In the task of differentially private (DP) continual counting, we receive a stream of increments and our goal is to output an approximate running total of these increments, without revealing too much about any specific increment. Despite its simplicity, differentially private continual counting has attracted significant attention both in theory and in practice. Existing algorithms for differentially private continual counting are either inefficient in terms of their space usage or add an excessive amount of noise, inducing suboptimal utility. The most practical DP continual counting algorithms add carefully correlated Gaussian noise to the values. The task of choosing the covariance for this noise can be expressed in terms of factoring the lower-triangular matrix of ones (which computes prefix sums). We present two approaches from this class (for different parameter regimes) that achieve near-optimal utility for DP continual counting and only require logarithmic or polylogarithmic space (and time). Our first approach is based on a space-efficient streaming matrix multiplication algorithm for a class of Toeplitz matrices. We show that to instantiate this algorithm for DP continual counting, it is sufficient to find a low-degree rational function that approximates the square root on a circle in the complex plane. We then apply and extend tools from approximation theory to achieve this. We also derive efficient closed-forms for the objective function for arbitrarily many steps, and show direct numerical optimization yields a highly practical solution to the problem. Our second approach combines our first approach with a recursive construction similar to the binary tree mechanism.

翻译：在差分隐私（DP）连续计数任务中，我们接收一个增量流，目标是输出这些增量的近似累加总数，同时不泄露任何特定增量的过多信息。尽管其形式简单，差分隐私连续计数在理论和实践中均引起了广泛关注。现有差分隐私连续计数算法要么在空间使用上效率低下，要么添加过多噪声导致次优效用。最实用的差分隐私连续计数算法向数值添加经过精心相关的高斯噪声。选择该噪声协方差的任务可表示为分解下三角单位矩阵（用于计算前缀和）。我们提出了该框架下的两种方法（适用于不同参数区间），这些方法实现了接近最优的DP连续计数效用，且仅需对数或多项式对数空间（及时间）。第一种方法基于一类Toeplitz矩阵的空间高效流式矩阵乘法算法。我们证明，为将该算法实例化用于DP连续计数，只需寻找一个低阶有理函数来逼近复平面上圆上的平方根。进而应用并扩展逼近论工具实现该目标。我们还推导了任意步长下目标函数的有效闭式解，并表明直接数值优化可得到该问题的高度实用解法。第二种方法将第一种方法与类似二叉树机制的递归构造相结合。