Gradient Perturbation is Underrated for Differentially Private Convex Optimization

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 non-private case . In this paper , we explore how the privacy noise affects the optimization property . We show that for differentially private convex optimization , the utility guarantee of both DP-GD and DP-SGD is determined by an expected curvature rather than the minimum curvature . The expected curvature represents the average curvature over the optimization path , which is usually much larger than the minimum curvature and hence can help us achieve a significantly improved utility guarantee . By using the expected curvature , our theory justifies the advantage of gradient perturbation over other perturbation methods and closes the gap between theory and practice . Extensive experiments on real world datasets corroborate our theoretical findings . 1 Introduction . Machine learning has become a powerful tool for many practical applications . The training process often needs access to some private dataset , e.g. , applications in financial and medical fields . Recent work has shown that the model learned from training data may leak unintended information of individual records ( Fredrikson et al. , 2015 ; Wu et al. , 2016 ; Shokri et al. , 2017 ; Hitaj et al. , 2017 ) . It is known that Differential privacy ( DP ) ( Dwork et al. , 2006a ; b ) is a golden standard for privacy preserving data analysis . It provides provable privacy guarantee by ensuring the influence of any individual record is negligible . It has been deployed into real world applications by large-scale corporations and U.S. Census Bureau ( Erlingsson et al. , 2014 ; McMillan , 2016 ; Abowd , 2016 ; Ding et al. , 2017 ) . We study the fundamental problem when differential privacy meets machine learning : the differentially private empirical risk minimization ( DP-ERM ) problem ( Chaudhuri & Monteleoni , 2009 ; Chaudhuri et al. , 2011 ; Kifer et al. , 2012 ; Bassily et al. , 2014 ; Talwar et al. , 2015 ; Wu et al. , 2017 ; Zhang et al. , 2017 ; Wang et al. , 2017 ; Smith et al. , 2017 ; Jayaraman et al. , 2018 ; Feldman et al. , 2018 ; Iyengar et al. , 2019 ; Wang & Gu , 2019 ) . DP-ERM minimizes the empirical risk while guaranteeing that the output of learning algorithm is differentially private with respect to the training data . Such privacy guarantee provides strong protection against potential adversaries ( Hitaj et al. , 2017 ; Rahman et al. , 2018 ) . In order to guarantee privacy , it is necessary to introduce randomness to the algorithm . There are usually three ways to introduce randomness according to the time of adding noise : output perturbation , objective perturbation and gradient perturbation . Output perturbation ( Wu et al. , 2017 ; Zhang et al. , 2017 ) first runs the learning algorithm the same as in the non-private case then adds noise to the output parameter . Objective perturbation ( Chaudhuri et al. , 2011 ; Kifer et al. , 2012 ; Iyengar et al. , 2019 ) perturbs the objective ( i.e. , the empirical loss ) then release the minimizer of the perturbed objective . Gradient perturbation ( Song et al. , 2013 ; Bassily et al. , 2014 ; Abadi et al. , 2016 ; Wang et al. , 2017 ; Lee & Kifer , 2018 ; Jayaraman et al. , 2018 ) perturbs each intermediate update . If each update is differentially private , the composition theorem of differential privacy ensures the whole learning procedure is differentially private . Gradient perturbation comes with several advantages over output/objective perturbations . Firstly , gradient perturbation does not require strong assumption on the objective because it only needs to bound the sensitivity of gradient update rather than the whole learning process . Secondly , gradient perturbation can release the noisy gradient at each iteration without damaging the privacy guarantee as differential privacy is immune to post processing ( Dwork et al. , 2014 ) . Thus , it is a more favorable choice for certain applications such as distributed optimization ( Rajkumar & Agarwal , 2012 ; Agarwal et al. , 2018 ; Jayaraman et al. , 2018 ) . At last , gradient perturbation often achieves better empirical utility than output/objective perturbations for DP-ERM . However , the existing theoretical utility guarantee for gradient perturbation is the same as or strictly inferior to that of other perturbation methods as shown in Table 1 . This motivates us to ask “ What is wrong with the theory for gradient perturbation ? Can we justify the empirical advantage of gradient perturbation theoretically ? ” We revisit the analysis for gradient perturbation approach . Previous work ( Bassily et al. , 2014 ; Wang et al. , 2017 ; Jayaraman et al. , 2018 ) derive the utility guarantee of gradient perturbation via two steps . They first determine the noise variance at each step that meets the privacy requirement and then derive the utility guarantee by using the convergence analysis the same as in non-private case . However , the noise to guarantee privacy naturally affects the optimization procedure , but previous approach