发表机构
Tippie College of Business, University of Iowa（爱荷华大学蒂皮商学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对凸Hölder光滑函数的最小化问题，证明了Active-Hull方法在高维情形下的最优线性最小化预言复杂度，并提出了无参数的元参数搜索算法来实现该复杂度。
AI 中文摘要
我们考虑凸问题$(P): {\min}_{y\in \mathbb{R}^n}\\, f(Ay)+g(y)$，其中$A:\mathbb{R}^n\to \mathbb{R}^m$是线性算子，$f$在$\mathbb{R}^m$上是凸的且$(M,\nu)$-Hölder光滑的，$g$是$\mathbb{R}^n$上的正常闭凸函数，并且允许（广义）线性最小化预言（LMO）。Active-Hull（AH）方法是指在每次迭代中调用LMO并输出位于所有过去LMO返回值的凸包中的点的方法。我们证明在高维情形下，对于求解$(P)$的AH方法类，最优LMO复杂度为$O\big(M^{{2}/({1+\nu})}D^2\varepsilon^{-{2}/({1+\nu})}\big)$，其中$D$表示可行域的直径，$\varepsilon$表示目标次优性间隙。特别地，最优LMO复杂度由单循环一阶原始-对偶分裂方法实现。由于该方法需要知道问题参数（例如$M$、$D$和$\nu$），我们提出了两种元参数搜索算法以解决此问题。这些算法是无参数的，但代价是其LMO复杂度中多了一个对数因子。
英文摘要
We consider the convex problem $(P): {\min}_{y\in \mathbb{R}^n}\, f(Ay)+g(y)$, where $A:\mathbb{R}^n\to \mathbb{R}^m$ is a linear operator, $f$ is convex and $(M,ν)$-Hölder smooth on $\mathbb{R}^m$, and $g$ is proper, closed convex on $\mathbb{R}^n$, and admits a (generalized) linear minimization oracle (LMO). An active-hull (AH) method calls the LMO in each iteration and outputs a point that lies in the convex hull of all the past LMO returns. We show that under the Euclidean geometry (and in the high-dimensional regime), the optimal LMO complexity for the class of AH methods to solve $(P)$ is $O\big(M^{{2}/({1+ν})}D^2\varepsilon^{-{2}/({1+ν})}\big)$, where $D$ denotes the diameter of the feasible region and $\varepsilon$ denotes the objective sub-optimality gap. In particular, the optimal LMO complexity is achieved by a single-loop first-order primal-dual splitting method. Since this method requires knowledge of the problem parameters (e.g., $M$, $D$ and $ν$), we propose two meta parameter-search algorithms that aim to resolve this issue. These algorithms are parameter-free, but come with the price of an additional log-factor in their LMO complexities.
Comments19 pages
