A logarithmic endpoint rate with explicit effective noise and tolerance control

Put $d_{\delta }^{+}:=\max \{d_{\delta },0\}$. The scalar inequality remains true with $d_{\delta }^{+}$ on its right, and $d_{\delta }^{+}=O({\delta }^p{L}_{\delta }^{-\nu })$. Since ${\alpha }_{\delta }\asymp {\delta }^p$, one has $\log (1\mathrm{/}(c{\alpha }_{\delta }))\sim p{L}_{\delta }$ for either fixed $c=C_{\ast }$ or $c=2C_{\ast }$. Thus both conjugate terms in [F1] are $O(L_{\delta }^{-\nu })$, while $d_{\delta }^{+}\mathrm{/}{\alpha }_{\delta }=O(L_{\delta }^{-\nu })$. Substitution into [F1] proves both error estimates. If the joint approximate-minimum comparison is made with actual noise ${\epsilon }_{\delta }$ and $p\ge 1$, its comparison term is $d_{\delta }=2{\epsilon }_{\delta }^p+{\eta }_{\delta }$; this follows by replacing the noise upper bound by its actual value in the triangle-power comparison. The explicitly stated effective noise hypothesis therefore suffices. The quadratic norm rate follows by taking square roots.

For the asserted counterexample, complete squares coordinatewise in $\mathrm{\parallel }Kx-y^{{\delta }_j}{\mathrm{\parallel }}^2+({\alpha }_j\mathrm{/}2)\mathrm{\parallel }x{\mathrm{\parallel }}^2$. Every coordinate $i\mathrm{\ne }j$ has unique minimizer zero. The $j$th coordinate has unique minimizer $\frac{j^{-1}{\delta }_j}{j^{-2}+{\alpha }_j\mathrm{/}2}=2\mathrm{/}3$. All quadratic coefficients are positive and the penalty is strictly positive definite, so this point is the unique global minimizer on ${\mathrm{\ell }}^2$. Its norm error is $2\mathrm{/}3$ for every $j$. At truth zero the regularizer subgradient is zero, so the variational source left side is zero and any nonnegative source index satisfies the inequality. Also $\mathrm{\parallel }{y}^{{\delta }_j}-K{x}^{†}\mathrm{\parallel }={\delta }_j$, so the ordinary deterministic noise bound holds exactly. This proves the false universal endpoint assertion fails and explains the need for an explicit additional noise condition in step 1.1.