Gradient Descent

Your Challenge: Lost in the Mountains

Imagine you are dropped onto a random spot in a vast, foggy mountain range at night.

You can’t see the bottom (the valley).
You can’t see more than 3 feet in front of you.
You have no map.

Your Goal: Find the absolute lowest point in the entire valley (the minimum). How do you do it? You can’t “solve” the mountain. You can’t just teleport to the bottom. You have to feel your way down.

You feel the slope under your feet.
You take a small step downhill.
You repeat this thousands of times.

Eventually, you reach the bottom.

The Algorithm Visualized

This is Gradient Descent. It’s the “blind hiker” algorithm.

The Mountain: Your Loss Function (Error).
Your Position: The current weights of your model.
The Slope: The Gradient.
The Step Size: The Learning Rate.
The Bottom: The Optimal Weights (Best Model).

🔗 ML Connection: Gradient descent is THE learning algorithm. Here’s where it runs:

System	Parameters	Gradient Steps
GPT-4	1.7 trillion	Billions of updates
Stable Diffusion	1 billion	~1 million steps
BERT	340 million	~1 million steps
Your first model	2 (w, b)	~100-1000 steps

The math is identical—only the scale changes!

The Core Intuition

Why We Need It

In high school math, you found the minimum by setting the derivative to zero:

f'(x) = 0

That works for simple parabolas like

x^2

. But in Deep Learning, your function looks like a crumpled piece of paper in 1,000,000 dimensions. You cannot solve

f'(x) = 0

algebraically. It’s impossible. So instead of solving for the answer directly, we search for it iteratively.

The Code: A Simple Descent

Let’s implement the “blind hiker” logic for a simple valley:

f(x) = x^2 - 4x + 5

import numpy as np

# 1. The Mountain (Loss Function)
def f(x):
    return x**2 - 4*x + 5

# 2. The Slope (Gradient)
def gradient(x):
    return 2*x - 4  # Derivative of x² - 4x + 5

# 3. The Hike
x = 0  # Start at random spot
learning_rate = 0.1  # Size of each step

print(f"Start at x={x}")

for step in range(20):
    # Feel the slope
    grad = gradient(x)
    
    # Take a step downhill (opposite to gradient)
    x = x - learning_rate * grad
    
    print(f"Step {step+1}: Moved to x={x:.4f}, Height={f(x):.4f}")

print(f"\nReached bottom at x={x:.4f}")

Output:

Start at x=0
Step 1: Moved to x=0.4000, Height=3.5600
Step 2: Moved to x=0.7200, Height=2.6384
...
Step 20: Moved to x=1.9769, Height=1.0005
Reached bottom at x=1.9769

Result: You started at 0 and walked your way to ~2 (the true minimum). You solved it without algebra!

🎮 Interactive Visualization: Explore gradient descent interactively!

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from IPython.display import HTML

def f(x):
    return x**2 - 4*x + 5

def gradient(x):
    return 2*x - 4

# Animate gradient descent
fig, ax = plt.subplots(figsize=(10, 6))
x_range = np.linspace(-1, 5, 100)
ax.plot(x_range, f(x_range), 'b-', linewidth=2, label='f(x) = x² - 4x + 5')
ax.set_xlabel('x')
ax.set_ylabel('f(x)')
ax.set_title('Gradient Descent Animation')
ax.legend()

# Initialize point
point, = ax.plot([], [], 'ro', markersize=15)
path_x, path_y = [], []
path_line, = ax.plot([], [], 'r--', alpha=0.5)

x_current = 0
lr = 0.1

def init():
    point.set_data([], [])
    path_line.set_data([], [])
    return point, path_line

def animate(frame):
    global x_current, path_x, path_y
    if frame == 0:
        x_current = 0
        path_x, path_y = [0], [f(0)]
    else:
        grad = gradient(x_current)
        x_current = x_current - lr * grad
        path_x.append(x_current)
        path_y.append(f(x_current))
    
    point.set_data([x_current], [f(x_current)])
    path_line.set_data(path_x, path_y)
    ax.set_title(f'Step {frame}: x = {x_current:.4f}, f(x) = {f(x_current):.4f}')
    return point, path_line

anim = FuncAnimation(fig, animate, init_func=init, frames=25, interval=300, blit=True)
HTML(anim.to_jshtml())  # Run in Jupyter!

The Algorithm

Mathematical Formulation

Update rule:

x_{new} = x_{old} - \alpha \cdot \nabla f(x_{old})

Where:

$\alpha$ = learning rate (step size)
$\nabla f$ = gradient (direction of steepest ascent)
We subtract because we want to go downhill

Pseudocode

1. Initialize parameters randomly
2. Repeat until convergence:
   a. Compute gradient at current point
   b. Update parameters: x = x - α × gradient
   c. Check if converged (gradient ≈ 0)
3. Return optimized parameters

Example 1: Training Your First Model

The Problem

You want to predict house prices based on square footage.

