I Introduction
Differential privacy, introduced by Dwork et al. (2006b), is a framework to quantify to what extent individual privacy in a statistical dataset is preserved while releasing useful aggregate information about the dataset. Differential privacy provides strong privacy guarantees by requiring the nearindistinguishability of whether an individual is in the dataset or not based on the released information. For more motivation and background of differential privacy, we refer the readers to the survey by Dwork (2008) and the book by Dwork and Roth (2014).
The classic differential privacy is called differential privacy, which imposes an upper bound on the multiplicative distance of the probability distributions of the randomized query outputs for any two neighboring datasets, and the standard approach for preserving
differential privacy is to add a Laplacian noise to the query output. Since its introduction, differential privacy has spawned a large body of research in differentially private datareleasing mechanism design, and the noiseadding mechanism has been applied in many machine learning algorithms to preserve differential privacy, e.g., logistic regression
(Chaudhuri and Monteleoni, 2008), empirical risk minimization (Chaudhuri et al., 2011), online learning (Jain et al., 2012), statistical risk minimization (Duchi et al., 2012)(Shokri and Shmatikov, 2015; Abadi et al., 2016; Phan et al., 2016; Agarwal et al., 2018), hypothesis testing (Sheffet, 2018), matrix completion (Jain et al., 2018)(Park et al., 2017), and principal component analysis
(Chaudhuri et al., 2012; Ge et al., 2018).To fully make use of the randomized query outputs, it is important to understand the fundamental tradeoff between privacy and utility (accuracy). Ghosh et al. (2009) studied a very general utilitymaximization framework for a single count query with sensitivity one under differential privacy. Gupte and Sundararajan (2010) derived the optimal noise probability distributions for a single count query with sensitivity one for minimax (riskaverse) users. Geng and Viswanath (2016b) derived the optimal differentially privacy noise adding mechanism for single realvalued query function with arbitrary query sensitivity, and show that the optimal noise distribution has a staircaseshaped probability density function. Geng et al. (2015) generalized the result in Geng and Viswanath (2016b) to twodimensional query output space for the cost function, and show the optimality of a twodimensional staircaseshaped probability density function. SoriaComas and DomingoFerrer (2013) also independently derived the staircaseshaped noise probability distribution under a different optimization framework.
A relaxed notion of differential privacy is differential privacy, introduced by Dwork et al. (2006a). The common interpretation of differential privacy is that it is differential privacy “except with probability ” (Mironov, 2017). The standard approach for preserving differential privacy is the Gaussian mechanism, which adds a Gaussian noise to the query output. Geng and Viswanath (2016a) studied the tradeoff between utility and privacy for a single integervalued query function in differential privacy. Geng and Viswanath (2016a) show that for and cost functions, the discrete uniform noise distribution is optimal for differential privacy when the query sensitivity is one, and is asymptotically optimal as for arbitrary query sensitivity. Balle and Wang (2018) improved the classic analysis of the Gaussian mechanism for differential in the high privacy regime (
), and develops an optimal Gaussian mechanism whose variance is calibrated directly using the Gaussian cumulative density function instead of a tail bound approximation.
Geng et al. (2018) derive the optimal noiseadding mechanism for single realvalued query function under differential privacy, and show that a uniform noise distribution with probability mass at the origin is optimal for a large class of cost functions.Ia Our Contributions
In this work, we derive a class of noise probability distributions to preserve differential privacy for single realvalued query function. The proposed noise distribution has a truncated exponential probability density function, which can be viewed as a truncated Laplacian distribution. We show the nearoptimality of the proposed truncated Laplacian
mechanism in various privacy regimes in the context of minimizing the noise amplitude and noise power. Our result closes the multiplicative gap between the lower bound and the upper bound (using uniform distribution) in the analysis of
Geng and Viswanath (2016a). Numeric experiments show the improvement of the truncated Laplacian mechanism over the optimal Gaussian mechanism by significantly reducing the noise amplitude and noise power.IB Organization
The paper is organized as follows. In Section II, we give some preliminaries on differential privacy, and formulate the tradeoff between privacy and utility under differential privacy for a single realvalued query function as a functional optimization problem. Section III presents the truncated Laplacian mechanism for preserving differential privacy. Section IV applies the truncated Laplacian mechanism to derive new upper bounds for the and cost functions, corresponding to noise amplitude and noise power. Section V derives new lower bounds on the minimum noise magnitude and noise power, and show that the lower bounds and upper bounds are close in various privacy regimes, which thus establishes the (asymptotic) optimality of the truncated Laplacian mechanism. Section VI conducts numeric experiments to compare the performance of the truncated Laplacian mechanism with the optimal Gaussian mechanisms in the context of minimizing noise amplitude and noise power.
Ii Problem Formulation
In this section, we first give some preliminaries on differential privacy, and then formulate the tradeoff between privacy and utility under differential privacy for a single realvalued query function as a functional optimization problem.
Iia Background on Differential Privacy
Consider a realvalued query function
where is the set of all possible datasets. The realvalued query function will be applied to a dataset, and the query output is a real number. Two datasets are called neighboring datasets if they differ in at most one element, i.e., one is a proper subset of the other and the larger dataset contains just one additional element Dwork (2008). A randomized queryanswering mechanism for the query function will randomly output a number with probability distribution depends on query output , where is the dataset.
Definition 1 (differential privacy (Dwork et al., 2006a)).
A randomized mechanism gives differential privacy if for all data sets and differing on at most one element, and all ,
(1) 
IiB Differential Privacy Constraint on the Noise Probability Distribution
The sensitivity of a realvalued query function is defined as:
Definition 2 (Query Sensitivity (Dwork, 2008)).
For a realvalued query function , the sensitivity of is defined as
for all differing in at most one element.
A standard approach for preserving differential privacy is queryoutput independent noiseadding mechanisms, where a random noise is added to the query output. Given a dataset , a queryoutput independent noiseadding mechanism will release the query output corrupted by an additive random noise with probability distribution :
The differential privacy constraint (1) on is that for any such that (corresponding to the query outputs for two neighboring datasets),
(2) 
where , is defined as the set .
Equivalently, the differential privacy constraint on the noise probability distribution is
(3) 
IiC Utility Model
Consider a cost function , which is a function of the additive noise in the queryoutput noiseadding mechanism. Given an additive noise , the cost is , and thus the expectation of the cost over is
(4) 
Our objective is to minimize the expectation of the cost over the noise probability distribution for preserving differential privacy.
IiD Optimization Problem
Iii Truncated Laplacian Mechanism
In this section, we present a class of noise probability distributions to preserve differential privacy constraint. The probability distribution can be viewed as a truncated Laplacian distribution.
Given and the query sensitivity , consider a probability distribution with a symmetric probability density function of defined as:
(7) 
where
We discuss some properties of the probability density function :

