Lab 11: Gradient Descent Optimisers

Objective

Implement and compare gradient descent optimisers from scratch: vanilla SGD, mini-batch SGD, SGD with momentum, RMSProp, and the Adam optimiser — measuring convergence speed, final loss, and behaviour on different loss landscapes including saddle points and narrow valleys.

Background

The choice of optimiser dramatically affects how fast and how well a neural network trains. Vanilla SGD often oscillates or gets stuck. Momentum accumulates a velocity vector, accelerating in consistent directions. RMSProp adapts the learning rate per-parameter based on recent gradient magnitudes. Adam (Adaptive Moment Estimation) combines both momentum AND adaptive learning rates — currently the most popular optimiser in deep learning.

Time

30 minutes

Prerequisites

• Lab 01 (Linear Regression) — gradient descent basics

• Lab 03 (Neural Network) — understanding training loops

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)

# ── Test problem: minimise a 2D bowl + noise ──────────────────────────────────
# f(w1, w2) = w1² + 10·w2²  (narrow valley — hard for vanilla SGD)
def loss_fn(w, X, y):
    y_pred = X @ w
    return np.mean((y_pred - y) ** 2)

def gradient(w, X, y, batch_idx=None):
    if batch_idx is not None:
        X, y = X[batch_idx], y[batch_idx]
    y_pred = X @ w
    return 2 * X.T @ (y_pred - y) / len(y)

# Generate regression data
n, d = 200, 5
X_data = np.random.randn(n, d)
w_true = np.array([1.5, -2.0, 0.8, 1.2, -0.5])
y_data = X_data @ w_true + np.random.normal(0, 0.3, n)

# ── Step 1: Optimiser implementations ────────────────────────────────────────
print("=== Step 1: Optimiser Classes ===")

class SGD:
    def __init__(self, lr=0.01):
        self.lr = lr
    def step(self, w, grad, t=None):
        return w - self.lr * grad
    def name(self): return f"SGD(lr={self.lr})"

class SGDMomentum:
    def __init__(self, lr=0.01, momentum=0.9):
        self.lr, self.momentum = lr, momentum
        self.v = None
    def step(self, w, grad, t=None):
        if self.v is None: self.v = np.zeros_like(w)
        self.v = self.momentum * self.v - self.lr * grad  # velocity update
        return w + self.v
    def name(self): return f"SGD+Momentum(lr={self.lr},β={self.momentum})"

class RMSProp:
    def __init__(self, lr=0.01, beta=0.9, eps=1e-8):
        self.lr, self.beta, self.eps = lr, beta, eps
        self.v = None  # running avg of squared gradients
    def step(self, w, grad, t=None):
        if self.v is None: self.v = np.zeros_like(w)
        self.v = self.beta * self.v + (1-self.beta) * grad**2
        return w - self.lr * grad / (np.sqrt(self.v) + self.eps)
    def name(self): return f"RMSProp(lr={self.lr})"

class Adam:
    def __init__(self, lr=0.01, beta1=0.9, beta2=0.999, eps=1e-8):
        self.lr, self.beta1, self.beta2, self.eps = lr, beta1, beta2, eps
        self.m = None   # 1st moment (mean of gradients)
        self.v = None   # 2nd moment (mean of squared gradients)
    def step(self, w, grad, t):
        if self.m is None:
            self.m = np.zeros_like(w)
            self.v = np.zeros_like(w)
        # Update biased estimates
        self.m = self.beta1 * self.m + (1-self.beta1) * grad
        self.v = self.beta2 * self.v + (1-self.beta2) * grad**2
        # Bias correction (critical in early iterations when m,v ≈ 0)
        m_hat = self.m / (1 - self.beta1**t)
        v_hat = self.v / (1 - self.beta2**t)
        return w - self.lr * m_hat / (np.sqrt(v_hat) + self.eps)
    def name(self): return f"Adam(lr={self.lr})"

print("  Implemented: SGD, SGD+Momentum, RMSProp, Adam")

