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.
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…
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…
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 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…
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…
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 $…
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…
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…
Northwest University（西北大学）；University of Birmingham（伯明翰大学）；State University of New York at Albany（纽约州立大学奥尔巴尼分校）
In this paper, we are concerned with differentially private {stochastic gradient descent (SGD)} algorithms in the setting of stochastic convex optimization (SCO). Most of the existing work requires the loss to be Lipschitz continuous and strongly smooth, and the model parameter to be uniformly bounded. However, these assumptions are restrictive as many popular losses violate these conditions including the hinge loss for SVM, the absolute loss in robust regression, and even the least square loss in an unbounded domain. We significantly relax these restrictive assumptions and establish privacy a…
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…
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）
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…
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 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\…
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）
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 …
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…
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…
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…
University of Science and Technology of China（中国科学技术大学）；Microsoft Research Asia（微软亚洲研究院）
By ensuring differential privacy in the learning algorithms, one can rigorously mitigate the risk of large models memorizing sensitive training data. In this paper, we study two algorithms for this purpose, i.e., DP-SGD and DP-NSGD, which first clip or normalize \textit{per-sample} gradients to bound the sensitivity and then add noise to obfuscate the exact information. We analyze the convergence behavior of these two algorithms in the non-convex optimization setting with two common assumptions and achieve a rate $\mathcal{O}\left(\sqrt[4]{\frac{d\log(1/δ)}{N^2ε^2}}\right)$ of the gradient nor…
We present new differentially private algorithms for learning a large-margin halfspace. In contrast to previous algorithms, which are based on either differentially private simulations of the statistical query model or on private convex optimization, the sample complexity of our algorithms depends only on the margin of the data, and not on the dimension. We complement our results with a lower bound, showing that the dependence of our upper bounds on the margin is optimal.
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…
Nanjing University（南京大学）；Microsoft Research Asia（微软亚洲研究院）
We investigate online convex optimization in changing environments, and choose the adaptive regret as the performance measure. The goal is to achieve a small regret over every interval so that the comparator is allowed to change over time. Different from previous works that only utilize the convexity condition, this paper further exploits smoothness to improve the adaptive regret. To this end, we develop novel adaptive algorithms for convex and smooth functions, and establish problem-dependent regret bounds over any interval. Our regret bounds are comparable to existing results in the worst ca…
In statistical learning theory, convex surrogates of the 0-1 loss are highly preferred because of the computational and theoretical virtues that convexity brings in. This is of more importance if we consider smooth surrogates as witnessed by the fact that the smoothness is further beneficial both computationally- by attaining an {\it optimal} convergence rate for optimization, and in a statistical sense- by providing an improved {\it optimistic} rate for generalization bound. In this paper we investigate the smoothness property from the viewpoint of statistical consistency and show how it affe…
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…
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 $({\sqrt{d}}/{n\varepsilon})^{\tfracκ{κ- 1}}$. …
Microsoft Research（微软研究院）；University of Washington（华盛顿大学）
In this paper, we study private optimization problems for non-smooth convex functions $F(x)=\mathbb{E}_i f_i(x)$ on $\mathbb{R}^d$. We show that modifying the exponential mechanism by adding an $\ell_2^2$ regularizer to $F(x)$ and sampling from $π(x)\propto \exp(-k(F(x)+μ\|x\|_2^2/2))$ recovers both the known optimal empirical risk and population loss under $(ε,δ)$-DP. Furthermore, we show how to implement this mechanism using $\widetilde{O}(n \min(d, n))$ queries to $f_i(x)$ for the DP-SCO where $n$ is the number of samples/users and $d$ is the ambient dimension. We also give a (nearly) match…
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…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We present a distributed solution to optimizing a convex function composed of several non-convex functions. Each non-convex function is privately stored with an agent while the agents communicate with neighbors to form a network. We show that coupled consensus and projected gradient descent algorithm proposed in [1] can optimize convex sum of non-convex functions under an additional assumption on gradient Lipschitzness. We further discuss the applications of this analysis in improving privacy in distributed optimization.
We present a private learner for halfspaces over an arbitrary finite domain $X\subset \mathbb{R}^d$ with sample complexity $mathrm{poly}(d,2^{\log^*|X|})$. The building block for this learner is a differentially private algorithm for locating an approximate center point of $m>\mathrm{poly}(d,2^{\log^*|X|})$ points -- a high dimensional generalization of the median function. Our construction establishes a relationship between these two problems that is reminiscent of the relation between the median and learning one-dimensional thresholds [Bun et al.\ FOCS '15]. This relationship suggests that t…
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 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} …
Differential privacy is concerned about the prediction quality while measuring the privacy impact on individuals whose information is contained in the data. We consider differentially private risk minimization problems with regularizers that induce structured sparsity. These regularizers are known to be convex but they are often non-differentiable. We analyze the standard differentially private algorithms, such as output perturbation, Frank-Wolfe and objective perturbation. Output perturbation is a differentially private algorithm that is known to perform well for minimizing risks that are str…
State University of New York at Buffalo（纽约州立大学布法罗分校）
In this paper we study the differentially private Empirical Risk Minimization (ERM) problem in different settings. For smooth (strongly) convex loss function with or without (non)-smooth regularization, we give algorithms that achieve either optimal or near optimal utility bounds with less gradient complexity compared with previous work. For ERM with smooth convex loss function in high-dimensional ($p\gg n$) setting, we give an algorithm which achieves the upper bound with less gradient complexity than previous ones. At last, we generalize the expected excess empirical risk from convex loss fu…
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…
