Provable data-driven calibration of the target confidence radius

Establish whether replacing the theoretical sparsity- and curvature-dependent calibration radius in the target-certified projection with an estimated data-driven radius preserves feasibility of the true target coefficient matrix and the resulting target-only error guarantee.

Background

The target-certified projection uses a confidence set whose radius is calibrated with the unknown target sparsity level and the unknown target restricted-curvature constant. This oracle calibration ensures that the true target coefficient matrix remains feasible, which is essential for both preserving the transfer guarantee and obtaining the target-only error cap.

The paper describes a possible plug-in based on an estimated active-row count and target validation, but does not establish that the estimated radius retains truth feasibility or the projection theorem’s guarantees. The issue is therefore an explicit unresolved problem concerning theoretically valid adaptive calibration.

References

Nothing above covers it: the argument rests on $\mathbf B0$ remaining feasible, and no result here shows that an estimated radius preserves that. We leave this direction for future research.

— Joint-Sparse Transfer Learning for High-Dimensional Multi-Output Regression  (2609.30879 - Lim et al., 25 Sep 2026) in Section 3, subsection “Closing the large-radius gap”; Appendix section “Calibration and a possible plug-in”