Separating Oblivious and Adaptive Models of Variable Selection

arXiv:2602.16568v2 Announce Type: replace-cross Abstract: 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 variable selection tasks, where the goal is to estimate the support of a k-sparse signal in R^d. Our main contribution is a provable separation between the oblivious (for each'') and 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 klog 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 partially-adaptive model, where we show nontrivial variable selection guarantees are possible with approx klog d measurements.

Source: arXiv cs.LG | 2026-06-24