Constraint Of Differential Privacy Research Articles

Minimax rates of convergence for sliced inverse regression with differential privacy

Sliced inverse regression (SIR) is a highly efficient paradigm used for the purpose of dimension reduction by replacing high-dimensional covariates with a limited number of linear combinations. This paper focuses on the implementation of the classical SIR approach integrated with a Gaussian differential privacy mechanism to estimate the central space while preserving privacy. We illustrate the tradeoff between statistical accuracy and privacy in sufficient dimension reduction problems under both the classical low- dimensional and modern high-dimensional settings. Additionally, we achieve the minimax rate of the proposed estimator with Gaussian differential privacy constraint and illustrate that this rate is also optimal for multiple index models with bounded dimension of the central space. Extensive numerical studies on synthetic data sets are conducted to assess the effectiveness of the proposed technique in finite sample scenarios, and a real data analysis is presented to showcase its practical application.

Differentially private distributed estimation and learning

We study distributed estimation and learning problems in a networked environment where agents exchange information to estimate unknown statistical properties of random variables from their privately observed samples. The agents can collectively estimate the unknown quantities by exchanging information about their private observations, but they also face privacy risks. Our novel algorithms extend the existing distributed estimation literature and enable the participating agents to estimate the expected value of a complete sufficient statistic from private signals acquired offline or online over time and to preserve the privacy of their signals and network neighborhoods. This is achieved through linear aggregation schemes with adjusted randomization schemes that add noise to the exchanged estimates subject to differential privacy constraints, both in an offline and online manner. We provide convergence rate analysis and tight finite-time convergence bounds. We show that the noise that minimizes the convergence time to the best estimates is the Laplace noise, with parameters corresponding to each agent’s sensitivity to their signal and network characteristics. Our algorithms are amenable to dynamic topologies and balancing privacy and accuracy trade-offs. Finally, to supplement and validate our theoretical results, we run experiments on real-world data from the US Power Grid Network and electric consumption data from German Households to estimate the average power consumption of power stations and households under all privacy regimes and show that our method outperforms existing first-order, privacy-aware, distributed optimization methods.

Utility-aware Privacy Perturbation for Training Data

Data perturbation under differential privacy constraint is an important approach of protecting data privacy. However, as the data dimensions increase, the privacy budget allocated to each dimension decreases and thus the amount of noise added increases, which eventually leads to lower data utility in training tasks. To protect the privacy of training data while enhancing data utility, we propose a Utility-aware training data Privacy Perturbation scheme based on attribute Partition and budget Allocation (UPPPA). UPPPA includes three procedures: the quantification of attribute privacy and attribute importance, attribute partition, and budget allocation. The quantification of attribute privacy and attribute importance based on information entropy and attribute correlation provide an arithmetic basis for attribute partition and budget allocation. During the attribute partition, all attributes of training data are classified into high and low classes to achieve privacy amplification and utility enhancement. During the budget allocation, a γ-privacy model is proposed to balance data privacy and data utility so as to provide privacy constraint and guide budget allocation. Three comprehensive sets of real-world data are applied to evaluate the performance of UPPPA. Experiments and privacy analysis show that our scheme can achieve the tradeoff between privacy and utility.

Statistical modelling under differential privacy constraints: a case study in fine-scale geographical analysis with Australian Bureau of Statistics TableBuilder data

ABSTRACT Consistent with the principles of differential privacy protection, the Australian Bureau of Statistics artificially perturbs all count data from the Australian Census prior to its release to researchers through the TableBuilder platform. This perturbation involves the addition of random noise to every non-zero cell count followed by the suppression of small values to zero. A consequent problem encountered by researchers working with geographical TableBuilder outputs is that the aggregate sums of cell counts over fine-scale units are often much lower than the direct query totals of their coarser enclosing units. Hierarchical Bayesian modelling is here proposed as a powerful methodology for achieving consistent multi-scale statistical reconstructions suitable for downstream spatial modelling. The utility of this framework is demonstrated for the toy problem of reconstructing Multiple Family Household counts by Mesh Block in the Perth Greater Capital City area.

Optimal algorithms for differentially private stochastic monotone variational inequalities and saddle-point problems

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 lower bounds of the same order on weak gap function. Hence, our algorithms are optimal. Key to our analysis is the investigation of algorithmic stability bounds, both of which are new even in the nonprivate case. The dependence of the running time of the sampling with replacement algorithms, with respect to the dataset size n, is n2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^2$$\\end{document} for NSEG and O~(n3/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{O}}(n^{3/2})$$\\end{document} for NISPP.

