Hyperparameter tuning#

In the previous section, we did not discuss the hyperparameters of random forest and histogram gradient-boosting. This notebook gives crucial information regarding how to set them.

Caution

For the sake of clarity, no nested cross-validation is used to estimate the variability of the testing error. We are only showing the effect of the parameters on the validation set.

We start by loading the california housing dataset.

from sklearn.datasets import fetch_california_housing
from sklearn.model_selection import train_test_split

data, target = fetch_california_housing(return_X_y=True, as_frame=True)
target *= 100  # rescale the target in k$
data_train, data_test, target_train, target_test = train_test_split(
    data, target, random_state=0
)

Random forest#

The main parameter to select in random forest is the n_estimators parameter. In general, the more trees in the forest, the better the generalization performance would be. However, adding trees slows down the fitting and prediction time. The goal is to balance computing time and generalization performance when setting the number of estimators. Here, we fix n_estimators=100, which is already the default value.

Caution

Tuning the n_estimators for random forests generally result in a waste of computer power. We just need to ensure that it is large enough so that doubling its value does not lead to a significant improvement of the validation error.

Instead, we can tune the hyperparameter max_features, which controls the size of the random subset of features to consider when looking for the best split when growing the trees: smaller values for max_features lead to more random trees with hopefully more uncorrelated prediction errors. However if max_features is too small, predictions can be too random, even after averaging with the trees in the ensemble.

If max_features is set to None, then this is equivalent to setting max_features=n_features which means that the only source of randomness in the random forest is the bagging procedure.

print(f"In this case, n_features={len(data.columns)}")
In this case, n_features=8

We can also tune the different parameters that control the depth of each tree in the forest. Two parameters are important for this: max_depth and max_leaf_nodes. They differ in the way they control the tree structure. Indeed, max_depth enforces growing symmetric trees, while max_leaf_nodes does not impose such constraint. If max_leaf_nodes=None then the number of leaf nodes is unlimited.

The hyperparameter min_samples_leaf controls the minimum number of samples required to be at a leaf node. This means that a split point (at any depth) is only done if it leaves at least min_samples_leaf training samples in each of the left and right branches. A small value for min_samples_leaf means that some samples can become isolated when a tree is deep, promoting overfitting. A large value would prevent deep trees, which can lead to underfitting.

Be aware that with random forest, trees are expected to be deep since we are seeking to overfit each tree on each bootstrap sample. Overfitting is mitigated when combining the trees altogether, whereas assembling underfitted trees (i.e. shallow trees) might also lead to an underfitted forest.

import pandas as pd
from sklearn.model_selection import RandomizedSearchCV
from sklearn.ensemble import RandomForestRegressor

param_distributions = {
    "max_features": [1, 2, 3, 5, None],
    "max_leaf_nodes": [10, 100, 1000, None],
    "min_samples_leaf": [1, 2, 5, 10, 20, 50, 100],
}
search_cv = RandomizedSearchCV(
    RandomForestRegressor(n_jobs=2),
    param_distributions=param_distributions,
    scoring="neg_mean_absolute_error",
    n_iter=10,
    random_state=0,
    n_jobs=2,
)
search_cv.fit(data_train, target_train)

columns = [f"param_{name}" for name in param_distributions.keys()]
columns += ["mean_test_error", "std_test_error"]
cv_results = pd.DataFrame(search_cv.cv_results_)
cv_results["mean_test_error"] = -cv_results["mean_test_score"]
cv_results["std_test_error"] = cv_results["std_test_score"]
cv_results[columns].sort_values(by="mean_test_error")
param_max_features param_max_leaf_nodes param_min_samples_leaf mean_test_error std_test_error
3 2 None 2 34.012374 0.518312
0 2 1000 10 36.758895 0.521091
7 None None 20 37.296113 0.332897
4 5 100 2 40.099047 0.598930
8 None 100 10 40.456815 0.592738
6 None 1000 50 40.899684 0.459898
9 1 100 2 50.049621 0.846382
2 1 100 1 50.297163 0.926681
5 1 None 100 54.602252 1.013700
1 3 10 10 55.050045 0.868824

We can observe in our search that we are required to have a large number of max_leaf_nodes and thus deep trees. This parameter seems particularly impactful with respect to the other tuning parameters, but large values of min_samples_leaf seem to reduce the performance of the model.

In practice, more iterations of random search would be necessary to precisely assert the role of each parameters. Using n_iter=10 is good enough to quickly inspect the hyperparameter combinations that yield models that work well enough without spending too much computational resources. Feel free to try more interations on your own.

Once the RandomizedSearchCV has found the best set of hyperparameters, it uses them to refit the model using the full training set. To estimate the generalization performance of the best model it suffices to call .score on the unseen data.

error = -search_cv.score(data_test, target_test)
print(
    f"On average, our random forest regressor makes an error of {error:.2f} k$"
)
On average, our random forest regressor makes an error of 33.48 k$

Histogram gradient-boosting decision trees#

For gradient-boosting, hyperparameters are coupled, so we cannot set them one after the other anymore. The important hyperparameters are max_iter, learning_rate, and max_depth or max_leaf_nodes (as previously discussed random forest).

