Diagnosing Bias and Variance Problems

Machine Learning with Tree-Based Models in Python

Elie Kawerk

Data Scientist

Estimating the Generalization Error

How do we estimate the generalization error of a model?
Cannot be done directly because:
- $f$ is unknown,
- usually you only have one dataset,
- noise is unpredictable.

Estimating the Generalization Error

Solution:

split the data to training and test sets,
fit $\hat{f}$ to the training set,
evaluate the error of $\hat{f}$ on the unseen test set.
generalization error of $\hat{f} \approx$ test set error of $\hat{f}$.

Better Model Evaluation with Cross-Validation

Test set should not be touched until we are confident about $\hat{f}$'s performance.
Evaluating $\hat{f}$ on training set: biased estimate, $\hat{f}$ has already seen all training points.
Solution $\rightarrow$ Cross-Validation (CV):
- K-Fold CV,
- Hold-Out CV.

K-Fold CV

KFoldCV

K-Fold CV

CVerror

Diagnose Variance Problems

If $\hat{f}$ suffers from high variance:

CV error of $\hat{f}$ > training set error of $\hat{f}$.

$\hat{f}$ is said to overfit the training set. To remedy overfitting:
- decrease model complexity,
- for ex: decrease max depth, increase min samples per leaf, ...
- gather more data, ..

Diagnose Bias Problems

if $\hat{f}$ suffers from high bias:

CV error of $\hat{f} \approx$ training set error of $\hat{f} >>$ desired error.
$\hat{f}$ is said to underfit the training set. To remedy underfitting:
- increase model complexity
- for ex: increase max depth, decrease min samples per leaf, ...
- gather more relevant features

K-Fold CV in sklearn on the Auto Dataset

from sklearn.tree import DecisionTreeRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error as MSE
from sklearn.model_selection import cross_val_score

# Set seed for reproducibility
SEED = 123
# Split data into 70% train and 30% test
X_train, X_test, y_train, y_test = train_test_split(X,y,
                                                    test_size=0.3, 
                                                    random_state=SEED)

# Instantiate decision tree regressor and assign it to 'dt'
dt = DecisionTreeRegressor(max_depth=4, 
                           min_samples_leaf=0.14,
                           random_state=SEED)

K-Fold CV in sklearn on the Auto Dataset

# Evaluate the list of MSE ontained by 10-fold CV 
# Set n_jobs to -1 in order to exploit all CPU cores in computation
MSE_CV = - cross_val_score(dt, X_train, y_train, cv= 10, 
                           scoring='neg_mean_squared_error',
                           n_jobs = -1)

# Fit 'dt' to the training set                        
dt.fit(X_train, y_train)
# Predict the labels of training set
y_predict_train = dt.predict(X_train)
# Predict the labels of test set
y_predict_test = dt.predict(X_test)

# CV MSE  
print('CV MSE: {:.2f}'.format(MSE_CV.mean()))

CV MSE: 20.51

# Training set MSE
print('Train MSE: {:.2f}'.format(MSE(y_train, y_predict_train)))

Train MSE: 15.30

# Test set MSE
print('Test MSE: {:.2f}'.format(MSE(y_test, y_predict_test)))

Test MSE: 20.92

Let's practice!

Machine Learning with Tree-Based Models in Python