Deep Learning with Gaussian Differential Privacy

TL;DR

This paper argues that $f$-differential privacy (with Gaussian differential privacy as a canonical case) provides a refined, composition-friendly privacy framework for training deep neural networks. By deriving closed-form, analytically tractable privacy guarantees for NoisySGD and NoisyAdam and leveraging a central-limit-type composition result, it demonstrates sharper privacy bounds than the traditional moments accountant and enables noise-tuning to improve utility under privacy constraints. Empirical evaluations on MNIST, Adult, IMDb, and MovieLens confirm that the GDP-based analysis yields favorable privacy-utility trade-offs, sometimes allowing significantly smaller privacy loss for similar or better accuracy. The work emphasizes practical impact through public TensorFlow Privacy implementations and highlights avenues for extending $f$-DP to time-varying hyperparameters, diverse architectures, and broader private-learning tasks.

Abstract

Deep learning models are often trained on datasets that contain sensitive information such as individuals' shopping transactions, personal contacts, and medical records. An increasingly important line of work therefore has sought to train neural networks subject to privacy constraints that are specified by differential privacy or its divergence-based relaxations. These privacy definitions, however, have weaknesses in handling certain important primitives (composition and subsampling), thereby giving loose or complicated privacy analyses of training neural networks. In this paper, we consider a recently proposed privacy definition termed \textit{$f$-differential privacy} [18] for a refined privacy analysis of training neural networks. Leveraging the appealing properties of $f$-differential privacy in handling composition and subsampling, this paper derives analytically tractable expressions for the privacy guarantees of both stochastic gradient descent and Adam used in training deep neural networks, without the need of developing sophisticated techniques as [3] did. Our results demonstrate that the $f$-differential privacy framework allows for a new privacy analysis that improves on the prior analysis~[3], which in turn suggests tuning certain parameters of neural networks for a better prediction accuracy without violating the privacy budget. These theoretically derived improvements are confirmed by our experiments in a range of tasks in image classification, text classification, and recommender systems. Python code to calculate the privacy cost for these experiments is publicly available in the \texttt{TensorFlow Privacy} library.

Figures (5)

• Figure 1: An illustration of the CLT approach in the $f$-DP framework and the moments accountant in the $(\varepsilon, \delta)$-DP framework. $\mathtt{NoisyOptimizer}(\sigma, \dots)$ using the moments accountant gives the same privacy guarantees in terms of $(\varepsilon, \delta)$-DP as $\mathtt{NoisyOptimizer}(\tilde{\sigma}, \dots)$ using the CLT approach (the ellipses denote omitted parameters). Note that the duality formula \ref{['eq:dual']} is used in solving $\tilde{\mu}_{\mathtt{CLT}}$ from \ref{['eq:tilde_sigma']}.
• Figure 2: Tradeoffs between $\varepsilon$ and $\delta$ for both CLT and MA, which henceforth denotes the moments accountant. The settings follows the MNIST experiment in \ref{['sec:results']} with $\sigma=0.7, p=256/60000$. The bottom two plots assume $\delta=10^{-5}$. Note $\varepsilon$ and $\delta$ in the CLT are related via \ref{['eq:dual']} with $\mu = \mu_{\mathtt{CLT}}$. The bottom right plot is consistent with the conclusion $\sigma > \tilde{\sigma}$ shown in the cloud icon of \ref{['fig:workflow']}.
• Figure 3: Comparisons between the two ways of privacy analysis on MNIST in terms of the trade-off between type I and type II errors, in the same setting as \ref{['tab:tradeoff']}. The plots are different from Figure 7 in dong2019gaussian. The $(\varepsilon, \delta)$-DP guarantees are plotted according to \ref{['eq:e_d_f']}. The blue regions in the plots from the second row correspond to all pairs of $(\varepsilon,\delta)$ computed by MA. The blue regions are not noticeable in the third row.
• Figure 4: Experimental results from one run of NoisySGD on MNIST with different noise scales but the same $(\varepsilon, \delta)$-DP guarantees. The top plots use $p=256/60000, \eta=0.15, R=1.5$, and $\sigma = 1.3, \tilde{\sigma}=1.06$. The CLT approach with $\tilde{\sigma}=1.06$ and the moments accountant with $\sigma=1.3$ give $(1.34, 10^{-5})$-DP at the 20th epoch ($\mu_{\mathtt{CLT}}=0.35$). The bottom plots use the same parameters except for $\sigma=0.7, \tilde{\sigma}=0.638$, and $\eta = 0.15$. Both approaches give $(8.68, 10^{-5})$-DP at epoch 70 ($\mu_{\mathtt{CLT}}=1.78$). The right plots show the privacy loss during the training process in terms of the $\varepsilon$ spending with respect to $\delta=10^{-5}$.
• Figure 5: $(pG_{1/\sigma}+(1-p)\mathrm{Id})^{\otimes T}$ (blue dashed) is numerically computed and compared with the GDP limit (red solid) predicted by CLT. The two are almost identical at merely epoch one.

Theorems & Definitions (27)

• Definition 2.1: DMNS06approxdp
• Definition 2.2: dong2019gaussian
• Definition 2.3: dong2019gaussian
• Theorem 1: Comparison in $f$-DP
• remark 1
• Theorem 2: Comparison in $(\varepsilon, \delta)$-DP
• Proposition A.1
• proof
• Lemma A.2
• proof : Proof of \ref{['lem:mixture']}