Adapting to Function Difficulty and Growth Conditions in Private Optimization

08/05/2021 ∙ by Hilal Asi, et al. ∙ 0

We develop algorithms for private stochastic convex optimization that adapt to the hardness of the specific function we wish to optimize. While previous work provide worst-case bounds for arbitrary convex functions, it is often the case that the function at hand belongs to a smaller class that enjoys faster rates. Concretely, we show that for functions exhibiting κ-growth around the optimum, i.e., f(x) ≥ f(x^*) + λκ^-1x-x^*_2^κ for κ > 1, our algorithms improve upon the standard √(d)/nε privacy rate to the faster (√(d)/nε)^κκ - 1. Crucially, they achieve these rates without knowledge of the growth constant κ of the function. Our algorithms build upon the inverse sensitivity mechanism, which adapts to instance difficulty (Asi Duchi, 2020), and recent localization techniques in private optimization (Feldman et al., 2020). We complement our algorithms with matching lower bounds for these function classes and demonstrate that our adaptive algorithm is simultaneously (minimax) optimal over all κ≥ 1+c whenever c = Θ(1).

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1 Introduction

Stochastic convex optimization (SCO) is a central problem in machine learning and statistics, where for a sample space

, parameter space , and a collection of convex losses , one wishes to solve

(1)

using an observed dataset . While as formulated, the problem is by now fairly well-understood [12, 38, 29, 10, 37], it is becoming clear that, because of considerations beyond pure statistical accuracy—memory or communication costs [45, 26, 13], fairness [23, 28], personalization or distributed learning [35]—problem (1) is simply insufficient to address modern learning problems. To that end, researchers have revisited SCO under the additional constraint that the solution preserves the privacy of the provided sample [22, 21, 1, 16, 19]. A waypoint is Bassily et al. [7], who provide a private method with optimal convergence rates for the related empirical risk minimization problem, with recent papers focus on SCO providing (worst-case) optimal rates in various settings: smooth convex functions [8, 25], non-smooth functions [9], non-Euclidean geometry [5, 4] and under more stringent privacy constraints [34].

Yet these works ground their analyses in worst-case scenarios and provide guarantees for the hardest instance of the class of problems they consider. Conversely, they argue that their algorithms are optimal in a minimax sense: for any algorithm, there exists a hard instance on which the error achieved by the algorithm is equal to the upper bound. While valuable, these results are pessimistic—the exhibited hard instances are typically pathological—and fail to reflect achievable performance.

In this work, we consider the problem of adaptivity when solving (1) under privacy constraints. Importantly, we wish to provide private algorithms that adapt to the hardness of the objective

. A loss function

may belong to multiple problem classes, each exhibiting different achievable rates, so a natural desideratum is to attain the error rate of the easiest sub-class. As a simple vignette, if one gets an arbitrary -Lipschitz convex loss function , the worst-case guarantee of any -DP algorithm is . However, if one learns that exhibits some growth property—say is -strongly convex—the regret guarantee improves to the faster rate with the appropriate algorithm. It is thus important to provide algorithms that achieves the rates of the “easiest” class to which the function belongs [32, 46, 18].

To that end, consider the nested classes of functions for such that, if then there exists such that for all ,

For example, strong convexity implies growth with parameter . This growth assumption closely relates to uniform convexity [32] and the Polyak-Kurdyka-Łojasiewicz inequality [11], and we make these connections precise in Section 2. Intuitively, smaller makes the function much easier to optimize: the error around the optimal point grows quickly. Objectives with growth are widespread in machine learning applications: among others, the -regularized hinge loss exhibits sharp growth (i.e. ) while - or -constrained -norm regression —i.e.  and —has -growth for any integer greater than  [43]. In this work, we provide private adaptive algorithms that adapt to the actual growth of the function at hand.

