Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds

Raef Bassily, Adam Smith, Abhradeep Thakurta

Introduction

Convex optimization is one of the most basic and powerful computational tools in statistics and machine learning. It is most commonly used for empirical risk minimization (ERM): the data set $\mathcal{D}=\{d_{1},...,d_{n}\}$ defines a convex loss function $\mathcal{L}(\cdot)$ which is minimized over a convex set $\mathcal{C}$ . When run on sensitive data, however, the results of convex ERM can leak sensitive information. For example, medians and support vector machine parameters can, in many cases, leak entire records in the clear (see “Motivation”, below).

In this paper, we provide new algorithms and matching lower bounds for differentially private convex ERM assuming only that each data point’s contribution to the loss function is Lipschitz and that the domain of optimization is bounded. This builds on a line of work started by Chaudhuri et al. .

Given a data set $\mathcal{D}=\{d_{1},...,d_{n}\}$ drawn from a universe $\mathcal{X}$ , and a closed, convex set $\mathcal{C}$ , our goal is to

We measure the success of our algorithms by the worst-case (over inputs) expected excess empirical risk, namely

where $\hat{\theta}$ is the output of the algorithm, $\theta^{*}=\arg\min_{\theta\in\mathcal{C}}\mathcal{L}(\theta;\mathcal{D})$ is the true minimizer, and the expectation is only over the coins of the algorithm. Expected risk guarantees can be converted to high-probability guarantees using standard amplification techniques (see Appendix D for details).

Another important measure of performance is an algorithm’s (excess) generalization error, where loss is measured with respect to the average over an unknown distribution from which the data are assumed to be drawn i.i.d.. Our upper bounds on empirical risk imply upper bounds on generalization error (via uniform convergence and similar ideas); the resulting bounds are only known to be tight in certain ranges of parameters, however. Detailed statements may be found in Appendix F.

Motivation.

Convex ERM is used for fitting models from simple least-squares regression to support vector machines, and their use may have significant implications to privacy. As a simple example, note that the Euclidean 1-median of a data set will typically be an actual data point, since the gradient of the loss function has discontinuities at each of the $d_{i}$ . (Thinking about the one-dimensional median, where there is always a data point that minimizes the loss, is helpful.) Thus, releasing the median may well reveal one of the data points in the clear. A more subtle example is the support vector machine (SVM). The solution to an SVM program is often presented in its dual form, whose coefficients typically consist of a set of $p+1$ exact data points. Kasiviswanathan et al. show how the results of many convex ERM problems can be combined to carry out reconstruction attacks in the spirit of Dinur and Nissim .

Differential privacy

is a rigorous notion of privacy that emerged from a line of work in theoretical computer science and cryptography . We say two data sets $\mathcal{D}$ and $\mathcal{D}^{\prime}$ of size $n$ are neighbors if they differ in one entry (that is, $\left|\mathcal{D}\triangle\mathcal{D}^{\prime}\right|=2$ ). A randomized algorithm $\mathcal{A}$ is $(\epsilon,\delta)$ -differentially private (Dwork et al. ) if, for all neighboring data sets $\mathcal{D}$ and $\mathcal{D}^{\prime}$ and for all events $S$ in the output space of $\mathcal{A}$ , we have

Algorithms that satisfy differential privacy for $\epsilon<1$ and $\delta\ll 1/n$ provide meaningful privacy guarantees, even in the presence of side information. In particular, they avoid the problems mentioned in “Motivation” above. See Dwork , Kasiviswanathan and Smith , Kifer and Machanavajjhala for discussion of the “semantics” of differential privacy.

Setting Parameters.

We will aim to quantify the role of several basic parameters on the excess risk of differentially private algorithms: the size of the data set $n$ , the dimension $p$ of the parameter space $\mathcal{C}$ , the Lipschitz constant $L$ of the loss functions, the diameter $\|\mathcal{C}\|_{2}$ of the constraint set and, when applicable, the strong convexity $\Delta$ .

We may take $L$ and $\|\mathcal{C}\|_{2}$ to be 1 without loss of generality: We can set $\|\mathcal{C}\|_{2}=1$ by rescaling $\theta$ (replacing by $\theta$ with $\theta\cdot\|\mathcal{C}\|_{2}$ ); we can then set $L=1$ by rescaling the loss function $\mathcal{L}$ (replacing $\mathcal{L}$ by $\mathcal{L}/L$ ). These two transformations change the excess risk by $L\|C\|_{2}$ . The parameter $\Delta$ cannot similarly be rescaled while keeping $L$ and $\|\mathcal{C}\|_{2}$ the same. However, we always have $\Delta\leq 2L/\|\mathcal{C}\|_{2}$ .

In the sequel, we thus focus on the setting where $L=\|\mathcal{C}\|_{2}=1$ and $\Delta\in$ . To convert excess risk bounds for $L=\|\mathcal{C}\|_{2}=1$ to the general setting, one can multiply the risk bounds by $L\|\mathcal{C}\|_{2}$ , and replace $\Delta$ by $\frac{\Delta\|\mathcal{C}\|_{2}}{L}$ .

1 Contributions

We give algorithms that significantly improve on the state of the art for optimizing non-smooth loss functions — for both the general case and strongly convex functions, we improve the excess risk bounds by a factor of $\sqrt{n}$ , asymptotically. The algorithms we give for $(\epsilon,0)$ - and $(\epsilon,\delta)$ -differential privacy work on very different principles. We group the algorithms below by technique: gradient descent, exponential sampling, and localization.

The gradient descent approach does not, to our knowledge, allow one to get optimal excess risk bounds for $(\epsilon,0)$ -differential privacy. The main obstacle is that “strong composition” of $(\epsilon,\delta)$ -privacy Dwork et al. appears necessary to allow a first-order method to run for sufficiently many steps.

Exponential Sampling-based Algorithms.

We give a polynomial time algorithm that achieves the optimal excess risk, namely $O(p/\epsilon)$ . Note that the achieved excess risk does not have any logarithmic factors which is shown to be the case using a “peeling-” type argument that is specific to convex functions. The idea of our algorithm is to sample efficiently from the continuous distribution on all points in $\mathcal{C}$ with density $\mathcal{P}(\theta)\propto e^{-\epsilon\mathcal{L}(\theta)}$ . Although the distribution we hope to sample from is log-concave, standard techniques do not work for our purposes: existing methods converge only in statistical difference, whereas we require a multiplicative convergence guarantee to provide $(\epsilon,0)$ -differential privacy. Previous solutions to this issue (Hardt and Talwar ) worked for the uniform distribution, but not for general log-concave distributions.

