SlowGuess/NeurIPS_2023_OCR huggingface dataset:
By Proposition 13.1, Mei and Montanari [2022], this equation is met precisely by choosing  $\bar{\lambda} \equiv \lambda^{opt}$ , only possible if  $\rho < \rho^{*}$  per the definition of Proposition 4. Otherwise (i.e., if  $\rho > \rho^{*}$ ), the equality is unsatisfiable, and furthermore, one can use the property of the map  $\omega \equiv \omega(\bar{\lambda}, \psi, \zeta)$  in Lemma 13.1, Mei and Montanari [2022], to show that  $\omega_{2}^{3} + (\rho \psi_{2} - \rho_{1})\omega_{2}^{2} - \psi_{2}\rho \omega_{2} > 0$ .
In the large sample regime, we similarly want to check the equality
$$
\frac {\omega_ {1} ^ {2} / \zeta^ {2} + \psi_ {1}}{\psi_ {1} - 2 \psi_ {2} \omega_ {1} + \psi_ {1} \omega_ {1} ^ {2} - \omega_ {1} ^ {2}} \stackrel {*} {=} \frac {1}{1 - \omega_ {1} ^ {2}}.
$$
$$
\Longleftrightarrow \omega_ {1} ^ {3} + (\zeta^ {2} \psi_ {1} - \zeta^ {2} - 1) \omega_ {1} ^ {2} - \psi_ {1} \zeta^ {2} \omega_ {1} \stackrel {*} {=} 0.
$$
In this case, the demanded equality does not have  $\rho$  in it. Utilizing symmetry and the same argument as in Proposition 13.1, Mei and Montanari [2022], one can show that the absence of  $\rho$  implies the equality is always satisfiable by choosing  $\bar{\lambda} = 0$ , which also corresponds to the minimizer of the asymptotic risk:  $\arg \min_{\bar{\lambda}\geq 0}\mathcal{R}_{lsamp} = 0$ .