Fast Privacy Loss Accounting for Random Allocation via Multiscale FFT

In the random allocation sampling scheme each user's data is used in $k$ steps chosen randomly and uniformly from a sequence of $t$ steps. This scheme is known to improve on Poisson sampling in private optimization and in communication-efficient aggregation. The existing privacy loss distribution (PLD) accountant for this scheme runs in time quadratic in the inverse of the accuracy parameter $α$ (Feldman and Shenfeld, 2026). We give an algorithm, \emph{multiscale FFT}, that represents likelihood ratios on a grid that is uniform within each dyadic interval and computes the exact rounding of every pairwise sum using FFTs within each interval. It keeps the guarantees of the accountant and improves the dependence of its running time on $α$ from $O(α^{-2})$ to $O(α^{-1}\log(1/α))$. In addition, we demonstrate that rounding can be performed via the mean-preserving connect-the-dots projection of (Doroshenko et al., 2026). This reduces the errors that rounding introduces in the mean of the privacy loss and in the $χ^2$ divergence of the computed pair from the order of $α$ to $O(α^2/\log(t))$. In our experiments, our algorithm achieves speedups of several orders of magnitude at high accuracy and gives improved privacy bounds for DP-SGD with PREAMBLE.