Audra McMillan

Vitaly Feldman, Audra McMillan, Kunal Talwar

The shuffle model of differential privacy has gained significant interest as an intermediate trust model between the standard local and central models [EFMRTT19; CSUZZ19]. A key result in this model is that randomly shuffling locally randomized data amplifies differential privacy guarantees. Such amplification implies substantially stronger privacy guarantees for systems in which data is contributed anonymously [BEMMRLRKTS17]. In this work, we improve the state of the art privacy amplification by shuffling results both theoretically and numerically. Our first contribution is the first asymptotically optimal analysis of the Rényi differential privacy parameters for the shuffled outputs of LDP randomizers. Our second contribution is a new analysis of privacy amplification by shuffling. This analysis improves on the techniques of [FMT20] and leads to tighter numerical bounds in all parameter settings.

Rachel Cummings, Vitaly Feldman, Audra McMillan et al.

A key challenge in many modern data analysis tasks is that user data are heterogeneous. Different users may possess vastly different numbers of data points. More importantly, it cannot be assumed that all users sample from the same underlying distribution. This is true, for example in language data, where different speech styles result in data heterogeneity. In this work we propose a simple model of heterogeneous user data that allows user data to differ in both distribution and quantity of data, and provide a method for estimating the population-level mean while preserving user-level differential privacy. We demonstrate asymptotic optimality of our estimator and also prove general lower bounds on the error achievable in the setting we introduce.

Kunal Talwar, Shan Wang, Audra McMillan et al.

We revisit the problem of designing scalable protocols for private statistics and private federated learning when each device holds its private data. Locally differentially private algorithms require little trust but are (provably) limited in their utility. Centrally differentially private algorithms can allow significantly better utility but require a trusted curator. This gap has led to significant interest in the design and implementation of simple cryptographic primitives, that can allow central-like utility guarantees without having to trust a central server. Our first contribution is to propose a new primitive that allows for efficient implementation of several commonly used algorithms, and allows for privacy accounting that is close to that in the central setting without requiring the strong trust assumptions it entails. {\em Shuffling} and {\em aggregation} primitives that have been proposed in earlier works enable this for some algorithms, but have significant limitations as primitives. We propose a {\em Samplable Anonymous Aggregation} primitive, which computes an aggregate over a random subset of the inputs and show that it leads to better privacy-utility trade-offs for various fundamental tasks. Secondly, we propose a system architecture that implements this primitive and perform a security analysis of the proposed system. Our design combines additive secret-sharing with anonymization and authentication infrastructures.

Karan Chadha, Junye Chen, John Duchi et al.

In this work, we study practical heuristics to improve the performance of prefix-tree based algorithms for differentially private heavy hitter detection. Our model assumes each user has multiple data points and the goal is to learn as many of the most frequent data points as possible across all users' data with aggregate and local differential privacy. We propose an adaptive hyperparameter tuning algorithm that improves the performance of the algorithm while satisfying computational, communication and privacy constraints. We explore the impact of different data-selection schemes as well as the impact of introducing deny lists during multiple runs of the algorithm. We test these improvements using extensive experimentation on the Reddit dataset~\cite{caldas2018leaf} on the task of learning the most frequent words.

Audra McMillan, Adam Smith, Jon Ullman

In this work, we study local minimax convergence estimation rates subject to $ε$-differential privacy. Unlike worst-case rates, which may be conservative, algorithms that are locally minimax optimal must adapt to easy instances of the problem. We construct locally minimax differentially private estimators for one-parameter exponential families and estimating the tail rate of a distribution. In these cases, we show that optimal algorithms for simple hypothesis testing, namely the recent optimal private testers of Canonne et al. (2019), directly inform the design of locally minimax estimation algorithms.