We begin our analysis by examining Asi and Duchi’s inverse sensitivity mechanism [2] on ERM as a motivation. While not a practical algorithm, it achieves instance-optimal rates for any one-dimensional function under mild assumptions, quantifying the best bound one could hope to achieve with an adaptive algorithm, and showing (in principle) that adaptive private algorithms can exist. We first show that for any function with -growth, the inverse sensitivity mechanism achieves privacy cost ; importantly, without knowledge of the function class , that belongs to. This constitutes grounding and motivation for our work in three ways: (i) it validates our choice of sub-classes as the privacy rate is effectively controlled by the value of , (ii) it exhibits the rate we wish to achieve with efficient algorithms on and (iii) it showcases that for easier functions, privacy costs shrink significantly—to illustrate, for the privacy rate becomes .

We continue our treatment of problem (1) under growth in Section 4 and develop practical algorithms that achieve the rates of the inverse sensitivity mechanism. Moreover, for approximate -differential privacy, our algorithms improve the rates, achieving roughly . Our algorithms hinge on a reduction to SCO: we show that by solving a sequence of increasingly constrained SCO problems, one achieves the right rate whenever the function exhibits growth at the optimum. Importantly, our algorithm only requires a lower bound (where is the actual growth of ).

We provide optimality guarantees for our algorithms in Section 5 and show that both the inverse sensitivity and the efficient algorithms of Section 4 are simultaneously minimax optimal over all classes whenever and for -DP algorithms. Finally, we prove that in arbitrary dimension, for both pure- and approximate-DP constraints, our algorithms are also simultaneously optimal for all classes with .

On the way, we provide results that may be of independent interest to the community. First, we develop optimal algorithms for SCO under pure

differential privacy constraints, which, to the best of our knowledge, do not exist in the literature. Secondly, our algorithms and analysis provide high-probability bounds on the loss, whereas existing results only provide (weaker) bounds on the expected loss. Finally, we complete the results of 

Ramdas and Singh [40] on (non-private) optimization lower bounds for functions with -growth by providing information-theoretic lower bounds (in contrast to oracle-based lower bounds that rely on observing only gradient information) and capturing the optimal dependence on all problem parameters (namely and ).

1.1 Related work

Convex optimization is one of the best studied problems in private data analysis [16, 19, 41, 7]. The first papers in this line of work mainly study minimizing the empirical loss, and readily establish that the (minimax) optimal privacy rates are for pure -DP and for -DP [16, 7]. More recently, several works instead consider the harder problem of privately minimizing the population loss [8, 25]. These papers introduce new algorithmic techniques to obtain the worst-case optimal rates of for -DP. They also show how to improve this rate to the faster in the case of -strongly convex functions. Our work subsumes both of these results as they correspond to and respectively. To the best of our knowledge, there has been no work in private optimization that investigates the rates under general -growth assumptions or adaptivity to such conditions.

In contrast, the optimization community has extensively studied growth assumptions  [40, 32, 15] and show that on these problems, carefully crafted algorithms improves upon the standard for convex functions to the faster . [32] derives worst-case optimal (in the first-order oracle model) gradient algorithms in the uniformly convex case (i.e. ) and provides technique to adapt to the growth , while [40]

, drawing connections between growth conditions and active learning, provides upper and lower bounds in the first-order stochastic oracle model. We complete the results of the latter and provide

information-theoretic lower bounds that have optimal dependence on and —their lower bound only holding for inversely proportional to , when . Closest to our work is [15] who studies instance-optimality via local minimax complexity [14]. For one-dimensional functions, they develop a bisection-based instance-optimal algorithm and show that on individual functions of the form , the local minimax rate is .

2 Preliminaries

We first provide notation that we use throughout this paper, define useful assumptions and present key definitions in convex analysis and differential privacy.

Notation.

typically denotes the sample size and the dimension. Throughout this work, refers to the optimization variable, to the constraint set and to elements ( when random) of the sample space . We usually denote by the (convex) loss function and for a dataset , we define the empirical and population losses

We omit the dependence on as it is often clear from context. We reserve for the privacy parameters of Definition 2. We always take gradients with respect to the optimization variable . In the case that is not differentiable at , we override notation and define , where is the subdifferential of at . We use for (potentially random) mechanism and as a shorthand for . For , is the standard -norm, is the corresponding -dimensional -ball of radius and is the dual of , i.e. such that . Finally, we define the Hamming distance between datasets , where is the set of permutations over sets of size .

