We study the task of $(\epsilon, \delta)$-differentially private online convex optimization (OCO). In the online setting, the release of each distinct decision or iterate carries with it the potential for privacy loss. This problem has a long history of research starting with Jain et al. [2012] and the best known results for the regime of {\epsilon} being very small are presented in Agarwal et al. [2023]. In this paper we improve upon the results of Agarwal et al. [2023] in terms of the dimension factors as well as removing the requirement of smoothness. Our results are now the best known rates for DP-OCO in this regime. Our algorithms builds upon the work of [Asi et al., 2023] which introduced the idea of explicitly limiting the number of switches via rejection sampling. The main innovation in our algorithm is the use of sampling from a strongly log-concave density which allows us to trade-off the dimension factors better leading to improved results.

翻译：我们研究 $(\epsilon, \delta)$-差分隐私在线凸优化（OCO）任务。在线场景中，每个不同决策或迭代的发布都伴随着隐私损失的风险。该问题自 Jain 等人 [2012] 起有长期研究历史，针对 $\epsilon$ 极小的情形，其已知最优结果由 Agarwal 等人 [2023] 给出。本文在维度因子方面改进了 Agarwal 等人 [2023] 的结果，并去除了光滑性要求。我们的结果现为该情形下 DP-OCO 的已知最优收敛速率。算法基于 Asi 等人 [2023] 的工作构建，后者通过拒绝采样显式限制切换次数。本算法的主要创新在于使用强对数凹密度采样，从而更优地权衡维度因子，进而实现改进结果。