Can Rényi Accountants Beat Lossless Accountants for DP-SGD?

We derive a tight analysis of the trade-off function for Differentially Private Stochastic Gradient Descent (DP-SGD) with subsampling based on random shuffling within the $f$-DP framework. Our analysis covers the regime $σ\geq \sqrt{3/\ln M}$, where $σ$ is the noise multiplier and $M$ is the number of rounds within a single epoch. Unlike $f$-DP analyses for Poisson subsampling, which yield non-closed implicit formulas that can be machine computed but are non-transparent, random shuffling admits a tight analysis yielding transparent and interpretable closed-form bounds. Our concrete bounds, derived via the Berry-Esseen theorem, are tight up to constant factors within the proof framework. We demonstrate worked parameter settings for a single epoch ($E=1$) with a corresponding trade-off function $\geq 1-a-δ$, that is, only $δ$ below the ideal random guessing diagonal $1-a$: For $δ= 1/100$ and $σ= 1$, roughly $M \approx 1.14\times 10^6$ rounds and $N \approx 1.14\times 10^7$ training samples suffice to achieve meaningful differential privacy. This is in contrast to recent negative results for the regime $σ\leq 1/\sqrt{2 \ln M}$. Our concrete bounds can be composed over multiple epochs leading to $δ$ having a linear in $E$ dependency, which restricts $E=O(\sqrt{M})$. To go beyond Berry--Esseen, we introduce a new proof technique based on a generalization of the law of large numbers that yields an asymptotic random guessing diagonal-limit result: if $E=c_M^2M$ with $c_M\to 0$, then the $E$-fold composed trade-off function satisfies $f^{\otimes E}(a)\to 1-a$ uniformly in $a\in[0,1]$ with $δ$ having only an $O(\sqrt{E})$ dependency. We compare this asymptotic regime with the corresponding Poisson subsampling asymptotic, and highlight the characterization of explicit convergence rates as an open question.

Understanding the relationship between generalization and privacy remains a challenge in modern machine learning theory, particularly for deep networks that are trained by variants of differentially private stochastic gradient descent (DP- SGD). In this work we make progress on this persistent open problem. First, we derive explicit upper bounds on the approximate max-information of any algorithm that fulfills $(ε, δ)$-differential privacy or Rényi differential privacy, thereby going beyond the classical results for pure $ε$-differential privacy. Subsequently, we show even stronger guarantees for two common private learning algorithms, output perturbation with the Gaussian mechanism, and streaming DP-SGD, by exploiting the structure of their internal randomization. As an application of our results, we demonstrate how to obtain non-vacuous PAC-Bayes generalization bounds for deep networks, in which the prior distribution is learned by DP-SGD instead of the classical way of choosing it in a data-independent way.

We consider the privacy amplification properties of a sampling scheme in which a user's data isused in $k$ steps chosen randomly and uniformly from a sequence (or set) of $t$ steps. This sampling scheme has been recently applied in the context of differentially private optimization (Chua et al., 2024a; Choquette-Choo et al., 2025) and communication-efficient high-dimensional private aggregation (Asi et al., 2026), where it was shown to have utility advantages over the standard Poisson sampling. Theoretical analyses of this sampling scheme (Feldman & Shenfeld, 2025; Dong et al., 2025) lead to bounds that are close to those of Poisson sampling, yet still have two significant shortcomings. First, in many practical settings, the resulting privacy parameters are not tight due to the approximation steps in the analysis. Second, the computed parameters are either the hockey stick or Renyi divergence, both of which introduce overheads when used in privacy loss accounting. In this work, we demonstrate that the privacy loss distribution (PLD) of random allocation applied to any differentially private algorithm can be computed efficiently. When applied to the Gaussian mechanism, our results demonstrate that the privacy-utility trade-off for random allocation is at least as good as that of Poisson subsampling. In particular, random allocation is better suited for training via DP-SGD. To support these computations, our work develops new tools for general privacy loss accounting based on a notion of PLD realization. This notion allows us to extend accurate privacy loss accounting to subsampling which previously required manual noise-mechanism-specific analysis.