Optimality of the ERM error bound with upper isometry remainder

Let $n$ be the sample size, let $\cF$ be a function class, and let $\mathbb P$ be a probability distribution. Write $\widehat f_n$ for the empirical risk minimizer, $\cI_L(n,\mathbb P)$ for the lower isometry remainder, $\varepsilon_U^2$ for the upper-isometry error, and $\varepsilon_*^2$ for the benchmark error. The notation $A\gg B$ means that $A$ is asymptotically much larger than $B$. Optimality conjecture. The bound of Theorem is optimal in the following way: there exist models $(n,\cF,\mathbb P)$ for which $\widehat f_n$ incurs an error

$$\varepsilon_U^2 \gg \cI_L(n,\mathbb P) \gtrsim \varepsilon_*^2$$

or an error

$$\cI_L(n,\mathbb P) \gg \varepsilon_U^2 \gg \varepsilon_*^2;$$

and therefore the bound cannot be improved in general. This concerns the sharpness of the random-design ERM error bound by exhibiting models in which either the upper-isometry contribution or the lower-isometry contribution dominates.