[Paper Review] Early stopping for kernel boosting algorithms: A general analysis with localized complexities

Yuting Wei, Fanny Yang+1 more|arXiv (Cornell University)|Jul 5, 2017

Statistical Methods and Inference32 references13 citations

TL;DR

This paper establishes a general theoretical connection between early stopping in kernel boosting algorithms and localized Gaussian complexity, demonstrating that optimal stopping rules can be derived by analyzing the complexity of the function class explored during iterative updates. For Sobolev and other regular kernel classes, the proposed data-dependent stopping rules achieve minimax optimal estimation rates.

ABSTRACT

Early stopping of iterative algorithms is a widely-used form of regularization in statistics, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including L2-boost, LogitBoost and AdaBoost, among others), we exhibit a direct connection between the performance of a stopped iterate and the localized Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis of Gaussian or Rademacher complexities, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

Motivation & Objective

• To establish a theoretical framework linking early stopping in kernel boosting to localized Gaussian complexity.
• To derive optimal stopping rules for a broad class of loss functions, including $L^2$-boost, LogitBoost, and AdaBoost.
• To show that these stopping rules achieve minimax optimal estimation rates for regular kernel classes such as Sobolev kernels.
• To bridge the gap in theoretical understanding between early stopping and penalized regularization, particularly in terms of complexity-based analysis.

Proposed method

• The authors analyze the effective function space explored by $T$ iterations of kernel boosting, characterizing its size via localized Gaussian complexity.
• They derive non-asymptotic risk bounds for the averaged estimator $\bar{\theta}^T$ using localized Gaussian width, linking estimation error to the complexity of the function class.
• The method relies on empirical process theory and fixed-point analysis of localized complexities, extending tools traditionally used in penalized estimation to early stopping.
• For kernel classes with $\gamma$-exponential or $\beta$-polynomial eigenvalue decay, the critical radius of the localized complexity is derived to determine optimal stopping times.
• The stopping rules are data-dependent, based on the eigenvalues of the empirical kernel matrix, and are computable from observed data.
• Theoretical guarantees are established via concentration inequalities, with high-probability bounds on estimation error.

Experimental results

Research questions

• RQ1 Can early stopping in kernel boosting be theoretically justified using localized complexity measures, similar to those used in penalized estimation?
• RQ2 What is the relationship between the stopping time of an iterative boosting algorithm and the localized Gaussian complexity of the function class it explores?
• RQ3 Do early stopping rules derived from localized complexity achieve minimax optimal rates for kernel classes such as Sobolev kernels?
• RQ4 How do the theoretical stopping rules compare to practical performance in finite-sample settings?
• RQ5 Can the same optimality results be extended from the averaged estimator $\bar{\theta}^T$ to the stopped estimator $f^T$?

Key findings

• For kernel classes with $\gamma$-exponential eigenvalue decay, the optimal stopping time scales as $T \asymp \frac{\log(n)^{1/\gamma}\sigma^2}{n}$, achieving estimation error $\lesssim \left(\frac{1}{\alpha m} + \frac{1}{m^2}\right)\frac{\log^{1/\gamma}n}{n}\sigma^2$ with high probability.
• For $\beta$-polynomial decay ($\mu_j \leq c_1 j^{-2\beta}$), the optimal stopping time is $T \asymp n^{-2\beta/(1+2\beta)}$, yielding estimation error $\lesssim \left(\frac{1}{\alpha m} + \frac{1}{m^2}\right)\left(\frac{\sigma^2}{n}\right)^{2\beta/(2\beta+1)}$ with high probability.
• The proposed stopping rules are data-dependent and computable from the empirical kernel matrix eigenvalues, enabling practical implementation.
• The theory establishes that early stopping based on localized complexity achieves minimax optimal rates for regular kernel classes, matching known statistical lower bounds.
• The connection between early stopping and localized Gaussian complexity provides a principled, general framework that extends beyond specific algorithms to a broad class of loss functions including $L^2$-boost, LogitBoost, and AdaBoost.
• Simulations confirm strong agreement between the theoretical predictions and empirical performance of the derived stopping rules.