is symmetric and monotonically decreasing in .

is exponentially decaying in with rate proportional to .

, i.e., . Indeed,

is a valid probability density function, and . Indeed,
Definition 3 (Truncated Laplacian mechanism).
Given the query sensitivity , and the privacy parameters , the truncated Laplacian mechanism adds a noise with probability distribution defined in (7) to the query output.
Theorem 1.
The truncated Laplacian mechanism preserves differential privacy.
Proof.
Equivalently, we need to show that the truncated Laplacian distribution defined in (7) satisfies the differential privacy constraint (6).
We are interested in maximizing in (6) and show that the maximum over is upper bounded by . Since is symmetric and monotonically decreasing in , without loss of generality, we can assume .
To maximize , shall not contain points in , as
shall not contain points in , as
Therefore, is maximized for some set . Since is monotonically decreasing in , is maximized at and the maximum value is
We conclude that satisfies the differential privacy constraint (6). ∎
Iv Upper Bound on Noise Amplitude and Noise Power
In this section, we apply the truncated Laplacian mechanism to derive new upper bounds on the minimum noise amplitude and noise power, corresponding to and cost functions, under differential privacy.
Let denote the set of noise probability distributions satisfying the differential privacy constraint (6).
Theorem 2 (Upper Bound on Minimum Noise Amplitude).
For the cost function, i.e., ,
(8) 
Proof.
We can compute the cost for the truncated Laplacian distribution defined in (7) via
Since the noise with probability density distribution can preserve differential privacy,this gives an upper bound on . ∎
Note that in Theorem 2:

Given , when , the upper bound converges to , and the truncated Laplacian mechanism will be reduced to the standard Laplacian mechanism.

