Exact minimax ratio throughout the intermediate-budget regime

Determine whether the weighted data-selection minimax ratio for linear regression satisfies $(d,d+k)=1+\Gamma_{d,k}$ for every dimension $d$ and every intermediate budget parameter $1\le k\le d-1$, thereby resolving the remaining cases of the open regime $d<n<2d$.

Background

The paper studies weighted selection of at most nn examples for linear regression, followed by minimum-norm empirical risk minimization, and denotes the worst-case full-data risk ratio by (d,n)(d,n). Previous work establishes the endpoint values (d,d)=d+1(d,d)=d+1 and (d,n)=1(d,n)=1 for n≥2dn\ge 2d, while this paper proves exact values at (d,2d−1)(d,2d-1), (3,4)(3,4), and (4,5)(4,5).

For an intermediate budget n=d+kn=d+k, the paper defines the harmonic quantity Γd,k\Gamma_{d,k} through balanced integer partitions and proves the lower bound (d,d+k)≥1+Γd,k(d,d+k)\ge 1+\Gamma_{d,k}. It also proves that this lower bound is exact for orthogonal circuit-block systems. The conjecture asserts that no general dataset has a larger minimax ratio; the paper explicitly notes that the smallest unresolved cell is (4,6)(4,6), conjectured to have value $3/2$.

References

For every intermediate budget $n=d+k$ we prove the lower bound $(d,d+k)\ge 1+\Gamma_{d,k}$, where $\Gamma_{d,k}$ is an explicit harmonic quantity over balanced partitions, and we show that this bound is the exact minimax value over the class of datasets whose whitened gradient systems carry an orthogonal circuit-block structure. All three exact values match $1+\Gamma_{d,k}$, and we conjecture that equality holds throughout the open regime.

— Exact Risk Ratios for Weighted Data Selection in Linear Regression  (2608.28007 - Zhang, 28 Aug 2026) in Section 6, Discussion and open problems, paragraph “The conjecture” (Conjecture 1); the smallest explicitly identified unresolved cell is $(d,n)=(4,6)$

Our sharpness result concerns one fixed leakage level and sufficiently large ambient dimensions. Optimality of the remainder for every prescribed leakage or regression benchmark, and uniform estimates as $k/d\uparrow1/2$, remain open.

— Subspace Uncertainty and Sharp Sampling Thresholds on the Boolean Cube  (2610.12358 - Weinberger, 8 Oct 2026) in Section Discussion, subsection “Limitations and open questions”