\text{Price} = w \cdot \text{sqft} + b

Your Goal: Find the best

w

(weight) and

b

(bias) that minimize the error.

The Data

# Training data
sqft = np.array([1.0, 1.5, 2.0, 2.5, 3.0]) # in 1000s sqft
prices = np.array([200, 250, 300, 350, 400])  # in $1000s

Gradient Descent Training

def predict(x, w, b):
    return w * x + b

def compute_gradients(w, b, x, y):
    n = len(x)
    pred = predict(x, w, b)
    error = pred - y
    
    # Gradients (Partial Derivatives of MSE)
    dL_dw = (2/n) * np.sum(error * x)
    dL_db = (2/n) * np.sum(error)
    return dL_dw, dL_db

# 1. Initialize randomly
w, b = 0.0, 0.0
learning_rate = 0.01

print("Training started...")
for epoch in range(1000):
    # 2. Compute Gradient
    grad_w, grad_b = compute_gradients(w, b, sqft, prices)
    
    # 3. Update Parameters (Step Downhill)
    w = w - learning_rate * grad_w
    b = b - learning_rate * grad_b
    
    if epoch % 100 == 0:
        loss = np.mean((predict(sqft, w, b) - prices)**2)
        print(f"Epoch {epoch}: Loss={loss:.2f}, w={w:.2f}, b={b:.2f}")

print(f"\nFinal Model: Price = {w:.2f} * sqft + {b:.2f}")
print("True Answer: Price = 100 * sqft + 100")

Result: Your model learned the relationship perfectly!

Example 2: Optimizing Your Prices

The Scenario

You run an e-commerce site. You control two things:

$x_1$ = Ad spend ($1000s)
$x_2$ = Discount percentage

Your Revenue Function (Unknown to you, but we simulate it):

R(x_1, x_2) = 100x_1 + 50x_2 - x_1^2 - x_2^2

Your Goal: Maximize Revenue (which means minimizing Negative Revenue).

Gradient Descent Optimization

def neg_revenue_gradient(x1, x2):
    # Gradient of -R (to minimize)
    # R = 100x + 50y - x^2 - y^2
    # dR/dx = 100 - 2x  ->  d(-R)/dx = 2x - 100
    # dR/dy = 50 - 2y   ->  d(-R)/dy = 2y - 50
    grad_x1 = 2*x1 - 100
    grad_x2 = 2*x2 - 50
    return np.array([grad_x1, grad_x2])

# Start with random strategy
strategy = np.array([0.0, 0.0]) # $0 ads, 0% discount
lr = 0.1

for step in range(50):
    grad = neg_revenue_gradient(strategy[0], strategy[1])
    strategy = strategy - lr * grad
    
    if step % 10 == 0:
        print(f"Step {step}: Ads=${strategy[0]:.2f}k, Discount={strategy[1]:.2f}%")

print(f"\nOptimal Strategy: Ads=${strategy[0]:.2f}k, Discount={strategy[1]:.2f}%")

Output:

Optimal Strategy: Ads=$50.00k, Discount=$25.00%

Insight: You found the optimal business strategy just by following the gradient!

Example 3: Training Your Neural Network

The Challenge

You want to train a simple neural network to solve a problem.

Input: $x$
Target: $y$
Model: $y_{pred} = \sigma(wx + b)$

The Code

def sigmoid(z): return 1 / (1 + np.exp(-z))

# Training data (XOR-like)
X = np.array([0, 1])
y = np.array([0, 1])

# Initialize
w, b = 0.5, 0.0
lr = 1.0

print("Training Neural Network...")
for epoch in range(100):
    total_grad_w = 0
    total_grad_b = 0
    
    for i in range(len(X)):
        # Forward
        z = w * X[i] + b
        pred = sigmoid(z)
        
        # Backward (Chain Rule!)
        error = pred - y[i]
        dL_dpred = 2 * error
        dpred_dz = pred * (1 - pred)
        dz_dw = X[i]
        dz_db = 1
        
        grad_w = dL_dpred * dpred_dz * dz_dw
        grad_b = dL_dpred * dpred_dz * dz_db
        
        # Accumulate gradients
        total_grad_w += grad_w
        total_grad_b += grad_b
    
    # Update weights (Gradient Descent Step)
    w = w - lr * (total_grad_w / len(X))
    b = b - lr * (total_grad_b / len(X))

print(f"Final Weights: w={w:.2f}, b={b:.2f}")
print(f"Prediction for 0: {sigmoid(w*0 + b):.2f}")
print(f"Prediction for 1: {sigmoid(w*1 + b):.2f}")

This is Deep Learning. It’s just Gradient Descent applied to a lot of weights!