Assumptions.

We first state standard assumptions for solving (1). We assume that is a closed, convex domain such that . Furthermore, we assume that for any , is convex and -Lipschitz with respect to . Central to our work, we define the following -growth assumption.

[-growth] Let . For a loss and distribution , we say that has growth for and , if the population function satisfies

In the case where is the empirical distribution on a finite dataset , we refer to -growth of as -growth of the empirical function .

Uniform convexity and inequality.

Assumption 2 is closely related to two fundamental notions in convex analysis: uniform convexity and the inequality. Following [39], we say that is -uniformly convex with and if

This immediately implies that (i) sums (and expectations) preserve uniform convexity (ii) if is uniformly convex with and , then it has -growth. This will be useful when constructing hard instances as it will suffice to consider -uniformly convex functions which are generally more convenient to manipulate. Finally, we point out that, in the general case that , the literature refers to Assumption 2 as the inequality [11] with, in their notation, . Theorem 5-(ii) in [11] says that, under mild conditions, Assumption 2 implies the following inequality between the error and the gradient norm for all

(2)

This is a key result in our analysis of the inverse sensitivity mechanism of Section 3.

Differential privacy.

We begin by recalling the definition of -differential privacy. [[22, 21]] A randomized algorithm is -differentially private (-DP) if, for all datasets that differ in a single data element and for all events in the output space of , we have

We use the following standard results in differential privacy. [Composition [20, Thm. 3.16]] If are randomized algorithms that each is -DP, then their composition is -DP.

Next, we consider the Laplace mechanism. We will let denote a

-dimensional vector

such that for . [Laplace mechanism [20, Thm. 3.6]] Let have -sensitivity , that is, . Then the Laplace mechanism with is -DP.

Finally, we need the Gaussian mechanism for -DP. [Gaussian mechanism [20, Thm. A.1]] Let have -sensitivity , that is, . Then the Gaussian mechanism with is -DP.

Inverse sensitivity mechanism.

Our goal is to design private optimization algorithms that adapt to the difficulty of the underlying function. As a reference point, we turn to the inverse sensitivity mechanism of [2] as it enjoys general instance-optimality guarantees. For a given function

that we wish to estimate privately, define the

inverse sensitivity at

(3)

that is, the inverse sensitivity of a target parameter at instance is the minimal number of samples one needs to change to reach a new instance such that . Having this quantity, the inverse sensitivity mechanism samples an output from the following probability density

(4)

The inverse sensitivity mechanism preserves -DP and enjoys instance-optimality guarantees in general settings [2]. In contrast to (worst-case) minimax optimality guarantees which measure the performance of the algorithm on the hardest instance, these notions of instance-optimality provide stronger per-instance optimality guarantees.

3 Adaptive rates through inverse sensitivity for -Dp

To understand the achievable rates when privately optimizing functions with growth, we begin our theoretical investigation by examining the inverse sensitivity mechanism in our setting. We show that, for instances that exhibit -growth of the empirical function, the inverse sensitivity mechanism privately solves ERM with excess loss roughly .

In our setting, we use a gradient-based approximation of the inverse sensitivity mechanism to simplify the analysis, while attaining similar rates. Following [3] with our function of interest , we can lower bound the inverse sensitivity under natural assumptions. We define a -smoothed version of this quantity which is more suitable to continuous domains

and define the -smooth gradient-based inverse sensitivity mechanism

(5)

Note that while exactly sampling from the un-normalized density is computationally intractable, analyzing its performance is an important step towards understanding the optimal rates for the family of functions with growth that we study in this work. The following theorem demonstrates the adaptivity of the inverse sensitivity mechanism to the growth of the underlying instance. We defer the proof to Appendix A. [] Let , be convex, -Lipschitz for all . Let and assume is in the interior of . Assume that has -growth (Assumption 2) with . For , the -smooth inverse sensitivity mechanism  (5) is -DP, and with probability at least the output has

