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，我们接收一系列增量输入，目标是输出这些增量的近似累加总和，同时不泄露过多关于任何单个增量的信息。尽管问题看似简单，差分隐私连续计数在理论与实践中均引起了广泛关注。现有的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性能次优。最实用的DP连续计数算法会向数值中精心添加具有相关性的高斯噪声。选择该噪声协方差的问题可转化为对元素全为1的下三角矩阵（用于计算前缀和）进行因式分解。我们提出了两类针对不同参数范围的此类方法，这些方法能在DP连续计数中实现近乎最优的效用，且仅需对数或次对数级空间（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，为将该算法应用于DP连续计数，只需找到在复平面单位圆上逼近平方根的低阶有理函数，并应用及扩展逼近论工具实现此目标。我们还推导了任意步数目标函数的闭合形式解，并表明直接数值优化可生成高度实用的解决方案。第二种方法将第一种方法与类似二叉树机制的递归结构相结合。