Bias-variance tradeoff in ML Python - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the mean squared error (MSE) between predictions and true values.

ML Python

mse = ((predictions - true_values) [1] 2).mean()

Drag options to blanks, or click blank then click option'

A-

B+

C*

D**

Attempts:

3 left

💡 Hint

Common Mistakes

Using '*' which multiplies by 2, '+' or '-' instead of '**'.

Confusing multiplication with exponentiation.

The mean squared error is calculated by squaring the difference between predictions and true values. The operator for element-wise exponentiation is **.

2fill in blank

medium

Complete the code to split data into training and testing sets using scikit-learn.

ML Python

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=[1], random_state=42)

Drag options to blanks, or click blank then click option'

A2

B0.2

C0.5

D20

Attempts:

3 left

💡 Hint

Common Mistakes

Using integers like 2 or 20 instead of a float between 0 and 1.

Using 0.5 which splits data evenly but is less common.

The test_size parameter expects a float between 0 and 1 representing the proportion of the dataset to include in the test split. 0.2 means 20% test data.

3fill in blank

hard

Fix the error in the code to train a linear regression model and make predictions.

ML Python

from sklearn.linear_model import LinearRegression
model = LinearRegression()
model.[1](X_train, y_train)
predictions = model.predict(X_test)

Drag options to blanks, or click blank then click option'

Afit

Bscore

Cpredict

Dtrain

Attempts:

3 left

💡 Hint

Common Mistakes

Using train instead of fit.

Calling predict before training.

The method to train a model in scikit-learn is fit. Using train or predict here is incorrect.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps model complexity to training error.

ML Python

errors = {complexity: model_errors[complexity] [1] for complexity in model_errors if model_errors[complexity] [2] 0}

Drag options to blanks, or click blank then click option'

A* 100

B>

C<

D+ 1

Attempts:

3 left

💡 Hint

Common Mistakes

Using '+' instead of '*' for scaling.

Using '<' instead of '>' for filtering.

The comprehension multiplies errors by 100 to convert to percentage and filters only positive errors using >.

5fill in blank

hard

Fill the blanks to calculate bias, variance, and total error from predictions.

ML Python

bias = (([1] - true_values) ** 2).mean()
variance = predictions.var(axis=0).mean()
total_error = bias + [2]

Drag options to blanks, or click blank then click option'

Apredictions.mean(axis=0)

Bvariance

Cbias

Dtrue_values

Attempts:

3 left

💡 Hint

Common Mistakes

Using true_values instead of mean predictions for bias.

Mixing up bias and variance in total error calculation.

Bias is the squared difference between the average prediction and true values. Total error is the sum of bias and variance.

Practice

1. What does high bias in a machine learning model usually indicate?

easy

A. The model is too complex and overfits noise

B. The model fits the training data perfectly

C. The model is too simple and misses important patterns

D. The model has low training error but high test error

Solution

1. Step 1: Understand bias in model context

   Bias refers to error from overly simple models that cannot capture the true patterns in data.

2. Step 2: Match bias description to options

   The model is too simple and misses important patterns correctly describes high bias as a simple model missing important patterns.

3. Final Answer:

   The model is too simple and misses important patterns -> Option C

4. Quick Check:

   High bias = Simple model misses patterns [OK]

Hint: High bias means model is too simple, misses patterns [OK]

Common Mistakes:

• Confusing bias with variance
• Thinking high bias means overfitting
• Mixing training and test error concepts

2. Which of the following statements about variance is correct?

easy

A. Variance measures error from models that are too simple

B. Variance decreases when model complexity increases

C. Variance is unrelated to model complexity

D. Variance is error caused by fitting noise in training data

Solution

1. Step 1: Define variance in machine learning

   Variance is the error caused by a model fitting noise or random fluctuations in training data.

2. Step 2: Identify correct statement

   Variance is error caused by fitting noise in training data correctly states variance is error from fitting noise in training data.