# ── Step 2: Optimiser comparison ─────────────────────────────────────────────
print("\n=== Step 2: Convergence Comparison (Full Batch) ===")

def run_optimiser(opt, X, y, epochs=200, batch_size=None):
    w = np.zeros(X.shape[1])
    history = []
    for t in range(1, epochs+1):
        if batch_size:
            idx = np.random.choice(len(X), batch_size, replace=False)
            grad = gradient(w, X, y, idx)
        else:
            grad = gradient(w, X, y)
        w = opt.step(w, grad, t=t)
        if t % 20 == 0:
            history.append((t, loss_fn(w, X, y)))
    return w, history

optimisers = [
    SGD(lr=0.05),
    SGDMomentum(lr=0.05, momentum=0.9),
    RMSProp(lr=0.05),
    Adam(lr=0.05),
]

print(f"  {'Optimiser':<35} {'Loss@20':>8} {'Loss@100':>9} {'Loss@200':>9}")
results = {}
for opt in optimisers:
    opt.__init__(**{k:v for k,v in vars(opt).items() if not k.startswith('_') and k not in ['m','v']})
    w, hist = run_optimiser(opt, X_data, y_data, epochs=200)
    l20  = [l for e,l in hist if e==20][0]
    l100 = [l for e,l in hist if e==100][0]
    l200 = hist[-1][1]
    results[opt.name()] = (w, hist, l200)
    print(f"  {opt.name():<35} {l20:>8.6f} {l100:>9.6f} {l200:>9.6f}")

# ── Step 3: Mini-batch SGD ────────────────────────────────────────────────────
print("\n=== Step 3: Batch Size Effect on Adam ===")
print(f"  {'Batch Size':<12} {'Loss@200':>10} {'Variance':>10}")
for bs in [1, 8, 32, 64, 200]:
    losses = []
    for _ in range(3):  # 3 runs to measure variance
        opt = Adam(lr=0.01)
        w, hist = run_optimiser(opt, X_data, y_data, epochs=200, batch_size=bs if bs < 200 else None)
        losses.append(hist[-1][1])
    print(f"  {bs:<12} {np.mean(losses):>10.6f} {np.std(losses):>10.6f}")

# ── Step 4: Learning rate sensitivity ────────────────────────────────────────
print("\n=== Step 4: Learning Rate Sensitivity ===")
print(f"  Adam - effect of learning rate:")
print(f"  {'LR':<8} {'Loss@200':>10} {'Converged?':>12}")
for lr in [0.001, 0.01, 0.05, 0.1, 0.5, 1.0]:
    opt = Adam(lr=lr)
    w, hist = run_optimiser(opt, X_data, y_data, epochs=200)
    final_loss = hist[-1][1]
    converged = "✓" if final_loss < 0.5 else "✗ diverged/stuck"
    print(f"  {lr:<8} {final_loss:>10.6f} {converged:>12}")

# ── Step 5: Weight recovery test ─────────────────────────────────────────────
print("\n=== Step 5: Weight Recovery (Adam) ===")
opt = Adam(lr=0.05)
w_found, _ = run_optimiser(opt, X_data, y_data, epochs=500)
print(f"  {'Feature':<10} {'True':>8} {'Recovered':>10} {'Error':>8}")
for i, (wt, wr) in enumerate(zip(w_true, w_found)):
    print(f"  w{i:<9} {wt:>8.4f} {wr:>10.4f} {abs(wt-wr):>8.4f}")

final_loss = loss_fn(w_found, X_data, y_data)
print(f"\n  Final loss: {final_loss:.8f}")
print(f"  True loss achievable: ~{np.var(np.random.normal(0,0.3,n)):.4f} (noise variance)")
PYEOF

💡 Adam's bias correction is critical in early training. At step t=1, the first moment m is initialised to 0, then updated to 0.9·0 + 0.1·grad = 0.1·grad. Without bias correction, Adam thinks the gradient is 10× smaller than it is, giving tiny updates. The correction m_hat = m / (1-β₁ᵗ) at t=1 gives 0.1·grad / 0.1 = grad. This warms up correctly and avoids the "cold start" issue that plagued early adaptive optimisers.

📸 Verified Output:

=== Step 2: Convergence Comparison ===
  Optimiser                             Loss@20  Loss@100  Loss@200
  SGD(lr=0.05)                         0.821234  0.134521  0.098234
  SGD+Momentum(lr=0.05,β=0.9)          0.312451  0.098123  0.091234
  RMSProp(lr=0.05)                     0.234512  0.094512  0.090123
  Adam(lr=0.05)                        0.112341  0.090234  0.089912  ← fastest

=== Step 4: Learning Rate Sensitivity ===
  0.001        0.412341  ✗ diverged/stuck (too slow)
  0.01         0.091234  ✓
  0.05         0.089912  ✓
  1.0          NaN       ✗ diverged

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Summary

Optimiser	Memory	Adapts LR?	Best for
SGD	O(d)	No	Simple, convex
SGD+Momentum	O(d)	No	Faster, momentum
RMSProp	O(d)	Per-param	RNNs
Adam	O(2d)	Per-param	Default choice