Learning Rate: The “Goldilocks” Problem

Choosing the step size (

\alpha

) is the most important decision you make.

1. Too Small (The Turtle)

Symptom: Loss decreases veeeery slowly.
Result: You run out of time/patience before reaching the bottom.

2. Too Large (The Grasshopper)

Symptom: Loss bounces around or even INCREASES.
Result: You overshoot the valley and never converge.

3. Just Right (Goldilocks)

Symptom: Loss decreases steadily and quickly.
Result: You reach the minimum efficiently.

How to Find It?

Start with 0.01 or 0.001. If loss is slow, increase it (0.1). If loss explodes, decrease it (0.0001).

Variants of Gradient Descent

Batch Gradient Descent

Uses all data points to compute gradient:

# Compute gradient using ALL data
gradient = compute_gradient(all_data)
params = params - lr * gradient

Pros: Stable, smooth convergence
Cons: Slow for large datasets

Stochastic Gradient Descent (SGD)

Uses one data point at a time:

for each data_point in dataset:
    gradient = compute_gradient(data_point)
    params = params - lr * gradient

Pros: Fast, can escape local minima
Cons: Noisy, unstable

Mini-Batch Gradient Descent

Uses small batches of data:

for batch in dataset.batches(batch_size=32):
    gradient = compute_gradient(batch)
    params = params - lr * gradient

Pros: Best of both worlds
Cons: Need to choose batch size This is what everyone uses in practice!

Convergence Criteria

When to Stop?

Option 1: Gradient is small

if np.linalg.norm(gradient) < 1e-6:
    break  # Converged!

Option 2: Loss stops improving

if abs(loss_new - loss_old) < 1e-6:
    break  # Converged!

Option 3: Maximum iterations

if epoch >= max_epochs:
    break  # Give up

Practice Exercises

Exercise 1: Implement Gradient Descent

# Minimize f(x) = x⁴ - 3x³ + 2
# TODO:
# 1. Compute the derivative
# 2. Implement gradient descent
# 3. Find the minimum
# 4. Try different learning rates

🎯 Practice Exercises & Real-World Applications

Challenge yourself! These exercises let you feel gradient descent working on real problems.

Exercise 1: Train a Linear Model by Hand 📈

Implement gradient descent to fit a line to data:

import numpy as np

# Dataset: House sizes and prices
sizes = np.array([1000, 1500, 2000, 2500, 3000])  # sq ft
prices = np.array([200, 280, 350, 400, 480])       # $1000s

# Model: price = w * size + b
# Loss: Mean Squared Error

# TODO:
# 1. Initialize w=0.1, b=50
# 2. Compute gradients dL/dw and dL/db
# 3. Run gradient descent for 1000 steps
# 4. Plot the learning curve (loss vs steps)

💡 Solution

import numpy as np

# Data
sizes = np.array([1000, 1500, 2000, 2500, 3000])
prices = np.array([200, 280, 350, 400, 480])
n = len(sizes)

def predict(w, b):
    return w * sizes + b

def mse_loss(w, b):
    preds = predict(w, b)
    return np.mean((preds - prices) ** 2)

def gradients(w, b):
    preds = predict(w, b)
    errors = preds - prices
    dw = 2 * np.mean(errors * sizes)
    db = 2 * np.mean(errors)
    return dw, db

print("📈 Linear Regression with Gradient Descent")
print("=" * 55)

# Initialize
w, b = 0.1, 50
lr = 1e-7  # Small learning rate (sizes are big numbers)
history = []

print("\n🚀 Training Progress:")
print(f"{'Step':<8} {'w':<12} {'b':<12} {'Loss':<12}")
print("-" * 44)

for step in range(1001):
    loss = mse_loss(w, b)
    history.append(loss)
    
    if step % 200 == 0:
        print(f"{step:<8} {w:<12.6f} {b:<12.2f} {loss:<12.2f}")
    
    dw, db = gradients(w, b)
    w = w - lr * dw
    b = b - lr * db

print("\n✅ Final Model:")
print(f"   price = {w:.4f} × size + {b:.2f}")
print(f"   Final loss: {mse_loss(w, b):.2f}")

# Test predictions
print("\n🏠 Predictions vs Actual:")
print(f"   {'Size':<8} {'Actual':<10} {'Predicted':<10} {'Error':<10}")
print("-" * 40)
preds = predict(w, b)
for s, actual, pred in zip(sizes, prices, preds):
    print(f"   {s:<8} ${actual:>6}k    ${pred:>6.1f}k    {abs(actual-pred):>6.1f}k")

# Learning curve visualization
print("\n📉 Loss Reduction:")
for i, s in enumerate([0, 200, 500, 1000]):
    bar_len = int(50 * history[s] / history[0])
    bar = "█" * bar_len
    print(f"   Step {s:4}: {bar} {history[s]:.1f}")

Real-World Insight: This is exactly how scikit-learn’s LinearRegression.fit() works under the hood (though it uses closed-form solution when possible for speed).

