arXiv:2409.15252v3 Announce Type: replace-cross
Abstract: We characterize the squared prediction risk of ensemble estimators obtained through subagging (subsample bootstrap aggregating) regularized M-estimators and construct a consistent estimator for the risk. Specifically, we consider a heterogeneous collection of $M ge 1$ regularized M-estimators, each trained with (possibly different) subsample sizes, convex differentiable losses, and convex regularizers. We operate under the proportional asymptotics regime, where the sample size $n$, feature size $p$, and subsample sizes $k_m$ for $m in [M]$ all diverge with fixed limiting ratios $n/p$ and $k_m/n$. Key to our analysis is a new result on the joint asymptotic behavior of correlations between the estimator and residual errors on overlapping subsamples, governed through a (provably) contractive nonlinear system of equations. Of independent interest, we also establish convergence of trace functionals related to degrees of freedom in the non-ensemble setting (with $M = 1$) along the way, extending previously known cases for squared loss with ridge and lasso regularizers.
When specialized to homogeneous ensembles trained with a common loss, regularizer, and subsample size, the risk characterization sheds some light on the implicit regularization effect due to the ensemble and subsample sizes $(M,k)$. For any ensemble size $M$, optimally tuning subsample size yields sample-wise monotonic risk. For the full-ensemble estimator (when $M to infty$), the optimal subsample size $k^star$ tends to be in the overparameterized regime $(k^star le min{n,p})$, when explicit regularization is vanishing. Finally, joint optimization of subsample size, ensemble size, and regularization can significantly outperform regularizer optimization alone on the full data (without any subagging).