We study online learning in the random-order model, where the multiset of loss functions is chosen adversarially but revealed in a uniformly random order. By extending the batch-to-online transformation of Dong and Yoshida (2023), we show that if an offline algorithm enjoys a $(1+\varepsilon)$-approximation guarantee, an average sensitivity bound controlled by a function $\varphi(\varepsilon)$, and stability with respect to $\varepsilon$, then we can obtain a small-loss regret bound typically of order $\tilde O(\varphi^{\star}(\mathrm{OPT}_T))$, where $\varphi^{\star}$ is the concave conjugate of $\varphi$, $\mathrm{OPT}_T$ is the offline optimum over $T$ rounds, and $\tilde O$ hides polylogarithmic factors in $T$. Our result refines their original $(1+\varepsilon)$-approximate regret guarantee and applies to a broad class of problems, including online $k$-means clustering and online low-rank approximation. We further apply our approach to online submodular function minimization using $(1\pm\varepsilon)$-cut sparsifiers of submodular hypergraphs, obtaining a small-loss regret bound of $\tilde O(n^3 + n^{3/4}\mathrm{OPT}_T^{3/4})$, where $n$ is the ground-set size; we also demonstrate its applicability to online $\ell_1$ regression. Our work sheds light on the power of sparsification and related algorithmic techniques in achieving small-loss regret bounds in the random-order model, without requiring structural assumptions on loss functions, such as linearity or smoothness.

翻译：我们研究随机序模型下的在线学习，其中损失函数的多重集由对抗性选择，但以均匀随机顺序揭示。通过扩展Dong和Yoshida（2023）的批量到在线转换，我们表明：若离线算法具有$(1+\varepsilon)$-近似保证、由函数$\varphi(\varepsilon)$控制的平均灵敏度界，以及关于$\varepsilon$的稳定性，则可获得典型为$\tilde O(\varphi^{\star}(\mathrm{OPT}_T))$阶的小损失遗憾界，其中$\varphi^{\star}$是$\varphi$的凹共轭，$\mathrm{OPT}_T$是$T$轮下的离线最优值，$\tilde O$隐藏$T$的多对数因子。我们的结果改进了其原始的$(1+\varepsilon)$-近似遗憾保证，并适用于广泛问题类别，包括在线$k$-均值聚类和在线低秩近似。我们进一步将方法应用于利用子模超图$(1\pm\varepsilon)$-切割稀疏化的在线子模函数最小化，获得$\tilde O(n^3 + n^{3/4}\mathrm{OPT}_T^{3/4})$的小损失遗憾界，其中$n$为基集大小；同时展示其对在线$\ell_1$回归的适用性。我们的工作揭示了稀疏化及相关算法技术在随机序模型下实现小损失遗憾界的能力，而无需对损失函数施加线性或光滑性等结构假设。