Differentially private (DP) machine learning algorithms incur many sources of randomness, such as random initialization, random batch subsampling, and shuffling. However, such randomness is difficult to take into account when proving differential privacy bounds because it induces mixture distributions for the algorithm's output that are difficult to analyze. This paper focuses on improving privacy bounds for shuffling models and one-iteration differentially private gradient descent (DP-GD) with random initializations using $f$-DP. We derive a closed-form expression of the trade-off function for shuffling models that outperforms the most up-to-date results based on $(\epsilon,\delta)$-DP. Moreover, we investigate the effects of random initialization on the privacy of one-iteration DP-GD. Our numerical computations of the trade-off function indicate that random initialization can enhance the privacy of DP-GD. Our analysis of $f$-DP guarantees for these mixture mechanisms relies on an inequality for trade-off functions introduced in this paper. This inequality implies the joint convexity of $F$-divergences. Finally, we study an $f$-DP analog of the advanced joint convexity of the hockey-stick divergence related to $(\epsilon,\delta)$-DP and apply it to analyze the privacy of mixture mechanisms.

该研究聚焦于利用f-DP改进随机初始化的洗牌模型和一次迭代的差分隐私梯度下降（DP-GD）算法的隐私界限，并得到了洗牌模型的交替函数的闭式表达式，同时研究了随机初始化对于DP-GD的隐私影响。

通过 $f$-差分隐私统一增强混合机制的隐私边界