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
针对重尾噪声下在线凸优化无参数通用动态悔值的开放问题，提出HT-PAder算法并证明其最优性，无需预先知晓问题参数且提供极小极大保证。
AI 中文摘要
我们研究重尾噪声下非平稳环境中的在线凸优化（OCO），其中随机梯度预言机仅存在某个$p \in (1,2]$的有限$p$阶中心矩。静态悔值已得到充分研究，但以无参数方式实现通用动态悔值仍是未解决的挑战。我们通过提出HT-PAder解决该问题，这是一种无参数算法，将重启的AdaGrad专家与块长度的几何池相结合，搭配路径元算法AdaGrad-Hedge，后者对元损失无矩条件要求。对于直径为$D$的域、Lipschitz常数$G$、噪声水平$\sigma$及比较器路径长度$P_T$，HT-PAder实现的期望通用动态悔值为$\widetilde O\left( GD\sqrt{T(1+P_T/D)} + \sigma D T^{1/p}(1+P_T/D)^{(p-1)/p} \right)$。该算法无需预先知晓任何问题参数。即使在有限方差（$p=2$）的特殊情况下，HT-PAder也提供首个无参数极小极大通用动态悔值保证。我们还证明了匹配的下界，确立了路径长度指数的最优性。
英文摘要
We study online convex optimization with one unbiased stochastic subgradient per round and noise having a finite $p$-th central moment, where $p\in(1,2]$ is unknown. For a bounded convex domain of diameter $D$, subgradients bounded by $G$, noise scale $σ$, and comparator path length $P_T$, let $Λ_T=1+P_T/D$. A single algorithm, using none of $G,σ,p,P_T$, attains expected dynamic regret $O_p\left(\min\{GD\sqrt{TΛ_T}+σDT^{1/p}Λ_T^{(p-1)/p},\,GDT\}\right)$ against every fixed comparator sequence. Restarted AdaGrad experts produce the noise-path exponent $(p-1)/p$, and a prior favoring longer restart intervals removes horizon-dependent logarithmic overhead. We give an explicit bound uniform in $p$; its logarithm-free form has noise coefficient $O(1+\log(p/(p-1)))$, while the static-regret constant is universal. The analysis requires only marginal noise moments and permits dependent errors. Complete pathwise proofs retain both the expert-loss range and the gradient energies preceding comparator movement. Matching lower bounds hold on every bounded convex domain of positive diameter, under the same gradient-only information model. Together with a path-budget-tuned upper bound, they characterize the minimax rate with universal constants, including its linear-regret saturation.
