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 algorithms both achieving the optimal bound on excess loss and using $O(\min\{n, n^2/d\})$ gradient computations. In particular, the algorithms match the running time of the optimal non-private algorithms. The first approach relies on the use of variable batch sizes and is analyzed using the privacy amplification by iteration technique of Feldman et al. (2018). The second approach is based on a general reduction to the problem of localizing an approximately optimal solution with differential privacy. Such localization, in turn, can be achieved using existing (non-private) uniformly stable optimization algorithms. As in the earlier work, our algorithms require a mild smoothness assumption. We also give a linear-time algorithm achieving the optimal bound on the excess loss for the strongly convex case, as well as a faster algorithm for the non-smooth case.
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…
Microsoft Research（微软研究院）；University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm $\|\cdot\|$. Our algorithms are based on a regularized exponential mechanism which samples from the density $\propto \exp(-k(F+μr))$ where $F$ is the empirical loss and $r$ is a regularizer which is strongly convex with respect to $\|\cdot\|$, generalizing a recent work of [Gopi, Lee, Liu '22] to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex…
Stanford University（斯坦福大学）；Apple（苹果公司）；Massachusetts Institute of Technology（麻省理工学院）
We study adaptive methods for differentially private convex optimization, proposing and analyzing differentially private variants of a Stochastic Gradient Descent (SGD) algorithm with adaptive stepsizes, as well as the AdaGrad algorithm. We provide upper bounds on the regret of both algorithms and show that the bounds are (worst-case) optimal. As a consequence of our development, we show that our private versions of AdaGrad outperform adaptive SGD, which in turn outperforms traditional SGD in scenarios with non-isotropic gradients where (non-private) Adagrad provably outperforms SGD. The major…
How to Make the Gradients Small Privately: Improved Rates for Differentially Private Non-Convex Optimization
Wisconsin Institute for Discovery, University of Wisconsin-Madison（威斯康星大学麦迪逊分校威斯康星发现研究院）；Khoury College of Computer Sciences, Northeastern University（东北大学卡尔计算机科学学院）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
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} …
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…
In this paper, we revisit the problem of private stochastic convex optimization. We propose an algorithm based on noisy mirror descent, which achieves optimal rates both in terms of statistical complexity and number of queries to a first-order stochastic oracle in the regime when the privacy parameter is inversely proportional to the number of samples.
We use gradient sparsification to reduce the adverse effect of differential privacy noise on performance of private machine learning models. To this aim, we employ compressed sensing and additive Laplace noise to evaluate differentially-private gradients. Noisy privacy-preserving gradients are used to perform stochastic gradient descent for training machine learning models. Sparsification, achieved by setting the smallest gradient entries to zero, can reduce the convergence speed of the training algorithm. However, by sparsification and compressed sensing, the dimension of communicated gradien…
We revisit the well-studied problem of differentially private empirical risk minimization (ERM). We show that for unconstrained convex generalized linear models (GLMs), one can obtain an excess empirical risk of $\tilde O\left(\sqrt{\texttt{rank}}/εn\right)$, where ${\texttt{rank}}$ is the rank of the feature matrix in the GLM problem, $n$ is the number of data samples, and $ε$ is the privacy parameter. This bound is attained via differentially private gradient descent (DP-GD). Furthermore, via the first lower bound for unconstrained private ERM, we show that our upper bound is tight. In sharp…
Technical University of Denmark（丹麦技术大学）；Syracuse University（雪城大学）；Georgia Institute of Technology（佐治亚理工学院）
This paper develops a novel differentially private framework to solve convex optimization problems with sensitive optimization data and complex physical or operational constraints. Unlike standard noise-additive algorithms, that act primarily on the problem data, objective or solution, and disregard the problem constraints, this framework requires the optimization variables to be a function of the noise and exploits a chance-constrained problem reformulation with formal feasibility guarantees. The noise is calibrated to provide differential privacy for identity and linear queries on the optimi…
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\…
University of Arizona（亚利桑那大学）；Boston University（波士顿大学）
We consider the problem of minimizing a convex risk with stochastic subgradients guaranteeing $ε$-locally differentially private ($ε$-LDP). While it has been shown that stochastic optimization is possible with $ε$-LDP via the standard SGD (Song et al., 2013), its convergence rate largely depends on the learning rate, which must be tuned via repeated runs. Further, tuning is detrimental to privacy loss since it significantly increases the number of gradient requests. In this work, we propose BANCO (Betting Algorithm for Noisy COins), the first $ε$-LDP SGD algorithm that essentially matches the …