Exercise 2: Learning Rate Experiments 🎛️

Explore how learning rate affects convergence:

import numpy as np

def f(x):
    """A simple valley: f(x) = x² - 4x + 5"""
    return x**2 - 4*x + 5

def gradient(x):
    return 2*x - 4

# TODO:
# 1. Try learning rates: 0.01, 0.1, 0.5, 1.0, 1.1
# 2. Run 20 steps from x=0
# 3. Observe: which converges? which diverges? which oscillates?
# 4. Find the "critical" learning rate where things break

💡 Solution

import numpy as np

def f(x):
    return x**2 - 4*x + 5

def gradient(x):
    return 2*x - 4

def run_gd(lr, steps=20, start=0):
    x = start
    history = [x]
    for _ in range(steps):
        x = x - lr * gradient(x)
        history.append(x)
        if abs(x) > 1000:  # Diverged
            break
    return history

print("🎛️ Learning Rate Experiments")
print("=" * 55)
print("Target: x* = 2.0 (minimum of f(x) = x² - 4x + 5)")

learning_rates = [0.01, 0.1, 0.5, 0.9, 1.0, 1.1, 1.5]

print("\n📊 Results after 20 steps:")
print(f"{'LR':<8} {'Final x':<12} {'Distance to optimal':<20} {'Status'}")
print("-" * 60)

for lr in learning_rates:
    history = run_gd(lr)
    final_x = history[-1]
    distance = abs(final_x - 2.0)
    
    if abs(final_x) > 1000:
        status = "💥 DIVERGED"
    elif abs(final_x - 2.0) < 0.01:
        status = "✅ Converged"
    elif len(set([round(x, 4) for x in history[-5:]])) > 1 and distance < 1:
        status = "🔄 Oscillating"
    else:
        status = "⏳ Still converging"
    
    if abs(final_x) < 1000:
        print(f"{lr:<8} {final_x:<12.4f} {distance:<20.6f} {status}")
    else:
        print(f"{lr:<8} {'inf':>12} {'∞':>20} {status}")

# Detailed trajectory for key learning rates
print("\n📈 Detailed Trajectories (first 10 steps):")
for lr in [0.1, 0.9, 1.1]:
    history = run_gd(lr, steps=10)
    print(f"\n   LR = {lr}:")
    print(f"   " + " → ".join([f"{x:.2f}" for x in history[:8]]) + 
          (" → 💥" if abs(history[-1]) > 100 else ""))

# Critical learning rate analysis
print("\n🎯 Critical Learning Rate Analysis:")
print("   For f(x) = x², the 2nd derivative f''(x) = 2")
print("   Critical LR = 2/f''(x) = 2/2 = 1.0")
print("   - LR < 1.0: Converges")
print("   - LR = 1.0: Oscillates forever (x → 0 → 4 → 0 → 4...)")
print("   - LR > 1.0: Diverges!")

Real-World Insight: This is why learning rate is the most important hyperparameter in deep learning! Too small = slow training, too large = divergence. That’s why Adam optimizer auto-adapts the learning rate.

Exercise 3: Escaping Local Minima 🕳️

Navigate a function with multiple minima:

import numpy as np

def f(x):
    """Function with local minima: f(x) = x⁴ - 8x² + x"""
    return x**4 - 8*x**2 + x

def gradient(x):
    return 4*x**3 - 16*x + 1

# This function has:
# - Local minimum near x ≈ -2
# - Global minimum near x ≈ 2  
# - Local maximum near x ≈ 0

# TODO:
# 1. Start at x = -3 and run gradient descent
# 2. Do you reach the global minimum?
# 3. Try adding "momentum" to escape local minima
# 4. Try random restarts - which works better?

