We introduce a new algorithm for numerical composition of privacy random variables, useful for computing the accurate differential privacy parameters for composition of mechanisms. Our algorithm achieves a running time and memory usage of $\mathrm{polylog}(k)$ for the task of self-composing a mechanism, from a broad class of mechanisms, $k$ times; this class, e.g., includes the sub-sampled Gaussian mechanism, that appears in the analysis of differentially private stochastic gradient descent. By comparison, recent work by Gopi et al. (NeurIPS 2021) has obtained a running time of $\widetilde{O}(\sqrt{k})$ for the same task. Our approach extends to the case of composing $k$ different mechanisms in the same class, improving upon their running time and memory usage from $\widetilde{O}(k^{1.5})$ to $\widetilde{O}(k)$.

翻译：我们引入了一个新的隐私随机变量数字构成算法, 用于计算机制构成的准确差异隐私参数。 我们的算法实现了运行时间和存储时间, 用于执行自我构建机制的任务, 由广泛的机制类别组成 $k$( k) ; 例如, 包括子标本高斯机制, 该机制出现在对差异性私人随机梯度下降的分析中 。 相比之下, Gopi 等人( NeurIPS 2021) 最近的工作( NeurIPS 2021) 为同一任务获得了运行时间和存储时间 $\ perpilde{ O} (\\\ qrt{k}) 。 我们的方法扩大到在同一类中将 $( k) 不同的机制组合在一起, 将运行时间和记忆使用从 $\ 全方位{O} ( k ⁇ 1.5} 改进到 $\ 全端T} (k) 美元。