Localization: Optimal Algorithms for Strongly Convex Functions.

The exponential-sampling-based technique discussed above does not take advantage of strong convexity of the loss function. We show, however, that a novel combination of two standard techniques—the exponential mechanism and Laplace-noise-based output perturbation—does yield an optimal algorithm. Chaudhuri et al. and showed that strongly convex functions have low-sensitivity minimizers, and hence that one can release the minimum of a strongly convex function with Laplace noise (with total Euclidean length about $\rho=\frac{p}{\Delta\epsilon n}$ if each loss function is $\Delta$ -strongly convex). Simply using this first estimate as a candidate output does not yield optimal utility in general; instead it gives a risk bound of roughly $\frac{p}{\Delta\epsilon}$ .

Lower Bounds.

We use techniques developed to bound the accuracy of releasing 1-way marginals (due to Hardt and Talwar for $(\epsilon,0)-$ and Bun et al. for $(\epsilon,\delta)$ -privacy) to show that our algorithms have essentially optimal risk bounds. The instances that arise in our lower bounds are simple: the functions can be linear (or quadratic, for the case of strong convexity) and the constraint set $\mathcal{C}$ can be either the unit ball or the hypercube. In particular, our lower bounds apply to special case of smooth functions, demonstrating the optimality of objective perturbation in that setting. The reduction to lower-bounds for 1-way marginals is not quite black-box; we exploit specific properties of the instances used by Hardt and Talwar , Bun et al. .

Finally, we provide a much stronger lower bound on the utility of a specific algorithm, the Huberization-based algorithm proposed by Chaudhuri et al. for support vector machines. In order to apply their algorithm to nonsmooth loss functions, they proposed smoothing the loss function by Huberization, and then running their algorithm (which requires smoothness for the privacy analysis) on the resulting, modified loss functions. We show that for any setting of the Huerization parameters, there are simple, one-dimensional nonsmooth loss functions for which the algorithm has error $\Omega(n)$ . This bound justifies the effort we put into designing new algorithms for nonsmooth loss functions.

Generalization Error.

2 Other Related Work

In addition to the previous work mentioned above, we mention several closely related works. A rich line of work seeks to characterize the optimal error of differentially private algorithms for learning and optimization Kasiviswanathan et al. , Beimel et al. , Chaudhuri and Hsu , Beimel et al. . In particular, our results on $(\epsilon,0)$ -differential privacy imply nearly-tight bounds on the “representation dimension” Beimel et al. of convex Lipschitz functions.

Efficient implementations of the exponential mechanism over infinite domains were discussed by Hardt and Talwar , Chaudhuri et al. and Kapralov and Talwar . The latter two works were specific to sampling (approximately) singular vectors of a matrix, and their techniques do not obviously apply here.

Differentially private convex learning in different models has also been studied: for example, Jain et al. , Duchi et al. , Smith and Thakurta study online optimization, Jain and Thakurta study an interactive model tailored to high-dimensional kernel learning. Convex optimization techniques have also played an important role in the development of algorithms for “simultaneous query release” (e.g., the line of work emerging from Hardt and Rothblum ). We do not know of a direct connection between those works and our setting.

3 Additional Definitions

For completeness, we state a few additional definitions related to convex sets and functions.

4 Organization of this Paper

Our upper bounds (efficient algorithms) are given in Sections 2, 3, and 4, whereas our lower bounds are given in Section 5. Namely, in Section 2, we give efficient construction for $(\epsilon,\delta)$ -differentially private algorithms for general convex loss as well as Lipschitz strongly convex loss. In Section 3, we discuss a pure $\epsilon$ -differentially private algorithm for general Lipschitz convex loss and outline an efficient construction for such algorithm. In Section 4, we discuss our localization technique and show how to construct efficient pure $\epsilon$ -differentially private algorithms for Lipschitz strongly convex loss. We derive our lower bound for general Lipschitz convex loss in Section 5.1 and our lower bound for Lipschitz strongly convex loss in Section 5.2. In Section 6, we discuss a generic construction of an efficient algorithm for sampling (with a multiplicative distance guarantee) from a logconcave distribution over an arbitrary convex bounded set. As a by-product of our generic construction, we give the details of the construction of our efficient $\epsilon$ -differentially private algorithm from Section 3.2.

The appendices contain proof details and supplementary material: Appendix A shows that smoothing a nonsmooth loss function in order to apply the objective perturbation technique of Chaudhuri et al. can introduce significant additional error. Appendix B gives details on the application of localization in the setting of $(\epsilon,\delta)$ -differential privacy. Appendix C provides additional details on the proofs of lower bounds. In Appendix D, we explain standard modifications that allow our algorithms to give high probability guarantees instead of expected risk guarantees. Finally, in Appendix F we discuss the how our algorithms can be adapted to provide guarantees on generalization error, rather than empirical error.

Gradient Descent and Optimal (ϵ,δ)italic-ϵ𝛿(\epsilon,\delta)-differentially private Optimization

In this section we provide an algorithm $\mathcal{A}_{\sf Noise-GD}$ (Algorithm 1) for computing $\theta^{priv}$ using a noisy stochastic variant of the classic gradient descent algorithm from the optimization literature . Our algorithm (and the utility analysis) was inspired by the approach of Williams and McSherry for logistic regression.

All the excess risk bounds (16) in this section and the rest of this paper, are presented in expectation over the randomness of the algorithm. In Section D we provide a generic tool to translate the expectation bounds into high probability bound albeit at a loss of extra logarithmic factor in the inverse of the failure probability.

Note(2): Instead of using the stochastic variant in Algorithm 1, one can use the complete gradient (i.e., $\bigtriangledown\mathcal{L}(\theta;\mathcal{D})$ ) in Step 5 and still have the same utility guarantee as Theorem 2.4. However, the running time goes up by a factor of $n$ .

Algorithm $\mathcal{A}_{\sf Noise-GD}$ (Algorithm 1) is $(\epsilon,\delta)$ -differentially private.

