Fundamental Limitations of Favorable Privacy-Utility Guarantees for DP-SGD
Murat Bilgehan Ertan, Marten van Dijk
TL;DR
This work analyzes DP-SGD under the f-DP privacy framework in a single-epoch shuffled setting, revealing a fundamental privacy–utility tension: achieving a minimal separation κ from the random-guessing line requires a non-negligible noise multiplier σ, and reducing σ to improve utility forces κ to remain sizable. By constructing an explicit suboptimal hypothesis test, the authors derive a lower bound on κ (and a corresponding σ bound) that cannot be simultaneously small, even as the number of updates M grows, with a slower-than-expected asymptotic decay. The results extend to Poisson subsampling via a mixture argument, yielding a unified worst-case characterization for both sampling schemes. Empirical evaluations on standard benchmarks confirm substantial accuracy degradation at the noise levels implied by the bounds, highlighting a bottleneck in DP-SGD under worst-case adversarial assumptions and motivating exploration of alternative privacy notions or algorithmic redesigns.
Abstract
Differentially Private Stochastic Gradient Descent (DP-SGD) is the dominant paradigm for private training, but its fundamental limitations under worst-case adversarial privacy definitions remain poorly understood. We analyze DP-SGD in the $f$-differential privacy framework, which characterizes privacy via hypothesis-testing trade-off curves, and study shuffled sampling over a single epoch with $M$ gradient updates. We derive an explicit suboptimal upper bound on the achievable trade-off curve. This result induces a geometric lower bound on the separation $κ$ which is the maximum distance between the mechanism's trade-off curve and the ideal random-guessing line. Because a large separation implies significant adversarial advantage, meaningful privacy requires small $κ$. However, we prove that enforcing a small separation imposes a strict lower bound on the Gaussian noise multiplier $σ$, which directly limits the achievable utility. In particular, under the standard worst-case adversarial model, shuffled DP-SGD must satisfy $σ\ge \frac{1}{\sqrt{2\ln M}}$ $\quad\text{or}\quad$ $κ\ge\ \frac{1}{\sqrt{8}}\!\left(1-\frac{1}{\sqrt{4π\ln M}}\right)$, and thus cannot simultaneously achieve strong privacy and high utility. Although this bound vanishes asymptotically as $M \to \infty$, the convergence is extremely slow: even for practically relevant numbers of updates the required noise magnitude remains substantial. We further show that the same limitation extends to Poisson subsampling up to constant factors. Our experiments confirm that the noise levels implied by this bound leads to significant accuracy degradation at realistic training settings, thus showing a critical bottleneck in DP-SGD under standard worst-case adversarial assumptions.