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 $\theta>1$ 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.

中文翻译：

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更快的微分私有随机凸优化速度

在本文中，我们重新审视了微分私有随机凸优化 (DP-SCO) 的问题，并为一些特殊类别的函数提供了超额人口风险，这些函数比之​​前的一般凸函数和强凸函数的结果更快。在论文的第一部分，我们研究了总体风险函数满足具有某些参数 $\theta>1$ 的 Tysbakov 噪声条件 (TNC) 的情况。具体来说，我们首先表明，在对损失函数的一些温和假设下，有一种算法的输出可以达到 $\tilde{O}((\frac{1}{\sqrt{n}}+\frac {\sqrt{d\log \frac{1}{\delta}}}{n\epsilon})^\frac{\theta}{\theta-1})$ 为 $(\epsilon, \delta)$- $\theta\geq 2$时的DP，这里$n$是样本大小，$d$是空间的维度。然后我们解决效率低下的问题，通过 $\text{Poly}(\log n)$ 因子改进上限，并扩展到 $\theta\geq \bar{\theta}>1$ 对于某些已知的 $\bar{\theta}$ 的情况。接下来我们证明参数$\theta>1$满足TNC的总体函数的超额总体风险总是下界$\Omega((\frac{d}{n\epsilon})^\frac{\theta}{ \theta-1}) $ 和 $\Omega((\frac{\sqrt{d\log \frac{1}{\delta}}}{n\epsilon})^\frac{\theta}{\theta- 1})$ 分别用于 $\epsilon$-DP 和 $(\epsilon, \delta)$-DP。在第二部分，我们关注人口风险函数是强凸的一个特殊情况。与之前的研究不同，这里我们假设损失函数是 {\em 非负} 并且 {\em 人口风险的最优值足够小}。有了这些额外的假设，