Over a domain of data sets $\mathcal{T}^{n}$ , if an algorithm $\mathcal{A}$ is $\epsilon^{\prime}\leq 1$ differentially private, then for any data set $\mathcal{D}\in\mathcal{T}^{n}$ , executing $\mathcal{A}$ on uniformly random $\gamma n$ entries of $\mathcal{D}$ ensures $2\gamma\epsilon^{\prime}$ -differential privacy.

To conclude the proof, we apply “strong composition” (Lemma 2.3) from . With probability at least $1-\delta$ , the privacy loss $W=\sum\limits_{t=1}^{n^{2}}W_{t}$ is at most $\epsilon$ . This concludes the proof.

Let $\epsilon,\delta^{\prime}\geq 0$ . The class of $\epsilon$ -differentially private algorithms satisfies $(\epsilon^{\prime},\delta^{\prime})$ -differential privacy under $T$ -fold adaptive composition for $\epsilon^{\prime}=\sqrt{2T\ln(1/\delta^{\prime})}\epsilon+T\epsilon(e^{\epsilon}-1)$ .

Let $\sigma^{2}=O\left(\frac{L^{2}n^{2}\log(n/\delta)\log(1/\delta)}{\epsilon^{2}}\right)$ . For $\theta^{priv}$ output by Algorithm $\mathcal{A}_{\sf Noise-GD}$ we have the following. (The expectation is over the randomness of the algorithm.)

Let $F(\theta)$ (for $\theta\in\mathcal{C}$ ) be a convex function and let $\theta^{*}=\arg\min\limits_{\theta\in\mathcal{C}}F(\theta)$ . Let $\theta_{1}$ be any arbitrary point from $\mathcal{C}$ . Consider the stochastic gradient descent algorithm $\theta_{t+1}=\Pi_{\mathcal{C}}\left[\theta_{t}-\eta(t)G_{t}(\theta_{t})\right]$ , where $E[G_{t}(\theta_{t})]=\bigtriangledown F(\theta_{t})$ , $E[\|G_{t}\|_{2}^{2}]\leq G^{2}$ and the learning rate function $\eta(t)=\frac{\|\mathcal{C}\|_{2}}{G\sqrt{t}}$ . Then for any $T>1$ , the following is true.

Using the bound from (2) in Lemma 2.5 (i.e., set $G=\sqrt{n^{2}L^{2}+p\sigma^{2}}$ ), and setting $T=n^{2}$ and the learning rate function $\eta(t)$ as in Lemma 2.5, gives us the required excess risk bound for Lipschitz convex functions. For Lipschitz and strongly convex functions we use the following result by .

Let $F(\theta)$ (for $\theta\in\mathcal{C}$ ) be a $\lambda$ -strongly convex function and let $\theta^{*}=\arg\min\limits_{\theta\in\mathcal{C}}F(\theta)$ . Let $\theta_{1}$ be any arbitrary point from $\mathcal{C}$ . Consider the stochastic gradient descent algorithm $\theta_{t+1}=\Pi_{\mathcal{C}}\left[\theta_{t}-\eta(t)G_{t}(\theta_{t})\right]$ , where $E[G_{t}(\theta_{t})]=\bigtriangledown F(\theta_{t})$ , $E[\|G_{t}\|_{2}^{2}]\leq G^{2}$ and the learning rate function $\eta(t)=\frac{1}{\lambda t}$ . Then for any $T>1$ , the following is true.

Using the bound from (2) in Lemma 2.6 (i.e., set $G=\sqrt{n^{2}L^{2}+p\sigma^{2}}$ ), $\lambda=n\Delta$ , and setting $T=n^{2}$ and the learning rate function $\eta(t)$ as in Lemma 2.6, gives us the required excess risk bound for Lipschitz and strongly convex convex functions. ∎

Exponential Sampling and Optimal (ϵ,0)italic-ϵ0(\epsilon,0)-private Optimization

In this section, we focus on the case of pure $\epsilon$ -differential privacy and provide an optimal efficient algorithm for empirical risk minimization for the general class of convex and Lipschitz loss functions. The main building block of this section is the well-known exponential mechanism .

First, we show that a variant of the exponential mechanism is optimal. A major technical contribution of this section is to make the exponential mechanism computationally efficient which is discussed in Section 3.2.

In this section we only deal with loss functions which are Lipschitz. We provide an $\epsilon$ -differentially private algorithm (Algorithm 2) which achieves the optimal excess risk for arbitrary convex bounded sets.

Algorithm 2 is $\epsilon$ -differentially private.

In the following we prove the utility guarantee for Algorithm $\mathcal{A}_{\sf exp-samp}$ .

Let $\theta^{priv}$ be the output of $\mathcal{A}_{\sf exp-samp}$ (Algorithm 2 above). Then, we have the following guarantee on the expected excess risk. (The expectation is over the randomness of the algorithm.)

Consider a differential cone $\Omega$ centered at $\theta^{*}$ (see Figure 1). We will bound the expected excess risk of $\theta^{priv}$ by $O\left(\frac{pL\|\mathcal{C}\|_{2}}{\epsilon}\right)$ conditioned on $\theta^{priv}\in\Omega\cap\mathcal{C}$ for every differential cone. This immediately implies the above theorem by the properties of conditional expectation.

Let $\Gamma$ be a fixed threshold (to be set later) and let ${\sf R}(\theta)=\mathcal{L}(\theta^{priv};\mathcal{D})-\mathcal{L}(\theta^{*};\mathcal{D})$ for the purposes of brevity. Let the marked sets $A_{i}$ ’s in Figure 1 be defined as

Instead of directly computing the probability of $\theta^{priv}$ being outside $A_{1}$ , we will analyze the probabilities for being in each of the $A_{i}$ ’s individually. This form of “peeling” arguments have been used for risk analysis of convex loss in the machine learning literature (e.g., see ) and will allow us to get rid of the extra logarithmic factor that would have otherwise shown up in the excess risk if we use the standard analysis of the exponential mechanism in .

Since $\Omega$ is a differential cone and since ${\sf R}(\theta)$ is continuous on $\mathcal{C}$ , it follows that within $\Omega\cap\mathcal{C}$ , ${\sf R}(\theta)$ only depends on $\|\theta-\theta^{*}\|_{2}$ . Therefore, let $r_{1},r_{2},\cdots$ be the distance of the set boundaries of $A_{1},A_{2},\cdots$ from $\theta^{*}$ . (See Figure 1.) One can equivalently write each $A_{i}$ as follows:

The following claim is the key part of the proof.

