非对偶设置下Lipschitz凸优化的一阶Oracle复杂度
The First-Order Oracle Complexity of Lipschitz Convex Optimization in Nondual Settings
- IMDEA Software Institute（IMDEA软件研究所）
- Purdue University（普渡大学）
- Tel Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究非对偶设置下Lipschitz凸优化的一阶Oracle复杂度，通过新在线学习博弈和序贯fat-shattering维数，证明了$\ell_1$-球上Euclidean-Lipschitz优化的$\widetilde O(1/T)$速率，并给出Wendel定理的定量推广。
AI中文摘要：
我们研究在$\ell_p$-球上目标函数关于$\ell_q$-范数Lipschitz的一阶黑箱凸优化问题，肯定地回答了COLT开放问题（Guz15b）中关于较小可行集（$p < q$）的几何结构能否改善凸优化收敛速度的非光滑版本，并在对数因子内匹配先前的下界。我们的收敛速率包括在$\ell_1$-球上的凸Euclidean-Lipschitz优化的$\widetilde O(1/T)$，在一般假设下改进了经典的$O(1/\sqrt{T})$速率。关键技术手段是一个新的在线学习博弈，其中比较器使用迄今为止观察到的仿射损失的最大值来评估。我们通过一个组合在线学习量——序贯fat-shattering维数——来界定该博弈值的上下界，并刻画了$\ell_p / \ell_q$情形下的该维数。我们的结果通常适用于可行集$X$和可能的次梯度集$H$为凸、中心对称且允许某种极小极大定理的情形，推进了Sridharan [Sri12, 第10.1.2节, Q3]提出的一个基本问题。作为我们分析的几何推论（具有独立意义），我们获得了若干Banach几何中样本凸包到其均值的期望距离的估计，这是著名的Wendel定理（Wen62）的一个定量版本，且适用于有界一般分布而非中心对称分布。
英文摘要：
We study first-order black-box convex optimization over an $\ell_p$-ball for objectives Lipschitz in the $\ell_q$-norm, solving in the affirmative the nonsmooth version of the COLT open question (Guz15b) on whether the geometry of a smaller feasible set ($p < q$) can improve convergence rates in convex optimization, and matching prior lower bounds up to logarithmic factors. Our rates include \(\widetilde O(1/T)\) for convex Euclidean-Lipschitz optimization over the $\ell_1$-ball, improving on the $O(1/\sqrt{T})$ classical rate under general assumptions. The key technical device is a new online learning game, where the comparator is evaluated using the maximum of affine losses observed so far. We bound the value of this game above and below in terms of a combinatorial online learning quantity: the sequential fat-shattering dimension, which we characterize for the $\ell_p / \ell_q$ case. Our results generally apply when the feasible set $X$ and the set of possible subgradients $H$ are convex, centrally symmetric, and admit a type of minmax theorem, advancing on a fundamental question by Sridharan [Sri12, Section 10.1.2, Q3]. As a geometric consequence of our analysis, of independent interest, we obtain estimates for the expected distance of a convex hull of samples to their mean in several Banach geometries, a version of the celebrated Wendel's theorem (Wen62), but quantitative and for bounded general distributions as opposed to centrally symmetric ones.
