[Paper Review] Bootstrapping the Cross-Validation Estimate

Bryan Cai, Luo, Yuanhui+7 more|arXiv (Cornell University)|Jul 1, 2023

Forecasting Techniques and Applications4 citations

TL;DR

This paper proposes a fast, computationally efficient bootstrap method to estimate the standard error and construct valid confidence intervals for cross-validation performance estimates in prediction models. By modeling variance components within a random effects framework, the method overcomes traditional computational bottlenecks while maintaining flexibility across diverse models and performance metrics.

ABSTRACT

Cross-validation is a widely used technique for evaluating the performance of prediction models, ranging from simple binary classification to complex precision medicine strategies. It helps correct for optimism bias in error estimates, which can be significant for models built using complex statistical learning algorithms. However, since the cross-validation estimate is a random value dependent on observed data, it is essential to accurately quantify the uncertainty associated with the estimate. This is especially important when comparing the performance of two models using cross-validation, as one must determine whether differences in estimated error are due to chance. Although various methods have been developed to make inferences on cross-validation estimates, they often have many limitations, such as requiring stringent model assumptions. This paper proposes a fast bootstrap method that quickly estimates the standard error of the cross-validation estimate and produces valid confidence intervals for a population parameter measuring average model performance. Our method overcomes the computational challenges inherent in bootstrapping a cross-validation estimate by estimating the variance component within a random-effects model. It is also as flexible as the cross-validation procedure itself. To showcase the effectiveness of our approach, we conducted comprehensive simulations and real-data analysis across two applications.

Motivation & Objective

• To address the lack of reliable uncertainty quantification for cross-validation estimates, which are inherently random and subject to sampling variability.
• To overcome the high computational cost of naively bootstrapping cross-validation, especially in high-dimensional or complex model settings.
• To develop a method that is both computationally efficient and statistically valid across a wide range of prediction models and performance metrics.
• To provide a practical solution for comparing model performance with appropriate confidence intervals, particularly in biomedical applications where model reliability is critical.
• To bridge the gap between theoretical asymptotic justification and empirical performance in finite-sample settings, especially for high-dimensional models like lasso-regularized logistic regression.

Proposed method

• The method models the cross-validation estimate as a random effect with a hierarchical structure, decomposing the total variance into components attributable to data splitting and model training.
• It estimates the variance component within a random effects model to approximate the sampling distribution of the cross-validation estimator without repeated resampling of the entire cross-validation process.
• The approach uses a parametric bootstrap framework that leverages the assumed random effects structure to simulate the distribution of the cross-validation estimate efficiently.
• The method is designed to be model-agnostic, allowing application to any performance metric (e.g., mean absolute error, c-index, individualized treatment response score) and any learning algorithm.
• It derives a consistent estimator of the standard error by approximating the influence of training set selection and prediction error estimation through a linearization of the loss function.
• The method ensures valid confidence intervals by accounting for both the randomness in data partitioning and the variability in model parameter estimation.

Experimental results

Research questions

• RQ1 How can we efficiently estimate the standard error of a cross-validation performance estimate without incurring prohibitive computational costs?
• RQ2 Can a bootstrap-based inference procedure for cross-validation be made both fast and statistically valid in finite samples, especially for high-dimensional models?
• RQ3 To what extent does the proposed method maintain correct coverage probability for confidence intervals across diverse performance metrics and model types?
• RQ4 How does the method perform in comparison to existing asymptotic and nested cross-validation approaches in terms of accuracy and computational efficiency?
• RQ5 Under what conditions does the bootstrap confidence interval for the cross-validation estimate correspond meaningfully to the true population performance parameter?

Key findings

• The proposed bootstrap method achieves accurate standard error estimation with significantly reduced computational cost compared to naive bootstrap resampling of the entire cross-validation process.
• Simulations and real data analyses demonstrate that the bootstrap confidence intervals maintain appropriate coverage levels even in moderate sample sizes and high-dimensional settings.
• For lasso-regularized logistic regression with high-dimensional covariates, the bootstrap-based intervals show good empirical coverage despite the lack of theoretical justification under current asymptotic frameworks.
• The method performs well across diverse performance metrics, including mean absolute prediction error, c-index, and individualized treatment response scores.
• The approach exhibits robustness to model complexity and data dimensionality, with coverage properties that remain stable even when the true error variance is comparable to the sampling variance of the cross-validation estimate.
• The method provides a practical alternative to nested cross-validation, offering similar inferential validity with substantially lower computational overhead.