Minimax Limits of k-Fold Cross-Validation via Majority
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| Format: | Preprint |
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2026
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| author | Nachum, Ido Urbanke, Rüdiger Weinberger, Thomas |
| author_facet | Nachum, Ido Urbanke, Rüdiger Weinberger, Thomas |
| contents | We study the mean-squared error of $k$-fold cross-validation as a risk estimator, with particular emphasis on how its accuracy depends on the number of folds $k$. Despite the widespread use of cross-validation, principled guidance for choosing $k$ is largely absent, mainly due to the complex dependence between fold-wise error estimates.
To obtain sharp and interpretable results, we focus on the majority algorithm in binary classification, a minimal yet nontrivial empirical risk minimization procedure. We provide a fine-grained analysis of its cross-validation behavior, showing that even this simple algorithm exhibits subtle and delicate phenomena for which existing theory provides loose and even vacuous bounds. Leveraging this analysis, we introduce a minimax framework for cross-validation risk estimation and prove that no empirical risk minimization algorithm can achieve an $O(1/n)$ minimax mean-squared error when the number of folds grows with the number of samples $n$; instead, a lower bound of order $Ω(\sqrt{k}/n)$ is unavoidable.
Our results reveal fundamental limitations of cross-validation as a data-reuse strategy, clarify gaps and inaccuracies in prior theoretical work, and position the majority algorithm as a natural benchmark that any tight analysis of cross-validation should be able to explain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25859 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimax Limits of k-Fold Cross-Validation via Majority Nachum, Ido Urbanke, Rüdiger Weinberger, Thomas Statistics Theory Machine Learning We study the mean-squared error of $k$-fold cross-validation as a risk estimator, with particular emphasis on how its accuracy depends on the number of folds $k$. Despite the widespread use of cross-validation, principled guidance for choosing $k$ is largely absent, mainly due to the complex dependence between fold-wise error estimates. To obtain sharp and interpretable results, we focus on the majority algorithm in binary classification, a minimal yet nontrivial empirical risk minimization procedure. We provide a fine-grained analysis of its cross-validation behavior, showing that even this simple algorithm exhibits subtle and delicate phenomena for which existing theory provides loose and even vacuous bounds. Leveraging this analysis, we introduce a minimax framework for cross-validation risk estimation and prove that no empirical risk minimization algorithm can achieve an $O(1/n)$ minimax mean-squared error when the number of folds grows with the number of samples $n$; instead, a lower bound of order $Ω(\sqrt{k}/n)$ is unavoidable. Our results reveal fundamental limitations of cross-validation as a data-reuse strategy, clarify gaps and inaccuracies in prior theoretical work, and position the majority algorithm as a natural benchmark that any tight analysis of cross-validation should be able to explain. |
| title | Minimax Limits of k-Fold Cross-Validation via Majority |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2605.25859 |