Exact Finite-Sample Variance Decomposition of Subagging: A Spectral Filtering Perspective

Standard resampling ratios, such as the bootstrap-inspired $\alpha \approx 0.632$, are widely used in ensemble learning, yet their finite-sample interaction with the functional complexity of the base learner remains poorly understood. We use the Hoeffding–ANOVA decomposition to derive the first exact finite-sample variance decomposition for subagging with any symmetric square-integrable base learner, where $n$ denotes the training sample size and $k$ the subsample size. The resulting identity shows that subagging acts as a deterministic spectral filter: the $c$-th order interaction component is attenuated by an explicit combinatorial factor ${\gamma }_c(n,k)$, bounded by and converging to ${\alpha }^c$ when the subsampling ratio $\alpha =k/n$ is held fixed. This explains why fixed resampling ratios can under-regularize high-capacity interpolators, whose variance is concentrated in high-order interactions, and why smaller $\alpha$ can suppress such noise exponentially. Building on this mechanism, we characterize how the optimal subsampling ratio shifts with the learner's interaction complexity and propose Complexity-Guided Adaptive Subsampling (CGAS), which empirically improves generalization over static resampling baselines.