EGD statistical-radius conjecture in the middle SNR regime

Let the generalized linear model have link-function power $p=2$, and let the true parameter $\theta^{*}$ satisfy

$$C_{1}(d/n)^{1/4}\leq \lVert\theta^{*}\rVert\leq C_{2},$$

where $C_{1}$ and $C_{2}$ are universal constants. Let the EGD initialization satisfy $\theta_n^0\in\mathbb{B}(\theta^{*},\rho)$ for a constant $\rho$.

EGD statistical-radius conjecture. There exist universal constants $C_{3},C_{4},C_{5}$ such that, whenever $n\geq C_{3}(d\log(d/\delta))^{4}$ and $t\geq C_{4}\log(n/(d+\log(1/\delta)))$, with probability $1-\delta$,

$$\min_{1\leq k\leq t}\lVert\theta_n^k-\theta^{*}\rVert\leq C_{5}\frac{1}{\lVert\theta^{*}\rVert}\sqrt{\frac{d}{n}}.$$

This conjecture would provide a logarithmic-iteration statistical guarantee for EGD in the middle SNR regime, matching the expected computational advantage over GD. The paper notes that establishing the required homogeneous assumption for the population loss is the main obstacle, so the claim remains open.