Vitaly Feldman, Audra McMillan, Guy N. Rothblum et al. · apple-ml

Pan-privacy was proposed by Dwork et al. as an approach to designing a private analytics system that retains its privacy properties in the face of intrusions that expose the system's internal state. Motivated by federated telemetry applications, we study local pan-privacy, where privacy should be retained under repeated unannounced intrusions on the local state. We consider the problem of monitoring the count of an event in a federated system, where event occurrences on a local device should be hidden even from an intruder on that device. We show that under reasonable constraints, the goal of providing information-theoretic differential privacy under intrusion is incompatible with collecting telemetry information. We then show that this problem can be solved in a scalable way using standard cryptographic primitives.

Daniela Antonova, Allegra Laro, Audra McMillan et al.

Differentially private (DP) selection involves choosing a high-scoring candidate from a finite candidate pool, where each score depends on a sensitive dataset. This problem arises naturally in a variety of contexts including model selection, hypothesis testing, and within many DP algorithms. Classical methods, such as Report Noisy Max (RNM), assume all candidates' scores are equally sensitive to changes in a single individual's data, but this often isn't the case. To address this, algorithms like the Generalised Exponential Mechanism (GEM) leverage variability in candidate sensitivities. However, we observe that while these algorithms can outperform RNM in some situations, they may underperform in others - they can even perform worse than random selection. In this work, we explore how the distribution of scores and sensitivities impacts DP selection mechanisms. In all settings we study, we find that there exists a mechanism that utilises heterogeneity in the candidate sensitivities that outperforms standard mechanisms like RNM. However, no single mechanism uniformly outperforms RNM. We propose using the correlation between the scores and sensitivities as the basis for deciding which DP selection mechanism to use. Further, we design a slight variant of GEM, modified GEM that generally performs well whenever GEM performs poorly. Relying on the correlation heuristic we propose combined GEM, which adaptively chooses between GEM and modified GEM and outperforms both in polarised settings.

Vitaly Feldman, Audra McMillan, Satchit Sivakumar et al.

Estimating the density of a distribution from samples is a fundamental problem in statistics. In many practical settings, the Wasserstein distance is an appropriate error metric for density estimation. For example, when estimating population densities in a geographic region, a small Wasserstein distance means that the estimate is able to capture roughly where the population mass is. In this work we study differentially private density estimation in the Wasserstein distance. We design and analyze instance-optimal algorithms for this problem that can adapt to easy instances. For distributions $P$ over $\mathbb{R}$, we consider a strong notion of instance-optimality: an algorithm that uniformly achieves the instance-optimal estimation rate is competitive with an algorithm that is told that the distribution is either $P$ or $Q_P$ for some distribution $Q_P$ whose probability density function (pdf) is within a factor of 2 of the pdf of $P$. For distributions over $\mathbb{R}^2$, we use a different notion of instance optimality. We say that an algorithm is instance-optimal if it is competitive with an algorithm that is given a constant-factor multiplicative approximation of the density of the distribution. We characterize the instance-optimal estimation rates in both these settings and show that they are uniformly achievable (up to polylogarithmic factors). Our approach for $\mathbb{R}^2$ extends to arbitrary metric spaces as it goes via hierarchically separated trees. As a special case our results lead to instance-optimal private learning in TV distance for discrete distributions.

Joerg Drechsler, Ira Globus-Harris, Audra McMillan et al.

Differential privacy is a restriction on data processing algorithms that provides strong confidentiality guarantees for individual records in the data. However, research on proper statistical inference, that is, research on properly quantifying the uncertainty of the (noisy) sample estimate regarding the true value in the population, is currently still limited. This paper proposes and evaluates several strategies to compute valid differentially private confidence intervals for the median. Instead of computing a differentially private point estimate and deriving its uncertainty, we directly estimate the interval bounds and discuss why this approach is superior if ensuring privacy is important. We also illustrate that addressing both sources of uncertainty--the error from sampling and the error from protecting the output--simultaneously should be preferred over simpler approaches that incorporate the uncertainty in a sequential fashion. We evaluate the performance of the different algorithms under various parameter settings in extensive simulation studies and demonstrate how the findings could be applied in practical settings using data from the 1940 Decennial Census.

