Universality beyond ridge penalties and squared test loss

Extend the matched-Gaussian universality theory for the proposed risk estimators beyond square-root ridge penalties and squared test loss, including nonlinear proximal maps such as those arising in square-root Lasso.

Background

The paper proves universality for square-root ridge with squared test loss by exploiting a finite-dimensional spectral and Krylov representation. The authors state that this argument does not currently cover nonlinear proximal maps or other test losses. Establishing universality in these broader settings would extend the applicability of the risk estimators to more complex nonsmooth regression procedures.

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 ...”