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

• URL: http://arxiv.org/abs/2603.03226v1
• Date: Tue, 03 Mar 2026 18:17:57 GMT
• Title: Adaptive Methods Are Preferable in High Privacy Settings: An SDE Perspective
• Authors: Enea Monzio Compagnoni, Alessandro Stanghellini, Rustem Islamov, Aurelien Lucchi, Anastasiia Koloskova,
• Abstract summary: Differential Privacy (DP) is becoming central to large-scale training as privacy regulations tighten.<n>We revisit how noise interacts with adaptivity in optimization through the lens of differential equations.<n>We show that DP-SGD converges at a Privacy-Utility Trade-Off of $mathcalO (1/varepsilon2)$ with speed independent of $varepsilon$, while DP-SignSGD converges at a speed linear in $varepsilon$ with speed independent of $varepsilon$.
• Score: 42.70658101277954
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 an $\mathcal{O}(1/\varepsilon)$ 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.

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