AI 中文总结
该研究解决了COLT 2025开放问题中向量值线性回归加权数据选择的阈值问题，确定了精确恢复全数据损失的最小样本预算及多个加权选择分布取值，给出了中间单元的取值范围并验证相关推测，同时纠正了预印本中的错误论断。
AI 中文摘要
我们解决了Hanneke、Moran、Shlimovich和Yehudayoff提出的COLT 2025开放问题“回归任务的数据选择”中问题4的阈值部分。在采用平方损失$\n\ell_{(x,y)}(W)=|Wx-y|_2^2\f$的向量值线性回归中，其中$x\in\mathbb{R}^d$、$y\in\mathbb{R}^m$，学习器是具有最小Frobenius范数的经验风险最小化器，我们证明了在每个有限数据集上恢复全数据损失所需的加权样本最小预算恰好为$n^*(d,m)=(m+1)d$。我们进一步确定了加权选择分布$F_w(d,m,n)$的另外两个取值：在接近阈值的预算下，$F_w(d,m,(m+1)d-1)=1+\frac{1}{dm^2}$；在生成张成空间的预算下，对任意$m$有$F_w(d,m,d)=d+1$，而当$n<d$时$F_w(d,m,n)=\infty$。对于最小的开放中间单元$(d,m)=(2,2)$，我们证明$F_w(2,2,3)\in[13/8,15/8]$且$F_w(2,2,4)\in[5/4,3/2]$，将推测的精确值13/8和5/4归约为圆上至多含7个原子的有限矩问题，并为该推测提供了强有力的结构证据。我们的上界技术（固定基锥压缩引理、最大证明的行列式面刚性定理，以及零均值加权点系统的精确稀疏化引理）具有独立研究价值。作为副产品，我们纠正了近期一篇未同行评审预印本中流传的错误论断，构造了一个$m=2$的显式数据集，其中无法通过加权选择$2d$个点恢复最优损失。所有结果仅在$m\ge2$时为新结论；标量情形$m=1$的结果由Hanneke等人得出。
英文摘要
We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. We study vector-valued linear regression with square loss $\ell_{(x,y)}(W)=\lVert Wx-y\rVert_2^2$, where $x\in\mathbb{R}^d$ and $y\in\mathbb{R}^m$. The learner returns the minimum-Frobenius-norm empirical risk minimizer. We prove that the minimal budget of weighted examples for recovering the full-data loss on every finite dataset is exactly $n^{\star}(d,m)=(m+1)d$. We determine the weighted selection profile $F_{\mathrm{weighted}}(d,m,n)$ at the near-threshold budget: $F_{\mathrm{weighted}}(d,m,(m+1)d-1)=1+1/(dm^2)$. We recover the known spanning-budget value $F_{\mathrm{weighted}}(d,m,d)=d+1$ for every $m$, and $F_{\mathrm{weighted}}(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_{\mathrm{weighted}}(2,2,3)\in[13/8,15/8]$ and $F_{\mathrm{weighted}}(2,2,4)\in[5/4,3/2]$. We reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms and assemble structural evidence for it. The upper bounds use a fixed-basis conic compression lemma, a determinant--facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems. These tools may be of independent interest. We also exhibit an explicit six-point integer dataset with $d=m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. Thus the scalar sufficient budget $2d$ does not extend to vector-valued outputs. Our new regression-profile results for $m\ge2$ extend the scalar theory for $m=1$.
Comments36 pages