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）
Microsoft Research (MSR) Redmond（微软研究院（雷德蒙德））；University of Washington（华盛顿大学）；Shanghai Qi Zhi Institute（上海期智研究院）
We study the differentially private Empirical Risk Minimization (ERM) and Stochastic Convex Optimization (SCO) problems for non-smooth convex functions. We get a (nearly) optimal bound on the excess empirical risk and excess population loss with subquadratic gradient complexity. More precisely, our differentially private algorithm requires $O(\frac{N^{3/2}}{d^{1/8}}+ \frac{N^2}{d})$ gradient queries for optimal excess empirical risk, which is achieved with the help of subsampling and smoothing the function via convolution. This is the first subquadratic algorithm for the non-smooth case when $…
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…
We develop theoretically guaranteed stochastic methods for outlier-robust PCA. Outlier-robust PCA seeks an underlying low-dimensional linear subspace from a dataset that is corrupted with outliers. We are able to show that our methods, which involve stochastic geodesic gradient descent over the Grassmannian manifold, converge and recover an underlying subspace in various regimes through the development of a novel convergence analysis. The main application of this method is an effective differentially private algorithm for outlier-robust PCA that uses a Gaussian noise mechanism within the stoch…
The Hong Kong University of Science and Technology（香港科技大学）
Differentially private (DP) stochastic convex optimization (SCO) is ubiquitous in trustworthy machine learning algorithm design. This paper studies the DP-SCO problem with streaming data sampled from a distribution and arrives sequentially. We also consider the continual release model where parameters related to private information are updated and released upon each new data, often known as the online algorithms. Despite that numerous algorithms have been developed to achieve the optimal excess risks in different $\ell_p$ norm geometries, yet none of the existing ones can be adapted to the str…
We study differentially private (DP) algorithms for stochastic non-convex optimization. In this problem, the goal is to minimize the population loss over a $p$-dimensional space given $n$ i.i.d. samples drawn from a distribution. We improve upon the population gradient bound of ${\sqrt{p}}/{\sqrt{n}}$ from prior work and obtain a sharper rate of $\sqrt[4]{p}/\sqrt{n}$. We obtain this rate by providing the first analyses on a collection of private gradient-based methods, including adaptive algorithms DP RMSProp and DP Adam. Our proof technique leverages the connection between differential priva…
In this paper we tackle the challenge of making the stochastic coordinate descent algorithm differentially private. Compared to the classical gradient descent algorithm where updates operate on a single model vector and controlled noise addition to this vector suffices to hide critical information about individuals, stochastic coordinate descent crucially relies on keeping auxiliary information in memory during training. This auxiliary information provides an additional privacy leak and poses the major challenge addressed in this work. Driven by the insight that under independent noise additio…
Oklahoma State University（俄克拉荷马州立大学）；University of Texas at San Antonio（德克萨斯大学圣安东尼奥分校）
Alternating Direction Method of Multipliers (ADMM) is a popular algorithm for distributed learning, where a network of nodes collaboratively solve a regularized empirical risk minimization by iterative local computation associated with distributed data and iterate exchanges. When the training data is sensitive, the exchanged iterates will cause serious privacy concern. In this paper, we aim to propose a new differentially private distributed ADMM algorithm with improved accuracy for a wide range of convex learning problems. In our proposed algorithm, we adopt the approximation of the objective…
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…
Google Research（谷歌研究院）；University of Washington（华盛顿大学）
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…
In shuffle privacy, each user sends a collection of randomized messages to a trusted shuffler, the shuffler randomly permutes these messages, and the resulting shuffled collection of messages must satisfy differential privacy. Prior work in this model has largely focused on protocols that use a single round of communication to compute algorithmic primitives like means, histograms, and counts. We present interactive shuffle protocols for stochastic convex optimization. Our protocols rely on a new noninteractive protocol for summing vectors of bounded $\ell_2$ norm. By combining this sum subrout…
