Universality at the correction-based projection boundary

Analyze the correction-based universality behavior when the limiting square-root projection argument satisfies \(\mathfrak m_\Sigma(a_s^2)=1\), thereby removing or resolving the no-boundary restriction at the nondifferentiability boundary.

Background

The universality results for the covariance-dependent and covariance-free risk estimators require a no-boundary condition ensuring that the square-root projection does not asymptotically land at norm one. At mΣ(as2)=1\mathfrak m_\Sigma(a_s^2)=1, the projection Jacobian is discontinuous, so the existing proof does not establish estimator universality. The authors explicitly identify this boundary case as unresolved.

References

Extending the universality argument beyond ridge or squared test loss remains open, as does the correction-based boundary case \mathfrak m_\Sigma(a_s2)=1.

— Generalization Error Estimation for Primal--Dual Algorithms in Non-Smooth Regression  (2608.13870 - Tan et al., 14 Aug 2026) in Remark following the proof of Theorem “Universality for square-root ridge,” Section “Proof of Theorem ...,” subsection “Completion of Theorem ...”