Differentially private (DP) stochastic convex optimization (SCO) is ubiquitous in trustworthy machine learning algorithm design. This paper studies the DP-SCO problem with streaming data sampled from a distribution and arrives sequentially. We also consider the continual release model where parameters related to private information are updated and released upon each new data, often known as the online algorithms. Despite that numerous algorithms have been developed to achieve the optimal excess risks in different $\ell_p$ norm geometries, yet none of the existing ones can be adapted to the streaming and continual release setting. To address such a challenge as the online convex optimization with privacy protection, we propose a private variant of online Frank-Wolfe algorithm with recursive gradients for variance reduction to update and reveal the parameters upon each data. Combined with the adaptive differential privacy analysis, our online algorithm achieves in linear time the optimal excess risk when $1<p\leq 2$ and the state-of-the-art excess risk meeting the non-private lower ones when $2<p\leq\infty$. Our algorithm can also be extended to the case $p=1$ to achieve nearly dimension-independent excess risk. While previous variance reduction results on recursive gradient have theoretical guarantee only in the independent and identically distributed sample setting, we establish such a guarantee in a non-stationary setting. To demonstrate the virtues of our method, we design the first DP algorithm for high-dimensional generalized linear bandits with logarithmic regret. Comparative experiments with a variety of DP-SCO and DP-Bandit algorithms exhibit the efficacy and utility of the proposed algorithms.

研究了使用差分隐私保护的在线随机凸优化问题，提出了一种具有递归梯度的私有在线Frank-Wolfe算法，可在线性时间内实现最优超额风险，并证明递归梯度的方差缩减结果在非平稳场景下也有理论保证。同时，该算法也被扩展到p=1的情况，可实现几乎与维度无关的超额风险。

关于私人在线凸优化：在$l_p$几何和高维情境赌徒中的最优算法