交叉验证方差理论：K-fold 一阶方差最优且与 K 无关

在该恒等式框架下，方差分解为两部分：仅取决于训练集大小的抽样方差，以及因每次划分不同权重变化带来的折划分方差。由此可推出明确结论：K-fold 交叉验证因各观测权重相等，在一阶意义上方差最优；Monte Carlo 和 bootstrap 交叉验证则不是。K-fold 的一阶方差与 K 无关，且在模型的合理条件下 leave-one-out 在所有 K-fold 方案中方差最小；对 AUC 指标，leave-pair-out 具有同样性质。论文还证明交叉验证估计量在完整方差下渐近正态，覆盖 MSE 这类按观测平均的指标和 AUC 这类按 case-control 对平均的指标。

这项工作为交叉验证性能推断提供了统一的理论基础，并给出了选择方差最优划分方案的普适原则。对做模型评估基准实验、置信区间构造或论文审稿中方法学判断的研究者有直接参考价值。

A Theory of Cross-Validation Variance

Cross-validation estimates predictive performance by repeatedly splitting the data into training and validation sets, fitting the model on the first and evaluating it on the second. Every observation influences the estimate in two roles, through its own evaluation when it is held out and through the models it helps fit when it trains. The random splits determine its weight in each role. From this representation, we prove an exact finite-sample identity for the variance of the cross-validated estimate. For a given splitting scheme with a common training set size, the variance is a function of these weights, with constants of the model, the metric, and the population. It covers metrics averaged over observations, such as mean squared error, and over case--control pairs, such as the AUC. The identity shows which schemes to use. The variance decomposes into sampling variance, which depends only on the training size, and variance due to the random fold splitting, which arises when the weights change from one draw of the splits to the next. K-fold cross-validation, whose weights are equal across observations by design, is therefore variance optimal at first order; Monte Carlo and bootstrap cross-validation are not. The first-order variance of K-fold does not depend on K, and under reasonable conditions on the model leave-one-out has the smallest variance of any K-fold scheme. For the AUC the same holds for leave-pair-out, which evaluates every case--control pair. Finally, we show the cross-validated estimate is asymptotically normal with the identity's full variance. Together, these results provide a general theoretical foundation for inference on cross-validated performance and establish principles for selecting variance-optimal cross-validation designs.

本条情报归属 AITOP「论文」核心追踪栏目。在跨区域产业观察坐标中，该动态直接关联「AI 开发者工具链、Coding Agent 与工程化实践」研究主线，反映出当前技术演进与商业落地的最新风向。涉及核心实体与技术议题包括：交叉验证、K-fold、AUC、统计学理论，读者可通过下方标签进入对应专题页查阅历史脉络与关联对比。

信源核验说明：本文原始线索采编自「arXiv cs.LG」（@Nathaniel S. O'Connell, Jaime L. Speiser），经 AITOP 多维评分管线结构化提炼。