Early-stopping optimality without a U-shape risk assumption

Consider the high-dimensional linear regression model with i.i.d. training data $D_n=\{(x_i,y_i)\}_{i=1}^n$, where for each $i$,

with $x_i\sim N(0,\Sigma)$ for some positive semidefinite $\Sigma\in\mathbb R^{p\times p}$, $\varepsilon_i\sim N(0,\sigma^2)$, and $\varepsilon_i$ independent of $x_i$. Let $p=p_n$ be comparable to $n$ (for example $p_n/n\to\kappa\in(0,\infty)$). Let an iterative algorithm (such as gradient descent, proximal gradient descent, or an accelerated variant) produce estimators $\hat\beta^0,\hat\beta^1,\dots,\hat\beta^{T_n}\in\mathbb R^p$, where each $\hat\beta^t$ is measurable with respect to $D_n$ (and any internal algorithmic randomness), and $T_n\ge 1$ may depend on $n$.

For each iteration $t\in\{1,\dots,T_n\}$, define the population prediction risk (generalization error) of $\hat\beta^t$ by

where $(X^{\mathrm{new}},Y^{\mathrm{new}})$ is an independent test pair distributed as $(x_i,y_i)$. Thus $(R_t)_{t=1}^{T_n}$ is a random sequence induced by $D_n$.