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
本文针对重尾噪声下的非光滑随机凸优化，提出了尖锐的高概率统计速率，通过在线不等式分离方向波动与二阶矩，在一般凸和强凸情形下达到已知统计下界，并优化了置信系数。
AI 中文摘要
我们为重尾噪声下的非光滑随机凸优化建立了尖锐的高概率统计界，在一般凸和强凸两种情形下填补了置信度依赖的空白。我们考虑具有Lipschitz凸损失和已知近端正则化项的复合目标，通过新的无偏随机次梯度访问。中心化噪声的条件范数和方向p阶矩分别以L^p和s^p为界，其中1<p≤2且0<s≤L。对于T次预言机调用和失败概率0<δ<1/10，设q=1-1/p，ℓ=log(3/δ)，A=s(L^2/s^2)^q。在初始距离界D下，非饱和统计区间的极小极大目标误差为Θ(D(A+sℓ^q)/T^q)。对于μ-强凸正则化项，相应的二次区间速率为Θ((A^2+s^2ℓ^{2q})/(μT^{2q}))。这些速率要求确定性和有限方差项占主导；匹配的下界需要d≥⌈L^2/s^2⌉，而上界在任意有限维中成立。新的上界达到已知的统计下界阶数，将先前的置信系数s^{1/p}L^{1-1/p}替换为s，在强凸情形下将其平方替换为s^2。关键在于一个在线不等式，它将方向波动与拒绝采样产生的总二阶矩分离。在强凸性下，局部化保持这种分离，无需额外的时间水平对数。结果刻画了具有无限制内部计算的预言机复杂度。
英文摘要
We establish sharp high-probability statistical rates for nonsmooth stochastic convex optimization under heavy-tailed noise, closing the confidence-dependence gap in both general convex and strongly convex settings. We consider composite objectives with a Lipschitz convex loss and a known proximal regularizer, accessed through fresh unbiased stochastic subgradients. The centered noise has conditional norm and directional $p$th moments bounded by $L^p$ and $s^p$, respectively, where $1<p\le2$ and $0<s\le L$. For $T$ oracle calls and failure probability $0<δ<1/10$, set $q=1-1/p$, $\ell=\log(3/δ)$, and $A=s(L^2/s^2)^q$. With initial distance bound $D$, the minimax objective error is $Θ(D(A+s\ell^q)/T^q)$ in the nonsaturated statistical regime. For a $μ$-strongly convex regularizer, the corresponding quadratic-regime rate is $Θ((A^2+s^2\ell^{2q})/(μT^{2q}))$. These rates require domination of the deterministic and finite-variance terms; matching lower bounds require $d\ge\lceil L^2/s^2\rceil$, while the upper bounds hold in every finite dimension. The new upper bounds attain known statistical lower-bound orders, replacing the previous confidence coefficient $s^{1/p}L^{1-1/p}$ by $s$, and its square by $s^2$ under strong convexity. The key is an online inequality that separates directional fluctuations from the total second moment created by rejection sampling. Localization preserves this separation under strong convexity without additional horizon logarithms. The results characterize oracle complexity with unrestricted internal computation.
