Uniform Inference for High-Dimensional Linear Models

For each sample size $n$, observe i.i.d. pairs $\{(y_i,x_i)\}_{i=1}^n$ generated by the linear model

where $x_i\in\mathbb{R}^{p_n}$, $\beta^*\in\mathbb{R}^{p_n}$ is unknown, and $\varepsilon_i$ is independent noise. Let $X\in\mathbb{R}^{n\times p_n}$ be the design matrix with rows $x_i^\top$, and $Y=(y_1,\dots,y_n)^\top$. Assume $p_n\gg n$ is allowed, $\mathbb{E}[x_i]=0$, $\mathrm{Cov}(x_i)=\Sigma$ with eigenvalues bounded away from $0$ and $\infty$, $x_i$ is sub-Gaussian, and $\varepsilon_i$ is mean-zero sub-Gaussian (often Gaussian with variance $\sigma^2$). Let the parameter space be the sparse class

Let $\hat\beta$ be the Lasso estimator

and define the debiased estimator

where $M\in\mathbb{R}^{p_n\times p_n}$ is a data-dependent approximate inverse of $\hat\Sigma=X^\top X/n$ (for example, row $m_j^\top$ solves $\min_{m} m^\top\hat\Sigma m$ subject to $\|\hat\Sigma m-e_j\|_\infty\le\mu_n$). For coordinate $j$, define the studentized pivot

with $\hat\tau_j^2$ a consistent estimator of the asymptotic variance (e.g., $\hat\tau_j^2=\hat\sigma^2\,m_j^\top\hat\Sigma m_j$).

A central unresolved question is to determine sharp conditions on $(n,p_n,s_n)$ (and on design/noise regularity) for regimes not already covered by existing positive results, under which inference based on $\hat\beta^d$ is uniformly valid over the whole sparse class and over all coordinates, namely

Several subcases are known (including 2017-era advances under stronger structural assumptions), but the exact frontier of achievable vs. non-achievable scaling remains open in general.

For fixed $\alpha\in(0,1)$, one related target is uniform coverage of coordinatewise intervals

in the sense

This fixed-$\alpha$ coverage criterion is weaker than full Kolmogorov convergence of the pivot law and does not by itself establish the full uniform distributional approximation above.