Improved rounding tolerance for structured band-membership matrices

Establish whether the two-band, extremely thin band-membership matrices arising in the cardinality analysis of non-monotone convex ridge losses admit a max-norm rounding theorem with tolerance larger than order $1/r$.

Background

The paper uses an elementary rounding lemma: a 0-1 matrix within max-norm distance $1/(4r)$ of a rank-rr matrix has rank at most $2r-1$. An appendix construction shows that no analogous theorem with uniform tolerance of order 1/r1/\sqrt r can hold for arbitrary 0-1 matrices.

However, the matrices used in the paper's proof have additional structure: they encode two projected bands, one of which is extremely thin. The arbitrary-matrix obstruction therefore does not settle whether a stronger rounding result is possible for these structured matrices. Such an improvement could reduce the dimension dependence of the regret bound.

References

The proposition concerns arbitrary $0$-$1$ matrices only; the band-membership matrices of Section~\ref{sec:size} have additional structure (two bands, one of them extremely thin), and whether that structure admits a rounding statement with a larger tolerance is open.

— Thompson Sampling for Non-Monotone Convex Ridge Bandits: Monotonicity Is Not Needed for Polynomial Regret  (2609.10981 - Li, 10 Sep 2026) in Appendix E, following Proposition \ref{prop:hadamard}