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	34.025977	0.391074
0	2	1000	10	36.682039	0.637373
7	None	None	20	37.299030	0.411896
4	5	100	2	40.108800	0.554809
8	None	100	10	40.468891	0.537382
6	None	1000	50	40.875430	0.498604
9	1	100	2	49.811945	0.764639
2	1	100	1	49.924762	0.688185
5	1	None	100	54.515705	1.065132
1	3	10	10	54.741823	0.818586

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 33.67 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	0.018640	30.919613	0.297558
6	300	20	0.047293	31.866929	0.290234
2	30	50	0.176656	32.531634	0.341191
13	300	10	0.297739	32.973404	0.717208
9	100	20	0.083745	33.173144	0.368474
19	100	10	0.215543	33.313344	0.325258
12	100	20	0.067503	33.661552	0.369654
16	300	5	0.059290	35.929702	0.321676
1	100	5	0.160519	36.428438	0.444111
0	1000	2	0.125207	40.696433	0.537806
7	1000	2	0.054511	42.143477	0.684640
8	3	5	0.906226	49.951073	0.736811
18	10	5	0.248463	50.344508	1.013917
5	10	100	0.061034	61.603376	0.658882
17	3	5	0.079415	81.493869	0.919585
4	10	2	0.035100	82.546814	0.904416
15	3	50	0.019923	87.660586	1.064093
3	3	2	0.039361	87.845803	1.048039
11	3	10	0.019351	88.413794	1.102170
10	3	5	0.017240	88.917710	1.124756

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 30.53 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