Convexity of ${\sf R}(\theta)$ for all $\theta\in\mathcal{C}$ implies that $r_{i}-r_{i-1}\leq r_{i-1}-r_{i-2}$ for all $i\geq 3$ .

Since by definition $\theta^{*}$ is the minimizer of ${\sf R}(\theta)$ within $\mathcal{C}$ and ${\sf R}(\theta)$ is convex, we have ${\sf R}(\theta_{2})\geq{\sf R}(\theta_{1})$ for any $\theta_{1},\theta_{2}\in\mathcal{C}\cap\Omega$ such that $\|\theta_{2}-\theta^{*}\|_{2}\geq\|\theta_{1}-\theta^{*}\|_{2}$ . This directly implies the required bound. ∎

Now, the volume of the set $A_{i}$ is given by ${{\sf Vol}(A_{i})}=\kappa\int\limits_{r_{i-1}}^{r_{i}}r^{p-1}dr$ for some fixed constant $\kappa$ . Hence,

where the last two inequalities follows from Claim 3.3. Hence, we get the following bound on the probability that the excess risk ${\sf R}(\theta^{priv})\geq 4\Gamma$ conditioned on $\theta^{priv}\in\mathcal{C}\cap\Omega$ (For brevity, we remove the conditioning sign from the probabilities below).

where the last inequality follows from the fact that $(i-1)^{p}\leq 3^{p}\cdot\left(2^{i-4}\right)^{p}$ for $i\geq 4$ . Hence, for every $t>0$ , if we choose $\Gamma=\frac{2L\|\mathcal{C}\|_{2}}{\epsilon}\left(\left(p+1\right)\ln 3+t\right)$ , then, conditioned on $\theta^{priv}\in\mathcal{C}\cap\Omega$ , we get

Since this is true for every $t>0$ , we have our required bound as a corollary.

In this section, we give a high-level description of a computationally efficient construction of Algorithm 2. Our algorithm runs in polynomial time in $n,p$ and outputs a sample $\theta\in\mathcal{C}$ from a distribution that is arbitrarily close (in the multiplicative sense) to the distribution of the output of Algorithm 2.

Since we are interested in an efficient pure $\epsilon$ -differentially private algorithm, we need an efficient sampler with a multiplicative distance guarantee. In fact, if we were interested in $(\epsilon,\delta)$ algorithms, efficient sampling with a total variation guarantee would have sufficed which would have made our task a lot easier as we could have used one of the exisiting algorithms, e.g., . In , it was shown how to sample efficiently with a multiplicative guarantee from the unifrom distribution over a convex bounded set. However, what we want to achieve here is more general, that is, to sample efficiently from any given logconcave distribution defined over a convex bounded set. To the best of our knowledge, this task has not been explicitly worked out before, nevertheless, all the ingredients needed to accomplish it are present in the literature, mainly .

We highlight here the main ideas of our constrution. Since such construction is not specific to our privacy problem and could be of independent interest, in this section, we only provide the high-level description of this construction, however all the details of such construction and the proof of our main result (Theorem 3.4 below) are deferred to Section 6.

There is an efficient version of Algorithm 2 that has the following guarantees.

Privacy: The algorithm is $\epsilon$ -differentially private.

Utility: The output $\theta^{priv}\in\mathcal{C}$ of the algorithm satisfies

Running time: Assuming $\mathcal{C}$ is in isotropic position, the algorithm runs in timeThe case where $\mathcal{C}$ is not in isotropic position is discussed below.

In fact, the running time of our algorithm depends on $\|\mathcal{C}\|_{\infty}$ rather than $\|\mathcal{C}\|_{2}$ . Namely, all the $\|\mathcal{C}\|_{2}$ terms in the running time can be replaced with $\|\mathcal{C}\|_{\infty}$ , however, we chose to write it in this less conservative way since all the bounds in this paper are expressed in terms of $\|\mathcal{C}\|_{2}$ .

Before describing our construction, we first introduce some useful notation and discuss some preliminaries.

where $\frac{d\mu(q)}{d\nu(q)}$ (resp., $\frac{d\nu(q)}{d\mu(q)}$ ) denotes the ratio of the two measures (more precisely, the Radon-Nikodym derivative).

3 Our construction

We use the grid-walk algorithm of for sampling from a logconcave distribution defined over a cube as a building block. Our construction is described as follows:

Enclose the set $\mathcal{C}$ with a cube $A$ with edges of length $\tau$ .

Obtain a convex Lipschitz extension $\bar{\mathcal{L}}(.;\mathcal{D})$ of the loss function $\mathcal{L}(.;\mathcal{D})$ over $A$ . This can be done efficiently using a projection oracle.

Define $F(\theta)\triangleq e^{-\frac{\epsilon}{6L\|\mathcal{C}\|_{2}}\bar{\mathcal{L}}(\theta;\mathcal{D})-\bar{\psi}_{\alpha}(\theta)},~{}\theta\in A$ , for a specific choice of $\alpha=O(\frac{\epsilon n}{\|C\|_{2}})$ (See Section 6 for details).

Run the grid-walk algorithm of with $F$ as the input weight function and $A$ as the input cube, and output a sample $\theta$ whose distribution is close, with respect to ${\sf Dist}_{\infty}$ , to the distribution induced by $F$ on $A$ which is given by $\frac{F(\theta)}{\int\limits_{v\in A}F(v)dv},~{}\theta\in A$ .

Localization and Optimal Private Algorithms for Strongly Convex Loss

It is unclear how to get a direct variant of Algorithm 2 in Section 3 for Lipschitz and strongly convex losses that can achieve optimal excess risk guarantees. The issue in extending Algorithm 2 directly is that the convex set $\mathcal{C}$ over which the exponential mechanism is defined is “too large” to provide tight guarantees.

Next, we instantiate the generic $\epsilon$ -differentially private algorithm in the second step with our efficient exponential mechanism of Section3.1 (Algorithm 2) to obtain an algorithm with optimal excess risk bound (Theorem 4.3).

Note: The localization technique is not specific to pure $\epsilon$ -differential privacy, and extends naturally to $(\epsilon,\delta)$ case. Although it is not relevant in our current context, since we already have gradient descent based algorithm which achieves optimal excess risk bound. We defer the details for the $(\epsilon,\delta)$ case to Appendix B.

