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≥3/lnM, 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 ≥1−a−δ, that is, only δ below the ideal random guessing diagonal 1−a: For δ=1/100 and σ=1, roughly M≈1.14×106 rounds and N≈1.14×107 training samples suffice to achieve meaningful differential privacy. This is in contrast to recent negative results for the regime σ≤1/2lnM. Our concrete bounds can be composed over multiple epochs leading to δ having a linear in E dependency, which restricts E=O(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=cM2M with cM→0, then the E-fold composed trade-off function satisfies f⊗E(a)→1−a uniformly in a∈[0,1] with δ having only an O(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.
Poisson subsampling is the default sampling scheme in differentially private machine learning, largely because its unstructured randomness yields tractable privacy amplification analyses. Yet this same randomness introduces substantial participation variance: each sample appears in very different numbers of training iterations. In this work, we show that this variance is not merely a practical artifact to be tolerated, but a fundamental source of suboptimal privacy amplification. We prove that Balanced Iteration Subsampling (BIS), a structured scheme in which each sample participates in exactly a fixed number of iterations, achieves stronger privacy amplification than Poisson subsampling and is optimal at both extremes of the noise spectrum (σ→0 and σ→∞). Our analysis reveals that the privacy-noise tradeoff is governed not by maximizing randomness, but by eliminating participation variance while preserving uniform marginal participation across iterations. To translate this asymptotic theory into finite-noise guarantees, we introduce a practical near-exact Monte Carlo accountant for BIS, which removes the analytical slack of existing RDP and composition-based PLD analyses. Evaluations across more than 60 practical DP-SGD configurations show that BIS consistently outperforms Poisson subsampling in the low-noise regimes most relevant for high-utility private training, reducing the required noise multiplier by up to 9.6%. These results overturn the common intuition that more sampling randomness necessarily yields stronger privacy amplification: in DP-SGD, structured participation can be both more practical and more private. Our implementation is available at https://github.com/dong-xin-ao-andy/bis-mc-accountant.
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.
Understanding the relationship between generalization and privacy remains a central challenge in modern machine learning theory, particularly for deep networks trained by variants of differentially private stochastic gradient descent (DP-SGD). In this work we make progress on this persistent open problem by proving a finite-sample bound on the approximate max-information of DP-SGD that exhibits scaling properties comparable with (Dwork et al, 2015)'s classic result for ε-differentially private algorithms, namely at most linear in the dataset size. From our result we obtain a general-purpose PAC-Bayes generalization bound in which the necessary prior distribution can be learned by DP-SGD, as well as a generalization bound for DP-SGD-trained models themselves, with a complexity term that is fully explicit and controlled by the optimization hyperparameters.