Multivariate density estimation from privatised data: universal consistency and minimax rates

We revisit the classical problem of nonparametric density estimation but impose local differential privacy constraints. Under such constraints, the original multivariate data cannot be directly observed, and all estimators are functions of the randomised output of a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and in this work we propose to add Laplace distributed noise to a discretisation of the location of an observed vector. Based on these randomised data, we propose a novel estimator of the density function, which can be viewed as a privatised version of the well-studied histogram density estimator. Our theoretical results include universal pointwise consistency and strong universal -consistency. In addition, a convergence rate for Lipschitz continuous functions is derived, which is complemented by a matching minimax lower bound. We illustrate the trade-off between data utility and privacy by means of a small simulation study.

A Unified Approach to Differentially Private Bayes Point Estimation

Parameter estimation in statistics and system identification relies on data that may contain sensitive information. To protect this sensitive information, the notion of differential privacy (DP) has been proposed, which enforces confidentiality by introducing randomization in the estimates. Standard algorithms for differentially private estimation are based on adding an appropriate amount of noise to the output of a traditional point estimation method. This leads to an accuracy-privacy trade off, as adding more noise reduces the accuracy while increasing privacy. In this paper, we propose a new Unified Bayes Private Point (UBaPP) approach to Bayes point estimation of the unknown parameters of a data generating mechanism under a DP constraint, that achieves a better accuracy-privacy trade off than traditional approaches. We verify the performance of our approach on a simple numerical example.

Quantitative Information Flow Techniques for Studying Optimality in Differential Privacy

Universal optimality for differential privacy seeks to characterise the "best" mechanisms, which are the ones that provide the most utility to some class of consumers whilst simultaneously satisfying a fixed ε-differential privacy constraint. The study of universal optimality was initiated in 2009 in two important papers: the first by Ghosh et al., who discovered a universally optimal mechanism for a large class of consumers within the differential privacy constraint context of "counting queries" on statistical datasets; the second by Brenner and Nissim who found that outside of this context, no such universally optimal mechanisms exist. Since those remarkable results, there have been few advances in this field. In this article we detail some recent work which sheds new light on these results using Quantitative Information Flow (QIF), a mathematical framework for quantifying information leaks under a Bayesian adversarial model. Using standard QIF reasoning we show how the earlier results can be both generalised and extended, opening up new avenues for exploring optimality in a wide variety of differential privacy contexts.

Cooperative Differentially Private LQG Control With Measurement Aggregation

When multiple agents solve cooperatively a joint optimal control problem, it is generally beneficial for them to coordinate their control signals. However, such strategies require that the agents share their local measurements, which may be privacy-sensitive. Motivated by this issue, this letter considers the Linear Quadratic Gaussian (LQG) control problem subject to differential privacy constraints. Differential privacy ensures that the published signals of an algorithm are not too sensitive to the data of any single participating agent. We propose a two-stage architecture for differentially private LQG control and show how to optimize it by leveraging a solution that we previously developed for the Kalman filtering problem. The first stage of this architecture is most easily implemented by a coordinator aggregating and perturbing the agents’ measurements appropriately, but it can also be implemented without a trusted aggregator by using a secure sum protocol. Numerical simulations illustrate the performance improvement of this architecture over simpler alternatives such as directly perturbing the agents’ measurements.

Locally Differentially Private Minimum Finding

We investigate a problem of finding the minimum, in which each user has a real value and we want to estimate the minimum of these values under the local differential privacy constraint. We reveal that this problem is fundamentally difficult, and we cannot construct a mechanism that is consistent in the worst case. Instead of considering the worst case, we aim to construct a private mechanism whose error rate is adaptive to the easiness of estimation of the minimum. As a measure of easiness, we introduce a parameter $\alpha$ that characterizes the fatness of the minimum-side tail of the user data distribution. As a result, we reveal that the mechanism can achieve $O((\ln^6N/\epsilon^2N)^{1/2\alpha})$ error without knowledge of $\alpha$ and the error rate is near-optimal in the sense that any mechanism incurs $\Omega((1/\epsilon^2N)^{1/2\alpha})$ error. Furthermore, we demonstrate that our mechanism outperforms a naive mechanism by empirical evaluations on synthetic datasets. Also, we conducted experiments on the MovieLens dataset and a purchase history dataset and demonstrate that our algorithm achieves $\tilde{O}((1/N)^{1/2\alpha})$ error adaptively to $\alpha$.