Details of the generic algorithm: We first give a simple algorithm that carries out the desired localization step. The crux of the algorithm is the same as to that of the output perturbation algorithm of . The high-level idea is to first compute $\theta^{*}=\arg\min\limits_{\theta\in\mathcal{C}}\mathcal{L}(\theta;\mathcal{D})$ and add noise according to the sensitivity of $\theta^{*}$ . The details of the algorithm are given in Algorithm 3.

Algorithm 4 is $\epsilon$ -differentially private.

The privacy guarantee follows directly from the composition theorem together with the fact that $\mathcal{A}^{\frac{\epsilon}{2}}_{\sf out-pert}$ is $\frac{\epsilon}{2}$ -differentially private (see ) and that $\mathcal{A}^{\frac{\epsilon}{2}}_{\sf gen-Lip}$ is $\frac{\epsilon}{2}$ -differentially private by assumption. ∎

In the following theorem, we provide a generic expression for the excess risk of Algorithm 4 in terms of the expected excess risk of any given algorithm $\mathcal{A}_{\sf gen-Lip}$ .

for some function $F$ , then the output $\theta^{priv}$ of Algorithm 4 satisfies

where $\theta^{*}=\arg\min\limits_{\theta\in\mathcal{C}}\mathcal{L}(\theta;\mathcal{D})$ .

The proof follows from the fact that, in Algorithm $\mathcal{A}^{\frac{\epsilon}{2}}_{\sf out-pert}$ , the norm of the noise vector $\|b\|_{2}$ is distributed according to Gamma distribution $\Gamma(p,\frac{4L}{\Delta\epsilon n})$ and hence satisfies

Note that the second term on the right-hand side above becomes $O(\frac{1}{n^{2}})$ . From our lower bound (Section 5.2 below), $F(.,n,.,.,.)$ must be at least $\Omega(\frac{1}{n})$ . Hence, we have

which completes the proof of the theorem. ∎

Instantiation of Algorithm $\mathcal{A}^{\frac{\epsilon}{2}}_{\sf gen-Lip}$ with the exponential sampling algorithm: Next, we give our optimal $\epsilon$ -differentially private algorithm for Lipschitz strongly convex loss functions. To do this, we instantiate the generic Algorithm $\mathcal{A}_{\sf gen-Lip}$ in Algorithm 4 with our exponential sampling algorithm from Section 3.1 (Algorithm 2), or its efficient version Algorithm $\mathcal{A}_{\sf eff-exp-samp}$ (See Section 3.2) to obtain the optimal excess risk bound. We formally state the bound in Theorem 4.3) below. The proof of Theorem 4.3 follows from Theorem 3.2 and Lemma 4.2 above.

Suppose we replace $\mathcal{A}^{\frac{\epsilon}{2}}_{\sf gen-Lip}$ in Algorithm 4 with Algorithm 2 (Section 3.1), or its efficient version Algorithm 7 (See Theorem 3.4 and Section 6 for details). Then, the output $\theta^{priv}$ satisfies

where $\theta^{*}=\arg\min\limits_{\theta\in\mathcal{C}}\mathcal{L}(\theta;\mathcal{D})$ .

Lower Bounds on Excess Risk

Before we state and prove our lower bounds, we first give the following useful lemma which gives lower bounds on the $L_{2}$ -error incurred by $\epsilon$ and $(\epsilon,\delta)$ -differentially private algorithms for estimating the $1$ -way marginals of datasets over $\{-\frac{1}{\sqrt{p}},\frac{1}{\sqrt{p}}\}^{p}$ . This lemma is based on the results of and , however, for the sake of completeness, we give a detailed proof of this lemma in Appendix C.

where $q(\mathcal{D})=\frac{1}{n}\sum_{i=1}^{n}d_{i}$ .

In this section, we give lower bounds for both $\epsilon$ and $(\epsilon,\delta)$ differentially private algorithms for minimizing any convex Lipschitz loss function $\mathcal{L}(\theta;\mathcal{D})$ . We consider the following loss function. Define

We use Part 2 of Lemma 5.1 and follow the same lines of the proof of Theorem 5.2. ∎

2 Lower bounds for Strongly Convex Functions

The proof follows directly from (7) and Part 1 of Lemma 5.1. ∎

The proof follows directly from (7) and Part 2 of Lemma 5.1. ∎

Hence, our upper bounds in Sections 2 and 4 imply that our lower bounds are tight for all values of $L,\|\mathcal{C}\|_{2},$ and $\Delta$ for which $\frac{\Delta\|\mathcal{C}\|_{2}}{L}=\Omega(1)$ . In other words, in the general case (where $\frac{\Delta\|\mathcal{C}\|_{2}}{L}$ is not necessarily $\Omega(1)$ ), these lower bounds are tight up to a factor of $\frac{\Delta\|\mathcal{C}\|_{2}}{L}$ .

Efficient Sampling from Logconcave Distributions over Convex Sets and The Proof of Theorem 3.4

In this section, we discuss a generic construction of an efficient algorithm for sampling from a logconcave distribution over an arbitrary convex bounded set. Such algorithm gives a multiplicative distance guarantee on the distribution of its output, that is, it outputs a sample from a distribution that is within a constant factor (close to 1) from the desired logconcave distribution. As a by-product of our generic construction, we give the construction of our efficient $\epsilon$ -differentially private algorithm $\mathcal{A}_{\sf eff-exp-samp}$ whose construction is outlined in Section 3.2 and prove Theorem 3.4. As argued in Section 3.2, we will assume that the convex set is already in isotropic position. The reader may refer to Section 3.2 for the details of dealing with the general case (where the set is not necessarily isotropic) and the effect of that on the running time.

We start by the following lemma which describes Algorithm $\mathcal{A}_{\sf cube-samp}$ for sampling from a distribution proprtional to a given logconcave function $F$ defined over a hypercube $A$ .

Let $P$ be a lazy, time reversible Markov chain over a finite state space $\Gamma$ . Then, the time $t_{\infty}$ required for relative $L_{\infty}$ convergence of $\epsilon^{\prime}$ is at most $1+\int_{4\pi^{*}}^{4/\epsilon^{\prime}}\frac{4dx}{x\Phi^{2}(x)}$ . Here, $\Phi(x)=\inf\{\phi_{S}:\pi(S)\leq x\}$ where $\phi_{S}$ denotes the conductance of the set $S\subseteq\Gamma$ and $\pi^{*}$ is the minimum probability assigned by the stationary distribution.

