Private Streaming SCO in $\ell_p$ Geometry with Applications in High Dimensional Online Decision Making

Abstract

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. Numerous algorithms have been developed to achieve optimal excess risks in different $\ell_p$ norm geometries, but none of the existing ones can be adapted to the streaming and continual release setting. We propose a private variant of the Frank-Wolfe algorithm with recursive gradients for variance reduction to update and reveal the parameters upon each data. Combined with the adaptive DP analysis, our algorithm achieves the first optimal excess risk in linear time in the case $1<p\leq 2$ 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 i.i.d. 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.