space into two regions.

The Decision Boundary

For two inputs, the perceptron draws a line through 2D space:

  x2
  ┤
  │  Class 1        /
  │    (0)          /
  │                /
  │               / w1·x1 + w2·x2 + b = 0
  │              /
  │             /     Class 2
  │            /        (1)
  ┼───────────/──────────── x1

Everything on one side of the line outputs 0. Everything on the other side outputs 1. Training moves this line until it correctly separates the classes.

The Learning Rule

The perceptron learning rule is simple:

For each training example (x, y_true):
    y_pred = predict(x)
    error = y_true - y_pred

    For each weight:
        w_i = w_i + learning_rate * error * x_i
    bias = bias + learning_rate * error

If the prediction is correct, error = 0, nothing changes. If it predicts 0 but should be 1, weights increase. If it predicts 1 but should be 0, weights decrease. The learning rate controls how big each adjustment is.

The XOR Problem

Here's where it breaks. Look at these logic gates:

AND gate:           OR gate:            XOR gate:
x1  x2  out         x1  x2  out         x1  x2  out
0   0   0           0   0   0           0   0   0
0   1   0           0   1   1           0   1   1
1   0   0           1   0   1           1   0   1
1   1   1           1   1   1           1   1   0

AND and OR are linearly separable: you can draw a single line to separate the 0s from the 1s. XOR is not. No single line can separate [0,1] and [1,0] from [0,0] and [1,1].

AND (separable):        XOR (not separable):

  x2                      x2
  1 ┤  0     1            1 ┤  1     0
    │     /                 │
  0 ┤  0 / 0              0 ┤  0     1
    ┼──/──────── x1         ┼──────────── x1
       line works!          no single line works!

This is a fundamental limit. A single perceptron can only solve linearly separable problems. Minsky and Papert proved this in 1969 and it nearly killed neural network research for a decade.

The fix: stack perceptrons into layers. A multi-layer perceptron can solve XOR by combining two linear decisions into a nonlinear one.

Build It

Step 1: The Perceptron class

class Perceptron:
    def __init__(self, n_inputs, learning_rate=0.1):
        self.weights = [0.0] * n_inputs
        self.bias = 0.0
        self.lr = learning_rate

    def predict(self, inputs):
        total = sum(w * x for w, x in zip(self.weights, inputs))
        total += self.bias
        return 1 if total >= 0 else 0

    def train(self, training_data, epochs=100):
        for epoch in range(epochs):
            errors = 0
            for inputs, target in training_data:
                prediction = self.predict(inputs)
                error = target - prediction
                if error != 0:
                    errors += 1
                    for i in range(len(self.weights)):
                        self.weights[i] += self.lr * error * inputs[i]
                    self.bias += self.lr * error
            if errors == 0:
                print(f"Converged at epoch {epoch + 1}")
                return
        print(f"Did not converge after {epochs} epochs")

Step 2: Train on logic gates

and_data = [
    ([0, 0], 0),
    ([0, 1], 0),
    ([1, 0], 0),
    ([1, 1], 1),
]

or_data = [
    ([0, 0], 0),
    ([0, 1], 1),
    ([1, 0], 1),
    ([1, 1], 1),
]

not_data = [
    ([0], 1),
    ([1], 0),
]

print("=== AND Gate ===")
p_and = Perceptron(2)
p_and.train(and_data)
for inputs, _ in and_data:
    print(f"  {inputs} -> {p_and.predict(inputs)}")

print("\n=== OR Gate ===")
p_or = Perceptron(2)
p_or.train(or_data)
for inputs, _ in or_data:
    print(f"  {inputs} -> {p_or.predict(inputs)}")

print("\n=== NOT Gate ===")
p_not = Perceptron(1)
p_not.train(not_data)
for inputs, _ in not_data:
    print(f"  {inputs} -> {p_not.predict(inputs)}")

Step 3: Watch XOR fail

xor_data = [
    ([0, 0], 0),
    ([0, 1], 1),
    ([1, 0], 1),
    ([1, 1], 0),
]

print("\n=== XOR Gate (single perceptron) ===")
p_xor = Perceptron(2)
p_xor.train(xor_data, epochs=1000)
for inputs, expected in xor_data:
    result = p_xor.predict(inputs)
    status = "OK" if result == expected else "WRONG"
    print(f"  {inputs} -> {result} (expected {expected}) {status}")

It will never converge. This is the hard proof that a single perceptron cannot learn XOR.

Step 4: Solve XOR with two layers

The trick: XOR = (x1 OR x2) AND NOT (x1 AND x2). Combine three perceptrons:

graph LR x1["x1"] --> OR["OR neuron"] x1 --> NAND["NAND neuron"] x2["x2"] --> OR x2 --> NAND OR --> AND["AND neuron"] NAND --> AND AND --> out["output"]
def xor_network(x1, x2):
    or_neuron = Perceptron(2)
    or_neuron.weights = [1.0, 1.0]
    or_neuron.bias = -0.5

    nand_neuron = Perceptron(2)
    nand_neuron.weights = [-1.0, -1.0]
    nand_neuron.bias = 1.5

    and_neuron = Perceptron(2)
    and_neuron.weights = [1.0, 1.0]
    and_neuron.bias = -1.5

    hidden1 = or_neuron.predict([x1, x2])
    hidden2 = nand_neuron.predict([x1, x2])
    output = and_neuron.predict([hidden1, hidden2])
    return output


print("\n=== XOR Gate (multi-layer network) ===")
for inputs, expected in xor_data:
    result = xor_network(inputs[0], inputs[1])
    print(f"  {inputs} -> {result} (expected {expected})")

All four cases correct. Stacking perceptrons into layers creates decision boundaries that no single perceptron can produce.

