Separating Oblivious and Adaptive Models of Variable Selection
This work addresses a theoretical gap in variable selection for high-dimensional statistics, providing a separation between oblivious and adaptive models, which is incremental but clarifies computational-statistical trade-offs.
The paper tackles the problem of sparse recovery with ℓ∞ error guarantees for variable selection, showing that in an oblivious model, optimal error is achievable with ≈k log d samples in near-linear time, while in an adaptive model, at least ≈k² samples are necessary, contrasting with the ℓ₂ setting.
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.