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Differentially private (DP) machine learning algorithms incur many sources ofrandomness, such as random initialization, random batch subsampling, andshuffling. However, such randomness is difficult to take into account whenproving differential privacy bounds because it induces mixture distributionsfor the algorithm's output that are difficult to analyze. This paper focuses onimproving privacy bounds for shuffling models and one-iteration differentiallyprivate gradient descent (DP-GD) with random initializations using $f$-DP. Wederive a closed-form expression of the trade-off function for shuffling modelsthat outperforms the most up-to-date results based on $(\epsilon,\delta)$-DP.Moreover, we investigate the effects of random initialization on the privacy ofone-iteration DP-GD. Our numerical computations of the trade-off functionindicate that random initialization can enhance the privacy of DP-GD. Ouranalysis of $f$-DP guarantees for these mixture mechanisms relies on aninequality for trade-off functions introduced in this paper. This inequalityimplies the joint convexity of $F$-divergences. Finally, we study an $f$-DPanalog of the advanced joint convexity of the hockey-stick divergence relatedto $(\epsilon,\delta)$-DP and apply it to analyze the privacy of mixturemechanisms.

Unified Enhancement of Privacy Bounds for Mixture Mechanisms via $f$-Differential Privacy