does not exploit the interaction between privacy noise and optimization of gradient perturbation . In this paper , we utilize the fact the privacy noise affects the optimization procedure and establish new and much tighter utility guarantees for gradient perturbation approaches . Our contribution can be summarized as follows . • We introduce an expected curvature that can characterize the optimization property accurately when there is perturbation noise at each gradient update . • We establish the utility guarantees for DP-GD for both convex and strongly convex objectives based on the expected curvature rather than the usual minimum curvature . • We also establish the the utility guarantees for DP-SGD for both convex and strongly convex objectives based on the expected curvature . To the best of our knowledge , this is the first work to remove the dependency on minimum curvature for DP-ERM algorithms . In DP-ERM literature , there is a gap between the utility guarantee of non-strongly convex objectives and that of strongly convex objectives . However , by using the expected curvature , we show that some of the non-strongly convex objectives can achieve the same order of utility guarantee as the strongly convex objectives , matching the empirical observation . This is because the expected curvature could be relatively large even for non-strongly convex objectives . As we mentioned earlier , prior to our work , there is a mismatch between theoretical guarantee and empirical observation of gradient perturbation approach compared with other two perturbation approaches . Our result theoretically justifies the advantage of gradient perturbation and close the mismatch . 1.1 Paper Organization . The rest of this paper is organized as follows . Section 2 introduces notations and the DP-ERM task . In Sections 3 , we first introduce the expected curvature and establish the utility guarantee of both DP-GD and DP-SGD based on such expected curvature . Then we give some discussion on three perturbation approaches . We conduct extensive experiments in Section 4 . Finally , we conclude in Section 5 . 2 Preliminary . We introduce notations and definitions in this section . Given dataset D = { d1 , . . . , dn } , the objective function F ( x ; D ) is defined as F ( x ; D ) =∆ 1n ∑n i=1 f ( x ; di ) , where f ( x ; di ) : Rp → R is the loss of model x ∈ Rp for the record di . For simplicity , we use F ( x ) to denote F ( x ; D ) . We use ‖v‖ to denote the l2 norm of a vector v. We use X ∗f = arg minx∈Rp f ( x ) to denote the set of optimal solutions of f ( x ) . Throughout this paper , we assume X ∗f non-empty . Definition 1 ( Objective properties ) . For any x , y ∈ Rp , a function f : Rp → R • is L-Lipschitz if |f ( x ) − f ( y ) | ≤ L ‖x− y‖ . • is β-smooth if f ( y ) ≤ f ( x ) + 〈∇f ( x ) , y − x〉+ β2 ‖y − x‖ 2 . • is convex if 〈∇f ( x ) −∇f ( y ) , x− y〉 ≥ 0 . • is µ-strongly convex ( or µ-SC ) if 〈∇f ( x ) −∇f ( y ) , x− y〉 ≥ µ ‖x− y‖2 . The strong convexity coefficient µ is the lower bound of the minimum curvature of function f over the domain . We say that two datasets D , D′ are neighboring datasets ( denoted as D ∼ D′ ) if D can be obtained by arbitrarily modifying one record in D′ ( or vice versa ) . In this paper we consider ( , δ ) -differential privacy as follows . Definition 2 ( ( , δ ) -DP ( Dwork et al. , 2006a ; b ) ) . A randomized mechanismM : D → R guarantees ( , δ ) -differential privacy if for any two neighboring input datasets D , D ′ and for any subset of outputs S ⊆ R it holds that Pr [ M ( D ) ∈ S ] ≤ e Pr [ M ( D′ ) ∈ S ] + δ . We note that δ can be viewed as the probability that original -DP fails and a meaningful setting requires δ 1n . By its definition , differential privacy controls the maximum influence that any individual record can produce . Smaller , δ implies less information leak but usually leads to worse utility . One can adjust , δ to trade off between privacy and utility . DP-ERM requires the output xout ∈ Rp is differentially private with respect to the input dataset D. Let x∗ ∈ X ∗F be one of the optimal solutions of F ( x ) , the utility of DP-ERM algorithm is measured by expected excess empirical risk : E [ F ( xout ) − F ( x∗ ) ] , where the expectation is taken over the algorithm randomness . 3 Main Results . In this section , we first define the expected curvature ν and explain why it depends only on the average curvature . We then use such expected curvature to improve the analysis of both DP-SGD and DP-GD . 