Separating Oblivious and Adaptive Models of Variable Selection

2026-02-18

AI Summary

This work investigates the fundamental distinction between oblivious and adaptive sampling models for support recovery of high-dimensional sparse signals under ℓ∞ error guarantees. Leveraging tools from high-dimensional statistical learning theory, minimax lower bound analysis, and sample complexity characterization, it establishes—for the first time—that in ℓ∞-norm sparse recovery, the oblivious model achieves optimal error rates with only ≈k log d samples in near-linear time, whereas the adaptive model requires ≳k² samples. This separation starkly contrasts with the ℓ₂ setting and rigorously demonstrates a provable gap between the two sampling paradigms in variable selection tasks. Furthermore, the study provides preliminary evidence that certain adaptive strategies can still attain nontrivial performance despite this inherent limitation.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Active Learning
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 \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.

Problem

• sparse recovery
• variable selection
• oblivious model
• adaptive model
• ℓ∞ error

Innovation

• sparse recovery
• variable selection
• oblivious vs adaptive
• ℓ∞ error
• sample complexity