We present a provably optimal differentially private algorithm for the stochastic multi-arm bandit problem, as opposed to the private analogue of the UCB-algorithm [Mishra and Thakurta, 2015; Tossou and Dimitrakakis, 2016] which doesn't meet the recently discovered lower-bound of $Ω\left(\frac{K\log(T)}ε \right)$ [Shariff and Sheffet, 2018]. Our construction is based on a different algorithm, Successive Elimination [Even-Dar et al. 2002], that repeatedly pulls all remaining arms until an arm is found to be suboptimal and is then eliminated. In order to devise a private analogue of Successive E…
Convex optimization with feedback is a framework where a learner relies on iterative queries and feedback to arrive at the minimizer of a convex function. It has gained considerable popularity thanks to its scalability in large-scale optimization and machine learning. The repeated interactions, however, expose the learner to privacy risks from eavesdropping adversaries that observe the submitted queries. In this paper, we study how to optimally obfuscate the learner's queries in convex optimization with first-order feedback, so that their learned optimal value is provably difficult to estimate…
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…
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…
We introduce new algorithms and convergence guarantees for privacy-preserving non-convex Empirical Risk Minimization (ERM) on smooth $d$-dimensional objectives. We develop an improved sensitivity analysis of stochastic gradient descent on smooth objectives that exploits the recurrence of examples in different epochs. By combining this new approach with recent analysis of momentum with private aggregation techniques, we provide an $(ε,δ)$-differential private algorithm that finds a gradient of norm $\tilde O\left(\frac{d^{1/3}}{(εN)^{2/3}}\right)$ in $O\left(\frac{N^{7/3}ε^{4/3}}{d^{2/3}}\right…
Stanford University（斯坦福大学）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）；Google Research（谷歌研究院）
Large pretrained models can be privately fine-tuned to achieve performance approaching that of non-private models. A common theme in these results is the surprising observation that high-dimensional models can achieve favorable privacy-utility trade-offs. This seemingly contradicts known results on the model-size dependence of differentially private convex learning and raises the following research question: When does the performance of differentially private learning not degrade with increasing model size? We identify that the magnitudes of gradients projected onto subspaces is a key factor t…
University of Electronic Science and Technology of China（电子科技大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
In this paper, we revisit the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) and provide excess population risks for some special classes of functions that are faster than the previous results of general convex and strongly convex functions. In the first part of the paper, we study the case where the population risk function satisfies the Tysbakov Noise Condition (TNC) with some parameter $θ>1$. Specifically, we first show that under some mild assumptions on the loss functions, there is an algorithm whose output could achieve an upper bound of $\tilde{O}((\frac{1}{\s…
Private machine learning involves addition of noise while training, resulting in lower accuracy. Intuitively, greater stability can imply greater privacy and improve this privacy-utility tradeoff. We study this role of stability in private empirical risk minimization, where differential privacy is achieved by output perturbation, and establish a corresponding theoretical result showing that for strongly-convex loss functions, an algorithm with uniform stability of $β$ implies a bound of $O(\sqrtβ)$ on the scale of noise required for differential privacy. The result applies to both explicit reg…
University of Technology Sydney（悉尼科技大学）；National University of Singapore（新加坡国立大学）；City University of Hong Kong（香港城市大学）；Global Business College of Australia（澳大利亚全球商学院）
It is challenging for stochastic optimizations to handle large-scale sensitive data safely. Recently, Duchi et al. proposed private sampling strategy to solve privacy leakage in stochastic optimizations. However, this strategy leads to robustness degeneration, since this strategy is equal to the noise injection on each gradient, which adversely affects updates of the primal variable. To address this challenge, we introduce a robust stochastic optimization under the framework of local privacy, which is called Privacy-pREserving StochasTIc Gradual lEarning (PRESTIGE). PRESTIGE bridges private up…
We develop algorithms for private stochastic convex optimization that adapt to the hardness of the specific function we wish to optimize. While previous work provide worst-case bounds for arbitrary convex functions, it is often the case that the function at hand belongs to a smaller class that enjoys faster rates. Concretely, we show that for functions exhibiting $κ$-growth around the optimum, i.e., $f(x) \ge f(x^*) + λκ^{-1} \|x-x^*\|_2^κ$ for $κ> 1$, our algorithms improve upon the standard ${\sqrt{d}}/{n\varepsilon}$ privacy rate to the faster $({\sqrt{d}}/{n\varepsilon})^{\tfracκ{κ- 1}}$. …
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）