The Role of Adaptive Optimizers for Honest Private Hyperparameter Selection

Hyperparameter optimization is a ubiquitous challenge in machine learning, and the performance of a trained model depends crucially upon their effective selection. While a rich set of tools exist for this purpose, there are currently no practical hyperparameter selection methods under the constraint of differential privacy (DP). We study honest hyperparameter selection for differentially private machine learning, in which the process of hyperparameter tuning is accounted for in the overall privacy budget. To this end, we i) show that standard composition tools outperform more advanced techniques in many settings, ii) empirically and theoretically demonstrate an intrinsic connection between the learning rate and clipping norm hyperparameters, iii) show that adaptive optimizers like DPAdam enjoy a significant advantage in the process of honest hyperparameter tuning, and iv) draw upon novel limiting behaviour of Adam in the DP setting to design a new and more efficient optimizer.

On the Impossibility of Non-trivial Accuracy in Presence of Fairness Constraints

One of the main concerns about fairness in machine learning (ML) is that, in order to achieve it, one may have to trade off some accuracy. To overcome this issue, Hardt et al. proposed the notion of equality of opportunity (EO), which is compatible with maximal accuracy when the target label is deterministic with respect to the input features. In the probabilistic case, however, the issue is more complicated: It has been shown that under differential privacy constraints, there are data sources for which EO can only be achieved at the total detriment of accuracy, in the sense that a classifier that satisfies EO cannot be more accurate than a trivial (random guessing) classifier. In our paper we strengthen this result by removing the privacy constraint. Namely, we show that for certain data sources, the most accurate classifier that satisfies EO is a trivial classifier. Furthermore, we study the trade-off between accuracy and EO loss (opportunity difference), and provide a sufficient condition on the data source under which EO and non-trivial accuracy are compatible.

Private Multi-Group Aggregation

We study the differentially private multi-group aggregation (PMGA) problem. This setting involves a single server and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> users. Each user belongs to one of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> distinct groups and holds a discrete value. The goal is to design schemes that allow the server to find the aggregate (sum) of the values in each group (with high accuracy) under communication and local differential privacy constraints. The privacy constraint guarantees that the user’s group remains private. This is motivated by applications where a user’s group can reveal sensitive information, such as his religious and political beliefs, health condition, or race. We propose a novel scheme, dubbed Query and Aggregate (Q&A) for PMGA. The novelty of Q&A is that it is an interactive aggregation scheme. In Q&A, each user is assigned a random query matrix, to which he sends the server an answer based on his group and value. We characterize the Q&A scheme’s performance in terms of accuracy (MSE), privacy, and communication. We compare Q&A to the Randomized Group (RG) scheme, which is non-interactive and adapts existing randomized response schemes to the PMGA setting. We observe that typically Q&A outperforms RG, in terms of privacy vs. utility, in the high privacy regime.

Minimax optimal goodness-of-fit testing for densities and multinomials under a local differential privacy constraint

Finding anonymization mechanisms to protect personal data is at the heart of recent machine learning research. Here, we consider the consequences of local differential privacy constraints on goodness-of-fit testing, that is, the statistical problem assessing whether sample points are generated from a fixed density f0, or not. The observations are kept hidden and replaced by a stochastic transformation satisfying the local differential privacy constraint. In this setting, we propose a testing procedure which is based on an estimation of the quadratic distance between the density f of the unobserved samples and f0. We establish an upper bound on the separation distance associated with this test, and a matching lower bound on the minimax separation rates of testing under non-interactive privacy in the case that f0 is uniform, in discrete and continuous settings. To the best of our knowledge, we provide the first minimax optimal test and associated private transformation under a local differential privacy constraint over Besov balls in the continuous setting, quantifying the price to pay for data privacy. We also present a test that is adaptive to the smoothness parameter of the unknown density and remains minimax optimal up to a logarithmic factor. Finally, we note that our results can be translated to the discrete case, where the treatment of probability vectors is shown to be equivalent to that of piecewise constant densities in our setting. That is why we work with a unified setting for both the continuous and the discrete cases.

Private Stochastic Dual Averaging for Decentralized Empirical Risk Minimization

In this work, we study the decentralized empirical risk minimization problem under the constraint of differential privacy (DP). Based on the algorithmic framework of dual averaging, we develop a novel decentralized stochastic optimization algorithm to solve the problem. The proposed algorithm features the following: i) it perturbs the stochastic subgradient evaluated over individual data samples, with which the information about the dataset can be released in a differentially private manner; ii) it employs hyperparameters that are more aggressive than conventional decentralized dual averaging algorithms to speed up convergence. The upper bound for the utility loss of the proposed algorithm is proven to be smaller than that of existing methods to achieve the same level of DP. As a by-product, when removing the perturbation, the non-private version of the proposed algorithm attains the optimal O(1/t) convergence rate for non-smooth stochastic optimization. Finally, experimental results are presented to demonstrate the effectiveness of the algorithm.