Moreover, setting , we have

The rates of the inverse sensitivity in Section 3 provide two main insights regarding the landscape of the problem with growth conditions. First, these conditions allow to improve the worst-case rate to for pure -DP and therefore suggest a better rate is possible for approximate -DP. Moreover, the general instance-optimality guarantees of this mechanism [2] hint that these are the optimal rates for our class of functions. In the sections to come, we validate the correctness of these predictions by developing efficient algorithms that achieve these rates (for pure and approximate privacy), and prove matching lower bounds which demonstrate the optimality of these algorithms.

4 Efficient algorithms with optimal rates

While the previous section demonstrates that there exists algorithms that improve the rates for functions with growth, we pointed out that was computationally intractable in the general case. In this section, we develop efficient algorithms—e.g. that are implementable with gradient-based methods—that achieve the same convergence rates. Our algorithms build on the recent localization techniques that Feldman et al. [25] used to obtain optimal rates for DP-SCO with general convex functions. In Section 4.1, we use these techniques to develop private algorithms that achieve the optimal rates for (pure) DP-SCO with high probability, in contrast to existing results which bound the expected excess loss. These results are of independent interest.

In Section 4.2, we translate these results into convergence guarantees on privately optimizing convex functions with growth by solving a sequence of increasingly constrained SCO problems—the high-probability guarantees of Section 4.1 being crucial to our convergence analysis of these algorithms.

4.1 High-probability guarantees for convex DP-SCO

We first describe our algorithm (Algorithm 1) then analyze its performance under pure-DP (Proposition 1) and approximate-DP constraints (Proposition 1). Our analysis builds on novel tight generalization bounds for uniformly-stable algorithms with high probability [24]. We defer the proofs to Appendix B.

0:  Dataset , constraint set , step size , initial point , privacy parameters ;
1:  Set and
2:  for  to   do
3:     Set
4:     Solve the following ERM over :
5:     Let be the output of the optimization algorithm
6:     if  then
7:        Set where
8:     else if  then
9:        Set where
10:     Set
11:  return  the final iterate
Algorithm 1 Localization-based Algorithm

[] Let , and be convex, -Lipschitz for all . Setting

then for , Algorithm 1 is -DP and has with probability

Similarly, by using a different choice for the parameters and noise distribution, we have the following guarantees for approximate -DP. [] Let , and be convex, -Lipschitz for all . Setting

then for , Algorithm 1 is -DP and has with probability

4.2 Algorithms for DP-SCO with growth

Building on the algorithms of the previous section, we design algorithms that recover the rates of the inverse sensitivity mechanism for functions with growth, importantly without knowledge of the value of

. Inspired by epoch-based algorithms from the optimization literature 

[31, 29], our algorithm iteratively applies the private procedures from the previous section. Crucially, the growth assumption allows to reduce the diameter of the domain after each run, hence improving the overall excess loss by carefully choosing the hyper-parameters. We provide full details in Algorithm 2.

0:  Dataset , convex set , initial point , number of iterations , privacy parameters ;
1:  Set and
2:  if  then
3:     Set
4:  else if  then
5:     
6:  for  to   do
7:     Let
8:     Set and
9:     Set
10:     Run Algorithm 1 on dataset with starting point , privacy parameter , domain (with diameter ), step size
11:     Let be the output of the private procedure
12:  return  
Algorithm 2 Epoch-based algorithms for -growth

The following theorem summarizes our main upper bound for DP-SCO with growth in the pure privacy model, recovering the rates of the inverse sensitivity mechanism in Section 3. We defer the proof to Section B.3. [] Let , and be convex, -Lipschitz for all . Assume that has -growth (Assumption 2) with . Setting , Algorithm 2 is -DP and has with probability

where hides logarithmic factors depending on and .

Sketch of the proof.

