Generalization-error estimation beyond Gaussian designs

Let $(x_i,y_i)_{i=1}^n$ be i.i.d. training data with $x_i\in\mathbb R^p$ and

where $\beta_0\in\mathbb R^p$ is deterministic (or independent of the sample), $\mathbb E[\varepsilon_i\mid x_i]=0$, and $\mathbb E[\varepsilon_i^2\mid x_i]<\infty$. Write $X\in\mathbb R^{n\times p}$ for the design matrix with rows $x_i^\top$, and $y=(y_1,\dots,y_n)^\top$.

Assume a high-dimensional regime $n,p\to\infty$ with $p/n\to\gamma\in(0,\infty)$ under non-Gaussian random designs (e.g., sub-Gaussian or elliptical families with conditions sufficient for asymptotic normality arguments). For fixed iteration index $t\in\mathbb N$, let $\hat\beta^t=\hat\beta^t(X,y)$ be the $t$-th iterate of a first-order method (such as GD, proximal GD, or an accelerated variant) applied to least squares, possibly with a proper closed convex penalty. With independent test covariate $x_{\mathrm{new}}\sim x_i$, define