In this paper, we revisit the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) and provide excess population risks for some special classes of functions that are faster than the previous results of general convex and strongly convex functions. In the first part of the paper, we study the case where the population risk function satisfies the Tysbakov Noise Condition (TNC) with some parameter $\theta>1$. Specifically, we first show that under some mild assumptions on the loss functions, there is an algorithm whose output could achieve an upper bound of $\tilde{O}((\frac{1}{\sqrt{n}}+\frac{\sqrt{d\log \frac{1}{\delta}}}{n\epsilon})^\frac{\theta}{\theta-1})$ for $(\epsilon, \delta)$-DP when $\theta\geq 2$, here $n$ is the sample size and $d$ is the dimension of the space. Then we address the inefficiency issue, improve the upper bounds by $\text{Poly}(\log n)$ factors and extend to the case where $\theta\geq \bar{\theta}>1$ for some known $\bar{\theta}$. Next we show that the excess population risk of population functions satisfying TNC with parameter $\theta\geq 2$ is always lower bounded by $\Omega((\frac{d}{n\epsilon})^\frac{\theta}{\theta-1}) $ and $\Omega((\frac{\sqrt{d\log \frac{1}{\delta}}}{n\epsilon})^\frac{\theta}{\theta-1})$ for $\epsilon$-DP and $(\epsilon, \delta)$-DP, respectively. In the second part, we focus on a special case where the population risk function is strongly convex. Unlike the previous studies, here we assume the loss function is {\em non-negative} and {\em the optimal value of population risk is sufficiently small}. With these additional assumptions, we propose a new method whose output could achieve an upper bound of $O(\frac{d\log\frac{1}{\delta}}{n^2\epsilon^2}+\frac{1}{n^{\tau}})$ for any $\tau\geq 1$ in $(\epsilon,\delta)$-DP model if the sample size $n$ is sufficiently large.

翻译：在本文的第一部分, 我们重新审视了“ 差异” 的私隐 Stochastic Contimization (DP- SCO) 问题, 并为某些特殊功能类别提供了超额人口风险, 这些功能比一般 convex 和强烈 convex 函数的先前结果更快。 在论文的第一部分, 我们研究人口风险功能满足 Tysbakov 噪音调控 (TNSC) 的某个参数 $\theta>1美元。 具体地说, 我们首先显示, 在对损失函数的某些微小假设下, 其样本大小和美元是空间的层面。 之后, 我们处理的是 divaltial $1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ laxxxxxxxxxxxxxxxxx美元 。