[xgboost] one idea why you may want to use polynomials in xgboost

In traditional linear models, polynomial features are often created to capture the non-linear relationships between the features and the target variable. However, tree-based models like XGBoost are inherently capable of modeling non-linear relationships, which raises the question: is there any benefit to creating polynomial features when using XGBoost?

Potential Benefits:

1. Complex Interactions: Even though XGBoost can handle non-linearities, polynomial features can sometimes make it easier for the model to learn complex interactions between variables without needing as many splits. This could make the model simpler and potentially faster to train.

2. Interpretable Features: Polynomial features can be more interpretable in certain contexts. For instance, a squared term in a regression equation could be more easily understood as a "growth effect" in fields like economics or biology.

Risks and Considerations:

1. Overfitting: Adding polynomial features increases the complexity of the model and the risk of overfitting. XGBoost has regularization to combat overfitting, but the risk still exists.

2. Computational Cost: The addition of polynomial features increases the number of features, thereby increasing computational cost in terms of both memory and CPU usage.

3. Collinearity: Polynomial features can be highly correlated with the original features, which might cause multicollinearity. While tree models are generally robust to multicollinearity, extreme cases can still degrade the model's performance and interpretability.

4. Dimensionality: The feature space can expand quite quickly when adding polynomial terms, potentially making the model harder to interpret and manage.

While it's generally less common to use polynomial features with XGBoost than with linear models, there are cases where it can be beneficial. It depends on the problem you're trying to solve, the data you have, and the complexity you're willing to introduce into your model. As with any technique, it's best to validate its utility through experimentation and cross-validation.

Addressing the Nature of Splits

In tree-based models like XGBoost, splits do indeed happen along individual variables, essentially carving the feature space into hyper-rectangles. Given that XGBoost already has the capability to model non-linear relationships through these splits, one might wonder what additional benefit, if any, would polynomial features bring.

Photo by Jackson Simmer on Unsplash. A well-split man.

The first thing to clarify is that while tree-based models like XGBoost make splits that can capture non-linear relationships between individual features and the target, these are piecewise constant approximations. That means for a given range of a feature, the prediction would be a constant, and a different constant for another range of the same feature.

Potential Scenarios for Polynomial Features:

1. Complex Interactions More Easily: Suppose you have an interaction term x1×x22x1​×x22​. The tree would have to make multiple nested splits on x1x1​ and x2x2​ to approximate this relationship, requiring more depth and potentially leading to overfitting. A polynomial feature could capture this complexity in a more straightforward manner.

2. Reduced Number of Splits: Sometimes introducing a polynomial feature can result in fewer splits to achieve the same level of predictive power, essentially making the model simpler. In some cases, this could lead to faster training and inference times.

3. Better Initial Splits: Early splits in the tree are crucial for model performance. A well-chosen polynomial feature could be selected for an early split, setting the stage for a more accurate model.

Trade-Offs:

1. Overfitting Risk: Introducing additional features, especially polynomial ones, increases the risk of overfitting, though XGBoost’s regularization parameters can mitigate this.

2. Computation Overhead: Calculating polynomial features adds a preprocessing step and increases the feature set size, which could be computationally expensive.

In summary, while XGBoost can capture non-linear relationships, polynomial features might still offer benefits in terms of capturing complex interactions more efficiently or enabling more accurate initial splits. However, these advantages should be weighed against the risks of overfitting and increased computational costs. Experimentation and cross-validation are key to determining the utility of polynomial features in any given XGBoost model.