💡 Solution

import numpy as np

def f(x):
    return x**4 - 8*x**2 + x

def gradient(x):
    return 4*x**3 - 16*x + 1

def gd_basic(start, lr=0.01, steps=200):
    x = start
    for _ in range(steps):
        x = x - lr * gradient(x)
    return x, f(x)

def gd_momentum(start, lr=0.01, momentum=0.9, steps=200):
    x = start
    velocity = 0
    for _ in range(steps):
        velocity = momentum * velocity + lr * gradient(x)
        x = x - velocity
    return x, f(x)

def random_restarts(n_restarts=10, lr=0.01, steps=200):
    best_x, best_f = None, float('inf')
    for _ in range(n_restarts):
        start = np.random.uniform(-4, 4)
        x, fx = gd_basic(start, lr, steps)
        if fx < best_f:
            best_x, best_f = x, fx
    return best_x, best_f

print("🕳️ Escaping Local Minima")
print("=" * 55)

# Find actual minima numerically
print("\n📍 Function Analysis:")
xs = np.linspace(-3, 3, 1000)
ys = [f(x) for x in xs]
local_min = xs[np.argmin(ys[:500])]  # Left half
global_min = xs[np.argmin(ys)]
print(f"   Local minimum: x ≈ {local_min:.2f}, f(x) = {f(local_min):.2f}")
print(f"   Global minimum: x ≈ {global_min:.2f}, f(x) = {f(global_min):.2f}")

# Test different methods
print("\n🧪 Experiment Results (starting at x = -3):")
print("-" * 55)

# Basic GD
x1, f1 = gd_basic(-3)
print(f"   Basic GD:      x = {x1:.4f}, f(x) = {f1:.4f}", 
      "✅ Global!" if x1 > 0 else "❌ Stuck local")

# Momentum
x2, f2 = gd_momentum(-3)
print(f"   With Momentum: x = {x2:.4f}, f(x) = {f2:.4f}",
      "✅ Global!" if x2 > 0 else "❌ Stuck local")

# Random restarts
x3, f3 = random_restarts(10)
print(f"   Random Restart: x = {x3:.4f}, f(x) = {f3:.4f}",
      "✅ Global!" if x3 > 0 else "❌ Stuck local")

# Starting position analysis
print("\n📊 Impact of Starting Position:")
print(f"   {'Start':<8} {'Basic GD':<15} {'Momentum':<15}")
print("-" * 40)
for start in [-3, -1, 0, 1, 3]:
    x_basic, f_basic = gd_basic(start)
    x_mom, f_mom = gd_momentum(start)
    print(f"   {start:<8} x={x_basic:>6.2f} ({f_basic:>6.2f})   x={x_mom:>6.2f} ({f_mom:>6.2f})")

print("\n💡 Key Insights:")
print("   - Starting position greatly affects which minimum you find")
print("   - Momentum helps escape shallow local minima")
print("   - Random restarts are simple but effective")
print("   - Real neural networks use all of these tricks!")

Real-World Insight: Deep learning uses all these tricks! Random initialization, momentum (in Adam optimizer), and sometimes explicit random restarts. That’s why training the same model twice can give different results!

Exercise 4: Mini-Batch Gradient Descent 📦

Implement mini-batch training on a larger dataset:

import numpy as np

# Generate a larger dataset
np.random.seed(42)
n_samples = 1000
X = np.random.randn(n_samples, 3)  # 3 features
true_w = np.array([2.0, -1.5, 0.5])
y = X @ true_w + np.random.randn(n_samples) * 0.5

# TODO:
# 1. Implement full-batch gradient descent
# 2. Implement mini-batch GD with batch_size=32
# 3. Compare convergence speed (iterations to reach loss < 1.0)
# 4. Compare wall-clock time (which is faster per epoch?)

💡 Solution

import numpy as np
import time

np.random.seed(42)
n_samples = 1000
X = np.random.randn(n_samples, 3)
true_w = np.array([2.0, -1.5, 0.5])
y = X @ true_w + np.random.randn(n_samples) * 0.5

def loss(w, X, y):
    return np.mean((X @ w - y) ** 2)

def gradient_full(w, X, y):
    """Gradient using all data"""
    errors = X @ w - y
    return 2 * X.T @ errors / len(y)

def gradient_batch(w, X_batch, y_batch):
    """Gradient using mini-batch"""
    errors = X_batch @ w - y_batch
    return 2 * X_batch.T @ errors / len(y_batch)

def full_batch_gd(X, y, lr=0.01, epochs=100):
    w = np.zeros(3)
    history = []
    
    start = time.time()
    for epoch in range(epochs):
        grad = gradient_full(w, X, y)
        w = w - lr * grad
        history.append(loss(w, X, y))
    elapsed = time.time() - start
    
    return w, history, elapsed

def mini_batch_gd(X, y, batch_size=32, lr=0.01, epochs=100):
    w = np.zeros(3)
    history = []
    n = len(y)
    
    start = time.time()
    for epoch in range(epochs):
        # Shuffle data
        indices = np.random.permutation(n)
        X_shuf = X[indices]
        y_shuf = y[indices]
        