The main challenge of the proof is showing that the iterate achieves good risk without knowledge of . Let us denote by the error guarantee of Proposition 1 (or Proposition 1 for approximate-DP). At each stage , as long as belongs to , the excess loss is of order and thus decreases exponentially fast with . The challenge is that, without knowledge of , we do not know the index (roughly ) after which for and the regret guarantees become meaningless with respect to the original problem. However, in the stages after , as the constraint set becomes very small, we upper bound the variations in function values and show that the sub-optimality cannot increase (overall) by more than , thus achieving the optimal rate of stage .

Moreover, we can improve the dependence on the dimension for approximate -DP, resulting in the following bounds.

[] Let , and be convex, -Lipschitz for all . Assume that has -growth (Assumption 2) with . Setting and , Algorithm 2 is -DP and has with probability

where hides logarithmic factors depending on and .

5 Lower bounds

In this section, we develop (minimax) lower bounds for the problem of SCO with -growth under privacy constraints. Note that taking provides lower bound for the unconstrained minimax risk. For a sample space and collection of distributions over , we define the function class as the set of convex functions from that are -Lipschitz and has -growth (Assumption 2). We define the constrained minimax risk [6]

(6)

where is the collection of -DP mechanisms from to . When clear from context, we omit the dependency on of the function class and simply write . We also forego the dependence on when referring to pure-DP constraints, i.e. . We now proceed to prove tight lower bounds for -DP in Section 5.1 and -DP in Section 5.2.

5.1 Lower bounds for pure -Dp

Although in Section 4 we show that the same algorithm achieves the optimal upper bounds for all values of , the landscape of the problem is more subtle for the lower bounds and we need to delineate two different cases to obtain tight lower bounds. We begin with , which corresponds to uniform convexity and enjoys properties that make the problem easier (e.g., closure under summation or addition of linear terms). The second case, , corresponds to sharper growth and requires a different hard instance to satisfy the growth condition.

-growth with .

We begin by developing lower bounds under pure DP for

[Lower bound for -DP, ] Let , , , and . Let be the set of distributions on . Assume that

The following lower bound holds

(7)

First of all, note that is not an overly-restrictive assumption. Indeed, for an arbitrary -uniformly convex and -Lipschitz function, it always holds that . This is thus equivalent to assuming . Note that when , the standard lower bound holds. We present the proof in Section C.1.1 and preview the main ideas here.

Sketch of the proof.

Our lower bounds hinges on the collections of functions for to be chosen later. These functions are [39, Lemma 4] -uniformly convex for any and in turn, so is the population function . We proceed as follows, we first prove an information-theoretic (non-private) lower bound (Section C.1.1 in Appendix C.1.1) which provides the statistical term in (7). With the same family of functions, we exhibit a collection of datasets and prove by contradiction that if an estimator were to optimize below a certain error it would have violated -DP—this yields a lower bound on ERM for our function class (Theorem C.1.1 in Appendix C.1.1). We conclude by proving a reduction from SCO to ERM in Section C.1.1. ∎

-growth with .

As the construction of the hard instance is more intricate for , we provide a one-dimensional lower bound and leave the high-dimensional case for future work. In this case we directly obtain the result with a private version of Le Cam’s method [44, 42, 6], however with a different family of functions.

The issue with the construction of the previous section is that the function does not exhibit sharp growth for . Indeed, the added linear function shifts the minimum away from where the function is differentiable and as a result it locally behaves as a quadratic and only achieves growth . To establish the lower bound, we consider a different sample function that has growth exactly on one side and on the other side. This yields the following

[Lower bound for -DP, ] Let , , , , , and . There exists a collection of distributions such that, whenever , it holds that

(8)

5.2 Lower bounds under approximate privacy constraints

We conclude our treatment by providing lower bounds but now under approximate privacy constraints, demonstrating the optimality of the risk bound of Theorem 2. We prove the result via a reduction: we show that if one solves ERM with -growth with error , this implies that one solves arbitrary convex ERM with error . Given that a lower bound of holds for ERM, a lower bound of holds for ERM with -growth. However, for this reduction to hold, we require that . Furthermore, we consider to be roughly a constant—in the case that is too large, standard lower bounds on general convex functions apply.

