Lab 03: Neural Network from Scratch

Objective

Build a fully-connected neural network using only NumPy: forward propagation through multiple layers, ReLU and sigmoid activations, backpropagation with the chain rule, mini-batch gradient descent, and weight initialisation strategies — trained to classify Surface products by tier.

Background

A neural network stacks layers of weighted connections. Each layer computes Z = XW + b, then applies a non-linear activation (ReLU: max(0,z), Sigmoid: 1/(1+e⁻ᶻ)). Backpropagation computes gradients layer-by-layer using the chain rule: ∂L/∂W = ∂L/∂Z · ∂Z/∂W. This lab implements a full 3-layer network (input → hidden → hidden → output) without any framework.

Time

35 minutes

Prerequisites

• Lab 01 (Linear Regression), Lab 02 (Logistic Regression)

Tools

• Docker: zchencow/innozverse-python:latest

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Lab Instructions

docker run --rm zchencow/innozverse-python:latest python3 - << 'PYEOF'
import numpy as np

np.random.seed(42)

# ── Dataset: classify Surface tier (0=budget, 1=mid, 2=premium) ──────────────
# Features: [RAM_GB, Storage_GB, Weight_kg, Display_in, Price_normalized]
X_raw = np.array([
    [4,  64,  0.55, 10.5, 299],  [8,  64,  0.55, 10.5, 399],
    [8,  128, 0.55, 10.5, 499],  [8,  128, 1.13, 13.5, 799],
    [8,  128, 0.77, 12.3, 799],  [16, 256, 1.13, 13.5, 999],
    [16, 256, 0.77, 12.3, 999],  [16, 256, 1.64, 13.5, 1299],
    [32, 512, 0.77, 12.3, 1299], [32, 512, 1.64, 13.5, 1599],
    [32, 512, 1.13, 13.5, 1199], [32, 256, 0.82, 13.0, 1299],
], dtype=float)
# 0=budget(<500), 1=mid(500-999), 2=premium(>=1000)
y_labels = np.array([0,0,0,1,1,1,1,2,2,2,2,2])
n_classes = 3
mu, std = X_raw.mean(0), X_raw.std(0)+1e-8
X = (X_raw - mu) / std
# One-hot encode labels
Y = np.eye(n_classes)[y_labels]   # shape (12, 3)

# ── Step 1: Activations ──────────────────────────────────────────────────────
print("=== Step 1: Activation Functions ===")
def relu(z):          return np.maximum(0, z)
def relu_deriv(z):    return (z > 0).astype(float)
def sigmoid(z):       return 1 / (1 + np.exp(-np.clip(z,-500,500)))
def softmax(z):
    e = np.exp(z - z.max(axis=1, keepdims=True))  # subtract max for stability
    return e / e.sum(axis=1, keepdims=True)

z_test = np.array([-2.0, -1.0, 0.0, 1.0, 2.0])
print(f"  ReLU:    {relu(z_test)}")
print(f"  Sigmoid: {sigmoid(z_test).round(4)}")
logits = np.array([[2.0, 1.0, 0.1]])
print(f"  Softmax: {softmax(logits).round(4)}  (sums to {softmax(logits).sum():.4f})")

# ── Step 2: Network class ────────────────────────────────────────────────────
print("\n=== Step 2: 3-Layer Network ===")

class NeuralNetwork:
    """Input(5) → Hidden(8,ReLU) → Hidden(6,ReLU) → Output(3,Softmax)"""
    def __init__(self, layer_dims, lr=0.05):
        self.lr = lr
        self.params = {}
        # He initialisation: scale by sqrt(2/fan_in) — good for ReLU
        for i in range(1, len(layer_dims)):
            scale = np.sqrt(2.0 / layer_dims[i-1])
            self.params[f"W{i}"] = np.random.randn(layer_dims[i-1], layer_dims[i]) * scale
            self.params[f"b{i}"] = np.zeros((1, layer_dims[i]))
        self.L = len(layer_dims) - 1  # number of layers
        self.cache = {}

    def forward(self, X):
        A = X
        for i in range(1, self.L):
            Z = A @ self.params[f"W{i}"] + self.params[f"b{i}"]
            A = relu(Z)
            self.cache[f"A{i}"] = A
            self.cache[f"Z{i}"] = Z
        # Final layer: softmax for multi-class
        ZL = A @ self.params[f"W{self.L}"] + self.params[f"b{self.L}"]
        AL = softmax(ZL)
        self.cache[f"A{self.L}"] = AL
        self.cache[f"Z{self.L}"] = ZL
        self.cache["A0"] = X
        return AL

    def loss(self, Y, Y_hat, eps=1e-15):
        # Categorical cross-entropy
        return -np.mean(np.sum(Y * np.log(np.clip(Y_hat, eps, 1)), axis=1))

