Minimax rate matching in the large-radius regime

Determine the minimax-optimal estimation rate for joint-sparse multitask transfer learning when source–target contrasts have large radius, and establish whether the available upper bounds match the corresponding minimax lower bound up to logarithmic factors.

Background

The paper compares upper bounds for TMTL(Fused) and TMTL(Debiased) with a minimax lower bound over a class of row-sparse multitask regression problems whose source coefficient matrices lie within a prescribed mixed-norm radius of the target coefficient matrix. The comparison shows matching results in some small- and intermediate-radius regimes, but the unprojected upper bounds contain heterogeneity terms that grow with the source–target radius.

For large radii, the minimax lower bound reaches the target-only rate, whereas the available unprojected upper bounds continue to grow with the contrast radius. A target-certified projection supplies a target-only cap and closes the gap for the projected estimators in the large-radius regime, but the abstract explicitly characterizes the broader question of matching bounds as unresolved.

References

Comparison with a minimax lower bound identifies regimes where the bounds match up to logarithmic factors, where matching remains unresolved, and where projection onto a target-based convex set closes the gap.

— Joint-Sparse Transfer Learning for High-Dimensional Multi-Output Regression  (2609.30879 - Lim et al., 25 Sep 2026) in Abstract; Section 3, subsection “Matching regimes and the large-radius gap”