Sharpen the finite-width coupling bounds

Derive sharper finite-width approximation and coupling bounds for the symmetry-based analysis of two-layer networks learning orthogonal multi-index targets, thereby improving the non-optimal polynomial width and sample requirements.

Background

The paper’s quantitative requirements, including polynomial width and sample size, arise largely from the finite-width coupling framework used to transfer the symmetrized population analysis to the actual empirical network. The authors explicitly state that these requirements are not intended to be optimal.

Improving the rates would require refining the finite-width approximation technique, particularly the control of coupling errors over the full training trajectory. The paper identifies this refinement as a concrete direction for future work.

References

The current quantitative requirements arise largely from this coupling framework, and obtaining sharper bounds would likely require a corresponding refinement of the finite-width approximation technique. We leave this as an interesting direction for future work.

— Learning Orthogonal Multi-Index Models Beyond Small Initialization: Incremental Learning, Competitive Dynamics and Symmetry  (2609.10879 - Zhou et al., 9 Sep 2026) in Appendix, Section Omitted results and proofs in Section 2.3, paragraph “Discussion of quantitative bounds and assumptions”