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）持续计数任务中，我们接收一个增量流，目标是输出这些增量的近似累计总值，同时不对任何特定增量泄露过多信息。尽管其结构简单，差分隐私持续计数在理论与实践中均引起了广泛关注。现有的差分隐私持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DP持续计数算法会对数值添加精心相关的加性高斯噪声。该噪声协方差的选择问题可转化为对全1下三角矩阵（用于计算前缀和）进行分解。我们针对不同参数区间提出了两种属于此类的方案，它们能实现DP持续计数近最优的效用，且仅需对数或多项式对数级的空间（与时间）。第一种方案基于一类Toeplitz矩阵的空间高效流式矩阵乘法算法。我们证明，为将该算法实例化至DP持续计数，只需在复平面上的圆上找到逼近平方根的低阶有理函数，并应用与拓展逼近论中的工具来实现此目标。我们还推导了任意步数下目标函数的封闭形式，表明直接数值优化能提供高度实用的解决方案。第二种方案将第一种方案与类似二叉树机制的递归结构相结合。