We study the problem of differentially private optimization with linear constraints when the right-hand-side of the constraints depends on private data. This type of problem appears in many applications, especially resource allocation. Previous research provided solutions that retained privacy but sometimes violated the constraints. In many settings, however, the constraints cannot be violated under any circumstances. To address this hard requirement, we present an algorithm that releases a nearly-optimal solution satisfying the constraints with probability 1. We also prove a lower bound demonstrating that the difference between the objective value of our algorithm's solution and the optimal solution is tight up to logarithmic factors among all differentially private algorithms. We conclude with experiments demonstrating that our algorithm can achieve nearly optimal performance while preserving privacy.
Booz Allen Hamilton（博思艾伦汉密尔顿公司）；University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）；Air Force Research Laboratory（美国空军研究实验室）
We present a framework to statistically audit the privacy guarantee conferred by a differentially private machine learner in practice. While previous works have taken steps toward evaluating privacy loss through poisoning attacks or membership inference, they have been tailored to specific models or have demonstrated low statistical power. Our work develops a general methodology to empirically evaluate the privacy of differentially private machine learning implementations, combining improved privacy search and verification methods with a toolkit of influence-based poisoning attacks. We demonst…
The Ohio State University（俄亥俄州立大学）；Google Research（谷歌研究院）；University of California Santa Cruz（加州大学圣克鲁兹分校）
We study differentially private (DP) algorithms for stochastic convex optimization (SCO). In this problem the goal is to approximately minimize the population loss given i.i.d. samples from a distribution over convex and Lipschitz loss functions. A long line of existing work on private convex optimization focuses on the empirical loss and derives asymptotically tight bounds on the excess empirical loss. However a significant gap exists in the known bounds for the population loss. We show that, up to logarithmic factors, the optimal excess population loss for DP algorithms is equal to the large…
Adaptive optimization methods have become the default solvers for many machine learning tasks. Unfortunately, the benefits of adaptivity may degrade when training with differential privacy, as the noise added to ensure privacy reduces the effectiveness of the adaptive preconditioner. To this end, we propose AdaDPS, a general framework that uses non-sensitive side information to precondition the gradients, allowing the effective use of adaptive methods in private settings. We formally show AdaDPS reduces the amount of noise needed to achieve similar privacy guarantees, thereby improving optimiz…
In this paper, we study the differentially private empirical risk minimization problem where the parameter is constrained to a Riemannian manifold. We introduce a framework of differentially private Riemannian optimization by adding noise to the Riemannian gradient on the tangent space. The noise follows a Gaussian distribution intrinsically defined with respect to the Riemannian metric. We adapt the Gaussian mechanism from the Euclidean space to the tangent space compatible to such generalized Gaussian distribution. We show that this strategy presents a simple analysis as compared to directly…
University of Pennsylvania（宾夕法尼亚大学）；University of Minnesota（明尼苏达大学）
One of the most effective algorithms for differentially private learning and optimization is objective perturbation. This technique augments a given optimization problem (e.g. deriving from an ERM problem) with a random linear term, and then exactly solves it. However, to date, analyses of this approach crucially rely on the convexity and smoothness of the objective function, limiting its generality. We give two algorithms that extend this approach substantially. The first algorithm requires nothing except boundedness of the loss function, and operates over a discrete domain. Its privacy and a…
The School of Data and Computer Science, Sun Yat-sen University（中山大学数据与计算机学院）；Microsoft Research Asia（微软亚洲研究院）；Guangdong Key Laboratory of Big Data Analysis and Processing（广东省大数据分析与处理重点实验室）
Gradient perturbation, widely used for differentially private optimization, injects noise at every iterative update to guarantee differential privacy. Previous work first determines the noise level that can satisfy the privacy requirement and then analyzes the utility of noisy gradient updates as in the non-private case. In contrast, we explore how privacy noise affects optimization property. We show that for differentially private convex optimization, the utility guarantee of differentially private (stochastic) gradient descent is determined by an \emph{expected curvature} rather than the min…
We present two classes of differentially private optimization algorithms derived from the well-known accelerated first-order methods. The first algorithm is inspired by Polyak's heavy ball method and employs a smoothing approach to decrease the accumulated noise on the gradient steps required for differential privacy. The second class of algorithms are based on Nesterov's accelerated gradient method and its recent multi-stage variant. We propose a noise dividing mechanism for the iterations of Nesterov's method in order to improve the error behavior of the algorithm. The convergence rate analy…
University of Waterloo（滑铁卢大学）；Meta；Cornell University（康奈尔大学）
We study stochastic convex optimization with heavy-tailed data under the constraint of differential privacy (DP). Most prior work on this problem is restricted to the case where the loss function is Lipschitz. Instead, as introduced by Wang, Xiao, Devadas, and Xu \cite{WangXDX20}, we study general convex loss functions with the assumption that the distribution of gradients has bounded $k$-th moments. We provide improved upper bounds on the excess population risk under concentrated DP for convex and strongly convex loss functions. Along the way, we derive new algorithms for private mean estimat…
We study differentially private (DP) algorithms for stochastic convex optimization: the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. (2019) has established the optimal bound on the excess population loss achievable given $n$ samples. Unfortunately, their algorithm achieving this bound is relatively inefficient: it requires $O(\min\{n^{3/2}, n^{5/2}/d\})$ gradient computations, where $d$ is the dimension of the optimization problem. We describe two new techniques for deriving DP convex optimization…
ETH Zurich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学习系统中心）；Amazon（亚马逊）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Washington（华盛顿大学）
We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy…
We develop a new reduction that converts any online convex optimization algorithm suffering $O(\sqrt{T})$ regret into an $ε$-differentially private stochastic convex optimization algorithm with the optimal convergence rate $\tilde O(1/\sqrt{T} + \sqrt{d}/εT)$ on smooth losses in linear time, forming a direct analogy to the classical non-private "online-to-batch" conversion. By applying our techniques to more advanced adaptive online algorithms, we produce adaptive differentially private counterparts whose convergence rates depend on apriori unknown variances or parameter norms.
