发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种无参数在线凸优化算法，在重尾噪声下实现区间动态遗憾界，通过区间自适应保持最优静态速率，并给出下界验证其最优性。
AI 中文摘要
我们研究在线凸优化问题，其中每轮仅有一个无偏随机次梯度，且未知有限的$p$阶噪声矩，$1<p\le2$。对于每个固定长度为$n$的区间$I$以及满足$\Lambda_I=1+P_I/D$的比较器路径，一个学习器实现了\\[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(\Lambda_I+\log^2(2T))} +\sigma Dn^{1/p}(\Lambda_I+\log^2(2T))^{(p-1)/p}]). \\] 该学习器不使用$G,\sigma,p,I,P_I$中的任何参数，且常数为通用的。区间自适应增加了比较器复杂度，同时保留了均值梯度和噪声指数的不同特征。分析控制了期望中的校准，并限制了观测尺度变化的成本。其一般定理与可预测可用的专家分布进行比较，相对熵依赖于非均匀先验。共同的先验偏好长窗口和长重启长度。在给定统计数据的情况下，区间成本变为$1+\log(T/n)$，包括最优的全时域静态速率。在显式条件下，对于保留全时域最优保证的学习器，一种测度变换下界识别了该对数的噪声幂。静态比较和确定性划分由相同的决策得出。
英文摘要
We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional $p$th noise moment, $1<p\le2$. For every fixed interval $I$ of length $n$ and comparator path with $Λ_I=1+P_I/D$, one learner achieves \[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(Λ_I+\log^2(2T))} +σDn^{1/p}(Λ_I+\log^2(2T))^{(p-1)/p}]). \] The learner uses none of $G,σ,p,I,P_I$, and the constant is universal. Interval adaptation adds to comparator complexity, preserving the distinct mean-gradient and noise exponents. The analysis controls calibration in expectation and limits the cost of observation-scale changes. Its general theorem compares to distributions over predictably available experts with relative-entropy dependence on a nonuniform prior. A common prior favors long windows and long restart lengths. With the statistics supplied, the interval cost becomes $1+\log(T/n)$, including the optimal full-horizon static rate. A change-of-measure lower bound identifies the noise power of this logarithm for learners retaining a full-horizon optimal guarantee, under explicit conditions. Static comparisons and deterministic partitions follow from the same decisions.