For clarity and completeness, we give a proof of this lemma here.

For the sake of simplicity, let’s assume that $\mathcal{C}$ is closed. Actually, this is no loss of generality since we can always redefine $f$ such that it is defined on the closure of $\mathcal{C}$ which is possible because $f$ is continuous on $\mathcal{C}$ . We use a standard extension in literature. Namely, define

where the inequality in the first line follows from the fact that $y_{\lambda}$ is the minimzer (w.r.t. $y$ ) of $g_{y}(x_{\lambda})$ and the inequality in the third line follows from the convexity of $f$ and the $L_{2}$ -norm. This completes the proof of the lemma. ∎

Now, we give the construction of Algorithm $\mathcal{A}_{\sf init-samp}$ followed by a Lemma asserting the probabilistic guarantee discussed above.

where the second inequality in the first line follows from the Lipschitz property of $\bar{f}$ and the second inequality in the second line follows from the fact that $e^{-x}\geq 1-x$ and $(1-x)^{p-1}\geq 1-px$ . On the other hand, we can upper bound $\mathcal{I}_{\sf out}$ as follows.

where the last inequality follows from the setting of $\alpha$ we made in Algorithm 5. Since this is true for any differential cone as described above, this proves that $\mathcal{A}_{\sf init-samp}$ outputs $\widetilde{\theta}\in\mathcal{C}$ with probability at least $1/2$ .

Let $\hat{\mu}_{\sf Good}$ denote the conditional distribution of $\widetilde{\theta}$ (the output of Algorithm $\mathcal{A}_{\sf eff-samp}$ ) conditioned on the event that $\mathcal{A}_{\sf init-samp}$ outputs a sample in $\mathcal{C}$ in one of the $m$ iterations of the for loop. From Lemma 6.4, it is easy to see that the probability measure of the output of $\mathcal{A}_{\sf eff-samp}$ can be expressed as

The running time of $\mathcal{A}_{\sf eff-samp}$ is at most $O(m\cdot T_{\mathcal{A}_{\sf init-samp}})$ where $T_{\mathcal{A}_{\sf init-samp}}$ is the running time of $\mathcal{A}_{\sf init-samp}$ (which is of the same order as that of $\mathcal{A}_{\sf cube-samp}$ given in Lemma 6.1). Note that Step 7 can be carried out in linear time using standard methods in literature. Finally, by plugging in our choice for the value of $m$ gives the expression in the lemma statement. This completes the proof. ∎

In this section, we show a straightforward construction for our efficient Algorithm $\mathcal{A}_{\sf eff-exp-samp}$ (referred to in Section 3.2) based on the construction established above for efficient logconcave sampling. Based on the results established above in this section, we give a proof of Theorem 3.4 which will also be fairly straightforward. First, fix a dataset $\mathcal{D}$ . Our goal is to construct an efficient version of Algorithm $\mathcal{A}_{\sf exp-samp}$ (Algorithm 2 from Section 3.1). To do this, we simply run Algorithm $\mathcal{A}_{\sf eff-samp}$ (Algorithm 6 above) with the function $f$ instantiated with the scaled decomposable loss function $\frac{\epsilon}{6L\|\mathcal{C}\|_{2}}\mathcal{L}(.;\mathcal{D})$ defined over the convex bounded set $\mathcal{C}$ (which is assumed to be in isotropic position as discussed in Section 3.2). Hence, $\eta$ in our case is $\frac{n\epsilon}{6\|\mathcal{C}\|_{2}}$ . Namely, as shown below, our $\epsilon$ -differentially private algorithm $\mathcal{A}_{\sf eff-exp-samp}$ is an instantiation of $\mathcal{A}_{\sf eff-samp}$ on inputs $\mathcal{C},~{}\frac{\epsilon}{6L\|\mathcal{C}\|_{2}}\mathcal{L}(.;\mathcal{D}),~{}\frac{n\epsilon}{6\|\mathcal{C}\|_{2}},$ and $\frac{\epsilon}{3}$ .

The choice of the scaling factor $\frac{\epsilon}{6L\|\mathcal{C}\|_{2}}$ of the loss function and the multiplicative distance guarantee of $\frac{\epsilon}{3}$ is tuned to yield an $\epsilon$ -differentially private algorithm. To see this, we rely on the following simple lemma given in .

Proof of Theorem 3.4: Having Lemmas 6.5 and 6.6 in hand, the proof of Theorem 3.4 becomes straightforward. First, we show differential privacy of Algorithm 7. For any dataset $\mathcal{D}$ , let $\mu^{\mathcal{D}}$ be the distribution of $\theta$ proportional to $e^{-\frac{\epsilon}{6L\|\mathcal{C}\|_{2}}\mathcal{L}(\theta;\mathcal{D})}$ . Note that $\mu^{\mathcal{D}}$ is the distribution of the output of Algorithm $\mathcal{A}_{\sf exp-samp}$ (Algorithm 2 from Section 3.1) when $\epsilon$ is replaced with $\frac{\epsilon}{3}$ . Let $\hat{\mu}^{\mathcal{D}}$ be the distribution of the output $\theta^{priv}$ of $\mathcal{A}_{\sf eff-exp-samp}$ (Algorithm 7 above). From Lemma 6.5, it follows that ${\sf Dist}_{\infty}\left(\hat{\mu}^{\mathcal{D}},\mu^{\mathcal{D}}\right)\leq\frac{\epsilon}{3}$ . Hence, from Theorem 3.1 and Lemma 6.6, we reach the fact that $\mathcal{A}_{\sf eff-exp-samp}$ is $\epsilon$ -differentially private.

To show the utility guarantee of $\mathcal{A}_{\sf eff-exp-samp}$ , we first observe that the distribution of the output $\theta^{priv}$ is close with respect to ${\sf Dist}_{\infty}$ to (i.e., within a constant factor of) the distribution of the output of $\mathcal{A}_{\sf exp-samp}$ (Algorithm 2 from Section 3.1), and hence, the utility analysis follows the same lines of Theorem 3.2.

Acknowledgments

We are grateful to Santosh Vempala and Ravi Kannan for discussions about efficient sampling algorithms for log-concave distributions over convex bodies. In particular, Ravi suggested the idea of using a penalty term to reduce from sampling over $\mathcal{C}$ to sampling over the cube.

