发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明两类归一化矩阵博弈的随机化矩阵向量查询下界，匹配确定性上界至对数因子，通过高斯核心和归约技术实现。
AI 中文摘要
我们证明了两种归一化矩阵博弈几何的随机化矩阵向量查询下界：一种是欧几里得单位球与概率单纯形之间的博弈，其中行范数至多为1；另一种是两个概率单纯形之间的博弈，其中元素绝对值至多为1。每次查询返回任意实数向量下的$(Ax,A^\top y)$。算法必须返回一个可行对，使得完全鞍点间隙至多为$\varepsilon$，且对每个可容许矩阵，概率至少为$2/3$。对于足够小的$\varepsilon$，最坏情况查询复杂度为：球-单纯形博弈为$\Omega(\varepsilon^{-2/3}/(\log^2(1/\varepsilon)\log\log(1/\varepsilon)))$，单纯形-单纯形博弈为$\Omega(\varepsilon^{-2/3}/(\log^{7/3}(1/\varepsilon)\log\log(1/\varepsilon)))$。困难实例的维度分别约为$\varepsilon^{-2/3}$和$\varepsilon^{-2/3}/\log^{1/3}(1/\varepsilon)$，且这些下界可扩展到更大维度。这些下界与Karmarkar、O'Carroll和Sidford的确定性上界在对数因子内匹配。证明在自适应双侧查询后提取一个新的高斯核心，并利用其最小奇异值的不确定性来建立线性系统求解的困难性。两个归约通过将小完全间隙转化为小残差，将这种困难性转移到矩阵博弈，仅对单纯形-单纯形博弈产生额外的对数归一化损失。
英文摘要
We prove randomized matrix-vector query lower bounds for two normalized matrix-game geometries: a Euclidean unit ball against a probability simplex, with row norms at most one, and two probability simplices, with entries of absolute value at most one. Each query returns $(Ax,A^\top y)$ for arbitrary real vectors. The algorithm must return a feasible pair with full saddle-point gap at most $\varepsilon$, with probability at least $2/3$ on every admissible matrix. For sufficiently small $\varepsilon$, the worst-case query complexities are $Ω(\varepsilon^{-2/3}/(\log^2(1/\varepsilon)\log\log(1/\varepsilon)))$ for ball-simplex games and $Ω(\varepsilon^{-2/3}/(\log^{7/3}(1/\varepsilon)\log\log(1/\varepsilon)))$ for simplex-simplex games. The hard instances have dimensions of order $\varepsilon^{-2/3}$ and $\varepsilon^{-2/3}/\log^{1/3}(1/\varepsilon)$, respectively, and the bounds extend to larger dimensions. These lower bounds match the deterministic upper bounds of Karmarkar, O'Carroll, and Sidford up to logarithmic factors. The proof extracts a fresh Gaussian core after adaptive two-sided queries and uses uncertainty in its smallest singular value to establish linear-system solve hardness. Two reductions transfer this hardness to matrix games by converting a small full gap into a small residual, with an additional logarithmic normalization loss only for simplex-simplex games.