Manipal Institute of Technology（马尼帕尔理工学院）；Carnegie Mellon University（卡内基梅隆大学）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Imperial College London（伦敦帝国学院）
Tuning the hyperparameters in the differentially private stochastic gradient descent (DPSGD) is a fundamental challenge. Unlike the typical SGD, private datasets cannot be used many times for hyperparameter search in DPSGD; e.g., via a grid search. Therefore, there is an essential need for algorithms that, within a given search space, can find near-optimal hyperparameters for the best achievable privacy-utility tradeoffs efficiently. We formulate this problem into a general optimization framework for establishing a desirable privacy-utility tradeoff, and systematically study three cost-effecti…
Bayesian optimization is a powerful tool for fine-tuning the hyper-parameters of a wide variety of machine learning models. The success of machine learning has led practitioners in diverse real-world settings to learn classifiers for practical problems. As machine learning becomes commonplace, Bayesian optimization becomes an attractive method for practitioners to automate the process of classifier hyper-parameter tuning. A key observation is that the data used for tuning models in these settings is often sensitive. Certain data such as genetic predisposition, personal email statistics, and ca…
Stochastic convex optimization over an $\ell_1$-bounded domain is ubiquitous in machine learning applications such as LASSO but remains poorly understood when learning with differential privacy. We show that, up to logarithmic factors the optimal excess population loss of any $(\varepsilon,δ)$-differentially private optimizer is $\sqrt{\log(d)/n} + \sqrt{d}/\varepsilon n.$ The upper bound is based on a new algorithm that combines the iterative localization approach of~\citet{FeldmanKoTa20} with a new analysis of private regularized mirror descent. It applies to $\ell_p$ bounded domains for $p\…
UCLA Electrical and Computer Engineering Department（加州大学洛杉矶分校电气与计算机工程系）；Amazon Web Services（亚马逊云科技）
Data privacy is an important concern in learning, when datasets contain sensitive information about individuals. This paper considers consensus-based distributed optimization under data privacy constraints. Consensus-based optimization consists of a set of computational nodes arranged in a graph, each having a local objective that depends on their local data, where in every step nodes take a linear combination of their neighbors' messages, as well as taking a new gradient step. Since the algorithm requires exchanging messages that depend on local data, private information gets leaked at every …
We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret ${O} \big( \varepsilon^{-1} \log^{1.5}{d} \big)$ where $d$ is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are ${O} \big( \varepsilon^{-1} \min\big\{d, T^{1/3}\log d\big\} \big)$. We also develop an adaptive algor…
State University of New York at Buffalo（纽约州立大学布法罗分校）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；MIT（麻省理工学院）
In this paper, we consider the problem of designing Differentially Private (DP) algorithms for Stochastic Convex Optimization (SCO) on heavy-tailed data. The irregularity of such data violates some key assumptions used in almost all existing DP-SCO and DP-ERM methods, resulting in failure to provide the DP guarantees. To better understand this type of challenges, we provide in this paper a comprehensive study of DP-SCO under various settings. First, we consider the case where the loss function is strongly convex and smooth. For this case, we propose a method based on the sample-and-aggregate f…
The performance of private gradient-based optimization algorithms is highly dependent on the choice of step size (or learning rate) which often requires non-trivial amount of tuning. In this paper, we introduce a stochastic variant of classic backtracking line search algorithm that satisfies Rényi differential privacy. Specifically, the proposed algorithm adaptively chooses the step size satsisfying the the Armijo condition (with high probability) using noisy gradients and function estimates. Furthermore, to improve the probability with which the chosen step size satisfies the condition, it ad…
We study the difficulties in learning that arise from robust and differentially private optimization. We first study convergence of gradient descent based adversarial training with differential privacy, taking a simple binary classification task on linearly separable data as an illustrative example. We compare the gap between adversarial and nominal risk in both private and non-private settings, showing that the data dimensionality dependent term introduced by private optimization compounds the difficulties of learning a robust model. After this, we discuss what parts of adversarial training a…