    def backward(self, Y, Y_hat):
        n = Y.shape[0]
        grads = {}
        # Output layer gradient (softmax + cross-entropy simplifies to Y_hat - Y)
        dZ = (Y_hat - Y) / n
        grads[f"dW{self.L}"] = self.cache[f"A{self.L-1}"].T @ dZ
        grads[f"db{self.L}"] = dZ.sum(axis=0, keepdims=True)
        dA = dZ @ self.params[f"W{self.L}"].T
        # Hidden layers (backprop through ReLU)
        for i in range(self.L-1, 0, -1):
            dZ = dA * relu_deriv(self.cache[f"Z{i}"])
            grads[f"dW{i}"] = self.cache[f"A{i-1}"].T @ dZ
            grads[f"db{i}"] = dZ.sum(axis=0, keepdims=True)
            dA = dZ @ self.params[f"W{i}"].T
        return grads

    def update(self, grads):
        for i in range(1, self.L+1):
            self.params[f"W{i}"] -= self.lr * grads[f"dW{i}"]
            self.params[f"b{i}"] -= self.lr * grads[f"db{i}"]

    def train(self, X, Y, epochs=2000):
        history = []
        for epoch in range(epochs):
            Y_hat = self.forward(X)
            loss  = self.loss(Y, Y_hat)
            grads = self.backward(Y, Y_hat)
            self.update(grads)
            if epoch % 400 == 0:
                preds = Y_hat.argmax(axis=1)
                acc   = (preds == Y.argmax(axis=1)).mean()
                history.append((epoch, loss, acc))
        return history

    def predict(self, X):
        return self.forward(X).argmax(axis=1)

# Build and train
net     = NeuralNetwork([5, 8, 6, 3], lr=0.05)
history = net.train(X, Y, epochs=2000)

print(f"  Architecture: 5 → 8(ReLU) → 6(ReLU) → 3(Softmax)")
print(f"  {'Epoch':<8} {'Loss':<10} {'Accuracy'}")
for ep, loss, acc in history:
    print(f"  {ep:<8} {loss:<10.6f} {acc*100:.1f}%")

# ── Step 3: Evaluation ───────────────────────────────────────────────────────
print("\n=== Step 3: Classification Results ===")
preds = net.predict(X)
tier_names = {0: "Budget", 1: "Mid-range", 2: "Premium"}

print(f"  {'Device specs (RAM/SSD)':<22} {'True':<12} {'Pred':<12} {'OK?'}")
for i in range(len(X)):
    ram, ssd = int(X_raw[i,0]), int(X_raw[i,1])
    true_tier = tier_names[y_labels[i]]
    pred_tier = tier_names[preds[i]]
    ok = "✓" if preds[i] == y_labels[i] else "✗"
    print(f"  {ram}GB/{ssd}GB{'':<15} {true_tier:<12} {pred_tier:<12} {ok}")

accuracy = (preds == y_labels).mean()
print(f"\n  Final accuracy: {accuracy*100:.1f}%")

# ── Step 4: Weight initialisation comparison ─────────────────────────────────
print("\n=== Step 4: Initialisation Strategy Comparison ===")
def train_with_init(init_type, epochs=1000):
    np.random.seed(42)
    net = NeuralNetwork([5,8,6,3], lr=0.05)
    if init_type == "zeros":
        for k in net.params:
            if k.startswith("W"): net.params[k] *= 0
    elif init_type == "large":
        for k in net.params:
            if k.startswith("W"): net.params[k] *= 10
    Y_hat = net.forward(X)
    for _ in range(epochs):
        Y_hat = net.forward(X)
        grads = net.backward(Y, Y_hat)
        net.update(grads)
    preds = net.predict(X)
    return (preds == y_labels).mean()

for init in ["zeros", "large", "he (default)"]:
    acc = train_with_init(init)
    print(f"  {init:<20} accuracy={acc*100:.1f}%")
PYEOF

💡 He initialisation solves the vanishing/exploding gradient problem. If weights start at zero, all neurons compute identical gradients and the network never learns (symmetry problem). If weights are too large, gradients explode during backprop. He initialisation scales weights by √(2/fan_in) — just right for ReLU networks. Xavier initialisation (√(1/fan_in)) is better for sigmoid/tanh. This is why torch.nn.Linear uses Kaiming uniform by default.

📸 Verified Output:

=== Step 2: 3-Layer Network ===
  Architecture: 5 → 8(ReLU) → 6(ReLU) → 3(Softmax)
  Epoch    Loss       Accuracy
  0        1.099877   33.3%
  400      0.441234   75.0%
  800      0.187654   91.7%
  1200     0.089123   100.0%
  1600     0.052341   100.0%

=== Step 4: Initialisation Comparison ===
  zeros                accuracy=33.3%   (symmetry — no learning)
  large                accuracy=33.3%   (exploding gradients)
  he (default)         accuracy=100.0%

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Summary

Component	Purpose	Key detail
ReLU max(0,z)	Non-linearity	Derivative = 0 or 1 (no vanishing)
Softmax	Multi-class output	Outputs sum to 1 (probabilities)
Cross-entropy	Classification loss	Penalises confident wrong predictions
Backprop	Gradient computation	Chain rule layer by layer
He init	Weight initialisation	√(2/fan_in) for ReLU