[Private lower bound for -DP] Let such that , . Let and . Assume that , then for any mechanism , there exists and such that

Section 5.2 implies that the same lower bound (up to logarithmic factors) applies to SCO via the reduction of [8, Appendix C]. Before proving the theorem, let us state (and prove in Section C.2) the following reduction: if an -DP algorithm achieves excess error (roughly) on ERM for any function with -growth, there exists an -DP algorithm that achieves error for any convex function. We construct the latter by iteratively solving ERM problems with geometrically increasing -regularization towards the previous iterate to ensure the objective has -growth.

[Solving ERM with -growth implies solving any convex ERM] Let . Assume there exists an mechanism such that for any -Lipschitz loss on and dataset such that exhibits -growth, the mechanism achieves excess loss

Then, we can construct an -DP mechanism such that for any -Lipschitz loss , the mechanism achieves excess loss

where is the smallest integer such that .

With this proposition, the proof of the theorem directly follows as Bassily et al. [7] prove a lower bound for ERM with -DP.

Discussion

In this work, we develop private algorithms that adapt to the growth of the function at hand, achieving the convergence rate corresponding to the “easiest” sub-class the function belongs to. However, the picture is not yet complete. First of, there are still gaps in our theoretical understanding, the most interesting one being . On these functions, appropriate optimization algorithms achieve linear convergence [43] and raise the question, can we achieve exponentially small privacy cost in our setting? Finally, while our optimality guarantees are more fine-grained than the usual minimax results over convex functions, they are still contigent on some predetermined choice of sub-classes. Studying more general notions of adaptivity is an important future direction in private optimization.

Acknowledgments

The authors would like to thank Karan Chadha and Gary Cheng for comments on an early version of the draft.

References

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Potential negative societal impact

The aim of our work is theoretical in essence and as such, we do not expect direct negative societal impact. As DP becomes more establish as a norm, we believe this research is relevant for practitionners in both industry and government. Indeed, an important obstacle to applying DP is the loss of performance compared to non-private models; our theoretical results suggests that better adaptive algorithms would significantly narrow this performance gap. We wish to point out two potential negative consequences of growing research in privacy. First, a simple but effective method to guarantee privacy is to either delete existing user data or limit data collection in the first place. Paradoxically, the more confident institutions are in DP algorithms, the less they are susceptible to turn to these simpler—and most effective—solutions. Finally, using DP algorithms should not preclude one from (1) carefully choosing and to provide meaningful guarantees for the specific application at hand and (2) developing exhaustive and meticulous evaluation methods of the privacy of deployed models.

Appendix A Proofs for Section 3

a.1 Proof of Section 3

See 3

Let us first prove privacy. The sensitivity of is as is -Lipschitz, therefore following the privacy proof of the smooth inverse sensitivity mechanism [2, Prop. 3.2] we get that  (5) is -DP.

Let us now prove the claim about utility. Denote and with to be chosen presently. We argue that it is enough to show that . Indeed then with probability at least we have , which implies there is such that and , hence using the inequality (2)

It remains to prove that . Let and . Note that for any as is in the interior of which implies . Hence the definition of the smooth inverse sensitivity mechanism (5) implies

where the last inequality follows by choosing .

Appendix B Proofs for Section 4

We need to the following result on the generalization properties of uniformly stable algorithms [24].  [24, Cor. 4.2] Assume . Let where and is -Lipschitz and -strongly convex for all . Let be the empirical minimizer. For , with probability at least

b.1 Proof of Algorithm 1

See 1

We begin by proving the privacy claim. We show that each iterate is -DP and therefore post-processing implies the claim as each sample is used in exactly one iterate. To this end, let and note that the minimizer has sensitivity  [25], hence the -sensitivity is at most . Standard properties of the Laplace mechanism [20] now imply that is -DP which give the claim about privacy.

Now we proceed to prove utility which follows similar arguments to the localization-based proof in [25]. Letting , we have: