The blog post explores the potential downsides of using polynomial features in machine learning, particularly focusing on their instability in high dimensions. While polynomial expansion can improve model fit by capturing non-linear relationships, it can also lead to extreme sensitivity to input changes, causing wild oscillations and poor generalization. The author demonstrates this issue with visualizations of simple polynomials raised to high powers and illustrates how even small perturbations in the input can drastically alter the output. They suggest Bernstein polynomials as a more stable alternative, highlighting their properties like non-negativity and partition of unity, which contribute to smoother behavior and better extrapolation. The post concludes that while polynomial features can be beneficial, their inherent instability requires careful consideration and potentially exploration of alternative basis functions like Bernstein polynomials.
The blog post "Are polynomial features the root of all evil? (2024)" by Alex Shtf explores the potential downsides of using polynomial features in machine learning, particularly focusing on their behavior when extrapolated beyond the training data distribution. The author meticulously dissects how polynomial features, while often beneficial within the training data's range, can lead to wildly unpredictable and undesirable extrapolations. This problematic behavior is exemplified through a series of illustrative examples and visualizations.
The core argument revolves around the inherent nature of polynomials. As the degree of the polynomial increases, the function becomes increasingly sensitive to changes in the input features, especially at larger magnitudes. This heightened sensitivity results in drastic changes in the output for even small deviations from the observed data, leading to unreliable predictions outside the training domain. The author visually demonstrates this phenomenon by showcasing how high-degree polynomial fits can oscillate dramatically and deviate significantly from the underlying true function they are intended to approximate, particularly in regions with sparse or no training data.
The post specifically highlights the dangers of employing polynomial features in combination with linear models, such as linear regression and logistic regression. While these models are typically favored for their interpretability and simplicity, their coupling with high-degree polynomials introduces a treacherous element of instability when extrapolating. The author argues that this combination can lead to overly confident and erroneous predictions in uncharted territories of the input space.
Furthermore, the post delves into the connection between polynomial features and the Bernstein basis polynomials. It explains how polynomial regression can be viewed as fitting a linear combination of Bernstein basis polynomials. This perspective sheds light on why polynomial features exhibit such extreme behavior during extrapolation: the individual Bernstein basis polynomials themselves exhibit pronounced oscillations and rapid growth outside the training range, which are then amplified when combined in a linear model.
Finally, the author suggests a more cautious and nuanced approach to utilizing polynomial features. While acknowledging their potential benefits within the training data's confines, the post emphasizes the importance of carefully considering the potential for erratic extrapolation. It advises practitioners to be mindful of the degree of the polynomial employed, the characteristics of the training data, and the intended use case of the model. The underlying message is that while polynomial features are not inherently "evil," their application requires judicious consideration and awareness of their limitations to avoid unintended and potentially harmful consequences.
Summary of Comments ( 1 )
https://news.ycombinator.com/item?id=43764101
HN users discuss potential downsides of polynomial features, particularly in the context of overfitting and interpretability issues. Some argue against their broad categorization as "evil," suggesting they can be valuable when applied judiciously and with proper regularization techniques. One commenter points out their usefulness in approximating non-linear functions and highlights the importance of understanding the underlying data and model behavior. Others discuss alternatives like splines, which offer more local control and flexibility, and the role of feature scaling in mitigating potential problems with polynomial features. The trade-off between complexity and interpretability is a recurring theme, with commenters emphasizing the importance of selecting the right tool for the specific problem and dataset.
The Hacker News post "Are polynomial features the root of all evil? (2024)" with ID 43764101 sparked a discussion with several interesting comments. The overall theme revolves around the author's claim that polynomial features often lead to overfitting and proposes Bernstein polynomials as a superior alternative. Commenters generally agree that overfitting is a valid concern with polynomial features but offer diverse perspectives on the proposed solution and the nuances of feature engineering in general.
One compelling comment points out that the core issue isn't polynomial features themselves, but rather the unchecked growth of the hypothesis space they create. This commenter argues that any basis expansion, including Bernstein polynomials, can lead to overfitting if not properly regularized. They suggest techniques like L1 or L2 regularization as effective ways to mitigate this risk, regardless of the specific polynomial basis used.
Another insightful comment highlights the importance of understanding the underlying data generating process. The commenter argues that if the true relationship is indeed polynomial, using polynomial features is perfectly reasonable. However, they caution against blindly applying polynomial transformations without considering the nature of the data. They propose exploring other basis functions, like trigonometric functions or splines, depending on the specific problem.
Several comments discuss the practical implications of using Bernstein polynomials. One commenter questions their computational efficiency, particularly for high-degree polynomials and large datasets. Another points out that while Bernstein polynomials might offer better extrapolation properties near the boundaries of the input space, they might not necessarily improve interpolation performance within the observed data range.
One commenter provides a more theoretical perspective, suggesting that the benefits of Bernstein polynomials might stem from their ability to form a partition of unity. This property ensures that the sum of the basis functions equals one, which can lead to more stable and predictable behavior, especially in the context of interpolation and approximation.
Finally, a recurring theme in the comments is the importance of cross-validation and proper evaluation metrics. Commenters emphasize that the effectiveness of any feature engineering technique, whether polynomial features or Bernstein polynomials, should be empirically assessed using robust evaluation procedures. Simply observing a good fit on the training data is not sufficient to guarantee generalization performance. Therefore, rigorous cross-validation is crucial for selecting the best approach and avoiding the pitfalls of overfitting.