Gradient descent optimization in ML Python - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

What does the learning rate control in gradient descent?

Imagine you are walking down a hill to reach the lowest point. In gradient descent, what does the learning rate control?

A The size of each step you take down the hill

B The speed at which the hill moves

C The total distance from the hill to the valley

D The direction you walk towards the hilltop

💡 Hint

Think about how big or small your steps are when walking down.

The learning rate controls how big each step is when moving towards the minimum. Too big steps can overshoot, too small steps slow progress.

❓ Predict Output

intermediate

Output of gradient descent update step

What is the value of w after one gradient descent update?

ML Python

w = 2.0
learning_rate = 0.1
gradient = 3.0
w = w - learning_rate * gradient
print(round(w, 2))

A 1.3

B 1.7

C 2.3

D 2.0

💡 Hint

Use the formula: new_w = old_w - learning_rate * gradient

Calculate: 2.0 - 0.1 * 3.0 = 2.0 - 0.3 = 1.7

❓ Hyperparameter

advanced

Choosing the right learning rate for convergence

Which learning rate is most likely to cause the gradient descent to converge smoothly to the minimum?

A 10.0

B 1.0

C 0.0001

D 0.1

💡 Hint

Too small means slow progress, too large means overshooting.

A learning rate of 0.1 is balanced: not too small to be slow, not too large to overshoot. 1.0 or 10.0 are too big and cause divergence. 0.0001 is very slow.

❓ Metrics

advanced

Effect of learning rate on training loss curve

Which training loss curve shape best represents a too-large learning rate during gradient descent?

A Loss oscillates up and down without settling

B Loss steadily decreases and flattens near zero

C Loss increases steadily over time

D Loss remains constant throughout training

💡 Hint

Think about what happens if steps are too big and jump around the minimum.

A too-large learning rate causes the loss to jump around the minimum, causing oscillations instead of smooth decrease.

🔧 Debug

expert

Identify the error in this gradient descent update code

What error will this code produce when running?

ML Python

w = 1.0
learning_rate = 0.05
gradient = None
w = w - learning_rate * gradient
print(w)

A SyntaxError: invalid syntax

B NameError: name 'gradient' is not defined

C TypeError: unsupported operand type(s) for *: 'float' and 'NoneType'

D No error, prints 1.0

💡 Hint

Check the type of gradient before multiplication.

Multiplying a float by None causes a TypeError because None is not a number.

Practice

1. What is the main goal of gradient descent in machine learning?

easy

A. To minimize the error or loss function

B. To increase the number of model parameters

C. To randomly change model weights

D. To maximize the training time

Solution

1. Step 1: Understand gradient descent purpose

   Gradient descent is used to find the best model parameters by reducing errors.

2. Step 2: Connect to error minimization

   The algorithm updates parameters to minimize the loss function, which measures error.

3. Final Answer:

   To minimize the error or loss function -> Option A

4. Quick Check:

   Goal = minimize loss [OK]

Hint: Gradient descent reduces errors step-by-step [OK]

Common Mistakes:

• Thinking it increases parameters
• Believing it changes weights randomly
• Confusing it with training time increase

2. Which of the following is the correct update rule for a parameter \(\theta\) using gradient descent with learning rate \(\alpha\) and gradient \(\nabla J(\theta)\)?

easy

A. \(\theta = \theta - \alpha \nabla J(\theta)\)

B. \(\theta = \theta + \alpha \nabla J(\theta)\)

C. \(\theta = \alpha \nabla J(\theta) - \theta\)

D. \(\theta = \theta \times \alpha \nabla J(\theta)\)

Solution

1. Step 1: Recall gradient descent formula

   Gradient descent updates parameters by moving opposite to the gradient to reduce loss.

2. Step 2: Identify correct sign and terms

   The update is \(\theta = \theta - \alpha \nabla J(\theta)\), subtracting the gradient scaled by learning rate.

3. Final Answer:

   \(\theta = \theta - \alpha \nabla J(\theta)\) -> Option A

4. Quick Check:

   Update = subtract gradient [OK]

Hint: Subtract gradient times learning rate [OK]

Common Mistakes:

• Adding instead of subtracting gradient
• Multiplying parameters incorrectly
• Swapping terms order

3. Consider this Python code snippet for one step of gradient descent on parameter \(\theta\):

theta = 2.0
learning_rate = 0.1
gradient = 3.0

theta = theta - learning_rate * gradient
print(round(theta, 2))

What will be the printed output?

medium

A. -0.1

B. 1.3

C. 2.3

D. 1.7

Solution

1. Step 1: Calculate the update amount

   Update = learning_rate * gradient = 0.1 * 3.0 = 0.3

2. Step 2: Apply update to theta

   New theta = 2.0 - 0.3 = 1.7

3. Step 3: Check printed value

   Printed value is rounded to 2 decimals: 1.7

4. Final Answer:

   1.7 -> Option D

5. Quick Check:

   2.0 - 0.3 = 1.7 [OK]

Hint: Subtract learning_rate * gradient from theta [OK]

Common Mistakes:

• Adding instead of subtracting update
• Mixing gradient and learning rate
• Rounding incorrectly

4. The following code tries to perform one gradient descent step but has a bug:

theta = 5.0
learning_rate = 0.01
gradient = -4.0

theta = theta + learning_rate * gradient
print(theta)

What is the bug and how to fix it?

medium

A. No bug; code is correct

B. Bug: learning rate too small; fix by increasing it

C. Bug: sign error; fix by changing '+' to '-' in update

D. Bug: gradient sign wrong; fix by negating gradient

Solution

1. Step 1: Recall standard gradient descent formula

   The correct update is always \(\theta = \theta - \alpha \nabla J(\theta)\).

2. Step 2: Identify the bug in code

   The code uses '+' instead of '-', performing gradient ascent. With gradient = -4.0, code updates to 5.0 - 0.04 = 4.96 (wrong direction); correct is 5.0 + 0.04 = 5.04.

3. Final Answer:

   Bug: sign error; fix by changing '+' to '-' in update -> Option C

4. Quick Check:

   Always subtract lr * gradient [OK]

Hint: Always use '-' for gradient descent [OK]

Common Mistakes:

• Thinking '+' works for negative gradients
• Confusing ascent (+) with descent (-)
• Ignoring standard update rule

5. You run gradient descent with a very large learning rate \(\alpha = 10\) on a simple quadratic loss \(J(\theta) = (\theta - 3)^2\). What is the most likely outcome after a few steps starting from \(\theta=0\)?

hard

A. The parameter \(\theta\) converges smoothly to 3

B. The parameter \(\theta\) oscillates or diverges away from 3

C. The parameter \(\theta\) stays at 0 without change

D. The parameter \(\theta\) becomes negative infinity

Solution

1. Step 1: Understand effect of large learning rate

   A large learning rate causes big jumps in parameter updates, possibly overshooting the minimum.

2. Step 2: Analyze behavior on quadratic loss

   Since \(J(\theta) = (\theta - 3)^2\), gradient is \(2(\theta - 3)\). Large steps can cause \(\theta\) to jump past 3 repeatedly, causing oscillation or divergence.

3. Final Answer:

   The parameter \(\theta\) oscillates or diverges away from 3 -> Option B

4. Quick Check:

   Large learning rate causes oscillation [OK]

Hint: Too big steps cause oscillation, not smooth learning [OK]

Common Mistakes:

• Assuming large learning rate helps convergence
• Thinking parameter stays fixed
• Confusing divergence with negative infinity