Highly Adaptive Ridge (2410.02680v1)

Abstract: In this paper we propose the Highly Adaptive Ridge (HAR): a regression method that achieves a $n^{-1/3}$ dimension-free L2 convergence rate in the class of right-continuous functions with square-integrable sectional derivatives. This is a large nonparametric function class that is particularly appropriate for tabular data. HAR is exactly kernel ridge regression with a specific data-adaptive kernel based on a saturated zero-order tensor-product spline basis expansion. We use simulation and real data to confirm our theory. We demonstrate empirical performance better than state-of-the-art algorithms for small datasets in particular.

• The paper presents a novel regression method that achieves a dimension‐free convergence rate of O_P(n^{-1/3}), enhancing high-dimensional nonparametric modeling.
• It employs an adaptive kernel derived from a saturated spline basis, enabling efficient closed-form solutions and scalable computation.
• Empirical and theoretical results validate HAR's competitive performance, bridging classical regularization techniques with modern machine learning approaches.

Highly Adaptive Ridge: An Overview

The paper presents the Highly Adaptive Ridge (HAR) regression method, targeting improved convergence in high-dimensional settings. HAR achieves an impressive dimension-free convergence rate of $n^{-1/3}\ \mathscr L_2$ within the class of right-continuous functions possessing square-integrable sectional derivatives. This focus on nonparametric function classes makes HAR particularly relevant for tabular data scenarios.

Methodology

HAR leverages kernel ridge regression with a sophisticated, data-adaptive kernel derived from a saturated zero-order tensor-product spline basis expansion. The algorithm is characterized by its computational efficiency in small datasets, leveraging closed-form solutions to kernel equations while using innovative kernel learning techniques.

This adaptive framework involves constructing a high-dimensional spline basis, which results in the HAR estimator minimizing the empirical risk under a specified norm constraint. The Flexibility of this model makes it competitive against state-of-the-art algorithms, especially under conditions of limited data availability.

Theoretical Contributions

The primary contribution is establishing a dimension-free asymptotic rate of $O_{\P}(n^{-1/3}(\log n)^{2(p-1)/3})$, fundamental for efficiency in high-dimensional nonparametric regression. The proofs employ empirical process methodologies, demonstrating that HAR maintains a tight fit within specified bounds by leveraging cross-validation to adapt regularization parameters dynamically.

By carefully blending aspects of kernel ridge regression with adaptive basis construction, HAR situates itself as a bridge between intuitive regularization techniques and advanced machine learning methodologies. The paper extends theoretical groundwork by invoking connections to the mixed Sobolev spaces and showing parallels in handling derivative functions beyond standard Sobolev classes.

Computational Advantages

A notable strength of HAR is its computational tractability. Utilizing the Woodbury matrix identity, it circumvents memory-intensive matrix operations, making it significantly more scalable than traditional methods which necessitate explicit computation over expanded design matrices. This advantage implicitly extends HAR's applicability to larger datasets without prohibitive computational overhead.

Moreover, HAR's innovative use of kernelization eliminates some significant limitations of large-scale lasso implementations, particularly in the squared error loss context. Cross-validation for regularization strength is efficiently addressed via exact leave-one-out expressions, optimizing the training process.

Empirical and Theoretical Implications

The experimental results validate HAR's empirical performance superiority across various benchmark datasets, with a noticeable advantage in lower-dimensional settings. This aligns with the theoretical predictions related to its convergence properties.

From a theoretical perspective, HAR challenges existing function class assumptions, offering a middle ground between entirely interactive and strictly additive models. This could significantly influence future AI and ML model design, suggesting more generalizable and computationally efficient paths through expansive, yet bounded, function classes.

Future Research Directions

Insights drawn from HAR suggest substantive potential for enhancing boosting algorithms, such as Lassoed Tree Boosting, to preserve statistical efficiency while alleviating computational complexities. Further exploration into alternative loss functions and extended basis adaptations could broaden HAR's applicability, especially in non-linear or highly interactive settings.

The dimension-independent qualities of HAR open pathways to reconsider assumptions in causal inference, potentially refining confidence interval construction with less stringent conditions. Future investigations could explore integrating HAR methodologies into diverse machine learning pipelines, enriching both theoretical understanding and practical deployment in high-dimensional data environments.