Nested cross-validation can be viewed as an extension of simpler cross-validation techniques. When performing model selection or model evaluation, $k$-fold cross-validation is a crucial method for estimating a particular model’s test error on unseen observations. However, as Cawley and Talbot discussed in a 2010 paper, when performing model selection and model evaluation, we should not use the same test sets to both select the hyperparameters of a model and evaluate a model. By doing so, we may optimistically bias our model evaluations and underestimate our test errors.

For clarity, hyperparameters are parameters that not determined directly within the model’s learning procedures, and thus must be defined prior to model fitting. In scikit-learn, hyperparameters are passed as arguments to the constructor of the estimator class. For example, when performing regularized logistic regression, we can define the regularization hyperparameter C via LogisticRegression(C=0.01).

Using nested cross-validation, we are able to both select the hyperparameters of a model and evaluate the model on the same initial dataset without optimistically biasing our model evaluations. We accomplish nested cross-validation by performing two sequential, or nested, $k$-fold cross-validation loops.

To begin the nested cross-validation procedure, we first split our entire dataset into $k$ outer folds, where one fold is a test set, and the remaining $k-1$ folds are our outer training set. We then take the outer training set and further split it into $l$ inner folds, where one fold is an inner validation set, and the remaining $l-1$ folds are our inner training set. Next, we train each individual particular model with different hyperparameters on the inner training set and evaluate it on the inner validation set. We then take the best performing model on the inner validation set, and evaluate it on the outer test set.

We then repeat the above process, looping through the outer $k$ folds until each $k_i$th fold is used as a test set. After doing this, we have performed an outer $k$-fold cross-validation loop on our entire dataset, and an inner $l$-fold cross-validation loop for each outer training set made up of $k-1$ folds.

After performing this nested cross-validation procedure, we typically report the mean and standard deviation of the test error estimates from the outer $k$-fold cross-validation procedure. For a great explanation of nested cross-validation, I would recommend this video by Dr. Cynthia Rudin of Duke University.

Next, let’s explore a working example to clarify the nested cross-validation technique.

Breast Cancer Data

For this working example, I’ll use the common breast cancer dataset available within scikit-learn’s datasets module. The breast cancer dataset was assembled by the University of Wisconsin and contains 569 observations and 30 predictor features. The target variable is encoded as 1 or 0, indicating whether the observation was found to have malignant or benign breast cancer.

from sklearn import datasets
X, y = datasets.load_breast_cancer(return_X_y=True)

On the breast cancer dataset, let’s train a regularized logistic regression model with multiple values of C. In the regularized logistic regression setting, C controls the penalty, or cost function, the penalizes the model for variable complexity. C commonly takes values ranging from 0.0001 to 10000.

Because we’re performing regularization, we should also scale our input features.

from sklearn.preprocessing import StandardScaler
standard_scaler = StandardScaler()

# Scaling features
X = standard_scaler.fit_transform(X)

Inner $l$-Fold Cross-Validation Loop (Hyperparameter Tuning)

To perform our inner $l$-fold Cross-Validation procedure, we’ll first construct a LogisticRegression estimator and a GridSearchCV object, both from scikit-learn.

We’ll construct our grid of Cs as 10 values between 0.0001 and 10000, evenly spaced on a logarithmic scale.

from sklearn.linear_model import LogisticRegression
from sklearn.model_selection import GridSearchCV

# Creating grid of potential C values
# Could have also used dict(C=(0.0001, 0.001, 0.01, ... , 1000)) to specify by hand
C_grid = dict(C=np.logspace(-4, 4, 10))

# Initializing estimator
log_reg = LogisticRegression()

# Creating GridSearch object with logistic regression classifier and candidate C values
# cv = 5, defaults to 5-fold cross-validation
log_reg_grid = GridSearchCV(estimator=log_reg, param_grid=C_grid)

Note, we could have also used RandomSearchCV instead of GridSearchCV above. While both objects perform cross-validation, GridSearchCV is an exhaustive search of all parameter combinations in param_grid, while RandomSearchCV implements a randomized search of the hyperparameter space. Within RandomSearchCV, the n_iter argument controls how many random combinations of parameters to search over. A random search, as opposed to an exhaustive grid search, can be understood as trading runtime for solution quality.

At this point, nothing else needs to be done to the log_reg_grid object that represents the inner loop of our nested cross-validation procedure. However, to demonstrate this object can be used to find the optimal value of C (of those defined in C_grid), we can fit the object on X and y.

# Fitting grid search object
log_reg_grid.fit(X=X, y=y)

# Getting optimal value of C
log_reg_grid.best_estimator_.C  # 0.3593813663804626

Outer $k$-Fold Cross-Validation Loop (Model Evaluation)

Now that we have constructed our inner $l$-fold cross-validation loop using GridSearchCV, we can implement it into an outer $k$-fold cross-validation loop using scikit-learn’s cross_val_score. Here, cross_val_score will iteratively perform cross-validation on the outer $k$ folds of our data to determine how well the optimal model with a particular C value found on the inner loop is performing.

Below, let’s perform an outer loop with cv=3 folds. cross_val_score will iteratively split the data into three folds, where one fold is a test set, and the other two folds are a training set. Then, our log_reg_grid search object will further perform 5-fold cross-validation on this training set to determine the optimal value of C, and then this model with the optimal value of C is evaluating on the test set. This process in then repeated until each outer fold is used as the test set once.

>>> from sklearn.model_selection import cross_val_score

>>> scores = cross_val_score(estimator=log_reg_grid, X=X, y=y, cv=3)
>>> scores
array([0.97894737, 0.97368421, 0.97883598])

Each of the above three values represent an unbiased test error estimate for our regularized logistic regression model. We can evaluate the mean of these values to evaluate how well our model is performing.

>>> scores.mean()
0.977155852594449

Note, this evaluation tells us how well our model is performing with hyperparameter tuning as a part of the fitting procedure. By definition, the nested cross-validation procedure allows us to evaluate model performance where hyperparameter turning is an integral part of the fitting procedure.

If we are interested in what the overall optimal value of C is - or any other hyperparameters in the general case - we should instead perform $k$-fold cross-validation on the entire dataset, and select the hyperparameter values that result in the lowest test error estimates.

Drawbacks of Nested Cross-Validation

While nested cross-validation is a useful procedure, it does have some drawbacks. The most obvious drawback is that it can be very computationally expensive. To fit a nested cross-validation procedure with $k$ outer folds, $l$ inner folds, and $m$ unique hyperparameter combinations, we need to perform $k \cdot l \cdot m$ model fits. In our above example with 3 outer folds, 5 inner folds, and 10 unique C candidates, we fit our logistic regression model 150 times. Of course, on our small dataset, and with a simple model fitting procedure, this was not too expensive. On larger datasets or models with more complex fitting procedure, nested cross-validation can be much more computationally expensive.

While more of a comment than a drawback, nested cross-validation is most useful when we have a few thousand observations or fewer. On larger datasets, we can perform model evaluation and hyperparameter tuning on the same folds without optimistically biasing our test error estimates as severely. That is not to say we will still not underestimate the test error of our model to some degree, and it is best practice to still perform nested cross-validation.