Stories with Tag bias-variance tradeoff

• Are polynomial features the root of all evil? (2024)

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  Posted: 2025-04-22 16:49:55

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  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.

  • polynomial features
  • feature engineering
  • machine learning
  • Overfitting
  • Bernstein polynomials
  • polynomial regression
  • model complexity
  • generalization
  • bias-variance tradeoff
  • numerical stability

  Summary of Comments ( 1 )
  https://news.ycombinator.com/item?id=43764101

  Verbose tl;dr

  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.

• Understanding Machine Learning: From Theory to Algorithms

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  Posted: 2025-04-04 18:25:23

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  "Understanding Machine Learning: From Theory to Algorithms" provides a comprehensive overview of machine learning, bridging the gap between theoretical principles and practical applications. The book covers a wide range of topics, from basic concepts like supervised and unsupervised learning to advanced techniques like Support Vector Machines, boosting, and dimensionality reduction. It emphasizes the theoretical foundations, including statistical learning theory and PAC learning, to provide a deep understanding of why and when different algorithms work. Practical aspects are also addressed through the presentation of efficient algorithms and their implementation considerations. The book aims to equip readers with the necessary tools to both analyze existing learning algorithms and design new ones.

  "Understanding Machine Learning: From Theory to Algorithms" by Shai Shalev-Shwartz and Shai Ben-David offers a comprehensive exploration of the fascinating field of machine learning, bridging the gap between theoretical foundations and practical algorithmic implementations. The book meticulously constructs a conceptual framework for understanding how machines learn from data, starting with fundamental concepts like the Probably Approximately Correct (PAC) learning model. This model provides a rigorous mathematical framework for analyzing the ability of learning algorithms to generalize from a limited set of training examples to unseen data, taking into account factors such as sample complexity, error rates, and computational efficiency.

  The authors delve into the core tenets of learnability, examining the conditions under which a concept can be effectively learned by a machine. They discuss various hypothesis classes and their representational power, highlighting the trade-off between expressiveness and the risk of overfitting, where a model learns the training data too well and fails to generalize to new instances. The book extensively covers key learning paradigms, including supervised learning, unsupervised learning, and reinforcement learning. Within supervised learning, specific techniques such as linear regression, logistic regression, support vector machines, and decision trees are explored in detail, both in terms of their mathematical underpinnings and practical implementation considerations.

  Unsupervised learning, which involves learning patterns from unlabeled data, is also given considerable attention. Clustering algorithms, dimensionality reduction techniques, and generative models are discussed, providing the reader with a diverse toolkit for extracting knowledge from unstructured data. Furthermore, the book touches upon the exciting field of reinforcement learning, where agents learn to interact with an environment to maximize rewards, introducing fundamental concepts like Markov Decision Processes and various reinforcement learning algorithms.

  A significant portion of the book is dedicated to a rigorous treatment of the theoretical foundations of machine learning. Concepts like Rademacher complexity, VC dimension, and stability are introduced and used to derive generalization bounds for different learning algorithms. These theoretical tools provide valuable insights into the behavior of learning algorithms and help explain why certain algorithms perform better than others in specific scenarios. The authors also address the computational aspects of machine learning, discussing optimization algorithms and their role in training complex models efficiently. They explore techniques such as gradient descent, stochastic gradient descent, and convex optimization, providing a thorough understanding of how these methods are used to find optimal model parameters.

  Beyond the core theoretical and algorithmic concepts, the book also touches upon more advanced topics, including online learning, multi-class classification, structured output prediction, and learning theory in the context of non-i.i.d. data. Throughout the text, the authors maintain a balance between theoretical rigor and practical applicability, providing numerous examples, illustrations, and exercises to help the reader solidify their understanding. This detailed and comprehensive approach makes the book a valuable resource for both students embarking on their machine learning journey and seasoned practitioners seeking to deepen their understanding of the field's theoretical foundations.

  • machine learning
  • algorithms
  • theory
  • supervised learning
  • Unsupervised Learning
  • reinforcement learning
  • statistical learning theory
  • PAC learning
  • VC dimension
  • bias-variance tradeoff
  • Overfitting
  • regularization
  • model selection
  • optimization
  • Gradient Descent
  • linear regression
  • logistic regression
  • support vector machines
  • Decision Trees
  • neural networks
  • deep learning
  • clustering
  • dimensionality reduction
  • Computer Science
  • artificial intelligence
  • Data Science

  Summary of Comments ( 45 )
  https://news.ycombinator.com/item?id=43586073

  Verbose tl;dr

  HN users largely praised Shai Shalev-Shwartz and Shai Ben-David's "Understanding Machine Learning" as a highly accessible and comprehensive introduction to the field. Commenters highlighted the book's clear explanations of fundamental concepts, its rigorous yet approachable mathematical treatment, and the helpful inclusion of exercises. Several pointed out its value for both beginners and those with prior ML experience seeking a deeper theoretical understanding. Some compared it favorably to other popular ML resources, noting its superior balance between theory and practice. A few commenters also shared specific chapters or sections they found particularly insightful, such as the treatment of PAC learning and the VC dimension. There was a brief discussion on the book's coverage (or lack thereof) of certain advanced topics like deep learning, but the overall sentiment remained strongly positive.

  The Hacker News post titled "Understanding Machine Learning: From Theory to Algorithms" linking to Shai Shalev-Shwartz and Shai Ben-David's book has a moderate number of comments, discussing various aspects of the book and machine learning education in general.

  Several commenters praise the book for its clarity and accessibility, especially for those with a stronger mathematical background. One user describes it as the "most digestible theory book," highlighting its helpful explanations of fundamental concepts. Another appreciates the book's focus on proving the theory behind ML algorithms, which they found lacking in other resources. The balance between theory and practical application is also commended, with some users noting how the book helped them bridge the gap between abstract concepts and real-world implementations. Specific chapters on PAC learning and VC dimension are singled out as particularly valuable.

  A recurring theme in the comments is the comparison of this book with other popular machine learning resources. "The Elements of Statistical Learning" is frequently mentioned as a more statistically-focused alternative, often considered more challenging. Some users suggest using both books in conjunction, leveraging Shalev-Shwartz and Ben-David's book as a starting point before tackling the more advanced "Elements of Statistical Learning." Another comparison is made with the "Hands-On Machine Learning" book, which is characterized as more practically oriented.

  Some commenters discuss the role of mathematical prerequisites in understanding machine learning. While the book is generally praised for its clarity, a few users acknowledge that a solid foundation in linear algebra, probability, and calculus is still necessary to fully grasp the material. One comment even suggests specific resources to brush up on these mathematical concepts before diving into the book.

  Beyond the book itself, the discussion touches upon broader topics in machine learning education. The importance of understanding the theoretical underpinnings of algorithms is emphasized, with several comments cautioning against relying solely on practical implementations without a deeper understanding of the underlying principles. The evolving nature of the field is also acknowledged, with some users mentioning more recent advancements that aren't covered in the book. Finally, there's a brief discussion about the role of online courses versus traditional textbooks in learning machine learning, with varying opinions on their respective merits.