ML Interview Q Series: How can you determine the best number of randomly selected features to consider at each decision split in a Random Forest?

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Comprehensive Explanation

In a Random Forest, each decision tree is built by choosing a subset of features at every node to split on, often referred to as mtry (the number of random features considered). The question is how to select this value in a way that leads to the best generalization performance. One classical approach is to tune this hyperparameter by leveraging some form of validation error or an out-of-bag (OOB) error estimate.

Intuition Behind Random Feature Selection

Randomly selecting a subset of features at each node helps decorrelate the individual trees. If we used all features at every node, the individual trees might look too similar to each other, weakening the ensemble’s overall benefit. On the other hand, if the subset of features considered is too small, each tree might become too weak or not sufficiently diverse in a beneficial way.

Approaches to Determine the Optimal Number of Features

1. Using Out-of-Bag (OOB) Error

When we train a Random Forest, each tree is built on a bootstrap sample from the training data. Any observation not included in a tree’s bootstrap sample is considered out-of-bag for that tree. We can measure the prediction accuracy on these OOB samples as an unbiased estimate of test error.

We try different values for mtry (ranging from 1 up to the total number of features) and compute the OOB error after training the Random Forest. The OOB error is calculated as the fraction of times the prediction for each data point is incorrect (for classification) or the average error (for regression) when only the trees that did not have that point in their bootstrap sample are used to predict its label/value.

Below is a central formula often used for classification to compute the OOB error:

Where:

• N is the total number of training examples.

• y_i is the true label of the i-th example.

• hat{y}_{OOB, i} is the predicted label for the i-th example, using only the trees that did not see this example in their bootstrap training set.

• 1(...) is the indicator function that is 1 if the condition is true (i.e., if y_i is not equal to hat{y}_{OOB, i}), and 0 otherwise.

By training multiple Random Forests with different mtry values, you can select the mtry that yields the lowest OOB error.

2. Using Cross-Validation

Another method is to use k-fold cross-validation. For each candidate mtry:

1. Partition the dataset into k folds.

2. For each fold, train a Random Forest on the remaining folds and evaluate its performance on the held-out fold.

3. Compute the average performance over all folds.

4. Choose the mtry that yields the best average performance.

Though cross-validation is more computationally expensive than OOB error, it can sometimes be more robust if you have sufficient computing resources.

3. Practical Heuristics

Many implementations use rules of thumb:

• For classification tasks, mtry is often set to the square root of the total number of features.

• For regression tasks, mtry is often set to one-third of the total number of features.

These defaults are good starting points, but in practice, tuning remains essential.

4. Trade-Offs in Tuning

• If mtry is set too large, trees might become too correlated, reducing the ensemble benefit.

• If mtry is set too small, individual trees might be weak and not capture important signals.

Hence, a systematic hyperparameter search (using either OOB error or cross-validation) is key.

What Are Some Possible Follow-Up Questions?

How do you balance OOB error with cross-validation approaches?

In practice, many Random Forest libraries allow you to retrieve an OOB error estimate for free, since it comes directly from the training procedure. Cross-validation requires additional training cycles, which can be computationally heavy for large datasets. OOB estimates are often sufficiently accurate for model tuning decisions. However, in some cases, cross-validation could provide a more stable and reliable estimate, especially if the dataset is not very large or if you want to compare Random Forest with other algorithms using the same evaluation procedure.

Would the method differ for extremely high-dimensional data?

When the number of features is very large compared to the number of samples, you might want to restrict the random subset to a smaller fraction of features. This can help avoid overfitting and reduce computational cost. You can attempt:

• Dimensionality reduction (like PCA) before training.

• Feature selection methods prior to Random Forest training.

• Adjusting mtry by testing values that are smaller than typical defaults.

Are there scenarios where you don’t need to tune the number of features?

In many real-world situations, the default heuristics (square root for classification, one-third for regression) may work well enough. But if you want the best performance, or if your dataset has special characteristics (e.g., highly correlated features, unusual distributions), it’s still better to tune. Extremely large or small mtry values might also suit specific data distributions. Therefore, relying solely on the default may not always give optimal results.

How does correlation among features affect the choice of mtry?

High correlation among features can reduce the effectiveness of random feature selection because many chosen features might essentially carry the same information. Sometimes, reducing mtry in the presence of highly correlated features can be beneficial. It increases the likelihood that your splits will consider genuinely different dimensions of the data, fostering diversity among the trees.

Does tuning mtry interact with other hyperparameters of Random Forest?

Yes. Other important hyperparameters include:

• Number of trees in the forest.

• Minimum samples per leaf (or minimum leaf size).

• Maximum tree depth (if your implementation allows a maximum depth).

• Criterion for splitting (e.g., Gini impurity, entropy, mean squared error).

Changes in these hyperparameters can affect the best choice of mtry. For instance, if each tree is grown very deep or is pruned in a certain way, the interplay with how many features are considered at each split can shift the optimal setting. A full hyperparameter search that includes mtry is often more thorough.

How does mtry affect model interpretability?

Random Forests are already more complex than a single decision tree. Changing mtry mostly affects how each tree is constructed internally, but it doesn't drastically change the overall interpretability approach. You can still employ feature importance methods (e.g., Gini importance or permutation importance) regardless of mtry. However, different mtry settings can lead to slightly different feature importance rankings, because different subsets of features might be used in more or fewer splits.

Could there be a risk of overfitting by tuning mtry too extensively?

Yes, if you have a very large search space for mtry and a small dataset, you might end up overfitting to the validation or OOB set. One mitigation strategy is to ensure you use either a robust cross-validation or a sufficiently large dataset for the OOB estimate. Using a separate hold-out test set to confirm the final choice of mtry is also a good practice if data allows.

Does the strategy differ for boosting-based methods?

In gradient boosting methods, we typically do not select a random subset of features at each split by default (unless specifically using a variant that does so). The concept of mtry is specific to Random Forest-like methods where decorrelation of trees is paramount. While boosting benefits from other forms of regularization (learning rate, number of estimators, subsampling of instances, etc.), the random subset of features is not as central as in a Random Forest.

These considerations demonstrate that finding the optimal number of random features for each split in a Random Forest is primarily a matter of practical experimentation, guided by standard procedures such as OOB error or cross-validation.