In this paper, we revisit the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) in Euclidean and general $\ell_p^d$ spaces. Specifically, we focus on three settings that are still far from well understood: (1) DP-SCO over a constrained and bounded (convex) set in Euclidean space; (2) unconstrained DP-SCO in $\ell_p^d$ space; (3) DP-SCO with heavy-tailed data over a constrained and bounded set in $\ell_p^d$ space. For problem (1), for both convex and strongly convex loss functions, we propose methods whose outputs could achieve (expected) excess population risks that are only dependent on the Gaussian width of the constraint set rather than the dimension of the space. Moreover, we also show the bound for strongly convex functions is optimal up to a logarithmic factor. For problems (2) and (3), we propose several novel algorithms and provide the first theoretical results for both cases when $1<p<2$ and $2\leq p\leq \infty$.

翻译：本文重新探讨了欧几里得空间与一般$\ell_p^d$空间中的差分私有随机凸优化（DP-SCO）问题。具体而言，我们聚焦于三个仍远未得到充分理解的场景：（1）欧几里得空间中约束且有界（凸）集上的DP-SCO；（2）$\ell_p^d$空间中的无约束DP-SCO；（3）$\ell_p^d$空间中基于重尾数据的约束有界凸集上的DP-SCO。针对问题（1），我们针对凸损失函数与强凸损失函数分别提出方法，其输出的（期望）超额总体风险仅依赖于约束集的高斯宽度而非空间维度。此外，我们还证明了强凸函数情形的界在忽略对数因子条件下是最优的。针对问题（2）和（3），我们提出若干新颖算法，并首次给出了$1<p<2$与$2\leq p\leq \infty$两种情形下的理论结果。