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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) contractible 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 square loss and ridge, 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)., Comment: 72 pages, 14 figures, 6 tables

This paper studies the asymptotics of resampling without replacement in the proportional regime where dimension $p$ and sample size $n$ are of the same order. For a given dataset $(X,y)\in \mathbb{R}^{n\times p}\times \mathbb{R}^n$ and fixed subsample ratio $q\in(0,1)$, the practitioner samples independently of $(X,y)$ iid subsets $I_1,...,I_M$ of $\{1,...,n\}$ of size $q n$ and trains estimators $\hat{\beta}(I_1),...,\hat{\beta}(I_M)$ on the corresponding subsets of rows of $(X, y)$. Understanding the performance of the bagged estimate $\bar{\beta} = \frac1M\sum_{m=1}^M \hat{\beta}(I_1),...,\hat{\beta}(I_M)$, for instance its squared error, requires us to understand correlations between two distinct $\hat{\beta}(I_m)$ and $\hat{\beta}(I_{m'})$ trained on different subsets $I_m$ and $I_{m'}$. In robust linear regression and logistic regression, we characterize the limit in probability of the correlation between two estimates trained on different subsets of the data. The limit is characterized as the unique solution of a simple nonlinear equation. We further provide data-driven estimators that are consistent for estimating this limit. These estimators of the limiting correlation allow us to estimate the squared error of the bagged estimate $\bar{\beta}$, and for instance perform parameter tuning to choose the optimal subsample ratio $q$. As a by-product of the proof argument, we obtain the limiting distribution of the bivariate pair $(x_i^T \hat{\beta}(I_m), x_i^T \hat{\beta}(I_{m'}))$ for observations $i\in I_m\cap I_{m'}$, i.e., for observations used to train both estimates., Comment: 25 pages, 10 figures

We consider unregularized robust M-estimators for linear models under Gaussian design and heavy-tailed noise, in the proportional asymptotics regime where the sample size n and the number of features p are both increasing such that $p/n \to \gamma\in (0,1)$. An estimator of the out-of-sample error of a robust M-estimator is analysed and proved to be consistent for a large family of loss functions that includes the Huber loss. As an application of this result, we propose an adaptive tuning procedure of the scale parameter $\lambda>0$ of a given loss function $\rho$: choosing$\hat \lambda$ in a given interval $I$ that minimizes the out-of-sample error estimate of the M-estimator constructed with loss $\rho_\lambda(\cdot) = \lambda^2 \rho(\cdot/\lambda)$ leads to the optimal out-of-sample error over $I$. The proof relies on a smoothing argument: the unregularized M-estimation objective function is perturbed, or smoothed, with a Ridge penalty that vanishes as $n\to+\infty$, and show that the unregularized M-estimator of interest inherits properties of its smoothed version., Comment: 33 pages, 10 figures

Major progress has been made in the previous decade to characterize the asymptotic behavior of regularized M-estimators in high-dimensional regression problems in the proportional asymptotic regime where the sample size $n$ and the number of features $p$ are increasing simultaneously such that $n/p\to \delta \in(0,\infty)$, using powerful tools such as Approximate Message Passing or the Convex Gaussian Min-Max Theorem (CGMT). The asymptotic error and behavior of the regularized M-estimator is then typically described by a system of nonlinear equations with a few scalar unknowns, and the solution to this system precisely characterizes the asymptotic error. Application of the CGMT and related machinery requires the existence and uniqueness of a solution to this low-dimensional system of equations or to a related scalar convex minimization problem. This paper resolves the question of existence and uniqueness of solution to this low-dimensional system for the case of linear models with independent additive noise, when both the data-fitting loss function and regularizer are separable and convex. Such existence result was previously known under strong convexity or for specific estimators such as the Lasso. The main idea behind this existence result is inspired by an argument developed by Montanari et al. [2023] and Celentano et al. [2023] in different contexts: By constructing an ad-hoc convex minimization problem in an infinite dimensional Hilbert space, the existence of the Lagrange multiplier for this optimization problem makes it possible to construct explicit solutions to the low-dimensional system of interest. The conditions under which we derive this existence result exactly correspond to the side of the phase transition where perfect recovery $\hat{x} = x_0$ fails, so that these conditions are optimal., Comment: 55 pages, 5 figures

Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least-squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of the ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. Furthermore, in the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV., Comment: 91 pages, 34 figures; this version adds general proof outlines (in Sections 4.3 and 5.3), add more experiments with non-Gaussian data (in Sections D and E), relaxes an assumption (in Section A.7), clarifies explanations at several places, and corrects minor typos at several places

We introduce a new deal of kernel density estimation using an exponentiated form of kernel density estimators. The density estimator has two hyperparameters flexibly controlling the smoothness of the resulting density. We tune them in a data-driven manner by minimizing an objective function based on the Hyv\"arinen score to avoid the optimization involving the intractable normalizing constant due to the exponentiation. We show the asymptotic properties of the proposed estimator and emphasize the importance of including the two hyperparameters for flexible density estimation. Our simulation studies and application to income data show that the proposed density estimator is appealing when the underlying density is multi-modal or observations contain outliers.

We derive the exact asymptotic distribution of the maximum likelihood estimator $(\hat{\alpha}_n, \hat{\theta}_n)$ of $(\alpha, \theta)$ for the Ewens--Pitman partition in the regime of $0<\alpha<1$ and $\theta>-\alpha$: we show that $\hat{\alpha}_n$ is $n^{\alpha/2}$-consistent and converges to a variance mixture of normal distributions, i.e., $\hat{\alpha}_n$ is asymptotically mixed normal, while $\hat{\theta}_n$ is not consistent and converges to a transformation of the generalized Mittag-Leffler distribution. As an application, we derive a confidence interval of $\alpha$ and propose a hypothesis testing of sparsity for network data. In our proof, we define an empirical measure induced by the Ewens--Pitman partition and prove a suitable convergence of the measure in some test functions, aiming to derive asymptotic behavior of the log likelihood., Comment: 58 pages