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Differentially private (DP) stochastic convex optimization (SCO) isubiquitous in trustworthy machine learning algorithm design. This paper studiesthe DP-SCO problem with streaming data sampled from a distribution and arrivessequentially. We also consider the continual release model where parametersrelated to private information are updated and released upon each new data,often known as the online algorithms. Despite that numerous algorithms havebeen developed to achieve the optimal excess risks in different $\ell_p$ normgeometries, yet none of the existing ones can be adapted to the streaming andcontinual release setting. To address such a challenge as the online convexoptimization with privacy protection, we propose a private variant of onlineFrank-Wolfe algorithm with recursive gradients for variance reduction to updateand reveal the parameters upon each data. Combined with the adaptivedifferential privacy analysis, our online algorithm achieves in linear time theoptimal excess risk when $1<p\leq 2$ and the state-of-the-art excess riskmeeting the non-private lower ones when $2<p\leq\infty$. Our algorithm can alsobe extended to the case $p=1$ to achieve nearly dimension-independent excessrisk. While previous variance reduction results on recursive gradient havetheoretical guarantee only in the independent and identically distributedsample setting, we establish such a guarantee in a non-stationary setting. Todemonstrate the virtues of our method, we design the first DP algorithm forhigh-dimensional generalized linear bandits with logarithmic regret.Comparative experiments with a variety of DP-SCO and DP-Bandit algorithmsexhibit the efficacy and utility of the proposed algorithms.

On Private Online Convex Optimization: Optimal Algorithms in $\ell_p$-Geometry and High Dimensional Contextual Bandits