Let’s first discuss max_iter which, similarly to the n_estimators hyperparameter in random forests, controls the number of trees in the estimator. The difference is that the actual number of trees trained by the model is not entirely set by the user, but depends also on the stopping criteria: the number of trees can be lower than max_iter if adding a new tree does not improve the model enough. We will give more details on this in the next exercise.

The depth of the trees is controlled by max_depth (or max_leaf_nodes). We saw in the section on gradient-boosting that boosting algorithms fit the error of the previous tree in the ensemble. Thus, fitting fully grown trees would be detrimental. Indeed, the first tree of the ensemble would perfectly fit (overfit) the data and thus no subsequent tree would be required, since there would be no residuals. Therefore, the tree used in gradient-boosting should have a low depth, typically between 3 to 8 levels, or few leaves (\(2^3=8\) to \(2^8=256\)). Having very weak learners at each step helps reducing overfitting.

With this consideration in mind, the deeper the trees, the faster the residuals are corrected and then less learners are required. Therefore, it can be beneficial to increase max_iter if max_depth is low.

Finally, we have overlooked the impact of the learning_rate parameter until now. When fitting the residuals, we would like the tree to try to correct all possible errors or only a fraction of them. The learning-rate allows you to control this behaviour. A small learning-rate value would only correct the residuals of very few samples. If a large learning-rate is set (e.g., 1), we would fit the residuals of all samples. So, with a very low learning-rate, we would need more estimators to correct the overall error. However, a too large learning-rate tends to obtain an overfitted ensemble, similar to having very deep trees.

from scipy.stats import loguniform
from sklearn.ensemble import HistGradientBoostingRegressor

param_distributions = {
    "max_iter": [3, 10, 30, 100, 300, 1000],
    "max_leaf_nodes": [2, 5, 10, 20, 50, 100],
    "learning_rate": loguniform(0.01, 1),
}
search_cv = RandomizedSearchCV(
    HistGradientBoostingRegressor(),
    param_distributions=param_distributions,
    scoring="neg_mean_absolute_error",
    n_iter=20,
    random_state=0,
    n_jobs=2,
)
search_cv.fit(data_train, target_train)

columns = [f"param_{name}" for name in param_distributions.keys()]
columns += ["mean_test_error", "std_test_error"]
cv_results = pd.DataFrame(search_cv.cv_results_)
cv_results["mean_test_error"] = -cv_results["mean_test_score"]
cv_results["std_test_error"] = cv_results["std_test_score"]
cv_results[columns].sort_values(by="mean_test_error")
param_max_iter param_max_leaf_nodes param_learning_rate mean_test_error std_test_error
14 300 100 0.01864 31.173686 0.299384
6 300 20 0.047293 31.868467 0.281284
2 30 50 0.176656 32.647583 0.358444
13 300 10 0.297739 32.783617 0.480020
9 100 20 0.083745 32.991498 0.356141
19 100 10 0.215543 33.212236 0.463788
12 100 20 0.067503 33.696071 0.486387
16 300 5 0.05929 35.849066 0.351205
1 100 5 0.160519 36.364972 0.625672
0 1000 2 0.125207 40.726425 0.561258
7 1000 2 0.054511 42.251947 0.646147
18 10 5 0.248463 50.077527 0.659438
8 3 5 0.906226 50.146583 0.731569
5 10 100 0.061034 61.637761 0.584822
17 3 5 0.079415 81.447068 0.865869
4 10 2 0.0351 82.545993 0.962567
15 3 50 0.019923 87.668108 1.076272
3 3 2 0.039361 87.745092 1.088164
11 3 10 0.019351 88.386979 1.090862
10 3 5 0.01724 88.951462 1.069568

Caution

Here, we tune max_iter but be aware that it is better to set max_iter to a fixed, large enough value and use parameters linked to early_stopping as we will do in Exercise M6.04.

In this search, we observe that for the best ranked models, having a smaller learning_rate, requires more trees or a larger number of leaves for each tree. However, it is particularly difficult to draw more detailed conclusions since the best value of each hyperparameter depends on the other hyperparameter values.

We can now estimate the generalization performance of the best model using the test set.

error = -search_cv.score(data_test, target_test)
print(f"On average, our HGBT regressor makes an error of {error:.2f} k$")
On average, our HGBT regressor makes an error of 30.52 k$

The mean test score in the held-out test set is slightly better than the score of the best model. The reason is that the final model is refitted on the whole training set and therefore, on more data than the cross-validated models of the grid search procedure.

We summarize these details in the following table:

Bagging & Random Forests

Boosting

fit trees independently

fit trees sequentially

each deep tree overfits

each shallow tree underfits

averaging the tree predictions reduces overfitting

sequentially adding trees reduces underfitting

generalization improves with the number of trees

too many trees may cause overfitting

does not have a learning_rate parameter

fitting the residuals is controlled by the learning_rate