The cost of privacy: Optimal rates of convergence for parameter estimation with differential privacy

Privacy-preserving data analysis is a rising challenge in contemporary statistics, as the privacy guarantees of statistical methods are often achieved at the expense of accuracy. In this paper, we investigate the tradeoff between statistical accuracy and privacy in mean estimation and linear regression, under both the classical low-dimensional and modern high-dimensional settings. A primary focus is to establish minimax optimality for statistical estimation with the (ε,δ)-differential privacy constraint. By refining the “tracing adversary” technique for lower bounds in the theoretical computer science literature, we improve existing minimax lower bound for low-dimensional mean estimation and establish new lower bounds for high-dimensional mean estimation and linear regression problems. We also design differentially private algorithms that attain the minimax lower bounds up to logarithmic factors. In particular, for high-dimensional linear regression, a novel private iterative hard thresholding algorithm is proposed. The numerical performance of differentially private algorithms is demonstrated by simulation studies and applications to real data sets.

Strongly universally consistent nonparametric regression and classification with privatised data

In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data (X1,Y1),...,(Xn,Yn), taking values in Rd×R, cannot be directly observed, and all estimators are functions of the randomised output from a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and here we add Laplace distributed noise to a discretisation of the location of a feature vector Xi and to the value of its response variable Yi. Based on this randomised data, we design a novel estimator of the regression function, which can be viewed as a privatised version of the well-studied partitioning regression estimator. The main result is that the estimator is strongly universally consistent, and we further establish an upper bound on the rate of convergence. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data.

Fisher Information Under Local Differential Privacy

We develop data processing inequalities that describe how Fisher information from statistical samples can scale with the privacy parameter ε under local differential privacy constraints. These bounds are valid under general conditions on the distribution of the score of the statistical model, and they elucidate under which conditions the dependence on ε is linear, quadratic, or exponential. We show how these inequalities imply order-optimal lower bounds for private estimation for both the Gaussian location model and discrete distribution estimation for all levels of privacy ε >0. We further apply these inequalities to sparse Bernoulli models and demonstrate privacy mechanisms and estimators with order-matching squared ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> error.

Geometrizing rates of convergence under local differential privacy constraints

We study the problem of estimating a functional $\theta ({\mathbb{P}})$ of an unknown probability distribution ${\mathbb{P}}\in {\mathcal{P}}$ in which the original iid sample $X_{1},\dots ,X_{n}$ is kept private even from the statistician via an $\alpha $-local differential privacy constraint. Let $\omega _{\mathrm{TV}}$ denote the modulus of continuity of the functional $\theta $ over ${\mathcal{P}}$ with respect to total variation distance. For a large class of loss functions $l$ and a fixed privacy level $\alpha $, we prove that the privatized minimax risk is equivalent to $l(\omega _{\mathrm{TV}}(n^{-1/2}))$ to within constants, under regularity conditions that are satisfied, in particular, if $\theta $ is linear and ${\mathcal{P}}$ is convex. Our results complement the theory developed by Donoho and Liu (1991) with the nowadays highly relevant case of privatized data. Somewhat surprisingly, the difficulty of the estimation problem in the private case is characterized by $\omega _{\mathrm{TV}}$, whereas, it is characterized by the Hellinger modulus of continuity if the original data $X_{1},\dots ,X_{n}$ are available. We also find that for locally private estimation of linear functionals over a convex model a simple sample mean estimator, based on independently and binary privatized observations, always achieves the minimax rate. We further provide a general recipe for choosing the functional parameter in the optimal binary privatization mechanisms and illustrate the general theory in numerous examples. Our theory allows us to quantify the price to be paid for local differential privacy in a large class of estimation problems. This price appears to be highly problem specific.

Protecting Your Shopping Preference With Differential Privacy

Online banks may disclose consumers’ shopping preferences due to various attacks. With differential privacy, each consumer can disturb his consumption amount locally before sending it to online banks. However, directly applying differential privacy in online banks will incur problems in reality because existing differential privacy schemes do not consider handling the noise boundary problem. In this paper, we propose an Optimized Differential prIvate Online tRansaction scheme (O-DIOR) for online banks to set boundaries of consumption amounts with added noises. We then revise O-DIOR to design a RO-DIOR scheme to select different boundaries while satisfying the differential privacy definition. Moreover, we provide in-depth theoretical analysis to prove that our schemes are capable to satisfy the differential privacy constraint. Finally, to evaluate the effectiveness, we have implemented our schemes in mobile payment experiments. Experimental results illustrate that the relevance between the consumption amount and online bank amount is reduced significantly, and the privacy losses are less than 0.5 in terms of mutual information.