The Last Iterate Advantage: Empirical Auditing and Principled Heuristic Analysis of Differentially Private SGD

We propose a simple heuristic privacy analysis of noisy clipped stochastic gradient descent (DP-SGD) in the setting where only the last iterate is released and the intermediate iterates remain hidden. Namely, our heuristic assumes a linear structure for the model. We show experimentally that our heuristic is predictive of the outcome of privacy auditing applied to various training procedures. Thus it can be used prior to training as a rough estimate of the final privacy leakage. We also probe the limitations of our heuristic by providing some artificial counterexamples where it underestimates the privacy leakage. The standard composition-based privacy analysis of DP-SGD effectively assumes that the adversary has access to all intermediate iterates, which is often unrealistic. However, this analysis remains the state of the art in practice. While our heuristic does not replace a rigorous privacy analysis, it illustrates the large gap between the best theoretical upper bounds and the privacy auditing lower bounds and sets a target for further work to improve the theoretical privacy analyses. We also empirically support our heuristic and show existing privacy auditing attacks are bounded by our heuristic analysis in both vision and language tasks.

Fine-Tuning Large Language Models with User-Level Differential Privacy

We investigate practical and scalable algorithms for training large language models (LLMs) with user-level differential privacy (DP) in order to provably safeguard all the examples contributed by each user. We study two variants of DP-SGD with: (1) example-level sampling (ELS) and per-example gradient clipping, and (2) user-level sampling (ULS) and per-user gradient clipping. We derive a novel user-level DP accountant that allows us to compute provably tight privacy guarantees for ELS. Using this, we show that while ELS can outperform ULS in specific settings, ULS generally yields better results when each user has a diverse collection of examples. We validate our findings through experiments in synthetic mean estimation and LLM fine-tuning tasks under fixed compute budgets. We find that ULS is significantly better in settings where either (1) strong privacy guarantees are required, or (2) the compute budget is large. Notably, our focus on LLM-compatible training algorithms allows us to scale to models with hundreds of millions of parameters and datasets with hundreds of thousands of users.

Optimal Rates for DP-SCO with a Single Epoch and Large Batches

The most common algorithms for differentially private (DP) machine learning (ML) are all based on stochastic gradient descent, for example, DP-SGD. These algorithms achieve DP by treating each gradient as an independent private query. However, this independence can cause us to overpay in privacy loss because we don't analyze the entire gradient trajectory. In this work, we propose a new DP algorithm, which we call Accelerated-DP-SRGD (DP stochastic recursive gradient descent), that enables us to break this independence and only pay for privacy in the gradient difference, i.e., in the new information at the current step. Our algorithm achieves the optimal DP-stochastic convex optimization (DP-SCO) error (up to polylog factors) using only a single epoch over the dataset, and converges at the Nesterov's accelerated rate. Our algorithm can be run in at most $\sqrt{n}$ batch gradient steps with batch size at least $\sqrt{n}$, unlike prior work which required $O(n)$ queries with mostly constant batch sizes. To achieve this, our algorithm combines three key ingredients, a variant of stochastic recursive gradients (SRG), accelerated gradient descent, and correlated noise generation from DP continual counting. Finally, we also show that our algorithm improves over existing SoTA on multi-class logistic regression on MNIST and CIFAR-10.

Tight Group-Level DP Guarantees for DP-SGD with Sampling via Mixture of Gaussians Mechanisms

We give a procedure for computing group-level $(\epsilon, \delta)$-DP guarantees for DP-SGD, when using Poisson sampling or fixed batch size sampling. Up to discretization errors in the implementation, the DP guarantees computed by this procedure are tight (assuming we release every intermediate iterate).

Privacy Amplification for Matrix Mechanisms

Privacy amplification exploits randomness in data selection to provide tighter differential privacy (DP) guarantees. This analysis is key to DP-SGD's success in machine learning, but, is not readily applicable to the newer state-of-the-art algorithms. This is because these algorithms, known as DP-FTRL, use the matrix mechanism to add correlated noise instead of independent noise as in DP-SGD. In this paper, we propose "MMCC", the first algorithm to analyze privacy amplification via sampling for any generic matrix mechanism. MMCC is nearly tight in that it approaches a lower bound as $\epsilon\to0$. To analyze correlated outputs in MMCC, we prove that they can be analyzed as if they were independent, by conditioning them on prior outputs. Our "conditional composition theorem" has broad utility: we use it to show that the noise added to binary-tree-DP-FTRL can asymptotically match the noise added to DP-SGD with amplification. Our amplification algorithm also has practical empirical utility: we show it leads to significant improvement in the privacy-utility trade-offs for DP-FTRL algorithms on standard benchmarks.

