发表机构
Moscow Independent Research Institute of Artificial Intelligence; Innopolis University; AI Institute MSU; Mohamed bin Zayed University of Artificial Intelligence; HSE University（莫斯科独立人工智能研究所; 因诺波利斯大学; 莫斯科国立大学人工智能研究所; 穆罕默德·本·扎耶德人工智能大学; 高等经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究几何不匹配的凸优化问题，证明极小极大间隙为 $LR^2/N^3$ 阶，并提出 Steiner 点水平方法达到最优速率，适用于多种目标和算子。
AI 中文摘要
在非欧几里得域（如 $\ell_1$ 球 $B_1^n(R)=\{x\in\mathbb R^n:\\|x\\|_1\le R\}$）上的最优一阶方法将 prox 函数与衡量光滑性的范数配对。当梯度仅在欧几里得范数下为 $L$-Lipschitz 时，使用欧几里得 prox 设置加速方法可将函数间隙 $f(x_N)-\min_{B_1^n(R)}f$ 在 $N$ 次一阶查询后降至 $O(LR^2/N^2)$（熵 $\ell_1$ prox 设置将 $L$ 替换为 $\ell_1\to\ell_\infty$ 常数 $L_1\le L$，代价是因子 $\log n$ 且指数相同）。Guzmán 和 Nemirovski 的下界为 $LR^2/N^3$ 阶，而上界能否降至该阶是 A. S. Nemirovski 提出的问题。我们证明，对于这个不匹配问题，间隙的极小极大值为 $LR^2/N^3$ 阶（忽略对数因子），对确定性和随机方法均成立，且已在维度与查询次数成比例的设置中实现。上界由 Steiner 点水平方法达到：每个已见支撑超平面被保留为水平切割，下一次查询在所得局部化多面体的 Steiner 点附近进行。分析基于一个几何事实：沿球的嵌套子集，Steiner 点移动的距离是维度的多对数，而欧几里得球则为 $\sqrt n$。所有查询保持可行，随机选择器保持内部工作多项式复杂度。相同的几何结构为非光滑目标、Hölder 梯度、高阶 oracle 和 Lipschitz 单调算子提供最优速率，后者具有匹配的确定性下界。对于凸二次函数，曲率学习方法在 $1\le p<2$ 的每个 $\ell_p$ 球上达到最优速率且无对数损失。实验证实了对欧几里得方法困难的目标上预测的 $N^{-3}$ 行为。
英文摘要
Optimal first-order methods on non-Euclidean domains such as the $\ell_1$ ball $B_1^n(R)=\{x\in\mathbb R^n:\|x\|_1\le R\}$ pair the prox-function with the norm in which smoothness is measured. When the gradient is $L$-Lipschitz in the Euclidean norm only, the accelerated method with a Euclidean prox-setup reduces the functional gap $f(x_N)-\min_{B_1^n(R)}f$ to $O(LR^2/N^2)$ after $N$ first-order queries (an entropic $\ell_1$ prox-setup replaces $L$ by the $\ell_1\to\ell_\infty$ constant $L_1\le L$, at the cost of a factor $\log n$ and with the same exponent). The lower bound of Guzmán and Nemirovski is of order $LR^2/N^3$, and whether the upper bound can be brought down to that order is a question of A. S. Nemirovski. We show that for this mismatched problem the minimax value of the gap is of order $LR^2/N^3$ up to logarithmic factors, for deterministic and for randomized methods alike, and already in dimension proportional to the number of queries. The upper bound is attained by a Steiner-point level method: every supporting hyperplane seen so far is kept as a level cut, and the next query is made near the Steiner point of the resulting localization polytope. The analysis rests on a single geometric fact: along nested subsets of the ball the Steiner points travel a distance that is polylogarithmic in the dimension, in contrast to $\sqrt n$ for the Euclidean ball. All queries stay feasible, and a randomized selector keeps the internal work polynomial. The same geometry yields optimal rates for nonsmooth objectives, Hölder gradients, higher-order oracles and Lipschitz monotone operators, the last with a matching deterministic lower bound. For convex quadratics, a curvature-learning method attains the optimal rate on every $\ell_p$ ball with $1\le p<2$ and without logarithmic loss. Experiments confirm the predicted $N^{-3}$ behaviour on objectives that are hard for Euclidean methods.
