Adaptive Methods Are Preferable in High Privacy Settings: An SDE Perspective

Differential Privacy (DP) is becoming central to large-scale training as privacy regulations tighten. We revisit how DP noise interacts with _adaptivity_ in optimization through the lens of _stochastic differential equations_, providing the first SDE-based analysis of private optimizers. Focusing on *DP-SGD* and *DP-SignSGD* under per-example clipping, we show a sharp contrast under fixed hyperparameters: *DP-SGD* converges at a Privacy-Utility Trade-Off of $\mathcal{O}(1/\varepsilon^2)$ with speed independent of $\varepsilon$, while *DP-SignSGD* converges at a speed *linear* in $\varepsilon$ with a $\mathcal{O}(1/\varepsilon)$ trade-off, dominating in high-privacy or large batch noise regimes. By contrast, under optimal learning rates, both methods achieve comparable theoretical asymptotic performance; however, the optimal learning rate of *DP-SGD* scales linearly with $\varepsilon$, while that of *DP-SignSGD* is essentially $\varepsilon$-independent. This makes adaptive methods far more practical, as their hyperparameters transfer across privacy levels with little or no re-tuning. Empirical results confirm our theory across training and test metrics, and empirically extend from *DP-SignSGD* to *DP-Adam*.