Vitaly Feldman, Audra McMillan, Kunal Talwar

Recent work of Erlingsson, Feldman, Mironov, Raghunathan, Talwar, and Thakurta [EFMRTT19] demonstrates that random shuffling amplifies differential privacy guarantees of locally randomized data. Such amplification implies substantially stronger privacy guarantees for systems in which data is contributed anonymously [BEMMRLRKTS17] and has lead to significant interest in the shuffle model of privacy [CSUZZ19; EFMRTT19]. We show that random shuffling of $n$ data records that are input to $\varepsilon_0$-differentially private local randomizers results in an $(O((1-e^{-\varepsilon_0})\sqrt{\frac{e^{\varepsilon_0}\log(1/δ)}{n}}), δ)$-differentially private algorithm. This significantly improves over previous work and achieves the asymptotically optimal dependence in $\varepsilon_0$. Our result is based on a new approach that is simpler than previous work and extends to approximate differential privacy with nearly the same guarantees. Importantly, our work also yields an algorithm for deriving tighter bounds on the resulting $\varepsilon$ and $δ$ as well as Rényi differential privacy guarantees. We show numerically that our algorithm gets to within a small constant factor of the optimal bound. As a direct corollary of our analysis we derive a simple and nearly optimal algorithm for frequency estimation in the shuffle model of privacy. We also observe that our result implies the first asymptotically optimal privacy analysis of noisy stochastic gradient descent that applies to sampling without replacement.

Mark Bun, Jörg Drechsler, Marco Gaboardi et al.

Sampling schemes are fundamental tools in statistics, survey design, and algorithm design. A fundamental result in differential privacy is that a differentially private mechanism run on a simple random sample of a population provides stronger privacy guarantees than the same algorithm run on the entire population. However, in practice, sampling designs are often more complex than the simple, data-independent sampling schemes that are addressed in prior work. In this work, we extend the study of privacy amplification results to more complex, data-dependent sampling schemes. We find that not only do these sampling schemes often fail to amplify privacy, they can actually result in privacy degradation. We analyze the privacy implications of the pervasive cluster sampling and stratified sampling paradigms, as well as provide some insight into the study of more general sampling designs.

Daniel Alabi, Audra McMillan, Jayshree Sarathy et al.

Economics and social science research often require analyzing datasets of sensitive personal information at fine granularity, with models fit to small subsets of the data. Unfortunately, such fine-grained analysis can easily reveal sensitive individual information. We study algorithms for simple linear regression that satisfy differential privacy, a constraint which guarantees that an algorithm's output reveals little about any individual input data record, even to an attacker with arbitrary side information about the dataset. We consider the design of differentially private algorithms for simple linear regression for small datasets, with tens to hundreds of datapoints, which is a particularly challenging regime for differential privacy. Focusing on a particular application to small-area analysis in economics research, we study the performance of a spectrum of algorithms we adapt to the setting. We identify key factors that affect their performance, showing through a range of experiments that algorithms based on robust estimators (in particular, the Theil-Sen estimator) perform well on the smallest datasets, but that other more standard algorithms do better as the dataset size increases.

Clément L. Canonne, Gautam Kamath, Audra McMillan et al.

In this work we present novel differentially private identity (goodness-of-fit) testers for natural and widely studied classes of multivariate product distributions: Gaussians in $\mathbb{R}^d$ with known covariance and product distributions over $\{\pm 1\}^{d}$. Our testers have improved sample complexity compared to those derived from previous techniques, and are the first testers whose sample complexity matches the order-optimal minimax sample complexity of $O(d^{1/2}/α^2)$ in many parameter regimes. We construct two types of testers, exhibiting tradeoffs between sample complexity and computational complexity. Finally, we provide a two-way reduction between testing a subclass of multivariate product distributions and testing univariate distributions, and thereby obtain upper and lower bounds for testing this subclass of product distributions.

