Exact Risk Ratios for Weighted Data Selection in Linear Regression

Abstract: Hanneke, Moran, Shlimovich and Yehudayoff (COLT 2025) posed the following open problem. A selector sees a finite dataset $$D \subseteq \mathbb{R}^d \times \mathbb{R}$$, picks at most $n$ examples together with nonnegative weights, and hands the weighted least squares objective to the minimum-norm ERM. Writing $F_w(d,n)$ for the worst-case ratio between the loss of the returned predictor on all of $D$ and the optimal loss, they proved $F_w(d,n)=\infty$ for $n&lt;d$, $F_w(d,d)=d+1$ and $F_w(d,n)=1$ for $n \ge 2d$, and asked for the value in the open regime $d&lt;n&lt;2d$. We determine this value in several cases. For every $d$ we prove $F_w(d,2d-1)=1+1/d$, which confirms a claim stated without proof in the original note. We further prove $F_w(3,4)=5/3$ and $F_w(4,5)=2$, the two smallest cells not covered by the endpoint formula. For every intermediate budget $n=d+k$ we prove the lower bound $F_w(d,d+k) \ge 1+Γ<em>{d,k}$, where $Γ</em>{d,k}$ 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 $1+Γ_{d,k}$, and we conjecture that equality holds throughout the open regime. The upper bound proofs run on a common geometric spine: a rigidity theorem for positive spanning configurations of loss gradients, classifications and structural reductions of small positive bases in $\mathbb{R}<sup>3$ and $\mathbb{R}<sup>4$, and a dimension-free extremal-basis argument that converts sign-cone geometry into five-point selections. We also give explicit counterexamples showing that several shorter routes fail, and constructive polynomial-time selection algorithms for all proved cases.