3. Final Answer:

   Variance is error caused by fitting noise in training data -> Option D

4. Quick Check:

   Variance = Overfitting noise [OK]

Hint: Variance means model fits noise, causing overfitting [OK]

Common Mistakes:

• Confusing variance with bias
• Thinking variance decreases with complexity
• Ignoring noise fitting in variance

3. Consider this Python code simulating model error components:

import numpy as np
np.random.seed(0)
true_pattern = 5
model_predictions = [4, 5, 6, 7, 8]
noise = np.random.normal(0, 1, 5)
errors = [(pred - true_pattern + n) ** 2 for pred, n in zip(model_predictions, noise)]
print(round(np.mean(errors), 2))

What is the printed output?

medium

A. 9.27

B. 6.24

C. 5.24

D. 3.24

Solution

1. Step 1: Calculate errors list values

   For each prediction, error = (prediction - 5 + noise)^2. Noise values from seed 0 are approx [1.764, 0.400, 0.978, 2.240, 1.867]. Calculate each error:
   1) (4 - 5 + 1.764)^2 = (0.764)^2 = 0.583
   2) (5 - 5 + 0.400)^2 = (0.400)^2 = 0.160
   3) (6 - 5 + 0.978)^2 = (1.978)^2 = 3.913
   4) (7 - 5 + 2.240)^2 = (4.240)^2 = 17.98
   5) (8 - 5 + 1.867)^2 = (4.867)^2 = 23.69

2. Step 2: Compute mean and round

   Mean error = (0.583 + 0.160 + 3.913 + 17.98 + 23.69) / 5 = 9.265. Rounded to 2 decimals = 9.27. Precise noise values [1.76405235, 0.40015721, 0.97873798, 2.2408932, 1.86755799] confirm np.mean(errors) rounds to 9.27.

3. Final Answer:

   9.27 -> Option A

4. Quick Check:

   Mean squared error with noise = 9.27 [OK]

Hint: Trace with seed(0): (bias + noise)^2 mean rounds to 9.27 [OK]

Common Mistakes:

• Ignoring noise in error calculation
• Not squaring the error terms
• Rounding errors too early

4. The following code tries to illustrate bias and variance but has a mistake:

import numpy as np
np.random.seed(1)
true_value = 10
predictions = [true_value + np.random.normal(0, 5) for _ in range(5)]
bias = np.mean(predictions) - true_value
variance = np.var(predictions)
print(f"Bias: {bias:.2f}, Variance: {variance:.2f}")

What is the error in this code?

medium

A. Bias should be absolute difference, not signed

B. Variance calculation should use ddof=1 for sample variance

C. Random seed is set incorrectly

D. Predictions should not include noise for bias calculation

Solution

1. Step 1: Understand variance calculation

   By default, np.var uses population variance (ddof=0). For sample variance, ddof=1 should be used to get an unbiased estimate.

2. Step 2: Identify mistake in code

   The code uses np.var(predictions) without ddof=1, so variance is underestimated for small samples.

3. Final Answer:

   Variance calculation should use ddof=1 for sample variance -> Option B

4. Quick Check:

   Use ddof=1 for unbiased variance estimate [OK]

Hint: Use ddof=1 in np.var for sample variance [OK]

Common Mistakes:

• Ignoring ddof parameter in variance
• Confusing bias sign with absolute error
• Thinking seed affects bias/variance calculation

5. You have a dataset with 1000 samples. You train two models:

• Model A: Simple linear regression
• Model B: Deep neural network with many layers

Model A has high training error and low variance. Model B has low training error but high variance.
Which strategy best balances the bias-variance tradeoff to improve test accuracy?

hard

A. Use Model B and increase training data or apply regularization

B. Use Model A and add more features to reduce bias

C. Use Model B without changes to maximize complexity

D. Use Model A and reduce training data to lower variance

Solution

1. Step 1: Analyze model characteristics

   Model A is simple with high bias (high training error) and low variance. Model B is complex with low bias but high variance (overfitting).

2. Step 2: Choose strategy to balance bias and variance

   To reduce variance in Model B, increase training data or apply regularization. This keeps low bias but controls overfitting.

3. Final Answer:

   Use Model B and increase training data or apply regularization -> Option A

4. Quick Check:

   Reduce variance by more data or regularization [OK]

Hint: Fix high variance with more data or regularization [OK]

Common Mistakes:

• Adding features to reduce variance instead of bias
• Reducing data to lower variance (actually increases variance)
• Ignoring regularization benefits