2D Stability Selection: design jittering for doubly stable feature selection

We study feature selection in high-dimensional regression under two distinct sources of instability: sampling variability and measurement error in the design matrix. Stability Selection addresses the former through sub-sampling and aggregation, but does not explicitly stress-test robustness to noisy predictors. We introduce doubly stable feature selection, a perturb-and-aggregate framework that targets features whose inclusion is stable both across randomization and across increasing levels of design noise. The method injects controlled additive noise into the design matrix, fits a fixed base selector such as the Lasso on the perturbed data, and aggregates selection frequencies. Sweeping over a grid of noise levels yields a stability path that summarizes robustness to measurement error while using the full sample size and isolating the effect of design perturbations. On the theory side, we show that classical model-selection conditions are preserved under sufficiently small perturbations, with a high-probability extension for Gaussian noise. Empirically, experiments on synthetic and real datasets show improved robustness compared with Stability Selection and standard base selectors.