        # Process mini-batches
        for i in range(0, n, batch_size):
            X_batch = X_shuf[i:i+batch_size]
            y_batch = y_shuf[i:i+batch_size]
            grad = gradient_batch(w, X_batch, y_batch)
            w = w - lr * grad
        
        history.append(loss(w, X, y))
    elapsed = time.time() - start
    
    return w, history, elapsed

print("📦 Mini-Batch vs Full-Batch Gradient Descent")
print("=" * 55)

# Run experiments
w_full, hist_full, time_full = full_batch_gd(X, y)
w_mini, hist_mini, time_mini = mini_batch_gd(X, y, batch_size=32)

print(f"\n📊 Results after 100 epochs:")
print("-" * 55)
print(f"{'Method':<15} {'Final Loss':<12} {'Time (s)':<10} {'Weights'}")
print("-" * 55)
print(f"{'Full Batch':<15} {hist_full[-1]:<12.4f} {time_full:<10.4f} {w_full.round(3)}")
print(f"{'Mini-Batch':<15} {hist_mini[-1]:<12.4f} {time_mini:<10.4f} {w_mini.round(3)}")
print(f"{'True Weights':<15} {'-':<12} {'-':<10} {true_w}")

# Convergence analysis
def epochs_to_threshold(history, threshold=1.0):
    for i, l in enumerate(history):
        if l < threshold:
            return i + 1
    return len(history)

e_full = epochs_to_threshold(hist_full)
e_mini = epochs_to_threshold(hist_mini)

print(f"\n⏱️ Epochs to reach loss < 1.0:")
print(f"   Full Batch: {e_full} epochs")
print(f"   Mini-Batch: {e_mini} epochs")

# Per-epoch analysis
print(f"\n📈 Loss at key epochs:")
print(f"   {'Epoch':<8} {'Full Batch':<15} {'Mini-Batch':<15}")
print("-" * 40)
for e in [1, 5, 10, 25, 50, 100]:
    print(f"   {e:<8} {hist_full[e-1]:<15.4f} {hist_mini[e-1]:<15.4f}")

print("\n💡 Key Insights:")
print("   - Mini-batch has noisier updates but often converges faster")
print("   - Mini-batch uses less memory (important for big data)")
print("   - The noise in mini-batch can help escape local minima")
print("   - This is why batch_size is a key hyperparameter!")

Real-World Insight: GPT-4 and other large language models are trained exclusively with mini-batch gradient descent. A single batch might process hundreds of examples, but it’s still a tiny fraction of the trillions of training tokens!

🎯 Optimizer Selection Guide: Which One Should You Use?

Common Mistake: Many beginners stick with vanilla SGD or randomly pick Adam. Understanding when to use which optimizer can save hours of training time!

Decision Flowchart

╔══════════════════════════════════════════╗
║ What's your use case?                         ║
╚══════════════════════════════════════════╝
           │
     ┌─────┴───────────────────────────┐
     │                                   │
  Just starting /            Computer Vision /
  Quick prototype            Image models
     │                                   │
     ↓                                   ↓
  ┌─────────┐                    ┌─────────────┐
  │  Adam    │                    │ SGD + Momentum│
  │ lr=1e-3 │                    │    + Nesterov │
  └─────────┘                    └─────────────┘
     │                                   │
     │                                   │
  NLP / Transformers          Fine-tuning 
     │                       pre-trained
     ↓                                   │
  ┌─────────────┐                    │
  │ AdamW        │                    ↓
  │ (Adam +      │            ┌─────────────┐
  │  weight decay)│            │   AdamW     │
  └─────────────┘            │ lr=1e-5 to  │
                               │   1e-4      │
                               └─────────────┘

Optimizer Comparison Table

Optimizer	Best For	Learning Rate	Pros	Cons
SGD	Understanding, simple models	0.01-0.1	Simple, predictable	Slow, gets stuck
SGD + Momentum	CV models (ResNet, etc.)	0.01-0.1	Fast, smooth	Needs tuning
Adam	Most cases, quick prototypes	1e-3 to 1e-4	Works out of box	Can overfit
AdamW	Transformers, NLP, fine-tuning	1e-4 to 1e-5	Best for LLMs	Slightly slower
RMSprop	RNNs, unstable gradients	1e-3 to 1e-4	Handles varying gradients	Less common now

Learning Rate Guidelines

# Starting learning rates by model type
lr_guidelines = {
    'linear_regression': 0.01,
    'small_neural_net': 0.001,
    'cnn_from_scratch': 0.01,  # with SGD
    'cnn_transfer_learning': 0.0001,
    'transformer_training': 0.0001,
    'llm_fine_tuning': 0.00001,  # Very small!
    'stable_diffusion': 0.00001,
}

print("Always start with these, then adjust based on:")
print("  - Loss not decreasing? Try 10x smaller")
print("  - Loss decreasing too slowly? Try 2-5x larger")
print("  - Loss oscillating? Try 2-10x smaller")

🚨 Real-World Challenge: Messy Training Data

Production Reality: Training data in the real world has issues that can completely break gradient descent!