References

Appendix A Straightforward Smoothing Does Not Yield Optimal Algorithms

In (9) the noise $b\sim\mathcal{N}(0,\frac{8\log(1/\delta)}{\epsilon^{2}})$ . In the results to follow, we show that for any choice of the Huberization parameter $h$ , there exists data sets of size $n$ from the domain above where the excess risk for objective perturbation will be provably worse than our results in this paper. We present the results for the $(\epsilon,\delta)$ -differential privacy case, but the same conclusions hold for the pure $\epsilon$ -differential privacy case.

For every $h>0$ , there exists $\mathcal{D}$ such the excess risk for the objective perturbation algorithm in (9) satisfies:

Consider the data set $\mathcal{D}_{1}$ with $\frac{n}{3}$ entries being $(x=-1,y=1)$ and $\frac{2n}{3}$ entries being $(x=1,y=-1)$ . In the following lemma we lower bound the excess risk on $\mathcal{D}_{1}$ for a given huberization parameter $h$ .

Let $\epsilon,\delta$ be the privacy parameters with $\epsilon$ being a constant ( $<1$ ) and $\delta=\Omega\left(\frac{1}{n^{4}}\right)$ . For the data set $\mathcal{D}_{1}$ mentioned above, the excess risk for objective perturbation (9) is as follows. For all $h>0$ , we have

Consider a data set $\mathcal{D}_{2}$ which has exactly $\max\{\frac{n}{2}-\frac{1}{32h},0\}$ entries with $(x=-1,y=1)$ and $\min\{\frac{n}{2}+\frac{1}{32h},n\}$ entries with $(x=1,y=1)$ . In the following lemma we lower bound the excess risk on $\mathcal{D}_{2}$ for a given huberization parameter $h$ .

Let $\epsilon,\delta$ be the privacy parameters with $\epsilon$ being a constant ( $<1$ ) and $\delta=\Omega\left(\frac{1}{n^{4}}\right)$ . Let $h<\frac{1}{\log n}$ be a fixed Huberization parameter. Then for the data set $\mathcal{D}_{2}$ mentioned above, the excess risk for objective perturbation (9) is as follows.

Solving for $\theta^{priv}$ , we have $\theta^{priv}=\min\left\{\frac{\epsilon}{4},4\epsilon nh\right\}+4b\epsilon h$ . By assumption $h<1/\log n$ and w.p. $\geq 2/3$ we have $|b|\leq\frac{8\sqrt{\log(1/\delta)}}{\epsilon}$ . Therefore, w.p. $\geq 2/3$ , we have $\theta^{priv}\leq\epsilon$ .

Finally combining Lemmas A.3 and A.2 completes the proof of Theorem A.1. ∎

Algorithm $\mathcal{A}^{(\epsilon,\delta)}_{\sf gen-str-convex}$ is $(\epsilon,\delta)$ -differentially private.

The privacy guarantee follows directly from the composition theorem together with the fact that $\mathcal{A}^{(\frac{\epsilon}{2},\frac{\delta}{2})}_{\sf out-pert}$ is $(\frac{\epsilon}{2},\frac{\delta}{2})$ -differentially private and that $\mathcal{A}^{(\frac{\epsilon}{2},\frac{\delta}{2})}_{\sf gen-Lip}$ is $(\frac{\epsilon}{2},\frac{\delta}{2})$ -differentially private by assumption. ∎

for some function $F$ , then the output $\theta^{priv}$ of $\mathcal{A}^{(\epsilon,\delta)}_{\sf gen-str-convex}$ satisfies

The proof follows the same lines of the proof of Theorem 4.2 except for the fact that, in Algorithm $\mathcal{A}^{(\frac{\epsilon}{2},\frac{\delta}{2})}_{\sf out-pert}$ , the noise vector $b$ is Gaussian and hence using the standard bounds on the norm of an i.i.d. Gaussian vector, we have

We set $\zeta=\sqrt{3\log(n)}$ and the rest of the proof follows in the same way as the proof of Theorem 4.2. ∎

Appendix C Proof of Lemma 5.1

We restate Part 1 of the lemma here for convenience.

where $q(\mathcal{D})=\frac{1}{n}\sum_{i=1}^{n}d_{i}$ .

We use a standard packing argument. Variants of such argument have appeared in several places in literature, e.g., and . We first construct $K=2^{p/2}$ points $d^{(1)},...,d^{(K)}$ in $\{-\frac{1}{\sqrt{p}},\frac{1}{\sqrt{p}}\}^{p}$ such that for every distinct pair $d^{(i)},d^{(j)}$ of these points, we have

It is easy to show the existence of such set of points using the probabilistic method (for example, the Gilbert-Varshamov construction of a linear random $(2^{p/2},p)$ -binary code over $\{-\frac{1}{\sqrt{p}},\frac{1}{\sqrt{p}}\}$ achieves this property).

Fix $\epsilon>0$ . Define $n^{*}=\frac{1}{20}\frac{p}{\epsilon}$ . Let’s first consider the case where $n\leq n^{*}$ . We construct $K$ datasets $\mathcal{D}^{(1)},...,\mathcal{D}^{(K)}$ where for each $i\in[K]$ , $\mathcal{D}^{(i)}$ contains n copies of $d^{(i)}$ . Note that for all $i\neq j$ ,

Let $\mathcal{A}$ be any $\epsilon$ -differentially private algorithm for answering $q$ . Suppose that for every $\mathcal{D}^{(i)},i\in[K]$ , with probability at least $1/2$ , $\|\mathcal{A}\left(\mathcal{D}^{(i)}\right)-q\left(\mathcal{D}^{(i)}\right)\|_{2}<\frac{1}{16}$ , i.e., for every $i\in[K]$ , $\Pr\left[\mathcal{A}\left(\mathcal{D}^{(i)}\right)\in\mathcal{B}\left(\mathcal{D}^{(i)}\right)\right]\geq\frac{1}{2}$ where for any dataset $\mathcal{D}$ , $\mathcal{B}(\mathcal{D})$ is defined as

Note that for all $i\neq j$ , $\mathcal{D}^{(i)}$ and $\mathcal{D}^{(j)}$ differ in all their $n$ entries. Since $\mathcal{A}$ is $\epsilon$ -differentially private, for all $i\in[K]$ , we have $\Pr\left[\mathcal{A}\left(\mathcal{D}^{(1)}\right)\in\mathcal{B}\left(\mathcal{D}^{(i)}\right)\right]\geq\frac{1}{2}e^{-\epsilon n}$ . Since, by (11) abd (12), all $\mathcal{B}\left(\mathcal{D}^{(i)}\right),~{}i\in[K],$ are mutually disjoint, then

which implies that $n>n^{*}$ for sufficiently large $p$ which is a contradiction to the fact that $n\leq n^{*}$ . Hence, there must exist a dataset $\mathcal{D}^{(i)}$ for some $i\in[K]$ on which $\mathcal{A}$ makes an $L_{2}$ -error which is at least $\frac{1}{16}$ with probability at least $\frac{1}{2}$ . Note also that the $L_{2}$ norm of the sum of the entries of such $\mathcal{D}^{(i)}$ is $n$ .