3.1 Expected Curvature . In non-private setting , the analysis of convex optimization relies on the strongly convex coefficient µ , which is the minimum curvature over the domain and can be extremely small for some common objectives . Previous work on DP-ERM uses the same analysis as in non-private case and therefore the resulting utility bounds rely on the minimum curvature . In our analysis , however , we avoid the dependency on the minimum curvature by exploiting how the privacy noise affects the optimization . With the perturbation noise , the expected curvature that the optimization path encounters is related to the average curvature instead of the minimum curvature . Definition 3 uses ν to capture such average curvature with Gaussian noise . We use x∗ = arg minx∈X∗ ‖x− x1‖ to denote the closest solution to the initial point . Definition 3 ( Expected curvature ) . A convex function F : Rp → R , has expected curvature ν with respect to noise N ( 0 , σ2Ip ) if for any x ∈ Rp and x̃ = x− z where z ∼ N ( 0 , σ2Ip ) , it holds that E [ 〈∇F ( x̃ ) , x̃− x∗〉 ] ≥ νE [ ‖x̃− x∗‖2 ] , ( 1 ) where the expectation is taken with respect to z . Claim 1 . If F is µ-strongly convex , we have ν ≥ µ . Proof . It can be verified that ν = µ always holds because of the strongly convex definition . In fact , ν represents the average curvature and is much larger than µ . We use x′ to denote the transpose of x . Let Hx = ∇2F ( x ) be the Hessian matrix evaluated at x . We use Taylor expansion to approximate the left hand side of Eq ( 1 ) as follows E [ 〈∇F ( x̃ ) , x̃− x∗〉 ] ≈ E [ 〈∇F ( x ) −Hxz , x− z − x∗〉 ] = 〈∇F ( x ) , x− x∗〉+ E [ z′Hxz ] = 〈∇F ( x ) , x− x∗〉+ σ2 tr ( Hx ) . For convex objective , the Hessian matrix is positive semi-definite and tr ( Hx ) is the sum of the eigenvalues of Hx . We can further express out the right hand side of Eq ( 1 ) as follows E [ ‖x̃− x∗‖2 ] = E [ ‖x− z − x∗‖2 ] = ν ( ‖x− x∗‖2 + pσ2 ) . Based on the above approximation , we can estimate the value of ν in Definition 3 : ν . tr ( Hx ) σ 2+µ‖x−x∗‖2 pσ2+‖x−x∗‖2 . For relatively large σ2 , this implies ν ≈ tr ( Hx ) p that is the average curvature at x . Large variance is a reasonable setting because meaningful differential privacy guarantee requires non-trivial amount of noise . The above analysis suggests that ν can be independent of and much larger than µ . This is indeed true for many convex objectives . Let us take the l2 regularized logistic regression as an example . The objective is strongly convex only due to the l2 regularizer . Thus , the minimum curvature ( strongly convex coefficient ) is the regularization coefficient λ. Sharmir et al . [ 1 ] shows the optimal choice of λ is Θ ( n−1/2 ) ( Section 4.3 in [ 1 ] ) . In practice , typical choice of λ is even smaller and could be on the order of n−1 . Figure 1 compares the minimum and average curvatures of regularized logistic regression during the training process . The average curvature is basically unaffected by the regularization term λ . In contrast , the minimum curvature reaches λ in first few steps . Therefore removing the dependence on minimum curvature is a significant improvement . We also plot the curvatures for another dataset KDDCup99 in the Appendix C. The resulting curvatures are similar to Figure 1 . Perturbation noise is necessary to attain ν > µ . We note that ν = µ when the training process does not involve perturbation noise ( corresponding to σ = 0 in Definition 3 ) . For example , objective/output perturbation can not utilize this expected curvature condition as no noise is injected in their training process . Therefore , among three existing perturbation methods , gradient perturbation is the only method can leverage such effect of noise . We note that µ = 0 does not necessarily lead to ν = 0 . A concrete example is given in Figure 2 ( from Negahban et al . ( 2012 ) ) . It provides an illustration of the loss function in the high-dimensional ( p > n ) setting , i.e. , the resticted strongly convex scenario : the loss is curved in certain directions but completely flat in others . The average curvature of such objective is always positive but the worst curvature is 0 . Though some recent work shows the utility guarantee of high dimensional DP-ERM task may not depend on the worst curvature ( Wang & Gu , 2019 ) , Figure 2 still provides a good illustration for the case of ν > µ = 0 . Moreover , as shown in Figure 1 , the average curvature of logistic regression on Adult dataset is above 0 during the training procedure even the regularization term is 0 . As we will show later , a positive ν over the optimization path is sufficient for our optimization analysis .

Summary: Differential privacy (DP) can be achieved by perturbing the objective function, the output or the gradient. In this paper, the authors consider gradient perturbation and claim that it is more advantageous than other methods. To prove this claim, they present a novel utility analysis by taking the noise into account. The previous papers (like Bassily et. al) present utility guarantees in DP context, but their analysis follow the same steps used for non-noisy setting. In non-noisy setting, the analysis is based on strong convexity parameter \mu which is the minimum curvature. However, in this study they present ”expected curvature” which is computed by considering the noise variance and based on averaging the curvatures along the number of iterations. The order of utility is given for both convex and strongly convex objectives and it has become smaller than the previous studies. Since other perturbation methods does not add noise at intermediate steps, expected curvature is the same with \mu and the utility advantage is not valid.