Handling Noisy Labels

import numpy as np

# Simulated noisy data (10% of labels are wrong)
np.random.seed(42)
X = np.random.randn(1000, 10)
y_true = (X[:, 0] + X[:, 1] > 0).astype(float)
y_noisy = y_true.copy()
noise_mask = np.random.rand(1000) < 0.10  # 10% noise
y_noisy[noise_mask] = 1 - y_noisy[noise_mask]  # Flip labels

print(f"Noisy labels: {noise_mask.sum()} out of {len(y_true)}")

# Strategies to handle noisy labels:
# 1. Label Smoothing
def label_smoothing(y, epsilon=0.1):
    """Soften hard labels to reduce overconfidence on wrong labels."""
    return y * (1 - epsilon) + 0.5 * epsilon

# 2. Early Stopping (networks learn clean patterns first)
# 3. Mixup / Data Augmentation
# 4. Confident Learning (identify and remove noisy samples)

Handling Missing Features

# Real data often has missing values
X_with_missing = X.copy()
missing_mask = np.random.rand(*X.shape) < 0.05  # 5% missing
X_with_missing[missing_mask] = np.nan

print(f"Missing values: {missing_mask.sum()} out of {X.size}")

# Strategies:
# 1. Imputation before training
from sklearn.impute import SimpleImputer
imputer = SimpleImputer(strategy='mean')
X_imputed = imputer.fit_transform(X_with_missing)

# 2. Mask and learn (for neural nets)
# Replace NaN with 0 and add a "missing indicator" feature
X_masked = np.nan_to_num(X_with_missing, nan=0)
missing_indicator = np.isnan(X_with_missing).astype(float)
X_augmented = np.hstack([X_masked, missing_indicator])

Handling Class Imbalance

# Example: Fraud detection (99% normal, 1% fraud)
y_imbalanced = np.zeros(10000)
y_imbalanced[:100] = 1  # Only 1% positive

print(f"Class distribution: {np.bincount(y_imbalanced.astype(int))}")

# Strategies:
# 1. Class Weights in Loss
class_weights = {0: 1.0, 1: 99.0}  # Weight minority class higher

# 2. Oversampling (SMOTE)
# 3. Undersampling
# 4. Focal Loss (reduces weight on easy examples)

def focal_loss(y_true, y_pred, gamma=2.0):
    """Focus on hard examples, ignore easy ones."""
    pt = y_true * y_pred + (1 - y_true) * (1 - y_pred)
    focal_weight = (1 - pt) ** gamma
    return -focal_weight * np.log(pt + 1e-8)

Rule of Thumb for Messy Data:

Always visualize your data before training
Check for outliers - they can dominate gradients
Monitor gradient norms - sudden spikes indicate problems
Use gradient clipping as a safety net

Key Takeaways

✅ Gradient descent = iterative optimization algorithm
✅ Follow gradient downhill = move opposite to gradient
✅ Learning rate = critical hyperparameter
✅ Mini-batch = best practice for large datasets
✅ Powers all ML = from linear regression to GPT

Learning Rate Schedulers: Advanced Techniques

Pro tip: The best learning rate changes during training! Start high, then decrease.

Popular Schedulers

Scheduler	Formula	When to Use
Step Decay	$\alpha_t = \alpha_0 \cdot \gamma^{\lfloor t/s \rfloor}$	Simple, predictable
Exponential	$\alpha_t = \alpha_0 \cdot e^{-\lambda t}$	Smooth decay
Cosine Annealing	$\alpha_t = \alpha_{min} + \frac{1}{2}(\alpha_0 - \alpha_{min})(1 + \cos(\frac{t\pi}{T}))$	State-of-the-art
Warmup + Decay	Linear increase then decrease	Transformers, LLMs

import torch
import torch.optim as optim
from torch.optim.lr_scheduler import StepLR, CosineAnnealingLR, OneCycleLR

# Example: Using schedulers in PyTorch
model = torch.nn.Linear(10, 1)
optimizer = optim.Adam(model.parameters(), lr=0.01)

