Joint Model and Data Sparsification via the Marginal Likelihood

Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with $\ell_\infty$ error guarantees. This variant of the problem is motivated by \emph{variable selection} tasks, where the goal is to estimate the support of a $k$-sparse signal in $\mathbb{R}^d$. Our main contribution is a provable separation between the \emph{oblivious} (for each'') and \emph{adaptive} (for all'') models of $\ell_\infty$ sparse recovery. We show that under an oblivious model, the optimal $\ell_\infty$ error is attainable in near-linear time with $\approx k\log d$ samples, whereas in an adaptive model, $\gtrsim k^2$ samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard $\ell_2$ setting, where $\approx k \log d$ samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a \emph{partially-adaptive} model, where we show nontrivial variable selection guarantees are possible with $\approx k\log d$ measurements.

Sparse additive models have attracted much attention in high-dimensional data analysis due to their flexible representation and strong interpretability. However, most existing models are limited to single-level learning under the mean-squared error criterion, whose empirical performance can degrade significantly in the presence of complex noise, such as non-Gaussian perturbations, outliers, noisy labels, and imbalanced categories. The sample reweighting strategy is widely used to reduce the model's sensitivity to atypical data; however, it typically requires prespecifying the weighting functions and manually selecting additional hyperparameters. To address this issue, we propose a new meta additive model (MAM) based on the bilevel optimization framework, which learns data-driven weighting of individual losses by parameterizing the weighting function via an MLP trained on meta data. MAM is capable of a variety of learning tasks, including variable selection, robust regression estimation, and imbalanced classification. Theoretically, MAM provides guarantees on convergence in computation, algorithmic generalization, and variable selection consistency under mild conditions. Empirically, MAM outperforms several state-of-the-art additive models on both synthetic and real-world data under various data corruptions.

Given a high-dimensional covariate matrix and a response vector, ridge-regularized sparse linear regression selects a subset of features that explains the relationship between covariates and the response in an interpretable manner. To choose hyperparameters that control the sparsity level and amount of regularization, practitioners commonly use k-fold cross-validation. However, cross-validation substantially increases the computational cost of sparse regression as it requires solving many mixed-integer optimization problems (MIOs) for each hyperparameter combination. To address this computational burden, we derive computationally tractable relaxations of the k-fold cross-validation loss, facilitating hyperparameter selection while solving $50$--$80\%$ fewer MIOs in practice. Our computational results demonstrate, across eleven real-world UCI datasets, that exact MIO-based cross-validation can be competitive with mature software packages such as glmnet and L0Learn.