发表机构
AI Institute MSU; HSE University; Moscow Independent Research Institute of Artificial Intelligence; Innopolis University; Mohamed bin Zayed University of Artificial Intelligence（莫斯科国立大学人工智能研究所; 高等经济大学; 莫斯科独立人工智能研究院; 因诺波利斯大学; 穆罕默德·本·扎耶德人工智能大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对精确标量值信息下的凸优化与凸-凹鞍点问题，证明了自适应随机化算法的查询复杂度下界，并扩展到强凸、双线性博弈及混合访问场景。
AI 中文摘要
我们研究了在精确标量值信息下优化的查询复杂度。对于定义在 $\mathbb R^d$ 上、且在半径为 $R$ 的欧几里得球内具有最小点的全局 $L$-光滑凸函数，我们在所述精度范围内证明了自适应随机化算法的下界 $\Omega(d\min\{d,\sqrt{LR^2/\varepsilon}\})$。查询不受限制，且该下界不含对数因子。它适用于二次函数，并匹配该类上两个基本上界的最小值。规定强凸性引入了额外的上限 $\sqrt{L/\mu}$。证明将矩阵的均值与其逆矩阵的均值在任意仿射切片上进行比较，并在自适应测量后限制后验矩。我们将这些信息下界扩展到具有独立块维度的鞍点问题。对于紧致双线性博弈，仅耦合就产生 $\Omega(r\min\{r,\sqrt{CR_xR_y/\varepsilon}\})$ 个查询，其中 $r=\min\{d_x,d_y\}$。一个强凸-强凹构造即使在已知各向同性对角黑塞矩阵的情况下也能产生依赖于条件的下界。单独的归约建立了在混合值和部分梯度访问下的标量查询下界，包括外部目标精度。欧几里得图像半径产生了对独立指定范数和已知惩罚的光滑和非光滑扩展。
英文摘要
We study the query complexity of optimization with exact scalar-value information. For globally $L$-smooth convex functions on $\mathbb R^d$ with a minimizer in a Euclidean ball of radius $R$, we prove the lower bound $Ω(d\min\{d,\sqrt{LR^2/\varepsilon}\})$ for adaptive randomized algorithms in the stated accuracy range. Queries are unrestricted, and the bound contains no logarithmic factor. It holds for quadratics and matches the minimum of two elementary upper bounds on that class. Prescribed strong convexity introduces the additional cap $\sqrt{L/μ}$. The proof compares the mean of a matrix with the mean of its inverse on arbitrary affine slices and bounds posterior moments after adaptive measurements. We extend these information bounds to saddle-point problems with independent block dimensions. For compact bilinear games, the coupling alone yields $Ω(r\min\{r,\sqrt{CR_xR_y/\varepsilon}\})$ queries, where $r=\min\lbrace d_x,d_y\rbrace$. A strongly convex--strongly concave construction yields a condition-dependent lower bound even with known isotropic diagonal Hessians. Separate reductions establish scalar query lower bounds under mixed value and partial-gradient access, including for outer objective accuracy. A Euclidean-image radius yields smooth and nonsmooth extensions to independently specified norms and known penalties.
