  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.
