We study privacy amplification for differentially private model training with matrix factorization under random allocation (also known as the balls-in-bins model). Recent work by Choquette-Choo et al. (2025) proposes a sampling-based Monte Carlo approach to compute amplification parameters in this setting. However, their guarantees either only hold with some high probability or require random abstention by the mechanism. Furthermore, the required number of samples for ensuring $(ε,δ)$-DP is inversely proportional to $δ$. In contrast, we develop sampling-free bounds based on Rényi divergence and conditional composition. The former is facilitated by a dynamic programming formulation to efficiently compute the bounds. The latter complements it by offering stronger privacy guarantees for small $ε$, where Rényi divergence bounds inherently lead to an over-approximation. Our framework applies to arbitrary banded and non-banded matrices. Through numerical comparisons, we demonstrate the efficacy of our approach across a broad range of matrix mechanisms used in research and practice.

翻译：本研究探讨了在随机分配（亦称球入箱模型）下，基于矩阵分解的差分隐私模型训练中的隐私放大问题。Choquette-Choo等人（2025）的最新工作提出了一种基于采样的蒙特卡洛方法来计算该场景下的放大参数。然而，其隐私保证要么仅以较高概率成立，要么要求机制随机弃权。此外，为确保满足$(ε,δ)$-差分隐私所需的样本数量与$δ$成反比。与此相反，我们基于Rényi散度和条件组合推导出了免采样的隐私边界。前者通过动态规划公式高效计算边界得以实现；后者则通过为小$ε$场景提供更强的隐私保证作为补充——在该场景下，Rényi散度边界本质上会导致过度近似。我们的框架适用于任意带状与非带状矩阵。通过数值比较，我们证明了该方法在研究与实践中广泛使用的各类矩阵机制上的有效性。