Given , when , the upper bound converges to . Indeed, when , , and thus
and the truncated Laplacian mechanism is reduced to a uniform distribution in the interval with probability density .

In the regime , the upper bound is
(9)
Similarly, we can derive an upper bound on the minimum noise power for the cost function.
Theorem 3 (Upper Bound on Minimum Noise Power).
For the cost function, i.e., ,
(10) 
Proof.
We can compute the cost for the truncated Laplacian distribution defined in (7) via
Since the noise with probability density distribution can preserve differential privacy, this gives an upper bound on . ∎
Note that in Theorem 3:

Given , when , the upper bound converges to , and the truncated Laplacian mechanism will be reduced to the standard Laplacian mechanism.

Given , when , the upper bound converges to . Indeed, let . When , , and thus
and the truncated Laplacian mechanism is reduced to a uniform distribution in the interval with probability density .

In the regime , the right hand side of (10) is
(11)
V Lower Bound on Noise Amplitude and Noise Power
In this section, we derive new lower bounds on the minimum noise amplitude and noise power, and show that they match the upper bounds in Section IV in various privacy regimes, and hence establish the (asymptotic) optimality of the truncated Laplacian mechanisms in these privacy regimes.
Geng and Viswanath (2016a) derived a lower bound for differential privacy for an integervalued query function. Extending this result to the continuous setting, we show a similar lower bound for realvalued query function under differential privacy. In particular, we show that the lower bound matches the upper bound (achieved by truncated Laplacian mechanism) in the high privacy regime when . Our results closes the multiplicative gap between the lower bound and the upper bound (using uniform distribution) in the analysis of Geng and Viswanath (2016a).
First, we give a lower bound for differential privacy for integervalued query function due to Geng and Viswanath (2016a).
Define
To avoid integer rounding issues, assume that there exists an integer such that
Theorem 4 (Theorem 8 in Geng and Viswanath (2016a)).
Consider a symmetric cost function . Given the privacy parameters and the discrete query sensitivity , if a discrete probability distribution satisfies that
(12) 
and the cost function satisfies that
(13) 
then we have
(14) 
Theorem 5 (Lower Bound on Minimum Noise Amplitude).
For the cost function, i.e., ,
(15) 
Proof.
Given , we can derive a lower bound on the cost by discretizing the probability distributions and applying the lower bound for integervalued query functions in Theorem 4.
We first discretize the probability distributions . Given a positive integer , define a discrete probability distribution via
Define the corresponding discrete cost function via
It is easy to see that
As the continuous probability distribution satisfies differential privacy constraint with query sensitivity , the discrete probability distribution satisfies the discrete differential privacy constraint (12) with query sensitivity , i.e., satisfies
We can verify that the condition (13) in Theorem 4 holds for and with query sensitivity when is sufficiently large. Indeed,
where the last step holds when .
The lower bound in (14) is
Therefore, for any , we have
and thus
∎
Next we compare the lower bound (15) in Theorem 5 with the upper bound (8) in Theorem 2 on minimum noise amplitude under differential privacy, and show that they are close in various privacy regions, which establishes the optimality of the truncated Laplacian mechanism in these privacy regimes. More precisely,
Corollary 6 (Comparison of Lower bound and Upper bound on Minimum Noise Amplitude).
(16)  
(17)  
(18) 
Proof.

Case : When , the upper bound (8) converges to , and the lower bound converges to , which matches the upper bound as . Therefore,

Case : When , the upper bound (8) converges to . For the lower bound, we have
and thus the lower bound converges to as . Therefore,

Case : The upper bound (8) converges to . For the lower bound, since , we have
As , , and thus
Note that as .
Therefore, as ,
Therefore, is lower bounded by in the regime . Since it is also upper bounded by by the truncated Laplacian mechanism (see Equation (9)), we conclude that
and thus the truncated Laplacian mechanism is asymptotically optimal in this regime in the context of minimizing noise amplitude.
∎
Theorem 7 (Lower Bound on Minimum Noise Power).
For the cost function, i.e., ,
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