发表机构
HSE University（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
本文提出采样最大子梯度方法（SMax-SGM）求解大规模凸有限最大问题，通过随机子集采样降低查询复杂度，理论分析给出复杂度界，实验验证其高效性。
AI 中文摘要
我们研究了用于大规模凸有限最大问题的采样最大子梯度方法（SMax-SGM）。每次迭代在$N$个分量中随机选取一个新鲜子集，并取采样最大值的一个子梯度。因此，该方法是在采样最大替代函数上的随机子梯度下降。我们将替代误差界定为平均top-$k$间隙加上错过所有top-$k$分量的概率。一个局部化论证要求这些量仅在接近最优的子水平集上成立。在有界子梯度矩假设下，这产生$O(\varepsilon^{-2})$次子梯度查询，并且当$k$个分量在该集合中保持接近活跃时，产生$\widetilde O((N/k)\varepsilon^{-2})$次分量值查询，上限为全扫描成本。相反，一个一维仿射构造表明，即使平均top-$k$间隙处处消失且子梯度查询不受限制，$\Omega(N/k)$次分量值查询也可能是必要的。一个张量网格特化解释了子集大小何时可以独立于网格基数。在具有$2\times10^5$和$5\times10^6$个分量的有限最大值上的实验表明，在报告协议下，SMax-SGM达到5%数值参考间隙所需的分量值查询更少，且优化器时间低于每个达到目标的比较方法。
英文摘要
We study the Sampled-Max Subgradient Method (SMax-SGM) for large convex finite-max problems. Each iteration maximizes over a fresh random subset of the $N$ components and takes one subgradient of the sampled maximizer. The method is therefore stochastic subgradient descent on a sampled-max surrogate. We bound the surrogate error by an average-top-$k$ gap plus the probability of missing all top-$k$ components. A localization argument requires these quantities only on a near-optimal sublevel set. Under a bounded subgradient-moment assumption, this yields $O(\varepsilon^{-2})$ subgradient queries and, when $k$ components remain nearly active in that set, $\widetilde O((N/k)\varepsilon^{-2})$ component-value queries, capped by the full-scan cost. Conversely, a one-dimensional affine construction shows that $Ω(N/k)$ component-value queries can be necessary even when the average-top-$k$ gap vanishes everywhere and subgradient queries are unlimited. A tensor-grid specialization explains when the subset size can become independent of grid cardinality. Experiments on finite maxima with $2\times10^5$ and $5\times10^6$ components show that, under the reported protocols, SMax-SGM reaches a 5% numerical-reference gap with fewer component-value queries and lower optimizer time than each comparison method that reaches the target.
Comments23 pages, 2 figures, 4 tables