Correlated Noise Provably Beats Independent Noise for Differentially Private Learning

Differentially private learning algorithms inject noise into the learning process. While the most common private learning algorithm, DP-SGD, adds independent Gaussian noise in each iteration, recent work on matrix factorization mechanisms has shown empirically that introducing correlations in the noise can greatly improve their utility. We characterize the asymptotic learning utility for any choice of the correlation function, giving precise analytical bounds for linear regression and as the solution to a convex program for general convex functions. We show, using these bounds, how correlated noise provably improves upon vanilla DP-SGD as a function of problem parameters such as the effective dimension and condition number. Moreover, our analytical expression for the near-optimal correlation function circumvents the cubic complexity of the semi-definite program used to optimize the noise correlation matrix in previous work. We validate our theory with experiments on private deep learning. Our work matches or outperforms prior work while being efficient both in terms of compute and memory.

(Amplified) Banded Matrix Factorization: A unified approach to private training

Matrix factorization (MF) mechanisms for differential privacy (DP) have substantially improved the state-of-the-art in privacy-utility-computation tradeoffs for ML applications in a variety of scenarios, but in both the centralized and federated settings there remain instances where either MF cannot be easily applied, or other algorithms provide better tradeoffs (typically, as $\epsilon$ becomes small). In this work, we show how MF can subsume prior state-of-the-art algorithms in both federated and centralized training settings, across all privacy budgets. The key technique throughout is the construction of MF mechanisms with banded matrices. For cross-device federated learning (FL), this enables multiple-participations with a relaxed device participation schema compatible with practical FL infrastructure (as demonstrated by a production deployment). In the centralized setting, we prove that banded matrices enjoy the same privacy amplification results as for the ubiquitous DP-SGD algorithm, but can provide strictly better performance in most scenarios -- this lets us always at least match DP-SGD, and often outperform it even at $\epsilon\ll2$. Finally, $\hat{b}$-banded matrices substantially reduce the memory and time complexity of per-step noise generation from $\mathcal{O}(n)$, $n$ the total number of iterations, to a constant $\mathcal{O}(\hat{b})$, compared to general MF mechanisms.

Faster Differentially Private Convex Optimization via Second-Order Methods

Differentially private (stochastic) gradient descent is the workhorse of DP private machine learning in both the convex and non-convex settings. Without privacy constraints, second-order methods, like Newton's method, converge faster than first-order methods like gradient descent. In this work, we investigate the prospect of using the second-order information from the loss function to accelerate DP convex optimization. We first develop a private variant of the regularized cubic Newton method of Nesterov and Polyak, and show that for the class of strongly convex loss functions, our algorithm has quadratic convergence and achieves the optimal excess loss. We then design a practical second-order DP algorithm for the unconstrained logistic regression problem. We theoretically and empirically study the performance of our algorithm. Empirical results show our algorithm consistently achieves the best excess loss compared to other baselines and is 10-40x faster than DP-GD/DP-SGD.

Private (Stochastic) Non-Convex Optimization Revisited: Second-Order Stationary Points and Excess Risks

We consider the problem of minimizing a non-convex objective while preserving the privacy of the examples in the training data. Building upon the previous variance-reduced algorithm SpiderBoost, we introduce a new framework that utilizes two different kinds of gradient oracles. The first kind of oracles can estimate the gradient of one point, and the second kind of oracles, less precise and more cost-effective, can estimate the gradient difference between two points. SpiderBoost uses the first kind periodically, once every few steps, while our framework proposes using the first oracle whenever the total drift has become large and relies on the second oracle otherwise. This new framework ensures the gradient estimations remain accurate all the time, resulting in improved rates for finding second-order stationary points. Moreover, we address a more challenging task of finding the global minima of a non-convex objective using the exponential mechanism. Our findings indicate that the regularized exponential mechanism can closely match previous empirical and population risk bounds, without requiring smoothness assumptions for algorithms with polynomial running time. Furthermore, by disregarding running time considerations, we show that the exponential mechanism can achieve a good population risk bound and provide a nearly matching lower bound.

Why Is Public Pretraining Necessary for Private Model Training?

In the privacy-utility tradeoff of a model trained on benchmark language and vision tasks, remarkable improvements have been widely reported with the use of pretraining on publicly available data. This is in part due to the benefits of transfer learning, which is the standard motivation for pretraining in non-private settings. However, the stark contrast in the improvement achieved through pretraining under privacy compared to non-private settings suggests that there may be a deeper, distinct cause driving these gains. To explain this phenomenon, we hypothesize that the non-convex loss landscape of a model training necessitates an optimization algorithm to go through two phases. In the first, the algorithm needs to select a good "basin" in the loss landscape. In the second, the algorithm solves an easy optimization within that basin. The former is a harder problem to solve with private data, while the latter is harder to solve with public data due to a distribution shift or data scarcity. Guided by this intuition, we provide theoretical constructions that provably demonstrate the separation between private training with and without public pretraining. Further, systematic experiments on CIFAR10 and LibriSpeech provide supporting evidence for our hypothesis.