Clément L. Canonne, Gautam Kamath, Audra McMillan et al.

Hypothesis testing plays a central role in statistical inference, and is used in many settings where privacy concerns are paramount. This work answers a basic question about privately testing simple hypotheses: given two distributions $P$ and $Q$, and a privacy level $\varepsilon$, how many i.i.d. samples are needed to distinguish $P$ from $Q$ subject to $\varepsilon$-differential privacy, and what sort of tests have optimal sample complexity? Specifically, we characterize this sample complexity up to constant factors in terms of the structure of $P$ and $Q$ and the privacy level $\varepsilon$, and show that this sample complexity is achieved by a certain randomized and clamped variant of the log-likelihood ratio test. Our result is an analogue of the classical Neyman-Pearson lemma in the setting of private hypothesis testing. We also give an application of our result to the private change-point detection. Our characterization applies more generally to hypothesis tests satisfying essentially any notion of algorithmic stability, which is known to imply strong generalization bounds in adaptive data analysis, and thus our results have applications even when privacy is not a primary concern.

Anna Gilbert, Audra McMillan

We consider the problem of property testing for differential privacy: with black-box access to a purportedly private algorithm, can we verify its privacy guarantees? In particular, we show that any privacy guarantee that can be efficiently verified is also efficiently breakable in the sense that there exist two databases between which we can efficiently distinguish. We give lower bounds on the query complexity of verifying pure differential privacy, approximate differential privacy, random pure differential privacy, and random approximate differential privacy. We also give algorithmic upper bounds. The lower bounds obtained in the work are infeasible for the scale of parameters that are typically considered reasonable in the differential privacy literature, even when we suppose that the verifier has access to an (untrusted) description of the algorithm. A central message of this work is that verifying privacy requires compromise by either the verifier or the algorithm owner. Either the verifier has to be satisfied with a weak privacy guarantee, or the algorithm owner has to compromise on side information or access to the algorithm.

Jacob Abernethy, Young Hun Jung, Chansoo Lee et al.

In this paper, we use differential privacy as a lens to examine online learning in both full and partial information settings. The differential privacy framework is, at heart, less about privacy and more about algorithmic stability, and thus has found application in domains well beyond those where information security is central. Here we develop an algorithmic property called one-step differential stability which facilitates a more refined regret analysis for online learning methods. We show that tools from the differential privacy literature can yield regret bounds for many interesting online learning problems including online convex optimization and online linear optimization. Our stability notion is particularly well-suited for deriving first-order regret bounds for follow-the-perturbed-leader algorithms, something that all previous analyses have struggled to achieve. We also generalize the standard max-divergence to obtain a broader class called Tsallis max-divergences. These define stronger notions of stability that are useful in deriving bounds in partial information settings such as multi-armed bandits and bandits with experts.

Anna C. Gilbert, Audra McMillan

In this work we explore the utility of locally differentially private thermal sensor data. We design a locally differentially private recovery algorithm for the 1-dimensional, discrete heat source location problem and analyse its performance in terms of the Earth Mover Distance error. Our work indicates that it is possible to produce locally private sensor measurements that both keep the exact locations of the heat sources private and permit recovery of the "general geographic vicinity" of the sources. We also discuss the relationship between the property of an inverse problem being ill-conditioned and the amount of noise needed to maintain privacy.

Audra McMillan, Adam Smith

Block graphons (also called stochastic block models) are an important and widely-studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $ρ$ on the values (connection probabilities) of the graphon, every estimator incurs error at least on the order of $\min(ρ, \sqrt{ρk^2/n^2})$ in the $δ_2$ metric with constant probability, in the worst case over graphons. In particular, our bound rules out any nontrivial estimation (that is, with $δ_2$ error substantially less than $ρ$) when $k\geq n\sqrtρ$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the minimax accuracy of graphon estimation in the $δ_2$ metric. A similar lower bound to ours was obtained independently by Klopp, Tsybakov and Verzelen (2016).