C.2 Proof of Part 2

where $q(\mathcal{D})=\frac{1}{n}\sum_{i=1}^{n}d_{i}$ .

Appendix D Converting Excess Risk Bounds in Expectation to High-probability Bounds

In this paper all of our utility guarantees are in terms of the expectation over the randomness of the algorithm. Although all the utility analysis except for the gradient descent based algorithm (Algorithm 1) provide high-probability guarantees directly, in this section we provide a generic approach for obtaining high-probability guarantee based on the expected risk bounds. The idea is to run the underlying differentially private algorithm $k$ -times, with the privacy parameters $\epsilon/k$ and $\delta/k$ for each run. Let $\theta^{priv}_{1},\cdots\theta^{priv}_{k}$ be the vectors output by the $k$ -runs. First notice that the vector $\theta^{priv}_{1},\cdots,\theta^{priv}_{k}$ is $(\epsilon,\delta)$ -differentially private. Moreover if the algorithm has expected excess risk of $F(\epsilon,\delta)$ (where $F$ is the specific excess risk function of $\epsilon$ and $\delta$ ), then by Markov’s inequality there exist an execution of the algorithm $i\in[k]$ for which the excess risk is $2F(\epsilon/k,\delta/k)$ with probability at least $1-1/2^{k}$ .

One can now use the exponential mechanism from Algorithm 2, to pick the best $\theta^{priv}_{i}$ from the list. By the same analysis of Theorem 3.2, one can show that with probability at least $1-\rho/2$ , the exponential mechanism will output a vector $\theta^{priv}$ that has excess risk of $\max\limits_{i}{\sf{Excess\_risk}}(\theta^{priv}_{i})-O\left(\frac{L\|\mathcal{C}\|_{2}}{\epsilon}\log(k/\rho)\right)$ . Setting $k=\log(2/\rho)$ , we have that with probability at lest $1-\rho$ , the excess risk for $\theta^{priv}$ is at most $O(F(\frac{\epsilon}{\log(1/\rho)},\frac{\delta}{\log(1/\rho)}))$ . Placing this bound in context of the paper, the high probability bounds are only a $\text{poly}\log(1/\rho)$ factor off from the expectation bounds.

Appendix E Excess Risk Bounds for Smooth Functions

Appendix F From Excess Empirical Risk to Generalization Error

In this section, provide a generic tool to interpret our ERM results in the context of generalization error (true risk) bounds. For a given distribution $\tau$ , let us define true risk for a model $\theta\in\mathcal{C}$ as follows.

Analogously, we define the excess risk for a given a given model $\theta$ by ${\sf ExcessRisk}(\theta)$ .

Let $\mathcal{D}$ be a data set of $n$ data samples drawn i.i.d. from the distribution $\tau$ . The following theorem from learning theory relates true excess risk to excess empirical risk.

Plugging in the utility guarantee for $\theta^{priv}$ from Theorems 2.4 and 4.3, and using the expectation to high-probability bound trick from Appendix D, we obtain the following.

There exists an $\epsilon$ -differentially private algorithm, that outputs $\theta^{priv}$ such that with probability at least $1-\gamma$ over the randomness of sampling the data set $\mathcal{D}$ and the risk minimization algorithm, the following is true:

There exists an $(\epsilon,\delta)$ -differentially private algorithm, that outputs $\theta^{priv}$ such that with probability at least $1-\gamma$ over the randomness of sampling the data set $\mathcal{D}$ and the risk minimization algorithm, the following is true:

Combining with Theorem F.2, plugging in (17), and optimizing for $\Delta$ , we have the following:

While the dependence on $n$ for our private algorithms in Theorem F.3 matches the bounds for the corresponding non-private algorithms (see), unlike the non-private counter parts, the private algorithms have an explicit dependence on $p$ . We leave it as an open problem to figure out the right dependence on $p$ w.r.t. excess risk for private algorithms.

Generalized Linear Models (GLM).

The above theorem follows from Theorem 2 in and the regularization trick above. Theorem F.4 shows that in case of GLM, we can essentially attain the non-private upper bound of $O(\frac{L\|C\|_{2}}{\sqrt{n}})$ which is known to be tight: for example, if we consider a linear loss function then using a standard Central Limit Theorem argument (or using standard lower bounds on the minimax error in parametric estimation), one can show that the there exists a distribution on $\mathcal{X}$ for which the true excess risk is $\Omega\left(\frac{L\|C\|_{2}}{\sqrt{n}}\right)$ .

For general Lipschitz loss functions, we provide a tool that can be used to obtain expectation (over algorithm’s random coins) guarantees on the excess risk (as opposed to the high probability guarantees given in Theorem F.3 above.)

Let $\theta^{priv}$ denote the output of an $(\epsilon,0)$ -differentially algorithm. We have

Let $\theta^{priv}$ denote the output of an $(\epsilon,\delta)$ -differentially algorithm. We have

where the expectation in both cases is over the random coins of the algorithm.

We can use this lemma together with our ERM upper bounds for general Lipschitz functions to give the following expectation guarantees on the excess risk:

There is an $\left(\Theta\left(\sqrt{\frac{p}{n}}\right),~{}0\right)$ -differentially private algorithm, that outputs $\theta^{priv}$ such that the following is true:

There exists an $\left(\Theta\left(\frac{p^{1/4}\log(n/\delta)}{\sqrt{n}}\right),\delta\right)$ -differentially private algorithm, that outputs $\theta^{priv}$ such that the following is true:

where the expectation in both cases is over the random coins of the algorithm.