Step 5: Train a Two-Layer Network

Step 4 hand-wired the weights. That works for XOR, but not for real problems where you don't know the right weights in advance. The fix: replace the step function with sigmoid and learn the weights automatically through backpropagation.

class TwoLayerNetwork:
    def __init__(self, learning_rate=0.5):
        import random
        random.seed(0)
        self.w_hidden = [[random.uniform(-1, 1), random.uniform(-1, 1)] for _ in range(2)]
        self.b_hidden = [random.uniform(-1, 1), random.uniform(-1, 1)]
        self.w_output = [random.uniform(-1, 1), random.uniform(-1, 1)]
        self.b_output = random.uniform(-1, 1)
        self.lr = learning_rate

    def sigmoid(self, x):
        import math
        x = max(-500, min(500, x))
        return 1.0 / (1.0 + math.exp(-x))

    def forward(self, inputs):
        self.inputs = inputs
        self.hidden_outputs = []
        for i in range(2):
            z = sum(w * x for w, x in zip(self.w_hidden[i], inputs)) + self.b_hidden[i]
            self.hidden_outputs.append(self.sigmoid(z))
        z_out = sum(w * h for w, h in zip(self.w_output, self.hidden_outputs)) + self.b_output
        self.output = self.sigmoid(z_out)
        return self.output

    def train(self, training_data, epochs=10000):
        for epoch in range(epochs):
            total_error = 0
            for inputs, target in training_data:
                output = self.forward(inputs)
                error = target - output
                total_error += error ** 2

                d_output = error * output * (1 - output)

                saved_w_output = self.w_output[:]
                hidden_deltas = []
                for i in range(2):
                    h = self.hidden_outputs[i]
                    hd = d_output * saved_w_output[i] * h * (1 - h)
                    hidden_deltas.append(hd)

                for i in range(2):
                    self.w_output[i] += self.lr * d_output * self.hidden_outputs[i]
                self.b_output += self.lr * d_output

                for i in range(2):
                    for j in range(len(inputs)):
                        self.w_hidden[i][j] += self.lr * hidden_deltas[i] * inputs[j]
                    self.b_hidden[i] += self.lr * hidden_deltas[i]
net = TwoLayerNetwork(learning_rate=2.0)
net.train(xor_data, epochs=10000)
for inputs, expected in xor_data:
    result = net.forward(inputs)
    predicted = 1 if result >= 0.5 else 0
    print(f"  {inputs} -> {result:.4f} (rounded: {predicted}, expected {expected})")

Two key differences from Step 4. First, sigmoid replaces the step function -- it's smooth, so gradients exist. Second, the train method propagates error backward from output to hidden layer, adjusting every weight proportionally to its contribution to the error. That's backpropagation in 20 lines.

This is the bridge to Lesson 03. The math behind d_output and hidden_deltas is the chain rule applied to the network graph. We'll derive it properly there.

Use It

Everything you just built from scratch exists in one import:

from sklearn.linear_model import Perceptron as SkPerceptron
import numpy as np

X = np.array([[0,0],[0,1],[1,0],[1,1]])
y = np.array([0, 0, 0, 1])

clf = SkPerceptron(max_iter=100, tol=1e-3)
clf.fit(X, y)
print([clf.predict([x])[0] for x in X])

Five lines. Your 30-line Perceptron class does the same thing. The sklearn version adds convergence checks, multiple loss functions, and sparse input support -- but the core loop is identical: weighted sum, step function, weight update on error.

The real gap shows up at scale. What changes in production networks:

A single perceptron can only draw straight lines. Stack them, and you can draw any shape.

Ship It

This lesson produces:

Exercises

  1. Train a perceptron on a NAND gate (the universal gate - any logic circuit can be built from NAND). Verify its weights and bias form a valid decision boundary.
  2. Modify the Perceptron class to track the decision boundary (w1*x1 + w2*x2 + b = 0) at each epoch. Print how the line shifts during training on the AND gate.
  3. Build a 3-input perceptron that outputs 1 only when at least 2 of the 3 inputs are 1 (a majority vote function). Is this linearly separable? Why?

Key Terms

Term What people say What it actually means
Perceptron "A fake neuron" A linear classifier: dot product of inputs and weights, plus bias, through a step function
Weight "How important an input is" A multiplier that scales each input's contribution to the decision
Bias "The threshold" A constant that shifts the decision boundary, letting the perceptron fire even with zero inputs
Activation function "The thing that squishes values" A function applied after the weighted sum - step function for perceptrons, sigmoid/ReLU for modern networks
Linearly separable "You can draw a line between them" A dataset where a single hyperplane can perfectly separate the classes
XOR problem "The thing perceptrons can't do" Proof that single-layer networks cannot learn non-linearly-separable functions
Decision boundary "Where the classifier switches" The hyperplane w*x + b = 0 that divides input space into two classes
Multi-layer perceptron "A real neural network" Perceptrons stacked in layers, where each layer's output feeds the next layer's input

Further Reading

← Feature SelectionMulti-Layer Networks and Forward Pass →