Extension beyond convex differentiable-loss framework

Let $(x_i,y_i)_{i=1}^n$ be training data with $x_i\in\mathbb R^p$ and $y_i\in\mathbb R$, and let $p=p_n$ and subsample size $k=k_n$ satisfy proportional asymptotics $p/n\to\delta\in(0,\infty)$ and $k/n\to\kappa\in(0,1]$. For each subsample index set $S\subseteq[n]$ with $|S|=k$, define the regularized empirical $M$-estimator

where $\ell$ is a loss function, $\rho$ is a regularizer, and $\lambda>0$ is a tuning parameter. The infinite-subagged estimator is

with expectation over uniformly sampled subsamples $S$.

This setup follows Koriyama et al. (2025).

In the source paper's baseline analyzed regime (Assumption A in Koriyama et al. (2025)), the loss is convex and differentiable and the regularizer is convex (with separable structure in the main development). Under the full assumption set used in the source, the main asymptotic results characterize deterministic limits for overlap/order-parameter quantities of independent subsample estimators and induced risk formulas for the subagged estimator, and provide a risk-estimation theorem under an additional stronger regularity condition.