# Option 1: Step decay (multiply by 0.1 every 10 epochs)
scheduler = StepLR(optimizer, step_size=10, gamma=0.1)

# Option 2: Cosine annealing (used by GPT, BERT)
scheduler = CosineAnnealingLR(optimizer, T_max=100, eta_min=1e-6)

# Option 3: One Cycle (super-convergence)
scheduler = OneCycleLR(optimizer, max_lr=0.1, total_steps=1000)

# Training loop
for epoch in range(100):
    # ... training code ...
    optimizer.step()
    scheduler.step()  # Update learning rate

When Training Gets Stuck: Debugging Checklist

🔴 Loss not decreasing

Learning rate too high? Try 10x smaller
Learning rate too low? Try 10x larger
Data issue? Check for NaN/Inf values
Bug in loss function? Verify on toy data

🔴 Loss oscillating wildly

Reduce learning rate by 2-10x
Add gradient clipping
Check for exploding gradients
Increase batch size for smoother gradients

🔴 Loss decreases then plateaus

Use learning rate scheduler
Add momentum or switch to Adam
Check if you’ve converged (that’s good!)
Try data augmentation

🔴 Training loss good, validation bad

Overfitting - add regularization
Reduce model complexity
Add dropout
Get more data

What’s Next?

Gradient descent is powerful, but basic. Can we do better? Can we converge faster? Can we escape local minima? Yes! Advanced optimization techniques like Momentum, Adam, and RMSprop!

Next: Optimization Techniques

Learn the optimizers that power modern deep learning

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​Gradient Descent

​Your Challenge: Lost in the Mountains

​The Algorithm Visualized

​The Core Intuition

​Why We Need It

​The Code: A Simple Descent

​The Algorithm

​Mathematical Formulation

​Pseudocode

​Example 1: Training Your First Model

​The Problem

​The Data

​Gradient Descent Training

​Example 2: Optimizing Your Prices

​The Scenario

​Gradient Descent Optimization

​Example 3: Training Your Neural Network

​The Challenge

​The Code

​Learning Rate: The “Goldilocks” Problem

​1. Too Small (The Turtle)

​2. Too Large (The Grasshopper)

​3. Just Right (Goldilocks)

​How to Find It?

​Variants of Gradient Descent

​Batch Gradient Descent

​Stochastic Gradient Descent (SGD)

​Mini-Batch Gradient Descent

​Convergence Criteria

​When to Stop?

​Practice Exercises

​Exercise 1: Implement Gradient Descent

​🎯 Practice Exercises & Real-World Applications

​Exercise 1: Train a Linear Model by Hand 📈

​Exercise 2: Learning Rate Experiments 🎛️

​Exercise 3: Escaping Local Minima 🕳️

​Exercise 4: Mini-Batch Gradient Descent 📦

​🎯 Optimizer Selection Guide: Which One Should You Use?

​Decision Flowchart

​Optimizer Comparison Table

​Learning Rate Guidelines

​🚨 Real-World Challenge: Messy Training Data

Gradient Descent

Your Challenge: Lost in the Mountains

The Algorithm Visualized

The Core Intuition

Why We Need It

The Code: A Simple Descent

The Algorithm

Mathematical Formulation

Pseudocode

Example 1: Training Your First Model

The Problem

The Data

Gradient Descent Training

Example 2: Optimizing Your Prices

The Scenario

Gradient Descent Optimization

Example 3: Training Your Neural Network

The Challenge

The Code

Learning Rate: The “Goldilocks” Problem

1. Too Small (The Turtle)

2. Too Large (The Grasshopper)

3. Just Right (Goldilocks)

How to Find It?

Variants of Gradient Descent

Batch Gradient Descent

Stochastic Gradient Descent (SGD)

Mini-Batch Gradient Descent

Convergence Criteria

When to Stop?

Practice Exercises

Exercise 1: Implement Gradient Descent

🎯 Practice Exercises & Real-World Applications

Exercise 1: Train a Linear Model by Hand 📈

Exercise 2: Learning Rate Experiments 🎛️

Exercise 3: Escaping Local Minima 🕳️

Exercise 4: Mini-Batch Gradient Descent 📦

🎯 Optimizer Selection Guide: Which One Should You Use?

Decision Flowchart

Optimizer Comparison Table

Learning Rate Guidelines

🚨 Real-World Challenge: Messy Training Data

Handling Noisy Labels

Handling Missing Features

Handling Class Imbalance

Key Takeaways

Learning Rate Schedulers: Advanced Techniques

Popular Schedulers

When Training Gets Stuck: Debugging Checklist

What’s Next?