Differentially private (DP) stochastic convex optimization (SCO) is a fundamental problem, where the goal is to approximately minimize the population risk with respect to a convex loss function, given a dataset of $n$ i.i.d. samples from a distribution, while satisfying differential privacy with respect to the dataset. Most of the existing works in the literature of private convex optimization focus on the Euclidean (i.e., $\ell_2$) setting, where the loss is assumed to be Lipschitz (and possibly smooth) w.r.t. the $\ell_2$ norm over a constraint set with bounded $\ell_2$ diameter. Algorithms …
When we enforce differential privacy in machine learning, the utility-privacy trade-off is different w.r.t. each group. Gradient clipping and random noise addition disproportionately affect underrepresented and complex classes and subgroups, which results in inequality in utility loss. In this work, we analyze the inequality in utility loss by differential privacy and propose a modified differentially private stochastic gradient descent (DPSGD), called DPSGD-F, to remove the potential disparate impact of differential privacy on the protected group. DPSGD-F adjusts the contribution of samples i…
In this paper, we consider efficient differentially private empirical risk minimization from the viewpoint of optimization algorithms. For strongly convex and smooth objectives, we prove that gradient descent with output perturbation not only achieves nearly optimal utility, but also significantly improves the running time of previous state-of-the-art private optimization algorithms, for both $ε$-DP and $(ε, δ)$-DP. For non-convex but smooth objectives, we propose an RRPSGD (Random Round Private Stochastic Gradient Descent) algorithm, which provably converges to a stationary point with privacy…
We give the first polynomial time and sample $(ε, δ)$-differentially private (DP) algorithm to estimate the mean, covariance and higher moments in the presence of a constant fraction of adversarial outliers. Our algorithm succeeds for families of distributions that satisfy two well-studied properties in prior works on robust estimation: certifiable subgaussianity of directional moments and certifiable hypercontractivity of degree 2 polynomials. Our recovery guarantees hold in the "right affine-invariant norms": Mahalanobis distance for mean, multiplicative spectral and relative Frobenius dista…
In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems -- problems where there exists a solution that simultaneously minimizes all of the sample losses -- than on non-interpolating ones; we show that generally similar improvements are impossible in the private setting. However, when the functions exhibit quadratic growth around the optimum, we show (near) exponential improvements in the private sample complexity. In particular, we propose an adaptive algorithm that improves the sample complexity to achieve expected error $α$ fr…
PASS: Private Attributes Protection with Stochastic Data Substitution
Department of Computer Science, University of Illinois Urbana-Champaign, USA（伊利诺伊大学厄巴纳-香槟分校计算机科学系）；Global Technology Applied Research, JPMorgan Chase, USA（摩根大通全球技术应用研究）
University of Houston（休斯顿大学）；Beijing University of Posts and Telecommunications（北京邮电大学）；University of Texas at San Antonio（德克萨斯大学圣安东尼奥分校）；University of Science and Technology of China（中国科学技术大学）
Due to massive amounts of data distributed across multiple locations, distributed machine learning has attracted a lot of research interests. Alternating Direction Method of Multipliers (ADMM) is a powerful method of designing distributed machine learning algorithm, whereby each agent computes over local datasets and exchanges computation results with its neighbor agents in an iterative procedure. There exists significant privacy leakage during this iterative process if the local data is sensitive. In this paper, we propose a differentially private ADMM algorithm (P-ADMM) to provide dynamic ze…
Southern Methodist University（南卫理公会大学）；University of Twente（特文特大学）；Pontificia Universidad Católica de Chile（智利天主教大学）
In this work, we conduct the first systematic study of stochastic variational inequality (SVI) and stochastic saddle point (SSP) problems under the constraint of differential privacy (DP). We propose two algorithms: Noisy Stochastic Extragradient (NSEG) and Noisy Inexact Stochastic Proximal Point (NISPP). We show that a stochastic approximation variant of these algorithms attains risk bounds vanishing as a function of the dataset size, with respect to the strong gap function; and a sampling with replacement variant achieves optimal risk bounds with respect to a weak gap function. We also show …
For graphs generated from stochastic blockmodels, adjacency spectral embedding is asymptotically consistent. Further, adjacency spectral embedding composed with universally consistent classifiers is universally consistent to achieve the Bayes error. However when the graph contains private or sensitive information, treating the data as non-private can potentially leak privacy and incur disclosure risks. In this paper, we propose a differentially private adjacency spectral embedding algorithm for stochastic blockmodels. We demonstrate that our proposed methodology can estimate the latent positio…
