Precise Asymptotics of Bagging Regularized M-estimators
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| Format: | Preprint |
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2024
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| author | Koriyama, Takuya Patil, Pratik Du, Jin-Hong Tan, Kai Bellec, Pierre C. |
| author_facet | Koriyama, Takuya Patil, Pratik Du, Jin-Hong Tan, Kai Bellec, Pierre C. |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_15252 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Precise Asymptotics of Bagging Regularized M-estimators Koriyama, Takuya Patil, Pratik Du, Jin-Hong Tan, Kai Bellec, Pierre C. Statistics Theory Machine Learning 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). |
| title | Precise Asymptotics of Bagging Regularized M-estimators |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2409.15252 |