In this paper we develop the first algorithms for online submodular minimization that preserve differential privacy under full information feedback and bandit feedback. A sequence of $T$ submodular functions over a collection of $n$ elements arrive online, and at each timestep the algorithm must choose a subset of $[n]$ before seeing the function. The algorithm incurs a cost equal to the function evaluated on the chosen set, and seeks to choose a sequence of sets that achieves low expected regret. Our first result is in the full information setting, where the algorithm can observe the entire f…
The Ohio State University（俄亥俄州立大学）；University of Twente（特文特大学）；Pontificia Universidad Católica de Chile（智利天主教大学）
We study differentially private stochastic optimization in convex and non-convex settings. For the convex case, we focus on the family of non-smooth generalized linear losses (GLLs). Our algorithm for the $\ell_2$ setting achieves optimal excess population risk in near-linear time, while the best known differentially private algorithms for general convex losses run in super-linear time. Our algorithm for the $\ell_1$ setting has nearly-optimal excess population risk $\tilde{O}\big(\sqrt{\frac{\log{d}}{n\varepsilon}}\big)$, and circumvents the dimension dependent lower bound of \cite{Asi:2021} …
National University of Singapore（新加坡国立大学）；MIT（麻省理工学院）
Bayesian optimization (BO) has recently been extended to the federated learning (FL) setting by the federated Thompson sampling (FTS) algorithm, which has promising applications such as federated hyperparameter tuning. However, FTS is not equipped with a rigorous privacy guarantee which is an important consideration in FL. Recent works have incorporated differential privacy (DP) into the training of deep neural networks through a general framework for adding DP to iterative algorithms. Following this general DP framework, our work here integrates DP into FTS to preserve user-level privacy. We …
In this paper we consider the problem of minimizing composite objective functions consisting of a convex differentiable loss function plus a non-smooth regularization term, such as $L_1$ norm or nuclear norm, under Rényi differential privacy (RDP). To solve the problem, we propose two stochastic alternating direction method of multipliers (ADMM) algorithms: ssADMM based on gradient perturbation and mpADMM based on output perturbation. Both algorithms decompose the original problem into sub-problems that have closed-form solutions. The first algorithm, ssADMM, applies the recent privacy amplifi…
Consider the following optimization problem: Given $n \times n$ matrices $A$ and $Λ$, maximize $\langle A, UΛU^*\rangle$ where $U$ varies over the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a matrix whose spectrum is the same as $Λ$ and, by setting $Λ$ to be appropriate diagonal matrices, one can recover matrix approximation problems such as PCA and rank-$k$ approximation. We study the problem of designing differentially private algorithms for this optimization problem in settings where the matrix $A$ is constructed using users' private data. We give efficient and …
We study differentially private distributed optimization under communication constraints. A server using SGD for optimization aggregates the client-side local gradients for model updates using distributed mean estimation (DME). We develop a communication-efficient private DME, using the recently developed multi-message shuffled (MMS) privacy framework. We analyze our proposed DME scheme to show that it achieves the order-optimal privacy-communication-performance tradeoff resolving an open question in [1], whether the shuffled models can improve the tradeoff obtained in Secure Aggregation. This…
Differentially Private Stochastic Convex Optimization in (Non)-Euclidean Space Revisited
Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；The Chinese University of Hong Kong（香港中文大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
University of Minnesota（明尼苏达大学）；Harvard University（哈佛大学）；Boston University（波士顿大学）；IBM Research（IBM研究院）
We present three new algorithms for constructing differentially private synthetic data---a sanitized version of a sensitive dataset that approximately preserves the answers to a large collection of statistical queries. All three algorithms are \emph{oracle-efficient} in the sense that they are computationally efficient when given access to an optimization oracle. Such an oracle can be implemented using many existing (non-private) optimization tools such as sophisticated integer program solvers. While the accuracy of the synthetic data is contingent on the oracle's optimization performance, the…
We consider machine learning, particularly regression, using locally-differentially private datasets. The Wasserstein distance is used to define an ambiguity set centered at the empirical distribution of the dataset corrupted by local differential privacy noise. The ambiguity set is shown to contain the probability distribution of unperturbed, clean data. The radius of the ambiguity set is a function of the privacy budget, spread of the data, and the size of the problem. Hence, machine learning with locally-differentially private datasets can be rewritten as a distributionally-robust optimizat…
